LMFDB - Embedded newform 2646.2.d.e.2645.15

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:	$16$
Coefficient field:	$\Q(\zeta_{48})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{8} + 1 $ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2645.15
Root			$-0.793353 + 0.608761i$ of defining polynomial
Character	$\chi$	$=$	2646.2645
Dual form			2646.2.d.e.2645.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +3.21486 q^{5} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +3.21486 q^{5} -1.00000i q^{8} +3.21486i q^{10} +0.674346i q^{11} -0.881075i q^{13} +1.00000 q^{16} +7.27533 q^{17} -7.76135i q^{19} -3.21486 q^{20} -0.674346 q^{22} -4.35449i q^{23} +5.33530 q^{25} +0.881075 q^{26} -5.15408i q^{29} -2.66105i q^{31} +1.00000i q^{32} +7.27533i q^{34} -10.2756 q^{37} +7.76135 q^{38} -3.21486i q^{40} -6.36193 q^{41} +9.96357 q^{43} -0.674346i q^{44} +4.35449 q^{46} +3.28264 q^{47} +5.33530i q^{50} +0.881075i q^{52} -2.60908i q^{53} +2.16792i q^{55} +5.15408 q^{58} -5.75484 q^{59} +6.98840i q^{61} +2.66105 q^{62} -1.00000 q^{64} -2.83253i q^{65} +7.94993 q^{67} -7.27533 q^{68} -4.11185i q^{71} +13.8105i q^{73} -10.2756i q^{74} +7.76135i q^{76} -1.21935 q^{79} +3.21486 q^{80} -6.36193i q^{82} +1.71958 q^{83} +23.3891 q^{85} +9.96357i q^{86} +0.674346 q^{88} +6.94806 q^{89} +4.35449i q^{92} +3.28264i q^{94} -24.9516i q^{95} +12.0129i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4}+O(q^{10}) $ 16 * q - 16 * q^4	$ 16 q - 16 q^{4} + 16 q^{16} - 32 q^{22} + 48 q^{25} - 32 q^{37} + 16 q^{43} - 48 q^{46} + 16 q^{58} - 16 q^{64} + 16 q^{67} + 32 q^{88}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^16 - 32 * q^22 + 48 * q^25 - 32 * q^37 + 16 * q^43 - 48 * q^46 + 16 * q^58 - 16 * q^64 + 16 * q^67 + 32 * q^88

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
$5$	3.21486	1.43773	0.718864	−	0.695151i	$-0.244662\pi$
$5$	3.21486	1.43773	0.718864	+	0.695151i	$0.244662\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	3.21486i	1.01663i
$11$	0.674346i	0.203323i	0.994819	+	0.101661i	$0.0324158\pi$
$11$	0.674346i	0.203323i	−0.994819	+	0.101661i	$0.967584\pi$
$12$	0	0
$13$	− 0.881075i	− 0.244366i	−0.992508	−	0.122183i	$-0.961010\pi$
$13$	− 0.881075i	− 0.244366i	0.992508	−	0.122183i	$-0.0389895\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	7.27533	1.76453	0.882263	−	0.470757i	$-0.156019\pi$
$17$	7.27533	1.76453	0.882263	+	0.470757i	$0.156019\pi$
$18$	0	0
$19$	− 7.76135i	− 1.78058i	−0.455398	−	0.890288i	$-0.650503\pi$
$19$	− 7.76135i	− 1.78058i	0.455398	−	0.890288i	$-0.349497\pi$
$20$	−3.21486	−0.718864
$21$	0	0
$22$	−0.674346	−0.143771
$23$	− 4.35449i	− 0.907975i	−0.891008	−	0.453987i	$-0.850001\pi$
$23$	− 4.35449i	− 0.907975i	0.891008	−	0.453987i	$-0.149999\pi$
$24$	0	0
$25$	5.33530	1.06706
$26$	0.881075	0.172793
$27$	0	0
$28$	0	0
$29$	− 5.15408i	− 0.957089i	−0.878063	−	0.478544i	$-0.841165\pi$
$29$	− 5.15408i	− 0.957089i	0.878063	−	0.478544i	$-0.158835\pi$
$30$	0	0
$31$	− 2.66105i	− 0.477939i	−0.971027	−	0.238970i	$-0.923190\pi$
$31$	− 2.66105i	− 0.477939i	0.971027	−	0.238970i	$-0.0768097\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	7.27533i	1.24771i
$35$	0	0
$36$	0	0
$37$	−10.2756	−1.68930	−0.844648	−	0.535322i	$-0.820190\pi$
$37$	−10.2756	−1.68930	−0.844648	+	0.535322i	$0.820190\pi$
$38$	7.76135	1.25906
$39$	0	0
$40$	− 3.21486i	− 0.508313i
$41$	−6.36193	−0.993567	−0.496783	−	0.867875i	$-0.665486\pi$
$41$	−6.36193	−0.993567	−0.496783	+	0.867875i	$0.665486\pi$
$42$	0	0
$43$	9.96357	1.51943	0.759715	−	0.650256i	$-0.225338\pi$
$43$	9.96357	1.51943	0.759715	+	0.650256i	$0.225338\pi$
$44$	− 0.674346i	− 0.101661i
$45$	0	0
$46$	4.35449	0.642035
$47$	3.28264	0.478822	0.239411	−	0.970918i	$-0.423046\pi$
$47$	3.28264	0.478822	0.239411	+	0.970918i	$0.423046\pi$
$48$	0	0
$49$	0	0
$50$	5.33530i	0.754526i
$51$	0	0
$52$	0.881075i	0.122183i
$53$	− 2.60908i	− 0.358385i	−0.983814	−	0.179192i	$-0.942652\pi$
$53$	− 2.60908i	− 0.358385i	0.983814	−	0.179192i	$-0.0573485\pi$
$54$	0	0
$55$	2.16792i	0.292323i
$56$	0	0
$57$	0	0
$58$	5.15408	0.676764
$59$	−5.75484	−0.749217	−0.374608	−	0.927183i	$-0.622223\pi$
$59$	−5.75484	−0.749217	−0.374608	+	0.927183i	$0.622223\pi$
$60$	0	0
$61$	6.98840i	0.894773i	0.894341	+	0.447387i	$0.147645\pi$
$61$	6.98840i	0.894773i	−0.894341	+	0.447387i	$0.852355\pi$
$62$	2.66105	0.337954
$63$	0	0
$64$	−1.00000	−0.125000
$65$	− 2.83253i	− 0.351332i
$66$	0	0
$67$	7.94993	0.971238	0.485619	−	0.874170i	$-0.338594\pi$
$67$	7.94993	0.971238	0.485619	+	0.874170i	$0.338594\pi$
$68$	−7.27533	−0.882263
$69$	0	0
$70$	0	0
$71$	− 4.11185i	− 0.487987i	−0.969777	−	0.243994i	$-0.921542\pi$
$71$	− 4.11185i	− 0.487987i	0.969777	−	0.243994i	$-0.0784576\pi$
$72$	0	0
$73$	13.8105i	1.61640i	0.588910	+	0.808199i	$0.299557\pi$
$73$	13.8105i	1.61640i	−0.588910	+	0.808199i	$0.700443\pi$
