LMFDB - Embedded newform 8007.2.a.g.1.11

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			1.11
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.93055 q^{2} -1.00000 q^{3} +1.72702 q^{4} -2.51216 q^{5} +1.93055 q^{6} -1.67367 q^{7} +0.526999 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.93055 q^{2} -1.00000 q^{3} +1.72702 q^{4} -2.51216 q^{5} +1.93055 q^{6} -1.67367 q^{7} +0.526999 q^{8} +1.00000 q^{9} +4.84984 q^{10} +3.46212 q^{11} -1.72702 q^{12} -3.72991 q^{13} +3.23111 q^{14} +2.51216 q^{15} -4.47144 q^{16} -1.00000 q^{17} -1.93055 q^{18} -6.65781 q^{19} -4.33855 q^{20} +1.67367 q^{21} -6.68379 q^{22} -2.05998 q^{23} -0.526999 q^{24} +1.31092 q^{25} +7.20078 q^{26} -1.00000 q^{27} -2.89047 q^{28} -1.98149 q^{29} -4.84984 q^{30} +4.55602 q^{31} +7.57834 q^{32} -3.46212 q^{33} +1.93055 q^{34} +4.20452 q^{35} +1.72702 q^{36} +6.72209 q^{37} +12.8532 q^{38} +3.72991 q^{39} -1.32390 q^{40} -8.70922 q^{41} -3.23111 q^{42} +0.0248505 q^{43} +5.97915 q^{44} -2.51216 q^{45} +3.97689 q^{46} -10.5151 q^{47} +4.47144 q^{48} -4.19882 q^{49} -2.53080 q^{50} +1.00000 q^{51} -6.44164 q^{52} -1.38373 q^{53} +1.93055 q^{54} -8.69738 q^{55} -0.882023 q^{56} +6.65781 q^{57} +3.82536 q^{58} -0.210577 q^{59} +4.33855 q^{60} -3.21782 q^{61} -8.79562 q^{62} -1.67367 q^{63} -5.68748 q^{64} +9.37012 q^{65} +6.68379 q^{66} +12.4029 q^{67} -1.72702 q^{68} +2.05998 q^{69} -8.11704 q^{70} -1.02758 q^{71} +0.526999 q^{72} -13.6556 q^{73} -12.9773 q^{74} -1.31092 q^{75} -11.4982 q^{76} -5.79445 q^{77} -7.20078 q^{78} -3.85826 q^{79} +11.2330 q^{80} +1.00000 q^{81} +16.8136 q^{82} -0.306799 q^{83} +2.89047 q^{84} +2.51216 q^{85} -0.0479751 q^{86} +1.98149 q^{87} +1.82453 q^{88} -11.0387 q^{89} +4.84984 q^{90} +6.24265 q^{91} -3.55762 q^{92} -4.55602 q^{93} +20.3000 q^{94} +16.7255 q^{95} -7.57834 q^{96} +4.61672 q^{97} +8.10603 q^{98} +3.46212 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + q^{2} - 56 q^{3} + 61 q^{4} + q^{5} - q^{6} + 19 q^{7} + 56 q^{9}+O(q^{10}) $ 56 * q + q^2 - 56 * q^3 + 61 * q^4 + q^5 - q^6 + 19 * q^7 + 56 * q^9	\( 56 q + q^{2} - 56 q^{3} + 61 q^{4} + q^{5} - q^{6} + 19 q^{7} + 56 q^{9} + 8 q^{10} - 7 q^{11} - 61 q^{12} + 8 q^{13} - 8 q^{14} - q^{15} + 71 q^{16} - 56 q^{17} + q^{18} - 2 q^{19} - 4 q^{20} - 19 q^{21} + 47 q^{22} + 16 q^{23} + 85 q^{25} - 11 q^{26} - 56 q^{27} + 52 q^{28} + 17 q^{29} - 8 q^{30} + 23 q^{31} + 11 q^{32} + 7 q^{33} - q^{34} - 41 q^{35} + 61 q^{36} + 58 q^{37} - 22 q^{38} - 8 q^{39} + 38 q^{40} - q^{41} + 8 q^{42} + 27 q^{43} + 2 q^{44} + q^{45} + 46 q^{46} + 5 q^{47} - 71 q^{48} + 59 q^{49} - 4 q^{50} + 56 q^{51} + 25 q^{52} + 15 q^{53} - q^{54} + 9 q^{55} - 36 q^{56} + 2 q^{57} + 89 q^{58} - 61 q^{59} + 4 q^{60} + 47 q^{61} + 8 q^{62} + 19 q^{63} + 88 q^{64} + 39 q^{65} - 47 q^{66} + 20 q^{67} - 61 q^{68} - 16 q^{69} + 36 q^{70} - 2 q^{71} + 93 q^{73} + 48 q^{74} - 85 q^{75} + 38 q^{76} + 26 q^{77} + 11 q^{78} + 72 q^{79} + 42 q^{80} + 56 q^{81} + 33 q^{82} - 11 q^{83} - 52 q^{84} - q^{85} - 4 q^{86} - 17 q^{87} + 130 q^{88} - 6 q^{89} + 8 q^{90} + 37 q^{91} + 132 q^{92} - 23 q^{93} - 32 q^{94} + 12 q^{95} - 11 q^{96} + 100 q^{97} + 42 q^{98} - 7 q^{99}+O(q^{100}) \) 56 * q + q^2 - 56 * q^3 + 61 * q^4 + q^5 - q^6 + 19 * q^7 + 56 * q^9 + 8 * q^10 - 7 * q^11 - 61 * q^12 + 8 * q^13 - 8 * q^14 - q^15 + 71 * q^16 - 56 * q^17 + q^18 - 2 * q^19 - 4 * q^20 - 19 * q^21 + 47 * q^22 + 16 * q^23 + 85 * q^25 - 11 * q^26 - 56 * q^27 + 52 * q^28 + 17 * q^29 - 8 * q^30 + 23 * q^31 + 11 * q^32 + 7 * q^33 - q^34 - 41 * q^35 + 61 * q^36 + 58 * q^37 - 22 * q^38 - 8 * q^39 + 38 * q^40 - q^41 + 8 * q^42 + 27 * q^43 + 2 * q^44 + q^45 + 46 * q^46 + 5 * q^47 - 71 * q^48 + 59 * q^49 - 4 * q^50 + 56 * q^51 + 25 * q^52 + 15 * q^53 - q^54 + 9 * q^55 - 36 * q^56 + 2 * q^57 + 89 * q^58 - 61 * q^59 + 4 * q^60 + 47 * q^61 + 8 * q^62 + 19 * q^63 + 88 * q^64 + 39 * q^65 - 47 * q^66 + 20 * q^67 - 61 * q^68 - 16 * q^69 + 36 * q^70 - 2 * q^71 + 93 * q^73 + 48 * q^74 - 85 * q^75 + 38 * q^76 + 26 * q^77 + 11 * q^78 + 72 * q^79 + 42 * q^80 + 56 * q^81 + 33 * q^82 - 11 * q^83 - 52 * q^84 - q^85 - 4 * q^86 - 17 * q^87 + 130 * q^88 - 6 * q^89 + 8 * q^90 + 37 * q^91 + 132 * q^92 - 23 * q^93 - 32 * q^94 + 12 * q^95 - 11 * q^96 + 100 * q^97 + 42 * q^98 - 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.93055	−1.36510	−0.682552	−	0.730837i	$-0.739130\pi$
$2$	−1.93055	−1.36510	−0.682552	+	0.730837i	$0.739130\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.72702	0.863511
$5$	−2.51216	−1.12347	−0.561735	−	0.827317i	$-0.689866\pi$
$5$	−2.51216	−1.12347	−0.561735	+	0.827317i	$0.689866\pi$
$6$	1.93055	0.788144
$7$	−1.67367	−0.632589	−0.316294	−	0.948661i	$-0.602439\pi$
$7$	−1.67367	−0.632589	−0.316294	+	0.948661i	$0.602439\pi$
$8$	0.526999	0.186322
$9$	1.00000	0.333333
$10$	4.84984	1.53365
$11$	3.46212	1.04387	0.521934	−	0.852986i	$-0.325211\pi$
$11$	3.46212	1.04387	0.521934	+	0.852986i	$0.325211\pi$
$12$	−1.72702	−0.498548
$13$	−3.72991	−1.03449	−0.517246	−	0.855837i	$-0.673043\pi$
$13$	−3.72991	−1.03449	−0.517246	+	0.855837i	$0.673043\pi$
$14$	3.23111	0.863550
$15$	2.51216	0.648636
$16$	−4.47144	−1.11786
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−1.93055	−0.455035
$19$	−6.65781	−1.52741	−0.763703	−	0.645567i	$-0.776621\pi$
$19$	−6.65781	−1.52741	−0.763703	+	0.645567i	$0.776621\pi$
$20$	−4.33855	−0.970128
$21$	1.67367	0.365225
$22$	−6.68379	−1.42499
$23$	−2.05998	−0.429535	−0.214767	−	0.976665i	$-0.568899\pi$
$23$	−2.05998	−0.429535	−0.214767	+	0.976665i	$0.568899\pi$
$24$	−0.526999	−0.107573
$25$	1.31092	0.262185
$26$	7.20078	1.41219
$27$	−1.00000	−0.192450
$28$	−2.89047	−0.546247
$29$	−1.98149	−0.367953	−0.183977	−	0.982931i	$-0.558897\pi$
$29$	−1.98149	−0.367953	−0.183977	+	0.982931i	$0.558897\pi$
$30$	−4.84984	−0.885456
$31$	4.55602	0.818285	0.409143	−	0.912470i	$-0.365828\pi$
$31$	4.55602	0.818285	0.409143	+	0.912470i	$0.365828\pi$
$32$	7.57834	1.33967
$33$	−3.46212	−0.602677
$34$	1.93055	0.331087
$35$	4.20452	0.710694
$36$	1.72702	0.287837
$37$	6.72209	1.10511	0.552553	−	0.833478i	$-0.313654\pi$
