LMFDB - Embedded newform 8007.2.a.h.1.3

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:	$0$
Dimension:	$56$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	$q-2.59527 q^{2} +1.00000 q^{3} +4.73543 q^{4} -1.29433 q^{5} -2.59527 q^{6} +2.82321 q^{7} -7.09919 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.59527 q^{2} +1.00000 q^{3} +4.73543 q^{4} -1.29433 q^{5} -2.59527 q^{6} +2.82321 q^{7} -7.09919 q^{8} +1.00000 q^{9} +3.35914 q^{10} +3.61231 q^{11} +4.73543 q^{12} -5.81262 q^{13} -7.32699 q^{14} -1.29433 q^{15} +8.95346 q^{16} -1.00000 q^{17} -2.59527 q^{18} -7.48624 q^{19} -6.12921 q^{20} +2.82321 q^{21} -9.37493 q^{22} +7.27184 q^{23} -7.09919 q^{24} -3.32471 q^{25} +15.0853 q^{26} +1.00000 q^{27} +13.3691 q^{28} +3.53187 q^{29} +3.35914 q^{30} -2.87338 q^{31} -9.03827 q^{32} +3.61231 q^{33} +2.59527 q^{34} -3.65416 q^{35} +4.73543 q^{36} +3.38174 q^{37} +19.4288 q^{38} -5.81262 q^{39} +9.18870 q^{40} -5.74628 q^{41} -7.32699 q^{42} -1.58489 q^{43} +17.1059 q^{44} -1.29433 q^{45} -18.8724 q^{46} -4.13004 q^{47} +8.95346 q^{48} +0.970492 q^{49} +8.62852 q^{50} -1.00000 q^{51} -27.5253 q^{52} +6.32322 q^{53} -2.59527 q^{54} -4.67552 q^{55} -20.0425 q^{56} -7.48624 q^{57} -9.16615 q^{58} +0.889294 q^{59} -6.12921 q^{60} +11.4810 q^{61} +7.45719 q^{62} +2.82321 q^{63} +5.54984 q^{64} +7.52345 q^{65} -9.37493 q^{66} -14.4850 q^{67} -4.73543 q^{68} +7.27184 q^{69} +9.48354 q^{70} -2.14486 q^{71} -7.09919 q^{72} -10.9195 q^{73} -8.77653 q^{74} -3.32471 q^{75} -35.4506 q^{76} +10.1983 q^{77} +15.0853 q^{78} +12.4188 q^{79} -11.5887 q^{80} +1.00000 q^{81} +14.9131 q^{82} +7.38217 q^{83} +13.3691 q^{84} +1.29433 q^{85} +4.11322 q^{86} +3.53187 q^{87} -25.6445 q^{88} -11.1213 q^{89} +3.35914 q^{90} -16.4102 q^{91} +34.4353 q^{92} -2.87338 q^{93} +10.7186 q^{94} +9.68966 q^{95} -9.03827 q^{96} +14.0376 q^{97} -2.51869 q^{98} +3.61231 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 7 q^{2} + 56 q^{3} + 61 q^{4} + 17 q^{5} + 7 q^{6} + 5 q^{7} + 18 q^{8} + 56 q^{9}+O(q^{10}) $ 56 * q + 7 * q^2 + 56 * q^3 + 61 * q^4 + 17 * q^5 + 7 * q^6 + 5 * q^7 + 18 * q^8 + 56 * q^9	\( 56 q + 7 q^{2} + 56 q^{3} + 61 q^{4} + 17 q^{5} + 7 q^{6} + 5 q^{7} + 18 q^{8} + 56 q^{9} - 2 q^{10} + 35 q^{11} + 61 q^{12} + 8 q^{13} + 36 q^{14} + 17 q^{15} + 71 q^{16} - 56 q^{17} + 7 q^{18} - 2 q^{19} + 58 q^{20} + 5 q^{21} + 27 q^{22} + 40 q^{23} + 18 q^{24} + 85 q^{25} + 15 q^{26} + 56 q^{27} - 4 q^{28} + 41 q^{29} - 2 q^{30} + q^{31} + 43 q^{32} + 35 q^{33} - 7 q^{34} + 57 q^{35} + 61 q^{36} + 34 q^{37} + 52 q^{38} + 8 q^{39} + 14 q^{40} + 49 q^{41} + 36 q^{42} + 27 q^{43} + 66 q^{44} + 17 q^{45} + 10 q^{46} + 43 q^{47} + 71 q^{48} + 51 q^{49} + 30 q^{50} - 56 q^{51} - 7 q^{52} + 73 q^{53} + 7 q^{54} + 15 q^{55} + 118 q^{56} - 2 q^{57} - q^{58} + 53 q^{59} + 58 q^{60} + 15 q^{61} + 16 q^{62} + 5 q^{63} + 124 q^{64} + 107 q^{65} + 27 q^{66} + 20 q^{67} - 61 q^{68} + 40 q^{69} + 16 q^{70} + 56 q^{71} + 18 q^{72} + 49 q^{73} + 28 q^{74} + 85 q^{75} - 38 q^{76} + 50 q^{77} + 15 q^{78} - 4 q^{79} + 74 q^{80} + 56 q^{81} + 59 q^{82} + 35 q^{83} - 4 q^{84} - 17 q^{85} + 38 q^{86} + 41 q^{87} + 64 q^{88} + 66 q^{89} - 2 q^{90} + 5 q^{91} + 96 q^{92} + q^{93} - 12 q^{94} + 70 q^{95} + 43 q^{96} + 60 q^{97} + 26 q^{98} + 35 q^{99}+O(q^{100}) \) 56 * q + 7 * q^2 + 56 * q^3 + 61 * q^4 + 17 * q^5 + 7 * q^6 + 5 * q^7 + 18 * q^8 + 56 * q^9 - 2 * q^10 + 35 * q^11 + 61 * q^12 + 8 * q^13 + 36 * q^14 + 17 * q^15 + 71 * q^16 - 56 * q^17 + 7 * q^18 - 2 * q^19 + 58 * q^20 + 5 * q^21 + 27 * q^22 + 40 * q^23 + 18 * q^24 + 85 * q^25 + 15 * q^26 + 56 * q^27 - 4 * q^28 + 41 * q^29 - 2 * q^30 + q^31 + 43 * q^32 + 35 * q^33 - 7 * q^34 + 57 * q^35 + 61 * q^36 + 34 * q^37 + 52 * q^38 + 8 * q^39 + 14 * q^40 + 49 * q^41 + 36 * q^42 + 27 * q^43 + 66 * q^44 + 17 * q^45 + 10 * q^46 + 43 * q^47 + 71 * q^48 + 51 * q^49 + 30 * q^50 - 56 * q^51 - 7 * q^52 + 73 * q^53 + 7 * q^54 + 15 * q^55 + 118 * q^56 - 2 * q^57 - q^58 + 53 * q^59 + 58 * q^60 + 15 * q^61 + 16 * q^62 + 5 * q^63 + 124 * q^64 + 107 * q^65 + 27 * q^66 + 20 * q^67 - 61 * q^68 + 40 * q^69 + 16 * q^70 + 56 * q^71 + 18 * q^72 + 49 * q^73 + 28 * q^74 + 85 * q^75 - 38 * q^76 + 50 * q^77 + 15 * q^78 - 4 * q^79 + 74 * q^80 + 56 * q^81 + 59 * q^82 + 35 * q^83 - 4 * q^84 - 17 * q^85 + 38 * q^86 + 41 * q^87 + 64 * q^88 + 66 * q^89 - 2 * q^90 + 5 * q^91 + 96 * q^92 + q^93 - 12 * q^94 + 70 * q^95 + 43 * q^96 + 60 * q^97 + 26 * q^98 + 35 * 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 $

$2$	−2.59527	−1.83513	−0.917567	−	0.397581i	$-0.869850\pi$
$2$	−2.59527	−1.83513	−0.917567	+	0.397581i	$0.869850\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.73543	2.36772
$5$	−1.29433	−0.578842	−0.289421	−	0.957202i	$-0.593463\pi$
$5$	−1.29433	−0.578842	−0.289421	+	0.957202i	$0.593463\pi$
$6$	−2.59527	−1.05952
$7$	2.82321	1.06707	0.533536	−	0.845777i	$-0.320863\pi$
$7$	2.82321	1.06707	0.533536	+	0.845777i	$0.320863\pi$
$8$	−7.09919	−2.50994
$9$	1.00000	0.333333
$10$	3.35914	1.06225
$11$	3.61231	1.08915	0.544576	−	0.838711i	$-0.316690\pi$
$11$	3.61231	1.08915	0.544576	+	0.838711i	$0.316690\pi$
$12$	4.73543	1.36700
$13$	−5.81262	−1.61213	−0.806066	−	0.591826i	$-0.798407\pi$
$13$	−5.81262	−1.61213	−0.806066	+	0.591826i	$0.798407\pi$
$14$	−7.32699	−1.95822
$15$	−1.29433	−0.334195
$16$	8.95346	2.23836
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−2.59527	−0.611711
$19$	−7.48624	−1.71746	−0.858730	−	0.512428i	$-0.828746\pi$
$19$	−7.48624	−1.71746	−0.858730	+	0.512428i	$0.828746\pi$
$20$	−6.12921	−1.37053
$21$	2.82321	0.616074
$22$	−9.37493	−1.99874
$23$	7.27184	1.51628	0.758142	−	0.652089i	$-0.226107\pi$
$23$	7.27184	1.51628	0.758142	+	0.652089i	$0.226107\pi$
$24$	−7.09919	−1.44912
$25$	−3.32471	−0.664942
$26$	15.0853	2.95848
$27$	1.00000	0.192450
$28$	13.3691	2.52652
$29$	3.53187	0.655851	0.327926	−	0.944704i	$-0.393650\pi$
$29$	3.53187	0.655851	0.327926	+	0.944704i	$0.393650\pi$
$30$	3.35914	0.613292
$31$	−2.87338	−0.516074	−0.258037	−	0.966135i	$-0.583076\pi$
$31$	−2.87338	−0.516074	−0.258037	+	0.966135i	$0.583076\pi$
