LMFDB - Embedded newform 7920.2.a.bs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7920,2,Mod(1,7920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7920, 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("7920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7920.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:	$63.2415184009$
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:	$ 2 $
Twist minimal:	no (minimal twist has level 1320)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	7920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -2.82843 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -2.82843 q^{7} -1.00000 q^{11} -0.828427 q^{13} -4.82843 q^{17} -5.65685 q^{19} -5.65685 q^{23} +1.00000 q^{25} -2.00000 q^{29} +5.65685 q^{31} +2.82843 q^{35} +6.00000 q^{37} +2.00000 q^{41} -6.82843 q^{43} -5.65685 q^{47} +1.00000 q^{49} -11.6569 q^{53} +1.00000 q^{55} -1.65685 q^{59} +11.6569 q^{61} +0.828427 q^{65} +1.65685 q^{67} -2.34315 q^{71} -2.48528 q^{73} +2.82843 q^{77} +1.65685 q^{79} +5.17157 q^{83} +4.82843 q^{85} -2.00000 q^{89} +2.34315 q^{91} +5.65685 q^{95} +15.6569 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5}+O(q^{10}) $ 2 * q - 2 * q^5	$ 2 q - 2 q^{5} - 2 q^{11} + 4 q^{13} - 4 q^{17} + 2 q^{25} - 4 q^{29} + 12 q^{37} + 4 q^{41} - 8 q^{43} + 2 q^{49} - 12 q^{53} + 2 q^{55} + 8 q^{59} + 12 q^{61} - 4 q^{65} - 8 q^{67} - 16 q^{71} + 12 q^{73} - 8 q^{79} + 16 q^{83} + 4 q^{85} - 4 q^{89} + 16 q^{91} + 20 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 + 12 * q^37 + 4 * q^41 - 8 * q^43 + 2 * q^49 - 12 * q^53 + 2 * q^55 + 8 * q^59 + 12 * q^61 - 4 * q^65 - 8 * q^67 - 16 * q^71 + 12 * q^73 - 8 * q^79 + 16 * q^83 + 4 * q^85 - 4 * q^89 + 16 * q^91 + 20 * q^97

Display 100 coefficients 10 coefficients

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
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.828427	−0.229764	−0.114882	−	0.993379i	$-0.536649\pi$
$13$	−0.828427	−0.229764	−0.114882	+	0.993379i	$0.536649\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.82843	−1.17107	−0.585533	−	0.810649i	$-0.699115\pi$
$17$	−4.82843	−1.17107	−0.585533	+	0.810649i	$0.699115\pi$
$18$	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
$23$	−5.65685	−1.17954	−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.82843	0.478091
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−6.82843	−1.04133	−0.520663	−	0.853762i	$-0.674315\pi$
$43$	−6.82843	−1.04133	−0.520663	+	0.853762i	$0.674315\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.65685	−0.825137	−0.412568	−	0.910927i	$-0.635368\pi$
$47$	−5.65685	−0.825137	−0.412568	+	0.910927i	$0.635368\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−11.6569	−1.60119	−0.800596	−	0.599204i	$-0.795484\pi$
$53$	−11.6569	−1.60119	−0.800596	+	0.599204i	$0.795484\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.65685	−0.215704	−0.107852	−	0.994167i	$-0.534397\pi$
$59$	−1.65685	−0.215704	−0.107852	+	0.994167i	$0.534397\pi$
$60$	0	0
$61$	11.6569	1.49251	0.746254	−	0.665662i	$-0.231851\pi$
$61$	11.6569	1.49251	0.746254	+	0.665662i	$0.231851\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.828427	0.102754
$66$	0	0
$67$	1.65685	0.202417	0.101208	−	0.994865i	$-0.467729\pi$
$67$	1.65685	0.202417	0.101208	+	0.994865i	$0.467729\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.34315	−0.278080	−0.139040	−	0.990287i	$-0.544402\pi$
$71$	−2.34315	−0.278080	−0.139040	+	0.990287i	$0.544402\pi$
$72$	0	0
$73$	−2.48528	−0.290880	−0.145440	−	0.989367i	$-0.546460\pi$
$73$	−2.48528	−0.290880	−0.145440	+	0.989367i	$0.546460\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.82843	0.322329
$78$	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
$83$	5.17157	0.567654	0.283827	−	0.958876i	$-0.408396\pi$
$83$	5.17157	0.567654	0.283827	+	0.958876i	$0.408396\pi$
$84$	0	0
$85$	4.82843	0.523716
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	2.34315	0.245628
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.65685	0.580381
$96$	0	0
