LMFDB - Embedded newform 6003.2.a.l.1.2

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			1.2
Root			$-1.57663$ 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)

$f(q)$	$=$	$q-1.57663 q^{2} +0.485749 q^{4} +1.88241 q^{5} -0.868755 q^{7} +2.38741 q^{8} +O(q^{10})$	\(q-1.57663 q^{2} +0.485749 q^{4} +1.88241 q^{5} -0.868755 q^{7} +2.38741 q^{8} -2.96785 q^{10} +4.15064 q^{11} -1.68508 q^{13} +1.36970 q^{14} -4.73555 q^{16} -6.19890 q^{17} -0.936729 q^{19} +0.914376 q^{20} -6.54400 q^{22} -1.00000 q^{23} -1.45655 q^{25} +2.65675 q^{26} -0.421997 q^{28} -1.00000 q^{29} +5.44674 q^{31} +2.69137 q^{32} +9.77334 q^{34} -1.63535 q^{35} -9.23764 q^{37} +1.47687 q^{38} +4.49407 q^{40} -8.57766 q^{41} +8.65313 q^{43} +2.01617 q^{44} +1.57663 q^{46} +2.84654 q^{47} -6.24526 q^{49} +2.29644 q^{50} -0.818527 q^{52} +9.40655 q^{53} +7.81318 q^{55} -2.07407 q^{56} +1.57663 q^{58} +13.3750 q^{59} +0.878868 q^{61} -8.58747 q^{62} +5.22781 q^{64} -3.17201 q^{65} +5.75008 q^{67} -3.01111 q^{68} +2.57833 q^{70} +11.0541 q^{71} -8.68524 q^{73} +14.5643 q^{74} -0.455015 q^{76} -3.60589 q^{77} -6.12127 q^{79} -8.91421 q^{80} +13.5238 q^{82} +2.70897 q^{83} -11.6688 q^{85} -13.6428 q^{86} +9.90926 q^{88} +1.59743 q^{89} +1.46393 q^{91} -0.485749 q^{92} -4.48792 q^{94} -1.76330 q^{95} +19.0596 q^{97} +9.84644 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.57663	−1.11484	−0.557421	−	0.830230i	$-0.688209\pi$
$2$	−1.57663	−1.11484	−0.557421	+	0.830230i	$0.688209\pi$
$3$	0	0
$3$	0	0
$4$	0.485749	0.242874
$5$	1.88241	0.841837	0.420919	−	0.907098i	$-0.361708\pi$
$5$	1.88241	0.841837	0.420919	+	0.907098i	$0.361708\pi$
$6$	0	0
$7$	−0.868755	−0.328359	−0.164179	−	0.986431i	$-0.552498\pi$
$7$	−0.868755	−0.328359	−0.164179	+	0.986431i	$0.552498\pi$
$8$	2.38741	0.844076
$9$	0	0
$10$	−2.96785	−0.938516
$11$	4.15064	1.25146	0.625732	−	0.780038i	$-0.284800\pi$
$11$	4.15064	1.25146	0.625732	+	0.780038i	$0.284800\pi$
$12$	0	0
$13$	−1.68508	−0.467358	−0.233679	−	0.972314i	$-0.575077\pi$
$13$	−1.68508	−0.467358	−0.233679	+	0.972314i	$0.575077\pi$
$14$	1.36970	0.366068
$15$	0	0
$16$	−4.73555	−1.18389
$17$	−6.19890	−1.50345	−0.751727	−	0.659474i	$-0.770779\pi$
$17$	−6.19890	−1.50345	−0.751727	+	0.659474i	$0.770779\pi$
$18$	0	0
$19$	−0.936729	−0.214900	−0.107450	−	0.994210i	$-0.534269\pi$
$19$	−0.936729	−0.214900	−0.107450	+	0.994210i	$0.534269\pi$
$20$	0.914376	0.204461
$21$	0	0
$22$	−6.54400	−1.39519
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−1.45655	−0.291310
$26$	2.65675	0.521031
$27$	0	0
$28$	−0.421997	−0.0797499
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	5.44674	0.978264	0.489132	−	0.872210i	$-0.337314\pi$
$31$	5.44674	0.978264	0.489132	+	0.872210i	$0.337314\pi$
$32$	2.69137	0.475771
$33$	0	0
$34$	9.77334	1.67611
$35$	−1.63535	−0.276424
$36$	0	0
$37$	−9.23764	−1.51866	−0.759329	−	0.650707i	$-0.774473\pi$
$37$	−9.23764	−1.51866	−0.759329	+	0.650707i	$0.774473\pi$
$38$	1.47687	0.239580
$39$	0	0
$40$	4.49407	0.710575
$41$	−8.57766	−1.33960	−0.669802	−	0.742539i	$-0.733621\pi$
$41$	−8.57766	−1.33960	−0.669802	+	0.742539i	$0.733621\pi$
$42$	0	0
$43$	8.65313	1.31959	0.659795	−	0.751446i	$-0.270643\pi$
$43$	8.65313	1.31959	0.659795	+	0.751446i	$0.270643\pi$
$44$	2.01617	0.303949
$45$	0	0
$46$	1.57663	0.232461
$47$	2.84654	0.415210	0.207605	−	0.978213i	$-0.433433\pi$
$47$	2.84654	0.415210	0.207605	+	0.978213i	$0.433433\pi$
$48$	0	0
$49$	−6.24526	−0.892181
$50$	2.29644	0.324765
$51$	0	0
$52$	−0.818527	−0.113509
$53$	9.40655	1.29209	0.646045	−	0.763300i	$-0.276422\pi$
$53$	9.40655	1.29209	0.646045	+	0.763300i	$0.276422\pi$
$54$	0	0
$55$	7.81318	1.05353
$56$	−2.07407	−0.277160
$57$	0	0
$58$	1.57663	0.207021
$59$	13.3750	1.74128	0.870640	−	0.491921i	$-0.163705\pi$
$59$	13.3750	1.74128	0.870640	+	0.491921i	$0.163705\pi$
$60$	0	0
$61$	0.878868	0.112528	0.0562638	−	0.998416i	$-0.482081\pi$
$61$	0.878868	0.112528	0.0562638	+	0.998416i	$0.482081\pi$
$62$	−8.58747	−1.09061
$63$	0	0
$64$	5.22781	0.653476
$65$	−3.17201	−0.393439
$66$	0	0
$67$	5.75008	0.702484	0.351242	−	0.936285i	$-0.385760\pi$
$67$	5.75008	0.702484	0.351242	+	0.936285i	$0.385760\pi$
$68$	−3.01111	−0.365150
$69$	0	0
$70$	2.57833	0.308170
$71$	11.0541	1.31188	0.655938	−	0.754815i	$-0.272273\pi$
$71$	11.0541	1.31188	0.655938	+	0.754815i	$0.272273\pi$
$72$	0	0
$73$	−8.68524	−1.01653	−0.508265	−	0.861201i	$-0.669713\pi$
$73$	−8.68524	−1.01653	−0.508265	+	0.861201i	$0.669713\pi$
$74$	14.5643	1.69307
$75$	0	0
$76$	−0.455015	−0.0521938
$77$	−3.60589	−0.410929
$78$	0	0
$79$	−6.12127	−0.688697	−0.344349	−	0.938842i	$-0.611900\pi$
$79$	−6.12127	−0.688697	−0.344349	+	0.938842i	$0.611900\pi$
$80$	−8.91421	−0.996640
$81$	0	0
$82$	13.5238	1.49345
$83$	2.70897	0.297348	0.148674	−	0.988886i	$-0.452499\pi$
