LMFDB - Embedded newform 2646.2.d.b.2645.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2646,2,Mod(2645,2646)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2646, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("2646.2645");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2646 = 2 \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2646.d (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:	$21.1284163748$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 378)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2645.4
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2646.2645
Dual form			2646.2.d.b.2645.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +3.46410 q^{5} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +3.46410 q^{5} -1.00000i q^{8} +3.46410i q^{10} -5.19615i q^{13} +1.00000 q^{16} -6.92820 q^{17} -3.46410i q^{19} -3.46410 q^{20} -6.00000i q^{23} +7.00000 q^{25} +5.19615 q^{26} -8.66025i q^{31} +1.00000i q^{32} -6.92820i q^{34} -5.00000 q^{37} +3.46410 q^{38} -3.46410i q^{40} +6.92820 q^{41} +1.00000 q^{43} +6.00000 q^{46} -6.92820 q^{47} +7.00000i q^{50} +5.19615i q^{52} -6.00000i q^{53} +6.92820 q^{59} -1.73205i q^{61} +8.66025 q^{62} -1.00000 q^{64} -18.0000i q^{65} -13.0000 q^{67} +6.92820 q^{68} +6.00000i q^{71} +6.92820i q^{73} -5.00000i q^{74} +3.46410i q^{76} -7.00000 q^{79} +3.46410 q^{80} +6.92820i q^{82} -3.46410 q^{83} -24.0000 q^{85} +1.00000i q^{86} +10.3923 q^{89} +6.00000i q^{92} -6.92820i q^{94} -12.0000i q^{95} +1.73205i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 4 q^{16} + 28 q^{25} - 20 q^{37} + 4 q^{43} + 24 q^{46} - 4 q^{64} - 52 q^{67} - 28 q^{79} - 96 q^{85}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^16 + 28 * q^25 - 20 * q^37 + 4 * q^43 + 24 * q^46 - 4 * q^64 - 52 * q^67 - 28 * q^79 - 96 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	3.46410	1.54919	0.774597	−	0.632456i	$-0.217953\pi$
$5$	3.46410	1.54919	0.774597	+	0.632456i	$0.217953\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	3.46410i	1.09545i
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	− 5.19615i	− 1.44115i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	− 5.19615i	− 1.44115i	0.693375	−	0.720577i	$-0.256123\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.92820	−1.68034	−0.840168	−	0.542326i	$-0.817544\pi$
$17$	−6.92820	−1.68034	−0.840168	+	0.542326i	$0.817544\pi$
$18$	0	0
$19$	− 3.46410i	− 0.794719i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	− 3.46410i	− 0.794719i	0.917663	−	0.397360i	$-0.130073\pi$
$20$	−3.46410	−0.774597
$21$	0	0
$22$	0	0
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	5.19615	1.01905
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	− 8.66025i	− 1.55543i	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	− 8.66025i	− 1.55543i	0.628619	−	0.777714i	$-0.283621\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	− 6.92820i	− 1.18818i
$35$	0	0
$36$	0	0
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	3.46410	0.561951
$39$	0	0
$40$	− 3.46410i	− 0.547723i
$41$	6.92820	1.08200	0.541002	−	0.841021i	$-0.318045\pi$
$41$	6.92820	1.08200	0.541002	+	0.841021i	$0.318045\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	0	0
$46$	6.00000	0.884652
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	0	0
$50$	7.00000i	0.989949i
$51$	0	0
$52$	5.19615i	0.720577i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.92820	0.901975	0.450988	−	0.892530i	$-0.351072\pi$
$59$	6.92820	0.901975	0.450988	+	0.892530i	$0.351072\pi$
$60$	0	0
$61$	− 1.73205i	− 0.221766i	−0.993833	−	0.110883i	$-0.964632\pi$
$61$	− 1.73205i	− 0.221766i	0.993833	−	0.110883i	$-0.0353679\pi$
$62$	8.66025	1.09985
$63$	0	0
$64$	−1.00000	−0.125000
$65$	− 18.0000i	− 2.23263i
$66$	0	0
$67$	−13.0000	−1.58820	−0.794101	−	0.607785i	$-0.792058\pi$
$67$	−13.0000	−1.58820	−0.794101	+	0.607785i	$0.792058\pi$
$68$	6.92820	0.840168
$69$	0	0
$70$	0	0
$71$	6.00000i	0.712069i	0.934473	+	0.356034i	$0.115871\pi$
$71$	6.00000i	0.712069i	−0.934473	+	0.356034i	$0.884129\pi$
$72$	0	0
$73$	6.92820i	0.810885i	0.914121	+	0.405442i	$0.132883\pi$
$73$	6.92820i	0.810885i	−0.914121	+	0.405442i	$0.867117\pi$
$74$	− 5.00000i	− 0.581238i
$75$	0	0
$76$	3.46410i	0.397360i
$77$	0	0
$78$	0	0
$79$	−7.00000	−0.787562	−0.393781	−	0.919204i	$-0.628833\pi$
$79$	−7.00000	−0.787562	−0.393781	+	0.919204i	$0.628833\pi$
$80$	3.46410	0.387298
$81$	0	0
$82$	6.92820i	0.765092i
$83$	−3.46410	−0.380235	−0.190117	−	0.981761i	$-0.560887\pi$
