LMFDB - Embedded newform 405.4.e.v.271.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,4,Mod(136,405)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("405.136"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	405.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,5,0,-17,15,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$23.8957735523$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.84779568.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 13x^{4} - 4x^{3} + 152x^{2} - 96x + 64 $ x^6 - x^5 + 13x^4 - 4x^3 + 152x^2 - 96x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			271.2
Root			$-1.66402 + 2.88216i$ of defining polynomial
Character	$\chi$	$=$	405.271
Dual form			405.4.e.v.136.2

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.06306 + 1.84127i) q^{2} +(1.73981 - 3.01344i) q^{4} +(2.50000 - 4.33013i) q^{5} +(-15.3500 - 26.5870i) q^{7} +24.4070 q^{8} +10.6306 q^{10} +(25.0774 + 43.4353i) q^{11} +(7.97962 - 13.8211i) q^{13} +(32.6360 - 56.5272i) q^{14} +(12.0277 + 20.8325i) q^{16} -105.668 q^{17} -21.3040 q^{19} +(-8.69904 - 15.0672i) q^{20} +(-53.3175 + 92.3487i) q^{22} +(68.0684 - 117.898i) q^{23} +(-12.5000 - 21.6506i) q^{25} +33.9312 q^{26} -106.824 q^{28} +(-112.162 - 194.270i) q^{29} +(112.991 - 195.706i) q^{31} +(72.0559 - 124.804i) q^{32} +(-112.332 - 194.564i) q^{34} -153.500 q^{35} -416.386 q^{37} +(-22.6474 - 39.2264i) q^{38} +(61.0176 - 105.686i) q^{40} +(38.0706 - 65.9402i) q^{41} +(-15.8686 - 27.4852i) q^{43} +174.519 q^{44} +289.443 q^{46} +(30.4013 + 52.6566i) q^{47} +(-299.746 + 519.176i) q^{49} +(26.5765 - 46.0318i) q^{50} +(-27.7660 - 48.0921i) q^{52} +466.532 q^{53} +250.774 q^{55} +(-374.649 - 648.910i) q^{56} +(238.469 - 413.040i) q^{58} +(47.7119 - 82.6395i) q^{59} +(178.587 + 309.322i) q^{61} +480.464 q^{62} +498.842 q^{64} +(-39.8981 - 69.1055i) q^{65} +(-43.9172 + 76.0668i) q^{67} +(-183.842 + 318.424i) q^{68} +(-163.180 - 282.636i) q^{70} -412.693 q^{71} -331.133 q^{73} +(-442.643 - 766.680i) q^{74} +(-37.0648 + 64.1981i) q^{76} +(769.877 - 1333.47i) q^{77} +(124.062 + 214.881i) q^{79} +120.277 q^{80} +161.885 q^{82} +(-276.253 - 478.484i) q^{83} +(-264.171 + 457.557i) q^{85} +(33.7386 - 58.4369i) q^{86} +(612.065 + 1060.13i) q^{88} +291.478 q^{89} -489.949 q^{91} +(-236.852 - 410.240i) q^{92} +(-64.6368 + 111.954i) q^{94} +(-53.2599 + 92.2489i) q^{95} +(-99.3032 - 171.998i) q^{97} -1274.59 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 5 q^{2} - 17 q^{4} + 15 q^{5} + 4 q^{7} - 150 q^{8} + 50 q^{10} + 5 q^{11} - 7 q^{13} + 60 q^{14} - 161 q^{16} - 310 q^{17} - 100 q^{19} + 85 q^{20} + 229 q^{22} + 285 q^{23} - 75 q^{25} - 370 q^{26}+ \cdots + 610 q^{98}+O(q^{100}) $ 6 * q + 5 * q^2 - 17 * q^4 + 15 * q^5 + 4 * q^7 - 150 * q^8 + 50 * q^10 + 5 * q^11 - 7 * q^13 + 60 * q^14 - 161 * q^16 - 310 * q^17 - 100 * q^19 + 85 * q^20 + 229 * q^22 + 285 * q^23 - 75 * q^25 - 370 * q^26 - 668 * q^28 + 115 * q^29 + 115 * q^31 + 775 * q^32 - 413 * q^34 + 40 * q^35 - 768 * q^37 - 1150 * q^38 - 375 * q^40 + 580 * q^41 + 797 * q^43 + 2830 * q^44 - 570 * q^46 - 145 * q^47 - 577 * q^49 + 125 * q^50 - 825 * q^52 + 800 * q^53 + 50 * q^55 - 2190 * q^56 + 59 * q^58 + 380 * q^59 + 152 * q^61 + 2010 * q^62 + 5874 * q^64 + 35 * q^65 - 2 * q^67 + 475 * q^68 - 300 * q^70 - 80 * q^71 - 1960 * q^73 - 2720 * q^74 + 3276 * q^76 + 1950 * q^77 - 1013 * q^79 - 1610 * q^80 + 8 * q^82 + 270 * q^83 - 775 * q^85 - 1555 * q^86 + 5193 * q^88 - 2040 * q^89 - 1264 * q^91 - 1215 * q^92 - 3833 * q^94 - 250 * q^95 - 720 * q^97 + 610 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.06306	+	1.84127i	0.375848	+	0.650988i	0.990454	−	0.137846i	$-0.0440180\pi$
$2$	1.06306	+	1.84127i	0.375848	+	0.650988i	−0.614605	+	0.788835i	$0.710685\pi$
$3$		0			0
$3$		0			0
$4$	1.73981	−	3.01344i	0.217476	−	0.376679i
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$6$		0			0
$7$	−15.3500	−	26.5870i	−0.828823	−	1.43556i	−0.898962	−	0.438027i	$-0.855678\pi$
$7$	−15.3500	−	26.5870i	−0.828823	−	1.43556i	0.0701387	−	0.997537i	$-0.477656\pi$
$8$	24.4070			1.07865
$9$		0			0
$10$	10.6306			0.336169
$11$	25.0774	+	43.4353i	0.687375	+	1.19057i	0.972684	+	0.232132i	$0.0745702\pi$
$11$	25.0774	+	43.4353i	0.687375	+	1.19057i	−0.285310	+	0.958435i	$0.592096\pi$
$12$		0			0
$13$	7.97962	−	13.8211i	0.170242	−	0.294868i	−0.768262	−	0.640135i	$-0.778878\pi$
$13$	7.97962	−	13.8211i	0.170242	−	0.294868i	0.938504	+	0.345267i	$0.112212\pi$
$14$	32.6360	−	56.5272i	0.623024	−	1.07911i
$15$		0			0
$16$	12.0277	+	20.8325i	0.187932	+	0.325508i
$17$	−105.668			−1.50755			−0.753774	−	0.657134i	$-0.771769\pi$
$17$	−105.668			−1.50755			−0.753774	+	0.657134i	$0.771769\pi$
$18$		0			0
$19$	−21.3040			−0.257235			−0.128618	−	0.991694i	$-0.541054\pi$
$19$	−21.3040			−0.257235			−0.128618	+	0.991694i	$0.541054\pi$
$20$	−8.69904	−	15.0672i	−0.0972582	−	0.168456i
$21$		0			0
$22$	−53.3175	+	92.3487i	−0.516697	+	0.894946i
$23$	68.0684	−	117.898i	0.617098	−	1.06884i	−0.372915	−	0.927866i	$-0.621642\pi$
$23$	68.0684	−	117.898i	0.617098	−	1.06884i	0.990013	−	0.140979i	$-0.0450251\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$	33.9312			0.255941
$27$		0			0
$28$	−106.824			−0.720997
$29$	−112.162	−	194.270i	−0.718203	−	1.24396i	−0.961711	−	0.274065i	$-0.911632\pi$
$29$	−112.162	−	194.270i	−0.718203	−	1.24396i	0.243508	−	0.969899i	$-0.421702\pi$
$30$		0			0
$31$	112.991	−	195.706i	0.654638	−	1.13387i	−0.327347	−	0.944904i	$-0.606155\pi$
$31$	112.991	−	195.706i	0.654638	−	1.13387i	0.981985	−	0.188961i	$-0.0605121\pi$
$32$	72.0559	−	124.804i	0.398056	−	0.689454i
$33$		0			0
$34$	−112.332	−	194.564i	−0.566609	−	0.981396i
$35$	−153.500			−0.741322
$36$		0			0
$37$	−416.386			−1.85009			−0.925046	−	0.379854i	$-0.875974\pi$
$37$	−416.386			−1.85009			−0.925046	+	0.379854i	$0.875974\pi$
$38$	−22.6474	−	39.2264i	−0.0966814	−	0.167457i
$39$		0			0
$40$	61.0176	−	105.686i	0.241193	−	0.417759i
$41$	38.0706	−	65.9402i	0.145015	−	0.251174i	−0.784363	−	0.620302i	$-0.787010\pi$
$41$	38.0706	−	65.9402i	0.145015	−	0.251174i	0.929379	+	0.369128i	$0.120344\pi$
$42$		0			0
$43$	−15.8686	−	27.4852i	−0.0562777	−	0.0974758i	0.836514	−	0.547945i	$-0.184590\pi$
$43$	−15.8686	−	27.4852i	−0.0562777	−	0.0974758i	−0.892792	+	0.450470i	$0.851257\pi$
$44$	174.519			0.597950
$45$		0			0
$46$	289.443			0.927741
$47$	30.4013	+	52.6566i	0.0943507	+	0.163420i	0.909337	−	0.416059i	$-0.136589\pi$
$47$	30.4013	+	52.6566i	0.0943507	+	0.163420i	−0.814987	+	0.579480i	$0.803256\pi$
$48$		0			0
$49$	−299.746	+	519.176i	−0.873896	+	1.51363i
$50$	26.5765	−	46.0318i	0.0751697	−	0.130198i
$51$		0			0
$52$	−27.7660	−	48.0921i	−0.0740471	−	0.128253i
$53$	466.532			1.20911			0.604557	−	0.796562i	$-0.293350\pi$
$53$	466.532			1.20911			0.604557	+	0.796562i	$0.293350\pi$
$54$		0			0
$55$	250.774			0.614806
$56$	−374.649	−	648.910i	−0.894009	−	1.54847i
$57$		0			0
$58$	238.469	−	413.040i	0.539871	−	0.935084i
$59$	47.7119	−	82.6395i	0.105281	−	0.182352i	−0.808572	−	0.588397i	$-0.799759\pi$
$59$	47.7119	−	82.6395i	0.105281	−	0.182352i	0.913853	+	0.406045i	$0.133093\pi$
$60$		0			0
$61$	178.587	+	309.322i	0.374848	+	0.649256i	0.990304	−	0.138915i	$-0.0443616\pi$
