LMFDB - Embedded newform 4400.2.b.t.4049.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4400,2,Mod(4049,4400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("4400.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.4
Root			$2.56155i$ of defining polynomial
Character	$\chi$	$=$	4400.4049
Dual form			4400.2.b.t.4049.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155i q^{3} -2.56155i q^{7} -3.56155 q^{9} +O(q^{10})$	$q+2.56155i q^{3} -2.56155i q^{7} -3.56155 q^{9} -1.00000 q^{11} +2.00000i q^{13} +0.561553i q^{17} +2.56155 q^{19} +6.56155 q^{21} -5.12311i q^{23} -1.43845i q^{27} -9.68466 q^{29} -6.56155 q^{31} -2.56155i q^{33} +5.68466i q^{37} -5.12311 q^{39} +2.00000 q^{41} -10.2462i q^{43} -13.1231i q^{47} +0.438447 q^{49} -1.43845 q^{51} +4.56155i q^{53} +6.56155i q^{57} +1.12311 q^{59} +2.31534 q^{61} +9.12311i q^{63} +6.24621i q^{67} +13.1231 q^{69} -3.68466 q^{71} -2.00000i q^{73} +2.56155i q^{77} -15.3693 q^{79} -7.00000 q^{81} -5.12311i q^{83} -24.8078i q^{87} +12.5616 q^{89} +5.12311 q^{91} -16.8078i q^{93} -7.12311i q^{97} +3.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{9}+O(q^{10}) $ 4 * q - 6 * q^9	$ 4 q - 6 q^{9} - 4 q^{11} + 2 q^{19} + 18 q^{21} - 14 q^{29} - 18 q^{31} - 4 q^{39} + 8 q^{41} + 10 q^{49} - 14 q^{51} - 12 q^{59} + 34 q^{61} + 36 q^{69} + 10 q^{71} - 12 q^{79} - 28 q^{81} + 42 q^{89} + 4 q^{91} + 6 q^{99}+O(q^{100}) $ 4 * q - 6 * q^9 - 4 * q^11 + 2 * q^19 + 18 * q^21 - 14 * q^29 - 18 * q^31 - 4 * q^39 + 8 * q^41 + 10 * q^49 - 14 * q^51 - 12 * q^59 + 34 * q^61 + 36 * q^69 + 10 * q^71 - 12 * q^79 - 28 * q^81 + 42 * q^89 + 4 * q^91 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$.

$n$	$177$	$1201$	$2751$	$3301$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	2.56155i	1.47891i	0.673204	+	0.739457i	$0.264917\pi$
$3$	2.56155i	1.47891i	−0.673204	+	0.739457i	$0.735083\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.56155i	− 0.968176i	−0.875019	−	0.484088i	$-0.839151\pi$
$7$	− 2.56155i	− 0.968176i	0.875019	−	0.484088i	$-0.160849\pi$
$8$	0	0
$9$	−3.56155	−1.18718
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.561553i	0.136197i	0.997679	+	0.0680983i	$0.0216931\pi$
$17$	0.561553i	0.136197i	−0.997679	+	0.0680983i	$0.978307\pi$
$18$	0	0
$19$	2.56155	0.587661	0.293830	−	0.955858i	$-0.405070\pi$
$19$	2.56155	0.587661	0.293830	+	0.955858i	$0.405070\pi$
$20$	0	0
$21$	6.56155	1.43185
$22$	0	0
$23$	− 5.12311i	− 1.06824i	−0.845408	−	0.534121i	$-0.820643\pi$
$23$	− 5.12311i	− 1.06824i	0.845408	−	0.534121i	$-0.179357\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.43845i	− 0.276829i
$28$	0	0
$29$	−9.68466	−1.79840	−0.899198	−	0.437542i	$-0.855849\pi$
$29$	−9.68466	−1.79840	−0.899198	+	0.437542i	$0.855849\pi$
$30$	0	0
$31$	−6.56155	−1.17849	−0.589245	−	0.807955i	$-0.700575\pi$
$31$	−6.56155	−1.17849	−0.589245	+	0.807955i	$0.700575\pi$
$32$	0	0
$33$	− 2.56155i	− 0.445909i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.68466i	0.934552i	0.884111	+	0.467276i	$0.154765\pi$
$37$	5.68466i	0.934552i	−0.884111	+	0.467276i	$0.845235\pi$
$38$	0	0
$39$	−5.12311	−0.820353
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	− 10.2462i	− 1.56253i	−0.624198	−	0.781266i	$-0.714574\pi$
$43$	− 10.2462i	− 1.56253i	0.624198	−	0.781266i	$-0.285426\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 13.1231i	− 1.91420i	−0.289755	−	0.957101i	$-0.593574\pi$
$47$	− 13.1231i	− 1.91420i	0.289755	−	0.957101i	$-0.406426\pi$
$48$	0	0
$49$	0.438447	0.0626353
$50$	0	0
$51$	−1.43845	−0.201423
$52$	0	0
$53$	4.56155i	0.626577i	0.949658	+	0.313289i	$0.101431\pi$
$53$	4.56155i	0.626577i	−0.949658	+	0.313289i	$0.898569\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.56155i	0.869099i
$58$	0	0
$59$	1.12311	0.146216	0.0731079	−	0.997324i	$-0.476708\pi$
$59$	1.12311	0.146216	0.0731079	+	0.997324i	$0.476708\pi$
$60$	0	0
$61$	2.31534	0.296449	0.148225	−	0.988954i	$-0.452644\pi$
$61$	2.31534	0.296449	0.148225	+	0.988954i	$0.452644\pi$
$62$	0	0
$63$	9.12311i	1.14940i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.24621i	0.763096i	0.924349	+	0.381548i	$0.124609\pi$
$67$	6.24621i	0.763096i	−0.924349	+	0.381548i	$0.875391\pi$
$68$	0	0
$69$	13.1231	1.57984
$70$	0	0
$71$	−3.68466	−0.437289	−0.218644	−	0.975805i	$-0.570163\pi$
$71$	−3.68466	−0.437289	−0.218644	+	0.975805i	$0.570163\pi$
$72$	0	0
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.56155i	0.291916i
$78$	0	0
$79$	−15.3693	−1.72918	−0.864592	−	0.502475i	$-0.832423\pi$
$79$	−15.3693	−1.72918	−0.864592	+	0.502475i	$0.832423\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	− 5.12311i	− 0.562334i	−0.959659	−	0.281167i	$-0.909279\pi$
