LMFDB - Embedded newform 3960.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3960,2,Mod(1,3960)]

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, 0, 0]))

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

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

chi := DirichletCharacter("3960.1");

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.a (trivial)

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:	yes
Analytic conductor:	$31.6207592004$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	3960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +1.41421 q^{7} +1.00000 q^{11} -0.585786 q^{13} +3.41421 q^{17} -5.65685 q^{19} +2.82843 q^{23} +1.00000 q^{25} -0.828427 q^{29} +6.48528 q^{31} -1.41421 q^{35} -7.65685 q^{37} +10.4853 q^{41} +2.58579 q^{43} -2.82843 q^{47} -5.00000 q^{49} +7.17157 q^{53} -1.00000 q^{55} +10.4853 q^{59} -3.17157 q^{61} +0.585786 q^{65} -8.48528 q^{67} +3.17157 q^{71} +7.41421 q^{73} +1.41421 q^{77} +1.65685 q^{79} -0.242641 q^{83} -3.41421 q^{85} +4.34315 q^{89} -0.828427 q^{91} +5.65685 q^{95} +0.828427 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 2 q^{11} - 4 q^{13} + 4 q^{17} + 2 q^{25} + 4 q^{29} - 4 q^{31} - 4 q^{37} + 4 q^{41} + 8 q^{43} - 10 q^{49} + 20 q^{53} - 2 q^{55} + 4 q^{59} - 12 q^{61} + 4 q^{65} + 12 q^{71} + 12 q^{73}+ \cdots - 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 2 * q^11 - 4 * q^13 + 4 * q^17 + 2 * q^25 + 4 * q^29 - 4 * q^31 - 4 * q^37 + 4 * q^41 + 8 * q^43 - 10 * q^49 + 20 * q^53 - 2 * q^55 + 4 * q^59 - 12 * q^61 + 4 * q^65 + 12 * q^71 + 12 * q^73 - 8 * q^79 + 8 * q^83 - 4 * q^85 + 20 * q^89 + 4 * q^91 - 4 * q^97

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.41421	0.534522	0.267261	−	0.963624i	$-0.413881\pi$
$7$	1.41421	0.534522	0.267261	+	0.963624i	$0.413881\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−0.585786	−0.162468	−0.0812340	−	0.996695i	$-0.525886\pi$
$13$	−0.585786	−0.162468	−0.0812340	+	0.996695i	$0.525886\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.41421	0.828068	0.414034	−	0.910261i	$-0.364119\pi$
$17$	3.41421	0.828068	0.414034	+	0.910261i	$0.364119\pi$
$18$	0	0
$19$	−5.65685	−1.29777	−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685	−1.29777	−0.648886	+	0.760886i	$0.724765\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.828427	−0.153835	−0.0769175	−	0.997037i	$-0.524508\pi$
$29$	−0.828427	−0.153835	−0.0769175	+	0.997037i	$0.524508\pi$
$30$	0	0
$31$	6.48528	1.16479	0.582395	−	0.812906i	$-0.302116\pi$
$31$	6.48528	1.16479	0.582395	+	0.812906i	$0.302116\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.41421	−0.239046
$36$	0	0
$37$	−7.65685	−1.25878	−0.629390	−	0.777090i	$-0.716695\pi$
$37$	−7.65685	−1.25878	−0.629390	+	0.777090i	$0.716695\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.4853	1.63753	0.818763	−	0.574132i	$-0.194660\pi$
$41$	10.4853	1.63753	0.818763	+	0.574132i	$0.194660\pi$
$42$	0	0
$43$	2.58579	0.394329	0.197164	−	0.980370i	$-0.436827\pi$
$43$	2.58579	0.394329	0.197164	+	0.980370i	$0.436827\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.17157	0.985091	0.492546	−	0.870287i	$-0.336066\pi$
$53$	7.17157	0.985091	0.492546	+	0.870287i	$0.336066\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.4853	1.36507	0.682534	−	0.730854i	$-0.260878\pi$
$59$	10.4853	1.36507	0.682534	+	0.730854i	$0.260878\pi$
$60$	0	0
$61$	−3.17157	−0.406078	−0.203039	−	0.979171i	$-0.565082\pi$
$61$	−3.17157	−0.406078	−0.203039	+	0.979171i	$0.565082\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.585786	0.0726579
$66$	0	0
$67$	−8.48528	−1.03664	−0.518321	−	0.855186i	$-0.673443\pi$
$67$	−8.48528	−1.03664	−0.518321	+	0.855186i	$0.673443\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.17157	0.376396	0.188198	−	0.982131i	$-0.439735\pi$
$71$	3.17157	0.376396	0.188198	+	0.982131i	$0.439735\pi$
$72$	0	0
$73$	7.41421	0.867768	0.433884	−	0.900969i	$-0.357143\pi$
$73$	7.41421	0.867768	0.433884	+	0.900969i	$0.357143\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.41421	0.161165
$78$	0	0
$79$	1.65685	0.186411	0.0932053	−	0.995647i	$-0.470289\pi$
$79$	1.65685	0.186411	0.0932053	+	0.995647i	$0.470289\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−0.242641	−0.0266333	−0.0133166	−	0.999911i	$-0.504239\pi$
