LMFDB - Embedded newform 5070.2.a.bq.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.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:	$40.4841538248$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{7}]$
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	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +0.643104 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +0.643104 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.40581 q^{11} +1.00000 q^{12} -0.643104 q^{14} +1.00000 q^{15} +1.00000 q^{16} +0.939001 q^{17} -1.00000 q^{18} +2.75302 q^{19} +1.00000 q^{20} +0.643104 q^{21} -4.40581 q^{22} -2.04892 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +0.643104 q^{28} -0.149145 q^{29} -1.00000 q^{30} +2.08815 q^{31} -1.00000 q^{32} +4.40581 q^{33} -0.939001 q^{34} +0.643104 q^{35} +1.00000 q^{36} +6.07069 q^{37} -2.75302 q^{38} -1.00000 q^{40} -5.74094 q^{41} -0.643104 q^{42} +4.69202 q^{43} +4.40581 q^{44} +1.00000 q^{45} +2.04892 q^{46} +7.00000 q^{47} +1.00000 q^{48} -6.58642 q^{49} -1.00000 q^{50} +0.939001 q^{51} +1.68664 q^{53} -1.00000 q^{54} +4.40581 q^{55} -0.643104 q^{56} +2.75302 q^{57} +0.149145 q^{58} +7.85623 q^{59} +1.00000 q^{60} -12.0761 q^{61} -2.08815 q^{62} +0.643104 q^{63} +1.00000 q^{64} -4.40581 q^{66} +1.45712 q^{67} +0.939001 q^{68} -2.04892 q^{69} -0.643104 q^{70} -7.03684 q^{71} -1.00000 q^{72} +14.8605 q^{73} -6.07069 q^{74} +1.00000 q^{75} +2.75302 q^{76} +2.83340 q^{77} +0.929312 q^{79} +1.00000 q^{80} +1.00000 q^{81} +5.74094 q^{82} +11.2959 q^{83} +0.643104 q^{84} +0.939001 q^{85} -4.69202 q^{86} -0.149145 q^{87} -4.40581 q^{88} -7.47650 q^{89} -1.00000 q^{90} -2.04892 q^{92} +2.08815 q^{93} -7.00000 q^{94} +2.75302 q^{95} -1.00000 q^{96} -5.05861 q^{97} +6.58642 q^{98} +4.40581 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 6 * q^7 - 3 * q^8 + 3 * q^9	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{10} + 3 q^{12} - 6 q^{14} + 3 q^{15} + 3 q^{16} - 7 q^{17} - 3 q^{18} + 13 q^{19} + 3 q^{20} + 6 q^{21} + 3 q^{23} - 3 q^{24} + 3 q^{25} + 3 q^{27} + 6 q^{28} - 14 q^{29} - 3 q^{30} + 10 q^{31} - 3 q^{32} + 7 q^{34} + 6 q^{35} + 3 q^{36} + 6 q^{37} - 13 q^{38} - 3 q^{40} - 3 q^{41} - 6 q^{42} + 9 q^{43} + 3 q^{45} - 3 q^{46} + 21 q^{47} + 3 q^{48} + 5 q^{49} - 3 q^{50} - 7 q^{51} + 3 q^{53} - 3 q^{54} - 6 q^{56} + 13 q^{57} + 14 q^{58} + 7 q^{59} + 3 q^{60} - 21 q^{61} - 10 q^{62} + 6 q^{63} + 3 q^{64} + 23 q^{67} - 7 q^{68} + 3 q^{69} - 6 q^{70} + 7 q^{71} - 3 q^{72} + 9 q^{73} - 6 q^{74} + 3 q^{75} + 13 q^{76} - 21 q^{77} + 15 q^{79} + 3 q^{80} + 3 q^{81} + 3 q^{82} + 20 q^{83} + 6 q^{84} - 7 q^{85} - 9 q^{86} - 14 q^{87} + 3 q^{89} - 3 q^{90} + 3 q^{92} + 10 q^{93} - 21 q^{94} + 13 q^{95} - 3 q^{96} + 16 q^{97} - 5 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 6 * q^7 - 3 * q^8 + 3 * q^9 - 3 * q^10 + 3 * q^12 - 6 * q^14 + 3 * q^15 + 3 * q^16 - 7 * q^17 - 3 * q^18 + 13 * q^19 + 3 * q^20 + 6 * q^21 + 3 * q^23 - 3 * q^24 + 3 * q^25 + 3 * q^27 + 6 * q^28 - 14 * q^29 - 3 * q^30 + 10 * q^31 - 3 * q^32 + 7 * q^34 + 6 * q^35 + 3 * q^36 + 6 * q^37 - 13 * q^38 - 3 * q^40 - 3 * q^41 - 6 * q^42 + 9 * q^43 + 3 * q^45 - 3 * q^46 + 21 * q^47 + 3 * q^48 + 5 * q^49 - 3 * q^50 - 7 * q^51 + 3 * q^53 - 3 * q^54 - 6 * q^56 + 13 * q^57 + 14 * q^58 + 7 * q^59 + 3 * q^60 - 21 * q^61 - 10 * q^62 + 6 * q^63 + 3 * q^64 + 23 * q^67 - 7 * q^68 + 3 * q^69 - 6 * q^70 + 7 * q^71 - 3 * q^72 + 9 * q^73 - 6 * q^74 + 3 * q^75 + 13 * q^76 - 21 * q^77 + 15 * q^79 + 3 * q^80 + 3 * q^81 + 3 * q^82 + 20 * q^83 + 6 * q^84 - 7 * q^85 - 9 * q^86 - 14 * q^87 + 3 * q^89 - 3 * q^90 + 3 * q^92 + 10 * q^93 - 21 * q^94 + 13 * q^95 - 3 * q^96 + 16 * q^97 - 5 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	0.643104	0.243071	0.121535	−	0.992587i	$-0.461218\pi$
$7$	0.643104	0.243071	0.121535	+	0.992587i	$0.461218\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	4.40581	1.32840	0.664201	−	0.747554i	$-0.268772\pi$
$11$	4.40581	1.32840	0.664201	+	0.747554i	$0.268772\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−0.643104	−0.171877
