LMFDB - Embedded newform 3960.2.d.f.3169.5

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:	$8$
Coefficient field:	8.0.47985531136.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 19x^{6} + 91x^{4} + 45x^{2} + 4 $ x^8 + 19x^6 + 91x^4 + 45*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3169.5
Root			$3.36007i$ of defining polynomial
Character	$\chi$	$=$	3960.3169
Dual form			3960.2.d.f.3169.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.256321 - 2.22133i) q^{5} -1.08258i q^{7} +O(q^{10})$	$q+(0.256321 - 2.22133i) q^{5} -1.08258i q^{7} -1.00000 q^{11} +4.00000i q^{13} -0.107866i q^{17} +6.61228 q^{19} +5.97235i q^{23} +(-4.86860 - 1.13874i) q^{25} +7.80273 q^{29} +1.12492 q^{31} +(-2.40477 - 0.277489i) q^{35} +7.05494i q^{37} -5.19045 q^{41} +5.52969i q^{43} -7.69486i q^{47} +5.82801 q^{49} +4.77745i q^{53} +(-0.256321 + 2.22133i) q^{55} -0.677809 q^{59} +0.197271 q^{61} +(8.88531 + 1.02528i) q^{65} +1.41737i q^{67} +6.15020 q^{71} +6.16517i q^{73} +1.08258i q^{77} +9.35562 q^{79} -13.7454i q^{83} +(-0.239607 - 0.0276484i) q^{85} -1.29009 q^{89} +4.33034 q^{91} +(1.69486 - 14.6880i) q^{95} +6.60782i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{11} + 18 q^{19} - 2 q^{25} + 6 q^{29} - 30 q^{31} - 30 q^{35} - 20 q^{41} - 18 q^{49} + 12 q^{59} + 58 q^{61} + 8 q^{65} + 2 q^{71} + 40 q^{79} - 26 q^{85} + 42 q^{89} + 8 q^{91} - 28 q^{95}+O(q^{100}) $ 8 * q - 8 * q^11 + 18 * q^19 - 2 * q^25 + 6 * q^29 - 30 * q^31 - 30 * q^35 - 20 * q^41 - 18 * q^49 + 12 * q^59 + 58 * q^61 + 8 * q^65 + 2 * q^71 + 40 * q^79 - 26 * q^85 + 42 * q^89 + 8 * q^91 - 28 * q^95

Display 100 coefficients 10 coefficients

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
$5$	0.256321	−	2.22133i	0.114630	−	0.993408i
$5$	0.256321	−	2.22133i	0.114630	−	0.993408i
$6$		0			0
$7$		−	1.08258i		−	0.409178i	−0.978848	−	0.204589i	$-0.934414\pi$
$7$		−	1.08258i		−	0.409178i	0.978848	−	0.204589i	$-0.0655858\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$		0			0
$13$			4.00000i			1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$			4.00000i			1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	0.107866i		−	0.0261614i	−0.999914	−	0.0130807i	$-0.995836\pi$
$17$		−	0.107866i		−	0.0261614i	0.999914	−	0.0130807i	$-0.00416384\pi$
$18$		0			0
$19$	6.61228			1.51696			0.758480	−	0.651696i	$-0.225942\pi$
$19$	6.61228			1.51696			0.758480	+	0.651696i	$0.225942\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			5.97235i			1.24532i	0.782492	+	0.622661i	$0.213948\pi$
$23$			5.97235i			1.24532i	−0.782492	+	0.622661i	$0.786052\pi$
$24$		0			0
$25$	−4.86860	−	1.13874i	−0.973720	−	0.227749i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7.80273			1.44893			0.724465	−	0.689311i	$-0.242087\pi$
$29$	7.80273			1.44893			0.724465	+	0.689311i	$0.242087\pi$
$30$		0			0
$31$	1.12492			0.202042			0.101021	−	0.994884i	$-0.467789\pi$
$31$	1.12492			0.202042			0.101021	+	0.994884i	$0.467789\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.40477	−	0.277489i	−0.406481	−	0.0469041i
$36$		0			0
$37$			7.05494i			1.15982i	0.814679	+	0.579912i	$0.196913\pi$
$37$			7.05494i			1.15982i	−0.814679	+	0.579912i	$0.803087\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.19045			−0.810612			−0.405306	−	0.914181i	$-0.632835\pi$
$41$	−5.19045			−0.810612			−0.405306	+	0.914181i	$0.632835\pi$
$42$		0			0
$43$			5.52969i			0.843271i	0.906766	+	0.421635i	$0.138544\pi$
$43$			5.52969i			0.843271i	−0.906766	+	0.421635i	$0.861456\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	7.69486i		−	1.12241i	−0.827676	−	0.561206i	$-0.810338\pi$
$47$		−	7.69486i		−	1.12241i	0.827676	−	0.561206i	$-0.189662\pi$
$48$		0			0
$49$	5.82801			0.832573
$50$		0			0
$51$		0			0
$52$		0			0
$53$			4.77745i			0.656233i	0.944637	+	0.328116i	$0.106414\pi$
$53$			4.77745i			0.656233i	−0.944637	+	0.328116i	$0.893586\pi$
$54$		0			0
$55$	−0.256321	+	2.22133i	−0.0345623	+	0.299524i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.677809			−0.0882433			−0.0441216	−	0.999026i	$-0.514049\pi$
$59$	−0.677809			−0.0882433			−0.0441216	+	0.999026i	$0.514049\pi$
$60$		0			0
