LMFDB - Embedded newform 6003.2.a.j.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:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 5x^{5} + 18x^{4} + 4x^{3} - 26x^{2} + x + 8 $ x^7 - 3x^6 - 5x^5 + 18x^4 + 4x^3 - 26*x^2 + x + 8
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			$-1.98973$ 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)

$f(q)$	$=$	$q-1.98973 q^{2} +1.95904 q^{4} +1.78703 q^{5} -0.840964 q^{7} +0.0814996 q^{8} +O(q^{10})$	\(q-1.98973 q^{2} +1.95904 q^{4} +1.78703 q^{5} -0.840964 q^{7} +0.0814996 q^{8} -3.55572 q^{10} +6.06998 q^{11} -4.54183 q^{13} +1.67330 q^{14} -4.08024 q^{16} -5.04736 q^{17} -0.331044 q^{19} +3.50087 q^{20} -12.0776 q^{22} -1.00000 q^{23} -1.80652 q^{25} +9.03702 q^{26} -1.64748 q^{28} +1.00000 q^{29} +0.306172 q^{31} +7.95560 q^{32} +10.0429 q^{34} -1.50283 q^{35} +8.76970 q^{37} +0.658690 q^{38} +0.145642 q^{40} +10.9832 q^{41} -3.35294 q^{43} +11.8913 q^{44} +1.98973 q^{46} +11.8030 q^{47} -6.29278 q^{49} +3.59449 q^{50} -8.89762 q^{52} -8.93872 q^{53} +10.8472 q^{55} -0.0685383 q^{56} -1.98973 q^{58} -14.0280 q^{59} -9.45160 q^{61} -0.609201 q^{62} -7.66903 q^{64} -8.11639 q^{65} -10.3723 q^{67} -9.88798 q^{68} +2.99023 q^{70} -7.91620 q^{71} +2.31986 q^{73} -17.4494 q^{74} -0.648529 q^{76} -5.10463 q^{77} -3.85026 q^{79} -7.29152 q^{80} -21.8536 q^{82} -9.11490 q^{83} -9.01980 q^{85} +6.67146 q^{86} +0.494701 q^{88} +0.431316 q^{89} +3.81951 q^{91} -1.95904 q^{92} -23.4848 q^{94} -0.591586 q^{95} +10.9444 q^{97} +12.5210 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8}+O(q^{10}) $ 7 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 5 * q^7 + 6 * q^8	$ 7 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8} + 3 q^{10} + 4 q^{11} - 18 q^{13} + 2 q^{14} - 7 q^{16} + 3 q^{17} - 4 q^{19} + 2 q^{20} - 26 q^{22} - 7 q^{23} - 8 q^{25} + 7 q^{26} - 6 q^{28} + 7 q^{29} - 22 q^{31} - 5 q^{32} + 9 q^{34} - 3 q^{35} - 25 q^{37} - 14 q^{38} - 10 q^{40} + 13 q^{41} - 2 q^{43} - 4 q^{44} - 3 q^{46} + 25 q^{47} - 8 q^{49} - 19 q^{50} - 12 q^{52} + 5 q^{53} - 15 q^{55} - 18 q^{56} + 3 q^{58} - 11 q^{59} - 33 q^{61} - 28 q^{62} - 14 q^{64} + 2 q^{65} + 8 q^{67} - 12 q^{68} - 22 q^{70} + 6 q^{71} + 15 q^{73} - 34 q^{74} - 28 q^{76} + q^{77} - 15 q^{79} + 12 q^{80} - 14 q^{82} - 21 q^{83} - 28 q^{85} + 12 q^{86} - 13 q^{88} - 8 q^{89} + 6 q^{91} - 5 q^{92} - 35 q^{94} + 25 q^{95} + 13 q^{97} - q^{98}+O(q^{100}) $ 7 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 5 * q^7 + 6 * q^8 + 3 * q^10 + 4 * q^11 - 18 * q^13 + 2 * q^14 - 7 * q^16 + 3 * q^17 - 4 * q^19 + 2 * q^20 - 26 * q^22 - 7 * q^23 - 8 * q^25 + 7 * q^26 - 6 * q^28 + 7 * q^29 - 22 * q^31 - 5 * q^32 + 9 * q^34 - 3 * q^35 - 25 * q^37 - 14 * q^38 - 10 * q^40 + 13 * q^41 - 2 * q^43 - 4 * q^44 - 3 * q^46 + 25 * q^47 - 8 * q^49 - 19 * q^50 - 12 * q^52 + 5 * q^53 - 15 * q^55 - 18 * q^56 + 3 * q^58 - 11 * q^59 - 33 * q^61 - 28 * q^62 - 14 * q^64 + 2 * q^65 + 8 * q^67 - 12 * q^68 - 22 * q^70 + 6 * q^71 + 15 * q^73 - 34 * q^74 - 28 * q^76 + q^77 - 15 * q^79 + 12 * q^80 - 14 * q^82 - 21 * q^83 - 28 * q^85 + 12 * q^86 - 13 * q^88 - 8 * q^89 + 6 * q^91 - 5 * q^92 - 35 * q^94 + 25 * q^95 + 13 * q^97 - 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 $

$2$	−1.98973	−1.40695	−0.703477	−	0.710718i	$-0.748370\pi$
$2$	−1.98973	−1.40695	−0.703477	+	0.710718i	$0.748370\pi$
$3$	0	0
$3$	0	0
$4$	1.95904	0.979520
$5$	1.78703	0.799185	0.399592	−	0.916693i	$-0.369152\pi$
$5$	1.78703	0.799185	0.399592	+	0.916693i	$0.369152\pi$
$6$	0	0
$7$	−0.840964	−0.317855	−0.158927	−	0.987290i	$-0.550804\pi$
$7$	−0.840964	−0.317855	−0.158927	+	0.987290i	$0.550804\pi$
$8$	0.0814996	0.0288145
$9$	0	0
$10$	−3.55572	−1.12442
$11$	6.06998	1.83017	0.915083	−	0.403265i	$-0.132125\pi$
$11$	6.06998	1.83017	0.915083	+	0.403265i	$0.132125\pi$
$12$	0	0
$13$	−4.54183	−1.25968	−0.629838	−	0.776727i	$-0.716879\pi$
$13$	−4.54183	−1.25968	−0.629838	+	0.776727i	$0.716879\pi$
$14$	1.67330	0.447207
$15$	0	0
$16$	−4.08024	−1.02006
$17$	−5.04736	−1.22417	−0.612083	−	0.790794i	$-0.709668\pi$
$17$	−5.04736	−1.22417	−0.612083	+	0.790794i	$0.709668\pi$
$18$	0	0
$19$	−0.331044	−0.0759467	−0.0379734	−	0.999279i	$-0.512090\pi$
$19$	−0.331044	−0.0759467	−0.0379734	+	0.999279i	$0.512090\pi$
$20$	3.50087	0.782817
$21$	0	0
$22$	−12.0776	−2.57496
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−1.80652	−0.361304
$26$	9.03702	1.77231
$27$	0	0
$28$	−1.64748	−0.311345
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	0.306172	0.0549902	0.0274951	−	0.999622i	$-0.491247\pi$
$31$	0.306172	0.0549902	0.0274951	+	0.999622i	$0.491247\pi$
$32$	7.95560	1.40636
$33$	0	0
$34$	10.0429	1.72234
$35$	−1.50283	−0.254025
$36$	0	0
$37$	8.76970	1.44173	0.720865	−	0.693075i	$-0.243745\pi$
$37$	8.76970	1.44173	0.720865	+	0.693075i	$0.243745\pi$
$38$	0.658690	0.106854
$39$	0	0
$40$	0.145642	0.0230281
$41$	10.9832	1.71528	0.857641	−	0.514249i	$-0.171929\pi$
$41$	10.9832	1.71528	0.857641	+	0.514249i	$0.171929\pi$
$42$	0	0
$43$	−3.35294	−0.511319	−0.255660	−	0.966767i	$-0.582293\pi$
$43$	−3.35294	−0.511319	−0.255660	+	0.966767i	$0.582293\pi$
$44$	11.8913	1.79268
$45$	0	0
$46$	1.98973	0.293370
$47$	11.8030	1.72164	0.860822	−	0.508905i	$-0.169950\pi$
