LMFDB - Embedded newform 3332.2.b.c.2549.2

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			2549.2
Root			$1.40126 - 0.809017i$ of defining polynomial
Character	$\chi$	$=$	3332.2549
Dual form			3332.2.b.c.2549.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61803i q^{3} +0.381966i q^{5} -3.85410 q^{9} +O(q^{10})$	$q-2.61803i q^{3} +0.381966i q^{5} -3.85410 q^{9} +3.46410i q^{11} -5.60503 q^{13} +1.00000 q^{15} +(3.46410 + 2.23607i) q^{17} -2.14093 q^{19} +2.14093i q^{23} +4.85410 q^{25} +2.23607i q^{27} +7.74597i q^{29} -4.85410i q^{31} +9.06914 q^{33} -9.06914i q^{37} +14.6742i q^{39} +2.61803i q^{41} +5.85410 q^{43} -1.47214i q^{45} +3.46410 q^{47} +(5.85410 - 9.06914i) q^{51} +4.85410 q^{53} -1.32317 q^{55} +5.60503i q^{57} +6.92820 q^{59} -11.5623i q^{61} -2.14093i q^{65} +12.5623 q^{67} +5.60503 q^{69} +9.06914i q^{71} +4.14590i q^{73} -12.7082i q^{75} +7.74597i q^{79} -5.70820 q^{81} +9.88690 q^{83} +(-0.854102 + 1.32317i) q^{85} +20.2792 q^{87} +10.3923 q^{89} -12.7082 q^{93} -0.817763i q^{95} -1.14590i q^{97} -13.3510i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{9}+O(q^{10}) $ 8 * q - 4 * q^9	$ 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

Display 100 coefficients 10 coefficients

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
$3$		−	2.61803i		−	1.51152i	−0.654847	−	0.755761i	$-0.727267\pi$
$3$		−	2.61803i		−	1.51152i	0.654847	−	0.755761i	$-0.272733\pi$
$4$		0			0
$5$			0.381966i			0.170820i	0.996346	+	0.0854102i	$0.0272201\pi$
$5$			0.381966i			0.170820i	−0.996346	+	0.0854102i	$0.972780\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−3.85410			−1.28470
$10$		0			0
$11$			3.46410i			1.04447i	0.852803	+	0.522233i	$0.174901\pi$
$11$			3.46410i			1.04447i	−0.852803	+	0.522233i	$0.825099\pi$
$12$		0			0
$13$	−5.60503			−1.55456			−0.777278	−	0.629157i	$-0.783400\pi$
$13$	−5.60503			−1.55456			−0.777278	+	0.629157i	$0.783400\pi$
$14$		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
$19$	−2.14093			−0.491164			−0.245582	−	0.969376i	$-0.578979\pi$
$19$	−2.14093			−0.491164			−0.245582	+	0.969376i	$0.578979\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			2.14093i			0.446415i	0.974771	+	0.223208i	$0.0716528\pi$
$23$			2.14093i			0.446415i	−0.974771	+	0.223208i	$0.928347\pi$
$24$		0			0
$25$	4.85410			0.970820
$26$		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
$31$		−	4.85410i		−	0.871822i	−0.899990	−	0.435911i	$-0.856426\pi$
$31$		−	4.85410i		−	0.871822i	0.899990	−	0.435911i	$-0.143574\pi$
$32$		0			0
$33$	9.06914			1.57873
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	9.06914i		−	1.49096i	−0.666530	−	0.745478i	$-0.732221\pi$
$37$		−	9.06914i		−	1.49096i	0.666530	−	0.745478i	$-0.267779\pi$
$38$		0			0
$39$			14.6742i			2.34975i
$40$		0			0
$41$			2.61803i			0.408868i	0.978880	+	0.204434i	$0.0655354\pi$
$41$			2.61803i			0.408868i	−0.978880	+	0.204434i	$0.934465\pi$
$42$		0			0
$43$	5.85410			0.892742			0.446371	−	0.894848i	$-0.352716\pi$
$43$	5.85410			0.892742			0.446371	+	0.894848i	$0.352716\pi$
$44$		0			0
$45$		−	1.47214i		−	0.219453i
$46$		0			0
$47$	3.46410			0.505291			0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410			0.505291			0.252646	+	0.967559i	$0.418699\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	5.85410	−	9.06914i	0.819738	−	1.26993i
$52$		0			0
$53$	4.85410			0.666762			0.333381	−	0.942792i	$-0.391810\pi$
$53$	4.85410			0.666762			0.333381	+	0.942792i	$0.391810\pi$
$54$		0			0
$55$	−1.32317			−0.178416
$56$		0			0
$57$			5.60503i			0.742405i
$58$		0			0
$59$	6.92820			0.901975			0.450988	−	0.892530i	$-0.351072\pi$
$59$	6.92820			0.901975			0.450988	+	0.892530i	$0.351072\pi$
$60$		0			0
$61$		−	11.5623i		−	1.48040i	−0.672386	−	0.740201i	$-0.734730\pi$
$61$		−	11.5623i		−	1.48040i	0.672386	−	0.740201i	$-0.265270\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	2.14093i		−	0.265550i
