LMFDB - Embedded newform 7920.2.a.br.1.2

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:	$1$
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:	no (minimal twist has level 3960)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-1.00000 q^{5} +1.41421 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +1.41421 q^{7} -1.00000 q^{11} -3.41421 q^{13} +0.585786 q^{17} -5.65685 q^{19} +2.82843 q^{23} +1.00000 q^{25} +4.82843 q^{29} +10.4853 q^{31} -1.41421 q^{35} +3.65685 q^{37} -6.48528 q^{41} -5.41421 q^{43} -2.82843 q^{47} -5.00000 q^{49} +12.8284 q^{53} +1.00000 q^{55} +6.48528 q^{59} -8.82843 q^{61} +3.41421 q^{65} -8.48528 q^{67} -8.82843 q^{71} +4.58579 q^{73} -1.41421 q^{77} +9.65685 q^{79} -8.24264 q^{83} -0.585786 q^{85} +15.6569 q^{89} -4.82843 q^{91} +5.65685 q^{95} -4.82843 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 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}+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

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
$7$	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
$13$	−3.41421	−0.946932	−0.473466	−	0.880812i	$-0.656997\pi$
$13$	−3.41421	−0.946932	−0.473466	+	0.880812i	$0.656997\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.585786	0.142074	0.0710370	−	0.997474i	$-0.477369\pi$
$17$	0.585786	0.142074	0.0710370	+	0.997474i	$0.477369\pi$
$18$	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
$29$	4.82843	0.896616	0.448308	−	0.893879i	$-0.352027\pi$
$29$	4.82843	0.896616	0.448308	+	0.893879i	$0.352027\pi$
$30$	0	0
$31$	10.4853	1.88321	0.941606	−	0.336717i	$-0.109316\pi$
$31$	10.4853	1.88321	0.941606	+	0.336717i	$0.109316\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.41421	−0.239046
$36$	0	0
$37$	3.65685	0.601183	0.300592	−	0.953753i	$-0.402816\pi$
$37$	3.65685	0.601183	0.300592	+	0.953753i	$0.402816\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.48528	−1.01283	−0.506415	−	0.862290i	$-0.669030\pi$
$41$	−6.48528	−1.01283	−0.506415	+	0.862290i	$0.669030\pi$
$42$	0	0
$43$	−5.41421	−0.825660	−0.412830	−	0.910808i	$-0.635460\pi$
$43$	−5.41421	−0.825660	−0.412830	+	0.910808i	$0.635460\pi$
$44$	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
$53$	12.8284	1.76212	0.881060	−	0.473005i	$-0.156831\pi$
$53$	12.8284	1.76212	0.881060	+	0.473005i	$0.156831\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.48528	0.844312	0.422156	−	0.906523i	$-0.361273\pi$
$59$	6.48528	0.844312	0.422156	+	0.906523i	$0.361273\pi$
$60$	0	0
$61$	−8.82843	−1.13036	−0.565182	−	0.824966i	$-0.691194\pi$
$61$	−8.82843	−1.13036	−0.565182	+	0.824966i	$0.691194\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.41421	0.423481
$66$	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
$71$	−8.82843	−1.04774	−0.523871	−	0.851798i	$-0.675513\pi$
$71$	−8.82843	−1.04774	−0.523871	+	0.851798i	$0.675513\pi$
$72$	0	0
$73$	4.58579	0.536726	0.268363	−	0.963318i	$-0.413517\pi$
$73$	4.58579	0.536726	0.268363	+	0.963318i	$0.413517\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.41421	−0.161165
$78$	0	0
$79$	9.65685	1.08648	0.543240	−	0.839577i	$-0.317197\pi$
$79$	9.65685	1.08648	0.543240	+	0.839577i	$0.317197\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.24264	−0.904747	−0.452374	−	0.891828i	$-0.649423\pi$
$83$	−8.24264	−0.904747	−0.452374	+	0.891828i	$0.649423\pi$
$84$	0	0
$85$	−0.585786	−0.0635375
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.6569	1.65962	0.829812	−	0.558044i	$-0.188448\pi$
$89$	15.6569	1.65962	0.829812	+	0.558044i	$0.188448\pi$
$90$	0	0
$91$	−4.82843	−0.506157
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.65685	0.580381
$96$	0	0
$97$	−4.82843	−0.490252	−0.245126	−	0.969491i	$-0.578829\pi$
$97$	−4.82843	−0.490252	−0.245126	+	0.969491i	$0.578829\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.65685	0.363871	0.181935	−	0.983311i	$-0.441764\pi$
$101$	3.65685	0.363871	0.181935	+	0.983311i	$0.441764\pi$
$102$	0	0
