LMFDB - Embedded newform 3680.2.i.b.1471.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3680,2,Mod(1471,3680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("3680.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3680 = 2^{5} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3680.i (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:	$29.3849479438$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.19
Character	$\chi$	$=$	3680.1471
Dual form			3680.2.i.b.1471.30

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.816645i q^{3} +1.00000i q^{5} +3.72066 q^{7} +2.33309 q^{9} +O(q^{10})$	$q-0.816645i q^{3} +1.00000i q^{5} +3.72066 q^{7} +2.33309 q^{9} -4.00419 q^{11} +0.870289 q^{13} +0.816645 q^{15} -1.55701i q^{17} -1.96862 q^{19} -3.03846i q^{21} +(4.66600 - 1.10834i) q^{23} -1.00000 q^{25} -4.35524i q^{27} +4.41251 q^{29} +1.57621i q^{31} +3.27000i q^{33} +3.72066i q^{35} -4.83946i q^{37} -0.710717i q^{39} +8.36388 q^{41} -4.40408 q^{43} +2.33309i q^{45} -6.69635i q^{47} +6.84335 q^{49} -1.27152 q^{51} -6.59474i q^{53} -4.00419i q^{55} +1.60766i q^{57} -2.86863i q^{59} +1.48217i q^{61} +8.68065 q^{63} +0.870289i q^{65} -2.81276 q^{67} +(-0.905121 - 3.81047i) q^{69} +12.6225i q^{71} +8.64489 q^{73} +0.816645i q^{75} -14.8983 q^{77} +1.43445 q^{79} +3.44259 q^{81} +9.55963 q^{83} +1.55701 q^{85} -3.60345i q^{87} -2.13714i q^{89} +3.23805 q^{91} +1.28721 q^{93} -1.96862i q^{95} +7.13091i q^{97} -9.34215 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 8 q^{7} - 48 q^{9}+O(q^{10}) $ 48 * q + 8 * q^7 - 48 * q^9	$ 48 q + 8 q^{7} - 48 q^{9} + 16 q^{11} + 20 q^{23} - 48 q^{25} - 8 q^{29} + 8 q^{41} + 56 q^{49} - 24 q^{51} - 120 q^{63} - 32 q^{67} - 20 q^{69} + 32 q^{77} + 72 q^{81} + 64 q^{83} + 40 q^{91} - 32 q^{93} - 144 q^{99}+O(q^{100}) $ 48 * q + 8 * q^7 - 48 * q^9 + 16 * q^11 + 20 * q^23 - 48 * q^25 - 8 * q^29 + 8 * q^41 + 56 * q^49 - 24 * q^51 - 120 * q^63 - 32 * q^67 - 20 * q^69 + 32 * q^77 + 72 * q^81 + 64 * q^83 + 40 * q^91 - 32 * q^93 - 144 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3680\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1381$	$3041$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$		−	0.816645i		−	0.471490i	−0.971815	−	0.235745i	$-0.924247\pi$
$3$		−	0.816645i		−	0.471490i	0.971815	−	0.235745i	$-0.0757530\pi$
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	3.72066			1.40628			0.703140	−	0.711052i	$-0.251781\pi$
$7$	3.72066			1.40628			0.703140	+	0.711052i	$0.251781\pi$
$8$		0			0
$9$	2.33309			0.777697
$10$		0			0
$11$	−4.00419			−1.20731			−0.603655	−	0.797246i	$-0.706290\pi$
$11$	−4.00419			−1.20731			−0.603655	+	0.797246i	$0.706290\pi$
$12$		0			0
$13$	0.870289			0.241375			0.120687	−	0.992691i	$-0.461490\pi$
$13$	0.870289			0.241375			0.120687	+	0.992691i	$0.461490\pi$
$14$		0			0
$15$	0.816645			0.210857
$16$		0			0
$17$		−	1.55701i		−	0.377630i	−0.982013	−	0.188815i	$-0.939535\pi$
$17$		−	1.55701i		−	0.377630i	0.982013	−	0.188815i	$-0.0604647\pi$
$18$		0			0
$19$	−1.96862			−0.451632			−0.225816	−	0.974170i	$-0.572505\pi$
$19$	−1.96862			−0.451632			−0.225816	+	0.974170i	$0.572505\pi$
$20$		0			0
$21$		−	3.03846i		−	0.663047i
$22$		0			0
$23$	4.66600	−	1.10834i	0.972929	−	0.231105i
$23$	4.66600	−	1.10834i	0.972929	−	0.231105i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		−	4.35524i		−	0.838167i
$28$		0			0
$29$	4.41251			0.819382			0.409691	−	0.912224i	$-0.365637\pi$
$29$	4.41251			0.819382			0.409691	+	0.912224i	$0.365637\pi$
$30$		0			0
$31$			1.57621i			0.283096i	0.989931	+	0.141548i	$0.0452080\pi$
$31$			1.57621i			0.283096i	−0.989931	+	0.141548i	$0.954792\pi$
$32$		0			0
$33$			3.27000i			0.569235i
$34$		0			0
$35$			3.72066i			0.628907i
$36$		0			0
$37$		−	4.83946i		−	0.795602i	−0.917472	−	0.397801i	$-0.869773\pi$
$37$		−	4.83946i		−	0.795602i	0.917472	−	0.397801i	$-0.130227\pi$
$38$		0			0
$39$		−	0.710717i		−	0.113806i
$40$		0			0
$41$	8.36388			1.30622			0.653109	−	0.757264i	$-0.273464\pi$
$41$	8.36388			1.30622			0.653109	+	0.757264i	$0.273464\pi$
$42$		0			0
$43$	−4.40408			−0.671616			−0.335808	−	0.941931i	$-0.609009\pi$
$43$	−4.40408			−0.671616			−0.335808	+	0.941931i	$0.609009\pi$
$44$		0			0
$45$			2.33309i			0.347797i
$46$		0			0
$47$		−	6.69635i		−	0.976763i	−0.872630	−	0.488381i	$-0.837588\pi$
$47$		−	6.69635i		−	0.976763i	0.872630	−	0.488381i	$-0.162412\pi$
$48$		0			0
$49$	6.84335			0.977621
$50$		0			0
$51$	−1.27152			−0.178049
$52$		0			0
$53$		−	6.59474i		−	0.905857i	−0.891547	−	0.452929i	$-0.850379\pi$
$53$		−	6.59474i		−	0.905857i	0.891547	−	0.452929i	$-0.149621\pi$
$54$		0			0
$55$		−	4.00419i		−	0.539926i
$56$		0			0
$57$			1.60766i			0.212940i
$58$		0			0
$59$		−	2.86863i		−	0.373464i	−0.982411	−	0.186732i	$-0.940210\pi$
$59$		−	2.86863i		−	0.373464i	0.982411	−	0.186732i	$-0.0597896\pi$
$60$		0			0
$61$			1.48217i			0.189772i	0.995488	+	0.0948860i	$0.0302487\pi$
$61$			1.48217i			0.189772i	−0.995488	+	0.0948860i	$0.969751\pi$
$62$		0			0
$63$	8.68065			1.09366
$64$		0			0
$65$			0.870289i			0.107946i
$66$		0			0
$67$	−2.81276			−0.343633			−0.171816	−	0.985129i	$-0.554964\pi$
$67$	−2.81276			−0.343633			−0.171816	+	0.985129i	$0.554964\pi$
$68$		0			0
$69$	−0.905121	−	3.81047i	−0.108964	−	0.458726i
$70$		0			0
