LMFDB - Embedded newform 6034.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6034,2,Mod(1,6034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6034, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6034 = 2 \cdot 7 \cdot 431 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6034.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:	$48.1817325796$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 3 x^{19} - 32 x^{18} + 106 x^{17} + 382 x^{16} - 1495 x^{15} - 1963 x^{14} + 10784 x^{13} + \cdots - 44 $ x^20 - 3x^19 - 32x^18 + 106x^17 + 382x^16 - 1495x^15 - 1963x^14 + 10784x^13 + 2170x^12 - 42069x^11 + 19553x^10 + 84697x^9 - 82713x^8 - 68589x^7 + 118845x^6 - 10051x^5 - 53572x^4 + 25598x^3 - 557x^2 - 1040*x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.73475$ of defining polynomial
Character	$\chi$	$=$	6034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.73475 q^{3} +1.00000 q^{4} +3.55748 q^{5} +2.73475 q^{6} +1.00000 q^{7} -1.00000 q^{8} +4.47885 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.73475 q^{3} +1.00000 q^{4} +3.55748 q^{5} +2.73475 q^{6} +1.00000 q^{7} -1.00000 q^{8} +4.47885 q^{9} -3.55748 q^{10} +1.87897 q^{11} -2.73475 q^{12} -2.49435 q^{13} -1.00000 q^{14} -9.72882 q^{15} +1.00000 q^{16} +3.64095 q^{17} -4.47885 q^{18} -2.29058 q^{19} +3.55748 q^{20} -2.73475 q^{21} -1.87897 q^{22} +3.29288 q^{23} +2.73475 q^{24} +7.65567 q^{25} +2.49435 q^{26} -4.04429 q^{27} +1.00000 q^{28} -3.63003 q^{29} +9.72882 q^{30} -6.48892 q^{31} -1.00000 q^{32} -5.13851 q^{33} -3.64095 q^{34} +3.55748 q^{35} +4.47885 q^{36} -9.17799 q^{37} +2.29058 q^{38} +6.82141 q^{39} -3.55748 q^{40} -1.26522 q^{41} +2.73475 q^{42} -4.96377 q^{43} +1.87897 q^{44} +15.9334 q^{45} -3.29288 q^{46} +3.93496 q^{47} -2.73475 q^{48} +1.00000 q^{49} -7.65567 q^{50} -9.95708 q^{51} -2.49435 q^{52} +7.65575 q^{53} +4.04429 q^{54} +6.68440 q^{55} -1.00000 q^{56} +6.26416 q^{57} +3.63003 q^{58} +2.09011 q^{59} -9.72882 q^{60} -6.35968 q^{61} +6.48892 q^{62} +4.47885 q^{63} +1.00000 q^{64} -8.87359 q^{65} +5.13851 q^{66} -15.9699 q^{67} +3.64095 q^{68} -9.00521 q^{69} -3.55748 q^{70} -12.7882 q^{71} -4.47885 q^{72} -13.5071 q^{73} +9.17799 q^{74} -20.9363 q^{75} -2.29058 q^{76} +1.87897 q^{77} -6.82141 q^{78} -14.1855 q^{79} +3.55748 q^{80} -2.37643 q^{81} +1.26522 q^{82} -4.68681 q^{83} -2.73475 q^{84} +12.9526 q^{85} +4.96377 q^{86} +9.92723 q^{87} -1.87897 q^{88} -10.8745 q^{89} -15.9334 q^{90} -2.49435 q^{91} +3.29288 q^{92} +17.7456 q^{93} -3.93496 q^{94} -8.14870 q^{95} +2.73475 q^{96} +10.8593 q^{97} -1.00000 q^{98} +8.41564 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 20 q^{2} + 3 q^{3} + 20 q^{4} - 3 q^{5} - 3 q^{6} + 20 q^{7} - 20 q^{8} + 13 q^{9}+O(q^{10}) $ 20 * q - 20 * q^2 + 3 * q^3 + 20 * q^4 - 3 * q^5 - 3 * q^6 + 20 * q^7 - 20 * q^8 + 13 * q^9	\( 20 q - 20 q^{2} + 3 q^{3} + 20 q^{4} - 3 q^{5} - 3 q^{6} + 20 q^{7} - 20 q^{8} + 13 q^{9} + 3 q^{10} - 8 q^{11} + 3 q^{12} - 4 q^{13} - 20 q^{14} - 25 q^{15} + 20 q^{16} + 9 q^{17} - 13 q^{18} - 14 q^{19} - 3 q^{20} + 3 q^{21} + 8 q^{22} - 23 q^{23} - 3 q^{24} + 31 q^{25} + 4 q^{26} - 21 q^{27} + 20 q^{28} - 48 q^{29} + 25 q^{30} - q^{31} - 20 q^{32} - 29 q^{33} - 9 q^{34} - 3 q^{35} + 13 q^{36} - q^{37} + 14 q^{38} - q^{39} + 3 q^{40} - 27 q^{41} - 3 q^{42} - 3 q^{43} - 8 q^{44} - 12 q^{45} + 23 q^{46} - 26 q^{47} + 3 q^{48} + 20 q^{49} - 31 q^{50} - 17 q^{51} - 4 q^{52} - 43 q^{53} + 21 q^{54} - 16 q^{55} - 20 q^{56} - 25 q^{57} + 48 q^{58} - 19 q^{59} - 25 q^{60} + 9 q^{61} + q^{62} + 13 q^{63} + 20 q^{64} - 87 q^{65} + 29 q^{66} + 32 q^{67} + 9 q^{68} - 23 q^{69} + 3 q^{70} - 63 q^{71} - 13 q^{72} + 2 q^{73} + q^{74} - 8 q^{75} - 14 q^{76} - 8 q^{77} + q^{78} - 51 q^{79} - 3 q^{80} + 4 q^{81} + 27 q^{82} - 24 q^{83} + 3 q^{84} + 31 q^{85} + 3 q^{86} - 33 q^{87} + 8 q^{88} - 35 q^{89} + 12 q^{90} - 4 q^{91} - 23 q^{92} + 17 q^{93} + 26 q^{94} - 30 q^{95} - 3 q^{96} + 5 q^{97} - 20 q^{98} - 31 q^{99}+O(q^{100}) \) 20 * q - 20 * q^2 + 3 * q^3 + 20 * q^4 - 3 * q^5 - 3 * q^6 + 20 * q^7 - 20 * q^8 + 13 * q^9 + 3 * q^10 - 8 * q^11 + 3 * q^12 - 4 * q^13 - 20 * q^14 - 25 * q^15 + 20 * q^16 + 9 * q^17 - 13 * q^18 - 14 * q^19 - 3 * q^20 + 3 * q^21 + 8 * q^22 - 23 * q^23 - 3 * q^24 + 31 * q^25 + 4 * q^26 - 21 * q^27 + 20 * q^28 - 48 * q^29 + 25 * q^30 - q^31 - 20 * q^32 - 29 * q^33 - 9 * q^34 - 3 * q^35 + 13 * q^36 - q^37 + 14 * q^38 - q^39 + 3 * q^40 - 27 * q^41 - 3 * q^42 - 3 * q^43 - 8 * q^44 - 12 * q^45 + 23 * q^46 - 26 * q^47 + 3 * q^48 + 20 * q^49 - 31 * q^50 - 17 * q^51 - 4 * q^52 - 43 * q^53 + 21 * q^54 - 16 * q^55 - 20 * q^56 - 25 * q^57 + 48 * q^58 - 19 * q^59 - 25 * q^60 + 9 * q^61 + q^62 + 13 * q^63 + 20 * q^64 - 87 * q^65 + 29 * q^66 + 32 * q^67 + 9 * q^68 - 23 * q^69 + 3 * q^70 - 63 * q^71 - 13 * q^72 + 2 * q^73 + q^74 - 8 * q^75 - 14 * q^76 - 8 * q^77 + q^78 - 51 * q^79 - 3 * q^80 + 4 * q^81 + 27 * q^82 - 24 * q^83 + 3 * q^84 + 31 * q^85 + 3 * q^86 - 33 * q^87 + 8 * q^88 - 35 * q^89 + 12 * q^90 - 4 * q^91 - 23 * q^92 + 17 * q^93 + 26 * q^94 - 30 * q^95 - 3 * q^96 + 5 * q^97 - 20 * q^98 - 31 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.73475	−1.57891	−0.789454	−	0.613810i	$-0.789636\pi$
$3$	−2.73475	−1.57891	−0.789454	+	0.613810i	$0.789636\pi$
$4$	1.00000	0.500000
$5$	3.55748	1.59095	0.795477	−	0.605984i	$-0.207220\pi$
$5$	3.55748	1.59095	0.795477	+	0.605984i	$0.207220\pi$
$6$	2.73475	1.11646
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	4.47885	1.49295
$10$	−3.55748	−1.12497
$11$	1.87897	0.566531	0.283266	−	0.959042i	$-0.408582\pi$
$11$	1.87897	0.566531	0.283266	+	0.959042i	$0.408582\pi$
$12$	−2.73475	−0.789454
$13$	−2.49435	−0.691807	−0.345903	−	0.938270i	$-0.612428\pi$
$13$	−2.49435	−0.691807	−0.345903	+	0.938270i	$0.612428\pi$
$14$	−1.00000	−0.267261
$15$	−9.72882	−2.51197
$16$	1.00000	0.250000
$17$	3.64095	0.883060	0.441530	−	0.897247i	$-0.354436\pi$
$17$	3.64095	0.883060	0.441530	+	0.897247i	$0.354436\pi$
$18$	−4.47885	−1.05568
$19$	−2.29058	−0.525495	−0.262748	−	0.964865i	$-0.584629\pi$
$19$	−2.29058	−0.525495	−0.262748	+	0.964865i	$0.584629\pi$
$20$	3.55748	0.795477
$21$	−2.73475	−0.596771
$22$	−1.87897	−0.400598
