LMFDB - Embedded newform 6040.2.a.s.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.49843 q^{3} +1.00000 q^{5} +1.73584 q^{7} -0.754697 q^{9} +O(q^{10})$	$q+1.49843 q^{3} +1.00000 q^{5} +1.73584 q^{7} -0.754697 q^{9} -6.27719 q^{11} -2.16376 q^{13} +1.49843 q^{15} +2.92391 q^{17} +5.08623 q^{19} +2.60104 q^{21} +9.07725 q^{23} +1.00000 q^{25} -5.62616 q^{27} +5.00134 q^{29} +1.20851 q^{31} -9.40596 q^{33} +1.73584 q^{35} +4.24830 q^{37} -3.24224 q^{39} -7.65235 q^{41} -5.21112 q^{43} -0.754697 q^{45} +5.34482 q^{47} -3.98686 q^{49} +4.38129 q^{51} +14.4133 q^{53} -6.27719 q^{55} +7.62137 q^{57} +8.09743 q^{59} +0.554623 q^{61} -1.31003 q^{63} -2.16376 q^{65} +5.27476 q^{67} +13.6017 q^{69} -8.57612 q^{71} +5.56709 q^{73} +1.49843 q^{75} -10.8962 q^{77} +4.31006 q^{79} -6.16634 q^{81} -4.84970 q^{83} +2.92391 q^{85} +7.49417 q^{87} -16.3618 q^{89} -3.75593 q^{91} +1.81087 q^{93} +5.08623 q^{95} +6.12811 q^{97} +4.73738 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9}+O(q^{10}) $ 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9	$ 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9} + 17 q^{11} + 16 q^{13} + 2 q^{15} + 22 q^{17} + 16 q^{19} - q^{21} + 7 q^{23} + 24 q^{25} - 4 q^{27} + 25 q^{29} + 28 q^{31} + 11 q^{33} + 3 q^{35} + 26 q^{37} + 13 q^{39} + 38 q^{41} - 13 q^{43} + 40 q^{45} + 12 q^{47} + 61 q^{49} + 53 q^{53} + 17 q^{55} + 30 q^{57} + 35 q^{59} + 44 q^{61} - 9 q^{63} + 16 q^{65} - 15 q^{67} + 9 q^{69} + 22 q^{71} + 31 q^{73} + 2 q^{75} + 26 q^{77} + 20 q^{79} + 88 q^{81} - 14 q^{83} + 22 q^{85} - 18 q^{87} + 37 q^{89} - 26 q^{91} + 13 q^{93} + 16 q^{95} + 21 q^{97} - 12 q^{99}+O(q^{100}) $ 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9 + 17 * q^11 + 16 * q^13 + 2 * q^15 + 22 * q^17 + 16 * q^19 - q^21 + 7 * q^23 + 24 * q^25 - 4 * q^27 + 25 * q^29 + 28 * q^31 + 11 * q^33 + 3 * q^35 + 26 * q^37 + 13 * q^39 + 38 * q^41 - 13 * q^43 + 40 * q^45 + 12 * q^47 + 61 * q^49 + 53 * q^53 + 17 * q^55 + 30 * q^57 + 35 * q^59 + 44 * q^61 - 9 * q^63 + 16 * q^65 - 15 * q^67 + 9 * q^69 + 22 * q^71 + 31 * q^73 + 2 * q^75 + 26 * q^77 + 20 * q^79 + 88 * q^81 - 14 * q^83 + 22 * q^85 - 18 * q^87 + 37 * q^89 - 26 * q^91 + 13 * q^93 + 16 * q^95 + 21 * q^97 - 12 * q^99

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
$3$	1.49843	0.865121	0.432561	−	0.901605i	$-0.357610\pi$
$3$	1.49843	0.865121	0.432561	+	0.901605i	$0.357610\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.73584	0.656086	0.328043	−	0.944663i	$-0.393611\pi$
$7$	1.73584	0.656086	0.328043	+	0.944663i	$0.393611\pi$
$8$	0	0
$9$	−0.754697	−0.251566
$10$	0	0
$11$	−6.27719	−1.89265	−0.946323	−	0.323223i	$-0.895233\pi$
$11$	−6.27719	−1.89265	−0.946323	+	0.323223i	$0.895233\pi$
$12$	0	0
$13$	−2.16376	−0.600118	−0.300059	−	0.953921i	$-0.597006\pi$
$13$	−2.16376	−0.600118	−0.300059	+	0.953921i	$0.597006\pi$
$14$	0	0
$15$	1.49843	0.386894
$16$	0	0
$17$	2.92391	0.709153	0.354577	−	0.935027i	$-0.384625\pi$
$17$	2.92391	0.709153	0.354577	+	0.935027i	$0.384625\pi$
$18$	0	0
$19$	5.08623	1.16686	0.583430	−	0.812163i	$-0.301710\pi$
$19$	5.08623	1.16686	0.583430	+	0.812163i	$0.301710\pi$
$20$	0	0
$21$	2.60104	0.567593
$22$	0	0
$23$	9.07725	1.89274	0.946368	−	0.323090i	$-0.104722\pi$
$23$	9.07725	1.89274	0.946368	+	0.323090i	$0.104722\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.62616	−1.08276
$28$	0	0
$29$	5.00134	0.928725	0.464362	−	0.885645i	$-0.346283\pi$
$29$	5.00134	0.928725	0.464362	+	0.885645i	$0.346283\pi$
$30$	0	0
$31$	1.20851	0.217054	0.108527	−	0.994093i	$-0.465387\pi$
$31$	1.20851	0.217054	0.108527	+	0.994093i	$0.465387\pi$
$32$	0	0
$33$	−9.40596	−1.63737
$34$	0	0
$35$	1.73584	0.293410
$36$	0	0
$37$	4.24830	0.698417	0.349208	−	0.937045i	$-0.386451\pi$
$37$	4.24830	0.698417	0.349208	+	0.937045i	$0.386451\pi$
$38$	0	0
$39$	−3.24224	−0.519175
$40$	0	0
$41$	−7.65235	−1.19510	−0.597548	−	0.801833i	$-0.703858\pi$
$41$	−7.65235	−1.19510	−0.597548	+	0.801833i	$0.703858\pi$
$42$	0	0
$43$	−5.21112	−0.794688	−0.397344	−	0.917670i	$-0.630068\pi$
$43$	−5.21112	−0.794688	−0.397344	+	0.917670i	$0.630068\pi$
$44$	0	0
$45$	−0.754697	−0.112504
$46$	0	0
$47$	5.34482	0.779622	0.389811	−	0.920895i	$-0.372540\pi$
$47$	5.34482	0.779622	0.389811	+	0.920895i	$0.372540\pi$
$48$	0	0
$49$	−3.98686	−0.569552
$50$	0	0
$51$	4.38129	0.613503
$52$	0	0
$53$	14.4133	1.97982	0.989912	−	0.141686i	$-0.0452523\pi$
$53$	14.4133	1.97982	0.989912	+	0.141686i	$0.0452523\pi$
$54$	0	0
$55$	−6.27719	−0.846417
$56$	0	0
$57$	7.62137	1.00948
$58$	0	0
$59$	8.09743	1.05420	0.527098	−	0.849805i	$-0.323280\pi$
$59$	8.09743	1.05420	0.527098	+	0.849805i	$0.323280\pi$
$60$	0	0
$61$	0.554623	0.0710122	0.0355061	−	0.999369i	$-0.488696\pi$
