LMFDB - Embedded newform 6422.2.a.bo.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.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:	$51.2799281781$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 31 x^{13} - 4 x^{12} + 373 x^{11} + 85 x^{10} - 2208 x^{9} - 636 x^{8} + 6791 x^{7} + \cdots + 392 $ x^15 - 31x^13 - 4x^12 + 373x^11 + 85x^10 - 2208x^9 - 636x^8 + 6791x^7 + 2005x^6 - 10779x^5 - 2515x^4 + 8205x^3 + 686x^2 - 2352*x + 392
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.98660$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.98660 q^{3} +1.00000 q^{4} +3.18940 q^{5} +2.98660 q^{6} -3.65892 q^{7} -1.00000 q^{8} +5.91977 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.98660 q^{3} +1.00000 q^{4} +3.18940 q^{5} +2.98660 q^{6} -3.65892 q^{7} -1.00000 q^{8} +5.91977 q^{9} -3.18940 q^{10} +5.85128 q^{11} -2.98660 q^{12} +3.65892 q^{14} -9.52545 q^{15} +1.00000 q^{16} -7.18356 q^{17} -5.91977 q^{18} +1.00000 q^{19} +3.18940 q^{20} +10.9277 q^{21} -5.85128 q^{22} +6.79565 q^{23} +2.98660 q^{24} +5.17225 q^{25} -8.72018 q^{27} -3.65892 q^{28} -3.51515 q^{29} +9.52545 q^{30} +3.97965 q^{31} -1.00000 q^{32} -17.4754 q^{33} +7.18356 q^{34} -11.6698 q^{35} +5.91977 q^{36} -6.38545 q^{37} -1.00000 q^{38} -3.18940 q^{40} -3.54209 q^{41} -10.9277 q^{42} -5.51708 q^{43} +5.85128 q^{44} +18.8805 q^{45} -6.79565 q^{46} +7.54609 q^{47} -2.98660 q^{48} +6.38772 q^{49} -5.17225 q^{50} +21.4544 q^{51} -2.99959 q^{53} +8.72018 q^{54} +18.6620 q^{55} +3.65892 q^{56} -2.98660 q^{57} +3.51515 q^{58} +1.42382 q^{59} -9.52545 q^{60} -4.50601 q^{61} -3.97965 q^{62} -21.6600 q^{63} +1.00000 q^{64} +17.4754 q^{66} -6.50838 q^{67} -7.18356 q^{68} -20.2959 q^{69} +11.6698 q^{70} -5.72777 q^{71} -5.91977 q^{72} -9.96517 q^{73} +6.38545 q^{74} -15.4474 q^{75} +1.00000 q^{76} -21.4094 q^{77} -2.77070 q^{79} +3.18940 q^{80} +8.28436 q^{81} +3.54209 q^{82} +9.64703 q^{83} +10.9277 q^{84} -22.9112 q^{85} +5.51708 q^{86} +10.4983 q^{87} -5.85128 q^{88} -3.27775 q^{89} -18.8805 q^{90} +6.79565 q^{92} -11.8856 q^{93} -7.54609 q^{94} +3.18940 q^{95} +2.98660 q^{96} +18.8504 q^{97} -6.38772 q^{98} +34.6382 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 15 q^{2} + 15 q^{4} - q^{5} - 18 q^{7} - 15 q^{8} + 17 q^{9}+O(q^{10}) $ 15 * q - 15 * q^2 + 15 * q^4 - q^5 - 18 * q^7 - 15 * q^8 + 17 * q^9	$ 15 q - 15 q^{2} + 15 q^{4} - q^{5} - 18 q^{7} - 15 q^{8} + 17 q^{9} + q^{10} - 4 q^{11} + 18 q^{14} - 23 q^{15} + 15 q^{16} + 2 q^{17} - 17 q^{18} + 15 q^{19} - q^{20} - 2 q^{21} + 4 q^{22} + 17 q^{23} + 8 q^{25} + 12 q^{27} - 18 q^{28} - 20 q^{29} + 23 q^{30} - 30 q^{31} - 15 q^{32} - 36 q^{33} - 2 q^{34} + 32 q^{35} + 17 q^{36} - 35 q^{37} - 15 q^{38} + q^{40} - 15 q^{41} + 2 q^{42} + q^{43} - 4 q^{44} + 11 q^{45} - 17 q^{46} + 29 q^{49} - 8 q^{50} - q^{51} - q^{53} - 12 q^{54} - 6 q^{55} + 18 q^{56} + 20 q^{58} + 7 q^{59} - 23 q^{60} - 2 q^{61} + 30 q^{62} - 42 q^{63} + 15 q^{64} + 36 q^{66} - 34 q^{67} + 2 q^{68} - 12 q^{69} - 32 q^{70} - 4 q^{71} - 17 q^{72} - 12 q^{73} + 35 q^{74} + 31 q^{75} + 15 q^{76} - 20 q^{77} + 23 q^{79} - q^{80} + 7 q^{81} + 15 q^{82} + 3 q^{83} - 2 q^{84} - 46 q^{85} - q^{86} + 22 q^{87} + 4 q^{88} - 17 q^{89} - 11 q^{90} + 17 q^{92} - 60 q^{93} - q^{95} - 18 q^{97} - 29 q^{98} + 6 q^{99}+O(q^{100}) $ 15 * q - 15 * q^2 + 15 * q^4 - q^5 - 18 * q^7 - 15 * q^8 + 17 * q^9 + q^10 - 4 * q^11 + 18 * q^14 - 23 * q^15 + 15 * q^16 + 2 * q^17 - 17 * q^18 + 15 * q^19 - q^20 - 2 * q^21 + 4 * q^22 + 17 * q^23 + 8 * q^25 + 12 * q^27 - 18 * q^28 - 20 * q^29 + 23 * q^30 - 30 * q^31 - 15 * q^32 - 36 * q^33 - 2 * q^34 + 32 * q^35 + 17 * q^36 - 35 * q^37 - 15 * q^38 + q^40 - 15 * q^41 + 2 * q^42 + q^43 - 4 * q^44 + 11 * q^45 - 17 * q^46 + 29 * q^49 - 8 * q^50 - q^51 - q^53 - 12 * q^54 - 6 * q^55 + 18 * q^56 + 20 * q^58 + 7 * q^59 - 23 * q^60 - 2 * q^61 + 30 * q^62 - 42 * q^63 + 15 * q^64 + 36 * q^66 - 34 * q^67 + 2 * q^68 - 12 * q^69 - 32 * q^70 - 4 * q^71 - 17 * q^72 - 12 * q^73 + 35 * q^74 + 31 * q^75 + 15 * q^76 - 20 * q^77 + 23 * q^79 - q^80 + 7 * q^81 + 15 * q^82 + 3 * q^83 - 2 * q^84 - 46 * q^85 - q^86 + 22 * q^87 + 4 * q^88 - 17 * q^89 - 11 * q^90 + 17 * q^92 - 60 * q^93 - q^95 - 18 * q^97 - 29 * q^98 + 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.98660	−1.72431	−0.862157	−	0.506642i	$-0.830887\pi$
$3$	−2.98660	−1.72431	−0.862157	+	0.506642i	$0.830887\pi$
$4$	1.00000	0.500000
$5$	3.18940	1.42634	0.713171	−	0.700990i	$-0.247258\pi$
$5$	3.18940	1.42634	0.713171	+	0.700990i	$0.247258\pi$
$6$	2.98660	1.21927
$7$	−3.65892	−1.38294	−0.691471	−	0.722404i	$-0.743037\pi$
$7$	−3.65892	−1.38294	−0.691471	+	0.722404i	$0.743037\pi$
$8$	−1.00000	−0.353553
$9$	5.91977	1.97326
$10$	−3.18940	−1.00858
$11$	5.85128	1.76423	0.882113	−	0.471037i	$-0.156120\pi$
$11$	5.85128	1.76423	0.882113	+	0.471037i	$0.156120\pi$
$12$	−2.98660	−0.862157
$13$	0	0
$13$	0	0
$14$	3.65892	0.977888
$15$	−9.52545	−2.45946
$16$	1.00000	0.250000
$17$	−7.18356	−1.74227	−0.871135	−	0.491043i	$-0.836616\pi$
$17$	−7.18356	−1.74227	−0.871135	+	0.491043i	$0.836616\pi$
$18$	−5.91977	−1.39530
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	3.18940	0.713171
$21$	10.9277	2.38463
$22$	−5.85128	−1.24750
$23$	6.79565	1.41699	0.708496	−	0.705715i	$-0.249374\pi$
$23$	6.79565	1.41699	0.708496	+	0.705715i	$0.249374\pi$
$24$	2.98660	0.609637
$25$	5.17225	1.03445
$26$	0	0
$27$	−8.72018	−1.67820
$28$	−3.65892	−0.691471
$29$	−3.51515	−0.652747	−0.326373	−	0.945241i	$-0.605827\pi$
$29$	−3.51515	−0.652747	−0.326373	+	0.945241i	$0.605827\pi$
$30$	9.52545	1.73910
$31$	3.97965	0.714766	0.357383	−	0.933958i	$-0.383669\pi$
$31$	3.97965	0.714766	0.357383	+	0.933958i	$0.383669\pi$
$32$	−1.00000	−0.176777
$33$	−17.4754	−3.04208
$34$	7.18356	1.23197
$35$	−11.6698	−1.97255
$36$	5.91977	0.986628
$37$	−6.38545	−1.04976	−0.524881	−	0.851176i	$-0.675890\pi$