$32$	−9.03827	−1.59776
$33$	3.61231	0.628823
$34$	2.59527	0.445085
$35$	−3.65416	−0.617666
$36$	4.73543	0.789239
$37$	3.38174	0.555954	0.277977	−	0.960588i	$-0.410336\pi$
$37$	3.38174	0.555954	0.277977	+	0.960588i	$0.410336\pi$
$38$	19.4288	3.15177
$39$	−5.81262	−0.930764
$40$	9.18870	1.45286
$41$	−5.74628	−0.897417	−0.448709	−	0.893678i	$-0.648116\pi$
$41$	−5.74628	−0.897417	−0.448709	+	0.893678i	$0.648116\pi$
$42$	−7.32699	−1.13058
$43$	−1.58489	−0.241694	−0.120847	−	0.992671i	$-0.538561\pi$
$43$	−1.58489	−0.241694	−0.120847	+	0.992671i	$0.538561\pi$
$44$	17.1059	2.57880
$45$	−1.29433	−0.192947
$46$	−18.8724	−2.78258
$47$	−4.13004	−0.602429	−0.301214	−	0.953556i	$-0.597392\pi$
$47$	−4.13004	−0.602429	−0.301214	+	0.953556i	$0.597392\pi$
$48$	8.95346	1.29232
$49$	0.970492	0.138642
$50$	8.62852	1.22026
$51$	−1.00000	−0.140028
$52$	−27.5253	−3.81707
$53$	6.32322	0.868561	0.434281	−	0.900778i	$-0.357003\pi$
$53$	6.32322	0.868561	0.434281	+	0.900778i	$0.357003\pi$
$54$	−2.59527	−0.353172
$55$	−4.67552	−0.630448
$56$	−20.0425	−2.67829
$57$	−7.48624	−0.991576
$58$	−9.16615	−1.20357
$59$	0.889294	0.115776	0.0578881	−	0.998323i	$-0.481563\pi$
$59$	0.889294	0.115776	0.0578881	+	0.998323i	$0.481563\pi$
$60$	−6.12921	−0.791278
$61$	11.4810	1.46999	0.734995	−	0.678073i	$-0.237184\pi$
$61$	11.4810	1.46999	0.734995	+	0.678073i	$0.237184\pi$
$62$	7.45719	0.947064
$63$	2.82321	0.355691
$64$	5.54984	0.693730
$65$	7.52345	0.933169
$66$	−9.37493	−1.15397
$67$	−14.4850	−1.76962	−0.884810	−	0.465953i	$-0.845712\pi$
$67$	−14.4850	−1.76962	−0.884810	+	0.465953i	$0.845712\pi$
$68$	−4.73543	−0.574256
$69$	7.27184	0.875427
$70$	9.48354	1.13350
$71$	−2.14486	−0.254548	−0.127274	−	0.991868i	$-0.540623\pi$
$71$	−2.14486	−0.254548	−0.127274	+	0.991868i	$0.540623\pi$
$72$	−7.09919	−0.836647
$73$	−10.9195	−1.27803	−0.639014	−	0.769195i	$-0.720658\pi$
$73$	−10.9195	−1.27803	−0.639014	+	0.769195i	$0.720658\pi$
$74$	−8.77653	−1.02025
$75$	−3.32471	−0.383904
$76$	−35.4506	−4.06646
$77$	10.1983	1.16220
$78$	15.0853	1.70808
$79$	12.4188	1.39722	0.698611	−	0.715502i	$-0.253802\pi$
$79$	12.4188	1.39722	0.698611	+	0.715502i	$0.253802\pi$
$80$	−11.5887	−1.29566
$81$	1.00000	0.111111
$82$	14.9131	1.64688
$83$	7.38217	0.810298	0.405149	−	0.914251i	$-0.367220\pi$
$83$	7.38217	0.810298	0.405149	+	0.914251i	$0.367220\pi$
$84$	13.3691	1.45869
$85$	1.29433	0.140390
$86$	4.11322	0.443540
$87$	3.53187	0.378656
$88$	−25.6445	−2.73371
$89$	−11.1213	−1.17885	−0.589427	−	0.807822i	$-0.700646\pi$
$89$	−11.1213	−1.17885	−0.589427	+	0.807822i	$0.700646\pi$
$90$	3.35914	0.354084
$91$	−16.4102	−1.72026
$92$	34.4353	3.59013
$93$	−2.87338	−0.297955
$94$	10.7186	1.10554
$95$	9.68966	0.994139
$96$	−9.03827	−0.922464
$97$	14.0376	1.42530	0.712651	−	0.701519i	$-0.247494\pi$
$97$	14.0376	1.42530	0.712651	+	0.701519i	$0.247494\pi$
$98$	−2.51869	−0.254426
$99$	3.61231	0.363051
$100$	−15.7439	−1.57439
$101$	0.0637420	0.00634257	0.00317128	−	0.999995i	$-0.498991\pi$
$101$	0.0637420	0.00634257	0.00317128	+	0.999995i	$0.498991\pi$
$102$	2.59527	0.256970
$103$	9.50778	0.936830	0.468415	−	0.883509i	$-0.344825\pi$
$103$	9.50778	0.936830	0.468415	+	0.883509i	$0.344825\pi$
$104$	41.2649	4.04636
$105$	−3.65416	−0.356610
$106$	−16.4105	−1.59393
$107$	3.18032	0.307453	0.153727	−	0.988113i	$-0.450872\pi$
$107$	3.18032	0.307453	0.153727	+	0.988113i	$0.450872\pi$
$108$	4.73543	0.455667
$109$	12.1132	1.16024	0.580119	−	0.814532i	$-0.303006\pi$
$109$	12.1132	1.16024	0.580119	+	0.814532i	$0.303006\pi$
$110$	12.1343	1.15696
$111$	3.38174	0.320980
$112$	25.2774	2.38849
$113$	11.5979	1.09104	0.545521	−	0.838097i	$-0.316332\pi$
$113$	11.5979	1.09104	0.545521	+	0.838097i	$0.316332\pi$
$114$	19.4288	1.81968
$115$	−9.41217	−0.877689
$116$	16.7249	1.55287
$117$	−5.81262	−0.537377
$118$	−2.30796	−0.212465
$119$	−2.82321	−0.258803
$120$	9.18870	0.838809
$121$	2.04879	0.186254
$122$	−29.7963	−2.69763
$123$	−5.74628	−0.518124
$124$	−13.6067	−1.22192
$125$	10.7749	0.963738
$126$	−7.32699	−0.652740
$127$	9.98213	0.885771	0.442885	−	0.896578i	$-0.353955\pi$
$127$	9.98213	0.885771	0.442885	+	0.896578i	$0.353955\pi$
$128$	3.67319	0.324667
$129$	−1.58489	−0.139542
$130$	−19.5254	−1.71249
$131$	10.8612	0.948943	0.474472	−	0.880271i	$-0.342639\pi$
$131$	10.8612	0.948943	0.474472	+	0.880271i	$0.342639\pi$
$132$	17.1059	1.48887
$133$	−21.1352	−1.83265
$134$	37.5924	3.24749
$135$	−1.29433	−0.111398
$136$	7.09919	0.608750
$137$	13.8085	1.17974	0.589870	−	0.807499i	$-0.299179\pi$
$137$	13.8085	1.17974	0.589870	+	0.807499i	$0.299179\pi$
$138$	−18.8724	−1.60653
$139$	13.9082	1.17968	0.589838	−	0.807521i	$-0.299191\pi$
$139$	13.9082	1.17968	0.589838	+	0.807521i	$0.299191\pi$
$140$	−17.3040	−1.46246
$141$	−4.13004	−0.347812
$142$	5.56650	0.467130
$143$	−20.9970	−1.75586
$144$	8.95346	0.746121
$145$	−4.57140	−0.379634
$146$	28.3390	2.34535
$147$	0.970492	0.0800448
$148$	16.0140	1.31634
$149$	3.20421	0.262499	0.131250	−	0.991349i	$-0.458101\pi$
$149$	3.20421	0.262499	0.131250	+	0.991349i	$0.458101\pi$
$150$	8.62852	0.704516
$151$	0.543275	0.0442111	0.0221056	−	0.999756i	$-0.492963\pi$
$151$	0.543275	0.0442111	0.0221056	+	0.999756i	$0.492963\pi$
$152$	53.1462	4.31073
$153$	−1.00000	−0.0808452
$154$	−26.4673	−2.13280
$155$	3.71910	0.298725
$156$	−27.5253	−2.20379
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	−32.2301	−2.56409
$159$	6.32322	0.501464
$160$	11.6985	0.924848
$161$	20.5299	1.61798
$162$	−2.59527	−0.203904
$163$	−17.5697	−1.37616	−0.688081	−	0.725634i	$-0.741547\pi$
$163$	−17.5697	−1.37616	−0.688081	+	0.725634i	$0.741547\pi$
$164$	−27.2111	−2.12483
$165$	−4.67552	−0.363989
$166$	−19.1587	−1.48701
$167$	−4.02439	−0.311417	−0.155708	−	0.987803i	$-0.549766\pi$
$167$	−4.02439	−0.311417	−0.155708	+	0.987803i	$0.549766\pi$
$168$	−20.0425	−1.54631
$169$	20.7866	1.59897
$170$	−3.35914	−0.257634
$171$	−7.48624	−0.572487
$172$	−7.50515	−0.572262
$173$	−11.7697	−0.894834	−0.447417	−	0.894326i	$-0.647656\pi$
$173$	−11.7697	−0.894834	−0.447417	+	0.894326i	$0.647656\pi$
$174$	−9.16615	−0.694884
$175$	−9.38634	−0.709540
$176$	32.3427	2.43792
