LMFDB - Embedded newform 441.3.b.e.197.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,3,Mod(197,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("441.197");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	441.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$12.0163796583$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.959512576.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 81 $ x^8 + 7*x^4 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.6
Root			$-1.52616 + 0.819051i$ of defining polynomial
Character	$\chi$	$=$	441.197
Dual form			441.3.b.e.197.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.63810i q^{2} +1.31662 q^{4} +1.68338i q^{5} +8.70917i q^{8} +O(q^{10})$	$q+1.63810i q^{2} +1.31662 q^{4} +1.68338i q^{5} +8.70917i q^{8} -2.75754 q^{10} +8.41439i q^{11} +20.1759 q^{13} -9.00000 q^{16} -12.3166i q^{17} -32.4560 q^{19} +2.21637i q^{20} -13.7836 q^{22} +24.4894i q^{23} +22.1662 q^{25} +33.0501i q^{26} +46.0795i q^{29} +7.89573 q^{31} +20.0938i q^{32} +20.1759 q^{34} +3.89975 q^{37} -53.1662i q^{38} -14.6608 q^{40} -54.3166i q^{41} -24.6332 q^{43} +11.0786i q^{44} -40.1161 q^{46} +77.7995i q^{47} +36.3106i q^{50} +26.5641 q^{52} -35.3553i q^{53} -14.1646 q^{55} -75.4829 q^{58} +64.3325i q^{59} -47.1168 q^{61} +12.9340i q^{62} -68.9156 q^{64} +33.9636i q^{65} -33.1662 q^{67} -16.2164i q^{68} -46.0795i q^{71} +83.4610 q^{73} +6.38818i q^{74} -42.7324 q^{76} +39.2665 q^{79} -15.1504i q^{80} +88.9761 q^{82} -138.765i q^{83} +20.7335 q^{85} -40.3518i q^{86} -73.2824 q^{88} -24.0159i q^{89} +32.2434i q^{92} -127.443 q^{94} -54.6357i q^{95} -32.0791 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4}+O(q^{10}) $ 8 * q - 16 * q^4	$ 8 q - 16 q^{4} - 72 q^{16} - 296 q^{22} - 88 q^{25} - 128 q^{37} - 144 q^{43} + 24 q^{46} - 312 q^{58} + 112 q^{64} + 208 q^{79} + 272 q^{85} + 24 q^{88}+O(q^{100}) $ 8 * q - 16 * q^4 - 72 * q^16 - 296 * q^22 - 88 * q^25 - 128 * q^37 - 144 * q^43 + 24 * q^46 - 312 * q^58 + 112 * q^64 + 208 * q^79 + 272 * q^85 + 24 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$.

$n$	$199$	$344$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.63810i	0.819051i	0.912299	+	0.409525i	$0.134306\pi$
$2$	1.63810i	0.819051i	−0.912299	+	0.409525i	$0.865694\pi$
$3$	0	0
$3$	0	0
$4$	1.31662	0.329156
$5$	1.68338i	0.336675i	0.985729	+	0.168338i	$0.0538399\pi$
$5$	1.68338i	0.336675i	−0.985729	+	0.168338i	$0.946160\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	8.70917i	1.08865i
$9$	0	0
$10$	−2.75754	−0.275754
$11$	8.41439i	0.764945i	0.923967	+	0.382472i	$0.124927\pi$
$11$	8.41439i	0.764945i	−0.923967	+	0.382472i	$0.875073\pi$
$12$	0	0
$13$	20.1759	1.55199	0.775995	−	0.630739i	$-0.217248\pi$
$13$	20.1759	1.55199	0.775995	+	0.630739i	$0.217248\pi$
$14$	0	0
$15$	0	0
$16$	−9.00000	−0.562500
$17$	− 12.3166i	− 0.724507i	−0.932080	−	0.362254i	$-0.882007\pi$
$17$	− 12.3166i	− 0.724507i	0.932080	−	0.362254i	$-0.117993\pi$
$18$	0	0
$19$	−32.4560	−1.70821	−0.854106	−	0.520099i	$-0.825895\pi$
$19$	−32.4560	−1.70821	−0.854106	+	0.520099i	$0.825895\pi$
$20$	2.21637i	0.110819i
$21$	0	0
$22$	−13.7836	−0.626528
$23$	24.4894i	1.06476i	0.846507	+	0.532378i	$0.178702\pi$
$23$	24.4894i	1.06476i	−0.846507	+	0.532378i	$0.821298\pi$
$24$	0	0
$25$	22.1662	0.886650
$26$	33.0501i	1.27116i
$27$	0	0
$28$	0	0
$29$	46.0795i	1.58895i	0.607298	+	0.794474i	$0.292253\pi$
$29$	46.0795i	1.58895i	−0.607298	+	0.794474i	$0.707747\pi$
$30$	0	0
$31$	7.89573	0.254701	0.127350	−	0.991858i	$-0.459353\pi$
$31$	7.89573	0.254701	0.127350	+	0.991858i	$0.459353\pi$
$32$	20.0938i	0.627930i
$33$	0	0
$34$	20.1759	0.593408
$35$	0	0
$36$	0	0
$37$	3.89975	0.105399	0.0526993	−	0.998610i	$-0.483218\pi$
$37$	3.89975	0.105399	0.0526993	+	0.998610i	$0.483218\pi$
$38$	− 53.1662i	− 1.39911i
$39$	0	0
$40$	−14.6608	−0.366520
$41$	− 54.3166i	− 1.32480i	−0.749152	−	0.662398i	$-0.769539\pi$
$41$	− 54.3166i	− 1.32480i	0.749152	−	0.662398i	$-0.230461\pi$
$42$	0	0
$43$	−24.6332	−0.572866	−0.286433	−	0.958100i	$-0.592470\pi$
$43$	−24.6332	−0.572866	−0.286433	+	0.958100i	$0.592470\pi$
$44$	11.0786i	0.251786i
$45$	0	0
$46$	−40.1161	−0.872090
$47$	77.7995i	1.65531i	0.561238	+	0.827654i	$0.310325\pi$
$47$	77.7995i	1.65531i	−0.561238	+	0.827654i	$0.689675\pi$
$48$	0	0
$49$	0	0
$50$	36.3106i	0.726211i
$51$	0	0
$52$	26.5641	0.510847
$53$	− 35.3553i	− 0.667082i	−0.942736	−	0.333541i	$-0.891756\pi$
$53$	− 35.3553i	− 0.667082i	0.942736	−	0.333541i	$-0.108244\pi$
$54$	0	0
$55$	−14.1646	−0.257538
$56$	0	0
$57$	0	0
$58$	−75.4829	−1.30143
$59$	64.3325i	1.09038i	0.838312	+	0.545191i	$0.183543\pi$
$59$	64.3325i	1.09038i	−0.838312	+	0.545191i	$0.816457\pi$
$60$	0	0
$61$	−47.1168	−0.772407	−0.386203	−	0.922414i	$-0.626214\pi$
$61$	−47.1168	−0.772407	−0.386203	+	0.922414i	$0.626214\pi$
$62$	12.9340i	0.208613i
$63$	0	0
$64$	−68.9156	−1.07681
$65$	33.9636i	0.522516i
$66$	0	0
$67$	−33.1662	−0.495019	−0.247509	−	0.968886i	$-0.579612\pi$
$67$	−33.1662	−0.495019	−0.247509	+	0.968886i	$0.579612\pi$