$71$			12.6225i			1.49802i	0.662561	+	0.749008i	$0.269469\pi$
$71$			12.6225i			1.49802i	−0.662561	+	0.749008i	$0.730531\pi$
$72$		0			0
$73$	8.64489			1.01181			0.505904	−	0.862590i	$-0.331159\pi$
$73$	8.64489			1.01181			0.505904	+	0.862590i	$0.331159\pi$
$74$		0			0
$75$			0.816645i			0.0942980i
$76$		0			0
$77$	−14.8983			−1.69782
$78$		0			0
$79$	1.43445			0.161388			0.0806941	−	0.996739i	$-0.474286\pi$
$79$	1.43445			0.161388			0.0806941	+	0.996739i	$0.474286\pi$
$80$		0			0
$81$	3.44259			0.382510
$82$		0			0
$83$	9.55963			1.04931			0.524653	−	0.851316i	$-0.324195\pi$
$83$	9.55963			1.04931			0.524653	+	0.851316i	$0.324195\pi$
$84$		0			0
$85$	1.55701			0.168881
$86$		0			0
$87$		−	3.60345i		−	0.386330i
$88$		0			0
$89$		−	2.13714i		−	0.226536i	−0.993564	−	0.113268i	$-0.963868\pi$
$89$		−	2.13714i		−	0.226536i	0.993564	−	0.113268i	$-0.0361319\pi$
$90$		0			0
$91$	3.23805			0.339440
$92$		0			0
$93$	1.28721			0.133477
$94$		0			0
$95$		−	1.96862i		−	0.201976i
$96$		0			0
$97$			7.13091i			0.724034i	0.932171	+	0.362017i	$0.117912\pi$
$97$			7.13091i			0.724034i	−0.932171	+	0.362017i	$0.882088\pi$
$98$		0			0
$99$	−9.34215			−0.938922
$100$		0			0
$101$	14.5826			1.45102			0.725509	−	0.688212i	$-0.241604\pi$
$101$	14.5826			1.45102			0.725509	+	0.688212i	$0.241604\pi$
$102$		0			0
$103$	−16.4865			−1.62447			−0.812234	−	0.583332i	$-0.801749\pi$
$103$	−16.4865			−1.62447			−0.812234	+	0.583332i	$0.801749\pi$
$104$		0			0
$105$	3.03846			0.296523
$106$		0			0
$107$	10.4882			1.01393			0.506965	−	0.861967i	$-0.330767\pi$
$107$	10.4882			1.01393			0.506965	+	0.861967i	$0.330767\pi$
$108$		0			0
$109$			20.0584i			1.92124i	0.277862	+	0.960621i	$0.410374\pi$
$109$			20.0584i			1.92124i	−0.277862	+	0.960621i	$0.589626\pi$
$110$		0			0
$111$	−3.95212			−0.375118
$112$		0			0
$113$			9.26841i			0.871899i	0.899971	+	0.435949i	$0.143587\pi$
$113$			9.26841i			0.871899i	−0.899971	+	0.435949i	$0.856413\pi$
$114$		0			0
$115$	1.10834	+	4.66600i	0.103353	+	0.435107i
$116$		0			0
$117$	2.03046			0.187716
$118$		0			0
$119$		−	5.79311i		−	0.531054i
$120$		0			0
$121$	5.03358			0.457598
$122$		0			0
$123$		−	6.83032i		−	0.615869i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	14.3498i		−	1.27334i	−0.771137	−	0.636669i	$-0.780312\pi$
$127$		−	14.3498i		−	1.27334i	0.771137	−	0.636669i	$-0.219688\pi$
$128$		0			0
$129$			3.59657i			0.316660i
$130$		0			0
$131$		−	1.41241i		−	0.123402i	−0.998095	−	0.0617012i	$-0.980347\pi$
$131$		−	1.41241i		−	0.123402i	0.998095	−	0.0617012i	$-0.0196526\pi$
$132$		0			0
$133$	−7.32457			−0.635121
$134$		0			0
$135$	4.35524			0.374839
$136$		0			0
$137$			6.33690i			0.541398i	0.962664	+	0.270699i	$0.0872549\pi$
$137$			6.33690i			0.541398i	−0.962664	+	0.270699i	$0.912745\pi$
$138$		0			0
$139$			6.94072i			0.588704i	0.955697	+	0.294352i	$0.0951038\pi$
$139$			6.94072i			0.588704i	−0.955697	+	0.294352i	$0.904896\pi$
$140$		0			0
$141$	−5.46854			−0.460534
$142$		0			0
$143$	−3.48481			−0.291414
$144$		0			0
$145$			4.41251i			0.366439i
$146$		0			0
$147$		−	5.58858i		−	0.460938i
$148$		0			0
$149$			1.00455i			0.0822962i	0.999153	+	0.0411481i	$0.0131015\pi$
$149$			1.00455i			0.0822962i	−0.999153	+	0.0411481i	$0.986898\pi$
$150$		0			0
$151$		−	22.0894i		−	1.79761i	−0.438353	−	0.898803i	$-0.644438\pi$
$151$		−	22.0894i		−	1.79761i	0.438353	−	0.898803i	$-0.355562\pi$
$152$		0			0
$153$		−	3.63265i		−	0.293682i
$154$		0			0
$155$	−1.57621			−0.126605
$156$		0			0
$157$			9.66319i			0.771206i	0.922665	+	0.385603i	$0.126007\pi$
$157$			9.66319i			0.771206i	−0.922665	+	0.385603i	$0.873993\pi$
$158$		0			0
$159$	−5.38556			−0.427103
$160$		0			0
$161$	17.3606	−	4.12376i	1.36821	−	0.324998i
$162$		0			0
$163$		−	7.10383i		−	0.556415i	−0.960521	−	0.278207i	$-0.910260\pi$
$163$		−	7.10383i		−	0.556415i	0.960521	−	0.278207i	$-0.0897402\pi$
$164$		0			0
$165$	−3.27000			−0.254570
$166$		0			0
$167$		−	13.2830i		−	1.02787i	−0.857828	−	0.513936i	$-0.828187\pi$
$167$		−	13.2830i		−	1.02787i	0.857828	−	0.513936i	$-0.171813\pi$
$168$		0			0
$169$	−12.2426			−0.941738
$170$		0			0
$171$	−4.59297			−0.351233
$172$		0			0
$173$	9.95442			0.756820			0.378410	−	0.925638i	$-0.376471\pi$
$173$	9.95442			0.756820			0.378410	+	0.925638i	$0.376471\pi$
$174$		0			0
$175$	−3.72066			−0.281256
$176$		0			0
$177$	−2.34265			−0.176084
$178$		0			0
$179$		−	14.1419i		−	1.05701i	−0.848930	−	0.528506i	$-0.822752\pi$
$179$		−	14.1419i		−	1.05701i	0.848930	−	0.528506i	$-0.177248\pi$
$180$		0			0
$181$			9.27465i			0.689379i	0.938717	+	0.344690i	$0.112016\pi$
$181$			9.27465i			0.689379i	−0.938717	+	0.344690i	$0.887984\pi$
$182$		0			0
$183$	1.21040			0.0894756
$184$		0			0
$185$	4.83946			0.355804
$186$		0			0
$187$			6.23457i			0.455917i
$188$		0			0
$189$		−	16.2044i		−	1.17870i
$190$		0			0
$191$	7.83976			0.567265			0.283632	−	0.958933i	$-0.408460\pi$
$191$	7.83976			0.567265			0.283632	+	0.958933i	$0.408460\pi$
$192$		0			0
$193$	15.3074			1.10185			0.550926	−	0.834554i	$-0.314275\pi$
$193$	15.3074			1.10185			0.550926	+	0.834554i	$0.314275\pi$
$194$		0			0
$195$	0.710717			0.0508955
$196$		0			0
$197$	5.73062			0.408290			0.204145	−	0.978941i	$-0.434559\pi$
$197$	5.73062			0.408290			0.204145	+	0.978941i	$0.434559\pi$
$198$		0			0
$199$	−14.8743			−1.05441			−0.527206	−	0.849737i	$-0.676760\pi$
