LMFDB - Embedded newform 6003.2.a.h.1.1

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:	$5$
Coefficient field:	5.5.312617.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 11x^{2} - x - 3 $ x^5 - 2x^4 - 5x^3 + 11*x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.447481$ of defining polynomial
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)

$f(q)$	$=$	$q-2.10891 q^{2} +2.44748 q^{4} -1.70032 q^{5} -4.14780 q^{7} -0.943696 q^{8} +O(q^{10})$	\(q-2.10891 q^{2} +2.44748 q^{4} -1.70032 q^{5} -4.14780 q^{7} -0.943696 q^{8} +3.58582 q^{10} +1.69086 q^{11} -1.39118 q^{13} +8.74732 q^{14} -2.90480 q^{16} +0.447481 q^{17} -1.61829 q^{19} -4.16151 q^{20} -3.56585 q^{22} -1.00000 q^{23} -2.10891 q^{25} +2.93386 q^{26} -10.1517 q^{28} -1.00000 q^{29} -9.36561 q^{31} +8.01333 q^{32} -0.943696 q^{34} +7.05260 q^{35} +4.24301 q^{37} +3.41282 q^{38} +1.60459 q^{40} -6.78622 q^{41} -8.81869 q^{43} +4.13834 q^{44} +2.10891 q^{46} +5.76510 q^{47} +10.2043 q^{49} +4.44748 q^{50} -3.40488 q^{52} +3.27411 q^{53} -2.87500 q^{55} +3.91426 q^{56} +2.10891 q^{58} -3.19131 q^{59} -6.01720 q^{61} +19.7512 q^{62} -11.0898 q^{64} +2.36545 q^{65} -0.789098 q^{67} +1.09520 q^{68} -14.8733 q^{70} +4.82663 q^{71} +1.62623 q^{73} -8.94810 q^{74} -3.96073 q^{76} -7.01333 q^{77} +2.87113 q^{79} +4.93909 q^{80} +14.3115 q^{82} -12.3502 q^{83} -0.760862 q^{85} +18.5978 q^{86} -1.59565 q^{88} +1.11874 q^{89} +5.77033 q^{91} -2.44748 q^{92} -12.1580 q^{94} +2.75161 q^{95} +13.3358 q^{97} -21.5198 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{2} + 8 q^{4} + 3 q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) $ 5 * q + 2 * q^2 + 8 * q^4 + 3 * q^5 - 5 * q^7 + 3 * q^8	$ 5 q + 2 q^{2} + 8 q^{4} + 3 q^{5} - 5 q^{7} + 3 q^{8} + 9 q^{10} + 8 q^{11} + 5 q^{13} + 2 q^{14} - 10 q^{16} - 2 q^{17} - 9 q^{19} + 12 q^{20} + 10 q^{22} - 5 q^{23} + 2 q^{25} + 3 q^{26} - 14 q^{28} - 5 q^{29} - 6 q^{31} + 8 q^{32} + 3 q^{34} + 15 q^{35} + 10 q^{37} + 10 q^{38} + 26 q^{40} + 11 q^{41} - 9 q^{43} + 16 q^{44} - 2 q^{46} + 13 q^{47} - 6 q^{49} + 18 q^{50} + 12 q^{52} - q^{53} + 13 q^{55} + 4 q^{56} - 2 q^{58} - 6 q^{59} + 23 q^{61} + 36 q^{62} - q^{64} + 20 q^{65} - 10 q^{67} + 10 q^{68} - 16 q^{70} + 11 q^{71} + 31 q^{73} + 18 q^{74} - 8 q^{76} - 3 q^{77} + 8 q^{79} + 8 q^{80} + 16 q^{82} - 7 q^{83} + 6 q^{85} + 36 q^{86} - 3 q^{88} - 3 q^{89} + 8 q^{91} - 8 q^{92} - 39 q^{94} + 11 q^{95} + 3 q^{97} - 38 q^{98}+O(q^{100}) $ 5 * q + 2 * q^2 + 8 * q^4 + 3 * q^5 - 5 * q^7 + 3 * q^8 + 9 * q^10 + 8 * q^11 + 5 * q^13 + 2 * q^14 - 10 * q^16 - 2 * q^17 - 9 * q^19 + 12 * q^20 + 10 * q^22 - 5 * q^23 + 2 * q^25 + 3 * q^26 - 14 * q^28 - 5 * q^29 - 6 * q^31 + 8 * q^32 + 3 * q^34 + 15 * q^35 + 10 * q^37 + 10 * q^38 + 26 * q^40 + 11 * q^41 - 9 * q^43 + 16 * q^44 - 2 * q^46 + 13 * q^47 - 6 * q^49 + 18 * q^50 + 12 * q^52 - q^53 + 13 * q^55 + 4 * q^56 - 2 * q^58 - 6 * q^59 + 23 * q^61 + 36 * q^62 - q^64 + 20 * q^65 - 10 * q^67 + 10 * q^68 - 16 * q^70 + 11 * q^71 + 31 * q^73 + 18 * q^74 - 8 * q^76 - 3 * q^77 + 8 * q^79 + 8 * q^80 + 16 * q^82 - 7 * q^83 + 6 * q^85 + 36 * q^86 - 3 * q^88 - 3 * q^89 + 8 * q^91 - 8 * q^92 - 39 * q^94 + 11 * q^95 + 3 * q^97 - 38 * 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 $

$2$	−2.10891	−1.49122	−0.745611	−	0.666382i	$-0.767842\pi$
$2$	−2.10891	−1.49122	−0.745611	+	0.666382i	$0.767842\pi$
$3$	0	0
$3$	0	0
$4$	2.44748	1.22374
$5$	−1.70032	−0.760407	−0.380204	−	0.924903i	$-0.624146\pi$
$5$	−1.70032	−0.760407	−0.380204	+	0.924903i	$0.624146\pi$
$6$	0	0
$7$	−4.14780	−1.56772	−0.783861	−	0.620936i	$-0.786753\pi$
$7$	−4.14780	−1.56772	−0.783861	+	0.620936i	$0.786753\pi$
$8$	−0.943696	−0.333647
$9$	0	0
$10$	3.58582	1.13394
$11$	1.69086	0.509812	0.254906	−	0.966966i	$-0.417955\pi$
$11$	1.69086	0.509812	0.254906	+	0.966966i	$0.417955\pi$
$12$	0	0
$13$	−1.39118	−0.385843	−0.192922	−	0.981214i	$-0.561796\pi$
$13$	−1.39118	−0.385843	−0.192922	+	0.981214i	$0.561796\pi$
$14$	8.74732	2.33782
$15$	0	0
$16$	−2.90480	−0.726199
$17$	0.447481	0.108530	0.0542651	−	0.998527i	$-0.482718\pi$
$17$	0.447481	0.108530	0.0542651	+	0.998527i	$0.482718\pi$
$18$	0	0
$19$	−1.61829	−0.371261	−0.185631	−	0.982620i	$-0.559433\pi$
$19$	−1.61829	−0.371261	−0.185631	+	0.982620i	$0.559433\pi$
$20$	−4.16151	−0.930541
$21$	0	0
$22$	−3.56585	−0.760242
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−2.10891	−0.421781
$26$	2.93386	0.575377
$27$	0	0
$28$	−10.1517	−1.91849
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−9.36561	−1.68211	−0.841057	−	0.540947i	$-0.818066\pi$
$31$	−9.36561	−1.68211	−0.841057	+	0.540947i	$0.818066\pi$
$32$	8.01333	1.41657
$33$	0	0
$34$	−0.943696	−0.161842
$35$	7.05260	1.19211
$36$	0	0
$37$	4.24301	0.697546	0.348773	−	0.937207i	$-0.386598\pi$
$37$	4.24301	0.697546	0.348773	+	0.937207i	$0.386598\pi$
$38$	3.41282	0.553632
$39$	0	0
$40$	1.60459	0.253707
$41$	−6.78622	−1.05983	−0.529915	−	0.848051i	$-0.677776\pi$
$41$	−6.78622	−1.05983	−0.529915	+	0.848051i	$0.677776\pi$
$42$	0	0
$43$	−8.81869	−1.34484	−0.672419	−	0.740171i	$-0.734745\pi$
$43$	−8.81869	−1.34484	−0.672419	+	0.740171i	$0.734745\pi$
$44$	4.13834	0.623878
$45$	0	0
$46$	2.10891	0.310941
