LMFDB - Embedded newform 342.10.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,10,Mod(1,342)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("342.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	342.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$176.142255968$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	342.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-16.0000 q^{2} +256.000 q^{4} +1581.00 q^{5} -4865.00 q^{7} -4096.00 q^{8} +O(q^{10})$

\(q-16.0000 q^{2} +256.000 q^{4} +1581.00 q^{5} -4865.00 q^{7} -4096.00 q^{8} -25296.0 q^{10} +64189.0 q^{11} -48516.0 q^{13} +77840.0 q^{14} +65536.0 q^{16} -314477. q^{17} +130321. q^{19} +404736. q^{20} -1.02702e6 q^{22} +51088.0 q^{23} +546436. q^{25} +776256. q^{26} -1.24544e6 q^{28} +1.54322e6 q^{29} +153108. q^{31} -1.04858e6 q^{32} +5.03163e6 q^{34} -7.69156e6 q^{35} +71578.0 q^{37} -2.08514e6 q^{38} -6.47578e6 q^{40} +2.41906e7 q^{41} -2.90653e6 q^{43} +1.64324e7 q^{44} -817408. q^{46} -1.46874e7 q^{47} -1.66854e7 q^{49} -8.74298e6 q^{50} -1.24201e7 q^{52} -1.07478e8 q^{53} +1.01483e8 q^{55} +1.99270e7 q^{56} -2.46915e7 q^{58} -1.38113e8 q^{59} -1.22366e8 q^{61} -2.44973e6 q^{62} +1.67772e7 q^{64} -7.67038e7 q^{65} +6.72966e7 q^{67} -8.05061e7 q^{68} +1.23065e8 q^{70} -2.53993e8 q^{71} +2.55181e7 q^{73} -1.14525e6 q^{74} +3.33622e7 q^{76} -3.12279e8 q^{77} -2.64202e8 q^{79} +1.03612e8 q^{80} -3.87050e8 q^{82} +7.24058e8 q^{83} -4.97188e8 q^{85} +4.65045e7 q^{86} -2.62918e8 q^{88} +1.07504e9 q^{89} +2.36030e8 q^{91} +1.30785e7 q^{92} +2.34998e8 q^{94} +2.06038e8 q^{95} +1.17323e9 q^{97} +2.66966e8 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 $

$2$	−16.0000	−0.707107
$2$	−16.0000	−0.707107
$3$	0	0
$3$	0	0
$4$	256.000	0.500000
$5$	1581.00	1.13127	0.565636	−	0.824655i	$-0.308631\pi$
$5$	1581.00	1.13127	0.565636	+	0.824655i	$0.308631\pi$
$6$	0	0
$7$	−4865.00	−0.765846	−0.382923	−	0.923780i	$-0.625083\pi$
$7$	−4865.00	−0.765846	−0.382923	+	0.923780i	$0.625083\pi$
$8$	−4096.00	−0.353553
$9$	0	0
$10$	−25296.0	−0.799930
$11$	64189.0	1.32188	0.660942	−	0.750437i	$-0.270157\pi$
$11$	64189.0	1.32188	0.660942	+	0.750437i	$0.270157\pi$
$12$	0	0
$13$	−48516.0	−0.471129	−0.235565	−	0.971859i	$-0.575694\pi$
$13$	−48516.0	−0.471129	−0.235565	+	0.971859i	$0.575694\pi$
$14$	77840.0	0.541535
$15$	0	0
$16$	65536.0	0.250000
$17$	−314477.	−0.913206	−0.456603	−	0.889671i	$-0.650934\pi$
$17$	−314477.	−0.913206	−0.456603	+	0.889671i	$0.650934\pi$
$18$	0	0
$19$	130321.	0.229416
$19$	130321.	0.229416
$20$	404736.	0.565636
$21$	0	0
$22$	−1.02702e6	−0.934714
$23$	51088.0	0.0380666	0.0190333	−	0.999819i	$-0.493941\pi$
$23$	51088.0	0.0380666	0.0190333	+	0.999819i	$0.493941\pi$
$24$	0	0
$25$	546436.	0.279775
$26$	776256.	0.333139
$27$	0	0
$28$	−1.24544e6	−0.382923
$29$	1.54322e6	0.405169	0.202585	−	0.979265i	$-0.435066\pi$
$29$	1.54322e6	0.405169	0.202585	+	0.979265i	$0.435066\pi$
$30$	0	0
$31$	153108.	0.0297763	0.0148881	−	0.999889i	$-0.495261\pi$
$31$	153108.	0.0297763	0.0148881	+	0.999889i	$0.495261\pi$
$32$	−1.04858e6	−0.176777
$33$	0	0
$34$	5.03163e6	0.645734
$35$	−7.69156e6	−0.866380
$36$	0	0
$37$	71578.0	0.00627873	0.00313936	−	0.999995i	$-0.499001\pi$
$37$	71578.0	0.00627873	0.00313936	+	0.999995i	$0.499001\pi$
$38$	−2.08514e6	−0.162221
$39$	0	0
$40$	−6.47578e6	−0.399965
$41$	2.41906e7	1.33696	0.668482	−	0.743729i	$-0.266945\pi$
$41$	2.41906e7	1.33696	0.668482	+	0.743729i	$0.266945\pi$
$42$	0	0
$43$	−2.90653e6	−0.129648	−0.0648241	−	0.997897i	$-0.520649\pi$
$43$	−2.90653e6	−0.129648	−0.0648241	+	0.997897i	$0.520649\pi$
$44$	1.64324e7	0.660942
$45$	0	0
$46$	−817408.	−0.0269171
$47$	−1.46874e7	−0.439041	−0.219520	−	0.975608i	$-0.570449\pi$
$47$	−1.46874e7	−0.439041	−0.219520	+	0.975608i	$0.570449\pi$
$48$	0	0
$49$	−1.66854e7	−0.413479
$50$	−8.74298e6	−0.197831
$51$	0	0
$52$	−1.24201e7	−0.235565
$53$	−1.07478e8	−1.87102	−0.935510	−	0.353301i	$-0.885059\pi$
$53$	−1.07478e8	−1.87102	−0.935510	+	0.353301i	$0.885059\pi$
$54$	0	0
$55$	1.01483e8	1.49541
$56$	1.99270e7	0.270768
$57$	0	0
$58$	−2.46915e7	−0.286498
$59$	−1.38113e8	−1.48388	−0.741941	−	0.670465i	$-0.766094\pi$
$59$	−1.38113e8	−1.48388	−0.741941	+	0.670465i	$0.766094\pi$
$60$	0	0
$61$	−1.22366e8	−1.13156	−0.565779	−	0.824557i	$-0.691424\pi$
$61$	−1.22366e8	−1.13156	−0.565779	+	0.824557i	$0.691424\pi$
$62$	−2.44973e6	−0.0210550
$63$	0	0
$64$	1.67772e7	0.125000
$65$	−7.67038e7	−0.532975
$66$	0	0
$67$	6.72966e7	0.407997	0.203998	−	0.978971i	$-0.434606\pi$
$67$	6.72966e7	0.407997	0.203998	+	0.978971i	$0.434606\pi$
$68$	−8.05061e7	−0.456603
$69$	0	0
$70$	1.23065e8	0.612623
$71$	−2.53993e8	−1.18620	−0.593101	−	0.805128i	$-0.702097\pi$
$71$	−2.53993e8	−1.18620	−0.593101	+	0.805128i	$0.702097\pi$
$72$	0	0
$73$	2.55181e7	0.105171	0.0525855	−	0.998616i	$-0.483254\pi$
$73$	2.55181e7	0.105171	0.0525855	+	0.998616i	$0.483254\pi$
$74$	−1.14525e6	−0.00443973
$75$	0	0
$76$	3.33622e7	0.114708
$77$	−3.12279e8	−1.01236
$78$	0	0
