LMFDB - Embedded newform 7440.2.a.bq.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7440,2,Mod(1,7440)] 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("7440.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7440.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,3,0,0,0,3,0,-6,0,0,0,-3,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$59.4086991038$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.7636.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 16x - 18 $ x^3 - 16*x - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1860)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.47467$ of defining polynomial
Character	$\chi$	$=$	7440.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +4.47467 q^{7} +1.00000 q^{9} -2.00000 q^{11} +4.47467 q^{13} -1.00000 q^{15} +1.07331 q^{17} -4.00000 q^{19} -4.47467 q^{21} +1.07331 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.54798 q^{29} +1.00000 q^{31} +2.00000 q^{33} +4.47467 q^{35} +2.47467 q^{37} -4.47467 q^{39} +4.00000 q^{41} -2.00000 q^{43} +1.00000 q^{45} +8.02265 q^{47} +13.0226 q^{49} -1.07331 q^{51} +10.0226 q^{53} -2.00000 q^{55} +4.00000 q^{57} -9.54798 q^{59} -0.146623 q^{61} +4.47467 q^{63} +4.47467 q^{65} +4.32804 q^{67} -1.07331 q^{69} -12.4973 q^{71} +15.4240 q^{73} -1.00000 q^{75} -8.94933 q^{77} -9.87602 q^{79} +1.00000 q^{81} +2.92669 q^{83} +1.07331 q^{85} +3.54798 q^{87} +15.4014 q^{89} +20.0226 q^{91} -1.00000 q^{93} -4.00000 q^{95} +1.85338 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{5} + 3 q^{9} - 6 q^{11} - 3 q^{15} + 2 q^{17} - 12 q^{19} + 2 q^{23} + 3 q^{25} - 3 q^{27} + 4 q^{29} + 3 q^{31} + 6 q^{33} - 6 q^{37} + 12 q^{41} - 6 q^{43} + 3 q^{45} - 4 q^{47} + 11 q^{49}+ \cdots - 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 + 3 * q^9 - 6 * q^11 - 3 * q^15 + 2 * q^17 - 12 * q^19 + 2 * q^23 + 3 * q^25 - 3 * q^27 + 4 * q^29 + 3 * q^31 + 6 * q^33 - 6 * q^37 + 12 * q^41 - 6 * q^43 + 3 * q^45 - 4 * q^47 + 11 * q^49 - 2 * q^51 + 2 * q^53 - 6 * q^55 + 12 * q^57 - 14 * q^59 + 2 * q^61 + 2 * q^67 - 2 * q^69 + 4 * q^71 + 6 * q^73 - 3 * q^75 - 4 * q^79 + 3 * q^81 + 10 * q^83 + 2 * q^85 - 4 * q^87 + 34 * q^89 + 32 * q^91 - 3 * q^93 - 12 * q^95 + 8 * q^97 - 6 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.47467	1.69127	0.845633	−	0.533765i	$-0.179223\pi$
$7$	4.47467	1.69127	0.845633	+	0.533765i	$0.179223\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	4.47467	1.24105	0.620525	−	0.784187i	$-0.286920\pi$
$13$	4.47467	1.24105	0.620525	+	0.784187i	$0.286920\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	1.07331	0.260316	0.130158	−	0.991493i	$-0.458451\pi$
$17$	1.07331	0.260316	0.130158	+	0.991493i	$0.458451\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−4.47467	−0.976452
$22$	0	0
$23$	1.07331	0.223801	0.111900	−	0.993719i	$-0.464306\pi$
$23$	1.07331	0.223801	0.111900	+	0.993719i	$0.464306\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.54798	−0.658843	−0.329422	−	0.944183i	$-0.606854\pi$
$29$	−3.54798	−0.658843	−0.329422	+	0.944183i	$0.606854\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	4.47467	0.756357
$36$	0	0
$37$	2.47467	0.406833	0.203416	−	0.979092i	$-0.434795\pi$
$37$	2.47467	0.406833	0.203416	+	0.979092i	$0.434795\pi$
$38$	0	0
$39$	−4.47467	−0.716520
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	8.02265	1.17022	0.585112	−	0.810953i	$-0.301051\pi$
$47$	8.02265	1.17022	0.585112	+	0.810953i	$0.301051\pi$
$48$	0	0
$49$	13.0226	1.86038
$50$	0	0
$51$	−1.07331	−0.150294
$52$	0	0
$53$	10.0226	1.37672	0.688358	−	0.725371i	$-0.258332\pi$
$53$	10.0226	1.37672	0.688358	+	0.725371i	$0.258332\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	−9.54798	−1.24304	−0.621520	−	0.783398i	$-0.713485\pi$
$59$	−9.54798	−1.24304	−0.621520	+	0.783398i	$0.713485\pi$
$60$	0	0
$61$	−0.146623	−0.0187731	−0.00938657	−	0.999956i	$-0.502988\pi$
$61$	−0.146623	−0.0187731	−0.00938657	+	0.999956i	$0.502988\pi$
$62$	0	0
$63$	4.47467	0.563755
$64$	0	0
$65$	4.47467	0.555014
$66$	0	0
$67$	4.32804	0.528755	0.264377	−	0.964419i	$-0.414834\pi$
$67$	4.32804	0.528755	0.264377	+	0.964419i	$0.414834\pi$
$68$	0	0
$69$	−1.07331	−0.129212
$70$	0	0
$71$	−12.4973	−1.48316	−0.741579	−	0.670865i	$-0.765923\pi$
$71$	−12.4973	−1.48316	−0.741579	+	0.670865i	$0.765923\pi$
$72$	0	0
$73$	15.4240	1.80524	0.902621	−	0.430435i	$-0.141640\pi$
$73$	15.4240	1.80524	0.902621	+	0.430435i	$0.141640\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−8.94933	−1.01987
$78$	0	0
$79$	−9.87602	−1.11114	−0.555570	−	0.831470i	$-0.687500\pi$
