LMFDB - Embedded newform 6046.2.a.e.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6046,2,Mod(1,6046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6046, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6046 = 2 \cdot 3023 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6046.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:	$48.2775530621$
Analytic rank:	$1$
Dimension:	$56$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	6046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.47034 q^{3} +1.00000 q^{4} +2.17542 q^{5} -1.47034 q^{6} -0.538935 q^{7} +1.00000 q^{8} -0.838111 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.47034 q^{3} +1.00000 q^{4} +2.17542 q^{5} -1.47034 q^{6} -0.538935 q^{7} +1.00000 q^{8} -0.838111 q^{9} +2.17542 q^{10} +3.74421 q^{11} -1.47034 q^{12} -5.29584 q^{13} -0.538935 q^{14} -3.19860 q^{15} +1.00000 q^{16} -1.99257 q^{17} -0.838111 q^{18} -2.32942 q^{19} +2.17542 q^{20} +0.792415 q^{21} +3.74421 q^{22} +1.97841 q^{23} -1.47034 q^{24} -0.267530 q^{25} -5.29584 q^{26} +5.64331 q^{27} -0.538935 q^{28} -6.50430 q^{29} -3.19860 q^{30} +8.42739 q^{31} +1.00000 q^{32} -5.50524 q^{33} -1.99257 q^{34} -1.17241 q^{35} -0.838111 q^{36} -6.11357 q^{37} -2.32942 q^{38} +7.78667 q^{39} +2.17542 q^{40} -1.81831 q^{41} +0.792415 q^{42} +1.03036 q^{43} +3.74421 q^{44} -1.82325 q^{45} +1.97841 q^{46} -9.51144 q^{47} -1.47034 q^{48} -6.70955 q^{49} -0.267530 q^{50} +2.92974 q^{51} -5.29584 q^{52} +13.2812 q^{53} +5.64331 q^{54} +8.14524 q^{55} -0.538935 q^{56} +3.42503 q^{57} -6.50430 q^{58} +3.26861 q^{59} -3.19860 q^{60} +3.63053 q^{61} +8.42739 q^{62} +0.451687 q^{63} +1.00000 q^{64} -11.5207 q^{65} -5.50524 q^{66} -15.4281 q^{67} -1.99257 q^{68} -2.90892 q^{69} -1.17241 q^{70} -16.6908 q^{71} -0.838111 q^{72} -4.28424 q^{73} -6.11357 q^{74} +0.393359 q^{75} -2.32942 q^{76} -2.01788 q^{77} +7.78667 q^{78} -15.4545 q^{79} +2.17542 q^{80} -5.78324 q^{81} -1.81831 q^{82} -6.93141 q^{83} +0.792415 q^{84} -4.33468 q^{85} +1.03036 q^{86} +9.56350 q^{87} +3.74421 q^{88} +10.0267 q^{89} -1.82325 q^{90} +2.85411 q^{91} +1.97841 q^{92} -12.3911 q^{93} -9.51144 q^{94} -5.06747 q^{95} -1.47034 q^{96} -12.5719 q^{97} -6.70955 q^{98} -3.13806 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 56 q^{2} - 18 q^{3} + 56 q^{4} - 17 q^{5} - 18 q^{6} - 35 q^{7} + 56 q^{8} + 34 q^{9}+O(q^{10}) $ 56 * q + 56 * q^2 - 18 * q^3 + 56 * q^4 - 17 * q^5 - 18 * q^6 - 35 * q^7 + 56 * q^8 + 34 * q^9	\( 56 q + 56 q^{2} - 18 q^{3} + 56 q^{4} - 17 q^{5} - 18 q^{6} - 35 q^{7} + 56 q^{8} + 34 q^{9} - 17 q^{10} - 53 q^{11} - 18 q^{12} - 21 q^{13} - 35 q^{14} - 36 q^{15} + 56 q^{16} - 22 q^{17} + 34 q^{18} - 31 q^{19} - 17 q^{20} - 23 q^{21} - 53 q^{22} - 59 q^{23} - 18 q^{24} + 41 q^{25} - 21 q^{26} - 63 q^{27} - 35 q^{28} - 88 q^{29} - 36 q^{30} - 44 q^{31} + 56 q^{32} + 4 q^{33} - 22 q^{34} - 51 q^{35} + 34 q^{36} - 60 q^{37} - 31 q^{38} - 62 q^{39} - 17 q^{40} - 39 q^{41} - 23 q^{42} - 66 q^{43} - 53 q^{44} - 34 q^{45} - 59 q^{46} - 51 q^{47} - 18 q^{48} + 41 q^{49} + 41 q^{50} - 48 q^{51} - 21 q^{52} - 75 q^{53} - 63 q^{54} - 41 q^{55} - 35 q^{56} - 12 q^{57} - 88 q^{58} - 77 q^{59} - 36 q^{60} - 43 q^{61} - 44 q^{62} - 88 q^{63} + 56 q^{64} - 54 q^{65} + 4 q^{66} - 62 q^{67} - 22 q^{68} - 48 q^{69} - 51 q^{70} - 122 q^{71} + 34 q^{72} - 7 q^{73} - 60 q^{74} - 63 q^{75} - 31 q^{76} - 39 q^{77} - 62 q^{78} - 91 q^{79} - 17 q^{80} + 8 q^{81} - 39 q^{82} - 51 q^{83} - 23 q^{84} - 72 q^{85} - 66 q^{86} - 19 q^{87} - 53 q^{88} - 62 q^{89} - 34 q^{90} - 48 q^{91} - 59 q^{92} - 41 q^{93} - 51 q^{94} - 120 q^{95} - 18 q^{96} + 6 q^{97} + 41 q^{98} - 128 q^{99}+O(q^{100}) \) 56 * q + 56 * q^2 - 18 * q^3 + 56 * q^4 - 17 * q^5 - 18 * q^6 - 35 * q^7 + 56 * q^8 + 34 * q^9 - 17 * q^10 - 53 * q^11 - 18 * q^12 - 21 * q^13 - 35 * q^14 - 36 * q^15 + 56 * q^16 - 22 * q^17 + 34 * q^18 - 31 * q^19 - 17 * q^20 - 23 * q^21 - 53 * q^22 - 59 * q^23 - 18 * q^24 + 41 * q^25 - 21 * q^26 - 63 * q^27 - 35 * q^28 - 88 * q^29 - 36 * q^30 - 44 * q^31 + 56 * q^32 + 4 * q^33 - 22 * q^34 - 51 * q^35 + 34 * q^36 - 60 * q^37 - 31 * q^38 - 62 * q^39 - 17 * q^40 - 39 * q^41 - 23 * q^42 - 66 * q^43 - 53 * q^44 - 34 * q^45 - 59 * q^46 - 51 * q^47 - 18 * q^48 + 41 * q^49 + 41 * q^50 - 48 * q^51 - 21 * q^52 - 75 * q^53 - 63 * q^54 - 41 * q^55 - 35 * q^56 - 12 * q^57 - 88 * q^58 - 77 * q^59 - 36 * q^60 - 43 * q^61 - 44 * q^62 - 88 * q^63 + 56 * q^64 - 54 * q^65 + 4 * q^66 - 62 * q^67 - 22 * q^68 - 48 * q^69 - 51 * q^70 - 122 * q^71 + 34 * q^72 - 7 * q^73 - 60 * q^74 - 63 * q^75 - 31 * q^76 - 39 * q^77 - 62 * q^78 - 91 * q^79 - 17 * q^80 + 8 * q^81 - 39 * q^82 - 51 * q^83 - 23 * q^84 - 72 * q^85 - 66 * q^86 - 19 * q^87 - 53 * q^88 - 62 * q^89 - 34 * q^90 - 48 * q^91 - 59 * q^92 - 41 * q^93 - 51 * q^94 - 120 * q^95 - 18 * q^96 + 6 * q^97 + 41 * q^98 - 128 * 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
$3$	−1.47034	−0.848899	−0.424450	−	0.905452i	$-0.639532\pi$
$3$	−1.47034	−0.848899	−0.424450	+	0.905452i	$0.639532\pi$
$4$	1.00000	0.500000
$5$	2.17542	0.972879	0.486440	−	0.873714i	$-0.338295\pi$
$5$	2.17542	0.972879	0.486440	+	0.873714i	$0.338295\pi$
$6$	−1.47034	−0.600262
$7$	−0.538935	−0.203698	−0.101849	−	0.994800i	$-0.532476\pi$
$7$	−0.538935	−0.203698	−0.101849	+	0.994800i	$0.532476\pi$
$8$	1.00000	0.353553
$9$	−0.838111	−0.279370
$10$	2.17542	0.687930
$11$	3.74421	1.12892	0.564461	−	0.825460i	$-0.309084\pi$
$11$	3.74421	1.12892	0.564461	+	0.825460i	$0.309084\pi$
$12$	−1.47034	−0.424450
$13$	−5.29584	−1.46880	−0.734401	−	0.678716i	$-0.762537\pi$
$13$	−5.29584	−1.46880	−0.734401	+	0.678716i	$0.762537\pi$
$14$	−0.538935	−0.144036
$15$	−3.19860	−0.825876
$16$	1.00000	0.250000
$17$	−1.99257	−0.483268	−0.241634	−	0.970367i	$-0.577683\pi$
$17$	−1.99257	−0.483268	−0.241634	+	0.970367i	$0.577683\pi$
$18$	−0.838111	−0.197545
$19$	−2.32942	−0.534405	−0.267203	−	0.963640i	$-0.586099\pi$
$19$	−2.32942	−0.534405	−0.267203	+	0.963640i	$0.586099\pi$
$20$	2.17542	0.486440
$21$	0.792415	0.172919
$22$	3.74421	0.798268
$23$	1.97841	0.412526	0.206263	−	0.978497i	$-0.433870\pi$
$23$	1.97841	0.412526	0.206263	+	0.978497i	$0.433870\pi$
$24$	−1.47034	−0.300131
$25$	−0.267530	−0.0535059
$26$	−5.29584	−1.03860
