LMFDB - Embedded newform 6003.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$47.9341963334$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 13 x^{10} + 41 x^{9} + 54 x^{8} - 188 x^{7} - 77 x^{6} + 342 x^{5} + 13 x^{4} + \cdots - 5 $ x^12 - 3x^11 - 13x^10 + 41x^9 + 54x^8 - 188x^7 - 77x^6 + 342x^5 + 13x^4 - 215x^3 + 9x^2 + 37*x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.52122$ 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)

$f(q)$	$=$	$q-2.52122 q^{2} +4.35654 q^{4} +3.14716 q^{5} -3.79117 q^{7} -5.94135 q^{8} +O(q^{10})$	\(q-2.52122 q^{2} +4.35654 q^{4} +3.14716 q^{5} -3.79117 q^{7} -5.94135 q^{8} -7.93467 q^{10} -2.98551 q^{11} -5.12356 q^{13} +9.55838 q^{14} +6.26637 q^{16} -1.85851 q^{17} -6.91251 q^{19} +13.7107 q^{20} +7.52712 q^{22} +1.00000 q^{23} +4.90459 q^{25} +12.9176 q^{26} -16.5164 q^{28} +1.00000 q^{29} +1.29487 q^{31} -3.91617 q^{32} +4.68572 q^{34} -11.9314 q^{35} -3.14889 q^{37} +17.4279 q^{38} -18.6984 q^{40} -9.94795 q^{41} +4.71417 q^{43} -13.0065 q^{44} -2.52122 q^{46} +12.1834 q^{47} +7.37300 q^{49} -12.3655 q^{50} -22.3210 q^{52} +5.29945 q^{53} -9.39586 q^{55} +22.5247 q^{56} -2.52122 q^{58} -8.92021 q^{59} -4.50085 q^{61} -3.26464 q^{62} -2.65922 q^{64} -16.1246 q^{65} -10.2328 q^{67} -8.09669 q^{68} +30.0817 q^{70} -13.6760 q^{71} +14.2658 q^{73} +7.93903 q^{74} -30.1146 q^{76} +11.3186 q^{77} -1.12804 q^{79} +19.7212 q^{80} +25.0810 q^{82} -8.19264 q^{83} -5.84903 q^{85} -11.8854 q^{86} +17.7380 q^{88} +10.0455 q^{89} +19.4243 q^{91} +4.35654 q^{92} -30.7171 q^{94} -21.7547 q^{95} -1.85984 q^{97} -18.5889 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{2} + 11 q^{4} + 16 q^{5} - 7 q^{7} + 9 q^{8}+O(q^{10}) $ 12 * q + 3 * q^2 + 11 * q^4 + 16 * q^5 - 7 * q^7 + 9 * q^8	$ 12 q + 3 q^{2} + 11 q^{4} + 16 q^{5} - 7 q^{7} + 9 q^{8} + 6 q^{11} - 15 q^{13} + 8 q^{14} + 17 q^{16} + 18 q^{17} - 6 q^{19} + 39 q^{20} - 5 q^{22} + 12 q^{23} + 14 q^{25} + 3 q^{26} - 19 q^{28} + 12 q^{29} + 16 q^{31} + 21 q^{32} - 7 q^{34} + 11 q^{35} - q^{37} + 24 q^{38} + 30 q^{40} - 3 q^{41} - 23 q^{43} - 23 q^{44} + 3 q^{46} + 35 q^{47} + 3 q^{49} + 2 q^{50} + 45 q^{53} + 17 q^{55} + 17 q^{56} + 3 q^{58} + 11 q^{59} + 4 q^{61} + 7 q^{62} + 15 q^{64} - 5 q^{65} - 19 q^{67} - q^{68} + 14 q^{70} - 19 q^{71} + 10 q^{73} + 15 q^{74} - 4 q^{76} + 39 q^{77} + 17 q^{79} + 90 q^{80} - 3 q^{82} + 12 q^{83} + 14 q^{85} - 17 q^{86} - 2 q^{88} + 20 q^{89} + 11 q^{91} + 11 q^{92} + 13 q^{94} - 12 q^{95} - 12 q^{97} - 75 q^{98}+O(q^{100}) $ 12 * q + 3 * q^2 + 11 * q^4 + 16 * q^5 - 7 * q^7 + 9 * q^8 + 6 * q^11 - 15 * q^13 + 8 * q^14 + 17 * q^16 + 18 * q^17 - 6 * q^19 + 39 * q^20 - 5 * q^22 + 12 * q^23 + 14 * q^25 + 3 * q^26 - 19 * q^28 + 12 * q^29 + 16 * q^31 + 21 * q^32 - 7 * q^34 + 11 * q^35 - q^37 + 24 * q^38 + 30 * q^40 - 3 * q^41 - 23 * q^43 - 23 * q^44 + 3 * q^46 + 35 * q^47 + 3 * q^49 + 2 * q^50 + 45 * q^53 + 17 * q^55 + 17 * q^56 + 3 * q^58 + 11 * q^59 + 4 * q^61 + 7 * q^62 + 15 * q^64 - 5 * q^65 - 19 * q^67 - q^68 + 14 * q^70 - 19 * q^71 + 10 * q^73 + 15 * q^74 - 4 * q^76 + 39 * q^77 + 17 * q^79 + 90 * q^80 - 3 * q^82 + 12 * q^83 + 14 * q^85 - 17 * q^86 - 2 * q^88 + 20 * q^89 + 11 * q^91 + 11 * q^92 + 13 * q^94 - 12 * q^95 - 12 * q^97 - 75 * 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 $

$2$	−2.52122	−1.78277	−0.891385	−	0.453247i	$-0.850266\pi$
$2$	−2.52122	−1.78277	−0.891385	+	0.453247i	$0.850266\pi$
$3$	0	0
$3$	0	0
$4$	4.35654	2.17827
$5$	3.14716	1.40745	0.703725	−	0.710472i	$-0.251519\pi$
$5$	3.14716	1.40745	0.703725	+	0.710472i	$0.251519\pi$
$6$	0	0
$7$	−3.79117	−1.43293	−0.716464	−	0.697624i	$-0.754241\pi$
$7$	−3.79117	−1.43293	−0.716464	+	0.697624i	$0.754241\pi$
$8$	−5.94135	−2.10059
$9$	0	0
$10$	−7.93467	−2.50916
$11$	−2.98551	−0.900164	−0.450082	−	0.892987i	$-0.648605\pi$
$11$	−2.98551	−0.900164	−0.450082	+	0.892987i	$0.648605\pi$
$12$	0	0
$13$	−5.12356	−1.42102	−0.710509	−	0.703688i	$-0.751535\pi$
$13$	−5.12356	−1.42102	−0.710509	+	0.703688i	$0.751535\pi$
$14$	9.55838	2.55458
$15$	0	0
$16$	6.26637	1.56659
$17$	−1.85851	−0.450756	−0.225378	−	0.974271i	$-0.572362\pi$
$17$	−1.85851	−0.450756	−0.225378	+	0.974271i	$0.572362\pi$
$18$	0	0
$19$	−6.91251	−1.58584	−0.792919	−	0.609327i	$-0.791440\pi$
$19$	−6.91251	−1.58584	−0.792919	+	0.609327i	$0.791440\pi$
$20$	13.7107	3.06581
$21$	0	0
$22$	7.52712	1.60479
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	4.90459	0.980918
$26$	12.9176	2.53335
$27$	0	0
$28$	−16.5164	−3.12131
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	1.29487	0.232565	0.116282	−	0.993216i	$-0.462902\pi$
$31$	1.29487	0.232565	0.116282	+	0.993216i	$0.462902\pi$
$32$	−3.91617	−0.692287
$33$	0	0
$34$	4.68572	0.803594
$35$	−11.9314	−2.01678
$36$	0	0
$37$	−3.14889	−0.517674	−0.258837	−	0.965921i	$-0.583339\pi$
$37$	−3.14889	−0.517674	−0.258837	+	0.965921i	$0.583339\pi$
$38$	17.4279	2.82718
$39$	0	0
$40$	−18.6984	−2.95647
$41$	−9.94795	−1.55361	−0.776805	−	0.629742i	$-0.783161\pi$
$41$	−9.94795	−1.55361	−0.776805	+	0.629742i	$0.783161\pi$
$42$	0	0
$43$	4.71417	0.718904	0.359452	−	0.933164i	$-0.382964\pi$
$43$	4.71417	0.718904	0.359452	+	0.933164i	$0.382964\pi$
$44$	−13.0065	−1.96080
