LMFDB - Embedded newform 6004.2.a.f.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$47.9421813736$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.642577 q^{3} -3.12522 q^{5} -2.66487 q^{7} -2.58709 q^{9} +O(q^{10})$	$q-0.642577 q^{3} -3.12522 q^{5} -2.66487 q^{7} -2.58709 q^{9} -1.14721 q^{11} +6.91058 q^{13} +2.00820 q^{15} -5.61384 q^{17} -1.00000 q^{19} +1.71239 q^{21} +2.10413 q^{23} +4.76703 q^{25} +3.59014 q^{27} +2.45823 q^{29} +8.32308 q^{31} +0.737173 q^{33} +8.32833 q^{35} +8.97602 q^{37} -4.44058 q^{39} +0.324952 q^{41} -8.03239 q^{43} +8.08525 q^{45} +5.54786 q^{47} +0.101551 q^{49} +3.60733 q^{51} +0.967459 q^{53} +3.58530 q^{55} +0.642577 q^{57} +0.254026 q^{59} -5.49050 q^{61} +6.89428 q^{63} -21.5971 q^{65} -3.52144 q^{67} -1.35207 q^{69} +13.6536 q^{71} +7.91969 q^{73} -3.06318 q^{75} +3.05718 q^{77} +1.00000 q^{79} +5.45434 q^{81} -9.28043 q^{83} +17.5445 q^{85} -1.57960 q^{87} -8.47077 q^{89} -18.4158 q^{91} -5.34823 q^{93} +3.12522 q^{95} +3.93411 q^{97} +2.96795 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) $ 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9	$ 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9} - 3 q^{11} + q^{13} - 5 q^{15} - 13 q^{17} - 25 q^{19} - 24 q^{21} - 31 q^{23} + 21 q^{25} + 7 q^{27} - 19 q^{29} - 7 q^{31} - 30 q^{33} - q^{35} - 29 q^{37} - 26 q^{39} - 40 q^{41} - 40 q^{45} - 8 q^{47} - 9 q^{49} + 12 q^{51} - 38 q^{53} - 29 q^{55} - 4 q^{57} + 18 q^{59} - 26 q^{61} - 40 q^{63} - 70 q^{65} - 13 q^{67} + q^{69} - 47 q^{71} - 8 q^{73} + 7 q^{75} - 19 q^{77} + 25 q^{79} - 19 q^{81} - 8 q^{83} - 33 q^{85} - 50 q^{87} - 54 q^{89} - 12 q^{91} - 24 q^{93} + 8 q^{95} - 4 q^{97} - 39 q^{99}+O(q^{100}) $ 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9 - 3 * q^11 + q^13 - 5 * q^15 - 13 * q^17 - 25 * q^19 - 24 * q^21 - 31 * q^23 + 21 * q^25 + 7 * q^27 - 19 * q^29 - 7 * q^31 - 30 * q^33 - q^35 - 29 * q^37 - 26 * q^39 - 40 * q^41 - 40 * q^45 - 8 * q^47 - 9 * q^49 + 12 * q^51 - 38 * q^53 - 29 * q^55 - 4 * q^57 + 18 * q^59 - 26 * q^61 - 40 * q^63 - 70 * q^65 - 13 * q^67 + q^69 - 47 * q^71 - 8 * q^73 + 7 * q^75 - 19 * q^77 + 25 * q^79 - 19 * q^81 - 8 * q^83 - 33 * q^85 - 50 * q^87 - 54 * q^89 - 12 * q^91 - 24 * q^93 + 8 * q^95 - 4 * q^97 - 39 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.642577	−0.370992	−0.185496	−	0.982645i	$-0.559389\pi$
$3$	−0.642577	−0.370992	−0.185496	+	0.982645i	$0.559389\pi$
$4$	0	0
$5$	−3.12522	−1.39764	−0.698821	−	0.715296i	$-0.746292\pi$
$5$	−3.12522	−1.39764	−0.698821	+	0.715296i	$0.746292\pi$
$6$	0	0
$7$	−2.66487	−1.00723	−0.503614	−	0.863929i	$-0.667997\pi$
$7$	−2.66487	−1.00723	−0.503614	+	0.863929i	$0.667997\pi$
$8$	0	0
$9$	−2.58709	−0.862365
$10$	0	0
$11$	−1.14721	−0.345898	−0.172949	−	0.984931i	$-0.555330\pi$
$11$	−1.14721	−0.345898	−0.172949	+	0.984931i	$0.555330\pi$
$12$	0	0
$13$	6.91058	1.91665	0.958324	−	0.285682i	$-0.0922202\pi$
$13$	6.91058	1.91665	0.958324	+	0.285682i	$0.0922202\pi$
$14$	0	0
$15$	2.00820	0.518515
$16$	0	0
$17$	−5.61384	−1.36156	−0.680778	−	0.732490i	$-0.738358\pi$
$17$	−5.61384	−1.36156	−0.680778	+	0.732490i	$0.738358\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.71239	0.373674
$22$	0	0
$23$	2.10413	0.438742	0.219371	−	0.975641i	$-0.429599\pi$
$23$	2.10413	0.438742	0.219371	+	0.975641i	$0.429599\pi$
$24$	0	0
$25$	4.76703	0.953405
$26$	0	0
$27$	3.59014	0.690923
$28$	0	0
$29$	2.45823	0.456482	0.228241	−	0.973605i	$-0.426703\pi$
$29$	2.45823	0.456482	0.228241	+	0.973605i	$0.426703\pi$
$30$	0	0
$31$	8.32308	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$31$	8.32308	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$32$	0	0
$33$	0.737173	0.128325
$34$	0	0
$35$	8.32833	1.40774
$36$	0	0
$37$	8.97602	1.47565	0.737824	−	0.674993i	$-0.235853\pi$
$37$	8.97602	1.47565	0.737824	+	0.674993i	$0.235853\pi$
$38$	0	0
$39$	−4.44058	−0.711062
$40$	0	0
$41$	0.324952	0.0507489	0.0253745	−	0.999678i	$-0.491922\pi$
$41$	0.324952	0.0507489	0.0253745	+	0.999678i	$0.491922\pi$
$42$	0	0
$43$	−8.03239	−1.22493	−0.612464	−	0.790498i	$-0.709822\pi$
$43$	−8.03239	−1.22493	−0.612464	+	0.790498i	$0.709822\pi$
$44$	0	0
$45$	8.08525	1.20528
$46$	0	0
$47$	5.54786	0.809239	0.404619	−	0.914485i	$-0.367404\pi$
$47$	5.54786	0.809239	0.404619	+	0.914485i	$0.367404\pi$
$48$	0	0
$49$	0.101551	0.0145072
$50$	0	0
$51$	3.60733	0.505127
$52$	0	0
$53$	0.967459	0.132891	0.0664454	−	0.997790i	$-0.478834\pi$
$53$	0.967459	0.132891	0.0664454	+	0.997790i	$0.478834\pi$
$54$	0	0
$55$	3.58530	0.483442
$56$	0	0
$57$	0.642577	0.0851115
$58$	0	0
$59$	0.254026	0.0330714	0.0165357	−	0.999863i	$-0.494736\pi$
$59$	0.254026	0.0330714	0.0165357	+	0.999863i	$0.494736\pi$
$60$	0	0
