LMFDB - Embedded newform 6003.2.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.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:	$47.9341963334$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-2.31658 q^{2} +3.36653 q^{4} +2.07685 q^{5} +1.30399 q^{7} -3.16567 q^{8} +O(q^{10})$	\(q-2.31658 q^{2} +3.36653 q^{4} +2.07685 q^{5} +1.30399 q^{7} -3.16567 q^{8} -4.81118 q^{10} +0.169489 q^{11} +0.705806 q^{13} -3.02080 q^{14} +0.600462 q^{16} +2.81253 q^{17} -7.73667 q^{19} +6.99178 q^{20} -0.392634 q^{22} -1.00000 q^{23} -0.686694 q^{25} -1.63505 q^{26} +4.38993 q^{28} -1.00000 q^{29} -4.11205 q^{31} +4.94033 q^{32} -6.51545 q^{34} +2.70820 q^{35} +4.93337 q^{37} +17.9226 q^{38} -6.57462 q^{40} +2.38205 q^{41} -1.36487 q^{43} +0.570590 q^{44} +2.31658 q^{46} +8.06261 q^{47} -5.29960 q^{49} +1.59078 q^{50} +2.37612 q^{52} -10.8473 q^{53} +0.352003 q^{55} -4.12802 q^{56} +2.31658 q^{58} -5.91579 q^{59} -2.67059 q^{61} +9.52588 q^{62} -12.6456 q^{64} +1.46585 q^{65} -4.10168 q^{67} +9.46847 q^{68} -6.27375 q^{70} -2.86912 q^{71} -9.80599 q^{73} -11.4285 q^{74} -26.0457 q^{76} +0.221013 q^{77} -3.16854 q^{79} +1.24707 q^{80} -5.51821 q^{82} +4.01310 q^{83} +5.84121 q^{85} +3.16183 q^{86} -0.536547 q^{88} +10.7736 q^{89} +0.920367 q^{91} -3.36653 q^{92} -18.6777 q^{94} -16.0679 q^{95} -8.28631 q^{97} +12.2769 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8}+O(q^{10}) $ 22 * q + 3 * q^2 + 17 * q^4 - 6 * q^7 + 6 * q^8	$ 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8} - 12 q^{10} - 28 q^{13} + q^{14} + 3 q^{16} + 10 q^{17} - 8 q^{19} - 11 q^{22} - 22 q^{23} - 11 q^{26} - 21 q^{28} - 22 q^{29} - 18 q^{31} - 5 q^{32} - 33 q^{34} - 2 q^{35} - 28 q^{37} - 14 q^{38} - 30 q^{40} + 10 q^{41} - 14 q^{43} - 37 q^{44} - 3 q^{46} + 18 q^{47} + 2 q^{49} - 7 q^{50} - 57 q^{52} - 20 q^{53} - 42 q^{55} + 2 q^{56} - 3 q^{58} + 20 q^{59} - 38 q^{61} - 4 q^{62} - 24 q^{64} - 12 q^{65} - 50 q^{67} - 11 q^{68} - 48 q^{70} - 12 q^{71} - 46 q^{73} + 6 q^{74} - 16 q^{76} + 14 q^{77} - 20 q^{79} + 58 q^{80} - 42 q^{82} - 22 q^{83} - 66 q^{85} - 22 q^{86} - 68 q^{88} + 14 q^{89} - 16 q^{91} - 17 q^{92} - 27 q^{94} + 20 q^{95} - 48 q^{97} + 28 q^{98}+O(q^{100}) $ 22 * q + 3 * q^2 + 17 * q^4 - 6 * q^7 + 6 * q^8 - 12 * q^10 - 28 * q^13 + q^14 + 3 * q^16 + 10 * q^17 - 8 * q^19 - 11 * q^22 - 22 * q^23 - 11 * q^26 - 21 * q^28 - 22 * q^29 - 18 * q^31 - 5 * q^32 - 33 * q^34 - 2 * q^35 - 28 * q^37 - 14 * q^38 - 30 * q^40 + 10 * q^41 - 14 * q^43 - 37 * q^44 - 3 * q^46 + 18 * q^47 + 2 * q^49 - 7 * q^50 - 57 * q^52 - 20 * q^53 - 42 * q^55 + 2 * q^56 - 3 * q^58 + 20 * q^59 - 38 * q^61 - 4 * q^62 - 24 * q^64 - 12 * q^65 - 50 * q^67 - 11 * q^68 - 48 * q^70 - 12 * q^71 - 46 * q^73 + 6 * q^74 - 16 * q^76 + 14 * q^77 - 20 * q^79 + 58 * q^80 - 42 * q^82 - 22 * q^83 - 66 * q^85 - 22 * q^86 - 68 * q^88 + 14 * q^89 - 16 * q^91 - 17 * q^92 - 27 * q^94 + 20 * q^95 - 48 * q^97 + 28 * q^98

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 $

$2$	−2.31658	−1.63807	−0.819034	−	0.573745i	$-0.805490\pi$
$2$	−2.31658	−1.63807	−0.819034	+	0.573745i	$0.805490\pi$
$3$	0	0
$3$	0	0
$4$	3.36653	1.68326
$5$	2.07685	0.928796	0.464398	−	0.885627i	$-0.346271\pi$
$5$	2.07685	0.928796	0.464398	+	0.885627i	$0.346271\pi$
$6$	0	0
$7$	1.30399	0.492863	0.246432	−	0.969160i	$-0.420742\pi$
$7$	1.30399	0.492863	0.246432	+	0.969160i	$0.420742\pi$
$8$	−3.16567	−1.11923
$9$	0	0
$10$	−4.81118	−1.52143
$11$	0.169489	0.0511029	0.0255514	−	0.999674i	$-0.491866\pi$
$11$	0.169489	0.0511029	0.0255514	+	0.999674i	$0.491866\pi$
$12$	0	0
$13$	0.705806	0.195755	0.0978777	−	0.995198i	$-0.468795\pi$
$13$	0.705806	0.195755	0.0978777	+	0.995198i	$0.468795\pi$
$14$	−3.02080	−0.807344
$15$	0	0
$16$	0.600462	0.150116
$17$	2.81253	0.682139	0.341070	−	0.940038i	$-0.389211\pi$
$17$	2.81253	0.682139	0.341070	+	0.940038i	$0.389211\pi$
$18$	0	0
$19$	−7.73667	−1.77491	−0.887457	−	0.460891i	$-0.847530\pi$
$19$	−7.73667	−1.77491	−0.887457	+	0.460891i	$0.847530\pi$
$20$	6.99178	1.56341
$21$	0	0
$22$	−0.392634	−0.0837100
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−0.686694	−0.137339
$26$	−1.63505	−0.320660
$27$	0	0
$28$	4.38993	0.829620
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−4.11205	−0.738546	−0.369273	−	0.929321i	$-0.620393\pi$
$31$	−4.11205	−0.738546	−0.369273	+	0.929321i	$0.620393\pi$
$32$	4.94033	0.873334
$33$	0	0
$34$	−6.51545	−1.11739
$35$	2.70820	0.457769
$36$	0	0
$37$	4.93337	0.811041	0.405520	−	0.914086i	$-0.367090\pi$
$37$	4.93337	0.811041	0.405520	+	0.914086i	$0.367090\pi$
$38$	17.9226	2.90743
$39$	0	0
$40$	−6.57462	−1.03954
$41$	2.38205	0.372014	0.186007	−	0.982548i	$-0.440445\pi$
$41$	2.38205	0.372014	0.186007	+	0.982548i	$0.440445\pi$
$42$	0	0
$43$	−1.36487	−0.208141	−0.104070	−	0.994570i	$-0.533187\pi$
$43$	−1.36487	−0.208141	−0.104070	+	0.994570i	$0.533187\pi$
$44$	0.570590	0.0860197
$45$	0	0
$46$	2.31658	0.341561
$47$	8.06261	1.17605	0.588026	−	0.808842i	$-0.299905\pi$
$47$	8.06261	1.17605	0.588026	+	0.808842i	$0.299905\pi$
$48$	0	0
$49$	−5.29960	−0.757086
$50$	1.59078	0.224970
$51$	0	0
$52$	2.37612	0.329508
$53$	−10.8473	−1.49000	−0.744998	−	0.667066i	$-0.767550\pi$