$83$	−3.46410	−0.380235	−0.190117	+	0.981761i	$0.560887\pi$
$84$	0	0
$85$	−24.0000	−2.60317
$86$	1.00000i	0.107833i
$87$	0	0
$88$	0	0
$89$	10.3923	1.10158	0.550791	−	0.834643i	$-0.314326\pi$
$89$	10.3923	1.10158	0.550791	+	0.834643i	$0.314326\pi$
$90$	0	0
$91$	0	0
$92$	6.00000i	0.625543i
$93$	0	0
$94$	− 6.92820i	− 0.714590i
$95$	− 12.0000i	− 1.23117i
$96$	0	0
$97$	1.73205i	0.175863i	0.996127	+	0.0879316i	$0.0280257\pi$
$97$	1.73205i	0.175863i	−0.996127	+	0.0879316i	$0.971974\pi$
$98$	0	0
$99$	0	0
$100$	−7.00000	−0.700000
$101$	10.3923	1.03407	0.517036	−	0.855963i	$-0.327035\pi$
$101$	10.3923	1.03407	0.517036	+	0.855963i	$0.327035\pi$
$102$	0	0
$103$	5.19615i	0.511992i	0.966678	+	0.255996i	$0.0824034\pi$
$103$	5.19615i	0.511992i	−0.966678	+	0.255996i	$0.917597\pi$
$104$	−5.19615	−0.509525
$105$	0	0
$106$	6.00000	0.582772
$107$	− 6.00000i	− 0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$	− 6.00000i	− 0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$	0	0
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	18.0000i	1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$	18.0000i	1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$	0	0
$115$	− 20.7846i	− 1.93817i
$116$	0	0
$117$	0	0
$118$	6.92820i	0.637793i
$119$	0	0
$120$	0	0
$121$	11.0000	1.00000
$122$	1.73205	0.156813
$123$	0	0
$124$	8.66025i	0.777714i
$125$	6.92820	0.619677
$126$	0	0
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	18.0000	1.57870
$131$	20.7846	1.81596	0.907980	−	0.419014i	$-0.137624\pi$
$131$	20.7846	1.81596	0.907980	+	0.419014i	$0.137624\pi$
$132$	0	0
$133$	0	0
$134$	− 13.0000i	− 1.12303i
$135$	0	0
$136$	6.92820i	0.594089i
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	0	0
$139$	− 5.19615i	− 0.440732i	−0.975417	−	0.220366i	$-0.929275\pi$
$139$	− 5.19615i	− 0.440732i	0.975417	−	0.220366i	$-0.0707252\pi$
$140$	0	0
$141$	0	0
$142$	−6.00000	−0.503509
$143$	0	0
$144$	0	0
$145$	0	0
$146$	−6.92820	−0.573382
$147$	0	0
$148$	5.00000	0.410997
$149$	6.00000i	0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$	6.00000i	0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$	0	0
$151$	1.00000	0.0813788	0.0406894	−	0.999172i	$-0.487045\pi$
$151$	1.00000	0.0813788	0.0406894	+	0.999172i	$0.487045\pi$
$152$	−3.46410	−0.280976
$153$	0	0
$154$	0	0
$155$	− 30.0000i	− 2.40966i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	− 7.00000i	− 0.556890i
$159$	0	0
$160$	3.46410i	0.273861i
$161$	0	0
$162$	0	0
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	−6.92820	−0.541002
$165$	0	0
$166$	− 3.46410i	− 0.268866i
$167$	3.46410	0.268060	0.134030	−	0.990977i	$-0.457208\pi$
$167$	3.46410	0.268060	0.134030	+	0.990977i	$0.457208\pi$
$168$	0	0
$169$	−14.0000	−1.07692
$170$	− 24.0000i	− 1.84072i
$171$	0	0
$172$	−1.00000	−0.0762493
$173$	20.7846	1.58022	0.790112	−	0.612962i	$-0.210022\pi$
$173$	20.7846	1.58022	0.790112	+	0.612962i	$0.210022\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	10.3923i	0.778936i
$179$	− 6.00000i	− 0.448461i	−0.974536	−	0.224231i	$-0.928013\pi$
$179$	− 6.00000i	− 0.448461i	0.974536	−	0.224231i	$-0.0719869\pi$
$180$	0	0
$181$	13.8564i	1.02994i	0.857209	+	0.514969i	$0.172197\pi$
$181$	13.8564i	1.02994i	−0.857209	+	0.514969i	$0.827803\pi$
$182$	0	0
$183$	0	0
$184$	−6.00000	−0.442326
$185$	−17.3205	−1.27343
$186$	0	0
$187$	0	0
$188$	6.92820	0.505291
$189$	0	0
$190$	12.0000	0.870572
$191$	− 6.00000i	− 0.434145i	−0.976156	−	0.217072i	$-0.930349\pi$
$191$	− 6.00000i	− 0.434145i	0.976156	−	0.217072i	$-0.0696508\pi$
$192$	0	0
$193$	1.00000	0.0719816	0.0359908	−	0.999352i	$-0.488541\pi$
$193$	1.00000	0.0719816	0.0359908	+	0.999352i	$0.488541\pi$
$194$	−1.73205	−0.124354
$195$	0	0
$196$	0	0
$197$	24.0000i	1.70993i	0.518686	+	0.854965i	$0.326421\pi$
$197$	24.0000i	1.70993i	−0.518686	+	0.854965i	$0.673579\pi$
$198$	0	0
$199$	1.73205i	0.122782i	0.998114	+	0.0613909i	$0.0195536\pi$
$199$	1.73205i	0.122782i	−0.998114	+	0.0613909i	$0.980446\pi$
$200$	− 7.00000i	− 0.494975i
$201$	0	0
$202$	10.3923i	0.731200i
$203$	0	0
$204$	0	0
$205$	24.0000	1.67623
$206$	−5.19615	−0.362033
$207$	0	0
$208$	− 5.19615i	− 0.360288i
$209$	0	0
$210$	0	0
$211$	25.0000	1.72107	0.860535	−	0.509390i	$-0.170129\pi$
$211$	25.0000	1.72107	0.860535	+	0.509390i	$0.170129\pi$
$212$	6.00000i	0.412082i
$213$	0	0
$214$	6.00000	0.410152
$215$	3.46410	0.236250
$216$	0	0
$217$	0	0
$218$	− 11.0000i	− 0.745014i
$219$	0	0
$220$	0	0
$221$	36.0000i	2.42162i
$222$	0	0
$223$	− 10.3923i	− 0.695920i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	− 10.3923i	− 0.695920i	0.937509	−	0.347960i	$-0.113126\pi$
