LMFDB - Embedded newform 3960.2.d.f.3169.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3960,2,Mod(3169,3960)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [8,0,0,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

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	$0.0655858\pi$
$7$			1.08258i			0.409178i	−0.978848	+	0.204589i	$0.934414\pi$
$8$		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	$-0.812833\pi$
$13$		−	4.00000i		−	1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			0.107866i			0.0261614i	0.999914	+	0.0130807i	$0.00416384\pi$
$17$			0.107866i			0.0261614i	−0.999914	+	0.0130807i	$0.995836\pi$
$18$		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	$-0.786052\pi$
$23$		−	5.97235i		−	1.24532i	0.782492	−	0.622661i	$-0.213948\pi$
$24$		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	$-0.803087\pi$
$37$		−	7.05494i		−	1.15982i	0.814679	−	0.579912i	$-0.196913\pi$
$38$		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	$-0.861456\pi$
$43$		−	5.52969i		−	0.843271i	0.906766	−	0.421635i	$-0.138544\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			7.69486i			1.12241i	0.827676	+	0.561206i	$0.189662\pi$
$47$			7.69486i			1.12241i	−0.827676	+	0.561206i	$0.810338\pi$
$48$		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	$-0.893586\pi$
$53$		−	4.77745i		−	0.656233i	0.944637	−	0.328116i	$-0.106414\pi$
$54$		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	$-0.972406\pi$
$67$		−	1.41737i		−	0.173160i	0.996245	−	0.0865799i	$-0.0275938\pi$
$68$		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	$-0.882507\pi$
$73$		−	6.16517i		−	0.721578i	0.932647	−	0.360789i	$-0.117493\pi$
$74$		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	$0.272062\pi$
$83$			13.7454i			1.50876i	−0.656440	+	0.754378i	$0.727938\pi$
$84$		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	$-0.891108\pi$
$97$		−	6.60782i		−	0.670923i	0.942054	−	0.335461i	$-0.108892\pi$
$98$		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	$0.0913485\pi$
$103$			5.74543i			0.566114i	−0.959103	+	0.283057i	$0.908651\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	4.30514i		−	0.416193i	−0.978108	−	0.208097i	$-0.933273\pi$
$107$		−	4.30514i		−	0.416193i	0.978108	−	0.208097i	$-0.0667269\pi$
$108$		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	$0.172173\pi$
$113$			10.9471i			1.02981i	−0.857246	+	0.514907i	$0.827827\pi$
$114$		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	$0.188491\pi$
$127$			12.5802i			1.11631i	−0.829737	+	0.558155i	$0.811509\pi$
$128$		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	$-0.785500\pi$
$137$		−	14.6078i		−	1.24803i	0.781412	−	0.624015i	$-0.214500\pi$
$138$		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	$0.0820108\pi$
$157$			6.38535i			0.509607i	−0.966993	+	0.254803i	$0.917989\pi$
$158$		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	$0.254468\pi$
$163$			18.3071i			1.43393i	−0.697111	+	0.716963i	$0.745532\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	0.197271i		−	0.0152653i	−0.999971	−	0.00763263i	$-0.997570\pi$
$167$		−	0.197271i		−	0.0152653i	0.999971	−	0.00763263i	$-0.00242956\pi$
$168$		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	$0.189036\pi$
$173$			14.7201i			1.11915i	−0.828779	+	0.559576i	$0.810964\pi$
$174$		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	$-0.872576\pi$
$193$		−	10.8280i		−	0.779417i	0.920938	−	0.389709i	$-0.127424\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			6.72015i			0.478791i	0.970922	+	0.239395i	$0.0769492\pi$
$197$			6.72015i			0.478791i	−0.970922	+	0.239395i	$0.923051\pi$
$198$		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	$0.204007\pi$
