LMFDB - Embedded newform 441.8.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [441,8,Mod(1,441)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("441.1"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	441.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-6,0,-92,390,0,0,1320,0,-2340,948,0,5098] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$137.761796238$
Analytic rank:	$0$
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	$\chi$	$=$	441.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-6.00000 q^{2} -92.0000 q^{4} +390.000 q^{5} +1320.00 q^{8} -2340.00 q^{10} +948.000 q^{11} +5098.00 q^{13} +3856.00 q^{16} +28386.0 q^{17} +8620.00 q^{19} -35880.0 q^{20} -5688.00 q^{22} +15288.0 q^{23} +73975.0 q^{25} -30588.0 q^{26} -36510.0 q^{29} +276808. q^{31} -192096. q^{32} -170316. q^{34} +268526. q^{37} -51720.0 q^{38} +514800. q^{40} -629718. q^{41} +685772. q^{43} -87216.0 q^{44} -91728.0 q^{46} +583296. q^{47} -443850. q^{50} -469016. q^{52} +428058. q^{53} +369720. q^{55} +219060. q^{58} +1.30638e6 q^{59} -300662. q^{61} -1.66085e6 q^{62} +659008. q^{64} +1.98822e6 q^{65} -507244. q^{67} -2.61151e6 q^{68} -5.56063e6 q^{71} -1.36908e6 q^{73} -1.61116e6 q^{74} -793040. q^{76} -6.91372e6 q^{79} +1.50384e6 q^{80} +3.77831e6 q^{82} -4.37675e6 q^{83} +1.10705e7 q^{85} -4.11463e6 q^{86} +1.25136e6 q^{88} -8.52831e6 q^{89} -1.40650e6 q^{92} -3.49978e6 q^{94} +3.36180e6 q^{95} +8.82681e6 q^{97} +O(q^{100})$

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 $

$2$	−6.00000	−0.530330	−0.265165	−	0.964203i	$-0.585426\pi$
$2$	−6.00000	−0.530330	−0.265165	+	0.964203i	$0.585426\pi$
$3$	0	0
$3$	0	0
$4$	−92.0000	−0.718750
$5$	390.000	1.39531	0.697653	−	0.716436i	$-0.254228\pi$
$5$	390.000	1.39531	0.697653	+	0.716436i	$0.254228\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1320.00	0.911505
$9$	0	0
$10$	−2340.00	−0.739973
$11$	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
$15$	0	0
$16$	3856.00	0.235352
$17$	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	$0.453953\pi$
$20$	−35880.0	−1.00288
$21$	0	0
$22$	−5688.00	−0.113889
$23$	15288.0	0.262001	0.131001	−	0.991382i	$-0.458181\pi$
$23$	15288.0	0.262001	0.131001	+	0.991382i	$0.458181\pi$
$24$	0	0
$25$	73975.0	0.946880
$26$	−30588.0	−0.341306
$27$	0	0
$28$	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	$-0.185805\pi$
$31$	276808.	1.66883	0.834416	+	0.551135i	$0.185805\pi$
$32$	−192096.	−1.03632
$33$	0	0
$34$	−170316.	−0.743155
$35$	0	0
$36$	0	0
$37$	268526.	0.871526	0.435763	−	0.900061i	$-0.356479\pi$
$37$	268526.	0.871526	0.435763	+	0.900061i	$0.356479\pi$
$38$	−51720.0	−0.152903
$39$	0	0
$40$	514800.	1.27183
$41$	−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	$-0.271541\pi$
$43$	685772.	1.31535	0.657673	+	0.753303i	$0.271541\pi$
$44$	−87216.0	−0.154352
$45$	0	0
$46$	−91728.0	−0.138947
$47$	583296.	0.819495	0.409748	−	0.912199i	$-0.365617\pi$
$47$	583296.	0.819495	0.409748	+	0.912199i	$0.365617\pi$
$48$	0	0
$49$	0	0
$50$	−443850.	−0.502159
$51$	0	0
$52$	−469016.	−0.462568
$53$	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
$57$	0	0
$58$	219060.	0.147423
$59$	1.30638e6	0.828109	0.414054	−	0.910252i	$-0.364112\pi$
$59$	1.30638e6	0.828109	0.414054	+	0.910252i	$0.364112\pi$
$60$	0	0
$61$	−300662.	−0.169599	−0.0847997	−	0.996398i	$-0.527025\pi$
$61$	−300662.	−0.169599	−0.0847997	+	0.996398i	$0.527025\pi$
$62$	−1.66085e6	−0.885032
$63$	0	0
$64$	659008.	0.314240
$65$	1.98822e6	0.897982
$66$	0	0
$67$	−507244.	−0.206042	−0.103021	−	0.994679i	$-0.532851\pi$
$67$	−507244.	−0.206042	−0.103021	+	0.994679i	$0.532851\pi$
$68$	−2.61151e6	−1.00719
$69$	0	0
$70$	0	0
$71$	−5.56063e6	−1.84383	−0.921913	−	0.387397i	$-0.873374\pi$
$71$	−5.56063e6	−1.84383	−0.921913	+	0.387397i	$0.873374\pi$
$72$	0	0
$73$	−1.36908e6	−0.411907	−0.205954	−	0.978562i	$-0.566030\pi$
$73$	−1.36908e6	−0.411907	−0.205954	+	0.978562i	$0.566030\pi$
$74$	−1.61116e6	−0.462196
$75$	0	0
$76$	−793040.	−0.207228
$77$	0	0
$78$	0	0
