LMFDB - Embedded newform 6003.2.a.u.1.14

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			1.14
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)

$f(q)$	$=$	$q+1.06922 q^{2} -0.856774 q^{4} -0.420836 q^{5} +1.13415 q^{7} -3.05451 q^{8} +O(q^{10})$	\(q+1.06922 q^{2} -0.856774 q^{4} -0.420836 q^{5} +1.13415 q^{7} -3.05451 q^{8} -0.449966 q^{10} +2.91779 q^{11} -1.70064 q^{13} +1.21266 q^{14} -1.55239 q^{16} -0.536110 q^{17} +0.585904 q^{19} +0.360562 q^{20} +3.11976 q^{22} -1.00000 q^{23} -4.82290 q^{25} -1.81835 q^{26} -0.971713 q^{28} -1.00000 q^{29} +7.49962 q^{31} +4.44918 q^{32} -0.573218 q^{34} -0.477293 q^{35} -1.22414 q^{37} +0.626459 q^{38} +1.28545 q^{40} -12.2002 q^{41} -2.78460 q^{43} -2.49989 q^{44} -1.06922 q^{46} -0.180494 q^{47} -5.71370 q^{49} -5.15673 q^{50} +1.45706 q^{52} +11.0675 q^{53} -1.22791 q^{55} -3.46429 q^{56} -1.06922 q^{58} +5.86452 q^{59} +12.1083 q^{61} +8.01872 q^{62} +7.86192 q^{64} +0.715690 q^{65} -14.3374 q^{67} +0.459325 q^{68} -0.510330 q^{70} -13.2457 q^{71} -4.56789 q^{73} -1.30887 q^{74} -0.501988 q^{76} +3.30923 q^{77} -6.89701 q^{79} +0.653302 q^{80} -13.0447 q^{82} +8.57178 q^{83} +0.225614 q^{85} -2.97734 q^{86} -8.91244 q^{88} -2.75169 q^{89} -1.92878 q^{91} +0.856774 q^{92} -0.192988 q^{94} -0.246570 q^{95} -7.28509 q^{97} -6.10918 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.06922	0.756051	0.378025	−	0.925795i	$-0.376603\pi$
$2$	1.06922	0.756051	0.378025	+	0.925795i	$0.376603\pi$
$3$	0	0
$3$	0	0
$4$	−0.856774	−0.428387
$5$	−0.420836	−0.188204	−0.0941019	−	0.995563i	$-0.529998\pi$
$5$	−0.420836	−0.188204	−0.0941019	+	0.995563i	$0.529998\pi$
$6$	0	0
$7$	1.13415	0.428670	0.214335	−	0.976760i	$-0.431242\pi$
$7$	1.13415	0.428670	0.214335	+	0.976760i	$0.431242\pi$
$8$	−3.05451	−1.07993
$9$	0	0
$10$	−0.449966	−0.142292
$11$	2.91779	0.879748	0.439874	−	0.898059i	$-0.355023\pi$
$11$	2.91779	0.879748	0.439874	+	0.898059i	$0.355023\pi$
$12$	0	0
$13$	−1.70064	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$13$	−1.70064	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$14$	1.21266	0.324096
$15$	0	0
$16$	−1.55239	−0.388097
$17$	−0.536110	−0.130026	−0.0650128	−	0.997884i	$-0.520709\pi$
$17$	−0.536110	−0.130026	−0.0650128	+	0.997884i	$0.520709\pi$
$18$	0	0
$19$	0.585904	0.134416	0.0672078	−	0.997739i	$-0.478591\pi$
$19$	0.585904	0.134416	0.0672078	+	0.997739i	$0.478591\pi$
$20$	0.360562	0.0806240
$21$	0	0
$22$	3.11976	0.665134
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.82290	−0.964579
$26$	−1.81835	−0.356608
$27$	0	0
$28$	−0.971713	−0.183637
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	7.49962	1.34697	0.673486	−	0.739200i	$-0.264796\pi$
$31$	7.49962	1.34697	0.673486	+	0.739200i	$0.264796\pi$
$32$	4.44918	0.786512
$33$	0	0
$34$	−0.573218	−0.0983060
$35$	−0.477293	−0.0806772
$36$	0	0
$37$	−1.22414	−0.201247	−0.100623	−	0.994925i	$-0.532084\pi$
$37$	−1.22414	−0.201247	−0.100623	+	0.994925i	$0.532084\pi$
$38$	0.626459	0.101625
$39$	0	0
$40$	1.28545	0.203247
$41$	−12.2002	−1.90535	−0.952676	−	0.303987i	$-0.901682\pi$
$41$	−12.2002	−1.90535	−0.952676	+	0.303987i	$0.901682\pi$
$42$	0	0
$43$	−2.78460	−0.424647	−0.212324	−	0.977199i	$-0.568103\pi$
$43$	−2.78460	−0.424647	−0.212324	+	0.977199i	$0.568103\pi$
$44$	−2.49989	−0.376873
$45$	0	0
$46$	−1.06922	−0.157648
$47$	−0.180494	−0.0263278	−0.0131639	−	0.999913i	$-0.504190\pi$
$47$	−0.180494	−0.0263278	−0.0131639	+	0.999913i	$0.504190\pi$
$48$	0	0
$49$	−5.71370	−0.816242
$50$	−5.15673	−0.729271
$51$	0	0
$52$	1.45706	0.202058
$53$	11.0675	1.52023	0.760116	−	0.649787i	$-0.225142\pi$
$53$	11.0675	1.52023	0.760116	+	0.649787i	$0.225142\pi$
$54$	0	0
$55$	−1.22791	−0.165572
$56$	−3.46429	−0.462935
$57$	0	0
$58$	−1.06922	−0.140395
$59$	5.86452	0.763496	0.381748	−	0.924266i	$-0.375322\pi$
$59$	5.86452	0.763496	0.381748	+	0.924266i	$0.375322\pi$
$60$	0	0
$61$	12.1083	1.55031	0.775153	−	0.631773i	$-0.217673\pi$
$61$	12.1083	1.55031	0.775153	+	0.631773i	$0.217673\pi$
$62$	8.01872	1.01838
$63$	0	0
$64$	7.86192	0.982740
$65$	0.715690	0.0887705
$66$	0	0
$67$	−14.3374	−1.75159	−0.875794	−	0.482686i	$-0.839661\pi$
$67$	−14.3374	−1.75159	−0.875794	+	0.482686i	$0.839661\pi$
$68$	0.459325	0.0557013
$69$	0	0
$70$	−0.510330	−0.0609961
$71$	−13.2457	−1.57198	−0.785988	−	0.618242i	$-0.787845\pi$
$71$	−13.2457	−1.57198	−0.785988	+	0.618242i	$0.787845\pi$
$72$	0	0
$73$	−4.56789	−0.534632	−0.267316	−	0.963609i	$-0.586137\pi$
$73$	−4.56789	−0.534632	−0.267316	+	0.963609i	$0.586137\pi$
$74$	−1.30887	−0.152153
$75$	0	0
$76$	−0.501988	−0.0575819
$77$	3.30923	0.377121
$78$	0	0
$79$	−6.89701	−0.775974	−0.387987	−	0.921665i	$-0.626829\pi$
$79$	−6.89701	−0.775974	−0.387987	+	0.921665i	$0.626829\pi$
$80$	0.653302	0.0730414
$81$	0	0
$82$	−13.0447	−1.44054
$83$	8.57178	0.940875	0.470438	−	0.882433i	$-0.344096\pi$
