LMFDB - Embedded newform 322.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [322,2,Mod(1,322)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(322, 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("322.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 322 = 2 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	322.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:	$2.57118294509$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	322.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.23607 q^{3} +1.00000 q^{4} +1.23607 q^{5} +3.23607 q^{6} -1.00000 q^{7} -1.00000 q^{8} +7.47214 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.23607 q^{3} +1.00000 q^{4} +1.23607 q^{5} +3.23607 q^{6} -1.00000 q^{7} -1.00000 q^{8} +7.47214 q^{9} -1.23607 q^{10} -3.23607 q^{12} +2.47214 q^{13} +1.00000 q^{14} -4.00000 q^{15} +1.00000 q^{16} -7.23607 q^{17} -7.47214 q^{18} -4.47214 q^{19} +1.23607 q^{20} +3.23607 q^{21} -1.00000 q^{23} +3.23607 q^{24} -3.47214 q^{25} -2.47214 q^{26} -14.4721 q^{27} -1.00000 q^{28} -2.00000 q^{29} +4.00000 q^{30} -1.23607 q^{31} -1.00000 q^{32} +7.23607 q^{34} -1.23607 q^{35} +7.47214 q^{36} +6.00000 q^{37} +4.47214 q^{38} -8.00000 q^{39} -1.23607 q^{40} -10.0000 q^{41} -3.23607 q^{42} -6.47214 q^{43} +9.23607 q^{45} +1.00000 q^{46} -6.76393 q^{47} -3.23607 q^{48} +1.00000 q^{49} +3.47214 q^{50} +23.4164 q^{51} +2.47214 q^{52} -6.00000 q^{53} +14.4721 q^{54} +1.00000 q^{56} +14.4721 q^{57} +2.00000 q^{58} +11.2361 q^{59} -4.00000 q^{60} +11.7082 q^{61} +1.23607 q^{62} -7.47214 q^{63} +1.00000 q^{64} +3.05573 q^{65} +4.00000 q^{67} -7.23607 q^{68} +3.23607 q^{69} +1.23607 q^{70} -2.47214 q^{71} -7.47214 q^{72} -13.4164 q^{73} -6.00000 q^{74} +11.2361 q^{75} -4.47214 q^{76} +8.00000 q^{78} +8.94427 q^{79} +1.23607 q^{80} +24.4164 q^{81} +10.0000 q^{82} -6.94427 q^{83} +3.23607 q^{84} -8.94427 q^{85} +6.47214 q^{86} +6.47214 q^{87} -16.1803 q^{89} -9.23607 q^{90} -2.47214 q^{91} -1.00000 q^{92} +4.00000 q^{93} +6.76393 q^{94} -5.52786 q^{95} +3.23607 q^{96} +8.18034 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - 2 * q^7 - 2 * q^8 + 6 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 6 q^{9} + 2 q^{10} - 2 q^{12} - 4 q^{13} + 2 q^{14} - 8 q^{15} + 2 q^{16} - 10 q^{17} - 6 q^{18} - 2 q^{20} + 2 q^{21} - 2 q^{23} + 2 q^{24} + 2 q^{25} + 4 q^{26} - 20 q^{27} - 2 q^{28} - 4 q^{29} + 8 q^{30} + 2 q^{31} - 2 q^{32} + 10 q^{34} + 2 q^{35} + 6 q^{36} + 12 q^{37} - 16 q^{39} + 2 q^{40} - 20 q^{41} - 2 q^{42} - 4 q^{43} + 14 q^{45} + 2 q^{46} - 18 q^{47} - 2 q^{48} + 2 q^{49} - 2 q^{50} + 20 q^{51} - 4 q^{52} - 12 q^{53} + 20 q^{54} + 2 q^{56} + 20 q^{57} + 4 q^{58} + 18 q^{59} - 8 q^{60} + 10 q^{61} - 2 q^{62} - 6 q^{63} + 2 q^{64} + 24 q^{65} + 8 q^{67} - 10 q^{68} + 2 q^{69} - 2 q^{70} + 4 q^{71} - 6 q^{72} - 12 q^{74} + 18 q^{75} + 16 q^{78} - 2 q^{80} + 22 q^{81} + 20 q^{82} + 4 q^{83} + 2 q^{84} + 4 q^{86} + 4 q^{87} - 10 q^{89} - 14 q^{90} + 4 q^{91} - 2 q^{92} + 8 q^{93} + 18 q^{94} - 20 q^{95} + 2 q^{96} - 6 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - 2 * q^7 - 2 * q^8 + 6 * q^9 + 2 * q^10 - 2 * q^12 - 4 * q^13 + 2 * q^14 - 8 * q^15 + 2 * q^16 - 10 * q^17 - 6 * q^18 - 2 * q^20 + 2 * q^21 - 2 * q^23 + 2 * q^24 + 2 * q^25 + 4 * q^26 - 20 * q^27 - 2 * q^28 - 4 * q^29 + 8 * q^30 + 2 * q^31 - 2 * q^32 + 10 * q^34 + 2 * q^35 + 6 * q^36 + 12 * q^37 - 16 * q^39 + 2 * q^40 - 20 * q^41 - 2 * q^42 - 4 * q^43 + 14 * q^45 + 2 * q^46 - 18 * q^47 - 2 * q^48 + 2 * q^49 - 2 * q^50 + 20 * q^51 - 4 * q^52 - 12 * q^53 + 20 * q^54 + 2 * q^56 + 20 * q^57 + 4 * q^58 + 18 * q^59 - 8 * q^60 + 10 * q^61 - 2 * q^62 - 6 * q^63 + 2 * q^64 + 24 * q^65 + 8 * q^67 - 10 * q^68 + 2 * q^69 - 2 * q^70 + 4 * q^71 - 6 * q^72 - 12 * q^74 + 18 * q^75 + 16 * q^78 - 2 * q^80 + 22 * q^81 + 20 * q^82 + 4 * q^83 + 2 * q^84 + 4 * q^86 + 4 * q^87 - 10 * q^89 - 14 * q^90 + 4 * q^91 - 2 * q^92 + 8 * q^93 + 18 * q^94 - 20 * q^95 + 2 * q^96 - 6 * q^97 - 2 * q^98

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
$3$	−3.23607	−1.86834	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	−3.23607	−1.86834	−0.934172	+	0.356822i	$0.883860\pi$
$4$	1.00000	0.500000
$5$	1.23607	0.552786	0.276393	−	0.961045i	$-0.410861\pi$
$5$	1.23607	0.552786	0.276393	+	0.961045i	$0.410861\pi$
$6$	3.23607	1.32112
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	7.47214	2.49071
$10$	−1.23607	−0.390879
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−3.23607	−0.934172
$13$	2.47214	0.685647	0.342824	−	0.939400i	$-0.388617\pi$
$13$	2.47214	0.685647	0.342824	+	0.939400i	$0.388617\pi$
$14$	1.00000	0.267261
$15$	−4.00000	−1.03280
$16$	1.00000	0.250000
$17$	−7.23607	−1.75500	−0.877502	−	0.479573i	$-0.840792\pi$
$17$	−7.23607	−1.75500	−0.877502	+	0.479573i	$0.840792\pi$
$18$	−7.47214	−1.76120
$19$	−4.47214	−1.02598	−0.512989	−	0.858395i	$-0.671462\pi$
$19$	−4.47214	−1.02598	−0.512989	+	0.858395i	$0.671462\pi$
$20$	1.23607	0.276393
$21$	3.23607	0.706168
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	3.23607	0.660560
$25$	−3.47214	−0.694427
$26$	−2.47214	−0.484826
$27$	−14.4721	−2.78516
$28$	−1.00000	−0.188982
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	4.00000	0.730297
$31$	−1.23607	−0.222004	−0.111002	−	0.993820i	$-0.535406\pi$
$31$	−1.23607	−0.222004	−0.111002	+	0.993820i	$0.535406\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	7.23607	1.24098
$35$	−1.23607	−0.208934
$36$	7.47214	1.24536
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	4.47214	0.725476
$39$	−8.00000	−1.28103
$40$	−1.23607	−0.195440
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	−3.23607	−0.499336
$43$	−6.47214	−0.986991	−0.493496	−	0.869748i	$-0.664281\pi$
