LMFDB - Embedded newform 8007.2.a.f.1.33

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.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:	$63.9362168984$
Analytic rank:	$1$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.33
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.04801 q^{2} -1.00000 q^{3} -0.901675 q^{4} +2.25717 q^{5} -1.04801 q^{6} -3.56643 q^{7} -3.04098 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.04801 q^{2} -1.00000 q^{3} -0.901675 q^{4} +2.25717 q^{5} -1.04801 q^{6} -3.56643 q^{7} -3.04098 q^{8} +1.00000 q^{9} +2.36554 q^{10} -0.421505 q^{11} +0.901675 q^{12} +1.27099 q^{13} -3.73765 q^{14} -2.25717 q^{15} -1.38363 q^{16} -1.00000 q^{17} +1.04801 q^{18} +7.63021 q^{19} -2.03524 q^{20} +3.56643 q^{21} -0.441741 q^{22} -3.06983 q^{23} +3.04098 q^{24} +0.0948310 q^{25} +1.33201 q^{26} -1.00000 q^{27} +3.21576 q^{28} +6.05741 q^{29} -2.36554 q^{30} +0.390628 q^{31} +4.63191 q^{32} +0.421505 q^{33} -1.04801 q^{34} -8.05005 q^{35} -0.901675 q^{36} -8.69864 q^{37} +7.99653 q^{38} -1.27099 q^{39} -6.86403 q^{40} -5.09143 q^{41} +3.73765 q^{42} +4.23838 q^{43} +0.380060 q^{44} +2.25717 q^{45} -3.21721 q^{46} -4.45646 q^{47} +1.38363 q^{48} +5.71941 q^{49} +0.0993838 q^{50} +1.00000 q^{51} -1.14602 q^{52} +5.96794 q^{53} -1.04801 q^{54} -0.951409 q^{55} +10.8455 q^{56} -7.63021 q^{57} +6.34823 q^{58} +13.2068 q^{59} +2.03524 q^{60} +3.31573 q^{61} +0.409382 q^{62} -3.56643 q^{63} +7.62155 q^{64} +2.86884 q^{65} +0.441741 q^{66} -9.42494 q^{67} +0.901675 q^{68} +3.06983 q^{69} -8.43653 q^{70} -3.49949 q^{71} -3.04098 q^{72} -2.53428 q^{73} -9.11626 q^{74} -0.0948310 q^{75} -6.87997 q^{76} +1.50327 q^{77} -1.33201 q^{78} +3.30310 q^{79} -3.12310 q^{80} +1.00000 q^{81} -5.33587 q^{82} +7.63814 q^{83} -3.21576 q^{84} -2.25717 q^{85} +4.44186 q^{86} -6.05741 q^{87} +1.28179 q^{88} -6.18787 q^{89} +2.36554 q^{90} -4.53290 q^{91} +2.76799 q^{92} -0.390628 q^{93} -4.67041 q^{94} +17.2227 q^{95} -4.63191 q^{96} -15.5958 q^{97} +5.99400 q^{98} -0.421505 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) $ 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * 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 $

$2$	1.04801	0.741055	0.370527	−	0.928822i	$-0.379177\pi$
$2$	1.04801	0.741055	0.370527	+	0.928822i	$0.379177\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.901675	−0.450838
$5$	2.25717	1.00944	0.504719	−	0.863284i	$-0.331596\pi$
$5$	2.25717	1.00944	0.504719	+	0.863284i	$0.331596\pi$
$6$	−1.04801	−0.427848
$7$	−3.56643	−1.34798	−0.673992	−	0.738739i	$-0.735422\pi$
$7$	−3.56643	−1.34798	−0.673992	+	0.738739i	$0.735422\pi$
$8$	−3.04098	−1.07515
$9$	1.00000	0.333333
$10$	2.36554	0.748049
$11$	−0.421505	−0.127088	−0.0635442	−	0.997979i	$-0.520240\pi$
$11$	−0.421505	−0.127088	−0.0635442	+	0.997979i	$0.520240\pi$
$12$	0.901675	0.260291
$13$	1.27099	0.352509	0.176255	−	0.984345i	$-0.443602\pi$
$13$	1.27099	0.352509	0.176255	+	0.984345i	$0.443602\pi$
$14$	−3.73765	−0.998930
$15$	−2.25717	−0.582800
$16$	−1.38363	−0.345908
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	1.04801	0.247018
$19$	7.63021	1.75049	0.875245	−	0.483680i	$-0.160700\pi$
$19$	7.63021	1.75049	0.875245	+	0.483680i	$0.160700\pi$
$20$	−2.03524	−0.455093
$21$	3.56643	0.778259
$22$	−0.441741	−0.0941795
$23$	−3.06983	−0.640103	−0.320052	−	0.947400i	$-0.603700\pi$
$23$	−3.06983	−0.640103	−0.320052	+	0.947400i	$0.603700\pi$
$24$	3.04098	0.620738
$25$	0.0948310	0.0189662
$26$	1.33201	0.261229
$27$	−1.00000	−0.192450
$28$	3.21576	0.607722
$29$	6.05741	1.12483	0.562417	−	0.826854i	$-0.309872\pi$
$29$	6.05741	1.12483	0.562417	+	0.826854i	$0.309872\pi$
$30$	−2.36554	−0.431887
$31$	0.390628	0.0701589	0.0350794	−	0.999385i	$-0.488832\pi$
$31$	0.390628	0.0701589	0.0350794	+	0.999385i	$0.488832\pi$
$32$	4.63191	0.818814
$33$	0.421505	0.0733745
$34$	−1.04801	−0.179732
$35$	−8.05005	−1.36071
$36$	−0.901675	−0.150279
$37$	−8.69864	−1.43005	−0.715024	−	0.699100i	$-0.753584\pi$
$37$	−8.69864	−1.43005	−0.715024	+	0.699100i	$0.753584\pi$
$38$	7.99653	1.29721
$39$	−1.27099	−0.203521
$40$	−6.86403	−1.08530
$41$	−5.09143	−0.795148	−0.397574	−	0.917570i	$-0.630148\pi$
$41$	−5.09143	−0.795148	−0.397574	+	0.917570i	$0.630148\pi$
$42$	3.73765	0.576732
$43$	4.23838	0.646346	0.323173	−	0.946340i	$-0.395250\pi$
$43$	4.23838	0.646346	0.323173	+	0.946340i	$0.395250\pi$
$44$	0.380060	0.0572962
$45$	2.25717	0.336480
$46$	−3.21721	−0.474352
$47$	−4.45646	−0.650041	−0.325020	−	0.945707i	$-0.605371\pi$
$47$	−4.45646	−0.650041	−0.325020	+	0.945707i	$0.605371\pi$
$48$	1.38363	0.199710
$49$	5.71941	0.817059
$50$	0.0993838	0.0140550
$51$	1.00000	0.140028
$52$	−1.14602	−0.158924
$53$	5.96794	0.819760	0.409880	−	0.912139i	$-0.365571\pi$
$53$	5.96794	0.819760	0.409880	+	0.912139i	$0.365571\pi$
$54$	−1.04801	−0.142616
$55$	−0.951409	−0.128288
$56$	10.8455	1.44928
$57$	−7.63021	−1.01065
$58$	6.34823	0.833563
$59$	13.2068	1.71938	0.859688	−	0.510819i	$-0.170658\pi$
$59$	13.2068	1.71938	0.859688	+	0.510819i	$0.170658\pi$
$60$	2.03524	0.262748
$61$	3.31573	0.424536	0.212268	−	0.977211i	$-0.431915\pi$
$61$	3.31573	0.424536	0.212268	+	0.977211i	$0.431915\pi$
$62$	0.409382	0.0519916
$63$	−3.56643	−0.449328
$64$	7.62155	0.952694
$65$	2.86884	0.355836
$66$	0.441741	0.0543746
$67$	−9.42494	−1.15144	−0.575720	−	0.817647i	$-0.695278\pi$
$67$	−9.42494	−1.15144	−0.575720	+	0.817647i	$0.695278\pi$
$68$	0.901675	0.109344
$69$	3.06983	0.369564
$70$	−8.43653	−1.00836
$71$	−3.49949	−0.415313	−0.207656	−	0.978202i	$-0.566584\pi$
$71$	−3.49949	−0.415313	−0.207656	+	0.978202i	$0.566584\pi$