$199$	−14.8743			−1.05441			−0.527206	+	0.849737i	$0.676760\pi$
$200$		0			0
$201$			2.29702i			0.162019i
$202$		0			0
$203$	16.4175			1.15228
$204$		0			0
$205$			8.36388i			0.584159i
$206$		0			0
$207$	10.8862	−	2.58586i	0.756644	−	0.179730i
$208$		0			0
$209$	7.88274			0.545260
$210$		0			0
$211$			7.40659i			0.509891i	0.966955	+	0.254945i	$0.0820575\pi$
$211$			7.40659i			0.509891i	−0.966955	+	0.254945i	$0.917942\pi$
$212$		0			0
$213$	10.3081			0.706299
$214$		0			0
$215$		−	4.40408i		−	0.300356i
$216$		0			0
$217$			5.86457i			0.398113i
$218$		0			0
$219$		−	7.05981i		−	0.477058i
$220$		0			0
$221$		−	1.35505i		−	0.0911504i
$222$		0			0
$223$			20.9543i			1.40320i	0.712571	+	0.701600i	$0.247531\pi$
$223$			20.9543i			1.40320i	−0.712571	+	0.701600i	$0.752469\pi$
$224$		0			0
$225$	−2.33309			−0.155539
$226$		0			0
$227$	15.5289			1.03069			0.515346	−	0.856982i	$-0.327664\pi$
$227$	15.5289			1.03069			0.515346	+	0.856982i	$0.327664\pi$
$228$		0			0
$229$		−	23.5821i		−	1.55835i	−0.626808	−	0.779174i	$-0.715639\pi$
$229$		−	23.5821i		−	1.55835i	0.626808	−	0.779174i	$-0.284361\pi$
$230$		0			0
$231$			12.1666i			0.800503i
$232$		0			0
$233$	10.1540			0.665214			0.332607	−	0.943066i	$-0.392072\pi$
$233$	10.1540			0.665214			0.332607	+	0.943066i	$0.392072\pi$
$234$		0			0
$235$	6.69635			0.436822
$236$		0			0
$237$		−	1.17144i		−	0.0760929i
$238$		0			0
$239$		−	23.7576i		−	1.53675i	−0.639998	−	0.768377i	$-0.721065\pi$
$239$		−	23.7576i		−	1.53675i	0.639998	−	0.768377i	$-0.278935\pi$
$240$		0			0
$241$		−	1.92674i		−	0.124112i	−0.998073	−	0.0620562i	$-0.980234\pi$
$241$		−	1.92674i		−	0.124112i	0.998073	−	0.0620562i	$-0.0197658\pi$
$242$		0			0
$243$		−	15.8771i		−	1.01852i
$244$		0			0
$245$			6.84335i			0.437205i
$246$		0			0
$247$	−1.71327			−0.109013
$248$		0			0
$249$		−	7.80682i		−	0.494737i
$250$		0			0
$251$	5.87699			0.370952			0.185476	−	0.982649i	$-0.440617\pi$
$251$	5.87699			0.370952			0.185476	+	0.982649i	$0.440617\pi$
$252$		0			0
$253$	−18.6836	+	4.43801i	−1.17463	+	0.279015i
$254$		0			0
$255$		−	1.27152i		−	0.0796259i
$256$		0			0
$257$	25.4709			1.58883			0.794416	−	0.607374i	$-0.207777\pi$
$257$	25.4709			1.58883			0.794416	+	0.607374i	$0.207777\pi$
$258$		0			0
$259$		−	18.0060i		−	1.11884i
$260$		0			0
$261$	10.2948			0.637231
$262$		0			0
$263$	−4.38302			−0.270269			−0.135134	−	0.990827i	$-0.543147\pi$
$263$	−4.38302			−0.270269			−0.135134	+	0.990827i	$0.543147\pi$
$264$		0			0
$265$	6.59474			0.405112
$266$		0			0
$267$	−1.74528			−0.106809
$268$		0			0
$269$	−19.7680			−1.20528			−0.602638	−	0.798014i	$-0.705884\pi$
$269$	−19.7680			−1.20528			−0.602638	+	0.798014i	$0.705884\pi$
$270$		0			0
$271$		−	3.52990i		−	0.214426i	−0.994236	−	0.107213i	$-0.965807\pi$
$271$		−	3.52990i		−	0.214426i	0.994236	−	0.107213i	$-0.0341927\pi$
$272$		0			0
$273$		−	2.64434i		−	0.160043i
$274$		0			0
$275$	4.00419			0.241462
$276$		0			0
$277$	−27.8012			−1.67041			−0.835206	−	0.549937i	$-0.814652\pi$
$277$	−27.8012			−1.67041			−0.835206	+	0.549937i	$0.814652\pi$
$278$		0			0
$279$			3.67745i			0.220163i
$280$		0			0
$281$		−	9.68603i		−	0.577820i	−0.957356	−	0.288910i	$-0.906707\pi$
$281$		−	9.68603i		−	0.577820i	0.957356	−	0.288910i	$-0.0932929\pi$
$282$		0			0
$283$	7.26714			0.431986			0.215993	−	0.976395i	$-0.430701\pi$
$283$	7.26714			0.431986			0.215993	+	0.976395i	$0.430701\pi$
$284$		0			0
$285$	−1.60766			−0.0952297
$286$		0			0
$287$	31.1192			1.83691
$288$		0			0
$289$	14.5757			0.857395
$290$		0			0
$291$	5.82342			0.341375
$292$		0			0
$293$		−	32.5004i		−	1.89870i	−0.314228	−	0.949348i	$-0.601746\pi$
$293$		−	32.5004i		−	1.89870i	0.314228	−	0.949348i	$-0.398254\pi$
$294$		0			0
$295$	2.86863			0.167018
$296$		0			0
$297$			17.4392i			1.01193i
$298$		0			0
$299$	4.06077	−	0.964577i	0.234840	−	0.0557829i
$300$		0			0
$301$	−16.3861			−0.944479
$302$		0			0
$303$		−	11.9088i		−	0.684141i
$304$		0			0
$305$	−1.48217			−0.0848686
$306$		0			0
$307$			19.6245i			1.12003i	0.828483	+	0.560014i	$0.189204\pi$
$307$			19.6245i			1.12003i	−0.828483	+	0.560014i	$0.810796\pi$
$308$		0			0
$309$			13.4636i			0.765920i
$310$		0			0
$311$			7.92276i			0.449258i	0.974444	+	0.224629i	$0.0721171\pi$
$311$			7.92276i			0.449258i	−0.974444	+	0.224629i	$0.927883\pi$
$312$		0			0
$313$		−	7.97441i		−	0.450740i	−0.974273	−	0.225370i	$-0.927641\pi$
$313$		−	7.97441i		−	0.450740i	0.974273	−	0.225370i	$-0.0723591\pi$
$314$		0			0
$315$			8.68065i			0.489099i
$316$		0			0
$317$	20.6089			1.15751			0.578757	−	0.815500i	$-0.303538\pi$
$317$	20.6089			1.15751			0.578757	+	0.815500i	$0.303538\pi$
$318$		0			0
$319$	−17.6685			−0.989248
$320$		0			0
$321$		−	8.56511i		−	0.478058i
$322$		0			0
$323$			3.06516i			0.170550i
$324$		0			0
$325$	−0.870289			−0.0482749
$326$		0			0
$327$	16.3805			0.905846
$328$		0			0
$329$		−	24.9149i		−	1.37360i
$330$		0			0
$331$		−	21.2956i		−	1.17051i	−0.810848	−	0.585257i	$-0.800994\pi$
$331$		−	21.2956i		−	1.17051i	0.810848	−	0.585257i	$-0.199006\pi$
$332$		0			0
$333$		−	11.2909i		−	0.618737i
$334$		0			0
$335$		−	2.81276i		−	0.153677i
$336$		0			0
$337$			8.53209i			0.464772i	0.972624	+	0.232386i	$0.0746533\pi$
$337$			8.53209i			0.464772i	−0.972624	+	0.232386i	$0.925347\pi$
$338$		0			0
