LMFDB - Embedded newform 6003.2.a.v.1.20

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:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q+1.12036 q^{2} -0.744798 q^{4} +4.16877 q^{5} +2.81319 q^{7} -3.07516 q^{8} +O(q^{10})$	\(q+1.12036 q^{2} -0.744798 q^{4} +4.16877 q^{5} +2.81319 q^{7} -3.07516 q^{8} +4.67051 q^{10} +4.43610 q^{11} +5.33557 q^{13} +3.15179 q^{14} -1.95568 q^{16} -2.48562 q^{17} +7.56547 q^{19} -3.10489 q^{20} +4.97002 q^{22} +1.00000 q^{23} +12.3786 q^{25} +5.97775 q^{26} -2.09526 q^{28} -1.00000 q^{29} -4.85295 q^{31} +3.95925 q^{32} -2.78478 q^{34} +11.7276 q^{35} -4.54985 q^{37} +8.47604 q^{38} -12.8196 q^{40} -10.9600 q^{41} +11.2922 q^{43} -3.30400 q^{44} +1.12036 q^{46} -5.94196 q^{47} +0.914065 q^{49} +13.8685 q^{50} -3.97392 q^{52} -9.72222 q^{53} +18.4931 q^{55} -8.65101 q^{56} -1.12036 q^{58} +9.78978 q^{59} -7.22802 q^{61} -5.43704 q^{62} +8.34714 q^{64} +22.2427 q^{65} -2.30813 q^{67} +1.85128 q^{68} +13.1391 q^{70} -1.63808 q^{71} -5.65822 q^{73} -5.09746 q^{74} -5.63475 q^{76} +12.4796 q^{77} -16.5474 q^{79} -8.15277 q^{80} -12.2791 q^{82} -3.24491 q^{83} -10.3620 q^{85} +12.6513 q^{86} -13.6417 q^{88} -0.972379 q^{89} +15.0100 q^{91} -0.744798 q^{92} -6.65712 q^{94} +31.5387 q^{95} -2.70159 q^{97} +1.02408 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8}+O(q^{10}) $ 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8	$ 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}+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

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 $

$2$	1.12036	0.792213	0.396106	−	0.918205i	$-0.370361\pi$
$2$	1.12036	0.792213	0.396106	+	0.918205i	$0.370361\pi$
$3$	0	0
$3$	0	0
$4$	−0.744798	−0.372399
$5$	4.16877	1.86433	0.932165	−	0.362034i	$-0.117918\pi$
$5$	4.16877	1.86433	0.932165	+	0.362034i	$0.117918\pi$
$6$	0	0
$7$	2.81319	1.06329	0.531644	−	0.846968i	$-0.321574\pi$
$7$	2.81319	1.06329	0.531644	+	0.846968i	$0.321574\pi$
$8$	−3.07516	−1.08723
$9$	0	0
$10$	4.67051	1.47695
$11$	4.43610	1.33754	0.668768	−	0.743471i	$-0.266822\pi$
$11$	4.43610	1.33754	0.668768	+	0.743471i	$0.266822\pi$
$12$	0	0
$13$	5.33557	1.47982	0.739910	−	0.672706i	$-0.234868\pi$
$13$	5.33557	1.47982	0.739910	+	0.672706i	$0.234868\pi$
$14$	3.15179	0.842350
$15$	0	0
$16$	−1.95568	−0.488920
$17$	−2.48562	−0.602851	−0.301426	−	0.953490i	$-0.597462\pi$
$17$	−2.48562	−0.602851	−0.301426	+	0.953490i	$0.597462\pi$
$18$	0	0
$19$	7.56547	1.73564	0.867819	−	0.496880i	$-0.165521\pi$
$19$	7.56547	1.73564	0.867819	+	0.496880i	$0.165521\pi$
$20$	−3.10489	−0.694275
$21$	0	0
$22$	4.97002	1.05961
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	12.3786	2.47572
$26$	5.97775	1.17233
$27$	0	0
$28$	−2.09526	−0.395967
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−4.85295	−0.871616	−0.435808	−	0.900040i	$-0.643537\pi$
$31$	−4.85295	−0.871616	−0.435808	+	0.900040i	$0.643537\pi$
$32$	3.95925	0.699903
$33$	0	0
$34$	−2.78478	−0.477586
$35$	11.7276	1.98232
$36$	0	0
$37$	−4.54985	−0.747991	−0.373996	−	0.927430i	$-0.622012\pi$
$37$	−4.54985	−0.747991	−0.373996	+	0.927430i	$0.622012\pi$
$38$	8.47604	1.37499
$39$	0	0
$40$	−12.8196	−2.02696
$41$	−10.9600	−1.71166	−0.855831	−	0.517256i	$-0.826954\pi$
$41$	−10.9600	−1.71166	−0.855831	+	0.517256i	$0.826954\pi$
$42$	0	0
$43$	11.2922	1.72204	0.861019	−	0.508573i	$-0.169827\pi$
$43$	11.2922	1.72204	0.861019	+	0.508573i	$0.169827\pi$
$44$	−3.30400	−0.498097
$45$	0	0
$46$	1.12036	0.165188
$47$	−5.94196	−0.866724	−0.433362	−	0.901220i	$-0.642673\pi$
$47$	−5.94196	−0.866724	−0.433362	+	0.901220i	$0.642673\pi$
$48$	0	0
$49$	0.914065	0.130581
$50$	13.8685	1.96130
$51$	0	0
$52$	−3.97392	−0.551084
$53$	−9.72222	−1.33545	−0.667725	−	0.744408i	$-0.732732\pi$
$53$	−9.72222	−1.33545	−0.667725	+	0.744408i	$0.732732\pi$
$54$	0	0
$55$	18.4931	2.49361
$56$	−8.65101	−1.15604
$57$	0	0
$58$	−1.12036	−0.147110
$59$	9.78978	1.27452	0.637261	−	0.770648i	$-0.280067\pi$
$59$	9.78978	1.27452	0.637261	+	0.770648i	$0.280067\pi$
$60$	0	0
$61$	−7.22802	−0.925454	−0.462727	−	0.886501i	$-0.653129\pi$
$61$	−7.22802	−0.925454	−0.462727	+	0.886501i	$0.653129\pi$
$62$	−5.43704	−0.690505
$63$	0	0
$64$	8.34714	1.04339
$65$	22.2427	2.75887
$66$	0	0
$67$	−2.30813	−0.281983	−0.140992	−	0.990011i	$-0.545029\pi$
$67$	−2.30813	−0.281983	−0.140992	+	0.990011i	$0.545029\pi$
$68$	1.85128	0.224501
$69$	0	0
$70$	13.1391	1.57042
$71$	−1.63808	−0.194405	−0.0972023	−	0.995265i	$-0.530989\pi$
$71$	−1.63808	−0.194405	−0.0972023	+	0.995265i	$0.530989\pi$
$72$	0	0
$73$	−5.65822	−0.662245	−0.331122	−	0.943588i	$-0.607427\pi$
$73$	−5.65822	−0.662245	−0.331122	+	0.943588i	$0.607427\pi$
$74$	−5.09746	−0.592568
$75$	0	0
$76$	−5.63475	−0.646350
$77$	12.4796	1.42219
$78$	0	0
$79$	−16.5474	−1.86173	−0.930865	−	0.365363i	$-0.880945\pi$
$79$	−16.5474	−1.86173	−0.930865	+	0.365363i	$0.880945\pi$
$80$	−8.15277	−0.911508
$81$	0	0
