LMFDB - Embedded newform 448.8.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [448,8,Mod(1,448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("448.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	448.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-66,0,400,0,343] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$139.948491417$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	448.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-66.0000 q^{3} +400.000 q^{5} +343.000 q^{7} +2169.00 q^{9} +40.0000 q^{11} +4452.00 q^{13} -26400.0 q^{15} +36502.0 q^{17} -46222.0 q^{19} -22638.0 q^{21} +105200. q^{23} +81875.0 q^{25} +1188.00 q^{27} +126334. q^{29} +170964. q^{31} -2640.00 q^{33} +137200. q^{35} -20954.0 q^{37} -293832. q^{39} +318486. q^{41} +77744.0 q^{43} +867600. q^{45} -703716. q^{47} +117649. q^{49} -2.40913e6 q^{51} -1.60328e6 q^{53} +16000.0 q^{55} +3.05065e6 q^{57} -1.17189e6 q^{59} +2.06887e6 q^{61} +743967. q^{63} +1.78080e6 q^{65} -994268. q^{67} -6.94320e6 q^{69} -33280.0 q^{71} -2.97145e6 q^{73} -5.40375e6 q^{75} +13720.0 q^{77} +2.37617e6 q^{79} -4.82201e6 q^{81} -2.12236e6 q^{83} +1.46008e7 q^{85} -8.33804e6 q^{87} +6.92035e6 q^{89} +1.52704e6 q^{91} -1.12836e7 q^{93} -1.84888e7 q^{95} +4.95271e6 q^{97} +86760.0 q^{99} +O(q^{100})$

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
$3$	−66.0000	−1.41130	−0.705650	−	0.708560i	$-0.749345\pi$
$3$	−66.0000	−1.41130	−0.705650	+	0.708560i	$0.749345\pi$
$4$	0	0
$5$	400.000	1.43108	0.715542	−	0.698570i	$-0.246180\pi$
$5$	400.000	1.43108	0.715542	+	0.698570i	$0.246180\pi$
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	0	0
$9$	2169.00	0.991770
$10$	0	0
$11$	40.0000	0.00906120	0.00453060	−	0.999990i	$-0.498558\pi$
$11$	40.0000	0.00906120	0.00453060	+	0.999990i	$0.498558\pi$
$12$	0	0
$13$	4452.00	0.562022	0.281011	−	0.959705i	$-0.409330\pi$
$13$	4452.00	0.562022	0.281011	+	0.959705i	$0.409330\pi$
$14$	0	0
$15$	−26400.0	−2.01969
$16$	0	0
$17$	36502.0	1.80196	0.900981	−	0.433859i	$-0.142849\pi$
$17$	36502.0	1.80196	0.900981	+	0.433859i	$0.142849\pi$
$18$	0	0
$19$	−46222.0	−1.54601	−0.773003	−	0.634402i	$-0.781246\pi$
$19$	−46222.0	−1.54601	−0.773003	+	0.634402i	$0.781246\pi$
$20$	0	0
$21$	−22638.0	−0.533422
$22$	0	0
$23$	105200.	1.80289	0.901443	−	0.432898i	$-0.142509\pi$
$23$	105200.	1.80289	0.901443	+	0.432898i	$0.142509\pi$
$24$	0	0
$25$	81875.0	1.04800
$26$	0	0
$27$	1188.00	0.0116156
$28$	0	0
$29$	126334.	0.961894	0.480947	−	0.876750i	$-0.340293\pi$
$29$	126334.	0.961894	0.480947	+	0.876750i	$0.340293\pi$
$30$	0	0
$31$	170964.	1.03072	0.515358	−	0.856975i	$-0.327659\pi$
$31$	170964.	1.03072	0.515358	+	0.856975i	$0.327659\pi$
$32$	0	0
$33$	−2640.00	−0.0127881
$34$	0	0
$35$	137200.	0.540899
$36$	0	0
$37$	−20954.0	−0.0680081	−0.0340041	−	0.999422i	$-0.510826\pi$
$37$	−20954.0	−0.0680081	−0.0340041	+	0.999422i	$0.510826\pi$
$38$	0	0
$39$	−293832.	−0.793182
$40$	0	0
$41$	318486.	0.721684	0.360842	−	0.932627i	$-0.382489\pi$
$41$	318486.	0.721684	0.360842	+	0.932627i	$0.382489\pi$
$42$	0	0
$43$	77744.0	0.149117	0.0745585	−	0.997217i	$-0.476245\pi$
$43$	77744.0	0.149117	0.0745585	+	0.997217i	$0.476245\pi$
$44$	0	0
$45$	867600.	1.41931
$46$	0	0
$47$	−703716.	−0.988678	−0.494339	−	0.869269i	$-0.664590\pi$
$47$	−703716.	−0.988678	−0.494339	+	0.869269i	$0.664590\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	−2.40913e6	−2.54311
$52$	0	0
$53$	−1.60328e6	−1.47926	−0.739628	−	0.673016i	$-0.764998\pi$
$53$	−1.60328e6	−1.47926	−0.739628	+	0.673016i	$0.764998\pi$
$54$	0	0
$55$	16000.0	0.0129673
$56$	0	0
$57$	3.05065e6	2.18188
$58$	0	0
$59$	−1.17189e6	−0.742859	−0.371429	−	0.928461i	$-0.621132\pi$
$59$	−1.17189e6	−0.742859	−0.371429	+	0.928461i	$0.621132\pi$
$60$	0	0
$61$	2.06887e6	1.16702	0.583511	−	0.812105i	$-0.301678\pi$
$61$	2.06887e6	1.16702	0.583511	+	0.812105i	$0.301678\pi$
$62$	0	0
$63$	743967.	0.374854
$64$	0	0
$65$	1.78080e6	0.804301
$66$	0	0
$67$	−994268.	−0.403870	−0.201935	−	0.979399i	$-0.564723\pi$
$67$	−994268.	−0.403870	−0.201935	+	0.979399i	$0.564723\pi$
$68$	0	0
$69$	−6.94320e6	−2.54441
$70$	0	0
$71$	−33280.0	−0.0110352	−0.00551759	−	0.999985i	$-0.501756\pi$
$71$	−33280.0	−0.0110352	−0.00551759	+	0.999985i	$0.501756\pi$
$72$	0	0
$73$	−2.97145e6	−0.894003	−0.447002	−	0.894533i	$-0.647508\pi$
$73$	−2.97145e6	−0.894003	−0.447002	+	0.894533i	$0.647508\pi$
$74$	0	0
$75$	−5.40375e6	−1.47904
$76$	0	0
$77$	13720.0	0.00342481
$78$	0	0
$79$	2.37617e6	0.542228	0.271114	−	0.962547i	$-0.412608\pi$
$79$	2.37617e6	0.542228	0.271114	+	0.962547i	$0.412608\pi$
$80$	0	0