$53$	−10.8473	−1.49000	−0.744998	+	0.667066i	$0.767550\pi$
$54$	0	0
$55$	0.352003	0.0474641
$56$	−4.12802	−0.551629
$57$	0	0
$58$	2.31658	0.304181
$59$	−5.91579	−0.770170	−0.385085	−	0.922881i	$-0.625828\pi$
$59$	−5.91579	−0.770170	−0.385085	+	0.922881i	$0.625828\pi$
$60$	0	0
$61$	−2.67059	−0.341934	−0.170967	−	0.985277i	$-0.554689\pi$
$61$	−2.67059	−0.341934	−0.170967	+	0.985277i	$0.554689\pi$
$62$	9.52588	1.20979
$63$	0	0
$64$	−12.6456	−1.58070
$65$	1.46585	0.181817
$66$	0	0
$67$	−4.10168	−0.501100	−0.250550	−	0.968104i	$-0.580612\pi$
$67$	−4.10168	−0.501100	−0.250550	+	0.968104i	$0.580612\pi$
$68$	9.46847	1.14822
$69$	0	0
$70$	−6.27375	−0.749857
$71$	−2.86912	−0.340502	−0.170251	−	0.985401i	$-0.554458\pi$
$71$	−2.86912	−0.340502	−0.170251	+	0.985401i	$0.554458\pi$
$72$	0	0
$73$	−9.80599	−1.14770	−0.573852	−	0.818959i	$-0.694552\pi$
$73$	−9.80599	−1.14770	−0.573852	+	0.818959i	$0.694552\pi$
$74$	−11.4285	−1.32854
$75$	0	0
$76$	−26.0457	−2.98765
$77$	0.221013	0.0251867
$78$	0	0
$79$	−3.16854	−0.356489	−0.178245	−	0.983986i	$-0.557042\pi$
$79$	−3.16854	−0.356489	−0.178245	+	0.983986i	$0.557042\pi$
$80$	1.24707	0.139427
$81$	0	0
$82$	−5.51821	−0.609384
$83$	4.01310	0.440495	0.220247	−	0.975444i	$-0.429314\pi$
$83$	4.01310	0.440495	0.220247	+	0.975444i	$0.429314\pi$
$84$	0	0
$85$	5.84121	0.633568
$86$	3.16183	0.340948
$87$	0	0
$88$	−0.536547	−0.0571961
$89$	10.7736	1.14200	0.570998	−	0.820951i	$-0.306556\pi$
$89$	10.7736	1.14200	0.570998	+	0.820951i	$0.306556\pi$
$90$	0	0
$91$	0.920367	0.0964806
$92$	−3.36653	−0.350985
$93$	0	0
$94$	−18.6777	−1.92645
$95$	−16.0679	−1.64853
$96$	0	0
$97$	−8.28631	−0.841347	−0.420673	−	0.907212i	$-0.638206\pi$
$97$	−8.28631	−0.841347	−0.420673	+	0.907212i	$0.638206\pi$
$98$	12.2769	1.24016
$99$	0	0
$100$	−2.31177	−0.231177
$101$	10.1111	1.00609	0.503045	−	0.864260i	$-0.332213\pi$
$101$	10.1111	1.00609	0.503045	+	0.864260i	$0.332213\pi$
$102$	0	0
$103$	17.7619	1.75013	0.875066	−	0.484004i	$-0.160818\pi$
$103$	17.7619	1.75013	0.875066	+	0.484004i	$0.160818\pi$
$104$	−2.23435	−0.219096
$105$	0	0
$106$	25.1287	2.44071
$107$	−9.90380	−0.957436	−0.478718	−	0.877969i	$-0.658898\pi$
$107$	−9.90380	−0.957436	−0.478718	+	0.877969i	$0.658898\pi$
$108$	0	0
$109$	−17.3128	−1.65826	−0.829130	−	0.559055i	$-0.811164\pi$
$109$	−17.3128	−1.65826	−0.829130	+	0.559055i	$0.811164\pi$
$110$	−0.815443	−0.0777494
$111$	0	0
$112$	0.782999	0.0739864
$113$	−16.8475	−1.58487	−0.792437	−	0.609953i	$-0.791188\pi$
$113$	−16.8475	−1.58487	−0.792437	+	0.609953i	$0.791188\pi$
$114$	0	0
$115$	−2.07685	−0.193667
$116$	−3.36653	−0.312574
$117$	0	0
$118$	13.7044	1.26159
$119$	3.66752	0.336201
$120$	0	0
$121$	−10.9713	−0.997388
$122$	6.18663	0.560112
$123$	0	0
$124$	−13.8433	−1.24317
$125$	−11.8104	−1.05636
$126$	0	0
$127$	13.2186	1.17296	0.586481	−	0.809963i	$-0.300513\pi$
$127$	13.2186	1.17296	0.586481	+	0.809963i	$0.300513\pi$
$128$	19.4138	1.71595
$129$	0	0
$130$	−3.39576	−0.297828
$131$	11.4306	0.998695	0.499347	−	0.866402i	$-0.333573\pi$
$131$	11.4306	0.998695	0.499347	+	0.866402i	$0.333573\pi$
$132$	0	0
$133$	−10.0886	−0.874790
$134$	9.50187	0.820836
$135$	0	0
$136$	−8.90355	−0.763473
$137$	−8.68483	−0.741995	−0.370998	−	0.928634i	$-0.620984\pi$
$137$	−8.68483	−0.741995	−0.370998	+	0.928634i	$0.620984\pi$
$138$	0	0
$139$	−12.5103	−1.06111	−0.530553	−	0.847652i	$-0.678016\pi$
$139$	−12.5103	−1.06111	−0.530553	+	0.847652i	$0.678016\pi$
$140$	9.11724	0.770547
$141$	0	0
$142$	6.64655	0.557766
$143$	0.119626	0.0100037
$144$	0	0
$145$	−2.07685	−0.172473
$146$	22.7163	1.88002
$147$	0	0
$148$	16.6083	1.36520
$149$	−15.1639	−1.24228	−0.621138	−	0.783701i	$-0.713330\pi$
$149$	−15.1639	−1.24228	−0.621138	+	0.783701i	$0.713330\pi$
$150$	0	0
$151$	−10.1266	−0.824091	−0.412045	−	0.911163i	$-0.635185\pi$
$151$	−10.1266	−0.824091	−0.412045	+	0.911163i	$0.635185\pi$
$152$	24.4918	1.98654
$153$	0	0
$154$	−0.511993	−0.0412576
$155$	−8.54011	−0.685958
$156$	0	0
$157$	0.740894	0.0591298	0.0295649	−	0.999563i	$-0.490588\pi$
$157$	0.740894	0.0591298	0.0295649	+	0.999563i	$0.490588\pi$
$158$	7.34018	0.583953
$159$	0	0
$160$	10.2603	0.811149
$161$	−1.30399	−0.102769
$162$	0	0
$163$	1.49045	0.116741	0.0583706	−	0.998295i	$-0.481409\pi$
$163$	1.49045	0.116741	0.0583706	+	0.998295i	$0.481409\pi$
$164$	8.01925	0.626198
$165$	0	0
$166$	−9.29665	−0.721560
$167$	10.1972	0.789079	0.394540	−	0.918879i	$-0.370904\pi$
$167$	10.1972	0.789079	0.394540	+	0.918879i	$0.370904\pi$
$168$	0	0
$169$	−12.5018	−0.961680
$170$	−13.5316	−1.03783
$171$	0	0
$172$	−4.59487	−0.350356
$173$	−14.4841	−1.10121	−0.550603	−	0.834767i	$-0.685602\pi$
$173$	−14.4841	−1.10121	−0.550603	+	0.834767i	$0.685602\pi$
$174$	0	0
$175$	−0.895444	−0.0676892
$176$	0.101772	0.00767133
$177$	0	0
$178$	−24.9578	−1.87067
$179$	15.8281	1.18305	0.591525	−	0.806287i	$-0.298526\pi$
$179$	15.8281	1.18305	0.591525	+	0.806287i	$0.298526\pi$
$180$	0	0
$181$	−10.5567	−0.784672	−0.392336	−	0.919822i	$-0.628333\pi$
$181$	−10.5567	−0.784672	−0.392336	+	0.919822i	$0.628333\pi$
$182$	−2.13210	−0.158042
$183$	0	0
$184$	3.16567	0.233376
$185$	10.2459	0.753291
$186$	0	0
$187$	0.476693	0.0348593
$188$	27.1430	1.97961
$189$	0	0
