LMFDB - Embedded newform 6422.2.a.v.1.2

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:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -0.445042 q^{3} +1.00000 q^{4} -0.753020 q^{5} -0.445042 q^{6} +1.60388 q^{7} +1.00000 q^{8} -2.80194 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.445042 q^{3} +1.00000 q^{4} -0.753020 q^{5} -0.445042 q^{6} +1.60388 q^{7} +1.00000 q^{8} -2.80194 q^{9} -0.753020 q^{10} +3.60388 q^{11} -0.445042 q^{12} +1.60388 q^{14} +0.335126 q^{15} +1.00000 q^{16} -1.40581 q^{17} -2.80194 q^{18} +1.00000 q^{19} -0.753020 q^{20} -0.713792 q^{21} +3.60388 q^{22} -8.59179 q^{23} -0.445042 q^{24} -4.43296 q^{25} +2.58211 q^{27} +1.60388 q^{28} -1.38404 q^{29} +0.335126 q^{30} -2.93900 q^{31} +1.00000 q^{32} -1.60388 q^{33} -1.40581 q^{34} -1.20775 q^{35} -2.80194 q^{36} -1.32975 q^{37} +1.00000 q^{38} -0.753020 q^{40} -2.61596 q^{41} -0.713792 q^{42} +1.78017 q^{43} +3.60388 q^{44} +2.10992 q^{45} -8.59179 q^{46} -0.219833 q^{47} -0.445042 q^{48} -4.42758 q^{49} -4.43296 q^{50} +0.625646 q^{51} -5.50604 q^{53} +2.58211 q^{54} -2.71379 q^{55} +1.60388 q^{56} -0.445042 q^{57} -1.38404 q^{58} -1.42327 q^{59} +0.335126 q^{60} -10.4547 q^{61} -2.93900 q^{62} -4.49396 q^{63} +1.00000 q^{64} -1.60388 q^{66} +2.66487 q^{67} -1.40581 q^{68} +3.82371 q^{69} -1.20775 q^{70} +10.2959 q^{71} -2.80194 q^{72} +1.40581 q^{73} -1.32975 q^{74} +1.97285 q^{75} +1.00000 q^{76} +5.78017 q^{77} +11.2838 q^{79} -0.753020 q^{80} +7.25667 q^{81} -2.61596 q^{82} -3.10992 q^{83} -0.713792 q^{84} +1.05861 q^{85} +1.78017 q^{86} +0.615957 q^{87} +3.60388 q^{88} +2.19567 q^{89} +2.10992 q^{90} -8.59179 q^{92} +1.30798 q^{93} -0.219833 q^{94} -0.753020 q^{95} -0.445042 q^{96} -13.0858 q^{97} -4.42758 q^{98} -10.0978 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} - 7 q^{5} - q^{6} - 4 q^{7} + 3 q^{8} - 4 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 - 7 * q^5 - q^6 - 4 * q^7 + 3 * q^8 - 4 * q^9	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} - 7 q^{5} - q^{6} - 4 q^{7} + 3 q^{8} - 4 q^{9} - 7 q^{10} + 2 q^{11} - q^{12} - 4 q^{14} + 3 q^{16} + 9 q^{17} - 4 q^{18} + 3 q^{19} - 7 q^{20} + 6 q^{21} + 2 q^{22} + 2 q^{23} - q^{24} + 6 q^{25} + 2 q^{27} - 4 q^{28} + 6 q^{29} + q^{31} + 3 q^{32} + 4 q^{33} + 9 q^{34} + 14 q^{35} - 4 q^{36} - 6 q^{37} + 3 q^{38} - 7 q^{40} - 18 q^{41} + 6 q^{42} + 4 q^{43} + 2 q^{44} + 7 q^{45} + 2 q^{46} - 2 q^{47} - q^{48} + 3 q^{49} + 6 q^{50} - 10 q^{51} - 26 q^{53} + 2 q^{54} - 4 q^{56} - q^{57} + 6 q^{58} - 7 q^{59} - 9 q^{61} + q^{62} - 4 q^{63} + 3 q^{64} + 4 q^{66} + 9 q^{67} + 9 q^{68} + 4 q^{69} + 14 q^{70} + 17 q^{71} - 4 q^{72} - 9 q^{73} - 6 q^{74} + 12 q^{75} + 3 q^{76} + 16 q^{77} + q^{79} - 7 q^{80} - 5 q^{81} - 18 q^{82} - 10 q^{83} + 6 q^{84} - 28 q^{85} + 4 q^{86} + 12 q^{87} + 2 q^{88} - 30 q^{89} + 7 q^{90} + 2 q^{92} + 9 q^{93} - 2 q^{94} - 7 q^{95} - q^{96} - 2 q^{97} + 3 q^{98} - 12 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 - 7 * q^5 - q^6 - 4 * q^7 + 3 * q^8 - 4 * q^9 - 7 * q^10 + 2 * q^11 - q^12 - 4 * q^14 + 3 * q^16 + 9 * q^17 - 4 * q^18 + 3 * q^19 - 7 * q^20 + 6 * q^21 + 2 * q^22 + 2 * q^23 - q^24 + 6 * q^25 + 2 * q^27 - 4 * q^28 + 6 * q^29 + q^31 + 3 * q^32 + 4 * q^33 + 9 * q^34 + 14 * q^35 - 4 * q^36 - 6 * q^37 + 3 * q^38 - 7 * q^40 - 18 * q^41 + 6 * q^42 + 4 * q^43 + 2 * q^44 + 7 * q^45 + 2 * q^46 - 2 * q^47 - q^48 + 3 * q^49 + 6 * q^50 - 10 * q^51 - 26 * q^53 + 2 * q^54 - 4 * q^56 - q^57 + 6 * q^58 - 7 * q^59 - 9 * q^61 + q^62 - 4 * q^63 + 3 * q^64 + 4 * q^66 + 9 * q^67 + 9 * q^68 + 4 * q^69 + 14 * q^70 + 17 * q^71 - 4 * q^72 - 9 * q^73 - 6 * q^74 + 12 * q^75 + 3 * q^76 + 16 * q^77 + q^79 - 7 * q^80 - 5 * q^81 - 18 * q^82 - 10 * q^83 + 6 * q^84 - 28 * q^85 + 4 * q^86 + 12 * q^87 + 2 * q^88 - 30 * q^89 + 7 * q^90 + 2 * q^92 + 9 * q^93 - 2 * q^94 - 7 * q^95 - q^96 - 2 * q^97 + 3 * q^98 - 12 * 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
$3$	−0.445042	−0.256945	−0.128473	−	0.991713i	$-0.541007\pi$
$3$	−0.445042	−0.256945	−0.128473	+	0.991713i	$0.541007\pi$
$4$	1.00000	0.500000
$5$	−0.753020	−0.336761	−0.168380	−	0.985722i	$-0.553854\pi$
$5$	−0.753020	−0.336761	−0.168380	+	0.985722i	$0.553854\pi$
$6$	−0.445042	−0.181688
$7$	1.60388	0.606208	0.303104	−	0.952957i	$-0.401977\pi$
$7$	1.60388	0.606208	0.303104	+	0.952957i	$0.401977\pi$
$8$	1.00000	0.353553
$9$	−2.80194	−0.933979
$10$	−0.753020	−0.238126
$11$	3.60388	1.08661	0.543305	−	0.839536i	$-0.317173\pi$
$11$	3.60388	1.08661	0.543305	+	0.839536i	$0.317173\pi$
$12$	−0.445042	−0.128473
$13$	0	0
$13$	0	0
$14$	1.60388	0.428654
$15$	0.335126	0.0865291
$16$	1.00000	0.250000
$17$	−1.40581	−0.340960	−0.170480	−	0.985361i	$-0.554532\pi$
$17$	−1.40581	−0.340960	−0.170480	+	0.985361i	$0.554532\pi$
$18$	−2.80194	−0.660423
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.753020	−0.168380
$21$	−0.713792	−0.155762
$22$	3.60388	0.768349
$23$	−8.59179	−1.79151	−0.895756	−	0.444545i	$-0.853365\pi$
$23$	−8.59179	−1.79151	−0.895756	+	0.444545i	$0.853365\pi$
$24$	−0.445042	−0.0908438
$25$	−4.43296	−0.886592
$26$	0	0
$27$	2.58211	0.496926
$28$	1.60388	0.303104
$29$	−1.38404	−0.257010	−0.128505	−	0.991709i	$-0.541018\pi$
$29$	−1.38404	−0.257010	−0.128505	+	0.991709i	$0.541018\pi$
$30$	0.335126	0.0611853
$31$	−2.93900	−0.527860	−0.263930	−	0.964542i	$-0.585019\pi$
$31$	−2.93900	−0.527860	−0.263930	+	0.964542i	$0.585019\pi$
$32$	1.00000	0.176777
$33$	−1.60388	−0.279199
$34$	−1.40581	−0.241095
$35$	−1.20775	−0.204147
$36$	−2.80194	−0.466990
$37$	−1.32975	−0.218609	−0.109305	−	0.994008i	$-0.534862\pi$
$37$	−1.32975	−0.218609	−0.109305	+	0.994008i	$0.534862\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	−0.753020	−0.119063
$41$	−2.61596	−0.408544	−0.204272	−	0.978914i	$-0.565483\pi$
$41$	−2.61596	−0.408544	−0.204272	+	0.978914i	$0.565483\pi$
$42$	−0.713792	−0.110140
$43$	1.78017	0.271473	0.135736	−	0.990745i	$-0.456660\pi$