$37$	6.72209	1.10511	0.552553	+	0.833478i	$0.313654\pi$
$38$	12.8532	2.08507
$39$	3.72991	0.597264
$40$	−1.32390	−0.209327
$41$	−8.70922	−1.36015	−0.680076	−	0.733141i	$-0.738053\pi$
$41$	−8.70922	−1.36015	−0.680076	+	0.733141i	$0.738053\pi$
$42$	−3.23111	−0.498571
$43$	0.0248505	0.00378966	0.00189483	−	0.999998i	$-0.499397\pi$
$43$	0.0248505	0.00378966	0.00189483	+	0.999998i	$0.499397\pi$
$44$	5.97915	0.901391
$45$	−2.51216	−0.374490
$46$	3.97689	0.586360
$47$	−10.5151	−1.53379	−0.766896	−	0.641771i	$-0.778200\pi$
$47$	−10.5151	−1.53379	−0.766896	+	0.641771i	$0.778200\pi$
$48$	4.47144	0.645397
$49$	−4.19882	−0.599832
$50$	−2.53080	−0.357910
$51$	1.00000	0.140028
$52$	−6.44164	−0.893295
$53$	−1.38373	−0.190071	−0.0950353	−	0.995474i	$-0.530296\pi$
$53$	−1.38373	−0.190071	−0.0950353	+	0.995474i	$0.530296\pi$
$54$	1.93055	0.262715
$55$	−8.69738	−1.17275
$56$	−0.882023	−0.117865
$57$	6.65781	0.881849
$58$	3.82536	0.502295
$59$	−0.210577	−0.0274148	−0.0137074	−	0.999906i	$-0.504363\pi$
$59$	−0.210577	−0.0274148	−0.0137074	+	0.999906i	$0.504363\pi$
$60$	4.33855	0.560104
$61$	−3.21782	−0.412000	−0.206000	−	0.978552i	$-0.566045\pi$
$61$	−3.21782	−0.412000	−0.206000	+	0.978552i	$0.566045\pi$
$62$	−8.79562	−1.11704
$63$	−1.67367	−0.210863
$64$	−5.68748	−0.710935
$65$	9.37012	1.16222
$66$	6.68379	0.822718
$67$	12.4029	1.51526	0.757628	−	0.652687i	$-0.226358\pi$
$67$	12.4029	1.51526	0.757628	+	0.652687i	$0.226358\pi$
$68$	−1.72702	−0.209432
$69$	2.05998	0.247992
$70$	−8.11704	−0.970172
$71$	−1.02758	−0.121952	−0.0609758	−	0.998139i	$-0.519421\pi$
$71$	−1.02758	−0.121952	−0.0609758	+	0.998139i	$0.519421\pi$
$72$	0.526999	0.0621074
$73$	−13.6556	−1.59827	−0.799136	−	0.601151i	$-0.794709\pi$
$73$	−13.6556	−1.59827	−0.799136	+	0.601151i	$0.794709\pi$
$74$	−12.9773	−1.50858
$75$	−1.31092	−0.151372
$76$	−11.4982	−1.31893
$77$	−5.79445	−0.660339
$78$	−7.20078	−0.815328
$79$	−3.85826	−0.434088	−0.217044	−	0.976162i	$-0.569642\pi$
$79$	−3.85826	−0.434088	−0.217044	+	0.976162i	$0.569642\pi$
$80$	11.2330	1.25588
$81$	1.00000	0.111111
$82$	16.8136	1.85675
$83$	−0.306799	−0.0336755	−0.0168378	−	0.999858i	$-0.505360\pi$
$83$	−0.306799	−0.0336755	−0.0168378	+	0.999858i	$0.505360\pi$
$84$	2.89047	0.315376
$85$	2.51216	0.272481
$86$	−0.0479751	−0.00517328
$87$	1.98149	0.212438
$88$	1.82453	0.194496
$89$	−11.0387	−1.17010	−0.585052	−	0.810996i	$-0.698926\pi$
$89$	−11.0387	−1.17010	−0.585052	+	0.810996i	$0.698926\pi$
$90$	4.84984	0.511218
$91$	6.24265	0.654408
$92$	−3.55762	−0.370908
$93$	−4.55602	−0.472437
$94$	20.3000	2.09379
$95$	16.7255	1.71600
$96$	−7.57834	−0.773461
$97$	4.61672	0.468757	0.234379	−	0.972145i	$-0.424694\pi$
$97$	4.61672	0.468757	0.234379	+	0.972145i	$0.424694\pi$
$98$	8.10603	0.818833
$99$	3.46212	0.347956
$100$	2.26399	0.226399
$101$	6.35949	0.632792	0.316396	−	0.948627i	$-0.397527\pi$
$101$	6.35949	0.632792	0.316396	+	0.948627i	$0.397527\pi$
$102$	−1.93055	−0.191153
$103$	14.2088	1.40003	0.700016	−	0.714127i	$-0.253176\pi$
$103$	14.2088	1.40003	0.700016	+	0.714127i	$0.253176\pi$
$104$	−1.96566	−0.192749
$105$	−4.20452	−0.410320
$106$	2.67137	0.259466
$107$	1.08825	0.105205	0.0526025	−	0.998616i	$-0.483248\pi$
$107$	1.08825	0.105205	0.0526025	+	0.998616i	$0.483248\pi$
$108$	−1.72702	−0.166183
$109$	−3.11324	−0.298195	−0.149097	−	0.988823i	$-0.547637\pi$
$109$	−3.11324	−0.298195	−0.149097	+	0.988823i	$0.547637\pi$
$110$	16.7907	1.60093
$111$	−6.72209	−0.638033
$112$	7.48372	0.707145
$113$	−5.89727	−0.554769	−0.277384	−	0.960759i	$-0.589468\pi$
$113$	−5.89727	−0.554769	−0.277384	+	0.960759i	$0.589468\pi$
$114$	−12.8532	−1.20382
$115$	5.17498	0.482569
$116$	−3.42208	−0.317732
$117$	−3.72991	−0.344831
$118$	0.406529	0.0374241
$119$	1.67367	0.153425
$120$	1.32390	0.120855
$121$	0.986262	0.0896602
$122$	6.21216	0.562422
$123$	8.70922	0.785284
$124$	7.86834	0.706598
$125$	9.26753	0.828913
$126$	3.23111	0.287850
$127$	−0.790930	−0.0701837	−0.0350918	−	0.999384i	$-0.511172\pi$
$127$	−0.790930	−0.0701837	−0.0350918	+	0.999384i	$0.511172\pi$
$128$	−4.17672	−0.369173
$129$	−0.0248505	−0.00218796
$130$	−18.0895	−1.58655
$131$	−17.0947	−1.49357	−0.746786	−	0.665064i	$-0.768404\pi$
$131$	−17.0947	−1.49357	−0.746786	+	0.665064i	$0.768404\pi$
$132$	−5.97915	−0.520418
$133$	11.1430	0.966220
$134$	−23.9444	−2.06848
$135$	2.51216	0.216212
$136$	−0.526999	−0.0451898
$137$	−2.09123	−0.178666	−0.0893330	−	0.996002i	$-0.528474\pi$
$137$	−2.09123	−0.178666	−0.0893330	+	0.996002i	$0.528474\pi$
$138$	−3.97689	−0.338535
$139$	−21.1566	−1.79448	−0.897239	−	0.441544i	$-0.854431\pi$
$139$	−21.1566	−1.79448	−0.897239	+	0.441544i	$0.854431\pi$
$140$	7.26130	0.613692
$141$	10.5151	0.885535
$142$	1.98380	0.166477
$143$	−12.9134	−1.07987
$144$	−4.47144	−0.372620
$145$	4.97781	0.413385
$146$	26.3629	2.18181
$147$	4.19882	0.346313
$148$	11.6092	0.954270
$149$	−3.05718	−0.250454	−0.125227	−	0.992128i	$-0.539966\pi$
$149$	−3.05718	−0.250454	−0.125227	+	0.992128i	$0.539966\pi$
$150$	2.53080	0.206639
$151$	−3.13032	−0.254741	−0.127371	−	0.991855i	$-0.540654\pi$
$151$	−3.13032	−0.254741	−0.127371	+	0.991855i	$0.540654\pi$
$152$	−3.50866	−0.284590
$153$	−1.00000	−0.0808452
$154$	11.1865	0.901432
$155$	−11.4454	−0.919319
$156$	6.44164	0.515744
$157$	1.00000	0.0798087
$157$	1.00000	0.0798087
$158$	7.44856	0.592576
$159$	1.38373	0.109737
$160$	−19.0380	−1.50508
$161$	3.44773	0.271719
$162$	−1.93055	−0.151678
$163$	5.83995	0.457420	0.228710	−	0.973495i	$-0.426549\pi$
$163$	5.83995	0.457420	0.228710	+	0.973495i	$0.426549\pi$
$164$	−15.0410	−1.17451
$165$	8.69738	0.677090
$166$	0.592290	0.0459706
$167$	−16.1611	−1.25058	−0.625291	−	0.780392i	$-0.715020\pi$
$167$	−16.1611	−1.25058	−0.625291	+	0.780392i	$0.715020\pi$
$168$	0.882023	0.0680496
$169$	0.912259	0.0701737
$170$	−4.84984	−0.371966
$171$	−6.65781	−0.509136
$172$	0.0429173	0.00327241
$173$	−25.6896	−1.95314	−0.976570	−	0.215199i	$-0.930960\pi$
$173$	−25.6896	−1.95314	−0.976570	+	0.215199i	$0.930960\pi$
$174$	−3.82536	−0.290000
$175$	−2.19406	−0.165855
$176$	−15.4807	−1.16690
$177$	0.210577	0.0158279
$178$	21.3108	1.59731
$179$	5.65554	0.422715	0.211358	−	0.977409i	$-0.432212\pi$
