LMFDB - Embedded newform 1232.2.e.d.769.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,2,Mod(769,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

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

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

chi := DirichletCharacter("1232.769");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1232 = 2^{4} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1232.e (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$9.83756952902$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 9x^{2} - 8x + 2 $ x^4 - 2x^3 + 9x^2 - 8*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 308)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.3
Root			$0.500000 - 0.0913379i$ of defining polynomial
Character	$\chi$	$=$	1232.769
Dual form			1232.2.e.d.769.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{3} +2.82843i q^{5} -2.64575i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.41421i q^{3} +2.82843i q^{5} -2.64575i q^{7} +1.00000 q^{9} +(-2.00000 - 2.64575i) q^{11} -3.74166 q^{13} -4.00000 q^{15} -3.74166 q^{17} -7.48331 q^{19} +3.74166 q^{21} -4.00000 q^{23} -3.00000 q^{25} +5.65685i q^{27} +1.41421i q^{31} +(3.74166 - 2.82843i) q^{33} +7.48331 q^{35} -4.00000 q^{37} -5.29150i q^{39} +3.74166 q^{41} +10.5830i q^{43} +2.82843i q^{45} -9.89949i q^{47} -7.00000 q^{49} -5.29150i q^{51} -4.00000 q^{53} +(7.48331 - 5.65685i) q^{55} -10.5830i q^{57} -1.41421i q^{59} -11.2250 q^{61} -2.64575i q^{63} -10.5830i q^{65} +12.0000 q^{67} -5.65685i q^{69} -8.00000 q^{71} +11.2250 q^{73} -4.24264i q^{75} +(-7.00000 + 5.29150i) q^{77} +10.5830i q^{79} -5.00000 q^{81} -7.48331 q^{83} -10.5830i q^{85} -16.9706i q^{89} +9.89949i q^{91} -2.00000 q^{93} -21.1660i q^{95} +5.65685i q^{97} +(-2.00000 - 2.64575i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^9	$ 4 q + 4 q^{9} - 8 q^{11} - 16 q^{15} - 16 q^{23} - 12 q^{25} - 16 q^{37} - 28 q^{49} - 16 q^{53} + 48 q^{67} - 32 q^{71} - 28 q^{77} - 20 q^{81} - 8 q^{93} - 8 q^{99}+O(q^{100}) $ 4 * q + 4 * q^9 - 8 * q^11 - 16 * q^15 - 16 * q^23 - 12 * q^25 - 16 * q^37 - 28 * q^49 - 16 * q^53 + 48 * q^67 - 32 * q^71 - 28 * q^77 - 20 * q^81 - 8 * q^93 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times$.

$n$	$309$	$353$	$463$	$673$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$			1.41421i			0.816497i	0.912871	+	0.408248i	$0.133860\pi$
$3$			1.41421i			0.816497i	−0.912871	+	0.408248i	$0.866140\pi$
$4$		0			0
$5$			2.82843i			1.26491i	0.774597	+	0.632456i	$0.217953\pi$
$5$			2.82843i			1.26491i	−0.774597	+	0.632456i	$0.782047\pi$
$6$		0			0
$7$		−	2.64575i		−	1.00000i
$7$		−	2.64575i		−	1.00000i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$	−2.00000	−	2.64575i	−0.603023	−	0.797724i
$11$	−2.00000	−	2.64575i	−0.603023	−	0.797724i
$12$		0			0
$13$	−3.74166			−1.03775			−0.518875	−	0.854850i	$-0.673649\pi$
$13$	−3.74166			−1.03775			−0.518875	+	0.854850i	$0.673649\pi$
$14$		0			0
$15$	−4.00000			−1.03280
$16$		0			0
$17$	−3.74166			−0.907485			−0.453743	−	0.891133i	$-0.649911\pi$
$17$	−3.74166			−0.907485			−0.453743	+	0.891133i	$0.649911\pi$
$18$		0			0
$19$	−7.48331			−1.71679			−0.858395	−	0.512989i	$-0.828538\pi$
$19$	−7.48331			−1.71679			−0.858395	+	0.512989i	$0.828538\pi$
$20$		0			0
$21$	3.74166			0.816497
$22$		0			0
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$		0			0
$25$	−3.00000			−0.600000
$26$		0			0
$27$			5.65685i			1.08866i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$			1.41421i			0.254000i	0.991903	+	0.127000i	$0.0405349\pi$
$31$			1.41421i			0.254000i	−0.991903	+	0.127000i	$0.959465\pi$
$32$		0			0
$33$	3.74166	−	2.82843i	0.651339	−	0.492366i
$34$		0			0
$35$	7.48331			1.26491
$36$		0			0
$37$	−4.00000			−0.657596			−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000			−0.657596			−0.328798	+	0.944400i	$0.606644\pi$
$38$		0			0
$39$		−	5.29150i		−	0.847319i
$40$		0			0
$41$	3.74166			0.584349			0.292174	−	0.956365i	$-0.405621\pi$
$41$	3.74166			0.584349			0.292174	+	0.956365i	$0.405621\pi$
$42$		0			0
$43$			10.5830i			1.61389i	0.590624	+	0.806947i	$0.298881\pi$
$43$			10.5830i			1.61389i	−0.590624	+	0.806947i	$0.701119\pi$
$44$		0			0
$45$			2.82843i			0.421637i
$46$		0			0
$47$		−	9.89949i		−	1.44399i	−0.691898	−	0.721995i	$-0.743225\pi$
$47$		−	9.89949i		−	1.44399i	0.691898	−	0.721995i	$-0.256775\pi$
$48$		0			0
$49$	−7.00000			−1.00000
$50$		0			0
$51$		−	5.29150i		−	0.740959i
$52$		0			0
$53$	−4.00000			−0.549442			−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000			−0.549442			−0.274721	+	0.961524i	$0.588586\pi$
$54$		0			0
$55$	7.48331	−	5.65685i	1.00905	−	0.762770i
$56$		0			0
$57$		−	10.5830i		−	1.40175i
$58$		0			0
$59$		−	1.41421i		−	0.184115i	−0.995754	−	0.0920575i	$-0.970656\pi$
$59$		−	1.41421i		−	0.184115i	0.995754	−	0.0920575i	$-0.0293443\pi$
$60$		0			0
$61$	−11.2250			−1.43721			−0.718605	−	0.695418i	$-0.755219\pi$
$61$	−11.2250			−1.43721			−0.718605	+	0.695418i	$0.755219\pi$
$62$		0			0
$63$		−	2.64575i		−	0.333333i
$64$		0			0
$65$		−	10.5830i		−	1.31266i
$66$		0			0
$67$	12.0000			1.46603			0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000			1.46603			0.733017	+	0.680211i	$0.238112\pi$
$68$		0			0
$69$		−	5.65685i		−	0.681005i
$70$		0			0
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$		0			0