$61$	−5.49050	−0.702986	−0.351493	−	0.936190i	$-0.614326\pi$
$61$	−5.49050	−0.702986	−0.351493	+	0.936190i	$0.614326\pi$
$62$	0	0
$63$	6.89428	0.868597
$64$	0	0
$65$	−21.5971	−2.67879
$66$	0	0
$67$	−3.52144	−0.430212	−0.215106	−	0.976591i	$-0.569010\pi$
$67$	−3.52144	−0.430212	−0.215106	+	0.976591i	$0.569010\pi$
$68$	0	0
$69$	−1.35207	−0.162770
$70$	0	0
$71$	13.6536	1.62038	0.810190	−	0.586168i	$-0.199364\pi$
$71$	13.6536	1.62038	0.810190	+	0.586168i	$0.199364\pi$
$72$	0	0
$73$	7.91969	0.926930	0.463465	−	0.886115i	$-0.346606\pi$
$73$	7.91969	0.926930	0.463465	+	0.886115i	$0.346606\pi$
$74$	0	0
$75$	−3.06318	−0.353706
$76$	0	0
$77$	3.05718	0.348398
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	5.45434	0.606038
$82$	0	0
$83$	−9.28043	−1.01866	−0.509330	−	0.860571i	$-0.670107\pi$
$83$	−9.28043	−1.01866	−0.509330	+	0.860571i	$0.670107\pi$
$84$	0	0
$85$	17.5445	1.90297
$86$	0	0
$87$	−1.57960	−0.169351
$88$	0	0
$89$	−8.47077	−0.897900	−0.448950	−	0.893557i	$-0.648202\pi$
$89$	−8.47077	−0.897900	−0.448950	+	0.893557i	$0.648202\pi$
$90$	0	0
$91$	−18.4158	−1.93050
$92$	0	0
$93$	−5.34823	−0.554585
$94$	0	0
$95$	3.12522	0.320641
$96$	0	0
$97$	3.93411	0.399449	0.199724	−	0.979852i	$-0.435995\pi$
$97$	3.93411	0.399449	0.199724	+	0.979852i	$0.435995\pi$
$98$	0	0
$99$	2.96795	0.298290
$100$	0	0
$101$	−14.0578	−1.39881	−0.699403	−	0.714727i	$-0.746551\pi$
$101$	−14.0578	−1.39881	−0.699403	+	0.714727i	$0.746551\pi$
$102$	0	0
$103$	−0.184985	−0.0182271	−0.00911354	−	0.999958i	$-0.502901\pi$
$103$	−0.184985	−0.0182271	−0.00911354	+	0.999958i	$0.502901\pi$
$104$	0	0
$105$	−5.35159	−0.522262
$106$	0	0
$107$	−8.04142	−0.777394	−0.388697	−	0.921366i	$-0.627075\pi$
$107$	−8.04142	−0.777394	−0.388697	+	0.921366i	$0.627075\pi$
$108$	0	0
$109$	−0.00287004	−0.000274900 0	−0.000137450	−	1.00000i	$-0.500044\pi$
$109$	−0.00287004	−0.000274900 0	−0.000137450		1.00000i	$0.500044\pi$
$110$	0	0
$111$	−5.76779	−0.547454
$112$	0	0
$113$	−1.58988	−0.149563	−0.0747815	−	0.997200i	$-0.523826\pi$
$113$	−1.58988	−0.149563	−0.0747815	+	0.997200i	$0.523826\pi$
$114$	0	0
$115$	−6.57589	−0.613205
$116$	0	0
$117$	−17.8783	−1.65285
$118$	0	0
$119$	14.9602	1.37140
$120$	0	0
$121$	−9.68390	−0.880355
$122$	0	0
$123$	−0.208807	−0.0188275
$124$	0	0
$125$	0.728096	0.0651229
$126$	0	0
$127$	6.89986	0.612264	0.306132	−	0.951989i	$-0.400965\pi$
$127$	6.89986	0.612264	0.306132	+	0.951989i	$0.400965\pi$
$128$	0	0
$129$	5.16144	0.454439
$130$	0	0
$131$	17.7260	1.54873	0.774365	−	0.632739i	$-0.218069\pi$
$131$	17.7260	1.54873	0.774365	+	0.632739i	$0.218069\pi$
$132$	0	0
$133$	2.66487	0.231074
$134$	0	0
$135$	−11.2200	−0.965663
$136$	0	0
$137$	−21.0892	−1.80177	−0.900884	−	0.434059i	$-0.857081\pi$
$137$	−21.0892	−1.80177	−0.900884	+	0.434059i	$0.857081\pi$
$138$	0	0
$139$	−11.4053	−0.967383	−0.483692	−	0.875239i	$-0.660704\pi$
$139$	−11.4053	−0.967383	−0.483692	+	0.875239i	$0.660704\pi$
$140$	0	0
$141$	−3.56493	−0.300221
$142$	0	0
$143$	−7.92790	−0.662965
$144$	0	0
$145$	−7.68252	−0.637999
$146$	0	0
$147$	−0.0652541	−0.00538207
$148$	0	0
$149$	17.0804	1.39928	0.699640	−	0.714496i	$-0.253344\pi$
$149$	17.0804	1.39928	0.699640	+	0.714496i	$0.253344\pi$
$150$	0	0
$151$	8.27109	0.673092	0.336546	−	0.941667i	$-0.390741\pi$
$151$	8.27109	0.673092	0.336546	+	0.941667i	$0.390741\pi$
$152$	0	0
$153$	14.5235	1.17416
$154$	0	0
$155$	−26.0115	−2.08929
$156$	0	0
$157$	−1.72233	−0.137457	−0.0687285	−	0.997635i	$-0.521894\pi$
$157$	−1.72233	−0.137457	−0.0687285	+	0.997635i	$0.521894\pi$
$158$	0	0
$159$	−0.621667	−0.0493014
$160$	0	0
$161$	−5.60725	−0.441913
$162$	0	0
$163$	20.0990	1.57428	0.787139	−	0.616776i	$-0.211561\pi$
$163$	20.0990	1.57428	0.787139	+	0.616776i	$0.211561\pi$
$164$	0	0
$165$	−2.30383	−0.179353
$166$	0	0
$167$	−15.4603	−1.19635	−0.598177	−	0.801364i	$-0.704108\pi$
$167$	−15.4603	−1.19635	−0.598177	+	0.801364i	$0.704108\pi$
$168$	0	0
$169$	34.7561	2.67354
$170$	0	0
$171$	2.58709	0.197840
$172$	0	0
$173$	−7.83587	−0.595750	−0.297875	−	0.954605i	$-0.596278\pi$
$173$	−7.83587	−0.595750	−0.297875	+	0.954605i	$0.596278\pi$
$174$	0	0
$175$	−12.7035	−0.960296
$176$	0	0
$177$	−0.163231	−0.0122692
$178$	0	0
$179$	−24.3065	−1.81675	−0.908377	−	0.418152i	$-0.862678\pi$
$179$	−24.3065	−1.81675	−0.908377	+	0.418152i	$0.862678\pi$
$180$	0	0
$181$	6.46577	0.480597	0.240298	−	0.970699i	$-0.422755\pi$
$181$	6.46577	0.480597	0.240298	+	0.970699i	$0.422755\pi$
$182$	0	0
$183$	3.52807	0.260802
$184$	0	0
$185$	−28.0521	−2.06243
$186$	0	0
$187$	6.44027	0.470959
$188$	0	0
$189$	−9.56727	−0.695917
$190$	0	0
$191$	3.68547	0.266671	0.133336	−	0.991071i	$-0.457431\pi$
$191$	3.68547	0.266671	0.133336	+	0.991071i	$0.457431\pi$
$192$	0	0
$193$	16.7740	1.20742	0.603708	−	0.797206i	$-0.293689\pi$
