LMFDB - Embedded newform 6003.2.a.k.1.5

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 17x^{8} + 23x^{7} + 69x^{6} - 88x^{5} - 106x^{4} + 101x^{3} + 60x^{2} - 23x - 11 $ x^10 - x^9 - 17x^8 + 23x^7 + 69x^6 - 88x^5 - 106x^4 + 101x^3 + 60x^2 - 23x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
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			1.5
Root			$-1.52497$ 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)

$f(q)$	$=$	$q-0.676519 q^{2} -1.54232 q^{4} +1.80585 q^{5} -4.44627 q^{7} +2.39645 q^{8} +O(q^{10})$	\(q-0.676519 q^{2} -1.54232 q^{4} +1.80585 q^{5} -4.44627 q^{7} +2.39645 q^{8} -1.22169 q^{10} +1.20345 q^{11} -3.05496 q^{13} +3.00799 q^{14} +1.46340 q^{16} +6.57035 q^{17} -2.24073 q^{19} -2.78521 q^{20} -0.814156 q^{22} +1.00000 q^{23} -1.73890 q^{25} +2.06674 q^{26} +6.85758 q^{28} -1.00000 q^{29} +2.34985 q^{31} -5.78292 q^{32} -4.44497 q^{34} -8.02931 q^{35} -8.63631 q^{37} +1.51589 q^{38} +4.32763 q^{40} +1.57477 q^{41} +6.47389 q^{43} -1.85610 q^{44} -0.676519 q^{46} +8.54965 q^{47} +12.7693 q^{49} +1.17640 q^{50} +4.71172 q^{52} -11.5800 q^{53} +2.17325 q^{55} -10.6553 q^{56} +0.676519 q^{58} +6.15579 q^{59} -0.352405 q^{61} -1.58972 q^{62} +0.985449 q^{64} -5.51680 q^{65} +1.02872 q^{67} -10.1336 q^{68} +5.43198 q^{70} +3.08000 q^{71} -8.06192 q^{73} +5.84263 q^{74} +3.45592 q^{76} -5.35086 q^{77} +4.13197 q^{79} +2.64269 q^{80} -1.06536 q^{82} +1.05082 q^{83} +11.8651 q^{85} -4.37971 q^{86} +2.88400 q^{88} -2.79007 q^{89} +13.5832 q^{91} -1.54232 q^{92} -5.78400 q^{94} -4.04642 q^{95} +8.49231 q^{97} -8.63870 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) $ 10 * q - 3 * q^2 + 17 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8	$ 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8} - 4 q^{10} - 9 q^{11} - 16 q^{13} - 16 q^{14} + 27 q^{16} + q^{19} - 21 q^{20} + 17 q^{22} + 10 q^{23} - 4 q^{25} - 28 q^{26} - 14 q^{28} - 10 q^{29} + 17 q^{31} - 21 q^{32} - 3 q^{34} - 29 q^{35} + q^{37} - 32 q^{38} + 13 q^{40} - 5 q^{43} - 33 q^{44} - 3 q^{46} - 15 q^{47} + 31 q^{49} + 22 q^{50} - 21 q^{52} - 35 q^{53} - 20 q^{55} - 18 q^{56} + 3 q^{58} - 49 q^{59} + 8 q^{61} - 15 q^{62} + 12 q^{64} + 3 q^{65} + 35 q^{67} + 18 q^{68} - 16 q^{70} - 30 q^{71} - 15 q^{73} - 23 q^{74} + 10 q^{76} - 23 q^{77} + 24 q^{79} - 23 q^{80} - 5 q^{82} - q^{83} + 10 q^{86} + 18 q^{88} - 15 q^{89} + 26 q^{91} + 17 q^{92} + 3 q^{94} - 7 q^{95} - 35 q^{97} - 3 q^{98}+O(q^{100}) $ 10 * q - 3 * q^2 + 17 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8 - 4 * q^10 - 9 * q^11 - 16 * q^13 - 16 * q^14 + 27 * q^16 + q^19 - 21 * q^20 + 17 * q^22 + 10 * q^23 - 4 * q^25 - 28 * q^26 - 14 * q^28 - 10 * q^29 + 17 * q^31 - 21 * q^32 - 3 * q^34 - 29 * q^35 + q^37 - 32 * q^38 + 13 * q^40 - 5 * q^43 - 33 * q^44 - 3 * q^46 - 15 * q^47 + 31 * q^49 + 22 * q^50 - 21 * q^52 - 35 * q^53 - 20 * q^55 - 18 * q^56 + 3 * q^58 - 49 * q^59 + 8 * q^61 - 15 * q^62 + 12 * q^64 + 3 * q^65 + 35 * q^67 + 18 * q^68 - 16 * q^70 - 30 * q^71 - 15 * q^73 - 23 * q^74 + 10 * q^76 - 23 * q^77 + 24 * q^79 - 23 * q^80 - 5 * q^82 - q^83 + 10 * q^86 + 18 * q^88 - 15 * q^89 + 26 * q^91 + 17 * q^92 + 3 * q^94 - 7 * q^95 - 35 * q^97 - 3 * 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 $

$2$	−0.676519	−0.478371	−0.239186	−	0.970974i	$-0.576880\pi$
$2$	−0.676519	−0.478371	−0.239186	+	0.970974i	$0.576880\pi$
$3$	0	0
$3$	0	0
$4$	−1.54232	−0.771161
$5$	1.80585	0.807601	0.403801	−	0.914847i	$-0.367689\pi$
$5$	1.80585	0.807601	0.403801	+	0.914847i	$0.367689\pi$
$6$	0	0
$7$	−4.44627	−1.68053	−0.840267	−	0.542173i	$-0.817602\pi$
$7$	−4.44627	−1.68053	−0.840267	+	0.542173i	$0.817602\pi$
$8$	2.39645	0.847272
$9$	0	0
$10$	−1.22169	−0.386333
$11$	1.20345	0.362853	0.181427	−	0.983404i	$-0.441928\pi$
$11$	1.20345	0.362853	0.181427	+	0.983404i	$0.441928\pi$
$12$	0	0
$13$	−3.05496	−0.847292	−0.423646	−	0.905828i	$-0.639250\pi$
$13$	−3.05496	−0.847292	−0.423646	+	0.905828i	$0.639250\pi$
$14$	3.00799	0.803919
$15$	0	0
$16$	1.46340	0.365850
$17$	6.57035	1.59354	0.796772	−	0.604280i	$-0.206539\pi$
$17$	6.57035	1.59354	0.796772	+	0.604280i	$0.206539\pi$
$18$	0	0
$19$	−2.24073	−0.514058	−0.257029	−	0.966404i	$-0.582744\pi$
$19$	−2.24073	−0.514058	−0.257029	+	0.966404i	$0.582744\pi$
$20$	−2.78521	−0.622791
$21$	0	0
$22$	−0.814156	−0.173579
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−1.73890	−0.347780
$26$	2.06674	0.405320
$27$	0	0
$28$	6.85758	1.29596
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	2.34985	0.422045	0.211022	−	0.977481i	$-0.432321\pi$
$31$	2.34985	0.422045	0.211022	+	0.977481i	$0.432321\pi$
$32$	−5.78292	−1.02228
$33$	0	0
$34$	−4.44497	−0.762306
$35$	−8.02931	−1.35720
$36$	0	0
$37$	−8.63631	−1.41980	−0.709900	−	0.704302i	$-0.751260\pi$
$37$	−8.63631	−1.41980	−0.709900	+	0.704302i	$0.751260\pi$
$38$	1.51589	0.245910
$39$	0	0
$40$	4.32763	0.684258
$41$	1.57477	0.245937	0.122969	−	0.992411i	$-0.460759\pi$
$41$	1.57477	0.245937	0.122969	+	0.992411i	$0.460759\pi$
$42$	0	0
$43$	6.47389	0.987259	0.493629	−	0.869672i	$-0.335670\pi$
$43$	6.47389	0.987259	0.493629	+	0.869672i	$0.335670\pi$
$44$	−1.85610	−0.279818
$45$	0	0
$46$	−0.676519	−0.0997473
$47$	8.54965	1.24709	0.623547	−	0.781786i	$-0.285691\pi$