$43$	−6.47214	−0.986991	−0.493496	+	0.869748i	$0.664281\pi$
$44$	0	0
$45$	9.23607	1.37683
$46$	1.00000	0.147442
$47$	−6.76393	−0.986621	−0.493310	−	0.869853i	$-0.664213\pi$
$47$	−6.76393	−0.986621	−0.493310	+	0.869853i	$0.664213\pi$
$48$	−3.23607	−0.467086
$49$	1.00000	0.142857
$50$	3.47214	0.491034
$51$	23.4164	3.27895
$52$	2.47214	0.342824
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	14.4721	1.96941
$55$	0	0
$56$	1.00000	0.133631
$57$	14.4721	1.91688
$58$	2.00000	0.262613
$59$	11.2361	1.46281	0.731406	−	0.681943i	$-0.238865\pi$
$59$	11.2361	1.46281	0.731406	+	0.681943i	$0.238865\pi$
$60$	−4.00000	−0.516398
$61$	11.7082	1.49908	0.749541	−	0.661958i	$-0.230274\pi$
$61$	11.7082	1.49908	0.749541	+	0.661958i	$0.230274\pi$
$62$	1.23607	0.156981
$63$	−7.47214	−0.941401
$64$	1.00000	0.125000
$65$	3.05573	0.379016
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−7.23607	−0.877502
$69$	3.23607	0.389577
$70$	1.23607	0.147738
$71$	−2.47214	−0.293389	−0.146694	−	0.989182i	$-0.546863\pi$
$71$	−2.47214	−0.293389	−0.146694	+	0.989182i	$0.546863\pi$
$72$	−7.47214	−0.880600
$73$	−13.4164	−1.57027	−0.785136	−	0.619324i	$-0.787407\pi$
$73$	−13.4164	−1.57027	−0.785136	+	0.619324i	$0.787407\pi$
$74$	−6.00000	−0.697486
$75$	11.2361	1.29743
$76$	−4.47214	−0.512989
$77$	0	0
$78$	8.00000	0.905822
$79$	8.94427	1.00631	0.503155	−	0.864196i	$-0.332173\pi$
$79$	8.94427	1.00631	0.503155	+	0.864196i	$0.332173\pi$
$80$	1.23607	0.138197
$81$	24.4164	2.71293
$82$	10.0000	1.10432
$83$	−6.94427	−0.762233	−0.381116	−	0.924527i	$-0.624460\pi$
$83$	−6.94427	−0.762233	−0.381116	+	0.924527i	$0.624460\pi$
$84$	3.23607	0.353084
$85$	−8.94427	−0.970143
$86$	6.47214	0.697908
$87$	6.47214	0.693886
$88$	0	0
$89$	−16.1803	−1.71511	−0.857556	−	0.514390i	$-0.828018\pi$
$89$	−16.1803	−1.71511	−0.857556	+	0.514390i	$0.828018\pi$
$90$	−9.23607	−0.973567
$91$	−2.47214	−0.259150
$92$	−1.00000	−0.104257
$93$	4.00000	0.414781
$94$	6.76393	0.697646
$95$	−5.52786	−0.567147
$96$	3.23607	0.330280
$97$	8.18034	0.830588	0.415294	−	0.909687i	$-0.363679\pi$
$97$	8.18034	0.830588	0.415294	+	0.909687i	$0.363679\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−3.47214	−0.347214
$101$	−14.4721	−1.44003	−0.720016	−	0.693958i	$-0.755865\pi$
$101$	−14.4721	−1.44003	−0.720016	+	0.693958i	$0.755865\pi$
$102$	−23.4164	−2.31857
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−2.47214	−0.242413
$105$	4.00000	0.390360
$106$	6.00000	0.582772
$107$	−10.4721	−1.01238	−0.506190	−	0.862422i	$-0.668946\pi$
$107$	−10.4721	−1.01238	−0.506190	+	0.862422i	$0.668946\pi$
$108$	−14.4721	−1.39258
$109$	3.52786	0.337908	0.168954	−	0.985624i	$-0.445961\pi$
$109$	3.52786	0.337908	0.168954	+	0.985624i	$0.445961\pi$
$110$	0	0
$111$	−19.4164	−1.84292
$112$	−1.00000	−0.0944911
$113$	3.52786	0.331874	0.165937	−	0.986136i	$-0.446935\pi$
$113$	3.52786	0.331874	0.165937	+	0.986136i	$0.446935\pi$
$114$	−14.4721	−1.35544
$115$	−1.23607	−0.115264
$116$	−2.00000	−0.185695
$117$	18.4721	1.70775
$118$	−11.2361	−1.03436
$119$	7.23607	0.663329
$120$	4.00000	0.365148
$121$	−11.0000	−1.00000
$122$	−11.7082	−1.06001
$123$	32.3607	2.91786
$124$	−1.23607	−0.111002
$125$	−10.4721	−0.936656
$126$	7.47214	0.665671
$127$	7.41641	0.658100	0.329050	−	0.944313i	$-0.393272\pi$
$127$	7.41641	0.658100	0.329050	+	0.944313i	$0.393272\pi$
$128$	−1.00000	−0.0883883
$129$	20.9443	1.84404
$130$	−3.05573	−0.268005
$131$	−12.7639	−1.11519	−0.557595	−	0.830113i	$-0.688276\pi$
$131$	−12.7639	−1.11519	−0.557595	+	0.830113i	$0.688276\pi$
$132$	0	0
$133$	4.47214	0.387783
$134$	−4.00000	−0.345547
$135$	−17.8885	−1.53960
$136$	7.23607	0.620488
$137$	17.4164	1.48798	0.743992	−	0.668188i	$-0.232930\pi$
$137$	17.4164	1.48798	0.743992	+	0.668188i	$0.232930\pi$
$138$	−3.23607	−0.275472
$139$	17.1246	1.45249	0.726245	−	0.687436i	$-0.241264\pi$
$139$	17.1246	1.45249	0.726245	+	0.687436i	$0.241264\pi$
$140$	−1.23607	−0.104467
$141$	21.8885	1.84335
$142$	2.47214	0.207457
$143$	0	0
$144$	7.47214	0.622678
$145$	−2.47214	−0.205300
$146$	13.4164	1.11035
$147$	−3.23607	−0.266906
$148$	6.00000	0.493197
$149$	−5.41641	−0.443729	−0.221865	−	0.975077i	$-0.571214\pi$
$149$	−5.41641	−0.443729	−0.221865	+	0.975077i	$0.571214\pi$
$150$	−11.2361	−0.917421
$151$	−18.4721	−1.50324	−0.751621	−	0.659596i	$-0.770728\pi$
$151$	−18.4721	−1.50324	−0.751621	+	0.659596i	$0.770728\pi$
$152$	4.47214	0.362738
$153$	−54.0689	−4.37121
$154$	0	0
$155$	−1.52786	−0.122721
$156$	−8.00000	−0.640513
$157$	12.6525	1.00978	0.504889	−	0.863184i	$-0.331534\pi$
$157$	12.6525	1.00978	0.504889	+	0.863184i	$0.331534\pi$
$158$	−8.94427	−0.711568
$159$	19.4164	1.53982
$160$	−1.23607	−0.0977198
$161$	1.00000	0.0788110
$162$	−24.4164	−1.91833
$163$	2.47214	0.193633	0.0968163	−	0.995302i	$-0.469134\pi$
$163$	2.47214	0.193633	0.0968163	+	0.995302i	$0.469134\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	6.94427	0.538980
$167$	24.6525	1.90767	0.953833	−	0.300338i	$-0.0970994\pi$
$167$	24.6525	1.90767	0.953833	+	0.300338i	$0.0970994\pi$
$168$	−3.23607	−0.249668
$169$	−6.88854	−0.529888
$170$	8.94427	0.685994
$171$	−33.4164	−2.55542
$172$	−6.47214	−0.493496
$173$	20.9443	1.59236	0.796182	−	0.605058i	$-0.206850\pi$
$173$	20.9443	1.59236	0.796182	+	0.605058i	$0.206850\pi$
$174$	−6.47214	−0.490651
$175$	3.47214	0.262469
$176$	0	0
$177$	−36.3607	−2.73304
$178$	16.1803	1.21277
$179$	−0.944272	−0.0705782	−0.0352891	−	0.999377i	$-0.511235\pi$
$179$	−0.944272	−0.0705782	−0.0352891	+	0.999377i	$0.511235\pi$
$180$	9.23607	0.688416
$181$	13.2361	0.983829	0.491915	−	0.870643i	$-0.336297\pi$
$181$	13.2361	0.983829	0.491915	+	0.870643i	$0.336297\pi$
$182$	2.47214	0.183247
$183$	−37.8885	−2.80080
$184$	1.00000	0.0737210
$185$	7.41641	0.545265
