LMFDB - Embedded newform 1840.2.a.q.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1840,2,Mod(1,1840)] 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("1840.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-2,0,3,0,-7,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$14.6924739719$
Analytic rank:	$1$
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:	$\Z[a_1, a_2, a_3]$
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	$\chi$	$=$	1840.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.47283 q^{3} +1.00000 q^{5} -0.527166 q^{7} +3.11491 q^{9} -3.11491 q^{11} +4.11491 q^{13} -2.47283 q^{15} -4.39905 q^{17} +3.70265 q^{19} +1.30359 q^{21} +1.00000 q^{23} +1.00000 q^{25} -0.284147 q^{27} -9.10170 q^{29} -4.83076 q^{31} +7.70265 q^{33} -0.527166 q^{35} +9.74378 q^{37} -10.1755 q^{39} +6.93246 q^{41} +4.45963 q^{43} +3.11491 q^{45} -0.642074 q^{47} -6.72210 q^{49} +10.8781 q^{51} -3.89134 q^{53} -3.11491 q^{55} -9.15604 q^{57} -8.79811 q^{59} -3.45339 q^{61} -1.64207 q^{63} +4.11491 q^{65} +8.60719 q^{67} -2.47283 q^{69} -12.3642 q^{71} -5.81756 q^{73} -2.47283 q^{75} +1.64207 q^{77} +3.17548 q^{79} -8.64207 q^{81} -4.71585 q^{83} -4.39905 q^{85} +22.5070 q^{87} -5.43171 q^{89} -2.16924 q^{91} +11.9457 q^{93} +3.70265 q^{95} -4.06058 q^{97} -9.70265 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} + 3 q^{5} - 7 q^{7} + 3 q^{9} - 3 q^{11} + 6 q^{13} - 2 q^{15} - 5 q^{17} - 7 q^{19} - 6 q^{21} + 3 q^{23} + 3 q^{25} + q^{27} - q^{29} - 10 q^{31} + 5 q^{33} - 7 q^{35} + 2 q^{37} - 7 q^{39}+ \cdots - 11 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 + 3 * q^5 - 7 * q^7 + 3 * q^9 - 3 * q^11 + 6 * q^13 - 2 * q^15 - 5 * q^17 - 7 * q^19 - 6 * q^21 + 3 * q^23 + 3 * q^25 + q^27 - q^29 - 10 * q^31 + 5 * q^33 - 7 * q^35 + 2 * q^37 - 7 * q^39 - 10 * q^41 - 12 * q^43 + 3 * q^45 - q^47 + 6 * q^49 - 9 * q^51 + 10 * q^53 - 3 * q^55 - 12 * q^57 - 10 * q^59 - 13 * q^61 - 4 * q^63 + 6 * q^65 + 6 * q^67 - 2 * q^69 - 10 * q^71 + 7 * q^73 - 2 * q^75 + 4 * q^77 - 14 * q^79 - 25 * q^81 - 16 * q^83 - 5 * q^85 + 5 * q^87 - 20 * q^89 - 11 * q^91 + 25 * q^93 - 7 * q^95 + 5 * q^97 - 11 * 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
$3$	−2.47283	−1.42769	−0.713846	−	0.700303i	$-0.753048\pi$
$3$	−2.47283	−1.42769	−0.713846	+	0.700303i	$0.753048\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.527166	−0.199250	−0.0996250	−	0.995025i	$-0.531764\pi$
$7$	−0.527166	−0.199250	−0.0996250	+	0.995025i	$0.531764\pi$
$8$	0	0
$9$	3.11491	1.03830
$10$	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
$15$	−2.47283	−0.638483
$16$	0	0
$17$	−4.39905	−1.06693	−0.533464	−	0.845823i	$-0.679110\pi$
$17$	−4.39905	−1.06693	−0.533464	+	0.845823i	$0.679110\pi$
$18$	0	0
$19$	3.70265	0.849446	0.424723	−	0.905323i	$-0.360372\pi$
$19$	3.70265	0.849446	0.424723	+	0.905323i	$0.360372\pi$
$20$	0	0
$21$	1.30359	0.284468
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.284147	−0.0546842
$28$	0	0
$29$	−9.10170	−1.69014	−0.845072	−	0.534653i	$-0.820442\pi$
$29$	−9.10170	−1.69014	−0.845072	+	0.534653i	$0.820442\pi$
$30$	0	0
$31$	−4.83076	−0.867630	−0.433815	−	0.901002i	$-0.642833\pi$
$31$	−4.83076	−0.867630	−0.433815	+	0.901002i	$0.642833\pi$
$32$	0	0
$33$	7.70265	1.34086
$34$	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
$39$	−10.1755	−1.62938
$40$	0	0
$41$	6.93246	1.08267	0.541334	−	0.840807i	$-0.317919\pi$
$41$	6.93246	1.08267	0.541334	+	0.840807i	$0.317919\pi$
$42$	0	0
$43$	4.45963	0.680087	0.340044	−	0.940410i	$-0.389558\pi$
$43$	4.45963	0.680087	0.340044	+	0.940410i	$0.389558\pi$
$44$	0	0
$45$	3.11491	0.464343
