LMFDB - Embedded newform 232.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [232,2,Mod(1,232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(232, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("232.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 232 = 2^{3} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	232.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:	$1.85252932689$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	232.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214 q^{3} -3.82843 q^{5} -4.00000 q^{7} -2.82843 q^{9} +O(q^{10})$	$q+0.414214 q^{3} -3.82843 q^{5} -4.00000 q^{7} -2.82843 q^{9} -2.41421 q^{11} +4.65685 q^{13} -1.58579 q^{15} +3.65685 q^{17} +2.00000 q^{19} -1.65685 q^{21} -4.82843 q^{23} +9.65685 q^{25} -2.41421 q^{27} +1.00000 q^{29} -8.41421 q^{31} -1.00000 q^{33} +15.3137 q^{35} -1.65685 q^{37} +1.92893 q^{39} -9.65685 q^{41} +1.58579 q^{43} +10.8284 q^{45} -12.0711 q^{47} +9.00000 q^{49} +1.51472 q^{51} -7.00000 q^{53} +9.24264 q^{55} +0.828427 q^{57} +10.4853 q^{59} +6.00000 q^{61} +11.3137 q^{63} -17.8284 q^{65} -5.65685 q^{67} -2.00000 q^{69} +6.48528 q^{71} +4.00000 q^{73} +4.00000 q^{75} +9.65685 q^{77} +9.72792 q^{79} +7.48528 q^{81} -8.82843 q^{83} -14.0000 q^{85} +0.414214 q^{87} -9.65685 q^{89} -18.6274 q^{91} -3.48528 q^{93} -7.65685 q^{95} -7.31371 q^{97} +6.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} - 8 q^{7}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 - 8 * q^7	$ 2 q - 2 q^{3} - 2 q^{5} - 8 q^{7} - 2 q^{11} - 2 q^{13} - 6 q^{15} - 4 q^{17} + 4 q^{19} + 8 q^{21} - 4 q^{23} + 8 q^{25} - 2 q^{27} + 2 q^{29} - 14 q^{31} - 2 q^{33} + 8 q^{35} + 8 q^{37} + 18 q^{39} - 8 q^{41} + 6 q^{43} + 16 q^{45} - 10 q^{47} + 18 q^{49} + 20 q^{51} - 14 q^{53} + 10 q^{55} - 4 q^{57} + 4 q^{59} + 12 q^{61} - 30 q^{65} - 4 q^{69} - 4 q^{71} + 8 q^{73} + 8 q^{75} + 8 q^{77} - 6 q^{79} - 2 q^{81} - 12 q^{83} - 28 q^{85} - 2 q^{87} - 8 q^{89} + 8 q^{91} + 10 q^{93} - 4 q^{95} + 8 q^{97} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 - 8 * q^7 - 2 * q^11 - 2 * q^13 - 6 * q^15 - 4 * q^17 + 4 * q^19 + 8 * q^21 - 4 * q^23 + 8 * q^25 - 2 * q^27 + 2 * q^29 - 14 * q^31 - 2 * q^33 + 8 * q^35 + 8 * q^37 + 18 * q^39 - 8 * q^41 + 6 * q^43 + 16 * q^45 - 10 * q^47 + 18 * q^49 + 20 * q^51 - 14 * q^53 + 10 * q^55 - 4 * q^57 + 4 * q^59 + 12 * q^61 - 30 * q^65 - 4 * q^69 - 4 * q^71 + 8 * q^73 + 8 * q^75 + 8 * q^77 - 6 * q^79 - 2 * q^81 - 12 * q^83 - 28 * q^85 - 2 * q^87 - 8 * q^89 + 8 * q^91 + 10 * q^93 - 4 * q^95 + 8 * q^97 + 8 * 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
$3$	0.414214	0.239146	0.119573	−	0.992825i	$-0.461847\pi$
$3$	0.414214	0.239146	0.119573	+	0.992825i	$0.461847\pi$
$4$	0	0
$5$	−3.82843	−1.71212	−0.856062	−	0.516873i	$-0.827096\pi$
$5$	−3.82843	−1.71212	−0.856062	+	0.516873i	$0.827096\pi$
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	−2.82843	−0.942809
$10$	0	0
$11$	−2.41421	−0.727913	−0.363956	−	0.931416i	$-0.618574\pi$
$11$	−2.41421	−0.727913	−0.363956	+	0.931416i	$0.618574\pi$
$12$	0	0
$13$	4.65685	1.29158	0.645789	−	0.763516i	$-0.276528\pi$
$13$	4.65685	1.29158	0.645789	+	0.763516i	$0.276528\pi$
$14$	0	0
$15$	−1.58579	−0.409448
$16$	0	0
$17$	3.65685	0.886917	0.443459	−	0.896295i	$-0.353751\pi$
$17$	3.65685	0.886917	0.443459	+	0.896295i	$0.353751\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	−1.65685	−0.361555
$22$	0	0
$23$	−4.82843	−1.00680	−0.503398	−	0.864054i	$-0.667917\pi$
$23$	−4.82843	−1.00680	−0.503398	+	0.864054i	$0.667917\pi$
$24$	0	0
$25$	9.65685	1.93137
$26$	0	0
$27$	−2.41421	−0.464616
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−8.41421	−1.51124	−0.755619	−	0.655012i	$-0.772664\pi$
$31$	−8.41421	−1.51124	−0.755619	+	0.655012i	$0.772664\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	15.3137	2.58849
$36$	0	0
$37$	−1.65685	−0.272385	−0.136193	−	0.990682i	$-0.543487\pi$
$37$	−1.65685	−0.272385	−0.136193	+	0.990682i	$0.543487\pi$
$38$	0	0
$39$	1.92893	0.308876
$40$	0	0
$41$	−9.65685	−1.50815	−0.754074	−	0.656790i	$-0.771914\pi$
$41$	−9.65685	−1.50815	−0.754074	+	0.656790i	$0.771914\pi$
$42$	0	0
$43$	1.58579	0.241830	0.120915	−	0.992663i	$-0.461417\pi$
$43$	1.58579	0.241830	0.120915	+	0.992663i	$0.461417\pi$
$44$	0	0
$45$	10.8284	1.61421
$46$	0	0
$47$	−12.0711	−1.76075	−0.880373	−	0.474282i	$-0.842708\pi$
$47$	−12.0711	−1.76075	−0.880373	+	0.474282i	$0.842708\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	1.51472	0.212103
$52$	0	0
$53$	−7.00000	−0.961524	−0.480762	−	0.876851i	$-0.659640\pi$
$53$	−7.00000	−0.961524	−0.480762	+	0.876851i	$0.659640\pi$
$54$	0	0
$55$	9.24264	1.24628