$83$	2.70897	0.297348	0.148674	+	0.988886i	$0.452499\pi$
$84$	0	0
$85$	−11.6688	−1.26566
$86$	−13.6428	−1.47114
$87$	0	0
$88$	9.90926	1.05633
$89$	1.59743	0.169327	0.0846635	−	0.996410i	$-0.473018\pi$
$89$	1.59743	0.169327	0.0846635	+	0.996410i	$0.473018\pi$
$90$	0	0
$91$	1.46393	0.153461
$92$	−0.485749	−0.0506428
$93$	0	0
$94$	−4.48792	−0.462894
$95$	−1.76330	−0.180911
$96$	0	0
$97$	19.0596	1.93521	0.967603	−	0.252478i	$-0.0812455\pi$
$97$	19.0596	1.93521	0.967603	+	0.252478i	$0.0812455\pi$
$98$	9.84644	0.994641
$99$	0	0
$100$	−0.707518	−0.0707518
$101$	4.74458	0.472103	0.236051	−	0.971741i	$-0.424147\pi$
$101$	4.74458	0.472103	0.236051	+	0.971741i	$0.424147\pi$
$102$	0	0
$103$	11.1015	1.09387	0.546933	−	0.837176i	$-0.315795\pi$
$103$	11.1015	1.09387	0.546933	+	0.837176i	$0.315795\pi$
$104$	−4.02298	−0.394486
$105$	0	0
$106$	−14.8306	−1.44048
$107$	6.99846	0.676566	0.338283	−	0.941044i	$-0.390154\pi$
$107$	6.99846	0.676566	0.338283	+	0.941044i	$0.390154\pi$
$108$	0	0
$109$	−6.42111	−0.615030	−0.307515	−	0.951543i	$-0.599498\pi$
$109$	−6.42111	−0.615030	−0.307515	+	0.951543i	$0.599498\pi$
$110$	−12.3185	−1.17452
$111$	0	0
$112$	4.11403	0.388739
$113$	14.6385	1.37708	0.688538	−	0.725200i	$-0.258253\pi$
$113$	14.6385	1.37708	0.688538	+	0.725200i	$0.258253\pi$
$114$	0	0
$115$	−1.88241	−0.175535
$116$	−0.485749	−0.0451006
$117$	0	0
$118$	−21.0874	−1.94125
$119$	5.38533	0.493672
$120$	0	0
$121$	6.22778	0.566162
$122$	−1.38565	−0.125451
$123$	0	0
$124$	2.64575	0.237595
$125$	−12.1538	−1.08707
$126$	0	0
$127$	12.0445	1.06878	0.534389	−	0.845239i	$-0.320542\pi$
$127$	12.0445	1.06878	0.534389	+	0.845239i	$0.320542\pi$
$128$	−13.6250	−1.20429
$129$	0	0
$130$	5.00107	0.438623
$131$	16.5949	1.44990	0.724951	−	0.688801i	$-0.241862\pi$
$131$	16.5949	1.44990	0.724951	+	0.688801i	$0.241862\pi$
$132$	0	0
$133$	0.813788	0.0705644
$134$	−9.06572	−0.783159
$135$	0	0
$136$	−14.7993	−1.26903
$137$	−16.3451	−1.39646	−0.698230	−	0.715874i	$-0.746029\pi$
$137$	−16.3451	−1.39646	−0.698230	+	0.715874i	$0.746029\pi$
$138$	0	0
$139$	−20.7245	−1.75783	−0.878915	−	0.476979i	$-0.841732\pi$
$139$	−20.7245	−1.75783	−0.878915	+	0.476979i	$0.841732\pi$
$140$	−0.794369	−0.0671364
$141$	0	0
$142$	−17.4281	−1.46254
$143$	−6.99417	−0.584882
$144$	0	0
$145$	−1.88241	−0.156325
$146$	13.6934	1.13327
$147$	0	0
$148$	−4.48717	−0.368843
$149$	−15.1727	−1.24300	−0.621500	−	0.783414i	$-0.713476\pi$
$149$	−15.1727	−1.24300	−0.621500	+	0.783414i	$0.713476\pi$
$150$	0	0
$151$	−7.21297	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$151$	−7.21297	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$152$	−2.23635	−0.181392
$153$	0	0
$154$	5.68514	0.458121
$155$	10.2530	0.823539
$156$	0	0
$157$	2.55512	0.203921	0.101961	−	0.994788i	$-0.467488\pi$
$157$	2.55512	0.203921	0.101961	+	0.994788i	$0.467488\pi$
$158$	9.65096	0.767789
$159$	0	0
$160$	5.06624	0.400522
$161$	0.868755	0.0684675
$162$	0	0
$163$	22.4906	1.76160	0.880798	−	0.473491i	$-0.157006\pi$
$163$	22.4906	1.76160	0.880798	+	0.473491i	$0.157006\pi$
$164$	−4.16659	−0.325356
$165$	0	0
$166$	−4.27104	−0.331497
$167$	7.59344	0.587598	0.293799	−	0.955867i	$-0.405080\pi$
$167$	7.59344	0.587598	0.293799	+	0.955867i	$0.405080\pi$
$168$	0	0
$169$	−10.1605	−0.781576
$170$	18.3974	1.41102
$171$	0	0
$172$	4.20325	0.320495
$173$	22.9798	1.74712	0.873561	−	0.486715i	$-0.161805\pi$
$173$	22.9798	1.74712	0.873561	+	0.486715i	$0.161805\pi$
$174$	0	0
$175$	1.26539	0.0956543
$176$	−19.6555	−1.48159
$177$	0	0
$178$	−2.51854	−0.188773
$179$	−0.974697	−0.0728523	−0.0364262	−	0.999336i	$-0.511597\pi$
$179$	−0.974697	−0.0728523	−0.0364262	+	0.999336i	$0.511597\pi$
$180$	0	0
$181$	11.6037	0.862494	0.431247	−	0.902234i	$-0.358074\pi$
$181$	11.6037	0.862494	0.431247	+	0.902234i	$0.358074\pi$
$182$	−2.30806	−0.171085
$183$	0	0
$184$	−2.38741	−0.176002
$185$	−17.3890	−1.27846
$186$	0	0
$187$	−25.7294	−1.88152
$188$	1.38270	0.100844
$189$	0	0
$190$	2.78007	0.201687
$191$	−5.08587	−0.368001	−0.184000	−	0.982926i	$-0.558905\pi$
$191$	−5.08587	−0.368001	−0.184000	+	0.982926i	$0.558905\pi$
$192$	0	0
$193$	3.36825	0.242452	0.121226	−	0.992625i	$-0.461317\pi$
$193$	3.36825	0.242452	0.121226	+	0.992625i	$0.461317\pi$
$194$	−30.0498	−2.15745
$195$	0	0
$196$	−3.03363	−0.216688
$197$	16.4811	1.17423	0.587114	−	0.809504i	$-0.300264\pi$
$197$	16.4811	1.17423	0.587114	+	0.809504i	$0.300264\pi$
$198$	0	0
$199$	2.15553	0.152801	0.0764007	−	0.997077i	$-0.475657\pi$
$199$	2.15553	0.152801	0.0764007	+	0.997077i	$0.475657\pi$
$200$	−3.47738	−0.245888
$201$	0	0
$202$	−7.48042	−0.526321
$203$	0.868755	0.0609747
$204$	0	0
$205$	−16.1466	−1.12773
$206$	−17.5030	−1.21949
$207$	0	0
$208$	7.97979	0.553299
$209$	−3.88802	−0.268940
$210$	0	0
$211$	−2.06623	−0.142245	−0.0711227	−	0.997468i	$-0.522658\pi$
