LMFDB - Embedded newform 3920.2.k.a.2351.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3920,2,Mod(2351,3920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3920, 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("3920.2351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3920 = 2^{4} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3920.k (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:	$31.3013575923$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2351.5
Root			$-0.923880 + 0.382683i$ of defining polynomial
Character	$\chi$	$=$	3920.2351
Dual form			3920.2.k.a.2351.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.43355 q^{3} -1.00000i q^{5} -0.944947 q^{9} +O(q^{10})$	$q-1.43355 q^{3} -1.00000i q^{5} -0.944947 q^{9} -6.10973i q^{11} -1.43355i q^{13} +1.43355i q^{15} -5.93015i q^{17} +1.55166 q^{19} -6.70353i q^{23} -1.00000 q^{25} +5.65526 q^{27} -8.46889 q^{29} +5.68297 q^{31} +8.75858i q^{33} -5.65980 q^{37} +2.05505i q^{39} -2.55007i q^{41} +10.6678i q^{43} +0.944947i q^{45} +3.86415 q^{47} +8.50114i q^{51} +6.81363 q^{53} -6.10973 q^{55} -2.22437 q^{57} +8.42222 q^{59} +4.57446i q^{61} -1.43355 q^{65} +1.33341i q^{67} +9.60981i q^{69} -13.2654i q^{71} -2.29316i q^{73} +1.43355 q^{75} -1.52395i q^{79} -5.27223 q^{81} -17.5172 q^{83} -5.93015 q^{85} +12.1405 q^{87} +4.07144i q^{89} -8.14680 q^{93} -1.55166i q^{95} +2.23238i q^{97} +5.77337i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 16 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 + 16 * q^9	$ 8 q - 8 q^{3} + 16 q^{9} + 16 q^{19} - 8 q^{25} - 8 q^{27} + 8 q^{29} + 48 q^{31} + 24 q^{47} + 32 q^{53} - 8 q^{55} - 32 q^{57} + 16 q^{59} - 8 q^{65} + 8 q^{75} - 8 q^{81} - 48 q^{83} - 24 q^{85} - 24 q^{87} - 32 q^{93}+O(q^{100}) $ 8 * q - 8 * q^3 + 16 * q^9 + 16 * q^19 - 8 * q^25 - 8 * q^27 + 8 * q^29 + 48 * q^31 + 24 * q^47 + 32 * q^53 - 8 * q^55 - 32 * q^57 + 16 * q^59 - 8 * q^65 + 8 * q^75 - 8 * q^81 - 48 * q^83 - 24 * q^85 - 24 * q^87 - 32 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3920\mathbb{Z}\right)^\times$.

$n$	$981$	$1471$	$3041$	$3137$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	−1.43355	−0.827658	−0.413829	−	0.910355i	$-0.635809\pi$
$3$	−1.43355	−0.827658	−0.413829	+	0.910355i	$0.635809\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−0.944947	−0.314982
$10$	0	0
$11$	− 6.10973i	− 1.84215i	−0.389381	−	0.921077i	$-0.627311\pi$
$11$	− 6.10973i	− 1.84215i	0.389381	−	0.921077i	$-0.372689\pi$
$12$	0	0
$13$	− 1.43355i	− 0.397594i	−0.980041	−	0.198797i	$-0.936297\pi$
$13$	− 1.43355i	− 0.397594i	0.980041	−	0.198797i	$-0.0637034\pi$
$14$	0	0
$15$	1.43355i	0.370140i
$16$	0	0
$17$	− 5.93015i	− 1.43827i	−0.694869	−	0.719136i	$-0.744538\pi$
$17$	− 5.93015i	− 1.43827i	0.694869	−	0.719136i	$-0.255462\pi$
$18$	0	0
$19$	1.55166	0.355975	0.177987	−	0.984033i	$-0.443041\pi$
$19$	1.55166	0.355975	0.177987	+	0.984033i	$0.443041\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 6.70353i	− 1.39778i	−0.715228	−	0.698891i	$-0.753677\pi$
$23$	− 6.70353i	− 1.39778i	0.715228	−	0.698891i	$-0.246323\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	5.65526	1.08836
$28$	0	0
$29$	−8.46889	−1.57263	−0.786317	−	0.617823i	$-0.788015\pi$
$29$	−8.46889	−1.57263	−0.786317	+	0.617823i	$0.788015\pi$
$30$	0	0
$31$	5.68297	1.02069	0.510346	−	0.859969i	$-0.329517\pi$
$31$	5.68297	1.02069	0.510346	+	0.859969i	$0.329517\pi$
$32$	0	0
$33$	8.75858i	1.52467i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.65980	−0.930465	−0.465232	−	0.885189i	$-0.654029\pi$
$37$	−5.65980	−0.930465	−0.465232	+	0.885189i	$0.654029\pi$
$38$	0	0
$39$	2.05505i	0.329072i
$40$	0	0
$41$	− 2.55007i	− 0.398253i	−0.979974	−	0.199127i	$-0.936189\pi$
$41$	− 2.55007i	− 0.398253i	0.979974	−	0.199127i	$-0.0638105\pi$
$42$	0	0
$43$	10.6678i	1.62682i	0.581687	+	0.813412i	$0.302393\pi$
$43$	10.6678i	1.62682i	−0.581687	+	0.813412i	$0.697607\pi$
$44$	0	0
$45$	0.944947i	0.140864i
$46$	0	0
$47$	3.86415	0.563644	0.281822	−	0.959467i	$-0.409061\pi$
$47$	3.86415	0.563644	0.281822	+	0.959467i	$0.409061\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	8.50114i	1.19040i
$52$	0	0
$53$	6.81363	0.935924	0.467962	−	0.883749i	$-0.344988\pi$
$53$	6.81363	0.935924	0.467962	+	0.883749i	$0.344988\pi$
$54$	0	0
$55$	−6.10973	−0.823836
$56$	0	0
$57$	−2.22437	−0.294625
$58$	0	0
$59$	8.42222	1.09648	0.548240	−	0.836321i	$-0.315298\pi$
$59$	8.42222	1.09648	0.548240	+	0.836321i	$0.315298\pi$
$60$	0	0
$61$	4.57446i	0.585700i	0.956158	+	0.292850i	$0.0946036\pi$
$61$	4.57446i	0.585700i	−0.956158	+	0.292850i	$0.905396\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.43355	−0.177809
$66$	0	0
$67$	1.33341i	0.162902i	0.996677	+	0.0814512i	$0.0259555\pi$