$339$	7.56900			0.411092
$340$		0			0
$341$		−	6.31147i		−	0.341785i
$342$		0			0
$343$	−0.582861			−0.0314715
$344$		0			0
$345$	3.81047	−	0.905121i	0.205149	−	0.0487301i
$346$		0			0
$347$			23.4085i			1.25663i	0.777957	+	0.628317i	$0.216256\pi$
$347$			23.4085i			1.25663i	−0.777957	+	0.628317i	$0.783744\pi$
$348$		0			0
$349$	−5.63919			−0.301859			−0.150930	−	0.988545i	$-0.548227\pi$
$349$	−5.63919			−0.301859			−0.150930	+	0.988545i	$0.548227\pi$
$350$		0			0
$351$		−	3.79032i		−	0.202312i
$352$		0			0
$353$	−24.8361			−1.32189			−0.660945	−	0.750434i	$-0.729844\pi$
$353$	−24.8361			−1.32189			−0.660945	+	0.750434i	$0.729844\pi$
$354$		0			0
$355$	−12.6225			−0.669933
$356$		0			0
$357$	−4.73091			−0.250386
$358$		0			0
$359$	2.47253			0.130495			0.0652476	−	0.997869i	$-0.479216\pi$
$359$	2.47253			0.130495			0.0652476	+	0.997869i	$0.479216\pi$
$360$		0			0
$361$	−15.1245			−0.796028
$362$		0			0
$363$		−	4.11064i		−	0.215753i
$364$		0			0
$365$			8.64489i			0.452494i
$366$		0			0
$367$	−0.423040			−0.0220825			−0.0110413	−	0.999939i	$-0.503515\pi$
$367$	−0.423040			−0.0220825			−0.0110413	+	0.999939i	$0.503515\pi$
$368$		0			0
$369$	19.5137			1.01584
$370$		0			0
$371$		−	24.5368i		−	1.27389i
$372$		0			0
$373$		−	5.87240i		−	0.304062i	−0.988376	−	0.152031i	$-0.951419\pi$
$373$		−	5.87240i		−	0.304062i	0.988376	−	0.152031i	$-0.0485813\pi$
$374$		0			0
$375$	−0.816645			−0.0421714
$376$		0			0
$377$	3.84016			0.197778
$378$		0			0
$379$	−27.3440			−1.40457			−0.702284	−	0.711897i	$-0.747836\pi$
$379$	−27.3440			−1.40457			−0.702284	+	0.711897i	$0.747836\pi$
$380$		0			0
$381$	−11.7187			−0.600367
$382$		0			0
$383$	−18.4603			−0.943275			−0.471637	−	0.881793i	$-0.656337\pi$
$383$	−18.4603			−0.943275			−0.471637	+	0.881793i	$0.656337\pi$
$384$		0			0
$385$		−	14.8983i		−	0.759286i
$386$		0			0
$387$	−10.2751			−0.522314
$388$		0			0
$389$			23.2151i			1.17705i	0.808478	+	0.588526i	$0.200292\pi$
$389$			23.2151i			1.17705i	−0.808478	+	0.588526i	$0.799708\pi$
$390$		0			0
$391$	−1.72570	−	7.26501i	−0.0872722	−	0.367407i
$392$		0			0
$393$	−1.15343			−0.0581830
$394$		0			0
$395$			1.43445i			0.0721750i
$396$		0			0
$397$	16.2512			0.815625			0.407813	−	0.913066i	$-0.366292\pi$
$397$	16.2512			0.815625			0.407813	+	0.913066i	$0.366292\pi$
$398$		0			0
$399$			5.98157i			0.299453i
$400$		0			0
$401$			12.1190i			0.605196i	0.953118	+	0.302598i	$0.0978540\pi$
$401$			12.1190i			0.605196i	−0.953118	+	0.302598i	$0.902146\pi$
$402$		0			0
$403$			1.37176i			0.0683323i
$404$		0			0
$405$			3.44259i			0.171064i
$406$		0			0
$407$			19.3781i			0.960539i
$408$		0			0
$409$	22.1108			1.09331			0.546655	−	0.837358i	$-0.315901\pi$
$409$	22.1108			1.09331			0.546655	+	0.837358i	$0.315901\pi$
$410$		0			0
$411$	5.17500			0.255264
$412$		0			0
$413$		−	10.6732i		−	0.525194i
$414$		0			0
$415$			9.55963i			0.469264i
$416$		0			0
$417$	5.66810			0.277568
$418$		0			0
$419$	−8.35006			−0.407927			−0.203964	−	0.978978i	$-0.565382\pi$
$419$	−8.35006			−0.407927			−0.203964	+	0.978978i	$0.565382\pi$
$420$		0			0
$421$		−	4.09428i		−	0.199543i	−0.995010	−	0.0997715i	$-0.968189\pi$
$421$		−	4.09428i		−	0.199543i	0.995010	−	0.0997715i	$-0.0318112\pi$
$422$		0			0
$423$		−	15.6232i		−	0.759625i
$424$		0			0
$425$			1.55701i			0.0755261i
$426$		0			0
$427$			5.51464i			0.266872i
$428$		0			0
$429$			2.84585i			0.137399i
$430$		0			0
$431$	12.6680			0.610196			0.305098	−	0.952321i	$-0.401311\pi$
$431$	12.6680			0.610196			0.305098	+	0.952321i	$0.401311\pi$
$432$		0			0
$433$			4.70392i			0.226056i	0.993592	+	0.113028i	$0.0360550\pi$
$433$			4.70392i			0.226056i	−0.993592	+	0.113028i	$0.963945\pi$
$434$		0			0
$435$	3.60345			0.172772
$436$		0			0
$437$	−9.18558	+	2.18190i	−0.439406	+	0.104374i
$438$		0			0
$439$			30.3433i			1.44820i	0.689693	+	0.724102i	$0.257746\pi$
$439$			30.3433i			1.44820i	−0.689693	+	0.724102i	$0.742254\pi$
$440$		0			0
$441$	15.9661			0.760293
$442$		0			0
$443$			2.36611i			0.112417i	0.998419	+	0.0562085i	$0.0179012\pi$
$443$			2.36611i			0.112417i	−0.998419	+	0.0562085i	$0.982099\pi$
$444$		0			0
$445$	2.13714			0.101310
$446$		0			0
$447$	0.820363			0.0388018
$448$		0			0
$449$	−1.98108			−0.0934931			−0.0467465	−	0.998907i	$-0.514885\pi$
$449$	−1.98108			−0.0934931			−0.0467465	+	0.998907i	$0.514885\pi$
$450$		0			0
$451$	−33.4906			−1.57701
$452$		0			0
$453$	−18.0392			−0.847553
$454$		0			0
$455$			3.23805i			0.151802i
$456$		0			0
$457$		−	3.93090i		−	0.183880i	−0.995765	−	0.0919399i	$-0.970693\pi$
$457$		−	3.93090i		−	0.183880i	0.995765	−	0.0919399i	$-0.0293068\pi$
$458$		0			0
$459$	−6.78115			−0.316517
$460$		0			0
$461$	−31.3730			−1.46119			−0.730594	−	0.682812i	$-0.760757\pi$
$461$	−31.3730			−1.46119			−0.730594	+	0.682812i	$0.760757\pi$
$462$		0			0
$463$		−	6.68999i		−	0.310910i	−0.987843	−	0.155455i	$-0.950316\pi$
$463$		−	6.68999i		−	0.310910i	0.987843	−	0.155455i	$-0.0496844\pi$
$464$		0			0
$465$			1.28721i			0.0596928i
$466$		0			0
$467$	−4.19291			−0.194025			−0.0970123	−	0.995283i	$-0.530929\pi$
$467$	−4.19291			−0.194025			−0.0970123	+	0.995283i	$0.530929\pi$
$468$		0			0
$469$	−10.4653			−0.483244
$470$		0			0
$471$	7.89139			0.363616
$472$		0			0
$473$	17.6348			0.810848
$474$		0			0
$475$	1.96862			0.0903265
$476$		0			0
