LMFDB - Embedded newform 3040.2.f.a.1521.7

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)

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);

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")

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 = [28] 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:	$28$
Twist minimal:	no (minimal twist has level 760)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1521.7
Character	$\chi$	$=$	3040.1521
Dual form			3040.2.f.a.1521.22

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.74129i q^{3} +1.00000i q^{5} +4.26759 q^{7} -0.0320903 q^{9} +3.47101i q^{11} -1.17033i q^{13} +1.74129 q^{15} -3.21425 q^{17} +1.00000i q^{19} -7.43111i q^{21} +1.30724 q^{23} -1.00000 q^{25} -5.16799i q^{27} -7.74967i q^{29} +4.01448 q^{31} +6.04404 q^{33} +4.26759i q^{35} -2.87596i q^{37} -2.03788 q^{39} +9.51830 q^{41} +8.38330i q^{43} -0.0320903i q^{45} -0.615265 q^{47} +11.2123 q^{49} +5.59693i q^{51} +2.95709i q^{53} -3.47101 q^{55} +1.74129 q^{57} +0.792754i q^{59} -3.37425i q^{61} -0.136948 q^{63} +1.17033 q^{65} +10.6847i q^{67} -2.27628i q^{69} +0.758738 q^{71} +10.0998 q^{73} +1.74129i q^{75} +14.8129i q^{77} +15.5194 q^{79} -9.09524 q^{81} +10.8166i q^{83} -3.21425i q^{85} -13.4944 q^{87} -2.55872 q^{89} -4.99447i q^{91} -6.99037i q^{93} -1.00000 q^{95} -15.1049 q^{97} -0.111386i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 4 q^{7} - 12 q^{9} - 24 q^{17} - 4 q^{23} - 28 q^{25} - 40 q^{33} + 24 q^{39} + 32 q^{41} + 20 q^{47} - 36 q^{49} - 8 q^{55} - 20 q^{63} - 12 q^{65} + 8 q^{71} + 88 q^{73} - 40 q^{79} - 28 q^{81}+ \cdots - 116 q^{97}+O(q^{100}) $ 28 * q - 4 * q^7 - 12 * q^9 - 24 * q^17 - 4 * q^23 - 28 * q^25 - 40 * q^33 + 24 * q^39 + 32 * q^41 + 20 * q^47 - 36 * q^49 - 8 * q^55 - 20 * q^63 - 12 * q^65 + 8 * q^71 + 88 * q^73 - 40 * q^79 - 28 * q^81 - 48 * q^87 + 48 * q^89 - 28 * 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
$3$	− 1.74129i	− 1.00533i	−0.864480	−	0.502667i	$-0.832352\pi$
$3$	− 1.74129i	− 1.00533i	0.864480	−	0.502667i	$-0.167648\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	4.26759	1.61300	0.806499	−	0.591235i	$-0.201360\pi$
$7$	4.26759	1.61300	0.806499	+	0.591235i	$0.201360\pi$
$8$	0	0
$9$	−0.0320903	−0.0106968
$10$	0	0
$11$	3.47101i	1.04655i	0.852164	+	0.523275i	$0.175290\pi$
$11$	3.47101i	1.04655i	−0.852164	+	0.523275i	$0.824710\pi$
$12$	0	0
$13$	− 1.17033i	− 0.324590i	−0.986742	−	0.162295i	$-0.948110\pi$
$13$	− 1.17033i	− 0.324590i	0.986742	−	0.162295i	$-0.0518896\pi$
$14$	0	0
$15$	1.74129	0.449599
$16$	0	0
$17$	−3.21425	−0.779569	−0.389784	−	0.920906i	$-0.627451\pi$
$17$	−3.21425	−0.779569	−0.389784	+	0.920906i	$0.627451\pi$
$18$	0	0
$19$	1.00000i	0.229416i
$19$	1.00000i	0.229416i
$20$	0	0
$21$	− 7.43111i	− 1.62160i
$22$	0	0
$23$	1.30724	0.272578	0.136289	−	0.990669i	$-0.456482\pi$
$23$	1.30724	0.272578	0.136289	+	0.990669i	$0.456482\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	− 5.16799i	− 0.994580i
$28$	0	0
$29$	− 7.74967i	− 1.43908i	−0.694452	−	0.719539i	$-0.744353\pi$
$29$	− 7.74967i	− 1.43908i	0.694452	−	0.719539i	$-0.255647\pi$
$30$	0	0
$31$	4.01448	0.721021	0.360511	−	0.932755i	$-0.382602\pi$
$31$	4.01448	0.721021	0.360511	+	0.932755i	$0.382602\pi$
$32$	0	0
$33$	6.04404	1.05213
$34$	0	0
$35$	4.26759i	0.721355i
$36$	0	0
$37$	− 2.87596i	− 0.472804i	−0.971655	−	0.236402i	$-0.924032\pi$
$37$	− 2.87596i	− 0.472804i	0.971655	−	0.236402i	$-0.0759683\pi$
$38$	0	0
$39$	−2.03788	−0.326321
$40$	0	0
$41$	9.51830	1.48651	0.743254	−	0.669009i	$-0.233281\pi$
$41$	9.51830	1.48651	0.743254	+	0.669009i	$0.233281\pi$
$42$	0	0
$43$	8.38330i	1.27844i	0.769024	+	0.639220i	$0.220743\pi$
$43$	8.38330i	1.27844i	−0.769024	+	0.639220i	$0.779257\pi$
$44$	0	0
$45$	− 0.0320903i	− 0.00478375i
$46$	0	0
$47$	−0.615265	−0.0897457	−0.0448728	−	0.998993i	$-0.514288\pi$
$47$	−0.615265	−0.0897457	−0.0448728	+	0.998993i	$0.514288\pi$
$48$	0	0
$49$	11.2123	1.60176
$50$	0	0
$51$	5.59693i	0.783727i
$52$	0	0
$53$	2.95709i	0.406187i	0.979159	+	0.203094i	$0.0650995\pi$
$53$	2.95709i	0.406187i	−0.979159	+	0.203094i	$0.934900\pi$
$54$	0	0
$55$	−3.47101	−0.468031
$56$	0	0
$57$	1.74129	0.230639
$58$	0	0
$59$	0.792754i	0.103208i	0.998668	+	0.0516039i	$0.0164333\pi$
$59$	0.792754i	0.103208i	−0.998668	+	0.0516039i	$0.983567\pi$
$60$	0	0
$61$	− 3.37425i	− 0.432028i	−0.976390	−	0.216014i	$-0.930694\pi$
$61$	− 3.37425i	− 0.432028i	0.976390	−	0.216014i	$-0.0693057\pi$
$62$	0	0
$63$	−0.136948	−0.0172539
$64$	0	0
$65$	1.17033	0.145161
$66$	0	0
$67$	10.6847i	1.30534i	0.757642	+	0.652670i	$0.226351\pi$
$67$	10.6847i	1.30534i	−0.757642	+	0.652670i	$0.773649\pi$
$68$	0	0
$69$	− 2.27628i	− 0.274032i
$70$	0	0
$71$	0.758738	0.0900456	0.0450228	−	0.998986i	$-0.485664\pi$