$223$			17.8577i			1.19584i	−0.801555	+	0.597922i	$0.795993\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	1.31396i		−	0.0872107i	−0.999049	−	0.0436054i	$-0.986116\pi$
$227$		−	1.31396i		−	0.0872107i	0.999049	−	0.0436054i	$-0.0138844\pi$
$228$		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	$-0.767294\pi$
$233$		−	20.3829i		−	1.33533i	0.744462	−	0.667664i	$-0.232706\pi$
$234$		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	$-0.766849\pi$
$257$		−	21.4403i		−	1.33741i	0.743528	−	0.668704i	$-0.233151\pi$
$258$		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	$-0.935546\pi$
$263$		−	6.52287i		−	0.402218i	0.979569	−	0.201109i	$-0.0644545\pi$
$264$		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	$0.276086\pi$
$277$			25.3850i			1.52524i	−0.646849	+	0.762618i	$0.723914\pi$
$278$		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	$0.227117\pi$
$283$			22.0205i			1.30898i	−0.756070	+	0.654490i	$0.772883\pi$
$284$		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	$0.0133957\pi$
$293$			1.44029i			0.0841427i	−0.999115	+	0.0420713i	$0.986604\pi$
$294$		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	$-0.974452\pi$
$307$		−	2.80955i		−	0.160349i	0.996781	−	0.0801747i	$-0.0255478\pi$
$308$		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	$-0.735171\pi$
$313$		−	26.1580i		−	1.47854i	0.673411	−	0.739268i	$-0.264829\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	29.3805i		−	1.65018i	−0.565005	−	0.825088i	$-0.691126\pi$
$317$		−	29.3805i		−	1.65018i	0.565005	−	0.825088i	$-0.308874\pi$
$318$		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	$-0.802652\pi$
$337$		−	21.3324i		−	1.16205i	0.813885	−	0.581026i	$-0.197348\pi$
$338$		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	$-0.909814\pi$
$347$		−	10.4150i		−	0.559107i	0.960130	−	0.279553i	$-0.0901864\pi$
$348$		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	$-0.890098\pi$
$353$		−	12.7177i		−	0.676895i	0.940985	−	0.338447i	$-0.109902\pi$
$354$		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	$0.0866594\pi$
$367$			10.3027i			0.537796i	−0.963169	+	0.268898i	$0.913341\pi$
$368$		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	$-0.793531\pi$
$373$		−	23.3344i		−	1.20821i	0.796904	−	0.604105i	$-0.206469\pi$
$374$		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	$-0.992091\pi$
$383$		−	0.972434i		−	0.0496891i	0.999691	−	0.0248445i	$-0.00790908\pi$
$384$		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	$-0.908690\pi$
$397$		−	11.2751i		−	0.565882i	0.959137	−	0.282941i	$-0.0913101\pi$
$398$		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	$0.219656\pi$
$433$			26.4932i			1.27318i	−0.771201	+	0.636591i	$0.780344\pi$
$434$		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	$-0.984662\pi$
$443$		−	2.02765i		−	0.0963365i	0.998839	−	0.0481682i	$-0.0153384\pi$
$444$		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	$-0.954818\pi$
$457$		−	6.04839i		−	0.282932i	0.989943	−	0.141466i	$-0.0451816\pi$
$458$		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	$-0.751907\pi$
$463$		−	30.2474i		−	1.40572i	0.711330	−	0.702858i	$-0.248093\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			31.3417i			1.45032i	0.688580	+	0.725160i	$0.258234\pi$
$467$			31.3417i			1.45032i	−0.688580	+	0.725160i	$0.741766\pi$
$468$		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	$-0.991825\pi$
$487$		−	1.13343i		−	0.0513605i	0.999670	−	0.0256802i	$-0.00817517\pi$
$488$		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	$0.0421638\pi$
$503$			5.92424i			0.264149i	−0.991240	+	0.132074i	$0.957836\pi$
$504$		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	$0.0362002\pi$
$523$			5.19045i			0.226963i	−0.993540	+	0.113481i	$0.963800\pi$
