LMFDB - Embedded newform 8280.2.a.bv.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1,8280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, 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("8280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.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:	$66.1161328736$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 18x^{5} + 46x^{4} + 60x^{3} - 76x^{2} - 51x - 7 $ x^7 - 3x^6 - 18x^5 + 46x^4 + 60x^3 - 76x^2 - 51x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.22540$ of defining polynomial
Character	$\chi$	$=$	8280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -2.82985 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -2.82985 q^{7} +4.94410 q^{11} +4.65799 q^{13} +2.16509 q^{17} -0.347949 q^{19} -1.00000 q^{23} +1.00000 q^{25} -1.32315 q^{29} +10.0343 q^{31} +2.82985 q^{35} -5.95454 q^{37} +4.98690 q^{41} +1.57815 q^{43} -13.0205 q^{47} +1.00804 q^{49} -5.12429 q^{53} -4.94410 q^{55} +12.4243 q^{59} +10.4630 q^{61} -4.65799 q^{65} -12.2600 q^{67} -4.84614 q^{71} +12.3742 q^{73} -13.9910 q^{77} -8.81099 q^{79} +14.0533 q^{83} -2.16509 q^{85} +13.2819 q^{89} -13.1814 q^{91} +0.347949 q^{95} -9.56802 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{5} - 6 q^{7}+O(q^{10}) $ 7 * q - 7 * q^5 - 6 * q^7	$ 7 q - 7 q^{5} - 6 q^{7} + 2 q^{11} - 6 q^{13} + 4 q^{17} - 8 q^{19} - 7 q^{23} + 7 q^{25} - 2 q^{29} - 8 q^{31} + 6 q^{35} - 16 q^{37} - 2 q^{41} - 10 q^{43} + 8 q^{47} + 19 q^{49} + 10 q^{53} - 2 q^{55} + 24 q^{59} + 8 q^{61} + 6 q^{65} - 20 q^{67} + 8 q^{71} + 2 q^{73} + 12 q^{77} - 2 q^{79} + 22 q^{83} - 4 q^{85} + 16 q^{89} - 20 q^{91} + 8 q^{95} + 4 q^{97}+O(q^{100}) $ 7 * q - 7 * q^5 - 6 * q^7 + 2 * q^11 - 6 * q^13 + 4 * q^17 - 8 * q^19 - 7 * q^23 + 7 * q^25 - 2 * q^29 - 8 * q^31 + 6 * q^35 - 16 * q^37 - 2 * q^41 - 10 * q^43 + 8 * q^47 + 19 * q^49 + 10 * q^53 - 2 * q^55 + 24 * q^59 + 8 * q^61 + 6 * q^65 - 20 * q^67 + 8 * q^71 + 2 * q^73 + 12 * q^77 - 2 * q^79 + 22 * q^83 - 4 * q^85 + 16 * q^89 - 20 * q^91 + 8 * q^95 + 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
$7$	−2.82985	−1.06958	−0.534791	−	0.844984i	$-0.679610\pi$
$7$	−2.82985	−1.06958	−0.534791	+	0.844984i	$0.679610\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.94410	1.49070	0.745350	−	0.666673i	$-0.232282\pi$
$11$	4.94410	1.49070	0.745350	+	0.666673i	$0.232282\pi$
$12$	0	0
$13$	4.65799	1.29189	0.645947	−	0.763382i	$-0.276463\pi$
$13$	4.65799	1.29189	0.645947	+	0.763382i	$0.276463\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.16509	0.525111	0.262556	−	0.964917i	$-0.415435\pi$
$17$	2.16509	0.525111	0.262556	+	0.964917i	$0.415435\pi$
$18$	0	0
$19$	−0.347949	−0.0798249	−0.0399125	−	0.999203i	$-0.512708\pi$
$19$	−0.347949	−0.0798249	−0.0399125	+	0.999203i	$0.512708\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.32315	−0.245702	−0.122851	−	0.992425i	$-0.539204\pi$
$29$	−1.32315	−0.245702	−0.122851	+	0.992425i	$0.539204\pi$
$30$	0	0
$31$	10.0343	1.80222	0.901109	−	0.433593i	$-0.142755\pi$
$31$	10.0343	1.80222	0.901109	+	0.433593i	$0.142755\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.82985	0.478332
$36$	0	0
$37$	−5.95454	−0.978920	−0.489460	−	0.872026i	$-0.662806\pi$
$37$	−5.95454	−0.978920	−0.489460	+	0.872026i	$0.662806\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.98690	0.778822	0.389411	−	0.921064i	$-0.372678\pi$
$41$	4.98690	0.778822	0.389411	+	0.921064i	$0.372678\pi$
$42$	0	0
$43$	1.57815	0.240665	0.120333	−	0.992734i	$-0.461604\pi$
$43$	1.57815	0.240665	0.120333	+	0.992734i	$0.461604\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−13.0205	−1.89924	−0.949619	−	0.313406i	$-0.898530\pi$
$47$	−13.0205	−1.89924	−0.949619	+	0.313406i	$0.898530\pi$
$48$	0	0
$49$	1.00804	0.144006
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.12429	−0.703876	−0.351938	−	0.936023i	$-0.614477\pi$
$53$	−5.12429	−0.703876	−0.351938	+	0.936023i	$0.614477\pi$
$54$	0	0
$55$	−4.94410	−0.666662
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.4243	1.61750	0.808752	−	0.588149i	$-0.200143\pi$
$59$	12.4243	1.61750	0.808752	+	0.588149i	$0.200143\pi$
$60$	0	0
$61$	10.4630	1.33965	0.669827	−	0.742517i	$-0.266368\pi$
$61$	10.4630	1.33965	0.669827	+	0.742517i	$0.266368\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.65799	−0.577753
$66$	0	0
$67$	−12.2600	−1.49780	−0.748898	−	0.662686i	$-0.769417\pi$
$67$	−12.2600	−1.49780	−0.748898	+	0.662686i	$0.769417\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.84614	−0.575131	−0.287566	−	0.957761i	$-0.592846\pi$
