LMFDB - Embedded newform 104.10.a.d.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [104,10,Mod(1,104)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(104, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("104.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 104 = 2^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	104.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,141] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$53.5637269610$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3 x^{7} - 135356 x^{6} - 24398 x^{5} + 5213582205 x^{4} + 598076469 x^{3} + \cdots + 15\!\cdots\!64 $ x^8 - 3x^7 - 135356x^6 - 24398x^5 + 5213582205x^4 + 598076469x^3 - 44131402568826x^2 + 60107523849756*x + 1558753465556664
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{25}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$6.68300$ of defining polynomial
Character	$\chi$	$=$	104.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+11.3170 q^{3} +2354.21 q^{5} -4597.91 q^{7} -19554.9 q^{9} +32973.7 q^{11} +28561.0 q^{13} +26642.6 q^{15} +429836. q^{17} +823596. q^{19} -52034.5 q^{21} -1.28007e6 q^{23} +3.58918e6 q^{25} -444056. q^{27} -3.75530e6 q^{29} -9.62778e6 q^{31} +373164. q^{33} -1.08244e7 q^{35} +1.56238e7 q^{37} +323225. q^{39} +2.86870e7 q^{41} +2.19320e7 q^{43} -4.60364e7 q^{45} +4.95057e7 q^{47} -1.92128e7 q^{49} +4.86446e6 q^{51} +9.82331e7 q^{53} +7.76271e7 q^{55} +9.32064e6 q^{57} -2.59979e7 q^{59} -8.44671e7 q^{61} +8.99117e7 q^{63} +6.72386e7 q^{65} -1.61951e8 q^{67} -1.44866e7 q^{69} +3.46667e8 q^{71} +2.95177e7 q^{73} +4.06188e7 q^{75} -1.51610e8 q^{77} -1.92628e8 q^{79} +3.79874e8 q^{81} +2.36393e8 q^{83} +1.01193e9 q^{85} -4.24987e7 q^{87} +1.08335e9 q^{89} -1.31321e8 q^{91} -1.08958e8 q^{93} +1.93892e9 q^{95} +5.84771e8 q^{97} -6.44799e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 141 q^{3} + 2051 q^{5} - 2417 q^{7} + 115741 q^{9} - 53118 q^{11} + 228488 q^{13} - 464555 q^{15} + 433095 q^{17} - 434954 q^{19} + 906875 q^{21} - 1124296 q^{23} + 5966065 q^{25} + 7820643 q^{27}+ \cdots - 641626736 q^{99}+O(q^{100}) $ 8 * q + 141 * q^3 + 2051 * q^5 - 2417 * q^7 + 115741 * q^9 - 53118 * q^11 + 228488 * q^13 - 464555 * q^15 + 433095 * q^17 - 434954 * q^19 + 906875 * q^21 - 1124296 * q^23 + 5966065 * q^25 + 7820643 * q^27 - 1913432 * q^29 + 5220148 * q^31 - 1626882 * q^33 + 950707 * q^35 - 419409 * q^37 + 4027101 * q^39 + 12148046 * q^41 + 32143187 * q^43 + 18893544 * q^45 + 57898411 * q^47 + 104302637 * q^49 + 198521477 * q^51 + 273106866 * q^53 + 141214662 * q^55 + 253032058 * q^57 + 50734502 * q^59 + 139520018 * q^61 - 20344158 * q^63 + 58578611 * q^65 + 528144786 * q^67 + 483506576 * q^69 + 334814973 * q^71 + 630864344 * q^73 + 2265796264 * q^75 + 939664146 * q^77 + 865037608 * q^79 + 3194281040 * q^81 + 1520229952 * q^83 + 2130158385 * q^85 + 1167867006 * q^87 + 894700240 * q^89 - 69031937 * q^91 + 3803209548 * q^93 - 2695926542 * q^95 + 980232164 * q^97 - 641626736 * q^99

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
$3$	11.3170	0.0806651	0.0403326	−	0.999186i	$-0.487158\pi$
$3$	11.3170	0.0806651	0.0403326	+	0.999186i	$0.487158\pi$
$4$	0	0
$5$	2354.21	1.68454	0.842268	−	0.539059i	$-0.181220\pi$
$5$	2354.21	1.68454	0.842268	+	0.539059i	$0.181220\pi$
$6$	0	0
$7$	−4597.91	−0.723801	−0.361900	−	0.932217i	$-0.617872\pi$
$7$	−4597.91	−0.723801	−0.361900	+	0.932217i	$0.617872\pi$
$8$	0	0
$9$	−19554.9	−0.993493
$10$	0	0
$11$	32973.7	0.679049	0.339525	−	0.940597i	$-0.389734\pi$
$11$	32973.7	0.679049	0.339525	+	0.940597i	$0.389734\pi$
$12$	0	0
$13$	28561.0	0.277350
$13$	28561.0	0.277350
$14$	0	0
$15$	26642.6	0.135883
$16$	0	0
$17$	429836.	1.24820	0.624098	−	0.781346i	$-0.285466\pi$
$17$	429836.	1.24820	0.624098	+	0.781346i	$0.285466\pi$
$18$	0	0
$19$	823596.	1.44985	0.724925	−	0.688828i	$-0.241874\pi$
$19$	823596.	1.44985	0.724925	+	0.688828i	$0.241874\pi$
$20$	0	0
$21$	−52034.5	−0.0583855
$22$	0	0
$23$	−1.28007e6	−0.953806	−0.476903	−	0.878956i	$-0.658241\pi$
$23$	−1.28007e6	−0.953806	−0.476903	+	0.878956i	$0.658241\pi$
$24$	0	0
$25$	3.58918e6	1.83766
$26$	0	0
$27$	−444056.	−0.160805
$28$	0	0
$29$	−3.75530e6	−0.985946	−0.492973	−	0.870045i	$-0.664090\pi$
$29$	−3.75530e6	−0.985946	−0.492973	+	0.870045i	$0.664090\pi$
$30$	0	0
$31$	−9.62778e6	−1.87240	−0.936200	−	0.351468i	$-0.885682\pi$
$31$	−9.62778e6	−1.87240	−0.936200	+	0.351468i	$0.885682\pi$
$32$	0	0
$33$	373164.	0.0547756
$34$	0	0
$35$	−1.08244e7	−1.21927
$36$	0	0
$37$	1.56238e7	1.37050	0.685249	−	0.728308i	$-0.259693\pi$
$37$	1.56238e7	1.37050	0.685249	+	0.728308i	$0.259693\pi$
$38$	0	0
$39$	323225.	0.0223725
$40$	0	0
$41$	2.86870e7	1.58547	0.792735	−	0.609567i	$-0.208657\pi$
$41$	2.86870e7	1.58547	0.792735	+	0.609567i	$0.208657\pi$
$42$	0	0
$43$	2.19320e7	0.978297	0.489148	−	0.872201i	$-0.337308\pi$
$43$	2.19320e7	0.978297	0.489148	+	0.872201i	$0.337308\pi$
$44$	0	0
$45$	−4.60364e7	−1.67357
$46$	0	0
$47$	4.95057e7	1.47984	0.739921	−	0.672694i	$-0.234863\pi$
$47$	4.95057e7	1.47984	0.739921	+	0.672694i	$0.234863\pi$
