LMFDB - Embedded newform 3680.2.f.b.1841.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3680,2,Mod(1841,3680)]

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([0, 1, 0, 0]))

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

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

chi := DirichletCharacter("3680.1841");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3680 = 2^{5} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3680.f (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:	$29.3849479438$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1871773696.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 31x^{4} + 81 $ x^8 + 31*x^4 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1841.4
Root			$0.921201 + 0.921201i$ of defining polynomial
Character	$\chi$	$=$	3680.1841
Dual form			3680.2.f.b.1841.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.30278i q^{3} +1.00000i q^{5} -2.30278 q^{7} +1.30278 q^{9} +O(q^{10})$	$q-1.30278i q^{3} +1.00000i q^{5} -2.30278 q^{7} +1.30278 q^{9} +5.97361i q^{11} +3.87459i q^{13} +1.30278 q^{15} -2.53963 q^{17} -4.95384i q^{19} +3.00000i q^{21} +1.00000 q^{23} -1.00000 q^{25} -5.60555i q^{27} +0.222876i q^{29} -10.9934 q^{31} +7.78227 q^{33} -2.30278i q^{35} +5.77712i q^{37} +5.04772 q^{39} +6.98758 q^{41} -1.07925i q^{43} +1.30278i q^{45} -13.0908 q^{47} -1.69722 q^{49} +3.30856i q^{51} -9.46193i q^{53} -5.97361 q^{55} -6.45374 q^{57} -6.51323i q^{59} -4.98758i q^{61} -3.00000 q^{63} -3.87459 q^{65} -6.51323i q^{67} -1.30278i q^{69} +2.53963 q^{71} +3.52630 q^{73} +1.30278i q^{75} -13.7559i q^{77} -1.87973 q^{79} -3.39445 q^{81} -15.3417i q^{83} -2.53963i q^{85} +0.290357 q^{87} -4.80048 q^{89} -8.92230i q^{91} +14.3219i q^{93} +4.95384 q^{95} -2.22443 q^{97} +7.78227i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{7} - 4 q^{9}+O(q^{10}) $ 8 * q - 4 * q^7 - 4 * q^9	$ 8 q - 4 q^{7} - 4 q^{9} - 4 q^{15} - 20 q^{17} + 8 q^{23} - 8 q^{25} + 4 q^{31} + 28 q^{33} + 16 q^{39} + 12 q^{41} - 8 q^{47} - 28 q^{49} + 4 q^{55} + 40 q^{57} - 24 q^{63} - 20 q^{65} + 20 q^{71} + 24 q^{79} - 56 q^{81} - 56 q^{87} + 28 q^{95} + 12 q^{97}+O(q^{100}) $ 8 * q - 4 * q^7 - 4 * q^9 - 4 * q^15 - 20 * q^17 + 8 * q^23 - 8 * q^25 + 4 * q^31 + 28 * q^33 + 16 * q^39 + 12 * q^41 - 8 * q^47 - 28 * q^49 + 4 * q^55 + 40 * q^57 - 24 * q^63 - 20 * q^65 + 20 * q^71 + 24 * q^79 - 56 * q^81 - 56 * q^87 + 28 * q^95 + 12 * q^97

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
$3$	− 1.30278i	− 0.752158i	−0.926588	−	0.376079i	$-0.877272\pi$
$3$	− 1.30278i	− 0.752158i	0.926588	−	0.376079i	$-0.122728\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	−2.30278	−0.870367	−0.435184	−	0.900342i	$-0.643317\pi$
$7$	−2.30278	−0.870367	−0.435184	+	0.900342i	$0.643317\pi$
$8$	0	0
$9$	1.30278	0.434259
$10$	0	0
$11$	5.97361i	1.80111i	0.434742	+	0.900555i	$0.356839\pi$
$11$	5.97361i	1.80111i	−0.434742	+	0.900555i	$0.643161\pi$
$12$	0	0
$13$	3.87459i	1.07462i	0.843386	+	0.537308i	$0.180559\pi$
$13$	3.87459i	1.07462i	−0.843386	+	0.537308i	$0.819441\pi$
$14$	0	0
$15$	1.30278	0.336375
$16$	0	0
$17$	−2.53963	−0.615950	−0.307975	−	0.951394i	$-0.599651\pi$
$17$	−2.53963	−0.615950	−0.307975	+	0.951394i	$0.599651\pi$
$18$	0	0
$19$	− 4.95384i	− 1.13649i	−0.822860	−	0.568245i	$-0.807623\pi$
$19$	− 4.95384i	− 1.13649i	0.822860	−	0.568245i	$-0.192377\pi$
$20$	0	0
$21$	3.00000i	0.654654i
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	− 5.60555i	− 1.07879i
$28$	0	0
$29$	0.222876i	0.0413870i	0.999786	+	0.0206935i	$0.00658742\pi$
$29$	0.222876i	0.0413870i	−0.999786	+	0.0206935i	$0.993413\pi$
$30$	0	0
$31$	−10.9934	−1.97447	−0.987234	−	0.159278i	$-0.949083\pi$
$31$	−10.9934	−1.97447	−0.987234	+	0.159278i	$0.949083\pi$
$32$	0	0
$33$	7.78227	1.35472
$34$	0	0
$35$	− 2.30278i	− 0.389240i
$36$	0	0
$37$	5.77712i	0.949753i	0.880052	+	0.474877i	$0.157507\pi$
$37$	5.77712i	0.949753i	−0.880052	+	0.474877i	$0.842493\pi$
$38$	0	0
$39$	5.04772	0.808282
$40$	0	0
$41$	6.98758	1.09128	0.545638	−	0.838021i	$-0.316287\pi$
$41$	6.98758	1.09128	0.545638	+	0.838021i	$0.316287\pi$
$42$	0	0
$43$	− 1.07925i	− 0.164585i	−0.996608	−	0.0822924i	$-0.973776\pi$
$43$	− 1.07925i	− 0.164585i	0.996608	−	0.0822924i	$-0.0262241\pi$
$44$	0	0
$45$	1.30278i	0.194206i
$46$	0	0
$47$	−13.0908	−1.90949	−0.954747	−	0.297419i	$-0.903874\pi$
$47$	−13.0908	−1.90949	−0.954747	+	0.297419i	$0.903874\pi$
$48$	0	0
$49$	−1.69722	−0.242461
$50$	0	0
$51$	3.30856i	0.463292i
$52$	0	0
$53$	− 9.46193i	− 1.29970i	−0.760064	−	0.649848i	$-0.774833\pi$
$53$	− 9.46193i	− 1.29970i	0.760064	−	0.649848i	$-0.225167\pi$
$54$	0	0
$55$	−5.97361	−0.805481
$56$	0	0
$57$	−6.45374	−0.854819
$58$	0	0
$59$	− 6.51323i	− 0.847951i	−0.905674	−	0.423975i	$-0.860634\pi$
$59$	− 6.51323i	− 0.847951i	0.905674	−	0.423975i	$-0.139366\pi$
$60$	0	0