$72$	−3.04098	−0.358383
$73$	−2.53428	−0.296615	−0.148307	−	0.988941i	$-0.547382\pi$
$73$	−2.53428	−0.296615	−0.148307	+	0.988941i	$0.547382\pi$
$74$	−9.11626	−1.05974
$75$	−0.0948310	−0.0109501
$76$	−6.87997	−0.789186
$77$	1.50327	0.171313
$78$	−1.33201	−0.150820
$79$	3.30310	0.371627	0.185814	−	0.982585i	$-0.440508\pi$
$79$	3.30310	0.371627	0.185814	+	0.982585i	$0.440508\pi$
$80$	−3.12310	−0.349173
$81$	1.00000	0.111111
$82$	−5.33587	−0.589248
$83$	7.63814	0.838394	0.419197	−	0.907895i	$-0.362312\pi$
$83$	7.63814	0.838394	0.419197	+	0.907895i	$0.362312\pi$
$84$	−3.21576	−0.350868
$85$	−2.25717	−0.244825
$86$	4.44186	0.478978
$87$	−6.05741	−0.649423
$88$	1.28179	0.136639
$89$	−6.18787	−0.655913	−0.327957	−	0.944693i	$-0.606360\pi$
$89$	−6.18787	−0.655913	−0.327957	+	0.944693i	$0.606360\pi$
$90$	2.36554	0.249350
$91$	−4.53290	−0.475177
$92$	2.76799	0.288583
$93$	−0.390628	−0.0405062
$94$	−4.67041	−0.481716
$95$	17.2227	1.76701
$96$	−4.63191	−0.472742
$97$	−15.5958	−1.58352	−0.791759	−	0.610834i	$-0.790834\pi$
$97$	−15.5958	−1.58352	−0.791759	+	0.610834i	$0.790834\pi$
$98$	5.99400	0.605486
$99$	−0.421505	−0.0423628
$100$	−0.0855067	−0.00855067
$101$	−6.17997	−0.614930	−0.307465	−	0.951559i	$-0.599481\pi$
$101$	−6.17997	−0.614930	−0.307465	+	0.951559i	$0.599481\pi$
$102$	1.04801	0.103768
$103$	8.29378	0.817210	0.408605	−	0.912711i	$-0.366015\pi$
$103$	8.29378	0.817210	0.408605	+	0.912711i	$0.366015\pi$
$104$	−3.86506	−0.379000
$105$	8.05005	0.785604
$106$	6.25446	0.607487
$107$	−8.12096	−0.785083	−0.392542	−	0.919734i	$-0.628404\pi$
$107$	−8.12096	−0.785083	−0.392542	+	0.919734i	$0.628404\pi$
$108$	0.901675	0.0867637
$109$	−4.46239	−0.427420	−0.213710	−	0.976897i	$-0.568555\pi$
$109$	−4.46239	−0.427420	−0.213710	+	0.976897i	$0.568555\pi$
$110$	−0.997086	−0.0950684
$111$	8.69864	0.825639
$112$	4.93462	0.466278
$113$	4.59014	0.431804	0.215902	−	0.976415i	$-0.430731\pi$
$113$	4.59014	0.431804	0.215902	+	0.976415i	$0.430731\pi$
$114$	−7.99653	−0.748944
$115$	−6.92913	−0.646145
$116$	−5.46182	−0.507117
$117$	1.27099	0.117503
$118$	13.8408	1.27415
$119$	3.56643	0.326934
$120$	6.86403	0.626597
$121$	−10.8223	−0.983849
$122$	3.47492	0.314605
$123$	5.09143	0.459079
$124$	−0.352220	−0.0316303
$125$	−11.0718	−0.990293
$126$	−3.73765	−0.332977
$127$	20.1734	1.79010	0.895049	−	0.445967i	$-0.147140\pi$
$127$	20.1734	1.79010	0.895049	+	0.445967i	$0.147140\pi$
$128$	−1.27636	−0.112815
$129$	−4.23838	−0.373168
$130$	3.00658	0.263694
$131$	−3.99782	−0.349291	−0.174645	−	0.984631i	$-0.555878\pi$
$131$	−3.99782	−0.349291	−0.174645	+	0.984631i	$0.555878\pi$
$132$	−0.380060	−0.0330800
$133$	−27.2126	−2.35963
$134$	−9.87744	−0.853280
$135$	−2.25717	−0.194267
$136$	3.04098	0.260762
$137$	−5.00848	−0.427903	−0.213952	−	0.976844i	$-0.568633\pi$
$137$	−5.00848	−0.427903	−0.213952	+	0.976844i	$0.568633\pi$
$138$	3.21721	0.273867
$139$	3.93158	0.333473	0.166736	−	0.986002i	$-0.446677\pi$
$139$	3.93158	0.333473	0.166736	+	0.986002i	$0.446677\pi$
$140$	7.25853	0.613458
$141$	4.45646	0.375301
$142$	−3.66750	−0.307769
$143$	−0.535728	−0.0447998
$144$	−1.38363	−0.115303
$145$	13.6726	1.13545
$146$	−2.65595	−0.219808
$147$	−5.71941	−0.471729
$148$	7.84335	0.644719
$149$	−7.57201	−0.620323	−0.310162	−	0.950684i	$-0.600383\pi$
$149$	−7.57201	−0.620323	−0.310162	+	0.950684i	$0.600383\pi$
$150$	−0.0993838	−0.00811465
$151$	−11.5120	−0.936834	−0.468417	−	0.883508i	$-0.655175\pi$
$151$	−11.5120	−0.936834	−0.468417	+	0.883508i	$0.655175\pi$
$152$	−23.2033	−1.88204
$153$	−1.00000	−0.0808452
$154$	1.57544	0.126952
$155$	0.881715	0.0708211
$156$	1.14602	0.0917550
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	3.46168	0.275396
$159$	−5.96794	−0.473289
$160$	10.4550	0.826542
$161$	10.9483	0.862849
$162$	1.04801	0.0823394
$163$	−1.21962	−0.0955283	−0.0477642	−	0.998859i	$-0.515210\pi$
$163$	−1.21962	−0.0955283	−0.0477642	+	0.998859i	$0.515210\pi$
$164$	4.59081	0.358482
$165$	0.951409	0.0740671
$166$	8.00484	0.621296
$167$	−0.879959	−0.0680933	−0.0340466	−	0.999420i	$-0.510839\pi$
$167$	−0.879959	−0.0680933	−0.0340466	+	0.999420i	$0.510839\pi$
$168$	−10.8455	−0.836745
$169$	−11.3846	−0.875737
$170$	−2.36554	−0.181429
$171$	7.63021	0.583496
$172$	−3.82164	−0.291397
$173$	2.61409	0.198746	0.0993728	−	0.995050i	$-0.468316\pi$
$173$	2.61409	0.198746	0.0993728	+	0.995050i	$0.468316\pi$
$174$	−6.34823	−0.481258
$175$	−0.338208	−0.0255661
$176$	0.583207	0.0439609
$177$	−13.2068	−0.992682
$178$	−6.48495	−0.486068
$179$	19.6040	1.46527	0.732637	−	0.680620i	$-0.238289\pi$
$179$	19.6040	1.46527	0.732637	+	0.680620i	$0.238289\pi$
$180$	−2.03524	−0.151698
$181$	−12.1854	−0.905733	−0.452866	−	0.891578i	$-0.649599\pi$
$181$	−12.1854	−0.905733	−0.452866	+	0.891578i	$0.649599\pi$
$182$	−4.75052	−0.352132
$183$	−3.31573	−0.245106
$184$	9.33530	0.688207
$185$	−19.6343	−1.44355
$186$	−0.409382	−0.0300174
$187$	0.421505	0.0308235
$188$	4.01828	0.293063
$189$	3.56643	0.259420
$190$	18.0496	1.30945
$191$	−23.0490	−1.66777	−0.833884	−	0.551939i	$-0.813888\pi$
$191$	−23.0490	−1.66777	−0.833884	+	0.551939i	$0.813888\pi$
$192$	−7.62155	−0.550038
$193$	−7.07541	−0.509299	−0.254650	−	0.967033i	$-0.581960\pi$
$193$	−7.07541	−0.509299	−0.254650	+	0.967033i	$0.581960\pi$
$194$	−16.3446	−1.17347
$195$	−2.86884	−0.205442
$196$	−5.15705	−0.368361
$197$	−23.2658	−1.65762	−0.828809	−	0.559532i	$-0.810981\pi$
$197$	−23.2658	−1.65762	−0.828809	+	0.559532i	$0.810981\pi$
$198$	−0.441741	−0.0313932
$199$	−11.4408	−0.811015	−0.405507	−	0.914092i	$-0.632905\pi$