$81$	−4.82201e6	−1.00816
$82$	0	0
$83$	−2.12236e6	−0.407423	−0.203711	−	0.979031i	$-0.565300\pi$
$83$	−2.12236e6	−0.407423	−0.203711	+	0.979031i	$0.565300\pi$
$84$	0	0
$85$	1.46008e7	2.57876
$86$	0	0
$87$	−8.33804e6	−1.35752
$88$	0	0
$89$	6.92035e6	1.04055	0.520275	−	0.853999i	$-0.325830\pi$
$89$	6.92035e6	1.04055	0.520275	+	0.853999i	$0.325830\pi$
$90$	0	0
$91$	1.52704e6	0.212424
$92$	0	0
$93$	−1.12836e7	−1.45465
$94$	0	0
$95$	−1.84888e7	−2.21246
$96$	0	0
$97$	4.95271e6	0.550988	0.275494	−	0.961303i	$-0.411159\pi$
$97$	4.95271e6	0.550988	0.275494	+	0.961303i	$0.411159\pi$
$98$	0	0
$99$	86760.0	0.00898662
$100$	0	0
$101$	−3.23000e6	−0.311945	−0.155972	−	0.987761i	$-0.549851\pi$
$101$	−3.23000e6	−0.311945	−0.155972	+	0.987761i	$0.549851\pi$
$102$	0	0
$103$	1.79909e6	0.162227	0.0811135	−	0.996705i	$-0.474152\pi$
$103$	1.79909e6	0.162227	0.0811135	+	0.996705i	$0.474152\pi$
$104$	0	0
$105$	−9.05520e6	−0.763371
$106$	0	0
$107$	−1.56429e7	−1.23445	−0.617225	−	0.786787i	$-0.711743\pi$
$107$	−1.56429e7	−1.23445	−0.617225	+	0.786787i	$0.711743\pi$
$108$	0	0
$109$	6.31890e6	0.467357	0.233679	−	0.972314i	$-0.424924\pi$
$109$	6.31890e6	0.467357	0.233679	+	0.972314i	$0.424924\pi$
$110$	0	0
$111$	1.38296e6	0.0959799
$112$	0	0
$113$	−1.02288e7	−0.666881	−0.333441	−	0.942771i	$-0.608210\pi$
$113$	−1.02288e7	−0.666881	−0.333441	+	0.942771i	$0.608210\pi$
$114$	0	0
$115$	4.20800e7	2.58008
$116$	0	0
$117$	9.65639e6	0.557396
$118$	0	0
$119$	1.25202e7	0.681077
$120$	0	0
$121$	−1.94856e7	−0.999918
$122$	0	0
$123$	−2.10201e7	−1.01851
$124$	0	0
$125$	1.50000e6	0.0686920
$126$	0	0
$127$	−6.00725e6	−0.260233	−0.130117	−	0.991499i	$-0.541535\pi$
$127$	−6.00725e6	−0.260233	−0.130117	+	0.991499i	$0.541535\pi$
$128$	0	0
$129$	−5.13110e6	−0.210449
$130$	0	0
$131$	2.06396e7	0.802144	0.401072	−	0.916047i	$-0.368638\pi$
$131$	2.06396e7	0.802144	0.401072	+	0.916047i	$0.368638\pi$
$132$	0	0
$133$	−1.58541e7	−0.584335
$134$	0	0
$135$	475200.	0.0166230
$136$	0	0
$137$	4.76199e6	0.158222	0.0791109	−	0.996866i	$-0.474792\pi$
$137$	4.76199e6	0.158222	0.0791109	+	0.996866i	$0.474792\pi$
$138$	0	0
$139$	−5.05723e6	−0.159721	−0.0798604	−	0.996806i	$-0.525447\pi$
$139$	−5.05723e6	−0.159721	−0.0798604	+	0.996806i	$0.525447\pi$
$140$	0	0
$141$	4.64453e7	1.39532
$142$	0	0
$143$	178080.	0.00509259
$144$	0	0
$145$	5.05336e7	1.37655
$146$	0	0
$147$	−7.76483e6	−0.201614
$148$	0	0
$149$	2.72736e7	0.675447	0.337723	−	0.941245i	$-0.390343\pi$
$149$	2.72736e7	0.675447	0.337723	+	0.941245i	$0.390343\pi$
$150$	0	0
$151$	−6.48921e6	−0.153381	−0.0766906	−	0.997055i	$-0.524435\pi$
$151$	−6.48921e6	−0.153381	−0.0766906	+	0.997055i	$0.524435\pi$
$152$	0	0
$153$	7.91728e7	1.78713
$154$	0	0
$155$	6.83856e7	1.47504
$156$	0	0
$157$	6.30810e7	1.30092	0.650459	−	0.759541i	$-0.274577\pi$
$157$	6.30810e7	1.30092	0.650459	+	0.759541i	$0.274577\pi$
$158$	0	0
$159$	1.05816e8	2.08767
$160$	0	0
$161$	3.60836e7	0.681427
$162$	0	0
$163$	8.32271e7	1.50525	0.752624	−	0.658450i	$-0.228788\pi$
$163$	8.32271e7	1.50525	0.752624	+	0.658450i	$0.228788\pi$
$164$	0	0
$165$	−1.05600e6	−0.0183008
$166$	0	0
$167$	−3.06916e7	−0.509931	−0.254965	−	0.966950i	$-0.582064\pi$
$167$	−3.06916e7	−0.509931	−0.254965	+	0.966950i	$0.582064\pi$
$168$	0	0
$169$	−4.29282e7	−0.684131
$170$	0	0
$171$	−1.00256e8	−1.53328
$172$	0	0
$173$	5.27338e7	0.774333	0.387167	−	0.922010i	$-0.373454\pi$
$173$	5.27338e7	0.774333	0.387167	+	0.922010i	$0.373454\pi$
$174$	0	0
$175$	2.80831e7	0.396107
$176$	0	0
$177$	7.73450e7	1.04840
$178$	0	0
$179$	8.42739e7	1.09827	0.549133	−	0.835735i	$-0.314958\pi$
$179$	8.42739e7	1.09827	0.549133	+	0.835735i	$0.314958\pi$
$180$	0	0
$181$	1.03956e8	1.30309	0.651547	−	0.758608i	$-0.274120\pi$
$181$	1.03956e8	1.30309	0.651547	+	0.758608i	$0.274120\pi$
$182$	0	0
$183$	−1.36546e8	−1.64702
$184$	0	0
$185$	−8.38160e6	−0.0973253
$186$	0	0
$187$	1.46008e6	0.0163279
$188$	0	0
$189$	407484.	0.00439030
$190$	0	0
$191$	1.24775e8	1.29572	0.647861	−	0.761759i	$-0.275664\pi$
$191$	1.24775e8	1.29572	0.647861	+	0.761759i	$0.275664\pi$
$192$	0	0
$193$	1.47589e8	1.47776	0.738878	−	0.673839i	$-0.235356\pi$
$193$	1.47589e8	1.47776	0.738878	+	0.673839i	$0.235356\pi$
$194$	0	0
$195$	−1.17533e8	−1.13511
$196$	0	0
$197$	−1.55812e8	−1.45200	−0.726002	−	0.687692i	$-0.758624\pi$
$197$	−1.55812e8	−1.45200	−0.726002	+	0.687692i	$0.758624\pi$
$198$	0	0
$199$	1.33193e7	0.119810	0.0599052	−	0.998204i	$-0.480920\pi$
$199$	1.33193e7	0.119810	0.0599052	+	0.998204i	$0.480920\pi$
$200$	0	0
$201$	6.56217e7	0.569982
$202$	0	0
$203$	4.33326e7	0.363562
$204$	0	0
$205$	1.27394e8	1.03279