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	0.939001	0.227741	0.113871	−	0.993496i	$-0.463675\pi$
$17$	0.939001	0.227741	0.113871	+	0.993496i	$0.463675\pi$
$18$	−1.00000	−0.235702
$19$	2.75302	0.631586	0.315793	−	0.948828i	$-0.397729\pi$
$19$	2.75302	0.631586	0.315793	+	0.948828i	$0.397729\pi$
$20$	1.00000	0.223607
$21$	0.643104	0.140337
$22$	−4.40581	−0.939323
$23$	−2.04892	−0.427229	−0.213614	−	0.976918i	$-0.568524\pi$
$23$	−2.04892	−0.427229	−0.213614	+	0.976918i	$0.568524\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0.643104	0.121535
$29$	−0.149145	−0.0276955	−0.0138478	−	0.999904i	$-0.504408\pi$
$29$	−0.149145	−0.0276955	−0.0138478	+	0.999904i	$0.504408\pi$
$30$	−1.00000	−0.182574
$31$	2.08815	0.375042	0.187521	−	0.982261i	$-0.439955\pi$
$31$	2.08815	0.375042	0.187521	+	0.982261i	$0.439955\pi$
$32$	−1.00000	−0.176777
$33$	4.40581	0.766954
$34$	−0.939001	−0.161037
$35$	0.643104	0.108704
$36$	1.00000	0.166667
$37$	6.07069	0.998015	0.499007	−	0.866598i	$-0.333698\pi$
$37$	6.07069	0.998015	0.499007	+	0.866598i	$0.333698\pi$
$38$	−2.75302	−0.446599
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−5.74094	−0.896584	−0.448292	−	0.893887i	$-0.647968\pi$
$41$	−5.74094	−0.896584	−0.448292	+	0.893887i	$0.647968\pi$
$42$	−0.643104	−0.0992331
$43$	4.69202	0.715527	0.357763	−	0.933812i	$-0.383539\pi$
$43$	4.69202	0.715527	0.357763	+	0.933812i	$0.383539\pi$
$44$	4.40581	0.664201
$45$	1.00000	0.149071
$46$	2.04892	0.302096
$47$	7.00000	1.02105	0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000	1.02105	0.510527	+	0.859861i	$0.329450\pi$
$48$	1.00000	0.144338
$49$	−6.58642	−0.940917
$50$	−1.00000	−0.141421
$51$	0.939001	0.131486
$52$	0	0
$53$	1.68664	0.231678	0.115839	−	0.993268i	$-0.463044\pi$
$53$	1.68664	0.231678	0.115839	+	0.993268i	$0.463044\pi$
$54$	−1.00000	−0.136083
$55$	4.40581	0.594080
$56$	−0.643104	−0.0859384
$57$	2.75302	0.364646
$58$	0.149145	0.0195837
$59$	7.85623	1.02279	0.511397	−	0.859344i	$-0.329128\pi$
$59$	7.85623	1.02279	0.511397	+	0.859344i	$0.329128\pi$
$60$	1.00000	0.129099
$61$	−12.0761	−1.54618	−0.773091	−	0.634295i	$-0.781290\pi$
$61$	−12.0761	−1.54618	−0.773091	+	0.634295i	$0.781290\pi$
$62$	−2.08815	−0.265195
$63$	0.643104	0.0810235
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.40581	−0.542318
$67$	1.45712	0.178016	0.0890080	−	0.996031i	$-0.471630\pi$
$67$	1.45712	0.178016	0.0890080	+	0.996031i	$0.471630\pi$
$68$	0.939001	0.113871
$69$	−2.04892	−0.246661
$70$	−0.643104	−0.0768656
$71$	−7.03684	−0.835119	−0.417559	−	0.908650i	$-0.637114\pi$
$71$	−7.03684	−0.835119	−0.417559	+	0.908650i	$0.637114\pi$
$72$	−1.00000	−0.117851
$73$	14.8605	1.73930	0.869648	−	0.493673i	$-0.164346\pi$
$73$	14.8605	1.73930	0.869648	+	0.493673i	$0.164346\pi$
$74$	−6.07069	−0.705703
$75$	1.00000	0.115470
$76$	2.75302	0.315793
$77$	2.83340	0.322896
$78$	0	0
$79$	0.929312	0.104556	0.0522779	−	0.998633i	$-0.483352\pi$
$79$	0.929312	0.104556	0.0522779	+	0.998633i	$0.483352\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	5.74094	0.633981
$83$	11.2959	1.23989	0.619943	−	0.784647i	$-0.287156\pi$
$83$	11.2959	1.23989	0.619943	+	0.784647i	$0.287156\pi$
$84$	0.643104	0.0701684
$85$	0.939001	0.101849
$86$	−4.69202	−0.505954
$87$	−0.149145	−0.0159900
$88$	−4.40581	−0.469661
$89$	−7.47650	−0.792508	−0.396254	−	0.918141i	$-0.629690\pi$
$89$	−7.47650	−0.792508	−0.396254	+	0.918141i	$0.629690\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−2.04892	−0.213614
$93$	2.08815	0.216531
$94$	−7.00000	−0.721995
$95$	2.75302	0.282454
$96$	−1.00000	−0.102062
$97$	−5.05861	−0.513624	−0.256812	−	0.966461i	$-0.582672\pi$
$97$	−5.05861	−0.513624	−0.256812	+	0.966461i	$0.582672\pi$
$98$	6.58642	0.665329
$99$	4.40581	0.442801
$100$	1.00000	0.100000
$101$	−6.81163	−0.677782	−0.338891	−	0.940826i	$-0.610052\pi$
$101$	−6.81163	−0.677782	−0.338891	+	0.940826i	$0.610052\pi$
$102$	−0.939001	−0.0929750
$103$	−12.0858	−1.19084	−0.595422	−	0.803413i	$-0.703015\pi$
$103$	−12.0858	−1.19084	−0.595422	+	0.803413i	$0.703015\pi$
$104$	0	0
$105$	0.643104	0.0627605
$106$	−1.68664	−0.163821
