LMFDB - Embedded newform 8007.2.a.e.1.46

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:	$46$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.46
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)

$f(q)$	$=$	$q+2.71698 q^{2} +1.00000 q^{3} +5.38198 q^{4} -1.62902 q^{5} +2.71698 q^{6} -4.45873 q^{7} +9.18876 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.71698 q^{2} +1.00000 q^{3} +5.38198 q^{4} -1.62902 q^{5} +2.71698 q^{6} -4.45873 q^{7} +9.18876 q^{8} +1.00000 q^{9} -4.42602 q^{10} -1.17747 q^{11} +5.38198 q^{12} -2.32445 q^{13} -12.1143 q^{14} -1.62902 q^{15} +14.2017 q^{16} -1.00000 q^{17} +2.71698 q^{18} -3.46662 q^{19} -8.76736 q^{20} -4.45873 q^{21} -3.19916 q^{22} -2.41966 q^{23} +9.18876 q^{24} -2.34629 q^{25} -6.31548 q^{26} +1.00000 q^{27} -23.9968 q^{28} -4.71574 q^{29} -4.42602 q^{30} -2.77564 q^{31} +20.2083 q^{32} -1.17747 q^{33} -2.71698 q^{34} +7.26337 q^{35} +5.38198 q^{36} +6.42528 q^{37} -9.41874 q^{38} -2.32445 q^{39} -14.9687 q^{40} -0.456417 q^{41} -12.1143 q^{42} -5.40658 q^{43} -6.33711 q^{44} -1.62902 q^{45} -6.57416 q^{46} -8.08398 q^{47} +14.2017 q^{48} +12.8803 q^{49} -6.37482 q^{50} -1.00000 q^{51} -12.5101 q^{52} +3.91837 q^{53} +2.71698 q^{54} +1.91812 q^{55} -40.9702 q^{56} -3.46662 q^{57} -12.8126 q^{58} +12.1047 q^{59} -8.76736 q^{60} +9.67494 q^{61} -7.54136 q^{62} -4.45873 q^{63} +26.5020 q^{64} +3.78658 q^{65} -3.19916 q^{66} -14.5543 q^{67} -5.38198 q^{68} -2.41966 q^{69} +19.7344 q^{70} -13.0914 q^{71} +9.18876 q^{72} -13.7004 q^{73} +17.4573 q^{74} -2.34629 q^{75} -18.6573 q^{76} +5.25002 q^{77} -6.31548 q^{78} -4.21560 q^{79} -23.1349 q^{80} +1.00000 q^{81} -1.24007 q^{82} +7.58654 q^{83} -23.9968 q^{84} +1.62902 q^{85} -14.6896 q^{86} -4.71574 q^{87} -10.8195 q^{88} +2.41565 q^{89} -4.42602 q^{90} +10.3641 q^{91} -13.0225 q^{92} -2.77564 q^{93} -21.9640 q^{94} +5.64720 q^{95} +20.2083 q^{96} -11.5460 q^{97} +34.9955 q^{98} -1.17747 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9}+O(q^{10}) $ 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9	\( 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9} - 10 q^{10} - 25 q^{11} + 43 q^{12} - 8 q^{13} - 28 q^{14} - 19 q^{15} + 33 q^{16} - 46 q^{17} - 5 q^{18} - 2 q^{19} - 56 q^{20} + q^{21} - 19 q^{22} - 64 q^{23} - 18 q^{24} + 11 q^{25} - 13 q^{26} + 46 q^{27} - 38 q^{28} - 51 q^{29} - 10 q^{30} - 19 q^{31} - 61 q^{32} - 25 q^{33} + 5 q^{34} - 39 q^{35} + 43 q^{36} - 46 q^{37} - 48 q^{38} - 8 q^{39} - 10 q^{40} - 53 q^{41} - 28 q^{42} - 33 q^{43} - 62 q^{44} - 19 q^{45} + 2 q^{46} - 45 q^{47} + 33 q^{48} + 21 q^{49} - 60 q^{50} - 46 q^{51} - 63 q^{52} - 47 q^{53} - 5 q^{54} + 5 q^{55} - 82 q^{56} - 2 q^{57} - 21 q^{58} - 65 q^{59} - 56 q^{60} - 37 q^{61} - 46 q^{62} + q^{63} + 74 q^{64} - 85 q^{65} - 19 q^{66} - 52 q^{67} - 43 q^{68} - 64 q^{69} - 20 q^{70} - 48 q^{71} - 18 q^{72} - 39 q^{73} - 16 q^{74} + 11 q^{75} + 42 q^{76} - 78 q^{77} - 13 q^{78} - 26 q^{79} - 78 q^{80} + 46 q^{81} + 3 q^{82} - 47 q^{83} - 38 q^{84} + 19 q^{85} - 6 q^{86} - 51 q^{87} - 58 q^{88} - 58 q^{89} - 10 q^{90} - 43 q^{91} - 68 q^{92} - 19 q^{93} - 78 q^{95} - 61 q^{96} - 44 q^{97} - 4 q^{98} - 25 q^{99}+O(q^{100}) \) 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9 - 10 * q^10 - 25 * q^11 + 43 * q^12 - 8 * q^13 - 28 * q^14 - 19 * q^15 + 33 * q^16 - 46 * q^17 - 5 * q^18 - 2 * q^19 - 56 * q^20 + q^21 - 19 * q^22 - 64 * q^23 - 18 * q^24 + 11 * q^25 - 13 * q^26 + 46 * q^27 - 38 * q^28 - 51 * q^29 - 10 * q^30 - 19 * q^31 - 61 * q^32 - 25 * q^33 + 5 * q^34 - 39 * q^35 + 43 * q^36 - 46 * q^37 - 48 * q^38 - 8 * q^39 - 10 * q^40 - 53 * q^41 - 28 * q^42 - 33 * q^43 - 62 * q^44 - 19 * q^45 + 2 * q^46 - 45 * q^47 + 33 * q^48 + 21 * q^49 - 60 * q^50 - 46 * q^51 - 63 * q^52 - 47 * q^53 - 5 * q^54 + 5 * q^55 - 82 * q^56 - 2 * q^57 - 21 * q^58 - 65 * q^59 - 56 * q^60 - 37 * q^61 - 46 * q^62 + q^63 + 74 * q^64 - 85 * q^65 - 19 * q^66 - 52 * q^67 - 43 * q^68 - 64 * q^69 - 20 * q^70 - 48 * q^71 - 18 * q^72 - 39 * q^73 - 16 * q^74 + 11 * q^75 + 42 * q^76 - 78 * q^77 - 13 * q^78 - 26 * q^79 - 78 * q^80 + 46 * q^81 + 3 * q^82 - 47 * q^83 - 38 * q^84 + 19 * q^85 - 6 * q^86 - 51 * q^87 - 58 * q^88 - 58 * q^89 - 10 * q^90 - 43 * q^91 - 68 * q^92 - 19 * q^93 - 78 * q^95 - 61 * q^96 - 44 * q^97 - 4 * q^98 - 25 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.71698	1.92119	0.960597	−	0.277944i	$-0.0896530\pi$
$2$	2.71698	1.92119	0.960597	+	0.277944i	$0.0896530\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.38198	2.69099
$5$	−1.62902	−0.728521	−0.364260	−	0.931297i	$-0.618678\pi$
$5$	−1.62902	−0.728521	−0.364260	+	0.931297i	$0.618678\pi$
$6$	2.71698	1.10920
$7$	−4.45873	−1.68524	−0.842621	−	0.538506i	$-0.818989\pi$
$7$	−4.45873	−1.68524	−0.842621	+	0.538506i	$0.818989\pi$
$8$	9.18876	3.24872
$9$	1.00000	0.333333
$10$	−4.42602	−1.39963
$11$	−1.17747	−0.355020	−0.177510	−	0.984119i	$-0.556804\pi$
$11$	−1.17747	−0.355020	−0.177510	+	0.984119i	$0.556804\pi$
$12$	5.38198	1.55364
$13$	−2.32445	−0.644686	−0.322343	−	0.946623i	$-0.604471\pi$
$13$	−2.32445	−0.644686	−0.322343	+	0.946623i	$0.604471\pi$
$14$	−12.1143	−3.23768
$15$	−1.62902	−0.420612
$16$	14.2017	3.55043
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	2.71698	0.640398
$19$	−3.46662	−0.795298	−0.397649	−	0.917538i	$-0.630174\pi$
$19$	−3.46662	−0.795298	−0.397649	+	0.917538i	$0.630174\pi$
$20$	−8.76736	−1.96044
$21$	−4.45873	−0.972975
$22$	−3.19916	−0.682062
$23$	−2.41966	−0.504534	−0.252267	−	0.967658i	$-0.581176\pi$
$23$	−2.41966	−0.504534	−0.252267	+	0.967658i	$0.581176\pi$
$24$	9.18876	1.87565
$25$	−2.34629	−0.469258
$26$	−6.31548	−1.23857
$27$	1.00000	0.192450
$28$	−23.9968	−4.53497
$29$	−4.71574	−0.875690	−0.437845	−	0.899050i	$-0.644258\pi$
$29$	−4.71574	−0.875690	−0.437845	+	0.899050i	$0.644258\pi$
$30$	−4.42602	−0.808077
$31$	−2.77564	−0.498520	−0.249260	−	0.968437i	$-0.580187\pi$
$31$	−2.77564	−0.498520	−0.249260	+	0.968437i	$0.580187\pi$
$32$	20.2083	3.57235
$33$	−1.17747	−0.204971
$34$	−2.71698	−0.465958
$35$	7.26337	1.22773