$66$		0			0
$67$	12.5623			1.53473			0.767365	−	0.641211i	$-0.221567\pi$
$67$	12.5623			1.53473			0.767365	+	0.641211i	$0.221567\pi$
$68$		0			0
$69$	5.60503			0.674767
$70$		0			0
$71$			9.06914i			1.07631i	0.842846	+	0.538154i	$0.180878\pi$
$71$			9.06914i			1.07631i	−0.842846	+	0.538154i	$0.819122\pi$
$72$		0			0
$73$			4.14590i			0.485241i	0.970121	+	0.242620i	$0.0780069\pi$
$73$			4.14590i			0.485241i	−0.970121	+	0.242620i	$0.921993\pi$
$74$		0			0
$75$		−	12.7082i		−	1.46742i
$76$		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
$81$	−5.70820			−0.634245
$82$		0			0
$83$	9.88690			1.08523			0.542614	−	0.839982i	$-0.317435\pi$
$83$	9.88690			1.08523			0.542614	+	0.839982i	$0.317435\pi$
$84$		0			0
$85$	−0.854102	+	1.32317i	−0.0926404	+	0.143518i
$86$		0			0
$87$	20.2792			2.17416
$88$		0			0
$89$	10.3923			1.10158			0.550791	−	0.834643i	$-0.314326\pi$
$89$	10.3923			1.10158			0.550791	+	0.834643i	$0.314326\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−12.7082			−1.31778
$94$		0			0
$95$		−	0.817763i		−	0.0839008i
$96$		0			0
$97$		−	1.14590i		−	0.116348i	−0.998306	−	0.0581742i	$-0.981472\pi$
$97$		−	1.14590i		−	0.116348i	0.998306	−	0.0581742i	$-0.0185279\pi$
$98$		0			0
$99$		−	13.3510i		−	1.34183i
$100$		0			0
$101$	−6.42280			−0.639092			−0.319546	−	0.947571i	$-0.603530\pi$
$101$	−6.42280			−0.639092			−0.319546	+	0.947571i	$0.603530\pi$
$102$		0			0
$103$	12.5332			1.23494			0.617468	−	0.786596i	$-0.288158\pi$
$103$	12.5332			1.23494			0.617468	+	0.786596i	$0.288158\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			8.25137i			0.797690i	0.917018	+	0.398845i	$0.130589\pi$
$107$			8.25137i			0.797690i	−0.917018	+	0.398845i	$0.869411\pi$
$108$		0			0
$109$		−	2.64634i		−	0.253473i	−0.991936	−	0.126737i	$-0.959550\pi$
$109$		−	2.64634i		−	0.253473i	0.991936	−	0.126737i	$-0.0404503\pi$
$110$		0			0
$111$	−23.7433			−2.25361
$112$		0			0
$113$			3.46410i			0.325875i	0.986636	+	0.162938i	$0.0520969\pi$
$113$			3.46410i			0.325875i	−0.986636	+	0.162938i	$0.947903\pi$
$114$		0			0
$115$	−0.817763			−0.0762568
$116$		0			0
$117$	21.6024			1.99714
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$	6.85410			0.618014
$124$		0			0
$125$			3.76393i			0.336656i
$126$		0			0
$127$	4.56231			0.404839			0.202420	−	0.979299i	$-0.435120\pi$
$127$	4.56231			0.404839			0.202420	+	0.979299i	$0.435120\pi$
$128$		0			0
$129$		−	15.3262i		−	1.34940i
$130$		0			0
$131$			1.52786i			0.133490i	0.997770	+	0.0667451i	$0.0212614\pi$
$131$			1.52786i			0.133490i	−0.997770	+	0.0667451i	$0.978739\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−0.854102			−0.0735094
$136$		0			0
$137$	−21.2705			−1.81726			−0.908631	−	0.417600i	$-0.862871\pi$
$137$	−21.2705			−1.81726			−0.908631	+	0.417600i	$0.862871\pi$
$138$		0			0
$139$			2.56231i			0.217332i	0.994078	+	0.108666i	$0.0346579\pi$
$139$			2.56231i			0.217332i	−0.994078	+	0.108666i	$0.965342\pi$
$140$		0			0
$141$		−	9.06914i		−	0.763759i
$142$		0			0
$143$		−	19.4164i		−	1.62368i
$144$		0			0
$145$	−2.95870			−0.245706
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.56231			0.455682			0.227841	−	0.973698i	$-0.426833\pi$
$149$	5.56231			0.455682			0.227841	+	0.973698i	$0.426833\pi$
$150$		0			0
$151$	−13.5623			−1.10368			−0.551842	−	0.833948i	$-0.686075\pi$
$151$	−13.5623			−1.10368			−0.551842	+	0.833948i	$0.686075\pi$
$152$		0			0
$153$	−13.3510	−	8.61803i	−1.07936	−	0.696727i
$154$		0			0
$155$	1.85410			0.148925
$156$		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
$159$		−	12.7082i		−	1.00783i
$160$		0			0
$161$		0			0
$162$		0			0
$163$			19.4614i			1.52434i	0.647378	+	0.762169i	$0.275865\pi$
$163$			19.4614i			1.52434i	−0.647378	+	0.762169i	$0.724135\pi$
$164$		0			0
$165$			3.46410i			0.269680i
$166$		0			0
$167$			16.7984i			1.29990i	0.759978	+	0.649949i	$0.225210\pi$
$167$			16.7984i			1.29990i	−0.759978	+	0.649949i	$0.774790\pi$
