LMFDB - Embedded newform 441.6.c.b.440.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,Mod(440,441)]

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, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("441.440");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	441.c (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:	$70.7292645375$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			440.3
Character	$\chi$	$=$	441.440
Dual form			441.6.c.b.440.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.9584i q^{2} -88.0874 q^{4} -82.6125 q^{5} +614.630i q^{8} +O(q^{10})$	$q-10.9584i q^{2} -88.0874 q^{4} -82.6125 q^{5} +614.630i q^{8} +905.305i q^{10} +113.816i q^{11} -329.288i q^{13} +3916.59 q^{16} +443.202 q^{17} +1679.45i q^{19} +7277.12 q^{20} +1247.24 q^{22} +224.060i q^{23} +3699.83 q^{25} -3608.48 q^{26} -2982.54i q^{29} +3376.55i q^{31} -23251.6i q^{32} -4856.80i q^{34} -11718.0 q^{37} +18404.1 q^{38} -50776.2i q^{40} +14554.2 q^{41} -10531.9 q^{43} -10025.7i q^{44} +2455.34 q^{46} +22092.2 q^{47} -40544.4i q^{50} +29006.1i q^{52} +4551.13i q^{53} -9402.59i q^{55} -32684.0 q^{58} -13333.2 q^{59} +50774.0i q^{61} +37001.8 q^{62} -129470. q^{64} +27203.3i q^{65} -51318.8 q^{67} -39040.5 q^{68} +1201.47i q^{71} +8835.52i q^{73} +128411. i q^{74} -147938. i q^{76} +43560.7 q^{79} -323560. q^{80} -159491. i q^{82} -58191.3 q^{83} -36614.0 q^{85} +115413. i q^{86} -69954.5 q^{88} +108510. q^{89} -19736.8i q^{92} -242095. i q^{94} -138743. i q^{95} +5444.51i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 608 q^{4}+O(q^{10}) $ 24 * q - 608 * q^4	$ 24 q - 608 q^{4} + 7288 q^{16} - 5648 q^{22} + 37704 q^{25} - 41096 q^{37} + 2200 q^{43} + 51424 q^{46} - 308600 q^{58} - 327880 q^{64} - 312648 q^{67} - 331512 q^{79} - 284448 q^{85} - 1164616 q^{88}+O(q^{100}) $ 24 * q - 608 * q^4 + 7288 * q^16 - 5648 * q^22 + 37704 * q^25 - 41096 * q^37 + 2200 * q^43 + 51424 * q^46 - 308600 * q^58 - 327880 * q^64 - 312648 * q^67 - 331512 * q^79 - 284448 * q^85 - 1164616 * 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 $

$2$	− 10.9584i	− 1.93720i	−0.248631	−	0.968598i	$-0.579981\pi$
$2$	− 10.9584i	− 1.93720i	0.248631	−	0.968598i	$-0.420019\pi$
$3$	0	0
$3$	0	0
$4$	−88.0874	−2.75273
$5$	−82.6125	−1.47782	−0.738909	−	0.673805i	$-0.764659\pi$
$5$	−82.6125	−1.47782	−0.738909	+	0.673805i	$0.764659\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	614.630i	3.39538i
$9$	0	0
$10$	905.305i	2.86282i
$11$	113.816i	0.283609i	0.989895	+	0.141804i	$0.0452904\pi$
$11$	113.816i	0.283609i	−0.989895	+	0.141804i	$0.954710\pi$
$12$	0	0
$13$	− 329.288i	− 0.540403i	−0.962804	−	0.270201i	$-0.912910\pi$
$13$	− 329.288i	− 0.540403i	0.962804	−	0.270201i	$-0.0870903\pi$
$14$	0	0
$15$	0	0
$16$	3916.59	3.82480
$17$	443.202	0.371946	0.185973	−	0.982555i	$-0.440456\pi$
$17$	443.202	0.371946	0.185973	+	0.982555i	$0.440456\pi$
$18$	0	0
$19$	1679.45i	1.06729i	0.845709	+	0.533644i	$0.179178\pi$
$19$	1679.45i	1.06729i	−0.845709	+	0.533644i	$0.820822\pi$
$20$	7277.12	4.06804
$21$	0	0
$22$	1247.24	0.549406
$23$	224.060i	0.0883169i	0.999025	+	0.0441584i	$0.0140606\pi$
$23$	224.060i	0.0883169i	−0.999025	+	0.0441584i	$0.985939\pi$
$24$	0	0
$25$	3699.83	1.18395
$26$	−3608.48	−1.04687
$27$	0	0
$28$	0	0
$29$	− 2982.54i	− 0.658554i	−0.944233	−	0.329277i	$-0.893195\pi$
$29$	− 2982.54i	− 0.658554i	0.944233	−	0.329277i	$-0.106805\pi$
$30$	0	0
$31$	3376.55i	0.631058i	0.948916	+	0.315529i	$0.102182\pi$
$31$	3376.55i	0.631058i	−0.948916	+	0.315529i	$0.897818\pi$
$32$	− 23251.6i	− 4.01400i
$33$	0	0
$34$	− 4856.80i	− 0.720532i
$35$	0	0
$36$	0	0
$37$	−11718.0	−1.40718	−0.703588	−	0.710608i	$-0.748420\pi$
$37$	−11718.0	−1.40718	−0.703588	+	0.710608i	$0.748420\pi$
$38$	18404.1	2.06755
$39$	0	0
$40$	− 50776.2i	− 5.01776i
$41$	14554.2	1.35216	0.676079	−	0.736829i	$-0.263678\pi$
$41$	14554.2	1.35216	0.676079	+	0.736829i	$0.263678\pi$
$42$	0	0
$43$	−10531.9	−0.868629	−0.434315	−	0.900761i	$-0.643009\pi$
$43$	−10531.9	−0.868629	−0.434315	+	0.900761i	$0.643009\pi$
$44$	− 10025.7i	− 0.780699i
$45$	0	0
$46$	2455.34	0.171087
$47$	22092.2	1.45879	0.729396	−	0.684092i	$-0.239801\pi$
$47$	22092.2	1.45879	0.729396	+	0.684092i	$0.239801\pi$
$48$	0	0
$49$	0	0
$50$	− 40544.4i	− 2.29354i
$51$	0	0
$52$	29006.1i	1.48758i
$53$	4551.13i	0.222551i	0.993790	+	0.111275i	$0.0354936\pi$
$53$	4551.13i	0.222551i	−0.993790	+	0.111275i	$0.964506\pi$
$54$	0	0
$55$	− 9402.59i	− 0.419122i
$56$	0	0
$57$	0	0
$58$	−32684.0	−1.27575
$59$	−13333.2	−0.498658	−0.249329	−	0.968419i	$-0.580210\pi$
$59$	−13333.2	−0.498658	−0.249329	+	0.968419i	$0.580210\pi$
$60$	0	0
$61$	50774.0i	1.74710i	0.486738	+	0.873548i	$0.338187\pi$
$61$	50774.0i	1.74710i	−0.486738	+	0.873548i	$0.661813\pi$
$62$	37001.8	1.22248
$63$	0	0
$64$	−129470.	−3.95111
$65$	27203.3i	0.798617i
$66$	0	0
$67$	−51318.8	−1.39666	−0.698328	−	0.715778i	$-0.746072\pi$
$67$	−51318.8	−1.39666	−0.698328	+	0.715778i	$0.746072\pi$
$68$	−39040.5	−1.02387
$69$	0	0
$70$	0	0
$71$	1201.47i	0.0282858i	0.999900	+	0.0141429i	$0.00450198\pi$