$36$	5.38198	0.896996
$37$	6.42528	1.05631	0.528155	−	0.849148i	$-0.322884\pi$
$37$	6.42528	1.05631	0.528155	+	0.849148i	$0.322884\pi$
$38$	−9.41874	−1.52792
$39$	−2.32445	−0.372210
$40$	−14.9687	−2.36676
$41$	−0.456417	−0.0712803	−0.0356401	−	0.999365i	$-0.511347\pi$
$41$	−0.456417	−0.0712803	−0.0356401	+	0.999365i	$0.511347\pi$
$42$	−12.1143	−1.86928
$43$	−5.40658	−0.824496	−0.412248	−	0.911072i	$-0.635256\pi$
$43$	−5.40658	−0.824496	−0.412248	+	0.911072i	$0.635256\pi$
$44$	−6.33711	−0.955355
$45$	−1.62902	−0.242840
$46$	−6.57416	−0.969307
$47$	−8.08398	−1.17917	−0.589585	−	0.807706i	$-0.700709\pi$
$47$	−8.08398	−1.17917	−0.589585	+	0.807706i	$0.700709\pi$
$48$	14.2017	2.04984
$49$	12.8803	1.84004
$50$	−6.37482	−0.901535
$51$	−1.00000	−0.140028
$52$	−12.5101	−1.73484
$53$	3.91837	0.538229	0.269115	−	0.963108i	$-0.413269\pi$
$53$	3.91837	0.538229	0.269115	+	0.963108i	$0.413269\pi$
$54$	2.71698	0.369734
$55$	1.91812	0.258639
$56$	−40.9702	−5.47488
$57$	−3.46662	−0.459165
$58$	−12.8126	−1.68237
$59$	12.1047	1.57590	0.787948	−	0.615741i	$-0.211143\pi$
$59$	12.1047	1.57590	0.787948	+	0.615741i	$0.211143\pi$
$60$	−8.76736	−1.13186
$61$	9.67494	1.23875	0.619375	−	0.785096i	$-0.287386\pi$
$61$	9.67494	1.23875	0.619375	+	0.785096i	$0.287386\pi$
$62$	−7.54136	−0.957753
$63$	−4.45873	−0.561748
$64$	26.5020	3.31275
$65$	3.78658	0.469667
$66$	−3.19916	−0.393789
$67$	−14.5543	−1.77810	−0.889048	−	0.457814i	$-0.848633\pi$
$67$	−14.5543	−1.77810	−0.889048	+	0.457814i	$0.848633\pi$
$68$	−5.38198	−0.652661
$69$	−2.41966	−0.291293
$70$	19.7344	2.35872
$71$	−13.0914	−1.55367	−0.776834	−	0.629705i	$-0.783176\pi$
$71$	−13.0914	−1.55367	−0.776834	+	0.629705i	$0.783176\pi$
$72$	9.18876	1.08291
$73$	−13.7004	−1.60352	−0.801758	−	0.597649i	$-0.796102\pi$
$73$	−13.7004	−1.60352	−0.801758	+	0.597649i	$0.796102\pi$
$74$	17.4573	2.02938
$75$	−2.34629	−0.270926
$76$	−18.6573	−2.14014
$77$	5.25002	0.598295
$78$	−6.31548	−0.715088
$79$	−4.21560	−0.474293	−0.237146	−	0.971474i	$-0.576212\pi$
$79$	−4.21560	−0.474293	−0.237146	+	0.971474i	$0.576212\pi$
$80$	−23.1349	−2.58656
$81$	1.00000	0.111111
$82$	−1.24007	−0.136943
$83$	7.58654	0.832730	0.416365	−	0.909197i	$-0.363304\pi$
$83$	7.58654	0.832730	0.416365	+	0.909197i	$0.363304\pi$
$84$	−23.9968	−2.61827
$85$	1.62902	0.176692
$86$	−14.6896	−1.58402
$87$	−4.71574	−0.505580
$88$	−10.8195	−1.15336
$89$	2.41565	0.256058	0.128029	−	0.991770i	$-0.459135\pi$
$89$	2.41565	0.256058	0.128029	+	0.991770i	$0.459135\pi$
$90$	−4.42602	−0.466543
$91$	10.3641	1.08645
$92$	−13.0225	−1.35769
$93$	−2.77564	−0.287821
$94$	−21.9640	−2.26541
$95$	5.64720	0.579391
$96$	20.2083	2.06250
$97$	−11.5460	−1.17232	−0.586158	−	0.810196i	$-0.699360\pi$
$97$	−11.5460	−1.17232	−0.586158	+	0.810196i	$0.699360\pi$
$98$	34.9955	3.53508
$99$	−1.17747	−0.118340
$100$	−12.6277	−1.26277
$101$	0.868849	0.0864537	0.0432269	−	0.999065i	$-0.486236\pi$
$101$	0.868849	0.0864537	0.0432269	+	0.999065i	$0.486236\pi$
$102$	−2.71698	−0.269021
$103$	16.4078	1.61671	0.808354	−	0.588697i	$-0.200359\pi$
$103$	16.4078	1.61671	0.808354	+	0.588697i	$0.200359\pi$
$104$	−21.3588	−2.09440
$105$	7.26337	0.708833
$106$	10.6461	1.03404
$107$	−6.39221	−0.617959	−0.308979	−	0.951069i	$-0.599987\pi$
$107$	−6.39221	−0.617959	−0.308979	+	0.951069i	$0.599987\pi$
$108$	5.38198	0.517881
$109$	−13.0322	−1.24826	−0.624129	−	0.781321i	$-0.714546\pi$
$109$	−13.0322	−1.24826	−0.624129	+	0.781321i	$0.714546\pi$
$110$	5.21150	0.496897
$111$	6.42528	0.609860
$112$	−63.3217	−5.98334
$113$	−12.2048	−1.14813	−0.574065	−	0.818809i	$-0.694635\pi$
$113$	−12.2048	−1.14813	−0.574065	+	0.818809i	$0.694635\pi$
$114$	−9.41874	−0.882146
$115$	3.94168	0.367563
$116$	−25.3800	−2.35647
$117$	−2.32445	−0.214895
$118$	32.8882	3.02760
$119$	4.45873	0.408731
$120$	−14.9687	−1.36645
$121$	−9.61357	−0.873961
$122$	26.2866	2.37988
$123$	−0.456417	−0.0411537
$124$	−14.9384	−1.34151
$125$	11.9673	1.07038
$126$	−12.1143	−1.07923
$127$	18.1364	1.60935	0.804674	−	0.593717i	$-0.202340\pi$
$127$	18.1364	1.60935	0.804674	+	0.593717i	$0.202340\pi$
$128$	31.5888	2.79208
$129$	−5.40658	−0.476023
$130$	10.2881	0.902323
$131$	5.14047	0.449125	0.224563	−	0.974460i	$-0.427905\pi$
$131$	5.14047	0.449125	0.224563	+	0.974460i	$0.427905\pi$
$132$	−6.33711	−0.551574
$133$	15.4567	1.34027
$134$	−39.5439	−3.41607
$135$	−1.62902	−0.140204
$136$	−9.18876	−0.787930
$137$	−17.2280	−1.47189	−0.735943	−	0.677044i	$-0.763261\pi$
$137$	−17.2280	−1.47189	−0.735943	+	0.677044i	$0.763261\pi$
$138$	−6.57416	−0.559630
$139$	2.76689	0.234684	0.117342	−	0.993092i	$-0.462563\pi$
$139$	2.76689	0.234684	0.117342	+	0.993092i	$0.462563\pi$
$140$	39.0913	3.30382
$141$	−8.08398	−0.680794
$142$	−35.5692	−2.98490
$143$	2.73697	0.228877
$144$	14.2017	1.18348
$145$	7.68204	0.637958
$146$	−37.2238	−3.08067
$147$	12.8803	1.06235
$148$	34.5807	2.84252
$149$	8.36834	0.685561	0.342780	−	0.939416i	$-0.388631\pi$
$149$	8.36834	0.685561	0.342780	+	0.939416i	$0.388631\pi$
$150$	−6.37482	−0.520502
$151$	13.6715	1.11257	0.556283	−	0.830993i	$-0.312227\pi$
$151$	13.6715	1.11257	0.556283	+	0.830993i	$0.312227\pi$
$152$	−31.8540	−2.58370
$153$	−1.00000	−0.0808452
$154$	14.2642	1.14944
$155$	4.52158	0.363182
$156$	−12.5101	−1.00161
$157$	1.00000	0.0798087
$157$	1.00000	0.0798087
$158$	−11.4537	−0.911208
$159$	3.91837	0.310747
$160$	−32.9197	−2.60253
$161$	10.7886	0.850262
$162$	2.71698	0.213466
$163$	0.170266	0.0133363	0.00666813	−	0.999978i	$-0.497877\pi$
$163$	0.170266	0.0133363	0.00666813	+	0.999978i	$0.497877\pi$
$164$	−2.45642	−0.191814
$165$	1.91812	0.149326
$166$	20.6125	1.59984
$167$	17.0729	1.32114	0.660570	−	0.750765i	$-0.270315\pi$
$167$	17.0729	1.32114	0.660570	+	0.750765i	$0.270315\pi$
$168$	−40.9702	−3.16092
$169$	−7.59693	−0.584379
$170$	4.42602	0.339460
$171$	−3.46662	−0.265099
$172$	−29.0981	−2.21871
$173$	1.37588	0.104606	0.0523030	−	0.998631i	$-0.483344\pi$
$173$	1.37588	0.104606	0.0523030	+	0.998631i	$0.483344\pi$
$174$	−12.8126	−0.971317
$175$	10.4615	0.790813
$176$	−16.7221	−1.26047
$177$	12.1047	0.909844
