LMFDB - Embedded newform 1232.2.e.c.769.1

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:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.1
Root			$-1.58114 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1232.769
Dual form			1232.2.e.c.769.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{3} -2.23607i q^{5} +(-1.58114 - 2.12132i) q^{7} -2.00000 q^{9} +O(q^{10})$	$q-2.23607i q^{3} -2.23607i q^{5} +(-1.58114 - 2.12132i) q^{7} -2.00000 q^{9} +(3.00000 - 1.41421i) q^{11} -6.32456 q^{13} -5.00000 q^{15} -3.16228 q^{19} +(-4.74342 + 3.53553i) q^{21} +3.00000 q^{23} -2.23607i q^{27} -1.41421i q^{29} +6.70820i q^{31} +(-3.16228 - 6.70820i) q^{33} +(-4.74342 + 3.53553i) q^{35} -1.00000 q^{37} +14.1421i q^{39} +9.48683 q^{41} -4.24264i q^{43} +4.47214i q^{45} -4.47214i q^{47} +(-2.00000 + 6.70820i) q^{49} +(-3.16228 - 6.70820i) q^{55} +7.07107i q^{57} +2.23607i q^{59} +3.16228 q^{61} +(3.16228 + 4.24264i) q^{63} +14.1421i q^{65} -11.0000 q^{67} -6.70820i q^{69} -9.00000 q^{71} +3.16228 q^{73} +(-7.74342 - 4.12789i) q^{77} +8.48528i q^{79} -11.0000 q^{81} +9.48683 q^{83} -3.16228 q^{87} -2.23607i q^{89} +(10.0000 + 13.4164i) q^{91} +15.0000 q^{93} +7.07107i q^{95} +6.70820i q^{97} +(-6.00000 + 2.82843i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{9}+O(q^{10}) $ 4 * q - 8 * q^9	$ 4 q - 8 q^{9} + 12 q^{11} - 20 q^{15} + 12 q^{23} - 4 q^{37} - 8 q^{49} - 44 q^{67} - 36 q^{71} - 12 q^{77} - 44 q^{81} + 40 q^{91} + 60 q^{93} - 24 q^{99}+O(q^{100}) $ 4 * q - 8 * q^9 + 12 * q^11 - 20 * q^15 + 12 * q^23 - 4 * q^37 - 8 * q^49 - 44 * q^67 - 36 * q^71 - 12 * q^77 - 44 * q^81 + 40 * q^91 + 60 * q^93 - 24 * 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
$3$		−	2.23607i		−	1.29099i	−0.763763	−	0.645497i	$-0.776650\pi$
$3$		−	2.23607i		−	1.29099i	0.763763	−	0.645497i	$-0.223350\pi$
$4$		0			0
$5$		−	2.23607i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$5$		−	2.23607i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$6$		0			0
$7$	−1.58114	−	2.12132i	−0.597614	−	0.801784i
$7$	−1.58114	−	2.12132i	−0.597614	−	0.801784i
$8$		0			0
$9$	−2.00000			−0.666667
$10$		0			0
$11$	3.00000	−	1.41421i	0.904534	−	0.426401i
$11$	3.00000	−	1.41421i	0.904534	−	0.426401i
$12$		0			0
$13$	−6.32456			−1.75412			−0.877058	−	0.480384i	$-0.840497\pi$
$13$	−6.32456			−1.75412			−0.877058	+	0.480384i	$0.840497\pi$
$14$		0			0
$15$	−5.00000			−1.29099
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	−3.16228			−0.725476			−0.362738	−	0.931891i	$-0.618158\pi$
$19$	−3.16228			−0.725476			−0.362738	+	0.931891i	$0.618158\pi$
$20$		0			0
$21$	−4.74342	+	3.53553i	−1.03510	+	0.771517i
$22$		0			0
$23$	3.00000			0.625543			0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000			0.625543			0.312772	+	0.949828i	$0.398743\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	2.23607i		−	0.430331i
$28$		0			0
$29$		−	1.41421i		−	0.262613i	−0.991342	−	0.131306i	$-0.958083\pi$
$29$		−	1.41421i		−	0.262613i	0.991342	−	0.131306i	$-0.0419172\pi$
$30$		0			0
$31$			6.70820i			1.20483i	0.798183	+	0.602414i	$0.205795\pi$
$31$			6.70820i			1.20483i	−0.798183	+	0.602414i	$0.794205\pi$
$32$		0			0
$33$	−3.16228	−	6.70820i	−0.550482	−	1.16775i
$34$		0			0
$35$	−4.74342	+	3.53553i	−0.801784	+	0.597614i
$36$		0			0
$37$	−1.00000			−0.164399			−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000			−0.164399			−0.0821995	+	0.996616i	$0.526194\pi$
$38$		0			0
$39$			14.1421i			2.26455i
$40$		0			0
$41$	9.48683			1.48159			0.740797	−	0.671729i	$-0.234448\pi$
$41$	9.48683			1.48159			0.740797	+	0.671729i	$0.234448\pi$
$42$		0			0
$43$		−	4.24264i		−	0.646997i	−0.946229	−	0.323498i	$-0.895141\pi$
$43$		−	4.24264i		−	0.646997i	0.946229	−	0.323498i	$-0.104859\pi$
$44$		0			0
$45$			4.47214i			0.666667i
$46$		0			0
$47$		−	4.47214i		−	0.652328i	−0.945313	−	0.326164i	$-0.894244\pi$
$47$		−	4.47214i		−	0.652328i	0.945313	−	0.326164i	$-0.105756\pi$
$48$		0			0
$49$	−2.00000	+	6.70820i	−0.285714	+	0.958315i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	−3.16228	−	6.70820i	−0.426401	−	0.904534i
$56$		0			0
$57$			7.07107i			0.936586i
$58$		0			0
$59$			2.23607i			0.291111i	0.989350	+	0.145556i	$0.0464970\pi$
$59$			2.23607i			0.291111i	−0.989350	+	0.145556i	$0.953503\pi$
$60$		0			0
$61$	3.16228			0.404888			0.202444	−	0.979294i	$-0.435112\pi$
$61$	3.16228			0.404888			0.202444	+	0.979294i	$0.435112\pi$
$62$		0			0
$63$	3.16228	+	4.24264i	0.398410	+	0.534522i
$64$		0			0
$65$			14.1421i			1.75412i
$66$		0			0
$67$	−11.0000			−1.34386			−0.671932	−	0.740613i	$-0.734535\pi$
$67$	−11.0000			−1.34386			−0.671932	+	0.740613i	$0.734535\pi$
$68$		0			0
$69$		−	6.70820i		−	0.807573i
$70$		0			0
$71$	−9.00000			−1.06810			−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000			−1.06810			−0.534052	+	0.845452i	$0.679331\pi$
$72$		0			0
$73$	3.16228			0.370117			0.185058	−	0.982728i	$-0.440753\pi$
$73$	3.16228			0.370117			0.185058	+	0.982728i	$0.440753\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−7.74342	−	4.12789i	−0.882444	−	0.470417i
$78$		0			0
$79$			8.48528i			0.954669i	0.878722	+	0.477334i	$0.158397\pi$
$79$			8.48528i			0.954669i	−0.878722	+	0.477334i	$0.841603\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$		0			0
$83$	9.48683			1.04132			0.520658	−	0.853766i	$-0.325687\pi$
$83$	9.48683			1.04132			0.520658	+	0.853766i	$0.325687\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−3.16228			−0.339032
$88$		0			0
$89$		−	2.23607i		−	0.237023i	−0.992953	−	0.118511i	$-0.962188\pi$
