LMFDB - Embedded newform 6003.2.a.l.1.7

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:	$10$
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} - 3x^{9} - 10x^{8} + 32x^{7} + 32x^{6} - 118x^{5} - 29x^{4} + 182x^{3} - 28x^{2} - 101x + 43 $ x^10 - 3x^9 - 10x^8 + 32x^7 + 32x^6 - 118x^5 - 29x^4 + 182x^3 - 28x^2 - 101*x + 43
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.21378$ 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)

$f(q)$	$=$	$q+1.21378 q^{2} -0.526738 q^{4} -2.40259 q^{5} -0.534912 q^{7} -3.06690 q^{8} +O(q^{10})$	\(q+1.21378 q^{2} -0.526738 q^{4} -2.40259 q^{5} -0.534912 q^{7} -3.06690 q^{8} -2.91621 q^{10} +0.714504 q^{11} -5.63589 q^{13} -0.649266 q^{14} -2.66907 q^{16} -4.81730 q^{17} +2.70156 q^{19} +1.26553 q^{20} +0.867251 q^{22} -1.00000 q^{23} +0.772423 q^{25} -6.84073 q^{26} +0.281759 q^{28} -1.00000 q^{29} -4.10740 q^{31} +2.89415 q^{32} -5.84714 q^{34} +1.28517 q^{35} -6.95130 q^{37} +3.27910 q^{38} +7.36850 q^{40} +7.10120 q^{41} -3.87302 q^{43} -0.376357 q^{44} -1.21378 q^{46} +10.0076 q^{47} -6.71387 q^{49} +0.937552 q^{50} +2.96864 q^{52} +11.7084 q^{53} -1.71666 q^{55} +1.64052 q^{56} -1.21378 q^{58} -10.4872 q^{59} -10.0627 q^{61} -4.98547 q^{62} +8.85099 q^{64} +13.5407 q^{65} +6.89573 q^{67} +2.53746 q^{68} +1.55992 q^{70} +4.11442 q^{71} +2.69054 q^{73} -8.43735 q^{74} -1.42301 q^{76} -0.382197 q^{77} -4.57015 q^{79} +6.41267 q^{80} +8.61929 q^{82} +4.36420 q^{83} +11.5740 q^{85} -4.70099 q^{86} -2.19132 q^{88} -5.25895 q^{89} +3.01471 q^{91} +0.526738 q^{92} +12.1470 q^{94} -6.49072 q^{95} +6.77751 q^{97} -8.14916 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 3 q^{2} + 9 q^{4} + 10 q^{5} + q^{7} + 9 q^{8}+O(q^{10}) $ 10 * q + 3 * q^2 + 9 * q^4 + 10 * q^5 + q^7 + 9 * q^8	$ 10 q + 3 q^{2} + 9 q^{4} + 10 q^{5} + q^{7} + 9 q^{8} - 6 q^{10} - 13 q^{13} + 12 q^{14} - 5 q^{16} + 22 q^{17} - 2 q^{19} - 3 q^{20} + 3 q^{22} - 10 q^{23} + 10 q^{25} + 25 q^{26} + 19 q^{28} - 10 q^{29} - 22 q^{31} + 31 q^{32} + 13 q^{34} + 15 q^{35} - 9 q^{37} + 10 q^{38} - 6 q^{40} + 25 q^{41} + 3 q^{43} + 27 q^{44} - 3 q^{46} + 17 q^{47} + 17 q^{49} - 2 q^{50} - 18 q^{52} + 43 q^{53} - 11 q^{55} + 7 q^{56} - 3 q^{58} + 7 q^{59} - 6 q^{61} - 3 q^{62} + 33 q^{64} - 11 q^{65} + 11 q^{67} + 51 q^{68} + 34 q^{70} + 17 q^{71} - 44 q^{73} - 9 q^{74} + 24 q^{76} + 71 q^{77} + 5 q^{79} - 38 q^{80} + 33 q^{82} + 32 q^{83} + 16 q^{85} + 9 q^{86} + 18 q^{88} + 10 q^{89} - 3 q^{91} - 9 q^{92} + 47 q^{94} + 8 q^{95} + 6 q^{97} + 73 q^{98}+O(q^{100}) $ 10 * q + 3 * q^2 + 9 * q^4 + 10 * q^5 + q^7 + 9 * q^8 - 6 * q^10 - 13 * q^13 + 12 * q^14 - 5 * q^16 + 22 * q^17 - 2 * q^19 - 3 * q^20 + 3 * q^22 - 10 * q^23 + 10 * q^25 + 25 * q^26 + 19 * q^28 - 10 * q^29 - 22 * q^31 + 31 * q^32 + 13 * q^34 + 15 * q^35 - 9 * q^37 + 10 * q^38 - 6 * q^40 + 25 * q^41 + 3 * q^43 + 27 * q^44 - 3 * q^46 + 17 * q^47 + 17 * q^49 - 2 * q^50 - 18 * q^52 + 43 * q^53 - 11 * q^55 + 7 * q^56 - 3 * q^58 + 7 * q^59 - 6 * q^61 - 3 * q^62 + 33 * q^64 - 11 * q^65 + 11 * q^67 + 51 * q^68 + 34 * q^70 + 17 * q^71 - 44 * q^73 - 9 * q^74 + 24 * q^76 + 71 * q^77 + 5 * q^79 - 38 * q^80 + 33 * q^82 + 32 * q^83 + 16 * q^85 + 9 * q^86 + 18 * q^88 + 10 * q^89 - 3 * q^91 - 9 * q^92 + 47 * q^94 + 8 * q^95 + 6 * q^97 + 73 * 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 $

$2$	1.21378	0.858272	0.429136	−	0.903240i	$-0.358818\pi$
$2$	1.21378	0.858272	0.429136	+	0.903240i	$0.358818\pi$
$3$	0	0
$3$	0	0
$4$	−0.526738	−0.263369
$5$	−2.40259	−1.07447	−0.537235	−	0.843433i	$-0.680531\pi$
$5$	−2.40259	−1.07447	−0.537235	+	0.843433i	$0.680531\pi$
$6$	0	0
$7$	−0.534912	−0.202178	−0.101089	−	0.994877i	$-0.532233\pi$
$7$	−0.534912	−0.202178	−0.101089	+	0.994877i	$0.532233\pi$
$8$	−3.06690	−1.08431
$9$	0	0
$10$	−2.91621	−0.922187
$11$	0.714504	0.215431	0.107716	−	0.994182i	$-0.465646\pi$
$11$	0.714504	0.215431	0.107716	+	0.994182i	$0.465646\pi$
$12$	0	0
$13$	−5.63589	−1.56311	−0.781557	−	0.623834i	$-0.785574\pi$
$13$	−5.63589	−1.56311	−0.781557	+	0.623834i	$0.785574\pi$
$14$	−0.649266	−0.173524
$15$	0	0
$16$	−2.66907	−0.667267
$17$	−4.81730	−1.16837	−0.584183	−	0.811622i	$-0.698585\pi$
$17$	−4.81730	−1.16837	−0.584183	+	0.811622i	$0.698585\pi$
$18$	0	0
$19$	2.70156	0.619780	0.309890	−	0.950772i	$-0.399708\pi$
$19$	2.70156	0.619780	0.309890	+	0.950772i	$0.399708\pi$
$20$	1.26553	0.282982
$21$	0	0
$22$	0.867251	0.184898
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0.772423	0.154485
$26$	−6.84073	−1.34158
$27$	0	0
$28$	0.281759	0.0532474
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−4.10740	−0.737710	−0.368855	−	0.929487i	$-0.620250\pi$
$31$	−4.10740	−0.737710	−0.368855	+	0.929487i	$0.620250\pi$
$32$	2.89415	0.511617
$33$	0	0
$34$	−5.84714	−1.00278
$35$	1.28517	0.217234
$36$	0	0
$37$	−6.95130	−1.14279	−0.571393	−	0.820676i	$-0.693597\pi$
$37$	−6.95130	−1.14279	−0.571393	+	0.820676i	$0.693597\pi$
$38$	3.27910	0.531940
$39$	0	0
$40$	7.36850	1.16506
$41$	7.10120	1.10902	0.554510	−	0.832177i	$-0.312906\pi$
$41$	7.10120	1.10902	0.554510	+	0.832177i	$0.312906\pi$
$42$	0	0
$43$	−3.87302	−0.590630	−0.295315	−	0.955400i	$-0.595425\pi$
$43$	−3.87302	−0.590630	−0.295315	+	0.955400i	$0.595425\pi$
$44$	−0.376357	−0.0567379
