LMFDB - Embedded newform 1170.2.b.f.181.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1170,2,Mod(181,1170)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1170, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1170.181");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1170.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$9.34249703649$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.2
Root			$2.56155i$ of defining polynomial
Character	$\chi$	$=$	1170.181
Dual form			1170.2.b.f.181.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} +5.12311i q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} +5.12311i q^{7} +1.00000i q^{8} +1.00000 q^{10} -3.12311i q^{11} +(-0.561553 - 3.56155i) q^{13} +5.12311 q^{14} +1.00000 q^{16} -2.00000 q^{17} +6.00000i q^{19} -1.00000i q^{20} -3.12311 q^{22} -3.12311 q^{23} -1.00000 q^{25} +(-3.56155 + 0.561553i) q^{26} -5.12311i q^{28} -2.00000 q^{29} +5.12311i q^{31} -1.00000i q^{32} +2.00000i q^{34} -5.12311 q^{35} -3.12311i q^{37} +6.00000 q^{38} -1.00000 q^{40} +9.12311i q^{41} -10.2462 q^{43} +3.12311i q^{44} +3.12311i q^{46} +10.2462i q^{47} -19.2462 q^{49} +1.00000i q^{50} +(0.561553 + 3.56155i) q^{52} -11.3693 q^{53} +3.12311 q^{55} -5.12311 q^{56} +2.00000i q^{58} +7.12311i q^{59} +10.0000 q^{61} +5.12311 q^{62} -1.00000 q^{64} +(3.56155 - 0.561553i) q^{65} -13.1231i q^{67} +2.00000 q^{68} +5.12311i q^{70} -6.24621i q^{71} +4.87689i q^{73} -3.12311 q^{74} -6.00000i q^{76} +16.0000 q^{77} +8.00000 q^{79} +1.00000i q^{80} +9.12311 q^{82} +10.2462i q^{83} -2.00000i q^{85} +10.2462i q^{86} +3.12311 q^{88} -5.12311i q^{89} +(18.2462 - 2.87689i) q^{91} +3.12311 q^{92} +10.2462 q^{94} -6.00000 q^{95} +4.87689i q^{97} +19.2462i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 4 q^{10} + 6 q^{13} + 4 q^{14} + 4 q^{16} - 8 q^{17} + 4 q^{22} + 4 q^{23} - 4 q^{25} - 6 q^{26} - 8 q^{29} - 4 q^{35} + 24 q^{38} - 4 q^{40} - 8 q^{43} - 44 q^{49} - 6 q^{52} + 4 q^{53} - 4 q^{55} - 4 q^{56} + 40 q^{61} + 4 q^{62} - 4 q^{64} + 6 q^{65} + 8 q^{68} + 4 q^{74} + 64 q^{77} + 32 q^{79} + 20 q^{82} - 4 q^{88} + 40 q^{91} - 4 q^{92} + 8 q^{94} - 24 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^10 + 6 * q^13 + 4 * q^14 + 4 * q^16 - 8 * q^17 + 4 * q^22 + 4 * q^23 - 4 * q^25 - 6 * q^26 - 8 * q^29 - 4 * q^35 + 24 * q^38 - 4 * q^40 - 8 * q^43 - 44 * q^49 - 6 * q^52 + 4 * q^53 - 4 * q^55 - 4 * q^56 + 40 * q^61 + 4 * q^62 - 4 * q^64 + 6 * q^65 + 8 * q^68 + 4 * q^74 + 64 * q^77 + 32 * q^79 + 20 * q^82 - 4 * q^88 + 40 * q^91 - 4 * q^92 + 8 * q^94 - 24 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times$.

$n$	$911$	$937$	$1081$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$			5.12311i			1.93635i	0.250270	+	0.968176i	$0.419480\pi$
$7$			5.12311i			1.93635i	−0.250270	+	0.968176i	$0.580520\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$	1.00000			0.316228
$11$		−	3.12311i		−	0.941652i	−0.882226	−	0.470826i	$-0.843956\pi$
$11$		−	3.12311i		−	0.941652i	0.882226	−	0.470826i	$-0.156044\pi$
$12$		0			0
$13$	−0.561553	−	3.56155i	−0.155747	−	0.987797i
$13$	−0.561553	−	3.56155i	−0.155747	−	0.987797i
$14$	5.12311			1.36921
$15$		0			0
$16$	1.00000			0.250000
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$			6.00000i			1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$			6.00000i			1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$		−	1.00000i		−	0.223607i
$21$		0			0
$22$	−3.12311			−0.665848
$23$	−3.12311			−0.651213			−0.325606	−	0.945505i	$-0.605568\pi$
$23$	−3.12311			−0.651213			−0.325606	+	0.945505i	$0.605568\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$	−3.56155	+	0.561553i	−0.698478	+	0.110130i
$27$		0			0
$28$		−	5.12311i		−	0.968176i
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$			5.12311i			0.920137i	0.887883	+	0.460068i	$0.152175\pi$
$31$			5.12311i			0.920137i	−0.887883	+	0.460068i	$0.847825\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$			2.00000i			0.342997i
$35$	−5.12311			−0.865963
$36$		0			0
$37$		−	3.12311i		−	0.513435i	−0.966486	−	0.256718i	$-0.917359\pi$
$37$		−	3.12311i		−	0.513435i	0.966486	−	0.256718i	$-0.0826411\pi$
$38$	6.00000			0.973329
$39$		0			0
$40$	−1.00000			−0.158114
$41$			9.12311i			1.42479i	0.701779	+	0.712395i	$0.252389\pi$
$41$			9.12311i			1.42479i	−0.701779	+	0.712395i	$0.747611\pi$
$42$		0			0
$43$	−10.2462			−1.56253			−0.781266	−	0.624198i	$-0.785426\pi$
$43$	−10.2462			−1.56253			−0.781266	+	0.624198i	$0.785426\pi$
$44$			3.12311i			0.470826i
$45$		0			0
$46$			3.12311i			0.460477i
$47$			10.2462i			1.49456i	0.664507	+	0.747282i	$0.268641\pi$
$47$			10.2462i			1.49456i	−0.664507	+	0.747282i	$0.731359\pi$
$48$		0			0
$49$	−19.2462			−2.74946
$50$			1.00000i			0.141421i
$51$		0			0
$52$	0.561553	+	3.56155i	0.0778734	+	0.493899i
$53$	−11.3693			−1.56170			−0.780848	−	0.624721i	$-0.785213\pi$
$53$	−11.3693			−1.56170			−0.780848	+	0.624721i	$0.785213\pi$
$54$		0			0
$55$	3.12311			0.421119
$56$	−5.12311			−0.684604
$57$		0			0
$58$			2.00000i			0.262613i
$59$			7.12311i			0.927349i	0.886006	+	0.463675i	$0.153469\pi$
$59$			7.12311i			0.927349i	−0.886006	+	0.463675i	$0.846531\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$	5.12311			0.650635
$63$		0			0
$64$	−1.00000			−0.125000
$65$	3.56155	−	0.561553i	0.441756	−	0.0696521i
$66$		0			0
$67$		−	13.1231i		−	1.60324i	−0.597832	−	0.801621i	$-0.703971\pi$
$67$		−	13.1231i		−	1.60324i	0.597832	−	0.801621i	$-0.296029\pi$
$68$	2.00000			0.242536
$69$		0			0
$70$			5.12311i			0.612328i
$71$		−	6.24621i		−	0.741289i	−0.928775	−	0.370644i	$-0.879137\pi$
$71$		−	6.24621i		−	0.741289i	0.928775	−	0.370644i	$-0.120863\pi$
$72$		0			0
$73$			4.87689i			0.570797i	0.958409	+	0.285399i	$0.0921260\pi$
$73$			4.87689i			0.570797i	−0.958409	+	0.285399i	$0.907874\pi$