$199$	−11.4408	−0.811015	−0.405507	+	0.914092i	$0.632905\pi$
$200$	−0.288380	−0.0203915
$201$	9.42494	0.664784
$202$	−6.47667	−0.455697
$203$	−21.6033	−1.51626
$204$	−0.901675	−0.0631299
$205$	−11.4922	−0.802653
$206$	8.69196	0.605598
$207$	−3.06983	−0.213368
$208$	−1.75858	−0.121936
$209$	−3.21617	−0.222467
$210$	8.43653	0.582176
$211$	0.124091	0.00854280	0.00427140	−	0.999991i	$-0.498640\pi$
$211$	0.124091	0.00854280	0.00427140	+	0.999991i	$0.498640\pi$
$212$	−5.38115	−0.369579
$213$	3.49949	0.239781
$214$	−8.51085	−0.581790
$215$	9.56675	0.652447
$216$	3.04098	0.206913
$217$	−1.39315	−0.0945730
$218$	−4.67663	−0.316741
$219$	2.53428	0.171251
$220$	0.857862	0.0578370
$221$	−1.27099	−0.0854960
$222$	9.11626	0.611843
$223$	−2.00533	−0.134287	−0.0671435	−	0.997743i	$-0.521389\pi$
$223$	−2.00533	−0.134287	−0.0671435	+	0.997743i	$0.521389\pi$
$224$	−16.5194	−1.10375
$225$	0.0948310	0.00632207
$226$	4.81051	0.319990
$227$	−12.3595	−0.820327	−0.410163	−	0.912012i	$-0.634528\pi$
$227$	−12.3595	−0.820327	−0.410163	+	0.912012i	$0.634528\pi$
$228$	6.87997	0.455637
$229$	8.69393	0.574511	0.287256	−	0.957854i	$-0.407257\pi$
$229$	8.69393	0.574511	0.287256	+	0.957854i	$0.407257\pi$
$230$	−7.26180	−0.478829
$231$	−1.50327	−0.0989076
$232$	−18.4205	−1.20937
$233$	−28.2536	−1.85095	−0.925477	−	0.378804i	$-0.876336\pi$
$233$	−28.2536	−1.85095	−0.925477	+	0.378804i	$0.876336\pi$
$234$	1.33201	0.0870762
$235$	−10.0590	−0.656176
$236$	−11.9082	−0.775159
$237$	−3.30310	−0.214559
$238$	3.73765	0.242276
$239$	−6.43478	−0.416232	−0.208116	−	0.978104i	$-0.566733\pi$
$239$	−6.43478	−0.416232	−0.208116	+	0.978104i	$0.566733\pi$
$240$	3.12310	0.201595
$241$	−5.43886	−0.350347	−0.175174	−	0.984538i	$-0.556049\pi$
$241$	−5.43886	−0.350347	−0.175174	+	0.984538i	$0.556049\pi$
$242$	−11.3419	−0.729086
$243$	−1.00000	−0.0641500
$244$	−2.98971	−0.191397
$245$	12.9097	0.824771
$246$	5.33587	0.340202
$247$	9.69792	0.617064
$248$	−1.18789	−0.0754313
$249$	−7.63814	−0.484047
$250$	−11.6034	−0.733862
$251$	−9.26943	−0.585081	−0.292541	−	0.956253i	$-0.594501\pi$
$251$	−9.26943	−0.585081	−0.292541	+	0.956253i	$0.594501\pi$
$252$	3.21576	0.202574
$253$	1.29395	0.0813497
$254$	21.1419	1.32656
$255$	2.25717	0.141350
$256$	−16.5807	−1.03630
$257$	−22.9228	−1.42989	−0.714943	−	0.699182i	$-0.753548\pi$
$257$	−22.9228	−1.42989	−0.714943	+	0.699182i	$0.753548\pi$
$258$	−4.44186	−0.276538
$259$	31.0231	1.92768
$260$	−2.58677	−0.160424
$261$	6.05741	0.374944
$262$	−4.18975	−0.258844
$263$	16.7423	1.03237	0.516186	−	0.856477i	$-0.327351\pi$
$263$	16.7423	1.03237	0.516186	+	0.856477i	$0.327351\pi$
$264$	−1.28179	−0.0788887
$265$	13.4707	0.827497
$266$	−28.5191	−1.74862
$267$	6.18787	0.378692
$268$	8.49824	0.519113
$269$	5.40082	0.329294	0.164647	−	0.986353i	$-0.447352\pi$
$269$	5.40082	0.329294	0.164647	+	0.986353i	$0.447352\pi$
$270$	−2.36554	−0.143962
$271$	4.80776	0.292050	0.146025	−	0.989281i	$-0.453352\pi$
$271$	4.80776	0.292050	0.146025	+	0.989281i	$0.453352\pi$
$272$	1.38363	0.0838950
$273$	4.53290	0.274343
$274$	−5.24894	−0.317100
$275$	−0.0399717	−0.00241038
$276$	−2.76799	−0.166613
$277$	−13.2220	−0.794431	−0.397215	−	0.917725i	$-0.630023\pi$
$277$	−13.2220	−0.794431	−0.397215	+	0.917725i	$0.630023\pi$
$278$	4.12034	0.247121
$279$	0.390628	0.0233863
$280$	24.4801	1.46296
$281$	10.7484	0.641195	0.320597	−	0.947216i	$-0.396116\pi$
$281$	10.7484	0.641195	0.320597	+	0.947216i	$0.396116\pi$
$282$	4.67041	0.278119
$283$	−11.3334	−0.673700	−0.336850	−	0.941558i	$-0.609362\pi$
$283$	−11.3334	−0.673700	−0.336850	+	0.941558i	$0.609362\pi$
$284$	3.15540	0.187238
$285$	−17.2227	−1.02018
$286$	−0.561448	−0.0331991
$287$	18.1582	1.07185
$288$	4.63191	0.272938
$289$	1.00000	0.0588235
$290$	14.3291	0.841431
$291$	15.5958	0.914245
$292$	2.28509	0.133725
$293$	−31.4410	−1.83680	−0.918402	−	0.395648i	$-0.870520\pi$
$293$	−31.4410	−1.83680	−0.918402	+	0.395648i	$0.870520\pi$
$294$	−5.99400	−0.349577
$295$	29.8100	1.73560
$296$	26.4524	1.53752
$297$	0.421505	0.0244582
$298$	−7.93554	−0.459694
$299$	−3.90172	−0.225642
$300$	0.0855067	0.00493673
$301$	−15.1159	−0.871264
$302$	−12.0647	−0.694245
$303$	6.17997	0.355030
$304$	−10.5574	−0.605508
$305$	7.48418	0.428543
$306$	−1.04801	−0.0599107
$307$	2.77979	0.158651	0.0793255	−	0.996849i	$-0.474723\pi$
$307$	2.77979	0.158651	0.0793255	+	0.996849i	$0.474723\pi$
$308$	−1.35546	−0.0772344
$309$	−8.29378	−0.471817
$310$	0.924046	0.0524823
$311$	18.3551	1.04082	0.520410	−	0.853917i	$-0.325779\pi$
$311$	18.3551	1.04082	0.520410	+	0.853917i	$0.325779\pi$
$312$	3.86506	0.218816
$313$	23.4219	1.32388	0.661941	−	0.749556i	$-0.269733\pi$
$313$	23.4219	1.32388	0.661941	+	0.749556i	$0.269733\pi$
$314$	−1.04801	−0.0591426
$315$	−8.05005	−0.453569
$316$	−2.97832	−0.167544
$317$	4.76548	0.267656	0.133828	−	0.991005i	$-0.457273\pi$
$317$	4.76548	0.267656	0.133828	+	0.991005i	$0.457273\pi$
$318$	−6.25446	−0.350733
$319$	−2.55323	−0.142953
$320$	17.2032	0.961686
$321$	8.12096	0.453268
$322$	11.4739	0.639418
$323$	−7.63021	−0.424556
$324$	−0.901675	−0.0500931
$325$	0.120529	0.00668576
$326$	−1.27818	−0.0707917
$327$	4.46239	0.246771
$328$	15.4830	0.854903
$329$	15.8936	0.876244
$330$	0.997086	0.0548878
$331$	6.21994	0.341879	0.170939	−	0.985282i	$-0.445320\pi$
$331$	6.21994	0.341879	0.170939	+	0.985282i	$0.445320\pi$
$332$	−6.88712	−0.377980
$333$	−8.69864	−0.476683
$334$	−0.922206	−0.0504609
$335$	−21.2737	−1.16231
$336$	−4.93462	−0.269206
$337$	−16.6725	−0.908208	−0.454104	−	0.890949i	$-0.650041\pi$
$337$	−16.6725	−0.908208	−0.454104	+	0.890949i	$0.650041\pi$