$82$	−12.2791	−1.35600
$83$	−3.24491	−0.356175	−0.178088	−	0.984015i	$-0.556991\pi$
$83$	−3.24491	−0.356175	−0.178088	+	0.984015i	$0.556991\pi$
$84$	0	0
$85$	−10.3620	−1.12391
$86$	12.6513	1.36422
$87$	0	0
$88$	−13.6417	−1.45421
$89$	−0.972379	−0.103072	−0.0515360	−	0.998671i	$-0.516412\pi$
$89$	−0.972379	−0.103072	−0.0515360	+	0.998671i	$0.516412\pi$
$90$	0	0
$91$	15.0100	1.57348
$92$	−0.744798	−0.0776506
$93$	0	0
$94$	−6.65712	−0.686630
$95$	31.5387	3.23580
$96$	0	0
$97$	−2.70159	−0.274305	−0.137152	−	0.990550i	$-0.543795\pi$
$97$	−2.70159	−0.274305	−0.137152	+	0.990550i	$0.543795\pi$
$98$	1.02408	0.103448
$99$	0	0
$100$	−9.21958	−0.921958
$101$	−13.8961	−1.38272	−0.691358	−	0.722513i	$-0.742987\pi$
$101$	−13.8961	−1.38272	−0.691358	+	0.722513i	$0.742987\pi$
$102$	0	0
$103$	3.98243	0.392401	0.196200	−	0.980564i	$-0.437140\pi$
$103$	3.98243	0.392401	0.196200	+	0.980564i	$0.437140\pi$
$104$	−16.4077	−1.60891
$105$	0	0
$106$	−10.8924	−1.05796
$107$	1.62159	0.156765	0.0783826	−	0.996923i	$-0.475024\pi$
$107$	1.62159	0.156765	0.0783826	+	0.996923i	$0.475024\pi$
$108$	0	0
$109$	0.991475	0.0949661	0.0474830	−	0.998872i	$-0.484880\pi$
$109$	0.991475	0.0949661	0.0474830	+	0.998872i	$0.484880\pi$
$110$	20.7189	1.97547
$111$	0	0
$112$	−5.50171	−0.519863
$113$	−13.3710	−1.25783	−0.628917	−	0.777472i	$-0.716502\pi$
$113$	−13.3710	−1.25783	−0.628917	+	0.777472i	$0.716502\pi$
$114$	0	0
$115$	4.16877	0.388740
$116$	0.744798	0.0691528
$117$	0	0
$118$	10.9681	1.00969
$119$	−6.99253	−0.641004
$120$	0	0
$121$	8.67901	0.789001
$122$	−8.09797	−0.733156
$123$	0	0
$124$	3.61447	0.324589
$125$	30.7598	2.75124
$126$	0	0
$127$	−18.9490	−1.68145	−0.840724	−	0.541464i	$-0.817870\pi$
$127$	−18.9490	−1.68145	−0.840724	+	0.541464i	$0.817870\pi$
$128$	1.43328	0.126685
$129$	0	0
$130$	24.9198	2.18561
$131$	17.7678	1.55238	0.776189	−	0.630501i	$-0.217150\pi$
$131$	17.7678	1.55238	0.776189	+	0.630501i	$0.217150\pi$
$132$	0	0
$133$	21.2831	1.84548
$134$	−2.58593	−0.223391
$135$	0	0
$136$	7.64367	0.655439
$137$	10.2205	0.873196	0.436598	−	0.899657i	$-0.356183\pi$
$137$	10.2205	0.873196	0.436598	+	0.899657i	$0.356183\pi$
$138$	0	0
$139$	−13.3021	−1.12827	−0.564135	−	0.825683i	$-0.690790\pi$
$139$	−13.3021	−1.12827	−0.564135	+	0.825683i	$0.690790\pi$
$140$	−8.73466	−0.738214
$141$	0	0
$142$	−1.83524	−0.154010
$143$	23.6691	1.97931
$144$	0	0
$145$	−4.16877	−0.346197
$146$	−6.33923	−0.524639
$147$	0	0
$148$	3.38872	0.278551
$149$	−4.69316	−0.384478	−0.192239	−	0.981348i	$-0.561575\pi$
$149$	−4.69316	−0.384478	−0.192239	+	0.981348i	$0.561575\pi$
$150$	0	0
$151$	7.04785	0.573546	0.286773	−	0.957999i	$-0.407417\pi$
$151$	7.04785	0.573546	0.286773	+	0.957999i	$0.407417\pi$
$152$	−23.2650	−1.88704
$153$	0	0
$154$	13.9816	1.12667
$155$	−20.2308	−1.62498
$156$	0	0
$157$	10.0485	0.801960	0.400980	−	0.916087i	$-0.368670\pi$
$157$	10.0485	0.801960	0.400980	+	0.916087i	$0.368670\pi$
$158$	−18.5390	−1.47489
$159$	0	0
$160$	16.5052	1.30485
$161$	2.81319	0.221711
$162$	0	0
$163$	−4.45128	−0.348651	−0.174326	−	0.984688i	$-0.555775\pi$
$163$	−4.45128	−0.348651	−0.174326	+	0.984688i	$0.555775\pi$
$164$	8.16297	0.637421
$165$	0	0
$166$	−3.63546	−0.282167
$167$	4.75376	0.367857	0.183928	−	0.982940i	$-0.441119\pi$
$167$	4.75376	0.367857	0.183928	+	0.982940i	$0.441119\pi$
$168$	0	0
$169$	15.4683	1.18987
$170$	−11.6091	−0.890378
$171$	0	0
$172$	−8.41038	−0.641285
$173$	0.0612630	0.00465774	0.00232887	−	0.999997i	$-0.499259\pi$
$173$	0.0612630	0.00465774	0.00232887	+	0.999997i	$0.499259\pi$
$174$	0	0
$175$	34.8235	2.63241
$176$	−8.67560	−0.653948
$177$	0	0
$178$	−1.08941	−0.0816549
$179$	−5.80083	−0.433575	−0.216787	−	0.976219i	$-0.569558\pi$
$179$	−5.80083	−0.433575	−0.216787	+	0.976219i	$0.569558\pi$
$180$	0	0
$181$	21.6061	1.60597	0.802986	−	0.595999i	$-0.203244\pi$
$181$	21.6061	1.60597	0.802986	+	0.595999i	$0.203244\pi$
$182$	16.8166	1.24653
$183$	0	0
$184$	−3.07516	−0.226704
$185$	−18.9673	−1.39450
$186$	0	0
$187$	−11.0265	−0.806335
$188$	4.42556	0.322767
$189$	0	0
$190$	35.3346	2.56344
$191$	−13.2691	−0.960115	−0.480058	−	0.877237i	$-0.659384\pi$
$191$	−13.2691	−0.960115	−0.480058	+	0.877237i	$0.659384\pi$
$192$	0	0
$193$	−15.6791	−1.12861	−0.564304	−	0.825567i	$-0.690855\pi$
$193$	−15.6791	−1.12861	−0.564304	+	0.825567i	$0.690855\pi$
$194$	−3.02674	−0.217308
$195$	0	0
$196$	−0.680794	−0.0486281
$197$	−11.4205	−0.813678	−0.406839	−	0.913500i	$-0.633369\pi$
$197$	−11.4205	−0.813678	−0.406839	+	0.913500i	$0.633369\pi$
$198$	0	0
$199$	−13.3317	−0.945061	−0.472530	−	0.881314i	$-0.656659\pi$
$199$	−13.3317	−0.945061	−0.472530	+	0.881314i	$0.656659\pi$
$200$	−38.0662	−2.69169
$201$	0	0
$202$	−15.5686	−1.09540
$203$	−2.81319	−0.197448
$204$	0	0
$205$	−45.6896	−3.19110
$206$	4.46175	0.310865
$207$	0	0
$208$	−10.4347	−0.723514
$209$	33.5612	2.32148
$210$	0	0
$211$	13.4414	0.925341	0.462671	−	0.886530i	$-0.346891\pi$