$46$	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
$51$	10.8781	1.52324
$52$	0	0
$53$	−3.89134	−0.534516	−0.267258	−	0.963625i	$-0.586118\pi$
$53$	−3.89134	−0.534516	−0.267258	+	0.963625i	$0.586118\pi$
$54$	0	0
$55$	−3.11491	−0.420014
$56$	0	0
$57$	−9.15604	−1.21275
$58$	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
$63$	−1.64207	−0.206882
$64$	0	0
$65$	4.11491	0.510391
$66$	0	0
$67$	8.60719	1.05154	0.525768	−	0.850628i	$-0.323778\pi$
$67$	8.60719	1.05154	0.525768	+	0.850628i	$0.323778\pi$
$68$	0	0
$69$	−2.47283	−0.297694
$70$	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
$75$	−2.47283	−0.285538
$76$	0	0
$77$	1.64207	0.187132
$78$	0	0
$79$	3.17548	0.357270	0.178635	−	0.983915i	$-0.442832\pi$
$79$	3.17548	0.357270	0.178635	+	0.983915i	$0.442832\pi$
$80$	0	0
$81$	−8.64207	−0.960230
$82$	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
$87$	22.5070	2.41300
$88$	0	0
$89$	−5.43171	−0.575760	−0.287880	−	0.957667i	$-0.592950\pi$
$89$	−5.43171	−0.575760	−0.287880	+	0.957667i	$0.592950\pi$
$90$	0	0
$91$	−2.16924	−0.227398
$92$	0	0
$93$	11.9457	1.23871
$94$	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
$99$	−9.70265	−0.975153
$100$	0	0
$101$	−15.5529	−1.54757	−0.773784	−	0.633450i	$-0.781638\pi$
$101$	−15.5529	−1.54757	−0.773784	+	0.633450i	$0.781638\pi$
$102$	0	0
$103$	−17.9519	−1.76885	−0.884427	−	0.466678i	$-0.845451\pi$
$103$	−17.9519	−1.76885	−0.884427	+	0.466678i	$0.845451\pi$
$104$	0	0
$105$	1.30359	0.127218
$106$	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
$111$	−24.0947	−2.28697
$112$	0	0
$113$	−13.1755	−1.23945	−0.619723	−	0.784821i	$-0.712755\pi$
$113$	−13.1755	−1.23945	−0.619723	+	0.784821i	$0.712755\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	12.8176	1.18498
$118$	0	0
$119$	2.31903	0.212585
$120$	0	0
$121$	−1.29735	−0.117941
$122$	0	0
$123$	−17.1428	−1.54572
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−19.5613	−1.73579	−0.867894	−	0.496750i	$-0.834527\pi$
$127$	−19.5613	−1.73579	−0.867894	+	0.496750i	$0.834527\pi$
$128$	0	0
$129$	−11.0279	−0.970955
$130$	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
$135$	−0.284147	−0.0244555
$136$	0	0
$137$	−11.1344	−0.951272	−0.475636	−	0.879642i	$-0.657782\pi$
$137$	−11.1344	−0.951272	−0.475636	+	0.879642i	$0.657782\pi$
$138$	0	0
$139$	6.49452	0.550858	0.275429	−	0.961321i	$-0.411180\pi$
$139$	6.49452	0.550858	0.275429	+	0.961321i	$0.411180\pi$
$140$	0	0
$141$	1.58774	0.133712
$142$	0	0
$143$	−12.8176	−1.07186
$144$	0	0
$145$	−9.10170	−0.755855
$146$	0	0
$147$	16.6226	1.37101
$148$	0	0
$149$	−5.47283	−0.448352	−0.224176	−	0.974549i	$-0.571969\pi$
$149$	−5.47283	−0.448352	−0.224176	+	0.974549i	$0.571969\pi$
$150$	0	0
$151$	0.0954606	0.00776848	0.00388424	−	0.999992i	$-0.498764\pi$
$151$	0.0954606	0.00776848	0.00388424	+	0.999992i	$0.498764\pi$
$152$	0	0
$153$	−13.7026	−1.10779
$154$	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
$159$	9.62263	0.763124
$160$	0	0
$161$	−0.527166	−0.0415465
$162$	0	0
$163$	−18.5745	−1.45487	−0.727435	−	0.686177i	$-0.759288\pi$
$163$	−18.5745	−1.45487	−0.727435	+	0.686177i	$0.759288\pi$
$164$	0	0
$165$	7.70265	0.599650
$166$	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
$171$	11.5334	0.881982
$172$	0	0
$173$	3.49228	0.265513	0.132757	−	0.991149i	$-0.457617\pi$
$173$	3.49228	0.265513	0.132757	+	0.991149i	$0.457617\pi$
$174$	0	0
$175$	−0.527166	−0.0398500
$176$	0	0
$177$	21.7563	1.63530
$178$	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
$183$	8.53965	0.631269
$184$	0	0
$185$	9.74378	0.716377
$186$	0	0
$187$	13.7026	1.00204
$188$	0	0
$189$	0.149793	0.0108958