$61$	178.587	+	309.322i	0.374848	+	0.649256i	−0.615456	+	0.788171i	$0.711028\pi$
$62$	480.464			0.984178
$63$		0			0
$64$	498.842			0.974300
$65$	−39.8981	−	69.1055i	−0.0761346	−	0.131869i
$66$		0			0
$67$	−43.9172	+	76.0668i	−0.0800797	+	0.138702i	−0.903284	−	0.429043i	$-0.858851\pi$
$67$	−43.9172	+	76.0668i	−0.0800797	+	0.138702i	0.823204	+	0.567745i	$0.192184\pi$
$68$	−183.842	+	318.424i	−0.327855	+	0.567862i
$69$		0			0
$70$	−163.180	−	282.636i	−0.278625	−	0.482592i
$71$	−412.693			−0.689826			−0.344913	−	0.938635i	$-0.612092\pi$
$71$	−412.693			−0.689826			−0.344913	+	0.938635i	$0.612092\pi$
$72$		0			0
$73$	−331.133			−0.530906			−0.265453	−	0.964124i	$-0.585522\pi$
$73$	−331.133			−0.530906			−0.265453	+	0.964124i	$0.585522\pi$
$74$	−442.643	−	766.680i	−0.695354	−	1.20439i
$75$		0			0
$76$	−37.0648	+	64.1981i	−0.0559424	+	0.0968952i
$77$	769.877	−	1333.47i	1.13942	−	1.97354i
$78$		0			0
$79$	124.062	+	214.881i	0.176684	+	0.306025i	0.940743	−	0.339121i	$-0.110130\pi$
$79$	124.062	+	214.881i	0.176684	+	0.306025i	−0.764059	+	0.645146i	$0.776796\pi$
$80$	120.277			0.168092
$81$		0			0
$82$	161.885			0.218015
$83$	−276.253	−	478.484i	−0.365333	−	0.632776i	0.623496	−	0.781826i	$-0.285712\pi$
$83$	−276.253	−	478.484i	−0.365333	−	0.632776i	−0.988830	+	0.149050i	$0.952378\pi$
$84$		0			0
$85$	−264.171	+	457.557i	−0.337098	+	0.583871i
$86$	33.7386	−	58.4369i	0.0423038	−	0.0732723i
$87$		0			0
$88$	612.065	+	1060.13i	0.741436	+	1.28420i
$89$	291.478			0.347153			0.173577	−	0.984820i	$-0.444468\pi$
$89$	291.478			0.347153			0.173577	+	0.984820i	$0.444468\pi$
$90$		0			0
$91$	−489.949			−0.564402
$92$	−236.852	−	410.240i	−0.268408	−	0.464896i
$93$		0			0
$94$	−64.6368	+	111.954i	−0.0709231	+	0.122842i
$95$	−53.2599	+	92.2489i	−0.0575195	+	0.0996267i
$96$		0			0
$97$	−99.3032	−	171.998i	−0.103946	−	0.180039i	0.809361	−	0.587311i	$-0.199813\pi$
$97$	−99.3032	−	171.998i	−0.103946	−	0.180039i	−0.913307	+	0.407272i	$0.866480\pi$
$98$	−1274.59			−1.31381
$99$		0			0
$100$	−86.9904			−0.0869904
$101$	408.117	+	706.880i	0.402071	+	0.696408i	0.993976	−	0.109600i	$-0.0349570\pi$
$101$	408.117	+	706.880i	0.402071	+	0.696408i	−0.591904	+	0.806008i	$0.701624\pi$
$102$		0			0
$103$	701.186	−	1214.49i	0.670776	−	1.16182i	−0.306908	−	0.951739i	$-0.599294\pi$
$103$	701.186	−	1214.49i	0.670776	−	1.16182i	0.977684	−	0.210079i	$-0.0673722\pi$
$104$	194.759	−	337.332i	0.183631	−	0.318059i
$105$		0			0
$106$	495.951	+	859.013i	0.454444	+	0.787120i
$107$	978.996			0.884515			0.442258	−	0.896888i	$-0.354178\pi$
$107$	978.996			0.884515			0.442258	+	0.896888i	$0.354178\pi$
$108$		0			0
$109$	2122.96			1.86553			0.932766	−	0.360484i	$-0.117388\pi$
$109$	2122.96			1.86553			0.932766	+	0.360484i	$0.117388\pi$
$110$	266.588	+	461.743i	0.231074	+	0.400232i
$111$		0			0
$112$	369.250	−	639.560i	0.311525	−	0.539578i
$113$	897.399	−	1554.34i	0.747082	−	1.29398i	−0.202134	−	0.979358i	$-0.564788\pi$
$113$	897.399	−	1554.34i	0.747082	−	1.29398i	0.949216	−	0.314626i	$-0.101879\pi$
$114$		0			0
$115$	−340.342	−	589.490i	−0.275975	−	0.478002i
$116$	−780.559			−0.624768
$117$		0			0
$118$	202.883			0.158278
$119$	1622.01	+	2809.40i	1.24949	+	2.16418i
$120$		0			0
$121$	−592.252	+	1025.81i	−0.444968	+	0.770706i
$122$	−379.697	+	657.655i	−0.281772	+	0.488043i
$123$		0			0
$124$	−393.165	−	680.982i	−0.284736	−	0.493177i
$125$	−125.000			−0.0894427
$126$		0			0
$127$	−748.894			−0.523257			−0.261628	−	0.965169i	$-0.584259\pi$
$127$	−748.894			−0.523257			−0.261628	+	0.965169i	$0.584259\pi$
$128$	−46.1486	−	79.9317i	−0.0318672	−	0.0551955i
$129$		0			0
$130$	84.8281	−	146.927i	0.0572301	−	0.0991255i
$131$	−1198.05	+	2075.09i	−0.799042	+	1.38398i	0.121198	+	0.992628i	$0.461326\pi$
$131$	−1198.05	+	2075.09i	−0.799042	+	1.38398i	−0.920240	+	0.391353i	$0.872007\pi$
$132$		0			0
$133$	327.016	+	566.409i	0.213202	+	0.369277i
$134$	−186.746			−0.120391
$135$		0			0
$136$	−2579.05			−1.62611
$137$	502.238	+	869.901i	0.313205	+	0.542486i	0.979054	−	0.203600i	$-0.0652641\pi$
$137$	502.238	+	869.901i	0.313205	+	0.542486i	−0.665850	+	0.746086i	$0.731931\pi$
$138$		0			0
$139$	−1201.51	+	2081.08i	−0.733172	+	1.26989i	0.222348	+	0.974967i	$0.428628\pi$
$139$	−1201.51	+	2081.08i	−0.733172	+	1.26989i	−0.955521	+	0.294924i	$0.904706\pi$
$140$	−267.061	+	462.563i	−0.161220	+	0.279241i
$141$		0			0
$142$	−438.717	−	759.881i	−0.259270	−	0.449069i
$143$	800.432			0.468080
$144$		0			0
$145$	−1121.62			−0.642380
$146$	−352.014	−	609.706i	−0.199540	−	0.345614i
$147$		0			0
$148$	−724.432	+	1254.75i	−0.402351	+	0.696892i
$149$	254.824	−	441.368i	0.140107	−	0.242673i	−0.787429	−	0.616405i	$-0.788589\pi$
$149$	254.824	−	441.368i	0.140107	−	0.242673i	0.927537	+	0.373732i	$0.121922\pi$
$150$		0			0
$151$	−721.648	−	1249.93i	−0.388920	−	0.673629i	0.603385	−	0.797450i	$-0.293818\pi$
$151$	−721.648	−	1249.93i	−0.388920	−	0.673629i	−0.992305	+	0.123821i	$0.960485\pi$
$152$	−519.967			−0.277466
$153$		0			0
$154$	3273.70			1.71300
$155$	−564.955	−	978.530i	−0.292763	−	0.507080i
$156$		0			0
$157$	1077.82	−	1866.84i	0.547895	−	0.948981i	−0.450524	−	0.892764i	$-0.648763\pi$
$157$	1077.82	−	1866.84i	0.547895	−	0.948981i	0.998419	−	0.0562170i	$-0.0179039\pi$
$158$	−263.770	+	456.863i	−0.132813	+	0.230038i
$159$		0			0
$160$	−360.279	−	624.022i	−0.178016	−	0.308333i
$161$	−4179.41			−2.04586
$162$		0			0
$163$	−529.909			−0.254636			−0.127318	−	0.991862i	$-0.540637\pi$
$163$	−529.909			−0.254636			−0.127318	+	0.991862i	$0.540637\pi$
$164$	−132.471	−	229.446i	−0.0630747	−	0.109249i
$165$		0			0
$166$	587.346	−	1017.31i	0.274620	−	0.475656i
$167$	−1489.64	+	2580.13i	−0.690250	+	1.19555i	0.281506	+	0.959559i	$0.409166\pi$
$167$	−1489.64	+	2580.13i	−0.690250	+	1.19555i	−0.971756	+	0.235988i	$0.924167\pi$
$168$		0			0
$169$	971.151	+	1682.08i	0.442035	+	0.765628i
$170$	−1123.32			−0.506791
$171$		0			0
$172$	−110.433			−0.0489562
$173$	889.939	+	1541.42i	0.391103	+	0.677410i	0.992595	−	0.121469i	$-0.0387604\pi$
$173$	889.939	+	1541.42i	0.391103	+	0.677410i	−0.601493	+	0.798878i	$0.705427\pi$
$174$		0			0
$175$	−383.751	+	664.675i	−0.165765	+	0.287113i
$176$	−603.246	+	1044.85i	−0.258360	+	0.447492i
$177$		0			0
$178$	309.859	+	536.692i	0.130477	+	0.225993i
$179$	2836.26			1.18431			0.592157	−	0.805823i	$-0.298276\pi$
$179$	2836.26			1.18431			0.592157	+	0.805823i	$0.298276\pi$
$180$		0			0
$181$	811.890			0.333410			0.166705	−	0.986007i	$-0.446687\pi$
$181$	811.890			0.333410			0.166705	+	0.986007i	$0.446687\pi$
$182$	−520.845	−	902.130i	−0.212130	−	0.367419i
$183$		0			0
$184$	1661.35	−	2877.54i	0.665632	−	1.15291i
$185$	−1040.96	+	1803.00i	−0.413693	+	0.716538i
$186$		0			0
$187$	−2649.88	−	4589.73i	−1.03625	−	1.79484i
$188$	211.570			0.0820761
$189$		0			0
$190$	−226.474			−0.0864745
$191$	574.231	+	994.597i	0.217539	+	0.376788i	0.954055	−	0.299632i	$-0.0968638\pi$
$191$	574.231	+	994.597i	0.217539	+	0.376788i	−0.736516	+	0.676420i	$0.763531\pi$
$192$		0			0
$193$	1075.47	−	1862.76i	0.401107	−	0.694739i	−0.592752	−	0.805385i	$-0.701959\pi$
$193$	1075.47	−	1862.76i	0.401107	−	0.694739i	0.993860	+	0.110646i	$0.0352921\pi$
$194$	211.131	−	365.689i	0.0781355	−	0.135335i
$195$		0			0
$196$	1043.00	+	1806.53i	0.380103	+	0.658357i