$168$		0			0
$169$	18.4164			1.41665
$170$		0			0
$171$	8.25137			0.630998
$172$		0			0
$173$			5.67376i			0.431368i	0.976463	+	0.215684i	$0.0691981\pi$
$173$			5.67376i			0.431368i	−0.976463	+	0.215684i	$0.930802\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	18.1383i		−	1.36336i
$178$		0			0
$179$	5.56231			0.415746			0.207873	−	0.978156i	$-0.433346\pi$
$179$	5.56231			0.415746			0.207873	+	0.978156i	$0.433346\pi$
$180$		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
$183$	−30.2705			−2.23766
$184$		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
$191$	8.56231			0.619547			0.309773	−	0.950810i	$-0.399747\pi$
$191$	8.56231			0.619547			0.309773	+	0.950810i	$0.399747\pi$
$192$		0			0
$193$		−	19.4614i		−	1.40087i	−0.713719	−	0.700433i	$-0.752990\pi$
$193$		−	19.4614i		−	1.40087i	0.713719	−	0.700433i	$-0.247010\pi$
$194$		0			0
$195$	−5.60503			−0.401385
$196$		0			0
$197$		−	12.5332i		−	0.892956i	−0.894795	−	0.446478i	$-0.852678\pi$
$197$		−	12.5332i		−	0.892956i	0.894795	−	0.446478i	$-0.147322\pi$
$198$		0			0
$199$		−	4.14590i		−	0.293895i	−0.989144	−	0.146947i	$-0.953055\pi$
$199$		−	4.14590i		−	0.293895i	0.989144	−	0.146947i	$-0.0469448\pi$
$200$		0			0
$201$		−	32.8885i		−	2.31978i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1.00000			−0.0698430
$206$		0			0
$207$		−	8.25137i		−	0.573510i
$208$		0			0
$209$		−	7.41641i		−	0.513004i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	23.7433			1.62686
$214$		0			0
$215$			2.23607i			0.152499i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	10.8541			0.733452
$220$		0			0
$221$	−19.4164	−	12.5332i	−1.30609	−	0.843077i
$222$		0			0
$223$	−22.9255			−1.53521			−0.767604	−	0.640924i	$-0.778551\pi$
$223$	−22.9255			−1.53521			−0.767604	+	0.640924i	$0.778551\pi$
$224$		0			0
$225$	−18.7082			−1.24721
$226$		0			0
$227$		−	11.5066i		−	0.763718i	−0.924221	−	0.381859i	$-0.875284\pi$
$227$		−	11.5066i		−	0.763718i	0.924221	−	0.381859i	$-0.124716\pi$
$228$		0			0
$229$	17.3205			1.14457			0.572286	−	0.820054i	$-0.306057\pi$
$229$	17.3205			1.14457			0.572286	+	0.820054i	$0.306057\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			22.9255i			1.50190i	0.660358	+	0.750951i	$0.270405\pi$
$233$			22.9255i			1.50190i	−0.660358	+	0.750951i	$0.729595\pi$
$234$		0			0
$235$			1.32317i			0.0863140i
$236$		0			0
$237$	20.2792			1.31728
$238$		0			0
$239$	−21.2705			−1.37587			−0.687937	−	0.725770i	$-0.741484\pi$
$239$	−21.2705			−1.37587			−0.687937	+	0.725770i	$0.741484\pi$
$240$		0			0
$241$			18.2705i			1.17691i	0.808531	+	0.588453i	$0.200263\pi$
$241$			18.2705i			1.17691i	−0.808531	+	0.588453i	$0.799737\pi$
$242$		0			0
$243$			21.6525i			1.38901i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	12.0000			0.763542
$248$		0			0
$249$		−	25.8842i		−	1.64035i
$250$		0			0
$251$	−20.2792			−1.28001			−0.640006	−	0.768370i	$-0.721068\pi$
$251$	−20.2792			−1.28001			−0.640006	+	0.768370i	$0.721068\pi$
$252$		0			0
$253$	−7.41641			−0.466266
$254$		0			0
$255$	3.46410	+	2.23607i	0.216930	+	0.140028i
$256$		0			0
$257$	−6.42280			−0.400643			−0.200322	−	0.979730i	$-0.564199\pi$
$257$	−6.42280			−0.400643			−0.200322	+	0.979730i	$0.564199\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		−	29.8537i		−	1.84790i
$262$		0			0
$263$	19.4164			1.19727			0.598633	−	0.801023i	$-0.295711\pi$
$263$	19.4164			1.19727			0.598633	+	0.801023i	$0.295711\pi$
$264$		0			0
$265$			1.85410i			0.113897i
$266$		0			0
$267$		−	27.2074i		−	1.66507i
$268$		0			0
$269$		−	26.9443i		−	1.64282i	−0.570337	−	0.821411i	$-0.693187\pi$
$269$		−	26.9443i		−	1.64282i	0.570337	−	0.821411i	$-0.306813\pi$
$270$		0			0
$271$	24.5611			1.49198			0.745989	−	0.665958i	$-0.231977\pi$
$271$	24.5611			1.49198			0.745989	+	0.665958i	$0.231977\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			16.8151i			1.01399i
$276$		0			0