$206$	0	0
$207$	2.28179e8	1.78805
$208$	0	0
$209$	−1.84888e6	−0.0140087
$210$	0	0
$211$	−2.04940e8	−1.50189	−0.750945	−	0.660365i	$-0.770402\pi$
$211$	−2.04940e8	−1.50189	−0.750945	+	0.660365i	$0.770402\pi$
$212$	0	0
$213$	2.19648e6	0.0155739
$214$	0	0
$215$	3.10976e7	0.213399
$216$	0	0
$217$	5.86407e7	0.389574
$218$	0	0
$219$	1.96116e8	1.26171
$220$	0	0
$221$	1.62507e8	1.01274
$222$	0	0
$223$	6.84858e7	0.413555	0.206778	−	0.978388i	$-0.433702\pi$
$223$	6.84858e7	0.413555	0.206778	+	0.978388i	$0.433702\pi$
$224$	0	0
$225$	1.77587e8	1.03937
$226$	0	0
$227$	−1.93627e7	−0.109869	−0.0549344	−	0.998490i	$-0.517495\pi$
$227$	−1.93627e7	−0.109869	−0.0549344	+	0.998490i	$0.517495\pi$
$228$	0	0
$229$	−3.08157e8	−1.69569	−0.847847	−	0.530241i	$-0.822102\pi$
$229$	−3.08157e8	−1.69569	−0.847847	+	0.530241i	$0.822102\pi$
$230$	0	0
$231$	−905520.	−0.00483344
$232$	0	0
$233$	3.55797e7	0.184271	0.0921355	−	0.995746i	$-0.470631\pi$
$233$	3.55797e7	0.184271	0.0921355	+	0.995746i	$0.470631\pi$
$234$	0	0
$235$	−2.81486e8	−1.41488
$236$	0	0
$237$	−1.56827e8	−0.765247
$238$	0	0
$239$	2.30056e8	1.09004	0.545018	−	0.838424i	$-0.316523\pi$
$239$	2.30056e8	1.09004	0.545018	+	0.838424i	$0.316523\pi$
$240$	0	0
$241$	5.03495e6	0.0231705	0.0115853	−	0.999933i	$-0.496312\pi$
$241$	5.03495e6	0.0231705	0.0115853	+	0.999933i	$0.496312\pi$
$242$	0	0
$243$	3.15655e8	1.41121
$244$	0	0
$245$	4.70596e7	0.204441
$246$	0	0
$247$	−2.05780e8	−0.868890
$248$	0	0
$249$	1.40076e8	0.574996
$250$	0	0
$251$	−1.03283e8	−0.412258	−0.206129	−	0.978525i	$-0.566087\pi$
$251$	−1.03283e8	−0.412258	−0.206129	+	0.978525i	$0.566087\pi$
$252$	0	0
$253$	4.20800e6	0.0163363
$254$	0	0
$255$	−9.63653e8	−3.63940
$256$	0	0
$257$	−2.32282e8	−0.853592	−0.426796	−	0.904348i	$-0.640358\pi$
$257$	−2.32282e8	−0.853592	−0.426796	+	0.904348i	$0.640358\pi$
$258$	0	0
$259$	−7.18722e6	−0.0257047
$260$	0	0
$261$	2.74018e8	0.953977
$262$	0	0
$263$	−4.16749e8	−1.41263	−0.706317	−	0.707896i	$-0.749644\pi$
$263$	−4.16749e8	−1.41263	−0.706317	+	0.707896i	$0.749644\pi$
$264$	0	0
$265$	−6.41311e8	−2.11694
$266$	0	0
$267$	−4.56743e8	−1.46853
$268$	0	0
$269$	3.14679e8	0.985676	0.492838	−	0.870121i	$-0.335959\pi$
$269$	3.14679e8	0.985676	0.492838	+	0.870121i	$0.335959\pi$
$270$	0	0
$271$	−1.92137e8	−0.586433	−0.293216	−	0.956046i	$-0.594726\pi$
$271$	−1.92137e8	−0.586433	−0.293216	+	0.956046i	$0.594726\pi$
$272$	0	0
$273$	−1.00784e8	−0.299795
$274$	0	0
$275$	3.27500e6	0.00949613
$276$	0	0
$277$	4.40393e8	1.24498	0.622489	−	0.782629i	$-0.286122\pi$
$277$	4.40393e8	1.24498	0.622489	+	0.782629i	$0.286122\pi$
$278$	0	0
$279$	3.70821e8	1.02223
$280$	0	0
$281$	3.59235e8	0.965842	0.482921	−	0.875664i	$-0.339576\pi$
$281$	3.59235e8	0.965842	0.482921	+	0.875664i	$0.339576\pi$
$282$	0	0
$283$	−8.11467e7	−0.212823	−0.106411	−	0.994322i	$-0.533936\pi$
$283$	−8.11467e7	−0.212823	−0.106411	+	0.994322i	$0.533936\pi$
$284$	0	0
$285$	1.22026e9	3.12245
$286$	0	0
$287$	1.09241e8	0.272771
$288$	0	0
$289$	9.22057e8	2.24706
$290$	0	0
$291$	−3.26879e8	−0.777609
$292$	0	0
$293$	−2.53416e8	−0.588569	−0.294285	−	0.955718i	$-0.595081\pi$
$293$	−2.53416e8	−0.588569	−0.294285	+	0.955718i	$0.595081\pi$
$294$	0	0
$295$	−4.68758e8	−1.06309
$296$	0	0
$297$	47520.0	0.000105252 0
$298$	0	0
$299$	4.68350e8	1.01326
$300$	0	0
$301$	2.66662e7	0.0563609
$302$	0	0
$303$	2.13180e8	0.440248
$304$	0	0
$305$	8.27549e8	1.67011
$306$	0	0
$307$	8.72706e8	1.72141	0.860703	−	0.509107i	$-0.170024\pi$
$307$	8.72706e8	1.72141	0.860703	+	0.509107i	$0.170024\pi$
$308$	0	0
$309$	−1.18740e8	−0.228951
$310$	0	0
$311$	6.71611e8	1.26607	0.633033	−	0.774124i	$-0.281810\pi$
$311$	6.71611e8	1.26607	0.633033	+	0.774124i	$0.281810\pi$
$312$	0	0
$313$	−1.92216e8	−0.354312	−0.177156	−	0.984183i	$-0.556690\pi$
$313$	−1.92216e8	−0.354312	−0.177156	+	0.984183i	$0.556690\pi$
$314$	0	0
$315$	2.97587e8	0.536447
$316$	0	0
$317$	1.33837e8	0.235977	0.117988	−	0.993015i	$-0.462355\pi$
$317$	1.33837e8	0.235977	0.117988	+	0.993015i	$0.462355\pi$
$318$	0	0
$319$	5.05336e6	0.00871591
$320$	0	0
$321$	1.03243e9	1.74218
$322$	0	0
$323$	−1.68720e9	−2.78584
$324$	0	0
$325$	3.64508e8	0.588999
$326$	0	0
$327$	−4.17048e8	−0.659581
$328$	0	0
$329$	−2.41375e8	−0.373685
$330$	0	0
$331$	4.25298e8	0.644608	0.322304	−	0.946636i	$-0.395543\pi$
$331$	4.25298e8	0.644608	0.322304	+	0.946636i	$0.395543\pi$
$332$	0	0
$333$	−4.54492e7	−0.0674484
$334$	0	0
$335$	−3.97707e8	−0.577972
$336$	0	0
$337$	1.07703e9	1.53293	0.766463	−	0.642288i	$-0.222015\pi$
$337$	1.07703e9	1.53293	0.766463	+	0.642288i	$0.222015\pi$
$338$	0	0
$339$	6.75098e8	0.941170
