LMFDB - Embedded newform 405.3.d.b.404.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,3,Mod(404,405)] 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("405.404"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	405.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24] 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:	$11.0354507066$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			404.1
Character	$\chi$	$=$	405.404
Dual form			405.3.d.b.404.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.57411 q^{2} +8.77429 q^{4} +(3.06744 - 3.94852i) q^{5} +6.02185i q^{7} -17.0638 q^{8} +(-10.9634 + 14.1125i) q^{10} -15.2008i q^{11} -11.0271i q^{13} -21.5228i q^{14} +25.8909 q^{16} -26.8426 q^{17} +15.8915 q^{19} +(26.9146 - 34.6455i) q^{20} +54.3292i q^{22} +11.5342 q^{23} +(-6.18165 - 24.2237i) q^{25} +39.4120i q^{26} +52.8374i q^{28} +54.4382i q^{29} -31.5722 q^{31} -24.2818 q^{32} +95.9384 q^{34} +(23.7774 + 18.4716i) q^{35} -38.1480i q^{37} -56.7980 q^{38} +(-52.3423 + 67.3769i) q^{40} -37.6636i q^{41} -4.53892i q^{43} -133.376i q^{44} -41.2246 q^{46} +3.71811 q^{47} +12.7374 q^{49} +(22.0939 + 86.5782i) q^{50} -96.7548i q^{52} -56.1230 q^{53} +(-60.0205 - 46.6274i) q^{55} -102.756i q^{56} -194.568i q^{58} -50.0428i q^{59} -48.4417 q^{61} +112.842 q^{62} -16.7778 q^{64} +(-43.5407 - 33.8249i) q^{65} -69.3702i q^{67} -235.524 q^{68} +(-84.9831 - 66.0197i) q^{70} -38.5522i q^{71} -27.2700i q^{73} +136.345i q^{74} +139.437 q^{76} +91.5366 q^{77} -78.8985 q^{79} +(79.4189 - 102.231i) q^{80} +134.614i q^{82} -150.602 q^{83} +(-82.3379 + 105.988i) q^{85} +16.2226i q^{86} +259.383i q^{88} +3.50521i q^{89} +66.4034 q^{91} +101.204 q^{92} -13.2890 q^{94} +(48.7462 - 62.7479i) q^{95} -27.6546i q^{97} -45.5248 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 48 q^{4} + 12 q^{10} + 96 q^{16} + 48 q^{25} + 144 q^{34} + 72 q^{40} - 168 q^{46} - 288 q^{49} - 132 q^{55} - 360 q^{61} - 72 q^{64} - 156 q^{70} + 48 q^{76} - 480 q^{79} - 456 q^{85} - 48 q^{91}+ \cdots + 384 q^{94}+O(q^{100}) $ 24 * q + 48 * q^4 + 12 * q^10 + 96 * q^16 + 48 * q^25 + 144 * q^34 + 72 * q^40 - 168 * q^46 - 288 * q^49 - 132 * q^55 - 360 * q^61 - 72 * q^64 - 156 * q^70 + 48 * q^76 - 480 * q^79 - 456 * q^85 - 48 * q^91 + 384 * q^94

Character values

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

$n$	$82$	$326$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−3.57411			−1.78706			−0.893528	−	0.449007i	$-0.851778\pi$
$2$	−3.57411			−1.78706			−0.893528	+	0.449007i	$0.851778\pi$
$3$		0			0
$3$		0			0
$4$	8.77429			2.19357
$5$	3.06744	−	3.94852i	0.613488	−	0.789704i
$5$	3.06744	−	3.94852i	0.613488	−	0.789704i
$6$		0			0
$7$			6.02185i			0.860264i	0.902766	+	0.430132i	$0.141533\pi$
$7$			6.02185i			0.860264i	−0.902766	+	0.430132i	$0.858467\pi$
$8$	−17.0638			−2.13298
$9$		0			0
$10$	−10.9634	+	14.1125i	−1.09634	+	1.41125i
$11$		−	15.2008i		−	1.38189i	−0.722909	−	0.690944i	$-0.757195\pi$
$11$		−	15.2008i		−	1.38189i	0.722909	−	0.690944i	$-0.242805\pi$
$12$		0			0
$13$		−	11.0271i		−	0.848237i	−0.905607	−	0.424119i	$-0.860584\pi$
$13$		−	11.0271i		−	0.848237i	0.905607	−	0.424119i	$-0.139416\pi$
$14$		−	21.5228i		−	1.53734i
$15$		0			0
$16$	25.8909			1.61818
$17$	−26.8426			−1.57898			−0.789488	−	0.613767i	$-0.789654\pi$
$17$	−26.8426			−1.57898			−0.789488	+	0.613767i	$0.789654\pi$
$18$		0			0
$19$	15.8915			0.836395			0.418197	−	0.908356i	$-0.362662\pi$
$19$	15.8915			0.836395			0.418197	+	0.908356i	$0.362662\pi$
$20$	26.9146	−	34.6455i	1.34573	−	1.73227i
$21$		0			0
$22$			54.3292i			2.46951i
$23$	11.5342			0.501488			0.250744	−	0.968053i	$-0.419325\pi$
$23$	11.5342			0.501488			0.250744	+	0.968053i	$0.419325\pi$
$24$		0			0
$25$	−6.18165	−	24.2237i	−0.247266	−	0.968948i
$26$			39.4120i			1.51585i
$27$		0			0
$28$			52.8374i			1.88705i
$29$			54.4382i			1.87718i	0.345037	+	0.938589i	$0.387866\pi$
$29$			54.4382i			1.87718i	−0.345037	+	0.938589i	$0.612134\pi$
$30$		0			0
$31$	−31.5722			−1.01846			−0.509228	−	0.860631i	$-0.670069\pi$
$31$	−31.5722			−1.01846			−0.509228	+	0.860631i	$0.670069\pi$
$32$	−24.2818			−0.758807
$33$		0			0
$34$	95.9384			2.82172
$35$	23.7774	+	18.4716i	0.679354	+	0.527761i
$36$		0			0
$37$		−	38.1480i		−	1.03103i	−0.856881	−	0.515514i	$-0.827601\pi$
$37$		−	38.1480i		−	1.03103i	0.856881	−	0.515514i	$-0.172399\pi$
$38$	−56.7980			−1.49468
$39$		0			0
$40$	−52.3423	+	67.3769i	−1.30856	+	1.68442i
$41$		−	37.6636i		−	0.918625i	−0.888275	−	0.459312i	$-0.848096\pi$
$41$		−	37.6636i		−	0.918625i	0.888275	−	0.459312i	$-0.151904\pi$
$42$		0			0
$43$		−	4.53892i		−	0.105556i	−0.998606	−	0.0527782i	$-0.983192\pi$
$43$		−	4.53892i		−	0.105556i	0.998606	−	0.0527782i	$-0.0168076\pi$
$44$		−	133.376i		−	3.03127i
$45$		0			0
$46$	−41.2246			−0.896187
$47$	3.71811			0.0791088			0.0395544	−	0.999217i	$-0.487406\pi$
$47$	3.71811			0.0791088			0.0395544	+	0.999217i	$0.487406\pi$
$48$		0			0
$49$	12.7374			0.259947
$50$	22.0939	+	86.5782i	0.441878	+	1.73156i
$51$		0			0
$52$		−	96.7548i		−	1.86067i
$53$	−56.1230			−1.05892			−0.529462	−	0.848333i	$-0.677606\pi$
$53$	−56.1230			−1.05892			−0.529462	+	0.848333i	$0.677606\pi$
$54$		0			0
$55$	−60.0205	−	46.6274i	−1.09128	−	0.847771i
$56$		−	102.756i		−	1.83492i
$57$		0			0
$58$		−	194.568i		−	3.35462i
$59$		−	50.0428i		−	0.848184i	−0.905619	−	0.424092i	$-0.860593\pi$
$59$		−	50.0428i		−	0.848184i	0.905619	−	0.424092i	$-0.139407\pi$
$60$		0			0
$61$	−48.4417			−0.794126			−0.397063	−	0.917791i	$-0.629971\pi$
$61$	−48.4417			−0.794126			−0.397063	+	0.917791i	$0.629971\pi$
$62$	112.842			1.82004
$63$		0			0
$64$	−16.7778			−0.262153
$65$	−43.5407	−	33.8249i	−0.669857	−	0.520383i
$66$		0			0
$67$		−	69.3702i		−	1.03538i	−0.855570	−	0.517688i	$-0.826793\pi$
$67$		−	69.3702i		−	1.03538i	0.855570	−	0.517688i	$-0.173207\pi$
$68$	−235.524			−3.46359
$69$		0			0
$70$	−84.9831	−	66.0197i	−1.21404	−	0.943139i
$71$		−	38.5522i		−	0.542989i	−0.962440	−	0.271494i	$-0.912482\pi$