$45$	0	0
$46$	−1.21378	−0.178962
$47$	10.0076	1.45976	0.729880	−	0.683575i	$-0.239576\pi$
$47$	10.0076	1.45976	0.729880	+	0.683575i	$0.239576\pi$
$48$	0	0
$49$	−6.71387	−0.959124
$50$	0.937552	0.132590
$51$	0	0
$52$	2.96864	0.411676
$53$	11.7084	1.60827	0.804134	−	0.594448i	$-0.202630\pi$
$53$	11.7084	1.60827	0.804134	+	0.594448i	$0.202630\pi$
$54$	0	0
$55$	−1.71666	−0.231474
$56$	1.64052	0.219224
$57$	0	0
$58$	−1.21378	−0.159377
$59$	−10.4872	−1.36532	−0.682658	−	0.730738i	$-0.739176\pi$
$59$	−10.4872	−1.36532	−0.682658	+	0.730738i	$0.739176\pi$
$60$	0	0
$61$	−10.0627	−1.28839	−0.644197	−	0.764860i	$-0.722808\pi$
$61$	−10.0627	−1.28839	−0.644197	+	0.764860i	$0.722808\pi$
$62$	−4.98547	−0.633156
$63$	0	0
$64$	8.85099	1.10637
$65$	13.5407	1.67952
$66$	0	0
$67$	6.89573	0.842448	0.421224	−	0.906957i	$-0.361601\pi$
$67$	6.89573	0.842448	0.421224	+	0.906957i	$0.361601\pi$
$68$	2.53746	0.307712
$69$	0	0
$70$	1.55992	0.186446
$71$	4.11442	0.488292	0.244146	−	0.969738i	$-0.421492\pi$
$71$	4.11442	0.488292	0.244146	+	0.969738i	$0.421492\pi$
$72$	0	0
$73$	2.69054	0.314904	0.157452	−	0.987527i	$-0.449672\pi$
$73$	2.69054	0.314904	0.157452	+	0.987527i	$0.449672\pi$
$74$	−8.43735	−0.980822
$75$	0	0
$76$	−1.42301	−0.163231
$77$	−0.382197	−0.0435554
$78$	0	0
$79$	−4.57015	−0.514182	−0.257091	−	0.966387i	$-0.582764\pi$
$79$	−4.57015	−0.514182	−0.257091	+	0.966387i	$0.582764\pi$
$80$	6.41267	0.716959
$81$	0	0
$82$	8.61929	0.951841
$83$	4.36420	0.479033	0.239517	−	0.970892i	$-0.423011\pi$
$83$	4.36420	0.479033	0.239517	+	0.970892i	$0.423011\pi$
$84$	0	0
$85$	11.5740	1.25537
$86$	−4.70099	−0.506921
$87$	0	0
$88$	−2.19132	−0.233595
$89$	−5.25895	−0.557448	−0.278724	−	0.960371i	$-0.589911\pi$
$89$	−5.25895	−0.557448	−0.278724	+	0.960371i	$0.589911\pi$
$90$	0	0
$91$	3.01471	0.316027
$92$	0.526738	0.0549163
$93$	0	0
$94$	12.1470	1.25287
$95$	−6.49072	−0.665934
$96$	0	0
$97$	6.77751	0.688152	0.344076	−	0.938942i	$-0.388192\pi$
$97$	6.77751	0.688152	0.344076	+	0.938942i	$0.388192\pi$
$98$	−8.14916	−0.823189
$99$	0	0
$100$	−0.406865	−0.0406865
$101$	13.1655	1.31002	0.655010	−	0.755620i	$-0.272664\pi$
$101$	13.1655	1.31002	0.655010	+	0.755620i	$0.272664\pi$
$102$	0	0
$103$	5.42406	0.534449	0.267224	−	0.963634i	$-0.413894\pi$
$103$	5.42406	0.534449	0.267224	+	0.963634i	$0.413894\pi$
$104$	17.2847	1.69491
$105$	0	0
$106$	14.2114	1.38033
$107$	16.4301	1.58836	0.794181	−	0.607682i	$-0.207900\pi$
$107$	16.4301	1.58836	0.794181	+	0.607682i	$0.207900\pi$
$108$	0	0
$109$	−4.86837	−0.466306	−0.233153	−	0.972440i	$-0.574904\pi$
$109$	−4.86837	−0.466306	−0.233153	+	0.972440i	$0.574904\pi$
$110$	−2.08364	−0.198668
$111$	0	0
$112$	1.42772	0.134907
$113$	−17.9726	−1.69072	−0.845361	−	0.534195i	$-0.820615\pi$
$113$	−17.9726	−1.69072	−0.845361	+	0.534195i	$0.820615\pi$
$114$	0	0
$115$	2.40259	0.224042
$116$	0.526738	0.0489064
$117$	0	0
$118$	−12.7291	−1.17181
$119$	2.57683	0.236218
$120$	0	0
$121$	−10.4895	−0.953589
$122$	−12.2139	−1.10579
$123$	0	0
$124$	2.16352	0.194290
$125$	10.1571	0.908480
$126$	0	0
$127$	−11.6836	−1.03675	−0.518377	−	0.855152i	$-0.673464\pi$
$127$	−11.6836	−1.03675	−0.518377	+	0.855152i	$0.673464\pi$
$128$	4.95487	0.437953
$129$	0	0
$130$	16.4354	1.44148
$131$	−5.45861	−0.476921	−0.238461	−	0.971152i	$-0.576643\pi$
$131$	−5.45861	−0.476921	−0.238461	+	0.971152i	$0.576643\pi$
$132$	0	0
$133$	−1.44510	−0.125306
$134$	8.36990	0.723050
$135$	0	0
$136$	14.7742	1.26688
$137$	12.5983	1.07634	0.538172	−	0.842835i	$-0.319115\pi$
$137$	12.5983	1.07634	0.538172	+	0.842835i	$0.319115\pi$
$138$	0	0
$139$	23.1565	1.96411	0.982056	−	0.188592i	$-0.0603922\pi$
$139$	23.1565	1.96411	0.982056	+	0.188592i	$0.0603922\pi$
$140$	−0.676950	−0.0572127
$141$	0	0
$142$	4.99400	0.419087
$143$	−4.02686	−0.336743
$144$	0	0
$145$	2.40259	0.199524
$146$	3.26572	0.270273
$147$	0	0
$148$	3.66152	0.300975
$149$	−6.55670	−0.537146	−0.268573	−	0.963259i	$-0.586552\pi$
$149$	−6.55670	−0.537146	−0.268573	+	0.963259i	$0.586552\pi$
$150$	0	0
$151$	19.9928	1.62699	0.813495	−	0.581572i	$-0.197562\pi$
$151$	19.9928	1.62699	0.813495	+	0.581572i	$0.197562\pi$
$152$	−8.28542	−0.672036
$153$	0	0
$154$	−0.463903	−0.0373824
$155$	9.86837	0.792647
$156$	0	0
$157$	−19.0161	−1.51765	−0.758825	−	0.651294i	$-0.774226\pi$
$157$	−19.0161	−1.51765	−0.758825	+	0.651294i	$0.774226\pi$
$158$	−5.54716	−0.441308
$159$	0	0
$160$	−6.95343	−0.549717
$161$	0.534912	0.0421570
$162$	0	0
$163$	−4.66760	−0.365595	−0.182797	−	0.983151i	$-0.558515\pi$
$163$	−4.66760	−0.365595	−0.182797	+	0.983151i	$0.558515\pi$
$164$	−3.74047	−0.292082
$165$	0	0
$166$	5.29718	0.411141
$167$	8.33614	0.645070	0.322535	−	0.946558i	$-0.395465\pi$
$167$	8.33614	0.645070	0.322535	+	0.946558i	$0.395465\pi$
$168$	0	0
$169$	18.7632	1.44332
$170$	14.0483	1.07745
$171$	0	0
$172$	2.04007	0.155554
$173$	9.82552	0.747020	0.373510	−	0.927626i	$-0.378154\pi$
$173$	9.82552	0.747020	0.373510	+	0.927626i	$0.378154\pi$
$174$	0	0
$175$	−0.413179	−0.0312334
$176$	−1.90706	−0.143750
$177$	0	0
$178$	−6.38321	−0.478442
$179$	−8.73945	−0.653217	−0.326608	−	0.945160i	$-0.605906\pi$
$179$	−8.73945	−0.653217	−0.326608	+	0.945160i	$0.605906\pi$
$180$	0	0
$181$	8.70109	0.646747	0.323373	−	0.946271i	$-0.395183\pi$