$71$	−4.84614	−0.575131	−0.287566	+	0.957761i	$0.592846\pi$
$72$	0	0
$73$	12.3742	1.44829	0.724147	−	0.689646i	$-0.242234\pi$
$73$	12.3742	1.44829	0.724147	+	0.689646i	$0.242234\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−13.9910	−1.59443
$78$	0	0
$79$	−8.81099	−0.991313	−0.495657	−	0.868519i	$-0.665073\pi$
$79$	−8.81099	−0.991313	−0.495657	+	0.868519i	$0.665073\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	14.0533	1.54255	0.771274	−	0.636503i	$-0.219620\pi$
$83$	14.0533	1.54255	0.771274	+	0.636503i	$0.219620\pi$
$84$	0	0
$85$	−2.16509	−0.234837
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.2819	1.40788	0.703940	−	0.710259i	$-0.251422\pi$
$89$	13.2819	1.40788	0.703940	+	0.710259i	$0.251422\pi$
$90$	0	0
$91$	−13.1814	−1.38179
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.347949	0.0356988
$96$	0	0
$97$	−9.56802	−0.971485	−0.485743	−	0.874102i	$-0.661451\pi$
$97$	−9.56802	−0.971485	−0.485743	+	0.874102i	$0.661451\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.1041	−1.00540	−0.502699	−	0.864462i	$-0.667660\pi$
$101$	−10.1041	−1.00540	−0.502699	+	0.864462i	$0.667660\pi$
$102$	0	0
$103$	−11.5213	−1.13523	−0.567613	−	0.823296i	$-0.692133\pi$
$103$	−11.5213	−1.13523	−0.567613	+	0.823296i	$0.692133\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.1192	1.65497	0.827486	−	0.561486i	$-0.189770\pi$
$107$	17.1192	1.65497	0.827486	+	0.561486i	$0.189770\pi$
$108$	0	0
$109$	10.6603	1.02107	0.510537	−	0.859856i	$-0.329447\pi$
$109$	10.6603	1.02107	0.510537	+	0.859856i	$0.329447\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.61589	−0.246082	−0.123041	−	0.992402i	$-0.539265\pi$
$113$	−2.61589	−0.246082	−0.123041	+	0.992402i	$0.539265\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.12687	−0.561650
$120$	0	0
$121$	13.4441	1.22219
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.23348	0.375660	0.187830	−	0.982202i	$-0.439855\pi$
$127$	4.23348	0.375660	0.187830	+	0.982202i	$0.439855\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−9.87807	−0.863051	−0.431525	−	0.902101i	$-0.642024\pi$
$131$	−9.87807	−0.863051	−0.431525	+	0.902101i	$0.642024\pi$
$132$	0	0
$133$	0.984642	0.0853793
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.2618	−0.962159	−0.481080	−	0.876677i	$-0.659755\pi$
$137$	−11.2618	−0.962159	−0.481080	+	0.876677i	$0.659755\pi$
$138$	0	0
$139$	−22.5163	−1.90981	−0.954905	−	0.296913i	$-0.904043\pi$
$139$	−22.5163	−1.90981	−0.954905	+	0.296913i	$0.904043\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	23.0296	1.92583
$144$	0	0
$145$	1.32315	0.109881
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.0435	1.64203	0.821015	−	0.570907i	$-0.193408\pi$
$149$	20.0435	1.64203	0.821015	+	0.570907i	$0.193408\pi$
$150$	0	0
$151$	−17.7448	−1.44405	−0.722025	−	0.691867i	$-0.756788\pi$
$151$	−17.7448	−1.44405	−0.722025	+	0.691867i	$0.756788\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−10.0343	−0.805976
$156$	0	0
$157$	3.47753	0.277537	0.138769	−	0.990325i	$-0.455686\pi$
$157$	3.47753	0.277537	0.138769	+	0.990325i	$0.455686\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.82985	0.223023
$162$	0	0
$163$	7.93108	0.621210	0.310605	−	0.950539i	$-0.399468\pi$
$163$	7.93108	0.621210	0.310605	+	0.950539i	$0.399468\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.59285	0.587552	0.293776	−	0.955874i	$-0.405088\pi$
$167$	7.59285	0.587552	0.293776	+	0.955874i	$0.405088\pi$
$168$	0	0
$169$	8.69689	0.668992
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.24531	0.550850	0.275425	−	0.961323i	$-0.411181\pi$
$173$	7.24531	0.550850	0.275425	+	0.961323i	$0.411181\pi$
$174$	0	0
$175$	−2.82985	−0.213916
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.54920	0.115793	0.0578964	−	0.998323i	$-0.481561\pi$
$179$	1.54920	0.115793	0.0578964	+	0.998323i	$0.481561\pi$
$180$	0	0
$181$	−1.41079	−0.104863	−0.0524316	−	0.998625i	$-0.516697\pi$
$181$	−1.41079	−0.104863	−0.0524316	+	0.998625i	$0.516697\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.95454	0.437786
$186$	0	0
$187$	10.7044	0.782784
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.5973	0.766795	0.383398	−	0.923583i	$-0.374754\pi$
$191$	10.5973	0.766795	0.383398	+	0.923583i	$0.374754\pi$
$192$	0	0
$193$	−20.7209	−1.49152	−0.745760	−	0.666215i	$-0.767914\pi$
$193$	−20.7209	−1.49152	−0.745760	+	0.666215i	$0.767914\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.8915	1.48846	0.744228	−	0.667926i	$-0.232818\pi$