$73$	11.2250			1.31378			0.656892	−	0.753985i	$-0.271871\pi$
$73$	11.2250			1.31378			0.656892	+	0.753985i	$0.271871\pi$
$74$		0			0
$75$		−	4.24264i		−	0.489898i
$76$		0			0
$77$	−7.00000	+	5.29150i	−0.797724	+	0.603023i
$78$		0			0
$79$			10.5830i			1.19068i	0.803473	+	0.595341i	$0.202983\pi$
$79$			10.5830i			1.19068i	−0.803473	+	0.595341i	$0.797017\pi$
$80$		0			0
$81$	−5.00000			−0.555556
$82$		0			0
$83$	−7.48331			−0.821401			−0.410700	−	0.911770i	$-0.634716\pi$
$83$	−7.48331			−0.821401			−0.410700	+	0.911770i	$0.634716\pi$
$84$		0			0
$85$		−	10.5830i		−	1.14789i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	16.9706i		−	1.79888i	−0.437048	−	0.899438i	$-0.643976\pi$
$89$		−	16.9706i		−	1.79888i	0.437048	−	0.899438i	$-0.356024\pi$
$90$		0			0
$91$			9.89949i			1.03775i
$92$		0			0
$93$	−2.00000			−0.207390
$94$		0			0
$95$		−	21.1660i		−	2.17159i
$96$		0			0
$97$			5.65685i			0.574367i	0.957876	+	0.287183i	$0.0927189\pi$
$97$			5.65685i			0.574367i	−0.957876	+	0.287183i	$0.907281\pi$
$98$		0			0
$99$	−2.00000	−	2.64575i	−0.201008	−	0.265908i
$100$		0			0
$101$	11.2250			1.11693			0.558463	−	0.829529i	$-0.311391\pi$
$101$	11.2250			1.11693			0.558463	+	0.829529i	$0.311391\pi$
$102$		0			0
$103$			9.89949i			0.975426i	0.873004	+	0.487713i	$0.162169\pi$
$103$			9.89949i			0.975426i	−0.873004	+	0.487713i	$0.837831\pi$
$104$		0			0
$105$			10.5830i			1.03280i
$106$		0			0
$107$			5.29150i			0.511549i	0.966736	+	0.255774i	$0.0823304\pi$
$107$			5.29150i			0.511549i	−0.966736	+	0.255774i	$0.917670\pi$
$108$		0			0
$109$		−	10.5830i		−	1.01367i	−0.862044	−	0.506834i	$-0.830816\pi$
$109$		−	10.5830i		−	1.01367i	0.862044	−	0.506834i	$-0.169184\pi$
$110$		0			0
$111$		−	5.65685i		−	0.536925i
$112$		0			0
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$		−	11.3137i		−	1.05501i
$116$		0			0
$117$	−3.74166			−0.345916
$118$		0			0
$119$			9.89949i			0.907485i
$120$		0			0
$121$	−3.00000	+	10.5830i	−0.272727	+	0.962091i
$122$		0			0
$123$			5.29150i			0.477119i
$124$		0			0
$125$			5.65685i			0.505964i
$126$		0			0
$127$			5.29150i			0.469545i	0.972050	+	0.234772i	$0.0754345\pi$
$127$			5.29150i			0.469545i	−0.972050	+	0.234772i	$0.924565\pi$
$128$		0			0
$129$	−14.9666			−1.31774
$130$		0			0
$131$	7.48331			0.653820			0.326910	−	0.945055i	$-0.393993\pi$
$131$	7.48331			0.653820			0.326910	+	0.945055i	$0.393993\pi$
$132$		0			0
$133$			19.7990i			1.71679i
$134$		0			0
$135$	−16.0000			−1.37706
$136$		0			0
$137$	12.0000			1.02523			0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000			1.02523			0.512615	+	0.858619i	$0.328677\pi$
$138$		0			0
$139$	−7.48331			−0.634726			−0.317363	−	0.948304i	$-0.602797\pi$
$139$	−7.48331			−0.634726			−0.317363	+	0.948304i	$0.602797\pi$
$140$		0			0
$141$	14.0000			1.17901
$142$		0			0
$143$	7.48331	+	9.89949i	0.625786	+	0.827837i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	9.89949i		−	0.816497i
$148$		0			0
$149$			10.5830i			0.866994i	0.901155	+	0.433497i	$0.142720\pi$
$149$			10.5830i			0.866994i	−0.901155	+	0.433497i	$0.857280\pi$
$150$		0			0
$151$		−	10.5830i		−	0.861233i	−0.902535	−	0.430616i	$-0.858296\pi$
$151$		−	10.5830i		−	0.861233i	0.902535	−	0.430616i	$-0.141704\pi$
$152$		0			0
$153$	−3.74166			−0.302495
$154$		0			0
$155$	−4.00000			−0.321288
$156$		0			0
$157$			8.48528i			0.677199i	0.940931	+	0.338600i	$0.109953\pi$
$157$			8.48528i			0.677199i	−0.940931	+	0.338600i	$0.890047\pi$
$158$		0			0
$159$		−	5.65685i		−	0.448618i
$160$		0			0
$161$			10.5830i			0.834058i
$162$		0			0
$163$	8.00000			0.626608			0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000			0.626608			0.313304	+	0.949653i	$0.398564\pi$
$164$		0			0
$165$	8.00000	+	10.5830i	0.622799	+	0.823886i
$166$		0			0
$167$	−14.9666			−1.15815			−0.579076	−	0.815273i	$-0.696587\pi$
$167$	−14.9666			−1.15815			−0.579076	+	0.815273i	$0.696587\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	−7.48331			−0.572263
$172$		0			0
$173$	−11.2250			−0.853419			−0.426709	−	0.904389i	$-0.640327\pi$
$173$	−11.2250			−0.853419			−0.426709	+	0.904389i	$0.640327\pi$
$174$		0			0
$175$			7.93725i			0.600000i
$176$		0			0
$177$	2.00000			0.150329
$178$		0			0
$179$	16.0000			1.19590			0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000			1.19590			0.597948	+	0.801535i	$0.295983\pi$
$180$		0			0
$181$		−	25.4558i		−	1.89212i	−0.323994	−	0.946059i	$-0.605026\pi$
$181$		−	25.4558i		−	1.89212i	0.323994	−	0.946059i	$-0.394974\pi$
$182$		0			0
$183$		−	15.8745i		−	1.17348i
$184$		0			0
$185$		−	11.3137i		−	0.831800i
$186$		0			0
$187$	7.48331	+	9.89949i	0.547234	+	0.723923i
$188$		0			0
$189$	14.9666			1.08866
$190$		0			0
$191$	−24.0000			−1.73658			−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000			−1.73658			−0.868290	+	0.496058i	$0.834780\pi$
$192$		0			0
$193$			10.5830i			0.761781i	0.924620	+	0.380891i	$0.124383\pi$
$193$			10.5830i			0.761781i	−0.924620	+	0.380891i	$0.875617\pi$
$194$		0			0
$195$	14.9666			1.07178
$196$		0			0
$197$		−	21.1660i		−	1.50802i	−0.656865	−	0.754008i	$-0.728118\pi$
$197$		−	21.1660i		−	1.50802i	0.656865	−	0.754008i	$-0.271882\pi$
$198$		0			0
$199$			9.89949i			0.701757i	0.936421	+	0.350878i	$0.114117\pi$
$199$			9.89949i			0.701757i	−0.936421	+	0.350878i	$0.885883\pi$
$200$		0			0