$45$	0	0
$46$	−2.52122	−0.371733
$47$	12.1834	1.77714	0.888568	−	0.458744i	$-0.151701\pi$
$47$	12.1834	1.77714	0.888568	+	0.458744i	$0.151701\pi$
$48$	0	0
$49$	7.37300	1.05329
$50$	−12.3655	−1.74875
$51$	0	0
$52$	−22.3210	−3.09536
$53$	5.29945	0.727935	0.363968	−	0.931412i	$-0.381422\pi$
$53$	5.29945	0.727935	0.363968	+	0.931412i	$0.381422\pi$
$54$	0	0
$55$	−9.39586	−1.26694
$56$	22.5247	3.00999
$57$	0	0
$58$	−2.52122	−0.331052
$59$	−8.92021	−1.16131	−0.580656	−	0.814149i	$-0.697204\pi$
$59$	−8.92021	−1.16131	−0.580656	+	0.814149i	$0.697204\pi$
$60$	0	0
$61$	−4.50085	−0.576274	−0.288137	−	0.957589i	$-0.593036\pi$
$61$	−4.50085	−0.576274	−0.288137	+	0.957589i	$0.593036\pi$
$62$	−3.26464	−0.414610
$63$	0	0
$64$	−2.65922	−0.332402
$65$	−16.1246	−2.00001
$66$	0	0
$67$	−10.2328	−1.25014	−0.625069	−	0.780569i	$-0.714929\pi$
$67$	−10.2328	−1.25014	−0.625069	+	0.780569i	$0.714929\pi$
$68$	−8.09669	−0.981868
$69$	0	0
$70$	30.0817	3.59545
$71$	−13.6760	−1.62305	−0.811524	−	0.584319i	$-0.801362\pi$
$71$	−13.6760	−1.62305	−0.811524	+	0.584319i	$0.801362\pi$
$72$	0	0
$73$	14.2658	1.66969	0.834846	−	0.550484i	$-0.185557\pi$
$73$	14.2658	1.66969	0.834846	+	0.550484i	$0.185557\pi$
$74$	7.93903	0.922893
$75$	0	0
$76$	−30.1146	−3.45438
$77$	11.3186	1.28987
$78$	0	0
$79$	−1.12804	−0.126914	−0.0634570	−	0.997985i	$-0.520213\pi$
$79$	−1.12804	−0.126914	−0.0634570	+	0.997985i	$0.520213\pi$
$80$	19.7212	2.20490
$81$	0	0
$82$	25.0810	2.76973
$83$	−8.19264	−0.899259	−0.449630	−	0.893215i	$-0.648444\pi$
$83$	−8.19264	−0.899259	−0.449630	+	0.893215i	$0.648444\pi$
$84$	0	0
$85$	−5.84903	−0.634417
$86$	−11.8854	−1.28164
$87$	0	0
$88$	17.7380	1.89087
$89$	10.0455	1.06482	0.532412	−	0.846486i	$-0.321286\pi$
$89$	10.0455	1.06482	0.532412	+	0.846486i	$0.321286\pi$
$90$	0	0
$91$	19.4243	2.03622
$92$	4.35654	0.454201
$93$	0	0
$94$	−30.7171	−3.16823
$95$	−21.7547	−2.23199
$96$	0	0
$97$	−1.85984	−0.188838	−0.0944188	−	0.995533i	$-0.530099\pi$
$97$	−1.85984	−0.188838	−0.0944188	+	0.995533i	$0.530099\pi$
$98$	−18.5889	−1.87777
$99$	0	0
$100$	21.3670	2.13670
$101$	15.2668	1.51910	0.759551	−	0.650447i	$-0.225418\pi$
$101$	15.2668	1.51910	0.759551	+	0.650447i	$0.225418\pi$
$102$	0	0
$103$	−7.80568	−0.769116	−0.384558	−	0.923101i	$-0.625646\pi$
$103$	−7.80568	−0.769116	−0.384558	+	0.923101i	$0.625646\pi$
$104$	30.4409	2.98497
$105$	0	0
$106$	−13.3611	−1.29774
$107$	3.52746	0.341012	0.170506	−	0.985357i	$-0.445460\pi$
$107$	3.52746	0.341012	0.170506	+	0.985357i	$0.445460\pi$
$108$	0	0
$109$	−15.9227	−1.52512	−0.762559	−	0.646918i	$-0.776058\pi$
$109$	−15.9227	−1.52512	−0.762559	+	0.646918i	$0.776058\pi$
$110$	23.6890	2.25866
$111$	0	0
$112$	−23.7569	−2.24481
$113$	13.1546	1.23749	0.618743	−	0.785594i	$-0.287642\pi$
$113$	13.1546	1.23749	0.618743	+	0.785594i	$0.287642\pi$
$114$	0	0
$115$	3.14716	0.293474
$116$	4.35654	0.404495
$117$	0	0
$118$	22.4898	2.07035
$119$	7.04595	0.645901
$120$	0	0
$121$	−2.08675	−0.189704
$122$	11.3476	1.02736
$123$	0	0
$124$	5.64114	0.506589
$125$	−0.300275	−0.0268574
$126$	0	0
$127$	−5.19246	−0.460756	−0.230378	−	0.973101i	$-0.573996\pi$
$127$	−5.19246	−0.460756	−0.230378	+	0.973101i	$0.573996\pi$
$128$	14.5368	1.28488
$129$	0	0
$130$	40.6537	3.56557
$131$	8.69892	0.760028	0.380014	−	0.924981i	$-0.375919\pi$
$131$	8.69892	0.760028	0.380014	+	0.924981i	$0.375919\pi$
$132$	0	0
$133$	26.2065	2.27239
$134$	25.7992	2.22871
$135$	0	0
$136$	11.0421	0.946852
$137$	13.7257	1.17267	0.586333	−	0.810070i	$-0.300571\pi$
$137$	13.7257	1.17267	0.586333	+	0.810070i	$0.300571\pi$
$138$	0	0
$139$	0.113011	0.00958547	0.00479273	−	0.999989i	$-0.498474\pi$
$139$	0.113011	0.00958547	0.00479273	+	0.999989i	$0.498474\pi$
$140$	−51.9797	−4.39309
$141$	0	0
$142$	34.4803	2.89352
$143$	15.2964	1.27915
$144$	0	0
$145$	3.14716	0.261357
$146$	−35.9673	−2.97668
$147$	0	0
$148$	−13.7183	−1.12763
$149$	13.9008	1.13880	0.569400	−	0.822061i	$-0.307176\pi$
$149$	13.9008	1.13880	0.569400	+	0.822061i	$0.307176\pi$
$150$	0	0
$151$	23.7251	1.93072	0.965360	−	0.260922i	$-0.0840266\pi$
$151$	23.7251	1.93072	0.965360	+	0.260922i	$0.0840266\pi$
$152$	41.0696	3.33119
$153$	0	0
$154$	−28.5366	−2.29954
$155$	4.07514	0.327323
$156$	0	0
$157$	4.70416	0.375433	0.187716	−	0.982223i	$-0.439891\pi$
$157$	4.70416	0.375433	0.187716	+	0.982223i	$0.439891\pi$
$158$	2.84403	0.226259
$159$	0	0
$160$	−12.3248	−0.974360
$161$	−3.79117	−0.298786
$162$	0	0
$163$	10.9750	0.859629	0.429814	−	0.902917i	$-0.358579\pi$
$163$	10.9750	0.859629	0.429814	+	0.902917i	$0.358579\pi$
$164$	−43.3387	−3.38418
$165$	0	0
$166$	20.6554	1.60317
$167$	8.02785	0.621213	0.310607	−	0.950539i	$-0.399468\pi$
$167$	8.02785	0.621213	0.310607	+	0.950539i	$0.399468\pi$
$168$	0	0
$169$	13.2508	1.01929
$170$	14.7467	1.13102
$171$	0	0
$172$	20.5375	1.56597
$173$	−13.2898	−1.01041	−0.505204	−	0.863000i	$-0.668583\pi$
$173$	−13.2898	−1.01041	−0.505204	+	0.863000i	$0.668583\pi$
$174$	0	0
$175$	−18.5941	−1.40559
$176$	−18.7083	−1.41019
$177$	0	0
$178$	−25.3270	−1.89834
$179$	−4.13541	−0.309095	−0.154548	−	0.987985i	$-0.549392\pi$
$179$	−4.13541	−0.309095	−0.154548	+	0.987985i	$0.549392\pi$
$180$	0	0
$181$	12.7399	0.946949	0.473474	−	0.880808i	$-0.343000\pi$