$193$	16.7740	1.20742	0.603708	+	0.797206i	$0.293689\pi$
$194$	0	0
$195$	13.8778	0.993810
$196$	0	0
$197$	−15.6096	−1.11213	−0.556067	−	0.831137i	$-0.687690\pi$
$197$	−15.6096	−1.11213	−0.556067	+	0.831137i	$0.687690\pi$
$198$	0	0
$199$	−14.1834	−1.00543	−0.502717	−	0.864451i	$-0.667666\pi$
$199$	−14.1834	−1.00543	−0.502717	+	0.864451i	$0.667666\pi$
$200$	0	0
$201$	2.26280	0.159605
$202$	0	0
$203$	−6.55087	−0.459781
$204$	0	0
$205$	−1.01555	−0.0709289
$206$	0	0
$207$	−5.44359	−0.378356
$208$	0	0
$209$	1.14721	0.0793544
$210$	0	0
$211$	−10.7314	−0.738782	−0.369391	−	0.929274i	$-0.620434\pi$
$211$	−10.7314	−0.738782	−0.369391	+	0.929274i	$0.620434\pi$
$212$	0	0
$213$	−8.77347	−0.601148
$214$	0	0
$215$	25.1030	1.71201
$216$	0	0
$217$	−22.1800	−1.50567
$218$	0	0
$219$	−5.08902	−0.343884
$220$	0	0
$221$	−38.7948	−2.60962
$222$	0	0
$223$	8.40625	0.562924	0.281462	−	0.959572i	$-0.409181\pi$
$223$	8.40625	0.562924	0.281462	+	0.959572i	$0.409181\pi$
$224$	0	0
$225$	−12.3327	−0.822183
$226$	0	0
$227$	10.3678	0.688132	0.344066	−	0.938945i	$-0.388196\pi$
$227$	10.3678	0.688132	0.344066	+	0.938945i	$0.388196\pi$
$228$	0	0
$229$	−1.07585	−0.0710944	−0.0355472	−	0.999368i	$-0.511317\pi$
$229$	−1.07585	−0.0710944	−0.0355472	+	0.999368i	$0.511317\pi$
$230$	0	0
$231$	−1.96447	−0.129253
$232$	0	0
$233$	−8.31836	−0.544954	−0.272477	−	0.962162i	$-0.587843\pi$
$233$	−8.31836	−0.544954	−0.272477	+	0.962162i	$0.587843\pi$
$234$	0	0
$235$	−17.3383	−1.13103
$236$	0	0
$237$	−0.642577	−0.0417399
$238$	0	0
$239$	0.482332	0.0311994	0.0155997	−	0.999878i	$-0.495034\pi$
$239$	0.482332	0.0311994	0.0155997	+	0.999878i	$0.495034\pi$
$240$	0	0
$241$	−8.12814	−0.523580	−0.261790	−	0.965125i	$-0.584313\pi$
$241$	−8.12814	−0.523580	−0.261790	+	0.965125i	$0.584313\pi$
$242$	0	0
$243$	−14.2753	−0.915758
$244$	0	0
$245$	−0.317368	−0.0202759
$246$	0	0
$247$	−6.91058	−0.439709
$248$	0	0
$249$	5.96340	0.377915
$250$	0	0
$251$	18.3517	1.15835	0.579176	−	0.815202i	$-0.303374\pi$
$251$	18.3517	1.15835	0.579176	+	0.815202i	$0.303374\pi$
$252$	0	0
$253$	−2.41389	−0.151760
$254$	0	0
$255$	−11.2737	−0.705986
$256$	0	0
$257$	3.63820	0.226944	0.113472	−	0.993541i	$-0.463803\pi$
$257$	3.63820	0.226944	0.113472	+	0.993541i	$0.463803\pi$
$258$	0	0
$259$	−23.9200	−1.48631
$260$	0	0
$261$	−6.35967	−0.393654
$262$	0	0
$263$	−24.9761	−1.54009	−0.770045	−	0.637989i	$-0.779766\pi$
$263$	−24.9761	−1.54009	−0.770045	+	0.637989i	$0.779766\pi$
$264$	0	0
$265$	−3.02353	−0.185734
$266$	0	0
$267$	5.44312	0.333114
$268$	0	0
$269$	17.5088	1.06753	0.533765	−	0.845633i	$-0.320777\pi$
$269$	17.5088	1.06753	0.533765	+	0.845633i	$0.320777\pi$
$270$	0	0
$271$	14.3795	0.873490	0.436745	−	0.899585i	$-0.356131\pi$
$271$	14.3795	0.873490	0.436745	+	0.899585i	$0.356131\pi$
$272$	0	0
$273$	11.8336	0.716201
$274$	0	0
$275$	−5.46880	−0.329781
$276$	0	0
$277$	−21.9950	−1.32155	−0.660777	−	0.750582i	$-0.729773\pi$
$277$	−21.9950	−1.32155	−0.660777	+	0.750582i	$0.729773\pi$
$278$	0	0
$279$	−21.5326	−1.28912
$280$	0	0
$281$	−21.3613	−1.27431	−0.637155	−	0.770736i	$-0.719889\pi$
$281$	−21.3613	−1.27431	−0.637155	+	0.770736i	$0.719889\pi$
$282$	0	0
$283$	−8.31919	−0.494525	−0.247262	−	0.968949i	$-0.579531\pi$
$283$	−8.31919	−0.494525	−0.247262	+	0.968949i	$0.579531\pi$
$284$	0	0
$285$	−2.00820	−0.118955
$286$	0	0
$287$	−0.865955	−0.0511157
$288$	0	0
$289$	14.5152	0.853833
$290$	0	0
$291$	−2.52797	−0.148192
$292$	0	0
$293$	−14.0336	−0.819850	−0.409925	−	0.912119i	$-0.634445\pi$
$293$	−14.0336	−0.819850	−0.409925	+	0.912119i	$0.634445\pi$
$294$	0	0
$295$	−0.793889	−0.0462220
$296$	0	0
$297$	−4.11866	−0.238989
$298$	0	0
$299$	14.5408	0.840915
$300$	0	0
$301$	21.4053	1.23378
$302$	0	0
$303$	9.03325	0.518947
$304$	0	0
$305$	17.1590	0.982524
$306$	0	0
$307$	8.37133	0.477777	0.238889	−	0.971047i	$-0.423217\pi$
$307$	8.37133	0.477777	0.238889	+	0.971047i	$0.423217\pi$
$308$	0	0
$309$	0.118867	0.00676211
$310$	0	0
$311$	−21.4176	−1.21448	−0.607241	−	0.794518i	$-0.707724\pi$
$311$	−21.4176	−1.21448	−0.607241	+	0.794518i	$0.707724\pi$
$312$	0	0
$313$	−5.52024	−0.312023	−0.156011	−	0.987755i	$-0.549864\pi$
$313$	−5.52024	−0.312023	−0.156011	+	0.987755i	$0.549864\pi$
$314$	0	0
$315$	−21.5462	−1.21399
$316$	0	0
$317$	−30.3748	−1.70602	−0.853010	−	0.521895i	$-0.825225\pi$
$317$	−30.3748	−1.70602	−0.853010	+	0.521895i	$0.825225\pi$
$318$	0	0
$319$	−2.82012	−0.157896
$320$	0	0
$321$	5.16724	0.288407
$322$	0	0
$323$	5.61384	0.312362
$324$	0	0
$325$	32.9429	1.82734
$326$	0	0
$327$	0.00184422	0.000101986 0
$328$	0	0
$329$	−14.7843	−0.815087
$330$	0	0
$331$	−18.6568	−1.02547	−0.512734	−	0.858548i	$-0.671367\pi$
$331$	−18.6568	−1.02547	−0.512734	+	0.858548i	$0.671367\pi$
$332$	0	0
$333$	−23.2218	−1.27255
$334$	0	0