$23$	3.29288	0.686614	0.343307	−	0.939223i	$-0.388453\pi$
$23$	3.29288	0.686614	0.343307	+	0.939223i	$0.388453\pi$
$24$	2.73475	0.558228
$25$	7.65567	1.53113
$26$	2.49435	0.489181
$27$	−4.04429	−0.778325
$28$	1.00000	0.188982
$29$	−3.63003	−0.674080	−0.337040	−	0.941490i	$-0.609426\pi$
$29$	−3.63003	−0.674080	−0.337040	+	0.941490i	$0.609426\pi$
$30$	9.72882	1.77623
$31$	−6.48892	−1.16544	−0.582722	−	0.812671i	$-0.698013\pi$
$31$	−6.48892	−1.16544	−0.582722	+	0.812671i	$0.698013\pi$
$32$	−1.00000	−0.176777
$33$	−5.13851	−0.894501
$34$	−3.64095	−0.624417
$35$	3.55748	0.601324
$36$	4.47885	0.746476
$37$	−9.17799	−1.50885	−0.754426	−	0.656385i	$-0.772085\pi$
$37$	−9.17799	−1.50885	−0.754426	+	0.656385i	$0.772085\pi$
$38$	2.29058	0.371581
$39$	6.82141	1.09230
$40$	−3.55748	−0.562487
$41$	−1.26522	−0.197595	−0.0987974	−	0.995108i	$-0.531500\pi$
$41$	−1.26522	−0.197595	−0.0987974	+	0.995108i	$0.531500\pi$
$42$	2.73475	0.421981
$43$	−4.96377	−0.756969	−0.378484	−	0.925608i	$-0.623555\pi$
$43$	−4.96377	−0.756969	−0.378484	+	0.925608i	$0.623555\pi$
$44$	1.87897	0.283266
$45$	15.9334	2.37522
$46$	−3.29288	−0.485509
$47$	3.93496	0.573973	0.286986	−	0.957935i	$-0.407347\pi$
$47$	3.93496	0.573973	0.286986	+	0.957935i	$0.407347\pi$
$48$	−2.73475	−0.394727
$49$	1.00000	0.142857
$50$	−7.65567	−1.08268
$51$	−9.95708	−1.39427
$52$	−2.49435	−0.345903
$53$	7.65575	1.05160	0.525799	−	0.850609i	$-0.323766\pi$
$53$	7.65575	1.05160	0.525799	+	0.850609i	$0.323766\pi$
$54$	4.04429	0.550359
$55$	6.68440	0.901325
$56$	−1.00000	−0.133631
$57$	6.26416	0.829709
$58$	3.63003	0.476646
$59$	2.09011	0.272109	0.136054	−	0.990701i	$-0.456558\pi$
$59$	2.09011	0.272109	0.136054	+	0.990701i	$0.456558\pi$
$60$	−9.72882	−1.25599
$61$	−6.35968	−0.814274	−0.407137	−	0.913367i	$-0.633473\pi$
$61$	−6.35968	−0.814274	−0.407137	+	0.913367i	$0.633473\pi$
$62$	6.48892	0.824094
$63$	4.47885	0.564283
$64$	1.00000	0.125000
$65$	−8.87359	−1.10063
$66$	5.13851	0.632507
$67$	−15.9699	−1.95104	−0.975519	−	0.219916i	$-0.929422\pi$
$67$	−15.9699	−1.95104	−0.975519	+	0.219916i	$0.929422\pi$
$68$	3.64095	0.441530
$69$	−9.00521	−1.08410
$70$	−3.55748	−0.425200
$71$	−12.7882	−1.51768	−0.758841	−	0.651276i	$-0.774234\pi$
$71$	−12.7882	−1.51768	−0.758841	+	0.651276i	$0.774234\pi$
$72$	−4.47885	−0.527838
$73$	−13.5071	−1.58089	−0.790445	−	0.612533i	$-0.790151\pi$
$73$	−13.5071	−1.58089	−0.790445	+	0.612533i	$0.790151\pi$
$74$	9.17799	1.06692
$75$	−20.9363	−2.41752
$76$	−2.29058	−0.262748
$77$	1.87897	0.214129
$78$	−6.82141	−0.772372
$79$	−14.1855	−1.59600	−0.797998	−	0.602660i	$-0.794108\pi$
$79$	−14.1855	−1.59600	−0.797998	+	0.602660i	$0.794108\pi$
$80$	3.55748	0.397738
$81$	−2.37643	−0.264048
$82$	1.26522	0.139721
$83$	−4.68681	−0.514444	−0.257222	−	0.966352i	$-0.582807\pi$
$83$	−4.68681	−0.514444	−0.257222	+	0.966352i	$0.582807\pi$
$84$	−2.73475	−0.298386
$85$	12.9526	1.40491
$86$	4.96377	0.535258
$87$	9.92723	1.06431
$88$	−1.87897	−0.200299
$89$	−10.8745	−1.15270	−0.576349	−	0.817204i	$-0.695523\pi$
$89$	−10.8745	−1.15270	−0.576349	+	0.817204i	$0.695523\pi$
$90$	−15.9334	−1.67953
$91$	−2.49435	−0.261478
$92$	3.29288	0.343307
$93$	17.7456	1.84013
$94$	−3.93496	−0.405860
$95$	−8.14870	−0.836038
$96$	2.73475	0.279114
$97$	10.8593	1.10260	0.551299	−	0.834308i	$-0.314133\pi$
$97$	10.8593	1.10260	0.551299	+	0.834308i	$0.314133\pi$
$98$	−1.00000	−0.101015
$99$	8.41564	0.845803
$100$	7.65567	0.765567
$101$	10.6848	1.06318	0.531588	−	0.847003i	$-0.321596\pi$
$101$	10.6848	1.06318	0.531588	+	0.847003i	$0.321596\pi$
$102$	9.95708	0.985898
$103$	−16.2116	−1.59737	−0.798687	−	0.601747i	$-0.794472\pi$
$103$	−16.2116	−1.59737	−0.798687	+	0.601747i	$0.794472\pi$
$104$	2.49435	0.244591
$105$	−9.72882	−0.949435
$106$	−7.65575	−0.743592
$107$	−0.936314	−0.0905168	−0.0452584	−	0.998975i	$-0.514411\pi$
$107$	−0.936314	−0.0905168	−0.0452584	+	0.998975i	$0.514411\pi$
$108$	−4.04429	−0.389162
$109$	9.39043	0.899440	0.449720	−	0.893170i	$-0.351524\pi$
$109$	9.39043	0.899440	0.449720	+	0.893170i	$0.351524\pi$
$110$	−6.68440	−0.637333
$111$	25.0995	2.38234
$112$	1.00000	0.0944911
$113$	−11.2614	−1.05939	−0.529693	−	0.848190i	$-0.677693\pi$
$113$	−11.2614	−1.05939	−0.529693	+	0.848190i	$0.677693\pi$
$114$	−6.26416	−0.586693
$115$	11.7144	1.09237
$116$	−3.63003	−0.337040
$117$	−11.1718	−1.03283
$118$	−2.09011	−0.192410
$119$	3.64095	0.333765
$120$	9.72882	0.888116
$121$	−7.46947	−0.679043
$122$	6.35968	0.575779
$123$	3.46007	0.311984
$124$	−6.48892	−0.582722
$125$	9.44750	0.845010
$126$	−4.47885	−0.399008
$127$	11.6524	1.03398	0.516991	−	0.855991i	$-0.327052\pi$
$127$	11.6524	1.03398	0.516991	+	0.855991i	$0.327052\pi$
$128$	−1.00000	−0.0883883
$129$	13.5747	1.19518
$130$	8.87359	0.778265
$131$	−14.1748	−1.23846	−0.619229	−	0.785211i	$-0.712555\pi$
$131$	−14.1748	−1.23846	−0.619229	+	0.785211i	$0.712555\pi$
$132$	−5.13851	−0.447250
$133$	−2.29058	−0.198618
$134$	15.9699	1.37959
$135$	−14.3875	−1.23828
$136$	−3.64095	−0.312209
$137$	3.23866	0.276697	0.138349	−	0.990384i	$-0.455821\pi$
$137$	3.23866	0.276697	0.138349	+	0.990384i	$0.455821\pi$
$138$	9.00521	0.766574
$139$	−8.17268	−0.693198	−0.346599	−	0.938013i	$-0.612663\pi$
$139$	−8.17268	−0.693198	−0.346599	+	0.938013i	$0.612663\pi$
$140$	3.55748	0.300662
$141$	−10.7611	−0.906250
$142$	12.7882	1.07316
$143$	−4.68680	−0.391930
$144$	4.47885	0.373238
$145$	−12.9138	−1.07243
$146$	13.5071	1.11786
$147$	−2.73475	−0.225558
$148$	−9.17799	−0.754426
$149$	20.5965	1.68733	0.843666	−	0.536868i	$-0.180393\pi$
$149$	20.5965	1.68733	0.843666	+	0.536868i	$0.180393\pi$
$150$	20.9363	1.70945
$151$	−9.88749	−0.804633	−0.402316	−	0.915501i	$-0.631795\pi$
$151$	−9.88749	−0.804633	−0.402316	+	0.915501i	$0.631795\pi$
$152$	2.29058	0.185791
$153$	16.3073	1.31836
$154$	−1.87897	−0.151412
$155$	−23.0842	−1.85417
$156$	6.82141	0.546150
$157$	11.9704	0.955344	0.477672	−	0.878538i	$-0.341481\pi$
$157$	11.9704	0.955344	0.477672	+	0.878538i	$0.341481\pi$
$158$	14.1855	1.12854
$159$	−20.9366	−1.66038
$160$	−3.55748	−0.281244
$161$	3.29288	0.259516
$162$	2.37643	0.186710
$163$	11.1024	0.869608	0.434804	−	0.900525i	$-0.356818\pi$
$163$	11.1024	0.869608	0.434804	+	0.900525i	$0.356818\pi$
$164$	−1.26522	−0.0987974