$211$	−2.06623	−0.142245	−0.0711227	+	0.997468i	$0.522658\pi$
$212$	4.56922	0.313815
$213$	0	0
$214$	−11.0339	−0.754265
$215$	16.2887	1.11088
$216$	0	0
$217$	−4.73189	−0.321221
$218$	10.1237	0.685662
$219$	0	0
$220$	3.79524	0.255875
$221$	10.4457	0.702652
$222$	0	0
$223$	17.2570	1.15561	0.577806	−	0.816174i	$-0.303909\pi$
$223$	17.2570	1.15561	0.577806	+	0.816174i	$0.303909\pi$
$224$	−2.33814	−0.156224
$225$	0	0
$226$	−23.0795	−1.53522
$227$	−26.9508	−1.78879	−0.894393	−	0.447282i	$-0.852392\pi$
$227$	−26.9508	−1.78879	−0.894393	+	0.447282i	$0.852392\pi$
$228$	0	0
$229$	2.02341	0.133711	0.0668554	−	0.997763i	$-0.478703\pi$
$229$	2.02341	0.133711	0.0668554	+	0.997763i	$0.478703\pi$
$230$	2.96785	0.195694
$231$	0	0
$232$	−2.38741	−0.156741
$233$	15.6722	1.02672	0.513360	−	0.858173i	$-0.328401\pi$
$233$	15.6722	1.02672	0.513360	+	0.858173i	$0.328401\pi$
$234$	0	0
$235$	5.35833	0.349539
$236$	6.49690	0.422912
$237$	0	0
$238$	−8.49065	−0.550367
$239$	−23.4791	−1.51873	−0.759367	−	0.650662i	$-0.774491\pi$
$239$	−23.4791	−1.51873	−0.759367	+	0.650662i	$0.774491\pi$
$240$	0	0
$241$	−4.20812	−0.271069	−0.135534	−	0.990773i	$-0.543275\pi$
$241$	−4.20812	−0.271069	−0.135534	+	0.990773i	$0.543275\pi$
$242$	−9.81888	−0.631182
$243$	0	0
$244$	0.426909	0.0273301
$245$	−11.7561	−0.751071
$246$	0	0
$247$	1.57847	0.100435
$248$	13.0036	0.825729
$249$	0	0
$250$	19.1621	1.21192
$251$	−1.80549	−0.113961	−0.0569807	−	0.998375i	$-0.518147\pi$
$251$	−1.80549	−0.113961	−0.0569807	+	0.998375i	$0.518147\pi$
$252$	0	0
$253$	−4.15064	−0.260948
$254$	−18.9897	−1.19152
$255$	0	0
$256$	11.0260	0.689123
$257$	22.1239	1.38005	0.690026	−	0.723784i	$-0.257599\pi$
$257$	22.1239	1.38005	0.690026	+	0.723784i	$0.257599\pi$
$258$	0	0
$259$	8.02525	0.498665
$260$	−1.54080	−0.0955564
$261$	0	0
$262$	−26.1639	−1.61641
$263$	15.9265	0.982070	0.491035	−	0.871140i	$-0.336619\pi$
$263$	15.9265	0.982070	0.491035	+	0.871140i	$0.336619\pi$
$264$	0	0
$265$	17.7069	1.08773
$266$	−1.28304	−0.0786682
$267$	0	0
$268$	2.79309	0.170615
$269$	22.6614	1.38169	0.690844	−	0.723004i	$-0.257239\pi$
$269$	22.6614	1.38169	0.690844	+	0.723004i	$0.257239\pi$
$270$	0	0
$271$	−10.8087	−0.656580	−0.328290	−	0.944577i	$-0.606472\pi$
$271$	−10.8087	−0.656580	−0.328290	+	0.944577i	$0.606472\pi$
$272$	29.3552	1.77992
$273$	0	0
$274$	25.7702	1.55683
$275$	−6.04562	−0.364564
$276$	0	0
$277$	−21.3226	−1.28115	−0.640574	−	0.767896i	$-0.721304\pi$
$277$	−21.3226	−1.28115	−0.640574	+	0.767896i	$0.721304\pi$
$278$	32.6748	1.95970
$279$	0	0
$280$	−3.90425	−0.233323
$281$	25.2749	1.50777	0.753887	−	0.657004i	$-0.228177\pi$
$281$	25.2749	1.50777	0.753887	+	0.657004i	$0.228177\pi$
$282$	0	0
$283$	−13.3764	−0.795143	−0.397572	−	0.917571i	$-0.630147\pi$
$283$	−13.3764	−0.795143	−0.397572	+	0.917571i	$0.630147\pi$
$284$	5.36950	0.318621
$285$	0	0
$286$	11.0272	0.652051
$287$	7.45188	0.439871
$288$	0	0
$289$	21.4264	1.26037
$290$	2.96785	0.174278
$291$	0	0
$292$	−4.21884	−0.246889
$293$	8.93925	0.522237	0.261118	−	0.965307i	$-0.415909\pi$
$293$	8.93925	0.522237	0.261118	+	0.965307i	$0.415909\pi$
$294$	0	0
$295$	25.1772	1.46587
$296$	−22.0540	−1.28186
$297$	0	0
$298$	23.9217	1.38575
$299$	1.68508	0.0974509
$300$	0	0
$301$	−7.51745	−0.433299
$302$	11.3722	0.654394
$303$	0	0
$304$	4.43592	0.254418
$305$	1.65439	0.0947299
$306$	0	0
$307$	12.3674	0.705845	0.352923	−	0.935652i	$-0.385188\pi$
$307$	12.3674	0.705845	0.352923	+	0.935652i	$0.385188\pi$
$308$	−1.75156	−0.0998041
$309$	0	0
$310$	−16.1651	−0.918116
$311$	−17.0409	−0.966299	−0.483150	−	0.875538i	$-0.660507\pi$
$311$	−17.0409	−0.966299	−0.483150	+	0.875538i	$0.660507\pi$
$312$	0	0
$313$	−2.71567	−0.153499	−0.0767494	−	0.997050i	$-0.524454\pi$
$313$	−2.71567	−0.153499	−0.0767494	+	0.997050i	$0.524454\pi$
$314$	−4.02847	−0.227340
$315$	0	0
$316$	−2.97340	−0.167267
$317$	−10.2522	−0.575823	−0.287912	−	0.957657i	$-0.592961\pi$
$317$	−10.2522	−0.575823	−0.287912	+	0.957657i	$0.592961\pi$
$318$	0	0
$319$	−4.15064	−0.232391
$320$	9.84086	0.550121
$321$	0	0
$322$	−1.36970	−0.0763305
$323$	5.80669	0.323093
$324$	0	0
$325$	2.45441	0.136146
$326$	−35.4592	−1.96390
$327$	0	0
$328$	−20.4784	−1.13073
$329$	−2.47294	−0.136338
$330$	0	0
$331$	15.7397	0.865130	0.432565	−	0.901603i	$-0.357609\pi$
$331$	15.7397	0.865130	0.432565	+	0.901603i	$0.357609\pi$
$332$	1.31588	0.0722183
$333$	0	0
$334$	−11.9720	−0.655079
$335$	10.8240	0.591377
$336$	0	0
$337$	34.4880	1.87868	0.939341	−	0.342984i	$-0.111438\pi$
$337$	34.4880	1.87868	0.939341	+	0.342984i	$0.111438\pi$
$338$	16.0193	0.871335
$339$	0	0
$340$	−5.66812	−0.307397
$341$	22.6074	1.22426
$342$	0	0
$343$	11.5069	0.621314
$344$	20.6586	1.11383
$345$	0	0
$346$	−36.2305	−1.94777
$347$	−16.0013	−0.858996	−0.429498	−	0.903068i	$-0.641309\pi$