$181$	8.70109	0.646747	0.323373	+	0.946271i	$0.395183\pi$
$182$	3.65919	0.271237
$183$	0	0
$184$	3.06690	0.226095
$185$	16.7011	1.22789
$186$	0	0
$187$	−3.44198	−0.251702
$188$	−5.27139	−0.384456
$189$	0	0
$190$	−7.87831	−0.571553
$191$	8.49082	0.614374	0.307187	−	0.951649i	$-0.400612\pi$
$191$	8.49082	0.614374	0.307187	+	0.951649i	$0.400612\pi$
$192$	0	0
$193$	−11.8821	−0.855292	−0.427646	−	0.903946i	$-0.640657\pi$
$193$	−11.8821	−0.855292	−0.427646	+	0.903946i	$0.640657\pi$
$194$	8.22640	0.590621
$195$	0	0
$196$	3.53645	0.252604
$197$	0.180384	0.0128518	0.00642592	−	0.999979i	$-0.497955\pi$
$197$	0.180384	0.0128518	0.00642592	+	0.999979i	$0.497955\pi$
$198$	0	0
$199$	20.0610	1.42209	0.711045	−	0.703147i	$-0.248222\pi$
$199$	20.0610	1.42209	0.711045	+	0.703147i	$0.248222\pi$
$200$	−2.36895	−0.167510
$201$	0	0
$202$	15.9801	1.12435
$203$	0.534912	0.0375435
$204$	0	0
$205$	−17.0612	−1.19161
$206$	6.58362	0.458702
$207$	0	0
$208$	15.0426	1.04301
$209$	1.93027	0.133520
$210$	0	0
$211$	15.2829	1.05212	0.526060	−	0.850448i	$-0.323669\pi$
$211$	15.2829	1.05212	0.526060	+	0.850448i	$0.323669\pi$
$212$	−6.16724	−0.423568
$213$	0	0
$214$	19.9426	1.36325
$215$	9.30526	0.634614
$216$	0	0
$217$	2.19710	0.149149
$218$	−5.90913	−0.400217
$219$	0	0
$220$	0.904230	0.0609631
$221$	27.1498	1.82629
$222$	0	0
$223$	19.5962	1.31226	0.656130	−	0.754648i	$-0.272192\pi$
$223$	19.5962	1.31226	0.656130	+	0.754648i	$0.272192\pi$
$224$	−1.54811	−0.103438
$225$	0	0
$226$	−21.8148	−1.45110
$227$	0.683407	0.0453593	0.0226797	−	0.999743i	$-0.492780\pi$
$227$	0.683407	0.0453593	0.0226797	+	0.999743i	$0.492780\pi$
$228$	0	0
$229$	14.2658	0.942712	0.471356	−	0.881943i	$-0.343765\pi$
$229$	14.2658	0.942712	0.471356	+	0.881943i	$0.343765\pi$
$230$	2.91621	0.192289
$231$	0	0
$232$	3.06690	0.201352
$233$	−28.2689	−1.85196	−0.925978	−	0.377577i	$-0.876757\pi$
$233$	−28.2689	−1.85196	−0.925978	+	0.377577i	$0.876757\pi$
$234$	0	0
$235$	−24.0442	−1.56847
$236$	5.52400	0.359582
$237$	0	0
$238$	3.12771	0.202739
$239$	−3.94219	−0.254999	−0.127499	−	0.991839i	$-0.540695\pi$
$239$	−3.94219	−0.254999	−0.127499	+	0.991839i	$0.540695\pi$
$240$	0	0
$241$	−14.9832	−0.965154	−0.482577	−	0.875854i	$-0.660299\pi$
$241$	−14.9832	−0.965154	−0.482577	+	0.875854i	$0.660299\pi$
$242$	−12.7319	−0.818439
$243$	0	0
$244$	5.30040	0.339323
$245$	16.1307	1.03055
$246$	0	0
$247$	−15.2257	−0.968786
$248$	12.5970	0.799910
$249$	0	0
$250$	12.3285	0.779723
$251$	−8.38660	−0.529358	−0.264679	−	0.964337i	$-0.585266\pi$
$251$	−8.38660	−0.529358	−0.264679	+	0.964337i	$0.585266\pi$
$252$	0	0
$253$	−0.714504	−0.0449205
$254$	−14.1813	−0.889817
$255$	0	0
$256$	−11.6879	−0.730492
$257$	−9.82691	−0.612986	−0.306493	−	0.951873i	$-0.599156\pi$
$257$	−9.82691	−0.612986	−0.306493	+	0.951873i	$0.599156\pi$
$258$	0	0
$259$	3.71834	0.231046
$260$	−7.13241	−0.442333
$261$	0	0
$262$	−6.62555	−0.409328
$263$	24.8730	1.53373	0.766866	−	0.641807i	$-0.221815\pi$
$263$	24.8730	1.53373	0.766866	+	0.641807i	$0.221815\pi$
$264$	0	0
$265$	−28.1304	−1.72803
$266$	−1.75403	−0.107546
$267$	0	0
$268$	−3.63225	−0.221875
$269$	−25.5522	−1.55795	−0.778974	−	0.627057i	$-0.784259\pi$
$269$	−25.5522	−1.55795	−0.778974	+	0.627057i	$0.784259\pi$
$270$	0	0
$271$	12.0902	0.734430	0.367215	−	0.930136i	$-0.380311\pi$
$271$	12.0902	0.734430	0.367215	+	0.930136i	$0.380311\pi$
$272$	12.8577	0.779613
$273$	0	0
$274$	15.2915	0.923796
$275$	0.551900	0.0332808
$276$	0	0
$277$	−2.94155	−0.176740	−0.0883702	−	0.996088i	$-0.528166\pi$
$277$	−2.94155	−0.176740	−0.0883702	+	0.996088i	$0.528166\pi$
$278$	28.1069	1.68574
$279$	0	0
$280$	−3.94150	−0.235550
$281$	−24.4105	−1.45621	−0.728103	−	0.685468i	$-0.759598\pi$
$281$	−24.4105	−1.45621	−0.728103	+	0.685468i	$0.759598\pi$
$282$	0	0
$283$	19.7264	1.17261	0.586306	−	0.810090i	$-0.300582\pi$
$283$	19.7264	1.17261	0.586306	+	0.810090i	$0.300582\pi$
$284$	−2.16722	−0.128601
$285$	0	0
$286$	−4.88773	−0.289017
$287$	−3.79852	−0.224219
$288$	0	0
$289$	6.20637	0.365081
$290$	2.91621	0.171246
$291$	0	0
$292$	−1.41721	−0.0829359
$293$	−6.30360	−0.368260	−0.184130	−	0.982902i	$-0.558947\pi$
$293$	−6.30360	−0.368260	−0.184130	+	0.982902i	$0.558947\pi$
$294$	0	0
$295$	25.1964	1.46699
$296$	21.3190	1.23914
$297$	0	0
$298$	−7.95839	−0.461017
$299$	5.63589	0.325932
$300$	0	0
$301$	2.07173	0.119412
$302$	24.2668	1.39640
$303$	0	0
$304$	−7.21064	−0.413559
$305$	24.1765	1.38434
$306$	0	0
$307$	−9.94702	−0.567706	−0.283853	−	0.958868i	$-0.591613\pi$
$307$	−9.94702	−0.567706	−0.283853	+	0.958868i	$0.591613\pi$
$308$	0.201318	0.0114711
$309$	0	0
$310$	11.9780	0.680307
$311$	9.82814	0.557303	0.278651	−	0.960392i	$-0.410113\pi$
$311$	9.82814	0.557303	0.278651	+	0.960392i	$0.410113\pi$
$312$	0	0
$313$	−33.5527	−1.89651	−0.948256	−	0.317508i	$-0.897154\pi$
$313$	−33.5527	−1.89651	−0.948256	+	0.317508i	$0.897154\pi$
$314$	−23.0814	−1.30256
$315$	0	0
$316$	2.40727	0.135420
$317$	18.5817	1.04365	0.521827	−	0.853051i	$-0.325250\pi$
$317$	18.5817	1.04365	0.521827	+	0.853051i	$0.325250\pi$
$318$	0	0
$319$	−0.714504	−0.0400045
$320$	−21.2653	−1.18877
$321$	0	0
$322$	0.649266	0.0361822
$323$	−13.0142	−0.724130
$324$	0	0
$325$	−4.35329	−0.241477
$326$	−5.66544	−0.313780
$327$	0	0
$328$	−21.7787	−1.20253