$97$	15.6569	1.58971	0.794856	−	0.606798i	$-0.207546\pi$
$97$	15.6569	1.58971	0.794856	+	0.606798i	$0.207546\pi$
$98$	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
$103$	−19.3137	−1.90304	−0.951518	−	0.307593i	$-0.900477\pi$
$103$	−19.3137	−1.90304	−0.951518	+	0.307593i	$0.900477\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.48528	0.820303	0.410152	−	0.912017i	$-0.365476\pi$
$107$	8.48528	0.820303	0.410152	+	0.912017i	$0.365476\pi$
$108$	0	0
$109$	17.3137	1.65835	0.829176	−	0.558987i	$-0.188810\pi$
$109$	17.3137	1.65835	0.829176	+	0.558987i	$0.188810\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.31371	0.876160	0.438080	−	0.898936i	$-0.355659\pi$
$113$	9.31371	0.876160	0.438080	+	0.898936i	$0.355659\pi$
$114$	0	0
$115$	5.65685	0.527504
$116$	0	0
$117$	0	0
$118$	0	0
$119$	13.6569	1.25192
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.82843	0.250982	0.125491	−	0.992095i	$-0.459949\pi$
$127$	2.82843	0.250982	0.125491	+	0.992095i	$0.459949\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−17.6569	−1.54269	−0.771343	−	0.636419i	$-0.780415\pi$
$131$	−17.6569	−1.54269	−0.771343	+	0.636419i	$0.780415\pi$
$132$	0	0
$133$	16.0000	1.38738
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.31371	−0.453981	−0.226990	−	0.973897i	$-0.572889\pi$
$137$	−5.31371	−0.453981	−0.226990	+	0.973897i	$0.572889\pi$
$138$	0	0
$139$	11.3137	0.959616	0.479808	−	0.877373i	$-0.340706\pi$
$139$	11.3137	0.959616	0.479808	+	0.877373i	$0.340706\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.828427	0.0692766
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.6569	0.954967	0.477483	−	0.878641i	$-0.341549\pi$
$149$	11.6569	0.954967	0.477483	+	0.878641i	$0.341549\pi$
$150$	0	0
$151$	20.9706	1.70656	0.853280	−	0.521453i	$-0.174610\pi$
$151$	20.9706	1.70656	0.853280	+	0.521453i	$0.174610\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5.65685	−0.454369
$156$	0	0
$157$	−10.9706	−0.875546	−0.437773	−	0.899085i	$-0.644233\pi$
$157$	−10.9706	−0.875546	−0.437773	+	0.899085i	$0.644233\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	16.0000	1.26098
$162$	0	0
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.1421	0.784822	0.392411	−	0.919790i	$-0.371641\pi$
$167$	10.1421	0.784822	0.392411	+	0.919790i	$0.371641\pi$
$168$	0	0
$169$	−12.3137	−0.947208
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18.4853	1.40541	0.702705	−	0.711481i	$-0.251975\pi$
$173$	18.4853	1.40541	0.702705	+	0.711481i	$0.251975\pi$
$174$	0	0
$175$	−2.82843	−0.213809
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	−13.3137	−0.989600	−0.494800	−	0.869007i	$-0.664759\pi$
$181$	−13.3137	−0.989600	−0.494800	+	0.869007i	$0.664759\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	4.82843	0.353090
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	8.82843	0.635484	0.317742	−	0.948177i	$-0.397075\pi$
$193$	8.82843	0.635484	0.317742	+	0.948177i	$0.397075\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.1716	1.08093	0.540465	−	0.841367i	$-0.318248\pi$
$197$	15.1716	1.08093	0.540465	+	0.841367i	$0.318248\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.65685	0.397033
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.65685	0.391293
$210$	0	0
$211$	−16.9706	−1.16830	−0.584151	−	0.811645i	$-0.698572\pi$
$211$	−16.9706	−1.16830	−0.584151	+	0.811645i	$0.698572\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.82843	0.465695
$216$	0	0
$217$	−16.0000	−1.08615
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	−8.97056	−0.600713	−0.300357	−	0.953827i	$-0.597106\pi$
$223$	−8.97056	−0.600713	−0.300357	+	0.953827i	$0.597106\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−8.48528	−0.563188	−0.281594	−	0.959534i	$-0.590863\pi$
$227$	−8.48528	−0.563188	−0.281594	+	0.959534i	$0.590863\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.48528	−0.162816	−0.0814081	−	0.996681i	$-0.525942\pi$
$233$	−2.48528	−0.162816	−0.0814081	+	0.996681i	$0.525942\pi$
$234$	0	0
$235$	5.65685	0.369012
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5.65685	−0.365911	−0.182956	−	0.983121i	$-0.558567\pi$
