LMFDB - Embedded newform 6045.2.a.w.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.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:	$48.2695680219$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 15 x^{9} + 29 x^{8} + 81 x^{7} - 151 x^{6} - 192 x^{5} + 345 x^{4} + 199 x^{3} + \cdots + 118 $ x^11 - 2x^10 - 15x^9 + 29x^8 + 81x^7 - 151x^6 - 192x^5 + 345x^4 + 199x^3 - 344x^2 - 70x + 118
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.849375$ of defining polynomial
Character	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.849375 q^{2} -1.00000 q^{3} -1.27856 q^{4} -1.00000 q^{5} +0.849375 q^{6} -1.45897 q^{7} +2.78473 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.849375 q^{2} -1.00000 q^{3} -1.27856 q^{4} -1.00000 q^{5} +0.849375 q^{6} -1.45897 q^{7} +2.78473 q^{8} +1.00000 q^{9} +0.849375 q^{10} +3.66455 q^{11} +1.27856 q^{12} -1.00000 q^{13} +1.23921 q^{14} +1.00000 q^{15} +0.191844 q^{16} -6.44928 q^{17} -0.849375 q^{18} +0.290013 q^{19} +1.27856 q^{20} +1.45897 q^{21} -3.11258 q^{22} +3.97579 q^{23} -2.78473 q^{24} +1.00000 q^{25} +0.849375 q^{26} -1.00000 q^{27} +1.86538 q^{28} -7.99714 q^{29} -0.849375 q^{30} -1.00000 q^{31} -5.73241 q^{32} -3.66455 q^{33} +5.47786 q^{34} +1.45897 q^{35} -1.27856 q^{36} -2.74033 q^{37} -0.246330 q^{38} +1.00000 q^{39} -2.78473 q^{40} +8.38792 q^{41} -1.23921 q^{42} +11.1660 q^{43} -4.68536 q^{44} -1.00000 q^{45} -3.37694 q^{46} +2.98604 q^{47} -0.191844 q^{48} -4.87142 q^{49} -0.849375 q^{50} +6.44928 q^{51} +1.27856 q^{52} +8.28287 q^{53} +0.849375 q^{54} -3.66455 q^{55} -4.06283 q^{56} -0.290013 q^{57} +6.79257 q^{58} -10.2823 q^{59} -1.27856 q^{60} -6.31354 q^{61} +0.849375 q^{62} -1.45897 q^{63} +4.48527 q^{64} +1.00000 q^{65} +3.11258 q^{66} +8.50951 q^{67} +8.24580 q^{68} -3.97579 q^{69} -1.23921 q^{70} -13.5594 q^{71} +2.78473 q^{72} +1.21624 q^{73} +2.32756 q^{74} -1.00000 q^{75} -0.370799 q^{76} -5.34646 q^{77} -0.849375 q^{78} -6.05820 q^{79} -0.191844 q^{80} +1.00000 q^{81} -7.12449 q^{82} +8.18137 q^{83} -1.86538 q^{84} +6.44928 q^{85} -9.48414 q^{86} +7.99714 q^{87} +10.2048 q^{88} -0.797748 q^{89} +0.849375 q^{90} +1.45897 q^{91} -5.08329 q^{92} +1.00000 q^{93} -2.53627 q^{94} -0.290013 q^{95} +5.73241 q^{96} +3.18136 q^{97} +4.13766 q^{98} +3.66455 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 2 q^{2} - 11 q^{3} + 12 q^{4} - 11 q^{5} - 2 q^{6} + 4 q^{7} + 3 q^{8} + 11 q^{9}+O(q^{10}) $ 11 * q + 2 * q^2 - 11 * q^3 + 12 * q^4 - 11 * q^5 - 2 * q^6 + 4 * q^7 + 3 * q^8 + 11 * q^9	\( 11 q + 2 q^{2} - 11 q^{3} + 12 q^{4} - 11 q^{5} - 2 q^{6} + 4 q^{7} + 3 q^{8} + 11 q^{9} - 2 q^{10} - 12 q^{12} - 11 q^{13} + 5 q^{14} + 11 q^{15} - 10 q^{16} - 3 q^{17} + 2 q^{18} - 8 q^{19} - 12 q^{20} - 4 q^{21} - 3 q^{22} + 11 q^{23} - 3 q^{24} + 11 q^{25} - 2 q^{26} - 11 q^{27} + 14 q^{28} - 14 q^{29} + 2 q^{30} - 11 q^{31} + 8 q^{32} - 11 q^{34} - 4 q^{35} + 12 q^{36} + 7 q^{37} + 8 q^{38} + 11 q^{39} - 3 q^{40} + 22 q^{41} - 5 q^{42} - 5 q^{43} - 13 q^{44} - 11 q^{45} - 22 q^{46} + 5 q^{47} + 10 q^{48} - 33 q^{49} + 2 q^{50} + 3 q^{51} - 12 q^{52} + 4 q^{53} - 2 q^{54} - 13 q^{56} + 8 q^{57} - 18 q^{58} - 3 q^{59} + 12 q^{60} - 28 q^{61} - 2 q^{62} + 4 q^{63} + 3 q^{64} + 11 q^{65} + 3 q^{66} - 11 q^{67} - 9 q^{68} - 11 q^{69} - 5 q^{70} + 5 q^{71} + 3 q^{72} - 3 q^{73} - 12 q^{74} - 11 q^{75} - 36 q^{76} + 18 q^{77} + 2 q^{78} - 43 q^{79} + 10 q^{80} + 11 q^{81} - 15 q^{82} - 28 q^{83} - 14 q^{84} + 3 q^{85} + 10 q^{86} + 14 q^{87} - 43 q^{88} - 25 q^{89} - 2 q^{90} - 4 q^{91} + 7 q^{92} + 11 q^{93} - 16 q^{94} + 8 q^{95} - 8 q^{96} - 6 q^{97} - 14 q^{98}+O(q^{100}) \) 11 * q + 2 * q^2 - 11 * q^3 + 12 * q^4 - 11 * q^5 - 2 * q^6 + 4 * q^7 + 3 * q^8 + 11 * q^9 - 2 * q^10 - 12 * q^12 - 11 * q^13 + 5 * q^14 + 11 * q^15 - 10 * q^16 - 3 * q^17 + 2 * q^18 - 8 * q^19 - 12 * q^20 - 4 * q^21 - 3 * q^22 + 11 * q^23 - 3 * q^24 + 11 * q^25 - 2 * q^26 - 11 * q^27 + 14 * q^28 - 14 * q^29 + 2 * q^30 - 11 * q^31 + 8 * q^32 - 11 * q^34 - 4 * q^35 + 12 * q^36 + 7 * q^37 + 8 * q^38 + 11 * q^39 - 3 * q^40 + 22 * q^41 - 5 * q^42 - 5 * q^43 - 13 * q^44 - 11 * q^45 - 22 * q^46 + 5 * q^47 + 10 * q^48 - 33 * q^49 + 2 * q^50 + 3 * q^51 - 12 * q^52 + 4 * q^53 - 2 * q^54 - 13 * q^56 + 8 * q^57 - 18 * q^58 - 3 * q^59 + 12 * q^60 - 28 * q^61 - 2 * q^62 + 4 * q^63 + 3 * q^64 + 11 * q^65 + 3 * q^66 - 11 * q^67 - 9 * q^68 - 11 * q^69 - 5 * q^70 + 5 * q^71 + 3 * q^72 - 3 * q^73 - 12 * q^74 - 11 * q^75 - 36 * q^76 + 18 * q^77 + 2 * q^78 - 43 * q^79 + 10 * q^80 + 11 * q^81 - 15 * q^82 - 28 * q^83 - 14 * q^84 + 3 * q^85 + 10 * q^86 + 14 * q^87 - 43 * q^88 - 25 * q^89 - 2 * q^90 - 4 * q^91 + 7 * q^92 + 11 * q^93 - 16 * q^94 + 8 * q^95 - 8 * q^96 - 6 * q^97 - 14 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.849375	−0.600599	−0.300299	−	0.953845i	$-0.597087\pi$
$2$	−0.849375	−0.600599	−0.300299	+	0.953845i	$0.597087\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.27856	−0.639281
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.849375	0.346756
$7$	−1.45897	−0.551437	−0.275719	−	0.961238i	$-0.588916\pi$
$7$	−1.45897	−0.551437	−0.275719	+	0.961238i	$0.588916\pi$
$8$	2.78473	0.984550
$9$	1.00000	0.333333
$10$	0.849375	0.268596
$11$	3.66455	1.10490	0.552452	−	0.833545i	$-0.313692\pi$
$11$	3.66455	1.10490	0.552452	+	0.833545i	$0.313692\pi$
$12$	1.27856	0.369089
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.23921	0.331193
$15$	1.00000	0.258199
$16$	0.191844	0.0479610
$17$	−6.44928	−1.56418	−0.782090	−	0.623165i	$-0.785846\pi$
$17$	−6.44928	−1.56418	−0.782090	+	0.623165i	$0.785846\pi$
$18$	−0.849375	−0.200200
$19$	0.290013	0.0665335	0.0332667	−	0.999447i	$-0.489409\pi$
$19$	0.290013	0.0665335	0.0332667	+	0.999447i	$0.489409\pi$
$20$	1.27856	0.285895
$21$	1.45897	0.318373
$22$	−3.11258	−0.663604
$23$	3.97579	0.829010	0.414505	−	0.910047i	$-0.363955\pi$
$23$	3.97579	0.829010	0.414505	+	0.910047i	$0.363955\pi$
$24$	−2.78473	−0.568430
$25$	1.00000	0.200000
$26$	0.849375	0.166576
$27$	−1.00000	−0.192450
$28$	1.86538	0.352523
$29$	−7.99714	−1.48503	−0.742516	−	0.669828i	$-0.766368\pi$
$29$	−7.99714	−1.48503	−0.742516	+	0.669828i	$0.766368\pi$
$30$	−0.849375	−0.155074