$67$	1.33341i	0.162902i	−0.996677	+	0.0814512i	$0.974045\pi$
$68$	0	0
$69$	9.60981i	1.15689i
$70$	0	0
$71$	− 13.2654i	− 1.57432i	−0.616750	−	0.787159i	$-0.711551\pi$
$71$	− 13.2654i	− 1.57432i	0.616750	−	0.787159i	$-0.288449\pi$
$72$	0	0
$73$	− 2.29316i	− 0.268394i	−0.990955	−	0.134197i	$-0.957155\pi$
$73$	− 2.29316i	− 0.268394i	0.990955	−	0.134197i	$-0.0428455\pi$
$74$	0	0
$75$	1.43355	0.165532
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 1.52395i	− 0.171457i	−0.996319	−	0.0857286i	$-0.972678\pi$
$79$	− 1.52395i	− 0.171457i	0.996319	−	0.0857286i	$-0.0273218\pi$
$80$	0	0
$81$	−5.27223	−0.585804
$82$	0	0
$83$	−17.5172	−1.92276	−0.961379	−	0.275228i	$-0.911247\pi$
$83$	−17.5172	−1.92276	−0.961379	+	0.275228i	$0.911247\pi$
$84$	0	0
$85$	−5.93015	−0.643215
$86$	0	0
$87$	12.1405	1.30160
$88$	0	0
$89$	4.07144i	0.431572i	0.976441	+	0.215786i	$0.0692313\pi$
$89$	4.07144i	0.431572i	−0.976441	+	0.215786i	$0.930769\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−8.14680	−0.844784
$94$	0	0
$95$	− 1.55166i	− 0.159197i
$96$	0	0
$97$	2.23238i	0.226664i	0.993557	+	0.113332i	$0.0361524\pi$
$97$	2.23238i	0.226664i	−0.993557	+	0.113332i	$0.963848\pi$
$98$	0	0
$99$	5.77337i	0.580246i
$100$	0	0
$101$	− 17.3314i	− 1.72454i	−0.506446	−	0.862272i	$-0.669041\pi$
$101$	− 17.3314i	− 1.72454i	0.506446	−	0.862272i	$-0.330959\pi$
$102$	0	0
$103$	4.70390	0.463489	0.231744	−	0.972777i	$-0.425557\pi$
$103$	4.70390	0.463489	0.231744	+	0.972777i	$0.425557\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.50981i	0.339306i	0.985504	+	0.169653i	$0.0542647\pi$
$107$	3.50981i	0.339306i	−0.985504	+	0.169653i	$0.945735\pi$
$108$	0	0
$109$	−9.28772	−0.889602	−0.444801	−	0.895629i	$-0.646726\pi$
$109$	−9.28772	−0.889602	−0.444801	+	0.895629i	$0.646726\pi$
$110$	0	0
$111$	8.11358	0.770107
$112$	0	0
$113$	−11.3376	−1.06655	−0.533275	−	0.845942i	$-0.679039\pi$
$113$	−11.3376	−1.06655	−0.533275	+	0.845942i	$0.679039\pi$
$114$	0	0
$115$	−6.70353	−0.625107
$116$	0	0
$117$	1.35462i	0.125235i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−26.3288	−2.39353
$122$	0	0
$123$	3.65564i	0.329618i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	15.2024i	1.34899i	0.738278	+	0.674497i	$0.235639\pi$
$127$	15.2024i	1.34899i	−0.738278	+	0.674497i	$0.764361\pi$
$128$	0	0
$129$	− 15.2928i	− 1.34645i
$130$	0	0
$131$	12.2707	1.07209	0.536047	−	0.844188i	$-0.319917\pi$
$131$	12.2707	1.07209	0.536047	+	0.844188i	$0.319917\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 5.65526i	− 0.486727i
$136$	0	0
$137$	8.63821	0.738012	0.369006	−	0.929427i	$-0.379698\pi$
$137$	8.63821	0.738012	0.369006	+	0.929427i	$0.379698\pi$
$138$	0	0
$139$	10.0463	0.852116	0.426058	−	0.904696i	$-0.359902\pi$
$139$	10.0463	0.852116	0.426058	+	0.904696i	$0.359902\pi$
$140$	0	0
$141$	−5.53943	−0.466504
$142$	0	0
$143$	−8.75858	−0.732429
$144$	0	0
$145$	8.46889i	0.703303i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13.2327	−1.08406	−0.542031	−	0.840358i	$-0.682345\pi$
$149$	−13.2327	−1.08406	−0.542031	+	0.840358i	$0.682345\pi$
$150$	0	0
$151$	− 4.00997i	− 0.326327i	−0.986599	−	0.163164i	$-0.947830\pi$
$151$	− 4.00997i	− 0.326327i	0.986599	−	0.163164i	$-0.0521698\pi$
$152$	0	0
$153$	5.60368i	0.453031i
$154$	0	0
$155$	− 5.68297i	− 0.456468i
$156$	0	0
$157$	− 2.34315i	− 0.187003i	−0.995619	−	0.0935017i	$-0.970194\pi$
$157$	− 2.34315i	− 0.187003i	0.995619	−	0.0935017i	$-0.0298061\pi$
$158$	0	0
$159$	−9.76765	−0.774625
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 2.77669i	− 0.217487i	−0.994070	−	0.108744i	$-0.965317\pi$
$163$	− 2.77669i	− 0.217487i	0.994070	−	0.108744i	$-0.0346828\pi$
$164$	0	0
$165$	8.75858	0.681854
$166$	0	0
$167$	10.4518	0.808788	0.404394	−	0.914585i	$-0.367482\pi$
$167$	10.4518	0.808788	0.404394	+	0.914585i	$0.367482\pi$
$168$	0	0
$169$	10.9449	0.841919
$170$	0	0
$171$	−1.46624	−0.112126
$172$	0	0
$173$	12.7563i	0.969845i	0.874557	+	0.484923i	$0.161152\pi$
$173$	12.7563i	0.969845i	−0.874557	+	0.484923i	$0.838848\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−12.0736	−0.907510
$178$	0	0
$179$	− 12.5143i	− 0.935366i	−0.883896	−	0.467683i	$-0.845089\pi$
$179$	− 12.5143i	− 0.935366i	0.883896	−	0.467683i	$-0.154911\pi$
$180$	0	0
$181$	17.3809i	1.29191i	0.763374	+	0.645957i	$0.223542\pi$
$181$	17.3809i	1.29191i	−0.763374	+	0.645957i	$0.776458\pi$
$182$	0	0
$183$	− 6.55770i	− 0.484759i
$184$	0	0
$185$	5.65980i	0.416117i
$186$	0	0
$187$	−36.2316	−2.64952
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 6.88029i	− 0.497841i	−0.968524	−	0.248920i	$-0.919924\pi$
$191$	− 6.88029i	− 0.497841i	0.968524	−	0.248920i	$-0.0800757\pi$
$192$	0	0
$193$	7.08067	0.509678	0.254839	−	0.966984i	$-0.417978\pi$
$193$	7.08067	0.509678	0.254839	+	0.966984i	$0.417978\pi$
$194$	0	0