$83$	−0.242641	−0.0266333	−0.0133166	+	0.999911i	$0.504239\pi$
$84$	0	0
$85$	−3.41421	−0.370323
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.34315	0.460373	0.230186	−	0.973147i	$-0.426066\pi$
$89$	4.34315	0.460373	0.230186	+	0.973147i	$0.426066\pi$
$90$	0	0
$91$	−0.828427	−0.0868428
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.65685	0.580381
$96$	0	0
$97$	0.828427	0.0841140	0.0420570	−	0.999115i	$-0.486609\pi$
$97$	0.828427	0.0841140	0.0420570	+	0.999115i	$0.486609\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.65685	−0.761885	−0.380943	−	0.924599i	$-0.624401\pi$
$101$	−7.65685	−0.761885	−0.380943	+	0.924599i	$0.624401\pi$
$102$	0	0
$103$	6.34315	0.625009	0.312504	−	0.949916i	$-0.398832\pi$
$103$	6.34315	0.625009	0.312504	+	0.949916i	$0.398832\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.89949	−0.570326	−0.285163	−	0.958479i	$-0.592048\pi$
$107$	−5.89949	−0.570326	−0.285163	+	0.958479i	$0.592048\pi$
$108$	0	0
$109$	14.9706	1.43392	0.716960	−	0.697114i	$-0.245533\pi$
$109$	14.9706	1.43392	0.716960	+	0.697114i	$0.245533\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0.343146	0.0322804	0.0161402	−	0.999870i	$-0.494862\pi$
$113$	0.343146	0.0322804	0.0161402	+	0.999870i	$0.494862\pi$
$114$	0	0
$115$	−2.82843	−0.263752
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.82843	0.442621
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	3.75736	0.333412	0.166706	−	0.986007i	$-0.446687\pi$
$127$	3.75736	0.333412	0.166706	+	0.986007i	$0.446687\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−0.485281	−0.0423992	−0.0211996	−	0.999775i	$-0.506749\pi$
$131$	−0.485281	−0.0423992	−0.0211996	+	0.999775i	$0.506749\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	0	0
$139$	2.82843	0.239904	0.119952	−	0.992780i	$-0.461726\pi$
$139$	2.82843	0.239904	0.119952	+	0.992780i	$0.461726\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.585786	−0.0489859
$144$	0	0
$145$	0.828427	0.0687971
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	10.3431	0.841713	0.420857	−	0.907127i	$-0.361730\pi$
$151$	10.3431	0.841713	0.420857	+	0.907127i	$0.361730\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.48528	−0.520910
$156$	0	0
$157$	−3.17157	−0.253119	−0.126560	−	0.991959i	$-0.540393\pi$
$157$	−3.17157	−0.253119	−0.126560	+	0.991959i	$0.540393\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	17.6569	1.38299	0.691496	−	0.722380i	$-0.256952\pi$
$163$	17.6569	1.38299	0.691496	+	0.722380i	$0.256952\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.24264	0.328305	0.164153	−	0.986435i	$-0.447511\pi$
$167$	4.24264	0.328305	0.164153	+	0.986435i	$0.447511\pi$
$168$	0	0
$169$	−12.6569	−0.973604
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18.7279	1.42386	0.711929	−	0.702252i	$-0.247822\pi$
$173$	18.7279	1.42386	0.711929	+	0.702252i	$0.247822\pi$
$174$	0	0
$175$	1.41421	0.106904
$176$	0	0
$177$	0	0
$178$	0	0
$179$	15.3137	1.14460	0.572300	−	0.820044i	$-0.306051\pi$
$179$	15.3137	1.14460	0.572300	+	0.820044i	$0.306051\pi$
$180$	0	0
$181$	−13.6569	−1.01511	−0.507553	−	0.861621i	$-0.669450\pi$
$181$	−13.6569	−1.01511	−0.507553	+	0.861621i	$0.669450\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.65685	0.562943
$186$	0	0
$187$	3.41421	0.249672
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.3431	0.748404	0.374202	−	0.927347i	$-0.377917\pi$
$191$	10.3431	0.748404	0.374202	+	0.927347i	$0.377917\pi$
$192$	0	0
$193$	−16.3848	−1.17940	−0.589701	−	0.807622i	$-0.700754\pi$
$193$	−16.3848	−1.17940	−0.589701	+	0.807622i	$0.700754\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.3848	1.16737	0.583683	−	0.811981i	$-0.301611\pi$
$197$	16.3848	1.16737	0.583683	+	0.811981i	$0.301611\pi$
$198$	0	0
$199$	11.3137	0.802008	0.401004	−	0.916076i	$-0.368661\pi$
$199$	11.3137	0.802008	0.401004	+	0.916076i	$0.368661\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.17157	−0.0822283
$204$	0	0
$205$	−10.4853	−0.732324
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.65685	−0.391293
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.58579	−0.176349
$216$	0	0
$217$	9.17157	0.622607
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	22.1421	1.48275	0.741374	−	0.671093i	$-0.234175\pi$