$74$	− 10.2756i	− 1.19451i
$75$	0	0
$76$	7.76135i	0.890288i
$77$	0	0
$78$	0	0
$79$	−1.21935	−0.137187	−0.0685936	−	0.997645i	$-0.521851\pi$
$79$	−1.21935	−0.137187	−0.0685936	+	0.997645i	$0.521851\pi$
$80$	3.21486	0.359432
$81$	0	0
$82$	− 6.36193i	− 0.702558i
$83$	1.71958	0.188749	0.0943744	−	0.995537i	$-0.469915\pi$
$83$	1.71958	0.188749	0.0943744	+	0.995537i	$0.469915\pi$
$84$	0	0
$85$	23.3891	2.53691
$86$	9.96357i	1.07440i
$87$	0	0
$88$	0.674346	0.0718855
$89$	6.94806	0.736492	0.368246	−	0.929728i	$-0.379958\pi$
$89$	6.94806	0.736492	0.368246	+	0.929728i	$0.379958\pi$
$90$	0	0
$91$	0	0
$92$	4.35449i	0.453987i
$93$	0	0
$94$	3.28264i	0.338578i
$95$	− 24.9516i	− 2.55998i
$96$	0	0
$97$	12.0129i	1.21972i	0.792507	+	0.609862i	$0.208775\pi$
$97$	12.0129i	1.21972i	−0.792507	+	0.609862i	$0.791225\pi$
$98$	0	0
$99$	0	0
$100$	−5.33530	−0.533530
$101$	−11.8595	−1.18006	−0.590032	−	0.807380i	$-0.700885\pi$
$101$	−11.8595	−1.18006	−0.590032	+	0.807380i	$0.700885\pi$
$102$	0	0
$103$	− 19.9459i	− 1.96533i	−0.185394	−	0.982664i	$-0.559356\pi$
$103$	− 19.9459i	− 1.96533i	0.185394	−	0.982664i	$-0.440644\pi$
$104$	−0.881075	−0.0863965
$105$	0	0
$106$	2.60908	0.253416
$107$	− 17.7961i	− 1.72041i	−0.509945	−	0.860207i	$-0.670334\pi$
$107$	− 17.7961i	− 1.72041i	0.509945	−	0.860207i	$-0.329666\pi$
$108$	0	0
$109$	4.09846	0.392562	0.196281	−	0.980548i	$-0.437114\pi$
$109$	4.09846	0.392562	0.196281	+	0.980548i	$0.437114\pi$
$110$	−2.16792	−0.206704
$111$	0	0
$112$	0	0
$113$	13.4756i	1.26768i	0.773464	+	0.633840i	$0.218522\pi$
$113$	13.4756i	1.26768i	−0.773464	+	0.633840i	$0.781478\pi$
$114$	0	0
$115$	− 13.9991i	− 1.30542i
$116$	5.15408i	0.478544i
$117$	0	0
$118$	− 5.75484i	− 0.529776i
$119$	0	0
$120$	0	0
$121$	10.5453	0.958660
$122$	−6.98840	−0.632700
$123$	0	0
$124$	2.66105i	0.238970i
$125$	1.07795	0.0964149
$126$	0	0
$127$	0.530170	0.0470450	0.0235225	−	0.999723i	$-0.492512\pi$
$127$	0.530170	0.0470450	0.0235225	+	0.999723i	$0.492512\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	2.83253	0.248429
$131$	−17.0302	−1.48794	−0.743970	−	0.668213i	$-0.767059\pi$
$131$	−17.0302	−1.48794	−0.743970	+	0.668213i	$0.767059\pi$
$132$	0	0
$133$	0	0
$134$	7.94993i	0.686769i
$135$	0	0
$136$	− 7.27533i	− 0.623854i
$137$	13.9365i	1.19068i	0.803474	+	0.595340i	$0.202982\pi$
$137$	13.9365i	1.19068i	−0.803474	+	0.595340i	$0.797018\pi$
$138$	0	0
$139$	6.55078i	0.555630i	0.960635	+	0.277815i	$0.0896102\pi$
$139$	6.55078i	0.555630i	−0.960635	+	0.277815i	$0.910390\pi$
$140$	0	0
$141$	0	0
$142$	4.11185	0.345059
$143$	0.594149	0.0496853
$144$	0	0
$145$	− 16.5696i	− 1.37603i
$146$	−13.8105	−1.14297
$147$	0	0
$148$	10.2756	0.844648
$149$	− 22.4410i	− 1.83844i	−0.393745	−	0.919220i	$-0.628821\pi$
$149$	− 22.4410i	− 1.83844i	0.393745	−	0.919220i	$-0.371179\pi$
$150$	0	0
$151$	19.6260	1.59714	0.798570	−	0.601903i	$-0.205590\pi$
$151$	19.6260	1.59714	0.798570	+	0.601903i	$0.205590\pi$
$152$	−7.76135	−0.629529
$153$	0	0
$154$	0	0
$155$	− 8.55491i	− 0.687147i
$156$	0	0
$157$	5.72677i	0.457046i	0.973539	+	0.228523i	$0.0733896\pi$
$157$	5.72677i	0.457046i	−0.973539	+	0.228523i	$0.926610\pi$
$158$	− 1.21935i	− 0.0970060i
$159$	0	0
$160$	3.21486i	0.254157i
$161$	0	0
$162$	0	0
$163$	20.0480	1.57028	0.785141	−	0.619316i	$-0.212590\pi$
$163$	20.0480	1.57028	0.785141	+	0.619316i	$0.212590\pi$
$164$	6.36193	0.496783
$165$	0	0
$166$	1.71958i	0.133466i
$167$	11.8314	0.915543	0.457772	−	0.889070i	$-0.348648\pi$
$167$	11.8314	0.915543	0.457772	+	0.889070i	$0.348648\pi$
$168$	0	0
$169$	12.2237	0.940285
$170$	23.3891i	1.79386i
$171$	0	0
$172$	−9.96357	−0.759715
$173$	9.44547	0.718126	0.359063	−	0.933313i	$-0.383096\pi$
$173$	9.44547	0.718126	0.359063	+	0.933313i	$0.383096\pi$
$174$	0	0
$175$	0	0
$176$	0.674346i	0.0508307i
$177$	0	0
$178$	6.94806i	0.520779i
$179$	17.4678i	1.30560i	0.757529	+	0.652802i	$0.226407\pi$
$179$	17.4678i	1.30560i	−0.757529	+	0.652802i	$0.773593\pi$
$180$	0	0
$181$	15.4317i	1.14703i	0.819195	+	0.573516i	$0.194421\pi$
$181$	15.4317i	1.14703i	−0.819195	+	0.573516i	$0.805579\pi$
$182$	0	0
$183$	0	0
$184$	−4.35449	−0.321018
$185$	−33.0345	−2.42875
$186$	0	0
$187$	4.90609i	0.358769i
$188$	−3.28264	−0.239411
$189$	0	0
$190$	24.9516	1.81018
$191$	17.2214i	1.24610i	0.782183	+	0.623049i	$0.214106\pi$
$191$	17.2214i	1.24610i	−0.782183	+	0.623049i	$0.785894\pi$
$192$	0	0
$193$	−18.6569	−1.34295	−0.671475	−	0.741027i	$-0.734339\pi$
$193$	−18.6569	−1.34295	−0.671475	+	0.741027i	$0.734339\pi$
$194$	−12.0129	−0.862475
$195$	0	0
$196$	0	0
$197$	− 7.70293i	− 0.548811i	−0.961614	−	0.274405i	$-0.911519\pi$
$197$	− 7.70293i	− 0.548811i	0.961614	−	0.274405i	$-0.0884810\pi$
$198$	0	0
$199$	3.02803i	0.214651i	0.994224	+	0.107326i	$0.0342288\pi$
$199$	3.02803i	0.214651i	−0.994224	+	0.107326i	$0.965771\pi$
$200$	− 5.33530i	− 0.377263i
$201$	0	0
$202$	− 11.8595i	− 0.834432i