$79$	−9.87602	−1.11114	−0.555570	+	0.831470i	$0.687500\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.92669	0.321246	0.160623	−	0.987016i	$-0.448650\pi$
$83$	2.92669	0.321246	0.160623	+	0.987016i	$0.448650\pi$
$84$	0	0
$85$	1.07331	0.116417
$86$	0	0
$87$	3.54798	0.380383
$88$	0	0
$89$	15.4014	1.63254	0.816270	−	0.577670i	$-0.196038\pi$
$89$	15.4014	1.63254	0.816270	+	0.577670i	$0.196038\pi$
$90$	0	0
$91$	20.0226	2.09894
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	1.85338	0.188182	0.0940910	−	0.995564i	$-0.470006\pi$
$97$	1.85338	0.188182	0.0940910	+	0.995564i	$0.470006\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−1.85338	−0.184418	−0.0922090	−	0.995740i	$-0.529393\pi$
$101$	−1.85338	−0.184418	−0.0922090	+	0.995740i	$0.529393\pi$
$102$	0	0
$103$	15.4240	1.51977	0.759886	−	0.650056i	$-0.225255\pi$
$103$	15.4240	1.51977	0.759886	+	0.650056i	$0.225255\pi$
$104$	0	0
$105$	−4.47467	−0.436683
$106$	0	0
$107$	1.21993	0.117935	0.0589677	−	0.998260i	$-0.481219\pi$
$107$	1.21993	0.117935	0.0589677	+	0.998260i	$0.481219\pi$
$108$	0	0
$109$	5.07331	0.485935	0.242968	−	0.970034i	$-0.421879\pi$
$109$	5.07331	0.485935	0.242968	+	0.970034i	$0.421879\pi$
$110$	0	0
$111$	−2.47467	−0.234885
$112$	0	0
$113$	1.05067	0.0988383	0.0494192	−	0.998778i	$-0.484263\pi$
$113$	1.05067	0.0988383	0.0494192	+	0.998778i	$0.484263\pi$
$114$	0	0
$115$	1.07331	0.100087
$116$	0	0
$117$	4.47467	0.413683
$118$	0	0
$119$	4.80271	0.440264
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−4.00000	−0.360668
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	3.05067	0.270703	0.135351	−	0.990798i	$-0.456784\pi$
$127$	3.05067	0.270703	0.135351	+	0.990798i	$0.456784\pi$
$128$	0	0
$129$	2.00000	0.176090
$130$	0	0
$131$	−7.40136	−0.646659	−0.323330	−	0.946286i	$-0.604802\pi$
$131$	−7.40136	−0.646659	−0.323330	+	0.946286i	$0.604802\pi$
$132$	0	0
$133$	−17.8987	−1.55201
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−14.9720	−1.27914	−0.639571	−	0.768732i	$-0.720888\pi$
$137$	−14.9720	−1.27914	−0.639571	+	0.768732i	$0.720888\pi$
$138$	0	0
$139$	14.8027	1.25555	0.627775	−	0.778395i	$-0.283966\pi$
$139$	14.8027	1.25555	0.627775	+	0.778395i	$0.283966\pi$
$140$	0	0
$141$	−8.02265	−0.675629
$142$	0	0
$143$	−8.94933	−0.748381
$144$	0	0
$145$	−3.54798	−0.294644
$146$	0	0
$147$	−13.0226	−1.07409
$148$	0	0
$149$	−9.89867	−0.810931	−0.405465	−	0.914110i	$-0.632890\pi$
$149$	−9.89867	−0.810931	−0.405465	+	0.914110i	$0.632890\pi$
$150$	0	0
$151$	4.92669	0.400928	0.200464	−	0.979701i	$-0.435755\pi$
$151$	4.92669	0.400928	0.200464	+	0.979701i	$0.435755\pi$
$152$	0	0
$153$	1.07331	0.0867721
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	0	0
$159$	−10.0226	−0.794848
$160$	0	0
$161$	4.80271	0.378507
$162$	0	0
$163$	14.4747	1.13374	0.566872	−	0.823806i	$-0.308154\pi$
$163$	14.4747	1.13374	0.566872	+	0.823806i	$0.308154\pi$
$164$	0	0
$165$	2.00000	0.155700
$166$	0	0
$167$	−17.2426	−1.33427	−0.667135	−	0.744936i	$-0.732480\pi$
$167$	−17.2426	−1.33427	−0.667135	+	0.744936i	$0.732480\pi$
$168$	0	0
$169$	7.02265	0.540204
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	−6.94933	−0.528348	−0.264174	−	0.964475i	$-0.585099\pi$
$173$	−6.94933	−0.528348	−0.264174	+	0.964475i	$0.585099\pi$
$174$	0	0
$175$	4.47467	0.338253
$176$	0	0
$177$	9.54798	0.717670
$178$	0	0
$179$	−5.85338	−0.437502	−0.218751	−	0.975781i	$-0.570198\pi$
$179$	−5.85338	−0.437502	−0.218751	+	0.975781i	$0.570198\pi$
$180$	0	0
$181$	−17.7520	−1.31950	−0.659750	−	0.751486i	$-0.729338\pi$
$181$	−17.7520	−1.31950	−0.659750	+	0.751486i	$0.729338\pi$
$182$	0	0
$183$	0.146623	0.0108387
$184$	0	0
$185$	2.47467	0.181941
$186$	0	0
$187$	−2.14662	−0.156977
$188$	0	0
$189$	−4.47467	−0.325484
$190$	0	0
$191$	−8.35069	−0.604235	−0.302117	−	0.953271i	$-0.597693\pi$
$191$	−8.35069	−0.604235	−0.302117	+	0.953271i	$0.597693\pi$
$192$	0	0
$193$	−8.00000	−0.575853	−0.287926	−	0.957653i	$-0.592966\pi$
$193$	−8.00000	−0.575853	−0.287926	+	0.957653i	$0.592966\pi$
$194$	0	0
$195$	−4.47467	−0.320438
$196$	0	0
$197$	6.97198	0.496733	0.248366	−	0.968666i	$-0.420106\pi$
$197$	6.97198	0.496733	0.248366	+	0.968666i	$0.420106\pi$
$198$	0	0
$199$	5.21993	0.370031	0.185016	−	0.982736i	$-0.440766\pi$
$199$	5.21993	0.370031	0.185016	+	0.982736i	$0.440766\pi$
$200$	0	0
$201$	−4.32804	−0.305277
$202$	0	0
$203$	−15.8760	−1.11428
$204$	0	0
$205$	4.00000	0.279372
$206$	0	0
$207$	1.07331	0.0746003
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	15.0960	1.03925	0.519624	−	0.854395i	$-0.326072\pi$
