LMFDB - Embedded newform 784.5.c.g.97.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [784,5,Mod(97,784)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("784.97"); S:= CuspForms(chi, 5); N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	784.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-352,0,-120] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$81.0420510577$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 66x^{6} + 212x^{5} + 2021x^{4} - 4400x^{3} - 25028x^{2} + 27264x + 127778 $ x^8 - 4x^7 - 66x^6 + 212x^5 + 2021x^4 - 4400x^3 - 25028x^2 + 27264*x + 127778
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 7^{6} $
Twist minimal:	no (minimal twist has level 49)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.2
Root			$-2.71722 - 0.765367i$ of defining polynomial
Character	$\chi$	$=$	784.97
Dual form			784.5.c.g.97.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-12.5695i q^{3} +33.6557i q^{5} -76.9935 q^{9} +149.005 q^{11} -58.8822i q^{13} +423.037 q^{15} +1.70522i q^{17} +169.127i q^{19} -581.203 q^{23} -507.708 q^{25} -50.3595i q^{27} +462.422 q^{29} -15.9075i q^{31} -1872.93i q^{33} -2250.84 q^{37} -740.123 q^{39} +3174.13i q^{41} +3013.25 q^{43} -2591.27i q^{45} -385.894i q^{47} +21.4339 q^{51} +1553.00 q^{53} +5014.87i q^{55} +2125.85 q^{57} +2921.92i q^{59} +5819.53i q^{61} +1981.72 q^{65} -10.5261 q^{67} +7305.46i q^{69} +4064.56 q^{71} +1062.62i q^{73} +6381.66i q^{75} -4268.53 q^{79} -6869.47 q^{81} -4406.56i q^{83} -57.3905 q^{85} -5812.44i q^{87} +7957.72i q^{89} -199.950 q^{93} -5692.10 q^{95} -9842.62i q^{97} -11472.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 352 q^{9} - 120 q^{11} + 632 q^{15} - 1752 q^{23} - 2192 q^{25} + 1248 q^{29} + 2368 q^{37} + 7672 q^{39} + 8552 q^{43} + 11976 q^{51} + 5496 q^{53} - 9200 q^{57} + 30240 q^{65} + 7440 q^{67} - 9984 q^{71}+ \cdots - 30144 q^{99}+O(q^{100}) $ 8 * q - 352 * q^9 - 120 * q^11 + 632 * q^15 - 1752 * q^23 - 2192 * q^25 + 1248 * q^29 + 2368 * q^37 + 7672 * q^39 + 8552 * q^43 + 11976 * q^51 + 5496 * q^53 - 9200 * q^57 + 30240 * q^65 + 7440 * q^67 - 9984 * q^71 + 14096 * q^79 + 3432 * q^81 + 11912 * q^85 + 9584 * q^93 - 22488 * q^95 - 30144 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 12.5695i	− 1.39662i	−0.715797	−	0.698308i	$-0.753937\pi$
$3$	− 12.5695i	− 1.39662i	0.715797	−	0.698308i	$-0.246063\pi$
$4$	0	0
$5$	33.6557i	1.34623i	0.739538	+	0.673115i	$0.235044\pi$
$5$	33.6557i	1.34623i	−0.739538	+	0.673115i	$0.764956\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−76.9935	−0.950537
$10$	0	0
$11$	149.005	1.23145	0.615723	−	0.787962i	$-0.288864\pi$
$11$	149.005	1.23145	0.615723	+	0.787962i	$0.288864\pi$
$12$	0	0
$13$	− 58.8822i	− 0.348416i	−0.984709	−	0.174208i	$-0.944264\pi$
$13$	− 58.8822i	− 0.348416i	0.984709	−	0.174208i	$-0.0557364\pi$
$14$	0	0
$15$	423.037	1.88017
$16$	0	0
$17$	1.70522i	0.00590042i	0.999996	+	0.00295021i	$0.000939082\pi$
$17$	1.70522i	0.00590042i	−0.999996	+	0.00295021i	$0.999061\pi$
$18$	0	0
$19$	169.127i	0.468497i	0.972177	+	0.234248i	$0.0752629\pi$
$19$	169.127i	0.468497i	−0.972177	+	0.234248i	$0.924737\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−581.203	−1.09868	−0.549341	−	0.835598i	$-0.685121\pi$
$23$	−581.203	−1.09868	−0.549341	+	0.835598i	$0.685121\pi$
$24$	0	0
$25$	−507.708	−0.812333
$26$	0	0
$27$	− 50.3595i	− 0.0690802i
$28$	0	0
$29$	462.422	0.549848	0.274924	−	0.961466i	$-0.411347\pi$
$29$	462.422	0.549848	0.274924	+	0.961466i	$0.411347\pi$
$30$	0	0
$31$	− 15.9075i	− 0.0165530i	−0.999966	−	0.00827651i	$-0.997365\pi$
$31$	− 15.9075i	− 0.0165530i	0.999966	−	0.00827651i	$-0.00263453\pi$
$32$	0	0
$33$	− 1872.93i	− 1.71986i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2250.84	−1.64415	−0.822073	−	0.569383i	$-0.807182\pi$
$37$	−2250.84	−1.64415	−0.822073	+	0.569383i	$0.807182\pi$
$38$	0	0
$39$	−740.123	−0.486603
$40$	0	0
$41$	3174.13i	1.88824i	0.329605	+	0.944119i	$0.393084\pi$
$41$	3174.13i	1.88824i	−0.329605	+	0.944119i	$0.606916\pi$
$42$	0	0
$43$	3013.25	1.62967	0.814833	−	0.579696i	$-0.196829\pi$
$43$	3013.25	1.62967	0.814833	+	0.579696i	$0.196829\pi$
$44$	0	0
$45$	− 2591.27i	− 1.27964i
$46$	0	0
$47$	− 385.894i	− 0.174692i	−0.996178	−	0.0873459i	$-0.972161\pi$
$47$	− 385.894i	− 0.174692i	0.996178	−	0.0873459i	$-0.0278385\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	21.4339	0.00824062
$52$	0	0
$53$	1553.00	0.552867	0.276434	−	0.961033i	$-0.410847\pi$
$53$	1553.00	0.552867	0.276434	+	0.961033i	$0.410847\pi$
$54$	0	0
$55$	5014.87i	1.65781i
$56$	0	0
$57$	2125.85	0.654310
$58$	0	0
$59$	2921.92i	0.839390i	0.907665	+	0.419695i	$0.137863\pi$
$59$	2921.92i	0.839390i	−0.907665	+	0.419695i	$0.862137\pi$
$60$	0	0
$61$	5819.53i	1.56397i	0.623297	+	0.781985i	$0.285793\pi$