$203$	0	0
$204$	0	0
$205$	−20.4527	−1.42848
$206$	19.9459	1.38970
$207$	0	0
$208$	− 0.881075i	− 0.0610916i
$209$	5.23383	0.362032
$210$	0	0
$211$	−11.0886	−0.763368	−0.381684	−	0.924293i	$-0.624656\pi$
$211$	−11.0886	−0.763368	−0.381684	+	0.924293i	$0.624656\pi$
$212$	2.60908i	0.179192i
$213$	0	0
$214$	17.7961	1.21652
$215$	32.0315	2.18453
$216$	0	0
$217$	0	0
$218$	4.09846i	0.277583i
$219$	0	0
$220$	− 2.16792i	− 0.146161i
$221$	− 6.41011i	− 0.431191i
$222$	0	0
$223$	14.4548i	0.967965i	0.875078	+	0.483983i	$0.160810\pi$
$223$	14.4548i	0.967965i	−0.875078	+	0.483983i	$0.839190\pi$
$224$	0	0
$225$	0	0
$226$	−13.4756	−0.896386
$227$	12.3627	0.820540	0.410270	−	0.911964i	$-0.365434\pi$
$227$	12.3627	0.820540	0.410270	+	0.911964i	$0.365434\pi$
$228$	0	0
$229$	7.74038i	0.511499i	0.966743	+	0.255749i	$0.0823222\pi$
$229$	7.74038i	0.511499i	−0.966743	+	0.255749i	$0.917678\pi$
$230$	13.9991	0.923072
$231$	0	0
$232$	−5.15408	−0.338382
$233$	10.3275i	0.676574i	0.941043	+	0.338287i	$0.109848\pi$
$233$	10.3275i	0.676574i	−0.941043	+	0.338287i	$0.890152\pi$
$234$	0	0
$235$	10.5532	0.688415
$236$	5.75484	0.374608
$237$	0	0
$238$	0	0
$239$	− 14.6167i	− 0.945475i	−0.881203	−	0.472738i	$-0.843266\pi$
$239$	− 14.6167i	− 0.945475i	0.881203	−	0.472738i	$-0.156734\pi$
$240$	0	0
$241$	20.8803i	1.34502i	0.740090	+	0.672508i	$0.234783\pi$
$241$	20.8803i	1.34502i	−0.740090	+	0.672508i	$0.765217\pi$
$242$	10.5453i	0.677875i
$243$	0	0
$244$	− 6.98840i	− 0.447387i
$245$	0	0
$246$	0	0
$247$	−6.83833	−0.435113
$248$	−2.66105	−0.168977
$249$	0	0
$250$	1.07795i	0.0681756i
$251$	2.38929	0.150811	0.0754054	−	0.997153i	$-0.475975\pi$
$251$	2.38929	0.150811	0.0754054	+	0.997153i	$0.475975\pi$
$252$	0	0
$253$	2.93643	0.184612
$254$	0.530170i	0.0332658i
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.40073	0.399267	0.199633	−	0.979871i	$-0.436025\pi$
$257$	6.40073	0.399267	0.199633	+	0.979871i	$0.436025\pi$
$258$	0	0
$259$	0	0
$260$	2.83253i	0.175666i
$261$	0	0
$262$	− 17.0302i	− 1.05213i
$263$	25.4911i	1.57185i	0.618323	+	0.785924i	$0.287812\pi$
$263$	25.4911i	1.57185i	−0.618323	+	0.785924i	$0.712188\pi$
$264$	0	0
$265$	− 8.38782i	− 0.515260i
$266$	0	0
$267$	0	0
$268$	−7.94993	−0.485619
$269$	15.8526	0.966552	0.483276	−	0.875468i	$-0.339447\pi$
$269$	15.8526	0.966552	0.483276	+	0.875468i	$0.339447\pi$
$270$	0	0
$271$	10.3945i	0.631422i	0.948855	+	0.315711i	$0.102243\pi$
$271$	10.3945i	0.631422i	−0.948855	+	0.315711i	$0.897757\pi$
$272$	7.27533	0.441132
$273$	0	0
$274$	−13.9365	−0.841937
$275$	3.59784i	0.216958i
$276$	0	0
$277$	−1.70585	−0.102494	−0.0512472	−	0.998686i	$-0.516320\pi$
$277$	−1.70585	−0.102494	−0.0512472	+	0.998686i	$0.516320\pi$
$278$	−6.55078	−0.392890
$279$	0	0
$280$	0	0
$281$	− 11.7087i	− 0.698484i	−0.937033	−	0.349242i	$-0.886439\pi$
$281$	− 11.7087i	− 0.698484i	0.937033	−	0.349242i	$-0.113561\pi$
$282$	0	0
$283$	1.09311i	0.0649789i	0.999472	+	0.0324894i	$0.0103435\pi$
$283$	1.09311i	0.0649789i	−0.999472	+	0.0324894i	$0.989656\pi$
$284$	4.11185i	0.243994i
$285$	0	0
$286$	0.594149i	0.0351328i
$287$	0	0
$288$	0	0
$289$	35.9304	2.11355
$290$	16.5696	0.973002
$291$	0	0
$292$	− 13.8105i	− 0.808199i
$293$	−4.40649	−0.257430	−0.128715	−	0.991682i	$-0.541085\pi$
$293$	−4.40649	−0.257430	−0.128715	+	0.991682i	$0.541085\pi$
$294$	0	0
$295$	−18.5010	−1.07717
$296$	10.2756i	0.597256i
$297$	0	0
$298$	22.4410	1.29997
$299$	−3.83664	−0.221878
$300$	0	0
$301$	0	0
$302$	19.6260i	1.12935i
$303$	0	0
$304$	− 7.76135i	− 0.445144i
$305$	22.4667i	1.28644i
$306$	0	0
$307$	26.5696i	1.51640i	0.652020	+	0.758202i	$0.273922\pi$
$307$	26.5696i	1.51640i	−0.652020	+	0.758202i	$0.726078\pi$
$308$	0	0
$309$	0	0
$310$	8.55491	0.485886
$311$	−17.0739	−0.968174	−0.484087	−	0.875020i	$-0.660848\pi$
$311$	−17.0739	−0.968174	−0.484087	+	0.875020i	$0.660848\pi$
$312$	0	0
$313$	− 10.9407i	− 0.618408i	−0.950996	−	0.309204i	$-0.899937\pi$
$313$	− 10.9407i	− 0.618408i	0.950996	−	0.309204i	$-0.100063\pi$
$314$	−5.72677	−0.323180
$315$	0	0
$316$	1.21935	0.0685936
$317$	− 15.3685i	− 0.863181i	−0.902070	−	0.431591i	$-0.857953\pi$
$317$	− 15.3685i	− 0.863181i	0.902070	−	0.431591i	$-0.142047\pi$
$318$	0	0
$319$	3.47563	0.194598
$320$	−3.21486	−0.179716
$321$	0	0
$322$	0	0
$323$	− 56.4664i	− 3.14187i
$324$	0	0
$325$	− 4.70080i	− 0.260754i
$326$	20.0480i	1.11036i
$327$	0	0
$328$	6.36193i	0.351279i
$329$	0	0
$330$	0	0
$331$	−23.7515	−1.30550	−0.652750	−	0.757574i	$-0.726385\pi$
$331$	−23.7515	−1.30550	−0.652750	+	0.757574i	$0.726385\pi$
$332$	−1.71958	−0.0943744
$333$	0	0
$334$	11.8314i	0.647387i
$335$	25.5579	1.39638
$336$	0	0
$337$	−2.59185	−0.141187	−0.0705934	−	0.997505i	$-0.522489\pi$
$337$	−2.59185	−0.141187	−0.0705934	+	0.997505i	$0.522489\pi$
$338$	12.2237i	0.664882i
$339$	0	0
$340$	−23.3891	−1.26845
$341$	1.79447	0.0971760
$342$	0	0
$343$	0	0
$344$	− 9.96357i	− 0.537200i