$190$	37.2225	2.70041
$191$	−12.5067	−0.904952	−0.452476	−	0.891777i	$-0.649459\pi$
$191$	−12.5067	−0.904952	−0.452476	+	0.891777i	$0.649459\pi$
$192$	0	0
$193$	−16.2747	−1.17148	−0.585739	−	0.810500i	$-0.699196\pi$
$193$	−16.2747	−1.17148	−0.585739	+	0.810500i	$0.699196\pi$
$194$	19.1959	1.37818
$195$	0	0
$196$	−17.8413	−1.27438
$197$	−17.8816	−1.27401	−0.637006	−	0.770859i	$-0.719827\pi$
$197$	−17.8816	−1.27401	−0.637006	+	0.770859i	$0.719827\pi$
$198$	0	0
$199$	11.4755	0.813479	0.406739	−	0.913544i	$-0.366666\pi$
$199$	11.4755	0.813479	0.406739	+	0.913544i	$0.366666\pi$
$200$	2.17385	0.153714
$201$	0	0
$202$	−23.4231	−1.64804
$203$	−1.30399	−0.0915224
$204$	0	0
$205$	4.94717	0.345525
$206$	−41.1468	−2.86683
$207$	0	0
$208$	0.423810	0.0293859
$209$	−1.31128	−0.0907032
$210$	0	0
$211$	−16.9126	−1.16431	−0.582154	−	0.813078i	$-0.697790\pi$
$211$	−16.9126	−1.16431	−0.582154	+	0.813078i	$0.697790\pi$
$212$	−36.5179	−2.50806
$213$	0	0
$214$	22.9429	1.56835
$215$	−2.83463	−0.193320
$216$	0	0
$217$	−5.36209	−0.364002
$218$	40.1063	2.71634
$219$	0	0
$220$	1.18503	0.0798947
$221$	1.98510	0.133532
$222$	0	0
$223$	25.4479	1.70412	0.852058	−	0.523447i	$-0.175354\pi$
$223$	25.4479	1.70412	0.852058	+	0.523447i	$0.175354\pi$
$224$	6.44216	0.430435
$225$	0	0
$226$	39.0284	2.59613
$227$	3.16010	0.209743	0.104871	−	0.994486i	$-0.466557\pi$
$227$	3.16010	0.209743	0.104871	+	0.994486i	$0.466557\pi$
$228$	0	0
$229$	4.29359	0.283728	0.141864	−	0.989886i	$-0.454690\pi$
$229$	4.29359	0.283728	0.141864	+	0.989886i	$0.454690\pi$
$230$	4.81118	0.317240
$231$	0	0
$232$	3.16567	0.207836
$233$	14.4837	0.948859	0.474430	−	0.880293i	$-0.342654\pi$
$233$	14.4837	0.948859	0.474430	+	0.880293i	$0.342654\pi$
$234$	0	0
$235$	16.7448	1.09231
$236$	−19.9157	−1.29640
$237$	0	0
$238$	−8.49610	−0.550721
$239$	−25.3341	−1.63873	−0.819364	−	0.573273i	$-0.805673\pi$
$239$	−25.3341	−1.63873	−0.819364	+	0.573273i	$0.805673\pi$
$240$	0	0
$241$	29.8407	1.92221	0.961105	−	0.276185i	$-0.0890701\pi$
$241$	29.8407	1.92221	0.961105	+	0.276185i	$0.0890701\pi$
$242$	25.4158	1.63379
$243$	0	0
$244$	−8.99063	−0.575566
$245$	−11.0065	−0.703178
$246$	0	0
$247$	−5.46059	−0.347449
$248$	13.0174	0.826606
$249$	0	0
$250$	27.3597	1.73038
$251$	13.9620	0.881273	0.440637	−	0.897686i	$-0.354753\pi$
$251$	13.9620	0.881273	0.440637	+	0.897686i	$0.354753\pi$
$252$	0	0
$253$	−0.169489	−0.0106557
$254$	−30.6219	−1.92139
$255$	0	0
$256$	−19.6824	−1.23015
$257$	−2.06288	−0.128679	−0.0643396	−	0.997928i	$-0.520494\pi$
$257$	−2.06288	−0.128679	−0.0643396	+	0.997928i	$0.520494\pi$
$258$	0	0
$259$	6.43308	0.399732
$260$	4.93484	0.306046
$261$	0	0
$262$	−26.4798	−1.63593
$263$	−6.51820	−0.401929	−0.200965	−	0.979599i	$-0.564408\pi$
$263$	−6.51820	−0.401929	−0.200965	+	0.979599i	$0.564408\pi$
$264$	0	0
$265$	−22.5283	−1.38390
$266$	23.3710	1.43297
$267$	0	0
$268$	−13.8084	−0.843485
$269$	22.3808	1.36458	0.682290	−	0.731082i	$-0.260984\pi$
$269$	22.3808	1.36458	0.682290	+	0.731082i	$0.260984\pi$
$270$	0	0
$271$	13.7657	0.836208	0.418104	−	0.908399i	$-0.362695\pi$
$271$	13.7657	0.836208	0.418104	+	0.908399i	$0.362695\pi$
$272$	1.68882	0.102400
$273$	0	0
$274$	20.1191	1.21544
$275$	−0.116387	−0.00701840
$276$	0	0
$277$	5.06087	0.304078	0.152039	−	0.988374i	$-0.451416\pi$
$277$	5.06087	0.304078	0.152039	+	0.988374i	$0.451416\pi$
$278$	28.9810	1.73816
$279$	0	0
$280$	−8.57327	−0.512351
$281$	24.2561	1.44700	0.723499	−	0.690326i	$-0.242533\pi$
$281$	24.2561	1.44700	0.723499	+	0.690326i	$0.242533\pi$
$282$	0	0
$283$	−5.35557	−0.318356	−0.159178	−	0.987250i	$-0.550884\pi$
$283$	−5.35557	−0.318356	−0.159178	+	0.987250i	$0.550884\pi$
$284$	−9.65899	−0.573156
$285$	0	0
$286$	−0.277124	−0.0163867
$287$	3.10618	0.183352
$288$	0	0
$289$	−9.08967	−0.534686
$290$	4.81118	0.282522
$291$	0	0
$292$	−33.0121	−1.93189
$293$	6.01450	0.351371	0.175685	−	0.984446i	$-0.443786\pi$
$293$	6.01450	0.351371	0.175685	+	0.984446i	$0.443786\pi$
$294$	0	0
$295$	−12.2862	−0.715331
$296$	−15.6174	−0.907744
$297$	0	0
$298$	35.1284	2.03493
$299$	−0.705806	−0.0408178
$300$	0	0
$301$	−1.77978	−0.102585
$302$	23.4590	1.34992
$303$	0	0
$304$	−4.64558	−0.266442
$305$	−5.54642	−0.317587
$306$	0	0
$307$	17.3807	0.991966	0.495983	−	0.868332i	$-0.334808\pi$
$307$	17.3807	0.991966	0.495983	+	0.868332i	$0.334808\pi$
$308$	0.744046	0.0423959
$309$	0	0
$310$	19.7838	1.12365
$311$	28.2892	1.60413	0.802067	−	0.597234i	$-0.203734\pi$
$311$	28.2892	1.60413	0.802067	+	0.597234i	$0.203734\pi$
$312$	0	0
$313$	−9.92102	−0.560769	−0.280385	−	0.959888i	$-0.590462\pi$
$313$	−9.92102	−0.560769	−0.280385	+	0.959888i	$0.590462\pi$
$314$	−1.71634	−0.0968586
$315$	0	0
$316$	−10.6670	−0.600066
$317$	20.6462	1.15961	0.579803	−	0.814756i	$-0.303129\pi$
$317$	20.6462	1.15961	0.579803	+	0.814756i	$0.303129\pi$
$318$	0	0
$319$	−0.169489	−0.00948957
$320$	−26.2630	−1.46814
$321$	0	0
$322$	3.02080	0.168343
$323$	−21.7596	−1.21074
$324$	0	0
$325$	−0.484672	−0.0268848
$326$	−3.45275	−0.191230
$327$	0	0
$328$	−7.54080	−0.416371
$329$	10.5136	0.579633
$330$	0	0
$331$	4.94691	0.271907	0.135953	−	0.990715i	$-0.456590\pi$
$331$	4.94691	0.271907	0.135953	+	0.990715i	$0.456590\pi$
$332$	13.5102	0.741469
