LMFDB - Embedded newform 6080.2.a.cb.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6080,2,Mod(1,6080)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6080.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6080, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6080 = 2^{6} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6080.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,4,0,3,0,-4,0,3,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.5490444289$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3040)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	6080.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.17009 q^{3} +1.00000 q^{5} -2.63090 q^{7} +7.04945 q^{9} +0.630898 q^{11} +5.95774 q^{13} +3.17009 q^{15} +3.26180 q^{17} +1.00000 q^{19} -8.34017 q^{21} +6.04945 q^{23} +1.00000 q^{25} +12.8371 q^{27} -8.34017 q^{29} -1.41855 q^{31} +2.00000 q^{33} -2.63090 q^{35} -6.29791 q^{37} +18.8865 q^{39} +11.1278 q^{43} +7.04945 q^{45} -4.78765 q^{47} -0.0783777 q^{49} +10.3402 q^{51} -11.3763 q^{53} +0.630898 q^{55} +3.17009 q^{57} +0.241276 q^{59} -4.63090 q^{61} -18.5464 q^{63} +5.95774 q^{65} +0.489741 q^{67} +19.1773 q^{69} +9.41855 q^{71} -2.18342 q^{73} +3.17009 q^{75} -1.65983 q^{77} +13.9155 q^{79} +19.5464 q^{81} -1.86603 q^{83} +3.26180 q^{85} -26.4391 q^{87} +12.3402 q^{89} -15.6742 q^{91} -4.49693 q^{93} +1.00000 q^{95} -17.7165 q^{97} +4.44748 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} + 4 q^{15} + 2 q^{17} + 3 q^{19} - 14 q^{21} + 3 q^{25} + 10 q^{27} - 14 q^{29} + 10 q^{31} + 6 q^{33} - 4 q^{35} + 8 q^{37} + 10 q^{39}+ \cdots + 14 q^{99}+O(q^{100}) $ 3 * q + 4 * q^3 + 3 * q^5 - 4 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 + 4 * q^15 + 2 * q^17 + 3 * q^19 - 14 * q^21 + 3 * q^25 + 10 * q^27 - 14 * q^29 + 10 * q^31 + 6 * q^33 - 4 * q^35 + 8 * q^37 + 10 * q^39 + 12 * q^43 + 3 * q^45 - 4 * q^47 + 3 * q^49 + 20 * q^51 - 4 * q^53 - 2 * q^55 + 4 * q^57 + 26 * q^59 - 10 * q^61 - 20 * q^63 + 2 * q^65 + 18 * q^67 + 18 * q^69 + 14 * q^71 - 2 * q^73 + 4 * q^75 - 16 * q^77 + 10 * q^79 + 23 * q^81 + 8 * q^83 + 2 * q^85 - 32 * q^87 + 26 * q^89 + 10 * q^91 + 4 * q^93 + 3 * q^95 - 12 * q^97 + 14 * q^99

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
$3$	3.17009	1.83025	0.915125	−	0.403170i	$-0.132092\pi$
$3$	3.17009	1.83025	0.915125	+	0.403170i	$0.132092\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.63090	−0.994386	−0.497193	−	0.867640i	$-0.665636\pi$
$7$	−2.63090	−0.994386	−0.497193	+	0.867640i	$0.665636\pi$
$8$	0	0
$9$	7.04945	2.34982
$10$	0	0
$11$	0.630898	0.190223	0.0951114	−	0.995467i	$-0.469679\pi$
$11$	0.630898	0.190223	0.0951114	+	0.995467i	$0.469679\pi$
$12$	0	0
$13$	5.95774	1.65238	0.826190	−	0.563392i	$-0.190504\pi$
$13$	5.95774	1.65238	0.826190	+	0.563392i	$0.190504\pi$
$14$	0	0
$15$	3.17009	0.818513
$16$	0	0
$17$	3.26180	0.791102	0.395551	−	0.918444i	$-0.370554\pi$
$17$	3.26180	0.791102	0.395551	+	0.918444i	$0.370554\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−8.34017	−1.81997
$22$	0	0
$23$	6.04945	1.26140	0.630699	−	0.776028i	$-0.282768\pi$
$23$	6.04945	1.26140	0.630699	+	0.776028i	$0.282768\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	12.8371	2.47050
$28$	0	0
$29$	−8.34017	−1.54873	−0.774366	−	0.632738i	$-0.781931\pi$
$29$	−8.34017	−1.54873	−0.774366	+	0.632738i	$0.781931\pi$
$30$	0	0
$31$	−1.41855	−0.254779	−0.127390	−	0.991853i	$-0.540660\pi$
$31$	−1.41855	−0.254779	−0.127390	+	0.991853i	$0.540660\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	−2.63090	−0.444703
$36$	0	0
$37$	−6.29791	−1.03537	−0.517685	−	0.855571i	$-0.673206\pi$
$37$	−6.29791	−1.03537	−0.517685	+	0.855571i	$0.673206\pi$
$38$	0	0
$39$	18.8865	3.02427
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	11.1278	1.69698	0.848489	−	0.529213i	$-0.177513\pi$
$43$	11.1278	1.69698	0.848489	+	0.529213i	$0.177513\pi$
$44$	0	0
$45$	7.04945	1.05087
$46$	0	0
$47$	−4.78765	−0.698351	−0.349175	−	0.937057i	$-0.613538\pi$
$47$	−4.78765	−0.698351	−0.349175	+	0.937057i	$0.613538\pi$
$48$	0	0
$49$	−0.0783777	−0.0111968
$50$	0	0
$51$	10.3402	1.44791
$52$	0	0
$53$	−11.3763	−1.56265	−0.781327	−	0.624122i	$-0.785457\pi$
$53$	−11.3763	−1.56265	−0.781327	+	0.624122i	$0.785457\pi$
$54$	0	0
$55$	0.630898	0.0850702
$56$	0	0
$57$	3.17009	0.419888
$58$	0	0
$59$	0.241276	0.0314115	0.0157057	−	0.999877i	$-0.495000\pi$
$59$	0.241276	0.0314115	0.0157057	+	0.999877i	$0.495000\pi$
$60$	0	0
$61$	−4.63090	−0.592926	−0.296463	−	0.955044i	$-0.595807\pi$
$61$	−4.63090	−0.592926	−0.296463	+	0.955044i	$0.595807\pi$
$62$	0	0
$63$	−18.5464	−2.33662
$64$	0	0
$65$	5.95774	0.738967