$71$	1201.47i	0.0282858i	−0.999900	+	0.0141429i	$0.995498\pi$
$72$	0	0
$73$	8835.52i	0.194055i	0.995282	+	0.0970276i	$0.0309335\pi$
$73$	8835.52i	0.194055i	−0.995282	+	0.0970276i	$0.969066\pi$
$74$	128411.i	2.72598i
$75$	0	0
$76$	− 147938.i	− 2.93796i
$77$	0	0
$78$	0	0
$79$	43560.7	0.785285	0.392642	−	0.919691i	$-0.371561\pi$
$79$	43560.7	0.785285	0.392642	+	0.919691i	$0.371561\pi$
$80$	−323560.	−5.65235
$81$	0	0
$82$	− 159491.i	− 2.61940i
$83$	−58191.3	−0.927177	−0.463589	−	0.886050i	$-0.653438\pi$
$83$	−58191.3	−0.927177	−0.463589	+	0.886050i	$0.653438\pi$
$84$	0	0
$85$	−36614.0	−0.549668
$86$	115413.i	1.68271i
$87$	0	0
$88$	−69954.5	−0.962961
$89$	108510.	1.45210	0.726049	−	0.687643i	$-0.241355\pi$
$89$	108510.	1.45210	0.726049	+	0.687643i	$0.241355\pi$
$90$	0	0
$91$	0	0
$92$	− 19736.8i	− 0.243113i
$93$	0	0
$94$	− 242095.i	− 2.82597i
$95$	− 138743.i	− 1.57726i
$96$	0	0
$97$	5444.51i	0.0587529i	0.999568	+	0.0293765i	$0.00935217\pi$
$97$	5444.51i	0.0587529i	−0.999568	+	0.0293765i	$0.990648\pi$
$98$	0	0
$99$	0	0
$100$	−325909.	−3.25909
$101$	−33070.4	−0.322579	−0.161289	−	0.986907i	$-0.551565\pi$
$101$	−33070.4	−0.322579	−0.161289	+	0.986907i	$0.551565\pi$
$102$	0	0
$103$	− 39040.8i	− 0.362598i	−0.983428	−	0.181299i	$-0.941970\pi$
$103$	− 39040.8i	− 0.362598i	0.983428	−	0.181299i	$-0.0580302\pi$
$104$	202390.	1.83488
$105$	0	0
$106$	49873.2	0.431125
$107$	− 126008.i	− 1.06400i	−0.846745	−	0.531998i	$-0.821441\pi$
$107$	− 126008.i	− 1.06400i	0.846745	−	0.531998i	$-0.178559\pi$
$108$	0	0
$109$	−134702.	−1.08595	−0.542974	−	0.839750i	$-0.682702\pi$
$109$	−134702.	−1.08595	−0.542974	+	0.839750i	$0.682702\pi$
$110$	−103038.	−0.811922
$111$	0	0
$112$	0	0
$113$	− 22220.5i	− 0.163703i	−0.996645	−	0.0818517i	$-0.973917\pi$
$113$	− 22220.5i	− 0.163703i	0.996645	−	0.0818517i	$-0.0260834\pi$
$114$	0	0
$115$	− 18510.1i	− 0.130516i
$116$	262724.i	1.81282i
$117$	0	0
$118$	146111.i	0.966000i
$119$	0	0
$120$	0	0
$121$	148097.	0.919566
$122$	556404.	3.38447
$123$	0	0
$124$	− 297432.i	− 1.73713i
$125$	−47488.4	−0.271840
$126$	0	0
$127$	−167321.	−0.920537	−0.460268	−	0.887780i	$-0.652247\pi$
$127$	−167321.	−0.920537	−0.460268	+	0.887780i	$0.652247\pi$
$128$	674738.i	3.64007i
$129$	0	0
$130$	298106.	1.54708
$131$	−347310.	−1.76823	−0.884117	−	0.467266i	$-0.845239\pi$
$131$	−347310.	−1.76823	−0.884117	+	0.467266i	$0.845239\pi$
$132$	0	0
$133$	0	0
$134$	562374.i	2.70560i
$135$	0	0
$136$	272405.i	1.26290i
$137$	− 369057.i	− 1.67993i	−0.542639	−	0.839966i	$-0.682575\pi$
$137$	− 369057.i	− 1.67993i	0.542639	−	0.839966i	$-0.317425\pi$
$138$	0	0
$139$	− 125256.i	− 0.549874i	−0.961462	−	0.274937i	$-0.911343\pi$
$139$	− 125256.i	− 0.549874i	0.961462	−	0.274937i	$-0.0886570\pi$
$140$	0	0
$141$	0	0
$142$	13166.3	0.0547952
$143$	37478.1	0.153263
$144$	0	0
$145$	246395.i	0.973224i
$146$	96823.6	0.375923
$147$	0	0
$148$	1.03221e6	3.87358
$149$	− 214190.i	− 0.790375i	−0.918600	−	0.395188i	$-0.870680\pi$
$149$	− 214190.i	− 0.790375i	0.918600	−	0.395188i	$-0.129320\pi$
$150$	0	0
$151$	152747.	0.545169	0.272584	−	0.962132i	$-0.412122\pi$
$151$	152747.	0.545169	0.272584	+	0.962132i	$0.412122\pi$
$152$	−1.03224e6	−3.62386
$153$	0	0
$154$	0	0
$155$	− 278946.i	− 0.932589i
$156$	0	0
$157$	116865.i	0.378386i	0.981940	+	0.189193i	$0.0605872\pi$
$157$	116865.i	0.378386i	−0.981940	+	0.189193i	$0.939413\pi$
$158$	− 477357.i	− 1.52125i
$159$	0	0
$160$	1.92087e6i	5.93196i
$161$	0	0
$162$	0	0
$163$	78585.0	0.231670	0.115835	−	0.993268i	$-0.463046\pi$
$163$	78585.0	0.231670	0.115835	+	0.993268i	$0.463046\pi$
$164$	−1.28204e6	−3.72213
$165$	0	0
$166$	637686.i	1.79613i
$167$	690721.	1.91651	0.958256	−	0.285911i	$-0.0922961\pi$
$167$	690721.	1.91651	0.958256	+	0.285911i	$0.0922961\pi$
$168$	0	0
$169$	262862.	0.707965
$170$	401233.i	1.06481i
$171$	0	0
$172$	927725.	2.39110
$173$	−177129.	−0.449960	−0.224980	−	0.974363i	$-0.572232\pi$
$173$	−177129.	−0.449960	−0.224980	+	0.974363i	$0.572232\pi$
$174$	0	0
$175$	0	0
$176$	445769.i	1.08475i
$177$	0	0
$178$	− 1.18910e6i	− 2.81300i
$179$	337632.i	0.787611i	0.919194	+	0.393805i	$0.128842\pi$
$179$	337632.i	0.787611i	−0.919194	+	0.393805i	$0.871158\pi$
$180$	0	0
$181$	− 514756.i	− 1.16790i	−0.811790	−	0.583949i	$-0.801507\pi$
$181$	− 514756.i	− 1.16790i	0.811790	−	0.583949i	$-0.198493\pi$
$182$	0	0
$183$	0	0
$184$	−137714.	−0.299870
$185$	968052.	2.07955
$186$	0	0
$187$	50443.3i	0.105487i
$188$	−1.94604e6	−4.01566
$189$	0	0
$190$	−1.52041e6	−3.05546
$191$	− 102194.i	− 0.202694i	−0.994851	−	0.101347i	$-0.967685\pi$
$191$	− 102194.i	− 0.202694i	0.994851	−	0.101347i	$-0.0323153\pi$
$192$	0	0
$193$	−709505.	−1.37108	−0.685539	−	0.728036i	$-0.740433\pi$
$193$	−709505.	−1.37108	−0.685539	+	0.728036i	$0.740433\pi$
$194$	59663.4	0.113816
$195$	0	0
$196$	0	0
$197$	86388.0i	0.158594i	0.996851	+	0.0792972i	$0.0252676\pi$
$197$	86388.0i	0.158594i	−0.996851	+	0.0792972i	$0.974732\pi$
$198$	0	0
$199$	− 721638.i	− 1.29178i	−0.763432	−	0.645888i	$-0.776487\pi$
$199$	− 721638.i	− 1.29178i	0.763432	−	0.645888i	$-0.223513\pi$
$200$	2.27403e6i	4.01995i
$201$	0	0
$202$	362400.i	0.624898i