$79$	−2.64202e8	−0.763158	−0.381579	−	0.924336i	$-0.624620\pi$
$79$	−2.64202e8	−0.763158	−0.381579	+	0.924336i	$0.624620\pi$
$80$	1.03612e8	0.282818
$81$	0	0
$82$	−3.87050e8	−0.945376
$83$	7.24058e8	1.67464	0.837321	−	0.546711i	$-0.184120\pi$
$83$	7.24058e8	1.67464	0.837321	+	0.546711i	$0.184120\pi$
$84$	0	0
$85$	−4.97188e8	−1.03308
$86$	4.65045e7	0.0916751
$87$	0	0
$88$	−2.62918e8	−0.467357
$89$	1.07504e9	1.81622	0.908110	−	0.418732i	$-0.137525\pi$
$89$	1.07504e9	1.81622	0.908110	+	0.418732i	$0.137525\pi$
$90$	0	0
$91$	2.36030e8	0.360812
$92$	1.30785e7	0.0190333
$93$	0	0
$94$	2.34998e8	0.310449
$95$	2.06038e8	0.259531
$96$	0	0
$97$	1.17323e9	1.34558	0.672792	−	0.739832i	$-0.265095\pi$
$97$	1.17323e9	1.34558	0.672792	+	0.739832i	$0.265095\pi$
$98$	2.66966e8	0.292374
$99$	0	0
$100$	1.39888e8	0.139888
$101$	1.20251e9	1.14985	0.574926	−	0.818205i	$-0.305031\pi$
$101$	1.20251e9	1.14985	0.574926	+	0.818205i	$0.305031\pi$
$102$	0	0
$103$	−1.30419e9	−1.14175	−0.570877	−	0.821036i	$-0.693397\pi$
$103$	−1.30419e9	−1.14175	−0.570877	+	0.821036i	$0.693397\pi$
$104$	1.98722e8	0.166569
$105$	0	0
$106$	1.71965e9	1.32301
$107$	−3.28531e8	−0.242297	−0.121149	−	0.992634i	$-0.538658\pi$
$107$	−3.28531e8	−0.242297	−0.121149	+	0.992634i	$0.538658\pi$
$108$	0	0
$109$	1.26912e9	0.861161	0.430581	−	0.902552i	$-0.358309\pi$
$109$	1.26912e9	0.861161	0.430581	+	0.902552i	$0.358309\pi$
$110$	−1.62372e9	−1.05741
$111$	0	0
$112$	−3.18833e8	−0.191462
$113$	6.03462e8	0.348175	0.174087	−	0.984730i	$-0.444302\pi$
$113$	6.03462e8	0.348175	0.174087	+	0.984730i	$0.444302\pi$
$114$	0	0
$115$	8.07701e7	0.0430636
$116$	3.95064e8	0.202585
$117$	0	0
$118$	2.20980e9	1.04926
$119$	1.52993e9	0.699375
$120$	0	0
$121$	1.76228e9	0.747379
$122$	1.95786e9	0.800132
$123$	0	0
$124$	3.91956e7	0.0148881
$125$	−2.22398e9	−0.814770
$126$	0	0
$127$	−9.19867e8	−0.313768	−0.156884	−	0.987617i	$-0.550145\pi$
$127$	−9.19867e8	−0.313768	−0.156884	+	0.987617i	$0.550145\pi$
$128$	−2.68435e8	−0.0883883
$129$	0	0
$130$	1.22726e9	0.376870
$131$	−2.13372e9	−0.633020	−0.316510	−	0.948589i	$-0.602511\pi$
$131$	−2.13372e9	−0.633020	−0.316510	+	0.948589i	$0.602511\pi$
$132$	0	0
$133$	−6.34012e8	−0.175697
$134$	−1.07675e9	−0.288497
$135$	0	0
$136$	1.28810e9	0.322867
$137$	4.83513e9	1.17264	0.586321	−	0.810079i	$-0.300576\pi$
$137$	4.83513e9	1.17264	0.586321	+	0.810079i	$0.300576\pi$
$138$	0	0
$139$	−5.89855e9	−1.34023	−0.670114	−	0.742258i	$-0.733755\pi$
$139$	−5.89855e9	−1.34023	−0.670114	+	0.742258i	$0.733755\pi$
$140$	−1.96904e9	−0.433190
$141$	0	0
$142$	4.06388e9	0.838772
$143$	−3.11419e9	−0.622778
$144$	0	0
$145$	2.43983e9	0.458356
$146$	−4.08290e8	−0.0743671
$147$	0	0
$148$	1.83240e7	0.00313936
$149$	1.64663e9	0.273690	0.136845	−	0.990592i	$-0.456304\pi$
$149$	1.64663e9	0.273690	0.136845	+	0.990592i	$0.456304\pi$
$150$	0	0
$151$	−1.21642e9	−0.190409	−0.0952046	−	0.995458i	$-0.530351\pi$
$151$	−1.21642e9	−0.190409	−0.0952046	+	0.995458i	$0.530351\pi$
$152$	−5.33795e8	−0.0811107
$153$	0	0
$154$	4.99647e9	0.715847
$155$	2.42064e8	0.0336850
$156$	0	0
$157$	−6.68596e9	−0.878244	−0.439122	−	0.898427i	$-0.644710\pi$
$157$	−6.68596e9	−0.878244	−0.439122	+	0.898427i	$0.644710\pi$
$158$	4.22723e9	0.539634
$159$	0	0
$160$	−1.65780e9	−0.199982
$161$	−2.48543e8	−0.0291531
$162$	0	0
$163$	−1.09773e10	−1.21802	−0.609008	−	0.793164i	$-0.708432\pi$
$163$	−1.09773e10	−1.21802	−0.609008	+	0.793164i	$0.708432\pi$
$164$	6.19280e9	0.668482
$165$	0	0
$166$	−1.15849e10	−1.18415
$167$	−2.11768e9	−0.210686	−0.105343	−	0.994436i	$-0.533594\pi$
$167$	−2.11768e9	−0.210686	−0.105343	+	0.994436i	$0.533594\pi$
$168$	0	0
$169$	−8.25070e9	−0.778037
$170$	7.95501e9	0.730501
$171$	0	0
$172$	−7.44071e8	−0.0648241
$173$	−1.84114e10	−1.56271	−0.781355	−	0.624087i	$-0.785471\pi$
$173$	−1.84114e10	−1.56271	−0.781355	+	0.624087i	$0.785471\pi$
$174$	0	0
$175$	−2.65841e9	−0.214265
$176$	4.20669e9	0.330471
$177$	0	0
$178$	−1.72006e10	−1.28426
$179$	1.82020e10	1.32520	0.662599	−	0.748974i	$-0.269453\pi$
$179$	1.82020e10	1.32520	0.662599	+	0.748974i	$0.269453\pi$
$180$	0	0
$181$	1.11247e10	0.770431	0.385216	−	0.922827i	$-0.374127\pi$
$181$	1.11247e10	0.770431	0.385216	+	0.922827i	$0.374127\pi$
$182$	−3.77649e9	−0.255133
$183$	0	0
$184$	−2.09256e8	−0.0134586
$185$	1.13165e8	0.00710295
$186$	0	0
$187$	−2.01860e10	−1.20715
$188$	−3.75998e9	−0.219520
$189$	0	0
$190$	−3.29660e9	−0.183516
$191$	2.23310e10	1.21411	0.607056	−	0.794659i	$-0.292351\pi$
$191$	2.23310e10	1.21411	0.607056	+	0.794659i	$0.292351\pi$
$192$	0	0
$193$	4.35980e9	0.226182	0.113091	−	0.993585i	$-0.463925\pi$
$193$	4.35980e9	0.226182	0.113091	+	0.993585i	$0.463925\pi$
$194$	−1.87717e10	−0.951471
$195$	0	0
$196$	−4.27146e9	−0.206740
$197$	−2.47577e10	−1.17115	−0.585574	−	0.810619i	$-0.699131\pi$
$197$	−2.47577e10	−1.17115	−0.585574	+	0.810619i	$0.699131\pi$
$198$	0	0
$199$	2.57114e10	1.16222	0.581108	−	0.813827i	$-0.302620\pi$
$199$	2.57114e10	1.16222	0.581108	+	0.813827i	$0.302620\pi$
