LMFDB - Embedded newform 3040.2.f.b.1521.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3040,2,Mod(1521,3040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3040.1521"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3040 = 2^{5} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3040.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$24.2745222145$
Analytic rank:	$0$
Dimension:	$44$
Twist minimal:	no (minimal twist has level 760)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1521.4
Character	$\chi$	$=$	3040.1521
Dual form			3040.2.f.b.1521.41

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.06281i q^{3} +1.00000i q^{5} +2.81229 q^{7} -6.38080 q^{9} +5.05668i q^{11} -3.98012i q^{13} +3.06281 q^{15} +1.62171 q^{17} -1.00000i q^{19} -8.61352i q^{21} -1.51916 q^{23} -1.00000 q^{25} +10.3548i q^{27} -0.617462i q^{29} +7.37672 q^{31} +15.4877 q^{33} +2.81229i q^{35} -6.69976i q^{37} -12.1904 q^{39} -10.2747 q^{41} -9.48263i q^{43} -6.38080i q^{45} +7.80972 q^{47} +0.908997 q^{49} -4.96700i q^{51} -8.96779i q^{53} -5.05668 q^{55} -3.06281 q^{57} -1.48840i q^{59} -7.73455i q^{61} -17.9447 q^{63} +3.98012 q^{65} -7.61986i q^{67} +4.65290i q^{69} +1.07002 q^{71} +10.8626 q^{73} +3.06281i q^{75} +14.2209i q^{77} -1.37931 q^{79} +12.5722 q^{81} -13.9294i q^{83} +1.62171i q^{85} -1.89117 q^{87} +0.709012 q^{89} -11.1933i q^{91} -22.5935i q^{93} +1.00000 q^{95} +14.3906 q^{97} -32.2657i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 4 q^{7} - 60 q^{9} + 24 q^{17} - 4 q^{23} - 44 q^{25} + 40 q^{33} + 24 q^{39} - 32 q^{41} + 20 q^{47} + 108 q^{49} - 8 q^{55} - 20 q^{63} + 12 q^{65} + 8 q^{71} - 88 q^{73} - 40 q^{79} + 116 q^{81}+ \cdots + 116 q^{97}+O(q^{100}) $ 44 * q - 4 * q^7 - 60 * q^9 + 24 * q^17 - 4 * q^23 - 44 * q^25 + 40 * q^33 + 24 * q^39 - 32 * q^41 + 20 * q^47 + 108 * q^49 - 8 * q^55 - 20 * q^63 + 12 * q^65 + 8 * q^71 - 88 * q^73 - 40 * q^79 + 116 * q^81 - 48 * q^87 - 64 * q^89 + 44 * q^95 + 116 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3040\mathbb{Z}\right)^\times$.

$n$	$191$	$1217$	$1921$	$2661$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 3.06281i	− 1.76831i	−0.467190	−	0.884157i	$-0.654734\pi$
$3$	− 3.06281i	− 1.76831i	0.467190	−	0.884157i	$-0.345266\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	2.81229	1.06295	0.531474	−	0.847075i	$-0.321638\pi$
$7$	2.81229	1.06295	0.531474	+	0.847075i	$0.321638\pi$
$8$	0	0
$9$	−6.38080	−2.12693
$10$	0	0
$11$	5.05668i	1.52465i	0.647196	+	0.762324i	$0.275942\pi$
$11$	5.05668i	1.52465i	−0.647196	+	0.762324i	$0.724058\pi$
$12$	0	0
$13$	− 3.98012i	− 1.10389i	−0.833881	−	0.551944i	$-0.813886\pi$
$13$	− 3.98012i	− 1.10389i	0.833881	−	0.551944i	$-0.186114\pi$
$14$	0	0
$15$	3.06281	0.790814
$16$	0	0
$17$	1.62171	0.393323	0.196662	−	0.980471i	$-0.436990\pi$
$17$	1.62171	0.393323	0.196662	+	0.980471i	$0.436990\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	− 8.61352i	− 1.87962i
$22$	0	0
$23$	−1.51916	−0.316767	−0.158384	−	0.987378i	$-0.550628\pi$
$23$	−1.51916	−0.316767	−0.158384	+	0.987378i	$0.550628\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	10.3548i	1.99277i
$28$	0	0
$29$	− 0.617462i	− 0.114660i	−0.998355	−	0.0573299i	$-0.981741\pi$
$29$	− 0.617462i	− 0.114660i	0.998355	−	0.0573299i	$-0.0182587\pi$
$30$	0	0
$31$	7.37672	1.32490	0.662449	−	0.749107i	$-0.269517\pi$
$31$	7.37672	1.32490	0.662449	+	0.749107i	$0.269517\pi$
$32$	0	0
$33$	15.4877	2.69606
$34$	0	0
$35$	2.81229i	0.475364i
$36$	0	0
$37$	− 6.69976i	− 1.10143i	−0.834692	−	0.550717i	$-0.814354\pi$
$37$	− 6.69976i	− 1.10143i	0.834692	−	0.550717i	$-0.185646\pi$
$38$	0	0
$39$	−12.1904	−1.95202
$40$	0	0
$41$	−10.2747	−1.60465	−0.802323	−	0.596890i	$-0.796403\pi$
$41$	−10.2747	−1.60465	−0.802323	+	0.596890i	$0.796403\pi$
$42$	0	0
$43$	− 9.48263i	− 1.44609i	−0.690802	−	0.723044i	$-0.742743\pi$
$43$	− 9.48263i	− 1.44609i	0.690802	−	0.723044i	$-0.257257\pi$
$44$	0	0
$45$	− 6.38080i	− 0.951194i
$46$	0	0
$47$	7.80972	1.13916	0.569582	−	0.821934i	$-0.307105\pi$
$47$	7.80972	1.13916	0.569582	+	0.821934i	$0.307105\pi$
$48$	0	0
$49$	0.908997	0.129857
$50$	0	0
$51$	− 4.96700i	− 0.695519i
$52$	0	0
$53$	− 8.96779i	− 1.23182i	−0.787816	−	0.615910i	$-0.788788\pi$
$53$	− 8.96779i	− 1.23182i	0.787816	−	0.615910i	$-0.211212\pi$
$54$	0	0
$55$	−5.05668	−0.681843
$56$	0	0
$57$	−3.06281	−0.405679
$58$	0	0
$59$	− 1.48840i	− 0.193773i	−0.995295	−	0.0968866i	$-0.969112\pi$
$59$	− 1.48840i	− 0.193773i	0.995295	−	0.0968866i	$-0.0308884\pi$
$60$	0	0
$61$	− 7.73455i	− 0.990307i	−0.868805	−	0.495154i	$-0.835112\pi$
$61$	− 7.73455i	− 0.990307i	0.868805	−	0.495154i	$-0.164888\pi$
$62$	0	0
$63$	−17.9447	−2.26082
$64$	0	0
$65$	3.98012	0.493674
$66$	0	0
$67$	− 7.61986i	− 0.930914i	−0.885071	−	0.465457i	$-0.845890\pi$