$224$	0	0
$225$	0	0
$226$	−18.0000	−1.19734
$227$	10.3923	0.689761	0.344881	−	0.938647i	$-0.387919\pi$
$227$	10.3923	0.689761	0.344881	+	0.938647i	$0.387919\pi$
$228$	0	0
$229$	1.73205i	0.114457i	0.998361	+	0.0572286i	$0.0182264\pi$
$229$	1.73205i	0.114457i	−0.998361	+	0.0572286i	$0.981774\pi$
$230$	20.7846	1.37050
$231$	0	0
$232$	0	0
$233$	12.0000i	0.786146i	0.919507	+	0.393073i	$0.128588\pi$
$233$	12.0000i	0.786146i	−0.919507	+	0.393073i	$0.871412\pi$
$234$	0	0
$235$	−24.0000	−1.56559
$236$	−6.92820	−0.450988
$237$	0	0
$238$	0	0
$239$	− 6.00000i	− 0.388108i	−0.980991	−	0.194054i	$-0.937836\pi$
$239$	− 6.00000i	− 0.388108i	0.980991	−	0.194054i	$-0.0621637\pi$
$240$	0	0
$241$	− 1.73205i	− 0.111571i	−0.998443	−	0.0557856i	$-0.982234\pi$
$241$	− 1.73205i	− 0.111571i	0.998443	−	0.0557856i	$-0.0177663\pi$
$242$	11.0000i	0.707107i
$243$	0	0
$244$	1.73205i	0.110883i
$245$	0	0
$246$	0	0
$247$	−18.0000	−1.14531
$248$	−8.66025	−0.549927
$249$	0	0
$250$	6.92820i	0.438178i
$251$	−17.3205	−1.09326	−0.546630	−	0.837374i	$-0.684090\pi$
$251$	−17.3205	−1.09326	−0.546630	+	0.837374i	$0.684090\pi$
$252$	0	0
$253$	0	0
$254$	− 7.00000i	− 0.439219i
$255$	0	0
$256$	1.00000	0.0625000
$257$	10.3923	0.648254	0.324127	−	0.946014i	$-0.394929\pi$
$257$	10.3923	0.648254	0.324127	+	0.946014i	$0.394929\pi$
$258$	0	0
$259$	0	0
$260$	18.0000i	1.11631i
$261$	0	0
$262$	20.7846i	1.28408i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	− 20.7846i	− 1.27679i
$266$	0	0
$267$	0	0
$268$	13.0000	0.794101
$269$	27.7128	1.68968	0.844840	−	0.535019i	$-0.179696\pi$
$269$	27.7128	1.68968	0.844840	+	0.535019i	$0.179696\pi$
$270$	0	0
$271$	− 29.4449i	− 1.78865i	−0.447420	−	0.894324i	$-0.647657\pi$
$271$	− 29.4449i	− 1.78865i	0.447420	−	0.894324i	$-0.352343\pi$
$272$	−6.92820	−0.420084
$273$	0	0
$274$	18.0000	1.08742
$275$	0	0
$276$	0	0
$277$	−1.00000	−0.0600842	−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000	−0.0600842	−0.0300421	+	0.999549i	$0.509564\pi$
$278$	5.19615	0.311645
$279$	0	0
$280$	0	0
$281$	− 18.0000i	− 1.07379i	−0.843649	−	0.536895i	$-0.819597\pi$
$281$	− 18.0000i	− 1.07379i	0.843649	−	0.536895i	$-0.180403\pi$
$282$	0	0
$283$	19.0526i	1.13256i	0.824214	+	0.566279i	$0.191617\pi$
$283$	19.0526i	1.13256i	−0.824214	+	0.566279i	$0.808383\pi$
$284$	− 6.00000i	− 0.356034i
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	31.0000	1.82353
$290$	0	0
$291$	0	0
$292$	− 6.92820i	− 0.405442i
$293$	27.7128	1.61900	0.809500	−	0.587120i	$-0.199738\pi$
$293$	27.7128	1.61900	0.809500	+	0.587120i	$0.199738\pi$
$294$	0	0
$295$	24.0000	1.39733
$296$	5.00000i	0.290619i
$297$	0	0
$298$	−6.00000	−0.347571
$299$	−31.1769	−1.80301
$300$	0	0
$301$	0	0
$302$	1.00000i	0.0575435i
$303$	0	0
$304$	− 3.46410i	− 0.198680i
$305$	− 6.00000i	− 0.343559i
$306$	0	0
$307$	− 8.66025i	− 0.494267i	−0.968981	−	0.247133i	$-0.920511\pi$
$307$	− 8.66025i	− 0.494267i	0.968981	−	0.247133i	$-0.0794886\pi$
$308$	0	0
$309$	0	0
$310$	30.0000	1.70389
$311$	3.46410	0.196431	0.0982156	−	0.995165i	$-0.468687\pi$
$311$	3.46410	0.196431	0.0982156	+	0.995165i	$0.468687\pi$
$312$	0	0
$313$	− 6.92820i	− 0.391605i	−0.980643	−	0.195803i	$-0.937269\pi$
$313$	− 6.92820i	− 0.391605i	0.980643	−	0.195803i	$-0.0627312\pi$
$314$	0	0
$315$	0	0
$316$	7.00000	0.393781
$317$	24.0000i	1.34797i	0.738743	+	0.673987i	$0.235420\pi$
$317$	24.0000i	1.34797i	−0.738743	+	0.673987i	$0.764580\pi$
$318$	0	0
$319$	0	0
$320$	−3.46410	−0.193649
$321$	0	0
$322$	0	0
$323$	24.0000i	1.33540i
$324$	0	0
$325$	− 36.3731i	− 2.01761i
$326$	11.0000i	0.609234i
$327$	0	0
$328$	− 6.92820i	− 0.382546i
$329$	0	0
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	3.46410	0.190117
$333$	0	0
$334$	3.46410i	0.189547i
$335$	−45.0333	−2.46043
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	− 14.0000i	− 0.761500i
$339$	0	0
$340$	24.0000	1.30158
$341$	0	0
$342$	0	0
$343$	0	0
$344$	− 1.00000i	− 0.0539164i
$345$	0	0
$346$	20.7846i	1.11739i
$347$	− 24.0000i	− 1.28839i	−0.764862	−	0.644194i	$-0.777193\pi$
$347$	− 24.0000i	− 1.28839i	0.764862	−	0.644194i	$-0.222807\pi$
$348$	0	0
$349$	19.0526i	1.01986i	0.860216	+	0.509930i	$0.170329\pi$
$349$	19.0526i	1.01986i	−0.860216	+	0.509930i	$0.829671\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.92820	−0.368751	−0.184376	−	0.982856i	$-0.559026\pi$
$353$	−6.92820	−0.368751	−0.184376	+	0.982856i	$0.559026\pi$
$354$	0	0
$355$	20.7846i	1.10313i
$356$	−10.3923	−0.550791
$357$	0	0
$358$	6.00000	0.317110
$359$	− 6.00000i	− 0.316668i	−0.987386	−	0.158334i	$-0.949388\pi$