$71$		−	38.5522i		−	0.542989i	0.962440	−	0.271494i	$-0.0875179\pi$
$72$		0			0
$73$		−	27.2700i		−	0.373561i	−0.982402	−	0.186781i	$-0.940195\pi$
$73$		−	27.2700i		−	0.373561i	0.982402	−	0.186781i	$-0.0598054\pi$
$74$			136.345i			1.84250i
$75$		0			0
$76$	139.437			1.83469
$77$	91.5366			1.18879
$78$		0			0
$79$	−78.8985			−0.998716			−0.499358	−	0.866396i	$-0.666431\pi$
$79$	−78.8985			−0.998716			−0.499358	+	0.866396i	$0.666431\pi$
$80$	79.4189	−	102.231i	0.992736	−	1.27789i
$81$		0			0
$82$			134.614i			1.64163i
$83$	−150.602			−1.81448			−0.907242	−	0.420609i	$-0.861816\pi$
$83$	−150.602			−1.81448			−0.907242	+	0.420609i	$0.861816\pi$
$84$		0			0
$85$	−82.3379	+	105.988i	−0.968682	+	1.24692i
$86$			16.2226i			0.188635i
$87$		0			0
$88$			259.383i			2.94754i
$89$			3.50521i			0.0393843i	0.999806	+	0.0196922i	$0.00626862\pi$
$89$			3.50521i			0.0393843i	−0.999806	+	0.0196922i	$0.993731\pi$
$90$		0			0
$91$	66.4034			0.729708
$92$	101.204			1.10005
$93$		0			0
$94$	−13.2890			−0.141372
$95$	48.7462	−	62.7479i	0.513118	−	0.660504i
$96$		0			0
$97$		−	27.6546i		−	0.285099i	−0.989788	−	0.142550i	$-0.954470\pi$
$97$		−	27.6546i		−	0.285099i	0.989788	−	0.142550i	$-0.0455300\pi$
$98$	−45.5248			−0.464539
$99$		0			0
$100$	−54.2396	−	212.546i	−0.542396	−	2.12546i
$101$			36.0907i			0.357333i	0.983910	+	0.178667i	$0.0571784\pi$
$101$			36.0907i			0.357333i	−0.983910	+	0.178667i	$0.942822\pi$
$102$		0			0
$103$		−	28.0612i		−	0.272439i	−0.990679	−	0.136219i	$-0.956505\pi$
$103$		−	28.0612i		−	0.272439i	0.990679	−	0.136219i	$-0.0434952\pi$
$104$			188.164i			1.80927i
$105$		0			0
$106$	200.590			1.89236
$107$	91.8455			0.858369			0.429184	−	0.903217i	$-0.358801\pi$
$107$	91.8455			0.858369			0.429184	+	0.903217i	$0.358801\pi$
$108$		0			0
$109$	78.1779			0.717228			0.358614	−	0.933486i	$-0.383249\pi$
$109$	78.1779			0.717228			0.358614	+	0.933486i	$0.383249\pi$
$110$	214.520	+	166.652i	1.95018	+	1.51501i
$111$		0			0
$112$			155.911i			1.39206i
$113$	6.56726			0.0581174			0.0290587	−	0.999578i	$-0.490749\pi$
$113$	6.56726			0.0581174			0.0290587	+	0.999578i	$0.490749\pi$
$114$		0			0
$115$	35.3805	−	45.5431i	0.307656	−	0.396027i
$116$			477.656i			4.11772i
$117$		0			0
$118$			178.859i			1.51575i
$119$		−	161.642i		−	1.35833i
$120$		0			0
$121$	−110.063			−0.909613
$122$	173.136			1.41915
$123$		0			0
$124$	−277.023			−2.23406
$125$	−114.610	−	49.8963i	−0.916877	−	0.399170i
$126$		0			0
$127$		−	247.486i		−	1.94871i	−0.225023	−	0.974353i	$-0.572246\pi$
$127$		−	247.486i		−	1.94871i	0.225023	−	0.974353i	$-0.427754\pi$
$128$	157.093			1.22729
$129$		0			0
$130$	155.619	+	120.894i	1.19707	+	0.929954i
$131$		−	142.174i		−	1.08530i	−0.839960	−	0.542648i	$-0.817422\pi$
$131$		−	142.174i		−	1.08530i	0.839960	−	0.542648i	$-0.182578\pi$
$132$		0			0
$133$			95.6961i			0.719520i
$134$			247.937i			1.85027i
$135$		0			0
$136$	458.037			3.36792
$137$	72.8900			0.532044			0.266022	−	0.963967i	$-0.414291\pi$
$137$	72.8900			0.532044			0.266022	+	0.963967i	$0.414291\pi$
$138$		0			0
$139$	−66.9989			−0.482007			−0.241003	−	0.970524i	$-0.577476\pi$
$139$	−66.9989			−0.482007			−0.241003	+	0.970524i	$0.577476\pi$
$140$	208.630	+	162.075i	1.49021	+	1.15768i
$141$		0			0
$142$			137.790i			0.970352i
$143$	−167.620			−1.17217
$144$		0			0
$145$	214.950	+	166.986i	1.48242	+	1.15163i
$146$			97.4660i			0.667575i
$147$		0			0
$148$		−	334.721i		−	2.26163i
$149$			76.5668i			0.513871i	0.966429	+	0.256936i	$0.0827128\pi$
$149$			76.5668i			0.513871i	−0.966429	+	0.256936i	$0.917287\pi$
$150$		0			0
$151$	−39.9051			−0.264272			−0.132136	−	0.991232i	$-0.542184\pi$
$151$	−39.9051			−0.264272			−0.132136	+	0.991232i	$0.542184\pi$
$152$	−271.170			−1.78401
$153$		0			0
$154$	−327.162			−2.12443
$155$	−96.8456	+	124.663i	−0.624810	+	0.804280i
$156$		0			0
$157$			135.575i			0.863538i	0.901984	+	0.431769i	$0.142110\pi$
$157$			135.575i			0.863538i	−0.901984	+	0.431769i	$0.857890\pi$
$158$	281.992			1.78476
$159$		0			0
$160$	−74.4830	+	95.8773i	−0.465518	+	0.599233i
$161$			69.4572i			0.431411i
$162$		0			0
$163$			85.9303i			0.527180i	0.964635	+	0.263590i	$0.0849066\pi$
$163$			85.9303i			0.527180i	−0.964635	+	0.263590i	$0.915093\pi$
$164$		−	330.471i		−	2.01507i
$165$		0			0
$166$	538.269			3.24258
$167$	−111.718			−0.668970			−0.334485	−	0.942401i	$-0.608562\pi$
$167$	−111.718			−0.668970			−0.334485	+	0.942401i	$0.608562\pi$
$168$		0			0
$169$	47.4034			0.280494
$170$	294.285	−	378.815i	1.73109	−	2.22832i
$171$		0			0
$172$		−	39.8258i		−	0.231545i
$173$	−8.68986			−0.0502304			−0.0251152	−	0.999685i	$-0.507995\pi$
$173$	−8.68986			−0.0502304			−0.0251152	+	0.999685i	$0.507995\pi$
$174$		0			0
$175$	145.871	−	37.2249i	0.833550	−	0.212714i
$176$		−	393.562i		−	2.23615i
$177$		0			0
$178$		−	12.5280i		−	0.0703821i
$179$		−	57.4651i		−	0.321034i	−0.987033	−	0.160517i	$-0.948684\pi$
$179$		−	57.4651i		−	0.321034i	0.987033	−	0.160517i	$-0.0513162\pi$
$180$		0			0
$181$	−139.711			−0.771884			−0.385942	−	0.922523i	$-0.626124\pi$
$181$	−139.711			−0.771884			−0.385942	+	0.922523i	$0.626124\pi$
$182$	−237.333			−1.30403
$183$		0			0
$184$	−196.818			−1.06966
$185$	−150.628	−	117.017i	−0.814207	−	0.632522i
$186$		0			0
$187$			408.028i			2.18197i
$188$	32.6238			0.173531
$189$		0			0
$190$	−174.224	+	224.268i	−0.916970	+	1.18036i
$191$		−	31.4760i		−	0.164796i	−0.996600	−	0.0823980i	$-0.973742\pi$
$191$		−	31.4760i		−	0.164796i	0.996600	−	0.0823980i	$-0.0262579\pi$
$192$		0			0
$193$			50.0752i			0.259457i	0.991550	+	0.129729i	$0.0414106\pi$
$193$			50.0752i			0.259457i	−0.991550	+	0.129729i	$0.958589\pi$
$194$			98.8407i			0.509488i
$195$		0			0
$196$	111.761			0.570211
$197$	−37.4908			−0.190308			−0.0951542	−	0.995463i	$-0.530334\pi$
$197$	−37.4908			−0.190308			−0.0951542	+	0.995463i	$0.530334\pi$
$198$		0			0
$199$	40.5173			0.203604			0.101802	−	0.994805i	$-0.467539\pi$
$199$	40.5173			0.203604			0.101802	+	0.994805i	$0.467539\pi$