$43$	1.78017	0.271473	0.135736	+	0.990745i	$0.456660\pi$
$44$	3.60388	0.543305
$45$	2.10992	0.314528
$46$	−8.59179	−1.26679
$47$	−0.219833	−0.0320659	−0.0160329	−	0.999871i	$-0.505104\pi$
$47$	−0.219833	−0.0320659	−0.0160329	+	0.999871i	$0.505104\pi$
$48$	−0.445042	−0.0642363
$49$	−4.42758	−0.632512
$50$	−4.43296	−0.626915
$51$	0.625646	0.0876079
$52$	0	0
$53$	−5.50604	−0.756313	−0.378156	−	0.925742i	$-0.623442\pi$
$53$	−5.50604	−0.756313	−0.378156	+	0.925742i	$0.623442\pi$
$54$	2.58211	0.351380
$55$	−2.71379	−0.365928
$56$	1.60388	0.214327
$57$	−0.445042	−0.0589472
$58$	−1.38404	−0.181734
$59$	−1.42327	−0.185294	−0.0926471	−	0.995699i	$-0.529533\pi$
$59$	−1.42327	−0.185294	−0.0926471	+	0.995699i	$0.529533\pi$
$60$	0.335126	0.0432645
$61$	−10.4547	−1.33859	−0.669296	−	0.742996i	$-0.733404\pi$
$61$	−10.4547	−1.33859	−0.669296	+	0.742996i	$0.733404\pi$
$62$	−2.93900	−0.373254
$63$	−4.49396	−0.566186
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.60388	−0.197423
$67$	2.66487	0.325566	0.162783	−	0.986662i	$-0.447953\pi$
$67$	2.66487	0.325566	0.162783	+	0.986662i	$0.447953\pi$
$68$	−1.40581	−0.170480
$69$	3.82371	0.460320
$70$	−1.20775	−0.144354
$71$	10.2959	1.22190	0.610949	−	0.791670i	$-0.290788\pi$
$71$	10.2959	1.22190	0.610949	+	0.791670i	$0.290788\pi$
$72$	−2.80194	−0.330212
$73$	1.40581	0.164538	0.0822690	−	0.996610i	$-0.473783\pi$
$73$	1.40581	0.164538	0.0822690	+	0.996610i	$0.473783\pi$
$74$	−1.32975	−0.154580
$75$	1.97285	0.227805
$76$	1.00000	0.114708
$77$	5.78017	0.658711
$78$	0	0
$79$	11.2838	1.26953	0.634764	−	0.772706i	$-0.281097\pi$
$79$	11.2838	1.26953	0.634764	+	0.772706i	$0.281097\pi$
$80$	−0.753020	−0.0841902
$81$	7.25667	0.806296
$82$	−2.61596	−0.288884
$83$	−3.10992	−0.341358	−0.170679	−	0.985327i	$-0.554596\pi$
$83$	−3.10992	−0.341358	−0.170679	+	0.985327i	$0.554596\pi$
$84$	−0.713792	−0.0778811
$85$	1.05861	0.114822
$86$	1.78017	0.191960
$87$	0.615957	0.0660375
$88$	3.60388	0.384174
$89$	2.19567	0.232740	0.116370	−	0.993206i	$-0.462874\pi$
$89$	2.19567	0.232740	0.116370	+	0.993206i	$0.462874\pi$
$90$	2.10992	0.222405
$91$	0	0
$92$	−8.59179	−0.895756
$93$	1.30798	0.135631
$94$	−0.219833	−0.0226740
$95$	−0.753020	−0.0772583
$96$	−0.445042	−0.0454219
$97$	−13.0858	−1.32866	−0.664328	−	0.747441i	$-0.731282\pi$
$97$	−13.0858	−1.32866	−0.664328	+	0.747441i	$0.731282\pi$
$98$	−4.42758	−0.447253
$99$	−10.0978	−1.01487
$100$	−4.43296	−0.443296
$101$	−7.16852	−0.713295	−0.356647	−	0.934239i	$-0.616080\pi$
$101$	−7.16852	−0.713295	−0.356647	+	0.934239i	$0.616080\pi$
$102$	0.625646	0.0619482
$103$	13.5646	1.33656	0.668282	−	0.743908i	$-0.267030\pi$
$103$	13.5646	1.33656	0.668282	+	0.743908i	$0.267030\pi$
$104$	0	0
$105$	0.537500	0.0524546
$106$	−5.50604	−0.534794
$107$	1.18598	0.114653	0.0573265	−	0.998355i	$-0.481742\pi$
$107$	1.18598	0.114653	0.0573265	+	0.998355i	$0.481742\pi$
$108$	2.58211	0.248463
$109$	−16.9879	−1.62715	−0.813574	−	0.581462i	$-0.802481\pi$
$109$	−16.9879	−1.62715	−0.813574	+	0.581462i	$0.802481\pi$
$110$	−2.71379	−0.258750
$111$	0.591794	0.0561706
$112$	1.60388	0.151552
$113$	4.89008	0.460020	0.230010	−	0.973188i	$-0.426124\pi$
$113$	4.89008	0.460020	0.230010	+	0.973188i	$0.426124\pi$
$114$	−0.445042	−0.0416820
$115$	6.46980	0.603312
$116$	−1.38404	−0.128505
$117$	0	0
$118$	−1.42327	−0.131023
$119$	−2.25475	−0.206693
$120$	0.335126	0.0305926
$121$	1.98792	0.180720
$122$	−10.4547	−0.946527
$123$	1.16421	0.104973
$124$	−2.93900	−0.263930
$125$	7.10321	0.635331
$126$	−4.49396	−0.400354
$127$	−13.9511	−1.23796	−0.618979	−	0.785407i	$-0.712453\pi$
$127$	−13.9511	−1.23796	−0.618979	+	0.785407i	$0.712453\pi$
$128$	1.00000	0.0883883
$129$	−0.792249	−0.0697536
$130$	0	0
$131$	−2.39612	−0.209350	−0.104675	−	0.994506i	$-0.533380\pi$
$131$	−2.39612	−0.209350	−0.104675	+	0.994506i	$0.533380\pi$
$132$	−1.60388	−0.139599
$133$	1.60388	0.139074
$134$	2.66487	0.230210
$135$	−1.94438	−0.167345
$136$	−1.40581	−0.120547
$137$	3.81940	0.326313	0.163157	−	0.986600i	$-0.447832\pi$
$137$	3.81940	0.326313	0.163157	+	0.986600i	$0.447832\pi$
$138$	3.82371	0.325496
$139$	−20.2935	−1.72127	−0.860636	−	0.509220i	$-0.829934\pi$
$139$	−20.2935	−1.72127	−0.860636	+	0.509220i	$0.829934\pi$
$140$	−1.20775	−0.102074
$141$	0.0978347	0.00823917
$142$	10.2959	0.864012
$143$	0	0
$144$	−2.80194	−0.233495
$145$	1.04221	0.0865510
$146$	1.40581	0.116346
$147$	1.97046	0.162521
$148$	−1.32975	−0.109305
$149$	−22.7114	−1.86059	−0.930295	−	0.366812i	$-0.880449\pi$
$149$	−22.7114	−1.86059	−0.930295	+	0.366812i	$0.880449\pi$
$150$	1.97285	0.161083
$151$	0.472189	0.0384262	0.0192131	−	0.999815i	$-0.493884\pi$
$151$	0.472189	0.0384262	0.0192131	+	0.999815i	$0.493884\pi$
$152$	1.00000	0.0811107
$153$	3.93900	0.318449
$154$	5.78017	0.465779
$155$	2.21313	0.177763
$156$	0	0
$157$	3.74764	0.299095	0.149547	−	0.988755i	$-0.452218\pi$
$157$	3.74764	0.299095	0.149547	+	0.988755i	$0.452218\pi$
$158$	11.2838	0.897692
$159$	2.45042	0.194331
$160$	−0.753020	−0.0595315
$161$	−13.7802	−1.08603
$162$	7.25667	0.570138
$163$	4.59179	0.359657	0.179829	−	0.983698i	$-0.442446\pi$
$163$	4.59179	0.359657	0.179829	+	0.983698i	$0.442446\pi$
$164$	−2.61596	−0.204272
$165$	1.20775	0.0940233
$166$	−3.10992	−0.241376
$167$	12.0248	0.930503	0.465252	−	0.885178i	$-0.345964\pi$
$167$	12.0248	0.930503	0.465252	+	0.885178i	$0.345964\pi$
$168$	−0.713792	−0.0550702
$169$	0	0
$170$	1.05861	0.0811914
$171$	−2.80194	−0.214270
$172$	1.78017	0.135736
$173$	4.84654	0.368476	0.184238	−	0.982882i	$-0.441018\pi$
$173$	4.84654	0.368476	0.184238	+	0.982882i	$0.441018\pi$
$174$	0.615957	0.0466956
$175$	−7.10992	−0.537459
$176$	3.60388	0.271652
$177$	0.633415	0.0476104
$178$	2.19567	0.164572
$179$	0.833397	0.0622910	0.0311455	−	0.999515i	$-0.490084\pi$
$179$	0.833397	0.0622910	0.0311455	+	0.999515i	$0.490084\pi$
$180$	2.10992	0.157264
$181$	24.1957	1.79845	0.899225	−	0.437487i	$-0.144131\pi$