$74$	−3.12311			−0.363054
$75$		0			0
$76$		−	6.00000i		−	0.688247i
$77$	16.0000			1.82337
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$			1.00000i			0.111803i
$81$		0			0
$82$	9.12311			1.00748
$83$			10.2462i			1.12467i	0.826910	+	0.562334i	$0.190096\pi$
$83$			10.2462i			1.12467i	−0.826910	+	0.562334i	$0.809904\pi$
$84$		0			0
$85$		−	2.00000i		−	0.216930i
$86$			10.2462i			1.10488i
$87$		0			0
$88$	3.12311			0.332924
$89$		−	5.12311i		−	0.543048i	−0.962432	−	0.271524i	$-0.912472\pi$
$89$		−	5.12311i		−	0.543048i	0.962432	−	0.271524i	$-0.0875277\pi$
$90$		0			0
$91$	18.2462	−	2.87689i	1.91272	−	0.301580i
$92$	3.12311			0.325606
$93$		0			0
$94$	10.2462			1.05682
$95$	−6.00000			−0.615587
$96$		0			0
$97$			4.87689i			0.495174i	0.968866	+	0.247587i	$0.0796375\pi$
$97$			4.87689i			0.495174i	−0.968866	+	0.247587i	$0.920362\pi$
$98$			19.2462i			1.94416i
$99$		0			0
$100$	1.00000			0.100000
$101$	−4.24621			−0.422514			−0.211257	−	0.977431i	$-0.567756\pi$
$101$	−4.24621			−0.422514			−0.211257	+	0.977431i	$0.567756\pi$
$102$		0			0
$103$	−4.87689			−0.480535			−0.240267	−	0.970707i	$-0.577235\pi$
$103$	−4.87689			−0.480535			−0.240267	+	0.970707i	$0.577235\pi$
$104$	3.56155	−	0.561553i	0.349239	−	0.0550648i
$105$		0			0
$106$			11.3693i			1.10429i
$107$	8.00000			0.773389			0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000			0.773389			0.386695	+	0.922208i	$0.373617\pi$
$108$		0			0
$109$		−	11.1231i		−	1.06540i	−0.846304	−	0.532700i	$-0.821177\pi$
$109$		−	11.1231i		−	1.06540i	0.846304	−	0.532700i	$-0.178823\pi$
$110$		−	3.12311i		−	0.297776i
$111$		0			0
$112$			5.12311i			0.484088i
$113$	4.24621			0.399450			0.199725	−	0.979852i	$-0.435995\pi$
$113$	4.24621			0.399450			0.199725	+	0.979852i	$0.435995\pi$
$114$		0			0
$115$		−	3.12311i		−	0.291231i
$116$	2.00000			0.185695
$117$		0			0
$118$	7.12311			0.655735
$119$		−	10.2462i		−	0.939269i
$120$		0			0
$121$	1.24621			0.113292
$122$		−	10.0000i		−	0.905357i
$123$		0			0
$124$		−	5.12311i		−	0.460068i
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	−4.87689			−0.432754			−0.216377	−	0.976310i	$-0.569424\pi$
$127$	−4.87689			−0.432754			−0.216377	+	0.976310i	$0.569424\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−0.561553	−	3.56155i	−0.0492514	−	0.312369i
$131$	−4.00000			−0.349482			−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000			−0.349482			−0.174741	+	0.984614i	$0.555909\pi$
$132$		0			0
$133$	−30.7386			−2.66538
$134$	−13.1231			−1.13366
$135$		0			0
$136$		−	2.00000i		−	0.171499i
$137$			22.4924i			1.92166i	0.277143	+	0.960829i	$0.410612\pi$
$137$			22.4924i			1.92166i	−0.277143	+	0.960829i	$0.589388\pi$
$138$		0			0
$139$	16.4924			1.39887			0.699435	−	0.714697i	$-0.253435\pi$
$139$	16.4924			1.39887			0.699435	+	0.714697i	$0.253435\pi$
$140$	5.12311			0.432981
$141$		0			0
$142$	−6.24621			−0.524170
$143$	−11.1231	+	1.75379i	−0.930161	+	0.146659i
$144$		0			0
$145$		−	2.00000i		−	0.166091i
$146$	4.87689			0.403615
$147$		0			0
$148$			3.12311i			0.256718i
$149$		−	14.0000i		−	1.14692i	−0.819232	−	0.573462i	$-0.805600\pi$
$149$		−	14.0000i		−	1.14692i	0.819232	−	0.573462i	$-0.194400\pi$
$150$		0			0
$151$			11.3693i			0.925222i	0.886561	+	0.462611i	$0.153087\pi$
$151$			11.3693i			0.925222i	−0.886561	+	0.462611i	$0.846913\pi$
$152$	−6.00000			−0.486664
$153$		0			0
$154$		−	16.0000i		−	1.28932i
$155$	−5.12311			−0.411498
$156$		0			0
$157$	−3.36932			−0.268901			−0.134450	−	0.990920i	$-0.542927\pi$
$157$	−3.36932			−0.268901			−0.134450	+	0.990920i	$0.542927\pi$
$158$		−	8.00000i		−	0.636446i
$159$		0			0
$160$	1.00000			0.0790569
$161$		−	16.0000i		−	1.26098i
$162$		0			0
$163$		−	1.12311i		−	0.0879684i	−0.999032	−	0.0439842i	$-0.985995\pi$
$163$		−	1.12311i		−	0.0879684i	0.999032	−	0.0439842i	$-0.0140051\pi$
$164$		−	9.12311i		−	0.712395i
$165$		0			0
$166$	10.2462			0.795260
$167$		−	5.75379i		−	0.445242i	−0.974905	−	0.222621i	$-0.928539\pi$
$167$		−	5.75379i		−	0.445242i	0.974905	−	0.222621i	$-0.0714612\pi$
$168$		0			0
$169$	−12.3693	+	4.00000i	−0.951486	+	0.307692i
$170$	−2.00000			−0.153393
$171$		0			0
$172$	10.2462			0.781266
$173$	−14.8769			−1.13107			−0.565535	−	0.824725i	$-0.691330\pi$
$173$	−14.8769			−1.13107			−0.565535	+	0.824725i	$0.691330\pi$
$174$		0			0
$175$		−	5.12311i		−	0.387270i
$176$		−	3.12311i		−	0.235413i
$177$		0			0
$178$	−5.12311			−0.383993
$179$	16.4924			1.23270			0.616351	−	0.787472i	$-0.288610\pi$
$179$	16.4924			1.23270			0.616351	+	0.787472i	$0.288610\pi$
$180$		0			0
$181$	−3.75379			−0.279017			−0.139508	−	0.990221i	$-0.544552\pi$
$181$	−3.75379			−0.279017			−0.139508	+	0.990221i	$0.544552\pi$
$182$	−2.87689	−	18.2462i	−0.213250	−	1.35250i
$183$		0			0
$184$		−	3.12311i		−	0.230238i
$185$	3.12311			0.229615
$186$		0			0
$187$			6.24621i			0.456768i
$188$		−	10.2462i		−	0.747282i
$189$		0			0
$190$			6.00000i			0.435286i
$191$	16.4924			1.19335			0.596675	−	0.802483i	$-0.296488\pi$
$191$	16.4924			1.19335			0.596675	+	0.802483i	$0.296488\pi$
$192$		0			0
$193$			16.8769i			1.21483i	0.794386	+	0.607413i	$0.207793\pi$
$193$			16.8769i			1.21483i	−0.794386	+	0.607413i	$0.792207\pi$
$194$	4.87689			0.350141
$195$		0			0
$196$	19.2462			1.37473
$197$		−	0.246211i		−	0.0175418i	−0.999962	−	0.00877091i	$-0.997208\pi$
$197$		−	0.246211i		−	0.0175418i	0.999962	−	0.00877091i	$-0.00279190\pi$
$198$		0			0
$199$	1.75379			0.124323			0.0621614	−	0.998066i	$-0.480201\pi$
$199$	1.75379			0.124323			0.0621614	+	0.998066i	$0.480201\pi$
$200$		−	1.00000i		−	0.0707107i
$201$		0			0
$202$			4.24621i			0.298762i
$203$		−	10.2462i		−	0.719143i
$204$		0			0
$205$	−9.12311			−0.637185
$206$			4.87689i			0.339789i
$207$		0			0
$208$	−0.561553	−	3.56155i	−0.0389367	−	0.246949i