$37$	−6.38545	−1.04976	−0.524881	+	0.851176i	$0.675890\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−3.18940	−0.504288
$41$	−3.54209	−0.553181	−0.276591	−	0.960988i	$-0.589205\pi$
$41$	−3.54209	−0.553181	−0.276591	+	0.960988i	$0.589205\pi$
$42$	−10.9277	−1.68619
$43$	−5.51708	−0.841347	−0.420673	−	0.907212i	$-0.638206\pi$
$43$	−5.51708	−0.841347	−0.420673	+	0.907212i	$0.638206\pi$
$44$	5.85128	0.882113
$45$	18.8805	2.81454
$46$	−6.79565	−1.00196
$47$	7.54609	1.10071	0.550355	−	0.834931i	$-0.314492\pi$
$47$	7.54609	1.10071	0.550355	+	0.834931i	$0.314492\pi$
$48$	−2.98660	−0.431078
$49$	6.38772	0.912531
$50$	−5.17225	−0.731467
$51$	21.4544	3.00422
$52$	0	0
$53$	−2.99959	−0.412025	−0.206013	−	0.978549i	$-0.566049\pi$
$53$	−2.99959	−0.412025	−0.206013	+	0.978549i	$0.566049\pi$
$54$	8.72018	1.18667
$55$	18.6620	2.51639
$56$	3.65892	0.488944
$57$	−2.98660	−0.395585
$58$	3.51515	0.461562
$59$	1.42382	0.185365	0.0926826	−	0.995696i	$-0.470456\pi$
$59$	1.42382	0.185365	0.0926826	+	0.995696i	$0.470456\pi$
$60$	−9.52545	−1.22973
$61$	−4.50601	−0.576936	−0.288468	−	0.957490i	$-0.593146\pi$
$61$	−4.50601	−0.576936	−0.288468	+	0.957490i	$0.593146\pi$
$62$	−3.97965	−0.505416
$63$	−21.6600	−2.72890
$64$	1.00000	0.125000
$65$	0	0
$66$	17.4754	2.15108
$67$	−6.50838	−0.795126	−0.397563	−	0.917575i	$-0.630144\pi$
$67$	−6.50838	−0.795126	−0.397563	+	0.917575i	$0.630144\pi$
$68$	−7.18356	−0.871135
$69$	−20.2959	−2.44334
$70$	11.6698	1.39480
$71$	−5.72777	−0.679761	−0.339880	−	0.940469i	$-0.610387\pi$
$71$	−5.72777	−0.679761	−0.339880	+	0.940469i	$0.610387\pi$
$72$	−5.91977	−0.697652
$73$	−9.96517	−1.16634	−0.583168	−	0.812352i	$-0.698187\pi$
$73$	−9.96517	−1.16634	−0.583168	+	0.812352i	$0.698187\pi$
$74$	6.38545	0.742293
$75$	−15.4474	−1.78372
$76$	1.00000	0.114708
$77$	−21.4094	−2.43982
$78$	0	0
$79$	−2.77070	−0.311728	−0.155864	−	0.987778i	$-0.549816\pi$
$79$	−2.77070	−0.311728	−0.155864	+	0.987778i	$0.549816\pi$
$80$	3.18940	0.356585
$81$	8.28436	0.920485
$82$	3.54209	0.391158
$83$	9.64703	1.05890	0.529450	−	0.848341i	$-0.322399\pi$
$83$	9.64703	1.05890	0.529450	+	0.848341i	$0.322399\pi$
$84$	10.9277	1.19231
$85$	−22.9112	−2.48507
$86$	5.51708	0.594922
$87$	10.4983	1.12554
$88$	−5.85128	−0.623748
$89$	−3.27775	−0.347441	−0.173721	−	0.984795i	$-0.555579\pi$
$89$	−3.27775	−0.347441	−0.173721	+	0.984795i	$0.555579\pi$
$90$	−18.8805	−1.99018
$91$	0	0
$92$	6.79565	0.708496
$93$	−11.8856	−1.23248
$94$	−7.54609	−0.778319
$95$	3.18940	0.327225
$96$	2.98660	0.304818
$97$	18.8504	1.91397	0.956983	−	0.290144i	$-0.0937030\pi$
$97$	18.8504	1.91397	0.956983	+	0.290144i	$0.0937030\pi$
$98$	−6.38772	−0.645257
$99$	34.6382	3.48127
$100$	5.17225	0.517225
$101$	−9.56609	−0.951862	−0.475931	−	0.879483i	$-0.657889\pi$
$101$	−9.56609	−0.951862	−0.475931	+	0.879483i	$0.657889\pi$
$102$	−21.4544	−2.12430
$103$	−15.2759	−1.50518	−0.752591	−	0.658488i	$-0.771196\pi$
$103$	−15.2759	−1.50518	−0.752591	+	0.658488i	$0.771196\pi$
$104$	0	0
$105$	34.8529	3.40129
$106$	2.99959	0.291346
$107$	−5.96947	−0.577091	−0.288545	−	0.957466i	$-0.593172\pi$
$107$	−5.96947	−0.577091	−0.288545	+	0.957466i	$0.593172\pi$
$108$	−8.72018	−0.839100
$109$	8.80290	0.843165	0.421582	−	0.906790i	$-0.361475\pi$
$109$	8.80290	0.843165	0.421582	+	0.906790i	$0.361475\pi$
$110$	−18.6620	−1.77936
$111$	19.0708	1.81012
$112$	−3.65892	−0.345736
$113$	−13.3554	−1.25637	−0.628186	−	0.778063i	$-0.716202\pi$
$113$	−13.3554	−1.25637	−0.628186	+	0.778063i	$0.716202\pi$
$114$	2.98660	0.279721
$115$	21.6740	2.02111
$116$	−3.51515	−0.326373
$117$	0	0
$118$	−1.42382	−0.131073
$119$	26.2841	2.40946
$120$	9.52545	0.869550
$121$	23.2375	2.11250
$122$	4.50601	0.407955
$123$	10.5788	0.953857
$124$	3.97965	0.357383
$125$	0.549386	0.0491386
$126$	21.6600	1.92962
$127$	13.8927	1.23277	0.616387	−	0.787443i	$-0.288596\pi$
$127$	13.8927	1.23277	0.616387	+	0.787443i	$0.288596\pi$
$128$	−1.00000	−0.0883883
$129$	16.4773	1.45075
$130$	0	0
$131$	−4.73759	−0.413925	−0.206962	−	0.978349i	$-0.566358\pi$
$131$	−4.73759	−0.413925	−0.206962	+	0.978349i	$0.566358\pi$
$132$	−17.4754	−1.52104
$133$	−3.65892	−0.317269
$134$	6.50838	0.562239
$135$	−27.8121	−2.39369
$136$	7.18356	0.615986
$137$	−0.809543	−0.0691639	−0.0345819	−	0.999402i	$-0.511010\pi$
$137$	−0.809543	−0.0691639	−0.0345819	+	0.999402i	$0.511010\pi$
$138$	20.2959	1.72770
$139$	12.5157	1.06156	0.530782	−	0.847508i	$-0.321898\pi$
$139$	12.5157	1.06156	0.530782	+	0.847508i	$0.321898\pi$
$140$	−11.6698	−0.986275
$141$	−22.5371	−1.89797
$142$	5.72777	0.480664
$143$	0	0
$144$	5.91977	0.493314
$145$	−11.2112	−0.931040
$146$	9.96517	0.824724
$147$	−19.0775	−1.57349
$148$	−6.38545	−0.524881
$149$	−20.9059	−1.71268	−0.856338	−	0.516416i	$-0.827266\pi$
$149$	−20.9059	−1.71268	−0.856338	+	0.516416i	$0.827266\pi$
$150$	15.4474	1.26128
$151$	−20.4029	−1.66037	−0.830184	−	0.557490i	$-0.811765\pi$
$151$	−20.4029	−1.66037	−0.830184	+	0.557490i	$0.811765\pi$
$152$	−1.00000	−0.0811107
$153$	−42.5250	−3.43795
$154$	21.4094	1.72522
$155$	12.6927	1.01950
$156$	0	0
$157$	19.7383	1.57529	0.787646	−	0.616129i	$-0.211300\pi$
$157$	19.7383	1.57529	0.787646	+	0.616129i	$0.211300\pi$
$158$	2.77070	0.220425
$159$	8.95856	0.710460
$160$	−3.18940	−0.252144
$161$	−24.8648	−1.95962
$162$	−8.28436	−0.650881
$163$	20.0620	1.57138	0.785688	−	0.618624i	$-0.212309\pi$
$163$	20.0620	1.57138	0.785688	+	0.618624i	$0.212309\pi$
$164$	−3.54209	−0.276591
$165$	−55.7360	−4.33905
$166$	−9.64703	−0.748755
$167$	−10.9928	−0.850645	−0.425322	−	0.905042i	$-0.639839\pi$
$167$	−10.9928	−0.850645	−0.425322	+	0.905042i	$0.639839\pi$
$168$	−10.9277	−0.843093
$169$	0	0
$170$	22.9112	1.75721
$171$	5.91977	0.452696
$172$	−5.51708	−0.420673
$173$	8.46539	0.643612	0.321806	−	0.946806i	$-0.395710\pi$
$173$	8.46539	0.643612	0.321806	+	0.946806i	$0.395710\pi$
$174$	−10.4983	−0.795877
$175$	−18.9249	−1.43059
$176$	5.85128	0.441057
$177$	−4.25237	−0.319628