$56$	0	0
$57$	0.828427	0.109728
$58$	0	0
$59$	10.4853	1.36507	0.682534	−	0.730854i	$-0.260878\pi$
$59$	10.4853	1.36507	0.682534	+	0.730854i	$0.260878\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	11.3137	1.42539
$64$	0	0
$65$	−17.8284	−2.21134
$66$	0	0
$67$	−5.65685	−0.691095	−0.345547	−	0.938401i	$-0.612307\pi$
$67$	−5.65685	−0.691095	−0.345547	+	0.938401i	$0.612307\pi$
$68$	0	0
$69$	−2.00000	−0.240772
$70$	0	0
$71$	6.48528	0.769661	0.384831	−	0.922987i	$-0.374260\pi$
$71$	6.48528	0.769661	0.384831	+	0.922987i	$0.374260\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	9.65685	1.10050
$78$	0	0
$79$	9.72792	1.09448	0.547238	−	0.836977i	$-0.315679\pi$
$79$	9.72792	1.09448	0.547238	+	0.836977i	$0.315679\pi$
$80$	0	0
$81$	7.48528	0.831698
$82$	0	0
$83$	−8.82843	−0.969046	−0.484523	−	0.874779i	$-0.661007\pi$
$83$	−8.82843	−0.969046	−0.484523	+	0.874779i	$0.661007\pi$
$84$	0	0
$85$	−14.0000	−1.51851
$86$	0	0
$87$	0.414214	0.0444084
$88$	0	0
$89$	−9.65685	−1.02362	−0.511812	−	0.859097i	$-0.671026\pi$
$89$	−9.65685	−1.02362	−0.511812	+	0.859097i	$0.671026\pi$
$90$	0	0
$91$	−18.6274	−1.95268
$92$	0	0
$93$	−3.48528	−0.361407
$94$	0	0
$95$	−7.65685	−0.785577
$96$	0	0
$97$	−7.31371	−0.742595	−0.371297	−	0.928514i	$-0.621087\pi$
$97$	−7.31371	−0.742595	−0.371297	+	0.928514i	$0.621087\pi$
$98$	0	0
$99$	6.82843	0.686283
$100$	0	0
$101$	11.3137	1.12576	0.562878	−	0.826540i	$-0.309694\pi$
$101$	11.3137	1.12576	0.562878	+	0.826540i	$0.309694\pi$
$102$	0	0
$103$	−16.8284	−1.65815	−0.829077	−	0.559134i	$-0.811134\pi$
$103$	−16.8284	−1.65815	−0.829077	+	0.559134i	$0.811134\pi$
$104$	0	0
$105$	6.34315	0.619028
$106$	0	0
$107$	2.34315	0.226520	0.113260	−	0.993565i	$-0.463871\pi$
$107$	2.34315	0.226520	0.113260	+	0.993565i	$0.463871\pi$
$108$	0	0
$109$	1.82843	0.175132	0.0875658	−	0.996159i	$-0.472091\pi$
$109$	1.82843	0.175132	0.0875658	+	0.996159i	$0.472091\pi$
$110$	0	0
$111$	−0.686292	−0.0651399
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	18.4853	1.72376
$116$	0	0
$117$	−13.1716	−1.21771
$118$	0	0
$119$	−14.6274	−1.34089
$120$	0	0
$121$	−5.17157	−0.470143
$122$	0	0
$123$	−4.00000	−0.360668
$124$	0	0
$125$	−17.8284	−1.59462
$126$	0	0
$127$	15.6569	1.38932	0.694661	−	0.719338i	$-0.255555\pi$
$127$	15.6569	1.38932	0.694661	+	0.719338i	$0.255555\pi$
$128$	0	0
$129$	0.656854	0.0578328
$130$	0	0
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	9.24264	0.795480
$136$	0	0
$137$	6.34315	0.541932	0.270966	−	0.962589i	$-0.412657\pi$
$137$	6.34315	0.541932	0.270966	+	0.962589i	$0.412657\pi$
$138$	0	0
$139$	8.82843	0.748817	0.374409	−	0.927264i	$-0.377846\pi$
$139$	8.82843	0.748817	0.374409	+	0.927264i	$0.377846\pi$
$140$	0	0
$141$	−5.00000	−0.421076
$142$	0	0
$143$	−11.2426	−0.940157
$144$	0	0
$145$	−3.82843	−0.317934
$146$	0	0
$147$	3.72792	0.307474
$148$	0	0
$149$	8.65685	0.709197	0.354599	−	0.935019i	$-0.384618\pi$
$149$	8.65685	0.709197	0.354599	+	0.935019i	$0.384618\pi$
$150$	0	0
$151$	6.34315	0.516198	0.258099	−	0.966118i	$-0.416904\pi$
$151$	6.34315	0.516198	0.258099	+	0.966118i	$0.416904\pi$
$152$	0	0
$153$	−10.3431	−0.836194
$154$	0	0
$155$	32.2132	2.58743
$156$	0	0
$157$	−11.3137	−0.902932	−0.451466	−	0.892288i	$-0.649099\pi$
$157$	−11.3137	−0.902932	−0.451466	+	0.892288i	$0.649099\pi$
$158$	0	0
$159$	−2.89949	−0.229945
$160$	0	0
$161$	19.3137	1.52213
$162$	0	0
$163$	6.89949	0.540410	0.270205	−	0.962803i	$-0.412908\pi$
$163$	6.89949	0.540410	0.270205	+	0.962803i	$0.412908\pi$
$164$	0	0
$165$	3.82843	0.298043
$166$	0	0
$167$	−1.51472	−0.117212	−0.0586062	−	0.998281i	$-0.518666\pi$
$167$	−1.51472	−0.117212	−0.0586062	+	0.998281i	$0.518666\pi$
$168$	0	0
$169$	8.68629	0.668176
$170$	0	0
$171$	−5.65685	−0.432590
$172$	0	0
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	−38.6274	−2.91996
$176$	0	0
$177$	4.34315	0.326451
$178$	0	0
$179$	5.51472	0.412189	0.206095	−	0.978532i	$-0.433925\pi$
$179$	5.51472	0.412189	0.206095	+	0.978532i	$0.433925\pi$
$180$	0	0
$181$	5.48528	0.407718	0.203859	−	0.979000i	$-0.434652\pi$
$181$	5.48528	0.407718	0.203859	+	0.979000i	$0.434652\pi$
$182$	0	0
$183$	2.48528	0.183717
$184$	0	0
$185$	6.34315	0.466357
$186$	0	0
$187$	−8.82843	−0.645599
$188$	0	0
$189$	9.65685	0.702433
$190$	0	0
$191$	−3.65685	−0.264601	−0.132300	−	0.991210i	$-0.542236\pi$
$191$	−3.65685	−0.264601	−0.132300	+	0.991210i	$0.542236\pi$
$192$	0	0
$193$	5.65685	0.407189	0.203595	−	0.979055i	$-0.434738\pi$
$193$	5.65685	0.407189	0.203595	+	0.979055i	$0.434738\pi$
$194$	0	0
$195$	−7.38478	−0.528835
$196$	0	0
$197$	13.3137	0.948562	0.474281	−	0.880373i	$-0.342708\pi$
$197$	13.3137	0.948562	0.474281	+	0.880373i	$0.342708\pi$