$61$	0.197271			0.0252579			0.0126290	−	0.999920i	$-0.495980\pi$
$61$	0.197271			0.0252579			0.0126290	+	0.999920i	$0.495980\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	8.88531	+	1.02528i	1.10209	+	0.127171i
$66$		0			0
$67$			1.41737i			0.173160i	0.996245	+	0.0865799i	$0.0275938\pi$
$67$			1.41737i			0.173160i	−0.996245	+	0.0865799i	$0.972406\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6.15020			0.729895			0.364947	−	0.931028i	$-0.381087\pi$
$71$	6.15020			0.729895			0.364947	+	0.931028i	$0.381087\pi$
$72$		0			0
$73$			6.16517i			0.721578i	0.932647	+	0.360789i	$0.117493\pi$
$73$			6.16517i			0.721578i	−0.932647	+	0.360789i	$0.882507\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			1.08258i			0.123372i
$78$		0			0
$79$	9.35562			1.05259			0.526295	−	0.850302i	$-0.323581\pi$
$79$	9.35562			1.05259			0.526295	+	0.850302i	$0.323581\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	13.7454i		−	1.50876i	−0.656440	−	0.754378i	$-0.727938\pi$
$83$		−	13.7454i		−	1.50876i	0.656440	−	0.754378i	$-0.272062\pi$
$84$		0			0
$85$	−0.239607	−	0.0276484i	−0.0259890	−	0.00299889i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.29009			−0.136749			−0.0683745	−	0.997660i	$-0.521781\pi$
$89$	−1.29009			−0.136749			−0.0683745	+	0.997660i	$0.521781\pi$
$90$		0			0
$91$	4.33034			0.453943
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.69486	−	14.6880i	0.173889	−	1.50696i
$96$		0			0
$97$			6.60782i			0.670923i	0.942054	+	0.335461i	$0.108892\pi$
$97$			6.60782i			0.670923i	−0.942054	+	0.335461i	$0.891108\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	15.4403			1.53637			0.768183	−	0.640230i	$-0.221161\pi$
$101$	15.4403			1.53637			0.768183	+	0.640230i	$0.221161\pi$
$102$		0			0
$103$		−	5.74543i		−	0.566114i	−0.959103	−	0.283057i	$-0.908651\pi$
$103$		−	5.74543i		−	0.566114i	0.959103	−	0.283057i	$-0.0913485\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			4.30514i			0.416193i	0.978108	+	0.208097i	$0.0667269\pi$
$107$			4.30514i			0.416193i	−0.978108	+	0.208097i	$0.933273\pi$
$108$		0			0
$109$	2.33034			0.223206			0.111603	−	0.993753i	$-0.464402\pi$
$109$	2.33034			0.223206			0.111603	+	0.993753i	$0.464402\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	10.9471i		−	1.02981i	−0.857246	−	0.514907i	$-0.827827\pi$
$113$		−	10.9471i		−	1.02981i	0.857246	−	0.514907i	$-0.172173\pi$
$114$		0			0
$115$	13.2666	+	1.53084i	1.23711	+	0.142751i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.116774			−0.0107047
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3.77745	+	10.5229i	−0.337865	+	0.941195i
$126$		0			0
$127$		−	12.5802i		−	1.11631i	−0.829737	−	0.558155i	$-0.811509\pi$
$127$		−	12.5802i		−	1.11631i	0.829737	−	0.558155i	$-0.188491\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−15.2430			−1.33179			−0.665894	−	0.746046i	$-0.731950\pi$
$131$	−15.2430			−1.33179			−0.665894	+	0.746046i	$0.731950\pi$
$132$		0			0
$133$		−	7.15835i		−	0.620707i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			14.6078i			1.24803i	0.781412	+	0.624015i	$0.214500\pi$
$137$			14.6078i			1.24803i	−0.781412	+	0.624015i	$0.785500\pi$
$138$		0			0
$139$	20.4150			1.73158			0.865789	−	0.500409i	$-0.166817\pi$
$139$	20.4150			1.73158			0.865789	+	0.500409i	$0.166817\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	4.00000i		−	0.334497i
$144$		0			0
$145$	2.00000	−	17.3324i	0.166091	−	1.43938i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−15.8874			−1.30155			−0.650773	−	0.759272i	$-0.725555\pi$
$149$	−15.8874			−1.30155			−0.650773	+	0.759272i	$0.725555\pi$
$150$		0			0
$151$	6.58018			0.535487			0.267744	−	0.963490i	$-0.413722\pi$
$151$	6.58018			0.535487			0.267744	+	0.963490i	$0.413722\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.288340	−	2.49882i	0.0231600	−	0.200710i
$156$		0			0
$157$		−	6.38535i		−	0.509607i	−0.966993	−	0.254803i	$-0.917989\pi$
$157$		−	6.38535i		−	0.509607i	0.966993	−	0.254803i	$-0.0820108\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.46557			0.509559
$162$		0			0
$163$		−	18.3071i		−	1.43393i	−0.697111	−	0.716963i	$-0.745532\pi$
