LMFDB - Embedded newform 3960.2.d.d.3169.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3960,2,Mod(3169,3960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3960, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3960.3169");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3960.d (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:	$31.6207592004$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1320)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3169.4
Root			$1.45161 - 1.45161i$ of defining polynomial
Character	$\chi$	$=$	3960.3169
Dual form			3960.2.d.d.3169.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.311108 + 2.21432i) q^{5} -1.52543i q^{7} -1.00000 q^{11} +0.903212i q^{13} -2.90321i q^{17} +2.00000 q^{19} +8.85728i q^{23} +(-4.80642 + 1.37778i) q^{25} -6.62222 q^{29} -10.2351 q^{31} +(3.37778 - 0.474572i) q^{35} -8.56199i q^{37} -2.23506 q^{41} -12.5763i q^{43} +8.56199i q^{47} +4.67307 q^{49} +3.18421i q^{53} +(-0.311108 - 2.21432i) q^{55} +12.0415 q^{59} -13.6128 q^{61} +(-2.00000 + 0.280996i) q^{65} +14.3684i q^{67} -6.91750 q^{71} -9.76049i q^{73} +1.52543i q^{77} +8.75557 q^{79} +7.09679i q^{83} +(6.42864 - 0.903212i) q^{85} -14.9906 q^{89} +1.37778 q^{91} +(0.622216 + 4.42864i) q^{95} +7.61285i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{5} - 6 q^{11} + 12 q^{19} - 2 q^{25} - 40 q^{29} - 8 q^{31} + 20 q^{35} + 40 q^{41} + 2 q^{49} - 2 q^{55} - 8 q^{59} - 28 q^{61} - 12 q^{65} - 16 q^{71} + 52 q^{79} + 12 q^{85} - 36 q^{89} + 8 q^{91}+ \cdots + 4 q^{95}+O(q^{100}) $ 6 * q + 2 * q^5 - 6 * q^11 + 12 * q^19 - 2 * q^25 - 40 * q^29 - 8 * q^31 + 20 * q^35 + 40 * q^41 + 2 * q^49 - 2 * q^55 - 8 * q^59 - 28 * q^61 - 12 * q^65 - 16 * q^71 + 52 * q^79 + 12 * q^85 - 36 * q^89 + 8 * q^91 + 4 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3960\mathbb{Z}\right)^\times$.

$n$	$991$	$1981$	$2377$	$2521$	$3521$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.311108	+	2.21432i	0.139132	+	0.990274i
$5$	0.311108	+	2.21432i	0.139132	+	0.990274i
$6$		0			0
$7$		−	1.52543i		−	0.576557i	−0.957547	−	0.288279i	$-0.906917\pi$
$7$		−	1.52543i		−	0.576557i	0.957547	−	0.288279i	$-0.0930830\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$		0			0
$13$			0.903212i			0.250506i	0.992125	+	0.125253i	$0.0399742\pi$
$13$			0.903212i			0.250506i	−0.992125	+	0.125253i	$0.960026\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	2.90321i		−	0.704132i	−0.935975	−	0.352066i	$-0.885479\pi$
$17$		−	2.90321i		−	0.704132i	0.935975	−	0.352066i	$-0.114521\pi$
$18$		0			0
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			8.85728i			1.84687i	0.383754	+	0.923435i	$0.374631\pi$
$23$			8.85728i			1.84687i	−0.383754	+	0.923435i	$0.625369\pi$
$24$		0			0
$25$	−4.80642	+	1.37778i	−0.961285	+	0.275557i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.62222			−1.22971			−0.614857	−	0.788638i	$-0.710786\pi$
$29$	−6.62222			−1.22971			−0.614857	+	0.788638i	$0.710786\pi$
$30$		0			0
$31$	−10.2351			−1.83827			−0.919136	−	0.393941i	$-0.871112\pi$
$31$	−10.2351			−1.83827			−0.919136	+	0.393941i	$0.871112\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.37778	−	0.474572i	0.570950	−	0.0802174i
$36$		0			0
$37$		−	8.56199i		−	1.40758i	−0.710407	−	0.703791i	$-0.751489\pi$
$37$		−	8.56199i		−	1.40758i	0.710407	−	0.703791i	$-0.248511\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.23506			−0.349058			−0.174529	−	0.984652i	$-0.555840\pi$
$41$	−2.23506			−0.349058			−0.174529	+	0.984652i	$0.555840\pi$
$42$		0			0
$43$		−	12.5763i		−	1.91787i	−0.283634	−	0.958933i	$-0.591540\pi$
$43$		−	12.5763i		−	1.91787i	0.283634	−	0.958933i	$-0.408460\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			8.56199i			1.24889i	0.781067	+	0.624447i	$0.214676\pi$
$47$			8.56199i			1.24889i	−0.781067	+	0.624447i	$0.785324\pi$
$48$		0			0
$49$	4.67307			0.667582
$50$		0			0
$51$		0			0
$52$		0			0
$53$			3.18421i			0.437385i	0.975794	+	0.218692i	$0.0701791\pi$
$53$			3.18421i			0.437385i	−0.975794	+	0.218692i	$0.929821\pi$
$54$		0			0
$55$	−0.311108	−	2.21432i	−0.0419498	−	0.298579i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	12.0415			1.56767			0.783834	−	0.620970i	$-0.213261\pi$
$59$	12.0415			1.56767			0.783834	+	0.620970i	$0.213261\pi$
$60$		0			0
$61$	−13.6128			−1.74295			−0.871473	−	0.490443i	$-0.836835\pi$
$61$	−13.6128			−1.74295			−0.871473	+	0.490443i	$0.836835\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.00000	+	0.280996i	−0.248069	+	0.0348533i
$66$		0			0
$67$			14.3684i			1.75538i	0.479227	+	0.877691i	$0.340917\pi$
$67$			14.3684i			1.75538i	−0.479227	+	0.877691i	$0.659083\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.91750			−0.820956			−0.410478	−	0.911870i	$-0.634638\pi$
$71$	−6.91750			−0.820956			−0.410478	+	0.911870i	$0.634638\pi$
$72$		0			0
$73$		−	9.76049i		−	1.14238i	−0.820818	−	0.571190i	$-0.806482\pi$
$73$		−	9.76049i		−	1.14238i	0.820818	−	0.571190i	$-0.193518\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			1.52543i			0.173839i
$78$		0			0
