LMFDB - Embedded newform 8280.2.a.bp.1.2

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:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.363328$ 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)

$f(q)$	$=$	$q+1.00000 q^{5} +0.636672 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +0.636672 q^{7} -1.50466 q^{11} -3.50466 q^{13} +0.636672 q^{17} -0.778008 q^{19} -1.00000 q^{23} +1.00000 q^{25} -4.14134 q^{29} +2.14134 q^{31} +0.636672 q^{35} +1.36333 q^{37} +1.85866 q^{41} -7.00933 q^{43} +3.22199 q^{47} -6.59465 q^{49} +0.636672 q^{53} -1.50466 q^{55} +11.1507 q^{59} +13.7873 q^{61} -3.50466 q^{65} +13.9287 q^{67} +8.97070 q^{71} +15.2406 q^{73} -0.957977 q^{77} -12.5653 q^{79} +9.64600 q^{83} +0.636672 q^{85} -2.44398 q^{89} -2.23132 q^{91} -0.778008 q^{95} -7.55602 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} + 4 q^{7}+O(q^{10}) $ 3 * q + 3 * q^5 + 4 * q^7	$ 3 q + 3 q^{5} + 4 q^{7} + 6 q^{11} + 4 q^{17} + 4 q^{19} - 3 q^{23} + 3 q^{25} - 4 q^{29} - 2 q^{31} + 4 q^{35} + 2 q^{37} + 14 q^{41} + 16 q^{47} - 3 q^{49} + 4 q^{53} + 6 q^{55} + 4 q^{59} + 14 q^{61} + 6 q^{67} + 10 q^{71} + 10 q^{73} + 16 q^{77} - 4 q^{79} + 10 q^{83} + 4 q^{85} - 20 q^{89} + 8 q^{91} + 4 q^{95} - 10 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 + 4 * q^7 + 6 * q^11 + 4 * q^17 + 4 * q^19 - 3 * q^23 + 3 * q^25 - 4 * q^29 - 2 * q^31 + 4 * q^35 + 2 * q^37 + 14 * q^41 + 16 * q^47 - 3 * q^49 + 4 * q^53 + 6 * q^55 + 4 * q^59 + 14 * q^61 + 6 * q^67 + 10 * q^71 + 10 * q^73 + 16 * q^77 - 4 * q^79 + 10 * q^83 + 4 * q^85 - 20 * q^89 + 8 * q^91 + 4 * q^95 - 10 * 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
$7$	0.636672	0.240639	0.120320	−	0.992735i	$-0.461608\pi$
$7$	0.636672	0.240639	0.120320	+	0.992735i	$0.461608\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.50466	−0.453673	−0.226837	−	0.973933i	$-0.572838\pi$
$11$	−1.50466	−0.453673	−0.226837	+	0.973933i	$0.572838\pi$
$12$	0	0
$13$	−3.50466	−0.972019	−0.486010	−	0.873954i	$-0.661548\pi$
$13$	−3.50466	−0.972019	−0.486010	+	0.873954i	$0.661548\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.636672	0.154416	0.0772078	−	0.997015i	$-0.475400\pi$
$17$	0.636672	0.154416	0.0772078	+	0.997015i	$0.475400\pi$
$18$	0	0
$19$	−0.778008	−0.178487	−0.0892436	−	0.996010i	$-0.528445\pi$
$19$	−0.778008	−0.178487	−0.0892436	+	0.996010i	$0.528445\pi$
$20$	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
$29$	−4.14134	−0.769027	−0.384513	−	0.923119i	$-0.625631\pi$
$29$	−4.14134	−0.769027	−0.384513	+	0.923119i	$0.625631\pi$
$30$	0	0
$31$	2.14134	0.384595	0.192298	−	0.981337i	$-0.438406\pi$
$31$	2.14134	0.384595	0.192298	+	0.981337i	$0.438406\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.636672	0.107617
$36$	0	0
$37$	1.36333	0.224130	0.112065	−	0.993701i	$-0.464254\pi$
$37$	1.36333	0.224130	0.112065	+	0.993701i	$0.464254\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.85866	0.290275	0.145137	−	0.989412i	$-0.453638\pi$
$41$	1.85866	0.290275	0.145137	+	0.989412i	$0.453638\pi$
$42$	0	0
$43$	−7.00933	−1.06891	−0.534456	−	0.845196i	$-0.679484\pi$
$43$	−7.00933	−1.06891	−0.534456	+	0.845196i	$0.679484\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.22199	0.469976	0.234988	−	0.971998i	$-0.424495\pi$
$47$	3.22199	0.469976	0.234988	+	0.971998i	$0.424495\pi$
$48$	0	0
$49$	−6.59465	−0.942093
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.636672	0.0874536	0.0437268	−	0.999044i	$-0.486077\pi$
$53$	0.636672	0.0874536	0.0437268	+	0.999044i	$0.486077\pi$
$54$	0	0
$55$	−1.50466	−0.202889
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.1507	1.45169	0.725846	−	0.687857i	$-0.241448\pi$
$59$	11.1507	1.45169	0.725846	+	0.687857i	$0.241448\pi$
$60$	0	0
$61$	13.7873	1.76529	0.882644	−	0.470043i	$-0.155761\pi$
$61$	13.7873	1.76529	0.882644	+	0.470043i	$0.155761\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.50466	−0.434700
$66$	0	0
$67$	13.9287	1.70166	0.850829	−	0.525443i	$-0.176100\pi$
$67$	13.9287	1.70166	0.850829	+	0.525443i	$0.176100\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.97070	1.06463	0.532313	−	0.846548i	$-0.321323\pi$
$71$	8.97070	1.06463	0.532313	+	0.846548i	$0.321323\pi$
$72$	0	0
$73$	15.2406	1.78378	0.891892	−	0.452249i	$-0.149378\pi$
$73$	15.2406	1.78378	0.891892	+	0.452249i	$0.149378\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.957977	−0.109172
$78$	0	0
$79$	−12.5653	−1.41371	−0.706856	−	0.707358i	$-0.749887\pi$