$71$	0.758738	0.0900456	0.0450228	+	0.998986i	$0.485664\pi$
$72$	0	0
$73$	10.0998	1.18209	0.591046	−	0.806638i	$-0.298715\pi$
$73$	10.0998	1.18209	0.591046	+	0.806638i	$0.298715\pi$
$74$	0	0
$75$	1.74129i	0.201067i
$76$	0	0
$77$	14.8129i	1.68808i
$78$	0	0
$79$	15.5194	1.74607	0.873033	−	0.487661i	$-0.162150\pi$
$79$	15.5194	1.74607	0.873033	+	0.487661i	$0.162150\pi$
$80$	0	0
$81$	−9.09524	−1.01058
$82$	0	0
$83$	10.8166i	1.18727i	0.804734	+	0.593636i	$0.202308\pi$
$83$	10.8166i	1.18727i	−0.804734	+	0.593636i	$0.797692\pi$
$84$	0	0
$85$	− 3.21425i	− 0.348634i
$86$	0	0
$87$	−13.4944	−1.44675
$88$	0	0
$89$	−2.55872	−0.271223	−0.135612	−	0.990762i	$-0.543300\pi$
$89$	−2.55872	−0.271223	−0.135612	+	0.990762i	$0.543300\pi$
$90$	0	0
$91$	− 4.99447i	− 0.523563i
$92$	0	0
$93$	− 6.99037i	− 0.724867i
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−15.1049	−1.53367	−0.766836	−	0.641843i	$-0.778170\pi$
$97$	−15.1049	−1.53367	−0.766836	+	0.641843i	$0.778170\pi$
$98$	0	0
$99$	− 0.111386i	− 0.0111947i
$100$	0	0
$101$	− 3.51450i	− 0.349706i	−0.984595	−	0.174853i	$-0.944055\pi$
$101$	− 3.51450i	− 0.349706i	0.984595	−	0.174853i	$-0.0559450\pi$
$102$	0	0
$103$	10.7032	1.05462	0.527309	−	0.849674i	$-0.323201\pi$
$103$	10.7032	1.05462	0.527309	+	0.849674i	$0.323201\pi$
$104$	0	0
$105$	7.43111	0.725202
$106$	0	0
$107$	− 13.3164i	− 1.28734i	−0.765301	−	0.643672i	$-0.777410\pi$
$107$	− 13.3164i	− 1.28734i	0.765301	−	0.643672i	$-0.222590\pi$
$108$	0	0
$109$	− 4.66724i	− 0.447041i	−0.974699	−	0.223520i	$-0.928245\pi$
$109$	− 4.66724i	− 0.447041i	0.974699	−	0.223520i	$-0.0717549\pi$
$110$	0	0
$111$	−5.00787	−0.475326
$112$	0	0
$113$	6.23106	0.586169	0.293084	−	0.956087i	$-0.405318\pi$
$113$	6.23106	0.586169	0.293084	+	0.956087i	$0.405318\pi$
$114$	0	0
$115$	1.30724i	0.121901i
$116$	0	0
$117$	0.0375562i	0.00347207i
$118$	0	0
$119$	−13.7171	−1.25744
$120$	0	0
$121$	−1.04794	−0.0952670
$122$	0	0
$123$	− 16.5741i	− 1.49444i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	12.5227	1.11121	0.555603	−	0.831448i	$-0.312488\pi$
$127$	12.5227	1.11121	0.555603	+	0.831448i	$0.312488\pi$
$128$	0	0
$129$	14.5977	1.28526
$130$	0	0
$131$	− 10.4192i	− 0.910330i	−0.890407	−	0.455165i	$-0.849580\pi$
$131$	− 10.4192i	− 0.910330i	0.890407	−	0.455165i	$-0.150420\pi$
$132$	0	0
$133$	4.26759i	0.370047i
$134$	0	0
$135$	5.16799	0.444790
$136$	0	0
$137$	−21.8395	−1.86587	−0.932936	−	0.360041i	$-0.882763\pi$
$137$	−21.8395	−1.86587	−0.932936	+	0.360041i	$0.882763\pi$
$138$	0	0
$139$	0.938309i	0.0795863i	0.999208	+	0.0397932i	$0.0126699\pi$
$139$	0.938309i	0.0795863i	−0.999208	+	0.0397932i	$0.987330\pi$
$140$	0	0
$141$	1.07136i	0.0902244i
$142$	0	0
$143$	4.06222	0.339700
$144$	0	0
$145$	7.74967	0.643575
$146$	0	0
$147$	− 19.5239i	− 1.61031i
$148$	0	0
$149$	4.84067i	0.396563i	0.980145	+	0.198281i	$0.0635360\pi$
$149$	4.84067i	0.396563i	−0.980145	+	0.198281i	$0.936464\pi$
$150$	0	0
$151$	16.9169	1.37668	0.688340	−	0.725388i	$-0.258340\pi$
$151$	16.9169	1.37668	0.688340	+	0.725388i	$0.258340\pi$
$152$	0	0
$153$	0.103146	0.00833888
$154$	0	0
$155$	4.01448i	0.322450i
$156$	0	0
$157$	− 2.31235i	− 0.184546i	−0.995734	−	0.0922728i	$-0.970587\pi$
$157$	− 2.31235i	− 0.184546i	0.995734	−	0.0922728i	$-0.0294132\pi$
$158$	0	0
$159$	5.14914	0.408354
$160$	0	0
$161$	5.57876	0.439668
$162$	0	0
$163$	13.3563i	1.04615i	0.852288	+	0.523073i	$0.175215\pi$
$163$	13.3563i	1.04615i	−0.852288	+	0.523073i	$0.824785\pi$
$164$	0	0
$165$	6.04404i	0.470528i
$166$	0	0
$167$	−17.5050	−1.35457	−0.677287	−	0.735719i	$-0.736844\pi$
$167$	−17.5050	−1.35457	−0.677287	+	0.735719i	$0.736844\pi$
$168$	0	0
$169$	11.6303	0.894641
$170$	0	0
$171$	− 0.0320903i	− 0.00245401i
$172$	0	0
$173$	19.6197i	1.49166i	0.666136	+	0.745830i	$0.267947\pi$
$173$	19.6197i	1.49166i	−0.666136	+	0.745830i	$0.732053\pi$
$174$	0	0
$175$	−4.26759	−0.322600
$176$	0	0
$177$	1.38041	0.103758
$178$	0	0
$179$	− 16.0479i	− 1.19948i	−0.800196	−	0.599739i	$-0.795271\pi$
$179$	− 16.0479i	− 1.19948i	0.800196	−	0.599739i	$-0.204729\pi$
$180$	0	0
$181$	− 18.4270i	− 1.36967i	−0.728698	−	0.684836i	$-0.759874\pi$
$181$	− 18.4270i	− 1.36967i	0.728698	−	0.684836i	$-0.240126\pi$
$182$	0	0
$183$	−5.87554	−0.434333
$184$	0	0
$185$	2.87596	0.211444
$186$	0	0
$187$	− 11.1567i	− 0.815858i
$188$	0	0
$189$	− 22.0549i	− 1.60426i
$190$	0	0
$191$	9.65264	0.698441	0.349220	−	0.937041i	$-0.386446\pi$
$191$	9.65264	0.698441	0.349220	+	0.937041i	$0.386446\pi$
$192$	0	0
$193$	12.4760	0.898042	0.449021	−	0.893521i	$-0.351773\pi$
$193$	12.4760	0.898042	0.449021	+	0.893521i	$0.351773\pi$
$194$	0	0
$195$	− 2.03788i	− 0.145935i
$196$	0	0
$197$	− 9.56205i	− 0.681268i	−0.940196	−	0.340634i	$-0.889358\pi$