$359$	− 6.00000i	− 0.316668i	0.987386	−	0.158334i	$-0.0506123\pi$
$360$	0	0
$361$	7.00000	0.368421
$362$	−13.8564	−0.728277
$363$	0	0
$364$	0	0
$365$	24.0000i	1.25622i
$366$	0	0
$367$	− 17.3205i	− 0.904123i	−0.891987	−	0.452062i	$-0.850689\pi$
$367$	− 17.3205i	− 0.904123i	0.891987	−	0.452062i	$-0.149311\pi$
$368$	− 6.00000i	− 0.312772i
$369$	0	0
$370$	− 17.3205i	− 0.900450i
$371$	0	0
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	0	0
$376$	6.92820i	0.357295i
$377$	0	0
$378$	0	0
$379$	−23.0000	−1.18143	−0.590715	−	0.806880i	$-0.701154\pi$
$379$	−23.0000	−1.18143	−0.590715	+	0.806880i	$0.701154\pi$
$380$	12.0000i	0.615587i
$381$	0	0
$382$	6.00000	0.306987
$383$	−3.46410	−0.177007	−0.0885037	−	0.996076i	$-0.528208\pi$
$383$	−3.46410	−0.177007	−0.0885037	+	0.996076i	$0.528208\pi$
$384$	0	0
$385$	0	0
$386$	1.00000i	0.0508987i
$387$	0	0
$388$	− 1.73205i	− 0.0879316i
$389$	12.0000i	0.608424i	0.952604	+	0.304212i	$0.0983931\pi$
$389$	12.0000i	0.608424i	−0.952604	+	0.304212i	$0.901607\pi$
$390$	0	0
$391$	41.5692i	2.10225i
$392$	0	0
$393$	0	0
$394$	−24.0000	−1.20910
$395$	−24.2487	−1.22009
$396$	0	0
$397$	8.66025i	0.434646i	0.976100	+	0.217323i	$0.0697324\pi$
$397$	8.66025i	0.434646i	−0.976100	+	0.217323i	$0.930268\pi$
$398$	−1.73205	−0.0868199
$399$	0	0
$400$	7.00000	0.350000
$401$	− 24.0000i	− 1.19850i	−0.800561	−	0.599251i	$-0.795465\pi$
$401$	− 24.0000i	− 1.19850i	0.800561	−	0.599251i	$-0.204535\pi$
$402$	0	0
$403$	−45.0000	−2.24161
$404$	−10.3923	−0.517036
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	15.5885i	0.770800i	0.922750	+	0.385400i	$0.125936\pi$
$409$	15.5885i	0.770800i	−0.922750	+	0.385400i	$0.874064\pi$
$410$	24.0000i	1.18528i
$411$	0	0
$412$	− 5.19615i	− 0.255996i
$413$	0	0
$414$	0	0
$415$	−12.0000	−0.589057
$416$	5.19615	0.254762
$417$	0	0
$418$	0	0
$419$	24.2487	1.18463	0.592314	−	0.805708i	$-0.298215\pi$
$419$	24.2487	1.18463	0.592314	+	0.805708i	$0.298215\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	25.0000i	1.21698i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	−48.4974	−2.35247
$426$	0	0
$427$	0	0
$428$	6.00000i	0.290021i
$429$	0	0
$430$	3.46410i	0.167054i
$431$	36.0000i	1.73406i	0.498257	+	0.867029i	$0.333974\pi$
$431$	36.0000i	1.73406i	−0.498257	+	0.867029i	$0.666026\pi$
$432$	0	0
$433$	39.8372i	1.91445i	0.289341	+	0.957226i	$0.406564\pi$
$433$	39.8372i	1.91445i	−0.289341	+	0.957226i	$0.593436\pi$
$434$	0	0
$435$	0	0
$436$	11.0000	0.526804
$437$	−20.7846	−0.994263
$438$	0	0
$439$	− 31.1769i	− 1.48799i	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	− 31.1769i	− 1.48799i	0.668184	−	0.743996i	$-0.267072\pi$
$440$	0	0
$441$	0	0
$442$	−36.0000	−1.71235
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	36.0000	1.70656
$446$	10.3923	0.492090
$447$	0	0
$448$	0	0
$449$	− 30.0000i	− 1.41579i	−0.706319	−	0.707894i	$-0.749646\pi$
$449$	− 30.0000i	− 1.41579i	0.706319	−	0.707894i	$-0.250354\pi$
$450$	0	0
$451$	0	0
$452$	− 18.0000i	− 0.846649i
$453$	0	0
$454$	10.3923i	0.487735i
$455$	0	0
$456$	0	0
$457$	31.0000	1.45012	0.725059	−	0.688686i	$-0.241812\pi$
$457$	31.0000	1.45012	0.725059	+	0.688686i	$0.241812\pi$
$458$	−1.73205	−0.0809334
$459$	0	0
$460$	20.7846i	0.969087i
$461$	−13.8564	−0.645357	−0.322679	−	0.946509i	$-0.604583\pi$
$461$	−13.8564	−0.645357	−0.322679	+	0.946509i	$0.604583\pi$
$462$	0	0
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	0	0
$465$	0	0
$466$	−12.0000	−0.555889
$467$	−10.3923	−0.480899	−0.240449	−	0.970662i	$-0.577295\pi$
$467$	−10.3923	−0.480899	−0.240449	+	0.970662i	$0.577295\pi$
$468$	0	0
$469$	0	0
$470$	− 24.0000i	− 1.10704i
$471$	0	0
$472$	− 6.92820i	− 0.318896i
$473$	0	0
$474$	0	0
$475$	− 24.2487i	− 1.11261i
$476$	0	0
$477$	0	0
$478$	6.00000	0.274434
$479$	−3.46410	−0.158279	−0.0791394	−	0.996864i	$-0.525217\pi$
$479$	−3.46410	−0.158279	−0.0791394	+	0.996864i	$0.525217\pi$
$480$	0	0
$481$	25.9808i	1.18462i
$482$	1.73205	0.0788928
$483$	0	0
$484$	−11.0000	−0.500000
$485$	6.00000i	0.272446i
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	−1.73205	−0.0784063
$489$	0	0
$490$	0	0
$491$	− 6.00000i	− 0.270776i	−0.990793	−	0.135388i	$-0.956772\pi$
$491$	− 6.00000i	− 0.270776i	0.990793	−	0.135388i	$-0.0432281\pi$
$492$	0	0
$493$	0	0
$494$	− 18.0000i	− 0.809858i
$495$	0	0
$496$	− 8.66025i	− 0.388857i
$497$	0	0
$498$	0	0
$499$	−23.0000	−1.02962	−0.514811	−	0.857304i	$-0.672138\pi$
$499$	−23.0000	−1.02962	−0.514811	+	0.857304i	$0.672138\pi$
$500$	−6.92820	−0.309839
$501$	0	0
$502$	− 17.3205i	− 0.773052i