$107$	−14.6679	−1.41800	−0.708998	−	0.705211i	$-0.750852\pi$
$107$	−14.6679	−1.41800	−0.708998	+	0.705211i	$0.750852\pi$
$108$	1.00000	0.0962250
$109$	8.15883	0.781475	0.390737	−	0.920502i	$-0.372220\pi$
$109$	8.15883	0.781475	0.390737	+	0.920502i	$0.372220\pi$
$110$	−4.40581	−0.420078
$111$	6.07069	0.576204
$112$	0.643104	0.0607676
$113$	16.8442	1.58456	0.792282	−	0.610155i	$-0.208893\pi$
$113$	16.8442	1.58456	0.792282	+	0.610155i	$0.208893\pi$
$114$	−2.75302	−0.257844
$115$	−2.04892	−0.191063
$116$	−0.149145	−0.0138478
$117$	0	0
$118$	−7.85623	−0.723225
$119$	0.603875	0.0553572
$120$	−1.00000	−0.0912871
$121$	8.41119	0.764654
$122$	12.0761	1.09332
$123$	−5.74094	−0.517643
$124$	2.08815	0.187521
$125$	1.00000	0.0894427
$126$	−0.643104	−0.0572923
$127$	4.85086	0.430444	0.215222	−	0.976565i	$-0.430953\pi$
$127$	4.85086	0.430444	0.215222	+	0.976565i	$0.430953\pi$
$128$	−1.00000	−0.0883883
$129$	4.69202	0.413109
$130$	0	0
$131$	1.52781	0.133485	0.0667427	−	0.997770i	$-0.478739\pi$
$131$	1.52781	0.133485	0.0667427	+	0.997770i	$0.478739\pi$
$132$	4.40581	0.383477
$133$	1.77048	0.153520
$134$	−1.45712	−0.125876
$135$	1.00000	0.0860663
$136$	−0.939001	−0.0805187
$137$	−6.41119	−0.547745	−0.273872	−	0.961766i	$-0.588305\pi$
$137$	−6.41119	−0.547745	−0.273872	+	0.961766i	$0.588305\pi$
$138$	2.04892	0.174415
$139$	−8.78017	−0.744724	−0.372362	−	0.928088i	$-0.621452\pi$
$139$	−8.78017	−0.744724	−0.372362	+	0.928088i	$0.621452\pi$
$140$	0.643104	0.0543522
$141$	7.00000	0.589506
$142$	7.03684	0.590518
$143$	0	0
$144$	1.00000	0.0833333
$145$	−0.149145	−0.0123858
$146$	−14.8605	−1.22987
$147$	−6.58642	−0.543239
$148$	6.07069	0.499007
$149$	−22.6504	−1.85559	−0.927797	−	0.373087i	$-0.878299\pi$
$149$	−22.6504	−1.85559	−0.927797	+	0.373087i	$0.878299\pi$
$150$	−1.00000	−0.0816497
$151$	10.2959	0.837868	0.418934	−	0.908017i	$-0.362404\pi$
$151$	10.2959	0.837868	0.418934	+	0.908017i	$0.362404\pi$
$152$	−2.75302	−0.223299
$153$	0.939001	0.0759137
$154$	−2.83340	−0.228322
$155$	2.08815	0.167724
$156$	0	0
$157$	0.994623	0.0793796	0.0396898	−	0.999212i	$-0.487363\pi$
$157$	0.994623	0.0793796	0.0396898	+	0.999212i	$0.487363\pi$
$158$	−0.929312	−0.0739321
$159$	1.68664	0.133760
$160$	−1.00000	−0.0790569
$161$	−1.31767	−0.103847
$162$	−1.00000	−0.0785674
$163$	21.6015	1.69196	0.845979	−	0.533216i	$-0.179017\pi$
$163$	21.6015	1.69196	0.845979	+	0.533216i	$0.179017\pi$
$164$	−5.74094	−0.448292
$165$	4.40581	0.342992
$166$	−11.2959	−0.876732
$167$	7.96615	0.616439	0.308220	−	0.951315i	$-0.400267\pi$
$167$	7.96615	0.616439	0.308220	+	0.951315i	$0.400267\pi$
$168$	−0.643104	−0.0496166
$169$	0	0
$170$	−0.939001	−0.0720181
$171$	2.75302	0.210529
$172$	4.69202	0.357763
$173$	4.11960	0.313208	0.156604	−	0.987661i	$-0.449945\pi$
$173$	4.11960	0.313208	0.156604	+	0.987661i	$0.449945\pi$
$174$	0.149145	0.0113066
$175$	0.643104	0.0486141
$176$	4.40581	0.332101
$177$	7.85623	0.590511
$178$	7.47650	0.560387
$179$	−8.78017	−0.656261	−0.328130	−	0.944632i	$-0.606418\pi$
$179$	−8.78017	−0.656261	−0.328130	+	0.944632i	$0.606418\pi$
$180$	1.00000	0.0745356
$181$	19.4741	1.44750	0.723750	−	0.690063i	$-0.242417\pi$
$181$	19.4741	1.44750	0.723750	+	0.690063i	$0.242417\pi$
$182$	0	0
$183$	−12.0761	−0.892688
$184$	2.04892	0.151048
$185$	6.07069	0.446326
$186$	−2.08815	−0.153110
$187$	4.13706	0.302532
$188$	7.00000	0.510527
$189$	0.643104	0.0467789
$190$	−2.75302	−0.199725
$191$	0.753020	0.0544866	0.0272433	−	0.999629i	$-0.491327\pi$
$191$	0.753020	0.0544866	0.0272433	+	0.999629i	$0.491327\pi$
$192$	1.00000	0.0721688
$193$	−7.84117	−0.564420	−0.282210	−	0.959353i	$-0.591067\pi$
$193$	−7.84117	−0.564420	−0.282210	+	0.959353i	$0.591067\pi$
$194$	5.05861	0.363187
$195$	0	0
$196$	−6.58642	−0.470458
$197$	10.2010	0.726794	0.363397	−	0.931634i	$-0.381617\pi$
$197$	10.2010	0.726794	0.363397	+	0.931634i	$0.381617\pi$
$198$	−4.40581	−0.313108
$199$	1.65279	0.117163	0.0585817	−	0.998283i	$-0.481342\pi$
$199$	1.65279	0.117163	0.0585817	+	0.998283i	$0.481342\pi$
$200$	−1.00000	−0.0707107
$201$	1.45712	0.102778
$202$	6.81163	0.479264
$203$	−0.0959157	−0.00673196