$203$	0	0
$204$	0	0
$205$	−1.20236e6	−1.99824
$206$	−427826.	−0.702424
$207$	0	0
$208$	− 1.28969e6i	− 2.06693i
$209$	−191147.	−0.302693
$210$	0	0
$211$	381389.	0.589743	0.294871	−	0.955537i	$-0.404723\pi$
$211$	381389.	0.589743	0.294871	+	0.955537i	$0.404723\pi$
$212$	− 400897.i	− 0.612623i
$213$	0	0
$214$	−1.38086e6	−2.06117
$215$	870065.	1.28368
$216$	0	0
$217$	0	0
$218$	1.47613e6i	2.10369i
$219$	0	0
$220$	828250.i	1.15373i
$221$	− 145941.i	− 0.201000i
$222$	0	0
$223$	273465.i	0.368247i	0.982903	+	0.184124i	$0.0589446\pi$
$223$	273465.i	0.368247i	−0.982903	+	0.184124i	$0.941055\pi$
$224$	0	0
$225$	0	0
$226$	−243502.	−0.317126
$227$	159539.	0.205496	0.102748	−	0.994707i	$-0.467236\pi$
$227$	159539.	0.205496	0.102748	+	0.994707i	$0.467236\pi$
$228$	0	0
$229$	1.02707e6i	1.29423i	0.762392	+	0.647116i	$0.224025\pi$
$229$	1.02707e6i	1.29423i	−0.762392	+	0.647116i	$0.775975\pi$
$230$	−202842.	−0.252836
$231$	0	0
$232$	1.83316e6	2.23605
$233$	595553.i	0.718672i	0.933208	+	0.359336i	$0.116997\pi$
$233$	595553.i	0.718672i	−0.933208	+	0.359336i	$0.883003\pi$
$234$	0	0
$235$	−1.82509e6	−2.15583
$236$	1.17448e6	1.37267
$237$	0	0
$238$	0	0
$239$	− 1.34378e6i	− 1.52171i	−0.648921	−	0.760856i	$-0.724780\pi$
$239$	− 1.34378e6i	− 1.52171i	0.648921	−	0.760856i	$-0.275220\pi$
$240$	0	0
$241$	− 1.61297e6i	− 1.78890i	−0.447173	−	0.894448i	$-0.647569\pi$
$241$	− 1.61297e6i	− 1.78890i	0.447173	−	0.894448i	$-0.352431\pi$
$242$	− 1.62291e6i	− 1.78138i
$243$	0	0
$244$	− 4.47255e6i	− 4.80929i
$245$	0	0
$246$	0	0
$247$	553021.	0.576766
$248$	−2.07533e6	−2.14269
$249$	0	0
$250$	520399.i	0.526607i
$251$	1.19624e6	1.19849	0.599244	−	0.800567i	$-0.295468\pi$
$251$	1.19624e6	1.19849	0.599244	+	0.800567i	$0.295468\pi$
$252$	0	0
$253$	−25501.5	−0.0250475
$254$	1.83358e6i	1.78326i
$255$	0	0
$256$	3.25104e6	3.10043
$257$	−125892.	−0.118895	−0.0594476	−	0.998231i	$-0.518934\pi$
$257$	−125892.	−0.118895	−0.0594476	+	0.998231i	$0.518934\pi$
$258$	0	0
$259$	0	0
$260$	− 2.39627e6i	− 2.19838i
$261$	0	0
$262$	3.80598e6i	3.42542i
$263$	− 31296.4i	− 0.0279000i	−0.999903	−	0.0139500i	$-0.995559\pi$
$263$	− 31296.4i	− 0.0279000i	0.999903	−	0.0139500i	$-0.00444057\pi$
$264$	0	0
$265$	− 375980.i	− 0.328890i
$266$	0	0
$267$	0	0
$268$	4.52054e6	3.84462
$269$	220143.	0.185492	0.0927459	−	0.995690i	$-0.470436\pi$
$269$	220143.	0.185492	0.0927459	+	0.995690i	$0.470436\pi$
$270$	0	0
$271$	− 1.16738e6i	− 0.965581i	−0.875736	−	0.482790i	$-0.839623\pi$
$271$	− 1.16738e6i	− 0.965581i	0.875736	−	0.482790i	$-0.160377\pi$
$272$	1.73584e6	1.42262
$273$	0	0
$274$	−4.04429e6	−3.25436
$275$	421098.i	0.335778i
$276$	0	0
$277$	819160.	0.641460	0.320730	−	0.947171i	$-0.396072\pi$
$277$	819160.	0.641460	0.320730	+	0.947171i	$0.396072\pi$
$278$	−1.37262e6	−1.06521
$279$	0	0
$280$	0	0
$281$	658409.i	0.497428i	0.968577	+	0.248714i	$0.0800079\pi$
$281$	658409.i	0.497428i	−0.968577	+	0.248714i	$0.919992\pi$
$282$	0	0
$283$	− 1.78291e6i	− 1.32332i	−0.749805	−	0.661659i	$-0.769853\pi$
$283$	− 1.78291e6i	− 1.32332i	0.749805	−	0.661659i	$-0.230147\pi$
$284$	− 105835.i	− 0.0778632i
$285$	0	0
$286$	− 410701.i	− 0.296901i
$287$	0	0
$288$	0	0
$289$	−1.22343e6	−0.861657
$290$	2.70011e6	1.88533
$291$	0	0
$292$	− 778298.i	− 0.534182i
$293$	693145.	0.471689	0.235844	−	0.971791i	$-0.424214\pi$
$293$	693145.	0.471689	0.235844	+	0.971791i	$0.424214\pi$
$294$	0	0
$295$	1.10149e6	0.736927
$296$	− 7.20222e6i	− 4.77790i
$297$	0	0
$298$	−2.34719e6	−1.53111
$299$	73780.1	0.0477267
$300$	0	0
$301$	0	0
$302$	− 1.67387e6i	− 1.05610i
$303$	0	0
$304$	6.57770e6i	4.08216i
$305$	− 4.19457e6i	− 2.58189i
$306$	0	0
$307$	− 1.35379e6i	− 0.819793i	−0.912132	−	0.409897i	$-0.865565\pi$
$307$	− 1.35379e6i	− 0.819793i	0.912132	−	0.409897i	$-0.134435\pi$
$308$	0	0
$309$	0	0
$310$	−3.05681e6	−1.80661
$311$	1.73952e6	1.01983	0.509915	−	0.860225i	$-0.329677\pi$
$311$	1.73952e6	1.01983	0.509915	+	0.860225i	$0.329677\pi$
$312$	0	0
$313$	1.98990e6i	1.14808i	0.818829	+	0.574038i	$0.194624\pi$
$313$	1.98990e6i	1.14808i	−0.818829	+	0.574038i	$0.805376\pi$
$314$	1.28066e6	0.733008
$315$	0	0
$316$	−3.83715e6	−2.16168
$317$	− 2.70064e6i	− 1.50945i	−0.656043	−	0.754723i	$-0.727771\pi$
$317$	− 2.70064e6i	− 1.50945i	0.656043	−	0.754723i	$-0.272229\pi$
$318$	0	0
$319$	339460.	0.186772
$320$	1.06958e7	5.83902
$321$	0	0
$322$	0	0
$323$	744334.i	0.396973i
$324$	0	0
$325$	− 1.21831e6i	− 0.639808i
$326$	− 861169.i	− 0.448791i
$327$	0	0
$328$	8.94543e6i	4.59110i
$329$	0	0
$330$	0	0
$331$	559430.	0.280657	0.140328	−	0.990105i	$-0.455184\pi$
$331$	559430.	0.280657	0.140328	+	0.990105i	$0.455184\pi$
$332$	5.12592e6	2.55227
$333$	0	0
$334$	− 7.56923e6i	− 3.71266i
$335$	4.23958e6	2.06400
$336$	0	0
$337$	−967665.	−0.464141	−0.232071	−	0.972699i	$-0.574550\pi$
$337$	−967665.	−0.464141	−0.232071	+	0.972699i	$0.574550\pi$
$338$	− 2.88056e6i	− 1.37147i
$339$	0	0
$340$	3.22524e6	1.51309
$341$	−384304.	−0.178974
$342$	0	0
$343$	0	0
$344$	− 6.47321e6i	− 2.94933i
$345$	0	0
$346$	1.94106e6i	0.871662i
$347$	2.22746e6i	0.993085i	0.868012	+	0.496543i	$0.165397\pi$