$61$	5819.53i	1.56397i	−0.623297	+	0.781985i	$0.714207\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1981.72	0.469047
$66$	0	0
$67$	−10.5261	−0.00234486	−0.00117243	−	0.999999i	$-0.500373\pi$
$67$	−10.5261	−0.00234486	−0.00117243	+	0.999999i	$0.500373\pi$
$68$	0	0
$69$	7305.46i	1.53444i
$70$	0	0
$71$	4064.56	0.806300	0.403150	−	0.915134i	$-0.367915\pi$
$71$	4064.56	0.806300	0.403150	+	0.915134i	$0.367915\pi$
$72$	0	0
$73$	1062.62i	0.199402i	0.995017	+	0.0997012i	$0.0317887\pi$
$73$	1062.62i	0.199402i	−0.995017	+	0.0997012i	$0.968211\pi$
$74$	0	0
$75$	6381.66i	1.13452i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4268.53	−0.683950	−0.341975	−	0.939709i	$-0.611096\pi$
$79$	−4268.53	−0.683950	−0.341975	+	0.939709i	$0.611096\pi$
$80$	0	0
$81$	−6869.47	−1.04702
$82$	0	0
$83$	− 4406.56i	− 0.639651i	−0.947476	−	0.319826i	$-0.896376\pi$
$83$	− 4406.56i	− 0.639651i	0.947476	−	0.319826i	$-0.103624\pi$
$84$	0	0
$85$	−57.3905	−0.00794332
$86$	0	0
$87$	− 5812.44i	− 0.767927i
$88$	0	0
$89$	7957.72i	1.00464i	0.864683	+	0.502318i	$0.167519\pi$
$89$	7957.72i	1.00464i	−0.864683	+	0.502318i	$0.832481\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−199.950	−0.0231182
$94$	0	0
$95$	−5692.10	−0.630704
$96$	0	0
$97$	− 9842.62i	− 1.04609i	−0.852306	−	0.523043i	$-0.824797\pi$
$97$	− 9842.62i	− 1.04609i	0.852306	−	0.523043i	$-0.175203\pi$
$98$	0	0
$99$	−11472.4	−1.17054
$100$	0	0
$101$	7398.19i	0.725242i	0.931937	+	0.362621i	$0.118118\pi$
$101$	7398.19i	0.725242i	−0.931937	+	0.362621i	$0.881882\pi$
$102$	0	0
$103$	15168.8i	1.42980i	0.699226	+	0.714900i	$0.253528\pi$
$103$	15168.8i	1.42980i	−0.699226	+	0.714900i	$0.746472\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13665.7	1.19362	0.596809	−	0.802383i	$-0.296435\pi$
$107$	13665.7	1.19362	0.596809	+	0.802383i	$0.296435\pi$
$108$	0	0
$109$	13896.0	1.16960	0.584799	−	0.811178i	$-0.301173\pi$
$109$	13896.0	1.16960	0.584799	+	0.811178i	$0.301173\pi$
$110$	0	0
$111$	28292.0i	2.29624i
$112$	0	0
$113$	4570.34	0.357925	0.178962	−	0.983856i	$-0.442726\pi$
$113$	4570.34	0.357925	0.178962	+	0.983856i	$0.442726\pi$
$114$	0	0
$115$	− 19560.8i	− 1.47908i
$116$	0	0
$117$	4533.55i	0.331182i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7561.50	0.516461
$122$	0	0
$123$	39897.3	2.63714
$124$	0	0
$125$	3947.54i	0.252643i
$126$	0	0
$127$	−9273.21	−0.574940	−0.287470	−	0.957790i	$-0.592814\pi$
$127$	−9273.21	−0.574940	−0.287470	+	0.957790i	$0.592814\pi$
$128$	0	0
$129$	− 37875.2i	− 2.27602i
$130$	0	0
$131$	− 3884.76i	− 0.226371i	−0.993574	−	0.113186i	$-0.963895\pi$
$131$	− 3884.76i	− 0.226371i	0.993574	−	0.113186i	$-0.0361055\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1694.89	0.0929978
$136$	0	0
$137$	26051.7	1.38802	0.694009	−	0.719966i	$-0.255843\pi$
$137$	26051.7	1.38802	0.694009	+	0.719966i	$0.255843\pi$
$138$	0	0
$139$	− 18399.0i	− 0.952278i	−0.879370	−	0.476139i	$-0.842036\pi$
$139$	− 18399.0i	− 0.952278i	0.879370	−	0.476139i	$-0.157964\pi$
$140$	0	0
$141$	−4850.51	−0.243977
$142$	0	0
$143$	− 8773.75i	− 0.429055i
$144$	0	0
$145$	15563.2i	0.740222i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11049.5	0.497701	0.248851	−	0.968542i	$-0.419947\pi$
$149$	11049.5	0.497701	0.248851	+	0.968542i	$0.419947\pi$
$150$	0	0
$151$	16759.4	0.735030	0.367515	−	0.930018i	$-0.380209\pi$
$151$	16759.4	0.735030	0.367515	+	0.930018i	$0.380209\pi$
$152$	0	0
$153$	− 131.291i	− 0.00560857i
$154$	0	0
$155$	535.377	0.0222842
$156$	0	0
$157$	36708.4i	1.48924i	0.667487	+	0.744622i	$0.267370\pi$
$157$	36708.4i	1.48924i	−0.667487	+	0.744622i	$0.732630\pi$
$158$	0	0
$159$	− 19520.6i	− 0.772144i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−8101.41	−0.304920	−0.152460	−	0.988310i	$-0.548719\pi$
$163$	−8101.41	−0.304920	−0.152460	+	0.988310i	$0.548719\pi$
$164$	0	0
$165$	63034.7	2.31532
$166$	0	0
$167$	− 19103.6i	− 0.684985i	−0.939521	−	0.342493i	$-0.888729\pi$
$167$	− 19103.6i	− 0.684985i	0.939521	−	0.342493i	$-0.111271\pi$
$168$	0	0
$169$	25093.9	0.878607
$170$	0	0
$171$	− 13021.7i	− 0.445324i
$172$	0	0
$173$	− 12558.0i	− 0.419594i	−0.977745	−	0.209797i	$-0.932720\pi$
$173$	− 12558.0i	− 0.419594i	0.977745	−	0.209797i	$-0.0672803\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	36727.2	1.17231
$178$	0	0
$179$	27973.8	0.873061	0.436531	−	0.899689i	$-0.356207\pi$
$179$	27973.8	0.873061	0.436531	+	0.899689i	$0.356207\pi$
$180$	0	0
$181$	− 10385.6i	− 0.317012i	−0.987358	−	0.158506i	$-0.949332\pi$
$181$	− 10385.6i	− 0.317012i	0.987358	−	0.158506i	$-0.0506677\pi$
$182$	0	0
$183$	73148.9	2.18427
$184$	0	0
$185$	− 75753.5i	− 2.21340i
$186$	0	0
$187$	254.086i	0.00726605i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17505.5	0.479854	0.239927	−	0.970791i	$-0.422877\pi$
$191$	17505.5	0.479854	0.239927	+	0.970791i	$0.422877\pi$
$192$	0	0
$193$	11449.7	0.307382	0.153691	−	0.988119i	$-0.450884\pi$