$277$		−	24.5611i		−	1.47573i	−0.674947	−	0.737866i	$-0.735834\pi$
$277$		−	24.5611i		−	1.47573i	0.674947	−	0.737866i	$-0.264166\pi$
$278$		0			0
$279$			18.7082i			1.12003i
$280$		0			0
$281$	26.5623			1.58457			0.792287	−	0.610148i	$-0.208890\pi$
$281$	26.5623			1.58457			0.792287	+	0.610148i	$0.208890\pi$
$282$		0			0
$283$			19.1459i			1.13811i	0.822301	+	0.569053i	$0.192690\pi$
$283$			19.1459i			1.13811i	−0.822301	+	0.569053i	$0.807310\pi$
$284$		0			0
$285$	−2.14093			−0.126818
$286$		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
$293$	20.2792			1.18472			0.592362	−	0.805672i	$-0.298196\pi$
$293$	20.2792			1.18472			0.592362	+	0.805672i	$0.298196\pi$
$294$		0			0
$295$			2.64634i			0.154076i
$296$		0			0
$297$	−7.74597			−0.449467
$298$		0			0
$299$		−	12.0000i		−	0.693978i
$300$		0			0
$301$		0			0
$302$		0			0
$303$			16.8151i			0.966002i
$304$		0			0
$305$	4.41641			0.252883
$306$		0			0
$307$	2.95870			0.168862			0.0844308	−	0.996429i	$-0.473093\pi$
$307$	2.95870			0.168862			0.0844308	+	0.996429i	$0.473093\pi$
$308$		0			0
$309$		−	32.8124i		−	1.86663i
$310$		0			0
$311$			30.3820i			1.72280i	0.507924	+	0.861402i	$0.330413\pi$
$311$			30.3820i			1.72280i	−0.507924	+	0.861402i	$0.669587\pi$
$312$		0			0
$313$		−	30.2705i		−	1.71099i	−0.517811	−	0.855495i	$-0.673253\pi$
$313$		−	30.2705i		−	1.71099i	0.517811	−	0.855495i	$-0.326747\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	24.5611i		−	1.37949i	−0.724054	−	0.689744i	$-0.757723\pi$
$317$		−	24.5611i		−	1.37949i	0.724054	−	0.689744i	$-0.242277\pi$
$318$		0			0
$319$	−26.8328			−1.50235
$320$		0			0
$321$	21.6024			1.20573
$322$		0			0
$323$	−7.41641	−	4.78727i	−0.412660	−	0.266371i
$324$		0			0
$325$	−27.2074			−1.50920
$326$		0			0
$327$	−6.92820			−0.383131
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−14.8541			−0.816455			−0.408228	−	0.912880i	$-0.633853\pi$
$331$	−14.8541			−0.816455			−0.408228	+	0.912880i	$0.633853\pi$
$332$		0			0
$333$			34.9534i			1.91543i
$334$		0			0
$335$			4.79837i			0.262163i
$336$		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
$339$	9.06914			0.492568
$340$		0			0
$341$	16.8151			0.910589
$342$		0			0
$343$		0			0
$344$		0			0
$345$			2.14093i			0.115264i
$346$		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
$349$	12.5332			0.670889			0.335444	−	0.942060i	$-0.391114\pi$
$349$	12.5332			0.670889			0.335444	+	0.942060i	$0.391114\pi$
$350$		0			0
$351$		−	12.5332i		−	0.668975i
$352$		0			0
$353$	30.1661			1.60558			0.802790	−	0.596262i	$-0.203348\pi$
$353$	30.1661			1.60558			0.802790	+	0.596262i	$0.203348\pi$
$354$		0			0
$355$	−3.46410			−0.183855
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.14590			0.0604782			0.0302391	−	0.999543i	$-0.490373\pi$
$359$	1.14590			0.0604782			0.0302391	+	0.999543i	$0.490373\pi$
$360$		0			0
$361$	−14.4164			−0.758758
$362$		0			0
$363$			2.61803i			0.137411i
$364$		0			0
$365$	−1.58359			−0.0828890
$366$		0			0
$367$		−	12.4377i		−	0.649242i	−0.945844	−	0.324621i	$-0.894763\pi$
$367$		−	12.4377i		−	0.649242i	0.945844	−	0.324621i	$-0.105237\pi$
$368$		0			0
$369$		−	10.0902i		−	0.525273i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−1.56231			−0.0808931			−0.0404466	−	0.999182i	$-0.512878\pi$
$373$	−1.56231			−0.0808931			−0.0404466	+	0.999182i	$0.512878\pi$
$374$		0			0
$375$	9.85410			0.508864
$376$		0			0
$377$		−	43.4164i		−	2.23606i
$378$		0			0
$379$			5.09963i			0.261950i	0.991386	+	0.130975i	$0.0418108\pi$
$379$			5.09963i			0.261950i	−0.991386	+	0.130975i	$0.958189\pi$
$380$		0			0
$381$		−	11.9443i		−	0.611924i
$382$		0			0
$383$	−17.3205			−0.885037			−0.442518	−	0.896759i	$-0.645915\pi$
$383$	−17.3205			−0.885037			−0.442518	+	0.896759i	$0.645915\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−22.5623			−1.14691
$388$		0			0
$389$	−7.14590			−0.362311			−0.181156	−	0.983454i	$-0.557984\pi$
