LMFDB - Embedded newform 6003.2.a.i.1.3

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} - x^{6} - 9x^{5} + 10x^{4} + 19x^{3} - 20x^{2} - 5x + 1 $ x^7 - x^6 - 9x^5 + 10x^4 + 19x^3 - 20x^2 - 5*x + 1
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			1.3
Root			$0.136094$ 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)

$f(q)$	$=$	$q-1.17088 q^{2} -0.629030 q^{4} -0.418864 q^{5} +1.98148 q^{7} +3.07829 q^{8} +O(q^{10})$	\(q-1.17088 q^{2} -0.629030 q^{4} -0.418864 q^{5} +1.98148 q^{7} +3.07829 q^{8} +0.490441 q^{10} +5.41188 q^{11} -1.43970 q^{13} -2.32008 q^{14} -2.34626 q^{16} +6.03721 q^{17} -6.20121 q^{19} +0.263478 q^{20} -6.33668 q^{22} +1.00000 q^{23} -4.82455 q^{25} +1.68572 q^{26} -1.24641 q^{28} -1.00000 q^{29} -6.35355 q^{31} -3.40938 q^{32} -7.06888 q^{34} -0.829970 q^{35} +4.47112 q^{37} +7.26090 q^{38} -1.28939 q^{40} -8.27534 q^{41} +6.28106 q^{43} -3.40424 q^{44} -1.17088 q^{46} -6.99741 q^{47} -3.07374 q^{49} +5.64899 q^{50} +0.905613 q^{52} -5.10879 q^{53} -2.26684 q^{55} +6.09957 q^{56} +1.17088 q^{58} -9.51627 q^{59} -4.79680 q^{61} +7.43927 q^{62} +8.68451 q^{64} +0.603037 q^{65} -12.2420 q^{67} -3.79759 q^{68} +0.971799 q^{70} +7.30808 q^{71} -4.61997 q^{73} -5.23516 q^{74} +3.90075 q^{76} +10.7235 q^{77} -4.07417 q^{79} +0.982764 q^{80} +9.68946 q^{82} -3.71419 q^{83} -2.52877 q^{85} -7.35439 q^{86} +16.6593 q^{88} +0.547294 q^{89} -2.85273 q^{91} -0.629030 q^{92} +8.19315 q^{94} +2.59746 q^{95} -8.58637 q^{97} +3.59900 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 7 q^{4} + 5 q^{5} - 5 q^{7} - 3 q^{8}+O(q^{10}) $ 7 * q + q^2 + 7 * q^4 + 5 * q^5 - 5 * q^7 - 3 * q^8	$ 7 q + q^{2} + 7 q^{4} + 5 q^{5} - 5 q^{7} - 3 q^{8} - 11 q^{10} + 12 q^{11} - 13 q^{13} + 3 q^{14} - 13 q^{16} + 12 q^{17} - 5 q^{19} + 8 q^{20} - q^{22} + 7 q^{23} - 4 q^{25} - 2 q^{26} - 21 q^{28} - 7 q^{29} - 8 q^{31} + 5 q^{32} - 28 q^{34} - 5 q^{35} - 24 q^{37} + 6 q^{38} - 20 q^{40} - 9 q^{41} - q^{43} + 23 q^{44} + q^{46} - 27 q^{47} - 14 q^{49} - 7 q^{50} - 9 q^{52} + q^{53} - 11 q^{55} + 20 q^{56} - q^{58} - 8 q^{59} + q^{61} + 3 q^{64} - 12 q^{65} - 16 q^{67} - 15 q^{68} + 40 q^{70} + 13 q^{71} - 23 q^{73} + 8 q^{74} - 2 q^{76} - 13 q^{77} - 44 q^{79} - 30 q^{80} - 10 q^{82} - 21 q^{83} - 6 q^{86} + 21 q^{88} + 5 q^{89} - 18 q^{91} + 7 q^{92} + 28 q^{94} - 9 q^{95} - 55 q^{97} - 36 q^{98}+O(q^{100}) $ 7 * q + q^2 + 7 * q^4 + 5 * q^5 - 5 * q^7 - 3 * q^8 - 11 * q^10 + 12 * q^11 - 13 * q^13 + 3 * q^14 - 13 * q^16 + 12 * q^17 - 5 * q^19 + 8 * q^20 - q^22 + 7 * q^23 - 4 * q^25 - 2 * q^26 - 21 * q^28 - 7 * q^29 - 8 * q^31 + 5 * q^32 - 28 * q^34 - 5 * q^35 - 24 * q^37 + 6 * q^38 - 20 * q^40 - 9 * q^41 - q^43 + 23 * q^44 + q^46 - 27 * q^47 - 14 * q^49 - 7 * q^50 - 9 * q^52 + q^53 - 11 * q^55 + 20 * q^56 - q^58 - 8 * q^59 + q^61 + 3 * q^64 - 12 * q^65 - 16 * q^67 - 15 * q^68 + 40 * q^70 + 13 * q^71 - 23 * q^73 + 8 * q^74 - 2 * q^76 - 13 * q^77 - 44 * q^79 - 30 * q^80 - 10 * q^82 - 21 * q^83 - 6 * q^86 + 21 * q^88 + 5 * q^89 - 18 * q^91 + 7 * q^92 + 28 * q^94 - 9 * q^95 - 55 * q^97 - 36 * 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 $

$2$	−1.17088	−0.827940	−0.413970	−	0.910291i	$-0.635858\pi$
$2$	−1.17088	−0.827940	−0.413970	+	0.910291i	$0.635858\pi$
$3$	0	0
$3$	0	0
$4$	−0.629030	−0.314515
$5$	−0.418864	−0.187322	−0.0936609	−	0.995604i	$-0.529857\pi$
$5$	−0.418864	−0.187322	−0.0936609	+	0.995604i	$0.529857\pi$
$6$	0	0
$7$	1.98148	0.748928	0.374464	−	0.927241i	$-0.377827\pi$
$7$	1.98148	0.748928	0.374464	+	0.927241i	$0.377827\pi$
$8$	3.07829	1.08834
$9$	0	0
$10$	0.490441	0.155091
$11$	5.41188	1.63174	0.815871	−	0.578234i	$-0.196258\pi$
$11$	5.41188	1.63174	0.815871	+	0.578234i	$0.196258\pi$
$12$	0	0
$13$	−1.43970	−0.399300	−0.199650	−	0.979867i	$-0.563980\pi$
$13$	−1.43970	−0.399300	−0.199650	+	0.979867i	$0.563980\pi$
$14$	−2.32008	−0.620068
$15$	0	0
$16$	−2.34626	−0.586565
$17$	6.03721	1.46424	0.732120	−	0.681176i	$-0.238531\pi$
$17$	6.03721	1.46424	0.732120	+	0.681176i	$0.238531\pi$
$18$	0	0
$19$	−6.20121	−1.42265	−0.711327	−	0.702861i	$-0.751906\pi$
$19$	−6.20121	−1.42265	−0.711327	+	0.702861i	$0.751906\pi$
$20$	0.263478	0.0589155
$21$	0	0
$22$	−6.33668	−1.35098
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.82455	−0.964911
$26$	1.68572	0.330596
$27$	0	0
$28$	−1.24641	−0.235549
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−6.35355	−1.14113	−0.570566	−	0.821252i	$-0.693276\pi$
$31$	−6.35355	−1.14113	−0.570566	+	0.821252i	$0.693276\pi$
$32$	−3.40938	−0.602699
$33$	0	0
$34$	−7.06888	−1.21230
$35$	−0.829970	−0.140291
$36$	0	0
$37$	4.47112	0.735048	0.367524	−	0.930014i	$-0.380206\pi$
$37$	4.47112	0.735048	0.367524	+	0.930014i	$0.380206\pi$
$38$	7.26090	1.17787
$39$	0	0
$40$	−1.28939	−0.203870
$41$	−8.27534	−1.29239	−0.646195	−	0.763172i	$-0.723641\pi$
$41$	−8.27534	−1.29239	−0.646195	+	0.763172i	$0.723641\pi$
$42$	0	0
$43$	6.28106	0.957852	0.478926	−	0.877855i	$-0.341026\pi$
$43$	6.28106	0.957852	0.478926	+	0.877855i	$0.341026\pi$
$44$	−3.40424	−0.513208
$45$	0	0
$46$	−1.17088	−0.172637
$47$	−6.99741	−1.02068	−0.510338	−	0.859974i	$-0.670480\pi$
$47$	−6.99741	−1.02068	−0.510338	+	0.859974i	$0.670480\pi$
$48$	0	0
$49$	−3.07374	−0.439106