$329$	−5.35319	−0.295131
$330$	0	0
$331$	18.9220	1.04005	0.520024	−	0.854151i	$-0.325923\pi$
$331$	18.9220	1.04005	0.520024	+	0.854151i	$0.325923\pi$
$332$	−2.29879	−0.126163
$333$	0	0
$334$	10.1182	0.553645
$335$	−16.5676	−0.905185
$336$	0	0
$337$	5.24377	0.285646	0.142823	−	0.989748i	$-0.454382\pi$
$337$	5.24377	0.285646	0.142823	+	0.989748i	$0.454382\pi$
$338$	22.7744	1.23876
$339$	0	0
$340$	−6.09646	−0.330627
$341$	−2.93475	−0.158926
$342$	0	0
$343$	7.33572	0.396092
$344$	11.8782	0.640428
$345$	0	0
$346$	11.9260	0.641146
$347$	−10.0113	−0.537434	−0.268717	−	0.963219i	$-0.586600\pi$
$347$	−10.0113	−0.537434	−0.268717	+	0.963219i	$0.586600\pi$
$348$	0	0
$349$	−11.9713	−0.640807	−0.320404	−	0.947281i	$-0.603819\pi$
$349$	−11.9713	−0.640807	−0.320404	+	0.947281i	$0.603819\pi$
$350$	−0.501508	−0.0268067
$351$	0	0
$352$	2.06788	0.110218
$353$	27.7617	1.47761	0.738803	−	0.673921i	$-0.235391\pi$
$353$	27.7617	1.47761	0.738803	+	0.673921i	$0.235391\pi$
$354$	0	0
$355$	−9.88525	−0.524655
$356$	2.77009	0.146815
$357$	0	0
$358$	−10.6078	−0.560638
$359$	6.14004	0.324059	0.162029	−	0.986786i	$-0.448196\pi$
$359$	6.14004	0.324059	0.162029	+	0.986786i	$0.448196\pi$
$360$	0	0
$361$	−11.7016	−0.615873
$362$	10.5612	0.555085
$363$	0	0
$364$	−1.58796	−0.0832318
$365$	−6.46425	−0.338354
$366$	0	0
$367$	−15.0054	−0.783277	−0.391639	−	0.920119i	$-0.628092\pi$
$367$	−15.0054	−0.783277	−0.391639	+	0.920119i	$0.628092\pi$
$368$	2.66907	0.139135
$369$	0	0
$370$	20.2715	1.05386
$371$	−6.26295	−0.325156
$372$	0	0
$373$	8.48124	0.439142	0.219571	−	0.975597i	$-0.429534\pi$
$373$	8.48124	0.439142	0.219571	+	0.975597i	$0.429534\pi$
$374$	−4.17781	−0.216029
$375$	0	0
$376$	−30.6924	−1.58284
$377$	5.63589	0.290263
$378$	0	0
$379$	−36.4192	−1.87073	−0.935364	−	0.353686i	$-0.884928\pi$
$379$	−36.4192	−1.87073	−0.935364	+	0.353686i	$0.884928\pi$
$380$	3.41891	0.175387
$381$	0	0
$382$	10.3060	0.527300
$383$	19.1703	0.979556	0.489778	−	0.871847i	$-0.337078\pi$
$383$	19.1703	0.979556	0.489778	+	0.871847i	$0.337078\pi$
$384$	0	0
$385$	0.918261	0.0467989
$386$	−14.4223	−0.734073
$387$	0	0
$388$	−3.56997	−0.181238
$389$	3.90670	0.198077	0.0990387	−	0.995084i	$-0.468423\pi$
$389$	3.90670	0.198077	0.0990387	+	0.995084i	$0.468423\pi$
$390$	0	0
$391$	4.81730	0.243621
$392$	20.5908	1.03999
$393$	0	0
$394$	0.218947	0.0110304
$395$	10.9802	0.552473
$396$	0	0
$397$	22.4392	1.12619	0.563096	−	0.826391i	$-0.309610\pi$
$397$	22.4392	1.12619	0.563096	+	0.826391i	$0.309610\pi$
$398$	24.3497	1.22054
$399$	0	0
$400$	−2.06165	−0.103083
$401$	29.9821	1.49723	0.748616	−	0.663004i	$-0.230719\pi$
$401$	29.9821	1.49723	0.748616	+	0.663004i	$0.230719\pi$
$402$	0	0
$403$	23.1488	1.15312
$404$	−6.93480	−0.345019
$405$	0	0
$406$	0.649266	0.0322225
$407$	−4.96673	−0.246192
$408$	0	0
$409$	−8.27762	−0.409302	−0.204651	−	0.978835i	$-0.565606\pi$
$409$	−8.27762	−0.409302	−0.204651	+	0.978835i	$0.565606\pi$
$410$	−20.7086	−1.02272
$411$	0	0
$412$	−2.85706	−0.140757
$413$	5.60973	0.276037
$414$	0	0
$415$	−10.4854	−0.514706
$416$	−16.3111	−0.799716
$417$	0	0
$418$	2.34293	0.114596
$419$	22.9671	1.12202	0.561008	−	0.827811i	$-0.310414\pi$
$419$	22.9671	1.12202	0.561008	+	0.827811i	$0.310414\pi$
$420$	0	0
$421$	−8.80438	−0.429099	−0.214550	−	0.976713i	$-0.568828\pi$
$421$	−8.80438	−0.429099	−0.214550	+	0.976713i	$0.568828\pi$
$422$	18.5501	0.903005
$423$	0	0
$424$	−35.9084	−1.74387
$425$	−3.72100	−0.180495
$426$	0	0
$427$	5.38265	0.260485
$428$	−8.65438	−0.418325
$429$	0	0
$430$	11.2945	0.544671
$431$	11.5543	0.556549	0.278274	−	0.960502i	$-0.410238\pi$
$431$	11.5543	0.556549	0.278274	+	0.960502i	$0.410238\pi$
$432$	0	0
$433$	5.49076	0.263869	0.131935	−	0.991258i	$-0.457881\pi$
$433$	5.49076	0.263869	0.131935	+	0.991258i	$0.457881\pi$
$434$	2.66679	0.128010
$435$	0	0
$436$	2.56436	0.122811
$437$	−2.70156	−0.129233
$438$	0	0
$439$	28.2337	1.34752	0.673762	−	0.738949i	$-0.264677\pi$
$439$	28.2337	1.34752	0.673762	+	0.738949i	$0.264677\pi$
$440$	5.26483	0.250991
$441$	0	0
$442$	32.9538	1.56745
$443$	16.0472	0.762424	0.381212	−	0.924488i	$-0.375507\pi$
$443$	16.0472	0.762424	0.381212	+	0.924488i	$0.375507\pi$
$444$	0	0
$445$	12.6351	0.598961
$446$	23.7855	1.12628
$447$	0	0
$448$	−4.73451	−0.223684
$449$	−40.7130	−1.92136	−0.960682	−	0.277651i	$-0.910444\pi$
$449$	−40.7130	−1.92136	−0.960682	+	0.277651i	$0.910444\pi$
$450$	0	0
$451$	5.07383	0.238917
$452$	9.46687	0.445284
$453$	0	0
$454$	0.829505	0.0389306
$455$	−7.24309	−0.339561
$456$	0	0
$457$	−18.2354	−0.853017	−0.426508	−	0.904484i	$-0.640257\pi$
$457$	−18.2354	−0.853017	−0.426508	+	0.904484i	$0.640257\pi$
$458$	17.3156	0.809103
$459$	0	0
$460$	−1.26553	−0.0590059
$461$	−1.78312	−0.0830482	−0.0415241	−	0.999138i	$-0.513221\pi$
$461$	−1.78312	−0.0830482	−0.0415241	+	0.999138i	$0.513221\pi$
$462$	0	0
$463$	1.03787	0.0482339	0.0241170	−	0.999709i	$-0.492323\pi$
$463$	1.03787	0.0482339	0.0241170	+	0.999709i	$0.492323\pi$
$464$	2.66907	0.123908
$465$	0	0
$466$	−34.3122	−1.58948
$467$	28.3153	1.31027	0.655137	−	0.755510i	$-0.272611\pi$
$467$	28.3153	1.31027	0.655137	+	0.755510i	$0.272611\pi$
$468$	0	0
$469$	−3.68861	−0.170324
$470$	−29.1843	−1.34617
$471$	0	0
$472$	32.1632	1.48043
$473$	−2.76729	−0.127240
$474$	0	0