$48$	0	0
$49$	−1.92128e7	−0.476112
$50$	0	0
$51$	4.86446e6	0.100686
$52$	0	0
$53$	9.82331e7	1.71008	0.855040	−	0.518562i	$-0.173532\pi$
$53$	9.82331e7	1.71008	0.855040	+	0.518562i	$0.173532\pi$
$54$	0	0
$55$	7.76271e7	1.14388
$56$	0	0
$57$	9.32064e6	0.116952
$58$	0	0
$59$	−2.59979e7	−0.279321	−0.139661	−	0.990199i	$-0.544601\pi$
$59$	−2.59979e7	−0.279321	−0.139661	+	0.990199i	$0.544601\pi$
$60$	0	0
$61$	−8.44671e7	−0.781094	−0.390547	−	0.920583i	$-0.627714\pi$
$61$	−8.44671e7	−0.781094	−0.390547	+	0.920583i	$0.627714\pi$
$62$	0	0
$63$	8.99117e7	0.719091
$64$	0	0
$65$	6.72386e7	0.467206
$66$	0	0
$67$	−1.61951e8	−0.981852	−0.490926	−	0.871201i	$-0.663342\pi$
$67$	−1.61951e8	−0.981852	−0.490926	+	0.871201i	$0.663342\pi$
$68$	0	0
$69$	−1.44866e7	−0.0769389
$70$	0	0
$71$	3.46667e8	1.61901	0.809505	−	0.587113i	$-0.199736\pi$
$71$	3.46667e8	1.61901	0.809505	+	0.587113i	$0.199736\pi$
$72$	0	0
$73$	2.95177e7	0.121655	0.0608276	−	0.998148i	$-0.480626\pi$
$73$	2.95177e7	0.121655	0.0608276	+	0.998148i	$0.480626\pi$
$74$	0	0
$75$	4.06188e7	0.148235
$76$	0	0
$77$	−1.51610e8	−0.491496
$78$	0	0
$79$	−1.92628e8	−0.556412	−0.278206	−	0.960521i	$-0.589740\pi$
$79$	−1.92628e8	−0.556412	−0.278206	+	0.960521i	$0.589740\pi$
$80$	0	0
$81$	3.79874e8	0.980522
$82$	0	0
$83$	2.36393e8	0.546744	0.273372	−	0.961908i	$-0.411861\pi$
$83$	2.36393e8	0.546744	0.273372	+	0.961908i	$0.411861\pi$
$84$	0	0
$85$	1.01193e9	2.10263
$86$	0	0
$87$	−4.24987e7	−0.0795314
$88$	0	0
$89$	1.08335e9	1.83026	0.915129	−	0.403162i	$-0.132089\pi$
$89$	1.08335e9	1.83026	0.915129	+	0.403162i	$0.132089\pi$
$90$	0	0
$91$	−1.31321e8	−0.200746
$92$	0	0
$93$	−1.08958e8	−0.151037
$94$	0	0
$95$	1.93892e9	2.44233
$96$	0	0
$97$	5.84771e8	0.670677	0.335338	−	0.942098i	$-0.391149\pi$
$97$	5.84771e8	0.670677	0.335338	+	0.942098i	$0.391149\pi$
$98$	0	0
$99$	−6.44799e8	−0.674631
$100$	0	0
$101$	−5.17234e8	−0.494585	−0.247292	−	0.968941i	$-0.579541\pi$
$101$	−5.17234e8	−0.494585	−0.247292	+	0.968941i	$0.579541\pi$
$102$	0	0
$103$	2.43025e8	0.212757	0.106378	−	0.994326i	$-0.466075\pi$
$103$	2.43025e8	0.212757	0.106378	+	0.994326i	$0.466075\pi$
$104$	0	0
$105$	−1.22500e8	−0.0983524
$106$	0	0
$107$	−1.92138e9	−1.41705	−0.708526	−	0.705684i	$-0.750640\pi$
$107$	−1.92138e9	−1.41705	−0.708526	+	0.705684i	$0.750640\pi$
$108$	0	0
$109$	9.22262e8	0.625799	0.312900	−	0.949786i	$-0.398700\pi$
$109$	9.22262e8	0.625799	0.312900	+	0.949786i	$0.398700\pi$
$110$	0	0
$111$	1.76814e8	0.110551
$112$	0	0
$113$	1.35214e9	0.780133	0.390066	−	0.920787i	$-0.372452\pi$
$113$	1.35214e9	0.780133	0.390066	+	0.920787i	$0.372452\pi$
$114$	0	0
$115$	−3.01357e9	−1.60672
$116$	0	0
$117$	−5.58508e8	−0.275545
$118$	0	0
$119$	−1.97635e9	−0.903446
$120$	0	0
$121$	−1.27068e9	−0.538892
$122$	0	0
$123$	3.24651e8	0.127892
$124$	0	0
$125$	3.85163e9	1.41107
$126$	0	0
$127$	−1.38052e9	−0.470898	−0.235449	−	0.971887i	$-0.575656\pi$
$127$	−1.38052e9	−0.470898	−0.235449	+	0.971887i	$0.575656\pi$
$128$	0	0
$129$	2.48205e8	0.0789144
$130$	0	0
$131$	5.66550e9	1.68081	0.840403	−	0.541963i	$-0.182319\pi$
$131$	5.66550e9	1.68081	0.840403	+	0.541963i	$0.182319\pi$
$132$	0	0
$133$	−3.78682e9	−1.04940
$134$	0	0
$135$	−1.04540e9	−0.270882
$136$	0	0
$137$	5.86061e9	1.42135	0.710674	−	0.703521i	$-0.248390\pi$
$137$	5.86061e9	1.42135	0.710674	+	0.703521i	$0.248390\pi$
$138$	0	0
$139$	−3.84126e9	−0.872784	−0.436392	−	0.899757i	$-0.643744\pi$
$139$	−3.84126e9	−0.872784	−0.436392	+	0.899757i	$0.643744\pi$
$140$	0	0
$141$	5.60256e8	0.119372
$142$	0	0
$143$	9.41763e8	0.188334
$144$	0	0
$145$	−8.84076e9	−1.66086
$146$	0	0
$147$	−2.17432e8	−0.0384057
$148$	0	0
$149$	3.50642e9	0.582809	0.291404	−	0.956600i	$-0.405877\pi$
$149$	3.50642e9	0.582809	0.291404	+	0.956600i	$0.405877\pi$
$150$	0	0
$151$	2.70489e9	0.423403	0.211701	−	0.977334i	$-0.432100\pi$
$151$	2.70489e9	0.423403	0.211701	+	0.977334i	$0.432100\pi$
$152$	0	0
$153$	−8.40542e9	−1.24007
$154$	0	0
$155$	−2.26658e10	−3.15412
$156$	0	0
$157$	3.50966e9	0.461017	0.230509	−	0.973070i	$-0.425961\pi$
$157$	3.50966e9	0.461017	0.230509	+	0.973070i	$0.425961\pi$
$158$	0	0
$159$	1.11170e9	0.137944
$160$	0	0
$161$	5.88567e9	0.690366
$162$	0	0
$163$	−1.32974e8	−0.0147545	−0.00737723	−	0.999973i	$-0.502348\pi$
$163$	−1.32974e8	−0.0147545	−0.00737723	+	0.999973i	$0.502348\pi$
$164$	0	0
$165$	8.78506e8	0.0922714
$166$	0	0
$167$	3.47780e9	0.346003	0.173002	−	0.984922i	$-0.444653\pi$
$167$	3.47780e9	0.346003	0.173002	+	0.984922i	$0.444653\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	−1.61054e10	−1.44042
$172$	0	0
$173$	−7.02978e9	−0.596670	−0.298335	−	0.954461i	$-0.596431\pi$
$173$	−7.02978e9	−0.596670	−0.298335	+	0.954461i	$0.596431\pi$
$174$	0	0
$175$	−1.65027e10	−1.33010
$176$	0	0
$177$	−2.94218e8	−0.0225315
$178$	0	0
$179$	−1.86437e10	−1.35736	−0.678678	−	0.734436i	$-0.737447\pi$
$179$	−1.86437e10	−1.35736	−0.678678	+	0.734436i	$0.737447\pi$
$180$	0	0
$181$	−3.40723e9	−0.235965	−0.117983	−	0.993016i	$-0.537643\pi$
$181$	−3.40723e9	−0.235965	−0.117983	+	0.993016i	$0.537643\pi$
$182$	0	0