$103$	−17.6569	−1.73978	−0.869891	−	0.493244i	$-0.835811\pi$
$103$	−17.6569	−1.73978	−0.869891	+	0.493244i	$0.835811\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.8995	−1.34371	−0.671857	−	0.740680i	$-0.734503\pi$
$107$	−13.8995	−1.34371	−0.671857	+	0.740680i	$0.734503\pi$
$108$	0	0
$109$	−18.9706	−1.81705	−0.908525	−	0.417830i	$-0.862791\pi$
$109$	−18.9706	−1.81705	−0.908525	+	0.417830i	$0.862791\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	11.6569	1.09658	0.548292	−	0.836287i	$-0.315278\pi$
$113$	11.6569	1.09658	0.548292	+	0.836287i	$0.315278\pi$
$114$	0	0
$115$	−2.82843	−0.263752
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.828427	0.0759418
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−12.2426	−1.08636	−0.543179	−	0.839617i	$-0.682780\pi$
$127$	−12.2426	−1.08636	−0.543179	+	0.839617i	$0.682780\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−16.4853	−1.44033	−0.720163	−	0.693805i	$-0.755933\pi$
$131$	−16.4853	−1.44033	−0.720163	+	0.693805i	$0.755933\pi$
$132$	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
$143$	3.41421	0.285511
$144$	0	0
$145$	−4.82843	−0.400979
$146$	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
$151$	−21.6569	−1.76241	−0.881205	−	0.472735i	$-0.843267\pi$
$151$	−21.6569	−1.76241	−0.881205	+	0.472735i	$0.843267\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−10.4853	−0.842198
$156$	0	0
$157$	−8.82843	−0.704585	−0.352293	−	0.935890i	$-0.614598\pi$
$157$	−8.82843	−0.704585	−0.352293	+	0.935890i	$0.614598\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	−6.34315	−0.496834	−0.248417	−	0.968653i	$-0.579910\pi$
$163$	−6.34315	−0.496834	−0.248417	+	0.968653i	$0.579910\pi$
$164$	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
$169$	−1.34315	−0.103319
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.72792	−0.511514	−0.255757	−	0.966741i	$-0.582325\pi$
$173$	−6.72792	−0.511514	−0.255757	+	0.966741i	$0.582325\pi$
$174$	0	0
$175$	1.41421	0.106904
$176$	0	0
$177$	0	0
$178$	0	0
$179$	7.31371	0.546652	0.273326	−	0.961921i	$-0.411876\pi$
$179$	7.31371	0.546652	0.273326	+	0.961921i	$0.411876\pi$
$180$	0	0
$181$	−2.34315	−0.174165	−0.0870823	−	0.996201i	$-0.527754\pi$
$181$	−2.34315	−0.174165	−0.0870823	+	0.996201i	$0.527754\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.65685	−0.268857
$186$	0	0
$187$	−0.585786	−0.0428369
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−21.6569	−1.56703	−0.783517	−	0.621370i	$-0.786577\pi$
$191$	−21.6569	−1.56703	−0.783517	+	0.621370i	$0.786577\pi$
$192$	0	0
$193$	20.3848	1.46733	0.733664	−	0.679512i	$-0.237809\pi$
$193$	20.3848	1.46733	0.733664	+	0.679512i	$0.237809\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−20.3848	−1.45236	−0.726178	−	0.687507i	$-0.758705\pi$
$197$	−20.3848	−1.45236	−0.726178	+	0.687507i	$0.758705\pi$
$198$	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
$203$	6.82843	0.479262
$204$	0	0
$205$	6.48528	0.452952
$206$	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	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.41421	0.369246
$216$	0	0
$217$	14.8284	1.00662
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	6.14214	0.411308	0.205654	−	0.978625i	$-0.434068\pi$
$223$	6.14214	0.411308	0.205654	+	0.978625i	$0.434068\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.727922	0.0483139	0.0241569	−	0.999708i	$-0.492310\pi$
$227$	0.727922	0.0483139	0.0241569	+	0.999708i	$0.492310\pi$
$228$	0	0
$229$	−2.68629	−0.177515	−0.0887576	−	0.996053i	$-0.528290\pi$
$229$	−2.68629	−0.177515	−0.0887576	+	0.996053i	$0.528290\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.7279	−0.964858	−0.482429	−	0.875935i	$-0.660245\pi$
$233$	−14.7279	−0.964858	−0.482429	+	0.875935i	$0.660245\pi$
$234$	0	0
$235$	2.82843	0.184506
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−27.7990	−1.79817	−0.899084	−	0.437777i	$-0.855766\pi$
$239$	−27.7990	−1.79817	−0.899084	+	0.437777i	$0.855766\pi$
$240$	0	0