$79$	−12.5653	−1.41371	−0.706856	+	0.707358i	$0.749887\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9.64600	1.05879	0.529393	−	0.848377i	$-0.322420\pi$
$83$	9.64600	1.05879	0.529393	+	0.848377i	$0.322420\pi$
$84$	0	0
$85$	0.636672	0.0690567
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.44398	−0.259062	−0.129531	−	0.991575i	$-0.541347\pi$
$89$	−2.44398	−0.259062	−0.129531	+	0.991575i	$0.541347\pi$
$90$	0	0
$91$	−2.23132	−0.233906
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.778008	−0.0798219
$96$	0	0
$97$	−7.55602	−0.767197	−0.383599	−	0.923500i	$-0.625315\pi$
$97$	−7.55602	−0.767197	−0.383599	+	0.923500i	$0.625315\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.8680	1.08141	0.540703	−	0.841214i	$-0.318158\pi$
$101$	10.8680	1.08141	0.540703	+	0.841214i	$0.318158\pi$
$102$	0	0
$103$	17.1893	1.69371	0.846856	−	0.531822i	$-0.178493\pi$
$103$	17.1893	1.69371	0.846856	+	0.531822i	$0.178493\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.9287	−1.34654	−0.673268	−	0.739399i	$-0.735110\pi$
$107$	−13.9287	−1.34654	−0.673268	+	0.739399i	$0.735110\pi$
$108$	0	0
$109$	−12.5140	−1.19862	−0.599312	−	0.800516i	$-0.704559\pi$
$109$	−12.5140	−1.19862	−0.599312	+	0.800516i	$0.704559\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.47536	−0.609151	−0.304575	−	0.952488i	$-0.598515\pi$
$113$	−6.47536	−0.609151	−0.304575	+	0.952488i	$0.598515\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.405351	0.0371585
$120$	0	0
$121$	−8.73599	−0.794180
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.60737	−0.852516	−0.426258	−	0.904602i	$-0.640168\pi$
$127$	−9.60737	−0.852516	−0.426258	+	0.904602i	$0.640168\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.00933	0.0881855	0.0440927	−	0.999027i	$-0.485960\pi$
$131$	1.00933	0.0881855	0.0440927	+	0.999027i	$0.485960\pi$
$132$	0	0
$133$	−0.495336	−0.0429510
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.2920	1.47736	0.738678	−	0.674059i	$-0.235451\pi$
$137$	17.2920	1.47736	0.738678	+	0.674059i	$0.235451\pi$
$138$	0	0
$139$	17.7160	1.50265	0.751326	−	0.659931i	$-0.229415\pi$
$139$	17.7160	1.50265	0.751326	+	0.659931i	$0.229415\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.27334	0.440979
$144$	0	0
$145$	−4.14134	−0.343919
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.2406	1.24856	0.624281	−	0.781200i	$-0.285392\pi$
$149$	15.2406	1.24856	0.624281	+	0.781200i	$0.285392\pi$
$150$	0	0
$151$	−4.28267	−0.348519	−0.174259	−	0.984700i	$-0.555753\pi$
$151$	−4.28267	−0.348519	−0.174259	+	0.984700i	$0.555753\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.14134	0.171996
$156$	0	0
$157$	11.2020	0.894018	0.447009	−	0.894529i	$-0.352489\pi$
$157$	11.2020	0.894018	0.447009	+	0.894529i	$0.352489\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.636672	−0.0501768
$162$	0	0
$163$	9.83869	0.770626	0.385313	−	0.922786i	$-0.374094\pi$
$163$	9.83869	0.770626	0.385313	+	0.922786i	$0.374094\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.6846	1.21371	0.606857	−	0.794811i	$-0.292430\pi$
$167$	15.6846	1.21371	0.606857	+	0.794811i	$0.292430\pi$
$168$	0	0
$169$	−0.717328	−0.0551791
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.8480	1.73710	0.868551	−	0.495599i	$-0.165052\pi$
$173$	22.8480	1.73710	0.868551	+	0.495599i	$0.165052\pi$
$174$	0	0
$175$	0.636672	0.0481279
$176$	0	0
$177$	0	0
$178$	0	0
$179$	21.1893	1.58376	0.791881	−	0.610675i	$-0.209102\pi$
$179$	21.1893	1.58376	0.791881	+	0.610675i	$0.209102\pi$
$180$	0	0
$181$	−0.161312	−0.0119902	−0.00599511	−	0.999982i	$-0.501908\pi$
$181$	−0.161312	−0.0119902	−0.00599511	+	0.999982i	$0.501908\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.36333	0.100234
$186$	0	0
$187$	−0.957977	−0.0700542
$188$	0	0
$189$	0	0
$190$	0	0
$191$	18.5140	1.33963	0.669813	−	0.742530i	$-0.266374\pi$
$191$	18.5140	1.33963	0.669813	+	0.742530i	$0.266374\pi$
$192$	0	0
$193$	−22.2827	−1.60394	−0.801971	−	0.597363i	$-0.796215\pi$
$193$	−22.2827	−1.60394	−0.801971	+	0.597363i	$0.796215\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−24.5840	−1.75154	−0.875769	−	0.482731i	$-0.839645\pi$
$197$	−24.5840	−1.75154	−0.875769	+	0.482731i	$0.839645\pi$
$198$	0	0
$199$	1.73599	0.123061	0.0615304	−	0.998105i	$-0.480402\pi$
$199$	1.73599	0.123061	0.0615304	+	0.998105i	$0.480402\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.63667	−0.185058
$204$	0	0
$205$	1.85866	0.129815
