LMFDB - Embedded newform 135.4.e.a.46.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [135,4,Mod(46,135)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(135, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("135.46"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	135.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$7.96525785077$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			46.2
Root			$1.68614 - 0.396143i$ of defining polynomial
Character	$\chi$	$=$	135.46
Dual form			135.4.e.a.91.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.18614 - 2.05446i) q^{2} +(1.18614 + 2.05446i) q^{4} +(2.50000 + 4.33013i) q^{5} +(-3.68614 + 6.38458i) q^{7} +24.6060 q^{8} +11.8614 q^{10} +(-2.06930 + 3.58413i) q^{11} +(39.4891 + 68.3972i) q^{13} +(8.74456 + 15.1460i) q^{14} +(19.6970 - 34.1162i) q^{16} +33.3070 q^{17} +89.3070 q^{19} +(-5.93070 + 10.2723i) q^{20} +(4.90895 + 8.50256i) q^{22} +(-99.7812 - 172.826i) q^{23} +(-12.5000 + 21.6506i) q^{25} +187.359 q^{26} -17.4891 q^{28} +(25.2405 - 43.7179i) q^{29} +(3.00000 + 5.19615i) q^{31} +(51.6970 + 89.5419i) q^{32} +(39.5068 - 68.4278i) q^{34} -36.8614 q^{35} -290.277 q^{37} +(105.931 - 183.477i) q^{38} +(61.5149 + 106.547i) q^{40} +(-26.6684 - 46.1911i) q^{41} +(149.194 - 258.412i) q^{43} -9.81791 q^{44} -473.418 q^{46} +(-209.452 + 362.782i) q^{47} +(144.325 + 249.978i) q^{49} +(29.6535 + 51.3614i) q^{50} +(-93.6793 + 162.257i) q^{52} -399.228 q^{53} -20.6930 q^{55} +(-90.7011 + 157.099i) q^{56} +(-59.8776 - 103.711i) q^{58} +(49.1209 + 85.0799i) q^{59} +(341.797 - 592.011i) q^{61} +14.2337 q^{62} +560.432 q^{64} +(-197.446 + 341.986i) q^{65} +(-112.750 - 195.289i) q^{67} +(39.5068 + 68.4278i) q^{68} +(-43.7228 + 75.7301i) q^{70} -512.951 q^{71} -994.318 q^{73} +(-344.310 + 596.362i) q^{74} +(105.931 + 183.477i) q^{76} +(-15.2554 - 26.4232i) q^{77} +(100.853 - 174.683i) q^{79} +196.970 q^{80} -126.530 q^{82} +(553.822 - 959.247i) q^{83} +(83.2676 + 144.224i) q^{85} +(-353.931 - 613.026i) q^{86} +(-50.9171 + 88.1909i) q^{88} -372.269 q^{89} -582.250 q^{91} +(236.709 - 409.992i) q^{92} +(496.880 + 860.622i) q^{94} +(223.268 + 386.711i) q^{95} +(69.6156 - 120.578i) q^{97} +684.758 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} - q^{4} + 10 q^{5} - 9 q^{7} + 18 q^{8} - 10 q^{10} - 37 q^{11} + 112 q^{13} + 12 q^{14} + 119 q^{16} - 154 q^{17} + 70 q^{19} + 5 q^{20} - 101 q^{22} - 267 q^{23} - 50 q^{25} + 152 q^{26}+ \cdots - 926 q^{98}+O(q^{100}) $ 4 * q - q^2 - q^4 + 10 * q^5 - 9 * q^7 + 18 * q^8 - 10 * q^10 - 37 * q^11 + 112 * q^13 + 12 * q^14 + 119 * q^16 - 154 * q^17 + 70 * q^19 + 5 * q^20 - 101 * q^22 - 267 * q^23 - 50 * q^25 + 152 * q^26 - 24 * q^28 + 325 * q^29 + 12 * q^31 + 247 * q^32 + 451 * q^34 - 90 * q^35 - 1276 * q^37 + 395 * q^38 + 45 * q^40 + 238 * q^41 + 97 * q^43 + 202 * q^44 - 492 * q^46 - 901 * q^47 + 629 * q^49 - 25 * q^50 - 76 * q^52 - 448 * q^53 - 370 * q^55 - 156 * q^56 + 806 * q^58 - 85 * q^59 + 247 * q^61 - 12 * q^62 + 1426 * q^64 - 560 * q^65 + 606 * q^67 + 451 * q^68 - 60 * q^70 - 788 * q^71 - 1622 * q^73 + 484 * q^74 + 395 * q^76 - 84 * q^77 + 840 * q^79 + 1190 * q^80 - 2218 * q^82 - 387 * q^83 - 385 * q^85 - 1387 * q^86 + 411 * q^88 + 2130 * q^89 - 1272 * q^91 + 246 * q^92 - 632 * q^94 + 175 * q^95 + 1031 * q^97 - 926 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/135\mathbb{Z}\right)^\times$.

$n$	$56$	$82$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$	1.18614	−	2.05446i	0.419364	−	0.726360i	−0.576512	−	0.817089i	$-0.695586\pi$
$2$	1.18614	−	2.05446i	0.419364	−	0.726360i	0.995876	+	0.0907292i	$0.0289198\pi$
$3$		0			0
$3$		0			0
$4$	1.18614	+	2.05446i	0.148268	+	0.256807i
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i
$6$		0			0
$7$	−3.68614	+	6.38458i	−0.199033	+	0.344735i	−0.948215	−	0.317629i	$-0.897113\pi$
$7$	−3.68614	+	6.38458i	−0.199033	+	0.344735i	0.749182	+	0.662364i	$0.230447\pi$
$8$	24.6060			1.08744
$9$		0			0
$10$	11.8614			0.375091
$11$	−2.06930	+	3.58413i	−0.0567197	+	0.0982414i	−0.892991	−	0.450074i	$-0.851397\pi$
$11$	−2.06930	+	3.58413i	−0.0567197	+	0.0982414i	0.836271	+	0.548316i	$0.184731\pi$
$12$		0			0
$13$	39.4891	+	68.3972i	0.842486	+	1.45923i	0.887787	+	0.460254i	$0.152242\pi$
$13$	39.4891	+	68.3972i	0.842486	+	1.45923i	−0.0453014	+	0.998973i	$0.514425\pi$
$14$	8.74456	+	15.1460i	0.166934	+	0.289139i
$15$		0			0
$16$	19.6970	−	34.1162i	0.307766	−	0.533066i
$17$	33.3070			0.475185			0.237592	−	0.971365i	$-0.423642\pi$
$17$	33.3070			0.475185			0.237592	+	0.971365i	$0.423642\pi$
$18$		0			0
$19$	89.3070			1.07834			0.539169	−	0.842197i	$-0.318738\pi$
$19$	89.3070			1.07834			0.539169	+	0.842197i	$0.318738\pi$
$20$	−5.93070	+	10.2723i	−0.0663073	+	0.114848i
$21$		0			0
$22$	4.90895	+	8.50256i	0.0475724	+	0.0823978i
$23$	−99.7812	−	172.826i	−0.904601	−	1.56682i	−0.821452	−	0.570278i	$-0.806835\pi$
$23$	−99.7812	−	172.826i	−0.904601	−	1.56682i	−0.0831494	−	0.996537i	$-0.526498\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.100000	+	0.173205i
$26$	187.359			1.41323
$27$		0			0
$28$	−17.4891			−0.118041
$29$	25.2405	−	43.7179i	0.161622	−	0.279938i	−0.773828	−	0.633395i	$-0.781661\pi$
$29$	25.2405	−	43.7179i	0.161622	−	0.279938i	0.935451	+	0.353457i	$0.114994\pi$
$30$		0			0
$31$	3.00000	+	5.19615i	0.0173812	+	0.0301050i	0.874585	−	0.484872i	$-0.161134\pi$
$31$	3.00000	+	5.19615i	0.0173812	+	0.0301050i	−0.857204	+	0.514977i	$0.827800\pi$
$32$	51.6970	+	89.5419i	0.285588	+	0.494654i
$33$		0			0
$34$	39.5068	−	68.4278i	0.199275	−	0.345155i
$35$	−36.8614			−0.178020
$36$		0			0
$37$	−290.277			−1.28976			−0.644882	−	0.764282i	$-0.723094\pi$
$37$	−290.277			−1.28976			−0.644882	+	0.764282i	$0.723094\pi$
$38$	105.931	−	183.477i	0.452217	−	0.783262i
$39$		0			0
$40$	61.5149	+	106.547i	0.243159	+	0.421164i
$41$	−26.6684	−	46.1911i	−0.101583	−	0.175947i	0.810754	−	0.585387i	$-0.199057\pi$
$41$	−26.6684	−	46.1911i	−0.101583	−	0.175947i	−0.912337	+	0.409440i	$0.865724\pi$
$42$		0			0
$43$	149.194	−	258.412i	0.529114	−	0.916453i	−0.470309	−	0.882502i	$-0.655858\pi$
$43$	149.194	−	258.412i	0.529114	−	0.916453i	0.999424	−	0.0339510i	$-0.0108090\pi$
$44$	−9.81791			−0.0336388
$45$		0			0
$46$	−473.418			−1.51743
$47$	−209.452	+	362.782i	−0.650038	+	1.12590i	0.333075	+	0.942900i	$0.391914\pi$
$47$	−209.452	+	362.782i	−0.650038	+	1.12590i	−0.983113	+	0.182998i	$0.941420\pi$
$48$		0			0
$49$	144.325	+	249.978i	0.420772	+	0.728798i
$50$	29.6535	+	51.3614i	0.0838728	+	0.145272i
$51$		0			0
$52$	−93.6793	+	162.257i	−0.249827	+	0.432712i
$53$	−399.228			−1.03468			−0.517342	−	0.855779i	$-0.673078\pi$
$53$	−399.228			−1.03468			−0.517342	+	0.855779i	$0.673078\pi$
$54$		0			0
$55$	−20.6930			−0.0507316
$56$	−90.7011	+	157.099i	−0.216436	+	0.374879i
$57$		0			0
$58$	−59.8776	−	103.711i	−0.135557	−	0.234792i
$59$	49.1209	+	85.0799i	0.108390	+	0.187737i	0.915118	−	0.403186i	$-0.132097\pi$
$59$	49.1209	+	85.0799i	0.108390	+	0.187737i	−0.806728	+	0.590923i	$0.798764\pi$
$60$		0			0
$61$	341.797	−	592.011i	0.717421	−	1.24261i	−0.244597	−	0.969625i	$-0.578656\pi$
$61$	341.797	−	592.011i	0.717421	−	1.24261i	0.962018	−	0.272985i	$-0.0880109\pi$
$62$	14.2337			0.0291561
$63$		0			0
$64$	560.432			1.09459