$50$	5.64899	0.798888
$51$	0	0
$52$	0.905613	0.125586
$53$	−5.10879	−0.701747	−0.350873	−	0.936423i	$-0.614115\pi$
$53$	−5.10879	−0.701747	−0.350873	+	0.936423i	$0.614115\pi$
$54$	0	0
$55$	−2.26684	−0.305661
$56$	6.09957	0.815089
$57$	0	0
$58$	1.17088	0.153745
$59$	−9.51627	−1.23891	−0.619457	−	0.785031i	$-0.712647\pi$
$59$	−9.51627	−1.23891	−0.619457	+	0.785031i	$0.712647\pi$
$60$	0	0
$61$	−4.79680	−0.614168	−0.307084	−	0.951682i	$-0.599353\pi$
$61$	−4.79680	−0.614168	−0.307084	+	0.951682i	$0.599353\pi$
$62$	7.43927	0.944788
$63$	0	0
$64$	8.68451	1.08556
$65$	0.603037	0.0747975
$66$	0	0
$67$	−12.2420	−1.49560	−0.747801	−	0.663923i	$-0.768890\pi$
$67$	−12.2420	−1.49560	−0.747801	+	0.663923i	$0.768890\pi$
$68$	−3.79759	−0.460526
$69$	0	0
$70$	0.971799	0.116152
$71$	7.30808	0.867309	0.433655	−	0.901079i	$-0.357224\pi$
$71$	7.30808	0.867309	0.433655	+	0.901079i	$0.357224\pi$
$72$	0	0
$73$	−4.61997	−0.540727	−0.270364	−	0.962758i	$-0.587144\pi$
$73$	−4.61997	−0.540727	−0.270364	+	0.962758i	$0.587144\pi$
$74$	−5.23516	−0.608576
$75$	0	0
$76$	3.90075	0.447447
$77$	10.7235	1.22206
$78$	0	0
$79$	−4.07417	−0.458380	−0.229190	−	0.973382i	$-0.573608\pi$
$79$	−4.07417	−0.458380	−0.229190	+	0.973382i	$0.573608\pi$
$80$	0.982764	0.109876
$81$	0	0
$82$	9.68946	1.07002
$83$	−3.71419	−0.407686	−0.203843	−	0.979004i	$-0.565343\pi$
$83$	−3.71419	−0.407686	−0.203843	+	0.979004i	$0.565343\pi$
$84$	0	0
$85$	−2.52877	−0.274284
$86$	−7.35439	−0.793044
$87$	0	0
$88$	16.6593	1.77589
$89$	0.547294	0.0580131	0.0290065	−	0.999579i	$-0.490766\pi$
$89$	0.547294	0.0580131	0.0290065	+	0.999579i	$0.490766\pi$
$90$	0	0
$91$	−2.85273	−0.299047
$92$	−0.629030	−0.0655810
$93$	0	0
$94$	8.19315	0.845059
$95$	2.59746	0.266494
$96$	0	0
$97$	−8.58637	−0.871814	−0.435907	−	0.899992i	$-0.643572\pi$
$97$	−8.58637	−0.871814	−0.435907	+	0.899992i	$0.643572\pi$
$98$	3.59900	0.363554
$99$	0	0
$100$	3.03479	0.303479
$101$	−15.6455	−1.55678	−0.778392	−	0.627779i	$-0.783964\pi$
$101$	−15.6455	−1.55678	−0.778392	+	0.627779i	$0.783964\pi$
$102$	0	0
$103$	−0.736275	−0.0725474	−0.0362737	−	0.999342i	$-0.511549\pi$
$103$	−0.736275	−0.0725474	−0.0362737	+	0.999342i	$0.511549\pi$
$104$	−4.43180	−0.434574
$105$	0	0
$106$	5.98180	0.581004
$107$	15.9928	1.54608	0.773040	−	0.634357i	$-0.218735\pi$
$107$	15.9928	1.54608	0.773040	+	0.634357i	$0.218735\pi$
$108$	0	0
$109$	−2.90116	−0.277881	−0.138940	−	0.990301i	$-0.544370\pi$
$109$	−2.90116	−0.277881	−0.138940	+	0.990301i	$0.544370\pi$
$110$	2.65421	0.253069
$111$	0	0
$112$	−4.64906	−0.439295
$113$	4.66285	0.438644	0.219322	−	0.975652i	$-0.429615\pi$
$113$	4.66285	0.438644	0.219322	+	0.975652i	$0.429615\pi$
$114$	0	0
$115$	−0.418864	−0.0390593
$116$	0.629030	0.0584040
$117$	0	0
$118$	11.1425	1.02575
$119$	11.9626	1.09661
$120$	0	0
$121$	18.2884	1.66258
$122$	5.61650	0.508494
$123$	0	0
$124$	3.99658	0.358903
$125$	4.11515	0.368070
$126$	0	0
$127$	2.45509	0.217854	0.108927	−	0.994050i	$-0.465259\pi$
$127$	2.45509	0.217854	0.108927	+	0.994050i	$0.465259\pi$
$128$	−3.34979	−0.296083
$129$	0	0
$130$	−0.706087	−0.0619279
$131$	6.30415	0.550796	0.275398	−	0.961330i	$-0.411190\pi$
$131$	6.30415	0.550796	0.275398	+	0.961330i	$0.411190\pi$
$132$	0	0
$133$	−12.2876	−1.06547
$134$	14.3340	1.23827
$135$	0	0
$136$	18.5843	1.59359
$137$	11.1580	0.953294	0.476647	−	0.879095i	$-0.341852\pi$
$137$	11.1580	0.953294	0.476647	+	0.879095i	$0.341852\pi$
$138$	0	0
$139$	−19.7370	−1.67407	−0.837036	−	0.547148i	$-0.815713\pi$
$139$	−19.7370	−1.67407	−0.837036	+	0.547148i	$0.815713\pi$
$140$	0.522077	0.0441235
$141$	0	0
$142$	−8.55691	−0.718080
$143$	−7.79146	−0.651555
$144$	0	0
$145$	0.418864	0.0347848
$146$	5.40945	0.447690
$147$	0	0
$148$	−2.81247	−0.231184
$149$	−22.6974	−1.85945	−0.929723	−	0.368260i	$-0.879954\pi$
$149$	−22.6974	−1.85945	−0.929723	+	0.368260i	$0.879954\pi$
$150$	0	0
$151$	10.5126	0.855507	0.427754	−	0.903895i	$-0.359305\pi$
$151$	10.5126	0.855507	0.427754	+	0.903895i	$0.359305\pi$
$152$	−19.0891	−1.54833
$153$	0	0
$154$	−12.5560	−1.01179
$155$	2.66127	0.213759
$156$	0	0
$157$	18.9413	1.51168	0.755840	−	0.654756i	$-0.227229\pi$
$157$	18.9413	1.51168	0.755840	+	0.654756i	$0.227229\pi$
$158$	4.77038	0.379511
$159$	0	0
$160$	1.42807	0.112899
$161$	1.98148	0.156162
$162$	0	0
$163$	1.36439	0.106868	0.0534338	−	0.998571i	$-0.482983\pi$
$163$	1.36439	0.106868	0.0534338	+	0.998571i	$0.482983\pi$
$164$	5.20544	0.406477
$165$	0	0
$166$	4.34889	0.337539
$167$	−14.9947	−1.16032	−0.580162	−	0.814501i	$-0.697011\pi$
$167$	−14.9947	−1.16032	−0.580162	+	0.814501i	$0.697011\pi$
$168$	0	0
$169$	−10.9273	−0.840560
$170$	2.96090	0.227091
$171$	0	0
$172$	−3.95098	−0.301259
$173$	24.0572	1.82903	0.914516	−	0.404551i	$-0.132572\pi$
$173$	24.0572	1.82903	0.914516	+	0.404551i	$0.132572\pi$
$174$	0	0
$175$	−9.55975	−0.722649
$176$	−12.6977	−0.957123
$177$	0	0
$178$	−0.640818	−0.0480313
$179$	15.0638	1.12592	0.562960	−	0.826484i	$-0.309663\pi$
$179$	15.0638	1.12592	0.562960	+	0.826484i	$0.309663\pi$
$180$	0	0
$181$	−4.16684	−0.309719	−0.154859	−	0.987937i	$-0.549492\pi$
$181$	−4.16684	−0.309719	−0.154859	+	0.987937i	$0.549492\pi$
$182$	3.34021	0.247593
$183$	0	0
$184$	3.07829	0.226935