$347$	−16.0013	−0.858996	−0.429498	+	0.903068i	$0.641309\pi$
$348$	0	0
$349$	−18.5463	−0.992759	−0.496380	−	0.868106i	$-0.665338\pi$
$349$	−18.5463	−0.992759	−0.496380	+	0.868106i	$0.665338\pi$
$350$	−1.99504	−0.106639
$351$	0	0
$352$	11.1709	0.595410
$353$	7.33444	0.390373	0.195186	−	0.980766i	$-0.437469\pi$
$353$	7.33444	0.390373	0.195186	+	0.980766i	$0.437469\pi$
$354$	0	0
$355$	20.8082	1.10439
$356$	0.775948	0.0411252
$357$	0	0
$358$	1.53673	0.0812189
$359$	−7.37549	−0.389263	−0.194632	−	0.980876i	$-0.562351\pi$
$359$	−7.37549	−0.389263	−0.194632	+	0.980876i	$0.562351\pi$
$360$	0	0
$361$	−18.1225	−0.953818
$362$	−18.2947	−0.961546
$363$	0	0
$364$	0.711100	0.0372718
$365$	−16.3491	−0.855753
$366$	0	0
$367$	−4.13238	−0.215708	−0.107854	−	0.994167i	$-0.534398\pi$
$367$	−4.13238	−0.215708	−0.107854	+	0.994167i	$0.534398\pi$
$368$	4.73555	0.246857
$369$	0	0
$370$	27.4159	1.42529
$371$	−8.17199	−0.424269
$372$	0	0
$373$	10.2899	0.532791	0.266396	−	0.963864i	$-0.414167\pi$
$373$	10.2899	0.532791	0.266396	+	0.963864i	$0.414167\pi$
$374$	40.5656	2.09760
$375$	0	0
$376$	6.79584	0.350469
$377$	1.68508	0.0867862
$378$	0	0
$379$	27.5989	1.41766	0.708830	−	0.705379i	$-0.249223\pi$
$379$	27.5989	1.41766	0.708830	+	0.705379i	$0.249223\pi$
$380$	−0.856522	−0.0439387
$381$	0	0
$382$	8.01852	0.410263
$383$	17.7391	0.906425	0.453212	−	0.891403i	$-0.350278\pi$
$383$	17.7391	0.906425	0.453212	+	0.891403i	$0.350278\pi$
$384$	0	0
$385$	−6.78774	−0.345935
$386$	−5.31046	−0.270295
$387$	0	0
$388$	9.25816	0.470012
$389$	19.9988	1.01398	0.506990	−	0.861952i	$-0.330758\pi$
$389$	19.9988	1.01398	0.506990	+	0.861952i	$0.330758\pi$
$390$	0	0
$391$	6.19890	0.313492
$392$	−14.9100	−0.753068
$393$	0	0
$394$	−25.9845	−1.30908
$395$	−11.5227	−0.579771
$396$	0	0
$397$	−18.0445	−0.905627	−0.452814	−	0.891605i	$-0.649580\pi$
$397$	−18.0445	−0.905627	−0.452814	+	0.891605i	$0.649580\pi$
$398$	−3.39846	−0.170350
$399$	0	0
$400$	6.89757	0.344878
$401$	23.2227	1.15969	0.579843	−	0.814728i	$-0.303114\pi$
$401$	23.2227	1.15969	0.579843	+	0.814728i	$0.303114\pi$
$402$	0	0
$403$	−9.17822	−0.457199
$404$	2.30467	0.114662
$405$	0	0
$406$	−1.36970	−0.0679772
$407$	−38.3421	−1.90055
$408$	0	0
$409$	27.6607	1.36773	0.683866	−	0.729608i	$-0.260297\pi$
$409$	27.6607	1.36773	0.683866	+	0.729608i	$0.260297\pi$
$410$	25.4572	1.25724
$411$	0	0
$412$	5.39255	0.265672
$413$	−11.6196	−0.571764
$414$	0	0
$415$	5.09938	0.250319
$416$	−4.53518	−0.222356
$417$	0	0
$418$	6.12995	0.299826
$419$	−23.6562	−1.15568	−0.577840	−	0.816150i	$-0.696104\pi$
$419$	−23.6562	−1.15568	−0.577840	+	0.816150i	$0.696104\pi$
$420$	0	0
$421$	−29.0298	−1.41483	−0.707414	−	0.706800i	$-0.750138\pi$
$421$	−29.0298	−1.41483	−0.707414	+	0.706800i	$0.750138\pi$
$422$	3.25768	0.158581
$423$	0	0
$424$	22.4573	1.09062
$425$	9.02902	0.437972
$426$	0	0
$427$	−0.763522	−0.0369494
$428$	3.39949	0.164321
$429$	0	0
$430$	−25.6812	−1.23846
$431$	−3.64444	−0.175547	−0.0877733	−	0.996140i	$-0.527975\pi$
$431$	−3.64444	−0.175547	−0.0877733	+	0.996140i	$0.527975\pi$
$432$	0	0
$433$	−34.4447	−1.65530	−0.827652	−	0.561241i	$-0.810324\pi$
$433$	−34.4447	−1.65530	−0.827652	+	0.561241i	$0.810324\pi$
$434$	7.46041	0.358111
$435$	0	0
$436$	−3.11904	−0.149375
$437$	0.936729	0.0448098
$438$	0	0
$439$	−3.96062	−0.189030	−0.0945149	−	0.995523i	$-0.530130\pi$
$439$	−3.96062	−0.189030	−0.0945149	+	0.995523i	$0.530130\pi$
$440$	18.6532	0.889258
$441$	0	0
$442$	−16.4689	−0.783346
$443$	−9.13741	−0.434132	−0.217066	−	0.976157i	$-0.569649\pi$
$443$	−9.13741	−0.434132	−0.217066	+	0.976157i	$0.569649\pi$
$444$	0	0
$445$	3.00700	0.142546
$446$	−27.2078	−1.28833
$447$	0	0
$448$	−4.54169	−0.214575
$449$	−8.12020	−0.383216	−0.191608	−	0.981472i	$-0.561370\pi$
$449$	−8.12020	−0.383216	−0.191608	+	0.981472i	$0.561370\pi$
$450$	0	0
$451$	−35.6027	−1.67647
$452$	7.11065	0.334457
$453$	0	0
$454$	42.4913	1.99421
$455$	2.75570	0.129189
$456$	0	0
$457$	−7.38780	−0.345587	−0.172793	−	0.984958i	$-0.555279\pi$
$457$	−7.38780	−0.345587	−0.172793	+	0.984958i	$0.555279\pi$
$458$	−3.19016	−0.149067
$459$	0	0
$460$	−0.914376	−0.0426330
$461$	−18.1761	−0.846548	−0.423274	−	0.906002i	$-0.639119\pi$
$461$	−18.1761	−0.846548	−0.423274	+	0.906002i	$0.639119\pi$
$462$	0	0
$463$	−30.1544	−1.40140	−0.700698	−	0.713458i	$-0.747128\pi$
$463$	−30.1544	−1.40140	−0.700698	+	0.713458i	$0.747128\pi$
$464$	4.73555	0.219842
$465$	0	0
$466$	−24.7092	−1.14463
$467$	2.22092	0.102772	0.0513859	−	0.998679i	$-0.483636\pi$
$467$	2.22092	0.102772	0.0513859	+	0.998679i	$0.483636\pi$
$468$	0	0
$469$	−4.99541	−0.230667
$470$	−8.44809	−0.389681
$471$	0	0
$472$	31.9316	1.46977
$473$	35.9160	1.65142
$474$	0	0
$475$	1.36439	0.0626027
$476$	2.61592	0.119900
$477$	0	0
$478$	37.0177	1.69315
$479$	15.6547	0.715281	0.357641	−	0.933859i	$-0.383581\pi$