$190$	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
$195$	−10.1755	−0.728681
$196$	0	0
$197$	0.850207	0.0605748	0.0302874	−	0.999541i	$-0.490358\pi$
$197$	0.850207	0.0605748	0.0302874	+	0.999541i	$0.490358\pi$
$198$	0	0
$199$	−8.56829	−0.607390	−0.303695	−	0.952769i	$-0.598220\pi$
$199$	−8.56829	−0.607390	−0.303695	+	0.952769i	$0.598220\pi$
$200$	0	0
$201$	−21.2841	−1.50127
$202$	0	0
$203$	4.79811	0.336761
$204$	0	0
$205$	6.93246	0.484184
$206$	0	0
$207$	3.11491	0.216501
$208$	0	0
$209$	−11.5334	−0.797783
$210$	0	0
$211$	−9.63511	−0.663309	−0.331654	−	0.943401i	$-0.607607\pi$
$211$	−9.63511	−0.663309	−0.331654	+	0.943401i	$0.607607\pi$
$212$	0	0
$213$	30.5745	2.09493
$214$	0	0
$215$	4.45963	0.304144
$216$	0	0
$217$	2.54661	0.172875
$218$	0	0
$219$	14.3859	0.972106
$220$	0	0
$221$	−18.1017	−1.21765
$222$	0	0
$223$	−9.43171	−0.631594	−0.315797	−	0.948827i	$-0.602272\pi$
$223$	−9.43171	−0.631594	−0.315797	+	0.948827i	$0.602272\pi$
$224$	0	0
$225$	3.11491	0.207661
$226$	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
$231$	−4.06058	−0.267166
$232$	0	0
$233$	−12.5334	−0.821091	−0.410545	−	0.911840i	$-0.634662\pi$
$233$	−12.5334	−0.821091	−0.410545	+	0.911840i	$0.634662\pi$
$234$	0	0
$235$	−0.642074	−0.0418843
$236$	0	0
$237$	−7.85244	−0.510071
$238$	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
$243$	22.2229	1.42560
$244$	0	0
$245$	−6.72210	−0.429459
$246$	0	0
$247$	15.2361	0.969447
$248$	0	0
$249$	11.6615	0.739019
$250$	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
$255$	10.8781	0.681215
$256$	0	0
$257$	22.2772	1.38961	0.694806	−	0.719197i	$-0.255490\pi$
$257$	22.2772	1.38961	0.694806	+	0.719197i	$0.255490\pi$
$258$	0	0
$259$	−5.13659	−0.319172
$260$	0	0
$261$	−28.3510	−1.75488
$262$	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
$267$	13.4317	0.822007
$268$	0	0
$269$	9.70889	0.591962	0.295981	−	0.955194i	$-0.404354\pi$
$269$	9.70889	0.591962	0.295981	+	0.955194i	$0.404354\pi$
$270$	0	0
$271$	6.75698	0.410457	0.205229	−	0.978714i	$-0.434206\pi$
$271$	6.75698	0.410457	0.205229	+	0.978714i	$0.434206\pi$
$272$	0	0
$273$	5.36417	0.324654
$274$	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
$279$	−15.0474	−0.900863
$280$	0	0
$281$	−32.8370	−1.95889	−0.979446	−	0.201708i	$-0.935351\pi$
$281$	−32.8370	−1.95889	−0.979446	+	0.201708i	$0.935351\pi$
$282$	0	0
$283$	−13.7827	−0.819295	−0.409647	−	0.912244i	$-0.634348\pi$
$283$	−13.7827	−0.819295	−0.409647	+	0.912244i	$0.634348\pi$
$284$	0	0
$285$	−9.15604	−0.542357
$286$	0	0
$287$	−3.65456	−0.215722
$288$	0	0
$289$	2.35168	0.138334
$290$	0	0
$291$	10.0411	0.588621
$292$	0	0
$293$	15.0668	0.880213	0.440106	−	0.897946i	$-0.354941\pi$
$293$	15.0668	0.880213	0.440106	+	0.897946i	$0.354941\pi$
$294$	0	0
$295$	−8.79811	−0.512246
$296$	0	0
$297$	0.885092	0.0513583
$298$	0	0
$299$	4.11491	0.237971
$300$	0	0
$301$	−2.35097	−0.135507
$302$	0	0
$303$	38.4596	2.20945
$304$	0	0
$305$	−3.45339	−0.197740
$306$	0	0
$307$	5.91302	0.337474	0.168737	−	0.985661i	$-0.446031\pi$
$307$	5.91302	0.337474	0.168737	+	0.985661i	$0.446031\pi$
$308$	0	0
$309$	44.3921	2.52538
$310$	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
$315$	−1.64207	−0.0925204
$316$	0	0
$317$	28.1623	1.58175	0.790876	−	0.611977i	$-0.209625\pi$
$317$	28.1623	1.58175	0.790876	+	0.611977i	$0.209625\pi$
$318$	0	0
$319$	28.3510	1.58735
$320$	0	0
$321$	−30.2423	−1.68796
$322$	0	0
$323$	−16.2882	−0.906297
$324$	0	0
$325$	4.11491	0.228254
$326$	0	0
$327$	−22.5224	−1.24549
$328$	0	0
$329$	0.338479	0.0186610
$330$	0	0
$331$	25.8176	1.41906	0.709531	−	0.704675i	$-0.248907\pi$
$331$	25.8176	1.41906	0.709531	+	0.704675i	$0.248907\pi$
