LMFDB - Embedded newform 6003.2.a.v.1.29

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6003,2,Mod(1,6003)] 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("6003.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6003, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.a (trivial)

Newform invariants

comment:select newform

sage:traces = [30,-1,0,37,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$47.9341963334$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.29
Character	$\chi$	$=$	6003.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.60906 q^{2} +4.80718 q^{4} -3.69173 q^{5} +1.51238 q^{7} +7.32410 q^{8} -9.63193 q^{10} -4.23707 q^{11} +0.701184 q^{13} +3.94588 q^{14} +9.49464 q^{16} -2.20964 q^{17} +3.51544 q^{19} -17.7468 q^{20} -11.0548 q^{22} +1.00000 q^{23} +8.62884 q^{25} +1.82943 q^{26} +7.27027 q^{28} -1.00000 q^{29} +8.97980 q^{31} +10.1239 q^{32} -5.76508 q^{34} -5.58328 q^{35} +9.90381 q^{37} +9.17198 q^{38} -27.0386 q^{40} +8.88026 q^{41} +11.2652 q^{43} -20.3684 q^{44} +2.60906 q^{46} +9.05117 q^{47} -4.71272 q^{49} +22.5131 q^{50} +3.37072 q^{52} -4.47941 q^{53} +15.6421 q^{55} +11.0768 q^{56} -2.60906 q^{58} +8.07095 q^{59} -0.602421 q^{61} +23.4288 q^{62} +7.42446 q^{64} -2.58858 q^{65} +2.51875 q^{67} -10.6221 q^{68} -14.5671 q^{70} +4.90800 q^{71} -0.365627 q^{73} +25.8396 q^{74} +16.8994 q^{76} -6.40805 q^{77} -1.39084 q^{79} -35.0516 q^{80} +23.1691 q^{82} -0.338109 q^{83} +8.15739 q^{85} +29.3915 q^{86} -31.0327 q^{88} +6.33212 q^{89} +1.06046 q^{91} +4.80718 q^{92} +23.6150 q^{94} -12.9780 q^{95} -5.60675 q^{97} -12.2957 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8} + 8 q^{10} + 36 q^{13} - 7 q^{14} + 47 q^{16} - 18 q^{17} + 16 q^{19} + 25 q^{22} + 30 q^{23} + 56 q^{25} - 11 q^{26} + 27 q^{28} - 30 q^{29} + 14 q^{31} + 7 q^{32}+ \cdots - 8 q^{98}+O(q^{100}) $ 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8 + 8 * q^10 + 36 * q^13 - 7 * q^14 + 47 * q^16 - 18 * q^17 + 16 * q^19 + 25 * q^22 + 30 * q^23 + 56 * q^25 - 11 * q^26 + 27 * q^28 - 30 * q^29 + 14 * q^31 + 7 * q^32 + 3 * q^34 + 22 * q^35 + 40 * q^37 - 6 * q^38 + 30 * q^40 - 14 * q^41 + 34 * q^43 - 5 * q^44 - q^46 + 2 * q^47 + 74 * q^49 + 21 * q^50 + 71 * q^52 - 16 * q^53 + 22 * q^55 - 14 * q^56 + q^58 + 32 * q^59 + 46 * q^61 - 20 * q^62 + 68 * q^64 - 12 * q^65 + 14 * q^67 - 27 * q^68 + 32 * q^71 + 50 * q^73 + 26 * q^74 + 56 * q^76 - 34 * q^77 + 16 * q^79 - 2 * q^80 + 38 * q^82 + 14 * q^83 + 38 * q^85 - 10 * q^86 + 40 * q^88 + 2 * q^89 + 32 * q^91 + 37 * q^92 + 29 * q^94 + 28 * q^95 + 56 * q^97 - 8 * q^98

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 $

$2$	2.60906	1.84488	0.922441	−	0.386137i	$-0.126191\pi$
$2$	2.60906	1.84488	0.922441	+	0.386137i	$0.126191\pi$
$3$	0	0
$3$	0	0
$4$	4.80718	2.40359
$5$	−3.69173	−1.65099	−0.825495	−	0.564409i	$-0.809104\pi$
$5$	−3.69173	−1.65099	−0.825495	+	0.564409i	$0.809104\pi$
$6$	0	0
$7$	1.51238	0.571625	0.285812	−	0.958286i	$-0.407737\pi$
$7$	1.51238	0.571625	0.285812	+	0.958286i	$0.407737\pi$
$8$	7.32410	2.58946
$9$	0	0
$10$	−9.63193	−3.04588
$11$	−4.23707	−1.27752	−0.638762	−	0.769404i	$-0.720553\pi$
$11$	−4.23707	−1.27752	−0.638762	+	0.769404i	$0.720553\pi$
$12$	0	0
$13$	0.701184	0.194474	0.0972368	−	0.995261i	$-0.469000\pi$
$13$	0.701184	0.194474	0.0972368	+	0.995261i	$0.469000\pi$
$14$	3.94588	1.05458
$15$	0	0
$16$	9.49464	2.37366
$17$	−2.20964	−0.535917	−0.267958	−	0.963431i	$-0.586349\pi$
$17$	−2.20964	−0.535917	−0.267958	+	0.963431i	$0.586349\pi$
$18$	0	0
$19$	3.51544	0.806497	0.403249	−	0.915091i	$-0.367881\pi$
$19$	3.51544	0.806497	0.403249	+	0.915091i	$0.367881\pi$
$20$	−17.7468	−3.96831
$21$	0	0
$22$	−11.0548	−2.35688
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	8.62884	1.72577
$26$	1.82943	0.358781
$27$	0	0
$28$	7.27027	1.37395
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	8.97980	1.61282	0.806410	−	0.591357i	$-0.201408\pi$
$31$	8.97980	1.61282	0.806410	+	0.591357i	$0.201408\pi$
$32$	10.1239	1.78966
$33$	0	0
$34$	−5.76508	−0.988703
$35$	−5.58328	−0.943747
$36$	0	0
$37$	9.90381	1.62818	0.814088	−	0.580741i	$-0.197237\pi$
$37$	9.90381	1.62818	0.814088	+	0.580741i	$0.197237\pi$
$38$	9.17198	1.48789
$39$	0	0
$40$	−27.0386	−4.27517
$41$	8.88026	1.38686	0.693431	−	0.720523i	$-0.256098\pi$
$41$	8.88026	1.38686	0.693431	+	0.720523i	$0.256098\pi$
$42$	0	0
$43$	11.2652	1.71792	0.858961	−	0.512041i	$-0.171110\pi$
$43$	11.2652	1.71792	0.858961	+	0.512041i	$0.171110\pi$
$44$	−20.3684	−3.07065
$45$	0	0
$46$	2.60906	0.384685
$47$	9.05117	1.32025	0.660125	−	0.751156i	$-0.270503\pi$
$47$	9.05117	1.32025	0.660125	+	0.751156i	$0.270503\pi$
$48$	0	0
$49$	−4.71272	−0.673245
$50$	22.5131	3.18384
$51$	0	0
$52$	3.37072	0.467435
$53$	−4.47941	−0.615294	−0.307647	−	0.951501i	$-0.599542\pi$
$53$	−4.47941	−0.615294	−0.307647	+	0.951501i	$0.599542\pi$
$54$	0	0
$55$	15.6421	2.10918
$56$	11.0768	1.48020
$57$	0	0
$58$	−2.60906	−0.342586
$59$	8.07095	1.05075	0.525374	−	0.850871i	$-0.323925\pi$