$201$			16.9706i			1.19701i
$202$		0			0
$203$		0			0
$204$		0			0
$205$			10.5830i			0.739149i
$206$		0			0
$207$	−4.00000			−0.278019
$208$		0			0
$209$	14.9666	+	19.7990i	1.03526	+	1.36952i
$210$		0			0
$211$			15.8745i			1.09285i	0.837509	+	0.546423i	$0.184011\pi$
$211$			15.8745i			1.09285i	−0.837509	+	0.546423i	$0.815989\pi$
$212$		0			0
$213$		−	11.3137i		−	0.775203i
$214$		0			0
$215$	−29.9333			−2.04143
$216$		0			0
$217$	3.74166			0.254000
$218$		0			0
$219$			15.8745i			1.07270i
$220$		0			0
$221$	14.0000			0.941742
$222$		0			0
$223$			4.24264i			0.284108i	0.989859	+	0.142054i	$0.0453707\pi$
$223$			4.24264i			0.284108i	−0.989859	+	0.142054i	$0.954629\pi$
$224$		0			0
$225$	−3.00000			−0.200000
$226$		0			0
$227$	22.4499			1.49006			0.745028	−	0.667034i	$-0.232436\pi$
$227$	22.4499			1.49006			0.745028	+	0.667034i	$0.232436\pi$
$228$		0			0
$229$			2.82843i			0.186908i	0.995624	+	0.0934539i	$0.0297908\pi$
$229$			2.82843i			0.186908i	−0.995624	+	0.0934539i	$0.970209\pi$
$230$		0			0
$231$	−7.48331	−	9.89949i	−0.492366	−	0.651339i
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$	28.0000			1.82652
$236$		0			0
$237$	−14.9666			−0.972187
$238$		0			0
$239$		−	5.29150i		−	0.342279i	−0.985247	−	0.171139i	$-0.945255\pi$
$239$		−	5.29150i		−	0.342279i	0.985247	−	0.171139i	$-0.0547449\pi$
$240$		0			0
$241$	−11.2250			−0.723064			−0.361532	−	0.932360i	$-0.617746\pi$
$241$	−11.2250			−0.723064			−0.361532	+	0.932360i	$0.617746\pi$
$242$		0			0
$243$			9.89949i			0.635053i
$244$		0			0
$245$		−	19.7990i		−	1.26491i
$246$		0			0
$247$	28.0000			1.78160
$248$		0			0
$249$		−	10.5830i		−	0.670671i
$250$		0			0
$251$		−	24.0416i		−	1.51749i	−0.651385	−	0.758747i	$-0.725812\pi$
$251$		−	24.0416i		−	1.51749i	0.651385	−	0.758747i	$-0.274188\pi$
$252$		0			0
$253$	8.00000	+	10.5830i	0.502956	+	0.665348i
$254$		0			0
$255$	14.9666			0.937247
$256$		0			0
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$			10.5830i			0.657596i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			5.29150i			0.326288i	0.986602	+	0.163144i	$0.0521635\pi$
$263$			5.29150i			0.326288i	−0.986602	+	0.163144i	$0.947836\pi$
$264$		0			0
$265$		−	11.3137i		−	0.694996i
$266$		0			0
$267$	24.0000			1.46878
$268$		0			0
$269$			19.7990i			1.20717i	0.797300	+	0.603583i	$0.206261\pi$
$269$			19.7990i			1.20717i	−0.797300	+	0.603583i	$0.793739\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$	−14.0000			−0.847319
$274$		0			0
$275$	6.00000	+	7.93725i	0.361814	+	0.478634i
$276$		0			0
$277$		−	10.5830i		−	0.635871i	−0.948112	−	0.317936i	$-0.897010\pi$
$277$		−	10.5830i		−	0.635871i	0.948112	−	0.317936i	$-0.102990\pi$
$278$		0			0
$279$			1.41421i			0.0846668i
$280$		0			0
$281$			21.1660i			1.26266i	0.775515	+	0.631329i	$0.217490\pi$
$281$			21.1660i			1.26266i	−0.775515	+	0.631329i	$0.782510\pi$
$282$		0			0
$283$	−22.4499			−1.33451			−0.667255	−	0.744829i	$-0.732531\pi$
$283$	−22.4499			−1.33451			−0.667255	+	0.744829i	$0.732531\pi$
$284$		0			0
$285$	29.9333			1.77309
$286$		0			0
$287$		−	9.89949i		−	0.584349i
$288$		0			0
$289$	−3.00000			−0.176471
$290$		0			0
$291$	−8.00000			−0.468968
$292$		0			0
$293$	−26.1916			−1.53013			−0.765065	−	0.643953i	$-0.777293\pi$
$293$	−26.1916			−1.53013			−0.765065	+	0.643953i	$0.777293\pi$
$294$		0			0
$295$	4.00000			0.232889
$296$		0			0
$297$	14.9666	−	11.3137i	0.868452	−	0.656488i
$298$		0			0
$299$	14.9666			0.865543
$300$		0			0
$301$	28.0000			1.61389
$302$		0			0
$303$			15.8745i			0.911967i
$304$		0			0
$305$		−	31.7490i		−	1.81794i
$306$		0			0
$307$	−7.48331			−0.427095			−0.213548	−	0.976933i	$-0.568502\pi$
$307$	−7.48331			−0.427095			−0.213548	+	0.976933i	$0.568502\pi$
$308$		0			0
$309$	−14.0000			−0.796432
$310$		0			0
$311$			9.89949i			0.561349i	0.959803	+	0.280674i	$0.0905581\pi$
$311$			9.89949i			0.561349i	−0.959803	+	0.280674i	$0.909442\pi$
$312$		0			0
$313$			16.9706i			0.959233i	0.877478	+	0.479616i	$0.159224\pi$
$313$			16.9706i			0.959233i	−0.877478	+	0.479616i	$0.840776\pi$
$314$		0			0
$315$	7.48331			0.421637
$316$		0			0
$317$	−18.0000			−1.01098			−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000			−1.01098			−0.505490	+	0.862832i	$0.668688\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−7.48331			−0.417678
$322$		0			0
$323$	28.0000			1.55796
$324$		0			0
$325$	11.2250			0.622649
$326$		0			0
$327$	14.9666			0.827657
$328$		0			0
$329$	−26.1916			−1.44399
$330$		0			0
$331$	4.00000			0.219860			0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000			0.219860			0.109930	+	0.993939i	$0.464937\pi$
$332$		0			0
$333$	−4.00000			−0.219199
$334$		0			0
$335$			33.9411i			1.85440i
$336$		0			0
$337$		−	10.5830i		−	0.576493i	−0.957556	−	0.288247i	$-0.906928\pi$
$337$		−	10.5830i		−	0.576493i	0.957556	−	0.288247i	$-0.0930723\pi$
$338$		0			0
$339$			8.48528i			0.460857i
$340$		0			0
$341$	3.74166	−	2.82843i	0.202622	−	0.153168i
$342$		0			0
$343$			18.5203i			1.00000i
$344$		0			0
$345$	16.0000			0.861411
$346$		0			0
$347$			10.5830i			0.568125i	0.958806	+	0.284063i	$0.0916824\pi$
$347$			10.5830i			0.568125i	−0.958806	+	0.284063i	$0.908318\pi$
$348$		0			0
$349$	18.7083			1.00143			0.500716	−	0.865612i	$-0.333070\pi$
$349$	18.7083			1.00143			0.500716	+	0.865612i	$0.333070\pi$
$350$		0			0
$351$		−	21.1660i		−	1.12976i