$67$	− 7.61986i	− 0.930914i	0.885071	−	0.465457i	$-0.154110\pi$
$68$	0	0
$69$	4.65290i	0.560144i
$70$	0	0
$71$	1.07002	0.126989	0.0634943	−	0.997982i	$-0.479776\pi$
$71$	1.07002	0.126989	0.0634943	+	0.997982i	$0.479776\pi$
$72$	0	0
$73$	10.8626	1.27137	0.635685	−	0.771949i	$-0.280718\pi$
$73$	10.8626	1.27137	0.635685	+	0.771949i	$0.280718\pi$
$74$	0	0
$75$	3.06281i	0.353663i
$76$	0	0
$77$	14.2209i	1.62062i
$78$	0	0
$79$	−1.37931	−0.155185	−0.0775924	−	0.996985i	$-0.524723\pi$
$79$	−1.37931	−0.155185	−0.0775924	+	0.996985i	$0.524723\pi$
$80$	0	0
$81$	12.5722	1.39691
$82$	0	0
$83$	− 13.9294i	− 1.52895i	−0.644653	−	0.764475i	$-0.722998\pi$
$83$	− 13.9294i	− 1.52895i	0.644653	−	0.764475i	$-0.277002\pi$
$84$	0	0
$85$	1.62171i	0.175899i
$86$	0	0
$87$	−1.89117	−0.202755
$88$	0	0
$89$	0.709012	0.0751551	0.0375776	−	0.999294i	$-0.488036\pi$
$89$	0.709012	0.0751551	0.0375776	+	0.999294i	$0.488036\pi$
$90$	0	0
$91$	− 11.1933i	− 1.17337i
$92$	0	0
$93$	− 22.5935i	− 2.34284i
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	14.3906	1.46115	0.730573	−	0.682835i	$-0.239253\pi$
$97$	14.3906	1.46115	0.730573	+	0.682835i	$0.239253\pi$
$98$	0	0
$99$	− 32.2657i	− 3.24282i
$100$	0	0
$101$	− 12.0028i	− 1.19432i	−0.802121	−	0.597161i	$-0.796295\pi$
$101$	− 12.0028i	− 1.19432i	0.802121	−	0.597161i	$-0.203705\pi$
$102$	0	0
$103$	17.9996	1.77356	0.886778	−	0.462195i	$-0.152938\pi$
$103$	17.9996	1.77356	0.886778	+	0.462195i	$0.152938\pi$
$104$	0	0
$105$	8.61352	0.840594
$106$	0	0
$107$	18.2082i	1.76025i	0.474738	+	0.880127i	$0.342543\pi$
$107$	18.2082i	1.76025i	−0.474738	+	0.880127i	$0.657457\pi$
$108$	0	0
$109$	− 1.41237i	− 0.135280i	−0.997710	−	0.0676402i	$-0.978453\pi$
$109$	− 1.41237i	− 0.135280i	0.997710	−	0.0676402i	$-0.0215470\pi$
$110$	0	0
$111$	−20.5201	−1.94768
$112$	0	0
$113$	−4.53336	−0.426463	−0.213231	−	0.977002i	$-0.568399\pi$
$113$	−4.53336	−0.426463	−0.213231	+	0.977002i	$0.568399\pi$
$114$	0	0
$115$	− 1.51916i	− 0.141663i
$116$	0	0
$117$	25.3964i	2.34790i
$118$	0	0
$119$	4.56073	0.418082
$120$	0	0
$121$	−14.5701	−1.32455
$122$	0	0
$123$	31.4696i	2.83752i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−16.3081	−1.44711	−0.723554	−	0.690268i	$-0.757493\pi$
$127$	−16.3081	−1.44711	−0.723554	+	0.690268i	$0.757493\pi$
$128$	0	0
$129$	−29.0435	−2.55714
$130$	0	0
$131$	8.26767i	0.722350i	0.932498	+	0.361175i	$0.117624\pi$
$131$	8.26767i	0.722350i	−0.932498	+	0.361175i	$0.882376\pi$
$132$	0	0
$133$	− 2.81229i	− 0.243857i
$134$	0	0
$135$	−10.3548	−0.891195
$136$	0	0
$137$	−17.5424	−1.49875	−0.749374	−	0.662147i	$-0.769646\pi$
$137$	−17.5424	−1.49875	−0.749374	+	0.662147i	$0.769646\pi$
$138$	0	0
$139$	− 0.878831i	− 0.0745414i	−0.999305	−	0.0372707i	$-0.988134\pi$
$139$	− 0.878831i	− 0.0745414i	0.999305	−	0.0372707i	$-0.0118664\pi$
$140$	0	0
$141$	− 23.9197i	− 2.01440i
$142$	0	0
$143$	20.1262	1.68304
$144$	0	0
$145$	0.617462	0.0512775
$146$	0	0
$147$	− 2.78409i	− 0.229627i
$148$	0	0
$149$	− 8.52396i	− 0.698310i	−0.937065	−	0.349155i	$-0.886469\pi$
$149$	− 8.52396i	− 0.698310i	0.937065	−	0.349155i	$-0.113531\pi$
$150$	0	0
$151$	15.8403	1.28906	0.644531	−	0.764578i	$-0.277053\pi$
$151$	15.8403	1.28906	0.644531	+	0.764578i	$0.277053\pi$
$152$	0	0
$153$	−10.3478	−0.836572
$154$	0	0
$155$	7.37672i	0.592513i
$156$	0	0
$157$	− 0.882769i	− 0.0704527i	−0.999379	−	0.0352263i	$-0.988785\pi$
$157$	− 0.882769i	− 0.0704527i	0.999379	−	0.0352263i	$-0.0112152\pi$
$158$	0	0
$159$	−27.4666	−2.17825
$160$	0	0
$161$	−4.27233	−0.336707
$162$	0	0
$163$	− 8.67648i	− 0.679594i	−0.940499	−	0.339797i	$-0.889642\pi$
$163$	− 8.67648i	− 0.679594i	0.940499	−	0.339797i	$-0.110358\pi$
$164$	0	0
$165$	15.4877i	1.20571i
$166$	0	0
$167$	21.7621	1.68400	0.842001	−	0.539476i	$-0.181378\pi$
$167$	21.7621	1.68400	0.842001	+	0.539476i	$0.181378\pi$
$168$	0	0
$169$	−2.84139	−0.218569
$170$	0	0
$171$	6.38080i	0.487952i
$172$	0	0
$173$	9.66571i	0.734870i	0.930049	+	0.367435i	$0.119764\pi$
$173$	9.66571i	0.734870i	−0.930049	+	0.367435i	$0.880236\pi$
$174$	0	0
$175$	−2.81229	−0.212589
$176$	0	0
$177$	−4.55868	−0.342652
$178$	0	0
$179$	20.0228i	1.49657i	0.663376	+	0.748286i	$0.269123\pi$
$179$	20.0228i	1.49657i	−0.663376	+	0.748286i	$0.730877\pi$
$180$	0	0
$181$	− 3.56175i	− 0.264742i	−0.991200	−	0.132371i	$-0.957741\pi$
$181$	− 3.56175i	− 0.264742i	0.991200	−	0.132371i	$-0.0422591\pi$
$182$	0	0
$183$	−23.6894	−1.75117
$184$	0	0
$185$	6.69976	0.492576
$186$	0	0
$187$	8.20049i	0.599679i
$188$	0	0
$189$	29.1206i	2.11821i
$190$	0	0
$191$	−18.5078	−1.33918	−0.669589	−	0.742732i	$-0.733530\pi$
$191$	−18.5078	−1.33918	−0.669589	+	0.742732i	$0.733530\pi$
$192$	0	0
$193$	−0.891942	−0.0642034	−0.0321017	−	0.999485i	$-0.510220\pi$