$211$	15.0960	1.03925	0.519624	+	0.854395i	$0.326072\pi$
$212$	0	0
$213$	12.4973	0.856302
$214$	0	0
$215$	−2.00000	−0.136399
$216$	0	0
$217$	4.47467	0.303760
$218$	0	0
$219$	−15.4240	−1.04226
$220$	0	0
$221$	4.80271	0.323065
$222$	0	0
$223$	18.9493	1.26894	0.634471	−	0.772947i	$-0.281218\pi$
$223$	18.9493	1.26894	0.634471	+	0.772947i	$0.281218\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	7.07331	0.469472	0.234736	−	0.972059i	$-0.424577\pi$
$227$	7.07331	0.469472	0.234736	+	0.972059i	$0.424577\pi$
$228$	0	0
$229$	−5.09596	−0.336750	−0.168375	−	0.985723i	$-0.553852\pi$
$229$	−5.09596	−0.336750	−0.168375	+	0.985723i	$0.553852\pi$
$230$	0	0
$231$	8.94933	0.588823
$232$	0	0
$233$	13.8760	0.909048	0.454524	−	0.890734i	$-0.349809\pi$
$233$	13.8760	0.909048	0.454524	+	0.890734i	$0.349809\pi$
$234$	0	0
$235$	8.02265	0.523340
$236$	0	0
$237$	9.87602	0.641517
$238$	0	0
$239$	1.09596	0.0708916	0.0354458	−	0.999372i	$-0.488715\pi$
$239$	1.09596	0.0708916	0.0354458	+	0.999372i	$0.488715\pi$
$240$	0	0
$241$	11.2426	0.724198	0.362099	−	0.932140i	$-0.382060\pi$
$241$	11.2426	0.724198	0.362099	+	0.932140i	$0.382060\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	13.0226	0.831986
$246$	0	0
$247$	−17.8987	−1.13886
$248$	0	0
$249$	−2.92669	−0.185471
$250$	0	0
$251$	−16.0453	−1.01277	−0.506385	−	0.862308i	$-0.669018\pi$
$251$	−16.0453	−1.01277	−0.506385	+	0.862308i	$0.669018\pi$
$252$	0	0
$253$	−2.14662	−0.134957
$254$	0	0
$255$	−1.07331	−0.0672134
$256$	0	0
$257$	4.02265	0.250926	0.125463	−	0.992098i	$-0.459958\pi$
$257$	4.02265	0.250926	0.125463	+	0.992098i	$0.459958\pi$
$258$	0	0
$259$	11.0733	0.688062
$260$	0	0
$261$	−3.54798	−0.219614
$262$	0	0
$263$	27.0960	1.67081	0.835404	−	0.549636i	$-0.185234\pi$
$263$	27.0960	1.67081	0.835404	+	0.549636i	$0.185234\pi$
$264$	0	0
$265$	10.0226	0.615686
$266$	0	0
$267$	−15.4014	−0.942548
$268$	0	0
$269$	0.745267	0.0454397	0.0227199	−	0.999742i	$-0.492767\pi$
$269$	0.745267	0.0454397	0.0227199	+	0.999742i	$0.492767\pi$
$270$	0	0
$271$	−20.9946	−1.27533	−0.637666	−	0.770313i	$-0.720100\pi$
$271$	−20.9946	−1.27533	−0.637666	+	0.770313i	$0.720100\pi$
$272$	0	0
$273$	−20.0226	−1.21183
$274$	0	0
$275$	−2.00000	−0.120605
$276$	0	0
$277$	−6.47467	−0.389025	−0.194513	−	0.980900i	$-0.562312\pi$
$277$	−6.47467	−0.389025	−0.194513	+	0.980900i	$0.562312\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	1.05067	0.0626775	0.0313387	−	0.999509i	$-0.490023\pi$
$281$	1.05067	0.0626775	0.0313387	+	0.999509i	$0.490023\pi$
$282$	0	0
$283$	−7.57062	−0.450027	−0.225013	−	0.974356i	$-0.572243\pi$
$283$	−7.57062	−0.450027	−0.225013	+	0.974356i	$0.572243\pi$
$284$	0	0
$285$	4.00000	0.236940
$286$	0	0
$287$	17.8987	1.05652
$288$	0	0
$289$	−15.8480	−0.932235
$290$	0	0
$291$	−1.85338	−0.108647
$292$	0	0
$293$	−5.87602	−0.343281	−0.171640	−	0.985160i	$-0.554907\pi$
$293$	−5.87602	−0.343281	−0.171640	+	0.985160i	$0.554907\pi$
$294$	0	0
$295$	−9.54798	−0.555905
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	4.80271	0.277748
$300$	0	0
$301$	−8.94933	−0.515831
$302$	0	0
$303$	1.85338	0.106474
$304$	0	0
$305$	−0.146623	−0.00839560
$306$	0	0
$307$	17.7172	1.01118	0.505588	−	0.862775i	$-0.331275\pi$
$307$	17.7172	1.01118	0.505588	+	0.862775i	$0.331275\pi$
$308$	0	0
$309$	−15.4240	−0.877441
$310$	0	0
$311$	14.3507	0.813753	0.406876	−	0.913483i	$-0.366618\pi$
$311$	14.3507	0.813753	0.406876	+	0.913483i	$0.366618\pi$
$312$	0	0
$313$	19.3227	1.09218	0.546091	−	0.837726i	$-0.316115\pi$
$313$	19.3227	1.09218	0.546091	+	0.837726i	$0.316115\pi$
$314$	0	0
$315$	4.47467	0.252119
$316$	0	0
$317$	−32.0226	−1.79857	−0.899285	−	0.437362i	$-0.855913\pi$
$317$	−32.0226	−1.79857	−0.899285	+	0.437362i	$0.855913\pi$
$318$	0	0
$319$	7.09596	0.397297
$320$	0	0
$321$	−1.21993	−0.0680901
$322$	0	0
$323$	−4.29325	−0.238883
$324$	0	0
$325$	4.47467	0.248210
$326$	0	0
$327$	−5.07331	−0.280555
$328$	0	0
$329$	35.8987	1.97916
$330$	0	0
$331$	29.8760	1.64213	0.821067	−	0.570831i	$-0.193379\pi$
$331$	29.8760	1.64213	0.821067	+	0.570831i	$0.193379\pi$
$332$	0	0
$333$	2.47467	0.135611
$334$	0	0
$335$	4.32804	0.236466
$336$	0	0
$337$	6.47467	0.352698	0.176349	−	0.984328i	$-0.443571\pi$
$337$	6.47467	0.352698	0.176349	+	0.984328i	$0.443571\pi$
$338$	0	0
$339$	−1.05067	−0.0570643
$340$	0	0
$341$	−2.00000	−0.108306
$342$	0	0
$343$	26.9493	1.45513
$344$	0	0
$345$	−1.07331	−0.0577851
$346$	0	0
$347$	−9.24258	−0.496168	−0.248084	−	0.968739i	$-0.579801\pi$
$347$	−9.24258	−0.496168	−0.248084	+	0.968739i	$0.579801\pi$
$348$	0	0
$349$	6.92669	0.370777	0.185389	−	0.982665i	$-0.440646\pi$