$186$	−4.00000	−0.293294
$187$	0	0
$188$	−6.76393	−0.493310
$189$	14.4721	1.05269
$190$	5.52786	0.401033
$191$	20.9443	1.51547	0.757737	−	0.652560i	$-0.226305\pi$
$191$	20.9443	1.51547	0.757737	+	0.652560i	$0.226305\pi$
$192$	−3.23607	−0.233543
$193$	−22.9443	−1.65156	−0.825782	−	0.563989i	$-0.809266\pi$
$193$	−22.9443	−1.65156	−0.825782	+	0.563989i	$0.809266\pi$
$194$	−8.18034	−0.587314
$195$	−9.88854	−0.708133
$196$	1.00000	0.0714286
$197$	15.8885	1.13201	0.566006	−	0.824401i	$-0.308488\pi$
$197$	15.8885	1.13201	0.566006	+	0.824401i	$0.308488\pi$
$198$	0	0
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	3.47214	0.245517
$201$	−12.9443	−0.913019
$202$	14.4721	1.01826
$203$	2.00000	0.140372
$204$	23.4164	1.63948
$205$	−12.3607	−0.863307
$206$	4.00000	0.278693
$207$	−7.47214	−0.519349
$208$	2.47214	0.171412
$209$	0	0
$210$	−4.00000	−0.276026
$211$	−0.944272	−0.0650064	−0.0325032	−	0.999472i	$-0.510348\pi$
$211$	−0.944272	−0.0650064	−0.0325032	+	0.999472i	$0.510348\pi$
$212$	−6.00000	−0.412082
$213$	8.00000	0.548151
$214$	10.4721	0.715860
$215$	−8.00000	−0.545595
$216$	14.4721	0.984704
$217$	1.23607	0.0839098
$218$	−3.52786	−0.238937
$219$	43.4164	2.93381
$220$	0	0
$221$	−17.8885	−1.20331
$222$	19.4164	1.30314
$223$	20.6525	1.38299	0.691496	−	0.722380i	$-0.256952\pi$
$223$	20.6525	1.38299	0.691496	+	0.722380i	$0.256952\pi$
$224$	1.00000	0.0668153
$225$	−25.9443	−1.72962
$226$	−3.52786	−0.234670
$227$	5.41641	0.359500	0.179750	−	0.983712i	$-0.442471\pi$
$227$	5.41641	0.359500	0.179750	+	0.983712i	$0.442471\pi$
$228$	14.4721	0.958441
$229$	−10.7639	−0.711301	−0.355650	−	0.934619i	$-0.615741\pi$
$229$	−10.7639	−0.711301	−0.355650	+	0.934619i	$0.615741\pi$
$230$	1.23607	0.0815039
$231$	0	0
$232$	2.00000	0.131306
$233$	−7.52786	−0.493167	−0.246583	−	0.969122i	$-0.579308\pi$
$233$	−7.52786	−0.493167	−0.246583	+	0.969122i	$0.579308\pi$
$234$	−18.4721	−1.20756
$235$	−8.36068	−0.545391
$236$	11.2361	0.731406
$237$	−28.9443	−1.88013
$238$	−7.23607	−0.469045
$239$	−2.47214	−0.159909	−0.0799546	−	0.996799i	$-0.525478\pi$
$239$	−2.47214	−0.159909	−0.0799546	+	0.996799i	$0.525478\pi$
$240$	−4.00000	−0.258199
$241$	13.7082	0.883023	0.441512	−	0.897256i	$-0.354442\pi$
$241$	13.7082	0.883023	0.441512	+	0.897256i	$0.354442\pi$
$242$	11.0000	0.707107
$243$	−35.5967	−2.28353
$244$	11.7082	0.749541
$245$	1.23607	0.0789695
$246$	−32.3607	−2.06324
$247$	−11.0557	−0.703459
$248$	1.23607	0.0784904
$249$	22.4721	1.42411
$250$	10.4721	0.662316
$251$	4.47214	0.282279	0.141139	−	0.989990i	$-0.454923\pi$
$251$	4.47214	0.282279	0.141139	+	0.989990i	$0.454923\pi$
$252$	−7.47214	−0.470700
$253$	0	0
$254$	−7.41641	−0.465347
$255$	28.9443	1.81256
$256$	1.00000	0.0625000
$257$	−9.05573	−0.564881	−0.282440	−	0.959285i	$-0.591144\pi$
$257$	−9.05573	−0.564881	−0.282440	+	0.959285i	$0.591144\pi$
$258$	−20.9443	−1.30393
$259$	−6.00000	−0.372822
$260$	3.05573	0.189508
$261$	−14.9443	−0.925027
$262$	12.7639	0.788558
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	−7.41641	−0.455586
$266$	−4.47214	−0.274204
$267$	52.3607	3.20442
$268$	4.00000	0.244339
$269$	20.9443	1.27699	0.638497	−	0.769624i	$-0.279556\pi$
$269$	20.9443	1.27699	0.638497	+	0.769624i	$0.279556\pi$
$270$	17.8885	1.08866
$271$	−10.7639	−0.653862	−0.326931	−	0.945048i	$-0.606015\pi$
$271$	−10.7639	−0.653862	−0.326931	+	0.945048i	$0.606015\pi$
$272$	−7.23607	−0.438751
$273$	8.00000	0.484182
$274$	−17.4164	−1.05216
$275$	0	0
$276$	3.23607	0.194788
$277$	−15.8885	−0.954650	−0.477325	−	0.878727i	$-0.658394\pi$
$277$	−15.8885	−0.954650	−0.477325	+	0.878727i	$0.658394\pi$
$278$	−17.1246	−1.02707
$279$	−9.23607	−0.552949
$280$	1.23607	0.0738692
$281$	−21.4164	−1.27760	−0.638798	−	0.769375i	$-0.720568\pi$
$281$	−21.4164	−1.27760	−0.638798	+	0.769375i	$0.720568\pi$
$282$	−21.8885	−1.30344
$283$	−4.47214	−0.265841	−0.132920	−	0.991127i	$-0.542435\pi$
$283$	−4.47214	−0.265841	−0.132920	+	0.991127i	$0.542435\pi$
$284$	−2.47214	−0.146694
$285$	17.8885	1.05963
$286$	0	0
$287$	10.0000	0.590281
$288$	−7.47214	−0.440300
$289$	35.3607	2.08004
$290$	2.47214	0.145169
$291$	−26.4721	−1.55182
$292$	−13.4164	−0.785136
$293$	11.1246	0.649907	0.324953	−	0.945730i	$-0.394651\pi$
$293$	11.1246	0.649907	0.324953	+	0.945730i	$0.394651\pi$
$294$	3.23607	0.188731
$295$	13.8885	0.808622
$296$	−6.00000	−0.348743
$297$	0	0
$298$	5.41641	0.313764
$299$	−2.47214	−0.142967
$300$	11.2361	0.648715
$301$	6.47214	0.373048
$302$	18.4721	1.06295
$303$	46.8328	2.69047
$304$	−4.47214	−0.256495
$305$	14.4721	0.828672
$306$	54.0689	3.09091
$307$	24.7639	1.41335	0.706676	−	0.707537i	$-0.250194\pi$
$307$	24.7639	1.41335	0.706676	+	0.707537i	$0.250194\pi$
$308$	0	0
$309$	12.9443	0.736374
$310$	1.52786	0.0867768
$311$	−15.1246	−0.857638	−0.428819	−	0.903390i	$-0.641070\pi$
$311$	−15.1246	−0.857638	−0.428819	+	0.903390i	$0.641070\pi$
$312$	8.00000	0.452911
$313$	3.81966	0.215900	0.107950	−	0.994156i	$-0.465571\pi$
$313$	3.81966	0.215900	0.107950	+	0.994156i	$0.465571\pi$
$314$	−12.6525	−0.714021
$315$	−9.23607	−0.520393
$316$	8.94427	0.503155
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	−19.4164	−1.08882
$319$	0	0
$320$	1.23607	0.0690983
$321$	33.8885	1.89147
$322$	−1.00000	−0.0557278
$323$	32.3607	1.80060
$324$	24.4164	1.35647
$325$	−8.58359	−0.476132
$326$	−2.47214	−0.136919
$327$	−11.4164	−0.631329
$328$	10.0000	0.552158
$329$	6.76393	0.372908
$330$	0	0
$331$	−21.5279	−1.18328	−0.591639	−	0.806203i	$-0.701519\pi$
$331$	−21.5279	−1.18328	−0.591639	+	0.806203i	$0.701519\pi$
$332$	−6.94427	−0.381116
$333$	44.8328	2.45682
$334$	−24.6525	−1.34892
$335$	4.94427	0.270134
$336$	3.23607	0.176542
$337$	−0.111456	−0.00607140	−0.00303570	−	0.999995i	$-0.500966\pi$