$178$	6.56326	0.491937
$179$	−1.74208	−0.130209	−0.0651045	−	0.997878i	$-0.520738\pi$
$179$	−1.74208	−0.130209	−0.0651045	+	0.997878i	$0.520738\pi$
$180$	−8.76736	−0.653480
$181$	−9.48881	−0.705297	−0.352649	−	0.935756i	$-0.614719\pi$
$181$	−9.48881	−0.705297	−0.352649	+	0.935756i	$0.614719\pi$
$182$	28.1591	2.08729
$183$	9.67494	0.715192
$184$	−22.2337	−1.63909
$185$	−10.4669	−0.769543
$186$	−7.54136	−0.552959
$187$	1.17747	0.0861050
$188$	−43.5078	−3.17313
$189$	−4.45873	−0.324325
$190$	15.3433	1.11312
$191$	−10.6658	−0.771750	−0.385875	−	0.922551i	$-0.626100\pi$
$191$	−10.6658	−0.771750	−0.385875	+	0.922551i	$0.626100\pi$
$192$	26.5020	1.91262
$193$	−10.6832	−0.768992	−0.384496	−	0.923127i	$-0.625625\pi$
$193$	−10.6832	−0.768992	−0.384496	+	0.923127i	$0.625625\pi$
$194$	−31.3702	−2.25225
$195$	3.78658	0.271163
$196$	69.3215	4.95154
$197$	1.53646	0.109468	0.0547342	−	0.998501i	$-0.482569\pi$
$197$	1.53646	0.109468	0.0547342	+	0.998501i	$0.482569\pi$
$198$	−3.19916	−0.227354
$199$	19.4156	1.37633	0.688166	−	0.725553i	$-0.258416\pi$
$199$	19.4156	1.37633	0.688166	+	0.725553i	$0.258416\pi$
$200$	−21.5595	−1.52449
$201$	−14.5543	−1.02658
$202$	2.36065	0.166094
$203$	21.0262	1.47575
$204$	−5.38198	−0.376814
$205$	0.743513	0.0519292
$206$	44.5796	3.10601
$207$	−2.41966	−0.168178
$208$	−33.0112	−2.28891
$209$	4.08184	0.282347
$210$	19.7344	1.36181
$211$	14.6593	1.00919	0.504595	−	0.863356i	$-0.331642\pi$
$211$	14.6593	1.00919	0.504595	+	0.863356i	$0.331642\pi$
$212$	21.0886	1.44837
$213$	−13.0914	−0.897011
$214$	−17.3675	−1.18722
$215$	8.80744	0.600662
$216$	9.18876	0.625216
$217$	12.3758	0.840127
$218$	−35.4082	−2.39815
$219$	−13.7004	−0.925790
$220$	10.3233	0.695996
$221$	2.32445	0.156359
$222$	17.4573	1.17166
$223$	9.80053	0.656292	0.328146	−	0.944627i	$-0.393576\pi$
$223$	9.80053	0.656292	0.328146	+	0.944627i	$0.393576\pi$
$224$	−90.1032	−6.02028
$225$	−2.34629	−0.156419
$226$	−33.1602	−2.20578
$227$	16.5888	1.10104	0.550520	−	0.834822i	$-0.314429\pi$
$227$	16.5888	1.10104	0.550520	+	0.834822i	$0.314429\pi$
$228$	−18.6573	−1.23561
$229$	−4.90950	−0.324429	−0.162214	−	0.986756i	$-0.551864\pi$
$229$	−4.90950	−0.324429	−0.162214	+	0.986756i	$0.551864\pi$
$230$	10.7095	0.706160
$231$	5.25002	0.345426
$232$	−43.3318	−2.84487
$233$	21.6344	1.41732	0.708660	−	0.705550i	$-0.249300\pi$
$233$	21.6344	1.41732	0.708660	+	0.705550i	$0.249300\pi$
$234$	−6.31548	−0.412856
$235$	13.1690	0.859050
$236$	65.1472	4.24072
$237$	−4.21560	−0.273833
$238$	12.1143	0.785253
$239$	8.22810	0.532231	0.266116	−	0.963941i	$-0.414260\pi$
$239$	8.22810	0.532231	0.266116	+	0.963941i	$0.414260\pi$
$240$	−23.1349	−1.49335
$241$	26.7610	1.72383	0.861914	−	0.507054i	$-0.169266\pi$
$241$	26.7610	1.72383	0.861914	+	0.507054i	$0.169266\pi$
$242$	−26.1199	−1.67905
$243$	1.00000	0.0641500
$244$	52.0703	3.33346
$245$	−20.9823	−1.34051
$246$	−1.24007	−0.0790643
$247$	8.05799	0.512718
$248$	−25.5047	−1.61955
$249$	7.58654	0.480777
$250$	32.5148	2.05642
$251$	−18.3228	−1.15652	−0.578262	−	0.815851i	$-0.696269\pi$
$251$	−18.3228	−1.15652	−0.578262	+	0.815851i	$0.696269\pi$
$252$	−23.9968	−1.51166
$253$	2.84907	0.179119
$254$	49.2763	3.09187
$255$	1.62902	0.102013
$256$	32.8222	2.05139
$257$	29.8851	1.86418	0.932091	−	0.362225i	$-0.117983\pi$
$257$	29.8851	1.86418	0.932091	+	0.362225i	$0.117983\pi$
$258$	−14.6896	−0.914532
$259$	−28.6486	−1.78014
$260$	20.3793	1.26387
$261$	−4.71574	−0.291897
$262$	13.9666	0.862857
$263$	−2.73712	−0.168778	−0.0843891	−	0.996433i	$-0.526894\pi$
$263$	−2.73712	−0.168778	−0.0843891	+	0.996433i	$0.526894\pi$
$264$	−10.8195	−0.665892
$265$	−6.38311	−0.392111
$266$	41.9957	2.57492
$267$	2.41565	0.147835
$268$	−78.3311	−4.78484
$269$	−10.7356	−0.654559	−0.327279	−	0.944928i	$-0.606132\pi$
$269$	−10.7356	−0.654559	−0.327279	+	0.944928i	$0.606132\pi$
$270$	−4.42602	−0.269359
$271$	13.3370	0.810168	0.405084	−	0.914279i	$-0.367242\pi$
$271$	13.3370	0.810168	0.405084	+	0.914279i	$0.367242\pi$
$272$	−14.2017	−0.861106
$273$	10.3641	0.627264
$274$	−46.8081	−2.82778
$275$	2.76268	0.166596
$276$	−13.0225	−0.783865
$277$	−12.4572	−0.748480	−0.374240	−	0.927332i	$-0.622096\pi$
$277$	−12.4572	−0.748480	−0.374240	+	0.927332i	$0.622096\pi$
$278$	7.51758	0.450874
$279$	−2.77564	−0.166173
$280$	66.7414	3.98856
$281$	26.2808	1.56778	0.783889	−	0.620900i	$-0.213233\pi$
$281$	26.2808	1.56778	0.783889	+	0.620900i	$0.213233\pi$
$282$	−21.9640	−1.30794
$283$	−29.0375	−1.72610	−0.863051	−	0.505116i	$-0.831450\pi$
$283$	−29.0375	−1.72610	−0.863051	+	0.505116i	$0.831450\pi$
$284$	−70.4579	−4.18090
$285$	5.64720	0.334511
$286$	7.43628	0.439716
$287$	2.03504	0.120125
$288$	20.2083	1.19078
$289$	1.00000	0.0588235
$290$	20.8719	1.22564
$291$	−11.5460	−0.676838
$292$	−73.7355	−4.31504
$293$	−23.3885	−1.36637	−0.683185	−	0.730245i	$-0.739406\pi$
$293$	−23.3885	−1.36637	−0.683185	+	0.730245i	$0.739406\pi$
$294$	34.9955	2.04098
$295$	−19.7188	−1.14807
$296$	59.0404	3.43165
$297$	−1.17747	−0.0683236
$298$	22.7366	1.31710
$299$	5.62438	0.325266
$300$	−12.6277	−0.729059
$301$	24.1065	1.38948
$302$	37.1451	2.13746
$303$	0.868849	0.0499141
$304$	−49.2320	−2.82365
$305$	−15.7607	−0.902454
$306$	−2.71698	−0.155319
$307$	13.4691	0.768721	0.384360	−	0.923183i	$-0.374422\pi$
$307$	13.4691	0.768721	0.384360	+	0.923183i	$0.374422\pi$
$308$	28.2555	1.61000
$309$	16.4078	0.933407
$310$	12.2850	0.697743
$311$	−28.4856	−1.61527	−0.807635	−	0.589683i	$-0.799253\pi$
$311$	−28.4856	−1.61527	−0.807635	+	0.589683i	$0.799253\pi$
$312$	−21.3588	−1.20920
$313$	−30.7352	−1.73726	−0.868628	−	0.495464i	$-0.834998\pi$
$313$	−30.7352	−1.73726	−0.868628	+	0.495464i	$0.834998\pi$
$314$	2.71698	0.153328
$315$	7.26337	0.409245
$316$	−22.6883	−1.27632
$317$	−8.53364	−0.479297	−0.239648	−	0.970860i	$-0.577032\pi$
$317$	−8.53364	−0.479297	−0.239648	+	0.970860i	$0.577032\pi$
$318$	10.6461	0.597005
$319$	5.55263	0.310887
$320$	−43.1723	−2.41341
$321$	−6.39221	−0.356779
$322$	29.3124	1.63352
$323$	3.46662	0.192888
$324$	5.38198	0.298999
$325$	5.45383	0.302524
$326$	0.462609	0.0256215
