LMFDB - Embedded newform 3332.2.b.c.2549.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,2,Mod(2549,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3332.2549");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3332 = 2^{2} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3332.b (of order $2$, degree $1$, 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:	$26.6061539535$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 $ x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2549.3
Root			$0.535233 + 0.309017i$ of defining polynomial
Character	$\chi$	$=$	3332.2549
Dual form			3332.2.b.c.2549.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.381966i q^{3} +2.61803i q^{5} +2.85410 q^{9} -3.46410i q^{11} -2.14093 q^{13} +1.00000 q^{15} +(-3.46410 - 2.23607i) q^{17} -5.60503 q^{19} +5.60503i q^{23} -1.85410 q^{25} -2.23607i q^{27} +7.74597i q^{29} +1.85410i q^{31} -1.32317 q^{33} +1.32317i q^{37} +0.817763i q^{39} +0.381966i q^{41} -0.854102 q^{43} +7.47214i q^{45} -3.46410 q^{47} +(-0.854102 + 1.32317i) q^{51} -1.85410 q^{53} +9.06914 q^{55} +2.14093i q^{57} -6.92820 q^{59} +8.56231i q^{61} -5.60503i q^{65} -7.56231 q^{67} +2.14093 q^{69} -1.32317i q^{71} +10.8541i q^{73} +0.708204i q^{75} +7.74597i q^{79} +7.70820 q^{81} +13.3510 q^{83} +(5.85410 - 9.06914i) q^{85} +2.95870 q^{87} -10.3923 q^{89} +0.708204 q^{93} -14.6742i q^{95} -7.85410i q^{97} -9.88690i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{9} + 8 q^{15} + 12 q^{25} + 20 q^{43} + 20 q^{51} + 12 q^{53} + 20 q^{67} + 8 q^{81} + 20 q^{85} - 48 q^{93}+O(q^{100}) $ 8 * q - 4 * q^9 + 8 * q^15 + 12 * q^25 + 20 * q^43 + 20 * q^51 + 12 * q^53 + 20 * q^67 + 8 * q^81 + 20 * q^85 - 48 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times$.

$n$	$785$	$885$	$1667$
$\chi(n)$	$-1$	$1$	$1$

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
$3$		−	0.381966i		−	0.220528i	−0.993902	−	0.110264i	$-0.964830\pi$
$3$		−	0.381966i		−	0.220528i	0.993902	−	0.110264i	$-0.0351697\pi$
$4$		0			0
$5$			2.61803i			1.17082i	0.810737	+	0.585410i	$0.199067\pi$
$5$			2.61803i			1.17082i	−0.810737	+	0.585410i	$0.800933\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	2.85410			0.951367
$10$		0			0
$11$		−	3.46410i		−	1.04447i	−0.852803	−	0.522233i	$-0.825099\pi$
$11$		−	3.46410i		−	1.04447i	0.852803	−	0.522233i	$-0.174901\pi$
$12$		0			0
$13$	−2.14093			−0.593788			−0.296894	−	0.954910i	$-0.595951\pi$
$13$	−2.14093			−0.593788			−0.296894	+	0.954910i	$0.595951\pi$
$14$		0			0
$15$	1.00000			0.258199
$16$		0			0
$17$	−3.46410	−	2.23607i	−0.840168	−	0.542326i
$17$	−3.46410	−	2.23607i	−0.840168	−	0.542326i
$18$		0			0
$19$	−5.60503			−1.28588			−0.642942	−	0.765915i	$-0.722286\pi$
$19$	−5.60503			−1.28588			−0.642942	+	0.765915i	$0.722286\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			5.60503i			1.16873i	0.811491	+	0.584365i	$0.198656\pi$
$23$			5.60503i			1.16873i	−0.811491	+	0.584365i	$0.801344\pi$
$24$		0			0
$25$	−1.85410			−0.370820
$26$		0			0
$27$		−	2.23607i		−	0.430331i
$28$		0			0
$29$			7.74597i			1.43839i	0.694808	+	0.719195i	$0.255489\pi$
$29$			7.74597i			1.43839i	−0.694808	+	0.719195i	$0.744511\pi$
$30$		0			0
$31$			1.85410i			0.333007i	0.986041	+	0.166503i	$0.0532476\pi$
$31$			1.85410i			0.333007i	−0.986041	+	0.166503i	$0.946752\pi$
$32$		0			0
$33$	−1.32317			−0.230334
$34$		0			0
$35$		0			0
$36$		0			0
$37$			1.32317i			0.217528i	0.994068	+	0.108764i	$0.0346892\pi$
$37$			1.32317i			0.217528i	−0.994068	+	0.108764i	$0.965311\pi$
$38$		0			0
$39$			0.817763i			0.130947i
$40$		0			0
$41$			0.381966i			0.0596531i	0.999555	+	0.0298265i	$0.00949549\pi$
$41$			0.381966i			0.0596531i	−0.999555	+	0.0298265i	$0.990505\pi$
$42$		0			0
$43$	−0.854102			−0.130249			−0.0651247	−	0.997877i	$-0.520745\pi$
$43$	−0.854102			−0.130249			−0.0651247	+	0.997877i	$0.520745\pi$
$44$		0			0
$45$			7.47214i			1.11388i
$46$		0			0
$47$	−3.46410			−0.505291			−0.252646	−	0.967559i	$-0.581301\pi$
$47$	−3.46410			−0.505291			−0.252646	+	0.967559i	$0.581301\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−0.854102	+	1.32317i	−0.119598	+	0.185281i
$52$		0			0
$53$	−1.85410			−0.254680			−0.127340	−	0.991859i	$-0.540644\pi$
$53$	−1.85410			−0.254680			−0.127340	+	0.991859i	$0.540644\pi$
$54$		0			0
$55$	9.06914			1.22288
$56$		0			0
$57$			2.14093i			0.283573i
$58$		0			0
$59$	−6.92820			−0.901975			−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820			−0.901975			−0.450988	+	0.892530i	$0.648928\pi$
$60$		0			0
$61$			8.56231i			1.09629i	0.836383	+	0.548145i	$0.184666\pi$
$61$			8.56231i			1.09629i	−0.836383	+	0.548145i	$0.815334\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	5.60503i		−	0.695219i
$66$		0			0
$67$	−7.56231			−0.923883			−0.461941	−	0.886910i	$-0.652847\pi$
$67$	−7.56231			−0.923883			−0.461941	+	0.886910i	$0.652847\pi$
$68$		0			0
$69$	2.14093			0.257738
$70$		0			0
$71$		−	1.32317i		−	0.157031i	−0.996913	−	0.0785156i	$-0.974982\pi$
$71$		−	1.32317i		−	0.157031i	0.996913	−	0.0785156i	$-0.0250181\pi$
$72$		0			0
$73$			10.8541i			1.27038i	0.772357	+	0.635188i	$0.219077\pi$