$66$	0	0
$67$	0.489741	0.0598313	0.0299157	−	0.999552i	$-0.490476\pi$
$67$	0.489741	0.0598313	0.0299157	+	0.999552i	$0.490476\pi$
$68$	0	0
$69$	19.1773	2.30867
$70$	0	0
$71$	9.41855	1.11778	0.558888	−	0.829243i	$-0.311228\pi$
$71$	9.41855	1.11778	0.558888	+	0.829243i	$0.311228\pi$
$72$	0	0
$73$	−2.18342	−0.255550	−0.127775	−	0.991803i	$-0.540783\pi$
$73$	−2.18342	−0.255550	−0.127775	+	0.991803i	$0.540783\pi$
$74$	0	0
$75$	3.17009	0.366050
$76$	0	0
$77$	−1.65983	−0.189155
$78$	0	0
$79$	13.9155	1.56561	0.782807	−	0.622265i	$-0.213787\pi$
$79$	13.9155	1.56561	0.782807	+	0.622265i	$0.213787\pi$
$80$	0	0
$81$	19.5464	2.17182
$82$	0	0
$83$	−1.86603	−0.204823	−0.102412	−	0.994742i	$-0.532656\pi$
$83$	−1.86603	−0.204823	−0.102412	+	0.994742i	$0.532656\pi$
$84$	0	0
$85$	3.26180	0.353791
$86$	0	0
$87$	−26.4391	−2.83457
$88$	0	0
$89$	12.3402	1.30806	0.654028	−	0.756470i	$-0.273078\pi$
$89$	12.3402	1.30806	0.654028	+	0.756470i	$0.273078\pi$
$90$	0	0
$91$	−15.6742	−1.64310
$92$	0	0
$93$	−4.49693	−0.466310
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−17.7165	−1.79883	−0.899417	−	0.437091i	$-0.856009\pi$
$97$	−17.7165	−1.79883	−0.899417	+	0.437091i	$0.856009\pi$
$98$	0	0
$99$	4.44748	0.446989
$100$	0	0
$101$	15.2846	1.52087	0.760436	−	0.649412i	$-0.224985\pi$
$101$	15.2846	1.52087	0.760436	+	0.649412i	$0.224985\pi$
$102$	0	0
$103$	6.82991	0.672971	0.336486	−	0.941689i	$-0.390762\pi$
$103$	6.82991	0.672971	0.336486	+	0.941689i	$0.390762\pi$
$104$	0	0
$105$	−8.34017	−0.813918
$106$	0	0
$107$	−8.68753	−0.839856	−0.419928	−	0.907558i	$-0.637945\pi$
$107$	−8.68753	−0.839856	−0.419928	+	0.907558i	$0.637945\pi$
$108$	0	0
$109$	−3.60197	−0.345006	−0.172503	−	0.985009i	$-0.555185\pi$
$109$	−3.60197	−0.345006	−0.172503	+	0.985009i	$0.555185\pi$
$110$	0	0
$111$	−19.9649	−1.89499
$112$	0	0
$113$	−7.61757	−0.716600	−0.358300	−	0.933606i	$-0.616644\pi$
$113$	−7.61757	−0.716600	−0.358300	+	0.933606i	$0.616644\pi$
$114$	0	0
$115$	6.04945	0.564114
$116$	0	0
$117$	41.9988	3.88279
$118$	0	0
$119$	−8.58145	−0.786660
$120$	0	0
$121$	−10.6020	−0.963815
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	10.4052	0.923314	0.461657	−	0.887059i	$-0.347255\pi$
$127$	10.4052	0.923314	0.461657	+	0.887059i	$0.347255\pi$
$128$	0	0
$129$	35.2762	3.10589
$130$	0	0
$131$	−6.34017	−0.553943	−0.276972	−	0.960878i	$-0.589331\pi$
$131$	−6.34017	−0.553943	−0.276972	+	0.960878i	$0.589331\pi$
$132$	0	0
$133$	−2.63090	−0.228128
$134$	0	0
$135$	12.8371	1.10484
$136$	0	0
$137$	−8.15676	−0.696879	−0.348439	−	0.937331i	$-0.613288\pi$
$137$	−8.15676	−0.696879	−0.348439	+	0.937331i	$0.613288\pi$
$138$	0	0
$139$	10.2062	0.865679	0.432839	−	0.901471i	$-0.357512\pi$
$139$	10.2062	0.865679	0.432839	+	0.901471i	$0.357512\pi$
$140$	0	0
$141$	−15.1773	−1.27816
$142$	0	0
$143$	3.75872	0.314320
$144$	0	0
$145$	−8.34017	−0.692614
$146$	0	0
$147$	−0.248464	−0.0204930
$148$	0	0
$149$	18.5730	1.52156	0.760781	−	0.649008i	$-0.224816\pi$
$149$	18.5730	1.52156	0.760781	+	0.649008i	$0.224816\pi$
$150$	0	0
$151$	−13.7587	−1.11967	−0.559835	−	0.828604i	$-0.689135\pi$
$151$	−13.7587	−1.11967	−0.559835	+	0.828604i	$0.689135\pi$
$152$	0	0
$153$	22.9939	1.85894
$154$	0	0
$155$	−1.41855	−0.113941
$156$	0	0
$157$	3.84324	0.306724	0.153362	−	0.988170i	$-0.450990\pi$
$157$	3.84324	0.306724	0.153362	+	0.988170i	$0.450990\pi$
$158$	0	0
$159$	−36.0638	−2.86005
$160$	0	0
$161$	−15.9155	−1.25432
$162$	0	0
$163$	24.3896	1.91034	0.955171	−	0.296054i	$-0.0956707\pi$
$163$	24.3896	1.91034	0.955171	+	0.296054i	$0.0956707\pi$
$164$	0	0
$165$	2.00000	0.155700
$166$	0	0
$167$	−21.2423	−1.64378	−0.821890	−	0.569646i	$-0.807080\pi$
$167$	−21.2423	−1.64378	−0.821890	+	0.569646i	$0.807080\pi$
$168$	0	0
$169$	22.4947	1.73036
$170$	0	0
$171$	7.04945	0.539085
$172$	0	0
$173$	−15.2195	−1.15712	−0.578560	−	0.815640i	$-0.696385\pi$
$173$	−15.2195	−1.15712	−0.578560	+	0.815640i	$0.696385\pi$
$174$	0	0
$175$	−2.63090	−0.198877
$176$	0	0
$177$	0.764867	0.0574909
$178$	0	0
$179$	−1.91548	−0.143170	−0.0715848	−	0.997435i	$-0.522806\pi$
$179$	−1.91548	−0.143170	−0.0715848	+	0.997435i	$0.522806\pi$
$180$	0	0
$181$	8.15676	0.606287	0.303143	−	0.952945i	$-0.401964\pi$
$181$	8.15676	0.606287	0.303143	+	0.952945i	$0.401964\pi$
$182$	0	0
$183$	−14.6803	−1.08520
$184$	0	0
$185$	−6.29791	−0.463032
$186$	0	0
$187$	2.05786	0.150486
$188$	0	0
$189$	−33.7731	−2.45663
$190$	0	0
$191$	−5.57531	−0.403415	−0.201707	−	0.979446i	$-0.564649\pi$
$191$	−5.57531	−0.403415	−0.201707	+	0.979446i	$0.564649\pi$
$192$	0	0
$193$	−14.8794	−1.07104	−0.535520	−	0.844523i	$-0.679884\pi$
$193$	−14.8794	−1.07104	−0.535520	+	0.844523i	$0.679884\pi$
$194$	0	0