$197$	5057.51			1.82910			0.914551	−	0.404471i	$-0.132544\pi$
$197$	5057.51			1.82910			0.914551	+	0.404471i	$0.132544\pi$
$198$		0			0
$199$	−3554.12			−1.26605			−0.633027	−	0.774130i	$-0.718188\pi$
$199$	−3554.12			−1.26605			−0.633027	+	0.774130i	$0.718188\pi$
$200$	−305.088	−	528.428i	−0.107865	−	0.186827i
$201$		0			0
$202$	−867.707	+	1502.91i	−0.302236	+	0.523488i
$203$	−3443.37	+	5964.09i	−1.19053	+	2.06205i
$204$		0			0
$205$	−190.353	−	329.701i	−0.0648528	−	0.112328i
$206$	2981.61			1.00844
$207$		0			0
$208$	383.905			0.127976
$209$	−534.248	−	925.345i	−0.176817	−	0.306256i
$210$		0			0
$211$	53.9547	−	93.4523i	0.0176038	−	0.0304906i	−0.857089	−	0.515168i	$-0.827730\pi$
$211$	53.9547	−	93.4523i	0.0176038	−	0.0304906i	0.874693	+	0.484677i	$0.161063\pi$
$212$	811.676	−	1405.86i	0.262953	−	0.455449i
$213$		0			0
$214$	1040.73	+	1802.60i	0.332444	+	0.575809i
$215$	−158.686			−0.0503363
$216$		0			0
$217$	−6937.65			−2.17032
$218$	2256.84	+	3908.95i	0.701157	+	1.21444i
$219$		0			0
$220$	436.299	−	755.691i	0.133706	−	0.231585i
$221$	−843.192	+	1460.45i	−0.256648	+	0.444528i
$222$		0			0
$223$	971.013	+	1681.84i	0.291587	+	0.505043i	0.974185	−	0.225751i	$-0.0724834\pi$
$223$	971.013	+	1681.84i	0.291587	+	0.505043i	−0.682598	+	0.730794i	$0.739150\pi$
$224$	−4424.24			−1.31967
$225$		0			0
$226$	3815.96			1.12316
$227$	1721.99	+	2982.58i	0.503492	+	0.872074i	0.999992	+	0.00403708i	$0.00128505\pi$
$227$	1721.99	+	2982.58i	0.503492	+	0.872074i	−0.496500	+	0.868037i	$0.665382\pi$
$228$		0			0
$229$	−1034.70	+	1792.15i	−0.298579	+	0.517155i	−0.975811	−	0.218615i	$-0.929846\pi$
$229$	−1034.70	+	1792.15i	−0.298579	+	0.517155i	0.677232	+	0.735770i	$0.263179\pi$
$230$	723.608	−	1253.33i	0.207449	−	0.359313i
$231$		0			0
$232$	−2737.53	−	4741.55i	−0.774689	−	1.34180i
$233$	2769.24			0.778622			0.389311	−	0.921106i	$-0.372713\pi$
$233$	2769.24			0.778622			0.389311	+	0.921106i	$0.372713\pi$
$234$		0			0
$235$	304.013			0.0843898
$236$	−166.019	−	287.554i	−0.0457921	−	0.0793142i
$237$		0			0
$238$	−3448.59	+	5973.13i	−0.939238	+	1.62681i
$239$	−1471.92	+	2549.44i	−0.398371	+	0.689999i	−0.993525	−	0.113613i	$-0.963758\pi$
$239$	−1471.92	+	2549.44i	−0.398371	+	0.689999i	0.595154	+	0.803612i	$0.297091\pi$
$240$		0			0
$241$	−2237.27	−	3875.06i	−0.597988	−	1.03575i	−0.993118	−	0.117120i	$-0.962634\pi$
$241$	−2237.27	−	3875.06i	−0.597988	−	1.03575i	0.395130	−	0.918625i	$-0.370700\pi$
$242$	−2518.40			−0.668961
$243$		0			0
$244$	1242.83			0.326082
$245$	1498.73	+	2595.88i	0.390818	+	0.676917i
$246$		0			0
$247$	−169.997	+	294.444i	−0.0437922	+	0.0758504i
$248$	2757.77	−	4776.60i	0.706124	−	1.22304i
$249$		0			0
$250$	−132.882	−	230.159i	−0.0336169	−	0.0582262i
$251$	2121.41			0.533474			0.266737	−	0.963769i	$-0.414055\pi$
$251$	2121.41			0.533474			0.266737	+	0.963769i	$0.414055\pi$
$252$		0			0
$253$	6827.92			1.69671
$254$	−796.119	−	1378.92i	−0.196665	−	0.340634i
$255$		0			0
$256$	2093.48	−	3626.02i	0.511104	−	0.885259i
$257$	2274.12	−	3938.89i	0.551967	−	0.956035i	−0.446165	−	0.894951i	$-0.647211\pi$
$257$	2274.12	−	3938.89i	0.551967	−	0.956035i	0.998133	−	0.0610849i	$-0.0194560\pi$
$258$		0			0
$259$	6391.53	+	11070.5i	1.53340	+	2.65593i
$260$	−277.660			−0.0662298
$261$		0			0
$262$	−5094.42			−1.20127
$263$	−1749.92	−	3030.95i	−0.410284	−	0.710632i	0.584637	−	0.811295i	$-0.301237\pi$
$263$	−1749.92	−	3030.95i	−0.410284	−	0.710632i	−0.994921	+	0.100663i	$0.967904\pi$
$264$		0			0
$265$	1166.33	−	2020.14i	0.270366	−	0.468288i
$266$	−695.276	+	1204.25i	−0.160264	+	0.277585i
$267$		0			0
$268$	152.815	+	264.683i	0.0348308	+	0.0603287i
$269$	−2594.25			−0.588008			−0.294004	−	0.955804i	$-0.594988\pi$
$269$	−2594.25			−0.588008			−0.294004	+	0.955804i	$0.594988\pi$
$270$		0			0
$271$	8518.20			1.90939			0.954693	−	0.297593i	$-0.0961838\pi$
$271$	8518.20			1.90939			0.954693	+	0.297593i	$0.0961838\pi$
$272$	−1270.94	−	2201.34i	−0.283317	−	0.490720i
$273$		0			0
$274$	−1067.82	+	1849.51i	−0.235435	+	0.407785i
$275$	626.935	−	1085.88i	0.137475	−	0.238114i
$276$		0			0
$277$	945.248	+	1637.22i	0.205034	+	0.355130i	0.950144	−	0.311813i	$-0.100936\pi$
$277$	945.248	+	1637.22i	0.205034	+	0.355130i	−0.745109	+	0.666942i	$0.767603\pi$
$278$	−5109.12			−1.10225
$279$		0			0
$280$	−3746.49			−0.799626
$281$	1569.34	+	2718.18i	0.333164	+	0.577057i	0.983130	−	0.182906i	$-0.0585504\pi$
$281$	1569.34	+	2718.18i	0.333164	+	0.577057i	−0.649967	+	0.759963i	$0.725217\pi$
$282$		0			0
$283$	−2034.55	+	3523.94i	−0.427354	+	0.740199i	−0.996637	−	0.0819429i	$-0.973888\pi$
$283$	−2034.55	+	3523.94i	−0.427354	+	0.740199i	0.569283	+	0.822142i	$0.307221\pi$
$284$	−718.007	+	1243.62i	−0.150021	+	0.259843i
$285$		0			0
$286$	850.907	+	1473.81i	0.175927	+	0.304715i
$287$	−2337.54			−0.480768
$288$		0			0
$289$	6252.77			1.27270
$290$	−1192.35	−	2065.20i	−0.241438	−	0.418182i
$291$		0			0
$292$	−576.108	+	997.848i	−0.115459	+	0.199982i
$293$	1818.97	−	3150.56i	0.362681	−	0.628182i	−0.625720	−	0.780048i	$-0.715195\pi$
$293$	1818.97	−	3150.56i	0.362681	−	0.628182i	0.988401	+	0.151865i	$0.0485281\pi$
$294$		0			0
$295$	−238.560	−	413.197i	−0.0470830	−	0.0815501i
$296$	−10162.7			−1.99560
$297$		0			0
$298$	1083.57			0.210637
$299$	−1086.32	−	1881.56i	−0.210112	−	0.363925i
$300$		0			0
$301$	−487.167	+	843.798i	−0.0932885	+	0.161580i
$302$	1534.31	−	2657.50i	0.292350	−	0.506365i
$303$		0			0
$304$	−256.237	−	443.816i	−0.0483428	−	0.0837322i
$305$	1785.87			0.335274
$306$		0			0
$307$	−6829.07			−1.26956			−0.634781	−	0.772692i	$-0.718910\pi$
$307$	−6829.07			−1.26956			−0.634781	+	0.772692i	$0.718910\pi$
$308$	−2678.88	−	4639.95i	−0.495595	−	0.858395i
$309$		0			0
$310$	1201.16	−	2080.47i	0.220069	−	0.381171i
$311$	3300.51	−	5716.66i	0.601785	−	1.04232i	−0.390766	−	0.920490i	$-0.627790\pi$
$311$	3300.51	−	5716.66i	0.601785	−	1.04232i	0.992551	−	0.121831i	$-0.0388767\pi$
$312$		0			0
$313$	−1383.30	−	2395.94i	−0.249803	−	0.432672i	0.713668	−	0.700484i	$-0.247033\pi$
$313$	−1383.30	−	2395.94i	−0.249803	−	0.432672i	−0.963471	+	0.267812i	$0.913699\pi$
$314$	4583.15			0.823701
$315$		0			0
$316$	863.373			0.153698
$317$	2282.21	+	3952.90i	0.404358	+	0.700368i	0.994247	−	0.107116i	$-0.0341617\pi$
$317$	2282.21	+	3952.90i	0.404358	+	0.700368i	−0.589889	+	0.807485i	$0.700828\pi$
$318$		0			0
$319$	5625.44	−	9743.55i	0.987349	−	1.71014i
$320$	1247.10	−	2160.05i	0.217860	−	0.377345i
$321$		0			0
$322$	−4442.96	−	7695.43i	−0.768933	−	1.33183i
$323$	2251.15			0.387794
$324$		0			0
$325$	−398.981			−0.0680968
$326$	−563.325	−	975.707i	−0.0957045	−	0.165765i
$327$		0			0
$328$	929.190	−	1609.40i	0.156421	−	0.270928i
$329$	933.321	−	1616.56i	0.156400	−	0.270893i
$330$		0			0
$331$	2332.81	+	4040.54i	0.387380	+	0.670962i	0.992096	−	0.125479i	$-0.0400468\pi$
$331$	2332.81	+	4040.54i	0.387380	+	0.670962i	−0.604716	+	0.796441i	$0.706713\pi$
$332$	−1922.51			−0.317805
$333$		0			0
$334$	−6334.30			−1.03772
$335$	219.586	+	380.334i	0.0358127	+	0.0620295i
$336$		0			0
$337$	1903.53	−	3297.01i	0.307691	−	0.532937i	−0.670166	−	0.742212i	$-0.733777\pi$
$337$	1903.53	−	3297.01i	0.307691	−	0.532937i	0.977857	+	0.209275i	$0.0671103\pi$
$338$	−2064.78	+	3576.31i	−0.332276	+	0.575520i