$179$	5.65554	0.422715	0.211358	+	0.977409i	$0.432212\pi$
$180$	−4.33855	−0.323376
$181$	−7.85662	−0.583978	−0.291989	−	0.956422i	$-0.594317\pi$
$181$	−7.85662	−0.583978	−0.291989	+	0.956422i	$0.594317\pi$
$182$	−12.0518	−0.893335
$183$	3.21782	0.237868
$184$	−1.08561	−0.0800319
$185$	−16.8869	−1.24155
$186$	8.79562	0.644926
$187$	−3.46212	−0.253175
$188$	−18.1599	−1.32445
$189$	1.67367	0.121742
$190$	−32.2893	−2.34251
$191$	−7.13454	−0.516237	−0.258119	−	0.966113i	$-0.583103\pi$
$191$	−7.13454	−0.516237	−0.258119	+	0.966113i	$0.583103\pi$
$192$	5.68748	0.410458
$193$	−0.281014	−0.0202278	−0.0101139	−	0.999949i	$-0.503219\pi$
$193$	−0.281014	−0.0202278	−0.0101139	+	0.999949i	$0.503219\pi$
$194$	−8.91281	−0.639903
$195$	−9.37012	−0.671008
$196$	−7.25146	−0.517961
$197$	−9.26499	−0.660103	−0.330052	−	0.943963i	$-0.607066\pi$
$197$	−9.26499	−0.660103	−0.330052	+	0.943963i	$0.607066\pi$
$198$	−6.68379	−0.474996
$199$	−9.96565	−0.706446	−0.353223	−	0.935539i	$-0.614914\pi$
$199$	−9.96565	−0.706446	−0.353223	+	0.935539i	$0.614914\pi$
$200$	0.690856	0.0488509
$201$	−12.4029	−0.874833
$202$	−12.2773	−0.863828
$203$	3.31636	0.232763
$204$	1.72702	0.120916
$205$	21.8789	1.52809
$206$	−27.4308	−1.91119
$207$	−2.05998	−0.143178
$208$	16.6781	1.15642
$209$	−23.0501	−1.59441
$210$	8.11704	0.560129
$211$	1.02867	0.0708168	0.0354084	−	0.999373i	$-0.488727\pi$
$211$	1.02867	0.0708168	0.0354084	+	0.999373i	$0.488727\pi$
$212$	−2.38974	−0.164128
$213$	1.02758	0.0704088
$214$	−2.10092	−0.143616
$215$	−0.0624282	−0.00425757
$216$	−0.526999	−0.0358577
$217$	−7.62528	−0.517638
$218$	6.01027	0.407067
$219$	13.6556	0.922762
$220$	−15.0206	−1.01269
$221$	3.72991	0.250901
$222$	12.9773	0.870982
$223$	1.70296	0.114039	0.0570193	−	0.998373i	$-0.481840\pi$
$223$	1.70296	0.114039	0.0570193	+	0.998373i	$0.481840\pi$
$224$	−12.6837	−0.847462
$225$	1.31092	0.0873949
$226$	11.3850	0.757317
$227$	−5.44891	−0.361657	−0.180829	−	0.983515i	$-0.557878\pi$
$227$	−5.44891	−0.361657	−0.180829	+	0.983515i	$0.557878\pi$
$228$	11.4982	0.761486
$229$	6.78858	0.448602	0.224301	−	0.974520i	$-0.427990\pi$
$229$	6.78858	0.448602	0.224301	+	0.974520i	$0.427990\pi$
$230$	−9.99056	−0.658758
$231$	5.79445	0.381247
$232$	−1.04424	−0.0685579
$233$	12.6977	0.831852	0.415926	−	0.909399i	$-0.363458\pi$
$233$	12.6977	0.831852	0.415926	+	0.909399i	$0.363458\pi$
$234$	7.20078	0.470730
$235$	26.4157	1.72317
$236$	−0.363671	−0.0236730
$237$	3.85826	0.250621
$238$	−3.23111	−0.209442
$239$	−17.1692	−1.11058	−0.555291	−	0.831656i	$-0.687393\pi$
$239$	−17.1692	−1.11058	−0.555291	+	0.831656i	$0.687393\pi$
$240$	−11.2330	−0.725084
$241$	−26.0110	−1.67552	−0.837759	−	0.546040i	$-0.816135\pi$
$241$	−26.0110	−1.67552	−0.837759	+	0.546040i	$0.816135\pi$
$242$	−1.90403	−0.122396
$243$	−1.00000	−0.0641500
$244$	−5.55724	−0.355766
$245$	10.5481	0.673893
$246$	−16.8136	−1.07200
$247$	24.8331	1.58009
$248$	2.40102	0.152465
$249$	0.306799	0.0194426
$250$	−17.8914	−1.13155
$251$	−15.9235	−1.00508	−0.502541	−	0.864553i	$-0.667601\pi$
$251$	−15.9235	−1.00508	−0.502541	+	0.864553i	$0.667601\pi$
$252$	−2.89047	−0.182082
$253$	−7.13188	−0.448378
$254$	1.52693	0.0958081
$255$	−2.51216	−0.157317
$256$	19.4383	1.21490
$257$	−2.48637	−0.155096	−0.0775479	−	0.996989i	$-0.524709\pi$
$257$	−2.48637	−0.155096	−0.0775479	+	0.996989i	$0.524709\pi$
$258$	0.0479751	0.00298680
$259$	−11.2506	−0.699077
$260$	16.1824	1.00359
$261$	−1.98149	−0.122651
$262$	33.0022	2.03888
$263$	−16.2319	−1.00090	−0.500452	−	0.865764i	$-0.666833\pi$
$263$	−16.2319	−1.00090	−0.500452	+	0.865764i	$0.666833\pi$
$264$	−1.82453	−0.112292
$265$	3.47616	0.213539
$266$	−21.5121	−1.31899
$267$	11.0387	0.675560
$268$	21.4201	1.30844
$269$	8.99471	0.548417	0.274209	−	0.961670i	$-0.411584\pi$
$269$	8.99471	0.548417	0.274209	+	0.961670i	$0.411584\pi$
$270$	−4.84984	−0.295152
$271$	11.7864	0.715974	0.357987	−	0.933727i	$-0.383463\pi$
$271$	11.7864	0.715974	0.357987	+	0.933727i	$0.383463\pi$
$272$	4.47144	0.271121
$273$	−6.24265	−0.377823
$274$	4.03723	0.243898
$275$	4.53857	0.273686
$276$	3.55762	0.214144
$277$	−3.36594	−0.202240	−0.101120	−	0.994874i	$-0.532243\pi$
$277$	−3.36594	−0.202240	−0.101120	+	0.994874i	$0.532243\pi$
$278$	40.8439	2.44965
$279$	4.55602	0.272762
$280$	2.21578	0.132418
$281$	−4.75777	−0.283825	−0.141912	−	0.989879i	$-0.545325\pi$
$281$	−4.75777	−0.283825	−0.141912	+	0.989879i	$0.545325\pi$
$282$	−20.3000	−1.20885
$283$	8.17862	0.486169	0.243084	−	0.970005i	$-0.421841\pi$
$283$	8.17862	0.486169	0.243084	+	0.970005i	$0.421841\pi$
$284$	−1.77466	−0.105307
$285$	−16.7255	−0.990731
$286$	24.9300	1.47414
$287$	14.5764	0.860417
$288$	7.57834	0.446558
$289$	1.00000	0.0588235
$290$	−9.60991	−0.564313
$291$	−4.61672	−0.270637
$292$	−23.5836	−1.38012
$293$	−28.2706	−1.65159	−0.825793	−	0.563974i	$-0.809272\pi$
$293$	−28.2706	−1.65159	−0.825793	+	0.563974i	$0.809272\pi$
$294$	−8.10603	−0.472753
$295$	0.529002	0.0307997
$296$	3.54254	0.205906
$297$	−3.46212	−0.200892
$298$	5.90204	0.341896
$299$	7.68354	0.444350
$300$	−2.26399	−0.130712
$301$	−0.0415915	−0.00239730
$302$	6.04323	0.347749
$303$	−6.35949	−0.365343
$304$	29.7700	1.70743
$305$	8.08366	0.462869
$306$	1.93055	0.110362
$307$	18.3624	1.04800	0.523998	−	0.851720i	$-0.324440\pi$
$307$	18.3624	1.04800	0.523998	+	0.851720i	$0.324440\pi$
$308$	−10.0071	−0.570210
$309$	−14.2088	−0.808309
$310$	22.0960	1.25497
$311$	−11.9053	−0.675087	−0.337543	−	0.941310i	$-0.609596\pi$
$311$	−11.9053	−0.675087	−0.337543	+	0.941310i	$0.609596\pi$
$312$	1.96566	0.111284
$313$	−22.4478	−1.26882	−0.634412	−	0.772995i	$-0.718758\pi$
$313$	−22.4478	−1.26882	−0.634412	+	0.772995i	$0.718758\pi$
$314$	−1.93055	−0.108947
$315$	4.20452	0.236898
$316$	−6.66329	−0.374840
$317$	−16.2101	−0.910451	−0.455226	−	0.890376i	$-0.650441\pi$
$317$	−16.2101	−0.910451	−0.455226	+	0.890376i	$0.650441\pi$
$318$	−2.67137	−0.149803
$319$	−6.86015	−0.384095
$320$	14.2878	0.798714
$321$	−1.08825	−0.0607402
$322$	−6.65600	−0.370925
$323$	6.65781	0.370451
$324$	1.72702	0.0959456
$325$	−4.88963	−0.271228
$326$	−11.2743	−0.624427
$327$	3.11324	0.172163
