LMFDB - Embedded newform 112.10.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,10,Mod(1,112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("112.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	112.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:	$57.6840136504$
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} - 823x - 4578 $ x^3 - 823*x - 4578
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-25.3451$ of defining polynomial
Character	$\chi$	$=$	112.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.32110 q^{3} +1191.17 q^{5} +2401.00 q^{7} -19629.4 q^{9} +O(q^{10})$	$q+7.32110 q^{3} +1191.17 q^{5} +2401.00 q^{7} -19629.4 q^{9} +18252.2 q^{11} +50333.5 q^{13} +8720.67 q^{15} +283542. q^{17} +273335. q^{19} +17578.0 q^{21} +1.06061e6 q^{23} -534240. q^{25} -287810. q^{27} -5.40121e6 q^{29} +1.75987e6 q^{31} +133626. q^{33} +2.86000e6 q^{35} -7.76468e6 q^{37} +368496. q^{39} +8.92276e6 q^{41} +3.46464e7 q^{43} -2.33819e7 q^{45} +3.66292e7 q^{47} +5.76480e6 q^{49} +2.07584e6 q^{51} -6.10693e7 q^{53} +2.17415e7 q^{55} +2.00111e6 q^{57} +9.02214e7 q^{59} +1.52451e8 q^{61} -4.71302e7 q^{63} +5.99557e7 q^{65} +2.18974e7 q^{67} +7.76485e6 q^{69} +2.33501e8 q^{71} +1.06705e8 q^{73} -3.91123e6 q^{75} +4.38236e7 q^{77} -2.89730e8 q^{79} +3.84258e8 q^{81} -4.73876e7 q^{83} +3.37747e8 q^{85} -3.95428e7 q^{87} +6.97049e8 q^{89} +1.20851e8 q^{91} +1.28842e7 q^{93} +3.25588e8 q^{95} +7.82411e8 q^{97} -3.58280e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 92 q^{3} + 274 q^{5} + 7203 q^{7} + 51871 q^{9}+O(q^{10}) $ 3 * q + 92 * q^3 + 274 * q^5 + 7203 * q^7 + 51871 * q^9	$ 3 q + 92 q^{3} + 274 q^{5} + 7203 q^{7} + 51871 q^{9} + 45364 q^{11} - 11158 q^{13} + 95584 q^{15} - 55866 q^{17} - 488772 q^{19} + 220892 q^{21} + 253888 q^{23} - 3872619 q^{25} + 10157192 q^{27} - 765318 q^{29} + 8189680 q^{31} - 7926240 q^{33} + 657874 q^{35} + 7208194 q^{37} + 44623664 q^{39} - 57891986 q^{41} + 37304708 q^{43} - 45527158 q^{45} + 29210992 q^{47} + 17294403 q^{49} + 142742952 q^{51} + 45140082 q^{53} - 1479800 q^{55} - 77053192 q^{57} + 205334804 q^{59} - 109264870 q^{61} + 124542271 q^{63} + 143056644 q^{65} + 52322348 q^{67} + 425002688 q^{69} + 150048136 q^{71} + 761429998 q^{73} - 260540684 q^{75} + 108918964 q^{77} + 214389152 q^{79} + 1527489907 q^{81} + 36110892 q^{83} + 675044052 q^{85} + 236662680 q^{87} + 1319848398 q^{89} - 26790358 q^{91} + 2846904208 q^{93} + 620260624 q^{95} + 101766310 q^{97} - 168732636 q^{99}+O(q^{100}) $ 3 * q + 92 * q^3 + 274 * q^5 + 7203 * q^7 + 51871 * q^9 + 45364 * q^11 - 11158 * q^13 + 95584 * q^15 - 55866 * q^17 - 488772 * q^19 + 220892 * q^21 + 253888 * q^23 - 3872619 * q^25 + 10157192 * q^27 - 765318 * q^29 + 8189680 * q^31 - 7926240 * q^33 + 657874 * q^35 + 7208194 * q^37 + 44623664 * q^39 - 57891986 * q^41 + 37304708 * q^43 - 45527158 * q^45 + 29210992 * q^47 + 17294403 * q^49 + 142742952 * q^51 + 45140082 * q^53 - 1479800 * q^55 - 77053192 * q^57 + 205334804 * q^59 - 109264870 * q^61 + 124542271 * q^63 + 143056644 * q^65 + 52322348 * q^67 + 425002688 * q^69 + 150048136 * q^71 + 761429998 * q^73 - 260540684 * q^75 + 108918964 * q^77 + 214389152 * q^79 + 1527489907 * q^81 + 36110892 * q^83 + 675044052 * q^85 + 236662680 * q^87 + 1319848398 * q^89 - 26790358 * q^91 + 2846904208 * q^93 + 620260624 * q^95 + 101766310 * q^97 - 168732636 * 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
$3$	7.32110	0.0521832	0.0260916	−	0.999660i	$-0.491694\pi$
$3$	7.32110	0.0521832	0.0260916	+	0.999660i	$0.491694\pi$
$4$	0	0
$5$	1191.17	0.852332	0.426166	−	0.904645i	$-0.359864\pi$
$5$	1191.17	0.852332	0.426166	+	0.904645i	$0.359864\pi$
$6$	0	0
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	0	0
$9$	−19629.4	−0.997277
$10$	0	0
$11$	18252.2	0.375880	0.187940	−	0.982181i	$-0.439819\pi$
$11$	18252.2	0.375880	0.187940	+	0.982181i	$0.439819\pi$
$12$	0	0
$13$	50333.5	0.488778	0.244389	−	0.969677i	$-0.421413\pi$
$13$	50333.5	0.488778	0.244389	+	0.969677i	$0.421413\pi$
$14$	0	0
$15$	8720.67	0.0444774
$16$	0	0
$17$	283542.	0.823375	0.411688	−	0.911325i	$-0.364939\pi$
$17$	283542.	0.823375	0.411688	+	0.911325i	$0.364939\pi$
$18$	0	0
$19$	273335.	0.481176	0.240588	−	0.970627i	$-0.422660\pi$
$19$	273335.	0.481176	0.240588	+	0.970627i	$0.422660\pi$
$20$	0	0
$21$	17578.0	0.0197234
$22$	0	0
$23$	1.06061e6	0.790281	0.395140	−	0.918621i	$-0.370696\pi$
$23$	1.06061e6	0.790281	0.395140	+	0.918621i	$0.370696\pi$
$24$	0	0
$25$	−534240.	−0.273531
$26$	0	0
$27$	−287810.	−0.104224
$28$	0	0
$29$	−5.40121e6	−1.41808	−0.709039	−	0.705170i	$-0.750871\pi$
$29$	−5.40121e6	−1.41808	−0.709039	+	0.705170i	$0.750871\pi$
$30$	0	0
$31$	1.75987e6	0.342257	0.171129	−	0.985249i	$-0.445259\pi$
$31$	1.75987e6	0.342257	0.171129	+	0.985249i	$0.445259\pi$
$32$	0	0
$33$	133626.	0.0196146
$34$	0	0
$35$	2.86000e6	0.322151
$36$	0	0
$37$	−7.76468e6	−0.681108	−0.340554	−	0.940225i	$-0.610615\pi$
$37$	−7.76468e6	−0.681108	−0.340554	+	0.940225i	$0.610615\pi$
$38$	0	0
$39$	368496.	0.0255060
$40$	0	0
$41$	8.92276e6	0.493142	0.246571	−	0.969125i	$-0.420696\pi$
$41$	8.92276e6	0.493142	0.246571	+	0.969125i	$0.420696\pi$
$42$	0	0
$43$	3.46464e7	1.54543	0.772716	−	0.634752i	$-0.218898\pi$