$239$	−5.65685	−0.365911	−0.182956	+	0.983121i	$0.558567\pi$
$240$	0	0
$241$	7.65685	0.493221	0.246611	−	0.969115i	$-0.420683\pi$
$241$	7.65685	0.493221	0.246611	+	0.969115i	$0.420683\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	4.68629	0.298182
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−20.9706	−1.32365	−0.661825	−	0.749658i	$-0.730218\pi$
$251$	−20.9706	−1.32365	−0.661825	+	0.749658i	$0.730218\pi$
$252$	0	0
$253$	5.65685	0.355643
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.97056	−0.185299	−0.0926493	−	0.995699i	$-0.529534\pi$
$257$	−2.97056	−0.185299	−0.0926493	+	0.995699i	$0.529534\pi$
$258$	0	0
$259$	−16.9706	−1.05450
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−14.8284	−0.914360	−0.457180	−	0.889374i	$-0.651140\pi$
$263$	−14.8284	−0.914360	−0.457180	+	0.889374i	$0.651140\pi$
$264$	0	0
$265$	11.6569	0.716075
$266$	0	0
$267$	0	0
$268$	0	0
$269$	29.3137	1.78729	0.893644	−	0.448776i	$-0.148140\pi$
$269$	29.3137	1.78729	0.893644	+	0.448776i	$0.148140\pi$
$270$	0	0
$271$	14.3431	0.871284	0.435642	−	0.900120i	$-0.356521\pi$
$271$	14.3431	0.871284	0.435642	+	0.900120i	$0.356521\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	7.17157	0.430898	0.215449	−	0.976515i	$-0.430878\pi$
$277$	7.17157	0.430898	0.215449	+	0.976515i	$0.430878\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	24.6274	1.46915	0.734574	−	0.678528i	$-0.237382\pi$
$281$	24.6274	1.46915	0.734574	+	0.678528i	$0.237382\pi$
$282$	0	0
$283$	−11.5147	−0.684479	−0.342239	−	0.939613i	$-0.611185\pi$
$283$	−11.5147	−0.684479	−0.342239	+	0.939613i	$0.611185\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.65685	−0.333914
$288$	0	0
$289$	6.31371	0.371395
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9.51472	0.555856	0.277928	−	0.960602i	$-0.410352\pi$
$293$	9.51472	0.555856	0.277928	+	0.960602i	$0.410352\pi$
$294$	0	0
$295$	1.65685	0.0964658
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.68629	0.271015
$300$	0	0
$301$	19.3137	1.11322
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−11.6569	−0.667470
$306$	0	0
$307$	20.4853	1.16916	0.584578	−	0.811337i	$-0.301260\pi$
$307$	20.4853	1.16916	0.584578	+	0.811337i	$0.301260\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−10.3431	−0.586506	−0.293253	−	0.956035i	$-0.594738\pi$
$311$	−10.3431	−0.586506	−0.293253	+	0.956035i	$0.594738\pi$
$312$	0	0
$313$	5.31371	0.300349	0.150174	−	0.988660i	$-0.452017\pi$
$313$	5.31371	0.300349	0.150174	+	0.988660i	$0.452017\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.6274	1.38321	0.691607	−	0.722274i	$-0.256903\pi$
$317$	24.6274	1.38321	0.691607	+	0.722274i	$0.256903\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	0	0
$321$	0	0
$322$	0	0
$323$	27.3137	1.51978
$324$	0	0
$325$	−0.828427	−0.0459529
$326$	0	0
$327$	0	0
$328$	0	0
$329$	16.0000	0.882109
$330$	0	0
$331$	−4.97056	−0.273207	−0.136603	−	0.990626i	$-0.543619\pi$
$331$	−4.97056	−0.273207	−0.136603	+	0.990626i	$0.543619\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.65685	−0.0905236
$336$	0	0
$337$	−3.85786	−0.210151	−0.105076	−	0.994464i	$-0.533508\pi$
$337$	−3.85786	−0.210151	−0.105076	+	0.994464i	$0.533508\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5.65685	−0.306336
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.4853	1.31444	0.657219	−	0.753699i	$-0.271732\pi$
$347$	24.4853	1.31444	0.657219	+	0.753699i	$0.271732\pi$
$348$	0	0
$349$	6.97056	0.373126	0.186563	−	0.982443i	$-0.440265\pi$
$349$	6.97056	0.373126	0.186563	+	0.982443i	$0.440265\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−34.9706	−1.86130	−0.930648	−	0.365917i	$-0.880756\pi$
$353$	−34.9706	−1.86130	−0.930648	+	0.365917i	$0.880756\pi$
$354$	0	0
$355$	2.34315	0.124361
$356$	0	0
$357$	0	0
$358$	0	0
$359$	28.2843	1.49279	0.746393	−	0.665505i	$-0.231784\pi$
$359$	28.2843	1.49279	0.746393	+	0.665505i	$0.231784\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.48528	0.130086
$366$	0	0
$367$	8.97056	0.468260	0.234130	−	0.972205i	$-0.424776\pi$