$197$	20.8915	1.48846	0.744228	+	0.667926i	$0.232818\pi$
$198$	0	0
$199$	−25.0562	−1.77618	−0.888092	−	0.459665i	$-0.847969\pi$
$199$	−25.0562	−1.77618	−0.888092	+	0.459665i	$0.847969\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.74430	0.262799
$204$	0	0
$205$	−4.98690	−0.348300
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.72029	−0.118995
$210$	0	0
$211$	11.6415	0.801436	0.400718	−	0.916201i	$-0.368761\pi$
$211$	11.6415	0.801436	0.400718	+	0.916201i	$0.368761\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.57815	−0.107629
$216$	0	0
$217$	−28.3956	−1.92762
$218$	0	0
$219$	0	0
$220$	0	0
$221$	10.0850	0.678388
$222$	0	0
$223$	19.4037	1.29937	0.649685	−	0.760204i	$-0.274901\pi$
$223$	19.4037	1.29937	0.649685	+	0.760204i	$0.274901\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.44027	0.427456	0.213728	−	0.976893i	$-0.431439\pi$
$227$	6.44027	0.427456	0.213728	+	0.976893i	$0.431439\pi$
$228$	0	0
$229$	−1.42608	−0.0942380	−0.0471190	−	0.998889i	$-0.515004\pi$
$229$	−1.42608	−0.0942380	−0.0471190	+	0.998889i	$0.515004\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.9705	0.718702	0.359351	−	0.933202i	$-0.382998\pi$
$233$	10.9705	0.718702	0.359351	+	0.933202i	$0.382998\pi$
$234$	0	0
$235$	13.0205	0.849365
$236$	0	0
$237$	0	0
$238$	0	0
$239$	14.7873	0.956514	0.478257	−	0.878220i	$-0.341269\pi$
$239$	14.7873	0.956514	0.478257	+	0.878220i	$0.341269\pi$
$240$	0	0
$241$	10.0504	0.647401	0.323700	−	0.946160i	$-0.395073\pi$
$241$	10.0504	0.647401	0.323700	+	0.946160i	$0.395073\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.00804	−0.0644013
$246$	0	0
$247$	−1.62074	−0.103125
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3.54142	−0.223532	−0.111766	−	0.993735i	$-0.535651\pi$
$251$	−3.54142	−0.223532	−0.111766	+	0.993735i	$0.535651\pi$
$252$	0	0
$253$	−4.94410	−0.310833
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.8020	1.23522	0.617608	−	0.786486i	$-0.288102\pi$
$257$	19.8020	1.23522	0.617608	+	0.786486i	$0.288102\pi$
$258$	0	0
$259$	16.8504	1.04703
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6.13769	−0.378466	−0.189233	−	0.981932i	$-0.560600\pi$
$263$	−6.13769	−0.378466	−0.189233	+	0.981932i	$0.560600\pi$
$264$	0	0
$265$	5.12429	0.314783
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.27775	0.138877	0.0694385	−	0.997586i	$-0.477879\pi$
$269$	2.27775	0.138877	0.0694385	+	0.997586i	$0.477879\pi$
$270$	0	0
$271$	30.8457	1.87374	0.936872	−	0.349674i	$-0.113707\pi$
$271$	30.8457	1.87374	0.936872	+	0.349674i	$0.113707\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.94410	0.298140
$276$	0	0
$277$	8.69943	0.522698	0.261349	−	0.965244i	$-0.415833\pi$
$277$	8.69943	0.522698	0.261349	+	0.965244i	$0.415833\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5.86657	−0.349970	−0.174985	−	0.984571i	$-0.555988\pi$
$281$	−5.86657	−0.349970	−0.174985	+	0.984571i	$0.555988\pi$
$282$	0	0
$283$	−5.30578	−0.315396	−0.157698	−	0.987487i	$-0.550407\pi$
$283$	−5.30578	−0.315396	−0.157698	+	0.987487i	$0.550407\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−14.1122	−0.833015
$288$	0	0
$289$	−12.3124	−0.724258
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.61142	0.0941401	0.0470701	−	0.998892i	$-0.485012\pi$
$293$	1.61142	0.0941401	0.0470701	+	0.998892i	$0.485012\pi$
$294$	0	0
$295$	−12.4243	−0.723370
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.65799	−0.269379
$300$	0	0
$301$	−4.46592	−0.257411
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−10.4630	−0.599112
$306$	0	0
$307$	−19.0852	−1.08925	−0.544626	−	0.838679i	$-0.683328\pi$
$307$	−19.0852	−1.08925	−0.544626	+	0.838679i	$0.683328\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	33.9003	1.92231	0.961155	−	0.276011i	$-0.0890125\pi$
$311$	33.9003	1.92231	0.961155	+	0.276011i	$0.0890125\pi$
$312$	0	0
$313$	23.8734	1.34940	0.674702	−	0.738090i	$-0.264272\pi$
$313$	23.8734	1.34940	0.674702	+	0.738090i	$0.264272\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−13.3541	−0.750041	−0.375021	−	0.927016i	$-0.622364\pi$
$317$	−13.3541	−0.750041	−0.375021	+	0.927016i	$0.622364\pi$
$318$	0	0
$319$	−6.54176	−0.366268
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−0.753340	−0.0419170
$324$	0	0
$325$	4.65799	0.258379
$326$	0	0
$327$	0	0
$328$	0	0
$329$	36.8461	2.03139
$330$	0	0
$331$	23.5923	1.29675	0.648376	−	0.761320i	$-0.275449\pi$
$331$	23.5923	1.29675	0.648376	+	0.761320i	$0.275449\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	12.2600	0.669834
$336$	0	0