$204$	0.939001	0.0657432
$205$	−5.74094	−0.400965
$206$	12.0858	0.842054
$207$	−2.04892	−0.142410
$208$	0	0
$209$	12.1293	0.839001
$210$	−0.643104	−0.0443784
$211$	11.7453	0.808576	0.404288	−	0.914632i	$-0.367519\pi$
$211$	11.7453	0.808576	0.404288	+	0.914632i	$0.367519\pi$
$212$	1.68664	0.115839
$213$	−7.03684	−0.482156
$214$	14.6679	1.00267
$215$	4.69202	0.319993
$216$	−1.00000	−0.0680414
$217$	1.34290	0.0911617
$218$	−8.15883	−0.552586
$219$	14.8605	1.00418
$220$	4.40581	0.297040
$221$	0	0
$222$	−6.07069	−0.407438
$223$	−9.73855	−0.652141	−0.326071	−	0.945345i	$-0.605725\pi$
$223$	−9.73855	−0.652141	−0.326071	+	0.945345i	$0.605725\pi$
$224$	−0.643104	−0.0429692
$225$	1.00000	0.0666667
$226$	−16.8442	−1.12046
$227$	−25.2500	−1.67590	−0.837949	−	0.545748i	$-0.816246\pi$
$227$	−25.2500	−1.67590	−0.837949	+	0.545748i	$0.816246\pi$
$228$	2.75302	0.182323
$229$	23.7845	1.57172	0.785861	−	0.618403i	$-0.212220\pi$
$229$	23.7845	1.57172	0.785861	+	0.618403i	$0.212220\pi$
$230$	2.04892	0.135102
$231$	2.83340	0.186424
$232$	0.149145	0.00979184
$233$	0.603875	0.0395612	0.0197806	−	0.999804i	$-0.493703\pi$
$233$	0.603875	0.0395612	0.0197806	+	0.999804i	$0.493703\pi$
$234$	0	0
$235$	7.00000	0.456630
$236$	7.85623	0.511397
$237$	0.929312	0.0603653
$238$	−0.603875	−0.0391434
$239$	−28.5579	−1.84726	−0.923630	−	0.383286i	$-0.874792\pi$
$239$	−28.5579	−1.84726	−0.923630	+	0.383286i	$0.874792\pi$
$240$	1.00000	0.0645497
$241$	6.31096	0.406525	0.203262	−	0.979124i	$-0.434846\pi$
$241$	6.31096	0.406525	0.203262	+	0.979124i	$0.434846\pi$
$242$	−8.41119	−0.540692
$243$	1.00000	0.0641500
$244$	−12.0761	−0.773091
$245$	−6.58642	−0.420791
$246$	5.74094	0.366029
$247$	0	0
$248$	−2.08815	−0.132597
$249$	11.2959	0.715848
$250$	−1.00000	−0.0632456
$251$	24.5972	1.55256	0.776280	−	0.630388i	$-0.217104\pi$
$251$	24.5972	1.55256	0.776280	+	0.630388i	$0.217104\pi$
$252$	0.643104	0.0405118
$253$	−9.02715	−0.567532
$254$	−4.85086	−0.304370
$255$	0.939001	0.0588025
$256$	1.00000	0.0625000
$257$	−26.9463	−1.68086	−0.840432	−	0.541917i	$-0.817699\pi$
$257$	−26.9463	−1.68086	−0.840432	+	0.541917i	$0.817699\pi$
$258$	−4.69202	−0.292112
$259$	3.90408	0.242588
$260$	0	0
$261$	−0.149145	−0.00923184
$262$	−1.52781	−0.0943885
$263$	−24.0465	−1.48277	−0.741386	−	0.671079i	$-0.765831\pi$
$263$	−24.0465	−1.48277	−0.741386	+	0.671079i	$0.765831\pi$
$264$	−4.40581	−0.271159
$265$	1.68664	0.103610
$266$	−1.77048	−0.108555
$267$	−7.47650	−0.457554
$268$	1.45712	0.0890080
$269$	−7.17523	−0.437481	−0.218741	−	0.975783i	$-0.570195\pi$
$269$	−7.17523	−0.437481	−0.218741	+	0.975783i	$0.570195\pi$
$270$	−1.00000	−0.0608581
$271$	10.9825	0.667142	0.333571	−	0.942725i	$-0.391746\pi$
$271$	10.9825	0.667142	0.333571	+	0.942725i	$0.391746\pi$
$272$	0.939001	0.0569353
$273$	0	0
$274$	6.41119	0.387314
$275$	4.40581	0.265681
$276$	−2.04892	−0.123330
$277$	16.2107	0.974009	0.487004	−	0.873400i	$-0.338090\pi$
$277$	16.2107	0.974009	0.487004	+	0.873400i	$0.338090\pi$
$278$	8.78017	0.526599
$279$	2.08815	0.125014
$280$	−0.643104	−0.0384328
$281$	−12.9584	−0.773032	−0.386516	−	0.922283i	$-0.626322\pi$
$281$	−12.9584	−0.773032	−0.386516	+	0.922283i	$0.626322\pi$
$282$	−7.00000	−0.416844
$283$	21.5415	1.28051	0.640256	−	0.768162i	$-0.278828\pi$
$283$	21.5415	1.28051	0.640256	+	0.768162i	$0.278828\pi$
$284$	−7.03684	−0.417559
$285$	2.75302	0.163075
$286$	0	0
$287$	−3.69202	−0.217933
$288$	−1.00000	−0.0589256
$289$	−16.1183	−0.948134
$290$	0.149145	0.00875809
$291$	−5.05861	−0.296541
$292$	14.8605	0.869648
$293$	9.57971	0.559653	0.279826	−	0.960051i	$-0.409723\pi$
$293$	9.57971	0.559653	0.279826	+	0.960051i	$0.409723\pi$
$294$	6.58642	0.384128
$295$	7.85623	0.457408
$296$	−6.07069	−0.352852
$297$	4.40581	0.255651
$298$	22.6504	1.31210
$299$	0	0
$300$	1.00000	0.0577350
$301$	3.01746	0.173923
$302$	−10.2959	−0.592462
$303$	−6.81163	−0.391318
$304$	2.75302	0.157897
$305$	−12.0761	−0.691473
$306$	−0.939001	−0.0536791
$307$	19.0140	1.08519	0.542593	−	0.839996i	$-0.317443\pi$
$307$	19.0140	1.08519	0.542593	+	0.839996i	$0.317443\pi$
$308$	2.83340	0.161448