$209$	18.7386			1.29618
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$	11.3693			0.780848
$213$		0			0
$214$		−	8.00000i		−	0.546869i
$215$		−	10.2462i		−	0.698786i
$216$		0			0
$217$	−26.2462			−1.78171
$218$	−11.1231			−0.753352
$219$		0			0
$220$	−3.12311			−0.210560
$221$	1.12311	+	7.12311i	0.0755483	+	0.479152i
$222$		0			0
$223$			15.3693i			1.02921i	0.857429	+	0.514603i	$0.172061\pi$
$223$			15.3693i			1.02921i	−0.857429	+	0.514603i	$0.827939\pi$
$224$	5.12311			0.342302
$225$		0			0
$226$		−	4.24621i		−	0.282454i
$227$			8.00000i			0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$			8.00000i			0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$		0			0
$229$		−	3.12311i		−	0.206381i	−0.994662	−	0.103190i	$-0.967095\pi$
$229$		−	3.12311i		−	0.206381i	0.994662	−	0.103190i	$-0.0329051\pi$
$230$	−3.12311			−0.205931
$231$		0			0
$232$		−	2.00000i		−	0.131306i
$233$	24.2462			1.58842			0.794211	−	0.607642i	$-0.207884\pi$
$233$	24.2462			1.58842			0.794211	+	0.607642i	$0.207884\pi$
$234$		0			0
$235$	−10.2462			−0.668389
$236$		−	7.12311i		−	0.463675i
$237$		0			0
$238$	−10.2462			−0.664163
$239$		−	28.4924i		−	1.84302i	−0.388353	−	0.921511i	$-0.626956\pi$
$239$		−	28.4924i		−	1.84302i	0.388353	−	0.921511i	$-0.373044\pi$
$240$		0			0
$241$		−	2.24621i		−	0.144691i	−0.997380	−	0.0723456i	$-0.976952\pi$
$241$		−	2.24621i		−	0.144691i	0.997380	−	0.0723456i	$-0.0230485\pi$
$242$		−	1.24621i		−	0.0801095i
$243$		0			0
$244$	−10.0000			−0.640184
$245$		−	19.2462i		−	1.22960i
$246$		0			0
$247$	21.3693	−	3.36932i	1.35970	−	0.214384i
$248$	−5.12311			−0.325318
$249$		0			0
$250$	−1.00000			−0.0632456
$251$	−9.75379			−0.615654			−0.307827	−	0.951442i	$-0.599602\pi$
$251$	−9.75379			−0.615654			−0.307827	+	0.951442i	$0.599602\pi$
$252$		0			0
$253$			9.75379i			0.613215i
$254$			4.87689i			0.306004i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−4.24621			−0.264871			−0.132436	−	0.991192i	$-0.542280\pi$
$257$	−4.24621			−0.264871			−0.132436	+	0.991192i	$0.542280\pi$
$258$		0			0
$259$	16.0000			0.994192
$260$	−3.56155	+	0.561553i	−0.220878	+	0.0348260i
$261$		0			0
$262$			4.00000i			0.247121i
$263$	2.63068			0.162215			0.0811074	−	0.996705i	$-0.474154\pi$
$263$	2.63068			0.162215			0.0811074	+	0.996705i	$0.474154\pi$
$264$		0			0
$265$		−	11.3693i		−	0.698412i
$266$			30.7386i			1.88471i
$267$		0			0
$268$			13.1231i			0.801621i
$269$	−0.246211			−0.0150118			−0.00750588	−	0.999972i	$-0.502389\pi$
$269$	−0.246211			−0.0150118			−0.00750588	+	0.999972i	$0.502389\pi$
$270$		0			0
$271$		−	14.8769i		−	0.903707i	−0.892092	−	0.451853i	$-0.850763\pi$
$271$		−	14.8769i		−	0.903707i	0.892092	−	0.451853i	$-0.149237\pi$
$272$	−2.00000			−0.121268
$273$		0			0
$274$	22.4924			1.35882
$275$			3.12311i			0.188330i
$276$		0			0
$277$	27.8617			1.67405			0.837025	−	0.547165i	$-0.184293\pi$
$277$	27.8617			1.67405			0.837025	+	0.547165i	$0.184293\pi$
$278$		−	16.4924i		−	0.989150i
$279$		0			0
$280$		−	5.12311i		−	0.306164i
$281$			5.12311i			0.305619i	0.988256	+	0.152809i	$0.0488321\pi$
$281$			5.12311i			0.305619i	−0.988256	+	0.152809i	$0.951168\pi$
$282$		0			0
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$			6.24621i			0.370644i
$285$		0			0
$286$	1.75379	+	11.1231i	0.103704	+	0.657723i
$287$	−46.7386			−2.75889
$288$		0			0
$289$	−13.0000			−0.764706
$290$	−2.00000			−0.117444
$291$		0			0
$292$		−	4.87689i		−	0.285399i
$293$		−	20.7386i		−	1.21156i	−0.795631	−	0.605782i	$-0.792860\pi$
$293$		−	20.7386i		−	1.21156i	0.795631	−	0.605782i	$-0.207140\pi$
$294$		0			0
$295$	−7.12311			−0.414723
$296$	3.12311			0.181527
$297$		0			0
$298$	−14.0000			−0.810998
$299$	1.75379	+	11.1231i	0.101424	+	0.643266i
$300$		0			0
$301$		−	52.4924i		−	3.02561i
$302$	11.3693			0.654231
$303$		0			0
$304$			6.00000i			0.344124i
$305$			10.0000i			0.572598i
$306$		0			0
$307$		−	22.8769i		−	1.30565i	−0.757507	−	0.652827i	$-0.773583\pi$
$307$		−	22.8769i		−	1.30565i	0.757507	−	0.652827i	$-0.226417\pi$
$308$	−16.0000			−0.911685
$309$		0			0
$310$			5.12311i			0.290973i
$311$	24.4924			1.38884			0.694419	−	0.719571i	$-0.255661\pi$
$311$	24.4924			1.38884			0.694419	+	0.719571i	$0.255661\pi$
$312$		0			0
$313$	−0.246211			−0.0139167			−0.00695834	−	0.999976i	$-0.502215\pi$
$313$	−0.246211			−0.0139167			−0.00695834	+	0.999976i	$0.502215\pi$
$314$			3.36932i			0.190142i
$315$		0			0
$316$	−8.00000			−0.450035
$317$			6.00000i			0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$			6.00000i			0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$		0			0
$319$			6.24621i			0.349721i
$320$		−	1.00000i		−	0.0559017i
$321$		0			0
$322$	−16.0000			−0.891645
$323$		−	12.0000i		−	0.667698i
$324$		0			0
$325$	0.561553	+	3.56155i	0.0311493	+	0.197559i
$326$	−1.12311			−0.0622031
$327$		0			0
$328$	−9.12311			−0.503739
$329$	−52.4924			−2.89400
$330$		0			0
$331$			24.2462i			1.33269i	0.745643	+	0.666346i	$0.232143\pi$
$331$			24.2462i			1.33269i	−0.745643	+	0.666346i	$0.767857\pi$
$332$		−	10.2462i		−	0.562334i
$333$		0			0
$334$	−5.75379			−0.314833
$335$	13.1231			0.716992
$336$		0			0
$337$	6.00000			0.326841			0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000			0.326841			0.163420	+	0.986557i	$0.447747\pi$
$338$	4.00000	+	12.3693i	0.217571	+	0.672802i
$339$		0			0
$340$			2.00000i			0.108465i
$341$	16.0000			0.866449
$342$		0			0
$343$		−	62.7386i		−	3.38757i
$344$		−	10.2462i		−	0.552439i
$345$		0			0
$346$			14.8769i			0.799787i
$347$	−2.24621			−0.120583			−0.0602915	−	0.998181i	$-0.519203\pi$
$347$	−2.24621			−0.120583			−0.0602915	+	0.998181i	$0.519203\pi$
$348$		0			0
$349$			5.36932i			0.287413i	0.989620	+	0.143706i	$0.0459021\pi$
$349$			5.36932i			0.287413i	−0.989620	+	0.143706i	$0.954098\pi$
$350$	−5.12311			−0.273842
$351$		0			0