$83$	− 5.12311i	− 0.562334i	0.959659	−	0.281167i	$-0.0907215\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 24.8078i	− 2.65967i
$88$	0	0
$89$	12.5616	1.33152	0.665761	−	0.746165i	$-0.268107\pi$
$89$	12.5616	1.33152	0.665761	+	0.746165i	$0.268107\pi$
$90$	0	0
$91$	5.12311	0.537047
$92$	0	0
$93$	− 16.8078i	− 1.74288i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 7.12311i	− 0.723242i	−0.932325	−	0.361621i	$-0.882223\pi$
$97$	− 7.12311i	− 0.723242i	0.932325	−	0.361621i	$-0.117777\pi$
$98$	0	0
$99$	3.56155	0.357950
$100$	0	0
$101$	−4.24621	−0.422514	−0.211257	−	0.977431i	$-0.567756\pi$
$101$	−4.24621	−0.422514	−0.211257	+	0.977431i	$0.567756\pi$
$102$	0	0
$103$	− 10.2462i	− 1.00959i	−0.863239	−	0.504795i	$-0.831568\pi$
$103$	− 10.2462i	− 1.00959i	0.863239	−	0.504795i	$-0.168432\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 16.0000i	− 1.54678i	−0.633932	−	0.773389i	$-0.718560\pi$
$107$	− 16.0000i	− 1.54678i	0.633932	−	0.773389i	$-0.281440\pi$
$108$	0	0
$109$	−18.4924	−1.77125	−0.885626	−	0.464398i	$-0.846271\pi$
$109$	−18.4924	−1.77125	−0.885626	+	0.464398i	$0.846271\pi$
$110$	0	0
$111$	−14.5616	−1.38212
$112$	0	0
$113$	− 19.1231i	− 1.79895i	−0.436972	−	0.899475i	$-0.643949\pi$
$113$	− 19.1231i	− 1.79895i	0.436972	−	0.899475i	$-0.356051\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 7.12311i	− 0.658531i
$118$	0	0
$119$	1.43845	0.131862
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	5.12311i	0.461935i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	14.2462i	1.26415i	0.774909	+	0.632073i	$0.217796\pi$
$127$	14.2462i	1.26415i	−0.774909	+	0.632073i	$0.782204\pi$
$128$	0	0
$129$	26.2462	2.31085
$130$	0	0
$131$	7.68466	0.671412	0.335706	−	0.941967i	$-0.391025\pi$
$131$	7.68466	0.671412	0.335706	+	0.941967i	$0.391025\pi$
$132$	0	0
$133$	− 6.56155i	− 0.568959i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 19.6155i	− 1.67587i	−0.545772	−	0.837934i	$-0.683763\pi$
$137$	− 19.6155i	− 1.67587i	0.545772	−	0.837934i	$-0.316237\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	33.6155	2.83094
$142$	0	0
$143$	− 2.00000i	− 0.167248i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.12311i	0.0926322i
$148$	0	0
$149$	−2.31534	−0.189680	−0.0948401	−	0.995493i	$-0.530234\pi$
$149$	−2.31534	−0.189680	−0.0948401	+	0.995493i	$0.530234\pi$
$150$	0	0
$151$	−13.1231	−1.06794	−0.533972	−	0.845502i	$-0.679301\pi$
$151$	−13.1231	−1.06794	−0.533972	+	0.845502i	$0.679301\pi$
$152$	0	0
$153$	− 2.00000i	− 0.161690i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 14.8078i	− 1.18179i	−0.806749	−	0.590894i	$-0.798775\pi$
$157$	− 14.8078i	− 1.18179i	0.806749	−	0.590894i	$-0.201225\pi$
$158$	0	0
$159$	−11.6847	−0.926654
$160$	0	0
$161$	−13.1231	−1.03425
$162$	0	0
$163$	− 3.19224i	− 0.250035i	−0.992155	−	0.125018i	$-0.960101\pi$
$163$	− 3.19224i	− 0.250035i	0.992155	−	0.125018i	$-0.0398987\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 10.5616i	− 0.817277i	−0.912696	−	0.408639i	$-0.866004\pi$
$167$	− 10.5616i	− 0.817277i	0.912696	−	0.408639i	$-0.133996\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	−9.12311	−0.697661
$172$	0	0
$173$	4.24621i	0.322833i	0.986886	+	0.161417i	$0.0516063\pi$
$173$	4.24621i	0.322833i	−0.986886	+	0.161417i	$0.948394\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.87689i	0.216241i
$178$	0	0
$179$	−6.24621	−0.466864	−0.233432	−	0.972373i	$-0.574996\pi$
$179$	−6.24621	−0.466864	−0.233432	+	0.972373i	$0.574996\pi$
$180$	0	0
$181$	−4.24621	−0.315618	−0.157809	−	0.987470i	$-0.550443\pi$
$181$	−4.24621	−0.315618	−0.157809	+	0.987470i	$0.550443\pi$
$182$	0	0
$183$	5.93087i	0.438423i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 0.561553i	− 0.0410648i
$188$	0	0
$189$	−3.68466	−0.268019
$190$	0	0
$191$	−10.2462	−0.741390	−0.370695	−	0.928755i	$-0.620880\pi$
$191$	−10.2462	−0.741390	−0.370695	+	0.928755i	$0.620880\pi$
$192$	0	0
$193$	5.19224i	0.373745i	0.982384	+	0.186873i	$0.0598352\pi$
$193$	5.19224i	0.373745i	−0.982384	+	0.186873i	$0.940165\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.36932i	0.382548i	0.981537	+	0.191274i	$0.0612619\pi$
$197$	5.36932i	0.382548i	−0.981537	+	0.191274i	$0.938738\pi$
$198$	0	0
$199$	−0.807764	−0.0572609	−0.0286304	−	0.999590i	$-0.509115\pi$
$199$	−0.807764	−0.0572609	−0.0286304	+	0.999590i	$0.509115\pi$
$200$	0	0
$201$	−16.0000	−1.12855
$202$	0	0
$203$	24.8078i	1.74116i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	18.2462i	1.26820i
$208$	0	0
$209$	−2.56155	−0.177186
$210$	0	0
$211$	−8.31534	−0.572452	−0.286226	−	0.958162i	$-0.592401\pi$