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−5.73241	−1.01336
$33$	−3.66455	−0.637916
$34$	5.47786	0.939445
$35$	1.45897	0.246610
$36$	−1.27856	−0.213094
$37$	−2.74033	−0.450507	−0.225253	−	0.974300i	$-0.572321\pi$
$37$	−2.74033	−0.450507	−0.225253	+	0.974300i	$0.572321\pi$
$38$	−0.246330	−0.0399599
$39$	1.00000	0.160128
$40$	−2.78473	−0.440304
$41$	8.38792	1.30997	0.654986	−	0.755641i	$-0.272674\pi$
$41$	8.38792	1.30997	0.654986	+	0.755641i	$0.272674\pi$
$42$	−1.23921	−0.191214
$43$	11.1660	1.70280	0.851401	−	0.524515i	$-0.175753\pi$
$43$	11.1660	1.70280	0.851401	+	0.524515i	$0.175753\pi$
$44$	−4.68536	−0.706344
$45$	−1.00000	−0.149071
$46$	−3.37694	−0.497902
$47$	2.98604	0.435559	0.217779	−	0.975998i	$-0.430119\pi$
$47$	2.98604	0.435559	0.217779	+	0.975998i	$0.430119\pi$
$48$	−0.191844	−0.0276903
$49$	−4.87142	−0.695917
$50$	−0.849375	−0.120120
$51$	6.44928	0.903080
$52$	1.27856	0.177305
$53$	8.28287	1.13774	0.568870	−	0.822428i	$-0.307381\pi$
$53$	8.28287	1.13774	0.568870	+	0.822428i	$0.307381\pi$
$54$	0.849375	0.115585
$55$	−3.66455	−0.494128
$56$	−4.06283	−0.542918
$57$	−0.290013	−0.0384131
$58$	6.79257	0.891909
$59$	−10.2823	−1.33864	−0.669319	−	0.742975i	$-0.733414\pi$
$59$	−10.2823	−1.33864	−0.669319	+	0.742975i	$0.733414\pi$
$60$	−1.27856	−0.165062
$61$	−6.31354	−0.808366	−0.404183	−	0.914678i	$-0.632444\pi$
$61$	−6.31354	−0.808366	−0.404183	+	0.914678i	$0.632444\pi$
$62$	0.849375	0.107871
$63$	−1.45897	−0.183812
$64$	4.48527	0.560659
$65$	1.00000	0.124035
$66$	3.11258	0.383132
$67$	8.50951	1.03960	0.519801	−	0.854287i	$-0.326006\pi$
$67$	8.50951	1.03960	0.519801	+	0.854287i	$0.326006\pi$
$68$	8.24580	0.999951
$69$	−3.97579	−0.478629
$70$	−1.23921	−0.148114
$71$	−13.5594	−1.60921	−0.804603	−	0.593813i	$-0.797622\pi$
$71$	−13.5594	−1.60921	−0.804603	+	0.593813i	$0.797622\pi$
$72$	2.78473	0.328183
$73$	1.21624	0.142350	0.0711751	−	0.997464i	$-0.477325\pi$
$73$	1.21624	0.142350	0.0711751	+	0.997464i	$0.477325\pi$
$74$	2.32756	0.270574
$75$	−1.00000	−0.115470
$76$	−0.370799	−0.0425336
$77$	−5.34646	−0.609285
$78$	−0.849375	−0.0961728
$79$	−6.05820	−0.681601	−0.340801	−	0.940136i	$-0.610698\pi$
$79$	−6.05820	−0.681601	−0.340801	+	0.940136i	$0.610698\pi$
$80$	−0.191844	−0.0214488
$81$	1.00000	0.111111
$82$	−7.12449	−0.786768
$83$	8.18137	0.898022	0.449011	−	0.893526i	$-0.351776\pi$
$83$	8.18137	0.898022	0.449011	+	0.893526i	$0.351776\pi$
$84$	−1.86538	−0.203529
$85$	6.44928	0.699523
$86$	−9.48414	−1.02270
$87$	7.99714	0.857384
$88$	10.2048	1.08783
$89$	−0.797748	−0.0845612	−0.0422806	−	0.999106i	$-0.513462\pi$
$89$	−0.797748	−0.0845612	−0.0422806	+	0.999106i	$0.513462\pi$
$90$	0.849375	0.0895320
$91$	1.45897	0.152941
$92$	−5.08329	−0.529970
$93$	1.00000	0.103695
$94$	−2.53627	−0.261596
$95$	−0.290013	−0.0297547
$96$	5.73241	0.585061
$97$	3.18136	0.323018	0.161509	−	0.986871i	$-0.448364\pi$
$97$	3.18136	0.323018	0.161509	+	0.986871i	$0.448364\pi$
$98$	4.13766	0.417967
$99$	3.66455	0.368301
$100$	−1.27856	−0.127856
$101$	6.01040	0.598057	0.299028	−	0.954244i	$-0.403337\pi$
$101$	6.01040	0.598057	0.299028	+	0.954244i	$0.403337\pi$
$102$	−5.47786	−0.542389
$103$	13.0170	1.28260	0.641302	−	0.767289i	$-0.278395\pi$
$103$	13.0170	1.28260	0.641302	+	0.767289i	$0.278395\pi$
$104$	−2.78473	−0.273065
$105$	−1.45897	−0.142381
$106$	−7.03526	−0.683325
$107$	18.9713	1.83403	0.917013	−	0.398856i	$-0.130593\pi$
$107$	18.9713	1.83403	0.917013	+	0.398856i	$0.130593\pi$
$108$	1.27856	0.123030
$109$	5.84518	0.559867	0.279934	−	0.960019i	$-0.409688\pi$
$109$	5.84518	0.559867	0.279934	+	0.960019i	$0.409688\pi$
$110$	3.11258	0.296773
$111$	2.74033	0.260100
$112$	−0.279894	−0.0264475
$113$	−7.39791	−0.695937	−0.347968	−	0.937506i	$-0.613128\pi$
$113$	−7.39791	−0.695937	−0.347968	+	0.937506i	$0.613128\pi$
$114$	0.246330	0.0230709
$115$	−3.97579	−0.370744
$116$	10.2248	0.949353
$117$	−1.00000	−0.0924500
$118$	8.73351	0.803985
$119$	9.40928	0.862547
$120$	2.78473	0.254210
$121$	2.42893	0.220812
$122$	5.36257	0.485504
$123$	−8.38792	−0.756313
$124$	1.27856	0.114818
$125$	−1.00000	−0.0894427
$126$	1.23921	0.110398
$127$	5.89331	0.522947	0.261473	−	0.965211i	$-0.415792\pi$
$127$	5.89331	0.522947	0.261473	+	0.965211i	$0.415792\pi$
$128$	7.65513	0.676624
$129$	−11.1660	−0.983113
$130$	−0.849375	−0.0744951
$131$	−5.08556	−0.444327	−0.222164	−	0.975009i	$-0.571312\pi$
$131$	−5.08556	−0.444327	−0.222164	+	0.975009i	$0.571312\pi$
$132$	4.68536	0.407808
$133$	−0.423119	−0.0366890
$134$	−7.22777	−0.624384
$135$	1.00000	0.0860663
$136$	−17.9595	−1.54001
$137$	0.335092	0.0286289	0.0143144	−	0.999898i	$-0.495443\pi$
$137$	0.335092	0.0286289	0.0143144	+	0.999898i	$0.495443\pi$
$138$	3.37694	0.287464
$139$	17.0543	1.44653	0.723263	−	0.690572i	$-0.242641\pi$
$139$	17.0543	1.44653	0.723263	+	0.690572i	$0.242641\pi$
$140$	−1.86538	−0.157653
$141$	−2.98604	−0.251470
$142$	11.5170	0.966488
$143$	−3.66455	−0.306445
$144$	0.191844	0.0159870
$145$	7.99714	0.664126
$146$	−1.03304	−0.0854953
$147$	4.87142	0.401788
$148$	3.50368	0.288000
$149$	7.91965	0.648803	0.324402	−	0.945919i	$-0.394837\pi$
$149$	7.91965	0.648803	0.324402	+	0.945919i	$0.394837\pi$
$150$	0.849375	0.0693512
$151$	4.08035	0.332054	0.166027	−	0.986121i	$-0.446906\pi$
$151$	4.08035	0.332054	0.166027	+	0.986121i	$0.446906\pi$
$152$	0.807607	0.0655055
$153$	−6.44928	−0.521393
$154$	4.54115	0.365936
$155$	1.00000	0.0803219
$156$	−1.27856	−0.102367
$157$	−5.12792	−0.409252	−0.204626	−	0.978840i	$-0.565598\pi$
$157$	−5.12792	−0.409252	−0.204626	+	0.978840i	$0.565598\pi$
$158$	5.14569	0.409369
$159$	−8.28287	−0.656874
$160$	5.73241	0.453186
$161$	−5.80054	−0.457147
$162$	−0.849375	−0.0667332
$163$	−10.6720	−0.835897	−0.417948	−	0.908471i	$-0.637251\pi$
$163$	−10.6720	−0.835897	−0.417948	+	0.908471i	$0.637251\pi$
$164$	−10.7245	−0.837440
$165$	3.66455	0.285285
$166$	−6.94905	−0.539351
$167$	2.57620	0.199352	0.0996762	−	0.995020i	$-0.468219\pi$
$167$	2.57620	0.199352	0.0996762	+	0.995020i	$0.468219\pi$
$168$	4.06283	0.313454
$169$	1.00000	0.0769231
$170$	−5.47786	−0.420133
$171$	0.290013	0.0221778
$172$	−14.2764	−1.08857
$173$	−5.48000	−0.416637	−0.208318	−	0.978061i	$-0.566799\pi$