$211$	13.4414	0.925341	0.462671	+	0.886530i	$0.346891\pi$
$212$	7.24109	0.497320
$213$	0	0
$214$	1.81676	0.124191
$215$	47.0744	3.21045
$216$	0	0
$217$	−13.6523	−0.926778
$218$	1.11081	0.0752333
$219$	0	0
$220$	−13.7736	−0.928617
$221$	−13.2622	−0.892112
$222$	0	0
$223$	1.49196	0.0999092	0.0499546	−	0.998751i	$-0.484092\pi$
$223$	1.49196	0.0999092	0.0499546	+	0.998751i	$0.484092\pi$
$224$	11.1381	0.744199
$225$	0	0
$226$	−14.9803	−0.996473
$227$	−6.69827	−0.444580	−0.222290	−	0.974981i	$-0.571353\pi$
$227$	−6.69827	−0.444580	−0.222290	+	0.974981i	$0.571353\pi$
$228$	0	0
$229$	21.6573	1.43115	0.715576	−	0.698535i	$-0.246164\pi$
$229$	21.6573	1.43115	0.715576	+	0.698535i	$0.246164\pi$
$230$	4.67051	0.307964
$231$	0	0
$232$	3.07516	0.201894
$233$	−4.85454	−0.318032	−0.159016	−	0.987276i	$-0.550832\pi$
$233$	−4.85454	−0.318032	−0.159016	+	0.987276i	$0.550832\pi$
$234$	0	0
$235$	−24.7707	−1.61586
$236$	−7.29141	−0.474631
$237$	0	0
$238$	−7.83414	−0.507812
$239$	6.21680	0.402131	0.201066	−	0.979578i	$-0.435560\pi$
$239$	6.21680	0.402131	0.201066	+	0.979578i	$0.435560\pi$
$240$	0	0
$241$	14.7423	0.949634	0.474817	−	0.880084i	$-0.342514\pi$
$241$	14.7423	0.949634	0.474817	+	0.880084i	$0.342514\pi$
$242$	9.72360	0.625057
$243$	0	0
$244$	5.38342	0.344638
$245$	3.81052	0.243445
$246$	0	0
$247$	40.3661	2.56843
$248$	14.9236	0.947648
$249$	0	0
$250$	34.4620	2.17957
$251$	−24.1843	−1.52650	−0.763248	−	0.646105i	$-0.776397\pi$
$251$	−24.1843	−1.52650	−0.763248	+	0.646105i	$0.776397\pi$
$252$	0	0
$253$	4.43610	0.278895
$254$	−21.2296	−1.33206
$255$	0	0
$256$	−15.0885	−0.943031
$257$	15.1135	0.942756	0.471378	−	0.881931i	$-0.343757\pi$
$257$	15.1135	0.942756	0.471378	+	0.881931i	$0.343757\pi$
$258$	0	0
$259$	−12.7996	−0.795330
$260$	−16.5664	−1.02740
$261$	0	0
$262$	19.9063	1.22981
$263$	−7.73656	−0.477057	−0.238528	−	0.971136i	$-0.576665\pi$
$263$	−7.73656	−0.477057	−0.238528	+	0.971136i	$0.576665\pi$
$264$	0	0
$265$	−40.5297	−2.48972
$266$	23.8447	1.46201
$267$	0	0
$268$	1.71909	0.105010
$269$	−22.9066	−1.39664	−0.698319	−	0.715786i	$-0.746068\pi$
$269$	−22.9066	−1.39664	−0.698319	+	0.715786i	$0.746068\pi$
$270$	0	0
$271$	17.4194	1.05816	0.529078	−	0.848573i	$-0.322538\pi$
$271$	17.4194	1.05816	0.529078	+	0.848573i	$0.322538\pi$
$272$	4.86108	0.294746
$273$	0	0
$274$	11.4506	0.691757
$275$	54.9129	3.31137
$276$	0	0
$277$	21.4223	1.28714	0.643570	−	0.765388i	$-0.277453\pi$
$277$	21.4223	1.28714	0.643570	+	0.765388i	$0.277453\pi$
$278$	−14.9031	−0.893830
$279$	0	0
$280$	−36.0641	−2.15524
$281$	22.2858	1.32946	0.664730	−	0.747083i	$-0.268546\pi$
$281$	22.2858	1.32946	0.664730	+	0.747083i	$0.268546\pi$
$282$	0	0
$283$	−15.7171	−0.934282	−0.467141	−	0.884183i	$-0.654716\pi$
$283$	−15.7171	−0.934282	−0.467141	+	0.884183i	$0.654716\pi$
$284$	1.22004	0.0723961
$285$	0	0
$286$	26.5179	1.56804
$287$	−30.8326	−1.81999
$288$	0	0
$289$	−10.8217	−0.636570
$290$	−4.67051	−0.274262
$291$	0	0
$292$	4.21423	0.246619
$293$	32.5262	1.90020	0.950101	−	0.311941i	$-0.100979\pi$
$293$	32.5262	1.90020	0.950101	+	0.311941i	$0.100979\pi$
$294$	0	0
$295$	40.8113	2.37613
$296$	13.9915	0.813240
$297$	0	0
$298$	−5.25802	−0.304589
$299$	5.33557	0.308564
$300$	0	0
$301$	31.7670	1.83102
$302$	7.89611	0.454370
$303$	0	0
$304$	−14.7956	−0.848588
$305$	−30.1320	−1.72535
$306$	0	0
$307$	−19.5901	−1.11806	−0.559032	−	0.829146i	$-0.688827\pi$
$307$	−19.5901	−1.11806	−0.559032	+	0.829146i	$0.688827\pi$
$308$	−9.29480	−0.529620
$309$	0	0
$310$	−22.6658	−1.28733
$311$	−7.05095	−0.399823	−0.199911	−	0.979814i	$-0.564065\pi$
$311$	−7.05095	−0.399823	−0.199911	+	0.979814i	$0.564065\pi$
$312$	0	0
$313$	−17.4206	−0.984672	−0.492336	−	0.870405i	$-0.663857\pi$
$313$	−17.4206	−0.984672	−0.492336	+	0.870405i	$0.663857\pi$
$314$	11.2580	0.635323
$315$	0	0
$316$	12.3245	0.693306
$317$	−15.8141	−0.888208	−0.444104	−	0.895975i	$-0.646478\pi$
$317$	−15.8141	−0.888208	−0.444104	+	0.895975i	$0.646478\pi$
$318$	0	0
$319$	−4.43610	−0.248374
$320$	34.7973	1.94523
$321$	0	0
$322$	3.15179	0.175642
$323$	−18.8049	−1.04633
$324$	0	0
$325$	66.0470	3.66363
$326$	−4.98703	−0.276206
$327$	0	0
$328$	33.7037	1.86097
$329$	−16.7159	−0.921577
$330$	0	0
$331$	−4.99944	−0.274794	−0.137397	−	0.990516i	$-0.543874\pi$
$331$	−4.99944	−0.274794	−0.137397	+	0.990516i	$0.543874\pi$
$332$	2.41680	0.132639
$333$	0	0
$334$	5.32591	0.291421
$335$	−9.62206	−0.525709
$336$	0	0
$337$	31.8629	1.73569	0.867843	−	0.496839i	$-0.165506\pi$
$337$	31.8629	1.73569	0.867843	+	0.496839i	$0.165506\pi$
$338$	17.3300	0.942629
$339$	0	0
$340$	7.71758	0.418544
$341$	−21.5282	−1.16582
$342$	0	0
$343$	−17.1209	−0.924443
$344$	−34.7252	−1.87225
$345$	0	0
$346$	0.0686365	0.00368992
$347$	19.7954	1.06267	0.531336	−	0.847161i	$-0.321690\pi$
$347$	19.7954	1.06267	0.531336	+	0.847161i	$0.321690\pi$
$348$	0	0
$349$	28.6600	1.53413	0.767066	−	0.641568i	$-0.221716\pi$