$503$	38.1051	1.69902	0.849512	−	0.527570i	$-0.176897\pi$
$503$	38.1051	1.69902	0.849512	+	0.527570i	$0.176897\pi$
$504$	0	0
$505$	36.0000	1.60198
$506$	0	0
$507$	0	0
$508$	7.00000	0.310575
$509$	−24.2487	−1.07481	−0.537403	−	0.843326i	$-0.680594\pi$
$509$	−24.2487	−1.07481	−0.537403	+	0.843326i	$0.680594\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	10.3923i	0.458385i
$515$	18.0000i	0.793175i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	−18.0000	−0.789352
$521$	−24.2487	−1.06236	−0.531178	−	0.847260i	$-0.678250\pi$
$521$	−24.2487	−1.06236	−0.531178	+	0.847260i	$0.678250\pi$
$522$	0	0
$523$	− 5.19615i	− 0.227212i	−0.993526	−	0.113606i	$-0.963760\pi$
$523$	− 5.19615i	− 0.227212i	0.993526	−	0.113606i	$-0.0362401\pi$
$524$	−20.7846	−0.907980
$525$	0	0
$526$	0	0
$527$	60.0000i	2.61364i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	20.7846	0.902826
$531$	0	0
$532$	0	0
$533$	− 36.0000i	− 1.55933i
$534$	0	0
$535$	− 20.7846i	− 0.898597i
$536$	13.0000i	0.561514i
$537$	0	0
$538$	27.7128i	1.19478i
$539$	0	0
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	29.4449	1.26477
$543$	0	0
$544$	− 6.92820i	− 0.297044i
$545$	−38.1051	−1.63224
$546$	0	0
$547$	−1.00000	−0.0427569	−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	−1.00000	−0.0427569	−0.0213785	+	0.999771i	$0.506805\pi$
$548$	18.0000i	0.768922i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	− 1.00000i	− 0.0424859i
$555$	0	0
$556$	5.19615i	0.220366i
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	0	0
$559$	− 5.19615i	− 0.219774i
$560$	0	0
$561$	0	0
$562$	18.0000	0.759284
$563$	10.3923	0.437983	0.218992	−	0.975727i	$-0.429723\pi$
$563$	10.3923	0.437983	0.218992	+	0.975727i	$0.429723\pi$
$564$	0	0
$565$	62.3538i	2.62325i
$566$	−19.0526	−0.800839
$567$	0	0
$568$	6.00000	0.251754
$569$	6.00000i	0.251533i	0.992060	+	0.125767i	$0.0401390\pi$
$569$	6.00000i	0.251533i	−0.992060	+	0.125767i	$0.959861\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 42.0000i	− 1.75152i
$576$	0	0
$577$	− 36.3731i	− 1.51423i	−0.653281	−	0.757115i	$-0.726608\pi$
$577$	− 36.3731i	− 1.51423i	0.653281	−	0.757115i	$-0.273392\pi$
$578$	31.0000i	1.28943i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	6.92820	0.286691
$585$	0	0
$586$	27.7128i	1.14481i
$587$	−10.3923	−0.428936	−0.214468	−	0.976731i	$-0.568802\pi$
$587$	−10.3923	−0.428936	−0.214468	+	0.976731i	$0.568802\pi$
$588$	0	0
$589$	−30.0000	−1.23613
$590$	24.0000i	0.988064i
$591$	0	0
$592$	−5.00000	−0.205499
$593$	17.3205	0.711268	0.355634	−	0.934625i	$-0.384265\pi$
$593$	17.3205	0.711268	0.355634	+	0.934625i	$0.384265\pi$
$594$	0	0
$595$	0	0
$596$	− 6.00000i	− 0.245770i
$597$	0	0
$598$	− 31.1769i	− 1.27492i
$599$	24.0000i	0.980613i	0.871550	+	0.490307i	$0.163115\pi$
$599$	24.0000i	0.980613i	−0.871550	+	0.490307i	$0.836885\pi$
$600$	0	0
$601$	− 19.0526i	− 0.777170i	−0.921413	−	0.388585i	$-0.872964\pi$
$601$	− 19.0526i	− 0.777170i	0.921413	−	0.388585i	$-0.127036\pi$
$602$	0	0
$603$	0	0
$604$	−1.00000	−0.0406894
$605$	38.1051	1.54919
$606$	0	0
$607$	− 17.3205i	− 0.703018i	−0.936185	−	0.351509i	$-0.885669\pi$
$607$	− 17.3205i	− 0.703018i	0.936185	−	0.351509i	$-0.114331\pi$
$608$	3.46410	0.140488
$609$	0	0
$610$	6.00000	0.242933
$611$	36.0000i	1.45640i
$612$	0	0
$613$	17.0000	0.686624	0.343312	−	0.939222i	$-0.388451\pi$
$613$	17.0000	0.686624	0.343312	+	0.939222i	$0.388451\pi$
$614$	8.66025	0.349499
$615$	0	0
$616$	0	0
$617$	12.0000i	0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$	12.0000i	0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$	0	0
$619$	15.5885i	0.626553i	0.949662	+	0.313276i	$0.101427\pi$
$619$	15.5885i	0.626553i	−0.949662	+	0.313276i	$0.898573\pi$
$620$	30.0000i	1.20483i
$621$	0	0
$622$	3.46410i	0.138898i
$623$	0	0
$624$	0	0
$625$	−11.0000	−0.440000
$626$	6.92820	0.276907
$627$	0	0
$628$	0	0
$629$	34.6410	1.38123
$630$	0	0
$631$	−25.0000	−0.995234	−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000	−0.995234	−0.497617	+	0.867397i	$0.665792\pi$
$632$	7.00000i	0.278445i
$633$	0	0
$634$	−24.0000	−0.953162
$635$	−24.2487	−0.962281
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	− 3.46410i	− 0.136931i
$641$	− 24.0000i	− 0.947943i	−0.880540	−	0.473972i	$-0.842820\pi$
$641$	− 24.0000i	− 0.947943i	0.880540	−	0.473972i	$-0.157180\pi$
$642$	0	0
$643$	− 43.3013i	− 1.70764i	−0.520572	−	0.853818i	$-0.674281\pi$
$643$	− 43.3013i	− 1.70764i	0.520572	−	0.853818i	$-0.325719\pi$
$644$	0	0
$645$	0	0
$646$	−24.0000	−0.944267