$479$	15.6547	0.715281	0.357641	+	0.933859i	$0.383581\pi$
$480$	0	0
$481$	15.5662	0.709757
$482$	6.63463	0.302199
$483$	0	0
$484$	3.02514	0.137506
$485$	35.8778	1.62913
$486$	0	0
$487$	24.6539	1.11718	0.558588	−	0.829445i	$-0.311343\pi$
$487$	24.6539	1.11718	0.558588	+	0.829445i	$0.311343\pi$
$488$	2.09822	0.0949818
$489$	0	0
$490$	18.5350	0.837326
$491$	9.69323	0.437449	0.218725	−	0.975787i	$-0.429810\pi$
$491$	9.69323	0.437449	0.218725	+	0.975787i	$0.429810\pi$
$492$	0	0
$493$	6.19890	0.279184
$494$	−2.48865	−0.111970
$495$	0	0
$496$	−25.7933	−1.15815
$497$	−9.60328	−0.430766
$498$	0	0
$499$	3.36493	0.150635	0.0753174	−	0.997160i	$-0.476003\pi$
$499$	3.36493	0.150635	0.0753174	+	0.997160i	$0.476003\pi$
$500$	−5.90371	−0.264022
$501$	0	0
$502$	2.84658	0.127049
$503$	19.9130	0.887876	0.443938	−	0.896058i	$-0.353581\pi$
$503$	19.9130	0.887876	0.443938	+	0.896058i	$0.353581\pi$
$504$	0	0
$505$	8.93121	0.397434
$506$	6.54400	0.290916
$507$	0	0
$508$	5.85061	0.259579
$509$	14.8402	0.657779	0.328890	−	0.944368i	$-0.393326\pi$
$509$	14.8402	0.657779	0.328890	+	0.944368i	$0.393326\pi$
$510$	0	0
$511$	7.54535	0.333787
$512$	9.86626	0.436031
$513$	0	0
$514$	−34.8811	−1.53854
$515$	20.8976	0.920857
$516$	0	0
$517$	11.8149	0.519620
$518$	−12.6528	−0.555933
$519$	0	0
$520$	−7.57288	−0.332093
$521$	14.5329	0.636699	0.318350	−	0.947973i	$-0.396871\pi$
$521$	14.5329	0.636699	0.318350	+	0.947973i	$0.396871\pi$
$522$	0	0
$523$	−38.9988	−1.70530	−0.852649	−	0.522485i	$-0.825005\pi$
$523$	−38.9988	−1.70530	−0.852649	+	0.522485i	$0.825005\pi$
$524$	8.06094	0.352144
$525$	0	0
$526$	−25.1101	−1.09485
$527$	−33.7638	−1.47077
$528$	0	0
$529$	1.00000	0.0434783
$530$	−27.9172	−1.21265
$531$	0	0
$532$	0.395297	0.0171383
$533$	14.4541	0.626075
$534$	0	0
$535$	13.1739	0.569559
$536$	13.7278	0.592950
$537$	0	0
$538$	−35.7285	−1.54036
$539$	−25.9218	−1.11653
$540$	0	0
$541$	26.3591	1.13327	0.566634	−	0.823970i	$-0.308245\pi$
$541$	26.3591	1.13327	0.566634	+	0.823970i	$0.308245\pi$
$542$	17.0412	0.731983
$543$	0	0
$544$	−16.6835	−0.715300
$545$	−12.0871	−0.517755
$546$	0	0
$547$	24.9433	1.06650	0.533250	−	0.845958i	$-0.320970\pi$
$547$	24.9433	1.06650	0.533250	+	0.845958i	$0.320970\pi$
$548$	−7.93963	−0.339164
$549$	0	0
$550$	9.53167	0.406432
$551$	0.936729	0.0399060
$552$	0	0
$553$	5.31789	0.226140
$554$	33.6177	1.42828
$555$	0	0
$556$	−10.0669	−0.426932
$557$	44.6582	1.89223	0.946114	−	0.323833i	$-0.104971\pi$
$557$	44.6582	1.89223	0.946114	+	0.323833i	$0.104971\pi$
$558$	0	0
$559$	−14.5813	−0.616721
$560$	7.74427	0.327255
$561$	0	0
$562$	−39.8490	−1.68093
$563$	−1.37744	−0.0580524	−0.0290262	−	0.999579i	$-0.509241\pi$
$563$	−1.37744	−0.0580524	−0.0290262	+	0.999579i	$0.509241\pi$
$564$	0	0
$565$	27.5556	1.15927
$566$	21.0895	0.886460
$567$	0	0
$568$	26.3906	1.10732
$569$	−35.8858	−1.50441	−0.752206	−	0.658928i	$-0.771010\pi$
$569$	−35.8858	−1.50441	−0.752206	+	0.658928i	$0.771010\pi$
$570$	0	0
$571$	6.82971	0.285814	0.142907	−	0.989736i	$-0.454355\pi$
$571$	6.82971	0.285814	0.142907	+	0.989736i	$0.454355\pi$
$572$	−3.39741	−0.142053
$573$	0	0
$574$	−11.7488	−0.490387
$575$	1.45655	0.0607424
$576$	0	0
$577$	41.7976	1.74006	0.870029	−	0.493000i	$-0.164100\pi$
$577$	41.7976	1.74006	0.870029	+	0.493000i	$0.164100\pi$
$578$	−33.7813	−1.40512
$579$	0	0
$580$	−0.914376	−0.0379674
$581$	−2.35343	−0.0976369
$582$	0	0
$583$	39.0432	1.61700
$584$	−20.7352	−0.858029
$585$	0	0
$586$	−14.0939	−0.582212
$587$	−36.6226	−1.51158	−0.755788	−	0.654816i	$-0.772746\pi$
$587$	−36.6226	−1.51158	−0.755788	+	0.654816i	$0.772746\pi$
$588$	0	0
$589$	−5.10212	−0.210229
$590$	−39.6950	−1.63422
$591$	0	0
$592$	43.7453	1.79792
$593$	20.1074	0.825711	0.412856	−	0.910796i	$-0.364531\pi$
$593$	20.1074	0.825711	0.412856	+	0.910796i	$0.364531\pi$
$594$	0	0
$595$	10.1374	0.415592
$596$	−7.37014	−0.301893
$597$	0	0
$598$	−2.65675	−0.108642
$599$	−12.2148	−0.499083	−0.249542	−	0.968364i	$-0.580280\pi$
$599$	−12.2148	−0.499083	−0.249542	+	0.968364i	$0.580280\pi$
$600$	0	0
$601$	3.18128	0.129767	0.0648835	−	0.997893i	$-0.479332\pi$
$601$	3.18128	0.129767	0.0648835	+	0.997893i	$0.479332\pi$
$602$	11.8522	0.483060
$603$	0	0
$604$	−3.50369	−0.142563
$605$	11.7232	0.476616
$606$	0	0
$607$	−24.9014	−1.01072	−0.505358	−	0.862910i	$-0.668639\pi$
$607$	−24.9014	−1.01072	−0.505358	+	0.862910i	$0.668639\pi$
$608$	−2.52108	−0.102243
$609$	0	0
$610$	−2.60835	−0.105609
$611$	−4.79665	−0.194052
$612$	0	0
$613$	31.6868	1.27982	0.639910	−	0.768450i	$-0.278972\pi$
$613$	31.6868	1.27982	0.639910	+	0.768450i	$0.278972\pi$
$614$	−19.4988	−0.786907
$615$	0	0
$616$	−8.60872	−0.346855
$617$	13.6332	0.548853	0.274427	−	0.961608i	$-0.411512\pi$
$617$	13.6332	0.548853	0.274427	+	0.961608i	$0.411512\pi$
$618$	0	0