$197$	− 9.56205i	− 0.681268i	0.940196	−	0.340634i	$-0.110642\pi$
$198$	0	0
$199$	−10.2805	−0.728766	−0.364383	−	0.931249i	$-0.618720\pi$
$199$	−10.2805	−0.728766	−0.364383	+	0.931249i	$0.618720\pi$
$200$	0	0
$201$	18.6051	1.31230
$202$	0	0
$203$	− 33.0724i	− 2.32123i
$204$	0	0
$205$	9.51830i	0.664787i
$206$	0	0
$207$	−0.0419498	−0.00291571
$208$	0	0
$209$	−3.47101	−0.240095
$210$	0	0
$211$	16.6190i	1.14410i	0.820218	+	0.572051i	$0.193852\pi$
$211$	16.6190i	1.14410i	−0.820218	+	0.572051i	$0.806148\pi$
$212$	0	0
$213$	− 1.32118i	− 0.0905259i
$214$	0	0
$215$	−8.38330	−0.571736
$216$	0	0
$217$	17.1321	1.16301
$218$	0	0
$219$	− 17.5867i	− 1.18840i
$220$	0	0
$221$	3.76171i	0.253040i
$222$	0	0
$223$	9.11396	0.610316	0.305158	−	0.952302i	$-0.401291\pi$
$223$	9.11396	0.610316	0.305158	+	0.952302i	$0.401291\pi$
$224$	0	0
$225$	0.0320903	0.00213936
$226$	0	0
$227$	21.0419i	1.39660i	0.715804	+	0.698302i	$0.246061\pi$
$227$	21.0419i	1.39660i	−0.715804	+	0.698302i	$0.753939\pi$
$228$	0	0
$229$	− 0.230519i	− 0.0152331i	−0.999971	−	0.00761655i	$-0.997576\pi$
$229$	− 0.230519i	− 0.0152331i	0.999971	−	0.00761655i	$-0.00242445\pi$
$230$	0	0
$231$	25.7935	1.69709
$232$	0	0
$233$	−2.23928	−0.146700	−0.0733501	−	0.997306i	$-0.523369\pi$
$233$	−2.23928	−0.146700	−0.0733501	+	0.997306i	$0.523369\pi$
$234$	0	0
$235$	− 0.615265i	− 0.0401355i
$236$	0	0
$237$	− 27.0237i	− 1.75538i
$238$	0	0
$239$	−18.5649	−1.20086	−0.600431	−	0.799677i	$-0.705004\pi$
$239$	−18.5649	−1.20086	−0.600431	+	0.799677i	$0.705004\pi$
$240$	0	0
$241$	−25.4868	−1.64175	−0.820874	−	0.571109i	$-0.806513\pi$
$241$	−25.4868	−1.64175	−0.820874	+	0.571109i	$0.806513\pi$
$242$	0	0
$243$	0.333478i	0.0213926i
$244$	0	0
$245$	11.2123i	0.716330i
$246$	0	0
$247$	1.17033	0.0744660
$248$	0	0
$249$	18.8348	1.19360
$250$	0	0
$251$	− 4.83471i	− 0.305164i	−0.988291	−	0.152582i	$-0.951241\pi$
$251$	− 4.83471i	− 0.305164i	0.988291	−	0.152582i	$-0.0487588\pi$
$252$	0	0
$253$	4.53745i	0.285267i
$254$	0	0
$255$	−5.59693	−0.350494
$256$	0	0
$257$	−28.4477	−1.77452	−0.887261	−	0.461268i	$-0.847395\pi$
$257$	−28.4477	−1.77452	−0.887261	+	0.461268i	$0.847395\pi$
$258$	0	0
$259$	− 12.2734i	− 0.762632i
$260$	0	0
$261$	0.248690i	0.0153935i
$262$	0	0
$263$	0.204621	0.0126175	0.00630874	−	0.999980i	$-0.497992\pi$
$263$	0.204621	0.0126175	0.00630874	+	0.999980i	$0.497992\pi$
$264$	0	0
$265$	−2.95709	−0.181652
$266$	0	0
$267$	4.45547i	0.272670i
$268$	0	0
$269$	13.0098i	0.793224i	0.917986	+	0.396612i	$0.129814\pi$
$269$	13.0098i	0.793224i	−0.917986	+	0.396612i	$0.870186\pi$
$270$	0	0
$271$	28.4792	1.72999	0.864995	−	0.501780i	$-0.167321\pi$
$271$	28.4792	1.72999	0.864995	+	0.501780i	$0.167321\pi$
$272$	0	0
$273$	−8.69682	−0.526356
$274$	0	0
$275$	− 3.47101i	− 0.209310i
$276$	0	0
$277$	− 16.6647i	− 1.00128i	−0.865654	−	0.500642i	$-0.833097\pi$
$277$	− 16.6647i	− 1.00128i	0.865654	−	0.500642i	$-0.166903\pi$
$278$	0	0
$279$	−0.128826	−0.00771260
$280$	0	0
$281$	−14.4794	−0.863768	−0.431884	−	0.901929i	$-0.642151\pi$
$281$	−14.4794	−0.863768	−0.431884	+	0.901929i	$0.642151\pi$
$282$	0	0
$283$	− 30.1537i	− 1.79245i	−0.443597	−	0.896227i	$-0.646298\pi$
$283$	− 30.1537i	− 1.79245i	0.443597	−	0.896227i	$-0.353702\pi$
$284$	0	0
$285$	1.74129i	0.103145i
$286$	0	0
$287$	40.6202	2.39773
$288$	0	0
$289$	−6.66863	−0.392272
$290$	0	0
$291$	26.3020i	1.54185i
$292$	0	0
$293$	− 26.5469i	− 1.55089i	−0.631418	−	0.775443i	$-0.717527\pi$
$293$	− 26.5469i	− 1.55089i	0.631418	−	0.775443i	$-0.282473\pi$
$294$	0	0
$295$	−0.792754	−0.0461559
$296$	0	0
$297$	17.9382	1.04088
$298$	0	0
$299$	− 1.52990i	− 0.0884762i
$300$	0	0
$301$	35.7765i	2.06212i
$302$	0	0
$303$	−6.11976	−0.351571
$304$	0	0
$305$	3.37425	0.193209
$306$	0	0
$307$	− 25.1571i	− 1.43579i	−0.696150	−	0.717896i	$-0.745105\pi$
$307$	− 25.1571i	− 1.43579i	0.696150	−	0.717896i	$-0.254895\pi$
$308$	0	0
$309$	− 18.6374i	− 1.06024i
$310$	0	0
$311$	−10.2706	−0.582394	−0.291197	−	0.956663i	$-0.594054\pi$
$311$	−10.2706	−0.582394	−0.291197	+	0.956663i	$0.594054\pi$
$312$	0	0
$313$	−5.85159	−0.330752	−0.165376	−	0.986231i	$-0.552884\pi$
$313$	−5.85159	−0.330752	−0.165376	+	0.986231i	$0.552884\pi$
$314$	0	0
$315$	− 0.136948i	− 0.00771617i
$316$	0	0
$317$	10.5400i	0.591985i	0.955190	+	0.295992i	$0.0956503\pi$
$317$	10.5400i	0.591985i	−0.955190	+	0.295992i	$0.904350\pi$
$318$	0	0
$319$	26.8992	1.50607
$320$	0	0
$321$	−23.1877	−1.29421
$322$	0	0
$323$	− 3.21425i	− 0.178845i
$324$	0	0
$325$	1.17033i	0.0649180i
$326$	0	0
$327$	−8.12702	−0.449425
$328$	0	0
$329$	−2.62570	−0.144760
$330$	0	0
$331$	− 6.16543i	− 0.338882i	−0.985540	−	0.169441i	$-0.945804\pi$
$331$	− 6.16543i	− 0.338882i	0.985540	−	0.169441i	$-0.0541963\pi$
$332$	0	0
$333$	0.0922904i	0.00505748i
$334$	0	0
$335$	−10.6847	−0.583766
$336$	0	0