$328$	−4.58975	−0.253427
$329$	17.5989	0.970259
$330$	−16.7907	−0.924299
$331$	8.59667	0.472516	0.236258	−	0.971690i	$-0.424079\pi$
$331$	8.59667	0.472516	0.236258	+	0.971690i	$0.424079\pi$
$332$	−0.529848	−0.0290792
$333$	6.72209	0.368369
$334$	31.1998	1.70718
$335$	−31.1580	−1.70234
$336$	−7.48372	−0.408271
$337$	10.4832	0.571057	0.285529	−	0.958370i	$-0.407831\pi$
$337$	10.4832	0.571057	0.285529	+	0.958370i	$0.407831\pi$
$338$	−1.76116	−0.0957945
$339$	5.89727	0.320296
$340$	4.33855	0.235291
$341$	15.7735	0.854182
$342$	12.8532	0.695023
$343$	18.7432	1.01204
$344$	0.0130962	0.000706098 0
$345$	−5.17498	−0.278612
$346$	49.5950	2.66624
$347$	−13.2059	−0.708928	−0.354464	−	0.935070i	$-0.615337\pi$
$347$	−13.2059	−0.708928	−0.354464	+	0.935070i	$0.615337\pi$
$348$	3.42208	0.183443
$349$	−10.4220	−0.557878	−0.278939	−	0.960309i	$-0.589983\pi$
$349$	−10.4220	−0.557878	−0.278939	+	0.960309i	$0.589983\pi$
$350$	4.23574	0.226410
$351$	3.72991	0.199088
$352$	26.2371	1.39844
$353$	11.5972	0.617256	0.308628	−	0.951183i	$-0.400130\pi$
$353$	11.5972	0.617256	0.308628	+	0.951183i	$0.400130\pi$
$354$	−0.406529	−0.0216068
$355$	2.58145	0.137009
$356$	−19.0641	−1.01040
$357$	−1.67367	−0.0885801
$358$	−10.9183	−0.577051
$359$	−27.1151	−1.43108	−0.715541	−	0.698571i	$-0.753820\pi$
$359$	−27.1151	−1.43108	−0.715541	+	0.698571i	$0.753820\pi$
$360$	−1.32390	−0.0697758
$361$	25.3265	1.33297
$362$	15.1676	0.797191
$363$	−0.986262	−0.0517653
$364$	10.7812	0.565088
$365$	34.3051	1.79561
$366$	−6.21216	−0.324715
$367$	14.3253	0.747775	0.373888	−	0.927474i	$-0.378025\pi$
$367$	14.3253	0.747775	0.373888	+	0.927474i	$0.378025\pi$
$368$	9.21106	0.480160
$369$	−8.70922	−0.453384
$370$	32.6011	1.69485
$371$	2.31592	0.120236
$372$	−7.86834	−0.407955
$373$	31.5554	1.63387	0.816937	−	0.576726i	$-0.195670\pi$
$373$	31.5554	1.63387	0.816937	+	0.576726i	$0.195670\pi$
$374$	6.68379	0.345611
$375$	−9.26753	−0.478573
$376$	−5.54147	−0.285780
$377$	7.39079	0.380645
$378$	−3.23111	−0.166190
$379$	6.42636	0.330100	0.165050	−	0.986285i	$-0.447221\pi$
$379$	6.42636	0.330100	0.165050	+	0.986285i	$0.447221\pi$
$380$	28.8852	1.48178
$381$	0.790930	0.0405206
$382$	13.7736	0.704718
$383$	17.8444	0.911805	0.455903	−	0.890030i	$-0.349317\pi$
$383$	17.8444	0.911805	0.455903	+	0.890030i	$0.349317\pi$
$384$	4.17672	0.213142
$385$	14.5566	0.741871
$386$	0.542512	0.0276131
$387$	0.0248505	0.00126322
$388$	7.97318	0.404777
$389$	−13.8673	−0.703100	−0.351550	−	0.936169i	$-0.614345\pi$
$389$	−13.8673	−0.703100	−0.351550	+	0.936169i	$0.614345\pi$
$390$	18.0895	0.915997
$391$	2.05998	0.104177
$392$	−2.21277	−0.111762
$393$	17.0947	0.862314
$394$	17.8865	0.901110
$395$	9.69254	0.487685
$396$	5.97915	0.300464
$397$	28.4216	1.42644	0.713219	−	0.700941i	$-0.247236\pi$
$397$	28.4216	1.42644	0.713219	+	0.700941i	$0.247236\pi$
$398$	19.2392	0.964373
$399$	−11.1430	−0.557847
$400$	−5.86172	−0.293086
$401$	1.69823	0.0848058	0.0424029	−	0.999101i	$-0.486499\pi$
$401$	1.69823	0.0848058	0.0424029	+	0.999101i	$0.486499\pi$
$402$	23.9444	1.19424
$403$	−16.9936	−0.846510
$404$	10.9830	0.546423
$405$	−2.51216	−0.124830
$406$	−6.40241	−0.317746
$407$	23.2727	1.15358
$408$	0.526999	0.0260903
$409$	0.580289	0.0286934	0.0143467	−	0.999897i	$-0.495433\pi$
$409$	0.580289	0.0286934	0.0143467	+	0.999897i	$0.495433\pi$
$410$	−42.2383	−2.08600
$411$	2.09123	0.103153
$412$	24.5389	1.20894
$413$	0.352437	0.0173423
$414$	3.97689	0.195453
$415$	0.770726	0.0378335
$416$	−28.2666	−1.38588
$417$	21.1566	1.03604
$418$	44.4994	2.17654
$419$	−1.32156	−0.0645626	−0.0322813	−	0.999479i	$-0.510277\pi$
$419$	−1.32156	−0.0645626	−0.0322813	+	0.999479i	$0.510277\pi$
$420$	−7.26130	−0.354315
$421$	−18.7972	−0.916118	−0.458059	−	0.888922i	$-0.651455\pi$
$421$	−18.7972	−0.916118	−0.458059	+	0.888922i	$0.651455\pi$
$422$	−1.98590	−0.0966723
$423$	−10.5151	−0.511264
$424$	−0.729226	−0.0354144
$425$	−1.31092	−0.0635892
$426$	−1.98380	−0.0961154
$427$	5.38557	0.260626
$428$	1.87943	0.0908457
$429$	12.9134	0.623465
$430$	0.120521	0.00581203
$431$	20.0725	0.966860	0.483430	−	0.875383i	$-0.339391\pi$
$431$	20.0725	0.966860	0.483430	+	0.875383i	$0.339391\pi$
$432$	4.47144	0.215132
$433$	13.6199	0.654529	0.327265	−	0.944933i	$-0.393873\pi$
$433$	13.6199	0.654529	0.327265	+	0.944933i	$0.393873\pi$
$434$	14.7210	0.706630
$435$	−4.97781	−0.238668
$436$	−5.37664	−0.257494
$437$	13.7149	0.656074
$438$	−26.3629	−1.25967
$439$	−29.1190	−1.38977	−0.694886	−	0.719119i	$-0.744545\pi$
$439$	−29.1190	−1.38977	−0.694886	+	0.719119i	$0.744545\pi$
$440$	−4.58351	−0.218510
$441$	−4.19882	−0.199944
$442$	−7.20078	−0.342506
$443$	−18.7153	−0.889190	−0.444595	−	0.895732i	$-0.646652\pi$
$443$	−18.7153	−0.889190	−0.444595	+	0.895732i	$0.646652\pi$
$444$	−11.6092	−0.550948
$445$	27.7310	1.31458
$446$	−3.28765	−0.155675
$447$	3.05718	0.144600
$448$	9.51897	0.449729
$449$	15.0678	0.711095	0.355548	−	0.934658i	$-0.384294\pi$
$449$	15.0678	0.711095	0.355548	+	0.934658i	$0.384294\pi$
$450$	−2.53080	−0.119303
$451$	−30.1524	−1.41982
$452$	−10.1847	−0.479049
$453$	3.13032	0.147075
$454$	10.5194	0.493700
$455$	−15.6825	−0.735208
$456$	3.50866	0.164308
$457$	35.9554	1.68192	0.840961	−	0.541096i	$-0.181991\pi$
$457$	35.9554	1.68192	0.840961	+	0.541096i	$0.181991\pi$
$458$	−13.1057	−0.612389
$459$	1.00000	0.0466760
$460$	8.93730	0.416704
$461$	12.7741	0.594951	0.297476	−	0.954729i	$-0.403855\pi$
$461$	12.7741	0.594951	0.297476	+	0.954729i	$0.403855\pi$
$462$	−11.1865	−0.520442
$463$	−30.3939	−1.41253	−0.706263	−	0.707950i	$-0.749620\pi$
$463$	−30.3939	−1.41253	−0.706263	+	0.707950i	$0.749620\pi$
$464$	8.86011	0.411320
$465$	11.4454	0.530769
$466$	−24.5135	−1.13556
$467$	−17.5440	−0.811838	−0.405919	−	0.913909i	$-0.633048\pi$
$467$	−17.5440	−0.811838	−0.405919	+	0.913909i	$0.633048\pi$
$468$	−6.44164	−0.297765
$469$	−20.7584	−0.958533
$470$	−50.9968	−2.35231
$471$	−1.00000	−0.0460776
$472$	−0.110974	−0.00510799
$473$	0.0860352	0.00395590
$474$	−7.44856	−0.342124
$475$	−8.72789	−0.400463
$476$	2.89047	0.132484
$477$	−1.38373	−0.0633568
$478$	33.1460	1.51606
$479$	−9.48653	−0.433451	−0.216725	−	0.976233i	$-0.569538\pi$
