LMFDB - Embedded newform 2352.2.b.k.1567.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,2,Mod(1567,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, 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("2352.1567");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$18.7808145554$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.339738624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 $ x^8 - 4x^6 + 14x^4 - 8*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1567.5
Root			$0.662827 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	2352.1567
Dual form			2352.2.b.k.1567.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.601731i q^{5} +1.00000 q^{9} +0.163636i q^{11} -3.37849i q^{13} -0.601731i q^{15} +2.13246i q^{17} -0.953669 q^{19} +6.99598i q^{23} +4.63792 q^{25} -1.00000 q^{27} -1.28897 q^{29} -3.35449 q^{31} -0.163636i q^{33} +5.16373 q^{37} +3.37849i q^{39} -6.67896i q^{41} -6.09172i q^{43} +0.601731i q^{45} +3.69764 q^{47} -2.13246i q^{51} +13.8806 q^{53} -0.0984649 q^{55} +0.953669 q^{57} -8.05188 q^{59} -0.181465i q^{61} +2.03294 q^{65} +9.09299i q^{67} -6.99598i q^{69} +3.36066i q^{71} +12.1442i q^{73} -4.63792 q^{75} +6.98840i q^{79} +1.00000 q^{81} +12.9788 q^{83} -1.28317 q^{85} +1.28897 q^{87} +14.8853i q^{89} +3.35449 q^{93} -0.573852i q^{95} -9.08274i q^{97} +0.163636i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 8 q^{9} - 24 q^{25} - 8 q^{27} + 16 q^{29} + 32 q^{31} + 16 q^{47} + 16 q^{53} + 64 q^{55} - 48 q^{59} - 16 q^{65} + 24 q^{75} + 8 q^{81} - 64 q^{85} - 16 q^{87} - 32 q^{93}+O(q^{100}) $ 8 * q - 8 * q^3 + 8 * q^9 - 24 * q^25 - 8 * q^27 + 16 * q^29 + 32 * q^31 + 16 * q^47 + 16 * q^53 + 64 * q^55 - 48 * q^59 - 16 * q^65 + 24 * q^75 + 8 * q^81 - 64 * q^85 - 16 * q^87 - 32 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0.601731i	0.269102i	0.990907	+	0.134551i	$0.0429592\pi$
$5$	0.601731i	0.269102i	−0.990907	+	0.134551i	$0.957041\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.163636i	0.0493382i	0.999696	+	0.0246691i	$0.00785321\pi$
$11$	0.163636i	0.0493382i	−0.999696	+	0.0246691i	$0.992147\pi$
$12$	0	0
$13$	− 3.37849i	− 0.937025i	−0.883457	−	0.468513i	$-0.844790\pi$
$13$	− 3.37849i	− 0.937025i	0.883457	−	0.468513i	$-0.155210\pi$
$14$	0	0
$15$	− 0.601731i	− 0.155366i
$16$	0	0
$17$	2.13246i	0.517199i	0.965985	+	0.258599i	$0.0832609\pi$
$17$	2.13246i	0.517199i	−0.965985	+	0.258599i	$0.916739\pi$
$18$	0	0
$19$	−0.953669	−0.218787	−0.109393	−	0.993999i	$-0.534891\pi$
$19$	−0.953669	−0.218787	−0.109393	+	0.993999i	$0.534891\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.99598i	1.45876i	0.684107	+	0.729382i	$0.260192\pi$
$23$	6.99598i	1.45876i	−0.684107	+	0.729382i	$0.739808\pi$
$24$	0	0
$25$	4.63792	0.927584
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.28897	−0.239356	−0.119678	−	0.992813i	$-0.538186\pi$
$29$	−1.28897	−0.239356	−0.119678	+	0.992813i	$0.538186\pi$
$30$	0	0
$31$	−3.35449	−0.602485	−0.301242	−	0.953548i	$-0.597401\pi$
$31$	−3.35449	−0.602485	−0.301242	+	0.953548i	$0.597401\pi$
$32$	0	0
$33$	− 0.163636i	− 0.0284854i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.16373	0.848912	0.424456	−	0.905449i	$-0.360465\pi$
$37$	5.16373	0.848912	0.424456	+	0.905449i	$0.360465\pi$
$38$	0	0
$39$	3.37849i	0.540992i
$40$	0	0
$41$	− 6.67896i	− 1.04308i	−0.853227	−	0.521539i	$-0.825358\pi$
$41$	− 6.67896i	− 1.04308i	0.853227	−	0.521539i	$-0.174642\pi$
$42$	0	0
$43$	− 6.09172i	− 0.928978i	−0.885579	−	0.464489i	$-0.846238\pi$
$43$	− 6.09172i	− 0.928978i	0.885579	−	0.464489i	$-0.153762\pi$
$44$	0	0
$45$	0.601731i	0.0897007i
$46$	0	0
$47$	3.69764	0.539356	0.269678	−	0.962951i	$-0.413083\pi$
$47$	3.69764	0.539356	0.269678	+	0.962951i	$0.413083\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 2.13246i	− 0.298605i
$52$	0	0
$53$	13.8806	1.90664	0.953321	−	0.301959i	$-0.0976407\pi$
$53$	13.8806	1.90664	0.953321	+	0.301959i	$0.0976407\pi$
$54$	0	0
$55$	−0.0984649	−0.0132770
$56$	0	0
$57$	0.953669	0.126317
$58$	0	0
$59$	−8.05188	−1.04827	−0.524133	−	0.851637i	$-0.675610\pi$
$59$	−8.05188	−1.04827	−0.524133	+	0.851637i	$0.675610\pi$
$60$	0	0
$61$	− 0.181465i	− 0.0232342i	−0.999933	−	0.0116171i	$-0.996302\pi$
$61$	− 0.181465i	− 0.0232342i	0.999933	−	0.0116171i	$-0.00369792\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.03294	0.252156
$66$	0	0
$67$	9.09299i	1.11089i	0.831555	+	0.555443i	$0.187451\pi$
$67$	9.09299i	1.11089i	−0.831555	+	0.555443i	$0.812549\pi$
$68$	0	0
$69$	− 6.99598i	− 0.842217i
$70$	0	0
$71$	3.36066i	0.398837i	0.979914	+	0.199419i	$0.0639054\pi$
$71$	3.36066i	0.398837i	−0.979914	+	0.199419i	$0.936095\pi$
$72$	0	0
$73$	12.1442i	1.42137i	0.703509	+	0.710686i	$0.251615\pi$
$73$	12.1442i	1.42137i	−0.703509	+	0.710686i	$0.748385\pi$
$74$	0	0
$75$	−4.63792	−0.535541