$477$		−	15.3861i		−	0.704482i
$478$		0			0
$479$	−26.4221			−1.20726			−0.603628	−	0.797266i	$-0.706279\pi$
$479$	−26.4221			−1.20726			−0.603628	+	0.797266i	$0.706279\pi$
$480$		0			0
$481$		−	4.21173i		−	0.192038i
$482$		0			0
$483$	−3.36765	−	14.1775i	−0.153233	−	0.645097i
$484$		0			0
$485$	−7.13091			−0.323798
$486$		0			0
$487$			33.0658i			1.49835i	0.662370	+	0.749177i	$0.269551\pi$
$487$			33.0658i			1.49835i	−0.662370	+	0.749177i	$0.730449\pi$
$488$		0			0
$489$	−5.80130			−0.262344
$490$		0			0
$491$			19.5196i			0.880908i	0.897775	+	0.440454i	$0.145182\pi$
$491$			19.5196i			0.880908i	−0.897775	+	0.440454i	$0.854818\pi$
$492$		0			0
$493$		−	6.87031i		−	0.309423i
$494$		0			0
$495$		−	9.34215i		−	0.419899i
$496$		0			0
$497$			46.9641i			2.10663i
$498$		0			0
$499$		−	0.995149i		−	0.0445490i	−0.999752	−	0.0222745i	$-0.992909\pi$
$499$		−	0.995149i		−	0.0445490i	0.999752	−	0.0222745i	$-0.00709078\pi$
$500$		0			0
$501$	−10.8475			−0.484632
$502$		0			0
$503$	−25.5047			−1.13720			−0.568600	−	0.822614i	$-0.692515\pi$
$503$	−25.5047			−1.13720			−0.568600	+	0.822614i	$0.692515\pi$
$504$		0			0
$505$			14.5826i			0.648915i
$506$		0			0
$507$			9.99785i			0.444020i
$508$		0			0
$509$	−27.3297			−1.21137			−0.605685	−	0.795705i	$-0.707101\pi$
$509$	−27.3297			−1.21137			−0.605685	+	0.795705i	$0.707101\pi$
$510$		0			0
$511$	32.1647			1.42288
$512$		0			0
$513$			8.57381i			0.378543i
$514$		0			0
$515$		−	16.4865i		−	0.726484i
$516$		0			0
$517$			26.8135i			1.17926i
$518$		0			0
$519$		−	8.12922i		−	0.356833i
$520$		0			0
$521$			34.7431i			1.52212i	0.648680	+	0.761061i	$0.275321\pi$
$521$			34.7431i			1.52212i	−0.648680	+	0.761061i	$0.724679\pi$
$522$		0			0
$523$	−6.32944			−0.276767			−0.138384	−	0.990379i	$-0.544191\pi$
$523$	−6.32944			−0.276767			−0.138384	+	0.990379i	$0.544191\pi$
$524$		0			0
$525$			3.03846i			0.132609i
$526$		0			0
$527$	2.45418			0.106906
$528$		0			0
$529$	20.5432	−	10.3430i	0.893181	−	0.449697i
$530$		0			0
$531$		−	6.69277i		−	0.290442i
$532$		0			0
$533$	7.27899			0.315288
$534$		0			0
$535$			10.4882i			0.453443i
$536$		0			0
$537$	−11.5489			−0.498371
$538$		0			0
$539$	−27.4021			−1.18029
$540$		0			0
$541$	−26.9145			−1.15715			−0.578573	−	0.815631i	$-0.696390\pi$
$541$	−26.9145			−1.15715			−0.578573	+	0.815631i	$0.696390\pi$
$542$		0			0
$543$	7.57410			0.325036
$544$		0			0
$545$	−20.0584			−0.859205
$546$		0			0
$547$		−	6.25556i		−	0.267468i	−0.991017	−	0.133734i	$-0.957303\pi$
$547$		−	6.25556i		−	0.267468i	0.991017	−	0.133734i	$-0.0426968\pi$
$548$		0			0
$549$			3.45803i			0.147585i
$550$		0			0
$551$	−8.68655			−0.370059
$552$		0			0
$553$	5.33711			0.226957
$554$		0			0
$555$		−	3.95212i		−	0.167758i
$556$		0			0
$557$			36.5060i			1.54681i	0.633913	+	0.773404i	$0.281448\pi$
$557$			36.5060i			1.54681i	−0.633913	+	0.773404i	$0.718552\pi$
$558$		0			0
$559$	−3.83282			−0.162111
$560$		0			0
$561$	5.09143			0.214960
$562$		0			0
$563$	−23.0717			−0.972357			−0.486178	−	0.873860i	$-0.661609\pi$
$563$	−23.0717			−0.972357			−0.486178	+	0.873860i	$0.661609\pi$
$564$		0			0
$565$	−9.26841			−0.389925
$566$		0			0
$567$	12.8087			0.537916
$568$		0			0
$569$		−	46.4285i		−	1.94638i	−0.229997	−	0.973191i	$-0.573872\pi$
$569$		−	46.4285i		−	1.94638i	0.229997	−	0.973191i	$-0.426128\pi$
$570$		0			0
$571$	2.01960			0.0845176			0.0422588	−	0.999107i	$-0.486545\pi$
$571$	2.01960			0.0845176			0.0422588	+	0.999107i	$0.486545\pi$
$572$		0			0
$573$		−	6.40230i		−	0.267460i
$574$		0			0
$575$	−4.66600	+	1.10834i	−0.194586	+	0.0462210i
$576$		0			0
$577$	−7.82559			−0.325783			−0.162892	−	0.986644i	$-0.552082\pi$
$577$	−7.82559			−0.325783			−0.162892	+	0.986644i	$0.552082\pi$
$578$		0			0
$579$		−	12.5007i		−	0.519512i
$580$		0			0
$581$	35.5682			1.47562
$582$		0			0
$583$			26.4066i			1.09365i
$584$		0			0
$585$			2.03046i			0.0839493i
$586$		0			0
$587$			29.5470i			1.21953i	0.792580	+	0.609767i	$0.208737\pi$
$587$			29.5470i			1.21953i	−0.792580	+	0.609767i	$0.791263\pi$
$588$		0			0
$589$		−	3.10297i		−	0.127856i
$590$		0			0
$591$		−	4.67988i		−	0.192505i
$592$		0			0
$593$	1.07744			0.0442450			0.0221225	−	0.999755i	$-0.492958\pi$
$593$	1.07744			0.0442450			0.0221225	+	0.999755i	$0.492958\pi$
$594$		0			0
$595$	5.79311			0.237494
$596$		0			0
$597$			12.1470i			0.497145i
$598$		0			0
$599$			6.34975i			0.259444i	0.991550	+	0.129722i	$0.0414085\pi$
$599$			6.34975i			0.259444i	−0.991550	+	0.129722i	$0.958592\pi$
$600$		0			0
$601$	−14.0712			−0.573977			−0.286988	−	0.957934i	$-0.592654\pi$
$601$	−14.0712			−0.573977			−0.286988	+	0.957934i	$0.592654\pi$
$602$		0			0
$603$	−6.56241			−0.267242
$604$		0			0
$605$			5.03358i			0.204644i
$606$		0			0
$607$		−	4.60650i		−	0.186972i	−0.995621	−	0.0934861i	$-0.970199\pi$
$607$		−	4.60650i		−	0.186972i	0.995621	−	0.0934861i	$-0.0298011\pi$
$608$		0			0
$609$		−	13.4072i		−	0.543288i
$610$		0			0
$611$		−	5.82776i		−	0.235766i
$612$		0			0
$613$		−	20.8122i		−	0.840599i	−0.907385	−	0.420299i	$-0.861925\pi$
$613$		−	20.8122i		−	0.840599i	0.907385	−	0.420299i	$-0.138075\pi$
$614$		0			0
$615$	6.83032			0.275425
$616$		0			0
$617$			7.53011i			0.303151i	0.988446	+	0.151575i	$0.0484346\pi$
$617$			7.53011i			0.303151i	−0.988446	+	0.151575i	$0.951565\pi$
