LMFDB - Embedded newform 6013.2.a.e.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6013,2,Mod(1,6013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6013, 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("6013.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6013 = 7 \cdot 859 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6013.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:	$48.0140467354$
Analytic rank:	$0$
Dimension:	$109$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	6013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.11721 q^{2} +0.0595888 q^{3} +2.48258 q^{4} -0.217642 q^{5} -0.126162 q^{6} +1.00000 q^{7} -1.02172 q^{8} -2.99645 q^{9} +O(q^{10})$	\(q-2.11721 q^{2} +0.0595888 q^{3} +2.48258 q^{4} -0.217642 q^{5} -0.126162 q^{6} +1.00000 q^{7} -1.02172 q^{8} -2.99645 q^{9} +0.460795 q^{10} -6.48236 q^{11} +0.147934 q^{12} -1.76262 q^{13} -2.11721 q^{14} -0.0129691 q^{15} -2.80196 q^{16} -3.36526 q^{17} +6.34411 q^{18} +2.28368 q^{19} -0.540315 q^{20} +0.0595888 q^{21} +13.7245 q^{22} +4.32025 q^{23} -0.0608833 q^{24} -4.95263 q^{25} +3.73184 q^{26} -0.357321 q^{27} +2.48258 q^{28} -9.07084 q^{29} +0.0274582 q^{30} -4.78988 q^{31} +7.97578 q^{32} -0.386276 q^{33} +7.12496 q^{34} -0.217642 q^{35} -7.43892 q^{36} -5.25246 q^{37} -4.83503 q^{38} -0.105032 q^{39} +0.222370 q^{40} +10.4507 q^{41} -0.126162 q^{42} -9.64737 q^{43} -16.0930 q^{44} +0.652155 q^{45} -9.14689 q^{46} -1.63389 q^{47} -0.166965 q^{48} +1.00000 q^{49} +10.4858 q^{50} -0.200532 q^{51} -4.37585 q^{52} +1.70108 q^{53} +0.756524 q^{54} +1.41084 q^{55} -1.02172 q^{56} +0.136082 q^{57} +19.2049 q^{58} -2.35256 q^{59} -0.0321967 q^{60} -9.12822 q^{61} +10.1412 q^{62} -2.99645 q^{63} -11.2825 q^{64} +0.383621 q^{65} +0.817828 q^{66} -5.08291 q^{67} -8.35453 q^{68} +0.257439 q^{69} +0.460795 q^{70} -6.42142 q^{71} +3.06154 q^{72} -11.3025 q^{73} +11.1206 q^{74} -0.295121 q^{75} +5.66942 q^{76} -6.48236 q^{77} +0.222376 q^{78} +7.17326 q^{79} +0.609825 q^{80} +8.96806 q^{81} -22.1264 q^{82} +3.43556 q^{83} +0.147934 q^{84} +0.732424 q^{85} +20.4255 q^{86} -0.540520 q^{87} +6.62318 q^{88} +1.17943 q^{89} -1.38075 q^{90} -1.76262 q^{91} +10.7254 q^{92} -0.285423 q^{93} +3.45928 q^{94} -0.497026 q^{95} +0.475267 q^{96} -17.0817 q^{97} -2.11721 q^{98} +19.4241 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 109 q + 19 q^{2} + 38 q^{3} + 111 q^{4} + 43 q^{5} + 14 q^{6} + 109 q^{7} + 48 q^{8} + 119 q^{9}+O(q^{10}) $ 109 * q + 19 * q^2 + 38 * q^3 + 111 * q^4 + 43 * q^5 + 14 * q^6 + 109 * q^7 + 48 * q^8 + 119 * q^9	\( 109 q + 19 q^{2} + 38 q^{3} + 111 q^{4} + 43 q^{5} + 14 q^{6} + 109 q^{7} + 48 q^{8} + 119 q^{9} + 15 q^{10} + 48 q^{11} + 72 q^{12} + 29 q^{13} + 19 q^{14} + 29 q^{15} + 115 q^{16} + 72 q^{17} + 33 q^{18} + 58 q^{19} + 88 q^{20} + 38 q^{21} + 4 q^{22} + 65 q^{23} + 46 q^{24} + 124 q^{25} + 49 q^{26} + 131 q^{27} + 111 q^{28} + 25 q^{29} + 2 q^{30} + 41 q^{31} + 75 q^{32} + 54 q^{33} + 23 q^{34} + 43 q^{35} + 111 q^{36} + 25 q^{37} + 54 q^{38} + 27 q^{39} + 30 q^{40} + 109 q^{41} + 14 q^{42} + 38 q^{43} + 68 q^{44} + 84 q^{45} - 9 q^{46} + 121 q^{47} + 106 q^{48} + 109 q^{49} + 14 q^{50} + 36 q^{51} + 38 q^{52} + 61 q^{53} + 31 q^{54} + 50 q^{55} + 48 q^{56} + 5 q^{57} - 20 q^{58} + 181 q^{59} + 25 q^{60} + 34 q^{61} + 75 q^{62} + 119 q^{63} + 96 q^{64} + 12 q^{65} + 19 q^{66} + 87 q^{67} + 150 q^{68} + 89 q^{69} + 15 q^{70} + 83 q^{71} + 65 q^{72} + 32 q^{73} - 19 q^{74} + 112 q^{75} + 84 q^{76} + 48 q^{77} - 34 q^{78} - 9 q^{79} + 137 q^{80} + 109 q^{81} - 19 q^{82} + 136 q^{83} + 72 q^{84} - 32 q^{85} - 24 q^{86} + 28 q^{87} - 24 q^{88} + 142 q^{89} + 19 q^{90} + 29 q^{91} + 96 q^{92} + 29 q^{93} + 9 q^{94} + 52 q^{95} + 88 q^{96} + 75 q^{97} + 19 q^{98} + 84 q^{99}+O(q^{100}) \) 109 * q + 19 * q^2 + 38 * q^3 + 111 * q^4 + 43 * q^5 + 14 * q^6 + 109 * q^7 + 48 * q^8 + 119 * q^9 + 15 * q^10 + 48 * q^11 + 72 * q^12 + 29 * q^13 + 19 * q^14 + 29 * q^15 + 115 * q^16 + 72 * q^17 + 33 * q^18 + 58 * q^19 + 88 * q^20 + 38 * q^21 + 4 * q^22 + 65 * q^23 + 46 * q^24 + 124 * q^25 + 49 * q^26 + 131 * q^27 + 111 * q^28 + 25 * q^29 + 2 * q^30 + 41 * q^31 + 75 * q^32 + 54 * q^33 + 23 * q^34 + 43 * q^35 + 111 * q^36 + 25 * q^37 + 54 * q^38 + 27 * q^39 + 30 * q^40 + 109 * q^41 + 14 * q^42 + 38 * q^43 + 68 * q^44 + 84 * q^45 - 9 * q^46 + 121 * q^47 + 106 * q^48 + 109 * q^49 + 14 * q^50 + 36 * q^51 + 38 * q^52 + 61 * q^53 + 31 * q^54 + 50 * q^55 + 48 * q^56 + 5 * q^57 - 20 * q^58 + 181 * q^59 + 25 * q^60 + 34 * q^61 + 75 * q^62 + 119 * q^63 + 96 * q^64 + 12 * q^65 + 19 * q^66 + 87 * q^67 + 150 * q^68 + 89 * q^69 + 15 * q^70 + 83 * q^71 + 65 * q^72 + 32 * q^73 - 19 * q^74 + 112 * q^75 + 84 * q^76 + 48 * q^77 - 34 * q^78 - 9 * q^79 + 137 * q^80 + 109 * q^81 - 19 * q^82 + 136 * q^83 + 72 * q^84 - 32 * q^85 - 24 * q^86 + 28 * q^87 - 24 * q^88 + 142 * q^89 + 19 * q^90 + 29 * q^91 + 96 * q^92 + 29 * q^93 + 9 * q^94 + 52 * q^95 + 88 * q^96 + 75 * q^97 + 19 * q^98 + 84 * 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 $

$2$	−2.11721	−1.49709	−0.748547	−	0.663082i	$-0.769248\pi$
$2$	−2.11721	−1.49709	−0.748547	+	0.663082i	$0.769248\pi$
$3$	0.0595888	0.0344036	0.0172018	−	0.999852i	$-0.494524\pi$
$3$	0.0595888	0.0344036	0.0172018	+	0.999852i	$0.494524\pi$
$4$	2.48258	1.24129
$5$	−0.217642	−0.0973327	−0.0486663	−	0.998815i	$-0.515497\pi$
$5$	−0.217642	−0.0973327	−0.0486663	+	0.998815i	$0.515497\pi$
$6$	−0.126162	−0.0515054
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.02172	−0.361234
$9$	−2.99645	−0.998816
$10$	0.460795	0.145716
$11$	−6.48236	−1.95451	−0.977253	−	0.212078i	$-0.931977\pi$
$11$	−6.48236	−1.95451	−0.977253	+	0.212078i	$0.931977\pi$
$12$	0.147934	0.0427048
$13$	−1.76262	−0.488863	−0.244432	−	0.969667i	$-0.578601\pi$
$13$	−1.76262	−0.488863	−0.244432	+	0.969667i	$0.578601\pi$
$14$	−2.11721	−0.565848
$15$	−0.0129691	−0.00334859
$16$	−2.80196	−0.700489
$17$	−3.36526	−0.816195	−0.408098	−	0.912938i	$-0.633808\pi$
$17$	−3.36526	−0.816195	−0.408098	+	0.912938i	$0.633808\pi$
$18$	6.34411	1.49532
$19$	2.28368	0.523912	0.261956	−	0.965080i	$-0.415632\pi$
$19$	2.28368	0.523912	0.261956	+	0.965080i	$0.415632\pi$
$20$	−0.540315	−0.120818
$21$	0.0595888	0.0130033
$22$	13.7245	2.92608
$23$	4.32025	0.900835	0.450418	−	0.892818i	$-0.351275\pi$