$47$	5.76510	0.840926	0.420463	−	0.907310i	$-0.361868\pi$
$47$	5.76510	0.840926	0.420463	+	0.907310i	$0.361868\pi$
$48$	0	0
$49$	10.2043	1.45775
$50$	4.44748	0.628969
$51$	0	0
$52$	−3.40488	−0.472172
$53$	3.27411	0.449734	0.224867	−	0.974389i	$-0.427805\pi$
$53$	3.27411	0.449734	0.224867	+	0.974389i	$0.427805\pi$
$54$	0	0
$55$	−2.87500	−0.387665
$56$	3.91426	0.523066
$57$	0	0
$58$	2.10891	0.276913
$59$	−3.19131	−0.415473	−0.207736	−	0.978185i	$-0.566610\pi$
$59$	−3.19131	−0.415473	−0.207736	+	0.978185i	$0.566610\pi$
$60$	0	0
$61$	−6.01720	−0.770424	−0.385212	−	0.922828i	$-0.625872\pi$
$61$	−6.01720	−0.770424	−0.385212	+	0.922828i	$0.625872\pi$
$62$	19.7512	2.50840
$63$	0	0
$64$	−11.0898	−1.38622
$65$	2.36545	0.293398
$66$	0	0
$67$	−0.789098	−0.0964037	−0.0482018	−	0.998838i	$-0.515349\pi$
$67$	−0.789098	−0.0964037	−0.0482018	+	0.998838i	$0.515349\pi$
$68$	1.09520	0.132813
$69$	0	0
$70$	−14.8733	−1.77770
$71$	4.82663	0.572816	0.286408	−	0.958108i	$-0.407539\pi$
$71$	4.82663	0.572816	0.286408	+	0.958108i	$0.407539\pi$
$72$	0	0
$73$	1.62623	0.190336	0.0951679	−	0.995461i	$-0.469661\pi$
$73$	1.62623	0.190336	0.0951679	+	0.995461i	$0.469661\pi$
$74$	−8.94810	−1.04020
$75$	0	0
$76$	−3.96073	−0.454327
$77$	−7.01333	−0.799244
$78$	0	0
$79$	2.87113	0.323027	0.161514	−	0.986870i	$-0.448362\pi$
$79$	2.87113	0.323027	0.161514	+	0.986870i	$0.448362\pi$
$80$	4.93909	0.552207
$81$	0	0
$82$	14.3115	1.58044
$83$	−12.3502	−1.35561	−0.677804	−	0.735242i	$-0.737068\pi$
$83$	−12.3502	−1.35561	−0.677804	+	0.735242i	$0.737068\pi$
$84$	0	0
$85$	−0.760862	−0.0825271
$86$	18.5978	2.00545
$87$	0	0
$88$	−1.59565	−0.170097
$89$	1.11874	0.118586	0.0592931	−	0.998241i	$-0.481115\pi$
$89$	1.11874	0.118586	0.0592931	+	0.998241i	$0.481115\pi$
$90$	0	0
$91$	5.77033	0.604895
$92$	−2.44748	−0.255168
$93$	0	0
$94$	−12.1580	−1.25401
$95$	2.75161	0.282310
$96$	0	0
$97$	13.3358	1.35405	0.677023	−	0.735961i	$-0.263270\pi$
$97$	13.3358	1.35405	0.677023	+	0.735961i	$0.263270\pi$
$98$	−21.5198	−2.17383
$99$	0	0
$100$	−5.16151	−0.516151
$101$	−4.12480	−0.410433	−0.205216	−	0.978717i	$-0.565790\pi$
$101$	−4.12480	−0.410433	−0.205216	+	0.978717i	$0.565790\pi$
$102$	0	0
$103$	−15.1520	−1.49297	−0.746487	−	0.665400i	$-0.768261\pi$
$103$	−15.1520	−1.49297	−0.746487	+	0.665400i	$0.768261\pi$
$104$	1.31285	0.128735
$105$	0	0
$106$	−6.90480	−0.670653
$107$	−19.3559	−1.87121	−0.935605	−	0.353049i	$-0.885145\pi$
$107$	−19.3559	−1.87121	−0.935605	+	0.353049i	$0.885145\pi$
$108$	0	0
$109$	−14.5279	−1.39153	−0.695763	−	0.718272i	$-0.744933\pi$
$109$	−14.5279	−1.39153	−0.695763	+	0.718272i	$0.744933\pi$
$110$	6.06310	0.578094
$111$	0	0
$112$	12.0485	1.13848
$113$	−11.9621	−1.12530	−0.562650	−	0.826695i	$-0.690218\pi$
$113$	−11.9621	−1.12530	−0.562650	+	0.826695i	$0.690218\pi$
$114$	0	0
$115$	1.70032	0.158556
$116$	−2.44748	−0.227243
$117$	0	0
$118$	6.73016	0.619562
$119$	−1.85606	−0.170145
$120$	0	0
$121$	−8.14101	−0.740092
$122$	12.6897	1.14887
$123$	0	0
$124$	−22.9222	−2.05847
$125$	12.0874	1.08113
$126$	0	0
$127$	−0.739002	−0.0655758	−0.0327879	−	0.999462i	$-0.510439\pi$
$127$	−0.739002	−0.0655758	−0.0327879	+	0.999462i	$0.510439\pi$
$128$	7.36060	0.650591
$129$	0	0
$130$	−4.98851	−0.437521
$131$	8.68511	0.758821	0.379411	−	0.925228i	$-0.376127\pi$
$131$	8.68511	0.758821	0.379411	+	0.925228i	$0.376127\pi$
$132$	0	0
$133$	6.71235	0.582034
$134$	1.66413	0.143759
$135$	0	0
$136$	−0.422286	−0.0362107
$137$	−3.99995	−0.341739	−0.170869	−	0.985294i	$-0.554658\pi$
$137$	−3.99995	−0.341739	−0.170869	+	0.985294i	$0.554658\pi$
$138$	0	0
$139$	−17.0685	−1.44773	−0.723865	−	0.689941i	$-0.757636\pi$
$139$	−17.0685	−1.44773	−0.723865	+	0.689941i	$0.757636\pi$
$140$	17.2611	1.45883
$141$	0	0
$142$	−10.1789	−0.854195
$143$	−2.35228	−0.196707
$144$	0	0
$145$	1.70032	0.141204
$146$	−3.42956	−0.283833
$147$	0	0
$148$	10.3847	0.853615
$149$	−5.30507	−0.434609	−0.217304	−	0.976104i	$-0.569726\pi$
$149$	−5.30507	−0.434609	−0.217304	+	0.976104i	$0.569726\pi$
$150$	0	0
$151$	−4.37769	−0.356252	−0.178126	−	0.984008i	$-0.557003\pi$
$151$	−4.37769	−0.356252	−0.178126	+	0.984008i	$0.557003\pi$
$152$	1.52717	0.123870
$153$	0	0
$154$	14.7905	1.19185
$155$	15.9246	1.27909
$156$	0	0
$157$	7.43735	0.593565	0.296783	−	0.954945i	$-0.404086\pi$
$157$	7.43735	0.593565	0.296783	+	0.954945i	$0.404086\pi$
$158$	−6.05494	−0.481705
$159$	0	0
$160$	−13.6252	−1.07717
$161$	4.14780	0.326893
$162$	0	0
$163$	7.68883	0.602236	0.301118	−	0.953587i	$-0.402640\pi$
$163$	7.68883	0.602236	0.301118	+	0.953587i	$0.402640\pi$
$164$	−16.6092	−1.29696
$165$	0	0
$166$	26.0454	2.02151
$167$	−19.9394	−1.54296	−0.771479	−	0.636254i	$-0.780483\pi$
$167$	−19.9394	−1.54296	−0.771479	+	0.636254i	$0.780483\pi$
$168$	0	0
$169$	−11.0646	−0.851125
$170$	1.60459	0.123066
$171$	0	0
$172$	−21.5836	−1.64573
$173$	12.0607	0.916958	0.458479	−	0.888705i	$-0.348394\pi$
$173$	12.0607	0.916958	0.458479	+	0.888705i	$0.348394\pi$
$174$	0	0
$175$	8.74732	0.661236
$176$	−4.91159	−0.370225
$177$	0	0
$178$	−2.35932	−0.176838
$179$	−1.21963	−0.0911596	−0.0455798	−	0.998961i	$-0.514514\pi$
$179$	−1.21963	−0.0911596	−0.0455798	+	0.998961i	$0.514514\pi$
$180$	0	0
$181$	−4.67942	−0.347818	−0.173909	−	0.984762i	$-0.555640\pi$