$352$	−3.12311			−0.166462
$353$			4.24621i			0.226003i	0.993595	+	0.113002i	$0.0360465\pi$
$353$			4.24621i			0.226003i	−0.993595	+	0.113002i	$0.963954\pi$
$354$		0			0
$355$	6.24621			0.331514
$356$			5.12311i			0.271524i
$357$		0			0
$358$		−	16.4924i		−	0.871652i
$359$			34.2462i			1.80745i	0.428118	+	0.903723i	$0.359177\pi$
$359$			34.2462i			1.80745i	−0.428118	+	0.903723i	$0.640823\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$			3.75379i			0.197295i
$363$		0			0
$364$	−18.2462	+	2.87689i	−0.956361	+	0.150790i
$365$	−4.87689			−0.255268
$366$		0			0
$367$	33.3693			1.74186			0.870932	−	0.491403i	$-0.163516\pi$
$367$	33.3693			1.74186			0.870932	+	0.491403i	$0.163516\pi$
$368$	−3.12311			−0.162803
$369$		0			0
$370$		−	3.12311i		−	0.162363i
$371$		−	58.2462i		−	3.02399i
$372$		0			0
$373$	−1.12311			−0.0581522			−0.0290761	−	0.999577i	$-0.509257\pi$
$373$	−1.12311			−0.0581522			−0.0290761	+	0.999577i	$0.509257\pi$
$374$	6.24621			0.322984
$375$		0			0
$376$	−10.2462			−0.528408
$377$	1.12311	+	7.12311i	0.0578429	+	0.366859i
$378$		0			0
$379$			6.00000i			0.308199i	0.988055	+	0.154100i	$0.0492477\pi$
$379$			6.00000i			0.308199i	−0.988055	+	0.154100i	$0.950752\pi$
$380$	6.00000			0.307794
$381$		0			0
$382$		−	16.4924i		−	0.843826i
$383$			18.2462i			0.932338i	0.884696	+	0.466169i	$0.154366\pi$
$383$			18.2462i			0.932338i	−0.884696	+	0.466169i	$0.845634\pi$
$384$		0			0
$385$			16.0000i			0.815436i
$386$	16.8769			0.859011
$387$		0			0
$388$		−	4.87689i		−	0.247587i
$389$	−16.2462			−0.823716			−0.411858	−	0.911248i	$-0.635120\pi$
$389$	−16.2462			−0.823716			−0.411858	+	0.911248i	$0.635120\pi$
$390$		0			0
$391$	6.24621			0.315884
$392$		−	19.2462i		−	0.972080i
$393$		0			0
$394$	−0.246211			−0.0124039
$395$			8.00000i			0.402524i
$396$		0			0
$397$		−	9.36932i		−	0.470233i	−0.971967	−	0.235116i	$-0.924453\pi$
$397$		−	9.36932i		−	0.470233i	0.971967	−	0.235116i	$-0.0755471\pi$
$398$		−	1.75379i		−	0.0879095i
$399$		0			0
$400$	−1.00000			−0.0500000
$401$			23.3693i			1.16701i	0.812110	+	0.583504i	$0.198319\pi$
$401$			23.3693i			1.16701i	−0.812110	+	0.583504i	$0.801681\pi$
$402$		0			0
$403$	18.2462	−	2.87689i	0.908909	−	0.143308i
$404$	4.24621			0.211257
$405$		0			0
$406$	−10.2462			−0.508511
$407$	−9.75379			−0.483477
$408$		0			0
$409$			24.4924i			1.21107i	0.795818	+	0.605536i	$0.207041\pi$
$409$			24.4924i			1.21107i	−0.795818	+	0.605536i	$0.792959\pi$
$410$			9.12311i			0.450558i
$411$		0			0
$412$	4.87689			0.240267
$413$	−36.4924			−1.79567
$414$		0			0
$415$	−10.2462			−0.502967
$416$	−3.56155	+	0.561553i	−0.174619	+	0.0275324i
$417$		0			0
$418$		−	18.7386i		−	0.916537i
$419$	28.0000			1.36789			0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000			1.36789			0.683945	+	0.729534i	$0.260263\pi$
$420$		0			0
$421$			25.3693i			1.23642i	0.786011	+	0.618212i	$0.212143\pi$
$421$			25.3693i			1.23642i	−0.786011	+	0.618212i	$0.787857\pi$
$422$		−	4.00000i		−	0.194717i
$423$		0			0
$424$		−	11.3693i		−	0.552143i
$425$	2.00000			0.0970143
$426$		0			0
$427$			51.2311i			2.47924i
$428$	−8.00000			−0.386695
$429$		0			0
$430$	−10.2462			−0.494116
$431$			0.492423i			0.0237192i	0.999930	+	0.0118596i	$0.00377511\pi$
$431$			0.492423i			0.0237192i	−0.999930	+	0.0118596i	$0.996225\pi$
$432$		0			0
$433$	−18.0000			−0.865025			−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000			−0.865025			−0.432512	+	0.901628i	$0.642373\pi$
$434$			26.2462i			1.25986i
$435$		0			0
$436$			11.1231i			0.532700i
$437$		−	18.7386i		−	0.896390i
$438$		0			0
$439$	−3.50758			−0.167408			−0.0837038	−	0.996491i	$-0.526675\pi$
$439$	−3.50758			−0.167408			−0.0837038	+	0.996491i	$0.526675\pi$
$440$			3.12311i			0.148888i
$441$		0			0
$442$	7.12311	−	1.12311i	0.338812	−	0.0534207i
$443$	36.4924			1.73381			0.866904	−	0.498476i	$-0.166107\pi$
$443$	36.4924			1.73381			0.866904	+	0.498476i	$0.166107\pi$
$444$		0			0
$445$	5.12311			0.242858
$446$	15.3693			0.727758
$447$		0			0
$448$		−	5.12311i		−	0.242044i
$449$		−	37.1231i		−	1.75195i	−0.482359	−	0.875974i	$-0.660220\pi$
$449$		−	37.1231i		−	1.75195i	0.482359	−	0.875974i	$-0.339780\pi$
$450$		0			0
$451$	28.4924			1.34166
$452$	−4.24621			−0.199725
$453$		0			0
$454$	8.00000			0.375459
$455$	2.87689	+	18.2462i	0.134871	+	0.855396i
$456$		0			0
$457$		−	6.63068i		−	0.310170i	−0.987901	−	0.155085i	$-0.950435\pi$
$457$		−	6.63068i		−	0.310170i	0.987901	−	0.155085i	$-0.0495652\pi$
$458$	−3.12311			−0.145933
$459$		0			0
$460$			3.12311i			0.145616i
$461$		−	14.4924i		−	0.674979i	−0.941329	−	0.337490i	$-0.890422\pi$
$461$		−	14.4924i		−	0.674979i	0.941329	−	0.337490i	$-0.109578\pi$
$462$		0			0
$463$		−	35.8617i		−	1.66664i	−0.552794	−	0.833318i	$-0.686438\pi$
$463$		−	35.8617i		−	1.66664i	0.552794	−	0.833318i	$-0.313562\pi$
$464$	−2.00000			−0.0928477
$465$		0			0
$466$		−	24.2462i		−	1.12318i
$467$	5.75379			0.266254			0.133127	−	0.991099i	$-0.457498\pi$
$467$	5.75379			0.266254			0.133127	+	0.991099i	$0.457498\pi$
$468$		0			0
$469$	67.2311			3.10444
$470$			10.2462i			0.472622i
$471$		0			0
$472$	−7.12311			−0.327868
$473$			32.0000i			1.47136i
$474$		0			0
$475$		−	6.00000i		−	0.275299i
$476$			10.2462i			0.469634i
$477$		0			0
$478$	−28.4924			−1.30321
$479$			20.4924i			0.936323i	0.883643	+	0.468161i	$0.155083\pi$
$479$			20.4924i			0.936323i	−0.883643	+	0.468161i	$0.844917\pi$
$480$		0			0
$481$	−11.1231	+	1.75379i	−0.507170	+	0.0799659i
$482$	−2.24621			−0.102312
$483$		0			0
$484$	−1.24621			−0.0566460
$485$	−4.87689			−0.221448
$486$		0			0
$487$			7.36932i			0.333936i	0.985962	+	0.166968i	$0.0533976\pi$
$487$			7.36932i			0.333936i	−0.985962	+	0.166968i	$0.946602\pi$
$488$			10.0000i			0.452679i
$489$		0			0
$490$	−19.2462			−0.869455