$337$	−0.111456	−0.00607140	−0.00303570	+	0.999995i	$0.500966\pi$
$338$	6.88854	0.374687
$339$	−11.4164	−0.620054
$340$	−8.94427	−0.485071
$341$	0	0
$342$	33.4164	1.80695
$343$	−1.00000	−0.0539949
$344$	6.47214	0.348954
$345$	4.00000	0.215353
$346$	−20.9443	−1.12597
$347$	−20.3607	−1.09302	−0.546509	−	0.837453i	$-0.684044\pi$
$347$	−20.3607	−1.09302	−0.546509	+	0.837453i	$0.684044\pi$
$348$	6.47214	0.346943
$349$	−20.9443	−1.12112	−0.560561	−	0.828113i	$-0.689414\pi$
$349$	−20.9443	−1.12112	−0.560561	+	0.828113i	$0.689414\pi$
$350$	−3.47214	−0.185593
$351$	−35.7771	−1.90964
$352$	0	0
$353$	−8.47214	−0.450926	−0.225463	−	0.974252i	$-0.572390\pi$
$353$	−8.47214	−0.450926	−0.225463	+	0.974252i	$0.572390\pi$
$354$	36.3607	1.93255
$355$	−3.05573	−0.162181
$356$	−16.1803	−0.857556
$357$	−23.4164	−1.23933
$358$	0.944272	0.0499063
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	−9.23607	−0.486784
$361$	1.00000	0.0526316
$362$	−13.2361	−0.695672
$363$	35.5967	1.86834
$364$	−2.47214	−0.129575
$365$	−16.5836	−0.868025
$366$	37.8885	1.98047
$367$	−20.3607	−1.06282	−0.531409	−	0.847115i	$-0.678337\pi$
$367$	−20.3607	−1.06282	−0.531409	+	0.847115i	$0.678337\pi$
$368$	−1.00000	−0.0521286
$369$	−74.7214	−3.88984
$370$	−7.41641	−0.385561
$371$	6.00000	0.311504
$372$	4.00000	0.207390
$373$	−37.4164	−1.93735	−0.968674	−	0.248336i	$-0.920116\pi$
$373$	−37.4164	−1.93735	−0.968674	+	0.248336i	$0.920116\pi$
$374$	0	0
$375$	33.8885	1.75000
$376$	6.76393	0.348823
$377$	−4.94427	−0.254643
$378$	−14.4721	−0.744366
$379$	17.8885	0.918873	0.459436	−	0.888211i	$-0.348051\pi$
$379$	17.8885	0.918873	0.459436	+	0.888211i	$0.348051\pi$
$380$	−5.52786	−0.283573
$381$	−24.0000	−1.22956
$382$	−20.9443	−1.07160
$383$	−27.4164	−1.40091	−0.700456	−	0.713695i	$-0.747020\pi$
$383$	−27.4164	−1.40091	−0.700456	+	0.713695i	$0.747020\pi$
$384$	3.23607	0.165140
$385$	0	0
$386$	22.9443	1.16783
$387$	−48.3607	−2.45831
$388$	8.18034	0.415294
$389$	−17.4164	−0.883047	−0.441523	−	0.897250i	$-0.645562\pi$
$389$	−17.4164	−0.883047	−0.441523	+	0.897250i	$0.645562\pi$
$390$	9.88854	0.500726
$391$	7.23607	0.365944
$392$	−1.00000	−0.0505076
$393$	41.3050	2.08356
$394$	−15.8885	−0.800453
$395$	11.0557	0.556274
$396$	0	0
$397$	5.52786	0.277436	0.138718	−	0.990332i	$-0.455702\pi$
$397$	5.52786	0.277436	0.138718	+	0.990332i	$0.455702\pi$
$398$	12.0000	0.601506
$399$	−14.4721	−0.724513
$400$	−3.47214	−0.173607
$401$	33.7771	1.68675	0.843374	−	0.537328i	$-0.180566\pi$
$401$	33.7771	1.68675	0.843374	+	0.537328i	$0.180566\pi$
$402$	12.9443	0.645602
$403$	−3.05573	−0.152217
$404$	−14.4721	−0.720016
$405$	30.1803	1.49967
$406$	−2.00000	−0.0992583
$407$	0	0
$408$	−23.4164	−1.15928
$409$	19.5279	0.965591	0.482795	−	0.875733i	$-0.339621\pi$
$409$	19.5279	0.965591	0.482795	+	0.875733i	$0.339621\pi$
$410$	12.3607	0.610450
$411$	−56.3607	−2.78007
$412$	−4.00000	−0.197066
$413$	−11.2361	−0.552891
$414$	7.47214	0.367235
$415$	−8.58359	−0.421352
$416$	−2.47214	−0.121206
$417$	−55.4164	−2.71375
$418$	0	0
$419$	−10.3607	−0.506152	−0.253076	−	0.967446i	$-0.581442\pi$
$419$	−10.3607	−0.506152	−0.253076	+	0.967446i	$0.581442\pi$
$420$	4.00000	0.195180
$421$	0.111456	0.00543204	0.00271602	−	0.999996i	$-0.499135\pi$
$421$	0.111456	0.00543204	0.00271602	+	0.999996i	$0.499135\pi$
$422$	0.944272	0.0459664
$423$	−50.5410	−2.45739
$424$	6.00000	0.291386
$425$	25.1246	1.21872
$426$	−8.00000	−0.387601
$427$	−11.7082	−0.566600
$428$	−10.4721	−0.506190
$429$	0	0
$430$	8.00000	0.385794
$431$	3.05573	0.147189	0.0735946	−	0.997288i	$-0.476553\pi$
$431$	3.05573	0.147189	0.0735946	+	0.997288i	$0.476553\pi$
$432$	−14.4721	−0.696291
$433$	11.2361	0.539971	0.269985	−	0.962864i	$-0.412981\pi$
$433$	11.2361	0.539971	0.269985	+	0.962864i	$0.412981\pi$
$434$	−1.23607	−0.0593332
$435$	8.00000	0.383571
$436$	3.52786	0.168954
$437$	4.47214	0.213931
$438$	−43.4164	−2.07452
$439$	−13.2361	−0.631723	−0.315862	−	0.948805i	$-0.602294\pi$
$439$	−13.2361	−0.631723	−0.315862	+	0.948805i	$0.602294\pi$
$440$	0	0
$441$	7.47214	0.355816
$442$	17.8885	0.850871
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	−19.4164	−0.921462
$445$	−20.0000	−0.948091
$446$	−20.6525	−0.977923
$447$	17.5279	0.829040
$448$	−1.00000	−0.0472456
$449$	−17.4164	−0.821931	−0.410966	−	0.911651i	$-0.634808\pi$
$449$	−17.4164	−0.821931	−0.410966	+	0.911651i	$0.634808\pi$
$450$	25.9443	1.22302
$451$	0	0
$452$	3.52786	0.165937
$453$	59.7771	2.80857
$454$	−5.41641	−0.254205
$455$	−3.05573	−0.143255
$456$	−14.4721	−0.677720
$457$	−22.3607	−1.04599	−0.522994	−	0.852336i	$-0.675185\pi$
$457$	−22.3607	−1.04599	−0.522994	+	0.852336i	$0.675185\pi$
$458$	10.7639	0.502966
$459$	104.721	4.88797
$460$	−1.23607	−0.0576320
$461$	20.9443	0.975472	0.487736	−	0.872991i	$-0.337823\pi$
$461$	20.9443	0.975472	0.487736	+	0.872991i	$0.337823\pi$
$462$	0	0
$463$	−28.9443	−1.34515	−0.672577	−	0.740027i	$-0.734813\pi$
$463$	−28.9443	−1.34515	−0.672577	+	0.740027i	$0.734813\pi$
$464$	−2.00000	−0.0928477
$465$	4.94427	0.229285
$466$	7.52786	0.348722
$467$	−20.4721	−0.947337	−0.473669	−	0.880703i	$-0.657071\pi$
$467$	−20.4721	−0.947337	−0.473669	+	0.880703i	$0.657071\pi$
$468$	18.4721	0.853875
$469$	−4.00000	−0.184703
$470$	8.36068	0.385649
$471$	−40.9443	−1.88661
$472$	−11.2361	−0.517182
$473$	0	0
$474$	28.9443	1.32945
$475$	15.5279	0.712467
$476$	7.23607	0.331665
$477$	−44.8328	−2.05275
$478$	2.47214	0.113073
$479$	18.8328	0.860493	0.430247	−	0.902711i	$-0.358427\pi$
$479$	18.8328	0.860493	0.430247	+	0.902711i	$0.358427\pi$
$480$	4.00000	0.182574
$481$	14.8328	0.676318
$482$	−13.7082	−0.624392
$483$	−3.23607	−0.147246
$484$	−11.0000	−0.500000
$485$	10.1115	0.459138
$486$	35.5967	1.61470
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	−11.7082	−0.530005