$178$	3.27775	0.245678
$179$	21.1209	1.57865	0.789326	−	0.613975i	$-0.210430\pi$
$179$	21.1209	1.57865	0.789326	+	0.613975i	$0.210430\pi$
$180$	18.8805	1.40727
$181$	16.6001	1.23388	0.616939	−	0.787011i	$-0.288373\pi$
$181$	16.6001	1.23388	0.616939	+	0.787011i	$0.288373\pi$
$182$	0	0
$183$	13.4577	0.994819
$184$	−6.79565	−0.500982
$185$	−20.3657	−1.49732
$186$	11.8856	0.871495
$187$	−42.0330	−3.07376
$188$	7.54609	0.550355
$189$	31.9065	2.32085
$190$	−3.18940	−0.231383
$191$	−11.6479	−0.842811	−0.421406	−	0.906872i	$-0.638463\pi$
$191$	−11.6479	−0.842811	−0.421406	+	0.906872i	$0.638463\pi$
$192$	−2.98660	−0.215539
$193$	−9.92439	−0.714373	−0.357187	−	0.934033i	$-0.616264\pi$
$193$	−9.92439	−0.714373	−0.357187	+	0.934033i	$0.616264\pi$
$194$	−18.8504	−1.35338
$195$	0	0
$196$	6.38772	0.456266
$197$	−4.05502	−0.288908	−0.144454	−	0.989511i	$-0.546143\pi$
$197$	−4.05502	−0.288908	−0.144454	+	0.989511i	$0.546143\pi$
$198$	−34.6382	−2.46163
$199$	2.78997	0.197776	0.0988880	−	0.995099i	$-0.468471\pi$
$199$	2.78997	0.197776	0.0988880	+	0.995099i	$0.468471\pi$
$200$	−5.17225	−0.365734
$201$	19.4379	1.37105
$202$	9.56609	0.673068
$203$	12.8617	0.902712
$204$	21.4544	1.50211
$205$	−11.2971	−0.789025
$206$	15.2759	1.06432
$207$	40.2287	2.79609
$208$	0	0
$209$	5.85128	0.404741
$210$	−34.8529	−2.40508
$211$	−8.76395	−0.603335	−0.301668	−	0.953413i	$-0.597543\pi$
$211$	−8.76395	−0.603335	−0.301668	+	0.953413i	$0.597543\pi$
$212$	−2.99959	−0.206013
$213$	17.1065	1.17212
$214$	5.96947	0.408065
$215$	−17.5962	−1.20005
$216$	8.72018	0.593333
$217$	−14.5612	−0.988480
$218$	−8.80290	−0.596208
$219$	29.7620	2.01113
$220$	18.6620	1.25820
$221$	0	0
$222$	−19.0708	−1.27995
$223$	5.82332	0.389958	0.194979	−	0.980807i	$-0.437536\pi$
$223$	5.82332	0.389958	0.194979	+	0.980807i	$0.437536\pi$
$224$	3.65892	0.244472
$225$	30.6185	2.04124
$226$	13.3554	0.888390
$227$	−16.9850	−1.12734	−0.563668	−	0.826001i	$-0.690610\pi$
$227$	−16.9850	−1.12734	−0.563668	+	0.826001i	$0.690610\pi$
$228$	−2.98660	−0.197792
$229$	7.28230	0.481228	0.240614	−	0.970621i	$-0.422651\pi$
$229$	7.28230	0.481228	0.240614	+	0.970621i	$0.422651\pi$
$230$	−21.6740	−1.42914
$231$	63.9412	4.20702
$232$	3.51515	0.230781
$233$	−1.50585	−0.0986515	−0.0493257	−	0.998783i	$-0.515707\pi$
$233$	−1.50585	−0.0986515	−0.0493257	+	0.998783i	$0.515707\pi$
$234$	0	0
$235$	24.0675	1.56999
$236$	1.42382	0.0926826
$237$	8.27498	0.537517
$238$	−26.2841	−1.70375
$239$	−2.22330	−0.143813	−0.0719065	−	0.997411i	$-0.522908\pi$
$239$	−2.22330	−0.143813	−0.0719065	+	0.997411i	$0.522908\pi$
$240$	−9.52545	−0.614865
$241$	−12.7483	−0.821188	−0.410594	−	0.911818i	$-0.634679\pi$
$241$	−12.7483	−0.821188	−0.410594	+	0.911818i	$0.634679\pi$
$242$	−23.2375	−1.49376
$243$	1.41848	0.0909952
$244$	−4.50601	−0.288468
$245$	20.3730	1.30158
$246$	−10.5788	−0.674479
$247$	0	0
$248$	−3.97965	−0.252708
$249$	−28.8118	−1.82587
$250$	−0.549386	−0.0347462
$251$	26.6689	1.68333	0.841664	−	0.540001i	$-0.181576\pi$
$251$	26.6689	1.68333	0.841664	+	0.540001i	$0.181576\pi$
$252$	−21.6600	−1.36445
$253$	39.7633	2.49990
$254$	−13.8927	−0.871704
$255$	68.4267	4.28504
$256$	1.00000	0.0625000
$257$	13.1090	0.817716	0.408858	−	0.912598i	$-0.365927\pi$
$257$	13.1090	0.817716	0.408858	+	0.912598i	$0.365927\pi$
$258$	−16.4773	−1.02583
$259$	23.3639	1.45176
$260$	0	0
$261$	−20.8089	−1.28804
$262$	4.73759	0.292689
$263$	7.45450	0.459664	0.229832	−	0.973230i	$-0.426182\pi$
$263$	7.45450	0.459664	0.229832	+	0.973230i	$0.426182\pi$
$264$	17.4754	1.07554
$265$	−9.56688	−0.587689
$266$	3.65892	0.224343
$267$	9.78933	0.599097
$268$	−6.50838	−0.397563
$269$	−15.5724	−0.949466	−0.474733	−	0.880130i	$-0.657455\pi$
$269$	−15.5724	−0.949466	−0.474733	+	0.880130i	$0.657455\pi$
$270$	27.8121	1.69259
$271$	−20.9796	−1.27442	−0.637210	−	0.770690i	$-0.719912\pi$
$271$	−20.9796	−1.27442	−0.637210	+	0.770690i	$0.719912\pi$
$272$	−7.18356	−0.435568
$273$	0	0
$274$	0.809543	0.0489063
$275$	30.2643	1.82501
$276$	−20.2959	−1.22167
$277$	0.267864	0.0160944	0.00804718	−	0.999968i	$-0.497438\pi$
$277$	0.267864	0.0160944	0.00804718	+	0.999968i	$0.497438\pi$
$278$	−12.5157	−0.750640
$279$	23.5586	1.41042
$280$	11.6698	0.697401
$281$	3.42450	0.204288	0.102144	−	0.994770i	$-0.467430\pi$
$281$	3.42450	0.204288	0.102144	+	0.994770i	$0.467430\pi$
$282$	22.5371	1.34207
$283$	0.711417	0.0422893	0.0211447	−	0.999776i	$-0.493269\pi$
$283$	0.711417	0.0422893	0.0211447	+	0.999776i	$0.493269\pi$
$284$	−5.72777	−0.339880
$285$	−9.52545	−0.564239
$286$	0	0
$287$	12.9602	0.765018
$288$	−5.91977	−0.348826
$289$	34.6036	2.03551
$290$	11.2112	0.658345
$291$	−56.2985	−3.30028
$292$	−9.96517	−0.583168
$293$	3.75388	0.219304	0.109652	−	0.993970i	$-0.465026\pi$
$293$	3.75388	0.219304	0.109652	+	0.993970i	$0.465026\pi$
$294$	19.0775	1.11263
$295$	4.54112	0.264394
$296$	6.38545	0.371147
$297$	−51.0242	−2.96072
$298$	20.9059	1.21104
$299$	0	0
$300$	−15.4474	−0.891859
$301$	20.1866	1.16353
$302$	20.4029	1.17406
$303$	28.5701	1.64131
$304$	1.00000	0.0573539
$305$	−14.3715	−0.822908
$306$	42.5250	2.43099
$307$	−4.08172	−0.232956	−0.116478	−	0.993193i	$-0.537160\pi$
$307$	−4.08172	−0.232956	−0.116478	+	0.993193i	$0.537160\pi$
$308$	−21.4094	−1.21991
$309$	45.6231	2.59541
$310$	−12.6927	−0.720896
$311$	20.5579	1.16573	0.582866	−	0.812568i	$-0.301931\pi$
$311$	20.5579	1.16573	0.582866	+	0.812568i	$0.301931\pi$
$312$	0	0
$313$	−26.7492	−1.51195	−0.755977	−	0.654599i	$-0.772838\pi$
$313$	−26.7492	−1.51195	−0.755977	+	0.654599i	$0.772838\pi$
$314$	−19.7383	−1.11390
$315$	−69.0823	−3.89235
$316$	−2.77070	−0.155864
$317$	11.0253	0.619244	0.309622	−	0.950860i	$-0.399798\pi$
$317$	11.0253	0.619244	0.309622	+	0.950860i	$0.399798\pi$
$318$	−8.95856	−0.502371
$319$	−20.5681	−1.15159
$320$	3.18940	0.178293
$321$	17.8284	0.995085
$322$	24.8648	1.38566
$323$	−7.18356	−0.399704
$324$	8.28436	0.460242
$325$	0	0