$193$	11449.7	0.307382	0.153691	+	0.988119i	$0.450884\pi$
$194$	0	0
$195$	− 24909.4i	− 0.655079i
$196$	0	0
$197$	−34525.0	−0.889612	−0.444806	−	0.895627i	$-0.646727\pi$
$197$	−34525.0	−0.889612	−0.444806	+	0.895627i	$0.646727\pi$
$198$	0	0
$199$	24395.1i	0.616022i	0.951383	+	0.308011i	$0.0996633\pi$
$199$	24395.1i	0.616022i	−0.951383	+	0.308011i	$0.900337\pi$
$200$	0	0
$201$	132.308i	0.00327487i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−106828.	−2.54200
$206$	0	0
$207$	44748.9	1.04434
$208$	0	0
$209$	25200.8i	0.576929i
$210$	0	0
$211$	−59793.2	−1.34303	−0.671517	−	0.740989i	$-0.734357\pi$
$211$	−59793.2	−1.34303	−0.671517	+	0.740989i	$0.734357\pi$
$212$	0	0
$213$	− 51089.6i	− 1.12609i
$214$	0	0
$215$	101413.i	2.19390i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	13356.6	0.278489
$220$	0	0
$221$	100.407	0.00205580
$222$	0	0
$223$	77418.3i	1.55680i	0.627766	+	0.778402i	$0.283969\pi$
$223$	77418.3i	1.55680i	−0.627766	+	0.778402i	$0.716031\pi$
$224$	0	0
$225$	39090.2	0.772153
$226$	0	0
$227$	− 76879.3i	− 1.49196i	−0.665968	−	0.745980i	$-0.731981\pi$
$227$	− 76879.3i	− 1.49196i	0.665968	−	0.745980i	$-0.268019\pi$
$228$	0	0
$229$	72838.6i	1.38896i	0.719511	+	0.694481i	$0.244366\pi$
$229$	72838.6i	1.38896i	−0.719511	+	0.694481i	$0.755634\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	73744.0	1.35836	0.679180	−	0.733972i	$-0.262336\pi$
$233$	73744.0	1.35836	0.679180	+	0.733972i	$0.262336\pi$
$234$	0	0
$235$	12987.5	0.235175
$236$	0	0
$237$	53653.5i	0.955216i
$238$	0	0
$239$	28067.7	0.491373	0.245686	−	0.969349i	$-0.420987\pi$
$239$	28067.7	0.491373	0.245686	+	0.969349i	$0.420987\pi$
$240$	0	0
$241$	− 8504.93i	− 0.146432i	−0.997316	−	0.0732161i	$-0.976674\pi$
$241$	− 8504.93i	− 0.146432i	0.997316	−	0.0732161i	$-0.0233263\pi$
$242$	0	0
$243$	82267.0i	1.39320i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	9958.59	0.163232
$248$	0	0
$249$	−55388.4	−0.893347
$250$	0	0
$251$	45956.3i	0.729453i	0.931115	+	0.364726i	$0.118837\pi$
$251$	45956.3i	0.729453i	−0.931115	+	0.364726i	$0.881163\pi$
$252$	0	0
$253$	−86602.2	−1.35297
$254$	0	0
$255$	721.372i	0.0110938i
$256$	0	0
$257$	− 42262.9i	− 0.639871i	−0.947439	−	0.319936i	$-0.896339\pi$
$257$	− 42262.9i	− 0.639871i	0.947439	−	0.319936i	$-0.103661\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−35603.5	−0.522651
$262$	0	0
$263$	−47638.2	−0.688722	−0.344361	−	0.938837i	$-0.611904\pi$
$263$	−47638.2	−0.688722	−0.344361	+	0.938837i	$0.611904\pi$
$264$	0	0
$265$	52267.5i	0.744286i
$266$	0	0
$267$	100025.	1.40309
$268$	0	0
$269$	131184.i	1.81291i	0.422303	+	0.906455i	$0.361222\pi$
$269$	131184.i	1.81291i	−0.422303	+	0.906455i	$0.638778\pi$
$270$	0	0
$271$	− 60576.0i	− 0.824825i	−0.910997	−	0.412412i	$-0.864686\pi$
$271$	− 60576.0i	− 0.824825i	0.910997	−	0.412412i	$-0.135314\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−75651.1	−1.00034
$276$	0	0
$277$	18431.8	0.240220	0.120110	−	0.992761i	$-0.461675\pi$
$277$	18431.8	0.240220	0.120110	+	0.992761i	$0.461675\pi$
$278$	0	0
$279$	1224.77i	0.0157343i
$280$	0	0
$281$	23061.9	0.292067	0.146033	−	0.989280i	$-0.453349\pi$
$281$	23061.9	0.292067	0.146033	+	0.989280i	$0.453349\pi$
$282$	0	0
$283$	92266.8i	1.15205i	0.817431	+	0.576027i	$0.195398\pi$
$283$	92266.8i	1.15205i	−0.817431	+	0.576027i	$0.804602\pi$
$284$	0	0
$285$	71547.2i	0.880852i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	83518.1	0.999965
$290$	0	0
$291$	−123717.	−1.46098
$292$	0	0
$293$	104037.i	1.21187i	0.795516	+	0.605933i	$0.207200\pi$
$293$	104037.i	1.21187i	−0.795516	+	0.605933i	$0.792800\pi$
$294$	0	0
$295$	−98339.3	−1.13001
$296$	0	0
$297$	− 7503.82i	− 0.0850686i
$298$	0	0
$299$	34222.5i	0.382798i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	92991.9	1.01288
$304$	0	0
$305$	−195861.	−2.10546
$306$	0	0
$307$	99690.4i	1.05773i	0.848705	+	0.528867i	$0.177383\pi$
$307$	99690.4i	1.05773i	−0.848705	+	0.528867i	$0.822617\pi$
$308$	0	0
$309$	190664.	1.99688
$310$	0	0
$311$	11076.9i	0.114524i	0.998359	+	0.0572620i	$0.0182370\pi$
$311$	11076.9i	0.114524i	−0.998359	+	0.0572620i	$0.981763\pi$
$312$	0	0
$313$	122237.i	1.24771i	0.781538	+	0.623857i	$0.214435\pi$
$313$	122237.i	1.24771i	−0.781538	+	0.623857i	$0.785565\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	32265.3	0.321082	0.160541	−	0.987029i	$-0.448676\pi$
$317$	32265.3	0.321082	0.160541	+	0.987029i	$0.448676\pi$
$318$	0	0
$319$	68903.3	0.677109
$320$	0	0
$321$	− 171772.i	− 1.66703i
$322$	0	0
$323$	−288.399	−0.00276433
$324$	0	0
$325$	29895.0i	0.283029i
$326$	0	0
$327$	− 174666.i	− 1.63348i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−104098.	−0.950136	−0.475068	−	0.879949i	$-0.657577\pi$
$331$	−104098.	−0.950136	−0.475068	+	0.879949i	$0.657577\pi$
$332$	0	0
$333$	173300.	1.56282
$334$	0	0
$335$	− 354.263i	− 0.00315672i
$336$	0	0