$61$	0.554623	0.0710122	0.0355061	+	0.999369i	$0.488696\pi$
$62$	0	0
$63$	−1.31003	−0.165049
$64$	0	0
$65$	−2.16376	−0.268381
$66$	0	0
$67$	5.27476	0.644414	0.322207	−	0.946669i	$-0.395575\pi$
$67$	5.27476	0.644414	0.322207	+	0.946669i	$0.395575\pi$
$68$	0	0
$69$	13.6017	1.63745
$70$	0	0
$71$	−8.57612	−1.01780	−0.508899	−	0.860826i	$-0.669947\pi$
$71$	−8.57612	−1.01780	−0.508899	+	0.860826i	$0.669947\pi$
$72$	0	0
$73$	5.56709	0.651579	0.325789	−	0.945442i	$-0.394370\pi$
$73$	5.56709	0.651579	0.325789	+	0.945442i	$0.394370\pi$
$74$	0	0
$75$	1.49843	0.173024
$76$	0	0
$77$	−10.8962	−1.24174
$78$	0	0
$79$	4.31006	0.484920	0.242460	−	0.970161i	$-0.422046\pi$
$79$	4.31006	0.484920	0.242460	+	0.970161i	$0.422046\pi$
$80$	0	0
$81$	−6.16634	−0.685149
$82$	0	0
$83$	−4.84970	−0.532324	−0.266162	−	0.963928i	$-0.585756\pi$
$83$	−4.84970	−0.532324	−0.266162	+	0.963928i	$0.585756\pi$
$84$	0	0
$85$	2.92391	0.317143
$86$	0	0
$87$	7.49417	0.803460
$88$	0	0
$89$	−16.3618	−1.73435	−0.867176	−	0.498002i	$-0.834067\pi$
$89$	−16.3618	−1.73435	−0.867176	+	0.498002i	$0.834067\pi$
$90$	0	0
$91$	−3.75593	−0.393729
$92$	0	0
$93$	1.81087	0.187778
$94$	0	0
$95$	5.08623	0.521836
$96$	0	0
$97$	6.12811	0.622215	0.311107	−	0.950375i	$-0.399300\pi$
$97$	6.12811	0.622215	0.311107	+	0.950375i	$0.399300\pi$
$98$	0	0
$99$	4.73738	0.476124
$100$	0	0
$101$	7.62130	0.758347	0.379174	−	0.925326i	$-0.376208\pi$
$101$	7.62130	0.758347	0.379174	+	0.925326i	$0.376208\pi$
$102$	0	0
$103$	−0.0593276	−0.00584572	−0.00292286	−	0.999996i	$-0.500930\pi$
$103$	−0.0593276	−0.00584572	−0.00292286	+	0.999996i	$0.500930\pi$
$104$	0	0
$105$	2.60104	0.253836
$106$	0	0
$107$	−16.5761	−1.60247	−0.801235	−	0.598350i	$-0.795823\pi$
$107$	−16.5761	−1.60247	−0.801235	+	0.598350i	$0.795823\pi$
$108$	0	0
$109$	−8.34353	−0.799166	−0.399583	−	0.916697i	$-0.630845\pi$
$109$	−8.34353	−0.799166	−0.399583	+	0.916697i	$0.630845\pi$
$110$	0	0
$111$	6.36580	0.604215
$112$	0	0
$113$	17.9803	1.69145	0.845724	−	0.533621i	$-0.179169\pi$
$113$	17.9803	1.69145	0.845724	+	0.533621i	$0.179169\pi$
$114$	0	0
$115$	9.07725	0.846458
$116$	0	0
$117$	1.63298	0.150969
$118$	0	0
$119$	5.07544	0.465265
$120$	0	0
$121$	28.4032	2.58211
$122$	0	0
$123$	−11.4665	−1.03390
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	19.9628	1.77141	0.885706	−	0.464247i	$-0.153675\pi$
$127$	19.9628	1.77141	0.885706	+	0.464247i	$0.153675\pi$
$128$	0	0
$129$	−7.80851	−0.687501
$130$	0	0
$131$	−1.88519	−0.164710	−0.0823551	−	0.996603i	$-0.526244\pi$
$131$	−1.88519	−0.164710	−0.0823551	+	0.996603i	$0.526244\pi$
$132$	0	0
$133$	8.82887	0.765560
$134$	0	0
$135$	−5.62616	−0.484223
$136$	0	0
$137$	15.9043	1.35880	0.679398	−	0.733770i	$-0.262241\pi$
$137$	15.9043	1.35880	0.679398	+	0.733770i	$0.262241\pi$
$138$	0	0
$139$	22.5539	1.91300	0.956500	−	0.291733i	$-0.0942318\pi$
$139$	22.5539	1.91300	0.956500	+	0.291733i	$0.0942318\pi$
$140$	0	0
$141$	8.00886	0.674468
$142$	0	0
$143$	13.5823	1.13581
$144$	0	0
$145$	5.00134	0.415338
$146$	0	0
$147$	−5.97405	−0.492731
$148$	0	0
$149$	11.3528	0.930057	0.465029	−	0.885296i	$-0.346044\pi$
$149$	11.3528	0.930057	0.465029	+	0.885296i	$0.346044\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	−2.20667	−0.178398
$154$	0	0
$155$	1.20851	0.0970697
$156$	0	0
$157$	7.41285	0.591610	0.295805	−	0.955248i	$-0.404412\pi$
$157$	7.41285	0.591610	0.295805	+	0.955248i	$0.404412\pi$
$158$	0	0
$159$	21.5974	1.71279
$160$	0	0
$161$	15.7566	1.24180
$162$	0	0
$163$	2.15827	0.169048	0.0845242	−	0.996421i	$-0.473063\pi$
$163$	2.15827	0.169048	0.0845242	+	0.996421i	$0.473063\pi$
$164$	0	0
$165$	−9.40596	−0.732253
$166$	0	0
$167$	22.7420	1.75983	0.879915	−	0.475131i	$-0.157599\pi$
$167$	22.7420	1.75983	0.879915	+	0.475131i	$0.157599\pi$
$168$	0	0
$169$	−8.31816	−0.639859
$170$	0	0
$171$	−3.83856	−0.293542
$172$	0	0
$173$	−1.17677	−0.0894682	−0.0447341	−	0.998999i	$-0.514244\pi$
$173$	−1.17677	−0.0894682	−0.0447341	+	0.998999i	$0.514244\pi$
$174$	0	0
$175$	1.73584	0.131217
$176$	0	0
$177$	12.1335	0.912007
$178$	0	0
$179$	−8.39112	−0.627182	−0.313591	−	0.949558i	$-0.601532\pi$
$179$	−8.39112	−0.627182	−0.313591	+	0.949558i	$0.601532\pi$
$180$	0	0
$181$	−24.9620	−1.85541	−0.927706	−	0.373312i	$-0.878222\pi$
$181$	−24.9620	−1.85541	−0.927706	+	0.373312i	$0.878222\pi$
$182$	0	0
$183$	0.831066	0.0614342
$184$	0	0
$185$	4.24830	0.312341
$186$	0	0
$187$	−18.3540	−1.34218
$188$	0	0
$189$	−9.76612	−0.710380
$190$	0	0
$191$	10.0409	0.726534	0.363267	−	0.931685i	$-0.381661\pi$
$191$	10.0409	0.726534	0.363267	+	0.931685i	$0.381661\pi$
$192$	0	0
$193$	17.9782	1.29410	0.647049	−	0.762449i	$-0.276003\pi$
$193$	17.9782	1.29410	0.647049	+	0.762449i	$0.276003\pi$
$194$	0	0
$195$	−3.24224	−0.232182