$65$	−197.446	+	341.986i	−0.376771	+	0.652587i
$66$		0			0
$67$	−112.750	−	195.289i	−0.205591	−	0.356094i	0.744730	−	0.667366i	$-0.232578\pi$
$67$	−112.750	−	195.289i	−0.205591	−	0.356094i	−0.950321	+	0.311272i	$0.899245\pi$
$68$	39.5068	+	68.4278i	0.0704545	+	0.122031i
$69$		0			0
$70$	−43.7228	+	75.7301i	−0.0746554	+	0.129307i
$71$	−512.951			−0.857410			−0.428705	−	0.903445i	$-0.641030\pi$
$71$	−512.951			−0.857410			−0.428705	+	0.903445i	$0.641030\pi$
$72$		0			0
$73$	−994.318			−1.59419			−0.797096	−	0.603852i	$-0.793632\pi$
$73$	−994.318			−1.59419			−0.797096	+	0.603852i	$0.793632\pi$
$74$	−344.310	+	596.362i	−0.540881	+	0.936833i
$75$		0			0
$76$	105.931	+	183.477i	0.159883	+	0.276925i
$77$	−15.2554	−	26.4232i	−0.0225782	−	0.0391065i
$78$		0			0
$79$	100.853	−	174.683i	0.143631	−	0.248777i	−0.785230	−	0.619204i	$-0.787455\pi$
$79$	100.853	−	174.683i	0.143631	−	0.248777i	0.928862	+	0.370427i	$0.120789\pi$
$80$	196.970			0.275274
$81$		0			0
$82$	−126.530			−0.170401
$83$	553.822	−	959.247i	0.732408	−	1.26857i	−0.223444	−	0.974717i	$-0.571730\pi$
$83$	553.822	−	959.247i	0.732408	−	1.26857i	0.955851	−	0.293850i	$-0.0949367\pi$
$84$		0			0
$85$	83.2676	+	144.224i	0.106255	+	0.184038i
$86$	−353.931	−	613.026i	−0.443783	−	0.768655i
$87$		0			0
$88$	−50.9171	+	88.1909i	−0.0616793	+	0.106832i
$89$	−372.269			−0.443375			−0.221688	−	0.975118i	$-0.571157\pi$
$89$	−372.269			−0.443375			−0.221688	+	0.975118i	$0.571157\pi$
$90$		0			0
$91$	−582.250			−0.670729
$92$	236.709	−	409.992i	0.268246	−	0.464616i
$93$		0			0
$94$	496.880	+	860.622i	0.545205	+	0.944323i
$95$	223.268	+	386.711i	0.241124	+	0.417639i
$96$		0			0
$97$	69.6156	−	120.578i	0.0728700	−	0.126215i	−0.827288	−	0.561778i	$-0.810118\pi$
$97$	69.6156	−	120.578i	0.0728700	−	0.126215i	0.900158	+	0.435563i	$0.143451\pi$
$98$	684.758			0.705826
$99$		0			0
$100$	−59.3070			−0.0593070
$101$	985.872	−	1707.58i	0.971267	−	1.68228i	0.279525	−	0.960139i	$-0.409823\pi$
$101$	985.872	−	1707.58i	0.971267	−	1.68228i	0.691742	−	0.722145i	$-0.256844\pi$
$102$		0			0
$103$	−906.353	−	1569.85i	−0.867045	−	1.50177i	−0.865002	−	0.501768i	$-0.832683\pi$
$103$	−906.353	−	1569.85i	−0.867045	−	1.50177i	−0.00204255	−	0.999998i	$-0.500650\pi$
$104$	971.668	+	1682.98i	0.916153	+	1.58682i
$105$		0			0
$106$	−473.541	+	820.197i	−0.433909	+	0.751552i
$107$	259.217			0.234201			0.117100	−	0.993120i	$-0.462640\pi$
$107$	259.217			0.234201			0.117100	+	0.993120i	$0.462640\pi$
$108$		0			0
$109$	775.556			0.681512			0.340756	−	0.940152i	$-0.389317\pi$
$109$	775.556			0.681512			0.340756	+	0.940152i	$0.389317\pi$
$110$	−24.5448	+	42.5128i	−0.0212750	+	0.0368494i
$111$		0			0
$112$	145.212	+	251.514i	0.122511	+	0.212195i
$113$	−107.628	−	186.417i	−0.0895997	−	0.155191i	0.817742	−	0.575585i	$-0.195225\pi$
$113$	−107.628	−	186.417i	−0.0895997	−	0.155191i	−0.907342	+	0.420393i	$0.861892\pi$
$114$		0			0
$115$	498.906	−	864.131i	0.404550	−	0.700701i
$116$	119.755			0.0958534
$117$		0			0
$118$	233.057			0.181819
$119$	−122.774	+	212.652i	−0.0945774	+	0.163813i
$120$		0			0
$121$	656.936	+	1137.85i	0.493566	+	0.854881i
$122$	−810.840	−	1404.42i	−0.601721	−	1.04221i
$123$		0			0
$124$	−7.11684	+	12.3267i	−0.00515412	+	0.00892721i
$125$	−125.000			−0.0894427
$126$		0			0
$127$	2424.76			1.69420			0.847098	−	0.531436i	$-0.178348\pi$
$127$	2424.76			1.69420			0.847098	+	0.531436i	$0.178348\pi$
$128$	251.175	−	435.048i	0.173445	−	0.300415i
$129$		0			0
$130$	468.397	+	811.287i	0.316008	+	0.547343i
$131$	130.247	+	225.595i	0.0868684	+	0.150461i	0.906186	−	0.422880i	$-0.138981\pi$
$131$	130.247	+	225.595i	0.0868684	+	0.150461i	−0.819317	+	0.573340i	$0.805647\pi$
$132$		0			0
$133$	−329.198	+	570.188i	−0.214625	+	0.371741i
$134$	−534.949			−0.344870
$135$		0			0
$136$	819.552			0.516735
$137$	−202.319	+	350.427i	−0.126170	+	0.218533i	−0.922190	−	0.386738i	$-0.873602\pi$
$137$	−202.319	+	350.427i	−0.126170	+	0.218533i	0.796020	+	0.605271i	$0.206935\pi$
$138$		0			0
$139$	883.841	+	1530.86i	0.539327	+	0.934141i	0.998940	+	0.0460221i	$0.0146545\pi$
$139$	883.841	+	1530.86i	0.539327	+	0.934141i	−0.459614	+	0.888119i	$0.652012\pi$
$140$	−43.7228	−	75.7301i	−0.0263947	−	0.0457169i
$141$		0			0
$142$	−608.432	+	1053.84i	−0.359567	+	0.622788i
$143$	−326.859			−0.191142
$144$		0			0
$145$	252.405			0.144559
$146$	−1179.40	+	2042.78i	−0.668547	+	1.15796i
$147$		0			0
$148$	−344.310	−	596.362i	−0.191230	−	0.331220i
$149$	960.344	+	1663.36i	0.528016	+	0.914551i	0.999467	+	0.0326584i	$0.0103973\pi$
$149$	960.344	+	1663.36i	0.528016	+	0.914551i	−0.471450	+	0.881893i	$0.656269\pi$
$150$		0			0
$151$	1262.28	−	2186.33i	0.680284	−	1.17829i	−0.294611	−	0.955617i	$-0.595190\pi$
$151$	1262.28	−	2186.33i	0.680284	−	1.17829i	0.974894	−	0.222668i	$-0.0714767\pi$
$152$	2197.49			1.17263
$153$		0			0
$154$	−72.3804			−0.0378739
$155$	−15.0000	+	25.9808i	−0.00777309	+	0.0134634i
$156$		0			0
$157$	−971.350	−	1682.43i	−0.493772	−	0.855238i	0.506202	−	0.862415i	$-0.331049\pi$
$157$	−971.350	−	1682.43i	−0.493772	−	0.855238i	−0.999974	+	0.00717683i	$0.997716\pi$
$158$	−239.252	−	414.397i	−0.120468	−	0.208656i
$159$		0			0
$160$	−258.485	+	447.709i	−0.127719	+	0.221216i
$161$	1471.23			0.720181
$162$		0			0
$163$	−1051.21			−0.505134			−0.252567	−	0.967579i	$-0.581275\pi$
$163$	−1051.21			−0.505134			−0.252567	+	0.967579i	$0.581275\pi$
$164$	63.2650	−	109.578i	0.0301230	−	0.0521745i
$165$		0			0
$166$	−1313.82	−	2275.60i	−0.614291	−	1.06398i
$167$	−1408.22	−	2439.11i	−0.652524	−	1.13020i	−0.982508	−	0.186218i	$-0.940377\pi$
$167$	−1408.22	−	2439.11i	−0.652524	−	1.13020i	0.329985	−	0.943986i	$-0.392956\pi$
$168$		0			0
$169$	−2020.28	+	3499.23i	−0.919564	+	1.59273i
$170$	395.068			0.178237
$171$		0			0
$172$	707.861			0.313802
$173$	−721.225	+	1249.20i	−0.316958	+	0.548987i	−0.979852	−	0.199726i	$-0.935995\pi$
$173$	−721.225	+	1249.20i	−0.316958	+	0.548987i	0.662894	+	0.748713i	$0.269328\pi$
$174$		0			0
$175$	−92.1535	−	159.615i	−0.0398066	−	0.0689470i
$176$	81.5179	+	141.193i	0.0349128	+	0.0604707i
$177$		0			0
$178$	−441.563	+	764.809i	−0.185936	+	0.322050i
$179$	−1534.89			−0.640912			−0.320456	−	0.947263i	$-0.603836\pi$
$179$	−1534.89			−0.640912			−0.320456	+	0.947263i	$0.603836\pi$
$180$		0			0
$181$	−3650.43			−1.49908			−0.749542	−	0.661956i	$-0.769726\pi$
$181$	−3650.43			−1.49908			−0.749542	+	0.661956i	$0.769726\pi$
$182$	−690.630	+	1196.21i	−0.281280	+	0.487191i
$183$		0			0
$184$	−2455.21	−	4252.56i	−0.983700	−	1.70382i
$185$	−725.693	−	1256.94i	−0.288400	−	0.499524i
$186$		0			0
$187$	−68.9221	+	119.377i	−0.0269523	+	0.0466828i
$188$	−993.760			−0.385518
$189$		0			0
$190$	1059.31			0.404475
$191$	678.644	−	1175.45i	0.257094	−	0.445300i	−0.708368	−	0.705843i	$-0.750568\pi$
$191$	678.644	−	1175.45i	0.257094	−	0.445300i	0.965462	+	0.260543i	$0.0839017\pi$
$192$		0			0
$193$	901.520	+	1561.48i	0.336232	+	0.582371i	0.983721	−	0.179704i	$-0.0575139\pi$
$193$	901.520	+	1561.48i	0.336232	+	0.582371i	−0.647488	+	0.762075i	$0.724181\pi$
$194$	−165.148	−	286.044i	−0.0611181	−	0.105860i
$195$		0			0
$196$	−342.379	+	593.018i	−0.124774	+	0.216114i
$197$	263.403			0.0952624			0.0476312	−	0.998865i	$-0.484833\pi$
$197$	263.403			0.0952624			0.0476312	+	0.998865i	$0.484833\pi$