$200$	105.483	+	413.349i	0.527413	+	2.06675i
$201$		0			0
$202$		−	128.992i		−	0.638575i
$203$	−327.818			−1.61487
$204$		0			0
$205$	−148.716	−	115.531i	−0.725442	−	0.563565i
$206$			100.294i			0.486864i
$207$		0			0
$208$		−	285.502i		−	1.37260i
$209$		−	241.563i		−	1.15580i
$210$		0			0
$211$	172.473			0.817406			0.408703	−	0.912668i	$-0.365981\pi$
$211$	172.473			0.817406			0.408703	+	0.912668i	$0.365981\pi$
$212$	−492.439			−2.32283
$213$		0			0
$214$	−328.266			−1.53395
$215$	−17.9220	−	13.9229i	−0.0833583	−	0.0647575i
$216$		0			0
$217$		−	190.123i		−	0.876141i
$218$	−279.417			−1.28173
$219$		0			0
$220$	−526.637	−	409.122i	−2.39381	−	1.85965i
$221$			295.995i			1.33935i
$222$		0			0
$223$		−	299.356i		−	1.34240i	−0.741276	−	0.671201i	$-0.765779\pi$
$223$		−	299.356i		−	1.34240i	0.741276	−	0.671201i	$-0.234221\pi$
$224$		−	146.221i		−	0.652774i
$225$		0			0
$226$	−23.4721			−0.103859
$227$	298.716			1.31593			0.657965	−	0.753049i	$-0.271418\pi$
$227$	298.716			1.31593			0.657965	+	0.753049i	$0.271418\pi$
$228$		0			0
$229$	−203.310			−0.887816			−0.443908	−	0.896072i	$-0.646408\pi$
$229$	−203.310			−0.887816			−0.443908	+	0.896072i	$0.646408\pi$
$230$	−126.454	+	162.776i	−0.549799	+	0.707722i
$231$		0			0
$232$		−	928.924i		−	4.00398i
$233$	436.398			1.87295			0.936476	−	0.350733i	$-0.114067\pi$
$233$	436.398			1.87295			0.936476	+	0.350733i	$0.114067\pi$
$234$		0			0
$235$	11.4051	−	14.6810i	0.0485322	−	0.0624725i
$236$		−	439.090i		−	1.86055i
$237$		0			0
$238$			577.726i			2.42742i
$239$			239.697i			1.00292i	0.865182	+	0.501458i	$0.167203\pi$
$239$			239.697i			1.00292i	−0.865182	+	0.501458i	$0.832797\pi$
$240$		0			0
$241$	205.348			0.852067			0.426034	−	0.904707i	$-0.359911\pi$
$241$	205.348			0.852067			0.426034	+	0.904707i	$0.359911\pi$
$242$	393.378			1.62553
$243$		0			0
$244$	−425.041			−1.74197
$245$	39.0711	−	50.2938i	0.159474	−	0.205281i
$246$		0			0
$247$		−	175.237i		−	0.709461i
$248$	538.742			2.17235
$249$		0			0
$250$	409.628	+	178.335i	1.63851	+	0.713340i
$251$		−	357.277i		−	1.42341i	−0.702477	−	0.711707i	$-0.747923\pi$
$251$		−	357.277i		−	1.42341i	0.702477	−	0.711707i	$-0.252077\pi$
$252$		0			0
$253$		−	175.329i		−	0.692999i
$254$			884.542i			3.48245i
$255$		0			0
$256$	−494.357			−1.93108
$257$	280.513			1.09149			0.545746	−	0.837951i	$-0.316246\pi$
$257$	280.513			1.09149			0.545746	+	0.837951i	$0.316246\pi$
$258$		0			0
$259$	229.721			0.886955
$260$	−382.038	−	296.789i	−1.46938	−	1.14150i
$261$		0			0
$262$			508.145i			1.93949i
$263$	120.494			0.458153			0.229077	−	0.973408i	$-0.426429\pi$
$263$	120.494			0.458153			0.229077	+	0.973408i	$0.426429\pi$
$264$		0			0
$265$	−172.154	+	221.603i	−0.649637	+	0.836237i
$266$		−	342.029i		−	1.28582i
$267$		0			0
$268$		−	608.674i		−	2.27117i
$269$		−	308.065i		−	1.14522i	−0.819827	−	0.572612i	$-0.805930\pi$
$269$		−	308.065i		−	1.14522i	0.819827	−	0.572612i	$-0.194070\pi$
$270$		0			0
$271$	354.684			1.30880			0.654399	−	0.756149i	$-0.272922\pi$
$271$	354.684			1.30880			0.654399	+	0.756149i	$0.272922\pi$
$272$	−694.980			−2.55507
$273$		0			0
$274$	−260.517			−0.950792
$275$	−368.219	+	93.9658i	−1.33898	+	0.341694i
$276$		0			0
$277$			480.306i			1.73396i	0.498346	+	0.866978i	$0.333941\pi$
$277$			480.306i			1.73396i	−0.498346	+	0.866978i	$0.666059\pi$
$278$	239.462			0.861373
$279$		0			0
$280$	−405.733	−	315.197i	−1.44905	−	1.12570i
$281$			126.250i			0.449287i	0.974441	+	0.224644i	$0.0721218\pi$
$281$			126.250i			0.449287i	−0.974441	+	0.224644i	$0.927878\pi$
$282$		0			0
$283$			365.044i			1.28991i	0.764221	+	0.644954i	$0.223124\pi$
$283$			365.044i			1.28991i	−0.764221	+	0.644954i	$0.776876\pi$
$284$		−	338.268i		−	1.19109i
$285$		0			0
$286$	599.093			2.09473
$287$	226.804			0.790259
$288$		0			0
$289$	431.524			1.49316
$290$	−768.257	−	596.826i	−2.64916	−	2.05802i
$291$		0			0
$292$		−	239.275i		−	0.819433i
$293$	−194.169			−0.662693			−0.331347	−	0.943509i	$-0.607503\pi$
$293$	−194.169			−0.662693			−0.331347	+	0.943509i	$0.607503\pi$
$294$		0			0
$295$	−197.595	−	153.503i	−0.669814	−	0.520350i
$296$			650.951i			2.19916i
$297$		0			0
$298$		−	273.658i		−	0.918317i
$299$		−	127.189i		−	0.425380i
$300$		0			0
$301$	27.3327			0.0908063
$302$	142.625			0.472269
$303$		0			0
$304$	411.446			1.35344
$305$	−148.592	+	191.273i	−0.487187	+	0.627125i
$306$		0			0
$307$			492.555i			1.60441i	0.597046	+	0.802207i	$0.296341\pi$
$307$			492.555i			1.60441i	−0.597046	+	0.802207i	$0.703659\pi$
$308$	803.168			2.60769
$309$		0			0
$310$	346.137	−	445.561i	1.11657	−	1.43729i
$311$		−	186.362i		−	0.599235i	−0.954059	−	0.299617i	$-0.903141\pi$
$311$		−	186.362i		−	0.599235i	0.954059	−	0.299617i	$-0.0968590\pi$
$312$		0			0
$313$		−	218.646i		−	0.698548i	−0.937021	−	0.349274i	$-0.886428\pi$
$313$		−	218.646i		−	0.698548i	0.937021	−	0.349274i	$-0.113572\pi$
$314$		−	484.562i		−	1.54319i
$315$		0			0
$316$	−692.278			−2.19075
$317$	204.603			0.645435			0.322718	−	0.946495i	$-0.395404\pi$
$317$	204.603			0.645435			0.322718	+	0.946495i	$0.395404\pi$
$318$		0			0
$319$	827.502			2.59405
$320$	−51.4649	+	66.2476i	−0.160828	+	0.207024i
$321$		0			0
$322$		−	248.248i		−	0.770957i
$323$	−426.569			−1.32065
$324$		0			0
$325$	−267.117	+	68.1656i	−0.821897	+	0.209740i
$326$		−	307.125i		−	0.942100i
$327$		0			0
$328$			642.686i			1.95941i
$329$			22.3899i			0.0680544i
$330$		0			0
$331$	−596.610			−1.80245			−0.901224	−	0.433353i	$-0.857330\pi$
$331$	−596.610			−1.80245			−0.901224	+	0.433353i	$0.857330\pi$
$332$	−1321.43			−3.98020
$333$		0			0
$334$	399.293			1.19549
$335$	−273.910	−	212.789i	−0.817641	−	0.635190i
$336$		0			0
$337$		−	332.596i		−	0.986931i	−0.869765	−	0.493466i	$-0.835730\pi$
$337$		−	332.596i		−	0.986931i	0.869765	−	0.493466i	$-0.164270\pi$
$338$	−169.425			−0.501258
$339$		0			0
$340$	−722.457	+	929.973i	−2.12487	+	2.73522i
$341$			479.921i			1.40739i
$342$		0			0
$343$			371.773i			1.08389i
$344$			77.4514i			0.225149i