$196$	0	0
$197$	14.1373	1.00724	0.503620	−	0.863925i	$-0.332001\pi$
$197$	14.1373	1.00724	0.503620	+	0.863925i	$0.332001\pi$
$198$	0	0
$199$	1.20016	0.0850771	0.0425386	−	0.999095i	$-0.486455\pi$
$199$	1.20016	0.0850771	0.0425386	+	0.999095i	$0.486455\pi$
$200$	0	0
$201$	7.90387	0.557496
$202$	0	0
$203$	8.68152	0.609323
$204$	0	0
$205$	−7.65235	−0.534463
$206$	0	0
$207$	−6.85057	−0.476147
$208$	0	0
$209$	−31.9272	−2.20845
$210$	0	0
$211$	−14.2151	−0.978610	−0.489305	−	0.872113i	$-0.662749\pi$
$211$	−14.2151	−0.978610	−0.489305	+	0.872113i	$0.662749\pi$
$212$	0	0
$213$	−12.8508	−0.880519
$214$	0	0
$215$	−5.21112	−0.355395
$216$	0	0
$217$	2.09777	0.142406
$218$	0	0
$219$	8.34192	0.563694
$220$	0	0
$221$	−6.32663	−0.425575
$222$	0	0
$223$	−5.38693	−0.360736	−0.180368	−	0.983599i	$-0.557729\pi$
$223$	−5.38693	−0.360736	−0.180368	+	0.983599i	$0.557729\pi$
$224$	0	0
$225$	−0.754697	−0.0503131
$226$	0	0
$227$	3.95017	0.262182	0.131091	−	0.991370i	$-0.458152\pi$
$227$	3.95017	0.262182	0.131091	+	0.991370i	$0.458152\pi$
$228$	0	0
$229$	−12.4651	−0.823716	−0.411858	−	0.911248i	$-0.635120\pi$
$229$	−12.4651	−0.823716	−0.411858	+	0.911248i	$0.635120\pi$
$230$	0	0
$231$	−16.3272	−1.07425
$232$	0	0
$233$	18.0760	1.18420	0.592100	−	0.805865i	$-0.298299\pi$
$233$	18.0760	1.18420	0.592100	+	0.805865i	$0.298299\pi$
$234$	0	0
$235$	5.34482	0.348658
$236$	0	0
$237$	6.45835	0.419515
$238$	0	0
$239$	−0.0986572	−0.00638160	−0.00319080	−	0.999995i	$-0.501016\pi$
$239$	−0.0986572	−0.00638160	−0.00319080	+	0.999995i	$0.501016\pi$
$240$	0	0
$241$	−19.5815	−1.26135	−0.630677	−	0.776045i	$-0.717223\pi$
$241$	−19.5815	−1.26135	−0.630677	+	0.776045i	$0.717223\pi$
$242$	0	0
$243$	7.63863	0.490019
$244$	0	0
$245$	−3.98686	−0.254711
$246$	0	0
$247$	−11.0054	−0.700254
$248$	0	0
$249$	−7.26696	−0.460525
$250$	0	0
$251$	−11.5621	−0.729791	−0.364896	−	0.931048i	$-0.618895\pi$
$251$	−11.5621	−0.729791	−0.364896	+	0.931048i	$0.618895\pi$
$252$	0	0
$253$	−56.9796	−3.58228
$254$	0	0
$255$	4.38129	0.274367
$256$	0	0
$257$	−20.8657	−1.30157	−0.650785	−	0.759262i	$-0.725560\pi$
$257$	−20.8657	−1.30157	−0.650785	+	0.759262i	$0.725560\pi$
$258$	0	0
$259$	7.37437	0.458221
$260$	0	0
$261$	−3.77449	−0.233635
$262$	0	0
$263$	14.1436	0.872134	0.436067	−	0.899914i	$-0.356371\pi$
$263$	14.1436	0.872134	0.436067	+	0.899914i	$0.356371\pi$
$264$	0	0
$265$	14.4133	0.885404
$266$	0	0
$267$	−24.5171	−1.50042
$268$	0	0
$269$	0.550232	0.0335482	0.0167741	−	0.999859i	$-0.494660\pi$
$269$	0.550232	0.0335482	0.0167741	+	0.999859i	$0.494660\pi$
$270$	0	0
$271$	−16.1153	−0.978934	−0.489467	−	0.872022i	$-0.662809\pi$
$271$	−16.1153	−0.978934	−0.489467	+	0.872022i	$0.662809\pi$
$272$	0	0
$273$	−5.62802	−0.340623
$274$	0	0
$275$	−6.27719	−0.378529
$276$	0	0
$277$	23.5526	1.41514	0.707569	−	0.706644i	$-0.249792\pi$
$277$	23.5526	1.41514	0.707569	+	0.706644i	$0.249792\pi$
$278$	0	0
$279$	−0.912057	−0.0546034
$280$	0	0
$281$	28.3259	1.68978	0.844890	−	0.534940i	$-0.179666\pi$
$281$	28.3259	1.68978	0.844890	+	0.534940i	$0.179666\pi$
$282$	0	0
$283$	−20.8831	−1.24137	−0.620686	−	0.784059i	$-0.713146\pi$
$283$	−20.8831	−1.24137	−0.620686	+	0.784059i	$0.713146\pi$
$284$	0	0
$285$	7.62137	0.451451
$286$	0	0
$287$	−13.2832	−0.784085
$288$	0	0
$289$	−8.45073	−0.497102
$290$	0	0
$291$	9.18256	0.538291
$292$	0	0
$293$	28.2232	1.64881	0.824407	−	0.565997i	$-0.191509\pi$
$293$	28.2232	1.64881	0.824407	+	0.565997i	$0.191509\pi$
$294$	0	0
$295$	8.09743	0.471451
$296$	0	0
$297$	35.3165	2.04927
$298$	0	0
$299$	−19.6409	−1.13587
$300$	0	0
$301$	−9.04566	−0.521383
$302$	0	0
$303$	11.4200	0.656062
$304$	0	0
$305$	0.554623	0.0317576
$306$	0	0
$307$	−9.40309	−0.536663	−0.268331	−	0.963327i	$-0.586472\pi$
$307$	−9.40309	−0.536663	−0.268331	+	0.963327i	$0.586472\pi$
$308$	0	0
$309$	−0.0888985	−0.00505726
$310$	0	0
$311$	−2.72699	−0.154633	−0.0773167	−	0.997007i	$-0.524635\pi$
$311$	−2.72699	−0.154633	−0.0773167	+	0.997007i	$0.524635\pi$
$312$	0	0
$313$	0.274816	0.0155335	0.00776677	−	0.999970i	$-0.497528\pi$
$313$	0.274816	0.0155335	0.00776677	+	0.999970i	$0.497528\pi$
$314$	0	0
$315$	−1.31003	−0.0738119
$316$	0	0
$317$	2.18015	0.122450	0.0612248	−	0.998124i	$-0.480499\pi$
$317$	2.18015	0.122450	0.0612248	+	0.998124i	$0.480499\pi$
$318$	0	0
$319$	−31.3944	−1.75775
$320$	0	0
$321$	−24.8381	−1.38633
$322$	0	0
$323$	14.8717	0.827483
$324$	0	0
$325$	−2.16376	−0.120024
$326$	0	0
$327$	−12.5022	−0.691375
$328$	0	0
$329$	9.27775	0.511499
$330$	0	0
$331$	6.38705	0.351064	0.175532	−	0.984474i	$-0.443835\pi$
$331$	6.38705	0.351064	0.175532	+	0.984474i	$0.443835\pi$
$332$	0	0
$333$	−3.20618	−0.175698
$334$	0	0
$335$	5.27476	0.288191
$336$	0	0