$73$			10.8541i			1.27038i	−0.772357	+	0.635188i	$0.780923\pi$
$74$		0			0
$75$			0.708204i			0.0817763i
$76$		0			0
$77$		0			0
$78$		0			0
$79$			7.74597i			0.871489i	0.900070	+	0.435745i	$0.143515\pi$
$79$			7.74597i			0.871489i	−0.900070	+	0.435745i	$0.856485\pi$
$80$		0			0
$81$	7.70820			0.856467
$82$		0			0
$83$	13.3510			1.46546			0.732731	−	0.680518i	$-0.238245\pi$
$83$	13.3510			1.46546			0.732731	+	0.680518i	$0.238245\pi$
$84$		0			0
$85$	5.85410	−	9.06914i	0.634967	−	0.983686i
$86$		0			0
$87$	2.95870			0.317206
$88$		0			0
$89$	−10.3923			−1.10158			−0.550791	−	0.834643i	$-0.685674\pi$
$89$	−10.3923			−1.10158			−0.550791	+	0.834643i	$0.685674\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	0.708204			0.0734373
$94$		0			0
$95$		−	14.6742i		−	1.50554i
$96$		0			0
$97$		−	7.85410i		−	0.797463i	−0.917068	−	0.398732i	$-0.869451\pi$
$97$		−	7.85410i		−	0.797463i	0.917068	−	0.398732i	$-0.130549\pi$
$98$		0			0
$99$		−	9.88690i		−	0.993671i
$100$		0			0
$101$	−16.8151			−1.67317			−0.836583	−	0.547841i	$-0.815450\pi$
$101$	−16.8151			−1.67317			−0.836583	+	0.547841i	$0.815450\pi$
$102$		0			0
$103$	−4.78727			−0.471704			−0.235852	−	0.971789i	$-0.575788\pi$
$103$	−4.78727			−0.471704			−0.235852	+	0.971789i	$0.575788\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	15.9973i		−	1.54652i	−0.634088	−	0.773261i	$-0.718624\pi$
$107$		−	15.9973i		−	1.54652i	0.634088	−	0.773261i	$-0.281376\pi$
$108$		0			0
$109$			18.1383i			1.73733i	0.495399	+	0.868666i	$0.335022\pi$
$109$			18.1383i			1.73733i	−0.495399	+	0.868666i	$0.664978\pi$
$110$		0			0
$111$	0.505406			0.0479710
$112$		0			0
$113$		−	3.46410i		−	0.325875i	−0.986636	−	0.162938i	$-0.947903\pi$
$113$		−	3.46410i		−	0.325875i	0.986636	−	0.162938i	$-0.0520969\pi$
$114$		0			0
$115$	−14.6742			−1.36837
$116$		0			0
$117$	−6.11044			−0.564910
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$	0.145898			0.0131552
$124$		0			0
$125$			8.23607i			0.736656i
$126$		0			0
$127$	−15.5623			−1.38093			−0.690466	−	0.723365i	$-0.742594\pi$
$127$	−15.5623			−1.38093			−0.690466	+	0.723365i	$0.742594\pi$
$128$		0			0
$129$			0.326238i			0.0287236i
$130$		0			0
$131$			10.4721i			0.914955i	0.889221	+	0.457477i	$0.151247\pi$
$131$			10.4721i			0.914955i	−0.889221	+	0.457477i	$0.848753\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	5.85410			0.503841
$136$		0			0
$137$	12.2705			1.04834			0.524170	−	0.851614i	$-0.324376\pi$
$137$	12.2705			1.04834			0.524170	+	0.851614i	$0.324376\pi$
$138$		0			0
$139$		−	17.5623i		−	1.48962i	−0.667279	−	0.744808i	$-0.732541\pi$
$139$		−	17.5623i		−	1.48962i	0.667279	−	0.744808i	$-0.267459\pi$
$140$		0			0
$141$			1.32317i			0.111431i
$142$		0			0
$143$			7.41641i			0.620191i
$144$		0			0
$145$	−20.2792			−1.68410
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−14.5623			−1.19299			−0.596495	−	0.802617i	$-0.703441\pi$
$149$	−14.5623			−1.19299			−0.596495	+	0.802617i	$0.703441\pi$
$150$		0			0
$151$	6.56231			0.534033			0.267016	−	0.963692i	$-0.413962\pi$
$151$	6.56231			0.534033			0.267016	+	0.963692i	$0.413962\pi$
$152$		0			0
$153$	−9.88690	−	6.38197i	−0.799308	−	0.515951i
$154$		0			0
$155$	−4.85410			−0.389891
$156$		0			0
$157$	7.74597			0.618195			0.309098	−	0.951030i	$-0.399973\pi$
$157$	7.74597			0.618195			0.309098	+	0.951030i	$0.399973\pi$
$158$		0			0
$159$			0.708204i			0.0561642i
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	11.7155i		−	0.917627i	−0.888533	−	0.458813i	$-0.848275\pi$
$163$		−	11.7155i		−	0.917627i	0.888533	−	0.458813i	$-0.151725\pi$
$164$		0			0
$165$		−	3.46410i		−	0.269680i
$166$		0			0
$167$		−	7.79837i		−	0.603456i	−0.953394	−	0.301728i	$-0.902437\pi$
$167$		−	7.79837i		−	0.603456i	0.953394	−	0.301728i	$-0.0975635\pi$
$168$		0			0
$169$	−8.41641			−0.647416
$170$		0			0
$171$	−15.9973			−1.22335
$172$		0			0
$173$			21.3262i			1.62140i	0.585459	+	0.810702i	$0.300914\pi$
$173$			21.3262i			1.62140i	−0.585459	+	0.810702i	$0.699086\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			2.64634i			0.198911i
$178$		0			0
$179$	−14.5623			−1.08844			−0.544219	−	0.838943i	$-0.683174\pi$
$179$	−14.5623			−1.08844			−0.544219	+	0.838943i	$0.683174\pi$
$180$		0			0
$181$			6.00000i			0.445976i	0.974821	+	0.222988i	$0.0715812\pi$
$181$			6.00000i			0.445976i	−0.974821	+	0.222988i	$0.928419\pi$
$182$		0			0
$183$	3.27051			0.241763
$184$		0			0
$185$	−3.46410			−0.254686
$186$		0			0
$187$	−7.74597	+	12.0000i	−0.566441	+	0.877527i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−11.5623			−0.836619			−0.418310	−	0.908305i	$-0.637377\pi$
$191$	−11.5623			−0.836619			−0.418310	+	0.908305i	$0.637377\pi$
$192$		0			0
$193$			11.7155i			0.843298i	0.906759	+	0.421649i	$0.138549\pi$
$193$			11.7155i			0.843298i	−0.906759	+	0.421649i	$0.861451\pi$
$194$		0			0
$195$	−2.14093			−0.153315
$196$		0			0
$197$			4.78727i			0.341079i	0.985351	+	0.170539i	$0.0545510\pi$
$197$			4.78727i			0.341079i	−0.985351	+	0.170539i	$0.945449\pi$
$198$		0			0
$199$		−	10.8541i		−	0.769427i	−0.923036	−	0.384713i	$-0.874300\pi$