$193$	−0.891942	−0.0642034	−0.0321017	+	0.999485i	$0.510220\pi$
$194$	0	0
$195$	− 12.1904i	− 0.872970i
$196$	0	0
$197$	12.9518i	0.922776i	0.887198	+	0.461388i	$0.152648\pi$
$197$	12.9518i	0.922776i	−0.887198	+	0.461388i	$0.847352\pi$
$198$	0	0
$199$	−6.52220	−0.462347	−0.231173	−	0.972913i	$-0.574256\pi$
$199$	−6.52220	−0.462347	−0.231173	+	0.972913i	$0.574256\pi$
$200$	0	0
$201$	−23.3382	−1.64615
$202$	0	0
$203$	− 1.73649i	− 0.121877i
$204$	0	0
$205$	− 10.2747i	− 0.717620i
$206$	0	0
$207$	9.69347	0.673743
$208$	0	0
$209$	5.05668	0.349778
$210$	0	0
$211$	26.7423i	1.84102i	0.390724	+	0.920508i	$0.372225\pi$
$211$	26.7423i	1.84102i	−0.390724	+	0.920508i	$0.627775\pi$
$212$	0	0
$213$	− 3.27728i	− 0.224556i
$214$	0	0
$215$	9.48263	0.646710
$216$	0	0
$217$	20.7455	1.40830
$218$	0	0
$219$	− 33.2700i	− 2.24818i
$220$	0	0
$221$	− 6.45462i	− 0.434185i
$222$	0	0
$223$	1.22439	0.0819910	0.0409955	−	0.999159i	$-0.486947\pi$
$223$	1.22439	0.0819910	0.0409955	+	0.999159i	$0.486947\pi$
$224$	0	0
$225$	6.38080	0.425387
$226$	0	0
$227$	− 15.1423i	− 1.00503i	−0.864568	−	0.502516i	$-0.832408\pi$
$227$	− 15.1423i	− 1.00503i	0.864568	−	0.502516i	$-0.167592\pi$
$228$	0	0
$229$	8.06769i	0.533128i	0.963817	+	0.266564i	$0.0858884\pi$
$229$	8.06769i	0.533128i	−0.963817	+	0.266564i	$0.914112\pi$
$230$	0	0
$231$	43.5559	2.86576
$232$	0	0
$233$	−18.1011	−1.18584	−0.592922	−	0.805260i	$-0.702026\pi$
$233$	−18.1011	−1.18584	−0.592922	+	0.805260i	$0.702026\pi$
$234$	0	0
$235$	7.80972i	0.509450i
$236$	0	0
$237$	4.22457i	0.274415i
$238$	0	0
$239$	22.5877	1.46107	0.730537	−	0.682873i	$-0.239270\pi$
$239$	22.5877	1.46107	0.730537	+	0.682873i	$0.239270\pi$
$240$	0	0
$241$	−1.66938	−0.107534	−0.0537671	−	0.998554i	$-0.517123\pi$
$241$	−1.66938	−0.107534	−0.0537671	+	0.998554i	$0.517123\pi$
$242$	0	0
$243$	− 7.44209i	− 0.477410i
$244$	0	0
$245$	0.908997i	0.0580737i
$246$	0	0
$247$	−3.98012	−0.253249
$248$	0	0
$249$	−42.6631	−2.70366
$250$	0	0
$251$	3.43312i	0.216697i	0.994113	+	0.108348i	$0.0345562\pi$
$251$	3.43312i	0.216697i	−0.994113	+	0.108348i	$0.965444\pi$
$252$	0	0
$253$	− 7.68192i	− 0.482958i
$254$	0	0
$255$	4.96700	0.311045
$256$	0	0
$257$	−4.54870	−0.283740	−0.141870	−	0.989885i	$-0.545312\pi$
$257$	−4.54870	−0.283740	−0.141870	+	0.989885i	$0.545312\pi$
$258$	0	0
$259$	− 18.8417i	− 1.17077i
$260$	0	0
$261$	3.93991i	0.243874i
$262$	0	0
$263$	−11.0906	−0.683878	−0.341939	−	0.939722i	$-0.611084\pi$
$263$	−11.0906	−0.683878	−0.341939	+	0.939722i	$0.611084\pi$
$264$	0	0
$265$	8.96779	0.550887
$266$	0	0
$267$	− 2.17157i	− 0.132898i
$268$	0	0
$269$	0.456872i	0.0278560i	0.999903	+	0.0139280i	$0.00443356\pi$
$269$	0.456872i	0.0278560i	−0.999903	+	0.0139280i	$0.995566\pi$
$270$	0	0
$271$	8.24792	0.501025	0.250513	−	0.968113i	$-0.419401\pi$
$271$	8.24792	0.501025	0.250513	+	0.968113i	$0.419401\pi$
$272$	0	0
$273$	−34.2829	−2.07489
$274$	0	0
$275$	− 5.05668i	− 0.304930i
$276$	0	0
$277$	17.0974i	1.02729i	0.858004	+	0.513643i	$0.171704\pi$
$277$	17.0974i	1.02729i	−0.858004	+	0.513643i	$0.828296\pi$
$278$	0	0
$279$	−47.0694	−2.81797
$280$	0	0
$281$	8.94353	0.533527	0.266763	−	0.963762i	$-0.414046\pi$
$281$	8.94353	0.533527	0.266763	+	0.963762i	$0.414046\pi$
$282$	0	0
$283$	− 32.6729i	− 1.94220i	−0.238671	−	0.971101i	$-0.576712\pi$
$283$	− 32.6729i	− 1.94220i	0.238671	−	0.971101i	$-0.423288\pi$
$284$	0	0
$285$	− 3.06281i	− 0.181425i
$286$	0	0
$287$	−28.8956	−1.70565
$288$	0	0
$289$	−14.3700	−0.845297
$290$	0	0
$291$	− 44.0757i	− 2.58376i
$292$	0	0
$293$	27.7877i	1.62338i	0.584091	+	0.811688i	$0.301451\pi$
$293$	27.7877i	1.62338i	−0.584091	+	0.811688i	$0.698549\pi$
$294$	0	0
$295$	1.48840	0.0866580
$296$	0	0
$297$	−52.3607	−3.03828
$298$	0	0
$299$	6.04645i	0.349675i
$300$	0	0
$301$	− 26.6679i	− 1.53711i
$302$	0	0
$303$	−36.7622	−2.11194
$304$	0	0
$305$	7.73455	0.442879
$306$	0	0
$307$	10.7593i	0.614067i	0.951699	+	0.307034i	$0.0993364\pi$
$307$	10.7593i	0.614067i	−0.951699	+	0.307034i	$0.900664\pi$
$308$	0	0
$309$	− 55.1294i	− 3.13620i
$310$	0	0
$311$	−14.8790	−0.843708	−0.421854	−	0.906664i	$-0.638621\pi$
$311$	−14.8790	−0.843708	−0.421854	+	0.906664i	$0.638621\pi$
$312$	0	0
$313$	−16.0734	−0.908523	−0.454261	−	0.890868i	$-0.650097\pi$
$313$	−16.0734	−0.908523	−0.454261	+	0.890868i	$0.650097\pi$
$314$	0	0
$315$	− 17.9447i	− 1.01107i
$316$	0	0
$317$	− 29.8614i	− 1.67718i	−0.544760	−	0.838592i	$-0.683379\pi$
$317$	− 29.8614i	− 1.67718i	0.544760	−	0.838592i	$-0.316621\pi$
$318$	0	0
$319$	3.12231	0.174816
$320$	0	0
$321$	55.7683	3.11268
$322$	0	0
$323$	− 1.62171i	− 0.0902345i
$324$	0	0
$325$	3.98012i	0.220778i
$326$	0	0
$327$	−4.32581	−0.239218
$328$	0	0
$329$	21.9632	1.21087
$330$	0	0
$331$	17.6580i	0.970573i	0.874355	+	0.485286i	$0.161285\pi$
$331$	17.6580i	0.970573i	−0.874355	+	0.485286i	$0.838715\pi$