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.17064	0.0809749
$210$	0	0
$211$	−3.69735	−0.254536	−0.127268	−	0.991868i	$-0.540621\pi$
$211$	−3.69735	−0.254536	−0.127268	+	0.991868i	$0.540621\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−7.00933	−0.478032
$216$	0	0
$217$	1.36333	0.0925488
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.23132	−0.150095
$222$	0	0
$223$	0.443984	0.0297314	0.0148657	−	0.999889i	$-0.495268\pi$
$223$	0.443984	0.0297314	0.0148657	+	0.999889i	$0.495268\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	22.0187	1.46143	0.730715	−	0.682683i	$-0.239187\pi$
$227$	22.0187	1.46143	0.730715	+	0.682683i	$0.239187\pi$
$228$	0	0
$229$	4.62395	0.305559	0.152780	−	0.988260i	$-0.451177\pi$
$229$	4.62395	0.305559	0.152780	+	0.988260i	$0.451177\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.83869	0.513530	0.256765	−	0.966474i	$-0.417343\pi$
$233$	7.83869	0.513530	0.256765	+	0.966474i	$0.417343\pi$
$234$	0	0
$235$	3.22199	0.210180
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−23.1693	−1.49870	−0.749349	−	0.662175i	$-0.769634\pi$
$239$	−23.1693	−1.49870	−0.749349	+	0.662175i	$0.769634\pi$
$240$	0	0
$241$	1.60737	0.103540	0.0517698	−	0.998659i	$-0.483514\pi$
$241$	1.60737	0.103540	0.0517698	+	0.998659i	$0.483514\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.59465	−0.421317
$246$	0	0
$247$	2.72666	0.173493
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.99067	0.188769	0.0943847	−	0.995536i	$-0.469912\pi$
$251$	2.99067	0.188769	0.0943847	+	0.995536i	$0.469912\pi$
$252$	0	0
$253$	1.50466	0.0945974
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.9486	0.745336	0.372668	−	0.927965i	$-0.378443\pi$
$257$	11.9486	0.745336	0.372668	+	0.927965i	$0.378443\pi$
$258$	0	0
$259$	0.867993	0.0539344
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−25.5033	−1.57260	−0.786302	−	0.617843i	$-0.788007\pi$
$263$	−25.5033	−1.57260	−0.786302	+	0.617843i	$0.788007\pi$
$264$	0	0
$265$	0.636672	0.0391104
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.33063	0.446957	0.223478	−	0.974709i	$-0.428259\pi$
$269$	7.33063	0.446957	0.223478	+	0.974709i	$0.428259\pi$
$270$	0	0
$271$	14.5267	0.882435	0.441217	−	0.897400i	$-0.354547\pi$
$271$	14.5267	0.882435	0.441217	+	0.897400i	$0.354547\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.50466	−0.0907347
$276$	0	0
$277$	2.72666	0.163829	0.0819145	−	0.996639i	$-0.473897\pi$
$277$	2.72666	0.163829	0.0819145	+	0.996639i	$0.473897\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−23.3620	−1.39366	−0.696830	−	0.717236i	$-0.745407\pi$
$281$	−23.3620	−1.39366	−0.696830	+	0.717236i	$0.745407\pi$
$282$	0	0
$283$	−18.2113	−1.08255	−0.541276	−	0.840845i	$-0.682059\pi$
$283$	−18.2113	−1.08255	−0.541276	+	0.840845i	$0.682059\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.18336	0.0698515
$288$	0	0
$289$	−16.5946	−0.976156
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3.26063	0.190488	0.0952439	−	0.995454i	$-0.469637\pi$
$293$	3.26063	0.190488	0.0952439	+	0.995454i	$0.469637\pi$
$294$	0	0
$295$	11.1507	0.649217
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.50466	0.202680
$300$	0	0
$301$	−4.46264	−0.257222
$302$	0	0
$303$	0	0
$304$	0	0
$305$	13.7873	0.789461
$306$	0	0
$307$	28.9580	1.65272	0.826360	−	0.563143i	$-0.190408\pi$
$307$	28.9580	1.65272	0.826360	+	0.563143i	$0.190408\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−12.3854	−0.702310	−0.351155	−	0.936317i	$-0.614211\pi$
$311$	−12.3854	−0.702310	−0.351155	+	0.936317i	$0.614211\pi$
$312$	0	0
$313$	−28.2886	−1.59897	−0.799483	−	0.600688i	$-0.794893\pi$
$313$	−28.2886	−1.59897	−0.799483	+	0.600688i	$0.794893\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−19.4206	−1.09077	−0.545385	−	0.838185i	$-0.683617\pi$
$317$	−19.4206	−1.09077	−0.545385	+	0.838185i	$0.683617\pi$
$318$	0	0
$319$	6.23132	0.348887
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−0.495336	−0.0275612
$324$	0	0
$325$	−3.50466	−0.194404
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.05135	0.113095
$330$	0	0
$331$	24.9707	1.37251	0.686257	−	0.727359i	$-0.259253\pi$
$331$	24.9707	1.37251	0.686257	+	0.727359i	$0.259253\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	13.9287	0.761005
$336$	0	0
$337$	−23.8387	−1.29858	−0.649288	−	0.760543i	$-0.724933\pi$
$337$	−23.8387	−1.29858	−0.649288	+	0.760543i	$0.724933\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.22199	−0.174481