$83$	8.57178	0.940875	0.470438	+	0.882433i	$0.344096\pi$
$84$	0	0
$85$	0.225614	0.0244713
$86$	−2.97734	−0.321055
$87$	0	0
$88$	−8.91244	−0.950069
$89$	−2.75169	−0.291678	−0.145839	−	0.989308i	$-0.546588\pi$
$89$	−2.75169	−0.291678	−0.145839	+	0.989308i	$0.546588\pi$
$90$	0	0
$91$	−1.92878	−0.202192
$92$	0.856774	0.0893249
$93$	0	0
$94$	−0.192988	−0.0199052
$95$	−0.246570	−0.0252975
$96$	0	0
$97$	−7.28509	−0.739689	−0.369844	−	0.929094i	$-0.620589\pi$
$97$	−7.28509	−0.739689	−0.369844	+	0.929094i	$0.620589\pi$
$98$	−6.10918	−0.617121
$99$	0	0
$100$	4.13213	0.413213
$101$	7.18183	0.714618	0.357309	−	0.933986i	$-0.383694\pi$
$101$	7.18183	0.714618	0.357309	+	0.933986i	$0.383694\pi$
$102$	0	0
$103$	3.02471	0.298034	0.149017	−	0.988835i	$-0.452389\pi$
$103$	3.02471	0.298034	0.149017	+	0.988835i	$0.452389\pi$
$104$	5.19462	0.509374
$105$	0	0
$106$	11.8335	1.14937
$107$	−9.61926	−0.929929	−0.464965	−	0.885329i	$-0.653933\pi$
$107$	−9.61926	−0.929929	−0.464965	+	0.885329i	$0.653933\pi$
$108$	0	0
$109$	10.0908	0.966521	0.483261	−	0.875477i	$-0.339452\pi$
$109$	10.0908	0.966521	0.483261	+	0.875477i	$0.339452\pi$
$110$	−1.31291	−0.125181
$111$	0	0
$112$	−1.76065	−0.166366
$113$	−6.82687	−0.642218	−0.321109	−	0.947042i	$-0.604056\pi$
$113$	−6.82687	−0.642218	−0.321109	+	0.947042i	$0.604056\pi$
$114$	0	0
$115$	0.420836	0.0392432
$116$	0.856774	0.0795495
$117$	0	0
$118$	6.27045	0.577242
$119$	−0.608031	−0.0557381
$120$	0	0
$121$	−2.48647	−0.226043
$122$	12.9464	1.17211
$123$	0	0
$124$	−6.42548	−0.577025
$125$	4.13383	0.369741
$126$	0	0
$127$	−5.53291	−0.490967	−0.245483	−	0.969401i	$-0.578947\pi$
$127$	−5.53291	−0.490967	−0.245483	+	0.969401i	$0.578947\pi$
$128$	−0.492261	−0.0435101
$129$	0	0
$130$	0.765229	0.0671150
$131$	−13.4974	−1.17927	−0.589637	−	0.807668i	$-0.700729\pi$
$131$	−13.4974	−1.17927	−0.589637	+	0.807668i	$0.700729\pi$
$132$	0	0
$133$	0.664505	0.0576199
$134$	−15.3298	−1.32429
$135$	0	0
$136$	1.63755	0.140419
$137$	−5.83987	−0.498934	−0.249467	−	0.968383i	$-0.580255\pi$
$137$	−5.83987	−0.498934	−0.249467	+	0.968383i	$0.580255\pi$
$138$	0	0
$139$	−22.3348	−1.89442	−0.947208	−	0.320619i	$-0.896109\pi$
$139$	−22.3348	−1.89442	−0.947208	+	0.320619i	$0.896109\pi$
$140$	0.408932	0.0345611
$141$	0	0
$142$	−14.1625	−1.18849
$143$	−4.96211	−0.414953
$144$	0	0
$145$	0.420836	0.0349486
$146$	−4.88407	−0.404209
$147$	0	0
$148$	1.04881	0.0862116
$149$	0.856186	0.0701415	0.0350707	−	0.999385i	$-0.488834\pi$
$149$	0.856186	0.0701415	0.0350707	+	0.999385i	$0.488834\pi$
$150$	0	0
$151$	−5.82116	−0.473720	−0.236860	−	0.971544i	$-0.576118\pi$
$151$	−5.82116	−0.473720	−0.236860	+	0.971544i	$0.576118\pi$
$152$	−1.78965	−0.145160
$153$	0	0
$154$	3.53828	0.285123
$155$	−3.15611	−0.253505
$156$	0	0
$157$	−1.59058	−0.126942	−0.0634709	−	0.997984i	$-0.520217\pi$
$157$	−1.59058	−0.126942	−0.0634709	+	0.997984i	$0.520217\pi$
$158$	−7.37440	−0.586676
$159$	0	0
$160$	−1.87238	−0.148024
$161$	−1.13415	−0.0893838
$162$	0	0
$163$	−9.80469	−0.767963	−0.383981	−	0.923341i	$-0.625447\pi$
$163$	−9.80469	−0.767963	−0.383981	+	0.923341i	$0.625447\pi$
$164$	10.4528	0.816228
$165$	0	0
$166$	9.16510	0.711349
$167$	−16.2385	−1.25658	−0.628288	−	0.777981i	$-0.716244\pi$
$167$	−16.2385	−1.25658	−0.628288	+	0.777981i	$0.716244\pi$
$168$	0	0
$169$	−10.1078	−0.777525
$170$	0.241231	0.0185016
$171$	0	0
$172$	2.38577	0.181913
$173$	14.2547	1.08376	0.541881	−	0.840455i	$-0.317712\pi$
$173$	14.2547	1.08376	0.541881	+	0.840455i	$0.317712\pi$
$174$	0	0
$175$	−5.46991	−0.413486
$176$	−4.52955	−0.341428
$177$	0	0
$178$	−2.94215	−0.220524
$179$	−0.143629	−0.0107353	−0.00536766	−	0.999986i	$-0.501709\pi$
$179$	−0.143629	−0.0107353	−0.00536766	+	0.999986i	$0.501709\pi$
$180$	0	0
$181$	−11.6508	−0.865997	−0.432998	−	0.901395i	$-0.642544\pi$
$181$	−11.6508	−0.865997	−0.432998	+	0.901395i	$0.642544\pi$
$182$	−2.06229	−0.152867
$183$	0	0
$184$	3.05451	0.225182
$185$	0.515161	0.0378754
$186$	0	0
$187$	−1.56426	−0.114390
$188$	0.154643	0.0112785
$189$	0	0
$190$	−0.263637	−0.0191262
$191$	−26.1784	−1.89420	−0.947100	−	0.320939i	$-0.896002\pi$
$191$	−26.1784	−1.89420	−0.947100	+	0.320939i	$0.896002\pi$
$192$	0	0
$193$	7.02480	0.505656	0.252828	−	0.967511i	$-0.418639\pi$
$193$	7.02480	0.505656	0.252828	+	0.967511i	$0.418639\pi$
$194$	−7.78934	−0.559242
$195$	0	0
$196$	4.89535	0.349668
$197$	−3.74205	−0.266610	−0.133305	−	0.991075i	$-0.542559\pi$
$197$	−3.74205	−0.266610	−0.133305	+	0.991075i	$0.542559\pi$
$198$	0	0
$199$	13.8087	0.978875	0.489437	−	0.872038i	$-0.337202\pi$
$199$	13.8087	0.978875	0.489437	+	0.872038i	$0.337202\pi$
$200$	14.7316	1.04168
$201$	0	0
$202$	7.67893	0.540288
$203$	−1.13415	−0.0796020
$204$	0	0
$205$	5.13429	0.358594
$206$	3.23407	0.225329
$207$	0	0
$208$	2.64005	0.183055
$209$	1.70955	0.118252