$339$		0			0
$340$	919.212	+	1592.12i	0.146621	+	0.253956i
$341$	11334.1			1.79993
$342$		0			0
$343$	7874.33			1.23957
$344$	−387.306	−	670.833i	−0.0607039	−	0.105142i
$345$		0			0
$346$	−1892.12	+	3277.24i	−0.293991	+	0.509207i
$347$	1155.63	−	2001.61i	0.178783	−	0.309660i	−0.762681	−	0.646775i	$-0.776117\pi$
$347$	1155.63	−	2001.61i	0.178783	−	0.309660i	0.941464	+	0.337114i	$0.109451\pi$
$348$		0			0
$349$	5436.61	+	9416.49i	0.833854	+	1.44428i	0.894960	+	0.446147i	$0.147204\pi$
$349$	5436.61	+	9416.49i	0.833854	+	1.44428i	−0.0611053	+	0.998131i	$0.519463\pi$
$350$	−1631.80			−0.249209
$351$		0			0
$352$	7227.89			1.09445
$353$	−1446.87	−	2506.05i	−0.218156	−	0.377857i	0.736088	−	0.676885i	$-0.236671\pi$
$353$	−1446.87	−	2506.05i	−0.218156	−	0.377857i	−0.954244	+	0.299029i	$0.903337\pi$
$354$		0			0
$355$	−1031.73	+	1787.01i	−0.154250	+	0.267168i
$356$	507.117	−	878.352i	0.0754975	−	0.130766i
$357$		0			0
$358$	3015.12	+	5222.34i	0.445123	+	0.770975i
$359$	9875.47			1.45183			0.725915	−	0.687784i	$-0.241416\pi$
$359$	9875.47			1.45183			0.725915	+	0.687784i	$0.241416\pi$
$360$		0			0
$361$	−6405.14			−0.933830
$362$	863.087	+	1494.91i	0.125312	+	0.217046i
$363$		0			0
$364$	−852.417	+	1476.43i	−0.122744	+	0.212599i
$365$	−827.832	+	1433.85i	−0.118714	+	0.205619i
$366$		0			0
$367$	−5361.08	−	9285.66i	−0.762523	−	1.32073i	−0.941546	−	0.336884i	$-0.890627\pi$
$367$	−5361.08	−	9285.66i	−0.762523	−	1.32073i	0.179023	−	0.983845i	$-0.442706\pi$
$368$	3274.82			0.463891
$369$		0			0
$370$	−4426.43			−0.621944
$371$	−7161.27	−	12403.7i	−1.00214	−	1.73576i
$372$		0			0
$373$	−2087.95	+	3616.44i	−0.289839	+	0.502016i	−0.973771	−	0.227530i	$-0.926935\pi$
$373$	−2087.95	+	3616.44i	−0.289839	+	0.502016i	0.683932	+	0.729546i	$0.260268\pi$
$374$	5633.97	−	9758.32i	0.778946	−	1.34917i
$375$		0			0
$376$	742.005	+	1285.19i	0.101771	+	0.176273i
$377$	−3580.03			−0.489074
$378$		0			0
$379$	1715.14			0.232457			0.116228	−	0.993223i	$-0.462920\pi$
$379$	1715.14			0.232457			0.116228	+	0.993223i	$0.462920\pi$
$380$	185.324	+	320.991i	0.0250182	+	0.0433328i
$381$		0			0
$382$	−1220.88	+	2114.63i	−0.163523	+	0.283230i
$383$	2569.53	−	4450.56i	0.342812	−	0.593768i	−0.642142	−	0.766586i	$-0.721954\pi$
$383$	2569.53	−	4450.56i	0.342812	−	0.593768i	0.984954	+	0.172818i	$0.0552873\pi$
$384$		0			0
$385$	−3849.39	−	6667.33i	−0.509566	−	0.882594i
$386$	4573.14			0.603022
$387$		0			0
$388$	−691.074			−0.0904226
$389$	4498.27	+	7791.23i	0.586301	+	1.01550i	0.994712	+	0.102705i	$0.0327498\pi$
$389$	4498.27	+	7791.23i	0.586301	+	1.01550i	−0.408411	+	0.912798i	$0.633917\pi$
$390$		0			0
$391$	−7192.67	+	12458.1i	−0.930304	+	1.61133i
$392$	−7315.92	+	12671.5i	−0.942627	+	1.63268i
$393$		0			0
$394$	5376.44	+	9312.27i	0.687465	+	1.19072i
$395$	1240.62			0.158031
$396$		0			0
$397$	−105.823			−0.0133781			−0.00668905	−	0.999978i	$-0.502129\pi$
$397$	−105.823			−0.0133781			−0.00668905	+	0.999978i	$0.502129\pi$
$398$	−3778.24	−	6544.10i	−0.475844	−	0.824186i
$399$		0			0
$400$	300.692	−	520.814i	0.0375865	−	0.0651017i
$401$	7240.91	−	12541.6i	0.901730	−	1.56184i	0.0764824	−	0.997071i	$-0.475631\pi$
$401$	7240.91	−	12541.6i	0.901730	−	1.56184i	0.825248	−	0.564771i	$-0.191036\pi$
$402$		0			0
$403$	−1803.25	−	3123.32i	−0.222894	−	0.386063i
$404$	2840.18			0.349763
$405$		0			0
$406$	−14642.0			−1.78983
$407$	−10441.9	−	18085.9i	−1.27171	−	2.20266i
$408$		0			0
$409$	430.532	−	745.704i	0.0520500	−	0.0901533i	−0.838826	−	0.544399i	$-0.816758\pi$
$409$	430.532	−	745.704i	0.0520500	−	0.0901533i	0.890876	+	0.454246i	$0.150091\pi$
$410$	404.713	−	700.983i	0.0487496	−	0.0844368i
$411$		0			0
$412$	−2439.86	−	4225.96i	−0.291755	−	0.505335i
$413$	−2929.52			−0.349037
$414$		0			0
$415$	−2762.53			−0.326764
$416$	−1149.96	−	1991.78i	−0.135532	−	0.234748i
$417$		0			0
$418$	1135.88	−	1967.39i	0.132913	−	0.230211i
$419$	−6508.87	+	11273.7i	−0.758900	+	1.31445i	0.184512	+	0.982830i	$0.440930\pi$
$419$	−6508.87	+	11273.7i	−0.758900	+	1.31445i	−0.943412	+	0.331623i	$0.892404\pi$
$420$		0			0
$421$	−7273.28	−	12597.7i	−0.841991	−	1.45837i	−0.888209	−	0.459439i	$-0.848050\pi$
$421$	−7273.28	−	12597.7i	−0.841991	−	1.45837i	0.0462188	−	0.998931i	$-0.485283\pi$
$422$	229.428			0.0264654
$423$		0			0
$424$	11386.7			1.30421
$425$	1320.85	+	2287.78i	0.150755	+	0.261115i
$426$		0			0
$427$	5482.63	−	9496.19i	0.621365	−	1.07624i
$428$	1703.27	−	2950.14i	0.192361	−	0.333179i
$429$		0			0
$430$	−168.693	−	292.185i	−0.0189188	−	0.0327683i
$431$	3539.94			0.395622			0.197811	−	0.980240i	$-0.436617\pi$
$431$	3539.94			0.395622			0.197811	+	0.980240i	$0.436617\pi$
$432$		0			0
$433$	669.471			0.0743019			0.0371509	−	0.999310i	$-0.488172\pi$
$433$	669.471			0.0743019			0.0371509	+	0.999310i	$0.488172\pi$
$434$	−7375.14	−	12774.1i	−0.815710	−	1.41285i
$435$		0			0
$436$	3693.55	−	6397.41i	0.405708	−	0.702707i
$437$	−1450.13	+	2511.70i	−0.158739	+	0.274944i
$438$		0			0
$439$	−6284.27	−	10884.7i	−0.683216	−	1.18337i	−0.973994	−	0.226575i	$-0.927247\pi$
$439$	−6284.27	−	10884.7i	−0.683216	−	1.18337i	0.290777	−	0.956791i	$-0.406086\pi$
$440$	6120.65			0.663160
$441$		0			0
$442$	−3585.45			−0.385843
$443$	5530.17	+	9578.53i	0.593106	+	1.02729i	0.993811	+	0.111083i	$0.0354320\pi$
$443$	5530.17	+	9578.53i	0.593106	+	1.02729i	−0.400705	+	0.916207i	$0.631235\pi$
$444$		0			0
$445$	728.696	−	1262.14i	0.0776259	−	0.134452i
$446$	−2064.49	+	3575.80i	−0.219185	+	0.379639i
$447$		0			0
$448$	−7657.23	−	13262.7i	−0.807522	−	1.39867i
$449$	−18553.9			−1.95014			−0.975072	−	0.221891i	$-0.928777\pi$
$449$	−18553.9			−1.95014			−0.975072	+	0.221891i	$0.928777\pi$
$450$		0			0
$451$	3818.84			0.398719
$452$	−3122.60	−	5408.51i	−0.324945	−	0.562821i
$453$		0			0
$454$	−3661.16	+	6341.32i	−0.378473	+	0.655535i
$455$	−1224.87	+	2121.54i	−0.126204	+	0.218592i
$456$		0			0
$457$	3401.13	+	5890.93i	0.348136	+	0.602989i	0.985918	−	0.167228i	$-0.0534814\pi$
$457$	3401.13	+	5890.93i	0.348136	+	0.602989i	−0.637782	+	0.770217i	$0.720148\pi$
$458$	−4399.78			−0.448882
$459$		0			0
$460$	−2368.52			−0.240071
$461$	7447.19	+	12898.9i	0.752386	+	1.30317i	0.946663	+	0.322224i	$0.104431\pi$
$461$	7447.19	+	12898.9i	0.752386	+	1.30317i	−0.194277	+	0.980947i	$0.562236\pi$
$462$		0			0
$463$	7144.33	−	12374.3i	0.717117	−	1.24208i	−0.245020	−	0.969518i	$-0.578794\pi$
$463$	7144.33	−	12374.3i	0.717117	−	1.24208i	0.962137	−	0.272566i	$-0.0878722\pi$
$464$	2698.09	−	4673.22i	0.269947	−	0.467562i
$465$		0			0
$466$	2943.87	+	5098.93i	0.292644	+	0.506874i
$467$	−13115.1			−1.29956			−0.649780	−	0.760122i	$-0.725139\pi$
$467$	−13115.1			−1.29956			−0.649780	+	0.760122i	$0.725139\pi$
$468$		0			0
$469$	2696.52			0.265488
$470$	323.184	+	559.771i	0.0317178	+	0.0549368i
$471$		0			0
$472$	1164.51	−	2016.99i	0.113561	−	0.196693i
$473$	795.887	−	1378.52i	0.0773677	−	0.134005i
$474$		0			0
$475$	266.300	+	461.244i	0.0257235	+	0.0445544i
$476$	11287.9			1.08694
$477$		0			0
$478$	−6258.96			−0.598908
$479$	3959.89	+	6858.74i	0.377729	+	0.654246i	0.990731	−	0.135835i	$-0.0433717\pi$
$479$	3959.89	+	6858.74i	0.377729	+	0.654246i	−0.613002	+	0.790081i	$0.710038\pi$
$480$		0			0
$481$	−3322.60	+	5754.91i	−0.314964	+	0.545533i