$647$	−20.7846	−0.817127	−0.408564	−	0.912730i	$-0.633970\pi$
$647$	−20.7846	−0.817127	−0.408564	+	0.912730i	$0.633970\pi$
$648$	0	0
$649$	0	0
$650$	36.3731	1.42667
$651$	0	0
$652$	−11.0000	−0.430793
$653$	− 6.00000i	− 0.234798i	−0.993085	−	0.117399i	$-0.962544\pi$
$653$	− 6.00000i	− 0.234798i	0.993085	−	0.117399i	$-0.0374557\pi$
$654$	0	0
$655$	72.0000	2.81327
$656$	6.92820	0.270501
$657$	0	0
$658$	0	0
$659$	18.0000i	0.701180i	0.936529	+	0.350590i	$0.114019\pi$
$659$	18.0000i	0.701180i	−0.936529	+	0.350590i	$0.885981\pi$
$660$	0	0
$661$	34.6410i	1.34738i	0.739014	+	0.673690i	$0.235292\pi$
$661$	34.6410i	1.34738i	−0.739014	+	0.673690i	$0.764708\pi$
$662$	− 8.00000i	− 0.310929i
$663$	0	0
$664$	3.46410i	0.134433i
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−3.46410	−0.134030
$669$	0	0
$670$	− 45.0333i	− 1.73979i
$671$	0	0
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	22.0000i	0.847408i
$675$	0	0
$676$	14.0000	0.538462
$677$	−41.5692	−1.59763	−0.798817	−	0.601574i	$-0.794541\pi$
$677$	−41.5692	−1.59763	−0.798817	+	0.601574i	$0.794541\pi$
$678$	0	0
$679$	0	0
$680$	24.0000i	0.920358i
$681$	0	0
$682$	0	0
$683$	18.0000i	0.688751i	0.938832	+	0.344375i	$0.111909\pi$
$683$	18.0000i	0.688751i	−0.938832	+	0.344375i	$0.888091\pi$
$684$	0	0
$685$	− 62.3538i	− 2.38242i
$686$	0	0
$687$	0	0
$688$	1.00000	0.0381246
$689$	−31.1769	−1.18775
$690$	0	0
$691$	− 32.9090i	− 1.25192i	−0.779857	−	0.625958i	$-0.784708\pi$
$691$	− 32.9090i	− 1.25192i	0.779857	−	0.625958i	$-0.215292\pi$
$692$	−20.7846	−0.790112
$693$	0	0
$694$	24.0000	0.911028
$695$	− 18.0000i	− 0.682779i
$696$	0	0
$697$	−48.0000	−1.81813
$698$	−19.0526	−0.721150
$699$	0	0
$700$	0	0
$701$	− 36.0000i	− 1.35970i	−0.733351	−	0.679851i	$-0.762045\pi$
$701$	− 36.0000i	− 1.35970i	0.733351	−	0.679851i	$-0.237955\pi$
$702$	0	0
$703$	17.3205i	0.653255i
$704$	0	0
$705$	0	0
$706$	− 6.92820i	− 0.260746i
$707$	0	0
$708$	0	0
$709$	1.00000	0.0375558	0.0187779	−	0.999824i	$-0.494022\pi$
$709$	1.00000	0.0375558	0.0187779	+	0.999824i	$0.494022\pi$
$710$	−20.7846	−0.780033
$711$	0	0
$712$	− 10.3923i	− 0.389468i
$713$	−51.9615	−1.94597
$714$	0	0
$715$	0	0
$716$	6.00000i	0.224231i
$717$	0	0
$718$	6.00000	0.223918
$719$	−3.46410	−0.129189	−0.0645946	−	0.997912i	$-0.520575\pi$
$719$	−3.46410	−0.129189	−0.0645946	+	0.997912i	$0.520575\pi$
$720$	0	0
$721$	0	0
$722$	7.00000i	0.260513i
$723$	0	0
$724$	− 13.8564i	− 0.514969i
$725$	0	0
$726$	0	0
$727$	19.0526i	0.706620i	0.935506	+	0.353310i	$0.114944\pi$
$727$	19.0526i	0.706620i	−0.935506	+	0.353310i	$0.885056\pi$
$728$	0	0
$729$	0	0
$730$	−24.0000	−0.888280
$731$	−6.92820	−0.256249
$732$	0	0
$733$	− 1.73205i	− 0.0639748i	−0.999488	−	0.0319874i	$-0.989816\pi$
$733$	− 1.73205i	− 0.0639748i	0.999488	−	0.0319874i	$-0.0101836\pi$
$734$	17.3205	0.639312
$735$	0	0
$736$	6.00000	0.221163
$737$	0	0
$738$	0	0
$739$	35.0000	1.28750	0.643748	−	0.765238i	$-0.277379\pi$
$739$	35.0000	1.28750	0.643748	+	0.765238i	$0.277379\pi$
$740$	17.3205	0.636715
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	20.7846i	0.761489i
$746$	− 22.0000i	− 0.805477i
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	−6.92820	−0.252646
$753$	0	0
$754$	0	0
$755$	3.46410	0.126072
$756$	0	0
$757$	25.0000	0.908640	0.454320	−	0.890838i	$-0.349882\pi$
$757$	25.0000	0.908640	0.454320	+	0.890838i	$0.349882\pi$
$758$	− 23.0000i	− 0.835398i
$759$	0	0
$760$	−12.0000	−0.435286
$761$	6.92820	0.251147	0.125574	−	0.992084i	$-0.459923\pi$
$761$	6.92820	0.251147	0.125574	+	0.992084i	$0.459923\pi$
$762$	0	0
$763$	0	0
$764$	6.00000i	0.217072i
$765$	0	0
$766$	− 3.46410i	− 0.125163i
$767$	− 36.0000i	− 1.29988i
$768$	0	0
$769$	6.92820i	0.249837i	0.992167	+	0.124919i	$0.0398670\pi$
$769$	6.92820i	0.249837i	−0.992167	+	0.124919i	$0.960133\pi$
$770$	0	0
$771$	0	0
$772$	−1.00000	−0.0359908
$773$	34.6410	1.24595	0.622975	−	0.782241i	$-0.285924\pi$
$773$	34.6410	1.24595	0.622975	+	0.782241i	$0.285924\pi$
$774$	0	0
$775$	− 60.6218i	− 2.17760i
$776$	1.73205	0.0621770
$777$	0	0
$778$	−12.0000	−0.430221
$779$	− 24.0000i	− 0.859889i
$780$	0	0
$781$	0	0
$782$	−41.5692	−1.48651
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.73205i	0.0617409i	0.999523	+	0.0308705i	$0.00982794\pi$
$787$	1.73205i	0.0617409i	−0.999523	+	0.0308705i	$0.990172\pi$
$788$	− 24.0000i	− 0.854965i
$789$	0	0
$790$	− 24.2487i	− 0.862730i
$791$	0	0
$792$	0	0
$793$	−9.00000	−0.319599
$794$	−8.66025	−0.307341
$795$	0	0
$796$	− 1.73205i	− 0.0613909i
$797$	−20.7846	−0.736229	−0.368114	−	0.929781i	$-0.619996\pi$