$349$	6.92669	0.370777	0.185389	+	0.982665i	$0.440646\pi$
$350$	0	0
$351$	−4.47467	−0.238840
$352$	0	0
$353$	0.780066	0.0415187	0.0207594	−	0.999785i	$-0.493392\pi$
$353$	0.780066	0.0415187	0.0207594	+	0.999785i	$0.493392\pi$
$354$	0	0
$355$	−12.4973	−0.663288
$356$	0	0
$357$	−4.80271	−0.254186
$358$	0	0
$359$	−32.3960	−1.70979	−0.854897	−	0.518797i	$-0.826380\pi$
$359$	−32.3960	−1.70979	−0.854897	+	0.518797i	$0.826380\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	15.4240	0.807329
$366$	0	0
$367$	−1.89867	−0.0991097	−0.0495548	−	0.998771i	$-0.515780\pi$
$367$	−1.89867	−0.0991097	−0.0495548	+	0.998771i	$0.515780\pi$
$368$	0	0
$369$	4.00000	0.208232
$370$	0	0
$371$	44.8480	2.32839
$372$	0	0
$373$	−29.0960	−1.50653	−0.753266	−	0.657716i	$-0.771523\pi$
$373$	−29.0960	−1.50653	−0.753266	+	0.657716i	$0.771523\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−15.8760	−0.817657
$378$	0	0
$379$	29.8987	1.53579	0.767896	−	0.640575i	$-0.221304\pi$
$379$	29.8987	1.53579	0.767896	+	0.640575i	$0.221304\pi$
$380$	0	0
$381$	−3.05067	−0.156290
$382$	0	0
$383$	32.8254	1.67730	0.838649	−	0.544673i	$-0.183346\pi$
$383$	32.8254	1.67730	0.838649	+	0.544673i	$0.183346\pi$
$384$	0	0
$385$	−8.94933	−0.456100
$386$	0	0
$387$	−2.00000	−0.101666
$388$	0	0
$389$	33.5480	1.70095	0.850475	−	0.526015i	$-0.176315\pi$
$389$	33.5480	1.70095	0.850475	+	0.526015i	$0.176315\pi$
$390$	0	0
$391$	1.15200	0.0582590
$392$	0	0
$393$	7.40136	0.373349
$394$	0	0
$395$	−9.87602	−0.496917
$396$	0	0
$397$	10.0453	0.504159	0.252079	−	0.967707i	$-0.418886\pi$
$397$	10.0453	0.504159	0.252079	+	0.967707i	$0.418886\pi$
$398$	0	0
$399$	17.8987	0.896054
$400$	0	0
$401$	−29.5933	−1.47782	−0.738909	−	0.673806i	$-0.764659\pi$
$401$	−29.5933	−1.47782	−0.738909	+	0.673806i	$0.764659\pi$
$402$	0	0
$403$	4.47467	0.222899
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−4.94933	−0.245329
$408$	0	0
$409$	5.75205	0.284420	0.142210	−	0.989836i	$-0.454579\pi$
$409$	5.75205	0.284420	0.142210	+	0.989836i	$0.454579\pi$
$410$	0	0
$411$	14.9720	0.738513
$412$	0	0
$413$	−42.7240	−2.10231
$414$	0	0
$415$	2.92669	0.143665
$416$	0	0
$417$	−14.8027	−0.724892
$418$	0	0
$419$	24.5426	1.19898	0.599492	−	0.800380i	$-0.295369\pi$
$419$	24.5426	1.19898	0.599492	+	0.800380i	$0.295369\pi$
$420$	0	0
$421$	−22.0679	−1.07553	−0.537763	−	0.843096i	$-0.680730\pi$
$421$	−22.0679	−1.07553	−0.537763	+	0.843096i	$0.680730\pi$
$422$	0	0
$423$	8.02265	0.390074
$424$	0	0
$425$	1.07331	0.0520633
$426$	0	0
$427$	−0.656088	−0.0317503
$428$	0	0
$429$	8.94933	0.432078
$430$	0	0
$431$	26.3507	1.26927	0.634634	−	0.772813i	$-0.281151\pi$
$431$	26.3507	1.26927	0.634634	+	0.772813i	$0.281151\pi$
$432$	0	0
$433$	29.6159	1.42325	0.711625	−	0.702559i	$-0.247960\pi$
$433$	29.6159	1.42325	0.711625	+	0.702559i	$0.247960\pi$
$434$	0	0
$435$	3.54798	0.170113
$436$	0	0
$437$	−4.29325	−0.205374
$438$	0	0
$439$	−27.7973	−1.32669	−0.663347	−	0.748312i	$-0.730865\pi$
$439$	−27.7973	−1.32669	−0.663347	+	0.748312i	$0.730865\pi$
$440$	0	0
$441$	13.0226	0.620126
$442$	0	0
$443$	17.2199	0.818144	0.409072	−	0.912502i	$-0.365853\pi$
$443$	17.2199	0.818144	0.409072	+	0.912502i	$0.365853\pi$
$444$	0	0
$445$	15.4014	0.730094
$446$	0	0
$447$	9.89867	0.468191
$448$	0	0
$449$	4.30540	0.203184	0.101592	−	0.994826i	$-0.467606\pi$
$449$	4.30540	0.203184	0.101592	+	0.994826i	$0.467606\pi$
$450$	0	0
$451$	−8.00000	−0.376705
$452$	0	0
$453$	−4.92669	−0.231476
$454$	0	0
$455$	20.0226	0.938676
$456$	0	0
$457$	11.3787	0.532274	0.266137	−	0.963935i	$-0.414253\pi$
$457$	11.3787	0.532274	0.266137	+	0.963935i	$0.414253\pi$
$458$	0	0
$459$	−1.07331	−0.0500979
$460$	0	0
$461$	3.54798	0.165246	0.0826229	−	0.996581i	$-0.473670\pi$
$461$	3.54798	0.165246	0.0826229	+	0.996581i	$0.473670\pi$
$462$	0	0
$463$	41.7520	1.94038	0.970191	−	0.242341i	$-0.0779154\pi$
$463$	41.7520	1.94038	0.970191	+	0.242341i	$0.0779154\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	−29.6281	−1.37102	−0.685512	−	0.728062i	$-0.740421\pi$
$467$	−29.6281	−1.37102	−0.685512	+	0.728062i	$0.740421\pi$
$468$	0	0
$469$	19.3666	0.894265
$470$	0	0
$471$	4.00000	0.184310
$472$	0	0
$473$	4.00000	0.183920
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	10.0226	0.458905
$478$	0	0
$479$	33.4466	1.52822	0.764108	−	0.645088i	$-0.223179\pi$
$479$	33.4466	1.52822	0.764108	+	0.645088i	$0.223179\pi$
$480$	0	0
$481$	11.0733	0.504900
$482$	0	0
$483$	−4.80271	−0.218531
$484$	0	0
$485$	1.85338	0.0841575
$486$	0	0
$487$	−23.3892	−1.05987	−0.529933	−	0.848040i	$-0.677783\pi$