$59$	8.07095	1.05075	0.525374	+	0.850871i	$0.323925\pi$
$60$	0	0
$61$	−0.602421	−0.0771320	−0.0385660	−	0.999256i	$-0.512279\pi$
$61$	−0.602421	−0.0771320	−0.0385660	+	0.999256i	$0.512279\pi$
$62$	23.4288	2.97546
$63$	0	0
$64$	7.42446	0.928057
$65$	−2.58858	−0.321074
$66$	0	0
$67$	2.51875	0.307714	0.153857	−	0.988093i	$-0.450831\pi$
$67$	2.51875	0.307714	0.153857	+	0.988093i	$0.450831\pi$
$68$	−10.6221	−1.28812
$69$	0	0
$70$	−14.5671	−1.74110
$71$	4.90800	0.582473	0.291236	−	0.956651i	$-0.405933\pi$
$71$	4.90800	0.582473	0.291236	+	0.956651i	$0.405933\pi$
$72$	0	0
$73$	−0.365627	−0.0427934	−0.0213967	−	0.999771i	$-0.506811\pi$
$73$	−0.365627	−0.0427934	−0.0213967	+	0.999771i	$0.506811\pi$
$74$	25.8396	3.00379
$75$	0	0
$76$	16.8994	1.93849
$77$	−6.40805	−0.730265
$78$	0	0
$79$	−1.39084	−0.156482	−0.0782411	−	0.996934i	$-0.524930\pi$
$79$	−1.39084	−0.156482	−0.0782411	+	0.996934i	$0.524930\pi$
$80$	−35.0516	−3.91889
$81$	0	0
$82$	23.1691	2.55860
$83$	−0.338109	−0.0371122	−0.0185561	−	0.999828i	$-0.505907\pi$
$83$	−0.338109	−0.0371122	−0.0185561	+	0.999828i	$0.505907\pi$
$84$	0	0
$85$	8.15739	0.884793
$86$	29.3915	3.16936
$87$	0	0
$88$	−31.0327	−3.30810
$89$	6.33212	0.671203	0.335602	−	0.942004i	$-0.391060\pi$
$89$	6.33212	0.671203	0.335602	+	0.942004i	$0.391060\pi$
$90$	0	0
$91$	1.06046	0.111166
$92$	4.80718	0.501183
$93$	0	0
$94$	23.6150	2.43570
$95$	−12.9780	−1.33152
$96$	0	0
$97$	−5.60675	−0.569279	−0.284639	−	0.958635i	$-0.591874\pi$
$97$	−5.60675	−0.569279	−0.284639	+	0.958635i	$0.591874\pi$
$98$	−12.2957	−1.24206
$99$	0	0
$100$	41.4804	4.14804
$101$	−8.51165	−0.846941	−0.423470	−	0.905910i	$-0.639188\pi$
$101$	−8.51165	−0.846941	−0.423470	+	0.905910i	$0.639188\pi$
$102$	0	0
$103$	−8.33916	−0.821682	−0.410841	−	0.911707i	$-0.634765\pi$
$103$	−8.33916	−0.821682	−0.410841	+	0.911707i	$0.634765\pi$
$104$	5.13555	0.503582
$105$	0	0
$106$	−11.6870	−1.13515
$107$	−6.51887	−0.630203	−0.315102	−	0.949058i	$-0.602039\pi$
$107$	−6.51887	−0.630203	−0.315102	+	0.949058i	$0.602039\pi$
$108$	0	0
$109$	−9.47639	−0.907673	−0.453837	−	0.891085i	$-0.649945\pi$
$109$	−9.47639	−0.907673	−0.453837	+	0.891085i	$0.649945\pi$
$110$	40.8111	3.89119
$111$	0	0
$112$	14.3595	1.35684
$113$	−18.4646	−1.73700	−0.868502	−	0.495685i	$-0.834917\pi$
$113$	−18.4646	−1.73700	−0.868502	+	0.495685i	$0.834917\pi$
$114$	0	0
$115$	−3.69173	−0.344255
$116$	−4.80718	−0.446336
$117$	0	0
$118$	21.0576	1.93851
$119$	−3.34181	−0.306343
$120$	0	0
$121$	6.95275	0.632068
$122$	−1.57175	−0.142300
$123$	0	0
$124$	43.1675	3.87656
$125$	−13.3967	−1.19824
$126$	0	0
$127$	−4.96044	−0.440168	−0.220084	−	0.975481i	$-0.570633\pi$
$127$	−4.96044	−0.440168	−0.220084	+	0.975481i	$0.570633\pi$
$128$	−0.876879	−0.0775059
$129$	0	0
$130$	−6.75376	−0.592344
$131$	13.2817	1.16043	0.580213	−	0.814464i	$-0.302969\pi$
$131$	13.2817	1.16043	0.580213	+	0.814464i	$0.302969\pi$
$132$	0	0
$133$	5.31667	0.461014
$134$	6.57156	0.567696
$135$	0	0
$136$	−16.1836	−1.38773
$137$	5.53897	0.473226	0.236613	−	0.971604i	$-0.423963\pi$
$137$	5.53897	0.473226	0.236613	+	0.971604i	$0.423963\pi$
$138$	0	0
$139$	−0.0482918	−0.00409606	−0.00204803	−	0.999998i	$-0.500652\pi$
$139$	−0.0482918	−0.00409606	−0.00204803	+	0.999998i	$0.500652\pi$
$140$	−26.8399	−2.26838
$141$	0	0
$142$	12.8053	1.07459
$143$	−2.97097	−0.248445
$144$	0	0
$145$	3.69173	0.306581
$146$	−0.953941	−0.0789487
$147$	0	0
$148$	47.6094	3.91347
$149$	−6.09994	−0.499726	−0.249863	−	0.968281i	$-0.580386\pi$
$149$	−6.09994	−0.499726	−0.249863	+	0.968281i	$0.580386\pi$
$150$	0	0
$151$	−5.40099	−0.439526	−0.219763	−	0.975553i	$-0.570528\pi$
$151$	−5.40099	−0.439526	−0.219763	+	0.975553i	$0.570528\pi$
$152$	25.7474	2.08839
$153$	0	0
$154$	−16.7190	−1.34725
$155$	−33.1510	−2.66275
$156$	0	0
$157$	13.8301	1.10376	0.551882	−	0.833922i	$-0.313910\pi$
$157$	13.8301	1.10376	0.551882	+	0.833922i	$0.313910\pi$
$158$	−3.62879	−0.288691
$159$	0	0
$160$	−37.3745	−2.95472
$161$	1.51238	0.119192
$162$	0	0
$163$	11.4078	0.893527	0.446764	−	0.894652i	$-0.352577\pi$
$163$	11.4078	0.893527	0.446764	+	0.894652i	$0.352577\pi$
$164$	42.6890	3.33345
$165$	0	0
$166$	−0.882145	−0.0684677
$167$	−14.7788	−1.14362	−0.571808	−	0.820387i	$-0.693758\pi$
$167$	−14.7788	−1.14362	−0.571808	+	0.820387i	$0.693758\pi$
$168$	0	0
$169$	−12.5083	−0.962180
$170$	21.2831	1.63234
$171$	0	0
$172$	54.1537	4.12918
$173$	19.9280	1.51510	0.757549	−	0.652779i	$-0.226397\pi$
$173$	19.9280	1.51510	0.757549	+	0.652779i	$0.226397\pi$
$174$	0	0
$175$	13.0501	0.986492
$176$	−40.2294	−3.03241
$177$	0	0
$178$	16.5209	1.23829
$179$	7.90810	0.591079	0.295540	−	0.955331i	$-0.404501\pi$
$179$	7.90810	0.591079	0.295540	+	0.955331i	$0.404501\pi$
$180$	0	0
$181$	−19.7661	−1.46920	−0.734602	−	0.678498i	$-0.762631\pi$
$181$	−19.7661	−1.46920	−0.734602	+	0.678498i	$0.762631\pi$
$182$	2.76679	0.205088
$183$	0	0
$184$	7.32410	0.539940
$185$	−36.5622	−2.68810
$186$	0	0
$187$	9.36240	0.684646
$188$	43.5106	3.17334
$189$	0	0
$190$	−33.8605	−2.45650
$191$	−13.2004	−0.955148	−0.477574	−	0.878592i	$-0.658484\pi$