$241$	22.4853	1.44840	0.724202	−	0.689588i	$-0.242208\pi$
$241$	22.4853	1.44840	0.724202	+	0.689588i	$0.242208\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5.00000	0.319438
$246$	0	0
$247$	19.3137	1.22890
$248$	0	0
$249$	0	0
$250$	0	0
$251$	9.79899	0.618507	0.309253	−	0.950980i	$-0.399921\pi$
$251$	9.79899	0.618507	0.309253	+	0.950980i	$0.399921\pi$
$252$	0	0
$253$	−2.82843	−0.177822
$254$	0	0
$255$	0	0
$256$	0	0
$257$	21.7990	1.35978	0.679892	−	0.733312i	$-0.262027\pi$
$257$	21.7990	1.35978	0.679892	+	0.733312i	$0.262027\pi$
$258$	0	0
$259$	5.17157	0.321346
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−17.4142	−1.07381	−0.536903	−	0.843644i	$-0.680406\pi$
$263$	−17.4142	−1.07381	−0.536903	+	0.843644i	$0.680406\pi$
$264$	0	0
$265$	−12.8284	−0.788044
$266$	0	0
$267$	0	0
$268$	0	0
$269$	17.6569	1.07656	0.538279	−	0.842767i	$-0.319075\pi$
$269$	17.6569	1.07656	0.538279	+	0.842767i	$0.319075\pi$
$270$	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
$277$	−10.2426	−0.615421	−0.307710	−	0.951480i	$-0.599563\pi$
$277$	−10.2426	−0.615421	−0.307710	+	0.951480i	$0.599563\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.1421	0.962959	0.481480	−	0.876457i	$-0.340100\pi$
$281$	16.1421	0.962959	0.481480	+	0.876457i	$0.340100\pi$
$282$	0	0
$283$	14.3848	0.855086	0.427543	−	0.903995i	$-0.359379\pi$
$283$	14.3848	0.855086	0.427543	+	0.903995i	$0.359379\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.17157	−0.541381
$288$	0	0
$289$	−16.6569	−0.979815
$290$	0	0
$291$	0	0
$292$	0	0
$293$	17.5563	1.02565	0.512826	−	0.858492i	$-0.328598\pi$
$293$	17.5563	1.02565	0.512826	+	0.858492i	$0.328598\pi$
$294$	0	0
$295$	−6.48528	−0.377588
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−9.65685	−0.558470
$300$	0	0
$301$	−7.65685	−0.441334
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8.82843	0.505514
$306$	0	0
$307$	−7.27208	−0.415039	−0.207520	−	0.978231i	$-0.566539\pi$
$307$	−7.27208	−0.415039	−0.207520	+	0.978231i	$0.566539\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	20.8284	1.18107	0.590536	−	0.807011i	$-0.298916\pi$
$311$	20.8284	1.18107	0.590536	+	0.807011i	$0.298916\pi$
$312$	0	0
$313$	−32.6274	−1.84421	−0.922105	−	0.386939i	$-0.873532\pi$
$313$	−32.6274	−1.84421	−0.922105	+	0.386939i	$0.873532\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	28.6274	1.60788	0.803938	−	0.594713i	$-0.202734\pi$
$317$	28.6274	1.60788	0.803938	+	0.594713i	$0.202734\pi$
$318$	0	0
$319$	−4.82843	−0.270340
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.31371	−0.184380
$324$	0	0
$325$	−3.41421	−0.189386
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	−3.17157	−0.174325	−0.0871627	−	0.996194i	$-0.527780\pi$
$331$	−3.17157	−0.174325	−0.0871627	+	0.996194i	$0.527780\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.48528	0.463600
$336$	0	0
$337$	1.07107	0.0583448	0.0291724	−	0.999574i	$-0.490713\pi$
$337$	1.07107	0.0583448	0.0291724	+	0.999574i	$0.490713\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−10.4853	−0.567810
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.89949	0.316701	0.158351	−	0.987383i	$-0.449382\pi$
$347$	5.89949	0.316701	0.158351	+	0.987383i	$0.449382\pi$
$348$	0	0
$349$	−9.51472	−0.509311	−0.254656	−	0.967032i	$-0.581962\pi$
$349$	−9.51472	−0.509311	−0.254656	+	0.967032i	$0.581962\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−13.7990	−0.734446	−0.367223	−	0.930133i	$-0.619691\pi$
$353$	−13.7990	−0.734446	−0.367223	+	0.930133i	$0.619691\pi$
$354$	0	0
$355$	8.82843	0.468564
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.17157	0.0618333	0.0309166	−	0.999522i	$-0.490157\pi$
$359$	1.17157	0.0618333	0.0309166	+	0.999522i	$0.490157\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.58579	−0.240031
$366$	0	0
$367$	−28.4853	−1.48692	−0.743460	−	0.668781i	$-0.766817\pi$
$367$	−28.4853	−1.48692	−0.743460	+	0.668781i	$0.766817\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	18.1421	0.941893
$372$	0	0