$47$	8.54965	1.24709	0.623547	+	0.781786i	$0.285691\pi$
$48$	0	0
$49$	12.7693	1.82419
$50$	1.17640	0.166368
$51$	0	0
$52$	4.71172	0.653399
$53$	−11.5800	−1.59063	−0.795315	−	0.606197i	$-0.792694\pi$
$53$	−11.5800	−1.59063	−0.795315	+	0.606197i	$0.792694\pi$
$54$	0	0
$55$	2.17325	0.293041
$56$	−10.6553	−1.42387
$57$	0	0
$58$	0.676519	0.0888313
$59$	6.15579	0.801416	0.400708	−	0.916206i	$-0.368764\pi$
$59$	6.15579	0.801416	0.400708	+	0.916206i	$0.368764\pi$
$60$	0	0
$61$	−0.352405	−0.0451209	−0.0225604	−	0.999745i	$-0.507182\pi$
$61$	−0.352405	−0.0451209	−0.0225604	+	0.999745i	$0.507182\pi$
$62$	−1.58972	−0.201894
$63$	0	0
$64$	0.985449	0.123181
$65$	−5.51680	−0.684274
$66$	0	0
$67$	1.02872	0.125678	0.0628389	−	0.998024i	$-0.479985\pi$
$67$	1.02872	0.125678	0.0628389	+	0.998024i	$0.479985\pi$
$68$	−10.1336	−1.22888
$69$	0	0
$70$	5.43198	0.649246
$71$	3.08000	0.365529	0.182764	−	0.983157i	$-0.441495\pi$
$71$	3.08000	0.365529	0.182764	+	0.983157i	$0.441495\pi$
$72$	0	0
$73$	−8.06192	−0.943577	−0.471788	−	0.881712i	$-0.656391\pi$
$73$	−8.06192	−0.943577	−0.471788	+	0.881712i	$0.656391\pi$
$74$	5.84263	0.679192
$75$	0	0
$76$	3.45592	0.396421
$77$	−5.35086	−0.609787
$78$	0	0
$79$	4.13197	0.464883	0.232441	−	0.972610i	$-0.425329\pi$
$79$	4.13197	0.464883	0.232441	+	0.972610i	$0.425329\pi$
$80$	2.64269	0.295461
$81$	0	0
$82$	−1.06536	−0.117649
$83$	1.05082	0.115343	0.0576715	−	0.998336i	$-0.481632\pi$
$83$	1.05082	0.115343	0.0576715	+	0.998336i	$0.481632\pi$
$84$	0	0
$85$	11.8651	1.28695
$86$	−4.37971	−0.472276
$87$	0	0
$88$	2.88400	0.307436
$89$	−2.79007	−0.295747	−0.147874	−	0.989006i	$-0.547243\pi$
$89$	−2.79007	−0.295747	−0.147874	+	0.989006i	$0.547243\pi$
$90$	0	0
$91$	13.5832	1.42390
$92$	−1.54232	−0.160798
$93$	0	0
$94$	−5.78400	−0.596574
$95$	−4.04642	−0.415154
$96$	0	0
$97$	8.49231	0.862264	0.431132	−	0.902289i	$-0.358114\pi$
$97$	8.49231	0.862264	0.431132	+	0.902289i	$0.358114\pi$
$98$	−8.63870	−0.872641
$99$	0	0
$100$	2.68194	0.268194
$101$	17.0960	1.70112	0.850558	−	0.525881i	$-0.176264\pi$
$101$	17.0960	1.70112	0.850558	+	0.525881i	$0.176264\pi$
$102$	0	0
$103$	−3.01098	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$103$	−3.01098	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$104$	−7.32104	−0.717887
$105$	0	0
$106$	7.83406	0.760911
$107$	−5.04070	−0.487302	−0.243651	−	0.969863i	$-0.578345\pi$
$107$	−5.04070	−0.487302	−0.243651	+	0.969863i	$0.578345\pi$
$108$	0	0
$109$	15.6993	1.50372	0.751859	−	0.659323i	$-0.229157\pi$
$109$	15.6993	1.50372	0.751859	+	0.659323i	$0.229157\pi$
$110$	−1.47024	−0.140182
$111$	0	0
$112$	−6.50668	−0.614824
$113$	−7.47730	−0.703405	−0.351702	−	0.936112i	$-0.614397\pi$
$113$	−7.47730	−0.703405	−0.351702	+	0.936112i	$0.614397\pi$
$114$	0	0
$115$	1.80585	0.168397
$116$	1.54232	0.143201
$117$	0	0
$118$	−4.16451	−0.383374
$119$	−29.2136	−2.67800
$120$	0	0
$121$	−9.55171	−0.868337
$122$	0.238409	0.0215845
$123$	0	0
$124$	−3.62422	−0.325465
$125$	−12.1695	−1.08847
$126$	0	0
$127$	−13.3040	−1.18054	−0.590271	−	0.807205i	$-0.700979\pi$
$127$	−13.3040	−1.18054	−0.590271	+	0.807205i	$0.700979\pi$
$128$	10.8992	0.963358
$129$	0	0
$130$	3.73222	0.327337
$131$	−2.38913	−0.208739	−0.104369	−	0.994539i	$-0.533282\pi$
$131$	−2.38913	−0.208739	−0.104369	+	0.994539i	$0.533282\pi$
$132$	0	0
$133$	9.96288	0.863891
$134$	−0.695947	−0.0601207
$135$	0	0
$136$	15.7455	1.35017
$137$	20.7731	1.77476	0.887381	−	0.461036i	$-0.152522\pi$
$137$	20.7731	1.77476	0.887381	+	0.461036i	$0.152522\pi$
$138$	0	0
$139$	−14.9484	−1.26790	−0.633952	−	0.773372i	$-0.718568\pi$
$139$	−14.9484	−1.26790	−0.633952	+	0.773372i	$0.718568\pi$
$140$	12.3838	1.04662
$141$	0	0
$142$	−2.08368	−0.174859
$143$	−3.67648	−0.307443
$144$	0	0
$145$	−1.80585	−0.149968
$146$	5.45404	0.451380
$147$	0	0
$148$	13.3200	1.09490
$149$	−4.47172	−0.366337	−0.183169	−	0.983082i	$-0.558635\pi$
$149$	−4.47172	−0.366337	−0.183169	+	0.983082i	$0.558635\pi$
$150$	0	0
$151$	9.66669	0.786664	0.393332	−	0.919396i	$-0.371322\pi$
$151$	9.66669	0.786664	0.393332	+	0.919396i	$0.371322\pi$
$152$	−5.36978	−0.435547
$153$	0	0
$154$	3.61996	0.291705
$155$	4.24347	0.340844
$156$	0	0
$157$	−5.80134	−0.462997	−0.231499	−	0.972835i	$-0.574363\pi$
$157$	−5.80134	−0.462997	−0.231499	+	0.972835i	$0.574363\pi$
$158$	−2.79536	−0.222387
$159$	0	0
$160$	−10.4431	−0.825599
$161$	−4.44627	−0.350415
$162$	0	0
$163$	−20.1694	−1.57979	−0.789895	−	0.613242i	$-0.789865\pi$
$163$	−20.1694	−1.57979	−0.789895	+	0.613242i	$0.789865\pi$
$164$	−2.42880	−0.189657
$165$	0	0
$166$	−0.710903	−0.0551768
$167$	−7.19069	−0.556432	−0.278216	−	0.960518i	$-0.589743\pi$
$167$	−7.19069	−0.556432	−0.278216	+	0.960518i	$0.589743\pi$
$168$	0	0
$169$	−3.66725	−0.282096
$170$	−8.02695	−0.615639
$171$	0	0
$172$	−9.98482	−0.761336
$173$	15.0384	1.14335	0.571675	−	0.820480i	$-0.306294\pi$
$173$	15.0384	1.14335	0.571675	+	0.820480i	$0.306294\pi$
$174$	0	0
$175$	7.73162	0.584456
$176$	1.76113	0.132750
$177$	0	0
$178$	1.88754	0.141477
$179$	−10.0005	−0.747471	−0.373736	−	0.927535i	$-0.621923\pi$
$179$	−10.0005	−0.747471	−0.373736	+	0.927535i	$0.621923\pi$
$180$	0	0
$181$	−11.6624	−0.866858	−0.433429	−	0.901188i	$-0.642696\pi$