$177$	0.889294	0.0668435
$178$	28.8628	2.16336
$179$	4.50796	0.336941	0.168470	−	0.985707i	$-0.446117\pi$
$179$	4.50796	0.336941	0.168470	+	0.985707i	$0.446117\pi$
$180$	−6.12921	−0.456845
$181$	−9.29124	−0.690613	−0.345306	−	0.938490i	$-0.612225\pi$
$181$	−9.29124	−0.690613	−0.345306	+	0.938490i	$0.612225\pi$
$182$	42.5890	3.15691
$183$	11.4810	0.848699
$184$	−51.6242	−3.80579
$185$	−4.37709	−0.321810
$186$	7.45719	0.546788
$187$	−3.61231	−0.264158
$188$	−19.5575	−1.42638
$189$	2.82321	0.205358
$190$	−25.1473	−1.82438
$191$	11.3729	0.822912	0.411456	−	0.911430i	$-0.365020\pi$
$191$	11.3729	0.822912	0.411456	+	0.911430i	$0.365020\pi$
$192$	5.54984	0.400525
$193$	8.48201	0.610548	0.305274	−	0.952264i	$-0.401252\pi$
$193$	8.48201	0.610548	0.305274	+	0.952264i	$0.401252\pi$
$194$	−36.4314	−2.61562
$195$	7.52345	0.538766
$196$	4.59570	0.328264
$197$	24.3679	1.73614	0.868071	−	0.496441i	$-0.165360\pi$
$197$	24.3679	1.73614	0.868071	+	0.496441i	$0.165360\pi$
$198$	−9.37493	−0.666247
$199$	12.3861	0.878030	0.439015	−	0.898480i	$-0.355327\pi$
$199$	12.3861	0.878030	0.439015	+	0.898480i	$0.355327\pi$
$200$	23.6027	1.66897
$201$	−14.4850	−1.02169
$202$	−0.165428	−0.0116395
$203$	9.97119	0.699840
$204$	−4.73543	−0.331547
$205$	7.43758	0.519463
$206$	−24.6753	−1.71921
$207$	7.27184	0.505428
$208$	−52.0431	−3.60854
$209$	−27.0426	−1.87058
$210$	9.48354	0.654426
$211$	−8.77784	−0.604291	−0.302146	−	0.953262i	$-0.597703\pi$
$211$	−8.77784	−0.604291	−0.302146	+	0.953262i	$0.597703\pi$
$212$	29.9432	2.05651
$213$	−2.14486	−0.146964
$214$	−8.25379	−0.564218
$215$	2.05137	0.139902
$216$	−7.09919	−0.483039
$217$	−8.11213	−0.550687
$218$	−31.4371	−2.12919
$219$	−10.9195	−0.737870
$220$	−22.1406	−1.49272
$221$	5.81262	0.390999
$222$	−8.77653	−0.589042
$223$	18.5431	1.24174	0.620870	−	0.783913i	$-0.286779\pi$
$223$	18.5431	1.24174	0.620870	+	0.783913i	$0.286779\pi$
$224$	−25.5169	−1.70492
$225$	−3.32471	−0.221647
$226$	−30.0998	−2.00221
$227$	15.5605	1.03278	0.516392	−	0.856352i	$-0.327275\pi$
$227$	15.5605	1.03278	0.516392	+	0.856352i	$0.327275\pi$
$228$	−35.4506	−2.34777
$229$	−14.1397	−0.934377	−0.467188	−	0.884158i	$-0.654733\pi$
$229$	−14.1397	−0.934377	−0.467188	+	0.884158i	$0.654733\pi$
$230$	24.4271	1.61068
$231$	10.1983	0.670999
$232$	−25.0734	−1.64615
$233$	1.71388	0.112280	0.0561399	−	0.998423i	$-0.482121\pi$
$233$	1.71388	0.112280	0.0561399	+	0.998423i	$0.482121\pi$
$234$	15.0853	0.986159
$235$	5.34564	0.348711
$236$	4.21119	0.274125
$237$	12.4188	0.806686
$238$	7.32699	0.474938
$239$	7.48247	0.484001	0.242000	−	0.970276i	$-0.422197\pi$
$239$	7.48247	0.484001	0.242000	+	0.970276i	$0.422197\pi$
$240$	−11.5887	−0.748049
$241$	9.67210	0.623035	0.311517	−	0.950240i	$-0.399163\pi$
$241$	9.67210	0.623035	0.311517	+	0.950240i	$0.399163\pi$
$242$	−5.31717	−0.341801
$243$	1.00000	0.0641500
$244$	54.3674	3.48052
$245$	−1.25614	−0.0802517
$246$	14.9131	0.950827
$247$	43.5147	2.76877
$248$	20.3986	1.29531
$249$	7.38217	0.467826
$250$	−27.9638	−1.76859
$251$	7.92989	0.500530	0.250265	−	0.968177i	$-0.419482\pi$
$251$	7.92989	0.500530	0.250265	+	0.968177i	$0.419482\pi$
$252$	13.3691	0.842174
$253$	26.2682	1.65147
$254$	−25.9063	−1.62551
$255$	1.29433	0.0810541
$256$	−20.6326	−1.28954
$257$	−12.1208	−0.756073	−0.378036	−	0.925791i	$-0.623401\pi$
$257$	−12.1208	−0.756073	−0.378036	+	0.925791i	$0.623401\pi$
$258$	4.11322	0.256078
$259$	9.54734	0.593243
$260$	35.6268	2.20948
$261$	3.53187	0.218617
$262$	−28.1876	−1.74144
$263$	6.13825	0.378501	0.189250	−	0.981929i	$-0.439394\pi$
$263$	6.13825	0.378501	0.189250	+	0.981929i	$0.439394\pi$
$264$	−25.6445	−1.57831
$265$	−8.18434	−0.502760
$266$	54.8516	3.36316
$267$	−11.1213	−0.680612
$268$	−68.5925	−4.18996
$269$	8.06986	0.492028	0.246014	−	0.969266i	$-0.420879\pi$
$269$	8.06986	0.492028	0.246014	+	0.969266i	$0.420879\pi$
$270$	3.35914	0.204431
$271$	−23.0801	−1.40202	−0.701010	−	0.713152i	$-0.747267\pi$
$271$	−23.0801	−1.40202	−0.701010	+	0.713152i	$0.747267\pi$
$272$	−8.95346	−0.542883
$273$	−16.4102	−0.993192
$274$	−35.8368	−2.16498
$275$	−12.0099	−0.724223
$276$	34.4353	2.07276
$277$	10.0238	0.602273	0.301137	−	0.953581i	$-0.402634\pi$
$277$	10.0238	0.602273	0.301137	+	0.953581i	$0.402634\pi$
$278$	−36.0955	−2.16486
$279$	−2.87338	−0.172025
$280$	25.9416	1.55031
$281$	0.274289	0.0163627	0.00818136	−	0.999967i	$-0.497396\pi$
$281$	0.274289	0.0163627	0.00818136	+	0.999967i	$0.497396\pi$
$282$	10.7186	0.638282
$283$	−12.4900	−0.742451	−0.371226	−	0.928543i	$-0.621062\pi$
$283$	−12.4900	−0.742451	−0.371226	+	0.928543i	$0.621062\pi$
$284$	−10.1569	−0.602698
$285$	9.68966	0.573966
$286$	54.4929	3.22223
$287$	−16.2229	−0.957609
$288$	−9.03827	−0.532585
$289$	1.00000	0.0588235
$290$	11.8640	0.696680
$291$	14.0376	0.822899
$292$	−51.7085	−3.02601
$293$	−13.6152	−0.795408	−0.397704	−	0.917514i	$-0.630193\pi$
$293$	−13.6152	−0.795408	−0.397704	+	0.917514i	$0.630193\pi$
$294$	−2.51869	−0.146893
$295$	−1.15104	−0.0670162
$296$	−24.0076	−1.39541
$297$	3.61231	0.209608
$298$	−8.31580	−0.481721
$299$	−42.2685	−2.44445
$300$	−15.7439	−0.908976
$301$	−4.47448	−0.257904
$302$	−1.40995	−0.0811333
$303$	0.0637420	0.00366188
$304$	−67.0277	−3.84430
$305$	−14.8602	−0.850892
$306$	2.59527	0.148362
$307$	4.07409	0.232520	0.116260	−	0.993219i	$-0.462909\pi$
$307$	4.07409	0.232520	0.116260	+	0.993219i	$0.462909\pi$
$308$	48.2934	2.75177
$309$	9.50778	0.540879
$310$	−9.65207	−0.548201
$311$	32.0659	1.81829	0.909146	−	0.416478i	$-0.136736\pi$
$311$	32.0659	1.81829	0.909146	+	0.416478i	$0.136736\pi$
$312$	41.2649	2.33617
$313$	31.7790	1.79626	0.898128	−	0.439735i	$-0.144928\pi$
$313$	31.7790	1.79626	0.898128	+	0.439735i	$0.144928\pi$
$314$	2.59527	0.146460
$315$	−3.65416	−0.205889
$316$	58.8083	3.30822
$317$	18.9762	1.06581	0.532906	−	0.846174i	$-0.321100\pi$
$317$	18.9762	1.06581	0.532906	+	0.846174i	$0.321100\pi$
$318$	−16.4105	−0.920254
$319$	12.7582	0.714322
$320$	−7.18333	−0.401560
$321$	3.18032	0.177508
$322$	−53.2807	−2.96922
$323$	7.48624	0.416545
$324$	4.73543	0.263080
$325$	19.3253	1.07197
$326$	45.5981	2.52544