$89$		−	2.23607i		−	0.237023i	0.992953	−	0.118511i	$-0.0378122\pi$
$90$		0			0
$91$	10.0000	+	13.4164i	1.04828	+	1.40642i
$92$		0			0
$93$	15.0000			1.55543
$94$		0			0
$95$			7.07107i			0.725476i
$96$		0			0
$97$			6.70820i			0.681115i	0.940224	+	0.340557i	$0.110616\pi$
$97$			6.70820i			0.681115i	−0.940224	+	0.340557i	$0.889384\pi$
$98$		0			0
$99$	−6.00000	+	2.82843i	−0.603023	+	0.284268i
$100$		0			0
$101$	−9.48683			−0.943975			−0.471988	−	0.881605i	$-0.656463\pi$
$101$	−9.48683			−0.943975			−0.471988	+	0.881605i	$0.656463\pi$
$102$		0			0
$103$		−	13.4164i		−	1.32196i	−0.750404	−	0.660979i	$-0.770141\pi$
$103$		−	13.4164i		−	1.32196i	0.750404	−	0.660979i	$-0.229859\pi$
$104$		0			0
$105$	7.90569	+	10.6066i	0.771517	+	1.03510i
$106$		0			0
$107$		−	2.82843i		−	0.273434i	−0.990610	−	0.136717i	$-0.956345\pi$
$107$		−	2.82843i		−	0.273434i	0.990610	−	0.136717i	$-0.0436552\pi$
$108$		0			0
$109$		−	12.7279i		−	1.21911i	−0.792742	−	0.609557i	$-0.791347\pi$
$109$		−	12.7279i		−	1.21911i	0.792742	−	0.609557i	$-0.208653\pi$
$110$		0			0
$111$			2.23607i			0.212238i
$112$		0			0
$113$	−3.00000			−0.282216			−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000			−0.282216			−0.141108	+	0.989994i	$0.545067\pi$
$114$		0			0
$115$		−	6.70820i		−	0.625543i
$116$		0			0
$117$	12.6491			1.16941
$118$		0			0
$119$		0			0
$120$		0			0
$121$	7.00000	−	8.48528i	0.636364	−	0.771389i
$122$		0			0
$123$		−	21.2132i		−	1.91273i
$124$		0			0
$125$		−	11.1803i		−	1.00000i
$126$		0			0
$127$			12.7279i			1.12942i	0.825289	+	0.564710i	$0.191012\pi$
$127$			12.7279i			1.12942i	−0.825289	+	0.564710i	$0.808988\pi$
$128$		0			0
$129$	−9.48683			−0.835269
$130$		0			0
$131$	18.9737			1.65774			0.828868	−	0.559444i	$-0.188985\pi$
$131$	18.9737			1.65774			0.828868	+	0.559444i	$0.188985\pi$
$132$		0			0
$133$	5.00000	+	6.70820i	0.433555	+	0.581675i
$134$		0			0
$135$	−5.00000			−0.430331
$136$		0			0
$137$	−15.0000			−1.28154			−0.640768	−	0.767734i	$-0.721384\pi$
$137$	−15.0000			−1.28154			−0.640768	+	0.767734i	$0.721384\pi$
$138$		0			0
$139$	−12.6491			−1.07288			−0.536442	−	0.843937i	$-0.680232\pi$
$139$	−12.6491			−1.07288			−0.536442	+	0.843937i	$0.680232\pi$
$140$		0			0
$141$	−10.0000			−0.842152
$142$		0			0
$143$	−18.9737	+	8.94427i	−1.58666	+	0.747958i
$144$		0			0
$145$	−3.16228			−0.262613
$146$		0			0
$147$	15.0000	+	4.47214i	1.23718	+	0.368856i
$148$		0			0
$149$			19.7990i			1.62200i	0.585049	+	0.810998i	$0.301075\pi$
$149$			19.7990i			1.62200i	−0.585049	+	0.810998i	$0.698925\pi$
$150$		0			0
$151$		−	16.9706i		−	1.38104i	−0.723311	−	0.690522i	$-0.757381\pi$
$151$		−	16.9706i		−	1.38104i	0.723311	−	0.690522i	$-0.242619\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	15.0000			1.20483
$156$		0			0
$157$		−	6.70820i		−	0.535373i	−0.963506	−	0.267686i	$-0.913741\pi$
$157$		−	6.70820i		−	0.535373i	0.963506	−	0.267686i	$-0.0862591\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−4.74342	−	6.36396i	−0.373834	−	0.501550i
$162$		0			0
$163$	10.0000			0.783260			0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000			0.783260			0.391630	+	0.920123i	$0.371911\pi$
$164$		0			0
$165$	−15.0000	+	7.07107i	−1.16775	+	0.550482i
$166$		0			0
$167$	−9.48683			−0.734113			−0.367057	−	0.930199i	$-0.619634\pi$
$167$	−9.48683			−0.734113			−0.367057	+	0.930199i	$0.619634\pi$
$168$		0			0
$169$	27.0000			2.07692
$170$		0			0
$171$	6.32456			0.483651
$172$		0			0
$173$	−9.48683			−0.721271			−0.360635	−	0.932707i	$-0.617440\pi$
$173$	−9.48683			−0.721271			−0.360635	+	0.932707i	$0.617440\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	5.00000			0.375823
$178$		0			0
$179$	−9.00000			−0.672692			−0.336346	−	0.941739i	$-0.609191\pi$
$179$	−9.00000			−0.672692			−0.336346	+	0.941739i	$0.609191\pi$
$180$		0			0
$181$		−	20.1246i		−	1.49585i	−0.663783	−	0.747925i	$-0.731050\pi$
$181$		−	20.1246i		−	1.49585i	0.663783	−	0.747925i	$-0.268950\pi$
$182$		0			0
$183$		−	7.07107i		−	0.522708i
$184$		0			0
$185$			2.23607i			0.164399i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−4.74342	+	3.53553i	−0.345033	+	0.257172i
$190$		0			0
$191$	−3.00000			−0.217072			−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000			−0.217072			−0.108536	+	0.994092i	$0.534616\pi$
$192$		0			0
$193$		−	4.24264i		−	0.305392i	−0.988273	−	0.152696i	$-0.951204\pi$
$193$		−	4.24264i		−	0.305392i	0.988273	−	0.152696i	$-0.0487955\pi$
$194$		0			0
$195$	31.6228			2.26455
$196$		0			0
$197$		−	5.65685i		−	0.403034i	−0.979485	−	0.201517i	$-0.935413\pi$
$197$		−	5.65685i		−	0.403034i	0.979485	−	0.201517i	$-0.0645872\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$			24.5967i			1.73492i
$202$		0			0
$203$	−3.00000	+	2.23607i	−0.210559	+	0.156941i
$204$		0			0
$205$		−	21.2132i		−	1.48159i
$206$		0			0
$207$	−6.00000			−0.417029
$208$		0			0
$209$	−9.48683	+	4.47214i	−0.656218	+	0.309344i
$210$		0			0
$211$		−	21.2132i		−	1.46038i	−0.683246	−	0.730189i	$-0.739432\pi$
$211$		−	21.2132i		−	1.46038i	0.683246	−	0.730189i	$-0.260568\pi$
$212$		0			0
$213$			20.1246i			1.37892i
$214$		0			0
$215$	−9.48683			−0.646997
$216$		0			0
$217$	14.2302	−	10.6066i	0.966012	−	0.720023i
$218$		0			0
$219$		−	7.07107i		−	0.477818i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	6.70820i		−	0.449215i	−0.974449	−	0.224607i	$-0.927890\pi$
$223$		−	6.70820i		−	0.449215i	0.974449	−	0.224607i	$-0.0721099\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.48683			0.629663			0.314832	−	0.949148i	$-0.398052\pi$