$210$	0	0
$211$	5.17516	0.356273	0.178136	−	0.984006i	$-0.442993\pi$
$211$	5.17516	0.356273	0.178136	+	0.984006i	$0.442993\pi$
$212$	−9.48231	−0.651248
$213$	0	0
$214$	−10.2851	−0.703074
$215$	1.17186	0.0799202
$216$	0	0
$217$	8.50572	0.577406
$218$	10.7892	0.730739
$219$	0	0
$220$	1.05205	0.0709289
$221$	0.911728	0.0613295
$222$	0	0
$223$	−12.8518	−0.860617	−0.430308	−	0.902682i	$-0.641595\pi$
$223$	−12.8518	−0.860617	−0.430308	+	0.902682i	$0.641595\pi$
$224$	5.04606	0.337154
$225$	0	0
$226$	−7.29941	−0.485550
$227$	12.5847	0.835278	0.417639	−	0.908613i	$-0.362858\pi$
$227$	12.5847	0.835278	0.417639	+	0.908613i	$0.362858\pi$
$228$	0	0
$229$	19.4879	1.28779	0.643897	−	0.765112i	$-0.277316\pi$
$229$	19.4879	1.28779	0.643897	+	0.765112i	$0.277316\pi$
$230$	0.449966	0.0296699
$231$	0	0
$232$	3.05451	0.200539
$233$	11.0040	0.720893	0.360446	−	0.932780i	$-0.382624\pi$
$233$	11.0040	0.720893	0.360446	+	0.932780i	$0.382624\pi$
$234$	0	0
$235$	0.0759586	0.00495499
$236$	−5.02457	−0.327072
$237$	0	0
$238$	−0.650117	−0.0421408
$239$	−9.61008	−0.621625	−0.310812	−	0.950471i	$-0.600601\pi$
$239$	−9.61008	−0.621625	−0.310812	+	0.950471i	$0.600601\pi$
$240$	0	0
$241$	−5.96508	−0.384244	−0.192122	−	0.981371i	$-0.561537\pi$
$241$	−5.96508	−0.384244	−0.192122	+	0.981371i	$0.561537\pi$
$242$	−2.65858	−0.170900
$243$	0	0
$244$	−10.3741	−0.664131
$245$	2.40453	0.153620
$246$	0	0
$247$	−0.996411	−0.0634001
$248$	−22.9077	−1.45464
$249$	0	0
$250$	4.41997	0.279543
$251$	9.33642	0.589310	0.294655	−	0.955604i	$-0.404795\pi$
$251$	9.33642	0.589310	0.294655	+	0.955604i	$0.404795\pi$
$252$	0	0
$253$	−2.91779	−0.183440
$254$	−5.91589	−0.371196
$255$	0	0
$256$	−16.2502	−1.01564
$257$	−27.9492	−1.74342	−0.871711	−	0.490021i	$-0.836989\pi$
$257$	−27.9492	−1.74342	−0.871711	+	0.490021i	$0.836989\pi$
$258$	0	0
$259$	−1.38836	−0.0862685
$260$	−0.613185	−0.0380281
$261$	0	0
$262$	−14.4317	−0.891591
$263$	27.8535	1.71752	0.858760	−	0.512379i	$-0.171236\pi$
$263$	27.8535	1.71752	0.858760	+	0.512379i	$0.171236\pi$
$264$	0	0
$265$	−4.65759	−0.286113
$266$	0.710501	0.0435636
$267$	0	0
$268$	12.2839	0.750357
$269$	−8.33222	−0.508025	−0.254012	−	0.967201i	$-0.581750\pi$
$269$	−8.33222	−0.508025	−0.254012	+	0.967201i	$0.581750\pi$
$270$	0	0
$271$	−25.0222	−1.51999	−0.759995	−	0.649929i	$-0.774799\pi$
$271$	−25.0222	−1.51999	−0.759995	+	0.649929i	$0.774799\pi$
$272$	0.832251	0.0504626
$273$	0	0
$274$	−6.24409	−0.377219
$275$	−14.0722	−0.848587
$276$	0	0
$277$	−17.5910	−1.05694	−0.528469	−	0.848952i	$-0.677234\pi$
$277$	−17.5910	−1.05694	−0.528469	+	0.848952i	$0.677234\pi$
$278$	−23.8808	−1.43228
$279$	0	0
$280$	1.45790	0.0871260
$281$	−2.63805	−0.157373	−0.0786864	−	0.996899i	$-0.525073\pi$
$281$	−2.63805	−0.157373	−0.0786864	+	0.996899i	$0.525073\pi$
$282$	0	0
$283$	−16.9340	−1.00662	−0.503312	−	0.864105i	$-0.667885\pi$
$283$	−16.9340	−1.00662	−0.503312	+	0.864105i	$0.667885\pi$
$284$	11.3486	0.673414
$285$	0	0
$286$	−5.30558	−0.313725
$287$	−13.8369	−0.816767
$288$	0	0
$289$	−16.7126	−0.983093
$290$	0.449966	0.0264229
$291$	0	0
$292$	3.91365	0.229029
$293$	−4.73836	−0.276818	−0.138409	−	0.990375i	$-0.544199\pi$
$293$	−4.73836	−0.276818	−0.138409	+	0.990375i	$0.544199\pi$
$294$	0	0
$295$	−2.46800	−0.143693
$296$	3.73914	0.217333
$297$	0	0
$298$	0.915449	0.0530305
$299$	1.70064	0.0983504
$300$	0	0
$301$	−3.15816	−0.182034
$302$	−6.22409	−0.358156
$303$	0	0
$304$	−0.909552	−0.0521664
$305$	−5.09561	−0.291774
$306$	0	0
$307$	−34.0679	−1.94436	−0.972178	−	0.234243i	$-0.924739\pi$
$307$	−34.0679	−1.94436	−0.972178	+	0.234243i	$0.924739\pi$
$308$	−2.83526	−0.161554
$309$	0	0
$310$	−3.37457	−0.191663
$311$	3.97157	0.225207	0.112604	−	0.993640i	$-0.464081\pi$
$311$	3.97157	0.225207	0.112604	+	0.993640i	$0.464081\pi$
$312$	0	0
$313$	21.4087	1.21009	0.605045	−	0.796192i	$-0.293155\pi$
$313$	21.4087	1.21009	0.605045	+	0.796192i	$0.293155\pi$
$314$	−1.70067	−0.0959744
$315$	0	0
$316$	5.90918	0.332417
$317$	28.2785	1.58828	0.794140	−	0.607735i	$-0.207922\pi$
$317$	28.2785	1.58828	0.794140	+	0.607735i	$0.207922\pi$
$318$	0	0
$319$	−2.91779	−0.163365
$320$	−3.30858	−0.184955
$321$	0	0
$322$	−1.21266	−0.0675787
$323$	−0.314109	−0.0174775
$324$	0	0
$325$	8.20200	0.454965
$326$	−10.4833	−0.580619
$327$	0	0
$328$	37.2657	2.05765
$329$	−0.204708	−0.0112859
$330$	0	0
$331$	−17.6922	−0.972450	−0.486225	−	0.873834i	$-0.661627\pi$
$331$	−17.6922	−0.972450	−0.486225	+	0.873834i	$0.661627\pi$
$332$	−7.34408	−0.403059
$333$	0	0
$334$	−17.3625	−0.950035
$335$	6.03368	0.329655
$336$	0	0
$337$	−19.4660	−1.06038	−0.530191	−	0.847878i	$-0.677880\pi$
$337$	−19.4660	−1.06038	−0.530191	+	0.847878i	$0.677880\pi$
$338$	−10.8075	−0.587849
$339$	0	0
$340$	−0.193301	−0.0104832
$341$	21.8823	1.18500
$342$	0	0
$343$	−14.4193	−0.778568
$344$	8.50559	0.458591
$345$	0	0