$337$	−27.7897	−1.51380	−0.756901	−	0.653529i	$-0.773288\pi$
$337$	−27.7897	−1.51380	−0.756901	+	0.653529i	$0.773288\pi$
$338$	0	0
$339$	26.9423	1.46331
$340$	0	0
$341$	−7.58604	−0.410807
$342$	0	0
$343$	−19.0714	−1.02976
$344$	0	0
$345$	13.6017	0.732288
$346$	0	0
$347$	−18.8765	−1.01334	−0.506671	−	0.862139i	$-0.669124\pi$
$347$	−18.8765	−1.01334	−0.506671	+	0.862139i	$0.669124\pi$
$348$	0	0
$349$	−12.1638	−0.651112	−0.325556	−	0.945523i	$-0.605552\pi$
$349$	−12.1638	−0.651112	−0.325556	+	0.945523i	$0.605552\pi$
$350$	0	0
$351$	12.1736	0.649781
$352$	0	0
$353$	29.9902	1.59622	0.798110	−	0.602512i	$-0.205834\pi$
$353$	29.9902	1.59622	0.798110	+	0.602512i	$0.205834\pi$
$354$	0	0
$355$	−8.57612	−0.455173
$356$	0	0
$357$	7.60522	0.402511
$358$	0	0
$359$	−10.0542	−0.530641	−0.265321	−	0.964160i	$-0.585478\pi$
$359$	−10.0542	−0.530641	−0.265321	+	0.964160i	$0.585478\pi$
$360$	0	0
$361$	6.86971	0.361564
$362$	0	0
$363$	42.5603	2.23383
$364$	0	0
$365$	5.56709	0.291395
$366$	0	0
$367$	−17.1447	−0.894947	−0.447473	−	0.894297i	$-0.647676\pi$
$367$	−17.1447	−0.894947	−0.447473	+	0.894297i	$0.647676\pi$
$368$	0	0
$369$	5.77520	0.300645
$370$	0	0
$371$	25.0192	1.29893
$372$	0	0
$373$	−15.5280	−0.804007	−0.402004	−	0.915638i	$-0.631686\pi$
$373$	−15.5280	−0.804007	−0.402004	+	0.915638i	$0.631686\pi$
$374$	0	0
$375$	1.49843	0.0773788
$376$	0	0
$377$	−10.8217	−0.557344
$378$	0	0
$379$	22.3246	1.14674	0.573369	−	0.819297i	$-0.305636\pi$
$379$	22.3246	1.14674	0.573369	+	0.819297i	$0.305636\pi$
$380$	0	0
$381$	29.9129	1.53249
$382$	0	0
$383$	−17.1954	−0.878645	−0.439322	−	0.898330i	$-0.644781\pi$
$383$	−17.1954	−0.878645	−0.439322	+	0.898330i	$0.644781\pi$
$384$	0	0
$385$	−10.8962	−0.555322
$386$	0	0
$387$	3.93281	0.199916
$388$	0	0
$389$	−37.5140	−1.90204	−0.951018	−	0.309136i	$-0.899960\pi$
$389$	−37.5140	−1.90204	−0.951018	+	0.309136i	$0.899960\pi$
$390$	0	0
$391$	26.5411	1.34224
$392$	0	0
$393$	−2.82484	−0.142494
$394$	0	0
$395$	4.31006	0.216863
$396$	0	0
$397$	−12.1737	−0.610980	−0.305490	−	0.952195i	$-0.598820\pi$
$397$	−12.1737	−0.610980	−0.305490	+	0.952195i	$0.598820\pi$
$398$	0	0
$399$	13.2295	0.662302
$400$	0	0
$401$	14.9027	0.744204	0.372102	−	0.928192i	$-0.378637\pi$
$401$	14.9027	0.744204	0.372102	+	0.928192i	$0.378637\pi$
$402$	0	0
$403$	−2.61492	−0.130258
$404$	0	0
$405$	−6.16634	−0.306408
$406$	0	0
$407$	−26.6674	−1.32186
$408$	0	0
$409$	−12.5646	−0.621278	−0.310639	−	0.950528i	$-0.600543\pi$
$409$	−12.5646	−0.621278	−0.310639	+	0.950528i	$0.600543\pi$
$410$	0	0
$411$	23.8315	1.17552
$412$	0	0
$413$	14.0558	0.691643
$414$	0	0
$415$	−4.84970	−0.238062
$416$	0	0
$417$	33.7956	1.65498
$418$	0	0
$419$	−32.2669	−1.57634	−0.788171	−	0.615456i	$-0.788972\pi$
$419$	−32.2669	−1.57634	−0.788171	+	0.615456i	$0.788972\pi$
$420$	0	0
$421$	20.3468	0.991641	0.495820	−	0.868425i	$-0.334867\pi$
$421$	20.3468	0.991641	0.495820	+	0.868425i	$0.334867\pi$
$422$	0	0
$423$	−4.03372	−0.196126
$424$	0	0
$425$	2.92391	0.141831
$426$	0	0
$427$	0.962737	0.0465901
$428$	0	0
$429$	20.3522	0.982613
$430$	0	0
$431$	−17.9224	−0.863294	−0.431647	−	0.902043i	$-0.642067\pi$
$431$	−17.9224	−0.863294	−0.431647	+	0.902043i	$0.642067\pi$
$432$	0	0
$433$	−40.2511	−1.93434	−0.967172	−	0.254124i	$-0.918213\pi$
$433$	−40.2511	−1.93434	−0.967172	+	0.254124i	$0.918213\pi$
$434$	0	0
$435$	7.49417	0.359318
$436$	0	0
$437$	46.1689	2.20856
$438$	0	0
$439$	−7.01430	−0.334774	−0.167387	−	0.985891i	$-0.553533\pi$
$439$	−7.01430	−0.334774	−0.167387	+	0.985891i	$0.553533\pi$
$440$	0	0
$441$	3.00887	0.143280
$442$	0	0
$443$	5.98105	0.284168	0.142084	−	0.989855i	$-0.454620\pi$
$443$	5.98105	0.284168	0.142084	+	0.989855i	$0.454620\pi$
$444$	0	0
$445$	−16.3618	−0.775626
$446$	0	0
$447$	17.0114	0.804612
$448$	0	0
$449$	−0.545274	−0.0257331	−0.0128665	−	0.999917i	$-0.504096\pi$
$449$	−0.545274	−0.0257331	−0.0128665	+	0.999917i	$0.504096\pi$
$450$	0	0
$451$	48.0353	2.26189
$452$	0	0
$453$	1.49843	0.0704026
$454$	0	0
$455$	−3.75593	−0.176081
$456$	0	0
$457$	30.6050	1.43164	0.715821	−	0.698284i	$-0.246053\pi$
$457$	30.6050	1.43164	0.715821	+	0.698284i	$0.246053\pi$
$458$	0	0
$459$	−16.4504	−0.767840
$460$	0	0
$461$	1.64045	0.0764033	0.0382016	−	0.999270i	$-0.487837\pi$
$461$	1.64045	0.0764033	0.0382016	+	0.999270i	$0.487837\pi$
$462$	0	0
$463$	12.0544	0.560217	0.280108	−	0.959968i	$-0.409630\pi$
$463$	12.0544	0.560217	0.280108	+	0.959968i	$0.409630\pi$
$464$	0	0
$465$	1.81087	0.0839770
$466$	0	0
$467$	−16.8978	−0.781937	−0.390969	−	0.920404i	$-0.627860\pi$
$467$	−16.8978	−0.781937	−0.390969	+	0.920404i	$0.627860\pi$
$468$	0	0
$469$	9.15613	0.422791
$470$	0	0
$471$	11.1077	0.511814
$472$	0	0
$473$	32.7112	1.50406
$474$	0	0
$475$	5.08623	0.233372
$476$	0	0