$335$	11.0053	0.601283
$336$	0	0
$337$	−4.65878	−0.253780	−0.126890	−	0.991917i	$-0.540500\pi$
$337$	−4.65878	−0.253780	−0.126890	+	0.991917i	$0.540500\pi$
$338$	0	0
$339$	1.02162	0.0554867
$340$	0	0
$341$	−9.54835	−0.517072
$342$	0	0
$343$	18.3835	0.992615
$344$	0	0
$345$	4.22552	0.227494
$346$	0	0
$347$	−13.7013	−0.735524	−0.367762	−	0.929920i	$-0.619876\pi$
$347$	−13.7013	−0.735524	−0.367762	+	0.929920i	$0.619876\pi$
$348$	0	0
$349$	−18.6369	−0.997610	−0.498805	−	0.866714i	$-0.666228\pi$
$349$	−18.6369	−0.997610	−0.498805	+	0.866714i	$0.666228\pi$
$350$	0	0
$351$	24.8099	1.32426
$352$	0	0
$353$	28.0231	1.49152	0.745759	−	0.666216i	$-0.232087\pi$
$353$	28.0231	1.49152	0.745759	+	0.666216i	$0.232087\pi$
$354$	0	0
$355$	−42.6704	−2.26471
$356$	0	0
$357$	−9.61306	−0.508777
$358$	0	0
$359$	−4.74760	−0.250569	−0.125284	−	0.992121i	$-0.539984\pi$
$359$	−4.74760	−0.250569	−0.125284	+	0.992121i	$0.539984\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	6.22266	0.326605
$364$	0	0
$365$	−24.7508	−1.29552
$366$	0	0
$367$	−21.6682	−1.13107	−0.565535	−	0.824724i	$-0.691330\pi$
$367$	−21.6682	−1.13107	−0.565535	+	0.824724i	$0.691330\pi$
$368$	0	0
$369$	−0.840681	−0.0437641
$370$	0	0
$371$	−2.57816	−0.133851
$372$	0	0
$373$	1.27086	0.0658026	0.0329013	−	0.999459i	$-0.489525\pi$
$373$	1.27086	0.0658026	0.0329013	+	0.999459i	$0.489525\pi$
$374$	0	0
$375$	−0.467858	−0.0241601
$376$	0	0
$377$	16.9878	0.874916
$378$	0	0
$379$	29.6427	1.52264	0.761321	−	0.648375i	$-0.224551\pi$
$379$	29.6427	1.52264	0.761321	+	0.648375i	$0.224551\pi$
$380$	0	0
$381$	−4.43369	−0.227145
$382$	0	0
$383$	9.10210	0.465096	0.232548	−	0.972585i	$-0.425294\pi$
$383$	9.10210	0.465096	0.232548	+	0.972585i	$0.425294\pi$
$384$	0	0
$385$	−9.55437	−0.486936
$386$	0	0
$387$	20.7806	1.05634
$388$	0	0
$389$	−2.14062	−0.108534	−0.0542668	−	0.998526i	$-0.517282\pi$
$389$	−2.14062	−0.108534	−0.0542668	+	0.998526i	$0.517282\pi$
$390$	0	0
$391$	−11.8123	−0.597372
$392$	0	0
$393$	−11.3903	−0.574567
$394$	0	0
$395$	−3.12522	−0.157247
$396$	0	0
$397$	0.817774	0.0410429	0.0205214	−	0.999789i	$-0.493467\pi$
$397$	0.817774	0.0410429	0.0205214	+	0.999789i	$0.493467\pi$
$398$	0	0
$399$	−1.71239	−0.0857266
$400$	0	0
$401$	4.98626	0.249002	0.124501	−	0.992219i	$-0.460267\pi$
$401$	4.98626	0.249002	0.124501	+	0.992219i	$0.460267\pi$
$402$	0	0
$403$	57.5173	2.86514
$404$	0	0
$405$	−17.0460	−0.847024
$406$	0	0
$407$	−10.2974	−0.510424
$408$	0	0
$409$	−34.4934	−1.70559	−0.852793	−	0.522249i	$-0.825093\pi$
$409$	−34.4934	−1.70559	−0.852793	+	0.522249i	$0.825093\pi$
$410$	0	0
$411$	13.5514	0.668442
$412$	0	0
$413$	−0.676948	−0.0333104
$414$	0	0
$415$	29.0034	1.42372
$416$	0	0
$417$	7.32877	0.358892
$418$	0	0
$419$	15.9381	0.778625	0.389313	−	0.921106i	$-0.372713\pi$
$419$	15.9381	0.778625	0.389313	+	0.921106i	$0.372713\pi$
$420$	0	0
$421$	37.3536	1.82050	0.910251	−	0.414056i	$-0.135888\pi$
$421$	37.3536	1.82050	0.910251	+	0.414056i	$0.135888\pi$
$422$	0	0
$423$	−14.3528	−0.697859
$424$	0	0
$425$	−26.7613	−1.29811
$426$	0	0
$427$	14.6315	0.708067
$428$	0	0
$429$	5.09429	0.245955
$430$	0	0
$431$	−2.01266	−0.0969465	−0.0484732	−	0.998824i	$-0.515436\pi$
$431$	−2.01266	−0.0969465	−0.0484732	+	0.998824i	$0.515436\pi$
$432$	0	0
$433$	23.4820	1.12847	0.564235	−	0.825614i	$-0.309171\pi$
$433$	23.4820	1.12847	0.564235	+	0.825614i	$0.309171\pi$
$434$	0	0
$435$	4.93662	0.236693
$436$	0	0
$437$	−2.10413	−0.100654
$438$	0	0
$439$	5.56896	0.265792	0.132896	−	0.991130i	$-0.457572\pi$
$439$	5.56896	0.265792	0.132896	+	0.991130i	$0.457572\pi$
$440$	0	0
$441$	−0.262721	−0.0125105
$442$	0	0
$443$	26.6718	1.26722	0.633609	−	0.773654i	$-0.281573\pi$
$443$	26.6718	1.26722	0.633609	+	0.773654i	$0.281573\pi$
$444$	0	0
$445$	26.4730	1.25494
$446$	0	0
$447$	−10.9755	−0.519122
$448$	0	0
$449$	−30.2901	−1.42948	−0.714740	−	0.699390i	$-0.753455\pi$
$449$	−30.2901	−1.42948	−0.714740	+	0.699390i	$0.753455\pi$
$450$	0	0
$451$	−0.372789	−0.0175539
$452$	0	0
$453$	−5.31481	−0.249712
$454$	0	0
$455$	57.5535	2.69815
$456$	0	0
$457$	−24.7611	−1.15828	−0.579138	−	0.815229i	$-0.696611\pi$
$457$	−24.7611	−1.15828	−0.579138	+	0.815229i	$0.696611\pi$
$458$	0	0
$459$	−20.1545	−0.940730
$460$	0	0
$461$	32.0204	1.49134	0.745670	−	0.666315i	$-0.232129\pi$
$461$	32.0204	1.49134	0.745670	+	0.666315i	$0.232129\pi$
$462$	0	0
$463$	32.4746	1.50922	0.754611	−	0.656172i	$-0.227826\pi$
$463$	32.4746	1.50922	0.754611	+	0.656172i	$0.227826\pi$
$464$	0	0
$465$	16.7144	0.775112
$466$	0	0
$467$	37.1914	1.72101	0.860506	−	0.509441i	$-0.170148\pi$
$467$	37.1914	1.72101	0.860506	+	0.509441i	$0.170148\pi$
$468$	0	0
$469$	9.38419	0.433321
$470$	0	0
$471$	1.10673	0.0509955
$472$	0	0
$473$	9.21487	0.423700
$474$	0	0