$227$	9.48683			0.629663			0.314832	+	0.949148i	$0.398052\pi$
$228$		0			0
$229$		−	6.70820i		−	0.443291i	−0.975127	−	0.221645i	$-0.928857\pi$
$229$		−	6.70820i		−	0.443291i	0.975127	−	0.221645i	$-0.0711427\pi$
$230$		0			0
$231$	−9.23025	+	17.3148i	−0.607306	+	1.13923i
$232$		0			0
$233$		−	18.3848i		−	1.20443i	−0.798335	−	0.602213i	$-0.794286\pi$
$233$		−	18.3848i		−	1.20443i	0.798335	−	0.602213i	$-0.205714\pi$
$234$		0			0
$235$	−10.0000			−0.652328
$236$		0			0
$237$	18.9737			1.23247
$238$		0			0
$239$		−	7.07107i		−	0.457389i	−0.973498	−	0.228695i	$-0.926554\pi$
$239$		−	7.07107i		−	0.457389i	0.973498	−	0.228695i	$-0.0734457\pi$
$240$		0			0
$241$	12.6491			0.814801			0.407400	−	0.913250i	$-0.366435\pi$
$241$	12.6491			0.814801			0.407400	+	0.913250i	$0.366435\pi$
$242$		0			0
$243$			17.8885i			1.14755i
$244$		0			0
$245$	15.0000	+	4.47214i	0.958315	+	0.285714i
$246$		0			0
$247$	20.0000			1.27257
$248$		0			0
$249$		−	21.2132i		−	1.34433i
$250$		0			0
$251$		−	11.1803i		−	0.705697i	−0.935681	−	0.352848i	$-0.885213\pi$
$251$		−	11.1803i		−	0.705697i	0.935681	−	0.352848i	$-0.114787\pi$
$252$		0			0
$253$	9.00000	−	4.24264i	0.565825	−	0.266733i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	22.3607i		−	1.39482i	−0.716672	−	0.697410i	$-0.754335\pi$
$257$		−	22.3607i		−	1.39482i	0.716672	−	0.697410i	$-0.245665\pi$
$258$		0			0
$259$	1.58114	+	2.12132i	0.0982472	+	0.131812i
$260$		0			0
$261$			2.82843i			0.175075i
$262$		0			0
$263$		−	11.3137i		−	0.697633i	−0.937191	−	0.348817i	$-0.886584\pi$
$263$		−	11.3137i		−	0.697633i	0.937191	−	0.348817i	$-0.113416\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−5.00000			−0.305995
$268$		0			0
$269$			17.8885i			1.09068i	0.838214	+	0.545342i	$0.183600\pi$
$269$			17.8885i			1.09068i	−0.838214	+	0.545342i	$0.816400\pi$
$270$		0			0
$271$	25.2982			1.53676			0.768379	−	0.639995i	$-0.221064\pi$
$271$	25.2982			1.53676			0.768379	+	0.639995i	$0.221064\pi$
$272$		0			0
$273$	30.0000	−	22.3607i	1.81568	−	1.35333i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			4.24264i			0.254916i	0.991844	+	0.127458i	$0.0406817\pi$
$277$			4.24264i			0.254916i	−0.991844	+	0.127458i	$0.959318\pi$
$278$		0			0
$279$		−	13.4164i		−	0.803219i
$280$		0			0
$281$			24.0416i			1.43420i	0.696969	+	0.717102i	$0.254532\pi$
$281$			24.0416i			1.43420i	−0.696969	+	0.717102i	$0.745468\pi$
$282$		0			0
$283$	6.32456			0.375956			0.187978	−	0.982173i	$-0.439807\pi$
$283$	6.32456			0.375956			0.187978	+	0.982173i	$0.439807\pi$
$284$		0			0
$285$	15.8114			0.936586
$286$		0			0
$287$	−15.0000	−	20.1246i	−0.885422	−	1.18792i
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$	15.0000			0.879316
$292$		0			0
$293$	9.48683			0.554227			0.277113	−	0.960837i	$-0.410622\pi$
$293$	9.48683			0.554227			0.277113	+	0.960837i	$0.410622\pi$
$294$		0			0
$295$	5.00000			0.291111
$296$		0			0
$297$	−3.16228	−	6.70820i	−0.183494	−	0.389249i
$298$		0			0
$299$	−18.9737			−1.09728
$300$		0			0
$301$	−9.00000	+	6.70820i	−0.518751	+	0.386654i
$302$		0			0
$303$			21.2132i			1.21867i
$304$		0			0
$305$		−	7.07107i		−	0.404888i
$306$		0			0
$307$	6.32456			0.360961			0.180481	−	0.983579i	$-0.442235\pi$
$307$	6.32456			0.360961			0.180481	+	0.983579i	$0.442235\pi$
$308$		0			0
$309$	−30.0000			−1.70664
$310$		0			0
$311$		−	4.47214i		−	0.253592i	−0.991929	−	0.126796i	$-0.959531\pi$
$311$		−	4.47214i		−	0.253592i	0.991929	−	0.126796i	$-0.0404693\pi$
$312$		0			0
$313$			6.70820i			0.379170i	0.981864	+	0.189585i	$0.0607143\pi$
$313$			6.70820i			0.379170i	−0.981864	+	0.189585i	$0.939286\pi$
$314$		0			0
$315$	9.48683	−	7.07107i	0.534522	−	0.398410i
$316$		0			0
$317$	21.0000			1.17948			0.589739	−	0.807594i	$-0.299231\pi$
$317$	21.0000			1.17948			0.589739	+	0.807594i	$0.299231\pi$
$318$		0			0
$319$	−2.00000	−	4.24264i	−0.111979	−	0.237542i
$320$		0			0
$321$	−6.32456			−0.353002
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−28.4605			−1.57387
$328$		0			0
$329$	−9.48683	+	7.07107i	−0.523026	+	0.389841i
$330$		0			0
$331$	19.0000			1.04433			0.522167	−	0.852843i	$-0.325124\pi$
$331$	19.0000			1.04433			0.522167	+	0.852843i	$0.325124\pi$
$332$		0			0
$333$	2.00000			0.109599
$334$		0			0
$335$			24.5967i			1.34386i
$336$		0			0
$337$		−	16.9706i		−	0.924445i	−0.886764	−	0.462223i	$-0.847052\pi$
$337$		−	16.9706i		−	0.924445i	0.886764	−	0.462223i	$-0.152948\pi$
$338$		0			0
$339$			6.70820i			0.364340i
$340$		0			0
$341$	9.48683	+	20.1246i	0.513741	+	1.08981i
$342$		0			0
$343$	17.3925	−	6.36396i	0.939108	−	0.343622i
$344$		0			0
$345$	−15.0000			−0.807573
$346$		0			0
$347$			1.41421i			0.0759190i	0.999279	+	0.0379595i	$0.0120858\pi$
$347$			1.41421i			0.0759190i	−0.999279	+	0.0379595i	$0.987914\pi$
$348$		0			0
$349$	12.6491			0.677091			0.338546	−	0.940950i	$-0.390065\pi$
$349$	12.6491			0.677091			0.338546	+	0.940950i	$0.390065\pi$
$350$		0			0
$351$			14.1421i			0.754851i
$352$		0			0
$353$		−	15.6525i		−	0.833097i	−0.909113	−	0.416549i	$-0.863240\pi$
$353$		−	15.6525i		−	0.833097i	0.909113	−	0.416549i	$-0.136760\pi$
$354$		0			0
$355$			20.1246i			1.06810i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			5.65685i			0.298557i	0.988795	+	0.149279i	$0.0476951\pi$
$359$			5.65685i			0.298557i	−0.988795	+	0.149279i	$0.952305\pi$
$360$		0			0
$361$	−9.00000			−0.473684
$362$		0			0
$363$	−18.9737	−	15.6525i	−0.995859	−	0.821542i
$364$		0			0
$365$		−	7.07107i		−	0.370117i
$366$		0			0
$367$			33.5410i			1.75083i	0.483375	+	0.875413i	$0.339411\pi$