$43$	3.46464e7	1.54543	0.772716	+	0.634752i	$0.218898\pi$
$44$	0	0
$45$	−2.33819e7	−0.850011
$46$	0	0
$47$	3.66292e7	1.09493	0.547466	−	0.836828i	$-0.315592\pi$
$47$	3.66292e7	1.09493	0.547466	+	0.836828i	$0.315592\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	0	0
$51$	2.07584e6	0.0429664
$52$	0	0
$53$	−6.10693e7	−1.06312	−0.531559	−	0.847021i	$-0.678394\pi$
$53$	−6.10693e7	−1.06312	−0.531559	+	0.847021i	$0.678394\pi$
$54$	0	0
$55$	2.17415e7	0.320374
$56$	0	0
$57$	2.00111e6	0.0251093
$58$	0	0
$59$	9.02214e7	0.969339	0.484669	−	0.874697i	$-0.338940\pi$
$59$	9.02214e7	0.969339	0.484669	+	0.874697i	$0.338940\pi$
$60$	0	0
$61$	1.52451e8	1.40976	0.704879	−	0.709327i	$-0.251001\pi$
$61$	1.52451e8	1.40976	0.704879	+	0.709327i	$0.251001\pi$
$62$	0	0
$63$	−4.71302e7	−0.376935
$64$	0	0
$65$	5.99557e7	0.416601
$66$	0	0
$67$	2.18974e7	0.132757	0.0663783	−	0.997795i	$-0.478856\pi$
$67$	2.18974e7	0.132757	0.0663783	+	0.997795i	$0.478856\pi$
$68$	0	0
$69$	7.76485e6	0.0412394
$70$	0	0
$71$	2.33501e8	1.09050	0.545251	−	0.838273i	$-0.316434\pi$
$71$	2.33501e8	1.09050	0.545251	+	0.838273i	$0.316434\pi$
$72$	0	0
$73$	1.06705e8	0.439775	0.219887	−	0.975525i	$-0.429431\pi$
$73$	1.06705e8	0.439775	0.219887	+	0.975525i	$0.429431\pi$
$74$	0	0
$75$	−3.91123e6	−0.0142737
$76$	0	0
$77$	4.38236e7	0.142069
$78$	0	0
$79$	−2.89730e8	−0.836896	−0.418448	−	0.908241i	$-0.637426\pi$
$79$	−2.89730e8	−0.836896	−0.418448	+	0.908241i	$0.637426\pi$
$80$	0	0
$81$	3.84258e8	0.991838
$82$	0	0
$83$	−4.73876e7	−0.109601	−0.0548004	−	0.998497i	$-0.517452\pi$
$83$	−4.73876e7	−0.109601	−0.0548004	+	0.998497i	$0.517452\pi$
$84$	0	0
$85$	3.37747e8	0.701789
$86$	0	0
$87$	−3.95428e7	−0.0739998
$88$	0	0
$89$	6.97049e8	1.17763	0.588815	−	0.808268i	$-0.299595\pi$
$89$	6.97049e8	1.17763	0.588815	+	0.808268i	$0.299595\pi$
$90$	0	0
$91$	1.20851e8	0.184741
$92$	0	0
$93$	1.28842e7	0.0178601
$94$	0	0
$95$	3.25588e8	0.410121
$96$	0	0
$97$	7.82411e8	0.897351	0.448676	−	0.893695i	$-0.351896\pi$
$97$	7.82411e8	0.897351	0.448676	+	0.893695i	$0.351896\pi$
$98$	0	0
$99$	−3.58280e8	−0.374856
$100$	0	0
$101$	1.04878e9	1.00285	0.501425	−	0.865201i	$-0.332809\pi$
$101$	1.04878e9	1.00285	0.501425	+	0.865201i	$0.332809\pi$
$102$	0	0
$103$	1.71802e9	1.50404	0.752022	−	0.659138i	$-0.229079\pi$
$103$	1.71802e9	1.50404	0.752022	+	0.659138i	$0.229079\pi$
$104$	0	0
$105$	2.09383e7	0.0168109
$106$	0	0
$107$	1.54453e8	0.113912	0.0569559	−	0.998377i	$-0.481861\pi$
$107$	1.54453e8	0.113912	0.0569559	+	0.998377i	$0.481861\pi$
$108$	0	0
$109$	1.25646e9	0.852570	0.426285	−	0.904589i	$-0.359822\pi$
$109$	1.25646e9	0.852570	0.426285	+	0.904589i	$0.359822\pi$
$110$	0	0
$111$	−5.68460e7	−0.0355424
$112$	0	0
$113$	−7.21610e8	−0.416341	−0.208171	−	0.978093i	$-0.566751\pi$
$113$	−7.21610e8	−0.416341	−0.208171	+	0.978093i	$0.566751\pi$
$114$	0	0
$115$	1.26337e9	0.673581
$116$	0	0
$117$	−9.88016e8	−0.487447
$118$	0	0
$119$	6.80785e8	0.311207
$120$	0	0
$121$	−2.02480e9	−0.858714
$122$	0	0
$123$	6.53244e7	0.0257337
$124$	0	0
$125$	−2.96287e9	−1.08547
$126$	0	0
$127$	−1.58994e9	−0.542332	−0.271166	−	0.962533i	$-0.587409\pi$
$127$	−1.58994e9	−0.542332	−0.271166	+	0.962533i	$0.587409\pi$
$128$	0	0
$129$	2.53650e8	0.0806456
$130$	0	0
$131$	−4.65327e8	−0.138050	−0.0690252	−	0.997615i	$-0.521989\pi$
$131$	−4.65327e8	−0.138050	−0.0690252	+	0.997615i	$0.521989\pi$
$132$	0	0
$133$	6.56277e8	0.181867
$134$	0	0
$135$	−3.42831e8	−0.0888337
$136$	0	0
$137$	−2.10445e8	−0.0510384	−0.0255192	−	0.999674i	$-0.508124\pi$
$137$	−2.10445e8	−0.0510384	−0.0255192	+	0.999674i	$0.508124\pi$
$138$	0	0
$139$	4.56712e9	1.03771	0.518855	−	0.854862i	$-0.326358\pi$
$139$	4.56712e9	1.03771	0.518855	+	0.854862i	$0.326358\pi$
$140$	0	0
$141$	2.68166e8	0.0571371
$142$	0	0
$143$	9.18698e8	0.183722
$144$	0	0
$145$	−6.43375e9	−1.20867
$146$	0	0
$147$	4.22047e7	0.00745474
$148$	0	0
$149$	3.78925e8	0.0629817	0.0314909	−	0.999504i	$-0.489974\pi$
$149$	3.78925e8	0.0629817	0.0314909	+	0.999504i	$0.489974\pi$
$150$	0	0
$151$	3.52786e9	0.552224	0.276112	−	0.961125i	$-0.410954\pi$
$151$	3.52786e9	0.552224	0.276112	+	0.961125i	$0.410954\pi$
$152$	0	0
$153$	−5.56577e9	−0.821133
$154$	0	0
$155$	2.09630e9	0.291717
$156$	0	0
$157$	2.89211e9	0.379897	0.189948	−	0.981794i	$-0.439168\pi$
$157$	2.89211e9	0.379897	0.189948	+	0.981794i	$0.439168\pi$
$158$	0	0
$159$	−4.47094e8	−0.0554769
$160$	0	0
$161$	2.54653e9	0.298698
$162$	0	0
$163$	−1.09988e10	−1.22040	−0.610199	−	0.792249i	$-0.708910\pi$
$163$	−1.09988e10	−1.22040	−0.610199	+	0.792249i	$0.708910\pi$
$164$	0	0
$165$	1.59172e8	0.0167181
$166$	0	0
$167$	−1.69151e10	−1.68287	−0.841434	−	0.540361i	$-0.818288\pi$
$167$	−1.69151e10	−1.68287	−0.841434	+	0.540361i	$0.818288\pi$
$168$	0	0
$169$	−8.07104e9	−0.761096
$170$	0	0
$171$	−5.36540e9	−0.479866
$172$	0	0
$173$	2.87029e9	0.243623	0.121812	−	0.992553i	$-0.461130\pi$
$173$	2.87029e9	0.243623	0.121812	+	0.992553i	$0.461130\pi$
$174$	0	0
$175$	−1.28271e9	−0.103385
$176$	0	0
$177$	6.60520e8	0.0505832
$178$	0	0
$179$	−1.99571e10	−1.45297	−0.726487	−	0.687180i	$-0.758848\pi$
$179$	−1.99571e10	−1.45297	−0.726487	+	0.687180i	$0.758848\pi$
$180$	0	0
$181$	8.41781e9	0.582969	0.291484	−	0.956576i	$-0.405851\pi$