$347$	2.22746e6i	0.993085i	−0.868012	+	0.496543i	$0.834603\pi$
$348$	0	0
$349$	− 1.98330e6i	− 0.871617i	−0.900040	−	0.435808i	$-0.856463\pi$
$349$	− 1.98330e6i	− 0.871617i	0.900040	−	0.435808i	$-0.143537\pi$
$350$	0	0
$351$	0	0
$352$	2.64639e6	1.13841
$353$	−1.65520e6	−0.706992	−0.353496	−	0.935436i	$-0.615007\pi$
$353$	−1.65520e6	−0.706992	−0.353496	+	0.935436i	$0.615007\pi$
$354$	0	0
$355$	− 99256.9i	− 0.0418013i
$356$	−9.55839e6	−3.99723
$357$	0	0
$358$	3.69992e6	1.52576
$359$	− 2.36035e6i	− 0.966587i	−0.875458	−	0.483294i	$-0.839440\pi$
$359$	− 2.36035e6i	− 0.966587i	0.875458	−	0.483294i	$-0.160560\pi$
$360$	0	0
$361$	−344439.	−0.139105
$362$	−5.64092e6	−2.26245
$363$	0	0
$364$	0	0
$365$	− 729925.i	− 0.286778i
$366$	0	0
$367$	− 3.60861e6i	− 1.39854i	−0.714858	−	0.699269i	$-0.753509\pi$
$367$	− 3.60861e6i	− 1.39854i	0.714858	−	0.699269i	$-0.246491\pi$
$368$	877550.i	0.337794i
$369$	0	0
$370$	− 1.06083e7i	− 4.02850i
$371$	0	0
$372$	0	0
$373$	−1.53621e6	−0.571713	−0.285856	−	0.958272i	$-0.592278\pi$
$373$	−1.53621e6	−0.571713	−0.285856	+	0.958272i	$0.592278\pi$
$374$	552780.	0.204349
$375$	0	0
$376$	1.35785e7i	4.95316i
$377$	−982116.	−0.355885
$378$	0	0
$379$	−4.61784e6	−1.65136	−0.825678	−	0.564141i	$-0.809207\pi$
$379$	−4.61784e6	−1.65136	−0.825678	+	0.564141i	$0.809207\pi$
$380$	1.22215e7i	4.34177i
$381$	0	0
$382$	−1.11989e6	−0.392659
$383$	−2.83027e6	−0.985896	−0.492948	−	0.870059i	$-0.664081\pi$
$383$	−2.83027e6	−0.985896	−0.492948	+	0.870059i	$0.664081\pi$
$384$	0	0
$385$	0	0
$386$	7.77507e6i	2.65605i
$387$	0	0
$388$	− 479593.i	− 0.161731i
$389$	− 5.05272e6i	− 1.69298i	−0.532408	−	0.846488i	$-0.678713\pi$
$389$	− 5.05272e6i	− 1.69298i	0.532408	−	0.846488i	$-0.321287\pi$
$390$	0	0
$391$	99303.6i	0.0328491i
$392$	0	0
$393$	0	0
$394$	946677.	0.307228
$395$	−3.59866e6	−1.16051
$396$	0	0
$397$	4.60555e6i	1.46658i	0.679917	+	0.733289i	$0.262016\pi$
$397$	4.60555e6i	1.46658i	−0.679917	+	0.733289i	$0.737984\pi$
$398$	−7.90803e6	−2.50242
$399$	0	0
$400$	1.44907e7	4.52835
$401$	1.91615e6i	0.595071i	0.954711	+	0.297535i	$0.0961646\pi$
$401$	1.91615e6i	0.595071i	−0.954711	+	0.297535i	$0.903835\pi$
$402$	0	0
$403$	1.11186e6	0.341026
$404$	2.91308e6	0.887973
$405$	0	0
$406$	0	0
$407$	− 1.33369e6i	− 0.399088i
$408$	0	0
$409$	2.48053e6i	0.733224i	0.930374	+	0.366612i	$0.119482\pi$
$409$	2.48053e6i	0.733224i	−0.930374	+	0.366612i	$0.880518\pi$
$410$	1.31760e7i	3.87099i
$411$	0	0
$412$	3.43900e6i	0.998135i
$413$	0	0
$414$	0	0
$415$	4.80733e6	1.37020
$416$	−7.65646e6	−2.16918
$417$	0	0
$418$	2.09467e6i	0.586375i
$419$	1.81159e6	0.504110	0.252055	−	0.967713i	$-0.418894\pi$
$419$	1.81159e6	0.504110	0.252055	+	0.967713i	$0.418894\pi$
$420$	0	0
$421$	6.66082e6	1.83157	0.915783	−	0.401673i	$-0.131571\pi$
$421$	6.66082e6	1.83157	0.915783	+	0.401673i	$0.131571\pi$
$422$	− 4.17943e6i	− 1.14245i
$423$	0	0
$424$	−2.79726e6	−0.755646
$425$	1.63977e6	0.440364
$426$	0	0
$427$	0	0
$428$	1.10998e7i	2.92890i
$429$	0	0
$430$	− 9.53455e6i	− 2.48673i
$431$	1.50214e6i	0.389509i	0.980852	+	0.194754i	$0.0623910\pi$
$431$	1.50214e6i	0.389509i	−0.980852	+	0.194754i	$0.937609\pi$
$432$	0	0
$433$	5.98390e6i	1.53379i	0.641775	+	0.766893i	$0.278198\pi$
$433$	5.98390e6i	1.53379i	−0.641775	+	0.766893i	$0.721802\pi$
$434$	0	0
$435$	0	0
$436$	1.18656e7	2.98932
$437$	−376296.	−0.0942596
$438$	0	0
$439$	− 4.20914e6i	− 1.04240i	−0.853436	−	0.521198i	$-0.825485\pi$
$439$	− 4.20914e6i	− 1.04240i	0.853436	−	0.521198i	$-0.174515\pi$
$440$	5.77912e6	1.42308
$441$	0	0
$442$	−1.59929e6	−0.389377
$443$	4.94962e6i	1.19829i	0.800640	+	0.599146i	$0.204493\pi$
$443$	4.94962e6i	1.19829i	−0.800640	+	0.599146i	$0.795507\pi$
$444$	0	0
$445$	−8.96431e6	−2.14594
$446$	2.99675e6	0.713367
$447$	0	0
$448$	0	0
$449$	− 4.75341e6i	− 1.11273i	−0.830939	−	0.556364i	$-0.812196\pi$
$449$	− 4.75341e6i	− 1.11273i	0.830939	−	0.556364i	$-0.187804\pi$
$450$	0	0
$451$	1.65649e6i	0.383484i
$452$	1.95735e6i	0.450631i
$453$	0	0
$454$	− 1.74830e6i	− 0.398086i
$455$	0	0
$456$	0	0
$457$	−7.61693e6	−1.70604	−0.853021	−	0.521877i	$-0.825232\pi$
$457$	−7.61693e6	−1.70604	−0.853021	+	0.521877i	$0.825232\pi$
$458$	1.12551e7	2.50718
$459$	0	0
$460$	1.63051e6i	0.359276i
$461$	−376986.	−0.0826176	−0.0413088	−	0.999146i	$-0.513153\pi$
$461$	−376986.	−0.0826176	−0.0413088	+	0.999146i	$0.513153\pi$
$462$	0	0
$463$	1.75954e6	0.381458	0.190729	−	0.981643i	$-0.438915\pi$
$463$	1.75954e6	0.381458	0.190729	+	0.981643i	$0.438915\pi$
$464$	− 1.16814e7i	− 2.51884i
$465$	0	0
$466$	6.52633e6	1.39221
$467$	8.39285e6	1.78081	0.890404	−	0.455171i	$-0.150422\pi$
$467$	8.39285e6	1.78081	0.890404	+	0.455171i	$0.150422\pi$
$468$	0	0
$469$	0	0
$470$	2.00001e7i	4.17627i
$471$	0	0
$472$	− 8.19497e6i	− 1.69314i
$473$	− 1.19869e6i	− 0.246351i
$474$	0	0
$475$	6.21367e6i	1.26361i
$476$	0	0
$477$	0	0
$478$	−1.47257e7	−2.94785
$479$	1.37288e6	0.273397	0.136699	−	0.990613i	$-0.456351\pi$
$479$	1.37288e6	0.273397	0.136699	+	0.990613i	$0.456351\pi$
$480$	0	0
$481$	3.85859e6i	0.760442i
$482$	−1.76757e7	−3.46544
$483$	0	0
$484$	−1.30455e7	−2.53132
$485$	− 449785.i	− 0.0868262i
