LMFDB - Embedded newform 6422.2.a.bp.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$51.2799281781$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 4 x^{14} - 23 x^{13} + 100 x^{12} + 185 x^{11} - 927 x^{10} - 584 x^{9} + 3886 x^{8} + 439 x^{7} + \cdots + 8 $ x^15 - 4x^14 - 23x^13 + 100x^12 + 185x^11 - 927x^10 - 584x^9 + 3886x^8 + 439x^7 - 7063x^6 + 535x^5 + 4067x^4 + 111x^3 - 446x^2 + 8x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.680104$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -0.680104 q^{3} +1.00000 q^{4} +3.16822 q^{5} +0.680104 q^{6} -4.20713 q^{7} -1.00000 q^{8} -2.53746 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.680104 q^{3} +1.00000 q^{4} +3.16822 q^{5} +0.680104 q^{6} -4.20713 q^{7} -1.00000 q^{8} -2.53746 q^{9} -3.16822 q^{10} +0.745314 q^{11} -0.680104 q^{12} +4.20713 q^{14} -2.15472 q^{15} +1.00000 q^{16} -0.781104 q^{17} +2.53746 q^{18} -1.00000 q^{19} +3.16822 q^{20} +2.86129 q^{21} -0.745314 q^{22} +6.32051 q^{23} +0.680104 q^{24} +5.03760 q^{25} +3.76605 q^{27} -4.20713 q^{28} -4.50954 q^{29} +2.15472 q^{30} -3.83358 q^{31} -1.00000 q^{32} -0.506891 q^{33} +0.781104 q^{34} -13.3291 q^{35} -2.53746 q^{36} +10.4977 q^{37} +1.00000 q^{38} -3.16822 q^{40} -4.69291 q^{41} -2.86129 q^{42} +0.647649 q^{43} +0.745314 q^{44} -8.03922 q^{45} -6.32051 q^{46} +3.51054 q^{47} -0.680104 q^{48} +10.7000 q^{49} -5.03760 q^{50} +0.531233 q^{51} +2.41765 q^{53} -3.76605 q^{54} +2.36132 q^{55} +4.20713 q^{56} +0.680104 q^{57} +4.50954 q^{58} +7.14765 q^{59} -2.15472 q^{60} -2.20374 q^{61} +3.83358 q^{62} +10.6754 q^{63} +1.00000 q^{64} +0.506891 q^{66} -5.51316 q^{67} -0.781104 q^{68} -4.29861 q^{69} +13.3291 q^{70} +12.4870 q^{71} +2.53746 q^{72} -8.29344 q^{73} -10.4977 q^{74} -3.42609 q^{75} -1.00000 q^{76} -3.13563 q^{77} +8.26925 q^{79} +3.16822 q^{80} +5.05107 q^{81} +4.69291 q^{82} +0.169234 q^{83} +2.86129 q^{84} -2.47471 q^{85} -0.647649 q^{86} +3.06696 q^{87} -0.745314 q^{88} -14.2885 q^{89} +8.03922 q^{90} +6.32051 q^{92} +2.60723 q^{93} -3.51054 q^{94} -3.16822 q^{95} +0.680104 q^{96} -15.5431 q^{97} -10.7000 q^{98} -1.89120 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 15 q^{2} + 4 q^{3} + 15 q^{4} - 3 q^{5} - 4 q^{6} - 10 q^{7} - 15 q^{8} + 17 q^{9}+O(q^{10}) $ 15 * q - 15 * q^2 + 4 * q^3 + 15 * q^4 - 3 * q^5 - 4 * q^6 - 10 * q^7 - 15 * q^8 + 17 * q^9	\( 15 q - 15 q^{2} + 4 q^{3} + 15 q^{4} - 3 q^{5} - 4 q^{6} - 10 q^{7} - 15 q^{8} + 17 q^{9} + 3 q^{10} - 4 q^{11} + 4 q^{12} + 10 q^{14} - 17 q^{15} + 15 q^{16} + 2 q^{17} - 17 q^{18} - 15 q^{19} - 3 q^{20} - 28 q^{21} + 4 q^{22} + 3 q^{23} - 4 q^{24} + 8 q^{25} + 16 q^{27} - 10 q^{28} + 12 q^{29} + 17 q^{30} - 4 q^{31} - 15 q^{32} + 2 q^{33} - 2 q^{34} + 4 q^{35} + 17 q^{36} - 17 q^{37} + 15 q^{38} + 3 q^{40} + 3 q^{41} + 28 q^{42} - 7 q^{43} - 4 q^{44} - 41 q^{45} - 3 q^{46} + 4 q^{48} + 9 q^{49} - 8 q^{50} + 49 q^{51} - 7 q^{53} - 16 q^{54} + 20 q^{55} + 10 q^{56} - 4 q^{57} - 12 q^{58} - 7 q^{59} - 17 q^{60} - 22 q^{61} + 4 q^{62} - 32 q^{63} + 15 q^{64} - 2 q^{66} - 58 q^{67} + 2 q^{68} - 10 q^{69} - 4 q^{70} - 2 q^{71} - 17 q^{72} - 64 q^{73} + 17 q^{74} - 11 q^{75} - 15 q^{76} - 14 q^{77} + 15 q^{79} - 3 q^{80} + 23 q^{81} - 3 q^{82} - 11 q^{83} - 28 q^{84} - 48 q^{85} + 7 q^{86} - 46 q^{87} + 4 q^{88} + 9 q^{89} + 41 q^{90} + 3 q^{92} - 22 q^{93} + 3 q^{95} - 4 q^{96} - 28 q^{97} - 9 q^{98} - 16 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 + 4 * q^3 + 15 * q^4 - 3 * q^5 - 4 * q^6 - 10 * q^7 - 15 * q^8 + 17 * q^9 + 3 * q^10 - 4 * q^11 + 4 * q^12 + 10 * q^14 - 17 * q^15 + 15 * q^16 + 2 * q^17 - 17 * q^18 - 15 * q^19 - 3 * q^20 - 28 * q^21 + 4 * q^22 + 3 * q^23 - 4 * q^24 + 8 * q^25 + 16 * q^27 - 10 * q^28 + 12 * q^29 + 17 * q^30 - 4 * q^31 - 15 * q^32 + 2 * q^33 - 2 * q^34 + 4 * q^35 + 17 * q^36 - 17 * q^37 + 15 * q^38 + 3 * q^40 + 3 * q^41 + 28 * q^42 - 7 * q^43 - 4 * q^44 - 41 * q^45 - 3 * q^46 + 4 * q^48 + 9 * q^49 - 8 * q^50 + 49 * q^51 - 7 * q^53 - 16 * q^54 + 20 * q^55 + 10 * q^56 - 4 * q^57 - 12 * q^58 - 7 * q^59 - 17 * q^60 - 22 * q^61 + 4 * q^62 - 32 * q^63 + 15 * q^64 - 2 * q^66 - 58 * q^67 + 2 * q^68 - 10 * q^69 - 4 * q^70 - 2 * q^71 - 17 * q^72 - 64 * q^73 + 17 * q^74 - 11 * q^75 - 15 * q^76 - 14 * q^77 + 15 * q^79 - 3 * q^80 + 23 * q^81 - 3 * q^82 - 11 * q^83 - 28 * q^84 - 48 * q^85 + 7 * q^86 - 46 * q^87 + 4 * q^88 + 9 * q^89 + 41 * q^90 + 3 * q^92 - 22 * q^93 + 3 * q^95 - 4 * q^96 - 28 * q^97 - 9 * q^98 - 16 * 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
$3$	−0.680104	−0.392658	−0.196329	−	0.980538i	$-0.562902\pi$
$3$	−0.680104	−0.392658	−0.196329	+	0.980538i	$0.562902\pi$
$4$	1.00000	0.500000
$5$	3.16822	1.41687	0.708435	−	0.705776i	$-0.249402\pi$
$5$	3.16822	1.41687	0.708435	+	0.705776i	$0.249402\pi$
$6$	0.680104	0.277651
$7$	−4.20713	−1.59015	−0.795073	−	0.606514i	$-0.792568\pi$
$7$	−4.20713	−1.59015	−0.795073	+	0.606514i	$0.792568\pi$
$8$	−1.00000	−0.353553
$9$	−2.53746	−0.845819
$10$	−3.16822	−1.00188
$11$	0.745314	0.224721	0.112360	−	0.993668i	$-0.464159\pi$
$11$	0.745314	0.224721	0.112360	+	0.993668i	$0.464159\pi$
$12$	−0.680104	−0.196329
$13$	0	0
$13$	0	0
$14$	4.20713	1.12440
$15$	−2.15472	−0.556346
$16$	1.00000	0.250000
$17$	−0.781104	−0.189446	−0.0947228	−	0.995504i	$-0.530196\pi$
$17$	−0.781104	−0.189446	−0.0947228	+	0.995504i	$0.530196\pi$
$18$	2.53746	0.598085
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	3.16822	0.708435
$21$	2.86129	0.624384
$22$	−0.745314	−0.158901
$23$	6.32051	1.31792	0.658959	−	0.752179i	$-0.270997\pi$
$23$	6.32051	1.31792	0.658959	+	0.752179i	$0.270997\pi$
$24$	0.680104	0.138826
$25$	5.03760	1.00752
$26$	0	0
$27$	3.76605	0.724777
$28$	−4.20713	−0.795073
$29$	−4.50954	−0.837401	−0.418701	−	0.908124i	$-0.637514\pi$
$29$	−4.50954	−0.837401	−0.418701	+	0.908124i	$0.637514\pi$
$30$	2.15472	0.393396
$31$	−3.83358	−0.688531	−0.344265	−	0.938872i	$-0.611872\pi$
$31$	−3.83358	−0.688531	−0.344265	+	0.938872i	$0.611872\pi$