$68$	− 16.2164i	− 0.238476i
$69$	0	0
$70$	0	0
$71$	− 46.0795i	− 0.649007i	−0.945884	−	0.324503i	$-0.894803\pi$
$71$	− 46.0795i	− 0.649007i	0.945884	−	0.324503i	$-0.105197\pi$
$72$	0	0
$73$	83.4610	1.14330	0.571651	−	0.820497i	$-0.306303\pi$
$73$	83.4610	1.14330	0.571651	+	0.820497i	$0.306303\pi$
$74$	6.38818i	0.0863268i
$75$	0	0
$76$	−42.7324	−0.562268
$77$	0	0
$78$	0	0
$79$	39.2665	0.497044	0.248522	−	0.968626i	$-0.420055\pi$
$79$	39.2665	0.497044	0.248522	+	0.968626i	$0.420055\pi$
$80$	− 15.1504i	− 0.189380i
$81$	0	0
$82$	88.9761	1.08507
$83$	− 138.765i	− 1.67187i	−0.548828	−	0.835935i	$-0.684926\pi$
$83$	− 138.765i	− 1.67187i	0.548828	−	0.835935i	$-0.315074\pi$
$84$	0	0
$85$	20.7335	0.243924
$86$	− 40.3518i	− 0.469206i
$87$	0	0
$88$	−73.2824	−0.832754
$89$	− 24.0159i	− 0.269841i	−0.990856	−	0.134921i	$-0.956922\pi$
$89$	− 24.0159i	− 0.269841i	0.990856	−	0.134921i	$-0.0430779\pi$
$90$	0	0
$91$	0	0
$92$	32.2434i	0.350471i
$93$	0	0
$94$	−127.443	−1.35578
$95$	− 54.6357i	− 0.575112i
$96$	0	0
$97$	−32.0791	−0.330713	−0.165356	−	0.986234i	$-0.552877\pi$
$97$	−32.0791	−0.330713	−0.165356	+	0.986234i	$0.552877\pi$
$98$	0	0
$99$	0	0
$100$	29.1846	0.291846
$101$	58.2164i	0.576400i	0.957570	+	0.288200i	$0.0930568\pi$
$101$	58.2164i	0.576400i	−0.957570	+	0.288200i	$0.906943\pi$
$102$	0	0
$103$	70.5465	0.684917	0.342459	−	0.939533i	$-0.388740\pi$
$103$	70.5465	0.684917	0.342459	+	0.939533i	$0.388740\pi$
$104$	175.715i	1.68957i
$105$	0	0
$106$	57.9156	0.546374
$107$	− 35.5196i	− 0.331959i	−0.986129	−	0.165979i	$-0.946922\pi$
$107$	− 35.5196i	− 0.331959i	0.986129	−	0.165979i	$-0.0530785\pi$
$108$	0	0
$109$	160.734	1.47462	0.737310	−	0.675555i	$-0.236096\pi$
$109$	160.734	1.47462	0.737310	+	0.675555i	$0.236096\pi$
$110$	− 23.2030i	− 0.210937i
$111$	0	0
$112$	0	0
$113$	− 195.397i	− 1.72917i	−0.502484	−	0.864587i	$-0.667580\pi$
$113$	− 195.397i	− 1.72917i	0.502484	−	0.864587i	$-0.332420\pi$
$114$	0	0
$115$	−41.2249	−0.358477
$116$	60.6694i	0.523012i
$117$	0	0
$118$	−105.383	−0.893077
$119$	0	0
$120$	0	0
$121$	50.1980	0.414859
$122$	− 77.1821i	− 0.632640i
$123$	0	0
$124$	10.3957	0.0838364
$125$	79.3985i	0.635188i
$126$	0	0
$127$	150.232	1.18293	0.591466	−	0.806330i	$-0.298550\pi$
$127$	150.232	1.18293	0.591466	+	0.806330i	$0.298550\pi$
$128$	− 32.5157i	− 0.254029i
$129$	0	0
$130$	−55.6358	−0.427967
$131$	− 228.264i	− 1.74247i	−0.490863	−	0.871237i	$-0.663319\pi$
$131$	− 228.264i	− 1.74247i	0.490863	−	0.871237i	$-0.336681\pi$
$132$	0	0
$133$	0	0
$134$	− 54.3297i	− 0.405445i
$135$	0	0
$136$	107.268	0.788732
$137$	− 41.7435i	− 0.304697i	−0.988327	−	0.152349i	$-0.951316\pi$
$137$	− 41.7435i	− 0.304697i	0.988327	−	0.152349i	$-0.0486837\pi$
$138$	0	0
$139$	234.334	1.68586	0.842928	−	0.538026i	$-0.180830\pi$
$139$	234.334	1.68586	0.842928	+	0.538026i	$0.180830\pi$
$140$	0	0
$141$	0	0
$142$	75.4829	0.531570
$143$	169.768i	1.18719i
$144$	0	0
$145$	−77.5691	−0.534959
$146$	136.718i	0.936422i
$147$	0	0
$148$	5.13451	0.0346926
$149$	− 231.129i	− 1.55120i	−0.631225	−	0.775600i	$-0.717447\pi$
$149$	− 231.129i	− 1.55120i	0.631225	−	0.775600i	$-0.282553\pi$
$150$	0	0
$151$	224.897	1.48939	0.744693	−	0.667407i	$-0.232596\pi$
$151$	224.897	1.48939	0.744693	+	0.667407i	$0.232596\pi$
$152$	− 282.665i	− 1.85964i
$153$	0	0
$154$	0	0
$155$	13.2915i	0.0857515i
$156$	0	0
$157$	−268.675	−1.71130	−0.855652	−	0.517552i	$-0.826843\pi$
$157$	−268.675	−1.71130	−0.855652	+	0.517552i	$0.826843\pi$
$158$	64.3225i	0.407104i
$159$	0	0
$160$	−33.8253	−0.211408
$161$	0	0
$162$	0	0
$163$	125.367	0.769121	0.384561	−	0.923100i	$-0.374353\pi$
$163$	125.367	0.769121	0.384561	+	0.923100i	$0.374353\pi$
$164$	− 71.5146i	− 0.436065i
$165$	0	0
$166$	227.311	1.36935
$167$	101.198i	0.605976i	0.952994	+	0.302988i	$0.0979842\pi$
$167$	101.198i	0.605976i	−0.952994	+	0.302988i	$0.902016\pi$
$168$	0	0
$169$	238.066	1.40867
$170$	33.9636i	0.199786i
$171$	0	0
$172$	−32.4327	−0.188562
$173$	166.681i	0.963473i	0.876316	+	0.481737i	$0.159994\pi$
$173$	166.681i	0.963473i	−0.876316	+	0.481737i	$0.840006\pi$
$174$	0	0
$175$	0	0
$176$	− 75.7295i	− 0.430281i
$177$	0	0
$178$	39.3404	0.221014
$179$	− 54.9901i	− 0.307207i	−0.988133	−	0.153604i	$-0.950912\pi$
$179$	− 54.9901i	− 0.307207i	0.988133	−	0.153604i	$-0.0490879\pi$
$180$	0	0
$181$	−34.4598	−0.190386	−0.0951928	−	0.995459i	$-0.530347\pi$
$181$	−34.4598	−0.190386	−0.0951928	+	0.995459i	$0.530347\pi$
$182$	0	0
$183$	0	0
$184$	−213.282	−1.15914
$185$	6.56474i	0.0354851i
$186$	0	0
$187$	103.637	0.554208
$188$	102.433i	0.544855i
$189$	0	0
$190$	89.4987	0.471046
$191$	205.695i	1.07694i	0.842645	+	0.538470i	$0.180997\pi$
$191$	205.695i	1.07694i	−0.842645	+	0.538470i	$0.819003\pi$
$192$	0	0
$193$	−72.6650	−0.376503	−0.188251	−	0.982121i	$-0.560282\pi$
$193$	−72.6650	−0.376503	−0.188251	+	0.982121i	$0.560282\pi$
$194$	− 52.5489i	− 0.270870i
$195$	0	0
$196$	0	0
$197$	− 105.357i	− 0.534808i	−0.963585	−	0.267404i	$-0.913834\pi$
$197$	− 105.357i	− 0.534808i	0.963585	−	0.267404i	$-0.0861658\pi$