$23$	4.32025	0.900835	0.450418	+	0.892818i	$0.351275\pi$
$24$	−0.0608833	−0.0124277
$25$	−4.95263	−0.990526
$26$	3.73184	0.731874
$27$	−0.357321	−0.0687665
$28$	2.48258	0.469164
$29$	−9.07084	−1.68441	−0.842207	−	0.539155i	$-0.818744\pi$
$29$	−9.07084	−1.68441	−0.842207	+	0.539155i	$0.818744\pi$
$30$	0.0274582	0.00501316
$31$	−4.78988	−0.860287	−0.430144	−	0.902760i	$-0.641537\pi$
$31$	−4.78988	−0.860287	−0.430144	+	0.902760i	$0.641537\pi$
$32$	7.97578	1.40993
$33$	−0.386276	−0.0672420
$34$	7.12496	1.22192
$35$	−0.217642	−0.0367883
$36$	−7.43892	−1.23982
$37$	−5.25246	−0.863499	−0.431750	−	0.901993i	$-0.642104\pi$
$37$	−5.25246	−0.863499	−0.431750	+	0.901993i	$0.642104\pi$
$38$	−4.83503	−0.784346
$39$	−0.105032	−0.0168187
$40$	0.222370	0.0351599
$41$	10.4507	1.63213	0.816066	−	0.577959i	$-0.196151\pi$
$41$	10.4507	1.63213	0.816066	+	0.577959i	$0.196151\pi$
$42$	−0.126162	−0.0194672
$43$	−9.64737	−1.47121	−0.735605	−	0.677410i	$-0.763102\pi$
$43$	−9.64737	−1.47121	−0.735605	+	0.677410i	$0.763102\pi$
$44$	−16.0930	−2.42611
$45$	0.652155	0.0972175
$46$	−9.14689	−1.34863
$47$	−1.63389	−0.238327	−0.119163	−	0.992875i	$-0.538021\pi$
$47$	−1.63389	−0.238327	−0.119163	+	0.992875i	$0.538021\pi$
$48$	−0.166965	−0.0240993
$49$	1.00000	0.142857
$50$	10.4858	1.48291
$51$	−0.200532	−0.0280801
$52$	−4.37585	−0.606821
$53$	1.70108	0.233661	0.116831	−	0.993152i	$-0.462727\pi$
$53$	1.70108	0.233661	0.116831	+	0.993152i	$0.462727\pi$
$54$	0.756524	0.102950
$55$	1.41084	0.190237
$56$	−1.02172	−0.136534
$57$	0.136082	0.0180245
$58$	19.2049	2.52172
$59$	−2.35256	−0.306278	−0.153139	−	0.988205i	$-0.548938\pi$
$59$	−2.35256	−0.306278	−0.153139	+	0.988205i	$0.548938\pi$
$60$	−0.0321967	−0.00415658
$61$	−9.12822	−1.16875	−0.584374	−	0.811484i	$-0.698660\pi$
$61$	−9.12822	−1.16875	−0.584374	+	0.811484i	$0.698660\pi$
$62$	10.1412	1.28793
$63$	−2.99645	−0.377517
$64$	−11.2825	−1.41031
$65$	0.383621	0.0475824
$66$	0.817828	0.100668
$67$	−5.08291	−0.620976	−0.310488	−	0.950577i	$-0.600492\pi$
$67$	−5.08291	−0.620976	−0.310488	+	0.950577i	$0.600492\pi$
$68$	−8.35453	−1.01314
$69$	0.257439	0.0309920
$70$	0.460795	0.0550755
$71$	−6.42142	−0.762083	−0.381041	−	0.924558i	$-0.624434\pi$
$71$	−6.42142	−0.762083	−0.381041	+	0.924558i	$0.624434\pi$
$72$	3.06154	0.360806
$73$	−11.3025	−1.32285	−0.661427	−	0.750010i	$-0.730049\pi$
$73$	−11.3025	−1.32285	−0.661427	+	0.750010i	$0.730049\pi$
$74$	11.1206	1.29274
$75$	−0.295121	−0.0340777
$76$	5.66942	0.650327
$77$	−6.48236	−0.738734
$78$	0.222376	0.0251791
$79$	7.17326	0.807055	0.403528	−	0.914968i	$-0.367784\pi$
$79$	7.17326	0.807055	0.403528	+	0.914968i	$0.367784\pi$
$80$	0.609825	0.0681805
$81$	8.96806	0.996451
$82$	−22.1264	−2.44345
$83$	3.43556	0.377101	0.188551	−	0.982063i	$-0.439621\pi$
$83$	3.43556	0.377101	0.188551	+	0.982063i	$0.439621\pi$
$84$	0.147934	0.0161409
$85$	0.732424	0.0794425
$86$	20.4255	2.20254
$87$	−0.540520	−0.0579499
$88$	6.62318	0.706033
$89$	1.17943	0.125019	0.0625094	−	0.998044i	$-0.480090\pi$
$89$	1.17943	0.125019	0.0625094	+	0.998044i	$0.480090\pi$
$90$	−1.38075	−0.145544
$91$	−1.76262	−0.184773
$92$	10.7254	1.11820
$93$	−0.285423	−0.0295970
$94$	3.45928	0.356798
$95$	−0.497026	−0.0509938
$96$	0.475267	0.0485067
$97$	−17.0817	−1.73439	−0.867193	−	0.497973i	$-0.834078\pi$
$97$	−17.0817	−1.73439	−0.867193	+	0.497973i	$0.834078\pi$
$98$	−2.11721	−0.213871
$99$	19.4241	1.95219
$100$	−12.2953	−1.22953
$101$	17.2815	1.71957	0.859784	−	0.510657i	$-0.170598\pi$
$101$	17.2815	1.71957	0.859784	+	0.510657i	$0.170598\pi$
$102$	0.424568	0.0420385
$103$	−7.02745	−0.692435	−0.346218	−	0.938154i	$-0.612534\pi$
$103$	−7.02745	−0.692435	−0.346218	+	0.938154i	$0.612534\pi$
$104$	1.80091	0.176594
$105$	−0.0129691	−0.00126565
$106$	−3.60155	−0.349813
$107$	9.30497	0.899545	0.449773	−	0.893143i	$-0.351505\pi$
$107$	9.30497	0.899545	0.449773	+	0.893143i	$0.351505\pi$
$108$	−0.887078	−0.0853591
$109$	7.46946	0.715444	0.357722	−	0.933828i	$-0.383554\pi$
$109$	7.46946	0.715444	0.357722	+	0.933828i	$0.383554\pi$
$110$	−2.98704	−0.284803
$111$	−0.312988	−0.0297075
$112$	−2.80196	−0.264760
$113$	11.3575	1.06842	0.534210	−	0.845352i	$-0.320609\pi$
$113$	11.3575	1.06842	0.534210	+	0.845352i	$0.320609\pi$
$114$	−0.288114	−0.0269843
$115$	−0.940271	−0.0876807
$116$	−22.5191	−2.09085
$117$	5.28161	0.488285
$118$	4.98087	0.458527
$119$	−3.36526	−0.308493
$120$	0.0132508	0.00120963
$121$	31.0210	2.82009
$122$	19.3264	1.74973
$123$	0.622747	0.0561512
$124$	−11.8913	−1.06787
$125$	2.16612	0.193743
$126$	6.34411	0.565179
$127$	−4.77108	−0.423365	−0.211682	−	0.977339i	$-0.567894\pi$
$127$	−4.77108	−0.423365	−0.211682	+	0.977339i	$0.567894\pi$
$128$	7.93585	0.701436
$129$	−0.574875	−0.0506149
$130$	−0.812207	−0.0712353
$131$	−9.59927	−0.838692	−0.419346	−	0.907827i	$-0.637741\pi$
$131$	−9.59927	−0.838692	−0.419346	+	0.907827i	$0.637741\pi$
$132$	−0.958961	−0.0834669
$133$	2.28368	0.198020
$134$	10.7616	0.929659
$135$	0.0777683	0.00669323
$136$	3.43837	0.294837
$137$	11.6660	0.996698	0.498349	−	0.866977i	$-0.333940\pi$
$137$	11.6660	0.996698	0.498349	+	0.866977i	$0.333940\pi$
$138$	−0.545052	−0.0463979
$139$	5.18473	0.439763	0.219882	−	0.975527i	$-0.429433\pi$
$139$	5.18473	0.439763	0.219882	+	0.975527i	$0.429433\pi$
$140$	−0.540315	−0.0456649
$141$	−0.0973613	−0.00819930
$142$	13.5955	1.14091
$143$	11.4259	0.955486
$144$	8.39592	0.699660
$145$	1.97420	0.163948
$146$	23.9297	1.98044
$147$	0.0595888	0.00491480
$148$	−13.0397	−1.07185
$149$	9.52641	0.780434	0.390217	−	0.920723i	$-0.372400\pi$
$149$	9.52641	0.780434	0.390217	+	0.920723i	$0.372400\pi$
$150$	0.624834	0.0510175
$151$	−8.59419	−0.699385	−0.349693	−	0.936864i	$-0.613714\pi$
$151$	−8.59419	−0.699385	−0.349693	+	0.936864i	$0.613714\pi$
$152$	−2.33329	−0.189255
$153$	10.0838	0.815229
$154$	13.7245	1.10595
$155$	1.04248	0.0837341
$156$	−0.260752	−0.0208768
$157$	−17.4332	−1.39132	−0.695660	−	0.718371i	$-0.744888\pi$
$157$	−17.4332	−1.39132	−0.695660	+	0.718371i	$0.744888\pi$
$158$	−15.1873	−1.20824
$159$	0.101365	0.00803879
$160$	−1.73587	−0.137232
$161$	4.32025	0.340484
$162$	−18.9873	−1.49178
$163$	7.80360	0.611225	0.305613	−	0.952156i	$-0.401139\pi$
$163$	7.80360	0.611225	0.305613	+	0.952156i	$0.401139\pi$
$164$	25.9448	2.02595
$165$	0.0840701	0.00654485
$166$	−7.27380	−0.564556