$32$	−1.00000	−0.176777
$33$	−0.506891	−0.0882384
$34$	0.781104	0.133958
$35$	−13.3291	−2.25303
$36$	−2.53746	−0.422910
$37$	10.4977	1.72581	0.862906	−	0.505364i	$-0.168642\pi$
$37$	10.4977	1.72581	0.862906	+	0.505364i	$0.168642\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	−3.16822	−0.500939
$41$	−4.69291	−0.732909	−0.366454	−	0.930436i	$-0.619428\pi$
$41$	−4.69291	−0.732909	−0.366454	+	0.930436i	$0.619428\pi$
$42$	−2.86129	−0.441506
$43$	0.647649	0.0987656	0.0493828	−	0.998780i	$-0.484275\pi$
$43$	0.647649	0.0987656	0.0493828	+	0.998780i	$0.484275\pi$
$44$	0.745314	0.112360
$45$	−8.03922	−1.19842
$46$	−6.32051	−0.931908
$47$	3.51054	0.512065	0.256032	−	0.966668i	$-0.417585\pi$
$47$	3.51054	0.512065	0.256032	+	0.966668i	$0.417585\pi$
$48$	−0.680104	−0.0981646
$49$	10.7000	1.52857
$50$	−5.03760	−0.712424
$51$	0.531233	0.0743874
$52$	0	0
$53$	2.41765	0.332090	0.166045	−	0.986118i	$-0.446900\pi$
$53$	2.41765	0.332090	0.166045	+	0.986118i	$0.446900\pi$
$54$	−3.76605	−0.512494
$55$	2.36132	0.318400
$56$	4.20713	0.562202
$57$	0.680104	0.0900820
$58$	4.50954	0.592132
$59$	7.14765	0.930545	0.465272	−	0.885168i	$-0.345956\pi$
$59$	7.14765	0.930545	0.465272	+	0.885168i	$0.345956\pi$
$60$	−2.15472	−0.278173
$61$	−2.20374	−0.282160	−0.141080	−	0.989998i	$-0.545057\pi$
$61$	−2.20374	−0.282160	−0.141080	+	0.989998i	$0.545057\pi$
$62$	3.83358	0.486865
$63$	10.6754	1.34498
$64$	1.00000	0.125000
$65$	0	0
$66$	0.506891	0.0623940
$67$	−5.51316	−0.673540	−0.336770	−	0.941587i	$-0.609335\pi$
$67$	−5.51316	−0.673540	−0.336770	+	0.941587i	$0.609335\pi$
$68$	−0.781104	−0.0947228
$69$	−4.29861	−0.517491
$70$	13.3291	1.59313
$71$	12.4870	1.48194	0.740970	−	0.671538i	$-0.234366\pi$
$71$	12.4870	1.48194	0.740970	+	0.671538i	$0.234366\pi$
$72$	2.53746	0.299042
$73$	−8.29344	−0.970674	−0.485337	−	0.874327i	$-0.661303\pi$
$73$	−8.29344	−0.970674	−0.485337	+	0.874327i	$0.661303\pi$
$74$	−10.4977	−1.22033
$75$	−3.42609	−0.395611
$76$	−1.00000	−0.114708
$77$	−3.13563	−0.357339
$78$	0	0
$79$	8.26925	0.930363	0.465182	−	0.885215i	$-0.345989\pi$
$79$	8.26925	0.930363	0.465182	+	0.885215i	$0.345989\pi$
$80$	3.16822	0.354217
$81$	5.05107	0.561230
$82$	4.69291	0.518245
$83$	0.169234	0.0185758	0.00928792	−	0.999957i	$-0.497044\pi$
$83$	0.169234	0.0185758	0.00928792	+	0.999957i	$0.497044\pi$
$84$	2.86129	0.312192
$85$	−2.47471	−0.268420
$86$	−0.647649	−0.0698378
$87$	3.06696	0.328813
$88$	−0.745314	−0.0794507
$89$	−14.2885	−1.51458	−0.757290	−	0.653079i	$-0.773477\pi$
$89$	−14.2885	−1.51458	−0.757290	+	0.653079i	$0.773477\pi$
$90$	8.03922	0.847408
$91$	0	0
$92$	6.32051	0.658959
$93$	2.60723	0.270357
$94$	−3.51054	−0.362085
$95$	−3.16822	−0.325052
$96$	0.680104	0.0694129
$97$	−15.5431	−1.57816	−0.789082	−	0.614288i	$-0.789443\pi$
$97$	−15.5431	−1.57816	−0.789082	+	0.614288i	$0.789443\pi$
$98$	−10.7000	−1.08086
$99$	−1.89120	−0.190073
$100$	5.03760	0.503760
$101$	−4.70133	−0.467800	−0.233900	−	0.972261i	$-0.575149\pi$
$101$	−4.70133	−0.467800	−0.233900	+	0.972261i	$0.575149\pi$
$102$	−0.531233	−0.0525999
$103$	11.2055	1.10411	0.552056	−	0.833807i	$-0.313843\pi$
$103$	11.2055	1.10411	0.552056	+	0.833807i	$0.313843\pi$
$104$	0	0
$105$	9.06518	0.884671
$106$	−2.41765	−0.234823
$107$	7.52809	0.727768	0.363884	−	0.931444i	$-0.381450\pi$
$107$	7.52809	0.727768	0.363884	+	0.931444i	$0.381450\pi$
$108$	3.76605	0.362388
$109$	−5.96995	−0.571817	−0.285909	−	0.958257i	$-0.592295\pi$
$109$	−5.96995	−0.571817	−0.285909	+	0.958257i	$0.592295\pi$
$110$	−2.36132	−0.225143
$111$	−7.13953	−0.677655
$112$	−4.20713	−0.397537
$113$	−2.61040	−0.245566	−0.122783	−	0.992434i	$-0.539182\pi$
$113$	−2.61040	−0.245566	−0.122783	+	0.992434i	$0.539182\pi$
$114$	−0.680104	−0.0636976
$115$	20.0247	1.86732
$116$	−4.50954	−0.418701
$117$	0	0
$118$	−7.14765	−0.657995
$119$	3.28621	0.301246
$120$	2.15472	0.196698
$121$	−10.4445	−0.949501
$122$	2.20374	0.199517
$123$	3.19167	0.287783
$124$	−3.83358	−0.344265
$125$	0.119110	0.0106535
$126$	−10.6754	−0.951042
$127$	−2.44779	−0.217207	−0.108603	−	0.994085i	$-0.534638\pi$
$127$	−2.44779	−0.217207	−0.108603	+	0.994085i	$0.534638\pi$
$128$	−1.00000	−0.0883883
$129$	−0.440469	−0.0387812
$130$	0	0
$131$	−16.4478	−1.43705	−0.718526	−	0.695500i	$-0.755183\pi$
$131$	−16.4478	−1.43705	−0.718526	+	0.695500i	$0.755183\pi$
$132$	−0.506891	−0.0441192
$133$	4.20713	0.364805
$134$	5.51316	0.476265
$135$	11.9317	1.02691
$136$	0.781104	0.0669792
$137$	−22.4405	−1.91722	−0.958612	−	0.284717i	$-0.908100\pi$
$137$	−22.4405	−1.91722	−0.958612	+	0.284717i	$0.908100\pi$
$138$	4.29861	0.365922
$139$	11.0206	0.934755	0.467378	−	0.884058i	$-0.345199\pi$
$139$	11.0206	0.934755	0.467378	+	0.884058i	$0.345199\pi$
$140$	−13.3291	−1.12651
$141$	−2.38753	−0.201067
$142$	−12.4870	−1.04789
$143$	0	0
$144$	−2.53746	−0.211455
$145$	−14.2872	−1.18649
$146$	8.29344	0.686370
$147$	−7.27709	−0.600204
$148$	10.4977	0.862906
$149$	−19.2311	−1.57547	−0.787737	−	0.616012i	$-0.788747\pi$
$149$	−19.2311	−1.57547	−0.787737	+	0.616012i	$0.788747\pi$
$150$	3.42609	0.279739
$151$	−7.20204	−0.586094	−0.293047	−	0.956098i	$-0.594669\pi$
$151$	−7.20204	−0.586094	−0.293047	+	0.956098i	$0.594669\pi$
$152$	1.00000	0.0811107
$153$	1.98202	0.160237
$154$	3.13563	0.252677
$155$	−12.1456	−0.975558
$156$	0	0
$157$	−13.9304	−1.11176	−0.555882	−	0.831261i	$-0.687619\pi$
$157$	−13.9304	−1.11176	−0.555882	+	0.831261i	$0.687619\pi$
$158$	−8.26925	−0.657866
$159$	−1.64426	−0.130398
$160$	−3.16822	−0.250470
$161$	−26.5912	−2.09568
$162$	−5.05107	−0.396849
$163$	−20.0085	−1.56718	−0.783592	−	0.621276i	$-0.786614\pi$
$163$	−20.0085	−1.56718	−0.783592	+	0.621276i	$0.786614\pi$
$164$	−4.69291	−0.366454
$165$	−1.60594	−0.125022
$166$	−0.169234	−0.0131351
$167$	0.829817	0.0642132	0.0321066	−	0.999484i	$-0.489778\pi$
$167$	0.829817	0.0642132	0.0321066	+	0.999484i	$0.489778\pi$
$168$	−2.86129	−0.220753
$169$	0	0
$170$	2.47471	0.189801
$171$	2.53746	0.194044
$172$	0.647649	0.0493828
$173$	−19.3374	−1.47019	−0.735096	−	0.677963i	$-0.762863\pi$