$618$		0			0
$619$	−17.6009			−0.707441			−0.353720	−	0.935351i	$-0.615084\pi$
$619$	−17.6009			−0.707441			−0.353720	+	0.935351i	$0.615084\pi$
$620$		0			0
$621$	−4.82709	−	20.3216i	−0.193704	−	0.815476i
$622$		0			0
$623$		−	7.95157i		−	0.318573i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	6.43739i		−	0.257085i
$628$		0			0
$629$	−7.53508			−0.300443
$630$		0			0
$631$	−36.2121			−1.44158			−0.720791	−	0.693152i	$-0.756221\pi$
$631$	−36.2121			−1.44158			−0.720791	+	0.693152i	$0.756221\pi$
$632$		0			0
$633$	6.04855			0.240408
$634$		0			0
$635$	14.3498			0.569454
$636$		0			0
$637$	5.95569			0.235973
$638$		0			0
$639$			29.4494i			1.16500i
$640$		0			0
$641$			34.9082i			1.37879i	0.724385	+	0.689396i	$0.242124\pi$
$641$			34.9082i			1.37879i	−0.724385	+	0.689396i	$0.757876\pi$
$642$		0			0
$643$	10.9741			0.432777			0.216388	−	0.976307i	$-0.430572\pi$
$643$	10.9741			0.432777			0.216388	+	0.976307i	$0.430572\pi$
$644$		0			0
$645$	−3.59657			−0.141615
$646$		0			0
$647$			29.4490i			1.15776i	0.815413	+	0.578880i	$0.196510\pi$
$647$			29.4490i			1.15776i	−0.815413	+	0.578880i	$0.803490\pi$
$648$		0			0
$649$			11.4866i			0.450887i
$650$		0			0
$651$	4.78927			0.187706
$652$		0			0
$653$	37.1927			1.45546			0.727731	−	0.685863i	$-0.240575\pi$
$653$	37.1927			1.45546			0.727731	+	0.685863i	$0.240575\pi$
$654$		0			0
$655$	1.41241			0.0551873
$656$		0			0
$657$	20.1693			0.786880
$658$		0			0
$659$	23.0904			0.899473			0.449737	−	0.893161i	$-0.351518\pi$
$659$	23.0904			0.899473			0.449737	+	0.893161i	$0.351518\pi$
$660$		0			0
$661$		−	19.0663i		−	0.741594i	−0.928714	−	0.370797i	$-0.879085\pi$
$661$		−	19.0663i		−	0.741594i	0.928714	−	0.370797i	$-0.120915\pi$
$662$		0			0
$663$	−1.10659			−0.0429765
$664$		0			0
$665$		−	7.32457i		−	0.284035i
$666$		0			0
$667$	20.5888	−	4.89056i	0.797200	−	0.189363i
$668$		0			0
$669$	17.1122			0.661595
$670$		0			0
$671$		−	5.93488i		−	0.229114i
$672$		0			0
$673$	−7.35958			−0.283691			−0.141845	−	0.989889i	$-0.545304\pi$
$673$	−7.35958			−0.283691			−0.141845	+	0.989889i	$0.545304\pi$
$674$		0			0
$675$			4.35524i			0.167633i
$676$		0			0
$677$			42.4770i			1.63252i	0.577683	+	0.816261i	$0.303957\pi$
$677$			42.4770i			1.63252i	−0.577683	+	0.816261i	$0.696043\pi$
$678$		0			0
$679$			26.5317i			1.01819i
$680$		0			0
$681$		−	12.6816i		−	0.485961i
$682$		0			0
$683$			37.0603i			1.41807i	0.705171	+	0.709037i	$0.250870\pi$
$683$			37.0603i			1.41807i	−0.705171	+	0.709037i	$0.749130\pi$
$684$		0			0
$685$	−6.33690			−0.242121
$686$		0			0
$687$	−19.2582			−0.734746
$688$		0			0
$689$		−	5.73933i		−	0.218651i
$690$		0			0
$691$			25.9119i			0.985735i	0.870105	+	0.492867i	$0.164051\pi$
$691$			25.9119i			0.985735i	−0.870105	+	0.492867i	$0.835949\pi$
$692$		0			0
$693$	−34.7590			−1.32039
$694$		0			0
$695$	−6.94072			−0.263276
$696$		0			0
$697$		−	13.0226i		−	0.493268i
$698$		0			0
$699$		−	8.29225i		−	0.313642i
$700$		0			0
$701$		−	31.0916i		−	1.17431i	−0.809473	−	0.587157i	$-0.800247\pi$
$701$		−	31.0916i		−	1.17431i	0.809473	−	0.587157i	$-0.199753\pi$
$702$		0			0
$703$			9.52705i			0.359320i
$704$		0			0
$705$		−	5.46854i		−	0.205957i
$706$		0			0
$707$	54.2568			2.04054
$708$		0			0
$709$		−	3.40001i		−	0.127690i	−0.997960	−	0.0638450i	$-0.979664\pi$
$709$		−	3.40001i		−	0.127690i	0.997960	−	0.0638450i	$-0.0203363\pi$
$710$		0			0
$711$	3.34670			0.125511
$712$		0			0
$713$	1.74698	+	7.35462i	0.0654250	+	0.275433i
$714$		0			0
$715$		−	3.48481i		−	0.130324i
$716$		0			0
$717$	−19.4015			−0.724564
$718$		0			0
$719$		−	31.4604i		−	1.17327i	−0.809850	−	0.586637i	$-0.800452\pi$
$719$		−	31.4604i		−	1.17327i	0.809850	−	0.586637i	$-0.199548\pi$
$720$		0			0
$721$	−61.3409			−2.28445
$722$		0			0
$723$	−1.57346			−0.0585178
$724$		0			0
$725$	−4.41251			−0.163876
$726$		0			0
$727$	38.8447			1.44067			0.720335	−	0.693627i	$-0.243988\pi$
$727$	38.8447			1.44067			0.720335	+	0.693627i	$0.243988\pi$
$728$		0			0
$729$	−2.63818			−0.0977103
$730$		0			0
$731$			6.85719i			0.253622i
$732$		0			0
$733$		−	40.5006i		−	1.49593i	−0.663741	−	0.747963i	$-0.731032\pi$
$733$		−	40.5006i		−	1.49593i	0.663741	−	0.747963i	$-0.268968\pi$
$734$		0			0
$735$	5.58858			0.206138
$736$		0			0
$737$	11.2628			0.414871
$738$		0			0
$739$			19.3099i			0.710327i	0.934804	+	0.355164i	$0.115575\pi$
$739$			19.3099i			0.710327i	−0.934804	+	0.355164i	$0.884425\pi$
$740$		0			0
$741$			1.39913i			0.0513984i
$742$		0			0
$743$	−16.6978			−0.612584			−0.306292	−	0.951938i	$-0.599088\pi$
$743$	−16.6978			−0.612584			−0.306292	+	0.951938i	$0.599088\pi$
$744$		0			0
$745$	−1.00455			−0.0368040
$746$		0			0
$747$	22.3035			0.816042
$748$		0			0
$749$	39.0230			1.42587
$750$		0			0
$751$	51.9859			1.89699			0.948497	−	0.316786i	$-0.102604\pi$
$751$	51.9859			1.89699			0.948497	+	0.316786i	$0.102604\pi$
$752$		0			0
$753$		−	4.79941i		−	0.174900i
$754$		0			0
$755$	22.0894			0.803914
$756$		0			0
$757$		−	1.43233i		−	0.0520590i	−0.999661	−	0.0260295i	$-0.991714\pi$
$757$		−	1.43233i		−	0.0520590i	0.999661	−	0.0260295i	$-0.00828638\pi$
$758$		0			0
$759$	3.62428	+	15.2579i	0.131553	+	0.553825i
$760$		0			0
$761$	−20.6626			−0.749020			−0.374510	−	0.927223i	$-0.622189\pi$
$761$	−20.6626			−0.749020			−0.374510	+	0.927223i	$0.622189\pi$
$762$		0			0