$326$	−20.0620	−1.11113
$327$	−26.2907	−1.45388
$328$	3.54209	0.195579
$329$	−27.6105	−1.52222
$330$	55.7360	3.06817
$331$	−22.4449	−1.23368	−0.616842	−	0.787087i	$-0.711588\pi$
$331$	−22.4449	−1.23368	−0.616842	+	0.787087i	$0.711588\pi$
$332$	9.64703	0.529450
$333$	−37.8004	−2.07145
$334$	10.9928	0.601497
$335$	−20.7578	−1.13412
$336$	10.9277	0.596157
$337$	−0.791589	−0.0431206	−0.0215603	−	0.999768i	$-0.506863\pi$
$337$	−0.791589	−0.0431206	−0.0215603	+	0.999768i	$0.506863\pi$
$338$	0	0
$339$	39.8873	2.16638
$340$	−22.9112	−1.24254
$341$	23.2860	1.26101
$342$	−5.91977	−0.320104
$343$	2.24029	0.120965
$344$	5.51708	0.297461
$345$	−64.7317	−3.48504
$346$	−8.46539	−0.455102
$347$	−9.99755	−0.536696	−0.268348	−	0.963322i	$-0.586478\pi$
$347$	−9.99755	−0.536696	−0.268348	+	0.963322i	$0.586478\pi$
$348$	10.4983	0.562770
$349$	15.5988	0.834987	0.417493	−	0.908680i	$-0.362909\pi$
$349$	15.5988	0.834987	0.417493	+	0.908680i	$0.362909\pi$
$350$	18.9249	1.01158
$351$	0	0
$352$	−5.85128	−0.311874
$353$	−7.79736	−0.415012	−0.207506	−	0.978234i	$-0.566535\pi$
$353$	−7.79736	−0.415012	−0.207506	+	0.978234i	$0.566535\pi$
$354$	4.25237	0.226011
$355$	−18.2681	−0.969571
$356$	−3.27775	−0.173721
$357$	−78.5001	−4.15466
$358$	−21.1209	−1.11628
$359$	−30.1863	−1.59317	−0.796587	−	0.604524i	$-0.793363\pi$
$359$	−30.1863	−1.59317	−0.796587	+	0.604524i	$0.793363\pi$
$360$	−18.8805	−0.995089
$361$	1.00000	0.0526316
$362$	−16.6001	−0.872483
$363$	−69.4009	−3.64261
$364$	0	0
$365$	−31.7829	−1.66359
$366$	−13.4577	−0.703443
$367$	25.2869	1.31997	0.659983	−	0.751281i	$-0.270564\pi$
$367$	25.2869	1.31997	0.659983	+	0.751281i	$0.270564\pi$
$368$	6.79565	0.354248
$369$	−20.9683	−1.09157
$370$	20.3657	1.05876
$371$	10.9753	0.569807
$372$	−11.8856	−0.616240
$373$	23.6463	1.22436	0.612180	−	0.790718i	$-0.290293\pi$
$373$	23.6463	1.22436	0.612180	+	0.790718i	$0.290293\pi$
$374$	42.0330	2.17348
$375$	−1.64079	−0.0847303
$376$	−7.54609	−0.389160
$377$	0	0
$378$	−31.9065	−1.64109
$379$	17.4455	0.896113	0.448057	−	0.894005i	$-0.352116\pi$
$379$	17.4455	0.896113	0.448057	+	0.894005i	$0.352116\pi$
$380$	3.18940	0.163613
$381$	−41.4918	−2.12569
$382$	11.6479	0.595958
$383$	−33.3062	−1.70187	−0.850933	−	0.525275i	$-0.823963\pi$
$383$	−33.3062	−1.70187	−0.850933	+	0.525275i	$0.823963\pi$
$384$	2.98660	0.152409
$385$	−68.2830	−3.48002
$386$	9.92439	0.505138
$387$	−32.6598	−1.66019
$388$	18.8504	0.956983
$389$	−31.1637	−1.58006	−0.790030	−	0.613068i	$-0.789935\pi$
$389$	−31.1637	−1.58006	−0.790030	+	0.613068i	$0.789935\pi$
$390$	0	0
$391$	−48.8170	−2.46878
$392$	−6.38772	−0.322628
$393$	14.1493	0.713736
$394$	4.05502	0.204289
$395$	−8.83687	−0.444631
$396$	34.6382	1.74064
$397$	−22.8856	−1.14860	−0.574299	−	0.818646i	$-0.694725\pi$
$397$	−22.8856	−1.14860	−0.574299	+	0.818646i	$0.694725\pi$
$398$	−2.78997	−0.139849
$399$	10.9277	0.547071
$400$	5.17225	0.258613
$401$	−18.2019	−0.908960	−0.454480	−	0.890757i	$-0.650175\pi$
$401$	−18.2019	−0.908960	−0.454480	+	0.890757i	$0.650175\pi$
$402$	−19.4379	−0.969476
$403$	0	0
$404$	−9.56609	−0.475931
$405$	26.4221	1.31293
$406$	−12.8617	−0.638314
$407$	−37.3630	−1.85202
$408$	−21.4544	−1.06215
$409$	9.28358	0.459044	0.229522	−	0.973304i	$-0.426284\pi$
$409$	9.28358	0.459044	0.229522	+	0.973304i	$0.426284\pi$
$410$	11.2971	0.557925
$411$	2.41778	0.119260
$412$	−15.2759	−0.752591
$413$	−5.20964	−0.256350
$414$	−40.2287	−1.97713
$415$	30.7682	1.51035
$416$	0	0
$417$	−37.3793	−1.83047
$418$	−5.85128	−0.286195
$419$	−29.4959	−1.44097	−0.720483	−	0.693472i	$-0.756080\pi$
$419$	−29.4959	−1.44097	−0.720483	+	0.693472i	$0.756080\pi$
$420$	34.8529	1.70065
$421$	−24.7158	−1.20458	−0.602288	−	0.798279i	$-0.705744\pi$
$421$	−24.7158	−1.20458	−0.602288	+	0.798279i	$0.705744\pi$
$422$	8.76395	0.426622
$423$	44.6711	2.17198
$424$	2.99959	0.145673
$425$	−37.1552	−1.80229
$426$	−17.1065	−0.828815
$427$	16.4872	0.797870
$428$	−5.96947	−0.288545
$429$	0	0
$430$	17.5962	0.848562
$431$	−10.9761	−0.528700	−0.264350	−	0.964427i	$-0.585157\pi$
$431$	−10.9761	−0.528700	−0.264350	+	0.964427i	$0.585157\pi$
$432$	−8.72018	−0.419550
$433$	−10.2075	−0.490540	−0.245270	−	0.969455i	$-0.578877\pi$
$433$	−10.2075	−0.490540	−0.245270	+	0.969455i	$0.578877\pi$
$434$	14.5612	0.698961
$435$	33.4834	1.60541
$436$	8.80290	0.421582
$437$	6.79565	0.325080
$438$	−29.7620	−1.42208
$439$	−0.964016	−0.0460100	−0.0230050	−	0.999735i	$-0.507323\pi$
$439$	−0.964016	−0.0460100	−0.0230050	+	0.999735i	$0.507323\pi$
$440$	−18.6620	−0.889678
$441$	37.8138	1.80066
$442$	0	0
$443$	−3.07930	−0.146302	−0.0731509	−	0.997321i	$-0.523305\pi$
$443$	−3.07930	−0.146302	−0.0731509	+	0.997321i	$0.523305\pi$
$444$	19.0708	0.905059
$445$	−10.4541	−0.495570
$446$	−5.82332	−0.275742
$447$	62.4374	2.95319
$448$	−3.65892	−0.172868
$449$	37.7506	1.78156	0.890781	−	0.454433i	$-0.150158\pi$
$449$	37.7506	1.78156	0.890781	+	0.454433i	$0.150158\pi$
$450$	−30.6185	−1.44337
$451$	−20.7257	−0.975937
$452$	−13.3554	−0.628186
$453$	60.9354	2.86299
$454$	16.9850	0.797147
$455$	0	0
$456$	2.98660	0.139860
$457$	−35.5399	−1.66249	−0.831243	−	0.555909i	$-0.812370\pi$
$457$	−35.5399	−1.66249	−0.831243	+	0.555909i	$0.812370\pi$
$458$	−7.28230	−0.340279
$459$	62.6420	2.92388
$460$	21.6740	1.01056
$461$	26.2650	1.22329	0.611643	−	0.791134i	$-0.290509\pi$
$461$	26.2650	1.22329	0.611643	+	0.791134i	$0.290509\pi$
$462$	−63.9412	−2.97481
$463$	21.0497	0.978262	0.489131	−	0.872210i	$-0.337314\pi$
$463$	21.0497	0.978262	0.489131	+	0.872210i	$0.337314\pi$
$464$	−3.51515	−0.163187
$465$	−37.9079	−1.75794
$466$	1.50585	0.0697571
$467$	0.843884	0.0390503	0.0195251	−	0.999809i	$-0.493785\pi$
$467$	0.843884	0.0390503	0.0195251	+	0.999809i	$0.493785\pi$
$468$	0	0
$469$	23.8137	1.09961
$470$	−24.0675	−1.11015
$471$	−58.9505	−2.71630
$472$	−1.42382	−0.0655365
$473$	−32.2820	−1.48433
$474$	−8.27498	−0.380082
$475$	5.17225	0.237319