$352$		0			0
$353$			28.2843i			1.50542i	0.658352	+	0.752710i	$0.271254\pi$
$353$			28.2843i			1.50542i	−0.658352	+	0.752710i	$0.728746\pi$
$354$		0			0
$355$		−	22.6274i		−	1.20094i
$356$		0			0
$357$	−14.0000			−0.740959
$358$		0			0
$359$		−	10.5830i		−	0.558550i	−0.960211	−	0.279275i	$-0.909906\pi$
$359$		−	10.5830i		−	0.558550i	0.960211	−	0.279275i	$-0.0900940\pi$
$360$		0			0
$361$	37.0000			1.94737
$362$		0			0
$363$	−14.9666	−	4.24264i	−0.785544	−	0.222681i
$364$		0			0
$365$			31.7490i			1.66182i
$366$		0			0
$367$		−	21.2132i		−	1.10732i	−0.832743	−	0.553660i	$-0.813231\pi$
$367$		−	21.2132i		−	1.10732i	0.832743	−	0.553660i	$-0.186769\pi$
$368$		0			0
$369$	3.74166			0.194783
$370$		0			0
$371$			10.5830i			0.549442i
$372$		0			0
$373$			21.1660i			1.09593i	0.836500	+	0.547967i	$0.184598\pi$
$373$			21.1660i			1.09593i	−0.836500	+	0.547967i	$0.815402\pi$
$374$		0			0
$375$	−8.00000			−0.413118
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−8.00000			−0.410932			−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000			−0.410932			−0.205466	+	0.978664i	$0.565871\pi$
$380$		0			0
$381$	−7.48331			−0.383382
$382$		0			0
$383$		−	7.07107i		−	0.361315i	−0.983546	−	0.180657i	$-0.942177\pi$
$383$		−	7.07107i		−	0.361315i	0.983546	−	0.180657i	$-0.0578225\pi$
$384$		0			0
$385$	−14.9666	−	19.7990i	−0.762770	−	1.00905i
$386$		0			0
$387$			10.5830i			0.537964i
$388$		0			0
$389$	−26.0000			−1.31825			−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000			−1.31825			−0.659126	+	0.752032i	$0.729074\pi$
$390$		0			0
$391$	14.9666			0.756895
$392$		0			0
$393$			10.5830i			0.533842i
$394$		0			0
$395$	−29.9333			−1.50611
$396$		0			0
$397$		−	19.7990i		−	0.993683i	−0.867841	−	0.496841i	$-0.834493\pi$
$397$		−	19.7990i		−	0.993683i	0.867841	−	0.496841i	$-0.165507\pi$
$398$		0			0
$399$	−28.0000			−1.40175
$400$		0			0
$401$	6.00000			0.299626			0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000			0.299626			0.149813	+	0.988714i	$0.452133\pi$
$402$		0			0
$403$		−	5.29150i		−	0.263589i
$404$		0			0
$405$		−	14.1421i		−	0.702728i
$406$		0			0
$407$	8.00000	+	10.5830i	0.396545	+	0.524580i
$408$		0			0
$409$	−18.7083			−0.925065			−0.462533	−	0.886602i	$-0.653059\pi$
$409$	−18.7083			−0.925065			−0.462533	+	0.886602i	$0.653059\pi$
$410$		0			0
$411$			16.9706i			0.837096i
$412$		0			0
$413$	−3.74166			−0.184115
$414$		0			0
$415$		−	21.1660i		−	1.03900i
$416$		0			0
$417$		−	10.5830i		−	0.518252i
$418$		0			0
$419$		−	9.89949i		−	0.483622i	−0.970323	−	0.241811i	$-0.922259\pi$
$419$		−	9.89949i		−	0.483622i	0.970323	−	0.241811i	$-0.0777414\pi$
$420$		0			0
$421$	12.0000			0.584844			0.292422	−	0.956289i	$-0.405539\pi$
$421$	12.0000			0.584844			0.292422	+	0.956289i	$0.405539\pi$
$422$		0			0
$423$		−	9.89949i		−	0.481330i
$424$		0			0
$425$	11.2250			0.544491
$426$		0			0
$427$			29.6985i			1.43721i
$428$		0			0
$429$	−14.0000	+	10.5830i	−0.675926	+	0.510952i
$430$		0			0
$431$		−	10.5830i		−	0.509765i	−0.966972	−	0.254883i	$-0.917963\pi$
$431$		−	10.5830i		−	0.509765i	0.966972	−	0.254883i	$-0.0820369\pi$
$432$		0			0
$433$			5.65685i			0.271851i	0.990719	+	0.135926i	$0.0434008\pi$
$433$			5.65685i			0.271851i	−0.990719	+	0.135926i	$0.956599\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	29.9333			1.43190
$438$		0			0
$439$	14.9666			0.714318			0.357159	−	0.934044i	$-0.383745\pi$
$439$	14.9666			0.714318			0.357159	+	0.934044i	$0.383745\pi$
$440$		0			0
$441$	−7.00000			−0.333333
$442$		0			0
$443$	−24.0000			−1.14027			−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000			−1.14027			−0.570137	+	0.821549i	$0.693110\pi$
$444$		0			0
$445$	48.0000			2.27542
$446$		0			0
$447$	−14.9666			−0.707897
$448$		0			0
$449$	−20.0000			−0.943858			−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000			−0.943858			−0.471929	+	0.881636i	$0.656442\pi$
$450$		0			0
$451$	−7.48331	−	9.89949i	−0.352376	−	0.466149i
$452$		0			0
$453$	14.9666			0.703194
$454$		0			0
$455$	−28.0000			−1.31266
$456$		0			0
$457$			31.7490i			1.48516i	0.669759	+	0.742578i	$0.266397\pi$
$457$			31.7490i			1.48516i	−0.669759	+	0.742578i	$0.733603\pi$
$458$		0			0
$459$		−	21.1660i		−	0.987945i
$460$		0			0
$461$	−3.74166			−0.174266			−0.0871332	−	0.996197i	$-0.527771\pi$
$461$	−3.74166			−0.174266			−0.0871332	+	0.996197i	$0.527771\pi$
$462$		0			0
$463$	−8.00000			−0.371792			−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000			−0.371792			−0.185896	+	0.982569i	$0.559519\pi$
$464$		0			0
$465$		−	5.65685i		−	0.262330i
$466$		0			0
$467$			12.7279i			0.588978i	0.955655	+	0.294489i	$0.0951494\pi$
$467$			12.7279i			0.588978i	−0.955655	+	0.294489i	$0.904851\pi$
$468$		0			0
$469$		−	31.7490i		−	1.46603i
$470$		0			0
$471$	−12.0000			−0.552931
$472$		0			0
$473$	28.0000	−	21.1660i	1.28744	−	0.973214i
$474$		0			0
$475$	22.4499			1.03007
$476$		0			0
$477$	−4.00000			−0.183147
$478$		0			0
$479$	14.9666			0.683843			0.341921	−	0.939729i	$-0.388922\pi$
$479$	14.9666			0.683843			0.341921	+	0.939729i	$0.388922\pi$
$480$		0			0
$481$	14.9666			0.682420
$482$		0			0
$483$	−14.9666			−0.681005
$484$		0			0
$485$	−16.0000			−0.726523
$486$		0			0
$487$	12.0000			0.543772			0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000			0.543772			0.271886	+	0.962329i	$0.412353\pi$