$79$	8.75557			0.985078			0.492539	−	0.870290i	$-0.336069\pi$
$79$	8.75557			0.985078			0.492539	+	0.870290i	$0.336069\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			7.09679i			0.778974i	0.921032	+	0.389487i	$0.127348\pi$
$83$			7.09679i			0.778974i	−0.921032	+	0.389487i	$0.872652\pi$
$84$		0			0
$85$	6.42864	−	0.903212i	0.697284	−	0.0979671i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−14.9906			−1.58900			−0.794502	−	0.607262i	$-0.792268\pi$
$89$	−14.9906			−1.58900			−0.794502	+	0.607262i	$0.792268\pi$
$90$		0			0
$91$	1.37778			0.144431
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.622216	+	4.42864i	0.0638380	+	0.454369i
$96$		0			0
$97$			7.61285i			0.772968i	0.922296	+	0.386484i	$0.126310\pi$
$97$			7.61285i			0.772968i	−0.922296	+	0.386484i	$0.873690\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−10.2351			−1.01843			−0.509213	−	0.860640i	$-0.670064\pi$
$101$	−10.2351			−1.01843			−0.509213	+	0.860640i	$0.670064\pi$
$102$		0			0
$103$			5.80642i			0.572124i	0.958211	+	0.286062i	$0.0923463\pi$
$103$			5.80642i			0.572124i	−0.958211	+	0.286062i	$0.907654\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			7.00492i			0.677191i	0.940932	+	0.338596i	$0.109952\pi$
$107$			7.00492i			0.677191i	−0.940932	+	0.338596i	$0.890048\pi$
$108$		0			0
$109$	−9.61285			−0.920744			−0.460372	−	0.887726i	$-0.652284\pi$
$109$	−9.61285			−0.920744			−0.460372	+	0.887726i	$0.652284\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	11.4795i		−	1.07990i	−0.841697	−	0.539950i	$-0.818443\pi$
$113$		−	11.4795i		−	1.07990i	0.841697	−	0.539950i	$-0.181557\pi$
$114$		0			0
$115$	−19.6128	+	2.75557i	−1.82891	+	0.256958i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.42864			−0.405973
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−4.54617	−	10.2143i	−0.406622	−	0.913597i
$126$		0			0
$127$		−	17.4336i		−	1.54698i	−0.633809	−	0.773489i	$-0.718510\pi$
$127$		−	17.4336i		−	1.54698i	0.633809	−	0.773489i	$-0.281490\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.24443			−0.108726			−0.0543632	−	0.998521i	$-0.517313\pi$
$131$	−1.24443			−0.108726			−0.0543632	+	0.998521i	$0.517313\pi$
$132$		0			0
$133$		−	3.05086i		−	0.264543i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			4.69535i			0.401150i	0.979678	+	0.200575i	$0.0642811\pi$
$137$			4.69535i			0.401150i	−0.979678	+	0.200575i	$0.935719\pi$
$138$		0			0
$139$	−14.4701			−1.22734			−0.613670	−	0.789563i	$-0.710307\pi$
$139$	−14.4701			−1.22734			−0.613670	+	0.789563i	$0.710307\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	0.903212i		−	0.0755304i
$144$		0			0
$145$	−2.06022	−	14.6637i	−0.171092	−	1.21775i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−11.4795			−0.940437			−0.470218	−	0.882550i	$-0.655825\pi$
$149$	−11.4795			−0.940437			−0.470218	+	0.882550i	$0.655825\pi$
$150$		0			0
$151$	−3.24443			−0.264028			−0.132014	−	0.991248i	$-0.542144\pi$
$151$	−3.24443			−0.264028			−0.132014	+	0.991248i	$0.542144\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.18421	−	22.6637i	−0.255762	−	1.82039i
$156$		0			0
$157$		−	7.90813i		−	0.631138i	−0.948903	−	0.315569i	$-0.897805\pi$
$157$		−	7.90813i		−	0.631138i	0.948903	−	0.315569i	$-0.102195\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	13.5111			1.06483
$162$		0			0
$163$			9.41927i			0.737774i	0.929474	+	0.368887i	$0.120261\pi$
$163$			9.41927i			0.737774i	−0.929474	+	0.368887i	$0.879739\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	17.3733i		−	1.34439i	−0.740374	−	0.672195i	$-0.765352\pi$
$167$		−	17.3733i		−	1.34439i	0.740374	−	0.672195i	$-0.234648\pi$
$168$		0			0
$169$	12.1842			0.937247
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	13.5669i		−	1.03147i	−0.856747	−	0.515737i	$-0.827518\pi$
$173$		−	13.5669i		−	1.03147i	0.856747	−	0.515737i	$-0.172482\pi$
$174$		0			0
$175$	2.10171	+	7.33185i	0.158874	+	0.554236i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	4.32693			0.321618			0.160809	−	0.986986i	$-0.448590\pi$
$181$	4.32693			0.321618			0.160809	+	0.986986i	$0.448590\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	18.9590	−	2.66370i	1.39389	−	0.195839i
$186$		0			0
$187$			2.90321i			0.212304i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$		0			0
$193$			15.5669i			1.12053i	0.828313	+	0.560266i	$0.189301\pi$
$193$			15.5669i			1.12053i	−0.828313	+	0.560266i	$0.810699\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			14.3412i			1.02177i	0.859649	+	0.510885i	$0.170682\pi$
$197$			14.3412i			1.02177i	−0.859649	+	0.510885i	$0.829318\pi$
$198$		0			0
$199$	−1.24443			−0.0882154			−0.0441077	−	0.999027i	$-0.514044\pi$
$199$	−1.24443			−0.0882154			−0.0441077	+	0.999027i	$0.514044\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			10.1017i			0.709001i
$204$		0			0
$205$	−0.695346	−	4.94914i	−0.0485650	−	0.345663i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.00000			−0.138343
$210$		0			0
$211$	−24.9590			−1.71825			−0.859124	−	0.511768i	$-0.828991\pi$
$211$	−24.9590			−1.71825			−0.859124	+	0.511768i	$0.828991\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	27.8479	−	3.91258i	1.89921	−	0.266836i