$200$	−2.23820e9	−0.0989155
$201$	0	0
$202$	−1.92401e10	−0.813068
$203$	−7.50776e9	−0.310297
$204$	0	0
$205$	3.82453e10	1.51247
$206$	2.08670e10	0.807341
$207$	0	0
$208$	−3.17954e9	−0.117782
$209$	8.36517e9	0.303261
$210$	0	0
$211$	−4.87650e10	−1.69370	−0.846852	−	0.531829i	$-0.821505\pi$
$211$	−4.87650e10	−1.69370	−0.846852	+	0.531829i	$0.821505\pi$
$212$	−2.75144e10	−0.935510
$213$	0	0
$214$	5.25649e9	0.171330
$215$	−4.59522e9	−0.146667
$216$	0	0
$217$	−7.44870e8	−0.0228040
$218$	−2.03060e10	−0.608933
$219$	0	0
$220$	2.59796e10	0.747705
$221$	1.52572e10	0.430238
$222$	0	0
$223$	−1.99213e10	−0.539443	−0.269722	−	0.962938i	$-0.586932\pi$
$223$	−1.99213e10	−0.539443	−0.269722	+	0.962938i	$0.586932\pi$
$224$	5.10132e9	0.135384
$225$	0	0
$226$	−9.65540e9	−0.246197
$227$	−6.30952e10	−1.57718	−0.788588	−	0.614922i	$-0.789188\pi$
$227$	−6.30952e10	−1.57718	−0.788588	+	0.614922i	$0.789188\pi$
$228$	0	0
$229$	−2.98031e10	−0.716147	−0.358073	−	0.933693i	$-0.616566\pi$
$229$	−2.98031e10	−0.716147	−0.358073	+	0.933693i	$0.616566\pi$
$230$	−1.29232e9	−0.0304506
$231$	0	0
$232$	−6.32102e9	−0.143249
$233$	−4.49049e9	−0.0998141	−0.0499070	−	0.998754i	$-0.515893\pi$
$233$	−4.49049e9	−0.0998141	−0.0499070	+	0.998754i	$0.515893\pi$
$234$	0	0
$235$	−2.32208e10	−0.496674
$236$	−3.53568e10	−0.741941
$237$	0	0
$238$	−2.44789e10	−0.494533
$239$	1.55900e10	0.309069	0.154534	−	0.987987i	$-0.450612\pi$
$239$	1.55900e10	0.309069	0.154534	+	0.987987i	$0.450612\pi$
$240$	0	0
$241$	1.02707e9	0.0196121	0.00980605	−	0.999952i	$-0.496879\pi$
$241$	1.02707e9	0.0196121	0.00980605	+	0.999952i	$0.496879\pi$
$242$	−2.81965e10	−0.528477
$243$	0	0
$244$	−3.13257e10	−0.565779
$245$	−2.63796e10	−0.467757
$246$	0	0
$247$	−6.32265e9	−0.108084
$248$	−6.27130e8	−0.0105275
$249$	0	0
$250$	3.55836e10	0.576129
$251$	8.57726e10	1.36401	0.682004	−	0.731349i	$-0.261109\pi$
$251$	8.57726e10	1.36401	0.682004	+	0.731349i	$0.261109\pi$
$252$	0	0
$253$	3.27929e9	0.0503196
$254$	1.47179e10	0.221867
$255$	0	0
$256$	4.29497e9	0.0625000
$257$	−2.45603e10	−0.351184	−0.175592	−	0.984463i	$-0.556184\pi$
$257$	−2.45603e10	−0.351184	−0.175592	+	0.984463i	$0.556184\pi$
$258$	0	0
$259$	−3.48227e8	−0.00480854
$260$	−1.96362e10	−0.266487
$261$	0	0
$262$	3.41396e10	0.447612
$263$	7.30924e10	0.942044	0.471022	−	0.882121i	$-0.343885\pi$
$263$	7.30924e10	0.942044	0.471022	+	0.882121i	$0.343885\pi$
$264$	0	0
$265$	−1.69923e11	−2.11663
$266$	1.01442e10	0.124237
$267$	0	0
$268$	1.72279e10	0.203998
$269$	−6.31008e10	−0.734767	−0.367384	−	0.930070i	$-0.619746\pi$
$269$	−6.31008e10	−0.734767	−0.367384	+	0.930070i	$0.619746\pi$
$270$	0	0
$271$	−9.88889e10	−1.11375	−0.556873	−	0.830598i	$-0.687999\pi$
$271$	−9.88889e10	−1.11375	−0.556873	+	0.830598i	$0.687999\pi$
$272$	−2.06096e10	−0.228301
$273$	0	0
$274$	−7.73621e10	−0.829183
$275$	3.50752e10	0.369831
$276$	0	0
$277$	−1.18609e11	−1.21048	−0.605242	−	0.796041i	$-0.706924\pi$
$277$	−1.18609e11	−1.21048	−0.605242	+	0.796041i	$0.706924\pi$
$278$	9.43768e10	0.947684
$279$	0	0
$280$	3.15047e10	0.306312
$281$	−1.93961e11	−1.85582	−0.927910	−	0.372805i	$-0.878396\pi$
$281$	−1.93961e11	−1.85582	−0.927910	+	0.372805i	$0.878396\pi$
$282$	0	0
$283$	1.11144e11	1.03003	0.515013	−	0.857183i	$-0.327787\pi$
$283$	1.11144e11	1.03003	0.515013	+	0.857183i	$0.327787\pi$
$284$	−6.50222e10	−0.593101
$285$	0	0
$286$	4.98271e10	0.440371
$287$	−1.17687e11	−1.02391
$288$	0	0
$289$	−1.96921e10	−0.166055
$290$	−3.90372e10	−0.324107
$291$	0	0
$292$	6.53264e9	0.0525855
$293$	−1.83473e11	−1.45435	−0.727175	−	0.686452i	$-0.759167\pi$
$293$	−1.83473e11	−1.45435	−0.727175	+	0.686452i	$0.759167\pi$
$294$	0	0
$295$	−2.18356e11	−1.67867
$296$	−2.93183e8	−0.00221987
$297$	0	0
$298$	−2.63461e10	−0.193528
$299$	−2.47859e9	−0.0179343
$300$	0	0
$301$	1.41403e10	0.0992906
$302$	1.94627e10	0.134640
$303$	0	0
$304$	8.54072e9	0.0573539
$305$	−1.93461e11	−1.28010
$306$	0	0
$307$	−1.04516e11	−0.671525	−0.335762	−	0.941947i	$-0.608994\pi$
$307$	−1.04516e11	−0.671525	−0.335762	+	0.941947i	$0.608994\pi$
$308$	−7.99435e10	−0.506180
$309$	0	0
$310$	−3.87302e9	−0.0238189
$311$	−2.51394e11	−1.52382	−0.761909	−	0.647685i	$-0.775737\pi$
$311$	−2.51394e11	−1.52382	−0.761909	+	0.647685i	$0.775737\pi$
$312$	0	0
$313$	−6.58635e10	−0.387878	−0.193939	−	0.981014i	$-0.562126\pi$
$313$	−6.58635e10	−0.387878	−0.193939	+	0.981014i	$0.562126\pi$
$314$	1.06975e11	0.621012
$315$	0	0
$316$	−6.76357e10	−0.381579
$317$	−3.05617e11	−1.69985	−0.849925	−	0.526904i	$-0.823353\pi$
$317$	−3.05617e11	−1.69985	−0.849925	+	0.526904i	$0.823353\pi$
$318$	0	0
$319$	9.90576e10	0.535587
$320$	2.65248e10	0.141409
$321$	0	0
$322$	3.97669e9	0.0206144
$323$	−4.09830e10	−0.209504
$324$	0	0
$325$	−2.65109e10	−0.131810
$326$	1.75637e11	0.861267
$327$	0	0
$328$	−9.90847e10	−0.472688
$329$	7.14542e10	0.336238
$330$	0	0
$331$	−1.42404e10	−0.0652073	−0.0326036	−	0.999468i	$-0.510380\pi$
$331$	−1.42404e10	−0.0652073	−0.0326036	+	0.999468i	$0.510380\pi$