$332$	0	0
$333$	30.3510	1.66322
$334$	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
$339$	32.5808	1.76955
$340$	0	0
$341$	15.0474	0.814861
$342$	0	0
$343$	7.23382	0.390590
$344$	0	0
$345$	−2.47283	−0.133133
$346$	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
$351$	−1.16924	−0.0624094
$352$	0	0
$353$	−25.1840	−1.34041	−0.670203	−	0.742177i	$-0.733793\pi$
$353$	−25.1840	−1.34041	−0.670203	+	0.742177i	$0.733793\pi$
$354$	0	0
$355$	−12.3642	−0.656222
$356$	0	0
$357$	−5.73458	−0.303506
$358$	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
$363$	3.20813	0.168383
$364$	0	0
$365$	−5.81756	−0.304505
$366$	0	0
$367$	−8.24230	−0.430245	−0.215122	−	0.976587i	$-0.569015\pi$
$367$	−8.24230	−0.430245	−0.215122	+	0.976587i	$0.569015\pi$
$368$	0	0
$369$	21.5940	1.12414
$370$	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
$375$	−2.47283	−0.127697
$376$	0	0
$377$	−37.4527	−1.92891
$378$	0	0
$379$	−30.1692	−1.54969	−0.774845	−	0.632151i	$-0.782172\pi$
$379$	−30.1692	−1.54969	−0.774845	+	0.632151i	$0.782172\pi$
$380$	0	0
$381$	48.3719	2.47817
$382$	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
$387$	13.8913	0.706136
$388$	0	0
$389$	−16.6414	−0.843750	−0.421875	−	0.906654i	$-0.638628\pi$
$389$	−16.6414	−0.843750	−0.421875	+	0.906654i	$0.638628\pi$
$390$	0	0
$391$	−4.39905	−0.222470
$392$	0	0
$393$	−10.2772	−0.518415
$394$	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
$399$	4.82675	0.241640
$400$	0	0
$401$	−5.62263	−0.280781	−0.140390	−	0.990096i	$-0.544836\pi$
$401$	−5.62263	−0.280781	−0.140390	+	0.990096i	$0.544836\pi$
$402$	0	0
$403$	−19.8781	−0.990200
$404$	0	0
$405$	−8.64207	−0.429428
$406$	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
$411$	27.5334	1.35812
$412$	0	0
$413$	4.63807	0.228224
$414$	0	0
$415$	−4.71585	−0.231492
$416$	0	0
$417$	−16.0599	−0.786455
$418$	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
$423$	−2.00000	−0.0972433
$424$	0	0
$425$	−4.39905	−0.213386
$426$	0	0
$427$	1.82051	0.0881006
$428$	0	0
$429$	31.6957	1.53028
$430$	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
$435$	22.5070	1.07913
$436$	0	0
$437$	3.70265	0.177122
$438$	0	0
$439$	−38.8168	−1.85263	−0.926313	−	0.376754i	$-0.877040\pi$
$439$	−38.8168	−1.85263	−0.926313	+	0.376754i	$0.877040\pi$
$440$	0	0
$441$	−20.9387	−0.997081
$442$	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
$447$	13.5334	0.640108
$448$	0	0
$449$	33.4589	1.57902	0.789512	−	0.613735i	$-0.210334\pi$
$449$	33.4589	1.57902	0.789512	+	0.613735i	$0.210334\pi$
$450$	0	0
$451$	−21.5940	−1.01682
$452$	0	0
$453$	−0.236058	−0.0110910
$454$	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
$459$	1.24998	0.0583440
$460$	0	0
$461$	36.3161	1.69141	0.845704	−	0.533652i	$-0.179181\pi$
$461$	36.3161	1.69141	0.845704	+	0.533652i	$0.179181\pi$
$462$	0	0
$463$	22.5544	1.04819	0.524095	−	0.851660i	$-0.324404\pi$
$463$	22.5544	1.04819	0.524095	+	0.851660i	$0.324404\pi$
$464$	0	0
$465$	11.9457	0.553967
$466$	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
$471$	8.42074	0.388007
$472$	0	0
$473$	−13.8913	−0.638724
$474$	0	0
$475$	3.70265	0.169889
$476$	0	0
$477$	−12.1212	−0.554989
$478$	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
$483$	1.30359	0.0593156
$484$	0	0
$485$	−4.06058	−0.184381
$486$	0	0
$487$	10.4945	0.475552	0.237776	−	0.971320i	$-0.423582\pi$
$487$	10.4945	0.475552	0.237776	+	0.971320i	$0.423582\pi$
$488$	0	0
$489$	45.9317	2.07711
$490$	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
$495$	−9.70265	−0.436102
$496$	0	0
$497$	6.51797	0.292371
$498$	0	0
$499$	−30.1560	−1.34997	−0.674985	−	0.737832i	$-0.735850\pi$