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.98840i	0.786257i	0.919484	+	0.393128i	$0.128607\pi$
$79$	6.98840i	0.786257i	−0.919484	+	0.393128i	$0.871393\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.9788	1.42460	0.712302	−	0.701873i	$-0.247652\pi$
$83$	12.9788	1.42460	0.712302	+	0.701873i	$0.247652\pi$
$84$	0	0
$85$	−1.28317	−0.139179
$86$	0	0
$87$	1.28897	0.138192
$88$	0	0
$89$	14.8853i	1.57784i	0.614496	+	0.788920i	$0.289359\pi$
$89$	14.8853i	1.57784i	−0.614496	+	0.788920i	$0.710641\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.35449	0.347845
$94$	0	0
$95$	− 0.573852i	− 0.0588760i
$96$	0	0
$97$	− 9.08274i	− 0.922213i	−0.887345	−	0.461106i	$-0.847453\pi$
$97$	− 9.08274i	− 0.922213i	0.887345	−	0.461106i	$-0.152547\pi$
$98$	0	0
$99$	0.163636i	0.0164461i
$100$	0	0
$101$	8.51647i	0.847420i	0.905798	+	0.423710i	$0.139273\pi$
$101$	8.51647i	0.847420i	−0.905798	+	0.423710i	$0.860727\pi$
$102$	0	0
$103$	−16.7609	−1.65150	−0.825749	−	0.564038i	$-0.809247\pi$
$103$	−16.7609	−1.65150	−0.825749	+	0.564038i	$0.809247\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.39735i	0.425108i	0.977149	+	0.212554i	$0.0681781\pi$
$107$	4.39735i	0.425108i	−0.977149	+	0.212554i	$0.931822\pi$
$108$	0	0
$109$	11.5453	1.10584	0.552918	−	0.833236i	$-0.313514\pi$
$109$	11.5453	1.10584	0.552918	+	0.833236i	$0.313514\pi$
$110$	0	0
$111$	−5.16373	−0.490120
$112$	0	0
$113$	−1.09821	−0.103311	−0.0516554	−	0.998665i	$-0.516450\pi$
$113$	−1.09821	−0.103311	−0.0516554	+	0.998665i	$0.516450\pi$
$114$	0	0
$115$	−4.20970	−0.392556
$116$	0	0
$117$	− 3.37849i	− 0.312342i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	10.9732	0.997566
$122$	0	0
$123$	6.67896i	0.602221i
$124$	0	0
$125$	5.79943i	0.518717i
$126$	0	0
$127$	5.22625i	0.463755i	0.972745	+	0.231877i	$0.0744868\pi$
$127$	5.22625i	0.463755i	−0.972745	+	0.231877i	$0.925513\pi$
$128$	0	0
$129$	6.09172i	0.536346i
$130$	0	0
$131$	22.5319	1.96862	0.984309	−	0.176452i	$-0.0564620\pi$
$131$	22.5319	1.96862	0.984309	+	0.176452i	$0.0564620\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 0.601731i	− 0.0517887i
$136$	0	0
$137$	14.5916	1.24664	0.623322	−	0.781965i	$-0.285783\pi$
$137$	14.5916	1.24664	0.623322	+	0.781965i	$0.285783\pi$
$138$	0	0
$139$	8.30816	0.704689	0.352345	−	0.935870i	$-0.385385\pi$
$139$	8.30816	0.704689	0.352345	+	0.935870i	$0.385385\pi$
$140$	0	0
$141$	−3.69764	−0.311397
$142$	0	0
$143$	0.552844	0.0462311
$144$	0	0
$145$	− 0.775614i	− 0.0644112i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.8017	0.884906	0.442453	−	0.896792i	$-0.354108\pi$
$149$	10.8017	0.884906	0.442453	+	0.896792i	$0.354108\pi$
$150$	0	0
$151$	14.1587i	1.15222i	0.817371	+	0.576111i	$0.195430\pi$
$151$	14.1587i	1.15222i	−0.817371	+	0.576111i	$0.804570\pi$
$152$	0	0
$153$	2.13246i	0.172400i
$154$	0	0
$155$	− 2.01850i	− 0.162130i
$156$	0	0
$157$	− 1.09264i	− 0.0872021i	−0.999049	−	0.0436011i	$-0.986117\pi$
$157$	− 1.09264i	− 0.0872021i	0.999049	−	0.0436011i	$-0.0138830\pi$
$158$	0	0
$159$	−13.8806	−1.10080
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 18.0857i	− 1.41658i	−0.705922	−	0.708290i	$-0.749467\pi$
$163$	− 18.0857i	− 1.41658i	0.705922	−	0.708290i	$-0.250533\pi$
$164$	0	0
$165$	0.0984649	0.00766548
$166$	0	0
$167$	19.0196	1.47178	0.735889	−	0.677103i	$-0.236765\pi$
$167$	19.0196	1.47178	0.735889	+	0.677103i	$0.236765\pi$
$168$	0	0
$169$	1.58579	0.121984
$170$	0	0
$171$	−0.953669	−0.0729289
$172$	0	0
$173$	8.25540i	0.627646i	0.949481	+	0.313823i	$0.101610\pi$
$173$	8.25540i	0.627646i	−0.949481	+	0.313823i	$0.898390\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.05188	0.605216
$178$	0	0
$179$	− 5.49647i	− 0.410825i	−0.978675	−	0.205413i	$-0.934146\pi$
$179$	− 5.49647i	− 0.410825i	0.978675	−	0.205413i	$-0.0658536\pi$
$180$	0	0
$181$	− 11.8519i	− 0.880946i	−0.897766	−	0.440473i	$-0.854811\pi$
$181$	− 11.8519i	− 0.880946i	0.897766	−	0.440473i	$-0.145189\pi$
$182$	0	0
$183$	0.181465i	0.0134143i
$184$	0	0
$185$	3.10717i	0.228444i
$186$	0	0
$187$	−0.348948	−0.0255176
$188$	0	0
$189$	0	0
$190$	0	0
$191$	23.1483i	1.67495i	0.546474	+	0.837476i	$0.315970\pi$
$191$	23.1483i	1.67495i	−0.546474	+	0.837476i	$0.684030\pi$
$192$	0	0
$193$	−0.512058	−0.0368588	−0.0184294	−	0.999830i	$-0.505867\pi$
$193$	−0.512058	−0.0368588	−0.0184294	+	0.999830i	$0.505867\pi$
$194$	0	0
$195$	−2.03294	−0.145582
$196$	0	0
$197$	−12.7934	−0.911495	−0.455748	−	0.890109i	$-0.650628\pi$
$197$	−12.7934	−0.911495	−0.455748	+	0.890109i	$0.650628\pi$
$198$	0	0
$199$	15.8072	1.12054	0.560271	−	0.828309i	$-0.310697\pi$
$199$	15.8072	1.12054	0.560271	+	0.828309i	$0.310697\pi$
$200$	0	0
$201$	− 9.09299i	− 0.641370i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.01893	0.280695
$206$	0	0
$207$	6.99598i	0.486254i
$208$	0	0
