LMFDB - Embedded newform 8.20.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,20,Mod(1,8)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 20, names="a")

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

chi := DirichletCharacter("8.1");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 20 $
Character orbit:	$[\chi]$	$=$	8.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:	$18.3053357245$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2519x + 43659 $ x^3 - x^2 - 2519*x + 43659
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$37.1696$ of defining polynomial
Character	$\chi$	$=$	8.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-34307.5 q^{3} +2.59881e6 q^{5} -1.93114e8 q^{7} +1.47441e7 q^{9} +O(q^{10})$	$q-34307.5 q^{3} +2.59881e6 q^{5} -1.93114e8 q^{7} +1.47441e7 q^{9} +8.32029e8 q^{11} +4.50621e10 q^{13} -8.91588e10 q^{15} +2.73062e11 q^{17} +9.22463e11 q^{19} +6.62527e12 q^{21} +1.26167e13 q^{23} -1.23197e13 q^{25} +3.93685e13 q^{27} +1.27884e14 q^{29} -1.96283e14 q^{31} -2.85448e13 q^{33} -5.01868e14 q^{35} -7.60758e14 q^{37} -1.54597e15 q^{39} +2.82783e15 q^{41} +9.55031e13 q^{43} +3.83173e13 q^{45} -4.05928e15 q^{47} +2.58942e16 q^{49} -9.36809e15 q^{51} +3.19955e16 q^{53} +2.16229e15 q^{55} -3.16474e16 q^{57} -5.99553e16 q^{59} -1.56065e17 q^{61} -2.84731e15 q^{63} +1.17108e17 q^{65} +1.35957e17 q^{67} -4.32847e17 q^{69} +1.18135e17 q^{71} +5.66518e17 q^{73} +4.22657e17 q^{75} -1.60677e17 q^{77} +1.38949e18 q^{79} -1.36777e18 q^{81} +1.99954e18 q^{83} +7.09638e17 q^{85} -4.38737e18 q^{87} -3.16978e18 q^{89} -8.70213e18 q^{91} +6.73399e18 q^{93} +2.39731e18 q^{95} +1.60346e18 q^{97} +1.22676e16 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 23732 q^{3} + 2140218 q^{5} + 55851720 q^{7} + 646753951 q^{9}+O(q^{10}) $ 3 * q + 23732 * q^3 + 2140218 * q^5 + 55851720 * q^7 + 646753951 * q^9	$ 3 q + 23732 q^{3} + 2140218 q^{5} + 55851720 q^{7} + 646753951 q^{9} - 297392964 q^{11} - 14862401022 q^{13} + 292635653528 q^{15} + 803332464534 q^{17} + 3212269666884 q^{19} + 11192319829728 q^{21} + 24948509305560 q^{23} + 72340360289109 q^{25} + 64092343553864 q^{27} + 77667139511058 q^{29} - 248431735193568 q^{31} - 252071696774128 q^{33} - 13\!\cdots\!08 q^{35}+ \cdots - 11\!\cdots\!60 q^{99}+O(q^{100}) $ 3 * q + 23732 * q^3 + 2140218 * q^5 + 55851720 * q^7 + 646753951 * q^9 - 297392964 * q^11 - 14862401022 * q^13 + 292635653528 * q^15 + 803332464534 * q^17 + 3212269666884 * q^19 + 11192319829728 * q^21 + 24948509305560 * q^23 + 72340360289109 * q^25 + 64092343553864 * q^27 + 77667139511058 * q^29 - 248431735193568 * q^31 - 252071696774128 * q^33 - 1384378114232208 * q^35 - 414866302559142 * q^37 - 5522045976027016 * q^39 + 2818737880869678 * q^41 - 6663230715469860 * q^43 + 22824972038174722 * q^45 + 1500497644728624 * q^47 + 39641123057321787 * q^49 - 1085042177122520 * q^51 + 56067344774978154 * q^53 - 56817855560205432 * q^55 - 6408983732955152 * q^57 - 154317270851496852 * q^59 - 134994376571654862 * q^61 - 78435010097874072 * q^63 - 563892790723886724 * q^65 + 151032181904450292 * q^67 - 153590366326625632 * q^69 + 1210541848845584136 * q^71 - 81876123599662770 * q^73 + 2698278154129173484 * q^75 + 97235023193490336 * q^77 + 1439028483035907408 * q^79 - 318667537468381733 * q^81 - 983438102798849916 * q^83 - 1618537843962618540 * q^85 - 4280411824494387144 * q^87 - 1312857528832070946 * q^89 - 11488143382330124496 * q^91 - 4308257105281185920 * q^93 - 10926153348157720968 * q^95 + 14033245412567998566 * q^97 - 11415749644087184660 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−34307.5	−1.00632	−0.503161	−	0.864192i	$-0.667830\pi$
$3$	−34307.5	−1.00632	−0.503161	+	0.864192i	$0.667830\pi$
$4$	0	0
$5$	2.59881e6	0.595059	0.297529	−	0.954713i	$-0.403837\pi$
$5$	2.59881e6	0.595059	0.297529	+	0.954713i	$0.403837\pi$
$6$	0	0
$7$	−1.93114e8	−1.80877	−0.904384	−	0.426719i	$-0.859669\pi$
$7$	−1.93114e8	−1.80877	−0.904384	+	0.426719i	$0.859669\pi$
$8$	0	0
$9$	1.47441e7	0.0126857
$10$	0	0
$11$	8.32029e8	0.106392	0.0531958	−	0.998584i	$-0.483059\pi$
$11$	8.32029e8	0.106392	0.0531958	+	0.998584i	$0.483059\pi$
$12$	0	0
$13$	4.50621e10	1.17855	0.589277	−	0.807931i	$-0.299413\pi$
$13$	4.50621e10	1.17855	0.589277	+	0.807931i	$0.299413\pi$
$14$	0	0
$15$	−8.91588e10	−0.598821
$16$	0	0
$17$	2.73062e11	0.558467	0.279233	−	0.960223i	$-0.409920\pi$
$17$	2.73062e11	0.558467	0.279233	+	0.960223i	$0.409920\pi$
$18$	0	0
$19$	9.22463e11	0.655828	0.327914	−	0.944708i	$-0.393654\pi$
$19$	9.22463e11	0.655828	0.327914	+	0.944708i	$0.393654\pi$
$20$	0	0
$21$	6.62527e12	1.82020
$22$	0	0
$23$	1.26167e13	1.46060	0.730299	−	0.683128i	$-0.239381\pi$
$23$	1.26167e13	1.46060	0.730299	+	0.683128i	$0.239381\pi$
$24$	0	0
$25$	−1.23197e13	−0.645905
$26$	0	0
$27$	3.93685e13	0.993557
$28$	0	0
$29$	1.27884e14	1.63695	0.818473	−	0.574545i	$-0.194821\pi$
$29$	1.27884e14	1.63695	0.818473	+	0.574545i	$0.194821\pi$
$30$	0	0
$31$	−1.96283e14	−1.33336	−0.666679	−	0.745345i	$-0.732285\pi$
$31$	−1.96283e14	−1.33336	−0.666679	+	0.745345i	$0.732285\pi$
$32$	0	0
$33$	−2.85448e13	−0.107064
$34$	0	0
$35$	−5.01868e14	−1.07632
$36$	0	0
$37$	−7.60758e14	−0.962343	−0.481172	−	0.876626i	$-0.659789\pi$
$37$	−7.60758e14	−0.962343	−0.481172	+	0.876626i	$0.659789\pi$