$475$	2.08675	0.0957465
$476$	−1.35732	−0.0622125
$477$	0	0
$478$	−4.78495	−0.218858
$479$	2.74699	0.125513	0.0627565	−	0.998029i	$-0.480011\pi$
$479$	2.74699	0.125513	0.0627565	+	0.998029i	$0.480011\pi$
$480$	0	0
$481$	39.1767	1.78631
$482$	−18.1863	−0.828364
$483$	0	0
$484$	5.52521	0.251146
$485$	−16.2835	−0.739398
$486$	0	0
$487$	18.8831	0.855676	0.427838	−	0.903855i	$-0.359275\pi$
$487$	18.8831	0.855676	0.427838	+	0.903855i	$0.359275\pi$
$488$	30.8613	1.39702
$489$	0	0
$490$	19.5791	0.884492
$491$	−14.0046	−0.632018	−0.316009	−	0.948756i	$-0.602343\pi$
$491$	−14.0046	−0.632018	−0.316009	+	0.948756i	$0.602343\pi$
$492$	0	0
$493$	4.81730	0.216960
$494$	−18.4806	−0.831482
$495$	0	0
$496$	10.9629	0.492250
$497$	−2.20085	−0.0987218
$498$	0	0
$499$	−31.6508	−1.41688	−0.708442	−	0.705769i	$-0.750602\pi$
$499$	−31.6508	−1.41688	−0.708442	+	0.705769i	$0.750602\pi$
$500$	−5.35014	−0.239266
$501$	0	0
$502$	−10.1795	−0.454333
$503$	−38.4971	−1.71650	−0.858251	−	0.513231i	$-0.828448\pi$
$503$	−38.4971	−1.71650	−0.858251	+	0.513231i	$0.828448\pi$
$504$	0	0
$505$	−31.6314	−1.40758
$506$	−0.867251	−0.0385540
$507$	0	0
$508$	6.15421	0.273049
$509$	−5.32451	−0.236005	−0.118002	−	0.993013i	$-0.537649\pi$
$509$	−5.32451	−0.236005	−0.118002	+	0.993013i	$0.537649\pi$
$510$	0	0
$511$	−1.43920	−0.0636665
$512$	−24.0962	−1.06491
$513$	0	0
$514$	−11.9277	−0.526108
$515$	−13.0318	−0.574249
$516$	0	0
$517$	7.15048	0.314478
$518$	4.51324	0.198300
$519$	0	0
$520$	−41.5280	−1.82113
$521$	−44.4646	−1.94803	−0.974015	−	0.226484i	$-0.927277\pi$
$521$	−44.4646	−1.94803	−0.974015	+	0.226484i	$0.927277\pi$
$522$	0	0
$523$	13.6749	0.597964	0.298982	−	0.954259i	$-0.403353\pi$
$523$	13.6749	0.597964	0.298982	+	0.954259i	$0.403353\pi$
$524$	2.87526	0.125606
$525$	0	0
$526$	30.1903	1.31636
$527$	19.7866	0.861916
$528$	0	0
$529$	1.00000	0.0434783
$530$	−34.1441	−1.48312
$531$	0	0
$532$	0.761188	0.0330017
$533$	−40.0215	−1.73352
$534$	0	0
$535$	−39.4748	−1.70665
$536$	−21.1486	−0.913479
$537$	0	0
$538$	−31.0148	−1.33714
$539$	−4.79709	−0.206625
$540$	0	0
$541$	−18.6167	−0.800396	−0.400198	−	0.916429i	$-0.631059\pi$
$541$	−18.6167	−0.800396	−0.400198	+	0.916429i	$0.631059\pi$
$542$	14.6749	0.630341
$543$	0	0
$544$	−13.9420	−0.597757
$545$	11.6967	0.501031
$546$	0	0
$547$	23.7789	1.01671	0.508355	−	0.861147i	$-0.330254\pi$
$547$	23.7789	1.01671	0.508355	+	0.861147i	$0.330254\pi$
$548$	−6.63600	−0.283476
$549$	0	0
$550$	0.669885	0.0285640
$551$	−2.70156	−0.115090
$552$	0	0
$553$	2.44463	0.103956
$554$	−3.57039	−0.151691
$555$	0	0
$556$	−12.1974	−0.517286
$557$	−14.5598	−0.616918	−0.308459	−	0.951238i	$-0.599813\pi$
$557$	−14.5598	−0.616918	−0.308459	+	0.951238i	$0.599813\pi$
$558$	0	0
$559$	21.8279	0.923222
$560$	−3.43022	−0.144953
$561$	0	0
$562$	−29.6289	−1.24982
$563$	11.7754	0.496276	0.248138	−	0.968725i	$-0.420181\pi$
$563$	11.7754	0.496276	0.248138	+	0.968725i	$0.420181\pi$
$564$	0	0
$565$	43.1808	1.81663
$566$	23.9435	1.00642
$567$	0	0
$568$	−12.6185	−0.529462
$569$	−36.7143	−1.53914	−0.769572	−	0.638561i	$-0.779530\pi$
$569$	−36.7143	−1.53914	−0.769572	+	0.638561i	$0.779530\pi$
$570$	0	0
$571$	−1.36078	−0.0569467	−0.0284733	−	0.999595i	$-0.509065\pi$
$571$	−1.36078	−0.0569467	−0.0284733	+	0.999595i	$0.509065\pi$
$572$	2.12110	0.0886878
$573$	0	0
$574$	−4.61056	−0.192441
$575$	−0.772423	−0.0322123
$576$	0	0
$577$	20.4913	0.853065	0.426532	−	0.904472i	$-0.359735\pi$
$577$	20.4913	0.853065	0.426532	+	0.904472i	$0.359735\pi$
$578$	7.53317	0.313339
$579$	0	0
$580$	−1.26553	−0.0525485
$581$	−2.33446	−0.0968499
$582$	0	0
$583$	8.36567	0.346471
$584$	−8.25162	−0.341455
$585$	0	0
$586$	−7.65119	−0.316068
$587$	9.48864	0.391638	0.195819	−	0.980640i	$-0.437263\pi$
$587$	9.48864	0.391638	0.195819	+	0.980640i	$0.437263\pi$
$588$	0	0
$589$	−11.0964	−0.457218
$590$	30.5829	1.25908
$591$	0	0
$592$	18.5535	0.762544
$593$	−4.16741	−0.171135	−0.0855675	−	0.996332i	$-0.527270\pi$
$593$	−4.16741	−0.171135	−0.0855675	+	0.996332i	$0.527270\pi$
$594$	0	0
$595$	−6.19106	−0.253809
$596$	3.45367	0.141468
$597$	0	0
$598$	6.84073	0.279738
$599$	−10.8256	−0.442321	−0.221160	−	0.975237i	$-0.570984\pi$
$599$	−10.8256	−0.442321	−0.221160	+	0.975237i	$0.570984\pi$
$600$	0	0
$601$	−1.30413	−0.0531965	−0.0265982	−	0.999646i	$-0.508467\pi$
$601$	−1.30413	−0.0531965	−0.0265982	+	0.999646i	$0.508467\pi$
$602$	2.51462	0.102488
$603$	0	0
$604$	−10.5310	−0.428499
$605$	25.2019	1.02460
$606$	0	0
$607$	−25.2419	−1.02454	−0.512268	−	0.858826i	$-0.671195\pi$
$607$	−25.2419	−1.02454	−0.512268	+	0.858826i	$0.671195\pi$
$608$	7.81870	0.317090
$609$	0	0
$610$	29.3449	1.18814
$611$	−56.4018	−2.28177
$612$	0	0
$613$	−18.0822	−0.730333	−0.365166	−	0.930942i	$-0.618988\pi$
$613$	−18.0822	−0.730333	−0.365166	+	0.930942i	$0.618988\pi$
$614$	−12.0735	−0.487247
$615$	0	0
$616$	1.17216	0.0472277
$617$	35.8406	1.44289	0.721444	−	0.692473i	$-0.243479\pi$
$617$	35.8406	1.44289	0.721444	+	0.692473i	$0.243479\pi$
$618$	0	0
$619$	−2.53005	−0.101691	−0.0508456	−	0.998707i	$-0.516192\pi$
$619$	−2.53005	−0.101691	−0.0508456	+	0.998707i	$0.516192\pi$
$620$	−5.19805	−0.208759
$621$	0	0
$622$	11.9292	0.478317
$623$	2.81308	0.112704
$624$	0	0
$625$	−28.2655	−1.13062
$626$	−40.7256	−1.62772
$627$	0	0