$173$	−19.3374	−1.47019	−0.735096	+	0.677963i	$0.762863\pi$
$174$	−3.06696	−0.232506
$175$	−21.1938	−1.60210
$176$	0.745314	0.0561802
$177$	−4.86115	−0.365386
$178$	14.2885	1.07097
$179$	−22.6261	−1.69115	−0.845577	−	0.533853i	$-0.820743\pi$
$179$	−22.6261	−1.69115	−0.845577	+	0.533853i	$0.820743\pi$
$180$	−8.03922	−0.599208
$181$	21.9289	1.62996	0.814981	−	0.579487i	$-0.196747\pi$
$181$	21.9289	1.62996	0.814981	+	0.579487i	$0.196747\pi$
$182$	0	0
$183$	1.49877	0.110792
$184$	−6.32051	−0.465954
$185$	33.2590	2.44525
$186$	−2.60723	−0.191172
$187$	−0.582168	−0.0425723
$188$	3.51054	0.256032
$189$	−15.8443	−1.15250
$190$	3.16822	0.229847
$191$	14.2598	1.03181	0.515903	−	0.856647i	$-0.327457\pi$
$191$	14.2598	1.03181	0.515903	+	0.856647i	$0.327457\pi$
$192$	−0.680104	−0.0490823
$193$	3.56077	0.256310	0.128155	−	0.991754i	$-0.459094\pi$
$193$	3.56077	0.256310	0.128155	+	0.991754i	$0.459094\pi$
$194$	15.5431	1.11593
$195$	0	0
$196$	10.7000	0.764283
$197$	−4.14163	−0.295079	−0.147540	−	0.989056i	$-0.547135\pi$
$197$	−4.14163	−0.295079	−0.147540	+	0.989056i	$0.547135\pi$
$198$	1.89120	0.134402
$199$	14.7854	1.04811	0.524056	−	0.851684i	$-0.324418\pi$
$199$	14.7854	1.04811	0.524056	+	0.851684i	$0.324418\pi$
$200$	−5.03760	−0.356212
$201$	3.74953	0.264471
$202$	4.70133	0.330785
$203$	18.9722	1.33159
$204$	0.531233	0.0371937
$205$	−14.8681	−1.03844
$206$	−11.2055	−0.780726
$207$	−16.0380	−1.11472
$208$	0	0
$209$	−0.745314	−0.0515544
$210$	−9.06518	−0.625557
$211$	19.9863	1.37591	0.687956	−	0.725753i	$-0.258508\pi$
$211$	19.9863	1.37591	0.687956	+	0.725753i	$0.258508\pi$
$212$	2.41765	0.166045
$213$	−8.49250	−0.581896
$214$	−7.52809	−0.514610
$215$	2.05189	0.139938
$216$	−3.76605	−0.256247
$217$	16.1284	1.09486
$218$	5.96995	0.404336
$219$	5.64040	0.381143
$220$	2.36132	0.159200
$221$	0	0
$222$	7.13953	0.479174
$223$	−1.53481	−0.102778	−0.0513891	−	0.998679i	$-0.516365\pi$
$223$	−1.53481	−0.102778	−0.0513891	+	0.998679i	$0.516365\pi$
$224$	4.20713	0.281101
$225$	−12.7827	−0.852179
$226$	2.61040	0.173641
$227$	−10.0133	−0.664603	−0.332301	−	0.943173i	$-0.607825\pi$
$227$	−10.0133	−0.664603	−0.332301	+	0.943173i	$0.607825\pi$
$228$	0.680104	0.0450410
$229$	14.8741	0.982909	0.491454	−	0.870903i	$-0.336465\pi$
$229$	14.8741	0.982909	0.491454	+	0.870903i	$0.336465\pi$
$230$	−20.0247	−1.32039
$231$	2.13256	0.140312
$232$	4.50954	0.296066
$233$	6.11477	0.400592	0.200296	−	0.979735i	$-0.435810\pi$
$233$	6.11477	0.400592	0.200296	+	0.979735i	$0.435810\pi$
$234$	0	0
$235$	11.1222	0.725529
$236$	7.14765	0.465272
$237$	−5.62395	−0.365315
$238$	−3.28621	−0.213013
$239$	−15.4630	−1.00022	−0.500109	−	0.865962i	$-0.666707\pi$
$239$	−15.4630	−1.00022	−0.500109	+	0.865962i	$0.666707\pi$
$240$	−2.15472	−0.139086
$241$	−8.07387	−0.520084	−0.260042	−	0.965597i	$-0.583736\pi$
$241$	−8.07387	−0.520084	−0.260042	+	0.965597i	$0.583736\pi$
$242$	10.4445	0.671398
$243$	−14.7334	−0.945148
$244$	−2.20374	−0.141080
$245$	33.8998	2.16578
$246$	−3.19167	−0.203493
$247$	0	0
$248$	3.83358	0.243432
$249$	−0.115097	−0.00729396
$250$	−0.119110	−0.00753318
$251$	24.6100	1.55337	0.776686	−	0.629888i	$-0.216899\pi$
$251$	24.6100	1.55337	0.776686	+	0.629888i	$0.216899\pi$
$252$	10.6754	0.672488
$253$	4.71076	0.296163
$254$	2.44779	0.153588
$255$	1.68306	0.105397
$256$	1.00000	0.0625000
$257$	−17.2013	−1.07299	−0.536493	−	0.843905i	$-0.680251\pi$
$257$	−17.2013	−1.07299	−0.536493	+	0.843905i	$0.680251\pi$
$258$	0.440469	0.0274224
$259$	−44.1652	−2.74429
$260$	0	0
$261$	11.4428	0.708290
$262$	16.4478	1.01615
$263$	−24.8206	−1.53050	−0.765252	−	0.643730i	$-0.777386\pi$
$263$	−24.8206	−1.53050	−0.765252	+	0.643730i	$0.777386\pi$
$264$	0.506891	0.0311970
$265$	7.65964	0.470528
$266$	−4.20713	−0.257956
$267$	9.71768	0.594712
$268$	−5.51316	−0.336770
$269$	18.8999	1.15234	0.576172	−	0.817328i	$-0.304546\pi$
$269$	18.8999	1.15234	0.576172	+	0.817328i	$0.304546\pi$
$270$	−11.9317	−0.726138
$271$	24.5529	1.49148	0.745741	−	0.666236i	$-0.232095\pi$
$271$	24.5529	1.49148	0.745741	+	0.666236i	$0.232095\pi$
$272$	−0.781104	−0.0473614
$273$	0	0
$274$	22.4405	1.35568
$275$	3.75459	0.226410
$276$	−4.29861	−0.258746
$277$	15.2316	0.915179	0.457590	−	0.889163i	$-0.348713\pi$
$277$	15.2316	0.915179	0.457590	+	0.889163i	$0.348713\pi$
$278$	−11.0206	−0.660972
$279$	9.72754	0.582373
$280$	13.3291	0.796566
$281$	−7.16822	−0.427620	−0.213810	−	0.976875i	$-0.568587\pi$
$281$	−7.16822	−0.427620	−0.213810	+	0.976875i	$0.568587\pi$
$282$	2.38753	0.142176
$283$	−27.0505	−1.60799	−0.803994	−	0.594638i	$-0.797296\pi$
$283$	−27.0505	−1.60799	−0.803994	+	0.594638i	$0.797296\pi$
$284$	12.4870	0.740970
$285$	2.15472	0.127634
$286$	0	0
$287$	19.7437	1.16543
$288$	2.53746	0.149521
$289$	−16.3899	−0.964110
$290$	14.2872	0.838974
$291$	10.5709	0.619679
$292$	−8.29344	−0.485337
$293$	−16.5283	−0.965595	−0.482797	−	0.875732i	$-0.660379\pi$
$293$	−16.5283	−0.965595	−0.482797	+	0.875732i	$0.660379\pi$
$294$	7.27709	0.424408
$295$	22.6453	1.31846
$296$	−10.4977	−0.610167
$297$	2.80689	0.162872
$298$	19.2311	1.11403
$299$	0	0
$300$	−3.42609	−0.197805
$301$	−2.72475	−0.157052
$302$	7.20204	0.414431
$303$	3.19740	0.183686
$304$	−1.00000	−0.0573539
$305$	−6.98192	−0.399784
$306$	−1.98202	−0.113305
$307$	−5.06112	−0.288853	−0.144427	−	0.989516i	$-0.546134\pi$
$307$	−5.06112	−0.288853	−0.144427	+	0.989516i	$0.546134\pi$
$308$	−3.13563	−0.178669
$309$	−7.62092	−0.433539
$310$	12.1456	0.689824
$311$	−23.7201	−1.34504	−0.672522	−	0.740077i	$-0.734789\pi$
$311$	−23.7201	−1.34504	−0.672522	+	0.740077i	$0.734789\pi$
$312$	0	0
$313$	10.7782	0.609222	0.304611	−	0.952477i	$-0.401474\pi$
$313$	10.7782	0.609222	0.304611	+	0.952477i	$0.401474\pi$
$314$	13.9304	0.786135
$315$	33.8220	1.90566
$316$	8.26925	0.465182
$317$	−19.1658	−1.07646	−0.538230	−	0.842798i	$-0.680907\pi$
$317$	−19.1658	−1.07646	−0.538230	+	0.842798i	$0.680907\pi$
$318$	1.64426	0.0922053
$319$	−3.36103	−0.188181
$320$	3.16822	0.177109