$27$	5.64331	1.08606
$28$	−0.538935	−0.101849
$29$	−6.50430	−1.20782	−0.603909	−	0.797053i	$-0.706391\pi$
$29$	−6.50430	−1.20782	−0.603909	+	0.797053i	$0.706391\pi$
$30$	−3.19860	−0.583983
$31$	8.42739	1.51360	0.756802	−	0.653645i	$-0.226761\pi$
$31$	8.42739	1.51360	0.756802	+	0.653645i	$0.226761\pi$
$32$	1.00000	0.176777
$33$	−5.50524	−0.958340
$34$	−1.99257	−0.341722
$35$	−1.17241	−0.198174
$36$	−0.838111	−0.139685
$37$	−6.11357	−1.00506	−0.502532	−	0.864558i	$-0.667598\pi$
$37$	−6.11357	−1.00506	−0.502532	+	0.864558i	$0.667598\pi$
$38$	−2.32942	−0.377882
$39$	7.78667	1.24686
$40$	2.17542	0.343965
$41$	−1.81831	−0.283973	−0.141986	−	0.989869i	$-0.545349\pi$
$41$	−1.81831	−0.283973	−0.141986	+	0.989869i	$0.545349\pi$
$42$	0.792415	0.122272
$43$	1.03036	0.157128	0.0785640	−	0.996909i	$-0.474967\pi$
$43$	1.03036	0.157128	0.0785640	+	0.996909i	$0.474967\pi$
$44$	3.74421	0.564461
$45$	−1.82325	−0.271794
$46$	1.97841	0.291700
$47$	−9.51144	−1.38739	−0.693693	−	0.720271i	$-0.744017\pi$
$47$	−9.51144	−1.38739	−0.693693	+	0.720271i	$0.744017\pi$
$48$	−1.47034	−0.212225
$49$	−6.70955	−0.958507
$50$	−0.267530	−0.0378344
$51$	2.92974	0.410246
$52$	−5.29584	−0.734401
$53$	13.2812	1.82432	0.912158	−	0.409838i	$-0.134415\pi$
$53$	13.2812	1.82432	0.912158	+	0.409838i	$0.134415\pi$
$54$	5.64331	0.767958
$55$	8.14524	1.09830
$56$	−0.538935	−0.0720182
$57$	3.42503	0.453656
$58$	−6.50430	−0.854056
$59$	3.26861	0.425536	0.212768	−	0.977103i	$-0.431752\pi$
$59$	3.26861	0.425536	0.212768	+	0.977103i	$0.431752\pi$
$60$	−3.19860	−0.412938
$61$	3.63053	0.464842	0.232421	−	0.972615i	$-0.425335\pi$
$61$	3.63053	0.464842	0.232421	+	0.972615i	$0.425335\pi$
$62$	8.42739	1.07028
$63$	0.451687	0.0569072
$64$	1.00000	0.125000
$65$	−11.5207	−1.42897
$66$	−5.50524	−0.677649
$67$	−15.4281	−1.88485	−0.942424	−	0.334421i	$-0.891459\pi$
$67$	−15.4281	−1.88485	−0.942424	+	0.334421i	$0.891459\pi$
$68$	−1.99257	−0.241634
$69$	−2.90892	−0.350193
$70$	−1.17241	−0.140130
$71$	−16.6908	−1.98083	−0.990416	−	0.138116i	$-0.955895\pi$
$71$	−16.6908	−1.98083	−0.990416	+	0.138116i	$0.955895\pi$
$72$	−0.838111	−0.0987724
$73$	−4.28424	−0.501432	−0.250716	−	0.968061i	$-0.580666\pi$
$73$	−4.28424	−0.501432	−0.250716	+	0.968061i	$0.580666\pi$
$74$	−6.11357	−0.710688
$75$	0.393359	0.0454211
$76$	−2.32942	−0.267203
$77$	−2.01788	−0.229959
$78$	7.78667	0.881667
$79$	−15.4545	−1.73877	−0.869384	−	0.494136i	$-0.835484\pi$
$79$	−15.4545	−1.73877	−0.869384	+	0.494136i	$0.835484\pi$
$80$	2.17542	0.243220
$81$	−5.78324	−0.642582
$82$	−1.81831	−0.200799
$83$	−6.93141	−0.760821	−0.380410	−	0.924818i	$-0.624217\pi$
$83$	−6.93141	−0.760821	−0.380410	+	0.924818i	$0.624217\pi$
$84$	0.792415	0.0864596
$85$	−4.33468	−0.470162
$86$	1.03036	0.111106
$87$	9.56350	1.02532
$88$	3.74421	0.399134
$89$	10.0267	1.06283	0.531416	−	0.847111i	$-0.321660\pi$
$89$	10.0267	1.06283	0.531416	+	0.847111i	$0.321660\pi$
$90$	−1.82325	−0.192187
$91$	2.85411	0.299192
$92$	1.97841	0.206263
$93$	−12.3911	−1.28490
$94$	−9.51144	−0.981030
$95$	−5.06747	−0.519912
$96$	−1.47034	−0.150066
$97$	−12.5719	−1.27648	−0.638241	−	0.769837i	$-0.720338\pi$
$97$	−12.5719	−1.27648	−0.638241	+	0.769837i	$0.720338\pi$
$98$	−6.70955	−0.677767
$99$	−3.13806	−0.315387
$100$	−0.267530	−0.0267530
$101$	−0.0333446	−0.00331791	−0.00165896	−	0.999999i	$-0.500528\pi$
$101$	−0.0333446	−0.00331791	−0.00165896	+	0.999999i	$0.500528\pi$
$102$	2.92974	0.290088
$103$	4.96877	0.489588	0.244794	−	0.969575i	$-0.421280\pi$
$103$	4.96877	0.489588	0.244794	+	0.969575i	$0.421280\pi$
$104$	−5.29584	−0.519300
$105$	1.72384	0.168229
$106$	13.2812	1.28999
$107$	18.4526	1.78388	0.891940	−	0.452153i	$-0.149344\pi$
$107$	18.4526	1.78388	0.891940	+	0.452153i	$0.149344\pi$
$108$	5.64331	0.543028
$109$	−2.49468	−0.238947	−0.119473	−	0.992837i	$-0.538121\pi$
$109$	−2.49468	−0.238947	−0.119473	+	0.992837i	$0.538121\pi$
$110$	8.14524	0.776618
$111$	8.98901	0.853199
$112$	−0.538935	−0.0509245
$113$	11.0610	1.04053	0.520266	−	0.854005i	$-0.325833\pi$
$113$	11.0610	1.04053	0.520266	+	0.854005i	$0.325833\pi$
$114$	3.42503	0.320783
$115$	4.30387	0.401338
$116$	−6.50430	−0.603909
$117$	4.43851	0.410340
$118$	3.26861	0.300900
$119$	1.07386	0.0984409
$120$	−3.19860	−0.291991
$121$	3.01909	0.274463
$122$	3.63053	0.328693
$123$	2.67353	0.241064
$124$	8.42739	0.756802
$125$	−11.4591	−1.02493
$126$	0.451687	0.0402395
$127$	19.2242	1.70588	0.852938	−	0.522013i	$-0.174819\pi$
$127$	19.2242	1.70588	0.852938	+	0.522013i	$0.174819\pi$
$128$	1.00000	0.0883883
$129$	−1.51497	−0.133386
$130$	−11.5207	−1.01043
$131$	−21.4852	−1.87717	−0.938586	−	0.345044i	$-0.887864\pi$
$131$	−21.4852	−1.87717	−0.938586	+	0.345044i	$0.887864\pi$
$132$	−5.50524	−0.479170
$133$	1.25540	0.108857
$134$	−15.4281	−1.33279
$135$	12.2766	1.05660
$136$	−1.99257	−0.170861
$137$	3.62244	0.309486	0.154743	−	0.987955i	$-0.450545\pi$
$137$	3.62244	0.309486	0.154743	+	0.987955i	$0.450545\pi$
$138$	−2.90892	−0.247624
$139$	10.7630	0.912904	0.456452	−	0.889748i	$-0.349120\pi$
$139$	10.7630	0.912904	0.456452	+	0.889748i	$0.349120\pi$
$140$	−1.17241	−0.0990869
$141$	13.9850	1.17775
$142$	−16.6908	−1.40066
$143$	−19.8287	−1.65816
$144$	−0.838111	−0.0698426
$145$	−14.1496	−1.17506
$146$	−4.28424	−0.354566
$147$	9.86529	0.813676
$148$	−6.11357	−0.502532
$149$	2.29721	0.188195	0.0940974	−	0.995563i	$-0.470003\pi$
$149$	2.29721	0.188195	0.0940974	+	0.995563i	$0.470003\pi$
$150$	0.393359	0.0321176
$151$	−14.0637	−1.14449	−0.572246	−	0.820082i	$-0.693928\pi$
$151$	−14.0637	−1.14449	−0.572246	+	0.820082i	$0.693928\pi$
$152$	−2.32942	−0.188941
$153$	1.66999	0.135011
$154$	−2.01788	−0.162606
$155$	18.3331	1.47255
$156$	7.78667	0.623432
$157$	−16.5574	−1.32142	−0.660712	−	0.750640i	$-0.729745\pi$
$157$	−16.5574	−1.32142	−0.660712	+	0.750640i	$0.729745\pi$
$158$	−15.4545	−1.22950
$159$	−19.5279	−1.54866
$160$	2.17542	0.171982
$161$	−1.06623	−0.0840309
$162$	−5.78324	−0.454374
$163$	−7.96697	−0.624021	−0.312011	−	0.950079i	$-0.601002\pi$
$163$	−7.96697	−0.624021	−0.312011	+	0.950079i	$0.601002\pi$
$164$	−1.81831	−0.141986
$165$	−11.9762	−0.932349
$166$	−6.93141	−0.537981
$167$	6.83737	0.529092	0.264546	−	0.964373i	$-0.414778\pi$
$167$	6.83737	0.529092	0.264546	+	0.964373i	$0.414778\pi$