$340$	0	0
$341$	6.83856e6	0.00933952
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	0	0
$345$	−2.77728e9	−3.64127
$346$	0	0
$347$	−7.23764e8	−0.929916	−0.464958	−	0.885333i	$-0.653931\pi$
$347$	−7.23764e8	−0.929916	−0.464958	+	0.885333i	$0.653931\pi$
$348$	0	0
$349$	4.48132e8	0.564310	0.282155	−	0.959369i	$-0.408951\pi$
$349$	4.48132e8	0.564310	0.282155	+	0.959369i	$0.408951\pi$
$350$	0	0
$351$	5.28898e6	0.00652825
$352$	0	0
$353$	−1.49946e9	−1.81435	−0.907177	−	0.420749i	$-0.861767\pi$
$353$	−1.49946e9	−1.81435	−0.907177	+	0.420749i	$0.861767\pi$
$354$	0	0
$355$	−1.33120e7	−0.0157923
$356$	0	0
$357$	−8.26332e8	−0.961205
$358$	0	0
$359$	2.56890e8	0.293033	0.146516	−	0.989208i	$-0.453194\pi$
$359$	2.56890e8	0.293033	0.146516	+	0.989208i	$0.453194\pi$
$360$	0	0
$361$	1.24260e9	1.39013
$362$	0	0
$363$	1.28605e9	1.41118
$364$	0	0
$365$	−1.18858e9	−1.27939
$366$	0	0
$367$	−6.50424e8	−0.686856	−0.343428	−	0.939179i	$-0.611588\pi$
$367$	−6.50424e8	−0.686856	−0.343428	+	0.939179i	$0.611588\pi$
$368$	0	0
$369$	6.90796e8	0.715744
$370$	0	0
$371$	−5.49924e8	−0.559106
$372$	0	0
$373$	4.66127e8	0.465076	0.232538	−	0.972587i	$-0.425297\pi$
$373$	4.66127e8	0.465076	0.232538	+	0.972587i	$0.425297\pi$
$374$	0	0
$375$	−9.90000e7	−0.0969451
$376$	0	0
$377$	5.62439e8	0.540606
$378$	0	0
$379$	2.85860e8	0.269721	0.134861	−	0.990865i	$-0.456941\pi$
$379$	2.85860e8	0.269721	0.134861	+	0.990865i	$0.456941\pi$
$380$	0	0
$381$	3.96478e8	0.367267
$382$	0	0
$383$	1.65075e9	1.50136	0.750681	−	0.660665i	$-0.229725\pi$
$383$	1.65075e9	1.50136	0.750681	+	0.660665i	$0.229725\pi$
$384$	0	0
$385$	5.48800e6	0.00490119
$386$	0	0
$387$	1.68627e8	0.147890
$388$	0	0
$389$	−1.51304e9	−1.30325	−0.651624	−	0.758542i	$-0.725912\pi$
$389$	−1.51304e9	−1.30325	−0.651624	+	0.758542i	$0.725912\pi$
$390$	0	0
$391$	3.84001e9	3.24873
$392$	0	0
$393$	−1.36221e9	−1.13207
$394$	0	0
$395$	9.50467e8	0.775974
$396$	0	0
$397$	8.63794e8	0.692857	0.346428	−	0.938076i	$-0.387394\pi$
$397$	8.63794e8	0.692857	0.346428	+	0.938076i	$0.387394\pi$
$398$	0	0
$399$	1.04637e9	0.824673
$400$	0	0
$401$	1.14042e8	0.0883199	0.0441599	−	0.999024i	$-0.485939\pi$
$401$	1.14042e8	0.0883199	0.0441599	+	0.999024i	$0.485939\pi$
$402$	0	0
$403$	7.61132e8	0.579285
$404$	0	0
$405$	−1.92880e9	−1.44277
$406$	0	0
$407$	−838160.	−0.000616235 0
$408$	0	0
$409$	−1.18328e9	−0.855176	−0.427588	−	0.903974i	$-0.640637\pi$
$409$	−1.18328e9	−0.855176	−0.427588	+	0.903974i	$0.640637\pi$
$410$	0	0
$411$	−3.14291e8	−0.223298
$412$	0	0
$413$	−4.01960e8	−0.280774
$414$	0	0
$415$	−8.48943e8	−0.583056
$416$	0	0
$417$	3.33777e8	0.225414
$418$	0	0
$419$	2.27959e8	0.151394	0.0756970	−	0.997131i	$-0.475882\pi$
$419$	2.27959e8	0.151394	0.0756970	+	0.997131i	$0.475882\pi$
$420$	0	0
$421$	3.90700e7	0.0255186	0.0127593	−	0.999919i	$-0.495938\pi$
$421$	3.90700e7	0.0255186	0.0127593	+	0.999919i	$0.495938\pi$
$422$	0	0
$423$	−1.52636e9	−0.980541
$424$	0	0
$425$	2.98860e9	1.88846
$426$	0	0
$427$	7.09623e8	0.441093
$428$	0	0
$429$	−1.17533e7	−0.00718718
$430$	0	0
$431$	2.58620e9	1.55594	0.777968	−	0.628304i	$-0.216251\pi$
$431$	2.58620e9	1.55594	0.777968	+	0.628304i	$0.216251\pi$
$432$	0	0
$433$	−1.78893e9	−1.05897	−0.529486	−	0.848318i	$-0.677615\pi$
$433$	−1.78893e9	−1.05897	−0.529486	+	0.848318i	$0.677615\pi$
$434$	0	0
$435$	−3.33522e9	−1.94273
$436$	0	0
$437$	−4.86255e9	−2.78727
$438$	0	0
$439$	4.58905e8	0.258879	0.129440	−	0.991587i	$-0.458682\pi$
$439$	4.58905e8	0.258879	0.129440	+	0.991587i	$0.458682\pi$
$440$	0	0
$441$	2.55181e8	0.141681
$442$	0	0
$443$	1.38459e9	0.756672	0.378336	−	0.925668i	$-0.376496\pi$
$443$	1.38459e9	0.756672	0.378336	+	0.925668i	$0.376496\pi$
$444$	0	0
$445$	2.76814e9	1.48911
$446$	0	0
$447$	−1.80006e9	−0.953259
$448$	0	0
$449$	−2.73611e9	−1.42650	−0.713248	−	0.700911i	$-0.752777\pi$
$449$	−2.73611e9	−1.42650	−0.713248	+	0.700911i	$0.752777\pi$
$450$	0	0
$451$	1.27394e7	0.00653932
$452$	0	0
$453$	4.28288e8	0.216467
$454$	0	0
$455$	6.10814e8	0.303997
$456$	0	0
$457$	2.43053e9	1.19123	0.595614	−	0.803271i	$-0.296909\pi$
$457$	2.43053e9	1.19123	0.595614	+	0.803271i	$0.296909\pi$
$458$	0	0
$459$	4.33644e7	0.0209309
$460$	0	0
$461$	−3.94884e9	−1.87723	−0.938613	−	0.344971i	$-0.887889\pi$
$461$	−3.94884e9	−1.87723	−0.938613	+	0.344971i	$0.887889\pi$
$462$	0	0
$463$	−2.57453e9	−1.20549	−0.602746	−	0.797933i	$-0.705927\pi$
$463$	−2.57453e9	−1.20549	−0.602746	+	0.797933i	$0.705927\pi$
$464$	0	0
$465$	−4.51345e9	−2.08172
$466$	0	0
$467$	2.98482e8	0.135616	0.0678078	−	0.997698i	$-0.478400\pi$
$467$	2.98482e8	0.135616	0.0678078	+	0.997698i	$0.478400\pi$