$79$	−6.91372e6	−1.57767	−0.788836	−	0.614603i	$-0.789316\pi$
$79$	−6.91372e6	−1.57767	−0.788836	+	0.614603i	$0.789316\pi$
$80$	1.50384e6	0.328388
$81$	0	0
$82$	3.77831e6	0.756744
$83$	−4.37675e6	−0.840191	−0.420096	−	0.907480i	$-0.638003\pi$
$83$	−4.37675e6	−0.840191	−0.420096	+	0.907480i	$0.638003\pi$
$84$	0	0
$85$	1.10705e7	1.95525
$86$	−4.11463e6	−0.697568
$87$	0	0
$88$	1.25136e6	0.195746
$89$	−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
$91$	0	0
$92$	−1.40650e6	−0.188313
$93$	0	0
$94$	−3.49978e6	−0.434603
$95$	3.36180e6	0.402290
$96$	0	0
$97$	8.82681e6	0.981981	0.490990	−	0.871165i	$-0.336635\pi$
$97$	8.82681e6	0.981981	0.490990	+	0.871165i	$0.336635\pi$
$98$	0	0
$99$	0	0
$100$	−6.80570e6	−0.680570
$101$	1.19864e7	1.15762	0.578808	−	0.815464i	$-0.303518\pi$
$101$	1.19864e7	1.15762	0.578808	+	0.815464i	$0.303518\pi$
$102$	0	0
$103$	−7.20939e6	−0.650082	−0.325041	−	0.945700i	$-0.605378\pi$
$103$	−7.20939e6	−0.650082	−0.325041	+	0.945700i	$0.605378\pi$
$104$	6.72936e6	0.586620
$105$	0	0
$106$	−2.56835e6	−0.209451
$107$	−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	$-0.452492\pi$
$109$	4.02095e6	0.297397	0.148698	+	0.988883i	$0.452492\pi$
$110$	−2.21832e6	−0.158909
$111$	0	0
$112$	0	0
$113$	1.77063e7	1.15439	0.577197	−	0.816605i	$-0.304147\pi$
$113$	1.77063e7	1.15439	0.577197	+	0.816605i	$0.304147\pi$
$114$	0	0
$115$	5.96232e6	0.365572
$116$	3.35892e6	0.199801
$117$	0	0
$118$	−7.83828e6	−0.439171
$119$	0	0
$120$	0	0
$121$	−1.85885e7	−0.953882
$122$	1.80397e6	0.0899436
$123$	0	0
$124$	−2.54663e7	−1.19947
$125$	−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	$0.381536\pi$
$128$	2.06342e7	0.869668
$129$	0	0
$130$	−1.19293e7	−0.476227
$131$	1.68268e7	0.653960	0.326980	−	0.945031i	$-0.393969\pi$
$131$	1.68268e7	0.653960	0.326980	+	0.945031i	$0.393969\pi$
$132$	0	0
$133$	0	0
$134$	3.04346e6	0.109270
$135$	0	0
$136$	3.74695e7	1.27730
$137$	−2.80449e7	−0.931820	−0.465910	−	0.884832i	$-0.654273\pi$
$137$	−2.80449e7	−0.931820	−0.465910	+	0.884832i	$0.654273\pi$
$138$	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
$141$	0	0
$142$	3.33638e7	0.977836
$143$	4.83290e6	0.138208
$144$	0	0
$145$	−1.42389e7	−0.387872
$146$	8.21449e6	0.218447
$147$	0	0
$148$	−2.47044e7	−0.626409
$149$	−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	$0.499714\pi$
$152$	1.13784e7	0.262802
$153$	0	0
$154$	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	$0.392328\pi$
$158$	4.14823e7	0.836687
$159$	0	0
$160$	−7.49174e7	−1.44598
$161$	0	0
$162$	0	0
$163$	5.83435e7	1.05520	0.527601	−	0.849492i	$-0.323092\pi$
$163$	5.83435e7	1.05520	0.527601	+	0.849492i	$0.323092\pi$
$164$	5.79341e7	1.02561
$165$	0	0
$166$	2.62605e7	0.445579
$167$	−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
$170$	−6.64232e7	−1.03693
$171$	0	0
$172$	−6.30910e7	−0.945405
$173$	6.35201e7	0.932716	0.466358	−	0.884596i	$-0.345566\pi$
$173$	6.35201e7	0.932716	0.466358	+	0.884596i	$0.345566\pi$
$174$	0	0
$175$	0	0
$176$	3.65549e6	0.0505418
$177$	0	0
$178$	5.11699e7	0.680055
$179$	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
$183$	0	0
$184$	2.01802e7	0.238815
$185$	1.04725e8	1.21605
$186$	0	0
$187$	2.69099e7	0.300931
$188$	−5.36632e7	−0.589012
$189$	0	0
$190$	−2.01708e7	−0.213346
$191$	−5.68274e7	−0.590121	−0.295060	−	0.955479i	$-0.595340\pi$
$191$	−5.68274e7	−0.590121	−0.295060	+	0.955479i	$0.595340\pi$
$192$	0	0
$193$	1.16377e8	1.16524	0.582621	−	0.812744i	$-0.302027\pi$
$193$	1.16377e8	1.16524	0.582621	+	0.812744i	$0.302027\pi$
$194$	−5.29609e7	−0.520774
$195$	0	0
$196$	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	$-0.359457\pi$
$199$	9.50106e7	0.854646	0.427323	+	0.904099i	$0.359457\pi$
$200$	9.76470e7	0.863086
$201$	0	0
$202$	−7.19185e7	−0.613919
$203$	0	0
$204$	0	0
$205$	−2.45590e8	−1.99100
$206$	4.32564e7	0.344758
$207$	0	0
$208$	1.96579e7	0.151466
$209$	8.17176e6	0.0619161
$210$	0	0
$211$	1.79246e8	1.31360	0.656798	−	0.754067i	$-0.271910\pi$
$211$	1.79246e8	1.31360	0.656798	+	0.754067i	$0.271910\pi$
$212$	−3.93813e7	−0.283867
$213$	0	0