$168$	0.792415	0.0611362
$169$	15.0459	1.15738
$170$	−4.33468	−0.332455
$171$	1.95231	0.149297
$172$	1.03036	0.0785640
$173$	26.0273	1.97882	0.989410	−	0.145144i	$-0.0463646\pi$
$173$	26.0273	1.97882	0.989410	+	0.145144i	$0.0463646\pi$
$174$	9.56350	0.725007
$175$	0.144181	0.0108991
$176$	3.74421	0.282230
$177$	−4.80595	−0.361237
$178$	10.0267	0.751535
$179$	−22.9389	−1.71453	−0.857267	−	0.514872i	$-0.827840\pi$
$179$	−22.9389	−1.71453	−0.857267	+	0.514872i	$0.827840\pi$
$180$	−1.82325	−0.135897
$181$	−5.21303	−0.387482	−0.193741	−	0.981053i	$-0.562062\pi$
$181$	−5.21303	−0.387482	−0.193741	+	0.981053i	$0.562062\pi$
$182$	2.85411	0.211561
$183$	−5.33811	−0.394604
$184$	1.97841	0.145850
$185$	−13.2996	−0.977807
$186$	−12.3911	−0.908559
$187$	−7.46058	−0.545572
$188$	−9.51144	−0.693693
$189$	−3.04138	−0.221228
$190$	−5.06747	−0.367633
$191$	2.24248	0.162260	0.0811301	−	0.996704i	$-0.474147\pi$
$191$	2.24248	0.162260	0.0811301	+	0.996704i	$0.474147\pi$
$192$	−1.47034	−0.106112
$193$	−6.96472	−0.501331	−0.250666	−	0.968074i	$-0.580649\pi$
$193$	−6.96472	−0.501331	−0.250666	+	0.968074i	$0.580649\pi$
$194$	−12.5719	−0.902609
$195$	16.9393	1.21305
$196$	−6.70955	−0.479254
$197$	3.17057	0.225894	0.112947	−	0.993601i	$-0.463971\pi$
$197$	3.17057	0.225894	0.112947	+	0.993601i	$0.463971\pi$
$198$	−3.13806	−0.223012
$199$	16.8173	1.19214	0.596072	−	0.802931i	$-0.296727\pi$
$199$	16.8173	1.19214	0.596072	+	0.802931i	$0.296727\pi$
$200$	−0.267530	−0.0189172
$201$	22.6846	1.60005
$202$	−0.0333446	−0.00234612
$203$	3.50539	0.246030
$204$	2.92974	0.205123
$205$	−3.95560	−0.276271
$206$	4.96877	0.346191
$207$	−1.65813	−0.115248
$208$	−5.29584	−0.367201
$209$	−8.72183	−0.603302
$210$	1.72384	0.118956
$211$	−1.45432	−0.100120	−0.0500598	−	0.998746i	$-0.515941\pi$
$211$	−1.45432	−0.100120	−0.0500598	+	0.998746i	$0.515941\pi$
$212$	13.2812	0.912158
$213$	24.5411	1.68153
$214$	18.4526	1.26139
$215$	2.24146	0.152867
$216$	5.64331	0.383979
$217$	−4.54181	−0.308318
$218$	−2.49468	−0.168961
$219$	6.29927	0.425665
$220$	8.14524	0.549152
$221$	10.5523	0.709826
$222$	8.98901	0.603303
$223$	−12.4620	−0.834520	−0.417260	−	0.908787i	$-0.637009\pi$
$223$	−12.4620	−0.834520	−0.417260	+	0.908787i	$0.637009\pi$
$224$	−0.538935	−0.0360091
$225$	0.224220	0.0149480
$226$	11.0610	0.735767
$227$	−7.95873	−0.528239	−0.264120	−	0.964490i	$-0.585081\pi$
$227$	−7.95873	−0.528239	−0.264120	+	0.964490i	$0.585081\pi$
$228$	3.42503	0.226828
$229$	−19.3523	−1.27883	−0.639417	−	0.768860i	$-0.720824\pi$
$229$	−19.3523	−1.27883	−0.639417	+	0.768860i	$0.720824\pi$
$230$	4.30387	0.283789
$231$	2.96697	0.195212
$232$	−6.50430	−0.427028
$233$	16.5616	1.08499	0.542494	−	0.840060i	$-0.317480\pi$
$233$	16.5616	1.08499	0.542494	+	0.840060i	$0.317480\pi$
$234$	4.43851	0.290154
$235$	−20.6914	−1.34976
$236$	3.26861	0.212768
$237$	22.7233	1.47604
$238$	1.07386	0.0696082
$239$	−27.5503	−1.78208	−0.891039	−	0.453926i	$-0.850023\pi$
$239$	−27.5503	−1.78208	−0.891039	+	0.453926i	$0.850023\pi$
$240$	−3.19860	−0.206469
$241$	−16.1289	−1.03895	−0.519477	−	0.854484i	$-0.673873\pi$
$241$	−16.1289	−1.03895	−0.519477	+	0.854484i	$0.673873\pi$
$242$	3.01909	0.194074
$243$	−8.42664	−0.540569
$244$	3.63053	0.232421
$245$	−14.5961	−0.932512
$246$	2.67353	0.170458
$247$	12.3362	0.784936
$248$	8.42739	0.535140
$249$	10.1915	0.645860
$250$	−11.4591	−0.724738
$251$	13.3914	0.845259	0.422630	−	0.906302i	$-0.361107\pi$
$251$	13.3914	0.845259	0.422630	+	0.906302i	$0.361107\pi$
$252$	0.451687	0.0284536
$253$	7.40757	0.465710
$254$	19.2242	1.20624
$255$	6.37343	0.399120
$256$	1.00000	0.0625000
$257$	−9.43376	−0.588462	−0.294231	−	0.955734i	$-0.595064\pi$
$257$	−9.43376	−0.588462	−0.294231	+	0.955734i	$0.595064\pi$
$258$	−1.51497	−0.0943180
$259$	3.29482	0.204730
$260$	−11.5207	−0.714484
$261$	5.45132	0.337429
$262$	−21.4852	−1.32736
$263$	2.65568	0.163756	0.0818782	−	0.996642i	$-0.473908\pi$
$263$	2.65568	0.163756	0.0818782	+	0.996642i	$0.473908\pi$
$264$	−5.50524	−0.338824
$265$	28.8923	1.77484
$266$	1.25540	0.0769738
$267$	−14.7427	−0.902236
$268$	−15.4281	−0.942424
$269$	−1.13853	−0.0694175	−0.0347088	−	0.999397i	$-0.511050\pi$
$269$	−1.13853	−0.0694175	−0.0347088	+	0.999397i	$0.511050\pi$
$270$	12.2766	0.747130
$271$	24.2202	1.47127	0.735635	−	0.677378i	$-0.236884\pi$
$271$	24.2202	1.47127	0.735635	+	0.677378i	$0.236884\pi$
$272$	−1.99257	−0.120817
$273$	−4.19651	−0.253984
$274$	3.62244	0.218839
$275$	−1.00169	−0.0604040
$276$	−2.90892	−0.175097
$277$	−22.4979	−1.35177	−0.675885	−	0.737007i	$-0.736239\pi$
$277$	−22.4979	−1.35177	−0.675885	+	0.737007i	$0.736239\pi$
$278$	10.7630	0.645520
$279$	−7.06309	−0.422856
$280$	−1.17241	−0.0700650
$281$	−10.5410	−0.628826	−0.314413	−	0.949286i	$-0.601808\pi$
$281$	−10.5410	−0.628826	−0.314413	+	0.949286i	$0.601808\pi$
$282$	13.9850	0.832795
$283$	−28.9099	−1.71851	−0.859257	−	0.511544i	$-0.829074\pi$
$283$	−28.9099	−1.71851	−0.859257	+	0.511544i	$0.829074\pi$
$284$	−16.6908	−0.990416
$285$	7.45089	0.441353
$286$	−19.8287	−1.17250
$287$	0.979951	0.0578447
$288$	−0.838111	−0.0493862
$289$	−13.0297	−0.766452
$290$	−14.1496	−0.830893
$291$	18.4849	1.08360
$292$	−4.28424	−0.250716
$293$	5.08108	0.296840	0.148420	−	0.988924i	$-0.452581\pi$
$293$	5.08108	0.296840	0.148420	+	0.988924i	$0.452581\pi$
$294$	9.86529	0.575356
$295$	7.11061	0.413996
$296$	−6.11357	−0.355344
$297$	21.1297	1.22607
$298$	2.29721	0.133074
$299$	−10.4773	−0.605920
$300$	0.393359	0.0227106
$301$	−0.555295	−0.0320067
$302$	−14.0637	−0.809278
$303$	0.0490278	0.00281657
$304$	−2.32942	−0.133601
$305$	7.89795	0.452235
$306$	1.66999	0.0954671
$307$	8.38760	0.478706	0.239353	−	0.970933i	$-0.423065\pi$
$307$	8.38760	0.478706	0.239353	+	0.970933i	$0.423065\pi$
$308$	−2.01788	−0.114980
$309$	−7.30576	−0.415610
$310$	18.3331	1.04125
$311$	−6.61374	−0.375031	−0.187515	−	0.982262i	$-0.560043\pi$
$311$	−6.61374	−0.375031	−0.187515	+	0.982262i	$0.560043\pi$
$312$	7.78667	0.440833
$313$	5.02795	0.284196	0.142098	−	0.989853i	$-0.454615\pi$
$313$	5.02795	0.284196	0.142098	+	0.989853i	$0.454615\pi$
$314$	−16.5574	−0.934388
$315$	0.982611	0.0553639
$316$	−15.4545	−0.869384
$317$	23.1194	1.29851	0.649257	−	0.760569i	$-0.275080\pi$