$373$	14.2426	0.737456	0.368728	−	0.929537i	$-0.379793\pi$
$373$	14.2426	0.737456	0.368728	+	0.929537i	$0.379793\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−16.4853	−0.849035
$378$	0	0
$379$	8.68629	0.446185	0.223092	−	0.974797i	$-0.428385\pi$
$379$	8.68629	0.446185	0.223092	+	0.974797i	$0.428385\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7.31371	−0.373713	−0.186857	−	0.982387i	$-0.559830\pi$
$383$	−7.31371	−0.373713	−0.186857	+	0.982387i	$0.559830\pi$
$384$	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	$-0.754501\pi$
$389$	−28.2843	−1.43407	−0.717035	+	0.697037i	$0.754501\pi$
$390$	0	0
$391$	1.65685	0.0837907
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−9.65685	−0.485889
$396$	0	0
$397$	−11.1716	−0.560685	−0.280343	−	0.959900i	$-0.590448\pi$
$397$	−11.1716	−0.560685	−0.280343	+	0.959900i	$0.590448\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−32.6274	−1.62934	−0.814668	−	0.579928i	$-0.803081\pi$
$401$	−32.6274	−1.62934	−0.814668	+	0.579928i	$0.803081\pi$
$402$	0	0
$403$	−35.7990	−1.78327
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.65685	−0.181264
$408$	0	0
$409$	1.31371	0.0649587	0.0324794	−	0.999472i	$-0.489660\pi$
$409$	1.31371	0.0649587	0.0324794	+	0.999472i	$0.489660\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9.17157	0.451304
$414$	0	0
$415$	8.24264	0.404615
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.65685	0.0809426	0.0404713	−	0.999181i	$-0.487114\pi$
$419$	1.65685	0.0809426	0.0404713	+	0.999181i	$0.487114\pi$
$420$	0	0
$421$	−27.3137	−1.33119	−0.665594	−	0.746314i	$-0.731822\pi$
$421$	−27.3137	−1.33119	−0.665594	+	0.746314i	$0.731822\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.585786	0.0284148
$426$	0	0
$427$	−12.4853	−0.604205
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−21.6569	−1.04317	−0.521587	−	0.853198i	$-0.674660\pi$
$431$	−21.6569	−1.04317	−0.521587	+	0.853198i	$0.674660\pi$
$432$	0	0
$433$	−0.142136	−0.00683060	−0.00341530	−	0.999994i	$-0.501087\pi$
$433$	−0.142136	−0.00683060	−0.00341530	+	0.999994i	$0.501087\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−16.0000	−0.765384
$438$	0	0
$439$	−10.1421	−0.484058	−0.242029	−	0.970269i	$-0.577813\pi$
$439$	−10.1421	−0.484058	−0.242029	+	0.970269i	$0.577813\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.9706	−0.616250	−0.308125	−	0.951346i	$-0.599702\pi$
$443$	−12.9706	−0.616250	−0.308125	+	0.951346i	$0.599702\pi$
$444$	0	0
$445$	−15.6569	−0.742206
$446$	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
$451$	6.48528	0.305380
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.82843	0.226360
$456$	0	0
$457$	−39.2132	−1.83432	−0.917158	−	0.398523i	$-0.869523\pi$
$457$	−39.2132	−1.83432	−0.917158	+	0.398523i	$0.869523\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.68629	−0.125113	−0.0625565	−	0.998041i	$-0.519925\pi$
$461$	−2.68629	−0.125113	−0.0625565	+	0.998041i	$0.519925\pi$
$462$	0	0
$463$	−13.1716	−0.612135	−0.306067	−	0.952010i	$-0.599013\pi$
$463$	−13.1716	−0.612135	−0.306067	+	0.952010i	$0.599013\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−0.485281	−0.0224561	−0.0112281	−	0.999937i	$-0.503574\pi$
$467$	−0.485281	−0.0224561	−0.0112281	+	0.999937i	$0.503574\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5.41421	0.248946
$474$	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
$481$	−12.4853	−0.569280
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.82843	0.219248
$486$	0	0
$487$	10.8284	0.490683	0.245341	−	0.969437i	$-0.421100\pi$
$487$	10.8284	0.490683	0.245341	+	0.969437i	$0.421100\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−36.2843	−1.63749	−0.818743	−	0.574160i	$-0.805329\pi$
$491$	−36.2843	−1.63749	−0.818743	+	0.574160i	$0.805329\pi$
$492$	0	0
$493$	2.82843	0.127386
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12.4853	−0.560041
$498$	0	0
$499$	−3.02944	−0.135616	−0.0678081	−	0.997698i	$-0.521601\pi$