$345$	0	0
$346$	9.44547i	0.507792i
$347$	− 4.78764i	− 0.257014i	−0.991709	−	0.128507i	$-0.958982\pi$
$347$	− 4.78764i	− 0.257014i	0.991709	−	0.128507i	$-0.0410185\pi$
$348$	0	0
$349$	− 30.1643i	− 1.61466i	−0.590101	−	0.807330i	$-0.700912\pi$
$349$	− 30.1643i	− 1.61466i	0.590101	−	0.807330i	$-0.299088\pi$
$350$	0	0
$351$	0	0
$352$	−0.674346	−0.0359427
$353$	−6.45715	−0.343679	−0.171840	−	0.985125i	$-0.554971\pi$
$353$	−6.45715	−0.343679	−0.171840	+	0.985125i	$0.554971\pi$
$354$	0	0
$355$	− 13.2190i	− 0.701593i
$356$	−6.94806	−0.368246
$357$	0	0
$358$	−17.4678	−0.923201
$359$	− 6.50159i	− 0.343141i	−0.985172	−	0.171570i	$-0.945116\pi$
$359$	− 6.50159i	− 0.343141i	0.985172	−	0.171570i	$-0.0548841\pi$
$360$	0	0
$361$	−41.2386	−2.17045
$362$	−15.4317	−0.811074
$363$	0	0
$364$	0	0
$365$	44.3988i	2.32394i
$366$	0	0
$367$	− 2.76854i	− 0.144517i	−0.997386	−	0.0722583i	$-0.976979\pi$
$367$	− 2.76854i	− 0.144517i	0.997386	−	0.0722583i	$-0.0230206\pi$
$368$	− 4.35449i	− 0.226994i
$369$	0	0
$370$	− 33.0345i	− 1.71738i
$371$	0	0
$372$	0	0
$373$	−18.6801	−0.967221	−0.483611	−	0.875283i	$-0.660675\pi$
$373$	−18.6801	−0.967221	−0.483611	+	0.875283i	$0.660675\pi$
$374$	−4.90609	−0.253688
$375$	0	0
$376$	− 3.28264i	− 0.169289i
$377$	−4.54113	−0.233880
$378$	0	0
$379$	−8.86137	−0.455178	−0.227589	−	0.973757i	$-0.573084\pi$
$379$	−8.86137	−0.455178	−0.227589	+	0.973757i	$0.573084\pi$
$380$	24.9516i	1.27999i
$381$	0	0
$382$	−17.2214	−0.881124
$383$	−30.8075	−1.57419	−0.787095	−	0.616832i	$-0.788416\pi$
$383$	−30.8075	−1.57419	−0.787095	+	0.616832i	$0.788416\pi$
$384$	0	0
$385$	0	0
$386$	− 18.6569i	− 0.949609i
$387$	0	0
$388$	− 12.0129i	− 0.609862i
$389$	8.41192i	0.426501i	0.976998	+	0.213251i	$0.0684051\pi$
$389$	8.41192i	0.426501i	−0.976998	+	0.213251i	$0.931595\pi$
$390$	0	0
$391$	− 31.6804i	− 1.60215i
$392$	0	0
$393$	0	0
$394$	7.70293	0.388068
$395$	−3.92002	−0.197238
$396$	0	0
$397$	23.7281i	1.19088i	0.803399	+	0.595441i	$0.203023\pi$
$397$	23.7281i	1.19088i	−0.803399	+	0.595441i	$0.796977\pi$
$398$	−3.02803	−0.151781
$399$	0	0
$400$	5.33530	0.266765
$401$	− 12.1940i	− 0.608940i	−0.952522	−	0.304470i	$-0.901521\pi$
$401$	− 12.1940i	− 0.608940i	0.952522	−	0.304470i	$-0.0984793\pi$
$402$	0	0
$403$	−2.34459	−0.116792
$404$	11.8595	0.590032
$405$	0	0
$406$	0	0
$407$	− 6.92930i	− 0.343472i
$408$	0	0
$409$	12.5274i	0.619442i	0.950828	+	0.309721i	$0.100236\pi$
$409$	12.5274i	0.619442i	−0.950828	+	0.309721i	$0.899764\pi$
$410$	− 20.4527i	− 1.01009i
$411$	0	0
$412$	19.9459i	0.982664i
$413$	0	0
$414$	0	0
$415$	5.52821	0.271369
$416$	0.881075	0.0431983
$417$	0	0
$418$	5.23383i	0.255995i
$419$	10.9782	0.536322	0.268161	−	0.963374i	$-0.413584\pi$
$419$	10.9782	0.536322	0.268161	+	0.963374i	$0.413584\pi$
$420$	0	0
$421$	−18.6015	−0.906581	−0.453291	−	0.891363i	$-0.649750\pi$
$421$	−18.6015	−0.906581	−0.453291	+	0.891363i	$0.649750\pi$
$422$	− 11.0886i	− 0.539783i
$423$	0	0
$424$	−2.60908	−0.126708
$425$	38.8161	1.88286
$426$	0	0
$427$	0	0
$428$	17.7961i	0.860207i
$429$	0	0
$430$	32.0315i	1.54469i
$431$	− 16.4856i	− 0.794086i	−0.917800	−	0.397043i	$-0.870036\pi$
$431$	− 16.4856i	− 0.794086i	0.917800	−	0.397043i	$-0.129964\pi$
$432$	0	0
$433$	− 11.0014i	− 0.528695i	−0.964428	−	0.264347i	$-0.914843\pi$
$433$	− 11.0014i	− 0.528695i	0.964428	−	0.264347i	$-0.0851565\pi$
$434$	0	0
$435$	0	0
$436$	−4.09846	−0.196281
$437$	−33.7968	−1.61672
$438$	0	0
$439$	1.17317i	0.0559921i	0.999608	+	0.0279961i	$0.00891259\pi$
$439$	1.17317i	0.0559921i	−0.999608	+	0.0279961i	$0.991087\pi$
$440$	2.16792	0.103352
$441$	0	0
$442$	6.41011	0.304898
$443$	− 17.0789i	− 0.811444i	−0.913997	−	0.405722i	$-0.867020\pi$
$443$	− 17.0789i	− 0.811444i	0.913997	−	0.405722i	$-0.132980\pi$
$444$	0	0
$445$	22.3370	1.05888
$446$	−14.4548	−0.684455
$447$	0	0
$448$	0	0
$449$	25.9288i	1.22366i	0.790990	+	0.611829i	$0.209566\pi$
$449$	25.9288i	1.22366i	−0.790990	+	0.611829i	$0.790434\pi$
$450$	0	0
$451$	− 4.29014i	− 0.202015i
$452$	− 13.4756i	− 0.633840i
$453$	0	0
$454$	12.3627i	0.580209i
$455$	0	0
$456$	0	0
$457$	3.24264	0.151684	0.0758422	−	0.997120i	$-0.475835\pi$
$457$	3.24264	0.151684	0.0758422	+	0.997120i	$0.475835\pi$
$458$	−7.74038	−0.361684
$459$	0	0
$460$	13.9991i	0.652710i
$461$	−14.4477	−0.672896	−0.336448	−	0.941702i	$-0.609226\pi$
$461$	−14.4477	−0.672896	−0.336448	+	0.941702i	$0.609226\pi$
$462$	0	0
$463$	−6.10257	−0.283610	−0.141805	−	0.989895i	$-0.545291\pi$
$463$	−6.10257	−0.283610	−0.141805	+	0.989895i	$0.545291\pi$
$464$	− 5.15408i	− 0.239272i
$465$	0	0
$466$	−10.3275	−0.478410
$467$	−2.40534	−0.111306	−0.0556530	−	0.998450i	$-0.517724\pi$
$467$	−2.40534	−0.111306	−0.0556530	+	0.998450i	$0.517724\pi$
$468$	0	0
$469$	0	0
$470$	10.5532i	0.486783i
$471$	0	0
$472$	5.75484i	0.264888i
$473$	6.71889i	0.308935i
$474$	0	0
$475$	− 41.4092i	− 1.89998i
$476$	0	0
$477$	0	0