$209$	− 0.156055i	− 0.0107945i
$210$	0	0
$211$	0.191712i	0.0131980i	0.999978	+	0.00659899i	$0.00210054\pi$
$211$	0.191712i	0.0131980i	−0.999978	+	0.00659899i	$0.997899\pi$
$212$	0	0
$213$	− 3.36066i	− 0.230269i
$214$	0	0
$215$	3.66557	0.249990
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 12.1442i	− 0.820630i
$220$	0	0
$221$	7.20452	0.484628
$222$	0	0
$223$	−10.0111	−0.670392	−0.335196	−	0.942148i	$-0.608803\pi$
$223$	−10.0111	−0.670392	−0.335196	+	0.942148i	$0.608803\pi$
$224$	0	0
$225$	4.63792	0.309195
$226$	0	0
$227$	−6.13633	−0.407283	−0.203641	−	0.979046i	$-0.565278\pi$
$227$	−6.13633	−0.407283	−0.203641	+	0.979046i	$0.565278\pi$
$228$	0	0
$229$	− 9.44500i	− 0.624143i	−0.950059	−	0.312072i	$-0.898977\pi$
$229$	− 9.44500i	− 0.624143i	0.950059	−	0.312072i	$-0.101023\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.2217	−1.45579	−0.727895	−	0.685689i	$-0.759501\pi$
$233$	−22.2217	−1.45579	−0.727895	+	0.685689i	$0.759501\pi$
$234$	0	0
$235$	2.22498i	0.145142i
$236$	0	0
$237$	− 6.98840i	− 0.453945i
$238$	0	0
$239$	20.7962i	1.34520i	0.740008	+	0.672598i	$0.234822\pi$
$239$	20.7962i	1.34520i	−0.740008	+	0.672598i	$0.765178\pi$
$240$	0	0
$241$	13.0554i	0.840971i	0.907299	+	0.420486i	$0.138140\pi$
$241$	13.0554i	0.840971i	−0.907299	+	0.420486i	$0.861860\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.22196i	0.205009i
$248$	0	0
$249$	−12.9788	−0.822496
$250$	0	0
$251$	17.8975	1.12968	0.564839	−	0.825201i	$-0.308938\pi$
$251$	17.8975	1.12968	0.564839	+	0.825201i	$0.308938\pi$
$252$	0	0
$253$	−1.14480	−0.0719727
$254$	0	0
$255$	1.28317	0.0803552
$256$	0	0
$257$	6.67896i	0.416622i	0.978063	+	0.208311i	$0.0667966\pi$
$257$	6.67896i	0.416622i	−0.978063	+	0.208311i	$0.933203\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.28897	−0.0797854
$262$	0	0
$263$	− 24.4164i	− 1.50558i	−0.658261	−	0.752790i	$-0.728708\pi$
$263$	− 24.4164i	− 1.50558i	0.658261	−	0.752790i	$-0.271292\pi$
$264$	0	0
$265$	8.35236i	0.513081i
$266$	0	0
$267$	− 14.8853i	− 0.910966i
$268$	0	0
$269$	1.14593i	0.0698685i	0.999390	+	0.0349342i	$0.0111222\pi$
$269$	1.14593i	0.0698685i	−0.999390	+	0.0349342i	$0.988878\pi$
$270$	0	0
$271$	−5.65157	−0.343308	−0.171654	−	0.985157i	$-0.554911\pi$
$271$	−5.65157	−0.343308	−0.171654	+	0.985157i	$0.554911\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.758932i	0.0457653i
$276$	0	0
$277$	−19.2758	−1.15817	−0.579087	−	0.815266i	$-0.696591\pi$
$277$	−19.2758	−1.15817	−0.579087	+	0.815266i	$0.696591\pi$
$278$	0	0
$279$	−3.35449	−0.200828
$280$	0	0
$281$	8.80369	0.525184	0.262592	−	0.964907i	$-0.415423\pi$
$281$	8.80369	0.525184	0.262592	+	0.964907i	$0.415423\pi$
$282$	0	0
$283$	−26.1451	−1.55416	−0.777081	−	0.629401i	$-0.783300\pi$
$283$	−26.1451	−1.55416	−0.777081	+	0.629401i	$0.783300\pi$
$284$	0	0
$285$	0.573852i	0.0339920i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	12.4526	0.732506
$290$	0	0
$291$	9.08274i	0.532440i
$292$	0	0
$293$	18.2640i	1.06699i	0.845802	+	0.533497i	$0.179123\pi$
$293$	18.2640i	1.06699i	−0.845802	+	0.533497i	$0.820877\pi$
$294$	0	0
$295$	− 4.84506i	− 0.282090i
$296$	0	0
$297$	− 0.163636i	− 0.00949513i
$298$	0	0
$299$	23.6359	1.36690
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 8.51647i	− 0.489258i
$304$	0	0
$305$	0.109193	0.00625237
$306$	0	0
$307$	−21.4472	−1.22405	−0.612027	−	0.790837i	$-0.709646\pi$
$307$	−21.4472	−1.22405	−0.612027	+	0.790837i	$0.709646\pi$
$308$	0	0
$309$	16.7609	0.953492
$310$	0	0
$311$	14.4800	0.821085	0.410543	−	0.911841i	$-0.365339\pi$
$311$	14.4800	0.821085	0.410543	+	0.911841i	$0.365339\pi$
$312$	0	0
$313$	− 1.48483i	− 0.0839275i	−0.999119	−	0.0419638i	$-0.986639\pi$
$313$	− 1.48483i	− 0.0839275i	0.999119	−	0.0419638i	$-0.0133614\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.4908	−0.645389	−0.322695	−	0.946503i	$-0.604589\pi$
$317$	−11.4908	−0.645389	−0.322695	+	0.946503i	$0.604589\pi$
$318$	0	0
$319$	− 0.210922i	− 0.0118094i
$320$	0	0
$321$	− 4.39735i	− 0.245436i
$322$	0	0
$323$	− 2.03366i	− 0.113156i
$324$	0	0
$325$	− 15.6692i	− 0.869170i
$326$	0	0
$327$	−11.5453	−0.638454
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 21.0967i	− 1.15958i	−0.814766	−	0.579790i	$-0.803134\pi$
$331$	− 21.0967i	− 1.15958i	0.814766	−	0.579790i	$-0.196866\pi$
$332$	0	0
$333$	5.16373	0.282971
$334$	0	0
$335$	−5.47153	−0.298942
$336$	0	0
$337$	−5.62683	−0.306513	−0.153256	−	0.988186i	$-0.548976\pi$
$337$	−5.62683	−0.306513	−0.153256	+	0.988186i	$0.548976\pi$
$338$	0	0
$339$	1.09821	0.0596465
$340$	0	0
$341$	− 0.548917i	− 0.0297255i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	4.20970	0.226643
$346$	0	0
$347$	− 24.3767i	− 1.30861i	−0.756231	−	0.654305i	$-0.772961\pi$
$347$	− 24.3767i	− 1.30861i	0.756231	−	0.654305i	$-0.227039\pi$
$348$	0	0