$491$	10.7386			0.484628			0.242314	−	0.970198i	$-0.422094\pi$
$491$	10.7386			0.484628			0.242314	+	0.970198i	$0.422094\pi$
$492$		0			0
$493$	4.00000			0.180151
$494$	−3.36932	−	21.3693i	−0.151593	−	0.961451i
$495$		0			0
$496$			5.12311i			0.230034i
$497$	32.0000			1.43540
$498$		0			0
$499$		−	1.50758i		−	0.0674884i	−0.999431	−	0.0337442i	$-0.989257\pi$
$499$		−	1.50758i		−	0.0674884i	0.999431	−	0.0337442i	$-0.0107432\pi$
$500$			1.00000i			0.0447214i
$501$		0			0
$502$			9.75379i			0.435333i
$503$	−10.6307			−0.473999			−0.236999	−	0.971510i	$-0.576164\pi$
$503$	−10.6307			−0.473999			−0.236999	+	0.971510i	$0.576164\pi$
$504$		0			0
$505$		−	4.24621i		−	0.188954i
$506$	9.75379			0.433609
$507$		0			0
$508$	4.87689			0.216377
$509$			15.7538i			0.698274i	0.937072	+	0.349137i	$0.113525\pi$
$509$			15.7538i			0.698274i	−0.937072	+	0.349137i	$0.886475\pi$
$510$		0			0
$511$	−24.9848			−1.10526
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$			4.24621i			0.187292i
$515$		−	4.87689i		−	0.214902i
$516$		0			0
$517$	32.0000			1.40736
$518$		−	16.0000i		−	0.703000i
$519$		0			0
$520$	0.561553	+	3.56155i	0.0246257	+	0.156184i
$521$	16.2462			0.711759			0.355880	−	0.934532i	$-0.384181\pi$
$521$	16.2462			0.711759			0.355880	+	0.934532i	$0.384181\pi$
$522$		0			0
$523$	22.7386			0.994291			0.497146	−	0.867667i	$-0.334382\pi$
$523$	22.7386			0.994291			0.497146	+	0.867667i	$0.334382\pi$
$524$	4.00000			0.174741
$525$		0			0
$526$		−	2.63068i		−	0.114703i
$527$		−	10.2462i		−	0.446332i
$528$		0			0
$529$	−13.2462			−0.575922
$530$	−11.3693			−0.493852
$531$		0			0
$532$	30.7386			1.33269
$533$	32.4924	−	5.12311i	1.40740	−	0.221906i
$534$		0			0
$535$			8.00000i			0.345870i
$536$	13.1231			0.566832
$537$		0			0
$538$			0.246211i			0.0106149i
$539$			60.1080i			2.58903i
$540$		0			0
$541$		−	19.1231i		−	0.822167i	−0.911598	−	0.411083i	$-0.865151\pi$
$541$		−	19.1231i		−	0.822167i	0.911598	−	0.411083i	$-0.134849\pi$
$542$	−14.8769			−0.639017
$543$		0			0
$544$			2.00000i			0.0857493i
$545$	11.1231			0.476461
$546$		0			0
$547$	−44.9848			−1.92341			−0.961707	−	0.274081i	$-0.911626\pi$
$547$	−44.9848			−1.92341			−0.961707	+	0.274081i	$0.911626\pi$
$548$		−	22.4924i		−	0.960829i
$549$		0			0
$550$	3.12311			0.133170
$551$		−	12.0000i		−	0.511217i
$552$		0			0
$553$			40.9848i			1.74285i
$554$		−	27.8617i		−	1.18373i
$555$		0			0
$556$	−16.4924			−0.699435
$557$			28.2462i			1.19683i	0.801186	+	0.598415i	$0.204203\pi$
$557$			28.2462i			1.19683i	−0.801186	+	0.598415i	$0.795797\pi$
$558$		0			0
$559$	5.75379	+	36.4924i	0.243359	+	1.54347i
$560$	−5.12311			−0.216491
$561$		0			0
$562$	5.12311			0.216105
$563$	−32.9848			−1.39015			−0.695073	−	0.718939i	$-0.744628\pi$
$563$	−32.9848			−1.39015			−0.695073	+	0.718939i	$0.744628\pi$
$564$		0			0
$565$			4.24621i			0.178639i
$566$			4.00000i			0.168133i
$567$		0			0
$568$	6.24621			0.262085
$569$	−36.7386			−1.54016			−0.770082	−	0.637945i	$-0.779785\pi$
$569$	−36.7386			−1.54016			−0.770082	+	0.637945i	$0.779785\pi$
$570$		0			0
$571$	24.4924			1.02498			0.512488	−	0.858694i	$-0.328724\pi$
$571$	24.4924			1.02498			0.512488	+	0.858694i	$0.328724\pi$
$572$	11.1231	−	1.75379i	0.465080	−	0.0733296i
$573$		0			0
$574$			46.7386i			1.95083i
$575$	3.12311			0.130243
$576$		0			0
$577$		−	2.63068i		−	0.109517i	−0.998500	−	0.0547584i	$-0.982561\pi$
$577$		−	2.63068i		−	0.109517i	0.998500	−	0.0547584i	$-0.0174389\pi$
$578$			13.0000i			0.540729i
$579$		0			0
$580$			2.00000i			0.0830455i
$581$	−52.4924			−2.17775
$582$		0			0
$583$			35.5076i			1.47057i
$584$	−4.87689			−0.201807
$585$		0			0
$586$	−20.7386			−0.856705
$587$			16.4924i			0.680715i	0.940296	+	0.340358i	$0.110548\pi$
$587$			16.4924i			0.680715i	−0.940296	+	0.340358i	$0.889452\pi$
$588$		0			0
$589$	−30.7386			−1.26656
$590$			7.12311i			0.293254i
$591$		0			0
$592$		−	3.12311i		−	0.128359i
$593$		−	38.4924i		−	1.58069i	−0.612659	−	0.790347i	$-0.709900\pi$
$593$		−	38.4924i		−	1.58069i	0.612659	−	0.790347i	$-0.290100\pi$
$594$		0			0
$595$	10.2462			0.420054
$596$			14.0000i			0.573462i
$597$		0			0
$598$	11.1231	−	1.75379i	0.454858	−	0.0717178i
$599$	3.50758			0.143316			0.0716579	−	0.997429i	$-0.477171\pi$
$599$	3.50758			0.143316			0.0716579	+	0.997429i	$0.477171\pi$
$600$		0			0
$601$	38.0000			1.55005			0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000			1.55005			0.775026	+	0.631929i	$0.217737\pi$
$602$	−52.4924			−2.13943
$603$		0			0
$604$		−	11.3693i		−	0.462611i
$605$			1.24621i			0.0506657i
$606$		0			0
$607$	−9.36932			−0.380289			−0.190144	−	0.981756i	$-0.560896\pi$
$607$	−9.36932			−0.380289			−0.190144	+	0.981756i	$0.560896\pi$
$608$	6.00000			0.243332
$609$		0			0
$610$	10.0000			0.404888
$611$	36.4924	−	5.75379i	1.47633	−	0.232773i
$612$		0			0
$613$		−	14.6307i		−	0.590928i	−0.955354	−	0.295464i	$-0.904526\pi$
$613$		−	14.6307i		−	0.590928i	0.955354	−	0.295464i	$-0.0954742\pi$
$614$	−22.8769			−0.923236
$615$		0			0
$616$			16.0000i			0.644658i
$617$			8.73863i			0.351804i	0.984408	+	0.175902i	$0.0562842\pi$
$617$			8.73863i			0.351804i	−0.984408	+	0.175902i	$0.943716\pi$
$618$		0			0
$619$			26.9848i			1.08461i	0.840181	+	0.542306i	$0.182449\pi$
$619$			26.9848i			1.08461i	−0.840181	+	0.542306i	$0.817551\pi$
$620$	5.12311			0.205749
$621$		0			0
$622$		−	24.4924i		−	0.982057i
$623$	26.2462			1.05153
$624$		0			0
$625$	1.00000			0.0400000
$626$			0.246211i			0.00984058i
$627$		0			0
$628$	3.36932			0.134450
$629$			6.24621i			0.249053i
$630$		0			0
$631$		−	5.61553i		−	0.223551i	−0.993734	−	0.111775i	$-0.964346\pi$
$631$		−	5.61553i		−	0.223551i	0.993734	−	0.111775i	$-0.0356537\pi$
$632$			8.00000i			0.318223i
$633$		0			0
$634$	6.00000			0.238290
$635$		−	4.87689i		−	0.193534i
$636$		0			0
$637$	10.8078	+	68.5464i	0.428219	+	2.71591i