$61$	− 4.98758i	− 0.638594i	−0.947655	−	0.319297i	$-0.896553\pi$
$61$	− 4.98758i	− 0.638594i	0.947655	−	0.319297i	$-0.103447\pi$
$62$	0	0
$63$	−3.00000	−0.377964
$64$	0	0
$65$	−3.87459	−0.480583
$66$	0	0
$67$	− 6.51323i	− 0.795718i	−0.917447	−	0.397859i	$-0.869753\pi$
$67$	− 6.51323i	− 0.795718i	0.917447	−	0.397859i	$-0.130247\pi$
$68$	0	0
$69$	− 1.30278i	− 0.156836i
$70$	0	0
$71$	2.53963	0.301398	0.150699	−	0.988580i	$-0.451848\pi$
$71$	2.53963	0.301398	0.150699	+	0.988580i	$0.451848\pi$
$72$	0	0
$73$	3.52630	0.412722	0.206361	−	0.978476i	$-0.433838\pi$
$73$	3.52630	0.412722	0.206361	+	0.978476i	$0.433838\pi$
$74$	0	0
$75$	1.30278i	0.150432i
$76$	0	0
$77$	− 13.7559i	− 1.56763i
$78$	0	0
$79$	−1.87973	−0.211486	−0.105743	−	0.994393i	$-0.533722\pi$
$79$	−1.87973	−0.211486	−0.105743	+	0.994393i	$0.533722\pi$
$80$	0	0
$81$	−3.39445	−0.377161
$82$	0	0
$83$	− 15.3417i	− 1.68397i	−0.539504	−	0.841983i	$-0.681388\pi$
$83$	− 15.3417i	− 1.68397i	0.539504	−	0.841983i	$-0.318612\pi$
$84$	0	0
$85$	− 2.53963i	− 0.275461i
$86$	0	0
$87$	0.290357	0.0311296
$88$	0	0
$89$	−4.80048	−0.508849	−0.254425	−	0.967093i	$-0.581886\pi$
$89$	−4.80048	−0.508849	−0.254425	+	0.967093i	$0.581886\pi$
$90$	0	0
$91$	− 8.92230i	− 0.935311i
$92$	0	0
$93$	14.3219i	1.48511i
$94$	0	0
$95$	4.95384	0.508253
$96$	0	0
$97$	−2.22443	−0.225857	−0.112928	−	0.993603i	$-0.536023\pi$
$97$	−2.22443	−0.225857	−0.112928	+	0.993603i	$0.536023\pi$
$98$	0	0
$99$	7.78227i	0.782147i
$100$	0	0
$101$	− 5.14362i	− 0.511809i	−0.966702	−	0.255905i	$-0.917627\pi$
$101$	− 5.14362i	− 0.511809i	0.966702	−	0.255905i	$-0.0823733\pi$
$102$	0	0
$103$	5.90833	0.582165	0.291082	−	0.956698i	$-0.405985\pi$
$103$	5.90833	0.582165	0.291082	+	0.956698i	$0.405985\pi$
$104$	0	0
$105$	−3.00000	−0.292770
$106$	0	0
$107$	− 7.81665i	− 0.755664i	−0.925874	−	0.377832i	$-0.876670\pi$
$107$	− 7.81665i	− 0.755664i	0.925874	−	0.377832i	$-0.123330\pi$
$108$	0	0
$109$	3.78954i	0.362972i	0.983393	+	0.181486i	$0.0580908\pi$
$109$	3.78954i	0.362972i	−0.983393	+	0.181486i	$0.941909\pi$
$110$	0	0
$111$	7.52630	0.714364
$112$	0	0
$113$	−17.0660	−1.60543	−0.802717	−	0.596360i	$-0.796613\pi$
$113$	−17.0660	−1.60543	−0.802717	+	0.596360i	$0.796613\pi$
$114$	0	0
$115$	1.00000i	0.0932505i
$116$	0	0
$117$	5.04772i	0.466662i
$118$	0	0
$119$	5.84819	0.536103
$120$	0	0
$121$	−24.6840	−2.24400
$122$	0	0
$123$	− 9.10325i	− 0.820813i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−18.5807	−1.64877	−0.824386	−	0.566028i	$-0.808479\pi$
$127$	−18.5807	−1.64877	−0.824386	+	0.566028i	$0.808479\pi$
$128$	0	0
$129$	−1.40603	−0.123794
$130$	0	0
$131$	13.5925i	1.18758i	0.804620	+	0.593791i	$0.202369\pi$
$131$	13.5925i	1.18758i	−0.804620	+	0.593791i	$0.797631\pi$
$132$	0	0
$133$	11.4076i	0.989163i
$134$	0	0
$135$	5.60555	0.482449
$136$	0	0
$137$	−0.703013	−0.0600625	−0.0300312	−	0.999549i	$-0.509561\pi$
$137$	−0.703013	−0.0600625	−0.0300312	+	0.999549i	$0.509561\pi$
$138$	0	0
$139$	17.8944i	1.51779i	0.651216	+	0.758893i	$0.274259\pi$
$139$	17.8944i	1.51779i	−0.651216	+	0.758893i	$0.725741\pi$
$140$	0	0
$141$	17.0544i	1.43624i
$142$	0	0
$143$	−23.1453	−1.93550
$144$	0	0
$145$	−0.222876	−0.0185088
$146$	0	0
$147$	2.21110i	0.182369i
$148$	0	0
$149$	− 2.10785i	− 0.172682i	−0.996266	−	0.0863410i	$-0.972483\pi$
$149$	− 2.10785i	− 0.172682i	0.996266	−	0.0863410i	$-0.0275174\pi$
$150$	0	0
$151$	2.98667	0.243052	0.121526	−	0.992588i	$-0.461221\pi$
$151$	2.98667	0.243052	0.121526	+	0.992588i	$0.461221\pi$
$152$	0	0
$153$	−3.30856	−0.267482
$154$	0	0
$155$	− 10.9934i	− 0.883009i
$156$	0	0
$157$	− 19.0265i	− 1.51848i	−0.650812	−	0.759239i	$-0.725572\pi$
$157$	− 19.0265i	− 1.51848i	0.650812	−	0.759239i	$-0.274428\pi$
$158$	0	0
$159$	−12.3268	−0.977577
$160$	0	0
$161$	−2.30278	−0.181484
$162$	0	0
$163$	16.0271i	1.25534i	0.778479	+	0.627670i	$0.215991\pi$
$163$	16.0271i	1.25534i	−0.778479	+	0.627670i	$0.784009\pi$
$164$	0	0
$165$	7.78227i	0.605849i
$166$	0	0
$167$	2.89591	0.224092	0.112046	−	0.993703i	$-0.464260\pi$
$167$	2.89591	0.224092	0.112046	+	0.993703i	$0.464260\pi$
$168$	0	0
$169$	−2.01242	−0.154801
$170$	0	0
$171$	− 6.45374i	− 0.493530i
$172$	0	0
$173$	− 0.950254i	− 0.0722465i	−0.999347	−	0.0361233i	$-0.988499\pi$
$173$	− 0.950254i	− 0.0722465i	0.999347	−	0.0361233i	$-0.0115009\pi$
$174$	0	0
$175$	2.30278	0.174073
$176$	0	0
$177$	−8.48528	−0.637793
$178$	0	0
$179$	19.8167i	1.48117i	0.671965	+	0.740583i	$0.265451\pi$
$179$	19.8167i	1.48117i	−0.671965	+	0.740583i	$0.734549\pi$
$180$	0	0
$181$	15.5139i	1.15314i	0.817049	+	0.576569i	$0.195609\pi$
$181$	15.5139i	1.15314i	−0.817049	+	0.576569i	$0.804391\pi$
$182$	0	0
$183$	−6.49770	−0.480324
$184$	0	0
$185$	−5.77712	−0.424743
$186$	0	0
$187$	− 15.1707i	− 1.10939i
$188$	0	0
$189$	12.9083i	0.938943i
$190$	0	0
$191$	−7.09083	−0.513075	−0.256537	−	0.966534i	$-0.582582\pi$