$349$	2.84502i	0.152290i	0.997097	+	0.0761451i	$0.0242612\pi$
$349$	2.84502i	0.152290i	−0.997097	+	0.0761451i	$0.975739\pi$
$350$	0	0
$351$	3.37849i	0.180331i
$352$	0	0
$353$	− 24.6581i	− 1.31242i	−0.754580	−	0.656208i	$-0.772159\pi$
$353$	− 24.6581i	− 1.31242i	0.754580	−	0.656208i	$-0.227841\pi$
$354$	0	0
$355$	−2.02221	−0.107328
$356$	0	0
$357$	0	0
$358$	0	0
$359$	35.6193i	1.87991i	0.341292	+	0.939957i	$0.389136\pi$
$359$	35.6193i	1.87991i	−0.341292	+	0.939957i	$0.610864\pi$
$360$	0	0
$361$	−18.0905	−0.952132
$362$	0	0
$363$	−10.9732	−0.575945
$364$	0	0
$365$	−7.30754	−0.382494
$366$	0	0
$367$	23.2676	1.21456	0.607280	−	0.794488i	$-0.292260\pi$
$367$	23.2676	1.21456	0.607280	+	0.794488i	$0.292260\pi$
$368$	0	0
$369$	− 6.67896i	− 0.347693i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−21.9732	−1.13773	−0.568865	−	0.822431i	$-0.692617\pi$
$373$	−21.9732	−1.13773	−0.568865	+	0.822431i	$0.692617\pi$
$374$	0	0
$375$	− 5.79943i	− 0.299481i
$376$	0	0
$377$	4.35478i	0.224283i
$378$	0	0
$379$	− 19.7876i	− 1.01642i	−0.861233	−	0.508211i	$-0.830307\pi$
$379$	− 19.7876i	− 1.01642i	0.861233	−	0.508211i	$-0.169693\pi$
$380$	0	0
$381$	− 5.22625i	− 0.267749i
$382$	0	0
$383$	13.6569	0.697833	0.348916	−	0.937154i	$-0.386550\pi$
$383$	13.6569	0.697833	0.348916	+	0.937154i	$0.386550\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 6.09172i	− 0.309659i
$388$	0	0
$389$	−13.0501	−0.661666	−0.330833	−	0.943689i	$-0.607330\pi$
$389$	−13.0501	−0.661666	−0.330833	+	0.943689i	$0.607330\pi$
$390$	0	0
$391$	−14.9187	−0.754470
$392$	0	0
$393$	−22.5319	−1.13658
$394$	0	0
$395$	−4.20514	−0.211583
$396$	0	0
$397$	− 6.52604i	− 0.327533i	−0.986499	−	0.163766i	$-0.947636\pi$
$397$	− 6.52604i	− 0.327533i	0.986499	−	0.163766i	$-0.0523643\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−31.4311	−1.56959	−0.784797	−	0.619753i	$-0.787233\pi$
$401$	−31.4311	−1.56959	−0.784797	+	0.619753i	$0.787233\pi$
$402$	0	0
$403$	11.3331i	0.564544i
$404$	0	0
$405$	0.601731i	0.0299002i
$406$	0	0
$407$	0.844973i	0.0418838i
$408$	0	0
$409$	− 34.8221i	− 1.72184i	−0.508739	−	0.860921i	$-0.669888\pi$
$409$	− 34.8221i	− 1.72184i	0.508739	−	0.860921i	$-0.330112\pi$
$410$	0	0
$411$	−14.5916	−0.719750
$412$	0	0
$413$	0	0
$414$	0	0
$415$	7.80972i	0.383364i
$416$	0	0
$417$	−8.30816	−0.406852
$418$	0	0
$419$	−11.9597	−0.584271	−0.292135	−	0.956377i	$-0.594366\pi$
$419$	−11.9597	−0.584271	−0.292135	+	0.956377i	$0.594366\pi$
$420$	0	0
$421$	22.6274	1.10279	0.551396	−	0.834243i	$-0.314095\pi$
$421$	22.6274	1.10279	0.551396	+	0.834243i	$0.314095\pi$
$422$	0	0
$423$	3.69764	0.179785
$424$	0	0
$425$	9.89020i	0.479745i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−0.552844	−0.0266915
$430$	0	0
$431$	− 19.1460i	− 0.922230i	−0.887340	−	0.461115i	$-0.847450\pi$
$431$	− 19.1460i	− 0.922230i	0.887340	−	0.461115i	$-0.152550\pi$
$432$	0	0
$433$	1.66205i	0.0798730i	0.999202	+	0.0399365i	$0.0127156\pi$
$433$	1.66205i	0.0798730i	−0.999202	+	0.0399365i	$0.987284\pi$
$434$	0	0
$435$	0.775614i	0.0371878i
$436$	0	0
$437$	− 6.67185i	− 0.319158i
$438$	0	0
$439$	−16.5586	−0.790301	−0.395151	−	0.918616i	$-0.629308\pi$
$439$	−16.5586	−0.790301	−0.395151	+	0.918616i	$0.629308\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 18.3344i	− 0.871096i	−0.900166	−	0.435548i	$-0.856555\pi$
$443$	− 18.3344i	− 0.871096i	0.900166	−	0.435548i	$-0.143445\pi$
$444$	0	0
$445$	−8.95695	−0.424600
$446$	0	0
$447$	−10.8017	−0.510901
$448$	0	0
$449$	18.9899	0.896187	0.448093	−	0.893987i	$-0.352103\pi$
$449$	18.9899	0.896187	0.448093	+	0.893987i	$0.352103\pi$
$450$	0	0
$451$	1.09292	0.0514636
$452$	0	0
$453$	− 14.1587i	− 0.665236i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−16.8395	−0.787719	−0.393860	−	0.919171i	$-0.628860\pi$
$457$	−16.8395	−0.787719	−0.393860	+	0.919171i	$0.628860\pi$
$458$	0	0
$459$	− 2.13246i	− 0.0995349i
$460$	0	0
$461$	36.8939i	1.71832i	0.511708	+	0.859160i	$0.329013\pi$
$461$	36.8939i	1.71832i	−0.511708	+	0.859160i	$0.670987\pi$
$462$	0	0
$463$	35.2394i	1.63771i	0.573997	+	0.818857i	$0.305392\pi$
$463$	35.2394i	1.63771i	−0.573997	+	0.818857i	$0.694608\pi$
$464$	0	0
$465$	2.01850i	0.0936058i
$466$	0	0
$467$	35.8246	1.65776	0.828882	−	0.559423i	$-0.188977\pi$
$467$	35.8246	1.65776	0.828882	+	0.559423i	$0.188977\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	1.09264i	0.0503462i
$472$	0	0
$473$	0.996826	0.0458341
$474$	0	0
$475$	−4.42304	−0.202943
$476$	0	0
$477$	13.8806	0.635547
$478$	0	0
$479$	6.10594	0.278988	0.139494	−	0.990223i	$-0.455452\pi$
$479$	6.10594	0.278988	0.139494	+	0.990223i	$0.455452\pi$
$480$	0	0
$481$	− 17.4456i	− 0.795452i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.46536	0.248169
$486$	0	0
$487$	− 17.0071i	− 0.770663i	−0.922778	−	0.385332i	$-0.874087\pi$