$211$	−8.31534	−0.572452	−0.286226	+	0.958162i	$0.592401\pi$
$212$	0	0
$213$	− 9.43845i	− 0.646712i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	16.8078i	1.14099i
$218$	0	0
$219$	5.12311	0.346187
$220$	0	0
$221$	−1.12311	−0.0755483
$222$	0	0
$223$	− 13.1231i	− 0.878788i	−0.898294	−	0.439394i	$-0.855193\pi$
$223$	− 13.1231i	− 0.878788i	0.898294	−	0.439394i	$-0.144807\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.87689i	0.190946i	0.995432	+	0.0954731i	$0.0304364\pi$
$227$	2.87689i	0.190946i	−0.995432	+	0.0954731i	$0.969564\pi$
$228$	0	0
$229$	24.7386	1.63477	0.817387	−	0.576088i	$-0.195422\pi$
$229$	24.7386	1.63477	0.817387	+	0.576088i	$0.195422\pi$
$230$	0	0
$231$	−6.56155	−0.431718
$232$	0	0
$233$	19.9309i	1.30571i	0.757481	+	0.652857i	$0.226430\pi$
$233$	19.9309i	1.30571i	−0.757481	+	0.652857i	$0.773570\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 39.3693i	− 2.55731i
$238$	0	0
$239$	25.6155	1.65693	0.828465	−	0.560040i	$-0.189214\pi$
$239$	25.6155	1.65693	0.828465	+	0.560040i	$0.189214\pi$
$240$	0	0
$241$	−16.2462	−1.04651	−0.523255	−	0.852176i	$-0.675283\pi$
$241$	−16.2462	−1.04651	−0.523255	+	0.852176i	$0.675283\pi$
$242$	0	0
$243$	− 22.2462i	− 1.42710i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.12311i	0.325975i
$248$	0	0
$249$	13.1231	0.831643
$250$	0	0
$251$	29.6155	1.86932	0.934658	−	0.355549i	$-0.115706\pi$
$251$	29.6155	1.86932	0.934658	+	0.355549i	$0.115706\pi$
$252$	0	0
$253$	5.12311i	0.322087i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.7386i	1.79267i	0.443381	+	0.896333i	$0.353779\pi$
$257$	28.7386i	1.79267i	−0.443381	+	0.896333i	$0.646221\pi$
$258$	0	0
$259$	14.5616	0.904811
$260$	0	0
$261$	34.4924	2.13503
$262$	0	0
$263$	− 4.80776i	− 0.296459i	−0.988953	−	0.148230i	$-0.952643\pi$
$263$	− 4.80776i	− 0.296459i	0.988953	−	0.148230i	$-0.0473575\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	32.1771i	1.96921i
$268$	0	0
$269$	19.6155	1.19598	0.597990	−	0.801504i	$-0.295966\pi$
$269$	19.6155	1.19598	0.597990	+	0.801504i	$0.295966\pi$
$270$	0	0
$271$	−28.4924	−1.73079	−0.865396	−	0.501089i	$-0.832933\pi$
$271$	−28.4924	−1.73079	−0.865396	+	0.501089i	$0.832933\pi$
$272$	0	0
$273$	13.1231i	0.794246i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 7.12311i	− 0.427986i	−0.976835	−	0.213993i	$-0.931353\pi$
$277$	− 7.12311i	− 0.427986i	0.976835	−	0.213993i	$-0.0686470\pi$
$278$	0	0
$279$	23.3693	1.39908
$280$	0	0
$281$	−8.24621	−0.491928	−0.245964	−	0.969279i	$-0.579104\pi$
$281$	−8.24621	−0.491928	−0.245964	+	0.969279i	$0.579104\pi$
$282$	0	0
$283$	23.3693i	1.38916i	0.719415	+	0.694581i	$0.244410\pi$
$283$	23.3693i	1.38916i	−0.719415	+	0.694581i	$0.755590\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 5.12311i	− 0.302407i
$288$	0	0
$289$	16.6847	0.981450
$290$	0	0
$291$	18.2462	1.06961
$292$	0	0
$293$	14.4924i	0.846656i	0.905976	+	0.423328i	$0.139138\pi$
$293$	14.4924i	0.846656i	−0.905976	+	0.423328i	$0.860862\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.43845i	0.0834672i
$298$	0	0
$299$	10.2462	0.592554
$300$	0	0
$301$	−26.2462	−1.51281
$302$	0	0
$303$	− 10.8769i	− 0.624861i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 25.6155i	− 1.46196i	−0.682401	−	0.730978i	$-0.739064\pi$
$307$	− 25.6155i	− 1.46196i	0.682401	−	0.730978i	$-0.260936\pi$
$308$	0	0
$309$	26.2462	1.49309
$310$	0	0
$311$	25.4384	1.44248	0.721241	−	0.692684i	$-0.243572\pi$
$311$	25.4384	1.44248	0.721241	+	0.692684i	$0.243572\pi$
$312$	0	0
$313$	30.4924i	1.72353i	0.507305	+	0.861767i	$0.330642\pi$
$313$	30.4924i	1.72353i	−0.507305	+	0.861767i	$0.669358\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 13.1922i	− 0.740950i	−0.928842	−	0.370475i	$-0.879195\pi$
$317$	− 13.1922i	− 0.740950i	0.928842	−	0.370475i	$-0.120805\pi$
$318$	0	0
$319$	9.68466	0.542237
$320$	0	0
$321$	40.9848	2.28755
$322$	0	0
$323$	1.43845i	0.0800373i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 47.3693i	− 2.61953i
$328$	0	0
$329$	−33.6155	−1.85328
$330$	0	0
$331$	8.49242	0.466786	0.233393	−	0.972383i	$-0.425017\pi$
$331$	8.49242	0.466786	0.233393	+	0.972383i	$0.425017\pi$
$332$	0	0
$333$	− 20.2462i	− 1.10949i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 2.31534i	− 0.126125i	−0.998010	−	0.0630623i	$-0.979913\pi$
$337$	− 2.31534i	− 0.126125i	0.998010	−	0.0630623i	$-0.0200867\pi$
$338$	0	0
$339$	48.9848	2.66049
$340$	0	0
$341$	6.56155	0.355328
$342$	0	0
$343$	− 19.0540i	− 1.02882i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 18.8769i	− 1.01336i	−0.862133	−	0.506682i	$-0.830872\pi$
$347$	− 18.8769i	− 1.01336i	0.862133	−	0.506682i	$-0.169128\pi$