$181$	24.1957	1.79845	0.899225	+	0.437487i	$0.144131\pi$
$182$	0	0
$183$	4.65279	0.343944
$184$	−8.59179	−0.633395
$185$	1.00133	0.0736191
$186$	1.30798	0.0959056
$187$	−5.06638	−0.370490
$188$	−0.219833	−0.0160329
$189$	4.14138	0.301241
$190$	−0.753020	−0.0546298
$191$	0.811626	0.0587272	0.0293636	−	0.999569i	$-0.490652\pi$
$191$	0.811626	0.0587272	0.0293636	+	0.999569i	$0.490652\pi$
$192$	−0.445042	−0.0321181
$193$	−20.9530	−1.50823	−0.754115	−	0.656742i	$-0.771934\pi$
$193$	−20.9530	−1.50823	−0.754115	+	0.656742i	$0.771934\pi$
$194$	−13.0858	−0.939502
$195$	0	0
$196$	−4.42758	−0.316256
$197$	−14.3153	−1.01992	−0.509961	−	0.860198i	$-0.670340\pi$
$197$	−14.3153	−1.01992	−0.509961	+	0.860198i	$0.670340\pi$
$198$	−10.0978	−0.717622
$199$	−3.83446	−0.271818	−0.135909	−	0.990721i	$-0.543395\pi$
$199$	−3.83446	−0.271818	−0.135909	+	0.990721i	$0.543395\pi$
$200$	−4.43296	−0.313458
$201$	−1.18598	−0.0836526
$202$	−7.16852	−0.504375
$203$	−2.21983	−0.155802
$204$	0.625646	0.0438040
$205$	1.96987	0.137582
$206$	13.5646	0.945094
$207$	24.0737	1.67324
$208$	0	0
$209$	3.60388	0.249285
$210$	0.537500	0.0370910
$211$	−13.0804	−0.900490	−0.450245	−	0.892905i	$-0.648663\pi$
$211$	−13.0804	−0.900490	−0.450245	+	0.892905i	$0.648663\pi$
$212$	−5.50604	−0.378156
$213$	−4.58211	−0.313961
$214$	1.18598	0.0810720
$215$	−1.34050	−0.0914215
$216$	2.58211	0.175690
$217$	−4.71379	−0.319993
$218$	−16.9879	−1.15057
$219$	−0.625646	−0.0422772
$220$	−2.71379	−0.182964
$221$	0	0
$222$	0.591794	0.0397186
$223$	−26.0084	−1.74165	−0.870824	−	0.491594i	$-0.836414\pi$
$223$	−26.0084	−1.74165	−0.870824	+	0.491594i	$0.836414\pi$
$224$	1.60388	0.107163
$225$	12.4209	0.828059
$226$	4.89008	0.325284
$227$	−11.5550	−0.766930	−0.383465	−	0.923556i	$-0.625269\pi$
$227$	−11.5550	−0.766930	−0.383465	+	0.923556i	$0.625269\pi$
$228$	−0.445042	−0.0294736
$229$	−4.03923	−0.266920	−0.133460	−	0.991054i	$-0.542609\pi$
$229$	−4.03923	−0.266920	−0.133460	+	0.991054i	$0.542609\pi$
$230$	6.46980	0.426606
$231$	−2.57242	−0.169253
$232$	−1.38404	−0.0908669
$233$	−8.09352	−0.530224	−0.265112	−	0.964218i	$-0.585409\pi$
$233$	−8.09352	−0.530224	−0.265112	+	0.964218i	$0.585409\pi$
$234$	0	0
$235$	0.165538	0.0107985
$236$	−1.42327	−0.0926471
$237$	−5.02177	−0.326199
$238$	−2.25475	−0.146154
$239$	22.9879	1.48696	0.743482	−	0.668755i	$-0.233173\pi$
$239$	22.9879	1.48696	0.743482	+	0.668755i	$0.233173\pi$
$240$	0.335126	0.0216323
$241$	−0.483206	−0.0311260	−0.0155630	−	0.999879i	$-0.504954\pi$
$241$	−0.483206	−0.0311260	−0.0155630	+	0.999879i	$0.504954\pi$
$242$	1.98792	0.127788
$243$	−10.9758	−0.704100
$244$	−10.4547	−0.669296
$245$	3.33406	0.213005
$246$	1.16421	0.0742273
$247$	0	0
$248$	−2.93900	−0.186627
$249$	1.38404	0.0877102
$250$	7.10321	0.449247
$251$	20.2198	1.27626	0.638132	−	0.769927i	$-0.279707\pi$
$251$	20.2198	1.27626	0.638132	+	0.769927i	$0.279707\pi$
$252$	−4.49396	−0.283093
$253$	−30.9638	−1.94667
$254$	−13.9511	−0.875369
$255$	−0.471124	−0.0295029
$256$	1.00000	0.0625000
$257$	−11.6823	−0.728724	−0.364362	−	0.931257i	$-0.618713\pi$
$257$	−11.6823	−0.728724	−0.364362	+	0.931257i	$0.618713\pi$
$258$	−0.792249	−0.0493233
$259$	−2.13275	−0.132523
$260$	0	0
$261$	3.87800	0.240042
$262$	−2.39612	−0.148033
$263$	−11.8345	−0.729744	−0.364872	−	0.931058i	$-0.618887\pi$
$263$	−11.8345	−0.729744	−0.364872	+	0.931058i	$0.618887\pi$
$264$	−1.60388	−0.0987117
$265$	4.14616	0.254697
$266$	1.60388	0.0983399
$267$	−0.977165	−0.0598015
$268$	2.66487	0.162783
$269$	−10.1414	−0.618331	−0.309165	−	0.951008i	$-0.600050\pi$
$269$	−10.1414	−0.618331	−0.309165	+	0.951008i	$0.600050\pi$
$270$	−1.94438	−0.118331
$271$	−5.40342	−0.328234	−0.164117	−	0.986441i	$-0.552478\pi$
$271$	−5.40342	−0.328234	−0.164117	+	0.986441i	$0.552478\pi$
$272$	−1.40581	−0.0852399
$273$	0	0
$274$	3.81940	0.230738
$275$	−15.9758	−0.963379
$276$	3.82371	0.230160
$277$	−0.204767	−0.0123033	−0.00615163	−	0.999981i	$-0.501958\pi$
$277$	−0.204767	−0.0123033	−0.00615163	+	0.999981i	$0.501958\pi$
$278$	−20.2935	−1.21712
$279$	8.23490	0.493010
$280$	−1.20775	−0.0721769
$281$	−20.5483	−1.22581	−0.612903	−	0.790158i	$-0.709998\pi$
$281$	−20.5483	−1.22581	−0.612903	+	0.790158i	$0.709998\pi$
$282$	0.0978347	0.00582597
$283$	−28.8418	−1.71446	−0.857232	−	0.514930i	$-0.827818\pi$
$283$	−28.8418	−1.71446	−0.857232	+	0.514930i	$0.827818\pi$
$284$	10.2959	0.610949
$285$	0.335126	0.0198511
$286$	0	0
$287$	−4.19567	−0.247663
$288$	−2.80194	−0.165106
$289$	−15.0237	−0.883746
$290$	1.04221	0.0612008
$291$	5.82371	0.341392
$292$	1.40581	0.0822690
$293$	−13.4819	−0.787620	−0.393810	−	0.919192i	$-0.628843\pi$
$293$	−13.4819	−0.787620	−0.393810	+	0.919192i	$0.628843\pi$
$294$	1.97046	0.114920
$295$	1.07175	0.0623998
$296$	−1.32975	−0.0772901
$297$	9.30559	0.539965
$298$	−22.7114	−1.31564
$299$	0	0
$300$	1.97285	0.113903
$301$	2.85517	0.164569
$302$	0.472189	0.0271714
$303$	3.19029	0.183278
$304$	1.00000	0.0573539
$305$	7.87263	0.450785
$306$	3.93900	0.225178
$307$	21.6286	1.23441	0.617206	−	0.786802i	$-0.288265\pi$
$307$	21.6286	1.23441	0.617206	+	0.786802i	$0.288265\pi$
$308$	5.78017	0.329356
$309$	−6.03684	−0.343424
$310$	2.21313	0.125697
$311$	22.2392	1.26107	0.630535	−	0.776161i	$-0.282836\pi$
$311$	22.2392	1.26107	0.630535	+	0.776161i	$0.282836\pi$
$312$	0	0
$313$	13.7952	0.779753	0.389876	−	0.920867i	$-0.372518\pi$
$313$	13.7952	0.779753	0.389876	+	0.920867i	$0.372518\pi$
$314$	3.74764	0.211492
$315$	3.38404	0.190669
$316$	11.2838	0.634764
$317$	20.1172	1.12990	0.564948	−	0.825127i	$-0.308896\pi$
$317$	20.1172	1.12990	0.564948	+	0.825127i	$0.308896\pi$
$318$	2.45042	0.137413
$319$	−4.98792	−0.279270
$320$	−0.753020	−0.0420951
$321$	−0.527811	−0.0294595
$322$	−13.7802	−0.767939
$323$	−1.40581	−0.0782215
$324$	7.25667	0.403148
$325$	0	0
$326$	4.59179	0.254316
$327$	7.56033	0.418087
$328$	−2.61596	−0.144442
$329$	−0.352584	−0.0194386
$330$	1.20775	0.0664845