$337$	55874.8	0.491990	0.245995	−	0.969271i	$-0.420885\pi$
$337$	55874.8	0.491990	0.245995	+	0.969271i	$0.420885\pi$
$338$	0	0
$339$	− 57447.1i	− 0.499883i
$340$	0	0
$341$	− 2370.29i	− 0.0203842i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−245871.	−2.06570
$346$	0	0
$347$	−158282.	−1.31454	−0.657269	−	0.753656i	$-0.728289\pi$
$347$	−158282.	−1.31454	−0.657269	+	0.753656i	$0.728289\pi$
$348$	0	0
$349$	19796.6i	0.162532i	0.996692	+	0.0812660i	$0.0258963\pi$
$349$	19796.6i	0.162532i	−0.996692	+	0.0812660i	$0.974104\pi$
$350$	0	0
$351$	−2965.28	−0.0240686
$352$	0	0
$353$	− 137832.i	− 1.10612i	−0.833143	−	0.553058i	$-0.813461\pi$
$353$	− 137832.i	− 1.10612i	0.833143	−	0.553058i	$-0.186539\pi$
$354$	0	0
$355$	136796.i	1.08546i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	35706.3	0.277049	0.138524	−	0.990359i	$-0.455764\pi$
$359$	35706.3	0.277049	0.138524	+	0.990359i	$0.455764\pi$
$360$	0	0
$361$	101717.	0.780511
$362$	0	0
$363$	− 95044.6i	− 0.721297i
$364$	0	0
$365$	−35763.1	−0.268441
$366$	0	0
$367$	17758.3i	0.131847i	0.997825	+	0.0659233i	$0.0209993\pi$
$367$	17758.3i	0.131847i	−0.997825	+	0.0659233i	$0.979001\pi$
$368$	0	0
$369$	− 244387.i	− 1.79484i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	30309.7	0.217853	0.108926	−	0.994050i	$-0.465259\pi$
$373$	30309.7	0.217853	0.108926	+	0.994050i	$0.465259\pi$
$374$	0	0
$375$	49618.8	0.352845
$376$	0	0
$377$	− 27228.5i	− 0.191576i
$378$	0	0
$379$	32246.5	0.224494	0.112247	−	0.993680i	$-0.464195\pi$
$379$	32246.5	0.224494	0.112247	+	0.993680i	$0.464195\pi$
$380$	0	0
$381$	116560.i	0.802970i
$382$	0	0
$383$	− 166129.i	− 1.13252i	−0.824225	−	0.566262i	$-0.808389\pi$
$383$	− 166129.i	− 1.13252i	0.824225	−	0.566262i	$-0.191611\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−232001.	−1.54906
$388$	0	0
$389$	−97062.4	−0.641434	−0.320717	−	0.947175i	$-0.603924\pi$
$389$	−97062.4	−0.641434	−0.320717	+	0.947175i	$0.603924\pi$
$390$	0	0
$391$	− 991.079i	− 0.00648269i
$392$	0	0
$393$	−48829.6	−0.316154
$394$	0	0
$395$	− 143661.i	− 0.920754i
$396$	0	0
$397$	− 278836.i	− 1.76916i	−0.466384	−	0.884582i	$-0.654444\pi$
$397$	− 278836.i	− 1.76916i	0.466384	−	0.884582i	$-0.345556\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	101047.	0.628398	0.314199	−	0.949357i	$-0.398264\pi$
$401$	101047.	0.628398	0.314199	+	0.949357i	$0.398264\pi$
$402$	0	0
$403$	−936.667	−0.00576733
$404$	0	0
$405$	− 231197.i	− 1.40952i
$406$	0	0
$407$	−335386.	−2.02468
$408$	0	0
$409$	238004.i	1.42278i	0.702797	+	0.711390i	$0.251934\pi$
$409$	238004.i	1.42278i	−0.702797	+	0.711390i	$0.748066\pi$
$410$	0	0
$411$	− 327458.i	− 1.93853i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	148306.	0.861117
$416$	0	0
$417$	−231267.	−1.32997
$418$	0	0
$419$	− 349836.i	− 1.99268i	−0.0855008	−	0.996338i	$-0.527249\pi$
$419$	− 349836.i	− 1.99268i	0.0855008	−	0.996338i	$-0.472751\pi$
$420$	0	0
$421$	−248575.	−1.40247	−0.701236	−	0.712929i	$-0.747368\pi$
$421$	−248575.	−1.40247	−0.701236	+	0.712929i	$0.747368\pi$
$422$	0	0
$423$	29711.3i	0.166051i
$424$	0	0
$425$	− 865.755i	− 0.00479311i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−110282.	−0.599225
$430$	0	0
$431$	131868.	0.709879	0.354940	−	0.934889i	$-0.384501\pi$
$431$	131868.	0.709879	0.354940	+	0.934889i	$0.384501\pi$
$432$	0	0
$433$	− 125370.i	− 0.668680i	−0.942453	−	0.334340i	$-0.891487\pi$
$433$	− 125370.i	− 0.668680i	0.942453	−	0.334340i	$-0.108513\pi$
$434$	0	0
$435$	195622.	1.03381
$436$	0	0
$437$	− 98297.3i	− 0.514729i
$438$	0	0
$439$	− 79030.8i	− 0.410079i	−0.978754	−	0.205039i	$-0.934268\pi$
$439$	− 79030.8i	− 0.410079i	0.978754	−	0.205039i	$-0.0657322\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−152348.	−0.776301	−0.388150	−	0.921596i	$-0.626886\pi$
$443$	−152348.	−0.776301	−0.388150	+	0.921596i	$0.626886\pi$
$444$	0	0
$445$	−267823.	−1.35247
$446$	0	0
$447$	− 138887.i	− 0.695098i
$448$	0	0
$449$	−125622.	−0.623124	−0.311562	−	0.950226i	$-0.600852\pi$
$449$	−125622.	−0.623124	−0.311562	+	0.950226i	$0.600852\pi$
$450$	0	0
$451$	472961.i	2.32526i
$452$	0	0
$453$	− 210658.i	− 1.02655i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	73357.2	0.351245	0.175622	−	0.984458i	$-0.443806\pi$
$457$	73357.2	0.351245	0.175622	+	0.984458i	$0.443806\pi$
$458$	0	0
$459$	85.8740	0.000407602 0
$460$	0	0
$461$	− 184410.i	− 0.867728i	−0.900978	−	0.433864i	$-0.857150\pi$
$461$	− 184410.i	− 0.867728i	0.900978	−	0.433864i	$-0.142850\pi$
$462$	0	0
$463$	219357.	1.02327	0.511635	−	0.859203i	$-0.329040\pi$
$463$	219357.	1.02327	0.511635	+	0.859203i	$0.329040\pi$
$464$	0	0
$465$	− 6729.45i	− 0.0311224i
$466$	0	0
$467$	363598.i	1.66720i	0.552369	+	0.833600i	$0.313724\pi$
$467$	363598.i	1.66720i	−0.552369	+	0.833600i	$0.686276\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	461407.	2.07990
$472$	0	0
$473$	448990.	2.00685