$216$		0			0
$217$			15.6128i			1.05987i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.62222			0.176389
$222$		0			0
$223$			1.53972i			0.103107i	0.998670	+	0.0515536i	$0.0164173\pi$
$223$			1.53972i			0.103107i	−0.998670	+	0.0515536i	$0.983583\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			15.3002i			1.01551i	0.861501	+	0.507755i	$0.169525\pi$
$227$			15.3002i			1.01551i	−0.861501	+	0.507755i	$0.830475\pi$
$228$		0			0
$229$	−15.7146			−1.03845			−0.519224	−	0.854638i	$-0.673779\pi$
$229$	−15.7146			−1.03845			−0.519224	+	0.854638i	$0.673779\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	29.0049i		−	1.90018i	−0.311983	−	0.950088i	$-0.600993\pi$
$233$		−	29.0049i		−	1.90018i	0.311983	−	0.950088i	$-0.399007\pi$
$234$		0			0
$235$	−18.9590	+	2.66370i	−1.23675	+	0.173761i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−15.6128			−1.00991			−0.504956	−	0.863145i	$-0.668491\pi$
$239$	−15.6128			−1.00991			−0.504956	+	0.863145i	$0.668491\pi$
$240$		0			0
$241$	5.61285			0.361555			0.180778	−	0.983524i	$-0.442139\pi$
$241$	5.61285			0.361555			0.180778	+	0.983524i	$0.442139\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.45383	+	10.3477i	0.0928817	+	0.661089i
$246$		0			0
$247$			1.80642i			0.114940i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.91750			0.184151			0.0920755	−	0.995752i	$-0.470650\pi$
$251$	2.91750			0.184151			0.0920755	+	0.995752i	$0.470650\pi$
$252$		0			0
$253$		−	8.85728i		−	0.556852i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	11.0923i		−	0.691921i	−0.938249	−	0.345961i	$-0.887553\pi$
$257$		−	11.0923i		−	0.691921i	0.938249	−	0.345961i	$-0.112447\pi$
$258$		0			0
$259$	−13.0607			−0.811552
$260$		0			0
$261$		0			0
$262$		0			0
$263$			5.46520i			0.336999i	0.985702	+	0.168499i	$0.0538921\pi$
$263$			5.46520i			0.336999i	−0.985702	+	0.168499i	$0.946108\pi$
$264$		0			0
$265$	−7.05086	+	0.990632i	−0.433131	+	0.0608540i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.8573			−0.661980			−0.330990	−	0.943634i	$-0.607383\pi$
$269$	−10.8573			−0.661980			−0.330990	+	0.943634i	$0.607383\pi$
$270$		0			0
$271$	−0.101710			−0.00617846			−0.00308923	−	0.999995i	$-0.500983\pi$
$271$	−0.101710			−0.00617846			−0.00308923	+	0.999995i	$0.500983\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.80642	−	1.37778i	0.289838	−	0.0830835i
$276$		0			0
$277$			9.19850i			0.552684i	0.961059	+	0.276342i	$0.0891223\pi$
$277$			9.19850i			0.552684i	−0.961059	+	0.276342i	$0.910878\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.11108			0.0662814			0.0331407	−	0.999451i	$-0.489449\pi$
$281$	1.11108			0.0662814			0.0331407	+	0.999451i	$0.489449\pi$
$282$		0			0
$283$		−	15.7190i		−	0.934398i	−0.884152	−	0.467199i	$-0.845263\pi$
$283$		−	15.7190i		−	0.934398i	0.884152	−	0.467199i	$-0.154737\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			3.40943i			0.201252i
$288$		0			0
$289$	8.57136			0.504198
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	6.43309i		−	0.375825i	−0.982186	−	0.187912i	$-0.939828\pi$
$293$		−	6.43309i		−	0.375825i	0.982186	−	0.187912i	$-0.0601721\pi$
$294$		0			0
$295$	3.74620	+	26.6637i	0.218112	+	1.55242i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−8.00000			−0.462652
$300$		0			0
$301$	−19.1842			−1.10576
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−4.23506	−	30.1432i	−0.242499	−	1.72599i
$306$		0			0
$307$			15.7190i			0.897131i	0.893750	+	0.448565i	$0.148065\pi$
$307$			15.7190i			0.897131i	−0.893750	+	0.448565i	$0.851935\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−24.8988			−1.41188			−0.705940	−	0.708272i	$-0.749475\pi$
$311$	−24.8988			−1.41188			−0.705940	+	0.708272i	$0.749475\pi$
$312$		0			0
$313$		−	17.1526i		−	0.969520i	−0.874647	−	0.484760i	$-0.838907\pi$
$313$		−	17.1526i		−	0.969520i	0.874647	−	0.484760i	$-0.161093\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	12.3368i		−	0.692902i	−0.938068	−	0.346451i	$-0.887387\pi$
$317$		−	12.3368i		−	0.692902i	0.938068	−	0.346451i	$-0.112613\pi$
$318$		0			0
$319$	6.62222			0.370773
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	5.80642i		−	0.323078i
$324$		0			0
$325$	−1.24443	−	4.34122i	−0.0690286	−	0.240808i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	13.0607			0.720060
$330$		0			0
$331$	−9.37778			−0.515450			−0.257725	−	0.966218i	$-0.582973\pi$
$331$	−9.37778			−0.515450			−0.257725	+	0.966218i	$0.582973\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−31.8163	+	4.47013i	−1.73831	+	0.244229i
$336$		0			0
$337$		−	18.0558i		−	0.983561i	−0.870719	−	0.491780i	$-0.836346\pi$
$337$		−	18.0558i		−	0.983561i	0.870719	−	0.491780i	$-0.163654\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	10.2351			0.554260
$342$		0			0
$343$		−	17.8064i		−	0.961457i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			16.5161i			0.886629i	0.896366	+	0.443314i	$0.146197\pi$
$347$			16.5161i			0.886629i	−0.896366	+	0.443314i	$0.853803\pi$
$348$		0			0
$349$	8.36842			0.447951			0.223976	−	0.974595i	$-0.428096\pi$