$183$	−9.55914e8	−0.0630070
$184$	0	0
$185$	3.67817e10	2.30865
$186$	0	0
$187$	1.41733e10	0.847587
$188$	0	0
$189$	2.04173e9	0.116391
$190$	0	0
$191$	−2.23903e10	−1.21733	−0.608666	−	0.793427i	$-0.708295\pi$
$191$	−2.23903e10	−1.21733	−0.608666	+	0.793427i	$0.708295\pi$
$192$	0	0
$193$	−1.22087e10	−0.633376	−0.316688	−	0.948530i	$-0.602571\pi$
$193$	−1.22087e10	−0.633376	−0.316688	+	0.948530i	$0.602571\pi$
$194$	0	0
$195$	7.60939e8	0.0376872
$196$	0	0
$197$	−2.49114e10	−1.17842	−0.589211	−	0.807979i	$-0.700561\pi$
$197$	−2.49114e10	−1.17842	−0.589211	+	0.807979i	$0.700561\pi$
$198$	0	0
$199$	−3.21750e10	−1.45439	−0.727194	−	0.686432i	$-0.759176\pi$
$199$	−3.21750e10	−1.45439	−0.727194	+	0.686432i	$0.759176\pi$
$200$	0	0
$201$	−1.83280e9	−0.0792012
$202$	0	0
$203$	1.72665e10	0.713628
$204$	0	0
$205$	6.75353e10	2.67078
$206$	0	0
$207$	2.50318e10	0.947600
$208$	0	0
$209$	2.71571e10	0.984520
$210$	0	0
$211$	−4.16591e10	−1.44690	−0.723451	−	0.690376i	$-0.757445\pi$
$211$	−4.16591e10	−1.44690	−0.723451	+	0.690376i	$0.757445\pi$
$212$	0	0
$213$	3.92323e9	0.130598
$214$	0	0
$215$	5.16326e10	1.64798
$216$	0	0
$217$	4.42676e10	1.35524
$218$	0	0
$219$	3.34052e8	0.00981332
$220$	0	0
$221$	1.22766e10	0.346187
$222$	0	0
$223$	−4.98678e10	−1.35036	−0.675178	−	0.737655i	$-0.735933\pi$
$223$	−4.98678e10	−1.35036	−0.675178	+	0.737655i	$0.735933\pi$
$224$	0	0
$225$	−7.01862e10	−1.82570
$226$	0	0
$227$	5.43841e10	1.35943	0.679713	−	0.733478i	$-0.262104\pi$
$227$	5.43841e10	1.35943	0.679713	+	0.733478i	$0.262104\pi$
$228$	0	0
$229$	2.27487e10	0.546634	0.273317	−	0.961924i	$-0.411879\pi$
$229$	2.27487e10	0.546634	0.273317	+	0.961924i	$0.411879\pi$
$230$	0	0
$231$	−1.71577e9	−0.0396466
$232$	0	0
$233$	−7.65100e10	−1.70066	−0.850328	−	0.526253i	$-0.823597\pi$
$233$	−7.65100e10	−1.70066	−0.850328	+	0.526253i	$0.823597\pi$
$234$	0	0
$235$	1.16547e11	2.49285
$236$	0	0
$237$	−2.17997e9	−0.0448830
$238$	0	0
$239$	7.77408e9	0.154120	0.0770599	−	0.997026i	$-0.475447\pi$
$239$	7.77408e9	0.154120	0.0770599	+	0.997026i	$0.475447\pi$
$240$	0	0
$241$	4.56052e9	0.0870838	0.0435419	−	0.999052i	$-0.486136\pi$
$241$	4.56052e9	0.0870838	0.0435419	+	0.999052i	$0.486136\pi$
$242$	0	0
$243$	1.30394e10	0.239899
$244$	0	0
$245$	−4.52311e10	−0.802028
$246$	0	0
$247$	2.35227e10	0.402116
$248$	0	0
$249$	2.67526e9	0.0441032
$250$	0	0
$251$	3.43672e10	0.546527	0.273264	−	0.961939i	$-0.411897\pi$
$251$	3.43672e10	0.546527	0.273264	+	0.961939i	$0.411897\pi$
$252$	0	0
$253$	−4.22089e10	−0.647681
$254$	0	0
$255$	1.14520e10	0.169609
$256$	0	0
$257$	−1.10777e9	−0.0158398	−0.00791989	−	0.999969i	$-0.502521\pi$
$257$	−1.10777e9	−0.0158398	−0.00791989	+	0.999969i	$0.502521\pi$
$258$	0	0
$259$	−7.18368e10	−0.991968
$260$	0	0
$261$	7.34345e10	0.979530
$262$	0	0
$263$	−1.40811e11	−1.81483	−0.907414	−	0.420237i	$-0.861947\pi$
$263$	−1.40811e11	−1.81483	−0.907414	+	0.420237i	$0.861947\pi$
$264$	0	0
$265$	2.31261e11	2.88069
$266$	0	0
$267$	1.22602e10	0.147638
$268$	0	0
$269$	5.53340e8	0.00644328	0.00322164	−	0.999995i	$-0.498975\pi$
$269$	5.53340e8	0.00644328	0.00322164	+	0.999995i	$0.498975\pi$
$270$	0	0
$271$	3.70252e10	0.417000	0.208500	−	0.978022i	$-0.433142\pi$
$271$	3.70252e10	0.417000	0.208500	+	0.978022i	$0.433142\pi$
$272$	0	0
$273$	−1.48616e9	−0.0161932
$274$	0	0
$275$	1.18349e11	1.24786
$276$	0	0
$277$	−1.39159e10	−0.142020	−0.0710102	−	0.997476i	$-0.522622\pi$
$277$	−1.39159e10	−0.142020	−0.0710102	+	0.997476i	$0.522622\pi$
$278$	0	0
$279$	1.88271e11	1.86022
$280$	0	0
$281$	3.03696e10	0.290577	0.145288	−	0.989389i	$-0.453589\pi$
$281$	3.03696e10	0.290577	0.145288	+	0.989389i	$0.453589\pi$
$282$	0	0
$283$	6.56957e10	0.608833	0.304416	−	0.952539i	$-0.401539\pi$
$283$	6.56957e10	0.608833	0.304416	+	0.952539i	$0.401539\pi$
$284$	0	0
$285$	2.19427e10	0.197010
$286$	0	0
$287$	−1.31900e11	−1.14756
$288$	0	0
$289$	6.61714e10	0.557995
$290$	0	0
$291$	6.61786e9	0.0541002
$292$	0	0
$293$	−1.35679e10	−0.107549	−0.0537746	−	0.998553i	$-0.517125\pi$
$293$	−1.35679e10	−0.107549	−0.0537746	+	0.998553i	$0.517125\pi$
$294$	0	0
$295$	−6.12045e10	−0.470527
$296$	0	0
$297$	−1.46422e10	−0.109195
$298$	0	0
$299$	−3.65602e10	−0.264538
$300$	0	0
$301$	−1.00841e11	−0.708092
$302$	0	0
$303$	−5.85353e9	−0.0398957
$304$	0	0
$305$	−1.98853e11	−1.31578
$306$	0	0
$307$	−7.65194e10	−0.491642	−0.245821	−	0.969315i	$-0.579058\pi$
$307$	−7.65194e10	−0.491642	−0.245821	+	0.969315i	$0.579058\pi$
$308$	0	0
$309$	2.75031e9	0.0171621
$310$	0	0
$311$	−7.77688e10	−0.471394	−0.235697	−	0.971827i	$-0.575737\pi$
$311$	−7.77688e10	−0.471394	−0.235697	+	0.971827i	$0.575737\pi$
$312$	0	0
$313$	−2.55951e11	−1.50733	−0.753664	−	0.657260i	$-0.771715\pi$
$313$	−2.55951e11	−1.50733	−0.753664	+	0.657260i	$0.771715\pi$
$314$	0	0
$315$	2.11671e11	1.21134
$316$	0	0
$317$	−2.81795e11	−1.56735	−0.783675	−	0.621171i	$-0.786657\pi$
$317$	−2.81795e11	−1.56735	−0.783675	+	0.621171i	$0.786657\pi$
$318$	0	0
$319$	−1.23826e11	−0.669506
$320$	0	0
$321$	−2.17442e10	−0.114307
$322$	0	0
$323$	3.54012e11	1.80970
$324$	0	0
$325$	1.02511e11	0.509676