$332$	1.85359e11	0.837321
$333$	0	0
$334$	3.38828e10	0.148977
$335$	1.06396e11	0.461555
$336$	0	0
$337$	1.41523e11	0.597714	0.298857	−	0.954298i	$-0.403395\pi$
$337$	1.41523e11	0.597714	0.298857	+	0.954298i	$0.403395\pi$
$338$	1.32011e11	0.550156
$339$	0	0
$340$	−1.27280e11	−0.516542
$341$	9.82785e9	0.0393608
$342$	0	0
$343$	2.77495e11	1.08251
$344$	1.19051e10	0.0458376
$345$	0	0
$346$	2.94582e11	1.10500
$347$	5.29148e11	1.95927	0.979635	−	0.200788i	$-0.0643502\pi$
$347$	5.29148e11	1.95927	0.979635	+	0.200788i	$0.0643502\pi$
$348$	0	0
$349$	5.76493e10	0.208008	0.104004	−	0.994577i	$-0.466835\pi$
$349$	5.76493e10	0.208008	0.104004	+	0.994577i	$0.466835\pi$
$350$	4.25346e10	0.151508
$351$	0	0
$352$	−6.73070e10	−0.233678
$353$	−3.89169e11	−1.33399	−0.666993	−	0.745064i	$-0.732419\pi$
$353$	−3.89169e11	−1.33399	−0.666993	+	0.745064i	$0.732419\pi$
$354$	0	0
$355$	−4.01563e11	−1.34192
$356$	2.75209e11	0.908110
$357$	0	0
$358$	−2.91232e11	−0.937057
$359$	−1.06143e11	−0.337262	−0.168631	−	0.985679i	$-0.553935\pi$
$359$	−1.06143e11	−0.337262	−0.168631	+	0.985679i	$0.553935\pi$
$360$	0	0
$361$	1.69836e10	0.0526316
$362$	−1.77995e11	−0.544777
$363$	0	0
$364$	6.04238e10	0.180406
$365$	4.03441e10	0.118977
$366$	0	0
$367$	−6.95837e10	−0.200221	−0.100111	−	0.994976i	$-0.531920\pi$
$367$	−6.95837e10	−0.200221	−0.100111	+	0.994976i	$0.531920\pi$
$368$	3.34810e9	0.00951664
$369$	0	0
$370$	−1.81064e9	−0.00502254
$371$	5.22881e11	1.43291
$372$	0	0
$373$	−1.74159e11	−0.465861	−0.232930	−	0.972493i	$-0.574831\pi$
$373$	−1.74159e11	−0.465861	−0.232930	+	0.972493i	$0.574831\pi$
$374$	3.22975e11	0.853586
$375$	0	0
$376$	6.01596e10	0.155224
$377$	−7.48708e10	−0.190887
$378$	0	0
$379$	8.64873e10	0.215316	0.107658	−	0.994188i	$-0.465665\pi$
$379$	8.64873e10	0.215316	0.107658	+	0.994188i	$0.465665\pi$
$380$	5.27456e10	0.129766
$381$	0	0
$382$	−3.57296e11	−0.858506
$383$	−8.93812e10	−0.212252	−0.106126	−	0.994353i	$-0.533845\pi$
$383$	−8.93812e10	−0.212252	−0.106126	+	0.994353i	$0.533845\pi$
$384$	0	0
$385$	−4.93714e11	−1.14525
$386$	−6.97568e10	−0.159935
$387$	0	0
$388$	3.00347e11	0.672792
$389$	4.19755e11	0.929444	0.464722	−	0.885457i	$-0.346154\pi$
$389$	4.19755e11	0.929444	0.464722	+	0.885457i	$0.346154\pi$
$390$	0	0
$391$	−1.60660e10	−0.0347626
$392$	6.83433e10	0.146187
$393$	0	0
$394$	3.96123e11	0.828126
$395$	−4.17704e11	−0.863339
$396$	0	0
$397$	−2.97495e11	−0.601065	−0.300533	−	0.953772i	$-0.597164\pi$
$397$	−2.97495e11	−0.601065	−0.300533	+	0.953772i	$0.597164\pi$
$398$	−4.11382e11	−0.821810
$399$	0	0
$400$	3.58112e10	0.0699438
$401$	9.27957e10	0.179217	0.0896083	−	0.995977i	$-0.471438\pi$
$401$	9.27957e10	0.179217	0.0896083	+	0.995977i	$0.471438\pi$
$402$	0	0
$403$	−7.42819e9	−0.0140285
$404$	3.07842e11	0.574926
$405$	0	0
$406$	1.20124e11	0.219413
$407$	4.59452e9	0.00829976
$408$	0	0
$409$	−1.05127e12	−1.85763	−0.928815	−	0.370543i	$-0.879172\pi$
$409$	−1.05127e12	−1.85763	−0.928815	+	0.370543i	$0.879172\pi$
$410$	−6.11926e11	−1.06948
$411$	0	0
$412$	−3.33872e11	−0.570877
$413$	6.71918e11	1.13643
$414$	0	0
$415$	1.14474e12	1.89448
$416$	5.08727e10	0.0832846
$417$	0	0
$418$	−1.33843e11	−0.214438
$419$	7.46398e11	1.18306	0.591531	−	0.806282i	$-0.298524\pi$
$419$	7.46398e11	1.18306	0.591531	+	0.806282i	$0.298524\pi$
$420$	0	0
$421$	−1.02671e12	−1.59286	−0.796432	−	0.604728i	$-0.793282\pi$
$421$	−1.02671e12	−1.59286	−0.796432	+	0.604728i	$0.793282\pi$
$422$	7.80240e11	1.19763
$423$	0	0
$424$	4.40230e11	0.661505
$425$	−1.71842e11	−0.255492
$426$	0	0
$427$	5.95311e11	0.866599
$428$	−8.41038e10	−0.121149
$429$	0	0
$430$	7.35236e10	0.103709
$431$	−6.01842e11	−0.840108	−0.420054	−	0.907499i	$-0.637989\pi$
$431$	−6.01842e11	−0.840108	−0.420054	+	0.907499i	$0.637989\pi$
$432$	0	0
$433$	3.80261e11	0.519860	0.259930	−	0.965627i	$-0.416300\pi$
$433$	3.80261e11	0.519860	0.259930	+	0.965627i	$0.416300\pi$
$434$	1.19179e10	0.0161249
$435$	0	0
$436$	3.24895e11	0.430581
$437$	6.65784e9	0.00873307
$438$	0	0
$439$	5.52278e11	0.709688	0.354844	−	0.934926i	$-0.384534\pi$
$439$	5.52278e11	0.709688	0.354844	+	0.934926i	$0.384534\pi$
$440$	−4.15674e11	−0.528707
$441$	0	0
$442$	−2.44115e11	−0.304224
$443$	−1.44599e12	−1.78381	−0.891906	−	0.452221i	$-0.850632\pi$
$443$	−1.44599e12	−1.78381	−0.891906	+	0.452221i	$0.850632\pi$
$444$	0	0
$445$	1.69963e12	2.05464
$446$	3.18741e11	0.381444
$447$	0	0
$448$	−8.16212e10	−0.0957308
$449$	4.28462e11	0.497513	0.248756	−	0.968566i	$-0.419978\pi$
$449$	4.28462e11	0.497513	0.248756	+	0.968566i	$0.419978\pi$
$450$	0	0
$451$	1.55277e12	1.76731
$452$	1.54486e11	0.174087
$453$	0	0
$454$	1.00952e12	1.11523
$455$	3.73164e11	0.408177
$456$	0	0
$457$	−1.40485e12	−1.50663	−0.753314	−	0.657661i	$-0.771546\pi$
$457$	−1.40485e12	−1.50663	−0.753314	+	0.657661i	$0.771546\pi$
$458$	4.76850e11	0.506392
$459$	0	0
$460$	2.06772e10	0.0215318
$461$	−1.37738e12	−1.42037	−0.710183	−	0.704018i	$-0.751388\pi$
$461$	−1.37738e12	−1.42037	−0.710183	+	0.704018i	$0.751388\pi$