$389$	−7.14590			−0.362311			−0.181156	+	0.983454i	$0.557984\pi$
$390$		0			0
$391$	−4.78727	+	7.41641i	−0.242103	+	0.375064i
$392$		0			0
$393$	4.00000			0.201773
$394$		0			0
$395$	−2.95870			−0.148868
$396$		0			0
$397$		−	10.8541i		−	0.544752i	−0.962191	−	0.272376i	$-0.912191\pi$
$397$		−	10.8541i		−	0.544752i	0.962191	−	0.272376i	$-0.0878094\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			4.78727i			0.239065i	0.992830	+	0.119532i	$0.0381396\pi$
$401$			4.78727i			0.239065i	−0.992830	+	0.119532i	$0.961860\pi$
$402$		0			0
$403$			27.2074i			1.35530i
$404$		0			0
$405$		−	2.18034i		−	0.108342i
$406$		0			0
$407$	31.4164			1.55725
$408$		0			0
$409$	−31.9947			−1.58204			−0.791018	−	0.611794i	$-0.790448\pi$
$409$	−31.9947			−1.58204			−0.791018	+	0.611794i	$0.790448\pi$
$410$		0			0
$411$			55.6869i			2.74683i
$412$		0			0
$413$		0			0
$414$		0			0
$415$			3.77646i			0.185379i
$416$		0			0
$417$	6.70820			0.328502
$418$		0			0
$419$		−	21.2148i		−	1.03641i	−0.855256	−	0.518205i	$-0.826600\pi$
$419$		−	21.2148i		−	1.03641i	0.855256	−	0.518205i	$-0.173400\pi$
$420$		0			0
$421$	−21.5623			−1.05088			−0.525441	−	0.850830i	$-0.676100\pi$
$421$	−21.5623			−1.05088			−0.525441	+	0.850830i	$0.676100\pi$
$422$		0			0
$423$	−13.3510			−0.649148
$424$		0			0
$425$	16.8151	+	10.8541i	0.815652	+	0.526501i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−50.8328			−2.45423
$430$		0			0
$431$		−	1.32317i		−	0.0637348i	−0.999492	−	0.0318674i	$-0.989855\pi$
$431$		−	1.32317i		−	0.0637348i	0.999492	−	0.0318674i	$-0.0101454\pi$
$432$		0			0
$433$	−14.6742			−0.705196			−0.352598	−	0.935775i	$-0.614702\pi$
$433$	−14.6742			−0.705196			−0.352598	+	0.935775i	$0.614702\pi$
$434$		0			0
$435$			7.74597i			0.371391i
$436$		0			0
$437$		−	4.58359i		−	0.219263i
$438$		0			0
$439$		−	39.2705i		−	1.87428i	−0.348955	−	0.937140i	$-0.613463\pi$
$439$		−	39.2705i		−	1.87428i	0.348955	−	0.937140i	$-0.386537\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	26.8328			1.27487			0.637433	−	0.770506i	$-0.279996\pi$
$443$	26.8328			1.27487			0.637433	+	0.770506i	$0.279996\pi$
$444$		0			0
$445$			3.96951i			0.188173i
$446$		0			0
$447$		−	14.5623i		−	0.688773i
$448$		0			0
$449$			37.5997i			1.77444i	0.461346	+	0.887220i	$0.347367\pi$
$449$			37.5997i			1.77444i	−0.461346	+	0.887220i	$0.652633\pi$
$450$		0			0
$451$	−9.06914			−0.427049
$452$		0			0
$453$			35.5066i			1.66824i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	30.5623			1.42964			0.714822	−	0.699306i	$-0.246508\pi$
$457$	30.5623			1.42964			0.714822	+	0.699306i	$0.246508\pi$
$458$		0			0
$459$	−5.00000	+	7.74597i	−0.233380	+	0.361551i
$460$		0			0
$461$	−9.38149			−0.436940			−0.218470	−	0.975844i	$-0.570107\pi$
$461$	−9.38149			−0.436940			−0.218470	+	0.975844i	$0.570107\pi$
$462$		0			0
$463$	−2.43769			−0.113289			−0.0566446	−	0.998394i	$-0.518040\pi$
$463$	−2.43769			−0.113289			−0.0566446	+	0.998394i	$0.518040\pi$
$464$		0			0
$465$		−	4.85410i		−	0.225104i
$466$		0			0
$467$	−31.1769			−1.44270			−0.721348	−	0.692573i	$-0.756477\pi$
$467$	−31.1769			−1.44270			−0.721348	+	0.692573i	$0.756477\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		−	20.2792i		−	0.934416i
$472$		0			0
$473$			20.2792i			0.932439i
$474$		0			0
$475$	−10.3923			−0.476832
$476$		0			0
$477$	−18.7082			−0.856590
$478$		0			0
$479$			17.4508i			0.797350i	0.917092	+	0.398675i	$0.130530\pi$
$479$			17.4508i			0.797350i	−0.917092	+	0.398675i	$0.869470\pi$
$480$		0			0
$481$			50.8328i			2.31778i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.437694			0.0198747
$486$		0			0
$487$			29.6607i			1.34405i	0.740526	+	0.672027i	$0.234576\pi$
$487$			29.6607i			1.34405i	−0.740526	+	0.672027i	$0.765424\pi$
$488$		0			0
$489$	50.9507			2.30407
$490$		0			0
$491$	34.8541			1.57294			0.786472	−	0.617626i	$-0.211906\pi$
$491$	34.8541			1.57294			0.786472	+	0.617626i	$0.211906\pi$
$492$		0			0