$198$		0			0
$199$	492.853			0.175565			0.0877824	−	0.996140i	$-0.472022\pi$
$199$	492.853			0.175565			0.0877824	+	0.996140i	$0.472022\pi$
$200$	−307.575	+	532.735i	−0.108744	+	0.188350i
$201$		0			0
$202$	−2338.77	−	4050.86i	−0.814629	−	1.41098i
$203$	186.080	+	322.300i	0.0643363	+	0.111434i
$204$		0			0
$205$	133.342	−	230.955i	0.0454294	−	0.0786860i
$206$	−4300.25			−1.45443
$207$		0			0
$208$	3111.27			1.03715
$209$	−184.803	+	320.088i	−0.0611630	+	0.105937i
$210$		0			0
$211$	500.772	+	867.362i	0.163387	+	0.282994i	0.936081	−	0.351784i	$-0.114425\pi$
$211$	500.772	+	867.362i	0.163387	+	0.282994i	−0.772695	+	0.634778i	$0.781092\pi$
$212$	−473.541	−	820.197i	−0.153410	−	0.265714i
$213$		0			0
$214$	307.468	−	532.551i	0.0982155	−	0.170114i
$215$	1491.94			0.473254
$216$		0			0
$217$	−44.2337			−0.0138377
$218$	919.919	−	1593.35i	0.285802	−	0.495023i
$219$		0			0
$220$	−24.5448	−	42.5128i	−0.00752185	−	0.0130282i
$221$	1315.27	+	2278.11i	0.400336	+	0.693403i
$222$		0			0
$223$	−280.574	+	485.969i	−0.0842540	+	0.145932i	−0.905073	−	0.425256i	$-0.860184\pi$
$223$	−280.574	+	485.969i	−0.0842540	+	0.145932i	0.820819	+	0.571188i	$0.193517\pi$
$224$	−762.250			−0.227366
$225$		0			0
$226$	−510.646			−0.150300
$227$	−1953.23	+	3383.09i	−0.571103	+	0.989180i	0.425350	+	0.905029i	$0.360151\pi$
$227$	−1953.23	+	3383.09i	−0.571103	+	0.989180i	−0.996453	+	0.0841506i	$0.973182\pi$
$228$		0			0
$229$	−1498.45	−	2595.40i	−0.432405	−	0.748947i	0.564675	−	0.825313i	$-0.309001\pi$
$229$	−1498.45	−	2595.40i	−0.432405	−	0.748947i	−0.997080	+	0.0763664i	$0.975668\pi$
$230$	−1183.55	−	2049.96i	−0.339307	−	0.587698i
$231$		0			0
$232$	621.067	−	1075.72i	0.175755	−	0.304416i
$233$	−4194.30			−1.17930			−0.589651	−	0.807658i	$-0.700735\pi$
$233$	−4194.30			−1.17930			−0.589651	+	0.807658i	$0.700735\pi$
$234$		0			0
$235$	−2094.52			−0.581412
$236$	−116.529	+	201.833i	−0.0321414	+	0.0556705i
$237$		0			0
$238$	291.255	+	504.469i	0.0793247	+	0.137394i
$239$	−83.2362	−	144.169i	−0.0225276	−	0.0390190i	0.854542	−	0.519383i	$-0.173838\pi$
$239$	−83.2362	−	144.169i	−0.0225276	−	0.0390190i	−0.877069	+	0.480364i	$0.840505\pi$
$240$		0			0
$241$	−307.272	+	532.210i	−0.0821291	+	0.142252i	−0.904164	−	0.427185i	$-0.859505\pi$
$241$	−307.272	+	532.210i	−0.0821291	+	0.142252i	0.822035	+	0.569437i	$0.192839\pi$
$242$	3116.87			0.827935
$243$		0			0
$244$	1621.68			0.425481
$245$	−721.624	+	1249.89i	−0.188175	+	0.325928i
$246$		0			0
$247$	3526.66	+	6108.35i	0.908485	+	1.57354i
$248$	73.8179	+	127.856i	0.0189010	+	0.0327374i
$249$		0			0
$250$	−148.268	+	256.807i	−0.0375091	+	0.0649676i
$251$	6136.16			1.54307			0.771536	−	0.636185i	$-0.219489\pi$
$251$	6136.16			1.54307			0.771536	+	0.636185i	$0.219489\pi$
$252$		0			0
$253$	825.908			0.205235
$254$	2876.11	−	4981.57i	0.710485	−	1.23060i
$255$		0			0
$256$	1645.87	+	2850.73i	0.401824	+	0.695979i
$257$	2725.20	+	4720.19i	0.661453	+	1.14567i	0.980234	+	0.197842i	$0.0633932\pi$
$257$	2725.20	+	4720.19i	0.661453	+	1.14567i	−0.318781	+	0.947828i	$0.603273\pi$
$258$		0			0
$259$	1070.00	−	1853.30i	0.256705	−	0.444627i
$260$	−936.793			−0.223452
$261$		0			0
$262$	617.967			0.145718
$263$	−391.620	+	678.305i	−0.0918186	+	0.159035i	−0.908276	−	0.418371i	$-0.862601\pi$
$263$	−391.620	+	678.305i	−0.0918186	+	0.159035i	0.816458	+	0.577405i	$0.195935\pi$
$264$		0			0
$265$	−998.070	−	1728.71i	−0.231362	−	0.400731i
$266$	780.951	+	1352.65i	0.180012	+	0.311790i
$267$		0			0
$268$	267.474	−	463.279i	0.0609649	−	0.105594i
$269$	−141.019			−0.0319632			−0.0159816	−	0.999872i	$-0.505087\pi$
$269$	−141.019			−0.0319632			−0.0159816	+	0.999872i	$0.505087\pi$
$270$		0			0
$271$	6375.83			1.42917			0.714583	−	0.699551i	$-0.246616\pi$
$271$	6375.83			1.42917			0.714583	+	0.699551i	$0.246616\pi$
$272$	656.049	−	1136.31i	0.146246	−	0.253305i
$273$		0			0
$274$	479.958	+	831.312i	0.105822	+	0.183290i
$275$	−51.7324	−	89.6032i	−0.0113439	−	0.0196483i
$276$		0			0
$277$	−3411.59	+	5909.04i	−0.740008	+	1.28173i	0.212482	+	0.977165i	$0.431845\pi$
$277$	−3411.59	+	5909.04i	−0.740008	+	1.28173i	−0.952491	+	0.304567i	$0.901488\pi$
$278$	4193.44			0.904697
$279$		0			0
$280$	−907.011			−0.193587
$281$	294.846	−	510.689i	0.0625945	−	0.108417i	−0.833030	−	0.553228i	$-0.813396\pi$
$281$	294.846	−	510.689i	0.0625945	−	0.108417i	0.895624	+	0.444811i	$0.146729\pi$
$282$		0			0
$283$	2988.01	+	5175.39i	0.627629	+	1.08709i	0.988026	+	0.154286i	$0.0493078\pi$
$283$	2988.01	+	5175.39i	0.627629	+	1.08709i	−0.360397	+	0.932799i	$0.617359\pi$
$284$	−608.432	−	1053.84i	−0.127126	−	0.220189i
$285$		0			0
$286$	−387.701	+	671.517i	−0.0801581	+	0.138838i
$287$	393.214			0.0808736
$288$		0			0
$289$	−3803.64			−0.774199
$290$	299.388	−	518.555i	0.0606230	−	0.105002i
$291$		0			0
$292$	−1179.40	−	2042.78i	−0.236367	−	0.409400i
$293$	4028.74	+	6977.99i	0.803282	+	1.39133i	0.917445	+	0.397864i	$0.130248\pi$
$293$	4028.74	+	6977.99i	0.803282	+	1.39133i	−0.114162	+	0.993462i	$0.536418\pi$
$294$		0			0
$295$	−245.604	+	425.399i	−0.0484734	+	0.0839583i
$296$	−7142.55			−1.40254
$297$		0			0
$298$	4556.41			0.885724
$299$	7880.55	−	13649.5i	1.52423	−	2.64004i
$300$		0			0
$301$	1099.90	+	1905.09i	0.210622	+	0.364808i
$302$	−2994.48	−	5186.59i	−0.570573	−	0.988261i
$303$		0			0
$304$	1759.08	−	3046.82i	0.331876	−	0.574826i
$305$	3417.97			0.641681
$306$		0			0
$307$	−4546.58			−0.845235			−0.422618	−	0.906308i	$-0.638889\pi$
$307$	−4546.58			−0.845235			−0.422618	+	0.906308i	$0.638889\pi$
$308$	36.1902	−	62.6832i	0.00669522	−	0.0115965i
$309$		0			0
$310$	35.5842	+	61.6337i	0.00651951	+	0.0112921i
$311$	−2101.40	−	3639.73i	−0.383149	−	0.663633i	0.608362	−	0.793660i	$-0.291827\pi$
$311$	−2101.40	−	3639.73i	−0.383149	−	0.663633i	−0.991510	+	0.130027i	$0.958494\pi$
$312$		0			0
$313$	3486.64	−	6039.04i	0.629637	−	1.09056i	−0.357987	−	0.933727i	$-0.616537\pi$
$313$	3486.64	−	6039.04i	0.629637	−	1.09056i	0.987624	−	0.156838i	$-0.0501299\pi$
$314$	−4608.63			−0.828281
$315$		0			0
$316$	478.505			0.0851835
$317$	−3680.02	+	6373.97i	−0.652020	+	1.12933i	0.330612	+	0.943767i	$0.392745\pi$
$317$	−3680.02	+	6373.97i	−0.652020	+	1.12933i	−0.982632	+	0.185565i	$0.940589\pi$
$318$		0			0
$319$	104.460	+	180.930i	0.0183343	+	0.0317560i
$320$	1401.08	+	2426.74i	0.244759	+	0.423934i
$321$		0			0
$322$	1745.09	−	3022.58i	0.302018	−	0.523111i
$323$	2974.55			0.512410
$324$		0			0
$325$	−1974.46			−0.336994
$326$	−1246.88	+	2159.66i	−0.211835	+	0.366909i
$327$		0			0
$328$	−656.203	−	1136.58i	−0.110466	−	0.191332i
$329$	−1544.14	−	2674.53i	−0.258758	−	0.448182i
$330$		0			0
$331$	3417.06	−	5918.52i	0.567428	−	0.982814i	−0.429391	−	0.903119i	$-0.641272\pi$
$331$	3417.06	−	5918.52i	0.567428	−	0.982814i	0.996819	−	0.0796957i	$-0.0253949\pi$
$332$	2627.64			0.434369
$333$		0			0
$334$	−6681.39			−1.09458
$335$	563.749	−	976.443i	0.0919430	−	0.159250i
$336$		0			0
$337$	3464.24	+	6000.24i	0.559968	+	0.969893i	0.997498	+	0.0706891i	$0.0225198\pi$
$337$	3464.24	+	6000.24i	0.559968	+	0.969893i	−0.437531	+	0.899204i	$0.644147\pi$
$338$	4792.68	+	8301.16i	0.771264	+	1.33587i
$339$		0			0
$340$	−197.534	+	342.139i	−0.0315082	+	0.0545738i
$341$	−24.8316			−0.00394341