$195$	2.05505	0.147165
$196$	0	0
$197$	−12.2980	−0.876197	−0.438098	−	0.898927i	$-0.644348\pi$
$197$	−12.2980	−0.876197	−0.438098	+	0.898927i	$0.644348\pi$
$198$	0	0
$199$	−20.6971	−1.46718	−0.733590	−	0.679593i	$-0.762157\pi$
$199$	−20.6971	−1.46718	−0.733590	+	0.679593i	$0.762157\pi$
$200$	0	0
$201$	− 1.91151i	− 0.134828i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−2.55007	−0.178104
$206$	0	0
$207$	6.33448i	0.440277i
$208$	0	0
$209$	− 9.48022i	− 0.655760i
$210$	0	0
$211$	− 16.3273i	− 1.12402i	−0.827132	−	0.562008i	$-0.810029\pi$
$211$	− 16.3273i	− 1.12402i	0.827132	−	0.562008i	$-0.189971\pi$
$212$	0	0
$213$	19.0166i	1.30300i
$214$	0	0
$215$	10.6678	0.727538
$216$	0	0
$217$	0	0
$218$	0	0
$219$	3.28735i	0.222138i
$220$	0	0
$221$	−8.50114	−0.571849
$222$	0	0
$223$	16.7563	1.12209	0.561043	−	0.827786i	$-0.310400\pi$
$223$	16.7563	1.12209	0.561043	+	0.827786i	$0.310400\pi$
$224$	0	0
$225$	0.944947	0.0629965
$226$	0	0
$227$	−16.0698	−1.06659	−0.533297	−	0.845928i	$-0.679047\pi$
$227$	−16.0698	−1.06659	−0.533297	+	0.845928i	$0.679047\pi$
$228$	0	0
$229$	17.7032i	1.16986i	0.811085	+	0.584929i	$0.198877\pi$
$229$	17.7032i	1.16986i	−0.811085	+	0.584929i	$0.801123\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.01774	−0.394235	−0.197118	−	0.980380i	$-0.563158\pi$
$233$	−6.01774	−0.394235	−0.197118	+	0.980380i	$0.563158\pi$
$234$	0	0
$235$	− 3.86415i	− 0.252069i
$236$	0	0
$237$	2.18464i	0.141908i
$238$	0	0
$239$	2.38913i	0.154540i	0.997010	+	0.0772699i	$0.0246203\pi$
$239$	2.38913i	0.154540i	−0.997010	+	0.0772699i	$0.975380\pi$
$240$	0	0
$241$	− 21.9453i	− 1.41362i	−0.707404	−	0.706809i	$-0.750134\pi$
$241$	− 21.9453i	− 1.41362i	0.707404	−	0.706809i	$-0.249866\pi$
$242$	0	0
$243$	−9.40780	−0.603511
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2.22437i	− 0.141533i
$248$	0	0
$249$	25.1116	1.59139
$250$	0	0
$251$	−30.0665	−1.89778	−0.948890	−	0.315608i	$-0.897791\pi$
$251$	−30.0665	−1.89778	−0.948890	+	0.315608i	$0.897791\pi$
$252$	0	0
$253$	−40.9567	−2.57493
$254$	0	0
$255$	8.50114	0.532362
$256$	0	0
$257$	11.4921i	0.716856i	0.933557	+	0.358428i	$0.116687\pi$
$257$	11.4921i	0.716856i	−0.933557	+	0.358428i	$0.883313\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	8.00266	0.495352
$262$	0	0
$263$	4.90819i	0.302652i	0.988484	+	0.151326i	$0.0483543\pi$
$263$	4.90819i	0.302652i	−0.988484	+	0.151326i	$0.951646\pi$
$264$	0	0
$265$	− 6.81363i	− 0.418558i
$266$	0	0
$267$	− 5.83660i	− 0.357194i
$268$	0	0
$269$	− 9.63246i	− 0.587301i	−0.955913	−	0.293651i	$-0.905130\pi$
$269$	− 9.63246i	− 0.587301i	0.955913	−	0.293651i	$-0.0948702\pi$
$270$	0	0
$271$	19.9118	1.20956	0.604778	−	0.796394i	$-0.293262\pi$
$271$	19.9118	1.20956	0.604778	+	0.796394i	$0.293262\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.10973i	0.368431i
$276$	0	0
$277$	−27.0668	−1.62629	−0.813145	−	0.582062i	$-0.802246\pi$
$277$	−27.0668	−1.62629	−0.813145	+	0.582062i	$0.802246\pi$
$278$	0	0
$279$	−5.37011	−0.321500
$280$	0	0
$281$	17.1229	1.02147	0.510734	−	0.859739i	$-0.329374\pi$
$281$	17.1229	1.02147	0.510734	+	0.859739i	$0.329374\pi$
$282$	0	0
$283$	17.5407	1.04268	0.521342	−	0.853348i	$-0.325432\pi$
$283$	17.5407	1.04268	0.521342	+	0.853348i	$0.325432\pi$
$284$	0	0
$285$	2.22437i	0.131760i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−18.1667	−1.06863
$290$	0	0
$291$	− 3.20022i	− 0.187600i
$292$	0	0
$293$	16.0585i	0.938149i	0.883159	+	0.469075i	$0.155412\pi$
$293$	16.0585i	0.938149i	−0.883159	+	0.469075i	$0.844588\pi$
$294$	0	0
$295$	− 8.42222i	− 0.490361i
$296$	0	0
$297$	− 34.5521i	− 2.00492i
$298$	0	0
$299$	−9.60981	−0.555750
$300$	0	0
$301$	0	0
$302$	0	0
$303$	24.8454i	1.42733i
$304$	0	0
$305$	4.57446	0.261933
$306$	0	0
$307$	−11.5233	−0.657667	−0.328833	−	0.944388i	$-0.606655\pi$
$307$	−11.5233	−0.657667	−0.328833	+	0.944388i	$0.606655\pi$
$308$	0	0
$309$	−6.74325	−0.383610
$310$	0	0
$311$	−34.5424	−1.95872	−0.979360	−	0.202124i	$-0.935215\pi$
$311$	−34.5424	−1.95872	−0.979360	+	0.202124i	$0.935215\pi$
$312$	0	0
$313$	25.8498i	1.46112i	0.682849	+	0.730559i	$0.260741\pi$
$313$	25.8498i	1.46112i	−0.682849	+	0.730559i	$0.739259\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−26.2860	−1.47637	−0.738184	−	0.674599i	$-0.764317\pi$
$317$	−26.2860	−1.47637	−0.738184	+	0.674599i	$0.764317\pi$
$318$	0	0
$319$	51.7427i	2.89703i
$320$	0	0
$321$	− 5.03147i	− 0.280829i
$322$	0	0
$323$	− 9.20157i	− 0.511989i
$324$	0	0
$325$	1.43355i	0.0795188i
$326$	0	0
$327$	13.3144	0.736286
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 8.73831i	− 0.480301i	−0.970736	−	0.240151i	$-0.922803\pi$
$331$	− 8.73831i	− 0.480301i	0.970736	−	0.240151i	$-0.0771968\pi$
$332$	0	0
$333$	5.34821	0.293080
$334$	0	0
$335$	1.33341	0.0728522
$336$	0	0