$475$	−4.76703	−0.218726
$476$	0	0
$477$	−2.50291	−0.114600
$478$	0	0
$479$	−30.6135	−1.39876	−0.699382	−	0.714748i	$-0.746541\pi$
$479$	−30.6135	−1.39876	−0.699382	+	0.714748i	$0.746541\pi$
$480$	0	0
$481$	62.0295	2.82830
$482$	0	0
$483$	3.60309	0.163946
$484$	0	0
$485$	−12.2950	−0.558286
$486$	0	0
$487$	20.5925	0.933135	0.466568	−	0.884486i	$-0.345490\pi$
$487$	20.5925	0.933135	0.466568	+	0.884486i	$0.345490\pi$
$488$	0	0
$489$	−12.9152	−0.584045
$490$	0	0
$491$	−0.0939364	−0.00423929	−0.00211964	−	0.999998i	$-0.500675\pi$
$491$	−0.0939364	−0.00423929	−0.00211964	+	0.999998i	$0.500675\pi$
$492$	0	0
$493$	−13.8001	−0.621526
$494$	0	0
$495$	−9.27551	−0.416903
$496$	0	0
$497$	−36.3850	−1.63209
$498$	0	0
$499$	−7.44571	−0.333316	−0.166658	−	0.986015i	$-0.553298\pi$
$499$	−7.44571	−0.333316	−0.166658	+	0.986015i	$0.553298\pi$
$500$	0	0
$501$	9.93443	0.443838
$502$	0	0
$503$	−0.550713	−0.0245551	−0.0122775	−	0.999925i	$-0.503908\pi$
$503$	−0.550713	−0.0245551	−0.0122775	+	0.999925i	$0.503908\pi$
$504$	0	0
$505$	43.9339	1.95503
$506$	0	0
$507$	−22.3335	−0.991864
$508$	0	0
$509$	39.2631	1.74031	0.870153	−	0.492782i	$-0.164020\pi$
$509$	39.2631	1.74031	0.870153	+	0.492782i	$0.164020\pi$
$510$	0	0
$511$	−21.1050	−0.933629
$512$	0	0
$513$	−3.59014	−0.158509
$514$	0	0
$515$	0.578119	0.0254750
$516$	0	0
$517$	−6.36458	−0.279914
$518$	0	0
$519$	5.03515	0.221019
$520$	0	0
$521$	−15.1772	−0.664927	−0.332463	−	0.943116i	$-0.607880\pi$
$521$	−15.1772	−0.664927	−0.332463	+	0.943116i	$0.607880\pi$
$522$	0	0
$523$	−24.5978	−1.07559	−0.537794	−	0.843076i	$-0.680742\pi$
$523$	−24.5978	−1.07559	−0.537794	+	0.843076i	$0.680742\pi$
$524$	0	0
$525$	8.16300	0.356262
$526$	0	0
$527$	−46.7244	−2.03535
$528$	0	0
$529$	−18.5726	−0.807505
$530$	0	0
$531$	−0.657190	−0.0285196
$532$	0	0
$533$	2.24560	0.0972679
$534$	0	0
$535$	25.1312	1.08652
$536$	0	0
$537$	15.6188	0.674002
$538$	0	0
$539$	−0.116500	−0.00501802
$540$	0	0
$541$	−22.7718	−0.979035	−0.489518	−	0.871993i	$-0.662827\pi$
$541$	−22.7718	−0.979035	−0.489518	+	0.871993i	$0.662827\pi$
$542$	0	0
$543$	−4.15476	−0.178298
$544$	0	0
$545$	0.00896951	0.000384212 0
$546$	0	0
$547$	25.8982	1.10733	0.553664	−	0.832740i	$-0.313229\pi$
$547$	25.8982	1.10733	0.553664	+	0.832740i	$0.313229\pi$
$548$	0	0
$549$	14.2044	0.606230
$550$	0	0
$551$	−2.45823	−0.104724
$552$	0	0
$553$	−2.66487	−0.113322
$554$	0	0
$555$	18.0256	0.765146
$556$	0	0
$557$	33.7332	1.42932	0.714660	−	0.699472i	$-0.246581\pi$
$557$	33.7332	1.42932	0.714660	+	0.699472i	$0.246581\pi$
$558$	0	0
$559$	−55.5085	−2.34776
$560$	0	0
$561$	−4.13837	−0.174722
$562$	0	0
$563$	−25.4185	−1.07126	−0.535630	−	0.844453i	$-0.679926\pi$
$563$	−25.4185	−1.07126	−0.535630	+	0.844453i	$0.679926\pi$
$564$	0	0
$565$	4.96872	0.209036
$566$	0	0
$567$	−14.5351	−0.610418
$568$	0	0
$569$	−27.9903	−1.17341	−0.586707	−	0.809799i	$-0.699576\pi$
$569$	−27.9903	−1.17341	−0.586707	+	0.809799i	$0.699576\pi$
$570$	0	0
$571$	−25.0203	−1.04707	−0.523534	−	0.852005i	$-0.675387\pi$
$571$	−25.0203	−1.04707	−0.523534	+	0.852005i	$0.675387\pi$
$572$	0	0
$573$	−2.36820	−0.0989330
$574$	0	0
$575$	10.0305	0.418299
$576$	0	0
$577$	−1.02754	−0.0427771	−0.0213886	−	0.999771i	$-0.506809\pi$
$577$	−1.02754	−0.0427771	−0.0213886	+	0.999771i	$0.506809\pi$
$578$	0	0
$579$	−10.7786	−0.447942
$580$	0	0
$581$	24.7312	1.02602
$582$	0	0
$583$	−1.10988	−0.0459666
$584$	0	0
$585$	55.8737	2.31009
$586$	0	0
$587$	−32.0069	−1.32107	−0.660533	−	0.750797i	$-0.729670\pi$
$587$	−32.0069	−1.32107	−0.660533	+	0.750797i	$0.729670\pi$
$588$	0	0
$589$	−8.32308	−0.342947
$590$	0	0
$591$	10.0303	0.412593
$592$	0	0
$593$	−25.3607	−1.04144	−0.520719	−	0.853728i	$-0.674336\pi$
$593$	−25.3607	−1.04144	−0.520719	+	0.853728i	$0.674336\pi$
$594$	0	0
$595$	−46.7539	−1.91672
$596$	0	0
$597$	9.11393	0.373008
$598$	0	0
$599$	15.7655	0.644160	0.322080	−	0.946712i	$-0.395618\pi$
$599$	15.7655	0.644160	0.322080	+	0.946712i	$0.395618\pi$
$600$	0	0
$601$	9.53815	0.389069	0.194535	−	0.980896i	$-0.437680\pi$
$601$	9.53815	0.389069	0.194535	+	0.980896i	$0.437680\pi$
$602$	0	0
$603$	9.11029	0.371000
$604$	0	0
$605$	30.2644	1.23042
$606$	0	0
$607$	−34.9392	−1.41814	−0.709068	−	0.705140i	$-0.750884\pi$
$607$	−34.9392	−1.41814	−0.709068	+	0.705140i	$0.750884\pi$
$608$	0	0
$609$	4.20944	0.170575
$610$	0	0
$611$	38.3389	1.55103
$612$	0	0
$613$	9.94799	0.401795	0.200898	−	0.979612i	$-0.435614\pi$
$613$	9.94799	0.401795	0.200898	+	0.979612i	$0.435614\pi$
$614$	0	0
$615$	0.652568	0.0263141
$616$	0	0
$617$	8.05924	0.324453	0.162226	−	0.986754i	$-0.448133\pi$
$617$	8.05924	0.324453	0.162226	+	0.986754i	$0.448133\pi$
$618$	0	0
$619$	−37.5794	−1.51044	−0.755222	−	0.655469i	$-0.772471\pi$
$619$	−37.5794	−1.51044	−0.755222	+	0.655469i	$0.772471\pi$