$191$	−13.2004	−0.955148	−0.477574	+	0.878592i	$0.658484\pi$
$192$	0	0
$193$	9.57445	0.689184	0.344592	−	0.938753i	$-0.388017\pi$
$193$	9.57445	0.689184	0.344592	+	0.938753i	$0.388017\pi$
$194$	−14.6283	−1.05025
$195$	0	0
$196$	−22.6549	−1.61821
$197$	9.61490	0.685033	0.342517	−	0.939512i	$-0.388721\pi$
$197$	9.61490	0.685033	0.342517	+	0.939512i	$0.388721\pi$
$198$	0	0
$199$	11.3326	0.803350	0.401675	−	0.915782i	$-0.368428\pi$
$199$	11.3326	0.803350	0.401675	+	0.915782i	$0.368428\pi$
$200$	63.1985	4.46881
$201$	0	0
$202$	−22.2074	−1.56251
$203$	−1.51238	−0.106148
$204$	0	0
$205$	−32.7835	−2.28970
$206$	−21.7573	−1.51591
$207$	0	0
$208$	6.65749	0.461614
$209$	−14.8952	−1.03032
$210$	0	0
$211$	−11.4631	−0.789154	−0.394577	−	0.918863i	$-0.629109\pi$
$211$	−11.4631	−0.789154	−0.394577	+	0.918863i	$0.629109\pi$
$212$	−21.5333	−1.47892
$213$	0	0
$214$	−17.0081	−1.16265
$215$	−41.5879	−2.83627
$216$	0	0
$217$	13.5808	0.921928
$218$	−24.7244	−1.67455
$219$	0	0
$220$	75.1944	5.06961
$221$	−1.54937	−0.104222
$222$	0	0
$223$	2.09705	0.140429	0.0702144	−	0.997532i	$-0.477632\pi$
$223$	2.09705	0.140429	0.0702144	+	0.997532i	$0.477632\pi$
$224$	15.3111	1.02302
$225$	0	0
$226$	−48.1752	−3.20457
$227$	25.0947	1.66560	0.832798	−	0.553578i	$-0.186738\pi$
$227$	25.0947	1.66560	0.832798	+	0.553578i	$0.186738\pi$
$228$	0	0
$229$	20.0430	1.32448	0.662238	−	0.749293i	$-0.269607\pi$
$229$	20.0430	1.32448	0.662238	+	0.749293i	$0.269607\pi$
$230$	−9.63193	−0.635110
$231$	0	0
$232$	−7.32410	−0.480851
$233$	−24.8577	−1.62848	−0.814240	−	0.580528i	$-0.802846\pi$
$233$	−24.8577	−1.62848	−0.814240	+	0.580528i	$0.802846\pi$
$234$	0	0
$235$	−33.4144	−2.17972
$236$	38.7985	2.52557
$237$	0	0
$238$	−8.71898	−0.565167
$239$	21.2735	1.37607	0.688036	−	0.725677i	$-0.258473\pi$
$239$	21.2735	1.37607	0.688036	+	0.725677i	$0.258473\pi$
$240$	0	0
$241$	−11.6991	−0.753604	−0.376802	−	0.926294i	$-0.622976\pi$
$241$	−11.6991	−0.753604	−0.376802	+	0.926294i	$0.622976\pi$
$242$	18.1401	1.16609
$243$	0	0
$244$	−2.89595	−0.185394
$245$	17.3981	1.11152
$246$	0	0
$247$	2.46497	0.156842
$248$	65.7690	4.17633
$249$	0	0
$250$	−34.9527	−2.21061
$251$	23.8704	1.50669	0.753345	−	0.657626i	$-0.228439\pi$
$251$	23.8704	1.50669	0.753345	+	0.657626i	$0.228439\pi$
$252$	0	0
$253$	−4.23707	−0.266382
$254$	−12.9421	−0.812059
$255$	0	0
$256$	−17.1367	−1.07105
$257$	−14.5546	−0.907889	−0.453945	−	0.891030i	$-0.649984\pi$
$257$	−14.5546	−0.907889	−0.453945	+	0.891030i	$0.649984\pi$
$258$	0	0
$259$	14.9783	0.930706
$260$	−12.4438	−0.771730
$261$	0	0
$262$	34.6527	2.14085
$263$	4.94078	0.304662	0.152331	−	0.988330i	$-0.451322\pi$
$263$	4.94078	0.304662	0.152331	+	0.988330i	$0.451322\pi$
$264$	0	0
$265$	16.5368	1.01584
$266$	13.8715	0.850516
$267$	0	0
$268$	12.1081	0.739619
$269$	20.1580	1.22906	0.614528	−	0.788895i	$-0.289346\pi$
$269$	20.1580	1.22906	0.614528	+	0.788895i	$0.289346\pi$
$270$	0	0
$271$	−12.7651	−0.775424	−0.387712	−	0.921781i	$-0.626734\pi$
$271$	−12.7651	−0.775424	−0.387712	+	0.921781i	$0.626734\pi$
$272$	−20.9797	−1.27208
$273$	0	0
$274$	14.4515	0.873046
$275$	−36.5610	−2.20471
$276$	0	0
$277$	−15.9763	−0.959925	−0.479963	−	0.877289i	$-0.659350\pi$
$277$	−15.9763	−0.959925	−0.479963	+	0.877289i	$0.659350\pi$
$278$	−0.125996	−0.00755675
$279$	0	0
$280$	−40.8925	−2.44380
$281$	16.3239	0.973803	0.486901	−	0.873457i	$-0.338127\pi$
$281$	16.3239	0.973803	0.486901	+	0.873457i	$0.338127\pi$
$282$	0	0
$283$	21.4250	1.27358	0.636791	−	0.771036i	$-0.280261\pi$
$283$	21.4250	1.27358	0.636791	+	0.771036i	$0.280261\pi$
$284$	23.5937	1.40003
$285$	0	0
$286$	−7.75142	−0.458351
$287$	13.4303	0.792765
$288$	0	0
$289$	−12.1175	−0.712793
$290$	9.63193	0.565606
$291$	0	0
$292$	−1.75763	−0.102858
$293$	30.6416	1.79010	0.895051	−	0.445963i	$-0.147139\pi$
$293$	30.6416	1.79010	0.895051	+	0.445963i	$0.147139\pi$
$294$	0	0
$295$	−29.7957	−1.73478
$296$	72.5365	4.21610
$297$	0	0
$298$	−15.9151	−0.921936
$299$	0.701184	0.0405505
$300$	0	0
$301$	17.0372	0.982007
$302$	−14.0915	−0.810874
$303$	0	0
$304$	33.3778	1.91435
$305$	2.22397	0.127344
$306$	0	0
$307$	−14.8878	−0.849692	−0.424846	−	0.905266i	$-0.639672\pi$
$307$	−14.8878	−0.849692	−0.424846	+	0.905266i	$0.639672\pi$
$308$	−30.8046	−1.75526
$309$	0	0
$310$	−86.4928	−4.91246
$311$	−5.96196	−0.338072	−0.169036	−	0.985610i	$-0.554065\pi$
$311$	−5.96196	−0.338072	−0.169036	+	0.985610i	$0.554065\pi$
$312$	0	0
$313$	3.19906	0.180822	0.0904108	−	0.995905i	$-0.471182\pi$
$313$	3.19906	0.180822	0.0904108	+	0.995905i	$0.471182\pi$
$314$	36.0836	2.03632
$315$	0	0
$316$	−6.68604	−0.376119
$317$	13.7326	0.771302	0.385651	−	0.922645i	$-0.373977\pi$
$317$	13.7326	0.771302	0.385651	+	0.922645i	$0.373977\pi$
$318$	0	0
$319$	4.23707	0.237230
$320$	−27.4091	−1.53221
$321$	0	0
$322$	3.94588	0.219895
$323$	−7.76786	−0.432215
$324$	0	0
$325$	6.05041	0.335616
$326$	29.7636	1.64845
$327$	0	0
$328$	65.0399	3.59123
$329$	13.6888	0.754687
$330$	0	0
$331$	26.8049	1.47333	0.736665	−	0.676258i	$-0.236400\pi$