$476$	26.2841	1.20473
$477$	−17.7569	−0.813031
$478$	2.22330	0.101691
$479$	7.66789	0.350355	0.175177	−	0.984537i	$-0.443950\pi$
$479$	7.66789	0.350355	0.175177	+	0.984537i	$0.443950\pi$
$480$	9.52545	0.434775
$481$	0	0
$482$	12.7483	0.580668
$483$	74.2611	3.37900
$484$	23.2375	1.05625
$485$	60.1214	2.72997
$486$	−1.41848	−0.0643433
$487$	−28.6932	−1.30021	−0.650107	−	0.759842i	$-0.725276\pi$
$487$	−28.6932	−1.30021	−0.650107	+	0.759842i	$0.725276\pi$
$488$	4.50601	0.203978
$489$	−59.9171	−2.70954
$490$	−20.3730	−0.920357
$491$	−14.5914	−0.658499	−0.329250	−	0.944243i	$-0.606796\pi$
$491$	−14.5914	−0.658499	−0.329250	+	0.944243i	$0.606796\pi$
$492$	10.5788	0.476929
$493$	25.2513	1.13726
$494$	0	0
$495$	110.475	4.96548
$496$	3.97965	0.178691
$497$	20.9575	0.940071
$498$	28.8118	1.29109
$499$	−38.1451	−1.70761	−0.853805	−	0.520592i	$-0.825711\pi$
$499$	−38.1451	−1.70761	−0.853805	+	0.520592i	$0.825711\pi$
$500$	0.549386	0.0245693
$501$	32.8309	1.46678
$502$	−26.6689	−1.19029
$503$	19.9147	0.887951	0.443976	−	0.896039i	$-0.353568\pi$
$503$	19.9147	0.887951	0.443976	+	0.896039i	$0.353568\pi$
$504$	21.6600	0.964812
$505$	−30.5101	−1.35768
$506$	−39.7633	−1.76769
$507$	0	0
$508$	13.8927	0.616387
$509$	−35.2394	−1.56196	−0.780981	−	0.624555i	$-0.785280\pi$
$509$	−35.2394	−1.56196	−0.780981	+	0.624555i	$0.785280\pi$
$510$	−68.4267	−3.02998
$511$	36.4618	1.61298
$512$	−1.00000	−0.0441942
$513$	−8.72018	−0.385005
$514$	−13.1090	−0.578213
$515$	−48.7210	−2.14690
$516$	16.4773	0.725373
$517$	44.1542	1.94190
$518$	−23.3639	−1.02655
$519$	−25.2827	−1.10979
$520$	0	0
$521$	−41.4829	−1.81740	−0.908698	−	0.417453i	$-0.862923\pi$
$521$	−41.4829	−1.81740	−0.908698	+	0.417453i	$0.862923\pi$
$522$	20.8089	0.910780
$523$	−13.9318	−0.609193	−0.304597	−	0.952481i	$-0.598522\pi$
$523$	−13.9318	−0.609193	−0.304597	+	0.952481i	$0.598522\pi$
$524$	−4.73759	−0.206962
$525$	56.5210	2.46678
$526$	−7.45450	−0.325032
$527$	−28.5881	−1.24532
$528$	−17.4754	−0.760520
$529$	23.1809	1.00787
$530$	9.56688	0.415559
$531$	8.42867	0.365773
$532$	−3.65892	−0.158634
$533$	0	0
$534$	−9.78933	−0.423626
$535$	−19.0390	−0.823129
$536$	6.50838	0.281119
$537$	−63.0797	−2.72209
$538$	15.5724	0.671374
$539$	37.3763	1.60991
$540$	−27.8121	−1.19684
$541$	−0.838455	−0.0360480	−0.0180240	−	0.999838i	$-0.505738\pi$
$541$	−0.838455	−0.0360480	−0.0180240	+	0.999838i	$0.505738\pi$
$542$	20.9796	0.901151
$543$	−49.5779	−2.12759
$544$	7.18356	0.307993
$545$	28.0759	1.20264
$546$	0	0
$547$	−27.2378	−1.16460	−0.582302	−	0.812972i	$-0.697848\pi$
$547$	−27.2378	−1.16460	−0.582302	+	0.812972i	$0.697848\pi$
$548$	−0.809543	−0.0345819
$549$	−26.6746	−1.13844
$550$	−30.2643	−1.29047
$551$	−3.51515	−0.149750
$552$	20.2959	0.863851
$553$	10.1378	0.431103
$554$	−0.267864	−0.0113804
$555$	60.8242	2.58185
$556$	12.5157	0.530782
$557$	−30.6730	−1.29966	−0.649829	−	0.760080i	$-0.725160\pi$
$557$	−30.6730	−1.29966	−0.649829	+	0.760080i	$0.725160\pi$
$558$	−23.5586	−0.997315
$559$	0	0
$560$	−11.6698	−0.493137
$561$	125.536	5.30012
$562$	−3.42450	−0.144454
$563$	27.4374	1.15635	0.578173	−	0.815914i	$-0.303766\pi$
$563$	27.4374	1.15635	0.578173	+	0.815914i	$0.303766\pi$
$564$	−22.5371	−0.948984
$565$	−42.5958	−1.79202
$566$	−0.711417	−0.0299031
$567$	−30.3118	−1.27298
$568$	5.72777	0.240332
$569$	−16.1793	−0.678273	−0.339136	−	0.940737i	$-0.610135\pi$
$569$	−16.1793	−0.678273	−0.339136	+	0.940737i	$0.610135\pi$
$570$	9.52545	0.398977
$571$	1.40846	0.0589421	0.0294710	−	0.999566i	$-0.490618\pi$
$571$	1.40846	0.0589421	0.0294710	+	0.999566i	$0.490618\pi$
$572$	0	0
$573$	34.7875	1.45327
$574$	−12.9602	−0.540949
$575$	35.1489	1.46581
$576$	5.91977	0.246657
$577$	−38.1906	−1.58990	−0.794948	−	0.606678i	$-0.792502\pi$
$577$	−38.1906	−1.58990	−0.794948	+	0.606678i	$0.792502\pi$
$578$	−34.6036	−1.43932
$579$	29.6402	1.23180
$580$	−11.2112	−0.465520
$581$	−35.2977	−1.46440
$582$	56.2985	2.33365
$583$	−17.5514	−0.726906
$584$	9.96517	0.412362
$585$	0	0
$586$	−3.75388	−0.155071
$587$	−23.3004	−0.961710	−0.480855	−	0.876800i	$-0.659674\pi$
$587$	−23.3004	−0.961710	−0.480855	+	0.876800i	$0.659674\pi$
$588$	−19.0775	−0.786745
$589$	3.97965	0.163979
$590$	−4.54112	−0.186955
$591$	12.1107	0.498169
$592$	−6.38545	−0.262440
$593$	−5.00088	−0.205361	−0.102681	−	0.994714i	$-0.532742\pi$
$593$	−5.00088	−0.205361	−0.102681	+	0.994714i	$0.532742\pi$
$594$	51.0242	2.09355
$595$	83.8305	3.43671
$596$	−20.9059	−0.856338
$597$	−8.33253	−0.341028
$598$	0	0
$599$	−26.7723	−1.09389	−0.546943	−	0.837170i	$-0.684209\pi$
$599$	−26.7723	−1.09389	−0.546943	+	0.837170i	$0.684209\pi$
$600$	15.4474	0.630639
$601$	−22.2383	−0.907120	−0.453560	−	0.891226i	$-0.649846\pi$
$601$	−22.2383	−0.907120	−0.453560	+	0.891226i	$0.649846\pi$
$602$	−20.1866	−0.822743
$603$	−38.5281	−1.56899
$604$	−20.4029	−0.830184
$605$	74.1135	3.01314
$606$	−28.5701	−1.16058
$607$	−24.1281	−0.979331	−0.489666	−	0.871910i	$-0.662881\pi$
$607$	−24.1281	−0.979331	−0.489666	+	0.871910i	$0.662881\pi$
$608$	−1.00000	−0.0405554
$609$	−38.4126	−1.55656
$610$	14.3715	0.581884
$611$	0	0
$612$	−42.5250	−1.71897
$613$	−6.09565	−0.246201	−0.123101	−	0.992394i	$-0.539284\pi$
$613$	−6.09565	−0.246201	−0.123101	+	0.992394i	$0.539284\pi$
$614$	4.08172	0.164725
$615$	33.7400	1.36053
$616$	21.4094	0.862608
$617$	23.0073	0.926239	0.463119	−	0.886296i	$-0.346730\pi$
$617$	23.0073	0.926239	0.463119	+	0.886296i	$0.346730\pi$
$618$	−45.6231	−1.83523
$619$	−0.915266	−0.0367877	−0.0183938	−	0.999831i	$-0.505855\pi$
$619$	−0.915266	−0.0367877	−0.0183938	+	0.999831i	$0.505855\pi$
$620$	12.6927	0.509750
$621$	−59.2593	−2.37799
$622$	−20.5579	−0.824297
$623$	11.9930	0.480491
$624$	0	0
$625$	−24.1091	−0.964362
$626$	26.7492	1.06911
$627$	−17.4754	−0.697901
$628$	19.7383	0.787646
$629$	45.8703	1.82897
$630$	69.0823	2.75230
$631$	−32.9834	−1.31305	−0.656523	−	0.754306i	$-0.727974\pi$