$167$	−4.36425	−0.337716	−0.168858	−	0.985640i	$-0.554008\pi$
$167$	−4.36425	−0.337716	−0.168858	+	0.985640i	$0.554008\pi$
$168$	−0.0608833	−0.00469725
$169$	−9.89317	−0.761013
$170$	−1.55069	−0.118933
$171$	−6.84293	−0.523292
$172$	−23.9504	−1.82620
$173$	−2.13481	−0.162307	−0.0811533	−	0.996702i	$-0.525860\pi$
$173$	−2.13481	−0.162307	−0.0811533	+	0.996702i	$0.525860\pi$
$174$	1.14440	0.0867564
$175$	−4.95263	−0.374384
$176$	18.1633	1.36911
$177$	−0.140186	−0.0105371
$178$	−2.49709	−0.187165
$179$	−6.27980	−0.469375	−0.234687	−	0.972071i	$-0.575407\pi$
$179$	−6.27980	−0.469375	−0.234687	+	0.972071i	$0.575407\pi$
$180$	1.61903	0.120675
$181$	8.83302	0.656553	0.328277	−	0.944582i	$-0.393532\pi$
$181$	8.83302	0.656553	0.328277	+	0.944582i	$0.393532\pi$
$182$	3.73184	0.276622
$183$	−0.543940	−0.0402092
$184$	−4.41410	−0.325412
$185$	1.14316	0.0840467
$186$	0.604300	0.0443095
$187$	21.8148	1.59526
$188$	−4.05626	−0.295833
$189$	−0.357321	−0.0259913
$190$	1.05231	0.0763425
$191$	4.74801	0.343554	0.171777	−	0.985136i	$-0.445049\pi$
$191$	4.74801	0.343554	0.171777	+	0.985136i	$0.445049\pi$
$192$	−0.672310	−0.0485198
$193$	−0.244290	−0.0175843	−0.00879217	−	0.999961i	$-0.502799\pi$
$193$	−0.244290	−0.0175843	−0.00879217	+	0.999961i	$0.502799\pi$
$194$	36.1656	2.59654
$195$	0.0228595	0.00163700
$196$	2.48258	0.177327
$197$	11.4463	0.815518	0.407759	−	0.913090i	$-0.366310\pi$
$197$	11.4463	0.815518	0.407759	+	0.913090i	$0.366310\pi$
$198$	−41.1248	−2.92261
$199$	−13.1190	−0.929982	−0.464991	−	0.885315i	$-0.653943\pi$
$199$	−13.1190	−0.929982	−0.464991	+	0.885315i	$0.653943\pi$
$200$	5.06022	0.357812
$201$	−0.302884	−0.0213638
$202$	−36.5885	−2.57436
$203$	−9.07084	−0.636648
$204$	−0.497836	−0.0348555
$205$	−2.27453	−0.158860
$206$	14.8786	1.03664
$207$	−12.9454	−0.899769
$208$	4.93879	0.342443
$209$	−14.8036	−1.02399
$210$	0.0274582	0.00189480
$211$	21.8108	1.50152	0.750760	−	0.660576i	$-0.229688\pi$
$211$	21.8108	1.50152	0.750760	+	0.660576i	$0.229688\pi$
$212$	4.22307	0.290042
$213$	−0.382645	−0.0262184
$214$	−19.7006	−1.34670
$215$	2.09968	0.143197
$216$	0.365083	0.0248408
$217$	−4.78988	−0.325158
$218$	−15.8144	−1.07109
$219$	−0.673500	−0.0455109
$220$	3.50252	0.236140
$221$	5.93168	0.399008
$222$	0.662661	0.0444749
$223$	2.63188	0.176244	0.0881220	−	0.996110i	$-0.471913\pi$
$223$	2.63188	0.176244	0.0881220	+	0.996110i	$0.471913\pi$
$224$	7.97578	0.532904
$225$	14.8403	0.989354
$226$	−24.0461	−1.59952
$227$	−19.5609	−1.29830	−0.649151	−	0.760660i	$-0.724876\pi$
$227$	−19.5609	−1.29830	−0.649151	+	0.760660i	$0.724876\pi$
$228$	0.337834	0.0223736
$229$	2.91229	0.192450	0.0962249	−	0.995360i	$-0.469323\pi$
$229$	2.91229	0.192450	0.0962249	+	0.995360i	$0.469323\pi$
$230$	1.99075	0.131266
$231$	−0.386276	−0.0254151
$232$	9.26789	0.608467
$233$	−13.8129	−0.904911	−0.452456	−	0.891787i	$-0.649452\pi$
$233$	−13.8129	−0.904911	−0.452456	+	0.891787i	$0.649452\pi$
$234$	−11.1823	−0.731008
$235$	0.355603	0.0231970
$236$	−5.84043	−0.380180
$237$	0.427446	0.0277656
$238$	7.12496	0.461843
$239$	−17.0656	−1.10388	−0.551940	−	0.833884i	$-0.686112\pi$
$239$	−17.0656	−1.10388	−0.551940	+	0.833884i	$0.686112\pi$
$240$	0.0363387	0.00234565
$241$	8.46587	0.545334	0.272667	−	0.962108i	$-0.412094\pi$
$241$	8.46587	0.545334	0.272667	+	0.962108i	$0.412094\pi$
$242$	−65.6780	−4.22194
$243$	1.60636	0.103048
$244$	−22.6615	−1.45076
$245$	−0.217642	−0.0139047
$246$	−1.31849	−0.0840636
$247$	−4.02526	−0.256121
$248$	4.89393	0.310765
$249$	0.204721	0.0129736
$250$	−4.58612	−0.290052
$251$	2.54112	0.160394	0.0801969	−	0.996779i	$-0.474445\pi$
$251$	2.54112	0.160394	0.0801969	+	0.996779i	$0.474445\pi$
$252$	−7.43892	−0.468608
$253$	−28.0054	−1.76069
$254$	10.1014	0.633817
$255$	0.0436442	0.00273311
$256$	5.76312	0.360195
$257$	19.0147	1.18610	0.593052	−	0.805164i	$-0.297923\pi$
$257$	19.0147	1.18610	0.593052	+	0.805164i	$0.297923\pi$
$258$	1.21713	0.0757753
$259$	−5.25246	−0.326372
$260$	0.952371	0.0590635
$261$	27.1803	1.68242
$262$	20.3237	1.25560
$263$	−19.9647	−1.23107	−0.615537	−	0.788108i	$-0.711061\pi$
$263$	−19.9647	−1.23107	−0.615537	+	0.788108i	$0.711061\pi$
$264$	0.394667	0.0242901
$265$	−0.370227	−0.0227429
$266$	−4.83503	−0.296455
$267$	0.0702805	0.00430110
$268$	−12.6187	−0.770811
$269$	−10.1215	−0.617119	−0.308560	−	0.951205i	$-0.599847\pi$
$269$	−10.1215	−0.617119	−0.308560	+	0.951205i	$0.599847\pi$
$270$	−0.164652	−0.0100204
$271$	−32.2546	−1.95933	−0.979663	−	0.200649i	$-0.935695\pi$
$271$	−32.2546	−1.95933	−0.979663	+	0.200649i	$0.935695\pi$
$272$	9.42931	0.571736
$273$	−0.105032	−0.00635685
$274$	−24.6995	−1.49215
$275$	32.1047	1.93599
$276$	0.639112	0.0384700
$277$	−16.6210	−0.998660	−0.499330	−	0.866412i	$-0.666420\pi$
$277$	−16.6210	−0.998660	−0.499330	+	0.866412i	$0.666420\pi$
$278$	−10.9772	−0.658367
$279$	14.3526	0.859269
$280$	0.222370	0.0132892
$281$	−2.60751	−0.155551	−0.0777755	−	0.996971i	$-0.524782\pi$
$281$	−2.60751	−0.155551	−0.0777755	+	0.996971i	$0.524782\pi$
$282$	0.206134	0.0122751
$283$	−2.48118	−0.147491	−0.0737455	−	0.997277i	$-0.523495\pi$
$283$	−2.48118	−0.147491	−0.0737455	+	0.997277i	$0.523495\pi$
$284$	−15.9417	−0.945966
$285$	−0.0296172	−0.00175437
$286$	−24.1911	−1.43045
$287$	10.4507	0.616888
$288$	−23.8990	−1.40826
$289$	−5.67503	−0.333825
$290$	−4.17980	−0.245446
$291$	−1.01788	−0.0596691
$292$	−28.0593	−1.64204
$293$	11.0299	0.644374	0.322187	−	0.946676i	$-0.395582\pi$
$293$	11.0299	0.644374	0.322187	+	0.946676i	$0.395582\pi$
$294$	−0.126162	−0.00735792
$295$	0.512018	0.0298108
$296$	5.36656	0.311925
$297$	2.31628	0.134404
$298$	−20.1694	−1.16838
$299$	−7.61497	−0.440385
$300$	−0.732662	−0.0423003
$301$	−9.64737	−0.556065
$302$	18.1957	1.04705
$303$	1.02978	0.0591593
$304$	−6.39877	−0.366995
$305$	1.98669	0.113757
$306$	−21.3496	−1.22047
$307$	21.7046	1.23875	0.619373	−	0.785097i	$-0.287387\pi$
$307$	21.7046	1.23875	0.619373	+	0.785097i	$0.287387\pi$
$308$	−16.0930	−0.916983
$309$	−0.418757	−0.0238223
$310$	−2.20715	−0.125358
$311$	3.32174	0.188359	0.0941794	−	0.995555i	$-0.469977\pi$
$311$	3.32174	0.188359	0.0941794	+	0.995555i	$0.469977\pi$
$312$	0.107314	0.00607547
$313$	17.1896	0.971614	0.485807	−	0.874066i	$-0.338526\pi$
$313$	17.1896	0.971614	0.485807	+	0.874066i	$0.338526\pi$
$314$	36.9097	2.08294
$315$	0.652155	0.0367448
$316$	17.8082	1.00179