$482$	4756.70	−	8238.85i	0.449506	−	0.778567i
$483$		0			0
$484$	2060.81	+	3569.43i	0.193540	+	0.335220i
$485$	−993.032			−0.0929717
$486$		0			0
$487$	−17003.1			−1.58210			−0.791051	−	0.611751i	$-0.790466\pi$
$487$	−17003.1			−1.58210			−0.791051	+	0.611751i	$0.790466\pi$
$488$	4358.78	+	7549.63i	0.404329	+	0.700319i
$489$		0			0
$490$	−3186.48	+	5519.15i	−0.293777	+	0.508836i
$491$	3195.87	−	5535.41i	0.293743	−	0.508777i	−0.680949	−	0.732331i	$-0.738432\pi$
$491$	3195.87	−	5535.41i	0.293743	−	0.508777i	0.974692	+	0.223554i	$0.0717658\pi$
$492$		0			0
$493$	11851.9	+	20528.1i	1.08273	+	1.87534i
$494$	−722.870			−0.0658370
$495$		0			0
$496$	5436.07			0.492111
$497$	6334.85	+	10972.3i	0.571744	+	0.990289i
$498$		0			0
$499$	1337.06	−	2315.85i	0.119950	−	0.207759i	−0.799798	−	0.600270i	$-0.795060\pi$
$499$	1337.06	−	2315.85i	0.119950	−	0.207759i	0.919748	+	0.392510i	$0.128393\pi$
$500$	−217.476	+	376.679i	−0.0194516	+	0.0336912i
$501$		0			0
$502$	2255.18	+	3906.09i	0.200505	+	0.347285i
$503$	−1263.13			−0.111969			−0.0559843	−	0.998432i	$-0.517830\pi$
$503$	−1263.13			−0.111969			−0.0559843	+	0.998432i	$0.517830\pi$
$504$		0			0
$505$	4081.17			0.359624
$506$	7258.48	+	12572.1i	0.637706	+	1.10454i
$507$		0			0
$508$	−1302.93	+	2256.74i	−0.113796	+	0.197100i
$509$	690.421	−	1195.84i	0.0601226	−	0.104135i	−0.834397	−	0.551163i	$-0.814184\pi$
$509$	690.421	−	1195.84i	0.0601226	−	0.104135i	0.894520	+	0.447028i	$0.147518\pi$
$510$		0			0
$511$	5082.90	+	8803.84i	0.440028	+	0.762150i
$512$	8163.62			0.704657
$513$		0			0
$514$	9670.09			0.829824
$515$	−3505.93	−	6072.45i	−0.299980	−	0.519581i
$516$		0			0
$517$	−1524.77	+	2640.98i	−0.129709	+	0.224662i
$518$	−13589.2	+	23537.1i	−1.15265	+	1.99645i
$519$		0			0
$520$	−973.794	−	1686.66i	−0.0821225	−	0.142240i
$521$	−2689.80			−0.226185			−0.113092	−	0.993584i	$-0.536076\pi$
$521$	−2689.80			−0.226185			−0.113092	+	0.993584i	$0.536076\pi$
$522$		0			0
$523$	7144.18			0.597310			0.298655	−	0.954361i	$-0.403462\pi$
$523$	7144.18			0.597310			0.298655	+	0.954361i	$0.403462\pi$
$524$	4168.77	+	7220.52i	0.347545	+	0.601966i
$525$		0			0
$526$	3720.54	−	6444.16i	0.308409	−	0.534180i
$527$	−11939.6	+	20679.9i	−0.986897	+	1.70936i
$528$		0			0
$529$	−3183.13	−	5513.34i	−0.261620	−	0.453138i
$530$	4959.51			0.406467
$531$		0			0
$532$	2275.78			0.185466
$533$	−607.577	−	1052.35i	−0.0493754	−	0.0855207i
$534$		0			0
$535$	2447.49	−	4239.18i	0.197784	−	0.342571i
$536$	−1071.89	+	1856.57i	−0.0863779	+	0.149611i
$537$		0			0
$538$	−2757.84	−	4776.72i	−0.221002	−	0.382786i
$539$	−30067.4			−2.40278
$540$		0			0
$541$	−3310.57			−0.263091			−0.131546	−	0.991310i	$-0.541994\pi$
$541$	−3310.57			−0.263091			−0.131546	+	0.991310i	$0.541994\pi$
$542$	9055.35	+	15684.3i	0.717640	+	1.24299i
$543$		0			0
$544$	−7614.02	+	13187.9i	−0.600089	+	1.03938i
$545$	5307.41	−	9192.70i	0.417145	−	0.722517i
$546$		0			0
$547$	−1143.26	−	1980.18i	−0.0893643	−	0.154784i	0.817878	−	0.575391i	$-0.195150\pi$
$547$	−1143.26	−	1980.18i	−0.0893643	−	0.154784i	−0.907243	+	0.420608i	$0.861817\pi$
$548$	3495.19			0.272458
$549$		0			0
$550$	2665.88			0.206679
$551$	2389.49	+	4138.71i	0.184747	+	0.319991i
$552$		0			0
$553$	3808.70	−	6596.85i	0.292879	−	0.507282i
$554$	−2009.71	+	3480.92i	−0.154124	+	0.266950i
$555$		0			0
$556$	4180.80	+	7241.36i	0.318895	+	0.552342i
$557$	13846.6			1.05332			0.526658	−	0.850077i	$-0.323445\pi$
$557$	13846.6			1.05332			0.526658	+	0.850077i	$0.323445\pi$
$558$		0			0
$559$	−506.502			−0.0383233
$560$	−1846.25	−	3197.80i	−0.139318	−	0.241307i
$561$		0			0
$562$	−3336.61	+	5779.17i	−0.250438	+	0.433772i
$563$	8082.21	−	13998.8i	0.605017	−	1.04792i	−0.387032	−	0.922066i	$-0.626500\pi$
$563$	8082.21	−	13998.8i	0.605017	−	1.04792i	0.992049	−	0.125853i	$-0.0401669\pi$
$564$		0			0
$565$	−4487.00	−	7771.71i	−0.334105	−	0.578687i
$566$	−8651.37			−0.642481
$567$		0			0
$568$	−10072.6			−0.744080
$569$	−248.167	−	429.838i	−0.0182842	−	0.0316691i	0.856739	−	0.515751i	$-0.172487\pi$
$569$	−248.167	−	429.838i	−0.0182842	−	0.0316691i	−0.875023	+	0.484082i	$0.839154\pi$
$570$		0			0
$571$	−2985.59	+	5171.20i	−0.218815	+	0.378998i	−0.954446	−	0.298384i	$-0.903552\pi$
$571$	−2985.59	+	5171.20i	−0.218815	+	0.378998i	0.735631	+	0.677382i	$0.236886\pi$
$572$	1392.60	−	2412.05i	0.101796	−	0.176316i
$573$		0			0
$574$	−2484.94	−	4304.04i	−0.180696	−	0.312974i
$575$	−3403.42			−0.246839
$576$		0			0
$577$	9571.82			0.690607			0.345304	−	0.938491i	$-0.387776\pi$
$577$	9571.82			0.690607			0.345304	+	0.938491i	$0.387776\pi$
$578$	6647.07	+	11513.1i	0.478342	+	0.828513i
$579$		0			0
$580$	−1951.40	+	3379.92i	−0.139702	+	0.241971i
$581$	−8480.97	+	14689.5i	−0.605594	+	1.04892i
$582$		0			0
$583$	11699.4	+	20264.0i	0.831115	+	1.43953i
$584$	−8081.97			−0.572662
$585$		0			0
$586$	7734.71			0.545253
$587$	−2661.09	−	4609.14i	−0.187112	−	0.324088i	0.757174	−	0.653213i	$-0.226580\pi$
$587$	−2661.09	−	4609.14i	−0.187112	−	0.324088i	−0.944286	+	0.329125i	$0.893246\pi$
$588$		0			0
$589$	−2407.16	+	4169.31i	−0.168396	+	0.291670i
$590$	507.206	−	878.507i	0.0353921	−	0.0613010i
$591$		0			0
$592$	−5008.15	−	8674.38i	−0.347692	−	0.602221i
$593$	−11066.5			−0.766354			−0.383177	−	0.923675i	$-0.625170\pi$
$593$	−11066.5			−0.766354			−0.383177	+	0.923675i	$0.625170\pi$
$594$		0			0
$595$	16220.1			1.11758
$596$	−886.690	−	1535.79i	−0.0609400	−	0.105551i
$597$		0			0
$598$	2309.65	−	4000.42i	0.157941	−	0.273561i
$599$	−9192.50	+	15921.9i	−0.627037	+	1.08606i	0.361106	+	0.932525i	$0.382399\pi$
$599$	−9192.50	+	15921.9i	−0.627037	+	1.08606i	−0.988143	+	0.153536i	$0.950934\pi$
$600$		0			0
$601$	308.915	+	535.057i	0.0209666	+	0.0363152i	0.876318	−	0.481733i	$-0.159992\pi$
$601$	308.915	+	535.057i	0.0209666	+	0.0363152i	−0.855352	+	0.518048i	$0.826659\pi$
$602$	−2071.55			−0.140249
$603$		0			0
$604$	−5022.12			−0.338323
$605$	2961.26	+	5129.05i	0.198996	+	0.344670i
$606$		0			0
$607$	−6919.89	+	11985.6i	−0.462718	+	0.801451i	−0.999095	−	0.0425272i	$-0.986459\pi$
$607$	−6919.89	+	11985.6i	−0.462718	+	0.801451i	0.536377	+	0.843978i	$0.319792\pi$
$608$	−1535.08	+	2658.83i	−0.102394	+	0.177352i
$609$		0			0
$610$	1898.49	+	3288.28i	0.126012	+	0.218260i
$611$	970.362			0.0642498
$612$		0			0
$613$	−25655.4			−1.69039			−0.845196	−	0.534457i	$-0.820516\pi$
$613$	−25655.4			−1.69039			−0.845196	+	0.534457i	$0.820516\pi$
$614$	−7259.71	−	12574.2i	−0.477163	−	0.826470i
$615$		0			0
$616$	18790.4	−	32546.0i	1.22904	−	2.12876i
$617$	−2109.13	+	3653.12i	−0.137618	+	0.238361i	−0.926594	−	0.376062i	$-0.877278\pi$
$617$	−2109.13	+	3653.12i	−0.137618	+	0.238361i	0.788977	+	0.614423i	$0.210611\pi$
$618$		0			0
$619$	−6091.28	−	10550.4i	−0.395524	−	0.685067i	0.597644	−	0.801761i	$-0.296104\pi$
$619$	−6091.28	−	10550.4i	−0.395524	−	0.685067i	−0.993168	+	0.116694i	$0.962770\pi$
$620$	−3931.65			−0.254676
$621$		0			0
$622$	14034.6			0.904719
$623$	−4474.20	−	7749.54i	−0.287729	−	0.498361i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$	2941.05	−	5094.05i	0.187776	−	0.325238i
$627$		0			0
$628$	−3750.40	−	6495.89i	−0.238308	−	0.412761i
$629$	43998.8			2.78910
$630$		0			0
$631$	16673.2			1.05190			0.525951	−	0.850515i	$-0.323709\pi$