$317$	23.1194	1.29851	0.649257	+	0.760569i	$0.275080\pi$
$318$	−19.5279	−1.09507
$319$	−24.3534	−1.36353
$320$	2.17542	0.121610
$321$	−27.1315	−1.51433
$322$	−1.06623	−0.0594188
$323$	4.64152	0.258261
$324$	−5.78324	−0.321291
$325$	1.41680	0.0785896
$326$	−7.96697	−0.441250
$327$	3.66801	0.202842
$328$	−1.81831	−0.100399
$329$	5.12604	0.282608
$330$	−11.9762	−0.659270
$331$	−19.4297	−1.06795	−0.533976	−	0.845500i	$-0.679303\pi$
$331$	−19.4297	−1.06795	−0.533976	+	0.845500i	$0.679303\pi$
$332$	−6.93141	−0.380410
$333$	5.12385	0.280785
$334$	6.83737	0.374124
$335$	−33.5627	−1.83373
$336$	0.792415	0.0432298
$337$	14.8285	0.807761	0.403881	−	0.914812i	$-0.367661\pi$
$337$	14.8285	0.807761	0.403881	+	0.914812i	$0.367661\pi$
$338$	15.0459	0.818391
$339$	−16.2634	−0.883306
$340$	−4.33468	−0.235081
$341$	31.5539	1.70874
$342$	1.95231	0.105569
$343$	7.38855	0.398944
$344$	1.03036	0.0555531
$345$	−6.32814	−0.340696
$346$	26.0273	1.39924
$347$	24.3072	1.30488	0.652438	−	0.757842i	$-0.273746\pi$
$347$	24.3072	1.30488	0.652438	+	0.757842i	$0.273746\pi$
$348$	9.56350	0.512658
$349$	16.1378	0.863839	0.431919	−	0.901912i	$-0.357836\pi$
$349$	16.1378	0.863839	0.431919	+	0.901912i	$0.357836\pi$
$350$	0.144181	0.00770680
$351$	−29.8861	−1.59520
$352$	3.74421	0.199567
$353$	17.8948	0.952444	0.476222	−	0.879325i	$-0.342006\pi$
$353$	17.8948	0.952444	0.476222	+	0.879325i	$0.342006\pi$
$354$	−4.80595	−0.255433
$355$	−36.3095	−1.92711
$356$	10.0267	0.531416
$357$	−1.57894	−0.0835664
$358$	−22.9389	−1.21236
$359$	−13.5716	−0.716284	−0.358142	−	0.933667i	$-0.616590\pi$
$359$	−13.5716	−0.716284	−0.358142	+	0.933667i	$0.616590\pi$
$360$	−1.82325	−0.0960936
$361$	−13.5738	−0.714411
$362$	−5.21303	−0.273991
$363$	−4.43908	−0.232991
$364$	2.85411	0.149596
$365$	−9.32003	−0.487833
$366$	−5.33811	−0.279027
$367$	−11.7646	−0.614109	−0.307055	−	0.951692i	$-0.599343\pi$
$367$	−11.7646	−0.614109	−0.307055	+	0.951692i	$0.599343\pi$
$368$	1.97841	0.103132
$369$	1.52395	0.0793336
$370$	−13.2996	−0.691414
$371$	−7.15771	−0.371610
$372$	−12.3911	−0.642448
$373$	−17.6789	−0.915379	−0.457689	−	0.889112i	$-0.651323\pi$
$373$	−17.6789	−0.915379	−0.457689	+	0.889112i	$0.651323\pi$
$374$	−7.46058	−0.385778
$375$	16.8487	0.870066
$376$	−9.51144	−0.490515
$377$	34.4457	1.77404
$378$	−3.04138	−0.156432
$379$	−29.0058	−1.48993	−0.744963	−	0.667106i	$-0.767533\pi$
$379$	−29.0058	−1.48993	−0.744963	+	0.667106i	$0.767533\pi$
$380$	−5.06747	−0.259956
$381$	−28.2661	−1.44812
$382$	2.24248	0.114735
$383$	−8.95716	−0.457689	−0.228845	−	0.973463i	$-0.573495\pi$
$383$	−8.95716	−0.457689	−0.228845	+	0.973463i	$0.573495\pi$
$384$	−1.47034	−0.0750328
$385$	−4.38975	−0.223722
$386$	−6.96472	−0.354495
$387$	−0.863554	−0.0438969
$388$	−12.5719	−0.638241
$389$	12.1691	0.616998	0.308499	−	0.951225i	$-0.400173\pi$
$389$	12.1691	0.616998	0.308499	+	0.951225i	$0.400173\pi$
$390$	16.9393	0.857755
$391$	−3.94211	−0.199361
$392$	−6.70955	−0.338883
$393$	31.5905	1.59353
$394$	3.17057	0.159731
$395$	−33.6201	−1.69161
$396$	−3.13806	−0.157694
$397$	34.5292	1.73297	0.866486	−	0.499201i	$-0.166373\pi$
$397$	34.5292	1.73297	0.866486	+	0.499201i	$0.166373\pi$
$398$	16.8173	0.842973
$399$	−1.84587	−0.0924089
$400$	−0.267530	−0.0133765
$401$	16.9355	0.845719	0.422859	−	0.906195i	$-0.361026\pi$
$401$	16.9355	0.845719	0.422859	+	0.906195i	$0.361026\pi$
$402$	22.6846	1.13140
$403$	−44.6301	−2.22318
$404$	−0.0333446	−0.00165896
$405$	−12.5810	−0.625154
$406$	3.50539	0.173970
$407$	−22.8905	−1.13464
$408$	2.92974	0.145044
$409$	−21.5229	−1.06424	−0.532120	−	0.846669i	$-0.678604\pi$
$409$	−21.5229	−1.06424	−0.532120	+	0.846669i	$0.678604\pi$
$410$	−3.95560	−0.195353
$411$	−5.32620	−0.262722
$412$	4.96877	0.244794
$413$	−1.76157	−0.0866810
$414$	−1.65813	−0.0814924
$415$	−15.0787	−0.740187
$416$	−5.29584	−0.259650
$417$	−15.8252	−0.774963
$418$	−8.72183	−0.426599
$419$	10.7224	0.523823	0.261911	−	0.965092i	$-0.415647\pi$
$419$	10.7224	0.523823	0.261911	+	0.965092i	$0.415647\pi$
$420$	1.72384	0.0841147
$421$	−22.6907	−1.10588	−0.552938	−	0.833222i	$-0.686494\pi$
$421$	−22.6907	−1.10588	−0.552938	+	0.833222i	$0.686494\pi$
$422$	−1.45432	−0.0707952
$423$	7.97164	0.387595
$424$	13.2812	0.644993
$425$	0.533071	0.0258577
$426$	24.5411	1.18902
$427$	−1.95662	−0.0946875
$428$	18.4526	0.891940
$429$	29.1549	1.40761
$430$	2.24146	0.108093
$431$	25.0787	1.20800	0.604000	−	0.796984i	$-0.293573\pi$
$431$	25.0787	1.20800	0.604000	+	0.796984i	$0.293573\pi$
$432$	5.64331	0.271514
$433$	−3.16813	−0.152251	−0.0761253	−	0.997098i	$-0.524255\pi$
$433$	−3.16813	−0.152251	−0.0761253	+	0.997098i	$0.524255\pi$
$434$	−4.54181	−0.218014
$435$	20.8047	0.997508
$436$	−2.49468	−0.119473
$437$	−4.60854	−0.220456
$438$	6.29927	0.300991
$439$	−26.3533	−1.25777	−0.628887	−	0.777496i	$-0.716489\pi$
$439$	−26.3533	−1.25777	−0.628887	+	0.777496i	$0.716489\pi$
$440$	8.14524	0.388309
$441$	5.62335	0.267779
$442$	10.5523	0.501923
$443$	16.2411	0.771637	0.385818	−	0.922575i	$-0.373919\pi$
$443$	16.2411	0.771637	0.385818	+	0.922575i	$0.373919\pi$
$444$	8.98901	0.426599
$445$	21.8124	1.03401
$446$	−12.4620	−0.590094
$447$	−3.37767	−0.159758
$448$	−0.538935	−0.0254623
$449$	−10.3249	−0.487263	−0.243632	−	0.969868i	$-0.578339\pi$
$449$	−10.3249	−0.487263	−0.243632	+	0.969868i	$0.578339\pi$
$450$	0.224220	0.0105698
$451$	−6.80814	−0.320583
$452$	11.0610	0.520266
$453$	20.6784	0.971558
$454$	−7.95873	−0.373522
$455$	6.20891	0.291078
$456$	3.42503	0.160392
$457$	0.342243	0.0160095	0.00800473	−	0.999968i	$-0.497452\pi$
$457$	0.342243	0.0160095	0.00800473	+	0.999968i	$0.497452\pi$
$458$	−19.3523	−0.904272
$459$	−11.2447	−0.524857
$460$	4.30387	0.200669
$461$	23.0273	1.07249	0.536244	−	0.844063i	$-0.319843\pi$
$461$	23.0273	1.07249	0.536244	+	0.844063i	$0.319843\pi$
$462$	2.96697	0.138036
$463$	−31.3639	−1.45760	−0.728802	−	0.684725i	$-0.759922\pi$
$463$	−31.3639	−1.45760	−0.728802	+	0.684725i	$0.759922\pi$
$464$	−6.50430	−0.301954
$465$	−26.9559	−1.25005
$466$	16.5616	0.767202
$467$	35.3321	1.63497	0.817487	−	0.575948i	$-0.195367\pi$
$467$	35.3321	1.63497	0.817487	+	0.575948i	$0.195367\pi$
$468$	4.43851	0.205170
$469$	8.31476	0.383940
$470$	−20.6914	−0.954424
$471$	24.3449	1.12176
$472$	3.26861	0.150450