$337$	26.1866	1.42647	0.713237	−	0.700923i	$-0.247228\pi$
$337$	26.1866	1.42647	0.713237	+	0.700923i	$0.247228\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	49.6106	2.68657
$342$	0	0
$343$	16.9563	0.915556
$344$	0	0
$345$	0	0
$346$	0	0
$347$	33.4133	1.79372	0.896859	−	0.442316i	$-0.145843\pi$
$347$	33.4133	1.79372	0.896859	+	0.442316i	$0.145843\pi$
$348$	0	0
$349$	13.7105	0.733904	0.366952	−	0.930240i	$-0.380401\pi$
$349$	13.7105	0.733904	0.366952	+	0.930240i	$0.380401\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	32.6135	1.73584	0.867920	−	0.496704i	$-0.165456\pi$
$353$	32.6135	1.73584	0.867920	+	0.496704i	$0.165456\pi$
$354$	0	0
$355$	4.84614	0.257207
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.30182	0.227042	0.113521	−	0.993536i	$-0.463787\pi$
$359$	4.30182	0.227042	0.113521	+	0.993536i	$0.463787\pi$
$360$	0	0
$361$	−18.8789	−0.993628
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−12.3742	−0.647697
$366$	0	0
$367$	−25.5453	−1.33345	−0.666727	−	0.745302i	$-0.732305\pi$
$367$	−25.5453	−1.33345	−0.666727	+	0.745302i	$0.732305\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	14.5010	0.752853
$372$	0	0
$373$	−1.08868	−0.0563697	−0.0281848	−	0.999603i	$-0.508973\pi$
$373$	−1.08868	−0.0563697	−0.0281848	+	0.999603i	$0.508973\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.16321	−0.317421
$378$	0	0
$379$	−21.5620	−1.10756	−0.553782	−	0.832662i	$-0.686816\pi$
$379$	−21.5620	−1.10756	−0.553782	+	0.832662i	$0.686816\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6.31231	0.322544	0.161272	−	0.986910i	$-0.448440\pi$
$383$	6.31231	0.322544	0.161272	+	0.986910i	$0.448440\pi$
$384$	0	0
$385$	13.9910	0.713049
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7.52936	−0.381754	−0.190877	−	0.981614i	$-0.561133\pi$
$389$	−7.52936	−0.381754	−0.190877	+	0.981614i	$0.561133\pi$
$390$	0	0
$391$	−2.16509	−0.109493
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.81099	0.443329
$396$	0	0
$397$	24.5533	1.23229	0.616147	−	0.787631i	$-0.288693\pi$
$397$	24.5533	1.23229	0.616147	+	0.787631i	$0.288693\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.4811	1.52215	0.761077	−	0.648662i	$-0.224671\pi$
$401$	30.4811	1.52215	0.761077	+	0.648662i	$0.224671\pi$
$402$	0	0
$403$	46.7398	2.32827
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−29.4398	−1.45928
$408$	0	0
$409$	37.8239	1.87027	0.935136	−	0.354290i	$-0.115277\pi$
$409$	37.8239	1.87027	0.935136	+	0.354290i	$0.115277\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−35.1589	−1.73005
$414$	0	0
$415$	−14.0533	−0.689848
$416$	0	0
$417$	0	0
$418$	0	0
$419$	21.9254	1.07112	0.535562	−	0.844496i	$-0.320100\pi$
$419$	21.9254	1.07112	0.535562	+	0.844496i	$0.320100\pi$
$420$	0	0
$421$	36.1125	1.76002	0.880008	−	0.474960i	$-0.157537\pi$
$421$	36.1125	1.76002	0.880008	+	0.474960i	$0.157537\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.16509	0.105022
$426$	0	0
$427$	−29.6088	−1.43287
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−33.4074	−1.60918	−0.804590	−	0.593831i	$-0.797615\pi$
$431$	−33.4074	−1.60918	−0.804590	+	0.593831i	$0.797615\pi$
$432$	0	0
$433$	−24.9892	−1.20090	−0.600451	−	0.799661i	$-0.705012\pi$
$433$	−24.9892	−1.20090	−0.600451	+	0.799661i	$0.705012\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.347949	0.0166446
$438$	0	0
$439$	20.5951	0.982950	0.491475	−	0.870892i	$-0.336458\pi$
$439$	20.5951	0.982950	0.491475	+	0.870892i	$0.336458\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	13.4349	0.638312	0.319156	−	0.947702i	$-0.396601\pi$
$443$	13.4349	0.638312	0.319156	+	0.947702i	$0.396601\pi$
$444$	0	0
$445$	−13.2819	−0.629623
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−10.0435	−0.473980	−0.236990	−	0.971512i	$-0.576161\pi$
$449$	−10.0435	−0.473980	−0.236990	+	0.971512i	$0.576161\pi$
$450$	0	0
$451$	24.6557	1.16099
$452$	0	0
$453$	0	0
$454$	0	0
$455$	13.1814	0.617954
$456$	0	0
$457$	5.60889	0.262373	0.131186	−	0.991358i	$-0.458121\pi$
$457$	5.60889	0.262373	0.131186	+	0.991358i	$0.458121\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	37.8370	1.76224	0.881121	−	0.472890i	$-0.156789\pi$
$461$	37.8370	1.76224	0.881121	+	0.472890i	$0.156789\pi$
$462$	0	0
$463$	−39.0326	−1.81400	−0.907000	−	0.421131i	$-0.861633\pi$
$463$	−39.0326	−1.81400	−0.907000	+	0.421131i	$0.861633\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.70089	0.217531	0.108766	−	0.994067i	$-0.465310\pi$
$467$	4.70089	0.217531	0.108766	+	0.994067i	$0.465310\pi$
$468$	0	0
$469$	34.6939	1.60201
