LMFDB - Embedded newform 8280.2.a.bl.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:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.11491$ 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)

$f(q)$	$=$	$q-1.00000 q^{5} +0.527166 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +0.527166 q^{7} -3.11491 q^{11} +4.11491 q^{13} +4.39905 q^{17} -3.70265 q^{19} +1.00000 q^{23} +1.00000 q^{25} +9.10170 q^{29} +4.83076 q^{31} -0.527166 q^{35} +9.74378 q^{37} -6.93246 q^{41} -4.45963 q^{43} -0.642074 q^{47} -6.72210 q^{49} +3.89134 q^{53} +3.11491 q^{55} -8.79811 q^{59} -3.45339 q^{61} -4.11491 q^{65} -8.60719 q^{67} -12.3642 q^{71} -5.81756 q^{73} -1.64207 q^{77} -3.17548 q^{79} -4.71585 q^{83} -4.39905 q^{85} +5.43171 q^{89} +2.16924 q^{91} +3.70265 q^{95} -4.06058 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 7 q^{7}+O(q^{10}) $ 3 * q - 3 * q^5 + 7 * q^7	$ 3 q - 3 q^{5} + 7 q^{7} - 3 q^{11} + 6 q^{13} + 5 q^{17} + 7 q^{19} + 3 q^{23} + 3 q^{25} + q^{29} + 10 q^{31} - 7 q^{35} + 2 q^{37} + 10 q^{41} + 12 q^{43} - q^{47} + 6 q^{49} - 10 q^{53} + 3 q^{55} - 10 q^{59} - 13 q^{61} - 6 q^{65} - 6 q^{67} - 10 q^{71} + 7 q^{73} - 4 q^{77} + 14 q^{79} - 16 q^{83} - 5 q^{85} + 20 q^{89} + 11 q^{91} - 7 q^{95} + 5 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 + 7 * q^7 - 3 * q^11 + 6 * q^13 + 5 * q^17 + 7 * q^19 + 3 * q^23 + 3 * q^25 + q^29 + 10 * q^31 - 7 * q^35 + 2 * q^37 + 10 * q^41 + 12 * q^43 - q^47 + 6 * q^49 - 10 * q^53 + 3 * q^55 - 10 * q^59 - 13 * q^61 - 6 * q^65 - 6 * q^67 - 10 * q^71 + 7 * q^73 - 4 * q^77 + 14 * q^79 - 16 * q^83 - 5 * q^85 + 20 * q^89 + 11 * q^91 - 7 * q^95 + 5 * 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
$7$	0.527166	0.199250	0.0996250	−	0.995025i	$-0.468236\pi$
$7$	0.527166	0.199250	0.0996250	+	0.995025i	$0.468236\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.11491	−0.939180	−0.469590	−	0.882885i	$-0.655598\pi$
$11$	−3.11491	−0.939180	−0.469590	+	0.882885i	$0.655598\pi$
$12$	0	0
$13$	4.11491	1.14127	0.570635	−	0.821204i	$-0.306697\pi$
$13$	4.11491	1.14127	0.570635	+	0.821204i	$0.306697\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.39905	1.06693	0.533464	−	0.845823i	$-0.320890\pi$
$17$	4.39905	1.06693	0.533464	+	0.845823i	$0.320890\pi$
$18$	0	0
$19$	−3.70265	−0.849446	−0.424723	−	0.905323i	$-0.639628\pi$
$19$	−3.70265	−0.849446	−0.424723	+	0.905323i	$0.639628\pi$
$20$	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
$29$	9.10170	1.69014	0.845072	−	0.534653i	$-0.179558\pi$
$29$	9.10170	1.69014	0.845072	+	0.534653i	$0.179558\pi$
$30$	0	0
$31$	4.83076	0.867630	0.433815	−	0.901002i	$-0.357167\pi$
$31$	4.83076	0.867630	0.433815	+	0.901002i	$0.357167\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.527166	−0.0891073
$36$	0	0
$37$	9.74378	1.60187	0.800934	−	0.598753i	$-0.204337\pi$
$37$	9.74378	1.60187	0.800934	+	0.598753i	$0.204337\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.93246	−1.08267	−0.541334	−	0.840807i	$-0.682081\pi$
$41$	−6.93246	−1.08267	−0.541334	+	0.840807i	$0.682081\pi$
$42$	0	0
$43$	−4.45963	−0.680087	−0.340044	−	0.940410i	$-0.610442\pi$
$43$	−4.45963	−0.680087	−0.340044	+	0.940410i	$0.610442\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.642074	−0.0936561	−0.0468280	−	0.998903i	$-0.514911\pi$
$47$	−0.642074	−0.0936561	−0.0468280	+	0.998903i	$0.514911\pi$
$48$	0	0
$49$	−6.72210	−0.960299
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.89134	0.534516	0.267258	−	0.963625i	$-0.413882\pi$
$53$	3.89134	0.534516	0.267258	+	0.963625i	$0.413882\pi$
$54$	0	0
$55$	3.11491	0.420014
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.79811	−1.14542	−0.572708	−	0.819759i	$-0.694107\pi$
$59$	−8.79811	−1.14542	−0.572708	+	0.819759i	$0.694107\pi$
$60$	0	0
$61$	−3.45339	−0.442161	−0.221080	−	0.975256i	$-0.570958\pi$
$61$	−3.45339	−0.442161	−0.221080	+	0.975256i	$0.570958\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.11491	−0.510391
$66$	0	0
$67$	−8.60719	−1.05154	−0.525768	−	0.850628i	$-0.676222\pi$
$67$	−8.60719	−1.05154	−0.525768	+	0.850628i	$0.676222\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.3642	−1.46736	−0.733678	−	0.679497i	$-0.762198\pi$
$71$	−12.3642	−1.46736	−0.733678	+	0.679497i	$0.762198\pi$
$72$	0	0
$73$	−5.81756	−0.680893	−0.340447	−	0.940264i	$-0.610578\pi$
$73$	−5.81756	−0.680893	−0.340447	+	0.940264i	$0.610578\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.64207	−0.187132
$78$	0	0
$79$	−3.17548	−0.357270	−0.178635	−	0.983915i	$-0.557168\pi$