$38$	0	0
$39$	−1.54597e15	−1.18601
$40$	0	0
$41$	2.82783e15	1.34898	0.674492	−	0.738282i	$-0.264363\pi$
$41$	2.82783e15	1.34898	0.674492	+	0.738282i	$0.264363\pi$
$42$	0	0
$43$	9.55031e13	0.0289779	0.0144890	−	0.999895i	$-0.495388\pi$
$43$	9.55031e13	0.0289779	0.0144890	+	0.999895i	$0.495388\pi$
$44$	0	0
$45$	3.83173e13	0.00754876
$46$	0	0
$47$	−4.05928e15	−0.529078	−0.264539	−	0.964375i	$-0.585220\pi$
$47$	−4.05928e15	−0.529078	−0.264539	+	0.964375i	$0.585220\pi$
$48$	0	0
$49$	2.58942e16	2.27164
$50$	0	0
$51$	−9.36809e15	−0.561998
$52$	0	0
$53$	3.19955e16	1.33189	0.665944	−	0.746002i	$-0.268029\pi$
$53$	3.19955e16	1.33189	0.665944	+	0.746002i	$0.268029\pi$
$54$	0	0
$55$	2.16229e15	0.0633093
$56$	0	0
$57$	−3.16474e16	−0.659975
$58$	0	0
$59$	−5.99553e16	−0.901019	−0.450509	−	0.892772i	$-0.648758\pi$
$59$	−5.99553e16	−0.901019	−0.450509	+	0.892772i	$0.648758\pi$
$60$	0	0
$61$	−1.56065e17	−1.70872	−0.854361	−	0.519679i	$-0.826051\pi$
$61$	−1.56065e17	−1.70872	−0.854361	+	0.519679i	$0.826051\pi$
$62$	0	0
$63$	−2.84731e15	−0.0229456
$64$	0	0
$65$	1.17108e17	0.701309
$66$	0	0
$67$	1.35957e17	0.610506	0.305253	−	0.952271i	$-0.401259\pi$
$67$	1.35957e17	0.610506	0.305253	+	0.952271i	$0.401259\pi$
$68$	0	0
$69$	−4.32847e17	−1.46983
$70$	0	0
$71$	1.18135e17	0.305790	0.152895	−	0.988242i	$-0.451140\pi$
$71$	1.18135e17	0.305790	0.152895	+	0.988242i	$0.451140\pi$
$72$	0	0
$73$	5.66518e17	1.12628	0.563140	−	0.826361i	$-0.309593\pi$
$73$	5.66518e17	1.12628	0.563140	+	0.826361i	$0.309593\pi$
$74$	0	0
$75$	4.22657e17	0.649989
$76$	0	0
$77$	−1.60677e17	−0.192438
$78$	0	0
$79$	1.38949e18	1.30437	0.652183	−	0.758061i	$-0.273853\pi$
$79$	1.38949e18	1.30437	0.652183	+	0.758061i	$0.273853\pi$
$80$	0	0
$81$	−1.36777e18	−1.01252
$82$	0	0
$83$	1.99954e18	1.17405	0.587027	−	0.809567i	$-0.300298\pi$
$83$	1.99954e18	1.17405	0.587027	+	0.809567i	$0.300298\pi$
$84$	0	0
$85$	7.09638e17	0.332320
$86$	0	0
$87$	−4.38737e18	−1.64730
$88$	0	0
$89$	−3.16978e18	−0.959012	−0.479506	−	0.877539i	$-0.659184\pi$
$89$	−3.16978e18	−0.959012	−0.479506	+	0.877539i	$0.659184\pi$
$90$	0	0
$91$	−8.70213e18	−2.13173
$92$	0	0
$93$	6.73399e18	1.34179
$94$	0	0
$95$	2.39731e18	0.390256
$96$	0	0
$97$	1.60346e18	0.214154	0.107077	−	0.994251i	$-0.465851\pi$
$97$	1.60346e18	0.214154	0.107077	+	0.994251i	$0.465851\pi$
$98$	0	0
$99$	1.22676e16	0.00134966
$100$	0	0
$101$	2.21136e18	0.201190	0.100595	−	0.994927i	$-0.467925\pi$
$101$	2.21136e18	0.201190	0.100595	+	0.994927i	$0.467925\pi$
$102$	0	0
$103$	−9.37696e17	−0.0708123	−0.0354061	−	0.999373i	$-0.511272\pi$
$103$	−9.37696e17	−0.0708123	−0.0354061	+	0.999373i	$0.511272\pi$
$104$	0	0
$105$	1.72178e19	1.08313
$106$	0	0
$107$	−4.07453e18	−0.214256	−0.107128	−	0.994245i	$-0.534165\pi$
$107$	−4.07453e18	−0.214256	−0.107128	+	0.994245i	$0.534165\pi$
$108$	0	0
$109$	5.17714e18	0.228317	0.114159	−	0.993463i	$-0.463583\pi$
$109$	5.17714e18	0.228317	0.114159	+	0.993463i	$0.463583\pi$
$110$	0	0
$111$	2.60997e19	0.968428
$112$	0	0
$113$	−1.96131e19	−0.614189	−0.307094	−	0.951679i	$-0.599357\pi$
$113$	−1.96131e19	−0.614189	−0.307094	+	0.951679i	$0.599357\pi$
$114$	0	0
$115$	3.27884e19	0.869142
$116$	0	0
$117$	6.64402e17	0.0149508
$118$	0	0
$119$	−5.27322e19	−1.01014
$120$	0	0
$121$	−6.04668e19	−0.988681
$122$	0	0
$123$	−9.70159e19	−1.35751
$124$	0	0
$125$	−8.15849e19	−0.979410
$126$	0	0
$127$	1.26799e20	1.30912	0.654560	−	0.756010i	$-0.272854\pi$
$127$	1.26799e20	1.30912	0.654560	+	0.756010i	$0.272854\pi$
$128$	0	0
$129$	−3.27647e18	−0.0291611
$130$	0	0
$131$	1.28991e20	0.991931	0.495966	−	0.868342i	$-0.334814\pi$
$131$	1.28991e20	0.991931	0.495966	+	0.868342i	$0.334814\pi$
$132$	0	0
$133$	−1.78141e20	−1.18624
$134$	0	0
$135$	1.02311e20	0.591225
$136$	0	0
$137$	2.85432e20	1.43436	0.717179	−	0.696889i	$-0.245433\pi$
$137$	2.85432e20	1.43436	0.717179	+	0.696889i	$0.245433\pi$
$138$	0	0
$139$	−1.44335e19	−0.0632022	−0.0316011	−	0.999501i	$-0.510061\pi$
$139$	−1.44335e19	−0.0632022	−0.0316011	+	0.999501i	$0.510061\pi$
$140$	0	0
$141$	1.39264e20	0.532423
$142$	0	0
$143$	3.74930e19	0.125388
$144$	0	0
$145$	3.32346e20	0.974079
$146$	0	0
$147$	−8.88366e20	−2.28601
$148$	0	0
$149$	4.57287e19	0.103495	0.0517475	−	0.998660i	$-0.483521\pi$
$149$	4.57287e19	0.103495	0.0517475	+	0.998660i	$0.483521\pi$
$150$	0	0
$151$	6.44110e20	1.28434	0.642169	−	0.766563i	$-0.278035\pi$
$151$	6.44110e20	1.28434	0.642169	+	0.766563i	$0.278035\pi$
$152$	0	0
$153$	4.02607e18	0.00708456
$154$	0	0
$155$	−5.10103e20	−0.793427
$156$	0	0
$157$	9.32864e20	1.28461	0.642306	−	0.766449i	$-0.277978\pi$
$157$	9.32864e20	1.28461	0.642306	+	0.766449i	$0.277978\pi$
$158$	0	0
$159$	−1.09768e21	−1.34031
$160$	0	0
$161$	−2.43646e21	−2.64188
$162$	0	0
$163$	−4.50271e20	−0.434202	−0.217101	−	0.976149i	$-0.569660\pi$
$163$	−4.50271e20	−0.434202	−0.217101	+	0.976149i	$0.569660\pi$
$164$	0	0
$165$	−7.41827e19	−0.0637096
$166$	0	0
$167$	1.83293e21	1.40391	0.701954	−	0.712223i	$-0.252311\pi$
$167$	1.83293e21	1.40391	0.701954	+	0.712223i	$0.252311\pi$
$168$	0	0
$169$	5.68672e20	0.388990
$170$	0	0
$171$	1.36009e19	0.00831966
$172$	0	0
$173$	7.77321e20	0.425758	0.212879	−	0.977079i	$-0.431716\pi$
$173$	7.77321e20	0.425758	0.212879	+	0.977079i	$0.431716\pi$
$174$	0	0