$477$	−10.8777	−0.498055
$478$	0	0
$479$	−31.8989	−1.45750	−0.728749	−	0.684781i	$-0.759898\pi$
$479$	−31.8989	−1.45750	−0.728749	+	0.684781i	$0.759898\pi$
$480$	0	0
$481$	−9.19229	−0.419132
$482$	0	0
$483$	23.6103	1.07430
$484$	0	0
$485$	6.12811	0.278263
$486$	0	0
$487$	35.7343	1.61928	0.809638	−	0.586929i	$-0.199663\pi$
$487$	35.7343	1.61928	0.809638	+	0.586929i	$0.199663\pi$
$488$	0	0
$489$	3.23402	0.146247
$490$	0	0
$491$	−16.4795	−0.743709	−0.371854	−	0.928291i	$-0.621278\pi$
$491$	−16.4795	−0.743709	−0.371854	+	0.928291i	$0.621278\pi$
$492$	0	0
$493$	14.6235	0.658608
$494$	0	0
$495$	4.73738	0.212929
$496$	0	0
$497$	−14.8868	−0.667763
$498$	0	0
$499$	−42.0961	−1.88448	−0.942239	−	0.334940i	$-0.891284\pi$
$499$	−42.0961	−1.88448	−0.942239	+	0.334940i	$0.891284\pi$
$500$	0	0
$501$	34.0774	1.52247
$502$	0	0
$503$	6.27308	0.279703	0.139851	−	0.990172i	$-0.455337\pi$
$503$	6.27308	0.279703	0.139851	+	0.990172i	$0.455337\pi$
$504$	0	0
$505$	7.62130	0.339143
$506$	0	0
$507$	−12.4642	−0.553555
$508$	0	0
$509$	9.80811	0.434737	0.217368	−	0.976090i	$-0.430253\pi$
$509$	9.80811	0.434737	0.217368	+	0.976090i	$0.430253\pi$
$510$	0	0
$511$	9.66357	0.427491
$512$	0	0
$513$	−28.6159	−1.26342
$514$	0	0
$515$	−0.0593276	−0.00261429
$516$	0	0
$517$	−33.5505	−1.47555
$518$	0	0
$519$	−1.76331	−0.0774008
$520$	0	0
$521$	11.3053	0.495292	0.247646	−	0.968851i	$-0.420343\pi$
$521$	11.3053	0.495292	0.247646	+	0.968851i	$0.420343\pi$
$522$	0	0
$523$	16.4129	0.717686	0.358843	−	0.933398i	$-0.383171\pi$
$523$	16.4129	0.717686	0.358843	+	0.933398i	$0.383171\pi$
$524$	0	0
$525$	2.60104	0.113519
$526$	0	0
$527$	3.53357	0.153925
$528$	0	0
$529$	59.3964	2.58245
$530$	0	0
$531$	−6.11110	−0.265199
$532$	0	0
$533$	16.5578	0.717198
$534$	0	0
$535$	−16.5761	−0.716646
$536$	0	0
$537$	−12.5735	−0.542588
$538$	0	0
$539$	25.0263	1.07796
$540$	0	0
$541$	29.3222	1.26066	0.630331	−	0.776327i	$-0.282919\pi$
$541$	29.3222	1.26066	0.630331	+	0.776327i	$0.282919\pi$
$542$	0	0
$543$	−37.4039	−1.60516
$544$	0	0
$545$	−8.34353	−0.357398
$546$	0	0
$547$	−18.3364	−0.784010	−0.392005	−	0.919963i	$-0.628218\pi$
$547$	−18.3364	−0.784010	−0.392005	+	0.919963i	$0.628218\pi$
$548$	0	0
$549$	−0.418572	−0.0178642
$550$	0	0
$551$	25.4379	1.08369
$552$	0	0
$553$	7.48158	0.318149
$554$	0	0
$555$	6.36580	0.270213
$556$	0	0
$557$	10.1488	0.430017	0.215009	−	0.976612i	$-0.431022\pi$
$557$	10.1488	0.430017	0.215009	+	0.976612i	$0.431022\pi$
$558$	0	0
$559$	11.2756	0.476906
$560$	0	0
$561$	−27.5022	−1.16114
$562$	0	0
$563$	−28.7527	−1.21178	−0.605891	−	0.795548i	$-0.707183\pi$
$563$	−28.7527	−1.21178	−0.605891	+	0.795548i	$0.707183\pi$
$564$	0	0
$565$	17.9803	0.756438
$566$	0	0
$567$	−10.7038	−0.449517
$568$	0	0
$569$	35.1969	1.47553	0.737765	−	0.675058i	$-0.235881\pi$
$569$	35.1969	1.47553	0.737765	+	0.675058i	$0.235881\pi$
$570$	0	0
$571$	36.0285	1.50775	0.753873	−	0.657020i	$-0.228183\pi$
$571$	36.0285	1.50775	0.753873	+	0.657020i	$0.228183\pi$
$572$	0	0
$573$	15.0456	0.628540
$574$	0	0
$575$	9.07725	0.378547
$576$	0	0
$577$	33.7003	1.40296	0.701481	−	0.712688i	$-0.252522\pi$
$577$	33.7003	1.40296	0.701481	+	0.712688i	$0.252522\pi$
$578$	0	0
$579$	26.9391	1.11955
$580$	0	0
$581$	−8.41830	−0.349250
$582$	0	0
$583$	−90.4753	−3.74710
$584$	0	0
$585$	1.63298	0.0675154
$586$	0	0
$587$	6.91253	0.285311	0.142655	−	0.989772i	$-0.454436\pi$
$587$	6.91253	0.285311	0.142655	+	0.989772i	$0.454436\pi$
$588$	0	0
$589$	6.14674	0.253272
$590$	0	0
$591$	21.1838	0.871385
$592$	0	0
$593$	−46.4371	−1.90694	−0.953472	−	0.301483i	$-0.902518\pi$
$593$	−46.4371	−1.90694	−0.953472	+	0.301483i	$0.902518\pi$
$594$	0	0
$595$	5.07544	0.208073
$596$	0	0
$597$	1.79836	0.0736020
$598$	0	0
$599$	−40.6986	−1.66290	−0.831449	−	0.555601i	$-0.812488\pi$
$599$	−40.6986	−1.66290	−0.831449	+	0.555601i	$0.812488\pi$
$600$	0	0
$601$	10.9574	0.446963	0.223482	−	0.974708i	$-0.428258\pi$
$601$	10.9574	0.446963	0.223482	+	0.974708i	$0.428258\pi$
$602$	0	0
$603$	−3.98084	−0.162112
$604$	0	0
$605$	28.4032	1.15475
$606$	0	0
$607$	−32.2931	−1.31074	−0.655368	−	0.755310i	$-0.727487\pi$
$607$	−32.2931	−1.31074	−0.655368	+	0.755310i	$0.727487\pi$
$608$	0	0
$609$	13.0087	0.527138
$610$	0	0
$611$	−11.5649	−0.467865
$612$	0	0
$613$	11.6066	0.468786	0.234393	−	0.972142i	$-0.424690\pi$
$613$	11.6066	0.468786	0.234393	+	0.972142i	$0.424690\pi$
$614$	0	0
$615$	−11.4665	−0.462375
$616$	0	0
$617$	24.2265	0.975321	0.487661	−	0.873033i	$-0.337850\pi$
$617$	24.2265	0.975321	0.487661	+	0.873033i	$0.337850\pi$
$618$	0	0
$619$	−10.0171	−0.402622	−0.201311	−	0.979527i	$-0.564520\pi$
$619$	−10.0171	−0.402622	−0.201311	+	0.979527i	$0.564520\pi$
$620$	0	0
$621$	−51.0701	−2.04937
$622$	0	0
$623$	−28.4015	−1.13788