$499$	−30.1560	−1.34997	−0.674985	+	0.737832i	$0.735850\pi$
$500$	0	0
$501$	−30.8106	−1.37652
$502$	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
$507$	−9.72433	−0.431873
$508$	0	0
$509$	11.8176	0.523804	0.261902	−	0.965094i	$-0.415650\pi$
$509$	11.8176	0.523804	0.261902	+	0.965094i	$0.415650\pi$
$510$	0	0
$511$	3.06682	0.135668
$512$	0	0
$513$	−1.05210	−0.0464512
$514$	0	0
$515$	−17.9519	−0.791056
$516$	0	0
$517$	2.00000	0.0879599
$518$	0	0
$519$	−8.63583	−0.379071
$520$	0	0
$521$	−35.7438	−1.56596	−0.782982	−	0.622045i	$-0.786302\pi$
$521$	−35.7438	−1.56596	−0.782982	+	0.622045i	$0.786302\pi$
$522$	0	0
$523$	−14.3899	−0.629225	−0.314612	−	0.949220i	$-0.601875\pi$
$523$	−14.3899	−0.629225	−0.314612	+	0.949220i	$0.601875\pi$
$524$	0	0
$525$	1.30359	0.0568935
$526$	0	0
$527$	21.2508	0.925698
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−27.4053	−1.18929
$532$	0	0
$533$	28.5264	1.23562
$534$	0	0
$535$	12.2298	0.528741
$536$	0	0
$537$	15.7911	0.681438
$538$	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
$543$	51.6282	2.21558
$544$	0	0
$545$	9.10795	0.390142
$546$	0	0
$547$	−19.6289	−0.839270	−0.419635	−	0.907693i	$-0.637842\pi$
$547$	−19.6289	−0.839270	−0.419635	+	0.907693i	$0.637842\pi$
$548$	0	0
$549$	−10.7570	−0.459097
$550$	0	0
$551$	−33.7004	−1.43569
$552$	0	0
$553$	−1.67401	−0.0711860
$554$	0	0
$555$	−24.0947	−1.02276
$556$	0	0
$557$	1.13659	0.0481588	0.0240794	−	0.999710i	$-0.492335\pi$
$557$	1.13659	0.0481588	0.0240794	+	0.999710i	$0.492335\pi$
$558$	0	0
$559$	18.3510	0.776163
$560$	0	0
$561$	−33.8844	−1.43060
$562$	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
$567$	4.55581	0.191326
$568$	0	0
$569$	−44.0558	−1.84692	−0.923459	−	0.383698i	$-0.874650\pi$
$569$	−44.0558	−1.84692	−0.923459	+	0.383698i	$0.874650\pi$
$570$	0	0
$571$	41.8991	1.75342	0.876711	−	0.481017i	$-0.159732\pi$
$571$	41.8991	1.75342	0.876711	+	0.481017i	$0.159732\pi$
$572$	0	0
$573$	−16.9068	−0.706291
$574$	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
$579$	−17.5613	−0.729824
$580$	0	0
$581$	2.48604	0.103138
$582$	0	0
$583$	12.1212	0.502007
$584$	0	0
$585$	12.8176	0.529941
$586$	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
$591$	−2.10242	−0.0864821
$592$	0	0
$593$	−22.5947	−0.927853	−0.463927	−	0.885874i	$-0.653560\pi$
$593$	−22.5947	−0.927853	−0.463927	+	0.885874i	$0.653560\pi$
$594$	0	0
$595$	2.31903	0.0950711
$596$	0	0
$597$	21.1880	0.867166
$598$	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
$603$	26.8106	1.09181
$604$	0	0
$605$	−1.29735	−0.0527448
$606$	0	0
$607$	20.7547	0.842409	0.421205	−	0.906966i	$-0.361607\pi$
$607$	20.7547	0.842409	0.421205	+	0.906966i	$0.361607\pi$
$608$	0	0
$609$	−11.8649	−0.480791
$610$	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
$615$	−17.1428	−0.691266
$616$	0	0
$617$	13.5621	0.545988	0.272994	−	0.962016i	$-0.411986\pi$
$617$	13.5621	0.545988	0.272994	+	0.962016i	$0.411986\pi$
$618$	0	0
$619$	12.9045	0.518677	0.259339	−	0.965786i	$-0.416495\pi$
$619$	12.9045	0.518677	0.259339	+	0.965786i	$0.416495\pi$
$620$	0	0
$621$	−0.284147	−0.0114024
$622$	0	0
$623$	2.86341	0.114720
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	28.5202	1.13899
$628$	0	0
$629$	−42.8634	−1.70908
$630$	0	0
$631$	−45.0404	−1.79303	−0.896515	−	0.443013i	$-0.853910\pi$
$631$	−45.0404	−1.79303	−0.896515	+	0.443013i	$0.853910\pi$
$632$	0	0
$633$	23.8260	0.947000
$634$	0	0
$635$	−19.5613	−0.776268
$636$	0	0
$637$	−27.6608	−1.09596
$638$	0	0
$639$	−38.5132	−1.52356
$640$	0	0
$641$	−3.08074	−0.121682	−0.0608410	−	0.998147i	$-0.519378\pi$
$641$	−3.08074	−0.121682	−0.0608410	+	0.998147i	$0.519378\pi$
