LMFDB - Embedded newform 192.8.a.i.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [192,8,Mod(1,192)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(192, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("192.1"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	192.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,27,0,-390] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$59.9779248930$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	192.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+27.0000 q^{3} -390.000 q^{5} -64.0000 q^{7} +729.000 q^{9} +948.000 q^{11} +5098.00 q^{13} -10530.0 q^{15} +28386.0 q^{17} +8620.00 q^{19} -1728.00 q^{21} -15288.0 q^{23} +73975.0 q^{25} +19683.0 q^{27} -36510.0 q^{29} -276808. q^{31} +25596.0 q^{33} +24960.0 q^{35} -268526. q^{37} +137646. q^{39} -629718. q^{41} -685772. q^{43} -284310. q^{45} +583296. q^{47} -819447. q^{49} +766422. q^{51} +428058. q^{53} -369720. q^{55} +232740. q^{57} -1.30638e6 q^{59} -300662. q^{61} -46656.0 q^{63} -1.98822e6 q^{65} +507244. q^{67} -412776. q^{69} +5.56063e6 q^{71} +1.36908e6 q^{73} +1.99733e6 q^{75} -60672.0 q^{77} -6.91372e6 q^{79} +531441. q^{81} +4.37675e6 q^{83} -1.10705e7 q^{85} -985770. q^{87} -8.52831e6 q^{89} -326272. q^{91} -7.47382e6 q^{93} -3.36180e6 q^{95} -8.82681e6 q^{97} +691092. q^{99} +O(q^{100})$

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
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	0	0
$5$	−390.000	−1.39531	−0.697653	−	0.716436i	$-0.745772\pi$
$5$	−390.000	−1.39531	−0.697653	+	0.716436i	$0.745772\pi$
$6$	0	0
$7$	−64.0000	−0.0705240	−0.0352620	−	0.999378i	$-0.511227\pi$
$7$	−64.0000	−0.0705240	−0.0352620	+	0.999378i	$0.511227\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	948.000	0.214750	0.107375	−	0.994219i	$-0.465755\pi$
$11$	948.000	0.214750	0.107375	+	0.994219i	$0.465755\pi$
$12$	0	0
$13$	5098.00	0.643573	0.321787	−	0.946812i	$-0.395717\pi$
$13$	5098.00	0.643573	0.321787	+	0.946812i	$0.395717\pi$
$14$	0	0
$15$	−10530.0	−0.805581
$16$	0	0
$17$	28386.0	1.40131	0.700653	−	0.713502i	$-0.252892\pi$
$17$	28386.0	1.40131	0.700653	+	0.713502i	$0.252892\pi$
$18$	0	0
$19$	8620.00	0.288317	0.144158	−	0.989555i	$-0.453953\pi$
$19$	8620.00	0.288317	0.144158	+	0.989555i	$0.453953\pi$
$20$	0	0
$21$	−1728.00	−0.0407170
$22$	0	0
$23$	−15288.0	−0.262001	−0.131001	−	0.991382i	$-0.541819\pi$
$23$	−15288.0	−0.262001	−0.131001	+	0.991382i	$0.541819\pi$
$24$	0	0
$25$	73975.0	0.946880
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	−36510.0	−0.277983	−0.138992	−	0.990294i	$-0.544386\pi$
$29$	−36510.0	−0.277983	−0.138992	+	0.990294i	$0.544386\pi$
$30$	0	0
$31$	−276808.	−1.66883	−0.834416	−	0.551135i	$-0.814195\pi$
$31$	−276808.	−1.66883	−0.834416	+	0.551135i	$0.814195\pi$
$32$	0	0
$33$	25596.0	0.123986
$34$	0	0
$35$	24960.0	0.0984026
$36$	0	0
$37$	−268526.	−0.871526	−0.435763	−	0.900061i	$-0.643521\pi$
$37$	−268526.	−0.871526	−0.435763	+	0.900061i	$0.643521\pi$
$38$	0	0
$39$	137646.	0.371567
$40$	0	0
$41$	−629718.	−1.42693	−0.713465	−	0.700691i	$-0.752875\pi$
$41$	−629718.	−1.42693	−0.713465	+	0.700691i	$0.752875\pi$
$42$	0	0
$43$	−685772.	−1.31535	−0.657673	−	0.753303i	$-0.728459\pi$
$43$	−685772.	−1.31535	−0.657673	+	0.753303i	$0.728459\pi$
$44$	0	0
$45$	−284310.	−0.465102
$46$	0	0
$47$	583296.	0.819495	0.409748	−	0.912199i	$-0.365617\pi$
$47$	583296.	0.819495	0.409748	+	0.912199i	$0.365617\pi$
$48$	0	0
$49$	−819447.	−0.995026
$50$	0	0
$51$	766422.	0.809044
$52$	0	0
$53$	428058.	0.394945	0.197473	−	0.980308i	$-0.436727\pi$
$53$	428058.	0.394945	0.197473	+	0.980308i	$0.436727\pi$
$54$	0	0
$55$	−369720.	−0.299643
$56$	0	0
$57$	232740.	0.166460
$58$	0	0
$59$	−1.30638e6	−0.828109	−0.414054	−	0.910252i	$-0.635888\pi$
$59$	−1.30638e6	−0.828109	−0.414054	+	0.910252i	$0.635888\pi$
$60$	0	0
$61$	−300662.	−0.169599	−0.0847997	−	0.996398i	$-0.527025\pi$
$61$	−300662.	−0.169599	−0.0847997	+	0.996398i	$0.527025\pi$
$62$	0	0
$63$	−46656.0	−0.0235080
$64$	0	0
$65$	−1.98822e6	−0.897982
$66$	0	0
$67$	507244.	0.206042	0.103021	−	0.994679i	$-0.467149\pi$
$67$	507244.	0.206042	0.103021	+	0.994679i	$0.467149\pi$
$68$	0	0
$69$	−412776.	−0.151266
$70$	0	0
$71$	5.56063e6	1.84383	0.921913	−	0.387397i	$-0.126626\pi$
$71$	5.56063e6	1.84383	0.921913	+	0.387397i	$0.126626\pi$
$72$	0	0
$73$	1.36908e6	0.411907	0.205954	−	0.978562i	$-0.433970\pi$
$73$	1.36908e6	0.411907	0.205954	+	0.978562i	$0.433970\pi$
$74$	0	0
$75$	1.99733e6	0.546681
$76$	0	0
$77$	−60672.0	−0.0151451
$78$	0	0
$79$	−6.91372e6	−1.57767	−0.788836	−	0.614603i	$-0.789316\pi$
$79$	−6.91372e6	−1.57767	−0.788836	+	0.614603i	$0.789316\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	4.37675e6	0.840191	0.420096	−	0.907480i	$-0.361997\pi$
