LMFDB - Embedded newform 8280.2.a.ba.1.1

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:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ 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)

$f(q)$	$=$	$q-1.00000 q^{5} -1.44949 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -1.44949 q^{7} -2.44949 q^{11} +0.449490 q^{13} +5.44949 q^{17} -2.44949 q^{19} +1.00000 q^{23} +1.00000 q^{25} -1.89898 q^{29} +1.89898 q^{31} +1.44949 q^{35} -6.34847 q^{37} -7.89898 q^{41} +8.89898 q^{43} +6.44949 q^{47} -4.89898 q^{49} -2.55051 q^{53} +2.44949 q^{55} +1.00000 q^{59} -0.449490 q^{61} -0.449490 q^{65} -4.55051 q^{67} -3.89898 q^{71} -8.44949 q^{73} +3.55051 q^{77} +10.5505 q^{83} -5.44949 q^{85} -0.898979 q^{89} -0.651531 q^{91} +2.44949 q^{95} -10.8990 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 + 2 * q^7	$ 2 q - 2 q^{5} + 2 q^{7} - 4 q^{13} + 6 q^{17} + 2 q^{23} + 2 q^{25} + 6 q^{29} - 6 q^{31} - 2 q^{35} + 2 q^{37} - 6 q^{41} + 8 q^{43} + 8 q^{47} - 10 q^{53} + 2 q^{59} + 4 q^{61} + 4 q^{65} - 14 q^{67} + 2 q^{71} - 12 q^{73} + 12 q^{77} + 26 q^{83} - 6 q^{85} + 8 q^{89} - 16 q^{91} - 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 2 * q^7 - 4 * q^13 + 6 * q^17 + 2 * q^23 + 2 * q^25 + 6 * q^29 - 6 * q^31 - 2 * q^35 + 2 * q^37 - 6 * q^41 + 8 * q^43 + 8 * q^47 - 10 * q^53 + 2 * q^59 + 4 * q^61 + 4 * q^65 - 14 * q^67 + 2 * q^71 - 12 * q^73 + 12 * q^77 + 26 * q^83 - 6 * q^85 + 8 * q^89 - 16 * q^91 - 12 * 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
$7$	−1.44949	−0.547856	−0.273928	−	0.961750i	$-0.588323\pi$
$7$	−1.44949	−0.547856	−0.273928	+	0.961750i	$0.588323\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.44949	−0.738549	−0.369274	−	0.929320i	$-0.620394\pi$
$11$	−2.44949	−0.738549	−0.369274	+	0.929320i	$0.620394\pi$
$12$	0	0
$13$	0.449490	0.124666	0.0623330	−	0.998055i	$-0.480146\pi$
$13$	0.449490	0.124666	0.0623330	+	0.998055i	$0.480146\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.44949	1.32170	0.660848	−	0.750520i	$-0.270197\pi$
$17$	5.44949	1.32170	0.660848	+	0.750520i	$0.270197\pi$
$18$	0	0
$19$	−2.44949	−0.561951	−0.280976	−	0.959715i	$-0.590658\pi$
$19$	−2.44949	−0.561951	−0.280976	+	0.959715i	$0.590658\pi$
$20$	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
$29$	−1.89898	−0.352632	−0.176316	−	0.984334i	$-0.556418\pi$
$29$	−1.89898	−0.352632	−0.176316	+	0.984334i	$0.556418\pi$
$30$	0	0
$31$	1.89898	0.341067	0.170533	−	0.985352i	$-0.445451\pi$
$31$	1.89898	0.341067	0.170533	+	0.985352i	$0.445451\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.44949	0.245008
$36$	0	0
$37$	−6.34847	−1.04368	−0.521841	−	0.853043i	$-0.674755\pi$
$37$	−6.34847	−1.04368	−0.521841	+	0.853043i	$0.674755\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.89898	−1.23361	−0.616807	−	0.787115i	$-0.711574\pi$
$41$	−7.89898	−1.23361	−0.616807	+	0.787115i	$0.711574\pi$
$42$	0	0
$43$	8.89898	1.35708	0.678541	−	0.734563i	$-0.262613\pi$
$43$	8.89898	1.35708	0.678541	+	0.734563i	$0.262613\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.44949	0.940755	0.470377	−	0.882465i	$-0.344118\pi$
$47$	6.44949	0.940755	0.470377	+	0.882465i	$0.344118\pi$
$48$	0	0
$49$	−4.89898	−0.699854
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.55051	−0.350340	−0.175170	−	0.984538i	$-0.556047\pi$
$53$	−2.55051	−0.350340	−0.175170	+	0.984538i	$0.556047\pi$
$54$	0	0
$55$	2.44949	0.330289
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.00000	0.130189	0.0650945	−	0.997879i	$-0.479265\pi$
$59$	1.00000	0.130189	0.0650945	+	0.997879i	$0.479265\pi$
$60$	0	0
$61$	−0.449490	−0.0575513	−0.0287756	−	0.999586i	$-0.509161\pi$
$61$	−0.449490	−0.0575513	−0.0287756	+	0.999586i	$0.509161\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.449490	−0.0557523
$66$	0	0
$67$	−4.55051	−0.555933	−0.277967	−	0.960591i	$-0.589660\pi$
$67$	−4.55051	−0.555933	−0.277967	+	0.960591i	$0.589660\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.89898	−0.462724	−0.231362	−	0.972868i	$-0.574318\pi$
$71$	−3.89898	−0.462724	−0.231362	+	0.972868i	$0.574318\pi$
$72$	0	0
$73$	−8.44949	−0.988938	−0.494469	−	0.869195i	$-0.664637\pi$
$73$	−8.44949	−0.988938	−0.494469	+	0.869195i	$0.664637\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.55051	0.404618