$349$	8.36842			0.447951			0.223976	+	0.974595i	$0.428096\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	11.7748i		−	0.626708i	−0.949636	−	0.313354i	$-0.898547\pi$
$353$		−	11.7748i		−	0.626708i	0.949636	−	0.313354i	$-0.101453\pi$
$354$		0			0
$355$	−2.15209	−	15.3176i	−0.114221	−	0.812972i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−12.2034			−0.644072			−0.322036	−	0.946727i	$-0.604367\pi$
$359$	−12.2034			−0.644072			−0.322036	+	0.946727i	$0.604367\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$	21.6128	−	3.03657i	1.13127	−	0.158941i
$366$		0			0
$367$			18.6637i			0.974237i	0.873336	+	0.487119i	$0.161952\pi$
$367$			18.6637i			0.974237i	−0.873336	+	0.487119i	$0.838048\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.85728			0.252177
$372$		0			0
$373$			34.3225i			1.77715i	0.458731	+	0.888575i	$0.348304\pi$
$373$			34.3225i			1.77715i	−0.458731	+	0.888575i	$0.651696\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	5.98126i		−	0.308051i
$378$		0			0
$379$	11.7333			0.602699			0.301349	−	0.953514i	$-0.402563\pi$
$379$	11.7333			0.602699			0.301349	+	0.953514i	$0.402563\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	18.6637i		−	0.953671i	−0.878993	−	0.476835i	$-0.841784\pi$
$383$		−	18.6637i		−	0.953671i	0.878993	−	0.476835i	$-0.158216\pi$
$384$		0			0
$385$	−3.37778	+	0.474572i	−0.172148	+	0.0241865i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−15.1240			−0.766816			−0.383408	−	0.923579i	$-0.625250\pi$
$389$	−15.1240			−0.766816			−0.383408	+	0.923579i	$0.625250\pi$
$390$		0			0
$391$	25.7146			1.30044
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.72393	+	19.3876i	0.137056	+	0.975497i
$396$		0			0
$397$			34.1847i			1.71568i	0.513917	+	0.857840i	$0.328194\pi$
$397$			34.1847i			1.71568i	−0.513917	+	0.857840i	$0.671806\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−17.0923			−0.853551			−0.426775	−	0.904358i	$-0.640351\pi$
$401$	−17.0923			−0.853551			−0.426775	+	0.904358i	$0.640351\pi$
$402$		0			0
$403$		−	9.24443i		−	0.460498i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			8.56199i			0.424402i
$408$		0			0
$409$	27.1240			1.34119			0.670597	−	0.741822i	$-0.266038\pi$
$409$	27.1240			1.34119			0.670597	+	0.741822i	$0.266038\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	18.3684i		−	0.903851i
$414$		0			0
$415$	−15.7146	+	2.20787i	−0.771397	+	0.108380i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	14.4889			0.707827			0.353914	−	0.935278i	$-0.384851\pi$
$419$	14.4889			0.707827			0.353914	+	0.935278i	$0.384851\pi$
$420$		0			0
$421$	−7.67307			−0.373963			−0.186981	−	0.982363i	$-0.559870\pi$
$421$	−7.67307			−0.373963			−0.186981	+	0.982363i	$0.559870\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.00000	+	13.9541i	0.194029	+	0.676872i
$426$		0			0
$427$			20.7654i			1.00491i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	8.38715			0.403995			0.201997	−	0.979386i	$-0.435257\pi$
$431$	8.38715			0.403995			0.201997	+	0.979386i	$0.435257\pi$
$432$		0			0
$433$			31.3461i			1.50640i	0.657792	+	0.753200i	$0.271491\pi$
$433$			31.3461i			1.50640i	−0.657792	+	0.753200i	$0.728509\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			17.7146i			0.847402i
$438$		0			0
$439$	37.5496			1.79214			0.896071	−	0.443910i	$-0.146409\pi$
$439$	37.5496			1.79214			0.896071	+	0.443910i	$0.146409\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			10.9304i			0.519319i	0.965700	+	0.259660i	$0.0836104\pi$
$443$			10.9304i			0.519319i	−0.965700	+	0.259660i	$0.916390\pi$
$444$		0			0
$445$	−4.66370	−	33.1941i	−0.221081	−	1.57355i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−11.2573			−0.531267			−0.265633	−	0.964074i	$-0.585581\pi$
$449$	−11.2573			−0.531267			−0.265633	+	0.964074i	$0.585581\pi$
$450$		0			0
$451$	2.23506			0.105245
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.428639	+	3.05086i	0.0200949	+	0.143026i
$456$		0			0
$457$		−	29.3733i		−	1.37403i	−0.726645	−	0.687013i	$-0.758921\pi$
$457$		−	29.3733i		−	1.37403i	0.726645	−	0.687013i	$-0.241079\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	31.9496			1.48804			0.744021	−	0.668156i	$-0.232916\pi$
$461$	31.9496			1.48804			0.744021	+	0.668156i	$0.232916\pi$
$462$		0			0
$463$			34.4800i			1.60242i	0.598383	+	0.801210i	$0.295810\pi$
$463$			34.4800i			1.60242i	−0.598383	+	0.801210i	$0.704190\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			11.6128i			0.537379i	0.963227	+	0.268689i	$0.0865905\pi$
$467$			11.6128i			0.537379i	−0.963227	+	0.268689i	$0.913410\pi$
$468$		0			0
$469$	21.9180			1.01208
$470$		0			0
$471$		0			0
$472$		0			0
$473$			12.5763i			0.578258i
$474$		0			0
$475$	−9.61285	+	2.75557i	−0.441068	+	0.126434i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−41.0607			−1.87611			−0.938056	−	0.346485i	$-0.887375\pi$
$479$	−41.0607			−1.87611			−0.938056	+	0.346485i	$0.887375\pi$
$480$		0			0
$481$	7.73329			0.352608
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−16.8573	+	2.36842i	−0.765450	+	0.107544i
$486$		0			0