$326$	0	0
$327$	1.04372e10	0.0504802
$328$	0	0
$329$	−2.27623e11	−1.07111
$330$	0	0
$331$	−1.74585e11	−0.799433	−0.399716	−	0.916639i	$-0.630891\pi$
$331$	−1.74585e11	−0.799433	−0.399716	+	0.916639i	$0.630891\pi$
$332$	0	0
$333$	−3.05522e11	−1.36158
$334$	0	0
$335$	−3.81266e11	−1.65397
$336$	0	0
$337$	3.50642e10	0.148091	0.0740456	−	0.997255i	$-0.476409\pi$
$337$	3.50642e10	0.148091	0.0740456	+	0.997255i	$0.476409\pi$
$338$	0	0
$339$	1.53022e10	0.0629295
$340$	0	0
$341$	−3.17464e11	−1.27145
$342$	0	0
$343$	2.73881e11	1.06841
$344$	0	0
$345$	−3.41045e10	−0.129606
$346$	0	0
$347$	−9.21912e10	−0.341355	−0.170678	−	0.985327i	$-0.554596\pi$
$347$	−9.21912e10	−0.341355	−0.170678	+	0.985327i	$0.554596\pi$
$348$	0	0
$349$	−8.25046e10	−0.297690	−0.148845	−	0.988861i	$-0.547556\pi$
$349$	−8.25046e10	−0.297690	−0.148845	+	0.988861i	$0.547556\pi$
$350$	0	0
$351$	−1.26827e10	−0.0445994
$352$	0	0
$353$	1.49140e11	0.511220	0.255610	−	0.966780i	$-0.417724\pi$
$353$	1.49140e11	0.511220	0.255610	+	0.966780i	$0.417724\pi$
$354$	0	0
$355$	8.16126e11	2.72728
$356$	0	0
$357$	−2.23663e10	−0.0728765
$358$	0	0
$359$	5.95504e11	1.89217	0.946084	−	0.323922i	$-0.105002\pi$
$359$	5.95504e11	1.89217	0.946084	+	0.323922i	$0.105002\pi$
$360$	0	0
$361$	3.55623e11	1.10207
$362$	0	0
$363$	−1.43803e10	−0.0434698
$364$	0	0
$365$	6.94910e10	0.204932
$366$	0	0
$367$	−1.92484e11	−0.553858	−0.276929	−	0.960890i	$-0.589317\pi$
$367$	−1.92484e11	−0.553858	−0.276929	+	0.960890i	$0.589317\pi$
$368$	0	0
$369$	−5.60972e11	−1.57515
$370$	0	0
$371$	−4.51667e11	−1.23776
$372$	0	0
$373$	−7.99931e10	−0.213975	−0.106987	−	0.994260i	$-0.534120\pi$
$373$	−7.99931e10	−0.213975	−0.106987	+	0.994260i	$0.534120\pi$
$374$	0	0
$375$	4.35888e10	0.113824
$376$	0	0
$377$	−1.07255e11	−0.273452
$378$	0	0
$379$	−7.76615e11	−1.93343	−0.966717	−	0.255847i	$-0.917646\pi$
$379$	−7.76615e11	−1.93343	−0.966717	+	0.255847i	$0.917646\pi$
$380$	0	0
$381$	−1.56234e10	−0.0379850
$382$	0	0
$383$	5.69019e11	1.35124	0.675619	−	0.737251i	$-0.263876\pi$
$383$	5.69019e11	1.35124	0.675619	+	0.737251i	$0.263876\pi$
$384$	0	0
$385$	−3.56922e11	−0.827943
$386$	0	0
$387$	−4.28879e11	−0.971931
$388$	0	0
$389$	8.19337e11	1.81422	0.907109	−	0.420895i	$-0.138284\pi$
$389$	8.19337e11	1.81422	0.907109	+	0.420895i	$0.138284\pi$
$390$	0	0
$391$	−5.50223e11	−1.19054
$392$	0	0
$393$	6.41165e10	0.135582
$394$	0	0
$395$	−4.53486e11	−0.937296
$396$	0	0
$397$	−8.93359e10	−0.180496	−0.0902482	−	0.995919i	$-0.528766\pi$
$397$	−8.93359e10	−0.180496	−0.0902482	+	0.995919i	$0.528766\pi$
$398$	0	0
$399$	−4.28554e10	−0.0846502
$400$	0	0
$401$	−7.57949e11	−1.46383	−0.731915	−	0.681396i	$-0.761373\pi$
$401$	−7.57949e11	−1.46383	−0.731915	+	0.681396i	$0.761373\pi$
$402$	0	0
$403$	−2.74979e11	−0.519310
$404$	0	0
$405$	8.94304e11	1.65172
$406$	0	0
$407$	5.15175e11	0.930636
$408$	0	0
$409$	−3.05185e10	−0.0539273	−0.0269637	−	0.999636i	$-0.508584\pi$
$409$	−3.05185e10	−0.0539273	−0.0269637	+	0.999636i	$0.508584\pi$
$410$	0	0
$411$	6.63246e10	0.114653
$412$	0	0
$413$	1.19536e11	0.202173
$414$	0	0
$415$	5.56520e11	0.921010
$416$	0	0
$417$	−4.34715e10	−0.0704032
$418$	0	0
$419$	2.18779e11	0.346771	0.173386	−	0.984854i	$-0.444529\pi$
$419$	2.18779e11	0.346771	0.173386	+	0.984854i	$0.444529\pi$
$420$	0	0
$421$	−2.83336e11	−0.439574	−0.219787	−	0.975548i	$-0.570536\pi$
$421$	−2.83336e11	−0.439574	−0.219787	+	0.975548i	$0.570536\pi$
$422$	0	0
$423$	−9.68081e11	−1.47021
$424$	0	0
$425$	1.54276e12	2.29376
$426$	0	0
$427$	3.88372e11	0.565356
$428$	0	0
$429$	1.06579e10	0.0151920
$430$	0	0
$431$	8.86220e11	1.23707	0.618534	−	0.785758i	$-0.287727\pi$
$431$	8.86220e11	1.23707	0.618534	+	0.785758i	$0.287727\pi$
$432$	0	0
$433$	−2.20953e11	−0.302068	−0.151034	−	0.988529i	$-0.548260\pi$
$433$	−2.20953e11	−0.302068	−0.151034	+	0.988529i	$0.548260\pi$
$434$	0	0
$435$	−1.00051e11	−0.133974
$436$	0	0
$437$	−1.05426e12	−1.38288
$438$	0	0
$439$	−3.29471e11	−0.423376	−0.211688	−	0.977337i	$-0.567896\pi$
$439$	−3.29471e11	−0.423376	−0.211688	+	0.977337i	$0.567896\pi$
$440$	0	0
$441$	3.75706e11	0.473014
$442$	0	0
$443$	−3.03581e11	−0.374505	−0.187253	−	0.982312i	$-0.559958\pi$
$443$	−3.03581e11	−0.374505	−0.187253	+	0.982312i	$0.559958\pi$
$444$	0	0
$445$	2.55042e12	3.08313
$446$	0	0
$447$	3.96822e10	0.0470123
$448$	0	0
$449$	4.98242e10	0.0578538	0.0289269	−	0.999582i	$-0.490791\pi$
$449$	4.98242e10	0.0578538	0.0289269	+	0.999582i	$0.490791\pi$
$450$	0	0
$451$	9.45918e11	1.07661
$452$	0	0
$453$	3.06113e10	0.0341538
$454$	0	0
$455$	−3.09157e11	−0.338164
$456$	0	0
$457$	1.80697e12	1.93788	0.968941	−	0.247292i	$-0.0795408\pi$
$457$	1.80697e12	1.93788	0.968941	+	0.247292i	$0.0795408\pi$
$458$	0	0
$459$	−1.90871e11	−0.200717
$460$	0	0
$461$	−4.46003e11	−0.459921	−0.229961	−	0.973200i	$-0.573860\pi$
$461$	−4.46003e11	−0.459921	−0.229961	+	0.973200i	$0.573860\pi$
$462$	0	0
$463$	−6.27698e11	−0.634799	−0.317399	−	0.948292i	$-0.602810\pi$
$463$	−6.27698e11	−0.634799	−0.317399	+	0.948292i	$0.602810\pi$
$464$	0	0
$465$	−2.56509e11	−0.254428
$466$	0	0