$331$	19.2131	1.05605	0.528025	−	0.849229i	$-0.322933\pi$
$331$	19.2131	1.05605	0.528025	+	0.849229i	$0.322933\pi$
$332$	−3.10992	−0.170679
$333$	3.72587	0.204177
$334$	12.0248	0.657965
$335$	−2.00670	−0.109638
$336$	−0.713792	−0.0389405
$337$	3.35988	0.183024	0.0915122	−	0.995804i	$-0.470830\pi$
$337$	3.35988	0.183024	0.0915122	+	0.995804i	$0.470830\pi$
$338$	0	0
$339$	−2.17629	−0.118200
$340$	1.05861	0.0574110
$341$	−10.5918	−0.573578
$342$	−2.80194	−0.151511
$343$	−18.3284	−0.989642
$344$	1.78017	0.0959802
$345$	−2.87933	−0.155018
$346$	4.84654	0.260552
$347$	−15.2513	−0.818732	−0.409366	−	0.912370i	$-0.634250\pi$
$347$	−15.2513	−0.818732	−0.409366	+	0.912370i	$0.634250\pi$
$348$	0.615957	0.0330188
$349$	−8.26875	−0.442616	−0.221308	−	0.975204i	$-0.571033\pi$
$349$	−8.26875	−0.442616	−0.221308	+	0.975204i	$0.571033\pi$
$350$	−7.10992	−0.380041
$351$	0	0
$352$	3.60388	0.192087
$353$	−18.0683	−0.961678	−0.480839	−	0.876809i	$-0.659668\pi$
$353$	−18.0683	−0.961678	−0.480839	+	0.876809i	$0.659668\pi$
$354$	0.633415	0.0336657
$355$	−7.75302	−0.411488
$356$	2.19567	0.116370
$357$	1.00346	0.0531086
$358$	0.833397	0.0440464
$359$	36.8116	1.94284	0.971422	−	0.237360i	$-0.0762822\pi$
$359$	36.8116	1.94284	0.971422	+	0.237360i	$0.0762822\pi$
$360$	2.10992	0.111202
$361$	1.00000	0.0526316
$362$	24.1957	1.27170
$363$	−0.884707	−0.0464351
$364$	0	0
$365$	−1.05861	−0.0554100
$366$	4.65279	0.243205
$367$	11.1099	0.579933	0.289966	−	0.957037i	$-0.406356\pi$
$367$	11.1099	0.579933	0.289966	+	0.957037i	$0.406356\pi$
$368$	−8.59179	−0.447878
$369$	7.32975	0.381571
$370$	1.00133	0.0520566
$371$	−8.83100	−0.458483
$372$	1.30798	0.0678155
$373$	−11.0750	−0.573442	−0.286721	−	0.958014i	$-0.592565\pi$
$373$	−11.0750	−0.573442	−0.286721	+	0.958014i	$0.592565\pi$
$374$	−5.06638	−0.261976
$375$	−3.16123	−0.163245
$376$	−0.219833	−0.0113370
$377$	0	0
$378$	4.14138	0.213009
$379$	−17.6233	−0.905246	−0.452623	−	0.891702i	$-0.649512\pi$
$379$	−17.6233	−0.905246	−0.452623	+	0.891702i	$0.649512\pi$
$380$	−0.753020	−0.0386291
$381$	6.20882	0.318087
$382$	0.811626	0.0415264
$383$	−13.2034	−0.674664	−0.337332	−	0.941386i	$-0.609525\pi$
$383$	−13.2034	−0.674664	−0.337332	+	0.941386i	$0.609525\pi$
$384$	−0.445042	−0.0227109
$385$	−4.35258	−0.221828
$386$	−20.9530	−1.06648
$387$	−4.98792	−0.253550
$388$	−13.0858	−0.664328
$389$	−26.9396	−1.36589	−0.682946	−	0.730469i	$-0.739301\pi$
$389$	−26.9396	−1.36589	−0.682946	+	0.730469i	$0.739301\pi$
$390$	0	0
$391$	12.0785	0.610834
$392$	−4.42758	−0.223627
$393$	1.06638	0.0537915
$394$	−14.3153	−0.721193
$395$	−8.49694	−0.427528
$396$	−10.0978	−0.507435
$397$	27.5211	1.38124	0.690622	−	0.723216i	$-0.257337\pi$
$397$	27.5211	1.38124	0.690622	+	0.723216i	$0.257337\pi$
$398$	−3.83446	−0.192204
$399$	−0.713792	−0.0357343
$400$	−4.43296	−0.221648
$401$	34.8068	1.73817	0.869085	−	0.494662i	$-0.164708\pi$
$401$	34.8068	1.73817	0.869085	+	0.494662i	$0.164708\pi$
$402$	−1.18598	−0.0591513
$403$	0	0
$404$	−7.16852	−0.356647
$405$	−5.46442	−0.271529
$406$	−2.21983	−0.110168
$407$	−4.79225	−0.237543
$408$	0.625646	0.0309741
$409$	−27.0858	−1.33930	−0.669652	−	0.742675i	$-0.733557\pi$
$409$	−27.0858	−1.33930	−0.669652	+	0.742675i	$0.733557\pi$
$410$	1.96987	0.0972849
$411$	−1.69979	−0.0838445
$412$	13.5646	0.668282
$413$	−2.28275	−0.112327
$414$	24.0737	1.18316
$415$	2.34183	0.114956
$416$	0	0
$417$	9.03146	0.442272
$418$	3.60388	0.176271
$419$	15.5690	0.760593	0.380297	−	0.924865i	$-0.375822\pi$
$419$	15.5690	0.760593	0.380297	+	0.924865i	$0.375822\pi$
$420$	0.537500	0.0262273
$421$	−10.5724	−0.515268	−0.257634	−	0.966243i	$-0.582943\pi$
$421$	−10.5724	−0.515268	−0.257634	+	0.966243i	$0.582943\pi$
$422$	−13.0804	−0.636743
$423$	0.615957	0.0299489
$424$	−5.50604	−0.267397
$425$	6.23191	0.302292
$426$	−4.58211	−0.222004
$427$	−16.7681	−0.811465
$428$	1.18598	0.0573265
$429$	0	0
$430$	−1.34050	−0.0646448
$431$	8.96316	0.431740	0.215870	−	0.976422i	$-0.430741\pi$
$431$	8.96316	0.431740	0.215870	+	0.976422i	$0.430741\pi$
$432$	2.58211	0.124232
$433$	28.6219	1.37548	0.687741	−	0.725956i	$-0.258602\pi$
$433$	28.6219	1.37548	0.687741	+	0.725956i	$0.258602\pi$
$434$	−4.71379	−0.226269
$435$	−0.463828	−0.0222389
$436$	−16.9879	−0.813574
$437$	−8.59179	−0.411001
$438$	−0.625646	−0.0298945
$439$	9.25667	0.441797	0.220898	−	0.975297i	$-0.429101\pi$
$439$	9.25667	0.441797	0.220898	+	0.975297i	$0.429101\pi$
$440$	−2.71379	−0.129375
$441$	12.4058	0.590753
$442$	0	0
$443$	12.8418	0.610130	0.305065	−	0.952331i	$-0.401322\pi$
$443$	12.8418	0.610130	0.305065	+	0.952331i	$0.401322\pi$
$444$	0.591794	0.0280853
$445$	−1.65338	−0.0783779
$446$	−26.0084	−1.23153
$447$	10.1075	0.478069
$448$	1.60388	0.0757760
$449$	−15.0858	−0.711941	−0.355970	−	0.934497i	$-0.615850\pi$
$449$	−15.0858	−0.711941	−0.355970	+	0.934497i	$0.615850\pi$
$450$	12.4209	0.585526
$451$	−9.42758	−0.443928
$452$	4.89008	0.230010
$453$	−0.210144	−0.00987342
$454$	−11.5550	−0.542301
$455$	0	0
$456$	−0.445042	−0.0208410
$457$	−13.0513	−0.610515	−0.305257	−	0.952270i	$-0.598742\pi$
$457$	−13.0513	−0.610515	−0.305257	+	0.952270i	$0.598742\pi$
$458$	−4.03923	−0.188741
$459$	−3.62996	−0.169432
$460$	6.46980	0.301656
$461$	−2.75973	−0.128533	−0.0642666	−	0.997933i	$-0.520471\pi$
$461$	−2.75973	−0.128533	−0.0642666	+	0.997933i	$0.520471\pi$
$462$	−2.57242	−0.119680
$463$	29.3551	1.36425	0.682123	−	0.731237i	$-0.261057\pi$
$463$	29.3551	1.36425	0.682123	+	0.731237i	$0.261057\pi$
$464$	−1.38404	−0.0642526
$465$	−0.984935	−0.0456752
$466$	−8.09352	−0.374925
$467$	14.0871	0.651872	0.325936	−	0.945392i	$-0.394321\pi$
$467$	14.0871	0.651872	0.325936	+	0.945392i	$0.394321\pi$
$468$	0	0
$469$	4.27413	0.197361
$470$	0.165538	0.00763572
$471$	−1.66786	−0.0768509
$472$	−1.42327	−0.0655114
$473$	6.41550	0.294985
$474$	−5.02177	−0.230658
$475$	−4.43296	−0.203398
$476$	−2.25475	−0.103346
$477$	15.4276	0.706381
$478$	22.9879	1.05144
$479$	−2.41550	−0.110367	−0.0551835	−	0.998476i	$-0.517574\pi$