$181$	−11.6624	−0.866858	−0.433429	+	0.901188i	$0.642696\pi$
$182$	−9.18927	−0.681154
$183$	0	0
$184$	2.39645	0.176669
$185$	−15.5959	−1.14663
$186$	0	0
$187$	7.90708	0.578223
$188$	−13.1863	−0.961710
$189$	0	0
$190$	2.73748	0.198598
$191$	−4.81810	−0.348626	−0.174313	−	0.984690i	$-0.555770\pi$
$191$	−4.81810	−0.348626	−0.174313	+	0.984690i	$0.555770\pi$
$192$	0	0
$193$	−26.0157	−1.87265	−0.936325	−	0.351133i	$-0.885796\pi$
$193$	−26.0157	−1.87265	−0.936325	+	0.351133i	$0.885796\pi$
$194$	−5.74521	−0.412482
$195$	0	0
$196$	−19.6944	−1.40675
$197$	−9.97511	−0.710697	−0.355349	−	0.934734i	$-0.615638\pi$
$197$	−9.97511	−0.710697	−0.355349	+	0.934734i	$0.615638\pi$
$198$	0	0
$199$	18.3470	1.30058	0.650292	−	0.759684i	$-0.274647\pi$
$199$	18.3470	1.30058	0.650292	+	0.759684i	$0.274647\pi$
$200$	−4.16718	−0.294664
$201$	0	0
$202$	−11.5658	−0.813765
$203$	4.44627	0.312067
$204$	0	0
$205$	2.84380	0.198619
$206$	2.03699	0.141923
$207$	0	0
$208$	−4.47063	−0.309982
$209$	−2.69660	−0.186528
$210$	0	0
$211$	−16.5709	−1.14079	−0.570394	−	0.821371i	$-0.693210\pi$
$211$	−16.5709	−1.14079	−0.570394	+	0.821371i	$0.693210\pi$
$212$	17.8600	1.22663
$213$	0	0
$214$	3.41013	0.233111
$215$	11.6909	0.797312
$216$	0	0
$217$	−10.4481	−0.709260
$218$	−10.6209	−0.719336
$219$	0	0
$220$	−3.35185	−0.225982
$221$	−20.0721	−1.35020
$222$	0	0
$223$	−9.89206	−0.662422	−0.331211	−	0.943557i	$-0.607457\pi$
$223$	−9.89206	−0.662422	−0.331211	+	0.943557i	$0.607457\pi$
$224$	25.7124	1.71798
$225$	0	0
$226$	5.05853	0.336489
$227$	−11.8612	−0.787258	−0.393629	−	0.919269i	$-0.628781\pi$
$227$	−11.8612	−0.787258	−0.393629	+	0.919269i	$0.628781\pi$
$228$	0	0
$229$	−26.6481	−1.76095	−0.880477	−	0.474089i	$-0.842777\pi$
$229$	−26.6481	−1.76095	−0.880477	+	0.474089i	$0.842777\pi$
$230$	−1.22169	−0.0805561
$231$	0	0
$232$	−2.39645	−0.157335
$233$	5.10509	0.334445	0.167223	−	0.985919i	$-0.446520\pi$
$233$	5.10509	0.334445	0.167223	+	0.985919i	$0.446520\pi$
$234$	0	0
$235$	15.4394	1.00716
$236$	−9.49422	−0.618021
$237$	0	0
$238$	19.7635	1.28108
$239$	−10.0944	−0.652950	−0.326475	−	0.945206i	$-0.605861\pi$
$239$	−10.0944	−0.652950	−0.326475	+	0.945206i	$0.605861\pi$
$240$	0	0
$241$	3.36654	0.216858	0.108429	−	0.994104i	$-0.465418\pi$
$241$	3.36654	0.216858	0.108429	+	0.994104i	$0.465418\pi$
$242$	6.46191	0.415388
$243$	0	0
$244$	0.543523	0.0347955
$245$	23.0595	1.47322
$246$	0	0
$247$	6.84532	0.435557
$248$	5.63128	0.357587
$249$	0	0
$250$	8.23287	0.520692
$251$	−4.68444	−0.295679	−0.147839	−	0.989011i	$-0.547232\pi$
$251$	−4.68444	−0.295679	−0.147839	+	0.989011i	$0.547232\pi$
$252$	0	0
$253$	1.20345	0.0756602
$254$	9.00044	0.564738
$255$	0	0
$256$	−9.34438	−0.584024
$257$	28.4842	1.77680	0.888398	−	0.459074i	$-0.151819\pi$
$257$	28.4842	1.77680	0.888398	+	0.459074i	$0.151819\pi$
$258$	0	0
$259$	38.3994	2.38602
$260$	8.50868	0.527686
$261$	0	0
$262$	1.61629	0.0998547
$263$	−15.3003	−0.943459	−0.471730	−	0.881743i	$-0.656370\pi$
$263$	−15.3003	−0.943459	−0.471730	+	0.881743i	$0.656370\pi$
$264$	0	0
$265$	−20.9117	−1.28459
$266$	−6.74008	−0.413261
$267$	0	0
$268$	−1.58661	−0.0969179
$269$	−11.6444	−0.709971	−0.354986	−	0.934872i	$-0.615514\pi$
$269$	−11.6444	−0.709971	−0.354986	+	0.934872i	$0.615514\pi$
$270$	0	0
$271$	24.5547	1.49159	0.745796	−	0.666174i	$-0.232069\pi$
$271$	24.5547	1.49159	0.745796	+	0.666174i	$0.232069\pi$
$272$	9.61506	0.582999
$273$	0	0
$274$	−14.0534	−0.848995
$275$	−2.09268	−0.126193
$276$	0	0
$277$	−12.1114	−0.727705	−0.363853	−	0.931457i	$-0.618539\pi$
$277$	−12.1114	−0.727705	−0.363853	+	0.931457i	$0.618539\pi$
$278$	10.1129	0.606529
$279$	0	0
$280$	−19.2418	−1.14992
$281$	18.1178	1.08082	0.540408	−	0.841403i	$-0.318270\pi$
$281$	18.1178	1.08082	0.540408	+	0.841403i	$0.318270\pi$
$282$	0	0
$283$	−0.417794	−0.0248352	−0.0124176	−	0.999923i	$-0.503953\pi$
$283$	−0.417794	−0.0248352	−0.0124176	+	0.999923i	$0.503953\pi$
$284$	−4.75035	−0.281882
$285$	0	0
$286$	2.48721	0.147072
$287$	−7.00185	−0.413306
$288$	0	0
$289$	26.1695	1.53938
$290$	1.22169	0.0717403
$291$	0	0
$292$	12.4341	0.727649
$293$	−25.6351	−1.49762	−0.748810	−	0.662785i	$-0.769374\pi$
$293$	−25.6351	−1.49762	−0.748810	+	0.662785i	$0.769374\pi$
$294$	0	0
$295$	11.1165	0.647225
$296$	−20.6965	−1.20296
$297$	0	0
$298$	3.02520	0.175245
$299$	−3.05496	−0.176673
$300$	0	0
$301$	−28.7847	−1.65912
$302$	−6.53970	−0.376318
$303$	0	0
$304$	−3.27908	−0.188068
$305$	−0.636392	−0.0364397
$306$	0	0
$307$	12.6821	0.723803	0.361901	−	0.932216i	$-0.382128\pi$
$307$	12.6821	0.723803	0.361901	+	0.932216i	$0.382128\pi$
$308$	8.25275	0.470244
$309$	0	0
$310$	−2.87079	−0.163050
$311$	4.59395	0.260499	0.130250	−	0.991481i	$-0.458422\pi$
$311$	4.59395	0.260499	0.130250	+	0.991481i	$0.458422\pi$
$312$	0	0
$313$	1.81844	0.102784	0.0513922	−	0.998679i	$-0.483634\pi$
$313$	1.81844	0.102784	0.0513922	+	0.998679i	$0.483634\pi$
$314$	3.92472	0.221485
$315$	0	0
$316$	−6.37283	−0.358500
$317$	1.90790	0.107158	0.0535790	−	0.998564i	$-0.482937\pi$
$317$	1.90790	0.107158	0.0535790	+	0.998564i	$0.482937\pi$
$318$	0	0
$319$	−1.20345	−0.0673802
$320$	1.77958	0.0994813
$321$	0	0
$322$	3.00799	0.167629
$323$	−14.7224	−0.819174
$324$	0	0
$325$	5.31226	0.294671
$326$	13.6450	0.755726
$327$	0	0
$328$	3.77385	0.208376