$198$	0	0
$199$	−25.6569	−1.81877	−0.909383	−	0.415960i	$-0.863446\pi$
$199$	−25.6569	−1.81877	−0.909383	+	0.415960i	$0.863446\pi$
$200$	0	0
$201$	−2.34315	−0.165273
$202$	0	0
$203$	−4.00000	−0.280745
$204$	0	0
$205$	36.9706	2.58214
$206$	0	0
$207$	13.6569	0.949217
$208$	0	0
$209$	−4.82843	−0.333989
$210$	0	0
$211$	−18.7574	−1.29131	−0.645655	−	0.763629i	$-0.723416\pi$
$211$	−18.7574	−1.29131	−0.645655	+	0.763629i	$0.723416\pi$
$212$	0	0
$213$	2.68629	0.184062
$214$	0	0
$215$	−6.07107	−0.414043
$216$	0	0
$217$	33.6569	2.28478
$218$	0	0
$219$	1.65685	0.111960
$220$	0	0
$221$	17.0294	1.14552
$222$	0	0
$223$	28.1421	1.88454	0.942268	−	0.334859i	$-0.108689\pi$
$223$	28.1421	1.88454	0.942268	+	0.334859i	$0.108689\pi$
$224$	0	0
$225$	−27.3137	−1.82091
$226$	0	0
$227$	−11.1716	−0.741483	−0.370742	−	0.928736i	$-0.620896\pi$
$227$	−11.1716	−0.741483	−0.370742	+	0.928736i	$0.620896\pi$
$228$	0	0
$229$	17.6569	1.16680	0.583399	−	0.812186i	$-0.301722\pi$
$229$	17.6569	1.16680	0.583399	+	0.812186i	$0.301722\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	4.65685	0.305081	0.152540	−	0.988297i	$-0.451255\pi$
$233$	4.65685	0.305081	0.152540	+	0.988297i	$0.451255\pi$
$234$	0	0
$235$	46.2132	3.01462
$236$	0	0
$237$	4.02944	0.261740
$238$	0	0
$239$	8.14214	0.526671	0.263335	−	0.964704i	$-0.415177\pi$
$239$	8.14214	0.526671	0.263335	+	0.964704i	$0.415177\pi$
$240$	0	0
$241$	−18.3137	−1.17969	−0.589845	−	0.807517i	$-0.700811\pi$
$241$	−18.3137	−1.17969	−0.589845	+	0.807517i	$0.700811\pi$
$242$	0	0
$243$	10.3431	0.663513
$244$	0	0
$245$	−34.4558	−2.20130
$246$	0	0
$247$	9.31371	0.592617
$248$	0	0
$249$	−3.65685	−0.231744
$250$	0	0
$251$	−18.2132	−1.14961	−0.574804	−	0.818291i	$-0.694922\pi$
$251$	−18.2132	−1.14961	−0.574804	+	0.818291i	$0.694922\pi$
$252$	0	0
$253$	11.6569	0.732860
$254$	0	0
$255$	−5.79899	−0.363147
$256$	0	0
$257$	−7.82843	−0.488324	−0.244162	−	0.969734i	$-0.578513\pi$
$257$	−7.82843	−0.488324	−0.244162	+	0.969734i	$0.578513\pi$
$258$	0	0
$259$	6.62742	0.411808
$260$	0	0
$261$	−2.82843	−0.175075
$262$	0	0
$263$	−13.3848	−0.825341	−0.412670	−	0.910880i	$-0.635404\pi$
$263$	−13.3848	−0.825341	−0.412670	+	0.910880i	$0.635404\pi$
$264$	0	0
$265$	26.7990	1.64625
$266$	0	0
$267$	−4.00000	−0.244796
$268$	0	0
$269$	22.9706	1.40054	0.700270	−	0.713878i	$-0.253063\pi$
$269$	22.9706	1.40054	0.700270	+	0.713878i	$0.253063\pi$
$270$	0	0
$271$	−3.58579	−0.217821	−0.108911	−	0.994052i	$-0.534736\pi$
$271$	−3.58579	−0.217821	−0.108911	+	0.994052i	$0.534736\pi$
$272$	0	0
$273$	−7.71573	−0.466977
$274$	0	0
$275$	−23.3137	−1.40587
$276$	0	0
$277$	−28.6274	−1.72005	−0.860027	−	0.510248i	$-0.829554\pi$
$277$	−28.6274	−1.72005	−0.860027	+	0.510248i	$0.829554\pi$
$278$	0	0
$279$	23.7990	1.42481
$280$	0	0
$281$	−1.00000	−0.0596550	−0.0298275	−	0.999555i	$-0.509496\pi$
$281$	−1.00000	−0.0596550	−0.0298275	+	0.999555i	$0.509496\pi$
$282$	0	0
$283$	3.17157	0.188530	0.0942652	−	0.995547i	$-0.469950\pi$
$283$	3.17157	0.188530	0.0942652	+	0.995547i	$0.469950\pi$
$284$	0	0
$285$	−3.17157	−0.187868
$286$	0	0
$287$	38.6274	2.28010
$288$	0	0
$289$	−3.62742	−0.213377
$290$	0	0
$291$	−3.02944	−0.177589
$292$	0	0
$293$	−10.6863	−0.624300	−0.312150	−	0.950033i	$-0.601049\pi$
$293$	−10.6863	−0.624300	−0.312150	+	0.950033i	$0.601049\pi$
$294$	0	0
$295$	−40.1421	−2.33717
$296$	0	0
$297$	5.82843	0.338200
$298$	0	0
$299$	−22.4853	−1.30036
$300$	0	0
$301$	−6.34315	−0.365613
$302$	0	0
$303$	4.68629	0.269220
$304$	0	0
$305$	−22.9706	−1.31529
$306$	0	0
$307$	−29.7279	−1.69666	−0.848331	−	0.529466i	$-0.822392\pi$
$307$	−29.7279	−1.69666	−0.848331	+	0.529466i	$0.822392\pi$
$308$	0	0
$309$	−6.97056	−0.396541
$310$	0	0
$311$	18.9706	1.07572	0.537861	−	0.843034i	$-0.319233\pi$
$311$	18.9706	1.07572	0.537861	+	0.843034i	$0.319233\pi$
$312$	0	0
$313$	−25.4853	−1.44051	−0.720257	−	0.693708i	$-0.755976\pi$
$313$	−25.4853	−1.44051	−0.720257	+	0.693708i	$0.755976\pi$
$314$	0	0
$315$	−43.3137	−2.44045
$316$	0	0
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	0	0
$319$	−2.41421	−0.135170
$320$	0	0
$321$	0.970563	0.0541715
$322$	0	0
$323$	7.31371	0.406946
$324$	0	0
$325$	44.9706	2.49452
$326$	0	0
$327$	0.757359	0.0418821
$328$	0	0
$329$	48.2843	2.66200
$330$	0	0
$331$	−6.07107	−0.333696	−0.166848	−	0.985983i	$-0.553359\pi$
$331$	−6.07107	−0.333696	−0.166848	+	0.985983i	$0.553359\pi$
$332$	0	0
$333$	4.68629	0.256807
$334$	0	0
$335$	21.6569	1.18324
$336$	0	0
$337$	13.3137	0.725244	0.362622	−	0.931936i	$-0.381882\pi$
$337$	13.3137	0.725244	0.362622	+	0.931936i	$0.381882\pi$
$338$	0	0
$339$	−4.14214	−0.224970
$340$	0	0
$341$	20.3137	1.10005
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	7.65685	0.412231