$83$	4.37675e6	0.840191	0.420096	+	0.907480i	$0.361997\pi$
$84$	0	0
$85$	−1.10705e7	−1.95525
$86$	0	0
$87$	−985770.	−0.160494
$88$	0	0
$89$	−8.52831e6	−1.28232	−0.641162	−	0.767405i	$-0.721547\pi$
$89$	−8.52831e6	−1.28232	−0.641162	+	0.767405i	$0.721547\pi$
$90$	0	0
$91$	−326272.	−0.0453874
$92$	0	0
$93$	−7.47382e6	−0.963501
$94$	0	0
$95$	−3.36180e6	−0.402290
$96$	0	0
$97$	−8.82681e6	−0.981981	−0.490990	−	0.871165i	$-0.663365\pi$
$97$	−8.82681e6	−0.981981	−0.490990	+	0.871165i	$0.663365\pi$
$98$	0	0
$99$	691092.	0.0715835
$100$	0	0
$101$	−1.19864e7	−1.15762	−0.578808	−	0.815464i	$-0.696482\pi$
$101$	−1.19864e7	−1.15762	−0.578808	+	0.815464i	$0.696482\pi$
$102$	0	0
$103$	7.20939e6	0.650082	0.325041	−	0.945700i	$-0.394622\pi$
$103$	7.20939e6	0.650082	0.325041	+	0.945700i	$0.394622\pi$
$104$	0	0
$105$	673920.	0.0568127
$106$	0	0
$107$	−1.14261e7	−0.901683	−0.450842	−	0.892604i	$-0.648876\pi$
$107$	−1.14261e7	−0.901683	−0.450842	+	0.892604i	$0.648876\pi$
$108$	0	0
$109$	−4.02095e6	−0.297397	−0.148698	−	0.988883i	$-0.547508\pi$
$109$	−4.02095e6	−0.297397	−0.148698	+	0.988883i	$0.547508\pi$
$110$	0	0
$111$	−7.25020e6	−0.503176
$112$	0	0
$113$	−1.77063e7	−1.15439	−0.577197	−	0.816605i	$-0.695853\pi$
$113$	−1.77063e7	−1.15439	−0.577197	+	0.816605i	$0.695853\pi$
$114$	0	0
$115$	5.96232e6	0.365572
$116$	0	0
$117$	3.71644e6	0.214524
$118$	0	0
$119$	−1.81670e6	−0.0988257
$120$	0	0
$121$	−1.85885e7	−0.953882
$122$	0	0
$123$	−1.70024e7	−0.823838
$124$	0	0
$125$	1.61850e6	0.0741187
$126$	0	0
$127$	1.67883e7	0.727267	0.363633	−	0.931542i	$-0.381536\pi$
$127$	1.67883e7	0.727267	0.363633	+	0.931542i	$0.381536\pi$
$128$	0	0
$129$	−1.85158e7	−0.759416
$130$	0	0
$131$	−1.68268e7	−0.653960	−0.326980	−	0.945031i	$-0.606031\pi$
$131$	−1.68268e7	−0.653960	−0.326980	+	0.945031i	$0.606031\pi$
$132$	0	0
$133$	−551680.	−0.0203332
$134$	0	0
$135$	−7.67637e6	−0.268527
$136$	0	0
$137$	2.80449e7	0.931820	0.465910	−	0.884832i	$-0.345727\pi$
$137$	2.80449e7	0.931820	0.465910	+	0.884832i	$0.345727\pi$
$138$	0	0
$139$	1.18273e7	0.373537	0.186769	−	0.982404i	$-0.440199\pi$
$139$	1.18273e7	0.373537	0.186769	+	0.982404i	$0.440199\pi$
$140$	0	0
$141$	1.57490e7	0.473136
$142$	0	0
$143$	4.83290e6	0.138208
$144$	0	0
$145$	1.42389e7	0.387872
$146$	0	0
$147$	−2.21251e7	−0.574479
$148$	0	0
$149$	−2.07846e7	−0.514743	−0.257371	−	0.966313i	$-0.582856\pi$
$149$	−2.07846e7	−0.514743	−0.257371	+	0.966313i	$0.582856\pi$
$150$	0	0
$151$	76112.0	0.00179901	0.000899505	−	1.00000i	$-0.499714\pi$
$151$	76112.0	0.00179901	0.000899505		1.00000i	$0.499714\pi$
$152$	0	0
$153$	2.06934e7	0.467102
$154$	0	0
$155$	1.07955e8	2.32853
$156$	0	0
$157$	3.21825e7	0.663698	0.331849	−	0.943332i	$-0.392328\pi$
$157$	3.21825e7	0.663698	0.331849	+	0.943332i	$0.392328\pi$
$158$	0	0
$159$	1.15576e7	0.228022
$160$	0	0
$161$	978432.	0.0184774
$162$	0	0
$163$	−5.83435e7	−1.05520	−0.527601	−	0.849492i	$-0.676908\pi$
$163$	−5.83435e7	−1.05520	−0.527601	+	0.849492i	$0.676908\pi$
$164$	0	0
$165$	−9.98244e6	−0.172999
$166$	0	0
$167$	−2.58365e7	−0.429266	−0.214633	−	0.976695i	$-0.568855\pi$
$167$	−2.58365e7	−0.429266	−0.214633	+	0.976695i	$0.568855\pi$
$168$	0	0
$169$	−3.67589e7	−0.585813
$170$	0	0
$171$	6.28398e6	0.0961055
$172$	0	0
$173$	−6.35201e7	−0.932716	−0.466358	−	0.884596i	$-0.654434\pi$
$173$	−6.35201e7	−0.932716	−0.466358	+	0.884596i	$0.654434\pi$
$174$	0	0
$175$	−4.73440e6	−0.0667777
$176$	0	0
$177$	−3.52723e7	−0.478109
$178$	0	0
$179$	8.09559e7	1.05503	0.527513	−	0.849547i	$-0.323125\pi$
$179$	8.09559e7	1.05503	0.527513	+	0.849547i	$0.323125\pi$
$180$	0	0
$181$	−6.45032e7	−0.808549	−0.404274	−	0.914638i	$-0.632476\pi$
$181$	−6.45032e7	−0.808549	−0.404274	+	0.914638i	$0.632476\pi$
$182$	0	0
$183$	−8.11787e6	−0.0979182
$184$	0	0
$185$	1.04725e8	1.21605
$186$	0	0
$187$	2.69099e7	0.300931
$188$	0	0
$189$	−1.25971e6	−0.0135723
$190$	0	0
$191$	5.68274e7	0.590121	0.295060	−	0.955479i	$-0.404660\pi$
$191$	5.68274e7	0.590121	0.295060	+	0.955479i	$0.404660\pi$
$192$	0	0
$193$	1.16377e8	1.16524	0.582621	−	0.812744i	$-0.302027\pi$
$193$	1.16377e8	1.16524	0.582621	+	0.812744i	$0.302027\pi$
$194$	0	0
$195$	−5.36819e7	−0.518450
$196$	0	0
$197$	1.18816e8	1.10724	0.553622	−	0.832768i	$-0.313245\pi$
$197$	1.18816e8	1.10724	0.553622	+	0.832768i	$0.313245\pi$
$198$	0	0
$199$	−9.50106e7	−0.854646	−0.427323	−	0.904099i	$-0.640543\pi$
$199$	−9.50106e7	−0.854646	−0.427323	+	0.904099i	$0.640543\pi$
$200$	0	0
$201$	1.36956e7	0.118958
$202$	0	0
$203$	2.33664e6	0.0196045
$204$	0	0
$205$	2.45590e8	1.99100
$206$	0	0
$207$	−1.11450e7	−0.0873337
$208$	0	0
$209$	8.17176e6	0.0619161
$210$	0	0
$211$	−1.79246e8	−1.31360	−0.656798	−	0.754067i	$-0.728090\pi$