$337$	−30.1337	−1.64149	−0.820745	−	0.571295i	$-0.806441\pi$
$337$	−30.1337	−1.64149	−0.820745	+	0.571295i	$0.806441\pi$
$338$	0	0
$339$	− 10.8501i	− 0.589296i
$340$	0	0
$341$	13.9343i	0.754585i
$342$	0	0
$343$	17.9765	0.970642
$344$	0	0
$345$	2.27628	0.122551
$346$	0	0
$347$	27.8202i	1.49347i	0.665123	+	0.746734i	$0.268379\pi$
$347$	27.8202i	1.49347i	−0.665123	+	0.746734i	$0.731621\pi$
$348$	0	0
$349$	7.91760i	0.423819i	0.977289	+	0.211910i	$0.0679683\pi$
$349$	7.91760i	0.423819i	−0.977289	+	0.211910i	$0.932032\pi$
$350$	0	0
$351$	−6.04823	−0.322831
$352$	0	0
$353$	27.1161	1.44324	0.721621	−	0.692288i	$-0.243397\pi$
$353$	27.1161	1.44324	0.721621	+	0.692288i	$0.243397\pi$
$354$	0	0
$355$	0.758738i	0.0402696i
$356$	0	0
$357$	23.8854i	1.26415i
$358$	0	0
$359$	−4.76453	−0.251462	−0.125731	−	0.992064i	$-0.540128\pi$
$359$	−4.76453	−0.251462	−0.125731	+	0.992064i	$0.540128\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	1.82476i	0.0957751i
$364$	0	0
$365$	10.0998i	0.528648i
$366$	0	0
$367$	−35.9120	−1.87459	−0.937296	−	0.348535i	$-0.886679\pi$
$367$	−35.9120	−1.87459	−0.937296	+	0.348535i	$0.886679\pi$
$368$	0	0
$369$	−0.305445	−0.0159009
$370$	0	0
$371$	12.6196i	0.655179i
$372$	0	0
$373$	22.3983i	1.15974i	0.814710	+	0.579869i	$0.196896\pi$
$373$	22.3983i	1.15974i	−0.814710	+	0.579869i	$0.803104\pi$
$374$	0	0
$375$	−1.74129	−0.0899198
$376$	0	0
$377$	−9.06964	−0.467110
$378$	0	0
$379$	4.29393i	0.220565i	0.993900	+	0.110282i	$0.0351755\pi$
$379$	4.29393i	0.220565i	−0.993900	+	0.110282i	$0.964825\pi$
$380$	0	0
$381$	− 21.8056i	− 1.11713i
$382$	0	0
$383$	−25.6760	−1.31198	−0.655991	−	0.754768i	$-0.727749\pi$
$383$	−25.6760	−1.31198	−0.655991	+	0.754768i	$0.727749\pi$
$384$	0	0
$385$	−14.8129	−0.754934
$386$	0	0
$387$	− 0.269023i	− 0.0136752i
$388$	0	0
$389$	− 21.3819i	− 1.08411i	−0.840344	−	0.542054i	$-0.817647\pi$
$389$	− 21.3819i	− 1.08411i	0.840344	−	0.542054i	$-0.182353\pi$
$390$	0	0
$391$	−4.20179	−0.212494
$392$	0	0
$393$	−18.1429	−0.915186
$394$	0	0
$395$	15.5194i	0.780864i
$396$	0	0
$397$	− 37.8072i	− 1.89749i	−0.316044	−	0.948745i	$-0.602355\pi$
$397$	− 37.8072i	− 1.89749i	0.316044	−	0.948745i	$-0.397645\pi$
$398$	0	0
$399$	7.43111	0.372021
$400$	0	0
$401$	9.31973	0.465405	0.232703	−	0.972548i	$-0.425243\pi$
$401$	9.31973	0.465405	0.232703	+	0.972548i	$0.425243\pi$
$402$	0	0
$403$	− 4.69824i	− 0.234036i
$404$	0	0
$405$	− 9.09524i	− 0.451946i
$406$	0	0
$407$	9.98248	0.494813
$408$	0	0
$409$	−5.74304	−0.283975	−0.141988	−	0.989868i	$-0.545349\pi$
$409$	−5.74304	−0.283975	−0.141988	+	0.989868i	$0.545349\pi$
$410$	0	0
$411$	38.0289i	1.87583i
$412$	0	0
$413$	3.38315i	0.166474i
$414$	0	0
$415$	−10.8166	−0.530964
$416$	0	0
$417$	1.63387	0.0800109
$418$	0	0
$419$	13.1417i	0.642014i	0.947077	+	0.321007i	$0.104021\pi$
$419$	13.1417i	0.642014i	−0.947077	+	0.321007i	$0.895979\pi$
$420$	0	0
$421$	29.3028i	1.42813i	0.700078	+	0.714066i	$0.253148\pi$
$421$	29.3028i	1.42813i	−0.700078	+	0.714066i	$0.746852\pi$
$422$	0	0
$423$	0.0197441	0.000959990 0
$424$	0	0
$425$	3.21425	0.155914
$426$	0	0
$427$	− 14.3999i	− 0.696861i
$428$	0	0
$429$	− 7.07350i	− 0.341512i
$430$	0	0
$431$	5.65353	0.272321	0.136160	−	0.990687i	$-0.456524\pi$
$431$	5.65353	0.272321	0.136160	+	0.990687i	$0.456524\pi$
$432$	0	0
$433$	−28.9462	−1.39106	−0.695532	−	0.718495i	$-0.744831\pi$
$433$	−28.9462	−1.39106	−0.695532	+	0.718495i	$0.744831\pi$
$434$	0	0
$435$	− 13.4944i	− 0.647008i
$436$	0	0
$437$	1.30724i	0.0625337i
$438$	0	0
$439$	20.1200	0.960274	0.480137	−	0.877193i	$-0.340587\pi$
$439$	20.1200	0.960274	0.480137	+	0.877193i	$0.340587\pi$
$440$	0	0
$441$	−0.359808	−0.0171337
$442$	0	0
$443$	− 21.9508i	− 1.04291i	−0.853278	−	0.521457i	$-0.825389\pi$
$443$	− 21.9508i	− 1.04291i	0.853278	−	0.521457i	$-0.174611\pi$
$444$	0	0
$445$	− 2.55872i	− 0.121295i
$446$	0	0
$447$	8.42900	0.398678
$448$	0	0
$449$	−13.6209	−0.642812	−0.321406	−	0.946941i	$-0.604155\pi$
$449$	−13.6209	−0.642812	−0.321406	+	0.946941i	$0.604155\pi$
$450$	0	0
$451$	33.0381i	1.55570i
$452$	0	0
$453$	− 29.4573i	− 1.38402i
$454$	0	0
$455$	4.99447	0.234144
$456$	0	0
$457$	27.7553	1.29834	0.649169	−	0.760644i	$-0.275117\pi$
$457$	27.7553	1.29834	0.649169	+	0.760644i	$0.275117\pi$
$458$	0	0
$459$	16.6112i	0.775344i
$460$	0	0
$461$	26.1891i	1.21975i	0.792499	+	0.609873i	$0.208780\pi$
$461$	26.1891i	1.21975i	−0.792499	+	0.609873i	$0.791220\pi$
$462$	0	0
$463$	−28.7874	−1.33786	−0.668932	−	0.743324i	$-0.733248\pi$
$463$	−28.7874	−1.33786	−0.668932	+	0.743324i	$0.733248\pi$
$464$	0	0
$465$	6.99037	0.324170
$466$	0	0
$467$	34.8523i	1.61277i	0.591390	+	0.806386i	$0.298579\pi$
$467$	34.8523i	1.61277i	−0.591390	+	0.806386i	$0.701421\pi$
$468$	0	0
$469$	45.5978i	2.10551i
$470$	0	0
$471$	−4.02647	−0.185530