$163$		−	18.3071i		−	1.43393i	0.697111	−	0.716963i	$-0.254468\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			0.197271i			0.0152653i	0.999971	+	0.00763263i	$0.00242956\pi$
$167$			0.197271i			0.0152653i	−0.999971	+	0.00763263i	$0.997570\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	14.7201i		−	1.11915i	−0.828779	−	0.559576i	$-0.810964\pi$
$173$		−	14.7201i		−	1.11915i	0.828779	−	0.559576i	$-0.189036\pi$
$174$		0			0
$175$	−1.23279	+	5.27067i	−0.0931899	+	0.398425i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	11.8518			0.885845			0.442923	−	0.896560i	$-0.353942\pi$
$179$	11.8518			0.885845			0.442923	+	0.896560i	$0.353942\pi$
$180$		0			0
$181$	−10.7625			−0.799969			−0.399984	−	0.916522i	$-0.630984\pi$
$181$	−10.7625			−0.799969			−0.399984	+	0.916522i	$0.630984\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	15.6713	+	1.80833i	1.15218	+	0.132951i
$186$		0			0
$187$			0.107866i			0.00788797i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.70309			−0.123231			−0.0616157	−	0.998100i	$-0.519625\pi$
$191$	−1.70309			−0.123231			−0.0616157	+	0.998100i	$0.519625\pi$
$192$		0			0
$193$			10.8280i			0.779417i	0.920938	+	0.389709i	$0.127424\pi$
$193$			10.8280i			0.779417i	−0.920938	+	0.389709i	$0.872576\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	6.72015i		−	0.478791i	−0.970922	−	0.239395i	$-0.923051\pi$
$197$		−	6.72015i		−	0.478791i	0.970922	−	0.239395i	$-0.0769492\pi$
$198$		0			0
$199$	10.8280			0.767577			0.383789	−	0.923421i	$-0.374619\pi$
$199$	10.8280			0.767577			0.383789	+	0.923421i	$0.374619\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	8.44711i		−	0.592871i
$204$		0			0
$205$	−1.33042	+	11.5297i	−0.0929205	+	0.805269i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.61228			−0.457381
$210$		0			0
$211$	0.447111			0.0307804			0.0153902	−	0.999882i	$-0.495101\pi$
$211$	0.447111			0.0307804			0.0153902	+	0.999882i	$0.495101\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	12.2833	+	1.41737i	0.837712	+	0.0966641i
$216$		0			0
$217$		−	1.21782i		−	0.0826710i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	0.431465			0.0290235
$222$		0			0
$223$		−	17.8577i		−	1.19584i	−0.801555	−	0.597922i	$-0.795993\pi$
$223$		−	17.8577i		−	1.19584i	0.801555	−	0.597922i	$-0.204007\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			1.31396i			0.0872107i	0.999049	+	0.0436054i	$0.0138844\pi$
$227$			1.31396i			0.0872107i	−0.999049	+	0.0436054i	$0.986116\pi$
$228$		0			0
$229$	−4.84298			−0.320033			−0.160016	−	0.987114i	$-0.551155\pi$
$229$	−4.84298			−0.320033			−0.160016	+	0.987114i	$0.551155\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			20.3829i			1.33533i	0.744462	+	0.667664i	$0.232706\pi$
$233$			20.3829i			1.33533i	−0.744462	+	0.667664i	$0.767294\pi$
$234$		0			0
$235$	−17.0928	−	1.97235i	−1.11501	−	0.128662i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−9.35562			−0.605165			−0.302582	−	0.953123i	$-0.597849\pi$
$239$	−9.35562			−0.605165			−0.302582	+	0.953123i	$0.597849\pi$
$240$		0			0
$241$	22.3004			1.43650			0.718248	−	0.695788i	$-0.244944\pi$
$241$	22.3004			1.43650			0.718248	+	0.695788i	$0.244944\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.49384	−	12.9459i	0.0954379	−	0.827085i
$246$		0			0
$247$			26.4491i			1.68292i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.9276			0.689747			0.344874	−	0.938649i	$-0.387922\pi$
$251$	10.9276			0.689747			0.344874	+	0.938649i	$0.387922\pi$
$252$		0			0
$253$		−	5.97235i		−	0.375479i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			21.4403i			1.33741i	0.743528	+	0.668704i	$0.233151\pi$
$257$			21.4403i			1.33741i	−0.743528	+	0.668704i	$0.766849\pi$
$258$		0			0
$259$	7.63756			0.474575
$260$		0			0
$261$		0			0
$262$		0			0
$263$			6.52287i			0.402218i	0.979569	+	0.201109i	$0.0644545\pi$
$263$			6.52287i			0.402218i	−0.979569	+	0.201109i	$0.935546\pi$
$264$		0			0
$265$	10.6123	+	1.22456i	0.651907	+	0.0752240i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5.60546			0.341771			0.170885	−	0.985291i	$-0.445337\pi$
$269$	5.60546			0.341771			0.170885	+	0.985291i	$0.445337\pi$