$309$	−12.0858	−0.687534
$310$	−2.08815	−0.118599
$311$	−3.74764	−0.212509	−0.106255	−	0.994339i	$-0.533886\pi$
$311$	−3.74764	−0.212509	−0.106255	+	0.994339i	$0.533886\pi$
$312$	0	0
$313$	25.1196	1.41984	0.709922	−	0.704280i	$-0.248730\pi$
$313$	25.1196	1.41984	0.709922	+	0.704280i	$0.248730\pi$
$314$	−0.994623	−0.0561298
$315$	0.643104	0.0362348
$316$	0.929312	0.0522779
$317$	12.6679	0.711498	0.355749	−	0.934582i	$-0.384226\pi$
$317$	12.6679	0.711498	0.355749	+	0.934582i	$0.384226\pi$
$318$	−1.68664	−0.0945823
$319$	−0.657105	−0.0367908
$320$	1.00000	0.0559017
$321$	−14.6679	−0.818680
$322$	1.31767	0.0734307
$323$	2.58509	0.143838
$324$	1.00000	0.0555556
$325$	0	0
$326$	−21.6015	−1.19640
$327$	8.15883	0.451185
$328$	5.74094	0.316990
$329$	4.50173	0.248188
$330$	−4.40581	−0.242532
$331$	−6.43057	−0.353456	−0.176728	−	0.984260i	$-0.556551\pi$
$331$	−6.43057	−0.353456	−0.176728	+	0.984260i	$0.556551\pi$
$332$	11.2959	0.619943
$333$	6.07069	0.332672
$334$	−7.96615	−0.435888
$335$	1.45712	0.0796112
$336$	0.643104	0.0350842
$337$	10.5496	0.574672	0.287336	−	0.957830i	$-0.407230\pi$
$337$	10.5496	0.574672	0.287336	+	0.957830i	$0.407230\pi$
$338$	0	0
$339$	16.8442	0.914849
$340$	0.939001	0.0509245
$341$	9.19998	0.498207
$342$	−2.75302	−0.148866
$343$	−8.73748	−0.471780
$344$	−4.69202	−0.252977
$345$	−2.04892	−0.110310
$346$	−4.11960	−0.221471
$347$	9.87130	0.529919	0.264960	−	0.964260i	$-0.414641\pi$
$347$	9.87130	0.529919	0.264960	+	0.964260i	$0.414641\pi$
$348$	−0.149145	−0.00799501
$349$	−30.5338	−1.63444	−0.817218	−	0.576329i	$-0.804485\pi$
$349$	−30.5338	−1.63444	−0.817218	+	0.576329i	$0.804485\pi$
$350$	−0.643104	−0.0343754
$351$	0	0
$352$	−4.40581	−0.234831
$353$	−24.7047	−1.31490	−0.657449	−	0.753499i	$-0.728365\pi$
$353$	−24.7047	−1.31490	−0.657449	+	0.753499i	$0.728365\pi$
$354$	−7.85623	−0.417554
$355$	−7.03684	−0.373476
$356$	−7.47650	−0.396254
$357$	0.603875	0.0319605
$358$	8.78017	0.464046
$359$	−11.5888	−0.611634	−0.305817	−	0.952090i	$-0.598930\pi$
$359$	−11.5888	−0.611634	−0.305817	+	0.952090i	$0.598930\pi$
$360$	−1.00000	−0.0527046
$361$	−11.4209	−0.601099
$362$	−19.4741	−1.02354
$363$	8.41119	0.441473
$364$	0	0
$365$	14.8605	0.777836
$366$	12.0761	0.631226
$367$	−20.5579	−1.07312	−0.536558	−	0.843863i	$-0.680276\pi$
$367$	−20.5579	−1.07312	−0.536558	+	0.843863i	$0.680276\pi$
$368$	−2.04892	−0.106807
$369$	−5.74094	−0.298861
$370$	−6.07069	−0.315600
$371$	1.08469	0.0563142
$372$	2.08815	0.108265
$373$	1.31037	0.0678485	0.0339242	−	0.999424i	$-0.489200\pi$
$373$	1.31037	0.0678485	0.0339242	+	0.999424i	$0.489200\pi$
$374$	−4.13706	−0.213922
$375$	1.00000	0.0516398
$376$	−7.00000	−0.360997
$377$	0	0
$378$	−0.643104	−0.0330777
$379$	28.1551	1.44623	0.723115	−	0.690727i	$-0.242709\pi$
$379$	28.1551	1.44623	0.723115	+	0.690727i	$0.242709\pi$
$380$	2.75302	0.141227
$381$	4.85086	0.248517
$382$	−0.753020	−0.0385279
$383$	−18.6679	−0.953883	−0.476941	−	0.878935i	$-0.658255\pi$
$383$	−18.6679	−0.953883	−0.476941	+	0.878935i	$0.658255\pi$
$384$	−1.00000	−0.0510310
$385$	2.83340	0.144403
$386$	7.84117	0.399105
$387$	4.69202	0.238509
$388$	−5.05861	−0.256812
$389$	−29.6752	−1.50459	−0.752295	−	0.658826i	$-0.771053\pi$
$389$	−29.6752	−1.50459	−0.752295	+	0.658826i	$0.771053\pi$
$390$	0	0
$391$	−1.92394	−0.0972976
$392$	6.58642	0.332664
$393$	1.52781	0.0770679
$394$	−10.2010	−0.513921
$395$	0.929312	0.0467588
$396$	4.40581	0.221400
$397$	−28.7536	−1.44310	−0.721551	−	0.692361i	$-0.756571\pi$
$397$	−28.7536	−1.44310	−0.721551	+	0.692361i	$0.756571\pi$
$398$	−1.65279	−0.0828470
$399$	1.77048	0.0886348
$400$	1.00000	0.0500000
$401$	18.6993	0.933799	0.466900	−	0.884310i	$-0.345371\pi$
$401$	18.6993	0.933799	0.466900	+	0.884310i	$0.345371\pi$
$402$	−1.45712	−0.0726747
$403$	0	0
$404$	−6.81163	−0.338891
$405$	1.00000	0.0496904
$406$	0.0959157	0.00476022
$407$	26.7463	1.32577
$408$	−0.939001	−0.0464875
$409$	10.6233	0.525286	0.262643	−	0.964893i	$-0.415406\pi$
$409$	10.6233	0.525286	0.262643	+	0.964893i	$0.415406\pi$
$410$	5.74094	0.283525
$411$	−6.41119	−0.316241
$412$	−12.0858	−0.595422
$413$	5.05238	0.248611