$223$	22.1421	1.48275	0.741374	+	0.671093i	$0.234175\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.7279	1.64125	0.820625	−	0.571467i	$-0.193626\pi$
$227$	24.7279	1.64125	0.820625	+	0.571467i	$0.193626\pi$
$228$	0	0
$229$	−25.3137	−1.67278	−0.836388	−	0.548137i	$-0.815337\pi$
$229$	−25.3137	−1.67278	−0.836388	+	0.548137i	$0.815337\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.7279	0.702810	0.351405	−	0.936224i	$-0.385704\pi$
$233$	10.7279	0.702810	0.351405	+	0.936224i	$0.385704\pi$
$234$	0	0
$235$	2.82843	0.184506
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−11.7990	−0.763213	−0.381607	−	0.924325i	$-0.624629\pi$
$239$	−11.7990	−0.763213	−0.381607	+	0.924325i	$0.624629\pi$
$240$	0	0
$241$	5.51472	0.355234	0.177617	−	0.984100i	$-0.443161\pi$
$241$	5.51472	0.355234	0.177617	+	0.984100i	$0.443161\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5.00000	0.319438
$246$	0	0
$247$	3.31371	0.210846
$248$	0	0
$249$	0	0
$250$	0	0
$251$	29.7990	1.88090	0.940448	−	0.339938i	$-0.110406\pi$
$251$	29.7990	1.88090	0.940448	+	0.339938i	$0.110406\pi$
$252$	0	0
$253$	2.82843	0.177822
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−17.7990	−1.11027	−0.555135	−	0.831760i	$-0.687334\pi$
$257$	−17.7990	−1.11027	−0.555135	+	0.831760i	$0.687334\pi$
$258$	0	0
$259$	−10.8284	−0.672846
$260$	0	0
$261$	0	0
$262$	0	0
$263$	14.5858	0.899398	0.449699	−	0.893180i	$-0.351531\pi$
$263$	14.5858	0.899398	0.449699	+	0.893180i	$0.351531\pi$
$264$	0	0
$265$	−7.17157	−0.440546
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6.34315	0.386748	0.193374	−	0.981125i	$-0.438057\pi$
$269$	6.34315	0.386748	0.193374	+	0.981125i	$0.438057\pi$
$270$	0	0
$271$	−16.9706	−1.03089	−0.515444	−	0.856923i	$-0.672373\pi$
$271$	−16.9706	−1.03089	−0.515444	+	0.856923i	$0.672373\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	−1.75736	−0.105589	−0.0527947	−	0.998605i	$-0.516813\pi$
$277$	−1.75736	−0.105589	−0.0527947	+	0.998605i	$0.516813\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.1421	−0.724339	−0.362170	−	0.932112i	$-0.617964\pi$
$281$	−12.1421	−0.724339	−0.362170	+	0.932112i	$0.617964\pi$
$282$	0	0
$283$	22.3848	1.33064	0.665318	−	0.746560i	$-0.268296\pi$
$283$	22.3848	1.33064	0.665318	+	0.746560i	$0.268296\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	14.8284	0.875294
$288$	0	0
$289$	−5.34315	−0.314303
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−13.5563	−0.791970	−0.395985	−	0.918257i	$-0.629597\pi$
$293$	−13.5563	−0.791970	−0.395985	+	0.918257i	$0.629597\pi$
$294$	0	0
$295$	−10.4853	−0.610477
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.65685	−0.0958184
$300$	0	0
$301$	3.65685	0.210778
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.17157	0.181604
$306$	0	0
$307$	32.7279	1.86788	0.933941	−	0.357428i	$-0.116346\pi$
$307$	32.7279	1.86788	0.933941	+	0.357428i	$0.116346\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−15.1716	−0.860301	−0.430150	−	0.902757i	$-0.641540\pi$
$311$	−15.1716	−0.860301	−0.430150	+	0.902757i	$0.641540\pi$
$312$	0	0
$313$	12.6274	0.713744	0.356872	−	0.934153i	$-0.383843\pi$
$313$	12.6274	0.713744	0.356872	+	0.934153i	$0.383843\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−16.6274	−0.933889	−0.466944	−	0.884287i	$-0.654645\pi$
$317$	−16.6274	−0.933889	−0.466944	+	0.884287i	$0.654645\pi$
$318$	0	0
$319$	−0.828427	−0.0463830
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−19.3137	−1.07464
$324$	0	0
$325$	−0.585786	−0.0324936
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	8.82843	0.485254	0.242627	−	0.970120i	$-0.421991\pi$
$331$	8.82843	0.485254	0.242627	+	0.970120i	$0.421991\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.48528	0.463600
$336$	0	0
$337$	−13.0711	−0.712026	−0.356013	−	0.934481i	$-0.615864\pi$
$337$	−13.0711	−0.712026	−0.356013	+	0.934481i	$0.615864\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6.48528	0.351198
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	13.8995	0.746164	0.373082	−	0.927798i	$-0.378301\pi$
$347$	13.8995	0.746164	0.373082	+	0.927798i	$0.378301\pi$
$348$	0	0
$349$	−26.4853	−1.41772	−0.708862	−	0.705347i	$-0.750791\pi$
$349$	−26.4853	−1.41772	−0.708862	+	0.705347i	$0.750791\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	25.7990	1.37314	0.686571	−	0.727063i	$-0.259115\pi$