$342$		0			0
$343$	−4656.70			−0.733055
$344$	3671.07	−	6358.48i	0.575380	−	0.996588i
$345$		0			0
$346$	1710.95	+	2963.45i	0.265842	+	0.460451i
$347$	−3770.30	−	6530.35i	−0.583286	−	1.01028i	−0.995087	−	0.0990071i	$-0.968433\pi$
$347$	−3770.30	−	6530.35i	−0.583286	−	1.01028i	0.411801	−	0.911274i	$-0.364900\pi$
$348$		0			0
$349$	922.084	−	1597.10i	0.141427	−	0.244959i	−0.786607	−	0.617454i	$-0.788164\pi$
$349$	922.084	−	1597.10i	0.141427	−	0.244959i	0.928034	+	0.372495i	$0.121498\pi$
$350$	−437.228			−0.0667738
$351$		0			0
$352$	−427.906			−0.0647939
$353$	−3548.74	+	6146.61i	−0.535073	+	0.926773i	0.464087	+	0.885789i	$0.346382\pi$
$353$	−3548.74	+	6146.61i	−0.535073	+	0.926773i	−0.999160	+	0.0409833i	$0.986951\pi$
$354$		0			0
$355$	−1282.38	−	2221.14i	−0.191723	−	0.332073i
$356$	−441.563	−	764.809i	−0.0657382	−	0.113862i
$357$		0			0
$358$	−1820.60	+	3153.37i	−0.268775	+	0.465532i
$359$	−7709.65			−1.13342			−0.566712	−	0.823916i	$-0.691785\pi$
$359$	−7709.65			−1.13342			−0.566712	+	0.823916i	$0.691785\pi$
$360$		0			0
$361$	1116.75			0.162815
$362$	−4329.92	+	7499.65i	−0.628662	+	1.08888i
$363$		0			0
$364$	−690.630	−	1196.21i	−0.0994474	−	0.172248i
$365$	−2485.79	−	4305.52i	−0.356472	−	0.617428i
$366$		0			0
$367$	2628.19	−	4552.17i	0.373816	−	0.647469i	−0.616333	−	0.787486i	$-0.711382\pi$
$367$	2628.19	−	4552.17i	0.373816	−	0.647469i	0.990149	+	0.140017i	$0.0447157\pi$
$368$	−7861.57			−1.11362
$369$		0			0
$370$	−3443.10			−0.483778
$371$	1471.61	−	2548.91i	0.205936	−	0.356692i
$372$		0			0
$373$	−498.227	−	862.955i	−0.0691615	−	0.119791i	0.829371	−	0.558698i	$-0.188699\pi$
$373$	−498.227	−	862.955i	−0.0691615	−	0.119791i	−0.898532	+	0.438907i	$0.855366\pi$
$374$	163.503	+	283.195i	0.0226057	+	0.0391542i
$375$		0			0
$376$	−5153.78	+	8926.61i	−0.706878	+	1.22435i
$377$	3986.90			0.544658
$378$		0			0
$379$	−2735.20			−0.370707			−0.185354	−	0.982672i	$-0.559343\pi$
$379$	−2735.20			−0.370707			−0.185354	+	0.982672i	$0.559343\pi$
$380$	−529.654	+	917.387i	−0.0715017	+	0.123845i
$381$		0			0
$382$	−1609.93	−	2788.49i	−0.215632	−	0.373485i
$383$	−4469.73	−	7741.80i	−0.596325	−	1.03287i	−0.993358	−	0.115062i	$-0.963293\pi$
$383$	−4469.73	−	7741.80i	−0.596325	−	1.03287i	0.397033	−	0.917804i	$-0.370040\pi$
$384$		0			0
$385$	76.2772	−	132.116i	0.0100973	−	0.0174890i
$386$	4277.32			0.564015
$387$		0			0
$388$	330.295			0.0432170
$389$	−5566.93	+	9642.21i	−0.725590	+	1.25676i	0.233140	+	0.972443i	$0.425100\pi$
$389$	−5566.93	+	9642.21i	−0.725590	+	1.25676i	−0.958731	+	0.284316i	$0.908233\pi$
$390$		0			0
$391$	−3323.42	−	5756.33i	−0.429853	−	0.744527i
$392$	3551.25	+	6150.95i	0.457564	+	0.792525i
$393$		0			0
$394$	312.433	−	541.150i	0.0399496	−	0.0691948i
$395$	1008.53			0.128468
$396$		0			0
$397$	−9479.40			−1.19838			−0.599191	−	0.800606i	$-0.704511\pi$
$397$	−9479.40			−1.19838			−0.599191	+	0.800606i	$0.704511\pi$
$398$	584.593	−	1012.54i	0.0736256	−	0.127523i
$399$		0			0
$400$	492.425	+	852.906i	0.0615532	+	0.106613i
$401$	4713.80	+	8164.54i	0.587022	+	1.01675i	0.994620	+	0.103591i	$0.0330333\pi$
$401$	4713.80	+	8164.54i	0.587022	+	1.01675i	−0.407598	+	0.913162i	$0.633633\pi$
$402$		0			0
$403$	−236.935	+	410.383i	−0.0292868	+	0.0507261i
$404$	4677.53			0.576029
$405$		0			0
$406$	882.869			0.107921
$407$	600.670	−	1040.39i	0.0731550	−	0.126708i
$408$		0			0
$409$	−204.093	−	353.500i	−0.0246742	−	0.0427370i	0.853425	−	0.521216i	$-0.174522\pi$
$409$	−204.093	−	353.500i	−0.0246742	−	0.0427370i	−0.878099	+	0.478479i	$0.841188\pi$
$410$	−316.325	−	547.891i	−0.0381029	−	0.0659962i
$411$		0			0
$412$	2150.12	−	3724.12i	0.257109	−	0.445326i
$413$	−724.266			−0.0862925
$414$		0			0
$415$	5538.22			0.655085
$416$	−4082.94	+	7071.86i	−0.481208	+	0.833477i
$417$		0			0
$418$	438.404	+	759.338i	0.0512992	+	0.0888527i
$419$	4406.94	+	7633.05i	0.513826	+	0.889973i	0.999871	+	0.0160393i	$0.00510569\pi$
$419$	4406.94	+	7633.05i	0.513826	+	0.889973i	−0.486045	+	0.873934i	$0.661561\pi$
$420$		0			0
$421$	1174.68	−	2034.61i	0.135987	−	0.235537i	−0.789987	−	0.613124i	$-0.789913\pi$
$421$	1174.68	−	2034.61i	0.135987	−	0.235537i	0.925974	+	0.377587i	$0.123246\pi$
$422$	2375.94			0.274074
$423$		0			0
$424$	−9823.40			−1.12516
$425$	−416.338	+	721.118i	−0.0475185	+	0.0823044i
$426$		0			0
$427$	2519.83	+	4364.47i	0.285581	+	0.494640i
$428$	307.468	+	532.551i	0.0347244	+	0.0601444i
$429$		0			0
$430$	1769.65	−	3065.13i	0.198466	−	0.343753i
$431$	−4481.16			−0.500812			−0.250406	−	0.968141i	$-0.580564\pi$
$431$	−4481.16			−0.500812			−0.250406	+	0.968141i	$0.580564\pi$
$432$		0			0
$433$	3422.69			0.379871			0.189935	−	0.981797i	$-0.439172\pi$
$433$	3422.69			0.379871			0.189935	+	0.981797i	$0.439172\pi$
$434$	−52.4674	+	90.8762i	−0.00580303	+	0.0100511i
$435$		0			0
$436$	919.919	+	1593.35i	0.101046	+	0.175017i
$437$	−8911.17	−	15434.6i	−0.975467	−	1.68956i
$438$		0			0
$439$	4064.59	−	7040.07i	0.441896	−	0.765386i	−0.555934	−	0.831226i	$-0.687639\pi$
$439$	4064.59	−	7040.07i	0.441896	−	0.765386i	0.997830	+	0.0658402i	$0.0209728\pi$
$440$	−509.171			−0.0551676
$441$		0			0
$442$	6240.36			0.671547
$443$	1175.39	−	2035.83i	0.126060	−	0.218342i	−0.796087	−	0.605182i	$-0.793100\pi$
$443$	1175.39	−	2035.83i	0.126060	−	0.218342i	0.922147	+	0.386840i	$0.126434\pi$
$444$		0			0
$445$	−930.672	−	1611.97i	−0.0991417	−	0.171718i
$446$	665.601	+	1152.86i	0.0706662	+	0.122397i
$447$		0			0
$448$	−2065.83	+	3578.12i	−0.217860	+	0.377345i
$449$	−4760.99			−0.500412			−0.250206	−	0.968193i	$-0.580498\pi$
$449$	−4760.99			−0.500412			−0.250206	+	0.968193i	$0.580498\pi$
$450$		0			0
$451$	220.740			0.0230471
$452$	255.323	−	442.233i	0.0265695	−	0.0460196i
$453$		0			0
$454$	4633.61	+	8025.65i	0.479000	+	0.829653i
$455$	−1455.62	−	2521.22i	−0.149980	−	0.259772i
$456$		0			0
$457$	−1114.54	+	1930.44i	−0.114083	+	0.197598i	−0.917413	−	0.397937i	$-0.869726\pi$
$457$	−1114.54	+	1930.44i	−0.114083	+	0.197598i	0.803330	+	0.595535i	$0.203060\pi$
$458$	−7109.51			−0.725340
$459$		0			0
$460$	2367.09			0.239927
$461$	2868.31	−	4968.06i	0.289784	−	0.501921i	−0.683974	−	0.729507i	$-0.739750\pi$
$461$	2868.31	−	4968.06i	0.289784	−	0.501921i	0.973758	+	0.227585i	$0.0730831\pi$
$462$		0			0
$463$	1801.99	+	3121.13i	0.180876	+	0.313286i	0.942179	−	0.335110i	$-0.108773\pi$
$463$	1801.99	+	3121.13i	0.180876	+	0.313286i	−0.761303	+	0.648396i	$0.775440\pi$
$464$	−994.326	−	1722.22i	−0.0994836	−	0.172311i
$465$		0			0
$466$	−4975.03	+	8617.00i	−0.494557	+	0.856598i
$467$	3780.37			0.374593			0.187296	−	0.982303i	$-0.440028\pi$
$467$	3780.37			0.374593			0.187296	+	0.982303i	$0.440028\pi$
$468$		0			0
$469$	1662.45			0.163677
$470$	−2484.40	+	4303.11i	−0.243823	+	0.422314i
$471$		0			0
$472$	1208.67	+	2093.47i	0.117867	+	0.204152i
$473$	617.454	+	1069.46i	0.0600224	+	0.103962i
$474$		0			0
$475$	−1116.34	+	1933.55i	−0.107834	+	0.186774i
$476$	−582.511			−0.0560911
$477$		0			0
$478$	−394.920			−0.0377891
$479$	−7230.58	+	12523.7i	−0.689715	+	1.19462i	0.282215	+	0.959351i	$0.408931\pi$
$479$	−7230.58	+	12523.7i	−0.689715	+	1.19462i	−0.971930	+	0.235271i	$0.924402\pi$
$480$		0			0
$481$	−11462.8	−	19854.1i	−1.08661	−	1.88206i
$482$	728.935	+	1262.55i	0.0688840	+	0.119311i
$483$		0			0