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.80251	0.358760
$474$	0	0
$475$	−0.347949	−0.0159650
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−15.8361	−0.723570	−0.361785	−	0.932262i	$-0.617833\pi$
$479$	−15.8361	−0.723570	−0.361785	+	0.932262i	$0.617833\pi$
$480$	0	0
$481$	−27.7362	−1.26466
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9.56802	0.434461
$486$	0	0
$487$	−16.6167	−0.752973	−0.376487	−	0.926422i	$-0.622868\pi$
$487$	−16.6167	−0.752973	−0.376487	+	0.926422i	$0.622868\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−26.9471	−1.21611	−0.608054	−	0.793896i	$-0.708049\pi$
$491$	−26.9471	−1.21611	−0.608054	+	0.793896i	$0.708049\pi$
$492$	0	0
$493$	−2.86473	−0.129021
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13.7139	0.615150
$498$	0	0
$499$	15.5034	0.694029	0.347015	−	0.937860i	$-0.387195\pi$
$499$	15.5034	0.694029	0.347015	+	0.937860i	$0.387195\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−31.4518	−1.40236	−0.701182	−	0.712982i	$-0.747344\pi$
$503$	−31.4518	−1.40236	−0.701182	+	0.712982i	$0.747344\pi$
$504$	0	0
$505$	10.1041	0.449628
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.46634	−0.0649943	−0.0324971	−	0.999472i	$-0.510346\pi$
$509$	−1.46634	−0.0649943	−0.0324971	+	0.999472i	$0.510346\pi$
$510$	0	0
$511$	−35.0172	−1.54907
$512$	0	0
$513$	0	0
$514$	0	0
$515$	11.5213	0.507688
$516$	0	0
$517$	−64.3747	−2.83120
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5.58182	0.244544	0.122272	−	0.992497i	$-0.460982\pi$
$521$	5.58182	0.244544	0.122272	+	0.992497i	$0.460982\pi$
$522$	0	0
$523$	−11.2870	−0.493548	−0.246774	−	0.969073i	$-0.579371\pi$
$523$	−11.2870	−0.493548	−0.246774	+	0.969073i	$0.579371\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	21.7252	0.946365
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	23.2289	1.00616
$534$	0	0
$535$	−17.1192	−0.740126
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.98385	0.214670
$540$	0	0
$541$	17.9045	0.769773	0.384887	−	0.922964i	$-0.374241\pi$
$541$	17.9045	0.769773	0.384887	+	0.922964i	$0.374241\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−10.6603	−0.456638
$546$	0	0
$547$	−16.1535	−0.690673	−0.345336	−	0.938479i	$-0.612235\pi$
$547$	−16.1535	−0.690673	−0.345336	+	0.938479i	$0.612235\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0.460387	0.0196132
$552$	0	0
$553$	24.9338	1.06029
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7.90248	−0.334839	−0.167419	−	0.985886i	$-0.553543\pi$
$557$	−7.90248	−0.334839	−0.167419	+	0.985886i	$0.553543\pi$
$558$	0	0
$559$	7.35100	0.310914
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7.36386	−0.310350	−0.155175	−	0.987887i	$-0.549594\pi$
$563$	−7.36386	−0.310350	−0.155175	+	0.987887i	$0.549594\pi$
$564$	0	0
$565$	2.61589	0.110051
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.18004	0.0494698	0.0247349	−	0.999694i	$-0.492126\pi$
$569$	1.18004	0.0494698	0.0247349	+	0.999694i	$0.492126\pi$
$570$	0	0
$571$	9.59220	0.401421	0.200711	−	0.979651i	$-0.435675\pi$
$571$	9.59220	0.401421	0.200711	+	0.979651i	$0.435675\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−21.2966	−0.886589	−0.443294	−	0.896376i	$-0.646190\pi$
$577$	−21.2966	−0.886589	−0.443294	+	0.896376i	$0.646190\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−39.7686	−1.64988
$582$	0	0
$583$	−25.3350	−1.04927
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−27.2149	−1.12328	−0.561640	−	0.827382i	$-0.689829\pi$
$587$	−27.2149	−1.12328	−0.561640	+	0.827382i	$0.689829\pi$
$588$	0	0
$589$	−3.49143	−0.143862
$590$	0	0
$591$	0	0
$592$	0	0
$593$	24.6044	1.01038	0.505191	−	0.863008i	$-0.331422\pi$
$593$	24.6044	1.01038	0.505191	+	0.863008i	$0.331422\pi$
$594$	0	0
$595$	6.12687	0.251177
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−20.9286	−0.855120	−0.427560	−	0.903987i	$-0.640627\pi$
$599$	−20.9286	−0.855120	−0.427560	+	0.903987i	$0.640627\pi$
$600$	0	0
$601$	28.1317	1.14752	0.573758	−	0.819025i	$-0.305485\pi$
$601$	28.1317	1.14752	0.573758	+	0.819025i	$0.305485\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−13.4441	−0.546580
$606$	0	0
$607$	4.02339	0.163304	0.0816522	−	0.996661i	$-0.473980\pi$
$607$	4.02339	0.163304	0.0816522	+	0.996661i	$0.473980\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−60.6495	−2.45362
$612$	0	0
$613$	−30.9604	−1.25048	−0.625240	−	0.780432i	$-0.714999\pi$
$613$	−30.9604	−1.25048	−0.625240	+	0.780432i	$0.714999\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−35.4413	−1.42681	−0.713407	−	0.700750i	$-0.752849\pi$