$175$	2.37910e21	1.16829
$176$	0	0
$177$	2.05692e21	0.906716
$178$	0	0
$179$	−2.26175e21	−0.896068	−0.448034	−	0.894016i	$-0.647876\pi$
$179$	−2.26175e21	−0.896068	−0.448034	+	0.894016i	$0.647876\pi$
$180$	0	0
$181$	−6.20820e20	−0.221319	−0.110660	−	0.993858i	$-0.535296\pi$
$181$	−6.20820e20	−0.221319	−0.110660	+	0.993858i	$0.535296\pi$
$182$	0	0
$183$	5.35420e21	1.71953
$184$	0	0
$185$	−1.97707e21	−0.572651
$186$	0	0
$187$	2.27196e20	0.0594162
$188$	0	0
$189$	−7.60261e21	−1.79711
$190$	0	0
$191$	1.01107e21	0.216253	0.108127	−	0.994137i	$-0.465515\pi$
$191$	1.01107e21	0.216253	0.108127	+	0.994137i	$0.465515\pi$
$192$	0	0
$193$	2.29835e21	0.445268	0.222634	−	0.974902i	$-0.428535\pi$
$193$	2.29835e21	0.445268	0.222634	+	0.974902i	$0.428535\pi$
$194$	0	0
$195$	−4.01768e21	−0.705743
$196$	0	0
$197$	1.67409e21	0.266900	0.133450	−	0.991056i	$-0.457394\pi$
$197$	1.67409e21	0.266900	0.133450	+	0.991056i	$0.457394\pi$
$198$	0	0
$199$	6.13391e21	0.888451	0.444225	−	0.895915i	$-0.353479\pi$
$199$	6.13391e21	0.888451	0.444225	+	0.895915i	$0.353479\pi$
$200$	0	0
$201$	−4.66434e21	−0.614366
$202$	0	0
$203$	−2.46962e22	−2.96086
$204$	0	0
$205$	7.34901e21	0.802725
$206$	0	0
$207$	1.86022e20	0.0185288
$208$	0	0
$209$	7.67516e20	0.0697746
$210$	0	0
$211$	8.82836e21	0.733156	0.366578	−	0.930387i	$-0.380529\pi$
$211$	8.82836e21	0.733156	0.366578	+	0.930387i	$0.380529\pi$
$212$	0	0
$213$	−4.05291e21	−0.307723
$214$	0	0
$215$	2.48195e20	0.0172436
$216$	0	0
$217$	3.79051e22	2.41174
$218$	0	0
$219$	−1.94358e22	−1.13340
$220$	0	0
$221$	1.23048e22	0.658183
$222$	0	0
$223$	1.70960e22	0.839456	0.419728	−	0.907650i	$-0.362125\pi$
$223$	1.70960e22	0.839456	0.419728	+	0.907650i	$0.362125\pi$
$224$	0	0
$225$	−1.81643e20	−0.00819378
$226$	0	0
$227$	2.50313e22	1.03810	0.519049	−	0.854745i	$-0.326286\pi$
$227$	2.50313e22	1.03810	0.519049	+	0.854745i	$0.326286\pi$
$228$	0	0
$229$	−3.28082e21	−0.125183	−0.0625915	−	0.998039i	$-0.519937\pi$
$229$	−3.28082e21	−0.125183	−0.0625915	+	0.998039i	$0.519937\pi$
$230$	0	0
$231$	5.51241e21	0.193655
$232$	0	0
$233$	−6.12997e22	−1.98416	−0.992080	−	0.125606i	$-0.959913\pi$
$233$	−6.12997e22	−1.98416	−0.992080	+	0.125606i	$0.959913\pi$
$234$	0	0
$235$	−1.05493e22	−0.314833
$236$	0	0
$237$	−4.76701e22	−1.31261
$238$	0	0
$239$	−2.05975e22	−0.523643	−0.261821	−	0.965116i	$-0.584323\pi$
$239$	−2.05975e22	−0.523643	−0.261821	+	0.965116i	$0.584323\pi$
$240$	0	0
$241$	8.23955e21	0.193527	0.0967635	−	0.995307i	$-0.469151\pi$
$241$	8.23955e21	0.193527	0.0967635	+	0.995307i	$0.469151\pi$
$242$	0	0
$243$	1.16837e21	0.0253700
$244$	0	0
$245$	6.72942e22	1.35176
$246$	0	0
$247$	4.15681e22	0.772929
$248$	0	0
$249$	−6.85993e22	−1.18148
$250$	0	0
$251$	−6.38251e22	−1.01881	−0.509403	−	0.860528i	$-0.670134\pi$
$251$	−6.38251e22	−1.01881	−0.509403	+	0.860528i	$0.670134\pi$
$252$	0	0
$253$	1.04974e22	0.155396
$254$	0	0
$255$	−2.43459e22	−0.334422
$256$	0	0
$257$	−8.06598e22	−1.02871	−0.514354	−	0.857578i	$-0.671968\pi$
$257$	−8.06598e22	−1.02871	−0.514354	+	0.857578i	$0.671968\pi$
$258$	0	0
$259$	1.46913e23	1.74066
$260$	0	0
$261$	1.88554e21	0.0207659
$262$	0	0
$263$	5.19992e22	0.532619	0.266310	−	0.963888i	$-0.414196\pi$
$263$	5.19992e22	0.532619	0.266310	+	0.963888i	$0.414196\pi$
$264$	0	0
$265$	8.31502e22	0.792551
$266$	0	0
$267$	1.08747e23	0.965075
$268$	0	0
$269$	4.34498e22	0.359204	0.179602	−	0.983739i	$-0.442519\pi$
$269$	4.34498e22	0.359204	0.179602	+	0.983739i	$0.442519\pi$
$270$	0	0
$271$	−2.03032e23	−1.56443	−0.782215	−	0.623009i	$-0.785910\pi$
$271$	−2.03032e23	−1.56443	−0.782215	+	0.623009i	$0.785910\pi$
$272$	0	0
$273$	2.98549e23	2.14521
$274$	0	0
$275$	−1.02503e22	−0.0687189
$276$	0	0
$277$	−1.84207e23	−1.15279	−0.576393	−	0.817173i	$-0.695540\pi$
$277$	−1.84207e23	−1.15279	−0.576393	+	0.817173i	$0.695540\pi$
$278$	0	0
$279$	−2.89403e21	−0.0169146
$280$	0	0
$281$	1.59747e23	0.872413	0.436206	−	0.899847i	$-0.356322\pi$
$281$	1.59747e23	0.872413	0.436206	+	0.899847i	$0.356322\pi$
$282$	0	0
$283$	−5.90709e22	−0.301580	−0.150790	−	0.988566i	$-0.548182\pi$
$283$	−5.90709e22	−0.301580	−0.150790	+	0.988566i	$0.548182\pi$
$284$	0	0
$285$	−8.22457e22	−0.392724
$286$	0	0
$287$	−5.46095e23	−2.44000
$288$	0	0
$289$	−1.64509e23	−0.688115
$290$	0	0
$291$	−5.50107e22	−0.215508
$292$	0	0
$293$	9.73152e22	0.357222	0.178611	−	0.983920i	$-0.442840\pi$
$293$	9.73152e22	0.357222	0.178611	+	0.983920i	$0.442840\pi$
$294$	0	0
$295$	−1.55813e23	−0.536159
$296$	0	0
$297$	3.27557e22	0.105706
$298$	0	0
$299$	5.68534e23	1.72139
$300$	0	0
$301$	−1.84430e22	−0.0524143
$302$	0	0
$303$	−7.58661e22	−0.202462
$304$	0	0
$305$	−4.05583e23	−1.01679
$306$	0	0
$307$	4.85956e23	1.14494	0.572470	−	0.819926i	$-0.305985\pi$
$307$	4.85956e23	1.14494	0.572470	+	0.819926i	$0.305985\pi$
$308$	0	0
$309$	3.21700e22	0.0712600
$310$	0	0
$311$	4.66721e23	0.972375	0.486188	−	0.873854i	$-0.338387\pi$
$311$	4.66721e23	0.972375	0.486188	+	0.873854i	$0.338387\pi$
$312$	0	0
$313$	−3.78777e23	−0.742526	−0.371263	−	0.928528i	$-0.621075\pi$
$313$	−3.78777e23	−0.742526	−0.371263	+	0.928528i	$0.621075\pi$
$314$	0	0
$315$	−7.39961e21	−0.0136540
$316$	0	0
$317$	6.09181e23	1.05848	0.529241	−	0.848472i	$-0.322477\pi$
$317$	6.09181e23	1.05848	0.529241	+	0.848472i	$0.322477\pi$
$318$	0	0