$487$		−	4.47013i		−	0.202561i	−0.994858	−	0.101280i	$-0.967706\pi$
$487$		−	4.47013i		−	0.202561i	0.994858	−	0.101280i	$-0.0322939\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	21.5941			0.974529			0.487264	−	0.873255i	$-0.337995\pi$
$491$	21.5941			0.974529			0.487264	+	0.873255i	$0.337995\pi$
$492$		0			0
$493$			19.2257i			0.865882i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			10.5521i			0.473329i
$498$		0			0
$499$	26.8385			1.20146			0.600729	−	0.799453i	$-0.294877\pi$
$499$	26.8385			1.20146			0.600729	+	0.799453i	$0.294877\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			17.7605i			0.791901i	0.918272	+	0.395951i	$0.129585\pi$
$503$			17.7605i			0.791901i	−0.918272	+	0.395951i	$0.870415\pi$
$504$		0			0
$505$	−3.18421	−	22.6637i	−0.141695	−	1.00852i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	18.0830			0.801514			0.400757	−	0.916184i	$-0.368747\pi$
$509$	18.0830			0.801514			0.400757	+	0.916184i	$0.368747\pi$
$510$		0			0
$511$	−14.8889			−0.658647
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−12.8573	+	1.80642i	−0.566559	+	0.0796005i
$516$		0			0
$517$		−	8.56199i		−	0.376556i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−2.40006			−0.105149			−0.0525743	−	0.998617i	$-0.516743\pi$
$521$	−2.40006			−0.105149			−0.0525743	+	0.998617i	$0.516743\pi$
$522$		0			0
$523$		−	19.7101i		−	0.861863i	−0.902385	−	0.430932i	$-0.858185\pi$
$523$		−	19.7101i		−	0.861863i	0.902385	−	0.430932i	$-0.141815\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			29.7146i			1.29439i
$528$		0			0
$529$	−55.4514			−2.41093
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	2.01874i		−	0.0874412i
$534$		0			0
$535$	−15.5111	+	2.17929i	−0.670605	+	0.0942188i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.67307			−0.201283
$540$		0			0
$541$	−2.59057			−0.111377			−0.0556887	−	0.998448i	$-0.517735\pi$
$541$	−2.59057			−0.111377			−0.0556887	+	0.998448i	$0.517735\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.99063	−	21.2859i	−0.128105	−	0.911789i
$546$		0			0
$547$			24.2810i			1.03818i	0.854719	+	0.519090i	$0.173729\pi$
$547$			24.2810i			1.03818i	−0.854719	+	0.519090i	$0.826271\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−13.2444			−0.564232
$552$		0			0
$553$		−	13.3560i		−	0.567954i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			45.1798i			1.91433i	0.289546	+	0.957164i	$0.406496\pi$
$557$			45.1798i			1.91433i	−0.289546	+	0.957164i	$0.593504\pi$
$558$		0			0
$559$	11.3590			0.480437
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	2.97634i		−	0.125438i	−0.998031	−	0.0627189i	$-0.980023\pi$
$563$		−	2.97634i		−	0.125438i	0.998031	−	0.0627189i	$-0.0199772\pi$
$564$		0			0
$565$	25.4193	−	3.57136i	1.06940	−	0.150248i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	39.1753			1.64231			0.821157	−	0.570702i	$-0.193329\pi$
$569$	39.1753			1.64231			0.821157	+	0.570702i	$0.193329\pi$
$570$		0			0
$571$	−11.1240			−0.465524			−0.232762	−	0.972534i	$-0.574776\pi$
$571$	−11.1240			−0.465524			−0.232762	+	0.972534i	$0.574776\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−12.2034	−	42.5718i	−0.508918	−	1.77537i
$576$		0			0
$577$			6.19358i			0.257842i	0.991655	+	0.128921i	$0.0411514\pi$
$577$			6.19358i			0.257842i	−0.991655	+	0.128921i	$0.958849\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	10.8256			0.449123
$582$		0			0
$583$		−	3.18421i		−	0.131876i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	7.22570i		−	0.298236i	−0.988819	−	0.149118i	$-0.952357\pi$
$587$		−	7.22570i		−	0.298236i	0.988819	−	0.149118i	$-0.0476435\pi$
$588$		0			0
$589$	−20.4701			−0.843457
$590$		0			0
$591$		0			0
$592$		0			0
$593$			21.5955i			0.886821i	0.896319	+	0.443410i	$0.146232\pi$
$593$			21.5955i			0.886821i	−0.896319	+	0.443410i	$0.853768\pi$
$594$		0			0
$595$	−1.37778	−	9.80642i	−0.0564837	−	0.402024i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	38.4514			1.57108			0.785541	−	0.618810i	$-0.212385\pi$
$599$	38.4514			1.57108			0.785541	+	0.618810i	$0.212385\pi$
$600$		0			0
$601$	22.5906			0.921489			0.460744	−	0.887533i	$-0.347583\pi$
$601$	22.5906			0.921489			0.460744	+	0.887533i	$0.347583\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.311108	+	2.21432i	0.0126483	+	0.0900249i
$606$		0			0
$607$			22.9733i			0.932457i	0.884664	+	0.466228i	$0.154387\pi$
$607$			22.9733i			0.932457i	−0.884664	+	0.466228i	$0.845613\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−7.73329			−0.312856
$612$		0			0
$613$		−	23.5383i		−	0.950704i	−0.879796	−	0.475352i	$-0.842321\pi$
$613$		−	23.5383i		−	0.950704i	0.879796	−	0.475352i	$-0.157679\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			22.8256i			0.918926i	0.888197	+	0.459463i	$0.151958\pi$
$617$			22.8256i			0.918926i	−0.888197	+	0.459463i	$0.848042\pi$
$618$		0			0
$619$	25.3778			1.02002			0.510010	−	0.860169i	$-0.329642\pi$
$619$	25.3778			1.02002			0.510010	+	0.860169i	$0.329642\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			22.8671i			0.916152i
$624$		0			0