$332$	0	0
$333$	42.7499i	2.34268i
$334$	0	0
$335$	7.61986	0.416317
$336$	0	0
$337$	−17.1181	−0.932482	−0.466241	−	0.884658i	$-0.654392\pi$
$337$	−17.1181	−0.932482	−0.466241	+	0.884658i	$0.654392\pi$
$338$	0	0
$339$	13.8848i	0.754120i
$340$	0	0
$341$	37.3018i	2.02000i
$342$	0	0
$343$	−17.1297	−0.924916
$344$	0	0
$345$	−4.65290	−0.250504
$346$	0	0
$347$	10.4086i	0.558761i	0.960180	+	0.279381i	$0.0901291\pi$
$347$	10.4086i	0.558761i	−0.960180	+	0.279381i	$0.909871\pi$
$348$	0	0
$349$	22.1049i	1.18325i	0.806214	+	0.591624i	$0.201513\pi$
$349$	22.1049i	1.18325i	−0.806214	+	0.591624i	$0.798487\pi$
$350$	0	0
$351$	41.2132	2.19980
$352$	0	0
$353$	−7.96563	−0.423968	−0.211984	−	0.977273i	$-0.567992\pi$
$353$	−7.96563	−0.423968	−0.211984	+	0.977273i	$0.567992\pi$
$354$	0	0
$355$	1.07002i	0.0567910i
$356$	0	0
$357$	− 13.9687i	− 0.739300i
$358$	0	0
$359$	32.8817	1.73543	0.867715	−	0.497062i	$-0.165588\pi$
$359$	32.8817	1.73543	0.867715	+	0.497062i	$0.165588\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	44.6253i	2.34222i
$364$	0	0
$365$	10.8626i	0.568574i
$366$	0	0
$367$	−8.35897	−0.436335	−0.218167	−	0.975911i	$-0.570008\pi$
$367$	−8.35897	−0.436335	−0.218167	+	0.975911i	$0.570008\pi$
$368$	0	0
$369$	65.5611	3.41298
$370$	0	0
$371$	− 25.2201i	− 1.30936i
$372$	0	0
$373$	− 5.73299i	− 0.296843i	−0.988924	−	0.148422i	$-0.952581\pi$
$373$	− 5.73299i	− 0.296843i	0.988924	−	0.148422i	$-0.0474192\pi$
$374$	0	0
$375$	−3.06281	−0.158163
$376$	0	0
$377$	−2.45758	−0.126572
$378$	0	0
$379$	− 26.5868i	− 1.36567i	−0.730571	−	0.682836i	$-0.760746\pi$
$379$	− 26.5868i	− 1.36567i	0.730571	−	0.682836i	$-0.239254\pi$
$380$	0	0
$381$	49.9485i	2.55894i
$382$	0	0
$383$	12.1889	0.622824	0.311412	−	0.950275i	$-0.399198\pi$
$383$	12.1889	0.622824	0.311412	+	0.950275i	$0.399198\pi$
$384$	0	0
$385$	−14.2209	−0.724763
$386$	0	0
$387$	60.5068i	3.07573i
$388$	0	0
$389$	− 17.2618i	− 0.875209i	−0.899168	−	0.437605i	$-0.855827\pi$
$389$	− 17.2618i	− 0.875209i	0.899168	−	0.437605i	$-0.144173\pi$
$390$	0	0
$391$	−2.46364	−0.124592
$392$	0	0
$393$	25.3223	1.27734
$394$	0	0
$395$	− 1.37931i	− 0.0694008i
$396$	0	0
$397$	0.694542i	0.0348581i	0.999848	+	0.0174290i	$0.00554812\pi$
$397$	0.694542i	0.0348581i	−0.999848	+	0.0174290i	$0.994452\pi$
$398$	0	0
$399$	−8.61352	−0.431215
$400$	0	0
$401$	24.2182	1.20940	0.604700	−	0.796454i	$-0.293293\pi$
$401$	24.2182	1.20940	0.604700	+	0.796454i	$0.293293\pi$
$402$	0	0
$403$	− 29.3603i	− 1.46254i
$404$	0	0
$405$	12.5722i	0.624719i
$406$	0	0
$407$	33.8786	1.67930
$408$	0	0
$409$	27.3171	1.35074	0.675371	−	0.737478i	$-0.263983\pi$
$409$	27.3171	1.35074	0.675371	+	0.737478i	$0.263983\pi$
$410$	0	0
$411$	53.7290i	2.65026i
$412$	0	0
$413$	− 4.18582i	− 0.205971i
$414$	0	0
$415$	13.9294	0.683767
$416$	0	0
$417$	−2.69169	−0.131813
$418$	0	0
$419$	7.94266i	0.388024i	0.980999	+	0.194012i	$0.0621501\pi$
$419$	7.94266i	0.388024i	−0.980999	+	0.194012i	$0.937850\pi$
$420$	0	0
$421$	24.3855i	1.18848i	0.804289	+	0.594239i	$0.202547\pi$
$421$	24.3855i	1.18848i	−0.804289	+	0.594239i	$0.797453\pi$
$422$	0	0
$423$	−49.8322	−2.42293
$424$	0	0
$425$	−1.62171	−0.0786646
$426$	0	0
$427$	− 21.7518i	− 1.05264i
$428$	0	0
$429$	− 61.6428i	− 2.97614i
$430$	0	0
$431$	18.7021	0.900850	0.450425	−	0.892814i	$-0.351272\pi$
$431$	18.7021	0.900850	0.450425	+	0.892814i	$0.351272\pi$
$432$	0	0
$433$	23.9396	1.15047	0.575233	−	0.817990i	$-0.304911\pi$
$433$	23.9396	1.15047	0.575233	+	0.817990i	$0.304911\pi$
$434$	0	0
$435$	− 1.89117i	− 0.0906746i
$436$	0	0
$437$	1.51916i	0.0726714i
$438$	0	0
$439$	7.24965	0.346007	0.173003	−	0.984921i	$-0.444653\pi$
$439$	7.24965	0.346007	0.173003	+	0.984921i	$0.444653\pi$
$440$	0	0
$441$	−5.80013	−0.276197
$442$	0	0
$443$	26.9821i	1.28196i	0.767558	+	0.640979i	$0.221471\pi$
$443$	26.9821i	1.28196i	−0.767558	+	0.640979i	$0.778529\pi$
$444$	0	0
$445$	0.709012i	0.0336104i
$446$	0	0
$447$	−26.1073	−1.23483
$448$	0	0
$449$	16.5865	0.782767	0.391384	−	0.920228i	$-0.371997\pi$
$449$	16.5865	0.782767	0.391384	+	0.920228i	$0.371997\pi$
$450$	0	0
$451$	− 51.9562i	− 2.44652i
$452$	0	0
$453$	− 48.5157i	− 2.27947i
$454$	0	0
$455$	11.1933	0.524749
$456$	0	0
$457$	5.05916	0.236657	0.118329	−	0.992974i	$-0.462246\pi$
$457$	5.05916	0.236657	0.118329	+	0.992974i	$0.462246\pi$
$458$	0	0
$459$	16.7924i	0.783804i
$460$	0	0
$461$	− 6.93588i	− 0.323036i	−0.986870	−	0.161518i	$-0.948361\pi$
$461$	− 6.93588i	− 0.323036i	0.986870	−	0.161518i	$-0.0516390\pi$
$462$	0	0
$463$	−7.62487	−0.354358	−0.177179	−	0.984179i	$-0.556697\pi$
$463$	−7.62487	−0.354358	−0.177179	+	0.984179i	$0.556697\pi$
$464$	0	0
$465$	22.5935	1.04775
$466$	0	0
$467$	3.20732i	0.148417i	0.997243	+	0.0742086i	$0.0236431\pi$
$467$	3.20732i	0.148417i	−0.997243	+	0.0742086i	$0.976357\pi$