$414$	2.04892	0.100699
$415$	11.2959	0.554494
$416$	0	0
$417$	−8.78017	−0.429967
$418$	−12.1293	−0.593263
$419$	19.4198	0.948720	0.474360	−	0.880331i	$-0.342680\pi$
$419$	19.4198	0.948720	0.474360	+	0.880331i	$0.342680\pi$
$420$	0.643104	0.0313803
$421$	20.5719	1.00262	0.501308	−	0.865269i	$-0.332853\pi$
$421$	20.5719	1.00262	0.501308	+	0.865269i	$0.332853\pi$
$422$	−11.7453	−0.571750
$423$	7.00000	0.340352
$424$	−1.68664	−0.0819107
$425$	0.939001	0.0455482
$426$	7.03684	0.340936
$427$	−7.76617	−0.375831
$428$	−14.6679	−0.708998
$429$	0	0
$430$	−4.69202	−0.226269
$431$	33.8079	1.62847	0.814235	−	0.580536i	$-0.197157\pi$
$431$	33.8079	1.62847	0.814235	+	0.580536i	$0.197157\pi$
$432$	1.00000	0.0481125
$433$	27.2838	1.31118	0.655588	−	0.755119i	$-0.272421\pi$
$433$	27.2838	1.31118	0.655588	+	0.755119i	$0.272421\pi$
$434$	−1.34290	−0.0644610
$435$	−0.149145	−0.00715095
$436$	8.15883	0.390737
$437$	−5.64071	−0.269832
$438$	−14.8605	−0.710064
$439$	−1.56943	−0.0749049	−0.0374525	−	0.999298i	$-0.511924\pi$
$439$	−1.56943	−0.0749049	−0.0374525	+	0.999298i	$0.511924\pi$
$440$	−4.40581	−0.210039
$441$	−6.58642	−0.313639
$442$	0	0
$443$	38.0489	1.80776	0.903879	−	0.427788i	$-0.140707\pi$
$443$	38.0489	1.80776	0.903879	+	0.427788i	$0.140707\pi$
$444$	6.07069	0.288102
$445$	−7.47650	−0.354420
$446$	9.73855	0.461134
$447$	−22.6504	−1.07133
$448$	0.643104	0.0303838
$449$	−24.2373	−1.14383	−0.571914	−	0.820313i	$-0.693799\pi$
$449$	−24.2373	−1.14383	−0.571914	+	0.820313i	$0.693799\pi$
$450$	−1.00000	−0.0471405
$451$	−25.2935	−1.19102
$452$	16.8442	0.792282
$453$	10.2959	0.483743
$454$	25.2500	1.18504
$455$	0	0
$456$	−2.75302	−0.128922
$457$	9.75063	0.456115	0.228058	−	0.973648i	$-0.426763\pi$
$457$	9.75063	0.456115	0.228058	+	0.973648i	$0.426763\pi$
$458$	−23.7845	−1.11138
$459$	0.939001	0.0438288
$460$	−2.04892	−0.0955313
$461$	31.0358	1.44548	0.722740	−	0.691120i	$-0.242882\pi$
$461$	31.0358	1.44548	0.722740	+	0.691120i	$0.242882\pi$
$462$	−2.83340	−0.131822
$463$	33.1618	1.54116	0.770580	−	0.637343i	$-0.219967\pi$
$463$	33.1618	1.54116	0.770580	+	0.637343i	$0.219967\pi$
$464$	−0.149145	−0.00692388
$465$	2.08815	0.0968355
$466$	−0.603875	−0.0279740
$467$	−16.1153	−0.745727	−0.372863	−	0.927886i	$-0.621624\pi$
$467$	−16.1153	−0.745727	−0.372863	+	0.927886i	$0.621624\pi$
$468$	0	0
$469$	0.937082	0.0432704
$470$	−7.00000	−0.322886
$471$	0.994623	0.0458298
$472$	−7.85623	−0.361612
$473$	20.6722	0.950507
$474$	−0.929312	−0.0426847
$475$	2.75302	0.126317
$476$	0.603875	0.0276786
$477$	1.68664	0.0772262
$478$	28.5579	1.30621
$479$	−30.6474	−1.40032	−0.700158	−	0.713988i	$-0.746887\pi$
$479$	−30.6474	−1.40032	−0.700158	+	0.713988i	$0.746887\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	−6.31096	−0.287456
$483$	−1.31767	−0.0599559
$484$	8.41119	0.382327
$485$	−5.05861	−0.229699
$486$	−1.00000	−0.0453609
$487$	17.6829	0.801290	0.400645	−	0.916233i	$-0.368786\pi$
$487$	17.6829	0.801290	0.400645	+	0.916233i	$0.368786\pi$
$488$	12.0761	0.546658
$489$	21.6015	0.976853
$490$	6.58642	0.297544
$491$	−28.5435	−1.28815	−0.644074	−	0.764963i	$-0.722757\pi$
$491$	−28.5435	−1.28815	−0.644074	+	0.764963i	$0.722757\pi$
$492$	−5.74094	−0.258822
$493$	−0.140047	−0.00630741
$494$	0	0
$495$	4.40581	0.198027
$496$	2.08815	0.0937605
$497$	−4.52542	−0.202993
$498$	−11.2959	−0.506181
$499$	−27.9409	−1.25081	−0.625404	−	0.780301i	$-0.715066\pi$
$499$	−27.9409	−1.25081	−0.625404	+	0.780301i	$0.715066\pi$
$500$	1.00000	0.0447214
$501$	7.96615	0.355901
$502$	−24.5972	−1.09783
$503$	31.0901	1.38624	0.693119	−	0.720823i	$-0.256236\pi$
$503$	31.0901	1.38624	0.693119	+	0.720823i	$0.256236\pi$
$504$	−0.643104	−0.0286461
$505$	−6.81163	−0.303113
$506$	9.02715	0.401306
$507$	0	0
$508$	4.85086	0.215222
$509$	0.0217703	0.000964950 0	0.000482475	−	1.00000i	$-0.499846\pi$
$509$	0.0217703	0.000964950 0	0.000482475		1.00000i	$0.499846\pi$
$510$	−0.939001	−0.0415797
$511$	9.55688	0.422771
$512$	−1.00000	−0.0441942
$513$	2.75302	0.121549
$514$	26.9463	1.18855
$515$	−12.0858	−0.532562
$516$	4.69202	0.206555
$517$	30.8407	1.35637