$367$	8.97056	0.468260	0.234130	+	0.972205i	$0.424776\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	32.9706	1.71175
$372$	0	0
$373$	1.51472	0.0784292	0.0392146	−	0.999231i	$-0.487514\pi$
$373$	1.51472	0.0784292	0.0392146	+	0.999231i	$0.487514\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.65685	0.0853323
$378$	0	0
$379$	23.3137	1.19754	0.598772	−	0.800919i	$-0.295655\pi$
$379$	23.3137	1.19754	0.598772	+	0.800919i	$0.295655\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−2.82843	−0.144150
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	0	0
$391$	27.3137	1.38131
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.65685	−0.0833654
$396$	0	0
$397$	11.6569	0.585041	0.292520	−	0.956259i	$-0.405506\pi$
$397$	11.6569	0.585041	0.292520	+	0.956259i	$0.405506\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.0000	0.699127	0.349563	−	0.936913i	$-0.386330\pi$
$401$	14.0000	0.699127	0.349563	+	0.936913i	$0.386330\pi$
$402$	0	0
$403$	−4.68629	−0.233441
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.00000	−0.297409
$408$	0	0
$409$	−25.3137	−1.25168	−0.625841	−	0.779951i	$-0.715244\pi$
$409$	−25.3137	−1.25168	−0.625841	+	0.779951i	$0.715244\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.68629	0.230597
$414$	0	0
$415$	−5.17157	−0.253863
$416$	0	0
$417$	0	0
$418$	0	0
$419$	31.3137	1.52977	0.764887	−	0.644164i	$-0.222795\pi$
$419$	31.3137	1.52977	0.764887	+	0.644164i	$0.222795\pi$
$420$	0	0
$421$	−8.62742	−0.420475	−0.210237	−	0.977650i	$-0.567424\pi$
$421$	−8.62742	−0.420475	−0.210237	+	0.977650i	$0.567424\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.82843	−0.234213
$426$	0	0
$427$	−32.9706	−1.59556
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.6863	0.611077	0.305539	−	0.952180i	$-0.401164\pi$
$431$	12.6863	0.611077	0.305539	+	0.952180i	$0.401164\pi$
$432$	0	0
$433$	−6.97056	−0.334984	−0.167492	−	0.985873i	$-0.553567\pi$
$433$	−6.97056	−0.334984	−0.167492	+	0.985873i	$0.553567\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	32.0000	1.53077
$438$	0	0
$439$	26.6274	1.27086	0.635429	−	0.772160i	$-0.280823\pi$
$439$	26.6274	1.27086	0.635429	+	0.772160i	$0.280823\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−37.9411	−1.80264	−0.901319	−	0.433157i	$-0.857400\pi$
$443$	−37.9411	−1.80264	−0.901319	+	0.433157i	$0.857400\pi$
$444$	0	0
$445$	2.00000	0.0948091
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−14.6863	−0.693089	−0.346544	−	0.938034i	$-0.612645\pi$
$449$	−14.6863	−0.693089	−0.346544	+	0.938034i	$0.612645\pi$
$450$	0	0
$451$	−2.00000	−0.0941763
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.34315	−0.109848
$456$	0	0
$457$	−32.1421	−1.50355	−0.751773	−	0.659422i	$-0.770801\pi$
$457$	−32.1421	−1.50355	−0.751773	+	0.659422i	$0.770801\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	19.6569	0.915511	0.457755	−	0.889078i	$-0.348654\pi$
$461$	19.6569	0.915511	0.457755	+	0.889078i	$0.348654\pi$
$462$	0	0
$463$	−2.34315	−0.108895	−0.0544476	−	0.998517i	$-0.517340\pi$
$463$	−2.34315	−0.108895	−0.0544476	+	0.998517i	$0.517340\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.97056	−0.230010	−0.115005	−	0.993365i	$-0.536688\pi$
$467$	−4.97056	−0.230010	−0.115005	+	0.993365i	$0.536688\pi$
$468$	0	0
$469$	−4.68629	−0.216393
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.82843	0.313971
$474$	0	0
$475$	−5.65685	−0.259554
$476$	0	0
$477$	0	0
$478$	0	0
$479$	37.6569	1.72059	0.860293	−	0.509800i	$-0.170281\pi$
$479$	37.6569	1.72059	0.860293	+	0.509800i	$0.170281\pi$
$480$	0	0
$481$	−4.97056	−0.226638
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−15.6569	−0.710941
$486$	0	0
$487$	−20.2843	−0.919168	−0.459584	−	0.888134i	$-0.652001\pi$
$487$	−20.2843	−0.919168	−0.459584	+	0.888134i	$0.652001\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	9.65685	0.434923
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.62742	0.297280
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−12.4853	−0.556691	−0.278346	−	0.960481i	$-0.589786\pi$
$503$	−12.4853	−0.556691	−0.278346	+	0.960481i	$0.589786\pi$