$185$	−1.87279	−0.137690
$186$	0	0
$187$	32.6727	2.38926
$188$	4.40158	0.321018
$189$	0	0
$190$	−3.04133	−0.220641
$191$	−21.6210	−1.56444	−0.782219	−	0.623003i	$-0.785912\pi$
$191$	−21.6210	−1.56444	−0.782219	+	0.623003i	$0.785912\pi$
$192$	0	0
$193$	20.6563	1.48688	0.743438	−	0.668805i	$-0.233194\pi$
$193$	20.6563	1.48688	0.743438	+	0.668805i	$0.233194\pi$
$194$	10.0536	0.721810
$195$	0	0
$196$	1.93348	0.138106
$197$	13.3639	0.952135	0.476068	−	0.879409i	$-0.342062\pi$
$197$	13.3639	0.952135	0.476068	+	0.879409i	$0.342062\pi$
$198$	0	0
$199$	0.555384	0.0393701	0.0196851	−	0.999806i	$-0.493734\pi$
$199$	0.555384	0.0393701	0.0196851	+	0.999806i	$0.493734\pi$
$200$	−14.8514	−1.05015
$201$	0	0
$202$	18.3190	1.28892
$203$	−1.98148	−0.139073
$204$	0	0
$205$	3.46624	0.242093
$206$	0.862093	0.0600649
$207$	0	0
$208$	3.37790	0.234215
$209$	−33.5602	−2.32141
$210$	0	0
$211$	−17.5831	−1.21047	−0.605236	−	0.796046i	$-0.706921\pi$
$211$	−17.5831	−1.21047	−0.605236	+	0.796046i	$0.706921\pi$
$212$	3.21359	0.220710
$213$	0	0
$214$	−18.7257	−1.28006
$215$	−2.63091	−0.179427
$216$	0	0
$217$	−12.5894	−0.854626
$218$	3.39692	0.230069
$219$	0	0
$220$	1.42591	0.0961350
$221$	−8.69176	−0.584671
$222$	0	0
$223$	−4.80810	−0.321974	−0.160987	−	0.986957i	$-0.551468\pi$
$223$	−4.80810	−0.321974	−0.160987	+	0.986957i	$0.551468\pi$
$224$	−6.75562	−0.451379
$225$	0	0
$226$	−5.45966	−0.363171
$227$	9.02664	0.599119	0.299560	−	0.954078i	$-0.403160\pi$
$227$	9.02664	0.599119	0.299560	+	0.954078i	$0.403160\pi$
$228$	0	0
$229$	−9.33240	−0.616702	−0.308351	−	0.951273i	$-0.599777\pi$
$229$	−9.33240	−0.616702	−0.308351	+	0.951273i	$0.599777\pi$
$230$	0.490441	0.0323387
$231$	0	0
$232$	−3.07829	−0.202100
$233$	−1.59220	−0.104309	−0.0521543	−	0.998639i	$-0.516609\pi$
$233$	−1.59220	−0.104309	−0.0521543	+	0.998639i	$0.516609\pi$
$234$	0	0
$235$	2.93096	0.191195
$236$	5.98603	0.389657
$237$	0	0
$238$	−14.0068	−0.907928
$239$	14.9035	0.964030	0.482015	−	0.876163i	$-0.339905\pi$
$239$	14.9035	0.964030	0.482015	+	0.876163i	$0.339905\pi$
$240$	0	0
$241$	−14.3190	−0.922370	−0.461185	−	0.887304i	$-0.652575\pi$
$241$	−14.3190	−0.922370	−0.461185	+	0.887304i	$0.652575\pi$
$242$	−21.4136	−1.37652
$243$	0	0
$244$	3.01734	0.193165
$245$	1.28748	0.0822541
$246$	0	0
$247$	8.92786	0.568066
$248$	−19.5581	−1.24194
$249$	0	0
$250$	−4.81837	−0.304740
$251$	−6.91890	−0.436717	−0.218358	−	0.975869i	$-0.570070\pi$
$251$	−6.91890	−0.436717	−0.218358	+	0.975869i	$0.570070\pi$
$252$	0	0
$253$	5.41188	0.340242
$254$	−2.87463	−0.180370
$255$	0	0
$256$	−13.4468	−0.840425
$257$	−22.2006	−1.38484	−0.692418	−	0.721497i	$-0.743455\pi$
$257$	−22.2006	−1.38484	−0.692418	+	0.721497i	$0.743455\pi$
$258$	0	0
$259$	8.85943	0.550498
$260$	−0.379329	−0.0235250
$261$	0	0
$262$	−7.38143	−0.456026
$263$	−20.0295	−1.23507	−0.617535	−	0.786544i	$-0.711868\pi$
$263$	−20.0295	−1.23507	−0.617535	+	0.786544i	$0.711868\pi$
$264$	0	0
$265$	2.13989	0.131452
$266$	14.3873	0.882143
$267$	0	0
$268$	7.70060	0.470389
$269$	−17.7058	−1.07954	−0.539772	−	0.841811i	$-0.681489\pi$
$269$	−17.7058	−1.07954	−0.539772	+	0.841811i	$0.681489\pi$
$270$	0	0
$271$	23.5873	1.43283	0.716414	−	0.697675i	$-0.245782\pi$
$271$	23.5873	1.43283	0.716414	+	0.697675i	$0.245782\pi$
$272$	−14.1649	−0.858872
$273$	0	0
$274$	−13.0647	−0.789270
$275$	−26.1099	−1.57449
$276$	0	0
$277$	9.80130	0.588903	0.294451	−	0.955666i	$-0.404863\pi$
$277$	9.80130	0.588903	0.294451	+	0.955666i	$0.404863\pi$
$278$	23.1098	1.38603
$279$	0	0
$280$	−2.55489	−0.152684
$281$	27.4425	1.63708	0.818542	−	0.574447i	$-0.194783\pi$
$281$	27.4425	1.63708	0.818542	+	0.574447i	$0.194783\pi$
$282$	0	0
$283$	0.0818159	0.00486345	0.00243173	−	0.999997i	$-0.499226\pi$
$283$	0.0818159	0.00486345	0.00243173	+	0.999997i	$0.499226\pi$
$284$	−4.59700	−0.272782
$285$	0	0
$286$	9.12290	0.539448
$287$	−16.3974	−0.967908
$288$	0	0
$289$	19.4480	1.14400
$290$	−0.490441	−0.0287997
$291$	0	0
$292$	2.90610	0.170067
$293$	−15.5146	−0.906370	−0.453185	−	0.891417i	$-0.649712\pi$
$293$	−15.5146	−0.906370	−0.453185	+	0.891417i	$0.649712\pi$
$294$	0	0
$295$	3.98603	0.232075
$296$	13.7634	0.799982
$297$	0	0
$298$	26.5761	1.53951
$299$	−1.43970	−0.0832598
$300$	0	0
$301$	12.4458	0.717363
$302$	−12.3091	−0.708309
$303$	0	0
$304$	14.5496	0.834479
$305$	2.00921	0.115047
$306$	0	0
$307$	−31.5596	−1.80120	−0.900600	−	0.434649i	$-0.856873\pi$
$307$	−31.5596	−1.80120	−0.900600	+	0.434649i	$0.856873\pi$
$308$	−6.74542	−0.384356
$309$	0	0
$310$	−3.11604	−0.176979
$311$	−5.49672	−0.311691	−0.155845	−	0.987781i	$-0.549810\pi$
$311$	−5.49672	−0.311691	−0.155845	+	0.987781i	$0.549810\pi$
$312$	0	0
$313$	−24.9161	−1.40834	−0.704170	−	0.710031i	$-0.748681\pi$
$313$	−24.9161	−1.40834	−0.704170	+	0.710031i	$0.748681\pi$
$314$	−22.1781	−1.25158
$315$	0	0
$316$	2.56278	0.144167
$317$	18.0990	1.01654	0.508270	−	0.861198i	$-0.330285\pi$
$317$	18.0990	1.01654	0.508270	+	0.861198i	$0.330285\pi$
$318$	0	0
$319$	−5.41188	−0.303007
$320$	−3.63763	−0.203350
$321$	0	0
$322$	−2.32008	−0.129293
$323$	−37.4380	−2.08311
$324$	0	0
$325$	6.94589	0.385289
$326$	−1.59755	−0.0884799
$327$	0	0
$328$	−25.4739	−1.40656
$329$	−13.8652	−0.764414
$330$	0	0