$181$	8.41781e9	0.582969	0.291484	+	0.956576i	$0.405851\pi$
$182$	0	0
$183$	1.11611e9	0.0735657
$184$	0	0
$185$	−9.24905e9	−0.580529
$186$	0	0
$187$	5.17528e9	0.309490
$188$	0	0
$189$	−6.91032e8	−0.0393931
$190$	0	0
$191$	−8.23905e9	−0.447948	−0.223974	−	0.974595i	$-0.571903\pi$
$191$	−8.23905e9	−0.447948	−0.223974	+	0.974595i	$0.571903\pi$
$192$	0	0
$193$	3.63295e9	0.188474	0.0942370	−	0.995550i	$-0.469959\pi$
$193$	3.63295e9	0.188474	0.0942370	+	0.995550i	$0.469959\pi$
$194$	0	0
$195$	4.38942e8	0.0217396
$196$	0	0
$197$	−1.62910e10	−0.770635	−0.385318	−	0.922784i	$-0.625908\pi$
$197$	−1.62910e10	−0.770635	−0.385318	+	0.922784i	$0.625908\pi$
$198$	0	0
$199$	−2.06004e10	−0.931185	−0.465592	−	0.884999i	$-0.654159\pi$
$199$	−2.06004e10	−0.931185	−0.465592	+	0.884999i	$0.654159\pi$
$200$	0	0
$201$	1.60313e8	0.00692766
$202$	0	0
$203$	−1.29683e10	−0.535983
$204$	0	0
$205$	1.06285e10	0.420320
$206$	0	0
$207$	−2.08192e10	−0.788129
$208$	0	0
$209$	4.98897e9	0.180864
$210$	0	0
$211$	1.38734e10	0.481850	0.240925	−	0.970544i	$-0.422549\pi$
$211$	1.38734e10	0.481850	0.240925	+	0.970544i	$0.422549\pi$
$212$	0	0
$213$	1.70949e9	0.0569059
$214$	0	0
$215$	4.12697e10	1.31722
$216$	0	0
$217$	4.22545e9	0.129361
$218$	0	0
$219$	7.81195e8	0.0229489
$220$	0	0
$221$	1.42717e10	0.402448
$222$	0	0
$223$	3.08350e10	0.834972	0.417486	−	0.908683i	$-0.362911\pi$
$223$	3.08350e10	0.834972	0.417486	+	0.908683i	$0.362911\pi$
$224$	0	0
$225$	1.04868e10	0.272786
$226$	0	0
$227$	−2.05939e10	−0.514780	−0.257390	−	0.966308i	$-0.582863\pi$
$227$	−2.05939e10	−0.514780	−0.257390	+	0.966308i	$0.582863\pi$
$228$	0	0
$229$	−5.22733e10	−1.25609	−0.628044	−	0.778178i	$-0.716144\pi$
$229$	−5.22733e10	−1.25609	−0.628044	+	0.778178i	$0.716144\pi$
$230$	0	0
$231$	3.20837e8	0.00741363
$232$	0	0
$233$	2.40946e10	0.535572	0.267786	−	0.963478i	$-0.413708\pi$
$233$	2.40946e10	0.535572	0.267786	+	0.963478i	$0.413708\pi$
$234$	0	0
$235$	4.36316e10	0.933245
$236$	0	0
$237$	−2.12114e9	−0.0436719
$238$	0	0
$239$	4.45198e10	0.882597	0.441298	−	0.897360i	$-0.354518\pi$
$239$	4.45198e10	0.882597	0.441298	+	0.897360i	$0.354518\pi$
$240$	0	0
$241$	−2.33464e10	−0.445804	−0.222902	−	0.974841i	$-0.571553\pi$
$241$	−2.33464e10	−0.445804	−0.222902	+	0.974841i	$0.571553\pi$
$242$	0	0
$243$	8.47816e9	0.155982
$244$	0	0
$245$	6.86686e9	0.121762
$246$	0	0
$247$	1.37579e10	0.235188
$248$	0	0
$249$	−3.46930e8	−0.00571932
$250$	0	0
$251$	1.17150e11	1.86300	0.931499	−	0.363744i	$-0.118502\pi$
$251$	1.17150e11	1.86300	0.931499	+	0.363744i	$0.118502\pi$
$252$	0	0
$253$	1.93585e10	0.297050
$254$	0	0
$255$	2.47268e9	0.0366216
$256$	0	0
$257$	−1.36110e11	−1.94621	−0.973106	−	0.230360i	$-0.926010\pi$
$257$	−1.36110e11	−1.94621	−0.973106	+	0.230360i	$0.926010\pi$
$258$	0	0
$259$	−1.86430e10	−0.257434
$260$	0	0
$261$	1.06022e11	1.41422
$262$	0	0
$263$	−1.05020e11	−1.35354	−0.676770	−	0.736194i	$-0.736621\pi$
$263$	−1.05020e11	−1.35354	−0.676770	+	0.736194i	$0.736621\pi$
$264$	0	0
$265$	−7.27438e10	−0.906128
$266$	0	0
$267$	5.10317e9	0.0614525
$268$	0	0
$269$	1.18762e11	1.38291	0.691453	−	0.722421i	$-0.256971\pi$
$269$	1.18762e11	1.38291	0.691453	+	0.722421i	$0.256971\pi$
$270$	0	0
$271$	1.45069e11	1.63385	0.816926	−	0.576742i	$-0.195676\pi$
$271$	1.45069e11	1.63385	0.816926	+	0.576742i	$0.195676\pi$
$272$	0	0
$273$	8.84760e8	0.00964037
$274$	0	0
$275$	−9.75108e9	−0.102815
$276$	0	0
$277$	1.38010e11	1.40848	0.704240	−	0.709962i	$-0.251288\pi$
$277$	1.38010e11	1.40848	0.704240	+	0.709962i	$0.251288\pi$
$278$	0	0
$279$	−3.45452e10	−0.341325
$280$	0	0
$281$	−2.20796e10	−0.211258	−0.105629	−	0.994406i	$-0.533686\pi$
$281$	−2.20796e10	−0.211258	−0.105629	+	0.994406i	$0.533686\pi$
$282$	0	0
$283$	−9.58947e9	−0.0888701	−0.0444350	−	0.999012i	$-0.514149\pi$
$283$	−9.58947e9	−0.0888701	−0.0444350	+	0.999012i	$0.514149\pi$
$284$	0	0
$285$	2.38366e9	0.0214014
$286$	0	0
$287$	2.14235e10	0.186390
$288$	0	0
$289$	−3.81916e10	−0.322053
$290$	0	0
$291$	5.72811e9	0.0468266
$292$	0	0
$293$	1.38336e11	1.09655	0.548277	−	0.836297i	$-0.315284\pi$
$293$	1.38336e11	1.09655	0.548277	+	0.836297i	$0.315284\pi$
$294$	0	0
$295$	1.07469e11	0.826198
$296$	0	0
$297$	−5.25318e9	−0.0391758
$298$	0	0
$299$	5.33843e10	0.386272
$300$	0	0
$301$	8.31859e10	0.584118
$302$	0	0
$303$	7.67819e9	0.0523319
$304$	0	0
$305$	1.81594e11	1.20158
$306$	0	0
$307$	−4.36744e10	−0.280611	−0.140305	−	0.990108i	$-0.544808\pi$
$307$	−4.36744e10	−0.280611	−0.140305	+	0.990108i	$0.544808\pi$
$308$	0	0
$309$	1.25778e10	0.0784859
$310$	0	0
$311$	−2.65756e11	−1.61088	−0.805438	−	0.592681i	$-0.798070\pi$
$311$	−2.65756e11	−1.61088	−0.805438	+	0.592681i	$0.798070\pi$
$312$	0	0
$313$	−2.73064e11	−1.60811	−0.804055	−	0.594556i	$-0.797328\pi$
$313$	−2.73064e11	−1.60811	−0.804055	+	0.594556i	$0.797328\pi$
$314$	0	0
$315$	−5.61400e10	−0.321274
$316$	0	0
$317$	2.17819e11	1.21151	0.605757	−	0.795650i	$-0.292870\pi$
$317$	2.17819e11	1.21151	0.605757	+	0.795650i	$0.292870\pi$
$318$	0	0
$319$	−9.85841e10	−0.533027
$320$	0	0
$321$	1.13076e9	0.00594428
$322$	0	0
$323$	7.75020e10	0.396188
$324$	0	0
$325$	−2.68902e10	−0.133696
$326$	0	0
$327$	9.19869e9	0.0444899
$328$	0	0
$329$	8.79468e10	0.413846
$330$	0	0