$327$	−13.0322	−0.720683
$328$	−4.19390	−0.231570
$329$	36.0443	1.98719
$330$	5.21150	0.286883
$331$	−0.229219	−0.0125990	−0.00629950	−	0.999980i	$-0.502005\pi$
$331$	−0.229219	−0.0125990	−0.00629950	+	0.999980i	$0.502005\pi$
$332$	40.8306	2.24087
$333$	6.42528	0.352103
$334$	46.3867	2.53817
$335$	23.7093	1.29538
$336$	−63.3217	−3.45448
$337$	−27.0287	−1.47235	−0.736174	−	0.676793i	$-0.763369\pi$
$337$	−27.0287	−1.47235	−0.736174	+	0.676793i	$0.763369\pi$
$338$	−20.6407	−1.12271
$339$	−12.2048	−0.662874
$340$	8.76736	0.475477
$341$	3.26823	0.176984
$342$	−9.41874	−0.509307
$343$	−26.2187	−1.41568
$344$	−49.6798	−2.67855
$345$	3.94168	0.212213
$346$	3.73823	0.200968
$347$	−5.30270	−0.284664	−0.142332	−	0.989819i	$-0.545460\pi$
$347$	−5.30270	−0.284664	−0.142332	+	0.989819i	$0.545460\pi$
$348$	−25.3800	−1.36051
$349$	−11.0840	−0.593311	−0.296655	−	0.954985i	$-0.595871\pi$
$349$	−11.0840	−0.593311	−0.296655	+	0.954985i	$0.595871\pi$
$350$	28.4236	1.51931
$351$	−2.32445	−0.124070
$352$	−23.7946	−1.26826
$353$	20.8930	1.11202	0.556011	−	0.831175i	$-0.312331\pi$
$353$	20.8930	1.11202	0.556011	+	0.831175i	$0.312331\pi$
$354$	32.8882	1.74799
$355$	21.3263	1.13188
$356$	13.0009	0.689049
$357$	4.45873	0.235981
$358$	−4.73319	−0.250157
$359$	9.75677	0.514942	0.257471	−	0.966286i	$-0.417111\pi$
$359$	9.75677	0.514942	0.257471	+	0.966286i	$0.417111\pi$
$360$	−14.9687	−0.788919
$361$	−6.98253	−0.367502
$362$	−25.7809	−1.35501
$363$	−9.61357	−0.504582
$364$	55.7794	2.92363
$365$	22.3183	1.16819
$366$	26.2866	1.37402
$367$	−1.30084	−0.0679032	−0.0339516	−	0.999423i	$-0.510809\pi$
$367$	−1.30084	−0.0679032	−0.0339516	+	0.999423i	$0.510809\pi$
$368$	−34.3633	−1.79131
$369$	−0.456417	−0.0237601
$370$	−28.4384	−1.47844
$371$	−17.4710	−0.907047
$372$	−14.9384	−0.774522
$373$	−32.8923	−1.70310	−0.851549	−	0.524274i	$-0.824337\pi$
$373$	−32.8923	−1.70310	−0.851549	+	0.524274i	$0.824337\pi$
$374$	3.19916	0.165424
$375$	11.9673	0.617987
$376$	−74.2818	−3.83079
$377$	10.9615	0.564546
$378$	−12.1143	−0.623092
$379$	−1.05124	−0.0539987	−0.0269994	−	0.999635i	$-0.508595\pi$
$379$	−1.05124	−0.0539987	−0.0269994	+	0.999635i	$0.508595\pi$
$380$	30.3931	1.55913
$381$	18.1364	0.929158
$382$	−28.9788	−1.48268
$383$	33.2874	1.70091	0.850453	−	0.526050i	$-0.176328\pi$
$383$	33.2874	1.70091	0.850453	+	0.526050i	$0.176328\pi$
$384$	31.5888	1.61201
$385$	−8.55239	−0.435870
$386$	−29.0260	−1.47738
$387$	−5.40658	−0.274832
$388$	−62.1402	−3.15469
$389$	−9.77582	−0.495654	−0.247827	−	0.968804i	$-0.579716\pi$
$389$	−9.77582	−0.495654	−0.247827	+	0.968804i	$0.579716\pi$
$390$	10.2881	0.520956
$391$	2.41966	0.122367
$392$	118.354	5.97778
$393$	5.14047	0.259303
$394$	4.17454	0.210310
$395$	6.86731	0.345532
$396$	−6.33711	−0.318452
$397$	−17.6707	−0.886868	−0.443434	−	0.896307i	$-0.646240\pi$
$397$	−17.6707	−0.886868	−0.443434	+	0.896307i	$0.646240\pi$
$398$	52.7517	2.64420
$399$	15.4567	0.773805
$400$	−33.3213	−1.66607
$401$	11.6996	0.584249	0.292125	−	0.956380i	$-0.405638\pi$
$401$	11.6996	0.584249	0.292125	+	0.956380i	$0.405638\pi$
$402$	−39.5439	−1.97227
$403$	6.45184	0.321389
$404$	4.67613	0.232646
$405$	−1.62902	−0.0809467
$406$	57.1278	2.83520
$407$	−7.56556	−0.375011
$408$	−9.18876	−0.454911
$409$	23.8791	1.18074	0.590372	−	0.807131i	$-0.298981\pi$
$409$	23.8791	1.18074	0.590372	+	0.807131i	$0.298981\pi$
$410$	2.02011	0.0997660
$411$	−17.2280	−0.849794
$412$	88.3064	4.35054
$413$	−53.9716	−2.65577
$414$	−6.57416	−0.323102
$415$	−12.3586	−0.606661
$416$	−46.9731	−2.30305
$417$	2.76689	0.135495
$418$	11.0903	0.542443
$419$	−15.9439	−0.778908	−0.389454	−	0.921046i	$-0.627336\pi$
$419$	−15.9439	−0.778908	−0.389454	+	0.921046i	$0.627336\pi$
$420$	39.0913	1.90746
$421$	−17.8026	−0.867646	−0.433823	−	0.900998i	$-0.642836\pi$
$421$	−17.8026	−0.867646	−0.433823	+	0.900998i	$0.642836\pi$
$422$	39.8291	1.93885
$423$	−8.08398	−0.393057
$424$	36.0049	1.74855
$425$	2.34629	0.113812
$426$	−35.5692	−1.72333
$427$	−43.1380	−2.08759
$428$	−34.4027	−1.66292
$429$	2.73697	0.132142
$430$	23.9296	1.15399
$431$	−17.8098	−0.857866	−0.428933	−	0.903336i	$-0.641110\pi$
$431$	−17.8098	−0.857866	−0.428933	+	0.903336i	$0.641110\pi$
$432$	14.2017	0.683281
$433$	23.9323	1.15011	0.575057	−	0.818113i	$-0.304980\pi$
$433$	23.9323	1.15011	0.575057	+	0.818113i	$0.304980\pi$
$434$	33.6249	1.61405
$435$	7.68204	0.368325
$436$	−70.1390	−3.35905
$437$	8.38804	0.401254
$438$	−37.2238	−1.77862
$439$	4.90238	0.233978	0.116989	−	0.993133i	$-0.462676\pi$
$439$	4.90238	0.233978	0.116989	+	0.993133i	$0.462676\pi$
$440$	17.6252	0.840246
$441$	12.8803	0.613348
$442$	6.31548	0.300397
$443$	−17.4211	−0.827703	−0.413851	−	0.910344i	$-0.635817\pi$
$443$	−17.4211	−0.827703	−0.413851	+	0.910344i	$0.635817\pi$
$444$	34.5807	1.64113
$445$	−3.93514	−0.186544
$446$	26.6278	1.26087
$447$	8.36834	0.395809
$448$	−118.165	−5.58278
$449$	11.4335	0.539581	0.269791	−	0.962919i	$-0.413046\pi$
$449$	11.4335	0.539581	0.269791	+	0.962919i	$0.413046\pi$
$450$	−6.37482	−0.300512
$451$	0.537416	0.0253059
$452$	−65.6859	−3.08961
$453$	13.6715	0.642341
$454$	45.0716	2.11531
$455$	−16.8834	−0.791504
$456$	−31.8540	−1.49170
$457$	−18.8474	−0.881645	−0.440822	−	0.897594i	$-0.645313\pi$
$457$	−18.8474	−0.881645	−0.440822	+	0.897594i	$0.645313\pi$
$458$	−13.3390	−0.623291
$459$	−1.00000	−0.0466760
$460$	21.2140	0.989108
$461$	26.8955	1.25265	0.626324	−	0.779563i	$-0.284559\pi$
$461$	26.8955	1.25265	0.626324	+	0.779563i	$0.284559\pi$
$462$	14.2642	0.663630
$463$	22.2150	1.03242	0.516210	−	0.856462i	$-0.327342\pi$
$463$	22.2150	1.03242	0.516210	+	0.856462i	$0.327342\pi$
$464$	−66.9716	−3.10908
$465$	4.52158	0.209683
$466$	58.7803	2.72295
$467$	0.368727	0.0170626	0.00853132	−	0.999964i	$-0.497284\pi$
$467$	0.368727	0.0170626	0.00853132	+	0.999964i	$0.497284\pi$
$468$	−12.5101	−0.578281
$469$	64.8939	2.99652
$470$	35.7799	1.65040
$471$	1.00000	0.0460776
$472$	111.227	5.11964
$473$	6.36607	0.292712
$474$	−11.4537	−0.526086
$475$	8.13369	0.373199
$476$	23.9968	1.09989
$477$	3.91837	0.179410
$478$	22.3556	1.02252