$337$	25.4013	1.38370	0.691849	−	0.722042i	$-0.256796\pi$
$337$	25.4013	1.38370	0.691849	+	0.722042i	$0.256796\pi$
$338$	0	0
$339$	16.2529	0.882738
$340$	0	0
$341$	− 34.7215i	− 1.88027i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	9.60981	0.517375
$346$	0	0
$347$	34.5798i	1.85634i	0.372152	+	0.928172i	$0.378620\pi$
$347$	34.5798i	1.85634i	−0.372152	+	0.928172i	$0.621380\pi$
$348$	0	0
$349$	− 7.91949i	− 0.423920i	−0.977278	−	0.211960i	$-0.932015\pi$
$349$	− 7.91949i	− 0.423920i	0.977278	−	0.211960i	$-0.0679847\pi$
$350$	0	0
$351$	− 8.10707i	− 0.432724i
$352$	0	0
$353$	21.0522i	1.12049i	0.828325	+	0.560247i	$0.189294\pi$
$353$	21.0522i	1.12049i	−0.828325	+	0.560247i	$0.810706\pi$
$354$	0	0
$355$	−13.2654	−0.704057
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.24317i	0.435058i	0.976054	+	0.217529i	$0.0697996\pi$
$359$	8.24317i	0.435058i	−0.976054	+	0.217529i	$0.930200\pi$
$360$	0	0
$361$	−16.5924	−0.873282
$362$	0	0
$363$	37.7436	1.98102
$364$	0	0
$365$	−2.29316	−0.120029
$366$	0	0
$367$	8.70315	0.454301	0.227150	−	0.973860i	$-0.427059\pi$
$367$	8.70315	0.454301	0.227150	+	0.973860i	$0.427059\pi$
$368$	0	0
$369$	2.40968i	0.125443i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	14.9528	0.774228	0.387114	−	0.922032i	$-0.373472\pi$
$373$	14.9528	0.774228	0.387114	+	0.922032i	$0.373472\pi$
$374$	0	0
$375$	− 1.43355i	− 0.0740280i
$376$	0	0
$377$	12.1405i	0.625270i
$378$	0	0
$379$	9.11292i	0.468099i	0.972225	+	0.234050i	$0.0751978\pi$
$379$	9.11292i	0.468099i	−0.972225	+	0.234050i	$0.924802\pi$
$380$	0	0
$381$	− 21.7933i	− 1.11651i
$382$	0	0
$383$	−22.2446	−1.13664	−0.568322	−	0.822806i	$-0.692407\pi$
$383$	−22.2446	−1.13664	−0.568322	+	0.822806i	$0.692407\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 10.0805i	− 0.512421i
$388$	0	0
$389$	−11.1357	−0.564603	−0.282302	−	0.959326i	$-0.591098\pi$
$389$	−11.1357	−0.564603	−0.282302	+	0.959326i	$0.591098\pi$
$390$	0	0
$391$	−39.7529	−2.01039
$392$	0	0
$393$	−17.5906	−0.887327
$394$	0	0
$395$	−1.52395	−0.0766780
$396$	0	0
$397$	− 28.2011i	− 1.41537i	−0.706527	−	0.707687i	$-0.749739\pi$
$397$	− 28.2011i	− 1.41537i	0.706527	−	0.707687i	$-0.250261\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.94214	0.146923	0.0734616	−	0.997298i	$-0.476595\pi$
$401$	2.94214	0.146923	0.0734616	+	0.997298i	$0.476595\pi$
$402$	0	0
$403$	− 8.14680i	− 0.405821i
$404$	0	0
$405$	5.27223i	0.261979i
$406$	0	0
$407$	34.5798i	1.71406i
$408$	0	0
$409$	15.8007i	0.781295i	0.920540	+	0.390648i	$0.127749\pi$
$409$	15.8007i	0.781295i	−0.920540	+	0.390648i	$0.872251\pi$
$410$	0	0
$411$	−12.3833	−0.610822
$412$	0	0
$413$	0	0
$414$	0	0
$415$	17.5172i	0.859883i
$416$	0	0
$417$	−14.4018	−0.705260
$418$	0	0
$419$	−3.38618	−0.165426	−0.0827129	−	0.996573i	$-0.526358\pi$
$419$	−3.38618	−0.165426	−0.0827129	+	0.996573i	$0.526358\pi$
$420$	0	0
$421$	0.524848	0.0255795	0.0127898	−	0.999918i	$-0.495929\pi$
$421$	0.524848	0.0255795	0.0127898	+	0.999918i	$0.495929\pi$
$422$	0	0
$423$	−3.65142	−0.177538
$424$	0	0
$425$	5.93015i	0.287655i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	12.5558	0.606201
$430$	0	0
$431$	− 14.5499i	− 0.700846i	−0.936592	−	0.350423i	$-0.886038\pi$
$431$	− 14.5499i	− 0.700846i	0.936592	−	0.350423i	$-0.113962\pi$
$432$	0	0
$433$	− 40.0603i	− 1.92518i	−0.270971	−	0.962588i	$-0.587345\pi$
$433$	− 40.0603i	− 1.92518i	0.270971	−	0.962588i	$-0.412655\pi$
$434$	0	0
$435$	− 12.1405i	− 0.582094i
$436$	0	0
$437$	− 10.4016i	− 0.497575i
$438$	0	0
$439$	17.3127	0.826289	0.413144	−	0.910666i	$-0.364430\pi$
$439$	17.3127	0.826289	0.413144	+	0.910666i	$0.364430\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	17.0023i	0.807803i	0.914803	+	0.403901i	$0.132346\pi$
$443$	17.0023i	0.807803i	−0.914803	+	0.403901i	$0.867654\pi$
$444$	0	0
$445$	4.07144	0.193005
$446$	0	0
$447$	18.9696	0.897233
$448$	0	0
$449$	−9.08104	−0.428561	−0.214280	−	0.976772i	$-0.568741\pi$
$449$	−9.08104	−0.428561	−0.214280	+	0.976772i	$0.568741\pi$
$450$	0	0
$451$	−15.5802	−0.733644
$452$	0	0
$453$	5.74848i	0.270087i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.8055	1.25391	0.626955	−	0.779055i	$-0.284301\pi$
$457$	26.8055	1.25391	0.626955	+	0.779055i	$0.284301\pi$
$458$	0	0
$459$	− 33.5366i	− 1.56535i
$460$	0	0
$461$	2.65498i	0.123655i	0.998087	+	0.0618273i	$0.0196928\pi$
$461$	2.65498i	0.123655i	−0.998087	+	0.0618273i	$0.980307\pi$
$462$	0	0
$463$	− 19.4753i	− 0.905094i	−0.891740	−	0.452547i	$-0.850515\pi$
$463$	− 19.4753i	− 0.905094i	0.891740	−	0.452547i	$-0.149485\pi$
$464$	0	0
$465$	8.14680i	0.377799i
$466$	0	0
$467$	11.2584	0.520975	0.260488	−	0.965477i	$-0.416117\pi$
$467$	11.2584	0.520975	0.260488	+	0.965477i	$0.416117\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	3.35901i	0.154775i