$524$		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	$-0.911346\pi$
$547$		−	12.8601i		−	0.549859i	0.961464	−	0.274929i	$-0.0886545\pi$
$548$		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	$0.231075\pi$
$557$			31.3344i			1.32768i	−0.747874	+	0.663841i	$0.768925\pi$
$558$		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	$-0.878406\pi$
$563$		−	17.6901i		−	0.745550i	0.927922	−	0.372775i	$-0.121594\pi$
$564$		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	$-0.762569\pi$
$577$		−	32.6031i		−	1.35728i	0.734469	−	0.678642i	$-0.237431\pi$
$578$		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	$-0.881477\pi$
$587$		−	17.6287i		−	0.727612i	0.931475	−	0.363806i	$-0.118523\pi$
$588$		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	$0.0875309\pi$
$593$			13.2246i			0.543068i	−0.962429	+	0.271534i	$0.912469\pi$
$594$		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	$0.0514262\pi$
$607$			7.92624i			0.321716i	−0.986978	+	0.160858i	$0.948574\pi$
$608$		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	$-0.935693\pi$
$613$		−	9.93596i		−	0.401310i	0.979662	−	0.200655i	$-0.0643070\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	38.1010i		−	1.53389i	−0.641715	−	0.766944i	$-0.721777\pi$
$617$		−	38.1010i		−	1.53389i	0.641715	−	0.766944i	$-0.278223\pi$
$618$		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	$0.166186\pi$
$643$			25.2911i			0.997385i	−0.866779	+	0.498693i	$0.833814\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	22.4262i		−	0.881665i	−0.897589	−	0.440832i	$-0.854683\pi$
$647$		−	22.4262i		−	0.881665i	0.897589	−	0.440832i	$-0.145317\pi$
$648$		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	$0.0744125\pi$
$653$			11.8391i			0.463301i	−0.972799	+	0.231650i	$0.925587\pi$
$654$		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	$0.0711279\pi$
$673$			11.4976i			0.443200i	−0.975138	+	0.221600i	$0.928872\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			21.9953i			0.845347i	0.906282	+	0.422673i	$0.138908\pi$
$677$			21.9953i			0.845347i	−0.906282	+	0.422673i	$0.861092\pi$
$678$		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	$0.00285194\pi$
$683$			0.468300i			0.0179190i	−0.999960	+	0.00895950i	$0.997148\pi$
$684$		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	$0.216798\pi$
$727$			33.9540i			1.25928i	−0.776886	+	0.629642i	$0.783202\pi$
$728$		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	$-0.872934\pi$
$733$		−	21.0457i		−	0.777342i	0.921377	−	0.388671i	$-0.127066\pi$
$734$		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	$0.143523\pi$
$743$			23.7563i			0.871536i	−0.900059	+	0.435768i	$0.856477\pi$
$744$		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	$0.270872\pi$
$757$			41.3761i			1.50384i	−0.659255	+	0.751920i	$0.729128\pi$
$758$		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	$0.0562355\pi$
$773$			9.77280i			0.351503i	−0.984435	+	0.175752i	$0.943764\pi$
$774$		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	$0.0298548\pi$
$787$			5.25466i			0.187308i	−0.995605	+	0.0936541i	$0.970145\pi$
$788$		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	$0.0809995\pi$
$797$			14.2133i			0.503460i	−0.967797	+	0.251730i	$0.919000\pi$
$798$		0			0
$799$	−0.830017			−0.0293639
$800$		0			0
$801$		0			0
$802$		0			0
$803$			6.16517i			0.217564i
$804$		0			0
$805$	1.65726	+	14.3622i	0.0584107	+	0.506200i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−12.8641			−0.452279			−0.226139	−	0.974095i	$-0.572610\pi$
$809$	−12.8641			−0.452279			−0.226139	+	0.974095i	$0.572610\pi$
$810$		0			0
$811$	4.31605			0.151557			0.0757785	−	0.997125i	$-0.475856\pi$
$811$	4.31605			0.151557			0.0757785	+	0.997125i	$0.475856\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−40.6662	+	4.69250i	−1.42447	+	0.164371i