$165$	−18.2802	−1.42311
$166$	4.68681	0.363767
$167$	9.92094	0.767706	0.383853	−	0.923394i	$-0.374597\pi$
$167$	9.92094	0.767706	0.383853	+	0.923394i	$0.374597\pi$
$168$	2.73475	0.210990
$169$	−6.77824	−0.521403
$170$	−12.9526	−0.993419
$171$	−10.2592	−0.784539
$172$	−4.96377	−0.378484
$173$	17.7586	1.35016	0.675082	−	0.737742i	$-0.264108\pi$
$173$	17.7586	1.35016	0.675082	+	0.737742i	$0.264108\pi$
$174$	−9.92723	−0.752581
$175$	7.65567	0.578714
$176$	1.87897	0.141633
$177$	−5.71591	−0.429634
$178$	10.8745	0.815081
$179$	−18.1576	−1.35716	−0.678582	−	0.734525i	$-0.737405\pi$
$179$	−18.1576	−1.35716	−0.678582	+	0.734525i	$0.737405\pi$
$180$	15.9334	1.18761
$181$	21.2430	1.57898	0.789490	−	0.613763i	$-0.210345\pi$
$181$	21.2430	1.57898	0.789490	+	0.613763i	$0.210345\pi$
$182$	2.49435	0.184893
$183$	17.3921	1.28566
$184$	−3.29288	−0.242755
$185$	−32.6505	−2.40052
$186$	−17.7456	−1.30117
$187$	6.84123	0.500281
$188$	3.93496	0.286986
$189$	−4.04429	−0.294179
$190$	8.14870	0.591168
$191$	2.78274	0.201352	0.100676	−	0.994919i	$-0.467899\pi$
$191$	2.78274	0.201352	0.100676	+	0.994919i	$0.467899\pi$
$192$	−2.73475	−0.197364
$193$	−3.10763	−0.223692	−0.111846	−	0.993726i	$-0.535676\pi$
$193$	−3.10763	−0.223692	−0.111846	+	0.993726i	$0.535676\pi$
$194$	−10.8593	−0.779654
$195$	24.2670	1.73780
$196$	1.00000	0.0714286
$197$	−3.54058	−0.252256	−0.126128	−	0.992014i	$-0.540255\pi$
$197$	−3.54058	−0.252256	−0.126128	+	0.992014i	$0.540255\pi$
$198$	−8.41564	−0.598073
$199$	−25.1238	−1.78098	−0.890490	−	0.455002i	$-0.849638\pi$
$199$	−25.1238	−1.78098	−0.890490	+	0.455002i	$0.849638\pi$
$200$	−7.65567	−0.541338
$201$	43.6738	3.08051
$202$	−10.6848	−0.751778
$203$	−3.63003	−0.254778
$204$	−9.95708	−0.697135
$205$	−4.50101	−0.314364
$206$	16.2116	1.12951
$207$	14.7483	1.02508
$208$	−2.49435	−0.172952
$209$	−4.30393	−0.297709
$210$	9.72882	0.671352
$211$	−14.5657	−1.00274	−0.501371	−	0.865232i	$-0.667171\pi$
$211$	−14.5657	−1.00274	−0.501371	+	0.865232i	$0.667171\pi$
$212$	7.65575	0.525799
$213$	34.9726	2.39628
$214$	0.936314	0.0640051
$215$	−17.6585	−1.20430
$216$	4.04429	0.275179
$217$	−6.48892	−0.440497
$218$	−9.39043	−0.636000
$219$	36.9386	2.49608
$220$	6.68440	0.450662
$221$	−9.08178	−0.610907
$222$	−25.0995	−1.68457
$223$	11.1695	0.747966	0.373983	−	0.927436i	$-0.377992\pi$
$223$	11.1695	0.747966	0.373983	+	0.927436i	$0.377992\pi$
$224$	−1.00000	−0.0668153
$225$	34.2886	2.28591
$226$	11.2614	0.749098
$227$	20.0586	1.33134	0.665669	−	0.746247i	$-0.268146\pi$
$227$	20.0586	1.33134	0.665669	+	0.746247i	$0.268146\pi$
$228$	6.26416	0.414854
$229$	6.74393	0.445651	0.222826	−	0.974858i	$-0.428472\pi$
$229$	6.74393	0.445651	0.222826	+	0.974858i	$0.428472\pi$
$230$	−11.7144	−0.772423
$231$	−5.13851	−0.338089
$232$	3.63003	0.238323
$233$	−0.858628	−0.0562506	−0.0281253	−	0.999604i	$-0.508954\pi$
$233$	−0.858628	−0.0562506	−0.0281253	+	0.999604i	$0.508954\pi$
$234$	11.1718	0.730324
$235$	13.9985	0.913164
$236$	2.09011	0.136054
$237$	38.7939	2.51993
$238$	−3.64095	−0.236008
$239$	24.3094	1.57245	0.786223	−	0.617943i	$-0.212034\pi$
$239$	24.3094	1.57245	0.786223	+	0.617943i	$0.212034\pi$
$240$	−9.72882	−0.627993
$241$	19.1486	1.23347	0.616733	−	0.787172i	$-0.288456\pi$
$241$	19.1486	1.23347	0.616733	+	0.787172i	$0.288456\pi$
$242$	7.46947	0.480156
$243$	18.6318	1.19523
$244$	−6.35968	−0.407137
$245$	3.55748	0.227279
$246$	−3.46007	−0.220606
$247$	5.71350	0.363541
$248$	6.48892	0.412047
$249$	12.8173	0.812261
$250$	−9.44750	−0.597512
$251$	21.3904	1.35015	0.675074	−	0.737750i	$-0.264112\pi$
$251$	21.3904	1.35015	0.675074	+	0.737750i	$0.264112\pi$
$252$	4.47885	0.282141
$253$	6.18723	0.388988
$254$	−11.6524	−0.731135
$255$	−35.4221	−2.21822
$256$	1.00000	0.0625000
$257$	10.9958	0.685899	0.342950	−	0.939354i	$-0.388574\pi$
$257$	10.9958	0.685899	0.342950	+	0.939354i	$0.388574\pi$
$258$	−13.5747	−0.845123
$259$	−9.17799	−0.570293
$260$	−8.87359	−0.550316
$261$	−16.2584	−1.00637
$262$	14.1748	0.875721
$263$	−29.2933	−1.80630	−0.903152	−	0.429322i	$-0.858753\pi$
$263$	−29.2933	−1.80630	−0.903152	+	0.429322i	$0.858753\pi$
$264$	5.13851	0.316254
$265$	27.2352	1.67304
$266$	2.29058	0.140444
$267$	29.7391	1.82000
$268$	−15.9699	−0.975519
$269$	−9.85673	−0.600975	−0.300488	−	0.953786i	$-0.597149\pi$
$269$	−9.85673	−0.600975	−0.300488	+	0.953786i	$0.597149\pi$
$270$	14.3875	0.875595
$271$	−22.5571	−1.37024	−0.685122	−	0.728428i	$-0.740251\pi$
$271$	−22.5571	−1.37024	−0.685122	+	0.728428i	$0.740251\pi$
$272$	3.64095	0.220765
$273$	6.82141	0.412850
$274$	−3.23866	−0.195655
$275$	14.3848	0.867435
$276$	−9.00521	−0.542050
$277$	30.0682	1.80662	0.903312	−	0.428984i	$-0.141128\pi$
$277$	30.0682	1.80662	0.903312	+	0.428984i	$0.141128\pi$
$278$	8.17268	0.490165
$279$	−29.0629	−1.73995
$280$	−3.55748	−0.212600
$281$	5.22327	0.311594	0.155797	−	0.987789i	$-0.450205\pi$
$281$	5.22327	0.311594	0.155797	+	0.987789i	$0.450205\pi$
$282$	10.7611	0.640816
$283$	−17.2046	−1.02271	−0.511353	−	0.859371i	$-0.670856\pi$
$283$	−17.2046	−1.02271	−0.511353	+	0.859371i	$0.670856\pi$
$284$	−12.7882	−0.758841
$285$	22.2846	1.32003
$286$	4.68680	0.277136
$287$	−1.26522	−0.0746838
$288$	−4.47885	−0.263919
$289$	−3.74350	−0.220206
$290$	12.9138	0.758323
$291$	−29.6975	−1.74090
$292$	−13.5071	−0.790445
$293$	−9.34484	−0.545931	−0.272966	−	0.962024i	$-0.588005\pi$
$293$	−9.34484	−0.545931	−0.272966	+	0.962024i	$0.588005\pi$
$294$	2.73475	0.159494
$295$	7.43551	0.432912
$296$	9.17799	0.533460
$297$	−7.59911	−0.440945
$298$	−20.5965	−1.19312
$299$	−8.21359	−0.475004
$300$	−20.9363	−1.20876
$301$	−4.96377	−0.286107
$302$	9.88749	0.568961
$303$	−29.2202	−1.67866
$304$	−2.29058	−0.131374
$305$	−22.6245	−1.29547
$306$	−16.3073	−0.932225
$307$	−1.47725	−0.0843112	−0.0421556	−	0.999111i	$-0.513423\pi$
$307$	−1.47725	−0.0843112	−0.0421556	+	0.999111i	$0.513423\pi$
$308$	1.87897	0.107064
$309$	44.3346	2.52211
$310$	23.0842	1.31110
$311$	11.0041	0.623982	0.311991	−	0.950085i	$-0.399004\pi$
$311$	11.0041	0.623982	0.311991	+	0.950085i	$0.399004\pi$
$312$	−6.82141	−0.386186
$313$	−1.48390	−0.0838749	−0.0419375	−	0.999120i	$-0.513353\pi$
$313$	−1.48390	−0.0838749	−0.0419375	+	0.999120i	$0.513353\pi$
$314$	−11.9704	−0.675530