$472$	0	0
$473$	65.1774	2.99686
$474$	0	0
$475$	−1.55166	−0.0711950
$476$	0	0
$477$	−6.43852	−0.294800
$478$	0	0
$479$	8.73087	0.398923	0.199462	−	0.979906i	$-0.436081\pi$
$479$	8.73087	0.398923	0.199462	+	0.979906i	$0.436081\pi$
$480$	0	0
$481$	8.11358i	0.369947i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.23238	0.101367
$486$	0	0
$487$	− 21.7371i	− 0.985003i	−0.870312	−	0.492502i	$-0.836083\pi$
$487$	− 21.7371i	− 0.985003i	0.870312	−	0.492502i	$-0.163917\pi$
$488$	0	0
$489$	3.98051i	0.180005i
$490$	0	0
$491$	9.17709i	0.414156i	0.978324	+	0.207078i	$0.0663954\pi$
$491$	9.17709i	0.414156i	−0.978324	+	0.207078i	$0.933605\pi$
$492$	0	0
$493$	50.2218i	2.26188i
$494$	0	0
$495$	5.77337	0.259494
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 30.2538i	− 1.35435i	−0.735824	−	0.677173i	$-0.763205\pi$
$499$	− 30.2538i	− 1.35435i	0.735824	−	0.677173i	$-0.236795\pi$
$500$	0	0
$501$	−14.9832	−0.669400
$502$	0	0
$503$	16.9566	0.756057	0.378028	−	0.925794i	$-0.376602\pi$
$503$	16.9566	0.756057	0.378028	+	0.925794i	$0.376602\pi$
$504$	0	0
$505$	−17.3314	−0.771239
$506$	0	0
$507$	−15.6901	−0.696821
$508$	0	0
$509$	− 30.5187i	− 1.35272i	−0.736571	−	0.676360i	$-0.763556\pi$
$509$	− 30.5187i	− 1.35272i	0.736571	−	0.676360i	$-0.236444\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	8.77503	0.387427
$514$	0	0
$515$	− 4.70390i	− 0.207279i
$516$	0	0
$517$	− 23.6089i	− 1.03832i
$518$	0	0
$519$	− 18.2868i	− 0.802700i
$520$	0	0
$521$	− 32.6165i	− 1.42896i	−0.699658	−	0.714478i	$-0.746664\pi$
$521$	− 32.6165i	− 1.42896i	0.699658	−	0.714478i	$-0.253336\pi$
$522$	0	0
$523$	31.1604	1.36255	0.681275	−	0.732027i	$-0.261426\pi$
$523$	31.1604	1.36255	0.681275	+	0.732027i	$0.261426\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 33.7009i	− 1.46803i
$528$	0	0
$529$	−21.9373	−0.953794
$530$	0	0
$531$	−7.95856	−0.345372
$532$	0	0
$533$	−3.65564	−0.158343
$534$	0	0
$535$	3.50981	0.151742
$536$	0	0
$537$	17.9399i	0.774163i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−37.5128	−1.61280	−0.806401	−	0.591369i	$-0.798588\pi$
$541$	−37.5128	−1.61280	−0.806401	+	0.591369i	$0.798588\pi$
$542$	0	0
$543$	− 24.9164i	− 1.06926i
$544$	0	0
$545$	9.28772i	0.397842i
$546$	0	0
$547$	18.3601i	0.785023i	0.919747	+	0.392511i	$0.128394\pi$
$547$	18.3601i	0.785023i	−0.919747	+	0.392511i	$0.871606\pi$
$548$	0	0
$549$	− 4.32263i	− 0.184485i
$550$	0	0
$551$	−13.1408	−0.559818
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 8.11358i	− 0.344402i
$556$	0	0
$557$	6.02052	0.255098	0.127549	−	0.991832i	$-0.459289\pi$
$557$	6.02052	0.255098	0.127549	+	0.991832i	$0.459289\pi$
$558$	0	0
$559$	15.2928	0.646816
$560$	0	0
$561$	51.9397	2.19290
$562$	0	0
$563$	31.6256	1.33286	0.666430	−	0.745568i	$-0.267822\pi$
$563$	31.6256	1.33286	0.666430	+	0.745568i	$0.267822\pi$
$564$	0	0
$565$	11.3376i	0.476975i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	22.4680	0.941907	0.470954	−	0.882158i	$-0.343910\pi$
$569$	22.4680	0.941907	0.470954	+	0.882158i	$0.343910\pi$
$570$	0	0
$571$	− 9.07695i	− 0.379859i	−0.981798	−	0.189929i	$-0.939174\pi$
$571$	− 9.07695i	− 0.379859i	0.981798	−	0.189929i	$-0.0608259\pi$
$572$	0	0
$573$	9.86321i	0.412042i
$574$	0	0
$575$	6.70353i	0.279556i
$576$	0	0
$577$	33.8255i	1.40817i	0.710114	+	0.704087i	$0.248643\pi$
$577$	33.8255i	1.40817i	−0.710114	+	0.704087i	$0.751357\pi$
$578$	0	0
$579$	−10.1505	−0.421839
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 41.6295i	− 1.72412i
$584$	0	0
$585$	1.35462	0.0560068
$586$	0	0
$587$	13.5049	0.557407	0.278703	−	0.960377i	$-0.410095\pi$
$587$	13.5049	0.557407	0.278703	+	0.960377i	$0.410095\pi$
$588$	0	0
$589$	8.81804	0.363341
$590$	0	0
$591$	17.6298	0.725191
$592$	0	0
$593$	22.5999i	0.928065i	0.885818	+	0.464033i	$0.153598\pi$
$593$	22.5999i	0.928065i	−0.885818	+	0.464033i	$0.846402\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	29.6703	1.21432
$598$	0	0
$599$	0.0369091i	0.00150806i	1.00000		0.000754032i	$0.000240016\pi$
$599$	0.0369091i	0.00150806i	−1.00000		0.000754032i	$0.999760\pi$
$600$	0	0
$601$	− 11.0051i	− 0.448906i	−0.974485	−	0.224453i	$-0.927940\pi$
$601$	− 11.0051i	− 0.448906i	0.974485	−	0.224453i	$-0.0720595\pi$
$602$	0	0
$603$	− 1.26001i	− 0.0513114i
$604$	0	0
$605$	26.3288i	1.07042i
$606$	0	0
$607$	−23.1170	−0.938290	−0.469145	−	0.883121i	$-0.655438\pi$
$607$	−23.1170	−0.938290	−0.469145	+	0.883121i	$0.655438\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 5.53943i	− 0.224101i
$612$	0	0
$613$	−5.77701	−0.233331	−0.116666	−	0.993171i	$-0.537221\pi$
$613$	−5.77701	−0.233331	−0.116666	+	0.993171i	$0.537221\pi$
$614$	0	0
$615$	3.65564	0.147409
$616$	0	0
$617$	46.2586	1.86230	0.931151	−	0.364635i	$-0.118806\pi$
$617$	46.2586	1.86230	0.931151	+	0.364635i	$0.118806\pi$