$493$	−17.3205	+	26.8328i	−0.780076	+	1.20849i
$494$		0			0
$495$	5.09963			0.229211
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	11.5224i		−	0.515815i	−0.966170	−	0.257907i	$-0.916967\pi$
$499$		−	11.5224i		−	0.515815i	0.966170	−	0.257907i	$-0.0830329\pi$
$500$		0			0
$501$	43.9787			1.96482
$502$		0			0
$503$			33.2148i			1.48097i	0.672071	+	0.740487i	$0.265405\pi$
$503$			33.2148i			1.48097i	−0.672071	+	0.740487i	$0.734595\pi$
$504$		0			0
$505$		−	2.45329i		−	0.109170i
$506$		0			0
$507$		−	48.2148i		−	2.14129i
$508$		0			0
$509$	−36.5889			−1.62177			−0.810887	−	0.585202i	$-0.801015\pi$
$509$	−36.5889			−1.62177			−0.810887	+	0.585202i	$0.801015\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		−	4.78727i		−	0.211363i
$514$		0			0
$515$			4.78727i			0.210952i
$516$		0			0
$517$			12.0000i			0.527759i
$518$		0			0
$519$	14.8541			0.652023
$520$		0			0
$521$			21.3820i			0.936761i	0.883527	+	0.468380i	$0.155162\pi$
$521$			21.3820i			0.936761i	−0.883527	+	0.468380i	$0.844838\pi$
$522$		0			0
$523$	−27.2074			−1.18970			−0.594848	−	0.803838i	$-0.702788\pi$
$523$	−27.2074			−1.18970			−0.594848	+	0.803838i	$0.702788\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	10.8541	−	16.8151i	0.472812	−	0.732477i
$528$		0			0
$529$	18.4164			0.800713
$530$		0			0
$531$	−26.7020			−1.15877
$532$		0			0
$533$		−	14.6742i		−	0.635609i
$534$		0			0
$535$	−3.15174			−0.136262
$536$		0			0
$537$		−	14.5623i		−	0.628410i
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	9.06914i		−	0.389913i	−0.980812	−	0.194956i	$-0.937543\pi$
$541$		−	9.06914i		−	0.389913i	0.980812	−	0.194956i	$-0.0624565\pi$
$542$		0			0
$543$	15.7082			0.674104
$544$		0			0
$545$	1.01081			0.0432984
$546$		0			0
$547$			46.6688i			1.99542i	0.0676676	+	0.997708i	$0.478444\pi$
$547$			46.6688i			1.99542i	−0.0676676	+	0.997708i	$0.521556\pi$
$548$		0			0
$549$			44.5623i			1.90187i
$550$		0			0
$551$		−	16.5836i		−	0.706485i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		−	9.06914i		−	0.384963i
$556$		0			0
$557$	25.4164			1.07693			0.538464	−	0.842649i	$-0.319005\pi$
$557$	25.4164			1.07693			0.538464	+	0.842649i	$0.319005\pi$
$558$		0			0
$559$	−32.8124			−1.38782
$560$		0			0
$561$	31.4164	+	20.2792i	1.32640	+	0.856189i
$562$		0			0
$563$	10.3923			0.437983			0.218992	−	0.975727i	$-0.429723\pi$
$563$	10.3923			0.437983			0.218992	+	0.975727i	$0.429723\pi$
$564$		0			0
$565$	−1.32317			−0.0556661
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.72949			−0.114426			−0.0572131	−	0.998362i	$-0.518221\pi$
$569$	−2.72949			−0.114426			−0.0572131	+	0.998362i	$0.518221\pi$
$570$		0			0
$571$		−	9.06914i		−	0.379532i	−0.981829	−	0.189766i	$-0.939227\pi$
$571$		−	9.06914i		−	0.379532i	0.981829	−	0.189766i	$-0.0607729\pi$
$572$		0			0
$573$		−	22.4164i		−	0.936459i
$574$		0			0
$575$			10.3923i			0.433389i
$576$		0			0
$577$	−6.61585			−0.275421			−0.137711	−	0.990473i	$-0.543974\pi$
$577$	−6.61585			−0.275421			−0.137711	+	0.990473i	$0.543974\pi$
$578$		0			0
$579$	−50.9507			−2.11744
$580$		0			0
$581$		0			0
$582$		0			0
$583$			16.8151i			0.696410i
$584$		0			0
$585$			8.25137i			0.341152i
$586$		0			0
$587$	10.3923			0.428936			0.214468	−	0.976731i	$-0.431198\pi$
$587$	10.3923			0.428936			0.214468	+	0.976731i	$0.431198\pi$
$588$		0			0
$589$			10.3923i			0.428207i
$590$		0			0
$591$	−32.8124			−1.34972
$592$		0			0
$593$	−7.43361			−0.305262			−0.152631	−	0.988283i	$-0.548775\pi$
$593$	−7.43361			−0.305262			−0.152631	+	0.988283i	$0.548775\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−10.8541			−0.444229
$598$		0			0
$599$	25.8541			1.05637			0.528185	−	0.849129i	$-0.322873\pi$
$599$	25.8541			1.05637			0.528185	+	0.849129i	$0.322873\pi$
$600$		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
$603$	−48.4164			−1.97167
$604$		0			0
$605$		−	0.381966i		−	0.0155291i
$606$		0			0