$467$	1.24133e12	1.20770	0.603852	−	0.797096i	$-0.293632\pi$
$467$	1.24133e12	1.20770	0.603852	+	0.797096i	$0.293632\pi$
$468$	0	0
$469$	7.44634e11	0.710666
$470$	0	0
$471$	3.97189e10	0.0371880
$472$	0	0
$473$	7.23181e11	0.664312
$474$	0	0
$475$	2.95604e12	2.66433
$476$	0	0
$477$	−1.92094e12	−1.69895
$478$	0	0
$479$	−4.80660e11	−0.417184	−0.208592	−	0.978003i	$-0.566888\pi$
$479$	−4.80660e11	−0.417184	−0.208592	+	0.978003i	$0.566888\pi$
$480$	0	0
$481$	4.46231e11	0.380108
$482$	0	0
$483$	6.66081e10	0.0556884
$484$	0	0
$485$	1.37668e12	1.12978
$486$	0	0
$487$	−7.76883e11	−0.625857	−0.312929	−	0.949777i	$-0.601310\pi$
$487$	−7.76883e11	−0.625857	−0.312929	+	0.949777i	$0.601310\pi$
$488$	0	0
$489$	−1.50487e9	−0.00119017
$490$	0	0
$491$	−1.13450e12	−0.880926	−0.440463	−	0.897771i	$-0.645186\pi$
$491$	−1.13450e12	−0.880926	−0.440463	+	0.897771i	$0.645186\pi$
$492$	0	0
$493$	−1.61416e12	−1.23065
$494$	0	0
$495$	−1.51799e12	−1.13644
$496$	0	0
$497$	−1.59394e12	−1.17184
$498$	0	0
$499$	1.86224e11	0.134457	0.0672286	−	0.997738i	$-0.478584\pi$
$499$	1.86224e11	0.134457	0.0672286	+	0.997738i	$0.478584\pi$
$500$	0	0
$501$	3.93583e10	0.0279104
$502$	0	0
$503$	−7.93901e10	−0.0552981	−0.0276491	−	0.999618i	$-0.508802\pi$
$503$	−7.93901e10	−0.0552981	−0.0276491	+	0.999618i	$0.508802\pi$
$504$	0	0
$505$	−1.21768e12	−0.833146
$506$	0	0
$507$	9.23163e9	0.00620501
$508$	0	0
$509$	2.83749e11	0.187372	0.0936858	−	0.995602i	$-0.470135\pi$
$509$	2.83749e11	0.187372	0.0936858	+	0.995602i	$0.470135\pi$
$510$	0	0
$511$	−1.35720e11	−0.0880541
$512$	0	0
$513$	−3.65723e11	−0.233144
$514$	0	0
$515$	5.72132e11	0.358397
$516$	0	0
$517$	1.63239e12	1.00489
$518$	0	0
$519$	−7.95560e10	−0.0481305
$520$	0	0
$521$	2.72237e12	1.61874	0.809371	−	0.587298i	$-0.199808\pi$
$521$	2.72237e12	1.61874	0.809371	+	0.587298i	$0.199808\pi$
$522$	0	0
$523$	8.68410e11	0.507536	0.253768	−	0.967265i	$-0.418330\pi$
$523$	8.68410e11	0.507536	0.253768	+	0.967265i	$0.418330\pi$
$524$	0	0
$525$	−1.86761e11	−0.107293
$526$	0	0
$527$	−4.13837e12	−2.33712
$528$	0	0
$529$	−1.62562e11	−0.0902542
$530$	0	0
$531$	5.08387e11	0.277504
$532$	0	0
$533$	8.19330e11	0.439730
$534$	0	0
$535$	−4.52333e12	−2.38708
$536$	0	0
$537$	−2.10991e11	−0.109491
$538$	0	0
$539$	−6.33520e11	−0.323304
$540$	0	0
$541$	−1.26830e12	−0.636552	−0.318276	−	0.947998i	$-0.603104\pi$
$541$	−1.26830e12	−0.636552	−0.318276	+	0.947998i	$0.603104\pi$
$542$	0	0
$543$	−3.85596e10	−0.0190341
$544$	0	0
$545$	2.17120e12	1.05418
$546$	0	0
$547$	2.42623e12	1.15875	0.579375	−	0.815061i	$-0.303297\pi$
$547$	2.42623e12	1.15875	0.579375	+	0.815061i	$0.303297\pi$
$548$	0	0
$549$	1.65175e12	0.776011
$550$	0	0
$551$	−3.09285e12	−1.42947
$552$	0	0
$553$	8.85684e11	0.402732
$554$	0	0
$555$	4.16259e11	0.186228
$556$	0	0
$557$	−8.75783e11	−0.385521	−0.192761	−	0.981246i	$-0.561744\pi$
$557$	−8.75783e11	−0.385521	−0.192761	+	0.981246i	$0.561744\pi$
$558$	0	0
$559$	6.26401e11	0.271331
$560$	0	0
$561$	1.60399e11	0.0683707
$562$	0	0
$563$	1.17721e12	0.493818	0.246909	−	0.969039i	$-0.420585\pi$
$563$	1.17721e12	0.493818	0.246909	+	0.969039i	$0.420585\pi$
$564$	0	0
$565$	3.18322e12	1.31416
$566$	0	0
$567$	−1.74663e12	−0.709702
$568$	0	0
$569$	−5.76718e11	−0.230652	−0.115326	−	0.993328i	$-0.536791\pi$
$569$	−5.76718e11	−0.230652	−0.115326	+	0.993328i	$0.536791\pi$
$570$	0	0
$571$	−3.78604e10	−0.0149047	−0.00745234	−	0.999972i	$-0.502372\pi$
$571$	−3.78604e10	−0.0149047	−0.00745234	+	0.999972i	$0.502372\pi$
$572$	0	0
$573$	−2.53391e11	−0.0981962
$574$	0	0
$575$	−4.59442e12	−1.75277
$576$	0	0
$577$	2.34089e12	0.879204	0.439602	−	0.898193i	$-0.355119\pi$
$577$	2.34089e12	0.879204	0.439602	+	0.898193i	$0.355119\pi$
$578$	0	0
$579$	−1.38166e11	−0.0510914
$580$	0	0
$581$	−1.08692e12	−0.395734
$582$	0	0
$583$	3.23911e12	1.16123
$584$	0	0
$585$	−1.31485e12	−0.464166
$586$	0	0
$587$	1.28235e12	0.445794	0.222897	−	0.974842i	$-0.428449\pi$
$587$	1.28235e12	0.445794	0.222897	+	0.974842i	$0.428449\pi$
$588$	0	0
$589$	−7.92940e12	−2.71470
$590$	0	0
$591$	−2.81923e11	−0.0950575
$592$	0	0
$593$	−4.04369e12	−1.34286	−0.671431	−	0.741067i	$-0.734320\pi$
$593$	−4.04369e12	−1.34286	−0.671431	+	0.741067i	$0.734320\pi$
$594$	0	0
$595$	−4.65274e12	−1.52189
$596$	0	0
$597$	−3.64125e11	−0.117318
$598$	0	0
$599$	−2.36350e12	−0.750128	−0.375064	−	0.926999i	$-0.622379\pi$
$599$	−2.36350e12	−0.750128	−0.375064	+	0.926999i	$0.622379\pi$
$600$	0	0
$601$	−2.49710e12	−0.780730	−0.390365	−	0.920660i	$-0.627651\pi$
$601$	−2.49710e12	−0.780730	−0.390365	+	0.920660i	$0.627651\pi$
$602$	0	0
$603$	3.16693e12	0.975464
$604$	0	0
$605$	−2.99145e12	−0.907783
$606$	0	0
$607$	2.07460e11	0.0620277	0.0310139	−	0.999519i	$-0.490126\pi$
$607$	2.07460e11	0.0620277	0.0310139	+	0.999519i	$0.490126\pi$
$608$	0	0
$609$	1.95405e11	0.0575649
$610$	0	0
$611$	1.41393e12	0.410434
$612$	0	0
$613$	−3.52577e12	−1.00851	−0.504257	−	0.863554i	$-0.668234\pi$
$613$	−3.52577e12	−1.00851	−0.504257	+	0.863554i	$0.668234\pi$
$614$	0	0
$615$	7.64297e11	0.215439
$616$	0	0
$617$	−1.94143e12	−0.539309	−0.269654	−	0.962957i	$-0.586909\pi$