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−47.8408	−1.91058
$628$	0	0
$629$	12.4217	0.495284
$630$	0	0
$631$	6.07699	0.241921	0.120961	−	0.992657i	$-0.461403\pi$
$631$	6.07699	0.241921	0.120961	+	0.992657i	$0.461403\pi$
$632$	0	0
$633$	−21.3004	−0.846616
$634$	0	0
$635$	19.9628	0.792200
$636$	0	0
$637$	8.62660	0.341798
$638$	0	0
$639$	6.47237	0.256043
$640$	0	0
$641$	45.4049	1.79339	0.896693	−	0.442654i	$-0.145963\pi$
$641$	45.4049	1.79339	0.896693	+	0.442654i	$0.145963\pi$
$642$	0	0
$643$	6.56960	0.259080	0.129540	−	0.991574i	$-0.458650\pi$
$643$	6.56960	0.259080	0.129540	+	0.991574i	$0.458650\pi$
$644$	0	0
$645$	−7.80851	−0.307460
$646$	0	0
$647$	21.5063	0.845501	0.422750	−	0.906246i	$-0.361065\pi$
$647$	21.5063	0.845501	0.422750	+	0.906246i	$0.361065\pi$
$648$	0	0
$649$	−50.8291	−1.99522
$650$	0	0
$651$	3.14338	0.123199
$652$	0	0
$653$	−33.7569	−1.32101	−0.660506	−	0.750821i	$-0.729658\pi$
$653$	−33.7569	−1.32101	−0.660506	+	0.750821i	$0.729658\pi$
$654$	0	0
$655$	−1.88519	−0.0736606
$656$	0	0
$657$	−4.20146	−0.163915
$658$	0	0
$659$	−9.77102	−0.380625	−0.190313	−	0.981724i	$-0.560950\pi$
$659$	−9.77102	−0.380625	−0.190313	+	0.981724i	$0.560950\pi$
$660$	0	0
$661$	0.144422	0.00561736	0.00280868	−	0.999996i	$-0.499106\pi$
$661$	0.144422	0.00561736	0.00280868	+	0.999996i	$0.499106\pi$
$662$	0	0
$663$	−9.48004	−0.368174
$664$	0	0
$665$	8.82887	0.342369
$666$	0	0
$667$	45.3984	1.75783
$668$	0	0
$669$	−8.07196	−0.312080
$670$	0	0
$671$	−3.48148	−0.134401
$672$	0	0
$673$	0.986677	0.0380336	0.0190168	−	0.999819i	$-0.493946\pi$
$673$	0.986677	0.0380336	0.0190168	+	0.999819i	$0.493946\pi$
$674$	0	0
$675$	−5.62616	−0.216551
$676$	0	0
$677$	14.6103	0.561520	0.280760	−	0.959778i	$-0.409413\pi$
$677$	14.6103	0.561520	0.280760	+	0.959778i	$0.409413\pi$
$678$	0	0
$679$	10.6374	0.408226
$680$	0	0
$681$	5.91906	0.226819
$682$	0	0
$683$	−35.4284	−1.35563	−0.677816	−	0.735232i	$-0.737073\pi$
$683$	−35.4284	−1.35563	−0.677816	+	0.735232i	$0.737073\pi$
$684$	0	0
$685$	15.9043	0.607672
$686$	0	0
$687$	−18.6781	−0.712614
$688$	0	0
$689$	−31.1869	−1.18813
$690$	0	0
$691$	−44.4400	−1.69058	−0.845288	−	0.534310i	$-0.820571\pi$
$691$	−44.4400	−1.69058	−0.845288	+	0.534310i	$0.820571\pi$
$692$	0	0
$693$	8.22332	0.312378
$694$	0	0
$695$	22.5539	0.855519
$696$	0	0
$697$	−22.3748	−0.847506
$698$	0	0
$699$	27.0857	1.02448
$700$	0	0
$701$	31.9465	1.20660	0.603302	−	0.797513i	$-0.293852\pi$
$701$	31.9465	1.20660	0.603302	+	0.797513i	$0.293852\pi$
$702$	0	0
$703$	21.6078	0.814955
$704$	0	0
$705$	8.00886	0.301631
$706$	0	0
$707$	13.2293	0.497541
$708$	0	0
$709$	−25.5493	−0.959525	−0.479763	−	0.877398i	$-0.659277\pi$
$709$	−25.5493	−0.959525	−0.479763	+	0.877398i	$0.659277\pi$
$710$	0	0
$711$	−3.25279	−0.121989
$712$	0	0
$713$	10.9699	0.410827
$714$	0	0
$715$	13.5823	0.507950
$716$	0	0
$717$	−0.147831	−0.00552086
$718$	0	0
$719$	−32.7577	−1.22166	−0.610828	−	0.791763i	$-0.709163\pi$
$719$	−32.7577	−1.22166	−0.610828	+	0.791763i	$0.709163\pi$
$720$	0	0
$721$	−0.102983	−0.00383529
$722$	0	0
$723$	−29.3416	−1.09122
$724$	0	0
$725$	5.00134	0.185745
$726$	0	0
$727$	−25.9372	−0.961956	−0.480978	−	0.876733i	$-0.659718\pi$
$727$	−25.9372	−0.961956	−0.480978	+	0.876733i	$0.659718\pi$
$728$	0	0
$729$	29.9450	1.10907
$730$	0	0
$731$	−15.2369	−0.563555
$732$	0	0
$733$	−52.6234	−1.94369	−0.971845	−	0.235623i	$-0.924287\pi$
$733$	−52.6234	−1.94369	−0.971845	+	0.235623i	$0.924287\pi$
$734$	0	0
$735$	−5.97405	−0.220356
$736$	0	0
$737$	−33.1107	−1.21965
$738$	0	0
$739$	−0.772166	−0.0284046	−0.0142023	−	0.999899i	$-0.504521\pi$
$739$	−0.772166	−0.0284046	−0.0142023	+	0.999899i	$0.504521\pi$
$740$	0	0
$741$	−16.4908	−0.605804
$742$	0	0
$743$	41.2478	1.51323	0.756617	−	0.653858i	$-0.226851\pi$
$743$	41.2478	1.51323	0.756617	+	0.653858i	$0.226851\pi$
$744$	0	0
$745$	11.3528	0.415934
$746$	0	0
$747$	3.66005	0.133914
$748$	0	0
$749$	−28.7734	−1.05136
$750$	0	0
$751$	49.0963	1.79155	0.895774	−	0.444509i	$-0.146622\pi$
$751$	49.0963	1.79155	0.895774	+	0.444509i	$0.146622\pi$
$752$	0	0
$753$	−17.3250	−0.631358
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	−26.1370	−0.949967	−0.474983	−	0.879995i	$-0.657546\pi$
$757$	−26.1370	−0.949967	−0.474983	+	0.879995i	$0.657546\pi$
$758$	0	0
$759$	−85.3802	−3.09911
$760$	0	0
$761$	13.8779	0.503072	0.251536	−	0.967848i	$-0.419064\pi$
$761$	13.8779	0.503072	0.251536	+	0.967848i	$0.419064\pi$
$762$	0	0
$763$	−14.4830	−0.524321
$764$	0	0
$765$	−2.20667	−0.0797822
$766$	0	0
$767$	−17.5209	−0.632642
$768$	0	0
$769$	−20.1751	−0.727531	−0.363766	−	0.931490i	$-0.618509\pi$
$769$	−20.1751	−0.727531	−0.363766	+	0.931490i	$0.618509\pi$
$770$	0	0
$771$	−31.2659	−1.12602
$772$	0	0