$331$	19.9928	1.09891	0.549453	−	0.835525i	$-0.314836\pi$
$331$	19.9928	1.09891	0.549453	+	0.835525i	$0.314836\pi$
$332$	2.33634	0.128223
$333$	0	0
$334$	17.5570	0.960678
$335$	5.12774	0.280159
$336$	0	0
$337$	6.05071	0.329603	0.164802	−	0.986327i	$-0.447302\pi$
$337$	6.05071	0.329603	0.164802	+	0.986327i	$0.447302\pi$
$338$	12.7946	0.695933
$339$	0	0
$340$	1.59067	0.0862665
$341$	−34.3846	−1.86203
$342$	0	0
$343$	−19.9609	−1.07779
$344$	19.3349	1.04247
$345$	0	0
$346$	−28.1681	−1.51433
$347$	5.26287	0.282526	0.141263	−	0.989972i	$-0.454884\pi$
$347$	5.26287	0.282526	0.141263	+	0.989972i	$0.454884\pi$
$348$	0	0
$349$	−14.6134	−0.782238	−0.391119	−	0.920340i	$-0.627912\pi$
$349$	−14.6134	−0.782238	−0.391119	+	0.920340i	$0.627912\pi$
$350$	11.1934	0.598310
$351$	0	0
$352$	−18.4512	−0.983450
$353$	21.6900	1.15444	0.577221	−	0.816588i	$-0.304137\pi$
$353$	21.6900	1.15444	0.577221	+	0.816588i	$0.304137\pi$
$354$	0	0
$355$	−3.06109	−0.162466
$356$	−0.344265	−0.0182460
$357$	0	0
$358$	−17.6379	−0.932194
$359$	−13.7538	−0.725899	−0.362949	−	0.931809i	$-0.618230\pi$
$359$	−13.7538	−0.725899	−0.362949	+	0.931809i	$0.618230\pi$
$360$	0	0
$361$	19.4550	1.02395
$362$	4.87889	0.256429
$363$	0	0
$364$	1.79445	0.0940549
$365$	1.93514	0.101290
$366$	0	0
$367$	−25.2233	−1.31664	−0.658322	−	0.752737i	$-0.728733\pi$
$367$	−25.2233	−1.31664	−0.658322	+	0.752737i	$0.728733\pi$
$368$	−2.34626	−0.122307
$369$	0	0
$370$	2.19282	0.113999
$371$	−10.1230	−0.525558
$372$	0	0
$373$	12.2050	0.631950	0.315975	−	0.948767i	$-0.397668\pi$
$373$	12.2050	0.631950	0.315975	+	0.948767i	$0.397668\pi$
$374$	−38.2559	−1.97817
$375$	0	0
$376$	−21.5401	−1.11084
$377$	1.43970	0.0741481
$378$	0	0
$379$	−24.4364	−1.25522	−0.627608	−	0.778530i	$-0.715966\pi$
$379$	−24.4364	−1.25522	−0.627608	+	0.778530i	$0.715966\pi$
$380$	−1.63388	−0.0838165
$381$	0	0
$382$	25.3157	1.29526
$383$	1.23449	0.0630793	0.0315397	−	0.999503i	$-0.489959\pi$
$383$	1.23449	0.0630793	0.0315397	+	0.999503i	$0.489959\pi$
$384$	0	0
$385$	−4.49170	−0.228918
$386$	−24.1862	−1.23104
$387$	0	0
$388$	5.40109	0.274199
$389$	−19.5460	−0.991023	−0.495511	−	0.868602i	$-0.665019\pi$
$389$	−19.5460	−0.991023	−0.495511	+	0.868602i	$0.665019\pi$
$390$	0	0
$391$	6.03721	0.305315
$392$	−9.46187	−0.477897
$393$	0	0
$394$	−15.6475	−0.788311
$395$	1.70652	0.0858645
$396$	0	0
$397$	2.21872	0.111354	0.0556771	−	0.998449i	$-0.482268\pi$
$397$	2.21872	0.111354	0.0556771	+	0.998449i	$0.482268\pi$
$398$	−0.650290	−0.0325961
$399$	0	0
$400$	11.3197	0.565983
$401$	1.79468	0.0896219	0.0448110	−	0.998995i	$-0.485731\pi$
$401$	1.79468	0.0896219	0.0448110	+	0.998995i	$0.485731\pi$
$402$	0	0
$403$	9.14718	0.455654
$404$	9.84148	0.489632
$405$	0	0
$406$	2.32008	0.115144
$407$	24.1972	1.19941
$408$	0	0
$409$	−20.6359	−1.02038	−0.510190	−	0.860062i	$-0.670425\pi$
$409$	−20.6359	−1.02038	−0.510190	+	0.860062i	$0.670425\pi$
$410$	−4.05857	−0.200438
$411$	0	0
$412$	0.463140	0.0228173
$413$	−18.8563	−0.927857
$414$	0	0
$415$	1.55574	0.0763684
$416$	4.90847	0.240658
$417$	0	0
$418$	39.2951	1.92198
$419$	13.8247	0.675379	0.337689	−	0.941258i	$-0.390355\pi$
$419$	13.8247	0.675379	0.337689	+	0.941258i	$0.390355\pi$
$420$	0	0
$421$	8.29062	0.404060	0.202030	−	0.979379i	$-0.435246\pi$
$421$	8.29062	0.404060	0.202030	+	0.979379i	$0.435246\pi$
$422$	20.5878	1.00220
$423$	0	0
$424$	−15.7263	−0.763739
$425$	−29.1269	−1.41286
$426$	0	0
$427$	−9.50477	−0.459968
$428$	−10.0599	−0.486266
$429$	0	0
$430$	3.08049	0.148554
$431$	28.2247	1.35953	0.679767	−	0.733428i	$-0.262081\pi$
$431$	28.2247	1.35953	0.679767	+	0.733428i	$0.262081\pi$
$432$	0	0
$433$	2.28751	0.109931	0.0549654	−	0.998488i	$-0.482495\pi$
$433$	2.28751	0.109931	0.0549654	+	0.998488i	$0.482495\pi$
$434$	14.7408	0.707579
$435$	0	0
$436$	1.82492	0.0873977
$437$	−6.20121	−0.296644
$438$	0	0
$439$	−9.84200	−0.469733	−0.234866	−	0.972028i	$-0.575465\pi$
$439$	−9.84200	−0.469733	−0.234866	+	0.972028i	$0.575465\pi$
$440$	−6.97799	−0.332663
$441$	0	0
$442$	10.1770	0.484072
$443$	12.8680	0.611376	0.305688	−	0.952132i	$-0.401114\pi$
$443$	12.8680	0.611376	0.305688	+	0.952132i	$0.401114\pi$
$444$	0	0
$445$	−0.229242	−0.0108671
$446$	5.62973	0.266575
$447$	0	0
$448$	17.2082	0.813010
$449$	−11.4040	−0.538189	−0.269094	−	0.963114i	$-0.586724\pi$
$449$	−11.4040	−0.538189	−0.269094	+	0.963114i	$0.586724\pi$
$450$	0	0
$451$	−44.7851	−2.10885
$452$	−2.93308	−0.137960
$453$	0	0
$454$	−10.5691	−0.496035
$455$	1.19491	0.0560180
$456$	0	0
$457$	−25.5939	−1.19723	−0.598617	−	0.801035i	$-0.704283\pi$
$457$	−25.5939	−1.19723	−0.598617	+	0.801035i	$0.704283\pi$
$458$	10.9272	0.510593
$459$	0	0
$460$	0.263478	0.0122847
$461$	−35.5432	−1.65541	−0.827707	−	0.561161i	$-0.810355\pi$
$461$	−35.5432	−1.65541	−0.827707	+	0.561161i	$0.810355\pi$
$462$	0	0
$463$	−8.53051	−0.396446	−0.198223	−	0.980157i	$-0.563517\pi$
$463$	−8.53051	−0.396446	−0.198223	+	0.980157i	$0.563517\pi$
$464$	2.34626	0.108922
$465$	0	0
$466$	1.86428	0.0863612
$467$	−14.1441	−0.654513	−0.327257	−	0.944936i	$-0.606124\pi$
$467$	−14.1441	−0.654513	−0.327257	+	0.944936i	$0.606124\pi$
$468$	0	0
$469$	−24.2573	−1.12010
$470$	−3.43182	−0.158298
$471$	0	0
$472$	−29.2938	−1.34836
$473$	33.9923	1.56297
$474$	0	0