$173$	−5.48000	−0.416637	−0.208318	+	0.978061i	$0.566799\pi$
$174$	−6.79257	−0.514944
$175$	−1.45897	−0.110287
$176$	0.703023	0.0529923
$177$	10.2823	0.772863
$178$	0.677588	0.0507873
$179$	−4.22330	−0.315664	−0.157832	−	0.987466i	$-0.550451\pi$
$179$	−4.22330	−0.315664	−0.157832	+	0.987466i	$0.550451\pi$
$180$	1.27856	0.0952984
$181$	−2.09007	−0.155353	−0.0776767	−	0.996979i	$-0.524750\pi$
$181$	−2.09007	−0.155353	−0.0776767	+	0.996979i	$0.524750\pi$
$182$	−1.23921	−0.0918563
$183$	6.31354	0.466710
$184$	11.0715	0.816202
$185$	2.74033	0.201473
$186$	−0.849375	−0.0622792
$187$	−23.6337	−1.72827
$188$	−3.81784	−0.278445
$189$	1.45897	0.106124
$190$	0.246330	0.0178706
$191$	22.5641	1.63268	0.816340	−	0.577572i	$-0.196000\pi$
$191$	22.5641	1.63268	0.816340	+	0.577572i	$0.196000\pi$
$192$	−4.48527	−0.323697
$193$	−15.3497	−1.10490	−0.552448	−	0.833547i	$-0.686306\pi$
$193$	−15.3497	−1.10490	−0.552448	+	0.833547i	$0.686306\pi$
$194$	−2.70216	−0.194004
$195$	−1.00000	−0.0716115
$196$	6.22841	0.444886
$197$	−5.62053	−0.400446	−0.200223	−	0.979750i	$-0.564167\pi$
$197$	−5.62053	−0.400446	−0.200223	+	0.979750i	$0.564167\pi$
$198$	−3.11258	−0.221201
$199$	−2.24238	−0.158958	−0.0794791	−	0.996837i	$-0.525326\pi$
$199$	−2.24238	−0.158958	−0.0794791	+	0.996837i	$0.525326\pi$
$200$	2.78473	0.196910
$201$	−8.50951	−0.600215
$202$	−5.10508	−0.359192
$203$	11.6676	0.818902
$204$	−8.24580	−0.577322
$205$	−8.38792	−0.585838
$206$	−11.0563	−0.770331
$207$	3.97579	0.276337
$208$	−0.191844	−0.0133020
$209$	1.06277	0.0735131
$210$	1.23921	0.0855136
$211$	0.561642	0.0386650	0.0193325	−	0.999813i	$-0.493846\pi$
$211$	0.561642	0.0386650	0.0193325	+	0.999813i	$0.493846\pi$
$212$	−10.5902	−0.727335
$213$	13.5594	0.929076
$214$	−16.1138	−1.10151
$215$	−11.1660	−0.761516
$216$	−2.78473	−0.189477
$217$	1.45897	0.0990411
$218$	−4.96475	−0.336256
$219$	−1.21624	−0.0821859
$220$	4.68536	0.315887
$221$	6.44928	0.433826
$222$	−2.32756	−0.156216
$223$	19.0140	1.27327	0.636635	−	0.771165i	$-0.280326\pi$
$223$	19.0140	1.27327	0.636635	+	0.771165i	$0.280326\pi$
$224$	8.36339	0.558802
$225$	1.00000	0.0666667
$226$	6.28360	0.417979
$227$	−17.1529	−1.13848	−0.569240	−	0.822172i	$-0.692762\pi$
$227$	−17.1529	−1.13848	−0.569240	+	0.822172i	$0.692762\pi$
$228$	0.370799	0.0245568
$229$	−8.08088	−0.534000	−0.267000	−	0.963697i	$-0.586032\pi$
$229$	−8.08088	−0.534000	−0.267000	+	0.963697i	$0.586032\pi$
$230$	3.37694	0.222669
$231$	5.34646	0.351771
$232$	−22.2699	−1.46209
$233$	−8.02411	−0.525677	−0.262838	−	0.964840i	$-0.584659\pi$
$233$	−8.02411	−0.525677	−0.262838	+	0.964840i	$0.584659\pi$
$234$	0.849375	0.0555254
$235$	−2.98604	−0.194788
$236$	13.1465	0.855766
$237$	6.05820	0.393523
$238$	−7.99201	−0.518045
$239$	−3.51946	−0.227655	−0.113828	−	0.993501i	$-0.536311\pi$
$239$	−3.51946	−0.227655	−0.113828	+	0.993501i	$0.536311\pi$
$240$	0.191844	0.0123835
$241$	−5.93713	−0.382444	−0.191222	−	0.981547i	$-0.561245\pi$
$241$	−5.93713	−0.382444	−0.191222	+	0.981547i	$0.561245\pi$
$242$	−2.06308	−0.132620
$243$	−1.00000	−0.0641500
$244$	8.07225	0.516773
$245$	4.87142	0.311223
$246$	7.12449	0.454241
$247$	−0.290013	−0.0184531
$248$	−2.78473	−0.176830
$249$	−8.18137	−0.518473
$250$	0.849375	0.0537192
$251$	−21.8173	−1.37710	−0.688549	−	0.725190i	$-0.741752\pi$
$251$	−21.8173	−1.37710	−0.688549	+	0.725190i	$0.741752\pi$
$252$	1.86538	0.117508
$253$	14.5695	0.915976
$254$	−5.00563	−0.314081
$255$	−6.44928	−0.403870
$256$	−15.4726	−0.967039
$257$	13.2698	0.827750	0.413875	−	0.910334i	$-0.364175\pi$
$257$	13.2698	0.827750	0.413875	+	0.910334i	$0.364175\pi$
$258$	9.48414	0.590457
$259$	3.99804	0.248426
$260$	−1.27856	−0.0792930
$261$	−7.99714	−0.495011
$262$	4.31955	0.266863
$263$	3.87429	0.238899	0.119449	−	0.992840i	$-0.461887\pi$
$263$	3.87429	0.238899	0.119449	+	0.992840i	$0.461887\pi$
$264$	−10.2048	−0.628061
$265$	−8.28287	−0.508812
$266$	0.359387	0.0220354
$267$	0.797748	0.0488214
$268$	−10.8799	−0.664598
$269$	7.77785	0.474224	0.237112	−	0.971482i	$-0.423799\pi$
$269$	7.77785	0.474224	0.237112	+	0.971482i	$0.423799\pi$
$270$	−0.849375	−0.0516913
$271$	−24.0901	−1.46337	−0.731684	−	0.681644i	$-0.761265\pi$
$271$	−24.0901	−1.46337	−0.731684	+	0.681644i	$0.761265\pi$
$272$	−1.23726	−0.0750197
$273$	−1.45897	−0.0883007
$274$	−0.284619	−0.0171945
$275$	3.66455	0.220981
$276$	5.08329	0.305978
$277$	−3.62531	−0.217824	−0.108912	−	0.994051i	$-0.534737\pi$
$277$	−3.62531	−0.217824	−0.108912	+	0.994051i	$0.534737\pi$
$278$	−14.4855	−0.868782
$279$	−1.00000	−0.0598684
$280$	4.06283	0.242800
$281$	−17.5171	−1.04498	−0.522492	−	0.852644i	$-0.674997\pi$
$281$	−17.5171	−1.04498	−0.522492	+	0.852644i	$0.674997\pi$
$282$	2.53627	0.151033
$283$	−25.4553	−1.51316	−0.756580	−	0.653901i	$-0.773131\pi$
$283$	−25.4553	−1.51316	−0.756580	+	0.653901i	$0.773131\pi$
$284$	17.3366	1.02874
$285$	0.290013	0.0171789
$286$	3.11258	0.184051
$287$	−12.2377	−0.722368
$288$	−5.73241	−0.337785
$289$	24.5932	1.44666
$290$	−6.79257	−0.398874
$291$	−3.18136	−0.186494
$292$	−1.55504	−0.0910017
$293$	−4.75975	−0.278067	−0.139034	−	0.990288i	$-0.544400\pi$
$293$	−4.75975	−0.278067	−0.139034	+	0.990288i	$0.544400\pi$
$294$	−4.13766	−0.241313
$295$	10.2823	0.598657
$296$	−7.63106	−0.443547
$297$	−3.66455	−0.212639
$298$	−6.72676	−0.389670
$299$	−3.97579	−0.229926
$300$	1.27856	0.0738178
$301$	−16.2908	−0.938989
$302$	−3.46574	−0.199431
$303$	−6.01040	−0.345288
$304$	0.0556372	0.00319101
$305$	6.31354	0.361512
$306$	5.47786	0.313148
$307$	−9.49405	−0.541854	−0.270927	−	0.962600i	$-0.587330\pi$
$307$	−9.49405	−0.541854	−0.270927	+	0.962600i	$0.587330\pi$
$308$	6.83578	0.389504
$309$	−13.0170	−0.740512
$310$	−0.849375	−0.0482413
$311$	29.9590	1.69882	0.849409	−	0.527734i	$-0.176958\pi$
$311$	29.9590	1.69882	0.849409	+	0.527734i	$0.176958\pi$
$312$	2.78473	0.157654
$313$	−29.2590	−1.65382	−0.826909	−	0.562335i	$-0.809903\pi$
$313$	−29.2590	−1.65382	−0.826909	+	0.562335i	$0.809903\pi$
$314$	4.35552	0.245796
$315$	1.45897	0.0822034
$316$	7.74579	0.435735
$317$	15.8573	0.890636	0.445318	−	0.895372i	$-0.353091\pi$
$317$	15.8573	0.890636	0.445318	+	0.895372i	$0.353091\pi$
$318$	7.03526	0.394518
$319$	−29.3059	−1.64082
$320$	−4.48527	−0.250734