$214$	6.85565e7	0.478190
$215$	2.67451e8	1.83531
$216$	0	0
$217$	0	0
$218$	−2.41257e7	−0.157718
$219$	0	0
$220$	−3.40142e7	−0.215368
$221$	1.44712e8	0.901843
$222$	0	0
$223$	2.06537e8	1.24718	0.623592	−	0.781750i	$-0.285673\pi$
$223$	2.06537e8	1.24718	0.623592	+	0.781750i	$0.285673\pi$
$224$	0	0
$225$	0	0
$226$	−1.06238e8	−0.612209
$227$	4.33954e7	0.246237	0.123118	−	0.992392i	$-0.460710\pi$
$227$	4.33954e7	0.246237	0.123118	+	0.992392i	$0.460710\pi$
$228$	0	0
$229$	3.61931e7	0.199160	0.0995799	−	0.995030i	$-0.468250\pi$
$229$	3.61931e7	0.199160	0.0995799	+	0.995030i	$0.468250\pi$
$230$	−3.57739e7	−0.193874
$231$	0	0
$232$	−4.81932e7	−0.253383
$233$	−9.22347e7	−0.477693	−0.238846	−	0.971057i	$-0.576769\pi$
$233$	−9.22347e7	−0.477693	−0.238846	+	0.971057i	$0.576769\pi$
$234$	0	0
$235$	2.27485e8	1.14345
$236$	−1.20187e8	−0.595203
$237$	0	0
$238$	0	0
$239$	−4.98468e7	−0.236181	−0.118090	−	0.993003i	$-0.537677\pi$
$239$	−4.98468e7	−0.236181	−0.118090	+	0.993003i	$0.537677\pi$
$240$	0	0
$241$	−1.99374e8	−0.917506	−0.458753	−	0.888564i	$-0.651704\pi$
$241$	−1.99374e8	−0.917506	−0.458753	+	0.888564i	$0.651704\pi$
$242$	1.11531e8	0.505872
$243$	0	0
$244$	2.76609e7	0.121900
$245$	0	0
$246$	0	0
$247$	4.39448e7	0.185553
$248$	3.65387e8	1.52115
$249$	0	0
$250$	9.71100e6	0.0393074
$251$	−3.94678e8	−1.57538	−0.787689	−	0.616073i	$-0.788723\pi$
$251$	−3.94678e8	−1.57538	−0.787689	+	0.616073i	$0.788723\pi$
$252$	0	0
$253$	1.44930e7	0.0562649
$254$	−1.00730e8	−0.385691
$255$	0	0
$256$	−2.08158e8	−0.775451
$257$	−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
$259$	0	0
$260$	−1.82916e8	−0.645425
$261$	0	0
$262$	−1.00961e8	−0.346815
$263$	−4.40241e8	−1.49226	−0.746131	−	0.665799i	$-0.768091\pi$
$263$	−4.40241e8	−1.49226	−0.746131	+	0.665799i	$0.768091\pi$
$264$	0	0
$265$	1.66943e8	0.551070
$266$	0	0
$267$	0	0
$268$	4.66664e7	0.148092
$269$	2.75405e8	0.862657	0.431329	−	0.902195i	$-0.358045\pi$
$269$	2.75405e8	0.862657	0.431329	+	0.902195i	$0.358045\pi$
$270$	0	0
$271$	4.24670e8	1.29616	0.648080	−	0.761572i	$-0.275572\pi$
$271$	4.24670e8	1.29616	0.648080	+	0.761572i	$0.275572\pi$
$272$	1.09456e8	0.329800
$273$	0	0
$274$	1.68269e8	0.494172
$275$	7.01283e7	0.203343
$276$	0	0
$277$	5.16158e8	1.45916	0.729581	−	0.683894i	$-0.239715\pi$
$277$	5.16158e8	1.45916	0.729581	+	0.683894i	$0.239715\pi$
$278$	−7.09638e7	−0.198098
$279$	0	0
$280$	0	0
$281$	3.11043e8	0.836273	0.418137	−	0.908384i	$-0.362683\pi$
$281$	3.11043e8	0.836273	0.418137	+	0.908384i	$0.362683\pi$
$282$	0	0
$283$	5.94308e8	1.55869	0.779344	−	0.626596i	$-0.215552\pi$
$283$	5.94308e8	1.55869	0.779344	+	0.626596i	$0.215552\pi$
$284$	5.11578e8	1.32525
$285$	0	0
$286$	−2.89974e7	−0.0732957
$287$	0	0
$288$	0	0
$289$	3.95426e8	0.963658
$290$	8.54334e7	0.205700
$291$	0	0
$292$	1.25956e8	0.296058
$293$	1.15515e8	0.268288	0.134144	−	0.990962i	$-0.457172\pi$
$293$	1.15515e8	0.268288	0.134144	+	0.990962i	$0.457172\pi$
$294$	0	0
$295$	5.09488e8	1.15547
$296$	3.54454e8	0.794400
$297$	0	0
$298$	1.24708e8	0.272984
$299$	7.79382e7	0.168617
$300$	0	0
$301$	0	0
$302$	−456672.	−0.000954070 0
$303$	0	0
$304$	3.32387e7	0.0678558
$305$	−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
$309$	0	0
$310$	−6.47731e8	−1.23489
$311$	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	$-0.360616\pi$
$313$	4.60074e8	0.848053	0.424026	+	0.905650i	$0.360616\pi$
$314$	−1.93095e8	−0.351979
$315$	0	0
$316$	6.36062e8	1.13395
$317$	−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
$320$	2.57013e8	0.438460
$321$	0	0
$322$	0	0
$323$	2.44687e8	0.404020
$324$	0	0
$325$	3.77125e8	0.609387
$326$	−3.50061e8	−0.559606
$327$	0	0
$328$	−8.31228e8	−1.30065
$329$	0	0
$330$	0	0
$331$	6.84236e8	1.03707	0.518535	−	0.855057i	$-0.326478\pi$
$331$	6.84236e8	1.03707	0.518535	+	0.855057i	$0.326478\pi$
$332$	4.02661e8	0.603888
$333$	0	0
$334$	1.55019e8	0.227652
$335$	−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	$0.647050\pi$
$338$	2.20553e8	0.310674
$339$	0	0
$340$	−1.01849e9	−1.40534
$341$	2.62414e8	0.358382