$468$	0	0
$469$	−3.41034e8	−0.152649
$470$	0	0
$471$	−4.16335e9	−1.83599
$472$	0	0
$473$	3.10976e6	0.00135118
$474$	0	0
$475$	−3.78443e9	−1.62021
$476$	0	0
$477$	−3.47751e9	−1.46708
$478$	0	0
$479$	−2.62000e9	−1.08925	−0.544625	−	0.838680i	$-0.683328\pi$
$479$	−2.62000e9	−1.08925	−0.544625	+	0.838680i	$0.683328\pi$
$480$	0	0
$481$	−9.32872e7	−0.0382221
$482$	0	0
$483$	−2.38152e9	−0.961698
$484$	0	0
$485$	1.98108e9	0.788509
$486$	0	0
$487$	4.16662e9	1.63468	0.817339	−	0.576157i	$-0.195448\pi$
$487$	4.16662e9	1.63468	0.817339	+	0.576157i	$0.195448\pi$
$488$	0	0
$489$	−5.49299e9	−2.12436
$490$	0	0
$491$	2.41300e9	0.919967	0.459983	−	0.887928i	$-0.347855\pi$
$491$	2.41300e9	0.919967	0.459983	+	0.887928i	$0.347855\pi$
$492$	0	0
$493$	4.61144e9	1.73330
$494$	0	0
$495$	3.47040e7	0.0128606
$496$	0	0
$497$	−1.14150e7	−0.00417090
$498$	0	0
$499$	−1.04092e9	−0.375029	−0.187515	−	0.982262i	$-0.560043\pi$
$499$	−1.04092e9	−0.375029	−0.187515	+	0.982262i	$0.560043\pi$
$500$	0	0
$501$	2.02564e9	0.719666
$502$	0	0
$503$	1.17273e9	0.410876	0.205438	−	0.978670i	$-0.434138\pi$
$503$	1.17273e9	0.410876	0.205438	+	0.978670i	$0.434138\pi$
$504$	0	0
$505$	−1.29200e9	−0.446419
$506$	0	0
$507$	2.83326e9	0.965515
$508$	0	0
$509$	8.13818e7	0.0273536	0.0136768	−	0.999906i	$-0.495646\pi$
$509$	8.13818e7	0.0273536	0.0136768	+	0.999906i	$0.495646\pi$
$510$	0	0
$511$	−1.01921e9	−0.337901
$512$	0	0
$513$	−5.49117e7	−0.0179579
$514$	0	0
$515$	7.19637e8	0.232160
$516$	0	0
$517$	−2.81486e7	−0.00895861
$518$	0	0
$519$	−3.48043e9	−1.09282
$520$	0	0
$521$	−2.77458e9	−0.859540	−0.429770	−	0.902939i	$-0.641405\pi$
$521$	−2.77458e9	−0.859540	−0.429770	+	0.902939i	$0.641405\pi$
$522$	0	0
$523$	4.99213e9	1.52591	0.762957	−	0.646449i	$-0.223747\pi$
$523$	4.99213e9	1.52591	0.762957	+	0.646449i	$0.223747\pi$
$524$	0	0
$525$	−1.85349e9	−0.559026
$526$	0	0
$527$	6.24053e9	1.85731
$528$	0	0
$529$	7.66221e9	2.25040
$530$	0	0
$531$	−2.54184e9	−0.736745
$532$	0	0
$533$	1.41790e9	0.405602
$534$	0	0
$535$	−6.25715e9	−1.76660
$536$	0	0
$537$	−5.56207e9	−1.54998
$538$	0	0
$539$	4.70596e6	0.00129446
$540$	0	0
$541$	−1.63095e9	−0.442844	−0.221422	−	0.975178i	$-0.571070\pi$
$541$	−1.63095e9	−0.442844	−0.221422	+	0.975178i	$0.571070\pi$
$542$	0	0
$543$	−6.86112e9	−1.83906
$544$	0	0
$545$	2.52756e9	0.668827
$546$	0	0
$547$	2.00950e9	0.524967	0.262484	−	0.964936i	$-0.415458\pi$
$547$	2.00950e9	0.524967	0.262484	+	0.964936i	$0.415458\pi$
$548$	0	0
$549$	4.48738e9	1.15742
$550$	0	0
$551$	−5.83941e9	−1.48709
$552$	0	0
$553$	8.15026e8	0.204943
$554$	0	0
$555$	5.53186e8	0.137355
$556$	0	0
$557$	4.47959e9	1.09836	0.549180	−	0.835704i	$-0.314940\pi$
$557$	4.47959e9	1.09836	0.549180	+	0.835704i	$0.314940\pi$
$558$	0	0
$559$	3.46116e8	0.0838071
$560$	0	0
$561$	−9.63653e7	−0.0230436
$562$	0	0
$563$	−1.50730e9	−0.355976	−0.177988	−	0.984033i	$-0.556959\pi$
$563$	−1.50730e9	−0.355976	−0.177988	+	0.984033i	$0.556959\pi$
$564$	0	0
$565$	−4.09150e9	−0.954363
$566$	0	0
$567$	−1.65395e9	−0.381050
$568$	0	0
$569$	2.33088e9	0.530428	0.265214	−	0.964190i	$-0.414557\pi$
$569$	2.33088e9	0.530428	0.265214	+	0.964190i	$0.414557\pi$
$570$	0	0
$571$	−2.91101e9	−0.654362	−0.327181	−	0.944962i	$-0.606099\pi$
$571$	−2.91101e9	−0.654362	−0.327181	+	0.944962i	$0.606099\pi$
$572$	0	0
$573$	−8.23517e9	−1.82865
$574$	0	0
$575$	8.61325e9	1.88942
$576$	0	0
$577$	−8.64805e9	−1.87414	−0.937072	−	0.349137i	$-0.886475\pi$
$577$	−8.64805e9	−1.87414	−0.937072	+	0.349137i	$0.886475\pi$
$578$	0	0
$579$	−9.74086e9	−2.08556
$580$	0	0
$581$	−7.27969e8	−0.153991
$582$	0	0
$583$	−6.41311e7	−0.0134038
$584$	0	0
$585$	3.86256e9	0.797681
$586$	0	0
$587$	−6.33513e9	−1.29277	−0.646387	−	0.763010i	$-0.723721\pi$
$587$	−6.33513e9	−1.29277	−0.646387	+	0.763010i	$0.723721\pi$
$588$	0	0
$589$	−7.90230e9	−1.59349
$590$	0	0
$591$	1.02836e10	2.04921
$592$	0	0
$593$	−1.70162e9	−0.335098	−0.167549	−	0.985864i	$-0.553585\pi$
$593$	−1.70162e9	−0.335098	−0.167549	+	0.985864i	$0.553585\pi$
$594$	0	0
$595$	5.00807e9	0.974679
$596$	0	0
$597$	−8.79071e8	−0.169088
$598$	0	0
$599$	−3.01977e9	−0.574090	−0.287045	−	0.957917i	$-0.592673\pi$
$599$	−3.01977e9	−0.574090	−0.287045	+	0.957917i	$0.592673\pi$
$600$	0	0
$601$	−5.92708e9	−1.11373	−0.556865	−	0.830603i	$-0.687996\pi$
$601$	−5.92708e9	−1.11373	−0.556865	+	0.830603i	$0.687996\pi$
$602$	0	0
$603$	−2.15657e9	−0.400546
$604$	0	0
$605$	−7.79423e9	−1.43097
$606$	0	0
$607$	1.45649e9	0.264331	0.132165	−	0.991228i	$-0.457807\pi$
$607$	1.45649e9	0.264331	0.132165	+	0.991228i	$0.457807\pi$
$608$	0	0
$609$	−2.85995e9	−0.513095
$610$	0	0