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.5505	1.15807	0.579034	−	0.815303i	$-0.303430\pi$
$83$	10.5505	1.15807	0.579034	+	0.815303i	$0.303430\pi$
$84$	0	0
$85$	−5.44949	−0.591080
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.898979	−0.0952916	−0.0476458	−	0.998864i	$-0.515172\pi$
$89$	−0.898979	−0.0952916	−0.0476458	+	0.998864i	$0.515172\pi$
$90$	0	0
$91$	−0.651531	−0.0682990
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.44949	0.251312
$96$	0	0
$97$	−10.8990	−1.10662	−0.553312	−	0.832974i	$-0.686636\pi$
$97$	−10.8990	−1.10662	−0.553312	+	0.832974i	$0.686636\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.0000	1.49256	0.746278	−	0.665635i	$-0.231839\pi$
$101$	15.0000	1.49256	0.746278	+	0.665635i	$0.231839\pi$
$102$	0	0
$103$	5.79796	0.571290	0.285645	−	0.958336i	$-0.407792\pi$
$103$	5.79796	0.571290	0.285645	+	0.958336i	$0.407792\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.2474	1.66737	0.833687	−	0.552237i	$-0.186226\pi$
$107$	17.2474	1.66737	0.833687	+	0.552237i	$0.186226\pi$
$108$	0	0
$109$	−2.44949	−0.234619	−0.117309	−	0.993095i	$-0.537427\pi$
$109$	−2.44949	−0.234619	−0.117309	+	0.993095i	$0.537427\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8.34847	−0.785358	−0.392679	−	0.919676i	$-0.628452\pi$
$113$	−8.34847	−0.785358	−0.392679	+	0.919676i	$0.628452\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.89898	−0.724098
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−4.24745	−0.376900	−0.188450	−	0.982083i	$-0.560346\pi$
$127$	−4.24745	−0.376900	−0.188450	+	0.982083i	$0.560346\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.10102	−0.0961966	−0.0480983	−	0.998843i	$-0.515316\pi$
$131$	−1.10102	−0.0961966	−0.0480983	+	0.998843i	$0.515316\pi$
$132$	0	0
$133$	3.55051	0.307868
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−0.898979	−0.0768050	−0.0384025	−	0.999262i	$-0.512227\pi$
$137$	−0.898979	−0.0768050	−0.0384025	+	0.999262i	$0.512227\pi$
$138$	0	0
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.10102	−0.0920720
$144$	0	0
$145$	1.89898	0.157702
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.4495	1.34759	0.673797	−	0.738916i	$-0.264662\pi$
$149$	16.4495	1.34759	0.673797	+	0.738916i	$0.264662\pi$
$150$	0	0
$151$	7.79796	0.634589	0.317294	−	0.948327i	$-0.397226\pi$
$151$	7.79796	0.634589	0.317294	+	0.948327i	$0.397226\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.89898	−0.152530
$156$	0	0
$157$	22.3485	1.78360	0.891801	−	0.452428i	$-0.149442\pi$
$157$	22.3485	1.78360	0.891801	+	0.452428i	$0.149442\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.44949	−0.114236
$162$	0	0
$163$	−6.89898	−0.540370	−0.270185	−	0.962808i	$-0.587085\pi$
$163$	−6.89898	−0.540370	−0.270185	+	0.962808i	$0.587085\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.55051	0.119982	0.0599910	−	0.998199i	$-0.480893\pi$
$167$	1.55051	0.119982	0.0599910	+	0.998199i	$0.480893\pi$
$168$	0	0
$169$	−12.7980	−0.984458
$170$	0	0
$171$	0	0
$172$	0	0
$173$	17.7980	1.35315	0.676577	−	0.736372i	$-0.263463\pi$
$173$	17.7980	1.35315	0.676577	+	0.736372i	$0.263463\pi$
$174$	0	0
$175$	−1.44949	−0.109571
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.79796	0.732334	0.366167	−	0.930549i	$-0.380670\pi$
$179$	9.79796	0.732334	0.366167	+	0.930549i	$0.380670\pi$
$180$	0	0
$181$	−16.8990	−1.25609	−0.628046	−	0.778177i	$-0.716145\pi$
$181$	−16.8990	−1.25609	−0.628046	+	0.778177i	$0.716145\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.34847	0.466749
$186$	0	0
$187$	−13.3485	−0.976137
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.2474	−0.741479	−0.370740	−	0.928737i	$-0.620896\pi$
$191$	−10.2474	−0.741479	−0.370740	+	0.928737i	$0.620896\pi$
$192$	0	0
$193$	5.79796	0.417346	0.208673	−	0.977985i	$-0.433086\pi$
$193$	5.79796	0.417346	0.208673	+	0.977985i	$0.433086\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.7980	1.12556	0.562779	−	0.826607i	$-0.309732\pi$
$197$	15.7980	1.12556	0.562779	+	0.826607i	$0.309732\pi$
$198$	0	0
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.75255	0.193191
$204$	0	0
$205$	7.89898	0.551689
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.00000	0.415029
$210$	0	0
$211$	20.7980	1.43179	0.715895	−	0.698208i	$-0.246019\pi$
$211$	20.7980	1.43179	0.715895	+	0.698208i	$0.246019\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.89898	−0.606905