$349$	28.6600	1.53413	0.767066	+	0.641568i	$0.221716\pi$
$350$	39.0148	2.08543
$351$	0	0
$352$	17.5636	0.936146
$353$	−8.16413	−0.434533	−0.217266	−	0.976112i	$-0.569714\pi$
$353$	−8.16413	−0.434533	−0.217266	+	0.976112i	$0.569714\pi$
$354$	0	0
$355$	−6.82879	−0.362434
$356$	0.724226	0.0383839
$357$	0	0
$358$	−6.49901	−0.343483
$359$	−17.6520	−0.931637	−0.465818	−	0.884880i	$-0.654240\pi$
$359$	−17.6520	−0.931637	−0.465818	+	0.884880i	$0.654240\pi$
$360$	0	0
$361$	38.2364	2.01244
$362$	24.2066	1.27227
$363$	0	0
$364$	−11.1794	−0.585961
$365$	−23.5878	−1.23464
$366$	0	0
$367$	10.5315	0.549741	0.274871	−	0.961481i	$-0.411365\pi$
$367$	10.5315	0.549741	0.274871	+	0.961481i	$0.411365\pi$
$368$	−1.95568	−0.101947
$369$	0	0
$370$	−21.2501	−1.10474
$371$	−27.3505	−1.41997
$372$	0	0
$373$	24.6025	1.27387	0.636934	−	0.770919i	$-0.280202\pi$
$373$	24.6025	1.27387	0.636934	+	0.770919i	$0.280202\pi$
$374$	−12.3536	−0.638789
$375$	0	0
$376$	18.2725	0.942330
$377$	−5.33557	−0.274796
$378$	0	0
$379$	19.9884	1.02673	0.513367	−	0.858169i	$-0.328398\pi$
$379$	19.9884	1.02673	0.513367	+	0.858169i	$0.328398\pi$
$380$	−23.4900	−1.20501
$381$	0	0
$382$	−14.8661	−0.760616
$383$	22.2611	1.13749	0.568744	−	0.822515i	$-0.307429\pi$
$383$	22.2611	1.13749	0.568744	+	0.822515i	$0.307429\pi$
$384$	0	0
$385$	52.0247	2.65142
$386$	−17.5662	−0.894098
$387$	0	0
$388$	2.01214	0.102151
$389$	2.79279	0.141600	0.0708002	−	0.997491i	$-0.477445\pi$
$389$	2.79279	0.141600	0.0708002	+	0.997491i	$0.477445\pi$
$390$	0	0
$391$	−2.48562	−0.125703
$392$	−2.81089	−0.141972
$393$	0	0
$394$	−12.7951	−0.644606
$395$	−68.9823	−3.47088
$396$	0	0
$397$	−4.29253	−0.215436	−0.107718	−	0.994181i	$-0.534354\pi$
$397$	−4.29253	−0.215436	−0.107718	+	0.994181i	$0.534354\pi$
$398$	−14.9363	−0.748689
$399$	0	0
$400$	−24.2086	−1.21043
$401$	25.9994	1.29835	0.649174	−	0.760640i	$-0.275115\pi$
$401$	25.9994	1.29835	0.649174	+	0.760640i	$0.275115\pi$
$402$	0	0
$403$	−25.8933	−1.28983
$404$	10.3498	0.514922
$405$	0	0
$406$	−3.15179	−0.156420
$407$	−20.1836	−1.00046
$408$	0	0
$409$	−27.7990	−1.37457	−0.687286	−	0.726387i	$-0.741198\pi$
$409$	−27.7990	−1.37457	−0.687286	+	0.726387i	$0.741198\pi$
$410$	−51.1887	−2.52803
$411$	0	0
$412$	−2.96611	−0.146130
$413$	27.5406	1.35518
$414$	0	0
$415$	−13.5273	−0.664028
$416$	21.1249	1.03573
$417$	0	0
$418$	37.6006	1.83910
$419$	−28.6153	−1.39795	−0.698974	−	0.715147i	$-0.746360\pi$
$419$	−28.6153	−1.39795	−0.698974	+	0.715147i	$0.746360\pi$
$420$	0	0
$421$	19.7918	0.964595	0.482298	−	0.876007i	$-0.339802\pi$
$421$	19.7918	0.964595	0.482298	+	0.876007i	$0.339802\pi$
$422$	15.0591	0.733067
$423$	0	0
$424$	29.8973	1.45194
$425$	−30.7685	−1.49249
$426$	0	0
$427$	−20.3338	−0.984023
$428$	−1.20776	−0.0583792
$429$	0	0
$430$	52.7402	2.54336
$431$	−26.1447	−1.25934	−0.629672	−	0.776861i	$-0.716811\pi$
$431$	−26.1447	−1.25934	−0.629672	+	0.776861i	$0.716811\pi$
$432$	0	0
$433$	37.3121	1.79311	0.896553	−	0.442937i	$-0.146064\pi$
$433$	37.3121	1.79311	0.896553	+	0.442937i	$0.146064\pi$
$434$	−15.2955	−0.734206
$435$	0	0
$436$	−0.738449	−0.0353653
$437$	7.56547	0.361906
$438$	0	0
$439$	−6.95189	−0.331796	−0.165898	−	0.986143i	$-0.553052\pi$
$439$	−6.95189	−0.331796	−0.165898	+	0.986143i	$0.553052\pi$
$440$	−56.8691	−2.71113
$441$	0	0
$442$	−14.8584	−0.706742
$443$	−39.5639	−1.87974	−0.939869	−	0.341536i	$-0.889053\pi$
$443$	−39.5639	−1.87974	−0.939869	+	0.341536i	$0.889053\pi$
$444$	0	0
$445$	−4.05362	−0.192160
$446$	1.67153	0.0791493
$447$	0	0
$448$	23.4821	1.10943
$449$	5.31517	0.250838	0.125419	−	0.992104i	$-0.459972\pi$
$449$	5.31517	0.250838	0.125419	+	0.992104i	$0.459972\pi$
$450$	0	0
$451$	−48.6196	−2.28941
$452$	9.95867	0.468416
$453$	0	0
$454$	−7.50446	−0.352202
$455$	62.5732	2.93348
$456$	0	0
$457$	14.3196	0.669843	0.334922	−	0.942246i	$-0.391290\pi$
$457$	14.3196	0.669843	0.334922	+	0.942246i	$0.391290\pi$
$458$	24.2639	1.13378
$459$	0	0
$460$	−3.10489	−0.144766
$461$	12.2764	0.571771	0.285886	−	0.958264i	$-0.407712\pi$
$461$	12.2764	0.571771	0.285886	+	0.958264i	$0.407712\pi$
$462$	0	0
$463$	−14.0444	−0.652696	−0.326348	−	0.945250i	$-0.605818\pi$
$463$	−14.0444	−0.652696	−0.326348	+	0.945250i	$0.605818\pi$
$464$	1.95568	0.0907902
$465$	0	0
$466$	−5.43883	−0.251949
$467$	33.3320	1.54242	0.771211	−	0.636580i	$-0.219652\pi$
$467$	33.3320	1.54242	0.771211	+	0.636580i	$0.219652\pi$
$468$	0	0
$469$	−6.49322	−0.299829
$470$	−27.7520	−1.28010
$471$	0	0
$472$	−30.1051	−1.38570
$473$	50.0932	2.30329
$474$	0	0
$475$	93.6501	4.29696
$476$	5.20802	0.238709
$477$	0	0
$478$	6.96504	0.318573
$479$	4.16701	0.190396	0.0951979	−	0.995458i	$-0.469652\pi$
$479$	4.16701	0.190396	0.0951979	+	0.995458i	$0.469652\pi$
$480$	0	0
$481$	−24.2761	−1.10689
$482$	16.5166	0.752312
$483$	0	0
$484$	−6.46411	−0.293823
$485$	−11.2623	−0.511394
$486$	0	0
$487$	23.2982	1.05574	0.527872	−	0.849324i	$-0.322990\pi$
$487$	23.2982	1.05574	0.527872	+	0.849324i	$0.322990\pi$