$199$		−	10.8541i		−	0.769427i	0.923036	−	0.384713i	$-0.125700\pi$
$200$		0			0
$201$			2.88854i			0.203742i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1.00000			−0.0698430
$206$		0			0
$207$			15.9973i			1.11189i
$208$		0			0
$209$			19.4164i			1.34306i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	−0.505406			−0.0346298
$214$		0			0
$215$		−	2.23607i		−	0.152499i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	4.14590			0.280154
$220$		0			0
$221$	7.41641	+	4.78727i	0.498882	+	0.322027i
$222$		0			0
$223$	15.1796			1.01650			0.508250	−	0.861210i	$-0.330293\pi$
$223$	15.1796			1.01650			0.508250	+	0.861210i	$0.330293\pi$
$224$		0			0
$225$	−5.29180			−0.352786
$226$		0			0
$227$			26.5066i			1.75930i	0.475618	+	0.879652i	$0.342224\pi$
$227$			26.5066i			1.75930i	−0.475618	+	0.879652i	$0.657776\pi$
$228$		0			0
$229$	−17.3205			−1.14457			−0.572286	−	0.820054i	$-0.693943\pi$
$229$	−17.3205			−1.14457			−0.572286	+	0.820054i	$0.693943\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	15.1796i		−	0.994447i	−0.867622	−	0.497224i	$-0.834353\pi$
$233$		−	15.1796i		−	0.994447i	0.867622	−	0.497224i	$-0.165647\pi$
$234$		0			0
$235$		−	9.06914i		−	0.591605i
$236$		0			0
$237$	2.95870			0.192188
$238$		0			0
$239$	12.2705			0.793713			0.396857	−	0.917881i	$-0.370101\pi$
$239$	12.2705			0.793713			0.396857	+	0.917881i	$0.370101\pi$
$240$		0			0
$241$		−	15.2705i		−	0.983660i	−0.870691	−	0.491830i	$-0.836328\pi$
$241$		−	15.2705i		−	0.983660i	0.870691	−	0.491830i	$-0.163672\pi$
$242$		0			0
$243$		−	9.65248i		−	0.619207i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	12.0000			0.763542
$248$		0			0
$249$		−	5.09963i		−	0.323176i
$250$		0			0
$251$	−2.95870			−0.186751			−0.0933756	−	0.995631i	$-0.529766\pi$
$251$	−2.95870			−0.186751			−0.0933756	+	0.995631i	$0.529766\pi$
$252$		0			0
$253$	19.4164			1.22070
$254$		0			0
$255$	−3.46410	−	2.23607i	−0.216930	−	0.140028i
$256$		0			0
$257$	−16.8151			−1.04890			−0.524449	−	0.851442i	$-0.675729\pi$
$257$	−16.8151			−1.04890			−0.524449	+	0.851442i	$0.675729\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$			22.1078i			1.36844i
$262$		0			0
$263$	−7.41641			−0.457315			−0.228658	−	0.973507i	$-0.573434\pi$
$263$	−7.41641			−0.457315			−0.228658	+	0.973507i	$0.573434\pi$
$264$		0			0
$265$		−	4.85410i		−	0.298185i
$266$		0			0
$267$			3.96951i			0.242930i
$268$		0			0
$269$		−	9.05573i		−	0.552137i	−0.961138	−	0.276069i	$-0.910968\pi$
$269$		−	9.05573i		−	0.552137i	0.961138	−	0.276069i	$-0.0890317\pi$
$270$		0			0
$271$	14.1688			0.860691			0.430346	−	0.902664i	$-0.358392\pi$
$271$	14.1688			0.860691			0.430346	+	0.902664i	$0.358392\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			6.42280i			0.387309i
$276$		0			0
$277$		−	14.1688i		−	0.851319i	−0.904884	−	0.425659i	$-0.860042\pi$
$277$		−	14.1688i		−	0.851319i	0.904884	−	0.425659i	$-0.139958\pi$
$278$		0			0
$279$			5.29180i			0.316812i
$280$		0			0
$281$	6.43769			0.384041			0.192020	−	0.981391i	$-0.438496\pi$
$281$	6.43769			0.384041			0.192020	+	0.981391i	$0.438496\pi$
$282$		0			0
$283$			25.8541i			1.53687i	0.639930	+	0.768433i	$0.278963\pi$
$283$			25.8541i			1.53687i	−0.639930	+	0.768433i	$0.721037\pi$
$284$		0			0
$285$	−5.60503			−0.332014
$286$		0			0
$287$		0			0
$288$		0			0
$289$	7.00000	+	15.4919i	0.411765	+	0.911290i
$290$		0			0
$291$	−3.00000			−0.175863
$292$		0			0
$293$	2.95870			0.172849			0.0864244	−	0.996258i	$-0.472456\pi$
$293$	2.95870			0.172849			0.0864244	+	0.996258i	$0.472456\pi$
$294$		0			0
$295$		−	18.1383i		−	1.05605i
$296$		0			0
$297$	−7.74597			−0.449467
$298$		0			0
$299$		−	12.0000i		−	0.693978i
$300$		0			0
$301$		0			0
$302$		0			0
$303$			6.42280i			0.368980i
$304$		0			0
$305$	−22.4164			−1.28356
$306$		0			0
$307$	20.2792			1.15740			0.578698	−	0.815542i	$-0.303561\pi$
$307$	20.2792			1.15740			0.578698	+	0.815542i	$0.303561\pi$
$308$		0			0
$309$			1.82857i			0.104024i
$310$		0			0
$311$			32.6180i			1.84960i	0.380455	+	0.924800i	$0.375767\pi$
$311$			32.6180i			1.84960i	−0.380455	+	0.924800i	$0.624233\pi$
$312$		0			0
$313$			3.27051i			0.184860i	0.995719	+	0.0924301i	$0.0294635\pi$
$313$			3.27051i			0.184860i	−0.995719	+	0.0924301i	$0.970537\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	14.1688i		−	0.795797i	−0.917429	−	0.397899i	$-0.869740\pi$
$317$		−	14.1688i		−	0.795797i	0.917429	−	0.397899i	$-0.130260\pi$
$318$		0			0
$319$	26.8328			1.50235
$320$		0			0
$321$	−6.11044			−0.341051
$322$		0			0
$323$	19.4164	+	12.5332i	1.08036	+	0.697368i
$324$		0			0
$325$	3.96951			0.220189
$326$		0			0
$327$	6.92820			0.383131
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−8.14590			−0.447739			−0.223870	−	0.974619i	$-0.571869\pi$
$331$	−8.14590			−0.447739			−0.223870	+	0.974619i	$0.571869\pi$
$332$		0			0
$333$			3.77646i			0.206949i
$334$		0			0
$335$		−	19.7984i		−	1.08170i
$336$		0			0
$337$			30.9839i			1.68780i	0.536501	+	0.843899i	$0.319746\pi$
$337$			30.9839i			1.68780i	−0.536501	+	0.843899i	$0.680254\pi$
$338$		0			0
$339$	−1.32317			−0.0718647