$47$	11.8030	1.72164	0.860822	+	0.508905i	$0.169950\pi$
$48$	0	0
$49$	−6.29278	−0.898968
$50$	3.59449	0.508338
$51$	0	0
$52$	−8.89762	−1.23388
$53$	−8.93872	−1.22783	−0.613914	−	0.789373i	$-0.710406\pi$
$53$	−8.93872	−1.22783	−0.613914	+	0.789373i	$0.710406\pi$
$54$	0	0
$55$	10.8472	1.46264
$56$	−0.0685383	−0.00915881
$57$	0	0
$58$	−1.98973	−0.261265
$59$	−14.0280	−1.82629	−0.913147	−	0.407630i	$-0.866355\pi$
$59$	−14.0280	−1.82629	−0.913147	+	0.407630i	$0.866355\pi$
$60$	0	0
$61$	−9.45160	−1.21015	−0.605077	−	0.796167i	$-0.706858\pi$
$61$	−9.45160	−1.21015	−0.605077	+	0.796167i	$0.706858\pi$
$62$	−0.609201	−0.0773686
$63$	0	0
$64$	−7.66903	−0.958629
$65$	−8.11639	−1.00671
$66$	0	0
$67$	−10.3723	−1.26718	−0.633589	−	0.773669i	$-0.718419\pi$
$67$	−10.3723	−1.26718	−0.633589	+	0.773669i	$0.718419\pi$
$68$	−9.88798	−1.19909
$69$	0	0
$70$	2.99023	0.357401
$71$	−7.91620	−0.939480	−0.469740	−	0.882805i	$-0.655652\pi$
$71$	−7.91620	−0.939480	−0.469740	+	0.882805i	$0.655652\pi$
$72$	0	0
$73$	2.31986	0.271519	0.135760	−	0.990742i	$-0.456653\pi$
$73$	2.31986	0.271519	0.135760	+	0.990742i	$0.456653\pi$
$74$	−17.4494	−2.02845
$75$	0	0
$76$	−0.648529	−0.0743913
$77$	−5.10463	−0.581727
$78$	0	0
$79$	−3.85026	−0.433188	−0.216594	−	0.976262i	$-0.569495\pi$
$79$	−3.85026	−0.433188	−0.216594	+	0.976262i	$0.569495\pi$
$80$	−7.29152	−0.815217
$81$	0	0
$82$	−21.8536	−2.41332
$83$	−9.11490	−1.00049	−0.500245	−	0.865884i	$-0.666757\pi$
$83$	−9.11490	−1.00049	−0.500245	+	0.865884i	$0.666757\pi$
$84$	0	0
$85$	−9.01980	−0.978334
$86$	6.67146	0.719402
$87$	0	0
$88$	0.494701	0.0527353
$89$	0.431316	0.0457194	0.0228597	−	0.999739i	$-0.492723\pi$
$89$	0.431316	0.0457194	0.0228597	+	0.999739i	$0.492723\pi$
$90$	0	0
$91$	3.81951	0.400394
$92$	−1.95904	−0.204244
$93$	0	0
$94$	−23.4848	−2.42228
$95$	−0.591586	−0.0606955
$96$	0	0
$97$	10.9444	1.11124	0.555618	−	0.831437i	$-0.312482\pi$
$97$	10.9444	1.11124	0.555618	+	0.831437i	$0.312482\pi$
$98$	12.5210	1.26481
$99$	0	0
$100$	−3.53904	−0.353904
$101$	−3.98635	−0.396657	−0.198328	−	0.980136i	$-0.563551\pi$
$101$	−3.98635	−0.396657	−0.198328	+	0.980136i	$0.563551\pi$
$102$	0	0
$103$	−0.382082	−0.0376476	−0.0188238	−	0.999823i	$-0.505992\pi$
$103$	−0.382082	−0.0376476	−0.0188238	+	0.999823i	$0.505992\pi$
$104$	−0.370157	−0.0362969
$105$	0	0
$106$	17.7857	1.72750
$107$	−1.94535	−0.188064	−0.0940320	−	0.995569i	$-0.529976\pi$
$107$	−1.94535	−0.188064	−0.0940320	+	0.995569i	$0.529976\pi$
$108$	0	0
$109$	17.3370	1.66058	0.830292	−	0.557328i	$-0.188173\pi$
$109$	17.3370	1.66058	0.830292	+	0.557328i	$0.188173\pi$
$110$	−21.5831	−2.05787
$111$	0	0
$112$	3.43134	0.324231
$113$	−2.51998	−0.237060	−0.118530	−	0.992950i	$-0.537818\pi$
$113$	−2.51998	−0.237060	−0.118530	+	0.992950i	$0.537818\pi$
$114$	0	0
$115$	−1.78703	−0.166642
$116$	1.95904	0.181892
$117$	0	0
$118$	27.9120	2.56951
$119$	4.24465	0.389107
$120$	0	0
$121$	25.8446	2.34951
$122$	18.8062	1.70263
$123$	0	0
$124$	0.599804	0.0538640
$125$	−12.1635	−1.08793
$126$	0	0
$127$	−18.5218	−1.64354	−0.821771	−	0.569818i	$-0.807014\pi$
$127$	−18.5218	−1.64354	−0.821771	+	0.569818i	$0.807014\pi$
$128$	−0.651860	−0.0576169
$129$	0	0
$130$	16.1494	1.41640
$131$	9.39538	0.820878	0.410439	−	0.911888i	$-0.365375\pi$
$131$	9.39538	0.820878	0.410439	+	0.911888i	$0.365375\pi$
$132$	0	0
$133$	0.278396	0.0241400
$134$	20.6381	1.78286
$135$	0	0
$136$	−0.411358	−0.0352737
$137$	3.44928	0.294691	0.147346	−	0.989085i	$-0.452927\pi$
$137$	3.44928	0.294691	0.147346	+	0.989085i	$0.452927\pi$
$138$	0	0
$139$	4.72420	0.400701	0.200351	−	0.979724i	$-0.435792\pi$
$139$	4.72420	0.400701	0.200351	+	0.979724i	$0.435792\pi$
$140$	−2.94410	−0.248822
$141$	0	0
$142$	15.7511	1.32180
$143$	−27.5688	−2.30542
$144$	0	0
$145$	1.78703	0.148405
$146$	−4.61590	−0.382015
$147$	0	0
$148$	17.1802	1.41220
$149$	2.90831	0.238258	0.119129	−	0.992879i	$-0.461990\pi$
$149$	2.90831	0.238258	0.119129	+	0.992879i	$0.461990\pi$
$150$	0	0
$151$	−14.6842	−1.19499	−0.597494	−	0.801874i	$-0.703837\pi$
$151$	−14.6842	−1.19499	−0.597494	+	0.801874i	$0.703837\pi$
$152$	−0.0269800	−0.00218836
$153$	0	0
$154$	10.1569	0.818463
$155$	0.547139	0.0439473
$156$	0	0
$157$	−23.6416	−1.88681	−0.943404	−	0.331645i	$-0.892396\pi$
$157$	−23.6416	−1.88681	−0.943404	+	0.331645i	$0.892396\pi$
$158$	7.66098	0.609475
$159$	0	0
$160$	14.2169	1.12394
$161$	0.840964	0.0662773
$162$	0	0
$163$	−17.3651	−1.36014	−0.680070	−	0.733147i	$-0.738051\pi$
$163$	−17.3651	−1.36014	−0.680070	+	0.733147i	$0.738051\pi$
$164$	21.5165	1.68015
$165$	0	0
$166$	18.1362	1.40764
$167$	11.4036	0.882436	0.441218	−	0.897400i	$-0.354546\pi$
$167$	11.4036	0.882436	0.441218	+	0.897400i	$0.354546\pi$
$168$	0	0
$169$	7.62818	0.586783
$170$	17.9470	1.37647
$171$	0	0
$172$	−6.56855	−0.500847
$173$	8.72254	0.663163	0.331581	−	0.943427i	$-0.392418\pi$
$173$	8.72254	0.663163	0.331581	+	0.943427i	$0.392418\pi$
$174$	0	0
$175$	1.51922	0.114842
$176$	−24.7670	−1.86688
$177$	0	0
$178$	−0.858203	−0.0643251
$179$	1.86971	0.139749	0.0698744	−	0.997556i	$-0.477740\pi$
$179$	1.86971	0.139749	0.0698744	+	0.997556i	$0.477740\pi$
$180$	0	0
$181$	−14.4493	−1.07401	−0.537004	−	0.843580i	$-0.680444\pi$
$181$	−14.4493	−1.07401	−0.537004	+	0.843580i	$0.680444\pi$