$618$	0	0
$619$	−30.5330	−1.22722	−0.613612	−	0.789608i	$-0.710284\pi$
$619$	−30.5330	−1.22722	−0.613612	+	0.789608i	$0.710284\pi$
$620$	0	0
$621$	− 37.9102i	− 1.52128i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	13.5903i	0.542745i
$628$	0	0
$629$	33.5635i	1.33826i
$630$	0	0
$631$	20.4290i	0.813264i	0.913592	+	0.406632i	$0.133297\pi$
$631$	20.4290i	0.813264i	−0.913592	+	0.406632i	$0.866703\pi$
$632$	0	0
$633$	23.4059i	0.930301i
$634$	0	0
$635$	15.2024	0.603288
$636$	0	0
$637$	0	0
$638$	0	0
$639$	12.5351i	0.495883i
$640$	0	0
$641$	1.22572	0.0484132	0.0242066	−	0.999707i	$-0.492294\pi$
$641$	1.22572	0.0484132	0.0242066	+	0.999707i	$0.492294\pi$
$642$	0	0
$643$	−41.6409	−1.64216	−0.821079	−	0.570814i	$-0.806628\pi$
$643$	−41.6409	−1.64216	−0.821079	+	0.570814i	$0.806628\pi$
$644$	0	0
$645$	−15.2928	−0.602153
$646$	0	0
$647$	16.6082	0.652936	0.326468	−	0.945208i	$-0.394141\pi$
$647$	16.6082	0.652936	0.326468	+	0.945208i	$0.394141\pi$
$648$	0	0
$649$	− 51.4575i	− 2.01988i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	36.2261	1.41764	0.708818	−	0.705392i	$-0.249229\pi$
$653$	36.2261	1.41764	0.708818	+	0.705392i	$0.249229\pi$
$654$	0	0
$655$	− 12.2707i	− 0.479455i
$656$	0	0
$657$	2.16691i	0.0845393i
$658$	0	0
$659$	48.2074i	1.87789i	0.344061	+	0.938947i	$0.388197\pi$
$659$	48.2074i	1.87789i	−0.344061	+	0.938947i	$0.611803\pi$
$660$	0	0
$661$	− 12.1379i	− 0.472109i	−0.971740	−	0.236055i	$-0.924146\pi$
$661$	− 12.1379i	− 0.472109i	0.971740	−	0.236055i	$-0.0758544\pi$
$662$	0	0
$663$	12.1868	0.473295
$664$	0	0
$665$	0	0
$666$	0	0
$667$	56.7714i	2.19820i
$668$	0	0
$669$	−24.0210	−0.928704
$670$	0	0
$671$	27.9487	1.07895
$672$	0	0
$673$	25.3353	0.976603	0.488301	−	0.872675i	$-0.337617\pi$
$673$	25.3353	0.976603	0.488301	+	0.872675i	$0.337617\pi$
$674$	0	0
$675$	−5.65526	−0.217671
$676$	0	0
$677$	− 49.0329i	− 1.88449i	−0.334928	−	0.942244i	$-0.608712\pi$
$677$	− 49.0329i	− 1.88449i	0.334928	−	0.942244i	$-0.391288\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	23.0369	0.882774
$682$	0	0
$683$	26.2222i	1.00337i	0.865051	+	0.501683i	$0.167286\pi$
$683$	26.2222i	1.00337i	−0.865051	+	0.501683i	$0.832714\pi$
$684$	0	0
$685$	− 8.63821i	− 0.330049i
$686$	0	0
$687$	− 25.3783i	− 0.968242i
$688$	0	0
$689$	− 9.76765i	− 0.372118i
$690$	0	0
$691$	35.1427	1.33689	0.668446	−	0.743761i	$-0.266960\pi$
$691$	35.1427	1.33689	0.668446	+	0.743761i	$0.266960\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 10.0463i	− 0.381078i
$696$	0	0
$697$	−15.1223	−0.572797
$698$	0	0
$699$	8.62670	0.326292
$700$	0	0
$701$	36.2937	1.37079	0.685397	−	0.728169i	$-0.259629\pi$
$701$	36.2937	1.37079	0.685397	+	0.728169i	$0.259629\pi$
$702$	0	0
$703$	−8.78207	−0.331222
$704$	0	0
$705$	5.53943i	0.208627i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	12.1940	0.457955	0.228978	−	0.973432i	$-0.426462\pi$
$709$	12.1940	0.457955	0.228978	+	0.973432i	$0.426462\pi$
$710$	0	0
$711$	1.44005i	0.0540060i
$712$	0	0
$713$	− 38.0960i	− 1.42671i
$714$	0	0
$715$	8.75858i	0.327552i
$716$	0	0
$717$	− 3.42492i	− 0.127906i
$718$	0	0
$719$	−18.0857	−0.674483	−0.337242	−	0.941418i	$-0.609494\pi$
$719$	−18.0857	−0.674483	−0.337242	+	0.941418i	$0.609494\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	31.4595i	1.16999i
$724$	0	0
$725$	8.46889	0.314527
$726$	0	0
$727$	−44.3729	−1.64570	−0.822850	−	0.568258i	$-0.807617\pi$
$727$	−44.3729	−1.64570	−0.822850	+	0.568258i	$0.807617\pi$
$728$	0	0
$729$	29.3032	1.08530
$730$	0	0
$731$	63.2617	2.33982
$732$	0	0
$733$	27.9409i	1.03202i	0.856582	+	0.516011i	$0.172583\pi$
$733$	27.9409i	1.03202i	−0.856582	+	0.516011i	$0.827417\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.14680	0.300091
$738$	0	0
$739$	− 6.73546i	− 0.247768i	−0.992297	−	0.123884i	$-0.960465\pi$
$739$	− 6.73546i	− 0.247768i	0.992297	−	0.123884i	$-0.0395350\pi$
$740$	0	0
$741$	3.18874i	0.117141i
$742$	0	0
$743$	38.5007i	1.41245i	0.707985	+	0.706227i	$0.249604\pi$
$743$	38.5007i	1.41245i	−0.707985	+	0.706227i	$0.750396\pi$
$744$	0	0
$745$	13.2327i	0.484807i
$746$	0	0
$747$	16.5528	0.605635
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 4.62726i	− 0.168851i	−0.996430	−	0.0844256i	$-0.973094\pi$
$751$	− 4.62726i	− 0.168851i	0.996430	−	0.0844256i	$-0.0269055\pi$
$752$	0	0
$753$	43.1017	1.57071
$754$	0	0
$755$	−4.00997	−0.145938
$756$	0	0
$757$	4.95946	0.180255	0.0901273	−	0.995930i	$-0.471273\pi$
$757$	4.95946	0.180255	0.0901273	+	0.995930i	$0.471273\pi$
$758$	0	0
$759$	58.7134	2.13116
$760$	0	0
$761$	− 29.1471i	− 1.05658i	−0.849063	−	0.528292i	$-0.822833\pi$
$761$	− 29.1471i	− 1.05658i	0.849063	−	0.528292i	$-0.177167\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	5.60368	0.202601
$766$	0	0
$767$	− 12.0736i	− 0.435954i
$768$	0	0