$631$	−32.9834	−1.31305	−0.656523	+	0.754306i	$0.727974\pi$
$632$	2.77070	0.110213
$633$	26.1744	1.04034
$634$	−11.0253	−0.437871
$635$	44.3092	1.75836
$636$	8.95856	0.355230
$637$	0	0
$638$	20.5681	0.814300
$639$	−33.9071	−1.34134
$640$	−3.18940	−0.126072
$641$	−31.5657	−1.24677	−0.623384	−	0.781916i	$-0.714243\pi$
$641$	−31.5657	−1.24677	−0.623384	+	0.781916i	$0.714243\pi$
$642$	−17.8284	−0.703631
$643$	−5.39149	−0.212620	−0.106310	−	0.994333i	$-0.533904\pi$
$643$	−5.39149	−0.212620	−0.106310	+	0.994333i	$0.533904\pi$
$644$	−24.8648	−0.979810
$645$	52.5527	2.06926
$646$	7.18356	0.282634
$647$	3.09899	0.121834	0.0609170	−	0.998143i	$-0.480598\pi$
$647$	3.09899	0.121834	0.0609170	+	0.998143i	$0.480598\pi$
$648$	−8.28436	−0.325440
$649$	8.33115	0.327026
$650$	0	0
$651$	43.4885	1.70445
$652$	20.0620	0.785688
$653$	4.77020	0.186672	0.0933362	−	0.995635i	$-0.470247\pi$
$653$	4.77020	0.186672	0.0933362	+	0.995635i	$0.470247\pi$
$654$	26.2907	1.02805
$655$	−15.1100	−0.590398
$656$	−3.54209	−0.138295
$657$	−58.9915	−2.30148
$658$	27.6105	1.07637
$659$	29.0713	1.13246	0.566229	−	0.824248i	$-0.308402\pi$
$659$	29.0713	1.13246	0.566229	+	0.824248i	$0.308402\pi$
$660$	−55.7360	−2.16952
$661$	6.34447	0.246771	0.123386	−	0.992359i	$-0.460625\pi$
$661$	6.34447	0.246771	0.123386	+	0.992359i	$0.460625\pi$
$662$	22.4449	0.872346
$663$	0	0
$664$	−9.64703	−0.374377
$665$	−11.6698	−0.452534
$666$	37.8004	1.46473
$667$	−23.8877	−0.924937
$668$	−10.9928	−0.425322
$669$	−17.3919	−0.672410
$670$	20.7578	0.801944
$671$	−26.3659	−1.01785
$672$	−10.9277	−0.421546
$673$	−41.9745	−1.61800	−0.808999	−	0.587811i	$-0.799990\pi$
$673$	−41.9745	−1.61800	−0.808999	+	0.587811i	$0.799990\pi$
$674$	0.791589	0.0304909
$675$	−45.1030	−1.73601
$676$	0	0
$677$	4.53817	0.174416	0.0872080	−	0.996190i	$-0.472206\pi$
$677$	4.53817	0.174416	0.0872080	+	0.996190i	$0.472206\pi$
$678$	−39.8873	−1.53186
$679$	−68.9721	−2.64691
$680$	22.9112	0.878606
$681$	50.7275	1.94388
$682$	−23.2860	−0.891668
$683$	−8.38882	−0.320989	−0.160495	−	0.987037i	$-0.551309\pi$
$683$	−8.38882	−0.320989	−0.160495	+	0.987037i	$0.551309\pi$
$684$	5.91977	0.226348
$685$	−2.58195	−0.0986513
$686$	−2.24029	−0.0855348
$687$	−21.7493	−0.829788
$688$	−5.51708	−0.210337
$689$	0	0
$690$	64.7317	2.46429
$691$	−11.0793	−0.421477	−0.210739	−	0.977542i	$-0.567587\pi$
$691$	−11.0793	−0.421477	−0.210739	+	0.977542i	$0.567587\pi$
$692$	8.46539	0.321806
$693$	−126.739	−4.81440
$694$	9.99755	0.379502
$695$	39.9174	1.51415
$696$	−10.4983	−0.397939
$697$	25.4448	0.963791
$698$	−15.5988	−0.590425
$699$	4.49737	0.170106
$700$	−18.9249	−0.715293
$701$	−16.2545	−0.613926	−0.306963	−	0.951721i	$-0.599313\pi$
$701$	−16.2545	−0.613926	−0.306963	+	0.951721i	$0.599313\pi$
$702$	0	0
$703$	−6.38545	−0.240832
$704$	5.85128	0.220528
$705$	−71.8798	−2.70715
$706$	7.79736	0.293458
$707$	35.0016	1.31637
$708$	−4.25237	−0.159814
$709$	6.42694	0.241369	0.120684	−	0.992691i	$-0.461491\pi$
$709$	6.42694	0.241369	0.120684	+	0.992691i	$0.461491\pi$
$710$	18.2681	0.685591
$711$	−16.4019	−0.615120
$712$	3.27775	0.122839
$713$	27.0443	1.01282
$714$	78.5001	2.93779
$715$	0	0
$716$	21.1209	0.789326
$717$	6.64009	0.247979
$718$	30.1863	1.12654
$719$	30.9196	1.15311	0.576553	−	0.817060i	$-0.304397\pi$
$719$	30.9196	1.15311	0.576553	+	0.817060i	$0.304397\pi$
$720$	18.8805	0.703635
$721$	55.8935	2.08158
$722$	−1.00000	−0.0372161
$723$	38.0739	1.41599
$724$	16.6001	0.616939
$725$	−18.1812	−0.675235
$726$	69.4009	2.57571
$727$	23.5368	0.872931	0.436466	−	0.899721i	$-0.356230\pi$
$727$	23.5368	0.872931	0.436466	+	0.899721i	$0.356230\pi$
$728$	0	0
$729$	−29.0895	−1.07739
$730$	31.7829	1.17634
$731$	39.6323	1.46585
$732$	13.4577	0.497409
$733$	−34.3396	−1.26836	−0.634180	−	0.773185i	$-0.718662\pi$
$733$	−34.3396	−1.26836	−0.634180	+	0.773185i	$0.718662\pi$
$734$	−25.2869	−0.933356
$735$	−60.8459	−2.24433
$736$	−6.79565	−0.250491
$737$	−38.0824	−1.40278
$738$	20.9683	0.771855
$739$	−19.0661	−0.701359	−0.350679	−	0.936496i	$-0.614049\pi$
$739$	−19.0661	−0.701359	−0.350679	+	0.936496i	$0.614049\pi$
$740$	−20.3657	−0.748659
$741$	0	0
$742$	−10.9753	−0.402915
$743$	1.00825	0.0369892	0.0184946	−	0.999829i	$-0.494113\pi$
$743$	1.00825	0.0369892	0.0184946	+	0.999829i	$0.494113\pi$
$744$	11.8856	0.435748
$745$	−66.6771	−2.44286
$746$	−23.6463	−0.865754
$747$	57.1082	2.08948
$748$	−42.0330	−1.53688
$749$	21.8418	0.798083
$750$	1.64079	0.0599133
$751$	13.8359	0.504879	0.252439	−	0.967613i	$-0.418767\pi$
$751$	13.8359	0.504879	0.252439	+	0.967613i	$0.418767\pi$
$752$	7.54609	0.275177
$753$	−79.6494	−2.90259
$754$	0	0
$755$	−65.0731	−2.36825
$756$	31.9065	1.16043
$757$	−31.8934	−1.15918	−0.579592	−	0.814907i	$-0.696788\pi$
$757$	−31.8934	−1.15918	−0.579592	+	0.814907i	$0.696788\pi$
$758$	−17.4455	−0.633648
$759$	−118.757	−4.31060
$760$	−3.18940	−0.115692
$761$	30.9320	1.12128	0.560642	−	0.828059i	$-0.310555\pi$
$761$	30.9320	1.12128	0.560642	+	0.828059i	$0.310555\pi$
$762$	41.4918	1.50309
$763$	−32.2091	−1.16605
$764$	−11.6479	−0.421406
$765$	−135.629	−4.90369
$766$	33.3062	1.20340
$767$	0	0
$768$	−2.98660	−0.107770
$769$	27.5442	0.993270	0.496635	−	0.867960i	$-0.334569\pi$
$769$	27.5442	0.993270	0.496635	+	0.867960i	$0.334569\pi$
$770$	68.2830	2.46075
$771$	−39.1513	−1.41000
$772$	−9.92439	−0.357187
$773$	16.6681	0.599511	0.299756	−	0.954016i	$-0.403095\pi$
$773$	16.6681	0.599511	0.299756	+	0.954016i	$0.403095\pi$
$774$	32.6598	1.17393
$775$	20.5837	0.739390
$776$	−18.8504	−0.676689
$777$	−69.7785	−2.50329
$778$	31.1637	1.11727
$779$	−3.54209	−0.126908
$780$	0	0
$781$	−33.5148	−1.19925
$782$	48.8170	1.74569
$783$	30.6527	1.09544
$784$	6.38772	0.228133
$785$	62.9534	2.24690
$786$	−14.1493	−0.504688
$787$	14.2973	0.509643	0.254821	−	0.966988i	$-0.417983\pi$
$787$	14.2973	0.509643	0.254821	+	0.966988i	$0.417983\pi$
$788$	−4.05502	−0.144454
$789$	−22.2636	−0.792605
$790$	8.83687	0.314402