$211$	−1.79246e8	−1.31360	−0.656798	+	0.754067i	$0.728090\pi$
$212$	0	0
$213$	1.50137e8	1.06453
$214$	0	0
$215$	2.67451e8	1.83531
$216$	0	0
$217$	1.77157e7	0.117693
$218$	0	0
$219$	3.69652e7	0.237815
$220$	0	0
$221$	1.44712e8	0.901843
$222$	0	0
$223$	−2.06537e8	−1.24718	−0.623592	−	0.781750i	$-0.714327\pi$
$223$	−2.06537e8	−1.24718	−0.623592	+	0.781750i	$0.714327\pi$
$224$	0	0
$225$	5.39278e7	0.315627
$226$	0	0
$227$	−4.33954e7	−0.246237	−0.123118	−	0.992392i	$-0.539290\pi$
$227$	−4.33954e7	−0.246237	−0.123118	+	0.992392i	$0.539290\pi$
$228$	0	0
$229$	3.61931e7	0.199160	0.0995799	−	0.995030i	$-0.468250\pi$
$229$	3.61931e7	0.199160	0.0995799	+	0.995030i	$0.468250\pi$
$230$	0	0
$231$	−1.63814e6	−0.00874400
$232$	0	0
$233$	9.22347e7	0.477693	0.238846	−	0.971057i	$-0.423231\pi$
$233$	9.22347e7	0.477693	0.238846	+	0.971057i	$0.423231\pi$
$234$	0	0
$235$	−2.27485e8	−1.14345
$236$	0	0
$237$	−1.86670e8	−0.910870
$238$	0	0
$239$	4.98468e7	0.236181	0.118090	−	0.993003i	$-0.462323\pi$
$239$	4.98468e7	0.236181	0.118090	+	0.993003i	$0.462323\pi$
$240$	0	0
$241$	1.99374e8	0.917506	0.458753	−	0.888564i	$-0.348296\pi$
$241$	1.99374e8	0.917506	0.458753	+	0.888564i	$0.348296\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	3.19584e8	1.38837
$246$	0	0
$247$	4.39448e7	0.185553
$248$	0	0
$249$	1.18172e8	0.485085
$250$	0	0
$251$	3.94678e8	1.57538	0.787689	−	0.616073i	$-0.211277\pi$
$251$	3.94678e8	1.57538	0.787689	+	0.616073i	$0.211277\pi$
$252$	0	0
$253$	−1.44930e7	−0.0562649
$254$	0	0
$255$	−2.98905e8	−1.12886
$256$	0	0
$257$	−1.42885e8	−0.525076	−0.262538	−	0.964922i	$-0.584559\pi$
$257$	−1.42885e8	−0.525076	−0.262538	+	0.964922i	$0.584559\pi$
$258$	0	0
$259$	1.71857e7	0.0614635
$260$	0	0
$261$	−2.66158e7	−0.0926611
$262$	0	0
$263$	4.40241e8	1.49226	0.746131	−	0.665799i	$-0.231909\pi$
$263$	4.40241e8	1.49226	0.746131	+	0.665799i	$0.231909\pi$
$264$	0	0
$265$	−1.66943e8	−0.551070
$266$	0	0
$267$	−2.30264e8	−0.740350
$268$	0	0
$269$	−2.75405e8	−0.862657	−0.431329	−	0.902195i	$-0.641955\pi$
$269$	−2.75405e8	−0.862657	−0.431329	+	0.902195i	$0.641955\pi$
$270$	0	0
$271$	−4.24670e8	−1.29616	−0.648080	−	0.761572i	$-0.724428\pi$
$271$	−4.24670e8	−1.29616	−0.648080	+	0.761572i	$0.724428\pi$
$272$	0	0
$273$	−8.80934e6	−0.0262044
$274$	0	0
$275$	7.01283e7	0.203343
$276$	0	0
$277$	−5.16158e8	−1.45916	−0.729581	−	0.683894i	$-0.760285\pi$
$277$	−5.16158e8	−1.45916	−0.729581	+	0.683894i	$0.760285\pi$
$278$	0	0
$279$	−2.01793e8	−0.556277
$280$	0	0
$281$	−3.11043e8	−0.836273	−0.418137	−	0.908384i	$-0.637317\pi$
$281$	−3.11043e8	−0.836273	−0.418137	+	0.908384i	$0.637317\pi$
$282$	0	0
$283$	5.94308e8	1.55869	0.779344	−	0.626596i	$-0.215552\pi$
$283$	5.94308e8	1.55869	0.779344	+	0.626596i	$0.215552\pi$
$284$	0	0
$285$	−9.07686e7	−0.232262
$286$	0	0
$287$	4.03020e7	0.100633
$288$	0	0
$289$	3.95426e8	0.963658
$290$	0	0
$291$	−2.38324e8	−0.566947
$292$	0	0
$293$	−1.15515e8	−0.268288	−0.134144	−	0.990962i	$-0.542828\pi$
$293$	−1.15515e8	−0.268288	−0.134144	+	0.990962i	$0.542828\pi$
$294$	0	0
$295$	5.09488e8	1.15547
$296$	0	0
$297$	1.86595e7	0.0413287
$298$	0	0
$299$	−7.79382e7	−0.168617
$300$	0	0
$301$	4.38894e7	0.0927635
$302$	0	0
$303$	−3.23633e8	−0.668350
$304$	0	0
$305$	1.17258e8	0.236643
$306$	0	0
$307$	2.60600e8	0.514032	0.257016	−	0.966407i	$-0.417261\pi$
$307$	2.60600e8	0.514032	0.257016	+	0.966407i	$0.417261\pi$
$308$	0	0
$309$	1.94654e8	0.375325
$310$	0	0
$311$	5.76795e8	1.08733	0.543663	−	0.839303i	$-0.317037\pi$
$311$	5.76795e8	1.08733	0.543663	+	0.839303i	$0.317037\pi$
$312$	0	0
$313$	−4.60074e8	−0.848053	−0.424026	−	0.905650i	$-0.639384\pi$
$313$	−4.60074e8	−0.848053	−0.424026	+	0.905650i	$0.639384\pi$
$314$	0	0
$315$	1.81958e7	0.0328009
$316$	0	0
$317$	−6.25561e7	−0.110297	−0.0551483	−	0.998478i	$-0.517563\pi$
$317$	−6.25561e7	−0.110297	−0.0551483	+	0.998478i	$0.517563\pi$
$318$	0	0
$319$	−3.46115e7	−0.0596970
$320$	0	0
$321$	−3.08504e8	−0.520587
$322$	0	0
$323$	2.44687e8	0.404020
$324$	0	0
$325$	3.77125e8	0.609387
$326$	0	0
$327$	−1.08566e8	−0.171702
$328$	0	0
$329$	−3.73309e7	−0.0577941
$330$	0	0
$331$	−6.84236e8	−1.03707	−0.518535	−	0.855057i	$-0.673522\pi$
$331$	−6.84236e8	−1.03707	−0.518535	+	0.855057i	$0.673522\pi$
$332$	0	0
$333$	−1.95755e8	−0.290509
$334$	0	0
$335$	−1.97825e8	−0.287491
$336$	0	0
$337$	−6.26313e8	−0.891429	−0.445714	−	0.895175i	$-0.647050\pi$
$337$	−6.26313e8	−0.891429	−0.445714	+	0.895175i	$0.647050\pi$
$338$	0	0
$339$	−4.78071e8	−0.666489
$340$	0	0
$341$	−2.62414e8	−0.358382
$342$	0	0
$343$	1.05151e8	0.140697
$344$	0	0
$345$	1.60983e8	0.211063
$346$	0	0
$347$	1.25340e9	1.61041	0.805203	−	0.593000i	$-0.202057\pi$
$347$	1.25340e9	1.61041	0.805203	+	0.593000i	$0.202057\pi$
$348$	0	0