$628$	10.0165	0.399702
$629$	33.4865	1.33519
$630$	0	0
$631$	41.3626	1.64662	0.823310	−	0.567592i	$-0.192125\pi$
$631$	41.3626	1.64662	0.823310	+	0.567592i	$0.192125\pi$
$632$	14.0162	0.557535
$633$	0	0
$634$	22.5542	0.895740
$635$	28.0709	1.11396
$636$	0	0
$637$	37.8386	1.49922
$638$	−0.867251	−0.0343348
$639$	0	0
$640$	−11.9045	−0.470567
$641$	−14.9533	−0.590620	−0.295310	−	0.955401i	$-0.595423\pi$
$641$	−14.9533	−0.590620	−0.295310	+	0.955401i	$0.595423\pi$
$642$	0	0
$643$	−28.2047	−1.11229	−0.556143	−	0.831087i	$-0.687719\pi$
$643$	−28.2047	−1.11229	−0.556143	+	0.831087i	$0.687719\pi$
$644$	−0.281759	−0.0111029
$645$	0	0
$646$	−15.7964	−0.621500
$647$	−27.8254	−1.09393	−0.546964	−	0.837156i	$-0.684217\pi$
$647$	−27.8254	−1.09393	−0.546964	+	0.837156i	$0.684217\pi$
$648$	0	0
$649$	−7.49314	−0.294131
$650$	−5.28394	−0.207253
$651$	0	0
$652$	2.45861	0.0962865
$653$	16.9982	0.665191	0.332595	−	0.943070i	$-0.392076\pi$
$653$	16.9982	0.665191	0.332595	+	0.943070i	$0.392076\pi$
$654$	0	0
$655$	13.1148	0.512437
$656$	−18.9536	−0.740013
$657$	0	0
$658$	−6.49760	−0.253303
$659$	−7.46898	−0.290950	−0.145475	−	0.989362i	$-0.546471\pi$
$659$	−7.46898	−0.290950	−0.145475	+	0.989362i	$0.546471\pi$
$660$	0	0
$661$	0.0462459	0.00179876	0.000899378	−	1.00000i	$-0.499714\pi$
$661$	0.0462459	0.00179876	0.000899378		1.00000i	$0.499714\pi$
$662$	22.9672	0.892645
$663$	0	0
$664$	−13.3846	−0.519422
$665$	3.47197	0.134637
$666$	0	0
$667$	1.00000	0.0387202
$668$	−4.39096	−0.169891
$669$	0	0
$670$	−20.1094	−0.776895
$671$	−7.18982	−0.277560
$672$	0	0
$673$	−47.9939	−1.85003	−0.925014	−	0.379933i	$-0.875947\pi$
$673$	−47.9939	−1.85003	−0.925014	+	0.379933i	$0.875947\pi$
$674$	6.36478	0.245162
$675$	0	0
$676$	−9.88331	−0.380127
$677$	23.9147	0.919115	0.459558	−	0.888148i	$-0.348008\pi$
$677$	23.9147	0.919115	0.459558	+	0.888148i	$0.348008\pi$
$678$	0	0
$679$	−3.62537	−0.139129
$680$	−35.4963	−1.36122
$681$	0	0
$682$	−3.56214	−0.136401
$683$	36.9990	1.41573	0.707864	−	0.706349i	$-0.249659\pi$
$683$	36.9990	1.41573	0.707864	+	0.706349i	$0.249659\pi$
$684$	0	0
$685$	−30.2685	−1.15650
$686$	8.90395	0.339954
$687$	0	0
$688$	10.3374	0.394108
$689$	−65.9870	−2.51390
$690$	0	0
$691$	−5.17244	−0.196769	−0.0983845	−	0.995148i	$-0.531367\pi$
$691$	−5.17244	−0.196769	−0.0983845	+	0.995148i	$0.531367\pi$
$692$	−5.17548	−0.196742
$693$	0	0
$694$	−12.1515	−0.461265
$695$	−55.6356	−2.11038
$696$	0	0
$697$	−34.2086	−1.29574
$698$	−14.5305	−0.549987
$699$	0	0
$700$	0.217637	0.00822591
$701$	−2.33923	−0.0883515	−0.0441757	−	0.999024i	$-0.514066\pi$
$701$	−2.33923	−0.0883515	−0.0441757	+	0.999024i	$0.514066\pi$
$702$	0	0
$703$	−18.7793	−0.708276
$704$	6.32407	0.238347
$705$	0	0
$706$	33.6966	1.26819
$707$	−7.04241	−0.264857
$708$	0	0
$709$	14.5621	0.546891	0.273445	−	0.961888i	$-0.411837\pi$
$709$	14.5621	0.546891	0.273445	+	0.961888i	$0.411837\pi$
$710$	−11.9985	−0.450297
$711$	0	0
$712$	16.1287	0.604449
$713$	4.10740	0.153823
$714$	0	0
$715$	9.67489	0.361820
$716$	4.60340	0.172037
$717$	0	0
$718$	7.45266	0.278131
$719$	−5.38346	−0.200769	−0.100385	−	0.994949i	$-0.532007\pi$
$719$	−5.38346	−0.200769	−0.100385	+	0.994949i	$0.532007\pi$
$720$	0	0
$721$	−2.90140	−0.108054
$722$	−14.2032	−0.528587
$723$	0	0
$724$	−4.58320	−0.170333
$725$	−0.772423	−0.0286871
$726$	0	0
$727$	2.41345	0.0895100	0.0447550	−	0.998998i	$-0.485749\pi$
$727$	2.41345	0.0895100	0.0447550	+	0.998998i	$0.485749\pi$
$728$	−9.24581	−0.342673
$729$	0	0
$730$	−7.84618	−0.290400
$731$	18.6575	0.690072
$732$	0	0
$733$	19.5748	0.723011	0.361506	−	0.932370i	$-0.382263\pi$
$733$	19.5748	0.723011	0.361506	+	0.932370i	$0.382263\pi$
$734$	−18.2133	−0.672265
$735$	0	0
$736$	−2.89415	−0.106680
$737$	4.92703	0.181489
$738$	0	0
$739$	−38.3596	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$739$	−38.3596	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$740$	−8.79711	−0.323388
$741$	0	0
$742$	−7.60184	−0.279072
$743$	−10.9823	−0.402901	−0.201450	−	0.979499i	$-0.564565\pi$
$743$	−10.9823	−0.402901	−0.201450	+	0.979499i	$0.564565\pi$
$744$	0	0
$745$	15.7530	0.577147
$746$	10.2944	0.376903
$747$	0	0
$748$	1.81302	0.0662907
$749$	−8.78868	−0.321131
$750$	0	0
$751$	−48.4059	−1.76636	−0.883178	−	0.469039i	$-0.844600\pi$
$751$	−48.4059	−1.76636	−0.883178	+	0.469039i	$0.844600\pi$
$752$	−26.7110	−0.974050
$753$	0	0
$754$	6.84073	0.249125
$755$	−48.0344	−1.74815
$756$	0	0
$757$	26.8464	0.975748	0.487874	−	0.872914i	$-0.337773\pi$
$757$	26.8464	0.975748	0.487874	+	0.872914i	$0.337773\pi$
$758$	−44.2049	−1.60559
$759$	0	0
$760$	19.9064	0.722082
$761$	53.7005	1.94664	0.973321	−	0.229448i	$-0.0736921\pi$
$761$	53.7005	1.94664	0.973321	+	0.229448i	$0.0736921\pi$
$762$	0	0
$763$	2.60415	0.0942767
$764$	−4.47244	−0.161807
$765$	0	0
$766$	23.2685	0.840725
$767$	59.1046	2.13414
$768$	0	0
$769$	14.7576	0.532174	0.266087	−	0.963949i	$-0.414269\pi$
$769$	14.7576	0.532174	0.266087	+	0.963949i	$0.414269\pi$
$770$	1.11457	0.0401662
$771$	0	0
$772$	6.25876	0.225258
$773$	40.8966	1.47095	0.735474	−	0.677553i	$-0.236960\pi$
$773$	40.8966	1.47095	0.735474	+	0.677553i	$0.236960\pi$
$774$	0	0
$775$	−3.17265	−0.113965
$776$	−20.7860	−0.746173
$777$	0	0
$778$	4.74187	0.170004
$779$	19.1843	0.687348