$346$	15.2413	0.819379
$347$	7.18999	0.385979	0.192989	−	0.981201i	$-0.438182\pi$
$347$	7.18999	0.385979	0.192989	+	0.981201i	$0.438182\pi$
$348$	0	0
$349$	19.0083	1.01749	0.508746	−	0.860917i	$-0.330109\pi$
$349$	19.0083	1.01749	0.508746	+	0.860917i	$0.330109\pi$
$350$	−5.84852	−0.312616
$351$	0	0
$352$	12.9818	0.691932
$353$	3.18574	0.169560	0.0847799	−	0.996400i	$-0.472981\pi$
$353$	3.18574	0.169560	0.0847799	+	0.996400i	$0.472981\pi$
$354$	0	0
$355$	5.57427	0.295852
$356$	2.35758	0.124951
$357$	0	0
$358$	−0.153571	−0.00811645
$359$	−10.3826	−0.547973	−0.273986	−	0.961734i	$-0.588342\pi$
$359$	−10.3826	−0.547973	−0.273986	+	0.961734i	$0.588342\pi$
$360$	0	0
$361$	−18.6567	−0.981932
$362$	−12.4572	−0.654737
$363$	0	0
$364$	1.65253	0.0866163
$365$	1.92234	0.100620
$366$	0	0
$367$	2.15378	0.112426	0.0562132	−	0.998419i	$-0.482097\pi$
$367$	2.15378	0.112426	0.0562132	+	0.998419i	$0.482097\pi$
$368$	1.55239	0.0809239
$369$	0	0
$370$	0.550820	0.0286357
$371$	12.5522	0.651678
$372$	0	0
$373$	10.6902	0.553517	0.276758	−	0.960940i	$-0.410740\pi$
$373$	10.6902	0.553517	0.276758	+	0.960940i	$0.410740\pi$
$374$	−1.67253	−0.0864846
$375$	0	0
$376$	0.551323	0.0284323
$377$	1.70064	0.0875873
$378$	0	0
$379$	9.12577	0.468759	0.234380	−	0.972145i	$-0.424694\pi$
$379$	9.12577	0.468759	0.234380	+	0.972145i	$0.424694\pi$
$380$	0.211255	0.0108371
$381$	0	0
$382$	−27.9904	−1.43211
$383$	11.0898	0.566663	0.283332	−	0.959022i	$-0.408560\pi$
$383$	11.0898	0.566663	0.283332	+	0.959022i	$0.408560\pi$
$384$	0	0
$385$	−1.39264	−0.0709757
$386$	7.51104	0.382302
$387$	0	0
$388$	6.24168	0.316873
$389$	7.52137	0.381349	0.190674	−	0.981653i	$-0.438933\pi$
$389$	7.52137	0.381349	0.190674	+	0.981653i	$0.438933\pi$
$390$	0	0
$391$	0.536110	0.0271122
$392$	17.4526	0.881487
$393$	0	0
$394$	−4.00106	−0.201571
$395$	2.90251	0.146041
$396$	0	0
$397$	7.78635	0.390786	0.195393	−	0.980725i	$-0.437402\pi$
$397$	7.78635	0.390786	0.195393	+	0.980725i	$0.437402\pi$
$398$	14.7645	0.740079
$399$	0	0
$400$	7.48702	0.374351
$401$	29.2302	1.45968	0.729842	−	0.683616i	$-0.239594\pi$
$401$	29.2302	1.45968	0.729842	+	0.683616i	$0.239594\pi$
$402$	0	0
$403$	−12.7541	−0.635329
$404$	−6.15320	−0.306133
$405$	0	0
$406$	−1.21266	−0.0601831
$407$	−3.57178	−0.177047
$408$	0	0
$409$	−13.3398	−0.659611	−0.329806	−	0.944049i	$-0.606983\pi$
$409$	−13.3398	−0.659611	−0.329806	+	0.944049i	$0.606983\pi$
$410$	5.48967	0.271116
$411$	0	0
$412$	−2.59149	−0.127674
$413$	6.65127	0.327288
$414$	0	0
$415$	−3.60732	−0.177076
$416$	−7.56645	−0.370976
$417$	0	0
$418$	1.82788	0.0894045
$419$	28.6408	1.39919	0.699597	−	0.714537i	$-0.253363\pi$
$419$	28.6408	1.39919	0.699597	+	0.714537i	$0.253363\pi$
$420$	0	0
$421$	−35.9225	−1.75075	−0.875377	−	0.483441i	$-0.839387\pi$
$421$	−35.9225	−1.75075	−0.875377	+	0.483441i	$0.839387\pi$
$422$	5.53337	0.269360
$423$	0	0
$424$	−33.8057	−1.64175
$425$	2.58560	0.125420
$426$	0	0
$427$	13.7327	0.664570
$428$	8.24154	0.398370
$429$	0	0
$430$	1.25297	0.0604238
$431$	27.2374	1.31198	0.655989	−	0.754770i	$-0.272252\pi$
$431$	27.2374	1.31198	0.655989	+	0.754770i	$0.272252\pi$
$432$	0	0
$433$	−23.7658	−1.14211	−0.571057	−	0.820910i	$-0.693466\pi$
$433$	−23.7658	−1.14211	−0.571057	+	0.820910i	$0.693466\pi$
$434$	9.09446	0.436548
$435$	0	0
$436$	−8.64552	−0.414045
$437$	−0.585904	−0.0280276
$438$	0	0
$439$	27.4214	1.30875	0.654377	−	0.756169i	$-0.272931\pi$
$439$	27.4214	1.30875	0.654377	+	0.756169i	$0.272931\pi$
$440$	3.75068	0.178807
$441$	0	0
$442$	0.974836	0.0463682
$443$	24.9966	1.18763	0.593813	−	0.804603i	$-0.297622\pi$
$443$	24.9966	1.18763	0.593813	+	0.804603i	$0.297622\pi$
$444$	0	0
$445$	1.15801	0.0548950
$446$	−13.7413	−0.650670
$447$	0	0
$448$	8.91663	0.421271
$449$	19.5495	0.922598	0.461299	−	0.887245i	$-0.347384\pi$
$449$	19.5495	0.922598	0.461299	+	0.887245i	$0.347384\pi$
$450$	0	0
$451$	−35.5977	−1.67623
$452$	5.84909	0.275118
$453$	0	0
$454$	13.4558	0.631512
$455$	0.811703	0.0380532
$456$	0	0
$457$	−11.7789	−0.550995	−0.275498	−	0.961302i	$-0.588843\pi$
$457$	−11.7789	−0.550995	−0.275498	+	0.961302i	$0.588843\pi$
$458$	20.8368	0.973638
$459$	0	0
$460$	−0.360562	−0.0168113
$461$	−2.35402	−0.109638	−0.0548189	−	0.998496i	$-0.517458\pi$
$461$	−2.35402	−0.109638	−0.0548189	+	0.998496i	$0.517458\pi$
$462$	0	0
$463$	33.5600	1.55967	0.779833	−	0.625988i	$-0.215304\pi$
$463$	33.5600	1.55967	0.779833	+	0.625988i	$0.215304\pi$
$464$	1.55239	0.0720679
$465$	0	0
$466$	11.7656	0.545032
$467$	−17.2653	−0.798945	−0.399472	−	0.916745i	$-0.630807\pi$
$467$	−17.2653	−0.798945	−0.399472	+	0.916745i	$0.630807\pi$
$468$	0	0
$469$	−16.2608	−0.750852
$470$	0.0812163	0.00374623
$471$	0	0
$472$	−17.9133	−0.824525
$473$	−8.12489	−0.373583
$474$	0	0
$475$	−2.82576	−0.129655
$476$	0.520945	0.0238775
$477$	0	0
$478$	−10.2753	−0.469980