$329$	−38.0141	−2.09578
$330$	0	0
$331$	−25.1550	−1.38264	−0.691321	−	0.722548i	$-0.742971\pi$
$331$	−25.1550	−1.38264	−0.691321	+	0.722548i	$0.742971\pi$
$332$	−1.62071	−0.0889480
$333$	0	0
$334$	4.86464	0.266181
$335$	1.85771	0.101498
$336$	0	0
$337$	0.556403	0.0303092	0.0151546	−	0.999885i	$-0.495176\pi$
$337$	0.556403	0.0303092	0.0151546	+	0.999885i	$0.495176\pi$
$338$	2.48096	0.134947
$339$	0	0
$340$	−18.2998	−0.992445
$341$	2.82792	0.153140
$342$	0	0
$343$	−25.6521	−1.38508
$344$	15.5143	0.836477
$345$	0	0
$346$	−10.1738	−0.546946
$347$	−10.0168	−0.537732	−0.268866	−	0.963178i	$-0.586649\pi$
$347$	−10.0168	−0.537732	−0.268866	+	0.963178i	$0.586649\pi$
$348$	0	0
$349$	−8.01232	−0.428889	−0.214445	−	0.976736i	$-0.568794\pi$
$349$	−8.01232	−0.428889	−0.214445	+	0.976736i	$0.568794\pi$
$350$	−5.23059	−0.279587
$351$	0	0
$352$	−6.95944	−0.370939
$353$	−8.43622	−0.449014	−0.224507	−	0.974472i	$-0.572077\pi$
$353$	−8.43622	−0.449014	−0.224507	+	0.974472i	$0.572077\pi$
$354$	0	0
$355$	5.56203	0.295202
$356$	4.30319	0.228069
$357$	0	0
$358$	6.76552	0.357569
$359$	−17.8109	−0.940023	−0.470011	−	0.882660i	$-0.655750\pi$
$359$	−17.8109	−0.940023	−0.470011	+	0.882660i	$0.655750\pi$
$360$	0	0
$361$	−13.9791	−0.735745
$362$	7.88982	0.414680
$363$	0	0
$364$	−20.9496	−1.09806
$365$	−14.5586	−0.762034
$366$	0	0
$367$	29.2520	1.52694	0.763471	−	0.645842i	$-0.223494\pi$
$367$	29.2520	1.52694	0.763471	+	0.645842i	$0.223494\pi$
$368$	1.46340	0.0762851
$369$	0	0
$370$	10.5509	0.548516
$371$	51.4876	2.67311
$372$	0	0
$373$	−29.6981	−1.53771	−0.768855	−	0.639423i	$-0.779173\pi$
$373$	−29.6981	−1.53771	−0.768855	+	0.639423i	$0.779173\pi$
$374$	−5.34929	−0.276605
$375$	0	0
$376$	20.4888	1.05663
$377$	3.05496	0.157338
$378$	0	0
$379$	8.27683	0.425152	0.212576	−	0.977144i	$-0.431815\pi$
$379$	8.27683	0.425152	0.212576	+	0.977144i	$0.431815\pi$
$380$	6.24088	0.320150
$381$	0	0
$382$	3.25954	0.166772
$383$	24.9707	1.27594	0.637972	−	0.770059i	$-0.279773\pi$
$383$	24.9707	1.27594	0.637972	+	0.770059i	$0.279773\pi$
$384$	0	0
$385$	−9.66286	−0.492465
$386$	17.6001	0.895822
$387$	0	0
$388$	−13.0979	−0.664944
$389$	−21.7567	−1.10311	−0.551554	−	0.834139i	$-0.685965\pi$
$389$	−21.7567	−1.10311	−0.551554	+	0.834139i	$0.685965\pi$
$390$	0	0
$391$	6.57035	0.332277
$392$	30.6011	1.54559
$393$	0	0
$394$	6.74835	0.339977
$395$	7.46172	0.375440
$396$	0	0
$397$	−25.4546	−1.27753	−0.638766	−	0.769401i	$-0.720555\pi$
$397$	−25.4546	−1.27753	−0.638766	+	0.769401i	$0.720555\pi$
$398$	−12.4121	−0.622162
$399$	0	0
$400$	−2.54471	−0.127235
$401$	−11.4283	−0.570702	−0.285351	−	0.958423i	$-0.592110\pi$
$401$	−11.4283	−0.570702	−0.285351	+	0.958423i	$0.592110\pi$
$402$	0	0
$403$	−7.17867	−0.357595
$404$	−26.3676	−1.31184
$405$	0	0
$406$	−3.00799	−0.149284
$407$	−10.3934	−0.515179
$408$	0	0
$409$	−1.27697	−0.0631420	−0.0315710	−	0.999502i	$-0.510051\pi$
$409$	−1.27697	−0.0631420	−0.0315710	+	0.999502i	$0.510051\pi$
$410$	−1.92388	−0.0950138
$411$	0	0
$412$	4.64390	0.228789
$413$	−27.3703	−1.34681
$414$	0	0
$415$	1.89763	0.0931512
$416$	17.6665	0.866174
$417$	0	0
$418$	1.82430	0.0892294
$419$	10.5176	0.513820	0.256910	−	0.966435i	$-0.417296\pi$
$419$	10.5176	0.513820	0.256910	+	0.966435i	$0.417296\pi$
$420$	0	0
$421$	−18.7938	−0.915952	−0.457976	−	0.888964i	$-0.651426\pi$
$421$	−18.7938	−0.915952	−0.457976	+	0.888964i	$0.651426\pi$
$422$	11.2105	0.545720
$423$	0	0
$424$	−27.7508	−1.34770
$425$	−11.4252	−0.554203
$426$	0	0
$427$	1.56689	0.0758272
$428$	7.77438	0.375789
$429$	0	0
$430$	−7.90911	−0.381411
$431$	−5.42412	−0.261270	−0.130635	−	0.991430i	$-0.541702\pi$
$431$	−5.42412	−0.261270	−0.130635	+	0.991430i	$0.541702\pi$
$432$	0	0
$433$	−8.03271	−0.386028	−0.193014	−	0.981196i	$-0.561826\pi$
$433$	−8.03271	−0.386028	−0.193014	+	0.981196i	$0.561826\pi$
$434$	7.06831	0.339290
$435$	0	0
$436$	−24.2134	−1.15961
$437$	−2.24073	−0.107188
$438$	0	0
$439$	−12.2945	−0.586786	−0.293393	−	0.955992i	$-0.594784\pi$
$439$	−12.2945	−0.586786	−0.293393	+	0.955992i	$0.594784\pi$
$440$	5.20808	0.248285
$441$	0	0
$442$	13.5792	0.645896
$443$	−5.21201	−0.247630	−0.123815	−	0.992305i	$-0.539513\pi$
$443$	−5.21201	−0.247630	−0.123815	+	0.992305i	$0.539513\pi$
$444$	0	0
$445$	−5.03846	−0.238846
$446$	6.69217	0.316883
$447$	0	0
$448$	−4.38158	−0.207010
$449$	−25.2315	−1.19075	−0.595373	−	0.803449i	$-0.702996\pi$
$449$	−25.2315	−1.19075	−0.595373	+	0.803449i	$0.702996\pi$
$450$	0	0
$451$	1.89515	0.0892392
$452$	11.5324	0.542438
$453$	0	0
$454$	8.02435	0.376601
$455$	24.5292	1.14995
$456$	0	0
$457$	−20.5153	−0.959664	−0.479832	−	0.877361i	$-0.659302\pi$
$457$	−20.5153	−0.959664	−0.479832	+	0.877361i	$0.659302\pi$
$458$	18.0279	0.842389
$459$	0	0
$460$	−2.78521	−0.129861
$461$	−17.7416	−0.826310	−0.413155	−	0.910661i	$-0.635573\pi$
$461$	−17.7416	−0.826310	−0.413155	+	0.910661i	$0.635573\pi$
$462$	0	0
$463$	25.4877	1.18451	0.592257	−	0.805749i	$-0.298237\pi$
$463$	25.4877	1.18451	0.592257	+	0.805749i	$0.298237\pi$
$464$	−1.46340	−0.0679367
$465$	0	0
$466$	−3.45369	−0.159989
$467$	3.94325	0.182472	0.0912358	−	0.995829i	$-0.470918\pi$
$467$	3.94325	0.182472	0.0912358	+	0.995829i	$0.470918\pi$
$468$	0	0
$469$	−4.57396	−0.211206