$181$	−4.67942	−0.347818	−0.173909	+	0.984762i	$0.555640\pi$
$182$	−12.1691	−0.902032
$183$	0	0
$184$	0.943696	0.0695702
$185$	−7.21448	−0.530419
$186$	0	0
$187$	0.756626	0.0553300
$188$	14.1100	1.02908
$189$	0	0
$190$	−5.80289	−0.420986
$191$	2.03591	0.147313	0.0736567	−	0.997284i	$-0.476533\pi$
$191$	2.03591	0.147313	0.0736567	+	0.997284i	$0.476533\pi$
$192$	0	0
$193$	−3.35495	−0.241495	−0.120747	−	0.992683i	$-0.538529\pi$
$193$	−3.35495	−0.241495	−0.120747	+	0.992683i	$0.538529\pi$
$194$	−28.1240	−2.01918
$195$	0	0
$196$	24.9748	1.78391
$197$	17.6069	1.25444	0.627219	−	0.778843i	$-0.284193\pi$
$197$	17.6069	1.25444	0.627219	+	0.778843i	$0.284193\pi$
$198$	0	0
$199$	18.0483	1.27941	0.639705	−	0.768620i	$-0.279056\pi$
$199$	18.0483	1.27941	0.639705	+	0.768620i	$0.279056\pi$
$200$	1.99016	0.140726
$201$	0	0
$202$	8.69881	0.612046
$203$	4.14780	0.291119
$204$	0	0
$205$	11.5388	0.805902
$206$	31.9542	2.22636
$207$	0	0
$208$	4.04109	0.280199
$209$	−2.73629	−0.189273
$210$	0	0
$211$	21.7200	1.49527	0.747634	−	0.664111i	$-0.231190\pi$
$211$	21.7200	1.49527	0.747634	+	0.664111i	$0.231190\pi$
$212$	8.01333	0.550358
$213$	0	0
$214$	40.8199	2.79039
$215$	14.9946	1.02262
$216$	0	0
$217$	38.8467	2.63709
$218$	30.6381	2.07507
$219$	0	0
$220$	−7.03650	−0.474401
$221$	−0.622526	−0.0418756
$222$	0	0
$223$	−1.71850	−0.115080	−0.0575398	−	0.998343i	$-0.518326\pi$
$223$	−1.71850	−0.115080	−0.0575398	+	0.998343i	$0.518326\pi$
$224$	−33.2377	−2.22079
$225$	0	0
$226$	25.2269	1.67807
$227$	6.95007	0.461292	0.230646	−	0.973038i	$-0.425916\pi$
$227$	6.95007	0.461292	0.230646	+	0.973038i	$0.425916\pi$
$228$	0	0
$229$	8.10462	0.535568	0.267784	−	0.963479i	$-0.413709\pi$
$229$	8.10462	0.535568	0.267784	+	0.963479i	$0.413709\pi$
$230$	−3.58582	−0.236442
$231$	0	0
$232$	0.943696	0.0619567
$233$	2.41871	0.158455	0.0792276	−	0.996857i	$-0.474755\pi$
$233$	2.41871	0.158455	0.0792276	+	0.996857i	$0.474755\pi$
$234$	0	0
$235$	−9.80252	−0.639446
$236$	−7.81066	−0.508431
$237$	0	0
$238$	3.91426	0.253724
$239$	−22.4055	−1.44929	−0.724646	−	0.689122i	$-0.757997\pi$
$239$	−22.4055	−1.44929	−0.724646	+	0.689122i	$0.757997\pi$
$240$	0	0
$241$	−19.2500	−1.24000	−0.620000	−	0.784602i	$-0.712867\pi$
$241$	−19.2500	−1.24000	−0.620000	+	0.784602i	$0.712867\pi$
$242$	17.1686	1.10364
$243$	0	0
$244$	−14.7270	−0.942799
$245$	−17.3505	−1.10849
$246$	0	0
$247$	2.25133	0.143249
$248$	8.83829	0.561232
$249$	0	0
$250$	−25.4912	−1.61221
$251$	−6.38401	−0.402955	−0.201478	−	0.979493i	$-0.564574\pi$
$251$	−6.38401	−0.402955	−0.201478	+	0.979493i	$0.564574\pi$
$252$	0	0
$253$	−1.69086	−0.106303
$254$	1.55849	0.0977881
$255$	0	0
$256$	6.65673	0.416045
$257$	−16.9585	−1.05784	−0.528922	−	0.848670i	$-0.677404\pi$
$257$	−16.9585	−1.05784	−0.528922	+	0.848670i	$0.677404\pi$
$258$	0	0
$259$	−17.5992	−1.09356
$260$	5.78939	0.359043
$261$	0	0
$262$	−18.3161	−1.13157
$263$	20.9475	1.29168	0.645840	−	0.763473i	$-0.276507\pi$
$263$	20.9475	1.29168	0.645840	+	0.763473i	$0.276507\pi$
$264$	0	0
$265$	−5.56705	−0.341981
$266$	−14.1557	−0.867942
$267$	0	0
$268$	−1.93130	−0.117973
$269$	21.3774	1.30341	0.651703	−	0.758475i	$-0.274055\pi$
$269$	21.3774	1.30341	0.651703	+	0.758475i	$0.274055\pi$
$270$	0	0
$271$	10.9086	0.662651	0.331326	−	0.943517i	$-0.392504\pi$
$271$	10.9086	0.662651	0.331326	+	0.943517i	$0.392504\pi$
$272$	−1.29984	−0.0788145
$273$	0	0
$274$	8.43551	0.509608
$275$	−3.56585	−0.215029
$276$	0	0
$277$	−18.2829	−1.09851	−0.549256	−	0.835654i	$-0.685089\pi$
$277$	−18.2829	−1.09851	−0.549256	+	0.835654i	$0.685089\pi$
$278$	35.9958	2.15889
$279$	0	0
$280$	−6.65551	−0.397743
$281$	12.5221	0.747005	0.373503	−	0.927629i	$-0.378157\pi$
$281$	12.5221	0.747005	0.373503	+	0.927629i	$0.378157\pi$
$282$	0	0
$283$	16.4946	0.980502	0.490251	−	0.871581i	$-0.336905\pi$
$283$	16.4946	0.980502	0.490251	+	0.871581i	$0.336905\pi$
$284$	11.8131	0.700978
$285$	0	0
$286$	4.96073	0.293334
$287$	28.1479	1.66152
$288$	0	0
$289$	−16.7998	−0.988221
$290$	−3.58582	−0.210566
$291$	0	0
$292$	3.98017	0.232922
$293$	−16.6292	−0.971486	−0.485743	−	0.874102i	$-0.661451\pi$
$293$	−16.6292	−0.971486	−0.485743	+	0.874102i	$0.661451\pi$
$294$	0	0
$295$	5.42625	0.315928
$296$	−4.00411	−0.232734
$297$	0	0
$298$	11.1879	0.648097
$299$	1.39118	0.0804539
$300$	0	0
$301$	36.5782	2.10833
$302$	9.23214	0.531250
$303$	0	0
$304$	4.70080	0.269610
$305$	10.2312	0.585836
$306$	0	0
$307$	−8.19558	−0.467747	−0.233873	−	0.972267i	$-0.575140\pi$
$307$	−8.19558	−0.467747	−0.233873	+	0.972267i	$0.575140\pi$
$308$	−17.1650	−0.978067
$309$	0	0
$310$	−33.5834	−1.90741
$311$	1.00832	0.0571767	0.0285883	−	0.999591i	$-0.490899\pi$
$311$	1.00832	0.0571767	0.0285883	+	0.999591i	$0.490899\pi$
$312$	0	0
$313$	20.3396	1.14966	0.574832	−	0.818271i	$-0.305067\pi$
$313$	20.3396	1.14966	0.574832	+	0.818271i	$0.305067\pi$
$314$	−15.6847	−0.885137
$315$	0	0
$316$	7.02704	0.395302
$317$	16.9983	0.954719	0.477359	−	0.878708i	$-0.341594\pi$
$317$	16.9983	0.954719	0.477359	+	0.878708i	$0.341594\pi$
$318$	0	0
$319$	−1.69086	−0.0946697
$320$	18.8562	1.05409
$321$	0	0
$322$	−8.74732	−0.487469
$323$	−0.724154	−0.0402930
$324$	0	0
$325$	2.93386	0.162741
$326$	−16.2150	−0.898067
$327$	0	0