$474$	0	0
$475$	− 85867.3i	− 0.380575i
$476$	0	0
$477$	−119571.	−0.525521
$478$	0	0
$479$	235276.i	1.02543i	0.858558	+	0.512716i	$0.171361\pi$
$479$	235276.i	1.02543i	−0.858558	+	0.512716i	$0.828639\pi$
$480$	0	0
$481$	132534.i	0.572846i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	331261.	1.40827
$486$	0	0
$487$	−58242.6	−0.245574	−0.122787	−	0.992433i	$-0.539183\pi$
$487$	−58242.6	−0.245574	−0.122787	+	0.992433i	$0.539183\pi$
$488$	0	0
$489$	101831.i	0.425856i
$490$	0	0
$491$	−221076.	−0.917020	−0.458510	−	0.888689i	$-0.651617\pi$
$491$	−221076.	−0.917020	−0.458510	+	0.888689i	$0.651617\pi$
$492$	0	0
$493$	788.532i	0.00324434i
$494$	0	0
$495$	− 386113.i	− 1.57581i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	351758.	1.41268	0.706338	−	0.707875i	$-0.250346\pi$
$499$	351758.	1.41268	0.706338	+	0.707875i	$0.250346\pi$
$500$	0	0
$501$	−240123.	−0.956662
$502$	0	0
$503$	78708.4i	0.311089i	0.987829	+	0.155545i	$0.0497133\pi$
$503$	78708.4i	0.311089i	−0.987829	+	0.155545i	$0.950287\pi$
$504$	0	0
$505$	−248992.	−0.976342
$506$	0	0
$507$	− 315419.i	− 1.22708i
$508$	0	0
$509$	98012.3i	0.378308i	0.981947	+	0.189154i	$0.0605745\pi$
$509$	98012.3i	0.378308i	−0.981947	+	0.189154i	$0.939426\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	8517.16	0.0323639
$514$	0	0
$515$	−510516.	−1.92484
$516$	0	0
$517$	− 57500.2i	− 0.215124i
$518$	0	0
$519$	−157849.	−0.586012
$520$	0	0
$521$	383220.i	1.41180i	0.708312	+	0.705899i	$0.249457\pi$
$521$	383220.i	1.41180i	−0.708312	+	0.705899i	$0.750543\pi$
$522$	0	0
$523$	261385.i	0.955602i	0.878468	+	0.477801i	$0.158566\pi$
$523$	261385.i	0.955602i	−0.878468	+	0.477801i	$0.841434\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	27.1257	9.76698e−5 0
$528$	0	0
$529$	57955.8	0.207103
$530$	0	0
$531$	− 224969.i	− 0.797872i
$532$	0	0
$533$	186900.	0.657891
$534$	0	0
$535$	459930.i	1.60688i
$536$	0	0
$537$	− 351617.i	− 1.21933i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−298637.	−1.02035	−0.510174	−	0.860071i	$-0.670419\pi$
$541$	−298637.	−1.02035	−0.510174	+	0.860071i	$0.670419\pi$
$542$	0	0
$543$	−130543.	−0.442744
$544$	0	0
$545$	467680.i	1.57455i
$546$	0	0
$547$	−504515.	−1.68616	−0.843082	−	0.537785i	$-0.819261\pi$
$547$	−504515.	−1.68616	−0.843082	+	0.537785i	$0.819261\pi$
$548$	0	0
$549$	− 448066.i	− 1.48661i
$550$	0	0
$551$	78208.3i	0.257602i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−952187.	−3.09127
$556$	0	0
$557$	−353616.	−1.13978	−0.569891	−	0.821720i	$-0.693015\pi$
$557$	−353616.	−1.13978	−0.569891	+	0.821720i	$0.693015\pi$
$558$	0	0
$559$	− 177427.i	− 0.567801i
$560$	0	0
$561$	3193.75	0.0101479
$562$	0	0
$563$	− 196934.i	− 0.621303i	−0.950524	−	0.310652i	$-0.899453\pi$
$563$	− 196934.i	− 0.621303i	0.950524	−	0.310652i	$-0.100547\pi$
$564$	0	0
$565$	153818.i	0.481849i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−206372.	−0.637422	−0.318711	−	0.947852i	$-0.603250\pi$
$569$	−206372.	−0.637422	−0.318711	+	0.947852i	$0.603250\pi$
$570$	0	0
$571$	−145009.	−0.444757	−0.222378	−	0.974960i	$-0.571382\pi$
$571$	−145009.	−0.444757	−0.222378	+	0.974960i	$0.571382\pi$
$572$	0	0
$573$	− 220037.i	− 0.670172i
$574$	0	0
$575$	295081.	0.892496
$576$	0	0
$577$	28699.5i	0.0862031i	0.999071	+	0.0431016i	$0.0137239\pi$
$577$	28699.5i	0.0862031i	−0.999071	+	0.0431016i	$0.986276\pi$
$578$	0	0
$579$	− 143917.i	− 0.429295i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	231405.	0.680827
$584$	0	0
$585$	−152580.	−0.445847
$586$	0	0
$587$	− 125084.i	− 0.363015i	−0.983390	−	0.181508i	$-0.941902\pi$
$587$	− 125084.i	− 0.363015i	0.983390	−	0.181508i	$-0.0580977\pi$
$588$	0	0
$589$	2690.39	0.00775504
$590$	0	0
$591$	433963.i	1.24245i
$592$	0	0
$593$	124044.i	0.352749i	0.984323	+	0.176375i	$0.0564370\pi$
$593$	124044.i	0.352749i	−0.984323	+	0.176375i	$0.943563\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	306635.	0.860346
$598$	0	0
$599$	−401469.	−1.11892	−0.559459	−	0.828858i	$-0.688991\pi$
$599$	−401469.	−1.11892	−0.559459	+	0.828858i	$0.688991\pi$
$600$	0	0
$601$	− 182796.i	− 0.506078i	−0.967456	−	0.253039i	$-0.918570\pi$
$601$	− 182796.i	− 0.506078i	0.967456	−	0.253039i	$-0.0814301\pi$
$602$	0	0
$603$	810.440	0.00222888
$604$	0	0
$605$	254488.i	0.695274i
$606$	0	0
$607$	− 211063.i	− 0.572843i	−0.958104	−	0.286421i	$-0.907534\pi$
$607$	− 211063.i	− 0.572843i	0.958104	−	0.286421i	$-0.0924657\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−22722.3	−0.0608653
$612$	0	0
$613$	−602935.	−1.60454	−0.802268	−	0.596964i	$-0.796374\pi$
$613$	−602935.	−1.60454	−0.802268	+	0.596964i	$0.796374\pi$
$614$	0	0
$615$	1.34277e6i	3.55020i
$616$	0	0
$617$	−408191.	−1.07224	−0.536122	−	0.844141i	$-0.680111\pi$
$617$	−408191.	−1.07224	−0.536122	+	0.844141i	$0.680111\pi$
$618$	0	0
$619$	− 647925.i	− 1.69100i	−0.533976	−	0.845499i	$-0.679303\pi$