$333$	0	0
$334$	−23.6225	−1.29257
$335$	−8.51858	−0.465420
$336$	0	0
$337$	−23.6415	−1.28784	−0.643918	−	0.765094i	$-0.722692\pi$
$337$	−23.6415	−1.28784	−0.643918	+	0.765094i	$0.722692\pi$
$338$	28.9615	1.57530
$339$	0	0
$340$	19.6646	1.06646
$341$	−0.696948	−0.0377418
$342$	0	0
$343$	−16.0386	−0.866003
$344$	4.32073	0.232958
$345$	0	0
$346$	33.5536	1.80385
$347$	6.29098	0.337717	0.168859	−	0.985640i	$-0.445992\pi$
$347$	6.29098	0.337717	0.168859	+	0.985640i	$0.445992\pi$
$348$	0	0
$349$	33.5081	1.79365	0.896823	−	0.442390i	$-0.145869\pi$
$349$	33.5081	1.79365	0.896823	+	0.442390i	$0.145869\pi$
$350$	2.07437	0.110880
$351$	0	0
$352$	0.837331	0.0446299
$353$	−28.8240	−1.53415	−0.767074	−	0.641559i	$-0.778288\pi$
$353$	−28.8240	−1.53415	−0.767074	+	0.641559i	$0.778288\pi$
$354$	0	0
$355$	−5.95874	−0.316257
$356$	36.2695	1.92228
$357$	0	0
$358$	−36.6671	−1.93791
$359$	30.9936	1.63578	0.817891	−	0.575373i	$-0.195143\pi$
$359$	30.9936	1.63578	0.817891	+	0.575373i	$0.195143\pi$
$360$	0	0
$361$	40.8561	2.15032
$362$	24.4554	1.28535
$363$	0	0
$364$	3.09844	0.162402
$365$	−20.3656	−1.06598
$366$	0	0
$367$	13.7921	0.719941	0.359971	−	0.932964i	$-0.382787\pi$
$367$	13.7921	0.719941	0.359971	+	0.932964i	$0.382787\pi$
$368$	−0.600462	−0.0313012
$369$	0	0
$370$	−23.7353	−1.23394
$371$	−14.1449	−0.734365
$372$	0	0
$373$	−1.58905	−0.0822781	−0.0411390	−	0.999153i	$-0.513099\pi$
$373$	−1.58905	−0.0822781	−0.0411390	+	0.999153i	$0.513099\pi$
$374$	−1.10430	−0.0571018
$375$	0	0
$376$	−25.5236	−1.31628
$377$	−0.705806	−0.0363509
$378$	0	0
$379$	−11.7068	−0.601338	−0.300669	−	0.953729i	$-0.597210\pi$
$379$	−11.7068	−0.601338	−0.300669	+	0.953729i	$0.597210\pi$
$380$	−54.0931	−2.77492
$381$	0	0
$382$	28.9727	1.48237
$383$	5.46335	0.279164	0.139582	−	0.990211i	$-0.455424\pi$
$383$	5.46335	0.279164	0.139582	+	0.990211i	$0.455424\pi$
$384$	0	0
$385$	0.459010	0.0233933
$386$	37.7016	1.91896
$387$	0	0
$388$	−27.8961	−1.41621
$389$	−5.66382	−0.287167	−0.143584	−	0.989638i	$-0.545863\pi$
$389$	−5.66382	−0.287167	−0.143584	+	0.989638i	$0.545863\pi$
$390$	0	0
$391$	−2.81253	−0.142236
$392$	16.7768	0.847356
$393$	0	0
$394$	41.4242	2.08692
$395$	−6.58059	−0.331106
$396$	0	0
$397$	−24.5959	−1.23443	−0.617215	−	0.786794i	$-0.711739\pi$
$397$	−24.5959	−1.23443	−0.617215	+	0.786794i	$0.711739\pi$
$398$	−26.5839	−1.33253
$399$	0	0
$400$	−0.412333	−0.0206167
$401$	−23.2123	−1.15916	−0.579582	−	0.814914i	$-0.696784\pi$
$401$	−23.2123	−1.15916	−0.579582	+	0.814914i	$0.696784\pi$
$402$	0	0
$403$	−2.90231	−0.144574
$404$	34.0393	1.69352
$405$	0	0
$406$	3.02080	0.149920
$407$	0.836152	0.0414465
$408$	0	0
$409$	−22.6437	−1.11966	−0.559829	−	0.828608i	$-0.689133\pi$
$409$	−22.6437	−1.11966	−0.559829	+	0.828608i	$0.689133\pi$
$410$	−11.4605	−0.565994
$411$	0	0
$412$	59.7959	2.94593
$413$	−7.71416	−0.379589
$414$	0	0
$415$	8.33460	0.409129
$416$	3.48691	0.170960
$417$	0	0
$418$	3.03768	0.148578
$419$	−19.3996	−0.947732	−0.473866	−	0.880597i	$-0.657142\pi$
$419$	−19.3996	−0.947732	−0.473866	+	0.880597i	$0.657142\pi$
$420$	0	0
$421$	−17.6305	−0.859259	−0.429630	−	0.903005i	$-0.641356\pi$
$421$	−17.6305	−0.859259	−0.429630	+	0.903005i	$0.641356\pi$
$422$	39.1792	1.90722
$423$	0	0
$424$	34.3391	1.66765
$425$	−1.93135	−0.0936841
$426$	0	0
$427$	−3.48244	−0.168527
$428$	−33.3414	−1.61162
$429$	0	0
$430$	6.56664	0.316671
$431$	3.82141	0.184071	0.0920355	−	0.995756i	$-0.470663\pi$
$431$	3.82141	0.184071	0.0920355	+	0.995756i	$0.470663\pi$
$432$	0	0
$433$	34.6578	1.66555	0.832774	−	0.553613i	$-0.186751\pi$
$433$	34.6578	1.66555	0.832774	+	0.553613i	$0.186751\pi$
$434$	12.4217	0.596261
$435$	0	0
$436$	−58.2839	−2.79129
$437$	7.73667	0.370095
$438$	0	0
$439$	18.4524	0.880686	0.440343	−	0.897830i	$-0.354857\pi$
$439$	18.4524	0.880686	0.440343	+	0.897830i	$0.354857\pi$
$440$	−1.11433	−0.0531234
$441$	0	0
$442$	−4.59864	−0.218735
$443$	−27.6628	−1.31430	−0.657150	−	0.753760i	$-0.728238\pi$
$443$	−27.6628	−1.31430	−0.657150	+	0.753760i	$0.728238\pi$
$444$	0	0
$445$	22.3751	1.06068
$446$	−58.9520	−2.79146
$447$	0	0
$448$	−16.4897	−0.779067
$449$	−27.7271	−1.30852	−0.654262	−	0.756268i	$-0.727021\pi$
$449$	−27.7271	−1.30852	−0.654262	+	0.756268i	$0.727021\pi$
$450$	0	0
$451$	0.403732	0.0190110
$452$	−56.7174	−2.66776
$453$	0	0
$454$	−7.32061	−0.343573
$455$	1.91146	0.0896108
$456$	0	0
$457$	−4.85874	−0.227282	−0.113641	−	0.993522i	$-0.536251\pi$
$457$	−4.85874	−0.227282	−0.113641	+	0.993522i	$0.536251\pi$
$458$	−9.94642	−0.464766
$459$	0	0
$460$	−6.99178	−0.325993
$461$	−19.6497	−0.915178	−0.457589	−	0.889164i	$-0.651287\pi$
$461$	−19.6497	−0.915178	−0.457589	+	0.889164i	$0.651287\pi$
$462$	0	0
$463$	−31.9707	−1.48580	−0.742902	−	0.669401i	$-0.766551\pi$
$463$	−31.9707	−1.48580	−0.742902	+	0.669401i	$0.766551\pi$
$464$	−0.600462	−0.0278757
$465$	0	0
$466$	−33.5526	−1.55430
$467$	−7.06027	−0.326710	−0.163355	−	0.986567i	$-0.552232\pi$
$467$	−7.06027	−0.326710	−0.163355	+	0.986567i	$0.552232\pi$
$468$	0	0
$469$	−5.34857	−0.246974
$470$	−38.7907	−1.78928
$471$	0	0
$472$	18.7274	0.862001
$473$	−0.231330	−0.0106366
$474$	0	0
$475$	5.31272	0.243764
$476$	12.3468	0.565916
$477$	0	0
$478$	58.6885	2.68435
