LMFDB - Embedded newform 3120.2.g.o.961.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3120,2,Mod(961,3120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("3120.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3120.g (of order $2$, degree $1$, not 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:	$24.9133254306$
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			961.4
Root			$2.56155i$ of defining polynomial
Character	$\chi$	$=$	3120.961
Dual form			3120.2.g.o.961.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000i q^{5} +5.12311i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000i q^{5} +5.12311i q^{7} +1.00000 q^{9} +3.12311i q^{11} +(-0.561553 + 3.56155i) q^{13} -1.00000i q^{15} +2.00000 q^{17} +6.00000i q^{19} -5.12311i q^{21} -3.12311 q^{23} -1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} +5.12311i q^{31} -3.12311i q^{33} -5.12311 q^{35} +3.12311i q^{37} +(0.561553 - 3.56155i) q^{39} +9.12311i q^{41} +10.2462 q^{43} +1.00000i q^{45} -10.2462i q^{47} -19.2462 q^{49} -2.00000 q^{51} +11.3693 q^{53} -3.12311 q^{55} -6.00000i q^{57} -7.12311i q^{59} +10.0000 q^{61} +5.12311i q^{63} +(-3.56155 - 0.561553i) q^{65} -13.1231i q^{67} +3.12311 q^{69} +6.24621i q^{71} -4.87689i q^{73} +1.00000 q^{75} -16.0000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -10.2462i q^{83} +2.00000i q^{85} -2.00000 q^{87} -5.12311i q^{89} +(-18.2462 - 2.87689i) q^{91} -5.12311i q^{93} -6.00000 q^{95} -4.87689i q^{97} +3.12311i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{9} + 6 q^{13} + 8 q^{17} + 4 q^{23} - 4 q^{25} - 4 q^{27} + 8 q^{29} - 4 q^{35} - 6 q^{39} + 8 q^{43} - 44 q^{49} - 8 q^{51} - 4 q^{53} + 4 q^{55} + 40 q^{61} - 6 q^{65} - 4 q^{69} + 4 q^{75} - 64 q^{77} - 32 q^{79} + 4 q^{81} - 8 q^{87} - 40 q^{91} - 24 q^{95}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^9 + 6 * q^13 + 8 * q^17 + 4 * q^23 - 4 * q^25 - 4 * q^27 + 8 * q^29 - 4 * q^35 - 6 * q^39 + 8 * q^43 - 44 * q^49 - 8 * q^51 - 4 * q^53 + 4 * q^55 + 40 * q^61 - 6 * q^65 - 4 * q^69 + 4 * q^75 - 64 * q^77 - 32 * q^79 + 4 * q^81 - 8 * q^87 - 40 * q^91 - 24 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times$.

$n$	$1951$	$2081$	$2341$	$2497$	$2641$
$\chi(n)$	$1$	$1$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$			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	$0.580520\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			3.12311i			0.941652i	0.882226	+	0.470826i	$0.156044\pi$
$11$			3.12311i			0.941652i	−0.882226	+	0.470826i	$0.843956\pi$
$12$		0			0
$13$	−0.561553	+	3.56155i	−0.155747	+	0.987797i
$13$	−0.561553	+	3.56155i	−0.155747	+	0.987797i
$14$		0			0
$15$		−	1.00000i		−	0.258199i
$16$		0			0
$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$		0			0
$19$			6.00000i			1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$			6.00000i			1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$		0			0
$21$		−	5.12311i		−	1.11795i
$22$		0			0
$23$	−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
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$			5.12311i			0.920137i	0.887883	+	0.460068i	$0.152175\pi$
$31$			5.12311i			0.920137i	−0.887883	+	0.460068i	$0.847825\pi$
$32$		0			0
$33$		−	3.12311i		−	0.543663i
$34$		0			0
$35$	−5.12311			−0.865963
$36$		0			0
$37$			3.12311i			0.513435i	0.966486	+	0.256718i	$0.0826411\pi$
$37$			3.12311i			0.513435i	−0.966486	+	0.256718i	$0.917359\pi$
$38$		0			0
$39$	0.561553	−	3.56155i	0.0899204	−	0.570305i
$40$		0			0
$41$			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	$-0.214574\pi$
$43$	10.2462			1.56253			0.781266	+	0.624198i	$0.214574\pi$
$44$		0			0
$45$			1.00000i			0.149071i
$46$		0			0
$47$		−	10.2462i		−	1.49456i	−0.664507	−	0.747282i	$-0.731359\pi$
$47$		−	10.2462i		−	1.49456i	0.664507	−	0.747282i	$-0.268641\pi$
$48$		0			0
$49$	−19.2462			−2.74946
$50$		0			0
$51$	−2.00000			−0.280056
$52$		0			0
$53$	11.3693			1.56170			0.780848	−	0.624721i	$-0.214787\pi$
$53$	11.3693			1.56170			0.780848	+	0.624721i	$0.214787\pi$
$54$		0			0
$55$	−3.12311			−0.421119
$56$		0			0
$57$		−	6.00000i		−	0.794719i
$58$		0			0
$59$		−	7.12311i		−	0.927349i	−0.886006	−	0.463675i	$-0.846531\pi$
$59$		−	7.12311i		−	0.927349i	0.886006	−	0.463675i	$-0.153469\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$			5.12311i			0.645451i
$64$		0			0