$195$	18.8865	1.35249
$196$	0	0
$197$	18.9627	1.35103	0.675517	−	0.737345i	$-0.263921\pi$
$197$	18.9627	1.35103	0.675517	+	0.737345i	$0.263921\pi$
$198$	0	0
$199$	15.2039	1.07778	0.538889	−	0.842377i	$-0.318844\pi$
$199$	15.2039	1.07778	0.538889	+	0.842377i	$0.318844\pi$
$200$	0	0
$201$	1.55252	0.109506
$202$	0	0
$203$	21.9421	1.54004
$204$	0	0
$205$	0	0
$206$	0	0
$207$	42.6453	2.96405
$208$	0	0
$209$	0.630898	0.0436401
$210$	0	0
$211$	−0.0845208	−0.00581865	−0.00290933	−	0.999996i	$-0.500926\pi$
$211$	−0.0845208	−0.00581865	−0.00290933	+	0.999996i	$0.500926\pi$
$212$	0	0
$213$	29.8576	2.04581
$214$	0	0
$215$	11.1278	0.758911
$216$	0	0
$217$	3.73206	0.253349
$218$	0	0
$219$	−6.92162	−0.467720
$220$	0	0
$221$	19.4329	1.30720
$222$	0	0
$223$	−3.66701	−0.245561	−0.122781	−	0.992434i	$-0.539181\pi$
$223$	−3.66701	−0.245561	−0.122781	+	0.992434i	$0.539181\pi$
$224$	0	0
$225$	7.04945	0.469963
$226$	0	0
$227$	28.7721	1.90967	0.954834	−	0.297139i	$-0.0960325\pi$
$227$	28.7721	1.90967	0.954834	+	0.297139i	$0.0960325\pi$
$228$	0	0
$229$	6.38962	0.422238	0.211119	−	0.977460i	$-0.432289\pi$
$229$	6.38962	0.422238	0.211119	+	0.977460i	$0.432289\pi$
$230$	0	0
$231$	−5.26180	−0.346201
$232$	0	0
$233$	−5.91548	−0.387536	−0.193768	−	0.981047i	$-0.562071\pi$
$233$	−5.91548	−0.387536	−0.193768	+	0.981047i	$0.562071\pi$
$234$	0	0
$235$	−4.78765	−0.312312
$236$	0	0
$237$	44.1133	2.86546
$238$	0	0
$239$	−24.9627	−1.61470	−0.807350	−	0.590073i	$-0.799099\pi$
$239$	−24.9627	−1.61470	−0.807350	+	0.590073i	$0.799099\pi$
$240$	0	0
$241$	−21.9421	−1.41342	−0.706709	−	0.707505i	$-0.749821\pi$
$241$	−21.9421	−1.41342	−0.706709	+	0.707505i	$0.749821\pi$
$242$	0	0
$243$	23.4524	1.50447
$244$	0	0
$245$	−0.0783777	−0.00500737
$246$	0	0
$247$	5.95774	0.379082
$248$	0	0
$249$	−5.91548	−0.374878
$250$	0	0
$251$	7.05172	0.445100	0.222550	−	0.974921i	$-0.428562\pi$
$251$	7.05172	0.445100	0.222550	+	0.974921i	$0.428562\pi$
$252$	0	0
$253$	3.81658	0.239946
$254$	0	0
$255$	10.3402	0.647527
$256$	0	0
$257$	8.45467	0.527388	0.263694	−	0.964606i	$-0.415059\pi$
$257$	8.45467	0.527388	0.263694	+	0.964606i	$0.415059\pi$
$258$	0	0
$259$	16.5692	1.02956
$260$	0	0
$261$	−58.7936	−3.63923
$262$	0	0
$263$	28.9854	1.78732	0.893660	−	0.448745i	$-0.148129\pi$
$263$	28.9854	1.78732	0.893660	+	0.448745i	$0.148129\pi$
$264$	0	0
$265$	−11.3763	−0.698840
$266$	0	0
$267$	39.1194	2.39407
$268$	0	0
$269$	−6.34017	−0.386567	−0.193284	−	0.981143i	$-0.561914\pi$
$269$	−6.34017	−0.386567	−0.193284	+	0.981143i	$0.561914\pi$
$270$	0	0
$271$	21.5259	1.30760	0.653801	−	0.756666i	$-0.273173\pi$
$271$	21.5259	1.30760	0.653801	+	0.756666i	$0.273173\pi$
$272$	0	0
$273$	−49.6886	−3.00729
$274$	0	0
$275$	0.630898	0.0380446
$276$	0	0
$277$	1.68649	0.101331	0.0506657	−	0.998716i	$-0.483866\pi$
$277$	1.68649	0.101331	0.0506657	+	0.998716i	$0.483866\pi$
$278$	0	0
$279$	−10.0000	−0.598684
$280$	0	0
$281$	17.6742	1.05435	0.527177	−	0.849755i	$-0.323250\pi$
$281$	17.6742	1.05435	0.527177	+	0.849755i	$0.323250\pi$
$282$	0	0
$283$	−9.33791	−0.555081	−0.277540	−	0.960714i	$-0.589519\pi$
$283$	−9.33791	−0.555081	−0.277540	+	0.960714i	$0.589519\pi$
$284$	0	0
$285$	3.17009	0.187780
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−6.36069	−0.374158
$290$	0	0
$291$	−56.1627	−3.29232
$292$	0	0
$293$	−8.63809	−0.504642	−0.252321	−	0.967644i	$-0.581194\pi$
$293$	−8.63809	−0.504642	−0.252321	+	0.967644i	$0.581194\pi$
$294$	0	0
$295$	0.241276	0.0140476
$296$	0	0
$297$	8.09890	0.469946
$298$	0	0
$299$	36.0410	2.08431
$300$	0	0
$301$	−29.2762	−1.68745
$302$	0	0
$303$	48.4534	2.78358
$304$	0	0
$305$	−4.63090	−0.265164
$306$	0	0
$307$	−33.6814	−1.92230	−0.961149	−	0.276029i	$-0.910981\pi$
$307$	−33.6814	−1.92230	−0.961149	+	0.276029i	$0.910981\pi$
$308$	0	0
$309$	21.6514	1.23171
$310$	0	0
$311$	12.3630	0.701039	0.350520	−	0.936555i	$-0.386005\pi$
$311$	12.3630	0.701039	0.350520	+	0.936555i	$0.386005\pi$
$312$	0	0
$313$	4.07223	0.230176	0.115088	−	0.993355i	$-0.463285\pi$
$313$	4.07223	0.230176	0.115088	+	0.993355i	$0.463285\pi$
$314$	0	0
$315$	−18.5464	−1.04497
$316$	0	0
$317$	−3.13501	−0.176080	−0.0880400	−	0.996117i	$-0.528060\pi$
$317$	−3.13501	−0.176080	−0.0880400	+	0.996117i	$0.528060\pi$
$318$	0	0
$319$	−5.26180	−0.294604
$320$	0	0
$321$	−27.5402	−1.53715
$322$	0	0
$323$	3.26180	0.181491
$324$	0	0
$325$	5.95774	0.330476
$326$	0	0
$327$	−11.4186	−0.631447
$328$	0	0
$329$	12.5958	0.694430
$330$	0	0
$331$	−29.1773	−1.60373	−0.801864	−	0.597507i	$-0.796158\pi$
$331$	−29.1773	−1.60373	−0.801864	+	0.597507i	$0.796158\pi$
$332$	0	0
$333$	−44.3968	−2.43293
$334$	0	0
$335$	0.489741	0.0267574
$336$	0	0