$270$		0			0
$271$	−28.9446			−1.75826			−0.879130	−	0.476582i	$-0.841876\pi$
$271$	−28.9446			−1.75826			−0.879130	+	0.476582i	$0.841876\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.86860	+	1.13874i	0.293588	+	0.0686689i
$276$		0			0
$277$		−	25.3850i		−	1.52524i	−0.646849	−	0.762618i	$-0.723914\pi$
$277$		−	25.3850i		−	1.52524i	0.646849	−	0.762618i	$-0.276086\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	27.1604			1.62025			0.810125	−	0.586257i	$-0.199399\pi$
$281$	27.1604			1.62025			0.810125	+	0.586257i	$0.199399\pi$
$282$		0			0
$283$		−	22.0205i		−	1.30898i	−0.756070	−	0.654490i	$-0.772883\pi$
$283$		−	22.0205i		−	1.30898i	0.756070	−	0.654490i	$-0.227117\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			5.61910i			0.331685i
$288$		0			0
$289$	16.9884			0.999316
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	1.44029i		−	0.0841427i	−0.999115	−	0.0420713i	$-0.986604\pi$
$293$		−	1.44029i		−	0.0841427i	0.999115	−	0.0420713i	$-0.0133957\pi$
$294$		0			0
$295$	−0.173736	+	1.50564i	−0.0101153	+	0.0876616i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−23.8894			−1.38156
$300$		0			0
$301$	5.98636			0.345048
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.0505645	−	0.438203i	0.00289532	−	0.0250914i
$306$		0			0
$307$			2.80955i			0.160349i	0.996781	+	0.0801747i	$0.0255478\pi$
$307$			2.80955i			0.160349i	−0.996781	+	0.0801747i	$0.974452\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	11.5529			0.655104			0.327552	−	0.944833i	$-0.393776\pi$
$311$	11.5529			0.655104			0.327552	+	0.944833i	$0.393776\pi$
$312$		0			0
$313$			26.1580i			1.47854i	0.673411	+	0.739268i	$0.264829\pi$
$313$			26.1580i			1.47854i	−0.673411	+	0.739268i	$0.735171\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			29.3805i			1.65018i	0.565005	+	0.825088i	$0.308874\pi$
$317$			29.3805i			1.65018i	−0.565005	+	0.825088i	$0.691126\pi$
$318$		0			0
$319$	−7.80273			−0.436869
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	0.713242i		−	0.0396859i
$324$		0			0
$325$	4.55498	−	19.4744i	0.252665	−	1.08025i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−8.33034			−0.459266
$330$		0			0
$331$	12.2833			0.675149			0.337575	−	0.941299i	$-0.390393\pi$
$331$	12.2833			0.675149			0.337575	+	0.941299i	$0.390393\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3.14845	+	0.363302i	0.172018	+	0.0198493i
$336$		0			0
$337$			21.3324i			1.16205i	0.813885	+	0.581026i	$0.197348\pi$
$337$			21.3324i			1.16205i	−0.813885	+	0.581026i	$0.802652\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.12492			−0.0609178
$342$		0			0
$343$		−	13.8874i		−	0.749849i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			10.4150i			0.559107i	0.960130	+	0.279553i	$0.0901864\pi$
$347$			10.4150i			0.559107i	−0.960130	+	0.279553i	$0.909814\pi$
$348$		0			0
$349$	13.4909			0.722149			0.361074	−	0.932537i	$-0.382410\pi$
$349$	13.4909			0.722149			0.361074	+	0.932537i	$0.382410\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			12.7177i			0.676895i	0.940985	+	0.338447i	$0.109902\pi$
$353$			12.7177i			0.676895i	−0.940985	+	0.338447i	$0.890098\pi$
$354$		0			0
$355$	1.57642	−	13.6616i	0.0836679	−	0.725083i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−19.8212			−1.04612			−0.523061	−	0.852295i	$-0.675210\pi$
$359$	−19.8212			−1.04612			−0.523061	+	0.852295i	$0.675210\pi$
$360$		0			0
$361$	24.7222			1.30117
$362$		0			0
$363$		0			0
$364$		0			0
$365$	13.6949	+	1.58026i	0.716822	+	0.0827146i
$366$		0			0
$367$		−	10.3027i		−	0.537796i	−0.963169	−	0.268898i	$-0.913341\pi$
$367$		−	10.3027i		−	0.537796i	0.963169	−	0.268898i	$-0.0866594\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	5.17199			0.268516
$372$		0			0
$373$			23.3344i			1.20821i	0.796904	+	0.604105i	$0.206469\pi$
$373$			23.3344i			1.20821i	−0.796904	+	0.604105i	$0.793531\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			31.2109i			1.60744i
$378$		0			0
$379$	21.2074			1.08935			0.544676	−	0.838647i	$-0.316653\pi$
$379$	21.2074			1.08935			0.544676	+	0.838647i	$0.316653\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			0.972434i			0.0496891i	0.999691	+	0.0248445i	$0.00790908\pi$