$470$	−10.4450	−0.481794
$471$	0	0
$472$	14.7520	0.679018
$473$	7.79099	0.358230
$474$	0	0
$475$	3.89640	0.178779
$476$	45.0567	2.06517
$477$	0	0
$478$	6.82903	0.312352
$479$	0.584497	0.0267064	0.0133532	−	0.999911i	$-0.495749\pi$
$479$	0.584497	0.0267064	0.0133532	+	0.999911i	$0.495749\pi$
$480$	0	0
$481$	26.3835	1.20299
$482$	−2.27753	−0.103738
$483$	0	0
$484$	14.7318	0.669628
$485$	15.3359	0.696365
$486$	0	0
$487$	−10.8422	−0.491309	−0.245655	−	0.969357i	$-0.579003\pi$
$487$	−10.8422	−0.491309	−0.245655	+	0.969357i	$0.579003\pi$
$488$	−0.844521	−0.0382297
$489$	0	0
$490$	−15.6002	−0.704746
$491$	−4.73316	−0.213604	−0.106802	−	0.994280i	$-0.534061\pi$
$491$	−4.73316	−0.213604	−0.106802	+	0.994280i	$0.534061\pi$
$492$	0	0
$493$	−6.57035	−0.295914
$494$	−4.63099	−0.208358
$495$	0	0
$496$	3.43877	0.154405
$497$	−13.6945	−0.614284
$498$	0	0
$499$	25.0439	1.12112	0.560560	−	0.828114i	$-0.310586\pi$
$499$	25.0439	1.12112	0.560560	+	0.828114i	$0.310586\pi$
$500$	18.7692	0.839385
$501$	0	0
$502$	3.16911	0.141444
$503$	−8.07677	−0.360125	−0.180063	−	0.983655i	$-0.557630\pi$
$503$	−8.07677	−0.360125	−0.180063	+	0.983655i	$0.557630\pi$
$504$	0	0
$505$	30.8729	1.37382
$506$	−0.814156	−0.0361936
$507$	0	0
$508$	20.5191	0.910388
$509$	13.6489	0.604975	0.302488	−	0.953153i	$-0.402183\pi$
$509$	13.6489	0.604975	0.302488	+	0.953153i	$0.402183\pi$
$510$	0	0
$511$	35.8455	1.58571
$512$	−15.4767	−0.683978
$513$	0	0
$514$	−19.2701	−0.849968
$515$	−5.43738	−0.239600
$516$	0	0
$517$	10.2891	0.452512
$518$	−25.9779	−1.14140
$519$	0	0
$520$	−13.2207	−0.579767
$521$	−22.6165	−0.990847	−0.495423	−	0.868652i	$-0.664987\pi$
$521$	−22.6165	−0.990847	−0.495423	+	0.868652i	$0.664987\pi$
$522$	0	0
$523$	2.40915	0.105345	0.0526724	−	0.998612i	$-0.483226\pi$
$523$	2.40915	0.105345	0.0526724	+	0.998612i	$0.483226\pi$
$524$	3.68480	0.160971
$525$	0	0
$526$	10.3510	0.451324
$527$	15.4393	0.672547
$528$	0	0
$529$	1.00000	0.0434783
$530$	14.1472	0.614513
$531$	0	0
$532$	−15.3660	−0.666199
$533$	−4.81084	−0.208381
$534$	0	0
$535$	−9.10275	−0.393546
$536$	2.46527	0.106483
$537$	0	0
$538$	7.87765	0.339630
$539$	15.3672	0.661914
$540$	0	0
$541$	−0.501966	−0.0215812	−0.0107906	−	0.999942i	$-0.503435\pi$
$541$	−0.501966	−0.0215812	−0.0107906	+	0.999942i	$0.503435\pi$
$542$	−16.6117	−0.713535
$543$	0	0
$544$	−37.9958	−1.62906
$545$	28.3506	1.21441
$546$	0	0
$547$	34.6720	1.48247	0.741235	−	0.671246i	$-0.234241\pi$
$547$	34.6720	1.48247	0.741235	+	0.671246i	$0.234241\pi$
$548$	−32.0388	−1.36863
$549$	0	0
$550$	1.41573	0.0603671
$551$	2.24073	0.0954581
$552$	0	0
$553$	−18.3719	−0.781251
$554$	8.19361	0.348113
$555$	0	0
$556$	23.0552	0.977758
$557$	−10.7212	−0.454272	−0.227136	−	0.973863i	$-0.572936\pi$
$557$	−10.7212	−0.454272	−0.227136	+	0.973863i	$0.572936\pi$
$558$	0	0
$559$	−19.7774	−0.836497
$560$	−11.7501	−0.496532
$561$	0	0
$562$	−12.2570	−0.517031
$563$	36.3492	1.53193	0.765967	−	0.642880i	$-0.222260\pi$
$563$	36.3492	1.53193	0.765967	+	0.642880i	$0.222260\pi$
$564$	0	0
$565$	−13.5029	−0.568071
$566$	0.282645	0.0118805
$567$	0	0
$568$	7.38106	0.309703
$569$	−18.6121	−0.780262	−0.390131	−	0.920759i	$-0.627570\pi$
$569$	−18.6121	−0.780262	−0.390131	+	0.920759i	$0.627570\pi$
$570$	0	0
$571$	12.1827	0.509832	0.254916	−	0.966963i	$-0.417952\pi$
$571$	12.1827	0.509832	0.254916	+	0.966963i	$0.417952\pi$
$572$	5.67032	0.237088
$573$	0	0
$574$	4.73688	0.197714
$575$	−1.73890	−0.0725171
$576$	0	0
$577$	−18.5881	−0.773835	−0.386917	−	0.922114i	$-0.626460\pi$
$577$	−18.5881	−0.773835	−0.386917	+	0.922114i	$0.626460\pi$
$578$	−17.7042	−0.736397
$579$	0	0
$580$	2.78521	0.115649
$581$	−4.67225	−0.193838
$582$	0	0
$583$	−13.9359	−0.577165
$584$	−19.3200	−0.799466
$585$	0	0
$586$	17.3426	0.716418
$587$	18.5251	0.764612	0.382306	−	0.924036i	$-0.375130\pi$
$587$	18.5251	0.764612	0.382306	+	0.924036i	$0.375130\pi$
$588$	0	0
$589$	−5.26536	−0.216955
$590$	−7.52049	−0.309614
$591$	0	0
$592$	−12.6384	−0.519435
$593$	22.0097	0.903829	0.451915	−	0.892061i	$-0.350741\pi$
$593$	22.0097	0.903829	0.451915	+	0.892061i	$0.350741\pi$
$594$	0	0
$595$	−52.7554	−2.16276
$596$	6.89683	0.282505
$597$	0	0
$598$	2.06674	0.0845151
$599$	−25.7854	−1.05356	−0.526782	−	0.850001i	$-0.676602\pi$
$599$	−25.7854	−1.05356	−0.526782	+	0.850001i	$0.676602\pi$
$600$	0	0
$601$	13.4458	0.548465	0.274232	−	0.961663i	$-0.411576\pi$
$601$	13.4458	0.548465	0.274232	+	0.961663i	$0.411576\pi$
$602$	19.4734	0.793676
$603$	0	0
$604$	−14.9092	−0.606645
$605$	−17.2490	−0.701271
$606$	0	0
$607$	−27.4175	−1.11284	−0.556420	−	0.830901i	$-0.687825\pi$
$607$	−27.4175	−1.11284	−0.556420	+	0.830901i	$0.687825\pi$
$608$	12.9579	0.525513
$609$	0	0
$610$	0.430531	0.0174317
$611$	−26.1188	−1.05665
$612$	0	0
$613$	−40.0774	−1.61871	−0.809356	−	0.587319i	$-0.800183\pi$
$613$	−40.0774	−1.61871	−0.809356	+	0.587319i	$0.800183\pi$
$614$	−8.57965	−0.346246
$615$	0	0
$616$	−12.8231	−0.516656
$617$	−4.99212	−0.200975	−0.100487	−	0.994938i	$-0.532040\pi$
$617$	−4.99212	−0.200975	−0.100487	+	0.994938i	$0.532040\pi$
$618$	0	0
$619$	−31.1905	−1.25365	−0.626826	−	0.779159i	$-0.715646\pi$
$619$	−31.1905	−1.25365	−0.626826	+	0.779159i	$0.715646\pi$
$620$	−6.54480	−0.262846
$621$	0	0
$622$	−3.10790	−0.124615