$473$	3.85787	0.177385
$474$	22.7233	1.04372
$475$	0.623189	0.0285939
$476$	1.07386	0.0492204
$477$	−11.1311	−0.509660
$478$	−27.5503	−1.26012
$479$	2.70686	0.123679	0.0618397	−	0.998086i	$-0.480303\pi$
$479$	2.70686	0.123679	0.0618397	+	0.998086i	$0.480303\pi$
$480$	−3.19860	−0.145996
$481$	32.3765	1.47624
$482$	−16.1289	−0.734652
$483$	1.56772	0.0713337
$484$	3.01909	0.137231
$485$	−27.3492	−1.24186
$486$	−8.42664	−0.382240
$487$	−15.9075	−0.720836	−0.360418	−	0.932791i	$-0.617366\pi$
$487$	−15.9075	−0.720836	−0.360418	+	0.932791i	$0.617366\pi$
$488$	3.63053	0.164347
$489$	11.7141	0.529731
$490$	−14.5961	−0.659385
$491$	−25.1029	−1.13288	−0.566439	−	0.824104i	$-0.691679\pi$
$491$	−25.1029	−1.13288	−0.566439	+	0.824104i	$0.691679\pi$
$492$	2.67353	0.120532
$493$	12.9602	0.583700
$494$	12.3362	0.555033
$495$	−6.82662	−0.306834
$496$	8.42739	0.378401
$497$	8.99524	0.403492
$498$	10.1915	0.456692
$499$	0.00691804	0.000309694 0	0.000154847	−	1.00000i	$-0.499951\pi$
$499$	0.00691804	0.000309694 0	0.000154847		1.00000i	$0.499951\pi$
$500$	−11.4591	−0.512467
$501$	−10.0532	−0.449145
$502$	13.3914	0.597689
$503$	−2.99044	−0.133337	−0.0666685	−	0.997775i	$-0.521237\pi$
$503$	−2.99044	−0.133337	−0.0666685	+	0.997775i	$0.521237\pi$
$504$	0.451687	0.0201198
$505$	−0.0725386	−0.00322793
$506$	7.40757	0.329307
$507$	−22.1226	−0.982499
$508$	19.2242	0.852938
$509$	−20.2578	−0.897911	−0.448955	−	0.893554i	$-0.648204\pi$
$509$	−20.2578	−0.897911	−0.448955	+	0.893554i	$0.648204\pi$
$510$	6.37343	0.282220
$511$	2.30892	0.102141
$512$	1.00000	0.0441942
$513$	−13.1456	−0.580394
$514$	−9.43376	−0.416106
$515$	10.8092	0.476310
$516$	−1.51497	−0.0666929
$517$	−35.6128	−1.56625
$518$	3.29482	0.144766
$519$	−38.2689	−1.67982
$520$	−11.5207	−0.505216
$521$	−21.6787	−0.949759	−0.474880	−	0.880051i	$-0.657508\pi$
$521$	−21.6787	−0.949759	−0.474880	+	0.880051i	$0.657508\pi$
$522$	5.45132	0.238598
$523$	−16.1494	−0.706165	−0.353082	−	0.935592i	$-0.614866\pi$
$523$	−16.1494	−0.706165	−0.353082	+	0.935592i	$0.614866\pi$
$524$	−21.4852	−0.938586
$525$	−0.211995	−0.00925220
$526$	2.65568	0.115793
$527$	−16.7921	−0.731477
$528$	−5.50524	−0.239585
$529$	−19.0859	−0.829822
$530$	28.8923	1.25500
$531$	−2.73946	−0.118882
$532$	1.25540	0.0544287
$533$	9.62949	0.417100
$534$	−14.7427	−0.637977
$535$	40.1422	1.73550
$536$	−15.4281	−0.666394
$537$	33.7279	1.45547
$538$	−1.13853	−0.0490856
$539$	−25.1219	−1.08208
$540$	12.2766	0.528301
$541$	23.8752	1.02648	0.513238	−	0.858246i	$-0.328446\pi$
$541$	23.8752	1.02648	0.513238	+	0.858246i	$0.328446\pi$
$542$	24.2202	1.04035
$543$	7.66491	0.328933
$544$	−1.99257	−0.0854306
$545$	−5.42698	−0.232466
$546$	−4.19651	−0.179594
$547$	−15.4102	−0.658894	−0.329447	−	0.944174i	$-0.606862\pi$
$547$	−15.4102	−0.658894	−0.329447	+	0.944174i	$0.606862\pi$
$548$	3.62244	0.154743
$549$	−3.04279	−0.129863
$550$	−1.00169	−0.0427121
$551$	15.1512	0.645464
$552$	−2.90892	−0.123812
$553$	8.32897	0.354184
$554$	−22.4979	−0.955846
$555$	19.5549	0.830059
$556$	10.7630	0.456452
$557$	−24.9296	−1.05630	−0.528150	−	0.849151i	$-0.677114\pi$
$557$	−24.9296	−1.05630	−0.528150	+	0.849151i	$0.677114\pi$
$558$	−7.06309	−0.299004
$559$	−5.45661	−0.230790
$560$	−1.17241	−0.0495434
$561$	10.9696	0.463135
$562$	−10.5410	−0.444647
$563$	18.3716	0.774269	0.387134	−	0.922023i	$-0.373465\pi$
$563$	18.3716	0.774269	0.387134	+	0.922023i	$0.373465\pi$
$564$	13.9850	0.588875
$565$	24.0624	1.01231
$566$	−28.9099	−1.21517
$567$	3.11679	0.130893
$568$	−16.6908	−0.700330
$569$	23.0159	0.964877	0.482438	−	0.875930i	$-0.339751\pi$
$569$	23.0159	0.964877	0.482438	+	0.875930i	$0.339751\pi$
$570$	7.45089	0.312084
$571$	21.9074	0.916797	0.458398	−	0.888747i	$-0.348423\pi$
$571$	21.9074	0.916797	0.458398	+	0.888747i	$0.348423\pi$
$572$	−19.8287	−0.829081
$573$	−3.29720	−0.137742
$574$	0.979951	0.0409024
$575$	−0.529283	−0.0220726
$576$	−0.838111	−0.0349213
$577$	−15.9145	−0.662531	−0.331266	−	0.943538i	$-0.607476\pi$
$577$	−15.9145	−0.662531	−0.331266	+	0.943538i	$0.607476\pi$
$578$	−13.0297	−0.541963
$579$	10.2405	0.425580
$580$	−14.1496	−0.587530
$581$	3.73558	0.154978
$582$	18.4849	0.766224
$583$	49.7277	2.05951
$584$	−4.28424	−0.177283
$585$	9.65563	0.399211
$586$	5.08108	0.209897
$587$	−7.72804	−0.318970	−0.159485	−	0.987200i	$-0.550983\pi$
$587$	−7.72804	−0.318970	−0.159485	+	0.987200i	$0.550983\pi$
$588$	9.86529	0.406838
$589$	−19.6309	−0.808878
$590$	7.11061	0.292739
$591$	−4.66181	−0.191761
$592$	−6.11357	−0.251266
$593$	28.0002	1.14983	0.574916	−	0.818212i	$-0.305035\pi$
$593$	28.0002	1.14983	0.574916	+	0.818212i	$0.305035\pi$
$594$	21.1297	0.866964
$595$	2.33611	0.0957711
$596$	2.29721	0.0940974
$597$	−24.7270	−1.01201
$598$	−10.4773	−0.428450
$599$	−17.9957	−0.735284	−0.367642	−	0.929967i	$-0.619835\pi$
$599$	−17.9957	−0.735284	−0.367642	+	0.929967i	$0.619835\pi$
$600$	0.393359	0.0160588
$601$	26.2393	1.07032	0.535161	−	0.844750i	$-0.320251\pi$
$601$	26.2393	1.07032	0.535161	+	0.844750i	$0.320251\pi$
$602$	−0.555295	−0.0226321
$603$	12.9305	0.526571
$604$	−14.0637	−0.572246
$605$	6.56780	0.267019
$606$	0.0490278	0.00199162
$607$	−11.7113	−0.475348	−0.237674	−	0.971345i	$-0.576385\pi$
$607$	−11.7113	−0.475348	−0.237674	+	0.971345i	$0.576385\pi$
$608$	−2.32942	−0.0944704
$609$	−5.15410	−0.208855
$610$	7.89795	0.319779
$611$	50.3711	2.03780
$612$	1.66999	0.0675055
$613$	23.5214	0.950020	0.475010	−	0.879980i	$-0.342444\pi$
$613$	23.5214	0.950020	0.475010	+	0.879980i	$0.342444\pi$
$614$	8.38760	0.338496
$615$	5.81606	0.234526
$616$	−2.01788	−0.0813028
$617$	26.6782	1.07403	0.537013	−	0.843574i	$-0.319553\pi$
$617$	26.6782	1.07403	0.537013	+	0.843574i	$0.319553\pi$
$618$	−7.30576	−0.293881
$619$	21.1782	0.851224	0.425612	−	0.904906i	$-0.360059\pi$
$619$	21.1782	0.851224	0.425612	+	0.904906i	$0.360059\pi$
$620$	18.3331	0.736277
$621$	11.1648	0.448027
$622$	−6.61374	−0.265187
$623$	−5.40375	−0.216497
$624$	7.78667	0.311716
$625$	−23.5908	−0.943631
$626$	5.02795	0.200957
$627$	12.8240	0.512142
$628$	−16.5574	−0.660712
$629$	12.1817	0.485716
$630$	0.982611	0.0391482
$631$	21.3409	0.849568	0.424784	−	0.905295i	$-0.360350\pi$
$631$	21.3409	0.849568	0.424784	+	0.905295i	$0.360350\pi$
$632$	−15.4545	−0.614748
$633$	2.13834	0.0849914