$468$	0	0
$469$	− 21.4293i	− 0.989512i
$470$	0	0
$471$	−2.70375	−0.124582
$472$	0	0
$473$	47.9507	2.20477
$474$	0	0
$475$	1.00000i	0.0458831i
$476$	0	0
$477$	57.2217i	2.62000i
$478$	0	0
$479$	−7.68052	−0.350932	−0.175466	−	0.984486i	$-0.556143\pi$
$479$	−7.68052	−0.350932	−0.175466	+	0.984486i	$0.556143\pi$
$480$	0	0
$481$	−26.6659	−1.21586
$482$	0	0
$483$	13.0853i	0.595403i
$484$	0	0
$485$	14.3906i	0.653444i
$486$	0	0
$487$	5.35340	0.242586	0.121293	−	0.992617i	$-0.461296\pi$
$487$	5.35340	0.242586	0.121293	+	0.992617i	$0.461296\pi$
$488$	0	0
$489$	−26.5744	−1.20174
$490$	0	0
$491$	− 28.4990i	− 1.28614i	−0.765807	−	0.643070i	$-0.777660\pi$
$491$	− 28.4990i	− 1.28614i	0.765807	−	0.643070i	$-0.222340\pi$
$492$	0	0
$493$	− 1.00135i	− 0.0450984i
$494$	0	0
$495$	32.2657	1.45024
$496$	0	0
$497$	3.00922	0.134982
$498$	0	0
$499$	− 7.57745i	− 0.339213i	−0.985512	−	0.169607i	$-0.945750\pi$
$499$	− 7.57745i	− 0.339213i	0.985512	−	0.169607i	$-0.0542497\pi$
$500$	0	0
$501$	− 66.6532i	− 2.97784i
$502$	0	0
$503$	−3.33433	−0.148670	−0.0743352	−	0.997233i	$-0.523683\pi$
$503$	−3.33433	−0.148670	−0.0743352	+	0.997233i	$0.523683\pi$
$504$	0	0
$505$	12.0028	0.534117
$506$	0	0
$507$	8.70264i	0.386498i
$508$	0	0
$509$	27.9961i	1.24091i	0.784243	+	0.620454i	$0.213051\pi$
$509$	27.9961i	1.24091i	−0.784243	+	0.620454i	$0.786949\pi$
$510$	0	0
$511$	30.5488	1.35140
$512$	0	0
$513$	10.3548	0.457174
$514$	0	0
$515$	17.9996i	0.793159i
$516$	0	0
$517$	39.4913i	1.73682i
$518$	0	0
$519$	29.6042	1.29948
$520$	0	0
$521$	13.5559	0.593897	0.296948	−	0.954894i	$-0.404031\pi$
$521$	13.5559	0.593897	0.296948	+	0.954894i	$0.404031\pi$
$522$	0	0
$523$	5.27502i	0.230660i	0.993327	+	0.115330i	$0.0367926\pi$
$523$	5.27502i	0.230660i	−0.993327	+	0.115330i	$0.963207\pi$
$524$	0	0
$525$	8.61352i	0.375925i
$526$	0	0
$527$	11.9629	0.521113
$528$	0	0
$529$	−20.6921	−0.899659
$530$	0	0
$531$	9.49718i	0.412143i
$532$	0	0
$533$	40.8948i	1.77135i
$534$	0	0
$535$	−18.2082	−0.787210
$536$	0	0
$537$	61.3260	2.64641
$538$	0	0
$539$	4.59651i	0.197986i
$540$	0	0
$541$	37.8198i	1.62600i	0.582265	+	0.812999i	$0.302167\pi$
$541$	37.8198i	1.62600i	−0.582265	+	0.812999i	$0.697833\pi$
$542$	0	0
$543$	−10.9089	−0.468148
$544$	0	0
$545$	1.41237	0.0604992
$546$	0	0
$547$	− 8.20562i	− 0.350847i	−0.984493	−	0.175423i	$-0.943871\pi$
$547$	− 8.20562i	− 0.350847i	0.984493	−	0.175423i	$-0.0561295\pi$
$548$	0	0
$549$	49.3526i	2.10632i
$550$	0	0
$551$	−0.617462	−0.0263048
$552$	0	0
$553$	−3.87903	−0.164953
$554$	0	0
$555$	− 20.5201i	− 0.871029i
$556$	0	0
$557$	− 38.8646i	− 1.64675i	−0.567500	−	0.823374i	$-0.692089\pi$
$557$	− 38.8646i	− 1.64675i	0.567500	−	0.823374i	$-0.307911\pi$
$558$	0	0
$559$	−37.7420	−1.59632
$560$	0	0
$561$	25.1165	1.06042
$562$	0	0
$563$	− 43.2675i	− 1.82351i	−0.410736	−	0.911754i	$-0.634728\pi$
$563$	− 43.2675i	− 1.82351i	0.410736	−	0.911754i	$-0.365272\pi$
$564$	0	0
$565$	− 4.53336i	− 0.190720i
$566$	0	0
$567$	35.3568	1.48485
$568$	0	0
$569$	−4.14459	−0.173750	−0.0868750	−	0.996219i	$-0.527688\pi$
$569$	−4.14459	−0.173750	−0.0868750	+	0.996219i	$0.527688\pi$
$570$	0	0
$571$	− 7.85216i	− 0.328603i	−0.986410	−	0.164301i	$-0.947463\pi$
$571$	− 7.85216i	− 0.328603i	0.986410	−	0.164301i	$-0.0525369\pi$
$572$	0	0
$573$	56.6859i	2.36809i
$574$	0	0
$575$	1.51916	0.0633534
$576$	0	0
$577$	8.01296	0.333584	0.166792	−	0.985992i	$-0.446659\pi$
$577$	8.01296	0.333584	0.166792	+	0.985992i	$0.446659\pi$
$578$	0	0
$579$	2.73185i	0.113532i
$580$	0	0
$581$	− 39.1736i	− 1.62519i
$582$	0	0
$583$	45.3473	1.87809
$584$	0	0
$585$	−25.3964	−1.05001
$586$	0	0
$587$	25.2668i	1.04287i	0.853290	+	0.521436i	$0.174604\pi$
$587$	25.2668i	1.04287i	−0.853290	+	0.521436i	$0.825396\pi$
$588$	0	0
$589$	− 7.37672i	− 0.303953i
$590$	0	0
$591$	39.6688	1.63176
$592$	0	0
$593$	10.1106	0.415192	0.207596	−	0.978215i	$-0.433436\pi$
$593$	10.1106	0.415192	0.207596	+	0.978215i	$0.433436\pi$
$594$	0	0
$595$	4.56073i	0.186972i
$596$	0	0
$597$	19.9763i	0.817574i
$598$	0	0
$599$	19.9087	0.813446	0.406723	−	0.913552i	$-0.366671\pi$
$599$	19.9087	0.813446	0.406723	+	0.913552i	$0.366671\pi$
$600$	0	0
$601$	−25.0367	−1.02127	−0.510634	−	0.859798i	$-0.670589\pi$
$601$	−25.0367	−1.02127	−0.510634	+	0.859798i	$0.670589\pi$
$602$	0	0
$603$	48.6208i	1.97999i
$604$	0	0
$605$	− 14.5701i	− 0.592357i
$606$	0	0
$607$	4.94906	0.200876	0.100438	−	0.994943i	$-0.467976\pi$
$607$	4.94906	0.200876	0.100438	+	0.994943i	$0.467976\pi$
$608$	0	0
$609$	−5.31852	−0.215518
$610$	0	0
$611$	− 31.0836i	− 1.25751i
$612$	0	0
$613$	28.5126i	1.15161i	0.817585	+	0.575807i	$0.195312\pi$
$613$	28.5126i	1.15161i	−0.817585	+	0.575807i	$0.804688\pi$
$614$	0	0
$615$	−31.4696	−1.26898
$616$	0	0
$617$	36.9906	1.48919	0.744593	−	0.667519i	$-0.232644\pi$