$315$	15.9334	0.897748
$316$	−14.1855	−0.797998
$317$	1.70030	0.0954983	0.0477492	−	0.998859i	$-0.484795\pi$
$317$	1.70030	0.0954983	0.0477492	+	0.998859i	$0.484795\pi$
$318$	20.9366	1.17406
$319$	−6.82072	−0.381887
$320$	3.55748	0.198869
$321$	2.56058	0.142918
$322$	−3.29288	−0.183505
$323$	−8.33988	−0.464043
$324$	−2.37643	−0.132024
$325$	−19.0959	−1.05925
$326$	−11.1024	−0.614906
$327$	−25.6805	−1.42013
$328$	1.26522	0.0698603
$329$	3.93496	0.216941
$330$	18.2802	1.00629
$331$	−21.6306	−1.18892	−0.594461	−	0.804124i	$-0.702635\pi$
$331$	−21.6306	−1.18892	−0.594461	+	0.804124i	$0.702635\pi$
$332$	−4.68681	−0.257222
$333$	−41.1069	−2.25264
$334$	−9.92094	−0.542850
$335$	−56.8127	−3.10401
$336$	−2.73475	−0.149193
$337$	1.48742	0.0810247	0.0405123	−	0.999179i	$-0.487101\pi$
$337$	1.48742	0.0810247	0.0405123	+	0.999179i	$0.487101\pi$
$338$	6.77824	0.368688
$339$	30.7972	1.67267
$340$	12.9526	0.702453
$341$	−12.1925	−0.660261
$342$	10.2592	0.554753
$343$	1.00000	0.0539949
$344$	4.96377	0.267629
$345$	−32.0359	−1.72475
$346$	−17.7586	−0.954711
$347$	−19.4075	−1.04185	−0.520925	−	0.853603i	$-0.674413\pi$
$347$	−19.4075	−1.04185	−0.520925	+	0.853603i	$0.674413\pi$
$348$	9.92723	0.532155
$349$	−9.60906	−0.514361	−0.257181	−	0.966363i	$-0.582794\pi$
$349$	−9.60906	−0.514361	−0.257181	+	0.966363i	$0.582794\pi$
$350$	−7.65567	−0.409213
$351$	10.0879	0.538451
$352$	−1.87897	−0.100149
$353$	26.7030	1.42126	0.710630	−	0.703566i	$-0.248410\pi$
$353$	26.7030	1.42126	0.710630	+	0.703566i	$0.248410\pi$
$354$	5.71591	0.303797
$355$	−45.4938	−2.41456
$356$	−10.8745	−0.576349
$357$	−9.95708	−0.526985
$358$	18.1576	0.959660
$359$	3.69303	0.194911	0.0974553	−	0.995240i	$-0.468930\pi$
$359$	3.69303	0.194911	0.0974553	+	0.995240i	$0.468930\pi$
$360$	−15.9334	−0.839766
$361$	−13.7532	−0.723855
$362$	−21.2430	−1.11651
$363$	20.4271	1.07215
$364$	−2.49435	−0.130739
$365$	−48.0514	−2.51512
$366$	−17.3921	−0.909102
$367$	−27.0811	−1.41362	−0.706812	−	0.707401i	$-0.749867\pi$
$367$	−27.0811	−1.41362	−0.706812	+	0.707401i	$0.749867\pi$
$368$	3.29288	0.171653
$369$	−5.66675	−0.294999
$370$	32.6505	1.69742
$371$	7.65575	0.397467
$372$	17.7456	0.920065
$373$	17.9578	0.929819	0.464909	−	0.885358i	$-0.346087\pi$
$373$	17.9578	0.929819	0.464909	+	0.885358i	$0.346087\pi$
$374$	−6.84123	−0.353752
$375$	−25.8365	−1.33419
$376$	−3.93496	−0.202930
$377$	9.05455	0.466333
$378$	4.04429	0.208016
$379$	37.9805	1.95093	0.975464	−	0.220158i	$-0.0706573\pi$
$379$	37.9805	1.95093	0.975464	+	0.220158i	$0.0706573\pi$
$380$	−8.14870	−0.418019
$381$	−31.8663	−1.63256
$382$	−2.78274	−0.142378
$383$	−0.195728	−0.0100012	−0.00500062	−	0.999987i	$-0.501592\pi$
$383$	−0.195728	−0.0100012	−0.00500062	+	0.999987i	$0.501592\pi$
$384$	2.73475	0.139557
$385$	6.68440	0.340669
$386$	3.10763	0.158174
$387$	−22.2320	−1.13012
$388$	10.8593	0.551299
$389$	−14.7887	−0.749817	−0.374909	−	0.927062i	$-0.622326\pi$
$389$	−14.7887	−0.749817	−0.374909	+	0.927062i	$0.622326\pi$
$390$	−24.2670	−1.22881
$391$	11.9892	0.606321
$392$	−1.00000	−0.0505076
$393$	38.7645	1.95541
$394$	3.54058	0.178372
$395$	−50.4648	−2.53916
$396$	8.41564	0.422902
$397$	0.828212	0.0415668	0.0207834	−	0.999784i	$-0.493384\pi$
$397$	0.828212	0.0415668	0.0207834	+	0.999784i	$0.493384\pi$
$398$	25.1238	1.25934
$399$	6.26416	0.313600
$400$	7.65567	0.382784
$401$	−17.2476	−0.861302	−0.430651	−	0.902519i	$-0.641716\pi$
$401$	−17.2476	−0.861302	−0.430651	+	0.902519i	$0.641716\pi$
$402$	−43.6738	−2.17825
$403$	16.1856	0.806263
$404$	10.6848	0.531588
$405$	−8.45410	−0.420088
$406$	3.63003	0.180155
$407$	−17.2452	−0.854812
$408$	9.95708	0.492949
$409$	−17.3823	−0.859499	−0.429749	−	0.902948i	$-0.641398\pi$
$409$	−17.3823	−0.859499	−0.429749	+	0.902948i	$0.641398\pi$
$410$	4.50101	0.222289
$411$	−8.85692	−0.436880
$412$	−16.2116	−0.798687
$413$	2.09011	0.102847
$414$	−14.7483	−0.724842
$415$	−16.6732	−0.818457
$416$	2.49435	0.122295
$417$	22.3502	1.09450
$418$	4.30393	0.210512
$419$	−21.4499	−1.04790	−0.523948	−	0.851750i	$-0.675541\pi$
$419$	−21.4499	−1.04790	−0.523948	+	0.851750i	$0.675541\pi$
$420$	−9.72882	−0.474718
$421$	−5.20290	−0.253574	−0.126787	−	0.991930i	$-0.540466\pi$
$421$	−5.20290	−0.253574	−0.126787	+	0.991930i	$0.540466\pi$
$422$	14.5657	0.709046
$423$	17.6241	0.856913
$424$	−7.65575	−0.371796
$425$	27.8739	1.35208
$426$	−34.9726	−1.69443
$427$	−6.35968	−0.307767
$428$	−0.936314	−0.0452584
$429$	12.8172	0.618822
$430$	17.6585	0.851570
$431$	1.00000	0.0481683
$431$	1.00000	0.0481683
$432$	−4.04429	−0.194581
$433$	−26.7157	−1.28387	−0.641937	−	0.766757i	$-0.721869\pi$
$433$	−26.7157	−1.28387	−0.641937	+	0.766757i	$0.721869\pi$
$434$	6.48892	0.311478
$435$	35.3159	1.69327
$436$	9.39043	0.449720
$437$	−7.54261	−0.360812
$438$	−36.9386	−1.76500
$439$	5.72597	0.273286	0.136643	−	0.990620i	$-0.456369\pi$
$439$	5.72597	0.273286	0.136643	+	0.990620i	$0.456369\pi$
$440$	−6.68440	−0.318666
$441$	4.47885	0.213279
$442$	9.08178	0.431976
$443$	22.7206	1.07949	0.539743	−	0.841830i	$-0.318521\pi$
$443$	22.7206	1.07949	0.539743	+	0.841830i	$0.318521\pi$
$444$	25.0995	1.19117
$445$	−38.6860	−1.83389
$446$	−11.1695	−0.528892
$447$	−56.3263	−2.66414
$448$	1.00000	0.0472456
$449$	−15.9861	−0.754429	−0.377214	−	0.926126i	$-0.623118\pi$
$449$	−15.9861	−0.754429	−0.377214	+	0.926126i	$0.623118\pi$
$450$	−34.2886	−1.61638
$451$	−2.37732	−0.111944
$452$	−11.2614	−0.529693
$453$	27.0398	1.27044
$454$	−20.0586	−0.941398
$455$	−8.87359	−0.416000
$456$	−6.26416	−0.293346
$457$	−12.4839	−0.583971	−0.291986	−	0.956423i	$-0.594316\pi$
$457$	−12.4839	−0.583971	−0.291986	+	0.956423i	$0.594316\pi$
$458$	−6.74393	−0.315123
$459$	−14.7251	−0.687307
$460$	11.7144	0.546185
$461$	1.76183	0.0820565	0.0410282	−	0.999158i	$-0.486937\pi$
$461$	1.76183	0.0820565	0.0410282	+	0.999158i	$0.486937\pi$
$462$	5.13851	0.239065
$463$	−3.26568	−0.151769	−0.0758845	−	0.997117i	$-0.524178\pi$
$463$	−3.26568	−0.151769	−0.0758845	+	0.997117i	$0.524178\pi$
$464$	−3.63003	−0.168520
$465$	63.1295	2.92756
$466$	0.858628	0.0397752
$467$	−23.5817	−1.09123	−0.545615	−	0.838036i	$-0.683704\pi$
$467$	−23.5817	−1.09123	−0.545615	+	0.838036i	$0.683704\pi$
$468$	−11.1718	−0.516417