$338$	−11.9312	−0.648969
$339$	−4.59014	−0.249302
$340$	2.03524	0.110376
$341$	−0.164652	−0.00891638
$342$	7.99653	0.432403
$343$	4.56712	0.246601
$344$	−12.8888	−0.694919
$345$	6.92913	0.373052
$346$	2.73959	0.147281
$347$	21.8226	1.17150	0.585750	−	0.810492i	$-0.300800\pi$
$347$	21.8226	1.17150	0.585750	+	0.810492i	$0.300800\pi$
$348$	5.46182	0.292784
$349$	−10.4391	−0.558794	−0.279397	−	0.960176i	$-0.590135\pi$
$349$	−10.4391	−0.558794	−0.279397	+	0.960176i	$0.590135\pi$
$350$	−0.354445	−0.0189459
$351$	−1.27099	−0.0678404
$352$	−1.95237	−0.104062
$353$	−20.4592	−1.08893	−0.544467	−	0.838783i	$-0.683268\pi$
$353$	−20.4592	−1.08893	−0.544467	+	0.838783i	$0.683268\pi$
$354$	−13.8408	−0.735632
$355$	−7.89895	−0.419232
$356$	5.57945	0.295710
$357$	−3.56643	−0.188755
$358$	20.5452	1.08585
$359$	−30.0263	−1.58473	−0.792364	−	0.610049i	$-0.791150\pi$
$359$	−30.0263	−1.58473	−0.792364	+	0.610049i	$0.791150\pi$
$360$	−6.86403	−0.361766
$361$	39.2200	2.06421
$362$	−12.7704	−0.671198
$363$	10.8223	0.568025
$364$	4.08720	0.214227
$365$	−5.72030	−0.299414
$366$	−3.47492	−0.181637
$367$	26.8682	1.40251	0.701255	−	0.712910i	$-0.252623\pi$
$367$	26.8682	1.40251	0.701255	+	0.712910i	$0.252623\pi$
$368$	4.24751	0.221417
$369$	−5.09143	−0.265049
$370$	−20.5770	−1.06975
$371$	−21.2842	−1.10502
$372$	0.352220	0.0182617
$373$	−35.9335	−1.86057	−0.930283	−	0.366842i	$-0.880439\pi$
$373$	−35.9335	−1.86057	−0.930283	+	0.366842i	$0.880439\pi$
$374$	0.441741	0.0228419
$375$	11.0718	0.571746
$376$	13.5520	0.698892
$377$	7.69891	0.396514
$378$	3.73765	0.192244
$379$	7.61080	0.390940	0.195470	−	0.980710i	$-0.437377\pi$
$379$	7.61080	0.390940	0.195470	+	0.980710i	$0.437377\pi$
$380$	−15.5293	−0.796635
$381$	−20.1734	−1.03351
$382$	−24.1556	−1.23591
$383$	−20.2083	−1.03260	−0.516299	−	0.856408i	$-0.672691\pi$
$383$	−20.2083	−1.03260	−0.516299	+	0.856408i	$0.672691\pi$
$384$	1.27636	0.0651339
$385$	3.39313	0.172930
$386$	−7.41510	−0.377419
$387$	4.23838	0.215449
$388$	14.0624	0.713909
$389$	5.63426	0.285668	0.142834	−	0.989747i	$-0.454378\pi$
$389$	5.63426	0.285668	0.142834	+	0.989747i	$0.454378\pi$
$390$	−3.00658	−0.152244
$391$	3.06983	0.155248
$392$	−17.3926	−0.878461
$393$	3.99782	0.201663
$394$	−24.3828	−1.22839
$395$	7.45566	0.375135
$396$	0.380060	0.0190987
$397$	−22.6177	−1.13515	−0.567574	−	0.823322i	$-0.692118\pi$
$397$	−22.6177	−1.13515	−0.567574	+	0.823322i	$0.692118\pi$
$398$	−11.9900	−0.601006
$399$	27.2126	1.36233
$400$	−0.131211	−0.00656056
$401$	−9.68940	−0.483866	−0.241933	−	0.970293i	$-0.577781\pi$
$401$	−9.68940	−0.483866	−0.241933	+	0.970293i	$0.577781\pi$
$402$	9.87744	0.492642
$403$	0.496484	0.0247316
$404$	5.57233	0.277234
$405$	2.25717	0.112160
$406$	−22.6405	−1.12363
$407$	3.66652	0.181743
$408$	−3.04098	−0.150551
$409$	11.0779	0.547768	0.273884	−	0.961763i	$-0.411692\pi$
$409$	11.0779	0.547768	0.273884	+	0.961763i	$0.411692\pi$
$410$	−12.0440	−0.594810
$411$	5.00848	0.247050
$412$	−7.47829	−0.368429
$413$	−47.1010	−2.31769
$414$	−3.21721	−0.158117
$415$	17.2406	0.846308
$416$	5.88711	0.288639
$417$	−3.93158	−0.192530
$418$	−3.37057	−0.164860
$419$	−17.7910	−0.869147	−0.434574	−	0.900636i	$-0.643101\pi$
$419$	−17.7910	−0.869147	−0.434574	+	0.900636i	$0.643101\pi$
$420$	−7.25853	−0.354180
$421$	9.33643	0.455030	0.227515	−	0.973775i	$-0.426940\pi$
$421$	9.33643	0.455030	0.227515	+	0.973775i	$0.426940\pi$
$422$	0.130049	0.00633068
$423$	−4.45646	−0.216680
$424$	−18.1484	−0.881365
$425$	−0.0948310	−0.00459998
$426$	3.66750	0.177691
$427$	−11.8253	−0.572267
$428$	7.32247	0.353945
$429$	0.535728	0.0258652
$430$	10.0260	0.483499
$431$	−8.79932	−0.423848	−0.211924	−	0.977286i	$-0.567973\pi$
$431$	−8.79932	−0.423848	−0.211924	+	0.977286i	$0.567973\pi$
$432$	1.38363	0.0665700
$433$	18.3743	0.883010	0.441505	−	0.897259i	$-0.354445\pi$
$433$	18.3743	0.883010	0.441505	+	0.897259i	$0.354445\pi$
$434$	−1.46003	−0.0700838
$435$	−13.6726	−0.655553
$436$	4.02363	0.192697
$437$	−23.4234	−1.12049
$438$	2.65595	0.126906
$439$	17.6191	0.840911	0.420456	−	0.907313i	$-0.361870\pi$
$439$	17.6191	0.840911	0.420456	+	0.907313i	$0.361870\pi$
$440$	2.89322	0.137929
$441$	5.71941	0.272353
$442$	−1.33201	−0.0633573
$443$	7.11162	0.337883	0.168942	−	0.985626i	$-0.445965\pi$
$443$	7.11162	0.337883	0.168942	+	0.985626i	$0.445965\pi$
$444$	−7.84335	−0.372229
$445$	−13.9671	−0.662104
$446$	−2.10161	−0.0995141
$447$	7.57201	0.358144
$448$	−27.1817	−1.28422
$449$	16.1775	0.763461	0.381731	−	0.924274i	$-0.375328\pi$
$449$	16.1775	0.763461	0.381731	+	0.924274i	$0.375328\pi$
$450$	0.0993838	0.00468500
$451$	2.14606	0.101054
$452$	−4.13881	−0.194673
$453$	11.5120	0.540881
$454$	−12.9528	−0.607907
$455$	−10.2315	−0.479661
$456$	23.2033	1.08660
$457$	−0.714642	−0.0334295	−0.0167148	−	0.999860i	$-0.505321\pi$
$457$	−0.714642	−0.0334295	−0.0167148	+	0.999860i	$0.505321\pi$
$458$	9.11133	0.425744
$459$	1.00000	0.0466760
$460$	6.24783	0.291306
$461$	33.0737	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$461$	33.0737	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$462$	−1.57544	−0.0732960
$463$	40.4768	1.88112	0.940559	−	0.339630i	$-0.110302\pi$
$463$	40.4768	1.88112	0.940559	+	0.339630i	$0.110302\pi$
$464$	−8.38123	−0.389089
$465$	−0.881715	−0.0408886
$466$	−29.6100	−1.37166
$467$	−8.17607	−0.378343	−0.189172	−	0.981944i	$-0.560580\pi$
$467$	−8.17607	−0.378343	−0.189172	+	0.981944i	$0.560580\pi$
$468$	−1.14602	−0.0529748
$469$	33.6134	1.55212
$470$	−10.5419	−0.486263
$471$	1.00000	0.0460776
$472$	−40.1616	−1.84859
$473$	−1.78649	−0.0821431
$474$	−3.46168	−0.159000