$763$			74.6304i			2.70180i
$764$		0			0
$765$	3.63265			0.131339
$766$		0			0
$767$		−	2.49654i		−	0.0901447i
$768$		0			0
$769$		−	33.4907i		−	1.20771i	−0.797095	−	0.603853i	$-0.793631\pi$
$769$		−	33.4907i		−	1.20771i	0.797095	−	0.603853i	$-0.206369\pi$
$770$		0			0
$771$		−	20.8007i		−	0.749118i
$772$		0			0
$773$		−	4.64785i		−	0.167172i	−0.996501	−	0.0835858i	$-0.973363\pi$
$773$		−	4.64785i		−	0.167172i	0.996501	−	0.0835858i	$-0.0266373\pi$
$774$		0			0
$775$		−	1.57621i		−	0.0566193i
$776$		0			0
$777$	−14.7045			−0.527521
$778$		0			0
$779$	−16.4653			−0.589930
$780$		0			0
$781$		−	50.5430i		−	1.80857i
$782$		0			0
$783$		−	19.2175i		−	0.686779i
$784$		0			0
$785$	−9.66319			−0.344894
$786$		0			0
$787$	32.6081			1.16235			0.581176	−	0.813778i	$-0.302593\pi$
$787$	32.6081			1.16235			0.581176	+	0.813778i	$0.302593\pi$
$788$		0			0
$789$			3.57937i			0.127429i
$790$		0			0
$791$			34.4846i			1.22613i
$792$		0			0
$793$			1.28991i			0.0458062i
$794$		0			0
$795$		−	5.38556i		−	0.191006i
$796$		0			0
$797$		−	24.2971i		−	0.860649i	−0.902674	−	0.430325i	$-0.858399\pi$
$797$		−	24.2971i		−	0.860649i	0.902674	−	0.430325i	$-0.141601\pi$
$798$		0			0
$799$	−10.4263			−0.368855
$800$		0			0
$801$		−	4.98613i		−	0.176176i
$802$		0			0
$803$	−34.6158			−1.22157
$804$		0			0
$805$	4.12376	+	17.3606i	0.145344	+	0.611882i
$806$		0			0
$807$			16.1434i			0.568276i
$808$		0			0
$809$	−13.0971			−0.460469			−0.230234	−	0.973135i	$-0.573949\pi$
$809$	−13.0971			−0.460469			−0.230234	+	0.973135i	$0.573949\pi$
$810$		0			0
$811$			37.4122i			1.31372i	0.754013	+	0.656860i	$0.228116\pi$
$811$			37.4122i			1.31372i	−0.754013	+	0.656860i	$0.771884\pi$
$812$		0			0
$813$	−2.88268			−0.101100
$814$		0			0
$815$	7.10383			0.248836
$816$		0			0
$817$	8.66995			0.303323
$818$		0			0
$819$	7.55467			0.263982
$820$		0			0
$821$	20.7596			0.724515			0.362258	−	0.932078i	$-0.382006\pi$
$821$	20.7596			0.724515			0.362258	+	0.932078i	$0.382006\pi$
$822$		0			0
$823$			32.6051i			1.13654i	0.822842	+	0.568271i	$0.192387\pi$
$823$			32.6051i			1.13654i	−0.822842	+	0.568271i	$0.807613\pi$
$824$		0			0
$825$		−	3.27000i		−	0.113847i
$826$		0			0
$827$	−24.4652			−0.850737			−0.425369	−	0.905020i	$-0.639856\pi$
$827$	−24.4652			−0.850737			−0.425369	+	0.905020i	$0.639856\pi$
$828$		0			0
$829$	−27.0651			−0.940011			−0.470006	−	0.882663i	$-0.655748\pi$
$829$	−27.0651			−0.940011			−0.470006	+	0.882663i	$0.655748\pi$
$830$		0			0
$831$			22.7037i			0.787583i
$832$		0			0
$833$		−	10.6552i		−	0.369179i
$834$		0			0
$835$	13.2830			0.459679
$836$		0			0
$837$	6.86479			0.237282
$838$		0			0
$839$	1.34426			0.0464089			0.0232044	−	0.999731i	$-0.492613\pi$
$839$	1.34426			0.0464089			0.0232044	+	0.999731i	$0.492613\pi$
$840$		0			0
$841$	−9.52978			−0.328613
$842$		0			0
$843$	−7.91005			−0.272437
$844$		0			0
$845$		−	12.2426i		−	0.421158i
$846$		0			0
$847$	18.7283			0.643510
$848$		0			0
$849$		−	5.93467i		−	0.203677i
$850$		0			0
$851$	−5.36377	−	22.5809i	−0.183868	−	0.774064i
$852$		0			0
$853$	15.0218			0.514338			0.257169	−	0.966366i	$-0.417210\pi$
$853$	15.0218			0.514338			0.257169	+	0.966366i	$0.417210\pi$
$854$		0			0
$855$		−	4.59297i		−	0.157076i
$856$		0			0
$857$	−55.8387			−1.90741			−0.953707	−	0.300739i	$-0.902767\pi$
$857$	−55.8387			−1.90741			−0.953707	+	0.300739i	$0.902767\pi$
$858$		0			0
$859$			21.1457i			0.721481i	0.932666	+	0.360740i	$0.117476\pi$
$859$			21.1457i			0.721481i	−0.932666	+	0.360740i	$0.882524\pi$
$860$		0			0
$861$		−	25.4133i		−	0.866084i
$862$		0			0
$863$		−	48.4704i		−	1.64995i	−0.565168	−	0.824976i	$-0.691189\pi$
$863$		−	48.4704i		−	1.64995i	0.565168	−	0.824976i	$-0.308811\pi$
$864$		0			0
$865$			9.95442i			0.338460i
$866$		0			0
$867$		−	11.9032i		−	0.404253i
$868$		0			0
$869$	−5.74382			−0.194846
$870$		0			0
$871$	−2.44791			−0.0829442
$872$		0			0
$873$			16.6371i			0.563079i
$874$		0			0
$875$		−	3.72066i		−	0.125781i
$876$		0			0
$877$	−51.6792			−1.74508			−0.872541	−	0.488541i	$-0.837529\pi$
$877$	−51.6792			−1.74508			−0.872541	+	0.488541i	$0.837529\pi$
$878$		0			0
$879$	−26.5413			−0.895216
$880$		0			0
$881$			22.2452i			0.749461i	0.927134	+	0.374731i	$0.122265\pi$
$881$			22.2452i			0.749461i	−0.927134	+	0.374731i	$0.877735\pi$
$882$		0			0
$883$			11.2185i			0.377531i	0.982022	+	0.188765i	$0.0604486\pi$
$883$			11.2185i			0.377531i	−0.982022	+	0.188765i	$0.939551\pi$
$884$		0			0
$885$		−	2.34265i		−	0.0787473i
$886$		0			0
$887$		−	1.11865i		−	0.0375605i	−0.999824	−	0.0187803i	$-0.994022\pi$
$887$		−	1.11865i		−	0.0375605i	0.999824	−	0.0187803i	$-0.00597829\pi$
$888$		0			0
$889$		−	53.3908i		−	1.79067i
$890$		0			0
$891$	−13.7848			−0.461808
$892$		0			0
$893$			13.1826i			0.441138i
$894$		0			0
$895$	14.1419			0.472710
$896$		0			0
$897$	−0.787716	−	3.31621i	−0.0263011	−	0.110725i
$898$		0			0
$899$			6.95506i			0.231964i
$900$		0			0
$901$	−10.2681			−0.342079
$902$		0			0
$903$			13.3816i			0.445312i
$904$		0			0
$905$	−9.27465			−0.308300
$906$		0			0
$907$	−10.6074			−0.352212			−0.176106	−	0.984371i	$-0.556350\pi$
$907$	−10.6074			−0.352212			−0.176106	+	0.984371i	$0.556350\pi$
$908$		0			0
$909$	34.0224			1.12845
$910$		0			0
$911$	−55.2320			−1.82992			−0.914959	−	0.403547i	$-0.867777\pi$