$181$	12.7399	0.946949	0.473474	+	0.880808i	$0.343000\pi$
$182$	−48.9729	−3.63011
$183$	0	0
$184$	−5.94135	−0.438002
$185$	−9.91004	−0.728600
$186$	0	0
$187$	5.54861	0.405754
$188$	53.0776	3.87108
$189$	0	0
$190$	54.8484	3.97912
$191$	−14.2667	−1.03230	−0.516152	−	0.856497i	$-0.672636\pi$
$191$	−14.2667	−1.03230	−0.516152	+	0.856497i	$0.672636\pi$
$192$	0	0
$193$	−9.20093	−0.662297	−0.331149	−	0.943579i	$-0.607436\pi$
$193$	−9.20093	−0.662297	−0.331149	+	0.943579i	$0.607436\pi$
$194$	4.68905	0.336654
$195$	0	0
$196$	32.1208	2.29434
$197$	11.6347	0.828935	0.414468	−	0.910064i	$-0.363968\pi$
$197$	11.6347	0.828935	0.414468	+	0.910064i	$0.363968\pi$
$198$	0	0
$199$	2.90995	0.206281	0.103140	−	0.994667i	$-0.467111\pi$
$199$	2.90995	0.206281	0.103140	+	0.994667i	$0.467111\pi$
$200$	−29.1399	−2.06050
$201$	0	0
$202$	−38.4909	−2.70821
$203$	−3.79117	−0.266088
$204$	0	0
$205$	−31.3078	−2.18663
$206$	19.6798	1.37116
$207$	0	0
$208$	−32.1061	−2.22616
$209$	20.6373	1.42751
$210$	0	0
$211$	3.20441	0.220601	0.110300	−	0.993898i	$-0.464819\pi$
$211$	3.20441	0.220601	0.110300	+	0.993898i	$0.464819\pi$
$212$	23.0873	1.58564
$213$	0	0
$214$	−8.89349	−0.607946
$215$	14.8362	1.01182
$216$	0	0
$217$	−4.90906	−0.333249
$218$	40.1446	2.71894
$219$	0	0
$220$	−40.9334	−2.75973
$221$	9.52220	0.640533
$222$	0	0
$223$	5.37585	0.359993	0.179997	−	0.983667i	$-0.442391\pi$
$223$	5.37585	0.359993	0.179997	+	0.983667i	$0.442391\pi$
$224$	14.8469	0.991999
$225$	0	0
$226$	−33.1657	−2.20615
$227$	2.91331	0.193363	0.0966815	−	0.995315i	$-0.469177\pi$
$227$	2.91331	0.193363	0.0966815	+	0.995315i	$0.469177\pi$
$228$	0	0
$229$	−6.82865	−0.451250	−0.225625	−	0.974214i	$-0.572442\pi$
$229$	−6.82865	−0.451250	−0.225625	+	0.974214i	$0.572442\pi$
$230$	−7.93467	−0.523196
$231$	0	0
$232$	−5.94135	−0.390069
$233$	19.4118	1.27171	0.635854	−	0.771810i	$-0.280648\pi$
$233$	19.4118	1.27171	0.635854	+	0.771810i	$0.280648\pi$
$234$	0	0
$235$	38.3432	2.50123
$236$	−38.8613	−2.52965
$237$	0	0
$238$	−17.7644	−1.15149
$239$	−17.4266	−1.12723	−0.563616	−	0.826037i	$-0.690590\pi$
$239$	−17.4266	−1.12723	−0.563616	+	0.826037i	$0.690590\pi$
$240$	0	0
$241$	−1.64339	−0.105860	−0.0529300	−	0.998598i	$-0.516856\pi$
$241$	−1.64339	−0.105860	−0.0529300	+	0.998598i	$0.516856\pi$
$242$	5.26114	0.338199
$243$	0	0
$244$	−19.6081	−1.25528
$245$	23.2040	1.48245
$246$	0	0
$247$	35.4166	2.25351
$248$	−7.69325	−0.488522
$249$	0	0
$250$	0.757058	0.0478805
$251$	−10.4267	−0.658130	−0.329065	−	0.944307i	$-0.606733\pi$
$251$	−10.4267	−0.658130	−0.329065	+	0.944307i	$0.606733\pi$
$252$	0	0
$253$	−2.98551	−0.187697
$254$	13.0913	0.821423
$255$	0	0
$256$	−31.3320	−1.95825
$257$	−1.99007	−0.124137	−0.0620687	−	0.998072i	$-0.519770\pi$
$257$	−1.99007	−0.124137	−0.0620687	+	0.998072i	$0.519770\pi$
$258$	0	0
$259$	11.9380	0.741790
$260$	−70.2476	−4.35657
$261$	0	0
$262$	−21.9319	−1.35496
$263$	−8.37646	−0.516515	−0.258257	−	0.966076i	$-0.583148\pi$
$263$	−8.37646	−0.516515	−0.258257	+	0.966076i	$0.583148\pi$
$264$	0	0
$265$	16.6782	1.02453
$266$	−66.0723	−4.05116
$267$	0	0
$268$	−44.5797	−2.72314
$269$	3.85784	0.235216	0.117608	−	0.993060i	$-0.462477\pi$
$269$	3.85784	0.235216	0.117608	+	0.993060i	$0.462477\pi$
$270$	0	0
$271$	−4.16788	−0.253180	−0.126590	−	0.991955i	$-0.540403\pi$
$271$	−4.16788	−0.253180	−0.126590	+	0.991955i	$0.540403\pi$
$272$	−11.6461	−0.706150
$273$	0	0
$274$	−34.6055	−2.09059
$275$	−14.6427	−0.882987
$276$	0	0
$277$	−2.23609	−0.134354	−0.0671769	−	0.997741i	$-0.521399\pi$
$277$	−2.23609	−0.134354	−0.0671769	+	0.997741i	$0.521399\pi$
$278$	−0.284925	−0.0170887
$279$	0	0
$280$	70.8887	4.23641
$281$	−10.5151	−0.627279	−0.313640	−	0.949542i	$-0.601548\pi$
$281$	−10.5151	−0.627279	−0.313640	+	0.949542i	$0.601548\pi$
$282$	0	0
$283$	1.60267	0.0952691	0.0476345	−	0.998865i	$-0.484832\pi$
$283$	1.60267	0.0952691	0.0476345	+	0.998865i	$0.484832\pi$
$284$	−59.5802	−3.53544
$285$	0	0
$286$	−38.5656	−2.28043
$287$	37.7144	2.22621
$288$	0	0
$289$	−13.5459	−0.796819
$290$	−7.93467	−0.465940
$291$	0	0
$292$	62.1498	3.63704
$293$	15.5340	0.907503	0.453752	−	0.891128i	$-0.350085\pi$
$293$	15.5340	0.907503	0.453752	+	0.891128i	$0.350085\pi$
$294$	0	0
$295$	−28.0733	−1.63449
$296$	18.7086	1.08742
$297$	0	0
$298$	−35.0470	−2.03022
$299$	−5.12356	−0.296303
$300$	0	0
$301$	−17.8722	−1.03014
$302$	−59.8161	−3.44203
$303$	0	0
$304$	−43.3163	−2.48436
$305$	−14.1649	−0.811078
$306$	0	0
$307$	−33.9723	−1.93890	−0.969449	−	0.245291i	$-0.921116\pi$
$307$	−33.9723	−1.93890	−0.969449	+	0.245291i	$0.921116\pi$
$308$	49.3098	2.80969
$309$	0	0
$310$	−10.2743	−0.583543
$311$	19.8179	1.12377	0.561886	−	0.827215i	$-0.310076\pi$
$311$	19.8179	1.12377	0.561886	+	0.827215i	$0.310076\pi$
$312$	0	0
$313$	11.5082	0.650484	0.325242	−	0.945631i	$-0.394554\pi$
$313$	11.5082	0.650484	0.325242	+	0.945631i	$0.394554\pi$
$314$	−11.8602	−0.669311
$315$	0	0
$316$	−4.91434	−0.276453
$317$	30.2031	1.69638	0.848188	−	0.529696i	$-0.177694\pi$
$317$	30.2031	1.69638	0.848188	+	0.529696i	$0.177694\pi$
$318$	0	0
$319$	−2.98551	−0.167156
$320$	−8.36897	−0.467840
$321$	0	0
$322$	9.55838	0.532667
$323$	12.8470	0.714826
$324$	0	0
$325$	−25.1289	−1.39390