$345$		0			0
$346$	31.0585			0.0897645
$347$	−69.2057			−0.199440			−0.0997201	−	0.995016i	$-0.531795\pi$
$347$	−69.2057			−0.199440			−0.0997201	+	0.995016i	$0.531795\pi$
$348$		0			0
$349$	554.851			1.58983			0.794915	−	0.606721i	$-0.207515\pi$
$349$	554.851			1.58983			0.794915	+	0.606721i	$0.207515\pi$
$350$	−521.361	+	133.046i	−1.48960	+	0.380132i
$351$		0			0
$352$			369.102i			1.04859i
$353$	176.268			0.499343			0.249672	−	0.968331i	$-0.419677\pi$
$353$	176.268			0.499343			0.249672	+	0.968331i	$0.419677\pi$
$354$		0			0
$355$	−152.224	−	118.257i	−0.428801	−	0.333117i
$356$			30.7557i			0.0863924i
$357$		0			0
$358$			205.387i			0.573706i
$359$		−	122.033i		−	0.339924i	−0.985451	−	0.169962i	$-0.945636\pi$
$359$		−	122.033i		−	0.339924i	0.985451	−	0.169962i	$-0.0543645\pi$
$360$		0			0
$361$	−108.460			−0.300444
$362$	499.343			1.37940
$363$		0			0
$364$	582.642			1.60067
$365$	−107.676	−	83.6490i	−0.295003	−	0.229175i
$366$		0			0
$367$			672.393i			1.83213i	0.401025	+	0.916067i	$0.368654\pi$
$367$			672.393i			1.83213i	−0.401025	+	0.916067i	$0.631346\pi$
$368$	298.632			0.811499
$369$		0			0
$370$	538.362	+	418.231i	1.45503	+	1.13035i
$371$		−	337.964i		−	0.910954i
$372$		0			0
$373$			513.816i			1.37752i	0.724988	+	0.688762i	$0.241845\pi$
$373$			513.816i			1.37752i	−0.724988	+	0.688762i	$0.758155\pi$
$374$		−	1458.34i		−	3.89930i
$375$		0			0
$376$	−63.4453			−0.168737
$377$	600.294			1.59229
$378$		0			0
$379$	−496.340			−1.30960			−0.654802	−	0.755800i	$-0.727248\pi$
$379$	−496.340			−1.30960			−0.654802	+	0.755800i	$0.727248\pi$
$380$	427.713	−	550.568i	1.12556	−	1.44886i
$381$		0			0
$382$			112.499i			0.294500i
$383$	595.923			1.55594			0.777968	−	0.628304i	$-0.216251\pi$
$383$	595.923			1.55594			0.777968	+	0.628304i	$0.216251\pi$
$384$		0			0
$385$	280.783	−	361.434i	0.729306	−	0.938791i
$386$		−	178.974i		−	0.463664i
$387$		0			0
$388$		−	242.650i		−	0.625385i
$389$			439.285i			1.12927i	0.825342	+	0.564633i	$0.190982\pi$
$389$			439.285i			1.12927i	−0.825342	+	0.564633i	$0.809018\pi$
$390$		0			0
$391$	−309.608			−0.791836
$392$	−217.349			−0.554461
$393$		0			0
$394$	133.996			0.340092
$395$	−242.016	+	311.533i	−0.612700	+	0.788690i
$396$		0			0
$397$		−	405.932i		−	1.02250i	−0.859432	−	0.511250i	$-0.829183\pi$
$397$		−	405.932i		−	1.02250i	0.859432	−	0.511250i	$-0.170817\pi$
$398$	−144.813			−0.363853
$399$		0			0
$400$	−160.049	−	627.174i	−0.400122	−	1.56794i
$401$			328.446i			0.819066i	0.912295	+	0.409533i	$0.134308\pi$
$401$			328.446i			0.819066i	−0.912295	+	0.409533i	$0.865692\pi$
$402$		0			0
$403$			348.149i			0.863893i
$404$			316.670i			0.783836i
$405$		0			0
$406$	1171.66			2.88586
$407$	−579.879			−1.42476
$408$		0			0
$409$	−16.3136			−0.0398866			−0.0199433	−	0.999801i	$-0.506349\pi$
$409$	−16.3136			−0.0398866			−0.0199433	+	0.999801i	$0.506349\pi$
$410$	531.526	+	412.920i	1.29641	+	1.00712i
$411$		0			0
$412$		−	246.217i		−	0.597614i
$413$	301.350			0.729662
$414$		0			0
$415$	−461.963	+	594.656i	−1.11316	+	1.43291i
$416$			267.758i			0.643648i
$417$		0			0
$418$			863.373i			2.06549i
$419$			381.549i			0.910618i	0.890333	+	0.455309i	$0.150471\pi$
$419$			381.549i			0.910618i	−0.890333	+	0.455309i	$0.849529\pi$
$420$		0			0
$421$	423.338			1.00555			0.502776	−	0.864416i	$-0.332312\pi$
$421$	423.338			1.00555			0.502776	+	0.864416i	$0.332312\pi$
$422$	−616.436			−1.46075
$423$		0			0
$424$	957.674			2.25866
$425$	165.931	+	650.226i	0.390427	+	1.52994i
$426$		0			0
$427$		−	291.708i		−	0.683158i
$428$	805.878			1.88289
$429$		0			0
$430$	64.0554	+	49.7619i	0.148966	+	0.115725i
$431$			48.3081i			0.112084i	0.998428	+	0.0560419i	$0.0178480\pi$
$431$			48.3081i			0.112084i	−0.998428	+	0.0560419i	$0.982152\pi$
$432$		0			0
$433$		−	36.0601i		−	0.0832797i	−0.999133	−	0.0416399i	$-0.986742\pi$
$433$		−	36.0601i		−	0.0832797i	0.999133	−	0.0416399i	$-0.0132582\pi$
$434$			679.520i			1.56571i
$435$		0			0
$436$	685.955			1.57329
$437$	183.296			0.419441
$438$		0			0
$439$	−260.086			−0.592452			−0.296226	−	0.955118i	$-0.595728\pi$
$439$	−260.086			−0.592452			−0.296226	+	0.955118i	$0.595728\pi$
$440$	1024.18	+	795.642i	2.32768	+	1.80828i
$441$		0			0
$442$		−	1057.92i		−	2.39349i
$443$	−22.5294			−0.0508565			−0.0254282	−	0.999677i	$-0.508095\pi$
$443$	−22.5294			−0.0508565			−0.0254282	+	0.999677i	$0.508095\pi$
$444$		0			0
$445$	13.8404	+	10.7520i	0.0311020	+	0.0241618i
$446$			1069.93i			2.39895i
$447$		0			0
$448$		−	101.033i		−	0.225521i
$449$			275.640i			0.613898i	0.951726	+	0.306949i	$0.0993080\pi$
$449$			275.640i			0.613898i	−0.951726	+	0.306949i	$0.900692\pi$
$450$		0			0
$451$	−572.516			−1.26944
$452$	57.6230			0.127485
$453$		0			0
$454$	−1067.64			−2.35164
$455$	203.688	−	262.195i	0.447667	−	0.576253i
$456$		0			0
$457$			119.408i			0.261286i	0.991429	+	0.130643i	$0.0417041\pi$
$457$			119.408i			0.261286i	−0.991429	+	0.130643i	$0.958296\pi$
$458$	726.653			1.58658
$459$		0			0
$460$	310.438	−	399.608i	0.674866	−	0.868713i
$461$			90.3566i			0.196001i	0.995186	+	0.0980007i	$0.0312447\pi$
$461$			90.3566i			0.196001i	−0.995186	+	0.0980007i	$0.968755\pi$
$462$		0			0
$463$		−	610.436i		−	1.31844i	−0.751951	−	0.659219i	$-0.770887\pi$
$463$		−	610.436i		−	1.31844i	0.751951	−	0.659219i	$-0.229113\pi$
$464$			1409.46i			3.03762i
$465$		0			0
$466$	−1559.73			−3.34707
$467$	65.6533			0.140585			0.0702926	−	0.997526i	$-0.477607\pi$
$467$	65.6533			0.140585			0.0702926	+	0.997526i	$0.477607\pi$
$468$		0			0
$469$	417.736			0.890696
$470$	−40.7630	+	52.4717i	−0.0867299	+	0.111642i
$471$		0			0
$472$			853.923i			1.80916i
$473$	−68.9951			−0.145867
$474$		0			0
$475$	−98.2357	−	384.951i	−0.206812	−	0.810423i
$476$		−	1418.29i		−	2.97960i
$477$		0			0
$478$		−	856.704i		−	1.79227i
$479$			93.9812i			0.196203i	0.995176	+	0.0981015i	$0.0312770\pi$
$479$			93.9812i			0.196203i	−0.995176	+	0.0981015i	$0.968723\pi$
$480$		0			0
$481$	−420.661			−0.874555
$482$	−733.938			−1.52269