$479$	31.1558	1.42354	0.711771	−	0.702411i	$-0.247893\pi$
$479$	31.1558	1.42354	0.711771	+	0.702411i	$0.247893\pi$
$480$	0	0
$481$	2.08181	0.0949226
$482$	−6.37796	−0.290508
$483$	0	0
$484$	2.13035	0.0968339
$485$	3.06583	0.139212
$486$	0	0
$487$	13.4246	0.608326	0.304163	−	0.952620i	$-0.401623\pi$
$487$	13.4246	0.608326	0.304163	+	0.952620i	$0.401623\pi$
$488$	−36.9849	−1.67423
$489$	0	0
$490$	2.57097	0.116144
$491$	4.34995	0.196310	0.0981552	−	0.995171i	$-0.468706\pi$
$491$	4.34995	0.196310	0.0981552	+	0.995171i	$0.468706\pi$
$492$	0	0
$493$	0.536110	0.0241452
$494$	−1.06538	−0.0479337
$495$	0	0
$496$	−11.6423	−0.522756
$497$	−15.0227	−0.673858
$498$	0	0
$499$	−34.3267	−1.53668	−0.768338	−	0.640044i	$-0.778916\pi$
$499$	−34.3267	−1.53668	−0.768338	+	0.640044i	$0.778916\pi$
$500$	−3.54176	−0.158392
$501$	0	0
$502$	9.98266	0.445548
$503$	−35.4539	−1.58081	−0.790406	−	0.612584i	$-0.790130\pi$
$503$	−35.4539	−1.58081	−0.790406	+	0.612584i	$0.790130\pi$
$504$	0	0
$505$	−3.02237	−0.134494
$506$	−3.11976	−0.138690
$507$	0	0
$508$	4.74046	0.210324
$509$	11.3342	0.502378	0.251189	−	0.967938i	$-0.419178\pi$
$509$	11.3342	0.502378	0.251189	+	0.967938i	$0.419178\pi$
$510$	0	0
$511$	−5.18069	−0.229180
$512$	−16.3905	−0.724363
$513$	0	0
$514$	−29.8837	−1.31812
$515$	−1.27291	−0.0560910
$516$	0	0
$517$	−0.526646	−0.0231619
$518$	−1.48446	−0.0652234
$519$	0	0
$520$	−2.18609	−0.0958662
$521$	−13.6651	−0.598679	−0.299339	−	0.954147i	$-0.596766\pi$
$521$	−13.6651	−0.598679	−0.299339	+	0.954147i	$0.596766\pi$
$522$	0	0
$523$	4.43578	0.193963	0.0969816	−	0.995286i	$-0.469081\pi$
$523$	4.43578	0.193963	0.0969816	+	0.995286i	$0.469081\pi$
$524$	11.5642	0.505186
$525$	0	0
$526$	29.7814	1.29853
$527$	−4.02062	−0.175141
$528$	0	0
$529$	1.00000	0.0434783
$530$	−4.97997	−0.216316
$531$	0	0
$532$	−0.569331	−0.0246836
$533$	20.7481	0.898702
$534$	0	0
$535$	4.04814	0.175016
$536$	43.7936	1.89160
$537$	0	0
$538$	−8.90896	−0.384092
$539$	−16.6714	−0.718088
$540$	0	0
$541$	−43.6790	−1.87791	−0.938954	−	0.344044i	$-0.888203\pi$
$541$	−43.6790	−1.87791	−0.938954	+	0.344044i	$0.888203\pi$
$542$	−26.7542	−1.14919
$543$	0	0
$544$	−2.38525	−0.102267
$545$	−4.24657	−0.181903
$546$	0	0
$547$	−22.1181	−0.945702	−0.472851	−	0.881142i	$-0.656775\pi$
$547$	−22.1181	−0.945702	−0.472851	+	0.881142i	$0.656775\pi$
$548$	5.00345	0.213737
$549$	0	0
$550$	−15.0463	−0.641575
$551$	−0.585904	−0.0249604
$552$	0	0
$553$	−7.82226	−0.332637
$554$	−18.8086	−0.799099
$555$	0	0
$556$	19.1359	0.811544
$557$	−4.37611	−0.185422	−0.0927108	−	0.995693i	$-0.529553\pi$
$557$	−4.37611	−0.185422	−0.0927108	+	0.995693i	$0.529553\pi$
$558$	0	0
$559$	4.73560	0.200294
$560$	0.740945	0.0313106
$561$	0	0
$562$	−2.82065	−0.118982
$563$	11.8239	0.498318	0.249159	−	0.968463i	$-0.419846\pi$
$563$	11.8239	0.498318	0.249159	+	0.968463i	$0.419846\pi$
$564$	0	0
$565$	2.87300	0.120868
$566$	−18.1062	−0.761058
$567$	0	0
$568$	40.4592	1.69763
$569$	38.5706	1.61696	0.808482	−	0.588521i	$-0.200290\pi$
$569$	38.5706	1.61696	0.808482	+	0.588521i	$0.200290\pi$
$570$	0	0
$571$	23.5913	0.987264	0.493632	−	0.869671i	$-0.335669\pi$
$571$	23.5913	0.987264	0.493632	+	0.869671i	$0.335669\pi$
$572$	4.25141	0.177760
$573$	0	0
$574$	−14.7947	−0.617517
$575$	4.82290	0.201129
$576$	0	0
$577$	1.28918	0.0536692	0.0268346	−	0.999640i	$-0.491457\pi$
$577$	1.28918	0.0536692	0.0268346	+	0.999640i	$0.491457\pi$
$578$	−17.8694	−0.743269
$579$	0	0
$580$	−0.360562	−0.0149715
$581$	9.72172	0.403325
$582$	0	0
$583$	32.2926	1.33742
$584$	13.9527	0.577366
$585$	0	0
$586$	−5.06634	−0.209288
$587$	−7.18244	−0.296451	−0.148226	−	0.988954i	$-0.547356\pi$
$587$	−7.18244	−0.296451	−0.148226	+	0.988954i	$0.547356\pi$
$588$	0	0
$589$	4.39406	0.181054
$590$	−2.63883	−0.108639
$591$	0	0
$592$	1.90034	0.0781034
$593$	29.1976	1.19900	0.599502	−	0.800373i	$-0.295365\pi$
$593$	29.1976	1.19900	0.599502	+	0.800373i	$0.295365\pi$
$594$	0	0
$595$	0.255881	0.0104901
$596$	−0.733558	−0.0300477
$597$	0	0
$598$	1.81835	0.0743579
$599$	−4.66540	−0.190623	−0.0953115	−	0.995448i	$-0.530385\pi$
$599$	−4.66540	−0.190623	−0.0953115	+	0.995448i	$0.530385\pi$
$600$	0	0
$601$	25.9769	1.05962	0.529810	−	0.848117i	$-0.322263\pi$
$601$	25.9769	1.05962	0.529810	+	0.848117i	$0.322263\pi$
$602$	−3.37676	−0.137627
$603$	0	0
$604$	4.98742	0.202935
$605$	1.04640	0.0425422
$606$	0	0
$607$	−14.6632	−0.595162	−0.297581	−	0.954697i	$-0.596180\pi$
$607$	−14.6632	−0.595162	−0.297581	+	0.954697i	$0.596180\pi$
$608$	2.60680	0.105720
$609$	0	0
$610$	−5.44831	−0.220596
$611$	0.306956	0.0124181
$612$	0	0
$613$	−15.1212	−0.610741	−0.305371	−	0.952234i	$-0.598780\pi$
$613$	−15.1212	−0.610741	−0.305371	+	0.952234i	$0.598780\pi$
$614$	−36.4260	−1.47003
$615$	0	0
$616$	−10.1081	−0.407266
$617$	−8.37624	−0.337215	−0.168607	−	0.985683i	$-0.553927\pi$