$65$	−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	$-0.296029\pi$
$68$		0			0
$69$	3.12311			0.375978
$70$		0			0
$71$			6.24621i			0.741289i	0.928775	+	0.370644i	$0.120863\pi$
$71$			6.24621i			0.741289i	−0.928775	+	0.370644i	$0.879137\pi$
$72$		0			0
$73$		−	4.87689i		−	0.570797i	−0.958409	−	0.285399i	$-0.907874\pi$
$73$		−	4.87689i		−	0.570797i	0.958409	−	0.285399i	$-0.0921260\pi$
$74$		0			0
$75$	1.00000			0.115470
$76$		0			0
$77$	−16.0000			−1.82337
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	10.2462i		−	1.12467i	−0.826910	−	0.562334i	$-0.809904\pi$
$83$		−	10.2462i		−	1.12467i	0.826910	−	0.562334i	$-0.190096\pi$
$84$		0			0
$85$			2.00000i			0.216930i
$86$		0			0
$87$	−2.00000			−0.214423
$88$		0			0
$89$		−	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
$92$		0			0
$93$		−	5.12311i		−	0.531241i
$94$		0			0
$95$	−6.00000			−0.615587
$96$		0			0
$97$		−	4.87689i		−	0.495174i	−0.968866	−	0.247587i	$-0.920362\pi$
$97$		−	4.87689i		−	0.495174i	0.968866	−	0.247587i	$-0.0796375\pi$
$98$		0			0
$99$			3.12311i			0.313884i
$100$		0			0
$101$	4.24621			0.422514			0.211257	−	0.977431i	$-0.432244\pi$
$101$	4.24621			0.422514			0.211257	+	0.977431i	$0.432244\pi$
$102$		0			0
$103$	4.87689			0.480535			0.240267	−	0.970707i	$-0.422765\pi$
$103$	4.87689			0.480535			0.240267	+	0.970707i	$0.422765\pi$
$104$		0			0
$105$	5.12311			0.499964
$106$		0			0
$107$	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	$0.178823\pi$
$109$			11.1231i			1.06540i	−0.846304	+	0.532700i	$0.821177\pi$
$110$		0			0
$111$		−	3.12311i		−	0.296432i
$112$		0			0
$113$	−4.24621			−0.399450			−0.199725	−	0.979852i	$-0.564005\pi$
$113$	−4.24621			−0.399450			−0.199725	+	0.979852i	$0.564005\pi$
$114$		0			0
$115$		−	3.12311i		−	0.291231i
$116$		0			0
$117$	−0.561553	+	3.56155i	−0.0519156	+	0.329266i
$118$		0			0
$119$			10.2462i			0.939269i
$120$		0			0
$121$	1.24621			0.113292
$122$		0			0
$123$		−	9.12311i		−	0.822603i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	4.87689			0.432754			0.216377	−	0.976310i	$-0.430576\pi$
$127$	4.87689			0.432754			0.216377	+	0.976310i	$0.430576\pi$
$128$		0			0
$129$	−10.2462			−0.902129
$130$		0			0
$131$	−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
$134$		0			0
$135$		−	1.00000i		−	0.0860663i
$136$		0			0
$137$			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	$-0.746565\pi$
$139$	−16.4924			−1.39887			−0.699435	+	0.714697i	$0.746565\pi$
$140$		0			0
$141$			10.2462i			0.862887i
$142$		0			0
$143$	−11.1231	−	1.75379i	−0.930161	−	0.146659i
$144$		0			0
$145$			2.00000i			0.166091i
$146$		0			0
$147$	19.2462			1.58740
$148$		0			0
$149$		−	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	$0.846913\pi$
$152$		0			0
$153$	2.00000			0.161690
$154$		0			0
$155$	−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	$0.542927\pi$
$158$		0			0
$159$	−11.3693			−0.901645
$160$		0			0
$161$		−	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	$-0.0140051\pi$
$164$		0			0
$165$	3.12311			0.243133
$166$		0			0
$167$			5.75379i			0.445242i	0.974905	+	0.222621i	$0.0714612\pi$
$167$			5.75379i			0.445242i	−0.974905	+	0.222621i	$0.928539\pi$
$168$		0			0
$169$	−12.3693	−	4.00000i	−0.951486	−	0.307692i
$170$		0			0
$171$			6.00000i			0.458831i
$172$		0			0
$173$	14.8769			1.13107			0.565535	−	0.824725i	$-0.308670\pi$
$173$	14.8769			1.13107			0.565535	+	0.824725i	$0.308670\pi$
$174$		0			0
$175$		−	5.12311i		−	0.387270i
$176$		0			0
$177$			7.12311i			0.535405i
$178$		0			0
$179$	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	$0.544552\pi$
$182$		0			0
$183$	−10.0000			−0.739221
$184$		0			0
$185$	−3.12311			−0.229615
$186$		0			0
$187$			6.24621i			0.456768i
$188$		0			0
$189$		−	5.12311i		−	0.372651i
$190$		0			0
$191$	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	$-0.792207\pi$
$193$		−	16.8769i		−	1.21483i	0.794386	−	0.607413i	$-0.207793\pi$
$194$		0			0
$195$	3.56155	+	0.561553i	0.255048	+	0.0402136i
$196$		0			0
$197$		−	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	$-0.519799\pi$
$199$	−1.75379			−0.124323			−0.0621614	+	0.998066i	$0.519799\pi$
$200$		0			0
$201$			13.1231i			0.925633i
$202$		0			0
$203$			10.2462i			0.719143i
$204$		0			0
$205$	−9.12311			−0.637185
$206$		0			0
$207$	−3.12311			−0.217071
$208$		0			0
$209$	−18.7386			−1.29618