$479$	−3.49893	−0.159870	−0.0799350	−	0.996800i	$-0.525471\pi$
$479$	−3.49893	−0.159870	−0.0799350	+	0.996800i	$0.525471\pi$
$480$	0	0
$481$	3.48200	0.158766
$482$	−69.1283	−3.14871
$483$	0	0
$484$	−36.9351	−1.67887
$485$	−17.2094	−0.781439
$486$	0	0
$487$	21.6955	0.983118	0.491559	−	0.870844i	$-0.336427\pi$
$487$	21.6955	0.983118	0.491559	+	0.870844i	$0.336427\pi$
$488$	8.45422	0.382704
$489$	0	0
$490$	25.4973	1.15185
$491$	−1.99184	−0.0898903	−0.0449452	−	0.998989i	$-0.514311\pi$
$491$	−1.99184	−0.0898903	−0.0449452	+	0.998989i	$0.514311\pi$
$492$	0	0
$493$	−2.81253	−0.126670
$494$	12.6499	0.569145
$495$	0	0
$496$	−2.46913	−0.110867
$497$	−3.74132	−0.167821
$498$	0	0
$499$	−27.0930	−1.21285	−0.606424	−	0.795142i	$-0.707396\pi$
$499$	−27.0930	−1.21285	−0.606424	+	0.795142i	$0.707396\pi$
$500$	−39.7601	−1.77813
$501$	0	0
$502$	−32.3440	−1.44358
$503$	12.4494	0.555094	0.277547	−	0.960712i	$-0.410479\pi$
$503$	12.4494	0.555094	0.277547	+	0.960712i	$0.410479\pi$
$504$	0	0
$505$	20.9992	0.934452
$506$	0.392634	0.0174547
$507$	0	0
$508$	44.5008	1.97441
$509$	−0.109995	−0.00487546	−0.00243773	−	0.999997i	$-0.500776\pi$
$509$	−0.109995	−0.00487546	−0.00243773	+	0.999997i	$0.500776\pi$
$510$	0	0
$511$	−12.7869	−0.565661
$512$	6.76821	0.299115
$513$	0	0
$514$	4.77883	0.210785
$515$	36.8888	1.62551
$516$	0	0
$517$	1.36652	0.0600997
$518$	−14.9027	−0.654788
$519$	0	0
$520$	−4.64041	−0.203495
$521$	−7.73135	−0.338717	−0.169358	−	0.985555i	$-0.554170\pi$
$521$	−7.73135	−0.338717	−0.169358	+	0.985555i	$0.554170\pi$
$522$	0	0
$523$	−3.75686	−0.164276	−0.0821381	−	0.996621i	$-0.526175\pi$
$523$	−3.75686	−0.164276	−0.0821381	+	0.996621i	$0.526175\pi$
$524$	38.4814	1.68107
$525$	0	0
$526$	15.0999	0.658387
$527$	−11.5653	−0.503791
$528$	0	0
$529$	1.00000	0.0434783
$530$	52.1885	2.26693
$531$	0	0
$532$	−33.9635	−1.47250
$533$	1.68127	0.0728238
$534$	0	0
$535$	−20.5687	−0.889263
$536$	12.9846	0.560849
$537$	0	0
$538$	−51.8468	−2.23527
$539$	−0.898224	−0.0386893
$540$	0	0
$541$	14.1131	0.606769	0.303385	−	0.952868i	$-0.401883\pi$
$541$	14.1131	0.606769	0.303385	+	0.952868i	$0.401883\pi$
$542$	−31.8894	−1.36977
$543$	0	0
$544$	13.8948	0.595735
$545$	−35.9560	−1.54019
$546$	0	0
$547$	−5.60250	−0.239546	−0.119773	−	0.992801i	$-0.538217\pi$
$547$	−5.60250	−0.239546	−0.119773	+	0.992801i	$0.538217\pi$
$548$	−29.2377	−1.24897
$549$	0	0
$550$	0.269620	0.0114966
$551$	7.73667	0.329593
$552$	0	0
$553$	−4.13176	−0.175700
$554$	−11.7239	−0.498101
$555$	0	0
$556$	−42.1162	−1.78612
$557$	−38.1737	−1.61747	−0.808735	−	0.588173i	$-0.799847\pi$
$557$	−38.1737	−1.61747	−0.808735	+	0.588173i	$0.799847\pi$
$558$	0	0
$559$	−0.963333	−0.0407446
$560$	1.62617	0.0687183
$561$	0	0
$562$	−56.1911	−2.37028
$563$	19.9152	0.839324	0.419662	−	0.907680i	$-0.362149\pi$
$563$	19.9152	0.839324	0.419662	+	0.907680i	$0.362149\pi$
$564$	0	0
$565$	−34.9896	−1.47202
$566$	12.4066	0.521488
$567$	0	0
$568$	9.08270	0.381102
$569$	−24.1599	−1.01283	−0.506417	−	0.862288i	$-0.669030\pi$
$569$	−24.1599	−1.01283	−0.506417	+	0.862288i	$0.669030\pi$
$570$	0	0
$571$	0.809648	0.0338827	0.0169414	−	0.999856i	$-0.494607\pi$
$571$	0.809648	0.0338827	0.0169414	+	0.999856i	$0.494607\pi$
$572$	0.402726	0.0168388
$573$	0	0
$574$	−7.19571	−0.300343
$575$	0.686694	0.0286371
$576$	0	0
$577$	−7.99326	−0.332764	−0.166382	−	0.986061i	$-0.553208\pi$
$577$	−7.99326	−0.332764	−0.166382	+	0.986061i	$0.553208\pi$
$578$	21.0569	0.875852
$579$	0	0
$580$	−6.99178	−0.290318
$581$	5.23305	0.217104
$582$	0	0
$583$	−1.83851	−0.0761431
$584$	31.0425	1.28455
$585$	0	0
$586$	−13.9330	−0.575569
$587$	−4.08246	−0.168501	−0.0842506	−	0.996445i	$-0.526850\pi$
$587$	−4.08246	−0.168501	−0.0842506	+	0.996445i	$0.526850\pi$
$588$	0	0
$589$	31.8136	1.31086
$590$	28.4620	1.17176
$591$	0	0
$592$	2.96230	0.121750
$593$	−8.41250	−0.345460	−0.172730	−	0.984969i	$-0.555259\pi$
$593$	−8.41250	−0.345460	−0.172730	+	0.984969i	$0.555259\pi$
$594$	0	0
$595$	7.61690	0.312262
$596$	−51.0498	−2.09108
$597$	0	0
$598$	1.63505	0.0668623
$599$	37.6547	1.53853	0.769265	−	0.638930i	$-0.220623\pi$
$599$	37.6547	1.53853	0.769265	+	0.638930i	$0.220623\pi$
$600$	0	0
$601$	11.6127	0.473693	0.236846	−	0.971547i	$-0.423886\pi$
$601$	11.6127	0.473693	0.236846	+	0.971547i	$0.423886\pi$
$602$	4.12300	0.168041
$603$	0	0
$604$	−34.0915	−1.38716
$605$	−22.7857	−0.926370
$606$	0	0
$607$	35.6191	1.44573	0.722867	−	0.690987i	$-0.242824\pi$
$607$	35.6191	1.44573	0.722867	+	0.690987i	$0.242824\pi$
$608$	−38.2217	−1.55009
$609$	0	0
$610$	12.8487	0.520229
$611$	5.69064	0.230219
$612$	0	0
$613$	36.4041	1.47035	0.735174	−	0.677878i	$-0.237100\pi$
$613$	36.4041	1.47035	0.735174	+	0.677878i	$0.237100\pi$
$614$	−40.2636	−1.62491
$615$	0	0
$616$	−0.699654	−0.0281898
$617$	−32.5101	−1.30881	−0.654404	−	0.756145i	$-0.727080\pi$
$617$	−32.5101	−1.30881	−0.654404	+	0.756145i	$0.727080\pi$
$618$	0	0
$619$	−3.18797	−0.128135	−0.0640677	−	0.997946i	$-0.520407\pi$
$619$	−3.18797	−0.128135	−0.0640677	+	0.997946i	$0.520407\pi$
$620$	−28.7505	−1.15465
$621$	0	0
$622$	−65.5341	−2.62768
$623$	14.0487	0.562848
$624$	0	0
$625$	−21.0950	−0.843799
$626$	22.9828	0.918578
$627$	0	0
$628$	2.49424	0.0995311