$487$	−23.3892	−1.05987	−0.529933	+	0.848040i	$0.677783\pi$
$488$	0	0
$489$	−14.4747	−0.654567
$490$	0	0
$491$	−32.7014	−1.47579	−0.737896	−	0.674914i	$-0.764181\pi$
$491$	−32.7014	−1.47579	−0.737896	+	0.674914i	$0.764181\pi$
$492$	0	0
$493$	−3.80809	−0.171508
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	0	0
$497$	−55.9213	−2.50841
$498$	0	0
$499$	−18.1919	−0.814382	−0.407191	−	0.913343i	$-0.633492\pi$
$499$	−18.1919	−0.814382	−0.407191	+	0.913343i	$0.633492\pi$
$500$	0	0
$501$	17.2426	0.770342
$502$	0	0
$503$	−36.0679	−1.60819	−0.804095	−	0.594501i	$-0.797350\pi$
$503$	−36.0679	−1.60819	−0.804095	+	0.594501i	$0.797350\pi$
$504$	0	0
$505$	−1.85338	−0.0824742
$506$	0	0
$507$	−7.02265	−0.311887
$508$	0	0
$509$	42.5426	1.88567	0.942834	−	0.333263i	$-0.108150\pi$
$509$	42.5426	1.88567	0.942834	+	0.333263i	$0.108150\pi$
$510$	0	0
$511$	69.0173	3.05314
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	15.4240	0.679663
$516$	0	0
$517$	−16.0453	−0.705671
$518$	0	0
$519$	6.94933	0.305042
$520$	0	0
$521$	8.29325	0.363334	0.181667	−	0.983360i	$-0.441851\pi$
$521$	8.29325	0.363334	0.181667	+	0.983360i	$0.441851\pi$
$522$	0	0
$523$	5.19729	0.227262	0.113631	−	0.993523i	$-0.463752\pi$
$523$	5.19729	0.227262	0.113631	+	0.993523i	$0.463752\pi$
$524$	0	0
$525$	−4.47467	−0.195290
$526$	0	0
$527$	1.07331	0.0467542
$528$	0	0
$529$	−21.8480	−0.949913
$530$	0	0
$531$	−9.54798	−0.414347
$532$	0	0
$533$	17.8987	0.775277
$534$	0	0
$535$	1.21993	0.0527424
$536$	0	0
$537$	5.85338	0.252592
$538$	0	0
$539$	−26.0453	−1.12185
$540$	0	0
$541$	43.6281	1.87572	0.937859	−	0.347018i	$-0.112806\pi$
$541$	43.6281	1.87572	0.937859	+	0.347018i	$0.112806\pi$
$542$	0	0
$543$	17.7520	0.761813
$544$	0	0
$545$	5.07331	0.217317
$546$	0	0
$547$	−5.52533	−0.236246	−0.118123	−	0.992999i	$-0.537688\pi$
$547$	−5.52533	−0.236246	−0.118123	+	0.992999i	$0.537688\pi$
$548$	0	0
$549$	−0.146623	−0.00625771
$550$	0	0
$551$	14.1919	0.604596
$552$	0	0
$553$	−44.1919	−1.87923
$554$	0	0
$555$	−2.47467	−0.105044
$556$	0	0
$557$	−17.0733	−0.723419	−0.361710	−	0.932291i	$-0.617807\pi$
$557$	−17.0733	−0.723419	−0.361710	+	0.932291i	$0.617807\pi$
$558$	0	0
$559$	−8.94933	−0.378517
$560$	0	0
$561$	2.14662	0.0906305
$562$	0	0
$563$	11.0733	0.466684	0.233342	−	0.972395i	$-0.425034\pi$
$563$	11.0733	0.466684	0.233342	+	0.972395i	$0.425034\pi$
$564$	0	0
$565$	1.05067	0.0442018
$566$	0	0
$567$	4.47467	0.187918
$568$	0	0
$569$	35.5933	1.49215	0.746074	−	0.665863i	$-0.231937\pi$
$569$	35.5933	1.49215	0.746074	+	0.665863i	$0.231937\pi$
$570$	0	0
$571$	−34.8933	−1.46024	−0.730119	−	0.683320i	$-0.760536\pi$
$571$	−34.8933	−1.46024	−0.730119	+	0.683320i	$0.760536\pi$
$572$	0	0
$573$	8.35069	0.348855
$574$	0	0
$575$	1.07331	0.0447602
$576$	0	0
$577$	−46.8933	−1.95219	−0.976097	−	0.217337i	$-0.930263\pi$
$577$	−46.8933	−1.95219	−0.976097	+	0.217337i	$0.930263\pi$
$578$	0	0
$579$	8.00000	0.332469
$580$	0	0
$581$	13.0960	0.543312
$582$	0	0
$583$	−20.0453	−0.830191
$584$	0	0
$585$	4.47467	0.185005
$586$	0	0
$587$	−7.09596	−0.292881	−0.146441	−	0.989219i	$-0.546782\pi$
$587$	−7.09596	−0.292881	−0.146441	+	0.989219i	$0.546782\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	−6.97198	−0.286789
$592$	0	0
$593$	18.9946	0.780016	0.390008	−	0.920812i	$-0.372472\pi$
$593$	18.9946	0.780016	0.390008	+	0.920812i	$0.372472\pi$
$594$	0	0
$595$	4.80271	0.196892
$596$	0	0
$597$	−5.21993	−0.213638
$598$	0	0
$599$	−21.4014	−0.874436	−0.437218	−	0.899356i	$-0.644036\pi$
$599$	−21.4014	−0.874436	−0.437218	+	0.899356i	$0.644036\pi$
$600$	0	0
$601$	17.0507	0.695511	0.347756	−	0.937585i	$-0.386944\pi$
$601$	17.0507	0.695511	0.347756	+	0.937585i	$0.386944\pi$
$602$	0	0
$603$	4.32804	0.176252
$604$	0	0
$605$	−7.00000	−0.284590
$606$	0	0
$607$	47.2213	1.91665	0.958327	−	0.285672i	$-0.0922167\pi$
$607$	47.2213	1.91665	0.958327	+	0.285672i	$0.0922167\pi$
$608$	0	0
$609$	15.8760	0.643329
$610$	0	0
$611$	35.8987	1.45230
$612$	0	0
$613$	13.1308	0.530346	0.265173	−	0.964201i	$-0.414571\pi$
$613$	13.1308	0.530346	0.265173	+	0.964201i	$0.414571\pi$
$614$	0	0
$615$	−4.00000	−0.161296
$616$	0	0
$617$	−31.6054	−1.27239	−0.636193	−	0.771530i	$-0.719492\pi$
$617$	−31.6054	−1.27239	−0.636193	+	0.771530i	$0.719492\pi$
$618$	0	0
$619$	−6.16927	−0.247964	−0.123982	−	0.992284i	$-0.539566\pi$
$619$	−6.16927	−0.247964	−0.123982	+	0.992284i	$0.539566\pi$
$620$	0	0
$621$	−1.07331	−0.0430705
$622$	0	0
$623$	68.9159	2.76106
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−8.00000	−0.319489
$628$	0	0
$629$	2.65609	0.105905