$489$	−8.00000	−0.361773
$490$	−1.23607	−0.0558399
$491$	−24.9443	−1.12572	−0.562860	−	0.826553i	$-0.690299\pi$
$491$	−24.9443	−1.12572	−0.562860	+	0.826553i	$0.690299\pi$
$492$	32.3607	1.45893
$493$	14.4721	0.651792
$494$	11.0557	0.497421
$495$	0	0
$496$	−1.23607	−0.0555011
$497$	2.47214	0.110890
$498$	−22.4721	−1.00700
$499$	8.94427	0.400401	0.200200	−	0.979755i	$-0.435841\pi$
$499$	8.94427	0.400401	0.200200	+	0.979755i	$0.435841\pi$
$500$	−10.4721	−0.468328
$501$	−79.7771	−3.56418
$502$	−4.47214	−0.199601
$503$	12.9443	0.577157	0.288578	−	0.957456i	$-0.406817\pi$
$503$	12.9443	0.577157	0.288578	+	0.957456i	$0.406817\pi$
$504$	7.47214	0.332835
$505$	−17.8885	−0.796030
$506$	0	0
$507$	22.2918	0.990013
$508$	7.41641	0.329050
$509$	−1.88854	−0.0837082	−0.0418541	−	0.999124i	$-0.513326\pi$
$509$	−1.88854	−0.0837082	−0.0418541	+	0.999124i	$0.513326\pi$
$510$	−28.9443	−1.28167
$511$	13.4164	0.593507
$512$	−1.00000	−0.0441942
$513$	64.7214	2.85752
$514$	9.05573	0.399431
$515$	−4.94427	−0.217871
$516$	20.9443	0.922020
$517$	0	0
$518$	6.00000	0.263625
$519$	−67.7771	−2.97508
$520$	−3.05573	−0.134003
$521$	−1.70820	−0.0748378	−0.0374189	−	0.999300i	$-0.511914\pi$
$521$	−1.70820	−0.0748378	−0.0374189	+	0.999300i	$0.511914\pi$
$522$	14.9443	0.654093
$523$	37.4164	1.63611	0.818053	−	0.575143i	$-0.195054\pi$
$523$	37.4164	1.63611	0.818053	+	0.575143i	$0.195054\pi$
$524$	−12.7639	−0.557595
$525$	−11.2361	−0.490382
$526$	−12.0000	−0.523225
$527$	8.94427	0.389619
$528$	0	0
$529$	1.00000	0.0434783
$530$	7.41641	0.322148
$531$	83.9574	3.64344
$532$	4.47214	0.193892
$533$	−24.7214	−1.07080
$534$	−52.3607	−2.26587
$535$	−12.9443	−0.559630
$536$	−4.00000	−0.172774
$537$	3.05573	0.131864
$538$	−20.9443	−0.902972
$539$	0	0
$540$	−17.8885	−0.769800
$541$	−31.8885	−1.37100	−0.685498	−	0.728075i	$-0.740415\pi$
$541$	−31.8885	−1.37100	−0.685498	+	0.728075i	$0.740415\pi$
$542$	10.7639	0.462350
$543$	−42.8328	−1.83813
$544$	7.23607	0.310244
$545$	4.36068	0.186791
$546$	−8.00000	−0.342368
$547$	−13.5279	−0.578410	−0.289205	−	0.957267i	$-0.593391\pi$
$547$	−13.5279	−0.578410	−0.289205	+	0.957267i	$0.593391\pi$
$548$	17.4164	0.743992
$549$	87.4853	3.73378
$550$	0	0
$551$	8.94427	0.381039
$552$	−3.23607	−0.137736
$553$	−8.94427	−0.380349
$554$	15.8885	0.675040
$555$	−24.0000	−1.01874
$556$	17.1246	0.726245
$557$	9.41641	0.398986	0.199493	−	0.979899i	$-0.436070\pi$
$557$	9.41641	0.398986	0.199493	+	0.979899i	$0.436070\pi$
$558$	9.23607	0.390994
$559$	−16.0000	−0.676728
$560$	−1.23607	−0.0522334
$561$	0	0
$562$	21.4164	0.903397
$563$	33.4164	1.40833	0.704167	−	0.710035i	$-0.251321\pi$
$563$	33.4164	1.40833	0.704167	+	0.710035i	$0.251321\pi$
$564$	21.8885	0.921674
$565$	4.36068	0.183455
$566$	4.47214	0.187978
$567$	−24.4164	−1.02539
$568$	2.47214	0.103729
$569$	−2.00000	−0.0838444	−0.0419222	−	0.999121i	$-0.513348\pi$
$569$	−2.00000	−0.0838444	−0.0419222	+	0.999121i	$0.513348\pi$
$570$	−17.8885	−0.749269
$571$	29.3050	1.22637	0.613187	−	0.789938i	$-0.289887\pi$
$571$	29.3050	1.22637	0.613187	+	0.789938i	$0.289887\pi$
$572$	0	0
$573$	−67.7771	−2.83143
$574$	−10.0000	−0.417392
$575$	3.47214	0.144798
$576$	7.47214	0.311339
$577$	1.41641	0.0589658	0.0294829	−	0.999565i	$-0.490614\pi$
$577$	1.41641	0.0589658	0.0294829	+	0.999565i	$0.490614\pi$
$578$	−35.3607	−1.47081
$579$	74.2492	3.08569
$580$	−2.47214	−0.102650
$581$	6.94427	0.288097
$582$	26.4721	1.09731
$583$	0	0
$584$	13.4164	0.555175
$585$	22.8328	0.944021
$586$	−11.1246	−0.459553
$587$	26.2918	1.08518	0.542589	−	0.839998i	$-0.317444\pi$
$587$	26.2918	1.08518	0.542589	+	0.839998i	$0.317444\pi$
$588$	−3.23607	−0.133453
$589$	5.52786	0.227772
$590$	−13.8885	−0.571782
$591$	−51.4164	−2.11499
$592$	6.00000	0.246598
$593$	−38.0000	−1.56047	−0.780236	−	0.625485i	$-0.784901\pi$
$593$	−38.0000	−1.56047	−0.780236	+	0.625485i	$0.784901\pi$
$594$	0	0
$595$	8.94427	0.366679
$596$	−5.41641	−0.221865
$597$	38.8328	1.58932
$598$	2.47214	0.101093
$599$	−14.8328	−0.606052	−0.303026	−	0.952982i	$-0.597997\pi$
$599$	−14.8328	−0.606052	−0.303026	+	0.952982i	$0.597997\pi$
$600$	−11.2361	−0.458711
$601$	4.47214	0.182422	0.0912111	−	0.995832i	$-0.470926\pi$
$601$	4.47214	0.182422	0.0912111	+	0.995832i	$0.470926\pi$
$602$	−6.47214	−0.263785
$603$	29.8885	1.21716
$604$	−18.4721	−0.751621
$605$	−13.5967	−0.552786
$606$	−46.8328	−1.90245
$607$	5.23607	0.212525	0.106263	−	0.994338i	$-0.466112\pi$
$607$	5.23607	0.212525	0.106263	+	0.994338i	$0.466112\pi$
$608$	4.47214	0.181369
$609$	−6.47214	−0.262264
$610$	−14.4721	−0.585960
$611$	−16.7214	−0.676474
$612$	−54.0689	−2.18561
$613$	44.2492	1.78721	0.893605	−	0.448855i	$-0.148168\pi$
$613$	44.2492	1.78721	0.893605	+	0.448855i	$0.148168\pi$
$614$	−24.7639	−0.999391
$615$	40.0000	1.61296
$616$	0	0
$617$	26.9443	1.08474	0.542368	−	0.840141i	$-0.317528\pi$
$617$	26.9443	1.08474	0.542368	+	0.840141i	$0.317528\pi$
$618$	−12.9443	−0.520695
$619$	18.9443	0.761435	0.380717	−	0.924691i	$-0.375677\pi$
$619$	18.9443	0.761435	0.380717	+	0.924691i	$0.375677\pi$
$620$	−1.52786	−0.0613605
$621$	14.4721	0.580747
$622$	15.1246	0.606442
$623$	16.1803	0.648252
$624$	−8.00000	−0.320256
$625$	4.41641	0.176656
$626$	−3.81966	−0.152664
$627$	0	0
$628$	12.6525	0.504889
$629$	−43.4164	−1.73113
$630$	9.23607	0.367974
$631$	−37.8885	−1.50832	−0.754160	−	0.656691i	$-0.771955\pi$
$631$	−37.8885	−1.50832	−0.754160	+	0.656691i	$0.771955\pi$
$632$	−8.94427	−0.355784
$633$	3.05573	0.121454
$634$	−10.0000	−0.397151
$635$	9.16718	0.363789
$636$	19.4164	0.769911
$637$	2.47214	0.0979496
$638$	0	0
$639$	−18.4721	−0.730746
$640$	−1.23607	−0.0488599
$641$	17.0557	0.673661	0.336830	−	0.941565i	$-0.390645\pi$
$641$	17.0557	0.673661	0.336830	+	0.941565i	$0.390645\pi$