$327$	12.1132	0.669863
$328$	40.7939	2.25247
$329$	−11.6600	−0.642835
$330$	12.1343	0.667969
$331$	25.0252	1.37551	0.687755	−	0.725943i	$-0.258596\pi$
$331$	25.0252	1.37551	0.687755	+	0.725943i	$0.258596\pi$
$332$	34.9578	1.91856
$333$	3.38174	0.185318
$334$	10.4444	0.571491
$335$	18.7483	1.02433
$336$	25.2774	1.37900
$337$	−6.87959	−0.374755	−0.187378	−	0.982288i	$-0.559999\pi$
$337$	−6.87959	−0.374755	−0.187378	+	0.982288i	$0.559999\pi$
$338$	−53.9468	−2.93432
$339$	11.5979	0.629913
$340$	6.12921	0.332403
$341$	−10.3795	−0.562083
$342$	19.4288	1.05059
$343$	−17.0225	−0.919131
$344$	11.2514	0.606637
$345$	−9.41217	−0.506734
$346$	30.5456	1.64214
$347$	−35.8581	−1.92496	−0.962481	−	0.271350i	$-0.912530\pi$
$347$	−35.8581	−1.92496	−0.962481	+	0.271350i	$0.912530\pi$
$348$	16.7249	0.896550
$349$	−29.0871	−1.55700	−0.778498	−	0.627647i	$-0.784018\pi$
$349$	−29.0871	−1.55700	−0.778498	+	0.627647i	$0.784018\pi$
$350$	24.3601	1.30210
$351$	−5.81262	−0.310255
$352$	−32.6490	−1.74020
$353$	−5.05833	−0.269228	−0.134614	−	0.990898i	$-0.542979\pi$
$353$	−5.05833	−0.269228	−0.134614	+	0.990898i	$0.542979\pi$
$354$	−2.30796	−0.122667
$355$	2.77616	0.147343
$356$	−52.6641	−2.79119
$357$	−2.82321	−0.149420
$358$	−11.6994	−0.618331
$359$	8.20114	0.432839	0.216420	−	0.976300i	$-0.430562\pi$
$359$	8.20114	0.432839	0.216420	+	0.976300i	$0.430562\pi$
$360$	9.18870	0.484287
$361$	37.0438	1.94967
$362$	24.1133	1.26737
$363$	2.04879	0.107534
$364$	−77.7095	−4.07309
$365$	14.1334	0.739777
$366$	−29.7963	−1.55748
$367$	24.3797	1.27261	0.636305	−	0.771437i	$-0.280462\pi$
$367$	24.3797	1.27261	0.636305	+	0.771437i	$0.280462\pi$
$368$	65.1081	3.39400
$369$	−5.74628	−0.299139
$370$	11.3597	0.590564
$371$	17.8518	0.926817
$372$	−13.6067	−0.705473
$373$	2.12895	0.110233	0.0551165	−	0.998480i	$-0.482447\pi$
$373$	2.12895	0.110233	0.0551165	+	0.998480i	$0.482447\pi$
$374$	9.37493	0.484766
$375$	10.7749	0.556415
$376$	29.3200	1.51206
$377$	−20.5294	−1.05732
$378$	−7.32699	−0.376859
$379$	5.78177	0.296989	0.148495	−	0.988913i	$-0.452557\pi$
$379$	5.78177	0.296989	0.148495	+	0.988913i	$0.452557\pi$
$380$	45.8848	2.35384
$381$	9.98213	0.511400
$382$	−29.5157	−1.51015
$383$	5.69557	0.291030	0.145515	−	0.989356i	$-0.453516\pi$
$383$	5.69557	0.291030	0.145515	+	0.989356i	$0.453516\pi$
$384$	3.67319	0.187446
$385$	−13.2000	−0.672733
$386$	−22.0131	−1.12044
$387$	−1.58489	−0.0805646
$388$	66.4741	3.37471
$389$	−15.4118	−0.781412	−0.390706	−	0.920516i	$-0.627769\pi$
$389$	−15.4118	−0.781412	−0.390706	+	0.920516i	$0.627769\pi$
$390$	−19.5254	−0.988707
$391$	−7.27184	−0.367753
$392$	−6.88971	−0.347983
$393$	10.8612	0.547873
$394$	−63.2413	−3.18605
$395$	−16.0740	−0.808771
$396$	17.1059	0.859602
$397$	−12.9190	−0.648385	−0.324192	−	0.945991i	$-0.605093\pi$
$397$	−12.9190	−0.648385	−0.324192	+	0.945991i	$0.605093\pi$
$398$	−32.1454	−1.61130
$399$	−21.1352	−1.05808
$400$	−29.7676	−1.48838
$401$	−14.7105	−0.734606	−0.367303	−	0.930101i	$-0.619719\pi$
$401$	−14.7105	−0.734606	−0.367303	+	0.930101i	$0.619719\pi$
$402$	37.5924	1.87494
$403$	16.7018	0.831978
$404$	0.301846	0.0150174
$405$	−1.29433	−0.0643158
$406$	−25.8779	−1.28430
$407$	12.2159	0.605519
$408$	7.09919	0.351462
$409$	−19.1704	−0.947914	−0.473957	−	0.880548i	$-0.657175\pi$
$409$	−19.1704	−0.947914	−0.473957	+	0.880548i	$0.657175\pi$
$410$	−19.3025	−0.953284
$411$	13.8085	0.681123
$412$	45.0235	2.21815
$413$	2.51066	0.123542
$414$	−18.8724	−0.927528
$415$	−9.55496	−0.469035
$416$	52.5360	2.57579
$417$	13.9082	0.681087
$418$	70.1829	3.43276
$419$	21.0610	1.02890	0.514448	−	0.857521i	$-0.327997\pi$
$419$	21.0610	1.02890	0.514448	+	0.857521i	$0.327997\pi$
$420$	−17.3040	−0.844350
$421$	14.8254	0.722546	0.361273	−	0.932460i	$-0.382342\pi$
$421$	14.8254	0.722546	0.361273	+	0.932460i	$0.382342\pi$
$422$	22.7809	1.10896
$423$	−4.13004	−0.200810
$424$	−44.8897	−2.18004
$425$	3.32471	0.161272
$426$	5.56650	0.269698
$427$	32.4132	1.56858
$428$	15.0602	0.727962
$429$	−20.9970	−1.01374
$430$	−5.32387	−0.256740
$431$	−34.7651	−1.67458	−0.837288	−	0.546762i	$-0.815860\pi$
$431$	−34.7651	−1.67458	−0.837288	+	0.546762i	$0.815860\pi$
$432$	8.95346	0.430773
$433$	−25.1982	−1.21095	−0.605473	−	0.795865i	$-0.707016\pi$
$433$	−25.1982	−1.21095	−0.605473	+	0.795865i	$0.707016\pi$
$434$	21.0532	1.01059
$435$	−4.57140	−0.219182
$436$	57.3614	2.74711
$437$	−54.4388	−2.60416
$438$	28.3390	1.35409
$439$	−28.0401	−1.33828	−0.669140	−	0.743137i	$-0.733337\pi$
$439$	−28.0401	−1.33828	−0.669140	+	0.743137i	$0.733337\pi$
$440$	33.1924	1.58239
$441$	0.970492	0.0462139
$442$	−15.0853	−0.717536
$443$	22.2825	1.05867	0.529336	−	0.848412i	$-0.322441\pi$
$443$	22.2825	1.05867	0.529336	+	0.848412i	$0.322441\pi$
$444$	16.0140	0.759990
$445$	14.3946	0.682370
$446$	−48.1245	−2.27876
$447$	3.20421	0.151554
$448$	15.6684	0.740260
$449$	−29.7771	−1.40527	−0.702633	−	0.711552i	$-0.747992\pi$
$449$	−29.7771	−1.40527	−0.702633	+	0.711552i	$0.747992\pi$
$450$	8.62852	0.406752
$451$	−20.7573	−0.977425
$452$	54.9212	2.58328
$453$	0.543275	0.0255253
$454$	−40.3836	−1.89530
$455$	21.2403	0.995759
$456$	53.1462	2.48880
$457$	28.0698	1.31305	0.656524	−	0.754305i	$-0.272026\pi$
$457$	28.0698	1.31305	0.656524	+	0.754305i	$0.272026\pi$
$458$	36.6963	1.71471
$459$	−1.00000	−0.0466760
$460$	−44.5707	−2.07812
$461$	−2.93265	−0.136587	−0.0682936	−	0.997665i	$-0.521755\pi$
$461$	−2.93265	−0.136587	−0.0682936	+	0.997665i	$0.521755\pi$
$462$	−26.4673	−1.23137
$463$	−0.0939542	−0.00436642	−0.00218321	−	0.999998i	$-0.500695\pi$
$463$	−0.0939542	−0.00436642	−0.00218321	+	0.999998i	$0.500695\pi$
$464$	31.6224	1.46803
$465$	3.71910	0.172469
$466$	−4.44797	−0.206048
$467$	32.0563	1.48339	0.741694	−	0.670738i	$-0.234022\pi$
$467$	32.0563	1.48339	0.741694	+	0.670738i	$0.234022\pi$
$468$	−27.5253	−1.27236
$469$	−40.8940	−1.88831
$470$	−13.8734	−0.639932
$471$	−1.00000	−0.0460776
$472$	−6.31327	−0.290592
$473$	−5.72512	−0.263241
$474$	−32.2301	−1.48038
$475$	24.8896	1.14201
$476$	−13.3691	−0.612772
$477$	6.32322	0.289520
$478$	−19.4190	−0.888206
$479$	15.4163	0.704389	0.352195	−	0.935927i	$-0.385436\pi$