$479$	−9.48653	−0.433451	−0.216725	+	0.976233i	$0.569538\pi$
$480$	19.0380	0.868960
$481$	−25.0728	−1.14322
$482$	50.2156	2.28726
$483$	−3.44773	−0.156877
$484$	1.70330	0.0774225
$485$	−11.5979	−0.526635
$486$	1.93055	0.0875715
$487$	35.6288	1.61450	0.807248	−	0.590212i	$-0.200956\pi$
$487$	35.6288	1.61450	0.807248	+	0.590212i	$0.200956\pi$
$488$	−1.69579	−0.0767647
$489$	−5.83995	−0.264092
$490$	−20.3636	−0.919934
$491$	14.7896	0.667443	0.333722	−	0.942672i	$-0.391695\pi$
$491$	14.7896	0.667443	0.333722	+	0.942672i	$0.391695\pi$
$492$	15.0410	0.678101
$493$	1.98149	0.0892418
$494$	−47.9415	−2.15699
$495$	−8.69738	−0.390918
$496$	−20.3720	−0.914728
$497$	1.71984	0.0771452
$498$	−0.592290	−0.0265412
$499$	−3.28950	−0.147258	−0.0736292	−	0.997286i	$-0.523458\pi$
$499$	−3.28950	−0.147258	−0.0736292	+	0.997286i	$0.523458\pi$
$500$	16.0052	0.715775
$501$	16.1611	0.722024
$502$	30.7411	1.37204
$503$	16.7004	0.744634	0.372317	−	0.928106i	$-0.378563\pi$
$503$	16.7004	0.744634	0.372317	+	0.928106i	$0.378563\pi$
$504$	−0.882023	−0.0392884
$505$	−15.9760	−0.710923
$506$	13.7685	0.612082
$507$	−0.912259	−0.0405148
$508$	−1.36595	−0.0606044
$509$	−31.1118	−1.37901	−0.689503	−	0.724282i	$-0.742171\pi$
$509$	−31.1118	−1.37901	−0.689503	+	0.724282i	$0.742171\pi$
$510$	4.84984	0.214755
$511$	22.8551	1.01105
$512$	−29.1732	−1.28929
$513$	6.65781	0.293950
$514$	4.80007	0.211722
$515$	−35.6947	−1.57289
$516$	−0.0429173	−0.00188933
$517$	−36.4047	−1.60108
$518$	21.7198	0.954313
$519$	25.6896	1.12765
$520$	4.93804	0.216548
$521$	35.8808	1.57197	0.785984	−	0.618247i	$-0.212157\pi$
$521$	35.8808	1.57197	0.785984	+	0.618247i	$0.212157\pi$
$522$	3.82536	0.167432
$523$	−18.0506	−0.789297	−0.394649	−	0.918832i	$-0.629134\pi$
$523$	−18.0506	−0.789297	−0.394649	+	0.918832i	$0.629134\pi$
$524$	−29.5229	−1.28972
$525$	2.19406	0.0957565
$526$	31.3366	1.36634
$527$	−4.55602	−0.198463
$528$	15.4807	0.673709
$529$	−18.7565	−0.815500
$530$	−6.71089	−0.291502
$531$	−0.210577	−0.00913826
$532$	19.2442	0.834341
$533$	32.4847	1.40707
$534$	−21.3108	−0.922209
$535$	−2.73385	−0.118195
$536$	6.53631	0.282326
$537$	−5.65554	−0.244055
$538$	−17.3647	−0.748647
$539$	−14.5368	−0.626145
$540$	4.33855	0.186701
$541$	−18.1550	−0.780546	−0.390273	−	0.920699i	$-0.627619\pi$
$541$	−18.1550	−0.780546	−0.390273	+	0.920699i	$0.627619\pi$
$542$	−22.7543	−0.977379
$543$	7.85662	0.337160
$544$	−7.57834	−0.324919
$545$	7.82095	0.335013
$546$	12.0518	0.515767
$547$	−3.92801	−0.167949	−0.0839747	−	0.996468i	$-0.526761\pi$
$547$	−3.92801	−0.167949	−0.0839747	+	0.996468i	$0.526761\pi$
$548$	−3.61160	−0.154280
$549$	−3.21782	−0.137333
$550$	−8.76194	−0.373611
$551$	13.1924	0.562015
$552$	1.08561	0.0462064
$553$	6.45746	0.274599
$554$	6.49811	0.276078
$555$	16.8869	0.716811
$556$	−36.5379	−1.54955
$557$	−2.70287	−0.114524	−0.0572621	−	0.998359i	$-0.518237\pi$
$557$	−2.70287	−0.114524	−0.0572621	+	0.998359i	$0.518237\pi$
$558$	−8.79562	−0.372348
$559$	−0.0926901	−0.00392037
$560$	−18.8003	−0.794457
$561$	3.46212	0.146171
$562$	9.18511	0.387450
$563$	16.0888	0.678063	0.339032	−	0.940775i	$-0.389901\pi$
$563$	16.0888	0.678063	0.339032	+	0.940775i	$0.389901\pi$
$564$	18.1599	0.764669
$565$	14.8149	0.623266
$566$	−15.7892	−0.663671
$567$	−1.67367	−0.0702876
$568$	−0.541535	−0.0227223
$569$	−9.96703	−0.417840	−0.208920	−	0.977933i	$-0.566995\pi$
$569$	−9.96703	−0.417840	−0.208920	+	0.977933i	$0.566995\pi$
$570$	32.2893	1.35245
$571$	−40.4208	−1.69156	−0.845779	−	0.533534i	$-0.820864\pi$
$571$	−40.4208	−1.69156	−0.845779	+	0.533534i	$0.820864\pi$
$572$	−22.3017	−0.932482
$573$	7.13454	0.298050
$574$	−28.1404	−1.17456
$575$	−2.70047	−0.112618
$576$	−5.68748	−0.236978
$577$	6.18503	0.257486	0.128743	−	0.991678i	$-0.458906\pi$
$577$	6.18503	0.257486	0.128743	+	0.991678i	$0.458906\pi$
$578$	−1.93055	−0.0803003
$579$	0.281014	0.0116786
$580$	8.59679	0.356962
$581$	0.513480	0.0213028
$582$	8.91281	0.369448
$583$	−4.79065	−0.198408
$584$	−7.19650	−0.297793
$585$	9.37012	0.387407
$586$	54.5778	2.25459
$587$	6.86081	0.283176	0.141588	−	0.989926i	$-0.454779\pi$
$587$	6.86081	0.283176	0.141588	+	0.989926i	$0.454779\pi$
$588$	7.25146	0.299045
$589$	−30.3331	−1.24985
$590$	−1.02126	−0.0420448
$591$	9.26499	0.381111
$592$	−30.0574	−1.23535
$593$	1.69648	0.0696662	0.0348331	−	0.999393i	$-0.488910\pi$
$593$	1.69648	0.0696662	0.0348331	+	0.999393i	$0.488910\pi$
$594$	6.68379	0.274239
$595$	−4.20452	−0.172369
$596$	−5.27982	−0.216270
$597$	9.96565	0.407867
$598$	−14.8334	−0.606585
$599$	22.5284	0.920485	0.460243	−	0.887793i	$-0.347762\pi$
$599$	22.5284	0.920485	0.460243	+	0.887793i	$0.347762\pi$
$600$	−0.690856	−0.0282041
$601$	8.03769	0.327864	0.163932	−	0.986472i	$-0.447582\pi$
$601$	8.03769	0.327864	0.163932	+	0.986472i	$0.447582\pi$
$602$	0.0802945	0.00327256
$603$	12.4029	0.505085
$604$	−5.40612	−0.219972
$605$	−2.47764	−0.100731
$606$	12.2773	0.498731
$607$	33.3826	1.35496	0.677479	−	0.735542i	$-0.263073\pi$
$607$	33.3826	1.35496	0.677479	+	0.735542i	$0.263073\pi$
$608$	−50.4551	−2.04623
$609$	−3.31636	−0.134386
$610$	−15.6059	−0.631865
$611$	39.2206	1.58670
$612$	−1.72702	−0.0698107
$613$	−33.4342	−1.35039	−0.675197	−	0.737638i	$-0.735941\pi$
$613$	−33.4342	−1.35039	−0.675197	+	0.737638i	$0.735941\pi$
$614$	−35.4495	−1.43062
$615$	−21.8789	−0.882243
$616$	−3.05367	−0.123036
$617$	−20.8953	−0.841214	−0.420607	−	0.907243i	$-0.638183\pi$
$617$	−20.8953	−0.841214	−0.420607	+	0.907243i	$0.638183\pi$
$618$	27.4308	1.10343
$619$	41.7831	1.67940	0.839702	−	0.543048i	$-0.182730\pi$
$619$	41.7831	1.67940	0.839702	+	0.543048i	$0.182730\pi$
$620$	−19.7665	−0.793842
$621$	2.05998	0.0826640
$622$	22.9837	0.921564
$623$	18.4752	0.740194
$624$	−16.6781	−0.667658
$625$	−29.8361	−1.19344
$626$	43.3366	1.73208
$627$	23.0501	0.920534
$628$	1.72702	0.0689157
$629$	−6.72209	−0.268027
$630$	−8.11704	−0.323391
$631$	9.59170	0.381839	0.190920	−	0.981606i	$-0.438853\pi$
$631$	9.59170	0.381839	0.190920	+	0.981606i	$0.438853\pi$
$632$	−2.03330	−0.0808802
$633$	−1.02867	−0.0408861
$634$	31.2944	1.24286
$635$	1.98694	0.0788493