$79$	−3.17548	−0.357270	−0.178635	+	0.983915i	$0.557168\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−4.71585	−0.517632	−0.258816	−	0.965927i	$-0.583332\pi$
$83$	−4.71585	−0.517632	−0.258816	+	0.965927i	$0.583332\pi$
$84$	0	0
$85$	−4.39905	−0.477144
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.43171	0.575760	0.287880	−	0.957667i	$-0.407050\pi$
$89$	5.43171	0.575760	0.287880	+	0.957667i	$0.407050\pi$
$90$	0	0
$91$	2.16924	0.227398
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.70265	0.379884
$96$	0	0
$97$	−4.06058	−0.412289	−0.206144	−	0.978522i	$-0.566092\pi$
$97$	−4.06058	−0.412289	−0.206144	+	0.978522i	$0.566092\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.5529	1.54757	0.773784	−	0.633450i	$-0.218362\pi$
$101$	15.5529	1.54757	0.773784	+	0.633450i	$0.218362\pi$
$102$	0	0
$103$	17.9519	1.76885	0.884427	−	0.466678i	$-0.154549\pi$
$103$	17.9519	1.76885	0.884427	+	0.466678i	$0.154549\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.2298	1.18230	0.591150	−	0.806561i	$-0.298674\pi$
$107$	12.2298	1.18230	0.591150	+	0.806561i	$0.298674\pi$
$108$	0	0
$109$	9.10795	0.872383	0.436192	−	0.899854i	$-0.356327\pi$
$109$	9.10795	0.872383	0.436192	+	0.899854i	$0.356327\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	13.1755	1.23945	0.619723	−	0.784821i	$-0.287245\pi$
$113$	13.1755	1.23945	0.619723	+	0.784821i	$0.287245\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.31903	0.212585
$120$	0	0
$121$	−1.29735	−0.117941
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	19.5613	1.73579	0.867894	−	0.496750i	$-0.165473\pi$
$127$	19.5613	1.73579	0.867894	+	0.496750i	$0.165473\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.15604	0.363115	0.181557	−	0.983380i	$-0.441886\pi$
$131$	4.15604	0.363115	0.181557	+	0.983380i	$0.441886\pi$
$132$	0	0
$133$	−1.95191	−0.169252
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.1344	0.951272	0.475636	−	0.879642i	$-0.342218\pi$
$137$	11.1344	0.951272	0.475636	+	0.879642i	$0.342218\pi$
$138$	0	0
$139$	−6.49452	−0.550858	−0.275429	−	0.961321i	$-0.588820\pi$
$139$	−6.49452	−0.550858	−0.275429	+	0.961321i	$0.588820\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−12.8176	−1.07186
$144$	0	0
$145$	−9.10170	−0.755855
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.47283	0.448352	0.224176	−	0.974549i	$-0.428031\pi$
$149$	5.47283	0.448352	0.224176	+	0.974549i	$0.428031\pi$
$150$	0	0
$151$	−0.0954606	−0.00776848	−0.00388424	−	0.999992i	$-0.501236\pi$
$151$	−0.0954606	−0.00776848	−0.00388424	+	0.999992i	$0.501236\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.83076	−0.388016
$156$	0	0
$157$	−3.40530	−0.271772	−0.135886	−	0.990724i	$-0.543388\pi$
$157$	−3.40530	−0.271772	−0.135886	+	0.990724i	$0.543388\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.527166	0.0415465
$162$	0	0
$163$	18.5745	1.45487	0.727435	−	0.686177i	$-0.240712\pi$
$163$	18.5745	1.45487	0.727435	+	0.686177i	$0.240712\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.4596	0.964155	0.482078	−	0.876129i	$-0.339882\pi$
$167$	12.4596	0.964155	0.482078	+	0.876129i	$0.339882\pi$
$168$	0	0
$169$	3.93246	0.302497
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3.49228	−0.265513	−0.132757	−	0.991149i	$-0.542383\pi$
$173$	−3.49228	−0.265513	−0.132757	+	0.991149i	$0.542383\pi$
$174$	0	0
$175$	0.527166	0.0398500
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−6.38585	−0.477301	−0.238650	−	0.971106i	$-0.576705\pi$
$179$	−6.38585	−0.477301	−0.238650	+	0.971106i	$0.576705\pi$
$180$	0	0
$181$	−20.8781	−1.55186	−0.775930	−	0.630819i	$-0.782719\pi$
$181$	−20.8781	−1.55186	−0.775930	+	0.630819i	$0.782719\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−9.74378	−0.716377
$186$	0	0
$187$	−13.7026	−1.00204
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.83700	0.494708	0.247354	−	0.968925i	$-0.420439\pi$
$191$	6.83700	0.494708	0.247354	+	0.968925i	$0.420439\pi$
$192$	0	0
$193$	7.10170	0.511192	0.255596	−	0.966784i	$-0.417728\pi$
$193$	7.10170	0.511192	0.255596	+	0.966784i	$0.417728\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.850207	−0.0605748	−0.0302874	−	0.999541i	$-0.509642\pi$
$197$	−0.850207	−0.0605748	−0.0302874	+	0.999541i	$0.509642\pi$
$198$	0	0
$199$	8.56829	0.607390	0.303695	−	0.952769i	$-0.401780\pi$
$199$	8.56829	0.607390	0.303695	+	0.952769i	$0.401780\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.79811	0.336761
$204$	0	0
$205$	6.93246	0.484184
$206$	0	0
$207$	0	0
$208$	0	0
$209$	11.5334	0.797783