$617$	−8.37624	−0.337215	−0.168607	+	0.985683i	$0.553927\pi$
$618$	0	0
$619$	24.4794	0.983912	0.491956	−	0.870620i	$-0.336282\pi$
$619$	24.4794	0.983912	0.491956	+	0.870620i	$0.336282\pi$
$620$	2.70408	0.108598
$621$	0	0
$622$	4.24648	0.170268
$623$	−3.12084	−0.125034
$624$	0	0
$625$	22.3748	0.894993
$626$	22.8905	0.914889
$627$	0	0
$628$	1.36276	0.0543802
$629$	0.656272	0.0261673
$630$	0	0
$631$	−19.8813	−0.791463	−0.395731	−	0.918366i	$-0.629509\pi$
$631$	−19.8813	−0.791463	−0.395731	+	0.918366i	$0.629509\pi$
$632$	21.0670	0.838000
$633$	0	0
$634$	30.2359	1.20082
$635$	2.32845	0.0924018
$636$	0	0
$637$	9.71693	0.384999
$638$	−3.11976	−0.123512
$639$	0	0
$640$	0.207161	0.00818876
$641$	30.4724	1.20359	0.601793	−	0.798652i	$-0.294453\pi$
$641$	30.4724	1.20359	0.601793	+	0.798652i	$0.294453\pi$
$642$	0	0
$643$	16.0989	0.634879	0.317440	−	0.948278i	$-0.397177\pi$
$643$	16.0989	0.634879	0.317440	+	0.948278i	$0.397177\pi$
$644$	0.971713	0.0382909
$645$	0	0
$646$	−0.335851	−0.0132139
$647$	−26.4180	−1.03860	−0.519299	−	0.854593i	$-0.673807\pi$
$647$	−26.4180	−1.03860	−0.519299	+	0.854593i	$0.673807\pi$
$648$	0	0
$649$	17.1115	0.671684
$650$	8.76972	0.343977
$651$	0	0
$652$	8.40041	0.328985
$653$	29.0738	1.13775	0.568873	−	0.822425i	$-0.307379\pi$
$653$	29.0738	1.13775	0.568873	+	0.822425i	$0.307379\pi$
$654$	0	0
$655$	5.68020	0.221944
$656$	18.9395	0.739462
$657$	0	0
$658$	−0.218878	−0.00853275
$659$	22.7601	0.886607	0.443303	−	0.896372i	$-0.353806\pi$
$659$	22.7601	0.886607	0.443303	+	0.896372i	$0.353806\pi$
$660$	0	0
$661$	35.5742	1.38368	0.691838	−	0.722053i	$-0.256801\pi$
$661$	35.5742	1.38368	0.691838	+	0.722053i	$0.256801\pi$
$662$	−18.9168	−0.735222
$663$	0	0
$664$	−26.1826	−1.01608
$665$	−0.279648	−0.0108443
$666$	0	0
$667$	1.00000	0.0387202
$668$	13.9128	0.538301
$669$	0	0
$670$	6.45132	0.249236
$671$	35.3295	1.36388
$672$	0	0
$673$	−45.9525	−1.77134	−0.885669	−	0.464318i	$-0.846300\pi$
$673$	−45.9525	−1.77134	−0.885669	+	0.464318i	$0.846300\pi$
$674$	−20.8134	−0.801703
$675$	0	0
$676$	8.66013	0.333082
$677$	−48.5051	−1.86420	−0.932102	−	0.362196i	$-0.882027\pi$
$677$	−48.5051	−1.86420	−0.932102	+	0.362196i	$0.882027\pi$
$678$	0	0
$679$	−8.26241	−0.317082
$680$	−0.689142	−0.0264274
$681$	0	0
$682$	23.3970	0.895917
$683$	1.90270	0.0728048	0.0364024	−	0.999337i	$-0.488410\pi$
$683$	1.90270	0.0728048	0.0364024	+	0.999337i	$0.488410\pi$
$684$	0	0
$685$	2.45763	0.0939012
$686$	−15.4173	−0.588637
$687$	0	0
$688$	4.32278	0.164805
$689$	−18.8217	−0.717051
$690$	0	0
$691$	4.71656	0.179426	0.0897132	−	0.995968i	$-0.471405\pi$
$691$	4.71656	0.179426	0.0897132	+	0.995968i	$0.471405\pi$
$692$	−12.2130	−0.464270
$693$	0	0
$694$	7.68766	0.291820
$695$	9.39931	0.356536
$696$	0	0
$697$	6.54065	0.247745
$698$	20.3240	0.769276
$699$	0	0
$700$	4.68647	0.177132
$701$	21.6293	0.816926	0.408463	−	0.912775i	$-0.366065\pi$
$701$	21.6293	0.816926	0.408463	+	0.912775i	$0.366065\pi$
$702$	0	0
$703$	−0.717227	−0.0270507
$704$	22.9395	0.864564
$705$	0	0
$706$	3.40625	0.128196
$707$	8.14529	0.306335
$708$	0	0
$709$	−9.74667	−0.366044	−0.183022	−	0.983109i	$-0.558588\pi$
$709$	−9.74667	−0.366044	−0.183022	+	0.983109i	$0.558588\pi$
$710$	5.96011	0.223679
$711$	0	0
$712$	8.40507	0.314993
$713$	−7.49962	−0.280863
$714$	0	0
$715$	2.08824	0.0780957
$716$	0.123058	0.00459888
$717$	0	0
$718$	−11.1013	−0.414295
$719$	−24.3097	−0.906599	−0.453300	−	0.891358i	$-0.649753\pi$
$719$	−24.3097	−0.906599	−0.453300	+	0.891358i	$0.649753\pi$
$720$	0	0
$721$	3.43049	0.127758
$722$	−19.9481	−0.742391
$723$	0	0
$724$	9.98210	0.370982
$725$	4.82290	0.179118
$726$	0	0
$727$	1.84325	0.0683625	0.0341812	−	0.999416i	$-0.489118\pi$
$727$	1.84325	0.0683625	0.0341812	+	0.999416i	$0.489118\pi$
$728$	5.89150	0.218353
$729$	0	0
$730$	2.05540	0.0760736
$731$	1.49285	0.0552151
$732$	0	0
$733$	−16.7848	−0.619961	−0.309981	−	0.950743i	$-0.600323\pi$
$733$	−16.7848	−0.619961	−0.309981	+	0.950743i	$0.600323\pi$
$734$	2.30286	0.0850001
$735$	0	0
$736$	−4.44918	−0.163999
$737$	−41.8335	−1.54096
$738$	0	0
$739$	43.0787	1.58467	0.792337	−	0.610083i	$-0.208864\pi$
$739$	43.0787	1.58467	0.792337	+	0.610083i	$0.208864\pi$
$740$	−0.441377	−0.0162253
$741$	0	0
$742$	13.4210	0.492701
$743$	−0.629086	−0.0230789	−0.0115395	−	0.999933i	$-0.503673\pi$
$743$	−0.629086	−0.0230789	−0.0115395	+	0.999933i	$0.503673\pi$
$744$	0	0
$745$	−0.360314	−0.0132009
$746$	11.4301	0.418487
$747$	0	0
$748$	1.34022	0.0490031
$749$	−10.9097	−0.398633
$750$	0	0
$751$	32.3302	1.17975	0.589873	−	0.807496i	$-0.299178\pi$
$751$	32.3302	1.17975	0.589873	+	0.807496i	$0.299178\pi$
$752$	0.280198	0.0102178
$753$	0	0
$754$	1.81835	0.0662205
$755$	2.44976	0.0891558
$756$	0	0
$757$	41.8728	1.52189	0.760946	−	0.648815i	$-0.224735\pi$
$757$	41.8728	1.52189	0.760946	+	0.648815i	$0.224735\pi$