$340$		0			0
$341$	6.42280			0.347814
$342$		0			0
$343$		0			0
$344$		0			0
$345$			5.60503i			0.301765i
$346$		0			0
$347$		−	15.4919i		−	0.831651i	−0.909445	−	0.415825i	$-0.863493\pi$
$347$		−	15.4919i		−	0.831651i	0.909445	−	0.415825i	$-0.136507\pi$
$348$		0			0
$349$	−4.78727			−0.256257			−0.128128	−	0.991758i	$-0.540897\pi$
$349$	−4.78727			−0.256257			−0.128128	+	0.991758i	$0.540897\pi$
$350$		0			0
$351$			4.78727i			0.255526i
$352$		0			0
$353$	16.3097			0.868078			0.434039	−	0.900894i	$-0.357088\pi$
$353$	16.3097			0.868078			0.434039	+	0.900894i	$0.357088\pi$
$354$		0			0
$355$	3.46410			0.183855
$356$		0			0
$357$		0			0
$358$		0			0
$359$	7.85410			0.414524			0.207262	−	0.978286i	$-0.433545\pi$
$359$	7.85410			0.414524			0.207262	+	0.978286i	$0.433545\pi$
$360$		0			0
$361$	12.4164			0.653495
$362$		0			0
$363$			0.381966i			0.0200480i
$364$		0			0
$365$	−28.4164			−1.48738
$366$		0			0
$367$		−	32.5623i		−	1.69974i	−0.526994	−	0.849869i	$-0.676681\pi$
$367$		−	32.5623i		−	1.69974i	0.526994	−	0.849869i	$-0.323319\pi$
$368$		0			0
$369$			1.09017i			0.0567520i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	18.5623			0.961120			0.480560	−	0.876962i	$-0.340433\pi$
$373$	18.5623			0.961120			0.480560	+	0.876962i	$0.340433\pi$
$374$		0			0
$375$	3.14590			0.162453
$376$		0			0
$377$		−	16.5836i		−	0.854098i
$378$		0			0
$379$			25.8842i			1.32958i	0.747029	+	0.664792i	$0.231480\pi$
$379$			25.8842i			1.32958i	−0.747029	+	0.664792i	$0.768520\pi$
$380$		0			0
$381$			5.94427i			0.304534i
$382$		0			0
$383$	17.3205			0.885037			0.442518	−	0.896759i	$-0.354085\pi$
$383$	17.3205			0.885037			0.442518	+	0.896759i	$0.354085\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−2.43769			−0.123915
$388$		0			0
$389$	−13.8541			−0.702431			−0.351215	−	0.936295i	$-0.614232\pi$
$389$	−13.8541			−0.702431			−0.351215	+	0.936295i	$0.614232\pi$
$390$		0			0
$391$	12.5332	−	19.4164i	0.633833	−	0.981930i
$392$		0			0
$393$	4.00000			0.201773
$394$		0			0
$395$	−20.2792			−1.02036
$396$		0			0
$397$		−	4.14590i		−	0.208077i	−0.994573	−	0.104038i	$-0.966824\pi$
$397$		−	4.14590i		−	0.208077i	0.994573	−	0.104038i	$-0.0331765\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	12.5332i		−	0.625880i	−0.949773	−	0.312940i	$-0.898686\pi$
$401$		−	12.5332i		−	0.625880i	0.949773	−	0.312940i	$-0.101314\pi$
$402$		0			0
$403$		−	3.96951i		−	0.197735i
$404$		0			0
$405$			20.1803i			1.00277i
$406$		0			0
$407$	4.58359			0.227200
$408$		0			0
$409$	16.5027			0.816008			0.408004	−	0.912980i	$-0.366225\pi$
$409$	16.5027			0.816008			0.408004	+	0.912980i	$0.366225\pi$
$410$		0			0
$411$		−	4.68692i		−	0.231189i
$412$		0			0
$413$		0			0
$414$		0			0
$415$			34.9534i			1.71579i
$416$		0			0
$417$	−6.70820			−0.328502
$418$		0			0
$419$			30.2148i			1.47609i	0.674752	+	0.738044i	$0.264250\pi$
$419$			30.2148i			1.47609i	−0.674752	+	0.738044i	$0.735750\pi$
$420$		0			0
$421$	−1.43769			−0.0700689			−0.0350345	−	0.999386i	$-0.511154\pi$
$421$	−1.43769			−0.0700689			−0.0350345	+	0.999386i	$0.511154\pi$
$422$		0			0
$423$	−9.88690			−0.480717
$424$		0			0
$425$	6.42280	+	4.14590i	0.311551	+	0.201106i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	2.83282			0.136770
$430$		0			0
$431$			9.06914i			0.436845i	0.975854	+	0.218422i	$0.0700911\pi$
$431$			9.06914i			0.436845i	−0.975854	+	0.218422i	$0.929909\pi$
$432$		0			0
$433$	−0.817763			−0.0392992			−0.0196496	−	0.999807i	$-0.506255\pi$
$433$	−0.817763			−0.0392992			−0.0196496	+	0.999807i	$0.506255\pi$
$434$		0			0
$435$			7.74597i			0.371391i
$436$		0			0
$437$		−	31.4164i		−	1.50285i
$438$		0			0
$439$		−	5.72949i		−	0.273454i	−0.990609	−	0.136727i	$-0.956342\pi$
$439$		−	5.72949i		−	0.273454i	0.990609	−	0.136727i	$-0.0436583\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−26.8328			−1.27487			−0.637433	−	0.770506i	$-0.720004\pi$
$443$	−26.8328			−1.27487			−0.637433	+	0.770506i	$0.720004\pi$
$444$		0			0
$445$		−	27.2074i		−	1.28975i
$446$		0			0
$447$			5.56231i			0.263088i
$448$		0			0
$449$		−	14.3618i		−	0.677776i	−0.940827	−	0.338888i	$-0.889949\pi$
$449$		−	14.3618i		−	0.677776i	0.940827	−	0.338888i	$-0.110051\pi$
$450$		0			0
$451$	1.32317			0.0623056
$452$		0			0
$453$		−	2.50658i		−	0.117769i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	10.4377			0.488255			0.244127	−	0.969743i	$-0.421499\pi$
$457$	10.4377			0.488255			0.244127	+	0.969743i	$0.421499\pi$
$458$		0			0
$459$	−5.00000	+	7.74597i	−0.233380	+	0.361551i
$460$		0			0
$461$	−37.0943			−1.72765			−0.863827	−	0.503788i	$-0.831939\pi$
$461$	−37.0943			−1.72765			−0.863827	+	0.503788i	$0.831939\pi$
$462$		0			0
$463$	−22.5623			−1.04856			−0.524280	−	0.851546i	$-0.675665\pi$
$463$	−22.5623			−1.04856			−0.524280	+	0.851546i	$0.675665\pi$
$464$		0			0
$465$			1.85410i			0.0859819i
$466$		0			0
$467$	31.1769			1.44270			0.721348	−	0.692573i	$-0.243523\pi$
$467$	31.1769			1.44270			0.721348	+	0.692573i	$0.243523\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		−	2.95870i		−	0.136330i
$472$		0			0
$473$			2.95870i			0.136041i