$486$	0	0
$487$	−1.79435e6	−0.342835	−0.171417	−	0.985198i	$-0.554835\pi$
$487$	−1.79435e6	−0.342835	−0.171417	+	0.985198i	$0.554835\pi$
$488$	−3.12072e7	−5.93206
$489$	0	0
$490$	0	0
$491$	− 1.00791e7i	− 1.88677i	−0.331698	−	0.943386i	$-0.607622\pi$
$491$	− 1.00791e7i	− 1.88677i	0.331698	−	0.943386i	$-0.392378\pi$
$492$	0	0
$493$	− 1.32187e6i	− 0.244946i
$494$	− 6.06025e6i	− 1.11731i
$495$	0	0
$496$	1.32246e7i	2.41367i
$497$	0	0
$498$	0	0
$499$	6.26310e6	1.12600	0.562999	−	0.826457i	$-0.309647\pi$
$499$	6.26310e6	1.12600	0.562999	+	0.826457i	$0.309647\pi$
$500$	4.18313e6	0.748301
$501$	0	0
$502$	− 1.31089e7i	− 2.32171i
$503$	−598654.	−0.105501	−0.0527504	−	0.998608i	$-0.516799\pi$
$503$	−598654.	−0.105501	−0.0527504	+	0.998608i	$0.516799\pi$
$504$	0	0
$505$	2.73203e6	0.476713
$506$	279456.i	0.0485219i
$507$	0	0
$508$	1.47389e7	2.53399
$509$	2.79756e6	0.478614	0.239307	−	0.970944i	$-0.423080\pi$
$509$	2.79756e6	0.478614	0.239307	+	0.970944i	$0.423080\pi$
$510$	0	0
$511$	0	0
$512$	− 1.40347e7i	− 2.36607i
$513$	0	0
$514$	1.37958e6i	0.230323i
$515$	3.22526e6i	0.535854i
$516$	0	0
$517$	2.51443e6i	0.413726i
$518$	0	0
$519$	0	0
$520$	−1.67200e7	−2.71161
$521$	1.88515e6	0.304265	0.152132	−	0.988360i	$-0.451386\pi$
$521$	1.88515e6	0.304265	0.152132	+	0.988360i	$0.451386\pi$
$522$	0	0
$523$	1.51525e6i	0.242231i	0.992638	+	0.121116i	$0.0386471\pi$
$523$	1.51525e6i	0.242231i	−0.992638	+	0.121116i	$0.961353\pi$
$524$	3.05937e7	4.86747
$525$	0	0
$526$	−342959.	−0.0540478
$527$	1.49650e6i	0.234719i
$528$	0	0
$529$	6.38614e6	0.992200
$530$	−4.12016e6	−0.637124
$531$	0	0
$532$	0	0
$533$	− 4.79251e6i	− 0.730710i
$534$	0	0
$535$	1.04099e7i	1.57239i
$536$	− 3.15421e7i	− 4.74219i
$537$	0	0
$538$	− 2.41243e6i	− 0.359334i
$539$	0	0
$540$	0	0
$541$	8.10157e6	1.19008	0.595040	−	0.803696i	$-0.297136\pi$
$541$	8.10157e6	1.19008	0.595040	+	0.803696i	$0.297136\pi$
$542$	−1.27926e7	−1.87052
$543$	0	0
$544$	− 1.03051e7i	− 1.49299i
$545$	1.11281e7	1.60483
$546$	0	0
$547$	8.84662e6	1.26418	0.632091	−	0.774895i	$-0.282197\pi$
$547$	8.84662e6	1.26418	0.632091	+	0.774895i	$0.282197\pi$
$548$	3.25093e7i	4.62440i
$549$	0	0
$550$	4.61458e6	0.650468
$551$	5.00902e6	0.702868
$552$	0	0
$553$	0	0
$554$	− 8.97672e6i	− 1.24263i
$555$	0	0
$556$	1.10335e7i	1.51365i
$557$	6.34394e6i	0.866405i	0.901297	+	0.433202i	$0.142616\pi$
$557$	6.34394e6i	0.866405i	−0.901297	+	0.433202i	$0.857384\pi$
$558$	0	0
$559$	3.46802e6i	0.469410i
$560$	0	0
$561$	0	0
$562$	7.21514e6	0.963616
$563$	−2.48304e6	−0.330152	−0.165076	−	0.986281i	$-0.552787\pi$
$563$	−2.48304e6	−0.330152	−0.165076	+	0.986281i	$0.552787\pi$
$564$	0	0
$565$	1.83569e6i	0.241924i
$566$	−1.95380e7	−2.56353
$567$	0	0
$568$	−738462.	−0.0960412
$569$	− 2.41831e6i	− 0.313135i	−0.987667	−	0.156568i	$-0.949957\pi$
$569$	− 2.41831e6i	− 0.313135i	0.987667	−	0.156568i	$-0.0500429\pi$
$570$	0	0
$571$	8.75488e6	1.12372	0.561862	−	0.827231i	$-0.310085\pi$
$571$	8.75488e6	1.12372	0.561862	+	0.827231i	$0.310085\pi$
$572$	−3.30135e6	−0.421892
$573$	0	0
$574$	0	0
$575$	828983.i	0.104562i
$576$	0	0
$577$	1.18818e7i	1.48574i	0.669438	+	0.742868i	$0.266535\pi$
$577$	1.18818e7i	1.48574i	−0.669438	+	0.742868i	$0.733465\pi$
$578$	1.34069e7i	1.66920i
$579$	0	0
$580$	− 2.17043e7i	− 2.67902i
$581$	0	0
$582$	0	0
$583$	−517989.	−0.0631174
$584$	−5.43058e6	−0.658892
$585$	0	0
$586$	− 7.59579e6i	− 0.913753i
$587$	−760681.	−0.0911186	−0.0455593	−	0.998962i	$-0.514507\pi$
$587$	−760681.	−0.0911186	−0.0455593	+	0.998962i	$0.514507\pi$
$588$	0	0
$589$	−5.67074e6	−0.673521
$590$	− 1.20706e7i	− 1.42757i
$591$	0	0
$592$	−4.58945e7	−5.38216
$593$	−9.07711e6	−1.06001	−0.530006	−	0.847994i	$-0.677810\pi$
$593$	−9.07711e6	−1.06001	−0.530006	+	0.847994i	$0.677810\pi$
$594$	0	0
$595$	0	0
$596$	1.88674e7i	2.17569i
$597$	0	0
$598$	− 808515.i	− 0.0924560i
$599$	− 3.46168e6i	− 0.394203i	−0.980383	−	0.197102i	$-0.936847\pi$
$599$	− 3.46168e6i	− 0.394203i	0.980383	−	0.197102i	$-0.0631529\pi$
$600$	0	0
$601$	− 3.99158e6i	− 0.450773i	−0.974269	−	0.225387i	$-0.927635\pi$
$601$	− 3.99158e6i	− 0.450773i	0.974269	−	0.225387i	$-0.0723646\pi$
$602$	0	0
$603$	0	0
$604$	−1.34551e7	−1.50070
$605$	−1.22347e7	−1.35895
$606$	0	0
$607$	83599.9i	0.00920946i	0.999989	+	0.00460473i	$0.00146574\pi$
$607$	83599.9i	0.00920946i	−0.999989	+	0.00460473i	$0.998534\pi$
$608$	3.90497e7	4.28410
$609$	0	0
$610$	−4.59659e7	−5.00163
$611$	− 7.27468e6i	− 0.788335i
$612$	0	0
$613$	7.71789e6	0.829559	0.414779	−	0.909922i	$-0.363859\pi$
$613$	7.71789e6	0.829559	0.414779	+	0.909922i	$0.363859\pi$
$614$	−1.48354e7	−1.58810
$615$	0	0
$616$	0	0
$617$	6.96962e6i	0.737049i	0.929618	+	0.368524i	$0.120137\pi$
$617$	6.96962e6i	0.737049i	−0.929618	+	0.368524i	$0.879863\pi$
$618$	0	0
$619$	− 3.65723e6i	− 0.383642i	−0.981430	−	0.191821i	$-0.938561\pi$
$619$	− 3.65723e6i	− 0.383642i	0.981430	−	0.191821i	$-0.0614393\pi$
$620$	2.45716e7i	2.56717i
$621$	0	0
$622$	− 1.90624e7i	− 1.97561i
$623$	0	0
$624$	0	0
$625$	−7.63884e6	−0.782217
$626$	2.18062e7	2.22405
$627$	0	0
$628$	− 1.02943e7i	− 1.04159i
$629$	−5.19343e6	−0.523393