$462$	0	0
$463$	1.35301e12	1.36832	0.684160	−	0.729332i	$-0.260169\pi$
$463$	1.35301e12	1.36832	0.684160	+	0.729332i	$0.260169\pi$
$464$	1.01136e11	0.101292
$465$	0	0
$466$	7.18478e10	0.0705792
$467$	−1.13537e12	−1.10462	−0.552310	−	0.833639i	$-0.686254\pi$
$467$	−1.13537e12	−1.10462	−0.552310	+	0.833639i	$0.686254\pi$
$468$	0	0
$469$	−3.27398e11	−0.312463
$470$	3.71533e11	0.351202
$471$	0	0
$472$	5.65709e11	0.524631
$473$	−1.86567e11	−0.171380
$474$	0	0
$475$	7.12121e10	0.0641848
$476$	3.91662e11	0.349688
$477$	0	0
$478$	−2.49440e11	−0.218545
$479$	1.03514e12	0.898440	0.449220	−	0.893421i	$-0.351702\pi$
$479$	1.03514e12	0.898440	0.449220	+	0.893421i	$0.351702\pi$
$480$	0	0
$481$	−3.47268e9	−0.00295809
$482$	−1.64332e10	−0.0138679
$483$	0	0
$484$	4.51144e11	0.373689
$485$	1.85488e12	1.52222
$486$	0	0
$487$	−1.77896e12	−1.43313	−0.716567	−	0.697519i	$-0.754287\pi$
$487$	−1.77896e12	−1.43313	−0.716567	+	0.697519i	$0.754287\pi$
$488$	5.01211e11	0.400066
$489$	0	0
$490$	4.22073e11	0.330754
$491$	2.22425e12	1.72710	0.863548	−	0.504267i	$-0.168237\pi$
$491$	2.22425e12	1.72710	0.863548	+	0.504267i	$0.168237\pi$
$492$	0	0
$493$	−4.85307e11	−0.370003
$494$	1.01162e11	0.0764272
$495$	0	0
$496$	1.00341e10	0.00744407
$497$	1.23567e12	0.908449
$498$	0	0
$499$	1.97300e12	1.42454	0.712268	−	0.701907i	$-0.247668\pi$
$499$	1.97300e12	1.42454	0.712268	+	0.701907i	$0.247668\pi$
$500$	−5.69338e11	−0.407385
$501$	0	0
$502$	−1.37236e12	−0.964499
$503$	−1.01903e12	−0.709795	−0.354897	−	0.934905i	$-0.615484\pi$
$503$	−1.01903e12	−0.709795	−0.354897	+	0.934905i	$0.615484\pi$
$504$	0	0
$505$	1.90117e12	1.30080
$506$	−5.24686e10	−0.0355813
$507$	0	0
$508$	−2.35486e11	−0.156884
$509$	−8.88315e11	−0.586593	−0.293297	−	0.956021i	$-0.594752\pi$
$509$	−8.88315e11	−0.586593	−0.293297	+	0.956021i	$0.594752\pi$
$510$	0	0
$511$	−1.24146e11	−0.0805448
$512$	−6.87195e10	−0.0441942
$513$	0	0
$514$	3.92965e11	0.248325
$515$	−2.06192e12	−1.29163
$516$	0	0
$517$	−9.42770e11	−0.580361
$518$	5.57163e9	0.00340015
$519$	0	0
$520$	3.14179e11	0.188435
$521$	−1.58630e12	−0.943228	−0.471614	−	0.881805i	$-0.656328\pi$
$521$	−1.58630e12	−0.943228	−0.471614	+	0.881805i	$0.656328\pi$
$522$	0	0
$523$	1.49701e11	0.0874915	0.0437458	−	0.999043i	$-0.486071\pi$
$523$	1.49701e11	0.0874915	0.0437458	+	0.999043i	$0.486071\pi$
$524$	−5.46233e11	−0.316510
$525$	0	0
$526$	−1.16948e12	−0.666126
$527$	−4.81489e10	−0.0271919
$528$	0	0
$529$	−1.79854e12	−0.998551
$530$	2.71876e12	1.49668
$531$	0	0
$532$	−1.62307e11	−0.0878486
$533$	−1.17363e12	−0.629882
$534$	0	0
$535$	−5.19407e11	−0.274104
$536$	−2.75647e11	−0.144249
$537$	0	0
$538$	1.00961e12	0.519559
$539$	−1.07102e12	−0.546572
$540$	0	0
$541$	3.19362e12	1.60286	0.801431	−	0.598088i	$-0.204073\pi$
$541$	3.19362e12	1.60286	0.801431	+	0.598088i	$0.204073\pi$
$542$	1.58222e12	0.787537
$543$	0	0
$544$	3.29753e11	0.161434
$545$	2.00648e12	0.974207
$546$	0	0
$547$	−4.19272e11	−0.200241	−0.100120	−	0.994975i	$-0.531923\pi$
$547$	−4.19272e11	−0.200241	−0.100120	+	0.994975i	$0.531923\pi$
$548$	1.23779e12	0.586321
$549$	0	0
$550$	−5.61203e11	−0.261510
$551$	2.01114e11	0.0929522
$552$	0	0
$553$	1.28534e12	0.584462
$554$	1.89775e12	0.855942
$555$	0	0
$556$	−1.51003e12	−0.670114
$557$	−3.22232e12	−1.41847	−0.709234	−	0.704973i	$-0.750959\pi$
$557$	−3.22232e12	−1.41847	−0.709234	+	0.704973i	$0.750959\pi$
$558$	0	0
$559$	1.41013e11	0.0610811
$560$	−5.04074e11	−0.216595
$561$	0	0
$562$	3.10337e12	1.31226
$563$	−3.54247e12	−1.48600	−0.743000	−	0.669291i	$-0.766598\pi$
$563$	−3.54247e12	−1.48600	−0.743000	+	0.669291i	$0.766598\pi$
$564$	0	0
$565$	9.54074e11	0.393880
$566$	−1.77831e12	−0.728338
$567$	0	0
$568$	1.04035e12	0.419386
$569$	−1.05951e12	−0.423742	−0.211871	−	0.977298i	$-0.567956\pi$
$569$	−1.05951e12	−0.423742	−0.211871	+	0.977298i	$0.567956\pi$
$570$	0	0
$571$	1.87884e12	0.739650	0.369825	−	0.929101i	$-0.379418\pi$
$571$	1.87884e12	0.739650	0.369825	+	0.929101i	$0.379418\pi$
$572$	−7.97234e11	−0.311389
$573$	0	0
$574$	1.88300e12	0.724012
$575$	2.79163e10	0.0106501
$576$	0	0
$577$	1.10245e12	0.414065	0.207032	−	0.978334i	$-0.433619\pi$
$577$	1.10245e12	0.414065	0.207032	+	0.978334i	$0.433619\pi$
$578$	3.15073e11	0.117419
$579$	0	0
$580$	6.24596e11	0.229178
$581$	−3.52254e12	−1.28252
$582$	0	0
$583$	−6.89891e12	−2.47327
$584$	−1.04522e11	−0.0371835
$585$	0	0
$586$	2.93558e12	1.02838
$587$	5.10026e12	1.77305	0.886525	−	0.462680i	$-0.153112\pi$
$587$	5.10026e12	1.77305	0.886525	+	0.462680i	$0.153112\pi$
$588$	0	0
$589$	1.99532e10	0.00683114
$590$	3.49370e12	1.18700
$591$	0	0
$592$	4.69094e9	0.00156968
$593$	2.73128e12	0.907026	0.453513	−	0.891250i	$-0.350171\pi$
$593$	2.73128e12	0.907026	0.453513	+	0.891250i	$0.350171\pi$
$594$	0	0
$595$	2.41882e12	0.791184
$596$	4.21538e11	0.136845
$597$	0	0
$598$	3.96574e10	0.0126814
$599$	−2.10529e11	−0.0668177	−0.0334089	−	0.999442i	$-0.510636\pi$
$599$	−2.10529e11	−0.0668177	−0.0334089	+	0.999442i	$0.510636\pi$