$331$	26.8049	1.47333	0.736665	+	0.676258i	$0.236400\pi$
$332$	−1.62535	−0.0892026
$333$	0	0
$334$	−38.5587	−2.10984
$335$	−9.29852	−0.508033
$336$	0	0
$337$	−20.3252	−1.10719	−0.553593	−	0.832787i	$-0.686744\pi$
$337$	−20.3252	−1.10719	−0.553593	+	0.832787i	$0.686744\pi$
$338$	−32.6350	−1.77511
$339$	0	0
$340$	39.2141	2.12668
$341$	−38.0480	−2.06042
$342$	0	0
$343$	−17.7140	−0.956468
$344$	82.5072	4.44849
$345$	0	0
$346$	51.9933	2.79518
$347$	9.89884	0.531398	0.265699	−	0.964056i	$-0.414397\pi$
$347$	9.89884	0.531398	0.265699	+	0.964056i	$0.414397\pi$
$348$	0	0
$349$	2.78360	0.149003	0.0745015	−	0.997221i	$-0.476263\pi$
$349$	2.78360	0.149003	0.0745015	+	0.997221i	$0.476263\pi$
$350$	34.0484	1.81996
$351$	0	0
$352$	−42.8955	−2.28634
$353$	−21.6010	−1.14970	−0.574852	−	0.818257i	$-0.694940\pi$
$353$	−21.6010	−1.14970	−0.574852	+	0.818257i	$0.694940\pi$
$354$	0	0
$355$	−18.1190	−0.961657
$356$	30.4397	1.61330
$357$	0	0
$358$	20.6327	1.09047
$359$	21.1444	1.11596	0.557980	−	0.829855i	$-0.311577\pi$
$359$	21.1444	1.11596	0.557980	+	0.829855i	$0.311577\pi$
$360$	0	0
$361$	−6.64169	−0.349563
$362$	−51.5710	−2.71051
$363$	0	0
$364$	5.09780	0.267197
$365$	1.34979	0.0706514
$366$	0	0
$367$	−36.4666	−1.90354	−0.951771	−	0.306809i	$-0.900739\pi$
$367$	−36.4666	−1.90354	−0.951771	+	0.306809i	$0.900739\pi$
$368$	9.49464	0.494942
$369$	0	0
$370$	−95.3928	−4.95924
$371$	−6.77456	−0.351717
$372$	0	0
$373$	−37.1573	−1.92393	−0.961966	−	0.273168i	$-0.911929\pi$
$373$	−37.1573	−1.92393	−0.961966	+	0.273168i	$0.911929\pi$
$374$	24.4270	1.26309
$375$	0	0
$376$	66.2917	3.41873
$377$	−0.701184	−0.0361128
$378$	0	0
$379$	−16.1181	−0.827934	−0.413967	−	0.910292i	$-0.635857\pi$
$379$	−16.1181	−0.827934	−0.413967	+	0.910292i	$0.635857\pi$
$380$	−62.3878	−3.20043
$381$	0	0
$382$	−34.4406	−1.76214
$383$	−4.94872	−0.252868	−0.126434	−	0.991975i	$-0.540353\pi$
$383$	−4.94872	−0.252868	−0.126434	+	0.991975i	$0.540353\pi$
$384$	0	0
$385$	23.6567	1.20566
$386$	24.9803	1.27146
$387$	0	0
$388$	−26.9527	−1.36831
$389$	−30.5289	−1.54788	−0.773939	−	0.633261i	$-0.781716\pi$
$389$	−30.5289	−1.54788	−0.773939	+	0.633261i	$0.781716\pi$
$390$	0	0
$391$	−2.20964	−0.111746
$392$	−34.5164	−1.74334
$393$	0	0
$394$	25.0858	1.26381
$395$	5.13461	0.258350
$396$	0	0
$397$	19.6614	0.986777	0.493388	−	0.869809i	$-0.335758\pi$
$397$	19.6614	0.986777	0.493388	+	0.869809i	$0.335758\pi$
$398$	29.5675	1.48209
$399$	0	0
$400$	81.9277	4.09639
$401$	2.20598	0.110161	0.0550807	−	0.998482i	$-0.482458\pi$
$401$	2.20598	0.110161	0.0550807	+	0.998482i	$0.482458\pi$
$402$	0	0
$403$	6.29650	0.313651
$404$	−40.9171	−2.03570
$405$	0	0
$406$	−3.94588	−0.195831
$407$	−41.9631	−2.08004
$408$	0	0
$409$	22.8667	1.13068	0.565342	−	0.824857i	$-0.308744\pi$
$409$	22.8667	1.13068	0.565342	+	0.824857i	$0.308744\pi$
$410$	−85.5340	−4.22422
$411$	0	0
$412$	−40.0879	−1.97499
$413$	12.2063	0.600634
$414$	0	0
$415$	1.24820	0.0612719
$416$	7.09869	0.348042
$417$	0	0
$418$	−38.8623	−1.90082
$419$	−29.4988	−1.44111	−0.720556	−	0.693397i	$-0.756113\pi$
$419$	−29.4988	−1.44111	−0.720556	+	0.693397i	$0.756113\pi$
$420$	0	0
$421$	13.4004	0.653096	0.326548	−	0.945181i	$-0.394114\pi$
$421$	13.4004	0.653096	0.326548	+	0.945181i	$0.394114\pi$
$422$	−29.9080	−1.45590
$423$	0	0
$424$	−32.8077	−1.59328
$425$	−19.0666	−0.924868
$426$	0	0
$427$	−0.911087	−0.0440906
$428$	−31.3374	−1.51475
$429$	0	0
$430$	−108.505	−5.23259
$431$	17.1152	0.824411	0.412206	−	0.911091i	$-0.364758\pi$
$431$	17.1152	0.824411	0.412206	+	0.911091i	$0.364758\pi$
$432$	0	0
$433$	12.5925	0.605159	0.302579	−	0.953124i	$-0.402152\pi$
$433$	12.5925	0.605159	0.302579	+	0.953124i	$0.402152\pi$
$434$	35.4332	1.70085
$435$	0	0
$436$	−45.5547	−2.18168
$437$	3.51544	0.168166
$438$	0	0
$439$	−23.2289	−1.10866	−0.554328	−	0.832298i	$-0.687025\pi$
$439$	−23.2289	−1.10866	−0.554328	+	0.832298i	$0.687025\pi$
$440$	114.564	5.46164
$441$	0	0
$442$	−4.04238	−0.192277
$443$	37.1009	1.76272	0.881359	−	0.472447i	$-0.156629\pi$
$443$	37.1009	1.76272	0.881359	+	0.472447i	$0.156629\pi$
$444$	0	0
$445$	−23.3764	−1.10815
$446$	5.47133	0.259075
$447$	0	0
$448$	11.2286	0.530501
$449$	−14.6343	−0.690635	−0.345318	−	0.938486i	$-0.612229\pi$
$449$	−14.6343	−0.690635	−0.345318	+	0.938486i	$0.612229\pi$
$450$	0	0
$451$	−37.6262	−1.77175
$452$	−88.7628	−4.17505
$453$	0	0
$454$	65.4736	3.07283
$455$	−3.91491	−0.183534
$456$	0	0
$457$	−11.8792	−0.555686	−0.277843	−	0.960626i	$-0.589620\pi$
$457$	−11.8792	−0.555686	−0.277843	+	0.960626i	$0.589620\pi$
$458$	52.2933	2.44350
$459$	0	0
$460$	−17.7468	−0.827449
$461$	33.7228	1.57063	0.785314	−	0.619097i	$-0.212501\pi$
$461$	33.7228	1.57063	0.785314	+	0.619097i	$0.212501\pi$
$462$	0	0
$463$	23.9819	1.11453	0.557267	−	0.830333i	$-0.311850\pi$
$463$	23.9819	1.11453	0.557267	+	0.830333i	$0.311850\pi$
$464$	−9.49464	−0.440778
$465$	0	0
$466$	−64.8551	−3.00436
$467$	7.83355	0.362494	0.181247	−	0.983438i	$-0.441987\pi$
$467$	7.83355	0.362494	0.181247	+	0.983438i	$0.441987\pi$
$468$	0	0
$469$	3.80930	0.175897
$470$	−87.1802	−4.02132