$367$			33.5410i			1.75083i	−0.483375	+	0.875413i	$0.660589\pi$
$368$		0			0
$369$	−18.9737			−0.987730
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	4.24264i		−	0.219676i	−0.993950	−	0.109838i	$-0.964967\pi$
$373$		−	4.24264i		−	0.219676i	0.993950	−	0.109838i	$-0.0350331\pi$
$374$		0			0
$375$	−25.0000			−1.29099
$376$		0			0
$377$			8.94427i			0.460653i
$378$		0			0
$379$	−17.0000			−0.873231			−0.436616	−	0.899648i	$-0.643823\pi$
$379$	−17.0000			−0.873231			−0.436616	+	0.899648i	$0.643823\pi$
$380$		0			0
$381$	28.4605			1.45808
$382$		0			0
$383$		−	24.5967i		−	1.25684i	−0.777876	−	0.628418i	$-0.783703\pi$
$383$		−	24.5967i		−	1.25684i	0.777876	−	0.628418i	$-0.216297\pi$
$384$		0			0
$385$	−9.23025	+	17.3148i	−0.470417	+	0.882444i
$386$		0			0
$387$			8.48528i			0.431331i
$388$		0			0
$389$	3.00000			0.152106			0.0760530	−	0.997104i	$-0.475768\pi$
$389$	3.00000			0.152106			0.0760530	+	0.997104i	$0.475768\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		−	42.4264i		−	2.14013i
$394$		0			0
$395$	18.9737			0.954669
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$	15.0000	−	11.1803i	0.750939	−	0.559717i
$400$		0			0
$401$	24.0000			1.19850			0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000			1.19850			0.599251	+	0.800561i	$0.295465\pi$
$402$		0			0
$403$		−	42.4264i		−	2.11341i
$404$		0			0
$405$			24.5967i			1.22222i
$406$		0			0
$407$	−3.00000	+	1.41421i	−0.148704	+	0.0701000i
$408$		0			0
$409$	12.6491			0.625458			0.312729	−	0.949842i	$-0.398757\pi$
$409$	12.6491			0.625458			0.312729	+	0.949842i	$0.398757\pi$
$410$		0			0
$411$			33.5410i			1.65446i
$412$		0			0
$413$	4.74342	−	3.53553i	0.233408	−	0.173972i
$414$		0			0
$415$		−	21.2132i		−	1.04132i
$416$		0			0
$417$			28.2843i			1.38509i
$418$		0			0
$419$			22.3607i			1.09239i	0.837658	+	0.546195i	$0.183924\pi$
$419$			22.3607i			1.09239i	−0.837658	+	0.546195i	$0.816076\pi$
$420$		0			0
$421$	−4.00000			−0.194948			−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000			−0.194948			−0.0974740	+	0.995238i	$0.531076\pi$
$422$		0			0
$423$			8.94427i			0.434885i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−5.00000	−	6.70820i	−0.241967	−	0.324633i
$428$		0			0
$429$	20.0000	+	42.4264i	0.965609	+	2.04837i
$430$		0			0
$431$			35.3553i			1.70301i	0.524349	+	0.851503i	$0.324309\pi$
$431$			35.3553i			1.70301i	−0.524349	+	0.851503i	$0.675691\pi$
$432$		0			0
$433$			20.1246i			0.967127i	0.875309	+	0.483564i	$0.160658\pi$
$433$			20.1246i			0.967127i	−0.875309	+	0.483564i	$0.839342\pi$
$434$		0			0
$435$			7.07107i			0.339032i
$436$		0			0
$437$	−9.48683			−0.453817
$438$		0			0
$439$	15.8114			0.754636			0.377318	−	0.926084i	$-0.376846\pi$
$439$	15.8114			0.754636			0.377318	+	0.926084i	$0.376846\pi$
$440$		0			0
$441$	4.00000	−	13.4164i	0.190476	−	0.638877i
$442$		0			0
$443$	15.0000			0.712672			0.356336	−	0.934358i	$-0.384026\pi$
$443$	15.0000			0.712672			0.356336	+	0.934358i	$0.384026\pi$
$444$		0			0
$445$	−5.00000			−0.237023
$446$		0			0
$447$	44.2719			2.09399
$448$		0			0
$449$	−27.0000			−1.27421			−0.637104	−	0.770778i	$-0.719868\pi$
$449$	−27.0000			−1.27421			−0.637104	+	0.770778i	$0.719868\pi$
$450$		0			0
$451$	28.4605	−	13.4164i	1.34015	−	0.631754i
$452$		0			0
$453$	−37.9473			−1.78292
$454$		0			0
$455$	30.0000	−	22.3607i	1.40642	−	1.04828i
$456$		0			0
$457$		−	8.48528i		−	0.396925i	−0.980109	−	0.198462i	$-0.936405\pi$
$457$		−	8.48528i		−	0.396925i	0.980109	−	0.198462i	$-0.0635948\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−9.48683			−0.441846			−0.220923	−	0.975291i	$-0.570907\pi$
$461$	−9.48683			−0.441846			−0.220923	+	0.975291i	$0.570907\pi$
$462$		0			0
$463$	25.0000			1.16185			0.580924	−	0.813958i	$-0.302691\pi$
$463$	25.0000			1.16185			0.580924	+	0.813958i	$0.302691\pi$
$464$		0			0
$465$		−	33.5410i		−	1.55543i
$466$		0			0
$467$		−	11.1803i		−	0.517364i	−0.965962	−	0.258682i	$-0.916712\pi$
$467$		−	11.1803i		−	0.517364i	0.965962	−	0.258682i	$-0.0832882\pi$
$468$		0			0
$469$	17.3925	+	23.3345i	0.803112	+	1.07749i
$470$		0			0
$471$	−15.0000			−0.691164
$472$		0			0
$473$	−6.00000	−	12.7279i	−0.275880	−	0.585230i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−18.9737			−0.866929			−0.433464	−	0.901171i	$-0.642709\pi$
$479$	−18.9737			−0.866929			−0.433464	+	0.901171i	$0.642709\pi$
$480$		0			0
$481$	6.32456			0.288375
$482$		0			0
$483$	−14.2302	+	10.6066i	−0.647499	+	0.482617i
$484$		0			0
$485$	15.0000			0.681115
$486$		0			0
$487$	−29.0000			−1.31412			−0.657058	−	0.753840i	$-0.728199\pi$
$487$	−29.0000			−1.31412			−0.657058	+	0.753840i	$0.728199\pi$
$488$		0			0
$489$		−	22.3607i		−	1.01118i
$490$		0			0
$491$		−	32.5269i		−	1.46792i	−0.679193	−	0.733959i	$-0.737670\pi$
$491$		−	32.5269i		−	1.46792i	0.679193	−	0.733959i	$-0.262330\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	6.32456	+	13.4164i	0.284268	+	0.603023i
$496$		0			0
$497$	14.2302	+	19.0919i	0.638314	+	0.856388i
$498$		0			0
$499$	16.0000			0.716258			0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000			0.716258			0.358129	+	0.933672i	$0.383415\pi$
$500$		0			0
$501$			21.2132i			0.947736i
$502$		0			0
$503$	−28.4605			−1.26899			−0.634495	−	0.772927i	$-0.718792\pi$
$503$	−28.4605			−1.26899			−0.634495	+	0.772927i	$0.718792\pi$
$504$		0			0
$505$			21.2132i			0.943975i
$506$		0			0
$507$		−	60.3738i		−	2.68130i
$508$		0			0
$509$		−	15.6525i		−	0.693784i	−0.937905	−	0.346892i	$-0.887237\pi$