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$		−	6.24621i		−	0.427983i
$214$		0			0
$215$			10.2462i			0.698786i
$216$		0			0
$217$	−26.2462			−1.78171
$218$		0			0
$219$			4.87689i			0.329550i
$220$		0			0
$221$	−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	$0.827939\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$		−	8.00000i		−	0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$		−	8.00000i		−	0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$		0			0
$229$			3.12311i			0.206381i	0.994662	+	0.103190i	$0.0329051\pi$
$229$			3.12311i			0.206381i	−0.994662	+	0.103190i	$0.967095\pi$
$230$		0			0
$231$	16.0000			1.05272
$232$		0			0
$233$	−24.2462			−1.58842			−0.794211	−	0.607642i	$-0.792116\pi$
$233$	−24.2462			−1.58842			−0.794211	+	0.607642i	$0.792116\pi$
$234$		0			0
$235$	10.2462			0.668389
$236$		0			0
$237$	8.00000			0.519656
$238$		0			0
$239$			28.4924i			1.84302i	0.388353	+	0.921511i	$0.373044\pi$
$239$			28.4924i			1.84302i	−0.388353	+	0.921511i	$0.626956\pi$
$240$		0			0
$241$			2.24621i			0.144691i	0.997380	+	0.0723456i	$0.0230485\pi$
$241$			2.24621i			0.144691i	−0.997380	+	0.0723456i	$0.976952\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$		−	19.2462i		−	1.22960i
$246$		0			0
$247$	−21.3693	−	3.36932i	−1.35970	−	0.214384i
$248$		0			0
$249$			10.2462i			0.649327i
$250$		0			0
$251$	−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
$254$		0			0
$255$		−	2.00000i		−	0.125245i
$256$		0			0
$257$	4.24621			0.264871			0.132436	−	0.991192i	$-0.457720\pi$
$257$	4.24621			0.264871			0.132436	+	0.991192i	$0.457720\pi$
$258$		0			0
$259$	−16.0000			−0.994192
$260$		0			0
$261$	2.00000			0.123797
$262$		0			0
$263$	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
$266$		0			0
$267$			5.12311i			0.313529i
$268$		0			0
$269$	0.246211			0.0150118			0.00750588	−	0.999972i	$-0.497611\pi$
$269$	0.246211			0.0150118			0.00750588	+	0.999972i	$0.497611\pi$
$270$		0			0
$271$		−	14.8769i		−	0.903707i	−0.892092	−	0.451853i	$-0.850763\pi$
$271$		−	14.8769i		−	0.903707i	0.892092	−	0.451853i	$-0.149237\pi$
$272$		0			0
$273$	18.2462	+	2.87689i	1.10431	+	0.174118i
$274$		0			0
$275$		−	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	$0.184293\pi$
$278$		0			0
$279$			5.12311i			0.306712i
$280$		0			0
$281$			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	$-0.462067\pi$
$283$	4.00000			0.237775			0.118888	+	0.992908i	$0.462067\pi$
$284$		0			0
$285$	6.00000			0.355409
$286$		0			0
$287$	−46.7386			−2.75889
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$			4.87689i			0.285889i
$292$		0			0
$293$		−	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
$296$		0			0
$297$		−	3.12311i		−	0.181221i
$298$		0			0
$299$	1.75379	−	11.1231i	0.101424	−	0.643266i
$300$		0			0
$301$			52.4924i			3.02561i
$302$		0			0
$303$	−4.24621			−0.243938
$304$		0			0
$305$			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	$-0.226417\pi$
$308$		0			0
$309$	−4.87689			−0.277437
$310$		0			0
$311$	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	$0.502215\pi$
$314$		0			0
$315$	−5.12311			−0.288654
$316$		0			0
$317$			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
$320$		0			0
$321$	−8.00000			−0.446516
$322$		0			0
$323$			12.0000i			0.667698i
$324$		0			0
$325$	0.561553	−	3.56155i	0.0311493	−	0.197559i
$326$		0			0
$327$		−	11.1231i		−	0.615109i
$328$		0			0
$329$	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	$0.767857\pi$
$332$		0			0
$333$			3.12311i			0.171145i
$334$		0			0
$335$	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	$0.447747\pi$
$338$		0			0
$339$	4.24621			0.230623
$340$		0			0
$341$	−16.0000			−0.866449
$342$		0			0
$343$		−	62.7386i		−	3.38757i
$344$		0			0
$345$			3.12311i			0.168142i
$346$		0			0
$347$	−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	$-0.954098\pi$
$349$		−	5.36932i		−	0.287413i	0.989620	−	0.143706i	$-0.0459021\pi$
$350$		0			0
$351$	0.561553	−	3.56155i	0.0299735	−	0.190102i
$352$		0			0
$353$			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
$356$		0			0
$357$		−	10.2462i		−	0.542287i
$358$		0			0
$359$		−	34.2462i		−	1.80745i	−0.428118	−	0.903723i	$-0.640823\pi$
$359$		−	34.2462i		−	1.80745i	0.428118	−	0.903723i	$-0.359177\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$		0			0
$363$	−1.24621			−0.0654091
$364$		0			0