$191$	−7.09083	−0.513075	−0.256537	+	0.966534i	$0.582582\pi$
$192$	0	0
$193$	−4.48528	−0.322858	−0.161429	−	0.986884i	$-0.551610\pi$
$193$	−4.48528	−0.322858	−0.161429	+	0.986884i	$0.551610\pi$
$194$	0	0
$195$	5.04772i	0.361474i
$196$	0	0
$197$	− 13.1555i	− 0.937288i	−0.883387	−	0.468644i	$-0.844743\pi$
$197$	− 13.1555i	− 0.937288i	0.883387	−	0.468644i	$-0.155257\pi$
$198$	0	0
$199$	23.8549	1.69103	0.845514	−	0.533953i	$-0.179294\pi$
$199$	23.8549	1.69103	0.845514	+	0.533953i	$0.179294\pi$
$200$	0	0
$201$	−8.48528	−0.598506
$202$	0	0
$203$	− 0.513233i	− 0.0360219i
$204$	0	0
$205$	6.98758i	0.488034i
$206$	0	0
$207$	1.30278	0.0905492
$208$	0	0
$209$	29.5923	2.04694
$210$	0	0
$211$	16.2655i	1.11976i	0.828572	+	0.559882i	$0.189154\pi$
$211$	16.2655i	1.11976i	−0.828572	+	0.559882i	$0.810846\pi$
$212$	0	0
$213$	− 3.30856i	− 0.226699i
$214$	0	0
$215$	1.07925	0.0736046
$216$	0	0
$217$	25.3153	1.71851
$218$	0	0
$219$	− 4.59397i	− 0.310432i
$220$	0	0
$221$	− 9.84001i	− 0.661910i
$222$	0	0
$223$	−5.33137	−0.357015	−0.178508	−	0.983939i	$-0.557127\pi$
$223$	−5.33137	−0.357015	−0.178508	+	0.983939i	$0.557127\pi$
$224$	0	0
$225$	−1.30278	−0.0868517
$226$	0	0
$227$	7.01306i	0.465473i	0.972540	+	0.232737i	$0.0747680\pi$
$227$	7.01306i	0.465473i	−0.972540	+	0.232737i	$0.925232\pi$
$228$	0	0
$229$	− 9.05259i	− 0.598212i	−0.954220	−	0.299106i	$-0.903312\pi$
$229$	− 9.05259i	− 0.598212i	0.954220	−	0.299106i	$-0.0966885\pi$
$230$	0	0
$231$	−17.9208	−1.17910
$232$	0	0
$233$	−12.6438	−0.828322	−0.414161	−	0.910204i	$-0.635925\pi$
$233$	−12.6438	−0.828322	−0.414161	+	0.910204i	$0.635925\pi$
$234$	0	0
$235$	− 13.0908i	− 0.853952i
$236$	0	0
$237$	2.44887i	0.159071i
$238$	0	0
$239$	0.158509	0.0102531	0.00512656	−	0.999987i	$-0.498368\pi$
$239$	0.158509	0.0102531	0.00512656	+	0.999987i	$0.498368\pi$
$240$	0	0
$241$	−16.6438	−1.07212	−0.536060	−	0.844180i	$-0.680088\pi$
$241$	−16.6438	−1.07212	−0.536060	+	0.844180i	$0.680088\pi$
$242$	0	0
$243$	− 12.3944i	− 0.795104i
$244$	0	0
$245$	− 1.69722i	− 0.108432i
$246$	0	0
$247$	19.1941	1.22129
$248$	0	0
$249$	−19.9867	−1.26661
$250$	0	0
$251$	6.35987i	0.401431i	0.979650	+	0.200716i	$0.0643267\pi$
$251$	6.35987i	0.401431i	−0.979650	+	0.200716i	$0.935673\pi$
$252$	0	0
$253$	5.97361i	0.375557i
$254$	0	0
$255$	−3.30856	−0.207190
$256$	0	0
$257$	−7.24934	−0.452202	−0.226101	−	0.974104i	$-0.572598\pi$
$257$	−7.24934	−0.452202	−0.226101	+	0.974104i	$0.572598\pi$
$258$	0	0
$259$	− 13.3034i	− 0.826634i
$260$	0	0
$261$	0.290357i	0.0179727i
$262$	0	0
$263$	−23.8263	−1.46919	−0.734596	−	0.678505i	$-0.762628\pi$
$263$	−23.8263	−1.46919	−0.734596	+	0.678505i	$0.762628\pi$
$264$	0	0
$265$	9.46193	0.581242
$266$	0	0
$267$	6.25394i	0.382735i
$268$	0	0
$269$	8.32989i	0.507882i	0.967220	+	0.253941i	$0.0817269\pi$
$269$	8.32989i	0.507882i	−0.967220	+	0.253941i	$0.918273\pi$
$270$	0	0
$271$	21.9524	1.33351	0.666755	−	0.745277i	$-0.267683\pi$
$271$	21.9524	1.33351	0.666755	+	0.745277i	$0.267683\pi$
$272$	0	0
$273$	−11.6238	−0.703502
$274$	0	0
$275$	− 5.97361i	− 0.360222i
$276$	0	0
$277$	25.6964i	1.54395i	0.635655	+	0.771973i	$0.280730\pi$
$277$	25.6964i	1.54395i	−0.635655	+	0.771973i	$0.719270\pi$
$278$	0	0
$279$	−14.3219	−0.857429
$280$	0	0
$281$	−3.48988	−0.208189	−0.104094	−	0.994567i	$-0.533194\pi$
$281$	−3.48988	−0.208189	−0.104094	+	0.994567i	$0.533194\pi$
$282$	0	0
$283$	20.4853i	1.21772i	0.793276	+	0.608862i	$0.208374\pi$
$283$	20.4853i	1.21772i	−0.793276	+	0.608862i	$0.791626\pi$
$284$	0	0
$285$	− 6.45374i	− 0.382287i
$286$	0	0
$287$	−16.0908	−0.949812
$288$	0	0
$289$	−10.5503	−0.620605
$290$	0	0
$291$	2.89794i	0.169880i
$292$	0	0
$293$	− 9.29184i	− 0.542835i	−0.962462	−	0.271418i	$-0.912508\pi$
$293$	− 9.29184i	− 0.542835i	0.962462	−	0.271418i	$-0.0874925\pi$
$294$	0	0
$295$	6.51323	0.379215
$296$	0	0
$297$	33.4854	1.94302
$298$	0	0
$299$	3.87459i	0.224073i
$300$	0	0
$301$	2.48528i	0.143249i
$302$	0	0
$303$	−6.70098	−0.384961
$304$	0	0
$305$	4.98758	0.285588
$306$	0	0
$307$	− 33.9581i	− 1.93809i	−0.246877	−	0.969047i	$-0.579404\pi$
$307$	− 33.9581i	− 1.93809i	0.246877	−	0.969047i	$-0.420596\pi$
$308$	0	0
$309$	− 7.69722i	− 0.437880i
$310$	0	0
$311$	25.3696	1.43858	0.719289	−	0.694711i	$-0.244468\pi$
$311$	25.3696	1.43858	0.719289	+	0.694711i	$0.244468\pi$
$312$	0	0
$313$	−32.2427	−1.82247	−0.911233	−	0.411891i	$-0.864868\pi$
$313$	−32.2427	−1.82247	−0.911233	+	0.411891i	$0.864868\pi$
$314$	0	0
$315$	− 3.00000i	− 0.169031i
$316$	0	0
$317$	− 9.91100i	− 0.556657i	−0.960486	−	0.278329i	$-0.910220\pi$
$317$	− 9.91100i	− 0.556657i	0.960486	−	0.278329i	$-0.0897804\pi$
$318$	0	0
$319$	−1.33137	−0.0745425
$320$	0	0
$321$	−10.1833	−0.568379
$322$	0	0
$323$	12.5809i	0.700021i
$324$	0	0
$325$	− 3.87459i	− 0.214923i
$326$	0	0
$327$	4.93692	0.273013
$328$	0	0
$329$	30.1453	1.66196
$330$	0	0