$631$	16673.2			1.05190			0.525951	+	0.850515i	$0.323709\pi$
$632$	3027.98	+	5244.61i	0.190580	+	0.330094i
$633$		0			0
$634$	−4852.24	+	8404.33i	−0.303954	+	0.526465i
$635$	−1872.23	+	3242.81i	−0.117004	+	0.202656i
$636$		0			0
$637$	4783.72	+	8285.65i	0.297548	+	0.515368i
$638$	23920.7			1.48437
$639$		0			0
$640$	−461.486			−0.0285028
$641$	−13135.4	−	22751.2i	−0.809388	−	1.40190i	−0.913289	−	0.407313i	$-0.866466\pi$
$641$	−13135.4	−	22751.2i	−0.809388	−	1.40190i	0.103901	−	0.994588i	$-0.466867\pi$
$642$		0			0
$643$	−5778.17	+	10008.1i	−0.354384	+	0.613810i	−0.987012	−	0.160645i	$-0.948643\pi$
$643$	−5778.17	+	10008.1i	−0.354384	+	0.613810i	0.632629	+	0.774455i	$0.281976\pi$
$644$	−7271.37	+	12594.4i	−0.444925	+	0.770634i
$645$		0			0
$646$	2393.11	+	4144.99i	0.145752	+	0.252449i
$647$	−25395.2			−1.54310			−0.771552	−	0.636166i	$-0.780519\pi$
$647$	−25395.2			−1.54310			−0.771552	+	0.636166i	$0.780519\pi$
$648$		0			0
$649$	4785.96			0.289469
$650$	−424.140	−	734.633i	−0.0255941	−	0.0443303i
$651$		0			0
$652$	−921.939	+	1596.85i	−0.0553772	+	0.0959161i
$653$	3091.42	−	5354.50i	0.185263	−	0.320885i	−0.758402	−	0.651787i	$-0.774020\pi$
$653$	3091.42	−	5354.50i	0.185263	−	0.320885i	0.943665	+	0.330902i	$0.107353\pi$
$654$		0			0
$655$	5990.27	+	10375.5i	0.357343	+	0.618935i
$656$	1831.60			0.109012
$657$		0			0
$658$	3968.70			0.235131
$659$	4442.66	+	7694.91i	0.262612	+	0.454858i	0.966935	−	0.255022i	$-0.0820827\pi$
$659$	4442.66	+	7694.91i	0.262612	+	0.454858i	−0.704323	+	0.709880i	$0.748749\pi$
$660$		0			0
$661$	3585.80	−	6210.79i	0.211001	−	0.365464i	−0.741027	−	0.671475i	$-0.765661\pi$
$661$	3585.80	−	6210.79i	0.211001	−	0.365464i	0.952028	+	0.306011i	$0.0989944\pi$
$662$	−4959.83	+	8590.68i	−0.291192	+	0.504360i
$663$		0			0
$664$	−6742.51	−	11678.4i	−0.394066	−	0.682543i
$665$	3270.16			0.190694
$666$		0			0
$667$	−30538.7			−1.77281
$668$	5183.37	+	8977.86i	0.300226	+	0.520006i
$669$		0			0
$670$	−466.866	+	808.636i	−0.0269203	+	0.0466273i
$671$	−8957.00	+	15514.0i	−0.515322	+	0.892564i
$672$		0			0
$673$	2125.26	+	3681.06i	0.121728	+	0.210839i	0.920449	−	0.390862i	$-0.127823\pi$
$673$	2125.26	+	3681.06i	0.121728	+	0.210839i	−0.798721	+	0.601701i	$0.794490\pi$
$674$	8094.27			0.462581
$675$		0			0
$676$	6758.47			0.384528
$677$	−1116.83	−	1934.40i	−0.0634019	−	0.109815i	0.832582	−	0.553902i	$-0.186862\pi$
$677$	−1116.83	−	1934.40i	−0.0634019	−	0.109815i	−0.895984	+	0.444086i	$0.853528\pi$
$678$		0			0
$679$	−3048.61	+	5280.35i	−0.172305	+	0.298441i
$680$	−6447.62	+	11167.6i	−0.363610	+	0.629791i
$681$		0			0
$682$	12048.8	+	20869.1i	0.676499	+	1.17173i
$683$	−6071.62			−0.340153			−0.170076	−	0.985431i	$-0.554401\pi$
$683$	−6071.62			−0.340153			−0.170076	+	0.985431i	$0.554401\pi$
$684$		0			0
$685$	5022.38			0.280139
$686$	8370.89	+	14498.8i	0.465892	+	0.806949i
$687$		0			0
$688$	381.725	−	661.167i	0.0211528	−	0.0366377i
$689$	3722.74	−	6447.98i	0.205842	−	0.356529i
$690$		0			0
$691$	382.892	+	663.188i	0.0210794	+	0.0365107i	0.876373	−	0.481634i	$-0.159956\pi$
$691$	382.892	+	663.188i	0.0210794	+	0.0365107i	−0.855293	+	0.518144i	$0.826623\pi$
$692$	6193.29			0.340222
$693$		0			0
$694$	4914.02			0.268780
$695$	6007.56	+	10405.4i	0.327885	+	0.567913i
$696$		0			0
$697$	−4022.85	+	6967.78i	−0.218617	+	0.378656i
$698$	−11558.9	+	20020.6i	−0.626806	+	1.08566i
$699$		0			0
$700$	1335.30	+	2312.82i	0.0720997	+	0.124880i
$701$	23564.3			1.26963			0.634814	−	0.772665i	$-0.281077\pi$
$701$	23564.3			1.26963			0.634814	+	0.772665i	$0.281077\pi$
$702$		0			0
$703$	8870.67			0.475909
$704$	12509.6	+	21667.3i	0.669709	+	1.15997i
$705$		0			0
$706$	3076.21	−	5328.16i	0.163987	−	0.284034i
$707$	12529.2	−	21701.3i	0.666492	−	1.15440i
$708$		0			0
$709$	−12086.1	−	20933.8i	−0.640204	−	1.10887i	−0.985387	−	0.170331i	$-0.945516\pi$
$709$	−12086.1	−	20933.8i	−0.640204	−	1.10887i	0.345183	−	0.938535i	$-0.387817\pi$
$710$	−4387.17			−0.231898
$711$		0			0
$712$	7114.13			0.374457
$713$	−15382.2	−	26642.8i	−0.807951	−	1.39941i
$714$		0			0
$715$	2001.08	−	3465.97i	0.104666	−	0.181287i
$716$	4934.55	−	8546.90i	0.257560	−	0.446107i
$717$		0			0
$718$	10498.2	+	18183.4i	0.545668	+	0.945125i
$719$	12829.0			0.665424			0.332712	−	0.943028i	$-0.392036\pi$
$719$	12829.0			0.665424			0.332712	+	0.943028i	$0.392036\pi$
$720$		0			0
$721$	−43052.9			−2.22382
$722$	−6809.05	−	11793.6i	−0.350979	−	0.607913i
$723$		0			0
$724$	1412.53	−	2446.58i	0.0725088	−	0.125589i
$725$	−2804.04	+	4856.74i	−0.143641	+	0.248793i
$726$		0			0
$727$	12362.4	+	21412.2i	0.630666	+	1.09235i	0.987416	+	0.158146i	$0.0505517\pi$
$727$	12362.4	+	21412.2i	0.630666	+	1.09235i	−0.356749	+	0.934200i	$0.616115\pi$
$728$	−11958.2			−0.608792
$729$		0			0
$730$	−3520.14			−0.178474
$731$	1676.81	+	2904.32i	0.0848413	+	0.146949i
$732$		0			0
$733$	10086.1	−	17469.7i	0.508239	−	0.880296i	−0.491715	−	0.870756i	$-0.663630\pi$
$733$	10086.1	−	17469.7i	0.508239	−	0.880296i	0.999954	−	0.00954010i	$-0.00303675\pi$
$734$	11398.3	−	19742.4i	0.573186	−	0.992788i
$735$		0			0
$736$	−9809.46	−	16990.5i	−0.491279	−	0.850921i
$737$	−4405.32			−0.220179
$738$		0			0
$739$	−16452.8			−0.818979			−0.409489	−	0.912315i	$-0.634293\pi$
$739$	−16452.8			−0.818979			−0.409489	+	0.912315i	$0.634293\pi$
$740$	3622.16	+	6273.76i	0.179937	+	0.311660i
$741$		0			0
$742$	15225.7	−	26371.7i	0.753307	−	1.30477i
$743$	−10432.3	+	18069.3i	−0.515107	+	0.892191i	0.484739	+	0.874659i	$0.338914\pi$
$743$	−10432.3	+	18069.3i	−0.515107	+	0.892191i	−0.999846	+	0.0175328i	$0.994419\pi$
$744$		0			0
$745$	−1274.12	−	2206.84i	−0.0626579	−	0.108527i
$746$	−8878.47			−0.435742
$747$		0			0
$748$	−18441.2			−0.901438
$749$	−15027.6	−	26028.6i	−0.733107	−	1.26978i
$750$		0			0
$751$	−7762.84	+	13445.6i	−0.377191	+	0.653313i	−0.990652	−	0.136411i	$-0.956443\pi$
$751$	−7762.84	+	13445.6i	−0.377191	+	0.653313i	0.613462	+	0.789725i	$0.289777\pi$
$752$	−731.313	+	1266.67i	−0.0354631	+	0.0614239i
$753$		0			0
$754$	−3805.78	−	6591.81i	−0.183818	−	0.318381i
$755$	−7216.48			−0.347861
$756$		0			0
$757$	30105.7			1.44546			0.722729	−	0.691132i	$-0.242888\pi$
$757$	30105.7			1.44546			0.722729	+	0.691132i	$0.242888\pi$
$758$	1823.30	+	3158.05i	0.0873684	+	0.151327i
$759$		0			0
$760$	−1299.92	+	2251.52i	−0.0620433	+	0.107462i
$761$	10869.9	−	18827.3i	0.517786	−	0.896831i	−0.482001	−	0.876171i	$-0.660090\pi$
$761$	10869.9	−	18827.3i	0.517786	−	0.896831i	0.999787	−	0.0206605i	$-0.00657692\pi$
$762$		0			0
$763$	−32587.5	−	56443.2i	−1.54620	−	2.67809i
$764$	3996.21			0.189238
$765$		0			0
$766$	10926.3			0.515381
$767$	−761.446	−	1318.86i	−0.0358464	−	0.0620878i
$768$		0			0
$769$	−971.112	+	1682.02i	−0.0455386	+	0.0788752i	−0.887896	−	0.460044i	$-0.847834\pi$
$769$	−971.112	+	1682.02i	−0.0455386	+	0.0788752i	0.842358	+	0.538919i	$0.181167\pi$
$770$	8184.25	−	14175.5i	0.383039	−	0.663443i
$771$		0			0
$772$	−3742.21	−	6481.70i	−0.174462	−	0.302178i
$773$	7921.02			0.368563			0.184282	−	0.982873i	$-0.441004\pi$
$773$	7921.02			0.368563			0.184282	+	0.982873i	$0.441004\pi$
$774$		0			0
$775$	−5649.55			−0.261855
$776$	−2423.70	−	4197.97i	−0.112121	−	0.194199i
$777$		0			0
$778$	−9563.85	+	16565.1i	−0.440721	+	0.763351i