$474$		0			0
$475$	10.3923			0.476832
$476$		0			0
$477$	−5.29180			−0.242295
$478$		0			0
$479$		−	38.4508i		−	1.75686i	−0.477867	−	0.878432i	$-0.658590\pi$
$479$		−	38.4508i		−	1.75686i	0.477867	−	0.878432i	$-0.341410\pi$
$480$		0			0
$481$		−	2.83282i		−	0.129165i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	20.5623			0.933686
$486$		0			0
$487$			40.0530i			1.81497i	0.420079	+	0.907487i	$0.362002\pi$
$487$			40.0530i			1.81497i	−0.420079	+	0.907487i	$0.637998\pi$
$488$		0			0
$489$	−4.47491			−0.202363
$490$		0			0
$491$	28.1459			1.27021			0.635103	−	0.772427i	$-0.280958\pi$
$491$	28.1459			1.27021			0.635103	+	0.772427i	$0.280958\pi$
$492$		0			0
$493$	17.3205	−	26.8328i	0.780076	−	1.20849i
$494$		0			0
$495$	25.8842			1.16341
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	42.6993i		−	1.91149i	−0.294204	−	0.955743i	$-0.595055\pi$
$499$		−	42.6993i		−	1.91149i	0.294204	−	0.955743i	$-0.404945\pi$
$500$		0			0
$501$	−2.97871			−0.133079
$502$		0			0
$503$		−	18.2148i		−	0.812157i	−0.913838	−	0.406078i	$-0.866896\pi$
$503$		−	18.2148i		−	0.812157i	0.913838	−	0.406078i	$-0.133104\pi$
$504$		0			0
$505$		−	44.0225i		−	1.95898i
$506$		0			0
$507$			3.21478i			0.142773i
$508$		0			0
$509$	−33.1248			−1.46823			−0.734115	−	0.679025i	$-0.762403\pi$
$509$	−33.1248			−1.46823			−0.734115	+	0.679025i	$0.762403\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$			12.5332i			0.553356i
$514$		0			0
$515$		−	12.5332i		−	0.552280i
$516$		0			0
$517$			12.0000i			0.527759i
$518$		0			0
$519$	8.14590			0.357565
$520$		0			0
$521$			23.6180i			1.03472i	0.855766	+	0.517362i	$0.173086\pi$
$521$			23.6180i			1.03472i	−0.855766	+	0.517362i	$0.826914\pi$
$522$		0			0
$523$	3.96951			0.173574			0.0867872	−	0.996227i	$-0.472340\pi$
$523$	3.96951			0.173574			0.0867872	+	0.996227i	$0.472340\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.14590	−	6.42280i	0.180598	−	0.279781i
$528$		0			0
$529$	−8.41641			−0.365931
$530$		0			0
$531$	−19.7738			−0.858110
$532$		0			0
$533$		−	0.817763i		−	0.0354213i
$534$		0			0
$535$	41.8816			1.81070
$536$		0			0
$537$			5.56231i			0.240031i
$538$		0			0
$539$		0			0
$540$		0			0
$541$			1.32317i			0.0568875i	0.999595	+	0.0284437i	$0.00905515\pi$
$541$			1.32317i			0.0568875i	−0.999595	+	0.0284437i	$0.990945\pi$
$542$		0			0
$543$	2.29180			0.0983504
$544$		0			0
$545$	−47.4866			−2.03410
$546$		0			0
$547$		−	15.6850i		−	0.670641i	−0.942104	−	0.335321i	$-0.891155\pi$
$547$		−	15.6850i		−	0.670641i	0.942104	−	0.335321i	$-0.108845\pi$
$548$		0			0
$549$			24.4377i			1.04298i
$550$		0			0
$551$		−	43.4164i		−	1.84960i
$552$		0			0
$553$		0			0
$554$		0			0
$555$			1.32317i			0.0561654i
$556$		0			0
$557$	−1.41641			−0.0600151			−0.0300076	−	0.999550i	$-0.509553\pi$
$557$	−1.41641			−0.0600151			−0.0300076	+	0.999550i	$0.509553\pi$
$558$		0			0
$559$	1.82857			0.0773405
$560$		0			0
$561$	4.58359	+	2.95870i	0.193519	+	0.124916i
$562$		0			0
$563$	−10.3923			−0.437983			−0.218992	−	0.975727i	$-0.570277\pi$
$563$	−10.3923			−0.437983			−0.218992	+	0.975727i	$0.570277\pi$
$564$		0			0
$565$	9.06914			0.381541
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−36.2705			−1.52054			−0.760269	−	0.649608i	$-0.774933\pi$
$569$	−36.2705			−1.52054			−0.760269	+	0.649608i	$0.774933\pi$
$570$		0			0
$571$			1.32317i			0.0553729i	0.999617	+	0.0276865i	$0.00881400\pi$
$571$			1.32317i			0.0553729i	−0.999617	+	0.0276865i	$0.991186\pi$
$572$		0			0
$573$			4.41641i			0.184498i
$574$		0			0
$575$		−	10.3923i		−	0.433389i
$576$		0			0
$577$	45.3457			1.88777			0.943883	−	0.330281i	$-0.107144\pi$
$577$	45.3457			1.88777			0.943883	+	0.330281i	$0.107144\pi$
$578$		0			0
$579$	4.47491			0.185971
$580$		0			0
$581$		0			0
$582$		0			0
$583$			6.42280i			0.266005i
$584$		0			0
$585$		−	15.9973i		−	0.661409i
$586$		0			0
$587$	−10.3923			−0.428936			−0.214468	−	0.976731i	$-0.568802\pi$
$587$	−10.3923			−0.428936			−0.214468	+	0.976731i	$0.568802\pi$
$588$		0			0
$589$		−	10.3923i		−	0.428207i
$590$		0			0
$591$	1.82857			0.0752175
$592$		0			0
$593$	30.6715			1.25953			0.629764	−	0.776787i	$-0.283152\pi$
$593$	30.6715			1.25953			0.629764	+	0.776787i	$0.283152\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−4.14590			−0.169680
$598$		0			0
$599$	19.1459			0.782280			0.391140	−	0.920331i	$-0.372081\pi$
$599$	19.1459			0.782280			0.391140	+	0.920331i	$0.372081\pi$
$600$		0			0
$601$			30.0000i			1.22373i	0.790964	+	0.611863i	$0.209580\pi$
$601$			30.0000i			1.22373i	−0.790964	+	0.611863i	$0.790420\pi$
$602$		0			0
$603$	−21.5836			−0.878952
$604$		0			0
$605$		−	2.61803i		−	0.106438i
$606$		0			0
$607$			22.1459i			0.898874i	0.893312	+	0.449437i	$0.148375\pi$
$607$			22.1459i			0.898874i	−0.893312	+	0.449437i	$0.851625\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	7.41641			0.300036
$612$		0			0
$613$	9.85410			0.398003			0.199002	−	0.979999i	$-0.436230\pi$
$613$	9.85410			0.398003			0.199002	+	0.979999i	$0.436230\pi$
$614$		0			0
$615$			0.381966i			0.0154024i
$616$		0			0
$617$		−	28.5306i		−	1.14860i	−0.818646	−	0.574299i	$-0.805275\pi$