$342$	0	0
$343$	0	0
$344$	9.05219e8	1.19894
$345$	0	0
$346$	−3.81120e8	−0.494647
$347$	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
$351$	0	0
$352$	−1.82107e8	−0.222550
$353$	−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
$356$	7.84605e8	0.921671
$357$	0	0
$358$	−4.85735e8	−0.559512
$359$	−9.32541e8	−1.06374	−0.531872	−	0.846825i	$-0.678511\pi$
$359$	−9.32541e8	−1.06374	−0.531872	+	0.846825i	$0.678511\pi$
$360$	0	0
$361$	−8.19567e8	−0.916874
$362$	3.87019e8	0.428798
$363$	0	0
$364$	0	0
$365$	−5.33942e8	−0.574737
$366$	0	0
$367$	8.52565e8	0.900318	0.450159	−	0.892948i	$-0.351367\pi$
$367$	8.52565e8	0.900318	0.450159	+	0.892948i	$0.351367\pi$
$368$	5.89505e7	0.0616624
$369$	0	0
$370$	−6.28351e8	−0.644906
$371$	0	0
$372$	0	0
$373$	3.81183e8	0.380323	0.190162	−	0.981753i	$-0.439099\pi$
$373$	3.81183e8	0.380323	0.190162	+	0.981753i	$0.439099\pi$
$374$	−1.61460e8	−0.159593
$375$	0	0
$376$	7.69951e8	0.746974
$377$	−1.86128e8	−0.178903
$378$	0	0
$379$	−1.48353e9	−1.39978	−0.699889	−	0.714251i	$-0.746767\pi$
$379$	−1.48353e9	−1.39978	−0.699889	+	0.714251i	$0.746767\pi$
$380$	−3.09286e8	−0.289146
$381$	0	0
$382$	3.40964e8	0.312959
$383$	−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
$385$	0	0
$386$	−6.98262e8	−0.617963
$387$	0	0
$388$	−8.12067e8	−0.705799
$389$	−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
$393$	0	0
$394$	−7.12896e8	−0.587205
$395$	−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	$0.771900\pi$
$398$	−5.70064e8	−0.453245
$399$	0	0
$400$	2.85248e8	0.222850
$401$	−2.68592e8	−0.208012	−0.104006	−	0.994577i	$-0.533166\pi$
$401$	−2.68592e8	−0.208012	−0.104006	+	0.994577i	$0.533166\pi$
$402$	0	0
$403$	1.41117e9	1.07402
$404$	−1.10275e9	−0.832037
$405$	0	0
$406$	0	0
$407$	2.54563e8	0.187161
$408$	0	0
$409$	−8.99478e7	−0.0650069	−0.0325034	−	0.999472i	$-0.510348\pi$
$409$	−8.99478e7	−0.0650069	−0.0325034	+	0.999472i	$0.510348\pi$
$410$	1.47354e9	1.05589
$411$	0	0
$412$	6.63264e8	0.467247
$413$	0	0
$414$	0	0
$415$	−1.70693e9	−1.17232
$416$	−9.79305e8	−0.666947
$417$	0	0
$418$	−4.90306e7	−0.0328360
$419$	1.69054e9	1.12273	0.561367	−	0.827567i	$-0.310276\pi$
$419$	1.69054e9	1.12273	0.561367	+	0.827567i	$0.310276\pi$
$420$	0	0
$421$	−1.13333e9	−0.740232	−0.370116	−	0.928985i	$-0.620682\pi$
$421$	−1.13333e9	−0.740232	−0.370116	+	0.928985i	$0.620682\pi$
$422$	−1.07548e9	−0.696639
$423$	0	0
$424$	5.65037e8	0.359995
$425$	2.09985e9	1.32687
$426$	0	0
$427$	0	0
$428$	1.05120e9	0.648085
$429$	0	0
$430$	−1.60471e9	−0.973321
$431$	−2.19943e9	−1.32324	−0.661621	−	0.749839i	$-0.730131\pi$
$431$	−2.19943e9	−1.32324	−0.661621	+	0.749839i	$0.730131\pi$
$432$	0	0
$433$	1.51738e8	0.0898227	0.0449114	−	0.998991i	$-0.485699\pi$
$433$	1.51738e8	0.0898227	0.0449114	+	0.998991i	$0.485699\pi$
$434$	0	0
$435$	0	0
$436$	−3.69927e8	−0.213754
$437$	1.31783e8	0.0755393
$438$	0	0
$439$	−9.90763e8	−0.558912	−0.279456	−	0.960158i	$-0.590154\pi$
$439$	−9.90763e8	−0.558912	−0.279456	+	0.960158i	$0.590154\pi$
$440$	4.88030e8	0.273126
$441$	0	0
$442$	−8.68271e8	−0.478275
$443$	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
$446$	−1.23922e9	−0.661419
$447$	0	0
$448$	0	0
$449$	2.77010e8	0.144422	0.0722110	−	0.997389i	$-0.476994\pi$
$449$	2.77010e8	0.144422	0.0722110	+	0.997389i	$0.476994\pi$
$450$	0	0
$451$	−5.96973e8	−0.306434
$452$	−1.62898e9	−0.829720
$453$	0	0
$454$	−2.60372e8	−0.130587
$455$	0	0
$456$	0	0
$457$	2.94758e9	1.44464	0.722320	−	0.691559i	$-0.243076\pi$
$457$	2.94758e9	1.44464	0.722320	+	0.691559i	$0.243076\pi$
$458$	−2.17159e8	−0.105620
$459$	0	0
$460$	−5.48533e8	−0.262755
$461$	−2.76687e9	−1.31533	−0.657667	−	0.753309i	$-0.728457\pi$
$461$	−2.76687e9	−1.31533	−0.657667	+	0.753309i	$0.728457\pi$
$462$	0	0
$463$	4.63553e8	0.217053	0.108527	−	0.994094i	$-0.465387\pi$
$463$	4.63553e8	0.217053	0.108527	+	0.994094i	$0.465387\pi$
$464$	−1.40783e8	−0.0654238
$465$	0	0
$466$	5.53408e8	0.253335
$467$	−4.17922e8	−0.189883	−0.0949415	−	0.995483i	$-0.530266\pi$