$321$	−18.9713	−1.05888
$322$	4.92684	0.274562
$323$	−1.87037	−0.104070
$324$	−1.27856	−0.0710312
$325$	−1.00000	−0.0554700
$326$	9.06455	0.502039
$327$	−5.84518	−0.323239
$328$	23.3581	1.28973
$329$	−4.35653	−0.240184
$330$	−3.11258	−0.171342
$331$	−34.4002	−1.89080	−0.945402	−	0.325906i	$-0.894331\pi$
$331$	−34.4002	−1.89080	−0.945402	+	0.325906i	$0.894331\pi$
$332$	−10.4604	−0.574088
$333$	−2.74033	−0.150169
$334$	−2.18816	−0.119731
$335$	−8.50951	−0.464924
$336$	0.279894	0.0152695
$337$	−31.3933	−1.71010	−0.855050	−	0.518546i	$-0.826473\pi$
$337$	−31.3933	−1.71010	−0.855050	+	0.518546i	$0.826473\pi$
$338$	−0.849375	−0.0461999
$339$	7.39791	0.401799
$340$	−8.24580	−0.447191
$341$	−3.66455	−0.198447
$342$	−0.246330	−0.0133200
$343$	17.3200	0.935192
$344$	31.0943	1.67649
$345$	3.97579	0.214049
$346$	4.65458	0.250232
$347$	21.3103	1.14400	0.571999	−	0.820255i	$-0.306168\pi$
$347$	21.3103	1.14400	0.571999	+	0.820255i	$0.306168\pi$
$348$	−10.2248	−0.548109
$349$	−6.23397	−0.333697	−0.166848	−	0.985983i	$-0.553359\pi$
$349$	−6.23397	−0.333697	−0.166848	+	0.985983i	$0.553359\pi$
$350$	1.23921	0.0662385
$351$	1.00000	0.0533761
$352$	−21.0067	−1.11966
$353$	−28.3976	−1.51145	−0.755727	−	0.654887i	$-0.772716\pi$
$353$	−28.3976	−1.51145	−0.755727	+	0.654887i	$0.772716\pi$
$354$	−8.73351	−0.464181
$355$	13.5594	0.719659
$356$	1.01997	0.0540583
$357$	−9.40928	−0.497992
$358$	3.58717	0.189588
$359$	0.719044	0.0379497	0.0189749	−	0.999820i	$-0.493960\pi$
$359$	0.719044	0.0379497	0.0189749	+	0.999820i	$0.493960\pi$
$360$	−2.78473	−0.146768
$361$	−18.9159	−0.995573
$362$	1.77525	0.0933051
$363$	−2.42893	−0.127486
$364$	−1.86538	−0.0977724
$365$	−1.21624	−0.0636609
$366$	−5.36257	−0.280306
$367$	2.39282	0.124904	0.0624520	−	0.998048i	$-0.480108\pi$
$367$	2.39282	0.124904	0.0624520	+	0.998048i	$0.480108\pi$
$368$	0.762732	0.0397602
$369$	8.38792	0.436658
$370$	−2.32756	−0.121004
$371$	−12.0844	−0.627392
$372$	−1.27856	−0.0662903
$373$	−21.1360	−1.09438	−0.547190	−	0.837008i	$-0.684302\pi$
$373$	−21.1360	−1.09438	−0.547190	+	0.837008i	$0.684302\pi$
$374$	20.0739	1.03800
$375$	1.00000	0.0516398
$376$	8.31532	0.428830
$377$	7.99714	0.411874
$378$	−1.23921	−0.0637381
$379$	−19.8582	−1.02005	−0.510025	−	0.860160i	$-0.670364\pi$
$379$	−19.8582	−1.02005	−0.510025	+	0.860160i	$0.670364\pi$
$380$	0.370799	0.0190216
$381$	−5.89331	−0.301924
$382$	−19.1654	−0.980585
$383$	36.4221	1.86108	0.930540	−	0.366190i	$-0.119338\pi$
$383$	36.4221	1.86108	0.930540	+	0.366190i	$0.119338\pi$
$384$	−7.65513	−0.390649
$385$	5.34646	0.272481
$386$	13.0377	0.663599
$387$	11.1660	0.567601
$388$	−4.06756	−0.206499
$389$	−23.8951	−1.21153	−0.605765	−	0.795643i	$-0.707133\pi$
$389$	−23.8951	−1.21153	−0.605765	+	0.795643i	$0.707133\pi$
$390$	0.849375	0.0430098
$391$	−25.6410	−1.29672
$392$	−13.5656	−0.685165
$393$	5.08556	0.256533
$394$	4.77394	0.240508
$395$	6.05820	0.304821
$396$	−4.68536	−0.235448
$397$	−4.52649	−0.227178	−0.113589	−	0.993528i	$-0.536235\pi$
$397$	−4.52649	−0.227178	−0.113589	+	0.993528i	$0.536235\pi$
$398$	1.90462	0.0954701
$399$	0.423119	0.0211824
$400$	0.191844	0.00959221
$401$	−7.53298	−0.376179	−0.188089	−	0.982152i	$-0.560229\pi$
$401$	−7.53298	−0.376179	−0.188089	+	0.982152i	$0.560229\pi$
$402$	7.22777	0.360488
$403$	1.00000	0.0498135
$404$	−7.68466	−0.382326
$405$	−1.00000	−0.0496904
$406$	−9.91014	−0.491832
$407$	−10.0421	−0.497767
$408$	17.9595	0.889128
$409$	−29.3731	−1.45240	−0.726202	−	0.687481i	$-0.758716\pi$
$409$	−29.3731	−1.45240	−0.726202	+	0.687481i	$0.758716\pi$
$410$	7.12449	0.351853
$411$	−0.335092	−0.0165289
$412$	−16.6431	−0.819944
$413$	15.0015	0.738175
$414$	−3.37694	−0.165967
$415$	−8.18137	−0.401608
$416$	5.73241	0.281054
$417$	−17.0543	−0.835153
$418$	−0.902687	−0.0441519
$419$	−1.57505	−0.0769463	−0.0384732	−	0.999260i	$-0.512249\pi$
$419$	−1.57505	−0.0769463	−0.0384732	+	0.999260i	$0.512249\pi$
$420$	1.86538	0.0910212
$421$	14.6823	0.715570	0.357785	−	0.933804i	$-0.383532\pi$
$421$	14.6823	0.715570	0.357785	+	0.933804i	$0.383532\pi$
$422$	−0.477044	−0.0232222
$423$	2.98604	0.145186
$424$	23.0655	1.12016
$425$	−6.44928	−0.312836
$426$	−11.5170	−0.558002
$427$	9.21124	0.445763
$428$	−24.2560	−1.17246
$429$	3.66455	0.176926
$430$	9.48414	0.457366
$431$	−40.3555	−1.94385	−0.971927	−	0.235283i	$-0.924398\pi$
$431$	−40.3555	−1.94385	−0.971927	+	0.235283i	$0.924398\pi$
$432$	−0.191844	−0.00923011
$433$	−29.7232	−1.42840	−0.714202	−	0.699939i	$-0.753210\pi$
$433$	−29.7232	−1.42840	−0.714202	+	0.699939i	$0.753210\pi$
$434$	−1.23921	−0.0594840
$435$	−7.99714	−0.383434
$436$	−7.47343	−0.357912
$437$	1.15303	0.0551569
$438$	1.03304	0.0493607
$439$	−38.0879	−1.81784	−0.908918	−	0.416976i	$-0.863090\pi$
$439$	−38.0879	−1.81784	−0.908918	+	0.416976i	$0.863090\pi$
$440$	−10.2048	−0.486494
$441$	−4.87142	−0.231972
$442$	−5.47786	−0.260555
$443$	40.1087	1.90562	0.952811	−	0.303564i	$-0.0981766\pi$
$443$	40.1087	1.90562	0.952811	+	0.303564i	$0.0981766\pi$
$444$	−3.50368	−0.166277
$445$	0.797748	0.0378169
$446$	−16.1500	−0.764725
$447$	−7.91965	−0.374587
$448$	−6.54386	−0.309169
$449$	−8.01132	−0.378077	−0.189039	−	0.981970i	$-0.560537\pi$
$449$	−8.01132	−0.378077	−0.189039	+	0.981970i	$0.560537\pi$
$450$	−0.849375	−0.0400399
$451$	30.7379	1.44739
$452$	9.45868	0.444899
$453$	−4.08035	−0.191711
$454$	14.5693	0.683769
$455$	−1.45897	−0.0683974
$456$	−0.807607	−0.0378196
$457$	−31.1251	−1.45597	−0.727985	−	0.685594i	$-0.759543\pi$
$457$	−31.1251	−1.45597	−0.727985	+	0.685594i	$0.759543\pi$
$458$	6.86370	0.320720
$459$	6.44928	0.301027
$460$	5.08329	0.237010
$461$	15.7329	0.732756	0.366378	−	0.930466i	$-0.380598\pi$
$461$	15.7329	0.732756	0.366378	+	0.930466i	$0.380598\pi$
$462$	−4.54115	−0.211273
$463$	30.1367	1.40057	0.700285	−	0.713864i	$-0.253056\pi$
$463$	30.1367	1.40057	0.700285	+	0.713864i	$0.253056\pi$
$464$	−1.53420	−0.0712237
$465$	−1.00000	−0.0463739
$466$	6.81548	0.315721
$467$	−4.06354	−0.188038	−0.0940190	−	0.995570i	$-0.529971\pi$
$467$	−4.06354	−0.188038	−0.0940190	+	0.995570i	$0.529971\pi$
$468$	1.27856	0.0591015
$469$	−12.4151	−0.573276
$470$	2.53627	0.116989
$471$	5.12792	0.236282
$472$	−28.6334	−1.31796
$473$	40.9184	1.88143