$620$	0	0
$621$	7.55414	0.303137
$622$	0	0
$623$	22.5735	0.904389
$624$	0	0
$625$	−26.1106	−1.04442
$626$	0	0
$627$	−0.737173	−0.0294399
$628$	0	0
$629$	−50.3899	−2.00918
$630$	0	0
$631$	21.6610	0.862310	0.431155	−	0.902278i	$-0.358106\pi$
$631$	21.6610	0.862310	0.431155	+	0.902278i	$0.358106\pi$
$632$	0	0
$633$	6.89577	0.274082
$634$	0	0
$635$	−21.5636	−0.855726
$636$	0	0
$637$	0.701773	0.0278053
$638$	0	0
$639$	−35.3231	−1.39736
$640$	0	0
$641$	28.2748	1.11679	0.558394	−	0.829576i	$-0.311418\pi$
$641$	28.2748	1.11679	0.558394	+	0.829576i	$0.311418\pi$
$642$	0	0
$643$	22.5740	0.890230	0.445115	−	0.895473i	$-0.353163\pi$
$643$	22.5740	0.890230	0.445115	+	0.895473i	$0.353163\pi$
$644$	0	0
$645$	−16.1306	−0.635143
$646$	0	0
$647$	0.0838253	0.00329551	0.00164776	−	0.999999i	$-0.499476\pi$
$647$	0.0838253	0.00329551	0.00164776	+	0.999999i	$0.499476\pi$
$648$	0	0
$649$	−0.291422	−0.0114393
$650$	0	0
$651$	14.2523	0.558593
$652$	0	0
$653$	1.13848	0.0445522	0.0222761	−	0.999752i	$-0.492909\pi$
$653$	1.13848	0.0445522	0.0222761	+	0.999752i	$0.492909\pi$
$654$	0	0
$655$	−55.3978	−2.16457
$656$	0	0
$657$	−20.4890	−0.799352
$658$	0	0
$659$	12.1543	0.473463	0.236732	−	0.971575i	$-0.423924\pi$
$659$	12.1543	0.473463	0.236732	+	0.971575i	$0.423924\pi$
$660$	0	0
$661$	−33.3480	−1.29709	−0.648544	−	0.761177i	$-0.724622\pi$
$661$	−33.3480	−1.29709	−0.648544	+	0.761177i	$0.724622\pi$
$662$	0	0
$663$	24.9287	0.968150
$664$	0	0
$665$	−8.32833	−0.322959
$666$	0	0
$667$	5.17245	0.200278
$668$	0	0
$669$	−5.40167	−0.208840
$670$	0	0
$671$	6.29877	0.243161
$672$	0	0
$673$	−43.7183	−1.68522	−0.842609	−	0.538526i	$-0.818981\pi$
$673$	−43.7183	−1.68522	−0.842609	+	0.538526i	$0.818981\pi$
$674$	0	0
$675$	17.1143	0.658729
$676$	0	0
$677$	−42.7453	−1.64284	−0.821418	−	0.570326i	$-0.806817\pi$
$677$	−42.7453	−1.64284	−0.821418	+	0.570326i	$0.806817\pi$
$678$	0	0
$679$	−10.4839	−0.402336
$680$	0	0
$681$	−6.66208	−0.255292
$682$	0	0
$683$	31.6681	1.21174	0.605872	−	0.795562i	$-0.292824\pi$
$683$	31.6681	1.21174	0.605872	+	0.795562i	$0.292824\pi$
$684$	0	0
$685$	65.9084	2.51823
$686$	0	0
$687$	0.691319	0.0263755
$688$	0	0
$689$	6.68570	0.254705
$690$	0	0
$691$	−19.9792	−0.760043	−0.380021	−	0.924978i	$-0.624083\pi$
$691$	−19.9792	−0.760043	−0.380021	+	0.924978i	$0.624083\pi$
$692$	0	0
$693$	−7.90921	−0.300446
$694$	0	0
$695$	35.6441	1.35206
$696$	0	0
$697$	−1.82423	−0.0690975
$698$	0	0
$699$	5.34519	0.202174
$700$	0	0
$701$	−4.54163	−0.171535	−0.0857675	−	0.996315i	$-0.527334\pi$
$701$	−4.54163	−0.171535	−0.0857675	+	0.996315i	$0.527334\pi$
$702$	0	0
$703$	−8.97602	−0.338537
$704$	0	0
$705$	11.1412	0.419602
$706$	0	0
$707$	37.4624	1.40892
$708$	0	0
$709$	−49.9629	−1.87640	−0.938198	−	0.346100i	$-0.887506\pi$
$709$	−49.9629	−1.87640	−0.938198	+	0.346100i	$0.887506\pi$
$710$	0	0
$711$	−2.58709	−0.0970236
$712$	0	0
$713$	17.5129	0.655863
$714$	0	0
$715$	24.7765	0.926588
$716$	0	0
$717$	−0.309935	−0.0115747
$718$	0	0
$719$	12.0897	0.450870	0.225435	−	0.974258i	$-0.427620\pi$
$719$	12.0897	0.450870	0.225435	+	0.974258i	$0.427620\pi$
$720$	0	0
$721$	0.492961	0.0183588
$722$	0	0
$723$	5.22296	0.194244
$724$	0	0
$725$	11.7185	0.435212
$726$	0	0
$727$	35.8526	1.32970	0.664850	−	0.746977i	$-0.268495\pi$
$727$	35.8526	1.32970	0.664850	+	0.746977i	$0.268495\pi$
$728$	0	0
$729$	−7.19006	−0.266299
$730$	0	0
$731$	45.0926	1.66781
$732$	0	0
$733$	−39.8404	−1.47154	−0.735769	−	0.677232i	$-0.763179\pi$
$733$	−39.8404	−1.47154	−0.735769	+	0.677232i	$0.763179\pi$
$734$	0	0
$735$	0.203934	0.00752221
$736$	0	0
$737$	4.03984	0.148809
$738$	0	0
$739$	43.6891	1.60713	0.803564	−	0.595218i	$-0.202934\pi$
$739$	43.6891	1.60713	0.803564	+	0.595218i	$0.202934\pi$
$740$	0	0
$741$	4.44058	0.163129
$742$	0	0
$743$	−3.33511	−0.122353	−0.0611767	−	0.998127i	$-0.519485\pi$
$743$	−3.33511	−0.122353	−0.0611767	+	0.998127i	$0.519485\pi$
$744$	0	0
$745$	−53.3800	−1.95569
$746$	0	0
$747$	24.0094	0.878456
$748$	0	0
$749$	21.4294	0.783012
$750$	0	0
$751$	−3.25651	−0.118832	−0.0594159	−	0.998233i	$-0.518924\pi$
$751$	−3.25651	−0.118832	−0.0594159	+	0.998233i	$0.518924\pi$
$752$	0	0
$753$	−11.7924	−0.429740
$754$	0	0
$755$	−25.8490	−0.940742
$756$	0	0
$757$	32.3875	1.17715	0.588573	−	0.808444i	$-0.299690\pi$
$757$	32.3875	1.17715	0.588573	+	0.808444i	$0.299690\pi$
$758$	0	0
$759$	1.55111	0.0563018
$760$	0	0
$761$	−35.7288	−1.29517	−0.647584	−	0.761994i	$-0.724220\pi$
$761$	−35.7288	−1.29517	−0.647584	+	0.761994i	$0.724220\pi$
$762$	0	0
$763$	0.00764829	0.000276887 0
$764$	0	0
$765$	−45.3893	−1.64105
$766$	0	0
$767$	1.75547	0.0633862
$768$	0	0
$769$	−14.6647	−0.528821	−0.264411	−	0.964410i	$-0.585177\pi$
$769$	−14.6647	−0.528821	−0.264411	+	0.964410i	$0.585177\pi$