$780$	0	0
$781$	2.93977	0.105193
$782$	5.84714	0.209093
$783$	0	0
$784$	17.9198	0.639992
$785$	45.6878	1.63067
$786$	0	0
$787$	−12.4083	−0.442309	−0.221155	−	0.975239i	$-0.570983\pi$
$787$	−12.4083	−0.442309	−0.221155	+	0.975239i	$0.570983\pi$
$788$	−0.0950153	−0.00338478
$789$	0	0
$790$	13.3275	0.474172
$791$	9.61378	0.341827
$792$	0	0
$793$	56.7121	2.01391
$794$	27.2363	0.966580
$795$	0	0
$796$	−10.5669	−0.374534
$797$	−31.8954	−1.12979	−0.564897	−	0.825161i	$-0.691084\pi$
$797$	−31.8954	−1.12979	−0.564897	+	0.825161i	$0.691084\pi$
$798$	0	0
$799$	−48.2097	−1.70553
$800$	2.23551	0.0790371
$801$	0	0
$802$	36.3916	1.28503
$803$	1.92240	0.0678400
$804$	0	0
$805$	−1.28517	−0.0452964
$806$	28.0976	0.989694
$807$	0	0
$808$	−40.3775	−1.42047
$809$	10.3553	0.364072	0.182036	−	0.983292i	$-0.441731\pi$
$809$	10.3553	0.364072	0.182036	+	0.983292i	$0.441731\pi$
$810$	0	0
$811$	40.1193	1.40878	0.704390	−	0.709813i	$-0.251220\pi$
$811$	40.1193	1.40878	0.704390	+	0.709813i	$0.251220\pi$
$812$	−0.281759	−0.00988780
$813$	0	0
$814$	−6.02852	−0.211299
$815$	11.2143	0.392821
$816$	0	0
$817$	−10.4632	−0.366060
$818$	−10.0472	−0.351292
$819$	0	0
$820$	8.98681	0.313833
$821$	20.2218	0.705744	0.352872	−	0.935672i	$-0.385205\pi$
$821$	20.2218	0.705744	0.352872	+	0.935672i	$0.385205\pi$
$822$	0	0
$823$	−39.4207	−1.37412	−0.687060	−	0.726601i	$-0.741099\pi$
$823$	−39.4207	−1.37412	−0.687060	+	0.726601i	$0.741099\pi$
$824$	−16.6351	−0.579510
$825$	0	0
$826$	6.80897	0.236914
$827$	46.5418	1.61842	0.809208	−	0.587522i	$-0.199897\pi$
$827$	46.5418	1.61842	0.809208	+	0.587522i	$0.199897\pi$
$828$	0	0
$829$	−30.6713	−1.06526	−0.532629	−	0.846349i	$-0.678796\pi$
$829$	−30.6713	−1.06526	−0.532629	+	0.846349i	$0.678796\pi$
$830$	−12.7269	−0.441758
$831$	0	0
$832$	−49.8832	−1.72939
$833$	32.3427	1.12061
$834$	0	0
$835$	−20.0283	−0.693108
$836$	−1.01675	−0.0351650
$837$	0	0
$838$	27.8770	0.962994
$839$	−41.3030	−1.42594	−0.712969	−	0.701195i	$-0.752650\pi$
$839$	−41.3030	−1.42594	−0.712969	+	0.701195i	$0.752650\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−10.6866	−0.368284
$843$	0	0
$844$	−8.05010	−0.277096
$845$	−45.0803	−1.55081
$846$	0	0
$847$	5.61095	0.192795
$848$	−31.2504	−1.07314
$849$	0	0
$850$	−4.51647	−0.154914
$851$	6.95130	0.238287
$852$	0	0
$853$	−22.6628	−0.775959	−0.387979	−	0.921668i	$-0.626827\pi$
$853$	−22.6628	−0.775959	−0.387979	+	0.921668i	$0.626827\pi$
$854$	6.53335	0.223567
$855$	0	0
$856$	−50.3897	−1.72228
$857$	−17.0466	−0.582301	−0.291150	−	0.956677i	$-0.594038\pi$
$857$	−17.0466	−0.582301	−0.291150	+	0.956677i	$0.594038\pi$
$858$	0	0
$859$	8.91688	0.304240	0.152120	−	0.988362i	$-0.451390\pi$
$859$	8.91688	0.304240	0.152120	+	0.988362i	$0.451390\pi$
$860$	−4.90144	−0.167138
$861$	0	0
$862$	14.0243	0.477670
$863$	−16.9782	−0.577944	−0.288972	−	0.957338i	$-0.593313\pi$
$863$	−16.9782	−0.577944	−0.288972	+	0.957338i	$0.593313\pi$
$864$	0	0
$865$	−23.6067	−0.802650
$866$	6.66457	0.226471
$867$	0	0
$868$	−1.15730	−0.0392812
$869$	−3.26539	−0.110771
$870$	0	0
$871$	−38.8636	−1.31684
$872$	14.9308	0.505622
$873$	0	0
$874$	−3.27910	−0.110917
$875$	−5.43317	−0.183675
$876$	0	0
$877$	43.4845	1.46837	0.734184	−	0.678950i	$-0.237565\pi$
$877$	43.4845	1.46837	0.734184	+	0.678950i	$0.237565\pi$
$878$	34.2696	1.15654
$879$	0	0
$880$	4.58188	0.154455
$881$	9.09207	0.306320	0.153160	−	0.988201i	$-0.451055\pi$
$881$	9.09207	0.306320	0.153160	+	0.988201i	$0.451055\pi$
$882$	0	0
$883$	−22.8983	−0.770590	−0.385295	−	0.922793i	$-0.625900\pi$
$883$	−22.8983	−0.770590	−0.385295	+	0.922793i	$0.625900\pi$
$884$	−14.3008	−0.480988
$885$	0	0
$886$	19.4777	0.654367
$887$	−18.4390	−0.619122	−0.309561	−	0.950880i	$-0.600182\pi$
$887$	−18.4390	−0.619122	−0.309561	+	0.950880i	$0.600182\pi$
$888$	0	0
$889$	6.24971	0.209609
$890$	15.3362	0.514071
$891$	0	0
$892$	−10.3221	−0.345609
$893$	27.0361	0.904730
$894$	0	0
$895$	20.9973	0.701862
$896$	−2.65042	−0.0885443
$897$	0	0
$898$	−49.4166	−1.64905
$899$	4.10740	0.136989
$900$	0	0
$901$	−56.4027	−1.87905
$902$	6.15852	0.205056
$903$	0	0
$904$	55.1203	1.83327
$905$	−20.9051	−0.694910
$906$	0	0
$907$	41.7936	1.38774	0.693868	−	0.720103i	$-0.255905\pi$
$907$	41.7936	1.38774	0.693868	+	0.720103i	$0.255905\pi$
$908$	−0.359977	−0.0119462
$909$	0	0
$910$	−8.79152	−0.291436
$911$	20.7922	0.688877	0.344439	−	0.938809i	$-0.388069\pi$
$911$	20.7922	0.688877	0.344439	+	0.938809i	$0.388069\pi$
$912$	0	0
$913$	3.11824	0.103199
$914$	−22.1338	−0.732120
$915$	0	0
$916$	−7.51436	−0.248281
$917$	2.91988	0.0964229
$918$	0	0
$919$	53.2209	1.75559	0.877797	−	0.479033i	$-0.159013\pi$
$919$	53.2209	1.75559	0.877797	+	0.479033i	$0.159013\pi$
$920$	−7.36850	−0.242932
$921$	0	0
$922$	−2.16432	−0.0712780
$923$	−23.1884	−0.763256
$924$	0	0
$925$	−5.36935	−0.176543
$926$	1.25975	0.0413978
$927$	0	0
$928$	−2.89415	−0.0950050
$929$	23.3956	0.767585	0.383793	−	0.923419i	$-0.374618\pi$
$929$	23.3956	0.767585	0.383793	+	0.923419i	$0.374618\pi$
$930$	0	0
$931$	−18.1379	−0.594446
$932$	14.8903	0.487748
$933$	0	0
$934$	34.3685	1.12457
$935$	8.26965	0.270447
$936$	0	0
$937$	41.6792	1.36160	0.680800	−	0.732469i	$-0.261632\pi$
$937$	41.6792	1.36160	0.680800	+	0.732469i	$0.261632\pi$
$938$	−4.47716	−0.146185
$939$	0	0