$504$	0	0
$505$	7.65685	0.340726
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−36.6274	−1.62348	−0.811741	−	0.584018i	$-0.801480\pi$
$509$	−36.6274	−1.62348	−0.811741	+	0.584018i	$0.801480\pi$
$510$	0	0
$511$	7.02944	0.310964
$512$	0	0
$513$	0	0
$514$	0	0
$515$	19.3137	0.851064
$516$	0	0
$517$	5.65685	0.248788
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−16.6274	−0.728460	−0.364230	−	0.931309i	$-0.618668\pi$
$521$	−16.6274	−0.728460	−0.364230	+	0.931309i	$0.618668\pi$
$522$	0	0
$523$	13.4558	0.588383	0.294191	−	0.955746i	$-0.404950\pi$
$523$	13.4558	0.588383	0.294191	+	0.955746i	$0.404950\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−27.3137	−1.18980
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.65685	−0.0717663
$534$	0	0
$535$	−8.48528	−0.366851
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−17.3137	−0.741638
$546$	0	0
$547$	−39.7990	−1.70168	−0.850841	−	0.525423i	$-0.823907\pi$
$547$	−39.7990	−1.70168	−0.850841	+	0.525423i	$0.823907\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	11.3137	0.481980
$552$	0	0
$553$	−4.68629	−0.199281
$554$	0	0
$555$	0	0
$556$	0	0
$557$	42.4853	1.80016	0.900080	−	0.435726i	$-0.143508\pi$
$557$	42.4853	1.80016	0.900080	+	0.435726i	$0.143508\pi$
$558$	0	0
$559$	5.65685	0.239259
$560$	0	0
$561$	0	0
$562$	0	0
$563$	33.4558	1.41000	0.704998	−	0.709209i	$-0.250948\pi$
$563$	33.4558	1.41000	0.704998	+	0.709209i	$0.250948\pi$
$564$	0	0
$565$	−9.31371	−0.391831
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−26.2843	−1.10189	−0.550947	−	0.834540i	$-0.685733\pi$
$569$	−26.2843	−1.10189	−0.550947	+	0.834540i	$0.685733\pi$
$570$	0	0
$571$	19.3137	0.808254	0.404127	−	0.914703i	$-0.367575\pi$
$571$	19.3137	0.808254	0.404127	+	0.914703i	$0.367575\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5.65685	−0.235907
$576$	0	0
$577$	−38.9706	−1.62237	−0.811183	−	0.584793i	$-0.801176\pi$
$577$	−38.9706	−1.62237	−0.811183	+	0.584793i	$0.801176\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−14.6274	−0.606848
$582$	0	0
$583$	11.6569	0.482778
$584$	0	0
$585$	0	0
$586$	0	0
$587$	41.6569	1.71936	0.859681	−	0.510831i	$-0.170662\pi$
$587$	41.6569	1.71936	0.859681	+	0.510831i	$0.170662\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	0	0
$592$	0	0
$593$	21.1127	0.866995	0.433497	−	0.901155i	$-0.357279\pi$
$593$	21.1127	0.866995	0.433497	+	0.901155i	$0.357279\pi$
$594$	0	0
$595$	−13.6569	−0.559876
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6.62742	−0.270789	−0.135394	−	0.990792i	$-0.543230\pi$
$599$	−6.62742	−0.270789	−0.135394	+	0.990792i	$0.543230\pi$
$600$	0	0
$601$	−24.3431	−0.992978	−0.496489	−	0.868043i	$-0.665378\pi$
$601$	−24.3431	−0.992978	−0.496489	+	0.868043i	$0.665378\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−19.7990	−0.803616	−0.401808	−	0.915724i	$-0.631618\pi$
$607$	−19.7990	−0.803616	−0.401808	+	0.915724i	$0.631618\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.68629	0.189587
$612$	0	0
$613$	−37.1127	−1.49897	−0.749484	−	0.662023i	$-0.769698\pi$
$613$	−37.1127	−1.49897	−0.749484	+	0.662023i	$0.769698\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	44.6274	1.79663	0.898316	−	0.439350i	$-0.144791\pi$
$617$	44.6274	1.79663	0.898316	+	0.439350i	$0.144791\pi$
$618$	0	0
$619$	17.6569	0.709689	0.354844	−	0.934925i	$-0.384534\pi$
$619$	17.6569	0.709689	0.354844	+	0.934925i	$0.384534\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.65685	0.226637
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−28.9706	−1.15513
$630$	0	0
$631$	12.6863	0.505033	0.252517	−	0.967593i	$-0.418742\pi$
$631$	12.6863	0.505033	0.252517	+	0.967593i	$0.418742\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.82843	−0.112243
$636$	0	0
$637$	−0.828427	−0.0328235
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	0	0
$643$	−42.6274	−1.68106	−0.840531	−	0.541764i	$-0.817757\pi$
$643$	−42.6274	−1.68106	−0.840531	+	0.541764i	$0.817757\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	0	0
$649$	1.65685	0.0650372
$650$	0	0
$651$	0	0