$365$	4.87689			0.255268
$366$		0			0
$367$	−33.3693			−1.74186			−0.870932	−	0.491403i	$-0.836484\pi$
$367$	−33.3693			−1.74186			−0.870932	+	0.491403i	$0.836484\pi$
$368$		0			0
$369$			9.12311i			0.474930i
$370$		0			0
$371$			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	$0.509257\pi$
$374$		0			0
$375$			1.00000i			0.0516398i
$376$		0			0
$377$	−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	$0.950752\pi$
$380$		0			0
$381$	−4.87689			−0.249851
$382$		0			0
$383$		−	18.2462i		−	0.932338i	−0.884696	−	0.466169i	$-0.845634\pi$
$383$		−	18.2462i		−	0.932338i	0.884696	−	0.466169i	$-0.154366\pi$
$384$		0			0
$385$		−	16.0000i		−	0.815436i
$386$		0			0
$387$	10.2462			0.520844
$388$		0			0
$389$	16.2462			0.823716			0.411858	−	0.911248i	$-0.364880\pi$
$389$	16.2462			0.823716			0.411858	+	0.911248i	$0.364880\pi$
$390$		0			0
$391$	−6.24621			−0.315884
$392$		0			0
$393$	4.00000			0.201773
$394$		0			0
$395$		−	8.00000i		−	0.402524i
$396$		0			0
$397$			9.36932i			0.470233i	0.971967	+	0.235116i	$0.0755471\pi$
$397$			9.36932i			0.470233i	−0.971967	+	0.235116i	$0.924453\pi$
$398$		0			0
$399$	30.7386			1.53886
$400$		0			0
$401$			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
$404$		0			0
$405$			1.00000i			0.0496904i
$406$		0			0
$407$	−9.75379			−0.483477
$408$		0			0
$409$		−	24.4924i		−	1.21107i	−0.795818	−	0.605536i	$-0.792959\pi$
$409$		−	24.4924i		−	1.21107i	0.795818	−	0.605536i	$-0.207041\pi$
$410$		0			0
$411$		−	22.4924i		−	1.10947i
$412$		0			0
$413$	36.4924			1.79567
$414$		0			0
$415$	10.2462			0.502967
$416$		0			0
$417$	16.4924			0.807637
$418$		0			0
$419$	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	$-0.787857\pi$
$421$		−	25.3693i		−	1.23642i	0.786011	−	0.618212i	$-0.212143\pi$
$422$		0			0
$423$		−	10.2462i		−	0.498188i
$424$		0			0
$425$	−2.00000			−0.0970143
$426$		0			0
$427$			51.2311i			2.47924i
$428$		0			0
$429$	11.1231	+	1.75379i	0.537029	+	0.0846737i
$430$		0			0
$431$		−	0.492423i		−	0.0237192i	−0.999930	−	0.0118596i	$-0.996225\pi$
$431$		−	0.492423i		−	0.0237192i	0.999930	−	0.0118596i	$-0.00377511\pi$
$432$		0			0
$433$	−18.0000			−0.865025			−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000			−0.865025			−0.432512	+	0.901628i	$0.642373\pi$
$434$		0			0
$435$		−	2.00000i		−	0.0958927i
$436$		0			0
$437$		−	18.7386i		−	0.896390i
$438$		0			0
$439$	3.50758			0.167408			0.0837038	−	0.996491i	$-0.473325\pi$
$439$	3.50758			0.167408			0.0837038	+	0.996491i	$0.473325\pi$
$440$		0			0
$441$	−19.2462			−0.916486
$442$		0			0
$443$	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
$446$		0			0
$447$			14.0000i			0.662177i
$448$		0			0
$449$		−	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
$452$		0			0
$453$		−	11.3693i		−	0.534177i
$454$		0			0
$455$	2.87689	−	18.2462i	0.134871	−	0.855396i
$456$		0			0
$457$			6.63068i			0.310170i	0.987901	+	0.155085i	$0.0495652\pi$
$457$			6.63068i			0.310170i	−0.987901	+	0.155085i	$0.950435\pi$
$458$		0			0
$459$	−2.00000			−0.0933520
$460$		0			0
$461$		−	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	$-0.313562\pi$
$464$		0			0
$465$	5.12311			0.237578
$466$		0			0
$467$	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
$470$		0			0
$471$	3.36932			0.155250
$472$		0			0
$473$			32.0000i			1.47136i
$474$		0			0
$475$		−	6.00000i		−	0.275299i
$476$		0			0
$477$	11.3693			0.520565
$478$		0			0
$479$		−	20.4924i		−	0.936323i	−0.883643	−	0.468161i	$-0.844917\pi$
$479$		−	20.4924i		−	0.936323i	0.883643	−	0.468161i	$-0.155083\pi$
$480$		0			0
$481$	−11.1231	−	1.75379i	−0.507170	−	0.0799659i
$482$		0			0
$483$			16.0000i			0.728025i
$484$		0			0
$485$	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	$0.946602\pi$
$488$		0			0
$489$			1.12311i			0.0507886i
$490$		0			0
$491$	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
$494$		0			0
$495$	−3.12311			−0.140373
$496$		0			0
$497$	−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	$-0.0107432\pi$
$500$		0			0
$501$		−	5.75379i		−	0.257060i
$502$		0			0
$503$	−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
$506$		0			0
$507$	12.3693	+	4.00000i	0.549341	+	0.177646i
$508$		0			0
$509$			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
$512$		0			0
$513$		−	6.00000i		−	0.264906i
$514$		0			0
$515$			4.87689i			0.214902i
$516$		0			0
$517$	32.0000			1.40736