$317$	14.9223	0.838120	0.419060	−	0.907959i	$-0.362360\pi$
$317$	14.9223	0.838120	0.419060	+	0.907959i	$0.362360\pi$
$318$	−0.214612	−0.0120348
$319$	58.8005	3.29219
$320$	2.45555	0.137269
$321$	0.554472	0.0309476
$322$	−9.14689	−0.509736
$323$	−7.68518	−0.427615
$324$	22.2639	1.23688
$325$	8.72962	0.484232
$326$	−16.5219	−0.915061
$327$	0.445096	0.0246139
$328$	−10.6778	−0.589581
$329$	−1.63389	−0.0900791
$330$	−0.177994	−0.00979825
$331$	24.0942	1.32433	0.662167	−	0.749356i	$-0.269637\pi$
$331$	24.0942	1.32433	0.662167	+	0.749356i	$0.269637\pi$
$332$	8.52905	0.468092
$333$	15.7387	0.862477
$334$	9.24003	0.505592
$335$	1.10626	0.0604412
$336$	−0.166965	−0.00910870
$337$	−4.75104	−0.258805	−0.129403	−	0.991592i	$-0.541306\pi$
$337$	−4.75104	−0.258805	−0.129403	+	0.991592i	$0.541306\pi$
$338$	20.9459	1.13931
$339$	0.676777	0.0367575
$340$	1.81830	0.0986112
$341$	31.0497	1.68144
$342$	14.4879	0.783417
$343$	1.00000	0.0539949
$344$	9.85695	0.531451
$345$	−0.0560296	−0.00301653
$346$	4.51984	0.242988
$347$	21.5673	1.15779	0.578896	−	0.815401i	$-0.303484\pi$
$347$	21.5673	1.15779	0.578896	+	0.815401i	$0.303484\pi$
$348$	−1.34189	−0.0719326
$349$	1.14669	0.0613810	0.0306905	−	0.999529i	$-0.490229\pi$
$349$	1.14669	0.0613810	0.0306905	+	0.999529i	$0.490229\pi$
$350$	10.4858	0.560488
$351$	0.629822	0.0336174
$352$	−51.7019	−2.75572
$353$	−36.1542	−1.92430	−0.962148	−	0.272527i	$-0.912141\pi$
$353$	−36.1542	−1.92430	−0.962148	+	0.272527i	$0.912141\pi$
$354$	0.296804	0.0157750
$355$	1.39757	0.0741755
$356$	2.92802	0.155185
$357$	−0.200532	−0.0106133
$358$	13.2957	0.702698
$359$	7.30795	0.385699	0.192849	−	0.981228i	$-0.438227\pi$
$359$	7.30795	0.385699	0.192849	+	0.981228i	$0.438227\pi$
$360$	−0.666322	−0.0351182
$361$	−13.7848	−0.725516
$362$	−18.7014	−0.982922
$363$	1.84850	0.0970213
$364$	−4.37585	−0.229357
$365$	2.45990	0.128757
$366$	1.15163	0.0601969
$367$	8.26315	0.431333	0.215667	−	0.976467i	$-0.430808\pi$
$367$	8.26315	0.431333	0.215667	+	0.976467i	$0.430808\pi$
$368$	−12.1052	−0.631025
$369$	−31.3151	−1.63020
$370$	−2.42031	−0.125826
$371$	1.70108	0.0883157
$372$	−0.708585	−0.0367384
$373$	9.42895	0.488213	0.244106	−	0.969748i	$-0.421505\pi$
$373$	9.42895	0.488213	0.244106	+	0.969748i	$0.421505\pi$
$374$	−46.1866	−2.38825
$375$	0.129076	0.00666547
$376$	1.66938	0.0860917
$377$	15.9885	0.823448
$378$	0.756524	0.0389114
$379$	−29.3093	−1.50552	−0.752758	−	0.658298i	$-0.771277\pi$
$379$	−29.3093	−1.50552	−0.752758	+	0.658298i	$0.771277\pi$
$380$	−1.23391	−0.0632981
$381$	−0.284303	−0.0145653
$382$	−10.0525	−0.514333
$383$	−3.08427	−0.157599	−0.0787994	−	0.996890i	$-0.525109\pi$
$383$	−3.08427	−0.157599	−0.0787994	+	0.996890i	$0.525109\pi$
$384$	0.472887	0.0241319
$385$	1.41084	0.0719029
$386$	0.517212	0.0263254
$387$	28.9079	1.46947
$388$	−42.4067	−2.15288
$389$	9.10273	0.461527	0.230763	−	0.973010i	$-0.425878\pi$
$389$	9.10273	0.461527	0.230763	+	0.973010i	$0.425878\pi$
$390$	−0.0483984	−0.00245075
$391$	−14.5388	−0.735258
$392$	−1.02172	−0.0516048
$393$	−0.572009	−0.0288540
$394$	−24.2343	−1.22091
$395$	−1.56121	−0.0785528
$396$	48.2218	2.42324
$397$	26.1887	1.31437	0.657186	−	0.753728i	$-0.271747\pi$
$397$	26.1887	1.31437	0.657186	+	0.753728i	$0.271747\pi$
$398$	27.7757	1.39227
$399$	0.136082	0.00681261
$400$	13.8771	0.693853
$401$	−3.59029	−0.179291	−0.0896453	−	0.995974i	$-0.528573\pi$
$401$	−3.59029	−0.179291	−0.0896453	+	0.995974i	$0.528573\pi$
$402$	0.641270	0.0319836
$403$	8.44274	0.420563
$404$	42.9026	2.13448
$405$	−1.95183	−0.0969872
$406$	19.2049	0.953122
$407$	34.0484	1.68771
$408$	0.204888	0.0101435
$409$	17.9430	0.887226	0.443613	−	0.896218i	$-0.353696\pi$
$409$	17.9430	0.887226	0.443613	+	0.896218i	$0.353696\pi$
$410$	4.81565	0.237828
$411$	0.695165	0.0342900
$412$	−17.4462	−0.859513
$413$	−2.35256	−0.115762
$414$	27.4082	1.34704
$415$	−0.747723	−0.0367043
$416$	−14.0583	−0.689264
$417$	0.308952	0.0151294
$418$	31.3424	1.53301
$419$	−31.8687	−1.55689	−0.778443	−	0.627715i	$-0.783990\pi$
$419$	−31.8687	−1.55689	−0.778443	+	0.627715i	$0.783990\pi$
$420$	−0.0321967	−0.00157104
$421$	22.7319	1.10788	0.553942	−	0.832555i	$-0.313123\pi$
$421$	22.7319	1.10788	0.553942	+	0.832555i	$0.313123\pi$
$422$	−46.1781	−2.24792
$423$	4.89586	0.238045
$424$	−1.73803	−0.0844064
$425$	16.6669	0.808463
$426$	0.810140	0.0392514
$427$	−9.12822	−0.441746
$428$	23.1003	1.11660
$429$	0.680858	0.0328722
$430$	−4.44546	−0.214379
$431$	−20.6156	−0.993017	−0.496509	−	0.868032i	$-0.665385\pi$
$431$	−20.6156	−0.993017	−0.496509	+	0.868032i	$0.665385\pi$
$432$	1.00120	0.0481702
$433$	−28.3262	−1.36127	−0.680635	−	0.732623i	$-0.738296\pi$
$433$	−28.3262	−1.36127	−0.680635	+	0.732623i	$0.738296\pi$
$434$	10.1412	0.486792
$435$	0.117640	0.00564042
$436$	18.5435	0.888074
$437$	9.86608	0.471959
$438$	1.42594	0.0681341
$439$	8.67275	0.413928	0.206964	−	0.978349i	$-0.433642\pi$
$439$	8.67275	0.413928	0.206964	+	0.978349i	$0.433642\pi$
$440$	−1.44149	−0.0687201
$441$	−2.99645	−0.142688
$442$	−12.5586	−0.597352
$443$	10.7649	0.511456	0.255728	−	0.966749i	$-0.417685\pi$
$443$	10.7649	0.511456	0.255728	+	0.966749i	$0.417685\pi$
$444$	−0.777017	−0.0368756
$445$	−0.256693	−0.0121684
$446$	−5.57225	−0.263854
$447$	0.567667	0.0268497
$448$	−11.2825	−0.533047
$449$	−20.9528	−0.988823	−0.494411	−	0.869228i	$-0.664616\pi$
$449$	−20.9528	−0.988823	−0.494411	+	0.869228i	$0.664616\pi$
$450$	−31.4201	−1.48116
$451$	−67.7455	−3.19001
$452$	28.1958	1.32622
$453$	−0.512117	−0.0240614
$454$	41.4145	1.94368
$455$	0.383621	0.0179844
$456$	−0.139038	−0.00651105
$457$	−17.8034	−0.832807	−0.416403	−	0.909180i	$-0.636710\pi$
$457$	−17.8034	−0.832807	−0.416403	+	0.909180i	$0.636710\pi$
$458$	−6.16594	−0.288115
$459$	1.20248	0.0561269
$460$	−2.33430	−0.108837
$461$	24.5697	1.14432	0.572162	−	0.820141i	$-0.306105\pi$
$461$	24.5697	1.14432	0.572162	+	0.820141i	$0.306105\pi$
$462$	0.817828	0.0380488
$463$	−7.91931	−0.368042	−0.184021	−	0.982922i	$-0.558911\pi$
$463$	−7.91931	−0.368042	−0.184021	+	0.982922i	$0.558911\pi$
$464$	25.4161	1.17991
$465$	0.0621202	0.00288075
$466$	29.2448	1.35474
$467$	−13.4611	−0.622904	−0.311452	−	0.950262i	$-0.600815\pi$
$467$	−13.4611	−0.622904	−0.311452	+	0.950262i	$0.600815\pi$
$468$	13.1120	0.606103
$469$	−5.08291	−0.234707
$470$	−0.752887	−0.0347281
$471$	−1.03882	−0.0478664