$488$		0			0
$489$			11.3137i			0.511624i
$490$		0			0
$491$		−	31.7490i		−	1.43281i	−0.697683	−	0.716407i	$-0.745786\pi$
$491$		−	31.7490i		−	1.43281i	0.697683	−	0.716407i	$-0.254214\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	7.48331	−	5.65685i	0.336350	−	0.254257i
$496$		0			0
$497$			21.1660i			0.949425i
$498$		0			0
$499$	−20.0000			−0.895323			−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000			−0.895323			−0.447661	+	0.894203i	$0.647743\pi$
$500$		0			0
$501$		−	21.1660i		−	0.945628i
$502$		0			0
$503$	14.9666			0.667329			0.333665	−	0.942692i	$-0.391715\pi$
$503$	14.9666			0.667329			0.333665	+	0.942692i	$0.391715\pi$
$504$		0			0
$505$			31.7490i			1.41281i
$506$		0			0
$507$			1.41421i			0.0628074i
$508$		0			0
$509$		−	2.82843i		−	0.125368i	−0.998033	−	0.0626839i	$-0.980034\pi$
$509$		−	2.82843i		−	0.125368i	0.998033	−	0.0626839i	$-0.0199660\pi$
$510$		0			0
$511$		−	29.6985i		−	1.31378i
$512$		0			0
$513$		−	42.3320i		−	1.86900i
$514$		0			0
$515$	−28.0000			−1.23383
$516$		0			0
$517$	−26.1916	+	19.7990i	−1.15191	+	0.870759i
$518$		0			0
$519$		−	15.8745i		−	0.696814i
$520$		0			0
$521$		−	11.3137i		−	0.495663i	−0.968803	−	0.247831i	$-0.920282\pi$
$521$		−	11.3137i		−	0.495663i	0.968803	−	0.247831i	$-0.0797179\pi$
$522$		0			0
$523$	37.4166			1.63611			0.818056	−	0.575138i	$-0.195052\pi$
$523$	37.4166			1.63611			0.818056	+	0.575138i	$0.195052\pi$
$524$		0			0
$525$	−11.2250			−0.489898
$526$		0			0
$527$		−	5.29150i		−	0.230501i
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$		−	1.41421i		−	0.0613716i
$532$		0			0
$533$	−14.0000			−0.606407
$534$		0			0
$535$	−14.9666			−0.647064
$536$		0			0
$537$			22.6274i			0.976445i
$538$		0			0
$539$	14.0000	+	18.5203i	0.603023	+	0.797724i
$540$		0			0
$541$			42.3320i			1.82000i	0.414613	+	0.909998i	$0.363917\pi$
$541$			42.3320i			1.82000i	−0.414613	+	0.909998i	$0.636083\pi$
$542$		0			0
$543$	36.0000			1.54491
$544$		0			0
$545$	29.9333			1.28220
$546$		0			0
$547$		−	26.4575i		−	1.13124i	−0.824665	−	0.565621i	$-0.808637\pi$
$547$		−	26.4575i		−	1.13124i	0.824665	−	0.565621i	$-0.191363\pi$
$548$		0			0
$549$	−11.2250			−0.479070
$550$		0			0
$551$		0			0
$552$		0			0
$553$	28.0000			1.19068
$554$		0			0
$555$	16.0000			0.679162
$556$		0			0
$557$		−	42.3320i		−	1.79367i	−0.442370	−	0.896833i	$-0.645862\pi$
$557$		−	42.3320i		−	1.79367i	0.442370	−	0.896833i	$-0.354138\pi$
$558$		0			0
$559$		−	39.5980i		−	1.67482i
$560$		0			0
$561$	−14.0000	+	10.5830i	−0.591080	+	0.446815i
$562$		0			0
$563$	−7.48331			−0.315384			−0.157692	−	0.987488i	$-0.550405\pi$
$563$	−7.48331			−0.315384			−0.157692	+	0.987488i	$0.550405\pi$
$564$		0			0
$565$			16.9706i			0.713957i
$566$		0			0
$567$			13.2288i			0.555556i
$568$		0			0
$569$		−	31.7490i		−	1.33099i	−0.746403	−	0.665494i	$-0.768221\pi$
$569$		−	31.7490i		−	1.33099i	0.746403	−	0.665494i	$-0.231779\pi$
$570$		0			0
$571$		−	37.0405i		−	1.55010i	−0.631901	−	0.775049i	$-0.717725\pi$
$571$		−	37.0405i		−	1.55010i	0.631901	−	0.775049i	$-0.282275\pi$
$572$		0			0
$573$		−	33.9411i		−	1.41791i
$574$		0			0
$575$	12.0000			0.500435
$576$		0			0
$577$		−	28.2843i		−	1.17749i	−0.808319	−	0.588745i	$-0.799622\pi$
$577$		−	28.2843i		−	1.17749i	0.808319	−	0.588745i	$-0.200378\pi$
$578$		0			0
$579$	−14.9666			−0.621992
$580$		0			0
$581$			19.7990i			0.821401i
$582$		0			0
$583$	8.00000	+	10.5830i	0.331326	+	0.438303i
$584$		0			0
$585$		−	10.5830i		−	0.437553i
$586$		0			0
$587$			29.6985i			1.22579i	0.790165	+	0.612894i	$0.209995\pi$
$587$			29.6985i			1.22579i	−0.790165	+	0.612894i	$0.790005\pi$
$588$		0			0
$589$		−	10.5830i		−	0.436065i
$590$		0			0
$591$	29.9333			1.23129
$592$		0			0
$593$	−41.1582			−1.69017			−0.845083	−	0.534635i	$-0.820449\pi$
$593$	−41.1582			−1.69017			−0.845083	+	0.534635i	$0.820449\pi$
$594$		0			0
$595$	−28.0000			−1.14789
$596$		0			0
$597$	−14.0000			−0.572982
$598$		0			0
$599$	−24.0000			−0.980613			−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000			−0.980613			−0.490307	+	0.871550i	$0.663115\pi$
$600$		0			0
$601$	33.6749			1.37363			0.686814	−	0.726833i	$-0.259009\pi$
$601$	33.6749			1.37363			0.686814	+	0.726833i	$0.259009\pi$
$602$		0			0
$603$	12.0000			0.488678
$604$		0			0
$605$	−29.9333	−	8.48528i	−1.21696	−	0.344976i
$606$		0			0
$607$	−44.8999			−1.82243			−0.911215	−	0.411931i	$-0.864855\pi$
$607$	−44.8999			−1.82243			−0.911215	+	0.411931i	$0.864855\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			37.0405i			1.49850i
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$	−14.9666			−0.603513
$616$		0			0
$617$	−12.0000			−0.483102			−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000			−0.483102			−0.241551	+	0.970388i	$0.577656\pi$
$618$		0			0
$619$			18.3848i			0.738947i	0.929241	+	0.369473i	$0.120462\pi$
$619$			18.3848i			0.738947i	−0.929241	+	0.369473i	$0.879538\pi$
$620$		0			0
$621$		−	22.6274i		−	0.908007i
$622$		0			0
$623$	−44.8999			−1.79888
$624$		0			0
$625$	−31.0000			−1.24000
$626$		0			0
$627$	−28.0000	+	21.1660i	−1.11821	+	0.845289i
$628$		0			0
$629$	14.9666			0.596759
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$		0			0
$633$	−22.4499			−0.892305
$634$		0			0
$635$	−14.9666			−0.593933
$636$		0			0
$637$	26.1916			1.03775
$638$		0			0
$639$	−8.00000			−0.316475