$469$	−15.9699	−0.737423
$470$	−13.9985	−0.645704
$471$	−32.7361	−1.50840
$472$	−2.09011	−0.0962049
$473$	−9.32679	−0.428846
$474$	−38.7939	−1.78186
$475$	−17.5359	−0.804604
$476$	3.64095	0.166883
$477$	34.2890	1.56999
$478$	−24.3094	−1.11189
$479$	33.1505	1.51468	0.757341	−	0.653019i	$-0.226498\pi$
$479$	33.1505	1.51468	0.757341	+	0.653019i	$0.226498\pi$
$480$	9.72882	0.444058
$481$	22.8931	1.04383
$482$	−19.1486	−0.872193
$483$	−9.00521	−0.409751
$484$	−7.46947	−0.339521
$485$	38.6318	1.75418
$486$	−18.6318	−0.845157
$487$	−19.2410	−0.871893	−0.435946	−	0.899973i	$-0.643586\pi$
$487$	−19.2410	−0.871893	−0.435946	+	0.899973i	$0.643586\pi$
$488$	6.35968	0.287889
$489$	−30.3623	−1.37303
$490$	−3.55748	−0.160711
$491$	−23.9522	−1.08095	−0.540474	−	0.841361i	$-0.681755\pi$
$491$	−23.9522	−1.08095	−0.540474	+	0.841361i	$0.681755\pi$
$492$	3.46007	0.155992
$493$	−13.2168	−0.595253
$494$	−5.71350	−0.257062
$495$	29.9385	1.34563
$496$	−6.48892	−0.291361
$497$	−12.7882	−0.573630
$498$	−12.8173	−0.574355
$499$	14.3945	0.644384	0.322192	−	0.946674i	$-0.395580\pi$
$499$	14.3945	0.644384	0.322192	+	0.946674i	$0.395580\pi$
$500$	9.44750	0.422505
$501$	−27.1313	−1.21214
$502$	−21.3904	−0.954698
$503$	6.47267	0.288602	0.144301	−	0.989534i	$-0.453907\pi$
$503$	6.47267	0.288602	0.144301	+	0.989534i	$0.453907\pi$
$504$	−4.47885	−0.199504
$505$	38.0109	1.69146
$506$	−6.18723	−0.275056
$507$	18.5368	0.823248
$508$	11.6524	0.516991
$509$	16.5067	0.731646	0.365823	−	0.930684i	$-0.380787\pi$
$509$	16.5067	0.731646	0.365823	+	0.930684i	$0.380787\pi$
$510$	35.4221	1.56852
$511$	−13.5071	−0.597520
$512$	−1.00000	−0.0441942
$513$	9.26378	0.409006
$514$	−10.9958	−0.485004
$515$	−57.6723	−2.54135
$516$	13.5747	0.597592
$517$	7.39367	0.325173
$518$	9.17799	0.403258
$519$	−48.5654	−2.13179
$520$	8.87359	0.389132
$521$	−21.5128	−0.942493	−0.471246	−	0.882002i	$-0.656196\pi$
$521$	−21.5128	−0.942493	−0.471246	+	0.882002i	$0.656196\pi$
$522$	16.2584	0.711610
$523$	12.4795	0.545690	0.272845	−	0.962058i	$-0.412035\pi$
$523$	12.4795	0.545690	0.272845	+	0.962058i	$0.412035\pi$
$524$	−14.1748	−0.619229
$525$	−20.9363	−0.913737
$526$	29.2933	1.27725
$527$	−23.6258	−1.02916
$528$	−5.13851	−0.223625
$529$	−12.1569	−0.528562
$530$	−27.2352	−1.18302
$531$	9.36128	0.406245
$532$	−2.29058	−0.0993092
$533$	3.15590	0.136697
$534$	−29.7391	−1.28694
$535$	−3.33092	−0.144008
$536$	15.9699	0.689796
$537$	49.6565	2.14284
$538$	9.85673	0.424954
$539$	1.87897	0.0809330
$540$	−14.3875	−0.619139
$541$	11.7749	0.506243	0.253122	−	0.967434i	$-0.418543\pi$
$541$	11.7749	0.506243	0.253122	+	0.967434i	$0.418543\pi$
$542$	22.5571	0.968909
$543$	−58.0943	−2.49306
$544$	−3.64095	−0.156104
$545$	33.4063	1.43097
$546$	−6.82141	−0.291929
$547$	16.8442	0.720204	0.360102	−	0.932913i	$-0.382742\pi$
$547$	16.8442	0.720204	0.360102	+	0.932913i	$0.382742\pi$
$548$	3.23866	0.138349
$549$	−28.4841	−1.21567
$550$	−14.3848	−0.613369
$551$	8.31488	0.354226
$552$	9.00521	0.383287
$553$	−14.1855	−0.603230
$554$	−30.0682	−1.27748
$555$	89.2910	3.79019
$556$	−8.17268	−0.346599
$557$	−2.66912	−0.113094	−0.0565472	−	0.998400i	$-0.518009\pi$
$557$	−2.66912	−0.113094	−0.0565472	+	0.998400i	$0.518009\pi$
$558$	29.0629	1.23033
$559$	12.3814	0.523676
$560$	3.55748	0.150331
$561$	−18.7091	−0.789897
$562$	−5.22327	−0.220330
$563$	−4.94443	−0.208383	−0.104191	−	0.994557i	$-0.533225\pi$
$563$	−4.94443	−0.208383	−0.104191	+	0.994557i	$0.533225\pi$
$564$	−10.7611	−0.453125
$565$	−40.0623	−1.68543
$566$	17.2046	0.723163
$567$	−2.37643	−0.0998006
$568$	12.7882	0.536582
$569$	−11.2207	−0.470395	−0.235197	−	0.971948i	$-0.575574\pi$
$569$	−11.2207	−0.470395	−0.235197	+	0.971948i	$0.575574\pi$
$570$	−22.2846	−0.933401
$571$	−28.5549	−1.19499	−0.597493	−	0.801874i	$-0.703836\pi$
$571$	−28.5549	−1.19499	−0.597493	+	0.801874i	$0.703836\pi$
$572$	−4.68680	−0.195965
$573$	−7.61011	−0.317917
$574$	1.26522	0.0528094
$575$	25.2092	1.05130
$576$	4.47885	0.186619
$577$	−5.61161	−0.233614	−0.116807	−	0.993155i	$-0.537266\pi$
$577$	−5.61161	−0.233614	−0.116807	+	0.993155i	$0.537266\pi$
$578$	3.74350	0.155709
$579$	8.49859	0.353189
$580$	−12.9138	−0.536215
$581$	−4.68681	−0.194442
$582$	29.6975	1.23100
$583$	14.3849	0.595763
$584$	13.5071	0.558929
$585$	−39.7435	−1.64319
$586$	9.34484	0.386032
$587$	42.1518	1.73979	0.869896	−	0.493235i	$-0.164186\pi$
$587$	42.1518	1.73979	0.869896	+	0.493235i	$0.164186\pi$
$588$	−2.73475	−0.112779
$589$	14.8634	0.612436
$590$	−7.43551	−0.306115
$591$	9.68260	0.398289
$592$	−9.17799	−0.377213
$593$	−0.801400	−0.0329096	−0.0164548	−	0.999865i	$-0.505238\pi$
$593$	−0.801400	−0.0329096	−0.0164548	+	0.999865i	$0.505238\pi$
$594$	7.59911	0.311795
$595$	12.9526	0.531005
$596$	20.5965	0.843666
$597$	68.7074	2.81201
$598$	8.21359	0.335879
$599$	5.24876	0.214459	0.107229	−	0.994234i	$-0.465802\pi$
$599$	5.24876	0.214459	0.107229	+	0.994234i	$0.465802\pi$
$600$	20.9363	0.854723
$601$	−30.0384	−1.22529	−0.612645	−	0.790358i	$-0.709894\pi$
$601$	−30.0384	−1.22529	−0.612645	+	0.790358i	$0.709894\pi$
$602$	4.96377	0.202308
$603$	−71.5270	−2.91280
$604$	−9.88749	−0.402316
$605$	−26.5725	−1.08033
$606$	29.2202	1.18699
$607$	−34.1693	−1.38689	−0.693444	−	0.720511i	$-0.743907\pi$
$607$	−34.1693	−1.38689	−0.693444	+	0.720511i	$0.743907\pi$
$608$	2.29058	0.0928953
$609$	9.92723	0.402272
$610$	22.6245	0.916037
$611$	−9.81514	−0.397078
$612$	16.3073	0.659182
$613$	12.5272	0.505971	0.252985	−	0.967470i	$-0.418588\pi$
$613$	12.5272	0.505971	0.252985	+	0.967470i	$0.418588\pi$
$614$	1.47725	0.0596170
$615$	12.3091	0.496352
$616$	−1.87897	−0.0757059
$617$	−21.0062	−0.845676	−0.422838	−	0.906205i	$-0.638966\pi$
$617$	−21.0062	−0.845676	−0.422838	+	0.906205i	$0.638966\pi$
$618$	−44.3346	−1.78340
$619$	16.0447	0.644892	0.322446	−	0.946588i	$-0.395495\pi$
$619$	16.0447	0.644892	0.322446	+	0.946588i	$0.395495\pi$
$620$	−23.0842	−0.927084
$621$	−13.3174	−0.534409
$622$	−11.0041	−0.441222
$623$	−10.8745	−0.435679
$624$	6.82141	0.273075
$625$	−4.66906	−0.186762
$626$	1.48390	0.0593085
$627$	11.7702	0.470056
$628$	11.9704	0.477672
$629$	−33.4166	−1.33241
$630$	−15.9334	−0.634803
$631$	11.6673	0.464466	0.232233	−	0.972660i	$-0.425397\pi$