$611$	−3.13294e9	−0.555659
$612$	0	0
$613$	6.71607e9	1.17762	0.588808	−	0.808273i	$-0.299597\pi$
$613$	6.71607e9	1.17762	0.588808	+	0.808273i	$0.299597\pi$
$614$	0	0
$615$	−8.40803e9	−1.45758
$616$	0	0
$617$	7.02027e9	1.20325	0.601625	−	0.798779i	$-0.294520\pi$
$617$	7.02027e9	1.20325	0.601625	+	0.798779i	$0.294520\pi$
$618$	0	0
$619$	5.14352e9	0.871652	0.435826	−	0.900031i	$-0.356456\pi$
$619$	5.14352e9	0.871652	0.435826	+	0.900031i	$0.356456\pi$
$620$	0	0
$621$	1.24978e8	0.0209417
$622$	0	0
$623$	2.37368e9	0.393291
$624$	0	0
$625$	−5.79648e9	−0.949696
$626$	0	0
$627$	1.22026e8	0.0197704
$628$	0	0
$629$	−7.64863e8	−0.122548
$630$	0	0
$631$	4.41574e9	0.699681	0.349841	−	0.936809i	$-0.386236\pi$
$631$	4.41574e9	0.699681	0.349841	+	0.936809i	$0.386236\pi$
$632$	0	0
$633$	1.35260e10	2.11962
$634$	0	0
$635$	−2.40290e9	−0.372415
$636$	0	0
$637$	5.23773e8	0.0802889
$638$	0	0
$639$	−7.21843e7	−0.0109443
$640$	0	0
$641$	6.94176e8	0.104104	0.0520519	−	0.998644i	$-0.483424\pi$
$641$	6.94176e8	0.104104	0.0520519	+	0.998644i	$0.483424\pi$
$642$	0	0
$643$	9.50809e9	1.41044	0.705220	−	0.708988i	$-0.250848\pi$
$643$	9.50809e9	1.41044	0.705220	+	0.708988i	$0.250848\pi$
$644$	0	0
$645$	−2.05244e9	−0.301170
$646$	0	0
$647$	7.73215e9	1.12237	0.561184	−	0.827691i	$-0.310346\pi$
$647$	7.73215e9	1.12237	0.561184	+	0.827691i	$0.310346\pi$
$648$	0	0
$649$	−4.68758e7	−0.00673119
$650$	0	0
$651$	−3.87028e9	−0.549806
$652$	0	0
$653$	5.06321e9	0.711590	0.355795	−	0.934564i	$-0.384210\pi$
$653$	5.06321e9	0.711590	0.355795	+	0.934564i	$0.384210\pi$
$654$	0	0
$655$	8.25585e9	1.14793
$656$	0	0
$657$	−6.44508e9	−0.886645
$658$	0	0
$659$	8.08113e9	1.09995	0.549975	−	0.835181i	$-0.314637\pi$
$659$	8.08113e9	1.09995	0.549975	+	0.835181i	$0.314637\pi$
$660$	0	0
$661$	−6.30089e9	−0.848588	−0.424294	−	0.905524i	$-0.639478\pi$
$661$	−6.30089e9	−0.848588	−0.424294	+	0.905524i	$0.639478\pi$
$662$	0	0
$663$	−1.07255e10	−1.42928
$664$	0	0
$665$	−6.34166e9	−0.836233
$666$	0	0
$667$	1.32903e10	1.73419
$668$	0	0
$669$	−4.52006e9	−0.583651
$670$	0	0
$671$	8.27549e7	0.0105746
$672$	0	0
$673$	−9.62624e9	−1.21732	−0.608659	−	0.793432i	$-0.708292\pi$
$673$	−9.62624e9	−1.21732	−0.608659	+	0.793432i	$0.708292\pi$
$674$	0	0
$675$	9.72675e7	0.0121732
$676$	0	0
$677$	9.45429e9	1.17103	0.585516	−	0.810661i	$-0.300892\pi$
$677$	9.45429e9	1.17103	0.585516	+	0.810661i	$0.300892\pi$
$678$	0	0
$679$	1.69878e9	0.208254
$680$	0	0
$681$	1.27794e9	0.155058
$682$	0	0
$683$	2.48879e8	0.0298893	0.0149447	−	0.999888i	$-0.495243\pi$
$683$	2.48879e8	0.0298893	0.0149447	+	0.999888i	$0.495243\pi$
$684$	0	0
$685$	1.90479e9	0.226429
$686$	0	0
$687$	2.03383e10	2.39313
$688$	0	0
$689$	−7.13779e9	−0.831375
$690$	0	0
$691$	−3.46412e9	−0.399411	−0.199705	−	0.979856i	$-0.563998\pi$
$691$	−3.46412e9	−0.399411	−0.199705	+	0.979856i	$0.563998\pi$
$692$	0	0
$693$	2.97587e7	0.00339662
$694$	0	0
$695$	−2.02289e9	−0.228574
$696$	0	0
$697$	1.16254e10	1.30045
$698$	0	0
$699$	−2.34826e9	−0.260062
$700$	0	0
$701$	−5.56322e9	−0.609976	−0.304988	−	0.952356i	$-0.598653\pi$
$701$	−5.56322e9	−0.609976	−0.304988	+	0.952356i	$0.598653\pi$
$702$	0	0
$703$	9.68536e8	0.105141
$704$	0	0
$705$	1.85781e10	1.99682
$706$	0	0
$707$	−1.10789e9	−0.117904
$708$	0	0
$709$	−8.23697e9	−0.867971	−0.433986	−	0.900920i	$-0.642893\pi$
$709$	−8.23697e9	−0.867971	−0.433986	+	0.900920i	$0.642893\pi$
$710$	0	0
$711$	5.15391e9	0.537766
$712$	0	0
$713$	1.79854e10	1.85826
$714$	0	0
$715$	7.12320e7	0.00728793
$716$	0	0
$717$	−1.51837e10	−1.53837
$718$	0	0
$719$	−5.85212e9	−0.587168	−0.293584	−	0.955933i	$-0.594848\pi$
$719$	−5.85212e9	−0.587168	−0.293584	+	0.955933i	$0.594848\pi$
$720$	0	0
$721$	6.17089e8	0.0613160
$722$	0	0
$723$	−3.32307e8	−0.0327005
$724$	0	0
$725$	1.03436e10	1.00807
$726$	0	0
$727$	−1.51706e10	−1.46431	−0.732154	−	0.681139i	$-0.761485\pi$
$727$	−1.51706e10	−1.46431	−0.732154	+	0.681139i	$0.761485\pi$
$728$	0	0
$729$	−1.02875e10	−0.983472
$730$	0	0
$731$	2.83781e9	0.268703
$732$	0	0
$733$	1.55969e10	1.46277	0.731383	−	0.681967i	$-0.238875\pi$
$733$	1.55969e10	1.46277	0.731383	+	0.681967i	$0.238875\pi$
$734$	0	0
$735$	−3.10593e9	−0.288527
$736$	0	0
$737$	−3.97707e7	−0.00365955
$738$	0	0
$739$	−6.95573e9	−0.633997	−0.316998	−	0.948426i	$-0.602675\pi$
$739$	−6.95573e9	−0.633997	−0.316998	+	0.948426i	$0.602675\pi$
$740$	0	0
$741$	1.35815e10	1.22626
$742$	0	0
$743$	1.17803e10	1.05365	0.526824	−	0.849975i	$-0.323383\pi$
$743$	1.17803e10	1.05365	0.526824	+	0.849975i	$0.323383\pi$
$744$	0	0
$745$	1.09095e10	0.966621
$746$	0	0
$747$	−4.60339e9	−0.404070
$748$	0	0