$342$	0	0
$343$	−8.65533	−0.467344
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	−21.8773	−1.17107	−0.585533	−	0.810649i	$-0.699115\pi$
$349$	−21.8773	−1.17107	−0.585533	+	0.810649i	$0.699115\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	35.9860	1.91534	0.957670	−	0.287869i	$-0.0929468\pi$
$353$	35.9860	1.91534	0.957670	+	0.287869i	$0.0929468\pi$
$354$	0	0
$355$	8.97070	0.476115
$356$	0	0
$357$	0	0
$358$	0	0
$359$	21.9486	1.15841	0.579203	−	0.815184i	$-0.303364\pi$
$359$	21.9486	1.15841	0.579203	+	0.815184i	$0.303364\pi$
$360$	0	0
$361$	−18.3947	−0.968142
$362$	0	0
$363$	0	0
$364$	0	0
$365$	15.2406	0.797732
$366$	0	0
$367$	2.79798	0.146054	0.0730268	−	0.997330i	$-0.476734\pi$
$367$	2.79798	0.146054	0.0730268	+	0.997330i	$0.476734\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.405351	0.0210448
$372$	0	0
$373$	36.8853	1.90985	0.954925	−	0.296847i	$-0.0959352\pi$
$373$	36.8853	1.90985	0.954925	+	0.296847i	$0.0959352\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14.5140	0.747509
$378$	0	0
$379$	−28.9253	−1.48579	−0.742896	−	0.669407i	$-0.766548\pi$
$379$	−28.9253	−1.48579	−0.742896	+	0.669407i	$0.766548\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	24.1927	1.23619	0.618094	−	0.786104i	$-0.287905\pi$
$383$	24.1927	1.23619	0.618094	+	0.786104i	$0.287905\pi$
$384$	0	0
$385$	−0.957977	−0.0488230
$386$	0	0
$387$	0	0
$388$	0	0
$389$	24.6867	1.25167	0.625833	−	0.779957i	$-0.284759\pi$
$389$	24.6867	1.25167	0.625833	+	0.779957i	$0.284759\pi$
$390$	0	0
$391$	−0.636672	−0.0321979
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−12.5653	−0.632231
$396$	0	0
$397$	−5.65872	−0.284003	−0.142001	−	0.989866i	$-0.545354\pi$
$397$	−5.65872	−0.284003	−0.142001	+	0.989866i	$0.545354\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	25.8387	1.29032	0.645161	−	0.764046i	$-0.276790\pi$
$401$	25.8387	1.29032	0.645161	+	0.764046i	$0.276790\pi$
$402$	0	0
$403$	−7.50466	−0.373834
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.05135	−0.101682
$408$	0	0
$409$	−25.9800	−1.28463	−0.642315	−	0.766441i	$-0.722026\pi$
$409$	−25.9800	−1.28463	−0.642315	+	0.766441i	$0.722026\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.09931	0.349334
$414$	0	0
$415$	9.64600	0.473504
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.96731	−0.0961092	−0.0480546	−	0.998845i	$-0.515302\pi$
$419$	−1.96731	−0.0961092	−0.0480546	+	0.998845i	$0.515302\pi$
$420$	0	0
$421$	−1.14473	−0.0557905	−0.0278953	−	0.999611i	$-0.508880\pi$
$421$	−1.14473	−0.0557905	−0.0278953	+	0.999611i	$0.508880\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.636672	0.0308831
$426$	0	0
$427$	8.77801	0.424798
$428$	0	0
$429$	0	0
$430$	0	0
$431$	19.4720	0.937932	0.468966	−	0.883216i	$-0.344627\pi$
$431$	19.4720	0.937932	0.468966	+	0.883216i	$0.344627\pi$
$432$	0	0
$433$	−11.1834	−0.537438	−0.268719	−	0.963219i	$-0.586600\pi$
$433$	−11.1834	−0.537438	−0.268719	+	0.963219i	$0.586600\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.778008	0.0372172
$438$	0	0
$439$	−13.2733	−0.633502	−0.316751	−	0.948509i	$-0.602592\pi$
$439$	−13.2733	−0.633502	−0.316751	+	0.948509i	$0.602592\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14.8994	0.707890	0.353945	−	0.935266i	$-0.384840\pi$
$443$	14.8994	0.707890	0.353945	+	0.935266i	$0.384840\pi$
$444$	0	0
$445$	−2.44398	−0.115856
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−23.1693	−1.09343	−0.546714	−	0.837319i	$-0.684121\pi$
$449$	−23.1693	−1.09343	−0.546714	+	0.837319i	$0.684121\pi$
$450$	0	0
$451$	−2.79667	−0.131690
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.23132	−0.104606
$456$	0	0
$457$	10.9380	0.511658	0.255829	−	0.966722i	$-0.417652\pi$
$457$	10.9380	0.511658	0.255829	+	0.966722i	$0.417652\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−13.2920	−0.619070	−0.309535	−	0.950888i	$-0.600173\pi$
$461$	−13.2920	−0.619070	−0.309535	+	0.950888i	$0.600173\pi$
$462$	0	0
$463$	0.436726	0.0202964	0.0101482	−	0.999949i	$-0.496770\pi$
$463$	0.436726	0.0202964	0.0101482	+	0.999949i	$0.496770\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−29.1180	−1.34742	−0.673709	−	0.738996i	$-0.735300\pi$
$467$	−29.1180	−1.34742	−0.673709	+	0.738996i	$0.735300\pi$
$468$	0	0
$469$	8.86799	0.409486
$470$	0	0
$471$	0	0
$472$	0	0
$473$	10.5467	0.484937
$474$	0	0
$475$	−0.778008	−0.0356974
$476$	0	0
$477$	0	0
$478$	0	0
$479$	10.1541	0.463951	0.231975	−	0.972722i	$-0.425481\pi$