$472$	0	0
$473$	−29.0985	−1.33795
$474$	0	0
$475$	− 1.00000i	− 0.0458831i
$476$	0	0
$477$	− 0.0948939i	− 0.00434489i
$478$	0	0
$479$	−27.5781	−1.26007	−0.630037	−	0.776565i	$-0.716960\pi$
$479$	−27.5781	−1.26007	−0.630037	+	0.776565i	$0.716960\pi$
$480$	0	0
$481$	−3.36581	−0.153468
$482$	0	0
$483$	− 9.71424i	− 0.442013i
$484$	0	0
$485$	− 15.1049i	− 0.685879i
$486$	0	0
$487$	−37.6560	−1.70636	−0.853179	−	0.521619i	$-0.825328\pi$
$487$	−37.6560	−1.70636	−0.853179	+	0.521619i	$0.825328\pi$
$488$	0	0
$489$	23.2572	1.05173
$490$	0	0
$491$	5.16332i	0.233018i	0.993190	+	0.116509i	$0.0371703\pi$
$491$	5.16332i	0.233018i	−0.993190	+	0.116509i	$0.962830\pi$
$492$	0	0
$493$	24.9093i	1.12186i
$494$	0	0
$495$	0.111386	0.00500643
$496$	0	0
$497$	3.23798	0.145243
$498$	0	0
$499$	− 31.2755i	− 1.40008i	−0.714102	−	0.700042i	$-0.753165\pi$
$499$	− 31.2755i	− 1.40008i	0.714102	−	0.700042i	$-0.246835\pi$
$500$	0	0
$501$	30.4812i	1.36180i
$502$	0	0
$503$	−34.1754	−1.52381	−0.761904	−	0.647690i	$-0.775735\pi$
$503$	−34.1754	−1.52381	−0.761904	+	0.647690i	$0.775735\pi$
$504$	0	0
$505$	3.51450	0.156393
$506$	0	0
$507$	− 20.2518i	− 0.899414i
$508$	0	0
$509$	7.59220i	0.336518i	0.985743	+	0.168259i	$0.0538145\pi$
$509$	7.59220i	0.336518i	−0.985743	+	0.168259i	$0.946185\pi$
$510$	0	0
$511$	43.1018	1.90671
$512$	0	0
$513$	5.16799	0.228172
$514$	0	0
$515$	10.7032i	0.471639i
$516$	0	0
$517$	− 2.13559i	− 0.0939233i
$518$	0	0
$519$	34.1636	1.49962
$520$	0	0
$521$	44.3595	1.94343	0.971713	−	0.236166i	$-0.0758909\pi$
$521$	44.3595	1.94343	0.971713	+	0.236166i	$0.0758909\pi$
$522$	0	0
$523$	13.6137i	0.595288i	0.954677	+	0.297644i	$0.0962008\pi$
$523$	13.6137i	0.595288i	−0.954677	+	0.297644i	$0.903799\pi$
$524$	0	0
$525$	7.43111i	0.324320i
$526$	0	0
$527$	−12.9035	−0.562086
$528$	0	0
$529$	−21.2911	−0.925701
$530$	0	0
$531$	− 0.0254397i	− 0.00110399i
$532$	0	0
$533$	− 11.1395i	− 0.482506i
$534$	0	0
$535$	13.3164	0.575718
$536$	0	0
$537$	−27.9441	−1.20588
$538$	0	0
$539$	38.9182i	1.67632i
$540$	0	0
$541$	7.68283i	0.330311i	0.986268	+	0.165155i	$0.0528126\pi$
$541$	7.68283i	0.330311i	−0.986268	+	0.165155i	$0.947187\pi$
$542$	0	0
$543$	−32.0868	−1.37698
$544$	0	0
$545$	4.66724	0.199923
$546$	0	0
$547$	9.48006i	0.405338i	0.979247	+	0.202669i	$0.0649615\pi$
$547$	9.48006i	0.405338i	−0.979247	+	0.202669i	$0.935038\pi$
$548$	0	0
$549$	0.108281i	0.00462131i
$550$	0	0
$551$	7.74967	0.330147
$552$	0	0
$553$	66.2303	2.81640
$554$	0	0
$555$	− 5.00787i	− 0.212572i
$556$	0	0
$557$	0.204570i	0.00866793i	0.999991	+	0.00433396i	$0.00137955\pi$
$557$	0.204570i	0.00866793i	−0.999991	+	0.00433396i	$0.998620\pi$
$558$	0	0
$559$	9.81119	0.414969
$560$	0	0
$561$	−19.4270	−0.820210
$562$	0	0
$563$	9.61089i	0.405051i	0.979277	+	0.202525i	$0.0649149\pi$
$563$	9.61089i	0.405051i	−0.979277	+	0.202525i	$0.935085\pi$
$564$	0	0
$565$	6.23106i	0.262143i
$566$	0	0
$567$	−38.8148	−1.63007
$568$	0	0
$569$	−28.0632	−1.17647	−0.588236	−	0.808689i	$-0.700177\pi$
$569$	−28.0632	−1.17647	−0.588236	+	0.808689i	$0.700177\pi$
$570$	0	0
$571$	34.0399i	1.42453i	0.701912	+	0.712264i	$0.252330\pi$
$571$	34.0399i	1.42453i	−0.701912	+	0.712264i	$0.747670\pi$
$572$	0	0
$573$	− 16.8081i	− 0.702166i
$574$	0	0
$575$	−1.30724	−0.0545156
$576$	0	0
$577$	24.5221	1.02087	0.510435	−	0.859917i	$-0.329485\pi$
$577$	24.5221	1.02087	0.510435	+	0.859917i	$0.329485\pi$
$578$	0	0
$579$	− 21.7243i	− 0.902832i
$580$	0	0
$581$	46.1607i	1.91507i
$582$	0	0
$583$	−10.2641	−0.425095
$584$	0	0
$585$	−0.0375562	−0.00155276
$586$	0	0
$587$	20.6206i	0.851102i	0.904935	+	0.425551i	$0.139920\pi$
$587$	20.6206i	0.851102i	−0.904935	+	0.425551i	$0.860080\pi$
$588$	0	0
$589$	4.01448i	0.165414i
$590$	0	0
$591$	−16.6503	−0.684902
$592$	0	0
$593$	5.04425	0.207143	0.103571	−	0.994622i	$-0.466973\pi$
$593$	5.04425	0.207143	0.103571	+	0.994622i	$0.466973\pi$
$594$	0	0
$595$	− 13.7171i	− 0.562346i
$596$	0	0
$597$	17.9013i	0.732653i
$598$	0	0
$599$	30.4668	1.24484	0.622420	−	0.782684i	$-0.286150\pi$
$599$	30.4668	1.24484	0.622420	+	0.782684i	$0.286150\pi$
$600$	0	0
$601$	9.78410	0.399102	0.199551	−	0.979887i	$-0.436052\pi$
$601$	9.78410	0.399102	0.199551	+	0.979887i	$0.436052\pi$
$602$	0	0
$603$	− 0.342875i	− 0.0139629i
$604$	0	0
$605$	− 1.04794i	− 0.0426047i
$606$	0	0
$607$	−25.7805	−1.04640	−0.523199	−	0.852211i	$-0.675262\pi$
$607$	−25.7805	−1.04640	−0.523199	+	0.852211i	$0.675262\pi$
$608$	0	0
$609$	−57.5887	−2.33361
$610$	0	0
$611$	0.720061i	0.0291305i
$612$	0	0
$613$	− 10.4870i	− 0.423567i	−0.977317	−	0.211784i	$-0.932073\pi$
$613$	− 10.4870i	− 0.423567i	0.977317	−	0.211784i	$-0.0679272\pi$
$614$	0	0
$615$	16.5741	0.668333
$616$	0	0
$617$	11.3355	0.456350	0.228175	−	0.973620i	$-0.426724\pi$