$487$	− 17.0071i	− 0.770663i	0.922778	−	0.385332i	$-0.125913\pi$
$488$	0	0
$489$	18.0857i	0.817863i
$490$	0	0
$491$	− 31.2897i	− 1.41209i	−0.708169	−	0.706043i	$-0.750479\pi$
$491$	− 31.2897i	− 1.41209i	0.708169	−	0.706043i	$-0.249521\pi$
$492$	0	0
$493$	− 2.74869i	− 0.123795i
$494$	0	0
$495$	−0.0984649	−0.00442567
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 21.2515i	− 0.951347i	−0.879622	−	0.475674i	$-0.842204\pi$
$499$	− 21.2515i	− 0.951347i	0.879622	−	0.475674i	$-0.157796\pi$
$500$	0	0
$501$	−19.0196	−0.849731
$502$	0	0
$503$	0.728285	0.0324726	0.0162363	−	0.999868i	$-0.494832\pi$
$503$	0.728285	0.0324726	0.0162363	+	0.999868i	$0.494832\pi$
$504$	0	0
$505$	−5.12462	−0.228043
$506$	0	0
$507$	−1.58579	−0.0704272
$508$	0	0
$509$	31.6424i	1.40253i	0.712903	+	0.701263i	$0.247380\pi$
$509$	31.6424i	1.40253i	−0.712903	+	0.701263i	$0.752620\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0.953669	0.0421055
$514$	0	0
$515$	− 10.0855i	− 0.444421i
$516$	0	0
$517$	0.605068i	0.0266108i
$518$	0	0
$519$	− 8.25540i	− 0.362372i
$520$	0	0
$521$	− 19.0336i	− 0.833878i	−0.908935	−	0.416939i	$-0.863103\pi$
$521$	− 19.0336i	− 0.833878i	0.908935	−	0.416939i	$-0.136897\pi$
$522$	0	0
$523$	4.91288	0.214825	0.107413	−	0.994215i	$-0.465743\pi$
$523$	4.91288	0.214825	0.107413	+	0.994215i	$0.465743\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 7.15334i	− 0.311604i
$528$	0	0
$529$	−25.9438	−1.12799
$530$	0	0
$531$	−8.05188	−0.349422
$532$	0	0
$533$	−22.5648	−0.977391
$534$	0	0
$535$	−2.64602	−0.114397
$536$	0	0
$537$	5.49647i	0.237190i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−28.0886	−1.20762	−0.603811	−	0.797127i	$-0.706352\pi$
$541$	−28.0886	−1.20762	−0.603811	+	0.797127i	$0.706352\pi$
$542$	0	0
$543$	11.8519i	0.508615i
$544$	0	0
$545$	6.94714i	0.297583i
$546$	0	0
$547$	− 18.5886i	− 0.794792i	−0.917647	−	0.397396i	$-0.869914\pi$
$547$	− 18.5886i	− 0.794792i	0.917647	−	0.397396i	$-0.130086\pi$
$548$	0	0
$549$	− 0.181465i	− 0.00774472i
$550$	0	0
$551$	1.22925	0.0523679
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 3.10717i	− 0.131892i
$556$	0	0
$557$	−35.9877	−1.52485	−0.762424	−	0.647078i	$-0.775991\pi$
$557$	−35.9877	−1.52485	−0.762424	+	0.647078i	$0.775991\pi$
$558$	0	0
$559$	−20.5808	−0.870476
$560$	0	0
$561$	0.348948	0.0147326
$562$	0	0
$563$	−25.3463	−1.06822	−0.534109	−	0.845415i	$-0.679353\pi$
$563$	−25.3463	−1.06822	−0.534109	+	0.845415i	$0.679353\pi$
$564$	0	0
$565$	− 0.660825i	− 0.0278011i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−17.6432	−0.739642	−0.369821	−	0.929103i	$-0.620581\pi$
$569$	−17.6432	−0.739642	−0.369821	+	0.929103i	$0.620581\pi$
$570$	0	0
$571$	− 29.9226i	− 1.25222i	−0.779733	−	0.626112i	$-0.784645\pi$
$571$	− 29.9226i	− 1.25222i	0.779733	−	0.626112i	$-0.215355\pi$
$572$	0	0
$573$	− 23.1483i	− 0.967034i
$574$	0	0
$575$	32.4468i	1.35313i
$576$	0	0
$577$	8.49884i	0.353811i	0.984228	+	0.176906i	$0.0566087\pi$
$577$	8.49884i	0.353811i	−0.984228	+	0.176906i	$0.943391\pi$
$578$	0	0
$579$	0.512058	0.0212804
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2.27136i	0.0940702i
$584$	0	0
$585$	2.03294	0.0840518
$586$	0	0
$587$	21.1156	0.871535	0.435767	−	0.900059i	$-0.356477\pi$
$587$	21.1156	0.871535	0.435767	+	0.900059i	$0.356477\pi$
$588$	0	0
$589$	3.19908	0.131816
$590$	0	0
$591$	12.7934	0.526252
$592$	0	0
$593$	− 35.8055i	− 1.47035i	−0.677875	−	0.735177i	$-0.737099\pi$
$593$	− 35.8055i	− 1.47035i	0.677875	−	0.735177i	$-0.262901\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−15.8072	−0.646945
$598$	0	0
$599$	− 10.0909i	− 0.412302i	−0.978520	−	0.206151i	$-0.933906\pi$
$599$	− 10.0909i	− 0.412302i	0.978520	−	0.206151i	$-0.0660938\pi$
$600$	0	0
$601$	31.0913i	1.26824i	0.773234	+	0.634120i	$0.218638\pi$
$601$	31.0913i	1.26824i	−0.773234	+	0.634120i	$0.781362\pi$
$602$	0	0
$603$	9.09299i	0.370295i
$604$	0	0
$605$	6.60293i	0.268447i
$606$	0	0
$607$	−21.7145	−0.881366	−0.440683	−	0.897663i	$-0.645264\pi$
$607$	−21.7145	−0.881366	−0.440683	+	0.897663i	$0.645264\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 12.4924i	− 0.505390i
$612$	0	0
$613$	−20.2810	−0.819143	−0.409571	−	0.912278i	$-0.634322\pi$
$613$	−20.2810	−0.819143	−0.409571	+	0.912278i	$0.634322\pi$
$614$	0	0
$615$	−4.01893	−0.162059
$616$	0	0
$617$	14.1442	0.569423	0.284712	−	0.958613i	$-0.408102\pi$
$617$	14.1442	0.569423	0.284712	+	0.958613i	$0.408102\pi$
$618$	0	0
$619$	42.5162	1.70887	0.854435	−	0.519559i	$-0.173904\pi$
$619$	42.5162	1.70887	0.854435	+	0.519559i	$0.173904\pi$
$620$	0	0
$621$	− 6.99598i	− 0.280739i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	19.6999	0.787996
$626$	0	0
$627$	0.156055i	0.00623223i
$628$	0	0
$629$	11.0115i	0.439056i
$630$	0	0
$631$	− 11.2426i	− 0.447561i	−0.974640	−	0.223781i	$-0.928160\pi$