$474$	−5.14569	−0.236349
$475$	0.290013	0.0133067
$476$	−12.0303	−0.551410
$477$	8.28287	0.379246
$478$	2.98934	0.136729
$479$	−17.7751	−0.812164	−0.406082	−	0.913837i	$-0.633105\pi$
$479$	−17.7751	−0.812164	−0.406082	+	0.913837i	$0.633105\pi$
$480$	−5.73241	−0.261647
$481$	2.74033	0.124948
$482$	5.04285	0.229696
$483$	5.80054	0.263934
$484$	−3.10554	−0.141161
$485$	−3.18136	−0.144458
$486$	0.849375	0.0385284
$487$	0.245678	0.0111327	0.00556636	−	0.999985i	$-0.498228\pi$
$487$	0.245678	0.0111327	0.00556636	+	0.999985i	$0.498228\pi$
$488$	−17.5815	−0.795877
$489$	10.6720	0.482605
$490$	−4.13766	−0.186920
$491$	−21.6712	−0.978008	−0.489004	−	0.872282i	$-0.662640\pi$
$491$	−21.6712	−0.978008	−0.489004	+	0.872282i	$0.662640\pi$
$492$	10.7245	0.483496
$493$	51.5758	2.32286
$494$	0.246330	0.0110829
$495$	−3.66455	−0.164709
$496$	−0.191844	−0.00861406
$497$	19.7827	0.887377
$498$	6.94905	0.311394
$499$	−12.1923	−0.545800	−0.272900	−	0.962042i	$-0.587983\pi$
$499$	−12.1923	−0.545800	−0.272900	+	0.962042i	$0.587983\pi$
$500$	1.27856	0.0571790
$501$	−2.57620	−0.115096
$502$	18.5311	0.827083
$503$	5.09321	0.227095	0.113548	−	0.993533i	$-0.463779\pi$
$503$	5.09321	0.227095	0.113548	+	0.993533i	$0.463779\pi$
$504$	−4.06283	−0.180973
$505$	−6.01040	−0.267459
$506$	−12.3750	−0.550134
$507$	−1.00000	−0.0444116
$508$	−7.53496	−0.334310
$509$	34.8939	1.54665	0.773323	−	0.634012i	$-0.218593\pi$
$509$	34.8939	1.54665	0.773323	+	0.634012i	$0.218593\pi$
$510$	5.47786	0.242564
$511$	−1.77445	−0.0784972
$512$	−2.16820	−0.0958216
$513$	−0.290013	−0.0128044
$514$	−11.2711	−0.497146
$515$	−13.0170	−0.573598
$516$	14.2764	0.628485
$517$	10.9425	0.481251
$518$	−3.39584	−0.149205
$519$	5.48000	0.240545
$520$	2.78473	0.122118
$521$	2.69286	0.117976	0.0589882	−	0.998259i	$-0.481213\pi$
$521$	2.69286	0.117976	0.0589882	+	0.998259i	$0.481213\pi$
$522$	6.79257	0.297303
$523$	27.1257	1.18613	0.593063	−	0.805156i	$-0.297919\pi$
$523$	27.1257	1.18613	0.593063	+	0.805156i	$0.297919\pi$
$524$	6.50220	0.284050
$525$	1.45897	0.0636745
$526$	−3.29073	−0.143482
$527$	6.44928	0.280935
$528$	−0.703023	−0.0305951
$529$	−7.19309	−0.312743
$530$	7.03526	0.305592
$531$	−10.2823	−0.446213
$532$	0.540983	0.0234546
$533$	−8.38792	−0.363321
$534$	−0.677588	−0.0293221
$535$	−18.9713	−0.820202
$536$	23.6967	1.02354
$537$	4.22330	0.182249
$538$	−6.60631	−0.284818
$539$	−17.8516	−0.768921
$540$	−1.27856	−0.0550205
$541$	37.0353	1.59227	0.796136	−	0.605118i	$-0.206874\pi$
$541$	37.0353	1.59227	0.796136	+	0.605118i	$0.206874\pi$
$542$	20.4615	0.878898
$543$	2.09007	0.0896934
$544$	36.9699	1.58507
$545$	−5.84518	−0.250380
$546$	1.23921	0.0530333
$547$	−27.6921	−1.18403	−0.592015	−	0.805927i	$-0.701667\pi$
$547$	−27.6921	−1.18403	−0.592015	+	0.805927i	$0.701667\pi$
$548$	−0.428436	−0.0183019
$549$	−6.31354	−0.269455
$550$	−3.11258	−0.132721
$551$	−2.31927	−0.0988043
$552$	−11.0715	−0.471234
$553$	8.83872	0.375860
$554$	3.07925	0.130825
$555$	−2.74033	−0.116320
$556$	−21.8050	−0.924737
$557$	37.0560	1.57011	0.785056	−	0.619425i	$-0.212634\pi$
$557$	37.0560	1.57011	0.785056	+	0.619425i	$0.212634\pi$
$558$	0.849375	0.0359569
$559$	−11.1660	−0.472272
$560$	0.279894	0.0118277
$561$	23.6337	0.997816
$562$	14.8786	0.627616
$563$	−25.3309	−1.06757	−0.533786	−	0.845620i	$-0.679231\pi$
$563$	−25.3309	−1.06757	−0.533786	+	0.845620i	$0.679231\pi$
$564$	3.81784	0.160760
$565$	7.39791	0.311232
$566$	21.6211	0.908803
$567$	−1.45897	−0.0612708
$568$	−37.7593	−1.58435
$569$	44.1877	1.85245	0.926223	−	0.376977i	$-0.123036\pi$
$569$	44.1877	1.85245	0.926223	+	0.376977i	$0.123036\pi$
$570$	−0.246330	−0.0103176
$571$	−21.7191	−0.908915	−0.454457	−	0.890769i	$-0.650167\pi$
$571$	−21.7191	−0.908915	−0.454457	+	0.890769i	$0.650167\pi$
$572$	4.68536	0.195905
$573$	−22.5641	−0.942628
$574$	10.3944	0.433853
$575$	3.97579	0.165802
$576$	4.48527	0.186886
$577$	−8.71810	−0.362939	−0.181470	−	0.983397i	$-0.558085\pi$
$577$	−8.71810	−0.362939	−0.181470	+	0.983397i	$0.558085\pi$
$578$	−20.8889	−0.868862
$579$	15.3497	0.637912
$580$	−10.2248	−0.424563
$581$	−11.9363	−0.495203
$582$	2.70216	0.112008
$583$	30.3530	1.25709
$584$	3.38690	0.140151
$585$	1.00000	0.0413449
$586$	4.04281	0.167007
$587$	−0.927863	−0.0382970	−0.0191485	−	0.999817i	$-0.506096\pi$
$587$	−0.927863	−0.0382970	−0.0191485	+	0.999817i	$0.506096\pi$
$588$	−6.22841	−0.256855
$589$	−0.290013	−0.0119498
$590$	−8.73351	−0.359553
$591$	5.62053	0.231198
$592$	−0.525715	−0.0216068
$593$	4.34842	0.178568	0.0892842	−	0.996006i	$-0.471542\pi$
$593$	4.34842	0.178568	0.0892842	+	0.996006i	$0.471542\pi$
$594$	3.11258	0.127711
$595$	−9.40928	−0.385743
$596$	−10.1258	−0.414768
$597$	2.24238	0.0917746
$598$	3.37694	0.138093
$599$	12.4868	0.510195	0.255097	−	0.966915i	$-0.417892\pi$
$599$	12.4868	0.510195	0.255097	+	0.966915i	$0.417892\pi$
$600$	−2.78473	−0.113686
$601$	3.04132	0.124058	0.0620290	−	0.998074i	$-0.480243\pi$
$601$	3.04132	0.124058	0.0620290	+	0.998074i	$0.480243\pi$
$602$	13.8370	0.563956
$603$	8.50951	0.346534
$604$	−5.21697	−0.212276
$605$	−2.42893	−0.0987502
$606$	5.10508	0.207380
$607$	32.8684	1.33409	0.667043	−	0.745019i	$-0.267560\pi$
$607$	32.8684	1.33409	0.667043	+	0.745019i	$0.267560\pi$
$608$	−1.66247	−0.0674221
$609$	−11.6676	−0.472793
$610$	−5.36257	−0.217124
$611$	−2.98604	−0.120802
$612$	8.24580	0.333317
$613$	33.1473	1.33881	0.669404	−	0.742899i	$-0.266550\pi$
$613$	33.1473	1.33881	0.669404	+	0.742899i	$0.266550\pi$
$614$	8.06401	0.325437
$615$	8.38792	0.338233
$616$	−14.8884	−0.599872
$617$	8.22725	0.331217	0.165608	−	0.986192i	$-0.447041\pi$
$617$	8.22725	0.331217	0.165608	+	0.986192i	$0.447041\pi$
$618$	11.0563	0.444751
$619$	−37.8396	−1.52090	−0.760451	−	0.649395i	$-0.775022\pi$
$619$	−37.8396	−1.52090	−0.760451	+	0.649395i	$0.775022\pi$
$620$	−1.27856	−0.0513483
$621$	−3.97579	−0.159543
$622$	−25.4464	−1.02031
$623$	1.16389	0.0466302
$624$	0.191844	0.00767991
$625$	1.00000	0.0400000
$626$	24.8519	0.993282
$627$	−1.06277	−0.0424428
$628$	6.55636	0.261627
$629$	17.6731	0.704674
$630$	−1.23921	−0.0493713
$631$	28.5814	1.13781	0.568903	−	0.822404i	$-0.307368\pi$
$631$	28.5814	1.13781	0.568903	+	0.822404i	$0.307368\pi$
$632$	−16.8705	−0.671071