$198$	0	0
$199$	−76.6960	−0.385407	−0.192703	−	0.981257i	$-0.561726\pi$
$199$	−76.6960	−0.385407	−0.192703	+	0.981257i	$0.561726\pi$
$200$	193.050i	0.965248i
$201$	0	0
$202$	−95.3643	−0.472101
$203$	0	0
$204$	0	0
$205$	91.4353	0.446026
$206$	115.562i	0.560982i
$207$	0	0
$208$	−181.583	−0.872995
$209$	− 273.098i	− 1.30669i
$210$	0	0
$211$	−132.201	−0.626543	−0.313271	−	0.949664i	$-0.601425\pi$
$211$	−132.201	−0.626543	−0.313271	+	0.949664i	$0.601425\pi$
$212$	− 46.5497i	− 0.219574i
$213$	0	0
$214$	58.1846	0.271891
$215$	− 41.4670i	− 0.192870i
$216$	0	0
$217$	0	0
$218$	263.298i	1.20779i
$219$	0	0
$220$	−18.6494	−0.0847702
$221$	− 248.499i	− 1.12443i
$222$	0	0
$223$	18.2914	0.0820244	0.0410122	−	0.999159i	$-0.486942\pi$
$223$	18.2914	0.0820244	0.0410122	+	0.999159i	$0.486942\pi$
$224$	0	0
$225$	0	0
$226$	320.079	1.41628
$227$	− 51.9315i	− 0.228773i	−0.993436	−	0.114387i	$-0.963510\pi$
$227$	− 51.9315i	− 0.228773i	0.993436	−	0.114387i	$-0.0364902\pi$
$228$	0	0
$229$	44.4975	0.194312	0.0971561	−	0.995269i	$-0.469025\pi$
$229$	44.4975	0.194312	0.0971561	+	0.995269i	$0.469025\pi$
$230$	− 67.5305i	− 0.293611i
$231$	0	0
$232$	−401.314	−1.72980
$233$	− 198.673i	− 0.852673i	−0.904565	−	0.426336i	$-0.859804\pi$
$233$	− 198.673i	− 0.852673i	0.904565	−	0.426336i	$-0.140196\pi$
$234$	0	0
$235$	−130.966	−0.557301
$236$	84.7018i	0.358906i
$237$	0	0
$238$	0	0
$239$	− 178.165i	− 0.745460i	−0.927940	−	0.372730i	$-0.878422\pi$
$239$	− 178.165i	− 0.745460i	0.927940	−	0.372730i	$-0.121578\pi$
$240$	0	0
$241$	324.064	1.34466	0.672332	−	0.740250i	$-0.265293\pi$
$241$	324.064	1.34466	0.672332	+	0.740250i	$0.265293\pi$
$242$	82.2294i	0.339791i
$243$	0	0
$244$	−62.0352	−0.254243
$245$	0	0
$246$	0	0
$247$	−654.829	−2.65113
$248$	68.7652i	0.277279i
$249$	0	0
$250$	−130.063	−0.520251
$251$	− 289.567i	− 1.15365i	−0.816866	−	0.576827i	$-0.804291\pi$
$251$	− 289.567i	− 1.15365i	0.816866	−	0.576827i	$-0.195709\pi$
$252$	0	0
$253$	−206.063	−0.814480
$254$	246.096i	0.968880i
$255$	0	0
$256$	−222.398	−0.868744
$257$	294.982i	1.14779i	0.818929	+	0.573894i	$0.194568\pi$
$257$	294.982i	1.14779i	−0.818929	+	0.573894i	$0.805432\pi$
$258$	0	0
$259$	0	0
$260$	44.7173i	0.171990i
$261$	0	0
$262$	373.919	1.42717
$263$	− 144.507i	− 0.549458i	−0.961522	−	0.274729i	$-0.911412\pi$
$263$	− 144.507i	− 0.549458i	0.961522	−	0.274729i	$-0.0885881\pi$
$264$	0	0
$265$	59.5163	0.224590
$266$	0	0
$267$	0	0
$268$	−43.6675	−0.162938
$269$	− 318.380i	− 1.18357i	−0.806096	−	0.591785i	$-0.798424\pi$
$269$	− 318.380i	− 1.18357i	0.806096	−	0.591785i	$-0.201576\pi$
$270$	0	0
$271$	−211.401	−0.780076	−0.390038	−	0.920799i	$-0.627538\pi$
$271$	−211.401	−0.780076	−0.390038	+	0.920799i	$0.627538\pi$
$272$	110.850i	0.407535i
$273$	0	0
$274$	68.3801	0.249562
$275$	186.516i	0.678238i
$276$	0	0
$277$	−243.129	−0.877724	−0.438862	−	0.898555i	$-0.644618\pi$
$277$	−243.129	−0.877724	−0.438862	+	0.898555i	$0.644618\pi$
$278$	383.863i	1.38080i
$279$	0	0
$280$	0	0
$281$	257.174i	0.915211i	0.889155	+	0.457605i	$0.151293\pi$
$281$	257.174i	0.915211i	−0.889155	+	0.457605i	$0.848707\pi$
$282$	0	0
$283$	−453.750	−1.60336	−0.801678	−	0.597756i	$-0.796059\pi$
$283$	−453.750	−1.60336	−0.801678	+	0.597756i	$0.796059\pi$
$284$	− 60.6694i	− 0.213625i
$285$	0	0
$286$	−278.097	−0.972366
$287$	0	0
$288$	0	0
$289$	137.301	0.475089
$290$	− 127.066i	− 0.438159i
$291$	0	0
$292$	109.887	0.376325
$293$	− 126.449i	− 0.431565i	−0.976441	−	0.215783i	$-0.930770\pi$
$293$	− 126.449i	− 0.431565i	0.976441	−	0.215783i	$-0.0692303\pi$
$294$	0	0
$295$	−108.296	−0.367104
$296$	33.9636i	0.114742i
$297$	0	0
$298$	378.612	1.27051
$299$	494.095i	1.65249i
$300$	0	0
$301$	0	0
$302$	368.404i	1.21988i
$303$	0	0
$304$	292.104	0.960869
$305$	− 79.3153i	− 0.260050i
$306$	0	0
$307$	72.5691	0.236381	0.118191	−	0.992991i	$-0.462291\pi$
$307$	72.5691	0.236381	0.118191	+	0.992991i	$0.462291\pi$
$308$	0	0
$309$	0	0
$310$	−21.7728	−0.0702348
$311$	274.865i	0.883812i	0.897061	+	0.441906i	$0.145698\pi$
$311$	274.865i	0.883812i	−0.897061	+	0.441906i	$0.854302\pi$
$312$	0	0
$313$	−194.359	−0.620956	−0.310478	−	0.950581i	$-0.600489\pi$
$313$	−194.359	−0.620956	−0.310478	+	0.950581i	$0.600489\pi$
$314$	− 440.116i	− 1.40164i
$315$	0	0
$316$	51.6992	0.163605
$317$	− 130.529i	− 0.411765i	−0.978577	−	0.205882i	$-0.933994\pi$
$317$	− 130.529i	− 0.411765i	0.978577	−	0.205882i	$-0.0660064\pi$
$318$	0	0
$319$	−387.731	−1.21546
$320$	− 116.011i	− 0.362534i
$321$	0	0
$322$	0	0
$323$	399.749i	1.23761i
$324$	0	0
$325$	447.223	1.37607
$326$	205.363i	0.629949i
$327$	0	0
$328$	473.053	1.44223
$329$	0	0
$330$	0	0
$331$	38.9975	0.117817	0.0589086	−	0.998263i	$-0.481238\pi$
$331$	38.9975	0.117817	0.0589086	+	0.998263i	$0.481238\pi$
$332$	− 182.702i	− 0.550307i
$333$	0	0
$334$	−165.773	−0.496325
$335$	− 55.8312i	− 0.166660i
$336$	0	0
$337$	−333.166	−0.988624	−0.494312	−	0.869285i	$-0.664580\pi$
$337$	−333.166	−0.988624	−0.494312	+	0.869285i	$0.664580\pi$
$338$	389.976i	1.15378i