$640$		0			0
$641$	−44.0000			−1.73790			−0.868948	−	0.494904i	$-0.835203\pi$
$641$	−44.0000			−1.73790			−0.868948	+	0.494904i	$0.835203\pi$
$642$		0			0
$643$			15.5563i			0.613483i	0.951793	+	0.306741i	$0.0992386\pi$
$643$			15.5563i			0.613483i	−0.951793	+	0.306741i	$0.900761\pi$
$644$		0			0
$645$		−	42.3320i		−	1.66682i
$646$		0			0
$647$			18.3848i			0.722780i	0.932415	+	0.361390i	$0.117698\pi$
$647$			18.3848i			0.722780i	−0.932415	+	0.361390i	$0.882302\pi$
$648$		0			0
$649$	−3.74166	+	2.82843i	−0.146873	+	0.111025i
$650$		0			0
$651$			5.29150i			0.207390i
$652$		0			0
$653$	18.0000			0.704394			0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000			0.704394			0.352197	+	0.935926i	$0.385435\pi$
$654$		0			0
$655$			21.1660i			0.827024i
$656$		0			0
$657$	11.2250			0.437928
$658$		0			0
$659$		−	31.7490i		−	1.23677i	−0.785877	−	0.618383i	$-0.787788\pi$
$659$		−	31.7490i		−	1.23677i	0.785877	−	0.618383i	$-0.212212\pi$
$660$		0			0
$661$			8.48528i			0.330039i	0.986290	+	0.165020i	$0.0527687\pi$
$661$			8.48528i			0.330039i	−0.986290	+	0.165020i	$0.947231\pi$
$662$		0			0
$663$			19.7990i			0.768929i
$664$		0			0
$665$	−56.0000			−2.17159
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−6.00000			−0.231973
$670$		0			0
$671$	22.4499	+	29.6985i	0.866670	+	1.14650i
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$		−	16.9706i		−	0.653197i
$676$		0			0
$677$	−18.7083			−0.719018			−0.359509	−	0.933142i	$-0.617056\pi$
$677$	−18.7083			−0.719018			−0.359509	+	0.933142i	$0.617056\pi$
$678$		0			0
$679$	14.9666			0.574367
$680$		0			0
$681$			31.7490i			1.21662i
$682$		0			0
$683$	24.0000			0.918334			0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000			0.918334			0.459167	+	0.888350i	$0.348148\pi$
$684$		0			0
$685$			33.9411i			1.29682i
$686$		0			0
$687$	−4.00000			−0.152610
$688$		0			0
$689$	14.9666			0.570183
$690$		0			0
$691$			46.6690i			1.77537i	0.460447	+	0.887687i	$0.347689\pi$
$691$			46.6690i			1.77537i	−0.460447	+	0.887687i	$0.652311\pi$
$692$		0			0
$693$	−7.00000	+	5.29150i	−0.265908	+	0.201008i
$694$		0			0
$695$		−	21.1660i		−	0.802873i
$696$		0			0
$697$	−14.0000			−0.530288
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	31.7490i		−	1.19914i	−0.800321	−	0.599572i	$-0.795338\pi$
$701$		−	31.7490i		−	1.19914i	0.800321	−	0.599572i	$-0.204662\pi$
$702$		0			0
$703$	29.9333			1.12895
$704$		0			0
$705$			39.5980i			1.49135i
$706$		0			0
$707$		−	29.6985i		−	1.11693i
$708$		0			0
$709$	−38.0000			−1.42712			−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000			−1.42712			−0.713560	+	0.700594i	$0.752918\pi$
$710$		0			0
$711$			10.5830i			0.396894i
$712$		0			0
$713$		−	5.65685i		−	0.211851i
$714$		0			0
$715$	−28.0000	+	21.1660i	−1.04714	+	0.791564i
$716$		0			0
$717$	7.48331			0.279470
$718$		0			0
$719$		−	7.07107i		−	0.263706i	−0.991269	−	0.131853i	$-0.957907\pi$
$719$		−	7.07107i		−	0.263706i	0.991269	−	0.131853i	$-0.0420927\pi$
$720$		0			0
$721$	26.1916			0.975426
$722$		0			0
$723$		−	15.8745i		−	0.590379i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	4.24264i		−	0.157351i	−0.996900	−	0.0786754i	$-0.974931\pi$
$727$		−	4.24264i		−	0.157351i	0.996900	−	0.0786754i	$-0.0250691\pi$
$728$		0			0
$729$	−29.0000			−1.07407
$730$		0			0
$731$		−	39.5980i		−	1.46458i
$732$		0			0
$733$	−11.2250			−0.414604			−0.207302	−	0.978277i	$-0.566468\pi$
$733$	−11.2250			−0.414604			−0.207302	+	0.978277i	$0.566468\pi$
$734$		0			0
$735$	28.0000			1.03280
$736$		0			0
$737$	−24.0000	−	31.7490i	−0.884051	−	1.16949i
$738$		0			0
$739$		−	15.8745i		−	0.583953i	−0.956425	−	0.291977i	$-0.905687\pi$
$739$		−	15.8745i		−	0.583953i	0.956425	−	0.291977i	$-0.0943129\pi$
$740$		0			0
$741$			39.5980i			1.45467i
$742$		0			0
$743$			15.8745i			0.582379i	0.956665	+	0.291190i	$0.0940511\pi$
$743$			15.8745i			0.582379i	−0.956665	+	0.291190i	$0.905949\pi$
$744$		0			0
$745$	−29.9333			−1.09667
$746$		0			0
$747$	−7.48331			−0.273800
$748$		0			0
$749$	14.0000			0.511549
$750$		0			0
$751$	−4.00000			−0.145962			−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000			−0.145962			−0.0729810	+	0.997333i	$0.523251\pi$
$752$		0			0
$753$	34.0000			1.23903
$754$		0			0
$755$	29.9333			1.08938
$756$		0			0
$757$	26.0000			0.944986			0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000			0.944986			0.472493	+	0.881334i	$0.343354\pi$
$758$		0			0
$759$	−14.9666	+	11.3137i	−0.543254	+	0.410662i
$760$		0			0
$761$	−18.7083			−0.678175			−0.339087	−	0.940755i	$-0.610118\pi$
$761$	−18.7083			−0.678175			−0.339087	+	0.940755i	$0.610118\pi$
$762$		0			0
$763$	−28.0000			−1.01367
$764$		0			0
$765$		−	10.5830i		−	0.382629i
$766$		0			0
$767$			5.29150i			0.191065i
$768$		0			0
$769$	−11.2250			−0.404783			−0.202391	−	0.979305i	$-0.564871\pi$
$769$	−11.2250			−0.404783			−0.202391	+	0.979305i	$0.564871\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			19.7990i			0.712120i	0.934463	+	0.356060i	$0.115880\pi$
$773$			19.7990i			0.712120i	−0.934463	+	0.356060i	$0.884120\pi$
$774$		0			0
$775$		−	4.24264i		−	0.152400i
$776$		0			0
$777$	−14.9666			−0.536925
$778$		0			0
$779$	−28.0000			−1.00320
$780$		0			0
$781$	16.0000	+	21.1660i	0.572525	+	0.757379i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−24.0000			−0.856597
$786$		0			0