$319$	1.06403e23	0.174157
$320$	0	0
$321$	1.39787e23	0.215610
$322$	0	0
$323$	2.51890e23	0.366258
$324$	0	0
$325$	−5.55150e23	−0.761234
$326$	0	0
$327$	−1.77615e23	−0.229761
$328$	0	0
$329$	7.83906e23	0.956980
$330$	0	0
$331$	−2.53458e22	−0.0292106	−0.0146053	−	0.999893i	$-0.504649\pi$
$331$	−2.53458e22	−0.0292106	−0.0146053	+	0.999893i	$0.504649\pi$
$332$	0	0
$333$	−1.12167e22	−0.0122080
$334$	0	0
$335$	3.53326e23	0.363287
$336$	0	0
$337$	−1.23470e24	−1.19971	−0.599856	−	0.800108i	$-0.704776\pi$
$337$	−1.23470e24	−1.19971	−0.599856	+	0.800108i	$0.704776\pi$
$338$	0	0
$339$	6.72878e23	0.618072
$340$	0	0
$341$	−1.63313e23	−0.141858
$342$	0	0
$343$	−2.79925e24	−2.30011
$344$	0	0
$345$	−1.12489e24	−0.874637
$346$	0	0
$347$	−2.52463e24	−1.85809	−0.929045	−	0.369966i	$-0.879369\pi$
$347$	−2.52463e24	−1.85809	−0.929045	+	0.369966i	$0.879369\pi$
$348$	0	0
$349$	−5.12360e23	−0.357054	−0.178527	−	0.983935i	$-0.557133\pi$
$349$	−5.12360e23	−0.357054	−0.178527	+	0.983935i	$0.557133\pi$
$350$	0	0
$351$	1.77403e24	1.17096
$352$	0	0
$353$	2.92915e24	1.83182	0.915909	−	0.401387i	$-0.131472\pi$
$353$	2.92915e24	1.83182	0.915909	+	0.401387i	$0.131472\pi$
$354$	0	0
$355$	3.07010e23	0.181963
$356$	0	0
$357$	1.80911e24	1.01652
$358$	0	0
$359$	−6.15112e23	−0.327761	−0.163880	−	0.986480i	$-0.552401\pi$
$359$	−6.15112e23	−0.327761	−0.163880	+	0.986480i	$0.552401\pi$
$360$	0	0
$361$	−1.12748e24	−0.569890
$362$	0	0
$363$	2.07447e24	0.994932
$364$	0	0
$365$	1.47227e24	0.670203
$366$	0	0
$367$	1.82304e24	0.787895	0.393947	−	0.919133i	$-0.371109\pi$
$367$	1.82304e24	0.787895	0.393947	+	0.919133i	$0.371109\pi$
$368$	0	0
$369$	4.16940e22	0.0171129
$370$	0	0
$371$	−6.17878e24	−2.40908
$372$	0	0
$373$	1.56558e24	0.580018	0.290009	−	0.957024i	$-0.406342\pi$
$373$	1.56558e24	0.580018	0.290009	+	0.957024i	$0.406342\pi$
$374$	0	0
$375$	2.79898e24	0.985603
$376$	0	0
$377$	5.76271e24	1.92923
$378$	0	0
$379$	−2.92164e23	−0.0930154	−0.0465077	−	0.998918i	$-0.514809\pi$
$379$	−2.92164e23	−0.0930154	−0.0465077	+	0.998918i	$0.514809\pi$
$380$	0	0
$381$	−4.35015e24	−1.31740
$382$	0	0
$383$	6.56509e23	0.189170	0.0945850	−	0.995517i	$-0.469848\pi$
$383$	6.56509e23	0.189170	0.0945850	+	0.995517i	$0.469848\pi$
$384$	0	0
$385$	−4.17568e23	−0.114512
$386$	0	0
$387$	1.40811e21	0.000367606 0
$388$	0	0
$389$	−7.21370e24	−1.79323	−0.896616	−	0.442808i	$-0.853982\pi$
$389$	−7.21370e24	−1.79323	−0.896616	+	0.442808i	$0.853982\pi$
$390$	0	0
$391$	3.44514e24	0.815695
$392$	0	0
$393$	−4.42535e24	−0.998203
$394$	0	0
$395$	3.61104e24	0.776175
$396$	0	0
$397$	−1.89515e24	−0.388270	−0.194135	−	0.980975i	$-0.562190\pi$
$397$	−1.89515e24	−0.388270	−0.194135	+	0.980975i	$0.562190\pi$
$398$	0	0
$399$	6.11157e24	1.19374
$400$	0	0
$401$	−1.39870e24	−0.260526	−0.130263	−	0.991479i	$-0.541582\pi$
$401$	−1.39870e24	−0.260526	−0.130263	+	0.991479i	$0.541582\pi$
$402$	0	0
$403$	−8.84493e24	−1.57144
$404$	0	0
$405$	−3.55458e24	−0.602512
$406$	0	0
$407$	−6.32972e23	−0.102385
$408$	0	0
$409$	−5.64124e24	−0.870969	−0.435485	−	0.900196i	$-0.643423\pi$
$409$	−5.64124e24	−0.870969	−0.435485	+	0.900196i	$0.643423\pi$
$410$	0	0
$411$	−9.79247e24	−1.44343
$412$	0	0
$413$	1.15782e25	1.62973
$414$	0	0
$415$	5.19643e24	0.698632
$416$	0	0
$417$	4.95179e23	0.0636018
$418$	0	0
$419$	−1.48234e24	−0.181935	−0.0909674	−	0.995854i	$-0.528996\pi$
$419$	−1.48234e24	−0.181935	−0.0909674	+	0.995854i	$0.528996\pi$
$420$	0	0
$421$	1.58366e25	1.85773	0.928863	−	0.370424i	$-0.120788\pi$
$421$	1.58366e25	1.85773	0.928863	+	0.370424i	$0.120788\pi$
$422$	0	0
$423$	−5.98507e22	−0.00671175
$424$	0	0
$425$	−3.36404e24	−0.360716
$426$	0	0
$427$	3.01383e25	3.09068
$428$	0	0
$429$	−1.28629e24	−0.126181
$430$	0	0
$431$	−1.40908e25	−1.32252	−0.661259	−	0.750157i	$-0.729978\pi$
$431$	−1.40908e25	−1.32252	−0.661259	+	0.750157i	$0.729978\pi$
$432$	0	0
$433$	4.35150e24	0.390845	0.195422	−	0.980719i	$-0.437392\pi$
$433$	4.35150e24	0.390845	0.195422	+	0.980719i	$0.437392\pi$
$434$	0	0
$435$	−1.14020e25	−0.980238
$436$	0	0
$437$	1.16384e25	0.957901
$438$	0	0
$439$	1.24030e25	0.977493	0.488746	−	0.872426i	$-0.337454\pi$
$439$	1.24030e25	0.977493	0.488746	+	0.872426i	$0.337454\pi$
$440$	0	0
$441$	3.81788e23	0.0288175
$442$	0	0
$443$	−2.71882e25	−1.96583	−0.982914	−	0.184064i	$-0.941075\pi$
$443$	−2.71882e25	−1.96583	−0.982914	+	0.184064i	$0.941075\pi$
$444$	0	0
$445$	−8.23766e24	−0.570668
$446$	0	0
$447$	−1.56884e24	−0.104149
$448$	0	0
$449$	5.59500e24	0.356008	0.178004	−	0.984030i	$-0.443036\pi$
$449$	5.59500e24	0.356008	0.178004	+	0.984030i	$0.443036\pi$
$450$	0	0
$451$	2.35284e24	0.143521
$452$	0	0
$453$	−2.20978e25	−1.29246
$454$	0	0
$455$	−2.26152e25	−1.26851
$456$	0	0
$457$	5.99055e24	0.322302	0.161151	−	0.986930i	$-0.448479\pi$
$457$	5.99055e24	0.322302	0.161151	+	0.986930i	$0.448479\pi$
$458$	0	0
$459$	1.07500e25	0.554868
$460$	0	0
$461$	3.18623e24	0.157804	0.0789019	−	0.996882i	$-0.474859\pi$
$461$	3.18623e24	0.157804	0.0789019	+	0.996882i	$0.474859\pi$
$462$	0	0
$463$	9.19655e24	0.437125	0.218562	−	0.975823i	$-0.429863\pi$
$463$	9.19655e24	0.437125	0.218562	+	0.975823i	$0.429863\pi$
$464$	0	0
$465$	1.75004e25	0.798444
$466$	0	0
$467$	5.34180e24	0.233979	0.116990	−	0.993133i	$-0.462676\pi$
$467$	5.34180e24	0.233979	0.116990	+	0.993133i	$0.462676\pi$