$339$	0	0
$340$	27.2982	0.0802889
$341$	66.4378i	0.194832i
$342$	0	0
$343$	0	0
$344$	− 214.535i	− 0.623649i
$345$	0	0
$346$	−273.040	−0.789133
$347$	536.718i	1.54674i	0.633956	+	0.773369i	$0.281430\pi$
$347$	536.718i	1.54674i	−0.633956	+	0.773369i	$0.718570\pi$
$348$	0	0
$349$	171.187	0.490508	0.245254	−	0.969459i	$-0.421129\pi$
$349$	171.187	0.490508	0.245254	+	0.969459i	$0.421129\pi$
$350$	0	0
$351$	0	0
$352$	−169.077	−0.480332
$353$	− 431.082i	− 1.22120i	−0.791941	−	0.610598i	$-0.790929\pi$
$353$	− 431.082i	− 1.22120i	0.791941	−	0.610598i	$-0.209071\pi$
$354$	0	0
$355$	77.5691	0.218504
$356$	− 31.6199i	− 0.0888199i
$357$	0	0
$358$	90.0794	0.251618
$359$	152.567i	0.424979i	0.977163	+	0.212489i	$0.0681571\pi$
$359$	152.567i	0.424979i	−0.977163	+	0.212489i	$0.931843\pi$
$360$	0	0
$361$	692.393	1.91799
$362$	− 56.4486i	− 0.155935i
$363$	0	0
$364$	0	0
$365$	140.496i	0.384921i
$366$	0	0
$367$	−355.904	−0.969767	−0.484884	−	0.874579i	$-0.661138\pi$
$367$	−355.904	−0.969767	−0.484884	+	0.874579i	$0.661138\pi$
$368$	− 220.405i	− 0.598926i
$369$	0	0
$370$	−10.7537	−0.0290641
$371$	0	0
$372$	0	0
$373$	−397.261	−1.06504	−0.532522	−	0.846416i	$-0.678756\pi$
$373$	−397.261	−1.06504	−0.532522	+	0.846416i	$0.678756\pi$
$374$	169.768i	0.453924i
$375$	0	0
$376$	−677.569	−1.80205
$377$	929.694i	2.46603i
$378$	0	0
$379$	7.46198	0.0196886	0.00984430	−	0.999952i	$-0.496866\pi$
$379$	7.46198	0.0196886	0.00984430	+	0.999952i	$0.496866\pi$
$380$	− 71.9347i	− 0.189302i
$381$	0	0
$382$	−336.950	−0.882068
$383$	− 256.602i	− 0.669978i	−0.942222	−	0.334989i	$-0.891267\pi$
$383$	− 256.602i	− 0.669978i	0.942222	−	0.334989i	$-0.108733\pi$
$384$	0	0
$385$	0	0
$386$	− 119.033i	− 0.308375i
$387$	0	0
$388$	−42.2362	−0.108856
$389$	− 246.472i	− 0.633605i	−0.948491	−	0.316803i	$-0.897391\pi$
$389$	− 246.472i	− 0.633605i	0.948491	−	0.316803i	$-0.102609\pi$
$390$	0	0
$391$	301.627	0.771424
$392$	0	0
$393$	0	0
$394$	172.586	0.438035
$395$	66.1003i	0.167342i
$396$	0	0
$397$	−48.6244	−0.122480	−0.0612398	−	0.998123i	$-0.519505\pi$
$397$	−48.6244	−0.122480	−0.0612398	+	0.998123i	$0.519505\pi$
$398$	− 125.636i	− 0.315668i
$399$	0	0
$400$	−199.496	−0.498741
$401$	65.3563i	0.162983i	0.996674	+	0.0814916i	$0.0259684\pi$
$401$	65.3563i	0.162983i	−0.996674	+	0.0814916i	$0.974032\pi$
$402$	0	0
$403$	159.303	0.395293
$404$	76.6491i	0.189726i
$405$	0	0
$406$	0	0
$407$	32.8140i	0.0806241i
$408$	0	0
$409$	−442.619	−1.08220	−0.541099	−	0.840959i	$-0.681992\pi$
$409$	−442.619	−1.08220	−0.541099	+	0.840959i	$0.681992\pi$
$410$	149.780i	0.365318i
$411$	0	0
$412$	92.8832	0.225445
$413$	0	0
$414$	0	0
$415$	233.594	0.562877
$416$	405.409i	0.974542i
$417$	0	0
$418$	447.362	1.07024
$419$	− 244.902i	− 0.584492i	−0.956343	−	0.292246i	$-0.905597\pi$
$419$	− 244.902i	− 0.584492i	0.956343	−	0.292246i	$-0.0944026\pi$
$420$	0	0
$421$	556.259	1.32128	0.660640	−	0.750703i	$-0.270285\pi$
$421$	556.259	1.32128	0.660640	+	0.750703i	$0.270285\pi$
$422$	− 216.558i	− 0.513170i
$423$	0	0
$424$	307.916	0.726216
$425$	− 273.013i	− 0.642384i
$426$	0	0
$427$	0	0
$428$	− 46.7659i	− 0.109266i
$429$	0	0
$430$	67.9271	0.157970
$431$	789.337i	1.83141i	0.401854	+	0.915704i	$0.368366\pi$
$431$	789.337i	1.83141i	−0.401854	+	0.915704i	$0.631634\pi$
$432$	0	0
$433$	−113.656	−0.262484	−0.131242	−	0.991350i	$-0.541897\pi$
$433$	−113.656	−0.262484	−0.131242	+	0.991350i	$0.541897\pi$
$434$	0	0
$435$	0	0
$436$	211.626	0.485380
$437$	− 794.829i	− 1.81883i
$438$	0	0
$439$	568.656	1.29534	0.647672	−	0.761920i	$-0.275743\pi$
$439$	568.656	1.29534	0.647672	+	0.761920i	$0.275743\pi$
$440$	− 123.362i	− 0.280368i
$441$	0	0
$442$	407.066	0.920964
$443$	− 53.8110i	− 0.121470i	−0.998154	−	0.0607348i	$-0.980656\pi$
$443$	− 53.8110i	− 0.121470i	0.998154	−	0.0607348i	$-0.0193444\pi$
$444$	0	0
$445$	40.4277	0.0908488
$446$	29.9632i	0.0671821i
$447$	0	0
$448$	0	0
$449$	− 148.022i	− 0.329671i	−0.986321	−	0.164835i	$-0.947291\pi$
$449$	− 148.022i	− 0.329671i	0.986321	−	0.164835i	$-0.0527093\pi$
$450$	0	0
$451$	457.041	1.01340
$452$	− 257.264i	− 0.569168i
$453$	0	0
$454$	85.0690	0.187377
$455$	0	0
$456$	0	0
$457$	567.799	1.24245	0.621225	−	0.783632i	$-0.286635\pi$
$457$	567.799	1.24245	0.621225	+	0.783632i	$0.286635\pi$
$458$	72.8914i	0.159152i
$459$	0	0
$460$	−54.2777	−0.117995
$461$	449.150i	0.974296i	0.873319	+	0.487148i	$0.161963\pi$
$461$	449.150i	0.974296i	−0.873319	+	0.487148i	$0.838037\pi$
$462$	0	0
$463$	295.636	0.638522	0.319261	−	0.947667i	$-0.396565\pi$
$463$	295.636	0.638522	0.319261	+	0.947667i	$0.396565\pi$
$464$	− 414.715i	− 0.893783i
$465$	0	0
$466$	325.446	0.698382
$467$	− 613.166i	− 1.31299i	−0.754330	−	0.656495i	$-0.772038\pi$
$467$	− 613.166i	− 1.31299i	0.754330	−	0.656495i	$-0.227962\pi$
$468$	0	0
$469$	0	0
$470$	− 214.535i	− 0.456458i
$471$	0	0
$472$	−560.283	−1.18704
$473$	− 207.274i	− 0.438211i
$474$	0	0
$475$	−719.428	−1.51459
$476$	0	0
$477$	0	0
$478$	291.852	0.610569
$479$	908.765i	1.89721i	0.316457	+	0.948607i	$0.397507\pi$