$636$	2.38974	0.0947593
$637$	15.6612	0.620521
$638$	13.2439	0.524330
$639$	−1.02758	−0.0406506
$640$	10.4926	0.414755
$641$	38.0209	1.50174	0.750868	−	0.660452i	$-0.229635\pi$
$641$	38.0209	1.50174	0.750868	+	0.660452i	$0.229635\pi$
$642$	2.10092	0.0829167
$643$	−1.99737	−0.0787685	−0.0393843	−	0.999224i	$-0.512540\pi$
$643$	−1.99737	−0.0787685	−0.0393843	+	0.999224i	$0.512540\pi$
$644$	5.95429	0.234632
$645$	0.0624282	0.00245811
$646$	−12.8532	−0.505704
$647$	10.8135	0.425124	0.212562	−	0.977148i	$-0.431819\pi$
$647$	10.8135	0.425124	0.212562	+	0.977148i	$0.431819\pi$
$648$	0.526999	0.0207025
$649$	−0.729043	−0.0286174
$650$	9.43968	0.370255
$651$	7.62528	0.298858
$652$	10.0857	0.394987
$653$	16.4297	0.642946	0.321473	−	0.946919i	$-0.395822\pi$
$653$	16.4297	0.642946	0.321473	+	0.946919i	$0.395822\pi$
$654$	−6.01027	−0.235020
$655$	42.9446	1.67798
$656$	38.9428	1.52046
$657$	−13.6556	−0.532757
$658$	−33.9756	−1.32451
$659$	9.99241	0.389249	0.194624	−	0.980878i	$-0.437651\pi$
$659$	9.99241	0.389249	0.194624	+	0.980878i	$0.437651\pi$
$660$	15.0206	0.584674
$661$	−8.60521	−0.334704	−0.167352	−	0.985897i	$-0.553522\pi$
$661$	−8.60521	−0.334704	−0.167352	+	0.985897i	$0.553522\pi$
$662$	−16.5963	−0.645033
$663$	−3.72991	−0.144858
$664$	−0.161683	−0.00627450
$665$	−27.9929	−1.08552
$666$	−12.9773	−0.502862
$667$	4.08182	0.158049
$668$	−27.9105	−1.07989
$669$	−1.70296	−0.0658402
$670$	60.1521	2.32388
$671$	−11.1405	−0.430073
$672$	12.6837	0.489283
$673$	43.1704	1.66410	0.832049	−	0.554703i	$-0.187168\pi$
$673$	43.1704	1.66410	0.832049	+	0.554703i	$0.187168\pi$
$674$	−20.2384	−0.779553
$675$	−1.31092	−0.0504575
$676$	1.57549	0.0605958
$677$	17.8509	0.686065	0.343032	−	0.939324i	$-0.388546\pi$
$677$	17.8509	0.686065	0.343032	+	0.939324i	$0.388546\pi$
$678$	−11.3850	−0.437237
$679$	−7.72688	−0.296531
$680$	1.32390	0.0507694
$681$	5.44891	0.208803
$682$	−30.4515	−1.16605
$683$	34.2263	1.30963	0.654816	−	0.755788i	$-0.272746\pi$
$683$	34.2263	1.30963	0.654816	+	0.755788i	$0.272746\pi$
$684$	−11.4982	−0.439644
$685$	5.25350	0.200726
$686$	−36.1846	−1.38153
$687$	−6.78858	−0.259001
$688$	−0.111117	−0.00423631
$689$	5.16121	0.196626
$690$	9.99056	0.380334
$691$	18.1166	0.689186	0.344593	−	0.938752i	$-0.388017\pi$
$691$	18.1166	0.689186	0.344593	+	0.938752i	$0.388017\pi$
$692$	−44.3664	−1.68656
$693$	−5.79445	−0.220113
$694$	25.4946	0.967760
$695$	53.1487	2.01604
$696$	1.04424	0.0395819
$697$	8.70922	0.329885
$698$	20.1202	0.761561
$699$	−12.6977	−0.480270
$700$	−3.78918	−0.143218
$701$	−17.9103	−0.676463	−0.338232	−	0.941063i	$-0.609829\pi$
$701$	−17.9103	−0.676463	−0.338232	+	0.941063i	$0.609829\pi$
$702$	−7.20078	−0.271776
$703$	−44.7544	−1.68795
$704$	−19.6907	−0.742122
$705$	−26.4157	−0.994872
$706$	−22.3890	−0.842619
$707$	−10.6437	−0.400297
$708$	0.363671	0.0136676
$709$	14.5563	0.546675	0.273338	−	0.961918i	$-0.411872\pi$
$709$	14.5563	0.546675	0.273338	+	0.961918i	$0.411872\pi$
$710$	−4.98361	−0.187032
$711$	−3.85826	−0.144696
$712$	−5.81740	−0.218016
$713$	−9.38529	−0.351482
$714$	3.23111	0.120921
$715$	32.4405	1.21320
$716$	9.76725	0.365019
$717$	17.1692	0.641195
$718$	52.3471	1.95358
$719$	33.9974	1.26789	0.633944	−	0.773379i	$-0.281435\pi$
$719$	33.9974	1.26789	0.633944	+	0.773379i	$0.281435\pi$
$720$	11.2330	0.418627
$721$	−23.7808	−0.885645
$722$	−48.8940	−1.81965
$723$	26.0110	0.967361
$724$	−13.5685	−0.504271
$725$	−2.59758	−0.0964718
$726$	1.90403	0.0706651
$727$	30.1627	1.11867	0.559337	−	0.828941i	$-0.311056\pi$
$727$	30.1627	1.11867	0.559337	+	0.828941i	$0.311056\pi$
$728$	3.28987	0.121931
$729$	1.00000	0.0370370
$730$	−66.2276	−2.45120
$731$	−0.0248505	−0.000919128 0
$732$	5.55724	0.205402
$733$	19.9292	0.736101	0.368051	−	0.929806i	$-0.380025\pi$
$733$	19.9292	0.736101	0.368051	+	0.929806i	$0.380025\pi$
$734$	−27.6557	−1.02079
$735$	−10.5481	−0.389072
$736$	−15.6112	−0.575436
$737$	42.9403	1.58173
$738$	16.8136	0.618917
$739$	29.7057	1.09274	0.546371	−	0.837543i	$-0.316009\pi$
$739$	29.7057	1.09274	0.546371	+	0.837543i	$0.316009\pi$
$740$	−29.1641	−1.07209
$741$	−24.8331	−0.912265
$742$	−4.47099	−0.164135
$743$	−20.1233	−0.738252	−0.369126	−	0.929379i	$-0.620343\pi$
$743$	−20.1233	−0.738252	−0.369126	+	0.929379i	$0.620343\pi$
$744$	−2.40102	−0.0880256
$745$	7.68012	0.281378
$746$	−60.9192	−2.23041
$747$	−0.306799	−0.0112252
$748$	−5.97915	−0.218619
$749$	−1.82137	−0.0665515
$750$	17.8914	0.653303
$751$	−19.0794	−0.696219	−0.348109	−	0.937454i	$-0.613176\pi$
$751$	−19.0794	−0.696219	−0.348109	+	0.937454i	$0.613176\pi$
$752$	47.0179	1.71456
$753$	15.9235	0.580284
$754$	−14.2683	−0.519620
$755$	7.86384	0.286194
$756$	2.89047	0.105125
$757$	−26.6074	−0.967061	−0.483531	−	0.875327i	$-0.660646\pi$
$757$	−26.6074	−0.967061	−0.483531	+	0.875327i	$0.660646\pi$
$758$	−12.4064	−0.450621
$759$	7.13188	0.258871
$760$	8.81430	0.319728
$761$	11.5460	0.418541	0.209270	−	0.977858i	$-0.432891\pi$
$761$	11.5460	0.418541	0.209270	+	0.977858i	$0.432891\pi$
$762$	−1.52693	−0.0553148
$763$	5.21055	0.188635
$764$	−12.3215	−0.445776
$765$	2.51216	0.0908272
$766$	−34.4495	−1.24471
$767$	0.785434	0.0283604
$768$	−19.4383	−0.701420
$769$	13.4707	0.485765	0.242883	−	0.970056i	$-0.421907\pi$
$769$	13.4707	0.485765	0.242883	+	0.970056i	$0.421907\pi$
$770$	−28.1022	−1.01273
$771$	2.48637	0.0895446
$772$	−0.485318	−0.0174670
$773$	−2.54625	−0.0915823	−0.0457911	−	0.998951i	$-0.514581\pi$
$773$	−2.54625	−0.0915823	−0.0457911	+	0.998951i	$0.514581\pi$
$774$	−0.0479751	−0.00172443
$775$	5.97260	0.214542
$776$	2.43301	0.0873399
$777$	11.2506	0.403612
$778$	26.7715	0.959806
$779$	57.9844	2.07751
$780$	−16.1824	−0.579423
$781$	−3.55761	−0.127301
$782$	−3.97689	−0.142213
$783$	1.98149	0.0708127
$784$	18.7748	0.670528
$785$	−2.51216	−0.0896627
$786$	−33.0022	−1.17715
$787$	−42.8048	−1.52583	−0.762913	−	0.646501i	$-0.776232\pi$
$787$	−42.8048	−1.52583	−0.762913	+	0.646501i	$0.776232\pi$
$788$	−16.0008	−0.570006
$789$	16.2319	0.577872
$790$	−18.7119	−0.665741
$791$	9.87010	0.350940
$792$	1.82453	0.0648319
$793$	12.0022	0.426210
$794$	−54.8693	−1.94724
$795$	−3.47616	−0.123287