$479$	10.1541	0.463951	0.231975	+	0.972722i	$0.425481\pi$
$480$	0	0
$481$	−4.77801	−0.217858
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.55602	−0.343101
$486$	0	0
$487$	18.5980	0.842758	0.421379	−	0.906885i	$-0.361546\pi$
$487$	18.5980	0.842758	0.421379	+	0.906885i	$0.361546\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	35.0080	1.57989	0.789945	−	0.613178i	$-0.210109\pi$
$491$	35.0080	1.57989	0.789945	+	0.613178i	$0.210109\pi$
$492$	0	0
$493$	−2.63667	−0.118750
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.71139	0.256191
$498$	0	0
$499$	−8.40535	−0.376275	−0.188138	−	0.982143i	$-0.560245\pi$
$499$	−8.40535	−0.376275	−0.188138	+	0.982143i	$0.560245\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−37.5806	−1.67564	−0.837818	−	0.545949i	$-0.816169\pi$
$503$	−37.5806	−1.67564	−0.837818	+	0.545949i	$0.816169\pi$
$504$	0	0
$505$	10.8680	0.483619
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−34.2427	−1.51778	−0.758891	−	0.651218i	$-0.774258\pi$
$509$	−34.2427	−1.51778	−0.758891	+	0.651218i	$0.774258\pi$
$510$	0	0
$511$	9.70329	0.429248
$512$	0	0
$513$	0	0
$514$	0	0
$515$	17.1893	0.757451
$516$	0	0
$517$	−4.84802	−0.213216
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0.572602	0.0250862	0.0125431	−	0.999921i	$-0.496007\pi$
$521$	0.572602	0.0250862	0.0125431	+	0.999921i	$0.496007\pi$
$522$	0	0
$523$	13.3760	0.584894	0.292447	−	0.956282i	$-0.405531\pi$
$523$	13.3760	0.584894	0.292447	+	0.956282i	$0.405531\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.36333	0.0593875
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−6.51399	−0.282152
$534$	0	0
$535$	−13.9287	−0.602189
$536$	0	0
$537$	0	0
$538$	0	0
$539$	9.92273	0.427402
$540$	0	0
$541$	−36.4040	−1.56513	−0.782566	−	0.622568i	$-0.786089\pi$
$541$	−36.4040	−1.56513	−0.782566	+	0.622568i	$0.786089\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−12.5140	−0.536041
$546$	0	0
$547$	−14.5840	−0.623567	−0.311783	−	0.950153i	$-0.600926\pi$
$547$	−14.5840	−0.623567	−0.311783	+	0.950153i	$0.600926\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.22199	0.137261
$552$	0	0
$553$	−8.00000	−0.340195
$554$	0	0
$555$	0	0
$556$	0	0
$557$	46.4100	1.96645	0.983227	−	0.182387i	$-0.0583824\pi$
$557$	46.4100	1.96645	0.983227	+	0.182387i	$0.0583824\pi$
$558$	0	0
$559$	24.5653	1.03900
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.36333	0.394617	0.197309	−	0.980341i	$-0.436780\pi$
$563$	9.36333	0.394617	0.197309	+	0.980341i	$0.436780\pi$
$564$	0	0
$565$	−6.47536	−0.272420
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−0.282672	−0.0118502	−0.00592512	−	0.999982i	$-0.501886\pi$
$569$	−0.282672	−0.0118502	−0.00592512	+	0.999982i	$0.501886\pi$
$570$	0	0
$571$	−24.1073	−1.00886	−0.504430	−	0.863453i	$-0.668297\pi$
$571$	−24.1073	−1.00886	−0.504430	+	0.863453i	$0.668297\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	22.4227	0.933469	0.466734	−	0.884398i	$-0.345430\pi$
$577$	22.4227	0.933469	0.466734	+	0.884398i	$0.345430\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.14134	0.254786
$582$	0	0
$583$	−0.957977	−0.0396754
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19.1893	0.792027	0.396014	−	0.918245i	$-0.370393\pi$
$587$	19.1893	0.792027	0.396014	+	0.918245i	$0.370393\pi$
$588$	0	0
$589$	−1.66598	−0.0686454
$590$	0	0
$591$	0	0
$592$	0	0
$593$	9.04202	0.371311	0.185656	−	0.982615i	$-0.440559\pi$
$593$	9.04202	0.371311	0.185656	+	0.982615i	$0.440559\pi$
$594$	0	0
$595$	0.405351	0.0166178
$596$	0	0
$597$	0	0
$598$	0	0
$599$	46.3386	1.89335	0.946673	−	0.322196i	$-0.104421\pi$
$599$	46.3386	1.89335	0.946673	+	0.322196i	$0.104421\pi$
$600$	0	0
$601$	−17.6974	−0.721890	−0.360945	−	0.932587i	$-0.617546\pi$
$601$	−17.6974	−0.721890	−0.360945	+	0.932587i	$0.617546\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−8.73599	−0.355168
$606$	0	0
$607$	42.5327	1.72635	0.863174	−	0.504907i	$-0.168473\pi$
$607$	42.5327	1.72635	0.863174	+	0.504907i	$0.168473\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−11.2920	−0.456825
$612$	0	0
$613$	−31.5747	−1.27529	−0.637645	−	0.770331i	$-0.720091\pi$
$613$	−31.5747	−1.27529	−0.637645	+	0.770331i	$0.720091\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	45.2393	1.82127	0.910633	−	0.413215i	$-0.135594\pi$
$617$	45.2393	1.82127	0.910633	+	0.413215i	$0.135594\pi$
$618$	0	0
$619$	29.4906	1.18533	0.592664	−	0.805450i	$-0.298076\pi$