$383$			0.972434i			0.0496891i	−0.999691	+	0.0248445i	$0.992091\pi$
$384$		0			0
$385$	2.40477	+	0.277489i	0.122559	+	0.0141421i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.76248			0.140063			0.0700317	−	0.997545i	$-0.477690\pi$
$389$	2.76248			0.140063			0.0700317	+	0.997545i	$0.477690\pi$
$390$		0			0
$391$	0.644216			0.0325794
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.39804	−	20.7819i	0.120658	−	1.04565i
$396$		0			0
$397$			11.2751i			0.565882i	0.959137	+	0.282941i	$0.0913101\pi$
$397$			11.2751i			0.565882i	−0.959137	+	0.282941i	$0.908690\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	10.4471			0.521704			0.260852	−	0.965379i	$-0.415997\pi$
$401$	10.4471			0.521704			0.260852	+	0.965379i	$0.415997\pi$
$402$		0			0
$403$			4.49968i			0.224145i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	7.05494i		−	0.349700i
$408$		0			0
$409$	1.27512			0.0630507			0.0315254	−	0.999503i	$-0.489963\pi$
$409$	1.27512			0.0630507			0.0315254	+	0.999503i	$0.489963\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			0.733786i			0.0361072i
$414$		0			0
$415$	−30.5331	−	3.52324i	−1.49881	−	0.172949i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	18.7795			0.917436			0.458718	−	0.888582i	$-0.348309\pi$
$419$	18.7795			0.917436			0.458718	+	0.888582i	$0.348309\pi$
$420$		0			0
$421$	−1.27512			−0.0621457			−0.0310728	−	0.999517i	$-0.509892\pi$
$421$	−1.27512			−0.0621457			−0.0310728	+	0.999517i	$0.509892\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.122832	+	0.525158i	−0.00595824	+	0.0254739i
$426$		0			0
$427$		−	0.213562i		−	0.0103350i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1.63556			−0.0787820			−0.0393910	−	0.999224i	$-0.512542\pi$
$431$	−1.63556			−0.0787820			−0.0393910	+	0.999224i	$0.512542\pi$
$432$		0			0
$433$		−	26.4932i		−	1.27318i	−0.771201	−	0.636591i	$-0.780344\pi$
$433$		−	26.4932i		−	1.27318i	0.771201	−	0.636591i	$-0.219656\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			39.4909i			1.88910i
$438$		0			0
$439$	−31.9358			−1.52421			−0.762106	−	0.647452i	$-0.775835\pi$
$439$	−31.9358			−1.52421			−0.762106	+	0.647452i	$0.775835\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			2.02765i			0.0963365i	0.998839	+	0.0481682i	$0.0153384\pi$
$443$			2.02765i			0.0963365i	−0.998839	+	0.0481682i	$0.984662\pi$
$444$		0			0
$445$	−0.330676	+	2.86571i	−0.0156756	+	0.135848i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−10.0675			−0.475116			−0.237558	−	0.971373i	$-0.576347\pi$
$449$	−10.0675			−0.475116			−0.237558	+	0.971373i	$0.576347\pi$
$450$		0			0
$451$	5.19045			0.244409
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.10995	−	9.61910i	0.0520355	−	0.450950i
$456$		0			0
$457$			6.04839i			0.282932i	0.989943	+	0.141466i	$0.0451816\pi$
$457$			6.04839i			0.282932i	−0.989943	+	0.141466i	$0.954818\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−31.8397			−1.48292			−0.741460	−	0.670997i	$-0.765866\pi$
$461$	−31.8397			−1.48292			−0.741460	+	0.670997i	$0.765866\pi$
$462$		0			0
$463$			30.2474i			1.40572i	0.711330	+	0.702858i	$0.248093\pi$
$463$			30.2474i			1.40572i	−0.711330	+	0.702858i	$0.751907\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	31.3417i		−	1.45032i	−0.688580	−	0.725160i	$-0.741766\pi$
$467$		−	31.3417i		−	1.45032i	0.688580	−	0.725160i	$-0.258234\pi$
$468$		0			0
$469$	1.53443			0.0708533
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	5.52969i		−	0.254256i
$474$		0			0
$475$	−32.1925	−	7.52969i	−1.47709	−	0.345486i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6.59663			0.301408			0.150704	−	0.988579i	$-0.451846\pi$
$479$	6.59663			0.301408			0.150704	+	0.988579i	$0.451846\pi$
$480$		0			0
$481$	−28.2197			−1.28671
$482$		0			0
$483$		0			0
$484$		0			0
$485$	14.6781	+	1.69372i	0.666500	+	0.0769079i
$486$		0			0
$487$			1.13343i			0.0513605i	0.999670	+	0.0256802i	$0.00817517\pi$
$487$			1.13343i			0.0513605i	−0.999670	+	0.0256802i	$0.991825\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−43.5569			−1.96570			−0.982848	−	0.184419i	$-0.940960\pi$