$475$	0.723580	0.0332001
$476$	−3.21576	−0.147394
$477$	5.96794	0.273253
$478$	−6.74372	−0.308450
$479$	−36.3104	−1.65907	−0.829533	−	0.558458i	$-0.811393\pi$
$479$	−36.3104	−1.65907	−0.829533	+	0.558458i	$0.811393\pi$
$480$	−10.4550	−0.477204
$481$	−11.0559	−0.504105
$482$	−5.69997	−0.259627
$483$	−10.9483	−0.498166
$484$	9.75823	0.443556
$485$	−35.2025	−1.59846
$486$	−1.04801	−0.0475387
$487$	14.7074	0.666458	0.333229	−	0.942846i	$-0.391862\pi$
$487$	14.7074	0.666458	0.333229	+	0.942846i	$0.391862\pi$
$488$	−10.0831	−0.456440
$489$	1.21962	0.0551533
$490$	13.5295	0.611201
$491$	36.6252	1.65287	0.826437	−	0.563029i	$-0.190364\pi$
$491$	36.6252	1.65287	0.826437	+	0.563029i	$0.190364\pi$
$492$	−4.59081	−0.206970
$493$	−6.05741	−0.272812
$494$	10.1635	0.457278
$495$	−0.951409	−0.0427626
$496$	−0.540485	−0.0242685
$497$	12.4807	0.559834
$498$	−8.00484	−0.358706
$499$	−23.3665	−1.04603	−0.523014	−	0.852324i	$-0.675192\pi$
$499$	−23.3665	−1.04603	−0.523014	+	0.852324i	$0.675192\pi$
$500$	9.98318	0.446461
$501$	0.879959	0.0393137
$502$	−9.71446	−0.433577
$503$	−36.0217	−1.60613	−0.803065	−	0.595892i	$-0.796799\pi$
$503$	−36.0217	−1.60613	−0.803065	+	0.595892i	$0.796799\pi$
$504$	10.8455	0.483095
$505$	−13.9493	−0.620734
$506$	1.35607	0.0602846
$507$	11.3846	0.505607
$508$	−18.1898	−0.807044
$509$	20.0525	0.888811	0.444405	−	0.895826i	$-0.353415\pi$
$509$	20.0525	0.888811	0.444405	+	0.895826i	$0.353415\pi$
$510$	2.36554	0.104748
$511$	9.03832	0.399832
$512$	−14.8241	−0.655137
$513$	−7.63021	−0.336882
$514$	−24.0234	−1.05962
$515$	18.7205	0.824923
$516$	3.82164	0.168238
$517$	1.87842	0.0826127
$518$	32.5125	1.42852
$519$	−2.61409	−0.114746
$520$	−8.72411	−0.382578
$521$	6.21976	0.272493	0.136246	−	0.990675i	$-0.456496\pi$
$521$	6.21976	0.272493	0.136246	+	0.990675i	$0.456496\pi$
$522$	6.34823	0.277854
$523$	−40.9217	−1.78938	−0.894690	−	0.446687i	$-0.852604\pi$
$523$	−40.9217	−1.78938	−0.894690	+	0.446687i	$0.852604\pi$
$524$	3.60473	0.157473
$525$	0.338208	0.0147606
$526$	17.5460	0.765044
$527$	−0.390628	−0.0170160
$528$	−0.583207	−0.0253808
$529$	−13.5762	−0.590268
$530$	14.1174	0.613221
$531$	13.2068	0.573125
$532$	24.5369	1.06381
$533$	−6.47115	−0.280297
$534$	6.48495	0.280631
$535$	−18.3304	−0.792493
$536$	28.6611	1.23797
$537$	−19.6040	−0.845976
$538$	5.66012	0.244025
$539$	−2.41076	−0.103839
$540$	2.03524	0.0875827
$541$	18.2912	0.786398	0.393199	−	0.919453i	$-0.371368\pi$
$541$	18.2912	0.786398	0.393199	+	0.919453i	$0.371368\pi$
$542$	5.03858	0.216425
$543$	12.1854	0.522925
$544$	−4.63191	−0.198591
$545$	−10.0724	−0.431454
$546$	4.75052	0.203303
$547$	−8.62032	−0.368578	−0.184289	−	0.982872i	$-0.558998\pi$
$547$	−8.62032	−0.368578	−0.184289	+	0.982872i	$0.558998\pi$
$548$	4.51602	0.192915
$549$	3.31573	0.141512
$550$	−0.0418907	−0.00178623
$551$	46.2193	1.96901
$552$	−9.33530	−0.397337
$553$	−11.7803	−0.500948
$554$	−13.8568	−0.588717
$555$	19.6343	0.833431
$556$	−3.54501	−0.150342
$557$	6.27485	0.265874	0.132937	−	0.991124i	$-0.457559\pi$
$557$	6.27485	0.265874	0.132937	+	0.991124i	$0.457559\pi$
$558$	0.409382	0.0173305
$559$	5.38693	0.227843
$560$	11.1383	0.470679
$561$	−0.421505	−0.0177959
$562$	11.2644	0.475161
$563$	−35.2018	−1.48358	−0.741789	−	0.670633i	$-0.766023\pi$
$563$	−35.2018	−1.48358	−0.741789	+	0.670633i	$0.766023\pi$
$564$	−4.01828	−0.169200
$565$	10.3607	0.435879
$566$	−11.8775	−0.499249
$567$	−3.56643	−0.149776
$568$	10.6419	0.446523
$569$	38.1849	1.60079	0.800397	−	0.599471i	$-0.204622\pi$
$569$	38.1849	1.60079	0.800397	+	0.599471i	$0.204622\pi$
$570$	−18.0496	−0.756013
$571$	−11.7239	−0.490632	−0.245316	−	0.969443i	$-0.578892\pi$
$571$	−11.7239	−0.490632	−0.245316	+	0.969443i	$0.578892\pi$
$572$	0.483053	0.0201974
$573$	23.0490	0.962887
$574$	19.0300	0.794296
$575$	−0.291115	−0.0121403
$576$	7.62155	0.317565
$577$	10.5620	0.439703	0.219852	−	0.975533i	$-0.429443\pi$
$577$	10.5620	0.439703	0.219852	+	0.975533i	$0.429443\pi$
$578$	1.04801	0.0435915
$579$	7.07541	0.294044
$580$	−12.3283	−0.511904
$581$	−27.2409	−1.13014
$582$	16.3446	0.677505
$583$	−2.51552	−0.104182
$584$	7.70670	0.318905
$585$	2.86884	0.118612
$586$	−32.9505	−1.36117
$587$	1.48901	0.0614579	0.0307290	−	0.999528i	$-0.490217\pi$
$587$	1.48901	0.0614579	0.0307290	+	0.999528i	$0.490217\pi$
$588$	5.15705	0.212673
$589$	2.98057	0.122812
$590$	31.2412	1.28618
$591$	23.2658	0.957026
$592$	12.0357	0.494665
$593$	−30.5617	−1.25502	−0.627509	−	0.778609i	$-0.715926\pi$
$593$	−30.5617	−1.25502	−0.627509	+	0.778609i	$0.715926\pi$
$594$	0.441741	0.0181249
$595$	8.05005	0.330020
$596$	6.82750	0.279665
$597$	11.4408	0.468240
$598$	−4.08904	−0.167213
$599$	−23.0670	−0.942493	−0.471246	−	0.882002i	$-0.656196\pi$
$599$	−23.0670	−0.942493	−0.471246	+	0.882002i	$0.656196\pi$
$600$	0.288380	0.0117730
$601$	−39.8889	−1.62710	−0.813551	−	0.581493i	$-0.802469\pi$
$601$	−39.8889	−1.62710	−0.813551	+	0.581493i	$0.802469\pi$
$602$	−15.8416	−0.645654
$603$	−9.42494	−0.383813
$604$	10.3801	0.422360
$605$	−24.4279	−0.993135
$606$	6.47667	0.263097
$607$	2.80259	0.113754	0.0568769	−	0.998381i	$-0.481886\pi$
$607$	2.80259	0.113754	0.0568769	+	0.998381i	$0.481886\pi$
$608$	35.3424	1.43332
$609$	21.6033	0.875411
$610$	7.84350	0.317574
$611$	−5.66411	−0.229145
$612$	0.901675	0.0364481
$613$	13.3848	0.540607	0.270303	−	0.962775i	$-0.412876\pi$
$613$	13.3848	0.540607	0.270303	+	0.962775i	$0.412876\pi$
$614$	2.91325	0.117569
$615$	11.4922	0.463412
$616$	−4.57141	−0.184187
$617$	44.7508	1.80160	0.900801	−	0.434233i	$-0.142981\pi$
$617$	44.7508	1.80160	0.900801	+	0.434233i	$0.142981\pi$