$518$		0			0
$519$	−14.8769			−0.653023
$520$		0			0
$521$	−16.2462			−0.711759			−0.355880	−	0.934532i	$-0.615819\pi$
$521$	−16.2462			−0.711759			−0.355880	+	0.934532i	$0.615819\pi$
$522$		0			0
$523$	−22.7386			−0.994291			−0.497146	−	0.867667i	$-0.665618\pi$
$523$	−22.7386			−0.994291			−0.497146	+	0.867667i	$0.665618\pi$
$524$		0			0
$525$			5.12311i			0.223591i
$526$		0			0
$527$			10.2462i			0.446332i
$528$		0			0
$529$	−13.2462			−0.575922
$530$		0			0
$531$		−	7.12311i		−	0.309116i
$532$		0			0
$533$	−32.4924	−	5.12311i	−1.40740	−	0.221906i
$534$		0			0
$535$			8.00000i			0.345870i
$536$		0			0
$537$	−16.4924			−0.711701
$538$		0			0
$539$		−	60.1080i		−	2.58903i
$540$		0			0
$541$			19.1231i			0.822167i	0.911598	+	0.411083i	$0.134849\pi$
$541$			19.1231i			0.822167i	−0.911598	+	0.411083i	$0.865151\pi$
$542$		0			0
$543$	3.75379			0.161090
$544$		0			0
$545$	−11.1231			−0.476461
$546$		0			0
$547$	44.9848			1.92341			0.961707	−	0.274081i	$-0.0883737\pi$
$547$	44.9848			1.92341			0.961707	+	0.274081i	$0.0883737\pi$
$548$		0			0
$549$	10.0000			0.426790
$550$		0			0
$551$			12.0000i			0.511217i
$552$		0			0
$553$		−	40.9848i		−	1.74285i
$554$		0			0
$555$	3.12311			0.132568
$556$		0			0
$557$			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
$560$		0			0
$561$		−	6.24621i		−	0.263715i
$562$		0			0
$563$	−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
$566$		0			0
$567$			5.12311i			0.215150i
$568$		0			0
$569$	36.7386			1.54016			0.770082	−	0.637945i	$-0.220215\pi$
$569$	36.7386			1.54016			0.770082	+	0.637945i	$0.220215\pi$
$570$		0			0
$571$	−24.4924			−1.02498			−0.512488	−	0.858694i	$-0.671276\pi$
$571$	−24.4924			−1.02498			−0.512488	+	0.858694i	$0.671276\pi$
$572$		0			0
$573$	−16.4924			−0.688981
$574$		0			0
$575$	3.12311			0.130243
$576$		0			0
$577$			2.63068i			0.109517i	0.998500	+	0.0547584i	$0.0174389\pi$
$577$			2.63068i			0.109517i	−0.998500	+	0.0547584i	$0.982561\pi$
$578$		0			0
$579$			16.8769i			0.701380i
$580$		0			0
$581$	52.4924			2.17775
$582$		0			0
$583$			35.5076i			1.47057i
$584$		0			0
$585$	−3.56155	−	0.561553i	−0.147252	−	0.0232174i
$586$		0			0
$587$		−	16.4924i		−	0.680715i	−0.940296	−	0.340358i	$-0.889452\pi$
$587$		−	16.4924i		−	0.680715i	0.940296	−	0.340358i	$-0.110548\pi$
$588$		0			0
$589$	−30.7386			−1.26656
$590$		0			0
$591$			0.246211i			0.0101278i
$592$		0			0
$593$		−	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
$596$		0			0
$597$	1.75379			0.0717778
$598$		0			0
$599$	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	$0.217737\pi$
$602$		0			0
$603$		−	13.1231i		−	0.534414i
$604$		0			0
$605$			1.24621i			0.0506657i
$606$		0			0
$607$	9.36932			0.380289			0.190144	−	0.981756i	$-0.439104\pi$
$607$	9.36932			0.380289			0.190144	+	0.981756i	$0.439104\pi$
$608$		0			0
$609$		−	10.2462i		−	0.415197i
$610$		0			0
$611$	36.4924	+	5.75379i	1.47633	+	0.232773i
$612$		0			0
$613$			14.6307i			0.590928i	0.955354	+	0.295464i	$0.0954742\pi$
$613$			14.6307i			0.590928i	−0.955354	+	0.295464i	$0.904526\pi$
$614$		0			0
$615$	9.12311			0.367879
$616$		0			0
$617$			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	$0.817551\pi$
$620$		0			0
$621$	3.12311			0.125326
$622$		0			0
$623$	26.2462			1.05153
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	18.7386			0.748349
$628$		0			0
$629$			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	$-0.0356537\pi$
$632$		0			0
$633$	4.00000			0.158986
$634$		0			0
$635$			4.87689i			0.193534i
$636$		0			0
$637$	10.8078	−	68.5464i	0.428219	−	2.71591i
$638$		0			0
$639$			6.24621i			0.247096i
$640$		0			0
$641$	−2.00000			−0.0789953			−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000			−0.0789953			−0.0394976	+	0.999220i	$0.512576\pi$
$642$		0			0
$643$			7.36932i			0.290617i	0.989386	+	0.145309i	$0.0464175\pi$
$643$			7.36932i			0.290617i	−0.989386	+	0.145309i	$0.953582\pi$
$644$		0			0
$645$		−	10.2462i		−	0.403444i
$646$		0			0
$647$	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
$650$		0			0
$651$	26.2462			1.02867
$652$		0			0
$653$	43.8617			1.71644			0.858221	−	0.513280i	$-0.171570\pi$
$653$	43.8617			1.71644			0.858221	+	0.513280i	$0.171570\pi$
$654$		0			0
$655$		−	4.00000i		−	0.156293i
$656$		0			0
$657$		−	4.87689i		−	0.190266i
$658$		0			0
$659$	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	$0.00542860\pi$