$331$	−4.26311e11	−1.95209	−0.976046	−	0.217565i	$-0.930189\pi$
$331$	−4.26311e11	−1.95209	−0.976046	+	0.217565i	$0.930189\pi$
$332$	0	0
$333$	1.52416e11	0.679253
$334$	0	0
$335$	2.60835e10	0.113153
$336$	0	0
$337$	−4.14030e11	−1.74863	−0.874314	−	0.485361i	$-0.838688\pi$
$337$	−4.14030e11	−1.74863	−0.874314	+	0.485361i	$0.838688\pi$
$338$	0	0
$339$	−5.28298e9	−0.0217260
$340$	0	0
$341$	3.21215e10	0.128648
$342$	0	0
$343$	1.38413e10	0.0539949
$344$	0	0
$345$	9.24925e9	0.0351496
$346$	0	0
$347$	−7.08800e10	−0.262447	−0.131223	−	0.991353i	$-0.541891\pi$
$347$	−7.08800e10	−0.262447	−0.131223	+	0.991353i	$0.541891\pi$
$348$	0	0
$349$	−3.11323e11	−1.12330	−0.561652	−	0.827374i	$-0.689834\pi$
$349$	−3.11323e11	−1.12330	−0.561652	+	0.827374i	$0.689834\pi$
$350$	0	0
$351$	−1.44865e10	−0.0509426
$352$	0	0
$353$	−1.47254e11	−0.504757	−0.252378	−	0.967629i	$-0.581213\pi$
$353$	−1.47254e11	−0.504757	−0.252378	+	0.967629i	$0.581213\pi$
$354$	0	0
$355$	2.78139e11	0.929469
$356$	0	0
$357$	4.98410e9	0.0162398
$358$	0	0
$359$	−1.73438e11	−0.551087	−0.275544	−	0.961289i	$-0.588858\pi$
$359$	−1.73438e11	−0.551087	−0.275544	+	0.961289i	$0.588858\pi$
$360$	0	0
$361$	−2.47976e11	−0.768470
$362$	0	0
$363$	−1.48238e10	−0.0448105
$364$	0	0
$365$	1.27103e11	0.374834
$366$	0	0
$367$	−1.92678e10	−0.0554415	−0.0277207	−	0.999616i	$-0.508825\pi$
$367$	−1.92678e10	−0.0554415	−0.0277207	+	0.999616i	$0.508825\pi$
$368$	0	0
$369$	−1.75148e11	−0.491799
$370$	0	0
$371$	−1.46627e11	−0.401821
$372$	0	0
$373$	−3.12662e11	−0.836344	−0.418172	−	0.908368i	$-0.637329\pi$
$373$	−3.12662e11	−0.836344	−0.418172	+	0.908368i	$0.637329\pi$
$374$	0	0
$375$	−2.16915e10	−0.0566433
$376$	0	0
$377$	−2.71862e11	−0.693125
$378$	0	0
$379$	6.96981e11	1.73518	0.867590	−	0.497280i	$-0.165668\pi$
$379$	6.96981e11	1.73518	0.867590	+	0.497280i	$0.165668\pi$
$380$	0	0
$381$	−1.16401e10	−0.0283006
$382$	0	0
$383$	4.23541e11	1.00577	0.502887	−	0.864352i	$-0.332271\pi$
$383$	4.23541e11	1.00577	0.502887	+	0.864352i	$0.332271\pi$
$384$	0	0
$385$	5.22014e10	0.121090
$386$	0	0
$387$	−6.80088e11	−1.54122
$388$	0	0
$389$	3.85703e11	0.854043	0.427022	−	0.904241i	$-0.359563\pi$
$389$	3.85703e11	0.854043	0.427022	+	0.904241i	$0.359563\pi$
$390$	0	0
$391$	3.00729e11	0.650698
$392$	0	0
$393$	−3.40671e9	−0.00720391
$394$	0	0
$395$	−3.45117e11	−0.713313
$396$	0	0
$397$	−5.55965e11	−1.12329	−0.561643	−	0.827380i	$-0.689830\pi$
$397$	−5.55965e11	−1.12329	−0.561643	+	0.827380i	$0.689830\pi$
$398$	0	0
$399$	4.80467e9	0.00949042
$400$	0	0
$401$	−7.97480e11	−1.54018	−0.770088	−	0.637938i	$-0.779788\pi$
$401$	−7.97480e11	−1.54018	−0.770088	+	0.637938i	$0.779788\pi$
$402$	0	0
$403$	8.85803e10	0.167288
$404$	0	0
$405$	4.57717e11	0.845375
$406$	0	0
$407$	−1.41723e11	−0.256015
$408$	0	0
$409$	−4.54805e11	−0.803657	−0.401828	−	0.915715i	$-0.631625\pi$
$409$	−4.54805e11	−0.803657	−0.401828	+	0.915715i	$0.631625\pi$
$410$	0	0
$411$	−1.54069e9	−0.00266335
$412$	0	0
$413$	2.16622e11	0.366376
$414$	0	0
$415$	−5.64467e10	−0.0934162
$416$	0	0
$417$	3.34364e10	0.0541510
$418$	0	0
$419$	1.01004e11	0.160095	0.0800473	−	0.996791i	$-0.474493\pi$
$419$	1.01004e11	0.160095	0.0800473	+	0.996791i	$0.474493\pi$
$420$	0	0
$421$	3.03777e11	0.471287	0.235643	−	0.971840i	$-0.424280\pi$
$421$	3.03777e11	0.471287	0.235643	+	0.971840i	$0.424280\pi$
$422$	0	0
$423$	−7.19010e11	−1.09195
$424$	0	0
$425$	−1.51480e11	−0.225219
$426$	0	0
$427$	3.66034e11	0.532838
$428$	0	0
$429$	6.72588e9	0.00958719
$430$	0	0
$431$	−2.08588e11	−0.291167	−0.145584	−	0.989346i	$-0.546506\pi$
$431$	−2.08588e11	−0.291167	−0.145584	+	0.989346i	$0.546506\pi$
$432$	0	0
$433$	−1.37965e11	−0.188614	−0.0943071	−	0.995543i	$-0.530064\pi$
$433$	−1.37965e11	−0.188614	−0.0943071	+	0.995543i	$0.530064\pi$
$434$	0	0
$435$	−4.71022e10	−0.0630724
$436$	0	0
$437$	2.89902e11	0.380264
$438$	0	0
$439$	4.56339e11	0.586404	0.293202	−	0.956051i	$-0.405279\pi$
$439$	4.56339e11	0.586404	0.293202	+	0.956051i	$0.405279\pi$
$440$	0	0
$441$	−1.13160e11	−0.142468
$442$	0	0
$443$	3.54325e11	0.437104	0.218552	−	0.975825i	$-0.429867\pi$
$443$	3.54325e11	0.437104	0.218552	+	0.975825i	$0.429867\pi$
$444$	0	0
$445$	8.30304e11	1.00373
$446$	0	0
$447$	2.77414e9	0.00328659
$448$	0	0
$449$	2.75864e11	0.320322	0.160161	−	0.987091i	$-0.448799\pi$
$449$	2.75864e11	0.320322	0.160161	+	0.987091i	$0.448799\pi$
$450$	0	0
$451$	1.62860e11	0.185362
$452$	0	0
$453$	2.58278e10	0.0288168
$454$	0	0
$455$	1.43954e11	0.157460
$456$	0	0
$457$	8.09394e11	0.868034	0.434017	−	0.900905i	$-0.357096\pi$
$457$	8.09394e11	0.868034	0.434017	+	0.900905i	$0.357096\pi$
$458$	0	0
$459$	−8.16064e10	−0.0858157
$460$	0	0
$461$	9.46929e11	0.976480	0.488240	−	0.872710i	$-0.337639\pi$
$461$	9.46929e11	0.976480	0.488240	+	0.872710i	$0.337639\pi$
$462$	0	0
$463$	−6.61863e11	−0.669351	−0.334675	−	0.942334i	$-0.608627\pi$
$463$	−6.61863e11	−0.669351	−0.334675	+	0.942334i	$0.608627\pi$
$464$	0	0
$465$	1.53472e10	0.0152227
$466$	0	0
$467$	−1.37167e12	−1.33451	−0.667257	−	0.744828i	$-0.732532\pi$
$467$	−1.37167e12	−1.33451	−0.667257	+	0.744828i	$0.732532\pi$
$468$	0	0
$469$	5.25757e10	0.0501773
$470$	0	0
$471$	2.11734e10	0.0198242
$472$	0	0
$473$	6.32374e11	0.580896
$474$	0	0