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	0	0
$633$	−15.0960	−0.600010
$634$	0	0
$635$	3.05067	0.121062
$636$	0	0
$637$	58.2720	2.30882
$638$	0	0
$639$	−12.4973	−0.494386
$640$	0	0
$641$	35.5933	1.40585	0.702925	−	0.711264i	$-0.251877\pi$
$641$	35.5933	1.40585	0.702925	+	0.711264i	$0.251877\pi$
$642$	0	0
$643$	−0.904043	−0.0356520	−0.0178260	−	0.999841i	$-0.505674\pi$
$643$	−0.904043	−0.0356520	−0.0178260	+	0.999841i	$0.505674\pi$
$644$	0	0
$645$	2.00000	0.0787499
$646$	0	0
$647$	4.24795	0.167004	0.0835022	−	0.996508i	$-0.473389\pi$
$647$	4.24795	0.167004	0.0835022	+	0.996508i	$0.473389\pi$
$648$	0	0
$649$	19.0960	0.749582
$650$	0	0
$651$	−4.47467	−0.175376
$652$	0	0
$653$	12.0679	0.472255	0.236127	−	0.971722i	$-0.424122\pi$
$653$	12.0679	0.472255	0.236127	+	0.971722i	$0.424122\pi$
$654$	0	0
$655$	−7.40136	−0.289195
$656$	0	0
$657$	15.4240	0.601748
$658$	0	0
$659$	−21.6946	−0.845102	−0.422551	−	0.906339i	$-0.638865\pi$
$659$	−21.6946	−0.845102	−0.422551	+	0.906339i	$0.638865\pi$
$660$	0	0
$661$	5.05067	0.196448	0.0982241	−	0.995164i	$-0.468684\pi$
$661$	5.05067	0.196448	0.0982241	+	0.995164i	$0.468684\pi$
$662$	0	0
$663$	−4.80271	−0.186522
$664$	0	0
$665$	−17.8987	−0.694081
$666$	0	0
$667$	−3.80809	−0.147450
$668$	0	0
$669$	−18.9493	−0.732624
$670$	0	0
$671$	0.293246	0.0113206
$672$	0	0
$673$	−33.7625	−1.30145	−0.650725	−	0.759313i	$-0.725535\pi$
$673$	−33.7625	−1.30145	−0.650725	+	0.759313i	$0.725535\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−35.8760	−1.37883	−0.689414	−	0.724368i	$-0.742132\pi$
$677$	−35.8760	−1.37883	−0.689414	+	0.724368i	$0.742132\pi$
$678$	0	0
$679$	8.29325	0.318266
$680$	0	0
$681$	−7.07331	−0.271050
$682$	0	0
$683$	−16.6787	−0.638194	−0.319097	−	0.947722i	$-0.603380\pi$
$683$	−16.6787	−0.638194	−0.319097	+	0.947722i	$0.603380\pi$
$684$	0	0
$685$	−14.9720	−0.572050
$686$	0	0
$687$	5.09596	0.194423
$688$	0	0
$689$	44.8480	1.70857
$690$	0	0
$691$	26.8027	1.01962	0.509812	−	0.860286i	$-0.329715\pi$
$691$	26.8027	1.01962	0.509812	+	0.860286i	$0.329715\pi$
$692$	0	0
$693$	−8.94933	−0.339957
$694$	0	0
$695$	14.8027	0.561499
$696$	0	0
$697$	4.29325	0.162618
$698$	0	0
$699$	−13.8760	−0.524839
$700$	0	0
$701$	12.9493	0.489090	0.244545	−	0.969638i	$-0.421361\pi$
$701$	12.9493	0.489090	0.244545	+	0.969638i	$0.421361\pi$
$702$	0	0
$703$	−9.89867	−0.373335
$704$	0	0
$705$	−8.02265	−0.302150
$706$	0	0
$707$	−8.29325	−0.311900
$708$	0	0
$709$	−19.6054	−0.736297	−0.368149	−	0.929767i	$-0.620008\pi$
$709$	−19.6054	−0.736297	−0.368149	+	0.929767i	$0.620008\pi$
$710$	0	0
$711$	−9.87602	−0.370380
$712$	0	0
$713$	1.07331	0.0401958
$714$	0	0
$715$	−8.94933	−0.334686
$716$	0	0
$717$	−1.09596	−0.0409293
$718$	0	0
$719$	22.0906	0.823840	0.411920	−	0.911220i	$-0.364858\pi$
$719$	22.0906	0.823840	0.411920	+	0.911220i	$0.364858\pi$
$720$	0	0
$721$	69.0173	2.57034
$722$	0	0
$723$	−11.2426	−0.418116
$724$	0	0
$725$	−3.54798	−0.131769
$726$	0	0
$727$	12.3280	0.457222	0.228611	−	0.973518i	$-0.426582\pi$
$727$	12.3280	0.457222	0.228611	+	0.973518i	$0.426582\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.14662	−0.0793957
$732$	0	0
$733$	−25.7973	−0.952846	−0.476423	−	0.879216i	$-0.658067\pi$
$733$	−25.7973	−0.952846	−0.476423	+	0.879216i	$0.658067\pi$
$734$	0	0
$735$	−13.0226	−0.480348
$736$	0	0
$737$	−8.65609	−0.318851
$738$	0	0
$739$	11.1186	0.409004	0.204502	−	0.978866i	$-0.434442\pi$
$739$	11.1186	0.409004	0.204502	+	0.978866i	$0.434442\pi$
$740$	0	0
$741$	17.8987	0.657524
$742$	0	0
$743$	22.8027	0.836550	0.418275	−	0.908320i	$-0.362635\pi$
$743$	22.8027	0.836550	0.418275	+	0.908320i	$0.362635\pi$
$744$	0	0
$745$	−9.89867	−0.362659
$746$	0	0
$747$	2.92669	0.107082
$748$	0	0
$749$	5.45880	0.199460
$750$	0	0
$751$	0.949334	0.0346417	0.0173208	−	0.999850i	$-0.494486\pi$
$751$	0.949334	0.0346417	0.0173208	+	0.999850i	$0.494486\pi$
$752$	0	0
$753$	16.0453	0.584723
$754$	0	0
$755$	4.92669	0.179301
$756$	0	0
$757$	36.8132	1.33800	0.668999	−	0.743263i	$-0.266723\pi$
$757$	36.8132	1.33800	0.668999	+	0.743263i	$0.266723\pi$
$758$	0	0
$759$	2.14662	0.0779175
$760$	0	0
$761$	35.5480	1.28861	0.644307	−	0.764767i	$-0.277146\pi$
$761$	35.5480	1.28861	0.644307	+	0.764767i	$0.277146\pi$
$762$	0	0
$763$	22.7014	0.821845
$764$	0	0
$765$	1.07331	0.0388057
$766$	0	0
$767$	−42.7240	−1.54268
$768$	0	0
$769$	−25.4118	−0.916375	−0.458187	−	0.888856i	$-0.651501\pi$
$769$	−25.4118	−0.916375	−0.458187	+	0.888856i	$0.651501\pi$
$770$	0	0
$771$	−4.02265	−0.144872
$772$	0	0
$773$	−30.7240	−1.10507	−0.552533	−	0.833491i	$-0.686339\pi$