$617$	−1.94143e12	−0.539309	−0.269654	+	0.962957i	$0.586909\pi$
$618$	0	0
$619$	5.94875e12	1.62861	0.814306	−	0.580436i	$-0.197118\pi$
$619$	5.94875e12	1.62861	0.814306	+	0.580436i	$0.197118\pi$
$620$	0	0
$621$	5.68424e11	0.153377
$622$	0	0
$623$	−4.98112e12	−1.32474
$624$	0	0
$625$	2.05741e12	0.539339
$626$	0	0
$627$	3.07336e11	0.0794164
$628$	0	0
$629$	6.71567e12	1.71065
$630$	0	0
$631$	4.40472e12	1.10608	0.553040	−	0.833155i	$-0.313468\pi$
$631$	4.40472e12	1.10608	0.553040	+	0.833155i	$0.313468\pi$
$632$	0	0
$633$	−4.71456e11	−0.116714
$634$	0	0
$635$	−3.25004e12	−0.793244
$636$	0	0
$637$	−5.48738e11	−0.132050
$638$	0	0
$639$	−6.77904e12	−1.60848
$640$	0	0
$641$	−2.30805e12	−0.539989	−0.269995	−	0.962862i	$-0.587022\pi$
$641$	−2.30805e12	−0.539989	−0.269995	+	0.962862i	$0.587022\pi$
$642$	0	0
$643$	6.27875e12	1.44852	0.724259	−	0.689528i	$-0.242182\pi$
$643$	6.27875e12	1.44852	0.724259	+	0.689528i	$0.242182\pi$
$644$	0	0
$645$	5.84326e11	0.132934
$646$	0	0
$647$	3.44904e12	0.773800	0.386900	−	0.922122i	$-0.373546\pi$
$647$	3.44904e12	0.773800	0.386900	+	0.922122i	$0.373546\pi$
$648$	0	0
$649$	−8.57248e11	−0.189673
$650$	0	0
$651$	5.00977e11	0.109321
$652$	0	0
$653$	−4.76677e12	−1.02592	−0.512962	−	0.858411i	$-0.671452\pi$
$653$	−4.76677e12	−1.02592	−0.512962	+	0.858411i	$0.671452\pi$
$654$	0	0
$655$	1.33378e13	2.83138
$656$	0	0
$657$	−5.77217e11	−0.120864
$658$	0	0
$659$	1.92345e12	0.397280	0.198640	−	0.980073i	$-0.436348\pi$
$659$	1.92345e12	0.397280	0.198640	+	0.980073i	$0.436348\pi$
$660$	0	0
$661$	7.20926e12	1.46887	0.734436	−	0.678678i	$-0.237447\pi$
$661$	7.20926e12	1.46887	0.734436	+	0.678678i	$0.237447\pi$
$662$	0	0
$663$	1.38934e11	0.0279252
$664$	0	0
$665$	−8.91497e12	−1.76776
$666$	0	0
$667$	4.80706e12	0.940401
$668$	0	0
$669$	−5.64354e11	−0.108927
$670$	0	0
$671$	−2.78520e12	−0.530401
$672$	0	0
$673$	−8.70236e12	−1.63519	−0.817597	−	0.575791i	$-0.804694\pi$
$673$	−8.70236e12	−1.63519	−0.817597	+	0.575791i	$0.804694\pi$
$674$	0	0
$675$	−1.59380e12	−0.295506
$676$	0	0
$677$	1.94340e12	0.355561	0.177780	−	0.984070i	$-0.443108\pi$
$677$	1.94340e12	0.355561	0.177780	+	0.984070i	$0.443108\pi$
$678$	0	0
$679$	−2.68873e12	−0.485436
$680$	0	0
$681$	6.15465e11	0.109658
$682$	0	0
$683$	2.36069e12	0.415094	0.207547	−	0.978225i	$-0.433452\pi$
$683$	2.36069e12	0.415094	0.207547	+	0.978225i	$0.433452\pi$
$684$	0	0
$685$	1.37971e13	2.39431
$686$	0	0
$687$	2.57447e11	0.0440943
$688$	0	0
$689$	2.80564e12	0.474291
$690$	0	0
$691$	1.34208e12	0.223937	0.111968	−	0.993712i	$-0.464284\pi$
$691$	1.34208e12	0.223937	0.111968	+	0.993712i	$0.464284\pi$
$692$	0	0
$693$	2.96473e12	0.488298
$694$	0	0
$695$	−9.04313e12	−1.47024
$696$	0	0
$697$	1.23307e13	1.97898
$698$	0	0
$699$	−8.65864e11	−0.137184
$700$	0	0
$701$	−1.25069e12	−0.195622	−0.0978112	−	0.995205i	$-0.531184\pi$
$701$	−1.25069e12	−0.195622	−0.0978112	+	0.995205i	$0.531184\pi$
$702$	0	0
$703$	1.28677e13	1.98702
$704$	0	0
$705$	1.31896e12	0.201086
$706$	0	0
$707$	2.37819e12	0.357981
$708$	0	0
$709$	−1.00959e13	−1.50050	−0.750252	−	0.661152i	$-0.770068\pi$
$709$	−1.00959e13	−1.50050	−0.750252	+	0.661152i	$0.770068\pi$
$710$	0	0
$711$	3.76682e12	0.552792
$712$	0	0
$713$	1.23243e13	1.78591
$714$	0	0
$715$	2.21711e12	0.317256
$716$	0	0
$717$	8.79792e10	0.0124321
$718$	0	0
$719$	−1.07531e13	−1.50057	−0.750284	−	0.661116i	$-0.770083\pi$
$719$	−1.07531e13	−1.50057	−0.750284	+	0.661116i	$0.770083\pi$
$720$	0	0
$721$	−1.11741e12	−0.153994
$722$	0	0
$723$	5.16114e10	0.00702462
$724$	0	0
$725$	−1.34784e13	−1.81183
$726$	0	0
$727$	9.11374e12	1.21002	0.605009	−	0.796219i	$-0.293169\pi$
$727$	9.11374e12	1.21002	0.605009	+	0.796219i	$0.293169\pi$
$728$	0	0
$729$	−7.32950e12	−0.961170
$730$	0	0
$731$	9.42718e12	1.22111
$732$	0	0
$733$	−6.10729e12	−0.781414	−0.390707	−	0.920515i	$-0.627769\pi$
$733$	−6.10729e12	−0.781414	−0.390707	+	0.920515i	$0.627769\pi$
$734$	0	0
$735$	−5.11880e11	−0.0646957
$736$	0	0
$737$	−5.34012e12	−0.666726
$738$	0	0
$739$	−1.63744e12	−0.201960	−0.100980	−	0.994888i	$-0.532198\pi$
$739$	−1.63744e12	−0.201960	−0.100980	+	0.994888i	$0.532198\pi$
$740$	0	0
$741$	2.66207e11	0.0324367
$742$	0	0
$743$	5.53753e12	0.666601	0.333301	−	0.942821i	$-0.391838\pi$
$743$	5.53753e12	0.666601	0.333301	+	0.942821i	$0.391838\pi$
$744$	0	0
$745$	8.25486e12	0.981762
$746$	0	0
$747$	−4.62266e12	−0.543186
$748$	0	0
$749$	8.83432e12	1.02566
$750$	0	0
$751$	5.42187e12	0.621970	0.310985	−	0.950415i	$-0.399341\pi$
$751$	5.42187e12	0.621970	0.310985	+	0.950415i	$0.399341\pi$
$752$	0	0
$753$	3.88933e11	0.0440857
$754$	0	0
$755$	6.36788e12	0.713237
$756$	0	0
$757$	1.11572e12	0.123487	0.0617437	−	0.998092i	$-0.480334\pi$
$757$	1.11572e12	0.123487	0.0617437	+	0.998092i	$0.480334\pi$
$758$	0	0
$759$	−4.77678e11	−0.0522453
$760$	0	0
$761$	1.63908e12	0.177162	0.0885809	−	0.996069i	$-0.471767\pi$
$761$	1.63908e12	0.177162	0.0885809	+	0.996069i	$0.471767\pi$
$762$	0	0
$763$	−4.24048e12	−0.452954
$764$	0	0
$765$	−1.97881e13	−2.08895
$766$	0	0
$767$	−7.42526e11	−0.0774698
$768$	0	0