$346$	0	0
$347$	6.48528	0.348148	0.174074	−	0.984733i	$-0.444307\pi$
$347$	6.48528	0.348148	0.174074	+	0.984733i	$0.444307\pi$
$348$	0	0
$349$	−18.3137	−0.980310	−0.490155	−	0.871635i	$-0.663060\pi$
$349$	−18.3137	−0.980310	−0.490155	+	0.871635i	$0.663060\pi$
$350$	0	0
$351$	−11.2426	−0.600088
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	−24.8284	−1.31776
$356$	0	0
$357$	−6.05887	−0.320670
$358$	0	0
$359$	37.7279	1.99120	0.995602	−	0.0936861i	$-0.0298650\pi$
$359$	37.7279	1.99120	0.995602	+	0.0936861i	$0.0298650\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−2.14214	−0.112433
$364$	0	0
$365$	−15.3137	−0.801556
$366$	0	0
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	0	0
$369$	27.3137	1.42189
$370$	0	0
$371$	28.0000	1.45369
$372$	0	0
$373$	−15.4853	−0.801797	−0.400899	−	0.916122i	$-0.631302\pi$
$373$	−15.4853	−0.801797	−0.400899	+	0.916122i	$0.631302\pi$
$374$	0	0
$375$	−7.38478	−0.381348
$376$	0	0
$377$	4.65685	0.239840
$378$	0	0
$379$	19.6569	1.00970	0.504852	−	0.863206i	$-0.331547\pi$
$379$	19.6569	1.00970	0.504852	+	0.863206i	$0.331547\pi$
$380$	0	0
$381$	6.48528	0.332251
$382$	0	0
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	0	0
$385$	−36.9706	−1.88419
$386$	0	0
$387$	−4.48528	−0.228000
$388$	0	0
$389$	−6.34315	−0.321610	−0.160805	−	0.986986i	$-0.551409\pi$
$389$	−6.34315	−0.321610	−0.160805	+	0.986986i	$0.551409\pi$
$390$	0	0
$391$	−17.6569	−0.892946
$392$	0	0
$393$	−4.14214	−0.208943
$394$	0	0
$395$	−37.2426	−1.87388
$396$	0	0
$397$	−32.4558	−1.62891	−0.814456	−	0.580225i	$-0.802965\pi$
$397$	−32.4558	−1.62891	−0.814456	+	0.580225i	$0.802965\pi$
$398$	0	0
$399$	−3.31371	−0.165893
$400$	0	0
$401$	−2.65685	−0.132677	−0.0663385	−	0.997797i	$-0.521132\pi$
$401$	−2.65685	−0.132677	−0.0663385	+	0.997797i	$0.521132\pi$
$402$	0	0
$403$	−39.1838	−1.95188
$404$	0	0
$405$	−28.6569	−1.42397
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	11.6569	0.576394	0.288197	−	0.957571i	$-0.406944\pi$
$409$	11.6569	0.576394	0.288197	+	0.957571i	$0.406944\pi$
$410$	0	0
$411$	2.62742	0.129601
$412$	0	0
$413$	−41.9411	−2.06379
$414$	0	0
$415$	33.7990	1.65913
$416$	0	0
$417$	3.65685	0.179077
$418$	0	0
$419$	9.51472	0.464824	0.232412	−	0.972617i	$-0.425338\pi$
$419$	9.51472	0.464824	0.232412	+	0.972617i	$0.425338\pi$
$420$	0	0
$421$	−34.9706	−1.70436	−0.852180	−	0.523248i	$-0.824720\pi$
$421$	−34.9706	−1.70436	−0.852180	+	0.523248i	$0.824720\pi$
$422$	0	0
$423$	34.1421	1.66005
$424$	0	0
$425$	35.3137	1.71297
$426$	0	0
$427$	−24.0000	−1.16144
$428$	0	0
$429$	−4.65685	−0.224835
$430$	0	0
$431$	25.7990	1.24269	0.621347	−	0.783536i	$-0.286586\pi$
$431$	25.7990	1.24269	0.621347	+	0.783536i	$0.286586\pi$
$432$	0	0
$433$	−8.00000	−0.384455	−0.192228	−	0.981350i	$-0.561571\pi$
$433$	−8.00000	−0.384455	−0.192228	+	0.981350i	$0.561571\pi$
$434$	0	0
$435$	−1.58579	−0.0760326
$436$	0	0
$437$	−9.65685	−0.461950
$438$	0	0
$439$	8.14214	0.388603	0.194301	−	0.980942i	$-0.437756\pi$
$439$	8.14214	0.388603	0.194301	+	0.980942i	$0.437756\pi$
$440$	0	0
$441$	−25.4558	−1.21218
$442$	0	0
$443$	36.6274	1.74022	0.870111	−	0.492857i	$-0.164047\pi$
$443$	36.6274	1.74022	0.870111	+	0.492857i	$0.164047\pi$
$444$	0	0
$445$	36.9706	1.75257
$446$	0	0
$447$	3.58579	0.169602
$448$	0	0
$449$	−16.6274	−0.784696	−0.392348	−	0.919817i	$-0.628337\pi$
$449$	−16.6274	−0.784696	−0.392348	+	0.919817i	$0.628337\pi$
$450$	0	0
$451$	23.3137	1.09780
$452$	0	0
$453$	2.62742	0.123447
$454$	0	0
$455$	71.3137	3.34324
$456$	0	0
$457$	16.6274	0.777798	0.388899	−	0.921280i	$-0.372856\pi$
$457$	16.6274	0.777798	0.388899	+	0.921280i	$0.372856\pi$
$458$	0	0
$459$	−8.82843	−0.412076
$460$	0	0
$461$	41.3137	1.92417	0.962086	−	0.272748i	$-0.0879324\pi$
$461$	41.3137	1.92417	0.962086	+	0.272748i	$0.0879324\pi$
$462$	0	0
$463$	−19.4558	−0.904190	−0.452095	−	0.891970i	$-0.649323\pi$
$463$	−19.4558	−0.904190	−0.452095	+	0.891970i	$0.649323\pi$
$464$	0	0
$465$	13.3431	0.618774
$466$	0	0
$467$	4.89949	0.226722	0.113361	−	0.993554i	$-0.463838\pi$
$467$	4.89949	0.226722	0.113361	+	0.993554i	$0.463838\pi$
$468$	0	0
$469$	22.6274	1.04484
$470$	0	0
$471$	−4.68629	−0.215933
$472$	0	0
$473$	−3.82843	−0.176031
$474$	0	0
$475$	19.3137	0.886174
$476$	0	0
$477$	19.7990	0.906533
$478$	0	0
$479$	−13.2426	−0.605072	−0.302536	−	0.953138i	$-0.597833\pi$
$479$	−13.2426	−0.605072	−0.302536	+	0.953138i	$0.597833\pi$
$480$	0	0
$481$	−7.71573	−0.351807
$482$	0	0
$483$	8.00000	0.364013
$484$	0	0
$485$	28.0000	1.27141
$486$	0	0
$487$	16.9706	0.769010	0.384505	−	0.923123i	$-0.374372\pi$
$487$	16.9706	0.769010	0.384505	+	0.923123i	$0.374372\pi$
$488$	0	0