$467$	−4.17922e8	−0.189883	−0.0949415	+	0.995483i	$0.530266\pi$
$468$	0	0
$469$	0	0
$470$	−1.36491e9	−0.606404
$471$	0	0
$472$	1.72442e9	0.754825
$473$	6.50112e8	0.282471
$474$	0	0
$475$	6.37664e8	0.273001
$476$	0	0
$477$	0	0
$478$	2.99081e8	0.125254
$479$	−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
$482$	1.19624e9	0.486581
$483$	0	0
$484$	1.71014e9	0.685603
$485$	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	$0.441637\pi$
$488$	−3.96874e8	−0.154591
$489$	0	0
$490$	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
$494$	−2.63669e8	−0.0984043
$495$	0	0
$496$	1.06737e9	0.392762
$497$	0	0
$498$	0	0
$499$	−4.10649e8	−0.147951	−0.0739757	−	0.997260i	$-0.523569\pi$
$499$	−4.10649e8	−0.147951	−0.0739757	+	0.997260i	$0.523569\pi$
$500$	1.48902e8	0.0532728
$501$	0	0
$502$	2.36807e9	0.835470
$503$	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
$506$	−8.69581e7	−0.0298389
$507$	0	0
$508$	−1.54452e9	−0.522723
$509$	−3.24926e9	−1.09212	−0.546062	−	0.837745i	$-0.683874\pi$
$509$	−3.24926e9	−1.09212	−0.546062	+	0.837745i	$0.683874\pi$
$510$	0	0
$511$	0	0
$512$	−1.39223e9	−0.458423
$513$	0	0
$514$	8.57312e8	0.278463
$515$	−2.81166e9	−0.907064
$516$	0	0
$517$	5.52965e8	0.175987
$518$	0	0
$519$	0	0
$520$	2.62445e9	0.818515
$521$	−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	$0.200364\pi$
$524$	−1.54806e9	−0.470034
$525$	0	0
$526$	2.64144e9	0.791391
$527$	7.85747e9	2.33854
$528$	0	0
$529$	−3.17110e9	−0.931355
$530$	−1.00166e9	−0.292249
$531$	0	0
$532$	0	0
$533$	−3.21030e9	−0.918334
$534$	0	0
$535$	−4.45617e9	−1.25812
$536$	−6.69562e8	−0.187808
$537$	0	0
$538$	−1.65243e9	−0.457493
$539$	0	0
$540$	0	0
$541$	3.04614e9	0.827101	0.413551	−	0.910481i	$-0.364288\pi$
$541$	3.04614e9	0.827101	0.413551	+	0.910481i	$0.364288\pi$
$542$	−2.54802e9	−0.687393
$543$	0	0
$544$	−5.45284e9	−1.45220
$545$	1.56817e9	0.414959
$546$	0	0
$547$	−4.85537e9	−1.26843	−0.634215	−	0.773157i	$-0.718677\pi$
$547$	−4.85537e9	−1.26843	−0.634215	+	0.773157i	$0.718677\pi$
$548$	2.58013e9	0.669746
$549$	0	0
$550$	−4.20770e8	−0.107839
$551$	−3.14716e8	−0.0801472
$552$	0	0
$553$	0	0
$554$	−3.09695e9	−0.773838
$555$	0	0
$556$	−1.08811e9	−0.268480
$557$	−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
$561$	0	0
$562$	−1.86626e9	−0.443501
$563$	4.71265e9	1.11297	0.556487	−	0.830856i	$-0.312149\pi$
$563$	4.71265e9	1.11297	0.556487	+	0.830856i	$0.312149\pi$
$564$	0	0
$565$	6.90546e9	1.61073
$566$	−3.56585e9	−0.826619
$567$	0	0
$568$	−7.34003e9	−1.68066
$569$	−4.57800e9	−1.04180	−0.520898	−	0.853619i	$-0.674403\pi$
$569$	−4.57800e9	−1.04180	−0.520898	+	0.853619i	$0.674403\pi$
$570$	0	0
$571$	4.95119e9	1.11297	0.556485	−	0.830858i	$-0.312150\pi$
$571$	4.95119e9	1.11297	0.556485	+	0.830858i	$0.312150\pi$
$572$	−4.44627e8	−0.0993367
$573$	0	0
$574$	0	0
$575$	1.13093e9	0.248084
$576$	0	0
$577$	−8.51847e9	−1.84606	−0.923031	−	0.384725i	$-0.874296\pi$
$577$	−8.51847e9	−1.84606	−0.923031	+	0.384725i	$0.874296\pi$
$578$	−2.37256e9	−0.511057
$579$	0	0
$580$	1.30998e9	0.278783
$581$	0	0
$582$	0	0
$583$	4.05799e8	0.0848147
$584$	−1.80719e9	−0.375455
$585$	0	0
$586$	−6.93088e8	−0.142281
$587$	−5.62247e8	−0.114735	−0.0573673	−	0.998353i	$-0.518271\pi$
$587$	−5.62247e8	−0.114735	−0.0573673	+	0.998353i	$0.518271\pi$
$588$	0	0
$589$	2.38608e9	0.481152
$590$	−3.05693e9	−0.612778
$591$	0	0
$592$	1.03544e9	0.205115
$593$	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
$595$	0	0
$596$	1.91219e9	0.369971
$597$	0	0
$598$	−4.67629e8	−0.0894227
$599$	7.48104e9	1.42222	0.711112	−	0.703079i	$-0.248192\pi$
$599$	7.48104e9	1.42222	0.711112	+	0.703079i	$0.248192\pi$
$600$	0	0
$601$	5.81270e9	1.09224	0.546119	−	0.837707i	$-0.316105\pi$
$601$	5.81270e9	1.09224	0.546119	+	0.837707i	$0.316105\pi$
$602$	0	0
$603$	0	0
$604$	−7.00230e6	−0.00129304
$605$	−7.24950e9	−1.33096
$606$	0	0
$607$	−3.84051e9	−0.696993	−0.348497	−	0.937310i	$-0.613308\pi$
$607$	−3.84051e9	−0.696993	−0.348497	+	0.937310i	$0.613308\pi$
$608$	−1.65587e9	−0.298788
$609$	0	0
$610$	7.03549e8	0.125499
$611$	2.97364e9	0.527405