$773$	17.0932	0.614801	0.307400	−	0.951580i	$-0.400541\pi$
$773$	17.0932	0.614801	0.307400	+	0.951580i	$0.400541\pi$
$774$	0	0
$775$	1.20851	0.0434109
$776$	0	0
$777$	11.0500	0.396417
$778$	0	0
$779$	−38.9216	−1.39451
$780$	0	0
$781$	53.8340	1.92633
$782$	0	0
$783$	−28.1383	−1.00558
$784$	0	0
$785$	7.41285	0.264576
$786$	0	0
$787$	−32.7456	−1.16725	−0.583627	−	0.812022i	$-0.698367\pi$
$787$	−32.7456	−1.16725	−0.583627	+	0.812022i	$0.698367\pi$
$788$	0	0
$789$	21.1933	0.754501
$790$	0	0
$791$	31.2110	1.10973
$792$	0	0
$793$	−1.20007	−0.0426157
$794$	0	0
$795$	21.5974	0.765982
$796$	0	0
$797$	11.7494	0.416186	0.208093	−	0.978109i	$-0.433274\pi$
$797$	11.7494	0.416186	0.208093	+	0.978109i	$0.433274\pi$
$798$	0	0
$799$	15.6278	0.552871
$800$	0	0
$801$	12.3482	0.436303
$802$	0	0
$803$	−34.9457	−1.23321
$804$	0	0
$805$	15.7566	0.555349
$806$	0	0
$807$	0.824486	0.0290233
$808$	0	0
$809$	−39.4465	−1.38687	−0.693433	−	0.720521i	$-0.743903\pi$
$809$	−39.4465	−1.38687	−0.693433	+	0.720521i	$0.743903\pi$
$810$	0	0
$811$	−9.99393	−0.350934	−0.175467	−	0.984485i	$-0.556144\pi$
$811$	−9.99393	−0.350934	−0.175467	+	0.984485i	$0.556144\pi$
$812$	0	0
$813$	−24.1477	−0.846896
$814$	0	0
$815$	2.15827	0.0756007
$816$	0	0
$817$	−26.5049	−0.927290
$818$	0	0
$819$	2.83459	0.0990486
$820$	0	0
$821$	8.66476	0.302402	0.151201	−	0.988503i	$-0.451686\pi$
$821$	8.66476	0.302402	0.151201	+	0.988503i	$0.451686\pi$
$822$	0	0
$823$	8.08780	0.281923	0.140961	−	0.990015i	$-0.454981\pi$
$823$	8.08780	0.281923	0.140961	+	0.990015i	$0.454981\pi$
$824$	0	0
$825$	−9.40596	−0.327473
$826$	0	0
$827$	11.4287	0.397415	0.198708	−	0.980059i	$-0.436326\pi$
$827$	11.4287	0.397415	0.198708	+	0.980059i	$0.436326\pi$
$828$	0	0
$829$	17.7262	0.615657	0.307828	−	0.951442i	$-0.400398\pi$
$829$	17.7262	0.615657	0.307828	+	0.951442i	$0.400398\pi$
$830$	0	0
$831$	35.2920	1.22427
$832$	0	0
$833$	−11.6572	−0.403899
$834$	0	0
$835$	22.7420	0.787020
$836$	0	0
$837$	−6.79926	−0.235017
$838$	0	0
$839$	8.24336	0.284592	0.142296	−	0.989824i	$-0.454551\pi$
$839$	8.24336	0.284592	0.142296	+	0.989824i	$0.454551\pi$
$840$	0	0
$841$	−3.98663	−0.137470
$842$	0	0
$843$	42.4445	1.46186
$844$	0	0
$845$	−8.31816	−0.286153
$846$	0	0
$847$	49.3033	1.69408
$848$	0	0
$849$	−31.2919	−1.07394
$850$	0	0
$851$	38.5629	1.32192
$852$	0	0
$853$	17.1826	0.588322	0.294161	−	0.955756i	$-0.404960\pi$
$853$	17.1826	0.588322	0.294161	+	0.955756i	$0.404960\pi$
$854$	0	0
$855$	−3.83856	−0.131276
$856$	0	0
$857$	50.2017	1.71486	0.857430	−	0.514601i	$-0.172060\pi$
$857$	50.2017	1.71486	0.857430	+	0.514601i	$0.172060\pi$
$858$	0	0
$859$	−46.6649	−1.59218	−0.796092	−	0.605175i	$-0.793103\pi$
$859$	−46.6649	−1.59218	−0.796092	+	0.605175i	$0.793103\pi$
$860$	0	0
$861$	−19.9041	−0.678328
$862$	0	0
$863$	−3.77134	−0.128378	−0.0641889	−	0.997938i	$-0.520446\pi$
$863$	−3.77134	−0.128378	−0.0641889	+	0.997938i	$0.520446\pi$
$864$	0	0
$865$	−1.17677	−0.0400114
$866$	0	0
$867$	−12.6629	−0.430053
$868$	0	0
$869$	−27.0551	−0.917782
$870$	0	0
$871$	−11.4133	−0.386724
$872$	0	0
$873$	−4.62486	−0.156528
$874$	0	0
$875$	1.73584	0.0586821
$876$	0	0
$877$	−3.84292	−0.129766	−0.0648832	−	0.997893i	$-0.520667\pi$
$877$	−3.84292	−0.129766	−0.0648832	+	0.997893i	$0.520667\pi$
$878$	0	0
$879$	42.2905	1.42642
$880$	0	0
$881$	13.7670	0.463822	0.231911	−	0.972737i	$-0.425502\pi$
$881$	13.7670	0.463822	0.231911	+	0.972737i	$0.425502\pi$
$882$	0	0
$883$	−29.4382	−0.990676	−0.495338	−	0.868700i	$-0.664956\pi$
$883$	−29.4382	−0.990676	−0.495338	+	0.868700i	$0.664956\pi$
$884$	0	0
$885$	12.1335	0.407862
$886$	0	0
$887$	−4.93901	−0.165836	−0.0829178	−	0.996556i	$-0.526424\pi$
$887$	−4.93901	−0.165836	−0.0829178	+	0.996556i	$0.526424\pi$
$888$	0	0
$889$	34.6522	1.16220
$890$	0	0
$891$	38.7073	1.29674
$892$	0	0
$893$	27.1850	0.909710
$894$	0	0
$895$	−8.39112	−0.280484
$896$	0	0
$897$	−29.4306	−0.982661
$898$	0	0
$899$	6.04415	0.201584
$900$	0	0
$901$	42.1433	1.40400
$902$	0	0
$903$	−13.5543	−0.451060
$904$	0	0
$905$	−24.9620	−0.829765
$906$	0	0
$907$	−31.1955	−1.03583	−0.517915	−	0.855432i	$-0.673292\pi$
$907$	−31.1955	−1.03583	−0.517915	+	0.855432i	$0.673292\pi$
$908$	0	0
$909$	−5.75177	−0.190774
$910$	0	0
$911$	−38.0889	−1.26194	−0.630971	−	0.775807i	$-0.717343\pi$
$911$	−38.0889	−1.26194	−0.630971	+	0.775807i	$0.717343\pi$
$912$	0	0
$913$	30.4425	1.00750
$914$	0	0
$915$	0.831066	0.0274742
$916$	0	0
$917$	−3.27239	−0.108064
$918$	0	0
$919$	−57.6481	−1.90164	−0.950818	−	0.309749i	$-0.899755\pi$
$919$	−57.6481	−1.90164	−0.950818	+	0.309749i	$0.899755\pi$
$920$	0	0
$921$	−14.0899	−0.464278
$922$	0	0
$923$	18.5566	0.610799
$924$	0	0
$925$	4.24830	0.139683
$926$	0	0
$927$	0.0447743	0.00147058
$928$	0	0