$631$	11.6673	0.464466	0.232233	+	0.972660i	$0.425397\pi$
$632$	14.1855	0.564270
$633$	39.8335	1.58324
$634$	−1.70030	−0.0675275
$635$	41.4531	1.64502
$636$	−20.9366	−0.830189
$637$	−2.49435	−0.0988296
$638$	6.82072	0.270035
$639$	−57.2766	−2.26583
$640$	−3.55748	−0.140622
$641$	39.3046	1.55244	0.776219	−	0.630463i	$-0.217135\pi$
$641$	39.3046	1.55244	0.776219	+	0.630463i	$0.217135\pi$
$642$	−2.56058	−0.101058
$643$	26.6361	1.05042	0.525212	−	0.850971i	$-0.323986\pi$
$643$	26.6361	1.05042	0.525212	+	0.850971i	$0.323986\pi$
$644$	3.29288	0.129758
$645$	48.2917	1.90148
$646$	8.33988	0.328128
$647$	−3.72270	−0.146354	−0.0731772	−	0.997319i	$-0.523314\pi$
$647$	−3.72270	−0.146354	−0.0731772	+	0.997319i	$0.523314\pi$
$648$	2.37643	0.0933550
$649$	3.92725	0.154158
$650$	19.0959	0.749002
$651$	17.7456	0.695504
$652$	11.1024	0.434804
$653$	−49.0384	−1.91902	−0.959509	−	0.281676i	$-0.909110\pi$
$653$	−49.0384	−1.91902	−0.959509	+	0.281676i	$0.909110\pi$
$654$	25.6805	1.00419
$655$	−50.4265	−1.97033
$656$	−1.26522	−0.0493987
$657$	−60.4965	−2.36019
$658$	−3.93496	−0.153401
$659$	1.79193	0.0698039	0.0349019	−	0.999391i	$-0.488888\pi$
$659$	1.79193	0.0698039	0.0349019	+	0.999391i	$0.488888\pi$
$660$	−18.2802	−0.711555
$661$	−21.2992	−0.828443	−0.414222	−	0.910176i	$-0.635946\pi$
$661$	−21.2992	−0.828443	−0.414222	+	0.910176i	$0.635946\pi$
$662$	21.6306	0.840695
$663$	24.8364	0.964566
$664$	4.68681	0.181884
$665$	−8.14870	−0.315993
$666$	41.1069	1.59286
$667$	−11.9533	−0.462833
$668$	9.92094	0.383853
$669$	−30.5458	−1.18097
$670$	56.8127	2.19487
$671$	−11.9497	−0.461312
$672$	2.73475	0.105495
$673$	27.3789	1.05538	0.527689	−	0.849438i	$-0.323059\pi$
$673$	27.3789	1.05538	0.527689	+	0.849438i	$0.323059\pi$
$674$	−1.48742	−0.0572931
$675$	−30.9618	−1.19172
$676$	−6.77824	−0.260702
$677$	9.86028	0.378962	0.189481	−	0.981884i	$-0.439320\pi$
$677$	9.86028	0.378962	0.189481	+	0.981884i	$0.439320\pi$
$678$	−30.7972	−1.18276
$679$	10.8593	0.416743
$680$	−12.9526	−0.496710
$681$	−54.8553	−2.10206
$682$	12.1925	0.466875
$683$	36.9750	1.41481	0.707404	−	0.706809i	$-0.249866\pi$
$683$	36.9750	1.41481	0.707404	+	0.706809i	$0.249866\pi$
$684$	−10.2592	−0.392269
$685$	11.5215	0.440213
$686$	−1.00000	−0.0381802
$687$	−18.4430	−0.703643
$688$	−4.96377	−0.189242
$689$	−19.0961	−0.727503
$690$	32.0359	1.21958
$691$	−18.9276	−0.720038	−0.360019	−	0.932945i	$-0.617230\pi$
$691$	−18.9276	−0.720038	−0.360019	+	0.932945i	$0.617230\pi$
$692$	17.7586	0.675082
$693$	8.41564	0.319684
$694$	19.4075	0.736699
$695$	−29.0742	−1.10285
$696$	−9.92723	−0.376291
$697$	−4.60661	−0.174488
$698$	9.60906	0.363708
$699$	2.34813	0.0888146
$700$	7.65567	0.289357
$701$	−40.3751	−1.52495	−0.762474	−	0.647019i	$-0.776015\pi$
$701$	−40.3751	−1.52495	−0.762474	+	0.647019i	$0.776015\pi$
$702$	−10.0879	−0.380742
$703$	21.0229	0.792895
$704$	1.87897	0.0708164
$705$	−38.2825	−1.44180
$706$	−26.7030	−1.00498
$707$	10.6848	0.401842
$708$	−5.71591	−0.214817
$709$	−14.6210	−0.549105	−0.274552	−	0.961572i	$-0.588530\pi$
$709$	−14.6210	−0.549105	−0.274552	+	0.961572i	$0.588530\pi$
$710$	45.4938	1.70735
$711$	−63.5349	−2.38275
$712$	10.8745	0.407540
$713$	−21.3673	−0.800210
$714$	9.95708	0.372634
$715$	−16.6732	−0.623543
$716$	−18.1576	−0.678582
$717$	−66.4802	−2.48275
$718$	−3.69303	−0.137823
$719$	10.5571	0.393713	0.196856	−	0.980432i	$-0.436927\pi$
$719$	10.5571	0.393713	0.196856	+	0.980432i	$0.436927\pi$
$720$	15.9334	0.593804
$721$	−16.2116	−0.603750
$722$	13.7532	0.511843
$723$	−52.3665	−1.94753
$724$	21.2430	0.789490
$725$	−27.7903	−1.03211
$726$	−20.4271	−0.758122
$727$	23.4054	0.868060	0.434030	−	0.900898i	$-0.357091\pi$
$727$	23.4054	0.868060	0.434030	+	0.900898i	$0.357091\pi$
$728$	2.49435	0.0924466
$729$	−43.8241	−1.62311
$730$	48.0514	1.77846
$731$	−18.0728	−0.668448
$732$	17.3921	0.642832
$733$	42.9991	1.58821	0.794103	−	0.607783i	$-0.207941\pi$
$733$	42.9991	1.58821	0.794103	+	0.607783i	$0.207941\pi$
$734$	27.0811	0.999583
$735$	−9.72882	−0.358853
$736$	−3.29288	−0.121377
$737$	−30.0070	−1.10532
$738$	5.66675	0.208596
$739$	5.09647	0.187477	0.0937384	−	0.995597i	$-0.470118\pi$
$739$	5.09647	0.187477	0.0937384	+	0.995597i	$0.470118\pi$
$740$	−32.6505	−1.20026
$741$	−15.6250	−0.573998
$742$	−7.65575	−0.281052
$743$	51.6454	1.89469	0.947343	−	0.320221i	$-0.103757\pi$
$743$	51.6454	1.89469	0.947343	+	0.320221i	$0.103757\pi$
$744$	−17.7456	−0.650584
$745$	73.2717	2.68447
$746$	−17.9578	−0.657481
$747$	−20.9915	−0.768041
$748$	6.84123	0.250140
$749$	−0.936314	−0.0342122
$750$	25.8365	0.943417
$751$	26.8959	0.981446	0.490723	−	0.871316i	$-0.336733\pi$
$751$	26.8959	0.981446	0.490723	+	0.871316i	$0.336733\pi$
$752$	3.93496	0.143493
$753$	−58.4973	−2.13176
$754$	−9.05455	−0.329747
$755$	−35.1746	−1.28013
$756$	−4.04429	−0.147090
$757$	13.0951	0.475950	0.237975	−	0.971271i	$-0.423516\pi$
$757$	13.0951	0.475950	0.237975	+	0.971271i	$0.423516\pi$
$758$	−37.9805	−1.37951
$759$	−16.9205	−0.614176
$760$	8.14870	0.295584
$761$	9.54940	0.346165	0.173083	−	0.984907i	$-0.444627\pi$
$761$	9.54940	0.346165	0.173083	+	0.984907i	$0.444627\pi$
$762$	31.8663	1.15440
$763$	9.39043	0.339956
$764$	2.78274	0.100676
$765$	58.0128	2.09746
$766$	0.195728	0.00707194
$767$	−5.21344	−0.188247
$768$	−2.73475	−0.0986818
$769$	10.0342	0.361843	0.180921	−	0.983498i	$-0.442092\pi$
$769$	10.0342	0.361843	0.180921	+	0.983498i	$0.442092\pi$
$770$	−6.68440	−0.240889
$771$	−30.0708	−1.08297
$772$	−3.10763	−0.111846
$773$	−26.7257	−0.961256	−0.480628	−	0.876925i	$-0.659591\pi$
$773$	−26.7257	−0.961256	−0.480628	+	0.876925i	$0.659591\pi$
$774$	22.2320	0.799114
$775$	−49.6771	−1.78445
$776$	−10.8593	−0.389827
$777$	25.0995	0.900440
$778$	14.7887	0.530201
$779$	2.89810	0.103835
$780$	24.2670	0.868899
$781$	−24.0287	−0.859814
$782$	−11.9892	−0.428734
$783$	14.6809	0.524653
$784$	1.00000	0.0357143
$785$	42.5846	1.51991
$786$	−38.7645	−1.38268
$787$	32.8987	1.17271	0.586355	−	0.810054i	$-0.300562\pi$
$787$	32.8987	1.17271	0.586355	+	0.810054i	$0.300562\pi$
$788$	−3.54058	−0.126128
$789$	80.1099	2.85199
$790$	50.4648	1.79546
$791$	−11.2614	−0.400410
$792$	−8.41564	−0.299037
$793$	15.8632	0.563320
$794$	−0.828212	−0.0293921
$795$	−74.4814	−2.64158