$612$	0	0
$613$	1.70484e9	0.298932	0.149466	−	0.988767i	$-0.452245\pi$
$613$	1.70484e9	0.298932	0.149466	+	0.988767i	$0.452245\pi$
$614$	−1.56360e9	−0.272606
$615$	0	0
$616$	0	0
$617$	2.80809e9	0.481297	0.240649	−	0.970612i	$-0.422640\pi$
$617$	2.80809e9	0.481297	0.240649	+	0.970612i	$0.422640\pi$
$618$	0	0
$619$	2.54365e9	0.431063	0.215532	−	0.976497i	$-0.430852\pi$
$619$	2.54365e9	0.431063	0.215532	+	0.976497i	$0.430852\pi$
$620$	−9.93187e9	−1.67363
$621$	0	0
$622$	−3.46077e9	−0.576642
$623$	0	0
$624$	0	0
$625$	−6.41051e9	−1.05030
$626$	−2.76045e9	−0.449748
$627$	0	0
$628$	−2.96079e9	−0.477033
$629$	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	$0.503812\pi$
$632$	−9.12611e9	−1.43806
$633$	0	0
$634$	3.75337e8	0.0584936
$635$	6.54744e9	1.01476
$636$	0	0
$637$	0	0
$638$	2.07669e8	0.0316591
$639$	0	0
$640$	8.04735e9	1.21345
$641$	1.23625e10	1.85397	0.926987	−	0.375094i	$-0.122390\pi$
$641$	1.23625e10	1.85397	0.926987	+	0.375094i	$0.122390\pi$
$642$	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
$645$	0	0
$646$	−1.46812e9	−0.214264
$647$	−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
$650$	−2.26275e9	−0.323176
$651$	0	0
$652$	−5.36760e9	−0.758427
$653$	−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
$656$	−2.42819e9	−0.335830
$657$	0	0
$658$	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	$0.349459\pi$
$662$	−4.10541e9	−0.549989
$663$	0	0
$664$	−5.77731e9	−0.765839
$665$	0	0
$666$	0	0
$667$	−5.58165e8	−0.0728320
$668$	2.37696e9	0.308535
$669$	0	0
$670$	1.18695e9	0.152465
$671$	−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	$0.535285\pi$
$674$	3.75788e9	0.472752
$675$	0	0
$676$	3.38182e9	0.421053
$677$	8.30011e9	1.02807	0.514036	−	0.857769i	$-0.328150\pi$
$677$	8.30011e9	1.02807	0.514036	+	0.857769i	$0.328150\pi$
$678$	0	0
$679$	0	0
$680$	1.46131e10	1.78222
$681$	0	0
$682$	−1.57448e9	−0.190061
$683$	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
$687$	0	0
$688$	2.64434e9	0.309569
$689$	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	$0.657115\pi$
$692$	−5.84385e9	−0.670390
$693$	0	0
$694$	−7.52038e9	−0.854046
$695$	4.61265e9	0.521199
$696$	0	0
$697$	−1.78752e10	−1.99957
$698$	1.59210e9	0.177205
$699$	0	0
$700$	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
$704$	6.24740e8	0.0674831
$705$	0	0
$706$	3.41781e9	0.365537
$707$	0	0
$708$	0	0
$709$	2.78975e9	0.293970	0.146985	−	0.989139i	$-0.453043\pi$
$709$	2.78975e9	0.293970	0.146985	+	0.989139i	$0.453043\pi$
$710$	1.30119e10	1.36438
$711$	0	0
$712$	−1.12574e10	−1.16885
$713$	4.23184e9	0.437236
$714$	0	0
$715$	1.88483e9	0.192842
$716$	−7.44794e9	−0.758299
$717$	0	0
$718$	5.59524e9	0.564136
$719$	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
$721$	0	0
$722$	4.91740e9	0.486246
$723$	0	0
$724$	5.93429e9	0.581144
$725$	−2.70083e9	−0.263217
$726$	0	0
$727$	8.11761e9	0.783534	0.391767	−	0.920065i	$-0.371864\pi$
$727$	8.11761e9	0.783534	0.391767	+	0.920065i	$0.371864\pi$
$728$	0	0
$729$	0	0
$730$	3.20365e9	0.304800
$731$	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	$0.339138\pi$
$734$	−5.11539e9	−0.477466
$735$	0	0
$736$	−2.93676e9	−0.271517
$737$	−4.80867e8	−0.0442475
$738$	0	0
$739$	−1.35365e10	−1.23382	−0.616908	−	0.787035i	$-0.711615\pi$
$739$	−1.35365e10	−1.23382	−0.616908	+	0.787035i	$0.711615\pi$
$740$	−9.63471e9	−0.874033
$741$	0	0
$742$	0	0
$743$	1.71936e10	1.53782	0.768910	−	0.639356i	$-0.220799\pi$
$743$	1.71936e10	1.53782	0.768910	+	0.639356i	$0.220799\pi$
$744$	0	0
$745$	−8.10601e9	−0.718224
$746$	−2.28710e9	−0.201697
$747$	0	0
$748$	−2.47571e9	−0.216294
$749$	0	0
$750$	0	0
$751$	1.12478e10	0.969013	0.484506	−	0.874788i	$-0.338999\pi$
$751$	1.12478e10	0.969013	0.484506	+	0.874788i	$0.338999\pi$
$752$	2.24919e9	0.192870
$753$	0	0
$754$	1.11677e9	0.0948775
$755$	2.96837e7	0.00251017
$756$	0	0
$757$	1.63068e10	1.36626	0.683131	−	0.730296i	$-0.260618\pi$
$757$	1.63068e10	1.36626	0.683131	+	0.730296i	$0.260618\pi$
$758$	8.90118e9	0.742345
$759$	0	0
$760$	4.43758e9	0.366689