$911$	−55.2320			−1.82992			−0.914959	+	0.403547i	$0.867777\pi$
$912$		0			0
$913$	−38.2786			−1.26684
$914$		0			0
$915$			1.21040i			0.0400147i
$916$		0			0
$917$		−	5.25509i		−	0.173538i
$918$		0			0
$919$	8.69503			0.286823			0.143411	−	0.989663i	$-0.454193\pi$
$919$	8.69503			0.286823			0.143411	+	0.989663i	$0.454193\pi$
$920$		0			0
$921$	16.0262			0.528082
$922$		0			0
$923$			10.9852i			0.361583i
$924$		0			0
$925$			4.83946i			0.159120i
$926$		0			0
$927$	−38.4646			−1.26334
$928$		0			0
$929$	−27.5239			−0.903030			−0.451515	−	0.892264i	$-0.649116\pi$
$929$	−27.5239			−0.903030			−0.451515	+	0.892264i	$0.649116\pi$
$930$		0			0
$931$	−13.4719			−0.441525
$932$		0			0
$933$	6.47008			0.211821
$934$		0			0
$935$	−6.23457			−0.203892
$936$		0			0
$937$			51.1033i			1.66947i	0.550650	+	0.834736i	$0.314380\pi$
$937$			51.1033i			1.66947i	−0.550650	+	0.834736i	$0.685620\pi$
$938$		0			0
$939$	−6.51226			−0.212519
$940$		0			0
$941$			24.0219i			0.783092i	0.920159	+	0.391546i	$0.128060\pi$
$941$			24.0219i			0.783092i	−0.920159	+	0.391546i	$0.871940\pi$
$942$		0			0
$943$	39.0259	−	9.27003i	1.27086	−	0.301874i
$944$		0			0
$945$	16.2044			0.527129
$946$		0			0
$947$		−	36.2712i		−	1.17866i	−0.807894	−	0.589328i	$-0.799393\pi$
$947$		−	36.2712i		−	1.17866i	0.807894	−	0.589328i	$-0.200607\pi$
$948$		0			0
$949$	7.52355			0.244225
$950$		0			0
$951$		−	16.8302i		−	0.545756i
$952$		0			0
$953$			13.0675i			0.423299i	0.977346	+	0.211649i	$0.0678835\pi$
$953$			13.0675i			0.423299i	−0.977346	+	0.211649i	$0.932117\pi$
$954$		0			0
$955$			7.83976i			0.253689i
$956$		0			0
$957$			14.4289i			0.466421i
$958$		0			0
$959$			23.5775i			0.761357i
$960$		0			0
$961$	28.5155			0.919856
$962$		0			0
$963$	24.4699			0.788530
$964$		0			0
$965$			15.3074i			0.492763i
$966$		0			0
$967$			11.9769i			0.385150i	0.981282	+	0.192575i	$0.0616838\pi$
$967$			11.9769i			0.385150i	−0.981282	+	0.192575i	$0.938316\pi$
$968$		0			0
$969$	2.50315			0.0804126
$970$		0			0
$971$	20.6005			0.661103			0.330552	−	0.943788i	$-0.392765\pi$
$971$	20.6005			0.661103			0.330552	+	0.943788i	$0.392765\pi$
$972$		0			0
$973$			25.8241i			0.827882i
$974$		0			0
$975$			0.710717i			0.0227612i
$976$		0			0
$977$		−	11.3727i		−	0.363844i	−0.983313	−	0.181922i	$-0.941768\pi$
$977$		−	11.3727i		−	0.363844i	0.983313	−	0.181922i	$-0.0582319\pi$
$978$		0			0
$979$			8.55751i			0.273499i
$980$		0			0
$981$			46.7980i			1.49414i
$982$		0			0
$983$	−27.8828			−0.889323			−0.444662	−	0.895699i	$-0.646676\pi$
$983$	−27.8828			−0.889323			−0.444662	+	0.895699i	$0.646676\pi$
$984$		0			0
$985$			5.73062i			0.182593i
$986$		0			0
$987$	−20.3466			−0.647639
$988$		0			0
$989$	−20.5494	+	4.88122i	−0.653434	+	0.155214i
$990$		0			0
$991$		−	33.3336i		−	1.05888i	−0.848349	−	0.529438i	$-0.822403\pi$
$991$		−	33.3336i		−	1.05888i	0.848349	−	0.529438i	$-0.177597\pi$
$992$		0			0
$993$	−17.3910			−0.551885
$994$		0			0
$995$		−	14.8743i		−	0.471547i
$996$		0			0
$997$	−10.6117			−0.336075			−0.168038	−	0.985781i	$-0.553743\pi$
$997$	−10.6117			−0.336075			−0.168038	+	0.985781i	$0.553743\pi$
$998$		0			0
$999$	−21.0770			−0.666847

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3680.2.i.b.1471.19	yes	48
4.3	odd	2		3680.2.i.a.1471.30	yes	48
23.22	odd	2		3680.2.i.a.1471.19	✓	48
92.91	even	2	inner	3680.2.i.b.1471.30	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.2.i.a.1471.19	✓	48	23.22	odd	2
3680.2.i.a.1471.30	yes	48	4.3	odd	2
3680.2.i.b.1471.19	yes	48	1.1	even	1	trivial
3680.2.i.b.1471.30	yes	48	92.91	even	2	inner

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:	\( N \)	\(=\)	\( 3680 = 2^{5} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3680.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.816645i q^{3} +1.00000i q^{5} +3.72066 q^{7} +2.33309 q^{9} +O(q^{10})\)	\(q-0.816645i q^{3} +1.00000i q^{5} +3.72066 q^{7} +2.33309 q^{9} -4.00419 q^{11} +0.870289 q^{13} +0.816645 q^{15} -1.55701i q^{17} -1.96862 q^{19} -3.03846i q^{21} +(4.66600 - 1.10834i) q^{23} -1.00000 q^{25} -4.35524i q^{27} +4.41251 q^{29} +1.57621i q^{31} +3.27000i q^{33} +3.72066i q^{35} -4.83946i q^{37} -0.710717i q^{39} +8.36388 q^{41} -4.40408 q^{43} +2.33309i q^{45} -6.69635i q^{47} +6.84335 q^{49} -1.27152 q^{51} -6.59474i q^{53} -4.00419i q^{55} +1.60766i q^{57} -2.86863i q^{59} +1.48217i q^{61} +8.68065 q^{63} +0.870289i q^{65} -2.81276 q^{67} +(-0.905121 - 3.81047i) q^{69} +12.6225i q^{71} +8.64489 q^{73} +0.816645i q^{75} -14.8983 q^{77} +1.43445 q^{79} +3.44259 q^{81} +9.55963 q^{83} +1.55701 q^{85} -3.60345i q^{87} -2.13714i q^{89} +3.23805 q^{91} +1.28721 q^{93} -1.96862i q^{95} +7.13091i q^{97} -9.34215 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 8 q^{7} - 48 q^{9}+O(q^{10}) \) 48 * q + 8 * q^7 - 48 * q^9	\( 48 q + 8 q^{7} - 48 q^{9} + 16 q^{11} + 20 q^{23} - 48 q^{25} - 8 q^{29} + 8 q^{41} + 56 q^{49} - 24 q^{51} - 120 q^{63} - 32 q^{67} - 20 q^{69} + 32 q^{77} + 72 q^{81} + 64 q^{83} + 40 q^{91} - 32 q^{93} - 144 q^{99}+O(q^{100}) \) 48 * q + 8 * q^7 - 48 * q^9 + 16 * q^11 + 20 * q^23 - 48 * q^25 - 8 * q^29 + 8 * q^41 + 56 * q^49 - 24 * q^51 - 120 * q^63 - 32 * q^67 - 20 * q^69 + 32 * q^77 + 72 * q^81 + 64 * q^83 + 40 * q^91 - 32 * q^93 - 144 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more