$787$	7.48331			0.266751			0.133376	−	0.991066i	$-0.457418\pi$
$787$	7.48331			0.266751			0.133376	+	0.991066i	$0.457418\pi$
$788$		0			0
$789$	−7.48331			−0.266413
$790$		0			0
$791$		−	15.8745i		−	0.564433i
$792$		0			0
$793$	42.0000			1.49146
$794$		0			0
$795$	16.0000			0.567462
$796$		0			0
$797$			25.4558i			0.901692i	0.892602	+	0.450846i	$0.148878\pi$
$797$			25.4558i			0.901692i	−0.892602	+	0.450846i	$0.851122\pi$
$798$		0			0
$799$			37.0405i			1.31040i
$800$		0			0
$801$		−	16.9706i		−	0.599625i
$802$		0			0
$803$	−22.4499	−	29.6985i	−0.792241	−	1.04804i
$804$		0			0
$805$	−29.9333			−1.05501
$806$		0			0
$807$	−28.0000			−0.985647
$808$		0			0
$809$			31.7490i			1.11624i	0.829762	+	0.558118i	$0.188476\pi$
$809$			31.7490i			1.11624i	−0.829762	+	0.558118i	$0.811524\pi$
$810$		0			0
$811$	−22.4499			−0.788324			−0.394162	−	0.919041i	$-0.628965\pi$
$811$	−22.4499			−0.788324			−0.394162	+	0.919041i	$0.628965\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			22.6274i			0.792604i
$816$		0			0
$817$		−	79.1960i		−	2.77072i
$818$		0			0
$819$			9.89949i			0.345916i
$820$		0			0
$821$		−	10.5830i		−	0.369349i	−0.982800	−	0.184675i	$-0.940877\pi$
$821$		−	10.5830i		−	0.369349i	0.982800	−	0.184675i	$-0.0591232\pi$
$822$		0			0
$823$	16.0000			0.557725			0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000			0.557725			0.278862	+	0.960331i	$0.410043\pi$
$824$		0			0
$825$	−11.2250	+	8.48528i	−0.390803	+	0.295420i
$826$		0			0
$827$		−	31.7490i		−	1.10402i	−0.833837	−	0.552011i	$-0.813861\pi$
$827$		−	31.7490i		−	1.10402i	0.833837	−	0.552011i	$-0.186139\pi$
$828$		0			0
$829$		−	48.0833i		−	1.67000i	−0.550249	−	0.835000i	$-0.685467\pi$
$829$		−	48.0833i		−	1.67000i	0.550249	−	0.835000i	$-0.314533\pi$
$830$		0			0
$831$	14.9666			0.519187
$832$		0			0
$833$	26.1916			0.907485
$834$		0			0
$835$		−	42.3320i		−	1.46496i
$836$		0			0
$837$	−8.00000			−0.276520
$838$		0			0
$839$		−	24.0416i		−	0.830009i	−0.909819	−	0.415005i	$-0.863780\pi$
$839$		−	24.0416i		−	0.830009i	0.909819	−	0.415005i	$-0.136220\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$	−29.9333			−1.03096
$844$		0			0
$845$			2.82843i			0.0973009i
$846$		0			0
$847$	28.0000	+	7.93725i	0.962091	+	0.272727i
$848$		0			0
$849$		−	31.7490i		−	1.08962i
$850$		0			0
$851$	16.0000			0.548473
$852$		0			0
$853$	41.1582			1.40923			0.704615	−	0.709589i	$-0.251120\pi$
$853$	41.1582			1.40923			0.704615	+	0.709589i	$0.251120\pi$
$854$		0			0
$855$		−	21.1660i		−	0.723862i
$856$		0			0
$857$	41.1582			1.40594			0.702969	−	0.711220i	$-0.251857\pi$
$857$	41.1582			1.40594			0.702969	+	0.711220i	$0.251857\pi$
$858$		0			0
$859$		−	46.6690i		−	1.59233i	−0.605081	−	0.796164i	$-0.706859\pi$
$859$		−	46.6690i		−	1.59233i	0.605081	−	0.796164i	$-0.293141\pi$
$860$		0			0
$861$	14.0000			0.477119
$862$		0			0
$863$	48.0000			1.63394			0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000			1.63394			0.816970	+	0.576681i	$0.195652\pi$
$864$		0			0
$865$		−	31.7490i		−	1.07950i
$866$		0			0
$867$		−	4.24264i		−	0.144088i
$868$		0			0
$869$	28.0000	−	21.1660i	0.949835	−	0.718008i
$870$		0			0
$871$	−44.8999			−1.52137
$872$		0			0
$873$			5.65685i			0.191456i
$874$		0			0
$875$	14.9666			0.505964
$876$		0			0
$877$			52.9150i			1.78681i	0.449249	+	0.893407i	$0.351692\pi$
$877$			52.9150i			1.78681i	−0.449249	+	0.893407i	$0.648308\pi$
$878$		0			0
$879$		−	37.0405i		−	1.24935i
$880$		0			0
$881$		−	45.2548i		−	1.52467i	−0.647180	−	0.762337i	$-0.724052\pi$
$881$		−	45.2548i		−	1.52467i	0.647180	−	0.762337i	$-0.275948\pi$
$882$		0			0
$883$	−16.0000			−0.538443			−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000			−0.538443			−0.269221	+	0.963078i	$0.586766\pi$
$884$		0			0
$885$			5.65685i			0.190153i
$886$		0			0
$887$	29.9333			1.00506			0.502530	−	0.864560i	$-0.332403\pi$
$887$	29.9333			1.00506			0.502530	+	0.864560i	$0.332403\pi$
$888$		0			0
$889$	14.0000			0.469545
$890$		0			0
$891$	10.0000	+	13.2288i	0.335013	+	0.443180i
$892$		0			0
$893$			74.0810i			2.47903i
$894$		0			0
$895$			45.2548i			1.51270i
$896$		0			0
$897$			21.1660i			0.706713i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	14.9666			0.498611
$902$		0			0
$903$			39.5980i			1.31774i
$904$		0			0
$905$	72.0000			2.39336
$906$		0			0
$907$	−52.0000			−1.72663			−0.863316	−	0.504664i	$-0.831616\pi$
$907$	−52.0000			−1.72663			−0.863316	+	0.504664i	$0.831616\pi$
$908$		0			0
$909$	11.2250			0.372309
$910$		0			0
$911$	−12.0000			−0.397578			−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000			−0.397578			−0.198789	+	0.980042i	$0.563701\pi$
$912$		0			0
$913$	14.9666	+	19.7990i	0.495323	+	0.655251i
$914$		0			0
$915$	44.8999			1.48434
$916$		0			0
$917$		−	19.7990i		−	0.653820i
$918$		0			0
$919$			5.29150i			0.174551i	0.996184	+	0.0872753i	$0.0278160\pi$
$919$			5.29150i			0.174551i	−0.996184	+	0.0872753i	$0.972184\pi$
$920$		0			0
$921$		−	10.5830i		−	0.348722i
$922$		0			0
$923$	29.9333			0.985265
$924$		0			0
$925$	12.0000			0.394558
$926$		0			0
$927$			9.89949i			0.325142i
$928$		0			0
$929$		−	39.5980i		−	1.29917i	−0.760290	−	0.649584i	$-0.774943\pi$
$929$		−	39.5980i		−	1.29917i	0.760290	−	0.649584i	$-0.225057\pi$
$930$		0			0
$931$	52.3832			1.71679
$932$		0			0
$933$	−14.0000			−0.458339
$934$		0			0