$642$	−33.8885	−1.33747
$643$	−36.2492	−1.42953	−0.714765	−	0.699365i	$-0.753466\pi$
$643$	−36.2492	−1.42953	−0.714765	+	0.699365i	$0.753466\pi$
$644$	1.00000	0.0394055
$645$	25.8885	1.01936
$646$	−32.3607	−1.27321
$647$	32.6525	1.28370	0.641851	−	0.766830i	$-0.278167\pi$
$647$	32.6525	1.28370	0.641851	+	0.766830i	$0.278167\pi$
$648$	−24.4164	−0.959167
$649$	0	0
$650$	8.58359	0.336676
$651$	−4.00000	−0.156772
$652$	2.47214	0.0968163
$653$	−41.7771	−1.63486	−0.817432	−	0.576025i	$-0.804603\pi$
$653$	−41.7771	−1.63486	−0.817432	+	0.576025i	$0.804603\pi$
$654$	11.4164	0.446417
$655$	−15.7771	−0.616462
$656$	−10.0000	−0.390434
$657$	−100.249	−3.91109
$658$	−6.76393	−0.263686
$659$	34.2492	1.33416	0.667080	−	0.744986i	$-0.267544\pi$
$659$	34.2492	1.33416	0.667080	+	0.744986i	$0.267544\pi$
$660$	0	0
$661$	−35.1246	−1.36619	−0.683095	−	0.730330i	$-0.739366\pi$
$661$	−35.1246	−1.36619	−0.683095	+	0.730330i	$0.739366\pi$
$662$	21.5279	0.836704
$663$	57.8885	2.24820
$664$	6.94427	0.269490
$665$	5.52786	0.214361
$666$	−44.8328	−1.73724
$667$	2.00000	0.0774403
$668$	24.6525	0.953833
$669$	−66.8328	−2.58391
$670$	−4.94427	−0.191014
$671$	0	0
$672$	−3.23607	−0.124834
$673$	−2.58359	−0.0995902	−0.0497951	−	0.998759i	$-0.515857\pi$
$673$	−2.58359	−0.0995902	−0.0497951	+	0.998759i	$0.515857\pi$
$674$	0.111456	0.00429313
$675$	50.2492	1.93409
$676$	−6.88854	−0.264944
$677$	18.7639	0.721156	0.360578	−	0.932729i	$-0.382579\pi$
$677$	18.7639	0.721156	0.360578	+	0.932729i	$0.382579\pi$
$678$	11.4164	0.438445
$679$	−8.18034	−0.313933
$680$	8.94427	0.342997
$681$	−17.5279	−0.671669
$682$	0	0
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	−33.4164	−1.27771
$685$	21.5279	0.822537
$686$	1.00000	0.0381802
$687$	34.8328	1.32895
$688$	−6.47214	−0.246748
$689$	−14.8328	−0.565085
$690$	−4.00000	−0.152277
$691$	−27.0132	−1.02763	−0.513814	−	0.857901i	$-0.671768\pi$
$691$	−27.0132	−1.02763	−0.513814	+	0.857901i	$0.671768\pi$
$692$	20.9443	0.796182
$693$	0	0
$694$	20.3607	0.772881
$695$	21.1672	0.802917
$696$	−6.47214	−0.245326
$697$	72.3607	2.74086
$698$	20.9443	0.792752
$699$	24.3607	0.921406
$700$	3.47214	0.131234
$701$	−38.9443	−1.47090	−0.735452	−	0.677576i	$-0.763030\pi$
$701$	−38.9443	−1.47090	−0.735452	+	0.677576i	$0.763030\pi$
$702$	35.7771	1.35032
$703$	−26.8328	−1.01202
$704$	0	0
$705$	27.0557	1.01898
$706$	8.47214	0.318853
$707$	14.4721	0.544281
$708$	−36.3607	−1.36652
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	3.05573	0.114679
$711$	66.8328	2.50643
$712$	16.1803	0.606384
$713$	1.23607	0.0462911
$714$	23.4164	0.876337
$715$	0	0
$716$	−0.944272	−0.0352891
$717$	8.00000	0.298765
$718$	−12.0000	−0.447836
$719$	34.5410	1.28816	0.644081	−	0.764957i	$-0.277240\pi$
$719$	34.5410	1.28816	0.644081	+	0.764957i	$0.277240\pi$
$720$	9.23607	0.344208
$721$	4.00000	0.148968
$722$	−1.00000	−0.0372161
$723$	−44.3607	−1.64979
$724$	13.2361	0.491915
$725$	6.94427	0.257904
$726$	−35.5967	−1.32112
$727$	24.9443	0.925132	0.462566	−	0.886585i	$-0.346929\pi$
$727$	24.9443	0.925132	0.462566	+	0.886585i	$0.346929\pi$
$728$	2.47214	0.0916235
$729$	41.9443	1.55349
$730$	16.5836	0.613786
$731$	46.8328	1.73217
$732$	−37.8885	−1.40040
$733$	−45.5967	−1.68415	−0.842077	−	0.539357i	$-0.818667\pi$
$733$	−45.5967	−1.68415	−0.842077	+	0.539357i	$0.818667\pi$
$734$	20.3607	0.751526
$735$	−4.00000	−0.147542
$736$	1.00000	0.0368605
$737$	0	0
$738$	74.7214	2.75053
$739$	−20.3607	−0.748980	−0.374490	−	0.927231i	$-0.622182\pi$
$739$	−20.3607	−0.748980	−0.374490	+	0.927231i	$0.622182\pi$
$740$	7.41641	0.272633
$741$	35.7771	1.31430
$742$	−6.00000	−0.220267
$743$	−31.0557	−1.13932	−0.569662	−	0.821879i	$-0.692926\pi$
$743$	−31.0557	−1.13932	−0.569662	+	0.821879i	$0.692926\pi$
$744$	−4.00000	−0.146647
$745$	−6.69505	−0.245288
$746$	37.4164	1.36991
$747$	−51.8885	−1.89850
$748$	0	0
$749$	10.4721	0.382644
$750$	−33.8885	−1.23743
$751$	−23.0557	−0.841315	−0.420658	−	0.907219i	$-0.638201\pi$
$751$	−23.0557	−0.841315	−0.420658	+	0.907219i	$0.638201\pi$
$752$	−6.76393	−0.246655
$753$	−14.4721	−0.527394
$754$	4.94427	0.180060
$755$	−22.8328	−0.830971
$756$	14.4721	0.526346
$757$	32.4721	1.18022	0.590110	−	0.807323i	$-0.299084\pi$
$757$	32.4721	1.18022	0.590110	+	0.807323i	$0.299084\pi$
$758$	−17.8885	−0.649741
$759$	0	0
$760$	5.52786	0.200517
$761$	8.47214	0.307115	0.153557	−	0.988140i	$-0.450927\pi$
$761$	8.47214	0.307115	0.153557	+	0.988140i	$0.450927\pi$
$762$	24.0000	0.869428
$763$	−3.52786	−0.127717
$764$	20.9443	0.757737
$765$	−66.8328	−2.41635
$766$	27.4164	0.990595
$767$	27.7771	1.00297
$768$	−3.23607	−0.116772
$769$	16.7639	0.604523	0.302261	−	0.953225i	$-0.402258\pi$
$769$	16.7639	0.604523	0.302261	+	0.953225i	$0.402258\pi$
$770$	0	0
$771$	29.3050	1.05539
$772$	−22.9443	−0.825782
$773$	−6.18034	−0.222291	−0.111146	−	0.993804i	$-0.535452\pi$
$773$	−6.18034	−0.222291	−0.111146	+	0.993804i	$0.535452\pi$
$774$	48.3607	1.73829
$775$	4.29180	0.154166
$776$	−8.18034	−0.293657
$777$	19.4164	0.696560
$778$	17.4164	0.624408
$779$	44.7214	1.60231
$780$	−9.88854	−0.354067
$781$	0	0
$782$	−7.23607	−0.258761
$783$	28.9443	1.03438
$784$	1.00000	0.0357143
$785$	15.6393	0.558191
$786$	−41.3050	−1.47330
$787$	−0.111456	−0.00397298	−0.00198649	−	0.999998i	$-0.500632\pi$
$787$	−0.111456	−0.00397298	−0.00198649	+	0.999998i	$0.500632\pi$
$788$	15.8885	0.566006
$789$	−38.8328	−1.38248
$790$	−11.0557	−0.393345
$791$	−3.52786	−0.125436
$792$	0	0
$793$	28.9443	1.02784
$794$	−5.52786	−0.196177
$795$	24.0000	0.851192
$796$	−12.0000	−0.425329
$797$	−44.6525	−1.58167	−0.790836	−	0.612028i	$-0.790354\pi$
$797$	−44.6525	−1.58167	−0.790836	+	0.612028i	$0.790354\pi$
$798$	14.4721	0.512308
$799$	48.9443	1.73152
$800$	3.47214	0.122759
$801$	−120.902	−4.27185