$475$	−1.46026e11	−0.131616
$476$	0	0
$477$	1.19875e12	1.06022
$478$	0	0
$479$	−1.51641e12	−1.31615	−0.658076	−	0.752952i	$-0.728629\pi$
$479$	−1.51641e12	−1.31615	−0.658076	+	0.752952i	$0.728629\pi$
$480$	0	0
$481$	−3.90823e11	−0.332910
$482$	0	0
$483$	1.86434e10	0.0155870
$484$	0	0
$485$	9.31985e11	0.764841
$486$	0	0
$487$	1.96046e12	1.57935	0.789673	−	0.613528i	$-0.210250\pi$
$487$	1.96046e12	1.57935	0.789673	+	0.613528i	$0.210250\pi$
$488$	0	0
$489$	−8.05233e10	−0.0636842
$490$	0	0
$491$	1.90574e12	1.47978	0.739890	−	0.672727i	$-0.234877\pi$
$491$	1.90574e12	1.47978	0.739890	+	0.672727i	$0.234877\pi$
$492$	0	0
$493$	−1.53147e12	−1.16761
$494$	0	0
$495$	−4.26773e11	−0.319502
$496$	0	0
$497$	5.60636e11	0.412171
$498$	0	0
$499$	8.05675e11	0.581711	0.290855	−	0.956767i	$-0.406060\pi$
$499$	8.05675e11	0.581711	0.290855	+	0.956767i	$0.406060\pi$
$500$	0	0
$501$	−1.23837e11	−0.0878174
$502$	0	0
$503$	2.21037e12	1.53961	0.769803	−	0.638282i	$-0.220355\pi$
$503$	2.21037e12	1.53961	0.769803	+	0.638282i	$0.220355\pi$
$504$	0	0
$505$	1.24927e12	0.854761
$506$	0	0
$507$	−5.90889e10	−0.0397164
$508$	0	0
$509$	1.97337e12	1.30310	0.651552	−	0.758604i	$-0.274118\pi$
$509$	1.97337e12	1.30310	0.651552	+	0.758604i	$0.274118\pi$
$510$	0	0
$511$	2.56198e11	0.166219
$512$	0	0
$513$	−7.86685e10	−0.0501502
$514$	0	0
$515$	2.04645e12	1.28194
$516$	0	0
$517$	6.68565e11	0.411563
$518$	0	0
$519$	2.10137e10	0.0127130
$520$	0	0
$521$	−1.86168e12	−1.10697	−0.553484	−	0.832860i	$-0.686702\pi$
$521$	−1.86168e12	−1.10697	−0.553484	+	0.832860i	$0.686702\pi$
$522$	0	0
$523$	−2.08208e12	−1.21686	−0.608430	−	0.793608i	$-0.708200\pi$
$523$	−2.08208e12	−1.21686	−0.608430	+	0.793608i	$0.708200\pi$
$524$	0	0
$525$	−9.39085e9	−0.00539496
$526$	0	0
$527$	4.98998e11	0.281806
$528$	0	0
$529$	−6.76255e11	−0.375457
$530$	0	0
$531$	−1.77099e12	−0.966699
$532$	0	0
$533$	4.49113e11	0.241037
$534$	0	0
$535$	1.83979e11	0.0970906
$536$	0	0
$537$	−1.46108e11	−0.0758209
$538$	0	0
$539$	1.05220e11	0.0536971
$540$	0	0
$541$	−2.69169e12	−1.35094	−0.675471	−	0.737387i	$-0.736060\pi$
$541$	−2.69169e12	−1.35094	−0.675471	+	0.737387i	$0.736060\pi$
$542$	0	0
$543$	6.16276e10	0.0304212
$544$	0	0
$545$	1.49666e12	0.726673
$546$	0	0
$547$	2.85496e12	1.36350	0.681752	−	0.731584i	$-0.261218\pi$
$547$	2.85496e12	1.36350	0.681752	+	0.731584i	$0.261218\pi$
$548$	0	0
$549$	−2.99251e12	−1.40592
$550$	0	0
$551$	−1.47634e12	−0.682344
$552$	0	0
$553$	−6.95641e11	−0.316317
$554$	0	0
$555$	−6.77132e10	−0.0302939
$556$	0	0
$557$	3.62399e11	0.159528	0.0797642	−	0.996814i	$-0.474583\pi$
$557$	3.62399e11	0.159528	0.0797642	+	0.996814i	$0.474583\pi$
$558$	0	0
$559$	1.74387e12	0.755373
$560$	0	0
$561$	3.78888e10	0.0161502
$562$	0	0
$563$	−1.58407e12	−0.664486	−0.332243	−	0.943194i	$-0.607805\pi$
$563$	−1.58407e12	−0.664486	−0.332243	+	0.943194i	$0.607805\pi$
$564$	0	0
$565$	−8.59560e11	−0.354861
$566$	0	0
$567$	9.22604e11	0.374880
$568$	0	0
$569$	−3.12579e12	−1.25013	−0.625064	−	0.780574i	$-0.714927\pi$
$569$	−3.12579e12	−1.25013	−0.625064	+	0.780574i	$0.714927\pi$
$570$	0	0
$571$	6.38509e11	0.251365	0.125682	−	0.992071i	$-0.459888\pi$
$571$	6.38509e11	0.251365	0.125682	+	0.992071i	$0.459888\pi$
$572$	0	0
$573$	−6.03189e10	−0.0233753
$574$	0	0
$575$	−5.66622e11	−0.216166
$576$	0	0
$577$	−2.96097e12	−1.11210	−0.556049	−	0.831149i	$-0.687683\pi$
$577$	−2.96097e12	−1.11210	−0.556049	+	0.831149i	$0.687683\pi$
$578$	0	0
$579$	2.65972e10	0.00983518
$580$	0	0
$581$	−1.13778e11	−0.0414252
$582$	0	0
$583$	−1.11465e12	−0.399604
$584$	0	0
$585$	−1.17689e12	−0.415467
$586$	0	0
$587$	−7.74417e11	−0.269218	−0.134609	−	0.990899i	$-0.542978\pi$
$587$	−7.74417e11	−0.269218	−0.134609	+	0.990899i	$0.542978\pi$
$588$	0	0
$589$	4.81034e11	0.164686
$590$	0	0
$591$	−1.19268e11	−0.0402142
$592$	0	0
$593$	4.77478e12	1.58565	0.792825	−	0.609450i	$-0.208610\pi$
$593$	4.77478e12	1.58565	0.792825	+	0.609450i	$0.208610\pi$
$594$	0	0
$595$	8.10931e11	0.265251
$596$	0	0
$597$	−1.50817e11	−0.0485922
$598$	0	0
$599$	2.61210e12	0.829028	0.414514	−	0.910043i	$-0.363952\pi$
$599$	2.61210e12	0.829028	0.414514	+	0.910043i	$0.363952\pi$
$600$	0	0
$601$	5.29832e12	1.65654	0.828272	−	0.560326i	$-0.189324\pi$
$601$	5.29832e12	1.65654	0.828272	+	0.560326i	$0.189324\pi$
$602$	0	0
$603$	−4.29833e11	−0.132395
$604$	0	0
$605$	−2.41188e12	−0.731909
$606$	0	0
$607$	−5.06614e12	−1.51470	−0.757352	−	0.653006i	$-0.773508\pi$
$607$	−5.06614e12	−1.51470	−0.757352	+	0.653006i	$0.773508\pi$
$608$	0	0
$609$	−9.49422e10	−0.0279693
$610$	0	0
$611$	1.84368e12	0.535179
$612$	0	0
$613$	−4.56765e12	−1.30653	−0.653267	−	0.757127i	$-0.726602\pi$
$613$	−4.56765e12	−1.30653	−0.653267	+	0.757127i	$0.726602\pi$
$614$	0	0
$615$	7.78125e10	0.0219337
$616$	0	0
$617$	5.57819e12	1.54957	0.774784	−	0.632226i	$-0.217859\pi$
$617$	5.57819e12	1.54957	0.774784	+	0.632226i	$0.217859\pi$
$618$	0	0
$619$	−5.50730e12	−1.50776	−0.753878	−	0.657015i	$-0.771819\pi$
$619$	−5.50730e12	−1.50776	−0.753878	+	0.657015i	$0.771819\pi$
$620$	0	0
$621$	−3.05255e11	−0.0823664
$622$	0	0
$623$	1.67362e12	0.445102
$624$	0	0
$625$	−2.48585e12	−0.651650
$626$	0	0
$627$	3.65248e10	0.00943808
$628$	0	0
$629$	−2.20162e12	−0.560807
$630$	0	0