$483$		0			0
$484$	−965.725			−1.99530
$485$	−109.195	−	84.8288i	−0.225144	−	0.174905i
$486$		0			0
$487$		−	805.464i		−	1.65393i	−0.562253	−	0.826965i	$-0.690065\pi$
$487$		−	805.464i		−	1.65393i	0.562253	−	0.826965i	$-0.309935\pi$
$488$	826.601			1.69386
$489$		0			0
$490$	−139.645	+	179.756i	−0.284989	+	0.366849i
$491$		−	940.783i		−	1.91605i	−0.286678	−	0.958027i	$-0.592551\pi$
$491$		−	940.783i		−	1.91605i	0.286678	−	0.958027i	$-0.407449\pi$
$492$		0			0
$493$		−	1461.26i		−	2.96402i
$494$			626.316i			1.26785i
$495$		0			0
$496$	−817.433			−1.64805
$497$	232.155			0.467114
$498$		0			0
$499$	379.929			0.761380			0.380690	−	0.924703i	$-0.375686\pi$
$499$	379.929			0.761380			0.380690	+	0.924703i	$0.375686\pi$
$500$	−1005.62	−	437.804i	−2.01123	−	0.875609i
$501$		0			0
$502$			1276.95i			2.54372i
$503$	−637.608			−1.26761			−0.633805	−	0.773493i	$-0.718508\pi$
$503$	−637.608			−1.26761			−0.633805	+	0.773493i	$0.718508\pi$
$504$		0			0
$505$	142.505	+	110.706i	0.282188	+	0.219220i
$506$			626.645i			1.23843i
$507$		0			0
$508$		−	2171.51i		−	4.27463i
$509$			963.113i			1.89217i	0.323922	+	0.946084i	$0.394998\pi$
$509$			963.113i			1.89217i	−0.323922	+	0.946084i	$0.605002\pi$
$510$		0			0
$511$	164.216			0.321361
$512$	1138.52			2.22366
$513$		0			0
$514$	−1002.59			−1.95056
$515$	−110.800	−	86.0760i	−0.215146	−	0.167138i
$516$		0			0
$517$		−	56.5181i		−	0.109319i
$518$	−821.050			−1.58504
$519$		0			0
$520$	742.971	+	577.182i	1.42879	+	1.10997i
$521$			36.7876i			0.0706095i	0.999377	+	0.0353048i	$0.0112402\pi$
$521$			36.7876i			0.0706095i	−0.999377	+	0.0353048i	$0.988760\pi$
$522$		0			0
$523$			441.292i			0.843770i	0.906649	+	0.421885i	$0.138631\pi$
$523$			441.292i			0.843770i	−0.906649	+	0.421885i	$0.861369\pi$
$524$		−	1247.47i		−	2.38067i
$525$		0			0
$526$	−430.660			−0.818746
$527$	847.478			1.60812
$528$		0			0
$529$	−395.962			−0.748510
$530$	615.297	−	792.034i	1.16094	−	1.49440i
$531$		0			0
$532$			839.665i			1.57832i
$533$	−415.320			−0.779212
$534$		0			0
$535$	281.730	−	362.654i	0.526599	−	0.677858i
$536$			1183.72i			2.20844i
$537$		0			0
$538$			1101.06i			2.04658i
$539$		−	193.618i		−	0.359217i
$540$		0			0
$541$	−426.631			−0.788598			−0.394299	−	0.918982i	$-0.629013\pi$
$541$	−426.631			−0.788598			−0.394299	+	0.918982i	$0.629013\pi$
$542$	−1267.68			−2.33890
$543$		0			0
$544$	651.786			1.19814
$545$	239.806	−	308.687i	0.440011	−	0.566398i
$546$		0			0
$547$		−	602.928i		−	1.10224i	−0.834424	−	0.551122i	$-0.814200\pi$
$547$		−	602.928i		−	1.10224i	0.834424	−	0.551122i	$-0.185800\pi$
$548$	639.558			1.16708
$549$		0			0
$550$	1316.05	−	335.844i	2.39283	−	0.610626i
$551$			865.104i			1.57006i
$552$		0			0
$553$		−	475.115i		−	0.859159i
$554$		−	1716.67i		−	3.09868i
$555$		0			0
$556$	−587.868			−1.05732
$557$	527.490			0.947020			0.473510	−	0.880788i	$-0.342987\pi$
$557$	527.490			0.947020			0.473510	+	0.880788i	$0.342987\pi$
$558$		0			0
$559$	−50.0511			−0.0895368
$560$	615.619	+	478.248i	1.09932	+	0.854014i
$561$		0			0
$562$		−	451.231i		−	0.802902i
$563$	14.2805			0.0253650			0.0126825	−	0.999920i	$-0.495963\pi$
$563$	14.2805			0.0253650			0.0126825	+	0.999920i	$0.495963\pi$
$564$		0			0
$565$	20.1447	−	25.9310i	0.0356543	−	0.0458955i
$566$		−	1304.71i		−	2.30514i
$567$		0			0
$568$			657.849i			1.15818i
$569$		−	672.782i		−	1.18239i	−0.806527	−	0.591197i	$-0.798656\pi$
$569$		−	672.782i		−	1.18239i	0.806527	−	0.591197i	$-0.201344\pi$
$570$		0			0
$571$	857.665			1.50204			0.751020	−	0.660280i	$-0.229562\pi$
$571$	857.665			1.50204			0.751020	+	0.660280i	$0.229562\pi$
$572$	−1470.75			−2.57123
$573$		0			0
$574$	−810.625			−1.41224
$575$	−71.3005	−	279.401i	−0.124001	−	0.485915i
$576$		0			0
$577$			134.308i			0.232769i	0.993204	+	0.116385i	$0.0371305\pi$
$577$			134.308i			0.232769i	−0.993204	+	0.116385i	$0.962869\pi$
$578$	−1542.32			−2.66837
$579$		0			0
$580$	1886.04	+	1465.18i	3.25179	+	2.52617i
$581$		−	906.903i		−	1.56093i
$582$		0			0
$583$			853.112i			1.46331i
$584$			465.330i			0.796799i
$585$		0			0
$586$	693.983			1.18427
$587$	900.274			1.53369			0.766843	−	0.641835i	$-0.221826\pi$
$587$	900.274			1.53369			0.766843	+	0.641835i	$0.221826\pi$
$588$		0			0
$589$	−501.729			−0.851832
$590$	706.228	+	548.638i	1.19700	+	0.929895i
$591$		0			0
$592$		−	987.688i		−	1.66839i
$593$	−282.338			−0.476118			−0.238059	−	0.971251i	$-0.576511\pi$
$593$	−282.338			−0.476118			−0.238059	+	0.971251i	$0.576511\pi$
$594$		0			0
$595$	−638.246	−	495.826i	−1.07268	−	0.833322i
$596$			671.819i			1.12721i
$597$		0			0
$598$			454.587i			0.760179i
$599$			96.2914i			0.160754i	0.996765	+	0.0803768i	$0.0256124\pi$
$599$			96.2914i			0.160754i	−0.996765	+	0.0803768i	$0.974388\pi$
$600$		0			0
$601$	−537.809			−0.894857			−0.447428	−	0.894320i	$-0.647660\pi$
$601$	−537.809			−0.894857			−0.447428	+	0.894320i	$0.647660\pi$
$602$	−97.6901			−0.162276
$603$		0			0
$604$	−350.138			−0.579699
$605$	−337.612	+	434.587i	−0.558036	+	0.718325i
$606$		0			0
$607$		−	473.265i		−	0.779679i	−0.920883	−	0.389839i	$-0.872530\pi$
$607$		−	473.265i		−	0.779679i	0.920883	−	0.389839i	$-0.127470\pi$
$608$	−385.874			−0.634662
$609$		0			0
$610$	531.084	−	683.632i	0.870630	−	1.12071i
$611$		−	40.9999i		−	0.0671030i
$612$		0			0
$613$			109.517i			0.178658i	0.996002	+	0.0893288i	$0.0284722\pi$
$613$			109.517i			0.178658i	−0.996002	+	0.0893288i	$0.971528\pi$
$614$		−	1760.45i		−	2.86718i
$615$		0			0
$616$	−1561.97			−2.53566
$617$	1132.89			1.83613			0.918064	−	0.396432i	$-0.129752\pi$
$617$	1132.89			1.83613			0.918064	+	0.396432i	$0.129752\pi$
$618$		0			0
$619$	153.529			0.248027			0.124014	−	0.992281i	$-0.460423\pi$
$619$	153.529			0.248027			0.124014	+	0.992281i	$0.460423\pi$
$620$	−849.751	+	1093.83i	−1.37057	+	1.76424i
$621$		0			0
$622$			666.079i			1.07087i
$623$	−21.1078			−0.0338809
$624$		0			0
$625$	−548.574	+	299.485i	−0.877719	+	0.479176i
$626$			781.464i			1.24835i
$627$		0			0