$348$	0	0
$349$	−10.4924	−0.561646	−0.280823	−	0.959760i	$-0.590607\pi$
$349$	−10.4924	−0.561646	−0.280823	+	0.959760i	$0.590607\pi$
$350$	0	0
$351$	2.87689	0.153557
$352$	0	0
$353$	− 34.4924i	− 1.83585i	−0.396758	−	0.917923i	$-0.629865\pi$
$353$	− 34.4924i	− 1.83585i	0.396758	−	0.917923i	$-0.370135\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.68466i	0.195013i
$358$	0	0
$359$	26.2462	1.38522	0.692611	−	0.721311i	$-0.256460\pi$
$359$	26.2462	1.38522	0.692611	+	0.721311i	$0.256460\pi$
$360$	0	0
$361$	−12.4384	−0.654655
$362$	0	0
$363$	2.56155i	0.134447i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 14.7386i	− 0.769350i	−0.923052	−	0.384675i	$-0.874313\pi$
$367$	− 14.7386i	− 0.769350i	0.923052	−	0.384675i	$-0.125687\pi$
$368$	0	0
$369$	−7.12311	−0.370814
$370$	0	0
$371$	11.6847	0.606637
$372$	0	0
$373$	2.00000i	0.103556i	0.998659	+	0.0517780i	$0.0164888\pi$
$373$	2.00000i	0.103556i	−0.998659	+	0.0517780i	$0.983511\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 19.3693i	− 0.997571i
$378$	0	0
$379$	−25.1231	−1.29049	−0.645244	−	0.763977i	$-0.723244\pi$
$379$	−25.1231	−1.29049	−0.645244	+	0.763977i	$0.723244\pi$
$380$	0	0
$381$	−36.4924	−1.86956
$382$	0	0
$383$	31.3693i	1.60290i	0.598064	+	0.801449i	$0.295937\pi$
$383$	31.3693i	1.60290i	−0.598064	+	0.801449i	$0.704063\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	36.4924i	1.85501i
$388$	0	0
$389$	−32.2462	−1.63495	−0.817474	−	0.575966i	$-0.804626\pi$
$389$	−32.2462	−1.63495	−0.817474	+	0.575966i	$0.804626\pi$
$390$	0	0
$391$	2.87689	0.145491
$392$	0	0
$393$	19.6847i	0.992960i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	18.0000i	0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$	18.0000i	0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$	0	0
$399$	16.8078	0.841441
$400$	0	0
$401$	−7.43845	−0.371458	−0.185729	−	0.982601i	$-0.559465\pi$
$401$	−7.43845	−0.371458	−0.185729	+	0.982601i	$0.559465\pi$
$402$	0	0
$403$	− 13.1231i	− 0.653708i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 5.68466i	− 0.281778i
$408$	0	0
$409$	−7.12311	−0.352215	−0.176107	−	0.984371i	$-0.556351\pi$
$409$	−7.12311	−0.352215	−0.176107	+	0.984371i	$0.556351\pi$
$410$	0	0
$411$	50.2462	2.47846
$412$	0	0
$413$	− 2.87689i	− 0.141563i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	30.7386i	1.50528i
$418$	0	0
$419$	−26.7386	−1.30627	−0.653134	−	0.757242i	$-0.726546\pi$
$419$	−26.7386	−1.30627	−0.653134	+	0.757242i	$0.726546\pi$
$420$	0	0
$421$	7.61553	0.371158	0.185579	−	0.982629i	$-0.440584\pi$
$421$	7.61553	0.371158	0.185579	+	0.982629i	$0.440584\pi$
$422$	0	0
$423$	46.7386i	2.27251i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 5.93087i	− 0.287015i
$428$	0	0
$429$	5.12311	0.247346
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	33.3693i	1.60363i	0.597574	+	0.801814i	$0.296131\pi$
$433$	33.3693i	1.60363i	−0.597574	+	0.801814i	$0.703869\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 13.1231i	− 0.627763i
$438$	0	0
$439$	−34.2462	−1.63448	−0.817241	−	0.576296i	$-0.804498\pi$
$439$	−34.2462	−1.63448	−0.817241	+	0.576296i	$0.804498\pi$
$440$	0	0
$441$	−1.56155	−0.0743597
$442$	0	0
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 5.93087i	− 0.280521i
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	−2.00000	−0.0941763
$452$	0	0
$453$	− 33.6155i	− 1.57940i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 28.5616i	− 1.33605i	−0.744137	−	0.668027i	$-0.767139\pi$
$457$	− 28.5616i	− 1.33605i	0.744137	−	0.668027i	$-0.232861\pi$
$458$	0	0
$459$	0.807764	0.0377032
$460$	0	0
$461$	20.5616	0.957647	0.478823	−	0.877911i	$-0.341063\pi$
$461$	20.5616	0.957647	0.478823	+	0.877911i	$0.341063\pi$
$462$	0	0
$463$	7.36932i	0.342481i	0.985229	+	0.171241i	$0.0547775\pi$
$463$	7.36932i	0.342481i	−0.985229	+	0.171241i	$0.945222\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9.93087i	0.459546i	0.973244	+	0.229773i	$0.0737984\pi$
$467$	9.93087i	0.459546i	−0.973244	+	0.229773i	$0.926202\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	37.9309	1.74776
$472$	0	0
$473$	10.2462i	0.471121i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 16.2462i	− 0.743863i
$478$	0	0
$479$	−32.9848	−1.50712	−0.753558	−	0.657381i	$-0.771664\pi$
$479$	−32.9848	−1.50712	−0.753558	+	0.657381i	$0.771664\pi$
$480$	0	0
$481$	−11.3693	−0.518396
$482$	0	0
$483$	− 33.6155i	− 1.52956i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 2.24621i	− 0.101786i	−0.998704	−	0.0508928i	$-0.983793\pi$
$487$	− 2.24621i	− 0.101786i	0.998704	−	0.0508928i	$-0.0162067\pi$
$488$	0	0
$489$	8.17708	0.369780
$490$	0	0