$479$	−2.41550	−0.110367	−0.0551835	+	0.998476i	$0.517574\pi$
$480$	0.335126	0.0152963
$481$	0	0
$482$	−0.483206	−0.0220094
$483$	6.13275	0.279050
$484$	1.98792	0.0903599
$485$	9.85384	0.447440
$486$	−10.9758	−0.497874
$487$	5.24591	0.237715	0.118858	−	0.992911i	$-0.462077\pi$
$487$	5.24591	0.237715	0.118858	+	0.992911i	$0.462077\pi$
$488$	−10.4547	−0.473263
$489$	−2.04354	−0.0924121
$490$	3.33406	0.150618
$491$	17.3491	0.782955	0.391478	−	0.920188i	$-0.371964\pi$
$491$	17.3491	0.782955	0.391478	+	0.920188i	$0.371964\pi$
$492$	1.16421	0.0524867
$493$	1.94571	0.0876302
$494$	0	0
$495$	7.60388	0.341769
$496$	−2.93900	−0.131965
$497$	16.5133	0.740724
$498$	1.38404	0.0620204
$499$	20.7573	0.929226	0.464613	−	0.885514i	$-0.346193\pi$
$499$	20.7573	0.929226	0.464613	+	0.885514i	$0.346193\pi$
$500$	7.10321	0.317665
$501$	−5.35152	−0.239088
$502$	20.2198	0.902455
$503$	−2.67887	−0.119445	−0.0597226	−	0.998215i	$-0.519022\pi$
$503$	−2.67887	−0.119445	−0.0597226	+	0.998215i	$0.519022\pi$
$504$	−4.49396	−0.200177
$505$	5.39804	0.240210
$506$	−30.9638	−1.37651
$507$	0	0
$508$	−13.9511	−0.618979
$509$	21.5060	0.953238	0.476619	−	0.879110i	$-0.341862\pi$
$509$	21.5060	0.953238	0.476619	+	0.879110i	$0.341862\pi$
$510$	−0.471124	−0.0208617
$511$	2.25475	0.0997442
$512$	1.00000	0.0441942
$513$	2.58211	0.114003
$514$	−11.6823	−0.515286
$515$	−10.2145	−0.450103
$516$	−0.792249	−0.0348768
$517$	−0.792249	−0.0348431
$518$	−2.13275	−0.0937077
$519$	−2.15691	−0.0946780
$520$	0	0
$521$	15.2513	0.668171	0.334086	−	0.942543i	$-0.391573\pi$
$521$	15.2513	0.668171	0.334086	+	0.942543i	$0.391573\pi$
$522$	3.87800	0.169736
$523$	−4.98792	−0.218106	−0.109053	−	0.994036i	$-0.534782\pi$
$523$	−4.98792	−0.218106	−0.109053	+	0.994036i	$0.534782\pi$
$524$	−2.39612	−0.104675
$525$	3.16421	0.138097
$526$	−11.8345	−0.516007
$527$	4.13169	0.179979
$528$	−1.60388	−0.0697997
$529$	50.8189	2.20952
$530$	4.14616	0.180098
$531$	3.98792	0.173061
$532$	1.60388	0.0695368
$533$	0	0
$534$	−0.977165	−0.0422861
$535$	−0.893068	−0.0386107
$536$	2.66487	0.115105
$537$	−0.370896	−0.0160054
$538$	−10.1414	−0.437226
$539$	−15.9565	−0.687293
$540$	−1.94438	−0.0836727
$541$	−6.85192	−0.294587	−0.147294	−	0.989093i	$-0.547056\pi$
$541$	−6.85192	−0.294587	−0.147294	+	0.989093i	$0.547056\pi$
$542$	−5.40342	−0.232097
$543$	−10.7681	−0.462103
$544$	−1.40581	−0.0602737
$545$	12.7922	0.547960
$546$	0	0
$547$	−15.4058	−0.658705	−0.329353	−	0.944207i	$-0.606830\pi$
$547$	−15.4058	−0.658705	−0.329353	+	0.944207i	$0.606830\pi$
$548$	3.81940	0.163157
$549$	29.2935	1.25022
$550$	−15.9758	−0.681212
$551$	−1.38404	−0.0589622
$552$	3.82371	0.162748
$553$	18.0978	0.769598
$554$	−0.204767	−0.00869972
$555$	−0.445633	−0.0189161
$556$	−20.2935	−0.860636
$557$	−17.5754	−0.744694	−0.372347	−	0.928094i	$-0.621447\pi$
$557$	−17.5754	−0.744694	−0.372347	+	0.928094i	$0.621447\pi$
$558$	8.23490	0.348611
$559$	0	0
$560$	−1.20775	−0.0510368
$561$	2.25475	0.0951956
$562$	−20.5483	−0.866776
$563$	37.0804	1.56275	0.781376	−	0.624061i	$-0.214518\pi$
$563$	37.0804	1.56275	0.781376	+	0.624061i	$0.214518\pi$
$564$	0.0978347	0.00411958
$565$	−3.68233	−0.154917
$566$	−28.8418	−1.21231
$567$	11.6388	0.488783
$568$	10.2959	0.432006
$569$	−6.49396	−0.272241	−0.136120	−	0.990692i	$-0.543463\pi$
$569$	−6.49396	−0.272241	−0.136120	+	0.990692i	$0.543463\pi$
$570$	0.335126	0.0140369
$571$	−13.4491	−0.562827	−0.281413	−	0.959587i	$-0.590803\pi$
$571$	−13.4491	−0.562827	−0.281413	+	0.959587i	$0.590803\pi$
$572$	0	0
$573$	−0.361208	−0.0150897
$574$	−4.19567	−0.175124
$575$	38.0871	1.58834
$576$	−2.80194	−0.116747
$577$	−6.91962	−0.288068	−0.144034	−	0.989573i	$-0.546007\pi$
$577$	−6.91962	−0.288068	−0.144034	+	0.989573i	$0.546007\pi$
$578$	−15.0237	−0.624903
$579$	9.32496	0.387532
$580$	1.04221	0.0432755
$581$	−4.98792	−0.206934
$582$	5.82371	0.241400
$583$	−19.8431	−0.821817
$584$	1.40581	0.0581730
$585$	0	0
$586$	−13.4819	−0.556931
$587$	−31.6340	−1.30568	−0.652838	−	0.757498i	$-0.726422\pi$
$587$	−31.6340	−1.30568	−0.652838	+	0.757498i	$0.726422\pi$
$588$	1.97046	0.0812604
$589$	−2.93900	−0.121099
$590$	1.07175	0.0441234
$591$	6.37090	0.262064
$592$	−1.32975	−0.0546523
$593$	−39.3793	−1.61711	−0.808556	−	0.588419i	$-0.799751\pi$
$593$	−39.3793	−1.61711	−0.808556	+	0.588419i	$0.799751\pi$
$594$	9.30559	0.381813
$595$	1.69787	0.0696060
$596$	−22.7114	−0.930295
$597$	1.70650	0.0698422
$598$	0	0
$599$	−7.18300	−0.293489	−0.146745	−	0.989174i	$-0.546880\pi$
$599$	−7.18300	−0.293489	−0.146745	+	0.989174i	$0.546880\pi$
$600$	1.97285	0.0805414
$601$	4.90084	0.199909	0.0999547	−	0.994992i	$-0.468130\pi$
$601$	4.90084	0.199909	0.0999547	+	0.994992i	$0.468130\pi$
$602$	2.85517	0.116368
$603$	−7.46681	−0.304072
$604$	0.472189	0.0192131
$605$	−1.49694	−0.0608594
$606$	3.19029	0.129597
$607$	35.2180	1.42946	0.714728	−	0.699403i	$-0.246551\pi$
$607$	35.2180	1.42946	0.714728	+	0.699403i	$0.246551\pi$
$608$	1.00000	0.0405554
$609$	0.987918	0.0400325
$610$	7.87263	0.318753
$611$	0	0
$612$	3.93900	0.159225
$613$	10.5381	0.425629	0.212815	−	0.977093i	$-0.431737\pi$
$613$	10.5381	0.425629	0.212815	+	0.977093i	$0.431737\pi$
$614$	21.6286	0.872860
$615$	−0.876674	−0.0353509
$616$	5.78017	0.232890
$617$	−30.9855	−1.24743	−0.623715	−	0.781652i	$-0.714377\pi$
$617$	−30.9855	−1.24743	−0.623715	+	0.781652i	$0.714377\pi$
$618$	−6.03684	−0.242837
$619$	−7.95167	−0.319605	−0.159802	−	0.987149i	$-0.551086\pi$
$619$	−7.95167	−0.319605	−0.159802	+	0.987149i	$0.551086\pi$
$620$	2.21313	0.0888813
$621$	−22.1849	−0.890250
$622$	22.2392	0.891711
$623$	3.52158	0.141089
$624$	0	0
$625$	16.8159	0.672638
$626$	13.7952	0.551368
$627$	−1.60388	−0.0640526
$628$	3.74764	0.149547
$629$	1.86938	0.0745370
$630$	3.38404	0.134823
$631$	−27.8345	−1.10807	−0.554036	−	0.832493i	$-0.686913\pi$
$631$	−27.8345	−1.10807	−0.554036	+	0.832493i	$0.686913\pi$
$632$	11.2838	0.448846