$629$	13.8753	0.553242
$630$	0	0
$631$	−18.6025	−0.740553	−0.370277	−	0.928922i	$-0.620737\pi$
$631$	−18.6025	−0.740553	−0.370277	+	0.928922i	$0.620737\pi$
$632$	10.0306	0.398995
$633$	0	0
$634$	−47.8285	−1.89951
$635$	27.4531	1.08944
$636$	0	0
$637$	−3.74049	−0.148204
$638$	0.392634	0.0155445
$639$	0	0
$640$	40.3195	1.59377
$641$	47.2445	1.86605	0.933023	−	0.359817i	$-0.117161\pi$
$641$	47.2445	1.86605	0.933023	+	0.359817i	$0.117161\pi$
$642$	0	0
$643$	26.7049	1.05314	0.526569	−	0.850133i	$-0.323478\pi$
$643$	26.7049	1.05314	0.526569	+	0.850133i	$0.323478\pi$
$644$	−4.38993	−0.172988
$645$	0	0
$646$	50.4078	1.98327
$647$	−7.37359	−0.289886	−0.144943	−	0.989440i	$-0.546300\pi$
$647$	−7.37359	−0.289886	−0.144943	+	0.989440i	$0.546300\pi$
$648$	0	0
$649$	−1.00266	−0.0393579
$650$	1.12278	0.0440391
$651$	0	0
$652$	5.01765	0.196506
$653$	−25.1850	−0.985567	−0.492783	−	0.870152i	$-0.664021\pi$
$653$	−25.1850	−0.985567	−0.492783	+	0.870152i	$0.664021\pi$
$654$	0	0
$655$	23.7396	0.927583
$656$	1.43033	0.0558451
$657$	0	0
$658$	−24.3556	−0.949479
$659$	−34.1093	−1.32871	−0.664355	−	0.747417i	$-0.731294\pi$
$659$	−34.1093	−1.32871	−0.664355	+	0.747417i	$0.731294\pi$
$660$	0	0
$661$	−18.1675	−0.706635	−0.353318	−	0.935503i	$-0.614947\pi$
$661$	−18.1675	−0.706635	−0.353318	+	0.935503i	$0.614947\pi$
$662$	−11.4599	−0.445401
$663$	0	0
$664$	−12.7041	−0.493016
$665$	−20.9525	−0.812501
$666$	0	0
$667$	1.00000	0.0387202
$668$	34.3290	1.32823
$669$	0	0
$670$	19.7340	0.762389
$671$	−0.452636	−0.0174738
$672$	0	0
$673$	−14.2757	−0.550286	−0.275143	−	0.961403i	$-0.588725\pi$
$673$	−14.2757	−0.550286	−0.275143	+	0.961403i	$0.588725\pi$
$674$	54.7675	2.10956
$675$	0	0
$676$	−42.0878	−1.61876
$677$	−47.2256	−1.81503	−0.907514	−	0.420021i	$-0.862023\pi$
$677$	−47.2256	−1.81503	−0.907514	+	0.420021i	$0.862023\pi$
$678$	0	0
$679$	−10.8053	−0.414669
$680$	−18.4913	−0.709110
$681$	0	0
$682$	1.61453	0.0618237
$683$	−14.4612	−0.553344	−0.276672	−	0.960964i	$-0.589232\pi$
$683$	−14.4612	−0.553344	−0.276672	+	0.960964i	$0.589232\pi$
$684$	0	0
$685$	−18.0371	−0.689162
$686$	37.1547	1.41857
$687$	0	0
$688$	−0.819552	−0.0312451
$689$	−7.65611	−0.291675
$690$	0	0
$691$	−43.7378	−1.66386	−0.831932	−	0.554878i	$-0.812765\pi$
$691$	−43.7378	−1.66386	−0.831932	+	0.554878i	$0.812765\pi$
$692$	−48.7612	−1.85362
$693$	0	0
$694$	−14.5735	−0.553204
$695$	−25.9819	−0.985551
$696$	0	0
$697$	6.69960	0.253765
$698$	−77.6240	−2.93811
$699$	0	0
$700$	−3.01454	−0.113939
$701$	−16.5091	−0.623540	−0.311770	−	0.950158i	$-0.600922\pi$
$701$	−16.5091	−0.623540	−0.311770	+	0.950158i	$0.600922\pi$
$702$	0	0
$703$	−38.1678	−1.43953
$704$	−2.14329	−0.0807781
$705$	0	0
$706$	66.7731	2.51304
$707$	13.1848	0.495865
$708$	0	0
$709$	44.8286	1.68357	0.841786	−	0.539811i	$-0.181504\pi$
$709$	44.8286	1.68357	0.841786	+	0.539811i	$0.181504\pi$
$710$	13.8039	0.518050
$711$	0	0
$712$	−34.1056	−1.27816
$713$	4.11205	0.153998
$714$	0	0
$715$	0.248446	0.00929136
$716$	53.2858	1.99139
$717$	0	0
$718$	−71.7992	−2.67952
$719$	−0.824697	−0.0307560	−0.0153780	−	0.999882i	$-0.504895\pi$
$719$	−0.824697	−0.0307560	−0.0153780	+	0.999882i	$0.504895\pi$
$720$	0	0
$721$	23.1614	0.862576
$722$	−94.6462	−3.52237
$723$	0	0
$724$	−35.5394	−1.32081
$725$	0.686694	0.0255032
$726$	0	0
$727$	−46.9159	−1.74002	−0.870008	−	0.493038i	$-0.835886\pi$
$727$	−46.9159	−1.74002	−0.870008	+	0.493038i	$0.835886\pi$
$728$	−2.91358	−0.107984
$729$	0	0
$730$	47.1784	1.74615
$731$	−3.83874	−0.141981
$732$	0	0
$733$	−34.3606	−1.26914	−0.634570	−	0.772866i	$-0.718823\pi$
$733$	−34.3606	−1.26914	−0.634570	+	0.772866i	$0.718823\pi$
$734$	−31.9504	−1.17931
$735$	0	0
$736$	−4.94033	−0.182103
$737$	−0.695191	−0.0256077
$738$	0	0
$739$	6.08214	0.223735	0.111868	−	0.993723i	$-0.464317\pi$
$739$	6.08214	0.223735	0.111868	+	0.993723i	$0.464317\pi$
$740$	34.4930	1.26799
$741$	0	0
$742$	32.7677	1.20294
$743$	−10.0165	−0.367471	−0.183735	−	0.982976i	$-0.558819\pi$
$743$	−10.0165	−0.367471	−0.183735	+	0.982976i	$0.558819\pi$
$744$	0	0
$745$	−31.4932	−1.15382
$746$	3.68117	0.134777
$747$	0	0
$748$	1.60480	0.0586774
$749$	−12.9145	−0.471885
$750$	0	0
$751$	13.3546	0.487318	0.243659	−	0.969861i	$-0.421652\pi$
$751$	13.3546	0.487318	0.243659	+	0.969861i	$0.421652\pi$
$752$	4.84129	0.176544
$753$	0	0
$754$	1.63505	0.0595451
$755$	−21.0314	−0.765412
$756$	0	0
$757$	2.19148	0.0796508	0.0398254	−	0.999207i	$-0.487320\pi$
$757$	2.19148	0.0796508	0.0398254	+	0.999207i	$0.487320\pi$
$758$	27.1197	0.985032
$759$	0	0
$760$	50.8657	1.84509
$761$	5.49565	0.199217	0.0996085	−	0.995027i	$-0.468241\pi$
$761$	5.49565	0.199217	0.0996085	+	0.995027i	$0.468241\pi$
$762$	0	0
$763$	−22.5757	−0.817296
$764$	−42.1041	−1.52327
$765$	0	0
$766$	−12.6563	−0.457289
$767$	−4.17540	−0.150765
$768$	0	0
$769$	−38.1501	−1.37573	−0.687863	−	0.725841i	$-0.741451\pi$
$769$	−38.1501	−1.37573	−0.687863	+	0.725841i	$0.741451\pi$
$770$	−1.06333	−0.0383199
$771$	0	0
$772$	−54.7892	−1.97191
$773$	33.6301	1.20959	0.604796	−	0.796381i	$-0.293255\pi$
$773$	33.6301	1.20959	0.604796	+	0.796381i	$0.293255\pi$
$774$	0	0
$775$	2.82372	0.101431
$776$	26.2317	0.941664
$777$	0	0
$778$	13.1207	0.470399