$802$	−33.7771	−1.19271
$803$	0	0
$804$	−12.9443	−0.456509
$805$	1.23607	0.0435657
$806$	3.05573	0.107633
$807$	−67.7771	−2.38587
$808$	14.4721	0.509128
$809$	20.8328	0.732443	0.366221	−	0.930528i	$-0.380651\pi$
$809$	20.8328	0.732443	0.366221	+	0.930528i	$0.380651\pi$
$810$	−30.1803	−1.06043
$811$	−27.5967	−0.969053	−0.484526	−	0.874777i	$-0.661008\pi$
$811$	−27.5967	−0.969053	−0.484526	+	0.874777i	$0.661008\pi$
$812$	2.00000	0.0701862
$813$	34.8328	1.22164
$814$	0	0
$815$	3.05573	0.107037
$816$	23.4164	0.819738
$817$	28.9443	1.01263
$818$	−19.5279	−0.682776
$819$	−18.4721	−0.645469
$820$	−12.3607	−0.431654
$821$	14.9443	0.521559	0.260779	−	0.965398i	$-0.416020\pi$
$821$	14.9443	0.521559	0.260779	+	0.965398i	$0.416020\pi$
$822$	56.3607	1.96580
$823$	15.4164	0.537382	0.268691	−	0.963226i	$-0.413409\pi$
$823$	15.4164	0.537382	0.268691	+	0.963226i	$0.413409\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	11.2361	0.390953
$827$	−39.4164	−1.37064	−0.685321	−	0.728241i	$-0.740338\pi$
$827$	−39.4164	−1.37064	−0.685321	+	0.728241i	$0.740338\pi$
$828$	−7.47214	−0.259675
$829$	−21.8885	−0.760221	−0.380110	−	0.924941i	$-0.624114\pi$
$829$	−21.8885	−0.760221	−0.380110	+	0.924941i	$0.624114\pi$
$830$	8.58359	0.297941
$831$	51.4164	1.78362
$832$	2.47214	0.0857059
$833$	−7.23607	−0.250715
$834$	55.4164	1.91891
$835$	30.4721	1.05453
$836$	0	0
$837$	17.8885	0.618319
$838$	10.3607	0.357904
$839$	−16.3607	−0.564833	−0.282417	−	0.959292i	$-0.591136\pi$
$839$	−16.3607	−0.564833	−0.282417	+	0.959292i	$0.591136\pi$
$840$	−4.00000	−0.138013
$841$	−25.0000	−0.862069
$842$	−0.111456	−0.00384103
$843$	69.3050	2.38699
$844$	−0.944272	−0.0325032
$845$	−8.51471	−0.292915
$846$	50.5410	1.73764
$847$	11.0000	0.377964
$848$	−6.00000	−0.206041
$849$	14.4721	0.496682
$850$	−25.1246	−0.861767
$851$	−6.00000	−0.205677
$852$	8.00000	0.274075
$853$	−15.4164	−0.527848	−0.263924	−	0.964544i	$-0.585017\pi$
$853$	−15.4164	−0.527848	−0.263924	+	0.964544i	$0.585017\pi$
$854$	11.7082	0.400646
$855$	−41.3050	−1.41260
$856$	10.4721	0.357930
$857$	23.8885	0.816017	0.408009	−	0.912978i	$-0.366223\pi$
$857$	23.8885	0.816017	0.408009	+	0.912978i	$0.366223\pi$
$858$	0	0
$859$	7.23607	0.246891	0.123446	−	0.992351i	$-0.460606\pi$
$859$	7.23607	0.246891	0.123446	+	0.992351i	$0.460606\pi$
$860$	−8.00000	−0.272798
$861$	−32.3607	−1.10285
$862$	−3.05573	−0.104079
$863$	23.4164	0.797104	0.398552	−	0.917146i	$-0.369513\pi$
$863$	23.4164	0.797104	0.398552	+	0.917146i	$0.369513\pi$
$864$	14.4721	0.492352
$865$	25.8885	0.880237
$866$	−11.2361	−0.381817
$867$	−114.430	−3.88623
$868$	1.23607	0.0419549
$869$	0	0
$870$	−8.00000	−0.271225
$871$	9.88854	0.335061
$872$	−3.52786	−0.119469
$873$	61.1246	2.06875
$874$	−4.47214	−0.151272
$875$	10.4721	0.354023
$876$	43.4164	1.46690
$877$	9.05573	0.305790	0.152895	−	0.988242i	$-0.451140\pi$
$877$	9.05573	0.305790	0.152895	+	0.988242i	$0.451140\pi$
$878$	13.2361	0.446696
$879$	−36.0000	−1.21425
$880$	0	0
$881$	−16.7639	−0.564791	−0.282396	−	0.959298i	$-0.591129\pi$
$881$	−16.7639	−0.564791	−0.282396	+	0.959298i	$0.591129\pi$
$882$	−7.47214	−0.251600
$883$	12.0000	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$884$	−17.8885	−0.601657
$885$	−44.9443	−1.51079
$886$	−4.00000	−0.134383
$887$	13.5967	0.456534	0.228267	−	0.973599i	$-0.426694\pi$
$887$	13.5967	0.456534	0.228267	+	0.973599i	$0.426694\pi$
$888$	19.4164	0.651572
$889$	−7.41641	−0.248738
$890$	20.0000	0.670402
$891$	0	0
$892$	20.6525	0.691496
$893$	30.2492	1.01225
$894$	−17.5279	−0.586219
$895$	−1.16718	−0.0390147
$896$	1.00000	0.0334077
$897$	8.00000	0.267112
$898$	17.4164	0.581193
$899$	2.47214	0.0824504
$900$	−25.9443	−0.864809
$901$	43.4164	1.44641
$902$	0	0
$903$	−20.9443	−0.696982
$904$	−3.52786	−0.117335
$905$	16.3607	0.543847
$906$	−59.7771	−1.98596
$907$	−10.4721	−0.347722	−0.173861	−	0.984770i	$-0.555624\pi$
$907$	−10.4721	−0.347722	−0.173861	+	0.984770i	$0.555624\pi$
$908$	5.41641	0.179750
$909$	−108.138	−3.58670
$910$	3.05573	0.101296
$911$	−13.8885	−0.460148	−0.230074	−	0.973173i	$-0.573897\pi$
$911$	−13.8885	−0.460148	−0.230074	+	0.973173i	$0.573897\pi$
$912$	14.4721	0.479220
$913$	0	0
$914$	22.3607	0.739626
$915$	−46.8328	−1.54825
$916$	−10.7639	−0.355650
$917$	12.7639	0.421502
$918$	−104.721	−3.45632
$919$	−45.8885	−1.51372	−0.756862	−	0.653575i	$-0.773268\pi$
$919$	−45.8885	−1.51372	−0.756862	+	0.653575i	$0.773268\pi$
$920$	1.23607	0.0407520
$921$	−80.1378	−2.64063
$922$	−20.9443	−0.689763
$923$	−6.11146	−0.201161
$924$	0	0
$925$	−20.8328	−0.684979
$926$	28.9443	0.951168
$927$	−29.8885	−0.981669
$928$	2.00000	0.0656532
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	−4.94427	−0.162129
$931$	−4.47214	−0.146568
$932$	−7.52786	−0.246583
$933$	48.9443	1.60236
$934$	20.4721	0.669869
$935$	0	0
$936$	−18.4721	−0.603781
$937$	55.2361	1.80448	0.902242	−	0.431230i	$-0.141920\pi$
$937$	55.2361	1.80448	0.902242	+	0.431230i	$0.141920\pi$
$938$	4.00000	0.130605
$939$	−12.3607	−0.403376
$940$	−8.36068	−0.272695
$941$	−29.5967	−0.964826	−0.482413	−	0.875944i	$-0.660240\pi$
$941$	−29.5967	−0.964826	−0.482413	+	0.875944i	$0.660240\pi$
$942$	40.9443	1.33404
$943$	10.0000	0.325645
$944$	11.2361	0.365703
$945$	17.8885	0.581914
$946$	0	0
$947$	−38.2492	−1.24293	−0.621466	−	0.783441i	$-0.713463\pi$
$947$	−38.2492	−1.24293	−0.621466	+	0.783441i	$0.713463\pi$
$948$	−28.9443	−0.940066
$949$	−33.1672	−1.07665
$950$	−15.5279	−0.503790
$951$	−32.3607	−1.04937
$952$	−7.23607	−0.234522
$953$	−25.7771	−0.835002	−0.417501	−	0.908677i	$-0.637094\pi$
$953$	−25.7771	−0.835002	−0.417501	+	0.908677i	$0.637094\pi$
$954$	44.8328	1.45152
$955$	25.8885	0.837734
$956$	−2.47214	−0.0799546
$957$	0	0
$958$	−18.8328	−0.608461