$619$	34.9084	1.40309	0.701544	−	0.712626i	$-0.252494\pi$
$619$	34.9084	1.40309	0.701544	+	0.712626i	$0.252494\pi$
$620$	4.98037	0.200016
$621$	0	0
$622$	26.8671	1.07727
$623$	−1.38777	−0.0556000
$624$	0	0
$625$	−15.5957	−0.623828
$626$	4.28159	0.171127
$627$	0	0
$628$	1.24115	0.0495272
$629$	57.2632	2.28323
$630$	0	0
$631$	19.7464	0.786092	0.393046	−	0.919519i	$-0.371421\pi$
$631$	19.7464	0.786092	0.393046	+	0.919519i	$0.371421\pi$
$632$	−14.6140	−0.581313
$633$	0	0
$634$	16.1640	0.641952
$635$	22.6727	0.899737
$636$	0	0
$637$	10.5238	0.416968
$638$	6.54400	0.259079
$639$	0	0
$640$	−25.6478	−1.01382
$641$	−40.0441	−1.58165	−0.790823	−	0.612044i	$-0.790347\pi$
$641$	−40.0441	−1.58165	−0.790823	+	0.612044i	$0.790347\pi$
$642$	0	0
$643$	34.5243	1.36151	0.680753	−	0.732513i	$-0.261653\pi$
$643$	34.5243	1.36151	0.680753	+	0.732513i	$0.261653\pi$
$644$	0.421997	0.0166290
$645$	0	0
$646$	−9.15497	−0.360198
$647$	44.7524	1.75940	0.879700	−	0.475529i	$-0.157743\pi$
$647$	44.7524	1.75940	0.879700	+	0.475529i	$0.157743\pi$
$648$	0	0
$649$	55.5149	2.17915
$650$	−3.86969	−0.151782
$651$	0	0
$652$	10.9248	0.427847
$653$	1.70807	0.0668421	0.0334210	−	0.999441i	$-0.489360\pi$
$653$	1.70807	0.0668421	0.0334210	+	0.999441i	$0.489360\pi$
$654$	0	0
$655$	31.2383	1.22058
$656$	40.6199	1.58594
$657$	0	0
$658$	3.89891	0.151995
$659$	15.3246	0.596963	0.298482	−	0.954415i	$-0.403520\pi$
$659$	15.3246	0.596963	0.298482	+	0.954415i	$0.403520\pi$
$660$	0	0
$661$	25.7316	1.00084	0.500422	−	0.865781i	$-0.333178\pi$
$661$	25.7316	1.00084	0.500422	+	0.865781i	$0.333178\pi$
$662$	−24.8156	−0.964484
$663$	0	0
$664$	6.46742	0.250985
$665$	1.53188	0.0594037
$666$	0	0
$667$	1.00000	0.0387202
$668$	3.68850	0.142712
$669$	0	0
$670$	−17.0654	−0.659292
$671$	3.64786	0.140824
$672$	0	0
$673$	42.3052	1.63075	0.815373	−	0.578935i	$-0.196532\pi$
$673$	42.3052	1.63075	0.815373	+	0.578935i	$0.196532\pi$
$674$	−54.3747	−2.09444
$675$	0	0
$676$	−4.93545	−0.189825
$677$	−43.5902	−1.67531	−0.837653	−	0.546202i	$-0.816073\pi$
$677$	−43.5902	−1.67531	−0.837653	+	0.546202i	$0.816073\pi$
$678$	0	0
$679$	−16.5581	−0.635441
$680$	−27.8583	−1.06832
$681$	0	0
$682$	−35.6435	−1.36486
$683$	−18.3291	−0.701343	−0.350672	−	0.936499i	$-0.614047\pi$
$683$	−18.3291	−0.701343	−0.350672	+	0.936499i	$0.614047\pi$
$684$	0	0
$685$	−30.7682	−1.17559
$686$	−18.1421	−0.692667
$687$	0	0
$688$	−40.9773	−1.56224
$689$	−15.8508	−0.603869
$690$	0	0
$691$	45.0053	1.71208	0.856041	−	0.516908i	$-0.172917\pi$
$691$	45.0053	1.71208	0.856041	+	0.516908i	$0.172917\pi$
$692$	11.1624	0.424331
$693$	0	0
$694$	25.2281	0.957646
$695$	−39.0119	−1.47981
$696$	0	0
$697$	53.1720	2.01403
$698$	29.2405	1.10677
$699$	0	0
$700$	0.614660	0.0232320
$701$	38.7789	1.46466	0.732329	−	0.680951i	$-0.238434\pi$
$701$	38.7789	1.46466	0.732329	+	0.680951i	$0.238434\pi$
$702$	0	0
$703$	8.65316	0.326360
$704$	21.6987	0.817802
$705$	0	0
$706$	−11.5637	−0.435204
$707$	−4.12188	−0.155019
$708$	0	0
$709$	−4.71179	−0.176955	−0.0884776	−	0.996078i	$-0.528200\pi$
$709$	−4.71179	−0.176955	−0.0884776	+	0.996078i	$0.528200\pi$
$710$	−32.8068	−1.23122
$711$	0	0
$712$	3.81371	0.142925
$713$	−5.44674	−0.203982
$714$	0	0
$715$	−13.1659	−0.492375
$716$	−0.473458	−0.0176940
$717$	0	0
$718$	11.6284	0.433967
$719$	37.5908	1.40190	0.700950	−	0.713211i	$-0.252760\pi$
$719$	37.5908	1.40190	0.700950	+	0.713211i	$0.252760\pi$
$720$	0	0
$721$	−9.64451	−0.359180
$722$	28.5725	1.06336
$723$	0	0
$724$	5.63647	0.209478
$725$	1.45655	0.0540950
$726$	0	0
$727$	15.6317	0.579747	0.289874	−	0.957065i	$-0.406387\pi$
$727$	15.6317	0.579747	0.289874	+	0.957065i	$0.406387\pi$
$728$	3.49499	0.129533
$729$	0	0
$730$	25.7765	0.954030
$731$	−53.6399	−1.98394
$732$	0	0
$733$	−40.0865	−1.48063	−0.740315	−	0.672260i	$-0.765324\pi$
$733$	−40.0865	−1.48063	−0.740315	+	0.672260i	$0.765324\pi$
$734$	6.51521	0.240481
$735$	0	0
$736$	−2.69137	−0.0992051
$737$	23.8665	0.879133
$738$	0	0
$739$	−15.5471	−0.571911	−0.285955	−	0.958243i	$-0.592311\pi$
$739$	−15.5471	−0.571911	−0.285955	+	0.958243i	$0.592311\pi$
$740$	−8.44667	−0.310506
$741$	0	0
$742$	12.8842	0.472993
$743$	−9.14826	−0.335617	−0.167809	−	0.985820i	$-0.553669\pi$
$743$	−9.14826	−0.335617	−0.167809	+	0.985820i	$0.553669\pi$
$744$	0	0
$745$	−28.5613	−1.04640
$746$	−16.2233	−0.593979
$747$	0	0
$748$	−12.4980	−0.456973
$749$	−6.07995	−0.222156
$750$	0	0
$751$	−25.0186	−0.912942	−0.456471	−	0.889738i	$-0.650887\pi$
$751$	−25.0186	−0.912942	−0.456471	+	0.889738i	$0.650887\pi$
$752$	−13.4799	−0.491561
$753$	0	0
$754$	−2.65675	−0.0967530
$755$	−13.5777	−0.494144
$756$	0	0
$757$	−6.91937	−0.251489	−0.125744	−	0.992063i	$-0.540132\pi$
$757$	−6.91937	−0.251489	−0.125744	+	0.992063i	$0.540132\pi$
$758$	−43.5131	−1.58047
$759$	0	0
$760$	−4.20972	−0.152703