$642$	0	0
$643$	−4.52493	−0.178446	−0.0892229	−	0.996012i	$-0.528438\pi$
$643$	−4.52493	−0.178446	−0.0892229	+	0.996012i	$0.528438\pi$
$644$	0	0
$645$	−11.0279	−0.434224
$646$	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
$651$	−6.29735	−0.246813
$652$	0	0
$653$	−35.3726	−1.38424	−0.692119	−	0.721783i	$-0.743323\pi$
$653$	−35.3726	−1.38424	−0.692119	+	0.721783i	$0.743323\pi$
$654$	0	0
$655$	4.15604	0.162390
$656$	0	0
$657$	−18.1212	−0.706973
$658$	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
$663$	44.7625	1.73843
$664$	0	0
$665$	−1.95191	−0.0756919
$666$	0	0
$667$	−9.10170	−0.352419
$668$	0	0
$669$	23.3230	0.901721
$670$	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
$675$	−0.284147	−0.0109368
$676$	0	0
$677$	−13.4876	−0.518369	−0.259184	−	0.965828i	$-0.583454\pi$
$677$	−13.4876	−0.518369	−0.259184	+	0.965828i	$0.583454\pi$
$678$	0	0
$679$	2.14060	0.0821486
$680$	0	0
$681$	−9.68793	−0.371242
$682$	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
$687$	54.6366	2.08452
$688$	0	0
$689$	−16.0125	−0.610027
$690$	0	0
$691$	−22.6506	−0.861668	−0.430834	−	0.902431i	$-0.641780\pi$
$691$	−22.6506	−0.861668	−0.430834	+	0.902431i	$0.641780\pi$
$692$	0	0
$693$	5.11491	0.194299
$694$	0	0
$695$	6.49452	0.246351
$696$	0	0
$697$	−30.4963	−1.15513
$698$	0	0
$699$	30.9930	1.17226
$700$	0	0
$701$	−41.8099	−1.57914	−0.789569	−	0.613662i	$-0.789696\pi$
$701$	−41.8099	−1.57914	−0.789569	+	0.613662i	$0.789696\pi$
$702$	0	0
$703$	36.0778	1.36070
$704$	0	0
$705$	1.58774	0.0597978
$706$	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
$711$	9.89134	0.370954
$712$	0	0
$713$	−4.83076	−0.180913
$714$	0	0
$715$	−12.8176	−0.479349
$716$	0	0
$717$	−30.6935	−1.14627
$718$	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
$723$	−14.2034	−0.528230
$724$	0	0
$725$	−9.10170	−0.338029
$726$	0	0
$727$	26.8198	0.994691	0.497345	−	0.867553i	$-0.334308\pi$
$727$	26.8198	0.994691	0.497345	+	0.867553i	$0.334308\pi$
$728$	0	0
$729$	−29.0272	−1.07508
$730$	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
$735$	16.6226	0.613135
$736$	0	0
$737$	−26.8106	−0.987581
$738$	0	0
$739$	−5.84396	−0.214974	−0.107487	−	0.994207i	$-0.534280\pi$
$739$	−5.84396	−0.214974	−0.107487	+	0.994207i	$0.534280\pi$
$740$	0	0
$741$	−37.6762	−1.38407
$742$	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
$747$	−14.6894	−0.537459
$748$	0	0
$749$	−6.44714	−0.235574
$750$	0	0
$751$	−33.0404	−1.20566	−0.602831	−	0.797869i	$-0.705961\pi$
$751$	−33.0404	−1.20566	−0.602831	+	0.797869i	$0.705961\pi$
$752$	0	0
$753$	−15.6685	−0.570991
$754$	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
$759$	7.70265	0.279588
$760$	0	0
$761$	34.1010	1.23616	0.618080	−	0.786115i	$-0.287911\pi$
$761$	34.1010	1.23616	0.618080	+	0.786115i	$0.287911\pi$
$762$	0	0
$763$	−4.80140	−0.173822
$764$	0	0
$765$	−13.7026	−0.495420
$766$	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
$771$	−55.0878	−1.98394
$772$	0	0
$773$	1.13659	0.0408803	0.0204401	−	0.999791i	$-0.493493\pi$
$773$	1.13659	0.0408803	0.0204401	+	0.999791i	$0.493493\pi$
$774$	0	0
$775$	−4.83076	−0.173526
$776$	0	0
$777$	12.7019	0.455679
$778$	0	0
$779$	25.6685	0.919669
$780$	0	0
$781$	38.5132	1.37811
$782$	0	0
$783$	2.58622	0.0924241
$784$	0	0
$785$	−3.40530	−0.121540
$786$	0	0
$787$	8.33848	0.297235	0.148617	−	0.988895i	$-0.452518\pi$
$787$	8.33848	0.297235	0.148617	+	0.988895i	$0.452518\pi$
$788$	0	0
$789$	−11.5426	−0.410928
$790$	0	0
$791$	6.94567	0.246960
$792$	0	0
$793$	−14.2104	−0.504625
$794$	0	0
$795$	9.62263	0.341279
$796$	0	0
$797$	−0.190921	−0.00676278	−0.00338139	−	0.999994i	$-0.501076\pi$