$796$	−17.2109	−0.610024
$797$	−3.50405	−0.124120	−0.0620599	−	0.998072i	$-0.519767\pi$
$797$	−3.50405	−0.124120	−0.0620599	+	0.998072i	$0.519767\pi$
$798$	21.5121	0.761520
$799$	10.5151	0.371999
$800$	9.93463	0.351242
$801$	−11.0387	−0.390034
$802$	−3.27852	−0.115769
$803$	−47.2774	−1.66838
$804$	−21.4201	−0.755428
$805$	−8.66122	−0.305268
$806$	32.8069	1.15557
$807$	−8.99471	−0.316629
$808$	3.35144	0.117903
$809$	23.5642	0.828473	0.414236	−	0.910169i	$-0.364049\pi$
$809$	23.5642	0.828473	0.414236	+	0.910169i	$0.364049\pi$
$810$	4.84984	0.170406
$811$	50.2965	1.76615	0.883074	−	0.469233i	$-0.155470\pi$
$811$	50.2965	1.76615	0.883074	+	0.469233i	$0.155470\pi$
$812$	5.72743	0.200993
$813$	−11.7864	−0.413368
$814$	−44.9291	−1.57476
$815$	−14.6709	−0.513898
$816$	−4.47144	−0.156532
$817$	−0.165450	−0.00578835
$818$	−1.12028	−0.0391695
$819$	6.24265	0.218136
$820$	37.7854	1.31952
$821$	25.6725	0.895975	0.447987	−	0.894040i	$-0.352141\pi$
$821$	25.6725	0.895975	0.447987	+	0.894040i	$0.352141\pi$
$822$	−4.03723	−0.140814
$823$	23.5756	0.821794	0.410897	−	0.911682i	$-0.365216\pi$
$823$	23.5756	0.821794	0.410897	+	0.911682i	$0.365216\pi$
$824$	7.48801	0.260857
$825$	−4.53857	−0.158013
$826$	−0.680397	−0.0236740
$827$	34.8835	1.21302	0.606510	−	0.795076i	$-0.292569\pi$
$827$	34.8835	1.21302	0.606510	+	0.795076i	$0.292569\pi$
$828$	−3.55762	−0.123636
$829$	−25.8913	−0.899244	−0.449622	−	0.893219i	$-0.648441\pi$
$829$	−25.8913	−0.899244	−0.449622	+	0.893219i	$0.648441\pi$
$830$	−1.48792	−0.0516466
$831$	3.36594	0.116763
$832$	21.2138	0.735456
$833$	4.19882	0.145481
$834$	−40.8439	−1.41431
$835$	40.5991	1.40499
$836$	−39.8081	−1.37679
$837$	−4.55602	−0.157479
$838$	2.55134	0.0881347
$839$	−22.2473	−0.768062	−0.384031	−	0.923320i	$-0.625464\pi$
$839$	−22.2473	−0.768062	−0.384031	+	0.923320i	$0.625464\pi$
$840$	−2.21578	−0.0764517
$841$	−25.0737	−0.864610
$842$	36.2889	1.25060
$843$	4.75777	0.163866
$844$	1.77654	0.0611510
$845$	−2.29174	−0.0788381
$846$	20.3000	0.697929
$847$	−1.65068	−0.0567180
$848$	6.18728	0.212472
$849$	−8.17862	−0.280690
$850$	2.53080	0.0868059
$851$	−13.8474	−0.474681
$852$	1.77466	0.0607988
$853$	−24.8293	−0.850138	−0.425069	−	0.905161i	$-0.639750\pi$
$853$	−24.8293	−0.850138	−0.425069	+	0.905161i	$0.639750\pi$
$854$	−10.3971	−0.355782
$855$	16.7255	0.571999
$856$	0.573507	0.0196020
$857$	38.6613	1.32064	0.660322	−	0.750983i	$-0.270420\pi$
$857$	38.6613	1.32064	0.660322	+	0.750983i	$0.270420\pi$
$858$	−24.9300	−0.851095
$859$	−29.3077	−0.999964	−0.499982	−	0.866036i	$-0.666660\pi$
$859$	−29.3077	−0.999964	−0.499982	+	0.866036i	$0.666660\pi$
$860$	−0.107815	−0.00367646
$861$	−14.5764	−0.496762
$862$	−38.7510	−1.31986
$863$	2.40031	0.0817075	0.0408537	−	0.999165i	$-0.486992\pi$
$863$	2.40031	0.0817075	0.0408537	+	0.999165i	$0.486992\pi$
$864$	−7.57834	−0.257820
$865$	64.5361	2.19429
$866$	−26.2938	−0.893501
$867$	−1.00000	−0.0339618
$868$	−13.1690	−0.446986
$869$	−13.3577	−0.453131
$870$	9.60991	0.325806
$871$	−46.2617	−1.56752
$872$	−1.64068	−0.0555603
$873$	4.61672	0.156252
$874$	−26.4774	−0.895610
$875$	−15.5108	−0.524361
$876$	23.5836	0.796815
$877$	−1.51670	−0.0512153	−0.0256077	−	0.999672i	$-0.508152\pi$
$877$	−1.51670	−0.0512153	−0.0256077	+	0.999672i	$0.508152\pi$
$878$	56.2156	1.89719
$879$	28.2706	0.953543
$880$	38.8898	1.31098
$881$	−17.1018	−0.576174	−0.288087	−	0.957604i	$-0.593019\pi$
$881$	−17.1018	−0.576174	−0.288087	+	0.957604i	$0.593019\pi$
$882$	8.10603	0.272944
$883$	−25.5324	−0.859233	−0.429616	−	0.903012i	$-0.641351\pi$
$883$	−25.5324	−0.859233	−0.429616	+	0.903012i	$0.641351\pi$
$884$	6.44164	0.216656
$885$	−0.529002	−0.0177822
$886$	36.1308	1.21384
$887$	−42.2577	−1.41888	−0.709438	−	0.704768i	$-0.751051\pi$
$887$	−42.2577	−1.41888	−0.709438	+	0.704768i	$0.751051\pi$
$888$	−3.54254	−0.118880
$889$	1.32376	0.0443974
$890$	−53.5361	−1.79453
$891$	3.46212	0.115985
$892$	2.94105	0.0984736
$893$	70.0079	2.34272
$894$	−5.90204	−0.197394
$895$	−14.2076	−0.474908
$896$	6.99046	0.233535
$897$	−7.68354	−0.256546
$898$	−29.0892	−0.970719
$899$	−9.02771	−0.301091
$900$	2.26399	0.0754665
$901$	1.38373	0.0460989
$902$	58.2106	1.93820
$903$	0.0415915	0.00138408
$904$	−3.10786	−0.103366
$905$	19.7370	0.656082
$906$	−6.04323	−0.200773
$907$	47.4660	1.57608	0.788041	−	0.615623i	$-0.211096\pi$
$907$	47.4660	1.57608	0.788041	+	0.615623i	$0.211096\pi$
$908$	−9.41039	−0.312295
$909$	6.35949	0.210931
$910$	30.2759	1.00364
$911$	−3.66128	−0.121303	−0.0606517	−	0.998159i	$-0.519318\pi$
$911$	−3.66128	−0.121303	−0.0606517	+	0.998159i	$0.519318\pi$
$912$	−29.7700	−0.985783
$913$	−1.06217	−0.0351528
$914$	−69.4136	−2.29600
$915$	−8.08366	−0.267238
$916$	11.7240	0.387373
$917$	28.6110	0.944817
$918$	−1.93055	−0.0637176
$919$	24.0100	0.792017	0.396008	−	0.918247i	$-0.370395\pi$
$919$	24.0100	0.792017	0.396008	+	0.918247i	$0.370395\pi$
$920$	2.72721	0.0899134
$921$	−18.3624	−0.605061
$922$	−24.6611	−0.812171
$923$	3.83280	0.126158
$924$	10.0071	0.329211
$925$	8.81216	0.289742
$926$	58.6770	1.92825
$927$	14.2088	0.466678
$928$	−15.0164	−0.492938
$929$	−8.22900	−0.269985	−0.134992	−	0.990847i	$-0.543101\pi$
$929$	−8.22900	−0.269985	−0.134992	+	0.990847i	$0.543101\pi$
$930$	−22.0960	−0.724555
$931$	27.9550	0.916187
$932$	21.9291	0.718313
$933$	11.9053	0.389762
$934$	33.8695	1.10824
$935$	8.69738	0.284435
$936$	−1.96566	−0.0642496
$937$	8.90621	0.290953	0.145477	−	0.989362i	$-0.453528\pi$
$937$	8.90621	0.290953	0.145477	+	0.989362i	$0.453528\pi$
$938$	40.0751	1.30850
$939$	22.4478	0.732556
$940$	45.6204	1.48797
$941$	−54.9261	−1.79054	−0.895270	−	0.445523i	$-0.853018\pi$
$941$	−54.9261	−1.79054	−0.895270	+	0.445523i	$0.853018\pi$
$942$	1.93055	0.0629007
$943$	17.9408	0.584233
$944$	0.941582	0.0306459
$945$	−4.20452	−0.136773
$946$	−0.166095	−0.00540022
$947$	14.3710	0.466996	0.233498	−	0.972357i	$-0.424983\pi$
$947$	14.3710	0.466996	0.233498	+	0.972357i	$0.424983\pi$
$948$	6.66329	0.216414
$949$	50.9343	1.65340
$950$	16.8496	0.546674
$951$	16.2101	0.525649
$952$	0.882023	0.0285865
$953$	44.5786	1.44404	0.722021	−	0.691871i	$-0.243213\pi$