$468$	0	0
$469$	−2.62552e25	−1.10426
$470$	0	0
$471$	−3.20043e25	−1.29273
$472$	0	0
$473$	7.94613e22	0.00308301
$474$	0	0
$475$	−1.13644e25	−0.423602
$476$	0	0
$477$	4.71746e23	0.0168960
$478$	0	0
$479$	5.41606e24	0.186422	0.0932108	−	0.995646i	$-0.470287\pi$
$479$	5.41606e24	0.186422	0.0932108	+	0.995646i	$0.470287\pi$
$480$	0	0
$481$	−3.42813e25	−1.13417
$482$	0	0
$483$	8.35889e25	2.65859
$484$	0	0
$485$	4.16709e24	0.127434
$486$	0	0
$487$	3.18277e25	0.936010	0.468005	−	0.883726i	$-0.344973\pi$
$487$	3.18277e25	0.936010	0.468005	+	0.883726i	$0.344973\pi$
$488$	0	0
$489$	1.54477e25	0.436947
$490$	0	0
$491$	1.75069e25	0.476360	0.238180	−	0.971221i	$-0.423449\pi$
$491$	1.75069e25	0.476360	0.238180	+	0.971221i	$0.423449\pi$
$492$	0	0
$493$	3.49202e25	0.914179
$494$	0	0
$495$	3.18811e22	0.000803126 0
$496$	0	0
$497$	−2.28135e25	−0.553103
$498$	0	0
$499$	5.62265e25	1.31216	0.656078	−	0.754693i	$-0.272214\pi$
$499$	5.62265e25	1.31216	0.656078	+	0.754693i	$0.272214\pi$
$500$	0	0
$501$	−6.28831e25	−1.41278
$502$	0	0
$503$	−2.26953e25	−0.490953	−0.245476	−	0.969403i	$-0.578944\pi$
$503$	−2.26953e25	−0.490953	−0.245476	+	0.969403i	$0.578944\pi$
$504$	0	0
$505$	5.74690e24	0.119720
$506$	0	0
$507$	−1.95097e25	−0.391449
$508$	0	0
$509$	6.77955e25	1.31033	0.655167	−	0.755484i	$-0.272598\pi$
$509$	6.77955e25	1.31033	0.655167	+	0.755484i	$0.272598\pi$
$510$	0	0
$511$	−1.09403e26	−2.03718
$512$	0	0
$513$	3.63160e25	0.651602
$514$	0	0
$515$	−2.43690e24	−0.0421375
$516$	0	0
$517$	−3.37744e24	−0.0562895
$518$	0	0
$519$	−2.66680e25	−0.428450
$520$	0	0
$521$	−1.02376e26	−1.58576	−0.792881	−	0.609377i	$-0.791420\pi$
$521$	−1.02376e26	−1.58576	−0.792881	+	0.609377i	$0.791420\pi$
$522$	0	0
$523$	−4.15135e25	−0.620045	−0.310023	−	0.950729i	$-0.600337\pi$
$523$	−4.15135e25	−0.620045	−0.310023	+	0.950729i	$0.600337\pi$
$524$	0	0
$525$	−8.16211e25	−1.17568
$526$	0	0
$527$	−5.35976e25	−0.744636
$528$	0	0
$529$	8.45652e25	1.13335
$530$	0	0
$531$	−8.83990e23	−0.0114301
$532$	0	0
$533$	1.27428e26	1.58985
$534$	0	0
$535$	−1.05889e25	−0.127495
$536$	0	0
$537$	7.75951e25	0.901734
$538$	0	0
$539$	2.15447e25	0.241684
$540$	0	0
$541$	−1.49513e25	−0.161922	−0.0809611	−	0.996717i	$-0.525799\pi$
$541$	−1.49513e25	−0.161922	−0.0809611	+	0.996717i	$0.525799\pi$
$542$	0	0
$543$	2.12988e25	0.222719
$544$	0	0
$545$	1.34544e25	0.135862
$546$	0	0
$547$	−5.62975e25	−0.549047	−0.274524	−	0.961580i	$-0.588520\pi$
$547$	−5.62975e25	−0.549047	−0.274524	+	0.961580i	$0.588520\pi$
$548$	0	0
$549$	−2.30104e24	−0.0216764
$550$	0	0
$551$	1.17968e26	1.07355
$552$	0	0
$553$	−2.68331e26	−2.35930
$554$	0	0
$555$	6.78282e25	0.576272
$556$	0	0
$557$	2.39450e26	1.96603	0.983014	−	0.183528i	$-0.0587519\pi$
$557$	2.39450e26	1.96603	0.983014	+	0.183528i	$0.0587519\pi$
$558$	0	0
$559$	4.30357e24	0.0341520
$560$	0	0
$561$	−7.79452e24	−0.0597919
$562$	0	0
$563$	1.51707e25	0.112506	0.0562531	−	0.998417i	$-0.482085\pi$
$563$	1.51707e25	0.112506	0.0562531	+	0.998417i	$0.482085\pi$
$564$	0	0
$565$	−5.09709e25	−0.365478
$566$	0	0
$567$	2.64136e26	1.83142
$568$	0	0
$569$	1.57702e26	1.05747	0.528737	−	0.848786i	$-0.322666\pi$
$569$	1.57702e26	1.05747	0.528737	+	0.848786i	$0.322666\pi$
$570$	0	0
$571$	−1.27956e26	−0.829882	−0.414941	−	0.909848i	$-0.636198\pi$
$571$	−1.27956e26	−0.829882	−0.414941	+	0.909848i	$0.636198\pi$
$572$	0	0
$573$	−3.46871e25	−0.217620
$574$	0	0
$575$	−1.55433e26	−0.943408
$576$	0	0
$577$	1.38761e26	0.814885	0.407442	−	0.913231i	$-0.366421\pi$
$577$	1.38761e26	0.814885	0.407442	+	0.913231i	$0.366421\pi$
$578$	0	0
$579$	−7.88506e25	−0.448083
$580$	0	0
$581$	−3.86140e26	−2.12359
$582$	0	0
$583$	2.66211e25	0.141702
$584$	0	0
$585$	1.72666e24	0.00889663
$586$	0	0
$587$	1.40537e26	0.701017	0.350509	−	0.936560i	$-0.386009\pi$
$587$	1.40537e26	0.701017	0.350509	+	0.936560i	$0.386009\pi$
$588$	0	0
$589$	−1.81064e26	−0.874454
$590$	0	0
$591$	−5.74338e25	−0.268588
$592$	0	0
$593$	−1.83090e26	−0.829171	−0.414585	−	0.910010i	$-0.636073\pi$
$593$	−1.83090e26	−0.829171	−0.414585	+	0.910010i	$0.636073\pi$
$594$	0	0
$595$	−1.37041e26	−0.601091
$596$	0	0
$597$	−2.10439e26	−0.894068
$598$	0	0
$599$	1.60928e26	0.662335	0.331167	−	0.943572i	$-0.392558\pi$
$599$	1.60928e26	0.662335	0.331167	+	0.943572i	$0.392558\pi$
$600$	0	0
$601$	−3.63963e26	−1.45127	−0.725637	−	0.688078i	$-0.758455\pi$
$601$	−3.63963e26	−1.45127	−0.725637	+	0.688078i	$0.758455\pi$
$602$	0	0
$603$	2.00457e24	0.00774472
$604$	0	0
$605$	−1.57142e26	−0.588323
$606$	0	0
$607$	5.19977e26	1.88665	0.943326	−	0.331869i	$-0.107679\pi$
$607$	5.19977e26	1.88665	0.943326	+	0.331869i	$0.107679\pi$
$608$	0	0
$609$	8.47264e26	2.97958
$610$	0	0
$611$	−1.82920e26	−0.623547
$612$	0	0
$613$	4.16714e26	1.37709	0.688547	−	0.725192i	$-0.258249\pi$
$613$	4.16714e26	1.37709	0.688547	+	0.725192i	$0.258249\pi$
$614$	0	0
$615$	−2.52126e26	−0.807801
$616$	0	0
$617$	2.46583e25	0.0766045	0.0383022	−	0.999266i	$-0.487805\pi$
$617$	2.46583e25	0.0766045	0.0383022	+	0.999266i	$0.487805\pi$
$618$	0	0
$619$	−6.19024e26	−1.86486	−0.932430	−	0.361351i	$-0.882316\pi$
$619$	−6.19024e26	−1.86486	−0.932430	+	0.361351i	$0.882316\pi$
$620$	0	0
$621$	4.96699e26	1.45119
$622$	0	0
$623$	6.12129e26	1.73463
$624$	0	0
$625$	2.29550e25	0.0630982
$626$	0	0