$770$	0	0
$771$	−2.33782	−0.0841946
$772$	0	0
$773$	44.1719	1.58875	0.794376	−	0.607426i	$-0.207798\pi$
$773$	44.1719	1.58875	0.794376	+	0.607426i	$0.207798\pi$
$774$	0	0
$775$	39.6764	1.42522
$776$	0	0
$777$	15.3704	0.551411
$778$	0	0
$779$	−0.324952	−0.0116426
$780$	0	0
$781$	−15.6635	−0.560486
$782$	0	0
$783$	8.82539	0.315394
$784$	0	0
$785$	5.38267	0.192116
$786$	0	0
$787$	−32.5057	−1.15870	−0.579351	−	0.815078i	$-0.696694\pi$
$787$	−32.5057	−1.15870	−0.579351	+	0.815078i	$0.696694\pi$
$788$	0	0
$789$	16.0491	0.571362
$790$	0	0
$791$	4.23682	0.150644
$792$	0	0
$793$	−37.9425	−1.34738
$794$	0	0
$795$	1.94285	0.0689058
$796$	0	0
$797$	−11.7071	−0.414688	−0.207344	−	0.978268i	$-0.566482\pi$
$797$	−11.7071	−0.414688	−0.207344	+	0.978268i	$0.566482\pi$
$798$	0	0
$799$	−31.1448	−1.10182
$800$	0	0
$801$	21.9147	0.774317
$802$	0	0
$803$	−9.08558	−0.320623
$804$	0	0
$805$	17.5239	0.617637
$806$	0	0
$807$	−11.2508	−0.396046
$808$	0	0
$809$	−47.2413	−1.66091	−0.830457	−	0.557082i	$-0.811921\pi$
$809$	−47.2413	−1.66091	−0.830457	+	0.557082i	$0.811921\pi$
$810$	0	0
$811$	1.26276	0.0443416	0.0221708	−	0.999754i	$-0.492942\pi$
$811$	1.26276	0.0443416	0.0221708	+	0.999754i	$0.492942\pi$
$812$	0	0
$813$	−9.23992	−0.324058
$814$	0	0
$815$	−62.8140	−2.20028
$816$	0	0
$817$	8.03239	0.281018
$818$	0	0
$819$	47.6434	1.66480
$820$	0	0
$821$	−23.4027	−0.816760	−0.408380	−	0.912812i	$-0.633906\pi$
$821$	−23.4027	−0.816760	−0.408380	+	0.912812i	$0.633906\pi$
$822$	0	0
$823$	41.9985	1.46397	0.731987	−	0.681319i	$-0.238593\pi$
$823$	41.9985	1.46397	0.731987	+	0.681319i	$0.238593\pi$
$824$	0	0
$825$	3.51412	0.122346
$826$	0	0
$827$	−28.8312	−1.00256	−0.501279	−	0.865286i	$-0.667137\pi$
$827$	−28.8312	−1.00256	−0.501279	+	0.865286i	$0.667137\pi$
$828$	0	0
$829$	−37.9637	−1.31853	−0.659266	−	0.751909i	$-0.729133\pi$
$829$	−37.9637	−1.31853	−0.659266	+	0.751909i	$0.729133\pi$
$830$	0	0
$831$	14.1335	0.490286
$832$	0	0
$833$	−0.570088	−0.0197524
$834$	0	0
$835$	48.3169	1.67207
$836$	0	0
$837$	29.8810	1.03284
$838$	0	0
$839$	−42.7100	−1.47451	−0.737256	−	0.675613i	$-0.763879\pi$
$839$	−42.7100	−1.47451	−0.737256	+	0.675613i	$0.763879\pi$
$840$	0	0
$841$	−22.9571	−0.791624
$842$	0	0
$843$	13.7263	0.472759
$844$	0	0
$845$	−108.620	−3.73666
$846$	0	0
$847$	25.8064	0.886717
$848$	0	0
$849$	5.34573	0.183465
$850$	0	0
$851$	18.8868	0.647430
$852$	0	0
$853$	−36.5064	−1.24996	−0.624978	−	0.780642i	$-0.714892\pi$
$853$	−36.5064	−1.24996	−0.624978	+	0.780642i	$0.714892\pi$
$854$	0	0
$855$	−8.08525	−0.276510
$856$	0	0
$857$	16.1247	0.550809	0.275405	−	0.961328i	$-0.411188\pi$
$857$	16.1247	0.550809	0.275405	+	0.961328i	$0.411188\pi$
$858$	0	0
$859$	37.1033	1.26595	0.632974	−	0.774173i	$-0.281834\pi$
$859$	37.1033	1.26595	0.632974	+	0.774173i	$0.281834\pi$
$860$	0	0
$861$	0.556443	0.0189635
$862$	0	0
$863$	−29.1835	−0.993419	−0.496709	−	0.867917i	$-0.665458\pi$
$863$	−29.1835	−0.993419	−0.496709	+	0.867917i	$0.665458\pi$
$864$	0	0
$865$	24.4888	0.832646
$866$	0	0
$867$	−9.32712	−0.316766
$868$	0	0
$869$	−1.14721	−0.0389165
$870$	0	0
$871$	−24.3352	−0.824566
$872$	0	0
$873$	−10.1779	−0.344470
$874$	0	0
$875$	−1.94028	−0.0655935
$876$	0	0
$877$	9.38219	0.316814	0.158407	−	0.987374i	$-0.449364\pi$
$877$	9.38219	0.316814	0.158407	+	0.987374i	$0.449364\pi$
$878$	0	0
$879$	9.01765	0.304158
$880$	0	0
$881$	7.88460	0.265639	0.132819	−	0.991140i	$-0.457597\pi$
$881$	7.88460	0.265639	0.132819	+	0.991140i	$0.457597\pi$
$882$	0	0
$883$	30.9719	1.04229	0.521144	−	0.853469i	$-0.325505\pi$
$883$	30.9719	1.04229	0.521144	+	0.853469i	$0.325505\pi$
$884$	0	0
$885$	0.510135	0.0171480
$886$	0	0
$887$	32.0298	1.07546	0.537728	−	0.843118i	$-0.319283\pi$
$887$	32.0298	1.07546	0.537728	+	0.843118i	$0.319283\pi$
$888$	0	0
$889$	−18.3873	−0.616689
$890$	0	0
$891$	−6.25729	−0.209627
$892$	0	0
$893$	−5.54786	−0.185652
$894$	0	0
$895$	75.9633	2.53917
$896$	0	0
$897$	−9.34358	−0.311973
$898$	0	0
$899$	20.4601	0.682381
$900$	0	0
$901$	−5.43116	−0.180938
$902$	0	0
$903$	−13.7546	−0.457724
$904$	0	0
$905$	−20.2070	−0.671702
$906$	0	0
$907$	1.36575	0.0453490	0.0226745	−	0.999743i	$-0.492782\pi$
$907$	1.36575	0.0453490	0.0226745	+	0.999743i	$0.492782\pi$
$908$	0	0
$909$	36.3689	1.20628
$910$	0	0
$911$	−0.548157	−0.0181612	−0.00908062	−	0.999959i	$-0.502890\pi$
$911$	−0.548157	−0.0181612	−0.00908062	+	0.999959i	$0.502890\pi$
$912$	0	0
$913$	10.6466	0.352352
$914$	0	0
$915$	−11.0260	−0.364509
$916$	0	0
$917$	−47.2376	−1.55992
$918$	0	0
$919$	−12.3856	−0.408565	−0.204282	−	0.978912i	$-0.565486\pi$
$919$	−12.3856	−0.408565	−0.204282	+	0.978912i	$0.565486\pi$
$920$	0	0
$921$	−5.37923	−0.177252
$922$	0	0
$923$	94.3540	3.10570
$924$	0	0
$925$	42.7889	1.40689
$926$	0	0