$652$	0	0
$653$	50.9706	1.99463	0.997316	−	0.0732157i	$-0.0233262\pi$
$653$	50.9706	1.99463	0.997316	+	0.0732157i	$0.0233262\pi$
$654$	0	0
$655$	17.6569	0.689910
$656$	0	0
$657$	0	0
$658$	0	0
$659$	7.31371	0.284902	0.142451	−	0.989802i	$-0.454502\pi$
$659$	7.31371	0.284902	0.142451	+	0.989802i	$0.454502\pi$
$660$	0	0
$661$	26.6863	1.03798	0.518988	−	0.854781i	$-0.326309\pi$
$661$	26.6863	1.03798	0.518988	+	0.854781i	$0.326309\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−16.0000	−0.620453
$666$	0	0
$667$	11.3137	0.438069
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−11.6569	−0.450008
$672$	0	0
$673$	15.4558	0.595779	0.297890	−	0.954600i	$-0.403717\pi$
$673$	15.4558	0.595779	0.297890	+	0.954600i	$0.403717\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.51472	0.0582154	0.0291077	−	0.999576i	$-0.490733\pi$
$677$	1.51472	0.0582154	0.0291077	+	0.999576i	$0.490733\pi$
$678$	0	0
$679$	−44.2843	−1.69947
$680$	0	0
$681$	0	0
$682$	0	0
$683$	33.6569	1.28784	0.643922	−	0.765091i	$-0.277306\pi$
$683$	33.6569	1.28784	0.643922	+	0.765091i	$0.277306\pi$
$684$	0	0
$685$	5.31371	0.203026
$686$	0	0
$687$	0	0
$688$	0	0
$689$	9.65685	0.367897
$690$	0	0
$691$	15.3137	0.582561	0.291280	−	0.956638i	$-0.405919\pi$
$691$	15.3137	0.582561	0.291280	+	0.956638i	$0.405919\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.3137	−0.429153
$696$	0	0
$697$	−9.65685	−0.365779
$698$	0	0
$699$	0	0
$700$	0	0
$701$	17.3137	0.653930	0.326965	−	0.945036i	$-0.393974\pi$
$701$	17.3137	0.653930	0.326965	+	0.945036i	$0.393974\pi$
$702$	0	0
$703$	−33.9411	−1.28011
$704$	0	0
$705$	0	0
$706$	0	0
$707$	21.6569	0.814490
$708$	0	0
$709$	−45.3137	−1.70179	−0.850896	−	0.525334i	$-0.823940\pi$
$709$	−45.3137	−1.70179	−0.850896	+	0.525334i	$0.823940\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	−0.828427	−0.0309814
$716$	0	0
$717$	0	0
$718$	0	0
$719$	43.3137	1.61533	0.807664	−	0.589642i	$-0.200731\pi$
$719$	43.3137	1.61533	0.807664	+	0.589642i	$0.200731\pi$
$720$	0	0
$721$	54.6274	2.03443
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	33.9411	1.25881	0.629403	−	0.777079i	$-0.283299\pi$
$727$	33.9411	1.25881	0.629403	+	0.777079i	$0.283299\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	32.9706	1.21946
$732$	0	0
$733$	−30.4853	−1.12600	−0.563000	−	0.826457i	$-0.690353\pi$
$733$	−30.4853	−1.12600	−0.563000	+	0.826457i	$0.690353\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.65685	−0.0610310
$738$	0	0
$739$	22.6274	0.832363	0.416181	−	0.909282i	$-0.363368\pi$
$739$	22.6274	0.832363	0.416181	+	0.909282i	$0.363368\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−38.8284	−1.42448	−0.712238	−	0.701938i	$-0.752319\pi$
$743$	−38.8284	−1.42448	−0.712238	+	0.701938i	$0.752319\pi$
$744$	0	0
$745$	−11.6569	−0.427074
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−24.0000	−0.876941
$750$	0	0
$751$	49.9411	1.82238	0.911189	−	0.411989i	$-0.135166\pi$
$751$	49.9411	1.82238	0.911189	+	0.411989i	$0.135166\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−20.9706	−0.763197
$756$	0	0
$757$	37.5980	1.36652	0.683261	−	0.730174i	$-0.260561\pi$
$757$	37.5980	1.36652	0.683261	+	0.730174i	$0.260561\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−8.34315	−0.302439	−0.151219	−	0.988500i	$-0.548320\pi$
$761$	−8.34315	−0.302439	−0.151219	+	0.988500i	$0.548320\pi$
$762$	0	0
$763$	−48.9706	−1.77285
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.37258	0.0495611
$768$	0	0
$769$	−29.5980	−1.06733	−0.533665	−	0.845696i	$-0.679186\pi$
$769$	−29.5980	−1.06733	−0.533665	+	0.845696i	$0.679186\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−22.9706	−0.826194	−0.413097	−	0.910687i	$-0.635553\pi$
$773$	−22.9706	−0.826194	−0.413097	+	0.910687i	$0.635553\pi$
$774$	0	0
$775$	5.65685	0.203200
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−11.3137	−0.405356
$780$	0	0
$781$	2.34315	0.0838443
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10.9706	0.391556
$786$	0	0
$787$	19.1127	0.681294	0.340647	−	0.940191i	$-0.389354\pi$