$328$	6.40413	0.353609
$329$	−23.9125	−1.31834
$330$	0	0
$331$	−29.8638	−1.64146	−0.820732	−	0.571314i	$-0.806434\pi$
$331$	−29.8638	−1.64146	−0.820732	+	0.571314i	$0.806434\pi$
$332$	−30.2268	−1.65891
$333$	0	0
$334$	42.0503	2.30089
$335$	1.34172	0.0733060
$336$	0	0
$337$	1.68783	0.0919420	0.0459710	−	0.998943i	$-0.485362\pi$
$337$	1.68783	0.0919420	0.0459710	+	0.998943i	$0.485362\pi$
$338$	23.3342	1.26922
$339$	0	0
$340$	−1.86220	−0.100992
$341$	−15.8359	−0.857562
$342$	0	0
$343$	−13.2907	−0.717630
$344$	8.32216	0.448701
$345$	0	0
$346$	−25.4349	−1.36739
$347$	12.4406	0.667848	0.333924	−	0.942600i	$-0.391627\pi$
$347$	12.4406	0.667848	0.333924	+	0.942600i	$0.391627\pi$
$348$	0	0
$349$	13.6676	0.731612	0.365806	−	0.930691i	$-0.380793\pi$
$349$	13.6676	0.731612	0.365806	+	0.930691i	$0.380793\pi$
$350$	−18.4473	−0.986048
$351$	0	0
$352$	13.5494	0.722185
$353$	−19.9111	−1.05976	−0.529881	−	0.848072i	$-0.677764\pi$
$353$	−19.9111	−1.05976	−0.529881	+	0.848072i	$0.677764\pi$
$354$	0	0
$355$	−8.20683	−0.435573
$356$	2.73810	0.145119
$357$	0	0
$358$	2.57209	0.135939
$359$	2.91592	0.153896	0.0769482	−	0.997035i	$-0.475482\pi$
$359$	2.91592	0.153896	0.0769482	+	0.997035i	$0.475482\pi$
$360$	0	0
$361$	−16.3811	−0.862165
$362$	9.86845	0.518674
$363$	0	0
$364$	14.1228	0.740234
$365$	−2.76511	−0.144733
$366$	0	0
$367$	−6.02666	−0.314589	−0.157294	−	0.987552i	$-0.550277\pi$
$367$	−6.02666	−0.314589	−0.157294	+	0.987552i	$0.550277\pi$
$368$	2.90480	0.151423
$369$	0	0
$370$	15.2146	0.790972
$371$	−13.5804	−0.705058
$372$	0	0
$373$	29.7097	1.53831	0.769155	−	0.639063i	$-0.220678\pi$
$373$	29.7097	1.53831	0.769155	+	0.639063i	$0.220678\pi$
$374$	−1.59565	−0.0825092
$375$	0	0
$376$	−5.44050	−0.280572
$377$	1.39118	0.0716493
$378$	0	0
$379$	−27.1750	−1.39589	−0.697943	−	0.716153i	$-0.745901\pi$
$379$	−27.1750	−1.39589	−0.697943	+	0.716153i	$0.745901\pi$
$380$	6.73452	0.345474
$381$	0	0
$382$	−4.29354	−0.219677
$383$	21.6492	1.10622	0.553111	−	0.833108i	$-0.313440\pi$
$383$	21.6492	1.10622	0.553111	+	0.833108i	$0.313440\pi$
$384$	0	0
$385$	11.9249	0.607751
$386$	7.07527	0.360122
$387$	0	0
$388$	32.6392	1.65700
$389$	−19.9748	−1.01276	−0.506382	−	0.862309i	$-0.669017\pi$
$389$	−19.9748	−1.01276	−0.506382	+	0.862309i	$0.669017\pi$
$390$	0	0
$391$	−0.447481	−0.0226301
$392$	−9.62973	−0.486375
$393$	0	0
$394$	−37.1313	−1.87065
$395$	−4.88185	−0.245632
$396$	0	0
$397$	10.0159	0.502683	0.251342	−	0.967898i	$-0.419128\pi$
$397$	10.0159	0.502683	0.251342	+	0.967898i	$0.419128\pi$
$398$	−38.0622	−1.90788
$399$	0	0
$400$	6.12594	0.306297
$401$	32.7020	1.63306	0.816529	−	0.577304i	$-0.195895\pi$
$401$	32.7020	1.63306	0.816529	+	0.577304i	$0.195895\pi$
$402$	0	0
$403$	13.0292	0.649032
$404$	−10.0954	−0.502263
$405$	0	0
$406$	−8.74732	−0.434122
$407$	7.17431	0.355617
$408$	0	0
$409$	1.79151	0.0885843	0.0442921	−	0.999019i	$-0.485897\pi$
$409$	1.79151	0.0885843	0.0442921	+	0.999019i	$0.485897\pi$
$410$	−24.3342	−1.20178
$411$	0	0
$412$	−37.0843	−1.82701
$413$	13.2369	0.651346
$414$	0	0
$415$	20.9993	1.03081
$416$	−11.1480	−0.546574
$417$	0	0
$418$	5.77058	0.282248
$419$	13.5809	0.663469	0.331734	−	0.943373i	$-0.392366\pi$
$419$	13.5809	0.663469	0.331734	+	0.943373i	$0.392366\pi$
$420$	0	0
$421$	21.9168	1.06816	0.534079	−	0.845435i	$-0.320658\pi$
$421$	21.9168	1.06816	0.534079	+	0.845435i	$0.320658\pi$
$422$	−45.8055	−2.22978
$423$	0	0
$424$	−3.08977	−0.150052
$425$	−0.943696	−0.0457760
$426$	0	0
$427$	24.9582	1.20781
$428$	−47.3733	−2.28988
$429$	0	0
$430$	−31.6222	−1.52496
$431$	−39.4774	−1.90156	−0.950780	−	0.309866i	$-0.899716\pi$
$431$	−39.4774	−1.90156	−0.950780	+	0.309866i	$0.899716\pi$
$432$	0	0
$433$	38.4149	1.84610	0.923051	−	0.384677i	$-0.125687\pi$
$433$	38.4149	1.84610	0.923051	+	0.384677i	$0.125687\pi$
$434$	−81.9241	−3.93248
$435$	0	0
$436$	−35.5569	−1.70287
$437$	1.61829	0.0774133
$438$	0	0
$439$	−24.6601	−1.17696	−0.588480	−	0.808512i	$-0.700274\pi$
$439$	−24.6601	−1.17696	−0.588480	+	0.808512i	$0.700274\pi$
$440$	2.71312	0.129343
$441$	0	0
$442$	1.31285	0.0624458
$443$	18.1723	0.863394	0.431697	−	0.902019i	$-0.357915\pi$
$443$	18.1723	0.863394	0.431697	+	0.902019i	$0.357915\pi$
$444$	0	0
$445$	−1.90222	−0.0901738
$446$	3.62416	0.171609
$447$	0	0
$448$	45.9982	2.17321
$449$	12.8184	0.604939	0.302469	−	0.953159i	$-0.402189\pi$
$449$	12.8184	0.604939	0.302469	+	0.953159i	$0.402189\pi$
$450$	0	0
$451$	−11.4745	−0.540314
$452$	−29.2770	−1.37707
$453$	0	0
$454$	−14.6570	−0.687889
$455$	−9.81142	−0.459966
$456$	0	0
$457$	−7.00643	−0.327747	−0.163873	−	0.986481i	$-0.552399\pi$
$457$	−7.00643	−0.327747	−0.163873	+	0.986481i	$0.552399\pi$
$458$	−17.0919	−0.798651
$459$	0	0
$460$	4.16151	0.194031
$461$	−9.09038	−0.423381	−0.211691	−	0.977337i	$-0.567897\pi$
$461$	−9.09038	−0.423381	−0.211691	+	0.977337i	$0.567897\pi$
$462$	0	0
$463$	−17.3440	−0.806045	−0.403023	−	0.915190i	$-0.632040\pi$
$463$	−17.3440	−0.806045	−0.403023	+	0.915190i	$0.632040\pi$
$464$	2.90480	0.134852
$465$	0	0
$466$	−5.10084	−0.236292
$467$	3.25749	0.150738	0.0753692	−	0.997156i	$-0.475986\pi$
$467$	3.25749	0.150738	0.0753692	+	0.997156i	$0.475986\pi$
$468$	0	0
$469$	3.27302	0.151134
$470$	20.6726	0.953556
$471$	0	0