$182$	−7.59982	−0.563336
$183$	0	0
$184$	−0.0814996	−0.00600823
$185$	15.6717	1.15221
$186$	0	0
$187$	−30.6374	−2.24043
$188$	23.1226	1.68639
$189$	0	0
$190$	1.17710	0.0853957
$191$	12.8649	0.930874	0.465437	−	0.885081i	$-0.345897\pi$
$191$	12.8649	0.930874	0.465437	+	0.885081i	$0.345897\pi$
$192$	0	0
$193$	−10.8178	−0.778679	−0.389339	−	0.921094i	$-0.627297\pi$
$193$	−10.8178	−0.778679	−0.389339	+	0.921094i	$0.627297\pi$
$194$	−21.7765	−1.56346
$195$	0	0
$196$	−12.3278	−0.880558
$197$	−12.1335	−0.864479	−0.432240	−	0.901759i	$-0.642276\pi$
$197$	−12.1335	−0.864479	−0.432240	+	0.901759i	$0.642276\pi$
$198$	0	0
$199$	6.74503	0.478142	0.239071	−	0.971002i	$-0.423157\pi$
$199$	6.74503	0.478142	0.239071	+	0.971002i	$0.423157\pi$
$200$	−0.147231	−0.0104108
$201$	0	0
$202$	7.93178	0.558078
$203$	−0.840964	−0.0590241
$204$	0	0
$205$	19.6273	1.37083
$206$	0.760241	0.0529685
$207$	0	0
$208$	18.5318	1.28495
$209$	−2.00943	−0.138995
$210$	0	0
$211$	16.2561	1.11911	0.559557	−	0.828792i	$-0.310971\pi$
$211$	16.2561	1.11911	0.559557	+	0.828792i	$0.310971\pi$
$212$	−17.5113	−1.20268
$213$	0	0
$214$	3.87073	0.264597
$215$	−5.99181	−0.408638
$216$	0	0
$217$	−0.257480	−0.0174789
$218$	−34.4960	−2.33637
$219$	0	0
$220$	21.2502	1.43269
$221$	22.9242	1.54205
$222$	0	0
$223$	−2.89594	−0.193926	−0.0969631	−	0.995288i	$-0.530913\pi$
$223$	−2.89594	−0.193926	−0.0969631	+	0.995288i	$0.530913\pi$
$224$	−6.69037	−0.447019
$225$	0	0
$226$	5.01409	0.333532
$227$	−21.8683	−1.45145	−0.725725	−	0.687985i	$-0.758496\pi$
$227$	−21.8683	−1.45145	−0.725725	+	0.687985i	$0.758496\pi$
$228$	0	0
$229$	7.52393	0.497195	0.248598	−	0.968607i	$-0.420030\pi$
$229$	7.52393	0.497195	0.248598	+	0.968607i	$0.420030\pi$
$230$	3.55572	0.234457
$231$	0	0
$232$	0.0814996	0.00535071
$233$	10.5587	0.691725	0.345863	−	0.938285i	$-0.387586\pi$
$233$	10.5587	0.691725	0.345863	+	0.938285i	$0.387586\pi$
$234$	0	0
$235$	21.0923	1.37591
$236$	−27.4815	−1.78889
$237$	0	0
$238$	−8.44573	−0.547455
$239$	−1.29331	−0.0836572	−0.0418286	−	0.999125i	$-0.513318\pi$
$239$	−1.29331	−0.0836572	−0.0418286	+	0.999125i	$0.513318\pi$
$240$	0	0
$241$	−1.93702	−0.124774	−0.0623871	−	0.998052i	$-0.519871\pi$
$241$	−1.93702	−0.124774	−0.0623871	+	0.998052i	$0.519871\pi$
$242$	−51.4239	−3.30565
$243$	0	0
$244$	−18.5161	−1.18537
$245$	−11.2454	−0.718442
$246$	0	0
$247$	1.50354	0.0956683
$248$	0.0249529	0.00158451
$249$	0	0
$250$	24.2021	1.53067
$251$	−3.48208	−0.219787	−0.109894	−	0.993943i	$-0.535051\pi$
$251$	−3.48208	−0.219787	−0.109894	+	0.993943i	$0.535051\pi$
$252$	0	0
$253$	−6.06998	−0.381616
$254$	36.8534	2.31239
$255$	0	0
$256$	16.6351	1.03969
$257$	22.3704	1.39543	0.697713	−	0.716378i	$-0.254201\pi$
$257$	22.3704	1.39543	0.697713	+	0.716378i	$0.254201\pi$
$258$	0	0
$259$	−7.37501	−0.458261
$260$	−15.9003	−0.986096
$261$	0	0
$262$	−18.6943	−1.15494
$263$	16.1945	0.998594	0.499297	−	0.866431i	$-0.333592\pi$
$263$	16.1945	0.998594	0.499297	+	0.866431i	$0.333592\pi$
$264$	0	0
$265$	−15.9738	−0.981261
$266$	−0.553934	−0.0339639
$267$	0	0
$268$	−20.3198	−1.24123
$269$	−4.61990	−0.281680	−0.140840	−	0.990032i	$-0.544980\pi$
$269$	−4.61990	−0.281680	−0.140840	+	0.990032i	$0.544980\pi$
$270$	0	0
$271$	19.8403	1.20521	0.602605	−	0.798040i	$-0.294130\pi$
$271$	19.8403	1.20521	0.602605	+	0.798040i	$0.294130\pi$
$272$	20.5945	1.24872
$273$	0	0
$274$	−6.86314	−0.414617
$275$	−10.9655	−0.661246
$276$	0	0
$277$	−9.66039	−0.580436	−0.290218	−	0.956960i	$-0.593728\pi$
$277$	−9.66039	−0.580436	−0.290218	+	0.956960i	$0.593728\pi$
$278$	−9.39990	−0.563769
$279$	0	0
$280$	−0.122480	−0.00731958
$281$	−23.9856	−1.43086	−0.715429	−	0.698685i	$-0.753769\pi$
$281$	−23.9856	−1.43086	−0.715429	+	0.698685i	$0.753769\pi$
$282$	0	0
$283$	−7.77103	−0.461940	−0.230970	−	0.972961i	$-0.574190\pi$
$283$	−7.77103	−0.461940	−0.230970	+	0.972961i	$0.574190\pi$
$284$	−15.5081	−0.920239
$285$	0	0
$286$	54.8545	3.24362
$287$	−9.23645	−0.545210
$288$	0	0
$289$	8.47587	0.498580
$290$	−3.55572	−0.208799
$291$	0	0
$292$	4.54470	0.265958
$293$	−13.9964	−0.817678	−0.408839	−	0.912607i	$-0.634066\pi$
$293$	−13.9964	−0.817678	−0.408839	+	0.912607i	$0.634066\pi$
$294$	0	0
$295$	−25.0685	−1.45955
$296$	0.714728	0.0415427
$297$	0	0
$298$	−5.78677	−0.335218
$299$	4.54183	0.262661
$300$	0	0
$301$	2.81971	0.162525
$302$	29.2177	1.68129
$303$	0	0
$304$	1.35074	0.0774703
$305$	−16.8903	−0.967137
$306$	0	0
$307$	5.68414	0.324411	0.162206	−	0.986757i	$-0.448139\pi$
$307$	5.68414	0.324411	0.162206	+	0.986757i	$0.448139\pi$
$308$	−10.0002	−0.569813
$309$	0	0
$310$	−1.08866	−0.0618318
$311$	12.7411	0.722484	0.361242	−	0.932472i	$-0.382353\pi$
$311$	12.7411	0.722484	0.361242	+	0.932472i	$0.382353\pi$
$312$	0	0
$313$	−8.05852	−0.455494	−0.227747	−	0.973720i	$-0.573136\pi$
$313$	−8.05852	−0.455494	−0.227747	+	0.973720i	$0.573136\pi$
$314$	47.0406	2.65465
$315$	0	0
$316$	−7.54281	−0.424316
$317$	7.68400	0.431576	0.215788	−	0.976440i	$-0.430768\pi$
$317$	7.68400	0.431576	0.215788	+	0.976440i	$0.430768\pi$
$318$	0	0
$319$	6.06998	0.339853
$320$	−13.7048	−0.766122
$321$	0	0
$322$	−1.67330	−0.0932491
$323$	1.67090	0.0929713
$324$	0	0
$325$	8.20489	0.455126
$326$	34.5519	1.91365
$327$	0	0
$328$	0.895124	0.0494249