$623$	12.4054	0.497013
$624$	0	0
$625$	−13.2817	−0.531269
$626$	−1.23021	−0.0491691
$627$	0	0
$628$	8.94753	0.357045
$629$	−56.7436	−2.26252
$630$	0	0
$631$	29.4354	1.17180	0.585902	−	0.810382i	$-0.300741\pi$
$631$	29.4354	1.17180	0.585902	+	0.810382i	$0.300741\pi$
$632$	9.90205	0.393882
$633$	0	0
$634$	−1.29073	−0.0512613
$635$	−24.0251	−0.953408
$636$	0	0
$637$	−39.0098	−1.54562
$638$	0.814156	0.0322327
$639$	0	0
$640$	19.6823	0.778010
$641$	−6.16488	−0.243498	−0.121749	−	0.992561i	$-0.538850\pi$
$641$	−6.16488	−0.243498	−0.121749	+	0.992561i	$0.538850\pi$
$642$	0	0
$643$	−17.8201	−0.702757	−0.351378	−	0.936234i	$-0.614287\pi$
$643$	−17.8201	−0.702757	−0.351378	+	0.936234i	$0.614287\pi$
$644$	6.85758	0.270227
$645$	0	0
$646$	9.95995	0.391869
$647$	−33.2373	−1.30669	−0.653347	−	0.757059i	$-0.726636\pi$
$647$	−33.2373	−1.30669	−0.653347	+	0.757059i	$0.726636\pi$
$648$	0	0
$649$	7.40818	0.290796
$650$	−3.59384	−0.140962
$651$	0	0
$652$	31.1077	1.21827
$653$	−2.86743	−0.112211	−0.0561056	−	0.998425i	$-0.517868\pi$
$653$	−2.86743	−0.112211	−0.0561056	+	0.998425i	$0.517868\pi$
$654$	0	0
$655$	−4.31441	−0.168578
$656$	2.30452	0.0899763
$657$	0	0
$658$	25.7172	1.00256
$659$	−48.5269	−1.89034	−0.945170	−	0.326580i	$-0.894104\pi$
$659$	−48.5269	−1.89034	−0.945170	+	0.326580i	$0.894104\pi$
$660$	0	0
$661$	−7.84015	−0.304946	−0.152473	−	0.988308i	$-0.548724\pi$
$661$	−7.84015	−0.304946	−0.152473	+	0.988308i	$0.548724\pi$
$662$	17.0178	0.661416
$663$	0	0
$664$	2.51825	0.0977269
$665$	17.9915	0.697680
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	11.0904	0.429099
$669$	0	0
$670$	−1.25678	−0.0485535
$671$	−0.424102	−0.0163723
$672$	0	0
$673$	43.6667	1.68323	0.841614	−	0.540079i	$-0.181606\pi$
$673$	43.6667	1.68323	0.841614	+	0.540079i	$0.181606\pi$
$674$	−0.376417	−0.0144991
$675$	0	0
$676$	5.65608	0.217542
$677$	−35.0350	−1.34650	−0.673252	−	0.739413i	$-0.735103\pi$
$677$	−35.0350	−1.34650	−0.673252	+	0.739413i	$0.735103\pi$
$678$	0	0
$679$	−37.7591	−1.44906
$680$	28.4341	1.09040
$681$	0	0
$682$	−1.91314	−0.0732579
$683$	−34.9144	−1.33596	−0.667982	−	0.744178i	$-0.732842\pi$
$683$	−34.9144	−1.33596	−0.667982	+	0.744178i	$0.732842\pi$
$684$	0	0
$685$	37.5131	1.43330
$686$	17.3541	0.662583
$687$	0	0
$688$	9.47390	0.361189
$689$	35.3762	1.34773
$690$	0	0
$691$	42.8296	1.62931	0.814657	−	0.579943i	$-0.196925\pi$
$691$	42.8296	1.62931	0.814657	+	0.579943i	$0.196925\pi$
$692$	−23.1941	−0.881708
$693$	0	0
$694$	6.77658	0.257235
$695$	−26.9945	−1.02396
$696$	0	0
$697$	10.3468	0.391912
$698$	5.42048	0.205168
$699$	0	0
$700$	−11.9246	−0.450709
$701$	31.6050	1.19370	0.596852	−	0.802351i	$-0.296418\pi$
$701$	31.6050	1.19370	0.596852	+	0.802351i	$0.296418\pi$
$702$	0	0
$703$	19.3516	0.729860
$704$	1.18594	0.0446967
$705$	0	0
$706$	5.70726	0.214796
$707$	−76.0135	−2.85878
$708$	0	0
$709$	−50.5318	−1.89776	−0.948881	−	0.315635i	$-0.897782\pi$
$709$	−50.5318	−1.89776	−0.948881	+	0.315635i	$0.897782\pi$
$710$	−3.76282	−0.141216
$711$	0	0
$712$	−6.68627	−0.250579
$713$	2.34985	0.0880024
$714$	0	0
$715$	−6.63918	−0.248291
$716$	15.4240	0.576421
$717$	0	0
$718$	12.0494	0.449680
$719$	−15.3215	−0.571394	−0.285697	−	0.958320i	$-0.592225\pi$
$719$	−15.3215	−0.571394	−0.285697	+	0.958320i	$0.592225\pi$
$720$	0	0
$721$	13.3876	0.498582
$722$	9.45716	0.351959
$723$	0	0
$724$	17.9871	0.668487
$725$	1.73890	0.0645811
$726$	0	0
$727$	42.4863	1.57573	0.787865	−	0.615848i	$-0.211186\pi$
$727$	42.4863	1.57573	0.787865	+	0.615848i	$0.211186\pi$
$728$	32.5513	1.20643
$729$	0	0
$730$	9.84919	0.364535
$731$	42.5357	1.57324
$732$	0	0
$733$	−20.4112	−0.753904	−0.376952	−	0.926233i	$-0.623028\pi$
$733$	−20.4112	−0.753904	−0.376952	+	0.926233i	$0.623028\pi$
$734$	−19.7895	−0.730445
$735$	0	0
$736$	−5.78292	−0.213161
$737$	1.23801	0.0456026
$738$	0	0
$739$	−39.5199	−1.45376	−0.726881	−	0.686763i	$-0.759031\pi$
$739$	−39.5199	−1.45376	−0.726881	+	0.686763i	$0.759031\pi$
$740$	24.0539	0.884239
$741$	0	0
$742$	−34.8324	−1.27874
$743$	1.06179	0.0389533	0.0194767	−	0.999810i	$-0.493800\pi$
$743$	1.06179	0.0389533	0.0194767	+	0.999810i	$0.493800\pi$
$744$	0	0
$745$	−8.07526	−0.295855
$746$	20.0914	0.735597
$747$	0	0
$748$	−12.1953	−0.445903
$749$	22.4123	0.818928
$750$	0	0
$751$	−12.3874	−0.452021	−0.226010	−	0.974125i	$-0.572568\pi$
$751$	−12.3874	−0.452021	−0.226010	+	0.974125i	$0.572568\pi$
$752$	12.5116	0.456250
$753$	0	0
$754$	−2.06674	−0.0752661
$755$	17.4566	0.635311
$756$	0	0
$757$	−5.54913	−0.201687	−0.100843	−	0.994902i	$-0.532154\pi$
$757$	−5.54913	−0.201687	−0.100843	+	0.994902i	$0.532154\pi$
$758$	−5.59943	−0.203381
$759$	0	0
$760$	−9.69703	−0.351748
$761$	42.7973	1.55140	0.775700	−	0.631102i	$-0.217397\pi$
$761$	42.7973	1.55140	0.775700	+	0.631102i	$0.217397\pi$
$762$	0	0
$763$	−69.8033	−2.52705
$764$	7.43106	0.268846
$765$	0	0
$766$	−16.8932	−0.610375
$767$	−18.8057	−0.679033
$768$	0	0
$769$	−19.0214	−0.685931	−0.342965	−	0.939348i	$-0.611431\pi$
$769$	−19.0214	−0.685931	−0.342965	+	0.939348i	$0.611431\pi$
$770$	6.53711	0.235581
$771$	0	0
$772$	40.1246	1.44412
$773$	−48.4032	−1.74094	−0.870472	−	0.492219i	$-0.836186\pi$
$773$	−48.4032	−1.74094	−0.870472	+	0.492219i	$0.836186\pi$
$774$	0	0
$775$	−4.08615	−0.146779