$627$	−2.63316e25	−0.0702158
$628$	0	0
$629$	−2.07734e26	−0.537436
$630$	0	0
$631$	−6.40082e26	−1.60678	−0.803390	−	0.595453i	$-0.796973\pi$
$631$	−6.40082e26	−1.60678	−0.803390	+	0.595453i	$0.796973\pi$
$632$	0	0
$633$	−3.02879e26	−0.737792
$634$	0	0
$635$	3.29526e26	0.779004
$636$	0	0
$637$	1.16685e27	2.67725
$638$	0	0
$639$	1.74180e24	0.00387917
$640$	0	0
$641$	−4.35543e26	−0.941629	−0.470815	−	0.882232i	$-0.656040\pi$
$641$	−4.35543e26	−0.941629	−0.470815	+	0.882232i	$0.656040\pi$
$642$	0	0
$643$	8.74903e25	0.183635	0.0918175	−	0.995776i	$-0.470732\pi$
$643$	8.74903e25	0.183635	0.0918175	+	0.995776i	$0.470732\pi$
$644$	0	0
$645$	−8.51494e24	−0.0173526
$646$	0	0
$647$	3.17146e26	0.627578	0.313789	−	0.949493i	$-0.398401\pi$
$647$	3.17146e26	0.627578	0.313789	+	0.949493i	$0.398401\pi$
$648$	0	0
$649$	−4.98846e25	−0.0958609
$650$	0	0
$651$	−1.30043e27	−2.42699
$652$	0	0
$653$	−1.61698e26	−0.293110	−0.146555	−	0.989203i	$-0.546818\pi$
$653$	−1.61698e26	−0.293110	−0.146555	+	0.989203i	$0.546818\pi$
$654$	0	0
$655$	3.35223e26	0.590257
$656$	0	0
$657$	8.35282e24	0.0142877
$658$	0	0
$659$	−1.45108e26	−0.241145	−0.120573	−	0.992705i	$-0.538473\pi$
$659$	−1.45108e26	−0.241145	−0.120573	+	0.992705i	$0.538473\pi$
$660$	0	0
$661$	−4.52310e26	−0.730336	−0.365168	−	0.930942i	$-0.618988\pi$
$661$	−4.52310e26	−0.730336	−0.365168	+	0.930942i	$0.618988\pi$
$662$	0	0
$663$	−4.22146e26	−0.662345
$664$	0	0
$665$	−4.62954e26	−0.705883
$666$	0	0
$667$	1.61347e27	2.39092
$668$	0	0
$669$	−5.86521e26	−0.844764
$670$	0	0
$671$	−1.29850e26	−0.181794
$672$	0	0
$673$	4.55615e26	0.620090	0.310045	−	0.950722i	$-0.399656\pi$
$673$	4.55615e26	0.620090	0.310045	+	0.950722i	$0.399656\pi$
$674$	0	0
$675$	−4.85006e26	−0.641743
$676$	0	0
$677$	−1.33349e27	−1.71553	−0.857766	−	0.514041i	$-0.828148\pi$
$677$	−1.33349e27	−1.71553	−0.857766	+	0.514041i	$0.828148\pi$
$678$	0	0
$679$	−3.09651e26	−0.387355
$680$	0	0
$681$	−8.58763e26	−1.04466
$682$	0	0
$683$	3.13697e26	0.371120	0.185560	−	0.982633i	$-0.440590\pi$
$683$	3.13697e26	0.371120	0.185560	+	0.982633i	$0.440590\pi$
$684$	0	0
$685$	7.41785e26	0.853527
$686$	0	0
$687$	1.12557e26	0.125974
$688$	0	0
$689$	1.44178e27	1.56970
$690$	0	0
$691$	1.07111e27	1.13446	0.567232	−	0.823558i	$-0.308014\pi$
$691$	1.07111e27	1.13446	0.567232	+	0.823558i	$0.308014\pi$
$692$	0	0
$693$	−2.36904e24	−0.00244122
$694$	0	0
$695$	−3.75101e25	−0.0376090
$696$	0	0
$697$	7.72175e26	0.753363
$698$	0	0
$699$	2.10304e27	1.99671
$700$	0	0
$701$	1.09656e27	1.01324	0.506620	−	0.862169i	$-0.330895\pi$
$701$	1.09656e27	1.01324	0.506620	+	0.862169i	$0.330895\pi$
$702$	0	0
$703$	−7.01771e26	−0.631131
$704$	0	0
$705$	3.61921e26	0.316823
$706$	0	0
$707$	−4.27044e26	−0.363905
$708$	0	0
$709$	−1.04320e27	−0.865421	−0.432710	−	0.901533i	$-0.642443\pi$
$709$	−1.04320e27	−0.865421	−0.432710	+	0.901533i	$0.642443\pi$
$710$	0	0
$711$	2.04869e25	0.0165469
$712$	0	0
$713$	−2.47644e27	−1.94750
$714$	0	0
$715$	9.74372e25	0.0746135
$716$	0	0
$717$	7.06650e26	0.526954
$718$	0	0
$719$	−9.53130e25	−0.0692193	−0.0346097	−	0.999401i	$-0.511019\pi$
$719$	−9.53130e25	−0.0692193	−0.0346097	+	0.999401i	$0.511019\pi$
$720$	0	0
$721$	1.81082e26	0.128083
$722$	0	0
$723$	−2.82678e26	−0.194751
$724$	0	0
$725$	−1.57548e27	−1.05731
$726$	0	0
$727$	−8.66622e26	−0.566569	−0.283284	−	0.959036i	$-0.591424\pi$
$727$	−8.66622e26	−0.566569	−0.283284	+	0.959036i	$0.591424\pi$
$728$	0	0
$729$	1.54962e27	0.986994
$730$	0	0
$731$	2.60783e25	0.0161832
$732$	0	0
$733$	1.89332e27	1.14481	0.572407	−	0.819969i	$-0.306010\pi$
$733$	1.89332e27	1.14481	0.572407	+	0.819969i	$0.306010\pi$
$734$	0	0
$735$	−2.30870e27	−1.36031
$736$	0	0
$737$	1.13120e26	0.0649528
$738$	0	0
$739$	−1.55052e27	−0.867673	−0.433837	−	0.900992i	$-0.642841\pi$
$739$	−1.55052e27	−0.867673	−0.433837	+	0.900992i	$0.642841\pi$
$740$	0	0
$741$	−1.42610e27	−0.777816
$742$	0	0
$743$	−3.71515e26	−0.197507	−0.0987534	−	0.995112i	$-0.531486\pi$
$743$	−3.71515e26	−0.197507	−0.0987534	+	0.995112i	$0.531486\pi$
$744$	0	0
$745$	1.18840e26	0.0615856
$746$	0	0
$747$	2.94815e25	0.0148938
$748$	0	0
$749$	7.86850e26	0.387539
$750$	0	0
$751$	−1.51500e27	−0.727501	−0.363751	−	0.931496i	$-0.618504\pi$
$751$	−1.51500e27	−0.727501	−0.363751	+	0.931496i	$0.618504\pi$
$752$	0	0
$753$	2.18968e27	1.02525
$754$	0	0
$755$	1.67392e27	0.764257
$756$	0	0
$757$	−5.53052e26	−0.246238	−0.123119	−	0.992392i	$-0.539290\pi$
$757$	−5.53052e26	−0.246238	−0.123119	+	0.992392i	$0.539290\pi$
$758$	0	0
$759$	−3.60141e26	−0.156378
$760$	0	0
$761$	−2.03558e27	−0.862054	−0.431027	−	0.902339i	$-0.641849\pi$
$761$	−2.03558e27	−0.862054	−0.431027	+	0.902339i	$0.641849\pi$
$762$	0	0
$763$	−9.99780e26	−0.412973
$764$	0	0
$765$	1.04630e25	0.00421573
$766$	0	0
$767$	−2.70171e27	−1.06190
$768$	0	0
$769$	2.79338e27	1.07110	0.535550	−	0.844503i	$-0.320104\pi$
$769$	2.79338e27	1.07110	0.535550	+	0.844503i	$0.320104\pi$
$770$	0	0
$771$	2.76724e27	1.03521
$772$	0	0
$773$	4.29473e27	1.56758	0.783791	−	0.621025i	$-0.213283\pi$
$773$	4.29473e27	1.56758	0.783791	+	0.621025i	$0.213283\pi$
$774$	0	0
$775$	2.41814e27	0.861223
$776$	0	0
$777$	−5.04023e27	−1.75166
$778$	0	0
$779$	2.60857e27	0.884702
$780$	0	0
$781$	9.82914e25	0.0325335
$782$	0	0
$783$	5.03459e27	1.62640
$784$	0	0
$785$	2.42434e27	0.764419