$634$	23.1194	0.918189
$635$	41.8209	1.65961
$636$	−19.5279	−0.774330
$637$	35.5327	1.40786
$638$	−24.3534	−0.964162
$639$	13.9887	0.553386
$640$	2.17542	0.0859912
$641$	24.7015	0.975651	0.487825	−	0.872941i	$-0.337790\pi$
$641$	24.7015	0.975651	0.487825	+	0.872941i	$0.337790\pi$
$642$	−27.1315	−1.07080
$643$	49.5352	1.95348	0.976738	−	0.214435i	$-0.0687909\pi$
$643$	49.5352	1.95348	0.976738	+	0.214435i	$0.0687909\pi$
$644$	−1.06623	−0.0420154
$645$	−3.29570	−0.129768
$646$	4.64152	0.182618
$647$	−16.8721	−0.663309	−0.331654	−	0.943401i	$-0.607607\pi$
$647$	−16.8721	−0.663309	−0.331654	+	0.943401i	$0.607607\pi$
$648$	−5.78324	−0.227187
$649$	12.2383	0.480397
$650$	1.41680	0.0555713
$651$	6.67799	0.261731
$652$	−7.96697	−0.312011
$653$	−23.7065	−0.927707	−0.463853	−	0.885912i	$-0.653534\pi$
$653$	−23.7065	−0.927707	−0.463853	+	0.885912i	$0.653534\pi$
$654$	3.66801	0.143431
$655$	−46.7395	−1.82626
$656$	−1.81831	−0.0709931
$657$	3.59067	0.140085
$658$	5.12604	0.199834
$659$	10.2463	0.399140	0.199570	−	0.979884i	$-0.436045\pi$
$659$	10.2463	0.399140	0.199570	+	0.979884i	$0.436045\pi$
$660$	−11.9762	−0.466175
$661$	−5.37562	−0.209088	−0.104544	−	0.994520i	$-0.533338\pi$
$661$	−5.37562	−0.209088	−0.104544	+	0.994520i	$0.533338\pi$
$662$	−19.4297	−0.755156
$663$	−15.5155	−0.602570
$664$	−6.93141	−0.268991
$665$	2.73104	0.105905
$666$	5.12385	0.198545
$667$	−12.8681	−0.498257
$668$	6.83737	0.264546
$669$	18.3234	0.708423
$670$	−33.5627	−1.29664
$671$	13.5935	0.524770
$672$	0.792415	0.0305681
$673$	45.2078	1.74263	0.871317	−	0.490720i	$-0.163266\pi$
$673$	45.2078	1.74263	0.871317	+	0.490720i	$0.163266\pi$
$674$	14.8285	0.571174
$675$	−1.50975	−0.0581105
$676$	15.0459	0.578690
$677$	28.2706	1.08653	0.543263	−	0.839563i	$-0.317189\pi$
$677$	28.2706	1.08653	0.543263	+	0.839563i	$0.317189\pi$
$678$	−16.2634	−0.624592
$679$	6.77542	0.260017
$680$	−4.33468	−0.166227
$681$	11.7020	0.448422
$682$	31.5539	1.20826
$683$	−8.18664	−0.313253	−0.156627	−	0.987658i	$-0.550062\pi$
$683$	−8.18664	−0.313253	−0.156627	+	0.987658i	$0.550062\pi$
$684$	1.95231	0.0746485
$685$	7.88034	0.301092
$686$	7.38855	0.282096
$687$	28.4543	1.08560
$688$	1.03036	0.0392820
$689$	−70.3353	−2.67956
$690$	−6.32814	−0.240908
$691$	9.81216	0.373272	0.186636	−	0.982429i	$-0.440241\pi$
$691$	9.81216	0.373272	0.186636	+	0.982429i	$0.440241\pi$
$692$	26.0273	0.989410
$693$	1.69121	0.0642438
$694$	24.3072	0.922687
$695$	23.4140	0.888145
$696$	9.56350	0.362504
$697$	3.62311	0.137235
$698$	16.1378	0.610826
$699$	−24.3511	−0.921045
$700$	0.144181	0.00544953
$701$	26.7309	1.00961	0.504807	−	0.863232i	$-0.331564\pi$
$701$	26.7309	1.00961	0.504807	+	0.863232i	$0.331564\pi$
$702$	−29.8861	−1.12798
$703$	14.2411	0.537112
$704$	3.74421	0.141115
$705$	30.4233	1.14581
$706$	17.8948	0.673480
$707$	0.0179706	0.000675852 0
$708$	−4.80595	−0.180619
$709$	−18.8963	−0.709664	−0.354832	−	0.934930i	$-0.615462\pi$
$709$	−18.8963	−0.709664	−0.354832	+	0.934930i	$0.615462\pi$
$710$	−36.3095	−1.36267
$711$	12.9526	0.485761
$712$	10.0267	0.375768
$713$	16.6728	0.624401
$714$	−1.57894	−0.0590903
$715$	−43.1359	−1.61319
$716$	−22.9389	−0.857267
$717$	40.5082	1.51281
$718$	−13.5716	−0.506489
$719$	4.46561	0.166539	0.0832696	−	0.996527i	$-0.473464\pi$
$719$	4.46561	0.166539	0.0832696	+	0.996527i	$0.473464\pi$
$720$	−1.82325	−0.0679484
$721$	−2.67784	−0.0997281
$722$	−13.5738	−0.505165
$723$	23.7149	0.881967
$724$	−5.21303	−0.193741
$725$	1.74009	0.0646254
$726$	−4.43908	−0.164750
$727$	40.0787	1.48644	0.743218	−	0.669050i	$-0.233299\pi$
$727$	40.0787	1.48644	0.743218	+	0.669050i	$0.233299\pi$
$728$	2.85411	0.105780
$729$	29.7397	1.10147
$730$	−9.32003	−0.344950
$731$	−2.05306	−0.0759350
$732$	−5.33811	−0.197302
$733$	−42.3520	−1.56431	−0.782153	−	0.623087i	$-0.785878\pi$
$733$	−42.3520	−1.56431	−0.782153	+	0.623087i	$0.785878\pi$
$734$	−11.7646	−0.434241
$735$	21.4612	0.791608
$736$	1.97841	0.0729251
$737$	−57.7662	−2.12784
$738$	1.52395	0.0560973
$739$	−52.8497	−1.94411	−0.972053	−	0.234760i	$-0.924570\pi$
$739$	−52.8497	−1.94411	−0.972053	+	0.234760i	$0.924570\pi$
$740$	−13.2996	−0.488903
$741$	−18.1384	−0.666331
$742$	−7.15771	−0.262768
$743$	−11.9054	−0.436766	−0.218383	−	0.975863i	$-0.570078\pi$
$743$	−11.9054	−0.436766	−0.218383	+	0.975863i	$0.570078\pi$
$744$	−12.3911	−0.454279
$745$	4.99741	0.183091
$746$	−17.6789	−0.647270
$747$	5.80929	0.212551
$748$	−7.46058	−0.272786
$749$	−9.94475	−0.363373
$750$	16.8487	0.615229
$751$	−52.6275	−1.92040	−0.960202	−	0.279307i	$-0.909895\pi$
$751$	−52.6275	−1.92040	−0.960202	+	0.279307i	$0.909895\pi$
$752$	−9.51144	−0.346846
$753$	−19.6899	−0.717540
$754$	34.4457	1.25444
$755$	−30.5946	−1.11345
$756$	−3.04138	−0.110614
$757$	−10.1632	−0.369388	−0.184694	−	0.982796i	$-0.559129\pi$
$757$	−10.1632	−0.369388	−0.184694	+	0.982796i	$0.559129\pi$
$758$	−29.0058	−1.05354
$759$	−10.8916	−0.395341
$760$	−5.06747	−0.183817
$761$	2.76167	0.100110	0.0500552	−	0.998746i	$-0.484060\pi$
$761$	2.76167	0.100110	0.0500552	+	0.998746i	$0.484060\pi$
$762$	−28.2661	−1.02397
$763$	1.34447	0.0486730
$764$	2.24248	0.0811301
$765$	3.63294	0.131349
$766$	−8.95716	−0.323635
$767$	−17.3100	−0.625029
$768$	−1.47034	−0.0530562
$769$	36.6348	1.32109	0.660543	−	0.750789i	$-0.270326\pi$
$769$	36.6348	1.32109	0.660543	+	0.750789i	$0.270326\pi$
$770$	−4.38975	−0.158196
$771$	13.8708	0.499545
$772$	−6.96472	−0.250666
$773$	23.5642	0.847547	0.423773	−	0.905768i	$-0.360705\pi$
$773$	23.5642	0.847547	0.423773	+	0.905768i	$0.360705\pi$
$774$	−0.863554	−0.0310398
$775$	−2.25458	−0.0809868
$776$	−12.5719	−0.451304
$777$	−4.84449	−0.173795
$778$	12.1691	0.436283
$779$	4.23561	0.151756
$780$	16.9393	0.606524
$781$	−62.4938	−2.23620
$782$	−3.94211	−0.140969
$783$	−36.7058	−1.31176
$784$	−6.70955	−0.239627
$785$	−36.0194	−1.28559
$786$	31.5905	1.12680
$787$	22.7795	0.812001	0.406000	−	0.913873i	$-0.366923\pi$
$787$	22.7795	0.812001	0.406000	+	0.913873i	$0.366923\pi$
$788$	3.17057	0.112947
$789$	−3.90475	−0.139013
$790$	−33.6201	−1.19615
$791$	−5.96116	−0.211954
$792$	−3.13806	−0.111506
$793$	−19.2267	−0.682761
$794$	34.5292	1.22540
$795$	−42.4814	−1.50666
$796$	16.8173	0.596072
$797$	−35.7128	−1.26501	−0.632506	−	0.774555i	$-0.717974\pi$