$499$	−3.02944	−0.135616	−0.0678081	+	0.997698i	$0.521601\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	39.0711	1.74209	0.871046	−	0.491201i	$-0.163442\pi$
$503$	39.0711	1.74209	0.871046	+	0.491201i	$0.163442\pi$
$504$	0	0
$505$	−3.65685	−0.162728
$506$	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
$511$	6.48528	0.286892
$512$	0	0
$513$	0	0
$514$	0	0
$515$	17.6569	0.778054
$516$	0	0
$517$	2.82843	0.124394
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.31371	0.0575546	0.0287773	−	0.999586i	$-0.490839\pi$
$521$	1.31371	0.0575546	0.0287773	+	0.999586i	$0.490839\pi$
$522$	0	0
$523$	8.24264	0.360426	0.180213	−	0.983628i	$-0.442321\pi$
$523$	8.24264	0.360426	0.180213	+	0.983628i	$0.442321\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.14214	0.267556
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	22.1421	0.959082
$534$	0	0
$535$	13.8995	0.600928
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.00000	0.215365
$540$	0	0
$541$	30.2843	1.30202	0.651011	−	0.759068i	$-0.274345\pi$
$541$	30.2843	1.30202	0.651011	+	0.759068i	$0.274345\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	18.9706	0.812610
$546$	0	0
$547$	−19.0711	−0.815420	−0.407710	−	0.913111i	$-0.633673\pi$
$547$	−19.0711	−0.815420	−0.407710	+	0.913111i	$0.633673\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−27.3137	−1.16360
$552$	0	0
$553$	13.6569	0.580749
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4.58579	−0.194306	−0.0971530	−	0.995269i	$-0.530974\pi$
$557$	−4.58579	−0.194306	−0.0971530	+	0.995269i	$0.530974\pi$
$558$	0	0
$559$	18.4853	0.781844
$560$	0	0
$561$	0	0
$562$	0	0
$563$	34.5858	1.45762	0.728809	−	0.684718i	$-0.240074\pi$
$563$	34.5858	1.45762	0.728809	+	0.684718i	$0.240074\pi$
$564$	0	0
$565$	−11.6569	−0.490408
$566$	0	0
$567$	0	0
$568$	0	0
$569$	36.6274	1.53550	0.767751	−	0.640749i	$-0.221376\pi$
$569$	36.6274	1.53550	0.767751	+	0.640749i	$0.221376\pi$
$570$	0	0
$571$	−17.1716	−0.718608	−0.359304	−	0.933221i	$-0.616986\pi$
$571$	−17.1716	−0.718608	−0.359304	+	0.933221i	$0.616986\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.82843	0.117954
$576$	0	0
$577$	46.7696	1.94704	0.973521	−	0.228598i	$-0.0734140\pi$
$577$	46.7696	1.94704	0.973521	+	0.228598i	$0.0734140\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−11.6569	−0.483608
$582$	0	0
$583$	−12.8284	−0.531299
$584$	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
$589$	−59.3137	−2.44398
$590$	0	0
$591$	0	0
$592$	0	0
$593$	16.3848	0.672842	0.336421	−	0.941712i	$-0.390783\pi$
$593$	16.3848	0.672842	0.336421	+	0.941712i	$0.390783\pi$
$594$	0	0
$595$	−0.828427	−0.0339622
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−8.97056	−0.366527	−0.183264	−	0.983064i	$-0.558666\pi$
$599$	−8.97056	−0.366527	−0.183264	+	0.983064i	$0.558666\pi$
$600$	0	0
$601$	6.20101	0.252944	0.126472	−	0.991970i	$-0.459635\pi$
$601$	6.20101	0.252944	0.126472	+	0.991970i	$0.459635\pi$
$602$	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
$611$	9.65685	0.390675
$612$	0	0
$613$	46.0416	1.85960	0.929802	−	0.368060i	$-0.119978\pi$
$613$	46.0416	1.85960	0.929802	+	0.368060i	$0.119978\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−19.6569	−0.791355	−0.395678	−	0.918389i	$-0.629490\pi$
$617$	−19.6569	−0.791355	−0.395678	+	0.918389i	$0.629490\pi$
$618$	0	0
$619$	17.5147	0.703976	0.351988	−	0.936005i	$-0.385506\pi$
$619$	17.5147	0.703976	0.351988	+	0.936005i	$0.385506\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	22.1421	0.887106
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.14214	0.0854125
$630$	0	0
$631$	−18.3431	−0.730229	−0.365115	−	0.930963i	$-0.618970\pi$
$631$	−18.3431	−0.730229	−0.365115	+	0.930963i	$0.618970\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	12.2426	0.485834
$636$	0	0
$637$	17.0711	0.676380
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−34.9706	−1.38125	−0.690627	−	0.723211i	$-0.742665\pi$
$641$	−34.9706	−1.38125	−0.690627	+	0.723211i	$0.742665\pi$