$210$	0	0
$211$	9.63511	0.663309	0.331654	−	0.943401i	$-0.392393\pi$
$211$	9.63511	0.663309	0.331654	+	0.943401i	$0.392393\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.45963	0.304144
$216$	0	0
$217$	2.54661	0.172875
$218$	0	0
$219$	0	0
$220$	0	0
$221$	18.1017	1.21765
$222$	0	0
$223$	9.43171	0.631594	0.315797	−	0.948827i	$-0.397728\pi$
$223$	9.43171	0.631594	0.315797	+	0.948827i	$0.397728\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.91774	0.260030	0.130015	−	0.991512i	$-0.458497\pi$
$227$	3.91774	0.260030	0.130015	+	0.991512i	$0.458497\pi$
$228$	0	0
$229$	−22.0947	−1.46006	−0.730031	−	0.683414i	$-0.760494\pi$
$229$	−22.0947	−1.46006	−0.730031	+	0.683414i	$0.760494\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.5334	0.821091	0.410545	−	0.911840i	$-0.365338\pi$
$233$	12.5334	0.821091	0.410545	+	0.911840i	$0.365338\pi$
$234$	0	0
$235$	0.642074	0.0418843
$236$	0	0
$237$	0	0
$238$	0	0
$239$	12.4123	0.802882	0.401441	−	0.915885i	$-0.368509\pi$
$239$	12.4123	0.802882	0.401441	+	0.915885i	$0.368509\pi$
$240$	0	0
$241$	5.74378	0.369989	0.184995	−	0.982740i	$-0.440773\pi$
$241$	5.74378	0.369989	0.184995	+	0.982740i	$0.440773\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.72210	0.429459
$246$	0	0
$247$	−15.2361	−0.969447
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.33624	0.399940	0.199970	−	0.979802i	$-0.435915\pi$
$251$	6.33624	0.399940	0.199970	+	0.979802i	$0.435915\pi$
$252$	0	0
$253$	−3.11491	−0.195833
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−22.2772	−1.38961	−0.694806	−	0.719197i	$-0.744510\pi$
$257$	−22.2772	−1.38961	−0.694806	+	0.719197i	$0.744510\pi$
$258$	0	0
$259$	5.13659	0.319172
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4.66776	0.287827	0.143913	−	0.989590i	$-0.454031\pi$
$263$	4.66776	0.287827	0.143913	+	0.989590i	$0.454031\pi$
$264$	0	0
$265$	−3.89134	−0.239043
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−9.70889	−0.591962	−0.295981	−	0.955194i	$-0.595646\pi$
$269$	−9.70889	−0.591962	−0.295981	+	0.955194i	$0.595646\pi$
$270$	0	0
$271$	−6.75698	−0.410457	−0.205229	−	0.978714i	$-0.565794\pi$
$271$	−6.75698	−0.410457	−0.205229	+	0.978714i	$0.565794\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.11491	−0.187836
$276$	0	0
$277$	19.0194	1.14277	0.571384	−	0.820683i	$-0.306407\pi$
$277$	19.0194	1.14277	0.571384	+	0.820683i	$0.306407\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	32.8370	1.95889	0.979446	−	0.201708i	$-0.0646492\pi$
$281$	32.8370	1.95889	0.979446	+	0.201708i	$0.0646492\pi$
$282$	0	0
$283$	13.7827	0.819295	0.409647	−	0.912244i	$-0.365652\pi$
$283$	13.7827	0.819295	0.409647	+	0.912244i	$0.365652\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.65456	−0.215722
$288$	0	0
$289$	2.35168	0.138334
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−15.0668	−0.880213	−0.440106	−	0.897946i	$-0.645059\pi$
$293$	−15.0668	−0.880213	−0.440106	+	0.897946i	$0.645059\pi$
$294$	0	0
$295$	8.79811	0.512246
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.11491	0.237971
$300$	0	0
$301$	−2.35097	−0.135507
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.45339	0.197740
$306$	0	0
$307$	−5.91302	−0.337474	−0.168737	−	0.985661i	$-0.553969\pi$
$307$	−5.91302	−0.337474	−0.168737	+	0.985661i	$0.553969\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	25.7911	1.46248	0.731241	−	0.682119i	$-0.238942\pi$
$311$	25.7911	1.46248	0.731241	+	0.682119i	$0.238942\pi$
$312$	0	0
$313$	16.7306	0.945668	0.472834	−	0.881152i	$-0.343231\pi$
$313$	16.7306	0.945668	0.472834	+	0.881152i	$0.343231\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−28.1623	−1.58175	−0.790876	−	0.611977i	$-0.790375\pi$
$317$	−28.1623	−1.58175	−0.790876	+	0.611977i	$0.790375\pi$
$318$	0	0
$319$	−28.3510	−1.58735
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−16.2882	−0.906297
$324$	0	0
$325$	4.11491	0.228254
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−0.338479	−0.0186610
$330$	0	0
$331$	−25.8176	−1.41906	−0.709531	−	0.704675i	$-0.751093\pi$
$331$	−25.8176	−1.41906	−0.709531	+	0.704675i	$0.751093\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.60719	0.470261
$336$	0	0
$337$	33.5676	1.82854	0.914271	−	0.405103i	$-0.132764\pi$
$337$	33.5676	1.82854	0.914271	+	0.405103i	$0.132764\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−15.0474	−0.814861
$342$	0	0
$343$	−7.23382	−0.390590
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7.63984	−0.410128	−0.205064	−	0.978749i	$-0.565740\pi$