$488$	22.2273	1.00618
$489$	0	0
$490$	4.26915	0.192861
$491$	−11.7314	−0.529429	−0.264714	−	0.964327i	$-0.585278\pi$
$491$	−11.7314	−0.529429	−0.264714	+	0.964327i	$0.585278\pi$
$492$	0	0
$493$	2.48562	0.111947
$494$	45.2245	2.03475
$495$	0	0
$496$	9.49082	0.426150
$497$	−4.60825	−0.206708
$498$	0	0
$499$	2.52259	0.112927	0.0564633	−	0.998405i	$-0.482018\pi$
$499$	2.52259	0.112927	0.0564633	+	0.998405i	$0.482018\pi$
$500$	−22.9098	−1.02456
$501$	0	0
$502$	−27.0950	−1.20931
$503$	23.2887	1.03839	0.519196	−	0.854655i	$-0.326231\pi$
$503$	23.2887	1.03839	0.519196	+	0.854655i	$0.326231\pi$
$504$	0	0
$505$	−57.9297	−2.57784
$506$	4.97002	0.220945
$507$	0	0
$508$	14.1131	0.626169
$509$	10.5768	0.468807	0.234404	−	0.972139i	$-0.424686\pi$
$509$	10.5768	0.468807	0.234404	+	0.972139i	$0.424686\pi$
$510$	0	0
$511$	−15.9177	−0.704157
$512$	−19.7711	−0.873766
$513$	0	0
$514$	16.9326	0.746863
$515$	16.6018	0.731564
$516$	0	0
$517$	−26.3592	−1.15927
$518$	−14.3402	−0.630070
$519$	0	0
$520$	−68.3999	−2.99954
$521$	−20.5162	−0.898830	−0.449415	−	0.893323i	$-0.648367\pi$
$521$	−20.5162	−0.898830	−0.449415	+	0.893323i	$0.648367\pi$
$522$	0	0
$523$	−29.3240	−1.28225	−0.641125	−	0.767437i	$-0.721532\pi$
$523$	−29.3240	−1.28225	−0.641125	+	0.767437i	$0.721532\pi$
$524$	−13.2334	−0.578104
$525$	0	0
$526$	−8.66772	−0.377930
$527$	12.0626	0.525455
$528$	0	0
$529$	1.00000	0.0434783
$530$	−45.4077	−1.97239
$531$	0	0
$532$	−15.8516	−0.687256
$533$	−58.4777	−2.53295
$534$	0	0
$535$	6.76004	0.292262
$536$	7.09786	0.306581
$537$	0	0
$538$	−25.6636	−1.10644
$539$	4.05489	0.174656
$540$	0	0
$541$	12.6903	0.545599	0.272800	−	0.962071i	$-0.412050\pi$
$541$	12.6903	0.545599	0.272800	+	0.962071i	$0.412050\pi$
$542$	19.5160	0.838285
$543$	0	0
$544$	−9.84119	−0.421938
$545$	4.13323	0.177048
$546$	0	0
$547$	−27.0199	−1.15529	−0.577645	−	0.816288i	$-0.696028\pi$
$547$	−27.0199	−1.15529	−0.577645	+	0.816288i	$0.696028\pi$
$548$	−7.61221	−0.325178
$549$	0	0
$550$	61.5221	2.62331
$551$	−7.56547	−0.322300
$552$	0	0
$553$	−46.5511	−1.97955
$554$	24.0006	1.01969
$555$	0	0
$556$	9.90738	0.420167
$557$	−9.96845	−0.422377	−0.211188	−	0.977445i	$-0.567733\pi$
$557$	−9.96845	−0.422377	−0.211188	+	0.977445i	$0.567733\pi$
$558$	0	0
$559$	60.2501	2.54831
$560$	−22.9353	−0.969195
$561$	0	0
$562$	24.9681	1.05322
$563$	−10.8810	−0.458578	−0.229289	−	0.973358i	$-0.573640\pi$
$563$	−10.8810	−0.458578	−0.229289	+	0.973358i	$0.573640\pi$
$564$	0	0
$565$	−55.7405	−2.34502
$566$	−17.6087	−0.740150
$567$	0	0
$568$	5.03736	0.211363
$569$	13.3460	0.559492	0.279746	−	0.960074i	$-0.409750\pi$
$569$	13.3460	0.559492	0.279746	+	0.960074i	$0.409750\pi$
$570$	0	0
$571$	22.1550	0.927159	0.463579	−	0.886055i	$-0.346565\pi$
$571$	22.1550	0.927159	0.463579	+	0.886055i	$0.346565\pi$
$572$	−17.6287	−0.737094
$573$	0	0
$574$	−34.5435	−1.44182
$575$	12.3786	0.516224
$576$	0	0
$577$	−27.3461	−1.13843	−0.569217	−	0.822187i	$-0.692754\pi$
$577$	−27.3461	−1.13843	−0.569217	+	0.822187i	$0.692754\pi$
$578$	−12.1242	−0.504299
$579$	0	0
$580$	3.10489	0.128924
$581$	−9.12857	−0.378717
$582$	0	0
$583$	−43.1288	−1.78621
$584$	17.3999	0.720014
$585$	0	0
$586$	36.4410	1.50536
$587$	−35.9367	−1.48327	−0.741633	−	0.670805i	$-0.765949\pi$
$587$	−35.9367	−1.48327	−0.741633	+	0.670805i	$0.765949\pi$
$588$	0	0
$589$	−36.7149	−1.51281
$590$	45.7233	1.88240
$591$	0	0
$592$	8.89805	0.365708
$593$	0.161975	0.00665150	0.00332575	−	0.999994i	$-0.498941\pi$
$593$	0.161975	0.00665150	0.00332575	+	0.999994i	$0.498941\pi$
$594$	0	0
$595$	−29.1502	−1.19504
$596$	3.49545	0.143179
$597$	0	0
$598$	5.97775	0.244448
$599$	33.4120	1.36518	0.682589	−	0.730802i	$-0.260854\pi$
$599$	33.4120	1.36518	0.682589	+	0.730802i	$0.260854\pi$
$600$	0	0
$601$	−36.6324	−1.49427	−0.747133	−	0.664674i	$-0.768570\pi$
$601$	−36.6324	−1.49427	−0.747133	+	0.664674i	$0.768570\pi$
$602$	35.5905	1.45056
$603$	0	0
$604$	−5.24922	−0.213588
$605$	36.1808	1.47096
$606$	0	0
$607$	11.0793	0.449696	0.224848	−	0.974394i	$-0.427812\pi$
$607$	11.0793	0.449696	0.224848	+	0.974394i	$0.427812\pi$
$608$	29.9536	1.21478
$609$	0	0
$610$	−33.7586	−1.36684
$611$	−31.7037	−1.28260
$612$	0	0
$613$	42.4276	1.71364	0.856818	−	0.515619i	$-0.172438\pi$
$613$	42.4276	1.71364	0.856818	+	0.515619i	$0.172438\pi$
$614$	−21.9479	−0.885745
$615$	0	0
$616$	−38.3768	−1.54625
$617$	37.9339	1.52716	0.763582	−	0.645711i	$-0.223439\pi$
$617$	37.9339	1.52716	0.763582	+	0.645711i	$0.223439\pi$
$618$	0	0
$619$	35.4038	1.42300	0.711498	−	0.702688i	$-0.248017\pi$
$619$	35.4038	1.42300	0.711498	+	0.702688i	$0.248017\pi$
$620$	15.0679	0.605141
$621$	0	0
$622$	−7.89959	−0.316745
$623$	−2.73549	−0.109595
$624$	0	0
$625$	66.3372	2.65349
$626$	−19.5173	−0.780069
$627$	0	0
$628$	−7.48413	−0.298649
$629$	11.3092	0.450927
$630$	0	0
$631$	4.72708	0.188182	0.0940910	−	0.995564i	$-0.470006\pi$
$631$	4.72708	0.188182	0.0940910	+	0.995564i	$0.470006\pi$