$491$	−43.5569			−1.96570			−0.982848	+	0.184419i	$0.940960\pi$
$492$		0			0
$493$		−	0.841652i		−	0.0379061i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	6.65811i		−	0.298657i
$498$		0			0
$499$	34.5502			1.54668			0.773341	−	0.633991i	$-0.218584\pi$
$499$	34.5502			1.54668			0.773341	+	0.633991i	$0.218584\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	5.92424i		−	0.264149i	−0.991240	−	0.132074i	$-0.957836\pi$
$503$		−	5.92424i		−	0.264149i	0.991240	−	0.132074i	$-0.0421638\pi$
$504$		0			0
$505$	3.95766	−	34.2980i	0.176114	−	1.52624i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−42.3679			−1.87793			−0.938963	−	0.344018i	$-0.888212\pi$
$509$	−42.3679			−1.87793			−0.938963	+	0.344018i	$0.888212\pi$
$510$		0			0
$511$	6.67431			0.295254
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−12.7625	−	1.47267i	−0.562382	−	0.0648936i
$516$		0			0
$517$			7.69486i			0.338420i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−21.2116			−0.929297			−0.464648	−	0.885495i	$-0.653819\pi$
$521$	−21.2116			−0.929297			−0.464648	+	0.885495i	$0.653819\pi$
$522$		0			0
$523$		−	5.19045i		−	0.226963i	−0.993540	−	0.113481i	$-0.963800\pi$
$523$		−	5.19045i		−	0.226963i	0.993540	−	0.113481i	$-0.0362002\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	0.121341i		−	0.00528570i
$528$		0			0
$529$	−12.6690			−0.550825
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	20.7618i		−	0.899293i
$534$		0			0
$535$	9.56312	+	1.10350i	0.413450	+	0.0477083i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−5.82801			−0.251030
$540$		0			0
$541$	−24.0185			−1.03263			−0.516317	−	0.856397i	$-0.672697\pi$
$541$	−24.0185			−1.03263			−0.516317	+	0.856397i	$0.672697\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.597313	−	5.17644i	0.0255861	−	0.221734i
$546$		0			0
$547$			12.8601i			0.549859i	0.961464	+	0.274929i	$0.0886545\pi$
$547$			12.8601i			0.549859i	−0.961464	+	0.274929i	$0.911346\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	51.5938			2.19797
$552$		0			0
$553$		−	10.1282i		−	0.430697i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	31.3344i		−	1.32768i	−0.747874	−	0.663841i	$-0.768925\pi$
$557$		−	31.3344i		−	1.32768i	0.747874	−	0.663841i	$-0.231075\pi$
$558$		0			0
$559$	−22.1188			−0.935525
$560$		0			0
$561$		0			0
$562$		0			0
$563$			17.6901i			0.745550i	0.927922	+	0.372775i	$0.121594\pi$
$563$			17.6901i			0.745550i	−0.927922	+	0.372775i	$0.878406\pi$
$564$		0			0
$565$	−24.3170	−	2.80596i	−1.02303	−	0.118048i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	26.3004			1.10257			0.551285	−	0.834317i	$-0.314138\pi$
$569$	26.3004			1.10257			0.551285	+	0.834317i	$0.314138\pi$
$570$		0			0
$571$	−35.9884			−1.50607			−0.753033	−	0.657983i	$-0.771410\pi$
$571$	−35.9884			−1.50607			−0.753033	+	0.657983i	$0.771410\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	6.80098	−	29.0770i	0.283621	−	1.21259i
$576$		0			0
$577$			32.6031i			1.35728i	0.734469	+	0.678642i	$0.237431\pi$
$577$			32.6031i			1.35728i	−0.734469	+	0.678642i	$0.762569\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−14.8806			−0.617351
$582$		0			0
$583$		−	4.77745i		−	0.197862i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			17.6287i			0.727612i	0.931475	+	0.363806i	$0.118523\pi$
$587$			17.6287i			0.727612i	−0.931475	+	0.363806i	$0.881477\pi$
$588$		0			0
$589$	7.43829			0.306489
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	13.2246i		−	0.543068i	−0.962429	−	0.271534i	$-0.912469\pi$
$593$		−	13.2246i		−	0.543068i	0.962429	−	0.271534i	$-0.0875309\pi$
$594$		0			0
$595$	−0.0299317	+	0.259394i	−0.00122708	+	0.0106341i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	37.2771			1.52310			0.761551	−	0.648105i	$-0.224438\pi$
$599$	37.2771			1.52310			0.761551	+	0.648105i	$0.224438\pi$
$600$		0			0
$601$	17.0594			0.695867			0.347934	−	0.937519i	$-0.386883\pi$
$601$	17.0594			0.695867			0.347934	+	0.937519i	$0.386883\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.256321	−	2.22133i	0.0104209	−	0.0903098i
$606$		0			0
$607$		−	7.92624i		−	0.321716i	−0.986978	−	0.160858i	$-0.948574\pi$