$489$	2.85786	0.129237
$490$	0	0
$491$	−11.2426	−0.507373	−0.253687	−	0.967286i	$-0.581643\pi$
$491$	−11.2426	−0.507373	−0.253687	+	0.967286i	$0.581643\pi$
$492$	0	0
$493$	3.65685	0.164696
$494$	0	0
$495$	−26.1421	−1.17500
$496$	0	0
$497$	−25.9411	−1.16362
$498$	0	0
$499$	11.8579	0.530831	0.265415	−	0.964134i	$-0.414491\pi$
$499$	11.8579	0.530831	0.265415	+	0.964134i	$0.414491\pi$
$500$	0	0
$501$	−0.627417	−0.0280309
$502$	0	0
$503$	2.75736	0.122945	0.0614723	−	0.998109i	$-0.480420\pi$
$503$	2.75736	0.122945	0.0614723	+	0.998109i	$0.480420\pi$
$504$	0	0
$505$	−43.3137	−1.92743
$506$	0	0
$507$	3.59798	0.159792
$508$	0	0
$509$	18.6569	0.826951	0.413475	−	0.910515i	$-0.364315\pi$
$509$	18.6569	0.826951	0.413475	+	0.910515i	$0.364315\pi$
$510$	0	0
$511$	−16.0000	−0.707798
$512$	0	0
$513$	−4.82843	−0.213180
$514$	0	0
$515$	64.4264	2.83897
$516$	0	0
$517$	29.1421	1.28167
$518$	0	0
$519$	−5.79899	−0.254547
$520$	0	0
$521$	43.8284	1.92016	0.960079	−	0.279729i	$-0.0902445\pi$
$521$	43.8284	1.92016	0.960079	+	0.279729i	$0.0902445\pi$
$522$	0	0
$523$	18.3431	0.802090	0.401045	−	0.916058i	$-0.368647\pi$
$523$	18.3431	0.802090	0.401045	+	0.916058i	$0.368647\pi$
$524$	0	0
$525$	−16.0000	−0.698297
$526$	0	0
$527$	−30.7696	−1.34034
$528$	0	0
$529$	0.313708	0.0136395
$530$	0	0
$531$	−29.6569	−1.28700
$532$	0	0
$533$	−44.9706	−1.94789
$534$	0	0
$535$	−8.97056	−0.387831
$536$	0	0
$537$	2.28427	0.0985736
$538$	0	0
$539$	−21.7279	−0.935888
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	0	0
$543$	2.27208	0.0975042
$544$	0	0
$545$	−7.00000	−0.299847
$546$	0	0
$547$	8.68629	0.371399	0.185700	−	0.982607i	$-0.440545\pi$
$547$	8.68629	0.371399	0.185700	+	0.982607i	$0.440545\pi$
$548$	0	0
$549$	−16.9706	−0.724286
$550$	0	0
$551$	2.00000	0.0852029
$552$	0	0
$553$	−38.9117	−1.65469
$554$	0	0
$555$	2.62742	0.111528
$556$	0	0
$557$	−9.31371	−0.394634	−0.197317	−	0.980340i	$-0.563223\pi$
$557$	−9.31371	−0.394634	−0.197317	+	0.980340i	$0.563223\pi$
$558$	0	0
$559$	7.38478	0.312343
$560$	0	0
$561$	−3.65685	−0.154393
$562$	0	0
$563$	−1.10051	−0.0463808	−0.0231904	−	0.999731i	$-0.507382\pi$
$563$	−1.10051	−0.0463808	−0.0231904	+	0.999731i	$0.507382\pi$
$564$	0	0
$565$	38.2843	1.61063
$566$	0	0
$567$	−29.9411	−1.25741
$568$	0	0
$569$	19.6569	0.824058	0.412029	−	0.911171i	$-0.364820\pi$
$569$	19.6569	0.824058	0.412029	+	0.911171i	$0.364820\pi$
$570$	0	0
$571$	24.9706	1.04499	0.522493	−	0.852644i	$-0.325002\pi$
$571$	24.9706	1.04499	0.522493	+	0.852644i	$0.325002\pi$
$572$	0	0
$573$	−1.51472	−0.0632783
$574$	0	0
$575$	−46.6274	−1.94450
$576$	0	0
$577$	10.6863	0.444876	0.222438	−	0.974947i	$-0.428598\pi$
$577$	10.6863	0.444876	0.222438	+	0.974947i	$0.428598\pi$
$578$	0	0
$579$	2.34315	0.0973778
$580$	0	0
$581$	35.3137	1.46506
$582$	0	0
$583$	16.8995	0.699906
$584$	0	0
$585$	50.4264	2.08488
$586$	0	0
$587$	−29.1127	−1.20161	−0.600805	−	0.799396i	$-0.705153\pi$
$587$	−29.1127	−1.20161	−0.600805	+	0.799396i	$0.705153\pi$
$588$	0	0
$589$	−16.8284	−0.693403
$590$	0	0
$591$	5.51472	0.226845
$592$	0	0
$593$	−5.82843	−0.239345	−0.119672	−	0.992813i	$-0.538184\pi$
$593$	−5.82843	−0.239345	−0.119672	+	0.992813i	$0.538184\pi$
$594$	0	0
$595$	56.0000	2.29578
$596$	0	0
$597$	−10.6274	−0.434951
$598$	0	0
$599$	7.38478	0.301734	0.150867	−	0.988554i	$-0.451794\pi$
$599$	7.38478	0.301734	0.150867	+	0.988554i	$0.451794\pi$
$600$	0	0
$601$	−12.0000	−0.489490	−0.244745	−	0.969587i	$-0.578704\pi$
$601$	−12.0000	−0.489490	−0.244745	+	0.969587i	$0.578704\pi$
$602$	0	0
$603$	16.0000	0.651570
$604$	0	0
$605$	19.7990	0.804943
$606$	0	0
$607$	−27.8701	−1.13121	−0.565605	−	0.824676i	$-0.691357\pi$
$607$	−27.8701	−1.13121	−0.565605	+	0.824676i	$0.691357\pi$
$608$	0	0
$609$	−1.65685	−0.0671391
$610$	0	0
$611$	−56.2132	−2.27414
$612$	0	0
$613$	−20.7990	−0.840063	−0.420032	−	0.907509i	$-0.637981\pi$
$613$	−20.7990	−0.840063	−0.420032	+	0.907509i	$0.637981\pi$
$614$	0	0
$615$	15.3137	0.617508
$616$	0	0
$617$	−3.02944	−0.121961	−0.0609803	−	0.998139i	$-0.519423\pi$
$617$	−3.02944	−0.121961	−0.0609803	+	0.998139i	$0.519423\pi$
$618$	0	0
$619$	−32.6985	−1.31426	−0.657132	−	0.753776i	$-0.728230\pi$
$619$	−32.6985	−1.31426	−0.657132	+	0.753776i	$0.728230\pi$
$620$	0	0
$621$	11.6569	0.467773
$622$	0	0
$623$	38.6274	1.54757
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	−2.00000	−0.0798723
$628$	0	0
$629$	−6.05887	−0.241583
$630$	0	0
$631$	−42.7696	−1.70263	−0.851315	−	0.524656i	$-0.824194\pi$
$631$	−42.7696	−1.70263	−0.851315	+	0.524656i	$0.824194\pi$
$632$	0	0
$633$	−7.76955	−0.308812
$634$	0	0
$635$	−59.9411	−2.37869
$636$	0	0
$637$	41.9117	1.66060
$638$	0	0
$639$	−18.3431	−0.725644
$640$	0	0