$471$	0	0
$472$	59.1125	2.72087
$473$	−47.7313	−2.19469
$474$	0	0
$475$	30.3342	1.39183
$476$	−16.0647	−0.736324
$477$	0	0
$478$	55.5039	2.53869
$479$	−37.0307	−1.69197	−0.845987	−	0.533203i	$-0.820988\pi$
$479$	−37.0307	−1.69197	−0.845987	+	0.533203i	$0.820988\pi$
$480$	0	0
$481$	6.94440	0.316637
$482$	−30.5236	−1.39031
$483$	0	0
$484$	33.4231	1.51923
$485$	20.6986	0.939874
$486$	0	0
$487$	−4.59789	−0.208350	−0.104175	−	0.994559i	$-0.533220\pi$
$487$	−4.59789	−0.208350	−0.104175	+	0.994559i	$0.533220\pi$
$488$	−4.41219	−0.199730
$489$	0	0
$490$	45.3925	2.05063
$491$	15.4946	0.699263	0.349631	−	0.936887i	$-0.386307\pi$
$491$	15.4946	0.699263	0.349631	+	0.936887i	$0.386307\pi$
$492$	0	0
$493$	2.20964	0.0995172
$494$	6.43125	0.289356
$495$	0	0
$496$	85.2600	3.82829
$497$	7.42275	0.332956
$498$	0	0
$499$	−20.3943	−0.912973	−0.456486	−	0.889730i	$-0.650892\pi$
$499$	−20.3943	−0.912973	−0.456486	+	0.889730i	$0.650892\pi$
$500$	−64.4003	−2.88007
$501$	0	0
$502$	62.2794	2.77966
$503$	−7.23057	−0.322395	−0.161198	−	0.986922i	$-0.551536\pi$
$503$	−7.23057	−0.322395	−0.161198	+	0.986922i	$0.551536\pi$
$504$	0	0
$505$	31.4227	1.39829
$506$	−11.0548	−0.491444
$507$	0	0
$508$	−23.8458	−1.05798
$509$	1.82765	0.0810092	0.0405046	−	0.999179i	$-0.487103\pi$
$509$	1.82765	0.0810092	0.0405046	+	0.999179i	$0.487103\pi$
$510$	0	0
$511$	−0.552965	−0.0244617
$512$	−42.9570	−1.89845
$513$	0	0
$514$	−37.9737	−1.67495
$515$	30.7859	1.35659
$516$	0	0
$517$	−38.3504	−1.68665
$518$	39.0793	1.71704
$519$	0	0
$520$	−18.9590	−0.831408
$521$	−40.4816	−1.77353	−0.886766	−	0.462219i	$-0.847053\pi$
$521$	−40.4816	−1.77353	−0.886766	+	0.462219i	$0.847053\pi$
$522$	0	0
$523$	19.5288	0.853935	0.426968	−	0.904267i	$-0.359582\pi$
$523$	19.5288	0.853935	0.426968	+	0.904267i	$0.359582\pi$
$524$	63.8475	2.78919
$525$	0	0
$526$	12.8908	0.562065
$527$	−19.8421	−0.864337
$528$	0	0
$529$	1.00000	0.0434783
$530$	43.1453	1.87411
$531$	0	0
$532$	25.5582	1.10809
$533$	6.22670	0.269708
$534$	0	0
$535$	24.0659	1.04046
$536$	18.4476	0.796813
$537$	0	0
$538$	52.5934	2.26746
$539$	19.9681	0.860087
$540$	0	0
$541$	40.4363	1.73849	0.869247	−	0.494378i	$-0.164604\pi$
$541$	40.4363	1.73849	0.869247	+	0.494378i	$0.164604\pi$
$542$	−33.3049	−1.43057
$543$	0	0
$544$	−22.3701	−0.959110
$545$	34.9842	1.49856
$546$	0	0
$547$	−45.5761	−1.94869	−0.974346	−	0.225054i	$-0.927744\pi$
$547$	−45.5761	−1.94869	−0.974346	+	0.225054i	$0.927744\pi$
$548$	26.6268	1.13744
$549$	0	0
$550$	−95.3897	−4.06743
$551$	−3.51544	−0.149763
$552$	0	0
$553$	−2.10348	−0.0894491
$554$	−41.6832	−1.77095
$555$	0	0
$556$	−0.232148	−0.00984525
$557$	−16.8910	−0.715693	−0.357847	−	0.933780i	$-0.616489\pi$
$557$	−16.8910	−0.715693	−0.357847	+	0.933780i	$0.616489\pi$
$558$	0	0
$559$	7.89896	0.334090
$560$	−53.0113	−2.24013
$561$	0	0
$562$	42.5900	1.79655
$563$	23.9174	1.00800	0.503998	−	0.863705i	$-0.331862\pi$
$563$	23.9174	1.00800	0.503998	+	0.863705i	$0.331862\pi$
$564$	0	0
$565$	68.1663	2.86778
$566$	55.8990	2.34961
$567$	0	0
$568$	35.9467	1.50829
$569$	−2.67119	−0.111982	−0.0559912	−	0.998431i	$-0.517832\pi$
$569$	−2.67119	−0.111982	−0.0559912	+	0.998431i	$0.517832\pi$
$570$	0	0
$571$	17.8008	0.744942	0.372471	−	0.928044i	$-0.378511\pi$
$571$	17.8008	0.744942	0.372471	+	0.928044i	$0.378511\pi$
$572$	−14.2820	−0.597159
$573$	0	0
$574$	35.0404	1.46256
$575$	8.62884	0.359848
$576$	0	0
$577$	43.4020	1.80685	0.903424	−	0.428747i	$-0.141045\pi$
$577$	43.4020	1.80685	0.903424	+	0.428747i	$0.141045\pi$
$578$	−31.6152	−1.31502
$579$	0	0
$580$	17.7468	0.736896
$581$	−0.511348	−0.0212143
$582$	0	0
$583$	18.9796	0.786053
$584$	−2.67789	−0.110812
$585$	0	0
$586$	79.9458	3.30253
$587$	−3.22180	−0.132978	−0.0664889	−	0.997787i	$-0.521180\pi$
$587$	−3.22180	−0.132978	−0.0664889	+	0.997787i	$0.521180\pi$
$588$	0	0
$589$	31.5679	1.30073
$590$	−77.7388	−3.20046
$591$	0	0
$592$	94.0331	3.86474
$593$	−15.5297	−0.637728	−0.318864	−	0.947800i	$-0.603301\pi$
$593$	−15.5297	−0.637728	−0.318864	+	0.947800i	$0.603301\pi$
$594$	0	0
$595$	12.3370	0.505770
$596$	−29.3235	−1.20114
$597$	0	0
$598$	1.82943	0.0748110
$599$	−0.373746	−0.0152709	−0.00763543	−	0.999971i	$-0.502430\pi$
$599$	−0.373746	−0.0152709	−0.00763543	+	0.999971i	$0.502430\pi$
$600$	0	0
$601$	−46.0054	−1.87660	−0.938300	−	0.345823i	$-0.887600\pi$
$601$	−46.0054	−1.87660	−0.938300	+	0.345823i	$0.887600\pi$
$602$	44.4510	1.81169
$603$	0	0
$604$	−25.9635	−1.05644
$605$	−25.6676	−1.04354
$606$	0	0
$607$	24.7346	1.00395	0.501973	−	0.864883i	$-0.332608\pi$
$607$	24.7346	1.00395	0.501973	+	0.864883i	$0.332608\pi$
$608$	35.5898	1.44336
$609$	0	0
$610$	5.80247	0.234935
$611$	6.34654	0.256754
$612$	0	0
$613$	13.2311	0.534398	0.267199	−	0.963641i	$-0.413902\pi$
$613$	13.2311	0.534398	0.267199	+	0.963641i	$0.413902\pi$
$614$	−38.8432	−1.56758
$615$	0	0
$616$	−46.9332	−1.89099
$617$	−14.7372	−0.593296	−0.296648	−	0.954987i	$-0.595869\pi$
$617$	−14.7372	−0.593296	−0.296648	+	0.954987i	$0.595869\pi$
$618$	0	0
$619$	29.8824	1.20107	0.600537	−	0.799597i	$-0.294953\pi$