$491$	−7.05398	−0.318341	−0.159171	−	0.987251i	$-0.550882\pi$
$491$	−7.05398	−0.318341	−0.159171	+	0.987251i	$0.550882\pi$
$492$	0	0
$493$	− 5.43845i	− 0.244935i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.43845i	0.423372i
$498$	0	0
$499$	16.4924	0.738302	0.369151	−	0.929369i	$-0.379648\pi$
$499$	16.4924	0.738302	0.369151	+	0.929369i	$0.379648\pi$
$500$	0	0
$501$	27.0540	1.20868
$502$	0	0
$503$	14.2462i	0.635207i	0.948224	+	0.317604i	$0.102878\pi$
$503$	14.2462i	0.635207i	−0.948224	+	0.317604i	$0.897122\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	23.0540i	1.02386i
$508$	0	0
$509$	−21.3693	−0.947178	−0.473589	−	0.880746i	$-0.657042\pi$
$509$	−21.3693	−0.947178	−0.473589	+	0.880746i	$0.657042\pi$
$510$	0	0
$511$	−5.12311	−0.226633
$512$	0	0
$513$	− 3.68466i	− 0.162682i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	13.1231i	0.577154i
$518$	0	0
$519$	−10.8769	−0.477443
$520$	0	0
$521$	8.73863	0.382846	0.191423	−	0.981508i	$-0.438690\pi$
$521$	8.73863	0.382846	0.191423	+	0.981508i	$0.438690\pi$
$522$	0	0
$523$	3.50758i	0.153376i	0.997055	+	0.0766878i	$0.0244345\pi$
$523$	3.50758i	0.153376i	−0.997055	+	0.0766878i	$0.975566\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 3.68466i	− 0.160506i
$528$	0	0
$529$	−3.24621	−0.141140
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	4.00000i	0.173259i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 16.0000i	− 0.690451i
$538$	0	0
$539$	−0.438447	−0.0188853
$540$	0	0
$541$	−33.5464	−1.44227	−0.721136	−	0.692793i	$-0.756380\pi$
$541$	−33.5464	−1.44227	−0.721136	+	0.692793i	$0.756380\pi$
$542$	0	0
$543$	− 10.8769i	− 0.466772i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 16.0000i	− 0.684111i	−0.939680	−	0.342055i	$-0.888877\pi$
$547$	− 16.0000i	− 0.684111i	0.939680	−	0.342055i	$-0.111123\pi$
$548$	0	0
$549$	−8.24621	−0.351940
$550$	0	0
$551$	−24.8078	−1.05685
$552$	0	0
$553$	39.3693i	1.67415i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 46.4924i	− 1.96995i	−0.172705	−	0.984974i	$-0.555251\pi$
$557$	− 46.4924i	− 1.96995i	0.172705	−	0.984974i	$-0.444749\pi$
$558$	0	0
$559$	20.4924	0.866737
$560$	0	0
$561$	1.43845	0.0607313
$562$	0	0
$563$	2.87689i	0.121247i	0.998161	+	0.0606233i	$0.0193088\pi$
$563$	2.87689i	0.121247i	−0.998161	+	0.0606233i	$0.980691\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	17.9309i	0.753026i
$568$	0	0
$569$	28.1080	1.17835	0.589173	−	0.808007i	$-0.299454\pi$
$569$	28.1080	1.17835	0.589173	+	0.808007i	$0.299454\pi$
$570$	0	0
$571$	−30.4233	−1.27318	−0.636588	−	0.771204i	$-0.719655\pi$
$571$	−30.4233	−1.27318	−0.636588	+	0.771204i	$0.719655\pi$
$572$	0	0
$573$	− 26.2462i	− 1.09645i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 24.7386i	− 1.02988i	−0.857225	−	0.514941i	$-0.827814\pi$
$577$	− 24.7386i	− 1.02988i	0.857225	−	0.514941i	$-0.172186\pi$
$578$	0	0
$579$	−13.3002	−0.552737
$580$	0	0
$581$	−13.1231	−0.544438
$582$	0	0
$583$	− 4.56155i	− 0.188920i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	32.3153i	1.33380i	0.745149	+	0.666898i	$0.232379\pi$
$587$	32.3153i	1.33380i	−0.745149	+	0.666898i	$0.767621\pi$
$588$	0	0
$589$	−16.8078	−0.692552
$590$	0	0
$591$	−13.7538	−0.565755
$592$	0	0
$593$	1.50758i	0.0619088i	0.999521	+	0.0309544i	$0.00985466\pi$
$593$	1.50758i	0.0619088i	−0.999521	+	0.0309544i	$0.990145\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 2.06913i	− 0.0846839i
$598$	0	0
$599$	−17.4384	−0.712516	−0.356258	−	0.934388i	$-0.615948\pi$
$599$	−17.4384	−0.712516	−0.356258	+	0.934388i	$0.615948\pi$
$600$	0	0
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	0	0
$603$	− 22.2462i	− 0.905936i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	20.8078i	0.844561i	0.906465	+	0.422281i	$0.138770\pi$
$607$	20.8078i	0.844561i	−0.906465	+	0.422281i	$0.861230\pi$
$608$	0	0
$609$	−63.5464	−2.57503
$610$	0	0
$611$	26.2462	1.06181
$612$	0	0
$613$	16.7386i	0.676067i	0.941134	+	0.338034i	$0.109762\pi$
$613$	16.7386i	0.676067i	−0.941134	+	0.338034i	$0.890238\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.63068i	0.266941i	0.991053	+	0.133471i	$0.0426122\pi$
$617$	6.63068i	0.266941i	−0.991053	+	0.133471i	$0.957388\pi$
$618$	0	0
$619$	−8.49242	−0.341339	−0.170670	−	0.985328i	$-0.554593\pi$
$619$	−8.49242	−0.341339	−0.170670	+	0.985328i	$0.554593\pi$
$620$	0	0
$621$	−7.36932	−0.295720
$622$	0	0
$623$	− 32.1771i	− 1.28915i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 6.56155i	− 0.262043i
$628$	0	0
$629$	−3.19224	−0.127283
$630$	0	0
$631$	16.8078	0.669107	0.334553	−	0.942377i	$-0.391415\pi$
$631$	16.8078	0.669107	0.334553	+	0.942377i	$0.391415\pi$
$632$	0	0