$617$	36.9906	1.48919	0.744593	+	0.667519i	$0.232644\pi$
$618$	0	0
$619$	14.0321i	0.563997i	0.959415	+	0.281998i	$0.0909973\pi$
$619$	14.0321i	0.563997i	−0.959415	+	0.281998i	$0.909003\pi$
$620$	0	0
$621$	− 15.7305i	− 0.631245i
$622$	0	0
$623$	1.99395	0.0798860
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	− 15.4877i	− 0.618518i
$628$	0	0
$629$	− 10.8651i	− 0.433219i
$630$	0	0
$631$	12.3580	0.491962	0.245981	−	0.969275i	$-0.420890\pi$
$631$	12.3580	0.491962	0.245981	+	0.969275i	$0.420890\pi$
$632$	0	0
$633$	81.9066	3.25549
$634$	0	0
$635$	− 16.3081i	− 0.647166i
$636$	0	0
$637$	− 3.61792i	− 0.143347i
$638$	0	0
$639$	−6.82762	−0.270096
$640$	0	0
$641$	−24.6420	−0.973301	−0.486651	−	0.873597i	$-0.661782\pi$
$641$	−24.6420	−0.973301	−0.486651	+	0.873597i	$0.661782\pi$
$642$	0	0
$643$	19.7785i	0.779990i	0.920817	+	0.389995i	$0.127523\pi$
$643$	19.7785i	0.779990i	−0.920817	+	0.389995i	$0.872477\pi$
$644$	0	0
$645$	− 29.0435i	− 1.14359i
$646$	0	0
$647$	9.04173	0.355467	0.177734	−	0.984079i	$-0.443123\pi$
$647$	9.04173	0.355467	0.177734	+	0.984079i	$0.443123\pi$
$648$	0	0
$649$	7.52637	0.295436
$650$	0	0
$651$	− 63.5396i	− 2.49031i
$652$	0	0
$653$	− 43.0547i	− 1.68486i	−0.538805	−	0.842430i	$-0.681124\pi$
$653$	− 43.0547i	− 1.68486i	0.538805	−	0.842430i	$-0.318876\pi$
$654$	0	0
$655$	−8.26767	−0.323045
$656$	0	0
$657$	−69.3120	−2.70412
$658$	0	0
$659$	49.7536i	1.93813i	0.246814	+	0.969063i	$0.420616\pi$
$659$	49.7536i	1.93813i	−0.246814	+	0.969063i	$0.579384\pi$
$660$	0	0
$661$	− 38.6735i	− 1.50423i	−0.659034	−	0.752113i	$-0.729035\pi$
$661$	− 38.6735i	− 1.50423i	0.659034	−	0.752113i	$-0.270965\pi$
$662$	0	0
$663$	−19.7693	−0.767775
$664$	0	0
$665$	2.81229	0.109056
$666$	0	0
$667$	0.938025i	0.0363205i
$668$	0	0
$669$	− 3.75006i	− 0.144986i
$670$	0	0
$671$	39.1112	1.50987
$672$	0	0
$673$	−30.4504	−1.17378	−0.586889	−	0.809668i	$-0.699647\pi$
$673$	−30.4504	−1.17378	−0.586889	+	0.809668i	$0.699647\pi$
$674$	0	0
$675$	− 10.3548i	− 0.398555i
$676$	0	0
$677$	− 21.2451i	− 0.816517i	−0.912866	−	0.408259i	$-0.866136\pi$
$677$	− 21.2451i	− 0.816517i	0.912866	−	0.408259i	$-0.133864\pi$
$678$	0	0
$679$	40.4706	1.55312
$680$	0	0
$681$	−46.3781	−1.77721
$682$	0	0
$683$	− 25.0990i	− 0.960388i	−0.877162	−	0.480194i	$-0.840566\pi$
$683$	− 25.0990i	− 0.960388i	0.877162	−	0.480194i	$-0.159434\pi$
$684$	0	0
$685$	− 17.5424i	− 0.670261i
$686$	0	0
$687$	24.7098	0.942738
$688$	0	0
$689$	−35.6929	−1.35979
$690$	0	0
$691$	− 43.6924i	− 1.66214i	−0.556171	−	0.831068i	$-0.687730\pi$
$691$	− 43.6924i	− 1.66214i	0.556171	−	0.831068i	$-0.312270\pi$
$692$	0	0
$693$	− 90.7406i	− 3.44695i
$694$	0	0
$695$	0.878831	0.0333359
$696$	0	0
$697$	−16.6627	−0.631144
$698$	0	0
$699$	55.4403i	2.09695i
$700$	0	0
$701$	5.13928i	0.194108i	0.995279	+	0.0970539i	$0.0309419\pi$
$701$	5.13928i	0.194108i	−0.995279	+	0.0970539i	$0.969058\pi$
$702$	0	0
$703$	−6.69976	−0.252686
$704$	0	0
$705$	23.9197	0.900867
$706$	0	0
$707$	− 33.7554i	− 1.26950i
$708$	0	0
$709$	− 23.4211i	− 0.879596i	−0.898097	−	0.439798i	$-0.855050\pi$
$709$	− 23.4211i	− 0.879596i	0.898097	−	0.439798i	$-0.144950\pi$
$710$	0	0
$711$	8.80112	0.330068
$712$	0	0
$713$	−11.2064	−0.419684
$714$	0	0
$715$	20.1262i	0.752678i
$716$	0	0
$717$	− 69.1817i	− 2.58364i
$718$	0	0
$719$	−22.6820	−0.845894	−0.422947	−	0.906154i	$-0.639004\pi$
$719$	−22.6820	−0.845894	−0.422947	+	0.906154i	$0.639004\pi$
$720$	0	0
$721$	50.6203	1.88520
$722$	0	0
$723$	5.11299i	0.190154i
$724$	0	0
$725$	0.617462i	0.0229320i
$726$	0	0
$727$	−34.2340	−1.26967	−0.634835	−	0.772648i	$-0.718932\pi$
$727$	−34.2340	−1.26967	−0.634835	+	0.772648i	$0.718932\pi$
$728$	0	0
$729$	14.9230	0.552704
$730$	0	0
$731$	− 15.3781i	− 0.568780i
$732$	0	0
$733$	35.0860i	1.29593i	0.761670	+	0.647965i	$0.224380\pi$
$733$	35.0860i	1.29593i	−0.761670	+	0.647965i	$0.775620\pi$
$734$	0	0
$735$	2.78409	0.102693
$736$	0	0
$737$	38.5312	1.41932
$738$	0	0
$739$	6.36634i	0.234190i	0.993121	+	0.117095i	$0.0373581\pi$
$739$	6.36634i	0.234190i	−0.993121	+	0.117095i	$0.962642\pi$
$740$	0	0
$741$	12.1904i	0.447824i
$742$	0	0
$743$	0.905818	0.0332312	0.0166156	−	0.999862i	$-0.494711\pi$
$743$	0.905818	0.0332312	0.0166156	+	0.999862i	$0.494711\pi$
$744$	0	0
$745$	8.52396	0.312294
$746$	0	0
$747$	88.8808i	3.25198i
$748$	0	0
$749$	51.2068i	1.87106i
$750$	0	0
$751$	17.1561	0.626035	0.313018	−	0.949747i	$-0.398660\pi$
$751$	17.1561	0.626035	0.313018	+	0.949747i	$0.398660\pi$
$752$	0	0
$753$	10.5150	0.383188
$754$	0	0
$755$	15.8403i	0.576486i
$756$	0	0
$757$	36.1374i	1.31344i	0.754136	+	0.656718i	$0.228056\pi$
$757$	36.1374i	1.31344i	−0.754136	+	0.656718i	$0.771944\pi$
$758$	0	0
$759$	−23.5283	−0.854022
$760$	0	0
$761$	−8.20143	−0.297302	−0.148651	−	0.988890i	$-0.547493\pi$