$331$	− 27.0660i	− 1.48768i	−0.668357	−	0.743841i	$-0.733002\pi$
$331$	− 27.0660i	− 1.48768i	0.668357	−	0.743841i	$-0.266998\pi$
$332$	0	0
$333$	7.52630i	0.412439i
$334$	0	0
$335$	6.51323	0.355856
$336$	0	0
$337$	−0.292386	−0.0159273	−0.00796365	−	0.999968i	$-0.502535\pi$
$337$	−0.292386	−0.0159273	−0.00796365	+	0.999968i	$0.502535\pi$
$338$	0	0
$339$	22.2332i	1.20754i
$340$	0	0
$341$	− 65.6701i	− 3.55623i
$342$	0	0
$343$	20.0278	1.08140
$344$	0	0
$345$	1.30278	0.0701391
$346$	0	0
$347$	25.2500i	1.35549i	0.735297	+	0.677745i	$0.237043\pi$
$347$	25.2500i	1.35549i	−0.735297	+	0.677745i	$0.762957\pi$
$348$	0	0
$349$	− 25.7639i	− 1.37911i	−0.724234	−	0.689554i	$-0.757807\pi$
$349$	− 25.7639i	− 1.37911i	0.724234	−	0.689554i	$-0.242193\pi$
$350$	0	0
$351$	21.7192	1.15928
$352$	0	0
$353$	28.4720	1.51541	0.757706	−	0.652596i	$-0.226320\pi$
$353$	28.4720	1.51541	0.757706	+	0.652596i	$0.226320\pi$
$354$	0	0
$355$	2.53963i	0.134789i
$356$	0	0
$357$	− 7.61888i	− 0.403234i
$358$	0	0
$359$	−28.5807	−1.50843	−0.754216	−	0.656626i	$-0.771983\pi$
$359$	−28.5807	−1.50843	−0.754216	+	0.656626i	$0.771983\pi$
$360$	0	0
$361$	−5.54054	−0.291607
$362$	0	0
$363$	32.1577i	1.68784i
$364$	0	0
$365$	3.52630i	0.184575i
$366$	0	0
$367$	−2.42221	−0.126438	−0.0632190	−	0.998000i	$-0.520137\pi$
$367$	−2.42221	−0.126438	−0.0632190	+	0.998000i	$0.520137\pi$
$368$	0	0
$369$	9.10325	0.473896
$370$	0	0
$371$	21.7887i	1.13121i
$372$	0	0
$373$	17.3448i	0.898078i	0.893512	+	0.449039i	$0.148234\pi$
$373$	17.3448i	0.898078i	−0.893512	+	0.449039i	$0.851766\pi$
$374$	0	0
$375$	−1.30278	−0.0672750
$376$	0	0
$377$	−0.863552	−0.0444752
$378$	0	0
$379$	11.6845i	0.600194i	0.953909	+	0.300097i	$0.0970191\pi$
$379$	11.6845i	0.600194i	−0.953909	+	0.300097i	$0.902981\pi$
$380$	0	0
$381$	24.2065i	1.24014i
$382$	0	0
$383$	−23.0526	−1.17793	−0.588966	−	0.808158i	$-0.700465\pi$
$383$	−23.0526	−1.17793	−0.588966	+	0.808158i	$0.700465\pi$
$384$	0	0
$385$	13.7559	0.701064
$386$	0	0
$387$	− 1.40603i	− 0.0714724i
$388$	0	0
$389$	13.1972i	0.669125i	0.942374	+	0.334562i	$0.108588\pi$
$389$	13.1972i	0.669125i	−0.942374	+	0.334562i	$0.891412\pi$
$390$	0	0
$391$	−2.53963	−0.128434
$392$	0	0
$393$	17.7080	0.893249
$394$	0	0
$395$	− 1.87973i	− 0.0945795i
$396$	0	0
$397$	18.0264i	0.904719i	0.891836	+	0.452359i	$0.149418\pi$
$397$	18.0264i	0.904719i	−0.891836	+	0.452359i	$0.850582\pi$
$398$	0	0
$399$	14.8615	0.744007
$400$	0	0
$401$	−29.5015	−1.47323	−0.736616	−	0.676311i	$-0.763578\pi$
$401$	−29.5015	−1.47323	−0.736616	+	0.676311i	$0.763578\pi$
$402$	0	0
$403$	− 42.5948i	− 2.12180i
$404$	0	0
$405$	− 3.39445i	− 0.168672i
$406$	0	0
$407$	−34.5103	−1.71061
$408$	0	0
$409$	5.25018	0.259605	0.129802	−	0.991540i	$-0.458566\pi$
$409$	5.25018	0.259605	0.129802	+	0.991540i	$0.458566\pi$
$410$	0	0
$411$	0.915869i	0.0451765i
$412$	0	0
$413$	14.9985i	0.738029i
$414$	0	0
$415$	15.3417	0.753092
$416$	0	0
$417$	23.3124	1.14161
$418$	0	0
$419$	− 0.923860i	− 0.0451335i	−0.999745	−	0.0225668i	$-0.992816\pi$
$419$	− 0.923860i	− 0.0451335i	0.999745	−	0.0225668i	$-0.00718383\pi$
$420$	0	0
$421$	− 27.2382i	− 1.32751i	−0.747950	−	0.663755i	$-0.768962\pi$
$421$	− 27.2382i	− 1.32751i	0.747950	−	0.663755i	$-0.231038\pi$
$422$	0	0
$423$	−17.0544	−0.829214
$424$	0	0
$425$	2.53963	0.123190
$426$	0	0
$427$	11.4853i	0.555812i
$428$	0	0
$429$	30.1531i	1.45580i
$430$	0	0
$431$	26.7756	1.28974	0.644869	−	0.764293i	$-0.276912\pi$
$431$	26.7756	1.28974	0.644869	+	0.764293i	$0.276912\pi$
$432$	0	0
$433$	5.33340	0.256307	0.128153	−	0.991754i	$-0.459095\pi$
$433$	5.33340	0.256307	0.128153	+	0.991754i	$0.459095\pi$
$434$	0	0
$435$	0.290357i	0.0139216i
$436$	0	0
$437$	− 4.95384i	− 0.236974i
$438$	0	0
$439$	−14.8233	−0.707477	−0.353738	−	0.935344i	$-0.615090\pi$
$439$	−14.8233	−0.707477	−0.353738	+	0.935344i	$0.615090\pi$
$440$	0	0
$441$	−2.21110	−0.105291
$442$	0	0
$443$	4.09330i	0.194479i	0.995261	+	0.0972393i	$0.0310012\pi$
$443$	4.09330i	0.194479i	−0.995261	+	0.0972393i	$0.968999\pi$
$444$	0	0
$445$	− 4.80048i	− 0.227564i
$446$	0	0
$447$	−2.74606	−0.129884
$448$	0	0
$449$	16.0622	0.758024	0.379012	−	0.925392i	$-0.376264\pi$
$449$	16.0622	0.758024	0.379012	+	0.925392i	$0.376264\pi$
$450$	0	0
$451$	41.7411i	1.96551i
$452$	0	0
$453$	− 3.89096i	− 0.182813i
$454$	0	0
$455$	8.92230	0.418284
$456$	0	0
$457$	−21.3830	−1.00026	−0.500128	−	0.865952i	$-0.666714\pi$
$457$	−21.3830	−1.00026	−0.500128	+	0.865952i	$0.666714\pi$
$458$	0	0
$459$	14.2360i	0.664480i
$460$	0	0
$461$	− 12.9737i	− 0.604244i	−0.953269	−	0.302122i	$-0.902305\pi$
$461$	− 12.9737i	− 0.604244i	0.953269	−	0.302122i	$-0.0976951\pi$
$462$	0	0
$463$	−17.2361	−0.801029	−0.400514	−	0.916290i	$-0.631169\pi$
$463$	−17.2361	−0.801029	−0.400514	+	0.916290i	$0.631169\pi$
$464$	0	0
$465$	−14.3219	−0.664162
$466$	0	0
$467$	− 13.3532i	− 0.617914i	−0.951076	−	0.308957i	$-0.900020\pi$