$472$	2.40367	0.110638
$473$	62.5378	2.87549
$474$	−0.904993	−0.0415677
$475$	−11.3102	−0.518949
$476$	−8.35453	−0.382929
$477$	−5.09720	−0.233385
$478$	36.1314	1.65261
$479$	−15.8429	−0.723882	−0.361941	−	0.932201i	$-0.617886\pi$
$479$	−15.8429	−0.723882	−0.361941	+	0.932201i	$0.617886\pi$
$480$	−0.103438	−0.00472129
$481$	9.25810	0.422133
$482$	−17.9240	−0.816417
$483$	0.257439	0.0117139
$484$	77.0121	3.50055
$485$	3.71771	0.168812
$486$	−3.40100	−0.154272
$487$	10.5448	0.477830	0.238915	−	0.971040i	$-0.423208\pi$
$487$	10.5448	0.477830	0.238915	+	0.971040i	$0.423208\pi$
$488$	9.32652	0.422192
$489$	0.465007	0.0210283
$490$	0.460795	0.0208166
$491$	8.13835	0.367279	0.183639	−	0.982994i	$-0.441212\pi$
$491$	8.13835	0.367279	0.183639	+	0.982994i	$0.441212\pi$
$492$	1.54602	0.0696999
$493$	30.5257	1.37481
$494$	8.52233	0.383438
$495$	−4.22750	−0.190012
$496$	13.4210	0.602622
$497$	−6.42142	−0.288040
$498$	−0.433437	−0.0194228
$499$	36.8607	1.65011	0.825055	−	0.565052i	$-0.191144\pi$
$499$	36.8607	1.65011	0.825055	+	0.565052i	$0.191144\pi$
$500$	5.37756	0.240492
$501$	−0.260060	−0.0116186
$502$	−5.38008	−0.240125
$503$	37.6896	1.68050	0.840248	−	0.542202i	$-0.182409\pi$
$503$	37.6896	1.68050	0.840248	+	0.542202i	$0.182409\pi$
$504$	3.06154	0.136372
$505$	−3.76118	−0.167370
$506$	59.2934	2.63591
$507$	−0.589522	−0.0261816
$508$	−11.8446	−0.525519
$509$	−20.2812	−0.898947	−0.449474	−	0.893294i	$-0.648388\pi$
$509$	−20.2812	−0.898947	−0.449474	+	0.893294i	$0.648388\pi$
$510$	−0.0924040	−0.00409172
$511$	−11.3025	−0.499992
$512$	−28.0734	−1.24068
$513$	−0.816007	−0.0360276
$514$	−40.2581	−1.77571
$515$	1.52947	0.0673966
$516$	−1.42717	−0.0628278
$517$	10.5914	0.465811
$518$	11.1206	0.488610
$519$	−0.127211	−0.00558393
$520$	−0.391955	−0.0171884
$521$	−19.0089	−0.832797	−0.416398	−	0.909182i	$-0.636708\pi$
$521$	−19.0089	−0.832797	−0.416398	+	0.909182i	$0.636708\pi$
$522$	−57.5465	−2.51874
$523$	27.4550	1.20052	0.600262	−	0.799803i	$-0.295063\pi$
$523$	27.4550	1.20052	0.600262	+	0.799803i	$0.295063\pi$
$524$	−23.8309	−1.04106
$525$	−0.295121	−0.0128801
$526$	42.2694	1.84303
$527$	16.1192	0.702163
$528$	1.08233	0.0471023
$529$	−4.33541	−0.188496
$530$	0.783849	0.0340482
$531$	7.04934	0.305915
$532$	5.66942	0.245801
$533$	−18.4207	−0.797889
$534$	−0.148799	−0.00643915
$535$	−2.02516	−0.0875552
$536$	5.19333	0.224318
$537$	−0.374206	−0.0161482
$538$	21.4294	0.923886
$539$	−6.48236	−0.279215
$540$	0.193066	0.00830823
$541$	31.2465	1.34339	0.671696	−	0.740827i	$-0.265566\pi$
$541$	31.2465	1.34339	0.671696	+	0.740827i	$0.265566\pi$
$542$	68.2897	2.93330
$543$	0.526349	0.0225878
$544$	−26.8406	−1.15078
$545$	−1.62567	−0.0696361
$546$	0.222376	0.00951681
$547$	−1.74443	−0.0745863	−0.0372932	−	0.999304i	$-0.511874\pi$
$547$	−1.74443	−0.0745863	−0.0372932	+	0.999304i	$0.511874\pi$
$548$	28.9619	1.23719
$549$	27.3522	1.16737
$550$	−67.9725	−2.89836
$551$	−20.7149	−0.882485
$552$	−0.263031	−0.0111953
$553$	7.17326	0.305038
$554$	35.1902	1.49509
$555$	0.0681194	0.00289151
$556$	12.8715	0.545874
$557$	−9.64378	−0.408620	−0.204310	−	0.978906i	$-0.565495\pi$
$557$	−9.64378	−0.408620	−0.204310	+	0.978906i	$0.565495\pi$
$558$	−30.3875	−1.28641
$559$	17.0047	0.719221
$560$	0.609825	0.0257698
$561$	1.29992	0.0548826
$562$	5.52065	0.232875
$563$	1.40870	0.0593698	0.0296849	−	0.999559i	$-0.490550\pi$
$563$	1.40870	0.0593698	0.0296849	+	0.999559i	$0.490550\pi$
$564$	−0.241707	−0.0101777
$565$	−2.47186	−0.103992
$566$	5.25319	0.220808
$567$	8.96806	0.376623
$568$	6.56092	0.275290
$569$	20.2389	0.848458	0.424229	−	0.905555i	$-0.360545\pi$
$569$	20.2389	0.848458	0.424229	+	0.905555i	$0.360545\pi$
$570$	0.0627058	0.00262646
$571$	4.64069	0.194207	0.0971033	−	0.995274i	$-0.469042\pi$
$571$	4.64069	0.194207	0.0971033	+	0.995274i	$0.469042\pi$
$572$	28.3658	1.18604
$573$	0.282928	0.0118195
$574$	−22.1264	−0.923539
$575$	−21.3966	−0.892301
$576$	33.8074	1.40864
$577$	−29.3050	−1.21998	−0.609991	−	0.792409i	$-0.708827\pi$
$577$	−29.3050	−1.21998	−0.609991	+	0.792409i	$0.708827\pi$
$578$	12.0152	0.499767
$579$	−0.0145569	−0.000604965 0
$580$	4.90111	0.203508
$581$	3.43556	0.142531
$582$	2.15506	0.0893302
$583$	−11.0270	−0.456692
$584$	11.5480	0.477859
$585$	−1.14950	−0.0475260
$586$	−23.3526	−0.964689
$587$	40.8916	1.68778	0.843888	−	0.536520i	$-0.180261\pi$
$587$	40.8916	1.68778	0.843888	+	0.536520i	$0.180261\pi$
$588$	0.147934	0.00610069
$589$	−10.9385	−0.450715
$590$	−1.08405	−0.0446296
$591$	0.682073	0.0280567
$592$	14.7172	0.604872
$593$	10.2466	0.420776	0.210388	−	0.977618i	$-0.432527\pi$
$593$	10.2466	0.420776	0.210388	+	0.977618i	$0.432527\pi$
$594$	−4.90406	−0.201216
$595$	0.732424	0.0300264
$596$	23.6501	0.968745
$597$	−0.781746	−0.0319947
$598$	16.1225	0.659298
$599$	−0.338809	−0.0138433	−0.00692167	−	0.999976i	$-0.502203\pi$
$599$	−0.338809	−0.0138433	−0.00692167	+	0.999976i	$0.502203\pi$
$600$	0.301532	0.0123100
$601$	−43.9287	−1.79189	−0.895944	−	0.444166i	$-0.853500\pi$
$601$	−43.9287	−1.79189	−0.895944	+	0.444166i	$0.853500\pi$
$602$	20.4255	0.832482
$603$	15.2307	0.620241
$604$	−21.3358	−0.868140
$605$	−6.75149	−0.274487
$606$	−2.18026	−0.0885671
$607$	39.2534	1.59325	0.796623	−	0.604476i	$-0.206617\pi$
$607$	39.2534	1.59325	0.796623	+	0.604476i	$0.206617\pi$
$608$	18.2141	0.738681
$609$	−0.540520	−0.0219030
$610$	−4.20624	−0.170306
$611$	2.87992	0.116509
$612$	25.0339	1.01194
$613$	−12.7386	−0.514506	−0.257253	−	0.966344i	$-0.582817\pi$
$613$	−12.7386	−0.514506	−0.257253	+	0.966344i	$0.582817\pi$
$614$	−45.9532	−1.85452
$615$	−0.135536	−0.00546535
$616$	6.62318	0.266856
$617$	−27.1520	−1.09310	−0.546549	−	0.837427i	$-0.684059\pi$
$617$	−27.1520	−1.09310	−0.546549	+	0.837427i	$0.684059\pi$
$618$	0.886597	0.0356642
$619$	−34.9457	−1.40459	−0.702293	−	0.711888i	$-0.747840\pi$
$619$	−34.9457	−1.40459	−0.702293	+	0.711888i	$0.747840\pi$
$620$	2.58804	0.103938
$621$	−1.54372	−0.0619473
$622$	−7.03283	−0.281991
$623$	1.17943	0.0472527
$624$	0.294296	0.0117813
$625$	24.2917	0.971669
$626$	−36.3940	−1.45460
$627$	−0.882131	−0.0352289
$628$	−43.2793	−1.72703
$629$	17.6759	0.704784
$630$	−1.38075	−0.0550103
$631$	15.5446	0.618821	0.309411	−	0.950929i	$-0.399868\pi$
$631$	15.5446	0.618821	0.309411	+	0.950929i	$0.399868\pi$
$632$	−7.32909	−0.291536
$633$	1.29968	0.0516577