$216$	0	0
$217$	−2.75255	−0.186855
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.44949	0.164771
$222$	0	0
$223$	−2.89898	−0.194130	−0.0970650	−	0.995278i	$-0.530945\pi$
$223$	−2.89898	−0.194130	−0.0970650	+	0.995278i	$0.530945\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−25.7980	−1.71227	−0.856135	−	0.516752i	$-0.827141\pi$
$227$	−25.7980	−1.71227	−0.856135	+	0.516752i	$0.827141\pi$
$228$	0	0
$229$	13.7980	0.911795	0.455897	−	0.890032i	$-0.349318\pi$
$229$	13.7980	0.911795	0.455897	+	0.890032i	$0.349318\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.8990	1.10709	0.553544	−	0.832820i	$-0.313275\pi$
$233$	16.8990	1.10709	0.553544	+	0.832820i	$0.313275\pi$
$234$	0	0
$235$	−6.44949	−0.420718
$236$	0	0
$237$	0	0
$238$	0	0
$239$	14.7980	0.957200	0.478600	−	0.878033i	$-0.341144\pi$
$239$	14.7980	0.957200	0.478600	+	0.878033i	$0.341144\pi$
$240$	0	0
$241$	30.0454	1.93539	0.967697	−	0.252114i	$-0.0811259\pi$
$241$	30.0454	1.93539	0.967697	+	0.252114i	$0.0811259\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.89898	0.312984
$246$	0	0
$247$	−1.10102	−0.0700563
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.6969	1.55886	0.779428	−	0.626491i	$-0.215510\pi$
$251$	24.6969	1.55886	0.779428	+	0.626491i	$0.215510\pi$
$252$	0	0
$253$	−2.44949	−0.153998
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.1464	0.820052	0.410026	−	0.912074i	$-0.365520\pi$
$257$	13.1464	0.820052	0.410026	+	0.912074i	$0.365520\pi$
$258$	0	0
$259$	9.20204	0.571787
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.24745	−0.200246	−0.100123	−	0.994975i	$-0.531924\pi$
$263$	−3.24745	−0.200246	−0.100123	+	0.994975i	$0.531924\pi$
$264$	0	0
$265$	2.55051	0.156677
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5.89898	−0.359667	−0.179834	−	0.983697i	$-0.557556\pi$
$269$	−5.89898	−0.359667	−0.179834	+	0.983697i	$0.557556\pi$
$270$	0	0
$271$	−2.79796	−0.169964	−0.0849820	−	0.996382i	$-0.527083\pi$
$271$	−2.79796	−0.169964	−0.0849820	+	0.996382i	$0.527083\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.44949	−0.147710
$276$	0	0
$277$	−9.10102	−0.546827	−0.273414	−	0.961897i	$-0.588153\pi$
$277$	−9.10102	−0.546827	−0.273414	+	0.961897i	$0.588153\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	14.4495	0.861984	0.430992	−	0.902356i	$-0.358164\pi$
$281$	14.4495	0.861984	0.430992	+	0.902356i	$0.358164\pi$
$282$	0	0
$283$	20.3485	1.20959	0.604795	−	0.796381i	$-0.293255\pi$
$283$	20.3485	1.20959	0.604795	+	0.796381i	$0.293255\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11.4495	0.675842
$288$	0	0
$289$	12.6969	0.746879
$290$	0	0
$291$	0	0
$292$	0	0
$293$	21.2474	1.24129	0.620645	−	0.784092i	$-0.286871\pi$
$293$	21.2474	1.24129	0.620645	+	0.784092i	$0.286871\pi$
$294$	0	0
$295$	−1.00000	−0.0582223
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.449490	0.0259947
$300$	0	0
$301$	−12.8990	−0.743485
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.449490	0.0257377
$306$	0	0
$307$	−2.24745	−0.128269	−0.0641343	−	0.997941i	$-0.520429\pi$
$307$	−2.24745	−0.128269	−0.0641343	+	0.997941i	$0.520429\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	14.8990	0.844844	0.422422	−	0.906399i	$-0.361180\pi$
$311$	14.8990	0.844844	0.422422	+	0.906399i	$0.361180\pi$
$312$	0	0
$313$	−33.0454	−1.86784	−0.933918	−	0.357486i	$-0.883634\pi$
$313$	−33.0454	−1.86784	−0.933918	+	0.357486i	$0.883634\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.44949	0.362239	0.181120	−	0.983461i	$-0.442028\pi$
$317$	6.44949	0.362239	0.181120	+	0.983461i	$0.442028\pi$
$318$	0	0
$319$	4.65153	0.260436
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−13.3485	−0.742729
$324$	0	0
$325$	0.449490	0.0249332
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9.34847	−0.515398
$330$	0	0
$331$	−3.69694	−0.203202	−0.101601	−	0.994825i	$-0.532397\pi$
$331$	−3.69694	−0.203202	−0.101601	+	0.994825i	$0.532397\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.55051	0.248621
$336$	0	0
$337$	−23.1010	−1.25839	−0.629196	−	0.777246i	$-0.716616\pi$
$337$	−23.1010	−1.25839	−0.629196	+	0.777246i	$0.716616\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4.65153	−0.251895
$342$	0	0
$343$	17.2474	0.931275
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	0	0