$321$	−5.11989	−0.285764
$322$	26.5912	1.48187
$323$	0.781104	0.0434618
$324$	5.05107	0.280615
$325$	0	0
$326$	20.0085	1.10817
$327$	4.06019	0.224529
$328$	4.69291	0.259122
$329$	−14.7693	−0.814258
$330$	1.60594	0.0884042
$331$	−30.2114	−1.66057	−0.830284	−	0.557341i	$-0.811822\pi$
$331$	−30.2114	−1.66057	−0.830284	+	0.557341i	$0.811822\pi$
$332$	0.169234	0.00928792
$333$	−26.6375	−1.45973
$334$	−0.829817	−0.0454056
$335$	−17.4669	−0.954319
$336$	2.86129	0.156096
$337$	18.0893	0.985388	0.492694	−	0.870203i	$-0.336012\pi$
$337$	18.0893	0.985388	0.492694	+	0.870203i	$0.336012\pi$
$338$	0	0
$339$	1.77534	0.0964234
$340$	−2.47471	−0.134210
$341$	−2.85722	−0.154727
$342$	−2.53746	−0.137210
$343$	−15.5662	−0.840496
$344$	−0.647649	−0.0349189
$345$	−13.6189	−0.733218
$346$	19.3374	1.03958
$347$	−25.4433	−1.36587	−0.682934	−	0.730480i	$-0.739296\pi$
$347$	−25.4433	−1.36587	−0.682934	+	0.730480i	$0.739296\pi$
$348$	3.06696	0.164406
$349$	−19.8056	−1.06017	−0.530084	−	0.847945i	$-0.677840\pi$
$349$	−19.8056	−1.06017	−0.530084	+	0.847945i	$0.677840\pi$
$350$	21.1938	1.13286
$351$	0	0
$352$	−0.745314	−0.0397254
$353$	−6.00777	−0.319762	−0.159881	−	0.987136i	$-0.551111\pi$
$353$	−6.00777	−0.319762	−0.159881	+	0.987136i	$0.551111\pi$
$354$	4.86115	0.258367
$355$	39.5617	2.09972
$356$	−14.2885	−0.757290
$357$	−2.23497	−0.118287
$358$	22.6261	1.19583
$359$	17.3761	0.917073	0.458537	−	0.888676i	$-0.348374\pi$
$359$	17.3761	0.917073	0.458537	+	0.888676i	$0.348374\pi$
$360$	8.03922	0.423704
$361$	1.00000	0.0526316
$362$	−21.9289	−1.15256
$363$	7.10335	0.372829
$364$	0	0
$365$	−26.2754	−1.37532
$366$	−1.49877	−0.0783421
$367$	−13.8136	−0.721066	−0.360533	−	0.932747i	$-0.617405\pi$
$367$	−13.8136	−0.721066	−0.360533	+	0.932747i	$0.617405\pi$
$368$	6.32051	0.329479
$369$	11.9081	0.619908
$370$	−33.2590	−1.72905
$371$	−10.1714	−0.528072
$372$	2.60723	0.135179
$373$	13.1012	0.678356	0.339178	−	0.940722i	$-0.389851\pi$
$373$	13.1012	0.678356	0.339178	+	0.940722i	$0.389851\pi$
$374$	0.582168	0.0301032
$375$	−0.0810073	−0.00418320
$376$	−3.51054	−0.181042
$377$	0	0
$378$	15.8443	0.814941
$379$	4.85229	0.249245	0.124623	−	0.992204i	$-0.460228\pi$
$379$	4.85229	0.249245	0.124623	+	0.992204i	$0.460228\pi$
$380$	−3.16822	−0.162526
$381$	1.66476	0.0852880
$382$	−14.2598	−0.729597
$383$	1.91133	0.0976644	0.0488322	−	0.998807i	$-0.484450\pi$
$383$	1.91133	0.0976644	0.0488322	+	0.998807i	$0.484450\pi$
$384$	0.680104	0.0347064
$385$	−9.93437	−0.506302
$386$	−3.56077	−0.181239
$387$	−1.64338	−0.0835379
$388$	−15.5431	−0.789082
$389$	36.7485	1.86323	0.931613	−	0.363453i	$-0.118402\pi$
$389$	36.7485	1.86323	0.931613	+	0.363453i	$0.118402\pi$
$390$	0	0
$391$	−4.93698	−0.249674
$392$	−10.7000	−0.540430
$393$	11.1862	0.564271
$394$	4.14163	0.208652
$395$	26.1988	1.31820
$396$	−1.89120	−0.0950365
$397$	12.4585	0.625277	0.312638	−	0.949872i	$-0.398787\pi$
$397$	12.4585	0.625277	0.312638	+	0.949872i	$0.398787\pi$
$398$	−14.7854	−0.741128
$399$	−2.86129	−0.143244
$400$	5.03760	0.251880
$401$	−20.9479	−1.04609	−0.523044	−	0.852306i	$-0.675204\pi$
$401$	−20.9479	−1.04609	−0.523044	+	0.852306i	$0.675204\pi$
$402$	−3.74953	−0.187009
$403$	0	0
$404$	−4.70133	−0.233900
$405$	16.0029	0.795189
$406$	−18.9722	−0.941577
$407$	7.82409	0.387826
$408$	−0.531233	−0.0262999
$409$	30.5960	1.51288	0.756438	−	0.654066i	$-0.226938\pi$
$409$	30.5960	1.51288	0.756438	+	0.654066i	$0.226938\pi$
$410$	14.8681	0.734285
$411$	15.2619	0.752814
$412$	11.2055	0.552056
$413$	−30.0711	−1.47970
$414$	16.0380	0.788226
$415$	0.536170	0.0263195
$416$	0	0
$417$	−7.49516	−0.367040
$418$	0.745314	0.0364545
$419$	23.7309	1.15933	0.579664	−	0.814855i	$-0.303184\pi$
$419$	23.7309	1.15933	0.579664	+	0.814855i	$0.303184\pi$
$420$	9.06518	0.442336
$421$	−6.06704	−0.295690	−0.147845	−	0.989011i	$-0.547234\pi$
$421$	−6.06704	−0.295690	−0.147845	+	0.989011i	$0.547234\pi$
$422$	−19.9863	−0.972916
$423$	−8.90785	−0.433114
$424$	−2.41765	−0.117412
$425$	−3.93489	−0.190870
$426$	8.49250	0.411463
$427$	9.27142	0.448676
$428$	7.52809	0.363884
$429$	0	0
$430$	−2.05189	−0.0989511
$431$	−31.3377	−1.50948	−0.754741	−	0.656023i	$-0.772238\pi$
$431$	−31.3377	−1.50948	−0.754741	+	0.656023i	$0.772238\pi$
$432$	3.76605	0.181194
$433$	7.67163	0.368675	0.184338	−	0.982863i	$-0.440986\pi$
$433$	7.67163	0.368675	0.184338	+	0.982863i	$0.440986\pi$
$434$	−16.1284	−0.774186
$435$	9.71679	0.465885
$436$	−5.96995	−0.285909
$437$	−6.32051	−0.302351
$438$	−5.64040	−0.269509
$439$	7.35012	0.350802	0.175401	−	0.984497i	$-0.443878\pi$
$439$	7.35012	0.350802	0.175401	+	0.984497i	$0.443878\pi$
$440$	−2.36132	−0.112571
$441$	−27.1507	−1.29289
$442$	0	0
$443$	−12.9937	−0.617348	−0.308674	−	0.951168i	$-0.599885\pi$
$443$	−12.9937	−0.617348	−0.308674	+	0.951168i	$0.599885\pi$
$444$	−7.13953	−0.338827
$445$	−45.2691	−2.14596
$446$	1.53481	0.0726752
$447$	13.0792	0.618623
$448$	−4.20713	−0.198768
$449$	35.1949	1.66095	0.830475	−	0.557055i	$-0.188069\pi$
$449$	35.1949	1.66095	0.830475	+	0.557055i	$0.188069\pi$
$450$	12.7827	0.602582
$451$	−3.49769	−0.164700
$452$	−2.61040	−0.122783
$453$	4.89814	0.230135
$454$	10.0133	0.469945
$455$	0	0
$456$	−0.680104	−0.0318488
$457$	−36.5708	−1.71071	−0.855354	−	0.518043i	$-0.826661\pi$
$457$	−36.5708	−1.71071	−0.855354	+	0.518043i	$0.826661\pi$
$458$	−14.8741	−0.695021
$459$	−2.94168	−0.137306
$460$	20.0247	0.933659
$461$	25.2919	1.17796	0.588981	−	0.808147i	$-0.299529\pi$
$461$	25.2919	1.17796	0.588981	+	0.808147i	$0.299529\pi$
$462$	−2.13256	−0.0992156
$463$	−10.6111	−0.493137	−0.246569	−	0.969125i	$-0.579303\pi$
$463$	−10.6111	−0.493137	−0.246569	+	0.969125i	$0.579303\pi$
$464$	−4.50954	−0.209350
$465$	8.26027	0.383061
$466$	−6.11477	−0.283261
$467$	7.58944	0.351197	0.175599	−	0.984462i	$-0.443814\pi$
$467$	7.58944	0.351197	0.175599	+	0.984462i	$0.443814\pi$
$468$	0	0
$469$	23.1946	1.07103
$470$	−11.1222	−0.513027
$471$	9.47409	0.436543
$472$	−7.14765	−0.328997
$473$	0.482702	0.0221947
$474$	5.62395	0.258317