$353$	25.7990	1.37314	0.686571	+	0.727063i	$0.259115\pi$
$354$	0	0
$355$	−3.17157	−0.168330
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.82843	−0.360391	−0.180195	−	0.983631i	$-0.557673\pi$
$359$	−6.82843	−0.360391	−0.180195	+	0.983631i	$0.557673\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7.41421	−0.388078
$366$	0	0
$367$	11.5147	0.601063	0.300532	−	0.953772i	$-0.402836\pi$
$367$	11.5147	0.601063	0.300532	+	0.953772i	$0.402836\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	10.1421	0.526553
$372$	0	0
$373$	5.75736	0.298105	0.149052	−	0.988829i	$-0.452378\pi$
$373$	5.75736	0.298105	0.149052	+	0.988829i	$0.452378\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.485281	0.0249933
$378$	0	0
$379$	−31.3137	−1.60848	−0.804239	−	0.594307i	$-0.797427\pi$
$379$	−31.3137	−1.60848	−0.804239	+	0.594307i	$0.797427\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−15.3137	−0.782494	−0.391247	−	0.920286i	$-0.627956\pi$
$383$	−15.3137	−0.782494	−0.391247	+	0.920286i	$0.627956\pi$
$384$	0	0
$385$	−1.41421	−0.0720750
$386$	0	0
$387$	0	0
$388$	0	0
$389$	28.2843	1.43407	0.717035	−	0.697037i	$-0.245499\pi$
$389$	28.2843	1.43407	0.717035	+	0.697037i	$0.245499\pi$
$390$	0	0
$391$	9.65685	0.488368
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.65685	−0.0833654
$396$	0	0
$397$	−16.8284	−0.844595	−0.422297	−	0.906457i	$-0.638776\pi$
$397$	−16.8284	−0.844595	−0.422297	+	0.906457i	$0.638776\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	12.6274	0.630583	0.315292	−	0.948995i	$-0.397898\pi$
$401$	12.6274	0.630583	0.315292	+	0.948995i	$0.397898\pi$
$402$	0	0
$403$	−3.79899	−0.189241
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.65685	−0.379536
$408$	0	0
$409$	−21.3137	−1.05390	−0.526948	−	0.849898i	$-0.676664\pi$
$409$	−21.3137	−1.05390	−0.526948	+	0.849898i	$0.676664\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	14.8284	0.729659
$414$	0	0
$415$	0.242641	0.0119108
$416$	0	0
$417$	0	0
$418$	0	0
$419$	9.65685	0.471768	0.235884	−	0.971781i	$-0.424201\pi$
$419$	9.65685	0.471768	0.235884	+	0.971781i	$0.424201\pi$
$420$	0	0
$421$	−4.68629	−0.228396	−0.114198	−	0.993458i	$-0.536430\pi$
$421$	−4.68629	−0.228396	−0.114198	+	0.993458i	$0.536430\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.41421	0.165614
$426$	0	0
$427$	−4.48528	−0.217058
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10.3431	0.498212	0.249106	−	0.968476i	$-0.419863\pi$
$431$	10.3431	0.498212	0.249106	+	0.968476i	$0.419863\pi$
$432$	0	0
$433$	28.1421	1.35242	0.676212	−	0.736707i	$-0.263620\pi$
$433$	28.1421	1.35242	0.676212	+	0.736707i	$0.263620\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−16.0000	−0.765384
$438$	0	0
$439$	−18.1421	−0.865877	−0.432938	−	0.901423i	$-0.642523\pi$
$439$	−18.1421	−0.865877	−0.432938	+	0.901423i	$0.642523\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−20.9706	−0.996342	−0.498171	−	0.867079i	$-0.665995\pi$
$443$	−20.9706	−0.996342	−0.498171	+	0.867079i	$0.665995\pi$
$444$	0	0
$445$	−4.34315	−0.205885
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	0	0
$451$	10.4853	0.493733
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.828427	0.0388373
$456$	0	0
$457$	3.21320	0.150307	0.0751537	−	0.997172i	$-0.476055\pi$
$457$	3.21320	0.150307	0.0751537	+	0.997172i	$0.476055\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−25.3137	−1.17898	−0.589488	−	0.807777i	$-0.700671\pi$
$461$	−25.3137	−1.17898	−0.589488	+	0.807777i	$0.700671\pi$
$462$	0	0
$463$	18.8284	0.875031	0.437516	−	0.899211i	$-0.355858\pi$
$463$	18.8284	0.875031	0.437516	+	0.899211i	$0.355858\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−16.4853	−0.762848	−0.381424	−	0.924400i	$-0.624566\pi$
$467$	−16.4853	−0.762848	−0.381424	+	0.924400i	$0.624566\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.58579	0.118895
$474$	0	0
$475$	−5.65685	−0.259554
$476$	0	0
$477$	0	0
$478$	0	0
$479$	25.4558	1.16311	0.581554	−	0.813508i	$-0.302445\pi$
$479$	25.4558	1.16311	0.581554	+	0.813508i	$0.302445\pi$
$480$	0	0
$481$	4.48528	0.204511
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.828427	−0.0376169
$486$	0	0
$487$	−5.17157	−0.234346	−0.117173	−	0.993111i	$-0.537383\pi$