$631$	−5.00751e12	−1.25745	−0.628724	−	0.777629i	$-0.716422\pi$
$631$	−5.00751e12	−1.25745	−0.628724	+	0.777629i	$0.716422\pi$
$632$	0	0
$633$	1.01568e11	0.0251445
$634$	0	0
$635$	−1.89389e12	−0.462247
$636$	0	0
$637$	2.90162e11	0.0698254
$638$	0	0
$639$	−4.58349e12	−1.08753
$640$	0	0
$641$	5.44242e12	1.27330	0.636650	−	0.771153i	$-0.280320\pi$
$641$	5.44242e12	1.27330	0.636650	+	0.771153i	$0.280320\pi$
$642$	0	0
$643$	−9.73637e11	−0.224620	−0.112310	−	0.993673i	$-0.535825\pi$
$643$	−9.73637e11	−0.224620	−0.112310	+	0.993673i	$0.535825\pi$
$644$	0	0
$645$	3.02140e11	0.0687368
$646$	0	0
$647$	−4.51341e12	−1.01259	−0.506297	−	0.862359i	$-0.668986\pi$
$647$	−4.51341e12	−1.01259	−0.506297	+	0.862359i	$0.668986\pi$
$648$	0	0
$649$	1.64674e12	0.364355
$650$	0	0
$651$	3.09349e10	0.00675048
$652$	0	0
$653$	−8.57043e12	−1.84456	−0.922281	−	0.386520i	$-0.873677\pi$
$653$	−8.57043e12	−1.84456	−0.922281	+	0.386520i	$0.873677\pi$
$654$	0	0
$655$	−5.54284e11	−0.117665
$656$	0	0
$657$	−2.09455e12	−0.438577
$658$	0	0
$659$	7.18378e10	0.0148378	0.00741889	−	0.999972i	$-0.497638\pi$
$659$	7.18378e10	0.0148378	0.00741889	+	0.999972i	$0.497638\pi$
$660$	0	0
$661$	−6.58236e12	−1.34114	−0.670572	−	0.741845i	$-0.733951\pi$
$661$	−6.58236e12	−1.34114	−0.670572	+	0.741845i	$0.733951\pi$
$662$	0	0
$663$	1.04484e11	0.0210010
$664$	0	0
$665$	7.81737e11	0.155011
$666$	0	0
$667$	−5.72859e12	−1.12068
$668$	0	0
$669$	2.25746e11	0.0435715
$670$	0	0
$671$	2.78256e12	0.529900
$672$	0	0
$673$	8.11955e12	1.52568	0.762841	−	0.646586i	$-0.223804\pi$
$673$	8.11955e12	1.52568	0.762841	+	0.646586i	$0.223804\pi$
$674$	0	0
$675$	1.53760e11	0.0285086
$676$	0	0
$677$	4.22756e12	0.773465	0.386733	−	0.922192i	$-0.373604\pi$
$677$	4.22756e12	0.773465	0.386733	+	0.922192i	$0.373604\pi$
$678$	0	0
$679$	1.87857e12	0.339167
$680$	0	0
$681$	−1.50770e11	−0.0268629
$682$	0	0
$683$	8.32796e12	1.46435	0.732176	−	0.681115i	$-0.238505\pi$
$683$	8.32796e12	1.46435	0.732176	+	0.681115i	$0.238505\pi$
$684$	0	0
$685$	−2.50676e11	−0.0435016
$686$	0	0
$687$	−3.82698e11	−0.0655467
$688$	0	0
$689$	−3.07383e12	−0.519629
$690$	0	0
$691$	−8.79408e12	−1.46737	−0.733684	−	0.679491i	$-0.762201\pi$
$691$	−8.79408e12	−1.46737	−0.733684	+	0.679491i	$0.762201\pi$
$692$	0	0
$693$	−8.60231e11	−0.141682
$694$	0	0
$695$	5.44022e12	0.884473
$696$	0	0
$697$	2.52998e12	0.406041
$698$	0	0
$699$	1.76399e11	0.0279478
$700$	0	0
$701$	3.99776e12	0.625296	0.312648	−	0.949869i	$-0.398784\pi$
$701$	3.99776e12	0.625296	0.312648	+	0.949869i	$0.398784\pi$
$702$	0	0
$703$	−2.12236e12	−0.327732
$704$	0	0
$705$	3.19431e11	0.0486997
$706$	0	0
$707$	2.51811e12	0.379042
$708$	0	0
$709$	−1.03261e12	−0.153471	−0.0767357	−	0.997051i	$-0.524450\pi$
$709$	−1.03261e12	−0.153471	−0.0767357	+	0.997051i	$0.524450\pi$
$710$	0	0
$711$	5.68722e12	0.834617
$712$	0	0
$713$	1.86654e12	0.270479
$714$	0	0
$715$	1.09433e12	0.156592
$716$	0	0
$717$	3.25934e11	0.0460567
$718$	0	0
$719$	5.05497e12	0.705405	0.352702	−	0.935736i	$-0.385263\pi$
$719$	5.05497e12	0.705405	0.352702	+	0.935736i	$0.385263\pi$
$720$	0	0
$721$	4.12497e12	0.568475
$722$	0	0
$723$	−1.70922e11	−0.0232635
$724$	0	0
$725$	2.88554e12	0.387888
$726$	0	0
$727$	−7.37305e12	−0.978908	−0.489454	−	0.872029i	$-0.662804\pi$
$727$	−7.37305e12	−0.978908	−0.489454	+	0.872029i	$0.662804\pi$
$728$	0	0
$729$	−7.50129e12	−0.983699
$730$	0	0
$731$	9.82372e12	1.27247
$732$	0	0
$733$	−7.03447e12	−0.900044	−0.450022	−	0.893018i	$-0.648584\pi$
$733$	−7.03447e12	−0.900044	−0.450022	+	0.893018i	$0.648584\pi$
$734$	0	0
$735$	5.02729e10	0.00635391
$736$	0	0
$737$	3.99676e11	0.0499005
$738$	0	0
$739$	5.70671e12	0.703859	0.351929	−	0.936027i	$-0.385526\pi$
$739$	5.70671e12	0.703859	0.351929	+	0.936027i	$0.385526\pi$
$740$	0	0
$741$	1.00723e11	0.0122729
$742$	0	0
$743$	1.59220e13	1.91668	0.958339	−	0.285634i	$-0.0922043\pi$
$743$	1.59220e13	1.91668	0.958339	+	0.285634i	$0.0922043\pi$
$744$	0	0
$745$	4.51363e11	0.0536813
$746$	0	0
$747$	9.30191e11	0.109302
$748$	0	0
$749$	3.70841e11	0.0430546
$750$	0	0
$751$	−2.25676e11	−0.0258884	−0.0129442	−	0.999916i	$-0.504120\pi$
$751$	−2.25676e11	−0.0258884	−0.0129442	+	0.999916i	$0.504120\pi$
$752$	0	0
$753$	8.57671e11	0.0972172
$754$	0	0
$755$	4.20228e12	0.470678
$756$	0	0
$757$	−3.81562e12	−0.422312	−0.211156	−	0.977452i	$-0.567723\pi$
$757$	−3.81562e12	−0.422312	−0.211156	+	0.977452i	$0.567723\pi$
$758$	0	0
$759$	1.41726e11	0.0155010
$760$	0	0
$761$	−1.69345e12	−0.183038	−0.0915190	−	0.995803i	$-0.529172\pi$
$761$	−1.69345e12	−0.183038	−0.0915190	+	0.995803i	$0.529172\pi$
$762$	0	0
$763$	3.01677e12	0.322241
$764$	0	0
$765$	−6.62977e12	−0.699878
$766$	0	0
$767$	4.54116e12	0.473792
$768$	0	0
$769$	8.73182e12	0.900402	0.450201	−	0.892927i	$-0.351352\pi$
$769$	8.73182e12	0.900402	0.450201	+	0.892927i	$0.351352\pi$
$770$	0	0
$771$	−9.96473e11	−0.101560
$772$	0	0
$773$	−6.50171e11	−0.0654968	−0.0327484	−	0.999464i	$-0.510426\pi$
$773$	−6.50171e11	−0.0654968	−0.0327484	+	0.999464i	$0.510426\pi$
$774$	0	0
$775$	−9.40193e11	−0.0936180
$776$	0	0
$777$	−1.36487e11	−0.0134338
$778$	0	0
$779$	2.43890e12	0.237288
$780$	0	0
$781$	4.26192e12	0.409898
$782$	0	0
$783$	1.55452e12	0.147798
$784$	0	0
$785$	3.44499e12	0.323798