$479$	25.9957	1.18777	0.593887	−	0.804548i	$-0.297592\pi$
$479$	25.9957	1.18777	0.593887	+	0.804548i	$0.297592\pi$
$480$	−32.9197	−1.50257
$481$	−14.9352	−0.680988
$482$	72.7091	3.31181
$483$	10.7886	0.490899
$484$	−51.7400	−2.35182
$485$	18.8087	0.854057
$486$	2.71698	0.123245
$487$	−3.22368	−0.146079	−0.0730395	−	0.997329i	$-0.523270\pi$
$487$	−3.22368	−0.146079	−0.0730395	+	0.997329i	$0.523270\pi$
$488$	88.9007	4.02435
$489$	0.170266	0.00769969
$490$	−57.0085	−2.57538
$491$	−26.3652	−1.18984	−0.594922	−	0.803783i	$-0.702817\pi$
$491$	−26.3652	−1.18984	−0.594922	+	0.803783i	$0.702817\pi$
$492$	−2.45642	−0.110744
$493$	4.71574	0.212386
$494$	21.8934	0.985030
$495$	1.91812	0.0862131
$496$	−39.4189	−1.76996
$497$	58.3713	2.61831
$498$	20.6125	0.923666
$499$	31.2626	1.39951	0.699753	−	0.714385i	$-0.253293\pi$
$499$	31.2626	1.39951	0.699753	+	0.714385i	$0.253293\pi$
$500$	64.4075	2.88039
$501$	17.0729	0.762760
$502$	−49.7826	−2.22191
$503$	−27.6387	−1.23235	−0.616175	−	0.787609i	$-0.711319\pi$
$503$	−27.6387	−1.23235	−0.616175	+	0.787609i	$0.711319\pi$
$504$	−40.9702	−1.82496
$505$	−1.41537	−0.0629833
$506$	7.74086	0.344123
$507$	−7.59693	−0.337392
$508$	97.6099	4.33074
$509$	−20.9531	−0.928731	−0.464366	−	0.885644i	$-0.653718\pi$
$509$	−20.9531	−0.928731	−0.464366	+	0.885644i	$0.653718\pi$
$510$	4.42602	0.195987
$511$	61.0867	2.70231
$512$	25.9996	1.14903
$513$	−3.46662	−0.153055
$514$	81.1972	3.58146
$515$	−26.7287	−1.17781
$516$	−29.0981	−1.28097
$517$	9.51863	0.418629
$518$	−77.8377	−3.41999
$519$	1.37588	0.0603943
$520$	34.7940	1.52582
$521$	4.44229	0.194620	0.0973102	−	0.995254i	$-0.468976\pi$
$521$	4.44229	0.194620	0.0973102	+	0.995254i	$0.468976\pi$
$522$	−12.8126	−0.560790
$523$	−27.5421	−1.20433	−0.602165	−	0.798371i	$-0.705695\pi$
$523$	−27.5421	−1.20433	−0.602165	+	0.798371i	$0.705695\pi$
$524$	27.6659	1.20859
$525$	10.4615	0.456576
$526$	−7.43671	−0.324256
$527$	2.77564	0.120909
$528$	−16.7221	−0.727735
$529$	−17.1453	−0.745446
$530$	−17.3428	−0.753322
$531$	12.1047	0.525299
$532$	83.1879	3.60665
$533$	1.06092	0.0459534
$534$	6.56326	0.284020
$535$	10.4131	0.450196
$536$	−133.736	−5.77653
$537$	−1.74208	−0.0751762
$538$	−29.1683	−1.25753
$539$	−15.1661	−0.653252
$540$	−8.76736	−0.377287
$541$	12.7564	0.548440	0.274220	−	0.961667i	$-0.411580\pi$
$541$	12.7564	0.548440	0.274220	+	0.961667i	$0.411580\pi$
$542$	36.2365	1.55649
$543$	−9.48881	−0.407204
$544$	−20.2083	−0.866422
$545$	21.2297	0.909382
$546$	28.1591	1.20510
$547$	8.74967	0.374109	0.187055	−	0.982350i	$-0.440106\pi$
$547$	8.74967	0.374109	0.187055	+	0.982350i	$0.440106\pi$
$548$	−92.7206	−3.96083
$549$	9.67494	0.412916
$550$	7.50614	0.320063
$551$	16.3477	0.696434
$552$	−22.2337	−0.946327
$553$	18.7963	0.799298
$554$	−33.8459	−1.43797
$555$	−10.4669	−0.444296
$556$	14.8913	0.631533
$557$	−9.67376	−0.409890	−0.204945	−	0.978773i	$-0.565702\pi$
$557$	−9.67376	−0.409890	−0.204945	+	0.978773i	$0.565702\pi$
$558$	−7.54136	−0.319251
$559$	12.5673	0.531541
$560$	103.152	4.35899
$561$	1.17747	0.0497127
$562$	71.4043	3.01201
$563$	−32.7591	−1.38063	−0.690317	−	0.723507i	$-0.742529\pi$
$563$	−32.7591	−1.38063	−0.690317	+	0.723507i	$0.742529\pi$
$564$	−43.5078	−1.83201
$565$	19.8819	0.836437
$566$	−78.8944	−3.31618
$567$	−4.45873	−0.187249
$568$	−120.294	−5.04743
$569$	−33.1588	−1.39009	−0.695045	−	0.718966i	$-0.744616\pi$
$569$	−33.1588	−1.39009	−0.695045	+	0.718966i	$0.744616\pi$
$570$	15.3433	0.642662
$571$	15.7899	0.660785	0.330393	−	0.943844i	$-0.392819\pi$
$571$	15.7899	0.660785	0.330393	+	0.943844i	$0.392819\pi$
$572$	14.7303	0.615904
$573$	−10.6658	−0.445570
$574$	5.52916	0.230783
$575$	5.67721	0.236756
$576$	26.5020	1.10425
$577$	9.93366	0.413544	0.206772	−	0.978389i	$-0.433704\pi$
$577$	9.93366	0.413544	0.206772	+	0.978389i	$0.433704\pi$
$578$	2.71698	0.113011
$579$	−10.6832	−0.443978
$580$	41.3445	1.71674
$581$	−33.8263	−1.40335
$582$	−31.3702	−1.30034
$583$	−4.61375	−0.191082
$584$	−125.890	−5.20937
$585$	3.78658	0.156556
$586$	−63.5461	−2.62506
$587$	17.0856	0.705197	0.352599	−	0.935775i	$-0.385298\pi$
$587$	17.0856	0.705197	0.352599	+	0.935775i	$0.385298\pi$
$588$	69.3215	2.85877
$589$	9.62210	0.396472
$590$	−53.5756	−2.20567
$591$	1.53646	0.0632016
$592$	91.2500	3.75035
$593$	−39.9373	−1.64003	−0.820015	−	0.572342i	$-0.806035\pi$
$593$	−39.9373	−1.64003	−0.820015	+	0.572342i	$0.806035\pi$
$594$	−3.19916	−0.131263
$595$	−7.26337	−0.297769
$596$	45.0382	1.84484
$597$	19.4156	0.794626
$598$	15.2813	0.624899
$599$	5.99270	0.244855	0.122428	−	0.992477i	$-0.460932\pi$
$599$	5.99270	0.244855	0.122428	+	0.992477i	$0.460932\pi$
$600$	−21.5595	−0.880162
$601$	−16.8832	−0.688681	−0.344340	−	0.938845i	$-0.611897\pi$
$601$	−16.8832	−0.688681	−0.344340	+	0.938845i	$0.611897\pi$
$602$	65.4969	2.66945
$603$	−14.5543	−0.592699
$604$	73.5794	2.99390
$605$	15.6607	0.636699
$606$	2.36065	0.0958947
$607$	−18.9388	−0.768704	−0.384352	−	0.923187i	$-0.625575\pi$
$607$	−18.9388	−0.768704	−0.384352	+	0.923187i	$0.625575\pi$
$608$	−70.0544	−2.84108
$609$	21.0262	0.852025
$610$	−42.8215	−1.73379
$611$	18.7908	0.760195
$612$	−5.38198	−0.217554
$613$	−35.5056	−1.43406	−0.717028	−	0.697044i	$-0.754498\pi$
$613$	−35.5056	−1.43406	−0.717028	+	0.697044i	$0.754498\pi$
$614$	36.5952	1.47686
$615$	0.743513	0.0299813
$616$	48.2411	1.94369
$617$	−8.23100	−0.331367	−0.165684	−	0.986179i	$-0.552983\pi$
$617$	−8.23100	−0.331367	−0.165684	+	0.986179i	$0.552983\pi$
$618$	44.5796	1.79326
$619$	15.8715	0.637930	0.318965	−	0.947767i	$-0.396665\pi$
$619$	15.8715	0.637930	0.318965	+	0.947767i	$0.396665\pi$
$620$	24.3350	0.977319
$621$	−2.41966	−0.0970975
$622$	−77.3948	−3.10325
$623$	−10.7707	−0.431520
$624$	−33.0112	−1.32151
$625$	−7.76349	−0.310540
$626$	−83.5069	−3.33761
$627$	4.08184	0.163013
$628$	5.38198	0.214764
$629$	−6.42528	−0.256193
$630$	19.7344	0.786239
$631$	20.7851	0.827442	0.413721	−	0.910404i	$-0.364229\pi$
$631$	20.7851	0.827442	0.413721	+	0.910404i	$0.364229\pi$
$632$	−38.7362	−1.54084
$633$	14.6593	0.582657
$634$	−23.1857	−0.920822
$635$	−29.5446	−1.17244