$617$		−	28.5306i		−	1.14860i	0.818646	−	0.574299i	$-0.194725\pi$
$618$		0			0
$619$		−	24.0000i		−	0.964641i	−0.875995	−	0.482321i	$-0.839794\pi$
$619$		−	24.0000i		−	0.964641i	0.875995	−	0.482321i	$-0.160206\pi$
$620$		0			0
$621$	12.5332			0.502941
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−30.8328			−1.23331
$626$		0			0
$627$	7.41641			0.296183
$628$		0			0
$629$	2.95870	−	4.58359i	0.117971	−	0.182760i
$630$		0			0
$631$	−7.27051			−0.289434			−0.144717	−	0.989473i	$-0.546227\pi$
$631$	−7.27051			−0.289434			−0.144717	+	0.989473i	$0.546227\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	40.7426i		−	1.61682i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		−	3.77646i		−	0.149394i
$640$		0			0
$641$		−	10.7047i		−	0.422809i	−0.977399	−	0.211404i	$-0.932196\pi$
$641$		−	10.7047i		−	0.422809i	0.977399	−	0.211404i	$-0.0678037\pi$
$642$		0			0
$643$		−	15.9787i		−	0.630139i	−0.949069	−	0.315069i	$-0.897972\pi$
$643$		−	15.9787i		−	0.630139i	0.949069	−	0.315069i	$-0.102028\pi$
$644$		0			0
$645$	−0.854102			−0.0336302
$646$		0			0
$647$	23.2379			0.913576			0.456788	−	0.889576i	$-0.349000\pi$
$647$	23.2379			0.913576			0.456788	+	0.889576i	$0.349000\pi$
$648$		0			0
$649$			24.0000i			0.942082i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	27.7128i		−	1.08449i	−0.840222	−	0.542243i	$-0.817575\pi$
$653$		−	27.7128i		−	1.08449i	0.840222	−	0.542243i	$-0.182425\pi$
$654$		0			0
$655$	−27.4164			−1.07125
$656$		0			0
$657$			30.9787i			1.20859i
$658$		0			0
$659$	−30.9787			−1.20676			−0.603380	−	0.797454i	$-0.706180\pi$
$659$	−30.9787			−1.20676			−0.603380	+	0.797454i	$0.706180\pi$
$660$		0			0
$661$	26.3896			1.02644			0.513219	−	0.858258i	$-0.328453\pi$
$661$	26.3896			1.02644			0.513219	+	0.858258i	$0.328453\pi$
$662$		0			0
$663$	1.82857	−	2.83282i	0.0710160	−	0.110017i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−43.4164			−1.68109
$668$		0			0
$669$		−	5.79808i		−	0.224167i
$670$		0			0
$671$	29.6607			1.14504
$672$		0			0
$673$		−	10.3923i		−	0.400594i	−0.979735	−	0.200297i	$-0.935809\pi$
$673$		−	10.3923i		−	0.400594i	0.979735	−	0.200297i	$-0.0641907\pi$
$674$		0			0
$675$			4.14590i			0.159576i
$676$		0			0
$677$		−	21.0557i		−	0.809237i	−0.914485	−	0.404619i	$-0.867404\pi$
$677$		−	21.0557i		−	0.809237i	0.914485	−	0.404619i	$-0.132596\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	10.1246			0.387976
$682$		0			0
$683$			20.2792i			0.775962i	0.921667	+	0.387981i	$0.126827\pi$
$683$			20.2792i			0.775962i	−0.921667	+	0.387981i	$0.873173\pi$
$684$		0			0
$685$			32.1246i			1.22742i
$686$		0			0
$687$			6.61585i			0.252410i
$688$		0			0
$689$	3.96951			0.151226
$690$		0			0
$691$		−	39.9787i		−	1.52086i	−0.649419	−	0.760431i	$-0.724988\pi$
$691$		−	39.9787i		−	1.52086i	0.649419	−	0.760431i	$-0.275012\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	45.9787			1.74407
$696$		0			0
$697$	0.854102	−	1.32317i	0.0323514	−	0.0501186i
$698$		0			0
$699$	−5.79808			−0.219304
$700$		0			0
$701$	−49.4164			−1.86643			−0.933216	−	0.359316i	$-0.883010\pi$
$701$	−49.4164			−1.86643			−0.933216	+	0.359316i	$0.883010\pi$
$702$		0			0
$703$		−	7.41641i		−	0.279715i
$704$		0			0
$705$	−3.46410			−0.130466
$706$		0			0
$707$		0			0
$708$		0			0
$709$			11.7155i			0.439984i	0.975502	+	0.219992i	$0.0706031\pi$
$709$			11.7155i			0.439984i	−0.975502	+	0.219992i	$0.929397\pi$
$710$		0			0
$711$			22.1078i			0.829106i
$712$		0			0
$713$	−10.3923			−0.389195
$714$		0			0
$715$	−19.4164			−0.726132
$716$		0			0
$717$		−	4.68692i		−	0.175036i
$718$		0			0
$719$		−	15.2148i		−	0.567416i	−0.958911	−	0.283708i	$-0.908435\pi$
$719$		−	15.2148i		−	0.567416i	0.958911	−	0.283708i	$-0.0915646\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−5.83282			−0.216925
$724$		0			0
$725$		−	14.3618i		−	0.533384i
$726$		0			0
$727$	41.0638			1.52297			0.761486	−	0.648181i	$-0.224470\pi$
$727$	41.0638			1.52297			0.761486	+	0.648181i	$0.224470\pi$
$728$		0			0
$729$	19.4377			0.719915
$730$		0			0
$731$	2.95870	+	1.90983i	0.109431	+	0.0706376i
$732$		0			0
$733$	30.1661			1.11421			0.557105	−	0.830442i	$-0.311912\pi$
$733$	30.1661			1.11421			0.557105	+	0.830442i	$0.311912\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			26.1966i			0.964964i
$738$		0			0
$739$	28.2705			1.03995			0.519974	−	0.854182i	$-0.325942\pi$
$739$	28.2705			1.03995			0.519974	+	0.854182i	$0.325942\pi$
$740$		0			0
$741$		−	4.58359i		−	0.168382i
$742$		0			0
$743$			12.8456i			0.471259i	0.971843	+	0.235630i	$0.0757152\pi$
$743$			12.8456i			0.471259i	−0.971843	+	0.235630i	$0.924285\pi$
$744$		0			0
$745$		−	38.1246i		−	1.39678i
$746$		0			0
$747$	38.1051			1.39419
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	29.6607i		−	1.08233i	−0.840915	−	0.541167i	$-0.817983\pi$
$751$		−	29.6607i		−	1.08233i	0.840915	−	0.541167i	$-0.182017\pi$
$752$		0			0
$753$			1.13012i			0.0411839i
$754$		0			0
$755$			17.1803i			0.625257i
$756$		0			0
$757$	44.2705			1.60904			0.804520	−	0.593926i	$-0.202423\pi$
$757$	44.2705			1.60904			0.804520	+	0.593926i	$0.202423\pi$