$484$	−1558.44	+	2699.29i	−0.146360	+	0.253502i
$485$	696.156			0.0651769
$486$		0			0
$487$	3581.74			0.333273			0.166636	−	0.986018i	$-0.446709\pi$
$487$	3581.74			0.333273			0.166636	+	0.986018i	$0.446709\pi$
$488$	8410.26	−	14567.0i	0.780153	−	1.35126i
$489$		0			0
$490$	1711.89	+	2965.09i	0.157828	+	0.273365i
$491$	−3507.69	−	6075.50i	−0.322403	−	0.558419i	0.658580	−	0.752511i	$-0.271157\pi$
$491$	−3507.69	−	6075.50i	−0.322403	−	0.558419i	−0.980983	+	0.194092i	$0.937824\pi$
$492$		0			0
$493$	840.687	−	1456.11i	0.0768005	−	0.133022i
$494$	16732.4			1.52394
$495$		0			0
$496$	236.364			0.0213973
$497$	1890.81	−	3274.98i	0.170653	−	0.295579i
$498$		0			0
$499$	−1259.90	−	2182.21i	−0.113028	−	0.195769i	0.803962	−	0.594681i	$-0.202721\pi$
$499$	−1259.90	−	2182.21i	−0.113028	−	0.195769i	−0.916990	+	0.398911i	$0.869388\pi$
$500$	−148.268	−	256.807i	−0.0132615	−	0.0229695i
$501$		0			0
$502$	7278.35	−	12606.5i	0.647109	−	1.12083i
$503$	4989.32			0.442272			0.221136	−	0.975243i	$-0.429024\pi$
$503$	4989.32			0.442272			0.221136	+	0.975243i	$0.429024\pi$
$504$		0			0
$505$	9858.72			0.868727
$506$	979.643	−	1696.79i	0.0860681	−	0.149074i
$507$		0			0
$508$	2876.11	+	4981.57i	0.251194	+	0.435081i
$509$	2359.76	+	4087.22i	0.205490	+	0.355919i	0.950289	−	0.311370i	$-0.100788\pi$
$509$	2359.76	+	4087.22i	0.205490	+	0.355919i	−0.744799	+	0.667289i	$0.767454\pi$
$510$		0			0
$511$	3665.19	−	6348.30i	0.317297	−	0.549574i
$512$	11827.7			1.02093
$513$		0			0
$514$	12929.9			1.10956
$515$	4531.77	−	7849.25i	0.387754	−	0.671610i
$516$		0			0
$517$	−866.839	−	1501.41i	−0.0737399	−	0.127721i
$518$	−2538.35	−	4396.55i	−0.215306	−	0.372921i
$519$		0			0
$520$	−4858.34	+	8414.89i	−0.409716	+	0.709649i
$521$	10711.0			0.900682			0.450341	−	0.892857i	$-0.351302\pi$
$521$	10711.0			0.900682			0.450341	+	0.892857i	$0.351302\pi$
$522$		0			0
$523$	−10566.4			−0.883433			−0.441717	−	0.897155i	$-0.645630\pi$
$523$	−10566.4			−0.883433			−0.441717	+	0.897155i	$0.645630\pi$
$524$	−308.983	+	535.175i	−0.0257595	+	0.0446168i
$525$		0			0
$526$	929.032	+	1609.13i	0.0770109	+	0.133387i
$527$	99.9211	+	173.068i	0.00825926	+	0.0143055i
$528$		0			0
$529$	−13829.1	+	23952.7i	−1.13661	+	1.96866i
$530$	−4735.41			−0.388100
$531$		0			0
$532$	−1561.90			−0.127288
$533$	2106.23	−	3648.09i	0.171165	−	0.296466i
$534$		0			0
$535$	648.044	+	1122.44i	0.0523689	+	0.0907056i
$536$	−2774.32	−	4805.26i	−0.223568	−	0.387231i
$537$		0			0
$538$	−167.269	+	289.718i	−0.0134042	+	0.0232168i
$539$	−1194.60			−0.0954642
$540$		0			0
$541$	−6595.81			−0.524170			−0.262085	−	0.965045i	$-0.584410\pi$
$541$	−6595.81			−0.524170			−0.262085	+	0.965045i	$0.584410\pi$
$542$	7562.63	−	13098.9i	0.599341	−	1.03809i
$543$		0			0
$544$	1721.87	+	2982.37i	0.135707	+	0.235052i
$545$	1938.89	+	3358.26i	0.152391	+	0.263949i
$546$		0			0
$547$	3194.39	−	5532.85i	0.249693	−	0.432482i	−0.713747	−	0.700403i	$-0.753003\pi$
$547$	3194.39	−	5532.85i	0.249693	−	0.432482i	0.963441	+	0.267922i	$0.0863368\pi$
$548$	−959.916			−0.0748277
$549$		0			0
$550$	−245.448			−0.0190290
$551$	2254.16	−	3904.31i	0.174284	−	0.301868i
$552$		0			0
$553$	743.519	+	1287.81i	0.0571748	+	0.0990296i
$554$	8093.24	+	14017.9i	0.620666	+	1.07502i
$555$		0			0
$556$	−2096.72	+	3631.62i	−0.159929	+	0.277006i
$557$	−15992.4			−1.21655			−0.608276	−	0.793726i	$-0.708138\pi$
$557$	−15992.4			−1.21655			−0.608276	+	0.793726i	$0.708138\pi$
$558$		0			0
$559$	23566.2			1.78308
$560$	−726.060	+	1257.57i	−0.0547886	+	0.0948967i
$561$		0			0
$562$	−699.459	−	1211.50i	−0.0524998	−	0.0909323i
$563$	−3175.56	−	5500.23i	−0.237716	−	0.411736i	0.722343	−	0.691535i	$-0.243065\pi$
$563$	−3175.56	−	5500.23i	−0.237716	−	0.411736i	−0.960058	+	0.279800i	$0.909732\pi$
$564$		0			0
$565$	538.139	−	932.083i	0.0400702	−	0.0694036i
$566$	14176.8			1.05282
$567$		0			0
$568$	−12621.7			−0.932382
$569$	4810.21	−	8331.53i	0.354402	−	0.613842i	−0.632614	−	0.774468i	$-0.718018\pi$
$569$	4810.21	−	8331.53i	0.354402	−	0.613842i	0.987015	+	0.160626i	$0.0513512\pi$
$570$		0			0
$571$	2532.30	+	4386.08i	0.185593	+	0.321456i	0.943776	−	0.330585i	$-0.107246\pi$
$571$	2532.30	+	4386.08i	0.185593	+	0.321456i	−0.758183	+	0.652042i	$0.773913\pi$
$572$	−387.701	−	671.517i	−0.0283402	−	0.0490866i
$573$		0			0
$574$	466.408	−	807.842i	0.0339155	−	0.0587433i
$575$	4989.06			0.361840
$576$		0			0
$577$	11355.1			0.819273			0.409637	−	0.912249i	$-0.365655\pi$
$577$	11355.1			0.819273			0.409637	+	0.912249i	$0.365655\pi$
$578$	−4511.65	+	7814.41i	−0.324671	+	0.562347i
$579$		0			0
$580$	299.388	+	518.555i	0.0214335	+	0.0371239i
$581$	4082.93	+	7071.84i	0.291546	+	0.504973i
$582$		0			0
$583$	826.121	−	1430.88i	0.0586869	−	0.101649i
$584$	−24466.2			−1.73359
$585$		0			0
$586$	19114.6			1.34747
$587$	−5016.00	+	8687.96i	−0.352696	+	0.610887i	−0.986721	−	0.162426i	$-0.948068\pi$
$587$	−5016.00	+	8687.96i	−0.352696	+	0.610887i	0.634025	+	0.773312i	$0.281402\pi$
$588$		0			0
$589$	267.921	+	464.053i	0.0187428	+	0.0324634i
$590$	582.643	+	1009.17i	0.0406560	+	0.0704182i
$591$		0			0
$592$	−5717.59	+	9903.16i	−0.396945	+	0.687530i
$593$	1325.12			0.0917643			0.0458821	−	0.998947i	$-0.485390\pi$
$593$	1325.12			0.0917643			0.0458821	+	0.998947i	$0.485390\pi$
$594$		0			0
$595$	−1227.74			−0.0845926
$596$	−2278.21	+	3945.97i	−0.156575	+	0.271197i
$597$		0			0
$598$	−18694.9	−	32380.5i	−1.27841	−	2.21427i
$599$	8285.78	+	14351.4i	0.565188	+	0.978935i	0.997032	+	0.0769865i	$0.0245298\pi$
$599$	8285.78	+	14351.4i	0.565188	+	0.978935i	−0.431844	+	0.901948i	$0.642137\pi$
$600$		0			0
$601$	−8604.49	+	14903.4i	−0.584001	+	1.01152i	0.410998	+	0.911636i	$0.365180\pi$
$601$	−8604.49	+	14903.4i	−0.584001	+	1.01152i	−0.994999	+	0.0998833i	$0.968153\pi$
$602$	5218.55			0.353310
$603$		0			0
$604$	5988.96			0.403456
$605$	−3284.68	+	5689.23i	−0.220729	+	0.382314i
$606$		0			0
$607$	−2088.99	−	3618.24i	−0.139686	−	0.241944i	0.787692	−	0.616070i	$-0.211276\pi$
$607$	−2088.99	−	3618.24i	−0.139686	−	0.241944i	−0.927378	+	0.374126i	$0.877943\pi$
$608$	4616.91	+	7996.72i	0.307961	+	0.533404i
$609$		0			0
$610$	4054.20	−	7022.08i	0.269098	−	0.466091i
$611$	−33084.4			−2.19059
$612$		0			0
$613$	14944.2			0.984649			0.492324	−	0.870412i	$-0.336147\pi$
$613$	14944.2			0.984649			0.492324	+	0.870412i	$0.336147\pi$
$614$	−5392.89	+	9340.75i	−0.354461	+	0.613945i
$615$		0			0
$616$	−375.375	−	650.168i	−0.0245524	−	0.0425260i
$617$	10251.4	+	17756.0i	0.668893	+	1.15856i	0.978214	+	0.207600i	$0.0665651\pi$
$617$	10251.4	+	17756.0i	0.668893	+	1.15856i	−0.309320	+	0.950958i	$0.600102\pi$
$618$		0			0
$619$	−1364.26	+	2362.97i	−0.0885854	+	0.153434i	−0.906913	−	0.421317i	$-0.861568\pi$
$619$	−1364.26	+	2362.97i	−0.0885854	+	0.153434i	0.818328	+	0.574751i	$0.194901\pi$
$620$	−71.1684			−0.00460999
$621$		0			0
$622$	−9970.21			−0.642715
$623$	1372.23	−	2376.78i	0.0882463	−	0.152847i
$624$		0			0
$625$	−312.500	−	541.266i	−0.0200000	−	0.0346410i
$626$	−8271.29	−	14326.3i	−0.528095	−	0.914687i
$627$		0			0
$628$	2304.32	−	3991.19i	0.146421	−	0.253608i
$629$	−9668.27			−0.612876
$630$		0			0
$631$	−3393.08			−0.214067			−0.107034	−	0.994255i	$-0.534135\pi$
$631$	−3393.08			−0.214067			−0.107034	+	0.994255i	$0.534135\pi$