$479$	908.765i	1.89721i	−0.316457	+	0.948607i	$0.602493\pi$
$480$	0	0
$481$	78.6808	0.163578
$482$	530.850i	1.10135i
$483$	0	0
$484$	66.0919	0.136554
$485$	− 54.0012i	− 0.111343i
$486$	0	0
$487$	−779.230	−1.60006	−0.800031	−	0.599959i	$-0.795183\pi$
$487$	−779.230	−1.60006	−0.800031	+	0.599959i	$0.795183\pi$
$488$	− 410.348i	− 0.840878i
$489$	0	0
$490$	0	0
$491$	− 207.815i	− 0.423248i	−0.977351	−	0.211624i	$-0.932125\pi$
$491$	− 207.815i	− 0.423248i	0.977351	−	0.211624i	$-0.0678753\pi$
$492$	0	0
$493$	567.544	1.15120
$494$	− 1072.68i	− 2.17141i
$495$	0	0
$496$	−71.0616	−0.143269
$497$	0	0
$498$	0	0
$499$	−874.565	−1.75263	−0.876317	−	0.481734i	$-0.840007\pi$
$499$	−874.565	−1.75263	−0.876317	+	0.481734i	$0.840007\pi$
$500$	104.538i	0.209076i
$501$	0	0
$502$	474.340	0.944901
$503$	198.301i	0.394236i	0.980380	+	0.197118i	$0.0631582\pi$
$503$	198.301i	0.394236i	−0.980380	+	0.197118i	$0.936842\pi$
$504$	0	0
$505$	−98.0000	−0.194059
$506$	− 337.553i	− 0.667100i
$507$	0	0
$508$	197.799	0.389369
$509$	252.280i	0.495638i	0.968806	+	0.247819i	$0.0797139\pi$
$509$	252.280i	0.495638i	−0.968806	+	0.247819i	$0.920286\pi$
$510$	0	0
$511$	0	0
$512$	− 494.374i	− 0.965574i
$513$	0	0
$514$	−483.210	−0.940097
$515$	118.756i	0.230594i
$516$	0	0
$517$	−654.636	−1.26622
$518$	0	0
$519$	0	0
$520$	−295.794	−0.568836
$521$	139.578i	0.267904i	0.990988	+	0.133952i	$0.0427668\pi$
$521$	139.578i	0.267904i	−0.990988	+	0.133952i	$0.957233\pi$
$522$	0	0
$523$	9.28394	0.0177513	0.00887566	−	0.999961i	$-0.497175\pi$
$523$	9.28394	0.0177513	0.00887566	+	0.999961i	$0.497175\pi$
$524$	− 300.538i	− 0.573546i
$525$	0	0
$526$	236.718	0.450034
$527$	− 97.2487i	− 0.184533i
$528$	0	0
$529$	−70.7310	−0.133707
$530$	97.4937i	0.183950i
$531$	0	0
$532$	0	0
$533$	− 1095.89i	− 2.05607i
$534$	0	0
$535$	59.7927	0.111762
$536$	− 288.850i	− 0.538900i
$537$	0	0
$538$	521.539	0.969403
$539$	0	0
$540$	0	0
$541$	1006.53	1.86050	0.930248	−	0.366932i	$-0.119592\pi$
$541$	1006.53	1.86050	0.930248	+	0.366932i	$0.119592\pi$
$542$	− 346.296i	− 0.638922i
$543$	0	0
$544$	247.487	0.454940
$545$	270.575i	0.496468i
$546$	0	0
$547$	−600.195	−1.09725	−0.548625	−	0.836069i	$-0.684848\pi$
$547$	−600.195	−1.09725	−0.548625	+	0.836069i	$0.684848\pi$
$548$	− 54.9606i	− 0.100293i
$549$	0	0
$550$	−305.531	−0.555511
$551$	− 1495.56i	− 2.71426i
$552$	0	0
$553$	0	0
$554$	− 398.271i	− 0.718900i
$555$	0	0
$556$	308.530	0.554910
$557$	464.702i	0.834295i	0.908839	+	0.417147i	$0.136970\pi$
$557$	464.702i	0.834295i	−0.908839	+	0.417147i	$0.863030\pi$
$558$	0	0
$559$	−496.997	−0.889083
$560$	0	0
$561$	0	0
$562$	−421.277	−0.749604
$563$	456.723i	0.811232i	0.914044	+	0.405616i	$0.132943\pi$
$563$	456.723i	0.811232i	−0.914044	+	0.405616i	$0.867057\pi$
$564$	0	0
$565$	328.926	0.582169
$566$	− 743.288i	− 1.31323i
$567$	0	0
$568$	401.314	0.706539
$569$	− 985.909i	− 1.73270i	−0.499433	−	0.866352i	$-0.666458\pi$
$569$	− 985.909i	− 1.73270i	0.499433	−	0.866352i	$-0.333542\pi$
$570$	0	0
$571$	−327.868	−0.574200	−0.287100	−	0.957901i	$-0.592691\pi$
$571$	−327.868	−0.574200	−0.287100	+	0.957901i	$0.592691\pi$
$572$	223.520i	0.390770i
$573$	0	0
$574$	0	0
$575$	542.838i	0.944066i
$576$	0	0
$577$	594.108	1.02965	0.514825	−	0.857295i	$-0.327857\pi$
$577$	594.108	1.02965	0.514825	+	0.857295i	$0.327857\pi$
$578$	224.913i	0.389122i
$579$	0	0
$580$	−102.129	−0.176085
$581$	0	0
$582$	0	0
$583$	297.494	0.510281
$584$	726.876i	1.24465i
$585$	0	0
$586$	207.136	0.353474
$587$	422.665i	0.720043i	0.932944	+	0.360021i	$0.117231\pi$
$587$	422.665i	0.720043i	−0.932944	+	0.360021i	$0.882769\pi$
$588$	0	0
$589$	−256.264	−0.435083
$590$	− 177.399i	− 0.300677i
$591$	0	0
$592$	−35.0977	−0.0592867
$593$	171.620i	0.289410i	0.989475	+	0.144705i	$0.0462233\pi$
$593$	171.620i	0.289410i	−0.989475	+	0.144705i	$0.953777\pi$
$594$	0	0
$595$	0	0
$596$	− 304.310i	− 0.510587i
$597$	0	0
$598$	−809.378	−1.35347
$599$	− 595.877i	− 0.994786i	−0.867525	−	0.497393i	$-0.834291\pi$
$599$	− 595.877i	− 0.994786i	0.867525	−	0.497393i	$-0.165709\pi$
$600$	0	0
$601$	−940.213	−1.56441	−0.782207	−	0.623018i	$-0.785906\pi$
$601$	−940.213	−1.56441	−0.782207	+	0.623018i	$0.785906\pi$
$602$	0	0
$603$	0	0
$604$	296.105	0.490241
$605$	84.5021i	0.139673i
$606$	0	0
$607$	715.301	1.17842	0.589210	−	0.807980i	$-0.299439\pi$
$607$	715.301	1.17842	0.589210	+	0.807980i	$0.299439\pi$
$608$	− 652.164i	− 1.07264i
$609$	0	0
$610$	129.926	0.212994
$611$	1569.67i	2.56902i
$612$	0	0
$613$	−35.8312	−0.0584523	−0.0292261	−	0.999573i	$-0.509304\pi$
$613$	−35.8312	−0.0584523	−0.0292261	+	0.999573i	$0.509304\pi$
$614$	118.876i	0.193608i
$615$	0	0
$616$	0	0
$617$	865.704i	1.40309i	0.712627	+	0.701543i	$0.247505\pi$
$617$	865.704i	1.40309i	−0.712627	+	0.701543i	$0.752495\pi$
$618$	0	0
$619$	−667.651	−1.07860	−0.539298	−	0.842115i	$-0.681310\pi$
$619$	−667.651	−1.07860	−0.539298	+	0.842115i	$0.681310\pi$
$620$	17.4999i	0.0282256i
$621$	0	0
$622$	−450.257	−0.723887
$623$	0	0
$624$	0	0
$625$	420.499	0.672798
$626$	− 318.380i	− 0.508594i