$337$	−28.4501	−1.54978	−0.774889	−	0.632098i	$-0.782194\pi$
$337$	−28.4501	−1.54978	−0.774889	+	0.632098i	$0.782194\pi$
$338$	0	0
$339$	−24.1483	−1.31156
$340$	0	0
$341$	−0.894960	−0.0484648
$342$	0	0
$343$	18.6225	1.00552
$344$	0	0
$345$	19.1773	1.03247
$346$	0	0
$347$	−18.2472	−0.979563	−0.489782	−	0.871845i	$-0.662923\pi$
$347$	−18.2472	−0.979563	−0.489782	+	0.871845i	$0.662923\pi$
$348$	0	0
$349$	−2.08452	−0.111582	−0.0557909	−	0.998442i	$-0.517768\pi$
$349$	−2.08452	−0.111582	−0.0557909	+	0.998442i	$0.517768\pi$
$350$	0	0
$351$	76.4801	4.08221
$352$	0	0
$353$	35.4596	1.88732	0.943662	−	0.330912i	$-0.107356\pi$
$353$	35.4596	1.88732	0.943662	+	0.330912i	$0.107356\pi$
$354$	0	0
$355$	9.41855	0.499885
$356$	0	0
$357$	−27.2039	−1.43978
$358$	0	0
$359$	−18.5197	−0.977433	−0.488717	−	0.872443i	$-0.662535\pi$
$359$	−18.5197	−0.977433	−0.488717	+	0.872443i	$0.662535\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−33.6092	−1.76402
$364$	0	0
$365$	−2.18342	−0.114285
$366$	0	0
$367$	24.8020	1.29466	0.647328	−	0.762212i	$-0.275887\pi$
$367$	24.8020	1.29466	0.647328	+	0.762212i	$0.275887\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	29.9299	1.55388
$372$	0	0
$373$	−25.1206	−1.30070	−0.650349	−	0.759636i	$-0.725377\pi$
$373$	−25.1206	−1.30070	−0.650349	+	0.759636i	$0.725377\pi$
$374$	0	0
$375$	3.17009	0.163703
$376$	0	0
$377$	−49.6886	−2.55909
$378$	0	0
$379$	10.8904	0.559404	0.279702	−	0.960087i	$-0.409764\pi$
$379$	10.8904	0.559404	0.279702	+	0.960087i	$0.409764\pi$
$380$	0	0
$381$	32.9854	1.68990
$382$	0	0
$383$	−8.19061	−0.418520	−0.209260	−	0.977860i	$-0.567106\pi$
$383$	−8.19061	−0.418520	−0.209260	+	0.977860i	$0.567106\pi$
$384$	0	0
$385$	−1.65983	−0.0845926
$386$	0	0
$387$	78.4450	3.98759
$388$	0	0
$389$	29.6020	1.50088	0.750440	−	0.660939i	$-0.229842\pi$
$389$	29.6020	1.50088	0.750440	+	0.660939i	$0.229842\pi$
$390$	0	0
$391$	19.7321	0.997893
$392$	0	0
$393$	−20.0989	−1.01386
$394$	0	0
$395$	13.9155	0.700164
$396$	0	0
$397$	1.60197	0.0804005	0.0402002	−	0.999192i	$-0.487200\pi$
$397$	1.60197	0.0804005	0.0402002	+	0.999192i	$0.487200\pi$
$398$	0	0
$399$	−8.34017	−0.417531
$400$	0	0
$401$	−31.1461	−1.55536	−0.777680	−	0.628660i	$-0.783604\pi$
$401$	−31.1461	−1.55536	−0.777680	+	0.628660i	$0.783604\pi$
$402$	0	0
$403$	−8.45136	−0.420992
$404$	0	0
$405$	19.5464	0.971267
$406$	0	0
$407$	−3.97334	−0.196951
$408$	0	0
$409$	−0.979481	−0.0484322	−0.0242161	−	0.999707i	$-0.507709\pi$
$409$	−0.979481	−0.0484322	−0.0242161	+	0.999707i	$0.507709\pi$
$410$	0	0
$411$	−25.8576	−1.27546
$412$	0	0
$413$	−0.634773	−0.0312352
$414$	0	0
$415$	−1.86603	−0.0915999
$416$	0	0
$417$	32.3545	1.58441
$418$	0	0
$419$	−19.9155	−0.972935	−0.486467	−	0.873699i	$-0.661715\pi$
$419$	−19.9155	−0.972935	−0.486467	+	0.873699i	$0.661715\pi$
$420$	0	0
$421$	3.05172	0.148732	0.0743658	−	0.997231i	$-0.476307\pi$
$421$	3.05172	0.148732	0.0743658	+	0.997231i	$0.476307\pi$
$422$	0	0
$423$	−33.7503	−1.64100
$424$	0	0
$425$	3.26180	0.158220
$426$	0	0
$427$	12.1834	0.589597
$428$	0	0
$429$	11.9155	0.575285
$430$	0	0
$431$	−16.3812	−0.789055	−0.394528	−	0.918884i	$-0.629092\pi$
$431$	−16.3812	−0.789055	−0.394528	+	0.918884i	$0.629092\pi$
$432$	0	0
$433$	12.2134	0.586938	0.293469	−	0.955969i	$-0.405190\pi$
$433$	12.2134	0.586938	0.293469	+	0.955969i	$0.405190\pi$
$434$	0	0
$435$	−26.4391	−1.26766
$436$	0	0
$437$	6.04945	0.289384
$438$	0	0
$439$	−25.6475	−1.22409	−0.612045	−	0.790823i	$-0.709653\pi$
$439$	−25.6475	−1.22409	−0.612045	+	0.790823i	$0.709653\pi$
$440$	0	0
$441$	−0.552520	−0.0263105
$442$	0	0
$443$	−8.57304	−0.407317	−0.203659	−	0.979042i	$-0.565283\pi$
$443$	−8.57304	−0.407317	−0.203659	+	0.979042i	$0.565283\pi$
$444$	0	0
$445$	12.3402	0.584980
$446$	0	0
$447$	58.8781	2.78484
$448$	0	0
$449$	−0.282314	−0.0133232	−0.00666161	−	0.999978i	$-0.502120\pi$
$449$	−0.282314	−0.0133232	−0.00666161	+	0.999978i	$0.502120\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−43.6163	−2.04927
$454$	0	0
$455$	−15.6742	−0.734818
$456$	0	0
$457$	−26.8781	−1.25731	−0.628653	−	0.777686i	$-0.716393\pi$
$457$	−26.8781	−1.25731	−0.628653	+	0.777686i	$0.716393\pi$
$458$	0	0
$459$	41.8720	1.95442
$460$	0	0
$461$	−39.8720	−1.85702	−0.928512	−	0.371302i	$-0.878911\pi$
$461$	−39.8720	−1.85702	−0.928512	+	0.371302i	$0.878911\pi$
$462$	0	0
$463$	−6.63090	−0.308164	−0.154082	−	0.988058i	$-0.549242\pi$
$463$	−6.63090	−0.308164	−0.154082	+	0.988058i	$0.549242\pi$
$464$	0	0
$465$	−4.49693	−0.208540
$466$	0	0
$467$	14.6309	0.677037	0.338519	−	0.940960i	$-0.390074\pi$
$467$	14.6309	0.677037	0.338519	+	0.940960i	$0.390074\pi$
$468$	0	0
$469$	−1.28846	−0.0594954
$470$	0	0