$797$	−35.7128	−1.26501	−0.632506	+	0.774555i	$0.717974\pi$
$798$	−1.84587	−0.0653430
$799$	18.9522	0.670480
$800$	−0.267530	−0.00945860
$801$	−8.40352	−0.296924
$802$	16.9355	0.598014
$803$	−16.0411	−0.566077
$804$	22.6846	0.800023
$805$	−2.31951	−0.0817519
$806$	−44.6301	−1.57203
$807$	1.67403	0.0589285
$808$	−0.0333446	−0.00117306
$809$	−38.3690	−1.34898	−0.674491	−	0.738283i	$-0.735637\pi$
$809$	−38.3690	−1.34898	−0.674491	+	0.738283i	$0.735637\pi$
$810$	−12.5810	−0.442051
$811$	44.6902	1.56928	0.784642	−	0.619949i	$-0.212847\pi$
$811$	44.6902	1.56928	0.784642	+	0.619949i	$0.212847\pi$
$812$	3.50539	0.123015
$813$	−35.6118	−1.24896
$814$	−22.8905	−0.802311
$815$	−17.3315	−0.607097
$816$	2.92974	0.102562
$817$	−2.40013	−0.0839700
$818$	−21.5229	−0.752531
$819$	−2.39206	−0.0835855
$820$	−3.95560	−0.138136
$821$	29.9717	1.04602	0.523009	−	0.852327i	$-0.324809\pi$
$821$	29.9717	1.04602	0.523009	+	0.852327i	$0.324809\pi$
$822$	−5.32620	−0.185773
$823$	−11.0819	−0.386290	−0.193145	−	0.981170i	$-0.561869\pi$
$823$	−11.0819	−0.386290	−0.193145	+	0.981170i	$0.561869\pi$
$824$	4.96877	0.173095
$825$	1.47282	0.0512769
$826$	−1.76157	−0.0612927
$827$	21.8707	0.760519	0.380260	−	0.924880i	$-0.375835\pi$
$827$	21.8707	0.760519	0.380260	+	0.924880i	$0.375835\pi$
$828$	−1.65813	−0.0576238
$829$	−18.6703	−0.648445	−0.324222	−	0.945981i	$-0.605103\pi$
$829$	−18.6703	−0.648445	−0.324222	+	0.945981i	$0.605103\pi$
$830$	−15.0787	−0.523391
$831$	33.0795	1.14752
$832$	−5.29584	−0.183600
$833$	13.3692	0.463216
$834$	−15.8252	−0.547982
$835$	14.8742	0.514742
$836$	−8.72183	−0.301651
$837$	47.5584	1.64386
$838$	10.7224	0.370398
$839$	19.4211	0.670491	0.335246	−	0.942131i	$-0.391181\pi$
$839$	19.4211	0.670491	0.335246	+	0.942131i	$0.391181\pi$
$840$	1.72384	0.0594781
$841$	13.3059	0.458823
$842$	−22.6907	−0.781973
$843$	15.4989	0.533810
$844$	−1.45432	−0.0500598
$845$	32.7313	1.12599
$846$	7.97164	0.274071
$847$	−1.62709	−0.0559076
$848$	13.2812	0.456079
$849$	42.5072	1.45884
$850$	0.533071	0.0182842
$851$	−12.0951	−0.414616
$852$	24.5411	0.840763
$853$	−22.7906	−0.780337	−0.390168	−	0.920744i	$-0.627583\pi$
$853$	−22.7906	−0.780337	−0.390168	+	0.920744i	$0.627583\pi$
$854$	−1.95662	−0.0669542
$855$	4.24711	0.145248
$856$	18.4526	0.630697
$857$	16.1607	0.552040	0.276020	−	0.961152i	$-0.410984\pi$
$857$	16.1607	0.552040	0.276020	+	0.961152i	$0.410984\pi$
$858$	29.1549	0.995332
$859$	18.8998	0.644851	0.322426	−	0.946595i	$-0.395502\pi$
$859$	18.8998	0.644851	0.322426	+	0.946595i	$0.395502\pi$
$860$	2.24146	0.0764333
$861$	−1.44086	−0.0491043
$862$	25.0787	0.854185
$863$	41.8347	1.42407	0.712035	−	0.702144i	$-0.247774\pi$
$863$	41.8347	1.42407	0.712035	+	0.702144i	$0.247774\pi$
$864$	5.64331	0.191989
$865$	56.6205	1.92515
$866$	−3.16813	−0.107657
$867$	19.1580	0.650640
$868$	−4.54181	−0.154159
$869$	−57.8649	−1.96293
$870$	20.8047	0.705345
$871$	81.7050	2.76847
$872$	−2.49468	−0.0844804
$873$	10.5366	0.356611
$874$	−4.60854	−0.155886
$875$	6.17571	0.208777
$876$	6.29927	0.212833
$877$	32.3876	1.09365	0.546826	−	0.837246i	$-0.315836\pi$
$877$	32.3876	1.09365	0.546826	+	0.837246i	$0.315836\pi$
$878$	−26.3533	−0.889381
$879$	−7.47089	−0.251987
$880$	8.14524	0.274576
$881$	53.4358	1.80030	0.900148	−	0.435583i	$-0.143458\pi$
$881$	53.4358	1.80030	0.900148	+	0.435583i	$0.143458\pi$
$882$	5.62335	0.189348
$883$	−18.3602	−0.617870	−0.308935	−	0.951083i	$-0.599973\pi$
$883$	−18.3602	−0.617870	−0.308935	+	0.951083i	$0.599973\pi$
$884$	10.5523	0.354913
$885$	−10.4550	−0.351440
$886$	16.2411	0.545630
$887$	−12.6834	−0.425868	−0.212934	−	0.977067i	$-0.568302\pi$
$887$	−12.6834	−0.425868	−0.212934	+	0.977067i	$0.568302\pi$
$888$	8.98901	0.301651
$889$	−10.3606	−0.347484
$890$	21.8124	0.731153
$891$	−21.6536	−0.725424
$892$	−12.4620	−0.417260
$893$	22.1561	0.741426
$894$	−3.37767	−0.112966
$895$	−49.9019	−1.66804
$896$	−0.538935	−0.0180045
$897$	15.4052	0.514365
$898$	−10.3249	−0.344547
$899$	−54.8142	−1.82816
$900$	0.224220	0.00747399
$901$	−26.4637	−0.881634
$902$	−6.80814	−0.226686
$903$	0.816470	0.0271704
$904$	11.0610	0.367883
$905$	−11.3406	−0.376973
$906$	20.6784	0.686995
$907$	−28.1335	−0.934157	−0.467078	−	0.884216i	$-0.654693\pi$
$907$	−28.1335	−0.934157	−0.467078	+	0.884216i	$0.654693\pi$
$908$	−7.95873	−0.264120
$909$	0.0279465	0.000926926 0
$910$	6.20891	0.205823
$911$	−57.2231	−1.89589	−0.947943	−	0.318440i	$-0.896841\pi$
$911$	−57.2231	−1.89589	−0.947943	+	0.318440i	$0.896841\pi$
$912$	3.42503	0.113414
$913$	−25.9526	−0.858906
$914$	0.342243	0.0113204
$915$	−11.6126	−0.383902
$916$	−19.3523	−0.639417
$917$	11.5791	0.382377
$918$	−11.2447	−0.371130
$919$	−29.0497	−0.958262	−0.479131	−	0.877743i	$-0.659048\pi$
$919$	−29.0497	−0.958262	−0.479131	+	0.877743i	$0.659048\pi$
$920$	4.30387	0.141895
$921$	−12.3326	−0.406373
$922$	23.0273	0.758364
$923$	88.3918	2.90945
$924$	2.96697	0.0976060
$925$	1.63556	0.0537769
$926$	−31.3639	−1.03068
$927$	−4.16438	−0.136776
$928$	−6.50430	−0.213514
$929$	13.8612	0.454772	0.227386	−	0.973805i	$-0.426982\pi$
$929$	13.8612	0.454772	0.227386	+	0.973805i	$0.426982\pi$
$930$	−26.9559	−0.883918
$931$	15.6294	0.512231
$932$	16.5616	0.542494
$933$	9.72442	0.318363
$934$	35.3321	1.15610
$935$	−16.2299	−0.530776
$936$	4.43851	0.145077
$937$	12.5950	0.411459	0.205730	−	0.978609i	$-0.434043\pi$
$937$	12.5950	0.411459	0.205730	+	0.978609i	$0.434043\pi$
$938$	8.31476	0.271487
$939$	−7.39277	−0.241254
$940$	−20.6914	−0.674879
$941$	−17.7925	−0.580019	−0.290010	−	0.957024i	$-0.593659\pi$
$941$	−17.7925	−0.580019	−0.290010	+	0.957024i	$0.593659\pi$
$942$	24.3449	0.793201
$943$	−3.59736	−0.117146
$944$	3.26861	0.106384
$945$	−6.61629	−0.215228
$946$	3.85787	0.125430
$947$	−36.9813	−1.20173	−0.600865	−	0.799351i	$-0.705177\pi$
$947$	−36.9813	−1.20173	−0.600865	+	0.799351i	$0.705177\pi$
$948$	22.7233	0.738020
$949$	22.6886	0.736504
$950$	0.623189	0.0202189
$951$	−33.9933	−1.10231
$952$	1.07386	0.0348041
$953$	2.64409	0.0856506	0.0428253	−	0.999083i	$-0.486364\pi$
$953$	2.64409	0.0856506	0.0428253	+	0.999083i	$0.486364\pi$
$954$	−11.1311	−0.360384
$955$	4.87834	0.157860
$956$	−27.5503	−0.891039
$957$	35.8077	1.15750
$958$	2.70686	0.0874546
$959$	−1.95226	−0.0630417