$619$	29.8824	1.20107	0.600537	+	0.799597i	$0.294953\pi$
$620$	−159.363	−6.40016
$621$	0	0
$622$	−15.5551	−0.623702
$623$	9.57655	0.383676
$624$	0	0
$625$	6.31270	0.252508
$626$	8.34653	0.333595
$627$	0	0
$628$	66.4839	2.65300
$629$	−21.8839	−0.872567
$630$	0	0
$631$	15.2973	0.608977	0.304488	−	0.952516i	$-0.401515\pi$
$631$	15.2973	0.608977	0.304488	+	0.952516i	$0.401515\pi$
$632$	−10.1867	−0.405204
$633$	0	0
$634$	35.8293	1.42296
$635$	18.3126	0.726713
$636$	0	0
$637$	−3.30448	−0.130928
$638$	11.0548	0.437662
$639$	0	0
$640$	3.23720	0.127961
$641$	6.11919	0.241694	0.120847	−	0.992671i	$-0.461439\pi$
$641$	6.11919	0.241694	0.120847	+	0.992671i	$0.461439\pi$
$642$	0	0
$643$	−3.52335	−0.138947	−0.0694736	−	0.997584i	$-0.522132\pi$
$643$	−3.52335	−0.138947	−0.0694736	+	0.997584i	$0.522132\pi$
$644$	7.27027	0.286489
$645$	0	0
$646$	−20.2668	−0.797386
$647$	−27.5242	−1.08209	−0.541043	−	0.840995i	$-0.681970\pi$
$647$	−27.5242	−1.08209	−0.541043	+	0.840995i	$0.681970\pi$
$648$	0	0
$649$	−34.1972	−1.34236
$650$	15.7859	0.619173
$651$	0	0
$652$	54.8393	2.14767
$653$	−19.7095	−0.771291	−0.385645	−	0.922647i	$-0.626021\pi$
$653$	−19.7095	−0.771291	−0.385645	+	0.922647i	$0.626021\pi$
$654$	0	0
$655$	−49.0324	−1.91585
$656$	84.3148	3.29194
$657$	0	0
$658$	35.7148	1.39231
$659$	−21.0981	−0.821865	−0.410933	−	0.911666i	$-0.634797\pi$
$659$	−21.0981	−0.821865	−0.410933	+	0.911666i	$0.634797\pi$
$660$	0	0
$661$	18.4792	0.718756	0.359378	−	0.933192i	$-0.382989\pi$
$661$	18.4792	0.718756	0.359378	+	0.933192i	$0.382989\pi$
$662$	69.9355	2.71812
$663$	0	0
$664$	−2.47634	−0.0961007
$665$	−19.6277	−0.761129
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−71.0443	−2.74879
$669$	0	0
$670$	−24.2604	−0.937261
$671$	2.55250	0.0985380
$672$	0	0
$673$	−32.0228	−1.23439	−0.617194	−	0.786811i	$-0.711731\pi$
$673$	−32.0228	−1.23439	−0.617194	+	0.786811i	$0.711731\pi$
$674$	−53.0297	−2.04263
$675$	0	0
$676$	−60.1299	−2.31269
$677$	−34.0940	−1.31034	−0.655170	−	0.755482i	$-0.727403\pi$
$677$	−34.0940	−1.31034	−0.655170	+	0.755482i	$0.727403\pi$
$678$	0	0
$679$	−8.47952	−0.325414
$680$	59.7455	2.29114
$681$	0	0
$682$	−99.2695	−3.80123
$683$	7.03001	0.268996	0.134498	−	0.990914i	$-0.457058\pi$
$683$	7.03001	0.268996	0.134498	+	0.990914i	$0.457058\pi$
$684$	0	0
$685$	−20.4484	−0.781291
$686$	−46.2170	−1.76457
$687$	0	0
$688$	106.959	4.07776
$689$	−3.14089	−0.119658
$690$	0	0
$691$	6.46982	0.246124	0.123062	−	0.992399i	$-0.460729\pi$
$691$	6.46982	0.246124	0.123062	+	0.992399i	$0.460729\pi$
$692$	95.7975	3.64168
$693$	0	0
$694$	25.8266	0.980366
$695$	0.178280	0.00676255
$696$	0	0
$697$	−19.6222	−0.743243
$698$	7.26259	0.274893
$699$	0	0
$700$	62.7340	2.37112
$701$	23.7050	0.895327	0.447664	−	0.894202i	$-0.352256\pi$
$701$	23.7050	0.895327	0.447664	+	0.894202i	$0.352256\pi$
$702$	0	0
$703$	34.8163	1.31312
$704$	−31.4579	−1.18562
$705$	0	0
$706$	−56.3582	−2.12107
$707$	−12.8728	−0.484132
$708$	0	0
$709$	−40.1025	−1.50608	−0.753041	−	0.657974i	$-0.771414\pi$
$709$	−40.1025	−1.50608	−0.753041	+	0.657974i	$0.771414\pi$
$710$	−47.2735	−1.77414
$711$	0	0
$712$	46.3771	1.73805
$713$	8.97980	0.336296
$714$	0	0
$715$	10.9680	0.410180
$716$	38.0157	1.42071
$717$	0	0
$718$	55.1670	2.05881
$719$	9.63733	0.359412	0.179706	−	0.983720i	$-0.442485\pi$
$719$	9.63733	0.359412	0.179706	+	0.983720i	$0.442485\pi$
$720$	0	0
$721$	−12.6120	−0.469694
$722$	−17.3285	−0.644902
$723$	0	0
$724$	−95.0194	−3.53137
$725$	−8.62884	−0.320467
$726$	0	0
$727$	−2.76929	−0.102707	−0.0513537	−	0.998681i	$-0.516354\pi$
$727$	−2.76929	−0.102707	−0.0513537	+	0.998681i	$0.516354\pi$
$728$	7.76688	0.287860
$729$	0	0
$730$	3.52169	0.130344
$731$	−24.8920	−0.920663
$732$	0	0
$733$	−16.2252	−0.599290	−0.299645	−	0.954051i	$-0.596868\pi$
$733$	−16.2252	−0.599290	−0.299645	+	0.954051i	$0.596868\pi$
$734$	−95.1435	−3.51181
$735$	0	0
$736$	10.1239	0.373170
$737$	−10.6721	−0.393112
$738$	0	0
$739$	−49.7801	−1.83119	−0.915595	−	0.402101i	$-0.868280\pi$
$739$	−49.7801	−1.83119	−0.915595	+	0.402101i	$0.868280\pi$
$740$	−175.761	−6.46110
$741$	0	0
$742$	−17.6752	−0.648877
$743$	14.2271	0.521943	0.260972	−	0.965346i	$-0.415957\pi$
$743$	14.2271	0.521943	0.260972	+	0.965346i	$0.415957\pi$
$744$	0	0
$745$	22.5193	0.825043
$746$	−96.9456	−3.54943
$747$	0	0
$748$	45.0068	1.64561
$749$	−9.85900	−0.360240
$750$	0	0
$751$	−25.0590	−0.914415	−0.457208	−	0.889360i	$-0.651150\pi$
$751$	−25.0590	−0.914415	−0.457208	+	0.889360i	$0.651150\pi$
$752$	85.9376	3.13382
$753$	0	0
$754$	−1.82943	−0.0666239
$755$	19.9390	0.725653
$756$	0	0
$757$	−3.05502	−0.111036	−0.0555182	−	0.998458i	$-0.517681\pi$
$757$	−3.05502	−0.111036	−0.0555182	+	0.998458i	$0.517681\pi$
$758$	−42.0532	−1.52744
$759$	0	0
$760$	−95.0525	−3.44792
$761$	−2.61322	−0.0947292	−0.0473646	−	0.998878i	$-0.515082\pi$
$761$	−2.61322	−0.0947292	−0.0473646	+	0.998878i	$0.515082\pi$
$762$	0	0
$763$	−14.3319	−0.518849
$764$	−63.4567	−2.29578
$765$	0	0
$766$	−12.9115	−0.466511
$767$	5.65923	0.204343
$768$	0	0