$475$	29.9181	1.37273
$476$	−7.52485	−0.344901
$477$	0	0
$478$	−17.4503	−0.798159
$479$	−15.6742	−0.716170	−0.358085	−	0.933689i	$-0.616570\pi$
$479$	−15.6742	−0.716170	−0.358085	+	0.933689i	$0.616570\pi$
$480$	0	0
$481$	−6.43706	−0.293504
$482$	16.7659	0.763667
$483$	0	0
$484$	−11.5040	−0.522908
$485$	3.59652	0.163310
$486$	0	0
$487$	40.9182	1.85418	0.927091	−	0.374836i	$-0.122301\pi$
$487$	40.9182	1.85418	0.927091	+	0.374836i	$0.122301\pi$
$488$	−14.7660	−0.668423
$489$	0	0
$490$	−1.50749	−0.0681015
$491$	−23.9426	−1.08052	−0.540258	−	0.841500i	$-0.681673\pi$
$491$	−23.9426	−1.08052	−0.540258	+	0.841500i	$0.681673\pi$
$492$	0	0
$493$	−6.03721	−0.271902
$494$	−10.4535	−0.470325
$495$	0	0
$496$	14.9071	0.669348
$497$	14.4808	0.649553
$498$	0	0
$499$	−17.0774	−0.764489	−0.382244	−	0.924061i	$-0.624849\pi$
$499$	−17.0774	−0.764489	−0.382244	+	0.924061i	$0.624849\pi$
$500$	−2.58856	−0.115764
$501$	0	0
$502$	8.10123	0.361575
$503$	24.9975	1.11458	0.557291	−	0.830317i	$-0.311841\pi$
$503$	24.9975	1.11458	0.557291	+	0.830317i	$0.311841\pi$
$504$	0	0
$505$	6.55333	0.291619
$506$	−6.33668	−0.281700
$507$	0	0
$508$	−1.54433	−0.0685184
$509$	30.2846	1.34234	0.671170	−	0.741303i	$-0.265792\pi$
$509$	30.2846	1.34234	0.671170	+	0.741303i	$0.265792\pi$
$510$	0	0
$511$	−9.15438	−0.404966
$512$	22.4442	0.991904
$513$	0	0
$514$	25.9943	1.14656
$515$	0.308399	0.0135897
$516$	0	0
$517$	−37.8691	−1.66548
$518$	−10.3734	−0.455780
$519$	0	0
$520$	1.85632	0.0814052
$521$	32.0208	1.40286	0.701429	−	0.712740i	$-0.252546\pi$
$521$	32.0208	1.40286	0.701429	+	0.712740i	$0.252546\pi$
$522$	0	0
$523$	−22.1259	−0.967500	−0.483750	−	0.875206i	$-0.660726\pi$
$523$	−22.1259	−0.967500	−0.483750	+	0.875206i	$0.660726\pi$
$524$	−3.96550	−0.173234
$525$	0	0
$526$	23.4522	1.02256
$527$	−38.3577	−1.67089
$528$	0	0
$529$	1.00000	0.0434783
$530$	−2.50556	−0.108835
$531$	0	0
$532$	7.72925	0.335105
$533$	11.9140	0.516052
$534$	0	0
$535$	−6.69880	−0.289614
$536$	−37.6845	−1.62772
$537$	0	0
$538$	20.7315	0.893797
$539$	−16.6347	−0.716508
$540$	0	0
$541$	15.9859	0.687289	0.343645	−	0.939100i	$-0.388338\pi$
$541$	15.9859	0.687289	0.343645	+	0.939100i	$0.388338\pi$
$542$	−27.6180	−1.18630
$543$	0	0
$544$	−20.5832	−0.882496
$545$	1.21519	0.0520531
$546$	0	0
$547$	−9.89012	−0.422871	−0.211436	−	0.977392i	$-0.567814\pi$
$547$	−9.89012	−0.422871	−0.211436	+	0.977392i	$0.567814\pi$
$548$	−7.01873	−0.299825
$549$	0	0
$550$	30.5717	1.30358
$551$	6.20121	0.264180
$552$	0	0
$553$	−8.07288	−0.343294
$554$	−11.4762	−0.487576
$555$	0	0
$556$	12.4152	0.526521
$557$	13.0049	0.551036	0.275518	−	0.961296i	$-0.411151\pi$
$557$	13.0049	0.551036	0.275518	+	0.961296i	$0.411151\pi$
$558$	0	0
$559$	−9.04282	−0.382470
$560$	1.94733	0.0822895
$561$	0	0
$562$	−32.1320	−1.35541
$563$	45.4359	1.91490	0.957448	−	0.288606i	$-0.0931917\pi$
$563$	45.4359	1.91490	0.957448	+	0.288606i	$0.0931917\pi$
$564$	0	0
$565$	−1.95310	−0.0821676
$566$	−0.0957970	−0.00402665
$567$	0	0
$568$	22.4964	0.943927
$569$	−33.2297	−1.39306	−0.696530	−	0.717527i	$-0.745274\pi$
$569$	−33.2297	−1.39306	−0.696530	+	0.717527i	$0.745274\pi$
$570$	0	0
$571$	−2.25155	−0.0942246	−0.0471123	−	0.998890i	$-0.515002\pi$
$571$	−2.25155	−0.0942246	−0.0471123	+	0.998890i	$0.515002\pi$
$572$	4.90106	0.204924
$573$	0	0
$574$	19.1995	0.801370
$575$	−4.82455	−0.201198
$576$	0	0
$577$	30.7244	1.27907	0.639537	−	0.768760i	$-0.279126\pi$
$577$	30.7244	1.27907	0.639537	+	0.768760i	$0.279126\pi$
$578$	−22.7713	−0.947162
$579$	0	0
$580$	−0.263478	−0.0109403
$581$	−7.35960	−0.305327
$582$	0	0
$583$	−27.6482	−1.14507
$584$	−14.2216	−0.588495
$585$	0	0
$586$	18.1657	0.750420
$587$	−11.0766	−0.457180	−0.228590	−	0.973523i	$-0.573411\pi$
$587$	−11.0766	−0.457180	−0.228590	+	0.973523i	$0.573411\pi$
$588$	0	0
$589$	39.3997	1.62344
$590$	−4.66717	−0.192145
$591$	0	0
$592$	−10.4904	−0.431153
$593$	0.411473	0.0168972	0.00844860	−	0.999964i	$-0.497311\pi$
$593$	0.411473	0.0168972	0.00844860	+	0.999964i	$0.497311\pi$
$594$	0	0
$595$	−5.01071	−0.205419
$596$	14.2774	0.584824
$597$	0	0
$598$	1.68572	0.0689341
$599$	16.1494	0.659845	0.329923	−	0.944008i	$-0.392977\pi$
$599$	16.1494	0.659845	0.329923	+	0.944008i	$0.392977\pi$
$600$	0	0
$601$	4.83308	0.197146	0.0985728	−	0.995130i	$-0.468572\pi$
$601$	4.83308	0.197146	0.0985728	+	0.995130i	$0.468572\pi$
$602$	−14.5726	−0.593934
$603$	0	0
$604$	−6.61277	−0.269070
$605$	−7.66036	−0.311438
$606$	0	0
$607$	28.1431	1.14229	0.571147	−	0.820848i	$-0.306499\pi$
$607$	28.1431	1.14229	0.571147	+	0.820848i	$0.306499\pi$
$608$	21.1423	0.857433
$609$	0	0
$610$	−2.35255	−0.0952520
$611$	10.0741	0.407556
$612$	0	0
$613$	−7.28446	−0.294216	−0.147108	−	0.989120i	$-0.546997\pi$
$613$	−7.28446	−0.294216	−0.147108	+	0.989120i	$0.546997\pi$
$614$	36.9526	1.49129
$615$	0	0
$616$	33.0101	1.33001
$617$	49.2391	1.98229	0.991145	−	0.132782i	$-0.0423911\pi$
$617$	49.2391	1.98229	0.991145	+	0.132782i	$0.0423911\pi$
$618$	0	0
$619$	14.5176	0.583513	0.291757	−	0.956493i	$-0.405760\pi$
$619$	14.5176	0.583513	0.291757	+	0.956493i	$0.405760\pi$
$620$	−1.67402	−0.0672303
$621$	0	0
$622$	6.43603	0.258061
$623$	1.08445	0.0434476
$624$	0	0
$625$	22.3991	0.895963
$626$	29.1739	1.16602