$617$	11.3355	0.456350	0.228175	+	0.973620i	$0.426724\pi$
$618$	0	0
$619$	− 22.1178i	− 0.888990i	−0.895781	−	0.444495i	$-0.853383\pi$
$619$	− 22.1178i	− 0.888990i	0.895781	−	0.444495i	$-0.146617\pi$
$620$	0	0
$621$	− 6.75580i	− 0.271101i
$622$	0	0
$623$	−10.9196	−0.437483
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	6.04404i	0.241376i
$628$	0	0
$629$	9.24403i	0.368584i
$630$	0	0
$631$	23.9941	0.955191	0.477596	−	0.878580i	$-0.341508\pi$
$631$	23.9941	0.955191	0.477596	+	0.878580i	$0.341508\pi$
$632$	0	0
$633$	28.9386	1.15020
$634$	0	0
$635$	12.5227i	0.496946i
$636$	0	0
$637$	− 13.1221i	− 0.519916i
$638$	0	0
$639$	−0.0243482	−0.000963198 0
$640$	0	0
$641$	9.70799	0.383442	0.191721	−	0.981449i	$-0.438593\pi$
$641$	9.70799	0.383442	0.191721	+	0.981449i	$0.438593\pi$
$642$	0	0
$643$	− 20.5326i	− 0.809727i	−0.914377	−	0.404863i	$-0.867319\pi$
$643$	− 20.5326i	− 0.809727i	0.914377	−	0.404863i	$-0.132681\pi$
$644$	0	0
$645$	14.5977i	0.574786i
$646$	0	0
$647$	44.4245	1.74651	0.873253	−	0.487266i	$-0.162006\pi$
$647$	44.4245	1.74651	0.873253	+	0.487266i	$0.162006\pi$
$648$	0	0
$649$	−2.75166	−0.108012
$650$	0	0
$651$	− 29.8320i	− 1.16921i
$652$	0	0
$653$	22.6543i	0.886530i	0.896391	+	0.443265i	$0.146180\pi$
$653$	22.6543i	0.886530i	−0.896391	+	0.443265i	$0.853820\pi$
$654$	0	0
$655$	10.4192	0.407112
$656$	0	0
$657$	−0.324106	−0.0126446
$658$	0	0
$659$	38.5716i	1.50254i	0.659998	+	0.751268i	$0.270557\pi$
$659$	38.5716i	1.50254i	−0.659998	+	0.751268i	$0.729443\pi$
$660$	0	0
$661$	24.3737i	0.948025i	0.880518	+	0.474013i	$0.157195\pi$
$661$	24.3737i	0.948025i	−0.880518	+	0.474013i	$0.842805\pi$
$662$	0	0
$663$	6.55023	0.254390
$664$	0	0
$665$	−4.26759	−0.165490
$666$	0	0
$667$	− 10.1307i	− 0.392261i
$668$	0	0
$669$	− 15.8700i	− 0.613572i
$670$	0	0
$671$	11.7121	0.452139
$672$	0	0
$673$	10.1067	0.389585	0.194793	−	0.980844i	$-0.437597\pi$
$673$	10.1067	0.389585	0.194793	+	0.980844i	$0.437597\pi$
$674$	0	0
$675$	5.16799i	0.198916i
$676$	0	0
$677$	− 28.7783i	− 1.10604i	−0.833167	−	0.553021i	$-0.813475\pi$
$677$	− 28.7783i	− 1.10604i	0.833167	−	0.553021i	$-0.186525\pi$
$678$	0	0
$679$	−64.4616	−2.47381
$680$	0	0
$681$	36.6401	1.40405
$682$	0	0
$683$	5.91909i	0.226488i	0.993567	+	0.113244i	$0.0361241\pi$
$683$	5.91909i	0.226488i	−0.993567	+	0.113244i	$0.963876\pi$
$684$	0	0
$685$	− 21.8395i	− 0.834444i
$686$	0	0
$687$	−0.401400	−0.0153144
$688$	0	0
$689$	3.46075	0.131844
$690$	0	0
$691$	− 34.4298i	− 1.30977i	−0.755727	−	0.654887i	$-0.772716\pi$
$691$	− 34.4298i	− 1.30977i	0.755727	−	0.654887i	$-0.227284\pi$
$692$	0	0
$693$	− 0.475350i	− 0.0180571i
$694$	0	0
$695$	−0.938309	−0.0355921
$696$	0	0
$697$	−30.5941	−1.15884
$698$	0	0
$699$	3.89924i	0.147483i
$700$	0	0
$701$	9.66489i	0.365038i	0.983202	+	0.182519i	$0.0584251\pi$
$701$	9.66489i	0.365038i	−0.983202	+	0.182519i	$0.941575\pi$
$702$	0	0
$703$	2.87596	0.108469
$704$	0	0
$705$	−1.07136	−0.0403496
$706$	0	0
$707$	− 14.9984i	− 0.564074i
$708$	0	0
$709$	− 39.6534i	− 1.48921i	−0.667503	−	0.744607i	$-0.732637\pi$
$709$	− 39.6534i	− 1.48921i	0.667503	−	0.744607i	$-0.267363\pi$
$710$	0	0
$711$	−0.498022	−0.0186773
$712$	0	0
$713$	5.24788	0.196535
$714$	0	0
$715$	4.06222i	0.151918i
$716$	0	0
$717$	32.3268i	1.20727i
$718$	0	0
$719$	6.63763	0.247542	0.123771	−	0.992311i	$-0.460501\pi$
$719$	6.63763	0.247542	0.123771	+	0.992311i	$0.460501\pi$
$720$	0	0
$721$	45.6769	1.70110
$722$	0	0
$723$	44.3799i	1.65051i
$724$	0	0
$725$	7.74967i	0.287816i
$726$	0	0
$727$	−30.3254	−1.12471	−0.562353	−	0.826898i	$-0.690104\pi$
$727$	−30.3254	−1.12471	−0.562353	+	0.826898i	$0.690104\pi$
$728$	0	0
$729$	−26.7050	−0.989076
$730$	0	0
$731$	− 26.9460i	− 0.996633i
$732$	0	0
$733$	44.5305i	1.64477i	0.568930	+	0.822386i	$0.307357\pi$
$733$	44.5305i	1.64477i	−0.568930	+	0.822386i	$0.692643\pi$
$734$	0	0
$735$	19.5239	0.720151
$736$	0	0
$737$	−37.0866	−1.36610
$738$	0	0
$739$	8.37611i	0.308120i	0.988061	+	0.154060i	$0.0492349\pi$
$739$	8.37611i	0.308120i	−0.988061	+	0.154060i	$0.950765\pi$
$740$	0	0
$741$	− 2.03788i	− 0.0748633i
$742$	0	0
$743$	14.1329	0.518485	0.259243	−	0.965812i	$-0.416527\pi$
$743$	14.1329	0.518485	0.259243	+	0.965812i	$0.416527\pi$
$744$	0	0
$745$	−4.84067	−0.177348
$746$	0	0
$747$	− 0.347107i	− 0.0127000i
$748$	0	0
$749$	− 56.8289i	− 2.07648i
$750$	0	0
$751$	4.23734	0.154623	0.0773114	−	0.997007i	$-0.475366\pi$
$751$	4.23734	0.154623	0.0773114	+	0.997007i	$0.475366\pi$
$752$	0	0
$753$	−8.41863	−0.306792
$754$	0	0
$755$	16.9169i	0.615670i
$756$	0	0
$757$	31.0769i	1.12951i	0.825258	+	0.564755i	$0.191029\pi$
$757$	31.0769i	1.12951i	−0.825258	+	0.564755i	$0.808971\pi$
$758$	0	0
$759$	7.90101	0.286788
$760$	0	0
$761$	14.8599	0.538671	0.269335	−	0.963046i	$-0.413196\pi$
$761$	14.8599	0.538671	0.269335	+	0.963046i	$0.413196\pi$