$758$		0			0
$759$		−	7.41641i		−	0.269199i
$760$		0			0
$761$	26.7020			0.967947			0.483973	−	0.875083i	$-0.339193\pi$
$761$	26.7020			0.967947			0.483973	+	0.875083i	$0.339193\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	16.7082	−	25.8842i	0.604086	−	0.935847i
$766$		0			0
$767$	14.8328			0.535582
$768$		0			0
$769$	−32.8124			−1.18325			−0.591623	−	0.806214i	$-0.701513\pi$
$769$	−32.8124			−1.18325			−0.591623	+	0.806214i	$0.701513\pi$
$770$		0			0
$771$			6.42280i			0.231311i
$772$		0			0
$773$	−16.3097			−0.586619			−0.293310	−	0.956018i	$-0.594757\pi$
$773$	−16.3097			−0.586619			−0.293310	+	0.956018i	$0.594757\pi$
$774$		0			0
$775$		−	3.43769i		−	0.123486i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	2.14093i		−	0.0767069i
$780$		0			0
$781$	−4.58359			−0.164014
$782$		0			0
$783$	17.3205			0.618984
$784$		0			0
$785$			20.2792i			0.723796i
$786$		0			0
$787$			16.5836i			0.591141i	0.955321	+	0.295571i	$0.0955098\pi$
$787$			16.5836i			0.591141i	−0.955321	+	0.295571i	$0.904490\pi$
$788$		0			0
$789$			2.83282i			0.100851i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	18.3313i		−	0.650964i
$794$		0			0
$795$	−1.85410			−0.0657582
$796$		0			0
$797$	−40.0530			−1.41875			−0.709375	−	0.704831i	$-0.751023\pi$
$797$	−40.0530			−1.41875			−0.709375	+	0.704831i	$0.751023\pi$
$798$		0			0
$799$	12.0000	+	7.74597i	0.424529	+	0.274033i
$800$		0			0
$801$	−29.6607			−1.04801
$802$		0			0
$803$	37.5997			1.32687
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−3.45898			−0.121762
$808$		0			0
$809$			39.4283i			1.38622i	0.720830	+	0.693112i	$0.243761\pi$
$809$			39.4283i			1.38622i	−0.720830	+	0.693112i	$0.756239\pi$
$810$		0			0
$811$		−	39.9787i		−	1.40384i	−0.712255	−	0.701921i	$-0.752326\pi$
$811$		−	39.9787i		−	1.40384i	0.712255	−	0.701921i	$-0.247674\pi$
$812$		0			0
$813$		−	5.41199i		−	0.189807i
$814$		0			0
$815$	30.6715			1.07438
$816$		0			0
$817$	4.78727			0.167485
$818$		0			0
$819$		0			0
$820$		0			0
$821$			23.7433i			0.828647i	0.910130	+	0.414324i	$0.135982\pi$
$821$			23.7433i			0.828647i	−0.910130	+	0.414324i	$0.864018\pi$
$822$		0			0
$823$		−	2.64634i		−	0.0922455i	−0.998936	−	0.0461228i	$-0.985313\pi$
$823$		−	2.64634i		−	0.0922455i	0.998936	−	0.0461228i	$-0.0146865\pi$
$824$		0			0
$825$	2.45329			0.0854126
$826$		0			0
$827$			31.4893i			1.09499i	0.836809	+	0.547495i	$0.184418\pi$
$827$			31.4893i			1.09499i	−0.836809	+	0.547495i	$0.815582\pi$
$828$		0			0
$829$	54.9202			1.90746			0.953729	−	0.300667i	$-0.0972094\pi$
$829$	54.9202			1.90746			0.953729	+	0.300667i	$0.0972094\pi$
$830$		0			0
$831$	−5.41199			−0.187740
$832$		0			0
$833$		0			0
$834$		0			0
$835$	20.4164			0.706539
$836$		0			0
$837$	4.14590			0.143303
$838$		0			0
$839$			16.3607i			0.564833i	0.959292	+	0.282417i	$0.0911361\pi$
$839$			16.3607i			0.564833i	−0.959292	+	0.282417i	$0.908864\pi$
$840$		0			0
$841$	−31.0000			−1.06897
$842$		0			0
$843$		−	2.45898i		−	0.0846918i
$844$		0			0
$845$		−	22.0344i		−	0.758008i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	9.87539			0.338922
$850$		0			0
$851$	−7.41641			−0.254231
$852$		0			0
$853$		−	49.4164i		−	1.69199i	−0.533194	−	0.845993i	$-0.679009\pi$
$853$		−	49.4164i		−	1.69199i	0.533194	−	0.845993i	$-0.320991\pi$
$854$		0			0
$855$		−	41.8816i		−	1.43232i
$856$		0			0
$857$			5.45085i			0.186197i	0.995657	+	0.0930987i	$0.0296772\pi$
$857$			5.45085i			0.186197i	−0.995657	+	0.0930987i	$0.970323\pi$
$858$		0			0
$859$	36.0835			1.23115			0.615576	−	0.788077i	$-0.288923\pi$
$859$	36.0835			1.23115			0.615576	+	0.788077i	$0.288923\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	6.97871			0.237558			0.118779	−	0.992921i	$-0.462102\pi$
$863$	6.97871			0.237558			0.118779	+	0.992921i	$0.462102\pi$
$864$		0			0
$865$	−55.8328			−1.89837
$866$		0			0
$867$	5.91739	−	2.67376i	0.200965	−	0.0908057i
$868$		0			0
$869$	26.8328			0.910241
$870$		0			0
$871$	16.1904			0.548590
$872$		0			0
$873$		−	22.4164i		−	0.758680i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			45.3457i			1.53121i	0.643308	+	0.765607i	$0.277561\pi$
$877$			45.3457i			1.53121i	−0.643308	+	0.765607i	$0.722439\pi$
$878$		0			0
$879$		−	1.13012i		−	0.0381180i
$880$		0			0
$881$			21.3820i			0.720377i	0.932880	+	0.360188i	$0.117288\pi$
$881$			21.3820i			0.720377i	−0.932880	+	0.360188i	$0.882712\pi$
$882$		0			0
$883$	54.3951			1.83054			0.915271	−	0.402839i	$-0.131977\pi$
$883$	54.3951			1.83054			0.915271	+	0.402839i	$0.131977\pi$
$884$		0			0
$885$	−6.92820			−0.232889
$886$		0			0
$887$		−	9.92299i		−	0.333181i	−0.986026	−	0.166591i	$-0.946724\pi$
$887$		−	9.92299i		−	0.333181i	0.986026	−	0.166591i	$-0.0532758\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		−	26.7020i		−	0.894551i
$892$		0			0
$893$	19.4164			0.649745
$894$		0			0
$895$		−	38.1246i		−	1.27437i
$896$		0			0
$897$	−4.58359			−0.153042
$898$		0			0
$899$	−14.3618			−0.478993
$900$		0			0
$901$	6.42280	+	4.14590i	0.213974	+	0.138120i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−15.7082			−0.522158
$906$		0			0