$349$	−3.69694	−0.197893	−0.0989463	−	0.995093i	$-0.531547\pi$
$349$	−3.69694	−0.197893	−0.0989463	+	0.995093i	$0.531547\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−30.0454	−1.59916	−0.799578	−	0.600562i	$-0.794943\pi$
$353$	−30.0454	−1.59916	−0.799578	+	0.600562i	$0.794943\pi$
$354$	0	0
$355$	3.89898	0.206936
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−18.2474	−0.963064	−0.481532	−	0.876429i	$-0.659919\pi$
$359$	−18.2474	−0.963064	−0.481532	+	0.876429i	$0.659919\pi$
$360$	0	0
$361$	−13.0000	−0.684211
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8.44949	0.442267
$366$	0	0
$367$	21.0454	1.09856	0.549281	−	0.835638i	$-0.314902\pi$
$367$	21.0454	1.09856	0.549281	+	0.835638i	$0.314902\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.69694	0.191935
$372$	0	0
$373$	0.202041	0.0104613	0.00523064	−	0.999986i	$-0.498335\pi$
$373$	0.202041	0.0104613	0.00523064	+	0.999986i	$0.498335\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.853572	−0.0439612
$378$	0	0
$379$	−13.7980	−0.708754	−0.354377	−	0.935103i	$-0.615307\pi$
$379$	−13.7980	−0.708754	−0.354377	+	0.935103i	$0.615307\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	23.9444	1.22350	0.611751	−	0.791051i	$-0.290466\pi$
$383$	23.9444	1.22350	0.611751	+	0.791051i	$0.290466\pi$
$384$	0	0
$385$	−3.55051	−0.180951
$386$	0	0
$387$	0	0
$388$	0	0
$389$	21.1010	1.06986	0.534932	−	0.844895i	$-0.320337\pi$
$389$	21.1010	1.06986	0.534932	+	0.844895i	$0.320337\pi$
$390$	0	0
$391$	5.44949	0.275593
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−15.5959	−0.782737	−0.391368	−	0.920234i	$-0.627998\pi$
$397$	−15.5959	−0.782737	−0.391368	+	0.920234i	$0.627998\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.4949	0.923591	0.461796	−	0.886986i	$-0.347205\pi$
$401$	18.4949	0.923591	0.461796	+	0.886986i	$0.347205\pi$
$402$	0	0
$403$	0.853572	0.0425194
$404$	0	0
$405$	0	0
$406$	0	0
$407$	15.5505	0.770810
$408$	0	0
$409$	−2.79796	−0.138350	−0.0691751	−	0.997605i	$-0.522037\pi$
$409$	−2.79796	−0.138350	−0.0691751	+	0.997605i	$0.522037\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.44949	−0.0713247
$414$	0	0
$415$	−10.5505	−0.517904
$416$	0	0
$417$	0	0
$418$	0	0
$419$	26.0454	1.27240	0.636201	−	0.771524i	$-0.280505\pi$
$419$	26.0454	1.27240	0.636201	+	0.771524i	$0.280505\pi$
$420$	0	0
$421$	−1.55051	−0.0755672	−0.0377836	−	0.999286i	$-0.512030\pi$
$421$	−1.55051	−0.0755672	−0.0377836	+	0.999286i	$0.512030\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.44949	0.264339
$426$	0	0
$427$	0.651531	0.0315298
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	−10.5505	−0.507025	−0.253513	−	0.967332i	$-0.581586\pi$
$433$	−10.5505	−0.507025	−0.253513	+	0.967332i	$0.581586\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.44949	−0.117175
$438$	0	0
$439$	6.89898	0.329270	0.164635	−	0.986355i	$-0.447355\pi$
$439$	6.89898	0.329270	0.164635	+	0.986355i	$0.447355\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.2474	0.581894	0.290947	−	0.956739i	$-0.406030\pi$
$443$	12.2474	0.581894	0.290947	+	0.956739i	$0.406030\pi$
$444$	0	0
$445$	0.898979	0.0426157
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.40408	0.0662627	0.0331314	−	0.999451i	$-0.489452\pi$
$449$	1.40408	0.0662627	0.0331314	+	0.999451i	$0.489452\pi$
$450$	0	0
$451$	19.3485	0.911084
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.651531	0.0305442
$456$	0	0
$457$	−1.24745	−0.0583532	−0.0291766	−	0.999574i	$-0.509289\pi$
$457$	−1.24745	−0.0583532	−0.0291766	+	0.999574i	$0.509289\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.30306	0.0606896	0.0303448	−	0.999539i	$-0.490339\pi$
$461$	1.30306	0.0606896	0.0303448	+	0.999539i	$0.490339\pi$
$462$	0	0
$463$	14.6515	0.680914	0.340457	−	0.940260i	$-0.389418\pi$
$463$	14.6515	0.680914	0.340457	+	0.940260i	$0.389418\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−26.1464	−1.20991	−0.604956	−	0.796259i	$-0.706809\pi$
$467$	−26.1464	−1.20991	−0.604956	+	0.796259i	$0.706809\pi$
$468$	0	0
$469$	6.59592	0.304571
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−21.7980	−1.00227
$474$	0	0
$475$	−2.44949	−0.112390
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−12.4495	−0.568832	−0.284416	−	0.958701i	$-0.591800\pi$