$633$	5.82132	0.231377
$634$	20.1172	0.798957
$635$	10.5054	0.416896
$636$	2.45042	0.0971654
$637$	0	0
$638$	−4.98792	−0.197474
$639$	−28.8485	−1.14123
$640$	−0.753020	−0.0297657
$641$	7.93230	0.313307	0.156653	−	0.987654i	$-0.449929\pi$
$641$	7.93230	0.313307	0.156653	+	0.987654i	$0.449929\pi$
$642$	−0.527811	−0.0208310
$643$	−17.6146	−0.694653	−0.347327	−	0.937744i	$-0.612910\pi$
$643$	−17.6146	−0.694653	−0.347327	+	0.937744i	$0.612910\pi$
$644$	−13.7802	−0.543015
$645$	0.596580	0.0234903
$646$	−1.40581	−0.0553110
$647$	26.6413	1.04738	0.523689	−	0.851910i	$-0.324555\pi$
$647$	26.6413	1.04738	0.523689	+	0.851910i	$0.324555\pi$
$648$	7.25667	0.285069
$649$	−5.12929	−0.201342
$650$	0	0
$651$	2.09783	0.0822206
$652$	4.59179	0.179829
$653$	−6.55927	−0.256684	−0.128342	−	0.991730i	$-0.540966\pi$
$653$	−6.55927	−0.256684	−0.128342	+	0.991730i	$0.540966\pi$
$654$	7.56033	0.295632
$655$	1.80433	0.0705010
$656$	−2.61596	−0.102136
$657$	−3.93900	−0.153675
$658$	−0.352584	−0.0137452
$659$	−39.9172	−1.55495	−0.777477	−	0.628911i	$-0.783501\pi$
$659$	−39.9172	−1.55495	−0.777477	+	0.628911i	$0.783501\pi$
$660$	1.20775	0.0470116
$661$	−29.6582	−1.15357	−0.576785	−	0.816896i	$-0.695693\pi$
$661$	−29.6582	−1.15357	−0.576785	+	0.816896i	$0.695693\pi$
$662$	19.2131	0.746739
$663$	0	0
$664$	−3.10992	−0.120688
$665$	−1.20775	−0.0468346
$666$	3.72587	0.144375
$667$	11.8914	0.460437
$668$	12.0248	0.465252
$669$	11.5748	0.447508
$670$	−2.00670	−0.0775258
$671$	−37.6775	−1.45453
$672$	−0.713792	−0.0275351
$673$	11.8974	0.458610	0.229305	−	0.973355i	$-0.426355\pi$
$673$	11.8974	0.458610	0.229305	+	0.973355i	$0.426355\pi$
$674$	3.35988	0.129418
$675$	−11.4464	−0.440571
$676$	0	0
$677$	25.3840	0.975588	0.487794	−	0.872959i	$-0.337802\pi$
$677$	25.3840	0.975588	0.487794	+	0.872959i	$0.337802\pi$
$678$	−2.17629	−0.0835800
$679$	−20.9879	−0.805442
$680$	1.05861	0.0405957
$681$	5.14244	0.197059
$682$	−10.5918	−0.405581
$683$	28.3655	1.08538	0.542688	−	0.839934i	$-0.317407\pi$
$683$	28.3655	1.08538	0.542688	+	0.839934i	$0.317407\pi$
$684$	−2.80194	−0.107135
$685$	−2.87608	−0.109889
$686$	−18.3284	−0.699782
$687$	1.79763	0.0685837
$688$	1.78017	0.0678682
$689$	0	0
$690$	−2.87933	−0.109614
$691$	−12.3854	−0.471162	−0.235581	−	0.971855i	$-0.575699\pi$
$691$	−12.3854	−0.471162	−0.235581	+	0.971855i	$0.575699\pi$
$692$	4.84654	0.184238
$693$	−16.1957	−0.615223
$694$	−15.2513	−0.578931
$695$	15.2814	0.579657
$696$	0.615957	0.0233478
$697$	3.67755	0.139297
$698$	−8.26875	−0.312977
$699$	3.60196	0.136239
$700$	−7.10992	−0.268730
$701$	−8.17092	−0.308611	−0.154306	−	0.988023i	$-0.549314\pi$
$701$	−8.17092	−0.308611	−0.154306	+	0.988023i	$0.549314\pi$
$702$	0	0
$703$	−1.32975	−0.0501524
$704$	3.60388	0.135826
$705$	−0.0736715	−0.00277463
$706$	−18.0683	−0.680009
$707$	−11.4974	−0.432405
$708$	0.633415	0.0238052
$709$	34.6679	1.30198	0.650989	−	0.759087i	$-0.274354\pi$
$709$	34.6679	1.30198	0.650989	+	0.759087i	$0.274354\pi$
$710$	−7.75302	−0.290966
$711$	−31.6165	−1.18571
$712$	2.19567	0.0822862
$713$	25.2513	0.945668
$714$	1.00346	0.0375535
$715$	0	0
$716$	0.833397	0.0311455
$717$	−10.2306	−0.382068
$718$	36.8116	1.37380
$719$	−1.14483	−0.0426951	−0.0213475	−	0.999772i	$-0.506796\pi$
$719$	−1.14483	−0.0426951	−0.0213475	+	0.999772i	$0.506796\pi$
$720$	2.10992	0.0786319
$721$	21.7560	0.810236
$722$	1.00000	0.0372161
$723$	0.215047	0.00799767
$724$	24.1957	0.899225
$725$	6.13541	0.227863
$726$	−0.884707	−0.0328346
$727$	3.98062	0.147633	0.0738166	−	0.997272i	$-0.476482\pi$
$727$	3.98062	0.147633	0.0738166	+	0.997272i	$0.476482\pi$
$728$	0	0
$729$	−16.8853	−0.625381
$730$	−1.05861	−0.0391808
$731$	−2.50258	−0.0925614
$732$	4.65279	0.171972
$733$	28.6872	1.05959	0.529794	−	0.848127i	$-0.322269\pi$
$733$	28.6872	1.05959	0.529794	+	0.848127i	$0.322269\pi$
$734$	11.1099	0.410074
$735$	−1.48380	−0.0547307
$736$	−8.59179	−0.316698
$737$	9.60388	0.353763
$738$	7.32975	0.269812
$739$	43.3986	1.59644	0.798222	−	0.602363i	$-0.205774\pi$
$739$	43.3986	1.59644	0.798222	+	0.602363i	$0.205774\pi$
$740$	1.00133	0.0368095
$741$	0	0
$742$	−8.83100	−0.324196
$743$	−5.14808	−0.188865	−0.0944324	−	0.995531i	$-0.530104\pi$
$743$	−5.14808	−0.188865	−0.0944324	+	0.995531i	$0.530104\pi$
$744$	1.30798	0.0479528
$745$	17.1021	0.626574
$746$	−11.0750	−0.405485
$747$	8.71379	0.318821
$748$	−5.06638	−0.185245
$749$	1.90217	0.0695036
$750$	−3.16123	−0.115432
$751$	36.9396	1.34794	0.673972	−	0.738756i	$-0.264587\pi$
$751$	36.9396	1.34794	0.673972	+	0.738756i	$0.264587\pi$
$752$	−0.219833	−0.00801647
$753$	−8.99867	−0.327930
$754$	0	0
$755$	−0.355568	−0.0129404
$756$	4.14138	0.150620
$757$	35.5599	1.29245	0.646223	−	0.763149i	$-0.276348\pi$
$757$	35.5599	1.29245	0.646223	+	0.763149i	$0.276348\pi$
$758$	−17.6233	−0.640105
$759$	13.7802	0.500188
$760$	−0.753020	−0.0273149
$761$	28.2825	1.02524	0.512620	−	0.858616i	$-0.328675\pi$
$761$	28.2825	1.02524	0.512620	+	0.858616i	$0.328675\pi$
$762$	6.20882	0.224922
$763$	−27.2465	−0.986390
$764$	0.811626	0.0293636
$765$	−2.96615	−0.107241
$766$	−13.2034	−0.477060
$767$	0	0
$768$	−0.445042	−0.0160591
$769$	−36.1021	−1.30188	−0.650938	−	0.759131i	$-0.725624\pi$
$769$	−36.1021	−1.30188	−0.650938	+	0.759131i	$0.725624\pi$
$770$	−4.35258	−0.156856
$771$	5.19913	0.187242
$772$	−20.9530	−0.754115
$773$	−2.05429	−0.0738878	−0.0369439	−	0.999317i	$-0.511762\pi$
$773$	−2.05429	−0.0738878	−0.0369439	+	0.999317i	$0.511762\pi$
$774$	−4.98792	−0.179287
$775$	13.0285	0.467997
$776$	−13.0858	−0.469751
$777$	0.949164	0.0340511
$778$	−26.9396	−0.965831
$779$	−2.61596	−0.0937264
$780$	0	0
$781$	37.1051	1.32773
$782$	12.0785	0.431925
$783$	−3.57374	−0.127715
$784$	−4.42758	−0.158128
$785$	−2.82205	−0.100723
$786$	1.06638	0.0380364
$787$	−48.8353	−1.74079	−0.870396	−	0.492353i	$-0.836137\pi$
$787$	−48.8353	−1.74079	−0.870396	+	0.492353i	$0.836137\pi$
$788$	−14.3153	−0.509961
$789$	5.26683	0.187504