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	0	0
$643$	−3.02944	−0.119469	−0.0597347	−	0.998214i	$-0.519025\pi$
$643$	−3.02944	−0.119469	−0.0597347	+	0.998214i	$0.519025\pi$
$644$	0	0
$645$	−2.51472	−0.0990169
$646$	0	0
$647$	−0.828427	−0.0325688	−0.0162844	−	0.999867i	$-0.505184\pi$
$647$	−0.828427	−0.0325688	−0.0162844	+	0.999867i	$0.505184\pi$
$648$	0	0
$649$	−25.3137	−0.993650
$650$	0	0
$651$	13.9411	0.546396
$652$	0	0
$653$	46.6274	1.82467	0.912336	−	0.409443i	$-0.134277\pi$
$653$	46.6274	1.82467	0.912336	+	0.409443i	$0.134277\pi$
$654$	0	0
$655$	38.2843	1.49589
$656$	0	0
$657$	−11.3137	−0.441390
$658$	0	0
$659$	2.07107	0.0806773	0.0403387	−	0.999186i	$-0.487156\pi$
$659$	2.07107	0.0806773	0.0403387	+	0.999186i	$0.487156\pi$
$660$	0	0
$661$	−29.3137	−1.14017	−0.570086	−	0.821585i	$-0.693090\pi$
$661$	−29.3137	−1.14017	−0.570086	+	0.821585i	$0.693090\pi$
$662$	0	0
$663$	7.05382	0.273948
$664$	0	0
$665$	30.6274	1.18768
$666$	0	0
$667$	−4.82843	−0.186957
$668$	0	0
$669$	11.6569	0.450680
$670$	0	0
$671$	−14.4853	−0.559198
$672$	0	0
$673$	21.2843	0.820448	0.410224	−	0.911985i	$-0.365450\pi$
$673$	21.2843	0.820448	0.410224	+	0.911985i	$0.365450\pi$
$674$	0	0
$675$	−23.3137	−0.897345
$676$	0	0
$677$	−22.9706	−0.882830	−0.441415	−	0.897303i	$-0.645523\pi$
$677$	−22.9706	−0.882830	−0.441415	+	0.897303i	$0.645523\pi$
$678$	0	0
$679$	29.2548	1.12270
$680$	0	0
$681$	−4.62742	−0.177323
$682$	0	0
$683$	4.97056	0.190193	0.0950966	−	0.995468i	$-0.469684\pi$
$683$	4.97056	0.190193	0.0950966	+	0.995468i	$0.469684\pi$
$684$	0	0
$685$	−24.2843	−0.927854
$686$	0	0
$687$	7.31371	0.279035
$688$	0	0
$689$	−32.5980	−1.24188
$690$	0	0
$691$	3.31371	0.126059	0.0630297	−	0.998012i	$-0.479924\pi$
$691$	3.31371	0.126059	0.0630297	+	0.998012i	$0.479924\pi$
$692$	0	0
$693$	−27.3137	−1.03756
$694$	0	0
$695$	−33.7990	−1.28207
$696$	0	0
$697$	−35.3137	−1.33760
$698$	0	0
$699$	1.92893	0.0729589
$700$	0	0
$701$	−7.62742	−0.288084	−0.144042	−	0.989572i	$-0.546010\pi$
$701$	−7.62742	−0.288084	−0.144042	+	0.989572i	$0.546010\pi$
$702$	0	0
$703$	−3.31371	−0.124979
$704$	0	0
$705$	19.1421	0.720935
$706$	0	0
$707$	−45.2548	−1.70198
$708$	0	0
$709$	−34.9411	−1.31224	−0.656121	−	0.754656i	$-0.727804\pi$
$709$	−34.9411	−1.31224	−0.656121	+	0.754656i	$0.727804\pi$
$710$	0	0
$711$	−27.5147	−1.03188
$712$	0	0
$713$	40.6274	1.52151
$714$	0	0
$715$	43.0416	1.60967
$716$	0	0
$717$	3.37258	0.125951
$718$	0	0
$719$	−17.5147	−0.653189	−0.326594	−	0.945165i	$-0.605901\pi$
$719$	−17.5147	−0.653189	−0.326594	+	0.945165i	$0.605901\pi$
$720$	0	0
$721$	67.3137	2.50689
$722$	0	0
$723$	−7.58579	−0.282118
$724$	0	0
$725$	9.65685	0.358647
$726$	0	0
$727$	38.2843	1.41989	0.709943	−	0.704260i	$-0.248721\pi$
$727$	38.2843	1.41989	0.709943	+	0.704260i	$0.248721\pi$
$728$	0	0
$729$	−18.1716	−0.673021
$730$	0	0
$731$	5.79899	0.214483
$732$	0	0
$733$	−42.6274	−1.57448	−0.787240	−	0.616647i	$-0.788491\pi$
$733$	−42.6274	−1.57448	−0.787240	+	0.616647i	$0.788491\pi$
$734$	0	0
$735$	−14.2721	−0.526434
$736$	0	0
$737$	13.6569	0.503057
$738$	0	0
$739$	−17.2426	−0.634281	−0.317140	−	0.948379i	$-0.602723\pi$
$739$	−17.2426	−0.634281	−0.317140	+	0.948379i	$0.602723\pi$
$740$	0	0
$741$	3.85786	0.141722
$742$	0	0
$743$	−49.5980	−1.81957	−0.909787	−	0.415076i	$-0.863755\pi$
$743$	−49.5980	−1.81957	−0.909787	+	0.415076i	$0.863755\pi$
$744$	0	0
$745$	−33.1421	−1.21423
$746$	0	0
$747$	24.9706	0.913625
$748$	0	0
$749$	−9.37258	−0.342467
$750$	0	0
$751$	32.6274	1.19059	0.595296	−	0.803507i	$-0.297035\pi$
$751$	32.6274	1.19059	0.595296	+	0.803507i	$0.297035\pi$
$752$	0	0
$753$	−7.54416	−0.274924
$754$	0	0
$755$	−24.2843	−0.883795
$756$	0	0
$757$	42.9706	1.56179	0.780896	−	0.624661i	$-0.214763\pi$
$757$	42.9706	1.56179	0.780896	+	0.624661i	$0.214763\pi$
$758$	0	0
$759$	4.82843	0.175261
$760$	0	0
$761$	−51.9411	−1.88286	−0.941432	−	0.337202i	$-0.890519\pi$
$761$	−51.9411	−1.88286	−0.941432	+	0.337202i	$0.890519\pi$
$762$	0	0
$763$	−7.31371	−0.264774
$764$	0	0
$765$	39.5980	1.43167
$766$	0	0
$767$	48.8284	1.76309
$768$	0	0
$769$	−16.6274	−0.599600	−0.299800	−	0.954002i	$-0.596920\pi$
$769$	−16.6274	−0.599600	−0.299800	+	0.954002i	$0.596920\pi$
$770$	0	0
$771$	−3.24264	−0.116781
$772$	0	0
$773$	25.6569	0.922813	0.461406	−	0.887189i	$-0.347345\pi$
$773$	25.6569	0.922813	0.461406	+	0.887189i	$0.347345\pi$
$774$	0	0
$775$	−81.2548	−2.91876
$776$	0	0
$777$	2.74517	0.0984823
$778$	0	0
$779$	−19.3137	−0.691985
$780$	0	0
$781$	−15.6569	−0.560246
$782$	0	0
$783$	−2.41421	−0.0862770
$784$	0	0
$785$	43.3137	1.54593
$786$	0	0
$787$	−8.14214	−0.290236	−0.145118	−	0.989414i	$-0.546356\pi$