$627$	0	0
$628$	−11.9147	−0.475446
$629$	26.9931	1.07629
$630$	0	0
$631$	2.60365	0.103649	0.0518247	−	0.998656i	$-0.483496\pi$
$631$	2.60365	0.103649	0.0518247	+	0.998656i	$0.483496\pi$
$632$	−12.5415	−0.498873
$633$	0	0
$634$	−21.1918	−0.841635
$635$	−1.02835	−0.0408088
$636$	0	0
$637$	4.42526	0.175335
$638$	6.33668	0.250872
$639$	0	0
$640$	1.40311	0.0554627
$641$	3.11790	0.123150	0.0615749	−	0.998102i	$-0.480388\pi$
$641$	3.11790	0.123150	0.0615749	+	0.998102i	$0.480388\pi$
$642$	0	0
$643$	−46.1556	−1.82020	−0.910100	−	0.414390i	$-0.863995\pi$
$643$	−46.1556	−1.82020	−0.910100	+	0.414390i	$0.863995\pi$
$644$	−1.24641	−0.0491154
$645$	0	0
$646$	43.8356	1.72469
$647$	−22.7429	−0.894114	−0.447057	−	0.894505i	$-0.647528\pi$
$647$	−22.7429	−0.894114	−0.447057	+	0.894505i	$0.647528\pi$
$648$	0	0
$649$	−51.5009	−2.02159
$650$	−8.13283	−0.318996
$651$	0	0
$652$	−0.858245	−0.0336115
$653$	−4.52151	−0.176940	−0.0884701	−	0.996079i	$-0.528198\pi$
$653$	−4.52151	−0.176940	−0.0884701	+	0.996079i	$0.528198\pi$
$654$	0	0
$655$	−2.64058	−0.103176
$656$	19.4161	0.758071
$657$	0	0
$658$	16.2346	0.632889
$659$	−10.2697	−0.400050	−0.200025	−	0.979791i	$-0.564102\pi$
$659$	−10.2697	−0.400050	−0.200025	+	0.979791i	$0.564102\pi$
$660$	0	0
$661$	8.53689	0.332047	0.166023	−	0.986122i	$-0.446907\pi$
$661$	8.53689	0.332047	0.166023	+	0.986122i	$0.446907\pi$
$662$	−23.4093	−0.909829
$663$	0	0
$664$	−11.4334	−0.443701
$665$	5.14682	0.199585
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	9.43211	0.364939
$669$	0	0
$670$	−6.00399	−0.231955
$671$	−25.9597	−1.00216
$672$	0	0
$673$	11.0930	0.427605	0.213803	−	0.976877i	$-0.431415\pi$
$673$	11.0930	0.427605	0.213803	+	0.976877i	$0.431415\pi$
$674$	−7.08468	−0.272892
$675$	0	0
$676$	6.87359	0.264369
$677$	−12.0257	−0.462186	−0.231093	−	0.972932i	$-0.574230\pi$
$677$	−12.0257	−0.462186	−0.231093	+	0.972932i	$0.574230\pi$
$678$	0	0
$679$	−17.0137	−0.652926
$680$	−7.78430	−0.298514
$681$	0	0
$682$	40.2604	1.54165
$683$	−36.4023	−1.39289	−0.696447	−	0.717608i	$-0.745237\pi$
$683$	−36.4023	−1.39289	−0.696447	+	0.717608i	$0.745237\pi$
$684$	0	0
$685$	−4.67369	−0.178573
$686$	23.3719	0.892344
$687$	0	0
$688$	−14.7370	−0.561843
$689$	7.35511	0.280207
$690$	0	0
$691$	42.5117	1.61722	0.808611	−	0.588344i	$-0.200220\pi$
$691$	42.5117	1.61722	0.808611	+	0.588344i	$0.200220\pi$
$692$	−15.1327	−0.575258
$693$	0	0
$694$	−6.16221	−0.233914
$695$	8.26713	0.313590
$696$	0	0
$697$	−49.9600	−1.89237
$698$	17.1106	0.647646
$699$	0	0
$700$	6.01337	0.227284
$701$	−25.5766	−0.966016	−0.483008	−	0.875616i	$-0.660456\pi$
$701$	−25.5766	−0.966016	−0.483008	+	0.875616i	$0.660456\pi$
$702$	0	0
$703$	−27.7264	−1.04572
$704$	46.9995	1.77136
$705$	0	0
$706$	−25.3965	−0.955809
$707$	−31.0012	−1.16592
$708$	0	0
$709$	35.7447	1.34242	0.671210	−	0.741267i	$-0.265775\pi$
$709$	35.7447	1.34242	0.671210	+	0.741267i	$0.265775\pi$
$710$	3.58418	0.134512
$711$	0	0
$712$	1.68473	0.0631379
$713$	−6.35355	−0.237942
$714$	0	0
$715$	3.26356	0.122050
$716$	−9.47557	−0.354119
$717$	0	0
$718$	16.1041	0.601001
$719$	10.3233	0.384994	0.192497	−	0.981298i	$-0.438341\pi$
$719$	10.3233	0.384994	0.192497	+	0.981298i	$0.438341\pi$
$720$	0	0
$721$	−1.45891	−0.0543328
$722$	−22.7795	−0.847766
$723$	0	0
$724$	2.62107	0.0974113
$725$	4.82455	0.179179
$726$	0	0
$727$	−35.6565	−1.32243	−0.661213	−	0.750199i	$-0.729958\pi$
$727$	−35.6565	−1.32243	−0.661213	+	0.750199i	$0.729958\pi$
$728$	−8.78152	−0.325465
$729$	0	0
$730$	−2.26583	−0.0838620
$731$	37.9201	1.40253
$732$	0	0
$733$	32.2388	1.19077	0.595385	−	0.803441i	$-0.297000\pi$
$733$	32.2388	1.19077	0.595385	+	0.803441i	$0.297000\pi$
$734$	29.5335	1.09010
$735$	0	0
$736$	−3.40938	−0.125671
$737$	−66.2523	−2.44044
$738$	0	0
$739$	−0.631756	−0.0232395	−0.0116198	−	0.999932i	$-0.503699\pi$
$739$	−0.631756	−0.0232395	−0.0116198	+	0.999932i	$0.503699\pi$
$740$	1.17804	0.0433057
$741$	0	0
$742$	11.8528	0.435131
$743$	19.9139	0.730570	0.365285	−	0.930896i	$-0.380972\pi$
$743$	19.9139	0.730570	0.365285	+	0.930896i	$0.380972\pi$
$744$	0	0
$745$	9.50714	0.348315
$746$	−14.2906	−0.523217
$747$	0	0
$748$	−20.5521	−0.751459
$749$	31.6893	1.15790
$750$	0	0
$751$	−26.2402	−0.957517	−0.478759	−	0.877947i	$-0.658913\pi$
$751$	−26.2402	−0.957517	−0.478759	+	0.877947i	$0.658913\pi$
$752$	16.4177	0.598693
$753$	0	0
$754$	−1.68572	−0.0613902
$755$	−4.40337	−0.160255
$756$	0	0
$757$	−20.9477	−0.761357	−0.380679	−	0.924707i	$-0.624310\pi$
$757$	−20.9477	−0.761357	−0.380679	+	0.924707i	$0.624310\pi$
$758$	28.6122	1.03924
$759$	0	0
$760$	7.99575	0.290036
$761$	16.9578	0.614720	0.307360	−	0.951593i	$-0.400554\pi$
$761$	16.9578	0.614720	0.307360	+	0.951593i	$0.400554\pi$
$762$	0	0
$763$	−5.74858	−0.208113
$764$	13.6003	0.492040
$765$	0	0
$766$	−1.44544	−0.0522259
$767$	13.7005	0.494698
$768$	0	0
$769$	−14.0287	−0.505889	−0.252944	−	0.967481i	$-0.581399\pi$
$769$	−14.0287	−0.505889	−0.252944	+	0.967481i	$0.581399\pi$
$770$	5.25926	0.189530
$771$	0	0
$772$	−12.9935	−0.467645
$773$	6.44609	0.231850	0.115925	−	0.993258i	$-0.463017\pi$
$773$	6.44609	0.231850	0.115925	+	0.993258i	$0.463017\pi$
$774$	0	0
$775$	30.6530	1.10109
$776$	−26.4313	−0.948830
$777$	0	0
$778$	22.8861	0.820507