$929$	23.9227	0.784877	0.392439	−	0.919778i	$-0.371632\pi$
$929$	23.9227	0.784877	0.392439	+	0.919778i	$0.371632\pi$
$930$	0	0
$931$	−20.2781	−0.664587
$932$	0	0
$933$	−4.08622	−0.133777
$934$	0	0
$935$	−18.3540	−0.600239
$936$	0	0
$937$	34.4282	1.12472	0.562360	−	0.826893i	$-0.309894\pi$
$937$	34.4282	1.12472	0.562360	+	0.826893i	$0.309894\pi$
$938$	0	0
$939$	0.411794	0.0134384
$940$	0	0
$941$	−51.6458	−1.68360	−0.841802	−	0.539787i	$-0.818505\pi$
$941$	−51.6458	−1.68360	−0.841802	+	0.539787i	$0.818505\pi$
$942$	0	0
$943$	−69.4622	−2.26200
$944$	0	0
$945$	−9.76612	−0.317692
$946$	0	0
$947$	45.7741	1.48746	0.743729	−	0.668481i	$-0.233055\pi$
$947$	45.7741	1.48746	0.743729	+	0.668481i	$0.233055\pi$
$948$	0	0
$949$	−12.0458	−0.391024
$950$	0	0
$951$	3.26681	0.105934
$952$	0	0
$953$	6.79561	0.220131	0.110066	−	0.993924i	$-0.464894\pi$
$953$	6.79561	0.220131	0.110066	+	0.993924i	$0.464894\pi$
$954$	0	0
$955$	10.0409	0.324916
$956$	0	0
$957$	−47.0424	−1.52066
$958$	0	0
$959$	27.6073	0.891486
$960$	0	0
$961$	−29.5395	−0.952887
$962$	0	0
$963$	12.5099	0.403126
$964$	0	0
$965$	17.9782	0.578738
$966$	0	0
$967$	−23.2428	−0.747438	−0.373719	−	0.927542i	$-0.621917\pi$
$967$	−23.2428	−0.747438	−0.373719	+	0.927542i	$0.621917\pi$
$968$	0	0
$969$	22.2842	0.715873
$970$	0	0
$971$	−31.7725	−1.01963	−0.509814	−	0.860285i	$-0.670286\pi$
$971$	−31.7725	−1.01963	−0.509814	+	0.860285i	$0.670286\pi$
$972$	0	0
$973$	39.1500	1.25509
$974$	0	0
$975$	−3.24224	−0.103835
$976$	0	0
$977$	−44.7555	−1.43185	−0.715927	−	0.698175i	$-0.753996\pi$
$977$	−44.7555	−1.43185	−0.715927	+	0.698175i	$0.753996\pi$
$978$	0	0
$979$	102.706	3.28251
$980$	0	0
$981$	6.29684	0.201043
$982$	0	0
$983$	11.5228	0.367520	0.183760	−	0.982971i	$-0.441173\pi$
$983$	11.5228	0.367520	0.183760	+	0.982971i	$0.441173\pi$
$984$	0	0
$985$	14.1373	0.450452
$986$	0	0
$987$	13.9021	0.442508
$988$	0	0
$989$	−47.3026	−1.50413
$990$	0	0
$991$	−12.5569	−0.398882	−0.199441	−	0.979910i	$-0.563913\pi$
$991$	−12.5569	−0.398882	−0.199441	+	0.979910i	$0.563913\pi$
$992$	0	0
$993$	9.57057	0.303713
$994$	0	0
$995$	1.20016	0.0380476
$996$	0	0
$997$	−53.9311	−1.70801	−0.854007	−	0.520261i	$-0.825834\pi$
$997$	−53.9311	−1.70801	−0.854007	+	0.520261i	$0.825834\pi$
$998$	0	0
$999$	−23.9016	−0.756215

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.s.1.17	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.s.1.17	✓	24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q+1.49843 q^{3} +1.00000 q^{5} +1.73584 q^{7} -0.754697 q^{9} +O(q^{10})\)	\(q+1.49843 q^{3} +1.00000 q^{5} +1.73584 q^{7} -0.754697 q^{9} -6.27719 q^{11} -2.16376 q^{13} +1.49843 q^{15} +2.92391 q^{17} +5.08623 q^{19} +2.60104 q^{21} +9.07725 q^{23} +1.00000 q^{25} -5.62616 q^{27} +5.00134 q^{29} +1.20851 q^{31} -9.40596 q^{33} +1.73584 q^{35} +4.24830 q^{37} -3.24224 q^{39} -7.65235 q^{41} -5.21112 q^{43} -0.754697 q^{45} +5.34482 q^{47} -3.98686 q^{49} +4.38129 q^{51} +14.4133 q^{53} -6.27719 q^{55} +7.62137 q^{57} +8.09743 q^{59} +0.554623 q^{61} -1.31003 q^{63} -2.16376 q^{65} +5.27476 q^{67} +13.6017 q^{69} -8.57612 q^{71} +5.56709 q^{73} +1.49843 q^{75} -10.8962 q^{77} +4.31006 q^{79} -6.16634 q^{81} -4.84970 q^{83} +2.92391 q^{85} +7.49417 q^{87} -16.3618 q^{89} -3.75593 q^{91} +1.81087 q^{93} +5.08623 q^{95} +6.12811 q^{97} +4.73738 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9}+O(q^{10}) \) 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9	\( 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9} + 17 q^{11} + 16 q^{13} + 2 q^{15} + 22 q^{17} + 16 q^{19} - q^{21} + 7 q^{23} + 24 q^{25} - 4 q^{27} + 25 q^{29} + 28 q^{31} + 11 q^{33} + 3 q^{35} + 26 q^{37} + 13 q^{39} + 38 q^{41} - 13 q^{43} + 40 q^{45} + 12 q^{47} + 61 q^{49} + 53 q^{53} + 17 q^{55} + 30 q^{57} + 35 q^{59} + 44 q^{61} - 9 q^{63} + 16 q^{65} - 15 q^{67} + 9 q^{69} + 22 q^{71} + 31 q^{73} + 2 q^{75} + 26 q^{77} + 20 q^{79} + 88 q^{81} - 14 q^{83} + 22 q^{85} - 18 q^{87} + 37 q^{89} - 26 q^{91} + 13 q^{93} + 16 q^{95} + 21 q^{97} - 12 q^{99}+O(q^{100}) \) 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9 + 17 * q^11 + 16 * q^13 + 2 * q^15 + 22 * q^17 + 16 * q^19 - q^21 + 7 * q^23 + 24 * q^25 - 4 * q^27 + 25 * q^29 + 28 * q^31 + 11 * q^33 + 3 * q^35 + 26 * q^37 + 13 * q^39 + 38 * q^41 - 13 * q^43 + 40 * q^45 + 12 * q^47 + 61 * q^49 + 53 * q^53 + 17 * q^55 + 30 * q^57 + 35 * q^59 + 44 * q^61 - 9 * q^63 + 16 * q^65 - 15 * q^67 + 9 * q^69 + 22 * q^71 + 31 * q^73 + 2 * q^75 + 26 * q^77 + 20 * q^79 + 88 * q^81 - 14 * q^83 + 22 * q^85 - 18 * q^87 + 37 * q^89 - 26 * q^91 + 13 * q^93 + 16 * q^95 + 21 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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