$487$	−5.17157	−0.234346	−0.117173	+	0.993111i	$0.537383\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−20.2843	−0.915416	−0.457708	−	0.889102i	$-0.651330\pi$
$491$	−20.2843	−0.915416	−0.457708	+	0.889102i	$0.651330\pi$
$492$	0	0
$493$	−2.82843	−0.127386
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.48528	0.201192
$498$	0	0
$499$	36.9706	1.65503	0.827515	−	0.561444i	$-0.189754\pi$
$499$	36.9706	1.65503	0.827515	+	0.561444i	$0.189754\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−24.9289	−1.11153	−0.555763	−	0.831341i	$-0.687574\pi$
$503$	−24.9289	−1.11153	−0.555763	+	0.831341i	$0.687574\pi$
$504$	0	0
$505$	7.65685	0.340726
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−8.00000	−0.354594	−0.177297	−	0.984157i	$-0.556735\pi$
$509$	−8.00000	−0.354594	−0.177297	+	0.984157i	$0.556735\pi$
$510$	0	0
$511$	10.4853	0.463842
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6.34315	−0.279512
$516$	0	0
$517$	−2.82843	−0.124394
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−21.3137	−0.933771	−0.466885	−	0.884318i	$-0.654624\pi$
$521$	−21.3137	−0.933771	−0.466885	+	0.884318i	$0.654624\pi$
$522$	0	0
$523$	0.242641	0.0106099	0.00530497	−	0.999986i	$-0.498311\pi$
$523$	0.242641	0.0106099	0.00530497	+	0.999986i	$0.498311\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	22.1421	0.964527
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−6.14214	−0.266045
$534$	0	0
$535$	5.89949	0.255057
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.00000	−0.215365
$540$	0	0
$541$	−26.2843	−1.13005	−0.565025	−	0.825074i	$-0.691133\pi$
$541$	−26.2843	−1.13005	−0.565025	+	0.825074i	$0.691133\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−14.9706	−0.641268
$546$	0	0
$547$	4.92893	0.210746	0.105373	−	0.994433i	$-0.466396\pi$
$547$	4.92893	0.210746	0.105373	+	0.994433i	$0.466396\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.68629	0.199643
$552$	0	0
$553$	2.34315	0.0996407
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7.41421	−0.314150	−0.157075	−	0.987587i	$-0.550206\pi$
$557$	−7.41421	−0.314150	−0.157075	+	0.987587i	$0.550206\pi$
$558$	0	0
$559$	−1.51472	−0.0640658
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−37.4142	−1.57682	−0.788411	−	0.615149i	$-0.789096\pi$
$563$	−37.4142	−1.57682	−0.788411	+	0.615149i	$0.789096\pi$
$564$	0	0
$565$	−0.343146	−0.0144363
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−8.62742	−0.361680	−0.180840	−	0.983513i	$-0.557882\pi$
$569$	−8.62742	−0.361680	−0.180840	+	0.983513i	$0.557882\pi$
$570$	0	0
$571$	22.8284	0.955340	0.477670	−	0.878539i	$-0.341481\pi$
$571$	22.8284	0.955340	0.477670	+	0.878539i	$0.341481\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.82843	0.117954
$576$	0	0
$577$	−26.7696	−1.11443	−0.557216	−	0.830368i	$-0.688130\pi$
$577$	−26.7696	−1.11443	−0.557216	+	0.830368i	$0.688130\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−0.343146	−0.0142361
$582$	0	0
$583$	7.17157	0.297016
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.82843	−0.116742	−0.0583708	−	0.998295i	$-0.518591\pi$
$587$	−2.82843	−0.116742	−0.0583708	+	0.998295i	$0.518591\pi$
$588$	0	0
$589$	−36.6863	−1.51163
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−20.3848	−0.837102	−0.418551	−	0.908193i	$-0.637462\pi$
$593$	−20.3848	−0.837102	−0.418551	+	0.908193i	$0.637462\pi$
$594$	0	0
$595$	−4.82843	−0.197946
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−24.9706	−1.02027	−0.510135	−	0.860094i	$-0.670404\pi$
$599$	−24.9706	−1.02027	−0.510135	+	0.860094i	$0.670404\pi$
$600$	0	0
$601$	45.7990	1.86818	0.934090	−	0.357038i	$-0.116213\pi$
$601$	45.7990	1.86818	0.934090	+	0.357038i	$0.116213\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−4.24264	−0.172203	−0.0861017	−	0.996286i	$-0.527441\pi$
$607$	−4.24264	−0.172203	−0.0861017	+	0.996286i	$0.527441\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.65685	0.0670291
$612$	0	0
$613$	−2.04163	−0.0824607	−0.0412303	−	0.999150i	$-0.513128\pi$
$613$	−2.04163	−0.0824607	−0.0412303	+	0.999150i	$0.513128\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8.34315	−0.335882	−0.167941	−	0.985797i	$-0.553712\pi$
$617$	−8.34315	−0.335882	−0.167941	+	0.985797i	$0.553712\pi$
$618$	0	0
$619$	−34.4853	−1.38608	−0.693040	−	0.720899i	$-0.743729\pi$