$472$	3.01162	0.138621
$473$	−14.9111	−0.685615
$474$	0	0
$475$	3.41282	0.156591
$476$	−4.54268	−0.208214
$477$	0	0
$478$	47.2511	2.16121
$479$	29.1623	1.33246	0.666230	−	0.745747i	$-0.267907\pi$
$479$	29.1623	1.33246	0.666230	+	0.745747i	$0.267907\pi$
$480$	0	0
$481$	−5.90277	−0.269143
$482$	40.5964	1.84911
$483$	0	0
$484$	−19.9250	−0.905680
$485$	−22.6752	−1.02963
$486$	0	0
$487$	−4.20908	−0.190732	−0.0953658	−	0.995442i	$-0.530402\pi$
$487$	−4.20908	−0.190732	−0.0953658	+	0.995442i	$0.530402\pi$
$488$	5.67841	0.257049
$489$	0	0
$490$	36.5907	1.65300
$491$	−3.86511	−0.174430	−0.0872150	−	0.996190i	$-0.527797\pi$
$491$	−3.86511	−0.174430	−0.0872150	+	0.996190i	$0.527797\pi$
$492$	0	0
$493$	−0.447481	−0.0201535
$494$	−4.74784	−0.213615
$495$	0	0
$496$	27.2052	1.22155
$497$	−20.0199	−0.898016
$498$	0	0
$499$	−2.52762	−0.113152	−0.0565759	−	0.998398i	$-0.518018\pi$
$499$	−2.52762	−0.113152	−0.0565759	+	0.998398i	$0.518018\pi$
$500$	29.5838	1.32303
$501$	0	0
$502$	13.4633	0.600896
$503$	31.9416	1.42420	0.712102	−	0.702076i	$-0.247743\pi$
$503$	31.9416	1.42420	0.712102	+	0.702076i	$0.247743\pi$
$504$	0	0
$505$	7.01349	0.312096
$506$	3.56585	0.158522
$507$	0	0
$508$	−1.80869	−0.0802478
$509$	−17.0955	−0.757745	−0.378873	−	0.925449i	$-0.623688\pi$
$509$	−17.0955	−0.757745	−0.378873	+	0.925449i	$0.623688\pi$
$510$	0	0
$511$	−6.74528	−0.298394
$512$	−28.7596	−1.27101
$513$	0	0
$514$	35.7640	1.57748
$515$	25.7633	1.13527
$516$	0	0
$517$	9.74795	0.428714
$518$	37.1149	1.63074
$519$	0	0
$520$	−2.23226	−0.0978913
$521$	−5.35710	−0.234699	−0.117349	−	0.993091i	$-0.537440\pi$
$521$	−5.35710	−0.234699	−0.117349	+	0.993091i	$0.537440\pi$
$522$	0	0
$523$	40.9503	1.79063	0.895316	−	0.445432i	$-0.146950\pi$
$523$	40.9503	1.79063	0.895316	+	0.445432i	$0.146950\pi$
$524$	21.2566	0.928600
$525$	0	0
$526$	−44.1763	−1.92618
$527$	−4.19094	−0.182560
$528$	0	0
$529$	1.00000	0.0434783
$530$	11.7404	0.509969
$531$	0	0
$532$	16.4283	0.712259
$533$	9.44084	0.408928
$534$	0	0
$535$	32.9113	1.42288
$536$	0.744668	0.0321648
$537$	0	0
$538$	−45.0830	−1.94367
$539$	17.2539	0.743180
$540$	0	0
$541$	−12.8469	−0.552333	−0.276166	−	0.961110i	$-0.589064\pi$
$541$	−12.8469	−0.552333	−0.276166	+	0.961110i	$0.589064\pi$
$542$	−23.0052	−0.988159
$543$	0	0
$544$	3.58582	0.153741
$545$	24.7022	1.05813
$546$	0	0
$547$	8.13191	0.347695	0.173848	−	0.984773i	$-0.444380\pi$
$547$	8.13191	0.347695	0.173848	+	0.984773i	$0.444380\pi$
$548$	−9.78980	−0.418199
$549$	0	0
$550$	7.52005	0.320656
$551$	1.61829	0.0689415
$552$	0	0
$553$	−11.9089	−0.506417
$554$	38.5569	1.63812
$555$	0	0
$556$	−41.7748	−1.77165
$557$	23.6634	1.00265	0.501326	−	0.865259i	$-0.332846\pi$
$557$	23.6634	1.00265	0.501326	+	0.865259i	$0.332846\pi$
$558$	0	0
$559$	12.2684	0.518897
$560$	−20.4864	−0.865708
$561$	0	0
$562$	−26.4079	−1.11395
$563$	3.36335	0.141748	0.0708742	−	0.997485i	$-0.477421\pi$
$563$	3.36335	0.141748	0.0708742	+	0.997485i	$0.477421\pi$
$564$	0	0
$565$	20.3394	0.855685
$566$	−34.7855	−1.46215
$567$	0	0
$568$	−4.55487	−0.191118
$569$	20.7465	0.869737	0.434868	−	0.900494i	$-0.356795\pi$
$569$	20.7465	0.869737	0.434868	+	0.900494i	$0.356795\pi$
$570$	0	0
$571$	−42.6361	−1.78427	−0.892133	−	0.451773i	$-0.850792\pi$
$571$	−42.6361	−1.78427	−0.892133	+	0.451773i	$0.850792\pi$
$572$	−5.75716	−0.240719
$573$	0	0
$574$	−59.3613	−2.47769
$575$	2.10891	0.0879474
$576$	0	0
$577$	−36.2162	−1.50770	−0.753851	−	0.657045i	$-0.771806\pi$
$577$	−36.2162	−1.50770	−0.753851	+	0.657045i	$0.771806\pi$
$578$	35.4291	1.47366
$579$	0	0
$580$	4.16151	0.172797
$581$	51.2261	2.12522
$582$	0	0
$583$	5.53605	0.229280
$584$	−1.53467	−0.0635049
$585$	0	0
$586$	35.0693	1.44870
$587$	5.87794	0.242609	0.121304	−	0.992615i	$-0.461292\pi$
$587$	5.87794	0.242609	0.121304	+	0.992615i	$0.461292\pi$
$588$	0	0
$589$	15.1563	0.624503
$590$	−11.4434	−0.471119
$591$	0	0
$592$	−12.3251	−0.506557
$593$	6.08323	0.249808	0.124904	−	0.992169i	$-0.460138\pi$
$593$	6.08323	0.249808	0.124904	+	0.992169i	$0.460138\pi$
$594$	0	0
$595$	3.15591	0.129380
$596$	−12.9841	−0.531848
$597$	0	0
$598$	−2.93386	−0.119974
$599$	31.2292	1.27599	0.637995	−	0.770040i	$-0.279764\pi$
$599$	31.2292	1.27599	0.637995	+	0.770040i	$0.279764\pi$
$600$	0	0
$601$	36.2896	1.48028	0.740142	−	0.672451i	$-0.234758\pi$
$601$	36.2896	1.48028	0.740142	+	0.672451i	$0.234758\pi$
$602$	−77.1400	−3.14399
$603$	0	0
$604$	−10.7143	−0.435959
$605$	13.8423	0.562771
$606$	0	0
$607$	31.8055	1.29095	0.645473	−	0.763783i	$-0.276660\pi$
$607$	31.8055	1.29095	0.645473	+	0.763783i	$0.276660\pi$
$608$	−12.9679	−0.525918
$609$	0	0
$610$	−21.5766	−0.873611
$611$	−8.02027	−0.324466
$612$	0	0
$613$	5.64787	0.228115	0.114058	−	0.993474i	$-0.463615\pi$
$613$	5.64787	0.228115	0.114058	+	0.993474i	$0.463615\pi$
$614$	17.2837	0.697514
$615$	0	0
$616$	6.61845	0.266665
$617$	9.45805	0.380767	0.190383	−	0.981710i	$-0.439027\pi$
$617$	9.45805	0.380767	0.190383	+	0.981710i	$0.439027\pi$
$618$	0	0
$619$	−30.5412	−1.22756	−0.613778	−	0.789479i	$-0.710351\pi$
$619$	−30.5412	−1.22756	−0.613778	+	0.789479i	$0.710351\pi$
$620$	38.9751	1.56528
$621$	0	0
$622$	−2.12646	−0.0852631
$623$	−4.64031	−0.185910
$624$	0	0
$625$	−10.0080	−0.400320
$626$	−42.8944	−1.71440