$661$			0.876894i			0.0341072i	−0.999855	+	0.0170536i	$0.994571\pi$
$662$		0			0
$663$	1.12311	−	7.12311i	0.0436178	−	0.276638i
$664$		0			0
$665$		−	30.7386i		−	1.19199i
$666$		0			0
$667$	−6.24621			−0.241854
$668$		0			0
$669$		−	15.3693i		−	0.594212i
$670$		0			0
$671$			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	$0.770610\pi$
$674$		0			0
$675$	1.00000			0.0384900
$676$		0			0
$677$	21.1231			0.811827			0.405913	−	0.913912i	$-0.366953\pi$
$677$	21.1231			0.811827			0.405913	+	0.913912i	$0.366953\pi$
$678$		0			0
$679$	24.9848			0.958830
$680$		0			0
$681$			8.00000i			0.306561i
$682$		0			0
$683$		−	48.9848i		−	1.87435i	−0.348856	−	0.937177i	$-0.613430\pi$
$683$		−	48.9848i		−	1.87435i	0.348856	−	0.937177i	$-0.386570\pi$
$684$		0			0
$685$	−22.4924			−0.859391
$686$		0			0
$687$		−	3.12311i		−	0.119154i
$688$		0			0
$689$	−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	$0.870929\pi$
$692$		0			0
$693$	−16.0000			−0.607790
$694$		0			0
$695$		−	16.4924i		−	0.625593i
$696$		0			0
$697$			18.2462i			0.691125i
$698$		0			0
$699$	24.2462			0.917076
$700$		0			0
$701$	−14.0000			−0.528773			−0.264386	−	0.964417i	$-0.585169\pi$
$701$	−14.0000			−0.528773			−0.264386	+	0.964417i	$0.585169\pi$
$702$		0			0
$703$	−18.7386			−0.706741
$704$		0			0
$705$	−10.2462			−0.385895
$706$		0			0
$707$			21.7538i			0.818135i
$708$		0			0
$709$			39.6155i			1.48779i	0.668295	+	0.743896i	$0.267024\pi$
$709$			39.6155i			1.48779i	−0.668295	+	0.743896i	$0.732976\pi$
$710$		0			0
$711$	−8.00000			−0.300023
$712$		0			0
$713$		−	16.0000i		−	0.599205i
$714$		0			0
$715$	1.75379	−	11.1231i	0.0655880	−	0.415981i
$716$		0			0
$717$		−	28.4924i		−	1.06407i
$718$		0			0
$719$	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
$722$		0			0
$723$		−	2.24621i		−	0.0835375i
$724$		0			0
$725$	−2.00000			−0.0742781
$726$		0			0
$727$	37.8617			1.40421			0.702107	−	0.712071i	$-0.252243\pi$
$727$	37.8617			1.40421			0.702107	+	0.712071i	$0.252243\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	20.4924			0.757940
$732$		0			0
$733$			10.6307i			0.392653i	0.980539	+	0.196327i	$0.0629013\pi$
$733$			10.6307i			0.392653i	−0.980539	+	0.196327i	$0.937099\pi$
$734$		0			0
$735$			19.2462i			0.709907i
$736$		0			0
$737$	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	$-0.312761\pi$
$740$		0			0
$741$	21.3693	+	3.36932i	0.785021	+	0.123775i
$742$		0			0
$743$			16.0000i			0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$			16.0000i			0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$		0			0
$745$	14.0000			0.512920
$746$		0			0
$747$		−	10.2462i		−	0.374889i
$748$		0			0
$749$			40.9848i			1.49755i
$750$		0			0
$751$	9.75379			0.355921			0.177960	−	0.984038i	$-0.443050\pi$
$751$	9.75379			0.355921			0.177960	+	0.984038i	$0.443050\pi$
$752$		0			0
$753$	9.75379			0.355448
$754$		0			0
$755$	−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	$0.470322\pi$
$758$		0			0
$759$			9.75379i			0.354040i
$760$		0			0
$761$			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
$764$		0			0
$765$			2.00000i			0.0723102i
$766$		0			0
$767$	25.3693	+	4.00000i	0.916033	+	0.144432i
$768$		0			0
$769$			32.9848i			1.18946i	0.803924	+	0.594732i	$0.202742\pi$
$769$			32.9848i			1.18946i	−0.803924	+	0.594732i	$0.797258\pi$
$770$		0			0
$771$	−4.24621			−0.152924
$772$		0			0
$773$			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
$776$		0			0
$777$	16.0000			0.573997
$778$		0			0
$779$	−54.7386			−1.96122
$780$		0			0
$781$	−19.5076			−0.698036
$782$		0			0
$783$	−2.00000			−0.0714742
$784$		0			0
$785$		−	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			0
$789$	−2.63068			−0.0936548
$790$		0			0
$791$		−	21.7538i		−	0.773476i
$792$		0			0
$793$	−5.61553	+	35.6155i	−0.199413	+	1.26474i
$794$		0			0
$795$		−	11.3693i		−	0.403228i
$796$		0			0
$797$	−31.8617			−1.12860			−0.564300	−	0.825570i	$-0.690854\pi$
$797$	−31.8617			−1.12860			−0.564300	+	0.825570i	$0.690854\pi$
$798$		0			0
$799$		−	20.4924i		−	0.724970i
$800$		0			0
$801$		−	5.12311i		−	0.181016i
$802$		0			0
$803$	15.2311			0.537492
$804$		0			0
$805$	16.0000			0.563926
$806$		0			0
$807$	−0.246211			−0.00866705
$808$		0			0
$809$	−46.4924			−1.63459			−0.817293	−	0.576222i	$-0.804526\pi$
$809$	−46.4924			−1.63459			−0.817293	+	0.576222i	$0.804526\pi$