$790$	−8.49694	−0.302308
$791$	7.84309	0.278868
$792$	−10.0978	−0.358811
$793$	0	0
$794$	27.5211	0.976688
$795$	−1.84522	−0.0654430
$796$	−3.83446	−0.135909
$797$	−34.4564	−1.22051	−0.610254	−	0.792206i	$-0.708933\pi$
$797$	−34.4564	−1.22051	−0.610254	+	0.792206i	$0.708933\pi$
$798$	−0.713792	−0.0252680
$799$	0.309043	0.0109332
$800$	−4.43296	−0.156729
$801$	−6.15213	−0.217375
$802$	34.8068	1.22907
$803$	5.06638	0.178789
$804$	−1.18598	−0.0418263
$805$	10.3767	0.365732
$806$	0	0
$807$	4.51334	0.158877
$808$	−7.16852	−0.252188
$809$	−3.80971	−0.133942	−0.0669711	−	0.997755i	$-0.521334\pi$
$809$	−3.80971	−0.133942	−0.0669711	+	0.997755i	$0.521334\pi$
$810$	−5.46442	−0.192000
$811$	4.38703	0.154049	0.0770247	−	0.997029i	$-0.475458\pi$
$811$	4.38703	0.154049	0.0770247	+	0.997029i	$0.475458\pi$
$812$	−2.21983	−0.0779009
$813$	2.40475	0.0843382
$814$	−4.79225	−0.167968
$815$	−3.45771	−0.121118
$816$	0.625646	0.0219020
$817$	1.78017	0.0622802
$818$	−27.0858	−0.947031
$819$	0	0
$820$	1.96987	0.0687908
$821$	9.30127	0.324617	0.162308	−	0.986740i	$-0.448106\pi$
$821$	9.30127	0.324617	0.162308	+	0.986740i	$0.448106\pi$
$822$	−1.69979	−0.0592870
$823$	−34.1849	−1.19161	−0.595806	−	0.803129i	$-0.703167\pi$
$823$	−34.1849	−1.19161	−0.595806	+	0.803129i	$0.703167\pi$
$824$	13.5646	0.472547
$825$	7.10992	0.247536
$826$	−2.28275	−0.0794270
$827$	−14.0170	−0.487418	−0.243709	−	0.969848i	$-0.578364\pi$
$827$	−14.0170	−0.487418	−0.243709	+	0.969848i	$0.578364\pi$
$828$	24.0737	0.836618
$829$	10.1870	0.353811	0.176905	−	0.984228i	$-0.443391\pi$
$829$	10.1870	0.353811	0.176905	+	0.984228i	$0.443391\pi$
$830$	2.34183	0.0812861
$831$	0.0911299	0.00316126
$832$	0	0
$833$	6.22436	0.215661
$834$	9.03146	0.312734
$835$	−9.05489	−0.313357
$836$	3.60388	0.124643
$837$	−7.58881	−0.262308
$838$	15.5690	0.537821
$839$	32.1758	1.11083	0.555416	−	0.831572i	$-0.312559\pi$
$839$	32.1758	1.11083	0.555416	+	0.831572i	$0.312559\pi$
$840$	0.537500	0.0185455
$841$	−27.0844	−0.933946
$842$	−10.5724	−0.364350
$843$	9.14483	0.314965
$844$	−13.0804	−0.450245
$845$	0	0
$846$	0.615957	0.0211770
$847$	3.18837	0.109554
$848$	−5.50604	−0.189078
$849$	12.8358	0.440523
$850$	6.23191	0.213753
$851$	11.4249	0.391641
$852$	−4.58211	−0.156980
$853$	−23.0331	−0.788639	−0.394319	−	0.918973i	$-0.629020\pi$
$853$	−23.0331	−0.788639	−0.394319	+	0.918973i	$0.629020\pi$
$854$	−16.7681	−0.573792
$855$	2.10992	0.0721576
$856$	1.18598	0.0405360
$857$	13.9457	0.476376	0.238188	−	0.971219i	$-0.423447\pi$
$857$	13.9457	0.476376	0.238188	+	0.971219i	$0.423447\pi$
$858$	0	0
$859$	55.4094	1.89054	0.945272	−	0.326283i	$-0.105796\pi$
$859$	55.4094	1.89054	0.945272	+	0.326283i	$0.105796\pi$
$860$	−1.34050	−0.0457108
$861$	1.86725	0.0636357
$862$	8.96316	0.305287
$863$	24.8793	0.846902	0.423451	−	0.905919i	$-0.360819\pi$
$863$	24.8793	0.846902	0.423451	+	0.905919i	$0.360819\pi$
$864$	2.58211	0.0878450
$865$	−3.64955	−0.124088
$866$	28.6219	0.972613
$867$	6.68617	0.227074
$868$	−4.71379	−0.159997
$869$	40.6655	1.37948
$870$	−0.463828	−0.0157252
$871$	0	0
$872$	−16.9879	−0.575284
$873$	36.6655	1.24094
$874$	−8.59179	−0.290622
$875$	11.3927	0.385142
$876$	−0.625646	−0.0211386
$877$	−8.12200	−0.274260	−0.137130	−	0.990553i	$-0.543788\pi$
$877$	−8.12200	−0.274260	−0.137130	+	0.990553i	$0.543788\pi$
$878$	9.25667	0.312397
$879$	6.00000	0.202375
$880$	−2.71379	−0.0914819
$881$	53.2277	1.79329	0.896644	−	0.442753i	$-0.145998\pi$
$881$	53.2277	1.79329	0.896644	+	0.442753i	$0.145998\pi$
$882$	12.4058	0.417725
$883$	−27.2862	−0.918254	−0.459127	−	0.888371i	$-0.651838\pi$
$883$	−27.2862	−0.918254	−0.459127	+	0.888371i	$0.651838\pi$
$884$	0	0
$885$	−0.476975	−0.0160333
$886$	12.8418	0.431427
$887$	−40.2204	−1.35047	−0.675235	−	0.737603i	$-0.735958\pi$
$887$	−40.2204	−1.35047	−0.675235	+	0.737603i	$0.735958\pi$
$888$	0.591794	0.0198593
$889$	−22.3758	−0.750460
$890$	−1.65338	−0.0554216
$891$	26.1521	0.876129
$892$	−26.0084	−0.870824
$893$	−0.219833	−0.00735642
$894$	10.1075	0.338046
$895$	−0.627565	−0.0209772
$896$	1.60388	0.0535817
$897$	0	0
$898$	−15.0858	−0.503418
$899$	4.06770	0.135666
$900$	12.4209	0.414029
$901$	7.74046	0.257872
$902$	−9.42758	−0.313904
$903$	−1.27067	−0.0422852
$904$	4.89008	0.162642
$905$	−18.2198	−0.605648
$906$	−0.210144	−0.00698156
$907$	43.0508	1.42948	0.714740	−	0.699390i	$-0.246545\pi$
$907$	43.0508	1.42948	0.714740	+	0.699390i	$0.246545\pi$
$908$	−11.5550	−0.383465
$909$	20.0858	0.666202
$910$	0	0
$911$	−25.1250	−0.832428	−0.416214	−	0.909267i	$-0.636643\pi$
$911$	−25.1250	−0.832428	−0.416214	+	0.909267i	$0.636643\pi$
$912$	−0.445042	−0.0147368
$913$	−11.2078	−0.370922
$914$	−13.0513	−0.431699
$915$	−3.50365	−0.115827
$916$	−4.03923	−0.133460
$917$	−3.84309	−0.126910
$918$	−3.62996	−0.119806
$919$	12.1220	0.399868	0.199934	−	0.979809i	$-0.435927\pi$
$919$	12.1220	0.399868	0.199934	+	0.979809i	$0.435927\pi$
$920$	6.46980	0.213303
$921$	−9.62565	−0.317176
$922$	−2.75973	−0.0908867
$923$	0	0
$924$	−2.57242	−0.0846263
$925$	5.89472	0.193817
$926$	29.3551	0.964668
$927$	−38.0073	−1.24832
$928$	−1.38404	−0.0454334
$929$	29.9597	0.982946	0.491473	−	0.870893i	$-0.336459\pi$
$929$	29.9597	0.982946	0.491473	+	0.870893i	$0.336459\pi$
$930$	−0.984935	−0.0322973
$931$	−4.42758	−0.145108
$932$	−8.09352	−0.265112
$933$	−9.89738	−0.324026
$934$	14.0871	0.460943
$935$	3.81508	0.124767
$936$	0	0
$937$	38.9691	1.27307	0.636533	−	0.771249i	$-0.280368\pi$
$937$	38.9691	1.27307	0.636533	+	0.771249i	$0.280368\pi$
$938$	4.27413	0.139555
$939$	−6.13946	−0.200354
$940$	0.165538	0.00539927
$941$	47.4905	1.54815	0.774073	−	0.633096i	$-0.218216\pi$
$941$	47.4905	1.54815	0.774073	+	0.633096i	$0.218216\pi$
$942$	−1.66786	−0.0543418
$943$	22.4758	0.731912
$944$	−1.42327	−0.0463235
$945$	−3.11854	−0.101446
$946$	6.41550	0.208586
$947$	−30.8659	−1.00301	−0.501504	−	0.865155i	$-0.667220\pi$
$947$	−30.8659	−1.00301	−0.501504	+	0.865155i	$0.667220\pi$