$762$	0	0
$763$	− 19.9179i	− 0.721076i
$764$	0	0
$765$	0.103146i	0.00372926i
$766$	0	0
$767$	0.927780	0.0335002
$768$	0	0
$769$	−7.04822	−0.254165	−0.127083	−	0.991892i	$-0.540561\pi$
$769$	−7.04822	−0.254165	−0.127083	+	0.991892i	$0.540561\pi$
$770$	0	0
$771$	49.5358i	1.78399i
$772$	0	0
$773$	− 23.7187i	− 0.853103i	−0.904463	−	0.426551i	$-0.859728\pi$
$773$	− 23.7187i	− 0.853103i	0.904463	−	0.426551i	$-0.140272\pi$
$774$	0	0
$775$	−4.01448	−0.144204
$776$	0	0
$777$	−21.3716	−0.766700
$778$	0	0
$779$	9.51830i	0.341028i
$780$	0	0
$781$	2.63359i	0.0942372i
$782$	0	0
$783$	−40.0502	−1.43128
$784$	0	0
$785$	2.31235	0.0825313
$786$	0	0
$787$	− 17.6151i	− 0.627910i	−0.949438	−	0.313955i	$-0.898346\pi$
$787$	− 17.6151i	− 0.627910i	0.949438	−	0.313955i	$-0.101654\pi$
$788$	0	0
$789$	− 0.356305i	− 0.0126848i
$790$	0	0
$791$	26.5916	0.945489
$792$	0	0
$793$	−3.94897	−0.140232
$794$	0	0
$795$	5.14914i	0.182621i
$796$	0	0
$797$	− 18.2108i	− 0.645059i	−0.946559	−	0.322530i	$-0.895467\pi$
$797$	− 18.2108i	− 0.645059i	0.946559	−	0.322530i	$-0.104533\pi$
$798$	0	0
$799$	1.97761	0.0699629
$800$	0	0
$801$	0.0821101	0.00290122
$802$	0	0
$803$	35.0565i	1.23712i
$804$	0	0
$805$	5.57876i	0.196626i
$806$	0	0
$807$	22.6539	0.797456
$808$	0	0
$809$	−28.3141	−0.995469	−0.497735	−	0.867329i	$-0.665835\pi$
$809$	−28.3141	−0.995469	−0.497735	+	0.867329i	$0.665835\pi$
$810$	0	0
$811$	3.81640i	0.134012i	0.997753	+	0.0670060i	$0.0213447\pi$
$811$	3.81640i	0.134012i	−0.997753	+	0.0670060i	$0.978655\pi$
$812$	0	0
$813$	− 49.5906i	− 1.73922i
$814$	0	0
$815$	−13.3563	−0.467851
$816$	0	0
$817$	−8.38330	−0.293294
$818$	0	0
$819$	0.160274i	0.00560044i
$820$	0	0
$821$	8.02224i	0.279978i	0.990153	+	0.139989i	$0.0447068\pi$
$821$	8.02224i	0.279978i	−0.990153	+	0.139989i	$0.955293\pi$
$822$	0	0
$823$	−27.9881	−0.975604	−0.487802	−	0.872954i	$-0.662201\pi$
$823$	−27.9881	−0.975604	−0.487802	+	0.872954i	$0.662201\pi$
$824$	0	0
$825$	−6.04404	−0.210427
$826$	0	0
$827$	− 35.0995i	− 1.22053i	−0.792197	−	0.610265i	$-0.791063\pi$
$827$	− 35.0995i	− 1.22053i	0.792197	−	0.610265i	$-0.208937\pi$
$828$	0	0
$829$	35.6312i	1.23752i	0.785578	+	0.618762i	$0.212366\pi$
$829$	35.6312i	1.23752i	−0.785578	+	0.618762i	$0.787634\pi$
$830$	0	0
$831$	−29.0181	−1.00663
$832$	0	0
$833$	−36.0392	−1.24868
$834$	0	0
$835$	− 17.5050i	− 0.605784i
$836$	0	0
$837$	− 20.7468i	− 0.717113i
$838$	0	0
$839$	−21.6230	−0.746507	−0.373254	−	0.927729i	$-0.621758\pi$
$839$	−21.6230	−0.746507	−0.373254	+	0.927729i	$0.621758\pi$
$840$	0	0
$841$	−31.0574	−1.07094
$842$	0	0
$843$	25.2128i	0.868376i
$844$	0	0
$845$	11.6303i	0.400096i
$846$	0	0
$847$	−4.47217	−0.153665
$848$	0	0
$849$	−52.5064	−1.80201
$850$	0	0
$851$	− 3.75956i	− 0.128876i
$852$	0	0
$853$	− 31.6534i	− 1.08379i	−0.840446	−	0.541895i	$-0.817707\pi$
$853$	− 31.6534i	− 1.08379i	0.840446	−	0.541895i	$-0.182293\pi$
$854$	0	0
$855$	0.0320903	0.00109747
$856$	0	0
$857$	−21.3449	−0.729128	−0.364564	−	0.931178i	$-0.618782\pi$
$857$	−21.3449	−0.729128	−0.364564	+	0.931178i	$0.618782\pi$
$858$	0	0
$859$	50.1606i	1.71146i	0.517423	+	0.855730i	$0.326891\pi$
$859$	50.1606i	1.71146i	−0.517423	+	0.855730i	$0.673109\pi$
$860$	0	0
$861$	− 70.7315i	− 2.41052i
$862$	0	0
$863$	−40.4392	−1.37657	−0.688283	−	0.725442i	$-0.741635\pi$
$863$	−40.4392	−1.37657	−0.688283	+	0.725442i	$0.741635\pi$
$864$	0	0
$865$	−19.6197	−0.667091
$866$	0	0
$867$	11.6120i	0.394365i
$868$	0	0
$869$	53.8680i	1.82735i
$870$	0	0
$871$	12.5045	0.423700
$872$	0	0
$873$	0.484722	0.0164054
$874$	0	0
$875$	− 4.26759i	− 0.144271i
$876$	0	0
$877$	− 40.6935i	− 1.37412i	−0.726599	−	0.687061i	$-0.758900\pi$
$877$	− 40.6935i	− 1.37412i	0.726599	−	0.687061i	$-0.241100\pi$
$878$	0	0
$879$	−46.2258	−1.55916
$880$	0	0
$881$	−50.5155	−1.70191	−0.850956	−	0.525238i	$-0.823976\pi$
$881$	−50.5155	−1.70191	−0.850956	+	0.525238i	$0.823976\pi$
$882$	0	0
$883$	36.2505i	1.21993i	0.792429	+	0.609964i	$0.208816\pi$
$883$	36.2505i	1.21993i	−0.792429	+	0.609964i	$0.791184\pi$
$884$	0	0
$885$	1.38041i	0.0464021i
$886$	0	0
$887$	11.6574	0.391418	0.195709	−	0.980662i	$-0.437299\pi$
$887$	11.6574	0.391418	0.195709	+	0.980662i	$0.437299\pi$
$888$	0	0
$889$	53.4416	1.79237
$890$	0	0
$891$	− 31.5697i	− 1.05763i
$892$	0	0
$893$	− 0.615265i	− 0.0205891i
$894$	0	0
$895$	16.0479	0.536423
$896$	0	0
$897$	−2.66399	−0.0889481
$898$	0	0
$899$	− 31.1109i	− 1.03761i
$900$	0	0
$901$	− 9.50480i	− 0.316651i
$902$	0	0
$903$	62.2972	2.07312
$904$	0	0
$905$	18.4270	0.612536
$906$	0	0
$907$	− 12.8607i	− 0.427033i	−0.976939	−	0.213516i	$-0.931508\pi$
$907$	− 12.8607i	− 0.427033i	0.976939	−	0.213516i	$-0.0684917\pi$
$908$	0	0
$909$	0.112781i	0.00374072i
$910$	0	0
$911$	−45.7183	−1.51471	−0.757357	−	0.653001i	$-0.773510\pi$