$636$	21.0886	0.836216
$637$	−29.9396	−1.18625
$638$	15.0864	0.597275
$639$	−13.0914	−0.517890
$640$	−51.4589	−2.03409
$641$	−27.9163	−1.10263	−0.551313	−	0.834298i	$-0.685873\pi$
$641$	−27.9163	−1.10263	−0.551313	+	0.834298i	$0.685873\pi$
$642$	−17.3675	−0.685441
$643$	26.5112	1.04550	0.522751	−	0.852486i	$-0.324906\pi$
$643$	26.5112	1.04550	0.522751	+	0.852486i	$0.324906\pi$
$644$	58.0641	2.28804
$645$	8.80744	0.346792
$646$	9.41874	0.370575
$647$	−8.04892	−0.316436	−0.158218	−	0.987404i	$-0.550575\pi$
$647$	−8.04892	−0.316436	−0.158218	+	0.987404i	$0.550575\pi$
$648$	9.18876	0.360969
$649$	−14.2529	−0.559475
$650$	14.8179	0.581207
$651$	12.3758	0.485048
$652$	0.916367	0.0358877
$653$	19.0020	0.743606	0.371803	−	0.928312i	$-0.378740\pi$
$653$	19.0020	0.743606	0.371803	+	0.928312i	$0.378740\pi$
$654$	−35.4082	−1.38457
$655$	−8.37394	−0.327197
$656$	−6.48190	−0.253076
$657$	−13.7004	−0.534505
$658$	97.9317	3.81777
$659$	−40.1346	−1.56342	−0.781710	−	0.623642i	$-0.785652\pi$
$659$	−40.1346	−1.56342	−0.781710	+	0.623642i	$0.785652\pi$
$660$	10.3233	0.401833
$661$	23.1011	0.898527	0.449264	−	0.893399i	$-0.351686\pi$
$661$	23.1011	0.898527	0.449264	+	0.893399i	$0.351686\pi$
$662$	−0.622783	−0.0242051
$663$	2.32445	0.0902742
$664$	69.7109	2.70531
$665$	−25.1794	−0.976414
$666$	17.4573	0.676459
$667$	11.4105	0.441815
$668$	91.8859	3.55517
$669$	9.80053	0.378911
$670$	64.4178	2.48868
$671$	−11.3919	−0.439781
$672$	−90.1032	−3.47581
$673$	−22.5940	−0.870935	−0.435468	−	0.900204i	$-0.643417\pi$
$673$	−22.5940	−0.870935	−0.435468	+	0.900204i	$0.643417\pi$
$674$	−73.4364	−2.82867
$675$	−2.34629	−0.0903087
$676$	−40.8865	−1.57256
$677$	24.6185	0.946167	0.473084	−	0.881018i	$-0.343141\pi$
$677$	24.6185	0.946167	0.473084	+	0.881018i	$0.343141\pi$
$678$	−33.1602	−1.27351
$679$	51.4805	1.97564
$680$	14.9687	0.574023
$681$	16.5888	0.635686
$682$	8.87971	0.340022
$683$	9.13214	0.349431	0.174716	−	0.984619i	$-0.444099\pi$
$683$	9.13214	0.349431	0.174716	+	0.984619i	$0.444099\pi$
$684$	−18.6573	−0.713379
$685$	28.0648	1.07230
$686$	−71.2357	−2.71979
$687$	−4.90950	−0.187309
$688$	−76.7827	−2.92731
$689$	−9.10805	−0.346989
$690$	10.7095	0.407702
$691$	−33.4417	−1.27218	−0.636091	−	0.771614i	$-0.719450\pi$
$691$	−33.4417	−1.27218	−0.636091	+	0.771614i	$0.719450\pi$
$692$	7.40494	0.281493
$693$	5.25002	0.199432
$694$	−14.4073	−0.546895
$695$	−4.50732	−0.170972
$696$	−43.3318	−1.64249
$697$	0.456417	0.0172880
$698$	−30.1149	−1.13987
$699$	21.6344	0.818290
$700$	56.3034	2.12807
$701$	−44.1737	−1.66842	−0.834209	−	0.551449i	$-0.814075\pi$
$701$	−44.1737	−1.66842	−0.834209	+	0.551449i	$0.814075\pi$
$702$	−6.31548	−0.238363
$703$	−22.2740	−0.840080
$704$	−31.2052	−1.17609
$705$	13.1690	0.495973
$706$	56.7658	2.13641
$707$	−3.87397	−0.145696
$708$	65.1472	2.44838
$709$	−3.31045	−0.124326	−0.0621632	−	0.998066i	$-0.519800\pi$
$709$	−3.31045	−0.124326	−0.0621632	+	0.998066i	$0.519800\pi$
$710$	57.9430	2.17456
$711$	−4.21560	−0.158098
$712$	22.1968	0.831860
$713$	6.71610	0.251520
$714$	12.1143	0.453366
$715$	−4.45858	−0.166741
$716$	−9.37582	−0.350391
$717$	8.22810	0.307284
$718$	26.5089	0.989304
$719$	8.46080	0.315535	0.157767	−	0.987476i	$-0.449570\pi$
$719$	8.46080	0.315535	0.157767	+	0.987476i	$0.449570\pi$
$720$	−23.1349	−0.862187
$721$	−73.1580	−2.72455
$722$	−18.9714	−0.706042
$723$	26.7610	0.995253
$724$	−51.0685	−1.89795
$725$	11.0645	0.410924
$726$	−26.1199	−0.969399
$727$	17.2218	0.638721	0.319361	−	0.947633i	$-0.396532\pi$
$727$	17.2218	0.638721	0.319361	+	0.947633i	$0.396532\pi$
$728$	95.2333	3.52958
$729$	1.00000	0.0370370
$730$	60.6384	2.24433
$731$	5.40658	0.199970
$732$	52.0703	1.92457
$733$	38.5344	1.42330	0.711650	−	0.702534i	$-0.247948\pi$
$733$	38.5344	1.42330	0.711650	+	0.702534i	$0.247948\pi$
$734$	−3.53435	−0.130455
$735$	−20.9823	−0.773944
$736$	−48.8971	−1.80237
$737$	17.1373	0.631260
$738$	−1.24007	−0.0456478
$739$	−0.692441	−0.0254719	−0.0127359	−	0.999919i	$-0.504054\pi$
$739$	−0.692441	−0.0254719	−0.0127359	+	0.999919i	$0.504054\pi$
$740$	−56.3327	−2.07083
$741$	8.05799	0.296018
$742$	−47.4682	−1.74261
$743$	30.5095	1.11929	0.559643	−	0.828734i	$-0.310938\pi$
$743$	30.5095	1.11929	0.559643	+	0.828734i	$0.310938\pi$
$744$	−25.5047	−0.935048
$745$	−13.6322	−0.499445
$746$	−89.3677	−3.27198
$747$	7.58654	0.277577
$748$	6.33711	0.231708
$749$	28.5012	1.04141
$750$	32.5148	1.18727
$751$	7.47963	0.272935	0.136468	−	0.990645i	$-0.456425\pi$
$751$	7.47963	0.272935	0.136468	+	0.990645i	$0.456425\pi$
$752$	−114.806	−4.18656
$753$	−18.3228	−0.667719
$754$	29.7821	1.08460
$755$	−22.2711	−0.810528
$756$	−23.9968	−0.872755
$757$	8.80229	0.319924	0.159962	−	0.987123i	$-0.448863\pi$
$757$	8.80229	0.319924	0.159962	+	0.987123i	$0.448863\pi$
$758$	−2.85621	−0.103742
$759$	2.84907	0.103415
$760$	51.8908	1.88228
$761$	30.6216	1.11003	0.555015	−	0.831840i	$-0.312712\pi$
$761$	30.6216	1.11003	0.555015	+	0.831840i	$0.312712\pi$
$762$	49.2763	1.78509
$763$	58.1071	2.10362
$764$	−57.4031	−2.07677
$765$	1.62902	0.0588974
$766$	90.4412	3.26777
$767$	−28.1368	−1.01596
$768$	32.8222	1.18437
$769$	19.1937	0.692142	0.346071	−	0.938208i	$-0.387516\pi$
$769$	19.1937	0.692142	0.346071	+	0.938208i	$0.387516\pi$
$770$	−23.2367	−0.837391
$771$	29.8851	1.07629
$772$	−57.4966	−2.06935
$773$	−6.98677	−0.251296	−0.125648	−	0.992075i	$-0.540101\pi$
$773$	−6.98677	−0.251296	−0.125648	+	0.992075i	$0.540101\pi$
$774$	−14.6896	−0.528006
$775$	6.51245	0.233934
$776$	−106.093	−3.80853
$777$	−28.6486	−1.02776
$778$	−26.5607	−0.952247
$779$	1.58222	0.0566891
$780$	20.3793	0.729695
$781$	15.4148	0.551583
$782$	6.57416	0.235092
$783$	−4.71574	−0.168527
$784$	182.923	6.53295
$785$	−1.62902	−0.0581423
$786$	13.9666	0.498171
$787$	3.92896	0.140052	0.0700261	−	0.997545i	$-0.477692\pi$
$787$	3.92896	0.140052	0.0700261	+	0.997545i	$0.477692\pi$
$788$	8.26921	0.294578
$789$	−2.73712	−0.0974442
$790$	18.6583	0.663834
$791$	54.4179	1.93488
$792$	−10.8195	−0.384453
$793$	−22.4889	−0.798605
$794$	−48.0110	−1.70385
$795$	−6.38311	−0.226385