$796$	−25.1238	−0.890490
$797$	40.1920	1.42367	0.711837	−	0.702345i	$-0.247863\pi$
$797$	40.1920	1.42367	0.711837	+	0.702345i	$0.247863\pi$
$798$	−6.26416	−0.221749
$799$	14.3270	0.506852
$800$	−7.65567	−0.270669
$801$	−48.7055	−1.72092
$802$	17.2476	0.609032
$803$	−25.3795	−0.895623
$804$	43.6738	1.54025
$805$	11.7144	0.412877
$806$	−16.1856	−0.570114
$807$	26.9557	0.948885
$808$	−10.6848	−0.375889
$809$	−25.8536	−0.908966	−0.454483	−	0.890756i	$-0.650176\pi$
$809$	−25.8536	−0.908966	−0.454483	+	0.890756i	$0.650176\pi$
$810$	8.45410	0.297047
$811$	27.3265	0.959562	0.479781	−	0.877388i	$-0.340716\pi$
$811$	27.3265	0.959562	0.479781	+	0.877388i	$0.340716\pi$
$812$	−3.63003	−0.127389
$813$	61.6879	2.16349
$814$	17.2452	0.604443
$815$	39.4966	1.38351
$816$	−9.95708	−0.348567
$817$	11.3699	0.397783
$818$	17.3823	0.607757
$819$	−11.1718	−0.390375
$820$	−4.50101	−0.157182
$821$	−33.5481	−1.17084	−0.585419	−	0.810731i	$-0.699070\pi$
$821$	−33.5481	−1.17084	−0.585419	+	0.810731i	$0.699070\pi$
$822$	8.85692	0.308921
$823$	−39.0168	−1.36004	−0.680020	−	0.733193i	$-0.738029\pi$
$823$	−39.0168	−1.36004	−0.680020	+	0.733193i	$0.738029\pi$
$824$	16.2116	0.564757
$825$	−39.3388	−1.36960
$826$	−2.09011	−0.0727241
$827$	30.7255	1.06843	0.534216	−	0.845348i	$-0.320607\pi$
$827$	30.7255	1.06843	0.534216	+	0.845348i	$0.320607\pi$
$828$	14.7483	0.512540
$829$	29.0293	1.00823	0.504115	−	0.863636i	$-0.331819\pi$
$829$	29.0293	1.00823	0.504115	+	0.863636i	$0.331819\pi$
$830$	16.6732	0.578737
$831$	−82.2291	−2.85249
$832$	−2.49435	−0.0864759
$833$	3.64095	0.126151
$834$	−22.3502	−0.773925
$835$	35.2936	1.22138
$836$	−4.30393	−0.148855
$837$	26.2431	0.907095
$838$	21.4499	0.740974
$839$	18.0056	0.621622	0.310811	−	0.950472i	$-0.399399\pi$
$839$	18.0056	0.621622	0.310811	+	0.950472i	$0.399399\pi$
$840$	9.72882	0.335676
$841$	−15.8229	−0.545616
$842$	5.20290	0.179304
$843$	−14.2843	−0.491979
$844$	−14.5657	−0.501371
$845$	−24.1135	−0.829529
$846$	−17.6241	−0.605929
$847$	−7.46947	−0.256654
$848$	7.65575	0.262900
$849$	47.0502	1.61476
$850$	−27.8739	−0.956067
$851$	−30.2221	−1.03600
$852$	34.9726	1.19814
$853$	−41.0237	−1.40463	−0.702313	−	0.711868i	$-0.747849\pi$
$853$	−41.0237	−1.40463	−0.702313	+	0.711868i	$0.747849\pi$
$854$	6.35968	0.217624
$855$	−36.4968	−1.24816
$856$	0.936314	0.0320025
$857$	32.4447	1.10829	0.554144	−	0.832421i	$-0.313046\pi$
$857$	32.4447	1.10829	0.554144	+	0.832421i	$0.313046\pi$
$858$	−12.8172	−0.437573
$859$	−12.5038	−0.426625	−0.213312	−	0.976984i	$-0.568425\pi$
$859$	−12.5038	−0.426625	−0.213312	+	0.976984i	$0.568425\pi$
$860$	−17.6585	−0.602151
$861$	3.46007	0.117919
$862$	−1.00000	−0.0340601
$863$	7.13263	0.242798	0.121399	−	0.992604i	$-0.461262\pi$
$863$	7.13263	0.242798	0.121399	+	0.992604i	$0.461262\pi$
$864$	4.04429	0.137590
$865$	63.1760	2.14805
$866$	26.7157	0.907836
$867$	10.2375	0.347685
$868$	−6.48892	−0.220248
$869$	−26.6542	−0.904182
$870$	−35.3159	−1.19732
$871$	39.8345	1.34974
$872$	−9.39043	−0.318000
$873$	48.6373	1.64612
$874$	7.54261	0.255133
$875$	9.44750	0.319384
$876$	36.9386	1.24804
$877$	11.9520	0.403590	0.201795	−	0.979428i	$-0.435323\pi$
$877$	11.9520	0.403590	0.201795	+	0.979428i	$0.435323\pi$
$878$	−5.72597	−0.193242
$879$	25.5558	0.861975
$880$	6.68440	0.225331
$881$	−33.4507	−1.12698	−0.563491	−	0.826122i	$-0.690542\pi$
$881$	−33.4507	−1.12698	−0.563491	+	0.826122i	$0.690542\pi$
$882$	−4.47885	−0.150811
$883$	−16.3593	−0.550534	−0.275267	−	0.961368i	$-0.588766\pi$
$883$	−16.3593	−0.550534	−0.275267	+	0.961368i	$0.588766\pi$
$884$	−9.08178	−0.305453
$885$	−20.3343	−0.683529
$886$	−22.7206	−0.763312
$887$	18.8279	0.632181	0.316090	−	0.948729i	$-0.397630\pi$
$887$	18.8279	0.632181	0.316090	+	0.948729i	$0.397630\pi$
$888$	−25.0995	−0.842284
$889$	11.6524	0.390808
$890$	38.6860	1.29676
$891$	−4.46524	−0.149591
$892$	11.1695	0.373983
$893$	−9.01334	−0.301620
$894$	56.3263	1.88383
$895$	−64.5954	−2.15919
$896$	−1.00000	−0.0334077
$897$	22.4621	0.749988
$898$	15.9861	0.533462
$899$	23.5550	0.785603
$900$	34.2886	1.14295
$901$	27.8742	0.928624
$902$	2.37732	0.0791560
$903$	13.5747	0.451737
$904$	11.2614	0.374549
$905$	75.5716	2.51208
$906$	−27.0398	−0.898338
$907$	29.8570	0.991386	0.495693	−	0.868498i	$-0.334914\pi$
$907$	29.8570	0.991386	0.495693	+	0.868498i	$0.334914\pi$
$908$	20.0586	0.665669
$909$	47.8556	1.58727
$910$	8.87359	0.294156
$911$	−31.0405	−1.02842	−0.514208	−	0.857665i	$-0.671914\pi$
$911$	−31.0405	−1.02842	−0.514208	+	0.857665i	$0.671914\pi$
$912$	6.26416	0.207427
$913$	−8.80638	−0.291449
$914$	12.4839	0.412930
$915$	61.8722	2.04543
$916$	6.74393	0.222826
$917$	−14.1748	−0.468093
$918$	14.7251	0.486000
$919$	8.73463	0.288129	0.144064	−	0.989568i	$-0.453983\pi$
$919$	8.73463	0.288129	0.144064	+	0.989568i	$0.453983\pi$
$920$	−11.7144	−0.386211
$921$	4.03991	0.133120
$922$	−1.76183	−0.0580227
$923$	31.8982	1.04994
$924$	−5.13851	−0.169045
$925$	−70.2637	−2.31026
$926$	3.26568	0.107317
$927$	−72.6093	−2.38480
$928$	3.63003	0.119162
$929$	10.7750	0.353515	0.176757	−	0.984254i	$-0.443439\pi$
$929$	10.7750	0.353515	0.176757	+	0.984254i	$0.443439\pi$
$930$	−63.1295	−2.07010
$931$	−2.29058	−0.0750707
$932$	−0.858628	−0.0281253
$933$	−30.0933	−0.985211
$934$	23.5817	0.771616
$935$	24.3376	0.795923
$936$	11.1718	0.365162
$937$	57.8211	1.88893	0.944466	−	0.328609i	$-0.106580\pi$
$937$	57.8211	1.88893	0.944466	+	0.328609i	$0.106580\pi$
$938$	15.9699	0.521437
$939$	4.05809	0.132431
$940$	13.9985	0.456582
$941$	−6.05892	−0.197515	−0.0987575	−	0.995112i	$-0.531487\pi$
$941$	−6.05892	−0.197515	−0.0987575	+	0.995112i	$0.531487\pi$
$942$	32.7361	1.06660
$943$	−4.16623	−0.135671
$944$	2.09011	0.0680271
$945$	−14.3875	−0.468025
$946$	9.32679	0.303240
$947$	−25.8427	−0.839774	−0.419887	−	0.907576i	$-0.637930\pi$
$947$	−25.8427	−0.839774	−0.419887	+	0.907576i	$0.637930\pi$
$948$	38.7939	1.25997
$949$	33.6914	1.09367
$950$	17.5359	0.568941
$951$	−4.64989	−0.150783
$952$	−3.64095	−0.118004
$953$	2.56145	0.0829735	0.0414868	−	0.999139i	$-0.486791\pi$
$953$	2.56145	0.0829735	0.0414868	+	0.999139i	$0.486791\pi$
$954$	−34.2890	−1.11015
$955$	9.89956	0.320342
$956$	24.3094	0.786223
$957$	18.6530	0.602965
$958$	−33.1505	−1.07104