$619$	−34.4853	−1.38608	−0.693040	+	0.720899i	$0.743729\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.14214	0.246079
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−26.1421	−1.04236
$630$	0	0
$631$	29.6569	1.18062	0.590310	−	0.807176i	$-0.299005\pi$
$631$	29.6569	1.18062	0.590310	+	0.807176i	$0.299005\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.75736	−0.149106
$636$	0	0
$637$	2.92893	0.116049
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.02944	−0.0406603	−0.0203302	−	0.999793i	$-0.506472\pi$
$641$	−1.02944	−0.0406603	−0.0203302	+	0.999793i	$0.506472\pi$
$642$	0	0
$643$	−21.9411	−0.865274	−0.432637	−	0.901568i	$-0.642417\pi$
$643$	−21.9411	−0.865274	−0.432637	+	0.901568i	$0.642417\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−28.9706	−1.13895	−0.569475	−	0.822009i	$-0.692854\pi$
$647$	−28.9706	−1.13895	−0.569475	+	0.822009i	$0.692854\pi$
$648$	0	0
$649$	10.4853	0.411583
$650$	0	0
$651$	0	0
$652$	0	0
$653$	29.1127	1.13927	0.569634	−	0.821899i	$-0.307085\pi$
$653$	29.1127	1.13927	0.569634	+	0.821899i	$0.307085\pi$
$654$	0	0
$655$	0.485281	0.0189615
$656$	0	0
$657$	0	0
$658$	0	0
$659$	18.3431	0.714548	0.357274	−	0.934000i	$-0.383706\pi$
$659$	18.3431	0.714548	0.357274	+	0.934000i	$0.383706\pi$
$660$	0	0
$661$	−16.3431	−0.635675	−0.317837	−	0.948145i	$-0.602957\pi$
$661$	−16.3431	−0.635675	−0.317837	+	0.948145i	$0.602957\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	−2.34315	−0.0907270
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3.17157	−0.122437
$672$	0	0
$673$	31.4142	1.21093	0.605464	−	0.795872i	$-0.292987\pi$
$673$	31.4142	1.21093	0.605464	+	0.795872i	$0.292987\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−31.8995	−1.22600	−0.612999	−	0.790084i	$-0.710037\pi$
$677$	−31.8995	−1.22600	−0.612999	+	0.790084i	$0.710037\pi$
$678$	0	0
$679$	1.17157	0.0449608
$680$	0	0
$681$	0	0
$682$	0	0
$683$	6.82843	0.261283	0.130641	−	0.991430i	$-0.458296\pi$
$683$	6.82843	0.261283	0.130641	+	0.991430i	$0.458296\pi$
$684$	0	0
$685$	−14.0000	−0.534913
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4.20101	−0.160046
$690$	0	0
$691$	6.34315	0.241305	0.120652	−	0.992695i	$-0.461501\pi$
$691$	6.34315	0.241305	0.120652	+	0.992695i	$0.461501\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.82843	−0.107288
$696$	0	0
$697$	35.7990	1.35598
$698$	0	0
$699$	0	0
$700$	0	0
$701$	3.17157	0.119789	0.0598943	−	0.998205i	$-0.480924\pi$
$701$	3.17157	0.119789	0.0598943	+	0.998205i	$0.480924\pi$
$702$	0	0
$703$	43.3137	1.63361
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10.8284	−0.407245
$708$	0	0
$709$	−38.3431	−1.44001	−0.720004	−	0.693970i	$-0.755860\pi$
$709$	−38.3431	−1.44001	−0.720004	+	0.693970i	$0.755860\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	18.3431	0.686956
$714$	0	0
$715$	0.585786	0.0219072
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	0	0
$721$	8.97056	0.334081
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.828427	−0.0307670
$726$	0	0
$727$	−18.3431	−0.680310	−0.340155	−	0.940369i	$-0.610479\pi$
$727$	−18.3431	−0.680310	−0.340155	+	0.940369i	$0.610479\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.82843	0.326531
$732$	0	0
$733$	14.7279	0.543988	0.271994	−	0.962299i	$-0.412317\pi$
$733$	14.7279	0.543988	0.271994	+	0.962299i	$0.412317\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.48528	−0.312559
$738$	0	0
$739$	−7.51472	−0.276433	−0.138217	−	0.990402i	$-0.544137\pi$
$739$	−7.51472	−0.276433	−0.138217	+	0.990402i	$0.544137\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	18.8701	0.692275	0.346138	−	0.938184i	$-0.387493\pi$
$743$	18.8701	0.692275	0.346138	+	0.938184i	$0.387493\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−8.34315	−0.304852
$750$	0	0
$751$	−6.62742	−0.241838	−0.120919	−	0.992662i	$-0.538584\pi$
$751$	−6.62742	−0.241838	−0.120919	+	0.992662i	$0.538584\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−10.3431	−0.376426
$756$	0	0
$757$	50.0833	1.82031	0.910154	−	0.414271i	$-0.135963\pi$
$757$	50.0833	1.82031	0.910154	+	0.414271i	$0.135963\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−38.9706	−1.41268	−0.706341	−	0.707872i	$-0.749655\pi$
$761$	−38.9706	−1.41268	−0.706341	+	0.707872i	$0.749655\pi$
$762$	0	0