$796$	104.494	3.70370
$797$	−7.88836	−0.279420	−0.139710	−	0.990192i	$-0.544617\pi$
$797$	−7.88836	−0.279420	−0.139710	+	0.990192i	$0.544617\pi$
$798$	41.9957	1.48663
$799$	8.08398	0.285991
$800$	−47.4144	−1.67635
$801$	2.41565	0.0853526
$802$	31.7875	1.12246
$803$	16.1318	0.569280
$804$	−78.3311	−2.76253
$805$	−17.5749	−0.619433
$806$	17.5295	0.617451
$807$	−10.7356	−0.377910
$808$	7.98365	0.280864
$809$	22.8904	0.804783	0.402391	−	0.915468i	$-0.368179\pi$
$809$	22.8904	0.804783	0.402391	+	0.915468i	$0.368179\pi$
$810$	−4.42602	−0.155514
$811$	22.1392	0.777412	0.388706	−	0.921362i	$-0.372922\pi$
$811$	22.1392	0.777412	0.388706	+	0.921362i	$0.372922\pi$
$812$	113.163	3.97123
$813$	13.3370	0.467751
$814$	−20.5555	−0.720469
$815$	−0.277367	−0.00971574
$816$	−14.2017	−0.497160
$817$	18.7426	0.655720
$818$	64.8790	2.26844
$819$	10.3641	0.362151
$820$	4.00157	0.139741
$821$	−22.5913	−0.788443	−0.394222	−	0.919015i	$-0.628986\pi$
$821$	−22.5913	−0.788443	−0.394222	+	0.919015i	$0.628986\pi$
$822$	−46.8081	−1.63262
$823$	−26.5495	−0.925458	−0.462729	−	0.886500i	$-0.653130\pi$
$823$	−26.5495	−0.925458	−0.462729	+	0.886500i	$0.653130\pi$
$824$	150.767	5.25223
$825$	2.76268	0.0961841
$826$	−146.640	−5.10225
$827$	−37.2700	−1.29600	−0.648002	−	0.761638i	$-0.724395\pi$
$827$	−37.2700	−1.29600	−0.648002	+	0.761638i	$0.724395\pi$
$828$	−13.0225	−0.452565
$829$	12.7048	0.441255	0.220628	−	0.975358i	$-0.429189\pi$
$829$	12.7048	0.441255	0.220628	+	0.975358i	$0.429189\pi$
$830$	−33.5781	−1.16551
$831$	−12.4572	−0.432135
$832$	−61.6025	−2.13568
$833$	−12.8803	−0.446276
$834$	7.51758	0.260312
$835$	−27.8121	−0.962477
$836$	21.9684	0.759791
$837$	−2.77564	−0.0959402
$838$	−43.3191	−1.49643
$839$	11.4703	0.395998	0.197999	−	0.980202i	$-0.436556\pi$
$839$	11.4703	0.395998	0.197999	+	0.980202i	$0.436556\pi$
$840$	66.7414	2.30280
$841$	−6.76184	−0.233167
$842$	−48.3693	−1.66692
$843$	26.2808	0.905158
$844$	78.8963	2.71572
$845$	12.3756	0.425732
$846$	−21.9640	−0.755138
$847$	42.8643	1.47284
$848$	55.6476	1.91095
$849$	−29.0375	−0.996566
$850$	6.37482	0.218654
$851$	−15.5470	−0.532944
$852$	−70.4579	−2.41385
$853$	−17.0673	−0.584373	−0.292187	−	0.956361i	$-0.594383\pi$
$853$	−17.0673	−0.584373	−0.292187	+	0.956361i	$0.594383\pi$
$854$	−117.205	−4.01067
$855$	5.64720	0.193130
$856$	−58.7365	−2.00757
$857$	−25.1899	−0.860470	−0.430235	−	0.902717i	$-0.641569\pi$
$857$	−25.1899	−0.860470	−0.430235	+	0.902717i	$0.641569\pi$
$858$	7.43628	0.253870
$859$	−37.8227	−1.29049	−0.645247	−	0.763974i	$-0.723246\pi$
$859$	−37.8227	−1.29049	−0.645247	+	0.763974i	$0.723246\pi$
$860$	47.4014	1.61637
$861$	2.03504	0.0693540
$862$	−48.3888	−1.64813
$863$	43.6482	1.48580	0.742900	−	0.669402i	$-0.233450\pi$
$863$	43.6482	1.48580	0.742900	+	0.669402i	$0.233450\pi$
$864$	20.2083	0.687499
$865$	−2.24133	−0.0762076
$866$	65.0236	2.20959
$867$	1.00000	0.0339618
$868$	66.6065	2.26077
$869$	4.96374	0.168383
$870$	20.8719	0.707625
$871$	33.8308	1.14631
$872$	−119.750	−4.05524
$873$	−11.5460	−0.390772
$874$	22.7901	0.770888
$875$	−53.3588	−1.80386
$876$	−73.7355	−2.49129
$877$	−4.14928	−0.140111	−0.0700556	−	0.997543i	$-0.522318\pi$
$877$	−4.14928	−0.140111	−0.0700556	+	0.997543i	$0.522318\pi$
$878$	13.3197	0.449517
$879$	−23.3885	−0.788875
$880$	27.2406	0.918281
$881$	−6.16234	−0.207614	−0.103807	−	0.994597i	$-0.533102\pi$
$881$	−6.16234	−0.207614	−0.103807	+	0.994597i	$0.533102\pi$
$882$	34.9955	1.17836
$883$	54.1765	1.82318	0.911591	−	0.411097i	$-0.134854\pi$
$883$	54.1765	1.82318	0.911591	+	0.411097i	$0.134854\pi$
$884$	12.5101	0.420761
$885$	−19.7188	−0.662841
$886$	−47.3328	−1.59018
$887$	−38.9153	−1.30665	−0.653324	−	0.757078i	$-0.726626\pi$
$887$	−38.9153	−1.30665	−0.653324	+	0.757078i	$0.726626\pi$
$888$	59.0404	1.98126
$889$	−80.8655	−2.71214
$890$	−10.6917	−0.358386
$891$	−1.17747	−0.0394467
$892$	52.7462	1.76607
$893$	28.0241	0.937791
$894$	22.7366	0.760426
$895$	2.83788	0.0948600
$896$	−140.846	−4.70534
$897$	5.62438	0.187792
$898$	31.0646	1.03664
$899$	13.0892	0.436549
$900$	−12.6277	−0.420922
$901$	−3.91837	−0.130540
$902$	1.46015	0.0486176
$903$	24.1065	0.802214
$904$	−112.147	−3.72995
$905$	15.4575	0.513824
$906$	37.1451	1.23406
$907$	−17.0673	−0.566711	−0.283356	−	0.959015i	$-0.591448\pi$
$907$	−17.0673	−0.566711	−0.283356	+	0.959015i	$0.591448\pi$
$908$	89.2808	2.96289
$909$	0.868849	0.0288179
$910$	−45.8717	−1.52063
$911$	14.9606	0.495667	0.247833	−	0.968803i	$-0.420281\pi$
$911$	14.9606	0.495667	0.247833	+	0.968803i	$0.420281\pi$
$912$	−49.2320	−1.63023
$913$	−8.93290	−0.295636
$914$	−51.2080	−1.69381
$915$	−15.7607	−0.521032
$916$	−26.4228	−0.873034
$917$	−22.9200	−0.756885
$918$	−2.71698	−0.0896737
$919$	31.8931	1.05205	0.526027	−	0.850468i	$-0.323681\pi$
$919$	31.8931	1.05205	0.526027	+	0.850468i	$0.323681\pi$
$920$	36.2191	1.19411
$921$	13.4691	0.443821
$922$	73.0745	2.40658
$923$	30.4304	1.00163
$924$	28.2555	0.929537
$925$	−15.0756	−0.495681
$926$	60.3578	1.98348
$927$	16.4078	0.538903
$928$	−95.2968	−3.12827
$929$	−2.72377	−0.0893641	−0.0446821	−	0.999001i	$-0.514227\pi$
$929$	−2.72377	−0.0893641	−0.0446821	+	0.999001i	$0.514227\pi$
$930$	12.2850	0.402842
$931$	−44.6512	−1.46338
$932$	116.436	3.81399
$933$	−28.4856	−0.932577
$934$	1.00182	0.0327806
$935$	−1.91812	−0.0627293
$936$	−21.3588	−0.698135
$937$	−42.0056	−1.37226	−0.686132	−	0.727477i	$-0.740693\pi$
$937$	−42.0056	−1.37226	−0.686132	+	0.727477i	$0.740693\pi$
$938$	176.316	5.75691
$939$	−30.7352	−1.00301
$940$	70.8752	2.31169
$941$	−26.1852	−0.853614	−0.426807	−	0.904343i	$-0.640362\pi$
$941$	−26.1852	−0.853614	−0.426807	+	0.904343i	$0.640362\pi$
$942$	2.71698	0.0885240
$943$	1.10437	0.0359633
$944$	171.907	5.59511
$945$	7.26337	0.236278
$946$	17.2965	0.562357
$947$	0.175761	0.00571147	0.00285574	−	0.999996i	$-0.499091\pi$
$947$	0.175761	0.00571147	0.00285574	+	0.999996i	$0.499091\pi$
$948$	−22.6883	−0.736881
$949$	31.8460	1.03377
$950$	22.0991	0.716989
$951$	−8.53364	−0.276722
$952$	40.9702	1.32785
$953$	24.4882	0.793249	0.396625	−	0.917981i	$-0.370181\pi$