$625$	21.2034	−	13.2444i	0.848137	−	0.529777i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−24.8573			−0.991125
$630$		0			0
$631$	28.6735			1.14148			0.570738	−	0.821132i	$-0.306657\pi$
$631$	28.6735			1.14148			0.570738	+	0.821132i	$0.306657\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	38.6035	−	5.42372i	1.53193	−	0.215234i
$636$		0			0
$637$			4.22077i			0.167233i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	18.6035			0.734793			0.367397	−	0.930064i	$-0.380249\pi$
$641$	18.6035			0.734793			0.367397	+	0.930064i	$0.380249\pi$
$642$		0			0
$643$		−	27.9081i		−	1.10059i	−0.834971	−	0.550295i	$-0.814515\pi$
$643$		−	27.9081i		−	1.10059i	0.834971	−	0.550295i	$-0.185485\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			20.5620i			0.808375i	0.914676	+	0.404188i	$0.132446\pi$
$647$			20.5620i			0.808375i	−0.914676	+	0.404188i	$0.867554\pi$
$648$		0			0
$649$	−12.0415			−0.472670
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	10.7685i		−	0.421403i	−0.977550	−	0.210702i	$-0.932425\pi$
$653$		−	10.7685i		−	0.421403i	0.977550	−	0.210702i	$-0.0675748\pi$
$654$		0			0
$655$	−0.387152	−	2.75557i	−0.0151273	−	0.107669i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−8.20342			−0.319560			−0.159780	−	0.987153i	$-0.551078\pi$
$659$	−8.20342			−0.319560			−0.159780	+	0.987153i	$0.551078\pi$
$660$		0			0
$661$	16.1017			0.626284			0.313142	−	0.949706i	$-0.398618\pi$
$661$	16.1017			0.626284			0.313142	+	0.949706i	$0.398618\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6.75557	−	0.949145i	0.261970	−	0.0368063i
$666$		0			0
$667$		−	58.6548i		−	2.27112i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	13.6128			0.525518
$672$		0			0
$673$		−	10.8845i		−	0.419566i	−0.977748	−	0.209783i	$-0.932724\pi$
$673$		−	10.8845i		−	0.419566i	0.977748	−	0.209783i	$-0.0672757\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	2.60793i		−	0.100231i	−0.998743	−	0.0501154i	$-0.984041\pi$
$677$		−	2.60793i		−	0.100231i	0.998743	−	0.0501154i	$-0.0159589\pi$
$678$		0			0
$679$	11.6128			0.445660
$680$		0			0
$681$		0			0
$682$		0			0
$683$			9.60300i			0.367449i	0.982978	+	0.183724i	$0.0588154\pi$
$683$			9.60300i			0.367449i	−0.982978	+	0.183724i	$0.941185\pi$
$684$		0			0
$685$	−10.3970	+	1.46076i	−0.397249	+	0.0558127i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2.87601			−0.109567
$690$		0			0
$691$	26.6351			1.01325			0.506624	−	0.862167i	$-0.330893\pi$
$691$	26.6351			1.01325			0.506624	+	0.862167i	$0.330893\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4.50177	−	32.0415i	−0.170762	−	1.21540i
$696$		0			0
$697$			6.48886i			0.245783i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	2.62222			0.0990397			0.0495199	−	0.998773i	$-0.484231\pi$
$701$	2.62222			0.0990397			0.0495199	+	0.998773i	$0.484231\pi$
$702$		0			0
$703$		−	17.1240i		−	0.645843i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			15.6128i			0.587182i
$708$		0			0
$709$	−29.1842			−1.09604			−0.548018	−	0.836467i	$-0.684617\pi$
$709$	−29.1842			−1.09604			−0.548018	+	0.836467i	$0.684617\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	90.6548i		−	3.39505i
$714$		0			0
$715$	2.00000	−	0.280996i	0.0747958	−	0.0105087i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	40.9403			1.52681			0.763407	−	0.645918i	$-0.223525\pi$
$719$	40.9403			1.52681			0.763407	+	0.645918i	$0.223525\pi$
$720$		0			0
$721$	8.85728			0.329862
$722$		0			0
$723$		0			0
$724$		0			0
$725$	31.8292	−	9.12399i	1.18211	−	0.338856i
$726$		0			0
$727$		−	11.2257i		−	0.416338i	−0.978093	−	0.208169i	$-0.933250\pi$
$727$		−	11.2257i		−	0.416338i	0.978093	−	0.208169i	$-0.0667504\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−36.5116			−1.35043
$732$		0			0
$733$		−	7.74176i		−	0.285948i	−0.989726	−	0.142974i	$-0.954333\pi$
$733$		−	7.74176i		−	0.285948i	0.989726	−	0.142974i	$-0.0456666\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	14.3684i		−	0.529267i
$738$		0			0
$739$	24.1017			0.886596			0.443298	−	0.896374i	$-0.353808\pi$
$739$	24.1017			0.886596			0.443298	+	0.896374i	$0.353808\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			8.22077i			0.301591i	0.988565	+	0.150795i	$0.0481835\pi$
$743$			8.22077i			0.301591i	−0.988565	+	0.150795i	$0.951817\pi$
$744$		0			0
$745$	−3.57136	−	25.4193i	−0.130845	−	0.931290i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	10.6855			0.390440
$750$		0			0
$751$	11.8163			0.431182			0.215591	−	0.976484i	$-0.430832\pi$
$751$	11.8163			0.431182			0.215591	+	0.976484i	$0.430832\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.00937	−	7.18421i	−0.0367347	−	0.261460i
$756$		0			0
$757$		−	10.6637i		−	0.387579i	−0.981043	−	0.193789i	$-0.937922\pi$
$757$		−	10.6637i		−	0.387579i	0.981043	−	0.193789i	$-0.0620779\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	42.7052			1.54806			0.774031	−	0.633148i	$-0.218237\pi$
$761$	42.7052			1.54806			0.774031	+	0.633148i	$0.218237\pi$
$762$		0			0
$763$			14.6637i			0.530862i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			10.8760i			0.392710i
$768$		0			0