$478$	14.6167	0.668552
$479$	0.509015	0.0232575	0.0116287	−	0.999932i	$-0.496298\pi$
$479$	0.509015	0.0232575	0.0116287	+	0.999932i	$0.496298\pi$
$480$	0	0
$481$	9.05356i	0.412807i
$482$	−20.8803	−0.951070
$483$	0	0
$484$	−10.5453	−0.479330
$485$	38.6197i	1.75363i
$486$	0	0
$487$	−32.9129	−1.49143	−0.745713	−	0.666267i	$-0.767891\pi$
$487$	−32.9129	−1.49143	−0.745713	+	0.666267i	$0.767891\pi$
$488$	6.98840	0.316350
$489$	0	0
$490$	0	0
$491$	35.2692i	1.59168i	0.605510	+	0.795838i	$0.292969\pi$
$491$	35.2692i	1.59168i	−0.605510	+	0.795838i	$0.707031\pi$
$492$	0	0
$493$	− 37.4976i	− 1.68881i
$494$	− 6.83833i	− 0.307671i
$495$	0	0
$496$	− 2.66105i	− 0.119485i
$497$	0	0
$498$	0	0
$499$	6.85933	0.307066	0.153533	−	0.988144i	$-0.450935\pi$
$499$	6.85933	0.307066	0.153533	+	0.988144i	$0.450935\pi$
$500$	−1.07795	−0.0482074
$501$	0	0
$502$	2.38929i	0.106639i
$503$	14.5716	0.649717	0.324858	−	0.945763i	$-0.394683\pi$
$503$	14.5716	0.649717	0.324858	+	0.945763i	$0.394683\pi$
$504$	0	0
$505$	−38.1266	−1.69661
$506$	2.93643i	0.130540i
$507$	0	0
$508$	−0.530170	−0.0235225
$509$	−4.90875	−0.217577	−0.108788	−	0.994065i	$-0.534697\pi$
$509$	−4.90875	−0.217577	−0.108788	+	0.994065i	$0.534697\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	6.40073i	0.282324i
$515$	− 64.1232i	− 2.82561i
$516$	0	0
$517$	2.21363i	0.0973554i
$518$	0	0
$519$	0	0
$520$	−2.83253	−0.124215
$521$	11.6749	0.511487	0.255744	−	0.966745i	$-0.417680\pi$
$521$	11.6749	0.511487	0.255744	+	0.966745i	$0.417680\pi$
$522$	0	0
$523$	19.1959i	0.839380i	0.907667	+	0.419690i	$0.137861\pi$
$523$	19.1959i	0.839380i	−0.907667	+	0.419690i	$0.862139\pi$
$524$	17.0302	0.743970
$525$	0	0
$526$	−25.4911	−1.11146
$527$	− 19.3600i	− 0.843337i
$528$	0	0
$529$	4.03838	0.175582
$530$	8.38782	0.364344
$531$	0	0
$532$	0	0
$533$	5.60534i	0.242794i
$534$	0	0
$535$	− 57.2119i	− 2.47349i
$536$	− 7.94993i	− 0.343385i
$537$	0	0
$538$	15.8526i	0.683456i
$539$	0	0
$540$	0	0
$541$	5.77836	0.248431	0.124215	−	0.992255i	$-0.460359\pi$
$541$	5.77836	0.248431	0.124215	+	0.992255i	$0.460359\pi$
$542$	−10.3945	−0.446483
$543$	0	0
$544$	7.27533i	0.311927i
$545$	13.1760	0.564397
$546$	0	0
$547$	5.75240	0.245955	0.122977	−	0.992409i	$-0.460756\pi$
$547$	5.75240	0.245955	0.122977	+	0.992409i	$0.460756\pi$
$548$	− 13.9365i	− 0.595340i
$549$	0	0
$550$	−3.59784	−0.153412
$551$	−40.0026	−1.70417
$552$	0	0
$553$	0	0
$554$	− 1.70585i	− 0.0724744i
$555$	0	0
$556$	− 6.55078i	− 0.277815i
$557$	22.4405i	0.950834i	0.879760	+	0.475417i	$0.157703\pi$
$557$	22.4405i	0.950834i	−0.879760	+	0.475417i	$0.842297\pi$
$558$	0	0
$559$	− 8.77866i	− 0.371298i
$560$	0	0
$561$	0	0
$562$	11.7087	0.493903
$563$	−31.5838	−1.33110	−0.665548	−	0.746355i	$-0.731802\pi$
$563$	−31.5838	−1.33110	−0.665548	+	0.746355i	$0.731802\pi$
$564$	0	0
$565$	43.3222i	1.82258i
$566$	−1.09311	−0.0459470
$567$	0	0
$568$	−4.11185	−0.172530
$569$	− 21.5934i	− 0.905242i	−0.891703	−	0.452621i	$-0.850489\pi$
$569$	− 21.5934i	− 0.905242i	0.891703	−	0.452621i	$-0.149511\pi$
$570$	0	0
$571$	−15.9561	−0.667742	−0.333871	−	0.942619i	$-0.608355\pi$
$571$	−15.9561	−0.667742	−0.333871	+	0.942619i	$0.608355\pi$
$572$	−0.594149	−0.0248426
$573$	0	0
$574$	0	0
$575$	− 23.2325i	− 0.968864i
$576$	0	0
$577$	− 23.0600i	− 0.960001i	−0.877268	−	0.480001i	$-0.840636\pi$
$577$	− 23.0600i	− 0.960001i	0.877268	−	0.480001i	$-0.159364\pi$
$578$	35.9304i	1.49451i
$579$	0	0
$580$	16.5696i	0.688017i
$581$	0	0
$582$	0	0
$583$	1.75942	0.0728678
$584$	13.8105	0.571483
$585$	0	0
$586$	− 4.40649i	− 0.182031i
$587$	−27.0023	−1.11450	−0.557252	−	0.830344i	$-0.688144\pi$
$587$	−27.0023	−1.11450	−0.557252	+	0.830344i	$0.688144\pi$
$588$	0	0
$589$	−20.6534	−0.851007
$590$	− 18.5010i	− 0.761674i
$591$	0	0
$592$	−10.2756	−0.422324
$593$	−8.86157	−0.363901	−0.181951	−	0.983308i	$-0.558241\pi$
$593$	−8.86157	−0.363901	−0.181951	+	0.983308i	$0.558241\pi$
$594$	0	0
$595$	0	0
$596$	22.4410i	0.919220i
$597$	0	0
$598$	− 3.83664i	− 0.156892i
$599$	− 32.0230i	− 1.30842i	−0.756312	−	0.654211i	$-0.773001\pi$
$599$	− 32.0230i	− 1.30842i	0.756312	−	0.654211i	$-0.226999\pi$
$600$	0	0
$601$	− 31.4075i	− 1.28114i	−0.767901	−	0.640569i	$-0.778699\pi$
$601$	− 31.4075i	− 1.28114i	0.767901	−	0.640569i	$-0.221301\pi$
$602$	0	0
$603$	0	0
$604$	−19.6260	−0.798570
$605$	33.9015	1.37829
$606$	0	0
$607$	37.4300i	1.51924i	0.650368	+	0.759619i	$0.274615\pi$
$607$	37.4300i	1.51924i	−0.650368	+	0.759619i	$0.725385\pi$
$608$	7.76135	0.314764
$609$	0	0
$610$	−22.4667	−0.909651
$611$	− 2.89225i	− 0.117008i
$612$	0	0
$613$	46.9883	1.89784	0.948920	−	0.315517i	$-0.102178\pi$
$613$	46.9883	1.89784	0.948920	+	0.315517i	$0.102178\pi$
$614$	−26.5696	−1.07226
$615$	0	0
$616$	0	0
$617$	− 29.2461i	− 1.17741i	−0.808350	−	0.588703i	$-0.799639\pi$
$617$	− 29.2461i	− 1.17741i	0.808350	−	0.588703i	$-0.200361\pi$
$618$	0	0