$509$		−	15.6525i		−	0.693784i	0.937905	−	0.346892i	$-0.112763\pi$
$510$		0			0
$511$	−5.00000	−	6.70820i	−0.221187	−	0.296753i
$512$		0			0
$513$			7.07107i			0.312195i
$514$		0			0
$515$	−30.0000			−1.32196
$516$		0			0
$517$	−6.32456	−	13.4164i	−0.278154	−	0.590053i
$518$		0			0
$519$			21.2132i			0.931156i
$520$		0			0
$521$			11.1803i			0.489820i	0.969546	+	0.244910i	$0.0787583\pi$
$521$			11.1803i			0.489820i	−0.969546	+	0.244910i	$0.921242\pi$
$522$		0			0
$523$	−31.6228			−1.38277			−0.691384	−	0.722488i	$-0.742999\pi$
$523$	−31.6228			−1.38277			−0.691384	+	0.722488i	$0.742999\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−14.0000			−0.608696
$530$		0			0
$531$		−	4.47214i		−	0.194074i
$532$		0			0
$533$	−60.0000			−2.59889
$534$		0			0
$535$	−6.32456			−0.273434
$536$		0			0
$537$			20.1246i			0.868441i
$538$		0			0
$539$	3.48683	+	22.9530i	0.150189	+	0.988657i
$540$		0			0
$541$			25.4558i			1.09443i	0.836991	+	0.547216i	$0.184312\pi$
$541$			25.4558i			1.09443i	−0.836991	+	0.547216i	$0.815688\pi$
$542$		0			0
$543$	−45.0000			−1.93113
$544$		0			0
$545$	−28.4605			−1.21911
$546$		0			0
$547$			8.48528i			0.362804i	0.983409	+	0.181402i	$0.0580636\pi$
$547$			8.48528i			0.362804i	−0.983409	+	0.181402i	$0.941936\pi$
$548$		0			0
$549$	−6.32456			−0.269925
$550$		0			0
$551$			4.47214i			0.190519i
$552$		0			0
$553$	18.0000	−	13.4164i	0.765438	−	0.570524i
$554$		0			0
$555$	5.00000			0.212238
$556$		0			0
$557$		−	22.6274i		−	0.958754i	−0.877609	−	0.479377i	$-0.840863\pi$
$557$		−	22.6274i		−	0.958754i	0.877609	−	0.479377i	$-0.159137\pi$
$558$		0			0
$559$			26.8328i			1.13491i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−37.9473			−1.59929			−0.799645	−	0.600473i	$-0.794979\pi$
$563$	−37.9473			−1.59929			−0.799645	+	0.600473i	$0.794979\pi$
$564$		0			0
$565$			6.70820i			0.282216i
$566$		0			0
$567$	17.3925	+	23.3345i	0.730417	+	0.979958i
$568$		0			0
$569$			15.5563i			0.652156i	0.945343	+	0.326078i	$0.105727\pi$
$569$			15.5563i			0.652156i	−0.945343	+	0.326078i	$0.894273\pi$
$570$		0			0
$571$			16.9706i			0.710196i	0.934829	+	0.355098i	$0.115552\pi$
$571$			16.9706i			0.710196i	−0.934829	+	0.355098i	$0.884448\pi$
$572$		0			0
$573$			6.70820i			0.280239i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			20.1246i			0.837799i	0.908033	+	0.418899i	$0.137584\pi$
$577$			20.1246i			0.837799i	−0.908033	+	0.418899i	$0.862416\pi$
$578$		0			0
$579$	−9.48683			−0.394259
$580$		0			0
$581$	−15.0000	−	20.1246i	−0.622305	−	0.834910i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		−	28.2843i		−	1.16941i
$586$		0			0
$587$		−	4.47214i		−	0.184585i	−0.995732	−	0.0922924i	$-0.970581\pi$
$587$		−	4.47214i		−	0.184585i	0.995732	−	0.0922924i	$-0.0294195\pi$
$588$		0			0
$589$		−	21.2132i		−	0.874075i
$590$		0			0
$591$	−12.6491			−0.520315
$592$		0			0
$593$	28.4605			1.16873			0.584366	−	0.811490i	$-0.301343\pi$
$593$	28.4605			1.16873			0.584366	+	0.811490i	$0.301343\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	12.0000			0.490307			0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000			0.490307			0.245153	+	0.969484i	$0.421162\pi$
$600$		0			0
$601$	−15.8114			−0.644960			−0.322480	−	0.946576i	$-0.604517\pi$
$601$	−15.8114			−0.644960			−0.322480	+	0.946576i	$0.604517\pi$
$602$		0			0
$603$	22.0000			0.895909
$604$		0			0
$605$	−18.9737	−	15.6525i	−0.771389	−	0.636364i
$606$		0			0
$607$	6.32456			0.256706			0.128353	−	0.991729i	$-0.459031\pi$
$607$	6.32456			0.256706			0.128353	+	0.991729i	$0.459031\pi$
$608$		0			0
$609$	5.00000	+	6.70820i	0.202610	+	0.271830i
$610$		0			0
$611$			28.2843i			1.14426i
$612$		0			0
$613$		−	25.4558i		−	1.02815i	−0.857745	−	0.514076i	$-0.828135\pi$
$613$		−	25.4558i		−	1.02815i	0.857745	−	0.514076i	$-0.171865\pi$
$614$		0			0
$615$	−47.4342			−1.91273
$616$		0			0
$617$	24.0000			0.966204			0.483102	−	0.875564i	$-0.339510\pi$
$617$	24.0000			0.966204			0.483102	+	0.875564i	$0.339510\pi$
$618$		0			0
$619$		−	46.9574i		−	1.88738i	−0.330833	−	0.943689i	$-0.607330\pi$
$619$		−	46.9574i		−	1.88738i	0.330833	−	0.943689i	$-0.392670\pi$
$620$		0			0
$621$		−	6.70820i		−	0.269191i
$622$		0			0
$623$	−4.74342	+	3.53553i	−0.190041	+	0.141648i
$624$		0			0
$625$	−25.0000			−1.00000
$626$		0			0
$627$	10.0000	+	21.2132i	0.399362	+	0.847174i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−11.0000			−0.437903			−0.218952	−	0.975736i	$-0.570264\pi$
$631$	−11.0000			−0.437903			−0.218952	+	0.975736i	$0.570264\pi$
$632$		0			0
$633$	−47.4342			−1.88534
$634$		0			0
$635$	28.4605			1.12942
$636$		0			0
$637$	12.6491	−	42.4264i	0.501176	−	1.68100i
$638$		0			0
$639$	18.0000			0.712069
$640$		0			0
$641$	21.0000			0.829450			0.414725	−	0.909947i	$-0.363878\pi$
$641$	21.0000			0.829450			0.414725	+	0.909947i	$0.363878\pi$
$642$		0			0
$643$			6.70820i			0.264546i	0.991213	+	0.132273i	$0.0422275\pi$
$643$			6.70820i			0.264546i	−0.991213	+	0.132273i	$0.957772\pi$
$644$		0			0
$645$			21.2132i			0.835269i
$646$		0			0
$647$			29.0689i			1.14282i	0.820666	+	0.571408i	$0.193603\pi$
$647$			29.0689i			1.14282i	−0.820666	+	0.571408i	$0.806397\pi$
$648$		0			0
$649$	3.16228	+	6.70820i	0.124130	+	0.263320i
$650$		0			0
$651$	−23.7171	−	31.8198i	−0.929546	−	1.24712i
$652$		0			0
$653$	15.0000			0.586995			0.293498	−	0.955960i	$-0.405181\pi$
$653$	15.0000			0.586995			0.293498	+	0.955960i	$0.405181\pi$
$654$		0			0
$655$		−	42.4264i		−	1.65774i
$656$		0			0
$657$	−6.32456			−0.246744
$658$		0			0
$659$			14.1421i			0.550899i	0.961315	+	0.275450i	$0.0888267\pi$