$634$	−31.5936	−1.25474
$635$	1.03839	0.0412072
$636$	0.251648	0.00997847
$637$	−1.76262	−0.0698376
$638$	−124.493	−4.92872
$639$	19.2415	0.761181
$640$	−1.72718	−0.0682727
$641$	−0.426687	−0.0168531	−0.00842657	−	0.999964i	$-0.502682\pi$
$641$	−0.426687	−0.0168531	−0.00842657	+	0.999964i	$0.502682\pi$
$642$	−1.17393	−0.0463315
$643$	7.50108	0.295814	0.147907	−	0.989001i	$-0.452746\pi$
$643$	7.50108	0.295814	0.147907	+	0.989001i	$0.452746\pi$
$644$	10.7254	0.422639
$645$	0.125117	0.00492649
$646$	16.2711	0.640179
$647$	−10.4776	−0.411915	−0.205958	−	0.978561i	$-0.566031\pi$
$647$	−10.4776	−0.411915	−0.205958	+	0.978561i	$0.566031\pi$
$648$	−9.16287	−0.359952
$649$	15.2502	0.598622
$650$	−18.4824	−0.724941
$651$	−0.285423	−0.0111866
$652$	19.3731	0.758708
$653$	25.6587	1.00410	0.502050	−	0.864838i	$-0.332579\pi$
$653$	25.6587	1.00410	0.502050	+	0.864838i	$0.332579\pi$
$654$	−0.942361	−0.0368492
$655$	2.08921	0.0816321
$656$	−29.2825	−1.14329
$657$	33.8673	1.32129
$658$	3.45928	0.134857
$659$	24.6107	0.958696	0.479348	−	0.877625i	$-0.340873\pi$
$659$	24.6107	0.958696	0.479348	+	0.877625i	$0.340873\pi$
$660$	0.208711	0.00812405
$661$	−7.59168	−0.295282	−0.147641	−	0.989041i	$-0.547168\pi$
$661$	−7.59168	−0.295282	−0.147641	+	0.989041i	$0.547168\pi$
$662$	−51.0124	−1.98265
$663$	0.353462	0.0137273
$664$	−3.51019	−0.136222
$665$	−0.497026	−0.0192738
$666$	−33.3222	−1.29121
$667$	−39.1883	−1.51738
$668$	−10.8346	−0.419203
$669$	0.156831	0.00606343
$670$	−2.34218	−0.0904862
$671$	59.1724	2.28433
$672$	0.475267	0.0183338
$673$	−0.166041	−0.00640042	−0.00320021	−	0.999995i	$-0.501019\pi$
$673$	−0.166041	−0.00640042	−0.00320021	+	0.999995i	$0.501019\pi$
$674$	10.0589	0.387456
$675$	1.76968	0.0681150
$676$	−24.5606	−0.944638
$677$	26.7521	1.02817	0.514084	−	0.857740i	$-0.328132\pi$
$677$	26.7521	1.02817	0.514084	+	0.857740i	$0.328132\pi$
$678$	−1.43288	−0.0550294
$679$	−17.0817	−0.655536
$680$	−0.748334	−0.0286973
$681$	−1.16561	−0.0446662
$682$	−65.7388	−2.51727
$683$	−4.78958	−0.183268	−0.0916340	−	0.995793i	$-0.529209\pi$
$683$	−4.78958	−0.183268	−0.0916340	+	0.995793i	$0.529209\pi$
$684$	−16.9881	−0.649557
$685$	−2.53903	−0.0970112
$686$	−2.11721	−0.0808355
$687$	0.173540	0.00662097
$688$	27.0315	1.03057
$689$	−2.99836	−0.114228
$690$	0.118626	0.00451603
$691$	24.7193	0.940365	0.470182	−	0.882569i	$-0.344188\pi$
$691$	24.7193	0.940365	0.470182	+	0.882569i	$0.344188\pi$
$692$	−5.29983	−0.201469
$693$	19.4241	0.737859
$694$	−45.6625	−1.73332
$695$	−1.12842	−0.0428033
$696$	0.552262	0.0209335
$697$	−35.1695	−1.33214
$698$	−2.42779	−0.0918931
$699$	−0.823092	−0.0311322
$700$	−12.2953	−0.464719
$701$	−21.8019	−0.823447	−0.411723	−	0.911309i	$-0.635073\pi$
$701$	−21.8019	−0.823447	−0.411723	+	0.911309i	$0.635073\pi$
$702$	−1.33347	−0.0503284
$703$	−11.9949	−0.452398
$704$	73.1372	2.75646
$705$	0.0211900	0.000798060 0
$706$	76.5461	2.88085
$707$	17.2815	0.649936
$708$	−0.348024	−0.0130795
$709$	8.49774	0.319139	0.159570	−	0.987187i	$-0.448989\pi$
$709$	8.49774	0.319139	0.159570	+	0.987187i	$0.448989\pi$
$710$	−2.95896	−0.111048
$711$	−21.4943	−0.806100
$712$	−1.20505	−0.0451610
$713$	−20.6935	−0.774977
$714$	0.424568	0.0158891
$715$	−2.48677	−0.0930000
$716$	−15.5901	−0.582630
$717$	−1.01692	−0.0379774
$718$	−15.4725	−0.577428
$719$	14.6774	0.547374	0.273687	−	0.961819i	$-0.411757\pi$
$719$	14.6774	0.547374	0.273687	+	0.961819i	$0.411757\pi$
$720$	−1.82731	−0.0680998
$721$	−7.02745	−0.261716
$722$	29.1853	1.08617
$723$	0.504471	0.0187615
$724$	21.9287	0.814973
$725$	44.9245	1.66846
$726$	−3.91367	−0.145250
$727$	−6.62906	−0.245858	−0.122929	−	0.992415i	$-0.539229\pi$
$727$	−6.62906	−0.245858	−0.122929	+	0.992415i	$0.539229\pi$
$728$	1.80091	0.0667462
$729$	−26.8084	−0.992905
$730$	−5.20812	−0.192761
$731$	32.4659	1.20080
$732$	−1.35037	−0.0499112
$733$	9.10716	0.336381	0.168190	−	0.985755i	$-0.446208\pi$
$733$	9.10716	0.336381	0.168190	+	0.985755i	$0.446208\pi$
$734$	−17.4948	−0.645746
$735$	−0.0129691	−0.000478371 0
$736$	34.4574	1.27012
$737$	32.9492	1.21370
$738$	66.3007	2.44056
$739$	11.0180	0.405305	0.202653	−	0.979251i	$-0.435044\pi$
$739$	11.0180	0.405305	0.202653	+	0.979251i	$0.435044\pi$
$740$	2.83798	0.104326
$741$	−0.239861	−0.00881150
$742$	−3.60155	−0.132217
$743$	−14.9881	−0.549861	−0.274930	−	0.961464i	$-0.588655\pi$
$743$	−14.9881	−0.549861	−0.274930	+	0.961464i	$0.588655\pi$
$744$	0.291623	0.0106914
$745$	−2.07335	−0.0759617
$746$	−19.9631	−0.730900
$747$	−10.2945	−0.376655
$748$	54.1571	1.98018
$749$	9.30497	0.339996
$750$	−0.273281	−0.00997883
$751$	−21.4583	−0.783026	−0.391513	−	0.920173i	$-0.628048\pi$
$751$	−21.4583	−0.783026	−0.391513	+	0.920173i	$0.628048\pi$
$752$	4.57808	0.166945
$753$	0.151422	0.00551812
$754$	−33.8509	−1.23278
$755$	1.87046	0.0680731
$756$	−0.887078	−0.0322627
$757$	40.9067	1.48678	0.743389	−	0.668859i	$-0.233217\pi$
$757$	40.9067	1.48678	0.743389	+	0.668859i	$0.233217\pi$
$758$	62.0539	2.25390
$759$	−1.66881	−0.0605740
$760$	0.507823	0.0184207
$761$	48.7549	1.76736	0.883682	−	0.468087i	$-0.155057\pi$
$761$	48.7549	1.76736	0.883682	+	0.468087i	$0.155057\pi$
$762$	0.601929	0.0218056
$763$	7.46946	0.270412
$764$	11.7873	0.426450
$765$	−2.19467	−0.0793485
$766$	6.53005	0.235940
$767$	4.14668	0.149728
$768$	0.343417	0.0123920
$769$	−35.2238	−1.27020	−0.635101	−	0.772429i	$-0.719041\pi$
$769$	−35.2238	−1.27020	−0.635101	+	0.772429i	$0.719041\pi$
$770$	−2.98704	−0.107645
$771$	1.13306	0.0408063
$772$	−0.606468	−0.0218273
$773$	−38.6719	−1.39093	−0.695465	−	0.718560i	$-0.744802\pi$
$773$	−38.6719	−1.39093	−0.695465	+	0.718560i	$0.744802\pi$
$774$	−61.2040	−2.19993
$775$	23.7225	0.852137
$776$	17.4528	0.626519
$777$	−0.312988	−0.0112284
$778$	−19.2724	−0.690949
$779$	23.8662	0.855094
$780$	0.0567506	0.00203200
$781$	41.6260	1.48949
$782$	30.7816	1.10075
$783$	3.24120	0.115831
$784$	−2.80196	−0.100070
$785$	3.79420	0.135421
$786$	1.21106	0.0431972
$787$	43.6273	1.55515	0.777573	−	0.628793i	$-0.216451\pi$
$787$	43.6273	1.55515	0.777573	+	0.628793i	$0.216451\pi$
$788$	28.4165	1.01229
$789$	−1.18967	−0.0423534
$790$	3.30540	0.117601
$791$	11.3575	0.403825
$792$	−19.8460	−0.705198
$793$	16.0896	0.571358
$794$	−55.4470	−1.96774
$795$	−0.0220614	−0.000782437 0
$796$	−32.5690	−1.15438
$797$	−0.673246	−0.0238476	−0.0119238	−	0.999929i	$-0.503796\pi$