$630$	0	0
$631$	−1.06183e6	−0.106165	−0.0530824	−	0.998590i	$-0.516905\pi$
$631$	−1.06183e6	−0.106165	−0.0530824	+	0.998590i	$0.516905\pi$
$632$	2.67737e7i	2.66634i
$633$	0	0
$634$	−2.95948e7	−2.92410
$635$	1.38228e7	1.36039
$636$	0	0
$637$	0	0
$638$	− 3.71995e6i	− 0.361814i
$639$	0	0
$640$	− 5.57418e7i	− 5.37937i
$641$	7.36200e6i	0.707703i	0.935302	+	0.353851i	$0.115128\pi$
$641$	7.36200e6i	0.707703i	−0.935302	+	0.353851i	$0.884872\pi$
$642$	0	0
$643$	1.39847e6i	0.133391i	0.997773	+	0.0666953i	$0.0212456\pi$
$643$	1.39847e6i	0.133391i	−0.997773	+	0.0666953i	$0.978754\pi$
$644$	0	0
$645$	0	0
$646$	8.15673e6	0.769015
$647$	−1.17625e7	−1.10469	−0.552345	−	0.833615i	$-0.686267\pi$
$647$	−1.17625e7	−1.10469	−0.552345	+	0.833615i	$0.686267\pi$
$648$	0	0
$649$	− 1.51752e6i	− 0.141424i
$650$	−1.33508e7	−1.23943
$651$	0	0
$652$	−6.92235e6	−0.637726
$653$	− 1.53090e7i	− 1.40496i	−0.711703	−	0.702481i	$-0.752076\pi$
$653$	− 1.53090e7i	− 1.40496i	0.711703	−	0.702481i	$-0.247924\pi$
$654$	0	0
$655$	2.86922e7	2.61313
$656$	5.70027e7	5.17173
$657$	0	0
$658$	0	0
$659$	− 6.80696e6i	− 0.610576i	−0.952260	−	0.305288i	$-0.901247\pi$
$659$	− 6.80696e6i	− 0.610576i	0.952260	−	0.305288i	$-0.0987527\pi$
$660$	0	0
$661$	− 3.98992e6i	− 0.355190i	−0.984104	−	0.177595i	$-0.943168\pi$
$661$	− 3.98992e6i	− 0.355190i	0.984104	−	0.177595i	$-0.0568317\pi$
$662$	− 6.13048e6i	− 0.543687i
$663$	0	0
$664$	− 3.57661e7i	− 3.14812i
$665$	0	0
$666$	0	0
$667$	668267.	0.0581615
$668$	−6.08438e7	−5.27564
$669$	0	0
$670$	− 4.64592e7i	− 3.99838i
$671$	−5.77887e6	−0.495492
$672$	0	0
$673$	1.33787e7	1.13861	0.569307	−	0.822125i	$-0.307212\pi$
$673$	1.33787e7	1.13861	0.569307	+	0.822125i	$0.307212\pi$
$674$	1.06041e7i	0.899133i
$675$	0	0
$676$	−2.31549e7	−1.94884
$677$	5.09283e6	0.427058	0.213529	−	0.976937i	$-0.431504\pi$
$677$	5.09283e6	0.427058	0.213529	+	0.976937i	$0.431504\pi$
$678$	0	0
$679$	0	0
$680$	− 2.25041e7i	− 1.86633i
$681$	0	0
$682$	4.21138e6i	0.346707i
$683$	2.40047e6i	0.196900i	0.995142	+	0.0984498i	$0.0313884\pi$
$683$	2.40047e6i	0.196900i	−0.995142	+	0.0984498i	$0.968612\pi$
$684$	0	0
$685$	3.04887e7i	2.48264i
$686$	0	0
$687$	0	0
$688$	−4.12490e7	−3.32233
$689$	1.49863e6	0.120267
$690$	0	0
$691$	− 4.89530e6i	− 0.390017i	−0.980802	−	0.195009i	$-0.937526\pi$
$691$	− 4.89530e6i	− 0.390017i	0.980802	−	0.195009i	$-0.0624735\pi$
$692$	1.56028e7	1.23862
$693$	0	0
$694$	2.44095e7	1.92380
$695$	1.03478e7i	0.812614i
$696$	0	0
$697$	6.45043e6	0.502929
$698$	−2.17339e7	−1.68849
$699$	0	0
$700$	0	0
$701$	− 263433.i	− 0.0202477i	−0.999949	−	0.0101238i	$-0.996777\pi$
$701$	− 263433.i	− 0.0202477i	0.999949	−	0.0101238i	$-0.00322258\pi$
$702$	0	0
$703$	− 1.96797e7i	− 1.50186i
$704$	− 1.47357e7i	− 1.12057i
$705$	0	0
$706$	1.81384e7i	1.36958i
$707$	0	0
$708$	0	0
$709$	2.12818e7	1.58998	0.794992	−	0.606620i	$-0.207475\pi$
$709$	2.12818e7	1.58998	0.794992	+	0.606620i	$0.207475\pi$
$710$	−1.08770e6	−0.0809773
$711$	0	0
$712$	6.66937e7i	4.93043i
$713$	−756549.	−0.0557331
$714$	0	0
$715$	−3.09616e6	−0.226495
$716$	− 2.97412e7i	− 2.16808i
$717$	0	0
$718$	−2.58658e7	−1.87247
$719$	1.01607e7	0.732998	0.366499	−	0.930418i	$-0.380556\pi$
$719$	1.01607e7	0.732998	0.366499	+	0.930418i	$0.380556\pi$
$720$	0	0
$721$	0	0
$722$	3.77451e6i	0.269475i
$723$	0	0
$724$	4.53435e7i	3.21491i
$725$	− 1.10349e7i	− 0.779693i
$726$	0	0
$727$	− 1.76047e7i	− 1.23536i	−0.786431	−	0.617678i	$-0.788074\pi$
$727$	− 1.76047e7i	− 1.23536i	0.786431	−	0.617678i	$-0.211926\pi$
$728$	0	0
$729$	0	0
$730$	−7.99884e6	−0.555546
$731$	−4.66775e6	−0.323083
$732$	0	0
$733$	− 1.21859e7i	− 0.837717i	−0.908052	−	0.418858i	$-0.862430\pi$
$733$	− 1.21859e7i	− 0.837717i	0.908052	−	0.418858i	$-0.137570\pi$
$734$	−3.95447e7	−2.70924
$735$	0	0
$736$	5.20973e6	0.354504
$737$	− 5.84088e6i	− 0.396104i
$738$	0	0
$739$	6.39657e6	0.430860	0.215430	−	0.976519i	$-0.430885\pi$
$739$	6.39657e6	0.430860	0.215430	+	0.976519i	$0.430885\pi$
$740$	−8.52732e7	−5.72444
$741$	0	0
$742$	0	0
$743$	2.66345e7i	1.77000i	0.465592	+	0.885000i	$0.345842\pi$
$743$	2.66345e7i	1.77000i	−0.465592	+	0.885000i	$0.654158\pi$
$744$	0	0
$745$	1.76948e7i	1.16803i
$746$	1.68344e7i	1.10752i
$747$	0	0
$748$	− 4.44342e6i	− 0.290378i
$749$	0	0
$750$	0	0
$751$	5.89775e6	0.381581	0.190790	−	0.981631i	$-0.438895\pi$
$751$	5.89775e6	0.381581	0.190790	+	0.981631i	$0.438895\pi$
$752$	8.65259e7	5.57958
$753$	0	0
$754$	1.07625e7i	0.689419i
$755$	−1.26188e7	−0.805660
$756$	0	0
$757$	1.22047e7	0.774082	0.387041	−	0.922063i	$-0.373497\pi$
$757$	1.22047e7	0.774082	0.387041	+	0.922063i	$0.373497\pi$
$758$	5.06043e7i	3.19900i
$759$	0	0
$760$	8.52758e7	5.35540
$761$	2.03127e7	1.27147	0.635736	−	0.771907i	$-0.280697\pi$
$761$	2.03127e7	1.27147	0.635736	+	0.771907i	$0.280697\pi$
$762$	0	0
$763$	0	0
$764$	9.00199e6i	0.557963i
$765$	0	0
$766$	3.10154e7i	1.90987i
$767$	4.39045e6i	0.269476i
$768$	0	0
$769$	1.83510e7i	1.11904i	0.828819	+	0.559518i	$0.189014\pi$
$769$	1.83510e7i	1.11904i	−0.828819	+	0.559518i	$0.810986\pi$
$770$	0	0
$771$	0	0
$772$	6.24985e7	3.77421