$475$	−5.03760	−0.231141
$476$	3.28621	0.150623
$477$	−6.13469	−0.280888
$478$	15.4630	0.707261
$479$	−33.3642	−1.52445	−0.762224	−	0.647313i	$-0.775893\pi$
$479$	−33.3642	−1.52445	−0.762224	+	0.647313i	$0.775893\pi$
$480$	2.15472	0.0983490
$481$	0	0
$482$	8.07387	0.367755
$483$	18.0848	0.822887
$484$	−10.4445	−0.474750
$485$	−49.2439	−2.23605
$486$	14.7334	0.668321
$487$	−20.7884	−0.942012	−0.471006	−	0.882130i	$-0.656109\pi$
$487$	−20.7884	−0.942012	−0.471006	+	0.882130i	$0.656109\pi$
$488$	2.20374	0.0997586
$489$	13.6078	0.615368
$490$	−33.8998	−1.53144
$491$	1.44691	0.0652983	0.0326491	−	0.999467i	$-0.489606\pi$
$491$	1.44691	0.0652983	0.0326491	+	0.999467i	$0.489606\pi$
$492$	3.19167	0.143891
$493$	3.52242	0.158642
$494$	0	0
$495$	−5.99174	−0.269309
$496$	−3.83358	−0.172133
$497$	−52.5347	−2.35650
$498$	0.115097	0.00515761
$499$	−17.5580	−0.786005	−0.393002	−	0.919537i	$-0.628564\pi$
$499$	−17.5580	−0.786005	−0.393002	+	0.919537i	$0.628564\pi$
$500$	0.119110	0.00532676
$501$	−0.564362	−0.0252138
$502$	−24.6100	−1.09840
$503$	29.8025	1.32883	0.664414	−	0.747365i	$-0.268681\pi$
$503$	29.8025	1.32883	0.664414	+	0.747365i	$0.268681\pi$
$504$	−10.6754	−0.475521
$505$	−14.8948	−0.662812
$506$	−4.71076	−0.209419
$507$	0	0
$508$	−2.44779	−0.108603
$509$	3.12239	0.138398	0.0691988	−	0.997603i	$-0.477956\pi$
$509$	3.12239	0.138398	0.0691988	+	0.997603i	$0.477956\pi$
$510$	−1.68306	−0.0745271
$511$	34.8916	1.54351
$512$	−1.00000	−0.0441942
$513$	−3.76605	−0.166275
$514$	17.2013	0.758715
$515$	35.5015	1.56438
$516$	−0.440469	−0.0193906
$517$	2.61645	0.115072
$518$	44.1652	1.94051
$519$	13.1514	0.577283
$520$	0	0
$521$	−4.49118	−0.196762	−0.0983810	−	0.995149i	$-0.531366\pi$
$521$	−4.49118	−0.196762	−0.0983810	+	0.995149i	$0.531366\pi$
$522$	−11.4428	−0.500837
$523$	−17.9673	−0.785656	−0.392828	−	0.919612i	$-0.628503\pi$
$523$	−17.9673	−0.785656	−0.392828	+	0.919612i	$0.628503\pi$
$524$	−16.4478	−0.718526
$525$	14.4140	0.629079
$526$	24.8206	1.08223
$527$	2.99442	0.130439
$528$	−0.506891	−0.0220596
$529$	16.9489	0.736907
$530$	−7.65964	−0.332714
$531$	−18.1369	−0.787073
$532$	4.20713	0.182402
$533$	0	0
$534$	−9.71768	−0.420525
$535$	23.8506	1.03115
$536$	5.51316	0.238132
$537$	15.3881	0.664046
$538$	−18.8999	−0.814830
$539$	7.97483	0.343500
$540$	11.9317	0.513457
$541$	21.1619	0.909821	0.454911	−	0.890537i	$-0.349671\pi$
$541$	21.1619	0.909821	0.454911	+	0.890537i	$0.349671\pi$
$542$	−24.5529	−1.05464
$543$	−14.9139	−0.640018
$544$	0.781104	0.0334896
$545$	−18.9141	−0.810191
$546$	0	0
$547$	−24.2674	−1.03760	−0.518800	−	0.854896i	$-0.673621\pi$
$547$	−24.2674	−1.03760	−0.518800	+	0.854896i	$0.673621\pi$
$548$	−22.4405	−0.958612
$549$	5.59190	0.238656
$550$	−3.75459	−0.160096
$551$	4.50954	0.192113
$552$	4.29861	0.182961
$553$	−34.7898	−1.47941
$554$	−15.2316	−0.647129
$555$	−22.6196	−0.960148
$556$	11.0206	0.467378
$557$	11.1528	0.472558	0.236279	−	0.971685i	$-0.424072\pi$
$557$	11.1528	0.472558	0.236279	+	0.971685i	$0.424072\pi$
$558$	−9.72754	−0.411800
$559$	0	0
$560$	−13.3291	−0.563257
$561$	0.395935	0.0167164
$562$	7.16822	0.302373
$563$	32.3746	1.36442	0.682212	−	0.731154i	$-0.261018\pi$
$563$	32.3746	1.36442	0.682212	+	0.731154i	$0.261018\pi$
$564$	−2.38753	−0.100533
$565$	−8.27031	−0.347934
$566$	27.0505	1.13702
$567$	−21.2505	−0.892437
$568$	−12.4870	−0.523945
$569$	−12.4235	−0.520822	−0.260411	−	0.965498i	$-0.583858\pi$
$569$	−12.4235	−0.520822	−0.260411	+	0.965498i	$0.583858\pi$
$570$	−2.15472	−0.0902512
$571$	−38.9926	−1.63179	−0.815895	−	0.578200i	$-0.803755\pi$
$571$	−38.9926	−1.63179	−0.815895	+	0.578200i	$0.803755\pi$
$572$	0	0
$573$	−9.69818	−0.405147
$574$	−19.7437	−0.824085
$575$	31.8402	1.32783
$576$	−2.53746	−0.105727
$577$	−21.6013	−0.899272	−0.449636	−	0.893212i	$-0.648446\pi$
$577$	−21.6013	−0.899272	−0.449636	+	0.893212i	$0.648446\pi$
$578$	16.3899	0.681729
$579$	−2.42170	−0.100642
$580$	−14.2872	−0.593244
$581$	−0.711989	−0.0295383
$582$	−10.5709	−0.438179
$583$	1.80191	0.0746275
$584$	8.29344	0.343185
$585$	0	0
$586$	16.5283	0.682779
$587$	17.9235	0.739783	0.369891	−	0.929075i	$-0.379395\pi$
$587$	17.9235	0.739783	0.369891	+	0.929075i	$0.379395\pi$
$588$	−7.27709	−0.300102
$589$	3.83358	0.157960
$590$	−22.6453	−0.932292
$591$	2.81674	0.115865
$592$	10.4977	0.431453
$593$	30.0859	1.23548	0.617739	−	0.786383i	$-0.288049\pi$
$593$	30.0859	1.23548	0.617739	+	0.786383i	$0.288049\pi$
$594$	−2.80689	−0.115168
$595$	10.4114	0.426827
$596$	−19.2311	−0.787737
$597$	−10.0556	−0.411550
$598$	0	0
$599$	25.8269	1.05526	0.527629	−	0.849475i	$-0.323081\pi$
$599$	25.8269	1.05526	0.527629	+	0.849475i	$0.323081\pi$
$600$	3.42609	0.139870
$601$	−34.7427	−1.41719	−0.708593	−	0.705617i	$-0.750670\pi$
$601$	−34.7427	−1.41719	−0.708593	+	0.705617i	$0.750670\pi$
$602$	2.72475	0.111052
$603$	13.9894	0.569693
$604$	−7.20204	−0.293047
$605$	−33.0905	−1.34532
$606$	−3.19740	−0.129885
$607$	−22.7745	−0.924389	−0.462194	−	0.886779i	$-0.652938\pi$
$607$	−22.7745	−0.924389	−0.462194	+	0.886779i	$0.652938\pi$
$608$	1.00000	0.0405554
$609$	−12.9031	−0.522860
$610$	6.98192	0.282690
$611$	0	0
$612$	1.98202	0.0801184
$613$	−41.2191	−1.66482	−0.832411	−	0.554158i	$-0.813040\pi$
$613$	−41.2191	−1.66482	−0.832411	+	0.554158i	$0.813040\pi$
$614$	5.06112	0.204250
$615$	10.1119	0.407751
$616$	3.13563	0.126338
$617$	−45.8707	−1.84669	−0.923343	−	0.383977i	$-0.874554\pi$
$617$	−45.8707	−1.84669	−0.923343	+	0.383977i	$0.874554\pi$
$618$	7.62092	0.306558
$619$	20.7965	0.835883	0.417942	−	0.908474i	$-0.362752\pi$
$619$	20.7965	0.835883	0.417942	+	0.908474i	$0.362752\pi$
$620$	−12.1456	−0.487779
$621$	23.8034	0.955196
$622$	23.7201	0.951090
$623$	60.1137	2.40840
$624$	0	0
$625$	−24.8106	−0.992424
$626$	−10.7782	−0.430785
$627$	0.506891	0.0202433
$628$	−13.9304	−0.555882
$629$	−8.19980	−0.326948
$630$	−33.8220	−1.34750
$631$	−33.7201	−1.34237	−0.671187	−	0.741288i	$-0.734215\pi$
$631$	−33.7201	−1.34237	−0.671187	+	0.741288i	$0.734215\pi$