$797$	−0.673246	−0.0238476	−0.0119238	+	0.999929i	$0.503796\pi$
$798$	−0.288114	−0.0101991
$799$	5.49845	0.194521
$800$	−39.5011	−1.39657
$801$	−3.53409	−0.124871
$802$	7.60140	0.268415
$803$	73.2667	2.58552
$804$	−0.751934	−0.0265187
$805$	−0.940271	−0.0331402
$806$	−17.8751	−0.629622
$807$	−0.603128	−0.0212311
$808$	−17.6569	−0.621166
$809$	36.8340	1.29502	0.647508	−	0.762059i	$-0.275811\pi$
$809$	36.8340	1.29502	0.647508	+	0.762059i	$0.275811\pi$
$810$	4.13243	0.145199
$811$	−1.64876	−0.0578957	−0.0289479	−	0.999581i	$-0.509216\pi$
$811$	−1.64876	−0.0578957	−0.0289479	+	0.999581i	$0.509216\pi$
$812$	−22.5191	−0.790265
$813$	−1.92201	−0.0674079
$814$	−72.0875	−2.52667
$815$	−1.69839	−0.0594922
$816$	0.561881	0.0196698
$817$	−22.0315	−0.770785
$818$	−37.9892	−1.32826
$819$	5.28161	0.184554
$820$	−5.64669	−0.197191
$821$	−0.143069	−0.00499314	−0.00249657	−	0.999997i	$-0.500795\pi$
$821$	−0.143069	−0.00499314	−0.00249657	+	0.999997i	$0.500795\pi$
$822$	−1.47181	−0.0513353
$823$	2.19352	0.0764613	0.0382307	−	0.999269i	$-0.487828\pi$
$823$	2.19352	0.0764613	0.0382307	+	0.999269i	$0.487828\pi$
$824$	7.18011	0.250131
$825$	1.91308	0.0666050
$826$	4.98087	0.173307
$827$	−20.7132	−0.720269	−0.360134	−	0.932900i	$-0.617269\pi$
$827$	−20.7132	−0.720269	−0.360134	+	0.932900i	$0.617269\pi$
$828$	−32.1380	−1.11687
$829$	41.2445	1.43248	0.716240	−	0.697854i	$-0.245862\pi$
$829$	41.2445	1.43248	0.716240	+	0.697854i	$0.245862\pi$
$830$	1.58309	0.0549498
$831$	−0.990426	−0.0343575
$832$	19.8868	0.689449
$833$	−3.36526	−0.116599
$834$	−0.654116	−0.0226502
$835$	0.949846	0.0328708
$836$	−36.7512	−1.27107
$837$	1.71152	0.0591589
$838$	67.4727	2.33080
$839$	2.79736	0.0965755	0.0482877	−	0.998833i	$-0.484624\pi$
$839$	2.79736	0.0965755	0.0482877	+	0.998833i	$0.484624\pi$
$840$	0.0132508	0.000457195 0
$841$	53.2802	1.83725
$842$	−48.1281	−1.65861
$843$	−0.155378	−0.00535152
$844$	54.1471	1.86382
$845$	2.15317	0.0740714
$846$	−10.3656	−0.356375
$847$	31.0210	1.06589
$848$	−4.76635	−0.163677
$849$	−0.147851	−0.00507422
$850$	−35.2873	−1.21035
$851$	−22.6920	−0.777871
$852$	−0.949946	−0.0325446
$853$	52.7964	1.80771	0.903857	−	0.427835i	$-0.140723\pi$
$853$	52.7964	1.80771	0.903857	+	0.427835i	$0.140723\pi$
$854$	19.3264	0.661335
$855$	1.48931	0.0509334
$856$	−9.50711	−0.324946
$857$	44.0863	1.50596	0.752980	−	0.658044i	$-0.228616\pi$
$857$	44.0863	1.50596	0.752980	+	0.658044i	$0.228616\pi$
$858$	−1.44152	−0.0492127
$859$	−1.00000	−0.0341196
$859$	−1.00000	−0.0341196
$860$	5.21262	0.177749
$861$	0.622747	0.0212232
$862$	43.6475	1.48664
$863$	5.54964	0.188912	0.0944560	−	0.995529i	$-0.469889\pi$
$863$	5.54964	0.188912	0.0944560	+	0.995529i	$0.469889\pi$
$864$	−2.84991	−0.0969560
$865$	0.464625	0.0157977
$866$	59.9725	2.03795
$867$	−0.338168	−0.0114848
$868$	−11.8913	−0.403615
$869$	−46.4997	−1.57739
$870$	−0.249069	−0.00844423
$871$	8.95924	0.303572
$872$	−7.63172	−0.258443
$873$	51.1845	1.73233
$874$	−20.8886	−0.706566
$875$	2.16612	0.0732281
$876$	−1.67202	−0.0564923
$877$	−33.9048	−1.14488	−0.572441	−	0.819946i	$-0.694004\pi$
$877$	−33.9048	−1.14488	−0.572441	+	0.819946i	$0.694004\pi$
$878$	−18.3620	−0.619688
$879$	0.657259	0.0221688
$880$	−3.95310	−0.133259
$881$	45.7117	1.54006	0.770032	−	0.638005i	$-0.220240\pi$
$881$	45.7117	1.54006	0.770032	+	0.638005i	$0.220240\pi$
$882$	6.34411	0.213617
$883$	36.8708	1.24080	0.620401	−	0.784285i	$-0.286970\pi$
$883$	36.8708	1.24080	0.620401	+	0.784285i	$0.286970\pi$
$884$	14.7259	0.495285
$885$	0.0305105	0.00102560
$886$	−22.7916	−0.765698
$887$	−36.9820	−1.24173	−0.620867	−	0.783916i	$-0.713219\pi$
$887$	−36.9820	−1.24173	−0.620867	+	0.783916i	$0.713219\pi$
$888$	0.319787	0.0107313
$889$	−4.77108	−0.160017
$890$	0.543473	0.0182173
$891$	−58.1342	−1.94757
$892$	6.53386	0.218770
$893$	−3.73128	−0.124862
$894$	−1.20187	−0.0401966
$895$	1.36675	0.0456855
$896$	7.93585	0.265118
$897$	−0.453767	−0.0151508
$898$	44.3614	1.48036
$899$	43.4482	1.44908
$900$	36.8423	1.22808
$901$	−5.72458	−0.190713
$902$	143.431	4.77574
$903$	−0.574875	−0.0191307
$904$	−11.6042	−0.385949
$905$	−1.92244	−0.0639041
$906$	1.08426	0.0360221
$907$	−20.4566	−0.679251	−0.339626	−	0.940561i	$-0.610300\pi$
$907$	−20.4566	−0.679251	−0.339626	+	0.940561i	$0.610300\pi$
$908$	−48.5615	−1.61157
$909$	−51.7830	−1.71753
$910$	−0.812207	−0.0269244
$911$	−26.4667	−0.876882	−0.438441	−	0.898760i	$-0.644469\pi$
$911$	−26.4667	−0.876882	−0.438441	+	0.898760i	$0.644469\pi$
$912$	−0.381295	−0.0126259
$913$	−22.2705	−0.737047
$914$	37.6935	1.24679
$915$	0.118384	0.00391367
$916$	7.23000	0.238886
$917$	−9.59927	−0.316996
$918$	−2.54590	−0.0840272
$919$	19.8580	0.655054	0.327527	−	0.944842i	$-0.393785\pi$
$919$	19.8580	0.655054	0.327527	+	0.944842i	$0.393785\pi$
$920$	0.960697	0.0316732
$921$	1.29335	0.0426173
$922$	−52.0191	−1.71316
$923$	11.3185	0.372554
$924$	−0.958961	−0.0315475
$925$	26.0135	0.855319
$926$	16.7668	0.550993
$927$	21.0574	0.691616
$928$	−72.3470	−2.37491
$929$	−22.6708	−0.743805	−0.371902	−	0.928272i	$-0.621294\pi$
$929$	−22.6708	−0.743805	−0.371902	+	0.928272i	$0.621294\pi$
$930$	−0.131521	−0.00431276
$931$	2.28368	0.0748446
$932$	−34.2916	−1.12326
$933$	0.197939	0.00648022
$934$	28.4999	0.932545
$935$	−4.74783	−0.155271
$936$	−5.39634	−0.176385
$937$	−22.6353	−0.739462	−0.369731	−	0.929139i	$-0.620550\pi$
$937$	−22.6353	−0.739462	−0.369731	+	0.929139i	$0.620550\pi$
$938$	10.7616	0.351378
$939$	1.02431	0.0334270
$940$	0.882814	0.0287942
$941$	−48.7597	−1.58952	−0.794760	−	0.606923i	$-0.792403\pi$
$941$	−48.7597	−1.58952	−0.794760	+	0.606923i	$0.792403\pi$
$942$	2.19941	0.0716605
$943$	45.1498	1.47028
$944$	6.59178	0.214544
$945$	0.0777683	0.00252980
$946$	−132.406	−4.30488
$947$	4.63379	0.150578	0.0752890	−	0.997162i	$-0.476012\pi$
$947$	4.63379	0.150578	0.0752890	+	0.997162i	$0.476012\pi$
$948$	1.06117	0.0344652
$949$	19.9220	0.646694
$950$	23.9461	0.776915
$951$	0.889202	0.0288343
$952$	3.43837	0.111438
$953$	−39.9812	−1.29512	−0.647559	−	0.762016i	$-0.724210\pi$
$953$	−39.9812	−1.29512	−0.647559	+	0.762016i	$0.724210\pi$
$954$	10.7918	0.349399
$955$	−1.03337	−0.0334390
$956$	−42.3666	−1.37023
$957$	3.50385	0.113263
$958$	33.5428	1.08372
$959$	11.6660	0.376716
$960$	0.146323	0.00472256
$961$	−8.05708	−0.259906
$962$	−19.6013	−0.631973