$763$	21.1716	0.766462
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.14214	−0.221780
$768$	0	0
$769$	2.20101	0.0793705	0.0396852	−	0.999212i	$-0.487364\pi$
$769$	2.20101	0.0793705	0.0396852	+	0.999212i	$0.487364\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7.17157	0.257944	0.128972	−	0.991648i	$-0.458832\pi$
$773$	7.17157	0.257944	0.128972	+	0.991648i	$0.458832\pi$
$774$	0	0
$775$	6.48528	0.232958
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−59.3137	−2.12513
$780$	0	0
$781$	3.17157	0.113488
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.17157	0.113198
$786$	0	0
$787$	35.5563	1.26745	0.633723	−	0.773560i	$-0.281526\pi$
$787$	35.5563	1.26745	0.633723	+	0.773560i	$0.281526\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0.485281	0.0172546
$792$	0	0
$793$	1.85786	0.0659747
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	−9.65685	−0.341635
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.41421	0.261642
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−25.5147	−0.897050	−0.448525	−	0.893770i	$-0.648050\pi$
$809$	−25.5147	−0.897050	−0.448525	+	0.893770i	$0.648050\pi$
$810$	0	0
$811$	−27.7990	−0.976155	−0.488077	−	0.872800i	$-0.662302\pi$
$811$	−27.7990	−0.976155	−0.488077	+	0.872800i	$0.662302\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−17.6569	−0.618493
$816$	0	0
$817$	−14.6274	−0.511749
$818$	0	0
$819$	0	0
$820$	0	0
$821$	29.7990	1.03999	0.519996	−	0.854169i	$-0.325933\pi$
$821$	29.7990	1.03999	0.519996	+	0.854169i	$0.325933\pi$
$822$	0	0
$823$	−16.9706	−0.591557	−0.295778	−	0.955257i	$-0.595579\pi$
$823$	−16.9706	−0.591557	−0.295778	+	0.955257i	$0.595579\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.5269	1.27017	0.635083	−	0.772444i	$-0.280966\pi$
$827$	36.5269	1.27017	0.635083	+	0.772444i	$0.280966\pi$
$828$	0	0
$829$	−35.5980	−1.23637	−0.618184	−	0.786033i	$-0.712132\pi$
$829$	−35.5980	−1.23637	−0.618184	+	0.786033i	$0.712132\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−17.0711	−0.591477
$834$	0	0
$835$	−4.24264	−0.146823
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−11.4558	−0.395500	−0.197750	−	0.980253i	$-0.563363\pi$
$839$	−11.4558	−0.395500	−0.197750	+	0.980253i	$0.563363\pi$
$840$	0	0
$841$	−28.3137	−0.976335
$842$	0	0
$843$	0	0
$844$	0	0
$845$	12.6569	0.435409
$846$	0	0
$847$	1.41421	0.0485930
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−21.6569	−0.742387
$852$	0	0
$853$	6.72792	0.230360	0.115180	−	0.993345i	$-0.463256\pi$
$853$	6.72792	0.230360	0.115180	+	0.993345i	$0.463256\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.21320	0.109761	0.0548805	−	0.998493i	$-0.482522\pi$
$857$	3.21320	0.109761	0.0548805	+	0.998493i	$0.482522\pi$
$858$	0	0
$859$	−50.4853	−1.72254	−0.861268	−	0.508151i	$-0.830329\pi$
$859$	−50.4853	−1.72254	−0.861268	+	0.508151i	$0.830329\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−19.0294	−0.647770	−0.323885	−	0.946097i	$-0.604989\pi$
$863$	−19.0294	−0.647770	−0.323885	+	0.946097i	$0.604989\pi$
$864$	0	0
$865$	−18.7279	−0.636768
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.65685	0.0562049
$870$	0	0
$871$	4.97056	0.168421
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.41421	−0.0478091
$876$	0	0
$877$	22.5269	0.760680	0.380340	−	0.924847i	$-0.375807\pi$
$877$	22.5269	0.760680	0.380340	+	0.924847i	$0.375807\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−35.3137	−1.18975	−0.594875	−	0.803819i	$-0.702798\pi$
$881$	−35.3137	−1.18975	−0.594875	+	0.803819i	$0.702798\pi$
$882$	0	0
$883$	−3.02944	−0.101949	−0.0509743	−	0.998700i	$-0.516233\pi$
$883$	−3.02944	−0.101949	−0.0509743	+	0.998700i	$0.516233\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−20.2426	−0.679681	−0.339841	−	0.940483i	$-0.610373\pi$
$887$	−20.2426	−0.679681	−0.339841	+	0.940483i	$0.610373\pi$
$888$	0	0
$889$	5.31371	0.178216
$890$	0	0
$891$	0	0
$892$	0	0
$893$	16.0000	0.535420
$894$	0	0
$895$	−15.3137	−0.511881
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−5.37258	−0.179186
$900$	0	0
$901$	24.4853	0.815723
$902$	0	0
$903$	0	0
$904$	0	0
$905$	13.6569	0.453969
$906$	0	0
$907$	24.7696	0.822459	0.411230	−	0.911532i	$-0.365099\pi$
$907$	24.7696	0.822459	0.411230	+	0.911532i	$0.365099\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	12.2843	0.406996	0.203498	−	0.979075i	$-0.434769\pi$