$467$	− 13.3532i	− 0.617914i	0.951076	−	0.308957i	$-0.0999799\pi$
$468$	0	0
$469$	14.9985i	0.692567i
$470$	0	0
$471$	−24.7872	−1.14213
$472$	0	0
$473$	6.44704	0.296435
$474$	0	0
$475$	4.95384i	0.227298i
$476$	0	0
$477$	− 12.3268i	− 0.564404i
$478$	0	0
$479$	−30.0498	−1.37301	−0.686506	−	0.727125i	$-0.740856\pi$
$479$	−30.0498	−1.37301	−0.686506	+	0.727125i	$0.740856\pi$
$480$	0	0
$481$	−22.3840	−1.02062
$482$	0	0
$483$	3.00000i	0.136505i
$484$	0	0
$485$	− 2.22443i	− 0.101006i
$486$	0	0
$487$	−18.8843	−0.855731	−0.427865	−	0.903842i	$-0.640734\pi$
$487$	−18.8843	−0.855731	−0.427865	+	0.903842i	$0.640734\pi$
$488$	0	0
$489$	20.8797	0.944214
$490$	0	0
$491$	31.4735i	1.42038i	0.704010	+	0.710190i	$0.251391\pi$
$491$	31.4735i	1.42038i	−0.704010	+	0.710190i	$0.748609\pi$
$492$	0	0
$493$	− 0.566022i	− 0.0254923i
$494$	0	0
$495$	−7.78227	−0.349787
$496$	0	0
$497$	−5.84819	−0.262327
$498$	0	0
$499$	29.0895i	1.30223i	0.758980	+	0.651113i	$0.225698\pi$
$499$	29.0895i	1.30223i	−0.758980	+	0.651113i	$0.774302\pi$
$500$	0	0
$501$	− 3.77272i	− 0.168553i
$502$	0	0
$503$	27.1033	1.20847	0.604237	−	0.796805i	$-0.293478\pi$
$503$	27.1033	1.20847	0.604237	+	0.796805i	$0.293478\pi$
$504$	0	0
$505$	5.14362	0.228888
$506$	0	0
$507$	2.62173i	0.116435i
$508$	0	0
$509$	17.9836i	0.797110i	0.917144	+	0.398555i	$0.130488\pi$
$509$	17.9836i	0.797110i	−0.917144	+	0.398555i	$0.869512\pi$
$510$	0	0
$511$	−8.12027	−0.359220
$512$	0	0
$513$	−27.7690	−1.22603
$514$	0	0
$515$	5.90833i	0.260352i
$516$	0	0
$517$	− 78.1995i	− 3.43921i
$518$	0	0
$519$	−1.23797	−0.0543408
$520$	0	0
$521$	23.5149	1.03020	0.515102	−	0.857129i	$-0.327754\pi$
$521$	23.5149	1.03020	0.515102	+	0.857129i	$0.327754\pi$
$522$	0	0
$523$	29.3284i	1.28244i	0.767356	+	0.641221i	$0.221572\pi$
$523$	29.3284i	1.28244i	−0.767356	+	0.641221i	$0.778428\pi$
$524$	0	0
$525$	− 3.00000i	− 0.130931i
$526$	0	0
$527$	27.9191	1.21617
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	− 8.48528i	− 0.368230i
$532$	0	0
$533$	27.0740i	1.17270i
$534$	0	0
$535$	7.81665	0.337943
$536$	0	0
$537$	25.8167	1.11407
$538$	0	0
$539$	− 10.1385i	− 0.436698i
$540$	0	0
$541$	− 13.6072i	− 0.585018i	−0.956263	−	0.292509i	$-0.905510\pi$
$541$	− 13.6072i	− 0.585018i	0.956263	−	0.292509i	$-0.0944902\pi$
$542$	0	0
$543$	20.2111	0.867341
$544$	0	0
$545$	−3.78954	−0.162326
$546$	0	0
$547$	− 7.45668i	− 0.318825i	−0.987212	−	0.159412i	$-0.949040\pi$
$547$	− 7.45668i	− 0.318825i	0.987212	−	0.159412i	$-0.0509600\pi$
$548$	0	0
$549$	− 6.49770i	− 0.277315i
$550$	0	0
$551$	1.10409	0.0470359
$552$	0	0
$553$	4.32860	0.184071
$554$	0	0
$555$	7.52630i	0.319473i
$556$	0	0
$557$	− 12.4294i	− 0.526650i	−0.964707	−	0.263325i	$-0.915181\pi$
$557$	− 12.4294i	− 0.526650i	0.964707	−	0.263325i	$-0.0848191\pi$
$558$	0	0
$559$	4.18167	0.176866
$560$	0	0
$561$	−19.7641	−0.834439
$562$	0	0
$563$	14.6407i	0.617031i	0.951219	+	0.308515i	$0.0998322\pi$
$563$	14.6407i	0.617031i	−0.951219	+	0.308515i	$0.900168\pi$
$564$	0	0
$565$	− 17.0660i	− 0.717972i
$566$	0	0
$567$	7.81665	0.328269
$568$	0	0
$569$	−31.0412	−1.30131	−0.650657	−	0.759372i	$-0.725506\pi$
$569$	−31.0412	−1.30131	−0.650657	+	0.759372i	$0.725506\pi$
$570$	0	0
$571$	− 34.8186i	− 1.45711i	−0.684987	−	0.728555i	$-0.740192\pi$
$571$	− 34.8186i	− 1.45711i	0.684987	−	0.728555i	$-0.259808\pi$
$572$	0	0
$573$	9.23776i	0.385913i
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	33.2992	1.38626	0.693131	−	0.720812i	$-0.256231\pi$
$577$	33.2992	1.38626	0.693131	+	0.720812i	$0.256231\pi$
$578$	0	0
$579$	5.84332i	0.242840i
$580$	0	0
$581$	35.3284i	1.46567i
$582$	0	0
$583$	56.5218	2.34090
$584$	0	0
$585$	−5.04772	−0.208697
$586$	0	0
$587$	− 20.1726i	− 0.832611i	−0.909225	−	0.416305i	$-0.863325\pi$
$587$	− 20.1726i	− 0.832611i	0.909225	−	0.416305i	$-0.136675\pi$
$588$	0	0
$589$	54.4594i	2.24396i
$590$	0	0
$591$	−17.1386	−0.704989
$592$	0	0
$593$	−17.6217	−0.723638	−0.361819	−	0.932248i	$-0.617844\pi$
$593$	−17.6217	−0.723638	−0.361819	+	0.932248i	$0.617844\pi$
$594$	0	0
$595$	5.84819i	0.239752i
$596$	0	0
$597$	− 31.0776i	− 1.27192i
$598$	0	0
$599$	34.3048	1.40166	0.700828	−	0.713331i	$-0.252814\pi$
$599$	34.3048	1.40166	0.700828	+	0.713331i	$0.252814\pi$
$600$	0	0
$601$	43.2601	1.76462	0.882308	−	0.470673i	$-0.155989\pi$
$601$	43.2601	1.76462	0.882308	+	0.470673i	$0.155989\pi$
$602$	0	0
$603$	− 8.48528i	− 0.345547i
$604$	0	0
$605$	− 24.6840i	− 1.00355i
$606$	0	0
$607$	8.66863	0.351849	0.175924	−	0.984404i	$-0.443709\pi$
$607$	8.66863	0.351849	0.175924	+	0.984404i	$0.443709\pi$
$608$	0	0
$609$	−0.668628	−0.0270942
$610$	0	0
$611$	− 50.7216i	− 2.05197i
$612$	0	0
$613$	− 41.9091i	− 1.69269i	−0.532633	−	0.846347i	$-0.678797\pi$
$613$	− 41.9091i	− 1.69269i	0.532633	−	0.846347i	$-0.321203\pi$
$614$	0	0
$615$	9.10325	0.367079
$616$	0	0