$471$	12.1834	0.561382
$472$	0	0
$473$	7.02052	0.322804
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	−80.1966	−3.67195
$478$	0	0
$479$	−35.2534	−1.61077	−0.805384	−	0.592753i	$-0.798041\pi$
$479$	−35.2534	−1.61077	−0.805384	+	0.592753i	$0.798041\pi$
$480$	0	0
$481$	−37.5213	−1.71083
$482$	0	0
$483$	−50.4534	−2.29571
$484$	0	0
$485$	−17.7165	−0.804463
$486$	0	0
$487$	−2.79872	−0.126822	−0.0634110	−	0.997987i	$-0.520198\pi$
$487$	−2.79872	−0.126822	−0.0634110	+	0.997987i	$0.520198\pi$
$488$	0	0
$489$	77.3172	3.49641
$490$	0	0
$491$	15.3028	0.690607	0.345304	−	0.938491i	$-0.387776\pi$
$491$	15.3028	0.690607	0.345304	+	0.938491i	$0.387776\pi$
$492$	0	0
$493$	−27.2039	−1.22520
$494$	0	0
$495$	4.44748	0.199899
$496$	0	0
$497$	−24.7792	−1.11150
$498$	0	0
$499$	−16.6765	−0.746541	−0.373271	−	0.927722i	$-0.621764\pi$
$499$	−16.6765	−0.746541	−0.373271	+	0.927722i	$0.621764\pi$
$500$	0	0
$501$	−67.3400	−3.00853
$502$	0	0
$503$	−15.9383	−0.710652	−0.355326	−	0.934742i	$-0.615630\pi$
$503$	−15.9383	−0.710652	−0.355326	+	0.934742i	$0.615630\pi$
$504$	0	0
$505$	15.2846	0.680155
$506$	0	0
$507$	71.3100	3.16699
$508$	0	0
$509$	−5.71769	−0.253432	−0.126716	−	0.991939i	$-0.540444\pi$
$509$	−5.71769	−0.253432	−0.126716	+	0.991939i	$0.540444\pi$
$510$	0	0
$511$	5.74435	0.254115
$512$	0	0
$513$	12.8371	0.566772
$514$	0	0
$515$	6.82991	0.300962
$516$	0	0
$517$	−3.02052	−0.132842
$518$	0	0
$519$	−48.2472	−2.11782
$520$	0	0
$521$	−10.8950	−0.477317	−0.238658	−	0.971104i	$-0.576708\pi$
$521$	−10.8950	−0.477317	−0.238658	+	0.971104i	$0.576708\pi$
$522$	0	0
$523$	6.67316	0.291797	0.145898	−	0.989300i	$-0.453393\pi$
$523$	6.67316	0.291797	0.145898	+	0.989300i	$0.453393\pi$
$524$	0	0
$525$	−8.34017	−0.363995
$526$	0	0
$527$	−4.62702	−0.201556
$528$	0	0
$529$	13.5958	0.591123
$530$	0	0
$531$	1.70086	0.0738112
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−8.68753	−0.375595
$536$	0	0
$537$	−6.07223	−0.262036
$538$	0	0
$539$	−0.0494483	−0.00212989
$540$	0	0
$541$	27.7503	1.19308	0.596540	−	0.802584i	$-0.296542\pi$
$541$	27.7503	1.19308	0.596540	+	0.802584i	$0.296542\pi$
$542$	0	0
$543$	25.8576	1.10966
$544$	0	0
$545$	−3.60197	−0.154291
$546$	0	0
$547$	33.8381	1.44681	0.723407	−	0.690421i	$-0.242575\pi$
$547$	33.8381	1.44681	0.723407	+	0.690421i	$0.242575\pi$
$548$	0	0
$549$	−32.6453	−1.39327
$550$	0	0
$551$	−8.34017	−0.355303
$552$	0	0
$553$	−36.6102	−1.55682
$554$	0	0
$555$	−19.9649	−0.847464
$556$	0	0
$557$	−18.5646	−0.786609	−0.393304	−	0.919408i	$-0.628668\pi$
$557$	−18.5646	−0.786609	−0.393304	+	0.919408i	$0.628668\pi$
$558$	0	0
$559$	66.2967	2.80405
$560$	0	0
$561$	6.52359	0.275426
$562$	0	0
$563$	−27.8504	−1.17376	−0.586878	−	0.809675i	$-0.699643\pi$
$563$	−27.8504	−1.17376	−0.586878	+	0.809675i	$0.699643\pi$
$564$	0	0
$565$	−7.61757	−0.320473
$566$	0	0
$567$	−51.4245	−2.15963
$568$	0	0
$569$	−46.8515	−1.96412	−0.982058	−	0.188579i	$-0.939612\pi$
$569$	−46.8515	−1.96412	−0.982058	+	0.188579i	$0.939612\pi$
$570$	0	0
$571$	−27.2001	−1.13829	−0.569144	−	0.822238i	$-0.692725\pi$
$571$	−27.2001	−1.13829	−0.569144	+	0.822238i	$0.692725\pi$
$572$	0	0
$573$	−17.6742	−0.738350
$574$	0	0
$575$	6.04945	0.252279
$576$	0	0
$577$	−42.4801	−1.76847	−0.884235	−	0.467042i	$-0.845320\pi$
$577$	−42.4801	−1.76847	−0.884235	+	0.467042i	$0.845320\pi$
$578$	0	0
$579$	−47.1689	−1.96027
$580$	0	0
$581$	4.90934	0.203674
$582$	0	0
$583$	−7.17727	−0.297252
$584$	0	0
$585$	41.9988	1.73644
$586$	0	0
$587$	9.49920	0.392074	0.196037	−	0.980597i	$-0.437193\pi$
$587$	9.49920	0.392074	0.196037	+	0.980597i	$0.437193\pi$
$588$	0	0
$589$	−1.41855	−0.0584504
$590$	0	0
$591$	60.1133	2.47273
$592$	0	0
$593$	−22.1978	−0.911554	−0.455777	−	0.890094i	$-0.650639\pi$
$593$	−22.1978	−0.911554	−0.455777	+	0.890094i	$0.650639\pi$
$594$	0	0
$595$	−8.58145	−0.351805
$596$	0	0
$597$	48.1978	1.97260
$598$	0	0
$599$	−12.7070	−0.519194	−0.259597	−	0.965717i	$-0.583590\pi$
$599$	−12.7070	−0.519194	−0.259597	+	0.965717i	$0.583590\pi$
$600$	0	0
$601$	27.4186	1.11843	0.559213	−	0.829024i	$-0.311103\pi$
$601$	27.4186	1.11843	0.559213	+	0.829024i	$0.311103\pi$
$602$	0	0
$603$	3.45240	0.140593
$604$	0	0
$605$	−10.6020	−0.431031
$606$	0	0
$607$	−15.1578	−0.615236	−0.307618	−	0.951510i	$-0.599532\pi$
$607$	−15.1578	−0.615236	−0.307618	+	0.951510i	$0.599532\pi$
$608$	0	0
$609$	69.5585	2.81865
$610$	0	0
$611$	−28.5236	−1.15394
$612$	0	0
$613$	−27.2306	−1.09983	−0.549917	−	0.835219i	$-0.685341\pi$
$613$	−27.2306	−1.09983	−0.549917	+	0.835219i	$0.685341\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	39.0082	1.57041	0.785206	−	0.619234i	$-0.212557\pi$
$617$	39.0082	1.57041	0.785206	+	0.619234i	$0.212557\pi$