$953$	44.5786	1.44404	0.722021	+	0.691871i	$0.243213\pi$
$954$	2.67137	0.0864887
$955$	17.9231	0.579977
$956$	−29.6516	−0.959000
$957$	6.86015	0.221757
$958$	18.3142	0.591705
$959$	3.50004	0.113022
$960$	−14.2878	−0.461138
$961$	−10.2427	−0.330409
$962$	48.4044	1.56062
$963$	1.08825	0.0350684
$964$	−44.9216	−1.44683
$965$	0.705951	0.0227254
$966$	6.65600	0.214153
$967$	−41.6799	−1.34034	−0.670168	−	0.742209i	$-0.733778\pi$
$967$	−41.6799	−1.34034	−0.670168	+	0.742209i	$0.733778\pi$
$968$	0.519759	0.0167057
$969$	−6.65781	−0.213880
$970$	22.3904	0.718912
$971$	39.7226	1.27476	0.637380	−	0.770550i	$-0.280018\pi$
$971$	39.7226	1.27476	0.637380	+	0.770550i	$0.280018\pi$
$972$	−1.72702	−0.0553942
$973$	35.4092	1.13517
$974$	−68.7832	−2.20396
$975$	4.88963	0.156594
$976$	14.3883	0.460558
$977$	48.7460	1.55952	0.779762	−	0.626076i	$-0.215340\pi$
$977$	48.7460	1.55952	0.779762	+	0.626076i	$0.215340\pi$
$978$	11.2743	0.360513
$979$	−38.2174	−1.22143
$980$	18.2168	0.581914
$981$	−3.11324	−0.0993982
$982$	−28.5520	−0.911130
$983$	−6.53825	−0.208538	−0.104269	−	0.994549i	$-0.533250\pi$
$983$	−6.53825	−0.208538	−0.104269	+	0.994549i	$0.533250\pi$
$984$	4.58975	0.146316
$985$	23.2751	0.741606
$986$	−3.82536	−0.121824
$987$	−17.5989	−0.560179
$988$	42.8872	1.36442
$989$	−0.0511914	−0.00162779
$990$	16.7907	0.533644
$991$	40.0919	1.27356	0.636781	−	0.771045i	$-0.280265\pi$
$991$	40.0919	1.27356	0.636781	+	0.771045i	$0.280265\pi$
$992$	34.5271	1.09624
$993$	−8.59667	−0.272807
$994$	−3.32023	−0.105311
$995$	25.0353	0.793671
$996$	0.529848	0.0167889
$997$	−12.3074	−0.389778	−0.194889	−	0.980825i	$-0.562435\pi$
$997$	−12.3074	−0.389778	−0.194889	+	0.980825i	$0.562435\pi$
$998$	6.35055	0.201023
$999$	−6.72209	−0.212678

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.g.1.11	✓	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.g.1.11	✓	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)

\(f(q)\)	\(=\)	\(q-1.93055 q^{2} -1.00000 q^{3} +1.72702 q^{4} -2.51216 q^{5} +1.93055 q^{6} -1.67367 q^{7} +0.526999 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.93055 q^{2} -1.00000 q^{3} +1.72702 q^{4} -2.51216 q^{5} +1.93055 q^{6} -1.67367 q^{7} +0.526999 q^{8} +1.00000 q^{9} +4.84984 q^{10} +3.46212 q^{11} -1.72702 q^{12} -3.72991 q^{13} +3.23111 q^{14} +2.51216 q^{15} -4.47144 q^{16} -1.00000 q^{17} -1.93055 q^{18} -6.65781 q^{19} -4.33855 q^{20} +1.67367 q^{21} -6.68379 q^{22} -2.05998 q^{23} -0.526999 q^{24} +1.31092 q^{25} +7.20078 q^{26} -1.00000 q^{27} -2.89047 q^{28} -1.98149 q^{29} -4.84984 q^{30} +4.55602 q^{31} +7.57834 q^{32} -3.46212 q^{33} +1.93055 q^{34} +4.20452 q^{35} +1.72702 q^{36} +6.72209 q^{37} +12.8532 q^{38} +3.72991 q^{39} -1.32390 q^{40} -8.70922 q^{41} -3.23111 q^{42} +0.0248505 q^{43} +5.97915 q^{44} -2.51216 q^{45} +3.97689 q^{46} -10.5151 q^{47} +4.47144 q^{48} -4.19882 q^{49} -2.53080 q^{50} +1.00000 q^{51} -6.44164 q^{52} -1.38373 q^{53} +1.93055 q^{54} -8.69738 q^{55} -0.882023 q^{56} +6.65781 q^{57} +3.82536 q^{58} -0.210577 q^{59} +4.33855 q^{60} -3.21782 q^{61} -8.79562 q^{62} -1.67367 q^{63} -5.68748 q^{64} +9.37012 q^{65} +6.68379 q^{66} +12.4029 q^{67} -1.72702 q^{68} +2.05998 q^{69} -8.11704 q^{70} -1.02758 q^{71} +0.526999 q^{72} -13.6556 q^{73} -12.9773 q^{74} -1.31092 q^{75} -11.4982 q^{76} -5.79445 q^{77} -7.20078 q^{78} -3.85826 q^{79} +11.2330 q^{80} +1.00000 q^{81} +16.8136 q^{82} -0.306799 q^{83} +2.89047 q^{84} +2.51216 q^{85} -0.0479751 q^{86} +1.98149 q^{87} +1.82453 q^{88} -11.0387 q^{89} +4.84984 q^{90} +6.24265 q^{91} -3.55762 q^{92} -4.55602 q^{93} +20.3000 q^{94} +16.7255 q^{95} -7.57834 q^{96} +4.61672 q^{97} +8.10603 q^{98} +3.46212 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + q^{2} - 56 q^{3} + 61 q^{4} + q^{5} - q^{6} + 19 q^{7} + 56 q^{9}+O(q^{10}) \) 56 * q + q^2 - 56 * q^3 + 61 * q^4 + q^5 - q^6 + 19 * q^7 + 56 * q^9	\( 56 q + q^{2} - 56 q^{3} + 61 q^{4} + q^{5} - q^{6} + 19 q^{7} + 56 q^{9} + 8 q^{10} - 7 q^{11} - 61 q^{12} + 8 q^{13} - 8 q^{14} - q^{15} + 71 q^{16} - 56 q^{17} + q^{18} - 2 q^{19} - 4 q^{20} - 19 q^{21} + 47 q^{22} + 16 q^{23} + 85 q^{25} - 11 q^{26} - 56 q^{27} + 52 q^{28} + 17 q^{29} - 8 q^{30} + 23 q^{31} + 11 q^{32} + 7 q^{33} - q^{34} - 41 q^{35} + 61 q^{36} + 58 q^{37} - 22 q^{38} - 8 q^{39} + 38 q^{40} - q^{41} + 8 q^{42} + 27 q^{43} + 2 q^{44} + q^{45} + 46 q^{46} + 5 q^{47} - 71 q^{48} + 59 q^{49} - 4 q^{50} + 56 q^{51} + 25 q^{52} + 15 q^{53} - q^{54} + 9 q^{55} - 36 q^{56} + 2 q^{57} + 89 q^{58} - 61 q^{59} + 4 q^{60} + 47 q^{61} + 8 q^{62} + 19 q^{63} + 88 q^{64} + 39 q^{65} - 47 q^{66} + 20 q^{67} - 61 q^{68} - 16 q^{69} + 36 q^{70} - 2 q^{71} + 93 q^{73} + 48 q^{74} - 85 q^{75} + 38 q^{76} + 26 q^{77} + 11 q^{78} + 72 q^{79} + 42 q^{80} + 56 q^{81} + 33 q^{82} - 11 q^{83} - 52 q^{84} - q^{85} - 4 q^{86} - 17 q^{87} + 130 q^{88} - 6 q^{89} + 8 q^{90} + 37 q^{91} + 132 q^{92} - 23 q^{93} - 32 q^{94} + 12 q^{95} - 11 q^{96} + 100 q^{97} + 42 q^{98} - 7 q^{99}+O(q^{100}) \) 56 * q + q^2 - 56 * q^3 + 61 * q^4 + q^5 - q^6 + 19 * q^7 + 56 * q^9 + 8 * q^10 - 7 * q^11 - 61 * q^12 + 8 * q^13 - 8 * q^14 - q^15 + 71 * q^16 - 56 * q^17 + q^18 - 2 * q^19 - 4 * q^20 - 19 * q^21 + 47 * q^22 + 16 * q^23 + 85 * q^25 - 11 * q^26 - 56 * q^27 + 52 * q^28 + 17 * q^29 - 8 * q^30 + 23 * q^31 + 11 * q^32 + 7 * q^33 - q^34 - 41 * q^35 + 61 * q^36 + 58 * q^37 - 22 * q^38 - 8 * q^39 + 38 * q^40 - q^41 + 8 * q^42 + 27 * q^43 + 2 * q^44 + q^45 + 46 * q^46 + 5 * q^47 - 71 * q^48 + 59 * q^49 - 4 * q^50 + 56 * q^51 + 25 * q^52 + 15 * q^53 - q^54 + 9 * q^55 - 36 * q^56 + 2 * q^57 + 89 * q^58 - 61 * q^59 + 4 * q^60 + 47 * q^61 + 8 * q^62 + 19 * q^63 + 88 * q^64 + 39 * q^65 - 47 * q^66 + 20 * q^67 - 61 * q^68 - 16 * q^69 + 36 * q^70 - 2 * q^71 + 93 * q^73 + 48 * q^74 - 85 * q^75 + 38 * q^76 + 26 * q^77 + 11 * q^78 + 72 * q^79 + 42 * q^80 + 56 * q^81 + 33 * q^82 - 11 * q^83 - 52 * q^84 - q^85 - 4 * q^86 - 17 * q^87 + 130 * q^88 - 6 * q^89 + 8 * q^90 + 37 * q^91 + 132 * q^92 - 23 * q^93 - 32 * q^94 + 12 * q^95 - 11 * q^96 + 100 * q^97 + 42 * q^98 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more