$810$		0			0
$811$			44.2462i			1.55369i	0.629689	+	0.776847i	$0.283182\pi$
$811$			44.2462i			1.55369i	−0.629689	+	0.776847i	$0.716818\pi$
$812$		0			0
$813$			14.8769i			0.521755i
$814$		0			0
$815$	1.12311			0.0393407
$816$		0			0
$817$			61.4773i			2.15082i
$818$		0			0
$819$	−18.2462	−	2.87689i	−0.637574	−	0.100527i
$820$		0			0
$821$			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	$-0.149991\pi$
$823$	51.1231			1.78204			0.891020	+	0.453965i	$0.149991\pi$
$824$		0			0
$825$			3.12311i			0.108733i
$826$		0			0
$827$		−	50.7386i		−	1.76436i	−0.470917	−	0.882178i	$-0.656077\pi$
$827$		−	50.7386i		−	1.76436i	0.470917	−	0.882178i	$-0.343923\pi$
$828$		0			0
$829$	−7.75379			−0.269300			−0.134650	−	0.990893i	$-0.542991\pi$
$829$	−7.75379			−0.269300			−0.134650	+	0.990893i	$0.542991\pi$
$830$		0			0
$831$	−27.8617			−0.966513
$832$		0			0
$833$	−38.4924			−1.33368
$834$		0			0
$835$	−5.75379			−0.199118
$836$		0			0
$837$		−	5.12311i		−	0.177080i
$838$		0			0
$839$		−	2.73863i		−	0.0945481i	−0.998882	−	0.0472741i	$-0.984947\pi$
$839$		−	2.73863i		−	0.0945481i	0.998882	−	0.0472741i	$-0.0150534\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$		−	5.12311i		−	0.176449i
$844$		0			0
$845$	4.00000	−	12.3693i	0.137604	−	0.425517i
$846$		0			0
$847$			6.38447i			0.219373i
$848$		0			0
$849$	−4.00000			−0.137280
$850$		0			0
$851$		−	9.75379i		−	0.334356i
$852$		0			0
$853$		−	21.8617i		−	0.748532i	−0.927321	−	0.374266i	$-0.877895\pi$
$853$		−	21.8617i		−	0.748532i	0.927321	−	0.374266i	$-0.122105\pi$
$854$		0			0
$855$	−6.00000			−0.205196
$856$		0			0
$857$	2.49242			0.0851395			0.0425698	−	0.999093i	$-0.486446\pi$
$857$	2.49242			0.0851395			0.0425698	+	0.999093i	$0.486446\pi$
$858$		0			0
$859$	12.0000			0.409435			0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000			0.409435			0.204717	+	0.978821i	$0.434372\pi$
$860$		0			0
$861$	46.7386			1.59285
$862$		0			0
$863$		−	10.2462i		−	0.348785i	−0.984676	−	0.174393i	$-0.944204\pi$
$863$		−	10.2462i		−	0.348785i	0.984676	−	0.174393i	$-0.0557962\pi$
$864$		0			0
$865$			14.8769i			0.505830i
$866$		0			0
$867$	13.0000			0.441503
$868$		0			0
$869$		−	24.9848i		−	0.847553i
$870$		0			0
$871$	46.7386	+	7.36932i	1.58368	+	0.249700i
$872$		0			0
$873$		−	4.87689i		−	0.165058i
$874$		0			0
$875$	5.12311			0.173193
$876$		0			0
$877$		−	27.1231i		−	0.915882i	−0.888983	−	0.457941i	$-0.848587\pi$
$877$		−	27.1231i		−	0.915882i	0.888983	−	0.457941i	$-0.151413\pi$
$878$		0			0
$879$			20.7386i			0.699497i
$880$		0			0
$881$	11.7538			0.395995			0.197998	−	0.980203i	$-0.436556\pi$
$881$	11.7538			0.395995			0.197998	+	0.980203i	$0.436556\pi$
$882$		0			0
$883$	26.2462			0.883255			0.441628	−	0.897198i	$-0.354401\pi$
$883$	26.2462			0.883255			0.441628	+	0.897198i	$0.354401\pi$
$884$		0			0
$885$	−7.12311			−0.239441
$886$		0			0
$887$	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
$890$		0			0
$891$			3.12311i			0.104628i
$892$		0			0
$893$	61.4773			2.05726
$894$		0			0
$895$			16.4924i			0.551281i
$896$		0			0
$897$	−1.75379	+	11.1231i	−0.0585573	+	0.371390i
$898$		0			0
$899$			10.2462i			0.341730i
$900$		0			0
$901$	22.7386			0.757534
$902$		0			0
$903$		−	52.4924i		−	1.74684i
$904$		0			0
$905$		−	3.75379i		−	0.124780i
$906$		0			0
$907$	−42.2462			−1.40276			−0.701381	−	0.712786i	$-0.747433\pi$
$907$	−42.2462			−1.40276			−0.701381	+	0.712786i	$0.747433\pi$
$908$		0			0
$909$	4.24621			0.140838
$910$		0			0
$911$	−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
$914$		0			0
$915$		−	10.0000i		−	0.330590i
$916$		0			0
$917$		−	20.4924i		−	0.676719i
$918$		0			0
$919$	38.2462			1.26163			0.630813	−	0.775935i	$-0.282721\pi$
$919$	38.2462			1.26163			0.630813	+	0.775935i	$0.282721\pi$
$920$		0			0
$921$			22.8769i			0.753819i
$922$		0			0
$923$	−22.2462	−	3.50758i	−0.732243	−	0.115453i
$924$		0			0
$925$		−	3.12311i		−	0.102687i
$926$		0			0
$927$	4.87689			0.160178
$928$		0			0
$929$		−	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
$932$		0			0
$933$	−24.4924			−0.801846
$934$		0			0
$935$	−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	$0.519530\pi$
$938$		0			0
$939$	0.246211			0.00803480
$940$		0			0
$941$		−	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
$944$		0			0
$945$	5.12311			0.166655
$946$		0			0