$619$	27.2036i	1.09341i	0.837327	+	0.546703i	$0.184117\pi$
$619$	27.2036i	1.09341i	−0.837327	+	0.546703i	$0.815883\pi$
$620$	8.55491i	0.343573i
$621$	0	0
$622$	− 17.0739i	− 0.684602i
$623$	0	0
$624$	0	0
$625$	−23.2111	−0.928442
$626$	10.9407	0.437280
$627$	0	0
$628$	− 5.72677i	− 0.228523i
$629$	−74.7582	−2.98081
$630$	0	0
$631$	−5.71335	−0.227445	−0.113722	−	0.993513i	$-0.536277\pi$
$631$	−5.71335	−0.227445	−0.113722	+	0.993513i	$0.536277\pi$
$632$	1.21935i	0.0485030i
$633$	0	0
$634$	15.3685	0.610361
$635$	1.70442	0.0676379
$636$	0	0
$637$	0	0
$638$	3.47563i	0.137602i
$639$	0	0
$640$	− 3.21486i	− 0.127078i
$641$	16.5644i	0.654256i	0.944980	+	0.327128i	$0.106081\pi$
$641$	16.5644i	0.654256i	−0.944980	+	0.327128i	$0.893919\pi$
$642$	0	0
$643$	− 14.7665i	− 0.582333i	−0.956672	−	0.291167i	$-0.905957\pi$
$643$	− 14.7665i	− 0.582333i	0.956672	−	0.291167i	$-0.0940434\pi$
$644$	0	0
$645$	0	0
$646$	56.4664	2.22164
$647$	3.54951	0.139546	0.0697729	−	0.997563i	$-0.477773\pi$
$647$	3.54951	0.139546	0.0697729	+	0.997563i	$0.477773\pi$
$648$	0	0
$649$	− 3.88075i	− 0.152333i
$650$	4.70080	0.184381
$651$	0	0
$652$	−20.0480	−0.785141
$653$	− 9.38344i	− 0.367202i	−0.983001	−	0.183601i	$-0.941225\pi$
$653$	− 9.38344i	− 0.367202i	0.983001	−	0.183601i	$-0.0587755\pi$
$654$	0	0
$655$	−54.7498	−2.13925
$656$	−6.36193	−0.248392
$657$	0	0
$658$	0	0
$659$	− 4.56685i	− 0.177899i	−0.996036	−	0.0889497i	$-0.971649\pi$
$659$	− 4.56685i	− 0.177899i	0.996036	−	0.0889497i	$-0.0283510\pi$
$660$	0	0
$661$	13.8593i	0.539063i	0.962992	+	0.269532i	$0.0868688\pi$
$661$	13.8593i	0.539063i	−0.962992	+	0.269532i	$0.913131\pi$
$662$	− 23.7515i	− 0.923127i
$663$	0	0
$664$	− 1.71958i	− 0.0667328i
$665$	0	0
$666$	0	0
$667$	−22.4434	−0.869013
$668$	−11.8314	−0.457772
$669$	0	0
$670$	25.5579i	0.987387i
$671$	−4.71260	−0.181928
$672$	0	0
$673$	36.7067	1.41494	0.707469	−	0.706744i	$-0.249837\pi$
$673$	36.7067	1.41494	0.707469	+	0.706744i	$0.249837\pi$
$674$	− 2.59185i	− 0.0998342i
$675$	0	0
$676$	−12.2237	−0.470143
$677$	49.6370	1.90770	0.953852	−	0.300279i	$-0.0970796\pi$
$677$	49.6370	1.90770	0.953852	+	0.300279i	$0.0970796\pi$
$678$	0	0
$679$	0	0
$680$	− 23.3891i	− 0.896932i
$681$	0	0
$682$	1.79447i	0.0687138i
$683$	− 17.5011i	− 0.669661i	−0.942278	−	0.334830i	$-0.891321\pi$
$683$	− 17.5011i	− 0.669661i	0.942278	−	0.334830i	$-0.108679\pi$
$684$	0	0
$685$	44.8040i	1.71187i
$686$	0	0
$687$	0	0
$688$	9.96357	0.379858
$689$	−2.29880	−0.0875771
$690$	0	0
$691$	2.89203i	0.110018i	0.998486	+	0.0550089i	$0.0175187\pi$
$691$	2.89203i	0.110018i	−0.998486	+	0.0550089i	$0.982481\pi$
$692$	−9.44547	−0.359063
$693$	0	0
$694$	4.78764	0.181736
$695$	21.0598i	0.798845i
$696$	0	0
$697$	−46.2851	−1.75318
$698$	30.1643	1.14174
$699$	0	0
$700$	0	0
$701$	37.6260i	1.42111i	0.703639	+	0.710557i	$0.251557\pi$
$701$	37.6260i	1.42111i	−0.703639	+	0.710557i	$0.748443\pi$
$702$	0	0
$703$	79.7524i	3.00792i
$704$	− 0.674346i	− 0.0254154i
$705$	0	0
$706$	− 6.45715i	− 0.243018i
$707$	0	0
$708$	0	0
$709$	10.6261	0.399071	0.199535	−	0.979891i	$-0.436057\pi$
$709$	10.6261	0.399071	0.199535	+	0.979891i	$0.436057\pi$
$710$	13.2190	0.496101
$711$	0	0
$712$	− 6.94806i	− 0.260389i
$713$	−11.5875	−0.433957
$714$	0	0
$715$	1.91010	0.0714339
$716$	− 17.4678i	− 0.652802i
$717$	0	0
$718$	6.50159	0.242637
$719$	10.8157	0.403358	0.201679	−	0.979452i	$-0.435360\pi$
$719$	10.8157	0.403358	0.201679	+	0.979452i	$0.435360\pi$
$720$	0	0
$721$	0	0
$722$	− 41.2386i	− 1.53474i
$723$	0	0
$724$	− 15.4317i	− 0.573516i
$725$	− 27.4986i	− 1.02127i
$726$	0	0
$727$	− 26.8706i	− 0.996574i	−0.867012	−	0.498287i	$-0.833963\pi$
$727$	− 26.8706i	− 0.996574i	0.867012	−	0.498287i	$-0.166037\pi$
$728$	0	0
$729$	0	0
$730$	−44.3988	−1.64327
$731$	72.4883	2.68108
$732$	0	0
$733$	34.9103i	1.28944i	0.764417	+	0.644722i	$0.223027\pi$
$733$	34.9103i	1.28944i	−0.764417	+	0.644722i	$0.776973\pi$
$734$	2.76854	0.102189
$735$	0	0
$736$	4.35449	0.160509
$737$	5.36100i	0.197475i
$738$	0	0
$739$	−28.1478	−1.03543	−0.517717	−	0.855552i	$-0.673218\pi$
$739$	−28.1478	−1.03543	−0.517717	+	0.855552i	$0.673218\pi$
$740$	33.0345	1.21437
$741$	0	0
$742$	0	0
$743$	− 12.0740i	− 0.442952i	−0.975166	−	0.221476i	$-0.928913\pi$
$743$	− 12.0740i	− 0.442952i	0.975166	−	0.221476i	$-0.0710874\pi$
$744$	0	0
$745$	− 72.1446i	− 2.64318i
$746$	− 18.6801i	− 0.683929i
$747$	0	0
$748$	− 4.90609i	− 0.179384i
$749$	0	0
$750$	0	0
$751$	34.0396	1.24212	0.621061	−	0.783762i	$-0.286702\pi$
$751$	34.0396	1.24212	0.621061	+	0.783762i	$0.286702\pi$
$752$	3.28264	0.119705
$753$	0	0
$754$	− 4.54113i	− 0.165378i
$755$	63.0947	2.29625
$756$	0	0
$757$	−51.6104	−1.87581	−0.937907	−	0.346888i	$-0.887238\pi$
$757$	−51.6104	−1.87581	−0.937907	+	0.346888i	$0.887238\pi$
$758$	− 8.86137i	− 0.321859i
$759$	0	0
$760$	−24.9516	−0.905091
$761$	−6.38623	−0.231501	−0.115750	−	0.993278i	$-0.536927\pi$