$618$	0	0
$619$	17.8927	0.719168	0.359584	−	0.933113i	$-0.382919\pi$
$619$	17.8927	0.719168	0.359584	+	0.933113i	$0.382919\pi$
$620$	0	0
$621$	77.6574	3.11628
$622$	0	0
$623$	−32.4657	−1.30071
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	2.00000	0.0798723
$628$	0	0
$629$	−20.5425	−0.819083
$630$	0	0
$631$	−42.1894	−1.67953	−0.839766	−	0.542948i	$-0.817308\pi$
$631$	−42.1894	−1.67953	−0.839766	+	0.542948i	$0.817308\pi$
$632$	0	0
$633$	−0.267938	−0.0106496
$634$	0	0
$635$	10.4052	0.412919
$636$	0	0
$637$	−0.466954	−0.0185014
$638$	0	0
$639$	66.3956	2.62657
$640$	0	0
$641$	−11.3028	−0.446435	−0.223218	−	0.974769i	$-0.571656\pi$
$641$	−11.3028	−0.446435	−0.223218	+	0.974769i	$0.571656\pi$
$642$	0	0
$643$	−29.4680	−1.16210	−0.581052	−	0.813866i	$-0.697359\pi$
$643$	−29.4680	−1.16210	−0.581052	+	0.813866i	$0.697359\pi$
$644$	0	0
$645$	35.2762	1.38900
$646$	0	0
$647$	37.7503	1.48412	0.742059	−	0.670335i	$-0.233849\pi$
$647$	37.7503	1.48412	0.742059	+	0.670335i	$0.233849\pi$
$648$	0	0
$649$	0.152221	0.00597518
$650$	0	0
$651$	11.8310	0.463692
$652$	0	0
$653$	9.13624	0.357529	0.178764	−	0.983892i	$-0.442790\pi$
$653$	9.13624	0.357529	0.178764	+	0.983892i	$0.442790\pi$
$654$	0	0
$655$	−6.34017	−0.247731
$656$	0	0
$657$	−15.3919	−0.600495
$658$	0	0
$659$	−24.8059	−0.966301	−0.483150	−	0.875537i	$-0.660508\pi$
$659$	−24.8059	−0.966301	−0.483150	+	0.875537i	$0.660508\pi$
$660$	0	0
$661$	30.0866	1.17023	0.585117	−	0.810949i	$-0.301049\pi$
$661$	30.0866	1.17023	0.585117	+	0.810949i	$0.301049\pi$
$662$	0	0
$663$	61.6041	2.39250
$664$	0	0
$665$	−2.63090	−0.102022
$666$	0	0
$667$	−50.4534	−1.95357
$668$	0	0
$669$	−11.6248	−0.449439
$670$	0	0
$671$	−2.92162	−0.112788
$672$	0	0
$673$	23.0894	0.890033	0.445016	−	0.895522i	$-0.353198\pi$
$673$	23.0894	0.890033	0.445016	+	0.895522i	$0.353198\pi$
$674$	0	0
$675$	12.8371	0.494100
$676$	0	0
$677$	−6.96388	−0.267644	−0.133822	−	0.991005i	$-0.542725\pi$
$677$	−6.96388	−0.267644	−0.133822	+	0.991005i	$0.542725\pi$
$678$	0	0
$679$	46.6102	1.78874
$680$	0	0
$681$	91.2099	3.49517
$682$	0	0
$683$	41.5103	1.58835	0.794173	−	0.607692i	$-0.207904\pi$
$683$	41.5103	1.58835	0.794173	+	0.607692i	$0.207904\pi$
$684$	0	0
$685$	−8.15676	−0.311654
$686$	0	0
$687$	20.2557	0.772801
$688$	0	0
$689$	−67.7770	−2.58210
$690$	0	0
$691$	11.4680	0.436263	0.218132	−	0.975919i	$-0.430004\pi$
$691$	11.4680	0.436263	0.218132	+	0.975919i	$0.430004\pi$
$692$	0	0
$693$	−11.7009	−0.444479
$694$	0	0
$695$	10.2062	0.387143
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−18.7526	−0.709288
$700$	0	0
$701$	2.16063	0.0816059	0.0408030	−	0.999167i	$-0.487008\pi$
$701$	2.16063	0.0816059	0.0408030	+	0.999167i	$0.487008\pi$
$702$	0	0
$703$	−6.29791	−0.237530
$704$	0	0
$705$	−15.1773	−0.571609
$706$	0	0
$707$	−40.2122	−1.51233
$708$	0	0
$709$	24.4924	0.919831	0.459916	−	0.887963i	$-0.347880\pi$
$709$	24.4924	0.919831	0.459916	+	0.887963i	$0.347880\pi$
$710$	0	0
$711$	98.0965	3.67890
$712$	0	0
$713$	−8.58145	−0.321378
$714$	0	0
$715$	3.75872	0.140568
$716$	0	0
$717$	−79.1338	−2.95531
$718$	0	0
$719$	−43.4680	−1.62108	−0.810541	−	0.585681i	$-0.800827\pi$
$719$	−43.4680	−1.62108	−0.810541	+	0.585681i	$0.800827\pi$
$720$	0	0
$721$	−17.9688	−0.669193
$722$	0	0
$723$	−69.5585	−2.58691
$724$	0	0
$725$	−8.34017	−0.309746
$726$	0	0
$727$	−30.7442	−1.14024	−0.570119	−	0.821562i	$-0.693103\pi$
$727$	−30.7442	−1.14024	−0.570119	+	0.821562i	$0.693103\pi$
$728$	0	0
$729$	15.7070	0.581741
$730$	0	0
$731$	36.2967	1.34248
$732$	0	0
$733$	−12.6004	−0.465405	−0.232702	−	0.972548i	$-0.574757\pi$
$733$	−12.6004	−0.465405	−0.232702	+	0.972548i	$0.574757\pi$
$734$	0	0
$735$	−0.248464	−0.00916474
$736$	0	0
$737$	0.308976	0.0113813
$738$	0	0
$739$	−52.1978	−1.92013	−0.960063	−	0.279782i	$-0.909738\pi$
$739$	−52.1978	−1.92013	−0.960063	+	0.279782i	$0.909738\pi$
$740$	0	0
$741$	18.8865	0.693815
$742$	0	0
$743$	19.0322	0.698225	0.349113	−	0.937081i	$-0.386483\pi$
$743$	19.0322	0.698225	0.349113	+	0.937081i	$0.386483\pi$
$744$	0	0
$745$	18.5730	0.680463
$746$	0	0
$747$	−13.1545	−0.481298
$748$	0	0
$749$	22.8560	0.835141
$750$	0	0
$751$	31.3874	1.14534	0.572670	−	0.819786i	$-0.305907\pi$
$751$	31.3874	1.14534	0.572670	+	0.819786i	$0.305907\pi$
$752$	0	0
$753$	22.3545	0.814645
$754$	0	0
$755$	−13.7587	−0.500731
$756$	0	0
$757$	33.0661	1.20181	0.600904	−	0.799321i	$-0.294807\pi$
$757$	33.0661	1.20181	0.600904	+	0.799321i	$0.294807\pi$
$758$	0	0
$759$	12.0989	0.439162
$760$	0	0
$761$	−18.1217	−0.656910	−0.328455	−	0.944520i	$-0.606528\pi$
$761$	−18.1217	−0.656910	−0.328455	+	0.944520i	$0.606528\pi$
$762$	0	0
$763$	9.47641	0.343069