$907$		−	28.3375i		−	0.940932i	−0.882418	−	0.470466i	$-0.844086\pi$
$907$		−	28.3375i		−	0.940932i	0.882418	−	0.470466i	$-0.155914\pi$
$908$		0			0
$909$	−47.9920			−1.59179
$910$		0			0
$911$			8.25137i			0.273380i	0.990614	+	0.136690i	$0.0436464\pi$
$911$			8.25137i			0.273380i	−0.990614	+	0.136690i	$0.956354\pi$
$912$		0			0
$913$		−	46.2492i		−	1.53063i
$914$		0			0
$915$			8.56231i			0.283061i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−20.8541			−0.687913			−0.343957	−	0.938986i	$-0.611767\pi$
$919$	−20.8541			−0.687913			−0.343957	+	0.938986i	$0.611767\pi$
$920$		0			0
$921$		−	7.74597i		−	0.255238i
$922$		0			0
$923$			2.83282i			0.0932433i
$924$		0			0
$925$		−	2.45329i		−	0.0806637i
$926$		0			0
$927$	−13.6634			−0.448764
$928$		0			0
$929$			17.6738i			0.579857i	0.957048	+	0.289929i	$0.0936316\pi$
$929$			17.6738i			0.579857i	−0.957048	+	0.289929i	$0.906368\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	12.4590			0.407889
$934$		0			0
$935$	−31.4164	−	20.2792i	−1.02743	−	0.663201i
$936$		0			0
$937$	26.7020			0.872316			0.436158	−	0.899870i	$-0.356339\pi$
$937$	26.7020			0.872316			0.436158	+	0.899870i	$0.356339\pi$
$938$		0			0
$939$	1.24922			0.0407669
$940$		0			0
$941$			5.50658i			0.179509i	0.995964	+	0.0897547i	$0.0286083\pi$
$941$			5.50658i			0.179509i	−0.995964	+	0.0897547i	$0.971392\pi$
$942$		0			0
$943$	−2.14093			−0.0697184
$944$		0			0
$945$		0			0
$946$		0			0
$947$			44.0225i			1.43054i	0.698849	+	0.715270i	$0.253696\pi$
$947$			44.0225i			1.43054i	−0.698849	+	0.715270i	$0.746304\pi$
$948$		0			0
$949$		−	23.2379i		−	0.754334i
$950$		0			0
$951$	−5.41199			−0.175496
$952$		0			0
$953$	−36.9787			−1.19786			−0.598929	−	0.800802i	$-0.704407\pi$
$953$	−36.9787			−1.19786			−0.598929	+	0.800802i	$0.704407\pi$
$954$		0			0
$955$		−	30.2705i		−	0.979531i
$956$		0			0
$957$		−	10.2492i		−	0.331310i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	27.5623			0.889107
$962$		0			0
$963$		−	45.6580i		−	1.47131i
$964$		0			0
$965$	−30.6715			−0.987351
$966$		0			0
$967$	51.1459			1.64474			0.822371	−	0.568952i	$-0.192651\pi$
$967$	51.1459			1.64474			0.822371	+	0.568952i	$0.192651\pi$
$968$		0			0
$969$	4.78727	−	7.41641i	0.153789	−	0.238249i
$970$		0			0
$971$	11.4031			0.365943			0.182972	−	0.983118i	$-0.441428\pi$
$971$	11.4031			0.365943			0.182972	+	0.983118i	$0.441428\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		−	1.51622i		−	0.0485578i
$976$		0			0
$977$	48.1033			1.53896			0.769481	−	0.638670i	$-0.220515\pi$
$977$	48.1033			1.53896			0.769481	+	0.638670i	$0.220515\pi$
$978$		0			0
$979$			36.0000i			1.15056i
$980$		0			0
$981$			51.7685i			1.65284i
$982$		0			0
$983$			0.326238i			0.0104054i	0.999986	+	0.00520269i	$0.00165607\pi$
$983$			0.326238i			0.0104054i	−0.999986	+	0.00520269i	$0.998344\pi$
$984$		0			0
$985$	−12.5332			−0.399342
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	4.78727i		−	0.152226i
$990$		0			0
$991$			59.7075i			1.89667i	0.317269	+	0.948335i	$0.397234\pi$
$991$			59.7075i			1.89667i	−0.317269	+	0.948335i	$0.602766\pi$
$992$		0			0
$993$			3.11146i			0.0987391i
$994$		0			0
$995$	28.4164			0.900861
$996$		0			0
$997$			5.72949i			0.181455i	0.995876	+	0.0907274i	$0.0289192\pi$
$997$			5.72949i			0.181455i	−0.995876	+	0.0907274i	$0.971081\pi$
$998$		0			0
$999$	2.95870			0.0936090

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3332.2.b.c.2549.3	✓	8
7.6	odd	2	inner	3332.2.b.c.2549.5	yes	8
17.16	even	2	inner	3332.2.b.c.2549.6	yes	8
119.118	odd	2	inner	3332.2.b.c.2549.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3332.2.b.c.2549.3	✓	8	1.1	even	1	trivial
3332.2.b.c.2549.4	yes	8	119.118	odd	2	inner
3332.2.b.c.2549.5	yes	8	7.6	odd	2	inner
3332.2.b.c.2549.6	yes	8	17.16	even	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:	\( N \)	\(=\)	\( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3332.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.381966i q^{3} +2.61803i q^{5} +2.85410 q^{9} -3.46410i q^{11} -2.14093 q^{13} +1.00000 q^{15} +(-3.46410 - 2.23607i) q^{17} -5.60503 q^{19} +5.60503i q^{23} -1.85410 q^{25} -2.23607i q^{27} +7.74597i q^{29} +1.85410i q^{31} -1.32317 q^{33} +1.32317i q^{37} +0.817763i q^{39} +0.381966i q^{41} -0.854102 q^{43} +7.47214i q^{45} -3.46410 q^{47} +(-0.854102 + 1.32317i) q^{51} -1.85410 q^{53} +9.06914 q^{55} +2.14093i q^{57} -6.92820 q^{59} +8.56231i q^{61} -5.60503i q^{65} -7.56231 q^{67} +2.14093 q^{69} -1.32317i q^{71} +10.8541i q^{73} +0.708204i q^{75} +7.74597i q^{79} +7.70820 q^{81} +13.3510 q^{83} +(5.85410 - 9.06914i) q^{85} +2.95870 q^{87} -10.3923 q^{89} +0.708204 q^{93} -14.6742i q^{95} -7.85410i q^{97} -9.88690i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{9} + 8 q^{15} + 12 q^{25} + 20 q^{43} + 20 q^{51} + 12 q^{53} + 20 q^{67} + 8 q^{81} + 20 q^{85} - 48 q^{93}+O(q^{100}) \) 8 * q - 4 * q^9 + 8 * q^15 + 12 * q^25 + 20 * q^43 + 20 * q^51 + 12 * q^53 + 20 * q^67 + 8 * q^81 + 20 * q^85 - 48 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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