$632$	−8.26925	−0.328933
$633$	−13.5927	−0.540263
$634$	19.1658	0.761172
$635$	−7.75514	−0.307753
$636$	−1.64426	−0.0651990
$637$	0	0
$638$	3.36103	0.133064
$639$	−31.6854	−1.25345
$640$	−3.16822	−0.125235
$641$	−1.04414	−0.0412409	−0.0206205	−	0.999787i	$-0.506564\pi$
$641$	−1.04414	−0.0412409	−0.0206205	+	0.999787i	$0.506564\pi$
$642$	5.11989	0.202066
$643$	19.1966	0.757039	0.378520	−	0.925593i	$-0.376433\pi$
$643$	19.1966	0.757039	0.378520	+	0.925593i	$0.376433\pi$
$644$	−26.5912	−1.04784
$645$	−1.39550	−0.0549478
$646$	−0.781104	−0.0307321
$647$	−35.1617	−1.38235	−0.691174	−	0.722688i	$-0.742906\pi$
$647$	−35.1617	−1.38235	−0.691174	+	0.722688i	$0.742906\pi$
$648$	−5.05107	−0.198425
$649$	5.32724	0.209113
$650$	0	0
$651$	−10.9690	−0.429908
$652$	−20.0085	−0.783592
$653$	−14.5759	−0.570399	−0.285199	−	0.958468i	$-0.592060\pi$
$653$	−14.5759	−0.570399	−0.285199	+	0.958468i	$0.592060\pi$
$654$	−4.06019	−0.158766
$655$	−52.1102	−2.03612
$656$	−4.69291	−0.183227
$657$	21.0443	0.821015
$658$	14.7693	0.575767
$659$	34.9056	1.35973	0.679864	−	0.733338i	$-0.262039\pi$
$659$	34.9056	1.35973	0.679864	+	0.733338i	$0.262039\pi$
$660$	−1.60594	−0.0625112
$661$	45.8903	1.78492	0.892462	−	0.451122i	$-0.148976\pi$
$661$	45.8903	1.78492	0.892462	+	0.451122i	$0.148976\pi$
$662$	30.2114	1.17420
$663$	0	0
$664$	−0.169234	−0.00656755
$665$	13.3291	0.516880
$666$	26.6375	1.03218
$667$	−28.5026	−1.10363
$668$	0.829817	0.0321066
$669$	1.04383	0.0403567
$670$	17.4669	0.674805
$671$	−1.64248	−0.0634071
$672$	−2.86129	−0.110377
$673$	35.2509	1.35882	0.679410	−	0.733759i	$-0.262236\pi$
$673$	35.2509	1.35882	0.679410	+	0.733759i	$0.262236\pi$
$674$	−18.0893	−0.696775
$675$	18.9718	0.730226
$676$	0	0
$677$	−27.0923	−1.04124	−0.520621	−	0.853788i	$-0.674299\pi$
$677$	−27.0923	−1.04124	−0.520621	+	0.853788i	$0.674299\pi$
$678$	−1.77534	−0.0681816
$679$	65.3919	2.50951
$680$	2.47471	0.0949007
$681$	6.81006	0.260962
$682$	2.85722	0.109409
$683$	33.9768	1.30008	0.650042	−	0.759898i	$-0.274751\pi$
$683$	33.9768	1.30008	0.650042	+	0.759898i	$0.274751\pi$
$684$	2.53746	0.0970221
$685$	−71.0964	−2.71645
$686$	15.5662	0.594321
$687$	−10.1159	−0.385947
$688$	0.647649	0.0246914
$689$	0	0
$690$	13.6189	0.518463
$691$	−18.8419	−0.716781	−0.358391	−	0.933572i	$-0.616674\pi$
$691$	−18.8419	−0.716781	−0.358391	+	0.933572i	$0.616674\pi$
$692$	−19.3374	−0.735096
$693$	7.95654	0.302244
$694$	25.4433	0.965815
$695$	34.9157	1.32443
$696$	−3.06696	−0.116253
$697$	3.66565	0.138846
$698$	19.8056	0.749652
$699$	−4.15868	−0.157296
$700$	−21.1938	−0.801051
$701$	32.2172	1.21683	0.608413	−	0.793620i	$-0.291806\pi$
$701$	32.2172	1.21683	0.608413	+	0.793620i	$0.291806\pi$
$702$	0	0
$703$	−10.4977	−0.395928
$704$	0.745314	0.0280901
$705$	−7.56422	−0.284885
$706$	6.00777	0.226106
$707$	19.7791	0.743870
$708$	−4.86115	−0.182693
$709$	−46.1357	−1.73266	−0.866332	−	0.499469i	$-0.833528\pi$
$709$	−46.1357	−1.73266	−0.866332	+	0.499469i	$0.833528\pi$
$710$	−39.5617	−1.48472
$711$	−20.9829	−0.786919
$712$	14.2885	0.535485
$713$	−24.2302	−0.907427
$714$	2.23497	0.0836415
$715$	0	0
$716$	−22.6261	−0.845577
$717$	10.5165	0.392744
$718$	−17.3761	−0.648469
$719$	−35.0775	−1.30817	−0.654084	−	0.756422i	$-0.726946\pi$
$719$	−35.0775	−1.30817	−0.654084	+	0.756422i	$0.726946\pi$
$720$	−8.03922	−0.299604
$721$	−47.1431	−1.75570
$722$	−1.00000	−0.0372161
$723$	5.49107	0.204215
$724$	21.9289	0.814981
$725$	−22.7173	−0.843698
$726$	−7.10335	−0.263630
$727$	−42.0585	−1.55986	−0.779932	−	0.625864i	$-0.784747\pi$
$727$	−42.0585	−1.55986	−0.779932	+	0.625864i	$0.784747\pi$
$728$	0	0
$729$	−5.13295	−0.190109
$730$	26.2754	0.972497
$731$	−0.505882	−0.0187107
$732$	1.49877	0.0553962
$733$	13.0427	0.481744	0.240872	−	0.970557i	$-0.422567\pi$
$733$	13.0427	0.481744	0.240872	+	0.970557i	$0.422567\pi$
$734$	13.8136	0.509870
$735$	−23.0554	−0.850411
$736$	−6.32051	−0.232977
$737$	−4.10904	−0.151358
$738$	−11.9081	−0.438341
$739$	49.3797	1.81646	0.908230	−	0.418471i	$-0.137434\pi$
$739$	49.3797	1.81646	0.908230	+	0.418471i	$0.137434\pi$
$740$	33.2590	1.22263
$741$	0	0
$742$	10.1714	0.373403
$743$	17.1543	0.629331	0.314666	−	0.949203i	$-0.398108\pi$
$743$	17.1543	0.629331	0.314666	+	0.949203i	$0.398108\pi$
$744$	−2.60723	−0.0955858
$745$	−60.9283	−2.23224
$746$	−13.1012	−0.479670
$747$	−0.429424	−0.0157118
$748$	−0.582168	−0.0212862
$749$	−31.6717	−1.15726
$750$	0.0810073	0.00295797
$751$	−41.8567	−1.52737	−0.763686	−	0.645588i	$-0.776612\pi$
$751$	−41.8567	−1.52737	−0.763686	+	0.645588i	$0.776612\pi$
$752$	3.51054	0.128016
$753$	−16.7374	−0.609944
$754$	0	0
$755$	−22.8176	−0.830418
$756$	−15.8443	−0.576250
$757$	−15.5236	−0.564214	−0.282107	−	0.959383i	$-0.591033\pi$
$757$	−15.5236	−0.564214	−0.282107	+	0.959383i	$0.591033\pi$
$758$	−4.85229	−0.176243
$759$	−3.20381	−0.116291
$760$	3.16822	0.114923
$761$	21.7920	0.789960	0.394980	−	0.918690i	$-0.370752\pi$
$761$	21.7920	0.789960	0.394980	+	0.918690i	$0.370752\pi$
$762$	−1.66476	−0.0603077
$763$	25.1164	0.909273
$764$	14.2598	0.515903
$765$	6.27947	0.227035
$766$	−1.91133	−0.0690592
$767$	0	0
$768$	−0.680104	−0.0245412
$769$	−42.5969	−1.53608	−0.768041	−	0.640401i	$-0.778768\pi$
$769$	−42.5969	−1.53608	−0.768041	+	0.640401i	$0.778768\pi$
$770$	9.93437	0.358010
$771$	11.6987	0.421317
$772$	3.56077	0.128155
$773$	41.8984	1.50698	0.753490	−	0.657459i	$-0.228369\pi$
$773$	41.8984	1.50698	0.753490	+	0.657459i	$0.228369\pi$
$774$	1.64338	0.0590702
$775$	−19.3120	−0.693708
$776$	15.5431	0.557965
$777$	30.0370	1.07757
$778$	−36.7485	−1.31750
$779$	4.69291	0.168141
$780$	0	0
$781$	9.30677	0.333022
$782$	4.93698	0.176546
$783$	−16.9832	−0.606929
$784$	10.7000	0.382141
$785$	−44.1344	−1.57522
$786$	−11.1862	−0.399000
$787$	−27.4413	−0.978178	−0.489089	−	0.872234i	$-0.662671\pi$
$787$	−27.4413	−0.978178	−0.489089	+	0.872234i	$0.662671\pi$
$788$	−4.14163	−0.147540
$789$	16.8806	0.600966
$790$	−26.1988	−0.932110
$791$	10.9823	0.390485
$792$	1.89120	0.0672010
$793$	0	0