$329$	−9.92590	−0.547233
$330$	0	0
$331$	0.642065	0.0352911	0.0176455	−	0.999844i	$-0.494383\pi$
$331$	0.642065	0.0352911	0.0176455	+	0.999844i	$0.494383\pi$
$332$	−17.8565	−0.980000
$333$	0	0
$334$	−22.6901	−1.24155
$335$	−18.5356	−1.01271
$336$	0	0
$337$	27.0913	1.47576	0.737879	−	0.674933i	$-0.235827\pi$
$337$	27.0913	1.47576	0.737879	+	0.674933i	$0.235827\pi$
$338$	−15.1781	−0.825577
$339$	0	0
$340$	−17.6701	−0.958298
$341$	1.85846	0.100641
$342$	0	0
$343$	11.1788	0.603596
$344$	−0.273264	−0.0147334
$345$	0	0
$346$	−17.3555	−0.933039
$347$	−1.86608	−0.100177	−0.0500883	−	0.998745i	$-0.515950\pi$
$347$	−1.86608	−0.100177	−0.0500883	+	0.998745i	$0.515950\pi$
$348$	0	0
$349$	4.27583	0.228880	0.114440	−	0.993430i	$-0.463493\pi$
$349$	4.27583	0.228880	0.114440	+	0.993430i	$0.463493\pi$
$350$	−3.02284	−0.161577
$351$	0	0
$352$	48.2903	2.57388
$353$	6.68507	0.355810	0.177905	−	0.984048i	$-0.443068\pi$
$353$	6.68507	0.355810	0.177905	+	0.984048i	$0.443068\pi$
$354$	0	0
$355$	−14.1465	−0.750818
$356$	0.844965	0.0447830
$357$	0	0
$358$	−3.72023	−0.196620
$359$	−19.2828	−1.01771	−0.508854	−	0.860853i	$-0.669931\pi$
$359$	−19.2828	−1.01771	−0.508854	+	0.860853i	$0.669931\pi$
$360$	0	0
$361$	−18.8904	−0.994232
$362$	28.7502	1.51108
$363$	0	0
$364$	7.48258	0.392194
$365$	4.14566	0.216994
$366$	0	0
$367$	−29.1585	−1.52206	−0.761030	−	0.648717i	$-0.775306\pi$
$367$	−29.1585	−1.52206	−0.761030	+	0.648717i	$0.775306\pi$
$368$	4.08024	0.212697
$369$	0	0
$370$	−31.1826	−1.62111
$371$	7.51715	0.390271
$372$	0	0
$373$	−16.1915	−0.838362	−0.419181	−	0.907903i	$-0.637683\pi$
$373$	−16.1915	−0.838362	−0.419181	+	0.907903i	$0.637683\pi$
$374$	60.9602	3.15218
$375$	0	0
$376$	0.961940	0.0496083
$377$	−4.54183	−0.233916
$378$	0	0
$379$	−38.5901	−1.98224	−0.991120	−	0.132970i	$-0.957549\pi$
$379$	−38.5901	−1.98224	−0.991120	+	0.132970i	$0.957549\pi$
$380$	−1.15894	−0.0594524
$381$	0	0
$382$	−25.5978	−1.30970
$383$	−19.3417	−0.988313	−0.494157	−	0.869373i	$-0.664523\pi$
$383$	−19.3417	−0.988313	−0.494157	+	0.869373i	$0.664523\pi$
$384$	0	0
$385$	−9.12214	−0.464907
$386$	21.5244	1.09557
$387$	0	0
$388$	21.4405	1.08848
$389$	5.30720	0.269086	0.134543	−	0.990908i	$-0.457043\pi$
$389$	5.30720	0.269086	0.134543	+	0.990908i	$0.457043\pi$
$390$	0	0
$391$	5.04736	0.255256
$392$	−0.512859	−0.0259033
$393$	0	0
$394$	24.1425	1.21628
$395$	−6.88053	−0.346197
$396$	0	0
$397$	−20.6163	−1.03470	−0.517351	−	0.855773i	$-0.673082\pi$
$397$	−20.6163	−1.03470	−0.517351	+	0.855773i	$0.673082\pi$
$398$	−13.4208	−0.672724
$399$	0	0
$400$	7.37103	0.368552
$401$	−25.5866	−1.27773	−0.638867	−	0.769317i	$-0.720597\pi$
$401$	−25.5866	−1.27773	−0.638867	+	0.769317i	$0.720597\pi$
$402$	0	0
$403$	−1.39058	−0.0692698
$404$	−7.80942	−0.388533
$405$	0	0
$406$	1.67330	0.0830442
$407$	53.2319	2.63861
$408$	0	0
$409$	8.93704	0.441908	0.220954	−	0.975284i	$-0.429083\pi$
$409$	8.93704	0.441908	0.220954	+	0.975284i	$0.429083\pi$
$410$	−39.0530	−1.92869
$411$	0	0
$412$	−0.748514	−0.0368766
$413$	11.7971	0.580496
$414$	0	0
$415$	−16.2886	−0.799577
$416$	−36.1329	−1.77156
$417$	0	0
$418$	3.99823	0.195560
$419$	18.3457	0.896247	0.448123	−	0.893972i	$-0.352092\pi$
$419$	18.3457	0.896247	0.448123	+	0.893972i	$0.352092\pi$
$420$	0	0
$421$	9.03442	0.440311	0.220155	−	0.975465i	$-0.429344\pi$
$421$	9.03442	0.440311	0.220155	+	0.975465i	$0.429344\pi$
$422$	−32.3452	−1.57454
$423$	0	0
$424$	−0.728503	−0.0353792
$425$	9.11815	0.442295
$426$	0	0
$427$	7.94846	0.384653
$428$	−3.81102	−0.184212
$429$	0	0
$430$	11.9221	0.574935
$431$	34.6927	1.67109	0.835544	−	0.549424i	$-0.185153\pi$
$431$	34.6927	1.67109	0.835544	+	0.549424i	$0.185153\pi$
$432$	0	0
$433$	−37.7622	−1.81474	−0.907368	−	0.420336i	$-0.861912\pi$
$433$	−37.7622	−1.81474	−0.907368	+	0.420336i	$0.861912\pi$
$434$	0.512317	0.0245920
$435$	0	0
$436$	33.9639	1.62658
$437$	0.331044	0.0158360
$438$	0	0
$439$	27.3792	1.30674	0.653368	−	0.757040i	$-0.273355\pi$
$439$	27.3792	1.30674	0.653368	+	0.757040i	$0.273355\pi$
$440$	0.884046	0.0421452
$441$	0	0
$442$	−45.6131	−2.16960
$443$	−25.0238	−1.18892	−0.594459	−	0.804126i	$-0.702634\pi$
$443$	−25.0238	−1.18892	−0.594459	+	0.804126i	$0.702634\pi$
$444$	0	0
$445$	0.770775	0.0365382
$446$	5.76214	0.272845
$447$	0	0
$448$	6.44938	0.304705
$449$	22.8023	1.07611	0.538053	−	0.842911i	$-0.319160\pi$
$449$	22.8023	1.07611	0.538053	+	0.842911i	$0.319160\pi$
$450$	0	0
$451$	66.6675	3.13925
$452$	−4.93674	−0.232205
$453$	0	0
$454$	43.5121	2.04212
$455$	6.82559	0.319989
$456$	0	0
$457$	17.1889	0.804064	0.402032	−	0.915626i	$-0.368304\pi$
$457$	17.1889	0.804064	0.402032	+	0.915626i	$0.368304\pi$
$458$	−14.9706	−0.699531
$459$	0	0
$460$	−3.50087	−0.163229
$461$	−21.9391	−1.02180	−0.510902	−	0.859639i	$-0.670689\pi$
$461$	−21.9391	−1.02180	−0.510902	+	0.859639i	$0.670689\pi$
$462$	0	0
$463$	−34.8195	−1.61820	−0.809100	−	0.587671i	$-0.800045\pi$
$463$	−34.8195	−1.61820	−0.809100	+	0.587671i	$0.800045\pi$
$464$	−4.08024	−0.189420
$465$	0	0
$466$	−21.0091	−0.973226
$467$	−33.4800	−1.54927	−0.774635	−	0.632409i	$-0.782066\pi$
$467$	−33.4800	−1.54927	−0.774635	+	0.632409i	$0.782066\pi$
$468$	0	0
$469$	8.72274	0.402779
$470$	−41.9681	−1.93585
$471$	0	0
$472$	−1.14328	−0.0526237