$633$	−0.561642	−0.0223232
$634$	−13.4688	−0.534915
$635$	−5.89331	−0.233869
$636$	10.5902	0.419927
$637$	4.87142	0.193013
$638$	24.8917	0.985473
$639$	−13.5594	−0.536402
$640$	−7.65513	−0.302596
$641$	0.562550	0.0222194	0.0111097	−	0.999938i	$-0.496464\pi$
$641$	0.562550	0.0222194	0.0111097	+	0.999938i	$0.496464\pi$
$642$	16.1138	0.635960
$643$	−2.69764	−0.106384	−0.0531922	−	0.998584i	$-0.516940\pi$
$643$	−2.69764	−0.106384	−0.0531922	+	0.998584i	$0.516940\pi$
$644$	7.41635	0.292245
$645$	11.1660	0.439662
$646$	1.58865	0.0625045
$647$	−34.4912	−1.35599	−0.677995	−	0.735067i	$-0.737151\pi$
$647$	−34.4912	−1.35599	−0.677995	+	0.735067i	$0.737151\pi$
$648$	2.78473	0.109394
$649$	−37.6799	−1.47907
$650$	0.849375	0.0333152
$651$	−1.45897	−0.0571814
$652$	13.6448	0.534373
$653$	16.9558	0.663531	0.331765	−	0.943362i	$-0.392356\pi$
$653$	16.9558	0.663531	0.331765	+	0.943362i	$0.392356\pi$
$654$	4.96475	0.194137
$655$	5.08556	0.198709
$656$	1.60917	0.0628276
$657$	1.21624	0.0474500
$658$	3.70033	0.144254
$659$	39.3575	1.53315	0.766576	−	0.642153i	$-0.221959\pi$
$659$	39.3575	1.53315	0.766576	+	0.642153i	$0.221959\pi$
$660$	−4.68536	−0.182377
$661$	−3.96307	−0.154146	−0.0770728	−	0.997025i	$-0.524557\pi$
$661$	−3.96307	−0.154146	−0.0770728	+	0.997025i	$0.524557\pi$
$662$	29.2187	1.13562
$663$	−6.44928	−0.250469
$664$	22.7829	0.884148
$665$	0.423119	0.0164078
$666$	2.32756	0.0901913
$667$	−31.7950	−1.23111
$668$	−3.29383	−0.127442
$669$	−19.0140	−0.735123
$670$	7.22777	0.279233
$671$	−23.1363	−0.893167
$672$	−8.36339	−0.322625
$673$	−0.970732	−0.0374190	−0.0187095	−	0.999825i	$-0.505956\pi$
$673$	−0.970732	−0.0374190	−0.0187095	+	0.999825i	$0.505956\pi$
$674$	26.6646	1.02708
$675$	−1.00000	−0.0384900
$676$	−1.27856	−0.0491755
$677$	−25.4722	−0.978975	−0.489487	−	0.872010i	$-0.662816\pi$
$677$	−25.4722	−0.978975	−0.489487	+	0.872010i	$0.662816\pi$
$678$	−6.28360	−0.241320
$679$	−4.64149	−0.178124
$680$	17.9595	0.688715
$681$	17.1529	0.657301
$682$	3.11258	0.119187
$683$	−40.9335	−1.56628	−0.783139	−	0.621846i	$-0.786383\pi$
$683$	−40.9335	−1.56628	−0.783139	+	0.621846i	$0.786383\pi$
$684$	−0.370799	−0.0141779
$685$	−0.335092	−0.0128032
$686$	−14.7112	−0.561675
$687$	8.08088	0.308305
$688$	2.14214	0.0816681
$689$	−8.28287	−0.315552
$690$	−3.37694	−0.128558
$691$	−6.98965	−0.265899	−0.132949	−	0.991123i	$-0.542445\pi$
$691$	−6.98965	−0.265899	−0.132949	+	0.991123i	$0.542445\pi$
$692$	7.00652	0.266348
$693$	−5.34646	−0.203095
$694$	−18.1004	−0.687083
$695$	−17.0543	−0.646906
$696$	22.2699	0.844137
$697$	−54.0960	−2.04903
$698$	5.29498	0.200418
$699$	8.02411	0.303500
$700$	1.86538	0.0705047
$701$	7.57673	0.286169	0.143085	−	0.989710i	$-0.454298\pi$
$701$	7.57673	0.286169	0.143085	+	0.989710i	$0.454298\pi$
$702$	−0.849375	−0.0320576
$703$	−0.794729	−0.0299738
$704$	16.4365	0.619475
$705$	2.98604	0.112461
$706$	24.1202	0.907777
$707$	−8.76897	−0.329791
$708$	−13.1465	−0.494077
$709$	4.97966	0.187015	0.0935075	−	0.995619i	$-0.470192\pi$
$709$	4.97966	0.187015	0.0935075	+	0.995619i	$0.470192\pi$
$710$	−11.5170	−0.432226
$711$	−6.05820	−0.227200
$712$	−2.22151	−0.0832547
$713$	−3.97579	−0.148895
$714$	7.99201	0.299093
$715$	3.66455	0.137046
$716$	5.39975	0.201798
$717$	3.51946	0.131437
$718$	−0.610738	−0.0227926
$719$	−14.3842	−0.536441	−0.268221	−	0.963358i	$-0.586436\pi$
$719$	−14.3842	−0.536441	−0.268221	+	0.963358i	$0.586436\pi$
$720$	−0.191844	−0.00714961
$721$	−18.9914	−0.707276
$722$	16.0667	0.597940
$723$	5.93713	0.220804
$724$	2.67228	0.0993145
$725$	−7.99714	−0.297006
$726$	2.06308	0.0765679
$727$	−16.7694	−0.621944	−0.310972	−	0.950419i	$-0.600654\pi$
$727$	−16.7694	−0.621944	−0.310972	+	0.950419i	$0.600654\pi$
$728$	4.06283	0.150578
$729$	1.00000	0.0370370
$730$	1.03304	0.0382347
$731$	−72.0128	−2.66349
$732$	−8.07225	−0.298359
$733$	−7.38876	−0.272910	−0.136455	−	0.990646i	$-0.543571\pi$
$733$	−7.38876	−0.272910	−0.136455	+	0.990646i	$0.543571\pi$
$734$	−2.03240	−0.0750172
$735$	−4.87142	−0.179685
$736$	−22.7908	−0.840082
$737$	31.1835	1.14866
$738$	−7.12449	−0.262256
$739$	8.25080	0.303510	0.151755	−	0.988418i	$-0.451507\pi$
$739$	8.25080	0.303510	0.151755	+	0.988418i	$0.451507\pi$
$740$	−3.50368	−0.128798
$741$	0.290013	0.0106539
$742$	10.2642	0.376811
$743$	10.4302	0.382647	0.191323	−	0.981527i	$-0.438722\pi$
$743$	10.4302	0.382647	0.191323	+	0.981527i	$0.438722\pi$
$744$	2.78473	0.102093
$745$	−7.91965	−0.290154
$746$	17.9524	0.657283
$747$	8.18137	0.299341
$748$	30.2172	1.10485
$749$	−27.6785	−1.01135
$750$	−0.849375	−0.0310148
$751$	17.7116	0.646306	0.323153	−	0.946347i	$-0.395257\pi$
$751$	17.7116	0.646306	0.323153	+	0.946347i	$0.395257\pi$
$752$	0.572855	0.0208899
$753$	21.8173	0.795068
$754$	−6.79257	−0.247371
$755$	−4.08035	−0.148499
$756$	−1.86538	−0.0678432
$757$	−44.8409	−1.62977	−0.814884	−	0.579624i	$-0.803200\pi$
$757$	−44.8409	−1.62977	−0.814884	+	0.579624i	$0.803200\pi$
$758$	16.8671	0.612641
$759$	−14.5695	−0.528839
$760$	−0.807607	−0.0292950
$761$	−23.4467	−0.849940	−0.424970	−	0.905207i	$-0.639715\pi$
$761$	−23.4467	−0.849940	−0.424970	+	0.905207i	$0.639715\pi$
$762$	5.00563	0.181335
$763$	−8.52793	−0.308732
$764$	−28.8496	−1.04374
$765$	6.44928	0.233174
$766$	−30.9360	−1.11776
$767$	10.2823	0.371271
$768$	15.4726	0.558320
$769$	29.9754	1.08094	0.540471	−	0.841363i	$-0.318246\pi$
$769$	29.9754	1.08094	0.540471	+	0.841363i	$0.318246\pi$
$770$	−4.54115	−0.163652
$771$	−13.2698	−0.477902
$772$	19.6255	0.706339
$773$	−34.3303	−1.23477	−0.617387	−	0.786659i	$-0.711809\pi$
$773$	−34.3303	−1.23477	−0.617387	+	0.786659i	$0.711809\pi$
$774$	−9.48414	−0.340900
$775$	−1.00000	−0.0359211
$776$	8.85921	0.318027
$777$	−3.99804	−0.143429
$778$	20.2959	0.727644
$779$	2.43260	0.0871570
$780$	1.27856	0.0457799
$781$	−49.6892	−1.77802
$782$	21.7788	0.778809
$783$	7.99714	0.285795
$784$	−0.934553	−0.0333769
$785$	5.12792	0.183023
$786$	−4.31955	−0.154073
$787$	16.1766	0.576632	0.288316	−	0.957535i	$-0.406905\pi$
$787$	16.1766	0.576632	0.288316	+	0.957535i	$0.406905\pi$
$788$	7.18620	0.255998
$789$	−3.87429	−0.137928
$790$	−5.14569	−0.183075
$791$	10.7933	0.383766
$792$	10.2048	0.362611
$793$	6.31354	0.224200
$794$	3.84469	0.136443