$349$	−2.65350e8	−0.334142	−0.167071	−	0.985945i	$-0.553431\pi$
$349$	−2.65350e8	−0.334142	−0.167071	+	0.985945i	$0.553431\pi$
$350$	0	0
$351$	1.00344e8	0.123856
$352$	0	0
$353$	−5.69636e8	−0.689264	−0.344632	−	0.938738i	$-0.611996\pi$
$353$	−5.69636e8	−0.689264	−0.344632	+	0.938738i	$0.611996\pi$
$354$	0	0
$355$	−2.16865e9	−2.57270
$356$	0	0
$357$	−4.90510e7	−0.0570570
$358$	0	0
$359$	9.32541e8	1.06374	0.531872	−	0.846825i	$-0.321489\pi$
$359$	9.32541e8	1.06374	0.531872	+	0.846825i	$0.321489\pi$
$360$	0	0
$361$	−8.19567e8	−0.916874
$362$	0	0
$363$	−5.01889e8	−0.550724
$364$	0	0
$365$	−5.33942e8	−0.574737
$366$	0	0
$367$	−8.52565e8	−0.900318	−0.450159	−	0.892948i	$-0.648633\pi$
$367$	−8.52565e8	−0.900318	−0.450159	+	0.892948i	$0.648633\pi$
$368$	0	0
$369$	−4.59064e8	−0.475643
$370$	0	0
$371$	−2.73957e7	−0.0278531
$372$	0	0
$373$	−3.81183e8	−0.380323	−0.190162	−	0.981753i	$-0.560901\pi$
$373$	−3.81183e8	−0.380323	−0.190162	+	0.981753i	$0.560901\pi$
$374$	0	0
$375$	4.36995e7	0.0427924
$376$	0	0
$377$	−1.86128e8	−0.178903
$378$	0	0
$379$	1.48353e9	1.39978	0.699889	−	0.714251i	$-0.253233\pi$
$379$	1.48353e9	1.39978	0.699889	+	0.714251i	$0.253233\pi$
$380$	0	0
$381$	4.53284e8	0.419888
$382$	0	0
$383$	−7.61930e8	−0.692978	−0.346489	−	0.938054i	$-0.612626\pi$
$383$	−7.61930e8	−0.692978	−0.346489	+	0.938054i	$0.612626\pi$
$384$	0	0
$385$	2.36621e7	0.0211320
$386$	0	0
$387$	−4.99928e8	−0.438449
$388$	0	0
$389$	−1.60902e9	−1.38592	−0.692959	−	0.720977i	$-0.743693\pi$
$389$	−1.60902e9	−1.38592	−0.692959	+	0.720977i	$0.743693\pi$
$390$	0	0
$391$	−4.33965e8	−0.367144
$392$	0	0
$393$	−4.54323e8	−0.377564
$394$	0	0
$395$	2.69635e9	2.20134
$396$	0	0
$397$	−1.88016e9	−1.50809	−0.754046	−	0.656822i	$-0.771900\pi$
$397$	−1.88016e9	−1.50809	−0.754046	+	0.656822i	$0.771900\pi$
$398$	0	0
$399$	−1.48954e7	−0.0117394
$400$	0	0
$401$	2.68592e8	0.208012	0.104006	−	0.994577i	$-0.466834\pi$
$401$	2.68592e8	0.208012	0.104006	+	0.994577i	$0.466834\pi$
$402$	0	0
$403$	−1.41117e9	−1.07402
$404$	0	0
$405$	−2.07262e8	−0.155034
$406$	0	0
$407$	−2.54563e8	−0.187161
$408$	0	0
$409$	8.99478e7	0.0650069	0.0325034	−	0.999472i	$-0.489652\pi$
$409$	8.99478e7	0.0650069	0.0325034	+	0.999472i	$0.489652\pi$
$410$	0	0
$411$	7.57212e8	0.537986
$412$	0	0
$413$	8.36083e7	0.0584015
$414$	0	0
$415$	−1.70693e9	−1.17232
$416$	0	0
$417$	3.19337e8	0.215662
$418$	0	0
$419$	−1.69054e9	−1.12273	−0.561367	−	0.827567i	$-0.689724\pi$
$419$	−1.69054e9	−1.12273	−0.561367	+	0.827567i	$0.689724\pi$
$420$	0	0
$421$	1.13333e9	0.740232	0.370116	−	0.928985i	$-0.379318\pi$
$421$	1.13333e9	0.740232	0.370116	+	0.928985i	$0.379318\pi$
$422$	0	0
$423$	4.25223e8	0.273165
$424$	0	0
$425$	2.09985e9	1.32687
$426$	0	0
$427$	1.92424e7	0.0119608
$428$	0	0
$429$	1.30488e8	0.0797942
$430$	0	0
$431$	2.19943e9	1.32324	0.661621	−	0.749839i	$-0.269869\pi$
$431$	2.19943e9	1.32324	0.661621	+	0.749839i	$0.269869\pi$
$432$	0	0
$433$	−1.51738e8	−0.0898227	−0.0449114	−	0.998991i	$-0.514301\pi$
$433$	−1.51738e8	−0.0898227	−0.0449114	+	0.998991i	$0.514301\pi$
$434$	0	0
$435$	3.84450e8	0.223938
$436$	0	0
$437$	−1.31783e8	−0.0755393
$438$	0	0
$439$	9.90763e8	0.558912	0.279456	−	0.960158i	$-0.409846\pi$
$439$	9.90763e8	0.558912	0.279456	+	0.960158i	$0.409846\pi$
$440$	0	0
$441$	−5.97377e8	−0.331675
$442$	0	0
$443$	1.77376e9	0.969351	0.484675	−	0.874694i	$-0.338938\pi$
$443$	1.77376e9	0.969351	0.484675	+	0.874694i	$0.338938\pi$
$444$	0	0
$445$	3.32604e9	1.78924
$446$	0	0
$447$	−5.61185e8	−0.297187
$448$	0	0
$449$	−2.77010e8	−0.144422	−0.0722110	−	0.997389i	$-0.523006\pi$
$449$	−2.77010e8	−0.144422	−0.0722110	+	0.997389i	$0.523006\pi$
$450$	0	0
$451$	−5.96973e8	−0.306434
$452$	0	0
$453$	2.05502e6	0.00103866
$454$	0	0
$455$	1.27246e8	0.0633293
$456$	0	0
$457$	2.94758e9	1.44464	0.722320	−	0.691559i	$-0.243076\pi$
$457$	2.94758e9	1.44464	0.722320	+	0.691559i	$0.243076\pi$
$458$	0	0
$459$	5.58722e8	0.269681
$460$	0	0
$461$	2.76687e9	1.31533	0.657667	−	0.753309i	$-0.271543\pi$
$461$	2.76687e9	1.31533	0.657667	+	0.753309i	$0.271543\pi$
$462$	0	0
$463$	4.63553e8	0.217053	0.108527	−	0.994094i	$-0.465387\pi$
$463$	4.63553e8	0.217053	0.108527	+	0.994094i	$0.465387\pi$
$464$	0	0
$465$	2.91479e9	1.34438
$466$	0	0
$467$	4.17922e8	0.189883	0.0949415	−	0.995483i	$-0.469734\pi$
$467$	4.17922e8	0.189883	0.0949415	+	0.995483i	$0.469734\pi$
$468$	0	0
$469$	−3.24636e7	−0.0145309
$470$	0	0
$471$	8.68927e8	0.383186
$472$	0	0
$473$	−6.50112e8	−0.282471
$474$	0	0
$475$	6.37664e8	0.273001
$476$	0	0
$477$	3.12054e8	0.131648
$478$	0	0
$479$	−1.50973e9	−0.627660	−0.313830	−	0.949479i	$-0.601612\pi$
$479$	−1.50973e9	−0.627660	−0.313830	+	0.949479i	$0.601612\pi$
$480$	0	0
$481$	−1.36895e9	−0.560891
$482$	0	0