$787$	−8.14214	−0.290236	−0.145118	+	0.989414i	$0.546356\pi$
$788$	0	0
$789$	−5.54416	−0.197377
$790$	0	0
$791$	40.0000	1.42224
$792$	0	0
$793$	27.9411	0.992218
$794$	0	0
$795$	11.1005	0.393694
$796$	0	0
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	0	0
$799$	−44.1421	−1.56164
$800$	0	0
$801$	27.3137	0.965082
$802$	0	0
$803$	−9.65685	−0.340783
$804$	0	0
$805$	−73.9411	−2.60608
$806$	0	0
$807$	9.51472	0.334934
$808$	0	0
$809$	−5.65685	−0.198884	−0.0994422	−	0.995043i	$-0.531706\pi$
$809$	−5.65685	−0.198884	−0.0994422	+	0.995043i	$0.531706\pi$
$810$	0	0
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	0	0
$813$	−1.48528	−0.0520911
$814$	0	0
$815$	−26.4142	−0.925250
$816$	0	0
$817$	3.17157	0.110959
$818$	0	0
$819$	52.6863	1.84101
$820$	0	0
$821$	4.65685	0.162525	0.0812627	−	0.996693i	$-0.474105\pi$
$821$	4.65685	0.162525	0.0812627	+	0.996693i	$0.474105\pi$
$822$	0	0
$823$	−22.9706	−0.800703	−0.400352	−	0.916362i	$-0.631112\pi$
$823$	−22.9706	−0.800703	−0.400352	+	0.916362i	$0.631112\pi$
$824$	0	0
$825$	−9.65685	−0.336209
$826$	0	0
$827$	−21.1005	−0.733736	−0.366868	−	0.930273i	$-0.619570\pi$
$827$	−21.1005	−0.733736	−0.366868	+	0.930273i	$0.619570\pi$
$828$	0	0
$829$	22.0000	0.764092	0.382046	−	0.924143i	$-0.375220\pi$
$829$	22.0000	0.764092	0.382046	+	0.924143i	$0.375220\pi$
$830$	0	0
$831$	−11.8579	−0.411345
$832$	0	0
$833$	32.9117	1.14032
$834$	0	0
$835$	5.79899	0.200682
$836$	0	0
$837$	20.3137	0.702144
$838$	0	0
$839$	38.8406	1.34093	0.670464	−	0.741942i	$-0.266095\pi$
$839$	38.8406	1.34093	0.670464	+	0.741942i	$0.266095\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−0.414214	−0.0142663
$844$	0	0
$845$	−33.2548	−1.14400
$846$	0	0
$847$	20.6863	0.710789
$848$	0	0
$849$	1.31371	0.0450864
$850$	0	0
$851$	8.00000	0.274236
$852$	0	0
$853$	51.9411	1.77843	0.889215	−	0.457489i	$-0.151251\pi$
$853$	51.9411	1.77843	0.889215	+	0.457489i	$0.151251\pi$
$854$	0	0
$855$	21.6569	0.740649
$856$	0	0
$857$	5.14214	0.175652	0.0878260	−	0.996136i	$-0.472008\pi$
$857$	5.14214	0.175652	0.0878260	+	0.996136i	$0.472008\pi$
$858$	0	0
$859$	18.4142	0.628285	0.314142	−	0.949376i	$-0.398283\pi$
$859$	18.4142	0.628285	0.314142	+	0.949376i	$0.398283\pi$
$860$	0	0
$861$	16.0000	0.545279
$862$	0	0
$863$	−15.1716	−0.516446	−0.258223	−	0.966085i	$-0.583137\pi$
$863$	−15.1716	−0.516446	−0.258223	+	0.966085i	$0.583137\pi$
$864$	0	0
$865$	53.5980	1.82239
$866$	0	0
$867$	−1.50253	−0.0510284
$868$	0	0
$869$	−23.4853	−0.796684
$870$	0	0
$871$	−26.3431	−0.892603
$872$	0	0
$873$	20.6863	0.700125
$874$	0	0
$875$	71.3137	2.41084
$876$	0	0
$877$	39.6274	1.33812	0.669061	−	0.743207i	$-0.266696\pi$
$877$	39.6274	1.33812	0.669061	+	0.743207i	$0.266696\pi$
$878$	0	0
$879$	−4.42641	−0.149299
$880$	0	0
$881$	16.3431	0.550615	0.275307	−	0.961356i	$-0.411220\pi$
$881$	16.3431	0.550615	0.275307	+	0.961356i	$0.411220\pi$
$882$	0	0
$883$	55.5980	1.87102	0.935510	−	0.353299i	$-0.114940\pi$
$883$	55.5980	1.87102	0.935510	+	0.353299i	$0.114940\pi$
$884$	0	0
$885$	−16.6274	−0.558925
$886$	0	0
$887$	19.8701	0.667171	0.333586	−	0.942720i	$-0.391741\pi$
$887$	19.8701	0.667171	0.333586	+	0.942720i	$0.391741\pi$
$888$	0	0
$889$	−62.6274	−2.10046
$890$	0	0
$891$	−18.0711	−0.605404
$892$	0	0
$893$	−24.1421	−0.807886
$894$	0	0
$895$	−21.1127	−0.705720
$896$	0	0
$897$	−9.31371	−0.310976
$898$	0	0
$899$	−8.41421	−0.280630
$900$	0	0
$901$	−25.5980	−0.852792
$902$	0	0
$903$	−2.62742	−0.0874350
$904$	0	0
$905$	−21.0000	−0.698064
$906$	0	0
$907$	−39.2548	−1.30344	−0.651718	−	0.758462i	$-0.725951\pi$
$907$	−39.2548	−1.30344	−0.651718	+	0.758462i	$0.725951\pi$
$908$	0	0
$909$	−32.0000	−1.06137
$910$	0	0
$911$	−5.52691	−0.183115	−0.0915574	−	0.995800i	$-0.529184\pi$
$911$	−5.52691	−0.183115	−0.0915574	+	0.995800i	$0.529184\pi$
$912$	0	0
$913$	21.3137	0.705381
$914$	0	0
$915$	−9.51472	−0.314547
$916$	0	0
$917$	40.0000	1.32092
$918$	0	0
$919$	−19.4558	−0.641789	−0.320895	−	0.947115i	$-0.603984\pi$
$919$	−19.4558	−0.641789	−0.320895	+	0.947115i	$0.603984\pi$
$920$	0	0
$921$	−12.3137	−0.405750
$922$	0	0
$923$	30.2010	0.994078
$924$	0	0
$925$	−16.0000	−0.526077
$926$	0	0
$927$	47.5980	1.56332
$928$	0	0
$929$	15.9411	0.523011	0.261506	−	0.965202i	$-0.415781\pi$
$929$	15.9411	0.523011	0.261506	+	0.965202i	$0.415781\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	0	0
$933$	7.85786	0.257255
$934$	0	0
$935$	33.7990	1.10535
$936$	0	0
$937$	−24.6274	−0.804543	−0.402271	−	0.915520i	$-0.631779\pi$
$937$	−24.6274	−0.804543	−0.402271	+	0.915520i	$0.631779\pi$
$938$	0	0
$939$	−10.5563	−0.344493
$940$	0	0
$941$	55.0833	1.79566	0.897831	−	0.440339i	$-0.145142\pi$
$941$	55.0833	1.79566	0.897831	+	0.440339i	$0.145142\pi$