$326$	−27.6704	−1.53252
$327$	0	0
$328$	59.1043	3.26349
$329$	−46.1895	−2.54651
$330$	0	0
$331$	5.13126	0.282040	0.141020	−	0.990007i	$-0.454962\pi$
$331$	5.13126	0.282040	0.141020	+	0.990007i	$0.454962\pi$
$332$	−35.6916	−1.95883
$333$	0	0
$334$	−20.2399	−1.10748
$335$	−32.2043	−1.75951
$336$	0	0
$337$	24.8690	1.35470	0.677350	−	0.735661i	$-0.263128\pi$
$337$	24.8690	1.35470	0.677350	+	0.735661i	$0.263128\pi$
$338$	−33.4082	−1.81717
$339$	0	0
$340$	−25.4816	−1.38193
$341$	−3.86583	−0.209346
$342$	0	0
$343$	−1.41410	−0.0763543
$344$	−28.0085	−1.51012
$345$	0	0
$346$	33.5066	1.80133
$347$	32.0746	1.72186	0.860928	−	0.508727i	$-0.169884\pi$
$347$	32.0746	1.72186	0.860928	+	0.508727i	$0.169884\pi$
$348$	0	0
$349$	18.3452	0.981996	0.490998	−	0.871161i	$-0.336632\pi$
$349$	18.3452	0.981996	0.490998	+	0.871161i	$0.336632\pi$
$350$	46.8799	2.50584
$351$	0	0
$352$	11.6918	0.623172
$353$	2.31281	0.123099	0.0615494	−	0.998104i	$-0.480396\pi$
$353$	2.31281	0.123099	0.0615494	+	0.998104i	$0.480396\pi$
$354$	0	0
$355$	−43.0406	−2.28436
$356$	43.7637	2.31947
$357$	0	0
$358$	10.4263	0.551046
$359$	−12.9321	−0.682529	−0.341264	−	0.939967i	$-0.610855\pi$
$359$	−12.9321	−0.682529	−0.341264	+	0.939967i	$0.610855\pi$
$360$	0	0
$361$	28.7828	1.51488
$362$	−32.1201	−1.68819
$363$	0	0
$364$	84.6227	4.43544
$365$	44.8968	2.35001
$366$	0	0
$367$	15.5060	0.809404	0.404702	−	0.914449i	$-0.367375\pi$
$367$	15.5060	0.809404	0.404702	+	0.914449i	$0.367375\pi$
$368$	6.26637	0.326657
$369$	0	0
$370$	24.9854	1.29893
$371$	−20.0911	−1.04308
$372$	0	0
$373$	−4.19872	−0.217401	−0.108701	−	0.994075i	$-0.534669\pi$
$373$	−4.19872	−0.217401	−0.108701	+	0.994075i	$0.534669\pi$
$374$	−13.9893	−0.723367
$375$	0	0
$376$	−72.3861	−3.73303
$377$	−5.12356	−0.263877
$378$	0	0
$379$	15.0031	0.770656	0.385328	−	0.922780i	$-0.374088\pi$
$379$	15.0031	0.770656	0.385328	+	0.922780i	$0.374088\pi$
$380$	−94.7754	−4.86188
$381$	0	0
$382$	35.9695	1.84036
$383$	−33.7224	−1.72314	−0.861568	−	0.507642i	$-0.830517\pi$
$383$	−33.7224	−1.72314	−0.861568	+	0.507642i	$0.830517\pi$
$384$	0	0
$385$	35.6213	1.81543
$386$	23.1975	1.18072
$387$	0	0
$388$	−8.10245	−0.411340
$389$	13.8532	0.702384	0.351192	−	0.936303i	$-0.385776\pi$
$389$	13.8532	0.702384	0.351192	+	0.936303i	$0.385776\pi$
$390$	0	0
$391$	−1.85851	−0.0939891
$392$	−43.8056	−2.21252
$393$	0	0
$394$	−29.3335	−1.47780
$395$	−3.55011	−0.178625
$396$	0	0
$397$	−13.9573	−0.700495	−0.350248	−	0.936657i	$-0.613903\pi$
$397$	−13.9573	−0.700495	−0.350248	+	0.936657i	$0.613903\pi$
$398$	−7.33662	−0.367752
$399$	0	0
$400$	30.7339	1.53670
$401$	−22.7239	−1.13478	−0.567389	−	0.823450i	$-0.692046\pi$
$401$	−22.7239	−1.13478	−0.567389	+	0.823450i	$0.692046\pi$
$402$	0	0
$403$	−6.63432	−0.330479
$404$	66.5104	3.30902
$405$	0	0
$406$	9.55838	0.474374
$407$	9.40102	0.465991
$408$	0	0
$409$	−35.3659	−1.74873	−0.874366	−	0.485267i	$-0.838722\pi$
$409$	−35.3659	−1.74873	−0.874366	+	0.485267i	$0.838722\pi$
$410$	78.9337	3.89826
$411$	0	0
$412$	−34.0058	−1.67534
$413$	33.8181	1.66408
$414$	0	0
$415$	−25.7835	−1.26566
$416$	20.0647	0.983753
$417$	0	0
$418$	−52.0312	−2.54493
$419$	−0.998464	−0.0487782	−0.0243891	−	0.999703i	$-0.507764\pi$
$419$	−0.998464	−0.0487782	−0.0243891	+	0.999703i	$0.507764\pi$
$420$	0	0
$421$	−1.64838	−0.0803372	−0.0401686	−	0.999193i	$-0.512789\pi$
$421$	−1.64838	−0.0803372	−0.0401686	+	0.999193i	$0.512789\pi$
$422$	−8.07902	−0.393281
$423$	0	0
$424$	−31.4859	−1.52909
$425$	−9.11525	−0.442155
$426$	0	0
$427$	17.0635	0.825760
$428$	15.3675	0.742817
$429$	0	0
$430$	−37.4053	−1.80385
$431$	14.4721	0.697094	0.348547	−	0.937291i	$-0.386675\pi$
$431$	14.4721	0.697094	0.348547	+	0.937291i	$0.386675\pi$
$432$	0	0
$433$	−5.70501	−0.274166	−0.137083	−	0.990560i	$-0.543773\pi$
$433$	−5.70501	−0.274166	−0.137083	+	0.990560i	$0.543773\pi$
$434$	12.3768	0.594106
$435$	0	0
$436$	−69.3679	−3.32212
$437$	−6.91251	−0.330670
$438$	0	0
$439$	−12.4838	−0.595819	−0.297910	−	0.954594i	$-0.596289\pi$
$439$	−12.4838	−0.595819	−0.297910	+	0.954594i	$0.596289\pi$
$440$	55.8241	2.66131
$441$	0	0
$442$	−24.0076	−1.14192
$443$	−31.6791	−1.50512	−0.752560	−	0.658524i	$-0.771181\pi$
$443$	−31.6791	−1.50512	−0.752560	+	0.658524i	$0.771181\pi$
$444$	0	0
$445$	31.6148	1.49869
$446$	−13.5537	−0.641785
$447$	0	0
$448$	10.0816	0.476309
$449$	19.3314	0.912303	0.456152	−	0.889902i	$-0.349227\pi$
$449$	19.3314	0.912303	0.456152	+	0.889902i	$0.349227\pi$
$450$	0	0
$451$	29.6997	1.39850
$452$	57.3088	2.69558
$453$	0	0
$454$	−7.34508	−0.344722
$455$	61.1313	2.86588
$456$	0	0
$457$	35.5691	1.66385	0.831927	−	0.554886i	$-0.187238\pi$
$457$	35.5691	1.66385	0.831927	+	0.554886i	$0.187238\pi$
$458$	17.2165	0.804475
$459$	0	0
$460$	13.7107	0.639265
$461$	−25.7075	−1.19732	−0.598659	−	0.801004i	$-0.704299\pi$
$461$	−25.7075	−1.19732	−0.598659	+	0.801004i	$0.704299\pi$
$462$	0	0
$463$	6.96767	0.323815	0.161908	−	0.986806i	$-0.448235\pi$
$463$	6.96767	0.323815	0.161908	+	0.986806i	$0.448235\pi$
$464$	6.26637	0.290909
$465$	0	0
$466$	−48.9413	−2.26716
$467$	34.3571	1.58986	0.794929	−	0.606702i	$-0.207508\pi$
$467$	34.3571	1.58986	0.794929	+	0.606702i	$0.207508\pi$
$468$	0	0
$469$	38.7944	1.79136
$470$	−96.6715	−4.45912