$518$	−3.90408	−0.171536
$519$	4.11960	0.180831
$520$	0	0
$521$	18.0747	0.791869	0.395934	−	0.918279i	$-0.370421\pi$
$521$	18.0747	0.791869	0.395934	+	0.918279i	$0.370421\pi$
$522$	0.149145	0.00652790
$523$	−15.3793	−0.672488	−0.336244	−	0.941775i	$-0.609157\pi$
$523$	−15.3793	−0.672488	−0.336244	+	0.941775i	$0.609157\pi$
$524$	1.52781	0.0667427
$525$	0.643104	0.0280674
$526$	24.0465	1.04848
$527$	1.96077	0.0854125
$528$	4.40581	0.191738
$529$	−18.8019	−0.817476
$530$	−1.68664	−0.0732632
$531$	7.85623	0.340931
$532$	1.77048	0.0767600
$533$	0	0
$534$	7.47650	0.323540
$535$	−14.6679	−0.634147
$536$	−1.45712	−0.0629381
$537$	−8.78017	−0.378892
$538$	7.17523	0.309346
$539$	−29.0185	−1.24992
$540$	1.00000	0.0430331
$541$	5.06339	0.217692	0.108846	−	0.994059i	$-0.465284\pi$
$541$	5.06339	0.217692	0.108846	+	0.994059i	$0.465284\pi$
$542$	−10.9825	−0.471741
$543$	19.4741	0.835714
$544$	−0.939001	−0.0402593
$545$	8.15883	0.349486
$546$	0	0
$547$	8.77107	0.375024	0.187512	−	0.982262i	$-0.439958\pi$
$547$	8.77107	0.375024	0.187512	+	0.982262i	$0.439958\pi$
$548$	−6.41119	−0.273872
$549$	−12.0761	−0.515394
$550$	−4.40581	−0.187865
$551$	−0.410599	−0.0174921
$552$	2.04892	0.0872077
$553$	0.597645	0.0254144
$554$	−16.2107	−0.688728
$555$	6.07069	0.257686
$556$	−8.78017	−0.372362
$557$	−8.10215	−0.343299	−0.171649	−	0.985158i	$-0.554910\pi$
$557$	−8.10215	−0.343299	−0.171649	+	0.985158i	$0.554910\pi$
$558$	−2.08815	−0.0883983
$559$	0	0
$560$	0.643104	0.0271761
$561$	4.13706	0.174667
$562$	12.9584	0.546616
$563$	−6.86699	−0.289409	−0.144704	−	0.989475i	$-0.546223\pi$
$563$	−6.86699	−0.289409	−0.144704	+	0.989475i	$0.546223\pi$
$564$	7.00000	0.294753
$565$	16.8442	0.708639
$566$	−21.5415	−0.905459
$567$	0.643104	0.0270078
$568$	7.03684	0.295259
$569$	6.93171	0.290592	0.145296	−	0.989388i	$-0.453587\pi$
$569$	6.93171	0.290592	0.145296	+	0.989388i	$0.453587\pi$
$570$	−2.75302	−0.115311
$571$	−24.2760	−1.01592	−0.507960	−	0.861380i	$-0.669600\pi$
$571$	−24.2760	−1.01592	−0.507960	+	0.861380i	$0.669600\pi$
$572$	0	0
$573$	0.753020	0.0314579
$574$	3.69202	0.154102
$575$	−2.04892	−0.0854458
$576$	1.00000	0.0416667
$577$	13.8189	0.575289	0.287645	−	0.957737i	$-0.407128\pi$
$577$	13.8189	0.575289	0.287645	+	0.957737i	$0.407128\pi$
$578$	16.1183	0.670432
$579$	−7.84117	−0.325868
$580$	−0.149145	−0.00619291
$581$	7.26444	0.301380
$582$	5.05861	0.209686
$583$	7.43104	0.307762
$584$	−14.8605	−0.614934
$585$	0	0
$586$	−9.57971	−0.395734
$587$	27.6165	1.13986	0.569928	−	0.821694i	$-0.306971\pi$
$587$	27.6165	1.13986	0.569928	+	0.821694i	$0.306971\pi$
$588$	−6.58642	−0.271619
$589$	5.74871	0.236871
$590$	−7.85623	−0.323436
$591$	10.2010	0.419615
$592$	6.07069	0.249504
$593$	−30.3913	−1.24802	−0.624011	−	0.781415i	$-0.714498\pi$
$593$	−30.3913	−1.24802	−0.624011	+	0.781415i	$0.714498\pi$
$594$	−4.40581	−0.180773
$595$	0.603875	0.0247565
$596$	−22.6504	−0.927797
$597$	1.65279	0.0676443
$598$	0	0
$599$	10.6480	0.435066	0.217533	−	0.976053i	$-0.430199\pi$
$599$	10.6480	0.435066	0.217533	+	0.976053i	$0.430199\pi$
$600$	−1.00000	−0.0408248
$601$	42.2495	1.72339	0.861696	−	0.507424i	$-0.169402\pi$
$601$	42.2495	1.72339	0.861696	+	0.507424i	$0.169402\pi$
$602$	−3.01746	−0.122982
$603$	1.45712	0.0593387
$604$	10.2959	0.418934
$605$	8.41119	0.341964
$606$	6.81163	0.276703
$607$	−17.6209	−0.715209	−0.357604	−	0.933873i	$-0.616406\pi$
$607$	−17.6209	−0.715209	−0.357604	+	0.933873i	$0.616406\pi$
$608$	−2.75302	−0.111650
$609$	−0.0959157	−0.00388670
$610$	12.0761	0.488946
$611$	0	0
$612$	0.939001	0.0379569
$613$	−27.9527	−1.12900	−0.564500	−	0.825433i	$-0.690931\pi$
$613$	−27.9527	−1.12900	−0.564500	+	0.825433i	$0.690931\pi$
$614$	−19.0140	−0.767343
$615$	−5.74094	−0.231497
$616$	−2.83340	−0.114161
$617$	14.9065	0.600112	0.300056	−	0.953922i	$-0.402995\pi$
$617$	14.9065	0.600112	0.300056	+	0.953922i	$0.402995\pi$
$618$	12.0858	0.486160
$619$	−33.7493	−1.35650	−0.678249	−	0.734832i	$-0.737261\pi$
$619$	−33.7493	−1.35650	−0.678249	+	0.734832i	$0.737261\pi$
$620$	2.08815	0.0838620
$621$	−2.04892	−0.0822202
$622$	3.74764	0.150267
$623$	−4.80817	−0.192635