$619$	− 647925.i	− 1.69100i	0.533976	−	0.845499i	$-0.320697\pi$
$620$	0	0
$621$	29269.1i	0.0758972i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−450175.	−1.15245
$626$	0	0
$627$	316763.	0.805748
$628$	0	0
$629$	− 3838.17i	− 0.00970115i
$630$	0	0
$631$	634575.	1.59377	0.796883	−	0.604134i	$-0.206481\pi$
$631$	634575.	1.59377	0.796883	+	0.604134i	$0.206481\pi$
$632$	0	0
$633$	751574.i	1.87570i
$634$	0	0
$635$	− 312096.i	− 0.774001i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−312945.	−0.766418
$640$	0	0
$641$	−51043.1	−0.124228	−0.0621142	−	0.998069i	$-0.519784\pi$
$641$	−51043.1	−0.124228	−0.0621142	+	0.998069i	$0.519784\pi$
$642$	0	0
$643$	− 212952.i	− 0.515061i	−0.966270	−	0.257531i	$-0.917091\pi$
$643$	− 212952.i	− 0.515061i	0.966270	−	0.257531i	$-0.0829088\pi$
$644$	0	0
$645$	1.27472e6	3.06404
$646$	0	0
$647$	− 323410.i	− 0.772582i	−0.922377	−	0.386291i	$-0.873756\pi$
$647$	− 323410.i	− 0.772582i	0.922377	−	0.386291i	$-0.126244\pi$
$648$	0	0
$649$	435380.i	1.03366i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	226839.	0.531976	0.265988	−	0.963976i	$-0.414302\pi$
$653$	226839.	0.531976	0.265988	+	0.963976i	$0.414302\pi$
$654$	0	0
$655$	130744.	0.304748
$656$	0	0
$657$	− 81814.5i	− 0.189539i
$658$	0	0
$659$	−280922.	−0.646867	−0.323433	−	0.946251i	$-0.604837\pi$
$659$	−280922.	−0.646867	−0.323433	+	0.946251i	$0.604837\pi$
$660$	0	0
$661$	− 409895.i	− 0.938145i	−0.883160	−	0.469073i	$-0.844588\pi$
$661$	− 409895.i	− 0.938145i	0.883160	−	0.469073i	$-0.155412\pi$
$662$	0	0
$663$	− 1262.07i	− 0.00287116i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−268761.	−0.604109
$668$	0	0
$669$	973113.	2.17426
$670$	0	0
$671$	867139.i	1.92595i
$672$	0	0
$673$	619982.	1.36883	0.684414	−	0.729094i	$-0.260058\pi$
$673$	619982.	1.36883	0.684414	+	0.729094i	$0.260058\pi$
$674$	0	0
$675$	25567.9i	0.0561162i
$676$	0	0
$677$	− 487445.i	− 1.06353i	−0.846893	−	0.531763i	$-0.821530\pi$
$677$	− 487445.i	− 1.06353i	0.846893	−	0.531763i	$-0.178470\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−966337.	−2.08370
$682$	0	0
$683$	629713.	1.34990	0.674949	−	0.737864i	$-0.264166\pi$
$683$	629713.	1.34990	0.674949	+	0.737864i	$0.264166\pi$
$684$	0	0
$685$	876790.i	1.86859i
$686$	0	0
$687$	915548.	1.93985
$688$	0	0
$689$	− 91444.3i	− 0.192628i
$690$	0	0
$691$	797932.i	1.67113i	0.549394	+	0.835564i	$0.314859\pi$
$691$	797932.i	1.67113i	−0.549394	+	0.835564i	$0.685141\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	619231.	1.28198
$696$	0	0
$697$	−5412.59	−0.0111414
$698$	0	0
$699$	− 926929.i	− 1.89711i
$700$	0	0
$701$	−104064.	−0.211769	−0.105885	−	0.994378i	$-0.533767\pi$
$701$	−104064.	−0.211769	−0.105885	+	0.994378i	$0.533767\pi$
$702$	0	0
$703$	− 380678.i	− 0.770277i
$704$	0	0
$705$	− 163248.i	− 0.328449i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	986633.	1.96274	0.981371	−	0.192123i	$-0.0615371\pi$
$709$	986633.	1.96274	0.981371	+	0.192123i	$0.0615371\pi$
$710$	0	0
$711$	328650.	0.650120
$712$	0	0
$713$	9245.46i	0.0181865i
$714$	0	0
$715$	295287.	0.577606
$716$	0	0
$717$	− 352798.i	− 0.686259i
$718$	0	0
$719$	− 319068.i	− 0.617199i	−0.951192	−	0.308600i	$-0.900140\pi$
$719$	− 319068.i	− 0.617199i	0.951192	−	0.308600i	$-0.0998603\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−106903.	−0.204510
$724$	0	0
$725$	−234776.	−0.446660
$726$	0	0
$727$	214869.i	0.406542i	0.979123	+	0.203271i	$0.0651572\pi$
$727$	214869.i	0.406542i	−0.979123	+	0.203271i	$0.934843\pi$
$728$	0	0
$729$	477632.	0.898749
$730$	0	0
$731$	5138.26i	0.00961571i
$732$	0	0
$733$	103272.i	0.192209i	0.995371	+	0.0961046i	$0.0306383\pi$
$733$	103272.i	0.192209i	−0.995371	+	0.0961046i	$0.969362\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1568.44	−0.00288757
$738$	0	0
$739$	−908518.	−1.66358	−0.831792	−	0.555087i	$-0.812685\pi$
$739$	−908518.	−1.66358	−0.831792	+	0.555087i	$0.812685\pi$
$740$	0	0
$741$	− 125175.i	− 0.227972i
$742$	0	0
$743$	−1.04905e6	−1.90028	−0.950142	−	0.311817i	$-0.899062\pi$
$743$	−1.04905e6	−1.90028	−0.950142	+	0.311817i	$0.899062\pi$
$744$	0	0
$745$	371878.i	0.670020i
$746$	0	0
$747$	339276.i	0.608012i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−404156.	−0.716588	−0.358294	−	0.933609i	$-0.616641\pi$
$751$	−404156.	−0.716588	−0.358294	+	0.933609i	$0.616641\pi$
$752$	0	0
$753$	577649.	1.01877
$754$	0	0
$755$	564050.i	0.989518i
$756$	0	0
$757$	−74517.7	−0.130037	−0.0650186	−	0.997884i	$-0.520711\pi$
$757$	−74517.7	−0.130037	−0.0650186	+	0.997884i	$0.520711\pi$
$758$	0	0
$759$	1.08855e6i	1.88958i
$760$	0	0
$761$	− 6428.89i	− 0.0111011i	−0.999985	−	0.00555056i	$-0.998233\pi$
$761$	− 6428.89i	− 0.0111011i	0.999985	−	0.00555056i	$-0.00176681\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	4418.69	0.00755042
$766$	0	0
$767$	172049.	0.292457
$768$	0	0