$632$	50.8859	2.02413
$633$	0	0
$634$	−17.7175	−0.703650
$635$	−78.9938	−3.13477
$636$	0	0
$637$	4.87706	0.193236
$638$	−4.97002	−0.196765
$639$	0	0
$640$	5.97501	0.236183
$641$	4.57076	0.180534	0.0902671	−	0.995918i	$-0.471228\pi$
$641$	4.57076	0.180534	0.0902671	+	0.995918i	$0.471228\pi$
$642$	0	0
$643$	29.5725	1.16623	0.583113	−	0.812391i	$-0.301834\pi$
$643$	29.5725	1.16623	0.583113	+	0.812391i	$0.301834\pi$
$644$	−2.09526	−0.0825649
$645$	0	0
$646$	−21.0682	−0.828917
$647$	27.7649	1.09155	0.545775	−	0.837932i	$-0.316235\pi$
$647$	27.7649	1.09155	0.545775	+	0.837932i	$0.316235\pi$
$648$	0	0
$649$	43.4285	1.70472
$650$	73.9963	2.90237
$651$	0	0
$652$	3.31531	0.129837
$653$	−44.5667	−1.74403	−0.872016	−	0.489478i	$-0.837187\pi$
$653$	−44.5667	−1.74403	−0.872016	+	0.489478i	$0.837187\pi$
$654$	0	0
$655$	74.0697	2.89414
$656$	21.4342	0.836866
$657$	0	0
$658$	−18.7278	−0.730085
$659$	10.8360	0.422112	0.211056	−	0.977474i	$-0.432310\pi$
$659$	10.8360	0.422112	0.211056	+	0.977474i	$0.432310\pi$
$660$	0	0
$661$	−45.4286	−1.76697	−0.883485	−	0.468460i	$-0.844809\pi$
$661$	−45.4286	−1.76697	−0.883485	+	0.468460i	$0.844809\pi$
$662$	−5.60116	−0.217695
$663$	0	0
$664$	9.97861	0.387245
$665$	88.7245	3.44059
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−3.54059	−0.136990
$669$	0	0
$670$	−10.7802	−0.416474
$671$	−32.0643	−1.23783
$672$	0	0
$673$	−26.0784	−1.00525	−0.502623	−	0.864505i	$-0.667632\pi$
$673$	−26.0784	−1.00525	−0.502623	+	0.864505i	$0.667632\pi$
$674$	35.6979	1.37503
$675$	0	0
$676$	−11.5208	−0.443106
$677$	34.3505	1.32020	0.660098	−	0.751179i	$-0.270515\pi$
$677$	34.3505	1.32020	0.660098	+	0.751179i	$0.270515\pi$
$678$	0	0
$679$	−7.60009	−0.291665
$680$	31.8647	1.22195
$681$	0	0
$682$	−24.1193	−0.923575
$683$	−47.1808	−1.80532	−0.902661	−	0.430352i	$-0.858390\pi$
$683$	−47.1808	−1.80532	−0.902661	+	0.430352i	$0.858390\pi$
$684$	0	0
$685$	42.6069	1.62793
$686$	−19.1816	−0.732355
$687$	0	0
$688$	−22.0838	−0.841939
$689$	−51.8736	−1.97623
$690$	0	0
$691$	22.4762	0.855036	0.427518	−	0.904007i	$-0.359388\pi$
$691$	22.4762	0.855036	0.427518	+	0.904007i	$0.359388\pi$
$692$	−0.0456286	−0.00173454
$693$	0	0
$694$	22.1779	0.841863
$695$	−55.4534	−2.10347
$696$	0	0
$697$	27.2423	1.03188
$698$	32.1094	1.21536
$699$	0	0
$700$	−25.9365	−0.980306
$701$	39.5104	1.49229	0.746144	−	0.665785i	$-0.231903\pi$
$701$	39.5104	1.49229	0.746144	+	0.665785i	$0.231903\pi$
$702$	0	0
$703$	−34.4218	−1.29824
$704$	37.0288	1.39557
$705$	0	0
$706$	−9.14675	−0.344242
$707$	−39.0925	−1.47022
$708$	0	0
$709$	10.4904	0.393975	0.196988	−	0.980406i	$-0.436884\pi$
$709$	10.4904	0.393975	0.196988	+	0.980406i	$0.436884\pi$
$710$	−7.65069	−0.287125
$711$	0	0
$712$	2.99022	0.112063
$713$	−4.85295	−0.181744
$714$	0	0
$715$	98.6711	3.69009
$716$	4.32045	0.161463
$717$	0	0
$718$	−19.7766	−0.738055
$719$	−12.9292	−0.482176	−0.241088	−	0.970503i	$-0.577504\pi$
$719$	−12.9292	−0.482176	−0.241088	+	0.970503i	$0.577504\pi$
$720$	0	0
$721$	11.2034	0.417235
$722$	42.8384	1.59428
$723$	0	0
$724$	−16.0922	−0.598062
$725$	−12.3786	−0.459731
$726$	0	0
$727$	28.8066	1.06838	0.534189	−	0.845365i	$-0.320617\pi$
$727$	28.8066	1.06838	0.534189	+	0.845365i	$0.320617\pi$
$728$	−46.1581	−1.71073
$729$	0	0
$730$	−26.4268	−0.978099
$731$	−28.0680	−1.03813
$732$	0	0
$733$	16.8473	0.622271	0.311136	−	0.950366i	$-0.399291\pi$
$733$	16.8473	0.622271	0.311136	+	0.950366i	$0.399291\pi$
$734$	11.7991	0.435512
$735$	0	0
$736$	3.95925	0.145940
$737$	−10.2391	−0.377162
$738$	0	0
$739$	7.86813	0.289434	0.144717	−	0.989473i	$-0.453773\pi$
$739$	7.86813	0.289434	0.144717	+	0.989473i	$0.453773\pi$
$740$	14.1268	0.519311
$741$	0	0
$742$	−30.6423	−1.12492
$743$	23.7669	0.871923	0.435962	−	0.899965i	$-0.356408\pi$
$743$	23.7669	0.871923	0.435962	+	0.899965i	$0.356408\pi$
$744$	0	0
$745$	−19.5647	−0.716794
$746$	27.5636	1.00917
$747$	0	0
$748$	8.21249	0.300278
$749$	4.56186	0.166687
$750$	0	0
$751$	−2.21512	−0.0808309	−0.0404155	−	0.999183i	$-0.512868\pi$
$751$	−2.21512	−0.0808309	−0.0404155	+	0.999183i	$0.512868\pi$
$752$	11.6206	0.423759
$753$	0	0
$754$	−5.97775	−0.217697
$755$	29.3808	1.06928
$756$	0	0
$757$	−4.67992	−0.170095	−0.0850474	−	0.996377i	$-0.527104\pi$
$757$	−4.67992	−0.170095	−0.0850474	+	0.996377i	$0.527104\pi$
$758$	22.3941	0.813391
$759$	0	0
$760$	−96.9864	−3.51807
$761$	−35.0206	−1.26950	−0.634748	−	0.772719i	$-0.718896\pi$
$761$	−35.0206	−1.26950	−0.634748	+	0.772719i	$0.718896\pi$
$762$	0	0
$763$	2.78921	0.100976
$764$	9.88277	0.357546
$765$	0	0
$766$	24.9404	0.901133
$767$	52.2341	1.88606
$768$	0	0
$769$	−7.18128	−0.258964	−0.129482	−	0.991582i	$-0.541331\pi$
$769$	−7.18128	−0.258964	−0.129482	+	0.991582i	$0.541331\pi$
$770$	58.2862	2.10049
$771$	0	0
$772$	11.6778	0.420293
$773$	18.0532	0.649329	0.324664	−	0.945829i	$-0.394749\pi$
$773$	18.0532	0.649329	0.324664	+	0.945829i	$0.394749\pi$