$628$			1189.58i			1.89423i
$629$			1023.99i			1.62797i
$630$		0			0
$631$	−289.486			−0.458774			−0.229387	−	0.973335i	$-0.573672\pi$
$631$	−289.486			−0.458774			−0.229387	+	0.973335i	$0.573672\pi$
$632$	1346.31			2.13024
$633$		0			0
$634$	−731.274			−1.15343
$635$	−977.203	−	759.147i	−1.53890	−	1.19551i
$636$		0			0
$637$		−	140.456i		−	0.220496i
$638$	−2957.58			−4.63571
$639$		0			0
$640$	481.873	−	620.285i	0.752927	−	0.969196i
$641$			777.452i			1.21287i	0.795132	+	0.606436i	$0.207402\pi$
$641$			777.452i			1.21287i	−0.795132	+	0.606436i	$0.792598\pi$
$642$		0			0
$643$		−	292.731i		−	0.455258i	−0.973748	−	0.227629i	$-0.926903\pi$
$643$		−	292.731i		−	0.455258i	0.973748	−	0.227629i	$-0.0730974\pi$
$644$			609.438i			0.946332i
$645$		0			0
$646$	1524.60			2.36007
$647$	−504.594			−0.779899			−0.389949	−	0.920836i	$-0.627507\pi$
$647$	−504.594			−0.779899			−0.389949	+	0.920836i	$0.627507\pi$
$648$		0			0
$649$	−760.689			−1.17209
$650$	954.705	−	243.631i	1.46878	−	0.374818i
$651$		0			0
$652$			753.977i			1.15641i
$653$	221.350			0.338974			0.169487	−	0.985532i	$-0.445789\pi$
$653$	221.350			0.338974			0.169487	+	0.985532i	$0.445789\pi$
$654$		0			0
$655$	−561.376	−	436.109i	−0.857063	−	0.665816i
$656$		−	975.146i		−	1.48650i
$657$		0			0
$658$		−	80.0240i		−	0.121617i
$659$		−	768.534i		−	1.16621i	−0.812396	−	0.583106i	$-0.801837\pi$
$659$		−	768.534i		−	1.16621i	0.812396	−	0.583106i	$-0.198163\pi$
$660$		0			0
$661$	−383.835			−0.580689			−0.290345	−	0.956922i	$-0.593770\pi$
$661$	−383.835			−0.580689			−0.290345	+	0.956922i	$0.593770\pi$
$662$	2132.35			3.22108
$663$		0			0
$664$	2569.85			3.87026
$665$	377.858	+	293.542i	0.568208	+	0.441416i
$666$		0			0
$667$			627.901i			0.941381i
$668$	−980.245			−1.46743
$669$		0			0
$670$	978.984	+	760.531i	1.46117	+	1.13512i
$671$			736.351i			1.09739i
$672$		0			0
$673$		−	394.850i		−	0.586702i	−0.956005	−	0.293351i	$-0.905230\pi$
$673$		−	394.850i		−	0.586702i	0.956005	−	0.293351i	$-0.0947704\pi$
$674$			1188.73i			1.76370i
$675$		0			0
$676$	415.931			0.615283
$677$	606.978			0.896571			0.448285	−	0.893890i	$-0.352035\pi$
$677$	606.978			0.896571			0.448285	+	0.893890i	$0.352035\pi$
$678$		0			0
$679$	166.532			0.245260
$680$	1405.00	−	1808.57i	2.06618	−	2.65966i
$681$		0			0
$682$		−	1715.29i		−	2.51509i
$683$	711.162			1.04123			0.520616	−	0.853791i	$-0.325702\pi$
$683$	711.162			1.04123			0.520616	+	0.853791i	$0.325702\pi$
$684$		0			0
$685$	223.586	−	287.808i	0.326402	−	0.420157i
$686$		−	1328.76i		−	1.93697i
$687$		0			0
$688$		−	117.517i		−	0.170810i
$689$			618.873i			0.898219i
$690$		0			0
$691$	850.879			1.23137			0.615687	−	0.787991i	$-0.288879\pi$
$691$	850.879			1.23137			0.615687	+	0.787991i	$0.288879\pi$
$692$	−76.2473			−0.110184
$693$		0			0
$694$	247.349			0.356411
$695$	−205.515	+	264.547i	−0.295705	+	0.380643i
$696$		0			0
$697$			1010.99i			1.45049i
$698$	−1983.10			−2.84112
$699$		0			0
$700$	1279.92	−	326.622i	1.82845	−	0.466603i
$701$			1073.69i			1.53166i	0.643043	+	0.765830i	$0.277672\pi$
$701$			1073.69i			1.53166i	−0.643043	+	0.765830i	$0.722328\pi$
$702$		0			0
$703$		−	606.229i		−	0.862345i
$704$			255.036i			0.362266i
$705$		0			0
$706$	−630.002			−0.892355
$707$	−217.332			−0.307401
$708$		0			0
$709$	833.608			1.17575			0.587876	−	0.808951i	$-0.299964\pi$
$709$	833.608			1.17575			0.587876	+	0.808951i	$0.299964\pi$
$710$	544.067	+	422.662i	0.766291	+	0.595299i
$711$		0			0
$712$		−	59.8123i		−	0.0840060i
$713$	−364.160			−0.510743
$714$		0			0
$715$	−514.164	+	661.851i	−0.719111	+	0.925666i
$716$		−	504.216i		−	0.704212i
$717$		0			0
$718$			436.159i			0.607463i
$719$		−	433.030i		−	0.602267i	−0.953582	−	0.301134i	$-0.902635\pi$
$719$		−	433.030i		−	0.602267i	0.953582	−	0.301134i	$-0.0973650\pi$
$720$		0			0
$721$	168.980			0.234369
$722$	387.649			0.536910
$723$		0			0
$724$	−1225.86			−1.69318
$725$	1318.69	−	336.518i	1.81889	−	0.464162i
$726$		0			0
$727$			678.559i			0.933369i	0.884424	+	0.466684i	$0.154552\pi$
$727$			678.559i			0.933369i	−0.884424	+	0.466684i	$0.845448\pi$
$728$	−1133.10			−1.55645
$729$		0			0
$730$	384.847	+	298.971i	0.527187	+	0.409549i
$731$			121.836i			0.166671i
$732$		0			0
$733$		−	182.056i		−	0.248371i	−0.992259	−	0.124186i	$-0.960368\pi$
$733$		−	182.056i		−	0.248371i	0.992259	−	0.124186i	$-0.0396318\pi$
$734$		−	2403.21i		−	3.27413i
$735$		0			0
$736$	−280.072			−0.380532
$737$	−1054.48			−1.43077
$738$		0			0
$739$	727.990			0.985101			0.492551	−	0.870284i	$-0.336065\pi$
$739$	727.990			0.985101			0.492551	+	0.870284i	$0.336065\pi$
$740$	−1321.65	−	1026.74i	−1.78602	−	1.38748i
$741$		0			0
$742$			1207.92i			1.62793i
$743$	−394.282			−0.530663			−0.265331	−	0.964157i	$-0.585481\pi$
$743$	−394.282			−0.530663			−0.265331	+	0.964157i	$0.585481\pi$
$744$		0			0
$745$	302.326	+	234.864i	0.405806	+	0.315254i
$746$		−	1836.44i		−	2.46171i
$747$		0			0
$748$			3580.15i			4.78630i
$749$			553.079i			0.738423i
$750$		0			0
$751$	1339.18			1.78320			0.891600	−	0.452824i	$-0.149583\pi$
$751$	1339.18			1.78320			0.891600	+	0.452824i	$0.149583\pi$
$752$	96.2654			0.128013
$753$		0			0
$754$	−2145.52			−2.84552
$755$	−122.406	+	157.566i	−0.162128	+	0.208697i
$756$		0			0
$757$			185.595i			0.245171i	0.992458	+	0.122586i	$0.0391186\pi$
$757$			185.595i			0.245171i	−0.992458	+	0.122586i	$0.960881\pi$
$758$	1773.98			2.34034
$759$		0			0
$760$	−831.797	+	1070.72i	−1.09447	+	1.40884i
$761$			933.732i			1.22698i	0.789703	+	0.613490i	$0.210235\pi$
$761$			933.732i			1.22698i	−0.789703	+	0.613490i	$0.789765\pi$
$762$		0			0
$763$			470.775i			0.617005i
$764$		−	276.180i		−	0.361492i
$765$		0			0
$766$	−2129.90			−2.78054
$767$	−551.827			−0.719461
$768$		0			0
$769$	1058.93			1.37702			0.688511	−	0.725226i	$-0.258265\pi$
$769$	1058.93			1.37702			0.688511	+	0.725226i	$0.258265\pi$
$770$	−1003.55	+	1291.81i	−1.30331	+	1.67767i
$771$		0			0
$772$			439.374i			0.569137i
$773$	−79.0622			−0.102280			−0.0511399	−	0.998692i	$-0.516285\pi$