$638$	6.24621			0.247290
$639$		0			0
$640$	−1.00000			−0.0395285
$641$	2.00000			0.0789953			0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000			0.0789953			0.0394976	+	0.999220i	$0.487424\pi$
$642$		0			0
$643$			7.36932i			0.290617i	0.989386	+	0.145309i	$0.0464175\pi$
$643$			7.36932i			0.290617i	−0.989386	+	0.145309i	$0.953582\pi$
$644$			16.0000i			0.630488i
$645$		0			0
$646$	−12.0000			−0.472134
$647$	11.6155			0.456654			0.228327	−	0.973585i	$-0.426675\pi$
$647$	11.6155			0.456654			0.228327	+	0.973585i	$0.426675\pi$
$648$		0			0
$649$	22.2462			0.873240
$650$	3.56155	−	0.561553i	0.139696	−	0.0220259i
$651$		0			0
$652$			1.12311i			0.0439842i
$653$	−43.8617			−1.71644			−0.858221	−	0.513280i	$-0.828430\pi$
$653$	−43.8617			−1.71644			−0.858221	+	0.513280i	$0.828430\pi$
$654$		0			0
$655$		−	4.00000i		−	0.156293i
$656$			9.12311i			0.356197i
$657$		0			0
$658$			52.4924i			2.04637i
$659$	38.2462			1.48986			0.744930	−	0.667142i	$-0.232483\pi$
$659$	38.2462			1.48986			0.744930	+	0.667142i	$0.232483\pi$
$660$		0			0
$661$		−	0.876894i		−	0.0341072i	−0.999855	−	0.0170536i	$-0.994571\pi$
$661$		−	0.876894i		−	0.0341072i	0.999855	−	0.0170536i	$-0.00542860\pi$
$662$	24.2462			0.942356
$663$		0			0
$664$	−10.2462			−0.397630
$665$		−	30.7386i		−	1.19199i
$666$		0			0
$667$	6.24621			0.241854
$668$			5.75379i			0.222621i
$669$		0			0
$670$		−	13.1231i		−	0.506990i
$671$		−	31.2311i		−	1.20566i
$672$		0			0
$673$	−38.9848			−1.50276			−0.751378	−	0.659872i	$-0.770610\pi$
$673$	−38.9848			−1.50276			−0.751378	+	0.659872i	$0.770610\pi$
$674$		−	6.00000i		−	0.231111i
$675$		0			0
$676$	12.3693	−	4.00000i	0.475743	−	0.153846i
$677$	−21.1231			−0.811827			−0.405913	−	0.913912i	$-0.633047\pi$
$677$	−21.1231			−0.811827			−0.405913	+	0.913912i	$0.633047\pi$
$678$		0			0
$679$	−24.9848			−0.958830
$680$	2.00000			0.0766965
$681$		0			0
$682$		−	16.0000i		−	0.612672i
$683$			48.9848i			1.87435i	0.348856	+	0.937177i	$0.386570\pi$
$683$			48.9848i			1.87435i	−0.348856	+	0.937177i	$0.613430\pi$
$684$		0			0
$685$	−22.4924			−0.859391
$686$	−62.7386			−2.39537
$687$		0			0
$688$	−10.2462			−0.390633
$689$	6.38447	+	40.4924i	0.243229	+	1.54264i
$690$		0			0
$691$			20.7386i			0.788935i	0.918910	+	0.394467i	$0.129071\pi$
$691$			20.7386i			0.788935i	−0.918910	+	0.394467i	$0.870929\pi$
$692$	14.8769			0.565535
$693$		0			0
$694$			2.24621i			0.0852650i
$695$			16.4924i			0.625593i
$696$		0			0
$697$		−	18.2462i		−	0.691125i
$698$	5.36932			0.203232
$699$		0			0
$700$			5.12311i			0.193635i
$701$	14.0000			0.528773			0.264386	−	0.964417i	$-0.414831\pi$
$701$	14.0000			0.528773			0.264386	+	0.964417i	$0.414831\pi$
$702$		0			0
$703$	18.7386			0.706741
$704$			3.12311i			0.117706i
$705$		0			0
$706$	4.24621			0.159808
$707$		−	21.7538i		−	0.818135i
$708$		0			0
$709$		−	39.6155i		−	1.48779i	−0.668295	−	0.743896i	$-0.732976\pi$
$709$		−	39.6155i		−	1.48779i	0.668295	−	0.743896i	$-0.267024\pi$
$710$		−	6.24621i		−	0.234416i
$711$		0			0
$712$	5.12311			0.191997
$713$		−	16.0000i		−	0.599205i
$714$		0			0
$715$	−1.75379	−	11.1231i	−0.0655880	−	0.415981i
$716$	−16.4924			−0.616351
$717$		0			0
$718$	34.2462			1.27806
$719$	20.0000			0.745874			0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000			0.745874			0.372937	+	0.927857i	$0.378351\pi$
$720$		0			0
$721$		−	24.9848i		−	0.930484i
$722$			17.0000i			0.632674i
$723$		0			0
$724$	3.75379			0.139508
$725$	2.00000			0.0742781
$726$		0			0
$727$	−37.8617			−1.40421			−0.702107	−	0.712071i	$-0.747757\pi$
$727$	−37.8617			−1.40421			−0.702107	+	0.712071i	$0.747757\pi$
$728$	2.87689	+	18.2462i	0.106625	+	0.676250i
$729$		0			0
$730$			4.87689i			0.180502i
$731$	20.4924			0.757940
$732$		0			0
$733$		−	10.6307i		−	0.392653i	−0.980539	−	0.196327i	$-0.937099\pi$
$733$		−	10.6307i		−	0.392653i	0.980539	−	0.196327i	$-0.0629013\pi$
$734$		−	33.3693i		−	1.23168i
$735$		0			0
$736$			3.12311i			0.115119i
$737$	−40.9848			−1.50970
$738$		0			0
$739$		−	45.2311i		−	1.66385i	−0.554887	−	0.831926i	$-0.687239\pi$
$739$		−	45.2311i		−	1.66385i	0.554887	−	0.831926i	$-0.312761\pi$
$740$	−3.12311			−0.114808
$741$		0			0
$742$	−58.2462			−2.13829
$743$		−	16.0000i		−	0.586983i	−0.955962	−	0.293492i	$-0.905183\pi$
$743$		−	16.0000i		−	0.586983i	0.955962	−	0.293492i	$-0.0948173\pi$
$744$		0			0
$745$	14.0000			0.512920
$746$			1.12311i			0.0411198i
$747$		0			0
$748$		−	6.24621i		−	0.228384i
$749$			40.9848i			1.49755i
$750$		0			0
$751$	−9.75379			−0.355921			−0.177960	−	0.984038i	$-0.556950\pi$
$751$	−9.75379			−0.355921			−0.177960	+	0.984038i	$0.556950\pi$
$752$			10.2462i			0.373641i
$753$		0			0
$754$	7.12311	−	1.12311i	0.259408	−	0.0409011i
$755$	−11.3693			−0.413772
$756$		0			0
$757$	5.12311			0.186202			0.0931012	−	0.995657i	$-0.470322\pi$
$757$	5.12311			0.186202			0.0931012	+	0.995657i	$0.470322\pi$
$758$	6.00000			0.217930
$759$		0			0
$760$		−	6.00000i		−	0.217643i
$761$			5.12311i			0.185712i	0.995680	+	0.0928562i	$0.0295997\pi$
$761$			5.12311i			0.185712i	−0.995680	+	0.0928562i	$0.970400\pi$
$762$		0			0
$763$	56.9848			2.06299
$764$	−16.4924			−0.596675
$765$		0			0
$766$	18.2462			0.659262
$767$	25.3693	−	4.00000i	0.916033	−	0.144432i
$768$		0			0
$769$		−	32.9848i		−	1.18946i	−0.803924	−	0.594732i	$-0.797258\pi$
$769$		−	32.9848i		−	1.18946i	0.803924	−	0.594732i	$-0.202742\pi$
$770$	16.0000			0.576600
$771$		0			0
$772$		−	16.8769i		−	0.607413i
$773$			0.246211i			0.00885560i	0.999990	+	0.00442780i	$0.00140942\pi$
$773$			0.246211i			0.00885560i	−0.999990	+	0.00442780i	$0.998591\pi$
$774$		0			0
$775$		−	5.12311i		−	0.184027i
$776$	−4.87689			−0.175070
$777$		0			0
$778$			16.2462i			0.582455i
$779$	−54.7386			−1.96122
$780$		0			0
$781$	−19.5076			−0.698036
$782$		−	6.24621i		−	0.223364i