$479$	15.4163	0.704389	0.352195	+	0.935927i	$0.385436\pi$
$480$	11.6985	0.533961
$481$	−19.6568	−0.896271
$482$	−25.1017	−1.14335
$483$	20.5299	0.934143
$484$	9.70192	0.440996
$485$	−18.1693	−0.825025
$486$	−2.59527	−0.117724
$487$	42.2304	1.91364	0.956821	−	0.290677i	$-0.0938805\pi$
$487$	42.2304	1.91364	0.956821	+	0.290677i	$0.0938805\pi$
$488$	−81.5057	−3.68959
$489$	−17.5697	−0.794528
$490$	3.26002	0.147273
$491$	−1.06759	−0.0481796	−0.0240898	−	0.999710i	$-0.507669\pi$
$491$	−1.06759	−0.0481796	−0.0240898	+	0.999710i	$0.507669\pi$
$492$	−27.2111	−1.22677
$493$	−3.53187	−0.159067
$494$	−112.932	−5.08107
$495$	−4.67552	−0.210149
$496$	−25.7266	−1.15516
$497$	−6.05539	−0.271621
$498$	−19.1587	−0.858523
$499$	12.4541	0.557522	0.278761	−	0.960361i	$-0.410076\pi$
$499$	12.4541	0.557522	0.278761	+	0.960361i	$0.410076\pi$
$500$	51.0239	2.28186
$501$	−4.02439	−0.179796
$502$	−20.5802	−0.918540
$503$	14.6062	0.651259	0.325629	−	0.945497i	$-0.394424\pi$
$503$	14.6062	0.651259	0.325629	+	0.945497i	$0.394424\pi$
$504$	−20.0425	−0.892763
$505$	−0.0825032	−0.00367135
$506$	−68.1730	−3.03066
$507$	20.7866	0.923164
$508$	47.2697	2.09725
$509$	−8.02771	−0.355822	−0.177911	−	0.984047i	$-0.556934\pi$
$509$	−8.02771	−0.355822	−0.177911	+	0.984047i	$0.556934\pi$
$510$	−3.35914	−0.148745
$511$	−30.8279	−1.36375
$512$	46.2008	2.04181
$513$	−7.48624	−0.330525
$514$	31.4567	1.38749
$515$	−12.3062	−0.542276
$516$	−7.50515	−0.330396
$517$	−14.9190	−0.656137
$518$	−24.7779	−1.08868
$519$	−11.7697	−0.516632
$520$	−53.4104	−2.34220
$521$	−36.0773	−1.58058	−0.790288	−	0.612735i	$-0.790069\pi$
$521$	−36.0773	−1.58058	−0.790288	+	0.612735i	$0.790069\pi$
$522$	−9.16615	−0.401192
$523$	−19.2575	−0.842070	−0.421035	−	0.907044i	$-0.638333\pi$
$523$	−19.2575	−0.842070	−0.421035	+	0.907044i	$0.638333\pi$
$524$	51.4323	2.24683
$525$	−9.38634	−0.409653
$526$	−15.9304	−0.694600
$527$	2.87338	0.125166
$528$	32.3427	1.40753
$529$	29.8797	1.29912
$530$	21.2406	0.922631
$531$	0.889294	0.0385921
$532$	−100.084	−4.33920
$533$	33.4009	1.44675
$534$	28.8628	1.24901
$535$	−4.11639	−0.177967
$536$	102.831	4.44164
$537$	4.50796	0.194533
$538$	−20.9435	−0.902937
$539$	3.50572	0.151002
$540$	−6.12921	−0.263759
$541$	−24.8452	−1.06818	−0.534090	−	0.845428i	$-0.679346\pi$
$541$	−24.8452	−1.06818	−0.534090	+	0.845428i	$0.679346\pi$
$542$	59.8992	2.57289
$543$	−9.29124	−0.398725
$544$	9.03827	0.387513
$545$	−15.6785	−0.671594
$546$	42.5890	1.82264
$547$	−30.2824	−1.29478	−0.647392	−	0.762157i	$-0.724140\pi$
$547$	−30.2824	−1.29478	−0.647392	+	0.762157i	$0.724140\pi$
$548$	65.3892	2.79329
$549$	11.4810	0.489997
$550$	31.1689	1.32905
$551$	−26.4404	−1.12640
$552$	−51.6242	−2.19727
$553$	35.0608	1.49094
$554$	−26.0145	−1.10525
$555$	−4.37709	−0.185797
$556$	65.8613	2.79314
$557$	2.64562	0.112098	0.0560492	−	0.998428i	$-0.482150\pi$
$557$	2.64562	0.112098	0.0560492	+	0.998428i	$0.482150\pi$
$558$	7.45719	0.315688
$559$	9.21238	0.389642
$560$	−32.7174	−1.38256
$561$	−3.61231	−0.152512
$562$	−0.711855	−0.0300278
$563$	−7.69666	−0.324375	−0.162188	−	0.986760i	$-0.551855\pi$
$563$	−7.69666	−0.324375	−0.162188	+	0.986760i	$0.551855\pi$
$564$	−19.5575	−0.823521
$565$	−15.0116	−0.631541
$566$	32.4148	1.36250
$567$	2.82321	0.118564
$568$	15.2268	0.638902
$569$	−31.3052	−1.31238	−0.656192	−	0.754594i	$-0.727834\pi$
$569$	−31.3052	−1.31238	−0.656192	+	0.754594i	$0.727834\pi$
$570$	−25.1473	−1.05330
$571$	−35.8050	−1.49839	−0.749197	−	0.662348i	$-0.769560\pi$
$571$	−35.8050	−1.49839	−0.749197	+	0.662348i	$0.769560\pi$
$572$	−99.4299	−4.15737
$573$	11.3729	0.475109
$574$	42.1029	1.75734
$575$	−24.1768	−1.00824
$576$	5.54984	0.231243
$577$	12.2738	0.510965	0.255482	−	0.966814i	$-0.417766\pi$
$577$	12.2738	0.510965	0.255482	+	0.966814i	$0.417766\pi$
$578$	−2.59527	−0.107949
$579$	8.48201	0.352500
$580$	−21.6476	−0.898867
$581$	20.8414	0.864646
$582$	−36.4314	−1.51013
$583$	22.8414	0.945996
$584$	77.5195	3.20778
$585$	7.52345	0.311056
$586$	35.3351	1.45968
$587$	−1.92171	−0.0793174	−0.0396587	−	0.999213i	$-0.512627\pi$
$587$	−1.92171	−0.0793174	−0.0396587	+	0.999213i	$0.512627\pi$
$588$	4.59570	0.189523
$589$	21.5108	0.886336
$590$	2.98726	0.122984
$591$	24.3679	1.00236
$592$	30.2782	1.24443
$593$	7.35301	0.301952	0.150976	−	0.988537i	$-0.451758\pi$
$593$	7.35301	0.301952	0.150976	+	0.988537i	$0.451758\pi$
$594$	−9.37493	−0.384658
$595$	3.65416	0.149806
$596$	15.1733	0.621524
$597$	12.3861	0.506931
$598$	109.698	4.48589
$599$	−32.1674	−1.31432	−0.657162	−	0.753750i	$-0.728243\pi$
$599$	−32.1674	−1.31432	−0.657162	+	0.753750i	$0.728243\pi$
$600$	23.6027	0.963578
$601$	−2.82880	−0.115389	−0.0576947	−	0.998334i	$-0.518375\pi$
$601$	−2.82880	−0.115389	−0.0576947	+	0.998334i	$0.518375\pi$
$602$	11.6125	0.473289
$603$	−14.4850	−0.589873
$604$	2.57264	0.104679
$605$	−2.65181	−0.107812
$606$	−0.165428	−0.00672005
$607$	5.83489	0.236831	0.118415	−	0.992964i	$-0.462219\pi$
$607$	5.83489	0.236831	0.118415	+	0.992964i	$0.462219\pi$
$608$	67.6626	2.74408
$609$	9.97119	0.404053
$610$	38.5662	1.56150
$611$	24.0064	0.971194
$612$	−4.73543	−0.191419
$613$	−1.56070	−0.0630360	−0.0315180	−	0.999503i	$-0.510034\pi$
$613$	−1.56070	−0.0630360	−0.0315180	+	0.999503i	$0.510034\pi$
$614$	−10.5734	−0.426706
$615$	7.43758	0.299912
$616$	−72.3996	−2.91706
$617$	−14.1999	−0.571668	−0.285834	−	0.958279i	$-0.592271\pi$
$617$	−14.1999	−0.571668	−0.285834	+	0.958279i	$0.592271\pi$
$618$	−24.6753	−0.992585
$619$	3.36048	0.135069	0.0675345	−	0.997717i	$-0.478487\pi$
$619$	3.36048	0.135069	0.0675345	+	0.997717i	$0.478487\pi$
$620$	17.6115	0.707296
$621$	7.27184	0.291809
$622$	−83.2198	−3.33681
$623$	−31.3977	−1.25792
$624$	−52.0431	−2.08339
$625$	2.67723	0.107089
$626$	−82.4751	−3.29637
$627$	−27.0426	−1.07998
$628$	−4.73543	−0.188964
$629$	−3.38174	−0.134839
$630$	9.48354	0.377833
$631$	−1.71110	−0.0681179	−0.0340589	−	0.999420i	$-0.510843\pi$
$631$	−1.71110	−0.0681179	−0.0340589	+	0.999420i	$0.510843\pi$
$632$	−88.1633	−3.50695
$633$	−8.77784	−0.348888
$634$	−49.2485	−1.95591
$635$	−12.9202	−0.512722
$636$	29.9432	1.18732