$779$	−811.054	+	1404.79i	−0.0373030	+	0.0646107i
$780$		0			0
$781$	−10349.3	−	17925.5i	−0.474169	−	0.821285i
$782$	−30585.0			−1.39861
$783$		0			0
$784$	−14421.0			−0.656933
$785$	−5389.10	−	9334.20i	−0.245026	−	0.424397i
$786$		0			0
$787$	680.702	−	1179.01i	0.0308315	−	0.0534017i	−0.850198	−	0.526463i	$-0.823518\pi$
$787$	680.702	−	1179.01i	0.0308315	−	0.0534017i	0.881029	+	0.473061i	$0.156851\pi$
$788$	8799.10	−	15240.5i	0.397786	−	0.688985i
$789$		0			0
$790$	1318.85	+	2284.31i	0.0593956	+	0.102876i
$791$	−55100.4			−2.47679
$792$		0			0
$793$	5700.22			0.255260
$794$	−112.496	−	194.849i	−0.00502814	−	0.00870899i
$795$		0			0
$796$	−6183.48	+	10710.1i	−0.275336	+	0.476896i
$797$	13958.7	−	24177.2i	0.620381	−	1.07453i	−0.369033	−	0.929416i	$-0.620311\pi$
$797$	13958.7	−	24177.2i	0.620381	−	1.07453i	0.989415	−	0.145116i	$-0.0463555\pi$
$798$		0			0
$799$	−3212.45	−	5564.13i	−0.142238	−	0.246364i
$800$	−3602.79			−0.159222
$801$		0			0
$802$	30790.1			1.35565
$803$	−8303.95	−	14382.9i	−0.364932	−	0.632080i
$804$		0			0
$805$	−10448.5	+	18097.4i	−0.457468	+	0.792358i
$806$	3833.92	−	6640.55i	0.167549	−	0.290203i
$807$		0			0
$808$	9960.94	+	17252.9i	0.433694	+	0.751180i
$809$	−5275.87			−0.229283			−0.114641	−	0.993407i	$-0.536572\pi$
$809$	−5275.87			−0.229283			−0.114641	+	0.993407i	$0.536572\pi$
$810$		0			0
$811$	−23769.1			−1.02916			−0.514579	−	0.857443i	$-0.672052\pi$
$811$	−23769.1			−1.02916			−0.514579	+	0.857443i	$0.672052\pi$
$812$	11981.6	+	20752.7i	0.517822	+	0.896894i
$813$		0			0
$814$	22200.7	−	38452.7i	0.955938	−	1.65573i
$815$	−1324.77	+	2294.57i	−0.0569383	+	0.0986201i
$816$		0			0
$817$	338.064	+	585.545i	0.0144766	+	0.0250742i
$818$	1830.73			0.0782516
$819$		0			0
$820$	−1324.71			−0.0564157
$821$	−17104.6	−	29626.1i	−0.727108	−	1.25939i	−0.958100	−	0.286432i	$-0.907531\pi$
$821$	−17104.6	−	29626.1i	−0.727108	−	1.25939i	0.230992	−	0.972956i	$-0.425803\pi$
$822$		0			0
$823$	−620.065	+	1073.98i	−0.0262626	+	0.0454882i	−0.878858	−	0.477083i	$-0.841694\pi$
$823$	−620.065	+	1073.98i	−0.0262626	+	0.0454882i	0.852595	+	0.522572i	$0.175027\pi$
$824$	17113.9	−	29642.1i	0.723532	−	1.25319i
$825$		0			0
$826$	−3114.25	−	5394.04i	−0.131185	−	0.227219i
$827$	26971.0			1.13407			0.567034	−	0.823694i	$-0.308091\pi$
$827$	26971.0			1.13407			0.567034	+	0.823694i	$0.308091\pi$
$828$		0			0
$829$	−26743.1			−1.12042			−0.560208	−	0.828352i	$-0.689279\pi$
$829$	−26743.1			−1.12042			−0.560208	+	0.828352i	$0.689279\pi$
$830$	−2936.73	−	5086.57i	−0.122814	−	0.212720i
$831$		0			0
$832$	3980.56	−	6894.54i	0.165867	−	0.287290i
$833$	31673.7	−	54860.4i	1.31744	−	2.28187i
$834$		0			0
$835$	7448.19	+	12900.7i	0.308689	+	0.534665i
$836$	−3717.96			−0.153814
$837$		0			0
$838$	−27677.3			−1.14093
$839$	−2636.57	−	4566.67i	−0.108492	−	0.187913i	0.806668	−	0.591005i	$-0.201269\pi$
$839$	−2636.57	−	4566.67i	−0.108492	−	0.187913i	−0.915159	+	0.403092i	$0.867935\pi$
$840$		0			0
$841$	−12966.0	+	22457.7i	−0.531631	+	0.920813i
$842$	15463.9	−	26784.2i	0.632922	−	1.09625i
$843$		0			0
$844$	−187.742	−	325.178i	−0.00765679	−	0.0132620i
$845$	9711.51			0.395368
$846$		0			0
$847$	36364.3			1.47520
$848$	5611.29	+	9719.04i	0.227232	+	0.393577i
$849$		0			0
$850$	−2808.29	+	4864.10i	−0.113322	+	0.196279i
$851$	−28342.7	+	49091.1i	−1.14169	+	1.97746i
$852$		0			0
$853$	−2521.73	−	4367.76i	−0.101222	−	0.175322i	0.810966	−	0.585093i	$-0.198942\pi$
$853$	−2521.73	−	4367.76i	−0.101222	−	0.175322i	−0.912188	+	0.409771i	$0.865609\pi$
$854$	23313.5			0.934157
$855$		0			0
$856$	23894.4			0.954081
$857$	−7239.27	−	12538.8i	−0.288552	−	0.499786i	0.684913	−	0.728625i	$-0.259840\pi$
$857$	−7239.27	−	12538.8i	−0.288552	−	0.499786i	−0.973464	+	0.228839i	$0.926507\pi$
$858$		0			0
$859$	3936.94	−	6818.98i	0.156376	−	0.270850i	−0.777183	−	0.629274i	$-0.783352\pi$
$859$	3936.94	−	6818.98i	0.156376	−	0.270850i	0.933559	+	0.358424i	$0.116686\pi$
$860$	−276.083	+	478.190i	−0.0109469	+	0.0189606i
$861$		0			0
$862$	3763.17	+	6518.01i	0.148694	+	0.257545i
$863$	−24166.0			−0.953209			−0.476604	−	0.879118i	$-0.658133\pi$
$863$	−24166.0			−0.953209			−0.476604	+	0.879118i	$0.658133\pi$
$864$		0			0
$865$	8899.39			0.349813
$866$	711.688	+	1232.68i	0.0279262	+	0.0483697i
$867$		0			0
$868$	−12070.2	+	20906.2i	−0.471992	+	0.817513i
$869$	−6222.28	+	10777.3i	−0.242896	+	0.420708i
$870$		0			0
$871$	700.885	+	1213.97i	0.0272659	+	0.0472259i
$872$	51815.2			2.01225
$873$		0			0
$874$	−6166.29			−0.238647
$875$	1918.75	+	3323.38i	0.0741322	+	0.128401i
$876$		0			0
$877$	−3205.91	+	5552.80i	−0.123439	+	0.213802i	−0.921122	−	0.389275i	$-0.872726\pi$
$877$	−3205.91	+	5552.80i	−0.123439	+	0.213802i	0.797683	+	0.603077i	$0.206059\pi$
$878$	13361.1	−	23142.1i	0.513572	−	0.889532i
$879$		0			0
$880$	3016.23	+	5224.26i	0.115542	+	0.200125i
$881$	−22908.5			−0.876059			−0.438029	−	0.898961i	$-0.644323\pi$
$881$	−22908.5			−0.876059			−0.438029	+	0.898961i	$0.644323\pi$
$882$		0			0
$883$	39198.4			1.49392			0.746960	−	0.664869i	$-0.231513\pi$
$883$	39198.4			1.49392			0.746960	+	0.664869i	$0.231513\pi$
$884$	2933.98	+	5081.81i	0.111630	+	0.193348i
$885$		0			0
$886$	−11757.8	+	20365.1i	−0.445836	+	0.772211i
$887$	2711.10	−	4695.75i	0.102626	−	0.177754i	−0.810140	−	0.586237i	$-0.800609\pi$
$887$	2711.10	−	4695.75i	0.102626	−	0.177754i	0.912766	+	0.408483i	$0.133942\pi$
$888$		0			0
$889$	11495.5	+	19910.9i	0.433687	+	0.751168i
$890$	3098.59			0.116702
$891$		0			0
$892$	6757.51			0.253653
$893$	−647.668	−	1121.79i	−0.0242703	−	0.0420374i
$894$		0			0
$895$	7090.66	−	12281.4i	0.264821	−	0.458683i
$896$	−1416.76	+	2453.91i	−0.0528245	+	0.0914947i
$897$		0			0
$898$	−19723.9	−	34162.9i	−0.732958	−	1.26952i
$899$	−50693.0			−1.88065
$900$		0			0
$901$	−49297.6			−1.82280
$902$	4059.66	+	7031.54i	0.149858	+	0.259562i
$903$		0			0
$904$	21902.9	−	37936.9i	0.805839	−	1.39575i
$905$	2029.72	−	3515.59i	0.0745528	−	0.129129i
$906$		0			0
$907$	15363.5	+	26610.3i	0.562444	+	0.974181i	0.997282	+	0.0736727i	$0.0234720\pi$
$907$	15363.5	+	26610.3i	0.562444	+	0.974181i	−0.434839	+	0.900508i	$0.643195\pi$
$908$	11983.8			0.437990
$909$		0			0
$910$	−5208.45			−0.189735
$911$	2050.67	+	3551.86i	0.0745793	+	0.129175i	0.900903	−	0.434020i	$-0.142905\pi$
$911$	2050.67	+	3551.86i	0.0745793	+	0.129175i	−0.826324	+	0.563195i	$0.809572\pi$
$912$		0			0
$913$	13855.4	−	23998.2i	0.502242	−	0.869908i
$914$	−7231.21	+	12524.8i	−0.261693	+	0.453265i
$915$		0			0
$916$	3600.35	+	6235.98i	0.129868	+	0.224937i
$917$	73560.7			2.64906
$918$		0			0
$919$	−50926.3			−1.82797			−0.913984	−	0.405750i	$-0.867010\pi$
$919$	−50926.3			−1.82797			−0.913984	+	0.405750i	$0.867010\pi$
$920$	−8306.74	−	14387.7i	−0.297680	−	0.515596i
$921$		0			0
$922$	−15833.6	+	27424.6i	−0.565566	+	0.979589i
$923$	−3293.13	+	5703.87i	−0.117437	+	0.203408i
$924$		0			0
$925$	5204.82	+	9015.02i	0.185009	+	0.320445i
$926$	30379.4			1.07811
$927$		0			0
$928$	−32327.6			−1.14354
$929$	11020.4	+	19087.9i	0.389201	+	0.674116i	0.992342	−	0.123519i	$-0.0394179\pi$
$929$	11020.4	+	19087.9i	0.389201	+	0.674116i	−0.603141	+	0.797634i	$0.706085\pi$
$930$		0			0
$931$	6385.79	−	11060.5i	0.224797	−	0.389359i
$932$	4817.95	−	8344.93i	0.169332	−	0.293291i