$940$	12.6650	0.413086
$941$	−0.914498	−0.0298118	−0.0149059	−	0.999889i	$-0.504745\pi$
$941$	−0.914498	−0.0298118	−0.0149059	+	0.999889i	$0.504745\pi$
$942$	0	0
$943$	−7.10120	−0.231247
$944$	27.9910	0.911031
$945$	0	0
$946$	−3.35888	−0.109207
$947$	−54.1267	−1.75888	−0.879441	−	0.476008i	$-0.842083\pi$
$947$	−54.1267	−1.75888	−0.879441	+	0.476008i	$0.842083\pi$
$948$	0	0
$949$	−15.1636	−0.492230
$950$	2.53285	0.0821765
$951$	0	0
$952$	−7.90290	−0.256134
$953$	−35.2352	−1.14138	−0.570691	−	0.821165i	$-0.693324\pi$
$953$	−35.2352	−1.14138	−0.570691	+	0.821165i	$0.693324\pi$
$954$	0	0
$955$	−20.3999	−0.660127
$956$	2.07650	0.0671589
$957$	0	0
$958$	3.33424	0.107724
$959$	−6.73897	−0.217613
$960$	0	0
$961$	−14.1293	−0.455784
$962$	47.5519	1.53314
$963$	0	0
$964$	7.89223	0.254192
$965$	28.5478	0.918985
$966$	0	0
$967$	36.1833	1.16358	0.581788	−	0.813341i	$-0.302353\pi$
$967$	36.1833	1.16358	0.581788	+	0.813341i	$0.302353\pi$
$968$	32.1702	1.03399
$969$	0	0
$970$	−19.7646	−0.634604
$971$	−7.16489	−0.229932	−0.114966	−	0.993369i	$-0.536676\pi$
$971$	−7.16489	−0.229932	−0.114966	+	0.993369i	$0.536676\pi$
$972$	0	0
$973$	−12.3867	−0.397100
$974$	22.9200	0.734403
$975$	0	0
$976$	26.8580	0.859703
$977$	−31.8131	−1.01779	−0.508895	−	0.860828i	$-0.669946\pi$
$977$	−31.8131	−1.01779	−0.508895	+	0.860828i	$0.669946\pi$
$978$	0	0
$979$	−3.75754	−0.120092
$980$	−8.49663	−0.271415
$981$	0	0
$982$	−16.9985	−0.542443
$983$	−2.37743	−0.0758283	−0.0379142	−	0.999281i	$-0.512071\pi$
$983$	−2.37743	−0.0758283	−0.0379142	+	0.999281i	$0.512071\pi$
$984$	0	0
$985$	−0.433389	−0.0138089
$986$	5.84714	0.186211
$987$	0	0
$988$	8.01994	0.255148
$989$	3.87302	0.123155
$990$	0	0
$991$	−19.5527	−0.621111	−0.310556	−	0.950555i	$-0.600515\pi$
$991$	−19.5527	−0.621111	−0.310556	+	0.950555i	$0.600515\pi$
$992$	−11.8874	−0.377425
$993$	0	0
$994$	−2.67135	−0.0847302
$995$	−48.1984	−1.52799
$996$	0	0
$997$	4.06506	0.128742	0.0643709	−	0.997926i	$-0.479496\pi$
$997$	4.06506	0.128742	0.0643709	+	0.997926i	$0.479496\pi$
$998$	−38.4171	−1.21607
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.l.1.7		10
3.2	odd	2		667.2.a.a.1.4	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.a.1.4	✓	10	3.2	odd	2
6003.2.a.l.1.7		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)

\(f(q)\)	\(=\)	\(q+1.21378 q^{2} -0.526738 q^{4} -2.40259 q^{5} -0.534912 q^{7} -3.06690 q^{8} +O(q^{10})\)	\(q+1.21378 q^{2} -0.526738 q^{4} -2.40259 q^{5} -0.534912 q^{7} -3.06690 q^{8} -2.91621 q^{10} +0.714504 q^{11} -5.63589 q^{13} -0.649266 q^{14} -2.66907 q^{16} -4.81730 q^{17} +2.70156 q^{19} +1.26553 q^{20} +0.867251 q^{22} -1.00000 q^{23} +0.772423 q^{25} -6.84073 q^{26} +0.281759 q^{28} -1.00000 q^{29} -4.10740 q^{31} +2.89415 q^{32} -5.84714 q^{34} +1.28517 q^{35} -6.95130 q^{37} +3.27910 q^{38} +7.36850 q^{40} +7.10120 q^{41} -3.87302 q^{43} -0.376357 q^{44} -1.21378 q^{46} +10.0076 q^{47} -6.71387 q^{49} +0.937552 q^{50} +2.96864 q^{52} +11.7084 q^{53} -1.71666 q^{55} +1.64052 q^{56} -1.21378 q^{58} -10.4872 q^{59} -10.0627 q^{61} -4.98547 q^{62} +8.85099 q^{64} +13.5407 q^{65} +6.89573 q^{67} +2.53746 q^{68} +1.55992 q^{70} +4.11442 q^{71} +2.69054 q^{73} -8.43735 q^{74} -1.42301 q^{76} -0.382197 q^{77} -4.57015 q^{79} +6.41267 q^{80} +8.61929 q^{82} +4.36420 q^{83} +11.5740 q^{85} -4.70099 q^{86} -2.19132 q^{88} -5.25895 q^{89} +3.01471 q^{91} +0.526738 q^{92} +12.1470 q^{94} -6.49072 q^{95} +6.77751 q^{97} -8.14916 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 3 q^{2} + 9 q^{4} + 10 q^{5} + q^{7} + 9 q^{8}+O(q^{10}) \) 10 * q + 3 * q^2 + 9 * q^4 + 10 * q^5 + q^7 + 9 * q^8	\( 10 q + 3 q^{2} + 9 q^{4} + 10 q^{5} + q^{7} + 9 q^{8} - 6 q^{10} - 13 q^{13} + 12 q^{14} - 5 q^{16} + 22 q^{17} - 2 q^{19} - 3 q^{20} + 3 q^{22} - 10 q^{23} + 10 q^{25} + 25 q^{26} + 19 q^{28} - 10 q^{29} - 22 q^{31} + 31 q^{32} + 13 q^{34} + 15 q^{35} - 9 q^{37} + 10 q^{38} - 6 q^{40} + 25 q^{41} + 3 q^{43} + 27 q^{44} - 3 q^{46} + 17 q^{47} + 17 q^{49} - 2 q^{50} - 18 q^{52} + 43 q^{53} - 11 q^{55} + 7 q^{56} - 3 q^{58} + 7 q^{59} - 6 q^{61} - 3 q^{62} + 33 q^{64} - 11 q^{65} + 11 q^{67} + 51 q^{68} + 34 q^{70} + 17 q^{71} - 44 q^{73} - 9 q^{74} + 24 q^{76} + 71 q^{77} + 5 q^{79} - 38 q^{80} + 33 q^{82} + 32 q^{83} + 16 q^{85} + 9 q^{86} + 18 q^{88} + 10 q^{89} - 3 q^{91} - 9 q^{92} + 47 q^{94} + 8 q^{95} + 6 q^{97} + 73 q^{98}+O(q^{100}) \) 10 * q + 3 * q^2 + 9 * q^4 + 10 * q^5 + q^7 + 9 * q^8 - 6 * q^10 - 13 * q^13 + 12 * q^14 - 5 * q^16 + 22 * q^17 - 2 * q^19 - 3 * q^20 + 3 * q^22 - 10 * q^23 + 10 * q^25 + 25 * q^26 + 19 * q^28 - 10 * q^29 - 22 * q^31 + 31 * q^32 + 13 * q^34 + 15 * q^35 - 9 * q^37 + 10 * q^38 - 6 * q^40 + 25 * q^41 + 3 * q^43 + 27 * q^44 - 3 * q^46 + 17 * q^47 + 17 * q^49 - 2 * q^50 - 18 * q^52 + 43 * q^53 - 11 * q^55 + 7 * q^56 - 3 * q^58 + 7 * q^59 - 6 * q^61 - 3 * q^62 + 33 * q^64 - 11 * q^65 + 11 * q^67 + 51 * q^68 + 34 * q^70 + 17 * q^71 - 44 * q^73 - 9 * q^74 + 24 * q^76 + 71 * q^77 + 5 * q^79 - 38 * q^80 + 33 * q^82 + 32 * q^83 + 16 * q^85 + 9 * q^86 + 18 * q^88 + 10 * q^89 - 3 * q^91 - 9 * q^92 + 47 * q^94 + 8 * q^95 + 6 * q^97 + 73 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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