$637$	−5.64110	−0.223509
$638$	−33.1110	−1.31088
$639$	−2.14486	−0.0848495
$640$	−4.75432	−0.187931
$641$	10.6998	0.422617	0.211308	−	0.977419i	$-0.432228\pi$
$641$	10.6998	0.422617	0.211308	+	0.977419i	$0.432228\pi$
$642$	−8.25379	−0.325751
$643$	14.7225	0.580600	0.290300	−	0.956936i	$-0.406245\pi$
$643$	14.7225	0.580600	0.290300	+	0.956936i	$0.406245\pi$
$644$	97.2180	3.83093
$645$	2.05137	0.0807727
$646$	−19.4288	−0.764417
$647$	11.6699	0.458793	0.229396	−	0.973333i	$-0.426325\pi$
$647$	11.6699	0.458793	0.229396	+	0.973333i	$0.426325\pi$
$648$	−7.09919	−0.278882
$649$	3.21241	0.126098
$650$	−50.1543	−1.96721
$651$	−8.11213	−0.317940
$652$	−83.2000	−3.25836
$653$	−32.0466	−1.25408	−0.627040	−	0.778987i	$-0.715734\pi$
$653$	−32.0466	−1.25408	−0.627040	+	0.778987i	$0.715734\pi$
$654$	−31.4371	−1.22929
$655$	−14.0579	−0.549288
$656$	−51.4490	−2.00875
$657$	−10.9195	−0.426010
$658$	30.2608	1.17969
$659$	−25.0546	−0.975987	−0.487993	−	0.872847i	$-0.662271\pi$
$659$	−25.0546	−0.975987	−0.487993	+	0.872847i	$0.662271\pi$
$660$	−22.1406	−0.861823
$661$	21.4200	0.833143	0.416572	−	0.909103i	$-0.363232\pi$
$661$	21.4200	0.833143	0.416572	+	0.909103i	$0.363232\pi$
$662$	−64.9472	−2.52425
$663$	5.81262	0.225744
$664$	−52.4074	−2.03380
$665$	27.3559	1.06082
$666$	−8.77653	−0.340083
$667$	25.6832	0.994457
$668$	−19.0572	−0.737346
$669$	18.5431	0.716919
$670$	−48.6570	−1.87978
$671$	41.4729	1.60104
$672$	−25.5169	−0.984335
$673$	44.7606	1.72539	0.862697	−	0.505720i	$-0.168773\pi$
$673$	44.7606	1.72539	0.862697	+	0.505720i	$0.168773\pi$
$674$	17.8544	0.687726
$675$	−3.32471	−0.127968
$676$	98.4334	3.78590
$677$	4.69525	0.180453	0.0902265	−	0.995921i	$-0.471241\pi$
$677$	4.69525	0.180453	0.0902265	+	0.995921i	$0.471241\pi$
$678$	−30.0998	−1.15597
$679$	39.6310	1.52090
$680$	−9.18870	−0.352370
$681$	15.5605	0.596278
$682$	26.9377	1.03150
$683$	47.1981	1.80598	0.902992	−	0.429658i	$-0.141366\pi$
$683$	47.1981	1.80598	0.902992	+	0.429658i	$0.141366\pi$
$684$	−35.4506	−1.35549
$685$	−17.8728	−0.682883
$686$	44.1781	1.68673
$687$	−14.1397	−0.539463
$688$	−14.1903	−0.540998
$689$	−36.7545	−1.40023
$690$	24.4271	0.929925
$691$	17.5693	0.668369	0.334184	−	0.942508i	$-0.391539\pi$
$691$	17.5693	0.668369	0.334184	+	0.942508i	$0.391539\pi$
$692$	−55.7346	−2.11871
$693$	10.1983	0.387401
$694$	93.0614	3.53256
$695$	−18.0018	−0.682847
$696$	−25.0734	−0.950405
$697$	5.74628	0.217656
$698$	75.4889	2.85730
$699$	1.71388	0.0648248
$700$	−44.4484	−1.67999
$701$	−7.25629	−0.274066	−0.137033	−	0.990566i	$-0.543757\pi$
$701$	−7.25629	−0.274066	−0.137033	+	0.990566i	$0.543757\pi$
$702$	15.0853	0.569359
$703$	−25.3165	−0.954829
$704$	20.0478	0.755578
$705$	5.34564	0.201329
$706$	13.1277	0.494069
$707$	0.179957	0.00676797
$708$	4.21119	0.158266
$709$	−42.0496	−1.57921	−0.789603	−	0.613618i	$-0.789713\pi$
$709$	−42.0496	−1.57921	−0.789603	+	0.613618i	$0.789713\pi$
$710$	−7.20489	−0.270395
$711$	12.4188	0.465741
$712$	78.9521	2.95886
$713$	−20.8947	−0.782514
$714$	7.32699	0.274206
$715$	27.1771	1.01636
$716$	21.3471	0.797780
$717$	7.48247	0.279438
$718$	−21.2842	−0.794318
$719$	38.8024	1.44709	0.723543	−	0.690279i	$-0.242512\pi$
$719$	38.8024	1.44709	0.723543	+	0.690279i	$0.242512\pi$
$720$	−11.5887	−0.431886
$721$	26.8424	0.999664
$722$	−96.1386	−3.57791
$723$	9.67210	0.359709
$724$	−43.9981	−1.63518
$725$	−11.7424	−0.436103
$726$	−5.31717	−0.197339
$727$	34.6091	1.28358	0.641791	−	0.766880i	$-0.278192\pi$
$727$	34.6091	1.28358	0.641791	+	0.766880i	$0.278192\pi$
$728$	116.499	4.31775
$729$	1.00000	0.0370370
$730$	−36.6801	−1.35759
$731$	1.58489	0.0586193
$732$	54.3674	2.00948
$733$	−1.12179	−0.0414341	−0.0207170	−	0.999785i	$-0.506595\pi$
$733$	−1.12179	−0.0414341	−0.0207170	+	0.999785i	$0.506595\pi$
$734$	−63.2720	−2.33541
$735$	−1.25614	−0.0463333
$736$	−65.7249	−2.42265
$737$	−52.3242	−1.92739
$738$	14.9131	0.548960
$739$	−15.8128	−0.581682	−0.290841	−	0.956771i	$-0.593935\pi$
$739$	−15.8128	−0.581682	−0.290841	+	0.956771i	$0.593935\pi$
$740$	−20.7274	−0.761954
$741$	43.5147	1.59855
$742$	−46.3301	−1.70083
$743$	−21.2476	−0.779500	−0.389750	−	0.920921i	$-0.627439\pi$
$743$	−21.2476	−0.779500	−0.389750	+	0.920921i	$0.627439\pi$
$744$	20.3986	0.747850
$745$	−4.14731	−0.151946
$746$	−5.52521	−0.202292
$747$	7.38217	0.270099
$748$	−17.1059	−0.625452
$749$	8.97870	0.328075
$750$	−27.9638	−1.02110
$751$	15.0078	0.547642	0.273821	−	0.961781i	$-0.411712\pi$
$751$	15.0078	0.547642	0.273821	+	0.961781i	$0.411712\pi$
$752$	−36.9782	−1.34846
$753$	7.92989	0.288981
$754$	53.2794	1.94032
$755$	−0.703178	−0.0255913
$756$	13.3691	0.486230
$757$	49.6998	1.80637	0.903185	−	0.429252i	$-0.141223\pi$
$757$	49.6998	1.80637	0.903185	+	0.429252i	$0.141223\pi$
$758$	−15.0052	−0.545015
$759$	26.2682	0.953474
$760$	−68.7888	−2.49523
$761$	−21.5094	−0.779716	−0.389858	−	0.920875i	$-0.627476\pi$
$761$	−21.5094	−0.779716	−0.389858	+	0.920875i	$0.627476\pi$
$762$	−25.9063	−0.938488
$763$	34.1982	1.23806
$764$	53.8555	1.94842
$765$	1.29433	0.0467966
$766$	−14.7815	−0.534079
$767$	−5.16913	−0.186647
$768$	−20.6326	−0.744515
$769$	−7.26860	−0.262113	−0.131056	−	0.991375i	$-0.541837\pi$
$769$	−7.26860	−0.262113	−0.131056	+	0.991375i	$0.541837\pi$
$770$	34.2575	1.23455
$771$	−12.1208	−0.436519
$772$	40.1660	1.44561
$773$	13.8343	0.497586	0.248793	−	0.968557i	$-0.419966\pi$
$773$	13.8343	0.497586	0.248793	+	0.968557i	$0.419966\pi$
$774$	4.11322	0.147847
$775$	9.55314	0.343159
$776$	−99.6556	−3.57743
$777$	9.54734	0.342509
$778$	39.9979	1.43400
$779$	43.0180	1.54128
$780$	35.6268	1.27564
$781$	−7.74791	−0.277242
$782$	18.8724	0.674876
$783$	3.53187	0.126219
$784$	8.68926	0.310331
$785$	1.29433	0.0461966
$786$	−28.1876	−1.00542
$787$	42.4968	1.51485	0.757424	−	0.652923i	$-0.226458\pi$
$787$	42.4968	1.51485	0.757424	+	0.652923i	$0.226458\pi$
$788$	115.393	4.11069
$789$	6.13825	0.218528
$790$	41.7164	1.48420
$791$	32.7433	1.16422
$792$	−25.6445	−0.911237
$793$	−66.7346	−2.36982
$794$	33.5282	1.18987
$795$	−8.18434	−0.290268
$796$	58.6537	2.07893
$797$	25.1374	0.890411	0.445206	−	0.895428i	$-0.353131\pi$