$783$		0			0
$784$	−19.2462			−0.687365
$785$		−	3.36932i		−	0.120256i
$786$		0			0
$787$			53.6155i			1.91119i	0.294688	+	0.955594i	$0.404784\pi$
$787$			53.6155i			1.91119i	−0.294688	+	0.955594i	$0.595216\pi$
$788$			0.246211i			0.00877091i
$789$		0			0
$790$	8.00000			0.284627
$791$			21.7538i			0.773476i
$792$		0			0
$793$	−5.61553	−	35.6155i	−0.199413	−	1.26474i
$794$	−9.36932			−0.332505
$795$		0			0
$796$	−1.75379			−0.0621614
$797$	31.8617			1.12860			0.564300	−	0.825570i	$-0.309146\pi$
$797$	31.8617			1.12860			0.564300	+	0.825570i	$0.309146\pi$
$798$		0			0
$799$		−	20.4924i		−	0.724970i
$800$			1.00000i			0.0353553i
$801$		0			0
$802$	23.3693			0.825199
$803$	15.2311			0.537492
$804$		0			0
$805$	16.0000			0.563926
$806$	−2.87689	−	18.2462i	−0.101334	−	0.642695i
$807$		0			0
$808$		−	4.24621i		−	0.149381i
$809$	46.4924			1.63459			0.817293	−	0.576222i	$-0.195474\pi$
$809$	46.4924			1.63459			0.817293	+	0.576222i	$0.195474\pi$
$810$		0			0
$811$			44.2462i			1.55369i	0.629689	+	0.776847i	$0.283182\pi$
$811$			44.2462i			1.55369i	−0.629689	+	0.776847i	$0.716818\pi$
$812$			10.2462i			0.359572i
$813$		0			0
$814$			9.75379i			0.341870i
$815$	1.12311			0.0393407
$816$		0			0
$817$		−	61.4773i		−	2.15082i
$818$	24.4924			0.856357
$819$		0			0
$820$	9.12311			0.318593
$821$			27.7538i			0.968614i	0.874898	+	0.484307i	$0.160928\pi$
$821$			27.7538i			0.968614i	−0.874898	+	0.484307i	$0.839072\pi$
$822$		0			0
$823$	−51.1231			−1.78204			−0.891020	−	0.453965i	$-0.850009\pi$
$823$	−51.1231			−1.78204			−0.891020	+	0.453965i	$0.850009\pi$
$824$		−	4.87689i		−	0.169895i
$825$		0			0
$826$			36.4924i			1.26973i
$827$			50.7386i			1.76436i	0.470917	+	0.882178i	$0.343923\pi$
$827$			50.7386i			1.76436i	−0.470917	+	0.882178i	$0.656077\pi$
$828$		0			0
$829$	−7.75379			−0.269300			−0.134650	−	0.990893i	$-0.542991\pi$
$829$	−7.75379			−0.269300			−0.134650	+	0.990893i	$0.542991\pi$
$830$			10.2462i			0.355651i
$831$		0			0
$832$	0.561553	+	3.56155i	0.0194683	+	0.123475i
$833$	38.4924			1.33368
$834$		0			0
$835$	5.75379			0.199118
$836$	−18.7386			−0.648089
$837$		0			0
$838$		−	28.0000i		−	0.967244i
$839$			2.73863i			0.0945481i	0.998882	+	0.0472741i	$0.0150534\pi$
$839$			2.73863i			0.0945481i	−0.998882	+	0.0472741i	$0.984947\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$	25.3693			0.874284
$843$		0			0
$844$	−4.00000			−0.137686
$845$	−4.00000	−	12.3693i	−0.137604	−	0.425517i
$846$		0			0
$847$			6.38447i			0.219373i
$848$	−11.3693			−0.390424
$849$		0			0
$850$		−	2.00000i		−	0.0685994i
$851$			9.75379i			0.334356i
$852$		0			0
$853$			21.8617i			0.748532i	0.927321	+	0.374266i	$0.122105\pi$
$853$			21.8617i			0.748532i	−0.927321	+	0.374266i	$0.877895\pi$
$854$	51.2311			1.75309
$855$		0			0
$856$			8.00000i			0.273434i
$857$	−2.49242			−0.0851395			−0.0425698	−	0.999093i	$-0.513554\pi$
$857$	−2.49242			−0.0851395			−0.0425698	+	0.999093i	$0.513554\pi$
$858$		0			0
$859$	−12.0000			−0.409435			−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000			−0.409435			−0.204717	+	0.978821i	$0.565628\pi$
$860$			10.2462i			0.349393i
$861$		0			0
$862$	0.492423			0.0167720
$863$			10.2462i			0.348785i	0.984676	+	0.174393i	$0.0557962\pi$
$863$			10.2462i			0.348785i	−0.984676	+	0.174393i	$0.944204\pi$
$864$		0			0
$865$		−	14.8769i		−	0.505830i
$866$			18.0000i			0.611665i
$867$		0			0
$868$	26.2462			0.890854
$869$		−	24.9848i		−	0.847553i
$870$		0			0
$871$	−46.7386	+	7.36932i	−1.58368	+	0.249700i
$872$	11.1231			0.376676
$873$		0			0
$874$	−18.7386			−0.633844
$875$	5.12311			0.173193
$876$		0			0
$877$			27.1231i			0.915882i	0.888983	+	0.457941i	$0.151413\pi$
$877$			27.1231i			0.915882i	−0.888983	+	0.457941i	$0.848587\pi$
$878$			3.50758i			0.118375i
$879$		0			0
$880$	3.12311			0.105280
$881$	−11.7538			−0.395995			−0.197998	−	0.980203i	$-0.563444\pi$
$881$	−11.7538			−0.395995			−0.197998	+	0.980203i	$0.563444\pi$
$882$		0			0
$883$	−26.2462			−0.883255			−0.441628	−	0.897198i	$-0.645599\pi$
$883$	−26.2462			−0.883255			−0.441628	+	0.897198i	$0.645599\pi$
$884$	−1.12311	−	7.12311i	−0.0377741	−	0.239576i
$885$		0			0
$886$		−	36.4924i		−	1.22599i
$887$	27.1231			0.910705			0.455352	−	0.890311i	$-0.349513\pi$
$887$	27.1231			0.910705			0.455352	+	0.890311i	$0.349513\pi$
$888$		0			0
$889$		−	24.9848i		−	0.837965i
$890$		−	5.12311i		−	0.171727i
$891$		0			0
$892$		−	15.3693i		−	0.514603i
$893$	−61.4773			−2.05726
$894$		0			0
$895$			16.4924i			0.551281i
$896$	−5.12311			−0.171151
$897$		0			0
$898$	−37.1231			−1.23881
$899$		−	10.2462i		−	0.341730i
$900$		0			0
$901$	22.7386			0.757534
$902$		−	28.4924i		−	0.948694i
$903$		0			0
$904$			4.24621i			0.141227i
$905$		−	3.75379i		−	0.124780i
$906$		0			0
$907$	42.2462			1.40276			0.701381	−	0.712786i	$-0.252567\pi$
$907$	42.2462			1.40276			0.701381	+	0.712786i	$0.252567\pi$
$908$		−	8.00000i		−	0.265489i
$909$		0			0
$910$	18.2462	−	2.87689i	0.604856	−	0.0953681i
$911$	−4.00000			−0.132526			−0.0662630	−	0.997802i	$-0.521108\pi$
$911$	−4.00000			−0.132526			−0.0662630	+	0.997802i	$0.521108\pi$
$912$		0			0
$913$	32.0000			1.05905
$914$	−6.63068			−0.219324
$915$		0			0
$916$			3.12311i			0.103190i
$917$		−	20.4924i		−	0.676719i
$918$		0			0
$919$	−38.2462			−1.26163			−0.630813	−	0.775935i	$-0.717279\pi$
$919$	−38.2462			−1.26163			−0.630813	+	0.775935i	$0.717279\pi$
$920$	3.12311			0.102966
$921$		0			0
$922$	−14.4924			−0.477283
$923$	−22.2462	+	3.50758i	−0.732243	+	0.115453i
$924$		0			0
$925$			3.12311i			0.102687i
$926$	−35.8617			−1.17849
$927$		0			0
$928$			2.00000i			0.0656532i
$929$		−	46.1080i		−	1.51275i	−0.654137	−	0.756376i	$-0.726968\pi$
$929$		−	46.1080i		−	1.51275i	0.654137	−	0.756376i	$-0.273032\pi$
$930$		0			0
$931$		−	115.477i		−	3.78461i