$659$			14.1421i			0.550899i	−0.961315	+	0.275450i	$0.911173\pi$
$660$		0			0
$661$		−	46.9574i		−	1.82643i	−0.407476	−	0.913216i	$-0.633591\pi$
$661$		−	46.9574i		−	1.82643i	0.407476	−	0.913216i	$-0.366409\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	15.0000	−	11.1803i	0.581675	−	0.433555i
$666$		0			0
$667$		−	4.24264i		−	0.164276i
$668$		0			0
$669$	−15.0000			−0.579934
$670$		0			0
$671$	9.48683	−	4.47214i	0.366235	−	0.172645i
$672$		0			0
$673$			29.6985i			1.14479i	0.819977	+	0.572396i	$0.193986\pi$
$673$			29.6985i			1.14479i	−0.819977	+	0.572396i	$0.806014\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−37.9473			−1.45843			−0.729217	−	0.684282i	$-0.760116\pi$
$677$	−37.9473			−1.45843			−0.729217	+	0.684282i	$0.760116\pi$
$678$		0			0
$679$	14.2302	−	10.6066i	0.546107	−	0.407044i
$680$		0			0
$681$		−	21.2132i		−	0.812892i
$682$		0			0
$683$	−18.0000			−0.688751			−0.344375	−	0.938832i	$-0.611909\pi$
$683$	−18.0000			−0.688751			−0.344375	+	0.938832i	$0.611909\pi$
$684$		0			0
$685$			33.5410i			1.28154i
$686$		0			0
$687$	−15.0000			−0.572286
$688$		0			0
$689$		0			0
$690$		0			0
$691$		−	6.70820i		−	0.255192i	−0.991826	−	0.127596i	$-0.959274\pi$
$691$		−	6.70820i		−	0.255192i	0.991826	−	0.127596i	$-0.0407261\pi$
$692$		0			0
$693$	15.4868	+	8.25579i	0.588296	+	0.313611i
$694$		0			0
$695$			28.2843i			1.07288i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−41.1096			−1.55491
$700$		0			0
$701$			7.07107i			0.267071i	0.991044	+	0.133535i	$0.0426329\pi$
$701$			7.07107i			0.267071i	−0.991044	+	0.133535i	$0.957367\pi$
$702$		0			0
$703$	3.16228			0.119268
$704$		0			0
$705$			22.3607i			0.842152i
$706$		0			0
$707$	15.0000	+	20.1246i	0.564133	+	0.756864i
$708$		0			0
$709$	41.0000			1.53979			0.769894	−	0.638172i	$-0.220309\pi$
$709$	41.0000			1.53979			0.769894	+	0.638172i	$0.220309\pi$
$710$		0			0
$711$		−	16.9706i		−	0.636446i
$712$		0			0
$713$			20.1246i			0.753673i
$714$		0			0
$715$	20.0000	+	42.4264i	0.747958	+	1.58666i
$716$		0			0
$717$	−15.8114			−0.590487
$718$		0			0
$719$		−	51.4296i		−	1.91800i	−0.283408	−	0.959000i	$-0.591465\pi$
$719$		−	51.4296i		−	1.91800i	0.283408	−	0.959000i	$-0.408535\pi$
$720$		0			0
$721$	−28.4605	+	21.2132i	−1.05992	+	0.790021i
$722$		0			0
$723$		−	28.2843i		−	1.05190i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			33.5410i			1.24397i	0.783030	+	0.621984i	$0.213673\pi$
$727$			33.5410i			1.24397i	−0.783030	+	0.621984i	$0.786327\pi$
$728$		0			0
$729$	7.00000			0.259259
$730$		0			0
$731$		0			0
$732$		0			0
$733$	12.6491			0.467206			0.233603	−	0.972332i	$-0.424948\pi$
$733$	12.6491			0.467206			0.233603	+	0.972332i	$0.424948\pi$
$734$		0			0
$735$	10.0000	−	33.5410i	0.368856	−	1.23718i
$736$		0			0
$737$	−33.0000	+	15.5563i	−1.21557	+	0.573025i
$738$		0			0
$739$			50.9117i			1.87282i	0.350912	+	0.936408i	$0.385872\pi$
$739$			50.9117i			1.87282i	−0.350912	+	0.936408i	$0.614128\pi$
$740$		0			0
$741$		−	44.7214i		−	1.64288i
$742$		0			0
$743$		−	19.7990i		−	0.726354i	−0.931720	−	0.363177i	$-0.881692\pi$
$743$		−	19.7990i		−	0.726354i	0.931720	−	0.363177i	$-0.118308\pi$
$744$		0			0
$745$	44.2719			1.62200
$746$		0			0
$747$	−18.9737			−0.694210
$748$		0			0
$749$	−6.00000	+	4.47214i	−0.219235	+	0.163408i
$750$		0			0
$751$	7.00000			0.255434			0.127717	−	0.991811i	$-0.459235\pi$
$751$	7.00000			0.255434			0.127717	+	0.991811i	$0.459235\pi$
$752$		0			0
$753$	−25.0000			−0.911051
$754$		0			0
$755$	−37.9473			−1.38104
$756$		0			0
$757$	44.0000			1.59921			0.799604	−	0.600528i	$-0.205043\pi$
$757$	44.0000			1.59921			0.799604	+	0.600528i	$0.205043\pi$
$758$		0			0
$759$	−9.48683	−	20.1246i	−0.344350	−	0.730477i
$760$		0			0
$761$	37.9473			1.37559			0.687795	−	0.725905i	$-0.258579\pi$
$761$	37.9473			1.37559			0.687795	+	0.725905i	$0.258579\pi$
$762$		0			0
$763$	−27.0000	+	20.1246i	−0.977466	+	0.728560i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	14.1421i		−	0.510643i
$768$		0			0
$769$	−6.32456			−0.228069			−0.114035	−	0.993477i	$-0.536377\pi$
$769$	−6.32456			−0.228069			−0.114035	+	0.993477i	$0.536377\pi$
$770$		0			0
$771$	−50.0000			−1.80071
$772$		0			0
$773$		−	8.94427i		−	0.321703i	−0.986979	−	0.160852i	$-0.948576\pi$
$773$		−	8.94427i		−	0.321703i	0.986979	−	0.160852i	$-0.0514240\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	4.74342	−	3.53553i	0.170169	−	0.126837i
$778$		0			0
$779$	−30.0000			−1.07486
$780$		0			0
$781$	−27.0000	+	12.7279i	−0.966136	+	0.455441i
$782$		0			0
$783$	−3.16228			−0.113011
$784$		0			0
$785$	−15.0000			−0.535373
$786$		0			0
$787$	−22.1359			−0.789061			−0.394531	−	0.918883i	$-0.629093\pi$
$787$	−22.1359			−0.789061			−0.394531	+	0.918883i	$0.629093\pi$
$788$		0			0
$789$	−25.2982			−0.900641
$790$		0			0
$791$	4.74342	+	6.36396i	0.168656	+	0.226276i
$792$		0			0
$793$	−20.0000			−0.710221
$794$		0			0
$795$		0			0
$796$		0			0
$797$			38.0132i			1.34650i	0.739417	+	0.673248i	$0.235101\pi$
$797$			38.0132i			1.34650i	−0.739417	+	0.673248i	$0.764899\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$			4.47214i			0.158015i
$802$		0			0
$803$	9.48683	−	4.47214i	0.334783	−	0.157818i
$804$		0			0
$805$	−14.2302	+	10.6066i	−0.501550	+	0.373834i
$806$		0			0
$807$	40.0000			1.40807
$808$		0			0
$809$			41.0122i			1.44191i	0.692981	+	0.720956i	$0.256297\pi$
$809$			41.0122i			1.44191i	−0.692981	+	0.720956i	$0.743703\pi$
$810$		0			0
$811$	25.2982			0.888341			0.444170	−	0.895942i	$-0.353498\pi$