$779$	−18.4292	−0.660293
$780$	0	0
$781$	−0.486285	−0.0174006
$782$	6.51545	0.232992
$783$	0	0
$784$	−3.18221	−0.113650
$785$	1.53873	0.0549195
$786$	0	0
$787$	41.8793	1.49284	0.746418	−	0.665477i	$-0.231772\pi$
$787$	41.8793	1.49284	0.746418	+	0.665477i	$0.231772\pi$
$788$	−60.1990	−2.14450
$789$	0	0
$790$	15.2445	0.542373
$791$	−21.9690	−0.781127
$792$	0	0
$793$	−1.88492	−0.0669355
$794$	56.9782	2.02208
$795$	0	0
$796$	38.6327	1.36930
$797$	22.3903	0.793104	0.396552	−	0.918012i	$-0.370207\pi$
$797$	22.3903	0.793104	0.396552	+	0.918012i	$0.370207\pi$
$798$	0	0
$799$	22.6763	0.802232
$800$	−3.39249	−0.119943
$801$	0	0
$802$	53.7730	1.89879
$803$	−1.66201	−0.0586510
$804$	0	0
$805$	−2.70820	−0.0954515
$806$	6.72342	0.236823
$807$	0	0
$808$	−32.0084	−1.12605
$809$	18.9095	0.664821	0.332411	−	0.943135i	$-0.392138\pi$
$809$	18.9095	0.664821	0.332411	+	0.943135i	$0.392138\pi$
$810$	0	0
$811$	19.8033	0.695387	0.347694	−	0.937608i	$-0.386965\pi$
$811$	19.8033	0.695387	0.347694	+	0.937608i	$0.386965\pi$
$812$	−4.38993	−0.154057
$813$	0	0
$814$	−1.93701	−0.0678922
$815$	3.09545	0.108429
$816$	0	0
$817$	10.5595	0.369432
$818$	52.4559	1.83408
$819$	0	0
$820$	16.6548	0.581610
$821$	28.6472	0.999794	0.499897	−	0.866085i	$-0.333371\pi$
$821$	28.6472	0.999794	0.499897	+	0.866085i	$0.333371\pi$
$822$	0	0
$823$	3.81800	0.133087	0.0665435	−	0.997784i	$-0.478803\pi$
$823$	3.81800	0.133087	0.0665435	+	0.997784i	$0.478803\pi$
$824$	−56.2283	−1.95881
$825$	0	0
$826$	17.8704	0.621792
$827$	−2.25983	−0.0785819	−0.0392909	−	0.999228i	$-0.512510\pi$
$827$	−2.25983	−0.0785819	−0.0392909	+	0.999228i	$0.512510\pi$
$828$	0	0
$829$	33.7692	1.17285	0.586426	−	0.810002i	$-0.300534\pi$
$829$	33.7692	1.17285	0.586426	+	0.810002i	$0.300534\pi$
$830$	−19.3077	−0.670181
$831$	0	0
$832$	−8.92532	−0.309430
$833$	−14.9053	−0.516438
$834$	0	0
$835$	21.1780	0.732894
$836$	−4.41447	−0.152677
$837$	0	0
$838$	44.9406	1.55245
$839$	−39.6474	−1.36878	−0.684390	−	0.729116i	$-0.739932\pi$
$839$	−39.6474	−1.36878	−0.684390	+	0.729116i	$0.739932\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	40.8425	1.40752
$843$	0	0
$844$	−56.9366	−1.95984
$845$	−25.9644	−0.893204
$846$	0	0
$847$	−14.3065	−0.491576
$848$	−6.51341	−0.223672
$849$	0	0
$850$	4.47411	0.153461
$851$	−4.93337	−0.169114
$852$	0	0
$853$	−20.9935	−0.718805	−0.359402	−	0.933183i	$-0.617019\pi$
$853$	−20.9935	−0.718805	−0.359402	+	0.933183i	$0.617019\pi$
$854$	8.06733	0.276059
$855$	0	0
$856$	31.3522	1.07160
$857$	11.0997	0.379158	0.189579	−	0.981865i	$-0.439288\pi$
$857$	11.0997	0.379158	0.189579	+	0.981865i	$0.439288\pi$
$858$	0	0
$859$	−8.62331	−0.294224	−0.147112	−	0.989120i	$-0.546998\pi$
$859$	−8.62331	−0.294224	−0.147112	+	0.989120i	$0.546998\pi$
$860$	−9.54286	−0.325409
$861$	0	0
$862$	−8.85260	−0.301521
$863$	−47.2590	−1.60872	−0.804358	−	0.594145i	$-0.797491\pi$
$863$	−47.2590	−1.60872	−0.804358	+	0.594145i	$0.797491\pi$
$864$	0	0
$865$	−30.0813	−1.02280
$866$	−80.2875	−2.72828
$867$	0	0
$868$	−18.0516	−0.612712
$869$	−0.537034	−0.0182176
$870$	0	0
$871$	−2.89499	−0.0980931
$872$	54.8065	1.85598
$873$	0	0
$874$	−17.9226	−0.606241
$875$	−15.4007	−0.520639
$876$	0	0
$877$	31.9405	1.07855	0.539277	−	0.842128i	$-0.318697\pi$
$877$	31.9405	1.07855	0.539277	+	0.842128i	$0.318697\pi$
$878$	−42.7464	−1.44262
$879$	0	0
$880$	0.211365	0.00712510
$881$	−7.21420	−0.243053	−0.121526	−	0.992588i	$-0.538779\pi$
$881$	−7.21420	−0.243053	−0.121526	+	0.992588i	$0.538779\pi$
$882$	0	0
$883$	−47.6682	−1.60416	−0.802082	−	0.597215i	$-0.796274\pi$
$883$	−47.6682	−1.60416	−0.802082	+	0.597215i	$0.796274\pi$
$884$	6.68290	0.224770
$885$	0	0
$886$	64.0830	2.15291
$887$	−21.0745	−0.707614	−0.353807	−	0.935318i	$-0.615113\pi$
$887$	−21.0745	−0.707614	−0.353807	+	0.935318i	$0.615113\pi$
$888$	0	0
$889$	17.2370	0.578110
$890$	−51.8336	−1.73747
$891$	0	0
$892$	85.6711	2.86848
$893$	−62.3778	−2.08739
$894$	0	0
$895$	32.8726	1.09881
$896$	25.3155	0.845730
$897$	0	0
$898$	64.2320	2.14345
$899$	4.11205	0.137145
$900$	0	0
$901$	−30.5085	−1.01638
$902$	−0.935276	−0.0311413
$903$	0	0
$904$	53.3335	1.77385
$905$	−21.9247	−0.728800
$906$	0	0
$907$	−57.2893	−1.90226	−0.951131	−	0.308788i	$-0.900077\pi$
$907$	−57.2893	−1.90226	−0.951131	+	0.308788i	$0.900077\pi$
$908$	10.6386	0.353053
$909$	0	0
$910$	−4.42805	−0.146789
$911$	9.02378	0.298971	0.149486	−	0.988764i	$-0.452238\pi$
$911$	9.02378	0.298971	0.149486	+	0.988764i	$0.452238\pi$
$912$	0	0
$913$	0.680176	0.0225105
$914$	11.2556	0.372304
$915$	0	0
$916$	14.4545	0.477590
$917$	14.9054	0.492220
$918$	0	0
$919$	−2.32312	−0.0766328	−0.0383164	−	0.999266i	$-0.512199\pi$
$919$	−2.32312	−0.0766328	−0.0383164	+	0.999266i	$0.512199\pi$
$920$	6.57462	0.216759
$921$	0	0
$922$	45.5200	1.49912
$923$	−2.02504	−0.0666551
$924$	0	0
$925$	−3.38771	−0.111387
$926$	74.0625	2.43385
$927$	0	0
$928$	−4.94033	−0.162174
$929$	51.3272	1.68399	0.841995	−	0.539486i	$-0.181381\pi$
$929$	51.3272	1.68399	0.841995	+	0.539486i	$0.181381\pi$
$930$	0	0
$931$	41.0013	1.34376
$932$	48.7598	1.59718
$933$	0	0
$934$	16.3557	0.535174
$935$	0.990020	0.0323771
$936$	0	0
$937$	−22.1298	−0.722948	−0.361474	−	0.932382i	$-0.617726\pi$