$963$	−27.8819	−0.898481
$964$	21.0172	0.676918
$965$	0.0531678	0.00171153
$966$	−0.545052	−0.0175368
$967$	7.80533	0.251002	0.125501	−	0.992093i	$-0.459946\pi$
$967$	7.80533	0.251002	0.125501	+	0.992093i	$0.459946\pi$
$968$	−31.6949	−1.01871
$969$	−0.457950	−0.0147115
$970$	−7.87117	−0.252728
$971$	−3.13358	−0.100562	−0.0502808	−	0.998735i	$-0.516012\pi$
$971$	−3.13358	−0.100562	−0.0502808	+	0.998735i	$0.516012\pi$
$972$	3.98791	0.127912
$973$	5.18473	0.166215
$974$	−22.3255	−0.715356
$975$	0.520187	0.0166593
$976$	25.5769	0.818696
$977$	−8.36532	−0.267630	−0.133815	−	0.991006i	$-0.542723\pi$
$977$	−8.36532	−0.267630	−0.133815	+	0.991006i	$0.542723\pi$
$978$	−0.984518	−0.0314814
$979$	−7.64546	−0.244350
$980$	−0.540315	−0.0172597
$981$	−22.3818	−0.714597
$982$	−17.2306	−0.549851
$983$	33.8004	1.07806	0.539032	−	0.842285i	$-0.318790\pi$
$983$	33.8004	1.07806	0.539032	+	0.842285i	$0.318790\pi$
$984$	−0.636275	−0.0202837
$985$	−2.49121	−0.0793765
$986$	−64.6294	−2.05822
$987$	−0.0973613	−0.00309905
$988$	−9.99304	−0.317921
$989$	−41.6791	−1.32532
$990$	8.95051	0.284466
$991$	12.2818	0.390144	0.195072	−	0.980789i	$-0.437506\pi$
$991$	12.2818	0.390144	0.195072	+	0.980789i	$0.437506\pi$
$992$	−38.2030	−1.21295
$993$	1.43574	0.0455619
$994$	13.5955	0.431223
$995$	2.85526	0.0905177
$996$	0.508235	0.0161041
$997$	−26.4235	−0.836840	−0.418420	−	0.908254i	$-0.637416\pi$
$997$	−26.4235	−0.836840	−0.418420	+	0.908254i	$0.637416\pi$
$998$	−78.0418	−2.47037
$999$	1.87682	0.0593798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6013.2.a.e.1.14	✓	109

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6013.2.a.e.1.14	✓	109	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:	\( N \)	\(=\)	\( 6013 = 7 \cdot 859 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6013.a (trivial)

\(f(q)\)	\(=\)	\(q-2.11721 q^{2} +0.0595888 q^{3} +2.48258 q^{4} -0.217642 q^{5} -0.126162 q^{6} +1.00000 q^{7} -1.02172 q^{8} -2.99645 q^{9} +O(q^{10})\)	\(q-2.11721 q^{2} +0.0595888 q^{3} +2.48258 q^{4} -0.217642 q^{5} -0.126162 q^{6} +1.00000 q^{7} -1.02172 q^{8} -2.99645 q^{9} +0.460795 q^{10} -6.48236 q^{11} +0.147934 q^{12} -1.76262 q^{13} -2.11721 q^{14} -0.0129691 q^{15} -2.80196 q^{16} -3.36526 q^{17} +6.34411 q^{18} +2.28368 q^{19} -0.540315 q^{20} +0.0595888 q^{21} +13.7245 q^{22} +4.32025 q^{23} -0.0608833 q^{24} -4.95263 q^{25} +3.73184 q^{26} -0.357321 q^{27} +2.48258 q^{28} -9.07084 q^{29} +0.0274582 q^{30} -4.78988 q^{31} +7.97578 q^{32} -0.386276 q^{33} +7.12496 q^{34} -0.217642 q^{35} -7.43892 q^{36} -5.25246 q^{37} -4.83503 q^{38} -0.105032 q^{39} +0.222370 q^{40} +10.4507 q^{41} -0.126162 q^{42} -9.64737 q^{43} -16.0930 q^{44} +0.652155 q^{45} -9.14689 q^{46} -1.63389 q^{47} -0.166965 q^{48} +1.00000 q^{49} +10.4858 q^{50} -0.200532 q^{51} -4.37585 q^{52} +1.70108 q^{53} +0.756524 q^{54} +1.41084 q^{55} -1.02172 q^{56} +0.136082 q^{57} +19.2049 q^{58} -2.35256 q^{59} -0.0321967 q^{60} -9.12822 q^{61} +10.1412 q^{62} -2.99645 q^{63} -11.2825 q^{64} +0.383621 q^{65} +0.817828 q^{66} -5.08291 q^{67} -8.35453 q^{68} +0.257439 q^{69} +0.460795 q^{70} -6.42142 q^{71} +3.06154 q^{72} -11.3025 q^{73} +11.1206 q^{74} -0.295121 q^{75} +5.66942 q^{76} -6.48236 q^{77} +0.222376 q^{78} +7.17326 q^{79} +0.609825 q^{80} +8.96806 q^{81} -22.1264 q^{82} +3.43556 q^{83} +0.147934 q^{84} +0.732424 q^{85} +20.4255 q^{86} -0.540520 q^{87} +6.62318 q^{88} +1.17943 q^{89} -1.38075 q^{90} -1.76262 q^{91} +10.7254 q^{92} -0.285423 q^{93} +3.45928 q^{94} -0.497026 q^{95} +0.475267 q^{96} -17.0817 q^{97} -2.11721 q^{98} +19.4241 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 109 q + 19 q^{2} + 38 q^{3} + 111 q^{4} + 43 q^{5} + 14 q^{6} + 109 q^{7} + 48 q^{8} + 119 q^{9}+O(q^{10}) \) 109 * q + 19 * q^2 + 38 * q^3 + 111 * q^4 + 43 * q^5 + 14 * q^6 + 109 * q^7 + 48 * q^8 + 119 * q^9	\( 109 q + 19 q^{2} + 38 q^{3} + 111 q^{4} + 43 q^{5} + 14 q^{6} + 109 q^{7} + 48 q^{8} + 119 q^{9} + 15 q^{10} + 48 q^{11} + 72 q^{12} + 29 q^{13} + 19 q^{14} + 29 q^{15} + 115 q^{16} + 72 q^{17} + 33 q^{18} + 58 q^{19} + 88 q^{20} + 38 q^{21} + 4 q^{22} + 65 q^{23} + 46 q^{24} + 124 q^{25} + 49 q^{26} + 131 q^{27} + 111 q^{28} + 25 q^{29} + 2 q^{30} + 41 q^{31} + 75 q^{32} + 54 q^{33} + 23 q^{34} + 43 q^{35} + 111 q^{36} + 25 q^{37} + 54 q^{38} + 27 q^{39} + 30 q^{40} + 109 q^{41} + 14 q^{42} + 38 q^{43} + 68 q^{44} + 84 q^{45} - 9 q^{46} + 121 q^{47} + 106 q^{48} + 109 q^{49} + 14 q^{50} + 36 q^{51} + 38 q^{52} + 61 q^{53} + 31 q^{54} + 50 q^{55} + 48 q^{56} + 5 q^{57} - 20 q^{58} + 181 q^{59} + 25 q^{60} + 34 q^{61} + 75 q^{62} + 119 q^{63} + 96 q^{64} + 12 q^{65} + 19 q^{66} + 87 q^{67} + 150 q^{68} + 89 q^{69} + 15 q^{70} + 83 q^{71} + 65 q^{72} + 32 q^{73} - 19 q^{74} + 112 q^{75} + 84 q^{76} + 48 q^{77} - 34 q^{78} - 9 q^{79} + 137 q^{80} + 109 q^{81} - 19 q^{82} + 136 q^{83} + 72 q^{84} - 32 q^{85} - 24 q^{86} + 28 q^{87} - 24 q^{88} + 142 q^{89} + 19 q^{90} + 29 q^{91} + 96 q^{92} + 29 q^{93} + 9 q^{94} + 52 q^{95} + 88 q^{96} + 75 q^{97} + 19 q^{98} + 84 q^{99}+O(q^{100}) \) 109 * q + 19 * q^2 + 38 * q^3 + 111 * q^4 + 43 * q^5 + 14 * q^6 + 109 * q^7 + 48 * q^8 + 119 * q^9 + 15 * q^10 + 48 * q^11 + 72 * q^12 + 29 * q^13 + 19 * q^14 + 29 * q^15 + 115 * q^16 + 72 * q^17 + 33 * q^18 + 58 * q^19 + 88 * q^20 + 38 * q^21 + 4 * q^22 + 65 * q^23 + 46 * q^24 + 124 * q^25 + 49 * q^26 + 131 * q^27 + 111 * q^28 + 25 * q^29 + 2 * q^30 + 41 * q^31 + 75 * q^32 + 54 * q^33 + 23 * q^34 + 43 * q^35 + 111 * q^36 + 25 * q^37 + 54 * q^38 + 27 * q^39 + 30 * q^40 + 109 * q^41 + 14 * q^42 + 38 * q^43 + 68 * q^44 + 84 * q^45 - 9 * q^46 + 121 * q^47 + 106 * q^48 + 109 * q^49 + 14 * q^50 + 36 * q^51 + 38 * q^52 + 61 * q^53 + 31 * q^54 + 50 * q^55 + 48 * q^56 + 5 * q^57 - 20 * q^58 + 181 * q^59 + 25 * q^60 + 34 * q^61 + 75 * q^62 + 119 * q^63 + 96 * q^64 + 12 * q^65 + 19 * q^66 + 87 * q^67 + 150 * q^68 + 89 * q^69 + 15 * q^70 + 83 * q^71 + 65 * q^72 + 32 * q^73 - 19 * q^74 + 112 * q^75 + 84 * q^76 + 48 * q^77 - 34 * q^78 - 9 * q^79 + 137 * q^80 + 109 * q^81 - 19 * q^82 + 136 * q^83 + 72 * q^84 - 32 * q^85 - 24 * q^86 + 28 * q^87 - 24 * q^88 + 142 * q^89 + 19 * q^90 + 29 * q^91 + 96 * q^92 + 29 * q^93 + 9 * q^94 + 52 * q^95 + 88 * q^96 + 75 * q^97 + 19 * q^98 + 84 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more