$633$	− 21.3002i	− 0.846606i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0.876894i	0.0347438i
$638$	0	0
$639$	13.1231	0.519142
$640$	0	0
$641$	−38.1771	−1.50790	−0.753952	−	0.656929i	$-0.771855\pi$
$641$	−38.1771	−1.50790	−0.753952	+	0.656929i	$0.771855\pi$
$642$	0	0
$643$	− 23.6847i	− 0.934032i	−0.884249	−	0.467016i	$-0.845329\pi$
$643$	− 23.6847i	− 0.934032i	0.884249	−	0.467016i	$-0.154671\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 13.1231i	− 0.515923i	−0.966155	−	0.257961i	$-0.916949\pi$
$647$	− 13.1231i	− 0.515923i	0.966155	−	0.257961i	$-0.0830507\pi$
$648$	0	0
$649$	−1.12311	−0.0440858
$650$	0	0
$651$	−43.0540	−1.68742
$652$	0	0
$653$	− 10.8078i	− 0.422940i	−0.977384	−	0.211470i	$-0.932175\pi$
$653$	− 10.8078i	− 0.422940i	0.977384	−	0.211470i	$-0.0678251\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	7.12311i	0.277899i
$658$	0	0
$659$	19.1922	0.747623	0.373812	−	0.927505i	$-0.378051\pi$
$659$	19.1922	0.747623	0.373812	+	0.927505i	$0.378051\pi$
$660$	0	0
$661$	16.2462	0.631904	0.315952	−	0.948775i	$-0.397676\pi$
$661$	16.2462	0.631904	0.315952	+	0.948775i	$0.397676\pi$
$662$	0	0
$663$	− 2.87689i	− 0.111729i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	49.6155i	1.92112i
$668$	0	0
$669$	33.6155	1.29965
$670$	0	0
$671$	−2.31534	−0.0893828
$672$	0	0
$673$	− 35.7926i	− 1.37970i	−0.723951	−	0.689852i	$-0.757676\pi$
$673$	− 35.7926i	− 1.37970i	0.723951	−	0.689852i	$-0.242324\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 27.6155i	− 1.06135i	−0.847575	−	0.530675i	$-0.821938\pi$
$677$	− 27.6155i	− 1.06135i	0.847575	−	0.530675i	$-0.178062\pi$
$678$	0	0
$679$	−18.2462	−0.700225
$680$	0	0
$681$	−7.36932	−0.282393
$682$	0	0
$683$	25.9309i	0.992217i	0.868260	+	0.496109i	$0.165238\pi$
$683$	25.9309i	0.992217i	−0.868260	+	0.496109i	$0.834762\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	63.3693i	2.41769i
$688$	0	0
$689$	−9.12311	−0.347563
$690$	0	0
$691$	−3.36932	−0.128175	−0.0640874	−	0.997944i	$-0.520414\pi$
$691$	−3.36932	−0.128175	−0.0640874	+	0.997944i	$0.520414\pi$
$692$	0	0
$693$	− 9.12311i	− 0.346558i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.12311i	0.0425407i
$698$	0	0
$699$	−51.0540	−1.93104
$700$	0	0
$701$	34.3153	1.29607	0.648036	−	0.761609i	$-0.275591\pi$
$701$	34.3153	1.29607	0.648036	+	0.761609i	$0.275591\pi$
$702$	0	0
$703$	14.5616i	0.549199i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.8769i	0.409068i
$708$	0	0
$709$	−1.50758	−0.0566183	−0.0283091	−	0.999599i	$-0.509012\pi$
$709$	−1.50758	−0.0566183	−0.0283091	+	0.999599i	$0.509012\pi$
$710$	0	0
$711$	54.7386	2.05286
$712$	0	0
$713$	33.6155i	1.25891i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	65.6155i	2.45046i
$718$	0	0
$719$	7.82292	0.291746	0.145873	−	0.989303i	$-0.453401\pi$
$719$	7.82292	0.291746	0.145873	+	0.989303i	$0.453401\pi$
$720$	0	0
$721$	−26.2462	−0.977460
$722$	0	0
$723$	− 41.6155i	− 1.54770i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	17.6155i	0.653324i	0.945141	+	0.326662i	$0.105924\pi$
$727$	17.6155i	0.653324i	−0.945141	+	0.326662i	$0.894076\pi$
$728$	0	0
$729$	35.9848	1.33277
$730$	0	0
$731$	5.75379	0.212812
$732$	0	0
$733$	− 19.1231i	− 0.706328i	−0.935561	−	0.353164i	$-0.885106\pi$
$733$	− 19.1231i	− 0.706328i	0.935561	−	0.353164i	$-0.114894\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 6.24621i	− 0.230082i
$738$	0	0
$739$	−42.7386	−1.57217	−0.786083	−	0.618121i	$-0.787894\pi$
$739$	−42.7386	−1.57217	−0.786083	+	0.618121i	$0.787894\pi$
$740$	0	0
$741$	−13.1231	−0.482089
$742$	0	0
$743$	2.56155i	0.0939743i	0.998895	+	0.0469871i	$0.0149620\pi$
$743$	2.56155i	0.0939743i	−0.998895	+	0.0469871i	$0.985038\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	18.2462i	0.667594i
$748$	0	0
$749$	−40.9848	−1.49755
$750$	0	0
$751$	16.1771	0.590310	0.295155	−	0.955449i	$-0.404629\pi$
$751$	16.1771	0.590310	0.295155	+	0.955449i	$0.404629\pi$
$752$	0	0
$753$	75.8617i	2.76456i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	26.0000i	0.944986i	0.881334	+	0.472493i	$0.156646\pi$
$757$	26.0000i	0.944986i	−0.881334	+	0.472493i	$0.843354\pi$
$758$	0	0
$759$	−13.1231	−0.476339
$760$	0	0
$761$	−36.7386	−1.33177	−0.665887	−	0.746052i	$-0.731947\pi$
$761$	−36.7386	−1.33177	−0.665887	+	0.746052i	$0.731947\pi$
$762$	0	0
$763$	47.3693i	1.71488i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.24621i	0.0811060i
$768$	0	0
$769$	−24.7386	−0.892098	−0.446049	−	0.895009i	$-0.647169\pi$
$769$	−24.7386	−0.892098	−0.446049	+	0.895009i	$0.647169\pi$
$770$	0	0
$771$	−73.6155	−2.65120
$772$	0	0
$773$	− 37.6847i	− 1.35542i	−0.735328	−	0.677711i	$-0.762972\pi$