$600$	0	0
$601$	3.31426e12	1.03622	0.518110	−	0.855314i	$-0.326636\pi$
$601$	3.31426e12	1.03622	0.518110	+	0.855314i	$0.326636\pi$
$602$	−2.26244e11	−0.0702091
$603$	0	0
$604$	−3.11404e11	−0.0952046
$605$	2.78616e12	0.845488
$606$	0	0
$607$	−3.75897e12	−1.12388	−0.561940	−	0.827178i	$-0.689945\pi$
$607$	−3.75897e12	−1.12388	−0.561940	+	0.827178i	$0.689945\pi$
$608$	−1.36651e11	−0.0405554
$609$	0	0
$610$	3.09537e12	0.905166
$611$	7.12574e11	0.206845
$612$	0	0
$613$	−4.35989e12	−1.24711	−0.623553	−	0.781781i	$-0.714311\pi$
$613$	−4.35989e12	−1.24711	−0.623553	+	0.781781i	$0.714311\pi$
$614$	1.67226e12	0.474840
$615$	0	0
$616$	1.27910e12	0.357923
$617$	−3.10182e11	−0.0861655	−0.0430827	−	0.999072i	$-0.513718\pi$
$617$	−3.10182e11	−0.0861655	−0.0430827	+	0.999072i	$0.513718\pi$
$618$	0	0
$619$	5.58231e11	0.152829	0.0764145	−	0.997076i	$-0.475653\pi$
$619$	5.58231e11	0.152829	0.0764145	+	0.997076i	$0.475653\pi$
$620$	6.19683e10	0.0168425
$621$	0	0
$622$	4.02230e12	1.07750
$623$	−5.23006e12	−1.39095
$624$	0	0
$625$	−4.58336e12	−1.20150
$626$	1.05382e12	0.274271
$627$	0	0
$628$	−1.71161e12	−0.439122
$629$	−2.25096e10	−0.00573377
$630$	0	0
$631$	−3.12950e12	−0.785856	−0.392928	−	0.919569i	$-0.628538\pi$
$631$	−3.12950e12	−0.785856	−0.392928	+	0.919569i	$0.628538\pi$
$632$	1.08217e12	0.269817
$633$	0	0
$634$	4.88987e12	1.20198
$635$	−1.45431e12	−0.354956
$636$	0	0
$637$	8.09508e11	0.194802
$638$	−1.58492e12	−0.378717
$639$	0	0
$640$	−4.24396e11	−0.0999912
$641$	−1.74234e12	−0.407635	−0.203817	−	0.979009i	$-0.565335\pi$
$641$	−1.74234e12	−0.407635	−0.203817	+	0.979009i	$0.565335\pi$
$642$	0	0
$643$	−2.16462e12	−0.499381	−0.249690	−	0.968326i	$-0.580329\pi$
$643$	−2.16462e12	−0.499381	−0.249690	+	0.968326i	$0.580329\pi$
$644$	−6.36270e10	−0.0145766
$645$	0	0
$646$	6.55727e11	0.148142
$647$	8.45274e12	1.89639	0.948197	−	0.317683i	$-0.102905\pi$
$647$	8.45274e12	1.89639	0.948197	+	0.317683i	$0.102905\pi$
$648$	0	0
$649$	−8.86531e12	−1.96152
$650$	4.24174e11	0.0932039
$651$	0	0
$652$	−2.81020e12	−0.609008
$653$	−1.94287e12	−0.418153	−0.209076	−	0.977899i	$-0.567046\pi$
$653$	−1.94287e12	−0.418153	−0.209076	+	0.977899i	$0.567046\pi$
$654$	0	0
$655$	−3.37342e12	−0.716117
$656$	1.58536e12	0.334241
$657$	0	0
$658$	−1.14327e12	−0.237756
$659$	1.78479e12	0.368640	0.184320	−	0.982866i	$-0.440992\pi$
$659$	1.78479e12	0.368640	0.184320	+	0.982866i	$0.440992\pi$
$660$	0	0
$661$	6.08589e12	1.23999	0.619994	−	0.784607i	$-0.287135\pi$
$661$	6.08589e12	1.23999	0.619994	+	0.784607i	$0.287135\pi$
$662$	2.27846e11	0.0461085
$663$	0	0
$664$	−2.96574e12	−0.592076
$665$	−1.00237e12	−0.198761
$666$	0	0
$667$	7.88399e10	0.0154234
$668$	−5.42125e11	−0.105343
$669$	0	0
$670$	−1.70234e12	−0.326369
$671$	−7.85455e12	−1.49579
$672$	0	0
$673$	4.24160e12	0.797007	0.398504	−	0.917167i	$-0.369530\pi$
$673$	4.24160e12	0.797007	0.398504	+	0.917167i	$0.369530\pi$
$674$	−2.26437e12	−0.422648
$675$	0	0
$676$	−2.11218e12	−0.389019
$677$	2.61978e12	0.479310	0.239655	−	0.970858i	$-0.422966\pi$
$677$	2.61978e12	0.479310	0.239655	+	0.970858i	$0.422966\pi$
$678$	0	0
$679$	−5.70777e12	−1.03051
$680$	2.03648e12	0.365250
$681$	0	0
$682$	−1.57246e11	−0.0278323
$683$	4.81312e12	0.846319	0.423159	−	0.906055i	$-0.360921\pi$
$683$	4.81312e12	0.846319	0.423159	+	0.906055i	$0.360921\pi$
$684$	0	0
$685$	7.64434e12	1.32658
$686$	−4.43991e12	−0.765449
$687$	0	0
$688$	−1.90482e11	−0.0324121
$689$	5.21441e12	0.881492
$690$	0	0
$691$	5.44127e12	0.907922	0.453961	−	0.891021i	$-0.350010\pi$
$691$	5.44127e12	0.907922	0.453961	+	0.891021i	$0.350010\pi$
$692$	−4.71331e12	−0.781355
$693$	0	0
$694$	−8.46636e12	−1.38541
$695$	−9.32561e12	−1.51616
$696$	0	0
$697$	−7.60739e12	−1.22092
$698$	−9.22389e11	−0.147084
$699$	0	0
$700$	−6.80553e11	−0.107132
$701$	−9.70124e12	−1.51739	−0.758693	−	0.651448i	$-0.774162\pi$
$701$	−9.70124e12	−1.51739	−0.758693	+	0.651448i	$0.774162\pi$
$702$	0	0
$703$	9.32812e9	0.00144044
$704$	1.07691e12	0.165236
$705$	0	0
$706$	6.22670e12	0.943271
$707$	−5.85021e12	−0.880610
$708$	0	0
$709$	1.08088e13	1.60646	0.803231	−	0.595668i	$-0.203113\pi$
$709$	1.08088e13	1.60646	0.803231	+	0.595668i	$0.203113\pi$
$710$	6.42500e12	0.948879
$711$	0	0
$712$	−4.40335e12	−0.642131
$713$	7.82198e9	0.00113348
$714$	0	0
$715$	−4.92354e12	−0.704531
$716$	4.65972e12	0.662599
$717$	0	0
$718$	1.69829e12	0.238480
$719$	1.28259e13	1.78981	0.894905	−	0.446256i	$-0.147243\pi$
$719$	1.28259e13	1.78981	0.894905	+	0.446256i	$0.147243\pi$
$720$	0	0
$721$	6.34487e12	0.874407
$722$	−2.71737e11	−0.0372161
$723$	0	0
$724$	2.84792e12	0.385216
$725$	8.43270e11	0.113356
$726$	0	0
$727$	−8.26428e12	−1.09724	−0.548618	−	0.836073i	$-0.684846\pi$
$727$	−8.26428e12	−1.09724	−0.548618	+	0.836073i	$0.684846\pi$
$728$	−9.66780e11	−0.127566
$729$	0	0
$730$	−6.45506e11	−0.0841294
$731$	9.14037e11	0.118396
$732$	0	0
$733$	9.23305e12	1.18135	0.590673	−	0.806911i	$-0.298862\pi$
$733$	9.23305e12	1.18135	0.590673	+	0.806911i	$0.298862\pi$