$749$	−5.36551e9	−0.466578
$750$	0	0
$751$	6.96200e9	0.599783	0.299892	−	0.953973i	$-0.403049\pi$
$751$	6.96200e9	0.599783	0.299892	+	0.953973i	$0.403049\pi$
$752$	0	0
$753$	6.81665e9	0.581820
$754$	0	0
$755$	−2.59568e9	−0.219501
$756$	0	0
$757$	2.07114e10	1.73530	0.867648	−	0.497179i	$-0.165631\pi$
$757$	2.07114e10	1.73530	0.867648	+	0.497179i	$0.165631\pi$
$758$	0	0
$759$	−2.77728e8	−0.0230554
$760$	0	0
$761$	1.65392e10	1.36041	0.680204	−	0.733023i	$-0.261891\pi$
$761$	1.65392e10	1.36041	0.680204	+	0.733023i	$0.261891\pi$
$762$	0	0
$763$	2.16738e9	0.176644
$764$	0	0
$765$	3.16691e10	2.55753
$766$	0	0
$767$	−5.21727e9	−0.417503
$768$	0	0
$769$	−2.33650e10	−1.85278	−0.926391	−	0.376562i	$-0.877106\pi$
$769$	−2.33650e10	−1.85278	−0.926391	+	0.376562i	$0.877106\pi$
$770$	0	0
$771$	1.53306e10	1.20467
$772$	0	0
$773$	−7.09263e9	−0.552305	−0.276152	−	0.961114i	$-0.589059\pi$
$773$	−7.09263e9	−0.552305	−0.276152	+	0.961114i	$0.589059\pi$
$774$	0	0
$775$	1.39977e10	1.08019
$776$	0	0
$777$	4.74357e8	0.0362770
$778$	0	0
$779$	−1.47211e10	−1.11573
$780$	0	0
$781$	−1.33120e6	−9.99919e−5 0
$782$	0	0
$783$	1.50085e8	0.0111730
$784$	0	0
$785$	2.52324e10	1.86172
$786$	0	0
$787$	−1.23030e10	−0.899703	−0.449851	−	0.893103i	$-0.648523\pi$
$787$	−1.23030e10	−0.899703	−0.449851	+	0.893103i	$0.648523\pi$
$788$	0	0
$789$	2.75054e10	1.99365
$790$	0	0
$791$	−3.50847e9	−0.252057
$792$	0	0
$793$	9.21062e9	0.655892
$794$	0	0
$795$	4.23265e10	2.98764
$796$	0	0
$797$	3.66650e9	0.256535	0.128268	−	0.991740i	$-0.459058\pi$
$797$	3.66650e9	0.256535	0.128268	+	0.991740i	$0.459058\pi$
$798$	0	0
$799$	−2.56870e10	−1.78156
$800$	0	0
$801$	1.50102e10	1.03199
$802$	0	0
$803$	−1.18858e8	−0.00810074
$804$	0	0
$805$	1.44334e10	0.975179
$806$	0	0
$807$	−2.07688e10	−1.39109
$808$	0	0
$809$	−2.96609e10	−1.96954	−0.984770	−	0.173861i	$-0.944376\pi$
$809$	−2.96609e10	−1.96954	−0.984770	+	0.173861i	$0.944376\pi$
$810$	0	0
$811$	2.51278e10	1.65417	0.827087	−	0.562073i	$-0.189996\pi$
$811$	2.51278e10	1.65417	0.827087	+	0.562073i	$0.189996\pi$
$812$	0	0
$813$	1.26810e10	0.827633
$814$	0	0
$815$	3.32908e10	2.15414
$816$	0	0
$817$	−3.59348e9	−0.230536
$818$	0	0
$819$	3.31214e9	0.210676
$820$	0	0
$821$	−4.57772e9	−0.288701	−0.144350	−	0.989527i	$-0.546109\pi$
$821$	−4.57772e9	−0.288701	−0.144350	+	0.989527i	$0.546109\pi$
$822$	0	0
$823$	−1.93133e9	−0.120769	−0.0603846	−	0.998175i	$-0.519233\pi$
$823$	−1.93133e9	−0.120769	−0.0603846	+	0.998175i	$0.519233\pi$
$824$	0	0
$825$	−2.16150e8	−0.0134019
$826$	0	0
$827$	1.58094e10	0.971958	0.485979	−	0.873971i	$-0.338463\pi$
$827$	1.58094e10	0.971958	0.485979	+	0.873971i	$0.338463\pi$
$828$	0	0
$829$	2.46536e9	0.150293	0.0751465	−	0.997173i	$-0.476058\pi$
$829$	2.46536e9	0.150293	0.0751465	+	0.997173i	$0.476058\pi$
$830$	0	0
$831$	−2.90660e10	−1.75704
$832$	0	0
$833$	4.29442e9	0.257423
$834$	0	0
$835$	−1.22766e10	−0.729754
$836$	0	0
$837$	2.03105e8	0.0119724
$838$	0	0
$839$	−2.51861e10	−1.47229	−0.736147	−	0.676822i	$-0.763357\pi$
$839$	−2.51861e10	−1.47229	−0.736147	+	0.676822i	$0.763357\pi$
$840$	0	0
$841$	−1.28960e9	−0.0747598
$842$	0	0
$843$	−2.37095e10	−1.36309
$844$	0	0
$845$	−1.71713e10	−0.979049
$846$	0	0
$847$	−6.68355e9	−0.377933
$848$	0	0
$849$	5.35568e9	0.300357
$850$	0	0
$851$	−2.20436e9	−0.122611
$852$	0	0
$853$	−1.07306e10	−0.591972	−0.295986	−	0.955192i	$-0.595648\pi$
$853$	−1.07306e10	−0.591972	−0.295986	+	0.955192i	$0.595648\pi$
$854$	0	0
$855$	−4.01022e10	−2.19425
$856$	0	0
$857$	−2.79332e10	−1.51596	−0.757979	−	0.652279i	$-0.773813\pi$
$857$	−2.79332e10	−1.51596	−0.757979	+	0.652279i	$0.773813\pi$
$858$	0	0
$859$	−1.94983e10	−1.04959	−0.524795	−	0.851229i	$-0.675858\pi$
$859$	−1.94983e10	−1.04959	−0.524795	+	0.851229i	$0.675858\pi$
$860$	0	0
$861$	−7.20989e9	−0.384962
$862$	0	0
$863$	1.63551e10	0.866193	0.433096	−	0.901348i	$-0.357421\pi$
$863$	1.63551e10	0.866193	0.433096	+	0.901348i	$0.357421\pi$
$864$	0	0
$865$	2.10935e10	1.10814
$866$	0	0
$867$	−6.08558e10	−3.17128
$868$	0	0
$869$	9.50467e7	0.00491324
$870$	0	0
$871$	−4.42648e9	−0.226984
$872$	0	0
$873$	1.07424e10	0.546453
$874$	0	0
$875$	5.14500e8	0.0259631
$876$	0	0
$877$	−2.68874e10	−1.34601	−0.673007	−	0.739636i	$-0.734998\pi$
$877$	−2.68874e10	−1.34601	−0.673007	+	0.739636i	$0.734998\pi$
$878$	0	0
$879$	1.67255e10	0.830648
$880$	0	0
$881$	1.08918e10	0.536644	0.268322	−	0.963329i	$-0.413531\pi$
$881$	1.08918e10	0.536644	0.268322	+	0.963329i	$0.413531\pi$
$882$	0	0
$883$	−3.99542e10	−1.95299	−0.976495	−	0.215542i	$-0.930848\pi$
$883$	−3.99542e10	−1.95299	−0.976495	+	0.215542i	$0.930848\pi$
$884$	0	0