$627$	0	0
$628$	−353.744	−0.563286
$629$	− 48.0317i	− 0.0763621i
$630$	0	0
$631$	48.8388	0.0773990	0.0386995	−	0.999251i	$-0.487678\pi$
$631$	48.8388	0.0773990	0.0386995	+	0.999251i	$0.487678\pi$
$632$	341.979i	0.541105i
$633$	0	0
$634$	213.820	0.337256
$635$	252.897i	0.398263i
$636$	0	0
$637$	0	0
$638$	− 635.143i	− 0.995521i
$639$	0	0
$640$	54.7361	0.0855252
$641$	− 11.6233i	− 0.0181330i	−0.999959	−	0.00906651i	$-0.997114\pi$
$641$	− 11.6233i	− 0.0181330i	0.999959	−	0.00906651i	$-0.00288600\pi$
$642$	0	0
$643$	483.348	0.751708	0.375854	−	0.926679i	$-0.377349\pi$
$643$	483.348	0.751708	0.375854	+	0.926679i	$0.377349\pi$
$644$	0	0
$645$	0	0
$646$	−654.829	−1.01367
$647$	− 513.789i	− 0.794110i	−0.917795	−	0.397055i	$-0.870032\pi$
$647$	− 513.789i	− 0.794110i	0.917795	−	0.397055i	$-0.129968\pi$
$648$	0	0
$649$	−541.319	−0.834082
$650$	732.597i	1.12707i
$651$	0	0
$652$	165.061	0.253161
$653$	− 275.988i	− 0.422646i	−0.977416	−	0.211323i	$-0.932223\pi$
$653$	− 275.988i	− 0.422646i	0.977416	−	0.211323i	$-0.0677772\pi$
$654$	0	0
$655$	384.254	0.586647
$656$	488.850i	0.745198i
$657$	0	0
$658$	0	0
$659$	− 150.164i	− 0.227867i	−0.993488	−	0.113933i	$-0.963655\pi$
$659$	− 150.164i	− 0.227867i	0.993488	−	0.113933i	$-0.0363450\pi$
$660$	0	0
$661$	866.928	1.31154	0.655770	−	0.754961i	$-0.272344\pi$
$661$	866.928	1.31154	0.655770	+	0.754961i	$0.272344\pi$
$662$	63.8818i	0.0964982i
$663$	0	0
$664$	1208.53	1.82008
$665$	0	0
$666$	0	0
$667$	−1128.46	−1.69184
$668$	133.240i	0.199461i
$669$	0	0
$670$	91.4572	0.136503
$671$	− 396.459i	− 0.590849i
$672$	0	0
$673$	479.731	0.712825	0.356412	−	0.934329i	$-0.384000\pi$
$673$	479.731	0.712825	0.356412	+	0.934329i	$0.384000\pi$
$674$	− 545.760i	− 0.809733i
$675$	0	0
$676$	313.444	0.463674
$677$	− 243.050i	− 0.359011i	−0.983757	−	0.179505i	$-0.942550\pi$
$677$	− 243.050i	− 0.359011i	0.983757	−	0.179505i	$-0.0574497\pi$
$678$	0	0
$679$	0	0
$680$	180.572i	0.265546i
$681$	0	0
$682$	−108.832	−0.159577
$683$	675.304i	0.988732i	0.869254	+	0.494366i	$0.164600\pi$
$683$	675.304i	0.988732i	−0.869254	+	0.494366i	$0.835400\pi$
$684$	0	0
$685$	70.2700	0.102584
$686$	0	0
$687$	0	0
$688$	221.699	0.322237
$689$	− 713.325i	− 1.03530i
$690$	0	0
$691$	−541.633	−0.783840	−0.391920	−	0.919999i	$-0.628189\pi$
$691$	−541.633	−0.783840	−0.391920	+	0.919999i	$0.628189\pi$
$692$	219.456i	0.317133i
$693$	0	0
$694$	−879.199	−1.26686
$695$	394.472i	0.567586i
$696$	0	0
$697$	−668.997	−0.959824
$698$	280.422i	0.401751i
$699$	0	0
$700$	0	0
$701$	873.152i	1.24558i	0.782389	+	0.622790i	$0.214001\pi$
$701$	873.152i	1.24558i	−0.782389	+	0.622790i	$0.785999\pi$
$702$	0	0
$703$	−126.570	−0.180043
$704$	− 579.883i	− 0.823698i
$705$	0	0
$706$	706.156	1.00022
$707$	0	0
$708$	0	0
$709$	488.259	0.688659	0.344329	−	0.938849i	$-0.388106\pi$
$709$	488.259	0.688659	0.344329	+	0.938849i	$0.388106\pi$
$710$	127.066i	0.178966i
$711$	0	0
$712$	209.158	0.293762
$713$	193.362i	0.271195i
$714$	0	0
$715$	−285.783	−0.399696
$716$	− 72.4013i	− 0.101119i
$717$	0	0
$718$	−249.921	−0.348079
$719$	0.728476i	0.00101318i	1.00000		0.000506589i	$0.000161252\pi$
$719$	0.728476i	0.00101318i	−1.00000		0.000506589i	$0.999839\pi$
$720$	0	0
$721$	0	0
$722$	1134.21i	1.57093i
$723$	0	0
$724$	−45.3706	−0.0626666
$725$	1021.41i	1.40884i
$726$	0	0
$727$	342.255	0.470777	0.235389	−	0.971901i	$-0.424364\pi$
$727$	342.255	0.470777	0.235389	+	0.971901i	$0.424364\pi$
$728$	0	0
$729$	0	0
$730$	−230.147	−0.315270
$731$	303.398i	0.415046i
$732$	0	0
$733$	−1400.87	−1.91114	−0.955571	−	0.294762i	$-0.904760\pi$
$733$	−1400.87	−1.91114	−0.955571	+	0.294762i	$0.904760\pi$
$734$	− 583.008i	− 0.794288i
$735$	0	0
$736$	−492.084	−0.668593
$737$	− 279.074i	− 0.378662i
$738$	0	0
$739$	792.755	1.07274	0.536370	−	0.843983i	$-0.319795\pi$
$739$	792.755	1.07274	0.536370	+	0.843983i	$0.319795\pi$
$740$	8.64330i	0.0116801i
$741$	0	0
$742$	0	0
$743$	586.406i	0.789241i	0.918844	+	0.394620i	$0.129124\pi$
$743$	586.406i	0.789241i	−0.918844	+	0.394620i	$0.870876\pi$
$744$	0	0
$745$	389.077	0.522250
$746$	− 650.754i	− 0.872325i
$747$	0	0
$748$	136.451	0.182421
$749$	0	0
$750$	0	0
$751$	477.457	0.635762	0.317881	−	0.948131i	$-0.397029\pi$
$751$	477.457	0.635762	0.317881	+	0.948131i	$0.397029\pi$
$752$	− 700.195i	− 0.931111i
$753$	0	0
$754$	−1522.93	−2.01981
$755$	378.586i	0.501439i
$756$	0	0
$757$	−95.8680	−0.126642	−0.0633210	−	0.997993i	$-0.520169\pi$
$757$	−95.8680	−0.126642	−0.0633210	+	0.997993i	$0.520169\pi$
$758$	12.2235i	0.0161260i
$759$	0	0
$760$	475.831	0.626094
$761$	1069.27i	1.40509i	0.711640	+	0.702544i	$0.247953\pi$
$761$	1069.27i	1.40509i	−0.711640	+	0.702544i	$0.752047\pi$
$762$	0	0
$763$	0	0
$764$	270.824i	0.354481i
$765$	0	0
$766$	420.339	0.548746
$767$	1297.96i	1.69226i
$768$	0	0
$769$	931.960	1.21191	0.605956	−	0.795499i	$-0.292791\pi$
$769$	931.960	1.21191	0.605956	+	0.795499i	$0.292791\pi$
$770$	0	0
$771$	0	0
$772$	−95.6725	−0.123928
$773$	− 1005.45i	− 1.30071i	−0.759632	−	0.650353i	$-0.774621\pi$