$773$	−30.7240	−1.10507	−0.552533	+	0.833491i	$0.686339\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	−11.0733	−0.397253
$778$	0	0
$779$	−16.0000	−0.573259
$780$	0	0
$781$	24.9946	0.894378
$782$	0	0
$783$	3.54798	0.126794
$784$	0	0
$785$	−4.00000	−0.142766
$786$	0	0
$787$	24.8933	0.887350	0.443675	−	0.896188i	$-0.353674\pi$
$787$	24.8933	0.887350	0.443675	+	0.896188i	$0.353674\pi$
$788$	0	0
$789$	−27.0960	−0.964642
$790$	0	0
$791$	4.70138	0.167162
$792$	0	0
$793$	−0.656088	−0.0232984
$794$	0	0
$795$	−10.0226	−0.355467
$796$	0	0
$797$	−17.0733	−0.604768	−0.302384	−	0.953186i	$-0.597782\pi$
$797$	−17.0733	−0.604768	−0.302384	+	0.953186i	$0.597782\pi$
$798$	0	0
$799$	8.61080	0.304628
$800$	0	0
$801$	15.4014	0.544180
$802$	0	0
$803$	−30.8480	−1.08860
$804$	0	0
$805$	4.80271	0.169273
$806$	0	0
$807$	−0.745267	−0.0262346
$808$	0	0
$809$	2.45202	0.0862085	0.0431042	−	0.999071i	$-0.486275\pi$
$809$	2.45202	0.0862085	0.0431042	+	0.999071i	$0.486275\pi$
$810$	0	0
$811$	15.1412	0.531681	0.265841	−	0.964017i	$-0.414351\pi$
$811$	15.1412	0.531681	0.265841	+	0.964017i	$0.414351\pi$
$812$	0	0
$813$	20.9946	0.736314
$814$	0	0
$815$	14.4747	0.507025
$816$	0	0
$817$	8.00000	0.279885
$818$	0	0
$819$	20.0226	0.699648
$820$	0	0
$821$	−2.45202	−0.0855761	−0.0427881	−	0.999084i	$-0.513624\pi$
$821$	−2.45202	−0.0855761	−0.0427881	+	0.999084i	$0.513624\pi$
$822$	0	0
$823$	−30.0453	−1.04731	−0.523657	−	0.851929i	$-0.675432\pi$
$823$	−30.0453	−1.04731	−0.523657	+	0.851929i	$0.675432\pi$
$824$	0	0
$825$	2.00000	0.0696311
$826$	0	0
$827$	56.9159	1.97916	0.989581	−	0.143980i	$-0.0459902\pi$
$827$	56.9159	1.97916	0.989581	+	0.143980i	$0.0459902\pi$
$828$	0	0
$829$	−32.8933	−1.14243	−0.571216	−	0.820800i	$-0.693528\pi$
$829$	−32.8933	−1.14243	−0.571216	+	0.820800i	$0.693528\pi$
$830$	0	0
$831$	6.47467	0.224604
$832$	0	0
$833$	13.9774	0.484287
$834$	0	0
$835$	−17.2426	−0.596704
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−38.7906	−1.33920	−0.669599	−	0.742722i	$-0.733534\pi$
$839$	−38.7906	−1.33920	−0.669599	+	0.742722i	$0.733534\pi$
$840$	0	0
$841$	−16.4118	−0.565926
$842$	0	0
$843$	−1.05067	−0.0361869
$844$	0	0
$845$	7.02265	0.241586
$846$	0	0
$847$	−31.3227	−1.07626
$848$	0	0
$849$	7.57062	0.259823
$850$	0	0
$851$	2.65609	0.0910495
$852$	0	0
$853$	48.9493	1.67599	0.837997	−	0.545675i	$-0.183727\pi$
$853$	48.9493	1.67599	0.837997	+	0.545675i	$0.183727\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	56.6453	1.93497	0.967484	−	0.252932i	$-0.0813950\pi$
$857$	56.6453	1.93497	0.967484	+	0.252932i	$0.0813950\pi$
$858$	0	0
$859$	−37.8987	−1.29309	−0.646543	−	0.762878i	$-0.723786\pi$
$859$	−37.8987	−1.29309	−0.646543	+	0.762878i	$0.723786\pi$
$860$	0	0
$861$	−17.8987	−0.609985
$862$	0	0
$863$	10.1013	0.343853	0.171927	−	0.985110i	$-0.445001\pi$
$863$	10.1013	0.343853	0.171927	+	0.985110i	$0.445001\pi$
$864$	0	0
$865$	−6.94933	−0.236284
$866$	0	0
$867$	15.8480	0.538226
$868$	0	0
$869$	19.7520	0.670042
$870$	0	0
$871$	19.3666	0.656211
$872$	0	0
$873$	1.85338	0.0627273
$874$	0	0
$875$	4.47467	0.151271
$876$	0	0
$877$	−12.7014	−0.428895	−0.214448	−	0.976736i	$-0.568795\pi$
$877$	−12.7014	−0.428895	−0.214448	+	0.976736i	$0.568795\pi$
$878$	0	0
$879$	5.87602	0.198193
$880$	0	0
$881$	−38.5426	−1.29853	−0.649267	−	0.760561i	$-0.724924\pi$
$881$	−38.5426	−1.29853	−0.649267	+	0.760561i	$0.724924\pi$
$882$	0	0
$883$	15.1973	0.511429	0.255715	−	0.966752i	$-0.417689\pi$
$883$	15.1973	0.511429	0.255715	+	0.966752i	$0.417689\pi$
$884$	0	0
$885$	9.54798	0.320952
$886$	0	0
$887$	−11.7294	−0.393835	−0.196917	−	0.980420i	$-0.563093\pi$
$887$	−11.7294	−0.393835	−0.196917	+	0.980420i	$0.563093\pi$
$888$	0	0
$889$	13.6507	0.457830
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	−32.0906	−1.07387
$894$	0	0
$895$	−5.85338	−0.195657
$896$	0	0
$897$	−4.80271	−0.160358
$898$	0	0
$899$	−3.54798	−0.118332
$900$	0	0
$901$	10.7574	0.358382
$902$	0	0
$903$	8.94933	0.297815
$904$	0	0
$905$	−17.7520	−0.590098
$906$	0	0
$907$	−42.3280	−1.40548	−0.702740	−	0.711447i	$-0.748040\pi$
$907$	−42.3280	−1.40548	−0.702740	+	0.711447i	$0.748040\pi$
$908$	0	0
$909$	−1.85338	−0.0614726
$910$	0	0
$911$	15.8987	0.526746	0.263373	−	0.964694i	$-0.415165\pi$
$911$	15.8987	0.526746	0.263373	+	0.964694i	$0.415165\pi$
$912$	0	0
$913$	−5.85338	−0.193719
$914$	0	0
$915$	0.146623	0.00484720
$916$	0	0
$917$	−33.1186	−1.09367
$918$	0	0
$919$	51.5494	1.70046	0.850229	−	0.526414i	$-0.176464\pi$
$919$	51.5494	1.70046	0.850229	+	0.526414i	$0.176464\pi$
$920$	0	0
$921$	−17.7172	−0.583803
$922$	0	0
$923$	−55.9213	−1.84067
$924$	0	0