$911$	−45.7183	−1.51471	−0.757357	+	0.653001i	$0.773510\pi$
$912$	0	0
$913$	−37.5444	−1.24254
$914$	0	0
$915$	− 5.87554i	− 0.194240i
$916$	0	0
$917$	− 44.4649i	− 1.46836i
$918$	0	0
$919$	−25.2194	−0.831910	−0.415955	−	0.909385i	$-0.636553\pi$
$919$	−25.2194	−0.831910	−0.415955	+	0.909385i	$0.636553\pi$
$920$	0	0
$921$	−43.8058	−1.44345
$922$	0	0
$923$	− 0.887970i	− 0.0292279i
$924$	0	0
$925$	2.87596i	0.0945608i
$926$	0	0
$927$	−0.343469	−0.0112810
$928$	0	0
$929$	43.1572	1.41594	0.707972	−	0.706241i	$-0.249610\pi$
$929$	43.1572	1.41594	0.707972	+	0.706241i	$0.249610\pi$
$930$	0	0
$931$	11.2123i	0.367470i
$932$	0	0
$933$	17.8841i	0.585501i
$934$	0	0
$935$	11.1567	0.364863
$936$	0	0
$937$	−5.42120	−0.177103	−0.0885515	−	0.996072i	$-0.528224\pi$
$937$	−5.42120	−0.177103	−0.0885515	+	0.996072i	$0.528224\pi$
$938$	0	0
$939$	10.1893i	0.332516i
$940$	0	0
$941$	− 3.35000i	− 0.109207i	−0.998508	−	0.0546035i	$-0.982611\pi$
$941$	− 3.35000i	− 0.109207i	0.998508	−	0.0546035i	$-0.0173895\pi$
$942$	0	0
$943$	12.4427	0.405190
$944$	0	0
$945$	22.0549	0.717445
$946$	0	0
$947$	19.1023i	0.620741i	0.950616	+	0.310370i	$0.100453\pi$
$947$	19.1023i	0.620741i	−0.950616	+	0.310370i	$0.899547\pi$
$948$	0	0
$949$	− 11.8201i	− 0.383695i
$950$	0	0
$951$	18.3532	0.595142
$952$	0	0
$953$	35.0195	1.13439	0.567196	−	0.823583i	$-0.308028\pi$
$953$	35.0195	1.13439	0.567196	+	0.823583i	$0.308028\pi$
$954$	0	0
$955$	9.65264i	0.312352i
$956$	0	0
$957$	− 46.8393i	− 1.51410i
$958$	0	0
$959$	−93.2020	−3.00965
$960$	0	0
$961$	−14.8840	−0.480129
$962$	0	0
$963$	0.427328i	0.0137704i
$964$	0	0
$965$	12.4760i	0.401617i
$966$	0	0
$967$	−35.2602	−1.13389	−0.566947	−	0.823755i	$-0.691875\pi$
$967$	−35.2602	−1.13389	−0.566947	+	0.823755i	$0.691875\pi$
$968$	0	0
$969$	−5.59693	−0.179799
$970$	0	0
$971$	− 30.0212i	− 0.963426i	−0.876329	−	0.481713i	$-0.840015\pi$
$971$	− 30.0212i	− 0.963426i	0.876329	−	0.481713i	$-0.159985\pi$
$972$	0	0
$973$	4.00432i	0.128373i
$974$	0	0
$975$	2.03788	0.0652643
$976$	0	0
$977$	3.41415	0.109228	0.0546142	−	0.998508i	$-0.482607\pi$
$977$	3.41415	0.109228	0.0546142	+	0.998508i	$0.482607\pi$
$978$	0	0
$979$	− 8.88134i	− 0.283849i
$980$	0	0
$981$	0.149773i	0.00478190i
$982$	0	0
$983$	−44.2342	−1.41085	−0.705426	−	0.708783i	$-0.749244\pi$
$983$	−44.2342	−1.41085	−0.705426	+	0.708783i	$0.749244\pi$
$984$	0	0
$985$	9.56205	0.304672
$986$	0	0
$987$	4.57211i	0.145532i
$988$	0	0
$989$	10.9590i	0.348475i
$990$	0	0
$991$	−38.5006	−1.22301	−0.611506	−	0.791240i	$-0.709436\pi$
$991$	−38.5006	−1.22301	−0.611506	+	0.791240i	$0.709436\pi$
$992$	0	0
$993$	−10.7358	−0.340690
$994$	0	0
$995$	− 10.2805i	− 0.325914i
$996$	0	0
$997$	− 38.3171i	− 1.21351i	−0.794887	−	0.606757i	$-0.792470\pi$
$997$	− 38.3171i	− 1.21351i	0.794887	−	0.606757i	$-0.207530\pi$
$998$	0	0
$999$	−14.8629	−0.470242

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3040.2.f.a.1521.7		28
4.3	odd	2		760.2.f.a.381.3	✓	28
8.3	odd	2		760.2.f.a.381.4	yes	28
8.5	even	2	inner	3040.2.f.a.1521.22		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.f.a.381.3	✓	28	4.3	odd	2
760.2.f.a.381.4	yes	28	8.3	odd	2
3040.2.f.a.1521.7		28	1.1	even	1	trivial
3040.2.f.a.1521.22		28	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)

\(f(q)\)	\(=\)	\(q-1.74129i q^{3} +1.00000i q^{5} +4.26759 q^{7} -0.0320903 q^{9} +3.47101i q^{11} -1.17033i q^{13} +1.74129 q^{15} -3.21425 q^{17} +1.00000i q^{19} -7.43111i q^{21} +1.30724 q^{23} -1.00000 q^{25} -5.16799i q^{27} -7.74967i q^{29} +4.01448 q^{31} +6.04404 q^{33} +4.26759i q^{35} -2.87596i q^{37} -2.03788 q^{39} +9.51830 q^{41} +8.38330i q^{43} -0.0320903i q^{45} -0.615265 q^{47} +11.2123 q^{49} +5.59693i q^{51} +2.95709i q^{53} -3.47101 q^{55} +1.74129 q^{57} +0.792754i q^{59} -3.37425i q^{61} -0.136948 q^{63} +1.17033 q^{65} +10.6847i q^{67} -2.27628i q^{69} +0.758738 q^{71} +10.0998 q^{73} +1.74129i q^{75} +14.8129i q^{77} +15.5194 q^{79} -9.09524 q^{81} +10.8166i q^{83} -3.21425i q^{85} -13.4944 q^{87} -2.55872 q^{89} -4.99447i q^{91} -6.99037i q^{93} -1.00000 q^{95} -15.1049 q^{97} -0.111386i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 4 q^{7} - 12 q^{9} - 24 q^{17} - 4 q^{23} - 28 q^{25} - 40 q^{33} + 24 q^{39} + 32 q^{41} + 20 q^{47} - 36 q^{49} - 8 q^{55} - 20 q^{63} - 12 q^{65} + 8 q^{71} + 88 q^{73} - 40 q^{79} - 28 q^{81}+ \cdots - 116 q^{97}+O(q^{100}) \) 28 * q - 4 * q^7 - 12 * q^9 - 24 * q^17 - 4 * q^23 - 28 * q^25 - 40 * q^33 + 24 * q^39 + 32 * q^41 + 20 * q^47 - 36 * q^49 - 8 * q^55 - 20 * q^63 - 12 * q^65 + 8 * q^71 + 88 * q^73 - 40 * q^79 - 28 * q^81 - 48 * q^87 + 48 * q^89 - 28 * q^95 - 116 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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