$779$	51.3171	1.83863
$780$	0	0
$781$	39.5504	1.41523
$782$	−7.06888	−0.252783
$783$	0	0
$784$	7.21180	0.257564
$785$	−7.93383	−0.283170
$786$	0	0
$787$	33.2754	1.18614	0.593069	−	0.805151i	$-0.297916\pi$
$787$	33.2754	1.18614	0.593069	+	0.805151i	$0.297916\pi$
$788$	−8.40627	−0.299461
$789$	0	0
$790$	−1.99814	−0.0710907
$791$	9.23934	0.328513
$792$	0	0
$793$	6.90594	0.245237
$794$	−2.59786	−0.0921946
$795$	0	0
$796$	−0.349353	−0.0123825
$797$	−17.1250	−0.606600	−0.303300	−	0.952895i	$-0.598088\pi$
$797$	−17.1250	−0.606600	−0.303300	+	0.952895i	$0.598088\pi$
$798$	0	0
$799$	−42.2449	−1.49452
$800$	16.4487	0.581551
$801$	0	0
$802$	−2.10136	−0.0742016
$803$	−25.0027	−0.882327
$804$	0	0
$805$	−0.829970	−0.0292526
$806$	−10.7103	−0.377254
$807$	0	0
$808$	−48.1613	−1.69431
$809$	−24.7592	−0.870487	−0.435244	−	0.900313i	$-0.643338\pi$
$809$	−24.7592	−0.870487	−0.435244	+	0.900313i	$0.643338\pi$
$810$	0	0
$811$	56.3750	1.97959	0.989797	−	0.142481i	$-0.0455081\pi$
$811$	56.3750	1.97959	0.989797	+	0.142481i	$0.0455081\pi$
$812$	1.24641	0.0437404
$813$	0	0
$814$	−28.3321	−0.993038
$815$	−0.571495	−0.0200186
$816$	0	0
$817$	−38.9501	−1.36269
$818$	24.1622	0.844813
$819$	0	0
$820$	−2.18037	−0.0761419
$821$	−29.7906	−1.03970	−0.519849	−	0.854258i	$-0.674012\pi$
$821$	−29.7906	−1.03970	−0.519849	+	0.854258i	$0.674012\pi$
$822$	0	0
$823$	−23.2267	−0.809631	−0.404816	−	0.914398i	$-0.632664\pi$
$823$	−23.2267	−0.809631	−0.404816	+	0.914398i	$0.632664\pi$
$824$	−2.26647	−0.0789562
$825$	0	0
$826$	22.0785	0.768210
$827$	−39.9652	−1.38973	−0.694863	−	0.719142i	$-0.744535\pi$
$827$	−39.9652	−1.38973	−0.694863	+	0.719142i	$0.744535\pi$
$828$	0	0
$829$	−12.4272	−0.431614	−0.215807	−	0.976436i	$-0.569238\pi$
$829$	−12.4272	−0.431614	−0.215807	+	0.976436i	$0.569238\pi$
$830$	−1.82159	−0.0632285
$831$	0	0
$832$	−12.5031	−0.433466
$833$	−18.5568	−0.642957
$834$	0	0
$835$	6.28073	0.217354
$836$	21.1104	0.730117
$837$	0	0
$838$	−16.1871	−0.559173
$839$	14.5525	0.502409	0.251205	−	0.967934i	$-0.419173\pi$
$839$	14.5525	0.502409	0.251205	+	0.967934i	$0.419173\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−9.70735	−0.334537
$843$	0	0
$844$	11.0603	0.380712
$845$	4.57704	0.157455
$846$	0	0
$847$	36.2381	1.24516
$848$	11.9866	0.411620
$849$	0	0
$850$	34.1042	1.16976
$851$	4.47112	0.153268
$852$	0	0
$853$	−12.2748	−0.420280	−0.210140	−	0.977671i	$-0.567392\pi$
$853$	−12.2748	−0.420280	−0.210140	+	0.977671i	$0.567392\pi$
$854$	11.1290	0.380826
$855$	0	0
$856$	49.2304	1.68266
$857$	56.8515	1.94201	0.971005	−	0.239059i	$-0.0768388\pi$
$857$	56.8515	1.94201	0.971005	+	0.239059i	$0.0768388\pi$
$858$	0	0
$859$	11.8379	0.403902	0.201951	−	0.979396i	$-0.435272\pi$
$859$	11.8379	0.403902	0.201951	+	0.979396i	$0.435272\pi$
$860$	1.65492	0.0564324
$861$	0	0
$862$	−33.0478	−1.12561
$863$	−24.4196	−0.831252	−0.415626	−	0.909536i	$-0.636437\pi$
$863$	−24.4196	−0.831252	−0.415626	+	0.909536i	$0.636437\pi$
$864$	0	0
$865$	−10.0767	−0.342617
$866$	−2.67841	−0.0910161
$867$	0	0
$868$	7.91913	0.268793
$869$	−22.0489	−0.747958
$870$	0	0
$871$	17.6248	0.597193
$872$	−8.93061	−0.302429
$873$	0	0
$874$	7.26090	0.245603
$875$	8.15409	0.275658
$876$	0	0
$877$	−32.9055	−1.11114	−0.555570	−	0.831470i	$-0.687500\pi$
$877$	−32.9055	−1.11114	−0.555570	+	0.831470i	$0.687500\pi$
$878$	11.5238	0.388911
$879$	0	0
$880$	5.31860	0.179290
$881$	−32.6832	−1.10113	−0.550563	−	0.834794i	$-0.685587\pi$
$881$	−32.6832	−1.10113	−0.550563	+	0.834794i	$0.685587\pi$
$882$	0	0
$883$	14.0627	0.473247	0.236624	−	0.971601i	$-0.423959\pi$
$883$	14.0627	0.473247	0.236624	+	0.971601i	$0.423959\pi$
$884$	5.46738	0.183888
$885$	0	0
$886$	−15.0669	−0.506182
$887$	−32.0083	−1.07473	−0.537367	−	0.843348i	$-0.680581\pi$
$887$	−32.0083	−1.07473	−0.537367	+	0.843348i	$0.680581\pi$
$888$	0	0
$889$	4.86471	0.163157
$890$	0.268416	0.00899731
$891$	0	0
$892$	3.02444	0.101266
$893$	43.3924	1.45207
$894$	0	0
$895$	−6.30967	−0.210909
$896$	−6.63754	−0.221745
$897$	0	0
$898$	13.3528	0.445588
$899$	6.35355	0.211903
$900$	0	0
$901$	−30.8429	−1.02753
$902$	52.4382	1.74600
$903$	0	0
$904$	14.3536	0.477394
$905$	1.74534	0.0580171
$906$	0	0
$907$	15.8519	0.526355	0.263177	−	0.964747i	$-0.415230\pi$
$907$	15.8519	0.526355	0.263177	+	0.964747i	$0.415230\pi$
$908$	−5.67803	−0.188432
$909$	0	0
$910$	−1.39910	−0.0463796
$911$	18.8578	0.624786	0.312393	−	0.949953i	$-0.398869\pi$
$911$	18.8578	0.624786	0.312393	+	0.949953i	$0.398869\pi$
$912$	0	0
$913$	−20.1008	−0.665238
$914$	29.9675	0.991238
$915$	0	0
$916$	5.87036	0.193962
$917$	12.4915	0.412507
$918$	0	0
$919$	−54.8856	−1.81051	−0.905254	−	0.424870i	$-0.860320\pi$
$919$	−54.8856	−1.81051	−0.905254	+	0.424870i	$0.860320\pi$
$920$	−1.28939	−0.0425098
$921$	0	0
$922$	41.6170	1.37058
$923$	−10.5214	−0.346317
$924$	0	0
$925$	−21.5712	−0.709255
$926$	9.98824	0.328234
$927$	0	0
$928$	3.40938	0.111918
$929$	−29.6410	−0.972488	−0.486244	−	0.873823i	$-0.661633\pi$
$929$	−29.6410	−0.972488	−0.486244	+	0.873823i	$0.661633\pi$
$930$	0	0
$931$	19.0609	0.624696
$932$	1.00154	0.0328066
$933$	0	0
$934$	16.5612	0.541898
$935$	−13.6854	−0.447561
$936$	0	0
$937$	7.03873	0.229945	0.114973	−	0.993369i	$-0.463322\pi$