$473$	−20.3523	−0.935799
$474$	0	0
$475$	0.598037	0.0274398
$476$	8.31544	0.381138
$477$	0	0
$478$	2.57334	0.117702
$479$	38.9319	1.77884	0.889422	−	0.457087i	$-0.151107\pi$
$479$	38.9319	1.77884	0.889422	+	0.457087i	$0.151107\pi$
$480$	0	0
$481$	−39.8305	−1.81611
$482$	3.85415	0.175552
$483$	0	0
$484$	50.6306	2.30139
$485$	19.5580	0.888083
$486$	0	0
$487$	−24.8392	−1.12557	−0.562786	−	0.826603i	$-0.690270\pi$
$487$	−24.8392	−1.12557	−0.562786	+	0.826603i	$0.690270\pi$
$488$	−0.770302	−0.0348699
$489$	0	0
$490$	22.3753	1.01081
$491$	−18.8404	−0.850256	−0.425128	−	0.905133i	$-0.639771\pi$
$491$	−18.8404	−0.850256	−0.425128	+	0.905133i	$0.639771\pi$
$492$	0	0
$493$	−5.04736	−0.227322
$494$	−2.99165	−0.134601
$495$	0	0
$496$	−1.24926	−0.0560933
$497$	6.65724	0.298618
$498$	0	0
$499$	−7.24305	−0.324243	−0.162122	−	0.986771i	$-0.551834\pi$
$499$	−7.24305	−0.324243	−0.162122	+	0.986771i	$0.551834\pi$
$500$	−23.8287	−1.06565
$501$	0	0
$502$	6.92842	0.309230
$503$	0.757920	0.0337940	0.0168970	−	0.999857i	$-0.494621\pi$
$503$	0.757920	0.0337940	0.0168970	+	0.999857i	$0.494621\pi$
$504$	0	0
$505$	−7.12373	−0.317002
$506$	12.0776	0.536916
$507$	0	0
$508$	−36.2849	−1.60988
$509$	−18.5923	−0.824087	−0.412044	−	0.911164i	$-0.635185\pi$
$509$	−18.5923	−0.824087	−0.412044	+	0.911164i	$0.635185\pi$
$510$	0	0
$511$	−1.95092	−0.0863036
$512$	−31.7957	−1.40518
$513$	0	0
$514$	−44.5111	−1.96330
$515$	−0.682792	−0.0300874
$516$	0	0
$517$	71.6439	3.15090
$518$	14.6743	0.644752
$519$	0	0
$520$	−0.661483	−0.0290079
$521$	19.5299	0.855621	0.427810	−	0.903869i	$-0.359285\pi$
$521$	19.5299	0.855621	0.427810	+	0.903869i	$0.359285\pi$
$522$	0	0
$523$	−41.2322	−1.80296	−0.901480	−	0.432821i	$-0.857518\pi$
$523$	−41.2322	−1.80296	−0.901480	+	0.432821i	$0.857518\pi$
$524$	18.4059	0.804067
$525$	0	0
$526$	−32.2227	−1.40498
$527$	−1.54536	−0.0673170
$528$	0	0
$529$	1.00000	0.0434783
$530$	31.7836	1.38059
$531$	0	0
$532$	0.545389	0.0236456
$533$	−49.8836	−2.16070
$534$	0	0
$535$	−3.47640	−0.150298
$536$	−0.845339	−0.0365131
$537$	0	0
$538$	9.19237	0.396311
$539$	−38.1970	−1.64526
$540$	0	0
$541$	−23.9140	−1.02814	−0.514071	−	0.857747i	$-0.671863\pi$
$541$	−23.9140	−1.02814	−0.514071	+	0.857747i	$0.671863\pi$
$542$	−39.4768	−1.69567
$543$	0	0
$544$	−40.1548	−1.72162
$545$	30.9818	1.32711
$546$	0	0
$547$	36.2775	1.55112	0.775558	−	0.631277i	$-0.217469\pi$
$547$	36.2775	1.55112	0.775558	+	0.631277i	$0.217469\pi$
$548$	6.75727	0.288656
$549$	0	0
$550$	21.8185	0.930343
$551$	−0.331044	−0.0141030
$552$	0	0
$553$	3.23793	0.137691
$554$	19.2216	0.816647
$555$	0	0
$556$	9.25490	0.392495
$557$	−18.0531	−0.764932	−0.382466	−	0.923969i	$-0.624925\pi$
$557$	−18.0531	−0.764932	−0.382466	+	0.923969i	$0.624925\pi$
$558$	0	0
$559$	15.2285	0.644096
$560$	6.13191	0.259120
$561$	0	0
$562$	47.7249	2.01315
$563$	−13.8794	−0.584947	−0.292473	−	0.956274i	$-0.594478\pi$
$563$	−13.8794	−0.584947	−0.292473	+	0.956274i	$0.594478\pi$
$564$	0	0
$565$	−4.50328	−0.189455
$566$	15.4623	0.649928
$567$	0	0
$568$	−0.645167	−0.0270706
$569$	11.2856	0.473117	0.236558	−	0.971617i	$-0.423981\pi$
$569$	11.2856	0.473117	0.236558	+	0.971617i	$0.423981\pi$
$570$	0	0
$571$	24.7160	1.03433	0.517167	−	0.855885i	$-0.326987\pi$
$571$	24.7160	1.03433	0.517167	+	0.855885i	$0.326987\pi$
$572$	−54.0083	−2.25820
$573$	0	0
$574$	18.3781	0.767086
$575$	1.80652	0.0753370
$576$	0	0
$577$	30.6508	1.27601	0.638004	−	0.770033i	$-0.279760\pi$
$577$	30.6508	1.27601	0.638004	+	0.770033i	$0.279760\pi$
$578$	−16.8647	−0.701480
$579$	0	0
$580$	3.50087	0.145366
$581$	7.66531	0.318011
$582$	0	0
$583$	−54.2578	−2.24713
$584$	0.189068	0.00782368
$585$	0	0
$586$	27.8491	1.15044
$587$	28.8782	1.19193	0.595965	−	0.803010i	$-0.296770\pi$
$587$	28.8782	1.19193	0.595965	+	0.803010i	$0.296770\pi$
$588$	0	0
$589$	−0.101357	−0.00417632
$590$	49.8797	2.05351
$591$	0	0
$592$	−35.7825	−1.47065
$593$	−41.8335	−1.71790	−0.858948	−	0.512063i	$-0.828882\pi$
$593$	−41.8335	−1.71790	−0.858948	+	0.512063i	$0.828882\pi$
$594$	0	0
$595$	7.58533	0.310968
$596$	5.69750	0.233379
$597$	0	0
$598$	−9.03702	−0.369551
$599$	20.2914	0.829082	0.414541	−	0.910031i	$-0.363942\pi$
$599$	20.2914	0.829082	0.414541	+	0.910031i	$0.363942\pi$
$600$	0	0
$601$	−30.7368	−1.25378	−0.626889	−	0.779108i	$-0.715672\pi$
$601$	−30.7368	−1.25378	−0.626889	+	0.779108i	$0.715672\pi$
$602$	−5.61046	−0.228665
$603$	0	0
$604$	−28.7670	−1.17051
$605$	46.1851	1.87769
$606$	0	0
$607$	−29.5455	−1.19922	−0.599608	−	0.800294i	$-0.704677\pi$
$607$	−29.5455	−1.19922	−0.599608	+	0.800294i	$0.704677\pi$
$608$	−2.63365	−0.106809
$609$	0	0
$610$	33.6072	1.36072
$611$	−53.6072	−2.16871
$612$	0	0
$613$	−45.6417	−1.84345	−0.921726	−	0.387841i	$-0.873221\pi$
$613$	−45.6417	−1.84345	−0.921726	+	0.387841i	$0.873221\pi$
$614$	−11.3099	−0.456432
$615$	0	0
$616$	−0.416026	−0.0167622
$617$	−5.67280	−0.228378	−0.114189	−	0.993459i	$-0.536427\pi$
$617$	−5.67280	−0.228378	−0.114189	+	0.993459i	$0.536427\pi$
$618$	0	0
$619$	5.67211	0.227981	0.113991	−	0.993482i	$-0.463637\pi$
$619$	5.67211	0.227981	0.113991	+	0.993482i	$0.463637\pi$
$620$	1.07187	0.0430473
$621$	0	0
$622$	−25.3515	−1.01650
$623$	−0.362721	−0.0145321
$624$	0	0
$625$	−12.7039	−0.508156
$626$	16.0343	0.640860