$769$	− 358812.i	− 0.606756i	−0.952870	−	0.303378i	$-0.901886\pi$
$769$	− 358812.i	− 0.606756i	0.952870	−	0.303378i	$-0.0981145\pi$
$770$	0	0
$771$	−531225.	−0.893655
$772$	0	0
$773$	− 560384.i	− 0.937835i	−0.883242	−	0.468918i	$-0.844644\pi$
$773$	− 560384.i	− 0.937835i	0.883242	−	0.468918i	$-0.155356\pi$
$774$	0	0
$775$	8076.35i	0.0134466i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−536832.	−0.884633
$780$	0	0
$781$	605639.	0.992915
$782$	0	0
$783$	− 23287.4i	− 0.0379836i
$784$	0	0
$785$	−1.23545e6	−2.00486
$786$	0	0
$787$	− 809526.i	− 1.30702i	−0.756919	−	0.653509i	$-0.773296\pi$
$787$	− 809526.i	− 1.30702i	0.756919	−	0.653509i	$-0.226704\pi$
$788$	0	0
$789$	598791.i	0.961881i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	342667.	0.544911
$794$	0	0
$795$	656979.	1.03948
$796$	0	0
$797$	367836.i	0.579078i	0.957166	+	0.289539i	$0.0935020\pi$
$797$	367836.i	0.579078i	−0.957166	+	0.289539i	$0.906498\pi$
$798$	0	0
$799$	658.035	0.00103075
$800$	0	0
$801$	− 612693.i	− 0.954944i
$802$	0	0
$803$	158335.i	0.245553i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.64892e6	2.53194
$808$	0	0
$809$	368305.	0.562744	0.281372	−	0.959599i	$-0.409210\pi$
$809$	368305.	0.562744	0.281372	+	0.959599i	$0.409210\pi$
$810$	0	0
$811$	− 1.19992e6i	− 1.82435i	−0.409798	−	0.912176i	$-0.634401\pi$
$811$	− 1.19992e6i	− 1.82435i	0.409798	−	0.912176i	$-0.365599\pi$
$812$	0	0
$813$	−761412.	−1.15196
$814$	0	0
$815$	− 272659.i	− 0.410492i
$816$	0	0
$817$	509623.i	0.763493i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.07580e6	−1.59604	−0.798022	−	0.602628i	$-0.794120\pi$
$821$	−1.07580e6	−1.59604	−0.798022	+	0.602628i	$0.794120\pi$
$822$	0	0
$823$	−114826.	−0.169528	−0.0847642	−	0.996401i	$-0.527014\pi$
$823$	−114826.	−0.169528	−0.0847642	+	0.996401i	$0.527014\pi$
$824$	0	0
$825$	950900.i	1.39710i
$826$	0	0
$827$	734458.	1.07388	0.536940	−	0.843620i	$-0.319580\pi$
$827$	734458.	1.07388	0.536940	+	0.843620i	$0.319580\pi$
$828$	0	0
$829$	− 1.10906e6i	− 1.61379i	−0.590693	−	0.806896i	$-0.701146\pi$
$829$	− 1.10906e6i	− 1.61379i	0.590693	−	0.806896i	$-0.298854\pi$
$830$	0	0
$831$	− 231680.i	− 0.335495i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	642944.	0.922147
$836$	0	0
$837$	−801.092	−0.00114349
$838$	0	0
$839$	308748.i	0.438612i	0.975656	+	0.219306i	$0.0703792\pi$
$839$	308748.i	0.438612i	−0.975656	+	0.219306i	$0.929621\pi$
$840$	0	0
$841$	−493446.	−0.697667
$842$	0	0
$843$	− 289877.i	− 0.407905i
$844$	0	0
$845$	844553.i	1.18281i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.15975e6	1.60898
$850$	0	0
$851$	1.30819e6	1.80639
$852$	0	0
$853$	− 1.02011e6i	− 1.40200i	−0.713160	−	0.701001i	$-0.752737\pi$
$853$	− 1.02011e6i	− 1.40200i	0.713160	−	0.701001i	$-0.247263\pi$
$854$	0	0
$855$	438255.	0.599508
$856$	0	0
$857$	− 545546.i	− 0.742797i	−0.928474	−	0.371398i	$-0.878878\pi$
$857$	− 545546.i	− 0.742797i	0.928474	−	0.371398i	$-0.121122\pi$
$858$	0	0
$859$	708517.i	0.960205i	0.877213	+	0.480102i	$0.159400\pi$
$859$	708517.i	0.960205i	−0.877213	+	0.480102i	$0.840600\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	178941.	0.240263	0.120132	−	0.992758i	$-0.461668\pi$
$863$	178941.	0.240263	0.120132	+	0.992758i	$0.461668\pi$
$864$	0	0
$865$	422649.	0.564869
$866$	0	0
$867$	− 1.04978e6i	− 1.39657i
$868$	0	0
$869$	−636033.	−0.842248
$870$	0	0
$871$	619.799i	0 0.000816985i
$872$	0	0
$873$	757818.i	0.994344i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	662846.	0.861814	0.430907	−	0.902396i	$-0.358194\pi$
$877$	662846.	0.861814	0.430907	+	0.902396i	$0.358194\pi$
$878$	0	0
$879$	1.30770e6	1.69251
$880$	0	0
$881$	687434.i	0.885685i	0.896599	+	0.442843i	$0.146030\pi$
$881$	687434.i	0.885685i	−0.896599	+	0.442843i	$0.853970\pi$
$882$	0	0
$883$	−670169.	−0.859534	−0.429767	−	0.902940i	$-0.641404\pi$
$883$	−670169.	−0.859534	−0.429767	+	0.902940i	$0.641404\pi$
$884$	0	0
$885$	1.23608e6i	1.57819i
$886$	0	0
$887$	1.33538e6i	1.69730i	0.528954	+	0.848650i	$0.322584\pi$
$887$	1.33538e6i	1.69730i	−0.528954	+	0.848650i	$0.677416\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.02359e6	−1.28934
$892$	0	0
$893$	65265.2	0.0818425
$894$	0	0
$895$	941477.i	1.17534i
$896$	0	0
$897$	430162.	0.534622
$898$	0	0
$899$	− 7355.97i	− 0.00910166i
$900$	0	0
$901$	2648.22i	0.00326215i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	349536.	0.426771
$906$	0	0
$907$	280160.	0.340559	0.170279	−	0.985396i	$-0.445533\pi$
$907$	280160.	0.340559	0.170279	+	0.985396i	$0.445533\pi$
$908$	0	0
$909$	− 569613.i	− 0.689370i
$910$	0	0
$911$	−872536.	−1.05135	−0.525674	−	0.850686i	$-0.676187\pi$
$911$	−872536.	−1.05135	−0.525674	+	0.850686i	$0.676187\pi$
$912$	0	0
$913$	− 656599.i	− 0.787696i
$914$	0	0
$915$	2.46188e6i	2.94052i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	924044.	1.09411	0.547056	−	0.837096i	$-0.315749\pi$