$617$	−35.4413	−1.42681	−0.713407	+	0.700750i	$0.752849\pi$
$618$	0	0
$619$	7.61916	0.306240	0.153120	−	0.988208i	$-0.451068\pi$
$619$	7.61916	0.306240	0.153120	+	0.988208i	$0.451068\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−37.5858	−1.50584
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−12.8921	−0.514042
$630$	0	0
$631$	−21.6842	−0.863235	−0.431618	−	0.902057i	$-0.642057\pi$
$631$	−21.6842	−0.863235	−0.431618	+	0.902057i	$0.642057\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4.23348	−0.168000
$636$	0	0
$637$	4.69544	0.186040
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−39.4616	−1.55864	−0.779320	−	0.626626i	$-0.784435\pi$
$641$	−39.4616	−1.55864	−0.779320	+	0.626626i	$0.784435\pi$
$642$	0	0
$643$	3.18996	0.125800	0.0628999	−	0.998020i	$-0.479965\pi$
$643$	3.18996	0.125800	0.0628999	+	0.998020i	$0.479965\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	18.2970	0.719330	0.359665	−	0.933081i	$-0.382891\pi$
$647$	18.2970	0.719330	0.359665	+	0.933081i	$0.382891\pi$
$648$	0	0
$649$	61.4269	2.41122
$650$	0	0
$651$	0	0
$652$	0	0
$653$	33.0406	1.29298	0.646490	−	0.762923i	$-0.276236\pi$
$653$	33.0406	1.29298	0.646490	+	0.762923i	$0.276236\pi$
$654$	0	0
$655$	9.87807	0.385968
$656$	0	0
$657$	0	0
$658$	0	0
$659$	27.7651	1.08157	0.540787	−	0.841160i	$-0.318127\pi$
$659$	27.7651	1.08157	0.540787	+	0.841160i	$0.318127\pi$
$660$	0	0
$661$	−30.2032	−1.17477	−0.587384	−	0.809308i	$-0.699842\pi$
$661$	−30.2032	−1.17477	−0.587384	+	0.809308i	$0.699842\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.984642	−0.0381828
$666$	0	0
$667$	1.32315	0.0512324
$668$	0	0
$669$	0	0
$670$	0	0
$671$	51.7303	1.99702
$672$	0	0
$673$	−23.3010	−0.898187	−0.449093	−	0.893485i	$-0.648253\pi$
$673$	−23.3010	−0.898187	−0.449093	+	0.893485i	$0.648253\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	43.2511	1.66228	0.831138	−	0.556066i	$-0.187690\pi$
$677$	43.2511	1.66228	0.831138	+	0.556066i	$0.187690\pi$
$678$	0	0
$679$	27.0760	1.03908
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−24.0271	−0.919372	−0.459686	−	0.888082i	$-0.652038\pi$
$683$	−24.0271	−0.919372	−0.459686	+	0.888082i	$0.652038\pi$
$684$	0	0
$685$	11.2618	0.430291
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−23.8689	−0.909333
$690$	0	0
$691$	−18.0742	−0.687574	−0.343787	−	0.939048i	$-0.611710\pi$
$691$	−18.0742	−0.687574	−0.343787	+	0.939048i	$0.611710\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	22.5163	0.854093
$696$	0	0
$697$	10.7971	0.408968
$698$	0	0
$699$	0	0
$700$	0	0
$701$	4.35858	0.164621	0.0823107	−	0.996607i	$-0.473770\pi$
$701$	4.35858	0.164621	0.0823107	+	0.996607i	$0.473770\pi$
$702$	0	0
$703$	2.07187	0.0781422
$704$	0	0
$705$	0	0
$706$	0	0
$707$	28.5931	1.07536
$708$	0	0
$709$	6.28641	0.236091	0.118045	−	0.993008i	$-0.462337\pi$
$709$	6.28641	0.236091	0.118045	+	0.993008i	$0.462337\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−10.0343	−0.375788
$714$	0	0
$715$	−23.0296	−0.861257
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−44.9514	−1.67640	−0.838202	−	0.545360i	$-0.816393\pi$
$719$	−44.9514	−1.67640	−0.838202	+	0.545360i	$0.816393\pi$
$720$	0	0
$721$	32.6035	1.21422
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.32315	−0.0491404
$726$	0	0
$727$	−22.6212	−0.838972	−0.419486	−	0.907762i	$-0.637790\pi$
$727$	−22.6212	−0.838972	−0.419486	+	0.907762i	$0.637790\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.41683	0.126376
$732$	0	0
$733$	45.9519	1.69727	0.848636	−	0.528977i	$-0.177424\pi$
$733$	45.9519	1.69727	0.848636	+	0.528977i	$0.177424\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−60.6145	−2.23276
$738$	0	0
$739$	12.1925	0.448509	0.224255	−	0.974531i	$-0.428005\pi$
$739$	12.1925	0.448509	0.224255	+	0.974531i	$0.428005\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2.03362	−0.0746063	−0.0373032	−	0.999304i	$-0.511877\pi$
$743$	−2.03362	−0.0746063	−0.0373032	+	0.999304i	$0.511877\pi$
$744$	0	0
$745$	−20.0435	−0.734338
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−48.4447	−1.77013
$750$	0	0
$751$	−0.935597	−0.0341404	−0.0170702	−	0.999854i	$-0.505434\pi$
$751$	−0.935597	−0.0341404	−0.0170702	+	0.999854i	$0.505434\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	17.7448	0.645798
$756$	0	0
$757$	−28.4242	−1.03310	−0.516548	−	0.856258i	$-0.672783\pi$
$757$	−28.4242	−1.03310	−0.516548	+	0.856258i	$0.672783\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	45.5123	1.64982	0.824909	−	0.565265i	$-0.191226\pi$