$631$	− 11.2426i	− 0.447561i	0.974640	−	0.223781i	$-0.0718399\pi$
$632$	0	0
$633$	− 0.191712i	− 0.00761986i
$634$	0	0
$635$	−3.14480	−0.124797
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3.36066i	0.132946i
$640$	0	0
$641$	0.471636	0.0186285	0.00931425	−	0.999957i	$-0.497035\pi$
$641$	0.471636	0.0186285	0.00931425	+	0.999957i	$0.497035\pi$
$642$	0	0
$643$	−1.10040	−0.0433955	−0.0216977	−	0.999765i	$-0.506907\pi$
$643$	−1.10040	−0.0433955	−0.0216977	+	0.999765i	$0.506907\pi$
$644$	0	0
$645$	−3.66557	−0.144332
$646$	0	0
$647$	31.7314	1.24749	0.623746	−	0.781627i	$-0.285610\pi$
$647$	31.7314	1.24749	0.623746	+	0.781627i	$0.285610\pi$
$648$	0	0
$649$	− 1.31758i	− 0.0517195i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8.98421	−0.351579	−0.175790	−	0.984428i	$-0.556248\pi$
$653$	−8.98421	−0.351579	−0.175790	+	0.984428i	$0.556248\pi$
$654$	0	0
$655$	13.5581i	0.529759i
$656$	0	0
$657$	12.1442i	0.473791i
$658$	0	0
$659$	26.0679i	1.01546i	0.861516	+	0.507731i	$0.169516\pi$
$659$	26.0679i	1.01546i	−0.861516	+	0.507731i	$0.830484\pi$
$660$	0	0
$661$	− 1.36376i	− 0.0530441i	−0.999648	−	0.0265221i	$-0.991557\pi$
$661$	− 1.36376i	− 0.0530441i	0.999648	−	0.0265221i	$-0.00844323\pi$
$662$	0	0
$663$	−7.20452	−0.279800
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 9.01763i	− 0.349164i
$668$	0	0
$669$	10.0111	0.387051
$670$	0	0
$671$	0.0296942	0.00114633
$672$	0	0
$673$	−17.6874	−0.681799	−0.340899	−	0.940100i	$-0.610732\pi$
$673$	−17.6874	−0.681799	−0.340899	+	0.940100i	$0.610732\pi$
$674$	0	0
$675$	−4.63792	−0.178514
$676$	0	0
$677$	− 7.95083i	− 0.305575i	−0.988259	−	0.152788i	$-0.951175\pi$
$677$	− 7.95083i	− 0.305575i	0.988259	−	0.152788i	$-0.0488250\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	6.13633	0.235145
$682$	0	0
$683$	− 18.0286i	− 0.689846i	−0.938631	−	0.344923i	$-0.887905\pi$
$683$	− 18.0286i	− 0.689846i	0.938631	−	0.344923i	$-0.112095\pi$
$684$	0	0
$685$	8.78021i	0.335474i
$686$	0	0
$687$	9.44500i	0.360349i
$688$	0	0
$689$	− 46.8954i	− 1.78657i
$690$	0	0
$691$	44.4976	1.69277	0.846384	−	0.532572i	$-0.178775\pi$
$691$	44.4976	1.69277	0.846384	+	0.532572i	$0.178775\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.99928i	0.189633i
$696$	0	0
$697$	14.2426	0.539478
$698$	0	0
$699$	22.2217	0.840501
$700$	0	0
$701$	−11.8785	−0.448646	−0.224323	−	0.974515i	$-0.572017\pi$
$701$	−11.8785	−0.448646	−0.224323	+	0.974515i	$0.572017\pi$
$702$	0	0
$703$	−4.92449	−0.185731
$704$	0	0
$705$	− 2.22498i	− 0.0837977i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	24.8643	0.933798	0.466899	−	0.884311i	$-0.345371\pi$
$709$	24.8643	0.933798	0.466899	+	0.884311i	$0.345371\pi$
$710$	0	0
$711$	6.98840i	0.262086i
$712$	0	0
$713$	− 23.4680i	− 0.878883i
$714$	0	0
$715$	0.332663i	0.0124409i
$716$	0	0
$717$	− 20.7962i	− 0.776650i
$718$	0	0
$719$	−0.604720	−0.0225523	−0.0112761	−	0.999936i	$-0.503589\pi$
$719$	−0.604720	−0.0225523	−0.0112761	+	0.999936i	$0.503589\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 13.0554i	− 0.485535i
$724$	0	0
$725$	−5.97815	−0.222023
$726$	0	0
$727$	17.7628	0.658786	0.329393	−	0.944193i	$-0.393156\pi$
$727$	17.7628	0.658786	0.329393	+	0.944193i	$0.393156\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.9904	0.480466
$732$	0	0
$733$	− 15.5269i	− 0.573501i	−0.958005	−	0.286750i	$-0.907425\pi$
$733$	− 15.5269i	− 0.573501i	0.958005	−	0.286750i	$-0.0925750\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.48794	−0.0548090
$738$	0	0
$739$	17.2042i	0.632865i	0.948615	+	0.316433i	$0.102485\pi$
$739$	17.2042i	0.632865i	−0.948615	+	0.316433i	$0.897515\pi$
$740$	0	0
$741$	− 3.22196i	− 0.118362i
$742$	0	0
$743$	− 13.1996i	− 0.484247i	−0.970245	−	0.242123i	$-0.922156\pi$
$743$	− 13.1996i	− 0.484247i	0.970245	−	0.242123i	$-0.0778439\pi$
$744$	0	0
$745$	6.49968i	0.238130i
$746$	0	0
$747$	12.9788	0.474868
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 20.2946i	− 0.740561i	−0.928920	−	0.370280i	$-0.879262\pi$
$751$	− 20.2946i	− 0.740561i	0.928920	−	0.370280i	$-0.120738\pi$
$752$	0	0
$753$	−17.8975	−0.652220
$754$	0	0
$755$	−8.51975	−0.310065
$756$	0	0
$757$	24.7011	0.897778	0.448889	−	0.893587i	$-0.351820\pi$
$757$	24.7011	0.897778	0.448889	+	0.893587i	$0.351820\pi$
$758$	0	0
$759$	1.14480	0.0415535
$760$	0	0
$761$	4.77524i	0.173102i	0.996247	+	0.0865512i	$0.0275846\pi$
$761$	4.77524i	0.173102i	−0.996247	+	0.0865512i	$0.972415\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1.28317	−0.0463931
$766$	0	0
$767$	27.2032i	0.982251i
$768$	0	0
$769$	14.7721i	0.532696i	0.963877	+	0.266348i	$0.0858170\pi$
$769$	14.7721i	0.532696i	−0.963877	+	0.266348i	$0.914183\pi$
$770$	0	0
$771$	− 6.67896i	− 0.240537i
$772$	0	0
$773$	36.9175i	1.32783i	0.747807	+	0.663916i	$0.231107\pi$
$773$	36.9175i	1.32783i	−0.747807	+	0.663916i	$0.768893\pi$
$774$	0	0