$761$	−8.20143	−0.297302	−0.148651	+	0.988890i	$0.547493\pi$
$762$	0	0
$763$	− 3.97199i	− 0.143796i
$764$	0	0
$765$	− 10.3478i	− 0.374127i
$766$	0	0
$767$	−5.92402	−0.213904
$768$	0	0
$769$	18.4357	0.664810	0.332405	−	0.943137i	$-0.392140\pi$
$769$	18.4357	0.664810	0.332405	+	0.943137i	$0.392140\pi$
$770$	0	0
$771$	13.9318i	0.501742i
$772$	0	0
$773$	20.9082i	0.752014i	0.926617	+	0.376007i	$0.122703\pi$
$773$	20.9082i	0.752014i	−0.926617	+	0.376007i	$0.877297\pi$
$774$	0	0
$775$	−7.37672	−0.264980
$776$	0	0
$777$	−57.7085	−2.07028
$778$	0	0
$779$	10.2747i	0.368131i
$780$	0	0
$781$	5.41078i	0.193613i
$782$	0	0
$783$	6.39367	0.228491
$784$	0	0
$785$	0.882769	0.0315074
$786$	0	0
$787$	34.8166i	1.24108i	0.784176	+	0.620539i	$0.213086\pi$
$787$	34.8166i	1.24108i	−0.784176	+	0.620539i	$0.786914\pi$
$788$	0	0
$789$	33.9685i	1.20931i
$790$	0	0
$791$	−12.7492	−0.453308
$792$	0	0
$793$	−30.7845	−1.09319
$794$	0	0
$795$	− 27.4666i	− 0.974141i
$796$	0	0
$797$	− 31.0580i	− 1.10013i	−0.835121	−	0.550066i	$-0.814603\pi$
$797$	− 31.0580i	− 1.10013i	0.835121	−	0.550066i	$-0.185397\pi$
$798$	0	0
$799$	12.6651	0.448060
$800$	0	0
$801$	−4.52407	−0.159850
$802$	0	0
$803$	54.9287i	1.93839i
$804$	0	0
$805$	− 4.27233i	− 0.150580i
$806$	0	0
$807$	1.39931	0.0492582
$808$	0	0
$809$	−18.1110	−0.636750	−0.318375	−	0.947965i	$-0.603137\pi$
$809$	−18.1110	−0.636750	−0.318375	+	0.947965i	$0.603137\pi$
$810$	0	0
$811$	− 26.9319i	− 0.945707i	−0.881141	−	0.472854i	$-0.843224\pi$
$811$	− 26.9319i	− 0.945707i	0.881141	−	0.472854i	$-0.156776\pi$
$812$	0	0
$813$	− 25.2618i	− 0.885970i
$814$	0	0
$815$	8.67648	0.303924
$816$	0	0
$817$	−9.48263	−0.331755
$818$	0	0
$819$	71.4221i	2.49569i
$820$	0	0
$821$	− 15.3754i	− 0.536604i	−0.963335	−	0.268302i	$-0.913537\pi$
$821$	− 15.3754i	− 0.536604i	0.963335	−	0.268302i	$-0.0864625\pi$
$822$	0	0
$823$	9.79320	0.341370	0.170685	−	0.985326i	$-0.445402\pi$
$823$	9.79320	0.341370	0.170685	+	0.985326i	$0.445402\pi$
$824$	0	0
$825$	−15.4877	−0.539211
$826$	0	0
$827$	13.5867i	0.472456i	0.971698	+	0.236228i	$0.0759113\pi$
$827$	13.5867i	0.472456i	−0.971698	+	0.236228i	$0.924089\pi$
$828$	0	0
$829$	4.13713i	0.143688i	0.997416	+	0.0718442i	$0.0228884\pi$
$829$	4.13713i	0.143688i	−0.997416	+	0.0718442i	$0.977112\pi$
$830$	0	0
$831$	52.3662	1.81656
$832$	0	0
$833$	1.47413	0.0510757
$834$	0	0
$835$	21.7621i	0.753108i
$836$	0	0
$837$	76.3841i	2.64022i
$838$	0	0
$839$	−3.97020	−0.137067	−0.0685333	−	0.997649i	$-0.521832\pi$
$839$	−3.97020	−0.137067	−0.0685333	+	0.997649i	$0.521832\pi$
$840$	0	0
$841$	28.6187	0.986853
$842$	0	0
$843$	− 27.3923i	− 0.943442i
$844$	0	0
$845$	− 2.84139i	− 0.0977468i
$846$	0	0
$847$	−40.9753	−1.40793
$848$	0	0
$849$	−100.071	−3.43442
$850$	0	0
$851$	10.1780i	0.348898i
$852$	0	0
$853$	36.8510i	1.26176i	0.775882	+	0.630878i	$0.217305\pi$
$853$	36.8510i	1.26176i	−0.775882	+	0.630878i	$0.782695\pi$
$854$	0	0
$855$	−6.38080	−0.218219
$856$	0	0
$857$	−21.6000	−0.737841	−0.368921	−	0.929461i	$-0.620273\pi$
$857$	−21.6000	−0.737841	−0.368921	+	0.929461i	$0.620273\pi$
$858$	0	0
$859$	5.01717i	0.171184i	0.996330	+	0.0855919i	$0.0272781\pi$
$859$	5.01717i	0.171184i	−0.996330	+	0.0855919i	$0.972722\pi$
$860$	0	0
$861$	88.5018i	3.01613i
$862$	0	0
$863$	48.4943	1.65077	0.825383	−	0.564573i	$-0.190959\pi$
$863$	48.4943	1.65077	0.825383	+	0.564573i	$0.190959\pi$
$864$	0	0
$865$	−9.66571	−0.328644
$866$	0	0
$867$	44.0127i	1.49475i
$868$	0	0
$869$	− 6.97475i	− 0.236602i
$870$	0	0
$871$	−30.3280	−1.02762
$872$	0	0
$873$	−91.8237	−3.10776
$874$	0	0
$875$	− 2.81229i	− 0.0950729i
$876$	0	0
$877$	− 9.26377i	− 0.312815i	−0.987693	−	0.156408i	$-0.950009\pi$
$877$	− 9.26377i	− 0.312815i	0.987693	−	0.156408i	$-0.0499913\pi$
$878$	0	0
$879$	85.1085	2.87064
$880$	0	0
$881$	14.0630	0.473793	0.236897	−	0.971535i	$-0.423870\pi$
$881$	14.0630	0.473793	0.236897	+	0.971535i	$0.423870\pi$
$882$	0	0
$883$	48.1637i	1.62084i	0.585850	+	0.810419i	$0.300761\pi$
$883$	48.1637i	1.62084i	−0.585850	+	0.810419i	$0.699239\pi$
$884$	0	0
$885$	− 4.55868i	− 0.153238i
$886$	0	0
$887$	−36.5264	−1.22644	−0.613218	−	0.789914i	$-0.710125\pi$
$887$	−36.5264	−1.22644	−0.613218	+	0.789914i	$0.710125\pi$
$888$	0	0
$889$	−45.8631	−1.53820
$890$	0	0
$891$	63.5738i	2.12980i
$892$	0	0
$893$	− 7.80972i	− 0.261342i
$894$	0	0
$895$	−20.0228	−0.669288
$896$	0	0
$897$	18.5191	0.618336
$898$	0	0
$899$	− 4.55485i	− 0.151913i
$900$	0	0
$901$	− 14.5432i	− 0.484504i
$902$	0	0
$903$	−81.6788	−2.71810
$904$	0	0
$905$	3.56175	0.118396
$906$	0	0
$907$	− 48.6955i	− 1.61691i	−0.588560	−	0.808454i	$-0.700305\pi$
$907$	− 48.6955i	− 1.61691i	0.588560	−	0.808454i	$-0.299695\pi$
$908$	0	0
$909$	76.5874i	2.54024i
$910$	0	0
$911$	37.7398	1.25038	0.625188	−	0.780474i	$-0.285022\pi$