$758$	9.75744	0.354406
$759$	0	0
$760$	0.753151	0.0273196
$761$	−22.7021	−0.822950	−0.411475	−	0.911421i	$-0.634986\pi$
$761$	−22.7021	−0.822950	−0.411475	+	0.911421i	$0.634986\pi$
$762$	0	0
$763$	11.4445	0.414318
$764$	22.4289	0.811451
$765$	0	0
$766$	11.8574	0.428426
$767$	−9.97343	−0.360120
$768$	0	0
$769$	−39.9953	−1.44227	−0.721134	−	0.692795i	$-0.756379\pi$
$769$	−39.9953	−1.44227	−0.721134	+	0.692795i	$0.756379\pi$
$770$	−1.48904	−0.0536612
$771$	0	0
$772$	−6.01867	−0.216617
$773$	−6.67792	−0.240188	−0.120094	−	0.992763i	$-0.538320\pi$
$773$	−6.67792	−0.240188	−0.120094	+	0.992763i	$0.538320\pi$
$774$	0	0
$775$	−36.1699	−1.29926
$776$	22.2524	0.798815
$777$	0	0
$778$	8.04198	0.288319
$779$	−7.14815	−0.256109
$780$	0	0
$781$	−38.6482	−1.38294
$782$	0.573218	0.0204982
$783$	0	0
$784$	8.86988	0.316782
$785$	0.669372	0.0238909
$786$	0	0
$787$	26.5400	0.946048	0.473024	−	0.881050i	$-0.343162\pi$
$787$	26.5400	0.946048	0.473024	+	0.881050i	$0.343162\pi$
$788$	3.20609	0.114212
$789$	0	0
$790$	3.10342	0.110415
$791$	−7.74272	−0.275299
$792$	0	0
$793$	−20.5918	−0.731237
$794$	8.32530	0.295454
$795$	0	0
$796$	−11.8310	−0.419337
$797$	−31.8075	−1.12668	−0.563340	−	0.826225i	$-0.690484\pi$
$797$	−31.8075	−1.12668	−0.563340	+	0.826225i	$0.690484\pi$
$798$	0	0
$799$	0.0967648	0.00342329
$800$	−21.4580	−0.758653
$801$	0	0
$802$	31.2534	1.10360
$803$	−13.3282	−0.470341
$804$	0	0
$805$	0.477293	0.0168224
$806$	−13.6369	−0.480341
$807$	0	0
$808$	−21.9370	−0.771740
$809$	17.4024	0.611836	0.305918	−	0.952058i	$-0.401037\pi$
$809$	17.4024	0.611836	0.305918	+	0.952058i	$0.401037\pi$
$810$	0	0
$811$	48.5297	1.70411	0.852055	−	0.523452i	$-0.175356\pi$
$811$	48.5297	1.70411	0.852055	+	0.523452i	$0.175356\pi$
$812$	0.971713	0.0341005
$813$	0	0
$814$	−3.81901	−0.133856
$815$	4.12617	0.144533
$816$	0	0
$817$	−1.63151	−0.0570793
$818$	−14.2632	−0.498700
$819$	0	0
$820$	−4.39893	−0.153617
$821$	21.8969	0.764207	0.382103	−	0.924120i	$-0.375200\pi$
$821$	21.8969	0.764207	0.382103	+	0.924120i	$0.375200\pi$
$822$	0	0
$823$	−12.2048	−0.425433	−0.212717	−	0.977114i	$-0.568231\pi$
$823$	−12.2048	−0.425433	−0.212717	+	0.977114i	$0.568231\pi$
$824$	−9.23902	−0.321856
$825$	0	0
$826$	7.11165	0.247446
$827$	36.2068	1.25903	0.629516	−	0.776987i	$-0.283253\pi$
$827$	36.2068	1.25903	0.629516	+	0.776987i	$0.283253\pi$
$828$	0	0
$829$	−2.38205	−0.0827320	−0.0413660	−	0.999144i	$-0.513171\pi$
$829$	−2.38205	−0.0827320	−0.0413660	+	0.999144i	$0.513171\pi$
$830$	−3.85701	−0.133879
$831$	0	0
$832$	−13.3703	−0.463531
$833$	3.06317	0.106132
$834$	0	0
$835$	6.83377	0.236492
$836$	−1.46470	−0.0506576
$837$	0	0
$838$	30.6232	1.05786
$839$	52.3933	1.80882	0.904408	−	0.426668i	$-0.140313\pi$
$839$	52.3933	1.80882	0.904408	+	0.426668i	$0.140313\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−38.4089	−1.32366
$843$	0	0
$844$	−4.43394	−0.152623
$845$	4.25374	0.146333
$846$	0	0
$847$	−2.82004	−0.0968978
$848$	−17.1810	−0.589998
$849$	0	0
$850$	2.76457	0.0948240
$851$	1.22414	0.0419629
$852$	0	0
$853$	−12.3989	−0.424529	−0.212265	−	0.977212i	$-0.568084\pi$
$853$	−12.3989	−0.424529	−0.212265	+	0.977212i	$0.568084\pi$
$854$	14.6832	0.502448
$855$	0	0
$856$	29.3822	1.00426
$857$	11.3105	0.386359	0.193179	−	0.981163i	$-0.438120\pi$
$857$	11.3105	0.386359	0.193179	+	0.981163i	$0.438120\pi$
$858$	0	0
$859$	−27.2778	−0.930706	−0.465353	−	0.885125i	$-0.654073\pi$
$859$	−27.2778	−0.930706	−0.465353	+	0.885125i	$0.654073\pi$
$860$	−1.00402	−0.0342368
$861$	0	0
$862$	29.1227	0.991923
$863$	−26.0532	−0.886863	−0.443431	−	0.896308i	$-0.646239\pi$
$863$	−26.0532	−0.886863	−0.443431	+	0.896308i	$0.646239\pi$
$864$	0	0
$865$	−5.99888	−0.203968
$866$	−25.4109	−0.863496
$867$	0	0
$868$	−7.28748	−0.247353
$869$	−20.1240	−0.682662
$870$	0	0
$871$	24.3827	0.826175
$872$	−30.8224	−1.04378
$873$	0	0
$874$	−0.626459	−0.0211903
$875$	4.68840	0.158497
$876$	0	0
$877$	−53.0592	−1.79168	−0.895841	−	0.444375i	$-0.853426\pi$
$877$	−53.0592	−1.79168	−0.895841	+	0.444375i	$0.853426\pi$
$878$	29.3195	0.989484
$879$	0	0
$880$	1.90620	0.0642580
$881$	−40.1909	−1.35406	−0.677032	−	0.735953i	$-0.736734\pi$
$881$	−40.1909	−1.35406	−0.677032	+	0.735953i	$0.736734\pi$
$882$	0	0
$883$	20.1420	0.677833	0.338916	−	0.940816i	$-0.389940\pi$
$883$	20.1420	0.677833	0.338916	+	0.940816i	$0.389940\pi$
$884$	−0.781145	−0.0262728
$885$	0	0
$886$	26.7268	0.897906
$887$	28.3924	0.953325	0.476662	−	0.879086i	$-0.341846\pi$
$887$	28.3924	0.953325	0.476662	+	0.879086i	$0.341846\pi$
$888$	0	0
$889$	−6.27517	−0.210463
$890$	1.23816	0.0415034
$891$	0	0
$892$	11.0110	0.368677
$893$	−0.105752	−0.00353887
$894$	0	0
$895$	0.0604443	0.00202043
$896$	−0.558299	−0.0186515
$897$	0	0
$898$	20.9027	0.697531
$899$	−7.49962	−0.250126
$900$	0	0
$901$	−5.93337	−0.197669
$902$	−38.0617	−1.26732
$903$	0	0