$619$	29.4906	1.18533	0.592664	+	0.805450i	$0.298076\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.55602	−0.0623405
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.867993	0.0346091
$630$	0	0
$631$	43.6447	1.73747	0.868734	−	0.495280i	$-0.164934\pi$
$631$	43.6447	1.73747	0.868734	+	0.495280i	$0.164934\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−9.60737	−0.381257
$636$	0	0
$637$	23.1120	0.915732
$638$	0	0
$639$	0	0
$640$	0	0
$641$	28.3527	1.11986	0.559932	−	0.828539i	$-0.310827\pi$
$641$	28.3527	1.11986	0.559932	+	0.828539i	$0.310827\pi$
$642$	0	0
$643$	22.1086	0.871880	0.435940	−	0.899976i	$-0.356416\pi$
$643$	22.1086	0.871880	0.435940	+	0.899976i	$0.356416\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.42062	0.370363	0.185181	−	0.982704i	$-0.440713\pi$
$647$	9.42062	0.370363	0.185181	+	0.982704i	$0.440713\pi$
$648$	0	0
$649$	−16.7780	−0.658594
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.76868	0.147480	0.0737399	−	0.997278i	$-0.476507\pi$
$653$	3.76868	0.147480	0.0737399	+	0.997278i	$0.476507\pi$
$654$	0	0
$655$	1.00933	0.0394377
$656$	0	0
$657$	0	0
$658$	0	0
$659$	16.9393	0.659862	0.329931	−	0.944005i	$-0.392974\pi$
$659$	16.9393	0.659862	0.329931	+	0.944005i	$0.392974\pi$
$660$	0	0
$661$	35.6960	1.38841	0.694207	−	0.719775i	$-0.255755\pi$
$661$	35.6960	1.38841	0.694207	+	0.719775i	$0.255755\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.495336	−0.0192083
$666$	0	0
$667$	4.14134	0.160353
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−20.7453	−0.800864
$672$	0	0
$673$	2.53265	0.0976265	0.0488133	−	0.998808i	$-0.484456\pi$
$673$	2.53265	0.0976265	0.0488133	+	0.998808i	$0.484456\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	31.3633	1.20539	0.602695	−	0.797971i	$-0.294093\pi$
$677$	31.3633	1.20539	0.602695	+	0.797971i	$0.294093\pi$
$678$	0	0
$679$	−4.81070	−0.184618
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−23.7873	−0.910197	−0.455099	−	0.890441i	$-0.650396\pi$
$683$	−23.7873	−0.910197	−0.455099	+	0.890441i	$0.650396\pi$
$684$	0	0
$685$	17.2920	0.660693
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.23132	−0.0850066
$690$	0	0
$691$	−22.6613	−0.862075	−0.431038	−	0.902334i	$-0.641852\pi$
$691$	−22.6613	−0.862075	−0.431038	+	0.902334i	$0.641852\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	17.7160	0.672007
$696$	0	0
$697$	1.18336	0.0448229
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−13.2593	−0.500797	−0.250399	−	0.968143i	$-0.580562\pi$
$701$	−13.2593	−0.500797	−0.250399	+	0.968143i	$0.580562\pi$
$702$	0	0
$703$	−1.06068	−0.0400043
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.91934	0.260229
$708$	0	0
$709$	−40.6940	−1.52829	−0.764147	−	0.645042i	$-0.776840\pi$
$709$	−40.6940	−1.52829	−0.764147	+	0.645042i	$0.776840\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.14134	−0.0801937
$714$	0	0
$715$	5.27334	0.197212
$716$	0	0
$717$	0	0
$718$	0	0
$719$	4.79073	0.178664	0.0893320	−	0.996002i	$-0.471527\pi$
$719$	4.79073	0.178664	0.0893320	+	0.996002i	$0.471527\pi$
$720$	0	0
$721$	10.9439	0.407574
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.14134	−0.153805
$726$	0	0
$727$	−17.9473	−0.665630	−0.332815	−	0.942992i	$-0.607998\pi$
$727$	−17.9473	−0.665630	−0.332815	+	0.942992i	$0.607998\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4.46264	−0.165057
$732$	0	0
$733$	−17.9287	−0.662211	−0.331105	−	0.943594i	$-0.607422\pi$
$733$	−17.9287	−0.662211	−0.331105	+	0.943594i	$0.607422\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−20.9580	−0.771997
$738$	0	0
$739$	16.5853	0.610101	0.305050	−	0.952336i	$-0.401327\pi$
$739$	16.5853	0.610101	0.305050	+	0.952336i	$0.401327\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.90663	0.106634	0.0533169	−	0.998578i	$-0.483021\pi$
$743$	2.90663	0.106634	0.0533169	+	0.998578i	$0.483021\pi$
$744$	0	0
$745$	15.2406	0.558374
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−8.86799	−0.324029
$750$	0	0
$751$	−18.8994	−0.689648	−0.344824	−	0.938667i	$-0.612061\pi$
$751$	−18.8994	−0.689648	−0.344824	+	0.938667i	$0.612061\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4.28267	−0.155862
$756$	0	0
$757$	3.34467	0.121564	0.0607821	−	0.998151i	$-0.480641\pi$
$757$	3.34467	0.121564	0.0607821	+	0.998151i	$0.480641\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	23.5946	0.855305	0.427653	−	0.903943i	$-0.359341\pi$
$761$	23.5946	0.855305	0.427653	+	0.903943i	$0.359341\pi$
$762$	0	0