$607$		−	7.92624i		−	0.321716i	0.986978	−	0.160858i	$-0.0514262\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	30.7795			1.24520
$612$		0			0
$613$			9.93596i			0.401310i	0.979662	+	0.200655i	$0.0643070\pi$
$613$			9.93596i			0.401310i	−0.979662	+	0.200655i	$0.935693\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			38.1010i			1.53389i	0.641715	+	0.766944i	$0.278223\pi$
$617$			38.1010i			1.53389i	−0.641715	+	0.766944i	$0.721777\pi$
$618$		0			0
$619$	−5.70309			−0.229227			−0.114613	−	0.993410i	$-0.536563\pi$
$619$	−5.70309			−0.229227			−0.114613	+	0.993410i	$0.536563\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			1.39663i			0.0559548i
$624$		0			0
$625$	22.4065	+	11.0882i	0.896261	+	0.443527i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	0.760990			0.0303427
$630$		0			0
$631$	17.3242			0.689665			0.344833	−	0.938664i	$-0.387936\pi$
$631$	17.3242			0.689665			0.344833	+	0.938664i	$0.387936\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−27.9447	−	3.22456i	−1.10895	−	0.127963i
$636$		0			0
$637$			23.3120i			0.923657i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	22.3523			0.882863			0.441431	−	0.897295i	$-0.354471\pi$
$641$	22.3523			0.882863			0.441431	+	0.897295i	$0.354471\pi$
$642$		0			0
$643$		−	25.2911i		−	0.997385i	−0.866779	−	0.498693i	$-0.833814\pi$
$643$		−	25.2911i		−	0.997385i	0.866779	−	0.498693i	$-0.166186\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			22.4262i			0.881665i	0.897589	+	0.440832i	$0.145317\pi$
$647$			22.4262i			0.881665i	−0.897589	+	0.440832i	$0.854683\pi$
$648$		0			0
$649$	0.677809			0.0266063
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	11.8391i		−	0.463301i	−0.972799	−	0.231650i	$-0.925587\pi$
$653$		−	11.8391i		−	0.463301i	0.972799	−	0.231650i	$-0.0744125\pi$
$654$		0			0
$655$	−3.90710	+	33.8598i	−0.152663	+	1.32301i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−15.7522			−0.613617			−0.306809	−	0.951771i	$-0.599261\pi$
$659$	−15.7522			−0.613617			−0.306809	+	0.951771i	$0.599261\pi$
$660$		0			0
$661$	−7.15702			−0.278376			−0.139188	−	0.990266i	$-0.544449\pi$
$661$	−7.15702			−0.278376			−0.139188	+	0.990266i	$0.544449\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−15.9010	−	1.83483i	−0.616616	−	0.0711517i
$666$		0			0
$667$			46.6006i			1.80438i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−0.197271			−0.00761555
$672$		0			0
$673$		−	11.4976i		−	0.443200i	−0.975138	−	0.221600i	$-0.928872\pi$
$673$		−	11.4976i		−	0.443200i	0.975138	−	0.221600i	$-0.0711279\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	21.9953i		−	0.845347i	−0.906282	−	0.422673i	$-0.861092\pi$
$677$		−	21.9953i		−	0.845347i	0.906282	−	0.422673i	$-0.138908\pi$
$678$		0			0
$679$	7.15353			0.274527
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	0.468300i		−	0.0179190i	−0.999960	−	0.00895950i	$-0.997148\pi$
$683$		−	0.468300i		−	0.0179190i	0.999960	−	0.00895950i	$-0.00285194\pi$
$684$		0			0
$685$	32.4488	+	3.74429i	1.23980	+	0.143062i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−19.1098			−0.728025
$690$		0			0
$691$	−36.9140			−1.40428			−0.702138	−	0.712041i	$-0.747771\pi$
$691$	−36.9140			−1.40428			−0.702138	+	0.712041i	$0.747771\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	5.23279	−	45.3484i	0.198491	−	1.72016i
$696$		0			0
$697$			0.559875i			0.0212068i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−18.6628			−0.704886			−0.352443	−	0.935833i	$-0.614649\pi$
$701$	−18.6628			−0.704886			−0.352443	+	0.935833i	$0.614649\pi$
$702$		0			0
$703$			46.6492i			1.75941i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	16.7154i		−	0.628648i
$708$		0			0
$709$	−6.66135			−0.250172			−0.125086	−	0.992146i	$-0.539921\pi$
$709$	−6.66135			−0.250172			−0.125086	+	0.992146i	$0.539921\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			6.71842i			0.251607i
$714$		0			0
$715$	−8.88531	−	1.02528i	−0.332292	−	0.0383434i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−3.11128			−0.116031			−0.0580156	−	0.998316i	$-0.518477\pi$
$719$	−3.11128			−0.116031			−0.0580156	+	0.998316i	$0.518477\pi$
$720$		0			0