$769$	22.1751	0.799656	0.399828	−	0.916590i	$-0.369070\pi$
$769$	22.1751	0.799656	0.399828	+	0.916590i	$0.369070\pi$
$770$	61.7218	2.22430
$771$	0	0
$772$	46.0261	1.65652
$773$	11.3168	0.407038	0.203519	−	0.979071i	$-0.434762\pi$
$773$	11.3168	0.407038	0.203519	+	0.979071i	$0.434762\pi$
$774$	0	0
$775$	77.4853	2.78335
$776$	−41.0644	−1.47413
$777$	0	0
$778$	−79.6517	−2.85565
$779$	31.2180	1.11850
$780$	0	0
$781$	−20.7955	−0.744123
$782$	−5.76508	−0.206159
$783$	0	0
$784$	−44.7455	−1.59805
$785$	−51.0570	−1.82230
$786$	0	0
$787$	37.1389	1.32386	0.661929	−	0.749567i	$-0.269738\pi$
$787$	37.1389	1.32386	0.661929	+	0.749567i	$0.269738\pi$
$788$	46.2206	1.64654
$789$	0	0
$790$	13.3965	0.476626
$791$	−27.9255	−0.992915
$792$	0	0
$793$	−0.422408	−0.0150001
$794$	51.2977	1.82049
$795$	0	0
$796$	54.4781	1.93093
$797$	−43.9748	−1.55767	−0.778834	−	0.627230i	$-0.784188\pi$
$797$	−43.9748	−1.55767	−0.778834	+	0.627230i	$0.784188\pi$
$798$	0	0
$799$	−19.9998	−0.707543
$800$	87.3572	3.08854
$801$	0	0
$802$	5.75553	0.203235
$803$	1.54918	0.0546695
$804$	0	0
$805$	−5.58328	−0.196785
$806$	16.4279	0.578649
$807$	0	0
$808$	−62.3402	−2.19312
$809$	41.4248	1.45642	0.728209	−	0.685355i	$-0.240353\pi$
$809$	41.4248	1.45642	0.728209	+	0.685355i	$0.240353\pi$
$810$	0	0
$811$	−10.3238	−0.362517	−0.181259	−	0.983435i	$-0.558017\pi$
$811$	−10.3238	−0.362517	−0.181259	+	0.983435i	$0.558017\pi$
$812$	−7.27027	−0.255137
$813$	0	0
$814$	−109.484	−3.83742
$815$	−42.1144	−1.47520
$816$	0	0
$817$	39.6020	1.38550
$818$	59.6604	2.08598
$819$	0	0
$820$	−157.596	−5.50350
$821$	−51.6597	−1.80294	−0.901468	−	0.432847i	$-0.857509\pi$
$821$	−51.6597	−1.80294	−0.901468	+	0.432847i	$0.857509\pi$
$822$	0	0
$823$	−40.6980	−1.41864	−0.709321	−	0.704885i	$-0.750999\pi$
$823$	−40.6980	−1.41864	−0.709321	+	0.704885i	$0.750999\pi$
$824$	−61.0769	−2.12771
$825$	0	0
$826$	31.8470	1.10810
$827$	50.8508	1.76825	0.884127	−	0.467246i	$-0.154754\pi$
$827$	50.8508	1.76825	0.884127	+	0.467246i	$0.154754\pi$
$828$	0	0
$829$	2.48767	0.0864002	0.0432001	−	0.999066i	$-0.486245\pi$
$829$	2.48767	0.0864002	0.0432001	+	0.999066i	$0.486245\pi$
$830$	3.25664	0.113040
$831$	0	0
$832$	5.20592	0.180483
$833$	10.4134	0.360803
$834$	0	0
$835$	54.5592	1.88810
$836$	−71.6037	−2.47647
$837$	0	0
$838$	−76.9642	−2.65868
$839$	−24.0680	−0.830920	−0.415460	−	0.909611i	$-0.636379\pi$
$839$	−24.0680	−0.830920	−0.415460	+	0.909611i	$0.636379\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	34.9625	1.20489
$843$	0	0
$844$	−55.1053	−1.89680
$845$	46.1774	1.58855
$846$	0	0
$847$	10.5152	0.361306
$848$	−42.5304	−1.46050
$849$	0	0
$850$	−49.7460	−1.70627
$851$	9.90381	0.339498
$852$	0	0
$853$	−29.3789	−1.00591	−0.502957	−	0.864311i	$-0.667755\pi$
$853$	−29.3789	−1.00591	−0.502957	+	0.864311i	$0.667755\pi$
$854$	−2.37708	−0.0813420
$855$	0	0
$856$	−47.7449	−1.63189
$857$	−36.0521	−1.23152	−0.615759	−	0.787935i	$-0.711150\pi$
$857$	−36.0521	−1.23152	−0.615759	+	0.787935i	$0.711150\pi$
$858$	0	0
$859$	1.20772	0.0412069	0.0206034	−	0.999788i	$-0.493441\pi$
$859$	1.20772	0.0412069	0.0206034	+	0.999788i	$0.493441\pi$
$860$	−199.921	−6.81724
$861$	0	0
$862$	44.6546	1.52094
$863$	32.7886	1.11614	0.558068	−	0.829795i	$-0.311543\pi$
$863$	32.7886	1.11614	0.558068	+	0.829795i	$0.311543\pi$
$864$	0	0
$865$	−73.5687	−2.50141
$866$	32.8547	1.11645
$867$	0	0
$868$	65.2856	2.21594
$869$	5.89310	0.199910
$870$	0	0
$871$	1.76611	0.0598422
$872$	−69.4060	−2.35038
$873$	0	0
$874$	9.17198	0.310247
$875$	−20.2608	−0.684942
$876$	0	0
$877$	−21.3463	−0.720814	−0.360407	−	0.932795i	$-0.617362\pi$
$877$	−21.3463	−0.720814	−0.360407	+	0.932795i	$0.617362\pi$
$878$	−60.6056	−2.04534
$879$	0	0
$880$	148.516	5.00648
$881$	9.84842	0.331802	0.165901	−	0.986142i	$-0.446947\pi$
$881$	9.84842	0.331802	0.165901	+	0.986142i	$0.446947\pi$
$882$	0	0
$883$	34.9158	1.17501	0.587505	−	0.809221i	$-0.300110\pi$
$883$	34.9158	1.17501	0.587505	+	0.809221i	$0.300110\pi$
$884$	−7.44808	−0.250506
$885$	0	0
$886$	96.7984	3.25201
$887$	0.630213	0.0211605	0.0105802	−	0.999944i	$-0.496632\pi$
$887$	0.630213	0.0211605	0.0105802	+	0.999944i	$0.496632\pi$
$888$	0	0
$889$	−7.50206	−0.251611
$890$	−60.9905	−2.04441
$891$	0	0
$892$	10.0809	0.337534
$893$	31.8188	1.06478
$894$	0	0
$895$	−29.1945	−0.975866
$896$	−1.32617	−0.0443043
$897$	0	0
$898$	−38.1817	−1.27414
$899$	−8.97980	−0.299493
$900$	0	0
$901$	9.89788	0.329746
$902$	−98.1691	−3.26867
$903$	0	0
$904$	−135.237	−4.49791
$905$	72.9711	2.42564
$906$	0	0
$907$	38.0210	1.26247	0.631233	−	0.775594i	$-0.282549\pi$
$907$	38.0210	1.26247	0.631233	+	0.775594i	$0.282549\pi$
$908$	120.635	4.00341
$909$	0	0
$910$	−10.2142	−0.338598
$911$	28.6837	0.950334	0.475167	−	0.879896i	$-0.342388\pi$
$911$	28.6837	0.950334	0.475167	+	0.879896i	$0.342388\pi$
$912$	0	0
$913$	1.43259	0.0474118
$914$	−30.9936	−1.02518
$915$	0	0
$916$	96.3502	3.18350
$917$	20.0869	0.663329
$918$	0	0
$919$	−39.2719	−1.29546	−0.647730	−	0.761870i	$-0.724281\pi$
$919$	−39.2719	−1.29546	−0.647730	+	0.761870i	$0.724281\pi$
$920$	−27.0386	−0.891436