$479$	−12.4495	−0.568832	−0.284416	+	0.958701i	$0.591800\pi$
$480$	0	0
$481$	−2.85357	−0.130112
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10.8990	0.494897
$486$	0	0
$487$	−30.4495	−1.37980	−0.689899	−	0.723906i	$-0.742345\pi$
$487$	−30.4495	−1.37980	−0.689899	+	0.723906i	$0.742345\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6.10102	0.275335	0.137668	−	0.990478i	$-0.456039\pi$
$491$	6.10102	0.275335	0.137668	+	0.990478i	$0.456039\pi$
$492$	0	0
$493$	−10.3485	−0.466072
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.65153	0.253506
$498$	0	0
$499$	−7.49490	−0.335518	−0.167759	−	0.985828i	$-0.553653\pi$
$499$	−7.49490	−0.335518	−0.167759	+	0.985828i	$0.553653\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.75255	0.301081	0.150541	−	0.988604i	$-0.451899\pi$
$503$	6.75255	0.301081	0.150541	+	0.988604i	$0.451899\pi$
$504$	0	0
$505$	−15.0000	−0.667491
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15.7980	0.700232	0.350116	−	0.936706i	$-0.386142\pi$
$509$	15.7980	0.700232	0.350116	+	0.936706i	$0.386142\pi$
$510$	0	0
$511$	12.2474	0.541795
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5.79796	−0.255489
$516$	0	0
$517$	−15.7980	−0.694793
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5.55051	0.243172	0.121586	−	0.992581i	$-0.461202\pi$
$521$	5.55051	0.243172	0.121586	+	0.992581i	$0.461202\pi$
$522$	0	0
$523$	39.7980	1.74024	0.870122	−	0.492837i	$-0.164040\pi$
$523$	39.7980	1.74024	0.870122	+	0.492837i	$0.164040\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.3485	0.450786
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3.55051	−0.153790
$534$	0	0
$535$	−17.2474	−0.745672
$536$	0	0
$537$	0	0
$538$	0	0
$539$	12.0000	0.516877
$540$	0	0
$541$	5.30306	0.227996	0.113998	−	0.993481i	$-0.463634\pi$
$541$	5.30306	0.227996	0.113998	+	0.993481i	$0.463634\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.44949	0.104925
$546$	0	0
$547$	−33.3939	−1.42782	−0.713910	−	0.700238i	$-0.753077\pi$
$547$	−33.3939	−1.42782	−0.713910	+	0.700238i	$0.753077\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.65153	0.198162
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−31.7423	−1.34497	−0.672483	−	0.740112i	$-0.734772\pi$
$557$	−31.7423	−1.34497	−0.672483	+	0.740112i	$0.734772\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18.3485	0.773296	0.386648	−	0.922227i	$-0.373633\pi$
$563$	18.3485	0.773296	0.386648	+	0.922227i	$0.373633\pi$
$564$	0	0
$565$	8.34847	0.351223
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15.7980	0.662285	0.331142	−	0.943581i	$-0.392566\pi$
$569$	15.7980	0.662285	0.331142	+	0.943581i	$0.392566\pi$
$570$	0	0
$571$	14.2474	0.596237	0.298119	−	0.954529i	$-0.403641\pi$
$571$	14.2474	0.596237	0.298119	+	0.954529i	$0.403641\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	4.89898	0.203947	0.101974	−	0.994787i	$-0.467484\pi$
$577$	4.89898	0.203947	0.101974	+	0.994787i	$0.467484\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−15.2929	−0.634455
$582$	0	0
$583$	6.24745	0.258743
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.20204	0.338534	0.169267	−	0.985570i	$-0.445860\pi$
$587$	8.20204	0.338534	0.169267	+	0.985570i	$0.445860\pi$
$588$	0	0
$589$	−4.65153	−0.191663
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.55051	0.0636718	0.0318359	−	0.999493i	$-0.489865\pi$
$593$	1.55051	0.0636718	0.0318359	+	0.999493i	$0.489865\pi$
$594$	0	0
$595$	7.89898	0.323827
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	0	0
$601$	32.5959	1.32962	0.664808	−	0.747015i	$-0.268514\pi$
$601$	32.5959	1.32962	0.664808	+	0.747015i	$0.268514\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5.00000	0.203279
$606$	0	0
$607$	−5.55051	−0.225288	−0.112644	−	0.993635i	$-0.535932\pi$
$607$	−5.55051	−0.225288	−0.112644	+	0.993635i	$0.535932\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.89898	0.117280
$612$	0	0
$613$	12.8990	0.520985	0.260492	−	0.965476i	$-0.416115\pi$
$613$	12.8990	0.520985	0.260492	+	0.965476i	$0.416115\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.0454	0.525188	0.262594	−	0.964906i	$-0.415422\pi$
$617$	13.0454	0.525188	0.262594	+	0.964906i	$0.415422\pi$
$618$	0	0
$619$	−6.20204	−0.249281	−0.124641	−	0.992202i	$-0.539778\pi$
$619$	−6.20204	−0.249281	−0.124641	+	0.992202i	$0.539778\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.30306	0.0522061