$761$	45.5123	1.64982	0.824909	+	0.565265i	$0.191226\pi$
$762$	0	0
$763$	−30.1671	−1.09212
$764$	0	0
$765$	0	0
$766$	0	0
$767$	57.8723	2.08965
$768$	0	0
$769$	4.93028	0.177790	0.0888952	−	0.996041i	$-0.471666\pi$
$769$	4.93028	0.177790	0.0888952	+	0.996041i	$0.471666\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−13.2800	−0.477648	−0.238824	−	0.971063i	$-0.576762\pi$
$773$	−13.2800	−0.477648	−0.238824	+	0.971063i	$0.576762\pi$
$774$	0	0
$775$	10.0343	0.360443
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.73518	−0.0621694
$780$	0	0
$781$	−23.9598	−0.857349
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3.47753	−0.124118
$786$	0	0
$787$	−9.94763	−0.354595	−0.177297	−	0.984157i	$-0.556735\pi$
$787$	−9.94763	−0.354595	−0.177297	+	0.984157i	$0.556735\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7.40256	0.263205
$792$	0	0
$793$	48.7368	1.73069
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30.1642	1.06847	0.534236	−	0.845335i	$-0.320599\pi$
$797$	30.1642	1.06847	0.534236	+	0.845335i	$0.320599\pi$
$798$	0	0
$799$	−28.1906	−0.997312
$800$	0	0
$801$	0	0
$802$	0	0
$803$	61.1794	2.15897
$804$	0	0
$805$	−2.82985	−0.0997390
$806$	0	0
$807$	0	0
$808$	0	0
$809$	21.2734	0.747935	0.373967	−	0.927442i	$-0.377997\pi$
$809$	21.2734	0.747935	0.373967	+	0.927442i	$0.377997\pi$
$810$	0	0
$811$	−0.697841	−0.0245045	−0.0122523	−	0.999925i	$-0.503900\pi$
$811$	−0.697841	−0.0245045	−0.0122523	+	0.999925i	$0.503900\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−7.93108	−0.277814
$816$	0	0
$817$	−0.549114	−0.0192111
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−30.8586	−1.07697	−0.538486	−	0.842635i	$-0.681003\pi$
$821$	−30.8586	−1.07697	−0.538486	+	0.842635i	$0.681003\pi$
$822$	0	0
$823$	19.6298	0.684253	0.342127	−	0.939654i	$-0.388853\pi$
$823$	19.6298	0.684253	0.342127	+	0.939654i	$0.388853\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.73006	0.338347	0.169174	−	0.985586i	$-0.445890\pi$
$827$	9.73006	0.338347	0.169174	+	0.985586i	$0.445890\pi$
$828$	0	0
$829$	−19.6529	−0.682573	−0.341287	−	0.939959i	$-0.610863\pi$
$829$	−19.6529	−0.682573	−0.341287	+	0.939959i	$0.610863\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.18250	0.0756191
$834$	0	0
$835$	−7.59285	−0.262761
$836$	0	0
$837$	0	0
$838$	0	0
$839$	32.1900	1.11132	0.555661	−	0.831409i	$-0.312465\pi$
$839$	32.1900	1.11132	0.555661	+	0.831409i	$0.312465\pi$
$840$	0	0
$841$	−27.2493	−0.939630
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−8.69689	−0.299182
$846$	0	0
$847$	−38.0447	−1.30723
$848$	0	0
$849$	0	0
$850$	0	0
$851$	5.95454	0.204119
$852$	0	0
$853$	−55.8766	−1.91318	−0.956590	−	0.291438i	$-0.905866\pi$
$853$	−55.8766	−1.91318	−0.956590	+	0.291438i	$0.905866\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.15146	0.210130	0.105065	−	0.994465i	$-0.466495\pi$
$857$	6.15146	0.210130	0.105065	+	0.994465i	$0.466495\pi$
$858$	0	0
$859$	−30.7220	−1.04822	−0.524111	−	0.851650i	$-0.675602\pi$
$859$	−30.7220	−1.04822	−0.524111	+	0.851650i	$0.675602\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−30.2001	−1.02802	−0.514012	−	0.857783i	$-0.671841\pi$
$863$	−30.2001	−1.02802	−0.514012	+	0.857783i	$0.671841\pi$
$864$	0	0
$865$	−7.24531	−0.246348
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−43.5624	−1.47775
$870$	0	0
$871$	−57.1069	−1.93499
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.82985	0.0956663
$876$	0	0
$877$	−19.8280	−0.669546	−0.334773	−	0.942299i	$-0.608660\pi$
$877$	−19.8280	−0.669546	−0.334773	+	0.942299i	$0.608660\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	55.1069	1.85660	0.928300	−	0.371833i	$-0.121271\pi$
$881$	55.1069	1.85660	0.928300	+	0.371833i	$0.121271\pi$
$882$	0	0
$883$	−26.8178	−0.902490	−0.451245	−	0.892400i	$-0.649020\pi$
$883$	−26.8178	−0.902490	−0.451245	+	0.892400i	$0.649020\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	40.8503	1.37162	0.685810	−	0.727781i	$-0.259448\pi$
$887$	40.8503	1.37162	0.685810	+	0.727781i	$0.259448\pi$
$888$	0	0
$889$	−11.9801	−0.401799
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4.53047	0.151607
$894$	0	0
$895$	−1.54920	−0.0517841
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−13.2769	−0.442809
$900$	0	0
$901$	−11.0945	−0.369613
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.41079	0.0468962
$906$	0	0
$907$	−9.16440	−0.304299	−0.152150	−	0.988357i	$-0.548620\pi$
$907$	−9.16440	−0.304299	−0.152150	+	0.988357i	$0.548620\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−7.72553	−0.255958	−0.127979	−	0.991777i	$-0.540849\pi$