$769$	−7.63158			−0.275202			−0.137601	−	0.990488i	$-0.543939\pi$
$769$	−7.63158			−0.275202			−0.137601	+	0.990488i	$0.543939\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	8.52051i		−	0.306461i	−0.988190	−	0.153231i	$-0.951032\pi$
$773$		−	8.52051i		−	0.306461i	0.988190	−	0.153231i	$-0.0489677\pi$
$774$		0			0
$775$	49.1941	−	14.1017i	1.76710	−	0.506548i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.47013			−0.160159
$780$		0			0
$781$	6.91750			0.247528
$782$		0			0
$783$		0			0
$784$		0			0
$785$	17.5111	−	2.46028i	0.624999	−	0.0878112i
$786$		0			0
$787$			41.3131i			1.47265i	0.676626	+	0.736327i	$0.263441\pi$
$787$			41.3131i			1.47265i	−0.676626	+	0.736327i	$0.736559\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−17.5111			−0.622624
$792$		0			0
$793$		−	12.2953i		−	0.436618i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			48.7239i			1.72589i	0.505298	+	0.862945i	$0.331383\pi$
$797$			48.7239i			1.72589i	−0.505298	+	0.862945i	$0.668617\pi$
$798$		0			0
$799$	24.8573			0.879387
$800$		0			0
$801$		0			0
$802$		0			0
$803$			9.76049i			0.344440i
$804$		0			0
$805$	4.20342	+	29.9180i	0.148151	+	1.05447i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−52.4197			−1.84298			−0.921490	−	0.388402i	$-0.873027\pi$
$809$	−52.4197			−1.84298			−0.921490	+	0.388402i	$0.873027\pi$
$810$		0			0
$811$	−22.8573			−0.802628			−0.401314	−	0.915941i	$-0.631446\pi$
$811$	−22.8573			−0.802628			−0.401314	+	0.915941i	$0.631446\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−20.8573	+	2.93041i	−0.730599	+	0.102648i
$816$		0			0
$817$		−	25.1526i		−	0.879977i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−52.5402			−1.83367			−0.916833	−	0.399272i	$-0.869263\pi$
$821$	−52.5402			−1.83367			−0.916833	+	0.399272i	$0.869263\pi$
$822$		0			0
$823$			24.8859i			0.867467i	0.901041	+	0.433733i	$0.142804\pi$
$823$			24.8859i			0.867467i	−0.901041	+	0.433733i	$0.857196\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	8.60793i		−	0.299327i	−0.988737	−	0.149663i	$-0.952181\pi$
$827$		−	8.60793i		−	0.299327i	0.988737	−	0.149663i	$-0.0478190\pi$
$828$		0			0
$829$	−6.16193			−0.214013			−0.107006	−	0.994258i	$-0.534127\pi$
$829$	−6.16193			−0.214013			−0.107006	+	0.994258i	$0.534127\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	13.5669i		−	0.470066i
$834$		0			0
$835$	38.4701	−	5.40498i	1.33131	−	0.187047i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−46.4929			−1.60511			−0.802556	−	0.596577i	$-0.796527\pi$
$839$	−46.4929			−1.60511			−0.802556	+	0.596577i	$0.796527\pi$
$840$		0			0
$841$	14.8537			0.512198
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.79060	+	26.9797i	0.130401	+	0.928131i
$846$		0			0
$847$		−	1.52543i		−	0.0524143i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	75.8360			2.59962
$852$		0			0
$853$		−	0.699791i		−	0.0239604i	−0.999928	−	0.0119802i	$-0.996186\pi$
$853$		−	0.699791i		−	0.0239604i	0.999928	−	0.0119802i	$-0.00381351\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			16.0558i			0.548455i	0.961665	+	0.274227i	$0.0884221\pi$
$857$			16.0558i			0.548455i	−0.961665	+	0.274227i	$0.911578\pi$
$858$		0			0
$859$	−1.11108			−0.0379095			−0.0189547	−	0.999820i	$-0.506034\pi$
$859$	−1.11108			−0.0379095			−0.0189547	+	0.999820i	$0.506034\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			43.0134i			1.46419i	0.681201	+	0.732096i	$0.261458\pi$
$863$			43.0134i			1.46419i	−0.681201	+	0.732096i	$0.738542\pi$
$864$		0			0
$865$	30.0415	−	4.22077i	1.02144	−	0.143511i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−8.75557			−0.297012
$870$		0			0
$871$	−12.9777			−0.439733
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−15.5812	+	6.93485i	−0.526741	+	0.234441i
$876$		0			0
$877$		−	8.03704i		−	0.271392i	−0.990751	−	0.135696i	$-0.956673\pi$
$877$		−	8.03704i		−	0.271392i	0.990751	−	0.135696i	$-0.0433270\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	17.9367			0.604303			0.302152	−	0.953260i	$-0.402295\pi$
$881$	17.9367			0.604303			0.302152	+	0.953260i	$0.402295\pi$
$882$		0			0
$883$			3.22570i			0.108553i	0.998526	+	0.0542766i	$0.0172853\pi$
$883$			3.22570i			0.108553i	−0.998526	+	0.0542766i	$0.982715\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	1.43662i		−	0.0482371i	−0.999709	−	0.0241186i	$-0.992322\pi$
$887$		−	1.43662i		−	0.0482371i	0.999709	−	0.0241186i	$-0.00767792\pi$
$888$		0			0
$889$	−26.5936			−0.891922
$890$		0			0
$891$		0			0
$892$		0			0
$893$			17.1240i			0.573032i
$894$		0			0
$895$	−3.73329	−	26.5718i	−0.124790	−	0.888199i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	67.7788			2.26055
$900$		0			0
$901$	9.24443			0.307977
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.34614	+	9.58120i	0.0447473	+	0.318490i
$906$		0			0
$907$			19.4380i			0.645428i	0.946496	+	0.322714i	$0.104595\pi$
$907$			19.4380i			0.645428i	−0.946496	+	0.322714i	$0.895405\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−16.6735			−0.552419			−0.276210	−	0.961097i	$-0.589078\pi$
$911$	−16.6735			−0.552419			−0.276210	+	0.961097i	$0.589078\pi$