$627$	0	0
$628$	−46.3149	−1.84817
$629$	−44.2639	−1.76492
$630$	0	0
$631$	−38.0554	−1.51496	−0.757481	−	0.652857i	$-0.773570\pi$
$631$	−38.0554	−1.51496	−0.757481	+	0.652857i	$0.773570\pi$
$632$	−0.313794	−0.0124821
$633$	0	0
$634$	−15.2891	−0.607208
$635$	−33.0990	−1.31349
$636$	0	0
$637$	28.5807	1.13241
$638$	−12.0776	−0.478158
$639$	0	0
$640$	−1.16489	−0.0460465
$641$	12.7894	0.505152	0.252576	−	0.967577i	$-0.418722\pi$
$641$	12.7894	0.505152	0.252576	+	0.967577i	$0.418722\pi$
$642$	0	0
$643$	17.1307	0.675570	0.337785	−	0.941223i	$-0.390322\pi$
$643$	17.1307	0.675570	0.337785	+	0.941223i	$0.390322\pi$
$644$	1.64748	0.0649199
$645$	0	0
$646$	−3.32464	−0.130806
$647$	9.77232	0.384190	0.192095	−	0.981376i	$-0.438472\pi$
$647$	9.77232	0.384190	0.192095	+	0.981376i	$0.438472\pi$
$648$	0	0
$649$	−85.1498	−3.34242
$650$	−16.3256	−0.640341
$651$	0	0
$652$	−34.0189	−1.33228
$653$	47.6351	1.86411	0.932054	−	0.362320i	$-0.118015\pi$
$653$	47.6351	1.86411	0.932054	+	0.362320i	$0.118015\pi$
$654$	0	0
$655$	16.7898	0.656033
$656$	−44.8140	−1.74969
$657$	0	0
$658$	19.7499	0.769932
$659$	−12.8147	−0.499190	−0.249595	−	0.968350i	$-0.580297\pi$
$659$	−12.8147	−0.499190	−0.249595	+	0.968350i	$0.580297\pi$
$660$	0	0
$661$	39.4077	1.53278	0.766392	−	0.642373i	$-0.222050\pi$
$661$	39.4077	1.53278	0.766392	+	0.642373i	$0.222050\pi$
$662$	−1.27754	−0.0496529
$663$	0	0
$664$	−0.742861	−0.0288286
$665$	0.497503	0.0192923
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	22.3401	0.864364
$669$	0	0
$670$	36.8810	1.42484
$671$	−57.3710	−2.21478
$672$	0	0
$673$	5.17575	0.199511	0.0997553	−	0.995012i	$-0.468194\pi$
$673$	5.17575	0.199511	0.0997553	+	0.995012i	$0.468194\pi$
$674$	−53.9045	−2.07632
$675$	0	0
$676$	14.9439	0.574766
$677$	15.1598	0.582639	0.291319	−	0.956626i	$-0.405906\pi$
$677$	15.1598	0.582639	0.291319	+	0.956626i	$0.405906\pi$
$678$	0	0
$679$	−9.20386	−0.353212
$680$	−0.735110	−0.0281902
$681$	0	0
$682$	−3.69784	−0.141597
$683$	−22.9972	−0.879963	−0.439981	−	0.898007i	$-0.645015\pi$
$683$	−22.9972	−0.879963	−0.439981	+	0.898007i	$0.645015\pi$
$684$	0	0
$685$	6.16396	0.235513
$686$	−22.2427	−0.849232
$687$	0	0
$688$	13.6808	0.521576
$689$	40.5981	1.54667
$690$	0	0
$691$	40.8354	1.55345	0.776725	−	0.629839i	$-0.216879\pi$
$691$	40.8354	1.55345	0.776725	+	0.629839i	$0.216879\pi$
$692$	17.0878	0.649581
$693$	0	0
$694$	3.71301	0.140944
$695$	8.44229	0.320235
$696$	0	0
$697$	−55.4360	−2.09979
$698$	−8.50777	−0.322024
$699$	0	0
$700$	2.97621	0.112490
$701$	33.4375	1.26292	0.631459	−	0.775409i	$-0.282457\pi$
$701$	33.4375	1.26292	0.631459	+	0.775409i	$0.282457\pi$
$702$	0	0
$703$	−2.90316	−0.109495
$704$	−46.5508	−1.75445
$705$	0	0
$706$	−13.3015	−0.500609
$707$	3.35238	0.126079
$708$	0	0
$709$	−14.1229	−0.530396	−0.265198	−	0.964194i	$-0.585437\pi$
$709$	−14.1229	−0.530396	−0.265198	+	0.964194i	$0.585437\pi$
$710$	28.1478	1.05637
$711$	0	0
$712$	0.0351521	0.00131738
$713$	−0.306172	−0.0114662
$714$	0	0
$715$	−49.2663	−1.84245
$716$	3.66284	0.136887
$717$	0	0
$718$	38.3676	1.43187
$719$	−31.6015	−1.17854	−0.589268	−	0.807938i	$-0.700584\pi$
$719$	−31.6015	−1.17854	−0.589268	+	0.807938i	$0.700584\pi$
$720$	0	0
$721$	0.321317	0.0119665
$722$	37.5869	1.39884
$723$	0	0
$724$	−28.3067	−1.05201
$725$	−1.80652	−0.0670924
$726$	0	0
$727$	25.1162	0.931509	0.465754	−	0.884914i	$-0.345783\pi$
$727$	25.1162	0.931509	0.465754	+	0.884914i	$0.345783\pi$
$728$	0.311289	0.0115371
$729$	0	0
$730$	−8.24876	−0.305300
$731$	16.9235	0.625939
$732$	0	0
$733$	23.9446	0.884414	0.442207	−	0.896913i	$-0.354196\pi$
$733$	23.9446	0.884414	0.442207	+	0.896913i	$0.354196\pi$
$734$	58.0176	2.14147
$735$	0	0
$736$	−7.95560	−0.293247
$737$	−62.9597	−2.31915
$738$	0	0
$739$	−2.62036	−0.0963917	−0.0481958	−	0.998838i	$-0.515347\pi$
$739$	−2.62036	−0.0963917	−0.0481958	+	0.998838i	$0.515347\pi$
$740$	30.7016	1.12861
$741$	0	0
$742$	−14.9571	−0.549093
$743$	17.4200	0.639079	0.319540	−	0.947573i	$-0.396472\pi$
$743$	17.4200	0.639079	0.319540	+	0.947573i	$0.396472\pi$
$744$	0	0
$745$	5.19725	0.190412
$746$	32.2167	1.17954
$747$	0	0
$748$	−60.0198	−2.19454
$749$	1.63597	0.0597770
$750$	0	0
$751$	−23.2221	−0.847388	−0.423694	−	0.905805i	$-0.639267\pi$
$751$	−23.2221	−0.847388	−0.423694	+	0.905805i	$0.639267\pi$
$752$	−48.1591	−1.75618
$753$	0	0
$754$	9.03702	0.329109
$755$	−26.2412	−0.955016
$756$	0	0
$757$	−51.0885	−1.85684	−0.928421	−	0.371529i	$-0.878834\pi$
$757$	−51.0885	−1.85684	−0.928421	+	0.371529i	$0.878834\pi$
$758$	76.7840	2.78892
$759$	0	0
$760$	−0.0482141	−0.00174891
$761$	−47.9760	−1.73913	−0.869564	−	0.493821i	$-0.835600\pi$
$761$	−47.9760	−1.73913	−0.869564	+	0.493821i	$0.835600\pi$
$762$	0	0
$763$	−14.5798	−0.527825
$764$	25.2029	0.911810
$765$	0	0
$766$	38.4848	1.39051
$767$	63.7129	2.30054
$768$	0	0
$769$	2.81281	0.101432	0.0507162	−	0.998713i	$-0.483850\pi$
$769$	2.81281	0.101432	0.0507162	+	0.998713i	$0.483850\pi$
$770$	18.1506	0.654103
$771$	0	0
$772$	−21.1924	−0.762731
$773$	−39.4338	−1.41834	−0.709168	−	0.705040i	$-0.750929\pi$
$773$	−39.4338	−1.41834	−0.709168	+	0.705040i	$0.750929\pi$
$774$	0	0
$775$	−0.553106	−0.0198681
$776$	0.891966	0.0320197
$777$	0	0
$778$	−10.5599	−0.378591
$779$	−3.63591	−0.130270
$780$	0	0