$773$	−814254.	−0.0490130	−0.0245065	−	0.999700i	$-0.507801\pi$
$773$	−814254.	−0.0490130	−0.0245065	+	0.999700i	$0.507801\pi$
$774$	0	0
$775$	1.24927e7i	0.747139i
$776$	−3.34636e6	−0.199489
$777$	0	0
$778$	−5.53699e7	−3.27963
$779$	2.44429e7i	1.44314i
$780$	0	0
$781$	−136746.	−0.00802211
$782$	1.08821e6	0.0636351
$783$	0	0
$784$	0	0
$785$	− 9.65451e6i	− 0.559186i
$786$	0	0
$787$	1.88779e7i	1.08647i	0.839582	+	0.543234i	$0.182800\pi$
$787$	1.88779e7i	1.08647i	−0.839582	+	0.543234i	$0.817200\pi$
$788$	− 7.60969e6i	− 0.436567i
$789$	0	0
$790$	3.94357e7i	2.24813i
$791$	0	0
$792$	0	0
$793$	1.67193e7	0.944136
$794$	5.04697e7	2.84105
$795$	0	0
$796$	6.35672e7i	3.55591i
$797$	3.37595e7	1.88257	0.941284	−	0.337616i	$-0.109621\pi$
$797$	3.37595e7	1.88257	0.941284	+	0.337616i	$0.109621\pi$
$798$	0	0
$799$	9.79128e6	0.542591
$800$	− 8.60269e7i	− 4.75236i
$801$	0	0
$802$	2.09980e7	1.15277
$803$	−1.00562e6	−0.0550358
$804$	0	0
$805$	0	0
$806$	− 1.21842e7i	− 0.660634i
$807$	0	0
$808$	− 2.03261e7i	− 1.09528i
$809$	1.86364e7i	1.00113i	0.865699	+	0.500565i	$0.166875\pi$
$809$	1.86364e7i	1.00113i	−0.865699	+	0.500565i	$0.833125\pi$
$810$	0	0
$811$	2.29132e7i	1.22330i	0.791128	+	0.611651i	$0.209494\pi$
$811$	2.29132e7i	1.22330i	−0.791128	+	0.611651i	$0.790506\pi$
$812$	0	0
$813$	0	0
$814$	−1.46151e7	−0.773111
$815$	−6.49211e6	−0.342367
$816$	0	0
$817$	− 1.76877e7i	− 0.927078i
$818$	2.71828e7	1.42040
$819$	0	0
$820$	1.05912e8	5.50063
$821$	− 1.01179e7i	− 0.523883i	−0.965084	−	0.261942i	$-0.915637\pi$
$821$	− 1.01179e7i	− 0.523883i	0.965084	−	0.261942i	$-0.0843628\pi$
$822$	0	0
$823$	−2.19484e7	−1.12954	−0.564771	−	0.825248i	$-0.691036\pi$
$823$	−2.19484e7	−1.12954	−0.564771	+	0.825248i	$0.691036\pi$
$824$	2.39957e7	1.23116
$825$	0	0
$826$	0	0
$827$	− 1.81637e7i	− 0.923509i	−0.887008	−	0.461754i	$-0.847220\pi$
$827$	− 1.81637e7i	− 0.923509i	0.887008	−	0.461754i	$-0.152780\pi$
$828$	0	0
$829$	− 2.23845e7i	− 1.13126i	−0.824660	−	0.565629i	$-0.808633\pi$
$829$	− 2.23845e7i	− 1.13126i	0.824660	−	0.565629i	$-0.191367\pi$
$830$	− 5.26809e7i	− 2.65435i
$831$	0	0
$832$	4.26329e7i	2.13519i
$833$	0	0
$834$	0	0
$835$	−5.70622e7	−2.83226
$836$	1.68376e7	0.833231
$837$	0	0
$838$	− 1.98522e7i	− 0.976560i
$839$	−1.08782e7	−0.533523	−0.266761	−	0.963763i	$-0.585954\pi$
$839$	−1.08782e7	−0.533523	−0.266761	+	0.963763i	$0.585954\pi$
$840$	0	0
$841$	1.16156e7	0.566306
$842$	− 7.29922e7i	− 3.54810i
$843$	0	0
$844$	−3.35956e7	−1.62340
$845$	−2.17157e7	−1.04624
$846$	0	0
$847$	0	0
$848$	1.78249e7i	0.851212i
$849$	0	0
$850$	− 1.79694e7i	− 0.853071i
$851$	− 2.62552e6i	− 0.124277i
$852$	0	0
$853$	1.14153e7i	0.537172i	0.963256	+	0.268586i	$0.0865564\pi$
$853$	1.14153e7i	0.537172i	−0.963256	+	0.268586i	$0.913444\pi$
$854$	0	0
$855$	0	0
$856$	7.74486e7	3.61268
$857$	1.29350e7	0.601611	0.300805	−	0.953686i	$-0.402745\pi$
$857$	1.29350e7	0.601611	0.300805	+	0.953686i	$0.402745\pi$
$858$	0	0
$859$	2.76240e7i	1.27733i	0.769484	+	0.638666i	$0.220513\pi$
$859$	2.76240e7i	1.27733i	−0.769484	+	0.638666i	$0.779487\pi$
$860$	−7.66417e7	−3.53361
$861$	0	0
$862$	1.64611e7	0.754555
$863$	1.67727e7i	0.766613i	0.923621	+	0.383307i	$0.125215\pi$
$863$	1.67727e7i	0.766613i	−0.923621	+	0.383307i	$0.874785\pi$
$864$	0	0
$865$	1.46331e7	0.664959
$866$	6.55742e7	2.97124
$867$	0	0
$868$	0	0
$869$	4.95789e6i	0.222714i
$870$	0	0
$871$	1.68987e7i	0.754757i
$872$	− 8.27921e7i	− 3.68721i
$873$	0	0
$874$	4.12361e6i	0.182599i
$875$	0	0
$876$	0	0
$877$	−1.88682e7	−0.828382	−0.414191	−	0.910190i	$-0.635935\pi$
$877$	−1.88682e7	−0.828382	−0.414191	+	0.910190i	$0.635935\pi$
$878$	−4.61256e7	−2.01932
$879$	0	0
$880$	− 3.68261e7i	− 1.60306i
$881$	−1.09803e7	−0.476621	−0.238311	−	0.971189i	$-0.576594\pi$
$881$	−1.09803e7	−0.476621	−0.238311	+	0.971189i	$0.576594\pi$
$882$	0	0
$883$	−1.31516e7	−0.567645	−0.283822	−	0.958877i	$-0.591603\pi$
$883$	−1.31516e7	−0.567645	−0.283822	+	0.958877i	$0.591603\pi$
$884$	1.28556e7i	0.553300i
$885$	0	0
$886$	5.42401e7	2.32133
$887$	1.35276e7	0.577313	0.288657	−	0.957433i	$-0.406791\pi$
$887$	1.35276e7	0.577313	0.288657	+	0.957433i	$0.406791\pi$
$888$	0	0
$889$	0	0
$890$	9.82349e7i	4.15710i
$891$	0	0
$892$	− 2.40888e7i	− 1.01369i
$893$	3.71026e7i	1.55695i
$894$	0	0
$895$	− 2.78927e7i	− 1.16395i
$896$	0	0
$897$	0	0
$898$	−5.20899e7	−2.15557
$899$	1.00707e7	0.415586
$900$	0	0
$901$	2.01707e6i	0.0827768i
$902$	1.81525e7	0.742884
$903$	0	0
$904$	1.36574e7	0.555836
$905$	4.25253e7i	1.72594i
$906$	0	0
$907$	−1.30814e7	−0.528001	−0.264001	−	0.964523i	$-0.585042\pi$
$907$	−1.30814e7	−0.528001	−0.264001	+	0.964523i	$0.585042\pi$
$908$	−1.40534e7	−0.565675
$909$	0	0
$910$	0	0
$911$	− 2.22270e7i	− 0.887329i	−0.896193	−	0.443665i	$-0.853678\pi$
$911$	− 2.22270e7i	− 0.887329i	0.896193	−	0.443665i	$-0.146322\pi$
$912$	0	0
$913$	− 6.62307e6i	− 0.262956i
$914$	8.34697e7i	3.30494i
$915$	0	0
$916$	− 9.04721e7i	− 3.56267i
$917$	0	0
$918$	0	0
$919$	1.31113e7	0.512103	0.256051	−	0.966663i	$-0.417578\pi$
$919$	1.31113e7	0.512103	0.256051	+	0.966663i	$0.417578\pi$
$920$	1.13769e7	0.443153
$921$	0	0
$922$	4.13117e6i	0.160046i