$911$	37.7398	1.25038	0.625188	+	0.780474i	$0.285022\pi$
$912$	0	0
$913$	70.4366	2.33111
$914$	0	0
$915$	− 23.6894i	− 0.783149i
$916$	0	0
$917$	23.2511i	0.767820i
$918$	0	0
$919$	−1.08745	−0.0358718	−0.0179359	−	0.999839i	$-0.505709\pi$
$919$	−1.08745	−0.0358718	−0.0179359	+	0.999839i	$0.505709\pi$
$920$	0	0
$921$	32.9538	1.08586
$922$	0	0
$923$	− 4.25883i	− 0.140181i
$924$	0	0
$925$	6.69976i	0.220287i
$926$	0	0
$927$	−114.852	−3.77224
$928$	0	0
$929$	7.94724	0.260740	0.130370	−	0.991465i	$-0.458383\pi$
$929$	7.94724	0.260740	0.130370	+	0.991465i	$0.458383\pi$
$930$	0	0
$931$	− 0.908997i	− 0.0297912i
$932$	0	0
$933$	45.5714i	1.49194i
$934$	0	0
$935$	−8.20049	−0.268185
$936$	0	0
$937$	−20.6423	−0.674356	−0.337178	−	0.941441i	$-0.609472\pi$
$937$	−20.6423	−0.674356	−0.337178	+	0.941441i	$0.609472\pi$
$938$	0	0
$939$	49.2298i	1.60655i
$940$	0	0
$941$	2.32808i	0.0758931i	0.999280	+	0.0379466i	$0.0120817\pi$
$941$	2.32808i	0.0758931i	−0.999280	+	0.0379466i	$0.987918\pi$
$942$	0	0
$943$	15.6090	0.508299
$944$	0	0
$945$	−29.1206	−0.947294
$946$	0	0
$947$	35.7286i	1.16102i	0.814252	+	0.580511i	$0.197147\pi$
$947$	35.7286i	1.16102i	−0.814252	+	0.580511i	$0.802853\pi$
$948$	0	0
$949$	− 43.2344i	− 1.40345i
$950$	0	0
$951$	−91.4598	−2.96579
$952$	0	0
$953$	31.4726	1.01950	0.509749	−	0.860323i	$-0.329738\pi$
$953$	31.4726	1.01950	0.509749	+	0.860323i	$0.329738\pi$
$954$	0	0
$955$	− 18.5078i	− 0.598898i
$956$	0	0
$957$	− 9.56305i	− 0.309129i
$958$	0	0
$959$	−49.3344	−1.59309
$960$	0	0
$961$	23.4160	0.755356
$962$	0	0
$963$	− 116.183i	− 3.74394i
$964$	0	0
$965$	− 0.891942i	− 0.0287126i
$966$	0	0
$967$	−18.4524	−0.593390	−0.296695	−	0.954972i	$-0.595884\pi$
$967$	−18.4524	−0.593390	−0.296695	+	0.954972i	$0.595884\pi$
$968$	0	0
$969$	−4.96700	−0.159563
$970$	0	0
$971$	13.8572i	0.444698i	0.974967	+	0.222349i	$0.0713724\pi$
$971$	13.8572i	0.444698i	−0.974967	+	0.222349i	$0.928628\pi$
$972$	0	0
$973$	− 2.47153i	− 0.0792336i
$974$	0	0
$975$	12.1904	0.390404
$976$	0	0
$977$	−12.4100	−0.397032	−0.198516	−	0.980098i	$-0.563612\pi$
$977$	−12.4100	−0.397032	−0.198516	+	0.980098i	$0.563612\pi$
$978$	0	0
$979$	3.58525i	0.114585i
$980$	0	0
$981$	9.01204i	0.287732i
$982$	0	0
$983$	−24.7262	−0.788643	−0.394322	−	0.918973i	$-0.629020\pi$
$983$	−24.7262	−0.788643	−0.394322	+	0.918973i	$0.629020\pi$
$984$	0	0
$985$	−12.9518	−0.412678
$986$	0	0
$987$	− 67.2691i	− 2.14120i
$988$	0	0
$989$	14.4056i	0.458073i
$990$	0	0
$991$	−25.6353	−0.814334	−0.407167	−	0.913354i	$-0.633483\pi$
$991$	−25.6353	−0.814334	−0.407167	+	0.913354i	$0.633483\pi$
$992$	0	0
$993$	54.0832	1.71628
$994$	0	0
$995$	− 6.52220i	− 0.206768i
$996$	0	0
$997$	14.5038i	0.459339i	0.973269	+	0.229669i	$0.0737645\pi$
$997$	14.5038i	0.459339i	−0.973269	+	0.229669i	$0.926236\pi$
$998$	0	0
$999$	69.3744	2.19491

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3040.2.f.b.1521.4		44
4.3	odd	2		760.2.f.b.381.12	yes	44
8.3	odd	2		760.2.f.b.381.11	✓	44
8.5	even	2	inner	3040.2.f.b.1521.41		44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.f.b.381.11	✓	44	8.3	odd	2
760.2.f.b.381.12	yes	44	4.3	odd	2
3040.2.f.b.1521.4		44	1.1	even	1	trivial
3040.2.f.b.1521.41		44	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3040.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.06281i q^{3} +1.00000i q^{5} +2.81229 q^{7} -6.38080 q^{9} +5.05668i q^{11} -3.98012i q^{13} +3.06281 q^{15} +1.62171 q^{17} -1.00000i q^{19} -8.61352i q^{21} -1.51916 q^{23} -1.00000 q^{25} +10.3548i q^{27} -0.617462i q^{29} +7.37672 q^{31} +15.4877 q^{33} +2.81229i q^{35} -6.69976i q^{37} -12.1904 q^{39} -10.2747 q^{41} -9.48263i q^{43} -6.38080i q^{45} +7.80972 q^{47} +0.908997 q^{49} -4.96700i q^{51} -8.96779i q^{53} -5.05668 q^{55} -3.06281 q^{57} -1.48840i q^{59} -7.73455i q^{61} -17.9447 q^{63} +3.98012 q^{65} -7.61986i q^{67} +4.65290i q^{69} +1.07002 q^{71} +10.8626 q^{73} +3.06281i q^{75} +14.2209i q^{77} -1.37931 q^{79} +12.5722 q^{81} -13.9294i q^{83} +1.62171i q^{85} -1.89117 q^{87} +0.709012 q^{89} -11.1933i q^{91} -22.5935i q^{93} +1.00000 q^{95} +14.3906 q^{97} -32.2657i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{7} - 60 q^{9} + 24 q^{17} - 4 q^{23} - 44 q^{25} + 40 q^{33} + 24 q^{39} - 32 q^{41} + 20 q^{47} + 108 q^{49} - 8 q^{55} - 20 q^{63} + 12 q^{65} + 8 q^{71} - 88 q^{73} - 40 q^{79} + 116 q^{81}+ \cdots + 116 q^{97}+O(q^{100}) \) 44 * q - 4 * q^7 - 60 * q^9 + 24 * q^17 - 4 * q^23 - 44 * q^25 + 40 * q^33 + 24 * q^39 - 32 * q^41 + 20 * q^47 + 108 * q^49 - 8 * q^55 - 20 * q^63 + 12 * q^65 + 8 * q^71 - 88 * q^73 - 40 * q^79 + 116 * q^81 - 48 * q^87 - 64 * q^89 + 44 * q^95 + 116 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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