$471$	0	0
$472$	52.9981	2.43944
$473$	−14.0742	−0.647131
$474$	0	0
$475$	−33.9030	−1.55558
$476$	30.6960	1.40695
$477$	0	0
$478$	43.9362	2.00959
$479$	−28.3536	−1.29551	−0.647755	−	0.761848i	$-0.724292\pi$
$479$	−28.3536	−1.29551	−0.647755	+	0.761848i	$0.724292\pi$
$480$	0	0
$481$	16.1335	0.735624
$482$	4.14334	0.188724
$483$	0	0
$484$	−9.09100	−0.413227
$485$	−5.85319	−0.265780
$486$	0	0
$487$	−20.7067	−0.938311	−0.469156	−	0.883116i	$-0.655442\pi$
$487$	−20.7067	−0.938311	−0.469156	+	0.883116i	$0.655442\pi$
$488$	26.7411	1.21051
$489$	0	0
$490$	−58.5023	−2.64286
$491$	18.2545	0.823814	0.411907	−	0.911226i	$-0.364863\pi$
$491$	18.2545	0.823814	0.411907	+	0.911226i	$0.364863\pi$
$492$	0	0
$493$	−1.85851	−0.0837033
$494$	−89.2930	−4.01748
$495$	0	0
$496$	8.11410	0.364334
$497$	51.8482	2.32571
$498$	0	0
$499$	16.6338	0.744629	0.372315	−	0.928107i	$-0.378564\pi$
$499$	16.6338	0.744629	0.372315	+	0.928107i	$0.378564\pi$
$500$	−1.30816	−0.0585026
$501$	0	0
$502$	26.2881	1.17329
$503$	−22.7414	−1.01399	−0.506995	−	0.861949i	$-0.669244\pi$
$503$	−22.7414	−1.01399	−0.506995	+	0.861949i	$0.669244\pi$
$504$	0	0
$505$	48.0470	2.13806
$506$	7.52712	0.334621
$507$	0	0
$508$	−22.6212	−1.00365
$509$	−25.1338	−1.11403	−0.557017	−	0.830501i	$-0.688054\pi$
$509$	−25.1338	−1.11403	−0.557017	+	0.830501i	$0.688054\pi$
$510$	0	0
$511$	−54.0843	−2.39255
$512$	49.9212	2.20623
$513$	0	0
$514$	5.01741	0.221308
$515$	−24.5657	−1.08249
$516$	0	0
$517$	−36.3737	−1.59972
$518$	−30.0982	−1.32244
$519$	0	0
$520$	95.8021	4.20120
$521$	25.4339	1.11428	0.557141	−	0.830418i	$-0.311898\pi$
$521$	25.4339	1.11428	0.557141	+	0.830418i	$0.311898\pi$
$522$	0	0
$523$	13.4013	0.586000	0.293000	−	0.956112i	$-0.405346\pi$
$523$	13.4013	0.586000	0.293000	+	0.956112i	$0.405346\pi$
$524$	37.8972	1.65555
$525$	0	0
$526$	21.1189	0.920827
$527$	−2.40653	−0.104830
$528$	0	0
$529$	1.00000	0.0434783
$530$	−42.0493	−1.82651
$531$	0	0
$532$	114.170	4.94989
$533$	50.9689	2.20771
$534$	0	0
$535$	11.1015	0.479958
$536$	60.7968	2.62602
$537$	0	0
$538$	−9.72644	−0.419337
$539$	−22.0121	−0.948130
$540$	0	0
$541$	−6.46250	−0.277844	−0.138922	−	0.990303i	$-0.544364\pi$
$541$	−6.46250	−0.277844	−0.138922	+	0.990303i	$0.544364\pi$
$542$	10.5081	0.451363
$543$	0	0
$544$	7.27826	0.312053
$545$	−50.1112	−2.14653
$546$	0	0
$547$	37.5725	1.60648	0.803242	−	0.595653i	$-0.203107\pi$
$547$	37.5725	1.60648	0.803242	+	0.595653i	$0.203107\pi$
$548$	59.7966	2.55438
$549$	0	0
$550$	36.9174	1.57416
$551$	−6.91251	−0.294483
$552$	0	0
$553$	4.27658	0.181859
$554$	5.63768	0.239522
$555$	0	0
$556$	0.492337	0.0208797
$557$	−39.5759	−1.67689	−0.838443	−	0.544989i	$-0.816534\pi$
$557$	−39.5759	−1.67689	−0.838443	+	0.544989i	$0.816534\pi$
$558$	0	0
$559$	−24.1533	−1.02158
$560$	−74.7666	−3.15947
$561$	0	0
$562$	26.5109	1.11829
$563$	28.4309	1.19822	0.599109	−	0.800667i	$-0.295522\pi$
$563$	28.4309	1.19822	0.599109	+	0.800667i	$0.295522\pi$
$564$	0	0
$565$	41.3997	1.74170
$566$	−4.04069	−0.169843
$567$	0	0
$568$	81.2542	3.40935
$569$	−29.4617	−1.23510	−0.617550	−	0.786532i	$-0.711875\pi$
$569$	−29.4617	−1.23510	−0.617550	+	0.786532i	$0.711875\pi$
$570$	0	0
$571$	−5.58193	−0.233597	−0.116798	−	0.993156i	$-0.537263\pi$
$571$	−5.58193	−0.233597	−0.116798	+	0.993156i	$0.537263\pi$
$572$	66.6395	2.78634
$573$	0	0
$574$	−95.0863	−3.96882
$575$	4.90459	0.204535
$576$	0	0
$577$	26.9142	1.12045	0.560226	−	0.828340i	$-0.310714\pi$
$577$	26.9142	1.12045	0.560226	+	0.828340i	$0.310714\pi$
$578$	34.1522	1.42055
$579$	0	0
$580$	13.7107	0.569306
$581$	31.0597	1.28857
$582$	0	0
$583$	−15.8215	−0.655261
$584$	−84.7585	−3.50733
$585$	0	0
$586$	−39.1645	−1.61787
$587$	35.4468	1.46305	0.731523	−	0.681816i	$-0.238810\pi$
$587$	35.4468	1.46305	0.731523	+	0.681816i	$0.238810\pi$
$588$	0	0
$589$	−8.95077	−0.368810
$590$	70.7789	2.91392
$591$	0	0
$592$	−19.7321	−0.810983
$593$	41.5701	1.70708	0.853539	−	0.521029i	$-0.174452\pi$
$593$	41.5701	1.70708	0.853539	+	0.521029i	$0.174452\pi$
$594$	0	0
$595$	22.1747	0.909074
$596$	60.5595	2.48061
$597$	0	0
$598$	12.9176	0.528240
$599$	−32.2783	−1.31886	−0.659428	−	0.751768i	$-0.729201\pi$
$599$	−32.2783	−1.31886	−0.659428	+	0.751768i	$0.729201\pi$
$600$	0	0
$601$	38.4660	1.56906	0.784531	−	0.620090i	$-0.212904\pi$
$601$	38.4660	1.56906	0.784531	+	0.620090i	$0.212904\pi$
$602$	45.0598	1.83650
$603$	0	0
$604$	103.359	4.20563
$605$	−6.56732	−0.266999
$606$	0	0
$607$	27.6689	1.12305	0.561524	−	0.827460i	$-0.310215\pi$
$607$	27.6689	1.12305	0.561524	+	0.827460i	$0.310215\pi$
$608$	27.0705	1.09786
$609$	0	0
$610$	35.7127	1.44597
$611$	−62.4225	−2.52534
$612$	0	0
$613$	−26.4829	−1.06963	−0.534817	−	0.844968i	$-0.679620\pi$
$613$	−26.4829	−1.06963	−0.534817	+	0.844968i	$0.679620\pi$
$614$	85.6515	3.45661
$615$	0	0
$616$	−67.2477	−2.70949
$617$	−35.2005	−1.41712	−0.708559	−	0.705651i	$-0.750654\pi$
$617$	−35.2005	−1.41712	−0.708559	+	0.705651i	$0.750654\pi$
$618$	0	0
$619$	37.5107	1.50768	0.753842	−	0.657056i	$-0.228199\pi$
$619$	37.5107	1.50768	0.753842	+	0.657056i	$0.228199\pi$
$620$	17.7535	0.712999
$621$	0	0
$622$	−49.9653	−2.00343
$623$	−38.0843	−1.52582
$624$	0	0
$625$	−25.4680	−1.01872
$626$	−29.0148	−1.15966