$923$	395631.	0.0152857
$924$	0	0
$925$	−4.33546e7	−1.66602
$926$	− 1.92818e7i	− 0.738960i
$927$	0	0
$928$	−6.93488e7	−2.64344
$929$	−2.88192e7	−1.09558	−0.547788	−	0.836617i	$-0.684530\pi$
$929$	−2.88192e7	−1.09558	−0.547788	+	0.836617i	$0.684530\pi$
$930$	0	0
$931$	0	0
$932$	− 5.24607e7i	− 1.97831i
$933$	0	0
$934$	− 9.19725e7i	− 3.44978i
$935$	− 4.16725e6i	− 0.155891i
$936$	0	0
$937$	− 3.33706e7i	− 1.24170i	−0.783931	−	0.620848i	$-0.786788\pi$
$937$	− 3.33706e7i	− 1.24170i	0.783931	−	0.620848i	$-0.213212\pi$
$938$	0	0
$939$	0	0
$940$	1.60767e8	5.93442
$941$	−3.20999e7	−1.18176	−0.590881	−	0.806758i	$-0.701220\pi$
$941$	−3.20999e7	−1.18176	−0.590881	+	0.806758i	$0.701220\pi$
$942$	0	0
$943$	3.26100e6i	0.119418i
$944$	−5.22206e7	−1.90727
$945$	0	0
$946$	−1.31358e7	−0.477230
$947$	− 4.28326e7i	− 1.55203i	−0.630716	−	0.776014i	$-0.717239\pi$
$947$	− 4.28326e7i	− 1.55203i	0.630716	−	0.776014i	$-0.282761\pi$
$948$	0	0
$949$	2.90943e6	0.104868
$950$	6.80921e7	2.44787
$951$	0	0
$952$	0	0
$953$	− 4.96089e7i	− 1.76940i	−0.466156	−	0.884702i	$-0.654362\pi$
$953$	− 4.96089e7i	− 1.76940i	0.466156	−	0.884702i	$-0.345638\pi$
$954$	0	0
$955$	8.44249e6i	0.299545i
$956$	1.18370e8i	4.18886i
$957$	0	0
$958$	− 1.50446e7i	− 0.529624i
$959$	0	0
$960$	0	0
$961$	1.72280e7	0.601765
$962$	4.22841e7	1.47313
$963$	0	0
$964$	1.42083e8i	4.92435i
$965$	5.86140e7	2.02620
$966$	0	0
$967$	3.78348e7	1.30114	0.650572	−	0.759445i	$-0.274529\pi$
$967$	3.78348e7	1.30114	0.650572	+	0.759445i	$0.274529\pi$
$968$	9.10249e7i	3.12228i
$969$	0	0
$970$	−4.92894e6	−0.168199
$971$	−4.25508e7	−1.44830	−0.724151	−	0.689641i	$-0.757768\pi$
$971$	−4.25508e7	−1.44830	−0.724151	+	0.689641i	$0.757768\pi$
$972$	0	0
$973$	0	0
$974$	1.96633e7i	0.664139i
$975$	0	0
$976$	1.98861e8i	6.68229i
$977$	− 2.82030e7i	− 0.945278i	−0.881256	−	0.472639i	$-0.843302\pi$
$977$	− 2.82030e7i	− 0.945278i	0.881256	−	0.472639i	$-0.156698\pi$
$978$	0	0
$979$	1.23502e7i	0.411828i
$980$	0	0
$981$	0	0
$982$	−1.10452e8	−3.65505
$983$	−2.72440e7	−0.899264	−0.449632	−	0.893214i	$-0.648445\pi$
$983$	−2.72440e7	−0.899264	−0.449632	+	0.893214i	$0.648445\pi$
$984$	0	0
$985$	− 7.13673e6i	− 0.234374i
$986$	−1.44856e7	−0.474509
$987$	0	0
$988$	−4.87142e7	−1.58768
$989$	− 2.35977e6i	− 0.0767146i
$990$	0	0
$991$	−3.94137e7	−1.27486	−0.637431	−	0.770508i	$-0.720003\pi$
$991$	−3.94137e7	−1.27486	−0.637431	+	0.770508i	$0.720003\pi$
$992$	7.85102e7	2.53307
$993$	0	0
$994$	0	0
$995$	5.96164e7i	1.90901i
$996$	0	0
$997$	3.01168e7i	0.959557i	0.877390	+	0.479779i	$0.159283\pi$
$997$	3.01168e7i	0.959557i	−0.877390	+	0.479779i	$0.840717\pi$
$998$	− 6.86338e7i	− 2.18128i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.c.b.440.3		24
3.2	odd	2	inner	441.6.c.b.440.22		24
7.4	even	3		63.6.p.b.26.12	yes	24
7.5	odd	6		63.6.p.b.17.1	✓	24
7.6	odd	2	inner	441.6.c.b.440.21		24
21.5	even	6		63.6.p.b.17.12	yes	24
21.11	odd	6		63.6.p.b.26.1	yes	24
21.20	even	2	inner	441.6.c.b.440.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.6.p.b.17.1	✓	24	7.5	odd	6
63.6.p.b.17.12	yes	24	21.5	even	6
63.6.p.b.26.1	yes	24	21.11	odd	6
63.6.p.b.26.12	yes	24	7.4	even	3
441.6.c.b.440.3		24	1.1	even	1	trivial
441.6.c.b.440.4		24	21.20	even	2	inner
441.6.c.b.440.21		24	7.6	odd	2	inner
441.6.c.b.440.22		24	3.2	odd	2	inner

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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	441.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-10.9584i q^{2} -88.0874 q^{4} -82.6125 q^{5} +614.630i q^{8} +O(q^{10})\)	\(q-10.9584i q^{2} -88.0874 q^{4} -82.6125 q^{5} +614.630i q^{8} +905.305i q^{10} +113.816i q^{11} -329.288i q^{13} +3916.59 q^{16} +443.202 q^{17} +1679.45i q^{19} +7277.12 q^{20} +1247.24 q^{22} +224.060i q^{23} +3699.83 q^{25} -3608.48 q^{26} -2982.54i q^{29} +3376.55i q^{31} -23251.6i q^{32} -4856.80i q^{34} -11718.0 q^{37} +18404.1 q^{38} -50776.2i q^{40} +14554.2 q^{41} -10531.9 q^{43} -10025.7i q^{44} +2455.34 q^{46} +22092.2 q^{47} -40544.4i q^{50} +29006.1i q^{52} +4551.13i q^{53} -9402.59i q^{55} -32684.0 q^{58} -13333.2 q^{59} +50774.0i q^{61} +37001.8 q^{62} -129470. q^{64} +27203.3i q^{65} -51318.8 q^{67} -39040.5 q^{68} +1201.47i q^{71} +8835.52i q^{73} +128411. i q^{74} -147938. i q^{76} +43560.7 q^{79} -323560. q^{80} -159491. i q^{82} -58191.3 q^{83} -36614.0 q^{85} +115413. i q^{86} -69954.5 q^{88} +108510. q^{89} -19736.8i q^{92} -242095. i q^{94} -138743. i q^{95} +5444.51i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 608 q^{4}+O(q^{10}) \) 24 * q - 608 * q^4	\( 24 q - 608 q^{4} + 7288 q^{16} - 5648 q^{22} + 37704 q^{25} - 41096 q^{37} + 2200 q^{43} + 51424 q^{46} - 308600 q^{58} - 327880 q^{64} - 312648 q^{67} - 331512 q^{79} - 284448 q^{85} - 1164616 q^{88}+O(q^{100}) \) 24 * q - 608 * q^4 + 7288 * q^16 - 5648 * q^22 + 37704 * q^25 - 41096 * q^37 + 2200 * q^43 + 51424 * q^46 - 308600 * q^58 - 327880 * q^64 - 312648 * q^67 - 331512 * q^79 - 284448 * q^85 - 1164616 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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