$912$		0			0
$913$		−	7.09679i		−	0.234869i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			1.89829i			0.0626871i
$918$		0			0
$919$	31.4479			1.03737			0.518684	−	0.854966i	$-0.326422\pi$
$919$	31.4479			1.03737			0.518684	+	0.854966i	$0.326422\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	6.24797i		−	0.205654i
$924$		0			0
$925$	11.7966	+	41.1526i	0.387869	+	1.35309i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−44.3881			−1.45633			−0.728163	−	0.685404i	$-0.759626\pi$
$929$	−44.3881			−1.45633			−0.728163	+	0.685404i	$0.759626\pi$
$930$		0			0
$931$	9.34614			0.306307
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−6.42864	+	0.903212i	−0.210239	+	0.0295382i
$936$		0			0
$937$			21.6686i			0.707883i	0.935268	+	0.353942i	$0.115159\pi$
$937$			21.6686i			0.707883i	−0.935268	+	0.353942i	$0.884841\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−15.4795			−0.504617			−0.252309	−	0.967647i	$-0.581190\pi$
$941$	−15.4795			−0.504617			−0.252309	+	0.967647i	$0.581190\pi$
$942$		0			0
$943$		−	19.7966i		−	0.644665i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	50.0098i		−	1.62510i	−0.582890	−	0.812551i	$-0.698078\pi$
$947$		−	50.0098i		−	1.62510i	0.582890	−	0.812551i	$-0.301922\pi$
$948$		0			0
$949$	8.81579			0.286173
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	10.9032i		−	0.353190i	−0.984284	−	0.176595i	$-0.943492\pi$
$953$		−	10.9032i		−	0.353190i	0.984284	−	0.176595i	$-0.0565082\pi$
$954$		0			0
$955$	2.48886	+	17.7146i	0.0805377	+	0.573230i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	7.16241			0.231286
$960$		0			0
$961$	73.7565			2.37924
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−34.4701	+	4.84299i	−1.10963	+	0.155901i
$966$		0			0
$967$		−	43.6815i		−	1.40470i	−0.711830	−	0.702352i	$-0.752133\pi$
$967$		−	43.6815i		−	1.40470i	0.711830	−	0.702352i	$-0.247867\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.55215			0.0819023			0.0409512	−	0.999161i	$-0.486961\pi$
$971$	2.55215			0.0819023			0.0409512	+	0.999161i	$0.486961\pi$
$972$		0			0
$973$			22.0731i			0.707632i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	6.40990i		−	0.205071i	−0.994729	−	0.102535i	$-0.967304\pi$
$977$		−	6.40990i		−	0.205071i	0.994729	−	0.102535i	$-0.0326955\pi$
$978$		0			0
$979$	14.9906			0.479103
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	40.1748i		−	1.28138i	−0.767800	−	0.640689i	$-0.778649\pi$
$983$		−	40.1748i		−	1.28138i	0.767800	−	0.640689i	$-0.221351\pi$
$984$		0			0
$985$	−31.7560	+	4.46167i	−1.01183	+	0.142160i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	111.392			3.54205
$990$		0			0
$991$	18.2351			0.579256			0.289628	−	0.957139i	$-0.406468\pi$
$991$	18.2351			0.579256			0.289628	+	0.957139i	$0.406468\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.387152	−	2.75557i	−0.0122736	−	0.0873574i
$996$		0			0
$997$			4.16638i			0.131951i	0.997821	+	0.0659753i	$0.0210158\pi$
$997$			4.16638i			0.131951i	−0.997821	+	0.0659753i	$0.978984\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3960.2.d.d.3169.4		6
3.2	odd	2		1320.2.d.b.529.5	yes	6
5.4	even	2	inner	3960.2.d.d.3169.3		6
12.11	even	2		2640.2.d.f.529.2		6
15.2	even	4		6600.2.a.bu.1.3		3
15.8	even	4		6600.2.a.bq.1.1		3
15.14	odd	2		1320.2.d.b.529.2	✓	6
60.59	even	2		2640.2.d.f.529.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1320.2.d.b.529.2	✓	6	15.14	odd	2
1320.2.d.b.529.5	yes	6	3.2	odd	2
2640.2.d.f.529.2		6	12.11	even	2
2640.2.d.f.529.5		6	60.59	even	2
3960.2.d.d.3169.3		6	5.4	even	2	inner
3960.2.d.d.3169.4		6	1.1	even	1	trivial
6600.2.a.bq.1.1		3	15.8	even	4
6600.2.a.bu.1.3		3	15.2	even	4

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:	\( N \)	\(=\)	\( 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3960.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.311108 + 2.21432i) q^{5} -1.52543i q^{7} -1.00000 q^{11} +0.903212i q^{13} -2.90321i q^{17} +2.00000 q^{19} +8.85728i q^{23} +(-4.80642 + 1.37778i) q^{25} -6.62222 q^{29} -10.2351 q^{31} +(3.37778 - 0.474572i) q^{35} -8.56199i q^{37} -2.23506 q^{41} -12.5763i q^{43} +8.56199i q^{47} +4.67307 q^{49} +3.18421i q^{53} +(-0.311108 - 2.21432i) q^{55} +12.0415 q^{59} -13.6128 q^{61} +(-2.00000 + 0.280996i) q^{65} +14.3684i q^{67} -6.91750 q^{71} -9.76049i q^{73} +1.52543i q^{77} +8.75557 q^{79} +7.09679i q^{83} +(6.42864 - 0.903212i) q^{85} -14.9906 q^{89} +1.37778 q^{91} +(0.622216 + 4.42864i) q^{95} +7.61285i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{5} - 6 q^{11} + 12 q^{19} - 2 q^{25} - 40 q^{29} - 8 q^{31} + 20 q^{35} + 40 q^{41} + 2 q^{49} - 2 q^{55} - 8 q^{59} - 28 q^{61} - 12 q^{65} - 16 q^{71} + 52 q^{79} + 12 q^{85} - 36 q^{89} + 8 q^{91}+ \cdots + 4 q^{95}+O(q^{100}) \) 6 * q + 2 * q^5 - 6 * q^11 + 12 * q^19 - 2 * q^25 - 40 * q^29 - 8 * q^31 + 20 * q^35 + 40 * q^41 + 2 * q^49 - 2 * q^55 - 8 * q^59 - 28 * q^61 - 12 * q^65 - 16 * q^71 + 52 * q^79 + 12 * q^85 - 36 * q^89 + 8 * q^91 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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