$794$	−12.4585	−0.442137
$795$	−5.20936	−0.184757
$796$	14.7854	0.524056
$797$	−39.0461	−1.38308	−0.691541	−	0.722337i	$-0.743068\pi$
$797$	−39.0461	−1.38308	−0.691541	+	0.722337i	$0.743068\pi$
$798$	2.86129	0.101289
$799$	−2.74210	−0.0970085
$800$	−5.03760	−0.178106
$801$	36.2565	1.28106
$802$	20.9479	0.739696
$803$	−6.18122	−0.218130
$804$	3.74953	0.132236
$805$	−84.2467	−2.96931
$806$	0	0
$807$	−12.8539	−0.452478
$808$	4.70133	0.165392
$809$	−16.2897	−0.572714	−0.286357	−	0.958123i	$-0.592444\pi$
$809$	−16.2897	−0.572714	−0.286357	+	0.958123i	$0.592444\pi$
$810$	−16.0029	−0.562284
$811$	−35.5659	−1.24889	−0.624445	−	0.781069i	$-0.714675\pi$
$811$	−35.5659	−1.24889	−0.624445	+	0.781069i	$0.714675\pi$
$812$	18.9722	0.665795
$813$	−16.6985	−0.585643
$814$	−7.82409	−0.274234
$815$	−63.3911	−2.22049
$816$	0.531233	0.0185969
$817$	−0.647649	−0.0226584
$818$	−30.5960	−1.06976
$819$	0	0
$820$	−14.8681	−0.519218
$821$	31.8102	1.11018	0.555092	−	0.831789i	$-0.312683\pi$
$821$	31.8102	1.11018	0.555092	+	0.831789i	$0.312683\pi$
$822$	−15.2619	−0.532320
$823$	−31.1240	−1.08492	−0.542458	−	0.840083i	$-0.682506\pi$
$823$	−31.1240	−1.08492	−0.542458	+	0.840083i	$0.682506\pi$
$824$	−11.2055	−0.390363
$825$	−2.55351	−0.0889019
$826$	30.0711	1.04631
$827$	21.2280	0.738169	0.369084	−	0.929396i	$-0.379671\pi$
$827$	21.2280	0.738169	0.369084	+	0.929396i	$0.379671\pi$
$828$	−16.0380	−0.557360
$829$	2.29390	0.0796706	0.0398353	−	0.999206i	$-0.487317\pi$
$829$	2.29390	0.0796706	0.0398353	+	0.999206i	$0.487317\pi$
$830$	−0.536170	−0.0186107
$831$	−10.3591	−0.359353
$832$	0	0
$833$	−8.35779	−0.289580
$834$	7.49516	0.259536
$835$	2.62904	0.0909817
$836$	−0.745314	−0.0257772
$837$	−14.4374	−0.499031
$838$	−23.7309	−0.819769
$839$	30.2364	1.04388	0.521939	−	0.852983i	$-0.325209\pi$
$839$	30.2364	1.04388	0.521939	+	0.852983i	$0.325209\pi$
$840$	−9.06518	−0.312778
$841$	−8.66401	−0.298759
$842$	6.06704	0.209084
$843$	4.87514	0.167909
$844$	19.9863	0.687956
$845$	0	0
$846$	8.90785	0.306258
$847$	43.9414	1.50984
$848$	2.41765	0.0830225
$849$	18.3972	0.631390
$850$	3.93489	0.134966
$851$	66.3509	2.27448
$852$	−8.49250	−0.290948
$853$	−26.8190	−0.918264	−0.459132	−	0.888368i	$-0.651840\pi$
$853$	−26.8190	−0.918264	−0.459132	+	0.888368i	$0.651840\pi$
$854$	−9.27142	−0.317262
$855$	8.03922	0.274935
$856$	−7.52809	−0.257305
$857$	−16.8138	−0.574348	−0.287174	−	0.957878i	$-0.592716\pi$
$857$	−16.8138	−0.574348	−0.287174	+	0.957878i	$0.592716\pi$
$858$	0	0
$859$	33.7398	1.15119	0.575593	−	0.817736i	$-0.304771\pi$
$859$	33.7398	1.15119	0.575593	+	0.817736i	$0.304771\pi$
$860$	2.05189	0.0699690
$861$	−13.4278	−0.457617
$862$	31.3377	1.06737
$863$	18.6033	0.633264	0.316632	−	0.948548i	$-0.397448\pi$
$863$	18.6033	0.633264	0.316632	+	0.948548i	$0.397448\pi$
$864$	−3.76605	−0.128124
$865$	−61.2649	−2.08307
$866$	−7.67163	−0.260693
$867$	11.1468	0.378566
$868$	16.1284	0.547432
$869$	6.16319	0.209072
$870$	−9.71679	−0.329430
$871$	0	0
$872$	5.96995	0.202168
$873$	39.4400	1.33484
$874$	6.32051	0.213794
$875$	−0.501112	−0.0169407
$876$	5.64040	0.190572
$877$	12.6952	0.428687	0.214344	−	0.976758i	$-0.431239\pi$
$877$	12.6952	0.428687	0.214344	+	0.976758i	$0.431239\pi$
$878$	−7.35012	−0.248055
$879$	11.2410	0.379149
$880$	2.36132	0.0795999
$881$	−37.2925	−1.25642	−0.628208	−	0.778046i	$-0.716211\pi$
$881$	−37.2925	−1.25642	−0.628208	+	0.778046i	$0.716211\pi$
$882$	27.1507	0.914211
$883$	−31.3281	−1.05427	−0.527136	−	0.849781i	$-0.676734\pi$
$883$	−31.3281	−1.05427	−0.527136	+	0.849781i	$0.676734\pi$
$884$	0	0
$885$	−15.4012	−0.517705
$886$	12.9937	0.436531
$887$	−41.7370	−1.40139	−0.700695	−	0.713461i	$-0.747127\pi$
$887$	−41.7370	−1.40139	−0.700695	+	0.713461i	$0.747127\pi$
$888$	7.13953	0.239587
$889$	10.2982	0.345390
$890$	45.2691	1.51742
$891$	3.76463	0.126120
$892$	−1.53481	−0.0513891
$893$	−3.51054	−0.117476
$894$	−13.0792	−0.437433
$895$	−71.6844	−2.39614
$896$	4.20713	0.140550
$897$	0	0
$898$	−35.1949	−1.17447
$899$	17.2877	0.576576
$900$	−12.7827	−0.426090
$901$	−1.88844	−0.0629130
$902$	3.49769	0.116460
$903$	1.85311	0.0616677
$904$	2.61040	0.0868206
$905$	69.4755	2.30944
$906$	−4.89814	−0.162730
$907$	−23.1624	−0.769094	−0.384547	−	0.923105i	$-0.625642\pi$
$907$	−23.1624	−0.769094	−0.384547	+	0.923105i	$0.625642\pi$
$908$	−10.0133	−0.332301
$909$	11.9294	0.395674
$910$	0	0
$911$	1.03252	0.0342090	0.0171045	−	0.999854i	$-0.494555\pi$
$911$	1.03252	0.0342090	0.0171045	+	0.999854i	$0.494555\pi$
$912$	0.680104	0.0225205
$913$	0.126132	0.00417437
$914$	36.5708	1.20965
$915$	4.74844	0.156978
$916$	14.8741	0.491454
$917$	69.1981	2.28512
$918$	2.94168	0.0970898
$919$	42.0020	1.38552	0.692758	−	0.721170i	$-0.256395\pi$
$919$	42.0020	1.38552	0.692758	+	0.721170i	$0.256395\pi$
$920$	−20.0247	−0.660196
$921$	3.44209	0.113421
$922$	−25.2919	−0.832945
$923$	0	0
$924$	2.13256	0.0701560
$925$	52.8832	1.73879
$926$	10.6111	0.348701
$927$	−28.4335	−0.933880
$928$	4.50954	0.148033
$929$	7.43069	0.243793	0.121896	−	0.992543i	$-0.461102\pi$
$929$	7.43069	0.243793	0.121896	+	0.992543i	$0.461102\pi$
$930$	−8.26027	−0.270865
$931$	−10.7000	−0.350677
$932$	6.11477	0.200296
$933$	16.1322	0.528143
$934$	−7.58944	−0.248334
$935$	−1.84443	−0.0603195
$936$	0	0
$937$	−53.8314	−1.75860	−0.879298	−	0.476273i	$-0.841987\pi$
$937$	−53.8314	−1.75860	−0.879298	+	0.476273i	$0.841987\pi$
$938$	−23.1946	−0.757331
$939$	−7.33033	−0.239216
$940$	11.1222	0.362765
$941$	−23.1591	−0.754967	−0.377483	−	0.926016i	$-0.623210\pi$
$941$	−23.1591	−0.754967	−0.377483	+	0.926016i	$0.623210\pi$
$942$	−9.47409	−0.308683
$943$	−29.6616	−0.965913
$944$	7.14765	0.232636
$945$	−50.1981	−1.63294
$946$	−0.482702	−0.0156940
$947$	50.1929	1.63105	0.815524	−	0.578723i	$-0.196449\pi$
$947$	50.1929	1.63105	0.815524	+	0.578723i	$0.196449\pi$
$948$	−5.62395	−0.182657
$949$	0	0
$950$	5.03760	0.163441
$951$	13.0348	0.422681
$952$	−3.28621	−0.106507
$953$	−40.6910	−1.31811	−0.659056	−	0.752094i	$-0.729044\pi$