$925$	2.47467	0.0813666
$926$	0	0
$927$	15.4240	0.506591
$928$	0	0
$929$	−27.2547	−0.894199	−0.447099	−	0.894484i	$-0.647543\pi$
$929$	−27.2547	−0.894199	−0.447099	+	0.894484i	$0.647543\pi$
$930$	0	0
$931$	−52.0906	−1.70720
$932$	0	0
$933$	−14.3507	−0.469820
$934$	0	0
$935$	−2.14662	−0.0702021
$936$	0	0
$937$	−51.5494	−1.68404	−0.842022	−	0.539442i	$-0.818635\pi$
$937$	−51.5494	−1.68404	−0.842022	+	0.539442i	$0.818635\pi$
$938$	0	0
$939$	−19.3227	−0.630571
$940$	0	0
$941$	−38.2946	−1.24837	−0.624185	−	0.781277i	$-0.714569\pi$
$941$	−38.2946	−1.24837	−0.624185	+	0.781277i	$0.714569\pi$
$942$	0	0
$943$	4.29325	0.139807
$944$	0	0
$945$	−4.47467	−0.145561
$946$	0	0
$947$	−25.0280	−0.813301	−0.406651	−	0.913584i	$-0.633303\pi$
$947$	−25.0280	−0.813301	−0.406651	+	0.913584i	$0.633303\pi$
$948$	0	0
$949$	69.0173	2.24040
$950$	0	0
$951$	32.0226	1.03841
$952$	0	0
$953$	−0.825357	−0.0267359	−0.0133680	−	0.999911i	$-0.504255\pi$
$953$	−0.825357	−0.0267359	−0.0133680	+	0.999911i	$0.504255\pi$
$954$	0	0
$955$	−8.35069	−0.270222
$956$	0	0
$957$	−7.09596	−0.229380
$958$	0	0
$959$	−66.9946	−2.16337
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	1.21993	0.0393118
$964$	0	0
$965$	−8.00000	−0.257529
$966$	0	0
$967$	10.9493	0.352107	0.176053	−	0.984381i	$-0.443667\pi$
$967$	10.9493	0.352107	0.176053	+	0.984381i	$0.443667\pi$
$968$	0	0
$969$	4.29325	0.137919
$970$	0	0
$971$	−35.3000	−1.13283	−0.566416	−	0.824120i	$-0.691670\pi$
$971$	−35.3000	−1.13283	−0.566416	+	0.824120i	$0.691670\pi$
$972$	0	0
$973$	66.2372	2.12347
$974$	0	0
$975$	−4.47467	−0.143304
$976$	0	0
$977$	39.6054	1.26709	0.633545	−	0.773706i	$-0.281599\pi$
$977$	39.6054	1.26709	0.633545	+	0.773706i	$0.281599\pi$
$978$	0	0
$979$	−30.8027	−0.984459
$980$	0	0
$981$	5.07331	0.161978
$982$	0	0
$983$	−43.5494	−1.38901	−0.694505	−	0.719488i	$-0.744376\pi$
$983$	−43.5494	−1.38901	−0.694505	+	0.719488i	$0.744376\pi$
$984$	0	0
$985$	6.97198	0.222146
$986$	0	0
$987$	−35.8987	−1.14267
$988$	0	0
$989$	−2.14662	−0.0682586
$990$	0	0
$991$	−38.8933	−1.23549	−0.617743	−	0.786380i	$-0.711953\pi$
$991$	−38.8933	−1.23549	−0.617743	+	0.786380i	$0.711953\pi$
$992$	0	0
$993$	−29.8760	−0.948087
$994$	0	0
$995$	5.21993	0.165483
$996$	0	0
$997$	39.2879	1.24426	0.622130	−	0.782914i	$-0.286268\pi$
$997$	39.2879	1.24426	0.622130	+	0.782914i	$0.286268\pi$
$998$	0	0
$999$	−2.47467	−0.0782950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bq.1.3		3
4.3	odd	2		1860.2.a.h.1.1	✓	3
12.11	even	2		5580.2.a.i.1.1		3
20.3	even	4		9300.2.g.r.3349.6		6
20.7	even	4		9300.2.g.r.3349.1		6
20.19	odd	2		9300.2.a.t.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.h.1.1	✓	3	4.3	odd	2
5580.2.a.i.1.1		3	12.11	even	2
7440.2.a.bq.1.3		3	1.1	even	1	trivial
9300.2.a.t.1.3		3	20.19	odd	2
9300.2.g.r.3349.1		6	20.7	even	4
9300.2.g.r.3349.6		6	20.3	even	4

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:	\( N \)	\(=\)	\( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7440.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +4.47467 q^{7} +1.00000 q^{9} -2.00000 q^{11} +4.47467 q^{13} -1.00000 q^{15} +1.07331 q^{17} -4.00000 q^{19} -4.47467 q^{21} +1.07331 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.54798 q^{29} +1.00000 q^{31} +2.00000 q^{33} +4.47467 q^{35} +2.47467 q^{37} -4.47467 q^{39} +4.00000 q^{41} -2.00000 q^{43} +1.00000 q^{45} +8.02265 q^{47} +13.0226 q^{49} -1.07331 q^{51} +10.0226 q^{53} -2.00000 q^{55} +4.00000 q^{57} -9.54798 q^{59} -0.146623 q^{61} +4.47467 q^{63} +4.47467 q^{65} +4.32804 q^{67} -1.07331 q^{69} -12.4973 q^{71} +15.4240 q^{73} -1.00000 q^{75} -8.94933 q^{77} -9.87602 q^{79} +1.00000 q^{81} +2.92669 q^{83} +1.07331 q^{85} +3.54798 q^{87} +15.4014 q^{89} +20.0226 q^{91} -1.00000 q^{93} -4.00000 q^{95} +1.85338 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{9} - 6 q^{11} - 3 q^{15} + 2 q^{17} - 12 q^{19} + 2 q^{23} + 3 q^{25} - 3 q^{27} + 4 q^{29} + 3 q^{31} + 6 q^{33} - 6 q^{37} + 12 q^{41} - 6 q^{43} + 3 q^{45} - 4 q^{47} + 11 q^{49}+ \cdots - 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 + 3 * q^9 - 6 * q^11 - 3 * q^15 + 2 * q^17 - 12 * q^19 + 2 * q^23 + 3 * q^25 - 3 * q^27 + 4 * q^29 + 3 * q^31 + 6 * q^33 - 6 * q^37 + 12 * q^41 - 6 * q^43 + 3 * q^45 - 4 * q^47 + 11 * q^49 - 2 * q^51 + 2 * q^53 - 6 * q^55 + 12 * q^57 - 14 * q^59 + 2 * q^61 + 2 * q^67 - 2 * q^69 + 4 * q^71 + 6 * q^73 - 3 * q^75 - 4 * q^79 + 3 * q^81 + 10 * q^83 + 2 * q^85 - 4 * q^87 + 34 * q^89 + 32 * q^91 - 3 * q^93 - 12 * q^95 + 8 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more