$932$	−24.2462			−0.794211
$933$		0			0
$934$		−	5.75379i		−	0.188270i
$935$	−6.24621			−0.204273
$936$		0			0
$937$	−3.75379			−0.122631			−0.0613155	−	0.998118i	$-0.519530\pi$
$937$	−3.75379			−0.122631			−0.0613155	+	0.998118i	$0.519530\pi$
$938$		−	67.2311i		−	2.19517i
$939$		0			0
$940$	10.2462			0.334195
$941$		−	14.0000i		−	0.456387i	−0.973616	−	0.228193i	$-0.926718\pi$
$941$		−	14.0000i		−	0.456387i	0.973616	−	0.228193i	$-0.0732819\pi$
$942$		0			0
$943$		−	28.4924i		−	0.927841i
$944$			7.12311i			0.231837i
$945$		0			0
$946$	32.0000			1.04041
$947$		−	24.4924i		−	0.795897i	−0.917408	−	0.397948i	$-0.869722\pi$
$947$		−	24.4924i		−	0.795897i	0.917408	−	0.397948i	$-0.130278\pi$
$948$		0			0
$949$	17.3693	−	2.73863i	0.563832	−	0.0888998i
$950$	−6.00000			−0.194666
$951$		0			0
$952$	10.2462			0.332082
$953$	42.9848			1.39242			0.696208	−	0.717840i	$-0.254869\pi$
$953$	42.9848			1.39242			0.696208	+	0.717840i	$0.254869\pi$
$954$		0			0
$955$			16.4924i			0.533682i
$956$			28.4924i			0.921511i
$957$		0			0
$958$	20.4924			0.662080
$959$	−115.231			−3.72100
$960$		0			0
$961$	4.75379			0.153348
$962$	1.75379	+	11.1231i	0.0565444	+	0.358623i
$963$		0			0
$964$			2.24621i			0.0723456i
$965$	−16.8769			−0.543286
$966$		0			0
$967$			6.38447i			0.205311i	0.994717	+	0.102655i	$0.0327339\pi$
$967$			6.38447i			0.205311i	−0.994717	+	0.102655i	$0.967266\pi$
$968$			1.24621i			0.0400547i
$969$		0			0
$970$			4.87689i			0.156588i
$971$	−54.2462			−1.74084			−0.870422	−	0.492307i	$-0.836154\pi$
$971$	−54.2462			−1.74084			−0.870422	+	0.492307i	$0.836154\pi$
$972$		0			0
$973$			84.4924i			2.70870i
$974$	7.36932			0.236128
$975$		0			0
$976$	10.0000			0.320092
$977$			44.7386i			1.43132i	0.698451	+	0.715658i	$0.253873\pi$
$977$			44.7386i			1.43132i	−0.698451	+	0.715658i	$0.746127\pi$
$978$		0			0
$979$	−16.0000			−0.511362
$980$			19.2462i			0.614798i
$981$		0			0
$982$		−	10.7386i		−	0.342684i
$983$		−	27.5076i		−	0.877355i	−0.898644	−	0.438678i	$-0.855447\pi$
$983$		−	27.5076i		−	0.877355i	0.898644	−	0.438678i	$-0.144553\pi$
$984$		0			0
$985$	0.246211			0.00784494
$986$		−	4.00000i		−	0.127386i
$987$		0			0
$988$	−21.3693	+	3.36932i	−0.679849	+	0.107192i
$989$	32.0000			1.01754
$990$		0			0
$991$	18.7386			0.595252			0.297626	−	0.954682i	$-0.403805\pi$
$991$	18.7386			0.595252			0.297626	+	0.954682i	$0.403805\pi$
$992$	5.12311			0.162659
$993$		0			0
$994$		−	32.0000i		−	1.01498i
$995$			1.75379i			0.0555988i
$996$		0			0
$997$	−52.3542			−1.65807			−0.829036	−	0.559195i	$-0.811110\pi$
$997$	−52.3542			−1.65807			−0.829036	+	0.559195i	$0.811110\pi$
$998$	−1.50758			−0.0477215
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.2.b.f.181.2		4
3.2	odd	2		390.2.b.d.181.4	yes	4
12.11	even	2		3120.2.g.o.961.1		4
13.12	even	2	inner	1170.2.b.f.181.3		4
15.2	even	4		1950.2.f.l.649.1		4
15.8	even	4		1950.2.f.o.649.4		4
15.14	odd	2		1950.2.b.h.1351.1		4
39.5	even	4		5070.2.a.bh.1.1		2
39.8	even	4		5070.2.a.bd.1.2		2
39.38	odd	2		390.2.b.d.181.1	✓	4
156.155	even	2		3120.2.g.o.961.4		4
195.38	even	4		1950.2.f.l.649.3		4
195.77	even	4		1950.2.f.o.649.2		4
195.194	odd	2		1950.2.b.h.1351.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.b.d.181.1	✓	4	39.38	odd	2
390.2.b.d.181.4	yes	4	3.2	odd	2
1170.2.b.f.181.2		4	1.1	even	1	trivial
1170.2.b.f.181.3		4	13.12	even	2	inner
1950.2.b.h.1351.1		4	15.14	odd	2
1950.2.b.h.1351.4		4	195.194	odd	2
1950.2.f.l.649.1		4	15.2	even	4
1950.2.f.l.649.3		4	195.38	even	4
1950.2.f.o.649.2		4	195.77	even	4
1950.2.f.o.649.4		4	15.8	even	4
3120.2.g.o.961.1		4	12.11	even	2
3120.2.g.o.961.4		4	156.155	even	2
5070.2.a.bd.1.2		2	39.8	even	4
5070.2.a.bh.1.1		2	39.5	even	4

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} +5.12311i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} +5.12311i q^{7} +1.00000i q^{8} +1.00000 q^{10} -3.12311i q^{11} +(-0.561553 - 3.56155i) q^{13} +5.12311 q^{14} +1.00000 q^{16} -2.00000 q^{17} +6.00000i q^{19} -1.00000i q^{20} -3.12311 q^{22} -3.12311 q^{23} -1.00000 q^{25} +(-3.56155 + 0.561553i) q^{26} -5.12311i q^{28} -2.00000 q^{29} +5.12311i q^{31} -1.00000i q^{32} +2.00000i q^{34} -5.12311 q^{35} -3.12311i q^{37} +6.00000 q^{38} -1.00000 q^{40} +9.12311i q^{41} -10.2462 q^{43} +3.12311i q^{44} +3.12311i q^{46} +10.2462i q^{47} -19.2462 q^{49} +1.00000i q^{50} +(0.561553 + 3.56155i) q^{52} -11.3693 q^{53} +3.12311 q^{55} -5.12311 q^{56} +2.00000i q^{58} +7.12311i q^{59} +10.0000 q^{61} +5.12311 q^{62} -1.00000 q^{64} +(3.56155 - 0.561553i) q^{65} -13.1231i q^{67} +2.00000 q^{68} +5.12311i q^{70} -6.24621i q^{71} +4.87689i q^{73} -3.12311 q^{74} -6.00000i q^{76} +16.0000 q^{77} +8.00000 q^{79} +1.00000i q^{80} +9.12311 q^{82} +10.2462i q^{83} -2.00000i q^{85} +10.2462i q^{86} +3.12311 q^{88} -5.12311i q^{89} +(18.2462 - 2.87689i) q^{91} +3.12311 q^{92} +10.2462 q^{94} -6.00000 q^{95} +4.87689i q^{97} +19.2462i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 4 q^{10} + 6 q^{13} + 4 q^{14} + 4 q^{16} - 8 q^{17} + 4 q^{22} + 4 q^{23} - 4 q^{25} - 6 q^{26} - 8 q^{29} - 4 q^{35} + 24 q^{38} - 4 q^{40} - 8 q^{43} - 44 q^{49} - 6 q^{52} + 4 q^{53} - 4 q^{55} - 4 q^{56} + 40 q^{61} + 4 q^{62} - 4 q^{64} + 6 q^{65} + 8 q^{68} + 4 q^{74} + 64 q^{77} + 32 q^{79} + 20 q^{82} - 4 q^{88} + 40 q^{91} - 4 q^{92} + 8 q^{94} - 24 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^10 + 6 * q^13 + 4 * q^14 + 4 * q^16 - 8 * q^17 + 4 * q^22 + 4 * q^23 - 4 * q^25 - 6 * q^26 - 8 * q^29 - 4 * q^35 + 24 * q^38 - 4 * q^40 - 8 * q^43 - 44 * q^49 - 6 * q^52 + 4 * q^53 - 4 * q^55 - 4 * q^56 + 40 * q^61 + 4 * q^62 - 4 * q^64 + 6 * q^65 + 8 * q^68 + 4 * q^74 + 64 * q^77 + 32 * q^79 + 20 * q^82 - 4 * q^88 + 40 * q^91 - 4 * q^92 + 8 * q^94 - 24 * q^95

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