$617$	−1.51757	−0.0610949	−0.0305475	−	0.999533i	$-0.509725\pi$
$617$	−1.51757	−0.0610949	−0.0305475	+	0.999533i	$0.509725\pi$
$618$	0	0
$619$	6.83669i	0.274790i	0.990516	+	0.137395i	$0.0438729\pi$
$619$	6.83669i	0.274790i	−0.990516	+	0.137395i	$0.956127\pi$
$620$	0	0
$621$	− 5.60555i	− 0.224943i
$622$	0	0
$623$	11.0544	0.442886
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	− 38.5521i	− 1.53962i
$628$	0	0
$629$	− 14.6717i	− 0.585001i
$630$	0	0
$631$	−5.39445	−0.214750	−0.107375	−	0.994219i	$-0.534244\pi$
$631$	−5.39445	−0.214750	−0.107375	+	0.994219i	$0.534244\pi$
$632$	0	0
$633$	21.1903	0.842240
$634$	0	0
$635$	− 18.5807i	− 0.737353i
$636$	0	0
$637$	− 6.57604i	− 0.260552i
$638$	0	0
$639$	3.30856	0.130885
$640$	0	0
$641$	−6.17009	−0.243704	−0.121852	−	0.992548i	$-0.538883\pi$
$641$	−6.17009	−0.243704	−0.121852	+	0.992548i	$0.538883\pi$
$642$	0	0
$643$	− 49.5016i	− 1.95215i	−0.217428	−	0.976076i	$-0.569767\pi$
$643$	− 49.5016i	− 1.95215i	0.217428	−	0.976076i	$-0.430233\pi$
$644$	0	0
$645$	− 1.40603i	− 0.0553622i
$646$	0	0
$647$	−37.0642	−1.45714	−0.728571	−	0.684970i	$-0.759815\pi$
$647$	−37.0642	−1.45714	−0.728571	+	0.684970i	$0.759815\pi$
$648$	0	0
$649$	38.9075	1.52725
$650$	0	0
$651$	− 32.9801i	− 1.29259i
$652$	0	0
$653$	13.1664i	0.515242i	0.966246	+	0.257621i	$0.0829386\pi$
$653$	13.1664i	0.515242i	−0.966246	+	0.257621i	$0.917061\pi$
$654$	0	0
$655$	−13.5925	−0.531102
$656$	0	0
$657$	4.59397	0.179228
$658$	0	0
$659$	− 36.6833i	− 1.42898i	−0.699646	−	0.714490i	$-0.746659\pi$
$659$	− 36.6833i	− 1.42898i	0.699646	−	0.714490i	$-0.253341\pi$
$660$	0	0
$661$	37.8495i	1.47217i	0.676887	+	0.736087i	$0.263328\pi$
$661$	37.8495i	1.47217i	−0.676887	+	0.736087i	$0.736672\pi$
$662$	0	0
$663$	−12.8193	−0.497861
$664$	0	0
$665$	−11.4076	−0.442367
$666$	0	0
$667$	0.222876i	0.00862979i
$668$	0	0
$669$	6.94558i	0.268532i
$670$	0	0
$671$	29.7938	1.15018
$672$	0	0
$673$	13.6028	0.524348	0.262174	−	0.965021i	$-0.415560\pi$
$673$	13.6028	0.524348	0.262174	+	0.965021i	$0.415560\pi$
$674$	0	0
$675$	5.60555i	0.215758i
$676$	0	0
$677$	− 43.5908i	− 1.67533i	−0.546184	−	0.837665i	$-0.683920\pi$
$677$	− 43.5908i	− 1.67533i	0.546184	−	0.837665i	$-0.316080\pi$
$678$	0	0
$679$	5.12237	0.196579
$680$	0	0
$681$	9.13645	0.350109
$682$	0	0
$683$	7.35537i	0.281445i	0.990049	+	0.140723i	$0.0449426\pi$
$683$	7.35537i	0.281445i	−0.990049	+	0.140723i	$0.955057\pi$
$684$	0	0
$685$	− 0.703013i	− 0.0268608i
$686$	0	0
$687$	−11.7935	−0.449950
$688$	0	0
$689$	36.6611	1.39668
$690$	0	0
$691$	− 23.8974i	− 0.909100i	−0.890721	−	0.454550i	$-0.849800\pi$
$691$	− 23.8974i	− 0.909100i	0.890721	−	0.454550i	$-0.150200\pi$
$692$	0	0
$693$	− 17.9208i	− 0.680756i
$694$	0	0
$695$	−17.8944	−0.678774
$696$	0	0
$697$	−17.7459	−0.672172
$698$	0	0
$699$	16.4720i	0.623029i
$700$	0	0
$701$	42.0151i	1.58689i	0.608642	+	0.793445i	$0.291715\pi$
$701$	42.0151i	1.58689i	−0.608642	+	0.793445i	$0.708285\pi$
$702$	0	0
$703$	28.6190	1.07938
$704$	0	0
$705$	−17.0544	−0.642307
$706$	0	0
$707$	11.8446i	0.445462i
$708$	0	0
$709$	− 5.17533i	− 0.194364i	−0.995267	−	0.0971819i	$-0.969017\pi$
$709$	− 5.17533i	− 0.194364i	0.995267	−	0.0971819i	$-0.0309828\pi$
$710$	0	0
$711$	−2.44887	−0.0918397
$712$	0	0
$713$	−10.9934	−0.411705
$714$	0	0
$715$	− 23.1453i	− 0.865583i
$716$	0	0
$717$	− 0.206502i	− 0.00771196i
$718$	0	0
$719$	−4.95696	−0.184863	−0.0924316	−	0.995719i	$-0.529464\pi$
$719$	−4.95696	−0.184863	−0.0924316	+	0.995719i	$0.529464\pi$
$720$	0	0
$721$	−13.6056	−0.506697
$722$	0	0
$723$	21.6831i	0.806404i
$724$	0	0
$725$	− 0.222876i	− 0.00827740i
$726$	0	0
$727$	7.69445	0.285371	0.142686	−	0.989768i	$-0.454426\pi$
$727$	7.69445	0.285371	0.142686	+	0.989768i	$0.454426\pi$
$728$	0	0
$729$	−26.3305	−0.975205
$730$	0	0
$731$	2.74090i	0.101376i
$732$	0	0
$733$	6.03641i	0.222960i	0.993767	+	0.111480i	$0.0355591\pi$
$733$	6.03641i	0.222960i	−0.993767	+	0.111480i	$0.964441\pi$
$734$	0	0
$735$	−2.21110	−0.0815577
$736$	0	0
$737$	38.9075	1.43318
$738$	0	0
$739$	− 16.1629i	− 0.594562i	−0.954790	−	0.297281i	$-0.903920\pi$
$739$	− 16.1629i	− 0.594562i	0.954790	−	0.297281i	$-0.0960799\pi$
$740$	0	0
$741$	− 25.0056i	− 0.918603i
$742$	0	0
$743$	0.486122	0.0178341	0.00891704	−	0.999960i	$-0.497162\pi$
$743$	0.486122	0.0178341	0.00891704	+	0.999960i	$0.497162\pi$
$744$	0	0
$745$	2.10785	0.0772257
$746$	0	0
$747$	− 19.9867i	− 0.731277i
$748$	0	0
$749$	18.0000i	0.657706i
$750$	0	0
$751$	1.82254	0.0665053	0.0332527	−	0.999447i	$-0.489413\pi$
$751$	1.82254	0.0665053	0.0332527	+	0.999447i	$0.489413\pi$
$752$	0	0
$753$	8.28548	0.301940
$754$	0	0
$755$	2.98667i	0.108696i
$756$	0	0
$757$	24.8698i	0.903908i	0.892041	+	0.451954i	$0.149273\pi$
$757$	24.8698i	0.903908i	−0.892041	+	0.451954i	$0.850727\pi$
$758$	0	0
$759$	7.78227	0.282478
$760$	0	0
$761$	20.4480	0.741240	0.370620	−	0.928785i	$-0.379145\pi$