$811$	25.2982			0.888341			0.444170	+	0.895942i	$0.353498\pi$
$812$		0			0
$813$		−	56.5685i		−	1.98395i
$814$		0			0
$815$		−	22.3607i		−	0.783260i
$816$		0			0
$817$			13.4164i			0.469381i
$818$		0			0
$819$	−20.0000	−	26.8328i	−0.698857	−	0.937614i
$820$		0			0
$821$		−	14.1421i		−	0.493564i	−0.969071	−	0.246782i	$-0.920627\pi$
$821$		−	14.1421i		−	0.493564i	0.969071	−	0.246782i	$-0.0793731\pi$
$822$		0			0
$823$	43.0000			1.49889			0.749443	−	0.662069i	$-0.230321\pi$
$823$	43.0000			1.49889			0.749443	+	0.662069i	$0.230321\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			22.6274i			0.786832i	0.919360	+	0.393416i	$0.128707\pi$
$827$			22.6274i			0.786832i	−0.919360	+	0.393416i	$0.871293\pi$
$828$		0			0
$829$			20.1246i			0.698957i	0.936944	+	0.349478i	$0.113641\pi$
$829$			20.1246i			0.698957i	−0.936944	+	0.349478i	$0.886359\pi$
$830$		0			0
$831$	9.48683			0.329095
$832$		0			0
$833$		0			0
$834$		0			0
$835$			21.2132i			0.734113i
$836$		0			0
$837$	15.0000			0.518476
$838$		0			0
$839$			15.6525i			0.540383i	0.962807	+	0.270192i	$0.0870871\pi$
$839$			15.6525i			0.540383i	−0.962807	+	0.270192i	$0.912913\pi$
$840$		0			0
$841$	27.0000			0.931034
$842$		0			0
$843$	53.7587			1.85155
$844$		0			0
$845$		−	60.3738i		−	2.07692i
$846$		0			0
$847$	−29.0680	−	1.43283i	−0.998787	−	0.0492328i
$848$		0			0
$849$		−	14.1421i		−	0.485357i
$850$		0			0
$851$	−3.00000			−0.102839
$852$		0			0
$853$	−25.2982			−0.866195			−0.433097	−	0.901347i	$-0.642579\pi$
$853$	−25.2982			−0.866195			−0.433097	+	0.901347i	$0.642579\pi$
$854$		0			0
$855$		−	14.1421i		−	0.483651i
$856$		0			0
$857$	18.9737			0.648128			0.324064	−	0.946035i	$-0.394951\pi$
$857$	18.9737			0.648128			0.324064	+	0.946035i	$0.394951\pi$
$858$		0			0
$859$			6.70820i			0.228881i	0.993430	+	0.114440i	$0.0365075\pi$
$859$			6.70820i			0.228881i	−0.993430	+	0.114440i	$0.963492\pi$
$860$		0			0
$861$	−45.0000	+	33.5410i	−1.53360	+	1.14307i
$862$		0			0
$863$	−18.0000			−0.612727			−0.306364	−	0.951915i	$-0.599112\pi$
$863$	−18.0000			−0.612727			−0.306364	+	0.951915i	$0.599112\pi$
$864$		0			0
$865$			21.2132i			0.721271i
$866$		0			0
$867$			38.0132i			1.29099i
$868$		0			0
$869$	12.0000	+	25.4558i	0.407072	+	0.863530i
$870$		0			0
$871$	69.5701			2.35729
$872$		0			0
$873$		−	13.4164i		−	0.454077i
$874$		0			0
$875$	−23.7171	+	17.6777i	−0.801784	+	0.597614i
$876$		0			0
$877$		−	4.24264i		−	0.143264i	−0.997431	−	0.0716319i	$-0.977179\pi$
$877$		−	4.24264i		−	0.143264i	0.997431	−	0.0716319i	$-0.0228207\pi$
$878$		0			0
$879$		−	21.2132i		−	0.715504i
$880$		0			0
$881$			51.4296i			1.73271i	0.499432	+	0.866353i	$0.333542\pi$
$881$			51.4296i			1.73271i	−0.499432	+	0.866353i	$0.666458\pi$
$882$		0			0
$883$	−8.00000			−0.269221			−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000			−0.269221			−0.134611	+	0.990899i	$0.542978\pi$
$884$		0			0
$885$		−	11.1803i		−	0.375823i
$886$		0			0
$887$	18.9737			0.637073			0.318537	−	0.947911i	$-0.396809\pi$
$887$	18.9737			0.637073			0.318537	+	0.947911i	$0.396809\pi$
$888$		0			0
$889$	27.0000	−	20.1246i	0.905551	−	0.674958i
$890$		0			0
$891$	−33.0000	+	15.5563i	−1.10554	+	0.521157i
$892$		0			0
$893$			14.1421i			0.473249i
$894$		0			0
$895$			20.1246i			0.672692i
$896$		0			0
$897$			42.4264i			1.41658i
$898$		0			0
$899$	9.48683			0.316404
$900$		0			0
$901$		0			0
$902$		0			0
$903$	15.0000	+	20.1246i	0.499169	+	0.669705i
$904$		0			0
$905$	−45.0000			−1.49585
$906$		0			0
$907$	10.0000			0.332045			0.166022	−	0.986122i	$-0.446908\pi$
$907$	10.0000			0.332045			0.166022	+	0.986122i	$0.446908\pi$
$908$		0			0
$909$	18.9737			0.629317
$910$		0			0
$911$	6.00000			0.198789			0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000			0.198789			0.0993944	+	0.995048i	$0.468309\pi$
$912$		0			0
$913$	28.4605	−	13.4164i	0.941905	−	0.444018i
$914$		0			0
$915$	−15.8114			−0.522708
$916$		0			0
$917$	−30.0000	−	40.2492i	−0.990687	−	1.32915i
$918$		0			0
$919$		−	12.7279i		−	0.419855i	−0.977717	−	0.209928i	$-0.932677\pi$
$919$		−	12.7279i		−	0.419855i	0.977717	−	0.209928i	$-0.0673229\pi$
$920$		0			0
$921$		−	14.1421i		−	0.465999i
$922$		0			0
$923$	56.9210			1.87358
$924$		0			0
$925$		0			0
$926$		0			0
$927$			26.8328i			0.881305i
$928$		0			0
$929$		−	35.7771i		−	1.17381i	−0.809656	−	0.586904i	$-0.800347\pi$
$929$		−	35.7771i		−	1.17381i	0.809656	−	0.586904i	$-0.199653\pi$
$930$		0			0
$931$	6.32456	−	21.2132i	0.207279	−	0.695235i
$932$		0			0
$933$	−10.0000			−0.327385
$934$		0			0
$935$		0			0
$936$		0			0
$937$	50.5964			1.65291			0.826457	−	0.563000i	$-0.190353\pi$
$937$	50.5964			1.65291			0.826457	+	0.563000i	$0.190353\pi$
$938$		0			0
$939$	15.0000			0.489506
$940$		0			0
$941$	37.9473			1.23705			0.618524	−	0.785766i	$-0.287731\pi$
$941$	37.9473			1.23705			0.618524	+	0.785766i	$0.287731\pi$
$942$		0			0
$943$	28.4605			0.926801
$944$		0			0
$945$	7.90569	+	10.6066i	0.257172	+	0.345033i
$946$		0			0
$947$	15.0000			0.487435			0.243717	−	0.969846i	$-0.421633\pi$
$947$	15.0000			0.487435			0.243717	+	0.969846i	$0.421633\pi$
$948$		0			0
$949$	−20.0000			−0.649227
$950$		0			0
$951$		−	46.9574i		−	1.52270i
$952$		0			0
$953$		−	9.89949i		−	0.320676i	−0.987062	−	0.160338i	$-0.948742\pi$
$953$		−	9.89949i		−	0.320676i	0.987062	−	0.160338i	$-0.0512584\pi$
$954$		0			0
$955$			6.70820i			0.217072i
$956$		0			0
$957$	−9.48683	+	4.47214i	−0.306666	+	0.144564i
$958$		0			0
$959$	23.7171	+	31.8198i	0.765865	+	1.02752i
$960$		0			0
$961$	−14.0000			−0.451613