$776$	20.3514	0.730572
$777$	0	0
$778$	14.7188	0.527695
$779$	−3.52862	−0.126426
$780$	0	0
$781$	3.70662	0.132633
$782$	−4.44497	−0.158952
$783$	0	0
$784$	18.6867	0.667381
$785$	−10.4764	−0.373917
$786$	0	0
$787$	10.9688	0.390995	0.195497	−	0.980704i	$-0.437368\pi$
$787$	10.9688	0.390995	0.195497	+	0.980704i	$0.437368\pi$
$788$	15.3848	0.548062
$789$	0	0
$790$	−5.04800	−0.179600
$791$	33.2461	1.18210
$792$	0	0
$793$	1.07658	0.0382306
$794$	17.2206	0.611135
$795$	0	0
$796$	−28.2970	−1.00296
$797$	−18.8889	−0.669080	−0.334540	−	0.942382i	$-0.608581\pi$
$797$	−18.8889	−0.669080	−0.334540	+	0.942382i	$0.608581\pi$
$798$	0	0
$799$	56.1742	1.98730
$800$	10.0559	0.355530
$801$	0	0
$802$	7.73146	0.273007
$803$	−9.70211	−0.342380
$804$	0	0
$805$	−8.02931	−0.282996
$806$	4.85651	0.171063
$807$	0	0
$808$	40.9697	1.44131
$809$	46.2240	1.62515	0.812574	−	0.582858i	$-0.198066\pi$
$809$	46.2240	1.62515	0.812574	+	0.582858i	$0.198066\pi$
$810$	0	0
$811$	−26.7571	−0.939567	−0.469784	−	0.882782i	$-0.655668\pi$
$811$	−26.7571	−0.939567	−0.469784	+	0.882782i	$0.655668\pi$
$812$	−6.85758	−0.240654
$813$	0	0
$814$	7.03130	0.246447
$815$	−36.4230	−1.27584
$816$	0	0
$817$	−14.5062	−0.507508
$818$	0.863894	0.0302053
$819$	0	0
$820$	−4.38605	−0.153168
$821$	−15.1449	−0.528560	−0.264280	−	0.964446i	$-0.585134\pi$
$821$	−15.1449	−0.528560	−0.264280	+	0.964446i	$0.585134\pi$
$822$	0	0
$823$	−39.7433	−1.38537	−0.692683	−	0.721242i	$-0.743571\pi$
$823$	−39.7433	−1.38537	−0.692683	+	0.721242i	$0.743571\pi$
$824$	−7.21566	−0.251369
$825$	0	0
$826$	18.5166	0.644273
$827$	52.1210	1.81242	0.906212	−	0.422824i	$-0.138961\pi$
$827$	52.1210	1.81242	0.906212	+	0.422824i	$0.138961\pi$
$828$	0	0
$829$	33.0195	1.14681	0.573407	−	0.819271i	$-0.305622\pi$
$829$	33.0195	1.14681	0.573407	+	0.819271i	$0.305622\pi$
$830$	−1.28379	−0.0445608
$831$	0	0
$832$	−3.01050	−0.104370
$833$	83.8991	2.90693
$834$	0	0
$835$	−12.9853	−0.449376
$836$	4.15902	0.143843
$837$	0	0
$838$	−7.11538	−0.245797
$839$	28.7654	0.993092	0.496546	−	0.868011i	$-0.334601\pi$
$839$	28.7654	0.993092	0.496546	+	0.868011i	$0.334601\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	12.7143	0.438165
$843$	0	0
$844$	25.5577	0.879732
$845$	−6.62251	−0.227821
$846$	0	0
$847$	42.4695	1.45927
$848$	−16.9461	−0.581932
$849$	0	0
$850$	7.72935	0.265115
$851$	−8.63631	−0.296049
$852$	0	0
$853$	36.1981	1.23940	0.619700	−	0.784839i	$-0.287254\pi$
$853$	36.1981	1.23940	0.619700	+	0.784839i	$0.287254\pi$
$854$	−1.06003	−0.0362735
$855$	0	0
$856$	−12.0798	−0.412878
$857$	−46.2205	−1.57886	−0.789432	−	0.613839i	$-0.789625\pi$
$857$	−46.2205	−1.57886	−0.789432	+	0.613839i	$0.789625\pi$
$858$	0	0
$859$	44.6583	1.52372	0.761860	−	0.647742i	$-0.224286\pi$
$859$	44.6583	1.52372	0.761860	+	0.647742i	$0.224286\pi$
$860$	−18.0311	−0.614856
$861$	0	0
$862$	3.66952	0.124984
$863$	5.47987	0.186537	0.0932685	−	0.995641i	$-0.470268\pi$
$863$	5.47987	0.186537	0.0932685	+	0.995641i	$0.470268\pi$
$864$	0	0
$865$	27.1572	0.923372
$866$	5.43428	0.184664
$867$	0	0
$868$	16.1143	0.546954
$869$	4.97261	0.168684
$870$	0	0
$871$	−3.14269	−0.106486
$872$	37.6225	1.27406
$873$	0	0
$874$	1.51589	0.0512759
$875$	54.1087	1.82921
$876$	0	0
$877$	20.0817	0.678111	0.339056	−	0.940766i	$-0.389893\pi$
$877$	20.0817	0.678111	0.339056	+	0.940766i	$0.389893\pi$
$878$	8.31748	0.280701
$879$	0	0
$880$	3.18034	0.107209
$881$	−3.95781	−0.133342	−0.0666709	−	0.997775i	$-0.521238\pi$
$881$	−3.95781	−0.133342	−0.0666709	+	0.997775i	$0.521238\pi$
$882$	0	0
$883$	−11.7593	−0.395731	−0.197866	−	0.980229i	$-0.563401\pi$
$883$	−11.7593	−0.395731	−0.197866	+	0.980229i	$0.563401\pi$
$884$	30.9577	1.04122
$885$	0	0
$886$	3.52603	0.118459
$887$	47.1597	1.58347	0.791734	−	0.610867i	$-0.209179\pi$
$887$	47.1597	1.58347	0.791734	+	0.610867i	$0.209179\pi$
$888$	0	0
$889$	59.1534	1.98394
$890$	3.40861	0.114257
$891$	0	0
$892$	15.2567	0.510834
$893$	−19.1574	−0.641078
$894$	0	0
$895$	−18.0594	−0.603659
$896$	−48.4606	−1.61896
$897$	0	0
$898$	17.0696	0.569619
$899$	−2.34985	−0.0783718
$900$	0	0
$901$	−76.0844	−2.53474
$902$	−1.28211	−0.0426895
$903$	0	0
$904$	−17.9190	−0.595976
$905$	−21.0605	−0.700076
$906$	0	0
$907$	10.5576	0.350559	0.175279	−	0.984519i	$-0.443917\pi$
$907$	10.5576	0.350559	0.175279	+	0.984519i	$0.443917\pi$
$908$	18.2938	0.607103
$909$	0	0
$910$	−16.5945	−0.550101
$911$	−4.94764	−0.163923	−0.0819614	−	0.996636i	$-0.526118\pi$
$911$	−4.94764	−0.163923	−0.0819614	+	0.996636i	$0.526118\pi$
$912$	0	0
$913$	1.26461	0.0418526
$914$	13.8790	0.459075
$915$	0	0
$916$	41.0999	1.35798
$917$	10.6227	0.350793
$918$	0	0
$919$	24.8410	0.819428	0.409714	−	0.912214i	$-0.365629\pi$
$919$	24.8410	0.819428	0.409714	+	0.912214i	$0.365629\pi$
$920$	4.32763	0.142678
$921$	0	0
$922$	12.0026	0.395283
$923$	−9.40927	−0.309710
$924$	0	0
$925$	15.0177	0.493778
$926$	−17.2429	−0.566638
$927$	0	0
$928$	5.78292	0.189833
$929$	11.1501	0.365822	0.182911	−	0.983130i	$-0.441448\pi$
$929$	11.1501	0.365822	0.182911	+	0.983130i	$0.441448\pi$
$930$	0	0
$931$	−28.6126	−0.937740
$932$	−7.87369	−0.257911
$933$	0	0
$934$	−2.66768	−0.0872892
$935$	14.2790	0.466974
$936$	0	0
$937$	−26.2983	−0.859129	−0.429564	−	0.903036i	$-0.641333\pi$