$774$	0	0
$775$	−60.0729	−2.15788
$776$	8.30780	0.298233
$777$	0	0
$778$	3.12893	0.112178
$779$	−82.9174	−2.97083
$780$	0	0
$781$	−7.26671	−0.260023
$782$	−2.78478	−0.0995837
$783$	0	0
$784$	−1.78762	−0.0638435
$785$	41.8900	1.49512
$786$	0	0
$787$	−0.234495	−0.00835883	−0.00417941	−	0.999991i	$-0.501330\pi$
$787$	−0.234495	−0.00835883	−0.00417941	+	0.999991i	$0.501330\pi$
$788$	8.50597	0.303013
$789$	0	0
$790$	−77.2849	−2.74967
$791$	−37.6151	−1.33744
$792$	0	0
$793$	−38.5656	−1.36951
$794$	−4.80917	−0.170671
$795$	0	0
$796$	9.92944	0.351940
$797$	−6.19317	−0.219373	−0.109687	−	0.993966i	$-0.534985\pi$
$797$	−6.19317	−0.219373	−0.109687	+	0.993966i	$0.534985\pi$
$798$	0	0
$799$	14.7695	0.522506
$800$	49.0101	1.73277
$801$	0	0
$802$	29.1286	1.02857
$803$	−25.1005	−0.885776
$804$	0	0
$805$	11.7276	0.413342
$806$	−29.0097	−1.02182
$807$	0	0
$808$	42.7327	1.50333
$809$	16.6106	0.583998	0.291999	−	0.956419i	$-0.405680\pi$
$809$	16.6106	0.583998	0.291999	+	0.956419i	$0.405680\pi$
$810$	0	0
$811$	−27.6740	−0.971765	−0.485883	−	0.874024i	$-0.661502\pi$
$811$	−27.6740	−0.971765	−0.485883	+	0.874024i	$0.661502\pi$
$812$	2.09526	0.0735293
$813$	0	0
$814$	−22.6129	−0.792581
$815$	−18.5564	−0.650001
$816$	0	0
$817$	85.4305	2.98884
$818$	−31.1448	−1.08895
$819$	0	0
$820$	34.0295	1.18836
$821$	−9.27779	−0.323797	−0.161899	−	0.986807i	$-0.551762\pi$
$821$	−9.27779	−0.323797	−0.161899	+	0.986807i	$0.551762\pi$
$822$	0	0
$823$	−36.0057	−1.25508	−0.627539	−	0.778585i	$-0.715938\pi$
$823$	−36.0057	−1.25508	−0.627539	+	0.778585i	$0.715938\pi$
$824$	−12.2466	−0.426631
$825$	0	0
$826$	30.8553	1.07359
$827$	23.4595	0.815767	0.407883	−	0.913034i	$-0.366267\pi$
$827$	23.4595	0.815767	0.407883	+	0.913034i	$0.366267\pi$
$828$	0	0
$829$	−9.67685	−0.336091	−0.168045	−	0.985779i	$-0.553746\pi$
$829$	−9.67685	−0.336091	−0.168045	+	0.985779i	$0.553746\pi$
$830$	−15.1554	−0.526052
$831$	0	0
$832$	44.5367	1.54403
$833$	−2.27202	−0.0787207
$834$	0	0
$835$	19.8173	0.685806
$836$	−24.9963	−0.864516
$837$	0	0
$838$	−32.0593	−1.10747
$839$	−18.7616	−0.647722	−0.323861	−	0.946105i	$-0.604981\pi$
$839$	−18.7616	−0.647722	−0.323861	+	0.946105i	$0.604981\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	22.1739	0.764165
$843$	0	0
$844$	−10.0111	−0.344596
$845$	64.4837	2.21831
$846$	0	0
$847$	24.4158	0.838935
$848$	19.0135	0.652928
$849$	0	0
$850$	−34.4718	−1.18237
$851$	−4.54985	−0.155967
$852$	0	0
$853$	−11.6480	−0.398821	−0.199411	−	0.979916i	$-0.563903\pi$
$853$	−11.6480	−0.398821	−0.199411	+	0.979916i	$0.563903\pi$
$854$	−22.7812	−0.779556
$855$	0	0
$856$	−4.98665	−0.170440
$857$	−11.5074	−0.393086	−0.196543	−	0.980495i	$-0.562971\pi$
$857$	−11.5074	−0.393086	−0.196543	+	0.980495i	$0.562971\pi$
$858$	0	0
$859$	45.0868	1.53834	0.769171	−	0.639043i	$-0.220670\pi$
$859$	45.0868	1.53834	0.769171	+	0.639043i	$0.220670\pi$
$860$	−35.0609	−1.19557
$861$	0	0
$862$	−29.2914	−0.997669
$863$	26.5473	0.903680	0.451840	−	0.892099i	$-0.350768\pi$
$863$	26.5473	0.903680	0.451840	+	0.892099i	$0.350768\pi$
$864$	0	0
$865$	0.255391	0.00868356
$866$	41.8029	1.42052
$867$	0	0
$868$	10.1682	0.345131
$869$	−73.4061	−2.49013
$870$	0	0
$871$	−12.3152	−0.417284
$872$	−3.04894	−0.103250
$873$	0	0
$874$	8.47604	0.286706
$875$	86.5332	2.92536
$876$	0	0
$877$	−4.23162	−0.142892	−0.0714458	−	0.997444i	$-0.522761\pi$
$877$	−4.23162	−0.142892	−0.0714458	+	0.997444i	$0.522761\pi$
$878$	−7.78861	−0.262853
$879$	0	0
$880$	−36.1666	−1.21917
$881$	−47.0731	−1.58593	−0.792966	−	0.609266i	$-0.791464\pi$
$881$	−47.0731	−1.58593	−0.792966	+	0.609266i	$0.791464\pi$
$882$	0	0
$883$	14.8161	0.498603	0.249301	−	0.968426i	$-0.419799\pi$
$883$	14.8161	0.498603	0.249301	+	0.968426i	$0.419799\pi$
$884$	9.87766	0.332222
$885$	0	0
$886$	−44.3257	−1.48915
$887$	−19.7056	−0.661650	−0.330825	−	0.943692i	$-0.607327\pi$
$887$	−19.7056	−0.661650	−0.330825	+	0.943692i	$0.607327\pi$
$888$	0	0
$889$	−53.3071	−1.78786
$890$	−4.54151	−0.152232
$891$	0	0
$892$	−1.11121	−0.0372061
$893$	−44.9537	−1.50432
$894$	0	0
$895$	−24.1823	−0.808326
$896$	4.03210	0.134703
$897$	0	0
$898$	5.95489	0.198717
$899$	4.85295	0.161855
$900$	0	0
$901$	24.1657	0.805077
$902$	−54.4714	−1.81370
$903$	0	0
$904$	41.1178	1.36756
$905$	90.0709	2.99406
$906$	0	0
$907$	−37.0971	−1.23179	−0.615894	−	0.787829i	$-0.711205\pi$
$907$	−37.0971	−1.23179	−0.615894	+	0.787829i	$0.711205\pi$
$908$	4.98886	0.165561
$909$	0	0
$910$	70.1044	2.32394
$911$	20.0888	0.665570	0.332785	−	0.943003i	$-0.392012\pi$
$911$	20.0888	0.665570	0.332785	+	0.943003i	$0.392012\pi$
$912$	0	0
$913$	−14.3948	−0.476397
$914$	16.0431	0.530658
$915$	0	0
$916$	−16.1303	−0.532960
$917$	49.9842	1.65062
$918$	0	0
$919$	47.4537	1.56535	0.782676	−	0.622430i	$-0.213854\pi$
$919$	47.4537	1.56535	0.782676	+	0.622430i	$0.213854\pi$
$920$	−12.8196	−0.422650
$921$	0	0
$922$	13.7540	0.452964
$923$	−8.74011	−0.287684
$924$	0	0