$769$	− 35.1089i	− 1.26606i	−0.774127	−	0.633030i	$-0.781811\pi$
$769$	− 35.1089i	− 1.26606i	0.774127	−	0.633030i	$-0.218189\pi$
$770$	0	0
$771$	− 16.4744i	− 0.593311i
$772$	0	0
$773$	39.5648i	1.42305i	0.702662	+	0.711524i	$0.251995\pi$
$773$	39.5648i	1.42305i	−0.702662	+	0.711524i	$0.748005\pi$
$774$	0	0
$775$	−5.68297	−0.204138
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 3.95683i	− 0.141768i
$780$	0	0
$781$	−81.0483	−2.90014
$782$	0	0
$783$	−47.8938	−1.71158
$784$	0	0
$785$	−2.34315	−0.0836305
$786$	0	0
$787$	27.9839	0.997518	0.498759	−	0.866741i	$-0.333789\pi$
$787$	27.9839	0.997518	0.498759	+	0.866741i	$0.333789\pi$
$788$	0	0
$789$	− 7.03612i	− 0.250492i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	6.55770	0.232871
$794$	0	0
$795$	9.76765i	0.346423i
$796$	0	0
$797$	41.2899i	1.46256i	0.682075	+	0.731282i	$0.261078\pi$
$797$	41.2899i	1.46256i	−0.682075	+	0.731282i	$0.738922\pi$
$798$	0	0
$799$	− 22.9150i	− 0.810674i
$800$	0	0
$801$	− 3.84730i	− 0.135938i
$802$	0	0
$803$	−14.0106	−0.494423
$804$	0	0
$805$	0	0
$806$	0	0
$807$	13.8086i	0.486085i
$808$	0	0
$809$	33.1217	1.16450	0.582248	−	0.813011i	$-0.302173\pi$
$809$	33.1217	1.16450	0.582248	+	0.813011i	$0.302173\pi$
$810$	0	0
$811$	44.8698	1.57559	0.787796	−	0.615936i	$-0.211222\pi$
$811$	44.8698	1.57559	0.787796	+	0.615936i	$0.211222\pi$
$812$	0	0
$813$	−28.5445	−1.00110
$814$	0	0
$815$	−2.77669	−0.0972633
$816$	0	0
$817$	16.5528i	0.579109i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	39.6771	1.38474	0.692370	−	0.721542i	$-0.256566\pi$
$821$	39.6771	1.38474	0.692370	+	0.721542i	$0.256566\pi$
$822$	0	0
$823$	18.7168i	0.652425i	0.945296	+	0.326213i	$0.105773\pi$
$823$	18.7168i	0.652425i	−0.945296	+	0.326213i	$0.894227\pi$
$824$	0	0
$825$	− 8.75858i	− 0.304935i
$826$	0	0
$827$	− 16.3900i	− 0.569935i	−0.958537	−	0.284968i	$-0.908017\pi$
$827$	− 16.3900i	− 0.569935i	0.958537	−	0.284968i	$-0.0919828\pi$
$828$	0	0
$829$	39.7482i	1.38051i	0.723565	+	0.690256i	$0.242502\pi$
$829$	39.7482i	1.38051i	−0.723565	+	0.690256i	$0.757498\pi$
$830$	0	0
$831$	38.8016	1.34601
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 10.4518i	− 0.361701i
$836$	0	0
$837$	32.1387	1.11088
$838$	0	0
$839$	−37.2269	−1.28522	−0.642608	−	0.766195i	$-0.722148\pi$
$839$	−37.2269	−1.28522	−0.642608	+	0.766195i	$0.722148\pi$
$840$	0	0
$841$	42.7221	1.47318
$842$	0	0
$843$	−24.5465	−0.845427
$844$	0	0
$845$	− 10.9449i	− 0.376518i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−25.1453	−0.862985
$850$	0	0
$851$	37.9406i	1.30059i
$852$	0	0
$853$	30.6688i	1.05008i	0.851078	+	0.525040i	$0.175949\pi$
$853$	30.6688i	1.05008i	−0.851078	+	0.525040i	$0.824051\pi$
$854$	0	0
$855$	1.46624i	0.0501442i
$856$	0	0
$857$	− 26.9410i	− 0.920286i	−0.887845	−	0.460143i	$-0.847798\pi$
$857$	− 26.9410i	− 0.920286i	0.887845	−	0.460143i	$-0.152202\pi$
$858$	0	0
$859$	48.1522	1.64293	0.821466	−	0.570258i	$-0.193157\pi$
$859$	48.1522	1.64293	0.821466	+	0.570258i	$0.193157\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 27.8219i	− 0.947068i	−0.880776	−	0.473534i	$-0.842978\pi$
$863$	− 27.8219i	− 0.947068i	0.880776	−	0.473534i	$-0.157022\pi$
$864$	0	0
$865$	12.7563	0.433728
$866$	0	0
$867$	26.0428	0.884459
$868$	0	0
$869$	−9.31090	−0.315851
$870$	0	0
$871$	1.91151	0.0647690
$872$	0	0
$873$	− 2.10948i	− 0.0713951i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−17.7875	−0.600643	−0.300321	−	0.953838i	$-0.597094\pi$
$877$	−17.7875	−0.600643	−0.300321	+	0.953838i	$0.597094\pi$
$878$	0	0
$879$	− 23.0206i	− 0.776466i
$880$	0	0
$881$	− 5.66968i	− 0.191016i	−0.995429	−	0.0955082i	$-0.969552\pi$
$881$	− 5.66968i	− 0.191016i	0.995429	−	0.0955082i	$-0.0304476\pi$
$882$	0	0
$883$	27.3679i	0.921003i	0.887659	+	0.460502i	$0.152330\pi$
$883$	27.3679i	0.921003i	−0.887659	+	0.460502i	$0.847670\pi$
$884$	0	0
$885$	12.0736i	0.405851i
$886$	0	0
$887$	−1.92492	−0.0646326	−0.0323163	−	0.999478i	$-0.510288\pi$
$887$	−1.92492	−0.0646326	−0.0323163	+	0.999478i	$0.510288\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	32.2119i	1.07914i
$892$	0	0
$893$	5.99584	0.200643
$894$	0	0
$895$	−12.5143	−0.418308
$896$	0	0
$897$	13.7761	0.459971
$898$	0	0
$899$	−48.1285	−1.60518
$900$	0	0
$901$	− 40.4059i	− 1.34611i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	17.3809	0.577762
$906$	0	0
$907$	− 58.4544i	− 1.94095i	−0.241206	−	0.970474i	$-0.577543\pi$
$907$	− 58.4544i	− 1.94095i	0.241206	−	0.970474i	$-0.422457\pi$
$908$	0	0
$909$	16.3773i	0.543201i
$910$	0	0
$911$	− 23.8881i	− 0.791448i	−0.918370	−	0.395724i	$-0.870494\pi$
$911$	− 23.8881i	− 0.791448i	0.918370	−	0.395724i	$-0.129506\pi$
$912$	0	0
$913$	107.025i	3.54201i
$914$	0	0
$915$	−6.55770	−0.216791
$916$	0	0
$917$	0	0
$918$	0	0