$937$	−26.2983	−0.859129	−0.429564	+	0.903036i	$0.641333\pi$
$938$	3.09437	0.101035
$939$	0	0
$940$	−23.8125	−0.776679
$941$	44.4861	1.45020	0.725102	−	0.688641i	$-0.241793\pi$
$941$	44.4861	1.45020	0.725102	+	0.688641i	$0.241793\pi$
$942$	0	0
$943$	1.57477	0.0512815
$944$	9.00840	0.293198
$945$	0	0
$946$	−5.27075	−0.171367
$947$	−52.8987	−1.71898	−0.859488	−	0.511157i	$-0.829217\pi$
$947$	−52.8987	−1.71898	−0.859488	+	0.511157i	$0.829217\pi$
$948$	0	0
$949$	24.6288	0.799485
$950$	−2.63599	−0.0855227
$951$	0	0
$952$	−70.0088	−2.26900
$953$	−51.7818	−1.67738	−0.838688	−	0.544612i	$-0.816677\pi$
$953$	−51.7818	−1.67738	−0.838688	+	0.544612i	$0.816677\pi$
$954$	0	0
$955$	−8.70077	−0.281550
$956$	15.5688	0.503530
$957$	0	0
$958$	−0.395424	−0.0127756
$959$	−92.3627	−2.98255
$960$	0	0
$961$	−25.4782	−0.821878
$962$	−17.8490	−0.575474
$963$	0	0
$964$	−5.19229	−0.167232
$965$	−46.9805	−1.51236
$966$	0	0
$967$	−16.0172	−0.515078	−0.257539	−	0.966268i	$-0.582912\pi$
$967$	−16.0172	−0.515078	−0.257539	+	0.966268i	$0.582912\pi$
$968$	−22.8902	−0.735718
$969$	0	0
$970$	−10.3750	−0.333121
$971$	21.5095	0.690272	0.345136	−	0.938553i	$-0.387833\pi$
$971$	21.5095	0.690272	0.345136	+	0.938553i	$0.387833\pi$
$972$	0	0
$973$	66.4645	2.13075
$974$	7.33499	0.235028
$975$	0	0
$976$	−0.515711	−0.0165075
$977$	48.4917	1.55139	0.775694	−	0.631109i	$-0.217400\pi$
$977$	48.4917	1.55139	0.775694	+	0.631109i	$0.217400\pi$
$978$	0	0
$979$	−3.35771	−0.107313
$980$	−35.5652	−1.13609
$981$	0	0
$982$	3.20207	0.102182
$983$	−23.9560	−0.764077	−0.382038	−	0.924146i	$-0.624778\pi$
$983$	−23.9560	−0.764077	−0.382038	+	0.924146i	$0.624778\pi$
$984$	0	0
$985$	−18.0136	−0.573960
$986$	4.44497	0.141557
$987$	0	0
$988$	−10.5577	−0.335885
$989$	6.47389	0.205858
$990$	0	0
$991$	54.4244	1.72885	0.864425	−	0.502762i	$-0.167683\pi$
$991$	54.4244	1.72885	0.864425	+	0.502762i	$0.167683\pi$
$992$	−13.5890	−0.431450
$993$	0	0
$994$	9.26461	0.293856
$995$	33.1320	1.05035
$996$	0	0
$997$	−13.9042	−0.440350	−0.220175	−	0.975460i	$-0.570663\pi$
$997$	−13.9042	−0.440350	−0.220175	+	0.975460i	$0.570663\pi$
$998$	−16.9427	−0.536312
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.k.1.5		10
3.2	odd	2		2001.2.a.k.1.6	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.k.1.6	✓	10	3.2	odd	2
6003.2.a.k.1.5		10	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)

\(f(q)\)	\(=\)	\(q-0.676519 q^{2} -1.54232 q^{4} +1.80585 q^{5} -4.44627 q^{7} +2.39645 q^{8} +O(q^{10})\)	\(q-0.676519 q^{2} -1.54232 q^{4} +1.80585 q^{5} -4.44627 q^{7} +2.39645 q^{8} -1.22169 q^{10} +1.20345 q^{11} -3.05496 q^{13} +3.00799 q^{14} +1.46340 q^{16} +6.57035 q^{17} -2.24073 q^{19} -2.78521 q^{20} -0.814156 q^{22} +1.00000 q^{23} -1.73890 q^{25} +2.06674 q^{26} +6.85758 q^{28} -1.00000 q^{29} +2.34985 q^{31} -5.78292 q^{32} -4.44497 q^{34} -8.02931 q^{35} -8.63631 q^{37} +1.51589 q^{38} +4.32763 q^{40} +1.57477 q^{41} +6.47389 q^{43} -1.85610 q^{44} -0.676519 q^{46} +8.54965 q^{47} +12.7693 q^{49} +1.17640 q^{50} +4.71172 q^{52} -11.5800 q^{53} +2.17325 q^{55} -10.6553 q^{56} +0.676519 q^{58} +6.15579 q^{59} -0.352405 q^{61} -1.58972 q^{62} +0.985449 q^{64} -5.51680 q^{65} +1.02872 q^{67} -10.1336 q^{68} +5.43198 q^{70} +3.08000 q^{71} -8.06192 q^{73} +5.84263 q^{74} +3.45592 q^{76} -5.35086 q^{77} +4.13197 q^{79} +2.64269 q^{80} -1.06536 q^{82} +1.05082 q^{83} +11.8651 q^{85} -4.37971 q^{86} +2.88400 q^{88} -2.79007 q^{89} +13.5832 q^{91} -1.54232 q^{92} -5.78400 q^{94} -4.04642 q^{95} +8.49231 q^{97} -8.63870 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) \) 10 * q - 3 * q^2 + 17 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8	\( 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8} - 4 q^{10} - 9 q^{11} - 16 q^{13} - 16 q^{14} + 27 q^{16} + q^{19} - 21 q^{20} + 17 q^{22} + 10 q^{23} - 4 q^{25} - 28 q^{26} - 14 q^{28} - 10 q^{29} + 17 q^{31} - 21 q^{32} - 3 q^{34} - 29 q^{35} + q^{37} - 32 q^{38} + 13 q^{40} - 5 q^{43} - 33 q^{44} - 3 q^{46} - 15 q^{47} + 31 q^{49} + 22 q^{50} - 21 q^{52} - 35 q^{53} - 20 q^{55} - 18 q^{56} + 3 q^{58} - 49 q^{59} + 8 q^{61} - 15 q^{62} + 12 q^{64} + 3 q^{65} + 35 q^{67} + 18 q^{68} - 16 q^{70} - 30 q^{71} - 15 q^{73} - 23 q^{74} + 10 q^{76} - 23 q^{77} + 24 q^{79} - 23 q^{80} - 5 q^{82} - q^{83} + 10 q^{86} + 18 q^{88} - 15 q^{89} + 26 q^{91} + 17 q^{92} + 3 q^{94} - 7 q^{95} - 35 q^{97} - 3 q^{98}+O(q^{100}) \) 10 * q - 3 * q^2 + 17 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8 - 4 * q^10 - 9 * q^11 - 16 * q^13 - 16 * q^14 + 27 * q^16 + q^19 - 21 * q^20 + 17 * q^22 + 10 * q^23 - 4 * q^25 - 28 * q^26 - 14 * q^28 - 10 * q^29 + 17 * q^31 - 21 * q^32 - 3 * q^34 - 29 * q^35 + q^37 - 32 * q^38 + 13 * q^40 - 5 * q^43 - 33 * q^44 - 3 * q^46 - 15 * q^47 + 31 * q^49 + 22 * q^50 - 21 * q^52 - 35 * q^53 - 20 * q^55 - 18 * q^56 + 3 * q^58 - 49 * q^59 + 8 * q^61 - 15 * q^62 + 12 * q^64 + 3 * q^65 + 35 * q^67 + 18 * q^68 - 16 * q^70 - 30 * q^71 - 15 * q^73 - 23 * q^74 + 10 * q^76 - 23 * q^77 + 24 * q^79 - 23 * q^80 - 5 * q^82 - q^83 + 10 * q^86 + 18 * q^88 - 15 * q^89 + 26 * q^91 + 17 * q^92 + 3 * q^94 - 7 * q^95 - 35 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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