$763$	−7.96731	−0.288436
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−39.0793	−1.41107
$768$	0	0
$769$	44.8340	1.61675	0.808377	−	0.588665i	$-0.200346\pi$
$769$	44.8340	1.61675	0.808377	+	0.588665i	$0.200346\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−19.9160	−0.716327	−0.358164	−	0.933659i	$-0.616597\pi$
$773$	−19.9160	−0.716327	−0.358164	+	0.933659i	$0.616597\pi$
$774$	0	0
$775$	2.14134	0.0769191
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.44606	−0.0518103
$780$	0	0
$781$	−13.4979	−0.482992
$782$	0	0
$783$	0	0
$784$	0	0
$785$	11.2020	0.399817
$786$	0	0
$787$	33.2207	1.18419	0.592095	−	0.805868i	$-0.298301\pi$
$787$	33.2207	1.18419	0.592095	+	0.805868i	$0.298301\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−4.12268	−0.146586
$792$	0	0
$793$	−48.3200	−1.71589
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−23.8633	−0.845281	−0.422640	−	0.906297i	$-0.638897\pi$
$797$	−23.8633	−0.845281	−0.422640	+	0.906297i	$0.638897\pi$
$798$	0	0
$799$	2.05135	0.0725716
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−22.9321	−0.809255
$804$	0	0
$805$	−0.636672	−0.0224397
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−35.1693	−1.23649	−0.618244	−	0.785986i	$-0.712156\pi$
$809$	−35.1693	−1.23649	−0.618244	+	0.785986i	$0.712156\pi$
$810$	0	0
$811$	29.0480	1.02001	0.510006	−	0.860171i	$-0.329643\pi$
$811$	29.0480	1.02001	0.510006	+	0.860171i	$0.329643\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	9.83869	0.344634
$816$	0	0
$817$	5.45331	0.190787
$818$	0	0
$819$	0	0
$820$	0	0
$821$	38.6027	1.34724	0.673621	−	0.739077i	$-0.264738\pi$
$821$	38.6027	1.34724	0.673621	+	0.739077i	$0.264738\pi$
$822$	0	0
$823$	−16.4040	−0.571809	−0.285904	−	0.958258i	$-0.592294\pi$
$823$	−16.4040	−0.571809	−0.285904	+	0.958258i	$0.592294\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	27.2393	0.947204	0.473602	−	0.880739i	$-0.342953\pi$
$827$	27.2393	0.947204	0.473602	+	0.880739i	$0.342953\pi$
$828$	0	0
$829$	31.1107	1.08052	0.540260	−	0.841498i	$-0.318326\pi$
$829$	31.1107	1.08052	0.540260	+	0.841498i	$0.318326\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−4.19863	−0.145474
$834$	0	0
$835$	15.6846	0.542789
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.71733	0.128336	0.0641682	−	0.997939i	$-0.479561\pi$
$839$	3.71733	0.128336	0.0641682	+	0.997939i	$0.479561\pi$
$840$	0	0
$841$	−11.8493	−0.408598
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−0.717328	−0.0246768
$846$	0	0
$847$	−5.56196	−0.191111
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.36333	−0.0467343
$852$	0	0
$853$	9.17064	0.313997	0.156998	−	0.987599i	$-0.449818\pi$
$853$	9.17064	0.313997	0.156998	+	0.987599i	$0.449818\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−14.9907	−0.512072	−0.256036	−	0.966667i	$-0.582416\pi$
$857$	−14.9907	−0.512072	−0.256036	+	0.966667i	$0.582416\pi$
$858$	0	0
$859$	−6.98935	−0.238474	−0.119237	−	0.992866i	$-0.538045\pi$
$859$	−6.98935	−0.238474	−0.119237	+	0.992866i	$0.538045\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	21.2080	0.721927	0.360964	−	0.932580i	$-0.382448\pi$
$863$	21.2080	0.721927	0.360964	+	0.932580i	$0.382448\pi$
$864$	0	0
$865$	22.8480	0.776856
$866$	0	0
$867$	0	0
$868$	0	0
$869$	18.9066	0.641363
$870$	0	0
$871$	−48.8153	−1.65404
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0.636672	0.0215234
$876$	0	0
$877$	29.3760	0.991959	0.495979	−	0.868334i	$-0.334809\pi$
$877$	29.3760	0.991959	0.495979	+	0.868334i	$0.334809\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	39.1820	1.32008	0.660038	−	0.751232i	$-0.270540\pi$
$881$	39.1820	1.32008	0.660038	+	0.751232i	$0.270540\pi$
$882$	0	0
$883$	1.13795	0.0382950	0.0191475	−	0.999817i	$-0.493905\pi$
$883$	1.13795	0.0382950	0.0191475	+	0.999817i	$0.493905\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−19.1520	−0.643061	−0.321530	−	0.946899i	$-0.604197\pi$
$887$	−19.1520	−0.643061	−0.321530	+	0.946899i	$0.604197\pi$
$888$	0	0
$889$	−6.11674	−0.205149
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.50674	−0.0838847
$894$	0	0
$895$	21.1893	0.708280
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−8.86799	−0.295764
$900$	0	0
$901$	0.405351	0.0135042
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.161312	−0.00536219
$906$	0	0
$907$	21.4406	0.711923	0.355962	−	0.934501i	$-0.384153\pi$