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−34.5959	−1.37943
$630$	0	0
$631$	36.0454	1.43495	0.717473	−	0.696587i	$-0.245299\pi$
$631$	36.0454	1.43495	0.717473	+	0.696587i	$0.245299\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.24745	0.168555
$636$	0	0
$637$	−2.20204	−0.0872480
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−30.9444	−1.22223	−0.611115	−	0.791542i	$-0.709279\pi$
$641$	−30.9444	−1.22223	−0.611115	+	0.791542i	$0.709279\pi$
$642$	0	0
$643$	−19.4495	−0.767013	−0.383507	−	0.923538i	$-0.625284\pi$
$643$	−19.4495	−0.767013	−0.383507	+	0.923538i	$0.625284\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−40.0454	−1.57435	−0.787174	−	0.616731i	$-0.788456\pi$
$647$	−40.0454	−1.57435	−0.787174	+	0.616731i	$0.788456\pi$
$648$	0	0
$649$	−2.44949	−0.0961509
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.6515	0.416827	0.208413	−	0.978041i	$-0.433170\pi$
$653$	10.6515	0.416827	0.208413	+	0.978041i	$0.433170\pi$
$654$	0	0
$655$	1.10102	0.0430204
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.55051	−0.0603993	−0.0301997	−	0.999544i	$-0.509614\pi$
$659$	−1.55051	−0.0603993	−0.0301997	+	0.999544i	$0.509614\pi$
$660$	0	0
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.55051	−0.137683
$666$	0	0
$667$	−1.89898	−0.0735288
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.10102	0.0425044
$672$	0	0
$673$	−5.55051	−0.213956	−0.106978	−	0.994261i	$-0.534118\pi$
$673$	−5.55051	−0.213956	−0.106978	+	0.994261i	$0.534118\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.55051	0.0980241	0.0490120	−	0.998798i	$-0.484393\pi$
$677$	2.55051	0.0980241	0.0490120	+	0.998798i	$0.484393\pi$
$678$	0	0
$679$	15.7980	0.606270
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−2.04541	−0.0782653	−0.0391327	−	0.999234i	$-0.512459\pi$
$683$	−2.04541	−0.0782653	−0.0391327	+	0.999234i	$0.512459\pi$
$684$	0	0
$685$	0.898979	0.0343482
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.14643	−0.0436754
$690$	0	0
$691$	−35.7980	−1.36182	−0.680909	−	0.732368i	$-0.738415\pi$
$691$	−35.7980	−1.36182	−0.680909	+	0.732368i	$0.738415\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−13.0000	−0.493118
$696$	0	0
$697$	−43.0454	−1.63046
$698$	0	0
$699$	0	0
$700$	0	0
$701$	5.75255	0.217271	0.108635	−	0.994082i	$-0.465352\pi$
$701$	5.75255	0.217271	0.108635	+	0.994082i	$0.465352\pi$
$702$	0	0
$703$	15.5505	0.586499
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−21.7423	−0.817705
$708$	0	0
$709$	36.0454	1.35371	0.676857	−	0.736115i	$-0.263342\pi$
$709$	36.0454	1.35371	0.676857	+	0.736115i	$0.263342\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.89898	0.0711173
$714$	0	0
$715$	1.10102	0.0411758
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−14.7980	−0.551871	−0.275935	−	0.961176i	$-0.588988\pi$
$719$	−14.7980	−0.551871	−0.275935	+	0.961176i	$0.588988\pi$
$720$	0	0
$721$	−8.40408	−0.312984
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.89898	−0.0705263
$726$	0	0
$727$	35.9444	1.33310	0.666552	−	0.745459i	$-0.267770\pi$
$727$	35.9444	1.33310	0.666552	+	0.745459i	$0.267770\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	48.4949	1.79365
$732$	0	0
$733$	6.75255	0.249411	0.124706	−	0.992194i	$-0.460201\pi$
$733$	6.75255	0.249411	0.124706	+	0.992194i	$0.460201\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.1464	0.410584
$738$	0	0
$739$	29.0000	1.06678	0.533391	−	0.845869i	$-0.320917\pi$
$739$	29.0000	1.06678	0.533391	+	0.845869i	$0.320917\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	39.1918	1.43781	0.718905	−	0.695109i	$-0.244644\pi$
$743$	39.1918	1.43781	0.718905	+	0.695109i	$0.244644\pi$
$744$	0	0
$745$	−16.4495	−0.602663
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−25.0000	−0.913480
$750$	0	0
$751$	20.6515	0.753585	0.376793	−	0.926298i	$-0.377027\pi$
$751$	20.6515	0.753585	0.376793	+	0.926298i	$0.377027\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−7.79796	−0.283797
$756$	0	0
$757$	−12.5505	−0.456156	−0.228078	−	0.973643i	$-0.573244\pi$
$757$	−12.5505	−0.456156	−0.228078	+	0.973643i	$0.573244\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.69694	0.0615140	0.0307570	−	0.999527i	$-0.490208\pi$
$761$	1.69694	0.0615140	0.0307570	+	0.999527i	$0.490208\pi$