$786$	0	0
$787$	−1.42694e13	−1.32593	−0.662965	−	0.748651i	$-0.730702\pi$
$787$	−1.42694e13	−1.32593	−0.662965	+	0.748651i	$0.730702\pi$
$788$	0	0
$789$	−7.68862e11	−0.0706321
$790$	0	0
$791$	−1.73259e12	−0.157362
$792$	0	0
$793$	7.67336e12	0.689059
$794$	0	0
$795$	−5.32565e11	−0.0472847
$796$	0	0
$797$	7.25418e12	0.636834	0.318417	−	0.947951i	$-0.396849\pi$
$797$	7.25418e12	0.636834	0.318417	+	0.947951i	$0.396849\pi$
$798$	0	0
$799$	1.03859e13	0.901540
$800$	0	0
$801$	−1.36827e13	−1.17442
$802$	0	0
$803$	1.94760e12	0.165302
$804$	0	0
$805$	3.03335e12	0.254590
$806$	0	0
$807$	8.69469e11	0.0721645
$808$	0	0
$809$	3.77473e12	0.309826	0.154913	−	0.987928i	$-0.450490\pi$
$809$	3.77473e12	0.309826	0.154913	+	0.987928i	$0.450490\pi$
$810$	0	0
$811$	1.87830e13	1.52465	0.762326	−	0.647194i	$-0.224058\pi$
$811$	1.87830e13	1.52465	0.762326	+	0.647194i	$0.224058\pi$
$812$	0	0
$813$	1.06206e12	0.0852596
$814$	0	0
$815$	−1.31014e13	−1.04018
$816$	0	0
$817$	9.47006e12	0.743624
$818$	0	0
$819$	−2.37223e12	−0.184238
$820$	0	0
$821$	1.09187e13	0.838741	0.419371	−	0.907815i	$-0.362251\pi$
$821$	1.09187e13	0.838741	0.419371	+	0.907815i	$0.362251\pi$
$822$	0	0
$823$	−1.65407e13	−1.25677	−0.628385	−	0.777902i	$-0.716284\pi$
$823$	−1.65407e13	−1.25677	−0.628385	+	0.777902i	$0.716284\pi$
$824$	0	0
$825$	−7.13886e10	−0.00536520
$826$	0	0
$827$	6.56644e12	0.488152	0.244076	−	0.969756i	$-0.421515\pi$
$827$	6.56644e12	0.488152	0.244076	+	0.969756i	$0.421515\pi$
$828$	0	0
$829$	−8.09828e12	−0.595522	−0.297761	−	0.954640i	$-0.596240\pi$
$829$	−8.09828e12	−0.595522	−0.297761	+	0.954640i	$0.596240\pi$
$830$	0	0
$831$	1.01038e12	0.0734990
$832$	0	0
$833$	1.63457e12	0.117625
$834$	0	0
$835$	−2.01487e13	−1.43436
$836$	0	0
$837$	−5.06508e11	−0.0356715
$838$	0	0
$839$	7.03849e12	0.490400	0.245200	−	0.969472i	$-0.421146\pi$
$839$	7.03849e12	0.490400	0.245200	+	0.969472i	$0.421146\pi$
$840$	0	0
$841$	1.46659e13	1.01094
$842$	0	0
$843$	−1.61647e11	−0.0110241
$844$	0	0
$845$	−9.61398e12	−0.648706
$846$	0	0
$847$	−4.86155e12	−0.324564
$848$	0	0
$849$	−7.02055e10	−0.00463753
$850$	0	0
$851$	−8.23531e12	−0.538266
$852$	0	0
$853$	−2.19614e12	−0.142033	−0.0710164	−	0.997475i	$-0.522624\pi$
$853$	−2.19614e12	−0.142033	−0.0710164	+	0.997475i	$0.522624\pi$
$854$	0	0
$855$	−6.39110e12	−0.409004
$856$	0	0
$857$	2.27488e13	1.44060	0.720301	−	0.693662i	$-0.244004\pi$
$857$	2.27488e13	1.44060	0.720301	+	0.693662i	$0.244004\pi$
$858$	0	0
$859$	1.67383e13	1.04892	0.524458	−	0.851436i	$-0.324268\pi$
$859$	1.67383e13	1.04892	0.524458	+	0.851436i	$0.324268\pi$
$860$	0	0
$861$	1.56844e11	0.00972643
$862$	0	0
$863$	−7.11605e12	−0.436707	−0.218354	−	0.975870i	$-0.570069\pi$
$863$	−7.11605e12	−0.436707	−0.218354	+	0.975870i	$0.570069\pi$
$864$	0	0
$865$	3.41900e12	0.207648
$866$	0	0
$867$	−2.79604e11	−0.0168057
$868$	0	0
$869$	−5.28822e12	−0.314572
$870$	0	0
$871$	1.10217e12	0.0648885
$872$	0	0
$873$	−1.53583e13	−0.894907
$874$	0	0
$875$	−7.11386e12	−0.410269
$876$	0	0
$877$	6.27864e12	0.358400	0.179200	−	0.983813i	$-0.442649\pi$
$877$	6.27864e12	0.358400	0.179200	+	0.983813i	$0.442649\pi$
$878$	0	0
$879$	1.01277e12	0.0572217
$880$	0	0
$881$	1.53264e12	0.0857133	0.0428566	−	0.999081i	$-0.486354\pi$
$881$	1.53264e12	0.0857133	0.0428566	+	0.999081i	$0.486354\pi$
$882$	0	0
$883$	−1.56128e13	−0.864289	−0.432144	−	0.901804i	$-0.642243\pi$
$883$	−1.56128e13	−0.864289	−0.432144	+	0.901804i	$0.642243\pi$
$884$	0	0
$885$	7.86791e11	0.0431137
$886$	0	0
$887$	6.62055e12	0.359119	0.179559	−	0.983747i	$-0.442533\pi$
$887$	6.62055e12	0.359119	0.179559	+	0.983747i	$0.442533\pi$
$888$	0	0
$889$	−3.81746e12	−0.204982
$890$	0	0
$891$	7.01357e12	0.372812
$892$	0	0
$893$	1.00120e13	0.526855
$894$	0	0
$895$	−2.37722e13	−1.23842
$896$	0	0
$897$	3.90832e11	0.0201569
$898$	0	0
$899$	−9.50542e12	−0.485347
$900$	0	0
$901$	−1.73157e13	−0.875345
$902$	0	0
$903$	6.09013e11	0.0304812
$904$	0	0
$905$	1.00270e13	0.496883
$906$	0	0
$907$	−2.87844e13	−1.41229	−0.706146	−	0.708066i	$-0.749568\pi$
$907$	−2.87844e13	−1.41229	−0.706146	+	0.708066i	$0.749568\pi$
$908$	0	0
$909$	−2.05868e13	−1.00012
$910$	0	0
$911$	1.82382e13	0.877304	0.438652	−	0.898657i	$-0.355456\pi$
$911$	1.82382e13	0.877304	0.438652	+	0.898657i	$0.355456\pi$
$912$	0	0
$913$	−8.64930e11	−0.0411967
$914$	0	0
$915$	1.32947e12	0.0627024
$916$	0	0
$917$	−1.11725e12	−0.0521782
$918$	0	0
$919$	−2.83922e12	−0.131304	−0.0656521	−	0.997843i	$-0.520913\pi$
$919$	−2.83922e12	−0.131304	−0.0656521	+	0.997843i	$0.520913\pi$
$920$	0	0
$921$	−3.19745e11	−0.0146432
$922$	0	0
$923$	1.17529e13	0.533014
$924$	0	0
$925$	4.14820e12	0.186304
$926$	0	0
$927$	−3.37237e13	−1.49995
$928$	0	0
$929$	−4.25676e13	−1.87503	−0.937514	−	0.347946i	$-0.886879\pi$
$929$	−4.25676e13	−1.87503	−0.937514	+	0.347946i	$0.886879\pi$
$930$	0	0
$931$	1.57572e12	0.0687394
$932$	0	0
$933$	−1.94563e12	−0.0840606
$934$	0	0
$935$	6.16464e12	0.263788
$936$	0	0
$937$	−2.54410e13	−1.07822	−0.539108	−	0.842237i	$-0.681238\pi$
$937$	−2.54410e13	−1.07822	−0.539108	+	0.842237i	$0.681238\pi$
$938$	0	0
$939$	−1.99913e12	−0.0839163
$940$	0	0
$941$	−2.05404e13	−0.853996	−0.426998	−	0.904253i	$-0.640429\pi$
$941$	−2.05404e13	−0.853996	−0.426998	+	0.904253i	$0.640429\pi$