$773$	− 37.6847i	− 1.35542i	0.735328	−	0.677711i	$-0.237028\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	37.3002i	1.33814i
$778$	0	0
$779$	5.12311	0.183554
$780$	0	0
$781$	3.68466	0.131847
$782$	0	0
$783$	13.9309i	0.497849i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 54.7386i	− 1.95122i	−0.219508	−	0.975611i	$-0.570445\pi$
$787$	− 54.7386i	− 1.95122i	0.219508	−	0.975611i	$-0.429555\pi$
$788$	0	0
$789$	12.3153	0.438438
$790$	0	0
$791$	−48.9848	−1.74170
$792$	0	0
$793$	4.63068i	0.164440i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23.7538i	0.841402i	0.907199	+	0.420701i	$0.138216\pi$
$797$	23.7538i	0.841402i	−0.907199	+	0.420701i	$0.861784\pi$
$798$	0	0
$799$	7.36932	0.260708
$800$	0	0
$801$	−44.7386	−1.58076
$802$	0	0
$803$	2.00000i	0.0705785i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	50.2462i	1.76875i
$808$	0	0
$809$	−6.49242	−0.228261	−0.114131	−	0.993466i	$-0.536408\pi$
$809$	−6.49242	−0.228261	−0.114131	+	0.993466i	$0.536408\pi$
$810$	0	0
$811$	−27.1922	−0.954849	−0.477424	−	0.878673i	$-0.658430\pi$
$811$	−27.1922	−0.954849	−0.477424	+	0.878673i	$0.658430\pi$
$812$	0	0
$813$	− 72.9848i	− 2.55969i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 26.2462i	− 0.918239i
$818$	0	0
$819$	−18.2462	−0.637574
$820$	0	0
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	0	0
$823$	52.4924i	1.82977i	0.403714	+	0.914885i	$0.367719\pi$
$823$	52.4924i	1.82977i	−0.403714	+	0.914885i	$0.632281\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 4.49242i	− 0.156217i	−0.996945	−	0.0781084i	$-0.975112\pi$
$827$	− 4.49242i	− 0.156217i	0.996945	−	0.0781084i	$-0.0248880\pi$
$828$	0	0
$829$	−5.36932	−0.186484	−0.0932420	−	0.995643i	$-0.529723\pi$
$829$	−5.36932	−0.186484	−0.0932420	+	0.995643i	$0.529723\pi$
$830$	0	0
$831$	18.2462	0.632954
$832$	0	0
$833$	0.246211i	0.00853071i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	9.43845i	0.326240i
$838$	0	0
$839$	11.5076	0.397286	0.198643	−	0.980072i	$-0.436347\pi$
$839$	11.5076	0.397286	0.198643	+	0.980072i	$0.436347\pi$
$840$	0	0
$841$	64.7926	2.23423
$842$	0	0
$843$	− 21.1231i	− 0.727518i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.56155i	− 0.0880160i
$848$	0	0
$849$	−59.8617	−2.05445
$850$	0	0
$851$	29.1231	0.998327
$852$	0	0
$853$	− 3.75379i	− 0.128527i	−0.997933	−	0.0642636i	$-0.979530\pi$
$853$	− 3.75379i	− 0.128527i	0.997933	−	0.0642636i	$-0.0204699\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 32.4233i	− 1.10756i	−0.832664	−	0.553779i	$-0.813185\pi$
$857$	− 32.4233i	− 1.10756i	0.832664	−	0.553779i	$-0.186815\pi$
$858$	0	0
$859$	−38.8769	−1.32646	−0.663231	−	0.748415i	$-0.730815\pi$
$859$	−38.8769	−1.32646	−0.663231	+	0.748415i	$0.730815\pi$
$860$	0	0
$861$	13.1231	0.447234
$862$	0	0
$863$	− 9.61553i	− 0.327316i	−0.986517	−	0.163658i	$-0.947671\pi$
$863$	− 9.61553i	− 0.327316i	0.986517	−	0.163658i	$-0.0523294\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	42.7386i	1.45148i
$868$	0	0
$869$	15.3693	0.521368
$870$	0	0
$871$	−12.4924	−0.423290
$872$	0	0
$873$	25.3693i	0.858621i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	32.2462i	1.08888i	0.838801	+	0.544439i	$0.183257\pi$
$877$	32.2462i	1.08888i	−0.838801	+	0.544439i	$0.816743\pi$
$878$	0	0
$879$	−37.1231	−1.25213
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	0	0
$883$	− 6.06913i	− 0.204242i	−0.994772	−	0.102121i	$-0.967437\pi$
$883$	− 6.06913i	− 0.204242i	0.994772	−	0.102121i	$-0.0325630\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 30.2462i	− 1.01557i	−0.861484	−	0.507784i	$-0.830465\pi$
$887$	− 30.2462i	− 1.01557i	0.861484	−	0.507784i	$-0.169535\pi$
$888$	0	0
$889$	36.4924	1.22392
$890$	0	0
$891$	7.00000	0.234509
$892$	0	0
$893$	− 33.6155i	− 1.12490i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	26.2462i	0.876335i
$898$	0	0
$899$	63.5464	2.11939
$900$	0	0
$901$	−2.56155	−0.0853377
$902$	0	0
$903$	− 67.2311i	− 2.23731i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	8.94602i	0.297048i	0.988909	+	0.148524i	$0.0474522\pi$
$907$	8.94602i	0.297048i	−0.988909	+	0.148524i	$0.952548\pi$
$908$	0	0
$909$	15.1231	0.501602
$910$	0	0
$911$	2.06913	0.0685533	0.0342767	−	0.999412i	$-0.489087\pi$
$911$	2.06913	0.0685533	0.0342767	+	0.999412i	$0.489087\pi$
$912$	0	0
$913$	5.12311i	0.169550i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 19.6847i	− 0.650045i
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	65.6155	2.16211
$922$	0	0
$923$	− 7.36932i	− 0.242564i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	36.4924i	1.19857i
$928$	0	0
$929$	−45.0540	−1.47817	−0.739086	−	0.673611i	$-0.764743\pi$