$761$	−6.38623	−0.231501	−0.115750	+	0.993278i	$0.536927\pi$
$762$	0	0
$763$	0	0
$764$	− 17.2214i	− 0.623049i
$765$	0	0
$766$	− 30.8075i	− 1.11312i
$767$	5.07045i	0.183083i
$768$	0	0
$769$	− 37.6316i	− 1.35703i	−0.734586	−	0.678516i	$-0.762623\pi$
$769$	− 37.6316i	− 1.35703i	0.734586	−	0.678516i	$-0.237377\pi$
$770$	0	0
$771$	0	0
$772$	18.6569	0.671475
$773$	40.0140	1.43920	0.719601	−	0.694387i	$-0.244325\pi$
$773$	40.0140	1.43920	0.719601	+	0.694387i	$0.244325\pi$
$774$	0	0
$775$	− 14.1975i	− 0.509990i
$776$	12.0129	0.431238
$777$	0	0
$778$	−8.41192	−0.301582
$779$	49.3772i	1.76912i
$780$	0	0
$781$	2.77281	0.0992190
$782$	31.6804	1.13289
$783$	0	0
$784$	0	0
$785$	18.4107i	0.657107i
$786$	0	0
$787$	− 26.4182i	− 0.941708i	−0.882211	−	0.470854i	$-0.843946\pi$
$787$	− 26.4182i	− 0.941708i	0.882211	−	0.470854i	$-0.156054\pi$
$788$	7.70293i	0.274405i
$789$	0	0
$790$	− 3.92002i	− 0.139468i
$791$	0	0
$792$	0	0
$793$	6.15731	0.218652
$794$	−23.7281	−0.842080
$795$	0	0
$796$	− 3.02803i	− 0.107326i
$797$	6.04129	0.213994	0.106997	−	0.994259i	$-0.465877\pi$
$797$	6.04129	0.213994	0.106997	+	0.994259i	$0.465877\pi$
$798$	0	0
$799$	23.8823	0.844894
$800$	5.33530i	0.188631i
$801$	0	0
$802$	12.1940	0.430586
$803$	−9.31305	−0.328651
$804$	0	0
$805$	0	0
$806$	− 2.34459i	− 0.0825846i
$807$	0	0
$808$	11.8595i	0.417216i
$809$	26.0910i	0.917310i	0.888614	+	0.458655i	$0.151669\pi$
$809$	26.0910i	0.917310i	−0.888614	+	0.458655i	$0.848331\pi$
$810$	0	0
$811$	− 20.3158i	− 0.713383i	−0.934222	−	0.356692i	$-0.883905\pi$
$811$	− 20.3158i	− 0.713383i	0.934222	−	0.356692i	$-0.116095\pi$
$812$	0	0
$813$	0	0
$814$	6.92930	0.242872
$815$	64.4515	2.25764
$816$	0	0
$817$	− 77.3308i	− 2.70546i
$818$	−12.5274	−0.438011
$819$	0	0
$820$	20.4527	0.714239
$821$	− 22.0129i	− 0.768255i	−0.923280	−	0.384128i	$-0.874502\pi$
$821$	− 22.0129i	− 0.768255i	0.923280	−	0.384128i	$-0.125498\pi$
$822$	0	0
$823$	43.0941	1.50217	0.751083	−	0.660207i	$-0.229532\pi$
$823$	43.0941	1.50217	0.751083	+	0.660207i	$0.229532\pi$
$824$	−19.9459	−0.694849
$825$	0	0
$826$	0	0
$827$	− 33.7439i	− 1.17339i	−0.809808	−	0.586695i	$-0.800429\pi$
$827$	− 33.7439i	− 1.17339i	0.809808	−	0.586695i	$-0.199571\pi$
$828$	0	0
$829$	9.20114i	0.319569i	0.987152	+	0.159784i	$0.0510799\pi$
$829$	9.20114i	0.319569i	−0.987152	+	0.159784i	$0.948920\pi$
$830$	5.52821i	0.191887i
$831$	0	0
$832$	0.881075i	0.0305458i
$833$	0	0
$834$	0	0
$835$	38.0363	1.31630
$836$	−5.23383	−0.181016
$837$	0	0
$838$	10.9782i	0.379237i
$839$	−13.8018	−0.476492	−0.238246	−	0.971205i	$-0.576572\pi$
$839$	−13.8018	−0.476492	−0.238246	+	0.971205i	$0.576572\pi$
$840$	0	0
$841$	2.43544	0.0839809
$842$	− 18.6015i	− 0.641050i
$843$	0	0
$844$	11.0886	0.381684
$845$	39.2975	1.35187
$846$	0	0
$847$	0	0
$848$	− 2.60908i	− 0.0895962i
$849$	0	0
$850$	38.8161i	1.33138i
$851$	44.7450i	1.53384i
$852$	0	0
$853$	18.0524i	0.618104i	0.951045	+	0.309052i	$0.100012\pi$
$853$	18.0524i	0.618104i	−0.951045	+	0.309052i	$0.899988\pi$
$854$	0	0
$855$	0	0
$856$	−17.7961	−0.608258
$857$	−30.0727	−1.02726	−0.513632	−	0.858010i	$-0.671700\pi$
$857$	−30.0727	−1.02726	−0.513632	+	0.858010i	$0.671700\pi$
$858$	0	0
$859$	21.3748i	0.729299i	0.931145	+	0.364649i	$0.118811\pi$
$859$	21.3748i	0.729299i	−0.931145	+	0.364649i	$0.881189\pi$
$860$	−32.0315	−1.09226
$861$	0	0
$862$	16.4856	0.561503
$863$	− 28.1613i	− 0.958623i	−0.877645	−	0.479311i	$-0.840886\pi$
$863$	− 28.1613i	− 0.958623i	0.877645	−	0.479311i	$-0.159114\pi$
$864$	0	0
$865$	30.3658	1.03247
$866$	11.0014	0.373843
$867$	0	0
$868$	0	0
$869$	− 0.822261i	− 0.0278933i
$870$	0	0
$871$	− 7.00448i	− 0.237338i
$872$	− 4.09846i	− 0.138792i
$873$	0	0
$874$	− 33.7968i	− 1.14319i
$875$	0	0
$876$	0	0
$877$	−24.0338	−0.811563	−0.405782	−	0.913970i	$-0.633001\pi$
$877$	−24.0338	−0.811563	−0.405782	+	0.913970i	$0.633001\pi$
$878$	−1.17317	−0.0395924
$879$	0	0
$880$	2.16792i	0.0730807i
$881$	13.4156	0.451984	0.225992	−	0.974129i	$-0.427438\pi$
$881$	13.4156	0.451984	0.225992	+	0.974129i	$0.427438\pi$
$882$	0	0
$883$	−47.2863	−1.59131	−0.795656	−	0.605749i	$-0.792874\pi$
$883$	−47.2863	−1.59131	−0.795656	+	0.605749i	$0.792874\pi$
$884$	6.41011i	0.215595i
$885$	0	0
$886$	17.0789	0.573777
$887$	−45.9051	−1.54134	−0.770671	−	0.637234i	$-0.780079\pi$
$887$	−45.9051	−1.54134	−0.770671	+	0.637234i	$0.780079\pi$
$888$	0	0
$889$	0	0
$890$	22.3370i	0.748738i
$891$	0	0
$892$	− 14.4548i	− 0.483983i
$893$	− 25.4777i	− 0.852579i
$894$	0	0
$895$	56.1564i	1.87710i
$896$	0	0
$897$	0	0
$898$	−25.9288	−0.865257
$899$	−13.7153	−0.457430
$900$	0	0
$901$	− 18.9819i	− 0.632379i
$902$	4.29014	0.142846
$903$	0	0
$904$	13.4756	0.448193
$905$	49.6108i	1.64912i
$906$	0	0
$907$	−32.6382	−1.08373	−0.541866	−	0.840465i	$-0.682282\pi$
$907$	−32.6382	−1.08373	−0.541866	+	0.840465i	$0.682282\pi$
$908$	−12.3627	−0.410270
$909$	0	0
$910$	0	0