$947$			24.4924i			0.795897i	0.917408	+	0.397948i	$0.130278\pi$
$947$			24.4924i			0.795897i	−0.917408	+	0.397948i	$0.869722\pi$
$948$		0			0
$949$	17.3693	+	2.73863i	0.563832	+	0.0888998i
$950$		0			0
$951$		−	6.00000i		−	0.194563i
$952$		0			0
$953$	−42.9848			−1.39242			−0.696208	−	0.717840i	$-0.745131\pi$
$953$	−42.9848			−1.39242			−0.696208	+	0.717840i	$0.745131\pi$
$954$		0			0
$955$			16.4924i			0.533682i
$956$		0			0
$957$		−	6.24621i		−	0.201911i
$958$		0			0
$959$	−115.231			−3.72100
$960$		0			0
$961$	4.75379			0.153348
$962$		0			0
$963$	8.00000			0.257796
$964$		0			0
$965$	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	$0.967266\pi$
$968$		0			0
$969$		−	12.0000i		−	0.385496i
$970$		0			0
$971$	−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
$974$		0			0
$975$	−0.561553	+	3.56155i	−0.0179841	+	0.114061i
$976$		0			0
$977$			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
$980$		0			0
$981$			11.1231i			0.355133i
$982$		0			0
$983$			27.5076i			0.877355i	0.898644	+	0.438678i	$0.144553\pi$
$983$			27.5076i			0.877355i	−0.898644	+	0.438678i	$0.855447\pi$
$984$		0			0
$985$	0.246211			0.00784494
$986$		0			0
$987$	−52.4924			−1.67085
$988$		0			0
$989$	−32.0000			−1.01754
$990$		0			0
$991$	−18.7386			−0.595252			−0.297626	−	0.954682i	$-0.596195\pi$
$991$	−18.7386			−0.595252			−0.297626	+	0.954682i	$0.596195\pi$
$992$		0			0
$993$		−	24.2462i		−	0.769430i
$994$		0			0
$995$		−	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	$0.811110\pi$
$998$		0			0
$999$		−	3.12311i		−	0.0988107i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.g.o.961.4		4
4.3	odd	2		390.2.b.d.181.1	✓	4
12.11	even	2		1170.2.b.f.181.3		4
13.12	even	2	inner	3120.2.g.o.961.1		4
20.3	even	4		1950.2.f.l.649.3		4
20.7	even	4		1950.2.f.o.649.2		4
20.19	odd	2		1950.2.b.h.1351.4		4
52.31	even	4		5070.2.a.bd.1.2		2
52.47	even	4		5070.2.a.bh.1.1		2
52.51	odd	2		390.2.b.d.181.4	yes	4
156.155	even	2		1170.2.b.f.181.2		4
260.103	even	4		1950.2.f.o.649.4		4
260.207	even	4		1950.2.f.l.649.1		4
260.259	odd	2		1950.2.b.h.1351.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.b.d.181.1	✓	4	4.3	odd	2
390.2.b.d.181.4	yes	4	52.51	odd	2
1170.2.b.f.181.2		4	156.155	even	2
1170.2.b.f.181.3		4	12.11	even	2
1950.2.b.h.1351.1		4	260.259	odd	2
1950.2.b.h.1351.4		4	20.19	odd	2
1950.2.f.l.649.1		4	260.207	even	4
1950.2.f.l.649.3		4	20.3	even	4
1950.2.f.o.649.2		4	20.7	even	4
1950.2.f.o.649.4		4	260.103	even	4
3120.2.g.o.961.1		4	13.12	even	2	inner
3120.2.g.o.961.4		4	1.1	even	1	trivial
5070.2.a.bd.1.2		2	52.31	even	4
5070.2.a.bh.1.1		2	52.47	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:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000i q^{5} +5.12311i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000i q^{5} +5.12311i q^{7} +1.00000 q^{9} +3.12311i q^{11} +(-0.561553 + 3.56155i) q^{13} -1.00000i q^{15} +2.00000 q^{17} +6.00000i q^{19} -5.12311i q^{21} -3.12311 q^{23} -1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} +5.12311i q^{31} -3.12311i q^{33} -5.12311 q^{35} +3.12311i q^{37} +(0.561553 - 3.56155i) q^{39} +9.12311i q^{41} +10.2462 q^{43} +1.00000i q^{45} -10.2462i q^{47} -19.2462 q^{49} -2.00000 q^{51} +11.3693 q^{53} -3.12311 q^{55} -6.00000i q^{57} -7.12311i q^{59} +10.0000 q^{61} +5.12311i q^{63} +(-3.56155 - 0.561553i) q^{65} -13.1231i q^{67} +3.12311 q^{69} +6.24621i q^{71} -4.87689i q^{73} +1.00000 q^{75} -16.0000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -10.2462i q^{83} +2.00000i q^{85} -2.00000 q^{87} -5.12311i q^{89} +(-18.2462 - 2.87689i) q^{91} -5.12311i q^{93} -6.00000 q^{95} -4.87689i q^{97} +3.12311i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{9} + 6 q^{13} + 8 q^{17} + 4 q^{23} - 4 q^{25} - 4 q^{27} + 8 q^{29} - 4 q^{35} - 6 q^{39} + 8 q^{43} - 44 q^{49} - 8 q^{51} - 4 q^{53} + 4 q^{55} + 40 q^{61} - 6 q^{65} - 4 q^{69} + 4 q^{75} - 64 q^{77} - 32 q^{79} + 4 q^{81} - 8 q^{87} - 40 q^{91} - 24 q^{95}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^9 + 6 * q^13 + 8 * q^17 + 4 * q^23 - 4 * q^25 - 4 * q^27 + 8 * q^29 - 4 * q^35 - 6 * q^39 + 8 * q^43 - 44 * q^49 - 8 * q^51 - 4 * q^53 + 4 * q^55 + 40 * q^61 - 6 * q^65 - 4 * q^69 + 4 * q^75 - 64 * q^77 - 32 * q^79 + 4 * q^81 - 8 * q^87 - 40 * q^91 - 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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