$816$		0			0
$817$		−	36.5639i		−	1.27921i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	42.9994			1.50069			0.750344	−	0.661048i	$-0.229888\pi$
$821$	42.9994			1.50069			0.750344	+	0.661048i	$0.229888\pi$
$822$		0			0
$823$			48.1834i			1.67957i	0.542922	+	0.839783i	$0.317318\pi$
$823$			48.1834i			1.67957i	−0.542922	+	0.839783i	$0.682682\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	32.2868i		−	1.12272i	−0.827571	−	0.561360i	$-0.810278\pi$
$827$		−	32.2868i		−	1.12272i	0.827571	−	0.561360i	$-0.189722\pi$
$828$		0			0
$829$	−54.9345			−1.90795			−0.953977	−	0.299881i	$-0.903053\pi$
$829$	−54.9345			−1.90795			−0.953977	+	0.299881i	$0.903053\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			0.628646i			0.0217813i
$834$		0			0
$835$	0.438203	−	0.0505645i	0.0151646	−	0.00174986i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	11.2281			0.387635			0.193818	−	0.981038i	$-0.437913\pi$
$839$	11.2281			0.387635			0.193818	+	0.981038i	$0.437913\pi$
$840$		0			0
$841$	31.8826			1.09940
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.768962	−	6.66399i	−0.0264531	−	0.229248i
$846$		0			0
$847$			1.08258i			0.0371980i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−42.1346			−1.44435
$852$		0			0
$853$		−	2.40328i		−	0.0822869i	−0.999153	−	0.0411434i	$-0.986900\pi$
$853$		−	2.40328i		−	0.0822869i	0.999153	−	0.0411434i	$-0.0131001\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	34.8280i		−	1.18970i	−0.803836	−	0.594851i	$-0.797211\pi$
$857$		−	34.8280i		−	1.18970i	0.803836	−	0.594851i	$-0.202789\pi$
$858$		0			0
$859$	−27.0627			−0.923368			−0.461684	−	0.887044i	$-0.652755\pi$
$859$	−27.0627			−0.923368			−0.461684	+	0.887044i	$0.652755\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			40.0157i			1.36215i	0.732213	+	0.681076i	$0.238488\pi$
$863$			40.0157i			1.36215i	−0.732213	+	0.681076i	$0.761512\pi$
$864$		0			0
$865$	−32.6983	+	3.77308i	−1.11177	+	0.128288i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−9.35562			−0.317368
$870$		0			0
$871$	−5.66950			−0.192104
$872$		0			0
$873$		0			0
$874$		0			0
$875$	11.3919	−	4.08940i	0.385116	−	0.138247i
$876$		0			0
$877$		−	18.3167i		−	0.618511i	−0.950979	−	0.309255i	$-0.899920\pi$
$877$		−	18.3167i		−	0.618511i	0.950979	−	0.309255i	$-0.100080\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−11.8560			−0.399438			−0.199719	−	0.979853i	$-0.564003\pi$
$881$	−11.8560			−0.399438			−0.199719	+	0.979853i	$0.564003\pi$
$882$		0			0
$883$		−	15.2478i		−	0.513128i	−0.966527	−	0.256564i	$-0.917410\pi$
$883$		−	15.2478i		−	0.513128i	0.966527	−	0.256564i	$-0.0825904\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	3.29631i		−	0.110679i	−0.998468	−	0.0553397i	$-0.982376\pi$
$887$		−	3.29631i		−	0.110679i	0.998468	−	0.0553397i	$-0.0176242\pi$
$888$		0			0
$889$	−13.6191			−0.456770
$890$		0			0
$891$		0			0
$892$		0			0
$893$			50.8806i			1.70265i
$894$		0			0
$895$	3.03786	+	26.3267i	0.101544	+	0.880006i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8.77745			0.292744
$900$		0			0
$901$	0.515326			0.0171680
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.75865	−	23.9070i	−0.0917005	−	0.794696i
$906$		0			0
$907$		−	16.3577i		−	0.543149i	−0.962417	−	0.271574i	$-0.912456\pi$
$907$		−	16.3577i		−	0.543149i	0.962417	−	0.271574i	$-0.0875443\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	8.34598			0.276515			0.138257	−	0.990396i	$-0.455850\pi$
$911$	8.34598			0.276515			0.138257	+	0.990396i	$0.455850\pi$
$912$		0			0
$913$		−	13.7454i		−	0.454907i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	16.5019i		−	0.544939i
$918$		0			0
$919$	12.9611			0.427546			0.213773	−	0.976883i	$-0.431425\pi$