$347$	−7.63984	−0.410128	−0.205064	+	0.978749i	$0.565740\pi$
$348$	0	0
$349$	3.01945	0.161627	0.0808136	−	0.996729i	$-0.474248\pi$
$349$	3.01945	0.161627	0.0808136	+	0.996729i	$0.474248\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	25.1840	1.34041	0.670203	−	0.742177i	$-0.266207\pi$
$353$	25.1840	1.34041	0.670203	+	0.742177i	$0.266207\pi$
$354$	0	0
$355$	12.3642	0.656222
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5.32304	0.280939	0.140470	−	0.990085i	$-0.455139\pi$
$359$	5.32304	0.280939	0.140470	+	0.990085i	$0.455139\pi$
$360$	0	0
$361$	−5.29039	−0.278442
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.81756	0.304505
$366$	0	0
$367$	8.24230	0.430245	0.215122	−	0.976587i	$-0.430985\pi$
$367$	8.24230	0.430245	0.215122	+	0.976587i	$0.430985\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.05138	0.106502
$372$	0	0
$373$	1.91774	0.0992970	0.0496485	−	0.998767i	$-0.484190\pi$
$373$	1.91774	0.0992970	0.0496485	+	0.998767i	$0.484190\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	37.4527	1.92891
$378$	0	0
$379$	30.1692	1.54969	0.774845	−	0.632151i	$-0.217828\pi$
$379$	30.1692	1.54969	0.774845	+	0.632151i	$0.217828\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.64903	0.0842617	0.0421309	−	0.999112i	$-0.486585\pi$
$383$	1.64903	0.0842617	0.0421309	+	0.999112i	$0.486585\pi$
$384$	0	0
$385$	1.64207	0.0836878
$386$	0	0
$387$	0	0
$388$	0	0
$389$	16.6414	0.843750	0.421875	−	0.906654i	$-0.361372\pi$
$389$	16.6414	0.843750	0.421875	+	0.906654i	$0.361372\pi$
$390$	0	0
$391$	4.39905	0.222470
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.17548	0.159776
$396$	0	0
$397$	36.0257	1.80808	0.904039	−	0.427450i	$-0.140588\pi$
$397$	36.0257	1.80808	0.904039	+	0.427450i	$0.140588\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.62263	0.280781	0.140390	−	0.990096i	$-0.455164\pi$
$401$	5.62263	0.280781	0.140390	+	0.990096i	$0.455164\pi$
$402$	0	0
$403$	19.8781	0.990200
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−30.3510	−1.50444
$408$	0	0
$409$	−2.76546	−0.136743	−0.0683716	−	0.997660i	$-0.521780\pi$
$409$	−2.76546	−0.136743	−0.0683716	+	0.997660i	$0.521780\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4.63807	−0.228224
$414$	0	0
$415$	4.71585	0.231492
$416$	0	0
$417$	0	0
$418$	0	0
$419$	17.5962	0.859632	0.429816	−	0.902917i	$-0.358579\pi$
$419$	17.5962	0.859632	0.429816	+	0.902917i	$0.358579\pi$
$420$	0	0
$421$	25.5676	1.24609	0.623044	−	0.782187i	$-0.285896\pi$
$421$	25.5676	1.24609	0.623044	+	0.782187i	$0.285896\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.39905	0.213386
$426$	0	0
$427$	−1.82051	−0.0881006
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−17.9736	−0.865757	−0.432879	−	0.901452i	$-0.642502\pi$
$431$	−17.9736	−0.865757	−0.432879	+	0.901452i	$0.642502\pi$
$432$	0	0
$433$	37.0382	1.77994	0.889971	−	0.456018i	$-0.150725\pi$
$433$	37.0382	1.77994	0.889971	+	0.456018i	$0.150725\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.70265	−0.177122
$438$	0	0
$439$	38.8168	1.85263	0.926313	−	0.376754i	$-0.122960\pi$
$439$	38.8168	1.85263	0.926313	+	0.376754i	$0.122960\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−25.3726	−1.20549	−0.602745	−	0.797934i	$-0.705927\pi$
$443$	−25.3726	−1.20549	−0.602745	+	0.797934i	$0.705927\pi$
$444$	0	0
$445$	−5.43171	−0.257488
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−33.4589	−1.57902	−0.789512	−	0.613735i	$-0.789666\pi$
$449$	−33.4589	−1.57902	−0.789512	+	0.613735i	$0.789666\pi$
$450$	0	0
$451$	21.5940	1.01682
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.16924	−0.101696
$456$	0	0
$457$	−27.5264	−1.28763	−0.643816	−	0.765180i	$-0.722650\pi$
$457$	−27.5264	−1.28763	−0.643816	+	0.765180i	$0.722650\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−36.3161	−1.69141	−0.845704	−	0.533652i	$-0.820819\pi$
$461$	−36.3161	−1.69141	−0.845704	+	0.533652i	$0.820819\pi$
$462$	0	0
$463$	−22.5544	−1.04819	−0.524095	−	0.851660i	$-0.675596\pi$
$463$	−22.5544	−1.04819	−0.524095	+	0.851660i	$0.675596\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−22.8370	−1.05677	−0.528385	−	0.849005i	$-0.677202\pi$
$467$	−22.8370	−1.05677	−0.528385	+	0.849005i	$0.677202\pi$
$468$	0	0
$469$	−4.53742	−0.209518
$470$	0	0
$471$	0	0
$472$	0	0
$473$	13.8913	0.638724
$474$	0	0
$475$	−3.70265	−0.169889
$476$	0	0
$477$	0	0
$478$	0	0
$479$	35.5264	1.62324	0.811622	−	0.584182i	$-0.198585\pi$
$479$	35.5264	1.62324	0.811622	+	0.584182i	$0.198585\pi$
$480$	0	0
$481$	40.0947	1.82816
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.06058	0.184381