$795$	8.28287	0.293763
$796$	2.86702	0.101619
$797$	8.11777	0.287546	0.143773	−	0.989611i	$-0.454076\pi$
$797$	8.11777	0.287546	0.143773	+	0.989611i	$0.454076\pi$
$798$	−0.359387	−0.0127221
$799$	−19.2578	−0.681293
$800$	−5.73241	−0.202671
$801$	−0.797748	−0.0281871
$802$	6.39832	0.225933
$803$	4.45697	0.157283
$804$	10.8799	0.383706
$805$	5.80054	0.204442
$806$	−0.849375	−0.0299180
$807$	−7.77785	−0.273793
$808$	16.7373	0.588817
$809$	13.2552	0.466029	0.233014	−	0.972473i	$-0.425141\pi$
$809$	13.2552	0.466029	0.233014	+	0.972473i	$0.425141\pi$
$810$	0.849375	0.0298440
$811$	−20.4212	−0.717086	−0.358543	−	0.933513i	$-0.616726\pi$
$811$	−20.4212	−0.717086	−0.358543	+	0.933513i	$0.616726\pi$
$812$	−14.9177	−0.523509
$813$	24.0901	0.844876
$814$	8.52948	0.298958
$815$	10.6720	0.373824
$816$	1.23726	0.0433126
$817$	3.23829	0.113293
$818$	24.9488	0.872313
$819$	1.45897	0.0509804
$820$	10.7245	0.374515
$821$	2.37020	0.0827205	0.0413603	−	0.999144i	$-0.486831\pi$
$821$	2.37020	0.0827205	0.0413603	+	0.999144i	$0.486831\pi$
$822$	0.284619	0.00992723
$823$	−52.1982	−1.81952	−0.909758	−	0.415139i	$-0.863733\pi$
$823$	−52.1982	−1.81952	−0.909758	+	0.415139i	$0.863733\pi$
$824$	36.2488	1.26279
$825$	−3.66455	−0.127583
$826$	−12.7419	−0.443347
$827$	13.3796	0.465254	0.232627	−	0.972566i	$-0.425268\pi$
$827$	13.3796	0.465254	0.232627	+	0.972566i	$0.425268\pi$
$828$	−5.08329	−0.176657
$829$	8.40975	0.292083	0.146041	−	0.989278i	$-0.453347\pi$
$829$	8.40975	0.292083	0.146041	+	0.989278i	$0.453347\pi$
$830$	6.94905	0.241205
$831$	3.62531	0.125761
$832$	−4.48527	−0.155499
$833$	31.4171	1.08854
$834$	14.4855	0.501592
$835$	−2.57620	−0.0891531
$836$	−1.35881	−0.0469955
$837$	1.00000	0.0345651
$838$	1.33781	0.0462139
$839$	−0.841800	−0.0290621	−0.0145311	−	0.999894i	$-0.504626\pi$
$839$	−0.841800	−0.0290621	−0.0145311	+	0.999894i	$0.504626\pi$
$840$	−4.06283	−0.140181
$841$	34.9543	1.20532
$842$	−12.4708	−0.429770
$843$	17.5171	0.603321
$844$	−0.718093	−0.0247178
$845$	−1.00000	−0.0344010
$846$	−2.53627	−0.0871987
$847$	−3.54373	−0.121764
$848$	1.58902	0.0545672
$849$	25.4553	0.873624
$850$	5.47786	0.187889
$851$	−10.8950	−0.373474
$852$	−17.3366	−0.593941
$853$	−31.0969	−1.06474	−0.532369	−	0.846512i	$-0.678698\pi$
$853$	−31.0969	−1.06474	−0.532369	+	0.846512i	$0.678698\pi$
$854$	−7.82380	−0.267725
$855$	−0.290013	−0.00991822
$856$	52.8300	1.80569
$857$	−38.0992	−1.30144	−0.650722	−	0.759316i	$-0.725534\pi$
$857$	−38.0992	−1.30144	−0.650722	+	0.759316i	$0.725534\pi$
$858$	−3.11258	−0.106262
$859$	−38.9421	−1.32869	−0.664343	−	0.747428i	$-0.731289\pi$
$859$	−38.9421	−1.32869	−0.664343	+	0.747428i	$0.731289\pi$
$860$	14.2764	0.486823
$861$	12.2377	0.417059
$862$	34.2769	1.16748
$863$	−54.8191	−1.86607	−0.933033	−	0.359792i	$-0.882848\pi$
$863$	−54.8191	−1.86607	−0.933033	+	0.359792i	$0.882848\pi$
$864$	5.73241	0.195020
$865$	5.48000	0.186326
$866$	25.2461	0.857898
$867$	−24.5932	−0.835229
$868$	−1.86538	−0.0633151
$869$	−22.2006	−0.753104
$870$	6.79257	0.230290
$871$	−8.50951	−0.288334
$872$	16.2773	0.551217
$873$	3.18136	0.107673
$874$	−0.979355	−0.0331272
$875$	1.45897	0.0493221
$876$	1.55504	0.0525399
$877$	12.8306	0.433259	0.216630	−	0.976254i	$-0.430494\pi$
$877$	12.8306	0.433259	0.216630	+	0.976254i	$0.430494\pi$
$878$	32.3509	1.09179
$879$	4.75975	0.160542
$880$	−0.703023	−0.0236989
$881$	34.3140	1.15607	0.578035	−	0.816012i	$-0.303820\pi$
$881$	34.3140	1.15607	0.578035	+	0.816012i	$0.303820\pi$
$882$	4.13766	0.139322
$883$	−44.6309	−1.50195	−0.750975	−	0.660331i	$-0.770416\pi$
$883$	−44.6309	−1.50195	−0.750975	+	0.660331i	$0.770416\pi$
$884$	−8.24580	−0.277336
$885$	−10.2823	−0.345635
$886$	−34.0673	−1.14451
$887$	−21.7311	−0.729658	−0.364829	−	0.931074i	$-0.618873\pi$
$887$	−21.7311	−0.729658	−0.364829	+	0.931074i	$0.618873\pi$
$888$	7.63106	0.256082
$889$	−8.59814	−0.288372
$890$	−0.677588	−0.0227128
$891$	3.66455	0.122767
$892$	−24.3106	−0.813977
$893$	0.865990	0.0289792
$894$	6.72676	0.224976
$895$	4.22330	0.141169
$896$	−11.1686	−0.373116
$897$	3.97579	0.132748
$898$	6.80461	0.227073
$899$	7.99714	0.266720
$900$	−1.27856	−0.0426187
$901$	−53.4185	−1.77963
$902$	−26.1080	−0.869303
$903$	16.2908	0.542125
$904$	−20.6012	−0.685185
$905$	2.09007	0.0694762
$906$	3.46574	0.115142
$907$	4.71564	0.156580	0.0782901	−	0.996931i	$-0.475054\pi$
$907$	4.71564	0.156580	0.0782901	+	0.996931i	$0.475054\pi$
$908$	21.9311	0.727808
$909$	6.01040	0.199352
$910$	1.23921	0.0410794
$911$	−13.3019	−0.440710	−0.220355	−	0.975420i	$-0.570722\pi$
$911$	−13.3019	−0.440710	−0.220355	+	0.975420i	$0.570722\pi$
$912$	−0.0556372	−0.00184233
$913$	29.9810	0.992228
$914$	26.4369	0.874453
$915$	−6.31354	−0.208719
$916$	10.3319	0.341376
$917$	7.41966	0.245019
$918$	−5.47786	−0.180796
$919$	−54.0633	−1.78338	−0.891692	−	0.452642i	$-0.850482\pi$
$919$	−54.0633	−1.78338	−0.891692	+	0.452642i	$0.850482\pi$
$920$	−11.0715	−0.365017
$921$	9.49405	0.312839
$922$	−13.3632	−0.440092
$923$	13.5594	0.446314
$924$	−6.83578	−0.224880
$925$	−2.74033	−0.0901014
$926$	−25.5973	−0.841181
$927$	13.0170	0.427535
$928$	45.8429	1.50487
$929$	−30.0366	−0.985469	−0.492735	−	0.870180i	$-0.664003\pi$
$929$	−30.0366	−0.985469	−0.492735	+	0.870180i	$0.664003\pi$
$930$	0.849375	0.0278521
$931$	−1.41277	−0.0463018
$932$	10.2593	0.336055
$933$	−29.9590	−0.980814
$934$	3.45147	0.112935
$935$	23.6337	0.772905
$936$	−2.78473	−0.0910217
$937$	−5.85122	−0.191151	−0.0955756	−	0.995422i	$-0.530469\pi$
$937$	−5.85122	−0.191151	−0.0955756	+	0.995422i	$0.530469\pi$
$938$	10.5451	0.344309
$939$	29.2590	0.954833
$940$	3.81784	0.124524
$941$	48.2014	1.57132	0.785660	−	0.618659i	$-0.212324\pi$
$941$	48.2014	1.57132	0.785660	+	0.618659i	$0.212324\pi$
$942$	−4.35552	−0.141911
$943$	33.3486	1.08598
$944$	−1.97259	−0.0642025
$945$	−1.45897	−0.0474602
$946$	−34.7551	−1.12999
$947$	−48.9267	−1.58990	−0.794951	−	0.606673i	$-0.792504\pi$
$947$	−48.9267	−1.58990	−0.794951	+	0.606673i	$0.792504\pi$
$948$	−7.74579	−0.251572
$949$	−1.21624	−0.0394808
$950$	−0.246330	−0.00799199
$951$	−15.8573	−0.514209
$952$	26.2023	0.849221
$953$	36.9452	1.19677	0.598386	−	0.801208i	$-0.295809\pi$
$953$	36.9452	1.19677	0.598386	+	0.801208i	$0.295809\pi$