$762$	0	0
$763$	3.55051	0.128537
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.449490	0.0162301
$768$	0	0
$769$	9.84337	0.354961	0.177480	−	0.984124i	$-0.443205\pi$
$769$	9.84337	0.354961	0.177480	+	0.984124i	$0.443205\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−19.3031	−0.694283	−0.347141	−	0.937813i	$-0.612848\pi$
$773$	−19.3031	−0.694283	−0.347141	+	0.937813i	$0.612848\pi$
$774$	0	0
$775$	1.89898	0.0682134
$776$	0	0
$777$	0	0
$778$	0	0
$779$	19.3485	0.693231
$780$	0	0
$781$	9.55051	0.341744
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−22.3485	−0.797651
$786$	0	0
$787$	14.3485	0.511468	0.255734	−	0.966747i	$-0.417683\pi$
$787$	14.3485	0.511468	0.255734	+	0.966747i	$0.417683\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12.1010	0.430263
$792$	0	0
$793$	−0.202041	−0.00717469
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9.24745	−0.327561	−0.163781	−	0.986497i	$-0.552369\pi$
$797$	−9.24745	−0.327561	−0.163781	+	0.986497i	$0.552369\pi$
$798$	0	0
$799$	35.1464	1.24339
$800$	0	0
$801$	0	0
$802$	0	0
$803$	20.6969	0.730379
$804$	0	0
$805$	1.44949	0.0510878
$806$	0	0
$807$	0	0
$808$	0	0
$809$	12.5959	0.442849	0.221424	−	0.975178i	$-0.428929\pi$
$809$	12.5959	0.442849	0.221424	+	0.975178i	$0.428929\pi$
$810$	0	0
$811$	−21.2929	−0.747693	−0.373847	−	0.927491i	$-0.621961\pi$
$811$	−21.2929	−0.747693	−0.373847	+	0.927491i	$0.621961\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6.89898	0.241661
$816$	0	0
$817$	−21.7980	−0.762614
$818$	0	0
$819$	0	0
$820$	0	0
$821$	33.1918	1.15840	0.579202	−	0.815184i	$-0.303364\pi$
$821$	33.1918	1.15840	0.579202	+	0.815184i	$0.303364\pi$
$822$	0	0
$823$	−14.6969	−0.512303	−0.256152	−	0.966637i	$-0.582455\pi$
$823$	−14.6969	−0.512303	−0.256152	+	0.966637i	$0.582455\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−41.9444	−1.45855	−0.729275	−	0.684221i	$-0.760142\pi$
$827$	−41.9444	−1.45855	−0.729275	+	0.684221i	$0.760142\pi$
$828$	0	0
$829$	26.3939	0.916697	0.458349	−	0.888772i	$-0.348441\pi$
$829$	26.3939	0.916697	0.458349	+	0.888772i	$0.348441\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−26.6969	−0.924994
$834$	0	0
$835$	−1.55051	−0.0536576
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−20.2020	−0.697452	−0.348726	−	0.937225i	$-0.613386\pi$
$839$	−20.2020	−0.697452	−0.348726	+	0.937225i	$0.613386\pi$
$840$	0	0
$841$	−25.3939	−0.875651
$842$	0	0
$843$	0	0
$844$	0	0
$845$	12.7980	0.440263
$846$	0	0
$847$	7.24745	0.249025
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−6.34847	−0.217623
$852$	0	0
$853$	49.1918	1.68430	0.842148	−	0.539246i	$-0.181291\pi$
$853$	49.1918	1.68430	0.842148	+	0.539246i	$0.181291\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.3031	0.386105	0.193053	−	0.981188i	$-0.438161\pi$
$857$	11.3031	0.386105	0.193053	+	0.981188i	$0.438161\pi$
$858$	0	0
$859$	33.0908	1.12904	0.564522	−	0.825418i	$-0.309061\pi$
$859$	33.0908	1.12904	0.564522	+	0.825418i	$0.309061\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	17.5959	0.598972	0.299486	−	0.954101i	$-0.403185\pi$
$863$	17.5959	0.598972	0.299486	+	0.954101i	$0.403185\pi$
$864$	0	0
$865$	−17.7980	−0.605149
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−2.04541	−0.0693060
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.44949	0.0490017
$876$	0	0
$877$	−32.2020	−1.08739	−0.543693	−	0.839284i	$-0.682974\pi$
$877$	−32.2020	−1.08739	−0.543693	+	0.839284i	$0.682974\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−13.3485	−0.449721	−0.224861	−	0.974391i	$-0.572193\pi$
$881$	−13.3485	−0.449721	−0.224861	+	0.974391i	$0.572193\pi$
$882$	0	0
$883$	−48.7423	−1.64031	−0.820155	−	0.572141i	$-0.806113\pi$
$883$	−48.7423	−1.64031	−0.820155	+	0.572141i	$0.806113\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−15.7980	−0.530444	−0.265222	−	0.964187i	$-0.585445\pi$
$887$	−15.7980	−0.530444	−0.265222	+	0.964187i	$0.585445\pi$
$888$	0	0
$889$	6.15663	0.206487
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−15.7980	−0.528659
$894$	0	0
$895$	−9.79796	−0.327510
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.60612	−0.120271
$900$	0	0
$901$	−13.8990	−0.463042
$902$	0	0
$903$	0	0
$904$	0	0
$905$	16.8990	0.561741
$906$	0	0
$907$	−10.5505	−0.350324	−0.175162	−	0.984540i	$-0.556045\pi$