$627$	0	0
$628$	18.2028	0.726370
$629$	1.89867	0.0757048
$630$	0	0
$631$	32.0399	1.27549	0.637743	−	0.770249i	$-0.279868\pi$
$631$	32.0399	1.27549	0.637743	+	0.770249i	$0.279868\pi$
$632$	−2.70947	−0.107777
$633$	0	0
$634$	−35.8478	−1.42370
$635$	1.25654	0.0498643
$636$	0	0
$637$	−14.1960	−0.562464
$638$	3.56585	0.141173
$639$	0	0
$640$	−12.5154	−0.494714
$641$	−11.6295	−0.459337	−0.229669	−	0.973269i	$-0.573764\pi$
$641$	−11.6295	−0.459337	−0.229669	+	0.973269i	$0.573764\pi$
$642$	0	0
$643$	22.2518	0.877527	0.438763	−	0.898603i	$-0.355417\pi$
$643$	22.2518	0.877527	0.438763	+	0.898603i	$0.355417\pi$
$644$	10.1517	0.400032
$645$	0	0
$646$	1.52717	0.0600858
$647$	49.8578	1.96011	0.980056	−	0.198721i	$-0.0636788\pi$
$647$	49.8578	1.96011	0.980056	+	0.198721i	$0.0636788\pi$
$648$	0	0
$649$	−5.39604	−0.211813
$650$	−6.18723	−0.242683
$651$	0	0
$652$	18.8183	0.736980
$653$	11.6765	0.456938	0.228469	−	0.973551i	$-0.426628\pi$
$653$	11.6765	0.456938	0.228469	+	0.973551i	$0.426628\pi$
$654$	0	0
$655$	−14.7675	−0.577013
$656$	19.7126	0.769648
$657$	0	0
$658$	50.4292	1.96593
$659$	4.04973	0.157755	0.0788775	−	0.996884i	$-0.474866\pi$
$659$	4.04973	0.157755	0.0788775	+	0.996884i	$0.474866\pi$
$660$	0	0
$661$	−7.81096	−0.303811	−0.151905	−	0.988395i	$-0.548541\pi$
$661$	−7.81096	−0.303811	−0.151905	+	0.988395i	$0.548541\pi$
$662$	62.9800	2.44779
$663$	0	0
$664$	11.6548	0.452294
$665$	−11.4132	−0.442583
$666$	0	0
$667$	1.00000	0.0387202
$668$	−48.8014	−1.88818
$669$	0	0
$670$	−2.82956	−0.109315
$671$	−10.1742	−0.392771
$672$	0	0
$673$	16.0017	0.616819	0.308410	−	0.951254i	$-0.400203\pi$
$673$	16.0017	0.616819	0.308410	+	0.951254i	$0.400203\pi$
$674$	−3.55948	−0.137106
$675$	0	0
$676$	−27.0805	−1.04156
$677$	34.7639	1.33609	0.668043	−	0.744123i	$-0.267132\pi$
$677$	34.7639	1.33609	0.668043	+	0.744123i	$0.267132\pi$
$678$	0	0
$679$	−55.3143	−2.12277
$680$	0.718023	0.0275349
$681$	0	0
$682$	33.3964	1.27881
$683$	18.2709	0.699118	0.349559	−	0.936914i	$-0.386331\pi$
$683$	18.2709	0.699118	0.349559	+	0.936914i	$0.386331\pi$
$684$	0	0
$685$	6.80120	0.259860
$686$	28.0288	1.07014
$687$	0	0
$688$	25.6165	0.976621
$689$	−4.55487	−0.173527
$690$	0	0
$691$	−22.3284	−0.849411	−0.424705	−	0.905332i	$-0.639622\pi$
$691$	−22.3284	−0.849411	−0.424705	+	0.905332i	$0.639622\pi$
$692$	29.5184	1.12212
$693$	0	0
$694$	−26.2361	−0.995909
$695$	29.0219	1.10086
$696$	0	0
$697$	−3.03671	−0.115024
$698$	−28.8237	−1.09099
$699$	0	0
$700$	21.4089	0.809181
$701$	8.51917	0.321765	0.160882	−	0.986974i	$-0.448566\pi$
$701$	8.51917	0.321765	0.160882	+	0.986974i	$0.448566\pi$
$702$	0	0
$703$	−6.86641	−0.258972
$704$	−18.7512	−0.706712
$705$	0	0
$706$	41.9907	1.58034
$707$	17.1089	0.643445
$708$	0	0
$709$	−35.3762	−1.32858	−0.664291	−	0.747474i	$-0.731266\pi$
$709$	−35.3762	−1.32858	−0.664291	+	0.747474i	$0.731266\pi$
$710$	17.3074	0.649536
$711$	0	0
$712$	−1.05575	−0.0395659
$713$	9.36561	0.350745
$714$	0	0
$715$	3.99963	0.149578
$716$	−2.98503	−0.111556
$717$	0	0
$718$	−6.14940	−0.229493
$719$	6.90935	0.257675	0.128838	−	0.991666i	$-0.458875\pi$
$719$	6.90935	0.257675	0.128838	+	0.991666i	$0.458875\pi$
$720$	0	0
$721$	62.8477	2.34057
$722$	34.5463	1.28568
$723$	0	0
$724$	−11.4528	−0.425639
$725$	2.10891	0.0783228
$726$	0	0
$727$	30.7011	1.13864	0.569321	−	0.822115i	$-0.307206\pi$
$727$	30.7011	1.13864	0.569321	+	0.822115i	$0.307206\pi$
$728$	−5.44544	−0.201821
$729$	0	0
$730$	5.83136	0.215828
$731$	−3.94620	−0.145956
$732$	0	0
$733$	20.6874	0.764108	0.382054	−	0.924140i	$-0.375217\pi$
$733$	20.6874	0.764108	0.382054	+	0.924140i	$0.375217\pi$
$734$	12.7096	0.469122
$735$	0	0
$736$	−8.01333	−0.295375
$737$	−1.33425	−0.0491477
$738$	0	0
$739$	−8.99509	−0.330890	−0.165445	−	0.986219i	$-0.552906\pi$
$739$	−8.99509	−0.330890	−0.165445	+	0.986219i	$0.552906\pi$
$740$	−17.6573	−0.649095
$741$	0	0
$742$	28.6397	1.05140
$743$	−48.0981	−1.76455	−0.882274	−	0.470737i	$-0.843988\pi$
$743$	−48.0981	−1.76455	−0.882274	+	0.470737i	$0.843988\pi$
$744$	0	0
$745$	9.02033	0.330479
$746$	−62.6549	−2.29396
$747$	0	0
$748$	1.85183	0.0677096
$749$	80.2846	2.93354
$750$	0	0
$751$	−24.7371	−0.902670	−0.451335	−	0.892355i	$-0.649052\pi$
$751$	−24.7371	−0.902670	−0.451335	+	0.892355i	$0.649052\pi$
$752$	−16.7464	−0.610680
$753$	0	0
$754$	−2.93386	−0.106845
$755$	7.44349	0.270896
$756$	0	0
$757$	−38.0418	−1.38265	−0.691327	−	0.722542i	$-0.742973\pi$
$757$	−38.0418	−1.38265	−0.691327	+	0.722542i	$0.742973\pi$
$758$	57.3095	2.08158
$759$	0	0
$760$	−2.59669	−0.0941917
$761$	33.9991	1.23247	0.616233	−	0.787564i	$-0.288658\pi$
$761$	33.9991	1.23247	0.616233	+	0.787564i	$0.288658\pi$
$762$	0	0
$763$	60.2591	2.18152
$764$	4.98285	0.180273
$765$	0	0
$766$	−45.6561	−1.64962
$767$	4.43967	0.160307
$768$	0	0
$769$	29.6812	1.07033	0.535167	−	0.844747i	$-0.320249\pi$
$769$	29.6812	1.07033	0.535167	+	0.844747i	$0.320249\pi$
$770$	−25.1485	−0.906291
$771$	0	0
$772$	−8.21118	−0.295527
$773$	18.1304	0.652104	0.326052	−	0.945352i	$-0.394282\pi$
$773$	18.1304	0.652104	0.326052	+	0.945352i	$0.394282\pi$
$774$	0	0
$775$	19.7512	0.709484
$776$	−12.5850	−0.451773
$777$	0	0
$778$	42.1250	1.51025
$779$	10.9821	0.393474
$780$	0	0
$781$	8.16114	0.292028