$486$	0	0
$487$	−10.4945	−0.475552	−0.237776	−	0.971320i	$-0.576418\pi$
$487$	−10.4945	−0.475552	−0.237776	+	0.971320i	$0.576418\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−42.1685	−1.90304	−0.951519	−	0.307589i	$-0.900478\pi$
$491$	−42.1685	−1.90304	−0.951519	+	0.307589i	$0.900478\pi$
$492$	0	0
$493$	40.0389	1.80326
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.51797	−0.292371
$498$	0	0
$499$	30.1560	1.34997	0.674985	−	0.737832i	$-0.264150\pi$
$499$	30.1560	1.34997	0.674985	+	0.737832i	$0.264150\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−22.9729	−1.02431	−0.512155	−	0.858893i	$-0.671153\pi$
$503$	−22.9729	−1.02431	−0.512155	+	0.858893i	$0.671153\pi$
$504$	0	0
$505$	−15.5529	−0.692093
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−11.8176	−0.523804	−0.261902	−	0.965094i	$-0.584350\pi$
$509$	−11.8176	−0.523804	−0.261902	+	0.965094i	$0.584350\pi$
$510$	0	0
$511$	−3.06682	−0.135668
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−17.9519	−0.791056
$516$	0	0
$517$	2.00000	0.0879599
$518$	0	0
$519$	0	0
$520$	0	0
$521$	35.7438	1.56596	0.782982	−	0.622045i	$-0.213698\pi$
$521$	35.7438	1.56596	0.782982	+	0.622045i	$0.213698\pi$
$522$	0	0
$523$	14.3899	0.629225	0.314612	−	0.949220i	$-0.398125\pi$
$523$	14.3899	0.629225	0.314612	+	0.949220i	$0.398125\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	21.2508	0.925698
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−28.5264	−1.23562
$534$	0	0
$535$	−12.2298	−0.528741
$536$	0	0
$537$	0	0
$538$	0	0
$539$	20.9387	0.901894
$540$	0	0
$541$	14.5209	0.624303	0.312152	−	0.950032i	$-0.398950\pi$
$541$	14.5209	0.624303	0.312152	+	0.950032i	$0.398950\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−9.10795	−0.390142
$546$	0	0
$547$	19.6289	0.839270	0.419635	−	0.907693i	$-0.362158\pi$
$547$	19.6289	0.839270	0.419635	+	0.907693i	$0.362158\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−33.7004	−1.43569
$552$	0	0
$553$	−1.67401	−0.0711860
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.13659	−0.0481588	−0.0240794	−	0.999710i	$-0.507665\pi$
$557$	−1.13659	−0.0481588	−0.0240794	+	0.999710i	$0.507665\pi$
$558$	0	0
$559$	−18.3510	−0.776163
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−25.8913	−1.09119	−0.545595	−	0.838049i	$-0.683696\pi$
$563$	−25.8913	−1.09119	−0.545595	+	0.838049i	$0.683696\pi$
$564$	0	0
$565$	−13.1755	−0.554297
$566$	0	0
$567$	0	0
$568$	0	0
$569$	44.0558	1.84692	0.923459	−	0.383698i	$-0.125350\pi$
$569$	44.0558	1.84692	0.923459	+	0.383698i	$0.125350\pi$
$570$	0	0
$571$	−41.8991	−1.75342	−0.876711	−	0.481017i	$-0.840268\pi$
$571$	−41.8991	−1.75342	−0.876711	+	0.481017i	$0.840268\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	1.34544	0.0560114	0.0280057	−	0.999608i	$-0.491084\pi$
$577$	1.34544	0.0560114	0.0280057	+	0.999608i	$0.491084\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.48604	−0.103138
$582$	0	0
$583$	−12.1212	−0.502007
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−25.0411	−1.03356	−0.516779	−	0.856119i	$-0.672869\pi$
$587$	−25.0411	−1.03356	−0.516779	+	0.856119i	$0.672869\pi$
$588$	0	0
$589$	−17.8866	−0.737005
$590$	0	0
$591$	0	0
$592$	0	0
$593$	22.5947	0.927853	0.463927	−	0.885874i	$-0.346440\pi$
$593$	22.5947	0.927853	0.463927	+	0.885874i	$0.346440\pi$
$594$	0	0
$595$	−2.31903	−0.0950711
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−14.7111	−0.601080	−0.300540	−	0.953769i	$-0.597167\pi$
$599$	−14.7111	−0.601080	−0.300540	+	0.953769i	$0.597167\pi$
$600$	0	0
$601$	19.2904	0.786871	0.393436	−	0.919352i	$-0.371286\pi$
$601$	19.2904	0.786871	0.393436	+	0.919352i	$0.371286\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.29735	0.0527448
$606$	0	0
$607$	−20.7547	−0.842409	−0.421205	−	0.906966i	$-0.638393\pi$
$607$	−20.7547	−0.842409	−0.421205	+	0.906966i	$0.638393\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.64207	−0.106887
$612$	0	0
$613$	−43.3091	−1.74924	−0.874619	−	0.484810i	$-0.838889\pi$
$613$	−43.3091	−1.74924	−0.874619	+	0.484810i	$0.838889\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−13.5621	−0.545988	−0.272994	−	0.962016i	$-0.588014\pi$
$617$	−13.5621	−0.545988	−0.272994	+	0.962016i	$0.588014\pi$
$618$	0	0
$619$	−12.9045	−0.518677	−0.259339	−	0.965786i	$-0.583505\pi$
$619$	−12.9045	−0.518677	−0.259339	+	0.965786i	$0.583505\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.86341	0.114720
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	42.8634	1.70908
$630$	0	0