$907$	−10.5505	−0.350324	−0.175162	+	0.984540i	$0.556045\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	28.4949	0.944078	0.472039	−	0.881578i	$-0.343518\pi$
$911$	28.4949	0.944078	0.472039	+	0.881578i	$0.343518\pi$
$912$	0	0
$913$	−25.8434	−0.855291
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.59592	0.0527019
$918$	0	0
$919$	42.6969	1.40844	0.704221	−	0.709981i	$-0.251296\pi$
$919$	42.6969	1.40844	0.704221	+	0.709981i	$0.251296\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.75255	−0.0576859
$924$	0	0
$925$	−6.34847	−0.208736
$926$	0	0
$927$	0	0
$928$	0	0
$929$	58.1918	1.90921	0.954606	−	0.297871i	$-0.0962766\pi$
$929$	58.1918	1.90921	0.954606	+	0.297871i	$0.0962766\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	0	0
$933$	0	0
$934$	0	0
$935$	13.3485	0.436542
$936$	0	0
$937$	−33.1010	−1.08136	−0.540682	−	0.841227i	$-0.681834\pi$
$937$	−33.1010	−1.08136	−0.540682	+	0.841227i	$0.681834\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−13.3485	−0.435148	−0.217574	−	0.976044i	$-0.569814\pi$
$941$	−13.3485	−0.435148	−0.217574	+	0.976044i	$0.569814\pi$
$942$	0	0
$943$	−7.89898	−0.257226
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20.8990	0.679126	0.339563	−	0.940583i	$-0.389721\pi$
$947$	20.8990	0.679126	0.339563	+	0.940583i	$0.389721\pi$
$948$	0	0
$949$	−3.79796	−0.123287
$950$	0	0
$951$	0	0
$952$	0	0
$953$	35.3939	1.14652	0.573260	−	0.819373i	$-0.305678\pi$
$953$	35.3939	1.14652	0.573260	+	0.819373i	$0.305678\pi$
$954$	0	0
$955$	10.2474	0.331600
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.30306	0.0420781
$960$	0	0
$961$	−27.3939	−0.883673
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−5.79796	−0.186643
$966$	0	0
$967$	−12.6515	−0.406846	−0.203423	−	0.979091i	$-0.565207\pi$
$967$	−12.6515	−0.406846	−0.203423	+	0.979091i	$0.565207\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−17.3031	−0.555282	−0.277641	−	0.960685i	$-0.589553\pi$
$971$	−17.3031	−0.555282	−0.277641	+	0.960685i	$0.589553\pi$
$972$	0	0
$973$	−18.8434	−0.604091
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−7.65153	−0.244794	−0.122397	−	0.992481i	$-0.539058\pi$
$977$	−7.65153	−0.244794	−0.122397	+	0.992481i	$0.539058\pi$
$978$	0	0
$979$	2.20204	0.0703775
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0.146428	0.00467033	0.00233517	−	0.999997i	$-0.499257\pi$
$983$	0.146428	0.00467033	0.00233517	+	0.999997i	$0.499257\pi$
$984$	0	0
$985$	−15.7980	−0.503365
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.89898	0.282971
$990$	0	0
$991$	−48.1918	−1.53086	−0.765432	−	0.643517i	$-0.777475\pi$
$991$	−48.1918	−1.53086	−0.765432	+	0.643517i	$0.777475\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−14.0000	−0.443830
$996$	0	0
$997$	−32.0000	−1.01345	−0.506725	−	0.862108i	$-0.669144\pi$
$997$	−32.0000	−1.01345	−0.506725	+	0.862108i	$0.669144\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.ba.1.1	✓	2
3.2	odd	2		8280.2.a.bg.1.1	yes	2

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -1.44949 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -1.44949 q^{7} -2.44949 q^{11} +0.449490 q^{13} +5.44949 q^{17} -2.44949 q^{19} +1.00000 q^{23} +1.00000 q^{25} -1.89898 q^{29} +1.89898 q^{31} +1.44949 q^{35} -6.34847 q^{37} -7.89898 q^{41} +8.89898 q^{43} +6.44949 q^{47} -4.89898 q^{49} -2.55051 q^{53} +2.44949 q^{55} +1.00000 q^{59} -0.449490 q^{61} -0.449490 q^{65} -4.55051 q^{67} -3.89898 q^{71} -8.44949 q^{73} +3.55051 q^{77} +10.5505 q^{83} -5.44949 q^{85} -0.898979 q^{89} -0.651531 q^{91} +2.44949 q^{95} -10.8990 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 + 2 * q^7	\( 2 q - 2 q^{5} + 2 q^{7} - 4 q^{13} + 6 q^{17} + 2 q^{23} + 2 q^{25} + 6 q^{29} - 6 q^{31} - 2 q^{35} + 2 q^{37} - 6 q^{41} + 8 q^{43} + 8 q^{47} - 10 q^{53} + 2 q^{59} + 4 q^{61} + 4 q^{65} - 14 q^{67} + 2 q^{71} - 12 q^{73} + 12 q^{77} + 26 q^{83} - 6 q^{85} + 8 q^{89} - 16 q^{91} - 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 2 * q^7 - 4 * q^13 + 6 * q^17 + 2 * q^23 + 2 * q^25 + 6 * q^29 - 6 * q^31 - 2 * q^35 + 2 * q^37 - 6 * q^41 + 8 * q^43 + 8 * q^47 - 10 * q^53 + 2 * q^59 + 4 * q^61 + 4 * q^65 - 14 * q^67 + 2 * q^71 - 12 * q^73 + 12 * q^77 + 26 * q^83 - 6 * q^85 + 8 * q^89 - 16 * q^91 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more