$919$	924044.	1.09411	0.547056	+	0.837096i	$0.315749\pi$
$920$	0	0
$921$	1.25306e6	1.47725
$922$	0	0
$923$	− 239330.i	− 0.280927i
$924$	0	0
$925$	1.14277e6	1.33559
$926$	0	0
$927$	− 1.16790e6i	− 1.35908i
$928$	0	0
$929$	− 895977.i	− 1.03816i	−0.854725	−	0.519082i	$-0.826274\pi$
$929$	− 895977.i	− 1.03816i	0.854725	−	0.519082i	$-0.173726\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	139231.	0.159946
$934$	0	0
$935$	−8551.47	−0.00978177
$936$	0	0
$937$	− 1.18796e6i	− 1.35308i	−0.736405	−	0.676541i	$-0.763478\pi$
$937$	− 1.18796e6i	− 1.35308i	0.736405	−	0.676541i	$-0.236522\pi$
$938$	0	0
$939$	1.53647e6	1.74258
$940$	0	0
$941$	62559.0i	0.0706498i	0.999376	+	0.0353249i	$0.0112466\pi$
$941$	62559.0i	0.0706498i	−0.999376	+	0.0353249i	$0.988753\pi$
$942$	0	0
$943$	− 1.84481e6i	− 2.07457i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	217615.	0.242655	0.121328	−	0.992613i	$-0.461285\pi$
$947$	217615.	0.242655	0.121328	+	0.992613i	$0.461285\pi$
$948$	0	0
$949$	62569.2	0.0694749
$950$	0	0
$951$	− 405560.i	− 0.448429i
$952$	0	0
$953$	291854.	0.321352	0.160676	−	0.987007i	$-0.448633\pi$
$953$	291854.	0.321352	0.160676	+	0.987007i	$0.448633\pi$
$954$	0	0
$955$	589162.i	0.645993i
$956$	0	0
$957$	− 866083.i	− 0.945661i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	923268.	0.999726
$962$	0	0
$963$	−1.05217e6	−1.13458
$964$	0	0
$965$	385348.i	0.413807i
$966$	0	0
$967$	15778.3	0.0168735	0.00843677	−	0.999964i	$-0.497314\pi$
$967$	15778.3	0.0168735	0.00843677	+	0.999964i	$0.497314\pi$
$968$	0	0
$969$	3625.05i	0.00386070i
$970$	0	0
$971$	− 1.25427e6i	− 1.33031i	−0.746705	−	0.665155i	$-0.768365\pi$
$971$	− 1.25427e6i	− 1.33031i	0.746705	−	0.665155i	$-0.231635\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	375766.	0.395284
$976$	0	0
$977$	−384179.	−0.402480	−0.201240	−	0.979542i	$-0.564497\pi$
$977$	−384179.	−0.402480	−0.201240	+	0.979542i	$0.564497\pi$
$978$	0	0
$979$	1.18574e6i	1.23716i
$980$	0	0
$981$	−1.06990e6	−1.11175
$982$	0	0
$983$	1.06846e6i	1.10574i	0.833268	+	0.552869i	$0.186467\pi$
$983$	1.06846e6i	1.10574i	−0.833268	+	0.552869i	$0.813533\pi$
$984$	0	0
$985$	− 1.16196e6i	− 1.19762i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.75131e6	−1.79049
$990$	0	0
$991$	1.84807e6	1.88179	0.940896	−	0.338694i	$-0.109985\pi$
$991$	1.84807e6	1.88179	0.940896	+	0.338694i	$0.109985\pi$
$992$	0	0
$993$	1.30846e6i	1.32698i
$994$	0	0
$995$	−821034.	−0.829306
$996$	0	0
$997$	1.61892e6i	1.62868i	0.580388	+	0.814340i	$0.302901\pi$
$997$	1.61892e6i	1.62868i	−0.580388	+	0.814340i	$0.697099\pi$
$998$	0	0
$999$	113351.i	0.113578i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.5.c.g.97.2		8
4.3	odd	2		49.5.b.b.48.4	yes	8
7.6	odd	2	inner	784.5.c.g.97.7		8
12.11	even	2		441.5.d.g.244.5		8
28.3	even	6		49.5.d.c.19.5		16
28.11	odd	6		49.5.d.c.19.6		16
28.19	even	6		49.5.d.c.31.6		16
28.23	odd	6		49.5.d.c.31.5		16
28.27	even	2		49.5.b.b.48.3	✓	8
84.83	odd	2		441.5.d.g.244.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.5.b.b.48.3	✓	8	28.27	even	2
49.5.b.b.48.4	yes	8	4.3	odd	2
49.5.d.c.19.5		16	28.3	even	6
49.5.d.c.19.6		16	28.11	odd	6
49.5.d.c.31.5		16	28.23	odd	6
49.5.d.c.31.6		16	28.19	even	6
441.5.d.g.244.5		8	12.11	even	2
441.5.d.g.244.6		8	84.83	odd	2
784.5.c.g.97.2		8	1.1	even	1	trivial
784.5.c.g.97.7		8	7.6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	784.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-12.5695i q^{3} +33.6557i q^{5} -76.9935 q^{9} +149.005 q^{11} -58.8822i q^{13} +423.037 q^{15} +1.70522i q^{17} +169.127i q^{19} -581.203 q^{23} -507.708 q^{25} -50.3595i q^{27} +462.422 q^{29} -15.9075i q^{31} -1872.93i q^{33} -2250.84 q^{37} -740.123 q^{39} +3174.13i q^{41} +3013.25 q^{43} -2591.27i q^{45} -385.894i q^{47} +21.4339 q^{51} +1553.00 q^{53} +5014.87i q^{55} +2125.85 q^{57} +2921.92i q^{59} +5819.53i q^{61} +1981.72 q^{65} -10.5261 q^{67} +7305.46i q^{69} +4064.56 q^{71} +1062.62i q^{73} +6381.66i q^{75} -4268.53 q^{79} -6869.47 q^{81} -4406.56i q^{83} -57.3905 q^{85} -5812.44i q^{87} +7957.72i q^{89} -199.950 q^{93} -5692.10 q^{95} -9842.62i q^{97} -11472.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 352 q^{9} - 120 q^{11} + 632 q^{15} - 1752 q^{23} - 2192 q^{25} + 1248 q^{29} + 2368 q^{37} + 7672 q^{39} + 8552 q^{43} + 11976 q^{51} + 5496 q^{53} - 9200 q^{57} + 30240 q^{65} + 7440 q^{67} - 9984 q^{71}+ \cdots - 30144 q^{99}+O(q^{100}) \) 8 * q - 352 * q^9 - 120 * q^11 + 632 * q^15 - 1752 * q^23 - 2192 * q^25 + 1248 * q^29 + 2368 * q^37 + 7672 * q^39 + 8552 * q^43 + 11976 * q^51 + 5496 * q^53 - 9200 * q^57 + 30240 * q^65 + 7440 * q^67 - 9984 * q^71 + 14096 * q^79 + 3432 * q^81 + 11912 * q^85 + 9584 * q^93 - 22488 * q^95 - 30144 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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