$631$	45.0404	1.79303	0.896515	−	0.443013i	$-0.146090\pi$
$631$	45.0404	1.79303	0.896515	+	0.443013i	$0.146090\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−19.5613	−0.776268
$636$	0	0
$637$	−27.6608	−1.09596
$638$	0	0
$639$	0	0
$640$	0	0
$641$	3.08074	0.121682	0.0608410	−	0.998147i	$-0.480622\pi$
$641$	3.08074	0.121682	0.0608410	+	0.998147i	$0.480622\pi$
$642$	0	0
$643$	4.52493	0.178446	0.0892229	−	0.996012i	$-0.471562\pi$
$643$	4.52493	0.178446	0.0892229	+	0.996012i	$0.471562\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−34.8719	−1.37096	−0.685478	−	0.728094i	$-0.740407\pi$
$647$	−34.8719	−1.37096	−0.685478	+	0.728094i	$0.740407\pi$
$648$	0	0
$649$	27.4053	1.07575
$650$	0	0
$651$	0	0
$652$	0	0
$653$	35.3726	1.38424	0.692119	−	0.721783i	$-0.256677\pi$
$653$	35.3726	1.38424	0.692119	+	0.721783i	$0.256677\pi$
$654$	0	0
$655$	−4.15604	−0.162390
$656$	0	0
$657$	0	0
$658$	0	0
$659$	2.10866	0.0821419	0.0410710	−	0.999156i	$-0.486923\pi$
$659$	2.10866	0.0821419	0.0410710	+	0.999156i	$0.486923\pi$
$660$	0	0
$661$	20.0342	0.779239	0.389619	−	0.920976i	$-0.372607\pi$
$661$	20.0342	0.779239	0.389619	+	0.920976i	$0.372607\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.95191	0.0756919
$666$	0	0
$667$	9.10170	0.352419
$668$	0	0
$669$	0	0
$670$	0	0
$671$	10.7570	0.415269
$672$	0	0
$673$	−5.07530	−0.195638	−0.0978191	−	0.995204i	$-0.531187\pi$
$673$	−5.07530	−0.195638	−0.0978191	+	0.995204i	$0.531187\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.4876	0.518369	0.259184	−	0.965828i	$-0.416546\pi$
$677$	13.4876	0.518369	0.259184	+	0.965828i	$0.416546\pi$
$678$	0	0
$679$	−2.14060	−0.0821486
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−25.4379	−0.973356	−0.486678	−	0.873581i	$-0.661792\pi$
$683$	−25.4379	−0.973356	−0.486678	+	0.873581i	$0.661792\pi$
$684$	0	0
$685$	−11.1344	−0.425422
$686$	0	0
$687$	0	0
$688$	0	0
$689$	16.0125	0.610027
$690$	0	0
$691$	22.6506	0.861668	0.430834	−	0.902431i	$-0.358220\pi$
$691$	22.6506	0.861668	0.430834	+	0.902431i	$0.358220\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.49452	0.246351
$696$	0	0
$697$	−30.4963	−1.15513
$698$	0	0
$699$	0	0
$700$	0	0
$701$	41.8099	1.57914	0.789569	−	0.613662i	$-0.210304\pi$
$701$	41.8099	1.57914	0.789569	+	0.613662i	$0.210304\pi$
$702$	0	0
$703$	−36.0778	−1.36070
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.19894	0.308353
$708$	0	0
$709$	2.87117	0.107829	0.0539146	−	0.998546i	$-0.482830\pi$
$709$	2.87117	0.107829	0.0539146	+	0.998546i	$0.482830\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	4.83076	0.180913
$714$	0	0
$715$	12.8176	0.479349
$716$	0	0
$717$	0	0
$718$	0	0
$719$	50.1623	1.87074	0.935369	−	0.353674i	$-0.115068\pi$
$719$	50.1623	1.87074	0.935369	+	0.353674i	$0.115068\pi$
$720$	0	0
$721$	9.46364	0.352444
$722$	0	0
$723$	0	0
$724$	0	0
$725$	9.10170	0.338029
$726$	0	0
$727$	−26.8198	−0.994691	−0.497345	−	0.867553i	$-0.665692\pi$
$727$	−26.8198	−0.994691	−0.497345	+	0.867553i	$0.665692\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−19.6182	−0.725604
$732$	0	0
$733$	35.9861	1.32918	0.664588	−	0.747210i	$-0.268607\pi$
$733$	35.9861	1.32918	0.664588	+	0.747210i	$0.268607\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	26.8106	0.987581
$738$	0	0
$739$	5.84396	0.214974	0.107487	−	0.994207i	$-0.465720\pi$
$739$	5.84396	0.214974	0.107487	+	0.994207i	$0.465720\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−14.2012	−0.520991	−0.260495	−	0.965475i	$-0.583886\pi$
$743$	−14.2012	−0.520991	−0.260495	+	0.965475i	$0.583886\pi$
$744$	0	0
$745$	−5.47283	−0.200509
$746$	0	0
$747$	0	0
$748$	0	0
$749$	6.44714	0.235574
$750$	0	0
$751$	33.0404	1.20566	0.602831	−	0.797869i	$-0.294039\pi$
$751$	33.0404	1.20566	0.602831	+	0.797869i	$0.294039\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0.0954606	0.00347417
$756$	0	0
$757$	18.2034	0.661614	0.330807	−	0.943698i	$-0.392679\pi$
$757$	18.2034	0.661614	0.330807	+	0.943698i	$0.392679\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−34.1010	−1.23616	−0.618080	−	0.786115i	$-0.712089\pi$
$761$	−34.1010	−1.23616	−0.618080	+	0.786115i	$0.712089\pi$
$762$	0	0
$763$	4.80140	0.173822
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−36.2034	−1.30723
$768$	0	0
$769$	35.1366	1.26706	0.633529	−	0.773719i	$-0.281606\pi$
$769$	35.1366	1.26706	0.633529	+	0.773719i	$0.281606\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.13659	−0.0408803	−0.0204401	−	0.999791i	$-0.506507\pi$
$773$	−1.13659	−0.0408803	−0.0204401	+	0.999791i	$0.506507\pi$