$954$	−7.03526	−0.227775
$955$	−22.5641	−0.730156
$956$	4.49985	0.145536
$957$	29.3059	0.947326
$958$	15.0977	0.487785
$959$	−0.488889	−0.0157870
$960$	4.48527	0.144762
$961$	1.00000	0.0322581
$962$	−2.32756	−0.0750437
$963$	18.9713	0.611342
$964$	7.59099	0.244489
$965$	15.3497	0.494124
$966$	−4.92684	−0.158518
$967$	25.5614	0.821998	0.410999	−	0.911636i	$-0.365180\pi$
$967$	25.5614	0.821998	0.410999	+	0.911636i	$0.365180\pi$
$968$	6.76392	0.217401
$969$	1.87037	0.0600850
$970$	2.70216	0.0867613
$971$	−9.93028	−0.318678	−0.159339	−	0.987224i	$-0.550936\pi$
$971$	−9.93028	−0.318678	−0.159339	+	0.987224i	$0.550936\pi$
$972$	1.27856	0.0410099
$973$	−24.8816	−0.797669
$974$	−0.208672	−0.00668630
$975$	1.00000	0.0320256
$976$	−1.21122	−0.0387701
$977$	45.5118	1.45605	0.728027	−	0.685549i	$-0.240438\pi$
$977$	45.5118	1.45605	0.728027	+	0.685549i	$0.240438\pi$
$978$	−9.06455	−0.289852
$979$	−2.92339	−0.0934319
$980$	−6.22841	−0.198959
$981$	5.84518	0.186622
$982$	18.4070	0.587390
$983$	−38.2489	−1.21995	−0.609975	−	0.792421i	$-0.708820\pi$
$983$	−38.2489	−1.21995	−0.609975	+	0.792421i	$0.708820\pi$
$984$	−23.3581	−0.744628
$985$	5.62053	0.179085
$986$	−43.8072	−1.39511
$987$	4.35653	0.138670
$988$	0.370799	0.0117967
$989$	44.3938	1.41164
$990$	3.11258	0.0989242
$991$	8.26876	0.262666	0.131333	−	0.991338i	$-0.458074\pi$
$991$	8.26876	0.262666	0.131333	+	0.991338i	$0.458074\pi$
$992$	5.73241	0.182004
$993$	34.4002	1.09166
$994$	−16.8030	−0.532958
$995$	2.24238	0.0710883
$996$	10.4604	0.331450
$997$	3.31359	0.104942	0.0524712	−	0.998622i	$-0.483290\pi$
$997$	3.31359	0.104942	0.0524712	+	0.998622i	$0.483290\pi$
$998$	10.3558	0.327807
$999$	2.74033	0.0867001

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.w.1.5	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.w.1.5	✓	11	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:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

\(f(q)\)	\(=\)	\(q-0.849375 q^{2} -1.00000 q^{3} -1.27856 q^{4} -1.00000 q^{5} +0.849375 q^{6} -1.45897 q^{7} +2.78473 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.849375 q^{2} -1.00000 q^{3} -1.27856 q^{4} -1.00000 q^{5} +0.849375 q^{6} -1.45897 q^{7} +2.78473 q^{8} +1.00000 q^{9} +0.849375 q^{10} +3.66455 q^{11} +1.27856 q^{12} -1.00000 q^{13} +1.23921 q^{14} +1.00000 q^{15} +0.191844 q^{16} -6.44928 q^{17} -0.849375 q^{18} +0.290013 q^{19} +1.27856 q^{20} +1.45897 q^{21} -3.11258 q^{22} +3.97579 q^{23} -2.78473 q^{24} +1.00000 q^{25} +0.849375 q^{26} -1.00000 q^{27} +1.86538 q^{28} -7.99714 q^{29} -0.849375 q^{30} -1.00000 q^{31} -5.73241 q^{32} -3.66455 q^{33} +5.47786 q^{34} +1.45897 q^{35} -1.27856 q^{36} -2.74033 q^{37} -0.246330 q^{38} +1.00000 q^{39} -2.78473 q^{40} +8.38792 q^{41} -1.23921 q^{42} +11.1660 q^{43} -4.68536 q^{44} -1.00000 q^{45} -3.37694 q^{46} +2.98604 q^{47} -0.191844 q^{48} -4.87142 q^{49} -0.849375 q^{50} +6.44928 q^{51} +1.27856 q^{52} +8.28287 q^{53} +0.849375 q^{54} -3.66455 q^{55} -4.06283 q^{56} -0.290013 q^{57} +6.79257 q^{58} -10.2823 q^{59} -1.27856 q^{60} -6.31354 q^{61} +0.849375 q^{62} -1.45897 q^{63} +4.48527 q^{64} +1.00000 q^{65} +3.11258 q^{66} +8.50951 q^{67} +8.24580 q^{68} -3.97579 q^{69} -1.23921 q^{70} -13.5594 q^{71} +2.78473 q^{72} +1.21624 q^{73} +2.32756 q^{74} -1.00000 q^{75} -0.370799 q^{76} -5.34646 q^{77} -0.849375 q^{78} -6.05820 q^{79} -0.191844 q^{80} +1.00000 q^{81} -7.12449 q^{82} +8.18137 q^{83} -1.86538 q^{84} +6.44928 q^{85} -9.48414 q^{86} +7.99714 q^{87} +10.2048 q^{88} -0.797748 q^{89} +0.849375 q^{90} +1.45897 q^{91} -5.08329 q^{92} +1.00000 q^{93} -2.53627 q^{94} -0.290013 q^{95} +5.73241 q^{96} +3.18136 q^{97} +4.13766 q^{98} +3.66455 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 2 q^{2} - 11 q^{3} + 12 q^{4} - 11 q^{5} - 2 q^{6} + 4 q^{7} + 3 q^{8} + 11 q^{9}+O(q^{10}) \) 11 * q + 2 * q^2 - 11 * q^3 + 12 * q^4 - 11 * q^5 - 2 * q^6 + 4 * q^7 + 3 * q^8 + 11 * q^9	\( 11 q + 2 q^{2} - 11 q^{3} + 12 q^{4} - 11 q^{5} - 2 q^{6} + 4 q^{7} + 3 q^{8} + 11 q^{9} - 2 q^{10} - 12 q^{12} - 11 q^{13} + 5 q^{14} + 11 q^{15} - 10 q^{16} - 3 q^{17} + 2 q^{18} - 8 q^{19} - 12 q^{20} - 4 q^{21} - 3 q^{22} + 11 q^{23} - 3 q^{24} + 11 q^{25} - 2 q^{26} - 11 q^{27} + 14 q^{28} - 14 q^{29} + 2 q^{30} - 11 q^{31} + 8 q^{32} - 11 q^{34} - 4 q^{35} + 12 q^{36} + 7 q^{37} + 8 q^{38} + 11 q^{39} - 3 q^{40} + 22 q^{41} - 5 q^{42} - 5 q^{43} - 13 q^{44} - 11 q^{45} - 22 q^{46} + 5 q^{47} + 10 q^{48} - 33 q^{49} + 2 q^{50} + 3 q^{51} - 12 q^{52} + 4 q^{53} - 2 q^{54} - 13 q^{56} + 8 q^{57} - 18 q^{58} - 3 q^{59} + 12 q^{60} - 28 q^{61} - 2 q^{62} + 4 q^{63} + 3 q^{64} + 11 q^{65} + 3 q^{66} - 11 q^{67} - 9 q^{68} - 11 q^{69} - 5 q^{70} + 5 q^{71} + 3 q^{72} - 3 q^{73} - 12 q^{74} - 11 q^{75} - 36 q^{76} + 18 q^{77} + 2 q^{78} - 43 q^{79} + 10 q^{80} + 11 q^{81} - 15 q^{82} - 28 q^{83} - 14 q^{84} + 3 q^{85} + 10 q^{86} + 14 q^{87} - 43 q^{88} - 25 q^{89} - 2 q^{90} - 4 q^{91} + 7 q^{92} + 11 q^{93} - 16 q^{94} + 8 q^{95} - 8 q^{96} - 6 q^{97} - 14 q^{98}+O(q^{100}) \) 11 * q + 2 * q^2 - 11 * q^3 + 12 * q^4 - 11 * q^5 - 2 * q^6 + 4 * q^7 + 3 * q^8 + 11 * q^9 - 2 * q^10 - 12 * q^12 - 11 * q^13 + 5 * q^14 + 11 * q^15 - 10 * q^16 - 3 * q^17 + 2 * q^18 - 8 * q^19 - 12 * q^20 - 4 * q^21 - 3 * q^22 + 11 * q^23 - 3 * q^24 + 11 * q^25 - 2 * q^26 - 11 * q^27 + 14 * q^28 - 14 * q^29 + 2 * q^30 - 11 * q^31 + 8 * q^32 - 11 * q^34 - 4 * q^35 + 12 * q^36 + 7 * q^37 + 8 * q^38 + 11 * q^39 - 3 * q^40 + 22 * q^41 - 5 * q^42 - 5 * q^43 - 13 * q^44 - 11 * q^45 - 22 * q^46 + 5 * q^47 + 10 * q^48 - 33 * q^49 + 2 * q^50 + 3 * q^51 - 12 * q^52 + 4 * q^53 - 2 * q^54 - 13 * q^56 + 8 * q^57 - 18 * q^58 - 3 * q^59 + 12 * q^60 - 28 * q^61 - 2 * q^62 + 4 * q^63 + 3 * q^64 + 11 * q^65 + 3 * q^66 - 11 * q^67 - 9 * q^68 - 11 * q^69 - 5 * q^70 + 5 * q^71 + 3 * q^72 - 3 * q^73 - 12 * q^74 - 11 * q^75 - 36 * q^76 + 18 * q^77 + 2 * q^78 - 43 * q^79 + 10 * q^80 + 11 * q^81 - 15 * q^82 - 28 * q^83 - 14 * q^84 + 3 * q^85 + 10 * q^86 + 14 * q^87 - 43 * q^88 - 25 * q^89 - 2 * q^90 - 4 * q^91 + 7 * q^92 + 11 * q^93 - 16 * q^94 + 8 * q^95 - 8 * q^96 - 6 * q^97 - 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more