$959$	−17.4164	−0.562405
$960$	−4.00000	−0.129099
$961$	−29.4721	−0.950714
$962$	−14.8328	−0.478229
$963$	−78.2492	−2.52155
$964$	13.7082	0.441512
$965$	−28.3607	−0.912963
$966$	3.23607	0.104119
$967$	25.8885	0.832519	0.416260	−	0.909246i	$-0.363341\pi$
$967$	25.8885	0.832519	0.416260	+	0.909246i	$0.363341\pi$
$968$	11.0000	0.353553
$969$	−104.721	−3.36413
$970$	−10.1115	−0.324659
$971$	−0.111456	−0.00357680	−0.00178840	−	0.999998i	$-0.500569\pi$
$971$	−0.111456	−0.00357680	−0.00178840	+	0.999998i	$0.500569\pi$
$972$	−35.5967	−1.14177
$973$	−17.1246	−0.548990
$974$	0	0
$975$	27.7771	0.889579
$976$	11.7082	0.374770
$977$	−3.52786	−0.112866	−0.0564332	−	0.998406i	$-0.517973\pi$
$977$	−3.52786	−0.112866	−0.0564332	+	0.998406i	$0.517973\pi$
$978$	8.00000	0.255812
$979$	0	0
$980$	1.23607	0.0394847
$981$	26.3607	0.841632
$982$	24.9443	0.796004
$983$	−22.4721	−0.716750	−0.358375	−	0.933578i	$-0.616669\pi$
$983$	−22.4721	−0.716750	−0.358375	+	0.933578i	$0.616669\pi$
$984$	−32.3607	−1.03162
$985$	19.6393	0.625761
$986$	−14.4721	−0.460887
$987$	−21.8885	−0.696720
$988$	−11.0557	−0.351730
$989$	6.47214	0.205802
$990$	0	0
$991$	−41.3050	−1.31210	−0.656048	−	0.754720i	$-0.727773\pi$
$991$	−41.3050	−1.31210	−0.656048	+	0.754720i	$0.727773\pi$
$992$	1.23607	0.0392452
$993$	69.6656	2.21077
$994$	−2.47214	−0.0784114
$995$	−14.8328	−0.470232
$996$	22.4721	0.712057
$997$	−42.8328	−1.35653	−0.678264	−	0.734818i	$-0.737268\pi$
$997$	−42.8328	−1.35653	−0.678264	+	0.734818i	$0.737268\pi$
$998$	−8.94427	−0.283126
$999$	−86.8328	−2.74727

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.a.e.1.1	✓	2
3.2	odd	2		2898.2.a.bd.1.1		2
4.3	odd	2		2576.2.a.t.1.2		2
5.4	even	2		8050.2.a.bf.1.2		2
7.6	odd	2		2254.2.a.k.1.2		2
23.22	odd	2		7406.2.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.a.e.1.1	✓	2	1.1	even	1	trivial
2254.2.a.k.1.2		2	7.6	odd	2
2576.2.a.t.1.2		2	4.3	odd	2
2898.2.a.bd.1.1		2	3.2	odd	2
7406.2.a.j.1.1		2	23.22	odd	2
8050.2.a.bf.1.2		2	5.4	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:	\( N \)	\(=\)	\( 322 = 2 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	322.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.23607 q^{3} +1.00000 q^{4} +1.23607 q^{5} +3.23607 q^{6} -1.00000 q^{7} -1.00000 q^{8} +7.47214 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.23607 q^{3} +1.00000 q^{4} +1.23607 q^{5} +3.23607 q^{6} -1.00000 q^{7} -1.00000 q^{8} +7.47214 q^{9} -1.23607 q^{10} -3.23607 q^{12} +2.47214 q^{13} +1.00000 q^{14} -4.00000 q^{15} +1.00000 q^{16} -7.23607 q^{17} -7.47214 q^{18} -4.47214 q^{19} +1.23607 q^{20} +3.23607 q^{21} -1.00000 q^{23} +3.23607 q^{24} -3.47214 q^{25} -2.47214 q^{26} -14.4721 q^{27} -1.00000 q^{28} -2.00000 q^{29} +4.00000 q^{30} -1.23607 q^{31} -1.00000 q^{32} +7.23607 q^{34} -1.23607 q^{35} +7.47214 q^{36} +6.00000 q^{37} +4.47214 q^{38} -8.00000 q^{39} -1.23607 q^{40} -10.0000 q^{41} -3.23607 q^{42} -6.47214 q^{43} +9.23607 q^{45} +1.00000 q^{46} -6.76393 q^{47} -3.23607 q^{48} +1.00000 q^{49} +3.47214 q^{50} +23.4164 q^{51} +2.47214 q^{52} -6.00000 q^{53} +14.4721 q^{54} +1.00000 q^{56} +14.4721 q^{57} +2.00000 q^{58} +11.2361 q^{59} -4.00000 q^{60} +11.7082 q^{61} +1.23607 q^{62} -7.47214 q^{63} +1.00000 q^{64} +3.05573 q^{65} +4.00000 q^{67} -7.23607 q^{68} +3.23607 q^{69} +1.23607 q^{70} -2.47214 q^{71} -7.47214 q^{72} -13.4164 q^{73} -6.00000 q^{74} +11.2361 q^{75} -4.47214 q^{76} +8.00000 q^{78} +8.94427 q^{79} +1.23607 q^{80} +24.4164 q^{81} +10.0000 q^{82} -6.94427 q^{83} +3.23607 q^{84} -8.94427 q^{85} +6.47214 q^{86} +6.47214 q^{87} -16.1803 q^{89} -9.23607 q^{90} -2.47214 q^{91} -1.00000 q^{92} +4.00000 q^{93} +6.76393 q^{94} -5.52786 q^{95} +3.23607 q^{96} +8.18034 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - 2 * q^7 - 2 * q^8 + 6 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 6 q^{9} + 2 q^{10} - 2 q^{12} - 4 q^{13} + 2 q^{14} - 8 q^{15} + 2 q^{16} - 10 q^{17} - 6 q^{18} - 2 q^{20} + 2 q^{21} - 2 q^{23} + 2 q^{24} + 2 q^{25} + 4 q^{26} - 20 q^{27} - 2 q^{28} - 4 q^{29} + 8 q^{30} + 2 q^{31} - 2 q^{32} + 10 q^{34} + 2 q^{35} + 6 q^{36} + 12 q^{37} - 16 q^{39} + 2 q^{40} - 20 q^{41} - 2 q^{42} - 4 q^{43} + 14 q^{45} + 2 q^{46} - 18 q^{47} - 2 q^{48} + 2 q^{49} - 2 q^{50} + 20 q^{51} - 4 q^{52} - 12 q^{53} + 20 q^{54} + 2 q^{56} + 20 q^{57} + 4 q^{58} + 18 q^{59} - 8 q^{60} + 10 q^{61} - 2 q^{62} - 6 q^{63} + 2 q^{64} + 24 q^{65} + 8 q^{67} - 10 q^{68} + 2 q^{69} - 2 q^{70} + 4 q^{71} - 6 q^{72} - 12 q^{74} + 18 q^{75} + 16 q^{78} - 2 q^{80} + 22 q^{81} + 20 q^{82} + 4 q^{83} + 2 q^{84} + 4 q^{86} + 4 q^{87} - 10 q^{89} - 14 q^{90} + 4 q^{91} - 2 q^{92} + 8 q^{93} + 18 q^{94} - 20 q^{95} + 2 q^{96} - 6 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - 2 * q^7 - 2 * q^8 + 6 * q^9 + 2 * q^10 - 2 * q^12 - 4 * q^13 + 2 * q^14 - 8 * q^15 + 2 * q^16 - 10 * q^17 - 6 * q^18 - 2 * q^20 + 2 * q^21 - 2 * q^23 + 2 * q^24 + 2 * q^25 + 4 * q^26 - 20 * q^27 - 2 * q^28 - 4 * q^29 + 8 * q^30 + 2 * q^31 - 2 * q^32 + 10 * q^34 + 2 * q^35 + 6 * q^36 + 12 * q^37 - 16 * q^39 + 2 * q^40 - 20 * q^41 - 2 * q^42 - 4 * q^43 + 14 * q^45 + 2 * q^46 - 18 * q^47 - 2 * q^48 + 2 * q^49 - 2 * q^50 + 20 * q^51 - 4 * q^52 - 12 * q^53 + 20 * q^54 + 2 * q^56 + 20 * q^57 + 4 * q^58 + 18 * q^59 - 8 * q^60 + 10 * q^61 - 2 * q^62 - 6 * q^63 + 2 * q^64 + 24 * q^65 + 8 * q^67 - 10 * q^68 + 2 * q^69 - 2 * q^70 + 4 * q^71 - 6 * q^72 - 12 * q^74 + 18 * q^75 + 16 * q^78 - 2 * q^80 + 22 * q^81 + 20 * q^82 + 4 * q^83 + 2 * q^84 + 4 * q^86 + 4 * q^87 - 10 * q^89 - 14 * q^90 + 4 * q^91 - 2 * q^92 + 8 * q^93 + 18 * q^94 - 20 * q^95 + 2 * q^96 - 6 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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