$791$	48.8665	1.73749
$792$	−34.6382	−1.23082
$793$	0	0
$794$	22.8856	0.812181
$795$	28.5724	1.01336
$796$	2.78997	0.0988880
$797$	25.4711	0.902231	0.451116	−	0.892466i	$-0.351026\pi$
$797$	25.4711	0.902231	0.451116	+	0.892466i	$0.351026\pi$
$798$	−10.9277	−0.386838
$799$	−54.2078	−1.91773
$800$	−5.17225	−0.182867
$801$	−19.4035	−0.685590
$802$	18.2019	0.642732
$803$	−58.3090	−2.05768
$804$	19.4379	0.685523
$805$	−79.3037	−2.79509
$806$	0	0
$807$	46.5085	1.63718
$808$	9.56609	0.336534
$809$	23.8866	0.839809	0.419904	−	0.907568i	$-0.362064\pi$
$809$	23.8866	0.839809	0.419904	+	0.907568i	$0.362064\pi$
$810$	−26.4221	−0.928379
$811$	−6.40769	−0.225005	−0.112502	−	0.993651i	$-0.535887\pi$
$811$	−6.40769	−0.225005	−0.112502	+	0.993651i	$0.535887\pi$
$812$	12.8617	0.451356
$813$	62.6577	2.19750
$814$	37.3630	1.30957
$815$	63.9856	2.24132
$816$	21.4544	0.751055
$817$	−5.51708	−0.193018
$818$	−9.28358	−0.324593
$819$	0	0
$820$	−11.2971	−0.394513
$821$	−38.2723	−1.33571	−0.667857	−	0.744290i	$-0.732788\pi$
$821$	−38.2723	−1.33571	−0.667857	+	0.744290i	$0.732788\pi$
$822$	−2.41778	−0.0843297
$823$	12.0793	0.421058	0.210529	−	0.977588i	$-0.432481\pi$
$823$	12.0793	0.421058	0.210529	+	0.977588i	$0.432481\pi$
$824$	15.2759	0.532162
$825$	−90.3873	−3.14688
$826$	5.20964	0.181267
$827$	24.7905	0.862051	0.431025	−	0.902340i	$-0.358152\pi$
$827$	24.7905	0.862051	0.431025	+	0.902340i	$0.358152\pi$
$828$	40.2287	1.39804
$829$	15.2791	0.530666	0.265333	−	0.964157i	$-0.414518\pi$
$829$	15.2791	0.530666	0.265333	+	0.964157i	$0.414518\pi$
$830$	−30.7682	−1.06798
$831$	−0.800001	−0.0277517
$832$	0	0
$833$	−45.8866	−1.58988
$834$	37.3793	1.29434
$835$	−35.0602	−1.21331
$836$	5.85128	0.202371
$837$	−34.7032	−1.19952
$838$	29.4959	1.01892
$839$	5.50135	0.189928	0.0949639	−	0.995481i	$-0.469726\pi$
$839$	5.50135	0.189928	0.0949639	+	0.995481i	$0.469726\pi$
$840$	−34.8529	−1.20254
$841$	−16.6437	−0.573921
$842$	24.7158	0.851764
$843$	−10.2276	−0.352257
$844$	−8.76395	−0.301668
$845$	0	0
$846$	−44.6711	−1.53582
$847$	−85.0241	−2.92146
$848$	−2.99959	−0.103006
$849$	−2.12472	−0.0729201
$850$	37.1552	1.27441
$851$	−43.3933	−1.48750
$852$	17.1065	0.586060
$853$	−8.55642	−0.292966	−0.146483	−	0.989213i	$-0.546795\pi$
$853$	−8.55642	−0.292966	−0.146483	+	0.989213i	$0.546795\pi$
$854$	−16.4872	−0.564179
$855$	18.8805	0.645699
$856$	5.96947	0.204032
$857$	−12.6462	−0.431986	−0.215993	−	0.976395i	$-0.569299\pi$
$857$	−12.6462	−0.431986	−0.215993	+	0.976395i	$0.569299\pi$
$858$	0	0
$859$	49.3725	1.68457	0.842284	−	0.539034i	$-0.181210\pi$
$859$	49.3725	1.68457	0.842284	+	0.539034i	$0.181210\pi$
$860$	−17.5962	−0.600024
$861$	−38.7070	−1.31913
$862$	10.9761	0.373848
$863$	−21.9233	−0.746279	−0.373140	−	0.927775i	$-0.621719\pi$
$863$	−21.9233	−0.746279	−0.373140	+	0.927775i	$0.621719\pi$
$864$	8.72018	0.296666
$865$	26.9995	0.918010
$866$	10.2075	0.346864
$867$	−103.347	−3.50985
$868$	−14.5612	−0.494240
$869$	−16.2122	−0.549960
$870$	−33.4834	−1.13519
$871$	0	0
$872$	−8.80290	−0.298104
$873$	111.590	3.77675
$874$	−6.79565	−0.229866
$875$	−2.01016	−0.0679558
$876$	29.7620	1.00556
$877$	3.28264	0.110847	0.0554235	−	0.998463i	$-0.482349\pi$
$877$	3.28264	0.110847	0.0554235	+	0.998463i	$0.482349\pi$
$878$	0.964016	0.0325340
$879$	−11.2113	−0.378148
$880$	18.6620	0.629098
$881$	32.2801	1.08754	0.543771	−	0.839233i	$-0.316996\pi$
$881$	32.2801	1.08754	0.543771	+	0.839233i	$0.316996\pi$
$882$	−37.8138	−1.27326
$883$	1.16032	0.0390478	0.0195239	−	0.999809i	$-0.493785\pi$
$883$	1.16032	0.0390478	0.0195239	+	0.999809i	$0.493785\pi$
$884$	0	0
$885$	−13.5625	−0.455898
$886$	3.07930	0.103451
$887$	32.8863	1.10421	0.552107	−	0.833773i	$-0.313824\pi$
$887$	32.8863	1.10421	0.552107	+	0.833773i	$0.313824\pi$
$888$	−19.0708	−0.639973
$889$	−50.8322	−1.70486
$890$	10.4541	0.350421
$891$	48.4741	1.62394
$892$	5.82332	0.194979
$893$	7.54609	0.252520
$894$	−62.4374	−2.08822
$895$	67.3630	2.25170
$896$	3.65892	0.122236
$897$	0	0
$898$	−37.7506	−1.25975
$899$	−13.9891	−0.466561
$900$	30.6185	1.02062
$901$	21.5477	0.717859
$902$	20.7257	0.690091
$903$	−60.2892	−2.00630
$904$	13.3554	0.444195
$905$	52.9444	1.75993
$906$	−60.9354	−2.02444
$907$	21.6289	0.718176	0.359088	−	0.933304i	$-0.383088\pi$
$907$	21.6289	0.718176	0.359088	+	0.933304i	$0.383088\pi$
$908$	−16.9850	−0.563668
$909$	−56.6291	−1.87827
$910$	0	0
$911$	−31.9276	−1.05781	−0.528905	−	0.848681i	$-0.677397\pi$
$911$	−31.9276	−1.05781	−0.528905	+	0.848681i	$0.677397\pi$
$912$	−2.98660	−0.0988962
$913$	56.4475	1.86814
$914$	35.5399	1.17555
$915$	42.9218	1.41895
$916$	7.28230	0.240614
$917$	17.3345	0.572434
$918$	−62.6420	−2.06749
$919$	1.93503	0.0638307	0.0319154	−	0.999491i	$-0.489839\pi$
$919$	1.93503	0.0638307	0.0319154	+	0.999491i	$0.489839\pi$
$920$	−21.6740	−0.714572
$921$	12.1904	0.401689
$922$	−26.2650	−0.864993
$923$	0	0
$924$	63.9412	2.10351
$925$	−33.0272	−1.08593
$926$	−21.0497	−0.691736
$927$	−90.4300	−2.97011
$928$	3.51515	0.115390
$929$	−31.0649	−1.01921	−0.509603	−	0.860410i	$-0.670208\pi$
$929$	−31.0649	−1.01921	−0.509603	+	0.860410i	$0.670208\pi$
$930$	37.9079	1.24305
$931$	6.38772	0.209349
$932$	−1.50585	−0.0493257
$933$	−61.3982	−2.01009
$934$	−0.843884	−0.0276127
$935$	−134.060	−4.38423
$936$	0	0
$937$	−21.6793	−0.708230	−0.354115	−	0.935202i	$-0.615218\pi$
$937$	−21.6793	−0.708230	−0.354115	+	0.935202i	$0.615218\pi$
$938$	−23.8137	−0.777544
$939$	79.8891	2.60708
$940$	24.0675	0.784994
$941$	−10.7927	−0.351832	−0.175916	−	0.984405i	$-0.556289\pi$
$941$	−10.7927	−0.351832	−0.175916	+	0.984405i	$0.556289\pi$
$942$	58.9505	1.92071
$943$	−24.0708	−0.783853
$944$	1.42382	0.0463413
$945$	101.762	3.31033
$946$	32.2820	1.04958
$947$	−4.44065	−0.144302	−0.0721509	−	0.997394i	$-0.522986\pi$
$947$	−4.44065	−0.144302	−0.0721509	+	0.997394i	$0.522986\pi$
$948$	8.27498	0.268759
$949$	0	0
$950$	−5.17225	−0.167810
$951$	−32.9282	−1.06777