$769$	3.81852e12	0.393756	0.196878	−	0.980428i	$-0.436920\pi$
$769$	3.81852e12	0.393756	0.196878	+	0.980428i	$0.436920\pi$
$770$	0	0
$771$	−1.25366e10	−0.00127772
$772$	0	0
$773$	3.33890e12	0.336354	0.168177	−	0.985757i	$-0.446212\pi$
$773$	3.33890e12	0.336354	0.168177	+	0.985757i	$0.446212\pi$
$774$	0	0
$775$	−3.45559e13	−3.44084
$776$	0	0
$777$	−8.12977e11	−0.0800172
$778$	0	0
$779$	2.36265e13	2.29869
$780$	0	0
$781$	1.14309e13	1.09939
$782$	0	0
$783$	1.66756e12	0.158545
$784$	0	0
$785$	8.26249e12	0.776600
$786$	0	0
$787$	−9.82101e12	−0.912577	−0.456289	−	0.889832i	$-0.650822\pi$
$787$	−9.82101e12	−0.912577	−0.456289	+	0.889832i	$0.650822\pi$
$788$	0	0
$789$	−1.59356e12	−0.146393
$790$	0	0
$791$	−6.21701e12	−0.564661
$792$	0	0
$793$	−2.41246e12	−0.216636
$794$	0	0
$795$	2.61719e12	0.232371
$796$	0	0
$797$	7.91721e12	0.695040	0.347520	−	0.937673i	$-0.387024\pi$
$797$	7.91721e12	0.695040	0.347520	+	0.937673i	$0.387024\pi$
$798$	0	0
$799$	2.12794e13	1.84713
$800$	0	0
$801$	−2.11847e13	−1.81835
$802$	0	0
$803$	9.73311e11	0.0826098
$804$	0	0
$805$	1.38561e13	1.16295
$806$	0	0
$807$	6.26215e9	0.000519748 0
$808$	0	0
$809$	−1.31850e12	−0.108221	−0.0541105	−	0.998535i	$-0.517232\pi$
$809$	−1.31850e12	−0.108221	−0.0541105	+	0.998535i	$0.517232\pi$
$810$	0	0
$811$	8.40957e12	0.682622	0.341311	−	0.939950i	$-0.389129\pi$
$811$	8.40957e12	0.682622	0.341311	+	0.939950i	$0.389129\pi$
$812$	0	0
$813$	4.19015e11	0.0336374
$814$	0	0
$815$	−3.13049e11	−0.0248544
$816$	0	0
$817$	1.80631e13	1.41838
$818$	0	0
$819$	2.56797e12	0.199440
$820$	0	0
$821$	−2.21766e13	−1.70354	−0.851768	−	0.523919i	$-0.824470\pi$
$821$	−2.21766e13	−1.70354	−0.851768	+	0.523919i	$0.824470\pi$
$822$	0	0
$823$	−5.30516e12	−0.403087	−0.201544	−	0.979480i	$-0.564596\pi$
$823$	−5.30516e12	−0.403087	−0.201544	+	0.979480i	$0.564596\pi$
$824$	0	0
$825$	1.33935e12	0.100659
$826$	0	0
$827$	−1.76310e13	−1.31069	−0.655347	−	0.755328i	$-0.727478\pi$
$827$	−1.76310e13	−1.31069	−0.655347	+	0.755328i	$0.727478\pi$
$828$	0	0
$829$	−5.12521e12	−0.376892	−0.188446	−	0.982084i	$-0.560345\pi$
$829$	−5.12521e12	−0.376892	−0.188446	+	0.982084i	$0.560345\pi$
$830$	0	0
$831$	−1.57486e11	−0.0114561
$832$	0	0
$833$	−8.25838e12	−0.594282
$834$	0	0
$835$	8.18747e12	0.582855
$836$	0	0
$837$	4.27527e12	0.301092
$838$	0	0
$839$	−3.75310e12	−0.261494	−0.130747	−	0.991416i	$-0.541737\pi$
$839$	−3.75310e12	−0.261494	−0.130747	+	0.991416i	$0.541737\pi$
$840$	0	0
$841$	−4.04905e11	−0.0279107
$842$	0	0
$843$	3.43693e11	0.0234394
$844$	0	0
$845$	1.92040e12	0.129580
$846$	0	0
$847$	5.84247e12	0.390051
$848$	0	0
$849$	7.43479e11	0.0491116
$850$	0	0
$851$	−1.99996e13	−1.30719
$852$	0	0
$853$	−2.38455e13	−1.54218	−0.771092	−	0.636724i	$-0.780289\pi$
$853$	−2.38455e13	−1.54218	−0.771092	+	0.636724i	$0.780289\pi$
$854$	0	0
$855$	−3.79154e13	−2.42643
$856$	0	0
$857$	1.02524e13	0.649249	0.324625	−	0.945843i	$-0.394762\pi$
$857$	1.02524e13	0.649249	0.324625	+	0.945843i	$0.394762\pi$
$858$	0	0
$859$	4.68549e12	0.293620	0.146810	−	0.989165i	$-0.453099\pi$
$859$	4.68549e12	0.293620	0.146810	+	0.989165i	$0.453099\pi$
$860$	0	0
$861$	−1.49271e12	−0.0925684
$862$	0	0
$863$	−3.22907e13	−1.98166	−0.990830	−	0.135116i	$-0.956859\pi$
$863$	−3.22907e13	−1.98166	−0.990830	+	0.135116i	$0.956859\pi$
$864$	0	0
$865$	−1.65496e13	−1.00511
$866$	0	0
$867$	7.48862e11	0.0450107
$868$	0	0
$869$	−6.35165e12	−0.377831
$870$	0	0
$871$	−4.62547e12	−0.272317
$872$	0	0
$873$	−1.14352e13	−0.666313
$874$	0	0
$875$	−1.77094e13	−1.02133
$876$	0	0
$877$	−1.82002e13	−1.03891	−0.519456	−	0.854497i	$-0.673865\pi$
$877$	−1.82002e13	−1.03891	−0.519456	+	0.854497i	$0.673865\pi$
$878$	0	0
$879$	−1.53548e11	−0.00867547
$880$	0	0
$881$	−1.19126e13	−0.666217	−0.333108	−	0.942889i	$-0.608098\pi$
$881$	−1.19126e13	−0.666217	−0.333108	+	0.942889i	$0.608098\pi$
$882$	0	0
$883$	−6.99095e11	−0.0387002	−0.0193501	−	0.999813i	$-0.506160\pi$
$883$	−6.99095e11	−0.0387002	−0.0193501	+	0.999813i	$0.506160\pi$
$884$	0	0
$885$	−6.92652e11	−0.0379551
$886$	0	0
$887$	1.03675e13	0.562363	0.281181	−	0.959655i	$-0.409274\pi$
$887$	1.03675e13	0.562363	0.281181	+	0.959655i	$0.409274\pi$
$888$	0	0
$889$	6.34751e12	0.340836
$890$	0	0
$891$	1.25259e13	0.665822
$892$	0	0
$893$	4.07727e13	2.14555
$894$	0	0
$895$	−4.38912e13	−2.28652
$896$	0	0
$897$	−4.13752e11	−0.0213390
$898$	0	0
$899$	3.61552e13	1.84608
$900$	0	0
$901$	4.22242e13	2.13452
$902$	0	0
$903$	−1.14122e12	−0.0571183
$904$	0	0
$905$	−8.02133e12	−0.397492
$906$	0	0
$907$	−1.58620e13	−0.778261	−0.389131	−	0.921183i	$-0.627225\pi$
$907$	−1.58620e13	−0.778261	−0.389131	+	0.921183i	$0.627225\pi$
$908$	0	0
$909$	1.01145e13	0.491367
$910$	0	0
$911$	−2.05101e13	−0.986586	−0.493293	−	0.869863i	$-0.664207\pi$
$911$	−2.05101e13	−0.986586	−0.493293	+	0.869863i	$0.664207\pi$
$912$	0	0
$913$	7.79478e12	0.371266
$914$	0	0
$915$	−2.25042e12	−0.106138
$916$	0	0
$917$	−2.60494e13	−1.21657
$918$	0	0
$919$	1.41684e13	0.655241	0.327620	−	0.944809i	$-0.393753\pi$
$919$	1.41684e13	0.655241	0.327620	+	0.944809i	$0.393753\pi$
$920$	0	0