$483$	2.64177e7	0.0106679
$484$	0	0
$485$	3.44246e9	1.37016
$486$	0	0
$487$	9.29460e8	0.364653	0.182326	−	0.983238i	$-0.441637\pi$
$487$	9.29460e8	0.364653	0.182326	+	0.983238i	$0.441637\pi$
$488$	0	0
$489$	−1.57527e9	−0.609221
$490$	0	0
$491$	−5.12803e9	−1.95508	−0.977541	−	0.210743i	$-0.932412\pi$
$491$	−5.12803e9	−1.95508	−0.977541	+	0.210743i	$0.932412\pi$
$492$	0	0
$493$	−1.03637e9	−0.389540
$494$	0	0
$495$	−2.69526e8	−0.0998809
$496$	0	0
$497$	−3.55880e8	−0.130034
$498$	0	0
$499$	4.10649e8	0.147951	0.0739757	−	0.997260i	$-0.476431\pi$
$499$	4.10649e8	0.147951	0.0739757	+	0.997260i	$0.476431\pi$
$500$	0	0
$501$	−6.97586e8	−0.247837
$502$	0	0
$503$	5.02041e9	1.75894	0.879470	−	0.475954i	$-0.157897\pi$
$503$	5.02041e9	1.75894	0.879470	+	0.475954i	$0.157897\pi$
$504$	0	0
$505$	4.67470e9	1.61523
$506$	0	0
$507$	−9.92491e8	−0.338219
$508$	0	0
$509$	3.24926e9	1.09212	0.546062	−	0.837745i	$-0.316126\pi$
$509$	3.24926e9	1.09212	0.546062	+	0.837745i	$0.316126\pi$
$510$	0	0
$511$	−8.76212e7	−0.0290493
$512$	0	0
$513$	1.69667e8	0.0554866
$514$	0	0
$515$	−2.81166e9	−0.907064
$516$	0	0
$517$	5.52965e8	0.175987
$518$	0	0
$519$	−1.71504e9	−0.538504
$520$	0	0
$521$	−2.10950e9	−0.653503	−0.326752	−	0.945110i	$-0.605954\pi$
$521$	−2.10950e9	−0.653503	−0.326752	+	0.945110i	$0.605954\pi$
$522$	0	0
$523$	5.28911e9	1.61669	0.808345	−	0.588709i	$-0.200364\pi$
$523$	5.28911e9	1.61669	0.808345	+	0.588709i	$0.200364\pi$
$524$	0	0
$525$	−1.27829e8	−0.0385542
$526$	0	0
$527$	−7.85747e9	−2.33854
$528$	0	0
$529$	−3.17110e9	−0.931355
$530$	0	0
$531$	−9.52351e8	−0.276036
$532$	0	0
$533$	−3.21030e9	−0.918334
$534$	0	0
$535$	4.45617e9	1.25812
$536$	0	0
$537$	2.18581e9	0.609119
$538$	0	0
$539$	−7.76836e8	−0.213682
$540$	0	0
$541$	−3.04614e9	−0.827101	−0.413551	−	0.910481i	$-0.635712\pi$
$541$	−3.04614e9	−0.827101	−0.413551	+	0.910481i	$0.635712\pi$
$542$	0	0
$543$	−1.74159e9	−0.466816
$544$	0	0
$545$	1.56817e9	0.414959
$546$	0	0
$547$	4.85537e9	1.26843	0.634215	−	0.773157i	$-0.281323\pi$
$547$	4.85537e9	1.26843	0.634215	+	0.773157i	$0.281323\pi$
$548$	0	0
$549$	−2.19183e8	−0.0565331
$550$	0	0
$551$	−3.14716e8	−0.0801472
$552$	0	0
$553$	4.42478e8	0.111264
$554$	0	0
$555$	2.82758e9	0.702084
$556$	0	0
$557$	−1.27762e9	−0.313263	−0.156631	−	0.987657i	$-0.550063\pi$
$557$	−1.27762e9	−0.313263	−0.156631	+	0.987657i	$0.550063\pi$
$558$	0	0
$559$	−3.49607e9	−0.846522
$560$	0	0
$561$	7.26568e8	0.173743
$562$	0	0
$563$	−4.71265e9	−1.11297	−0.556487	−	0.830856i	$-0.687851\pi$
$563$	−4.71265e9	−1.11297	−0.556487	+	0.830856i	$0.687851\pi$
$564$	0	0
$565$	6.90546e9	1.61073
$566$	0	0
$567$	−3.40122e7	−0.00783600
$568$	0	0
$569$	4.57800e9	1.04180	0.520898	−	0.853619i	$-0.325597\pi$
$569$	4.57800e9	1.04180	0.520898	+	0.853619i	$0.325597\pi$
$570$	0	0
$571$	−4.95119e9	−1.11297	−0.556485	−	0.830858i	$-0.687850\pi$
$571$	−4.95119e9	−1.11297	−0.556485	+	0.830858i	$0.687850\pi$
$572$	0	0
$573$	1.53434e9	0.340706
$574$	0	0
$575$	−1.13093e9	−0.248084
$576$	0	0
$577$	8.51847e9	1.84606	0.923031	−	0.384725i	$-0.125704\pi$
$577$	8.51847e9	1.84606	0.923031	+	0.384725i	$0.125704\pi$
$578$	0	0
$579$	3.14218e9	0.672753
$580$	0	0
$581$	−2.80112e8	−0.0592536
$582$	0	0
$583$	4.05799e8	0.0848147
$584$	0	0
$585$	−1.44941e9	−0.299327
$586$	0	0
$587$	5.62247e8	0.114735	0.0573673	−	0.998353i	$-0.481729\pi$
$587$	5.62247e8	0.114735	0.0573673	+	0.998353i	$0.481729\pi$
$588$	0	0
$589$	−2.38608e9	−0.481152
$590$	0	0
$591$	3.20803e9	0.639268
$592$	0	0
$593$	3.62110e9	0.713099	0.356549	−	0.934277i	$-0.383953\pi$
$593$	3.62110e9	0.713099	0.356549	+	0.934277i	$0.383953\pi$
$594$	0	0
$595$	7.08515e8	0.137892
$596$	0	0
$597$	−2.56529e9	−0.493430
$598$	0	0
$599$	−7.48104e9	−1.42222	−0.711112	−	0.703079i	$-0.751808\pi$
$599$	−7.48104e9	−1.42222	−0.711112	+	0.703079i	$0.751808\pi$
$600$	0	0
$601$	−5.81270e9	−1.09224	−0.546119	−	0.837707i	$-0.683895\pi$
$601$	−5.81270e9	−1.09224	−0.546119	+	0.837707i	$0.683895\pi$
$602$	0	0
$603$	3.69781e8	0.0686806
$604$	0	0
$605$	7.24950e9	1.33096
$606$	0	0
$607$	3.84051e9	0.696993	0.348497	−	0.937310i	$-0.386692\pi$
$607$	3.84051e9	0.696993	0.348497	+	0.937310i	$0.386692\pi$
$608$	0	0
$609$	6.30893e7	0.0113187
$610$	0	0
$611$	2.97364e9	0.527405
$612$	0	0
$613$	−1.70484e9	−0.298932	−0.149466	−	0.988767i	$-0.547755\pi$
$613$	−1.70484e9	−0.298932	−0.149466	+	0.988767i	$0.547755\pi$
$614$	0	0
$615$	6.63093e9	1.14951
$616$	0	0
$617$	−2.80809e9	−0.481297	−0.240649	−	0.970612i	$-0.577360\pi$
$617$	−2.80809e9	−0.481297	−0.240649	+	0.970612i	$0.577360\pi$
$618$	0	0
$619$	2.54365e9	0.431063	0.215532	−	0.976497i	$-0.430852\pi$
$619$	2.54365e9	0.431063	0.215532	+	0.976497i	$0.430852\pi$
$620$	0	0
$621$	−3.00914e8	−0.0504222
$622$	0	0