$627$	0	0
$628$	20.4939	0.817794
$629$	5.85225	0.233345
$630$	0	0
$631$	−21.6524	−0.861968	−0.430984	−	0.902360i	$-0.641833\pi$
$631$	−21.6524	−0.861968	−0.430984	+	0.902360i	$0.641833\pi$
$632$	6.70206	0.266594
$633$	0	0
$634$	−76.1486	−3.02425
$635$	−16.3415	−0.648492
$636$	0	0
$637$	−37.7760	−1.49674
$638$	7.52712	0.298001
$639$	0	0
$640$	45.7496	1.80841
$641$	13.8060	0.545305	0.272652	−	0.962113i	$-0.412099\pi$
$641$	13.8060	0.545305	0.272652	+	0.962113i	$0.412099\pi$
$642$	0	0
$643$	−46.2843	−1.82527	−0.912637	−	0.408770i	$-0.865958\pi$
$643$	−46.2843	−1.82527	−0.912637	+	0.408770i	$0.865958\pi$
$644$	−16.5164	−0.650837
$645$	0	0
$646$	−32.3901	−1.27437
$647$	−40.8008	−1.60404	−0.802022	−	0.597294i	$-0.796242\pi$
$647$	−40.8008	−1.60404	−0.802022	+	0.597294i	$0.796242\pi$
$648$	0	0
$649$	26.6314	1.04537
$650$	63.3555	2.48501
$651$	0	0
$652$	47.8131	1.87250
$653$	0.845450	0.0330850	0.0165425	−	0.999863i	$-0.494734\pi$
$653$	0.845450	0.0330850	0.0165425	+	0.999863i	$0.494734\pi$
$654$	0	0
$655$	27.3768	1.06970
$656$	−62.3375	−2.43387
$657$	0	0
$658$	116.454	4.53984
$659$	42.0249	1.63706	0.818528	−	0.574466i	$-0.194790\pi$
$659$	42.0249	1.63706	0.818528	+	0.574466i	$0.194790\pi$
$660$	0	0
$661$	40.6806	1.58229	0.791146	−	0.611627i	$-0.209485\pi$
$661$	40.6806	1.58229	0.791146	+	0.611627i	$0.209485\pi$
$662$	−12.9370	−0.502812
$663$	0	0
$664$	48.6754	1.88897
$665$	82.4760	3.19828
$666$	0	0
$667$	1.00000	0.0387202
$668$	34.9736	1.35317
$669$	0	0
$670$	81.1941	3.13680
$671$	13.4373	0.518741
$672$	0	0
$673$	−27.7880	−1.07115	−0.535574	−	0.844488i	$-0.679905\pi$
$673$	−27.7880	−1.07115	−0.535574	+	0.844488i	$0.679905\pi$
$674$	−62.7002	−2.41512
$675$	0	0
$676$	57.7278	2.22030
$677$	8.99651	0.345764	0.172882	−	0.984943i	$-0.444692\pi$
$677$	8.99651	0.345764	0.172882	+	0.984943i	$0.444692\pi$
$678$	0	0
$679$	7.05096	0.270591
$680$	34.7512	1.33265
$681$	0	0
$682$	9.74660	0.373217
$683$	38.4783	1.47233	0.736165	−	0.676802i	$-0.236635\pi$
$683$	38.4783	1.47233	0.736165	+	0.676802i	$0.236635\pi$
$684$	0	0
$685$	43.1969	1.65047
$686$	3.56526	0.136122
$687$	0	0
$688$	29.5407	1.12623
$689$	−27.1520	−1.03441
$690$	0	0
$691$	25.2155	0.959244	0.479622	−	0.877475i	$-0.340774\pi$
$691$	25.2155	0.959244	0.479622	+	0.877475i	$0.340774\pi$
$692$	−57.8977	−2.20094
$693$	0	0
$694$	−80.8671	−3.06967
$695$	0.355663	0.0134911
$696$	0	0
$697$	18.4884	0.700299
$698$	−46.2522	−1.75067
$699$	0	0
$700$	−81.0062	−3.06175
$701$	39.2413	1.48212	0.741061	−	0.671438i	$-0.234323\pi$
$701$	39.2413	1.48212	0.741061	+	0.671438i	$0.234323\pi$
$702$	0	0
$703$	21.7667	0.820947
$704$	7.93911	0.299216
$705$	0	0
$706$	−5.83111	−0.219457
$707$	−57.8791	−2.17677
$708$	0	0
$709$	3.50075	0.131474	0.0657368	−	0.997837i	$-0.479060\pi$
$709$	3.50075	0.131474	0.0657368	+	0.997837i	$0.479060\pi$
$710$	108.515	4.07249
$711$	0	0
$712$	−59.6840	−2.23675
$713$	1.29487	0.0484931
$714$	0	0
$715$	48.1402	1.80034
$716$	−18.0161	−0.673293
$717$	0	0
$718$	32.6046	1.21679
$719$	40.4831	1.50976	0.754882	−	0.655860i	$-0.227694\pi$
$719$	40.4831	1.50976	0.754882	+	0.655860i	$0.227694\pi$
$720$	0	0
$721$	29.5927	1.10209
$722$	−72.5676	−2.70069
$723$	0	0
$724$	55.5019	2.06271
$725$	4.90459	0.182152
$726$	0	0
$727$	−6.41280	−0.237838	−0.118919	−	0.992904i	$-0.537943\pi$
$727$	−6.41280	−0.237838	−0.118919	+	0.992904i	$0.537943\pi$
$728$	−115.407	−4.27725
$729$	0	0
$730$	−113.195	−4.18953
$731$	−8.76135	−0.324050
$732$	0	0
$733$	15.2150	0.561980	0.280990	−	0.959711i	$-0.409337\pi$
$733$	15.2150	0.561980	0.280990	+	0.959711i	$0.409337\pi$
$734$	−39.0939	−1.44298
$735$	0	0
$736$	−3.91617	−0.144352
$737$	30.5502	1.12533
$738$	0	0
$739$	21.4186	0.787898	0.393949	−	0.919132i	$-0.371109\pi$
$739$	21.4186	0.787898	0.393949	+	0.919132i	$0.371109\pi$
$740$	−43.1735	−1.58709
$741$	0	0
$742$	50.6541	1.85957
$743$	−30.0028	−1.10069	−0.550347	−	0.834936i	$-0.685505\pi$
$743$	−30.0028	−1.10069	−0.550347	+	0.834936i	$0.685505\pi$
$744$	0	0
$745$	43.7481	1.60280
$746$	10.5859	0.387577
$747$	0	0
$748$	24.1727	0.883843
$749$	−13.3732	−0.488646
$750$	0	0
$751$	−33.1491	−1.20963	−0.604813	−	0.796367i	$-0.706752\pi$
$751$	−33.1491	−1.20963	−0.604813	+	0.796367i	$0.706752\pi$
$752$	76.3459	2.78405
$753$	0	0
$754$	12.9176	0.470431
$755$	74.6665	2.71739
$756$	0	0
$757$	34.0644	1.23809	0.619046	−	0.785354i	$-0.287519\pi$
$757$	34.0644	1.23809	0.619046	+	0.785354i	$0.287519\pi$
$758$	−37.8260	−1.37390
$759$	0	0
$760$	129.253	4.68848
$761$	13.1114	0.475289	0.237644	−	0.971352i	$-0.423625\pi$
$761$	13.1114	0.475289	0.237644	+	0.971352i	$0.423625\pi$
$762$	0	0
$763$	60.3658	2.18539
$764$	−62.1535	−2.24864
$765$	0	0
$766$	85.0216	3.07196
$767$	45.7032	1.65025
$768$	0	0
$769$	−42.7804	−1.54270	−0.771351	−	0.636410i	$-0.780419\pi$
$769$	−42.7804	−1.54270	−0.771351	+	0.636410i	$0.780419\pi$
$770$	−89.8091	−3.23650
$771$	0	0
$772$	−40.0842	−1.44266
$773$	32.4200	1.16607	0.583033	−	0.812449i	$-0.301866\pi$
$773$	32.4200	1.16607	0.583033	+	0.812449i	$0.301866\pi$
$774$	0	0
$775$	6.35078	0.228127
$776$	11.0499	0.396670
$777$	0	0
$778$	−34.9269	−1.25219
$779$	68.7653	2.46377
$780$	0	0
$781$	40.8299	1.46101
$782$	4.68572	0.167561
$783$	0	0