$774$	0	0
$775$	4.83076	0.173526
$776$	0	0
$777$	0	0
$778$	0	0
$779$	25.6685	0.919669
$780$	0	0
$781$	38.5132	1.37811
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.40530	0.121540
$786$	0	0
$787$	−8.33848	−0.297235	−0.148617	−	0.988895i	$-0.547482\pi$
$787$	−8.33848	−0.297235	−0.148617	+	0.988895i	$0.547482\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	6.94567	0.246960
$792$	0	0
$793$	−14.2104	−0.504625
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0.190921	0.00676278	0.00338139	−	0.999994i	$-0.498924\pi$
$797$	0.190921	0.00676278	0.00338139	+	0.999994i	$0.498924\pi$
$798$	0	0
$799$	−2.82452	−0.0999242
$800$	0	0
$801$	0	0
$802$	0	0
$803$	18.1212	0.639482
$804$	0	0
$805$	−0.527166	−0.0185802
$806$	0	0
$807$	0	0
$808$	0	0
$809$	14.5202	0.510503	0.255252	−	0.966875i	$-0.417842\pi$
$809$	14.5202	0.510503	0.255252	+	0.966875i	$0.417842\pi$
$810$	0	0
$811$	−51.5474	−1.81007	−0.905037	−	0.425332i	$-0.860157\pi$
$811$	−51.5474	−1.81007	−0.905037	+	0.425332i	$0.860157\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−18.5745	−0.650638
$816$	0	0
$817$	16.5124	0.577697
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−13.7827	−0.481019	−0.240509	−	0.970647i	$-0.577314\pi$
$821$	−13.7827	−0.481019	−0.240509	+	0.970647i	$0.577314\pi$
$822$	0	0
$823$	−37.1531	−1.29508	−0.647538	−	0.762034i	$-0.724201\pi$
$823$	−37.1531	−1.29508	−0.647538	+	0.762034i	$0.724201\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	39.8510	1.38576	0.692878	−	0.721055i	$-0.256343\pi$
$827$	39.8510	1.38576	0.692878	+	0.721055i	$0.256343\pi$
$828$	0	0
$829$	20.5933	0.715234	0.357617	−	0.933868i	$-0.383589\pi$
$829$	20.5933	0.715234	0.357617	+	0.933868i	$0.383589\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−29.5709	−1.02457
$834$	0	0
$835$	−12.4596	−0.431183
$836$	0	0
$837$	0	0
$838$	0	0
$839$	36.5669	1.26243	0.631214	−	0.775609i	$-0.282557\pi$
$839$	36.5669	1.26243	0.631214	+	0.775609i	$0.282557\pi$
$840$	0	0
$841$	53.8410	1.85659
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3.93246	−0.135281
$846$	0	0
$847$	−0.683919	−0.0234998
$848$	0	0
$849$	0	0
$850$	0	0
$851$	9.74378	0.334012
$852$	0	0
$853$	−19.3208	−0.661532	−0.330766	−	0.943713i	$-0.607307\pi$
$853$	−19.3208	−0.661532	−0.330766	+	0.943713i	$0.607307\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−54.6017	−1.86516	−0.932580	−	0.360963i	$-0.882448\pi$
$857$	−54.6017	−1.86516	−0.932580	+	0.360963i	$0.882448\pi$
$858$	0	0
$859$	−32.9541	−1.12438	−0.562190	−	0.827008i	$-0.690041\pi$
$859$	−32.9541	−1.12438	−0.562190	+	0.827008i	$0.690041\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	54.6840	1.86147	0.930733	−	0.365701i	$-0.119171\pi$
$863$	54.6840	1.86147	0.930733	+	0.365701i	$0.119171\pi$
$864$	0	0
$865$	3.49228	0.118741
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9.89134	0.335541
$870$	0	0
$871$	−35.4178	−1.20009
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−0.527166	−0.0178215
$876$	0	0
$877$	16.7500	0.565608	0.282804	−	0.959178i	$-0.408735\pi$
$877$	16.7500	0.565608	0.282804	+	0.959178i	$0.408735\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	31.7049	1.06816	0.534082	−	0.845432i	$-0.320657\pi$
$881$	31.7049	1.06816	0.534082	+	0.845432i	$0.320657\pi$
$882$	0	0
$883$	51.9185	1.74720	0.873599	−	0.486646i	$-0.161780\pi$
$883$	51.9185	1.74720	0.873599	+	0.486646i	$0.161780\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−36.2897	−1.21849	−0.609244	−	0.792983i	$-0.708527\pi$
$887$	−36.2897	−1.21849	−0.609244	+	0.792983i	$0.708527\pi$
$888$	0	0
$889$	10.3121	0.345856
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.37737	0.0795558
$894$	0	0
$895$	6.38585	0.213455
$896$	0	0
$897$	0	0
$898$	0	0
$899$	43.9681	1.46642
$900$	0	0
$901$	17.1182	0.570290
$902$	0	0
$903$	0	0
$904$	0	0
$905$	20.8781	0.694013
$906$	0	0
$907$	−25.4442	−0.844861	−0.422430	−	0.906395i	$-0.638823\pi$
$907$	−25.4442	−0.844861	−0.422430	+	0.906395i	$0.638823\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−47.9597	−1.58897	−0.794487	−	0.607281i	$-0.792260\pi$
$911$	−47.9597	−1.58897	−0.794487	+	0.607281i	$0.792260\pi$
$912$	0	0
$913$	14.6894	0.486150
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.19092	0.0723506
$918$	0	0
$919$	2.78562	0.0918892	0.0459446	−	0.998944i	$-0.485370\pi$
$919$	2.78562	0.0918892	0.0459446	+	0.998944i	$0.485370\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−50.8774	−1.67465
$924$	0	0
$925$	9.74378	0.320373