$948$	−5.02177	−0.163100
$949$	0	0
$950$	−4.43296	−0.143824
$951$	−8.95300	−0.290321
$952$	−2.25475	−0.0730768
$953$	5.61729	0.181962	0.0909809	−	0.995853i	$-0.471000\pi$
$953$	5.61729	0.181962	0.0909809	+	0.995853i	$0.471000\pi$
$954$	15.4276	0.499486
$955$	−0.611171	−0.0197770
$956$	22.9879	0.743482
$957$	2.21983	0.0717570
$958$	−2.41550	−0.0780413
$959$	6.12584	0.197814
$960$	0.335126	0.0108161
$961$	−22.3623	−0.721364
$962$	0	0
$963$	−3.32304	−0.107084
$964$	−0.483206	−0.0155630
$965$	15.7780	0.507913
$966$	6.13275	0.197318
$967$	7.40821	0.238232	0.119116	−	0.992880i	$-0.461994\pi$
$967$	7.40821	0.238232	0.119116	+	0.992880i	$0.461994\pi$
$968$	1.98792	0.0638941
$969$	0.625646	0.0200986
$970$	9.85384	0.316388
$971$	4.55363	0.146133	0.0730665	−	0.997327i	$-0.476721\pi$
$971$	4.55363	0.146133	0.0730665	+	0.997327i	$0.476721\pi$
$972$	−10.9758	−0.352050
$973$	−32.5483	−1.04345
$974$	5.24591	0.168090
$975$	0	0
$976$	−10.4547	−0.334648
$977$	58.9724	1.88669	0.943347	−	0.331808i	$-0.107659\pi$
$977$	58.9724	1.88669	0.943347	+	0.331808i	$0.107659\pi$
$978$	−2.04354	−0.0653452
$979$	7.91292	0.252898
$980$	3.33406	0.106503
$981$	47.5991	1.51972
$982$	17.3491	0.553633
$983$	−46.4432	−1.48131	−0.740655	−	0.671886i	$-0.765485\pi$
$983$	−46.4432	−1.48131	−0.740655	+	0.671886i	$0.765485\pi$
$984$	1.16421	0.0371137
$985$	10.7797	0.343470
$986$	1.94571	0.0619639
$987$	0.156915	0.00499465
$988$	0	0
$989$	−15.2948	−0.486347
$990$	7.60388	0.241667
$991$	−46.0828	−1.46387	−0.731934	−	0.681376i	$-0.761382\pi$
$991$	−46.0828	−1.46387	−0.731934	+	0.681376i	$0.761382\pi$
$992$	−2.93900	−0.0933134
$993$	−8.55065	−0.271347
$994$	16.5133	0.523771
$995$	2.88743	0.0915376
$996$	1.38404	0.0438551
$997$	−3.04759	−0.0965181	−0.0482591	−	0.998835i	$-0.515367\pi$
$997$	−3.04759	−0.0965181	−0.0482591	+	0.998835i	$0.515367\pi$
$998$	20.7573	0.657062
$999$	−3.43355	−0.108633

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.v.1.2	yes	3
13.12	even	2		6422.2.a.n.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.n.1.2	✓	3	13.12	even	2
6422.2.a.v.1.2	yes	3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.445042 q^{3} +1.00000 q^{4} -0.753020 q^{5} -0.445042 q^{6} +1.60388 q^{7} +1.00000 q^{8} -2.80194 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.445042 q^{3} +1.00000 q^{4} -0.753020 q^{5} -0.445042 q^{6} +1.60388 q^{7} +1.00000 q^{8} -2.80194 q^{9} -0.753020 q^{10} +3.60388 q^{11} -0.445042 q^{12} +1.60388 q^{14} +0.335126 q^{15} +1.00000 q^{16} -1.40581 q^{17} -2.80194 q^{18} +1.00000 q^{19} -0.753020 q^{20} -0.713792 q^{21} +3.60388 q^{22} -8.59179 q^{23} -0.445042 q^{24} -4.43296 q^{25} +2.58211 q^{27} +1.60388 q^{28} -1.38404 q^{29} +0.335126 q^{30} -2.93900 q^{31} +1.00000 q^{32} -1.60388 q^{33} -1.40581 q^{34} -1.20775 q^{35} -2.80194 q^{36} -1.32975 q^{37} +1.00000 q^{38} -0.753020 q^{40} -2.61596 q^{41} -0.713792 q^{42} +1.78017 q^{43} +3.60388 q^{44} +2.10992 q^{45} -8.59179 q^{46} -0.219833 q^{47} -0.445042 q^{48} -4.42758 q^{49} -4.43296 q^{50} +0.625646 q^{51} -5.50604 q^{53} +2.58211 q^{54} -2.71379 q^{55} +1.60388 q^{56} -0.445042 q^{57} -1.38404 q^{58} -1.42327 q^{59} +0.335126 q^{60} -10.4547 q^{61} -2.93900 q^{62} -4.49396 q^{63} +1.00000 q^{64} -1.60388 q^{66} +2.66487 q^{67} -1.40581 q^{68} +3.82371 q^{69} -1.20775 q^{70} +10.2959 q^{71} -2.80194 q^{72} +1.40581 q^{73} -1.32975 q^{74} +1.97285 q^{75} +1.00000 q^{76} +5.78017 q^{77} +11.2838 q^{79} -0.753020 q^{80} +7.25667 q^{81} -2.61596 q^{82} -3.10992 q^{83} -0.713792 q^{84} +1.05861 q^{85} +1.78017 q^{86} +0.615957 q^{87} +3.60388 q^{88} +2.19567 q^{89} +2.10992 q^{90} -8.59179 q^{92} +1.30798 q^{93} -0.219833 q^{94} -0.753020 q^{95} -0.445042 q^{96} -13.0858 q^{97} -4.42758 q^{98} -10.0978 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - 7 q^{5} - q^{6} - 4 q^{7} + 3 q^{8} - 4 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 - 7 * q^5 - q^6 - 4 * q^7 + 3 * q^8 - 4 * q^9	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - 7 q^{5} - q^{6} - 4 q^{7} + 3 q^{8} - 4 q^{9} - 7 q^{10} + 2 q^{11} - q^{12} - 4 q^{14} + 3 q^{16} + 9 q^{17} - 4 q^{18} + 3 q^{19} - 7 q^{20} + 6 q^{21} + 2 q^{22} + 2 q^{23} - q^{24} + 6 q^{25} + 2 q^{27} - 4 q^{28} + 6 q^{29} + q^{31} + 3 q^{32} + 4 q^{33} + 9 q^{34} + 14 q^{35} - 4 q^{36} - 6 q^{37} + 3 q^{38} - 7 q^{40} - 18 q^{41} + 6 q^{42} + 4 q^{43} + 2 q^{44} + 7 q^{45} + 2 q^{46} - 2 q^{47} - q^{48} + 3 q^{49} + 6 q^{50} - 10 q^{51} - 26 q^{53} + 2 q^{54} - 4 q^{56} - q^{57} + 6 q^{58} - 7 q^{59} - 9 q^{61} + q^{62} - 4 q^{63} + 3 q^{64} + 4 q^{66} + 9 q^{67} + 9 q^{68} + 4 q^{69} + 14 q^{70} + 17 q^{71} - 4 q^{72} - 9 q^{73} - 6 q^{74} + 12 q^{75} + 3 q^{76} + 16 q^{77} + q^{79} - 7 q^{80} - 5 q^{81} - 18 q^{82} - 10 q^{83} + 6 q^{84} - 28 q^{85} + 4 q^{86} + 12 q^{87} + 2 q^{88} - 30 q^{89} + 7 q^{90} + 2 q^{92} + 9 q^{93} - 2 q^{94} - 7 q^{95} - q^{96} - 2 q^{97} + 3 q^{98} - 12 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 - 7 * q^5 - q^6 - 4 * q^7 + 3 * q^8 - 4 * q^9 - 7 * q^10 + 2 * q^11 - q^12 - 4 * q^14 + 3 * q^16 + 9 * q^17 - 4 * q^18 + 3 * q^19 - 7 * q^20 + 6 * q^21 + 2 * q^22 + 2 * q^23 - q^24 + 6 * q^25 + 2 * q^27 - 4 * q^28 + 6 * q^29 + q^31 + 3 * q^32 + 4 * q^33 + 9 * q^34 + 14 * q^35 - 4 * q^36 - 6 * q^37 + 3 * q^38 - 7 * q^40 - 18 * q^41 + 6 * q^42 + 4 * q^43 + 2 * q^44 + 7 * q^45 + 2 * q^46 - 2 * q^47 - q^48 + 3 * q^49 + 6 * q^50 - 10 * q^51 - 26 * q^53 + 2 * q^54 - 4 * q^56 - q^57 + 6 * q^58 - 7 * q^59 - 9 * q^61 + q^62 - 4 * q^63 + 3 * q^64 + 4 * q^66 + 9 * q^67 + 9 * q^68 + 4 * q^69 + 14 * q^70 + 17 * q^71 - 4 * q^72 - 9 * q^73 - 6 * q^74 + 12 * q^75 + 3 * q^76 + 16 * q^77 + q^79 - 7 * q^80 - 5 * q^81 - 18 * q^82 - 10 * q^83 + 6 * q^84 - 28 * q^85 + 4 * q^86 + 12 * q^87 + 2 * q^88 - 30 * q^89 + 7 * q^90 + 2 * q^92 + 9 * q^93 - 2 * q^94 - 7 * q^95 - q^96 - 2 * q^97 + 3 * q^98 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more