$942$	0	0
$943$	9.46359e12	0.389720
$944$	0	0
$945$	−8.23136e11	−0.0335760
$946$	0	0
$947$	−6.15960e12	−0.248873	−0.124436	−	0.992228i	$-0.539712\pi$
$947$	−6.15960e12	−0.248873	−0.124436	+	0.992228i	$0.539712\pi$
$948$	0	0
$949$	5.37081e12	0.214952
$950$	0	0
$951$	1.59467e12	0.0632206
$952$	0	0
$953$	−3.37083e13	−1.32379	−0.661895	−	0.749597i	$-0.730247\pi$
$953$	−3.37083e13	−1.32379	−0.661895	+	0.749597i	$0.730247\pi$
$954$	0	0
$955$	−9.81411e12	−0.381800
$956$	0	0
$957$	−7.21744e11	−0.0278150
$958$	0	0
$959$	−5.05280e11	−0.0192907
$960$	0	0
$961$	−2.33425e13	−0.882860
$962$	0	0
$963$	−3.03182e12	−0.113602
$964$	0	0
$965$	4.32746e12	0.160642
$966$	0	0
$967$	2.89462e13	1.06456	0.532282	−	0.846567i	$-0.321335\pi$
$967$	2.89462e13	1.06456	0.532282	+	0.846567i	$0.321335\pi$
$968$	0	0
$969$	5.67400e11	0.0206744
$970$	0	0
$971$	3.65630e13	1.31994	0.659971	−	0.751291i	$-0.270569\pi$
$971$	3.65630e13	1.31994	0.659971	+	0.751291i	$0.270569\pi$
$972$	0	0
$973$	1.09657e13	0.392217
$974$	0	0
$975$	−1.96866e11	−0.00697668
$976$	0	0
$977$	1.23119e13	0.432313	0.216156	−	0.976359i	$-0.430648\pi$
$977$	1.23119e13	0.432313	0.216156	+	0.976359i	$0.430648\pi$
$978$	0	0
$979$	1.27227e13	0.442647
$980$	0	0
$981$	−2.46636e13	−0.850249
$982$	0	0
$983$	4.82186e13	1.64711	0.823556	−	0.567234i	$-0.191987\pi$
$983$	4.82186e13	1.64711	0.823556	+	0.567234i	$0.191987\pi$
$984$	0	0
$985$	−1.94053e13	−0.656837
$986$	0	0
$987$	6.43867e11	0.0215958
$988$	0	0
$989$	3.67464e13	1.22132
$990$	0	0
$991$	−4.42370e13	−1.45698	−0.728491	−	0.685055i	$-0.759778\pi$
$991$	−4.42370e13	−1.45698	−0.728491	+	0.685055i	$0.759778\pi$
$992$	0	0
$993$	−3.12106e12	−0.101866
$994$	0	0
$995$	−2.45385e13	−0.793678
$996$	0	0
$997$	6.21063e13	1.99071	0.995354	−	0.0962782i	$-0.0306939\pi$
$997$	6.21063e13	1.99071	0.995354	+	0.0962782i	$0.0306939\pi$
$998$	0	0
$999$	2.23475e12	0.0709880

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.a.i.1.2		3
4.3	odd	2		56.10.a.a.1.2	✓	3
28.27	even	2		392.10.a.d.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.10.a.a.1.2	✓	3	4.3	odd	2
112.10.a.i.1.2		3	1.1	even	1	trivial
392.10.a.d.1.2		3	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	112.a (trivial)

\(f(q)\)	\(=\)	\(q+7.32110 q^{3} +1191.17 q^{5} +2401.00 q^{7} -19629.4 q^{9} +O(q^{10})\)	\(q+7.32110 q^{3} +1191.17 q^{5} +2401.00 q^{7} -19629.4 q^{9} +18252.2 q^{11} +50333.5 q^{13} +8720.67 q^{15} +283542. q^{17} +273335. q^{19} +17578.0 q^{21} +1.06061e6 q^{23} -534240. q^{25} -287810. q^{27} -5.40121e6 q^{29} +1.75987e6 q^{31} +133626. q^{33} +2.86000e6 q^{35} -7.76468e6 q^{37} +368496. q^{39} +8.92276e6 q^{41} +3.46464e7 q^{43} -2.33819e7 q^{45} +3.66292e7 q^{47} +5.76480e6 q^{49} +2.07584e6 q^{51} -6.10693e7 q^{53} +2.17415e7 q^{55} +2.00111e6 q^{57} +9.02214e7 q^{59} +1.52451e8 q^{61} -4.71302e7 q^{63} +5.99557e7 q^{65} +2.18974e7 q^{67} +7.76485e6 q^{69} +2.33501e8 q^{71} +1.06705e8 q^{73} -3.91123e6 q^{75} +4.38236e7 q^{77} -2.89730e8 q^{79} +3.84258e8 q^{81} -4.73876e7 q^{83} +3.37747e8 q^{85} -3.95428e7 q^{87} +6.97049e8 q^{89} +1.20851e8 q^{91} +1.28842e7 q^{93} +3.25588e8 q^{95} +7.82411e8 q^{97} -3.58280e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 92 q^{3} + 274 q^{5} + 7203 q^{7} + 51871 q^{9}+O(q^{10}) \) 3 * q + 92 * q^3 + 274 * q^5 + 7203 * q^7 + 51871 * q^9	\( 3 q + 92 q^{3} + 274 q^{5} + 7203 q^{7} + 51871 q^{9} + 45364 q^{11} - 11158 q^{13} + 95584 q^{15} - 55866 q^{17} - 488772 q^{19} + 220892 q^{21} + 253888 q^{23} - 3872619 q^{25} + 10157192 q^{27} - 765318 q^{29} + 8189680 q^{31} - 7926240 q^{33} + 657874 q^{35} + 7208194 q^{37} + 44623664 q^{39} - 57891986 q^{41} + 37304708 q^{43} - 45527158 q^{45} + 29210992 q^{47} + 17294403 q^{49} + 142742952 q^{51} + 45140082 q^{53} - 1479800 q^{55} - 77053192 q^{57} + 205334804 q^{59} - 109264870 q^{61} + 124542271 q^{63} + 143056644 q^{65} + 52322348 q^{67} + 425002688 q^{69} + 150048136 q^{71} + 761429998 q^{73} - 260540684 q^{75} + 108918964 q^{77} + 214389152 q^{79} + 1527489907 q^{81} + 36110892 q^{83} + 675044052 q^{85} + 236662680 q^{87} + 1319848398 q^{89} - 26790358 q^{91} + 2846904208 q^{93} + 620260624 q^{95} + 101766310 q^{97} - 168732636 q^{99}+O(q^{100}) \) 3 * q + 92 * q^3 + 274 * q^5 + 7203 * q^7 + 51871 * q^9 + 45364 * q^11 - 11158 * q^13 + 95584 * q^15 - 55866 * q^17 - 488772 * q^19 + 220892 * q^21 + 253888 * q^23 - 3872619 * q^25 + 10157192 * q^27 - 765318 * q^29 + 8189680 * q^31 - 7926240 * q^33 + 657874 * q^35 + 7208194 * q^37 + 44623664 * q^39 - 57891986 * q^41 + 37304708 * q^43 - 45527158 * q^45 + 29210992 * q^47 + 17294403 * q^49 + 142742952 * q^51 + 45140082 * q^53 - 1479800 * q^55 - 77053192 * q^57 + 205334804 * q^59 - 109264870 * q^61 + 124542271 * q^63 + 143056644 * q^65 + 52322348 * q^67 + 425002688 * q^69 + 150048136 * q^71 + 761429998 * q^73 - 260540684 * q^75 + 108918964 * q^77 + 214389152 * q^79 + 1527489907 * q^81 + 36110892 * q^83 + 675044052 * q^85 + 236662680 * q^87 + 1319848398 * q^89 - 26790358 * q^91 + 2846904208 * q^93 + 620260624 * q^95 + 101766310 * q^97 - 168732636 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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