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−40.2508	−1.32059	−0.660293	−	0.751008i	$-0.729568\pi$
$929$	−40.2508	−1.32059	−0.660293	+	0.751008i	$0.729568\pi$
$930$	0	0
$931$	24.8896	0.815722
$932$	0	0
$933$	0	0
$934$	0	0
$935$	13.7026	0.448125
$936$	0	0
$937$	−38.9868	−1.27364	−0.636822	−	0.771011i	$-0.719751\pi$
$937$	−38.9868	−1.27364	−0.636822	+	0.771011i	$0.719751\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−38.3682	−1.25077	−0.625383	−	0.780318i	$-0.715057\pi$
$941$	−38.3682	−1.25077	−0.625383	+	0.780318i	$0.715057\pi$
$942$	0	0
$943$	−6.93246	−0.225752
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−15.0217	−0.488139	−0.244070	−	0.969758i	$-0.578483\pi$
$947$	−15.0217	−0.488139	−0.244070	+	0.969758i	$0.578483\pi$
$948$	0	0
$949$	−23.9387	−0.777083
$950$	0	0
$951$	0	0
$952$	0	0
$953$	12.1303	0.392940	0.196470	−	0.980510i	$-0.437052\pi$
$953$	12.1303	0.392940	0.196470	+	0.980510i	$0.437052\pi$
$954$	0	0
$955$	−6.83700	−0.221240
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.86965	0.189541
$960$	0	0
$961$	−7.66376	−0.247218
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−7.10170	−0.228612
$966$	0	0
$967$	21.2882	0.684581	0.342290	−	0.939594i	$-0.388797\pi$
$967$	21.2882	0.684581	0.342290	+	0.939594i	$0.388797\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−11.4006	−0.365862	−0.182931	−	0.983126i	$-0.558558\pi$
$971$	−11.4006	−0.365862	−0.182931	+	0.983126i	$0.558558\pi$
$972$	0	0
$973$	−3.42369	−0.109758
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−41.2849	−1.32082	−0.660411	−	0.750904i	$-0.729618\pi$
$977$	−41.2849	−1.32082	−0.660411	+	0.750904i	$0.729618\pi$
$978$	0	0
$979$	−16.9193	−0.540742
$980$	0	0
$981$	0	0
$982$	0	0
$983$	42.8448	1.36654	0.683268	−	0.730168i	$-0.260558\pi$
$983$	42.8448	1.36654	0.683268	+	0.730168i	$0.260558\pi$
$984$	0	0
$985$	0.850207	0.0270899
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.45963	−0.141808
$990$	0	0
$991$	8.73307	0.277415	0.138707	−	0.990333i	$-0.455705\pi$
$991$	8.73307	0.277415	0.138707	+	0.990333i	$0.455705\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−8.56829	−0.271633
$996$	0	0
$997$	−44.0808	−1.39605	−0.698027	−	0.716072i	$-0.745938\pi$
$997$	−44.0808	−1.39605	−0.698027	+	0.716072i	$0.745938\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.bl.1.1		3
3.2	odd	2		920.2.a.i.1.3	✓	3
12.11	even	2		1840.2.a.q.1.1		3
15.2	even	4		4600.2.e.q.4049.1		6
15.8	even	4		4600.2.e.q.4049.6		6
15.14	odd	2		4600.2.a.v.1.1		3
24.5	odd	2		7360.2.a.bw.1.1		3
24.11	even	2		7360.2.a.cf.1.3		3
60.59	even	2		9200.2.a.ci.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.i.1.3	✓	3	3.2	odd	2
1840.2.a.q.1.1		3	12.11	even	2
4600.2.a.v.1.1		3	15.14	odd	2
4600.2.e.q.4049.1		6	15.2	even	4
4600.2.e.q.4049.6		6	15.8	even	4
7360.2.a.bw.1.1		3	24.5	odd	2
7360.2.a.cf.1.3		3	24.11	even	2
8280.2.a.bl.1.1		3	1.1	even	1	trivial
9200.2.a.ci.1.3		3	60.59	even	2

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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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +0.527166 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +0.527166 q^{7} -3.11491 q^{11} +4.11491 q^{13} +4.39905 q^{17} -3.70265 q^{19} +1.00000 q^{23} +1.00000 q^{25} +9.10170 q^{29} +4.83076 q^{31} -0.527166 q^{35} +9.74378 q^{37} -6.93246 q^{41} -4.45963 q^{43} -0.642074 q^{47} -6.72210 q^{49} +3.89134 q^{53} +3.11491 q^{55} -8.79811 q^{59} -3.45339 q^{61} -4.11491 q^{65} -8.60719 q^{67} -12.3642 q^{71} -5.81756 q^{73} -1.64207 q^{77} -3.17548 q^{79} -4.71585 q^{83} -4.39905 q^{85} +5.43171 q^{89} +2.16924 q^{91} +3.70265 q^{95} -4.06058 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 7 q^{7}+O(q^{10}) \) 3 * q - 3 * q^5 + 7 * q^7	\( 3 q - 3 q^{5} + 7 q^{7} - 3 q^{11} + 6 q^{13} + 5 q^{17} + 7 q^{19} + 3 q^{23} + 3 q^{25} + q^{29} + 10 q^{31} - 7 q^{35} + 2 q^{37} + 10 q^{41} + 12 q^{43} - q^{47} + 6 q^{49} - 10 q^{53} + 3 q^{55} - 10 q^{59} - 13 q^{61} - 6 q^{65} - 6 q^{67} - 10 q^{71} + 7 q^{73} - 4 q^{77} + 14 q^{79} - 16 q^{83} - 5 q^{85} + 20 q^{89} + 11 q^{91} - 7 q^{95} + 5 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 + 7 * q^7 - 3 * q^11 + 6 * q^13 + 5 * q^17 + 7 * q^19 + 3 * q^23 + 3 * q^25 + q^29 + 10 * q^31 - 7 * q^35 + 2 * q^37 + 10 * q^41 + 12 * q^43 - q^47 + 6 * q^49 - 10 * q^53 + 3 * q^55 - 10 * q^59 - 13 * q^61 - 6 * q^65 - 6 * q^67 - 10 * q^71 + 7 * q^73 - 4 * q^77 + 14 * q^79 - 16 * q^83 - 5 * q^85 + 20 * q^89 + 11 * q^91 - 7 * q^95 + 5 * q^97

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