$786$	0	0
$787$	2.18659e27	0.672987	0.336494	−	0.941686i	$-0.390759\pi$
$787$	2.18659e27	0.672987	0.336494	+	0.941686i	$0.390759\pi$
$788$	0	0
$789$	−1.78396e27	−0.535987
$790$	0	0
$791$	3.78758e27	1.11093
$792$	0	0
$793$	−7.03261e27	−2.01382
$794$	0	0
$795$	−2.85268e27	−0.797563
$796$	0	0
$797$	1.65866e27	0.452796	0.226398	−	0.974035i	$-0.427305\pi$
$797$	1.65866e27	0.452796	0.226398	+	0.974035i	$0.427305\pi$
$798$	0	0
$799$	−1.10844e27	−0.295472
$800$	0	0
$801$	−4.67357e25	−0.0121658
$802$	0	0
$803$	4.71359e26	0.119827
$804$	0	0
$805$	−6.33191e27	−1.57208
$806$	0	0
$807$	−1.49066e27	−0.361475
$808$	0	0
$809$	−2.56250e27	−0.606951	−0.303475	−	0.952839i	$-0.598147\pi$
$809$	−2.56250e27	−0.606951	−0.303475	+	0.952839i	$0.598147\pi$
$810$	0	0
$811$	−4.85027e27	−1.12219	−0.561096	−	0.827751i	$-0.689620\pi$
$811$	−4.85027e27	−1.12219	−0.561096	+	0.827751i	$0.689620\pi$
$812$	0	0
$813$	6.96552e27	1.57432
$814$	0	0
$815$	−1.17017e27	−0.258376
$816$	0	0
$817$	8.80981e25	0.0190045
$818$	0	0
$819$	−1.28306e26	−0.0270426
$820$	0	0
$821$	2.50335e26	0.0515540	0.0257770	−	0.999668i	$-0.491794\pi$
$821$	2.50335e26	0.0515540	0.0257770	+	0.999668i	$0.491794\pi$
$822$	0	0
$823$	−5.35775e27	−1.07816	−0.539081	−	0.842254i	$-0.681228\pi$
$823$	−5.35775e27	−1.07816	−0.539081	+	0.842254i	$0.681228\pi$
$824$	0	0
$825$	3.51663e26	0.0691534
$826$	0	0
$827$	−8.46014e27	−1.62583	−0.812915	−	0.582382i	$-0.802121\pi$
$827$	−8.46014e27	−1.62583	−0.812915	+	0.582382i	$0.802121\pi$
$828$	0	0
$829$	−6.64078e27	−1.24724	−0.623621	−	0.781727i	$-0.714339\pi$
$829$	−6.64078e27	−1.24724	−0.623621	+	0.781727i	$0.714339\pi$
$830$	0	0
$831$	6.31969e27	1.16007
$832$	0	0
$833$	7.07074e27	1.26864
$834$	0	0
$835$	4.76343e27	0.835407
$836$	0	0
$837$	−7.72737e27	−1.32477
$838$	0	0
$839$	8.48056e27	1.42130	0.710650	−	0.703546i	$-0.248401\pi$
$839$	8.48056e27	1.42130	0.710650	+	0.703546i	$0.248401\pi$
$840$	0	0
$841$	1.02510e28	1.67959
$842$	0	0
$843$	−5.48052e27	−0.877929
$844$	0	0
$845$	1.47787e27	0.231472
$846$	0	0
$847$	1.16770e28	1.78829
$848$	0	0
$849$	2.02658e27	0.303487
$850$	0	0
$851$	−9.59824e27	−1.40560
$852$	0	0
$853$	−7.67513e27	−1.09918	−0.549591	−	0.835434i	$-0.685217\pi$
$853$	−7.67513e27	−1.09918	−0.549591	+	0.835434i	$0.685217\pi$
$854$	0	0
$855$	3.53463e25	0.00495069
$856$	0	0
$857$	1.77524e27	0.243187	0.121593	−	0.992580i	$-0.461200\pi$
$857$	1.77524e27	0.243187	0.121593	+	0.992580i	$0.461200\pi$
$858$	0	0
$859$	−8.09486e27	−1.08461	−0.542306	−	0.840181i	$-0.682449\pi$
$859$	−8.09486e27	−1.08461	−0.542306	+	0.840181i	$0.682449\pi$
$860$	0	0
$861$	1.87352e28	2.45543
$862$	0	0
$863$	−1.31037e28	−1.67993	−0.839963	−	0.542644i	$-0.817423\pi$
$863$	−1.31037e28	−1.67993	−0.839963	+	0.542644i	$0.817423\pi$
$864$	0	0
$865$	2.02011e27	0.253351
$866$	0	0
$867$	5.64391e27	0.692466
$868$	0	0
$869$	1.15610e27	0.138774
$870$	0	0
$871$	6.12649e27	0.719514
$872$	0	0
$873$	2.36416e25	0.00271670
$874$	0	0
$875$	1.57552e28	1.77153
$876$	0	0
$877$	−8.69739e26	−0.0956958	−0.0478479	−	0.998855i	$-0.515236\pi$
$877$	−8.69739e26	−0.0956958	−0.0478479	+	0.998855i	$0.515236\pi$
$878$	0	0
$879$	−3.33864e27	−0.359481
$880$	0	0
$881$	9.28888e27	0.978796	0.489398	−	0.872061i	$-0.337217\pi$
$881$	9.28888e27	0.978796	0.489398	+	0.872061i	$0.337217\pi$
$882$	0	0
$883$	−9.02161e27	−0.930374	−0.465187	−	0.885213i	$-0.654013\pi$
$883$	−9.02161e27	−0.930374	−0.465187	+	0.885213i	$0.654013\pi$
$884$	0	0
$885$	5.34555e27	0.539549
$886$	0	0
$887$	6.25902e27	0.618346	0.309173	−	0.951006i	$-0.399948\pi$
$887$	6.25902e27	0.618346	0.309173	+	0.951006i	$0.399948\pi$
$888$	0	0
$889$	−2.44866e28	−2.36790
$890$	0	0
$891$	−1.13802e27	−0.107724
$892$	0	0
$893$	−3.74454e27	−0.346984
$894$	0	0
$895$	−5.87787e27	−0.533213
$896$	0	0
$897$	−1.95050e28	−1.73228
$898$	0	0
$899$	−2.51014e28	−2.18264
$900$	0	0
$901$	8.73675e27	0.743814
$902$	0	0
$903$	6.32734e26	0.0527457
$904$	0	0
$905$	−1.61340e27	−0.131698
$906$	0	0
$907$	1.07075e28	0.855891	0.427946	−	0.903804i	$-0.359237\pi$
$907$	1.07075e28	0.855891	0.427946	+	0.903804i	$0.359237\pi$
$908$	0	0
$909$	3.26046e25	0.00255224
$910$	0	0
$911$	4.49055e27	0.344251	0.172125	−	0.985075i	$-0.444937\pi$
$911$	4.49055e27	0.344251	0.172125	+	0.985075i	$0.444937\pi$
$912$	0	0
$913$	1.66367e27	0.124910
$914$	0	0
$915$	1.39146e28	1.02322
$916$	0	0
$917$	−2.49100e28	−1.79417
$918$	0	0
$919$	1.03334e27	0.0729029	0.0364514	−	0.999335i	$-0.488395\pi$
$919$	1.03334e27	0.0729029	0.0364514	+	0.999335i	$0.488395\pi$
$920$	0	0
$921$	−1.66720e28	−1.15218
$922$	0	0
$923$	5.32340e27	0.360390
$924$	0	0
$925$	9.37228e27	0.621582
$926$	0	0
$927$	−1.38255e25	−0.000898306 0
$928$	0	0
$929$	2.40721e28	1.53237	0.766186	−	0.642619i	$-0.222152\pi$
$929$	2.40721e28	1.53237	0.766186	+	0.642619i	$0.222152\pi$
$930$	0	0
$931$	2.38865e28	1.48981
$932$	0	0
$933$	−1.60120e28	−0.978523
$934$	0	0
$935$	5.90439e26	0.0353561
$936$	0	0
$937$	9.20092e27	0.539890	0.269945	−	0.962876i	$-0.412995\pi$
$937$	9.20092e27	0.539890	0.269945	+	0.962876i	$0.412995\pi$
$938$	0	0
$939$	1.29949e28	0.747221
$940$	0	0
$941$	4.25377e27	0.239703	0.119851	−	0.992792i	$-0.461758\pi$
$941$	4.25377e27	0.239703	0.119851	+	0.992792i	$0.461758\pi$
$942$	0	0
$943$	3.56779e28	1.97032
$944$	0	0
$945$	−1.97578e28	−1.06939