$921$	−8.65970e11	−0.0396584
$922$	0	0
$923$	9.90115e12	0.449033
$924$	0	0
$925$	5.60766e13	2.51851
$926$	0	0
$927$	−4.75234e12	−0.211372
$928$	0	0
$929$	1.80998e13	0.797265	0.398632	−	0.917111i	$-0.369485\pi$
$929$	1.80998e13	0.797265	0.398632	+	0.917111i	$0.369485\pi$
$930$	0	0
$931$	−1.58236e13	−0.690292
$932$	0	0
$933$	−8.80110e11	−0.0380250
$934$	0	0
$935$	3.33670e13	1.42779
$936$	0	0
$937$	−2.22382e12	−0.0942477	−0.0471239	−	0.998889i	$-0.515006\pi$
$937$	−2.22382e12	−0.0942477	−0.0471239	+	0.998889i	$0.515006\pi$
$938$	0	0
$939$	−2.89660e12	−0.121589
$940$	0	0
$941$	6.13618e12	0.255120	0.127560	−	0.991831i	$-0.459285\pi$
$941$	6.13618e12	0.255120	0.127560	+	0.991831i	$0.459285\pi$
$942$	0	0
$943$	−3.67215e13	−1.51223
$944$	0	0
$945$	4.80666e12	0.196065
$946$	0	0
$947$	9.33151e12	0.377031	0.188516	−	0.982070i	$-0.439632\pi$
$947$	9.33151e12	0.377031	0.188516	+	0.982070i	$0.439632\pi$
$948$	0	0
$949$	8.43056e11	0.0337411
$950$	0	0
$951$	−3.18907e12	−0.126430
$952$	0	0
$953$	4.28393e13	1.68238	0.841190	−	0.540740i	$-0.181856\pi$
$953$	4.28393e13	1.68238	0.841190	+	0.540740i	$0.181856\pi$
$954$	0	0
$955$	−5.27114e13	−2.05064
$956$	0	0
$957$	−1.40134e12	−0.0540057
$958$	0	0
$959$	−2.69466e13	−1.02877
$960$	0	0
$961$	6.62545e13	2.50588
$962$	0	0
$963$	3.75724e13	1.40783
$964$	0	0
$965$	−2.87419e13	−1.06695
$966$	0	0
$967$	2.77098e13	1.01910	0.509548	−	0.860442i	$-0.329813\pi$
$967$	2.77098e13	1.01910	0.509548	+	0.860442i	$0.329813\pi$
$968$	0	0
$969$	4.00635e12	0.145980
$970$	0	0
$971$	2.01303e13	0.726715	0.363357	−	0.931650i	$-0.381630\pi$
$971$	2.01303e13	0.726715	0.363357	+	0.931650i	$0.381630\pi$
$972$	0	0
$973$	1.76618e13	0.631722
$974$	0	0
$975$	1.16011e12	0.0411130
$976$	0	0
$977$	−2.93311e13	−1.02992	−0.514959	−	0.857215i	$-0.672193\pi$
$977$	−2.93311e13	−1.02992	−0.514959	+	0.857215i	$0.672193\pi$
$978$	0	0
$979$	3.57220e13	1.24283
$980$	0	0
$981$	−1.80348e13	−0.621728
$982$	0	0
$983$	−4.03013e13	−1.37667	−0.688333	−	0.725395i	$-0.741657\pi$
$983$	−4.03013e13	−1.37667	−0.688333	+	0.725395i	$0.741657\pi$
$984$	0	0
$985$	−5.86468e13	−1.98509
$986$	0	0
$987$	−2.57601e12	−0.0864012
$988$	0	0
$989$	−2.80746e13	−0.933105
$990$	0	0
$991$	5.56880e13	1.83413	0.917064	−	0.398739i	$-0.130552\pi$
$991$	5.56880e13	1.83413	0.917064	+	0.398739i	$0.130552\pi$
$992$	0	0
$993$	−1.97578e12	−0.0644863
$994$	0	0
$995$	−7.57468e13	−2.44997
$996$	0	0
$997$	−7.91253e12	−0.253622	−0.126811	−	0.991927i	$-0.540474\pi$
$997$	−7.91253e12	−0.253622	−0.126811	+	0.991927i	$0.540474\pi$
$998$	0	0
$999$	−6.93783e12	−0.220384

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	104.10.a.d.1.4	✓	8
4.3	odd	2		208.10.a.m.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.a.d.1.4	✓	8	1.1	even	1	trivial
208.10.a.m.1.5		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 104 = 2^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	104.a (trivial)

\(f(q)\)	\(=\)	\(q+11.3170 q^{3} +2354.21 q^{5} -4597.91 q^{7} -19554.9 q^{9} +32973.7 q^{11} +28561.0 q^{13} +26642.6 q^{15} +429836. q^{17} +823596. q^{19} -52034.5 q^{21} -1.28007e6 q^{23} +3.58918e6 q^{25} -444056. q^{27} -3.75530e6 q^{29} -9.62778e6 q^{31} +373164. q^{33} -1.08244e7 q^{35} +1.56238e7 q^{37} +323225. q^{39} +2.86870e7 q^{41} +2.19320e7 q^{43} -4.60364e7 q^{45} +4.95057e7 q^{47} -1.92128e7 q^{49} +4.86446e6 q^{51} +9.82331e7 q^{53} +7.76271e7 q^{55} +9.32064e6 q^{57} -2.59979e7 q^{59} -8.44671e7 q^{61} +8.99117e7 q^{63} +6.72386e7 q^{65} -1.61951e8 q^{67} -1.44866e7 q^{69} +3.46667e8 q^{71} +2.95177e7 q^{73} +4.06188e7 q^{75} -1.51610e8 q^{77} -1.92628e8 q^{79} +3.79874e8 q^{81} +2.36393e8 q^{83} +1.01193e9 q^{85} -4.24987e7 q^{87} +1.08335e9 q^{89} -1.31321e8 q^{91} -1.08958e8 q^{93} +1.93892e9 q^{95} +5.84771e8 q^{97} -6.44799e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 141 q^{3} + 2051 q^{5} - 2417 q^{7} + 115741 q^{9} - 53118 q^{11} + 228488 q^{13} - 464555 q^{15} + 433095 q^{17} - 434954 q^{19} + 906875 q^{21} - 1124296 q^{23} + 5966065 q^{25} + 7820643 q^{27}+ \cdots - 641626736 q^{99}+O(q^{100}) \) 8 * q + 141 * q^3 + 2051 * q^5 - 2417 * q^7 + 115741 * q^9 - 53118 * q^11 + 228488 * q^13 - 464555 * q^15 + 433095 * q^17 - 434954 * q^19 + 906875 * q^21 - 1124296 * q^23 + 5966065 * q^25 + 7820643 * q^27 - 1913432 * q^29 + 5220148 * q^31 - 1626882 * q^33 + 950707 * q^35 - 419409 * q^37 + 4027101 * q^39 + 12148046 * q^41 + 32143187 * q^43 + 18893544 * q^45 + 57898411 * q^47 + 104302637 * q^49 + 198521477 * q^51 + 273106866 * q^53 + 141214662 * q^55 + 253032058 * q^57 + 50734502 * q^59 + 139520018 * q^61 - 20344158 * q^63 + 58578611 * q^65 + 528144786 * q^67 + 483506576 * q^69 + 334814973 * q^71 + 630864344 * q^73 + 2265796264 * q^75 + 939664146 * q^77 + 865037608 * q^79 + 3194281040 * q^81 + 1520229952 * q^83 + 2130158385 * q^85 + 1167867006 * q^87 + 894700240 * q^89 - 69031937 * q^91 + 3803209548 * q^93 - 2695926542 * q^95 + 980232164 * q^97 - 641626736 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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