$907$	21.4406	0.711923	0.355962	+	0.934501i	$0.384153\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−39.9346	−1.32309	−0.661546	−	0.749904i	$-0.730100\pi$
$911$	−39.9346	−1.32309	−0.661546	+	0.749904i	$0.730100\pi$
$912$	0	0
$913$	−14.5140	−0.480343
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.642611	0.0212209
$918$	0	0
$919$	−47.4066	−1.56380	−0.781899	−	0.623405i	$-0.785749\pi$
$919$	−47.4066	−1.56380	−0.781899	+	0.623405i	$0.785749\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−31.4393	−1.03484
$924$	0	0
$925$	1.36333	0.0448260
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−40.9053	−1.34206	−0.671030	−	0.741430i	$-0.734148\pi$
$929$	−40.9053	−1.34206	−0.671030	+	0.741430i	$0.734148\pi$
$930$	0	0
$931$	5.13069	0.168152
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−0.957977	−0.0313292
$936$	0	0
$937$	−25.6120	−0.836707	−0.418354	−	0.908284i	$-0.637393\pi$
$937$	−25.6120	−0.836707	−0.418354	+	0.908284i	$0.637393\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−6.21266	−0.202527	−0.101264	−	0.994860i	$-0.532289\pi$
$941$	−6.21266	−0.202527	−0.101264	+	0.994860i	$0.532289\pi$
$942$	0	0
$943$	−1.85866	−0.0605264
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.7987	1.22829	0.614147	−	0.789192i	$-0.289500\pi$
$947$	37.7987	1.22829	0.614147	+	0.789192i	$0.289500\pi$
$948$	0	0
$949$	−53.4134	−1.73387
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.32262	0.0752372	0.0376186	−	0.999292i	$-0.488023\pi$
$953$	2.32262	0.0752372	0.0376186	+	0.999292i	$0.488023\pi$
$954$	0	0
$955$	18.5140	0.599099
$956$	0	0
$957$	0	0
$958$	0	0
$959$	11.0093	0.355510
$960$	0	0
$961$	−26.4147	−0.852086
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−22.2827	−0.717305
$966$	0	0
$967$	9.14473	0.294075	0.147037	−	0.989131i	$-0.453026\pi$
$967$	9.14473	0.294075	0.147037	+	0.989131i	$0.453026\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−38.7453	−1.24340	−0.621698	−	0.783257i	$-0.713557\pi$
$971$	−38.7453	−1.24340	−0.621698	+	0.783257i	$0.713557\pi$
$972$	0	0
$973$	11.2793	0.361597
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−32.8794	−1.05190	−0.525952	−	0.850514i	$-0.676291\pi$
$977$	−32.8794	−1.05190	−0.525952	+	0.850514i	$0.676291\pi$
$978$	0	0
$979$	3.67738	0.117529
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.13408	0.0680665	0.0340333	−	0.999421i	$-0.489165\pi$
$983$	2.13408	0.0680665	0.0340333	+	0.999421i	$0.489165\pi$
$984$	0	0
$985$	−24.5840	−0.783311
$986$	0	0
$987$	0	0
$988$	0	0
$989$	7.00933	0.222884
$990$	0	0
$991$	58.3586	1.85382	0.926911	−	0.375280i	$-0.122454\pi$
$991$	58.3586	1.85382	0.926911	+	0.375280i	$0.122454\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.73599	0.0550344
$996$	0	0
$997$	11.4347	0.362139	0.181070	−	0.983470i	$-0.442044\pi$
$997$	11.4347	0.362139	0.181070	+	0.983470i	$0.442044\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.bp.1.2	yes	3
3.2	odd	2		8280.2.a.bk.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.a.bk.1.2	✓	3	3.2	odd	2
8280.2.a.bp.1.2	yes	3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +0.636672 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +0.636672 q^{7} -1.50466 q^{11} -3.50466 q^{13} +0.636672 q^{17} -0.778008 q^{19} -1.00000 q^{23} +1.00000 q^{25} -4.14134 q^{29} +2.14134 q^{31} +0.636672 q^{35} +1.36333 q^{37} +1.85866 q^{41} -7.00933 q^{43} +3.22199 q^{47} -6.59465 q^{49} +0.636672 q^{53} -1.50466 q^{55} +11.1507 q^{59} +13.7873 q^{61} -3.50466 q^{65} +13.9287 q^{67} +8.97070 q^{71} +15.2406 q^{73} -0.957977 q^{77} -12.5653 q^{79} +9.64600 q^{83} +0.636672 q^{85} -2.44398 q^{89} -2.23132 q^{91} -0.778008 q^{95} -7.55602 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} + 4 q^{7}+O(q^{10}) \) 3 * q + 3 * q^5 + 4 * q^7	\( 3 q + 3 q^{5} + 4 q^{7} + 6 q^{11} + 4 q^{17} + 4 q^{19} - 3 q^{23} + 3 q^{25} - 4 q^{29} - 2 q^{31} + 4 q^{35} + 2 q^{37} + 14 q^{41} + 16 q^{47} - 3 q^{49} + 4 q^{53} + 6 q^{55} + 4 q^{59} + 14 q^{61} + 6 q^{67} + 10 q^{71} + 10 q^{73} + 16 q^{77} - 4 q^{79} + 10 q^{83} + 4 q^{85} - 20 q^{89} + 8 q^{91} + 4 q^{95} - 10 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 + 4 * q^7 + 6 * q^11 + 4 * q^17 + 4 * q^19 - 3 * q^23 + 3 * q^25 - 4 * q^29 - 2 * q^31 + 4 * q^35 + 2 * q^37 + 14 * q^41 + 16 * q^47 - 3 * q^49 + 4 * q^53 + 6 * q^55 + 4 * q^59 + 14 * q^61 + 6 * q^67 + 10 * q^71 + 10 * q^73 + 16 * q^77 - 4 * q^79 + 10 * q^83 + 4 * q^85 - 20 * q^89 + 8 * q^91 + 4 * q^95 - 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more