$761$	−8.82260	−0.319819	−0.159909	−	0.987132i	$-0.551120\pi$
$761$	−8.82260	−0.319819	−0.159909	+	0.987132i	$0.551120\pi$
$762$	0	0
$763$	5.57837	0.201951
$764$	−2.47046	−0.0893780
$765$	0	0
$766$	−27.9679	−1.01052
$767$	−22.5380	−0.813801
$768$	0	0
$769$	1.32325	0.0477176	0.0238588	−	0.999715i	$-0.492405\pi$
$769$	1.32325	0.0477176	0.0238588	+	0.999715i	$0.492405\pi$
$770$	10.7017	0.385663
$771$	0	0
$772$	1.63612	0.0588853
$773$	5.40626	0.194450	0.0972249	−	0.995262i	$-0.469003\pi$
$773$	5.40626	0.194450	0.0972249	+	0.995262i	$0.469003\pi$
$774$	0	0
$775$	−7.93346	−0.284978
$776$	45.5029	1.63346
$777$	0	0
$778$	−31.5307	−1.13043
$779$	8.03494	0.287881
$780$	0	0
$781$	45.8814	1.64177
$782$	−9.77334	−0.349494
$783$	0	0
$784$	29.5747	1.05624
$785$	4.80978	0.171668
$786$	0	0
$787$	41.8981	1.49351	0.746753	−	0.665102i	$-0.231612\pi$
$787$	41.8981	1.49351	0.746753	+	0.665102i	$0.231612\pi$
$788$	8.00566	0.285190
$789$	0	0
$790$	18.1670	0.646353
$791$	−12.7173	−0.452175
$792$	0	0
$793$	−1.48097	−0.0525907
$794$	28.4494	1.00963
$795$	0	0
$796$	1.04705	0.0371115
$797$	5.09257	0.180388	0.0901940	−	0.995924i	$-0.471251\pi$
$797$	5.09257	0.180388	0.0901940	+	0.995924i	$0.471251\pi$
$798$	0	0
$799$	−17.6454	−0.624249
$800$	−3.92012	−0.138597
$801$	0	0
$802$	−36.6135	−1.29287
$803$	−36.0493	−1.27215
$804$	0	0
$805$	1.63535	0.0576385
$806$	14.4706	0.509706
$807$	0	0
$808$	11.3272	0.398491
$809$	−29.1875	−1.02618	−0.513089	−	0.858335i	$-0.671499\pi$
$809$	−29.1875	−1.02618	−0.513089	+	0.858335i	$0.671499\pi$
$810$	0	0
$811$	22.9329	0.805283	0.402641	−	0.915358i	$-0.368092\pi$
$811$	22.9329	0.805283	0.402641	+	0.915358i	$0.368092\pi$
$812$	0.421997	0.0148092
$813$	0	0
$814$	60.4511	2.11881
$815$	42.3364	1.48298
$816$	0	0
$817$	−8.10564	−0.283580
$818$	−43.6105	−1.52481
$819$	0	0
$820$	−7.84320	−0.273896
$821$	−7.21749	−0.251892	−0.125946	−	0.992037i	$-0.540197\pi$
$821$	−7.21749	−0.251892	−0.125946	+	0.992037i	$0.540197\pi$
$822$	0	0
$823$	35.6266	1.24186	0.620932	−	0.783865i	$-0.286754\pi$
$823$	35.6266	1.24186	0.620932	+	0.783865i	$0.286754\pi$
$824$	26.5039	0.923306
$825$	0	0
$826$	18.3198	0.637427
$827$	36.4344	1.26695	0.633473	−	0.773765i	$-0.281629\pi$
$827$	36.4344	1.26695	0.633473	+	0.773765i	$0.281629\pi$
$828$	0	0
$829$	−25.5740	−0.888223	−0.444112	−	0.895972i	$-0.646481\pi$
$829$	−25.5740	−0.888223	−0.444112	+	0.895972i	$0.646481\pi$
$830$	−8.03982	−0.279066
$831$	0	0
$832$	−8.80930	−0.305408
$833$	38.7138	1.34135
$834$	0	0
$835$	14.2939	0.494662
$836$	−1.88860	−0.0653186
$837$	0	0
$838$	37.2969	1.28840
$839$	−40.2505	−1.38960	−0.694801	−	0.719202i	$-0.744508\pi$
$839$	−40.2505	−1.38960	−0.694801	+	0.719202i	$0.744508\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	45.7692	1.57731
$843$	0	0
$844$	−1.00367	−0.0345478
$845$	−19.1262	−0.657960
$846$	0	0
$847$	−5.41042	−0.185904
$848$	−44.5452	−1.52969
$849$	0	0
$850$	−14.2354	−0.488269
$851$	9.23764	0.316662
$852$	0	0
$853$	−41.6762	−1.42697	−0.713484	−	0.700672i	$-0.752884\pi$
$853$	−41.6762	−1.42697	−0.713484	+	0.700672i	$0.752884\pi$
$854$	1.20379	0.0411928
$855$	0	0
$856$	16.7082	0.571073
$857$	12.9088	0.440955	0.220477	−	0.975392i	$-0.429239\pi$
$857$	12.9088	0.440955	0.220477	+	0.975392i	$0.429239\pi$
$858$	0	0
$859$	−16.0206	−0.546614	−0.273307	−	0.961927i	$-0.588118\pi$
$859$	−16.0206	−0.546614	−0.273307	+	0.961927i	$0.588118\pi$
$860$	7.91221	0.269804
$861$	0	0
$862$	5.74592	0.195707
$863$	51.2650	1.74508	0.872541	−	0.488541i	$-0.162471\pi$
$863$	51.2650	1.74508	0.872541	+	0.488541i	$0.162471\pi$
$864$	0	0
$865$	43.2573	1.47079
$866$	54.3063	1.84540
$867$	0	0
$868$	−2.29851	−0.0780164
$869$	−25.4072	−0.861880
$870$	0	0
$871$	−9.68936	−0.328311
$872$	−15.3298	−0.519132
$873$	0	0
$874$	−1.47687	−0.0499559
$875$	10.5587	0.356950
$876$	0	0
$877$	−35.0554	−1.18374	−0.591868	−	0.806035i	$-0.701610\pi$
$877$	−35.0554	−1.18374	−0.591868	+	0.806035i	$0.701610\pi$
$878$	6.24441	0.210739
$879$	0	0
$880$	−36.9997	−1.24726
$881$	−7.68816	−0.259021	−0.129510	−	0.991578i	$-0.541341\pi$
$881$	−7.68816	−0.259021	−0.129510	+	0.991578i	$0.541341\pi$
$882$	0	0
$883$	−21.6048	−0.727058	−0.363529	−	0.931583i	$-0.618428\pi$
$883$	−21.6048	−0.727058	−0.363529	+	0.931583i	$0.618428\pi$
$884$	5.07397	0.170656
$885$	0	0
$886$	14.4063	0.483989
$887$	13.9394	0.468040	0.234020	−	0.972232i	$-0.424812\pi$
$887$	13.9394	0.468040	0.234020	+	0.972232i	$0.424812\pi$
$888$	0	0
$889$	−10.4637	−0.350942
$890$	−4.74092	−0.158916
$891$	0	0
$892$	8.38256	0.280669
$893$	−2.66643	−0.0892288
$894$	0	0
$895$	−1.83478	−0.0613298
$896$	11.8368	0.395441
$897$	0	0
$898$	12.8025	0.427226
$899$	−5.44674	−0.181659
$900$	0	0
$901$	−58.3103	−1.94260
$902$	56.1322	1.86900
$903$	0	0
$904$	34.9481	1.16236
$905$	21.8428	0.726080
$906$	0	0