$911$	12.2843	0.406996	0.203498	+	0.979075i	$0.434769\pi$
$912$	0	0
$913$	−0.242641	−0.00803023
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.686292	−0.0226633
$918$	0	0
$919$	−43.7990	−1.44480	−0.722398	−	0.691478i	$-0.756960\pi$
$919$	−43.7990	−1.44480	−0.722398	+	0.691478i	$0.756960\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.85786	−0.0611524
$924$	0	0
$925$	−7.65685	−0.251756
$926$	0	0
$927$	0	0
$928$	0	0
$929$	34.6274	1.13609	0.568044	−	0.822998i	$-0.307700\pi$
$929$	34.6274	1.13609	0.568044	+	0.822998i	$0.307700\pi$
$930$	0	0
$931$	28.2843	0.926980
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3.41421	−0.111657
$936$	0	0
$937$	22.7279	0.742489	0.371244	−	0.928535i	$-0.378931\pi$
$937$	22.7279	0.742489	0.371244	+	0.928535i	$0.378931\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−25.3137	−0.825203	−0.412602	−	0.910912i	$-0.635380\pi$
$941$	−25.3137	−0.825203	−0.412602	+	0.910912i	$0.635380\pi$
$942$	0	0
$943$	29.6569	0.965760
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.48528	0.275735	0.137867	−	0.990451i	$-0.455975\pi$
$947$	8.48528	0.275735	0.137867	+	0.990451i	$0.455975\pi$
$948$	0	0
$949$	−4.34315	−0.140984
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3.89949	0.126317	0.0631585	−	0.998004i	$-0.479883\pi$
$953$	3.89949	0.126317	0.0631585	+	0.998004i	$0.479883\pi$
$954$	0	0
$955$	−10.3431	−0.334696
$956$	0	0
$957$	0	0
$958$	0	0
$959$	19.7990	0.639343
$960$	0	0
$961$	11.0589	0.356738
$962$	0	0
$963$	0	0
$964$	0	0
$965$	16.3848	0.527445
$966$	0	0
$967$	19.7574	0.635354	0.317677	−	0.948199i	$-0.397097\pi$
$967$	19.7574	0.635354	0.317677	+	0.948199i	$0.397097\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0.686292	0.0220241	0.0110121	−	0.999939i	$-0.496495\pi$
$971$	0.686292	0.0220241	0.0110121	+	0.999939i	$0.496495\pi$
$972$	0	0
$973$	4.00000	0.128234
$974$	0	0
$975$	0	0
$976$	0	0
$977$	22.9706	0.734893	0.367447	−	0.930045i	$-0.380232\pi$
$977$	22.9706	0.734893	0.367447	+	0.930045i	$0.380232\pi$
$978$	0	0
$979$	4.34315	0.138808
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−31.3137	−0.998752	−0.499376	−	0.866385i	$-0.666437\pi$
$983$	−31.3137	−0.998752	−0.499376	+	0.866385i	$0.666437\pi$
$984$	0	0
$985$	−16.3848	−0.522062
$986$	0	0
$987$	0	0
$988$	0	0
$989$	7.31371	0.232562
$990$	0	0
$991$	−32.8284	−1.04283	−0.521415	−	0.853303i	$-0.674596\pi$
$991$	−32.8284	−1.04283	−0.521415	+	0.853303i	$0.674596\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−11.3137	−0.358669
$996$	0	0
$997$	−26.0416	−0.824747	−0.412373	−	0.911015i	$-0.635300\pi$
$997$	−26.0416	−0.824747	−0.412373	+	0.911015i	$0.635300\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3960.2.a.y.1.2	✓	2
3.2	odd	2		3960.2.a.bd.1.2	yes	2
4.3	odd	2		7920.2.a.br.1.1		2
12.11	even	2		7920.2.a.cc.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3960.2.a.y.1.2	✓	2	1.1	even	1	trivial
3960.2.a.bd.1.2	yes	2	3.2	odd	2
7920.2.a.br.1.1		2	4.3	odd	2
7920.2.a.cc.1.1		2	12.11	even	2

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.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +1.41421 q^{7} +1.00000 q^{11} -0.585786 q^{13} +3.41421 q^{17} -5.65685 q^{19} +2.82843 q^{23} +1.00000 q^{25} -0.828427 q^{29} +6.48528 q^{31} -1.41421 q^{35} -7.65685 q^{37} +10.4853 q^{41} +2.58579 q^{43} -2.82843 q^{47} -5.00000 q^{49} +7.17157 q^{53} -1.00000 q^{55} +10.4853 q^{59} -3.17157 q^{61} +0.585786 q^{65} -8.48528 q^{67} +3.17157 q^{71} +7.41421 q^{73} +1.41421 q^{77} +1.65685 q^{79} -0.242641 q^{83} -3.41421 q^{85} +4.34315 q^{89} -0.828427 q^{91} +5.65685 q^{95} +0.828427 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 2 q^{11} - 4 q^{13} + 4 q^{17} + 2 q^{25} + 4 q^{29} - 4 q^{31} - 4 q^{37} + 4 q^{41} + 8 q^{43} - 10 q^{49} + 20 q^{53} - 2 q^{55} + 4 q^{59} - 12 q^{61} + 4 q^{65} + 12 q^{71} + 12 q^{73}+ \cdots - 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 2 * q^11 - 4 * q^13 + 4 * q^17 + 2 * q^25 + 4 * q^29 - 4 * q^31 - 4 * q^37 + 4 * q^41 + 8 * q^43 - 10 * q^49 + 20 * q^53 - 2 * q^55 + 4 * q^59 - 12 * q^61 + 4 * q^65 + 12 * q^71 + 12 * q^73 - 8 * q^79 + 8 * q^83 - 4 * q^85 + 20 * q^89 + 4 * q^91 - 4 * q^97

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