$775$	−15.5579	−0.558855
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.36951i	0.228212i
$780$	0	0
$781$	−0.549926	−0.0196779
$782$	0	0
$783$	1.28897	0.0460641
$784$	0	0
$785$	0.657475	0.0234663
$786$	0	0
$787$	−37.0398	−1.32033	−0.660164	−	0.751122i	$-0.729513\pi$
$787$	−37.0398	−1.32033	−0.660164	+	0.751122i	$0.729513\pi$
$788$	0	0
$789$	24.4164i	0.869247i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−0.613077	−0.0217710
$794$	0	0
$795$	− 8.35236i	− 0.296228i
$796$	0	0
$797$	36.9752i	1.30973i	0.755746	+	0.654865i	$0.227275\pi$
$797$	36.9752i	1.30973i	−0.755746	+	0.654865i	$0.772725\pi$
$798$	0	0
$799$	7.88509i	0.278954i
$800$	0	0
$801$	14.8853i	0.525947i
$802$	0	0
$803$	−1.98723	−0.0701279
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 1.14593i	− 0.0403386i
$808$	0	0
$809$	5.34100	0.187780	0.0938898	−	0.995583i	$-0.470070\pi$
$809$	5.34100	0.187780	0.0938898	+	0.995583i	$0.470070\pi$
$810$	0	0
$811$	37.4180	1.31392	0.656961	−	0.753924i	$-0.271841\pi$
$811$	37.4180	1.31392	0.656961	+	0.753924i	$0.271841\pi$
$812$	0	0
$813$	5.65157	0.198209
$814$	0	0
$815$	10.8827	0.381205
$816$	0	0
$817$	5.80948i	0.203248i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	20.3460	0.710081	0.355041	−	0.934851i	$-0.384467\pi$
$821$	20.3460	0.710081	0.355041	+	0.934851i	$0.384467\pi$
$822$	0	0
$823$	− 46.3367i	− 1.61520i	−0.589734	−	0.807598i	$-0.700767\pi$
$823$	− 46.3367i	− 1.61520i	0.589734	−	0.807598i	$-0.299233\pi$
$824$	0	0
$825$	− 0.758932i	− 0.0264226i
$826$	0	0
$827$	2.20358i	0.0766260i	0.999266	+	0.0383130i	$0.0121984\pi$
$827$	2.20358i	0.0766260i	−0.999266	+	0.0383130i	$0.987802\pi$
$828$	0	0
$829$	− 17.2639i	− 0.599599i	−0.954002	−	0.299800i	$-0.903080\pi$
$829$	− 17.2639i	− 0.599599i	0.954002	−	0.299800i	$-0.0969199\pi$
$830$	0	0
$831$	19.2758	0.668671
$832$	0	0
$833$	0	0
$834$	0	0
$835$	11.4446i	0.396058i
$836$	0	0
$837$	3.35449	0.115948
$838$	0	0
$839$	−17.4781	−0.603410	−0.301705	−	0.953401i	$-0.597556\pi$
$839$	−17.4781	−0.603410	−0.301705	+	0.953401i	$0.597556\pi$
$840$	0	0
$841$	−27.3386	−0.942709
$842$	0	0
$843$	−8.80369	−0.303215
$844$	0	0
$845$	0.954216i	0.0328260i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	26.1451	0.897296
$850$	0	0
$851$	36.1254i	1.23836i
$852$	0	0
$853$	− 30.7781i	− 1.05382i	−0.849920	−	0.526912i	$-0.823350\pi$
$853$	− 30.7781i	− 1.05382i	0.849920	−	0.526912i	$-0.176650\pi$
$854$	0	0
$855$	− 0.573852i	− 0.0196253i
$856$	0	0
$857$	23.4865i	0.802283i	0.916016	+	0.401142i	$0.131386\pi$
$857$	23.4865i	0.802283i	−0.916016	+	0.401142i	$0.868614\pi$
$858$	0	0
$859$	−42.2249	−1.44070	−0.720348	−	0.693613i	$-0.756018\pi$
$859$	−42.2249	−1.44070	−0.720348	+	0.693613i	$0.756018\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.37615i	0.217047i	0.994094	+	0.108523i	$0.0346122\pi$
$863$	6.37615i	0.217047i	−0.994094	+	0.108523i	$0.965388\pi$
$864$	0	0
$865$	−4.96753	−0.168901
$866$	0	0
$867$	−12.4526	−0.422912
$868$	0	0
$869$	−1.14356	−0.0387925
$870$	0	0
$871$	30.7206	1.04093
$872$	0	0
$873$	− 9.08274i	− 0.307404i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−12.2199	−0.412638	−0.206319	−	0.978485i	$-0.566148\pi$
$877$	−12.2199	−0.412638	−0.206319	+	0.978485i	$0.566148\pi$
$878$	0	0
$879$	− 18.2640i	− 0.616030i
$880$	0	0
$881$	− 45.0545i	− 1.51793i	−0.651134	−	0.758963i	$-0.725706\pi$
$881$	− 45.0545i	− 1.51793i	0.651134	−	0.758963i	$-0.274294\pi$
$882$	0	0
$883$	21.8481i	0.735246i	0.929975	+	0.367623i	$0.119828\pi$
$883$	21.8481i	0.735246i	−0.929975	+	0.367623i	$0.880172\pi$
$884$	0	0
$885$	4.84506i	0.162865i
$886$	0	0
$887$	−32.2604	−1.08320	−0.541599	−	0.840637i	$-0.682181\pi$
$887$	−32.2604	−1.08320	−0.541599	+	0.840637i	$0.682181\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0.163636i	0.00548202i
$892$	0	0
$893$	−3.52632	−0.118004
$894$	0	0
$895$	3.30739	0.110554
$896$	0	0
$897$	−23.6359	−0.789179
$898$	0	0
$899$	4.32385	0.144208
$900$	0	0
$901$	29.5998i	0.986112i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7.13166	0.237065
$906$	0	0
$907$	17.5626i	0.583158i	0.956547	+	0.291579i	$0.0941806\pi$
$907$	17.5626i	0.583158i	−0.956547	+	0.291579i	$0.905819\pi$
$908$	0	0
$909$	8.51647i	0.282473i
$910$	0	0
$911$	0.475746i	0.0157622i	0.999969	+	0.00788108i	$0.00250865\pi$
$911$	0.475746i	0.0157622i	−0.999969	+	0.00788108i	$0.997491\pi$
$912$	0	0
$913$	2.12380i	0.0702874i
$914$	0	0
$915$	−0.109193	−0.00360980
$916$	0	0
$917$	0	0
$918$	0	0
$919$	32.8870i	1.08484i	0.840107	+	0.542420i	$0.182492\pi$
$919$	32.8870i	1.08484i	−0.840107	+	0.542420i	$0.817508\pi$
$920$	0	0
$921$	21.4472	0.706708
$922$	0	0
$923$	11.3540	0.373721
$924$	0	0
$925$	23.9490	0.787437
$926$	0	0
$927$	−16.7609	−0.550499
$928$	0	0
$929$	12.7798i	0.419293i	0.977777	+	0.209646i	$0.0672313\pi$