$623$	5.45812e8	0.0904346
$624$	0	0
$625$	−6.41051e9	−1.05030
$626$	0	0
$627$	2.20638e8	0.0357473
$628$	0	0
$629$	−7.62238e9	−1.22127
$630$	0	0
$631$	−1.51146e8	−0.0239494	−0.0119747	−	0.999928i	$-0.503812\pi$
$631$	−1.51146e8	−0.0239494	−0.0119747	+	0.999928i	$0.503812\pi$
$632$	0	0
$633$	−4.83965e9	−0.758405
$634$	0	0
$635$	−6.54744e9	−1.01476
$636$	0	0
$637$	−4.17754e9	−0.640373
$638$	0	0
$639$	4.05370e9	0.614609
$640$	0	0
$641$	−1.23625e10	−1.85397	−0.926987	−	0.375094i	$-0.877610\pi$
$641$	−1.23625e10	−1.85397	−0.926987	+	0.375094i	$0.877610\pi$
$642$	0	0
$643$	−2.86744e9	−0.425359	−0.212680	−	0.977122i	$-0.568219\pi$
$643$	−2.86744e9	−0.425359	−0.212680	+	0.977122i	$0.568219\pi$
$644$	0	0
$645$	7.22118e9	1.05962
$646$	0	0
$647$	−4.10640e9	−0.596068	−0.298034	−	0.954555i	$-0.596331\pi$
$647$	−4.10640e9	−0.596068	−0.298034	+	0.954555i	$0.596331\pi$
$648$	0	0
$649$	−1.23845e9	−0.177837
$650$	0	0
$651$	4.78324e8	0.0679499
$652$	0	0
$653$	−6.91100e9	−0.971280	−0.485640	−	0.874159i	$-0.661413\pi$
$653$	−6.91100e9	−0.971280	−0.485640	+	0.874159i	$0.661413\pi$
$654$	0	0
$655$	6.56244e9	0.912475
$656$	0	0
$657$	9.98061e8	0.137302
$658$	0	0
$659$	−3.42444e9	−0.466112	−0.233056	−	0.972463i	$-0.574873\pi$
$659$	−3.42444e9	−0.466112	−0.233056	+	0.972463i	$0.574873\pi$
$660$	0	0
$661$	6.76437e9	0.911008	0.455504	−	0.890234i	$-0.349459\pi$
$661$	6.76437e9	0.911008	0.455504	+	0.890234i	$0.349459\pi$
$662$	0	0
$663$	3.90722e9	0.520679
$664$	0	0
$665$	2.15155e8	0.0283711
$666$	0	0
$667$	5.58165e8	0.0728320
$668$	0	0
$669$	−5.57650e9	−0.720062
$670$	0	0
$671$	−2.85028e8	−0.0364215
$672$	0	0
$673$	−1.74959e9	−0.221250	−0.110625	−	0.993862i	$-0.535285\pi$
$673$	−1.74959e9	−0.221250	−0.110625	+	0.993862i	$0.535285\pi$
$674$	0	0
$675$	1.45605e9	0.182227
$676$	0	0
$677$	−8.30011e9	−1.02807	−0.514036	−	0.857769i	$-0.671850\pi$
$677$	−8.30011e9	−1.02807	−0.514036	+	0.857769i	$0.671850\pi$
$678$	0	0
$679$	5.64916e8	0.0692532
$680$	0	0
$681$	−1.17168e9	−0.142165
$682$	0	0
$683$	1.21232e10	1.45594	0.727969	−	0.685610i	$-0.240464\pi$
$683$	1.21232e10	1.45594	0.727969	+	0.685610i	$0.240464\pi$
$684$	0	0
$685$	−1.09375e10	−1.30017
$686$	0	0
$687$	9.77213e8	0.114985
$688$	0	0
$689$	2.18224e9	0.254176
$690$	0	0
$691$	−8.21846e9	−0.947583	−0.473791	−	0.880637i	$-0.657115\pi$
$691$	−8.21846e9	−0.947583	−0.473791	+	0.880637i	$0.657115\pi$
$692$	0	0
$693$	−4.42299e7	−0.00504835
$694$	0	0
$695$	−4.61265e9	−0.521199
$696$	0	0
$697$	−1.78752e10	−1.99957
$698$	0	0
$699$	2.49034e9	0.275796
$700$	0	0
$701$	−4.72231e9	−0.517775	−0.258888	−	0.965907i	$-0.583356\pi$
$701$	−4.72231e9	−0.517775	−0.258888	+	0.965907i	$0.583356\pi$
$702$	0	0
$703$	−2.31469e9	−0.251275
$704$	0	0
$705$	−6.14211e9	−0.660170
$706$	0	0
$707$	7.67131e8	0.0816397
$708$	0	0
$709$	−2.78975e9	−0.293970	−0.146985	−	0.989139i	$-0.546957\pi$
$709$	−2.78975e9	−0.293970	−0.146985	+	0.989139i	$0.546957\pi$
$710$	0	0
$711$	−5.04010e9	−0.525891
$712$	0	0
$713$	4.23184e9	0.437236
$714$	0	0
$715$	−1.88483e9	−0.192842
$716$	0	0
$717$	1.34586e9	0.136359
$718$	0	0
$719$	1.51985e9	0.152493	0.0762463	−	0.997089i	$-0.475706\pi$
$719$	1.51985e9	0.152493	0.0762463	+	0.997089i	$0.475706\pi$
$720$	0	0
$721$	−4.61401e8	−0.0458464
$722$	0	0
$723$	5.38310e9	0.529722
$724$	0	0
$725$	−2.70083e9	−0.263217
$726$	0	0
$727$	−8.11761e9	−0.783534	−0.391767	−	0.920065i	$-0.628136\pi$
$727$	−8.11761e9	−0.783534	−0.391767	+	0.920065i	$0.628136\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−1.94663e10	−1.84320
$732$	0	0
$733$	1.03241e10	0.968249	0.484124	−	0.874999i	$-0.339138\pi$
$733$	1.03241e10	0.968249	0.484124	+	0.874999i	$0.339138\pi$
$734$	0	0
$735$	8.62878e9	0.801574
$736$	0	0
$737$	4.80867e8	0.0442475
$738$	0	0
$739$	1.35365e10	1.23382	0.616908	−	0.787035i	$-0.288385\pi$
$739$	1.35365e10	1.23382	0.616908	+	0.787035i	$0.288385\pi$
$740$	0	0
$741$	1.18651e9	0.107129
$742$	0	0
$743$	−1.71936e10	−1.53782	−0.768910	−	0.639356i	$-0.779201\pi$
$743$	−1.71936e10	−1.53782	−0.768910	+	0.639356i	$0.779201\pi$
$744$	0	0
$745$	8.10601e9	0.718224
$746$	0	0
$747$	3.19065e9	0.280064
$748$	0	0
$749$	7.31269e8	0.0635903
$750$	0	0
$751$	1.12478e10	0.969013	0.484506	−	0.874788i	$-0.338999\pi$
$751$	1.12478e10	0.969013	0.484506	+	0.874788i	$0.338999\pi$
$752$	0	0
$753$	1.06563e10	0.909545
$754$	0	0
$755$	−2.96837e7	−0.00251017
$756$	0	0
$757$	−1.63068e10	−1.36626	−0.683131	−	0.730296i	$-0.739382\pi$
$757$	−1.63068e10	−1.36626	−0.683131	+	0.730296i	$0.739382\pi$
$758$	0	0
$759$	−3.91312e8	−0.0324845
$760$	0	0
$761$	6.14069e9	0.505093	0.252546	−	0.967585i	$-0.418732\pi$
$761$	6.14069e9	0.505093	0.252546	+	0.967585i	$0.418732\pi$
$762$	0	0
$763$	2.57341e8	0.0209736
$764$	0	0
$765$	−8.07042e9	−0.651750