$764$	0	0
$765$	22.9939	0.831345
$766$	0	0
$767$	1.43746	0.0519037
$768$	0	0
$769$	−12.0806	−0.435639	−0.217820	−	0.975989i	$-0.569894\pi$
$769$	−12.0806	−0.435639	−0.217820	+	0.975989i	$0.569894\pi$
$770$	0	0
$771$	26.8020	0.965251
$772$	0	0
$773$	15.5864	0.560603	0.280301	−	0.959912i	$-0.409566\pi$
$773$	15.5864	0.560603	0.280301	+	0.959912i	$0.409566\pi$
$774$	0	0
$775$	−1.41855	−0.0509558
$776$	0	0
$777$	52.5257	1.88435
$778$	0	0
$779$	0	0
$780$	0	0
$781$	5.94214	0.212627
$782$	0	0
$783$	−107.064	−3.82614
$784$	0	0
$785$	3.84324	0.137171
$786$	0	0
$787$	24.4775	0.872527	0.436264	−	0.899819i	$-0.356302\pi$
$787$	24.4775	0.872527	0.436264	+	0.899819i	$0.356302\pi$
$788$	0	0
$789$	91.8864	3.27124
$790$	0	0
$791$	20.0410	0.712577
$792$	0	0
$793$	−27.5897	−0.979738
$794$	0	0
$795$	−36.0638	−1.27905
$796$	0	0
$797$	−22.2134	−0.786839	−0.393419	−	0.919359i	$-0.628708\pi$
$797$	−22.2134	−0.786839	−0.393419	+	0.919359i	$0.628708\pi$
$798$	0	0
$799$	−15.6163	−0.552467
$800$	0	0
$801$	86.9914	3.07369
$802$	0	0
$803$	−1.37751	−0.0486114
$804$	0	0
$805$	−15.9155	−0.560947
$806$	0	0
$807$	−20.0989	−0.707515
$808$	0	0
$809$	8.55479	0.300770	0.150385	−	0.988627i	$-0.451949\pi$
$809$	8.55479	0.300770	0.150385	+	0.988627i	$0.451949\pi$
$810$	0	0
$811$	13.3028	0.467126	0.233563	−	0.972342i	$-0.424962\pi$
$811$	13.3028	0.467126	0.233563	+	0.972342i	$0.424962\pi$
$812$	0	0
$813$	68.2388	2.39324
$814$	0	0
$815$	24.3896	0.854331
$816$	0	0
$817$	11.1278	0.389313
$818$	0	0
$819$	−110.494	−3.86099
$820$	0	0
$821$	19.1317	0.667701	0.333850	−	0.942626i	$-0.391652\pi$
$821$	19.1317	0.667701	0.333850	+	0.942626i	$0.391652\pi$
$822$	0	0
$823$	−40.3728	−1.40731	−0.703654	−	0.710543i	$-0.748449\pi$
$823$	−40.3728	−1.40731	−0.703654	+	0.710543i	$0.748449\pi$
$824$	0	0
$825$	2.00000	0.0696311
$826$	0	0
$827$	4.46308	0.155196	0.0775982	−	0.996985i	$-0.475275\pi$
$827$	4.46308	0.155196	0.0775982	+	0.996985i	$0.475275\pi$
$828$	0	0
$829$	−27.4329	−0.952785	−0.476392	−	0.879233i	$-0.658056\pi$
$829$	−27.4329	−0.952785	−0.476392	+	0.879233i	$0.658056\pi$
$830$	0	0
$831$	5.34632	0.185462
$832$	0	0
$833$	−0.255652	−0.00885782
$834$	0	0
$835$	−21.2423	−0.735121
$836$	0	0
$837$	−18.2101	−0.629432
$838$	0	0
$839$	25.8987	0.894121	0.447060	−	0.894504i	$-0.352471\pi$
$839$	25.8987	0.894121	0.447060	+	0.894504i	$0.352471\pi$
$840$	0	0
$841$	40.5585	1.39857
$842$	0	0
$843$	56.0288	1.92973
$844$	0	0
$845$	22.4947	0.773840
$846$	0	0
$847$	27.8927	0.958404
$848$	0	0
$849$	−29.6020	−1.01594
$850$	0	0
$851$	−38.0989	−1.30601
$852$	0	0
$853$	2.58145	0.0883871	0.0441936	−	0.999023i	$-0.485928\pi$
$853$	2.58145	0.0883871	0.0441936	+	0.999023i	$0.485928\pi$
$854$	0	0
$855$	7.04945	0.241086
$856$	0	0
$857$	38.9783	1.33147	0.665736	−	0.746187i	$-0.268118\pi$
$857$	38.9783	1.33147	0.665736	+	0.746187i	$0.268118\pi$
$858$	0	0
$859$	−1.87444	−0.0639551	−0.0319776	−	0.999489i	$-0.510181\pi$
$859$	−1.87444	−0.0639551	−0.0319776	+	0.999489i	$0.510181\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.00104	−0.102157	−0.0510784	−	0.998695i	$-0.516266\pi$
$863$	−3.00104	−0.102157	−0.0510784	+	0.998695i	$0.516266\pi$
$864$	0	0
$865$	−15.2195	−0.517480
$866$	0	0
$867$	−20.1639	−0.684803
$868$	0	0
$869$	8.77924	0.297815
$870$	0	0
$871$	2.91775	0.0988641
$872$	0	0
$873$	−124.891	−4.22693
$874$	0	0
$875$	−2.63090	−0.0889406
$876$	0	0
$877$	4.40910	0.148885	0.0744423	−	0.997225i	$-0.476282\pi$
$877$	4.40910	0.148885	0.0744423	+	0.997225i	$0.476282\pi$
$878$	0	0
$879$	−27.3835	−0.923622
$880$	0	0
$881$	2.09049	0.0704303	0.0352151	−	0.999380i	$-0.488788\pi$
$881$	2.09049	0.0704303	0.0352151	+	0.999380i	$0.488788\pi$
$882$	0	0
$883$	17.2534	0.580623	0.290311	−	0.956932i	$-0.406241\pi$
$883$	17.2534	0.580623	0.290311	+	0.956932i	$0.406241\pi$
$884$	0	0
$885$	0.764867	0.0257107
$886$	0	0
$887$	−47.9760	−1.61088	−0.805438	−	0.592680i	$-0.798070\pi$
$887$	−47.9760	−1.61088	−0.805438	+	0.592680i	$0.798070\pi$
$888$	0	0
$889$	−27.3751	−0.918130
$890$	0	0
$891$	12.3318	0.413130
$892$	0	0
$893$	−4.78765	−0.160213
$894$	0	0
$895$	−1.91548	−0.0640274
$896$	0	0
$897$	114.253	3.81480
$898$	0	0
$899$	11.8310	0.394585
$900$	0	0
$901$	−37.1071	−1.23622
$902$	0	0
$903$	−92.8080	−3.08846
$904$	0	0
$905$	8.15676	0.271140
$906$	0	0
$907$	7.99281	0.265397	0.132698	−	0.991156i	$-0.457636\pi$
$907$	7.99281	0.265397	0.132698	+	0.991156i	$0.457636\pi$
$908$	0	0
$909$	107.748	3.57377
$910$	0	0
$911$	−6.02666	−0.199672	−0.0998361	−	0.995004i	$-0.531832\pi$
$911$	−6.02666	−0.199672	−0.0998361	+	0.995004i	$0.531832\pi$
$912$	0	0
$913$	−1.17727	−0.0389621
$914$	0	0
$915$	−14.6803	−0.485317
$916$	0	0