$721$	−6.21991			−0.231641
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−37.9884	−	8.88531i	−1.41085	−	0.329992i
$726$		0			0
$727$		−	33.9540i		−	1.25928i	−0.776886	−	0.629642i	$-0.783202\pi$
$727$		−	33.9540i		−	1.25928i	0.776886	−	0.629642i	$-0.216798\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	0.596468			0.0220612
$732$		0			0
$733$			21.0457i			0.777342i	0.921377	+	0.388671i	$0.127066\pi$
$733$			21.0457i			0.777342i	−0.921377	+	0.388671i	$0.872934\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	1.41737i		−	0.0522097i
$738$		0			0
$739$	17.0253			0.626285			0.313143	−	0.949706i	$-0.398618\pi$
$739$	17.0253			0.626285			0.313143	+	0.949706i	$0.398618\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	23.7563i		−	0.871536i	−0.900059	−	0.435768i	$-0.856477\pi$
$743$		−	23.7563i		−	0.871536i	0.900059	−	0.435768i	$-0.143523\pi$
$744$		0			0
$745$	−4.07227	+	35.2911i	−0.149196	+	1.29297i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.66067			0.170297
$750$		0			0
$751$	−44.8654			−1.63716			−0.818582	−	0.574390i	$-0.805239\pi$
$751$	−44.8654			−1.63716			−0.818582	+	0.574390i	$0.805239\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.68663	−	14.6167i	0.0613829	−	0.531957i
$756$		0			0
$757$		−	41.3761i		−	1.50384i	−0.659255	−	0.751920i	$-0.729128\pi$
$757$		−	41.3761i		−	1.50384i	0.659255	−	0.751920i	$-0.270872\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−16.3002			−0.590883			−0.295442	−	0.955361i	$-0.595467\pi$
$761$	−16.3002			−0.590883			−0.295442	+	0.955361i	$0.595467\pi$
$762$		0			0
$763$		−	2.52279i		−	0.0913310i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	2.71124i		−	0.0978971i
$768$		0			0
$769$	−41.1262			−1.48305			−0.741525	−	0.670925i	$-0.765897\pi$
$769$	−41.1262			−1.48305			−0.741525	+	0.670925i	$0.765897\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	9.77280i		−	0.351503i	−0.984435	−	0.175752i	$-0.943764\pi$
$773$		−	9.77280i		−	0.351503i	0.984435	−	0.175752i	$-0.0562355\pi$
$774$		0			0
$775$	−5.47679	−	1.28100i	−0.196732	−	0.0460147i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−34.3207			−1.22967
$780$		0			0
$781$	−6.15020			−0.220072
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−14.1840	−	1.63670i	−0.506248	−	0.0584162i
$786$		0			0
$787$		−	5.25466i		−	0.187308i	−0.995605	−	0.0936541i	$-0.970145\pi$
$787$		−	5.25466i		−	0.187308i	0.995605	−	0.0936541i	$-0.0298548\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−11.8511			−0.421377
$792$		0			0
$793$			0.789082i			0.0280211i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	14.2133i		−	0.503460i	−0.967797	−	0.251730i	$-0.919000\pi$
$797$		−	14.2133i		−	0.503460i	0.967797	−	0.251730i	$-0.0809995\pi$
$798$		0			0
$799$	−0.830017			−0.0293639
$800$		0			0
$801$		0

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)

\(f(q)\)	\(=\)	\(q+(0.256321 - 2.22133i) q^{5} -1.08258i q^{7} +O(q^{10})\)	\(q+(0.256321 - 2.22133i) q^{5} -1.08258i q^{7} -1.00000 q^{11} +4.00000i q^{13} -0.107866i q^{17} +6.61228 q^{19} +5.97235i q^{23} +(-4.86860 - 1.13874i) q^{25} +7.80273 q^{29} +1.12492 q^{31} +(-2.40477 - 0.277489i) q^{35} +7.05494i q^{37} -5.19045 q^{41} +5.52969i q^{43} -7.69486i q^{47} +5.82801 q^{49} +4.77745i q^{53} +(-0.256321 + 2.22133i) q^{55} -0.677809 q^{59} +0.197271 q^{61} +(8.88531 + 1.02528i) q^{65} +1.41737i q^{67} +6.15020 q^{71} +6.16517i q^{73} +1.08258i q^{77} +9.35562 q^{79} -13.7454i q^{83} +(-0.239607 - 0.0276484i) q^{85} -1.29009 q^{89} +4.33034 q^{91} +(1.69486 - 14.6880i) q^{95} +6.60782i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{11} + 18 q^{19} - 2 q^{25} + 6 q^{29} - 30 q^{31} - 30 q^{35} - 20 q^{41} - 18 q^{49} + 12 q^{59} + 58 q^{61} + 8 q^{65} + 2 q^{71} + 40 q^{79} - 26 q^{85} + 42 q^{89} + 8 q^{91} - 28 q^{95}+O(q^{100}) \) 8 * q - 8 * q^11 + 18 * q^19 - 2 * q^25 + 6 * q^29 - 30 * q^31 - 30 * q^35 - 20 * q^41 - 18 * q^49 + 12 * q^59 + 58 * q^61 + 8 * q^65 + 2 * q^71 + 40 * q^79 - 26 * q^85 + 42 * q^89 + 8 * q^91 - 28 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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