$904$	20.8528	0.693553
$905$	4.90308	0.162984
$906$	0	0
$907$	−24.3598	−0.808854	−0.404427	−	0.914570i	$-0.632529\pi$
$907$	−24.3598	−0.808854	−0.404427	+	0.914570i	$0.632529\pi$
$908$	−10.7823	−0.357822
$909$	0	0
$910$	0.867887	0.0287702
$911$	33.0626	1.09541	0.547707	−	0.836670i	$-0.315501\pi$
$911$	33.0626	1.09541	0.547707	+	0.836670i	$0.315501\pi$
$912$	0	0
$913$	25.0107	0.827733
$914$	−12.5942	−0.416580
$915$	0	0
$916$	−16.6967	−0.551675
$917$	−15.3081	−0.505519
$918$	0	0
$919$	47.7236	1.57426	0.787128	−	0.616789i	$-0.211567\pi$
$919$	47.7236	1.57426	0.787128	+	0.616789i	$0.211567\pi$
$920$	−1.28545	−0.0423800
$921$	0	0
$922$	−2.51696	−0.0828918
$923$	22.5261	0.741457
$924$	0	0
$925$	5.90389	0.194119
$926$	35.8829	1.17919
$927$	0	0
$928$	−4.44918	−0.146052
$929$	21.6699	0.710968	0.355484	−	0.934682i	$-0.384316\pi$
$929$	21.6699	0.710968	0.355484	+	0.934682i	$0.384316\pi$
$930$	0	0
$931$	−3.34768	−0.109716
$932$	−9.42790	−0.308821
$933$	0	0
$934$	−18.4604	−0.604043
$935$	0.658297	0.0215286
$936$	0	0
$937$	11.3492	0.370764	0.185382	−	0.982667i	$-0.440648\pi$
$937$	11.3492	0.370764	0.185382	+	0.982667i	$0.440648\pi$
$938$	−17.3863	−0.567683
$939$	0	0
$940$	−0.0650794	−0.00212266
$941$	−12.8107	−0.417618	−0.208809	−	0.977956i	$-0.566959\pi$
$941$	−12.8107	−0.417618	−0.208809	+	0.977956i	$0.566959\pi$
$942$	0	0
$943$	12.2002	0.397293
$944$	−9.10403	−0.296311
$945$	0	0
$946$	−8.68727	−0.282448
$947$	17.0001	0.552428	0.276214	−	0.961096i	$-0.410920\pi$
$947$	17.0001	0.552428	0.276214	+	0.961096i	$0.410920\pi$
$948$	0	0
$949$	7.76834	0.252171
$950$	−3.02135	−0.0980255
$951$	0	0
$952$	1.85724	0.0601934
$953$	36.4300	1.18008	0.590042	−	0.807373i	$-0.299111\pi$
$953$	36.4300	1.18008	0.590042	+	0.807373i	$0.299111\pi$
$954$	0	0
$955$	11.0168	0.356495
$956$	8.23367	0.266296
$957$	0	0
$958$	33.3123	1.07627
$959$	−6.62331	−0.213878
$960$	0	0
$961$	25.2443	0.814332
$962$	2.22591	0.0717663
$963$	0	0
$964$	5.11072	0.164605
$965$	−2.95629	−0.0951664
$966$	0	0
$967$	−23.2204	−0.746719	−0.373359	−	0.927687i	$-0.621794\pi$
$967$	−23.2204	−0.746719	−0.373359	+	0.927687i	$0.621794\pi$
$968$	7.59497	0.244111
$969$	0	0
$970$	3.27804	0.105252
$971$	−25.4051	−0.815289	−0.407644	−	0.913141i	$-0.633650\pi$
$971$	−25.4051	−0.815289	−0.407644	+	0.913141i	$0.633650\pi$
$972$	0	0
$973$	−25.3311	−0.812079
$974$	14.3538	0.459926
$975$	0	0
$976$	−18.7968	−0.601670
$977$	35.6930	1.14192	0.570961	−	0.820978i	$-0.306571\pi$
$977$	35.6930	1.14192	0.570961	+	0.820978i	$0.306571\pi$
$978$	0	0
$979$	−8.02886	−0.256604
$980$	−2.06014	−0.0658088
$981$	0	0
$982$	4.65104	0.148421
$983$	−30.1919	−0.962973	−0.481486	−	0.876454i	$-0.659903\pi$
$983$	−30.1919	−0.962973	−0.481486	+	0.876454i	$0.659903\pi$
$984$	0	0
$985$	1.57479	0.0501770
$986$	0.573218	0.0182550
$987$	0	0
$988$	0.853699	0.0271598
$989$	2.78460	0.0885451
$990$	0	0
$991$	−18.5980	−0.590784	−0.295392	−	0.955376i	$-0.595450\pi$
$991$	−18.5980	−0.590784	−0.295392	+	0.955376i	$0.595450\pi$
$992$	33.3672	1.05941
$993$	0	0
$994$	−16.0625	−0.509471
$995$	−5.81122	−0.184228
$996$	0	0
$997$	−57.0452	−1.80664	−0.903319	−	0.428969i	$-0.858877\pi$
$997$	−57.0452	−1.80664	−0.903319	+	0.428969i	$0.858877\pi$
$998$	−36.7027	−1.16181
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.u.1.14	yes	22
3.2	odd	2		6003.2.a.t.1.9	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.t.1.9	✓	22	3.2	odd	2
6003.2.a.u.1.14	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)

\(f(q)\)	\(=\)	\(q+1.06922 q^{2} -0.856774 q^{4} -0.420836 q^{5} +1.13415 q^{7} -3.05451 q^{8} +O(q^{10})\)	\(q+1.06922 q^{2} -0.856774 q^{4} -0.420836 q^{5} +1.13415 q^{7} -3.05451 q^{8} -0.449966 q^{10} +2.91779 q^{11} -1.70064 q^{13} +1.21266 q^{14} -1.55239 q^{16} -0.536110 q^{17} +0.585904 q^{19} +0.360562 q^{20} +3.11976 q^{22} -1.00000 q^{23} -4.82290 q^{25} -1.81835 q^{26} -0.971713 q^{28} -1.00000 q^{29} +7.49962 q^{31} +4.44918 q^{32} -0.573218 q^{34} -0.477293 q^{35} -1.22414 q^{37} +0.626459 q^{38} +1.28545 q^{40} -12.2002 q^{41} -2.78460 q^{43} -2.49989 q^{44} -1.06922 q^{46} -0.180494 q^{47} -5.71370 q^{49} -5.15673 q^{50} +1.45706 q^{52} +11.0675 q^{53} -1.22791 q^{55} -3.46429 q^{56} -1.06922 q^{58} +5.86452 q^{59} +12.1083 q^{61} +8.01872 q^{62} +7.86192 q^{64} +0.715690 q^{65} -14.3374 q^{67} +0.459325 q^{68} -0.510330 q^{70} -13.2457 q^{71} -4.56789 q^{73} -1.30887 q^{74} -0.501988 q^{76} +3.30923 q^{77} -6.89701 q^{79} +0.653302 q^{80} -13.0447 q^{82} +8.57178 q^{83} +0.225614 q^{85} -2.97734 q^{86} -8.91244 q^{88} -2.75169 q^{89} -1.92878 q^{91} +0.856774 q^{92} -0.192988 q^{94} -0.246570 q^{95} -7.28509 q^{97} -6.10918 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8}+O(q^{10}) \) 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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