$642$	0	0
$643$	−45.9411	−1.81174	−0.905871	−	0.423555i	$-0.860782\pi$
$643$	−45.9411	−1.81174	−0.905871	+	0.423555i	$0.860782\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−4.97056	−0.195413	−0.0977065	−	0.995215i	$-0.531151\pi$
$647$	−4.97056	−0.195413	−0.0977065	+	0.995215i	$0.531151\pi$
$648$	0	0
$649$	−6.48528	−0.254570
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−33.1127	−1.29580	−0.647900	−	0.761725i	$-0.724353\pi$
$653$	−33.1127	−1.29580	−0.647900	+	0.761725i	$0.724353\pi$
$654$	0	0
$655$	16.4853	0.644133
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−29.6569	−1.15527	−0.577634	−	0.816296i	$-0.696024\pi$
$659$	−29.6569	−1.15527	−0.577634	+	0.816296i	$0.696024\pi$
$660$	0	0
$661$	−27.6569	−1.07573	−0.537863	−	0.843032i	$-0.680768\pi$
$661$	−27.6569	−1.07573	−0.537863	+	0.843032i	$0.680768\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	13.6569	0.528796
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8.82843	0.340818
$672$	0	0
$673$	28.5858	1.10190	0.550951	−	0.834538i	$-0.314265\pi$
$673$	28.5858	1.10190	0.550951	+	0.834538i	$0.314265\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−12.1005	−0.465060	−0.232530	−	0.972589i	$-0.574700\pi$
$677$	−12.1005	−0.465060	−0.232530	+	0.972589i	$0.574700\pi$
$678$	0	0
$679$	−6.82843	−0.262051
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.17157	−0.0448290	−0.0224145	−	0.999749i	$-0.507135\pi$
$683$	−1.17157	−0.0448290	−0.0224145	+	0.999749i	$0.507135\pi$
$684$	0	0
$685$	−14.0000	−0.534913
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−43.7990	−1.66861
$690$	0	0
$691$	−17.6569	−0.671698	−0.335849	−	0.941916i	$-0.609023\pi$
$691$	−17.6569	−0.671698	−0.335849	+	0.941916i	$0.609023\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.82843	−0.107288
$696$	0	0
$697$	−3.79899	−0.143897
$698$	0	0
$699$	0	0
$700$	0	0
$701$	8.82843	0.333445	0.166723	−	0.986004i	$-0.446682\pi$
$701$	8.82843	0.333445	0.166723	+	0.986004i	$0.446682\pi$
$702$	0	0
$703$	−20.6863	−0.780198
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.17157	0.194497
$708$	0	0
$709$	−49.6569	−1.86490	−0.932451	−	0.361296i	$-0.882334\pi$
$709$	−49.6569	−1.86490	−0.932451	+	0.361296i	$0.882334\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	29.6569	1.11066
$714$	0	0
$715$	−3.41421	−0.127684
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	0	0
$721$	−24.9706	−0.929952
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.82843	0.179323
$726$	0	0
$727$	29.6569	1.09991	0.549956	−	0.835194i	$-0.314645\pi$
$727$	29.6569	1.09991	0.549956	+	0.835194i	$0.314645\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−3.17157	−0.117305
$732$	0	0
$733$	−10.7279	−0.396245	−0.198122	−	0.980177i	$-0.563484\pi$
$733$	−10.7279	−0.396245	−0.198122	+	0.980177i	$0.563484\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.48528	0.312559
$738$	0	0
$739$	24.4853	0.900706	0.450353	−	0.892851i	$-0.351298\pi$
$739$	24.4853	0.900706	0.450353	+	0.892851i	$0.351298\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	34.8701	1.27926	0.639629	−	0.768684i	$-0.279088\pi$
$743$	34.8701	1.27926	0.639629	+	0.768684i	$0.279088\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−19.6569	−0.718246
$750$	0	0
$751$	−38.6274	−1.40953	−0.704767	−	0.709439i	$-0.748949\pi$
$751$	−38.6274	−1.40953	−0.704767	+	0.709439i	$0.748949\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	21.6569	0.788174
$756$	0	0
$757$	−46.0833	−1.67492	−0.837462	−	0.546495i	$-0.815962\pi$
$757$	−46.0833	−1.67492	−0.837462	+	0.546495i	$0.815962\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−5.02944	−0.182317	−0.0911585	−	0.995836i	$-0.529057\pi$
$761$	−5.02944	−0.182317	−0.0911585	+	0.995836i	$0.529057\pi$
$762$	0	0
$763$	−26.8284	−0.971254
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−22.1421	−0.799506
$768$	0	0
$769$	41.7990	1.50731	0.753655	−	0.657270i	$-0.228289\pi$