$734$	1.11334e12	0.141578
$735$	0	0
$736$	−5.35697e10	−0.00672928
$737$	4.31970e12	0.539324
$738$	0	0
$739$	4.26380e12	0.525892	0.262946	−	0.964811i	$-0.415306\pi$
$739$	4.26380e12	0.525892	0.262946	+	0.964811i	$0.415306\pi$
$740$	2.89702e10	0.00355147
$741$	0	0
$742$	−8.36609e12	−1.01322
$743$	5.50358e12	0.662514	0.331257	−	0.943541i	$-0.392527\pi$
$743$	5.50358e12	0.662514	0.331257	+	0.943541i	$0.392527\pi$
$744$	0	0
$745$	2.60333e12	0.309617
$746$	2.78654e12	0.329413
$747$	0	0
$748$	−5.16761e12	−0.603576
$749$	1.59830e12	0.185563
$750$	0	0
$751$	1.00995e13	1.15857	0.579285	−	0.815125i	$-0.303332\pi$
$751$	1.00995e13	1.15857	0.579285	+	0.815125i	$0.303332\pi$
$752$	−9.62554e11	−0.109760
$753$	0	0
$754$	1.19793e12	0.134977
$755$	−1.92316e12	−0.215404
$756$	0	0
$757$	1.26401e13	1.39900	0.699502	−	0.714630i	$-0.253405\pi$
$757$	1.26401e13	1.39900	0.699502	+	0.714630i	$0.253405\pi$
$758$	−1.38380e12	−0.152251
$759$	0	0
$760$	−8.43930e11	−0.0917582
$761$	−1.25016e13	−1.35125	−0.675625	−	0.737245i	$-0.736126\pi$
$761$	−1.25016e13	−1.35125	−0.675625	+	0.737245i	$0.736126\pi$
$762$	0	0
$763$	−6.17428e12	−0.659517
$764$	5.71674e12	0.607056
$765$	0	0
$766$	1.43010e12	0.150085
$767$	6.70067e12	0.699100
$768$	0	0
$769$	−6.06695e12	−0.625607	−0.312804	−	0.949818i	$-0.601268\pi$
$769$	−6.06695e12	−0.625607	−0.312804	+	0.949818i	$0.601268\pi$
$770$	7.89942e12	0.809817
$771$	0	0
$772$	1.11611e12	0.113091
$773$	−1.36551e13	−1.37559	−0.687794	−	0.725906i	$-0.741421\pi$
$773$	−1.36551e13	−1.37559	−0.687794	+	0.725906i	$0.741421\pi$
$774$	0	0
$775$	8.36637e10	0.00833066
$776$	−4.80555e12	−0.475736
$777$	0	0
$778$	−6.71609e12	−0.657216
$779$	3.15254e12	0.306720
$780$	0	0
$781$	−1.63035e13	−1.56802
$782$	2.57056e11	0.0245809
$783$	0	0
$784$	−1.09349e12	−0.103370
$785$	−1.05705e13	−0.993532
$786$	0	0
$787$	−4.83993e11	−0.0449731	−0.0224865	−	0.999747i	$-0.507158\pi$
$787$	−4.83993e11	−0.0449731	−0.0224865	+	0.999747i	$0.507158\pi$
$788$	−6.33796e12	−0.585574
$789$	0	0
$790$	6.68326e12	0.610473
$791$	−2.93584e12	−0.266648
$792$	0	0
$793$	5.93671e12	0.533110
$794$	4.75991e12	0.425017
$795$	0	0
$796$	6.58211e12	0.581108
$797$	−1.27488e13	−1.11919	−0.559597	−	0.828765i	$-0.689044\pi$
$797$	−1.27488e13	−1.11919	−0.559597	+	0.828765i	$0.689044\pi$
$798$	0	0
$799$	4.61885e12	0.400935
$800$	−5.72980e11	−0.0494577
$801$	0	0
$802$	−1.48473e12	−0.126725
$803$	1.63798e12	0.139024
$804$	0	0
$805$	−3.92947e11	−0.0329801
$806$	1.18851e11	0.00991962
$807$	0	0
$808$	−4.92548e12	−0.406534
$809$	1.62239e13	1.33164	0.665821	−	0.746112i	$-0.268082\pi$
$809$	1.62239e13	1.33164	0.665821	+	0.746112i	$0.268082\pi$
$810$	0	0
$811$	1.64958e11	0.0133899	0.00669497	−	0.999978i	$-0.497869\pi$
$811$	1.64958e11	0.0133899	0.00669497	+	0.999978i	$0.497869\pi$
$812$	−1.92199e12	−0.155149
$813$	0	0
$814$	−7.35123e10	−0.00586881
$815$	−1.73552e13	−1.37791
$816$	0	0
$817$	−3.78782e11	−0.0297433
$818$	1.68203e13	1.31354
$819$	0	0
$820$	9.79081e12	0.756234
$821$	6.60429e12	0.507320	0.253660	−	0.967293i	$-0.418366\pi$
$821$	6.60429e12	0.507320	0.253660	+	0.967293i	$0.418366\pi$
$822$	0	0
$823$	−1.65599e13	−1.25823	−0.629113	−	0.777314i	$-0.716582\pi$
$823$	−1.65599e13	−1.25823	−0.629113	+	0.777314i	$0.716582\pi$
$824$	5.34195e12	0.403671
$825$	0	0
$826$	−1.07507e13	−0.803574
$827$	8.02404e12	0.596511	0.298255	−	0.954486i	$-0.403595\pi$
$827$	8.02404e12	0.596511	0.298255	+	0.954486i	$0.403595\pi$
$828$	0	0
$829$	−2.19013e13	−1.61055	−0.805276	−	0.592901i	$-0.797983\pi$
$829$	−2.19013e13	−1.61055	−0.805276	+	0.592901i	$0.797983\pi$
$830$	−1.83158e13	−1.33960
$831$	0	0
$832$	−8.13963e11	−0.0588911
$833$	5.24717e12	0.377592
$834$	0	0
$835$	−3.34805e12	−0.238343
$836$	2.14148e12	0.151631
$837$	0	0
$838$	−1.19424e13	−0.836551
$839$	3.11694e12	0.217170	0.108585	−	0.994087i	$-0.465368\pi$
$839$	3.11694e12	0.217170	0.108585	+	0.994087i	$0.465368\pi$
$840$	0	0
$841$	−1.21256e13	−0.835838
$842$	1.64274e13	1.12632
$843$	0	0
$844$	−1.24838e13	−0.846852
$845$	−1.30444e13	−0.880172
$846$	0	0
$847$	−8.57349e12	−0.572377
$848$	−7.04368e12	−0.467755
$849$	0	0
$850$	2.74946e12	0.180660
$851$	3.65678e9	0.000239010 0
$852$	0	0
$853$	−5.54035e12	−0.358316	−0.179158	−	0.983820i	$-0.557337\pi$
$853$	−5.54035e12	−0.358316	−0.179158	+	0.983820i	$0.557337\pi$
$854$	−9.52497e12	−0.612778
$855$	0	0
$856$	1.34566e12	0.0856651
$857$	6.97547e12	0.441733	0.220867	−	0.975304i	$-0.429111\pi$
$857$	6.97547e12	0.441733	0.220867	+	0.975304i	$0.429111\pi$
$858$	0	0
$859$	−4.19926e11	−0.0263150	−0.0131575	−	0.999913i	$-0.504188\pi$
$859$	−4.19926e11	−0.0263150	−0.0131575	+	0.999913i	$0.504188\pi$
$860$	−1.17638e12	−0.0733337
$861$	0	0
$862$	9.62947e12	0.594046
$863$	1.42650e13	0.875434	0.437717	−	0.899113i	$-0.355787\pi$
$863$	1.42650e13	0.875434	0.437717	+	0.899113i	$0.355787\pi$
$864$	0	0
$865$	−2.91084e13	−1.76785
$866$	−6.08418e12	−0.367597
$867$	0	0
$868$	−1.90687e11	−0.0114020
$869$	−1.69589e13	−1.00881
$870$	0	0
$871$	−3.26496e12	−0.192219