$784$	46.2019	1.65007
$785$	14.8047	0.528403
$786$	0	0
$787$	−52.6751	−1.87766	−0.938832	−	0.344375i	$-0.888091\pi$
$787$	−52.6751	−1.87766	−0.938832	+	0.344375i	$0.888091\pi$
$788$	50.6869	1.80564
$789$	0	0
$790$	8.95059	0.318448
$791$	−49.8716	−1.77323
$792$	0	0
$793$	23.0603	0.818896
$794$	35.1893	1.24882
$795$	0	0
$796$	12.6773	0.449336
$797$	7.19178	0.254746	0.127373	−	0.991855i	$-0.459345\pi$
$797$	7.19178	0.254746	0.127373	+	0.991855i	$0.459345\pi$
$798$	0	0
$799$	−22.6431	−0.801055
$800$	−19.2072	−0.679077
$801$	0	0
$802$	57.2919	2.02305
$803$	−42.5908	−1.50300
$804$	0	0
$805$	−11.9314	−0.420527
$806$	16.7266	0.589168
$807$	0	0
$808$	−90.7054	−3.19101
$809$	−2.66177	−0.0935828	−0.0467914	−	0.998905i	$-0.514900\pi$
$809$	−2.66177	−0.0935828	−0.0467914	+	0.998905i	$0.514900\pi$
$810$	0	0
$811$	−5.48270	−0.192524	−0.0962618	−	0.995356i	$-0.530689\pi$
$811$	−5.48270	−0.192524	−0.0962618	+	0.995356i	$0.530689\pi$
$812$	−16.5164	−0.579612
$813$	0	0
$814$	−23.7020	−0.830756
$815$	34.5400	1.20989
$816$	0	0
$817$	−32.5867	−1.14006
$818$	89.1652	3.11759
$819$	0	0
$820$	−136.394	−4.76307
$821$	55.1738	1.92558	0.962789	−	0.270255i	$-0.0871081\pi$
$821$	55.1738	1.92558	0.962789	+	0.270255i	$0.0871081\pi$
$822$	0	0
$823$	5.70940	0.199017	0.0995086	−	0.995037i	$-0.468273\pi$
$823$	5.70940	0.199017	0.0995086	+	0.995037i	$0.468273\pi$
$824$	46.3763	1.61559
$825$	0	0
$826$	−85.2628	−2.96667
$827$	−28.5949	−0.994340	−0.497170	−	0.867653i	$-0.665628\pi$
$827$	−28.5949	−0.994340	−0.497170	+	0.867653i	$0.665628\pi$
$828$	0	0
$829$	13.3296	0.462956	0.231478	−	0.972840i	$-0.425644\pi$
$829$	13.3296	0.462956	0.231478	+	0.972840i	$0.425644\pi$
$830$	65.0059	2.25639
$831$	0	0
$832$	13.6246	0.472350
$833$	−13.7028	−0.474775
$834$	0	0
$835$	25.2649	0.874327
$836$	89.9074	3.10951
$837$	0	0
$838$	2.51735	0.0869603
$839$	−16.2462	−0.560882	−0.280441	−	0.959871i	$-0.590481\pi$
$839$	−16.2462	−0.560882	−0.280441	+	0.959871i	$0.590481\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	4.15593	0.143223
$843$	0	0
$844$	13.9602	0.480528
$845$	41.7024	1.43461
$846$	0	0
$847$	7.91122	0.271833
$848$	33.2083	1.14038
$849$	0	0
$850$	22.9815	0.788260
$851$	−3.14889	−0.107942
$852$	0	0
$853$	−35.1839	−1.20467	−0.602337	−	0.798242i	$-0.705764\pi$
$853$	−35.1839	−1.20467	−0.602337	+	0.798242i	$0.705764\pi$
$854$	−43.0208	−1.47214
$855$	0	0
$856$	−20.9579	−0.716325
$857$	2.97223	0.101529	0.0507647	−	0.998711i	$-0.483834\pi$
$857$	2.97223	0.101529	0.0507647	+	0.998711i	$0.483834\pi$
$858$	0	0
$859$	−37.1582	−1.26782	−0.633910	−	0.773407i	$-0.718551\pi$
$859$	−37.1582	−1.26782	−0.633910	+	0.773407i	$0.718551\pi$
$860$	64.6346	2.20402
$861$	0	0
$862$	−36.4872	−1.24276
$863$	−31.9088	−1.08619	−0.543095	−	0.839671i	$-0.682748\pi$
$863$	−31.9088	−1.08619	−0.543095	+	0.839671i	$0.682748\pi$
$864$	0	0
$865$	−41.8252	−1.42210
$866$	14.3836	0.488774
$867$	0	0
$868$	−21.3865	−0.725906
$869$	3.36776	0.114243
$870$	0	0
$871$	52.4285	1.77647
$872$	94.6024	3.20364
$873$	0	0
$874$	17.4279	0.589509
$875$	1.13839	0.0384847
$876$	0	0
$877$	16.5843	0.560012	0.280006	−	0.959998i	$-0.409664\pi$
$877$	16.5843	0.560012	0.280006	+	0.959998i	$0.409664\pi$
$878$	31.4744	1.06221
$879$	0	0
$880$	−58.8779	−1.98477
$881$	32.1781	1.08411	0.542054	−	0.840344i	$-0.317647\pi$
$881$	32.1781	1.08411	0.542054	+	0.840344i	$0.317647\pi$
$882$	0	0
$883$	−42.4818	−1.42963	−0.714813	−	0.699316i	$-0.753488\pi$
$883$	−42.4818	−1.42963	−0.714813	+	0.699316i	$0.753488\pi$
$884$	41.4839	1.39525
$885$	0	0
$886$	79.8699	2.68328
$887$	8.45687	0.283954	0.141977	−	0.989870i	$-0.454654\pi$
$887$	8.45687	0.283954	0.141977	+	0.989870i	$0.454654\pi$
$888$	0	0
$889$	19.6855	0.660231
$890$	−79.7079	−2.67181
$891$	0	0
$892$	23.4201	0.784163
$893$	−84.2181	−2.81825
$894$	0	0
$895$	−13.0148	−0.435036
$896$	−55.1115	−1.84115
$897$	0	0
$898$	−48.7386	−1.62643
$899$	1.29487	0.0431862
$900$	0	0
$901$	−9.84910	−0.328121
$902$	−74.8794	−2.49321
$903$	0	0
$904$	−78.1564	−2.59944
$905$	40.0944	1.33278
$906$	0	0
$907$	−2.92173	−0.0970146	−0.0485073	−	0.998823i	$-0.515446\pi$
$907$	−2.92173	−0.0970146	−0.0485073	+	0.998823i	$0.515446\pi$
$908$	12.6919	0.421197
$909$	0	0
$910$	−154.125	−5.10920
$911$	−23.4621	−0.777335	−0.388668	−	0.921378i	$-0.627065\pi$
$911$	−23.4621	−0.777335	−0.388668	+	0.921378i	$0.627065\pi$
$912$	0	0
$913$	24.4592	0.809481
$914$	−89.6775	−2.96627
$915$	0	0
$916$	−29.7493	−0.982944
$917$	−32.9791	−1.08907
$918$	0	0
$919$	−43.1543	−1.42353	−0.711764	−	0.702418i	$-0.752104\pi$
$919$	−43.1543	−1.42353	−0.711764	+	0.702418i	$0.752104\pi$
$920$	−18.6984	−0.616467
$921$	0	0
$922$	64.8142	2.13454
$923$	70.0700	2.30638
$924$	0	0
$925$	−15.4440	−0.507795
$926$	−17.5670	−0.577288
$927$	0	0
$928$	−3.91617	−0.128555
$929$	−22.4996	−0.738189	−0.369094	−	0.929392i	$-0.620332\pi$
$929$	−22.4996	−0.738189	−0.369094	+	0.929392i	$0.620332\pi$
$930$	0	0
$931$	−50.9659	−1.67034
$932$	84.5682	2.77012
$933$	0	0
$934$	−86.6218	−2.83435
$935$	17.4623	0.571079
$936$	0	0
$937$	−9.74922	−0.318493	−0.159247	−	0.987239i	$-0.550906\pi$
$937$	−9.74922	−0.318493	−0.159247	+	0.987239i	$0.550906\pi$
$938$	−97.8092	−3.19358
$939$	0	0
$940$	167.044	5.44836