$782$	0.943696	0.0337465
$783$	0	0
$784$	−29.6413	−1.05862
$785$	−12.6459	−0.451351
$786$	0	0
$787$	−27.4897	−0.979904	−0.489952	−	0.871750i	$-0.662986\pi$
$787$	−27.4897	−0.979904	−0.489952	+	0.871750i	$0.662986\pi$
$788$	43.0925	1.53511
$789$	0	0
$790$	10.2954	0.366292
$791$	49.6164	1.76416
$792$	0	0
$793$	8.37099	0.297263
$794$	−21.1226	−0.749612
$795$	0	0
$796$	44.1729	1.56567
$797$	18.9492	0.671217	0.335608	−	0.942002i	$-0.391058\pi$
$797$	18.9492	0.671217	0.335608	+	0.942002i	$0.391058\pi$
$798$	0	0
$799$	2.57977	0.0912658
$800$	−16.8994	−0.597483
$801$	0	0
$802$	−68.9653	−2.43525
$803$	2.74972	0.0970354
$804$	0	0
$805$	−7.05260	−0.248572
$806$	−27.4774	−0.967850
$807$	0	0
$808$	3.89255	0.136940
$809$	31.3371	1.10176	0.550878	−	0.834586i	$-0.314293\pi$
$809$	31.3371	1.10176	0.550878	+	0.834586i	$0.314293\pi$
$810$	0	0
$811$	−2.22969	−0.0782948	−0.0391474	−	0.999233i	$-0.512464\pi$
$811$	−2.22969	−0.0782948	−0.0391474	+	0.999233i	$0.512464\pi$
$812$	10.1517	0.356254
$813$	0	0
$814$	−15.1299	−0.530304
$815$	−13.0735	−0.457944
$816$	0	0
$817$	14.2712	0.499286
$818$	−3.77812	−0.132099
$819$	0	0
$820$	28.2409	0.986215
$821$	8.93914	0.311978	0.155989	−	0.987759i	$-0.450144\pi$
$821$	8.93914	0.311978	0.155989	+	0.987759i	$0.450144\pi$
$822$	0	0
$823$	−1.05732	−0.0368559	−0.0184279	−	0.999830i	$-0.505866\pi$
$823$	−1.05732	−0.0368559	−0.0184279	+	0.999830i	$0.505866\pi$
$824$	14.2989	0.498126
$825$	0	0
$826$	−27.9154	−0.971301
$827$	1.63075	0.0567066	0.0283533	−	0.999598i	$-0.490974\pi$
$827$	1.63075	0.0567066	0.0283533	+	0.999598i	$0.490974\pi$
$828$	0	0
$829$	−48.4947	−1.68429	−0.842145	−	0.539252i	$-0.818707\pi$
$829$	−48.4947	−1.68429	−0.842145	+	0.539252i	$0.818707\pi$
$830$	−44.2855	−1.53717
$831$	0	0
$832$	15.4278	0.534864
$833$	4.56622	0.158210
$834$	0	0
$835$	33.9034	1.17328
$836$	−6.69703	−0.231622
$837$	0	0
$838$	−28.6408	−0.989379
$839$	11.4182	0.394201	0.197100	−	0.980383i	$-0.436848\pi$
$839$	11.4182	0.394201	0.197100	+	0.980383i	$0.436848\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−46.2204	−1.59286
$843$	0	0
$844$	53.1594	1.82982
$845$	18.8134	0.647202
$846$	0	0
$847$	33.7673	1.16026
$848$	−9.51064	−0.326597
$849$	0	0
$850$	1.99016	0.0682621
$851$	−4.24301	−0.145448
$852$	0	0
$853$	−8.33060	−0.285234	−0.142617	−	0.989778i	$-0.545552\pi$
$853$	−8.33060	−0.285234	−0.142617	+	0.989778i	$0.545552\pi$
$854$	−52.6344	−1.80111
$855$	0	0
$856$	18.2661	0.624323
$857$	16.7918	0.573597	0.286799	−	0.957991i	$-0.407409\pi$
$857$	16.7918	0.573597	0.286799	+	0.957991i	$0.407409\pi$
$858$	0	0
$859$	19.9260	0.679867	0.339934	−	0.940449i	$-0.389595\pi$
$859$	19.9260	0.679867	0.339934	+	0.940449i	$0.389595\pi$
$860$	36.6991	1.25143
$861$	0	0
$862$	83.2541	2.83565
$863$	6.97593	0.237463	0.118732	−	0.992926i	$-0.462117\pi$
$863$	6.97593	0.237463	0.118732	+	0.992926i	$0.462117\pi$
$864$	0	0
$865$	−20.5071	−0.697262
$866$	−81.0134	−2.75295
$867$	0	0
$868$	95.0766	3.22711
$869$	4.85467	0.164683
$870$	0	0
$871$	1.09777	0.0371967
$872$	13.7100	0.464278
$873$	0	0
$874$	−3.41282	−0.115440
$875$	−50.1363	−1.69492
$876$	0	0
$877$	−37.4415	−1.26431	−0.632154	−	0.774843i	$-0.717829\pi$
$877$	−37.4415	−1.26431	−0.632154	+	0.774843i	$0.717829\pi$
$878$	52.0057	1.75511
$879$	0	0
$880$	8.35129	0.281522
$881$	25.2931	0.852147	0.426073	−	0.904689i	$-0.359897\pi$
$881$	25.2931	0.852147	0.426073	+	0.904689i	$0.359897\pi$
$882$	0	0
$883$	22.2727	0.749536	0.374768	−	0.927119i	$-0.377722\pi$
$883$	22.2727	0.749536	0.374768	+	0.927119i	$0.377722\pi$
$884$	−1.52362	−0.0512449
$885$	0	0
$886$	−38.3237	−1.28751
$887$	31.6882	1.06399	0.531993	−	0.846749i	$-0.321443\pi$
$887$	31.6882	1.06399	0.531993	+	0.846749i	$0.321443\pi$
$888$	0	0
$889$	3.06524	0.102805
$890$	4.01160	0.134469
$891$	0	0
$892$	−4.20601	−0.140828
$893$	−9.32960	−0.312203
$894$	0	0
$895$	2.07377	0.0693184
$896$	−30.5303	−1.01995
$897$	0	0
$898$	−27.0328	−0.902098
$899$	9.36561	0.312361
$900$	0	0
$901$	1.46511	0.0488097
$902$	24.1987	0.805728
$903$	0	0
$904$	11.2886	0.375453
$905$	7.95651	0.264483
$906$	0	0
$907$	−31.1899	−1.03564	−0.517822	−	0.855489i	$-0.673257\pi$
$907$	−31.1899	−1.03564	−0.517822	+	0.855489i	$0.673257\pi$
$908$	17.0102	0.564502
$909$	0	0
$910$	20.6913	0.685912
$911$	−22.9836	−0.761481	−0.380740	−	0.924682i	$-0.624331\pi$
$911$	−22.9836	−0.761481	−0.380740	+	0.924682i	$0.624331\pi$
$912$	0	0
$913$	−20.8824	−0.691105
$914$	14.7759	0.488743
$915$	0	0
$916$	19.8359	0.655396
$917$	−36.0241	−1.18962
$918$	0	0
$919$	9.47938	0.312696	0.156348	−	0.987702i	$-0.450028\pi$
$919$	9.47938	0.312696	0.156348	+	0.987702i	$0.450028\pi$
$920$	−1.60459	−0.0529017
$921$	0	0
$922$	19.1708	0.631355
$923$	−6.71470	−0.221017
$924$	0	0
$925$	−8.94810	−0.294212
$926$	36.5769	1.20199
$927$	0	0
$928$	−8.01333	−0.263051
$929$	−32.5207	−1.06697	−0.533485	−	0.845810i	$-0.679118\pi$
$929$	−32.5207	−1.06697	−0.533485	+	0.845810i	$0.679118\pi$
$930$	0	0
$931$	−16.5135	−0.541207
$932$	5.91975	0.193908
$933$	0	0
$934$	−6.86973	−0.224784
$935$	−1.28651	−0.0420733
$936$	0	0
$937$	22.9482	0.749684	0.374842	−	0.927089i	$-0.377697\pi$
$937$	22.9482	0.749684	0.374842	+	0.927089i	$0.377697\pi$
$938$	−6.90250	−0.225374
$939$	0	0