$797$	−20.7846	−0.736229	−0.368114	+	0.929781i	$0.619996\pi$
$798$	0	0
$799$	48.0000	1.69812
$800$	7.00000i	0.247487i
$801$	0	0
$802$	24.0000	0.847469
$803$	0	0
$804$	0	0
$805$	0	0
$806$	− 45.0000i	− 1.58506i
$807$	0	0
$808$	− 10.3923i	− 0.365600i
$809$	− 30.0000i	− 1.05474i	−0.849635	−	0.527372i	$-0.823177\pi$
$809$	− 30.0000i	− 1.05474i	0.849635	−	0.527372i	$-0.176823\pi$
$810$	0	0
$811$	− 10.3923i	− 0.364923i	−0.983213	−	0.182462i	$-0.941593\pi$
$811$	− 10.3923i	− 0.364923i	0.983213	−	0.182462i	$-0.0584065\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	38.1051	1.33476
$816$	0	0
$817$	− 3.46410i	− 0.121194i
$818$	−15.5885	−0.545038
$819$	0	0
$820$	−24.0000	−0.838116
$821$	18.0000i	0.628204i	0.949389	+	0.314102i	$0.101703\pi$
$821$	18.0000i	0.628204i	−0.949389	+	0.314102i	$0.898297\pi$
$822$	0	0
$823$	−49.0000	−1.70803	−0.854016	−	0.520246i	$-0.825840\pi$
$823$	−49.0000	−1.70803	−0.854016	+	0.520246i	$0.825840\pi$
$824$	5.19615	0.181017
$825$	0	0
$826$	0	0
$827$	48.0000i	1.66912i	0.550914	+	0.834562i	$0.314279\pi$
$827$	48.0000i	1.66912i	−0.550914	+	0.834562i	$0.685721\pi$
$828$	0	0
$829$	13.8564i	0.481253i	0.970618	+	0.240626i	$0.0773529\pi$
$829$	13.8564i	0.481253i	−0.970618	+	0.240626i	$0.922647\pi$
$830$	− 12.0000i	− 0.416526i
$831$	0	0
$832$	5.19615i	0.180144i
$833$	0	0
$834$	0	0
$835$	12.0000	0.415277
$836$	0	0
$837$	0	0
$838$	24.2487i	0.837658i
$839$	45.0333	1.55472	0.777361	−	0.629054i	$-0.216558\pi$
$839$	45.0333	1.55472	0.777361	+	0.629054i	$0.216558\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	− 22.0000i	− 0.758170i
$843$	0	0
$844$	−25.0000	−0.860535
$845$	−48.4974	−1.66836
$846$	0	0
$847$	0	0
$848$	− 6.00000i	− 0.206041i
$849$	0	0
$850$	− 48.4974i	− 1.66345i
$851$	30.0000i	1.02839i
$852$	0	0
$853$	13.8564i	0.474434i	0.971457	+	0.237217i	$0.0762353\pi$
$853$	13.8564i	0.474434i	−0.971457	+	0.237217i	$0.923765\pi$
$854$	0	0
$855$	0	0
$856$	−6.00000	−0.205076
$857$	−17.3205	−0.591657	−0.295829	−	0.955241i	$-0.595596\pi$
$857$	−17.3205	−0.591657	−0.295829	+	0.955241i	$0.595596\pi$
$858$	0	0
$859$	− 8.66025i	− 0.295484i	−0.989026	−	0.147742i	$-0.952799\pi$
$859$	− 8.66025i	− 0.295484i	0.989026	−	0.147742i	$-0.0472005\pi$
$860$	−3.46410	−0.118125
$861$	0	0
$862$	−36.0000	−1.22616
$863$	− 36.0000i	− 1.22545i	−0.790295	−	0.612727i	$-0.790072\pi$
$863$	− 36.0000i	− 1.22545i	0.790295	−	0.612727i	$-0.209928\pi$
$864$	0	0
$865$	72.0000	2.44807
$866$	−39.8372	−1.35372
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	67.5500i	2.28884i
$872$	11.0000i	0.372507i
$873$	0	0
$874$	− 20.7846i	− 0.703050i
$875$	0	0
$876$	0	0
$877$	41.0000	1.38447	0.692236	−	0.721671i	$-0.256626\pi$
$877$	41.0000	1.38447	0.692236	+	0.721671i	$0.256626\pi$
$878$	31.1769	1.05217
$879$	0	0
$880$	0	0
$881$	−31.1769	−1.05038	−0.525188	−	0.850986i	$-0.676005\pi$
$881$	−31.1769	−1.05038	−0.525188	+	0.850986i	$0.676005\pi$
$882$	0	0
$883$	52.0000	1.74994	0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000	1.74994	0.874970	+	0.484178i	$0.160881\pi$
$884$	− 36.0000i	− 1.21081i
$885$	0	0
$886$	0	0
$887$	3.46410	0.116313	0.0581566	−	0.998307i	$-0.481478\pi$
$887$	3.46410	0.116313	0.0581566	+	0.998307i	$0.481478\pi$
$888$	0	0
$889$	0	0
$890$	36.0000i	1.20672i
$891$	0	0
$892$	10.3923i	0.347960i
$893$	24.0000i	0.803129i
$894$	0	0
$895$	− 20.7846i	− 0.694753i
$896$	0	0
$897$	0	0
$898$	30.0000	1.00111
$899$	0	0
$900$	0	0
$901$	41.5692i	1.38487i
$902$	0	0
$903$	0	0
$904$	18.0000	0.598671
$905$	48.0000i	1.59557i
$906$	0	0
$907$	−47.0000	−1.56061	−0.780305	−	0.625400i	$-0.784936\pi$
$907$	−47.0000	−1.56061	−0.780305	+	0.625400i	$0.784936\pi$
$908$	−10.3923	−0.344881
$909$	0	0
$910$	0	0
$911$	48.0000i	1.59031i	0.606406	+	0.795155i	$0.292611\pi$
$911$	48.0000i	1.59031i	−0.606406	+	0.795155i	$0.707389\pi$
$912$	0	0
$913$	0	0
$914$	31.0000i	1.02539i
$915$	0	0
$916$	− 1.73205i	− 0.0572286i
$917$	0	0
$918$	0	0
$919$	25.0000	0.824674	0.412337	−	0.911031i	$-0.364713\pi$
$919$	25.0000	0.824674	0.412337	+	0.911031i	$0.364713\pi$
$920$	−20.7846	−0.685248
$921$	0	0
$922$	− 13.8564i	− 0.456336i
$923$	31.1769	1.02620
$924$	0	0
$925$	−35.0000	−1.15079
$926$	− 40.0000i	− 1.31448i
$927$	0	0
$928$	0	0
$929$	45.0333	1.47750	0.738748	−	0.673982i	$-0.235418\pi$
$929$	45.0333	1.47750	0.738748	+	0.673982i	$0.235418\pi$
$930$	0	0
$931$	0	0
$932$	− 12.0000i	− 0.393073i
$933$	0	0
$934$	− 10.3923i	− 0.340047i
$935$	0	0
$936$	0	0
$937$	5.19615i	0.169751i	0.996392	+	0.0848755i	$0.0270492\pi$
$937$	5.19615i	0.169751i	−0.996392	+	0.0848755i	$0.972951\pi$
$938$	0	0
$939$	0	0