$925$	−56.3209	−1.85182
$926$	−15.7347	−0.517074
$927$	0	0
$928$	−3.95925	−0.129969
$929$	−12.1762	−0.399487	−0.199744	−	0.979848i	$-0.564011\pi$
$929$	−12.1762	−0.399487	−0.199744	+	0.979848i	$0.564011\pi$
$930$	0	0
$931$	6.91533	0.226641
$932$	3.61566	0.118435
$933$	0	0
$934$	37.3438	1.22193
$935$	−45.9668	−1.50327
$936$	0	0
$937$	48.0196	1.56873	0.784365	−	0.620299i	$-0.212989\pi$
$937$	48.0196	1.56873	0.784365	+	0.620299i	$0.212989\pi$
$938$	−7.27473	−0.237528
$939$	0	0
$940$	18.4491	0.601744
$941$	−7.94925	−0.259138	−0.129569	−	0.991570i	$-0.541359\pi$
$941$	−7.94925	−0.259138	−0.129569	+	0.991570i	$0.541359\pi$
$942$	0	0
$943$	−10.9600	−0.356906
$944$	−19.1457	−0.623139
$945$	0	0
$946$	56.1223	1.82469
$947$	−8.91699	−0.289763	−0.144882	−	0.989449i	$-0.546280\pi$
$947$	−8.91699	−0.289763	−0.144882	+	0.989449i	$0.546280\pi$
$948$	0	0
$949$	−30.1898	−0.980003
$950$	104.922	3.40411
$951$	0	0
$952$	21.5031	0.696920
$953$	−44.5830	−1.44419	−0.722093	−	0.691796i	$-0.756820\pi$
$953$	−44.5830	−1.44419	−0.722093	+	0.691796i	$0.756820\pi$
$954$	0	0
$955$	−55.3156	−1.78997
$956$	−4.63026	−0.149753
$957$	0	0
$958$	4.66855	0.150834
$959$	28.7523	0.928459
$960$	0	0
$961$	−7.44887	−0.240286
$962$	−27.1979	−0.876894
$963$	0	0
$964$	−10.9800	−0.353643
$965$	−65.3626	−2.10410
$966$	0	0
$967$	−29.4771	−0.947919	−0.473959	−	0.880547i	$-0.657176\pi$
$967$	−29.4771	−0.947919	−0.473959	+	0.880547i	$0.657176\pi$
$968$	−26.6893	−0.857827
$969$	0	0
$970$	−12.6178	−0.405133
$971$	−22.9764	−0.737346	−0.368673	−	0.929559i	$-0.620188\pi$
$971$	−22.9764	−0.737346	−0.368673	+	0.929559i	$0.620188\pi$
$972$	0	0
$973$	−37.4214	−1.19968
$974$	26.1024	0.836374
$975$	0	0
$976$	14.1357	0.452473
$977$	48.0107	1.53600	0.768000	−	0.640450i	$-0.221252\pi$
$977$	48.0107	1.53600	0.768000	+	0.640450i	$0.221252\pi$
$978$	0	0
$979$	−4.31357	−0.137862
$980$	−2.83807	−0.0906589
$981$	0	0
$982$	−13.1433	−0.419420
$983$	−42.9474	−1.36981	−0.684904	−	0.728633i	$-0.740156\pi$
$983$	−42.9474	−1.36981	−0.684904	+	0.728633i	$0.740156\pi$
$984$	0	0
$985$	−47.6094	−1.51696
$986$	2.78478	0.0886856
$987$	0	0
$988$	−30.0646	−0.956482
$989$	11.2922	0.359070
$990$	0	0
$991$	15.6821	0.498160	0.249080	−	0.968483i	$-0.419872\pi$
$991$	15.6821	0.498160	0.249080	+	0.968483i	$0.419872\pi$
$992$	−19.2141	−0.610047
$993$	0	0
$994$	−5.16289	−0.163757
$995$	−55.5769	−1.76190
$996$	0	0
$997$	−53.0668	−1.68064	−0.840321	−	0.542089i	$-0.817634\pi$
$997$	−53.0668	−1.68064	−0.840321	+	0.542089i	$0.817634\pi$
$998$	2.82620	0.0894619
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.v.1.20	✓	30
3.2	odd	2		6003.2.a.w.1.11	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.v.1.20	✓	30	1.1	even	1	trivial
6003.2.a.w.1.11	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.12036 q^{2} -0.744798 q^{4} +4.16877 q^{5} +2.81319 q^{7} -3.07516 q^{8} +O(q^{10})\)	\(q+1.12036 q^{2} -0.744798 q^{4} +4.16877 q^{5} +2.81319 q^{7} -3.07516 q^{8} +4.67051 q^{10} +4.43610 q^{11} +5.33557 q^{13} +3.15179 q^{14} -1.95568 q^{16} -2.48562 q^{17} +7.56547 q^{19} -3.10489 q^{20} +4.97002 q^{22} +1.00000 q^{23} +12.3786 q^{25} +5.97775 q^{26} -2.09526 q^{28} -1.00000 q^{29} -4.85295 q^{31} +3.95925 q^{32} -2.78478 q^{34} +11.7276 q^{35} -4.54985 q^{37} +8.47604 q^{38} -12.8196 q^{40} -10.9600 q^{41} +11.2922 q^{43} -3.30400 q^{44} +1.12036 q^{46} -5.94196 q^{47} +0.914065 q^{49} +13.8685 q^{50} -3.97392 q^{52} -9.72222 q^{53} +18.4931 q^{55} -8.65101 q^{56} -1.12036 q^{58} +9.78978 q^{59} -7.22802 q^{61} -5.43704 q^{62} +8.34714 q^{64} +22.2427 q^{65} -2.30813 q^{67} +1.85128 q^{68} +13.1391 q^{70} -1.63808 q^{71} -5.65822 q^{73} -5.09746 q^{74} -5.63475 q^{76} +12.4796 q^{77} -16.5474 q^{79} -8.15277 q^{80} -12.2791 q^{82} -3.24491 q^{83} -10.3620 q^{85} +12.6513 q^{86} -13.6417 q^{88} -0.972379 q^{89} +15.0100 q^{91} -0.744798 q^{92} -6.65712 q^{94} +31.5387 q^{95} -2.70159 q^{97} +1.02408 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8}+O(q^{10}) \) 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8	\( 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8} + 8 q^{10} + 36 q^{13} - 7 q^{14} + 47 q^{16} - 18 q^{17} + 16 q^{19} + 25 q^{22} + 30 q^{23} + 56 q^{25} - 11 q^{26} + 27 q^{28} - 30 q^{29} + 14 q^{31} + 7 q^{32} + 3 q^{34} + 22 q^{35} + 40 q^{37} - 6 q^{38} + 30 q^{40} - 14 q^{41} + 34 q^{43} - 5 q^{44} - q^{46} + 2 q^{47} + 74 q^{49} + 21 q^{50} + 71 q^{52} - 16 q^{53} + 22 q^{55} - 14 q^{56} + q^{58} + 32 q^{59} + 46 q^{61} - 20 q^{62} + 68 q^{64} - 12 q^{65} + 14 q^{67} - 27 q^{68} + 32 q^{71} + 50 q^{73} + 26 q^{74} + 56 q^{76} - 34 q^{77} + 16 q^{79} - 2 q^{80} + 38 q^{82} + 14 q^{83} + 38 q^{85} - 10 q^{86} + 40 q^{88} + 2 q^{89} + 32 q^{91} + 37 q^{92} + 29 q^{94} + 28 q^{95} + 56 q^{97} - 8 q^{98}+O(q^{100}) \) 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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