$769$	41.7990	1.50731	0.753655	+	0.657270i	$0.228289\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	12.8284	0.461406	0.230703	−	0.973024i	$-0.425897\pi$
$773$	12.8284	0.461406	0.230703	+	0.973024i	$0.425897\pi$
$774$	0	0
$775$	10.4853	0.376642
$776$	0	0
$777$	0	0
$778$	0	0
$779$	36.6863	1.31442
$780$	0	0
$781$	8.82843	0.315906
$782$	0	0
$783$	0	0
$784$	0	0
$785$	8.82843	0.315100
$786$	0	0
$787$	−4.44365	−0.158399	−0.0791995	−	0.996859i	$-0.525236\pi$
$787$	−4.44365	−0.158399	−0.0791995	+	0.996859i	$0.525236\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	16.4853	0.586149
$792$	0	0
$793$	30.1421	1.07038
$794$	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
$799$	−1.65685	−0.0586153
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−4.58579	−0.161829
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−42.4853	−1.49370	−0.746851	−	0.664991i	$-0.768435\pi$
$809$	−42.4853	−1.49370	−0.746851	+	0.664991i	$0.768435\pi$
$810$	0	0
$811$	−11.7990	−0.414319	−0.207159	−	0.978307i	$-0.566422\pi$
$811$	−11.7990	−0.414319	−0.207159	+	0.978307i	$0.566422\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6.34315	0.222191
$816$	0	0
$817$	30.6274	1.07152
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−9.79899	−0.341987	−0.170994	−	0.985272i	$-0.554698\pi$
$821$	−9.79899	−0.341987	−0.170994	+	0.985272i	$0.554698\pi$
$822$	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
$827$	28.5269	0.991978	0.495989	−	0.868329i	$-0.334806\pi$
$827$	28.5269	0.991978	0.495989	+	0.868329i	$0.334806\pi$
$828$	0	0
$829$	43.5980	1.51422	0.757110	−	0.653287i	$-0.226611\pi$
$829$	43.5980	1.51422	0.757110	+	0.653287i	$0.226611\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.92893	−0.101481
$834$	0	0
$835$	−4.24264	−0.146823
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−39.4558	−1.36217	−0.681084	−	0.732206i	$-0.738491\pi$
$839$	−39.4558	−1.36217	−0.681084	+	0.732206i	$0.738491\pi$
$840$	0	0
$841$	−5.68629	−0.196079
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.34315	0.0462056
$846$	0	0
$847$	1.41421	0.0485930
$848$	0	0
$849$	0	0
$850$	0	0
$851$	10.3431	0.354558
$852$	0	0
$853$	−18.7279	−0.641232	−0.320616	−	0.947209i	$-0.603890\pi$
$853$	−18.7279	−0.641232	−0.320616	+	0.947209i	$0.603890\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−39.2132	−1.33950	−0.669749	−	0.742588i	$-0.733598\pi$
$857$	−39.2132	−1.33950	−0.669749	+	0.742588i	$0.733598\pi$
$858$	0	0
$859$	33.5147	1.14351	0.571754	−	0.820425i	$-0.306263\pi$
$859$	33.5147	1.14351	0.571754	+	0.820425i	$0.306263\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	52.9706	1.80314	0.901569	−	0.432634i	$-0.142416\pi$
$863$	52.9706	1.80314	0.901569	+	0.432634i	$0.142416\pi$
$864$	0	0
$865$	6.72792	0.228756
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−9.65685	−0.327586
$870$	0	0
$871$	28.9706	0.981630
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.41421	−0.0478091
$876$	0	0
$877$	−42.5269	−1.43603	−0.718016	−	0.696027i	$-0.754950\pi$
$877$	−42.5269	−1.43603	−0.718016	+	0.696027i	$0.754950\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−12.6863	−0.427412	−0.213706	−	0.976898i	$-0.568553\pi$
$881$	−12.6863	−0.427412	−0.213706	+	0.976898i	$0.568553\pi$
$882$	0	0
$883$	36.9706	1.24416	0.622079	−	0.782954i	$-0.286288\pi$
$883$	36.9706	1.24416	0.622079	+	0.782954i	$0.286288\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.7574	0.394773	0.197387	−	0.980326i	$-0.436755\pi$
$887$	11.7574	0.394773	0.197387	+	0.980326i	$0.436755\pi$
$888$	0	0
$889$	−17.3137	−0.580683
$890$	0	0
$891$	0	0
$892$	0	0
$893$	16.0000	0.535420
$894$	0	0
$895$	−7.31371	−0.244470
$896$	0	0
$897$	0	0
$898$	0	0
$899$	50.6274	1.68852
$900$	0	0
$901$	7.51472	0.250352
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.34315	0.0778888
$906$	0	0
$907$	48.7696	1.61937	0.809683	−	0.586867i	$-0.199639\pi$
$907$	48.7696	1.61937	0.809683	+	0.586867i	$0.199639\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	44.2843	1.46720	0.733602	−	0.679580i	$-0.237838\pi$