$761$	20.4480	0.741240	0.370620	+	0.928785i	$0.379145\pi$
$762$	0	0
$763$	− 8.72647i	− 0.315919i
$764$	0	0
$765$	− 3.30856i	− 0.119621i
$766$	0	0
$767$	25.2361	0.911222
$768$	0	0
$769$	−24.2407	−0.874141	−0.437071	−	0.899427i	$-0.643984\pi$
$769$	−24.2407	−0.874141	−0.437071	+	0.899427i	$0.643984\pi$
$770$	0	0
$771$	9.44427i	0.340127i
$772$	0	0
$773$	6.06619i	0.218186i	0.994032	+	0.109093i	$0.0347946\pi$
$773$	6.06619i	0.218186i	−0.994032	+	0.109093i	$0.965205\pi$
$774$	0	0
$775$	10.9934	0.394894
$776$	0	0
$777$	−17.3314	−0.621760
$778$	0	0
$779$	− 34.6154i	− 1.24022i
$780$	0	0
$781$	15.1707i	0.542851i
$782$	0	0
$783$	1.24934	0.0446478
$784$	0	0
$785$	19.0265	0.679084
$786$	0	0
$787$	11.8446i	0.422215i	0.977463	+	0.211107i	$0.0677069\pi$
$787$	11.8446i	0.422215i	−0.977463	+	0.211107i	$0.932293\pi$
$788$	0	0
$789$	31.0403i	1.10506i
$790$	0	0
$791$	39.2992	1.39732
$792$	0	0
$793$	19.3248	0.686244
$794$	0	0
$795$	− 12.3268i	− 0.437186i
$796$	0	0
$797$	− 17.0016i	− 0.602229i	−0.953588	−	0.301114i	$-0.902641\pi$
$797$	− 17.0016i	− 0.602229i	0.953588	−	0.301114i	$-0.0973586\pi$
$798$	0	0
$799$	33.2458	1.17615
$800$	0	0
$801$	−6.25394	−0.220972
$802$	0	0
$803$	21.0647i	0.743357i
$804$	0	0
$805$	− 2.30278i	− 0.0811622i
$806$	0	0
$807$	10.8520	0.382008
$808$	0	0
$809$	−21.4595	−0.754474	−0.377237	−	0.926117i	$-0.623126\pi$
$809$	−21.4595	−0.754474	−0.377237	+	0.926117i	$0.623126\pi$
$810$	0	0
$811$	− 38.2083i	− 1.34168i	−0.741604	−	0.670838i	$-0.765935\pi$
$811$	− 38.2083i	− 1.34168i	0.741604	−	0.670838i	$-0.234065\pi$
$812$	0	0
$813$	− 28.5990i	− 1.00301i
$814$	0	0
$815$	−16.0271	−0.561405
$816$	0	0
$817$	−5.34646	−0.187049
$818$	0	0
$819$	− 11.6238i	− 0.406167i
$820$	0	0
$821$	− 22.4235i	− 0.782585i	−0.920266	−	0.391293i	$-0.872028\pi$
$821$	− 22.4235i	− 0.782585i	0.920266	−	0.391293i	$-0.127972\pi$
$822$	0	0
$823$	39.3199	1.37061	0.685303	−	0.728258i	$-0.259670\pi$
$823$	39.3199	1.37061	0.685303	+	0.728258i	$0.259670\pi$
$824$	0	0
$825$	−7.78227	−0.270944
$826$	0	0
$827$	21.3564i	0.742633i	0.928506	+	0.371317i	$0.121094\pi$
$827$	21.3564i	0.742633i	−0.928506	+	0.371317i	$0.878906\pi$
$828$	0	0
$829$	24.3930i	0.847203i	0.905849	+	0.423601i	$0.139234\pi$
$829$	24.3930i	0.847203i	−0.905849	+	0.423601i	$0.860766\pi$
$830$	0	0
$831$	33.4766	1.16129
$832$	0	0
$833$	4.31032	0.149344
$834$	0	0
$835$	2.89591i	0.100217i
$836$	0	0
$837$	61.6239i	2.13003i
$838$	0	0
$839$	−10.6669	−0.368264	−0.184132	−	0.982902i	$-0.558947\pi$
$839$	−10.6669	−0.368264	−0.184132	+	0.982902i	$0.558947\pi$
$840$	0	0
$841$	28.9503	0.998287
$842$	0	0
$843$	4.54653i	0.156591i
$844$	0	0
$845$	− 2.01242i	− 0.0692293i
$846$	0	0
$847$	56.8416	1.95310
$848$	0	0
$849$	26.6877	0.915920
$850$	0	0
$851$	5.77712i	0.198037i
$852$	0	0
$853$	− 7.33523i	− 0.251153i	−0.992084	−	0.125577i	$-0.959922\pi$
$853$	− 7.33523i	− 0.251153i	0.992084	−	0.125577i	$-0.0400781\pi$
$854$	0	0
$855$	6.45374	0.220713
$856$	0	0
$857$	−38.9075	−1.32905	−0.664527	−	0.747264i	$-0.731367\pi$
$857$	−38.9075	−1.32905	−0.664527	+	0.747264i	$0.731367\pi$
$858$	0	0
$859$	46.4179i	1.58376i	0.610677	+	0.791880i	$0.290897\pi$
$859$	46.4179i	1.58376i	−0.610677	+	0.791880i	$0.709103\pi$
$860$	0	0
$861$	20.9627i	0.714408i
$862$	0	0
$863$	9.86647	0.335859	0.167929	−	0.985799i	$-0.446292\pi$
$863$	9.86647	0.335859	0.167929	+	0.985799i	$0.446292\pi$
$864$	0	0
$865$	0.950254	0.0323096
$866$	0	0
$867$	13.7447i	0.466793i
$868$	0	0
$869$	− 11.2288i	− 0.380910i
$870$	0	0
$871$	25.2361	0.855092
$872$	0	0
$873$	−2.89794	−0.0980803
$874$	0	0
$875$	2.30278i	0.0778480i
$876$	0	0
$877$	3.86918i	0.130653i	0.997864	+	0.0653264i	$0.0208089\pi$
$877$	3.86918i	0.130653i	−0.997864	+	0.0653264i	$0.979191\pi$
$878$	0	0
$879$	−12.1052	−0.408298
$880$	0	0
$881$	−5.40603	−0.182134	−0.0910668	−	0.995845i	$-0.529028\pi$
$881$	−5.40603	−0.182134	−0.0910668	+	0.995845i	$0.529028\pi$
$882$	0	0
$883$	34.3660i	1.15651i	0.815857	+	0.578254i	$0.196266\pi$
$883$	34.3660i	1.15651i	−0.815857	+	0.578254i	$0.803734\pi$
$884$	0	0
$885$	− 8.48528i	− 0.285230i
$886$	0	0
$887$	−23.9867	−0.805396	−0.402698	−	0.915333i	$-0.631928\pi$
$887$	−23.9867	−0.805396	−0.402698	+	0.915333i	$0.631928\pi$
$888$	0	0
$889$	42.7872	1.43504
$890$	0	0
$891$	− 20.2771i	− 0.679308i
$892$	0	0
$893$	64.8499i	2.17012i
$894$	0	0
$895$	−19.8167	−0.662398
$896$	0	0
$897$	5.04772	0.168538
$898$	0	0
$899$	− 2.45016i	− 0.0817173i
$900$	0	0
$901$	24.0298i	0.800548i
$902$	0	0
$903$	3.23776	0.107746
$904$	0	0
$905$	−15.5139	−0.515699
$906$	0	0
$907$	− 7.11438i	− 0.236229i	−0.993000	−	0.118115i	$-0.962315\pi$
$907$	− 7.11438i	− 0.236229i	0.993000	−	0.118115i	$-0.0376850\pi$
$908$	0	0
$909$	− 6.70098i	− 0.222258i
$910$	0	0
$911$	20.8328	0.690223	0.345111	−	0.938562i	$-0.387841\pi$
$911$	20.8328	0.690223	0.345111	+	0.938562i	$0.387841\pi$
$912$	0	0