$940$	−23.9915	−0.782516
$941$	40.6049	1.32368	0.661840	−	0.749645i	$-0.269776\pi$
$941$	40.6049	1.32368	0.661840	+	0.749645i	$0.269776\pi$
$942$	0	0
$943$	6.78622	0.220990
$944$	9.27010	0.301716
$945$	0	0
$946$	31.4462	1.02240
$947$	5.50625	0.178929	0.0894646	−	0.995990i	$-0.471484\pi$
$947$	5.50625	0.178929	0.0894646	+	0.995990i	$0.471484\pi$
$948$	0	0
$949$	−2.26237	−0.0734397
$950$	−7.19731	−0.233512
$951$	0	0
$952$	1.75156	0.0567684
$953$	−0.342739	−0.0111024	−0.00555120	−	0.999985i	$-0.501767\pi$
$953$	−0.342739	−0.0111024	−0.00555120	+	0.999985i	$0.501767\pi$
$954$	0	0
$955$	−3.46170	−0.112018
$956$	−54.8371	−1.77356
$957$	0	0
$958$	−61.5005	−1.98699
$959$	16.5910	0.535751
$960$	0	0
$961$	56.7147	1.82951
$962$	12.4484	0.401352
$963$	0	0
$964$	−47.1139	−1.51744
$965$	5.70450	0.183634
$966$	0	0
$967$	20.4218	0.656722	0.328361	−	0.944552i	$-0.393504\pi$
$967$	20.4218	0.656722	0.328361	+	0.944552i	$0.393504\pi$
$968$	7.68264	0.246929
$969$	0	0
$970$	47.8198	1.53540
$971$	−61.1290	−1.96172	−0.980861	−	0.194710i	$-0.937623\pi$
$971$	−61.1290	−1.96172	−0.980861	+	0.194710i	$0.937623\pi$
$972$	0	0
$973$	70.7968	2.26964
$974$	8.87655	0.284423
$975$	0	0
$976$	17.4788	0.559481
$977$	−3.33305	−0.106634	−0.0533169	−	0.998578i	$-0.516979\pi$
$977$	−3.33305	−0.106634	−0.0533169	+	0.998578i	$0.516979\pi$
$978$	0	0
$979$	1.89163	0.0604567
$980$	−42.4651	−1.35650
$981$	0	0
$982$	8.15115	0.260114
$983$	15.6569	0.499377	0.249688	−	0.968326i	$-0.419672\pi$
$983$	15.6569	0.499377	0.249688	+	0.968326i	$0.419672\pi$
$984$	0	0
$985$	−29.9374	−0.953884
$986$	0.943696	0.0300534
$987$	0	0
$988$	5.51008	0.175299
$989$	8.81869	0.280418
$990$	0	0
$991$	−17.7559	−0.564036	−0.282018	−	0.959409i	$-0.591004\pi$
$991$	−17.7559	−0.564036	−0.282018	+	0.959409i	$0.591004\pi$
$992$	−75.0498	−2.38283
$993$	0	0
$994$	42.2201	1.33914
$995$	−30.6879	−0.972873
$996$	0	0
$997$	−37.3955	−1.18433	−0.592164	−	0.805818i	$-0.701726\pi$
$997$	−37.3955	−1.18433	−0.592164	+	0.805818i	$0.701726\pi$
$998$	5.33051	0.168734
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.h.1.1		5
3.2	odd	2		2001.2.a.h.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.h.1.5	✓	5	3.2	odd	2
6003.2.a.h.1.1		5	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)

\(f(q)\)	\(=\)	\(q-2.10891 q^{2} +2.44748 q^{4} -1.70032 q^{5} -4.14780 q^{7} -0.943696 q^{8} +O(q^{10})\)	\(q-2.10891 q^{2} +2.44748 q^{4} -1.70032 q^{5} -4.14780 q^{7} -0.943696 q^{8} +3.58582 q^{10} +1.69086 q^{11} -1.39118 q^{13} +8.74732 q^{14} -2.90480 q^{16} +0.447481 q^{17} -1.61829 q^{19} -4.16151 q^{20} -3.56585 q^{22} -1.00000 q^{23} -2.10891 q^{25} +2.93386 q^{26} -10.1517 q^{28} -1.00000 q^{29} -9.36561 q^{31} +8.01333 q^{32} -0.943696 q^{34} +7.05260 q^{35} +4.24301 q^{37} +3.41282 q^{38} +1.60459 q^{40} -6.78622 q^{41} -8.81869 q^{43} +4.13834 q^{44} +2.10891 q^{46} +5.76510 q^{47} +10.2043 q^{49} +4.44748 q^{50} -3.40488 q^{52} +3.27411 q^{53} -2.87500 q^{55} +3.91426 q^{56} +2.10891 q^{58} -3.19131 q^{59} -6.01720 q^{61} +19.7512 q^{62} -11.0898 q^{64} +2.36545 q^{65} -0.789098 q^{67} +1.09520 q^{68} -14.8733 q^{70} +4.82663 q^{71} +1.62623 q^{73} -8.94810 q^{74} -3.96073 q^{76} -7.01333 q^{77} +2.87113 q^{79} +4.93909 q^{80} +14.3115 q^{82} -12.3502 q^{83} -0.760862 q^{85} +18.5978 q^{86} -1.59565 q^{88} +1.11874 q^{89} +5.77033 q^{91} -2.44748 q^{92} -12.1580 q^{94} +2.75161 q^{95} +13.3358 q^{97} -21.5198 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} + 8 q^{4} + 3 q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) \) 5 * q + 2 * q^2 + 8 * q^4 + 3 * q^5 - 5 * q^7 + 3 * q^8	\( 5 q + 2 q^{2} + 8 q^{4} + 3 q^{5} - 5 q^{7} + 3 q^{8} + 9 q^{10} + 8 q^{11} + 5 q^{13} + 2 q^{14} - 10 q^{16} - 2 q^{17} - 9 q^{19} + 12 q^{20} + 10 q^{22} - 5 q^{23} + 2 q^{25} + 3 q^{26} - 14 q^{28} - 5 q^{29} - 6 q^{31} + 8 q^{32} + 3 q^{34} + 15 q^{35} + 10 q^{37} + 10 q^{38} + 26 q^{40} + 11 q^{41} - 9 q^{43} + 16 q^{44} - 2 q^{46} + 13 q^{47} - 6 q^{49} + 18 q^{50} + 12 q^{52} - q^{53} + 13 q^{55} + 4 q^{56} - 2 q^{58} - 6 q^{59} + 23 q^{61} + 36 q^{62} - q^{64} + 20 q^{65} - 10 q^{67} + 10 q^{68} - 16 q^{70} + 11 q^{71} + 31 q^{73} + 18 q^{74} - 8 q^{76} - 3 q^{77} + 8 q^{79} + 8 q^{80} + 16 q^{82} - 7 q^{83} + 6 q^{85} + 36 q^{86} - 3 q^{88} - 3 q^{89} + 8 q^{91} - 8 q^{92} - 39 q^{94} + 11 q^{95} + 3 q^{97} - 38 q^{98}+O(q^{100}) \) 5 * q + 2 * q^2 + 8 * q^4 + 3 * q^5 - 5 * q^7 + 3 * q^8 + 9 * q^10 + 8 * q^11 + 5 * q^13 + 2 * q^14 - 10 * q^16 - 2 * q^17 - 9 * q^19 + 12 * q^20 + 10 * q^22 - 5 * q^23 + 2 * q^25 + 3 * q^26 - 14 * q^28 - 5 * q^29 - 6 * q^31 + 8 * q^32 + 3 * q^34 + 15 * q^35 + 10 * q^37 + 10 * q^38 + 26 * q^40 + 11 * q^41 - 9 * q^43 + 16 * q^44 - 2 * q^46 + 13 * q^47 - 6 * q^49 + 18 * q^50 + 12 * q^52 - q^53 + 13 * q^55 + 4 * q^56 - 2 * q^58 - 6 * q^59 + 23 * q^61 + 36 * q^62 - q^64 + 20 * q^65 - 10 * q^67 + 10 * q^68 - 16 * q^70 + 11 * q^71 + 31 * q^73 + 18 * q^74 - 8 * q^76 - 3 * q^77 + 8 * q^79 + 8 * q^80 + 16 * q^82 - 7 * q^83 + 6 * q^85 + 36 * q^86 - 3 * q^88 - 3 * q^89 + 8 * q^91 - 8 * q^92 - 39 * q^94 + 11 * q^95 + 3 * q^97 - 38 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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