$962$		0			0
$963$			5.65685i			0.182290i
$964$		0			0
$965$	−9.48683			−0.305392
$966$		0			0
$967$		−	46.6690i		−	1.50078i	−0.660998	−	0.750388i	$-0.729867\pi$
$967$		−	46.6690i		−	1.50078i	0.660998	−	0.750388i	$-0.270133\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			29.0689i			0.932865i	0.884557	+	0.466432i	$0.154461\pi$
$971$			29.0689i			0.932865i	−0.884557	+	0.466432i	$0.845539\pi$
$972$		0			0
$973$	20.0000	+	26.8328i	0.641171	+	0.860221i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−21.0000			−0.671850			−0.335925	−	0.941889i	$-0.609049\pi$
$977$	−21.0000			−0.671850			−0.335925	+	0.941889i	$0.609049\pi$
$978$		0			0
$979$	−3.16228	−	6.70820i	−0.101067	−	0.214395i
$980$		0			0
$981$			25.4558i			0.812743i
$982$		0			0
$983$			29.0689i			0.927153i	0.886057	+	0.463577i	$0.153434\pi$
$983$			29.0689i			0.927153i	−0.886057	+	0.463577i	$0.846566\pi$
$984$		0			0
$985$	−12.6491			−0.403034
$986$		0			0
$987$	15.8114	+	21.2132i	0.503282	+	0.675224i
$988$		0			0
$989$		−	12.7279i		−	0.404724i
$990$		0			0
$991$	46.0000			1.46124			0.730619	−	0.682785i	$-0.239232\pi$
$991$	46.0000			1.46124			0.730619	+	0.682785i	$0.239232\pi$
$992$		0			0
$993$		−	42.4853i		−	1.34823i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−53.7587			−1.70256			−0.851278	−	0.524715i	$-0.824172\pi$
$997$	−53.7587			−1.70256			−0.851278	+	0.524715i	$0.824172\pi$
$998$		0			0
$999$			2.23607i			0.0707461i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.2.e.c.769.1		4
4.3	odd	2		77.2.b.b.76.2	yes	4
7.6	odd	2	inner	1232.2.e.c.769.4		4
11.10	odd	2	inner	1232.2.e.c.769.2		4
12.11	even	2		693.2.c.b.307.4		4
28.3	even	6		539.2.i.b.362.3		8
28.11	odd	6		539.2.i.b.362.4		8
28.19	even	6		539.2.i.b.472.2		8
28.23	odd	6		539.2.i.b.472.1		8
28.27	even	2		77.2.b.b.76.1	✓	4
44.3	odd	10		847.2.l.g.475.1		16
44.7	even	10		847.2.l.g.699.2		16
44.15	odd	10		847.2.l.g.699.4		16
44.19	even	10		847.2.l.g.475.3		16
44.27	odd	10		847.2.l.g.118.3		16
44.31	odd	10		847.2.l.g.524.2		16
44.35	even	10		847.2.l.g.524.4		16
44.39	even	10		847.2.l.g.118.1		16
44.43	even	2		77.2.b.b.76.4	yes	4
77.76	even	2	inner	1232.2.e.c.769.3		4
84.83	odd	2		693.2.c.b.307.3		4
132.131	odd	2		693.2.c.b.307.2		4
308.27	even	10		847.2.l.g.118.4		16
308.83	odd	10		847.2.l.g.118.2		16
308.87	odd	6		539.2.i.b.362.1		8
308.131	odd	6		539.2.i.b.472.4		8
308.139	odd	10		847.2.l.g.699.1		16
308.167	odd	10		847.2.l.g.524.3		16
308.195	odd	10		847.2.l.g.475.4		16
308.219	even	6		539.2.i.b.472.3		8
308.223	even	10		847.2.l.g.475.2		16
308.251	even	10		847.2.l.g.524.1		16
308.263	even	6		539.2.i.b.362.2		8
308.279	even	10		847.2.l.g.699.3		16
308.307	odd	2		77.2.b.b.76.3	yes	4
924.923	even	2		693.2.c.b.307.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.b.b.76.1	✓	4	28.27	even	2
77.2.b.b.76.2	yes	4	4.3	odd	2
77.2.b.b.76.3	yes	4	308.307	odd	2
77.2.b.b.76.4	yes	4	44.43	even	2
539.2.i.b.362.1		8	308.87	odd	6
539.2.i.b.362.2		8	308.263	even	6
539.2.i.b.362.3		8	28.3	even	6
539.2.i.b.362.4		8	28.11	odd	6
539.2.i.b.472.1		8	28.23	odd	6
539.2.i.b.472.2		8	28.19	even	6
539.2.i.b.472.3		8	308.219	even	6
539.2.i.b.472.4		8	308.131	odd	6
693.2.c.b.307.1		4	924.923	even	2
693.2.c.b.307.2		4	132.131	odd	2
693.2.c.b.307.3		4	84.83	odd	2
693.2.c.b.307.4		4	12.11	even	2
847.2.l.g.118.1		16	44.39	even	10
847.2.l.g.118.2		16	308.83	odd	10
847.2.l.g.118.3		16	44.27	odd	10
847.2.l.g.118.4		16	308.27	even	10
847.2.l.g.475.1		16	44.3	odd	10
847.2.l.g.475.2		16	308.223	even	10
847.2.l.g.475.3		16	44.19	even	10
847.2.l.g.475.4		16	308.195	odd	10
847.2.l.g.524.1		16	308.251	even	10
847.2.l.g.524.2		16	44.31	odd	10
847.2.l.g.524.3		16	308.167	odd	10
847.2.l.g.524.4		16	44.35	even	10
847.2.l.g.699.1		16	308.139	odd	10
847.2.l.g.699.2		16	44.7	even	10
847.2.l.g.699.3		16	308.279	even	10
847.2.l.g.699.4		16	44.15	odd	10
1232.2.e.c.769.1		4	1.1	even	1	trivial
1232.2.e.c.769.2		4	11.10	odd	2	inner
1232.2.e.c.769.3		4	77.76	even	2	inner
1232.2.e.c.769.4		4	7.6	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-2.23607i q^{3} -2.23607i q^{5} +(-1.58114 - 2.12132i) q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q-2.23607i q^{3} -2.23607i q^{5} +(-1.58114 - 2.12132i) q^{7} -2.00000 q^{9} +(3.00000 - 1.41421i) q^{11} -6.32456 q^{13} -5.00000 q^{15} -3.16228 q^{19} +(-4.74342 + 3.53553i) q^{21} +3.00000 q^{23} -2.23607i q^{27} -1.41421i q^{29} +6.70820i q^{31} +(-3.16228 - 6.70820i) q^{33} +(-4.74342 + 3.53553i) q^{35} -1.00000 q^{37} +14.1421i q^{39} +9.48683 q^{41} -4.24264i q^{43} +4.47214i q^{45} -4.47214i q^{47} +(-2.00000 + 6.70820i) q^{49} +(-3.16228 - 6.70820i) q^{55} +7.07107i q^{57} +2.23607i q^{59} +3.16228 q^{61} +(3.16228 + 4.24264i) q^{63} +14.1421i q^{65} -11.0000 q^{67} -6.70820i q^{69} -9.00000 q^{71} +3.16228 q^{73} +(-7.74342 - 4.12789i) q^{77} +8.48528i q^{79} -11.0000 q^{81} +9.48683 q^{83} -3.16228 q^{87} -2.23607i q^{89} +(10.0000 + 13.4164i) q^{91} +15.0000 q^{93} +7.07107i q^{95} +6.70820i q^{97} +(-6.00000 + 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{9}+O(q^{10}) \) 4 * q - 8 * q^9	\( 4 q - 8 q^{9} + 12 q^{11} - 20 q^{15} + 12 q^{23} - 4 q^{37} - 8 q^{49} - 44 q^{67} - 36 q^{71} - 12 q^{77} - 44 q^{81} + 40 q^{91} + 60 q^{93} - 24 q^{99}+O(q^{100}) \) 4 * q - 8 * q^9 + 12 * q^11 - 20 * q^15 + 12 * q^23 - 4 * q^37 - 8 * q^49 - 44 * q^67 - 36 * q^71 - 12 * q^77 - 44 * q^81 + 40 * q^91 + 60 * q^93 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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