$797$	−0.190921	−0.00676278	−0.00338139	+	0.999994i	$0.501076\pi$
$798$	0	0
$799$	2.82452	0.0999242
$800$	0	0
$801$	−16.9193	−0.597813
$802$	0	0
$803$	18.1212	0.639482
$804$	0	0
$805$	−0.527166	−0.0185802
$806$	0	0
$807$	−24.0085	−0.845138
$808$	0	0
$809$	−14.5202	−0.510503	−0.255252	−	0.966875i	$-0.582158\pi$
$809$	−14.5202	−0.510503	−0.255252	+	0.966875i	$0.582158\pi$
$810$	0	0
$811$	51.5474	1.81007	0.905037	−	0.425332i	$-0.139843\pi$
$811$	51.5474	1.81007	0.905037	+	0.425332i	$0.139843\pi$
$812$	0	0
$813$	−16.7089	−0.586006
$814$	0	0
$815$	−18.5745	−0.650638
$816$	0	0
$817$	16.5124	0.577697
$818$	0	0
$819$	−6.75698	−0.236108
$820$	0	0
$821$	13.7827	0.481019	0.240509	−	0.970647i	$-0.422686\pi$
$821$	13.7827	0.481019	0.240509	+	0.970647i	$0.422686\pi$
$822$	0	0
$823$	37.1531	1.29508	0.647538	−	0.762034i	$-0.275799\pi$
$823$	37.1531	1.29508	0.647538	+	0.762034i	$0.275799\pi$
$824$	0	0
$825$	7.70265	0.268172
$826$	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
$831$	−47.0319	−1.63152
$832$	0	0
$833$	29.5709	1.02457
$834$	0	0
$835$	12.4596	0.431183
$836$	0	0
$837$	1.37265	0.0474456
$838$	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
$843$	81.2005	2.79669
$844$	0	0
$845$	3.93246	0.135281
$846$	0	0
$847$	0.683919	0.0234998
$848$	0	0
$849$	34.0823	1.16970
$850$	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
$855$	11.5334	0.394434
$856$	0	0
$857$	54.6017	1.86516	0.932580	−	0.360963i	$-0.117552\pi$
$857$	54.6017	1.86516	0.932580	+	0.360963i	$0.117552\pi$
$858$	0	0
$859$	32.9541	1.12438	0.562190	−	0.827008i	$-0.309959\pi$
$859$	32.9541	1.12438	0.562190	+	0.827008i	$0.309959\pi$
$860$	0	0
$861$	9.03712	0.307984
$862$	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
$867$	−5.81532	−0.197499
$868$	0	0
$869$	−9.89134	−0.335541
$870$	0	0
$871$	35.4178	1.20009
$872$	0	0
$873$	−12.6483	−0.428081
$874$	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
$879$	−37.2577	−1.25667
$880$	0	0
$881$	−31.7049	−1.06816	−0.534082	−	0.845432i	$-0.679343\pi$
$881$	−31.7049	−1.06816	−0.534082	+	0.845432i	$0.679343\pi$
$882$	0	0
$883$	−51.9185	−1.74720	−0.873599	−	0.486646i	$-0.838220\pi$
$883$	−51.9185	−1.74720	−0.873599	+	0.486646i	$0.838220\pi$
$884$	0	0
$885$	21.7563	0.731329
$886$	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
$891$	26.9193	0.901829
$892$	0	0
$893$	−2.37737	−0.0795558
$894$	0	0
$895$	−6.38585	−0.213455
$896$	0	0
$897$	−10.1755	−0.339749
$898$	0	0
$899$	43.9681	1.46642
$900$	0	0
$901$	17.1182	0.570290
$902$	0	0
$903$	5.81355	0.193463
$904$	0	0
$905$	−20.8781	−0.694013
$906$	0	0
$907$	25.4442	0.844861	0.422430	−	0.906395i	$-0.361177\pi$
$907$	25.4442	0.844861	0.422430	+	0.906395i	$0.361177\pi$
$908$	0	0
$909$	−48.4457	−1.60684
$910$	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
$915$	8.53965	0.282312
$916$	0	0
$917$	−2.19092	−0.0723506
$918$	0	0
$919$	−2.78562	−0.0918892	−0.0459446	−	0.998944i	$-0.514630\pi$
$919$	−2.78562	−0.0918892	−0.0459446	+	0.998944i	$0.514630\pi$
$920$	0	0
$921$	−14.6219	−0.481808
$922$	0	0
$923$	−50.8774	−1.67465
$924$	0	0
$925$	9.74378	0.320373
$926$	0	0
$927$	−55.9185	−1.83661
$928$	0	0
$929$	40.2508	1.32059	0.660293	−	0.751008i	$-0.270432\pi$
$929$	40.2508	1.32059	0.660293	+	0.751008i	$0.270432\pi$
$930$	0	0
$931$	−24.8896	−0.815722
$932$	0	0
$933$	−63.7772	−2.08797
$934$	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
$939$	−41.3719	−1.35012
$940$	0	0
$941$	38.3682	1.25077	0.625383	−	0.780318i	$-0.284943\pi$
$941$	38.3682	1.25077	0.625383	+	0.780318i	$0.284943\pi$
$942$	0	0
$943$	6.93246	0.225752
$944$	0	0
$945$	0.149793	0.00487276