$761$	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
$763$	0	0
$764$	5.22812e9	0.424149
$765$	0	0
$766$	4.57158e9	0.367507
$767$	6.65993e9	0.532949
$768$	0	0
$769$	−2.45069e10	−1.94333	−0.971664	−	0.236368i	$-0.924043\pi$
$769$	−2.45069e10	−1.94333	−0.971664	+	0.236368i	$0.924043\pi$
$770$	0	0
$771$	0	0
$772$	−1.07067e10	−0.837518
$773$	−1.01722e10	−0.792110	−0.396055	−	0.918227i	$-0.629621\pi$
$773$	−1.01722e10	−0.792110	−0.396055	+	0.918227i	$0.629621\pi$
$774$	0	0
$775$	2.04769e10	1.58018
$776$	1.16514e10	0.895080
$777$	0	0
$778$	9.65411e9	0.734994
$779$	−5.42817e9	−0.411408
$780$	0	0
$781$	−5.27148e9	−0.395962
$782$	−2.60379e9	−0.194707
$783$	0	0
$784$	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	$0.383454\pi$
$788$	−1.09311e10	−0.795832
$789$	0	0
$790$	1.61781e10	1.16744
$791$	0	0
$792$	0	0
$793$	−1.53277e9	−0.109150
$794$	1.12809e10	0.799786
$795$	0	0
$796$	−8.74098e9	−0.614277
$797$	−9.75782e9	−0.682729	−0.341365	−	0.939931i	$-0.610889\pi$
$797$	−9.75782e9	−0.682729	−0.341365	+	0.939931i	$0.610889\pi$
$798$	0	0
$799$	1.65574e10	1.14836
$800$	−1.42103e10	−0.981270
$801$	0	0
$802$	1.61155e9	0.110315
$803$	−1.29789e9	−0.0884572
$804$	0	0
$805$	0	0
$806$	−8.46700e9	−0.569583
$807$	0	0
$808$	1.58221e10	1.05517
$809$	2.78706e9	0.185066	0.0925330	−	0.995710i	$-0.470504\pi$
$809$	2.78706e9	0.185066	0.0925330	+	0.995710i	$0.470504\pi$
$810$	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
$813$	0	0
$814$	−1.52738e9	−0.0992569
$815$	2.27540e10	1.47233
$816$	0	0
$817$	5.91135e9	0.379236
$818$	5.39687e8	0.0344751
$819$	0	0
$820$	2.25943e10	1.43103
$821$	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	$0.163262\pi$
$824$	−9.51640e9	−0.592553
$825$	0	0
$826$	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	$0.371790\pi$
$830$	1.02416e10	0.621719
$831$	0	0
$832$	3.35962e9	0.202236
$833$	0	0
$834$	0	0
$835$	−1.00762e10	−0.598957
$836$	−7.51802e8	−0.0445022
$837$	0	0
$838$	−1.01433e10	−0.595420
$839$	−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
$842$	6.79996e9	0.392567
$843$	0	0
$844$	−1.64907e10	−0.944147
$845$	−1.43360e10	−0.817389
$846$	0	0
$847$	0	0
$848$	1.65059e9	0.0929510
$849$	0	0
$850$	−1.25991e10	−0.703678
$851$	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
$855$	0	0
$856$	−1.50824e10	−0.821888
$857$	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	$0.699594\pi$
$860$	−2.46055e10	−1.31913
$861$	0	0
$862$	1.31966e10	0.701755
$863$	−1.04728e10	−0.554657	−0.277329	−	0.960775i	$-0.589449\pi$
$863$	−1.04728e10	−0.554657	−0.277329	+	0.960775i	$0.589449\pi$
$864$	0	0
$865$	2.47728e10	1.30142
$866$	−9.10427e8	−0.0476357
$867$	0	0
$868$	0	0
$869$	−6.55421e9	−0.338806
$870$	0	0
$871$	−2.58593e9	−0.132603
$872$	5.30765e9	0.271078
$873$	0	0
$874$	−7.90695e8	−0.0400608
$875$	0	0
$876$	0	0
$877$	−1.77787e10	−0.890024	−0.445012	−	0.895525i	$-0.646801\pi$
$877$	−1.77787e10	−0.890024	−0.445012	+	0.895525i	$0.646801\pi$
$878$	5.94458e9	0.296408
$879$	0	0
$880$	1.42564e9	0.0705213
$881$	−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	$-0.736060\pi$
$883$	−2.76375e10	−1.35094	−0.675472	+	0.737386i	$0.736060\pi$
$884$	−1.33135e10	−0.648200
$885$	0	0
$886$	−1.06425e10	−0.514076
$887$	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
$889$	0	0
$890$	1.99562e10	0.948886
$891$	0	0
$892$	−1.90014e10	−0.896414
$893$	5.02801e9	0.236274
$894$	0	0
$895$	3.15728e10	1.47208
$896$	0	0
$897$	0	0
$898$	−1.66206e9	−0.0765914
$899$	−1.01063e10	−0.463908
$900$	0	0
$901$	1.21509e10	0.553439
$902$	3.58184e9	0.162511
$903$	0	0
$904$	2.33723e10	1.05223
$905$	−2.51562e10	−1.12817
$906$	0	0
$907$	2.27142e10	1.01082	0.505409	−	0.862880i	$-0.331342\pi$
$907$	2.27142e10	1.01082	0.505409	+	0.862880i	$0.331342\pi$
$908$	−3.99238e9	−0.176983
$909$	0	0
$910$	0	0
$911$	−7.50925e9	−0.329065	−0.164533	−	0.986372i	$-0.552612\pi$
$911$	−7.50925e9	−0.329065	−0.164533	+	0.986372i	$0.552612\pi$
$912$	0	0
$913$	−4.14916e9	−0.180431
$914$	−1.76855e10	−0.766136
$915$	0	0
$916$	−3.32976e9	−0.143146
$917$	0	0