$787$	19.1127	0.681294	0.340647	+	0.940191i	$0.389354\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−26.3431	−0.936654
$792$	0	0
$793$	−9.65685	−0.342925
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−20.6274	−0.730661	−0.365330	−	0.930878i	$-0.619044\pi$
$797$	−20.6274	−0.730661	−0.365330	+	0.930878i	$0.619044\pi$
$798$	0	0
$799$	27.3137	0.966290
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.48528	0.0877037
$804$	0	0
$805$	−16.0000	−0.563926
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−22.0000	−0.773479	−0.386739	−	0.922189i	$-0.626399\pi$
$809$	−22.0000	−0.773479	−0.386739	+	0.922189i	$0.626399\pi$
$810$	0	0
$811$	−43.3137	−1.52095	−0.760475	−	0.649367i	$-0.775034\pi$
$811$	−43.3137	−1.52095	−0.760475	+	0.649367i	$0.775034\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	20.0000	0.700569
$816$	0	0
$817$	38.6274	1.35140
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	−51.3137	−1.78868	−0.894342	−	0.447385i	$-0.852356\pi$
$823$	−51.3137	−1.78868	−0.894342	+	0.447385i	$0.852356\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.85786	0.0646043	0.0323021	−	0.999478i	$-0.489716\pi$
$827$	1.85786	0.0646043	0.0323021	+	0.999478i	$0.489716\pi$
$828$	0	0
$829$	39.9411	1.38721	0.693606	−	0.720354i	$-0.256021\pi$
$829$	39.9411	1.38721	0.693606	+	0.720354i	$0.256021\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−4.82843	−0.167295
$834$	0	0
$835$	−10.1421	−0.350983
$836$	0	0
$837$	0	0
$838$	0	0
$839$	10.3431	0.357085	0.178543	−	0.983932i	$-0.442862\pi$
$839$	10.3431	0.357085	0.178543	+	0.983932i	$0.442862\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	12.3137	0.423604
$846$	0	0
$847$	−2.82843	−0.0971859
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−33.9411	−1.16349
$852$	0	0
$853$	3.45584	0.118326	0.0591629	−	0.998248i	$-0.481157\pi$
$853$	3.45584	0.118326	0.0591629	+	0.998248i	$0.481157\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−30.7696	−1.05107	−0.525534	−	0.850773i	$-0.676134\pi$
$857$	−30.7696	−1.05107	−0.525534	+	0.850773i	$0.676134\pi$
$858$	0	0
$859$	11.0294	0.376320	0.188160	−	0.982138i	$-0.439748\pi$
$859$	11.0294	0.376320	0.188160	+	0.982138i	$0.439748\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.3137	−0.385123	−0.192562	−	0.981285i	$-0.561680\pi$
$863$	−11.3137	−0.385123	−0.192562	+	0.981285i	$0.561680\pi$
$864$	0	0
$865$	−18.4853	−0.628518
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.65685	−0.0562049
$870$	0	0
$871$	−1.37258	−0.0465082
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.82843	0.0956183
$876$	0	0
$877$	−38.4853	−1.29956	−0.649778	−	0.760124i	$-0.725138\pi$
$877$	−38.4853	−1.29956	−0.649778	+	0.760124i	$0.725138\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−14.6863	−0.494794	−0.247397	−	0.968914i	$-0.579575\pi$
$881$	−14.6863	−0.494794	−0.247397	+	0.968914i	$0.579575\pi$
$882$	0	0
$883$	58.6274	1.97297	0.986485	−	0.163853i	$-0.0523921\pi$
$883$	58.6274	1.97297	0.986485	+	0.163853i	$0.0523921\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−31.7990	−1.06771	−0.533853	−	0.845577i	$-0.679256\pi$
$887$	−31.7990	−1.06771	−0.533853	+	0.845577i	$0.679256\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	0	0
$892$	0	0
$893$	32.0000	1.07084
$894$	0	0
$895$	−4.00000	−0.133705
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−11.3137	−0.377333
$900$	0	0
$901$	56.2843	1.87510
$902$	0	0
$903$	0	0
$904$	0	0
$905$	13.3137	0.442563
$906$	0	0
$907$	−9.65685	−0.320651	−0.160325	−	0.987064i	$-0.551254\pi$
$907$	−9.65685	−0.320651	−0.160325	+	0.987064i	$0.551254\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	17.9411	0.594416	0.297208	−	0.954813i	$-0.403945\pi$
$911$	17.9411	0.594416	0.297208	+	0.954813i	$0.403945\pi$
$912$	0	0
$913$	−5.17157	−0.171154
$914$	0	0
$915$	0	0
$916$	0	0
$917$	49.9411	1.64920
$918$	0	0
$919$	24.6863	0.814326	0.407163	−	0.913356i	$-0.366518\pi$
$919$	24.6863	0.814326	0.407163	+	0.913356i	$0.366518\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.94113	0.0638929
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−30.6863	−1.00678	−0.503392	−	0.864058i	$-0.667915\pi$