$937$	−22.1298	−0.722948	−0.361474	+	0.932382i	$0.617726\pi$
$938$	12.3904	0.404560
$939$	0	0
$940$	56.3720	1.83865
$941$	45.4073	1.48024	0.740118	−	0.672477i	$-0.234770\pi$
$941$	45.4073	1.48024	0.740118	+	0.672477i	$0.234770\pi$
$942$	0	0
$943$	−2.38205	−0.0775703
$944$	−3.55221	−0.115615
$945$	0	0
$946$	0.535895	0.0174234
$947$	1.93103	0.0627502	0.0313751	−	0.999508i	$-0.490011\pi$
$947$	1.93103	0.0627502	0.0313751	+	0.999508i	$0.490011\pi$
$948$	0	0
$949$	−6.92112	−0.224669
$950$	−12.3073	−0.399302
$951$	0	0
$952$	−11.6102	−0.376288
$953$	18.7399	0.607046	0.303523	−	0.952824i	$-0.401837\pi$
$953$	18.7399	0.607046	0.303523	+	0.952824i	$0.401837\pi$
$954$	0	0
$955$	−25.9745	−0.840516
$956$	−85.2881	−2.75841
$957$	0	0
$958$	8.10554	0.261878
$959$	−11.3250	−0.365702
$960$	0	0
$961$	−14.0910	−0.454549
$962$	−8.06632	−0.260069
$963$	0	0
$964$	100.460	3.23559
$965$	−33.8001	−1.08806
$966$	0	0
$967$	8.69701	0.279677	0.139839	−	0.990174i	$-0.455342\pi$
$967$	8.69701	0.279677	0.139839	+	0.990174i	$0.455342\pi$
$968$	34.7314	1.11631
$969$	0	0
$970$	39.8669	1.28005
$971$	48.9916	1.57221	0.786107	−	0.618090i	$-0.212093\pi$
$971$	48.9916	1.57221	0.786107	+	0.618090i	$0.212093\pi$
$972$	0	0
$973$	−16.3133	−0.522981
$974$	−50.2594	−1.61041
$975$	0	0
$976$	−1.60359	−0.0513297
$977$	−13.6751	−0.437505	−0.218753	−	0.975780i	$-0.570199\pi$
$977$	−13.6751	−0.437505	−0.218753	+	0.975780i	$0.570199\pi$
$978$	0	0
$979$	1.82600	0.0583593
$980$	−37.0536	−1.18363
$981$	0	0
$982$	4.61424	0.147246
$983$	−51.7903	−1.65185	−0.825927	−	0.563777i	$-0.809348\pi$
$983$	−51.7903	−1.65185	−0.825927	+	0.563777i	$0.809348\pi$
$984$	0	0
$985$	−37.1374	−1.18330
$986$	6.51545	0.207494
$987$	0	0
$988$	−18.3832	−0.584848
$989$	1.36487	0.0434003
$990$	0	0
$991$	−32.8944	−1.04492	−0.522462	−	0.852663i	$-0.674986\pi$
$991$	−32.8944	−1.04492	−0.522462	+	0.852663i	$0.674986\pi$
$992$	−20.3149	−0.644998
$993$	0	0
$994$	8.66706	0.274902
$995$	23.8329	0.755555
$996$	0	0
$997$	−46.7778	−1.48147	−0.740733	−	0.671799i	$-0.765522\pi$
$997$	−46.7778	−1.48147	−0.740733	+	0.671799i	$0.765522\pi$
$998$	62.7629	1.98673
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.u.1.2	yes	22
3.2	odd	2		6003.2.a.t.1.21	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.t.1.21	✓	22	3.2	odd	2
6003.2.a.u.1.2	yes	22	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q-2.31658 q^{2} +3.36653 q^{4} +2.07685 q^{5} +1.30399 q^{7} -3.16567 q^{8} +O(q^{10})\)	\(q-2.31658 q^{2} +3.36653 q^{4} +2.07685 q^{5} +1.30399 q^{7} -3.16567 q^{8} -4.81118 q^{10} +0.169489 q^{11} +0.705806 q^{13} -3.02080 q^{14} +0.600462 q^{16} +2.81253 q^{17} -7.73667 q^{19} +6.99178 q^{20} -0.392634 q^{22} -1.00000 q^{23} -0.686694 q^{25} -1.63505 q^{26} +4.38993 q^{28} -1.00000 q^{29} -4.11205 q^{31} +4.94033 q^{32} -6.51545 q^{34} +2.70820 q^{35} +4.93337 q^{37} +17.9226 q^{38} -6.57462 q^{40} +2.38205 q^{41} -1.36487 q^{43} +0.570590 q^{44} +2.31658 q^{46} +8.06261 q^{47} -5.29960 q^{49} +1.59078 q^{50} +2.37612 q^{52} -10.8473 q^{53} +0.352003 q^{55} -4.12802 q^{56} +2.31658 q^{58} -5.91579 q^{59} -2.67059 q^{61} +9.52588 q^{62} -12.6456 q^{64} +1.46585 q^{65} -4.10168 q^{67} +9.46847 q^{68} -6.27375 q^{70} -2.86912 q^{71} -9.80599 q^{73} -11.4285 q^{74} -26.0457 q^{76} +0.221013 q^{77} -3.16854 q^{79} +1.24707 q^{80} -5.51821 q^{82} +4.01310 q^{83} +5.84121 q^{85} +3.16183 q^{86} -0.536547 q^{88} +10.7736 q^{89} +0.920367 q^{91} -3.36653 q^{92} -18.6777 q^{94} -16.0679 q^{95} -8.28631 q^{97} +12.2769 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8}+O(q^{10}) \) 22 * q + 3 * q^2 + 17 * q^4 - 6 * q^7 + 6 * q^8	\( 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8} - 12 q^{10} - 28 q^{13} + q^{14} + 3 q^{16} + 10 q^{17} - 8 q^{19} - 11 q^{22} - 22 q^{23} - 11 q^{26} - 21 q^{28} - 22 q^{29} - 18 q^{31} - 5 q^{32} - 33 q^{34} - 2 q^{35} - 28 q^{37} - 14 q^{38} - 30 q^{40} + 10 q^{41} - 14 q^{43} - 37 q^{44} - 3 q^{46} + 18 q^{47} + 2 q^{49} - 7 q^{50} - 57 q^{52} - 20 q^{53} - 42 q^{55} + 2 q^{56} - 3 q^{58} + 20 q^{59} - 38 q^{61} - 4 q^{62} - 24 q^{64} - 12 q^{65} - 50 q^{67} - 11 q^{68} - 48 q^{70} - 12 q^{71} - 46 q^{73} + 6 q^{74} - 16 q^{76} + 14 q^{77} - 20 q^{79} + 58 q^{80} - 42 q^{82} - 22 q^{83} - 66 q^{85} - 22 q^{86} - 68 q^{88} + 14 q^{89} - 16 q^{91} - 17 q^{92} - 27 q^{94} + 20 q^{95} - 48 q^{97} + 28 q^{98}+O(q^{100}) \) 22 * q + 3 * q^2 + 17 * q^4 - 6 * q^7 + 6 * q^8 - 12 * q^10 - 28 * q^13 + q^14 + 3 * q^16 + 10 * q^17 - 8 * q^19 - 11 * q^22 - 22 * q^23 - 11 * q^26 - 21 * q^28 - 22 * q^29 - 18 * q^31 - 5 * q^32 - 33 * q^34 - 2 * q^35 - 28 * q^37 - 14 * q^38 - 30 * q^40 + 10 * q^41 - 14 * q^43 - 37 * q^44 - 3 * q^46 + 18 * q^47 + 2 * q^49 - 7 * q^50 - 57 * q^52 - 20 * q^53 - 42 * q^55 + 2 * q^56 - 3 * q^58 + 20 * q^59 - 38 * q^61 - 4 * q^62 - 24 * q^64 - 12 * q^65 - 50 * q^67 - 11 * q^68 - 48 * q^70 - 12 * q^71 - 46 * q^73 + 6 * q^74 - 16 * q^76 + 14 * q^77 - 20 * q^79 + 58 * q^80 - 42 * q^82 - 22 * q^83 - 66 * q^85 - 22 * q^86 - 68 * q^88 + 14 * q^89 - 16 * q^91 - 17 * q^92 - 27 * q^94 + 20 * q^95 - 48 * q^97 + 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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