$911$	−7.72553	−0.255958	−0.127979	+	0.991777i	$0.540849\pi$
$912$	0	0
$913$	69.4808	2.29948
$914$	0	0
$915$	0	0
$916$	0	0
$917$	27.9534	0.923103
$918$	0	0
$919$	23.8924	0.788139	0.394069	−	0.919081i	$-0.371067\pi$
$919$	23.8924	0.788139	0.394069	+	0.919081i	$0.371067\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−22.5733	−0.743009
$924$	0	0
$925$	−5.95454	−0.195784
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−19.6787	−0.645636	−0.322818	−	0.946461i	$-0.604630\pi$
$929$	−19.6787	−0.645636	−0.322818	+	0.946461i	$0.604630\pi$
$930$	0	0
$931$	−0.350746	−0.0114952
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−10.7044	−0.350072
$936$	0	0
$937$	−52.6536	−1.72012	−0.860058	−	0.510196i	$-0.829573\pi$
$937$	−52.6536	−1.72012	−0.860058	+	0.510196i	$0.829573\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−4.34001	−0.141480	−0.0707401	−	0.997495i	$-0.522536\pi$
$941$	−4.34001	−0.141480	−0.0707401	+	0.997495i	$0.522536\pi$
$942$	0	0
$943$	−4.98690	−0.162396
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18.0278	0.585825	0.292912	−	0.956139i	$-0.405376\pi$
$947$	18.0278	0.585825	0.292912	+	0.956139i	$0.405376\pi$
$948$	0	0
$949$	57.6391	1.87104
$950$	0	0
$951$	0	0
$952$	0	0
$953$	43.3464	1.40413	0.702064	−	0.712113i	$-0.252262\pi$
$953$	43.3464	1.40413	0.702064	+	0.712113i	$0.252262\pi$
$954$	0	0
$955$	−10.5973	−0.342921
$956$	0	0
$957$	0	0
$958$	0	0
$959$	31.8691	1.02911
$960$	0	0
$961$	69.6876	2.24799
$962$	0	0
$963$	0	0
$964$	0	0
$965$	20.7209	0.667028
$966$	0	0
$967$	32.5215	1.04582	0.522911	−	0.852387i	$-0.324846\pi$
$967$	32.5215	1.04582	0.522911	+	0.852387i	$0.324846\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	34.0432	1.09250	0.546249	−	0.837623i	$-0.316055\pi$
$971$	34.0432	1.09250	0.546249	+	0.837623i	$0.316055\pi$
$972$	0	0
$973$	63.7178	2.04270
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−31.8626	−1.01937	−0.509687	−	0.860360i	$-0.670239\pi$
$977$	−31.8626	−1.01937	−0.509687	+	0.860360i	$0.670239\pi$
$978$	0	0
$979$	65.6671	2.09873
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−17.9450	−0.572357	−0.286178	−	0.958176i	$-0.592385\pi$
$983$	−17.9450	−0.572357	−0.286178	+	0.958176i	$0.592385\pi$
$984$	0	0
$985$	−20.8915	−0.665658
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.57815	−0.0501822
$990$	0	0
$991$	−42.1482	−1.33888	−0.669441	−	0.742865i	$-0.733466\pi$
$991$	−42.1482	−1.33888	−0.669441	+	0.742865i	$0.733466\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	25.0562	0.794334
$996$	0	0
$997$	−50.0852	−1.58622	−0.793108	−	0.609082i	$-0.791538\pi$
$997$	−50.0852	−1.58622	−0.793108	+	0.609082i	$0.791538\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.bv.1.3	✓	7
3.2	odd	2		8280.2.a.bw.1.3	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.a.bv.1.3	✓	7	1.1	even	1	trivial
8280.2.a.bw.1.3	yes	7	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:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -2.82985 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -2.82985 q^{7} +4.94410 q^{11} +4.65799 q^{13} +2.16509 q^{17} -0.347949 q^{19} -1.00000 q^{23} +1.00000 q^{25} -1.32315 q^{29} +10.0343 q^{31} +2.82985 q^{35} -5.95454 q^{37} +4.98690 q^{41} +1.57815 q^{43} -13.0205 q^{47} +1.00804 q^{49} -5.12429 q^{53} -4.94410 q^{55} +12.4243 q^{59} +10.4630 q^{61} -4.65799 q^{65} -12.2600 q^{67} -4.84614 q^{71} +12.3742 q^{73} -13.9910 q^{77} -8.81099 q^{79} +14.0533 q^{83} -2.16509 q^{85} +13.2819 q^{89} -13.1814 q^{91} +0.347949 q^{95} -9.56802 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{5} - 6 q^{7}+O(q^{10}) \) 7 * q - 7 * q^5 - 6 * q^7	\( 7 q - 7 q^{5} - 6 q^{7} + 2 q^{11} - 6 q^{13} + 4 q^{17} - 8 q^{19} - 7 q^{23} + 7 q^{25} - 2 q^{29} - 8 q^{31} + 6 q^{35} - 16 q^{37} - 2 q^{41} - 10 q^{43} + 8 q^{47} + 19 q^{49} + 10 q^{53} - 2 q^{55} + 24 q^{59} + 8 q^{61} + 6 q^{65} - 20 q^{67} + 8 q^{71} + 2 q^{73} + 12 q^{77} - 2 q^{79} + 22 q^{83} - 4 q^{85} + 16 q^{89} - 20 q^{91} + 8 q^{95} + 4 q^{97}+O(q^{100}) \) 7 * q - 7 * q^5 - 6 * q^7 + 2 * q^11 - 6 * q^13 + 4 * q^17 - 8 * q^19 - 7 * q^23 + 7 * q^25 - 2 * q^29 - 8 * q^31 + 6 * q^35 - 16 * q^37 - 2 * q^41 - 10 * q^43 + 8 * q^47 + 19 * q^49 + 10 * q^53 - 2 * q^55 + 24 * q^59 + 8 * q^61 + 6 * q^65 - 20 * q^67 + 8 * q^71 + 2 * q^73 + 12 * q^77 - 2 * q^79 + 22 * q^83 - 4 * q^85 + 16 * q^89 - 20 * q^91 + 8 * q^95 + 4 * q^97

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