$929$	−30.6863	−1.00678	−0.503392	+	0.864058i	$0.667915\pi$
$930$	0	0
$931$	−5.65685	−0.185396
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4.82843	−0.157906
$936$	0	0
$937$	37.1127	1.21242	0.606210	−	0.795305i	$-0.292689\pi$
$937$	37.1127	1.21242	0.606210	+	0.795305i	$0.292689\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	16.3431	0.532771	0.266386	−	0.963867i	$-0.414171\pi$
$941$	16.3431	0.532771	0.266386	+	0.963867i	$0.414171\pi$
$942$	0	0
$943$	−11.3137	−0.368425
$944$	0	0
$945$	0	0
$946$	0	0
$947$	59.5980	1.93667	0.968337	−	0.249646i	$-0.0803144\pi$
$947$	59.5980	1.93667	0.968337	+	0.249646i	$0.0803144\pi$
$948$	0	0
$949$	2.05887	0.0668339
$950$	0	0
$951$	0	0
$952$	0	0
$953$	21.5147	0.696930	0.348465	−	0.937322i	$-0.386703\pi$
$953$	21.5147	0.696930	0.348465	+	0.937322i	$0.386703\pi$
$954$	0	0
$955$	16.0000	0.517748
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15.0294	0.485326
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−8.82843	−0.284197
$966$	0	0
$967$	−11.7990	−0.379430	−0.189715	−	0.981839i	$-0.560756\pi$
$967$	−11.7990	−0.379430	−0.189715	+	0.981839i	$0.560756\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	37.9411	1.21759	0.608794	−	0.793328i	$-0.291653\pi$
$971$	37.9411	1.21759	0.608794	+	0.793328i	$0.291653\pi$
$972$	0	0
$973$	−32.0000	−1.02587
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−34.0000	−1.08776	−0.543878	−	0.839164i	$-0.683045\pi$
$977$	−34.0000	−1.08776	−0.543878	+	0.839164i	$0.683045\pi$
$978$	0	0
$979$	2.00000	0.0639203
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−35.3137	−1.12633	−0.563166	−	0.826344i	$-0.690417\pi$
$983$	−35.3137	−1.12633	−0.563166	+	0.826344i	$0.690417\pi$
$984$	0	0
$985$	−15.1716	−0.483407
$986$	0	0
$987$	0	0
$988$	0	0
$989$	38.6274	1.22828
$990$	0	0
$991$	15.0294	0.477426	0.238713	−	0.971090i	$-0.423275\pi$
$991$	15.0294	0.477426	0.238713	+	0.971090i	$0.423275\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−8.00000	−0.253617
$996$	0	0
$997$	24.1421	0.764589	0.382295	−	0.924041i	$-0.375134\pi$
$997$	24.1421	0.764589	0.382295	+	0.924041i	$0.375134\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7920.2.a.bs.1.1		2
3.2	odd	2		2640.2.a.bd.1.1		2
4.3	odd	2		3960.2.a.z.1.2		2
12.11	even	2		1320.2.a.p.1.2	✓	2
60.23	odd	4		6600.2.d.z.1849.1		4
60.47	odd	4		6600.2.d.z.1849.4		4
60.59	even	2		6600.2.a.bk.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1320.2.a.p.1.2	✓	2	12.11	even	2
2640.2.a.bd.1.1		2	3.2	odd	2
3960.2.a.z.1.2		2	4.3	odd	2
6600.2.a.bk.1.1		2	60.59	even	2
6600.2.d.z.1849.1		4	60.23	odd	4
6600.2.d.z.1849.4		4	60.47	odd	4
7920.2.a.bs.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7920.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -2.82843 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -2.82843 q^{7} -1.00000 q^{11} -0.828427 q^{13} -4.82843 q^{17} -5.65685 q^{19} -5.65685 q^{23} +1.00000 q^{25} -2.00000 q^{29} +5.65685 q^{31} +2.82843 q^{35} +6.00000 q^{37} +2.00000 q^{41} -6.82843 q^{43} -5.65685 q^{47} +1.00000 q^{49} -11.6569 q^{53} +1.00000 q^{55} -1.65685 q^{59} +11.6569 q^{61} +0.828427 q^{65} +1.65685 q^{67} -2.34315 q^{71} -2.48528 q^{73} +2.82843 q^{77} +1.65685 q^{79} +5.17157 q^{83} +4.82843 q^{85} -2.00000 q^{89} +2.34315 q^{91} +5.65685 q^{95} +15.6569 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^5	\( 2 q - 2 q^{5} - 2 q^{11} + 4 q^{13} - 4 q^{17} + 2 q^{25} - 4 q^{29} + 12 q^{37} + 4 q^{41} - 8 q^{43} + 2 q^{49} - 12 q^{53} + 2 q^{55} + 8 q^{59} + 12 q^{61} - 4 q^{65} - 8 q^{67} - 16 q^{71} + 12 q^{73} - 8 q^{79} + 16 q^{83} + 4 q^{85} - 4 q^{89} + 16 q^{91} + 20 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^11 + 4 * q^13 - 4 * q^17 + 2 * q^25 - 4 * q^29 + 12 * q^37 + 4 * q^41 - 8 * q^43 + 2 * q^49 - 12 * q^53 + 2 * q^55 + 8 * q^59 + 12 * q^61 - 4 * q^65 - 8 * q^67 - 16 * q^71 + 12 * q^73 - 8 * q^79 + 16 * q^83 + 4 * q^85 - 4 * q^89 + 16 * q^91 + 20 * q^97

Introduction

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