$618$	−8.69196	−0.349642
$619$	−9.06908	−0.364517	−0.182259	−	0.983251i	$-0.558341\pi$
$619$	−9.06908	−0.364517	−0.182259	+	0.983251i	$0.558341\pi$
$620$	−0.795021	−0.0319288
$621$	3.06983	0.123188
$622$	19.2363	0.771304
$623$	22.0686	0.884160
$624$	1.75858	0.0703996
$625$	−25.4652	−1.01861
$626$	24.5463	0.981069
$627$	3.21617	0.128441
$628$	0.901675	0.0359808
$629$	8.69864	0.346838
$630$	−8.43653	−0.336119
$631$	−16.1839	−0.644271	−0.322135	−	0.946694i	$-0.604401\pi$
$631$	−16.1839	−0.644271	−0.322135	+	0.946694i	$0.604401\pi$
$632$	−10.0447	−0.399555
$633$	−0.124091	−0.00493219
$634$	4.99427	0.198348
$635$	45.5348	1.80699
$636$	5.38115	0.213376
$637$	7.26932	0.288021
$638$	−2.67581	−0.105936
$639$	−3.49949	−0.138438
$640$	−2.88096	−0.113880
$641$	26.2476	1.03672	0.518358	−	0.855164i	$-0.326543\pi$
$641$	26.2476	1.03672	0.518358	+	0.855164i	$0.326543\pi$
$642$	8.51085	0.335896
$643$	1.97964	0.0780695	0.0390347	−	0.999238i	$-0.487572\pi$
$643$	1.97964	0.0780695	0.0390347	+	0.999238i	$0.487572\pi$
$644$	−9.87183	−0.389005
$645$	−9.56675	−0.376690
$646$	−7.99653	−0.314619
$647$	−33.1377	−1.30278	−0.651388	−	0.758744i	$-0.725813\pi$
$647$	−33.1377	−1.30278	−0.651388	+	0.758744i	$0.725813\pi$
$648$	−3.04098	−0.119461
$649$	−5.56672	−0.218513
$650$	0.126316	0.00495451
$651$	1.39315	0.0546017
$652$	1.09970	0.0430678
$653$	35.6631	1.39561	0.697803	−	0.716290i	$-0.254161\pi$
$653$	35.6631	1.39561	0.697803	+	0.716290i	$0.254161\pi$
$654$	4.67663	0.182871
$655$	−9.02376	−0.352588
$656$	7.04466	0.275048
$657$	−2.53428	−0.0988716
$658$	16.6567	0.649345
$659$	−37.0274	−1.44238	−0.721191	−	0.692736i	$-0.756405\pi$
$659$	−37.0274	−1.44238	−0.721191	+	0.692736i	$0.756405\pi$
$660$	−0.857862	−0.0333922
$661$	−46.0223	−1.79006	−0.895029	−	0.446008i	$-0.852845\pi$
$661$	−46.0223	−1.79006	−0.895029	+	0.446008i	$0.852845\pi$
$662$	6.51856	0.253351
$663$	1.27099	0.0493612
$664$	−23.2275	−0.901400
$665$	−61.4235	−2.38190
$666$	−9.11626	−0.353248
$667$	−18.5952	−0.720010
$668$	0.793437	0.0306990
$669$	2.00533	0.0775307
$670$	−22.2951	−0.861334
$671$	−1.39760	−0.0539536
$672$	16.5194	0.637249
$673$	−26.9682	−1.03955	−0.519774	−	0.854304i	$-0.673984\pi$
$673$	−26.9682	−1.03955	−0.519774	+	0.854304i	$0.673984\pi$
$674$	−17.4729	−0.673032
$675$	−0.0948310	−0.00365005
$676$	10.2652	0.394815
$677$	−12.6308	−0.485440	−0.242720	−	0.970096i	$-0.578040\pi$
$677$	−12.6308	−0.485440	−0.242720	+	0.970096i	$0.578040\pi$
$678$	−4.81051	−0.184747
$679$	55.6215	2.13456
$680$	6.86403	0.263223
$681$	12.3595	0.473616
$682$	−0.172556	−0.00660753
$683$	−24.6097	−0.941664	−0.470832	−	0.882223i	$-0.656046\pi$
$683$	−24.6097	−0.941664	−0.470832	+	0.882223i	$0.656046\pi$
$684$	−6.87997	−0.263062
$685$	−11.3050	−0.431942
$686$	4.78639	0.182745
$687$	−8.69393	−0.331694
$688$	−5.86435	−0.223576
$689$	7.58520	0.288973
$690$	7.26180	0.276452
$691$	−10.6772	−0.406179	−0.203090	−	0.979160i	$-0.565098\pi$
$691$	−10.6772	−0.406179	−0.203090	+	0.979160i	$0.565098\pi$
$692$	−2.35706	−0.0896020
$693$	1.50327	0.0571044
$694$	22.8703	0.868145
$695$	8.87426	0.336620
$696$	18.4205	0.698227
$697$	5.09143	0.192852
$698$	−10.9403	−0.414097
$699$	28.2536	1.06865
$700$	0.304954	0.0115262
$701$	−12.3930	−0.468076	−0.234038	−	0.972227i	$-0.575194\pi$
$701$	−12.3930	−0.468076	−0.234038	+	0.972227i	$0.575194\pi$
$702$	−1.33201	−0.0502735
$703$	−66.3724	−2.50328
$704$	−3.21252	−0.121076
$705$	10.0590	0.378844
$706$	−21.4414	−0.806959
$707$	22.0404	0.828916
$708$	11.9082	0.447538
$709$	38.0417	1.42868	0.714342	−	0.699796i	$-0.246726\pi$
$709$	38.0417	1.42868	0.714342	+	0.699796i	$0.246726\pi$
$710$	−8.27817	−0.310674
$711$	3.30310	0.123876
$712$	18.8172	0.705205
$713$	−1.19916	−0.0449089
$714$	−3.73765	−0.139878
$715$	−1.20923	−0.0452227
$716$	−17.6765	−0.660600
$717$	6.43478	0.240311
$718$	−31.4679	−1.17437
$719$	35.8137	1.33562	0.667812	−	0.744330i	$-0.267231\pi$
$719$	35.8137	1.33562	0.667812	+	0.744330i	$0.267231\pi$
$720$	−3.12310	−0.116391
$721$	−29.5792	−1.10159
$722$	41.1030	1.52970
$723$	5.43886	0.202273
$724$	10.9873	0.408338
$725$	0.574430	0.0213338
$726$	11.3419	0.420938
$727$	42.4575	1.57466	0.787330	−	0.616531i	$-0.211463\pi$
$727$	42.4575	1.57466	0.787330	+	0.616531i	$0.211463\pi$
$728$	13.7845	0.510886
$729$	1.00000	0.0370370
$730$	−5.99493	−0.221882
$731$	−4.23838	−0.156762
$732$	2.98971	0.110503
$733$	−9.19777	−0.339728	−0.169864	−	0.985468i	$-0.554333\pi$
$733$	−9.19777	−0.339728	−0.169864	+	0.985468i	$0.554333\pi$
$734$	28.1582	1.03934
$735$	−12.9097	−0.476182
$736$	−14.2192	−0.524125
$737$	3.97266	0.146335
$738$	−5.33587	−0.196416
$739$	1.19669	0.0440211	0.0220106	−	0.999758i	$-0.492993\pi$
$739$	1.19669	0.0440211	0.0220106	+	0.999758i	$0.492993\pi$
$740$	17.7038	0.650805
$741$	−9.69792	−0.356262
$742$	−22.3061	−0.818883
$743$	10.8113	0.396629	0.198314	−	0.980138i	$-0.436453\pi$
$743$	10.8113	0.396629	0.198314	+	0.980138i	$0.436453\pi$
$744$	1.18789	0.0435503
$745$	−17.0913	−0.626178
$746$	−37.6587	−1.37878
$747$	7.63814	0.279465
$748$	−0.380060	−0.0138964
$749$	28.9628	1.05828
$750$	11.6034	0.423695
$751$	17.4646	0.637291	0.318646	−	0.947874i	$-0.396772\pi$
$751$	17.4646	0.637291	0.318646	+	0.947874i	$0.396772\pi$
$752$	6.16609	0.224854
$753$	9.26943	0.337797
$754$	8.06854	0.293839
$755$	−25.9846	−0.945676
$756$	−3.21576	−0.116956
$757$	−7.17784	−0.260883	−0.130441	−	0.991456i	$-0.541639\pi$
$757$	−7.17784	−0.260883	−0.130441	+	0.991456i	$0.541639\pi$
$758$	7.97619	0.289708
$759$	−1.29395	−0.0469673
$760$	−52.3740	−1.89980