$781$	−48.0511	−1.71940
$782$	−10.0429	−0.359134
$783$	0	0
$784$	25.6761	0.917002
$785$	−42.2484	−1.50791
$786$	0	0
$787$	2.85927	0.101922	0.0509609	−	0.998701i	$-0.483772\pi$
$787$	2.85927	0.101922	0.0509609	+	0.998701i	$0.483772\pi$
$788$	−23.7701	−0.846774
$789$	0	0
$790$	13.6904	0.487083
$791$	2.11921	0.0753506
$792$	0	0
$793$	42.9275	1.52440
$794$	41.0210	1.45578
$795$	0	0
$796$	13.2138	0.468350
$797$	−30.5582	−1.08243	−0.541213	−	0.840885i	$-0.682035\pi$
$797$	−30.5582	−1.08243	−0.541213	+	0.840885i	$0.682035\pi$
$798$	0	0
$799$	−59.5740	−2.10758
$800$	−14.3719	−0.508124
$801$	0	0
$802$	50.9105	1.79771
$803$	14.0815	0.496925
$804$	0	0
$805$	1.50283	0.0529678
$806$	2.76689	0.0974594
$807$	0	0
$808$	−0.324886	−0.0114295
$809$	−5.16063	−0.181438	−0.0907191	−	0.995877i	$-0.528917\pi$
$809$	−5.16063	−0.181438	−0.0907191	+	0.995877i	$0.528917\pi$
$810$	0	0
$811$	3.03686	0.106639	0.0533193	−	0.998578i	$-0.483020\pi$
$811$	3.03686	0.106639	0.0533193	+	0.998578i	$0.483020\pi$
$812$	−1.64748	−0.0578153
$813$	0	0
$814$	−105.917	−3.71240
$815$	−31.0320	−1.08700
$816$	0	0
$817$	1.10997	0.0388330
$818$	−17.7823	−0.621745
$819$	0	0
$820$	38.4506	1.34275
$821$	41.0190	1.43157	0.715787	−	0.698318i	$-0.246068\pi$
$821$	41.0190	1.43157	0.715787	+	0.698318i	$0.246068\pi$
$822$	0	0
$823$	6.93047	0.241581	0.120791	−	0.992678i	$-0.461457\pi$
$823$	6.93047	0.241581	0.120791	+	0.992678i	$0.461457\pi$
$824$	−0.0311395	−0.00108480
$825$	0	0
$826$	−23.4730	−0.816731
$827$	−13.3949	−0.465786	−0.232893	−	0.972502i	$-0.574819\pi$
$827$	−13.3949	−0.465786	−0.232893	+	0.972502i	$0.574819\pi$
$828$	0	0
$829$	−21.3896	−0.742890	−0.371445	−	0.928455i	$-0.621138\pi$
$829$	−21.3896	−0.742890	−0.371445	+	0.928455i	$0.621138\pi$
$830$	32.4100	1.12497
$831$	0	0
$832$	34.8314	1.20756
$833$	31.7619	1.10049
$834$	0	0
$835$	20.3786	0.705230
$836$	−3.93655	−0.136149
$837$	0	0
$838$	−36.5031	−1.26098
$839$	46.0069	1.58834	0.794168	−	0.607698i	$-0.207907\pi$
$839$	46.0069	1.58834	0.794168	+	0.607698i	$0.207907\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−17.9761	−0.619497
$843$	0	0
$844$	31.8463	1.09619
$845$	13.6318	0.468948
$846$	0	0
$847$	−21.7344	−0.746803
$848$	36.4722	1.25246
$849$	0	0
$850$	−18.1427	−0.622289
$851$	−8.76970	−0.300622
$852$	0	0
$853$	1.43402	0.0490999	0.0245500	−	0.999699i	$-0.492185\pi$
$853$	1.43402	0.0490999	0.0245500	+	0.999699i	$0.492185\pi$
$854$	−15.8153	−0.541189
$855$	0	0
$856$	−0.158545	−0.00541897
$857$	−36.3290	−1.24098	−0.620488	−	0.784216i	$-0.713065\pi$
$857$	−36.3290	−1.24098	−0.620488	+	0.784216i	$0.713065\pi$
$858$	0	0
$859$	−32.0791	−1.09452	−0.547262	−	0.836962i	$-0.684330\pi$
$859$	−32.0791	−1.09452	−0.547262	+	0.836962i	$0.684330\pi$
$860$	−11.7382	−0.400269
$861$	0	0
$862$	−69.0292	−2.35114
$863$	−10.7672	−0.366519	−0.183259	−	0.983065i	$-0.558665\pi$
$863$	−10.7672	−0.366519	−0.183259	+	0.983065i	$0.558665\pi$
$864$	0	0
$865$	15.5875	0.529990
$866$	75.1368	2.55325
$867$	0	0
$868$	−0.504413	−0.0171209
$869$	−23.3710	−0.792806
$870$	0	0
$871$	47.1092	1.59623
$872$	1.41296	0.0478489
$873$	0	0
$874$	−0.658690	−0.0222805
$875$	10.2290	0.345805
$876$	0	0
$877$	49.3097	1.66507	0.832535	−	0.553973i	$-0.186889\pi$
$877$	49.3097	1.66507	0.832535	+	0.553973i	$0.186889\pi$
$878$	−54.4772	−1.83852
$879$	0	0
$880$	−44.2594	−1.49198
$881$	22.3006	0.751325	0.375663	−	0.926756i	$-0.377415\pi$
$881$	22.3006	0.751325	0.375663	+	0.926756i	$0.377415\pi$
$882$	0	0
$883$	−4.70286	−0.158264	−0.0791319	−	0.996864i	$-0.525215\pi$
$883$	−4.70286	−0.158264	−0.0791319	+	0.996864i	$0.525215\pi$
$884$	44.9095	1.51047
$885$	0	0
$886$	49.7907	1.67275
$887$	11.1307	0.373732	0.186866	−	0.982385i	$-0.440167\pi$
$887$	11.1307	0.373732	0.186866	+	0.982385i	$0.440167\pi$
$888$	0	0
$889$	15.5761	0.522407
$890$	−1.53364	−0.0514076
$891$	0	0
$892$	−5.67326	−0.189955
$893$	−3.90731	−0.130753
$894$	0	0
$895$	3.34123	0.111685
$896$	0.548191	0.0183138
$897$	0	0
$898$	−45.3705	−1.51403
$899$	0.306172	0.0102114
$900$	0	0
$901$	45.1170	1.50306
$902$	−132.651	−4.41678
$903$	0	0
$904$	−0.205377	−0.00683075
$905$	−25.8213	−0.858330
$906$	0	0
$907$	25.0562	0.831979	0.415990	−	0.909369i	$-0.363435\pi$
$907$	25.0562	0.831979	0.415990	+	0.909369i	$0.363435\pi$
$908$	−42.8409	−1.42172
$909$	0	0
$910$	−13.5811	−0.450209
$911$	41.0905	1.36139	0.680695	−	0.732567i	$-0.261678\pi$
$911$	41.0905	1.36139	0.680695	+	0.732567i	$0.261678\pi$
$912$	0	0
$913$	−55.3272	−1.83106
$914$	−34.2014	−1.13128
$915$	0	0
$916$	14.7397	0.487013
$917$	−7.90118	−0.260920
$918$	0	0
$919$	−42.9882	−1.41805	−0.709025	−	0.705183i	$-0.750865\pi$
$919$	−42.9882	−1.41805	−0.709025	+	0.705183i	$0.750865\pi$
$920$	−0.145642	−0.00480169
$921$	0	0
$922$	43.6529	1.43763
$923$	35.9540	1.18344
$924$	0	0
$925$	−15.8426	−0.520902
$926$	69.2816	2.27673
$927$	0	0
$928$	7.95560	0.261155
$929$	−41.3667	−1.35720	−0.678599	−	0.734509i	$-0.737413\pi$
$929$	−41.3667	−1.35720	−0.678599	+	0.734509i	$0.737413\pi$
$930$	0	0
$931$	2.08319	0.0682737
$932$	20.6850	0.677559
$933$	0	0
$934$	66.6163	2.17975
$935$	−54.7499	−1.79051
$936$	0	0
$937$	20.3980	0.666373	0.333186	−	0.942861i	$-0.391876\pi$
$937$	20.3980	0.666373	0.333186	+	0.942861i	$0.391876\pi$
$938$	−17.3559	−0.566691
$939$	0	0