$917$	16.6803	0.550834
$918$	0	0
$919$	42.0098	1.38578	0.692888	−	0.721045i	$-0.256338\pi$
$919$	42.0098	1.38578	0.692888	+	0.721045i	$0.256338\pi$
$920$	0	0
$921$	−106.773	−3.51829
$922$	0	0
$923$	56.1133	1.84699
$924$	0	0
$925$	−6.29791	−0.207074
$926$	0	0
$927$	48.1471	1.58136
$928$	0	0
$929$	−40.4079	−1.32574	−0.662870	−	0.748735i	$-0.730662\pi$
$929$	−40.4079	−1.32574	−0.662870	+	0.748735i	$0.730662\pi$
$930$	0	0
$931$	−0.0783777	−0.00256873
$932$	0	0
$933$	39.1917	1.28308
$934$	0	0
$935$	2.05786	0.0672992
$936$	0	0
$937$	−20.8227	−0.680249	−0.340124	−	0.940380i	$-0.610469\pi$
$937$	−20.8227	−0.680249	−0.340124	+	0.940380i	$0.610469\pi$
$938$	0	0
$939$	12.9093	0.421280
$940$	0	0
$941$	−38.3591	−1.25047	−0.625235	−	0.780436i	$-0.714997\pi$
$941$	−38.3591	−1.25047	−0.625235	+	0.780436i	$0.714997\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−33.7731	−1.09864
$946$	0	0
$947$	25.9337	0.842733	0.421367	−	0.906890i	$-0.361551\pi$
$947$	25.9337	0.842733	0.421367	+	0.906890i	$0.361551\pi$
$948$	0	0
$949$	−13.0082	−0.422265
$950$	0	0
$951$	−9.93827	−0.322270
$952$	0	0
$953$	−38.8482	−1.25842	−0.629208	−	0.777237i	$-0.716621\pi$
$953$	−38.8482	−1.25842	−0.629208	+	0.777237i	$0.716621\pi$
$954$	0	0
$955$	−5.57531	−0.180413
$956$	0	0
$957$	−16.6803	−0.539199
$958$	0	0
$959$	21.4596	0.692966
$960$	0	0
$961$	−28.9877	−0.935088
$962$	0	0
$963$	−61.2423	−1.97351
$964$	0	0
$965$	−14.8794	−0.478984
$966$	0	0
$967$	10.0183	0.322165	0.161083	−	0.986941i	$-0.448501\pi$
$967$	10.0183	0.322165	0.161083	+	0.986941i	$0.448501\pi$
$968$	0	0
$969$	10.3402	0.332174
$970$	0	0
$971$	−23.8432	−0.765166	−0.382583	−	0.923921i	$-0.624965\pi$
$971$	−23.8432	−0.765166	−0.382583	+	0.923921i	$0.624965\pi$
$972$	0	0
$973$	−26.8515	−0.860819
$974$	0	0
$975$	18.8865	0.604854
$976$	0	0
$977$	−0.481330	−0.0153991	−0.00769956	−	0.999970i	$-0.502451\pi$
$977$	−0.481330	−0.0153991	−0.00769956	+	0.999970i	$0.502451\pi$
$978$	0	0
$979$	7.78539	0.248822
$980$	0	0
$981$	−25.3919	−0.810701
$982$	0	0
$983$	−1.05681	−0.0337071	−0.0168536	−	0.999858i	$-0.505365\pi$
$983$	−1.05681	−0.0337071	−0.0168536	+	0.999858i	$0.505365\pi$
$984$	0	0
$985$	18.9627	0.604201
$986$	0	0
$987$	39.9299	1.27098
$988$	0	0
$989$	67.3172	2.14056
$990$	0	0
$991$	46.8059	1.48684	0.743419	−	0.668826i	$-0.233203\pi$
$991$	46.8059	1.48684	0.743419	+	0.668826i	$0.233203\pi$
$992$	0	0
$993$	−92.4945	−2.93522
$994$	0	0
$995$	15.2039	0.481997
$996$	0	0
$997$	9.53427	0.301953	0.150977	−	0.988537i	$-0.451758\pi$
$997$	9.53427	0.301953	0.150977	+	0.988537i	$0.451758\pi$
$998$	0	0
$999$	−80.8469	−2.55788

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6080.2.a.cb.1.3		3
4.3	odd	2		6080.2.a.bn.1.1		3
8.3	odd	2		3040.2.a.o.1.3	yes	3
8.5	even	2		3040.2.a.i.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.i.1.1	✓	3	8.5	even	2
3040.2.a.o.1.3	yes	3	8.3	odd	2
6080.2.a.bn.1.1		3	4.3	odd	2
6080.2.a.cb.1.3		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6080.a (trivial)

\(f(q)\)	\(=\)	\(q+3.17009 q^{3} +1.00000 q^{5} -2.63090 q^{7} +7.04945 q^{9} +0.630898 q^{11} +5.95774 q^{13} +3.17009 q^{15} +3.26180 q^{17} +1.00000 q^{19} -8.34017 q^{21} +6.04945 q^{23} +1.00000 q^{25} +12.8371 q^{27} -8.34017 q^{29} -1.41855 q^{31} +2.00000 q^{33} -2.63090 q^{35} -6.29791 q^{37} +18.8865 q^{39} +11.1278 q^{43} +7.04945 q^{45} -4.78765 q^{47} -0.0783777 q^{49} +10.3402 q^{51} -11.3763 q^{53} +0.630898 q^{55} +3.17009 q^{57} +0.241276 q^{59} -4.63090 q^{61} -18.5464 q^{63} +5.95774 q^{65} +0.489741 q^{67} +19.1773 q^{69} +9.41855 q^{71} -2.18342 q^{73} +3.17009 q^{75} -1.65983 q^{77} +13.9155 q^{79} +19.5464 q^{81} -1.86603 q^{83} +3.26180 q^{85} -26.4391 q^{87} +12.3402 q^{89} -15.6742 q^{91} -4.49693 q^{93} +1.00000 q^{95} -17.7165 q^{97} +4.44748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} + 4 q^{15} + 2 q^{17} + 3 q^{19} - 14 q^{21} + 3 q^{25} + 10 q^{27} - 14 q^{29} + 10 q^{31} + 6 q^{33} - 4 q^{35} + 8 q^{37} + 10 q^{39}+ \cdots + 14 q^{99}+O(q^{100}) \) 3 * q + 4 * q^3 + 3 * q^5 - 4 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 + 4 * q^15 + 2 * q^17 + 3 * q^19 - 14 * q^21 + 3 * q^25 + 10 * q^27 - 14 * q^29 + 10 * q^31 + 6 * q^33 - 4 * q^35 + 8 * q^37 + 10 * q^39 + 12 * q^43 + 3 * q^45 - 4 * q^47 + 3 * q^49 + 20 * q^51 - 4 * q^53 - 2 * q^55 + 4 * q^57 + 26 * q^59 - 10 * q^61 - 20 * q^63 + 2 * q^65 + 18 * q^67 + 18 * q^69 + 14 * q^71 - 2 * q^73 + 4 * q^75 - 16 * q^77 + 10 * q^79 + 23 * q^81 + 8 * q^83 + 2 * q^85 - 32 * q^87 + 26 * q^89 + 10 * q^91 + 4 * q^93 + 3 * q^95 - 12 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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