$761$	25.1432	0.911441	0.455721	−	0.890123i	$-0.349382\pi$
$761$	25.1432	0.911441	0.455721	+	0.890123i	$0.349382\pi$
$762$	−21.1419	−0.765891
$763$	15.9148	0.576154
$764$	20.7827	0.751893
$765$	−2.25717	−0.0816083
$766$	−21.1785	−0.765212
$767$	16.7857	0.606096
$768$	16.5807	0.598306
$769$	28.5661	1.03012	0.515061	−	0.857154i	$-0.327769\pi$
$769$	28.5661	1.03012	0.515061	+	0.857154i	$0.327769\pi$
$770$	3.55604	0.128151
$771$	22.9228	0.825546
$772$	6.37972	0.229611
$773$	−16.5370	−0.594795	−0.297398	−	0.954754i	$-0.596119\pi$
$773$	−16.5370	−0.594795	−0.297398	+	0.954754i	$0.596119\pi$
$774$	4.44186	0.159659
$775$	0.0370436	0.00133065
$776$	47.4267	1.70252
$777$	−31.0231	−1.11295
$778$	5.90476	0.211696
$779$	−38.8486	−1.39190
$780$	2.58677	0.0926211
$781$	1.47505	0.0527814
$782$	3.21721	0.115047
$783$	−6.05741	−0.216474
$784$	−7.91356	−0.282627
$785$	−2.25717	−0.0805620
$786$	4.18975	0.149443
$787$	26.6057	0.948389	0.474195	−	0.880420i	$-0.342739\pi$
$787$	26.6057	0.948389	0.474195	+	0.880420i	$0.342739\pi$
$788$	20.9782	0.747316
$789$	−16.7423	−0.596040
$790$	7.81361	0.277996
$791$	−16.3704	−0.582064
$792$	1.28179	0.0455464
$793$	4.21426	0.149653
$794$	−23.7035	−0.841207
$795$	−13.4707	−0.477756
$796$	10.3159	0.365636
$797$	1.21962	0.0432010	0.0216005	−	0.999767i	$-0.493124\pi$
$797$	1.21962	0.0432010	0.0216005	+	0.999767i	$0.493124\pi$
$798$	28.5191	1.00956
$799$	4.45646	0.157658
$800$	0.439248	0.0155298
$801$	−6.18787	−0.218638
$802$	−10.1546	−0.358571
$803$	1.06821	0.0376963
$804$	−8.49824	−0.299710
$805$	24.7123	0.870993
$806$	0.520321	0.0183275
$807$	−5.40082	−0.190118
$808$	18.7932	0.661142
$809$	−37.8086	−1.32928	−0.664640	−	0.747163i	$-0.731415\pi$
$809$	−37.8086	−1.32928	−0.664640	+	0.747163i	$0.731415\pi$
$810$	2.36554	0.0831166
$811$	−17.3871	−0.610543	−0.305271	−	0.952265i	$-0.598747\pi$
$811$	−17.3871	−0.610543	−0.305271	+	0.952265i	$0.598747\pi$
$812$	19.4792	0.683586
$813$	−4.80776	−0.168615
$814$	3.84255	0.134681
$815$	−2.75290	−0.0964300
$816$	−1.38363	−0.0484368
$817$	32.3397	1.13142
$818$	11.6098	0.405926
$819$	−4.53290	−0.158392
$820$	10.3623	0.361866
$821$	44.8647	1.56579	0.782895	−	0.622154i	$-0.213742\pi$
$821$	44.8647	1.56579	0.782895	+	0.622154i	$0.213742\pi$
$822$	5.24894	0.183078
$823$	−48.8419	−1.70252	−0.851261	−	0.524743i	$-0.824161\pi$
$823$	−48.8419	−1.70252	−0.851261	+	0.524743i	$0.824161\pi$
$824$	−25.2212	−0.878624
$825$	0.0399717	0.00139164
$826$	−49.3623	−1.71754
$827$	23.4411	0.815128	0.407564	−	0.913177i	$-0.366378\pi$
$827$	23.4411	0.815128	0.407564	+	0.913177i	$0.366378\pi$
$828$	2.76799	0.0961942
$829$	−0.896816	−0.0311477	−0.0155739	−	0.999879i	$-0.504958\pi$
$829$	−0.896816	−0.0311477	−0.0155739	+	0.999879i	$0.504958\pi$
$830$	18.0683	0.627160
$831$	13.2220	0.458665
$832$	9.68691	0.335833
$833$	−5.71941	−0.198166
$834$	−4.12034	−0.142676
$835$	−1.98622	−0.0687360
$836$	2.89994	0.100296
$837$	−0.390628	−0.0135021
$838$	−18.6451	−0.644086
$839$	−18.3829	−0.634648	−0.317324	−	0.948317i	$-0.602784\pi$
$839$	−18.3829	−0.634648	−0.317324	+	0.948317i	$0.602784\pi$
$840$	−24.4801	−0.844643
$841$	7.69226	0.265250
$842$	9.78467	0.337202
$843$	−10.7484	−0.370194
$844$	−0.111890	−0.00385141
$845$	−25.6970	−0.884003
$846$	−4.67041	−0.160572
$847$	38.5971	1.32621
$848$	−8.25744	−0.283561
$849$	11.3334	0.388961
$850$	−0.0993838	−0.00340884
$851$	26.7033	0.915378
$852$	−3.15540	−0.108102
$853$	−41.8665	−1.43348	−0.716741	−	0.697339i	$-0.754367\pi$
$853$	−41.8665	−1.43348	−0.716741	+	0.697339i	$0.754367\pi$
$854$	−12.3931	−0.424082
$855$	17.2227	0.589004
$856$	24.6957	0.844082
$857$	15.4003	0.526063	0.263031	−	0.964787i	$-0.415278\pi$
$857$	15.4003	0.526063	0.263031	+	0.964787i	$0.415278\pi$
$858$	0.561448	0.0191675
$859$	18.8746	0.643992	0.321996	−	0.946741i	$-0.395646\pi$
$859$	18.8746	0.643992	0.321996	+	0.946741i	$0.395646\pi$
$860$	−8.62610	−0.294148
$861$	−18.1582	−0.618830
$862$	−9.22177	−0.314095
$863$	45.1180	1.53584	0.767918	−	0.640548i	$-0.221293\pi$
$863$	45.1180	1.53584	0.767918	+	0.640548i	$0.221293\pi$
$864$	−4.63191	−0.157581
$865$	5.90045	0.200621
$866$	19.2564	0.654359
$867$	−1.00000	−0.0339618
$868$	1.25617	0.0426371
$869$	−1.39227	−0.0472295
$870$	−14.3291	−0.485800
$871$	−11.9790	−0.405893
$872$	13.5701	0.459540
$873$	−15.5958	−0.527839
$874$	−24.5480	−0.830348
$875$	39.4868	1.33490
$876$	−2.28509	−0.0772062
$877$	27.5437	0.930084	0.465042	−	0.885289i	$-0.346039\pi$
$877$	27.5437	0.930084	0.465042	+	0.885289i	$0.346039\pi$
$878$	18.4649	0.623162
$879$	31.4410	1.06048
$880$	1.31640	0.0443758
$881$	7.79557	0.262639	0.131320	−	0.991340i	$-0.458079\pi$
$881$	7.79557	0.262639	0.131320	+	0.991340i	$0.458079\pi$
$882$	5.99400	0.201829
$883$	19.4591	0.654852	0.327426	−	0.944877i	$-0.393819\pi$
$883$	19.4591	0.654852	0.327426	+	0.944877i	$0.393819\pi$
$884$	1.14602	0.0385448
$885$	−29.8100	−1.00205
$886$	7.45305	0.250390
$887$	−31.2118	−1.04799	−0.523996	−	0.851721i	$-0.675559\pi$
$887$	−31.2118	−1.04799	−0.523996	+	0.851721i	$0.675559\pi$
$888$	−26.4524	−0.887686
$889$	−71.9469	−2.41302
$890$	−14.6377	−0.490656
$891$	−0.421505	−0.0141209
$892$	1.80816	0.0605417
$893$	−34.0037	−1.13789
$894$	7.93554	0.265404
$895$	44.2497	1.47910
$896$	4.55204	0.152073
$897$	3.90172	0.130275
$898$	16.9541	0.565767
$899$	2.36620	0.0789170
$900$	−0.0855067	−0.00285022
$901$	−5.96794	−0.198821
$902$	2.24909	0.0748866
$903$	15.1159	0.503025
$904$	−13.9585	−0.464254
$905$	−27.5045	−0.914281
$906$	12.0647	0.400823
$907$	−8.35956	−0.277575	−0.138787	−	0.990322i	$-0.544320\pi$
$907$	−8.35956	−0.277575	−0.138787	+	0.990322i	$0.544320\pi$