$773$	−79.0622			−0.102280			−0.0511399	+	0.998692i	$0.516285\pi$
$774$		0			0
$775$	195.168	+	764.794i	0.251830	+	0.986831i
$776$			471.894i			0.608111i
$777$		0			0
$778$		−	1570.05i		−	2.01806i
$779$		−	598.531i		−	0.768333i
$780$		0			0
$781$	−586.023			−0.750350
$782$	1106.57			1.41506
$783$		0			0
$784$	329.783			0.420641
$785$	535.323	+	415.869i	0.681940	+	0.529770i
$786$		0			0
$787$		−	966.245i		−	1.22776i	−0.789401	−	0.613878i	$-0.789609\pi$
$787$		−	966.245i		−	1.22776i	0.789401	−	0.613878i	$-0.210391\pi$
$788$	−328.955			−0.417455
$789$		0			0
$790$	864.994	−	1113.45i	1.09493	−	1.40943i
$791$			39.5470i			0.0499962i
$792$		0			0
$793$			534.171i			0.673608i
$794$			1450.85i			1.82726i
$795$		0			0
$796$	355.510			0.446621
$797$	−1275.81			−1.60077			−0.800384	−	0.599488i	$-0.795371\pi$
$797$	−1275.81			−1.60077			−0.800384	+	0.599488i	$0.795371\pi$
$798$		0			0
$799$	−99.8037			−0.124911
$800$	150.102	+	588.195i	0.187627	+	0.735244i
$801$		0			0
$802$		−	1173.90i		−	1.46372i
$803$	−414.524			−0.516220
$804$		0			0
$805$	274.253	+	213.056i	0.340688	+	0.264666i
$806$		−	1244.32i		−	1.54383i
$807$		0			0
$808$		−	615.845i		−	0.762185i
$809$		−	1533.89i		−	1.89603i	−0.318220	−	0.948017i	$-0.603085\pi$
$809$		−	1533.89i		−	1.89603i	0.318220	−	0.948017i	$-0.396915\pi$
$810$		0			0
$811$	−753.037			−0.928530			−0.464265	−	0.885696i	$-0.653681\pi$
$811$	−753.037			−0.928530			−0.464265	+	0.885696i	$0.653681\pi$
$812$	−2876.37			−3.54233
$813$		0			0
$814$	2072.55			2.54613
$815$	339.298	+	263.586i	0.416316	+	0.323418i
$816$		0			0
$817$		−	72.1303i		−	0.0882867i
$818$	58.3067			0.0712796
$819$		0			0
$820$	−1304.87	−	1013.70i	−1.59131	−	1.23622i
$821$		−	1543.26i		−	1.87973i	−0.341551	−	0.939863i	$-0.610952\pi$
$821$		−	1543.26i		−	1.87973i	0.341551	−	0.939863i	$-0.389048\pi$
$822$		0			0
$823$		−	1167.96i		−	1.41915i	−0.704632	−	0.709573i	$-0.748888\pi$
$823$		−	1167.96i		−	1.41915i	0.704632	−	0.709573i	$-0.251112\pi$
$824$			478.832i			0.581107i
$825$		0			0
$826$	−1077.06			−1.30395
$827$	432.942			0.523509			0.261755	−	0.965134i	$-0.415699\pi$
$827$	432.942			0.523509			0.261755	+	0.965134i	$0.415699\pi$
$828$		0			0
$829$	470.069			0.567031			0.283515	−	0.958968i	$-0.408499\pi$
$829$	470.069			0.567031			0.283515	+	0.958968i	$0.408499\pi$
$830$	1651.11	−	2125.37i	1.98929	−	2.56068i
$831$		0			0
$832$			185.010i			0.222368i
$833$	−341.904			−0.410449
$834$		0			0
$835$	−342.688	+	441.121i	−0.410405	+	0.528288i
$836$		−	2119.54i		−	2.53534i
$837$		0			0
$838$		−	1363.70i		−	1.62733i
$839$		−	1009.21i		−	1.20288i	−0.798919	−	0.601439i	$-0.794594\pi$
$839$		−	1009.21i		−	1.20288i	0.798919	−	0.601439i	$-0.205406\pi$
$840$		0			0
$841$	−2122.51			−2.52380
$842$	−1513.06			−1.79698
$843$		0			0
$844$	1513.32			1.79304
$845$	145.407	−	187.173i	0.172079	−	0.221507i
$846$		0			0
$847$		−	662.783i		−	0.782507i
$848$	−1453.08			−1.71353
$849$		0			0
$850$	−593.058	−	2323.98i	−0.697715	−	2.73410i
$851$		−	440.007i		−	0.517047i
$852$		0			0
$853$			900.755i			1.05598i	0.849249	+	0.527992i	$0.177055\pi$
$853$			900.755i			1.05598i	−0.849249	+	0.527992i	$0.822945\pi$
$854$			1042.60i			1.22084i
$855$		0			0
$856$	−1567.24			−1.83088
$857$	−111.824			−0.130483			−0.0652417	−	0.997869i	$-0.520782\pi$
$857$	−111.824			−0.130483			−0.0652417	+	0.997869i	$0.520782\pi$
$858$		0			0
$859$	180.456			0.210076			0.105038	−	0.994468i	$-0.466504\pi$
$859$	180.456			0.210076			0.105038	+	0.994468i	$0.466504\pi$
$860$	−157.253	−	122.163i	−0.182852	−	0.142050i
$861$		0			0
$862$		−	172.659i		−	0.200300i
$863$	451.735			0.523447			0.261724	−	0.965143i	$-0.415709\pi$
$863$	451.735			0.523447			0.261724	+	0.965143i	$0.415709\pi$
$864$		0			0
$865$	−26.6556	+	34.3121i	−0.0308157	+	0.0396672i
$866$			128.883i			0.148826i
$867$		0			0
$868$		−	1668.19i		−	1.92188i
$869$			1199.32i			1.38011i
$870$		0			0
$871$	−764.951			−0.878244
$872$	−1334.01			−1.52983
$873$		0			0
$874$	−655.120			−0.749566
$875$	300.468	−	690.161i	0.343392	−	0.788756i
$876$		0			0
$877$			736.225i			0.839481i	0.907644	+	0.419741i	$0.137879\pi$
$877$			736.225i			0.839481i	−0.907644	+	0.419741i	$0.862121\pi$
$878$	929.579			1.05875
$879$		0			0
$880$	−1553.99	−	1207.23i	−1.76590	−	1.37185i
$881$		−	52.0576i		−	0.0590892i	−0.999563	−	0.0295446i	$-0.990594\pi$
$881$		−	52.0576i		−	0.0590892i	0.999563	−	0.0295446i	$-0.00940571\pi$
$882$		0			0
$883$			637.785i			0.722294i	0.932509	+	0.361147i	$0.117615\pi$
$883$			637.785i			0.722294i	−0.932509	+	0.361147i	$0.882385\pi$
$884$			2597.15i			2.93795i
$885$		0			0
$886$	80.5227			0.0908834
$887$	−1346.54			−1.51809			−0.759043	−	0.651041i	$-0.774333\pi$
$887$	−1346.54			−1.51809			−0.759043	+	0.651041i	$0.774333\pi$
$888$		0			0
$889$	1490.32			1.67640
$890$	−49.4671	−	38.4289i	−0.0555810	−	0.0431785i
$891$		0			0
$892$		−	2626.63i		−	2.94465i
$893$	59.0864			0.0661661
$894$		0			0
$895$	−226.902	−	176.271i	−0.253522	−	0.196951i
$896$			945.990i			1.05579i
$897$		0			0
$898$		−	985.169i		−	1.09707i
$899$		−	1718.73i		−	1.91182i
$900$		0			0
$901$	1506.49			1.67202
$902$	2046.24			2.26855
$903$		0			0
$904$	−112.063			−0.123963
$905$	−428.555	+	551.652i	−0.473541	+	0.609560i
$906$		0			0
$907$		−	1073.34i		−	1.18339i	−0.806161	−	0.591696i	$-0.798459\pi$
$907$		−	1073.34i		−	1.18339i	0.806161	−	0.591696i	$-0.201541\pi$
$908$	2621.02			2.88658
$909$		0			0
$910$	−728.005	+	937.115i	−0.800005	+	1.02980i
$911$			83.7182i			0.0918970i	0.998944	+	0.0459485i	$0.0146310\pi$
$911$			83.7182i			0.0918970i	−0.998944	+	0.0459485i	$0.985369\pi$
$912$		0			0
$913$			2289.27i			2.50741i
$914$		−	426.776i		−	0.466932i
$915$		0			0
$916$	−1783.90			−1.94749
$917$	856.149			0.933641
$918$		0			0
$919$	1235.72			1.34463			0.672317	−	0.740264i	$-0.265299\pi$
$919$	1235.72			1.34463			0.672317	+	0.740264i	$0.265299\pi$
$920$	−603.727	+	777.140i	−0.656225	+	0.844717i
$921$		0			0
$922$		−	322.945i		−	0.350266i