$632$	2481.59	−	4298.25i	0.156191	−	0.270530i
$633$		0			0
$634$	8730.03	+	15120.9i	0.546867	+	0.947202i
$635$	6061.91	+	10499.5i	0.378834	+	0.656159i
$636$		0			0
$637$	−11398.5	+	19742.8i	−0.708988	+	1.22800i
$638$	495.618			0.0307550
$639$		0			0
$640$	2511.75			0.155134
$641$	6780.88	−	11744.8i	0.417830	−	0.723702i	−0.577891	−	0.816114i	$-0.696124\pi$
$641$	6780.88	−	11744.8i	0.417830	−	0.723702i	0.995721	+	0.0924116i	$0.0294576\pi$
$642$		0			0
$643$	−6483.44	−	11229.7i	−0.397639	−	0.688731i	0.595795	−	0.803137i	$-0.296837\pi$
$643$	−6483.44	−	11229.7i	−0.397639	−	0.688731i	−0.993434	+	0.114405i	$0.963504\pi$
$644$	1745.09	+	3022.58i	0.106780	+	0.184948i
$645$		0			0
$646$	3528.24	−	6111.09i	0.214886	−	0.372194i
$647$	−25837.9			−1.57000			−0.785001	−	0.619495i	$-0.787337\pi$
$647$	−25837.9			−1.57000			−0.785001	+	0.619495i	$0.787337\pi$
$648$		0			0
$649$	−406.583			−0.0245913
$650$	−2341.98	+	4056.43i	−0.141323	+	0.244779i
$651$		0			0
$652$	−1246.88	−	2159.66i	−0.0748950	−	0.129722i
$653$	−14133.4	−	24479.8i	−0.846989	−	1.46703i	−0.883883	−	0.467709i	$-0.845080\pi$
$653$	−14133.4	−	24479.8i	−0.846989	−	1.46703i	0.0368938	−	0.999319i	$-0.488254\pi$
$654$		0			0
$655$	−651.237	+	1127.98i	−0.0388487	+	0.0672880i
$656$	−2101.15			−0.125055
$657$		0			0
$658$	−7326.28			−0.434055
$659$	−7978.92	+	13819.9i	−0.471646	+	0.816914i	−0.999474	−	0.0324369i	$-0.989673\pi$
$659$	−7978.92	+	13819.9i	−0.471646	+	0.816914i	0.527828	+	0.849351i	$0.323007\pi$
$660$		0			0
$661$	−12028.9	−	20834.6i	−0.707821	−	1.22598i	−0.965664	−	0.259795i	$-0.916345\pi$
$661$	−12028.9	−	20834.6i	−0.707821	−	1.22598i	0.257843	−	0.966187i	$-0.416988\pi$
$662$	−8106.23	−	14040.4i	−0.475918	−	0.824314i
$663$		0			0
$664$	13627.3	−	23603.2i	0.796450	−	1.37949i
$665$	−3291.98			−0.191966
$666$		0			0
$667$	−10074.1			−0.584815
$668$	3340.70	−	5786.26i	0.193496	−	0.335145i
$669$		0			0
$670$	−1337.37	−	2316.40i	−0.0771152	−	0.133567i
$671$	1414.56	+	2450.09i	0.0813838	+	0.140961i
$672$		0			0
$673$	−626.519	+	1085.16i	−0.0358849	+	0.0621545i	−0.883410	−	0.468601i	$-0.844758\pi$
$673$	−626.519	+	1085.16i	−0.0358849	+	0.0621545i	0.847525	+	0.530755i	$0.178092\pi$
$674$	16436.3			0.939321
$675$		0			0
$676$	−9585.35			−0.545366
$677$	440.364	−	762.733i	0.0249994	−	0.0433002i	−0.853255	−	0.521494i	$-0.825375\pi$
$677$	440.364	−	762.733i	0.0249994	−	0.0433002i	0.878254	+	0.478194i	$0.158708\pi$
$678$		0			0
$679$	513.226	+	888.933i	0.0290071	+	0.0502417i
$680$	2048.88	+	3548.76i	0.115546	+	0.200131i
$681$		0			0
$682$	−29.4537	+	51.0153i	−0.00165373	+	0.00286434i
$683$	26686.4			1.49506			0.747531	−	0.664227i	$-0.231239\pi$
$683$	26686.4			1.49506			0.747531	+	0.664227i	$0.231239\pi$
$684$		0			0
$685$	−2023.19			−0.112850
$686$	−5523.50	+	9566.98i	−0.307417	+	0.532462i
$687$		0			0
$688$	−5877.36	−	10179.9i	−0.325687	−	0.564106i
$689$	−15765.2	−	27306.1i	−0.871706	−	1.50984i
$690$		0			0
$691$	85.7060	−	148.447i	0.00471839	−	0.00817249i	−0.863657	−	0.504081i	$-0.831831\pi$
$691$	85.7060	−	148.447i	0.00471839	−	0.00817249i	0.868375	+	0.495908i	$0.165165\pi$
$692$	−3421.90			−0.187978
$693$		0			0
$694$	−17888.4			−0.978437
$695$	−4419.20	+	7654.28i	−0.241194	+	0.417761i
$696$		0			0
$697$	−888.247	−	1538.49i	−0.0482708	−	0.0836075i
$698$	−2187.44	−	3788.76i	−0.118619	−	0.205454i
$699$		0			0
$700$	218.614	−	378.651i	0.0118041	−	0.0204452i
$701$	14229.6			0.766684			0.383342	−	0.923607i	$-0.374773\pi$
$701$	14229.6			0.766684			0.383342	+	0.923607i	$0.374773\pi$
$702$		0			0
$703$	−25923.8			−1.39080
$704$	−1159.70	+	2008.66i	−0.0620850	+	0.107534i
$705$		0			0
$706$	8418.62	+	14581.5i	0.448780	+	0.777310i
$707$	7268.13	+	12588.8i	0.386628	+	0.669659i
$708$		0			0
$709$	3887.04	−	6732.55i	0.205897	−	0.356624i	−0.744521	−	0.667599i	$-0.767322\pi$
$709$	3887.04	−	6732.55i	0.205897	−	0.356624i	0.950418	+	0.310975i	$0.100656\pi$
$710$	−6084.32			−0.321606
$711$		0			0
$712$	−9160.03			−0.482144
$713$	598.687	−	1036.96i	0.0314460	−	0.0544661i
$714$		0			0
$715$	−817.147	−	1415.34i	−0.0427407	−	0.0740290i
$716$	−1820.60	−	3153.37i	−0.0950264	−	0.164591i
$717$		0			0
$718$	−9144.72	+	15839.1i	−0.475318	+	0.823274i
$719$	22091.8			1.14588			0.572939	−	0.819598i	$-0.305803\pi$
$719$	22091.8			1.14588			0.572939	+	0.819598i	$0.305803\pi$
$720$		0			0
$721$	13363.8			0.690282
$722$	1324.62	−	2294.31i	0.0682786	−	0.118262i
$723$		0			0
$724$	−4329.92	−	7499.65i	−0.222266	−	0.384975i
$725$	631.013	+	1092.95i	0.0323245	+	0.0559876i
$726$		0			0
$727$	−3321.74	+	5753.42i	−0.169459	+	0.293511i	−0.938230	−	0.346013i	$-0.887535\pi$
$727$	−3321.74	+	5753.42i	−0.169459	+	0.293511i	0.768771	+	0.639524i	$0.220869\pi$
$728$	−14326.8			−0.729378
$729$		0			0
$730$	−11794.0			−0.597967
$731$	4969.22	−	8606.94i	0.251427	−	0.435484i
$732$		0			0
$733$	1561.40	+	2704.43i	0.0786791	+	0.136276i	0.902680	−	0.430312i	$-0.141596\pi$
$733$	1561.40	+	2704.43i	0.0786791	+	0.136276i	−0.824001	+	0.566588i	$0.808263\pi$
$734$	−6234.82	−	10799.0i	−0.313530	−	0.543050i
$735$		0			0
$736$	10316.8	−	17869.2i	0.516687	−	0.894928i
$737$	933.252			0.0466442
$738$		0			0
$739$	19549.5			0.973127			0.486563	−	0.873645i	$-0.338250\pi$
$739$	19549.5			0.973127			0.486563	+	0.873645i	$0.338250\pi$
$740$	1721.55	−	2981.81i	0.0855208	−	0.148126i
$741$		0			0
$742$	−3491.08	−	6046.72i	−0.172724	−	0.299167i
$743$	4695.92	+	8133.58i	0.231866	+	0.401604i	0.958357	−	0.285572i	$-0.0921834\pi$
$743$	4695.92	+	8133.58i	0.231866	+	0.401604i	−0.726491	+	0.687176i	$0.758850\pi$
$744$		0			0
$745$	−4801.72	+	8316.82i	−0.236136	+	0.409000i
$746$	−2363.87			−0.116015
$747$		0			0
$748$	−327.005			−0.0159846
$749$	−955.512	+	1655.00i	−0.0466137	+	0.0807373i
$750$		0			0
$751$	9136.01	+	15824.0i	0.443912	+	0.768878i	0.997976	−	0.0635962i	$-0.0202570\pi$
$751$	9136.01	+	15824.0i	0.443912	+	0.768878i	−0.554064	+	0.832474i	$0.686924\pi$
$752$	8251.18	+	14291.5i	0.400119	+	0.693026i
$753$		0			0
$754$	4729.03	−	8190.92i	0.228410	−	0.395618i
$755$	12622.8			0.608464
$756$		0			0
$757$	2016.30			0.0968082			0.0484041	−	0.998828i	$-0.484586\pi$
$757$	2016.30			0.0968082			0.0484041	+	0.998828i	$0.484586\pi$
$758$	−3244.34	+	5619.36i	−0.155461	+	0.269267i
$759$		0			0
$760$	5493.72	+	9515.39i	0.262208	+	0.454157i
$761$	−8846.39	−	15322.4i	−0.421395	−	0.729877i	0.574681	−	0.818377i	$-0.305126\pi$
$761$	−8846.39	−	15322.4i	−0.421395	−	0.729877i	−0.996076	+	0.0885001i	$0.971793\pi$
$762$		0			0
$763$	−2858.81	+	4951.60i	−0.135643	+	0.234941i
$764$	3219.87			0.152475
$765$		0			0
$766$	−21206.9			−1.00031
$767$	−3879.48	+	6719.46i	−0.182634	+	0.316331i
$768$		0			0
$769$	4879.86	+	8452.16i	0.228833	+	0.396350i	0.957462	−	0.288558i	$-0.0931759\pi$
$769$	4879.86	+	8452.16i	0.228833	+	0.396350i	−0.728630	+	0.684908i	$0.759843\pi$
$770$	−180.951	−	313.416i	−0.00846886	−	0.0146685i
$771$		0			0
$772$	−2138.66	+	3704.27i	−0.0997047	+	0.172694i
$773$	10338.3			0.481038			0.240519	−	0.970644i	$-0.422682\pi$
$773$	10338.3			0.481038			0.240519	+	0.970644i	$0.422682\pi$
$774$		0			0
$775$	−150.000			−0.00695246
$776$	1712.96	−	2966.93i	0.0792418	−	0.137251i
$777$		0			0
$778$	13206.3	+	22874.0i	0.608573	+	1.05408i
$779$	−2381.68	−	4125.19i	−0.109541	−	0.189731i