$773$	− 1005.45i	− 1.30071i	0.759632	−	0.650353i	$-0.225379\pi$
$774$	0	0
$775$	175.019	0.225831
$776$	− 279.383i	− 0.360029i
$777$	0	0
$778$	403.747	0.518955
$779$	1762.90i	2.26303i
$780$	0	0
$781$	387.731	0.496455
$782$	494.095i	0.631835i
$783$	0	0
$784$	0	0
$785$	− 452.280i	− 0.576153i
$786$	0	0
$787$	43.8442	0.0557105	0.0278553	−	0.999612i	$-0.491132\pi$
$787$	43.8442	0.0557105	0.0278553	+	0.999612i	$0.491132\pi$
$788$	− 138.716i	− 0.176035i
$789$	0	0
$790$	−108.279	−0.137062
$791$	0	0
$792$	0	0
$793$	−950.623	−1.19877
$794$	− 79.6516i	− 0.100317i
$795$	0	0
$796$	−100.980	−0.126859
$797$	564.517i	0.708303i	0.935188	+	0.354151i	$0.115230\pi$
$797$	564.517i	0.708303i	−0.935188	+	0.354151i	$0.884770\pi$
$798$	0	0
$799$	958.227	1.19928
$800$	445.403i	0.556754i
$801$	0	0
$802$	−107.060	−0.133491
$803$	702.274i	0.874563i
$804$	0	0
$805$	0	0
$806$	260.955i	0.323765i
$807$	0	0
$808$	−507.016	−0.627495
$809$	− 665.662i	− 0.822821i	−0.911450	−	0.411410i	$-0.865036\pi$
$809$	− 665.662i	− 0.822821i	0.911450	−	0.411410i	$-0.134964\pi$
$810$	0	0
$811$	635.829	0.784006	0.392003	−	0.919964i	$-0.371782\pi$
$811$	635.829	0.784006	0.392003	+	0.919964i	$0.371782\pi$
$812$	0	0
$813$	0	0
$814$	−53.7527	−0.0660352
$815$	211.039i	0.258944i
$816$	0	0
$817$	799.497	0.978577
$818$	− 725.055i	− 0.886375i
$819$	0	0
$820$	120.386	0.146812
$821$	− 94.5208i	− 0.115129i	−0.998342	−	0.0575644i	$-0.981667\pi$
$821$	− 94.5208i	− 0.115129i	0.998342	−	0.0575644i	$-0.0183335\pi$
$822$	0	0
$823$	42.2005	0.0512764	0.0256382	−	0.999671i	$-0.491838\pi$
$823$	42.2005	0.0512764	0.0256382	+	0.999671i	$0.491838\pi$
$824$	614.401i	0.745632i
$825$	0	0
$826$	0	0
$827$	1105.74i	1.33705i	0.743690	+	0.668525i	$0.233074\pi$
$827$	1105.74i	1.33705i	−0.743690	+	0.668525i	$0.766926\pi$
$828$	0	0
$829$	−492.852	−0.594513	−0.297257	−	0.954798i	$-0.596072\pi$
$829$	−492.852	−0.594513	−0.297257	+	0.954798i	$0.596072\pi$
$830$	382.651i	0.461025i
$831$	0	0
$832$	−1390.43	−1.67119
$833$	0	0
$834$	0	0
$835$	−170.354	−0.204017
$836$	− 359.567i	− 0.430104i
$837$	0	0
$838$	401.175	0.478729
$839$	− 940.692i	− 1.12121i	−0.828085	−	0.560603i	$-0.810569\pi$
$839$	− 940.692i	− 1.12121i	0.828085	−	0.560603i	$-0.189431\pi$
$840$	0	0
$841$	−1282.32	−1.52476
$842$	911.208i	1.08220i
$843$	0	0
$844$	−174.058	−0.206230
$845$	400.754i	0.474266i
$846$	0	0
$847$	0	0
$848$	318.198i	0.375234i
$849$	0	0
$850$	447.223	0.526145
$851$	95.5025i	0.112224i
$852$	0	0
$853$	−338.071	−0.396332	−0.198166	−	0.980168i	$-0.563499\pi$
$853$	−338.071	−0.396332	−0.198166	+	0.980168i	$0.563499\pi$
$854$	0	0
$855$	0	0
$856$	309.346	0.361385
$857$	− 875.520i	− 1.02161i	−0.859697	−	0.510805i	$-0.829347\pi$
$857$	− 875.520i	− 1.02161i	0.859697	−	0.510805i	$-0.170653\pi$
$858$	0	0
$859$	−877.544	−1.02159	−0.510794	−	0.859703i	$-0.670648\pi$
$859$	−877.544	−1.02159	−0.510794	+	0.859703i	$0.670648\pi$
$860$	− 54.5965i	− 0.0634843i
$861$	0	0
$862$	−1293.01	−1.50002
$863$	− 1070.52i	− 1.24047i	−0.784417	−	0.620233i	$-0.787038\pi$
$863$	− 1070.52i	− 1.24047i	0.784417	−	0.620233i	$-0.212962\pi$
$864$	0	0
$865$	−280.586	−0.324377
$866$	− 186.180i	− 0.214988i
$867$	0	0
$868$	0	0
$869$	330.404i	0.380211i
$870$	0	0
$871$	−669.158	−0.768264
$872$	1399.86i	1.60534i
$873$	0	0
$874$	1302.01	1.48971
$875$	0	0
$876$	0	0
$877$	−190.037	−0.216690	−0.108345	−	0.994113i	$-0.534555\pi$
$877$	−190.037	−0.216690	−0.108345	+	0.994113i	$0.534555\pi$
$878$	931.515i	1.06095i
$879$	0	0
$880$	127.481	0.144865
$881$	− 382.322i	− 0.433963i	−0.976176	−	0.216982i	$-0.930379\pi$
$881$	− 382.322i	− 0.433963i	0.976176	−	0.216982i	$-0.0696212\pi$
$882$	0	0
$883$	−1507.96	−1.70777	−0.853883	−	0.520464i	$-0.825759\pi$
$883$	−1507.96	−1.70777	−0.853883	+	0.520464i	$0.825759\pi$
$884$	− 327.180i	− 0.370113i
$885$	0	0
$886$	88.1479	0.0994897
$887$	− 808.997i	− 0.912060i	−0.889964	−	0.456030i	$-0.849271\pi$
$887$	− 808.997i	− 0.912060i	0.889964	−	0.456030i	$-0.150729\pi$
$888$	0	0
$889$	0	0
$890$	66.2247i	0.0744098i
$891$	0	0
$892$	24.0830	0.0269988
$893$	− 2525.06i	− 2.82762i
$894$	0	0
$895$	92.5690	0.103429
$896$	0	0
$897$	0	0
$898$	242.475	0.270017
$899$	363.831i	0.404707i
$900$	0	0
$901$	−435.458	−0.483306
$902$	748.680i	0.830022i
$903$	0	0
$904$	1701.74	1.88246
$905$	− 58.0088i	− 0.0640981i
$906$	0	0
$907$	−1599.08	−1.76305	−0.881523	−	0.472141i	$-0.843481\pi$
$907$	−1599.08	−1.76305	−0.881523	+	0.472141i	$0.843481\pi$
$908$	− 68.3743i	− 0.0753021i
$909$	0	0
$910$	0	0
$911$	− 737.021i	− 0.809025i	−0.914533	−	0.404512i	$-0.867441\pi$
$911$	− 737.021i	− 0.809025i	0.914533	−	0.404512i	$-0.132559\pi$
$912$	0	0
$913$	1167.63	1.27889
$914$	930.113i	1.01763i
$915$	0	0
$916$	58.5865	0.0639591
$917$	0	0
$918$	0	0
$919$	−351.430	−0.382405	−0.191203	−	0.981551i	$-0.561239\pi$
$919$	−351.430	−0.382405	−0.191203	+	0.981551i	$0.561239\pi$
$920$	− 359.034i	− 0.390255i
$921$	0	0
$922$	−735.754	−0.797998
$923$	− 929.694i	− 1.00725i
$924$	0	0
$925$	86.4428	0.0934517
$926$	484.281i	0.522982i
$927$	0	0