$937$	7.03873	0.229945	0.114973	+	0.993369i	$0.463322\pi$
$938$	28.4025	0.927374
$939$	0	0
$940$	−1.84367	−0.0601337
$941$	19.4424	0.633804	0.316902	−	0.948458i	$-0.397357\pi$
$941$	19.4424	0.633804	0.316902	+	0.948458i	$0.397357\pi$
$942$	0	0
$943$	−8.27534	−0.269482
$944$	22.3277	0.726703
$945$	0	0
$946$	−39.8011	−1.29404
$947$	60.2582	1.95813	0.979065	−	0.203550i	$-0.0652479\pi$
$947$	60.2582	1.95813	0.979065	+	0.203550i	$0.0652479\pi$
$948$	0	0
$949$	6.65136	0.215912
$950$	−35.0306	−1.13654
$951$	0	0
$952$	36.8244	1.19349
$953$	−34.8557	−1.12909	−0.564543	−	0.825404i	$-0.690948\pi$
$953$	−34.8557	−1.12909	−0.564543	+	0.825404i	$0.690948\pi$
$954$	0	0
$955$	9.05625	0.293053
$956$	−9.37479	−0.303202
$957$	0	0
$958$	18.3526	0.592946
$959$	22.1094	0.713949
$960$	0	0
$961$	9.36758	0.302180
$962$	7.53705	0.243004
$963$	0	0
$964$	9.00710	0.290099
$965$	−8.65220	−0.278524
$966$	0	0
$967$	−50.9108	−1.63718	−0.818591	−	0.574377i	$-0.805245\pi$
$967$	−50.9108	−1.63718	−0.818591	+	0.574377i	$0.805245\pi$
$968$	56.2970	1.80946
$969$	0	0
$970$	−4.21111	−0.135211
$971$	−9.00540	−0.288997	−0.144498	−	0.989505i	$-0.546157\pi$
$971$	−9.00540	−0.288997	−0.144498	+	0.989505i	$0.546157\pi$
$972$	0	0
$973$	−39.1085	−1.25376
$974$	−47.9105	−1.53515
$975$	0	0
$976$	11.2546	0.360249
$977$	13.1636	0.421142	0.210571	−	0.977579i	$-0.432468\pi$
$977$	13.1636	0.421142	0.210571	+	0.977579i	$0.432468\pi$
$978$	0	0
$979$	2.96189	0.0946624
$980$	−0.809864	−0.0258702
$981$	0	0
$982$	28.0340	0.894602
$983$	49.6509	1.58362	0.791809	−	0.610769i	$-0.209139\pi$
$983$	49.6509	1.58362	0.791809	+	0.610769i	$0.209139\pi$
$984$	0	0
$985$	−5.59764	−0.178356
$986$	7.06888	0.225119
$987$	0	0
$988$	−5.61589	−0.178665
$989$	6.28106	0.199726
$990$	0	0
$991$	−11.4473	−0.363634	−0.181817	−	0.983332i	$-0.558198\pi$
$991$	−11.4473	−0.363634	−0.181817	+	0.983332i	$0.558198\pi$
$992$	21.6617	0.687759
$993$	0	0
$994$	−16.9553	−0.537791
$995$	−0.232630	−0.00737488
$996$	0	0
$997$	−3.23372	−0.102413	−0.0512064	−	0.998688i	$-0.516307\pi$
$997$	−3.23372	−0.102413	−0.0512064	+	0.998688i	$0.516307\pi$
$998$	19.9956	0.632951
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.i.1.3		7
3.2	odd	2		2001.2.a.j.1.5	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.j.1.5	✓	7	3.2	odd	2
6003.2.a.i.1.3		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)

\(f(q)\)	\(=\)	\(q-1.17088 q^{2} -0.629030 q^{4} -0.418864 q^{5} +1.98148 q^{7} +3.07829 q^{8} +O(q^{10})\)	\(q-1.17088 q^{2} -0.629030 q^{4} -0.418864 q^{5} +1.98148 q^{7} +3.07829 q^{8} +0.490441 q^{10} +5.41188 q^{11} -1.43970 q^{13} -2.32008 q^{14} -2.34626 q^{16} +6.03721 q^{17} -6.20121 q^{19} +0.263478 q^{20} -6.33668 q^{22} +1.00000 q^{23} -4.82455 q^{25} +1.68572 q^{26} -1.24641 q^{28} -1.00000 q^{29} -6.35355 q^{31} -3.40938 q^{32} -7.06888 q^{34} -0.829970 q^{35} +4.47112 q^{37} +7.26090 q^{38} -1.28939 q^{40} -8.27534 q^{41} +6.28106 q^{43} -3.40424 q^{44} -1.17088 q^{46} -6.99741 q^{47} -3.07374 q^{49} +5.64899 q^{50} +0.905613 q^{52} -5.10879 q^{53} -2.26684 q^{55} +6.09957 q^{56} +1.17088 q^{58} -9.51627 q^{59} -4.79680 q^{61} +7.43927 q^{62} +8.68451 q^{64} +0.603037 q^{65} -12.2420 q^{67} -3.79759 q^{68} +0.971799 q^{70} +7.30808 q^{71} -4.61997 q^{73} -5.23516 q^{74} +3.90075 q^{76} +10.7235 q^{77} -4.07417 q^{79} +0.982764 q^{80} +9.68946 q^{82} -3.71419 q^{83} -2.52877 q^{85} -7.35439 q^{86} +16.6593 q^{88} +0.547294 q^{89} -2.85273 q^{91} -0.629030 q^{92} +8.19315 q^{94} +2.59746 q^{95} -8.58637 q^{97} +3.59900 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 7 q^{4} + 5 q^{5} - 5 q^{7} - 3 q^{8}+O(q^{10}) \) 7 * q + q^2 + 7 * q^4 + 5 * q^5 - 5 * q^7 - 3 * q^8	\( 7 q + q^{2} + 7 q^{4} + 5 q^{5} - 5 q^{7} - 3 q^{8} - 11 q^{10} + 12 q^{11} - 13 q^{13} + 3 q^{14} - 13 q^{16} + 12 q^{17} - 5 q^{19} + 8 q^{20} - q^{22} + 7 q^{23} - 4 q^{25} - 2 q^{26} - 21 q^{28} - 7 q^{29} - 8 q^{31} + 5 q^{32} - 28 q^{34} - 5 q^{35} - 24 q^{37} + 6 q^{38} - 20 q^{40} - 9 q^{41} - q^{43} + 23 q^{44} + q^{46} - 27 q^{47} - 14 q^{49} - 7 q^{50} - 9 q^{52} + q^{53} - 11 q^{55} + 20 q^{56} - q^{58} - 8 q^{59} + q^{61} + 3 q^{64} - 12 q^{65} - 16 q^{67} - 15 q^{68} + 40 q^{70} + 13 q^{71} - 23 q^{73} + 8 q^{74} - 2 q^{76} - 13 q^{77} - 44 q^{79} - 30 q^{80} - 10 q^{82} - 21 q^{83} - 6 q^{86} + 21 q^{88} + 5 q^{89} - 18 q^{91} + 7 q^{92} + 28 q^{94} - 9 q^{95} - 55 q^{97} - 36 q^{98}+O(q^{100}) \) 7 * q + q^2 + 7 * q^4 + 5 * q^5 - 5 * q^7 - 3 * q^8 - 11 * q^10 + 12 * q^11 - 13 * q^13 + 3 * q^14 - 13 * q^16 + 12 * q^17 - 5 * q^19 + 8 * q^20 - q^22 + 7 * q^23 - 4 * q^25 - 2 * q^26 - 21 * q^28 - 7 * q^29 - 8 * q^31 + 5 * q^32 - 28 * q^34 - 5 * q^35 - 24 * q^37 + 6 * q^38 - 20 * q^40 - 9 * q^41 - q^43 + 23 * q^44 + q^46 - 27 * q^47 - 14 * q^49 - 7 * q^50 - 9 * q^52 + q^53 - 11 * q^55 + 20 * q^56 - q^58 - 8 * q^59 + q^61 + 3 * q^64 - 12 * q^65 - 16 * q^67 - 15 * q^68 + 40 * q^70 + 13 * q^71 - 23 * q^73 + 8 * q^74 - 2 * q^76 - 13 * q^77 - 44 * q^79 - 30 * q^80 - 10 * q^82 - 21 * q^83 - 6 * q^86 + 21 * q^88 + 5 * q^89 - 18 * q^91 + 7 * q^92 + 28 * q^94 - 9 * q^95 - 55 * q^97 - 36 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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