$766$	0	0
$767$	−6.65993e9	−0.532949
$768$	0	0
$769$	2.45069e10	1.94333	0.971664	−	0.236368i	$-0.0759569\pi$
$769$	2.45069e10	1.94333	0.971664	+	0.236368i	$0.0759569\pi$
$770$	0	0
$771$	−3.85791e9	−0.303153
$772$	0	0
$773$	1.01722e10	0.792110	0.396055	−	0.918227i	$-0.370379\pi$
$773$	1.01722e10	0.792110	0.396055	+	0.918227i	$0.370379\pi$
$774$	0	0
$775$	−2.04769e10	−1.58018
$776$	0	0
$777$	4.64013e8	0.0354860
$778$	0	0
$779$	−5.42817e9	−0.411408
$780$	0	0
$781$	5.27148e9	0.395962
$782$	0	0
$783$	−7.18626e8	−0.0534979
$784$	0	0
$785$	−1.25512e10	−0.926062
$786$	0	0
$787$	9.79135e9	0.716030	0.358015	−	0.933716i	$-0.383454\pi$
$787$	9.79135e9	0.716030	0.358015	+	0.933716i	$0.383454\pi$
$788$	0	0
$789$	1.18865e10	0.861558
$790$	0	0
$791$	1.13320e9	0.0814124
$792$	0	0
$793$	−1.53277e9	−0.109150
$794$	0	0
$795$	−4.50745e9	−0.318160
$796$	0	0
$797$	9.75782e9	0.682729	0.341365	−	0.939931i	$-0.389111\pi$
$797$	9.75782e9	0.682729	0.341365	+	0.939931i	$0.389111\pi$
$798$	0	0
$799$	1.65574e10	1.14836
$800$	0	0
$801$	−6.21714e9	−0.427442
$802$	0	0
$803$	1.29789e9	0.0884572
$804$	0	0
$805$	−3.81588e8	−0.0257816
$806$	0	0
$807$	−7.43592e9	−0.498055
$808$	0	0
$809$	−2.78706e9	−0.185066	−0.0925330	−	0.995710i	$-0.529496\pi$
$809$	−2.78706e9	−0.185066	−0.0925330	+	0.995710i	$0.529496\pi$
$810$	0	0
$811$	7.99983e9	0.526633	0.263316	−	0.964710i	$-0.415184\pi$
$811$	7.99983e9	0.526633	0.263316	+	0.964710i	$0.415184\pi$
$812$	0	0
$813$	−1.14661e10	−0.748339
$814$	0	0
$815$	2.27540e10	1.47233
$816$	0	0
$817$	−5.91135e9	−0.379236
$818$	0	0
$819$	−2.37852e8	−0.0151291
$820$	0	0
$821$	1.02402e10	0.645813	0.322906	−	0.946431i	$-0.395340\pi$
$821$	1.02402e10	0.645813	0.322906	+	0.946431i	$0.395340\pi$
$822$	0	0
$823$	2.78682e10	1.74265	0.871324	−	0.490707i	$-0.163262\pi$
$823$	2.78682e10	1.74265	0.871324	+	0.490707i	$0.163262\pi$
$824$	0	0
$825$	1.89346e9	0.117400
$826$	0	0
$827$	−2.35125e10	−1.44554	−0.722769	−	0.691090i	$-0.757131\pi$
$827$	−2.35125e10	−1.44554	−0.722769	+	0.691090i	$0.757131\pi$
$828$	0	0
$829$	1.28598e10	0.783960	0.391980	−	0.919974i	$-0.371790\pi$
$829$	1.28598e10	0.783960	0.391980	+	0.919974i	$0.371790\pi$
$830$	0	0
$831$	−1.39363e10	−0.842448
$832$	0	0
$833$	−2.32608e10	−1.39434
$834$	0	0
$835$	1.00762e10	0.598957
$836$	0	0
$837$	−5.44841e9	−0.321167
$838$	0	0
$839$	−7.99832e9	−0.467554	−0.233777	−	0.972290i	$-0.575109\pi$
$839$	−7.99832e9	−0.467554	−0.233777	+	0.972290i	$0.575109\pi$
$840$	0	0
$841$	−1.59169e10	−0.922725
$842$	0	0
$843$	−8.39816e9	−0.482822
$844$	0	0
$845$	1.43360e10	0.817389
$846$	0	0
$847$	1.18966e9	0.0672716
$848$	0	0
$849$	1.60463e10	0.899909
$850$	0	0
$851$	4.10523e9	0.228341
$852$	0	0
$853$	−4.20827e9	−0.232157	−0.116079	−	0.993240i	$-0.537032\pi$
$853$	−4.20827e9	−0.232157	−0.116079	+	0.993240i	$0.537032\pi$
$854$	0	0
$855$	−2.45075e9	−0.134097
$856$	0	0
$857$	3.19307e10	1.73291	0.866453	−	0.499259i	$-0.166394\pi$
$857$	3.19307e10	1.73291	0.866453	+	0.499259i	$0.166394\pi$
$858$	0	0
$859$	−2.18002e10	−1.17350	−0.586752	−	0.809767i	$-0.699594\pi$
$859$	−2.18002e10	−1.17350	−0.586752	+	0.809767i	$0.699594\pi$
$860$	0	0
$861$	1.08815e9	0.0581004
$862$	0	0
$863$	1.04728e10	0.554657	0.277329	−	0.960775i	$-0.410551\pi$
$863$	1.04728e10	0.554657	0.277329	+	0.960775i	$0.410551\pi$
$864$	0	0
$865$	2.47728e10	1.30142
$866$	0	0
$867$	1.06765e10	0.556368
$868$	0	0
$869$	−6.55421e9	−0.338806
$870$	0	0
$871$	2.58593e9	0.132603
$872$	0	0
$873$	−6.43475e9	−0.327327
$874$	0	0
$875$	−1.03584e8	−0.00522714
$876$	0	0
$877$	1.77787e10	0.890024	0.445012	−	0.895525i	$-0.353199\pi$
$877$	1.77787e10	0.890024	0.445012	+	0.895525i	$0.353199\pi$
$878$	0	0
$879$	−3.11890e9	−0.154896
$880$	0	0
$881$	−7.64253e9	−0.376549	−0.188274	−	0.982116i	$-0.560289\pi$
$881$	−7.64253e9	−0.376549	−0.188274	+	0.982116i	$0.560289\pi$
$882$	0	0
$883$	2.76375e10	1.35094	0.675472	−	0.737386i	$-0.263940\pi$
$883$	2.76375e10	1.35094	0.675472	+	0.737386i	$0.263940\pi$
$884$	0	0
$885$	1.37562e10	0.667108
$886$	0	0
$887$	3.23087e10	1.55449	0.777243	−	0.629200i	$-0.216617\pi$
$887$	3.23087e10	1.55449	0.777243	+	0.629200i	$0.216617\pi$
$888$	0	0
$889$	−1.07445e9	−0.0512897
$890$	0	0
$891$	5.03806e8	0.0238612
$892$	0	0
$893$	5.02801e9	0.236274
$894$	0	0
$895$	−3.15728e10	−1.47208
$896$	0	0
$897$	−2.10433e9	−0.0973511
$898$	0	0
$899$	1.01063e10	0.463908
$900$	0	0
$901$	1.21509e10	0.553439
$902$	0	0
$903$	1.18501e9	0.0535570
$904$	0	0
$905$	2.51562e10	1.12817
$906$	0	0
$907$	−2.27142e10	−1.01082	−0.505409	−	0.862880i	$-0.668658\pi$
$907$	−2.27142e10	−1.01082	−0.505409	+	0.862880i	$0.668658\pi$
$908$	0	0
$909$	−8.73810e9	−0.385872
$910$	0	0
$911$	7.50925e9	0.329065	0.164533	−	0.986372i	$-0.447388\pi$