$908$	11.1442	0.369834
$909$	−6.17997	−0.204977
$910$	−10.7227	−0.355456
$911$	26.9822	0.893961	0.446981	−	0.894544i	$-0.352499\pi$
$911$	26.9822	0.893961	0.446981	+	0.894544i	$0.352499\pi$
$912$	10.5574	0.349590
$913$	−3.21951	−0.106550
$914$	−0.748952	−0.0247731
$915$	−7.48418	−0.247419
$916$	−7.83910	−0.259011
$917$	14.2579	0.470838
$918$	1.04801	0.0345895
$919$	−47.6301	−1.57117	−0.785586	−	0.618753i	$-0.787638\pi$
$919$	−47.6301	−1.57117	−0.785586	+	0.618753i	$0.787638\pi$
$920$	21.0714	0.694703
$921$	−2.77979	−0.0915972
$922$	34.6615	1.14152
$923$	−4.44781	−0.146401
$924$	1.35546	0.0445913
$925$	−0.824901	−0.0271226
$926$	42.4201	1.39401
$927$	8.29378	0.272403
$928$	28.0574	0.921029
$929$	36.8521	1.20908	0.604538	−	0.796576i	$-0.293358\pi$
$929$	36.8521	1.20908	0.604538	+	0.796576i	$0.293358\pi$
$930$	−0.924046	−0.0303007
$931$	43.6403	1.43025
$932$	25.4756	0.834480
$933$	−18.3551	−0.600917
$934$	−8.56860	−0.280373
$935$	0.951409	0.0311144
$936$	−3.86506	−0.126333
$937$	−52.1678	−1.70425	−0.852124	−	0.523341i	$-0.824686\pi$
$937$	−52.1678	−1.70425	−0.852124	+	0.523341i	$0.824686\pi$
$938$	35.2272	1.15021
$939$	−23.4219	−0.764344
$940$	9.06994	0.295829
$941$	−22.8975	−0.746439	−0.373219	−	0.927743i	$-0.621746\pi$
$941$	−22.8975	−0.746439	−0.373219	+	0.927743i	$0.621746\pi$
$942$	1.04801	0.0341460
$943$	15.6298	0.508977
$944$	−18.2733	−0.594746
$945$	8.05005	0.261868
$946$	−1.87226	−0.0608726
$947$	−10.2886	−0.334334	−0.167167	−	0.985929i	$-0.553462\pi$
$947$	−10.2886	−0.334334	−0.167167	+	0.985929i	$0.553462\pi$
$948$	2.97832	0.0967313
$949$	−3.22104	−0.104559
$950$	0.758319	0.0246031
$951$	−4.76548	−0.154531
$952$	−10.8455	−0.351503
$953$	37.8773	1.22697	0.613483	−	0.789708i	$-0.289768\pi$
$953$	37.8773	1.22697	0.613483	+	0.789708i	$0.289768\pi$
$954$	6.25446	0.202496
$955$	−52.0256	−1.68351
$956$	5.80208	0.187653
$957$	2.55323	0.0825341
$958$	−38.0537	−1.22946
$959$	17.8624	0.576806
$960$	−17.2032	−0.555230
$961$	−30.8474	−0.995078
$962$	−11.5867	−0.373569
$963$	−8.12096	−0.261694
$964$	4.90408	0.157950
$965$	−15.9704	−0.514106
$966$	−11.4739	−0.369168
$967$	0.957426	0.0307888	0.0153944	−	0.999881i	$-0.495100\pi$
$967$	0.957426	0.0307888	0.0153944	+	0.999881i	$0.495100\pi$
$968$	32.9105	1.05779
$969$	7.63021	0.245118
$970$	−36.8926	−1.18455
$971$	−22.4966	−0.721950	−0.360975	−	0.932576i	$-0.617556\pi$
$971$	−22.4966	−0.721950	−0.360975	+	0.932576i	$0.617556\pi$
$972$	0.901675	0.0289212
$973$	−14.0217	−0.449515
$974$	15.4135	0.493882
$975$	−0.120529	−0.00386002
$976$	−4.58775	−0.146850
$977$	−10.2955	−0.329384	−0.164692	−	0.986345i	$-0.552663\pi$
$977$	−10.2955	−0.329384	−0.164692	+	0.986345i	$0.552663\pi$
$978$	1.27818	0.0408716
$979$	2.60822	0.0833590
$980$	−11.6404	−0.371838
$981$	−4.46239	−0.142473
$982$	38.3836	1.22487
$983$	−11.7517	−0.374822	−0.187411	−	0.982282i	$-0.560010\pi$
$983$	−11.7517	−0.374822	−0.187411	+	0.982282i	$0.560010\pi$
$984$	−15.4830	−0.493579
$985$	−52.5149	−1.67326
$986$	−6.34823	−0.202169
$987$	−15.8936	−0.505900
$988$	−8.74437	−0.278195
$989$	−13.0111	−0.413728
$990$	−0.997086	−0.0316895
$991$	31.0682	0.986913	0.493457	−	0.869770i	$-0.335733\pi$
$991$	31.0682	0.986913	0.493457	+	0.869770i	$0.335733\pi$
$992$	1.80935	0.0574470
$993$	−6.21994	−0.197384
$994$	13.0799	0.414868
$995$	−25.8238	−0.818670
$996$	6.88712	0.218227
$997$	2.74773	0.0870215	0.0435108	−	0.999053i	$-0.486146\pi$
$997$	2.74773	0.0870215	0.0435108	+	0.999053i	$0.486146\pi$
$998$	−24.4883	−0.775164
$999$	8.69864	0.275213

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.f.1.33	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.f.1.33	✓	48	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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q+1.04801 q^{2} -1.00000 q^{3} -0.901675 q^{4} +2.25717 q^{5} -1.04801 q^{6} -3.56643 q^{7} -3.04098 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.04801 q^{2} -1.00000 q^{3} -0.901675 q^{4} +2.25717 q^{5} -1.04801 q^{6} -3.56643 q^{7} -3.04098 q^{8} +1.00000 q^{9} +2.36554 q^{10} -0.421505 q^{11} +0.901675 q^{12} +1.27099 q^{13} -3.73765 q^{14} -2.25717 q^{15} -1.38363 q^{16} -1.00000 q^{17} +1.04801 q^{18} +7.63021 q^{19} -2.03524 q^{20} +3.56643 q^{21} -0.441741 q^{22} -3.06983 q^{23} +3.04098 q^{24} +0.0948310 q^{25} +1.33201 q^{26} -1.00000 q^{27} +3.21576 q^{28} +6.05741 q^{29} -2.36554 q^{30} +0.390628 q^{31} +4.63191 q^{32} +0.421505 q^{33} -1.04801 q^{34} -8.05005 q^{35} -0.901675 q^{36} -8.69864 q^{37} +7.99653 q^{38} -1.27099 q^{39} -6.86403 q^{40} -5.09143 q^{41} +3.73765 q^{42} +4.23838 q^{43} +0.380060 q^{44} +2.25717 q^{45} -3.21721 q^{46} -4.45646 q^{47} +1.38363 q^{48} +5.71941 q^{49} +0.0993838 q^{50} +1.00000 q^{51} -1.14602 q^{52} +5.96794 q^{53} -1.04801 q^{54} -0.951409 q^{55} +10.8455 q^{56} -7.63021 q^{57} +6.34823 q^{58} +13.2068 q^{59} +2.03524 q^{60} +3.31573 q^{61} +0.409382 q^{62} -3.56643 q^{63} +7.62155 q^{64} +2.86884 q^{65} +0.441741 q^{66} -9.42494 q^{67} +0.901675 q^{68} +3.06983 q^{69} -8.43653 q^{70} -3.49949 q^{71} -3.04098 q^{72} -2.53428 q^{73} -9.11626 q^{74} -0.0948310 q^{75} -6.87997 q^{76} +1.50327 q^{77} -1.33201 q^{78} +3.30310 q^{79} -3.12310 q^{80} +1.00000 q^{81} -5.33587 q^{82} +7.63814 q^{83} -3.21576 q^{84} -2.25717 q^{85} +4.44186 q^{86} -6.05741 q^{87} +1.28179 q^{88} -6.18787 q^{89} +2.36554 q^{90} -4.53290 q^{91} +2.76799 q^{92} -0.390628 q^{93} -4.67041 q^{94} +17.2227 q^{95} -4.63191 q^{96} -15.5958 q^{97} +5.99400 q^{98} -0.421505 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more