$953$	−40.6910	−1.31811	−0.659056	+	0.752094i	$0.729044\pi$
$954$	6.13469	0.198618
$955$	45.1783	1.46193
$956$	−15.4630	−0.500109
$957$	2.28585	0.0738910
$958$	33.3642	1.07795
$959$	94.4102	3.04867
$960$	−2.15472	−0.0695432
$961$	−16.3037	−0.525926
$962$	0	0
$963$	−19.1022	−0.615560
$964$	−8.07387	−0.260042
$965$	11.2813	0.363158
$966$	−18.0848	−0.581869
$967$	9.55821	0.307371	0.153686	−	0.988120i	$-0.450886\pi$
$967$	9.55821	0.307371	0.153686	+	0.988120i	$0.450886\pi$
$968$	10.4445	0.335699
$969$	−0.531233	−0.0170656
$970$	49.2439	1.58113
$971$	16.1245	0.517461	0.258731	−	0.965950i	$-0.416696\pi$
$971$	16.1245	0.517461	0.258731	+	0.965950i	$0.416696\pi$
$972$	−14.7334	−0.472574
$973$	−46.3651	−1.48640
$974$	20.7884	0.666103
$975$	0	0
$976$	−2.20374	−0.0705400
$977$	46.7284	1.49497	0.747487	−	0.664277i	$-0.231260\pi$
$977$	46.7284	1.49497	0.747487	+	0.664277i	$0.231260\pi$
$978$	−13.6078	−0.435131
$979$	−10.6494	−0.340357
$980$	33.8998	1.08289
$981$	15.1485	0.483654
$982$	−1.44691	−0.0461729
$983$	11.1258	0.354858	0.177429	−	0.984134i	$-0.443222\pi$
$983$	11.1258	0.354858	0.177429	+	0.984134i	$0.443222\pi$
$984$	−3.19167	−0.101747
$985$	−13.1216	−0.418089
$986$	−3.52242	−0.112177
$987$	10.0447	0.319725
$988$	0	0
$989$	4.09348	0.130165
$990$	5.99174	0.190430
$991$	16.1402	0.512709	0.256355	−	0.966583i	$-0.417479\pi$
$991$	16.1402	0.512709	0.256355	+	0.966583i	$0.417479\pi$
$992$	3.83358	0.121716
$993$	20.5469	0.652036
$994$	52.5347	1.66630
$995$	46.8435	1.48504
$996$	−0.115097	−0.00364698
$997$	28.1476	0.891444	0.445722	−	0.895171i	$-0.352947\pi$
$997$	28.1476	0.891444	0.445722	+	0.895171i	$0.352947\pi$
$998$	17.5580	0.555789
$999$	39.5349	1.25083

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bp.1.5	✓	15
13.12	even	2		6422.2.a.br.1.5	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.bp.1.5	✓	15	1.1	even	1	trivial
6422.2.a.br.1.5	yes	15	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.680104 q^{3} +1.00000 q^{4} +3.16822 q^{5} +0.680104 q^{6} -4.20713 q^{7} -1.00000 q^{8} -2.53746 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.680104 q^{3} +1.00000 q^{4} +3.16822 q^{5} +0.680104 q^{6} -4.20713 q^{7} -1.00000 q^{8} -2.53746 q^{9} -3.16822 q^{10} +0.745314 q^{11} -0.680104 q^{12} +4.20713 q^{14} -2.15472 q^{15} +1.00000 q^{16} -0.781104 q^{17} +2.53746 q^{18} -1.00000 q^{19} +3.16822 q^{20} +2.86129 q^{21} -0.745314 q^{22} +6.32051 q^{23} +0.680104 q^{24} +5.03760 q^{25} +3.76605 q^{27} -4.20713 q^{28} -4.50954 q^{29} +2.15472 q^{30} -3.83358 q^{31} -1.00000 q^{32} -0.506891 q^{33} +0.781104 q^{34} -13.3291 q^{35} -2.53746 q^{36} +10.4977 q^{37} +1.00000 q^{38} -3.16822 q^{40} -4.69291 q^{41} -2.86129 q^{42} +0.647649 q^{43} +0.745314 q^{44} -8.03922 q^{45} -6.32051 q^{46} +3.51054 q^{47} -0.680104 q^{48} +10.7000 q^{49} -5.03760 q^{50} +0.531233 q^{51} +2.41765 q^{53} -3.76605 q^{54} +2.36132 q^{55} +4.20713 q^{56} +0.680104 q^{57} +4.50954 q^{58} +7.14765 q^{59} -2.15472 q^{60} -2.20374 q^{61} +3.83358 q^{62} +10.6754 q^{63} +1.00000 q^{64} +0.506891 q^{66} -5.51316 q^{67} -0.781104 q^{68} -4.29861 q^{69} +13.3291 q^{70} +12.4870 q^{71} +2.53746 q^{72} -8.29344 q^{73} -10.4977 q^{74} -3.42609 q^{75} -1.00000 q^{76} -3.13563 q^{77} +8.26925 q^{79} +3.16822 q^{80} +5.05107 q^{81} +4.69291 q^{82} +0.169234 q^{83} +2.86129 q^{84} -2.47471 q^{85} -0.647649 q^{86} +3.06696 q^{87} -0.745314 q^{88} -14.2885 q^{89} +8.03922 q^{90} +6.32051 q^{92} +2.60723 q^{93} -3.51054 q^{94} -3.16822 q^{95} +0.680104 q^{96} -15.5431 q^{97} -10.7000 q^{98} -1.89120 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 15 q^{2} + 4 q^{3} + 15 q^{4} - 3 q^{5} - 4 q^{6} - 10 q^{7} - 15 q^{8} + 17 q^{9}+O(q^{10}) \) 15 * q - 15 * q^2 + 4 * q^3 + 15 * q^4 - 3 * q^5 - 4 * q^6 - 10 * q^7 - 15 * q^8 + 17 * q^9	\( 15 q - 15 q^{2} + 4 q^{3} + 15 q^{4} - 3 q^{5} - 4 q^{6} - 10 q^{7} - 15 q^{8} + 17 q^{9} + 3 q^{10} - 4 q^{11} + 4 q^{12} + 10 q^{14} - 17 q^{15} + 15 q^{16} + 2 q^{17} - 17 q^{18} - 15 q^{19} - 3 q^{20} - 28 q^{21} + 4 q^{22} + 3 q^{23} - 4 q^{24} + 8 q^{25} + 16 q^{27} - 10 q^{28} + 12 q^{29} + 17 q^{30} - 4 q^{31} - 15 q^{32} + 2 q^{33} - 2 q^{34} + 4 q^{35} + 17 q^{36} - 17 q^{37} + 15 q^{38} + 3 q^{40} + 3 q^{41} + 28 q^{42} - 7 q^{43} - 4 q^{44} - 41 q^{45} - 3 q^{46} + 4 q^{48} + 9 q^{49} - 8 q^{50} + 49 q^{51} - 7 q^{53} - 16 q^{54} + 20 q^{55} + 10 q^{56} - 4 q^{57} - 12 q^{58} - 7 q^{59} - 17 q^{60} - 22 q^{61} + 4 q^{62} - 32 q^{63} + 15 q^{64} - 2 q^{66} - 58 q^{67} + 2 q^{68} - 10 q^{69} - 4 q^{70} - 2 q^{71} - 17 q^{72} - 64 q^{73} + 17 q^{74} - 11 q^{75} - 15 q^{76} - 14 q^{77} + 15 q^{79} - 3 q^{80} + 23 q^{81} - 3 q^{82} - 11 q^{83} - 28 q^{84} - 48 q^{85} + 7 q^{86} - 46 q^{87} + 4 q^{88} + 9 q^{89} + 41 q^{90} + 3 q^{92} - 22 q^{93} + 3 q^{95} - 4 q^{96} - 28 q^{97} - 9 q^{98} - 16 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 + 4 * q^3 + 15 * q^4 - 3 * q^5 - 4 * q^6 - 10 * q^7 - 15 * q^8 + 17 * q^9 + 3 * q^10 - 4 * q^11 + 4 * q^12 + 10 * q^14 - 17 * q^15 + 15 * q^16 + 2 * q^17 - 17 * q^18 - 15 * q^19 - 3 * q^20 - 28 * q^21 + 4 * q^22 + 3 * q^23 - 4 * q^24 + 8 * q^25 + 16 * q^27 - 10 * q^28 + 12 * q^29 + 17 * q^30 - 4 * q^31 - 15 * q^32 + 2 * q^33 - 2 * q^34 + 4 * q^35 + 17 * q^36 - 17 * q^37 + 15 * q^38 + 3 * q^40 + 3 * q^41 + 28 * q^42 - 7 * q^43 - 4 * q^44 - 41 * q^45 - 3 * q^46 + 4 * q^48 + 9 * q^49 - 8 * q^50 + 49 * q^51 - 7 * q^53 - 16 * q^54 + 20 * q^55 + 10 * q^56 - 4 * q^57 - 12 * q^58 - 7 * q^59 - 17 * q^60 - 22 * q^61 + 4 * q^62 - 32 * q^63 + 15 * q^64 - 2 * q^66 - 58 * q^67 + 2 * q^68 - 10 * q^69 - 4 * q^70 - 2 * q^71 - 17 * q^72 - 64 * q^73 + 17 * q^74 - 11 * q^75 - 15 * q^76 - 14 * q^77 + 15 * q^79 - 3 * q^80 + 23 * q^81 - 3 * q^82 - 11 * q^83 - 28 * q^84 - 48 * q^85 + 7 * q^86 - 46 * q^87 + 4 * q^88 + 9 * q^89 + 41 * q^90 + 3 * q^92 - 22 * q^93 + 3 * q^95 - 4 * q^96 - 28 * q^97 - 9 * q^98 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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