$607$			28.8541i			1.17115i	0.810618	+	0.585576i	$0.199132\pi$
$607$			28.8541i			1.17115i	−0.810618	+	0.585576i	$0.800868\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−19.4164			−0.785504
$612$		0			0
$613$	3.14590			0.127062			0.0635308	−	0.997980i	$-0.479764\pi$
$613$	3.14590			0.127062			0.0635308	+	0.997980i	$0.479764\pi$
$614$		0			0
$615$			2.61803i			0.105569i
$616$		0			0
$617$			13.0386i			0.524916i	0.964943	+	0.262458i	$0.0845331\pi$
$617$			13.0386i			0.524916i	−0.964943	+	0.262458i	$0.915467\pi$
$618$		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
$621$	−4.78727			−0.192107
$622$		0			0
$623$		0			0
$624$		0			0
$625$	22.8328			0.913313
$626$		0			0
$627$	−19.4164			−0.775417
$628$		0			0
$629$	20.2792	−	31.4164i	0.808585	−	1.25265i
$630$		0			0
$631$	26.2705			1.04581			0.522906	−	0.852390i	$-0.324848\pi$
$631$	26.2705			1.04581			0.522906	+	0.852390i	$0.324848\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			1.74265i			0.0691548i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		−	34.9534i		−	1.38273i
$640$		0			0
$641$		−	28.0252i		−	1.10693i	−0.832873	−	0.553464i	$-0.813306\pi$
$641$		−	28.0252i		−	1.10693i	0.832873	−	0.553464i	$-0.186694\pi$
$642$		0			0
$643$			30.9787i			1.22168i	0.791754	+	0.610841i	$0.209168\pi$
$643$			30.9787i			1.22168i	−0.791754	+	0.610841i	$0.790832\pi$
$644$		0			0
$645$	5.85410			0.230505
$646$		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	$0.182425\pi$
$653$			27.7128i			1.08449i	−0.840222	+	0.542243i	$0.817575\pi$
$654$		0			0
$655$	−0.583592			−0.0228028
$656$		0			0
$657$		−	15.9787i		−	0.623389i
$658$		0			0
$659$	15.9787			0.622442			0.311221	−	0.950338i	$-0.399262\pi$
$659$	15.9787			0.622442			0.311221	+	0.950338i	$0.399262\pi$
$660$		0			0
$661$	−18.6437			−0.725155			−0.362577	−	0.931954i	$-0.618103\pi$
$661$	−18.6437			−0.725155			−0.362577	+	0.931954i	$0.618103\pi$
$662$		0			0
$663$	−32.8124	+	50.8328i	−1.27433	+	1.97418i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−16.5836			−0.642119
$668$		0			0
$669$			60.0198i			2.32050i
$670$		0			0
$671$	40.0530			1.54623
$672$		0			0
$673$			10.3923i			0.400594i	0.979735	+	0.200297i	$0.0641907\pi$
$673$			10.3923i			0.400594i	−0.979735	+	0.200297i	$0.935809\pi$
$674$		0			0
$675$			10.8541i			0.417775i
$676$		0			0
$677$		−	38.9443i		−	1.49675i	−0.663276	−	0.748375i	$-0.730834\pi$
$677$		−	38.9443i		−	1.49675i	0.663276	−	0.748375i	$-0.269166\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−30.1246			−1.15438
$682$		0			0
$683$			2.95870i			0.113211i	0.998397	+	0.0566057i	$0.0180278\pi$
$683$			2.95870i			0.113211i	−0.998397	+	0.0566057i	$0.981972\pi$
$684$		0			0
$685$		−	8.12461i		−	0.310425i
$686$		0			0
$687$		−	45.3457i		−	1.73005i
$688$		0			0
$689$	−27.2074			−1.03652
$690$		0			0
$691$			6.97871i			0.265483i	0.991151	+	0.132741i	$0.0423780\pi$
$691$			6.97871i			0.265483i	−0.991151	+	0.132741i	$0.957622\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−0.978714			−0.0371247
$696$		0			0
$697$	−5.85410	+	9.06914i	−0.221740	+	0.343518i
$698$		0			0
$699$	60.0198			2.27016
$700$		0			0
$701$	−22.5836			−0.852971			−0.426485	−	0.904495i	$-0.640248\pi$
$701$	−22.5836			−0.852971			−0.426485	+	0.904495i	$0.640248\pi$
$702$		0			0
$703$			19.4164i			0.732304i
$704$		0			0
$705$	3.46410			0.130466
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	19.4614i		−	0.730890i	−0.930833	−	0.365445i	$-0.880917\pi$
$709$		−	19.4614i		−	0.730890i	0.930833	−	0.365445i	$-0.119083\pi$
$710$		0			0
$711$		−	29.8537i		−	1.11960i
$712$		0			0
$713$	10.3923			0.389195
$714$		0			0
$715$	7.41641			0.277358
$716$		0			0
$717$			55.6869i			2.07967i
$718$		0			0
$719$			36.2148i			1.35058i	0.737551	+	0.675292i	$0.235982\pi$
$719$			36.2148i			1.35058i	−0.737551	+	0.675292i	$0.764018\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	47.8328			1.77892
$724$		0			0
$725$			37.5997i			1.39642i