$959$	3.23866	0.104582
$960$	−9.72882	−0.313996
$961$	11.1061	0.358262
$962$	−22.8931	−0.738103
$963$	−4.19361	−0.135137
$964$	19.1486	0.616733
$965$	−11.0553	−0.355884
$966$	9.00521	0.289738
$967$	19.9062	0.640141	0.320071	−	0.947394i	$-0.396293\pi$
$967$	19.9062	0.640141	0.320071	+	0.947394i	$0.396293\pi$
$968$	7.46947	0.240078
$969$	22.8075	0.732682
$970$	−38.6318	−1.24039
$971$	−27.0371	−0.867663	−0.433831	−	0.900994i	$-0.642839\pi$
$971$	−27.0371	−0.867663	−0.433831	+	0.900994i	$0.642839\pi$
$972$	18.6318	0.597616
$973$	−8.17268	−0.262004
$974$	19.2410	0.616521
$975$	52.2225	1.67246
$976$	−6.35968	−0.203569
$977$	−52.6676	−1.68499	−0.842493	−	0.538707i	$-0.818913\pi$
$977$	−52.6676	−1.68499	−0.842493	+	0.538707i	$0.818913\pi$
$978$	30.3623	0.970880
$979$	−20.4329	−0.653039
$980$	3.55748	0.113640
$981$	42.0584	1.34282
$982$	23.9522	0.764346
$983$	2.51616	0.0802529	0.0401265	−	0.999195i	$-0.487224\pi$
$983$	2.51616	0.0802529	0.0401265	+	0.999195i	$0.487224\pi$
$984$	−3.46007	−0.110303
$985$	−12.5956	−0.401328
$986$	13.2168	0.420907
$987$	−10.7611	−0.342530
$988$	5.71350	0.181771
$989$	−16.3451	−0.519745
$990$	−29.9385	−0.951507
$991$	0.302891	0.00962165	0.00481082	−	0.999988i	$-0.498469\pi$
$991$	0.302891	0.00962165	0.00481082	+	0.999988i	$0.498469\pi$
$992$	6.48892	0.206023
$993$	59.1541	1.87720
$994$	12.7882	0.405618
$995$	−89.3775	−2.83346
$996$	12.8173	0.406130
$997$	−31.7823	−1.00655	−0.503277	−	0.864125i	$-0.667873\pi$
$997$	−31.7823	−1.00655	−0.503277	+	0.864125i	$0.667873\pi$
$998$	−14.3945	−0.455648
$999$	37.1185	1.17438

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6034.2.a.k.1.2	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6034.2.a.k.1.2	✓	20	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:	\( N \)	\(=\)	\( 6034 = 2 \cdot 7 \cdot 431 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6034.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.73475 q^{3} +1.00000 q^{4} +3.55748 q^{5} +2.73475 q^{6} +1.00000 q^{7} -1.00000 q^{8} +4.47885 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.73475 q^{3} +1.00000 q^{4} +3.55748 q^{5} +2.73475 q^{6} +1.00000 q^{7} -1.00000 q^{8} +4.47885 q^{9} -3.55748 q^{10} +1.87897 q^{11} -2.73475 q^{12} -2.49435 q^{13} -1.00000 q^{14} -9.72882 q^{15} +1.00000 q^{16} +3.64095 q^{17} -4.47885 q^{18} -2.29058 q^{19} +3.55748 q^{20} -2.73475 q^{21} -1.87897 q^{22} +3.29288 q^{23} +2.73475 q^{24} +7.65567 q^{25} +2.49435 q^{26} -4.04429 q^{27} +1.00000 q^{28} -3.63003 q^{29} +9.72882 q^{30} -6.48892 q^{31} -1.00000 q^{32} -5.13851 q^{33} -3.64095 q^{34} +3.55748 q^{35} +4.47885 q^{36} -9.17799 q^{37} +2.29058 q^{38} +6.82141 q^{39} -3.55748 q^{40} -1.26522 q^{41} +2.73475 q^{42} -4.96377 q^{43} +1.87897 q^{44} +15.9334 q^{45} -3.29288 q^{46} +3.93496 q^{47} -2.73475 q^{48} +1.00000 q^{49} -7.65567 q^{50} -9.95708 q^{51} -2.49435 q^{52} +7.65575 q^{53} +4.04429 q^{54} +6.68440 q^{55} -1.00000 q^{56} +6.26416 q^{57} +3.63003 q^{58} +2.09011 q^{59} -9.72882 q^{60} -6.35968 q^{61} +6.48892 q^{62} +4.47885 q^{63} +1.00000 q^{64} -8.87359 q^{65} +5.13851 q^{66} -15.9699 q^{67} +3.64095 q^{68} -9.00521 q^{69} -3.55748 q^{70} -12.7882 q^{71} -4.47885 q^{72} -13.5071 q^{73} +9.17799 q^{74} -20.9363 q^{75} -2.29058 q^{76} +1.87897 q^{77} -6.82141 q^{78} -14.1855 q^{79} +3.55748 q^{80} -2.37643 q^{81} +1.26522 q^{82} -4.68681 q^{83} -2.73475 q^{84} +12.9526 q^{85} +4.96377 q^{86} +9.92723 q^{87} -1.87897 q^{88} -10.8745 q^{89} -15.9334 q^{90} -2.49435 q^{91} +3.29288 q^{92} +17.7456 q^{93} -3.93496 q^{94} -8.14870 q^{95} +2.73475 q^{96} +10.8593 q^{97} -1.00000 q^{98} +8.41564 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 20 q^{2} + 3 q^{3} + 20 q^{4} - 3 q^{5} - 3 q^{6} + 20 q^{7} - 20 q^{8} + 13 q^{9}+O(q^{10}) \) 20 * q - 20 * q^2 + 3 * q^3 + 20 * q^4 - 3 * q^5 - 3 * q^6 + 20 * q^7 - 20 * q^8 + 13 * q^9	\( 20 q - 20 q^{2} + 3 q^{3} + 20 q^{4} - 3 q^{5} - 3 q^{6} + 20 q^{7} - 20 q^{8} + 13 q^{9} + 3 q^{10} - 8 q^{11} + 3 q^{12} - 4 q^{13} - 20 q^{14} - 25 q^{15} + 20 q^{16} + 9 q^{17} - 13 q^{18} - 14 q^{19} - 3 q^{20} + 3 q^{21} + 8 q^{22} - 23 q^{23} - 3 q^{24} + 31 q^{25} + 4 q^{26} - 21 q^{27} + 20 q^{28} - 48 q^{29} + 25 q^{30} - q^{31} - 20 q^{32} - 29 q^{33} - 9 q^{34} - 3 q^{35} + 13 q^{36} - q^{37} + 14 q^{38} - q^{39} + 3 q^{40} - 27 q^{41} - 3 q^{42} - 3 q^{43} - 8 q^{44} - 12 q^{45} + 23 q^{46} - 26 q^{47} + 3 q^{48} + 20 q^{49} - 31 q^{50} - 17 q^{51} - 4 q^{52} - 43 q^{53} + 21 q^{54} - 16 q^{55} - 20 q^{56} - 25 q^{57} + 48 q^{58} - 19 q^{59} - 25 q^{60} + 9 q^{61} + q^{62} + 13 q^{63} + 20 q^{64} - 87 q^{65} + 29 q^{66} + 32 q^{67} + 9 q^{68} - 23 q^{69} + 3 q^{70} - 63 q^{71} - 13 q^{72} + 2 q^{73} + q^{74} - 8 q^{75} - 14 q^{76} - 8 q^{77} + q^{78} - 51 q^{79} - 3 q^{80} + 4 q^{81} + 27 q^{82} - 24 q^{83} + 3 q^{84} + 31 q^{85} + 3 q^{86} - 33 q^{87} + 8 q^{88} - 35 q^{89} + 12 q^{90} - 4 q^{91} - 23 q^{92} + 17 q^{93} + 26 q^{94} - 30 q^{95} - 3 q^{96} + 5 q^{97} - 20 q^{98} - 31 q^{99}+O(q^{100}) \) 20 * q - 20 * q^2 + 3 * q^3 + 20 * q^4 - 3 * q^5 - 3 * q^6 + 20 * q^7 - 20 * q^8 + 13 * q^9 + 3 * q^10 - 8 * q^11 + 3 * q^12 - 4 * q^13 - 20 * q^14 - 25 * q^15 + 20 * q^16 + 9 * q^17 - 13 * q^18 - 14 * q^19 - 3 * q^20 + 3 * q^21 + 8 * q^22 - 23 * q^23 - 3 * q^24 + 31 * q^25 + 4 * q^26 - 21 * q^27 + 20 * q^28 - 48 * q^29 + 25 * q^30 - q^31 - 20 * q^32 - 29 * q^33 - 9 * q^34 - 3 * q^35 + 13 * q^36 - q^37 + 14 * q^38 - q^39 + 3 * q^40 - 27 * q^41 - 3 * q^42 - 3 * q^43 - 8 * q^44 - 12 * q^45 + 23 * q^46 - 26 * q^47 + 3 * q^48 + 20 * q^49 - 31 * q^50 - 17 * q^51 - 4 * q^52 - 43 * q^53 + 21 * q^54 - 16 * q^55 - 20 * q^56 - 25 * q^57 + 48 * q^58 - 19 * q^59 - 25 * q^60 + 9 * q^61 + q^62 + 13 * q^63 + 20 * q^64 - 87 * q^65 + 29 * q^66 + 32 * q^67 + 9 * q^68 - 23 * q^69 + 3 * q^70 - 63 * q^71 - 13 * q^72 + 2 * q^73 + q^74 - 8 * q^75 - 14 * q^76 - 8 * q^77 + q^78 - 51 * q^79 - 3 * q^80 + 4 * q^81 + 27 * q^82 - 24 * q^83 + 3 * q^84 + 31 * q^85 + 3 * q^86 - 33 * q^87 + 8 * q^88 - 35 * q^89 + 12 * q^90 - 4 * q^91 - 23 * q^92 + 17 * q^93 + 26 * q^94 - 30 * q^95 - 3 * q^96 + 5 * q^97 - 20 * q^98 - 31 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more