$946$	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
$951$	−69.6406	−2.25825
$952$	0	0
$953$	−12.1303	−0.392940	−0.196470	−	0.980510i	$-0.562948\pi$
$953$	−12.1303	−0.392940	−0.196470	+	0.980510i	$0.562948\pi$
$954$	0	0
$955$	6.83700	0.221240
$956$	0	0
$957$	−70.1072	−2.26624
$958$	0	0
$959$	5.86965	0.189541
$960$	0	0
$961$	−7.66376	−0.247218
$962$	0	0
$963$	38.0947	1.22759
$964$	0	0
$965$	7.10170	0.228612
$966$	0	0
$967$	−21.2882	−0.684581	−0.342290	−	0.939594i	$-0.611203\pi$
$967$	−21.2882	−0.684581	−0.342290	+	0.939594i	$0.611203\pi$
$968$	0	0
$969$	40.2779	1.29391
$970$	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
$975$	−10.1755	−0.325876
$976$	0	0
$977$	41.2849	1.32082	0.660411	−	0.750904i	$-0.270382\pi$
$977$	41.2849	1.32082	0.660411	+	0.750904i	$0.270382\pi$
$978$	0	0
$979$	16.9193	0.540742
$980$	0	0
$981$	28.3704	0.905798
$982$	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
$987$	−0.837003	−0.0266421
$988$	0	0
$989$	4.45963	0.141808
$990$	0	0
$991$	−8.73307	−0.277415	−0.138707	−	0.990333i	$-0.544295\pi$
$991$	−8.73307	−0.277415	−0.138707	+	0.990333i	$0.544295\pi$
$992$	0	0
$993$	−63.8425	−2.02598
$994$	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
$999$	−2.76867	−0.0875968

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.a.q.1.1		3
4.3	odd	2		920.2.a.i.1.3	✓	3
5.4	even	2		9200.2.a.ci.1.3		3
8.3	odd	2		7360.2.a.bw.1.1		3
8.5	even	2		7360.2.a.cf.1.3		3
12.11	even	2		8280.2.a.bl.1.1		3
20.3	even	4		4600.2.e.q.4049.6		6
20.7	even	4		4600.2.e.q.4049.1		6
20.19	odd	2		4600.2.a.v.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.i.1.3	✓	3	4.3	odd	2
1840.2.a.q.1.1		3	1.1	even	1	trivial
4600.2.a.v.1.1		3	20.19	odd	2
4600.2.e.q.4049.1		6	20.7	even	4
4600.2.e.q.4049.6		6	20.3	even	4
7360.2.a.bw.1.1		3	8.3	odd	2
7360.2.a.cf.1.3		3	8.5	even	2
8280.2.a.bl.1.1		3	12.11	even	2
9200.2.a.ci.1.3		3	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

\(f(q)\)	\(=\)	\(q-2.47283 q^{3} +1.00000 q^{5} -0.527166 q^{7} +3.11491 q^{9} -3.11491 q^{11} +4.11491 q^{13} -2.47283 q^{15} -4.39905 q^{17} +3.70265 q^{19} +1.30359 q^{21} +1.00000 q^{23} +1.00000 q^{25} -0.284147 q^{27} -9.10170 q^{29} -4.83076 q^{31} +7.70265 q^{33} -0.527166 q^{35} +9.74378 q^{37} -10.1755 q^{39} +6.93246 q^{41} +4.45963 q^{43} +3.11491 q^{45} -0.642074 q^{47} -6.72210 q^{49} +10.8781 q^{51} -3.89134 q^{53} -3.11491 q^{55} -9.15604 q^{57} -8.79811 q^{59} -3.45339 q^{61} -1.64207 q^{63} +4.11491 q^{65} +8.60719 q^{67} -2.47283 q^{69} -12.3642 q^{71} -5.81756 q^{73} -2.47283 q^{75} +1.64207 q^{77} +3.17548 q^{79} -8.64207 q^{81} -4.71585 q^{83} -4.39905 q^{85} +22.5070 q^{87} -5.43171 q^{89} -2.16924 q^{91} +11.9457 q^{93} +3.70265 q^{95} -4.06058 q^{97} -9.70265 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} + 3 q^{5} - 7 q^{7} + 3 q^{9} - 3 q^{11} + 6 q^{13} - 2 q^{15} - 5 q^{17} - 7 q^{19} - 6 q^{21} + 3 q^{23} + 3 q^{25} + q^{27} - q^{29} - 10 q^{31} + 5 q^{33} - 7 q^{35} + 2 q^{37} - 7 q^{39}+ \cdots - 11 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 + 3 * q^5 - 7 * q^7 + 3 * q^9 - 3 * q^11 + 6 * q^13 - 2 * q^15 - 5 * q^17 - 7 * q^19 - 6 * q^21 + 3 * q^23 + 3 * q^25 + q^27 - q^29 - 10 * q^31 + 5 * q^33 - 7 * q^35 + 2 * q^37 - 7 * q^39 - 10 * q^41 - 12 * q^43 + 3 * q^45 - q^47 + 6 * q^49 - 9 * q^51 + 10 * q^53 - 3 * q^55 - 12 * q^57 - 10 * q^59 - 13 * q^61 - 4 * q^63 + 6 * q^65 + 6 * q^67 - 2 * q^69 - 10 * q^71 + 7 * q^73 - 2 * q^75 + 4 * q^77 - 14 * q^79 - 25 * q^81 - 16 * q^83 - 5 * q^85 + 5 * q^87 - 20 * q^89 - 11 * q^91 + 25 * q^93 - 7 * q^95 + 5 * q^97 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more