$919$	12.9611			0.427546			0.213773	+	0.976883i	$0.431425\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	24.6008i		−	0.809746i
$924$		0			0
$925$	8.03377	+	34.3477i	0.264149	+	1.12934i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	37.2635			1.22258			0.611288	−	0.791408i	$-0.290652\pi$
$929$	37.2635			1.22258			0.611288	+	0.791408i	$0.290652\pi$
$930$		0			0
$931$	38.5364			1.26298
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.239607	−	0.0276484i	0.00783597	−	0.000904198i
$936$		0			0
$937$		−	4.28395i		−	0.139950i	−0.997549	−	0.0699752i	$-0.977708\pi$
$937$		−	4.28395i		−	0.139950i	0.997549	−	0.0699752i	$-0.0222920\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−23.2089			−0.756589			−0.378294	−	0.925685i	$-0.623489\pi$
$941$	−23.2089			−0.756589			−0.378294	+	0.925685i	$0.623489\pi$
$942$		0			0
$943$			30.9992i			1.00947i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	20.8646i		−	0.678007i	−0.940785	−	0.339004i	$-0.889910\pi$
$947$		−	20.8646i		−	0.678007i	0.940785	−	0.339004i	$-0.110090\pi$
$948$		0			0
$949$	−24.6607			−0.800519
$950$		0			0
$951$		0			0
$952$		0			0
$953$			45.0061i			1.45789i	0.684572	+	0.728945i	$0.259989\pi$
$953$			45.0061i			1.45789i	−0.684572	+	0.728945i	$0.740011\pi$
$954$		0			0
$955$	−0.436537	−	3.78313i	−0.0141260	−	0.122419i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	15.8142			0.510667
$960$		0			0
$961$	−29.7346			−0.959179
$962$		0			0
$963$		0			0
$964$		0			0
$965$	24.0526	−	2.77544i	0.774280	−	0.0893446i
$966$		0			0
$967$		−	21.9679i		−	0.706440i	−0.935540	−	0.353220i	$-0.885087\pi$
$967$		−	21.9679i		−	0.706440i	0.935540	−	0.353220i	$-0.114913\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	16.5837			0.532195			0.266098	−	0.963946i	$-0.414266\pi$
$971$	16.5837			0.532195			0.266098	+	0.963946i	$0.414266\pi$
$972$		0			0
$973$			22.1010i			0.708524i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	19.8189i		−	0.634063i	−0.948415	−	0.317032i	$-0.897314\pi$
$977$		−	19.8189i		−	0.634063i	0.948415	−	0.317032i	$-0.102686\pi$
$978$		0			0
$979$	1.29009			0.0412314
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	13.9634i		−	0.445365i	−0.974891	−	0.222682i	$-0.928519\pi$
$983$		−	13.9634i		−	0.445365i	0.974891	−	0.222682i	$-0.0714813\pi$
$984$		0			0
$985$	−14.9276	+	1.72251i	−0.475634	+	0.0548838i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−33.0253			−1.05014
$990$		0			0
$991$	4.10113			0.130277			0.0651383	−	0.997876i	$-0.479251\pi$
$991$	4.10113			0.130277			0.0651383	+	0.997876i	$0.479251\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2.77544	+	24.0526i	0.0879874	+	0.762518i
$996$		0			0
$997$			33.3208i			1.05528i	0.849468	+	0.527640i	$0.176923\pi$
$997$			33.3208i			1.05528i	−0.849468	+	0.527640i	$0.823077\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3960.2.d.f.3169.6		8
3.2	odd	2		440.2.b.d.89.1	✓	8
5.4	even	2	inner	3960.2.d.f.3169.5		8
12.11	even	2		880.2.b.j.529.8		8
15.2	even	4		2200.2.a.x.1.1		4
15.8	even	4		2200.2.a.y.1.4		4
15.14	odd	2		440.2.b.d.89.8	yes	8
60.23	odd	4		4400.2.a.cb.1.1		4
60.47	odd	4		4400.2.a.ce.1.4		4
60.59	even	2		880.2.b.j.529.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.b.d.89.1	✓	8	3.2	odd	2
440.2.b.d.89.8	yes	8	15.14	odd	2
880.2.b.j.529.1		8	60.59	even	2
880.2.b.j.529.8		8	12.11	even	2
2200.2.a.x.1.1		4	15.2	even	4
2200.2.a.y.1.4		4	15.8	even	4
3960.2.d.f.3169.5		8	5.4	even	2	inner
3960.2.d.f.3169.6		8	1.1	even	1	trivial
4400.2.a.cb.1.1		4	60.23	odd	4
4400.2.a.ce.1.4		4	60.47	odd	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 $\F_{q}$
Belyi maps

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)

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