$960$	−3.19860	−0.103235
$961$	40.0208	1.29100
$962$	32.3765	1.04386
$963$	−15.4653	−0.498364
$964$	−16.1289	−0.519477
$965$	−15.1512	−0.487735
$966$	1.56772	0.0504406
$967$	−44.9907	−1.44680	−0.723402	−	0.690427i	$-0.757423\pi$
$967$	−44.9907	−1.44680	−0.723402	+	0.690427i	$0.757423\pi$
$968$	3.01909	0.0970372
$969$	−6.82460	−0.219238
$970$	−27.3492	−0.878129
$971$	−26.9405	−0.864563	−0.432282	−	0.901739i	$-0.642291\pi$
$971$	−26.9405	−0.864563	−0.432282	+	0.901739i	$0.642291\pi$
$972$	−8.42664	−0.270285
$973$	−5.80054	−0.185957
$974$	−15.9075	−0.509708
$975$	−2.08317	−0.0667147
$976$	3.63053	0.116211
$977$	12.0051	0.384078	0.192039	−	0.981387i	$-0.438490\pi$
$977$	12.0051	0.384078	0.192039	+	0.981387i	$0.438490\pi$
$978$	11.7141	0.374576
$979$	37.5422	1.19985
$980$	−14.5961	−0.466256
$981$	2.09082	0.0667546
$982$	−25.1029	−0.801066
$983$	17.6764	0.563789	0.281894	−	0.959445i	$-0.409037\pi$
$983$	17.6764	0.563789	0.281894	+	0.959445i	$0.409037\pi$
$984$	2.67353	0.0852290
$985$	6.89734	0.219767
$986$	12.9602	0.412738
$987$	−7.53701	−0.239906
$988$	12.3362	0.392468
$989$	2.03847	0.0648194
$990$	−6.82662	−0.216964
$991$	36.1980	1.14987	0.574934	−	0.818200i	$-0.305028\pi$
$991$	36.1980	1.14987	0.574934	+	0.818200i	$0.305028\pi$
$992$	8.42739	0.267570
$993$	28.5682	0.906584
$994$	8.99524	0.285312
$995$	36.5847	1.15981
$996$	10.1915	0.322930
$997$	36.2255	1.14727	0.573636	−	0.819110i	$-0.305532\pi$
$997$	36.2255	1.14727	0.573636	+	0.819110i	$0.305532\pi$
$998$	0.00691804	0.000218987 0
$999$	−34.5008	−1.09156

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6046.2.a.e.1.19	✓	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6046.2.a.e.1.19	✓	56	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:	\( N \)	\(=\)	\( 6046 = 2 \cdot 3023 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.47034 q^{3} +1.00000 q^{4} +2.17542 q^{5} -1.47034 q^{6} -0.538935 q^{7} +1.00000 q^{8} -0.838111 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.47034 q^{3} +1.00000 q^{4} +2.17542 q^{5} -1.47034 q^{6} -0.538935 q^{7} +1.00000 q^{8} -0.838111 q^{9} +2.17542 q^{10} +3.74421 q^{11} -1.47034 q^{12} -5.29584 q^{13} -0.538935 q^{14} -3.19860 q^{15} +1.00000 q^{16} -1.99257 q^{17} -0.838111 q^{18} -2.32942 q^{19} +2.17542 q^{20} +0.792415 q^{21} +3.74421 q^{22} +1.97841 q^{23} -1.47034 q^{24} -0.267530 q^{25} -5.29584 q^{26} +5.64331 q^{27} -0.538935 q^{28} -6.50430 q^{29} -3.19860 q^{30} +8.42739 q^{31} +1.00000 q^{32} -5.50524 q^{33} -1.99257 q^{34} -1.17241 q^{35} -0.838111 q^{36} -6.11357 q^{37} -2.32942 q^{38} +7.78667 q^{39} +2.17542 q^{40} -1.81831 q^{41} +0.792415 q^{42} +1.03036 q^{43} +3.74421 q^{44} -1.82325 q^{45} +1.97841 q^{46} -9.51144 q^{47} -1.47034 q^{48} -6.70955 q^{49} -0.267530 q^{50} +2.92974 q^{51} -5.29584 q^{52} +13.2812 q^{53} +5.64331 q^{54} +8.14524 q^{55} -0.538935 q^{56} +3.42503 q^{57} -6.50430 q^{58} +3.26861 q^{59} -3.19860 q^{60} +3.63053 q^{61} +8.42739 q^{62} +0.451687 q^{63} +1.00000 q^{64} -11.5207 q^{65} -5.50524 q^{66} -15.4281 q^{67} -1.99257 q^{68} -2.90892 q^{69} -1.17241 q^{70} -16.6908 q^{71} -0.838111 q^{72} -4.28424 q^{73} -6.11357 q^{74} +0.393359 q^{75} -2.32942 q^{76} -2.01788 q^{77} +7.78667 q^{78} -15.4545 q^{79} +2.17542 q^{80} -5.78324 q^{81} -1.81831 q^{82} -6.93141 q^{83} +0.792415 q^{84} -4.33468 q^{85} +1.03036 q^{86} +9.56350 q^{87} +3.74421 q^{88} +10.0267 q^{89} -1.82325 q^{90} +2.85411 q^{91} +1.97841 q^{92} -12.3911 q^{93} -9.51144 q^{94} -5.06747 q^{95} -1.47034 q^{96} -12.5719 q^{97} -6.70955 q^{98} -3.13806 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 56 q^{2} - 18 q^{3} + 56 q^{4} - 17 q^{5} - 18 q^{6} - 35 q^{7} + 56 q^{8} + 34 q^{9}+O(q^{10}) \) 56 * q + 56 * q^2 - 18 * q^3 + 56 * q^4 - 17 * q^5 - 18 * q^6 - 35 * q^7 + 56 * q^8 + 34 * q^9	\( 56 q + 56 q^{2} - 18 q^{3} + 56 q^{4} - 17 q^{5} - 18 q^{6} - 35 q^{7} + 56 q^{8} + 34 q^{9} - 17 q^{10} - 53 q^{11} - 18 q^{12} - 21 q^{13} - 35 q^{14} - 36 q^{15} + 56 q^{16} - 22 q^{17} + 34 q^{18} - 31 q^{19} - 17 q^{20} - 23 q^{21} - 53 q^{22} - 59 q^{23} - 18 q^{24} + 41 q^{25} - 21 q^{26} - 63 q^{27} - 35 q^{28} - 88 q^{29} - 36 q^{30} - 44 q^{31} + 56 q^{32} + 4 q^{33} - 22 q^{34} - 51 q^{35} + 34 q^{36} - 60 q^{37} - 31 q^{38} - 62 q^{39} - 17 q^{40} - 39 q^{41} - 23 q^{42} - 66 q^{43} - 53 q^{44} - 34 q^{45} - 59 q^{46} - 51 q^{47} - 18 q^{48} + 41 q^{49} + 41 q^{50} - 48 q^{51} - 21 q^{52} - 75 q^{53} - 63 q^{54} - 41 q^{55} - 35 q^{56} - 12 q^{57} - 88 q^{58} - 77 q^{59} - 36 q^{60} - 43 q^{61} - 44 q^{62} - 88 q^{63} + 56 q^{64} - 54 q^{65} + 4 q^{66} - 62 q^{67} - 22 q^{68} - 48 q^{69} - 51 q^{70} - 122 q^{71} + 34 q^{72} - 7 q^{73} - 60 q^{74} - 63 q^{75} - 31 q^{76} - 39 q^{77} - 62 q^{78} - 91 q^{79} - 17 q^{80} + 8 q^{81} - 39 q^{82} - 51 q^{83} - 23 q^{84} - 72 q^{85} - 66 q^{86} - 19 q^{87} - 53 q^{88} - 62 q^{89} - 34 q^{90} - 48 q^{91} - 59 q^{92} - 41 q^{93} - 51 q^{94} - 120 q^{95} - 18 q^{96} + 6 q^{97} + 41 q^{98} - 128 q^{99}+O(q^{100}) \) 56 * q + 56 * q^2 - 18 * q^3 + 56 * q^4 - 17 * q^5 - 18 * q^6 - 35 * q^7 + 56 * q^8 + 34 * q^9 - 17 * q^10 - 53 * q^11 - 18 * q^12 - 21 * q^13 - 35 * q^14 - 36 * q^15 + 56 * q^16 - 22 * q^17 + 34 * q^18 - 31 * q^19 - 17 * q^20 - 23 * q^21 - 53 * q^22 - 59 * q^23 - 18 * q^24 + 41 * q^25 - 21 * q^26 - 63 * q^27 - 35 * q^28 - 88 * q^29 - 36 * q^30 - 44 * q^31 + 56 * q^32 + 4 * q^33 - 22 * q^34 - 51 * q^35 + 34 * q^36 - 60 * q^37 - 31 * q^38 - 62 * q^39 - 17 * q^40 - 39 * q^41 - 23 * q^42 - 66 * q^43 - 53 * q^44 - 34 * q^45 - 59 * q^46 - 51 * q^47 - 18 * q^48 + 41 * q^49 + 41 * q^50 - 48 * q^51 - 21 * q^52 - 75 * q^53 - 63 * q^54 - 41 * q^55 - 35 * q^56 - 12 * q^57 - 88 * q^58 - 77 * q^59 - 36 * q^60 - 43 * q^61 - 44 * q^62 - 88 * q^63 + 56 * q^64 - 54 * q^65 + 4 * q^66 - 62 * q^67 - 22 * q^68 - 48 * q^69 - 51 * q^70 - 122 * q^71 + 34 * q^72 - 7 * q^73 - 60 * q^74 - 63 * q^75 - 31 * q^76 - 39 * q^77 - 62 * q^78 - 91 * q^79 - 17 * q^80 + 8 * q^81 - 39 * q^82 - 51 * q^83 - 23 * q^84 - 72 * q^85 - 66 * q^86 - 19 * q^87 - 53 * q^88 - 62 * q^89 - 34 * q^90 - 48 * q^91 - 59 * q^92 - 41 * q^93 - 51 * q^94 - 120 * q^95 - 18 * q^96 + 6 * q^97 + 41 * q^98 - 128 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more