$940$	41.3207	1.34773
$941$	−2.01928	−0.0658267	−0.0329133	−	0.999458i	$-0.510479\pi$
$941$	−2.01928	−0.0658267	−0.0329133	+	0.999458i	$0.510479\pi$
$942$	0	0
$943$	−10.9832	−0.357661
$944$	57.2378	1.86293
$945$	0	0
$946$	40.4956	1.31663
$947$	22.3878	0.727506	0.363753	−	0.931495i	$-0.381495\pi$
$947$	22.3878	0.727506	0.363753	+	0.931495i	$0.381495\pi$
$948$	0	0
$949$	−10.5364	−0.342026
$950$	−1.18993	−0.0386066
$951$	0	0
$952$	0.345938	0.0112119
$953$	−24.1537	−0.782414	−0.391207	−	0.920303i	$-0.627942\pi$
$953$	−24.1537	−0.782414	−0.391207	+	0.920303i	$0.627942\pi$
$954$	0	0
$955$	22.9900	0.743941
$956$	−2.53364	−0.0819439
$957$	0	0
$958$	−77.4641	−2.50275
$959$	−2.90072	−0.0936691
$960$	0	0
$961$	−30.9063	−0.996976
$962$	79.2520	2.55519
$963$	0	0
$964$	−3.79470	−0.122219
$965$	−19.3317	−0.622308
$966$	0	0
$967$	−20.2043	−0.649725	−0.324863	−	0.945761i	$-0.605318\pi$
$967$	−20.2043	−0.649725	−0.324863	+	0.945761i	$0.605318\pi$
$968$	2.10633	0.0676999
$969$	0	0
$970$	−38.9152	−1.24949
$971$	−26.3808	−0.846599	−0.423300	−	0.905990i	$-0.639128\pi$
$971$	−26.3808	−0.846599	−0.423300	+	0.905990i	$0.639128\pi$
$972$	0	0
$973$	−3.97288	−0.127365
$974$	49.4234	1.58363
$975$	0	0
$976$	38.5648	1.23443
$977$	27.5852	0.882530	0.441265	−	0.897377i	$-0.354530\pi$
$977$	27.5852	0.882530	0.441265	+	0.897377i	$0.354530\pi$
$978$	0	0
$979$	2.61808	0.0836741
$980$	−22.0302	−0.703728
$981$	0	0
$982$	37.4874	1.19627
$983$	−23.3775	−0.745626	−0.372813	−	0.927906i	$-0.621607\pi$
$983$	−23.3775	−0.745626	−0.372813	+	0.927906i	$0.621607\pi$
$984$	0	0
$985$	−21.6830	−0.690878
$986$	10.0429	0.319831
$987$	0	0
$988$	2.94550	0.0937090
$989$	3.35294	0.106617
$990$	0	0
$991$	−36.8734	−1.17132	−0.585662	−	0.810556i	$-0.699165\pi$
$991$	−36.8734	−1.17132	−0.585662	+	0.810556i	$0.699165\pi$
$992$	2.43578	0.0773362
$993$	0	0
$994$	−13.2461	−0.420142
$995$	12.0536	0.382124
$996$	0	0
$997$	40.5687	1.28482	0.642412	−	0.766359i	$-0.277934\pi$
$997$	40.5687	1.28482	0.642412	+	0.766359i	$0.277934\pi$
$998$	14.4117	0.456196
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.j.1.1		7
3.2	odd	2		2001.2.a.i.1.7	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.i.1.7	✓	7	3.2	odd	2
6003.2.a.j.1.1		7	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)

\(f(q)\)	\(=\)	\(q-1.98973 q^{2} +1.95904 q^{4} +1.78703 q^{5} -0.840964 q^{7} +0.0814996 q^{8} +O(q^{10})\)	\(q-1.98973 q^{2} +1.95904 q^{4} +1.78703 q^{5} -0.840964 q^{7} +0.0814996 q^{8} -3.55572 q^{10} +6.06998 q^{11} -4.54183 q^{13} +1.67330 q^{14} -4.08024 q^{16} -5.04736 q^{17} -0.331044 q^{19} +3.50087 q^{20} -12.0776 q^{22} -1.00000 q^{23} -1.80652 q^{25} +9.03702 q^{26} -1.64748 q^{28} +1.00000 q^{29} +0.306172 q^{31} +7.95560 q^{32} +10.0429 q^{34} -1.50283 q^{35} +8.76970 q^{37} +0.658690 q^{38} +0.145642 q^{40} +10.9832 q^{41} -3.35294 q^{43} +11.8913 q^{44} +1.98973 q^{46} +11.8030 q^{47} -6.29278 q^{49} +3.59449 q^{50} -8.89762 q^{52} -8.93872 q^{53} +10.8472 q^{55} -0.0685383 q^{56} -1.98973 q^{58} -14.0280 q^{59} -9.45160 q^{61} -0.609201 q^{62} -7.66903 q^{64} -8.11639 q^{65} -10.3723 q^{67} -9.88798 q^{68} +2.99023 q^{70} -7.91620 q^{71} +2.31986 q^{73} -17.4494 q^{74} -0.648529 q^{76} -5.10463 q^{77} -3.85026 q^{79} -7.29152 q^{80} -21.8536 q^{82} -9.11490 q^{83} -9.01980 q^{85} +6.67146 q^{86} +0.494701 q^{88} +0.431316 q^{89} +3.81951 q^{91} -1.95904 q^{92} -23.4848 q^{94} -0.591586 q^{95} +10.9444 q^{97} +12.5210 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8}+O(q^{10}) \) 7 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 5 * q^7 + 6 * q^8	\( 7 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8} + 3 q^{10} + 4 q^{11} - 18 q^{13} + 2 q^{14} - 7 q^{16} + 3 q^{17} - 4 q^{19} + 2 q^{20} - 26 q^{22} - 7 q^{23} - 8 q^{25} + 7 q^{26} - 6 q^{28} + 7 q^{29} - 22 q^{31} - 5 q^{32} + 9 q^{34} - 3 q^{35} - 25 q^{37} - 14 q^{38} - 10 q^{40} + 13 q^{41} - 2 q^{43} - 4 q^{44} - 3 q^{46} + 25 q^{47} - 8 q^{49} - 19 q^{50} - 12 q^{52} + 5 q^{53} - 15 q^{55} - 18 q^{56} + 3 q^{58} - 11 q^{59} - 33 q^{61} - 28 q^{62} - 14 q^{64} + 2 q^{65} + 8 q^{67} - 12 q^{68} - 22 q^{70} + 6 q^{71} + 15 q^{73} - 34 q^{74} - 28 q^{76} + q^{77} - 15 q^{79} + 12 q^{80} - 14 q^{82} - 21 q^{83} - 28 q^{85} + 12 q^{86} - 13 q^{88} - 8 q^{89} + 6 q^{91} - 5 q^{92} - 35 q^{94} + 25 q^{95} + 13 q^{97} - q^{98}+O(q^{100}) \) 7 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 5 * q^7 + 6 * q^8 + 3 * q^10 + 4 * q^11 - 18 * q^13 + 2 * q^14 - 7 * q^16 + 3 * q^17 - 4 * q^19 + 2 * q^20 - 26 * q^22 - 7 * q^23 - 8 * q^25 + 7 * q^26 - 6 * q^28 + 7 * q^29 - 22 * q^31 - 5 * q^32 + 9 * q^34 - 3 * q^35 - 25 * q^37 - 14 * q^38 - 10 * q^40 + 13 * q^41 - 2 * q^43 - 4 * q^44 - 3 * q^46 + 25 * q^47 - 8 * q^49 - 19 * q^50 - 12 * q^52 + 5 * q^53 - 15 * q^55 - 18 * q^56 + 3 * q^58 - 11 * q^59 - 33 * q^61 - 28 * q^62 - 14 * q^64 + 2 * q^65 + 8 * q^67 - 12 * q^68 - 22 * q^70 + 6 * q^71 + 15 * q^73 - 34 * q^74 - 28 * q^76 + q^77 - 15 * q^79 + 12 * q^80 - 14 * q^82 - 21 * q^83 - 28 * q^85 + 12 * q^86 - 13 * q^88 - 8 * q^89 + 6 * q^91 - 5 * q^92 - 35 * q^94 + 25 * q^95 + 13 * q^97 - q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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