$942$	0	0
$943$	46.6274	1.51840
$944$	0	0
$945$	−36.9706	−1.20265
$946$	0	0
$947$	−33.7279	−1.09601	−0.548005	−	0.836475i	$-0.684613\pi$
$947$	−33.7279	−1.09601	−0.548005	+	0.836475i	$0.684613\pi$
$948$	0	0
$949$	18.6274	0.604672
$950$	0	0
$951$	−5.79899	−0.188045
$952$	0	0
$953$	−37.0000	−1.19855	−0.599274	−	0.800544i	$-0.704544\pi$
$953$	−37.0000	−1.19855	−0.599274	+	0.800544i	$0.704544\pi$
$954$	0	0
$955$	14.0000	0.453029
$956$	0	0
$957$	−1.00000	−0.0323254
$958$	0	0
$959$	−25.3726	−0.819323
$960$	0	0
$961$	39.7990	1.28384
$962$	0	0
$963$	−6.62742	−0.213566
$964$	0	0
$965$	−21.6569	−0.697159
$966$	0	0
$967$	−5.44365	−0.175056	−0.0875280	−	0.996162i	$-0.527897\pi$
$967$	−5.44365	−0.175056	−0.0875280	+	0.996162i	$0.527897\pi$
$968$	0	0
$969$	3.02944	0.0973195
$970$	0	0
$971$	−33.3137	−1.06909	−0.534544	−	0.845141i	$-0.679517\pi$
$971$	−33.3137	−1.06909	−0.534544	+	0.845141i	$0.679517\pi$
$972$	0	0
$973$	−35.3137	−1.13211
$974$	0	0
$975$	18.6274	0.596555
$976$	0	0
$977$	−2.85786	−0.0914312	−0.0457156	−	0.998954i	$-0.514557\pi$
$977$	−2.85786	−0.0914312	−0.0457156	+	0.998954i	$0.514557\pi$
$978$	0	0
$979$	23.3137	0.745109
$980$	0	0
$981$	−5.17157	−0.165116
$982$	0	0
$983$	1.87006	0.0596456	0.0298228	−	0.999555i	$-0.490506\pi$
$983$	1.87006	0.0596456	0.0298228	+	0.999555i	$0.490506\pi$
$984$	0	0
$985$	−50.9706	−1.62406
$986$	0	0
$987$	20.0000	0.636607
$988$	0	0
$989$	−7.65685	−0.243474
$990$	0	0
$991$	−38.4853	−1.22253	−0.611263	−	0.791428i	$-0.709338\pi$
$991$	−38.4853	−1.22253	−0.611263	+	0.791428i	$0.709338\pi$
$992$	0	0
$993$	−2.51472	−0.0798022
$994$	0	0
$995$	98.2254	3.11395
$996$	0	0
$997$	−24.9706	−0.790826	−0.395413	−	0.918504i	$-0.629398\pi$
$997$	−24.9706	−0.790826	−0.395413	+	0.918504i	$0.629398\pi$
$998$	0	0
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	232.2.a.c.1.2	✓	2
3.2	odd	2		2088.2.a.q.1.2		2
4.3	odd	2		464.2.a.i.1.1		2
5.4	even	2		5800.2.a.o.1.1		2
8.3	odd	2		1856.2.a.s.1.2		2
8.5	even	2		1856.2.a.v.1.1		2
12.11	even	2		4176.2.a.br.1.2		2
29.28	even	2		6728.2.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.2.a.c.1.2	✓	2	1.1	even	1	trivial
464.2.a.i.1.1		2	4.3	odd	2
1856.2.a.s.1.2		2	8.3	odd	2
1856.2.a.v.1.1		2	8.5	even	2
2088.2.a.q.1.2		2	3.2	odd	2
4176.2.a.br.1.2		2	12.11	even	2
5800.2.a.o.1.1		2	5.4	even	2
6728.2.a.f.1.1		2	29.28	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:	\( N \)	\(=\)	\( 232 = 2^{3} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	232.a (trivial)

\(f(q)\)	\(=\)	\(q+0.414214 q^{3} -3.82843 q^{5} -4.00000 q^{7} -2.82843 q^{9} +O(q^{10})\)	\(q+0.414214 q^{3} -3.82843 q^{5} -4.00000 q^{7} -2.82843 q^{9} -2.41421 q^{11} +4.65685 q^{13} -1.58579 q^{15} +3.65685 q^{17} +2.00000 q^{19} -1.65685 q^{21} -4.82843 q^{23} +9.65685 q^{25} -2.41421 q^{27} +1.00000 q^{29} -8.41421 q^{31} -1.00000 q^{33} +15.3137 q^{35} -1.65685 q^{37} +1.92893 q^{39} -9.65685 q^{41} +1.58579 q^{43} +10.8284 q^{45} -12.0711 q^{47} +9.00000 q^{49} +1.51472 q^{51} -7.00000 q^{53} +9.24264 q^{55} +0.828427 q^{57} +10.4853 q^{59} +6.00000 q^{61} +11.3137 q^{63} -17.8284 q^{65} -5.65685 q^{67} -2.00000 q^{69} +6.48528 q^{71} +4.00000 q^{73} +4.00000 q^{75} +9.65685 q^{77} +9.72792 q^{79} +7.48528 q^{81} -8.82843 q^{83} -14.0000 q^{85} +0.414214 q^{87} -9.65685 q^{89} -18.6274 q^{91} -3.48528 q^{93} -7.65685 q^{95} -7.31371 q^{97} +6.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - 8 q^{7}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 - 8 * q^7	\( 2 q - 2 q^{3} - 2 q^{5} - 8 q^{7} - 2 q^{11} - 2 q^{13} - 6 q^{15} - 4 q^{17} + 4 q^{19} + 8 q^{21} - 4 q^{23} + 8 q^{25} - 2 q^{27} + 2 q^{29} - 14 q^{31} - 2 q^{33} + 8 q^{35} + 8 q^{37} + 18 q^{39} - 8 q^{41} + 6 q^{43} + 16 q^{45} - 10 q^{47} + 18 q^{49} + 20 q^{51} - 14 q^{53} + 10 q^{55} - 4 q^{57} + 4 q^{59} + 12 q^{61} - 30 q^{65} - 4 q^{69} - 4 q^{71} + 8 q^{73} + 8 q^{75} + 8 q^{77} - 6 q^{79} - 2 q^{81} - 12 q^{83} - 28 q^{85} - 2 q^{87} - 8 q^{89} + 8 q^{91} + 10 q^{93} - 4 q^{95} + 8 q^{97} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - 8 * q^7 - 2 * q^11 - 2 * q^13 - 6 * q^15 - 4 * q^17 + 4 * q^19 + 8 * q^21 - 4 * q^23 + 8 * q^25 - 2 * q^27 + 2 * q^29 - 14 * q^31 - 2 * q^33 + 8 * q^35 + 8 * q^37 + 18 * q^39 - 8 * q^41 + 6 * q^43 + 16 * q^45 - 10 * q^47 + 18 * q^49 + 20 * q^51 - 14 * q^53 + 10 * q^55 - 4 * q^57 + 4 * q^59 + 12 * q^61 - 30 * q^65 - 4 * q^69 - 4 * q^71 + 8 * q^73 + 8 * q^75 + 8 * q^77 - 6 * q^79 - 2 * q^81 - 12 * q^83 - 28 * q^85 - 2 * q^87 - 8 * q^89 + 8 * q^91 + 10 * q^93 - 4 * q^95 + 8 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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