$929$	12.7798i	0.419293i	−0.977777	+	0.209646i	$0.932769\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−14.4800	−0.474054
$934$	0	0
$935$	− 0.209973i	− 0.00686685i
$936$	0	0
$937$	4.03304i	0.131754i	0.997828	+	0.0658768i	$0.0209844\pi$
$937$	4.03304i	0.131754i	−0.997828	+	0.0658768i	$0.979016\pi$
$938$	0	0
$939$	1.48483i	0.0484556i
$940$	0	0
$941$	58.7586i	1.91548i	0.287640	+	0.957739i	$0.407129\pi$
$941$	58.7586i	1.91548i	−0.287640	+	0.957739i	$0.592871\pi$
$942$	0	0
$943$	46.7259	1.52160
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 13.4052i	− 0.435611i	−0.975992	−	0.217805i	$-0.930110\pi$
$947$	− 13.4052i	− 0.435611i	0.975992	−	0.217805i	$-0.0698898\pi$
$948$	0	0
$949$	41.0291	1.33186
$950$	0	0
$951$	11.4908	0.372616
$952$	0	0
$953$	−1.89697	−0.0614490	−0.0307245	−	0.999528i	$-0.509781\pi$
$953$	−1.89697	−0.0614490	−0.0307245	+	0.999528i	$0.509781\pi$
$954$	0	0
$955$	−13.9290	−0.450733
$956$	0	0
$957$	0.210922i	0.00681815i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−19.7474	−0.637012
$962$	0	0
$963$	4.39735i	0.141703i
$964$	0	0
$965$	− 0.308121i	− 0.00991877i
$966$	0	0
$967$	− 4.84855i	− 0.155919i	−0.996957	−	0.0779595i	$-0.975160\pi$
$967$	− 4.84855i	− 0.155919i	0.996957	−	0.0779595i	$-0.0248405\pi$
$968$	0	0
$969$	2.03366i	0.0653307i
$970$	0	0
$971$	−37.7418	−1.21119	−0.605596	−	0.795772i	$-0.707065\pi$
$971$	−37.7418	−1.21119	−0.605596	+	0.795772i	$0.707065\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	15.6692i	0.501815i
$976$	0	0
$977$	24.6686	0.789217	0.394609	−	0.918849i	$-0.370880\pi$
$977$	24.6686	0.789217	0.394609	+	0.918849i	$0.370880\pi$
$978$	0	0
$979$	−2.43578	−0.0778477
$980$	0	0
$981$	11.5453	0.368612
$982$	0	0
$983$	−37.7500	−1.20404	−0.602019	−	0.798481i	$-0.705637\pi$
$983$	−37.7500	−1.20404	−0.602019	+	0.798481i	$0.705637\pi$
$984$	0	0
$985$	− 7.69821i	− 0.245285i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	42.6176	1.35516
$990$	0	0
$991$	55.5501i	1.76461i	0.470682	+	0.882303i	$0.344008\pi$
$991$	55.5501i	1.76461i	−0.470682	+	0.882303i	$0.655992\pi$
$992$	0	0
$993$	21.0967i	0.669484i
$994$	0	0
$995$	9.51167i	0.301540i
$996$	0	0
$997$	24.1419i	0.764583i	0.924042	+	0.382291i	$0.124865\pi$
$997$	24.1419i	0.764583i	−0.924042	+	0.382291i	$0.875135\pi$
$998$	0	0
$999$	−5.16373	−0.163373

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.b.k.1567.5	yes	8
3.2	odd	2		7056.2.b.x.1567.4		8
4.3	odd	2		2352.2.b.l.1567.5	yes	8
7.2	even	3		2352.2.bl.t.31.3		8
7.3	odd	6		2352.2.bl.o.607.3		8
7.4	even	3		2352.2.bl.r.607.2		8
7.5	odd	6		2352.2.bl.q.31.2		8
7.6	odd	2		2352.2.b.l.1567.4	yes	8
12.11	even	2		7056.2.b.w.1567.4		8
21.20	even	2		7056.2.b.w.1567.5		8
28.3	even	6		2352.2.bl.t.607.3		8
28.11	odd	6		2352.2.bl.q.607.2		8
28.19	even	6		2352.2.bl.r.31.2		8
28.23	odd	6		2352.2.bl.o.31.3		8
28.27	even	2	inner	2352.2.b.k.1567.4	✓	8
84.83	odd	2		7056.2.b.x.1567.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2352.2.b.k.1567.4	✓	8	28.27	even	2	inner
2352.2.b.k.1567.5	yes	8	1.1	even	1	trivial
2352.2.b.l.1567.4	yes	8	7.6	odd	2
2352.2.b.l.1567.5	yes	8	4.3	odd	2
2352.2.bl.o.31.3		8	28.23	odd	6
2352.2.bl.o.607.3		8	7.3	odd	6
2352.2.bl.q.31.2		8	7.5	odd	6
2352.2.bl.q.607.2		8	28.11	odd	6
2352.2.bl.r.31.2		8	28.19	even	6
2352.2.bl.r.607.2		8	7.4	even	3
2352.2.bl.t.31.3		8	7.2	even	3
2352.2.bl.t.607.3		8	28.3	even	6
7056.2.b.w.1567.4		8	12.11	even	2
7056.2.b.w.1567.5		8	21.20	even	2
7056.2.b.x.1567.4		8	3.2	odd	2
7056.2.b.x.1567.5		8	84.83	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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.601731i q^{5} +1.00000 q^{9} +0.163636i q^{11} -3.37849i q^{13} -0.601731i q^{15} +2.13246i q^{17} -0.953669 q^{19} +6.99598i q^{23} +4.63792 q^{25} -1.00000 q^{27} -1.28897 q^{29} -3.35449 q^{31} -0.163636i q^{33} +5.16373 q^{37} +3.37849i q^{39} -6.67896i q^{41} -6.09172i q^{43} +0.601731i q^{45} +3.69764 q^{47} -2.13246i q^{51} +13.8806 q^{53} -0.0984649 q^{55} +0.953669 q^{57} -8.05188 q^{59} -0.181465i q^{61} +2.03294 q^{65} +9.09299i q^{67} -6.99598i q^{69} +3.36066i q^{71} +12.1442i q^{73} -4.63792 q^{75} +6.98840i q^{79} +1.00000 q^{81} +12.9788 q^{83} -1.28317 q^{85} +1.28897 q^{87} +14.8853i q^{89} +3.35449 q^{93} -0.573852i q^{95} -9.08274i q^{97} +0.163636i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 8 q^{9} - 24 q^{25} - 8 q^{27} + 16 q^{29} + 32 q^{31} + 16 q^{47} + 16 q^{53} + 64 q^{55} - 48 q^{59} - 16 q^{65} + 24 q^{75} + 8 q^{81} - 64 q^{85} - 16 q^{87} - 32 q^{93}+O(q^{100}) \) 8 * q - 8 * q^3 + 8 * q^9 - 24 * q^25 - 8 * q^27 + 16 * q^29 + 32 * q^31 + 16 * q^47 + 16 * q^53 + 64 * q^55 - 48 * q^59 - 16 * q^65 + 24 * q^75 + 8 * q^81 - 64 * q^85 - 16 * q^87 - 32 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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