$952$	−26.2841	−0.851873
$953$	53.9153	1.74649	0.873244	−	0.487284i	$-0.162012\pi$
$953$	53.9153	1.74649	0.873244	+	0.487284i	$0.162012\pi$
$954$	17.7569	0.574900
$955$	−37.1497	−1.20214
$956$	−2.22330	−0.0719065
$957$	61.4287	1.98571
$958$	−7.66789	−0.247738
$959$	2.96205	0.0956497
$960$	−9.52545	−0.307433
$961$	−15.1624	−0.489110
$962$	0	0
$963$	−35.3379	−1.13875
$964$	−12.7483	−0.410594
$965$	−31.6528	−1.01894
$966$	−74.2611	−2.38931
$967$	18.0909	0.581766	0.290883	−	0.956759i	$-0.406051\pi$
$967$	18.0909	0.581766	0.290883	+	0.956759i	$0.406051\pi$
$968$	−23.2375	−0.746880
$969$	21.4544	0.689215
$970$	−60.1214	−1.93038
$971$	−16.9440	−0.543760	−0.271880	−	0.962331i	$-0.587645\pi$
$971$	−16.9440	−0.543760	−0.271880	+	0.962331i	$0.587645\pi$
$972$	1.41848	0.0454976
$973$	−45.7938	−1.46808
$974$	28.6932	0.919391
$975$	0	0
$976$	−4.50601	−0.144234
$977$	19.3019	0.617523	0.308761	−	0.951140i	$-0.400086\pi$
$977$	19.3019	0.617523	0.308761	+	0.951140i	$0.400086\pi$
$978$	59.9171	1.91594
$979$	−19.1790	−0.612965
$980$	20.3730	0.650791
$981$	52.1111	1.66378
$982$	14.5914	0.465629
$983$	33.8881	1.08086	0.540432	−	0.841388i	$-0.318261\pi$
$983$	33.8881	1.08086	0.540432	+	0.841388i	$0.318261\pi$
$984$	−10.5788	−0.337240
$985$	−12.9331	−0.412082
$986$	−25.2513	−0.804165
$987$	82.4616	2.62478
$988$	0	0
$989$	−37.4922	−1.19218
$990$	−110.475	−3.51113
$991$	0.966118	0.0306897	0.0153449	−	0.999882i	$-0.495115\pi$
$991$	0.966118	0.0306897	0.0153449	+	0.999882i	$0.495115\pi$
$992$	−3.97965	−0.126354
$993$	67.0339	2.12726
$994$	−20.9575	−0.664730
$995$	8.89833	0.282096
$996$	−28.8118	−0.912937
$997$	−36.3987	−1.15276	−0.576380	−	0.817182i	$-0.695535\pi$
$997$	−36.3987	−1.15276	−0.576380	+	0.817182i	$0.695535\pi$
$998$	38.1451	1.20746
$999$	55.6822	1.76171

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bo.1.1	✓	15
13.12	even	2		6422.2.a.bq.1.1	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.bo.1.1	✓	15	1.1	even	1	trivial
6422.2.a.bq.1.1	yes	15	13.12	even	2

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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.98660 q^{3} +1.00000 q^{4} +3.18940 q^{5} +2.98660 q^{6} -3.65892 q^{7} -1.00000 q^{8} +5.91977 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.98660 q^{3} +1.00000 q^{4} +3.18940 q^{5} +2.98660 q^{6} -3.65892 q^{7} -1.00000 q^{8} +5.91977 q^{9} -3.18940 q^{10} +5.85128 q^{11} -2.98660 q^{12} +3.65892 q^{14} -9.52545 q^{15} +1.00000 q^{16} -7.18356 q^{17} -5.91977 q^{18} +1.00000 q^{19} +3.18940 q^{20} +10.9277 q^{21} -5.85128 q^{22} +6.79565 q^{23} +2.98660 q^{24} +5.17225 q^{25} -8.72018 q^{27} -3.65892 q^{28} -3.51515 q^{29} +9.52545 q^{30} +3.97965 q^{31} -1.00000 q^{32} -17.4754 q^{33} +7.18356 q^{34} -11.6698 q^{35} +5.91977 q^{36} -6.38545 q^{37} -1.00000 q^{38} -3.18940 q^{40} -3.54209 q^{41} -10.9277 q^{42} -5.51708 q^{43} +5.85128 q^{44} +18.8805 q^{45} -6.79565 q^{46} +7.54609 q^{47} -2.98660 q^{48} +6.38772 q^{49} -5.17225 q^{50} +21.4544 q^{51} -2.99959 q^{53} +8.72018 q^{54} +18.6620 q^{55} +3.65892 q^{56} -2.98660 q^{57} +3.51515 q^{58} +1.42382 q^{59} -9.52545 q^{60} -4.50601 q^{61} -3.97965 q^{62} -21.6600 q^{63} +1.00000 q^{64} +17.4754 q^{66} -6.50838 q^{67} -7.18356 q^{68} -20.2959 q^{69} +11.6698 q^{70} -5.72777 q^{71} -5.91977 q^{72} -9.96517 q^{73} +6.38545 q^{74} -15.4474 q^{75} +1.00000 q^{76} -21.4094 q^{77} -2.77070 q^{79} +3.18940 q^{80} +8.28436 q^{81} +3.54209 q^{82} +9.64703 q^{83} +10.9277 q^{84} -22.9112 q^{85} +5.51708 q^{86} +10.4983 q^{87} -5.85128 q^{88} -3.27775 q^{89} -18.8805 q^{90} +6.79565 q^{92} -11.8856 q^{93} -7.54609 q^{94} +3.18940 q^{95} +2.98660 q^{96} +18.8504 q^{97} -6.38772 q^{98} +34.6382 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 15 q^{2} + 15 q^{4} - q^{5} - 18 q^{7} - 15 q^{8} + 17 q^{9}+O(q^{10}) \) 15 * q - 15 * q^2 + 15 * q^4 - q^5 - 18 * q^7 - 15 * q^8 + 17 * q^9	\( 15 q - 15 q^{2} + 15 q^{4} - q^{5} - 18 q^{7} - 15 q^{8} + 17 q^{9} + q^{10} - 4 q^{11} + 18 q^{14} - 23 q^{15} + 15 q^{16} + 2 q^{17} - 17 q^{18} + 15 q^{19} - q^{20} - 2 q^{21} + 4 q^{22} + 17 q^{23} + 8 q^{25} + 12 q^{27} - 18 q^{28} - 20 q^{29} + 23 q^{30} - 30 q^{31} - 15 q^{32} - 36 q^{33} - 2 q^{34} + 32 q^{35} + 17 q^{36} - 35 q^{37} - 15 q^{38} + q^{40} - 15 q^{41} + 2 q^{42} + q^{43} - 4 q^{44} + 11 q^{45} - 17 q^{46} + 29 q^{49} - 8 q^{50} - q^{51} - q^{53} - 12 q^{54} - 6 q^{55} + 18 q^{56} + 20 q^{58} + 7 q^{59} - 23 q^{60} - 2 q^{61} + 30 q^{62} - 42 q^{63} + 15 q^{64} + 36 q^{66} - 34 q^{67} + 2 q^{68} - 12 q^{69} - 32 q^{70} - 4 q^{71} - 17 q^{72} - 12 q^{73} + 35 q^{74} + 31 q^{75} + 15 q^{76} - 20 q^{77} + 23 q^{79} - q^{80} + 7 q^{81} + 15 q^{82} + 3 q^{83} - 2 q^{84} - 46 q^{85} - q^{86} + 22 q^{87} + 4 q^{88} - 17 q^{89} - 11 q^{90} + 17 q^{92} - 60 q^{93} - q^{95} - 18 q^{97} - 29 q^{98} + 6 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 + 15 * q^4 - q^5 - 18 * q^7 - 15 * q^8 + 17 * q^9 + q^10 - 4 * q^11 + 18 * q^14 - 23 * q^15 + 15 * q^16 + 2 * q^17 - 17 * q^18 + 15 * q^19 - q^20 - 2 * q^21 + 4 * q^22 + 17 * q^23 + 8 * q^25 + 12 * q^27 - 18 * q^28 - 20 * q^29 + 23 * q^30 - 30 * q^31 - 15 * q^32 - 36 * q^33 - 2 * q^34 + 32 * q^35 + 17 * q^36 - 35 * q^37 - 15 * q^38 + q^40 - 15 * q^41 + 2 * q^42 + q^43 - 4 * q^44 + 11 * q^45 - 17 * q^46 + 29 * q^49 - 8 * q^50 - q^51 - q^53 - 12 * q^54 - 6 * q^55 + 18 * q^56 + 20 * q^58 + 7 * q^59 - 23 * q^60 - 2 * q^61 + 30 * q^62 - 42 * q^63 + 15 * q^64 + 36 * q^66 - 34 * q^67 + 2 * q^68 - 12 * q^69 - 32 * q^70 - 4 * q^71 - 17 * q^72 - 12 * q^73 + 35 * q^74 + 31 * q^75 + 15 * q^76 - 20 * q^77 + 23 * q^79 - q^80 + 7 * q^81 + 15 * q^82 + 3 * q^83 - 2 * q^84 - 46 * q^85 - q^86 + 22 * q^87 + 4 * q^88 - 17 * q^89 - 11 * q^90 + 17 * q^92 - 60 * q^93 - q^95 - 18 * q^97 - 29 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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