$923$	−425.119			−0.460583
$924$		0			0
$925$	−924.085	+	235.818i	−0.999011	+	0.254938i
$926$			2181.77i			2.35612i
$927$		0			0
$928$		−	1321.86i		−	1.42442i
$929$			745.068i			0.802011i	0.916076	+	0.401006i	$0.131339\pi$
$929$			745.068i			0.802011i	−0.916076	+	0.401006i	$0.868661\pi$
$930$		0			0
$931$	202.416			0.217418
$932$	3829.08			4.10845
$933$		0			0
$934$	−234.652			−0.251234
$935$	1611.11	+	1251.60i	1.72311	+	1.33861i
$936$		0			0
$937$		−	353.460i		−	0.377225i	−0.982052	−	0.188613i	$-0.939601\pi$
$937$		−	353.460i		−	0.377225i	0.982052	−	0.188613i	$-0.0603991\pi$
$938$	−1493.04			−1.59172
$939$		0			0
$940$	100.071	−	128.816i	0.106459	−	0.137038i
$941$		−	253.779i		−	0.269690i	−0.990867	−	0.134845i	$-0.956946\pi$
$941$		−	253.779i		−	0.269690i	0.990867	−	0.134845i	$-0.0430537\pi$
$942$		0			0
$943$		−	434.420i		−	0.460679i
$944$		−	1295.66i		−	1.37252i
$945$		0			0
$946$	246.596			0.260673
$947$	−1574.72			−1.66285			−0.831424	−	0.555639i	$-0.812474\pi$
$947$	−1574.72			−1.66285			−0.831424	+	0.555639i	$0.812474\pi$
$948$		0			0
$949$	−300.708			−0.316869
$950$	351.105	+	1375.86i	0.369585	+	1.44827i
$951$		0			0
$952$			2758.23i			2.89730i
$953$	137.098			0.143860			0.0719300	−	0.997410i	$-0.477084\pi$
$953$	137.098			0.143860			0.0719300	+	0.997410i	$0.477084\pi$
$954$		0			0
$955$	−124.284	−	96.5508i	−0.130140	−	0.101100i
$956$			2103.17i			2.19997i
$957$		0			0
$958$		−	335.900i		−	0.350626i
$959$			438.932i			0.457698i
$960$		0			0
$961$	35.8009			0.0372538
$962$	1503.49			1.56288
$963$		0			0
$964$	1801.78			1.86907
$965$	197.723	+	153.603i	0.204894	+	0.159174i
$966$		0			0
$967$			377.253i			0.390127i	0.980791	+	0.195064i	$0.0624913\pi$
$967$			377.253i			0.390127i	−0.980791	+	0.195064i	$0.937509\pi$
$968$	1878.10			1.94019
$969$		0			0
$970$	390.275	+	303.188i	0.402345	+	0.312565i
$971$		−	855.490i		−	0.881040i	−0.897743	−	0.440520i	$-0.854794\pi$
$971$		−	855.490i		−	0.881040i	0.897743	−	0.440520i	$-0.145206\pi$
$972$		0			0
$973$		−	403.457i		−	0.414653i
$974$			2878.82i			2.95567i
$975$		0			0
$976$	−1254.20			−1.28504
$977$	−1074.71			−1.10001			−0.550004	−	0.835162i	$-0.685374\pi$
$977$	−1074.71			−1.10001			−0.550004	+	0.835162i	$0.685374\pi$
$978$		0			0
$979$	53.2818			0.0544247
$980$	342.821	−	441.292i	0.349818	−	0.450298i
$981$		0			0
$982$			3362.46i			3.42410i
$983$	1161.55			1.18164			0.590821	−	0.806803i	$-0.298804\pi$
$983$	1161.55			1.18164			0.590821	+	0.806803i	$0.298804\pi$
$984$		0			0
$985$	−115.001	+	148.033i	−0.116752	+	0.150287i
$986$			5222.71i			5.29687i
$987$		0			0
$988$		−	1537.58i		−	1.55625i
$989$		−	52.3529i		−	0.0529352i
$990$		0			0
$991$	463.882			0.468095			0.234047	−	0.972225i	$-0.424803\pi$
$991$	463.882			0.468095			0.234047	+	0.972225i	$0.424803\pi$
$992$	766.629			0.772812
$993$		0			0
$994$	−829.750			−0.834759
$995$	124.284	−	159.983i	0.124909	−	0.160787i
$996$		0			0
$997$		−	186.488i		−	0.187050i	−0.995617	−	0.0935248i	$-0.970187\pi$
$997$		−	186.488i		−	0.187050i	0.995617	−	0.0935248i	$-0.0298134\pi$
$998$	−1357.91			−1.36063
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.3.d.b.404.1	✓	24
3.2	odd	2	inner	405.3.d.b.404.24	yes	24
5.4	even	2	inner	405.3.d.b.404.23	yes	24
9.2	odd	6		405.3.h.k.134.1		48
9.4	even	3		405.3.h.k.269.23		48
9.5	odd	6		405.3.h.k.269.2		48
9.7	even	3		405.3.h.k.134.24		48
15.14	odd	2	inner	405.3.d.b.404.2	yes	24
45.4	even	6		405.3.h.k.269.1		48
45.14	odd	6		405.3.h.k.269.24		48
45.29	odd	6		405.3.h.k.134.23		48
45.34	even	6		405.3.h.k.134.2		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.3.d.b.404.1	✓	24	1.1	even	1	trivial
405.3.d.b.404.2	yes	24	15.14	odd	2	inner
405.3.d.b.404.23	yes	24	5.4	even	2	inner
405.3.d.b.404.24	yes	24	3.2	odd	2	inner
405.3.h.k.134.1		48	9.2	odd	6
405.3.h.k.134.2		48	45.34	even	6
405.3.h.k.134.23		48	45.29	odd	6
405.3.h.k.134.24		48	9.7	even	3
405.3.h.k.269.1		48	45.4	even	6
405.3.h.k.269.2		48	9.5	odd	6
405.3.h.k.269.23		48	9.4	even	3
405.3.h.k.269.24		48	45.14	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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	405.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.57411 q^{2} +8.77429 q^{4} +(3.06744 - 3.94852i) q^{5} +6.02185i q^{7} -17.0638 q^{8} +(-10.9634 + 14.1125i) q^{10} -15.2008i q^{11} -11.0271i q^{13} -21.5228i q^{14} +25.8909 q^{16} -26.8426 q^{17} +15.8915 q^{19} +(26.9146 - 34.6455i) q^{20} +54.3292i q^{22} +11.5342 q^{23} +(-6.18165 - 24.2237i) q^{25} +39.4120i q^{26} +52.8374i q^{28} +54.4382i q^{29} -31.5722 q^{31} -24.2818 q^{32} +95.9384 q^{34} +(23.7774 + 18.4716i) q^{35} -38.1480i q^{37} -56.7980 q^{38} +(-52.3423 + 67.3769i) q^{40} -37.6636i q^{41} -4.53892i q^{43} -133.376i q^{44} -41.2246 q^{46} +3.71811 q^{47} +12.7374 q^{49} +(22.0939 + 86.5782i) q^{50} -96.7548i q^{52} -56.1230 q^{53} +(-60.0205 - 46.6274i) q^{55} -102.756i q^{56} -194.568i q^{58} -50.0428i q^{59} -48.4417 q^{61} +112.842 q^{62} -16.7778 q^{64} +(-43.5407 - 33.8249i) q^{65} -69.3702i q^{67} -235.524 q^{68} +(-84.9831 - 66.0197i) q^{70} -38.5522i q^{71} -27.2700i q^{73} +136.345i q^{74} +139.437 q^{76} +91.5366 q^{77} -78.8985 q^{79} +(79.4189 - 102.231i) q^{80} +134.614i q^{82} -150.602 q^{83} +(-82.3379 + 105.988i) q^{85} +16.2226i q^{86} +259.383i q^{88} +3.50521i q^{89} +66.4034 q^{91} +101.204 q^{92} -13.2890 q^{94} +(48.7462 - 62.7479i) q^{95} -27.6546i q^{97} -45.5248 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 48 q^{4} + 12 q^{10} + 96 q^{16} + 48 q^{25} + 144 q^{34} + 72 q^{40} - 168 q^{46} - 288 q^{49} - 132 q^{55} - 360 q^{61} - 72 q^{64} - 156 q^{70} + 48 q^{76} - 480 q^{79} - 456 q^{85} - 48 q^{91}+ \cdots + 384 q^{94}+O(q^{100}) \) 24 * q + 48 * q^4 + 12 * q^10 + 96 * q^16 + 48 * q^25 + 144 * q^34 + 72 * q^40 - 168 * q^46 - 288 * q^49 - 132 * q^55 - 360 * q^61 - 72 * q^64 - 156 * q^70 + 48 * q^76 - 480 * q^79 - 456 * q^85 - 48 * q^91 + 384 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more