$929$	−45.0540	−1.47817	−0.739086	+	0.673611i	$0.764743\pi$
$930$	0	0
$931$	1.12311	0.0368083
$932$	0	0
$933$	65.1619i	2.13331i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	42.0000i	1.37208i	0.727564	+	0.686040i	$0.240653\pi$
$937$	42.0000i	1.37208i	−0.727564	+	0.686040i	$0.759347\pi$
$938$	0	0
$939$	−78.1080	−2.54896
$940$	0	0
$941$	−4.06913	−0.132650	−0.0663249	−	0.997798i	$-0.521127\pi$
$941$	−4.06913	−0.132650	−0.0663249	+	0.997798i	$0.521127\pi$
$942$	0	0
$943$	− 10.2462i	− 0.333663i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 39.6847i	− 1.28958i	−0.764361	−	0.644789i	$-0.776945\pi$
$947$	− 39.6847i	− 1.28958i	0.764361	−	0.644789i	$-0.223055\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	33.7926	1.09580
$952$	0	0
$953$	− 34.1771i	− 1.10710i	−0.832815	−	0.553552i	$-0.813272\pi$
$953$	− 34.1771i	− 1.10710i	0.832815	−	0.553552i	$-0.186728\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	24.8078i	0.801921i
$958$	0	0
$959$	−50.2462	−1.62253
$960$	0	0
$961$	12.0540	0.388838
$962$	0	0
$963$	56.9848i	1.83631i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	11.5464i	0.371307i	0.982615	+	0.185654i	$0.0594402\pi$
$967$	11.5464i	0.371307i	−0.982615	+	0.185654i	$0.940560\pi$
$968$	0	0
$969$	−3.68466	−0.118368
$970$	0	0
$971$	−6.24621	−0.200450	−0.100225	−	0.994965i	$-0.531956\pi$
$971$	−6.24621	−0.200450	−0.100225	+	0.994965i	$0.531956\pi$
$972$	0	0
$973$	− 30.7386i	− 0.985435i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.8769i	0.795882i	0.917411	+	0.397941i	$0.130275\pi$
$977$	24.8769i	0.795882i	−0.917411	+	0.397941i	$0.869725\pi$
$978$	0	0
$979$	−12.5616	−0.401469
$980$	0	0
$981$	65.8617	2.10280
$982$	0	0
$983$	− 18.8769i	− 0.602079i	−0.953612	−	0.301040i	$-0.902666\pi$
$983$	− 18.8769i	− 0.602079i	0.953612	−	0.301040i	$-0.0973337\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 86.1080i	− 2.74085i
$988$	0	0
$989$	−52.4924	−1.66916
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	21.7538i	0.690336i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 13.8617i	− 0.439006i	−0.975612	−	0.219503i	$-0.929556\pi$
$997$	− 13.8617i	− 0.439006i	0.975612	−	0.219503i	$-0.0704435\pi$
$998$	0	0
$999$	8.17708	0.258711

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.b.t.4049.4		4
4.3	odd	2		2200.2.b.i.1849.1		4
5.2	odd	4		880.2.a.o.1.2		2
5.3	odd	4		4400.2.a.bj.1.1		2
5.4	even	2	inner	4400.2.b.t.4049.1		4
15.2	even	4		7920.2.a.bu.1.2		2
20.3	even	4		2200.2.a.s.1.2		2
20.7	even	4		440.2.a.e.1.1	✓	2
20.19	odd	2		2200.2.b.i.1849.4		4
40.27	even	4		3520.2.a.bp.1.2		2
40.37	odd	4		3520.2.a.bk.1.1		2
55.32	even	4		9680.2.a.bs.1.2		2
60.47	odd	4		3960.2.a.w.1.1		2
220.87	odd	4		4840.2.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.a.e.1.1	✓	2	20.7	even	4
880.2.a.o.1.2		2	5.2	odd	4
2200.2.a.s.1.2		2	20.3	even	4
2200.2.b.i.1849.1		4	4.3	odd	2
2200.2.b.i.1849.4		4	20.19	odd	2
3520.2.a.bk.1.1		2	40.37	odd	4
3520.2.a.bp.1.2		2	40.27	even	4
3960.2.a.w.1.1		2	60.47	odd	4
4400.2.a.bj.1.1		2	5.3	odd	4
4400.2.b.t.4049.1		4	5.4	even	2	inner
4400.2.b.t.4049.4		4	1.1	even	1	trivial
4840.2.a.j.1.1		2	220.87	odd	4
7920.2.a.bu.1.2		2	15.2	even	4
9680.2.a.bs.1.2		2	55.32	even	4

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:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.56155i q^{3} -2.56155i q^{7} -3.56155 q^{9} +O(q^{10})\)	\(q+2.56155i q^{3} -2.56155i q^{7} -3.56155 q^{9} -1.00000 q^{11} +2.00000i q^{13} +0.561553i q^{17} +2.56155 q^{19} +6.56155 q^{21} -5.12311i q^{23} -1.43845i q^{27} -9.68466 q^{29} -6.56155 q^{31} -2.56155i q^{33} +5.68466i q^{37} -5.12311 q^{39} +2.00000 q^{41} -10.2462i q^{43} -13.1231i q^{47} +0.438447 q^{49} -1.43845 q^{51} +4.56155i q^{53} +6.56155i q^{57} +1.12311 q^{59} +2.31534 q^{61} +9.12311i q^{63} +6.24621i q^{67} +13.1231 q^{69} -3.68466 q^{71} -2.00000i q^{73} +2.56155i q^{77} -15.3693 q^{79} -7.00000 q^{81} -5.12311i q^{83} -24.8078i q^{87} +12.5616 q^{89} +5.12311 q^{91} -16.8078i q^{93} -7.12311i q^{97} +3.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{9}+O(q^{10}) \) 4 * q - 6 * q^9	\( 4 q - 6 q^{9} - 4 q^{11} + 2 q^{19} + 18 q^{21} - 14 q^{29} - 18 q^{31} - 4 q^{39} + 8 q^{41} + 10 q^{49} - 14 q^{51} - 12 q^{59} + 34 q^{61} + 36 q^{69} + 10 q^{71} - 12 q^{79} - 28 q^{81} + 42 q^{89} + 4 q^{91} + 6 q^{99}+O(q^{100}) \) 4 * q - 6 * q^9 - 4 * q^11 + 2 * q^19 + 18 * q^21 - 14 * q^29 - 18 * q^31 - 4 * q^39 + 8 * q^41 + 10 * q^49 - 14 * q^51 - 12 * q^59 + 34 * q^61 + 36 * q^69 + 10 * q^71 - 12 * q^79 - 28 * q^81 + 42 * q^89 + 4 * q^91 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more