$940$	24.0000	0.782794
$941$	−13.8564	−0.451706	−0.225853	−	0.974161i	$-0.572517\pi$
$941$	−13.8564	−0.451706	−0.225853	+	0.974161i	$0.572517\pi$
$942$	0	0
$943$	− 41.5692i	− 1.35368i
$944$	6.92820	0.225494
$945$	0	0
$946$	0	0
$947$	36.0000i	1.16984i	0.811090	+	0.584921i	$0.198875\pi$
$947$	36.0000i	1.16984i	−0.811090	+	0.584921i	$0.801125\pi$
$948$	0	0
$949$	36.0000	1.16861
$950$	24.2487	0.786732
$951$	0	0
$952$	0	0
$953$	− 6.00000i	− 0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$	− 6.00000i	− 0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$	0	0
$955$	− 20.7846i	− 0.672574i
$956$	6.00000i	0.194054i
$957$	0	0
$958$	− 3.46410i	− 0.111920i
$959$	0	0
$960$	0	0
$961$	−44.0000	−1.41935
$962$	−25.9808	−0.837653
$963$	0	0
$964$	1.73205i	0.0557856i
$965$	3.46410	0.111513
$966$	0	0
$967$	−31.0000	−0.996893	−0.498446	−	0.866921i	$-0.666096\pi$
$967$	−31.0000	−0.996893	−0.498446	+	0.866921i	$0.666096\pi$
$968$	− 11.0000i	− 0.353553i
$969$	0	0
$970$	−6.00000	−0.192648
$971$	27.7128	0.889346	0.444673	−	0.895693i	$-0.353320\pi$
$971$	27.7128	0.889346	0.444673	+	0.895693i	$0.353320\pi$
$972$	0	0
$973$	0	0
$974$	− 8.00000i	− 0.256337i
$975$	0	0
$976$	− 1.73205i	− 0.0554416i
$977$	− 6.00000i	− 0.191957i	−0.995383	−	0.0959785i	$-0.969402\pi$
$977$	− 6.00000i	− 0.191957i	0.995383	−	0.0959785i	$-0.0305980\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	6.00000	0.191468
$983$	−24.2487	−0.773414	−0.386707	−	0.922203i	$-0.626387\pi$
$983$	−24.2487	−0.773414	−0.386707	+	0.922203i	$0.626387\pi$
$984$	0	0
$985$	83.1384i	2.64901i
$986$	0	0
$987$	0	0
$988$	18.0000	0.572656
$989$	− 6.00000i	− 0.190789i
$990$	0	0
$991$	13.0000	0.412959	0.206479	−	0.978451i	$-0.433799\pi$
$991$	13.0000	0.412959	0.206479	+	0.978451i	$0.433799\pi$
$992$	8.66025	0.274963
$993$	0	0
$994$	0	0
$995$	6.00000i	0.190213i
$996$	0	0
$997$	− 32.9090i	− 1.04224i	−0.853484	−	0.521119i	$-0.825515\pi$
$997$	− 32.9090i	− 1.04224i	0.853484	−	0.521119i	$-0.174485\pi$
$998$	− 23.0000i	− 0.728052i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2646.2.d.b.2645.4		4
3.2	odd	2	inner	2646.2.d.b.2645.1		4
7.2	even	3		378.2.k.b.269.2	yes	4
7.3	odd	6		378.2.k.b.215.1	✓	4
7.6	odd	2	inner	2646.2.d.b.2645.3		4
21.2	odd	6		378.2.k.b.269.1	yes	4
21.17	even	6		378.2.k.b.215.2	yes	4
21.20	even	2	inner	2646.2.d.b.2645.2		4
63.2	odd	6		1134.2.t.b.1025.2		4
63.16	even	3		1134.2.t.b.1025.1		4
63.23	odd	6		1134.2.l.a.269.1		4
63.31	odd	6		1134.2.t.b.593.2		4
63.38	even	6		1134.2.l.a.215.1		4
63.52	odd	6		1134.2.l.a.215.2		4
63.58	even	3		1134.2.l.a.269.2		4
63.59	even	6		1134.2.t.b.593.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.k.b.215.1	✓	4	7.3	odd	6
378.2.k.b.215.2	yes	4	21.17	even	6
378.2.k.b.269.1	yes	4	21.2	odd	6
378.2.k.b.269.2	yes	4	7.2	even	3
1134.2.l.a.215.1		4	63.38	even	6
1134.2.l.a.215.2		4	63.52	odd	6
1134.2.l.a.269.1		4	63.23	odd	6
1134.2.l.a.269.2		4	63.58	even	3
1134.2.t.b.593.1		4	63.59	even	6
1134.2.t.b.593.2		4	63.31	odd	6
1134.2.t.b.1025.1		4	63.16	even	3
1134.2.t.b.1025.2		4	63.2	odd	6
2646.2.d.b.2645.1		4	3.2	odd	2	inner
2646.2.d.b.2645.2		4	21.20	even	2	inner
2646.2.d.b.2645.3		4	7.6	odd	2	inner
2646.2.d.b.2645.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2646.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +3.46410 q^{5} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +3.46410 q^{5} -1.00000i q^{8} +3.46410i q^{10} -5.19615i q^{13} +1.00000 q^{16} -6.92820 q^{17} -3.46410i q^{19} -3.46410 q^{20} -6.00000i q^{23} +7.00000 q^{25} +5.19615 q^{26} -8.66025i q^{31} +1.00000i q^{32} -6.92820i q^{34} -5.00000 q^{37} +3.46410 q^{38} -3.46410i q^{40} +6.92820 q^{41} +1.00000 q^{43} +6.00000 q^{46} -6.92820 q^{47} +7.00000i q^{50} +5.19615i q^{52} -6.00000i q^{53} +6.92820 q^{59} -1.73205i q^{61} +8.66025 q^{62} -1.00000 q^{64} -18.0000i q^{65} -13.0000 q^{67} +6.92820 q^{68} +6.00000i q^{71} +6.92820i q^{73} -5.00000i q^{74} +3.46410i q^{76} -7.00000 q^{79} +3.46410 q^{80} +6.92820i q^{82} -3.46410 q^{83} -24.0000 q^{85} +1.00000i q^{86} +10.3923 q^{89} +6.00000i q^{92} -6.92820i q^{94} -12.0000i q^{95} +1.73205i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 4 q^{16} + 28 q^{25} - 20 q^{37} + 4 q^{43} + 24 q^{46} - 4 q^{64} - 52 q^{67} - 28 q^{79} - 96 q^{85}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^16 + 28 * q^25 - 20 * q^37 + 4 * q^43 + 24 * q^46 - 4 * q^64 - 52 * q^67 - 28 * q^79 - 96 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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