$927$	0.478573	0.0157184
$928$	0	0
$929$	−10.6047	−0.347928	−0.173964	−	0.984752i	$-0.555658\pi$
$929$	−10.6047	−0.347928	−0.173964	+	0.984752i	$0.555658\pi$
$930$	0	0
$931$	−0.101551	−0.00332819
$932$	0	0
$933$	13.7625	0.450563
$934$	0	0
$935$	−20.1273	−0.658233
$936$	0	0
$937$	41.0902	1.34236	0.671180	−	0.741295i	$-0.265788\pi$
$937$	41.0902	1.34236	0.671180	+	0.741295i	$0.265788\pi$
$938$	0	0
$939$	3.54718	0.115758
$940$	0	0
$941$	50.0053	1.63013	0.815063	−	0.579372i	$-0.196702\pi$
$941$	50.0053	1.63013	0.815063	+	0.579372i	$0.196702\pi$
$942$	0	0
$943$	0.683742	0.0222657
$944$	0	0
$945$	29.8999	0.972643
$946$	0	0
$947$	−20.3054	−0.659838	−0.329919	−	0.944009i	$-0.607021\pi$
$947$	−20.3054	−0.659838	−0.329919	+	0.944009i	$0.607021\pi$
$948$	0	0
$949$	54.7296	1.77660
$950$	0	0
$951$	19.5182	0.632920
$952$	0	0
$953$	−32.6644	−1.05811	−0.529053	−	0.848589i	$-0.677453\pi$
$953$	−32.6644	−1.05811	−0.529053	+	0.848589i	$0.677453\pi$
$954$	0	0
$955$	−11.5179	−0.372711
$956$	0	0
$957$	1.81214	0.0585782
$958$	0	0
$959$	56.2000	1.81479
$960$	0	0
$961$	38.2737	1.23464
$962$	0	0
$963$	20.8039	0.670397
$964$	0	0
$965$	−52.4224	−1.68754
$966$	0	0
$967$	−1.76683	−0.0568174	−0.0284087	−	0.999596i	$-0.509044\pi$
$967$	−1.76683	−0.0568174	−0.0284087	+	0.999596i	$0.509044\pi$
$968$	0	0
$969$	−3.60733	−0.115884
$970$	0	0
$971$	13.0530	0.418891	0.209446	−	0.977820i	$-0.432834\pi$
$971$	13.0530	0.418891	0.209446	+	0.977820i	$0.432834\pi$
$972$	0	0
$973$	30.3936	0.974375
$974$	0	0
$975$	−21.1684	−0.677930
$976$	0	0
$977$	−32.4757	−1.03899	−0.519494	−	0.854474i	$-0.673880\pi$
$977$	−32.4757	−1.03899	−0.519494	+	0.854474i	$0.673880\pi$
$978$	0	0
$979$	9.71778	0.310582
$980$	0	0
$981$	0.00742506	0.000237064 0
$982$	0	0
$983$	5.61637	0.179134	0.0895672	−	0.995981i	$-0.471452\pi$
$983$	5.61637	0.179134	0.0895672	+	0.995981i	$0.471452\pi$
$984$	0	0
$985$	48.7833	1.55437
$986$	0	0
$987$	9.50009	0.302391
$988$	0	0
$989$	−16.9012	−0.537428
$990$	0	0
$991$	53.6140	1.70310	0.851552	−	0.524270i	$-0.175662\pi$
$991$	53.6140	1.70310	0.851552	+	0.524270i	$0.175662\pi$
$992$	0	0
$993$	11.9884	0.380441
$994$	0	0
$995$	44.3263	1.40524
$996$	0	0
$997$	5.60637	0.177556	0.0887778	−	0.996051i	$-0.471704\pi$
$997$	5.60637	0.177556	0.0887778	+	0.996051i	$0.471704\pi$
$998$	0	0
$999$	32.2252	1.01956

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.f.1.11	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.f.1.11	✓	25	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q-0.642577 q^{3} -3.12522 q^{5} -2.66487 q^{7} -2.58709 q^{9} +O(q^{10})\)	\(q-0.642577 q^{3} -3.12522 q^{5} -2.66487 q^{7} -2.58709 q^{9} -1.14721 q^{11} +6.91058 q^{13} +2.00820 q^{15} -5.61384 q^{17} -1.00000 q^{19} +1.71239 q^{21} +2.10413 q^{23} +4.76703 q^{25} +3.59014 q^{27} +2.45823 q^{29} +8.32308 q^{31} +0.737173 q^{33} +8.32833 q^{35} +8.97602 q^{37} -4.44058 q^{39} +0.324952 q^{41} -8.03239 q^{43} +8.08525 q^{45} +5.54786 q^{47} +0.101551 q^{49} +3.60733 q^{51} +0.967459 q^{53} +3.58530 q^{55} +0.642577 q^{57} +0.254026 q^{59} -5.49050 q^{61} +6.89428 q^{63} -21.5971 q^{65} -3.52144 q^{67} -1.35207 q^{69} +13.6536 q^{71} +7.91969 q^{73} -3.06318 q^{75} +3.05718 q^{77} +1.00000 q^{79} +5.45434 q^{81} -9.28043 q^{83} +17.5445 q^{85} -1.57960 q^{87} -8.47077 q^{89} -18.4158 q^{91} -5.34823 q^{93} +3.12522 q^{95} +3.93411 q^{97} +2.96795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) \) 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9	\( 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9} - 3 q^{11} + q^{13} - 5 q^{15} - 13 q^{17} - 25 q^{19} - 24 q^{21} - 31 q^{23} + 21 q^{25} + 7 q^{27} - 19 q^{29} - 7 q^{31} - 30 q^{33} - q^{35} - 29 q^{37} - 26 q^{39} - 40 q^{41} - 40 q^{45} - 8 q^{47} - 9 q^{49} + 12 q^{51} - 38 q^{53} - 29 q^{55} - 4 q^{57} + 18 q^{59} - 26 q^{61} - 40 q^{63} - 70 q^{65} - 13 q^{67} + q^{69} - 47 q^{71} - 8 q^{73} + 7 q^{75} - 19 q^{77} + 25 q^{79} - 19 q^{81} - 8 q^{83} - 33 q^{85} - 50 q^{87} - 54 q^{89} - 12 q^{91} - 24 q^{93} + 8 q^{95} - 4 q^{97} - 39 q^{99}+O(q^{100}) \) 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9 - 3 * q^11 + q^13 - 5 * q^15 - 13 * q^17 - 25 * q^19 - 24 * q^21 - 31 * q^23 + 21 * q^25 + 7 * q^27 - 19 * q^29 - 7 * q^31 - 30 * q^33 - q^35 - 29 * q^37 - 26 * q^39 - 40 * q^41 - 40 * q^45 - 8 * q^47 - 9 * q^49 + 12 * q^51 - 38 * q^53 - 29 * q^55 - 4 * q^57 + 18 * q^59 - 26 * q^61 - 40 * q^63 - 70 * q^65 - 13 * q^67 + q^69 - 47 * q^71 - 8 * q^73 + 7 * q^75 - 19 * q^77 + 25 * q^79 - 19 * q^81 - 8 * q^83 - 33 * q^85 - 50 * q^87 - 54 * q^89 - 12 * q^91 - 24 * q^93 + 8 * q^95 - 4 * q^97 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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