$780$		0			0
$781$	1061.45	−	1838.48i	0.0486320	−	0.0842331i
$782$	−15768.2			−0.721059
$783$		0			0
$784$	11371.1			0.517997
$785$	4856.75	−	8412.14i	0.220821	−	0.382474i
$786$		0			0
$787$	−2363.64	−	4093.95i	−0.107058	−	0.185430i	0.807519	−	0.589842i	$-0.200810\pi$
$787$	−2363.64	−	4093.95i	−0.107058	−	0.185430i	−0.914577	+	0.404411i	$0.867476\pi$
$788$	312.433	+	541.150i	0.0141243	+	0.0244640i
$789$		0			0
$790$	1196.26	−	2071.99i	0.0538748	−	0.0933139i
$791$	1586.92			0.0713331
$792$		0			0
$793$	53989.1			2.41767
$794$	−11243.9	+	19475.0i	−0.502558	+	0.870456i
$795$		0			0
$796$	584.593	+	1012.54i	0.0260306	+	0.0450863i
$797$	14099.4	+	24420.9i	0.626634	+	1.08536i	0.988222	+	0.153025i	$0.0489014\pi$
$797$	14099.4	+	24420.9i	0.626634	+	1.08536i	−0.361588	+	0.932338i	$0.617765\pi$
$798$		0			0
$799$	−6976.24	+	12083.2i	−0.308888	+	0.535010i
$800$	−2584.85			−0.114235
$801$		0			0
$802$	22364.9			0.984704
$803$	2057.54	−	3563.76i	0.0904221	−	0.156616i
$804$		0			0
$805$	3678.08	+	6370.62i	0.161037	+	0.278925i
$806$	562.076	+	973.544i	0.0245636	+	0.0425454i
$807$		0			0
$808$	24258.3	−	42016.7i	1.05619	−	1.82938i
$809$	27310.5			1.18688			0.593439	−	0.804879i	$-0.297770\pi$
$809$	27310.5			1.18688			0.593439	+	0.804879i	$0.297770\pi$
$810$		0			0
$811$	−18045.0			−0.781312			−0.390656	−	0.920537i	$-0.627752\pi$
$811$	−18045.0			−0.781312			−0.390656	+	0.920537i	$0.627752\pi$
$812$	−441.435	+	764.587i	−0.0190780	+	0.0330440i
$813$		0			0
$814$	−1424.96	−	2468.10i	−0.0613572	−	0.106274i
$815$	−2628.02	−	4551.86i	−0.112951	−	0.195638i
$816$		0			0
$817$	13324.1	−	23078.0i	0.570564	−	0.988246i
$818$	−968.333			−0.0413899
$819$		0			0
$820$	632.650			0.0269428
$821$	−6288.85	+	10892.6i	−0.267335	+	0.463038i	−0.968173	−	0.250283i	$-0.919476\pi$
$821$	−6288.85	+	10892.6i	−0.267335	+	0.463038i	0.700837	+	0.713321i	$0.252810\pi$
$822$		0			0
$823$	17484.2	+	30283.6i	0.740537	+	1.28265i	0.952251	+	0.305316i	$0.0987622\pi$
$823$	17484.2	+	30283.6i	0.740537	+	1.28265i	−0.211714	+	0.977332i	$0.567904\pi$
$824$	−22301.7	−	38627.7i	−0.942860	−	1.63308i
$825$		0			0
$826$	−859.081	+	1487.97i	−0.0361880	+	0.0626794i
$827$	6735.01			0.283191			0.141596	−	0.989925i	$-0.454777\pi$
$827$	6735.01			0.283191			0.141596	+	0.989925i	$0.454777\pi$
$828$		0			0
$829$	−2867.97			−0.120155			−0.0600777	−	0.998194i	$-0.519135\pi$
$829$	−2867.97			−0.120155			−0.0600777	+	0.998194i	$0.519135\pi$
$830$	6569.10	−	11378.0i	0.274719	−	0.475828i
$831$		0			0
$832$	22131.0	+	38332.0i	0.922179	+	1.59726i
$833$	4807.03	+	8326.02i	0.199944	+	0.346314i
$834$		0			0
$835$	7041.11	−	12195.6i	0.291817	−	0.505443i
$836$	−876.808			−0.0362740
$837$		0			0
$838$	20909.0			0.861921
$839$	13927.9	−	24123.8i	0.573116	−	0.992666i	−0.423128	−	0.906070i	$-0.639068\pi$
$839$	13927.9	−	24123.8i	0.573116	−	0.992666i	0.996244	−	0.0865958i	$-0.0275989\pi$
$840$		0			0
$841$	10920.3	+	18914.6i	0.447756	+	0.775537i
$842$	−2786.68	−	4826.67i	−0.114056	−	0.197551i
$843$		0			0
$844$	−1187.97	+	2057.63i	−0.0484499	+	0.0839176i
$845$	−20202.8			−0.822483
$846$		0			0
$847$	−9686.23			−0.392943
$848$	−7863.60	+	13620.2i	−0.318440	+	0.551554i
$849$		0			0
$850$	987.671	+	1710.70i	0.0398551	+	0.0690310i
$851$	28964.2	+	50167.5i	1.16672	+	2.02082i
$852$		0			0
$853$	−5463.05	+	9462.29i	−0.219287	+	0.379815i	−0.954590	−	0.297923i	$-0.903706\pi$
$853$	−5463.05	+	9462.29i	−0.219287	+	0.379815i	0.735304	+	0.677738i	$0.237040\pi$
$854$	11955.5			0.479049
$855$		0			0
$856$	6378.30			0.254680
$857$	22584.6	−	39117.7i	0.900206	−	1.55920i	0.0729799	−	0.997333i	$-0.476749\pi$
$857$	22584.6	−	39117.7i	0.900206	−	1.55920i	0.827226	−	0.561869i	$-0.189918\pi$
$858$		0			0
$859$	3525.95	+	6107.12i	0.140051	+	0.242576i	0.927516	−	0.373784i	$-0.121940\pi$
$859$	3525.95	+	6107.12i	0.140051	+	0.242576i	−0.787465	+	0.616360i	$0.788607\pi$
$860$	1769.65	+	3065.13i	0.0701682	+	0.121535i
$861$		0			0
$862$	−5315.29	+	9206.35i	−0.210022	+	0.363770i
$863$	−6882.52			−0.271476			−0.135738	−	0.990745i	$-0.543341\pi$
$863$	−6882.52			−0.271476			−0.135738	+	0.990745i	$0.543341\pi$
$864$		0			0
$865$	−7212.25			−0.283496
$866$	4059.80	−	7031.77i	0.159304	−	0.275923i
$867$		0			0
$868$	−52.4674	−	90.8762i	−0.00205168	−	0.00355362i
$869$	417.391	+	722.942i	0.0162935	+	0.0282211i
$870$		0			0
$871$	8904.79	−	15423.5i	0.346415	−	0.600008i
$872$	19083.3			0.741104
$873$		0			0
$874$	−42279.6			−1.63630
$875$	460.768	−	798.073i	0.0178020	−	0.0308340i
$876$		0			0
$877$	−18684.3	−	32362.2i	−0.719413	−	1.24606i	−0.961233	−	0.275738i	$-0.911078\pi$
$877$	−18684.3	−	32362.2i	−0.719413	−	1.24606i	0.241820	−	0.970321i	$-0.422256\pi$
$878$	−9642.35	−	16701.0i	−0.370630	−	0.641951i
$879$		0			0
$880$	−407.590	+	705.966i	−0.0156135	+	0.0270433i
$881$	23880.5			0.913229			0.456614	−	0.889665i	$-0.349062\pi$
$881$	23880.5			0.913229			0.456614	+	0.889665i	$0.349062\pi$
$882$		0			0
$883$	−33107.0			−1.26177			−0.630883	−	0.775878i	$-0.717307\pi$
$883$	−33107.0			−1.26177			−0.630883	+	0.775878i	$0.717307\pi$
$884$	−3120.18	+	5404.31i	−0.118714	+	0.205618i
$885$		0			0
$886$	−2788.36	−	4829.57i	−0.105730	−	0.183129i
$887$	5651.07	+	9787.95i	0.213917	+	0.370515i	0.952937	−	0.303168i	$-0.0980444\pi$
$887$	5651.07	+	9787.95i	0.213917	+	0.370515i	−0.739020	+	0.673684i	$0.764711\pi$
$888$		0			0
$889$	−8938.02	+	15481.1i	−0.337201	+	0.584049i
$890$	−4415.63			−0.166306
$891$		0			0
$892$	−1331.20			−0.0499686
$893$	−18705.6	+	32399.0i	−0.700961	+	1.21410i
$894$		0			0
$895$	−3837.23	−	6646.27i	−0.143312	−	0.248224i
$896$	1851.73	+	3207.30i	0.0690425	+	0.119585i
$897$		0			0
$898$	−5647.21	+	9781.25i	−0.209855	+	0.363479i
$899$	302.886			0.0112367
$900$		0			0
$901$	−13297.1			−0.491666
$902$	261.828	−	453.500i	0.00966511	−	0.0167405i
$903$		0			0
$904$	−2648.28	−	4586.96i	−0.0974343	−	0.168761i
$905$	−9126.08	−	15806.8i	−0.335206	−	0.580593i
$906$		0			0
$907$	−1981.12	+	3431.40i	−0.0725270	+	0.125620i	−0.900008	−	0.435873i	$-0.856440\pi$
$907$	−1981.12	+	3431.40i	−0.0725270	+	0.125620i	0.827481	+	0.561493i	$0.189773\pi$
$908$	−9267.22			−0.338704
$909$		0			0
$910$	−6906.30			−0.251584
$911$	−17023.1	+	29484.8i	−0.619100	+	1.07231i	0.370551	+	0.928812i	$0.379169\pi$
$911$	−17023.1	+	29484.8i	−0.619100	+	1.07231i	−0.989650	+	0.143500i	$0.954164\pi$
$912$		0			0
$913$	2292.04	+	3969.93i	0.0830838	+	0.143905i
$914$	2644.01	+	4579.56i	0.0956849	+	0.165731i
$915$		0			0
$916$	3554.76	−	6157.02i	0.128223	−	0.222089i
$917$	−1920.44			−0.0691587
$918$		0			0
$919$	−35121.5			−1.26066			−0.630332	−	0.776326i	$-0.717081\pi$
$919$	−35121.5			−1.26066			−0.630332	+	0.776326i	$0.717081\pi$
$920$	12276.1	−	21262.8i	0.439924	−	0.761971i
$921$		0			0
$922$	−6804.44	−	11785.6i	−0.243050	−	0.420975i
$923$	−20256.0	−	35084.4i	−0.722355	−	1.25116i
$924$		0			0
$925$	3628.46	−	6284.69i	0.128976	−	0.223394i
$926$	8549.64			0.303411
$927$		0			0
$928$	5219.44			0.184630
$929$	12228.9	−	21181.1i	0.431881	−	0.748039i	−0.565155	−	0.824985i	$-0.691184\pi$
$929$	12228.9	−	21181.1i	0.431881	−	0.748039i	0.997035	+	0.0769459i	$0.0245169\pi$
$930$		0			0
$931$	12889.2	+	22324.8i	0.453735	+	0.785891i
$932$	−4975.03	−	8617.00i	−0.174852	−	0.302853i