$885$	3.09380e10	1.50034
$886$	0	0
$887$	−1.30306e10	−0.626946	−0.313473	−	0.949597i	$-0.601493\pi$
$887$	−1.30306e10	−0.626946	−0.313473	+	0.949597i	$0.601493\pi$
$888$	0	0
$889$	−2.06049e9	−0.0983589
$890$	0	0
$891$	−1.92880e8	−0.00913516
$892$	0	0
$893$	3.25272e10	1.52850
$894$	0	0
$895$	3.37095e10	1.57171
$896$	0	0
$897$	−3.09111e10	−1.43002
$898$	0	0
$899$	2.15986e10	0.991439
$900$	0	0
$901$	−5.85229e10	−2.66556
$902$	0	0
$903$	−1.75997e9	−0.0795422
$904$	0	0
$905$	4.15825e10	1.86484
$906$	0	0
$907$	4.04015e10	1.79792	0.898962	−	0.438026i	$-0.144322\pi$
$907$	4.04015e10	1.79792	0.898962	+	0.438026i	$0.144322\pi$
$908$	0	0
$909$	−7.00587e9	−0.309377
$910$	0	0
$911$	−1.98919e10	−0.871690	−0.435845	−	0.900022i	$-0.643550\pi$
$911$	−1.98919e10	−0.871690	−0.435845	+	0.900022i	$0.643550\pi$
$912$	0	0
$913$	−8.48943e7	−0.00369174
$914$	0	0
$915$	−5.46182e10	−2.35702
$916$	0	0
$917$	7.07939e9	0.303182
$918$	0	0
$919$	4.10990e10	1.74674	0.873368	−	0.487061i	$-0.161931\pi$
$919$	4.10990e10	1.74674	0.873368	+	0.487061i	$0.161931\pi$
$920$	0	0
$921$	−5.75986e10	−2.42942
$922$	0	0
$923$	−1.48163e8	−0.00620201
$924$	0	0
$925$	−1.71561e9	−0.0712725
$926$	0	0
$927$	3.90223e9	0.160892
$928$	0	0
$929$	−1.90374e10	−0.779027	−0.389513	−	0.921021i	$-0.627357\pi$
$929$	−1.90374e10	−0.779027	−0.389513	+	0.921021i	$0.627357\pi$
$930$	0	0
$931$	−5.43797e9	−0.220858
$932$	0	0
$933$	−4.43264e10	−1.78680
$934$	0	0
$935$	5.84032e8	0.0233666
$936$	0	0
$937$	3.93830e10	1.56394	0.781971	−	0.623315i	$-0.214214\pi$
$937$	3.93830e10	1.56394	0.781971	+	0.623315i	$0.214214\pi$
$938$	0	0
$939$	1.26863e10	0.500040
$940$	0	0
$941$	1.59186e10	0.622791	0.311395	−	0.950280i	$-0.399204\pi$
$941$	1.59186e10	0.622791	0.311395	+	0.950280i	$0.399204\pi$
$942$	0	0
$943$	3.35047e10	1.30111
$944$	0	0
$945$	1.62994e8	0.00628289
$946$	0	0
$947$	3.57237e9	0.136688	0.0683442	−	0.997662i	$-0.478228\pi$
$947$	3.57237e9	0.136688	0.0683442	+	0.997662i	$0.478228\pi$
$948$	0	0
$949$	−1.32289e10	−0.502450
$950$	0	0
$951$	−8.83326e9	−0.333034
$952$	0	0
$953$	4.05101e9	0.151614	0.0758068	−	0.997123i	$-0.475847\pi$
$953$	4.05101e9	0.151614	0.0758068	+	0.997123i	$0.475847\pi$
$954$	0	0
$955$	4.99101e10	1.85429
$956$	0	0
$957$	−3.33522e8	−0.0123008
$958$	0	0
$959$	1.63336e9	0.0598022
$960$	0	0
$961$	1.71608e9	0.0623741
$962$	0	0
$963$	−3.39294e10	−1.22429
$964$	0	0
$965$	5.90355e10	2.11479
$966$	0	0
$967$	−2.55791e10	−0.909689	−0.454844	−	0.890571i	$-0.650305\pi$
$967$	−2.55791e10	−0.909689	−0.454844	+	0.890571i	$0.650305\pi$
$968$	0	0
$969$	1.11355e11	3.93166
$970$	0	0
$971$	4.10323e10	1.43833	0.719165	−	0.694840i	$-0.244525\pi$
$971$	4.10323e10	1.43833	0.719165	+	0.694840i	$0.244525\pi$
$972$	0	0
$973$	−1.73463e9	−0.0603688
$974$	0	0
$975$	−2.40575e10	−0.831255
$976$	0	0
$977$	4.87277e10	1.67165	0.835824	−	0.548998i	$-0.184990\pi$
$977$	4.87277e10	1.67165	0.835824	+	0.548998i	$0.184990\pi$
$978$	0	0
$979$	2.76814e8	0.00942863
$980$	0	0
$981$	1.37057e10	0.463511
$982$	0	0
$983$	6.94762e9	0.233291	0.116646	−	0.993174i	$-0.462786\pi$
$983$	6.94762e9	0.233291	0.116646	+	0.993174i	$0.462786\pi$
$984$	0	0
$985$	−6.23246e10	−2.07794
$986$	0	0
$987$	1.59307e10	0.527382
$988$	0	0
$989$	8.17867e9	0.268841
$990$	0	0
$991$	1.83565e10	0.599144	0.299572	−	0.954074i	$-0.403156\pi$
$991$	1.83565e10	0.599144	0.299572	+	0.954074i	$0.403156\pi$
$992$	0	0
$993$	−2.80697e10	−0.909735
$994$	0	0
$995$	5.32770e9	0.171459
$996$	0	0
$997$	−3.44954e10	−1.10237	−0.551185	−	0.834383i	$-0.685824\pi$
$997$	−3.44954e10	−1.10237	−0.551185	+	0.834383i	$0.685824\pi$
$998$	0	0
$999$	−2.48934e7	−0.000789958 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.8.a.b.1.1		1
4.3	odd	2		448.8.a.i.1.1		1
8.3	odd	2		14.8.a.b.1.1	✓	1
8.5	even	2		112.8.a.d.1.1		1
24.11	even	2		126.8.a.c.1.1		1
40.3	even	4		350.8.c.b.99.1		2
40.19	odd	2		350.8.a.d.1.1		1
40.27	even	4		350.8.c.b.99.2		2
56.3	even	6		98.8.c.a.79.1		2
56.11	odd	6		98.8.c.b.79.1		2
56.19	even	6		98.8.c.a.67.1		2
56.27	even	2		98.8.a.c.1.1		1
56.51	odd	6		98.8.c.b.67.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.8.a.b.1.1	✓	1	8.3	odd	2
98.8.a.c.1.1		1	56.27	even	2
98.8.c.a.67.1		2	56.19	even	6
98.8.c.a.79.1		2	56.3	even	6
98.8.c.b.67.1		2	56.51	odd	6
98.8.c.b.79.1		2	56.11	odd	6
112.8.a.d.1.1		1	8.5	even	2
126.8.a.c.1.1		1	24.11	even	2
350.8.a.d.1.1		1	40.19	odd	2
350.8.c.b.99.1		2	40.3	even	4
350.8.c.b.99.2		2	40.27	even	4
448.8.a.b.1.1		1	1.1	even	1	trivial
448.8.a.i.1.1		1	4.3	odd	2

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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