$726$		0			0
$727$	−17.8259			−0.661127			−0.330563	−	0.943784i	$-0.607239\pi$
$727$	−17.8259			−0.661127			−0.330563	+	0.943784i	$0.607239\pi$
$728$		0			0
$729$	39.5623			1.46527
$730$		0			0
$731$	20.2792	+	13.0902i	0.750053	+	0.484157i
$732$		0			0
$733$	16.3097			0.602412			0.301206	−	0.953559i	$-0.402611\pi$
$733$	16.3097			0.602412			0.301206	+	0.953559i	$0.402611\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			43.5171i			1.60297i
$738$		0			0
$739$	−5.27051			−0.193879			−0.0969394	−	0.995290i	$-0.530905\pi$
$739$	−5.27051			−0.193879			−0.0969394	+	0.995290i	$0.530905\pi$
$740$		0			0
$741$		−	31.4164i		−	1.15411i
$742$		0			0
$743$			33.6302i			1.23377i	0.787052	+	0.616886i	$0.211606\pi$
$743$			33.6302i			1.23377i	−0.787052	+	0.616886i	$0.788394\pi$
$744$		0			0
$745$			2.12461i			0.0778398i
$746$		0			0
$747$	−38.1051			−1.39419
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	40.0530i		−	1.46155i	−0.682616	−	0.730777i	$-0.739158\pi$
$751$		−	40.0530i		−	1.46155i	0.682616	−	0.730777i	$-0.260842\pi$
$752$		0			0
$753$			53.0916i			1.93477i
$754$		0			0
$755$		−	5.18034i		−	0.188532i
$756$		0			0
$757$	10.7295			0.389970			0.194985	−	0.980806i	$-0.437534\pi$
$757$	10.7295			0.389970			0.194985	+	0.980806i	$0.437534\pi$
$758$		0			0
$759$			19.4164i			0.704771i
$760$		0			0
$761$	19.7738			0.716800			0.358400	−	0.933568i	$-0.383322\pi$
$761$	19.7738			0.716800			0.358400	+	0.933568i	$0.383322\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	3.29180	−	5.09963i	0.119015	−	0.184377i
$766$		0			0
$767$	−38.8328			−1.40217
$768$		0			0
$769$	1.82857			0.0659401			0.0329700	−	0.999456i	$-0.489503\pi$
$769$	1.82857			0.0659401			0.0329700	+	0.999456i	$0.489503\pi$
$770$		0			0
$771$			16.8151i			0.605581i
$772$		0			0
$773$	−30.1661			−1.08500			−0.542500	−	0.840056i	$-0.682522\pi$
$773$	−30.1661			−1.08500			−0.542500	+	0.840056i	$0.682522\pi$
$774$		0			0
$775$		−	23.5623i		−	0.846383i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	5.60503i		−	0.200821i
$780$		0			0
$781$	−31.4164			−1.12417
$782$		0			0
$783$	−17.3205			−0.618984
$784$		0			0
$785$			2.95870i			0.105600i
$786$		0			0
$787$			43.4164i			1.54763i	0.633413	+	0.773814i	$0.281653\pi$
$787$			43.4164i			1.54763i	−0.633413	+	0.773814i	$0.718347\pi$
$788$		0			0
$789$		−	50.8328i		−	1.80970i
$790$		0			0
$791$		0			0
$792$		0			0
$793$			64.8071i			2.30137i
$794$		0			0
$795$	4.85410			0.172157
$796$		0			0
$797$	−29.6607			−1.05064			−0.525318	−	0.850906i	$-0.676054\pi$
$797$	−29.6607			−1.05064			−0.525318	+	0.850906i	$0.676054\pi$
$798$		0			0
$799$	12.0000	+	7.74597i	0.424529	+	0.274033i
$800$		0			0
$801$	−40.0530			&min

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)

\(f(q)\)	\(=\)	\(q-2.61803i q^{3} +0.381966i q^{5} -3.85410 q^{9} +O(q^{10})\)	\(q-2.61803i q^{3} +0.381966i q^{5} -3.85410 q^{9} +3.46410i q^{11} -5.60503 q^{13} +1.00000 q^{15} +(3.46410 + 2.23607i) q^{17} -2.14093 q^{19} +2.14093i q^{23} +4.85410 q^{25} +2.23607i q^{27} +7.74597i q^{29} -4.85410i q^{31} +9.06914 q^{33} -9.06914i q^{37} +14.6742i q^{39} +2.61803i q^{41} +5.85410 q^{43} -1.47214i q^{45} +3.46410 q^{47} +(5.85410 - 9.06914i) q^{51} +4.85410 q^{53} -1.32317 q^{55} +5.60503i q^{57} +6.92820 q^{59} -11.5623i q^{61} -2.14093i q^{65} +12.5623 q^{67} +5.60503 q^{69} +9.06914i q^{71} +4.14590i q^{73} -12.7082i q^{75} +7.74597i q^{79} -5.70820 q^{81} +9.88690 q^{83} +(-0.854102 + 1.32317i) q^{85} +20.2792 q^{87} +10.3923 q^{89} -12.7082 q^{93} -0.817763i q^{95} -1.14590i q^{97} -13.3510i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{9}+O(q^{10}) \) 8 * q - 4 * q^9	\( 8 q - 4 q^{9} + 8 q^{15} + 12 q^{25} + 20 q^{43} + 20 q^{51} + 12 q^{53} + 20 q^{67} + 8 q^{81} + 20 q^{85} - 48 q^{93}+O(q^{100}) \) 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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