$919$	30.4268i	1.00369i	0.864958	+	0.501843i	$0.167345\pi$
$919$	30.4268i	1.00369i	−0.864958	+	0.501843i	$0.832655\pi$
$920$	0	0
$921$	16.5191	0.544323
$922$	0	0
$923$	−19.0166	−0.625940
$924$	0	0
$925$	5.65980	0.186093
$926$	0	0
$927$	−4.44494	−0.145991
$928$	0	0
$929$	19.6566i	0.644911i	0.946585	+	0.322456i	$0.104508\pi$
$929$	19.6566i	0.644911i	−0.946585	+	0.322456i	$0.895492\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	49.5181	1.62115
$934$	0	0
$935$	36.2316i	1.18490i
$936$	0	0
$937$	− 37.4452i	− 1.22328i	−0.791136	−	0.611640i	$-0.790510\pi$
$937$	− 37.4452i	− 1.22328i	0.791136	−	0.611640i	$-0.209490\pi$
$938$	0	0
$939$	− 37.0569i	− 1.20931i
$940$	0	0
$941$	15.8158i	0.515579i	0.966201	+	0.257789i	$0.0829941\pi$
$941$	15.8158i	0.515579i	−0.966201	+	0.257789i	$0.917006\pi$
$942$	0	0
$943$	−17.0944	−0.556671
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 34.9504i	− 1.13573i	−0.823120	−	0.567867i	$-0.807769\pi$
$947$	− 34.9504i	− 1.13573i	0.823120	−	0.567867i	$-0.192231\pi$
$948$	0	0
$949$	−3.28735	−0.106712
$950$	0	0
$951$	37.6822	1.22193
$952$	0	0
$953$	−52.7042	−1.70726	−0.853629	−	0.520882i	$-0.825603\pi$
$953$	−52.7042	−1.70726	−0.853629	+	0.520882i	$0.825603\pi$
$954$	0	0
$955$	−6.88029	−0.222641
$956$	0	0
$957$	− 74.1755i	− 2.39775i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.29620	0.0418129
$962$	0	0
$963$	− 3.31658i	− 0.106875i
$964$	0	0
$965$	− 7.08067i	− 0.227935i
$966$	0	0
$967$	− 13.4095i	− 0.431221i	−0.976479	−	0.215611i	$-0.930826\pi$
$967$	− 13.4095i	− 0.431221i	0.976479	−	0.215611i	$-0.0691742\pi$
$968$	0	0
$969$	13.1909i	0.423752i
$970$	0	0
$971$	−40.0132	−1.28408	−0.642042	−	0.766669i	$-0.721913\pi$
$971$	−40.0132	−1.28408	−0.642042	+	0.766669i	$0.721913\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 2.05505i	− 0.0658144i
$976$	0	0
$977$	−49.1353	−1.57198	−0.785990	−	0.618240i	$-0.787846\pi$
$977$	−49.1353	−1.57198	−0.785990	+	0.618240i	$0.787846\pi$
$978$	0	0
$979$	24.8754	0.795022
$980$	0	0
$981$	8.77641	0.280209
$982$	0	0
$983$	−37.8087	−1.20591	−0.602955	−	0.797775i	$-0.706010\pi$
$983$	−37.8087	−1.20591	−0.602955	+	0.797775i	$0.706010\pi$
$984$	0	0
$985$	12.2980i	0.391847i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	71.5119	2.27395
$990$	0	0
$991$	− 4.11333i	− 0.130664i	−0.997864	−	0.0653322i	$-0.979189\pi$
$991$	− 4.11333i	− 0.130664i	0.997864	−	0.0653322i	$-0.0208107\pi$
$992$	0	0
$993$	12.5268i	0.397525i
$994$	0	0
$995$	20.6971i	0.656143i
$996$	0	0
$997$	− 27.9830i	− 0.886231i	−0.896464	−	0.443116i	$-0.853873\pi$
$997$	− 27.9830i	− 0.886231i	0.896464	−	0.443116i	$-0.146127\pi$
$998$	0	0
$999$	−32.0076	−1.01268

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3920.2.k.a.2351.5	✓	8
4.3	odd	2		3920.2.k.c.2351.3	yes	8
7.6	odd	2		3920.2.k.c.2351.4	yes	8
28.27	even	2	inner	3920.2.k.a.2351.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3920.2.k.a.2351.5	✓	8	1.1	even	1	trivial
3920.2.k.a.2351.6	yes	8	28.27	even	2	inner
3920.2.k.c.2351.3	yes	8	4.3	odd	2
3920.2.k.c.2351.4	yes	8	7.6	odd	2

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:	\( N \)	\(=\)	\( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3920.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.43355 q^{3} -1.00000i q^{5} -0.944947 q^{9} +O(q^{10})\)	\(q-1.43355 q^{3} -1.00000i q^{5} -0.944947 q^{9} -6.10973i q^{11} -1.43355i q^{13} +1.43355i q^{15} -5.93015i q^{17} +1.55166 q^{19} -6.70353i q^{23} -1.00000 q^{25} +5.65526 q^{27} -8.46889 q^{29} +5.68297 q^{31} +8.75858i q^{33} -5.65980 q^{37} +2.05505i q^{39} -2.55007i q^{41} +10.6678i q^{43} +0.944947i q^{45} +3.86415 q^{47} +8.50114i q^{51} +6.81363 q^{53} -6.10973 q^{55} -2.22437 q^{57} +8.42222 q^{59} +4.57446i q^{61} -1.43355 q^{65} +1.33341i q^{67} +9.60981i q^{69} -13.2654i q^{71} -2.29316i q^{73} +1.43355 q^{75} -1.52395i q^{79} -5.27223 q^{81} -17.5172 q^{83} -5.93015 q^{85} +12.1405 q^{87} +4.07144i q^{89} -8.14680 q^{93} -1.55166i q^{95} +2.23238i q^{97} +5.77337i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 16 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 + 16 * q^9	\( 8 q - 8 q^{3} + 16 q^{9} + 16 q^{19} - 8 q^{25} - 8 q^{27} + 8 q^{29} + 48 q^{31} + 24 q^{47} + 32 q^{53} - 8 q^{55} - 32 q^{57} + 16 q^{59} - 8 q^{65} + 8 q^{75} - 8 q^{81} - 48 q^{83} - 24 q^{85} - 24 q^{87} - 32 q^{93}+O(q^{100}) \) 8 * q - 8 * q^3 + 16 * q^9 + 16 * q^19 - 8 * q^25 - 8 * q^27 + 8 * q^29 + 48 * q^31 + 24 * q^47 + 32 * q^53 - 8 * q^55 - 32 * q^57 + 16 * q^59 - 8 * q^65 + 8 * q^75 - 8 * q^81 - 48 * q^83 - 24 * q^85 - 24 * q^87 - 32 * q^93

Introduction

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Modular forms

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Database

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