$928$	−925.911	−0.997748
$929$	612.744i	0.659574i	0.944055	+	0.329787i	$0.106977\pi$
$929$	612.744i	0.659574i	−0.944055	+	0.329787i	$0.893023\pi$
$930$	0	0
$931$	0	0
$932$	− 261.578i	− 0.280663i
$933$	0	0
$934$	1004.43	1.07541
$935$	174.460i	0.186588i
$936$	0	0
$937$	−889.070	−0.948848	−0.474424	−	0.880297i	$-0.657344\pi$
$937$	−889.070	−0.948848	−0.474424	+	0.880297i	$0.657344\pi$
$938$	0	0
$939$	0	0
$940$	−172.433	−0.183439
$941$	1225.38i	1.30221i	0.758989	+	0.651104i	$0.225694\pi$
$941$	1225.38i	1.30221i	−0.758989	+	0.651104i	$0.774306\pi$
$942$	0	0
$943$	1330.18	1.41059
$944$	− 578.992i	− 0.613339i
$945$	0	0
$946$	339.536	0.358917
$947$	− 63.3714i	− 0.0669181i	−0.999440	−	0.0334590i	$-0.989348\pi$
$947$	− 63.3714i	− 0.0669181i	0.999440	−	0.0334590i	$-0.0106523\pi$
$948$	0	0
$949$	1683.90	1.77439
$950$	− 1178.50i	− 1.24052i
$951$	0	0
$952$	0	0
$953$	− 1238.42i	− 1.29950i	−0.760149	−	0.649749i	$-0.774874\pi$
$953$	− 1238.42i	− 1.29950i	0.760149	−	0.649749i	$-0.225126\pi$
$954$	0	0
$955$	−346.263	−0.362579
$956$	− 234.576i	− 0.245373i
$957$	0	0
$958$	−1488.65	−1.55391
$959$	0	0
$960$	0	0
$961$	−898.657	−0.935127
$962$	128.887i	0.133978i
$963$	0	0
$964$	426.671	0.442604
$965$	− 122.322i	− 0.126759i
$966$	0	0
$967$	−718.423	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$967$	−718.423	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$968$	437.183i	0.451635i
$969$	0	0
$970$	88.4595	0.0911953
$971$	737.425i	0.759449i	0.925100	+	0.379725i	$0.123981\pi$
$971$	737.425i	0.759449i	−0.925100	+	0.379725i	$0.876019\pi$
$972$	0	0
$973$	0	0
$974$	− 1276.46i	− 1.31053i
$975$	0	0
$976$	424.051	0.434479
$977$	− 332.729i	− 0.340562i	−0.985396	−	0.170281i	$-0.945532\pi$
$977$	− 332.729i	− 0.340562i	0.985396	−	0.170281i	$-0.0544675\pi$
$978$	0	0
$979$	202.079	0.206414
$980$	0	0
$981$	0	0
$982$	340.422	0.346662
$983$	1593.36i	1.62092i	0.585796	+	0.810459i	$0.300782\pi$
$983$	1593.36i	1.62092i	−0.585796	+	0.810459i	$0.699218\pi$
$984$	0	0
$985$	177.356	0.180056
$986$	929.694i	0.942895i
$987$	0	0
$988$	−862.164	−0.872635
$989$	− 603.254i	− 0.609963i
$990$	0	0
$991$	−1435.68	−1.44872	−0.724361	−	0.689421i	$-0.757865\pi$
$991$	−1435.68	−1.44872	−0.724361	+	0.689421i	$0.757865\pi$
$992$	158.655i	0.159934i
$993$	0	0
$994$	0	0
$995$	− 129.108i	− 0.129757i
$996$	0	0
$997$	−286.690	−0.287552	−0.143776	−	0.989610i	$-0.545925\pi$
$997$	−286.690	−0.287552	−0.143776	+	0.989610i	$0.545925\pi$
$998$	− 1432.63i	− 1.43550i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.3.b.e.197.6	yes	8
3.2	odd	2	inner	441.3.b.e.197.3	✓	8
7.2	even	3		441.3.q.e.116.6		16
7.3	odd	6		441.3.q.e.422.4		16
7.4	even	3		441.3.q.e.422.3		16
7.5	odd	6		441.3.q.e.116.5		16
7.6	odd	2	inner	441.3.b.e.197.5	yes	8
21.2	odd	6		441.3.q.e.116.3		16
21.5	even	6		441.3.q.e.116.4		16
21.11	odd	6		441.3.q.e.422.6		16
21.17	even	6		441.3.q.e.422.5		16
21.20	even	2	inner	441.3.b.e.197.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.3.b.e.197.3	✓	8	3.2	odd	2	inner
441.3.b.e.197.4	yes	8	21.20	even	2	inner
441.3.b.e.197.5	yes	8	7.6	odd	2	inner
441.3.b.e.197.6	yes	8	1.1	even	1	trivial
441.3.q.e.116.3		16	21.2	odd	6
441.3.q.e.116.4		16	21.5	even	6
441.3.q.e.116.5		16	7.5	odd	6
441.3.q.e.116.6		16	7.2	even	3
441.3.q.e.422.3		16	7.4	even	3
441.3.q.e.422.4		16	7.3	odd	6
441.3.q.e.422.5		16	21.17	even	6
441.3.q.e.422.6		16	21.11	odd	6

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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	441.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.63810i q^{2} +1.31662 q^{4} +1.68338i q^{5} +8.70917i q^{8} +O(q^{10})\)	\(q+1.63810i q^{2} +1.31662 q^{4} +1.68338i q^{5} +8.70917i q^{8} -2.75754 q^{10} +8.41439i q^{11} +20.1759 q^{13} -9.00000 q^{16} -12.3166i q^{17} -32.4560 q^{19} +2.21637i q^{20} -13.7836 q^{22} +24.4894i q^{23} +22.1662 q^{25} +33.0501i q^{26} +46.0795i q^{29} +7.89573 q^{31} +20.0938i q^{32} +20.1759 q^{34} +3.89975 q^{37} -53.1662i q^{38} -14.6608 q^{40} -54.3166i q^{41} -24.6332 q^{43} +11.0786i q^{44} -40.1161 q^{46} +77.7995i q^{47} +36.3106i q^{50} +26.5641 q^{52} -35.3553i q^{53} -14.1646 q^{55} -75.4829 q^{58} +64.3325i q^{59} -47.1168 q^{61} +12.9340i q^{62} -68.9156 q^{64} +33.9636i q^{65} -33.1662 q^{67} -16.2164i q^{68} -46.0795i q^{71} +83.4610 q^{73} +6.38818i q^{74} -42.7324 q^{76} +39.2665 q^{79} -15.1504i q^{80} +88.9761 q^{82} -138.765i q^{83} +20.7335 q^{85} -40.3518i q^{86} -73.2824 q^{88} -24.0159i q^{89} +32.2434i q^{92} -127.443 q^{94} -54.6357i q^{95} -32.0791 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4}+O(q^{10}) \) 8 * q - 16 * q^4	\( 8 q - 16 q^{4} - 72 q^{16} - 296 q^{22} - 88 q^{25} - 128 q^{37} - 144 q^{43} + 24 q^{46} - 312 q^{58} + 112 q^{64} + 208 q^{79} + 272 q^{85} + 24 q^{88}+O(q^{100}) \) 8 * q - 16 * q^4 - 72 * q^16 - 296 * q^22 - 88 * q^25 - 128 * q^37 - 144 * q^43 + 24 * q^46 - 312 * q^58 + 112 * q^64 + 208 * q^79 + 272 * q^85 + 24 * q^88

Introduction

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