$913$	91.6450	3.03301
$914$	0	0
$915$	− 6.49770i	− 0.214807i
$916$	0	0
$917$	− 31.3004i	− 1.03363i
$918$	0	0
$919$	−47.0527	−1.55213	−0.776063	−	0.630655i	$-0.782786\pi$
$919$	−47.0527	−1.55213	−0.776063	+	0.630655i	$0.782786\pi$
$920$	0	0
$921$	−44.2398	−1.45775
$922$	0	0
$923$	9.84001i	0.323888i
$924$	0	0
$925$	− 5.77712i	− 0.189951i
$926$	0	0
$927$	7.69722	0.252810
$928$	0	0
$929$	−57.4924	−1.88626	−0.943132	−	0.332418i	$-0.892136\pi$
$929$	−57.4924	−1.88626	−0.943132	+	0.332418i	$0.892136\pi$
$930$	0	0
$931$	8.40778i	0.275554i
$932$	0	0
$933$	− 33.0509i	− 1.08204i
$934$	0	0
$935$	15.1707	0.496136
$936$	0	0
$937$	−11.9713	−0.391086	−0.195543	−	0.980695i	$-0.562647\pi$
$937$	−11.9713	−0.391086	−0.195543	+	0.980695i	$0.562647\pi$
$938$	0	0
$939$	42.0050i	1.37078i
$940$	0	0
$941$	− 3.22204i	− 0.105035i	−0.998620	−	0.0525177i	$-0.983275\pi$
$941$	− 3.22204i	− 0.105035i	0.998620	−	0.0525177i	$-0.0167246\pi$
$942$	0	0
$943$	6.98758	0.227547
$944$	0	0
$945$	−12.9083	−0.419908
$946$	0	0
$947$	6.73527i	0.218867i	0.993994	+	0.109433i	$0.0349036\pi$
$947$	6.73527i	0.218867i	−0.993994	+	0.109433i	$0.965096\pi$
$948$	0	0
$949$	13.6629i	0.443518i
$950$	0	0
$951$	−12.9118	−0.418694
$952$	0	0
$953$	−29.1683	−0.944855	−0.472428	−	0.881370i	$-0.656622\pi$
$953$	−29.1683	−0.944855	−0.472428	+	0.881370i	$0.656622\pi$
$954$	0	0
$955$	− 7.09083i	− 0.229454i
$956$	0	0
$957$	1.73448i	0.0560678i
$958$	0	0
$959$	1.61888	0.0522764
$960$	0	0
$961$	89.8542	2.89852
$962$	0	0
$963$	− 10.1833i	− 0.328154i
$964$	0	0
$965$	− 4.48528i	− 0.144386i
$966$	0	0
$967$	48.0614	1.54555	0.772775	−	0.634680i	$-0.218868\pi$
$967$	48.0614	1.54555	0.772775	+	0.634680i	$0.218868\pi$
$968$	0	0
$969$	16.3901	0.526526
$970$	0	0
$971$	10.9254i	0.350613i	0.984514	+	0.175307i	$0.0560917\pi$
$971$	10.9254i	0.350613i	−0.984514	+	0.175307i	$0.943908\pi$
$972$	0	0
$973$	− 41.2068i	− 1.32103i
$974$	0	0
$975$	−5.04772	−0.161656
$976$	0	0
$977$	34.7394	1.11141	0.555706	−	0.831379i	$-0.312448\pi$
$977$	34.7394	1.11141	0.555706	+	0.831379i	$0.312448\pi$
$978$	0	0
$979$	− 28.6761i	− 0.916494i
$980$	0	0
$981$	4.93692i	0.157624i
$982$	0	0
$983$	−12.0419	−0.384076	−0.192038	−	0.981388i	$-0.561510\pi$
$983$	−12.0419	−0.384076	−0.192038	+	0.981388i	$0.561510\pi$
$984$	0	0
$985$	13.1555	0.419168
$986$	0	0
$987$	− 39.2725i	− 1.25006i
$988$	0	0
$989$	− 1.07925i	− 0.0343183i
$990$	0	0
$991$	−52.9868	−1.68318	−0.841591	−	0.540116i	$-0.818380\pi$
$991$	−52.9868	−1.68318	−0.841591	+	0.540116i	$0.818380\pi$
$992$	0	0
$993$	−35.2609	−1.11897
$994$	0	0
$995$	23.8549i	0.756251i
$996$	0	0
$997$	22.5382i	0.713792i	0.934144	+	0.356896i	$0.116165\pi$
$997$	22.5382i	0.713792i	−0.934144	+	0.356896i	$0.883835\pi$
$998$	0	0
$999$	32.3840	1.02458

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3680.2.f.b.1841.4		8
4.3	odd	2		920.2.f.b.461.3	✓	8
8.3	odd	2		920.2.f.b.461.6	yes	8
8.5	even	2	inner	3680.2.f.b.1841.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.f.b.461.3	✓	8	4.3	odd	2
920.2.f.b.461.6	yes	8	8.3	odd	2
3680.2.f.b.1841.4		8	1.1	even	1	trivial
3680.2.f.b.1841.5		8	8.5	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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.30278i q^{3} +1.00000i q^{5} -2.30278 q^{7} +1.30278 q^{9} +O(q^{10})\)	\(q-1.30278i q^{3} +1.00000i q^{5} -2.30278 q^{7} +1.30278 q^{9} +5.97361i q^{11} +3.87459i q^{13} +1.30278 q^{15} -2.53963 q^{17} -4.95384i q^{19} +3.00000i q^{21} +1.00000 q^{23} -1.00000 q^{25} -5.60555i q^{27} +0.222876i q^{29} -10.9934 q^{31} +7.78227 q^{33} -2.30278i q^{35} +5.77712i q^{37} +5.04772 q^{39} +6.98758 q^{41} -1.07925i q^{43} +1.30278i q^{45} -13.0908 q^{47} -1.69722 q^{49} +3.30856i q^{51} -9.46193i q^{53} -5.97361 q^{55} -6.45374 q^{57} -6.51323i q^{59} -4.98758i q^{61} -3.00000 q^{63} -3.87459 q^{65} -6.51323i q^{67} -1.30278i q^{69} +2.53963 q^{71} +3.52630 q^{73} +1.30278i q^{75} -13.7559i q^{77} -1.87973 q^{79} -3.39445 q^{81} -15.3417i q^{83} -2.53963i q^{85} +0.290357 q^{87} -4.80048 q^{89} -8.92230i q^{91} +14.3219i q^{93} +4.95384 q^{95} -2.22443 q^{97} +7.78227i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{7} - 4 q^{9}+O(q^{10}) \) 8 * q - 4 * q^7 - 4 * q^9	\( 8 q - 4 q^{7} - 4 q^{9} - 4 q^{15} - 20 q^{17} + 8 q^{23} - 8 q^{25} + 4 q^{31} + 28 q^{33} + 16 q^{39} + 12 q^{41} - 8 q^{47} - 28 q^{49} + 4 q^{55} + 40 q^{57} - 24 q^{63} - 20 q^{65} + 20 q^{71} + 24 q^{79} - 56 q^{81} - 56 q^{87} + 28 q^{95} + 12 q^{97}+O(q^{100}) \) 8 * q - 4 * q^7 - 4 * q^9 - 4 * q^15 - 20 * q^17 + 8 * q^23 - 8 * q^25 + 4 * q^31 + 28 * q^33 + 16 * q^39 + 12 * q^41 - 8 * q^47 - 28 * q^49 + 4 * q^55 + 40 * q^57 - 24 * q^63 - 20 * q^65 + 20 * q^71 + 24 * q^79 - 56 * q^81 - 56 * q^87 + 28 * q^95 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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