$933$		0			0
$934$	−13942.1	−	24148.5i	−0.488438	−	0.845999i
$935$	−26498.8			−0.926850
$936$		0			0
$937$	28111.2			0.980097			0.490049	−	0.871695i	$-0.336979\pi$
$937$	28111.2			0.980097			0.490049	+	0.871695i	$0.336979\pi$
$938$	2866.56	+	4965.03i	0.0997831	+	0.172829i
$939$		0			0
$940$	528.924	−	916.123i	0.0183528	−	0.0317879i
$941$	26405.6	−	45735.9i	0.914770	−	1.58443i	0.107533	−	0.994201i	$-0.465705\pi$
$941$	26405.6	−	45735.9i	0.914770	−	1.58443i	0.807237	−	0.590227i	$-0.200962\pi$
$942$		0			0
$943$	−5182.81	−	8976.89i	−0.178977	−	0.309998i
$944$	2295.45			0.0791427
$945$		0			0
$946$	3384.30			0.116314
$947$	−913.114	−	1581.56i	−0.0313329	−	0.0542701i	0.849934	−	0.526890i	$-0.176642\pi$
$947$	−913.114	−	1581.56i	−0.0313329	−	0.0542701i	−0.881267	+	0.472619i	$0.843309\pi$
$948$		0			0
$949$	−2642.31	+	4576.62i	−0.0903826	+	0.156547i
$950$	−566.185	+	980.661i	−0.0193363	+	0.0334914i
$951$		0			0
$952$	39588.4	+	68569.2i	1.34776	+	2.33439i
$953$	41164.2			1.39920			0.699601	−	0.714534i	$-0.253361\pi$
$953$	41164.2			1.39920			0.699601	+	0.714534i	$0.253361\pi$
$954$		0			0
$955$	5742.31			0.194573
$956$	5121.72	+	8871.08i	0.173272	+	0.300116i
$957$		0			0
$958$	−8419.21	+	14582.5i	−0.283938	+	0.491795i
$959$	15418.7	−	26706.0i	0.519183	−	0.899251i
$960$		0			0
$961$	−10638.4	−	18426.2i	−0.357101	−	0.618517i
$962$	−14128.5			−0.473514
$963$		0			0
$964$	−15569.7			−0.520192
$965$	−5377.33	−	9313.81i	−0.179381	−	0.310697i
$966$		0			0
$967$	3544.39	−	6139.06i	0.117870	−	0.204156i	−0.801054	−	0.598593i	$-0.795727\pi$
$967$	3544.39	−	6139.06i	0.117870	−	0.204156i	0.918923	+	0.394436i	$0.129060\pi$
$968$	−14455.1	+	25037.0i	−0.479964	+	0.831321i
$969$		0			0
$970$	−1055.65	−	1828.44i	−0.0349433	−	0.0605235i
$971$	−2355.90			−0.0778625			−0.0389312	−	0.999242i	$-0.512395\pi$
$971$	−2355.90			−0.0778625			−0.0389312	+	0.999242i	$0.512395\pi$
$972$		0			0
$973$	73773.0			2.43068
$974$	−18075.3	−	31307.3i	−0.594630	−	1.02993i
$975$		0			0
$976$	−4295.97	+	7440.84i	−0.140892	+	0.244032i
$977$	−8397.55	+	14545.0i	−0.274986	+	0.476290i	−0.970132	−	0.242579i	$-0.922007\pi$
$977$	−8397.55	+	14545.0i	−0.274986	+	0.476290i	0.695146	+	0.718869i	$0.255340\pi$
$978$		0			0
$979$	7309.52	+	12660.5i	0.238624	+	0.413310i
$980$	10430.0			0.339974
$981$		0			0
$982$	13589.6			0.441611
$983$	−7366.92	−	12759.9i	−0.239032	−	0.414015i	0.721405	−	0.692513i	$-0.243497\pi$
$983$	−7366.92	−	12759.9i	−0.239032	−	0.414015i	−0.960437	+	0.278498i	$0.910163\pi$
$984$		0			0
$985$	12643.8	−	21899.7i	0.409000	−	0.708408i
$986$	−25198.6	+	43645.3i	−0.813881	+	1.40968i
$987$		0			0
$988$	591.526	+	1024.55i	0.0190475	+	0.0329913i
$989$	−4320.61			−0.138915
$990$		0			0
$991$	36809.2			1.17990			0.589950	−	0.807440i	$-0.299147\pi$
$991$	36809.2			1.17990			0.589950	+	0.807440i	$0.299147\pi$
$992$	−16283.3	−	28203.5i	−0.521165	−	0.902685i
$993$		0			0
$994$	−13468.6	+	23328.4i	−0.429778	+	0.744397i
$995$	−8885.29	+	15389.8i	−0.283098	+	0.490340i
$996$		0			0
$997$	10605.0	+	18368.4i	0.336875	+	0.583484i	0.983843	−	0.179033i	$-0.0572968\pi$
$997$	10605.0	+	18368.4i	0.336875	+	0.583484i	−0.646969	+	0.762517i	$0.723964\pi$
$998$	5685.49			0.180332
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.e.v.271.2		6
3.2	odd	2		405.4.e.q.271.2		6
9.2	odd	6		405.4.e.q.136.2		6
9.4	even	3		135.4.a.e.1.2	✓	3
9.5	odd	6		135.4.a.h.1.2	yes	3
9.7	even	3	inner	405.4.e.v.136.2		6
36.23	even	6		2160.4.a.bq.1.1		3
36.31	odd	6		2160.4.a.bi.1.1		3
45.4	even	6		675.4.a.s.1.2		3
45.13	odd	12		675.4.b.m.649.4		6
45.14	odd	6		675.4.a.p.1.2		3
45.22	odd	12		675.4.b.m.649.3		6
45.23	even	12		675.4.b.n.649.3		6
45.32	even	12		675.4.b.n.649.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.e.1.2	✓	3	9.4	even	3
135.4.a.h.1.2	yes	3	9.5	odd	6
405.4.e.q.136.2		6	9.2	odd	6
405.4.e.q.271.2		6	3.2	odd	2
405.4.e.v.136.2		6	9.7	even	3	inner
405.4.e.v.271.2		6	1.1	even	1	trivial
675.4.a.p.1.2		3	45.14	odd	6
675.4.a.s.1.2		3	45.4	even	6
675.4.b.m.649.3		6	45.22	odd	12
675.4.b.m.649.4		6	45.13	odd	12
675.4.b.n.649.3		6	45.23	even	12
675.4.b.n.649.4		6	45.32	even	12
2160.4.a.bi.1.1		3	36.31	odd	6
2160.4.a.bq.1.1		3	36.23	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.06306 + 1.84127i) q^{2} +(1.73981 - 3.01344i) q^{4} +(2.50000 - 4.33013i) q^{5} +(-15.3500 - 26.5870i) q^{7} +24.4070 q^{8} +10.6306 q^{10} +(25.0774 + 43.4353i) q^{11} +(7.97962 - 13.8211i) q^{13} +(32.6360 - 56.5272i) q^{14} +(12.0277 + 20.8325i) q^{16} -105.668 q^{17} -21.3040 q^{19} +(-8.69904 - 15.0672i) q^{20} +(-53.3175 + 92.3487i) q^{22} +(68.0684 - 117.898i) q^{23} +(-12.5000 - 21.6506i) q^{25} +33.9312 q^{26} -106.824 q^{28} +(-112.162 - 194.270i) q^{29} +(112.991 - 195.706i) q^{31} +(72.0559 - 124.804i) q^{32} +(-112.332 - 194.564i) q^{34} -153.500 q^{35} -416.386 q^{37} +(-22.6474 - 39.2264i) q^{38} +(61.0176 - 105.686i) q^{40} +(38.0706 - 65.9402i) q^{41} +(-15.8686 - 27.4852i) q^{43} +174.519 q^{44} +289.443 q^{46} +(30.4013 + 52.6566i) q^{47} +(-299.746 + 519.176i) q^{49} +(26.5765 - 46.0318i) q^{50} +(-27.7660 - 48.0921i) q^{52} +466.532 q^{53} +250.774 q^{55} +(-374.649 - 648.910i) q^{56} +(238.469 - 413.040i) q^{58} +(47.7119 - 82.6395i) q^{59} +(178.587 + 309.322i) q^{61} +480.464 q^{62} +498.842 q^{64} +(-39.8981 - 69.1055i) q^{65} +(-43.9172 + 76.0668i) q^{67} +(-183.842 + 318.424i) q^{68} +(-163.180 - 282.636i) q^{70} -412.693 q^{71} -331.133 q^{73} +(-442.643 - 766.680i) q^{74} +(-37.0648 + 64.1981i) q^{76} +(769.877 - 1333.47i) q^{77} +(124.062 + 214.881i) q^{79} +120.277 q^{80} +161.885 q^{82} +(-276.253 - 478.484i) q^{83} +(-264.171 + 457.557i) q^{85} +(33.7386 - 58.4369i) q^{86} +(612.065 + 1060.13i) q^{88} +291.478 q^{89} -489.949 q^{91} +(-236.852 - 410.240i) q^{92} +(-64.6368 + 111.954i) q^{94} +(-53.2599 + 92.2489i) q^{95} +(-99.3032 - 171.998i) q^{97} -1274.59 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{2} - 17 q^{4} + 15 q^{5} + 4 q^{7} - 150 q^{8} + 50 q^{10} + 5 q^{11} - 7 q^{13} + 60 q^{14} - 161 q^{16} - 310 q^{17} - 100 q^{19} + 85 q^{20} + 229 q^{22} + 285 q^{23} - 75 q^{25} - 370 q^{26}+ \cdots + 610 q^{98}+O(q^{100}) \) 6 * q + 5 * q^2 - 17 * q^4 + 15 * q^5 + 4 * q^7 - 150 * q^8 + 50 * q^10 + 5 * q^11 - 7 * q^13 + 60 * q^14 - 161 * q^16 - 310 * q^17 - 100 * q^19 + 85 * q^20 + 229 * q^22 + 285 * q^23 - 75 * q^25 - 370 * q^26 - 668 * q^28 + 115 * q^29 + 115 * q^31 + 775 * q^32 - 413 * q^34 + 40 * q^35 - 768 * q^37 - 1150 * q^38 - 375 * q^40 + 580 * q^41 + 797 * q^43 + 2830 * q^44 - 570 * q^46 - 145 * q^47 - 577 * q^49 + 125 * q^50 - 825 * q^52 + 800 * q^53 + 50 * q^55 - 2190 * q^56 + 59 * q^58 + 380 * q^59 + 152 * q^61 + 2010 * q^62 + 5874 * q^64 + 35 * q^65 - 2 * q^67 + 475 * q^68 - 300 * q^70 - 80 * q^71 - 1960 * q^73 - 2720 * q^74 + 3276 * q^76 + 1950 * q^77 - 1013 * q^79 - 1610 * q^80 + 8 * q^82 + 270 * q^83 - 775 * q^85 - 1555 * q^86 + 5193 * q^88 - 2040 * q^89 - 1264 * q^91 - 1215 * q^92 - 3833 * q^94 - 250 * q^95 - 720 * q^97 + 610 * q^98

Introduction

L-functions

Modular forms

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Representations

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Database

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