$797$	25.1374	0.890411	0.445206	+	0.895428i	$0.353131\pi$
$798$	54.8516	1.94172
$799$	4.13004	0.146110
$800$	30.0496	1.06241
$801$	−11.1213	−0.392951
$802$	38.1777	1.34810
$803$	−39.4446	−1.39197
$804$	−68.5925	−2.41907
$805$	−26.5725	−0.936557
$806$	−43.3458	−1.52679
$807$	8.06986	0.284073
$808$	−0.452517	−0.0159195
$809$	38.8299	1.36519	0.682593	−	0.730798i	$-0.260852\pi$
$809$	38.8299	1.36519	0.682593	+	0.730798i	$0.260852\pi$
$810$	3.35914	0.118028
$811$	0.808399	0.0283867	0.0141934	−	0.999899i	$-0.495482\pi$
$811$	0.808399	0.0283867	0.0141934	+	0.999899i	$0.495482\pi$
$812$	47.2179	1.65702
$813$	−23.0801	−0.809456
$814$	−31.7035	−1.11121
$815$	22.7410	0.796581
$816$	−8.95346	−0.313434
$817$	11.8649	0.415099
$818$	49.7523	1.73955
$819$	−16.4102	−0.573420
$820$	35.2202	1.22994
$821$	9.75485	0.340446	0.170223	−	0.985406i	$-0.445551\pi$
$821$	9.75485	0.340446	0.170223	+	0.985406i	$0.445551\pi$
$822$	−35.8368	−1.24995
$823$	−9.46164	−0.329812	−0.164906	−	0.986309i	$-0.552732\pi$
$823$	−9.46164	−0.329812	−0.164906	+	0.986309i	$0.552732\pi$
$824$	−67.4975	−2.35139
$825$	−12.0099	−0.418130
$826$	−6.51585	−0.226715
$827$	−9.47357	−0.329428	−0.164714	−	0.986341i	$-0.552670\pi$
$827$	−9.47357	−0.329428	−0.164714	+	0.986341i	$0.552670\pi$
$828$	34.4353	1.19671
$829$	−4.16372	−0.144612	−0.0723059	−	0.997383i	$-0.523036\pi$
$829$	−4.16372	−0.144612	−0.0723059	+	0.997383i	$0.523036\pi$
$830$	24.7977	0.860741
$831$	10.0238	0.347723
$832$	−32.2591	−1.11838
$833$	−0.970492	−0.0336256
$834$	−36.0955	−1.24989
$835$	5.20889	0.180261
$836$	−128.058	−4.42900
$837$	−2.87338	−0.0993184
$838$	−54.6590	−1.88816
$839$	23.9392	0.826473	0.413237	−	0.910624i	$-0.364398\pi$
$839$	23.9392	0.826473	0.413237	+	0.910624i	$0.364398\pi$
$840$	25.9416	0.895070
$841$	−16.5259	−0.569859
$842$	−38.4759	−1.32597
$843$	0.274289	0.00944702
$844$	−41.5669	−1.43079
$845$	−26.9047	−0.925550
$846$	10.7186	0.368513
$847$	5.78416	0.198746
$848$	56.6147	1.94416
$849$	−12.4900	−0.428655
$850$	−8.62852	−0.295956
$851$	24.5915	0.842985
$852$	−10.1569	−0.347968
$853$	36.8137	1.26048	0.630239	−	0.776402i	$-0.282957\pi$
$853$	36.8137	1.26048	0.630239	+	0.776402i	$0.282957\pi$
$854$	−84.1210	−2.87856
$855$	9.68966	0.331380
$856$	−22.5777	−0.771690
$857$	−54.5359	−1.86291	−0.931455	−	0.363857i	$-0.881460\pi$
$857$	−54.5359	−1.86291	−0.931455	+	0.363857i	$0.881460\pi$
$858$	54.4929	1.86036
$859$	29.4693	1.00548	0.502739	−	0.864438i	$-0.332326\pi$
$859$	29.4693	1.00548	0.502739	+	0.864438i	$0.332326\pi$
$860$	9.71414	0.331249
$861$	−16.2229	−0.552876
$862$	90.2249	3.07307
$863$	−7.37658	−0.251102	−0.125551	−	0.992087i	$-0.540070\pi$
$863$	−7.37658	−0.251102	−0.125551	+	0.992087i	$0.540070\pi$
$864$	−9.03827	−0.307488
$865$	15.2339	0.517967
$866$	65.3961	2.22225
$867$	1.00000	0.0339618
$868$	−38.4145	−1.30387
$869$	44.8605	1.52179
$870$	11.8640	0.402228
$871$	84.1956	2.85286
$872$	−85.9941	−2.91213
$873$	14.0376	0.475101
$874$	141.283	4.77898
$875$	30.4198	1.02838
$876$	−51.7085	−1.74707
$877$	−18.1335	−0.612326	−0.306163	−	0.951979i	$-0.599045\pi$
$877$	−18.1335	−0.612326	−0.306163	+	0.951979i	$0.599045\pi$
$878$	72.7716	2.45592
$879$	−13.6152	−0.459229
$880$	−41.8621	−1.41117
$881$	−10.3405	−0.348382	−0.174191	−	0.984712i	$-0.555731\pi$
$881$	−10.3405	−0.348382	−0.174191	+	0.984712i	$0.555731\pi$
$882$	−2.51869	−0.0848087
$883$	50.9050	1.71309	0.856546	−	0.516071i	$-0.172606\pi$
$883$	50.9050	1.71309	0.856546	+	0.516071i	$0.172606\pi$
$884$	27.5253	0.925775
$885$	−1.15104	−0.0386918
$886$	−57.8291	−1.94281
$887$	42.4458	1.42519	0.712595	−	0.701575i	$-0.247520\pi$
$887$	42.4458	1.42519	0.712595	+	0.701575i	$0.247520\pi$
$888$	−24.0076	−0.805642
$889$	28.1816	0.945181
$890$	−37.3579	−1.25224
$891$	3.61231	0.121017
$892$	87.8098	2.94009
$893$	30.9185	1.03465
$894$	−8.31580	−0.278122
$895$	−5.83479	−0.195035
$896$	10.3702	0.346443
$897$	−42.2685	−1.41130
$898$	77.2795	2.57885
$899$	−10.1484	−0.338468
$900$	−15.7439	−0.524798
$901$	−6.32322	−0.210657
$902$	53.8709	1.79371
$903$	−4.47448	−0.148901
$904$	−82.3359	−2.73845
$905$	12.0259	0.399756
$906$	−1.40995	−0.0468424
$907$	24.5417	0.814893	0.407446	−	0.913229i	$-0.366419\pi$
$907$	24.5417	0.814893	0.407446	+	0.913229i	$0.366419\pi$
$908$	73.6855	2.44534
$909$	0.0637420	0.00211419
$910$	−55.1242	−1.82735
$911$	32.8705	1.08905	0.544525	−	0.838745i	$-0.316710\pi$
$911$	32.8705	1.08905	0.544525	+	0.838745i	$0.316710\pi$
$912$	−67.0277	−2.21951
$913$	26.6667	0.882538
$914$	−72.8487	−2.40962
$915$	−14.8602	−0.491263
$916$	−66.9575	−2.21234
$917$	30.6633	1.01259
$918$	2.59527	0.0856567
$919$	16.4430	0.542406	0.271203	−	0.962522i	$-0.412579\pi$
$919$	16.4430	0.542406	0.271203	+	0.962522i	$0.412579\pi$
$920$	66.8188	2.20295
$921$	4.07409	0.134246
$922$	7.61103	0.250656
$923$	12.4673	0.410365
$924$	48.2934	1.58873
$925$	−11.2433	−0.369677
$926$	0.243837	0.00801296
$927$	9.50778	0.312277
$928$	−31.9220	−1.04789
$929$	57.9519	1.90134	0.950670	−	0.310206i	$-0.100398\pi$
$929$	57.9519	1.90134	0.950670	+	0.310206i	$0.100398\pi$
$930$	−9.65207	−0.316504
$931$	−7.26533	−0.238112
$932$	8.11594	0.265847
$933$	32.0659	1.04979
$934$	−83.1948	−2.72222
$935$	4.67552	0.152906
$936$	41.2649	1.34879
$937$	−42.4066	−1.38536	−0.692681	−	0.721244i	$-0.743571\pi$
$937$	−42.4066	−1.38536	−0.692681	+	0.721244i	$0.743571\pi$
$938$	106.131	3.46530
$939$	31.7790	1.03707
$940$	25.3139	0.825649
$941$	38.4670	1.25399	0.626994	−	0.779024i	$-0.284285\pi$
$941$	38.4670	1.25399	0.626994	+	0.779024i	$0.284285\pi$
$942$	2.59527	0.0845585
$943$	−41.7860	−1.36074
$944$	7.96226	0.259149
$945$	−3.65416	−0.118870
$946$	14.8582	0.483083
$947$	46.4605	1.50976	0.754882	−	0.655860i	$-0.227694\pi$
$947$	46.4605	1.50976	0.754882	+	0.655860i	$0.227694\pi$
$948$	58.8083	1.91000
$949$	63.4708	2.06035
$950$	−64.5952	−2.09574
$951$	18.9762	0.615347
$952$	20.0425	0.649580
$953$	6.28133	0.203472	0.101736	−	0.994811i	$-0.467560\pi$
$953$	6.28133	0.203472	0.101736	+	0.994811i	$0.467560\pi$
$954$	−16.4105	−0.531309
$955$	−14.7203	−0.476336
$956$	35.4327	1.14598
$957$	12.7582	0.412414