$872$	−5.19833e12	−0.304466
$873$	0	0
$874$	−1.06525e11	−0.00617521
$875$	1.08196e13	0.623988
$876$	0	0
$877$	1.63714e12	0.0934519	0.0467259	−	0.998908i	$-0.485121\pi$
$877$	1.63714e12	0.0934519	0.0467259	+	0.998908i	$0.485121\pi$
$878$	−8.83645e12	−0.501825
$879$	0	0
$880$	6.65078e12	0.373853
$881$	4.20163e11	0.0234977	0.0117489	−	0.999931i	$-0.496260\pi$
$881$	4.20163e11	0.0234977	0.0117489	+	0.999931i	$0.496260\pi$
$882$	0	0
$883$	−2.88540e13	−1.59729	−0.798644	−	0.601803i	$-0.794449\pi$
$883$	−2.88540e13	−1.59729	−0.798644	+	0.601803i	$0.794449\pi$
$884$	3.90583e12	0.215119
$885$	0	0
$886$	2.31359e13	1.26135
$887$	−8.99827e12	−0.488093	−0.244047	−	0.969764i	$-0.578475\pi$
$887$	−8.99827e12	−0.488093	−0.244047	+	0.969764i	$0.578475\pi$
$888$	0	0
$889$	4.47515e12	0.240298
$890$	−2.71941e13	−1.45285
$891$	0	0
$892$	−5.09985e12	−0.269722
$893$	−1.91408e12	−0.100723
$894$	0	0
$895$	2.87774e13	1.49916
$896$	1.30594e12	0.0676919
$897$	0	0
$898$	−6.85540e12	−0.351794
$899$	2.36279e11	0.0120644
$900$	0	0
$901$	3.37994e13	1.70863
$902$	−2.48443e13	−1.24968
$903$	0	0
$904$	−2.47178e12	−0.123098
$905$	1.75881e13	0.871567
$906$	0	0
$907$	−2.57709e13	−1.26444	−0.632218	−	0.774790i	$-0.717855\pi$
$907$	−2.57709e13	−1.26444	−0.632218	+	0.774790i	$0.717855\pi$
$908$	−1.61524e13	−0.788588
$909$	0	0
$910$	−5.97062e12	−0.288625
$911$	1.30097e12	0.0625798	0.0312899	−	0.999510i	$-0.490038\pi$
$911$	1.30097e12	0.0625798	0.0312899	+	0.999510i	$0.490038\pi$
$912$	0	0
$913$	4.64766e13	2.21368
$914$	2.24776e13	1.06535
$915$	0	0
$916$	−7.62960e12	−0.358073
$917$	1.03806e13	0.484796
$918$	0	0
$919$	−2.22215e13	−1.02767	−0.513835	−	0.857889i	$-0.671776\pi$
$919$	−2.22215e13	−1.02767	−0.513835	+	0.857889i	$0.671776\pi$
$920$	−3.30834e11	−0.0152253
$921$	0	0
$922$	2.20381e13	1.00435
$923$	1.23227e13	0.558854
$924$	0	0
$925$	3.91128e10	0.00175663
$926$	−2.16482e13	−0.967548
$927$	0	0
$928$	−1.61818e12	−0.0716245
$929$	−3.02539e13	−1.33263	−0.666316	−	0.745670i	$-0.732130\pi$
$929$	−3.02539e13	−1.33263	−0.666316	+	0.745670i	$0.732130\pi$
$930$	0	0
$931$	−2.17446e12	−0.0948587
$932$	−1.14956e12	−0.0499070
$933$	0	0
$934$	1.81660e13	0.781084
$935$	−3.19140e13	−1.36562
$936$	0	0
$937$	−3.10153e13	−1.31446	−0.657231	−	0.753689i	$-0.728273\pi$
$937$	−3.10153e13	−1.31446	−0.657231	+	0.753689i	$0.728273\pi$
$938$	5.23837e12	0.220945
$939$	0	0
$940$	−5.94452e12	−0.248337
$941$	−3.24185e13	−1.34784	−0.673922	−	0.738803i	$-0.735392\pi$
$941$	−3.24185e13	−1.34784	−0.673922	+	0.738803i	$0.735392\pi$
$942$	0	0
$943$	1.23585e12	0.0508936
$944$	−9.05135e12	−0.370970
$945$	0	0
$946$	2.98508e12	0.121184
$947$	2.07659e13	0.839028	0.419514	−	0.907749i	$-0.362200\pi$
$947$	2.07659e13	0.839028	0.419514	+	0.907749i	$0.362200\pi$
$948$	0	0
$949$	−1.23804e12	−0.0495491
$950$	−1.13939e12	−0.0453855
$951$	0	0
$952$	−6.26660e12	−0.247267
$953$	3.46692e13	1.36153	0.680763	−	0.732504i	$-0.261648\pi$
$953$	3.46692e13	1.36153	0.680763	+	0.732504i	$0.261648\pi$
$954$	0	0
$955$	3.53053e13	1.37349
$956$	3.99103e12	0.154534
$957$	0	0
$958$	−1.65622e13	−0.635293
$959$	−2.35229e13	−0.898064
$960$	0	0
$961$	−2.64162e13	−0.999113
$962$	5.55629e10	0.00209169
$963$	0	0
$964$	2.62930e11	0.00980605
$965$	6.89285e12	0.255874
$966$	0	0
$967$	−4.45702e13	−1.63918	−0.819589	−	0.572952i	$-0.805798\pi$
$967$	−4.45702e13	−1.63918	−0.819589	+	0.572952i	$0.805798\pi$
$968$	−7.21830e12	−0.264238
$969$	0	0
$970$	−2.96780e13	−1.07637
$971$	−4.61615e13	−1.66645	−0.833227	−	0.552931i	$-0.813509\pi$
$971$	−4.61615e13	−1.66645	−0.833227	+	0.552931i	$0.813509\pi$
$972$	0	0
$973$	2.86964e13	1.02641
$974$	2.84634e13	1.01338
$975$	0	0
$976$	−8.01938e12	−0.282889
$977$	1.27596e13	0.448036	0.224018	−	0.974585i	$-0.428083\pi$
$977$	1.27596e13	0.448036	0.224018	+	0.974585i	$0.428083\pi$
$978$	0	0
$979$	6.90056e13	2.40083
$980$	−6.75317e12	−0.233879
$981$	0	0
$982$	−3.55880e13	−1.22124
$983$	2.49925e13	0.853725	0.426863	−	0.904317i	$-0.359619\pi$
$983$	2.49925e13	0.853725	0.426863	+	0.904317i	$0.359619\pi$
$984$	0	0
$985$	−3.91419e13	−1.32489
$986$	7.76491e12	0.261632
$987$	0	0
$988$	−1.61860e12	−0.0540422
$989$	−1.48489e11	−0.00493526
$990$	0	0
$991$	7.39764e12	0.243647	0.121824	−	0.992552i	$-0.461126\pi$
$991$	7.39764e12	0.243647	0.121824	+	0.992552i	$0.461126\pi$
$992$	−1.60545e11	−0.00526375
$993$	0	0
$994$	−1.97708e13	−0.642370
$995$	4.06497e13	1.31478
$996$	0	0
$997$	−5.44084e13	−1.74396	−0.871982	−	0.489537i	$-0.837166\pi$
$997$	−5.44084e13	−1.74396	−0.871982	+	0.489537i	$0.837166\pi$
$998$	−3.15679e13	−1.00730
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.10.a.b.1.1		1
3.2	odd	2		38.10.a.b.1.1	✓	1
12.11	even	2		304.10.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.10.a.b.1.1	✓	1	3.2	odd	2
304.10.a.a.1.1		1	12.11	even	2
342.10.a.b.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	342.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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