$933$		0			0
$934$	4484.05	−	7766.61i	0.157091	−	0.272089i
$935$	−689.221			−0.0241069
$936$		0			0
$937$	41877.5			1.46006			0.730032	−	0.683413i	$-0.239505\pi$
$937$	41877.5			1.46006			0.730032	+	0.683413i	$0.239505\pi$
$938$	1971.90	−	3415.43i	0.0686404	−	0.118889i
$939$		0			0
$940$	−2484.40	−	4303.11i	−0.0862045	−	0.149311i
$941$	−4599.48	−	7966.54i	−0.159340	−	0.275985i	0.775291	−	0.631604i	$-0.217603\pi$
$941$	−4599.48	−	7966.54i	−0.159340	−	0.275985i	−0.934631	+	0.355619i	$0.884270\pi$
$942$		0			0
$943$	−5322.02	+	9218.01i	−0.183785	+	0.318324i
$944$	3870.14			0.133435
$945$		0			0
$946$	2929.55			0.100685
$947$	10481.8	−	18155.0i	0.359675	−	0.622976i	−0.628231	−	0.778027i	$-0.716221\pi$
$947$	10481.8	−	18155.0i	0.359675	−	0.622976i	0.987906	+	0.155051i	$0.0495541\pi$
$948$		0			0
$949$	−39264.7	−	68008.5i	−1.34308	−	2.32629i
$950$	2648.27	+	4586.93i	0.0904433	+	0.156652i
$951$		0			0
$952$	−3020.98	+	5232.50i	−0.102847	+	0.178137i
$953$	27943.7			0.949828			0.474914	−	0.880032i	$-0.342479\pi$
$953$	27943.7			0.949828			0.474914	+	0.880032i	$0.342479\pi$
$954$		0			0
$955$	6786.44			0.229952
$956$	197.460	−	342.010i	0.00668024	−	0.0115705i
$957$		0			0
$958$	17153.0	+	29709.8i	0.578484	+	1.00196i
$959$	−1491.55	−	2583.45i	−0.0502240	−	0.0869905i
$960$		0			0
$961$	14877.5	−	25768.6i	0.499396	−	0.864979i
$962$	−54385.9			−1.82274
$963$		0			0
$964$	−1457.87			−0.0487084
$965$	−4507.60	+	7807.39i	−0.150368	+	0.260444i
$966$		0			0
$967$	−1121.76	−	1942.95i	−0.0373045	−	0.0646132i	0.846770	−	0.531959i	$-0.178544\pi$
$967$	−1121.76	−	1942.95i	−0.0373045	−	0.0646132i	−0.884075	+	0.467345i	$0.845210\pi$
$968$	16164.5	+	27997.8i	0.536723	+	0.929632i
$969$		0			0
$970$	825.739	−	1430.22i	0.0273329	−	0.0473419i
$971$	−57345.0			−1.89525			−0.947626	−	0.319381i	$-0.896525\pi$
$971$	−57345.0			−1.89525			−0.947626	+	0.319381i	$0.896525\pi$
$972$		0			0
$973$	−13031.8			−0.429375
$974$	4248.44	−	7358.52i	0.139763	−	0.242076i
$975$		0			0
$976$	−13464.8	−	23321.7i	−0.441595	−	0.764866i
$977$	9395.44	+	16273.4i	0.307663	+	0.532888i	0.977851	−	0.209304i	$-0.0671197\pi$
$977$	9395.44	+	16273.4i	0.307663	+	0.532888i	−0.670188	+	0.742192i	$0.733786\pi$
$978$		0			0
$979$	770.334	−	1334.26i	0.0251481	−	0.0435578i
$980$	−3423.79			−0.111601
$981$		0			0
$982$	−16642.5			−0.540817
$983$	−17298.8	+	29962.4i	−0.561289	+	0.972181i	0.436096	+	0.899900i	$0.356361\pi$
$983$	−17298.8	+	29962.4i	−0.561289	+	0.972181i	−0.997384	+	0.0722803i	$0.976972\pi$
$984$		0			0
$985$	658.508	+	1140.57i	0.0213013	+	0.0368950i
$986$	−1994.35	−	3454.31i	−0.0644147	−	0.111570i
$987$		0			0
$988$	−8366.22	+	14490.7i	−0.269398	+	0.466611i
$989$	−59547.1			−1.91455
$990$		0			0
$991$	−45026.3			−1.44330			−0.721649	−	0.692259i	$-0.756616\pi$
$991$	−45026.3			−1.44330			−0.721649	+	0.692259i	$0.756616\pi$
$992$	−310.182	+	537.251i	−0.00992771	+	0.0171953i
$993$		0			0
$994$	−4485.53	−	7769.17i	−0.143131	−	0.247911i
$995$	1232.13	+	2134.12i	0.0392575	+	0.0679960i
$996$		0			0
$997$	5986.04	−	10368.1i	0.190150	−	0.329350i	−0.755150	−	0.655552i	$-0.772436\pi$
$997$	5986.04	−	10368.1i	0.190150	−	0.329350i	0.945300	+	0.326202i	$0.105769\pi$
$998$	−5977.66			−0.189599
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.e.a.46.2		4
3.2	odd	2		45.4.e.a.16.1	✓	4
9.2	odd	6		405.4.a.d.1.2		2
9.4	even	3	inner	135.4.e.a.91.2		4
9.5	odd	6		45.4.e.a.31.1	yes	4
9.7	even	3		405.4.a.e.1.1		2
15.2	even	4		225.4.k.a.124.2		8
15.8	even	4		225.4.k.a.124.3		8
15.14	odd	2		225.4.e.a.151.2		4
45.14	odd	6		225.4.e.a.76.2		4
45.23	even	12		225.4.k.a.49.2		8
45.29	odd	6		2025.4.a.l.1.1		2
45.32	even	12		225.4.k.a.49.3		8
45.34	even	6		2025.4.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.a.16.1	✓	4	3.2	odd	2
45.4.e.a.31.1	yes	4	9.5	odd	6
135.4.e.a.46.2		4	1.1	even	1	trivial
135.4.e.a.91.2		4	9.4	even	3	inner
225.4.e.a.76.2		4	45.14	odd	6
225.4.e.a.151.2		4	15.14	odd	2
225.4.k.a.49.2		8	45.23	even	12
225.4.k.a.49.3		8	45.32	even	12
225.4.k.a.124.2		8	15.2	even	4
225.4.k.a.124.3		8	15.8	even	4
405.4.a.d.1.2		2	9.2	odd	6
405.4.a.e.1.1		2	9.7	even	3
2025.4.a.j.1.2		2	45.34	even	6
2025.4.a.l.1.1		2	45.29	odd	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:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	135.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.18614 - 2.05446i) q^{2} +(1.18614 + 2.05446i) q^{4} +(2.50000 + 4.33013i) q^{5} +(-3.68614 + 6.38458i) q^{7} +24.6060 q^{8} +11.8614 q^{10} +(-2.06930 + 3.58413i) q^{11} +(39.4891 + 68.3972i) q^{13} +(8.74456 + 15.1460i) q^{14} +(19.6970 - 34.1162i) q^{16} +33.3070 q^{17} +89.3070 q^{19} +(-5.93070 + 10.2723i) q^{20} +(4.90895 + 8.50256i) q^{22} +(-99.7812 - 172.826i) q^{23} +(-12.5000 + 21.6506i) q^{25} +187.359 q^{26} -17.4891 q^{28} +(25.2405 - 43.7179i) q^{29} +(3.00000 + 5.19615i) q^{31} +(51.6970 + 89.5419i) q^{32} +(39.5068 - 68.4278i) q^{34} -36.8614 q^{35} -290.277 q^{37} +(105.931 - 183.477i) q^{38} +(61.5149 + 106.547i) q^{40} +(-26.6684 - 46.1911i) q^{41} +(149.194 - 258.412i) q^{43} -9.81791 q^{44} -473.418 q^{46} +(-209.452 + 362.782i) q^{47} +(144.325 + 249.978i) q^{49} +(29.6535 + 51.3614i) q^{50} +(-93.6793 + 162.257i) q^{52} -399.228 q^{53} -20.6930 q^{55} +(-90.7011 + 157.099i) q^{56} +(-59.8776 - 103.711i) q^{58} +(49.1209 + 85.0799i) q^{59} +(341.797 - 592.011i) q^{61} +14.2337 q^{62} +560.432 q^{64} +(-197.446 + 341.986i) q^{65} +(-112.750 - 195.289i) q^{67} +(39.5068 + 68.4278i) q^{68} +(-43.7228 + 75.7301i) q^{70} -512.951 q^{71} -994.318 q^{73} +(-344.310 + 596.362i) q^{74} +(105.931 + 183.477i) q^{76} +(-15.2554 - 26.4232i) q^{77} +(100.853 - 174.683i) q^{79} +196.970 q^{80} -126.530 q^{82} +(553.822 - 959.247i) q^{83} +(83.2676 + 144.224i) q^{85} +(-353.931 - 613.026i) q^{86} +(-50.9171 + 88.1909i) q^{88} -372.269 q^{89} -582.250 q^{91} +(236.709 - 409.992i) q^{92} +(496.880 + 860.622i) q^{94} +(223.268 + 386.711i) q^{95} +(69.6156 - 120.578i) q^{97} +684.758 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - q^{4} + 10 q^{5} - 9 q^{7} + 18 q^{8} - 10 q^{10} - 37 q^{11} + 112 q^{13} + 12 q^{14} + 119 q^{16} - 154 q^{17} + 70 q^{19} + 5 q^{20} - 101 q^{22} - 267 q^{23} - 50 q^{25} + 152 q^{26}+ \cdots - 926 q^{98}+O(q^{100}) \) 4 * q - q^2 - q^4 + 10 * q^5 - 9 * q^7 + 18 * q^8 - 10 * q^10 - 37 * q^11 + 112 * q^13 + 12 * q^14 + 119 * q^16 - 154 * q^17 + 70 * q^19 + 5 * q^20 - 101 * q^22 - 267 * q^23 - 50 * q^25 + 152 * q^26 - 24 * q^28 + 325 * q^29 + 12 * q^31 + 247 * q^32 + 451 * q^34 - 90 * q^35 - 1276 * q^37 + 395 * q^38 + 45 * q^40 + 238 * q^41 + 97 * q^43 + 202 * q^44 - 492 * q^46 - 901 * q^47 + 629 * q^49 - 25 * q^50 - 76 * q^52 - 448 * q^53 - 370 * q^55 - 156 * q^56 + 806 * q^58 - 85 * q^59 + 247 * q^61 - 12 * q^62 + 1426 * q^64 - 560 * q^65 + 606 * q^67 + 451 * q^68 - 60 * q^70 - 788 * q^71 - 1622 * q^73 + 484 * q^74 + 395 * q^76 - 84 * q^77 + 840 * q^79 + 1190 * q^80 - 2218 * q^82 - 387 * q^83 - 385 * q^85 - 1387 * q^86 + 411 * q^88 + 2130 * q^89 - 1272 * q^91 + 246 * q^92 - 632 * q^94 + 175 * q^95 + 1031 * q^97 - 926 * q^98

Introduction

L-functions

Modular forms

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Database

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