$958$	−40.0095	−1.29265
$959$	38.9842	1.25887
$960$	−7.18333	−0.231841
$961$	−22.7437	−0.733668
$962$	51.0146	1.64478
$963$	3.18032	0.102484
$964$	45.8016	1.47517
$965$	−10.9785	−0.353411
$966$	−53.2807	−1.71428
$967$	49.8623	1.60346	0.801732	−	0.597683i	$-0.203912\pi$
$967$	49.8623	1.60346	0.801732	+	0.597683i	$0.203912\pi$
$968$	−14.5448	−0.467486
$969$	7.48624	0.240493
$970$	47.1542	1.51403
$971$	40.4682	1.29869	0.649344	−	0.760495i	$-0.275044\pi$
$971$	40.4682	1.29869	0.649344	+	0.760495i	$0.275044\pi$
$972$	4.73543	0.151889
$973$	39.2657	1.25880
$974$	−109.599	−3.51179
$975$	19.3253	0.618904
$976$	102.795	3.29037
$977$	40.6955	1.30196	0.650982	−	0.759094i	$-0.274358\pi$
$977$	40.6955	1.30196	0.650982	+	0.759094i	$0.274358\pi$
$978$	45.5981	1.45806
$979$	−40.1736	−1.28395
$980$	−5.94835	−0.190013
$981$	12.1132	0.386746
$982$	2.77068	0.0884161
$983$	−17.5672	−0.560306	−0.280153	−	0.959955i	$-0.590385\pi$
$983$	−17.5672	−0.560306	−0.280153	+	0.959955i	$0.590385\pi$
$984$	40.7939	1.30046
$985$	−31.5401	−1.00495
$986$	9.16615	0.291910
$987$	−11.6600	−0.371141
$988$	206.061	6.55567
$989$	−11.5251	−0.366476
$990$	12.1343	0.385652
$991$	−52.6209	−1.67156	−0.835779	−	0.549066i	$-0.814983\pi$
$991$	−52.6209	−1.67156	−0.835779	+	0.549066i	$0.814983\pi$
$992$	25.9703	0.824559
$993$	25.0252	0.794151
$994$	15.7154	0.498462
$995$	−16.0318	−0.508241
$996$	34.9578	1.10768
$997$	−44.7132	−1.41608	−0.708041	−	0.706171i	$-0.750421\pi$
$997$	−44.7132	−1.41608	−0.708041	+	0.706171i	$0.750421\pi$
$998$	−32.3217	−1.02313
$999$	3.38174	0.106993

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.h.1.3	✓	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.h.1.3	✓	56	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q-2.59527 q^{2} +1.00000 q^{3} +4.73543 q^{4} -1.29433 q^{5} -2.59527 q^{6} +2.82321 q^{7} -7.09919 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.59527 q^{2} +1.00000 q^{3} +4.73543 q^{4} -1.29433 q^{5} -2.59527 q^{6} +2.82321 q^{7} -7.09919 q^{8} +1.00000 q^{9} +3.35914 q^{10} +3.61231 q^{11} +4.73543 q^{12} -5.81262 q^{13} -7.32699 q^{14} -1.29433 q^{15} +8.95346 q^{16} -1.00000 q^{17} -2.59527 q^{18} -7.48624 q^{19} -6.12921 q^{20} +2.82321 q^{21} -9.37493 q^{22} +7.27184 q^{23} -7.09919 q^{24} -3.32471 q^{25} +15.0853 q^{26} +1.00000 q^{27} +13.3691 q^{28} +3.53187 q^{29} +3.35914 q^{30} -2.87338 q^{31} -9.03827 q^{32} +3.61231 q^{33} +2.59527 q^{34} -3.65416 q^{35} +4.73543 q^{36} +3.38174 q^{37} +19.4288 q^{38} -5.81262 q^{39} +9.18870 q^{40} -5.74628 q^{41} -7.32699 q^{42} -1.58489 q^{43} +17.1059 q^{44} -1.29433 q^{45} -18.8724 q^{46} -4.13004 q^{47} +8.95346 q^{48} +0.970492 q^{49} +8.62852 q^{50} -1.00000 q^{51} -27.5253 q^{52} +6.32322 q^{53} -2.59527 q^{54} -4.67552 q^{55} -20.0425 q^{56} -7.48624 q^{57} -9.16615 q^{58} +0.889294 q^{59} -6.12921 q^{60} +11.4810 q^{61} +7.45719 q^{62} +2.82321 q^{63} +5.54984 q^{64} +7.52345 q^{65} -9.37493 q^{66} -14.4850 q^{67} -4.73543 q^{68} +7.27184 q^{69} +9.48354 q^{70} -2.14486 q^{71} -7.09919 q^{72} -10.9195 q^{73} -8.77653 q^{74} -3.32471 q^{75} -35.4506 q^{76} +10.1983 q^{77} +15.0853 q^{78} +12.4188 q^{79} -11.5887 q^{80} +1.00000 q^{81} +14.9131 q^{82} +7.38217 q^{83} +13.3691 q^{84} +1.29433 q^{85} +4.11322 q^{86} +3.53187 q^{87} -25.6445 q^{88} -11.1213 q^{89} +3.35914 q^{90} -16.4102 q^{91} +34.4353 q^{92} -2.87338 q^{93} +10.7186 q^{94} +9.68966 q^{95} -9.03827 q^{96} +14.0376 q^{97} -2.51869 q^{98} +3.61231 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 7 q^{2} + 56 q^{3} + 61 q^{4} + 17 q^{5} + 7 q^{6} + 5 q^{7} + 18 q^{8} + 56 q^{9}+O(q^{10}) \) 56 * q + 7 * q^2 + 56 * q^3 + 61 * q^4 + 17 * q^5 + 7 * q^6 + 5 * q^7 + 18 * q^8 + 56 * q^9	\( 56 q + 7 q^{2} + 56 q^{3} + 61 q^{4} + 17 q^{5} + 7 q^{6} + 5 q^{7} + 18 q^{8} + 56 q^{9} - 2 q^{10} + 35 q^{11} + 61 q^{12} + 8 q^{13} + 36 q^{14} + 17 q^{15} + 71 q^{16} - 56 q^{17} + 7 q^{18} - 2 q^{19} + 58 q^{20} + 5 q^{21} + 27 q^{22} + 40 q^{23} + 18 q^{24} + 85 q^{25} + 15 q^{26} + 56 q^{27} - 4 q^{28} + 41 q^{29} - 2 q^{30} + q^{31} + 43 q^{32} + 35 q^{33} - 7 q^{34} + 57 q^{35} + 61 q^{36} + 34 q^{37} + 52 q^{38} + 8 q^{39} + 14 q^{40} + 49 q^{41} + 36 q^{42} + 27 q^{43} + 66 q^{44} + 17 q^{45} + 10 q^{46} + 43 q^{47} + 71 q^{48} + 51 q^{49} + 30 q^{50} - 56 q^{51} - 7 q^{52} + 73 q^{53} + 7 q^{54} + 15 q^{55} + 118 q^{56} - 2 q^{57} - q^{58} + 53 q^{59} + 58 q^{60} + 15 q^{61} + 16 q^{62} + 5 q^{63} + 124 q^{64} + 107 q^{65} + 27 q^{66} + 20 q^{67} - 61 q^{68} + 40 q^{69} + 16 q^{70} + 56 q^{71} + 18 q^{72} + 49 q^{73} + 28 q^{74} + 85 q^{75} - 38 q^{76} + 50 q^{77} + 15 q^{78} - 4 q^{79} + 74 q^{80} + 56 q^{81} + 59 q^{82} + 35 q^{83} - 4 q^{84} - 17 q^{85} + 38 q^{86} + 41 q^{87} + 64 q^{88} + 66 q^{89} - 2 q^{90} + 5 q^{91} + 96 q^{92} + q^{93} - 12 q^{94} + 70 q^{95} + 43 q^{96} + 60 q^{97} + 26 q^{98} + 35 q^{99}+O(q^{100}) \) 56 * q + 7 * q^2 + 56 * q^3 + 61 * q^4 + 17 * q^5 + 7 * q^6 + 5 * q^7 + 18 * q^8 + 56 * q^9 - 2 * q^10 + 35 * q^11 + 61 * q^12 + 8 * q^13 + 36 * q^14 + 17 * q^15 + 71 * q^16 - 56 * q^17 + 7 * q^18 - 2 * q^19 + 58 * q^20 + 5 * q^21 + 27 * q^22 + 40 * q^23 + 18 * q^24 + 85 * q^25 + 15 * q^26 + 56 * q^27 - 4 * q^28 + 41 * q^29 - 2 * q^30 + q^31 + 43 * q^32 + 35 * q^33 - 7 * q^34 + 57 * q^35 + 61 * q^36 + 34 * q^37 + 52 * q^38 + 8 * q^39 + 14 * q^40 + 49 * q^41 + 36 * q^42 + 27 * q^43 + 66 * q^44 + 17 * q^45 + 10 * q^46 + 43 * q^47 + 71 * q^48 + 51 * q^49 + 30 * q^50 - 56 * q^51 - 7 * q^52 + 73 * q^53 + 7 * q^54 + 15 * q^55 + 118 * q^56 - 2 * q^57 - q^58 + 53 * q^59 + 58 * q^60 + 15 * q^61 + 16 * q^62 + 5 * q^63 + 124 * q^64 + 107 * q^65 + 27 * q^66 + 20 * q^67 - 61 * q^68 + 40 * q^69 + 16 * q^70 + 56 * q^71 + 18 * q^72 + 49 * q^73 + 28 * q^74 + 85 * q^75 - 38 * q^76 + 50 * q^77 + 15 * q^78 - 4 * q^79 + 74 * q^80 + 56 * q^81 + 59 * q^82 + 35 * q^83 - 4 * q^84 - 17 * q^85 + 38 * q^86 + 41 * q^87 + 64 * q^88 + 66 * q^89 - 2 * q^90 + 5 * q^91 + 96 * q^92 + q^93 - 12 * q^94 + 70 * q^95 + 43 * q^96 + 60 * q^97 + 26 * q^98 + 35 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more