$921$	0	0
$922$	87.9848	2.89763
$923$	3.44141	0.113276
$924$	0	0
$925$	85.4584	2.80986
$926$	62.5702	2.05619
$927$	0	0
$928$	−10.1239	−0.332332
$929$	−4.31768	−0.141658	−0.0708292	−	0.997488i	$-0.522565\pi$
$929$	−4.31768	−0.141658	−0.0708292	+	0.997488i	$0.522565\pi$
$930$	0	0
$931$	−16.5673	−0.542970
$932$	−119.495	−3.91420
$933$	0	0
$934$	20.4382	0.668758
$935$	−34.5634	−1.13034
$936$	0	0
$937$	−52.6718	−1.72071	−0.860357	−	0.509692i	$-0.829759\pi$
$937$	−52.6718	−1.72071	−0.860357	+	0.509692i	$0.829759\pi$
$938$	9.93867	0.324509
$939$	0	0
$940$	−160.629	−5.23915
$941$	−42.8842	−1.39799	−0.698993	−	0.715129i	$-0.746368\pi$
$941$	−42.8842	−1.39799	−0.698993	+	0.715129i	$0.746368\pi$
$942$	0	0
$943$	8.88026	0.289181
$944$	76.6308	2.49412
$945$	0	0
$946$	−124.534	−4.04894
$947$	2.44951	0.0795985	0.0397992	−	0.999208i	$-0.487328\pi$
$947$	2.44951	0.0795985	0.0397992	+	0.999208i	$0.487328\pi$
$948$	0	0
$949$	−0.256372	−0.00832218
$950$	79.1436	2.56776
$951$	0	0
$952$	−24.4758	−0.793264
$953$	−56.1592	−1.81917	−0.909587	−	0.415514i	$-0.863602\pi$
$953$	−56.1592	−1.81917	−0.909587	+	0.415514i	$0.863602\pi$
$954$	0	0
$955$	48.7323	1.57694
$956$	102.266	3.30751
$957$	0	0
$958$	−96.6151	−3.12149
$959$	8.37701	0.270508
$960$	0	0
$961$	49.6368	1.60119
$962$	18.1183	0.584159
$963$	0	0
$964$	−56.2396	−1.81136
$965$	−35.3463	−1.13784
$966$	0	0
$967$	4.76873	0.153352	0.0766759	−	0.997056i	$-0.475569\pi$
$967$	4.76873	0.153352	0.0766759	+	0.997056i	$0.475569\pi$
$968$	50.9226	1.63672
$969$	0	0
$970$	54.0038	1.73396
$971$	−3.50955	−0.112627	−0.0563135	−	0.998413i	$-0.517935\pi$
$971$	−3.50955	−0.112627	−0.0563135	+	0.998413i	$0.517935\pi$
$972$	0	0
$973$	−0.0730355	−0.00234141
$974$	−11.9962	−0.384382
$975$	0	0
$976$	−5.71977	−0.183085
$977$	−9.87694	−0.315991	−0.157996	−	0.987440i	$-0.550503\pi$
$977$	−9.87694	−0.315991	−0.157996	+	0.987440i	$0.550503\pi$
$978$	0	0
$979$	−26.8296	−0.857478
$980$	83.6356	2.67164
$981$	0	0
$982$	40.4264	1.29006
$983$	−33.0274	−1.05341	−0.526705	−	0.850048i	$-0.676573\pi$
$983$	−33.0274	−1.05341	−0.526705	+	0.850048i	$0.676573\pi$
$984$	0	0
$985$	−35.4956	−1.13098
$986$	5.76508	0.183598
$987$	0	0
$988$	11.8496	0.376985
$989$	11.2652	0.358211
$990$	0	0
$991$	44.4376	1.41161	0.705803	−	0.708408i	$-0.250586\pi$
$991$	44.4376	1.41161	0.705803	+	0.708408i	$0.250586\pi$
$992$	90.9103	2.88640
$993$	0	0
$994$	19.3664	0.614264
$995$	−41.8370	−1.32632
$996$	0	0
$997$	29.6782	0.939919	0.469960	−	0.882688i	$-0.344268\pi$
$997$	29.6782	0.939919	0.469960	+	0.882688i	$0.344268\pi$
$998$	−53.2098	−1.68433
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.v.1.29	✓	30
3.2	odd	2		6003.2.a.w.1.2	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.v.1.29	✓	30	1.1	even	1	trivial
6003.2.a.w.1.2	yes	30	3.2	odd	2

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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q+2.60906 q^{2} +4.80718 q^{4} -3.69173 q^{5} +1.51238 q^{7} +7.32410 q^{8} -9.63193 q^{10} -4.23707 q^{11} +0.701184 q^{13} +3.94588 q^{14} +9.49464 q^{16} -2.20964 q^{17} +3.51544 q^{19} -17.7468 q^{20} -11.0548 q^{22} +1.00000 q^{23} +8.62884 q^{25} +1.82943 q^{26} +7.27027 q^{28} -1.00000 q^{29} +8.97980 q^{31} +10.1239 q^{32} -5.76508 q^{34} -5.58328 q^{35} +9.90381 q^{37} +9.17198 q^{38} -27.0386 q^{40} +8.88026 q^{41} +11.2652 q^{43} -20.3684 q^{44} +2.60906 q^{46} +9.05117 q^{47} -4.71272 q^{49} +22.5131 q^{50} +3.37072 q^{52} -4.47941 q^{53} +15.6421 q^{55} +11.0768 q^{56} -2.60906 q^{58} +8.07095 q^{59} -0.602421 q^{61} +23.4288 q^{62} +7.42446 q^{64} -2.58858 q^{65} +2.51875 q^{67} -10.6221 q^{68} -14.5671 q^{70} +4.90800 q^{71} -0.365627 q^{73} +25.8396 q^{74} +16.8994 q^{76} -6.40805 q^{77} -1.39084 q^{79} -35.0516 q^{80} +23.1691 q^{82} -0.338109 q^{83} +8.15739 q^{85} +29.3915 q^{86} -31.0327 q^{88} +6.33212 q^{89} +1.06046 q^{91} +4.80718 q^{92} +23.6150 q^{94} -12.9780 q^{95} -5.60675 q^{97} -12.2957 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8} + 8 q^{10} + 36 q^{13} - 7 q^{14} + 47 q^{16} - 18 q^{17} + 16 q^{19} + 25 q^{22} + 30 q^{23} + 56 q^{25} - 11 q^{26} + 27 q^{28} - 30 q^{29} + 14 q^{31} + 7 q^{32}+ \cdots - 8 q^{98}+O(q^{100}) \) 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8 + 8 * q^10 + 36 * q^13 - 7 * q^14 + 47 * q^16 - 18 * q^17 + 16 * q^19 + 25 * q^22 + 30 * q^23 + 56 * q^25 - 11 * q^26 + 27 * q^28 - 30 * q^29 + 14 * q^31 + 7 * q^32 + 3 * q^34 + 22 * q^35 + 40 * q^37 - 6 * q^38 + 30 * q^40 - 14 * q^41 + 34 * q^43 - 5 * q^44 - q^46 + 2 * q^47 + 74 * q^49 + 21 * q^50 + 71 * q^52 - 16 * q^53 + 22 * q^55 - 14 * q^56 + q^58 + 32 * q^59 + 46 * q^61 - 20 * q^62 + 68 * q^64 - 12 * q^65 + 14 * q^67 - 27 * q^68 + 32 * q^71 + 50 * q^73 + 26 * q^74 + 56 * q^76 - 34 * q^77 + 16 * q^79 - 2 * q^80 + 38 * q^82 + 14 * q^83 + 38 * q^85 - 10 * q^86 + 40 * q^88 + 2 * q^89 + 32 * q^91 + 37 * q^92 + 29 * q^94 + 28 * q^95 + 56 * q^97 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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