$911$	7.50925e9	0.329065	0.164533	+	0.986372i	$0.447388\pi$
$912$	0	0
$913$	4.14916e9	0.180431
$914$	0	0
$915$	3.16597e9	0.136626
$916$	0	0
$917$	1.07691e9	0.0461199
$918$	0	0
$919$	−2.49374e10	−1.05986	−0.529928	−	0.848043i	$-0.677781\pi$
$919$	−2.49374e10	−1.05986	−0.529928	+	0.848043i	$0.677781\pi$
$920$	0	0
$921$	7.03619e9	0.296776
$922$	0	0
$923$	2.83481e10	1.18664
$924$	0	0
$925$	−1.98642e10	−0.825230
$926$	0	0
$927$	5.25565e9	0.216694
$928$	0	0
$929$	−8.66205e9	−0.354459	−0.177229	−	0.984170i	$-0.556713\pi$
$929$	−8.66205e9	−0.354459	−0.177229	+	0.984170i	$0.556713\pi$
$930$	0	0
$931$	−7.06363e9	−0.286883
$932$	0	0
$933$	1.55735e10	0.627768
$934$	0	0
$935$	−1.04949e10	−0.419891
$936$	0	0
$937$	2.82655e10	1.12245	0.561226	−	0.827663i	$-0.310330\pi$
$937$	2.82655e10	1.12245	0.561226	+	0.827663i	$0.310330\pi$
$938$	0	0
$939$	−1.24220e10	−0.489623
$940$	0	0
$941$	4.67082e10	1.82738	0.913691	−	0.406410i	$-0.133220\pi$
$941$	4.67082e10	1.82738	0.913691	+	0.406410i	$0.133220\pi$
$942$	0	0
$943$	9.62713e9	0.373857
$944$	0	0
$945$	4.91288e8	0.0189376
$946$	0	0
$947$	4.67392e10	1.78837	0.894184	−	0.447701i	$-0.147757\pi$
$947$	4.67392e10	1.78837	0.894184	+	0.447701i	$0.147757\pi$
$948$	0	0
$949$	6.97958e9	0.265093
$950$	0	0
$951$	−1.68902e9	−0.0636798
$952$	0	0
$953$	3.82420e10	1.43125	0.715625	−	0.698484i	$-0.246142\pi$
$953$	3.82420e10	1.43125	0.715625	+	0.698484i	$0.246142\pi$
$954$	0	0
$955$	−2.21627e10	−0.823399
$956$	0	0
$957$	−9.34510e8	−0.0344661
$958$	0	0
$959$	−1.79487e9	−0.0657157
$960$	0	0
$961$	4.91101e10	1.78500
$962$	0	0
$963$	−8.32961e9	−0.300561
$964$	0	0
$965$	−4.53870e10	−1.62587
$966$	0	0
$967$	−4.90012e10	−1.74267	−0.871333	−	0.490692i	$-0.836744\pi$
$967$	−4.90012e10	−1.74267	−0.871333	+	0.490692i	$0.836744\pi$
$968$	0	0
$969$	6.60656e9	0.233261
$970$	0	0
$971$	−2.72929e10	−0.956713	−0.478357	−	0.878166i	$-0.658767\pi$
$971$	−2.72929e10	−0.956713	−0.478357	+	0.878166i	$0.658767\pi$
$972$	0	0
$973$	−7.56947e8	−0.0263433
$974$	0	0
$975$	1.01824e10	0.351830
$976$	0	0
$977$	3.94482e9	0.135331	0.0676653	−	0.997708i	$-0.478445\pi$
$977$	3.94482e9	0.135331	0.0676653	+	0.997708i	$0.478445\pi$
$978$	0	0
$979$	−8.08484e9	−0.275380
$980$	0	0
$981$	−2.93127e9	−0.0991322
$982$	0	0
$983$	4.74320e8	0.0159270	0.00796351	−	0.999968i	$-0.497465\pi$
$983$	4.74320e8	0.0159270	0.00796351	+	0.999968i	$0.497465\pi$
$984$	0	0
$985$	−4.63383e10	−1.54494
$986$	0	0
$987$	−1.00794e9	−0.0333674
$988$	0	0
$989$	1.04841e10	0.344622
$990$	0	0
$991$	1.22197e10	0.398843	0.199421	−	0.979914i	$-0.436094\pi$
$991$	1.22197e10	0.398843	0.199421	+	0.979914i	$0.436094\pi$
$992$	0	0
$993$	−1.84744e10	−0.598752
$994$	0	0
$995$	3.70541e10	1.19249
$996$	0	0
$997$	3.60690e10	1.15266	0.576330	−	0.817217i	$-0.304484\pi$
$997$	3.60690e10	1.15266	0.576330	+	0.817217i	$0.304484\pi$
$998$	0	0
$999$	−5.28540e9	−0.167725

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.8.a.i.1.1		1
3.2	odd	2		576.8.a.w.1.1		1
4.3	odd	2		192.8.a.a.1.1		1
8.3	odd	2		48.8.a.g.1.1		1
8.5	even	2		3.8.a.a.1.1	✓	1
12.11	even	2		576.8.a.x.1.1		1
24.5	odd	2		9.8.a.a.1.1		1
24.11	even	2		144.8.a.b.1.1		1
40.13	odd	4		75.8.b.c.49.1		2
40.29	even	2		75.8.a.a.1.1		1
40.37	odd	4		75.8.b.c.49.2		2
56.5	odd	6		147.8.e.a.67.1		2
56.13	odd	2		147.8.a.b.1.1		1
56.37	even	6		147.8.e.b.67.1		2
56.45	odd	6		147.8.e.a.79.1		2
56.53	even	6		147.8.e.b.79.1		2
72.5	odd	6		81.8.c.c.55.1		2
72.13	even	6		81.8.c.a.55.1		2
72.29	odd	6		81.8.c.c.28.1		2
72.61	even	6		81.8.c.a.28.1		2
88.21	odd	2		363.8.a.b.1.1		1
104.77	even	2		507.8.a.a.1.1		1
120.29	odd	2		225.8.a.i.1.1		1
120.53	even	4		225.8.b.f.199.2		2
120.77	even	4		225.8.b.f.199.1		2
168.125	even	2		441.8.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.8.a.a.1.1	✓	1	8.5	even	2
9.8.a.a.1.1		1	24.5	odd	2
48.8.a.g.1.1		1	8.3	odd	2
75.8.a.a.1.1		1	40.29	even	2
75.8.b.c.49.1		2	40.13	odd	4
75.8.b.c.49.2		2	40.37	odd	4
81.8.c.a.28.1		2	72.61	even	6
81.8.c.a.55.1		2	72.13	even	6
81.8.c.c.28.1		2	72.29	odd	6
81.8.c.c.55.1		2	72.5	odd	6
144.8.a.b.1.1		1	24.11	even	2
147.8.a.b.1.1		1	56.13	odd	2
147.8.e.a.67.1		2	56.5	odd	6
147.8.e.a.79.1		2	56.45	odd	6
147.8.e.b.67.1		2	56.37	even	6
147.8.e.b.79.1		2	56.53	even	6
192.8.a.a.1.1		1	4.3	odd	2
192.8.a.i.1.1		1	1.1	even	1	trivial
225.8.a.i.1.1		1	120.29	odd	2
225.8.b.f.199.1		2	120.77	even	4
225.8.b.f.199.2		2	120.53	even	4
363.8.a.b.1.1		1	88.21	odd	2
441.8.a.a.1.1		1	168.125	even	2
507.8.a.a.1.1		1	104.77	even	2
576.8.a.w.1.1		1	3.2	odd	2
576.8.a.x.1.1		1	12.11	even	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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	192.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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