$911$	44.2843	1.46720	0.733602	+	0.679580i	$0.237838\pi$
$912$	0	0
$913$	8.24264	0.272792
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−23.3137	−0.769886
$918$	0	0
$919$	4.20101	0.138579	0.0692893	−	0.997597i	$-0.477927\pi$
$919$	4.20101	0.138579	0.0692893	+	0.997597i	$0.477927\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	30.1421	0.992140
$924$	0	0
$925$	3.65685	0.120237
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−10.6274	−0.348674	−0.174337	−	0.984686i	$-0.555778\pi$
$929$	−10.6274	−0.348674	−0.174337	+	0.984686i	$0.555778\pi$
$930$	0	0
$931$	28.2843	0.926980
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.585786	0.0191573
$936$	0	0
$937$	−2.72792	−0.0891173	−0.0445587	−	0.999007i	$-0.514188\pi$
$937$	−2.72792	−0.0891173	−0.0445587	+	0.999007i	$0.514188\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−2.68629	−0.0875706	−0.0437853	−	0.999041i	$-0.513942\pi$
$941$	−2.68629	−0.0875706	−0.0437853	+	0.999041i	$0.513942\pi$
$942$	0	0
$943$	−18.3431	−0.597335
$944$	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
$949$	−15.6569	−0.508243
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−15.8995	−0.515035	−0.257518	−	0.966274i	$-0.582905\pi$
$953$	−15.8995	−0.515035	−0.257518	+	0.966274i	$0.582905\pi$
$954$	0	0
$955$	21.6569	0.700799
$956$	0	0
$957$	0	0
$958$	0	0
$959$	19.7990	0.639343
$960$	0	0
$961$	78.9411	2.54649
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−20.3848	−0.656209
$966$	0	0
$967$	−28.2426	−0.908222	−0.454111	−	0.890945i	$-0.650043\pi$
$967$	−28.2426	−0.908222	−0.454111	+	0.890945i	$0.650043\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−23.3137	−0.748173	−0.374086	−	0.927394i	$-0.622044\pi$
$971$	−23.3137	−0.748173	−0.374086	+	0.927394i	$0.622044\pi$
$972$	0	0
$973$	4.00000	0.128234
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10.9706	−0.350979	−0.175490	−	0.984481i	$-0.556151\pi$
$977$	−10.9706	−0.350979	−0.175490	+	0.984481i	$0.556151\pi$
$978$	0	0
$979$	−15.6569	−0.500395
$980$	0	0
$981$	0	0
$982$	0	0
$983$	8.68629	0.277050	0.138525	−	0.990359i	$-0.455764\pi$
$983$	8.68629	0.277050	0.138525	+	0.990359i	$0.455764\pi$
$984$	0	0
$985$	20.3848	0.649513
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−15.3137	−0.486948
$990$	0	0
$991$	27.1716	0.863133	0.431567	−	0.902081i	$-0.357961\pi$
$991$	27.1716	0.863133	0.431567	+	0.902081i	$0.357961\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−11.3137	−0.358669
$996$	0	0
$997$	22.0416	0.698065	0.349033	−	0.937111i	$-0.386510\pi$
$997$	22.0416	0.698065	0.349033	+	0.937111i	$0.386510\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3960.2.a.y.1.1	✓	2	4.3	odd	2
3960.2.a.bd.1.1	yes	2	12.11	even	2
7920.2.a.br.1.2		2	1.1	even	1	trivial
7920.2.a.cc.1.2		2	3.2	odd	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:	\( N \)	\(=\)	\( 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7920.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +1.41421 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +1.41421 q^{7} -1.00000 q^{11} -3.41421 q^{13} +0.585786 q^{17} -5.65685 q^{19} +2.82843 q^{23} +1.00000 q^{25} +4.82843 q^{29} +10.4853 q^{31} -1.41421 q^{35} +3.65685 q^{37} -6.48528 q^{41} -5.41421 q^{43} -2.82843 q^{47} -5.00000 q^{49} +12.8284 q^{53} +1.00000 q^{55} +6.48528 q^{59} -8.82843 q^{61} +3.41421 q^{65} -8.48528 q^{67} -8.82843 q^{71} +4.58579 q^{73} -1.41421 q^{77} +9.65685 q^{79} -8.24264 q^{83} -0.585786 q^{85} +15.6569 q^{89} -4.82843 q^{91} +5.65685 q^{95} -4.82843 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^5	\( 2 q - 2 q^{5} - 2 q^{11} - 4 q^{13} + 4 q^{17} + 2 q^{25} + 4 q^{29} + 4 q^{31} - 4 q^{37} + 4 q^{41} - 8 q^{43} - 10 q^{49} + 20 q^{53} + 2 q^{55} - 4 q^{59} - 12 q^{61} + 4 q^{65} - 12 q^{71} + 12 q^{73} + 8 q^{79} - 8 q^{83} - 4 q^{85} + 20 q^{89} - 4 q^{91} - 4 q^{97}+O(q^{100}) \) 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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