$941$	18.4198	0.600469	0.300234	−	0.953865i	$-0.402935\pi$
$941$	18.4198	0.600469	0.300234	+	0.953865i	$0.402935\pi$
$942$	0	0
$943$	−9.94795	−0.323950
$944$	−55.8973	−1.81930
$945$	0	0
$946$	35.4841	1.15369
$947$	−17.8237	−0.579194	−0.289597	−	0.957149i	$-0.593521\pi$
$947$	−17.8237	−0.579194	−0.289597	+	0.957149i	$0.593521\pi$
$948$	0	0
$949$	−73.0919	−2.37266
$950$	85.4769	2.77324
$951$	0	0
$952$	−41.8625	−1.35677
$953$	−12.0895	−0.391616	−0.195808	−	0.980642i	$-0.562733\pi$
$953$	−12.0895	−0.391616	−0.195808	+	0.980642i	$0.562733\pi$
$954$	0	0
$955$	−44.8996	−1.45292
$956$	−75.9196	−2.45541
$957$	0	0
$958$	71.4857	2.30960
$959$	−52.0365	−1.68035
$960$	0	0
$961$	−29.3233	−0.945914
$962$	−40.6761	−1.31145
$963$	0	0
$964$	−7.15949	−0.230592
$965$	−28.9568	−0.932151
$966$	0	0
$967$	−13.9735	−0.449356	−0.224678	−	0.974433i	$-0.572133\pi$
$967$	−13.9735	−0.449356	−0.224678	+	0.974433i	$0.572133\pi$
$968$	12.3981	0.398490
$969$	0	0
$970$	14.7572	0.473824
$971$	22.1718	0.711528	0.355764	−	0.934576i	$-0.384221\pi$
$971$	22.1718	0.711528	0.355764	+	0.934576i	$0.384221\pi$
$972$	0	0
$973$	−0.428444	−0.0137353
$974$	52.2062	1.67279
$975$	0	0
$976$	−28.2039	−0.902786
$977$	−43.3371	−1.38648	−0.693238	−	0.720709i	$-0.743817\pi$
$977$	−43.3371	−1.38648	−0.693238	+	0.720709i	$0.743817\pi$
$978$	0	0
$979$	−29.9910	−0.958516
$980$	101.089	3.22917
$981$	0	0
$982$	−46.0236	−1.46867
$983$	26.7308	0.852580	0.426290	−	0.904587i	$-0.359820\pi$
$983$	26.7308	0.852580	0.426290	+	0.904587i	$0.359820\pi$
$984$	0	0
$985$	36.6161	1.16669
$986$	4.68572	0.149224
$987$	0	0
$988$	154.294	4.90874
$989$	4.71417	0.149902
$990$	0	0
$991$	−19.3587	−0.614951	−0.307475	−	0.951556i	$-0.599484\pi$
$991$	−19.3587	−0.614951	−0.307475	+	0.951556i	$0.599484\pi$
$992$	−5.07091	−0.161002
$993$	0	0
$994$	−130.721	−4.14621
$995$	9.15807	0.290330
$996$	0	0
$997$	−2.87179	−0.0909504	−0.0454752	−	0.998965i	$-0.514480\pi$
$997$	−2.87179	−0.0909504	−0.0454752	+	0.998965i	$0.514480\pi$
$998$	−41.9373	−1.32750
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.n.1.1		12
3.2	odd	2		667.2.a.b.1.12	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.b.1.12	✓	12	3.2	odd	2
6003.2.a.n.1.1		12	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-2.52122 q^{2} +4.35654 q^{4} +3.14716 q^{5} -3.79117 q^{7} -5.94135 q^{8} +O(q^{10})\)	\(q-2.52122 q^{2} +4.35654 q^{4} +3.14716 q^{5} -3.79117 q^{7} -5.94135 q^{8} -7.93467 q^{10} -2.98551 q^{11} -5.12356 q^{13} +9.55838 q^{14} +6.26637 q^{16} -1.85851 q^{17} -6.91251 q^{19} +13.7107 q^{20} +7.52712 q^{22} +1.00000 q^{23} +4.90459 q^{25} +12.9176 q^{26} -16.5164 q^{28} +1.00000 q^{29} +1.29487 q^{31} -3.91617 q^{32} +4.68572 q^{34} -11.9314 q^{35} -3.14889 q^{37} +17.4279 q^{38} -18.6984 q^{40} -9.94795 q^{41} +4.71417 q^{43} -13.0065 q^{44} -2.52122 q^{46} +12.1834 q^{47} +7.37300 q^{49} -12.3655 q^{50} -22.3210 q^{52} +5.29945 q^{53} -9.39586 q^{55} +22.5247 q^{56} -2.52122 q^{58} -8.92021 q^{59} -4.50085 q^{61} -3.26464 q^{62} -2.65922 q^{64} -16.1246 q^{65} -10.2328 q^{67} -8.09669 q^{68} +30.0817 q^{70} -13.6760 q^{71} +14.2658 q^{73} +7.93903 q^{74} -30.1146 q^{76} +11.3186 q^{77} -1.12804 q^{79} +19.7212 q^{80} +25.0810 q^{82} -8.19264 q^{83} -5.84903 q^{85} -11.8854 q^{86} +17.7380 q^{88} +10.0455 q^{89} +19.4243 q^{91} +4.35654 q^{92} -30.7171 q^{94} -21.7547 q^{95} -1.85984 q^{97} -18.5889 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{2} + 11 q^{4} + 16 q^{5} - 7 q^{7} + 9 q^{8}+O(q^{10}) \) 12 * q + 3 * q^2 + 11 * q^4 + 16 * q^5 - 7 * q^7 + 9 * q^8	\( 12 q + 3 q^{2} + 11 q^{4} + 16 q^{5} - 7 q^{7} + 9 q^{8} + 6 q^{11} - 15 q^{13} + 8 q^{14} + 17 q^{16} + 18 q^{17} - 6 q^{19} + 39 q^{20} - 5 q^{22} + 12 q^{23} + 14 q^{25} + 3 q^{26} - 19 q^{28} + 12 q^{29} + 16 q^{31} + 21 q^{32} - 7 q^{34} + 11 q^{35} - q^{37} + 24 q^{38} + 30 q^{40} - 3 q^{41} - 23 q^{43} - 23 q^{44} + 3 q^{46} + 35 q^{47} + 3 q^{49} + 2 q^{50} + 45 q^{53} + 17 q^{55} + 17 q^{56} + 3 q^{58} + 11 q^{59} + 4 q^{61} + 7 q^{62} + 15 q^{64} - 5 q^{65} - 19 q^{67} - q^{68} + 14 q^{70} - 19 q^{71} + 10 q^{73} + 15 q^{74} - 4 q^{76} + 39 q^{77} + 17 q^{79} + 90 q^{80} - 3 q^{82} + 12 q^{83} + 14 q^{85} - 17 q^{86} - 2 q^{88} + 20 q^{89} + 11 q^{91} + 11 q^{92} + 13 q^{94} - 12 q^{95} - 12 q^{97} - 75 q^{98}+O(q^{100}) \) 12 * q + 3 * q^2 + 11 * q^4 + 16 * q^5 - 7 * q^7 + 9 * q^8 + 6 * q^11 - 15 * q^13 + 8 * q^14 + 17 * q^16 + 18 * q^17 - 6 * q^19 + 39 * q^20 - 5 * q^22 + 12 * q^23 + 14 * q^25 + 3 * q^26 - 19 * q^28 + 12 * q^29 + 16 * q^31 + 21 * q^32 - 7 * q^34 + 11 * q^35 - q^37 + 24 * q^38 + 30 * q^40 - 3 * q^41 - 23 * q^43 - 23 * q^44 + 3 * q^46 + 35 * q^47 + 3 * q^49 + 2 * q^50 + 45 * q^53 + 17 * q^55 + 17 * q^56 + 3 * q^58 + 11 * q^59 + 4 * q^61 + 7 * q^62 + 15 * q^64 - 5 * q^65 - 19 * q^67 - q^68 + 14 * q^70 - 19 * q^71 + 10 * q^73 + 15 * q^74 - 4 * q^76 + 39 * q^77 + 17 * q^79 + 90 * q^80 - 3 * q^82 + 12 * q^83 + 14 * q^85 - 17 * q^86 - 2 * q^88 + 20 * q^89 + 11 * q^91 + 11 * q^92 + 13 * q^94 - 12 * q^95 - 12 * q^97 - 75 * q^98

Introduction

L-functions

Modular forms

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Database

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