$911$	− 7.99143i	− 0.264768i	−0.991199	−	0.132384i	$-0.957737\pi$
$911$	− 7.99143i	− 0.264768i	0.991199	−	0.132384i	$-0.0422632\pi$
$912$	0	0
$913$	1.15959i	0.0383769i
$914$	3.24264i	0.107257i
$915$	0	0
$916$	− 7.74038i	− 0.255749i
$917$	0	0
$918$	0	0
$919$	−14.1404	−0.466450	−0.233225	−	0.972423i	$-0.574928\pi$
$919$	−14.1404	−0.466450	−0.233225	+	0.972423i	$0.574928\pi$
$920$	−13.9991	−0.461536
$921$	0	0
$922$	− 14.4477i	− 0.475809i
$923$	−3.62285	−0.119248
$924$	0	0
$925$	−54.8234	−1.80258
$926$	− 6.10257i	− 0.200543i
$927$	0	0
$928$	5.15408	0.169191
$929$	−18.9909	−0.623070	−0.311535	−	0.950235i	$-0.600843\pi$
$929$	−18.9909	−0.623070	−0.311535	+	0.950235i	$0.600843\pi$
$930$	0	0
$931$	0	0
$932$	− 10.3275i	− 0.338287i
$933$	0	0
$934$	− 2.40534i	− 0.0787053i
$935$	15.7724i	0.515811i
$936$	0	0
$937$	10.9240i	0.356872i	0.983951	+	0.178436i	$0.0571038\pi$
$937$	10.9240i	0.356872i	−0.983951	+	0.178436i	$0.942896\pi$
$938$	0	0
$939$	0	0
$940$	−10.5532	−0.344208
$941$	24.3553	0.793960	0.396980	−	0.917827i	$-0.370058\pi$
$941$	24.3553	0.793960	0.396980	+	0.917827i	$0.370058\pi$
$942$	0	0
$943$	27.7030i	0.902134i
$944$	−5.75484	−0.187304
$945$	0	0
$946$	−6.71889	−0.218450
$947$	34.2229i	1.11209i	0.831151	+	0.556047i	$0.187682\pi$
$947$	34.2229i	1.11209i	−0.831151	+	0.556047i	$0.812318\pi$
$948$	0	0
$949$	12.1681	0.394993
$950$	41.4092	1.34349
$951$	0	0
$952$	0	0
$953$	41.8036i	1.35415i	0.735913	+	0.677076i	$0.236753\pi$
$953$	41.8036i	1.35415i	−0.735913	+	0.677076i	$0.763247\pi$
$954$	0	0
$955$	55.3644i	1.79155i
$956$	14.6167i	0.472738i
$957$	0	0
$958$	0.509015i	0.0164455i
$959$	0	0
$960$	0	0
$961$	23.9188	0.771574
$962$	−9.05356	−0.291899
$963$	0	0
$964$	− 20.8803i	− 0.672508i
$965$	−59.9791	−1.93080
$966$	0	0
$967$	11.1057	0.357136	0.178568	−	0.983928i	$-0.442854\pi$
$967$	11.1057	0.357136	0.178568	+	0.983928i	$0.442854\pi$
$968$	− 10.5453i	− 0.338937i
$969$	0	0
$970$	−38.6197	−1.24000
$971$	−37.7489	−1.21142	−0.605710	−	0.795685i	$-0.707111\pi$
$971$	−37.7489	−1.21142	−0.605710	+	0.795685i	$0.707111\pi$
$972$	0	0
$973$	0	0
$974$	− 32.9129i	− 1.05460i
$975$	0	0
$976$	6.98840i	0.223693i
$977$	31.6108i	1.01132i	0.862733	+	0.505660i	$0.168751\pi$
$977$	31.6108i	1.01132i	−0.862733	+	0.505660i	$0.831249\pi$
$978$	0	0
$979$	4.68539i	0.149746i
$980$	0	0
$981$	0	0
$982$	−35.2692	−1.12548
$983$	4.52580	0.144351	0.0721754	−	0.997392i	$-0.477006\pi$
$983$	4.52580	0.144351	0.0721754	+	0.997392i	$0.477006\pi$
$984$	0	0
$985$	− 24.7638i	− 0.789041i
$986$	37.4976	1.19417
$987$	0	0
$988$	6.83833	0.217556
$989$	− 43.3863i	− 1.37960i
$990$	0	0
$991$	−2.72416	−0.0865358	−0.0432679	−	0.999064i	$-0.513777\pi$
$991$	−2.72416	−0.0865358	−0.0432679	+	0.999064i	$0.513777\pi$
$992$	2.66105	0.0844885
$993$	0	0
$994$	0	0
$995$	9.73469i	0.308610i
$996$	0	0
$997$	2.91474i	0.0923107i	0.998934	+	0.0461554i	$0.0146969\pi$
$997$	2.91474i	0.0923107i	−0.998934	+	0.0461554i	$0.985303\pi$
$998$	6.85933i	0.217128i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2646.2.d.e.2645.15	yes	16
3.2	odd	2	inner	2646.2.d.e.2645.2	✓	16
7.6	odd	2	inner	2646.2.d.e.2645.10	yes	16
21.20	even	2	inner	2646.2.d.e.2645.7	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2646.2.d.e.2645.2	✓	16	3.2	odd	2	inner
2646.2.d.e.2645.7	yes	16	21.20	even	2	inner
2646.2.d.e.2645.10	yes	16	7.6	odd	2	inner
2646.2.d.e.2645.15	yes	16	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +3.21486 q^{5} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +3.21486 q^{5} -1.00000i q^{8} +3.21486i q^{10} +0.674346i q^{11} -0.881075i q^{13} +1.00000 q^{16} +7.27533 q^{17} -7.76135i q^{19} -3.21486 q^{20} -0.674346 q^{22} -4.35449i q^{23} +5.33530 q^{25} +0.881075 q^{26} -5.15408i q^{29} -2.66105i q^{31} +1.00000i q^{32} +7.27533i q^{34} -10.2756 q^{37} +7.76135 q^{38} -3.21486i q^{40} -6.36193 q^{41} +9.96357 q^{43} -0.674346i q^{44} +4.35449 q^{46} +3.28264 q^{47} +5.33530i q^{50} +0.881075i q^{52} -2.60908i q^{53} +2.16792i q^{55} +5.15408 q^{58} -5.75484 q^{59} +6.98840i q^{61} +2.66105 q^{62} -1.00000 q^{64} -2.83253i q^{65} +7.94993 q^{67} -7.27533 q^{68} -4.11185i q^{71} +13.8105i q^{73} -10.2756i q^{74} +7.76135i q^{76} -1.21935 q^{79} +3.21486 q^{80} -6.36193i q^{82} +1.71958 q^{83} +23.3891 q^{85} +9.96357i q^{86} +0.674346 q^{88} +6.94806 q^{89} +4.35449i q^{92} +3.28264i q^{94} -24.9516i q^{95} +12.0129i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 16 * q - 16 * q^4	\( 16 q - 16 q^{4} + 16 q^{16} - 32 q^{22} + 48 q^{25} - 32 q^{37} + 16 q^{43} - 48 q^{46} + 16 q^{58} - 16 q^{64} + 16 q^{67} + 32 q^{88}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^16 - 32 * q^22 + 48 * q^25 - 32 * q^37 + 16 * q^43 - 48 * q^46 + 16 * q^58 - 16 * q^64 + 16 * q^67 + 32 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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