$918$	0	0
$919$	−2.49374e10	−1.05986	−0.529928	−	0.848043i	$-0.677781\pi$
$919$	−2.49374e10	−1.05986	−0.529928	+	0.848043i	$0.677781\pi$
$920$	7.87026e9	0.333221
$921$	0	0
$922$	1.66012e10	0.697561
$923$	−2.83481e10	−1.18664
$924$	0	0
$925$	1.98642e10	0.825230
$926$	−2.78132e9	−0.115110
$927$	0	0
$928$	7.01342e9	0.288079
$929$	−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
$931$	0	0
$932$	8.48559e9	0.343342
$933$	0	0
$934$	2.50753e9	0.100701
$935$	1.04949e10	0.419891
$936$	0	0
$937$	−2.82655e10	−1.12245	−0.561226	−	0.827663i	$-0.689670\pi$
$937$	−2.82655e10	−1.12245	−0.561226	+	0.827663i	$0.689670\pi$
$938$	0	0
$939$	0	0
$940$	−2.09287e10	−0.821853
$941$	−4.67082e10	−1.82738	−0.913691	−	0.406410i	$-0.866780\pi$
$941$	−4.67082e10	−1.82738	−0.913691	+	0.406410i	$0.866780\pi$
$942$	0	0
$943$	−9.62713e9	−0.373857
$944$	5.03740e9	0.194897
$945$	0	0
$946$	−3.90067e9	−0.149803
$947$	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
$950$	−3.82599e9	−0.144781
$951$	0	0
$952$	0	0
$953$	−3.82420e10	−1.43125	−0.715625	−	0.698484i	$-0.753858\pi$
$953$	−3.82420e10	−1.43125	−0.715625	+	0.698484i	$0.753858\pi$
$954$	0	0
$955$	−2.21627e10	−0.823399
$956$	4.58591e9	0.169755
$957$	0	0
$958$	9.05837e9	0.332867
$959$	0	0
$960$	0	0
$961$	4.91101e10	1.78500
$962$	−8.21367e9	−0.297457
$963$	0	0
$964$	1.83424e10	0.659457
$965$	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	$0.836744\pi$
$968$	−2.45368e10	−0.869468
$969$	0	0
$970$	−2.06547e10	−0.726639
$971$	2.72929e10	0.956713	0.478357	−	0.878166i	$-0.341233\pi$
$971$	2.72929e10	0.956713	0.478357	+	0.878166i	$0.341233\pi$
$972$	0	0
$973$	0	0
$974$	−5.57676e9	−0.193386
$975$	0	0
$976$	−1.15935e9	−0.0399155
$977$	−3.94482e9	−0.135331	−0.0676653	−	0.997708i	$-0.521555\pi$
$977$	−3.94482e9	−0.135331	−0.0676653	+	0.997708i	$0.521555\pi$
$978$	0	0
$979$	−8.08484e9	−0.275380
$980$	0	0
$981$	0	0
$982$	3.07682e10	1.03684
$983$	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
$986$	6.21824e9	0.206585
$987$	0	0
$988$	−4.04292e9	−0.133366
$989$	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	$0.436094\pi$
$992$	−5.31737e10	−1.72944
$993$	0	0
$994$	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	$0.304484\pi$
$998$	2.46390e9	0.0784631
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.8.a.a.1.1		1
3.2	odd	2		147.8.a.b.1.1		1
7.6	odd	2		9.8.a.a.1.1		1
21.2	odd	6		147.8.e.a.67.1		2
21.5	even	6		147.8.e.b.67.1		2
21.11	odd	6		147.8.e.a.79.1		2
21.17	even	6		147.8.e.b.79.1		2
21.20	even	2		3.8.a.a.1.1	✓	1
28.27	even	2		144.8.a.b.1.1		1
35.13	even	4		225.8.b.f.199.2		2
35.27	even	4		225.8.b.f.199.1		2
35.34	odd	2		225.8.a.i.1.1		1
56.13	odd	2		576.8.a.w.1.1		1
56.27	even	2		576.8.a.x.1.1		1
63.13	odd	6		81.8.c.c.55.1		2
63.20	even	6		81.8.c.a.28.1		2
63.34	odd	6		81.8.c.c.28.1		2
63.41	even	6		81.8.c.a.55.1		2
84.83	odd	2		48.8.a.g.1.1		1
105.62	odd	4		75.8.b.c.49.2		2
105.83	odd	4		75.8.b.c.49.1		2
105.104	even	2		75.8.a.a.1.1		1
168.83	odd	2		192.8.a.a.1.1		1
168.125	even	2		192.8.a.i.1.1		1
231.230	odd	2		363.8.a.b.1.1		1
273.272	even	2		507.8.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.8.a.a.1.1	✓	1	21.20	even	2
9.8.a.a.1.1		1	7.6	odd	2
48.8.a.g.1.1		1	84.83	odd	2
75.8.a.a.1.1		1	105.104	even	2
75.8.b.c.49.1		2	105.83	odd	4
75.8.b.c.49.2		2	105.62	odd	4
81.8.c.a.28.1		2	63.20	even	6
81.8.c.a.55.1		2	63.41	even	6
81.8.c.c.28.1		2	63.34	odd	6
81.8.c.c.55.1		2	63.13	odd	6
144.8.a.b.1.1		1	28.27	even	2
147.8.a.b.1.1		1	3.2	odd	2
147.8.e.a.67.1		2	21.2	odd	6
147.8.e.a.79.1		2	21.11	odd	6
147.8.e.b.67.1		2	21.5	even	6
147.8.e.b.79.1		2	21.17	even	6
192.8.a.a.1.1		1	168.83	odd	2
192.8.a.i.1.1		1	168.125	even	2
225.8.a.i.1.1		1	35.34	odd	2
225.8.b.f.199.1		2	35.27	even	4
225.8.b.f.199.2		2	35.13	even	4
363.8.a.b.1.1		1	231.230	odd	2
441.8.a.a.1.1		1	1.1	even	1	trivial
507.8.a.a.1.1		1	273.272	even	2
576.8.a.w.1.1		1	56.13	odd	2
576.8.a.x.1.1		1	56.27	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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