$907$	−52.9629	−1.75860	−0.879302	−	0.476264i	$-0.841991\pi$
$907$	−52.9629	−1.75860	−0.879302	+	0.476264i	$0.841991\pi$
$908$	−13.0913	−0.434450
$909$	0	0
$910$	−4.34471	−0.144026
$911$	−45.0081	−1.49118	−0.745592	−	0.666403i	$-0.767833\pi$
$911$	−45.0081	−1.49118	−0.745592	+	0.666403i	$0.767833\pi$
$912$	0	0
$913$	11.2440	0.372121
$914$	11.6478	0.385275
$915$	0	0
$916$	0.982870	0.0324749
$917$	−14.4169	−0.476088
$918$	0	0
$919$	−4.73468	−0.156183	−0.0780913	−	0.996946i	$-0.524883\pi$
$919$	−4.73468	−0.156183	−0.0780913	+	0.996946i	$0.524883\pi$
$920$	−4.49407	−0.148165
$921$	0	0
$922$	28.6570	0.943768
$923$	−18.6270	−0.613116
$924$	0	0
$925$	13.4551	0.442401
$926$	47.5423	1.56234
$927$	0	0
$928$	−2.69137	−0.0883485
$929$	11.8684	0.389391	0.194695	−	0.980864i	$-0.437628\pi$
$929$	11.8684	0.389391	0.194695	+	0.980864i	$0.437628\pi$
$930$	0	0
$931$	5.85012	0.191730
$932$	7.61275	0.249364
$933$	0	0
$934$	−3.50156	−0.114574
$935$	−48.4331	−1.58393
$936$	0	0
$937$	−4.23328	−0.138295	−0.0691476	−	0.997606i	$-0.522028\pi$
$937$	−4.23328	−0.138295	−0.0691476	+	0.997606i	$0.522028\pi$
$938$	7.87589	0.257157
$939$	0	0
$940$	2.60280	0.0848941
$941$	−23.5729	−0.768454	−0.384227	−	0.923239i	$-0.625532\pi$
$941$	−23.5729	−0.768454	−0.384227	+	0.923239i	$0.625532\pi$
$942$	0	0
$943$	8.57766	0.279327
$944$	−63.3380	−2.06148
$945$	0	0
$946$	−56.6261	−1.84107
$947$	−0.731434	−0.0237684	−0.0118842	−	0.999929i	$-0.503783\pi$
$947$	−0.731434	−0.0237684	−0.0118842	+	0.999929i	$0.503783\pi$
$948$	0	0
$949$	14.6354	0.475084
$950$	−2.15114	−0.0697921
$951$	0	0
$952$	12.8570	0.416697
$953$	51.8022	1.67804	0.839020	−	0.544101i	$-0.183129\pi$
$953$	51.8022	1.67804	0.839020	+	0.544101i	$0.183129\pi$
$954$	0	0
$955$	−9.57367	−0.309797
$956$	−11.4049	−0.368862
$957$	0	0
$958$	−24.6816	−0.797426
$959$	14.1999	0.458540
$960$	0	0
$961$	−1.33301	−0.0430004
$962$	−24.5421	−0.791268
$963$	0	0
$964$	−2.04409	−0.0658357
$965$	6.34040	0.204105
$966$	0	0
$967$	15.9898	0.514198	0.257099	−	0.966385i	$-0.417233\pi$
$967$	15.9898	0.514198	0.257099	+	0.966385i	$0.417233\pi$
$968$	14.8683	0.477884
$969$	0	0
$970$	−56.5659	−1.81622
$971$	33.5583	1.07694	0.538469	−	0.842645i	$-0.319003\pi$
$971$	33.5583	1.07694	0.538469	+	0.842645i	$0.319003\pi$
$972$	0	0
$973$	18.0045	0.577198
$974$	−38.8700	−1.24547
$975$	0	0
$976$	−4.16192	−0.133220
$977$	−44.1508	−1.41251	−0.706255	−	0.707958i	$-0.749617\pi$
$977$	−44.1508	−1.41251	−0.706255	+	0.707958i	$0.749617\pi$
$978$	0	0
$979$	6.63034	0.211907
$980$	−5.71052	−0.182416
$981$	0	0
$982$	−15.2826	−0.487687
$983$	24.0011	0.765518	0.382759	−	0.923848i	$-0.374974\pi$
$983$	24.0011	0.765518	0.382759	+	0.923848i	$0.374974\pi$
$984$	0	0
$985$	31.0241	0.988508
$986$	−9.77334	−0.311247
$987$	0	0
$988$	0.766738	0.0243932
$989$	−8.65313	−0.275154
$990$	0	0
$991$	−46.4921	−1.47687	−0.738436	−	0.674324i	$-0.764435\pi$
$991$	−46.4921	−1.47687	−0.738436	+	0.674324i	$0.764435\pi$
$992$	14.6592	0.465430
$993$	0	0
$994$	15.1408	0.480236
$995$	4.05758	0.128634
$996$	0	0
$997$	−19.7245	−0.624680	−0.312340	−	0.949970i	$-0.601113\pi$
$997$	−19.7245	−0.624680	−0.312340	+	0.949970i	$0.601113\pi$
$998$	−5.30523	−0.167934
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.l.1.2		10
3.2	odd	2		667.2.a.a.1.9	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.a.1.9	✓	10	3.2	odd	2
6003.2.a.l.1.2		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)

\(f(q)\)	\(=\)	\(q-1.57663 q^{2} +0.485749 q^{4} +1.88241 q^{5} -0.868755 q^{7} +2.38741 q^{8} +O(q^{10})\)	\(q-1.57663 q^{2} +0.485749 q^{4} +1.88241 q^{5} -0.868755 q^{7} +2.38741 q^{8} -2.96785 q^{10} +4.15064 q^{11} -1.68508 q^{13} +1.36970 q^{14} -4.73555 q^{16} -6.19890 q^{17} -0.936729 q^{19} +0.914376 q^{20} -6.54400 q^{22} -1.00000 q^{23} -1.45655 q^{25} +2.65675 q^{26} -0.421997 q^{28} -1.00000 q^{29} +5.44674 q^{31} +2.69137 q^{32} +9.77334 q^{34} -1.63535 q^{35} -9.23764 q^{37} +1.47687 q^{38} +4.49407 q^{40} -8.57766 q^{41} +8.65313 q^{43} +2.01617 q^{44} +1.57663 q^{46} +2.84654 q^{47} -6.24526 q^{49} +2.29644 q^{50} -0.818527 q^{52} +9.40655 q^{53} +7.81318 q^{55} -2.07407 q^{56} +1.57663 q^{58} +13.3750 q^{59} +0.878868 q^{61} -8.58747 q^{62} +5.22781 q^{64} -3.17201 q^{65} +5.75008 q^{67} -3.01111 q^{68} +2.57833 q^{70} +11.0541 q^{71} -8.68524 q^{73} +14.5643 q^{74} -0.455015 q^{76} -3.60589 q^{77} -6.12127 q^{79} -8.91421 q^{80} +13.5238 q^{82} +2.70897 q^{83} -11.6688 q^{85} -13.6428 q^{86} +9.90926 q^{88} +1.59743 q^{89} +1.46393 q^{91} -0.485749 q^{92} -4.48792 q^{94} -1.76330 q^{95} +19.0596 q^{97} +9.84644 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 3 q^{2} + 9 q^{4} + 10 q^{5} + q^{7} + 9 q^{8}+O(q^{10}) \) 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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