$935$	−28.0000	+	21.1660i	−0.915698	+	0.692203i
$936$		0			0
$937$	3.74166			0.122235			0.0611173	−	0.998131i	$-0.480534\pi$
$937$	3.74166			0.122235			0.0611173	+	0.998131i	$0.480534\pi$
$938$		0			0
$939$	−24.0000			−0.783210
$940$		0			0
$941$	18.7083			0.609873			0.304936	−	0.952373i	$-0.401365\pi$
$941$	18.7083			0.609873			0.304936	+	0.952373i	$0.401365\pi$
$942$		0			0
$943$	−14.9666			−0.487381
$944$		0			0
$945$			42.3320i			1.37706i
$946$		0			0
$947$	4.00000			0.129983			0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000			0.129983			0.0649913	+	0.997886i	$0.479298\pi$
$948$		0			0
$949$	−42.0000			−1.36338
$950$		0			0
$951$		−	25.4558i		−	0.825462i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		−	67.8823i		−	2.19662i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	31.7490i		−	1.02523i
$960$		0			0
$961$	29.0000			0.935484
$962$		0			0
$963$			5.29150i			0.170516i
$964$		0			0
$965$	−29.9333			−0.963586
$966$		0			0
$967$			31.7490i			1.02098i	0.859884	+	0.510490i	$0.170536\pi$
$967$			31.7490i			1.02098i	−0.859884	+	0.510490i	$0.829464\pi$
$968$		0			0
$969$			39.5980i			1.27207i
$970$		0			0
$971$			7.07107i			0.226921i	0.993542	+	0.113461i	$0.0361936\pi$
$971$			7.07107i			0.226921i	−0.993542	+	0.113461i	$0.963806\pi$
$972$		0			0
$973$			19.7990i			0.634726i
$974$		0			0
$975$			15.8745i			0.508391i
$976$		0			0
$977$	30.0000			0.959785			0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000			0.959785			0.479893	+	0.877327i	$0.340676\pi$
$978$		0			0
$979$	−44.8999	+	33.9411i	−1.43501	+	1.08476i
$980$		0			0
$981$		−	10.5830i		−	0.337889i
$982$		0			0
$983$		−	60.8112i		−	1.93958i	−0.243950	−	0.969788i	$-0.578443\pi$
$983$		−	60.8112i		−	1.93958i	0.243950	−	0.969788i	$-0.421557\pi$
$984$		0			0
$985$	59.8665			1.90751
$986$		0			0
$987$		−	37.0405i		−	1.17901i
$988$		0			0
$989$		−	42.3320i		−	1.34608i
$990$		0			0
$991$	32.0000			1.01651			0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000			1.01651			0.508257	+	0.861206i	$0.330290\pi$
$992$		0			0
$993$			5.65685i			0.179515i
$994$		0			0
$995$	−28.0000			−0.887660
$996$		0			0
$997$	−56.1249			−1.77749			−0.888746	−	0.458400i	$-0.848423\pi$
$997$	−56.1249			−1.77749			−0.888746	+	0.458400i	$0.848423\pi$
$998$		0			0
$999$		−	22.6274i		−	0.715900i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.2.e.d.769.3		4
4.3	odd	2		308.2.c.b.153.2	yes	4
7.6	odd	2	inner	1232.2.e.d.769.1		4
11.10	odd	2	inner	1232.2.e.d.769.4		4
12.11	even	2		2772.2.i.a.1693.2		4
28.3	even	6		2156.2.q.a.901.4		8
28.11	odd	6		2156.2.q.a.901.2		8
28.19	even	6		2156.2.q.a.2089.1		8
28.23	odd	6		2156.2.q.a.2089.3		8
28.27	even	2		308.2.c.b.153.4	yes	4
44.43	even	2		308.2.c.b.153.1	✓	4
77.76	even	2	inner	1232.2.e.d.769.2		4
84.83	odd	2		2772.2.i.a.1693.4		4
132.131	odd	2		2772.2.i.a.1693.1		4
308.87	odd	6		2156.2.q.a.901.3		8
308.131	odd	6		2156.2.q.a.2089.2		8
308.219	even	6		2156.2.q.a.2089.4		8
308.263	even	6		2156.2.q.a.901.1		8
308.307	odd	2		308.2.c.b.153.3	yes	4
924.923	even	2		2772.2.i.a.1693.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.2.c.b.153.1	✓	4	44.43	even	2
308.2.c.b.153.2	yes	4	4.3	odd	2
308.2.c.b.153.3	yes	4	308.307	odd	2
308.2.c.b.153.4	yes	4	28.27	even	2
1232.2.e.d.769.1		4	7.6	odd	2	inner
1232.2.e.d.769.2		4	77.76	even	2	inner
1232.2.e.d.769.3		4	1.1	even	1	trivial
1232.2.e.d.769.4		4	11.10	odd	2	inner
2156.2.q.a.901.1		8	308.263	even	6
2156.2.q.a.901.2		8	28.11	odd	6
2156.2.q.a.901.3		8	308.87	odd	6
2156.2.q.a.901.4		8	28.3	even	6
2156.2.q.a.2089.1		8	28.19	even	6
2156.2.q.a.2089.2		8	308.131	odd	6
2156.2.q.a.2089.3		8	28.23	odd	6
2156.2.q.a.2089.4		8	308.219	even	6
2772.2.i.a.1693.1		4	132.131	odd	2
2772.2.i.a.1693.2		4	12.11	even	2
2772.2.i.a.1693.3		4	924.923	even	2
2772.2.i.a.1693.4		4	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1232.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{3} +2.82843i q^{5} -2.64575i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.41421i q^{3} +2.82843i q^{5} -2.64575i q^{7} +1.00000 q^{9} +(-2.00000 - 2.64575i) q^{11} -3.74166 q^{13} -4.00000 q^{15} -3.74166 q^{17} -7.48331 q^{19} +3.74166 q^{21} -4.00000 q^{23} -3.00000 q^{25} +5.65685i q^{27} +1.41421i q^{31} +(3.74166 - 2.82843i) q^{33} +7.48331 q^{35} -4.00000 q^{37} -5.29150i q^{39} +3.74166 q^{41} +10.5830i q^{43} +2.82843i q^{45} -9.89949i q^{47} -7.00000 q^{49} -5.29150i q^{51} -4.00000 q^{53} +(7.48331 - 5.65685i) q^{55} -10.5830i q^{57} -1.41421i q^{59} -11.2250 q^{61} -2.64575i q^{63} -10.5830i q^{65} +12.0000 q^{67} -5.65685i q^{69} -8.00000 q^{71} +11.2250 q^{73} -4.24264i q^{75} +(-7.00000 + 5.29150i) q^{77} +10.5830i q^{79} -5.00000 q^{81} -7.48331 q^{83} -10.5830i q^{85} -16.9706i q^{89} +9.89949i q^{91} -2.00000 q^{93} -21.1660i q^{95} +5.65685i q^{97} +(-2.00000 - 2.64575i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^9	\( 4 q + 4 q^{9} - 8 q^{11} - 16 q^{15} - 16 q^{23} - 12 q^{25} - 16 q^{37} - 28 q^{49} - 16 q^{53} + 48 q^{67} - 32 q^{71} - 28 q^{77} - 20 q^{81} - 8 q^{93} - 8 q^{99}+O(q^{100}) \) 4 * q + 4 * q^9 - 8 * q^11 - 16 * q^15 - 16 * q^23 - 12 * q^25 - 16 * q^37 - 28 * q^49 - 16 * q^53 + 48 * q^67 - 32 * q^71 - 28 * q^77 - 20 * q^81 - 8 * q^93 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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