$946$	0	0
$947$	−3.12257e28	−1.65649	−0.828243	−	0.560369i	$-0.810659\pi$
$947$	−3.12257e28	−1.65649	−0.828243	+	0.560369i	$0.810659\pi$
$948$	0	0
$949$	2.55285e28	1.32738
$950$	0	0
$951$	−2.08995e28	−1.06517
$952$	0	0
$953$	3.43857e28	1.71789	0.858945	−	0.512068i	$-0.171120\pi$
$953$	3.43857e28	1.71789	0.858945	+	0.512068i	$0.171120\pi$
$954$	0	0
$955$	2.62757e27	0.128683
$956$	0	0
$957$	−3.65042e27	−0.175259
$958$	0	0
$959$	−5.51210e28	−2.59442
$960$	0	0
$961$	1.68564e28	0.777845
$962$	0	0
$963$	−6.00755e25	−0.00271799
$964$	0	0
$965$	5.97297e27	0.264961
$966$	0	0
$967$	−1.49897e28	−0.651990	−0.325995	−	0.945371i	$-0.605699\pi$
$967$	−1.49897e28	−0.651990	−0.325995	+	0.945371i	$0.605699\pi$
$968$	0	0
$969$	−8.64172e27	−0.368574
$970$	0	0
$971$	4.03020e28	1.68556	0.842780	−	0.538258i	$-0.180917\pi$
$971$	4.03020e28	1.68556	0.842780	+	0.538258i	$0.180917\pi$
$972$	0	0
$973$	2.78732e27	0.114318
$974$	0	0
$975$	1.90458e28	0.766047
$976$	0	0
$977$	−2.15857e28	−0.851468	−0.425734	−	0.904848i	$-0.639984\pi$
$977$	−2.15857e28	−0.851468	−0.425734	+	0.904848i	$0.639984\pi$
$978$	0	0
$979$	−2.63735e27	−0.102031
$980$	0	0
$981$	7.63326e25	0.00289637
$982$	0	0
$983$	−3.69372e28	−1.37469	−0.687346	−	0.726330i	$-0.741225\pi$
$983$	−3.69372e28	−1.37469	−0.687346	+	0.726330i	$0.741225\pi$
$984$	0	0
$985$	4.35064e27	0.158821
$986$	0	0
$987$	−2.68939e28	−0.963031
$988$	0	0
$989$	1.20493e27	0.0423251
$990$	0	0
$991$	−9.20845e27	−0.317312	−0.158656	−	0.987334i	$-0.550716\pi$
$991$	−9.20845e27	−0.317312	−0.158656	+	0.987334i	$0.550716\pi$
$992$	0	0
$993$	8.69551e26	0.0293953
$994$	0	0
$995$	1.59409e28	0.528680
$996$	0	0
$997$	−3.39300e28	−1.10403	−0.552013	−	0.833835i	$-0.686140\pi$
$997$	−3.39300e28	−1.10403	−0.552013	+	0.833835i	$0.686140\pi$
$998$	0	0
$999$	−2.99499e28	−0.956143

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.20.a.b.1.1	✓	3
3.2	odd	2		72.20.a.f.1.2		3
4.3	odd	2		16.20.a.f.1.3		3
8.3	odd	2		64.20.a.m.1.1		3
8.5	even	2		64.20.a.l.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.20.a.b.1.1	✓	3	1.1	even	1	trivial
16.20.a.f.1.3		3	4.3	odd	2
64.20.a.l.1.3		3	8.5	even	2
64.20.a.m.1.1		3	8.3	odd	2
72.20.a.f.1.2		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	8.a (trivial)

\(f(q)\)	\(=\)	\(q-34307.5 q^{3} +2.59881e6 q^{5} -1.93114e8 q^{7} +1.47441e7 q^{9} +O(q^{10})\)	\(q-34307.5 q^{3} +2.59881e6 q^{5} -1.93114e8 q^{7} +1.47441e7 q^{9} +8.32029e8 q^{11} +4.50621e10 q^{13} -8.91588e10 q^{15} +2.73062e11 q^{17} +9.22463e11 q^{19} +6.62527e12 q^{21} +1.26167e13 q^{23} -1.23197e13 q^{25} +3.93685e13 q^{27} +1.27884e14 q^{29} -1.96283e14 q^{31} -2.85448e13 q^{33} -5.01868e14 q^{35} -7.60758e14 q^{37} -1.54597e15 q^{39} +2.82783e15 q^{41} +9.55031e13 q^{43} +3.83173e13 q^{45} -4.05928e15 q^{47} +2.58942e16 q^{49} -9.36809e15 q^{51} +3.19955e16 q^{53} +2.16229e15 q^{55} -3.16474e16 q^{57} -5.99553e16 q^{59} -1.56065e17 q^{61} -2.84731e15 q^{63} +1.17108e17 q^{65} +1.35957e17 q^{67} -4.32847e17 q^{69} +1.18135e17 q^{71} +5.66518e17 q^{73} +4.22657e17 q^{75} -1.60677e17 q^{77} +1.38949e18 q^{79} -1.36777e18 q^{81} +1.99954e18 q^{83} +7.09638e17 q^{85} -4.38737e18 q^{87} -3.16978e18 q^{89} -8.70213e18 q^{91} +6.73399e18 q^{93} +2.39731e18 q^{95} +1.60346e18 q^{97} +1.22676e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 23732 q^{3} + 2140218 q^{5} + 55851720 q^{7} + 646753951 q^{9}+O(q^{10}) \) 3 * q + 23732 * q^3 + 2140218 * q^5 + 55851720 * q^7 + 646753951 * q^9	\( 3 q + 23732 q^{3} + 2140218 q^{5} + 55851720 q^{7} + 646753951 q^{9} - 297392964 q^{11} - 14862401022 q^{13} + 292635653528 q^{15} + 803332464534 q^{17} + 3212269666884 q^{19} + 11192319829728 q^{21} + 24948509305560 q^{23} + 72340360289109 q^{25} + 64092343553864 q^{27} + 77667139511058 q^{29} - 248431735193568 q^{31} - 252071696774128 q^{33} - 13\!\cdots\!08 q^{35}+ \cdots - 11\!\cdots\!60 q^{99}+O(q^{100}) \) 3 * q + 23732 * q^3 + 2140218 * q^5 + 55851720 * q^7 + 646753951 * q^9 - 297392964 * q^11 - 14862401022 * q^13 + 292635653528 * q^15 + 803332464534 * q^17 + 3212269666884 * q^19 + 11192319829728 * q^21 + 24948509305560 * q^23 + 72340360289109 * q^25 + 64092343553864 * q^27 + 77667139511058 * q^29 - 248431735193568 * q^31 - 252071696774128 * q^33 - 1384378114232208 * q^35 - 414866302559142 * q^37 - 5522045976027016 * q^39 + 2818737880869678 * q^41 - 6663230715469860 * q^43 + 22824972038174722 * q^45 + 1500497644728624 * q^47 + 39641123057321787 * q^49 - 1085042177122520 * q^51 + 56067344774978154 * q^53 - 56817855560205432 * q^55 - 6408983732955152 * q^57 - 154317270851496852 * q^59 - 134994376571654862 * q^61 - 78435010097874072 * q^63 - 563892790723886724 * q^65 + 151032181904450292 * q^67 - 153590366326625632 * q^69 + 1210541848845584136 * q^71 - 81876123599662770 * q^73 + 2698278154129173484 * q^75 + 97235023193490336 * q^77 + 1439028483035907408 * q^79 - 318667537468381733 * q^81 - 983438102798849916 * q^83 - 1618537843962618540 * q^85 - 4280411824494387144 * q^87 - 1312857528832070946 * q^89 - 11488143382330124496 * q^91 - 4308257105281185920 * q^93 - 10926153348157720968 * q^95 + 14033245412567998566 * q^97 - 11415749644087184660 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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