$953$	24.4882	0.793249	0.396625	+	0.917981i	$0.370181\pi$
$954$	10.6461	0.344681
$955$	17.3748	0.562236
$956$	44.2834	1.43223
$957$	5.55263	0.179491
$958$	70.6298	2.28195
$959$	76.8150	2.48048
$960$	−43.1723	−1.39338
$961$	−23.2958	−0.751478
$962$	−40.5787	−1.30831
$963$	−6.39221	−0.205986
$964$	144.027	4.63880
$965$	17.4031	0.560226
$966$	29.3124	0.943112
$967$	40.7147	1.30930	0.654648	−	0.755934i	$-0.272817\pi$
$967$	40.7147	1.30930	0.654648	+	0.755934i	$0.272817\pi$
$968$	−88.3368	−2.83925
$969$	3.46662	0.111364
$970$	51.1027	1.64081
$971$	−41.3550	−1.32715	−0.663573	−	0.748112i	$-0.730961\pi$
$971$	−41.3550	−1.32715	−0.663573	+	0.748112i	$0.730961\pi$
$972$	5.38198	0.172627
$973$	−12.3368	−0.395500
$974$	−8.75868	−0.280646
$975$	5.45383	0.174662
$976$	137.401	4.39809
$977$	17.1854	0.549811	0.274905	−	0.961471i	$-0.411353\pi$
$977$	17.1854	0.549811	0.274905	+	0.961471i	$0.411353\pi$
$978$	0.462609	0.0147926
$979$	−2.84435	−0.0909057
$980$	−112.926	−3.60730
$981$	−13.0322	−0.416086
$982$	−71.6337	−2.28592
$983$	−46.7909	−1.49240	−0.746199	−	0.665723i	$-0.768123\pi$
$983$	−46.7909	−1.49240	−0.746199	+	0.665723i	$0.768123\pi$
$984$	−4.19390	−0.133697
$985$	−2.50293	−0.0797500
$986$	12.8126	0.408035
$987$	36.0443	1.14730
$988$	43.3679	1.37972
$989$	13.0821	0.415986
$990$	5.21150	0.165632
$991$	41.6979	1.32458	0.662288	−	0.749249i	$-0.269585\pi$
$991$	41.6979	1.32458	0.662288	+	0.749249i	$0.269585\pi$
$992$	−56.0909	−1.78089
$993$	−0.229219	−0.00727404
$994$	158.594	5.03028
$995$	−31.6284	−1.00269
$996$	40.8306	1.29377
$997$	55.2288	1.74911	0.874556	−	0.484924i	$-0.161153\pi$
$997$	55.2288	1.74911	0.874556	+	0.484924i	$0.161153\pi$
$998$	84.9398	2.68872
$999$	6.42528	0.203287

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.e.1.46	✓	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.e.1.46	✓	46	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)

\(f(q)\)	\(=\)	\(q+2.71698 q^{2} +1.00000 q^{3} +5.38198 q^{4} -1.62902 q^{5} +2.71698 q^{6} -4.45873 q^{7} +9.18876 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.71698 q^{2} +1.00000 q^{3} +5.38198 q^{4} -1.62902 q^{5} +2.71698 q^{6} -4.45873 q^{7} +9.18876 q^{8} +1.00000 q^{9} -4.42602 q^{10} -1.17747 q^{11} +5.38198 q^{12} -2.32445 q^{13} -12.1143 q^{14} -1.62902 q^{15} +14.2017 q^{16} -1.00000 q^{17} +2.71698 q^{18} -3.46662 q^{19} -8.76736 q^{20} -4.45873 q^{21} -3.19916 q^{22} -2.41966 q^{23} +9.18876 q^{24} -2.34629 q^{25} -6.31548 q^{26} +1.00000 q^{27} -23.9968 q^{28} -4.71574 q^{29} -4.42602 q^{30} -2.77564 q^{31} +20.2083 q^{32} -1.17747 q^{33} -2.71698 q^{34} +7.26337 q^{35} +5.38198 q^{36} +6.42528 q^{37} -9.41874 q^{38} -2.32445 q^{39} -14.9687 q^{40} -0.456417 q^{41} -12.1143 q^{42} -5.40658 q^{43} -6.33711 q^{44} -1.62902 q^{45} -6.57416 q^{46} -8.08398 q^{47} +14.2017 q^{48} +12.8803 q^{49} -6.37482 q^{50} -1.00000 q^{51} -12.5101 q^{52} +3.91837 q^{53} +2.71698 q^{54} +1.91812 q^{55} -40.9702 q^{56} -3.46662 q^{57} -12.8126 q^{58} +12.1047 q^{59} -8.76736 q^{60} +9.67494 q^{61} -7.54136 q^{62} -4.45873 q^{63} +26.5020 q^{64} +3.78658 q^{65} -3.19916 q^{66} -14.5543 q^{67} -5.38198 q^{68} -2.41966 q^{69} +19.7344 q^{70} -13.0914 q^{71} +9.18876 q^{72} -13.7004 q^{73} +17.4573 q^{74} -2.34629 q^{75} -18.6573 q^{76} +5.25002 q^{77} -6.31548 q^{78} -4.21560 q^{79} -23.1349 q^{80} +1.00000 q^{81} -1.24007 q^{82} +7.58654 q^{83} -23.9968 q^{84} +1.62902 q^{85} -14.6896 q^{86} -4.71574 q^{87} -10.8195 q^{88} +2.41565 q^{89} -4.42602 q^{90} +10.3641 q^{91} -13.0225 q^{92} -2.77564 q^{93} -21.9640 q^{94} +5.64720 q^{95} +20.2083 q^{96} -11.5460 q^{97} +34.9955 q^{98} -1.17747 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9}+O(q^{10}) \) 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9	\( 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9} - 10 q^{10} - 25 q^{11} + 43 q^{12} - 8 q^{13} - 28 q^{14} - 19 q^{15} + 33 q^{16} - 46 q^{17} - 5 q^{18} - 2 q^{19} - 56 q^{20} + q^{21} - 19 q^{22} - 64 q^{23} - 18 q^{24} + 11 q^{25} - 13 q^{26} + 46 q^{27} - 38 q^{28} - 51 q^{29} - 10 q^{30} - 19 q^{31} - 61 q^{32} - 25 q^{33} + 5 q^{34} - 39 q^{35} + 43 q^{36} - 46 q^{37} - 48 q^{38} - 8 q^{39} - 10 q^{40} - 53 q^{41} - 28 q^{42} - 33 q^{43} - 62 q^{44} - 19 q^{45} + 2 q^{46} - 45 q^{47} + 33 q^{48} + 21 q^{49} - 60 q^{50} - 46 q^{51} - 63 q^{52} - 47 q^{53} - 5 q^{54} + 5 q^{55} - 82 q^{56} - 2 q^{57} - 21 q^{58} - 65 q^{59} - 56 q^{60} - 37 q^{61} - 46 q^{62} + q^{63} + 74 q^{64} - 85 q^{65} - 19 q^{66} - 52 q^{67} - 43 q^{68} - 64 q^{69} - 20 q^{70} - 48 q^{71} - 18 q^{72} - 39 q^{73} - 16 q^{74} + 11 q^{75} + 42 q^{76} - 78 q^{77} - 13 q^{78} - 26 q^{79} - 78 q^{80} + 46 q^{81} + 3 q^{82} - 47 q^{83} - 38 q^{84} + 19 q^{85} - 6 q^{86} - 51 q^{87} - 58 q^{88} - 58 q^{89} - 10 q^{90} - 43 q^{91} - 68 q^{92} - 19 q^{93} - 78 q^{95} - 61 q^{96} - 44 q^{97} - 4 q^{98} - 25 q^{99}+O(q^{100}) \) 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9 - 10 * q^10 - 25 * q^11 + 43 * q^12 - 8 * q^13 - 28 * q^14 - 19 * q^15 + 33 * q^16 - 46 * q^17 - 5 * q^18 - 2 * q^19 - 56 * q^20 + q^21 - 19 * q^22 - 64 * q^23 - 18 * q^24 + 11 * q^25 - 13 * q^26 + 46 * q^27 - 38 * q^28 - 51 * q^29 - 10 * q^30 - 19 * q^31 - 61 * q^32 - 25 * q^33 + 5 * q^34 - 39 * q^35 + 43 * q^36 - 46 * q^37 - 48 * q^38 - 8 * q^39 - 10 * q^40 - 53 * q^41 - 28 * q^42 - 33 * q^43 - 62 * q^44 - 19 * q^45 + 2 * q^46 - 45 * q^47 + 33 * q^48 + 21 * q^49 - 60 * q^50 - 46 * q^51 - 63 * q^52 - 47 * q^53 - 5 * q^54 + 5 * q^55 - 82 * q^56 - 2 * q^57 - 21 * q^58 - 65 * q^59 - 56 * q^60 - 37 * q^61 - 46 * q^62 + q^63 + 74 * q^64 - 85 * q^65 - 19 * q^66 - 52 * q^67 - 43 * q^68 - 64 * q^69 - 20 * q^70 - 48 * q^71 - 18 * q^72 - 39 * q^73 - 16 * q^74 + 11 * q^75 + 42 * q^76 - 78 * q^77 - 13 * q^78 - 26 * q^79 - 78 * q^80 + 46 * q^81 + 3 * q^82 - 47 * q^83 - 38 * q^84 + 19 * q^85 - 6 * q^86 - 51 * q^87 - 58 * q^88 - 58 * q^89 - 10 * q^90 - 43 * q^91 - 68 * q^92 - 19 * q^93 - 78 * q^95 - 61 * q^96 - 44 * q^97 - 4 * q^98 - 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more