$624$	0	0
$625$	1.00000	0.0400000
$626$	−25.1196	−1.00398
$627$	12.1293	0.484397
$628$	0.994623	0.0396898
$629$	5.70038	0.227289
$630$	−0.643104	−0.0256219
$631$	−37.2519	−1.48297	−0.741487	−	0.670967i	$-0.765879\pi$
$631$	−37.2519	−1.48297	−0.741487	+	0.670967i	$0.765879\pi$
$632$	−0.929312	−0.0369661
$633$	11.7453	0.466832
$634$	−12.6679	−0.503105
$635$	4.85086	0.192500
$636$	1.68664	0.0668798
$637$	0	0
$638$	0.657105	0.0260150
$639$	−7.03684	−0.278373
$640$	−1.00000	−0.0395285
$641$	24.8203	0.980341	0.490170	−	0.871627i	$-0.336935\pi$
$641$	24.8203	0.980341	0.490170	+	0.871627i	$0.336935\pi$
$642$	14.6679	0.578894
$643$	−5.66248	−0.223306	−0.111653	−	0.993747i	$-0.535615\pi$
$643$	−5.66248	−0.223306	−0.111653	+	0.993747i	$0.535615\pi$
$644$	−1.31767	−0.0519234
$645$	4.69202	0.184748
$646$	−2.58509	−0.101709
$647$	−27.0465	−1.06331	−0.531654	−	0.846961i	$-0.678429\pi$
$647$	−27.0465	−1.06331	−0.531654	+	0.846961i	$0.678429\pi$
$648$	−1.00000	−0.0392837
$649$	34.6131	1.35868
$650$	0	0
$651$	1.34290	0.0526322
$652$	21.6015	0.845979
$653$	42.7741	1.67388	0.836939	−	0.547296i	$-0.184343\pi$
$653$	42.7741	1.67388	0.836939	+	0.547296i	$0.184343\pi$
$654$	−8.15883	−0.319036
$655$	1.52781	0.0596965
$656$	−5.74094	−0.224146
$657$	14.8605	0.579765
$658$	−4.50173	−0.175496
$659$	21.5623	0.839946	0.419973	−	0.907537i	$-0.362040\pi$
$659$	21.5623	0.839946	0.419973	+	0.907537i	$0.362040\pi$
$660$	4.40581	0.171496
$661$	48.5002	1.88644	0.943219	−	0.332170i	$-0.107781\pi$
$661$	48.5002	1.88644	0.943219	+	0.332170i	$0.107781\pi$
$662$	6.43057	0.249931
$663$	0	0
$664$	−11.2959	−0.438366
$665$	1.77048	0.0686562
$666$	−6.07069	−0.235234
$667$	0.305586	0.0118323
$668$	7.96615	0.308220
$669$	−9.73855	−0.376514
$670$	−1.45712	−0.0562936
$671$	−53.2049	−2.05395
$672$	−0.643104	−0.0248083
$673$	30.4711	1.17458	0.587288	−	0.809378i	$-0.300196\pi$
$673$	30.4711	1.17458	0.587288	+	0.809378i	$0.300196\pi$
$674$	−10.5496	−0.406355
$675$	1.00000	0.0384900
$676$	0	0
$677$	−8.57971	−0.329745	−0.164873	−	0.986315i	$-0.552721\pi$
$677$	−8.57971	−0.329745	−0.164873	+	0.986315i	$0.552721\pi$
$678$	−16.8442	−0.646896
$679$	−3.25321	−0.124847
$680$	−0.939001	−0.0360090
$681$	−25.2500	−0.967581
$682$	−9.19998	−0.352285
$683$	27.8890	1.06714	0.533572	−	0.845755i	$-0.320849\pi$
$683$	27.8890	1.06714	0.533572	+	0.845755i	$0.320849\pi$
$684$	2.75302	0.105264
$685$	−6.41119	−0.244959
$686$	8.73748	0.333599
$687$	23.7845	0.907434
$688$	4.69202	0.178882
$689$	0	0
$690$	2.04892	0.0780010
$691$	39.4198	1.49960	0.749800	−	0.661664i	$-0.230150\pi$
$691$	39.4198	1.49960	0.749800	+	0.661664i	$0.230150\pi$
$692$	4.11960	0.156604
$693$	2.83340	0.107632
$694$	−9.87130	−0.374709
$695$	−8.78017	−0.333051
$696$	0.149145	0.00565332
$697$	−5.39075	−0.204189
$698$	30.5338	1.15572
$699$	0.603875	0.0228407
$700$	0.643104	0.0243071
$701$	−51.7101	−1.95306	−0.976531	−	0.215376i	$-0.930902\pi$
$701$	−51.7101	−1.95306	−0.976531	+	0.215376i	$0.930902\pi$
$702$	0	0
$703$	16.7127	0.630332
$704$	4.40581	0.166050
$705$	7.00000	0.263635
$706$	24.7047	0.929773
$707$	−4.38059	−0.164749
$708$	7.85623	0.295255
$709$	−12.7278	−0.478002	−0.239001	−	0.971019i	$-0.576820\pi$
$709$	−12.7278	−0.478002	−0.239001	+	0.971019i	$0.576820\pi$
$710$	7.03684	0.264088
$711$	0.929312	0.0348519
$712$	7.47650	0.280194
$713$	−4.27844	−0.160229
$714$	−0.603875	−0.0225995
$715$	0	0
$716$	−8.78017	−0.328130
$717$	−28.5579	−1.06652
$718$	11.5888	0.432491
$719$	−11.7342	−0.437613	−0.218807	−	0.975768i	$-0.570216\pi$
$719$	−11.7342	−0.437613	−0.218807	+	0.975768i	$0.570216\pi$
$720$	1.00000	0.0372678
$721$	−7.77240	−0.289459
$722$	11.4209	0.425041
$723$	6.31096	0.234707
$724$	19.4741	0.723750
$725$	−0.149145	−0.00553910
$726$	−8.41119	−0.312169
$727$	−6.84979	−0.254045	−0.127022	−	0.991900i	$-0.540542\pi$
$727$	−6.84979	−0.254045	−0.127022	+	0.991900i	$0.540542\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−14.8605	−0.550013
$731$	4.40581	0.162955
$732$	−12.0761	−0.446344
$733$	−0.224144	−0.00827896	−0.00413948	−	0.999991i	$-0.501318\pi$
$733$	−0.224144	−0.00827896	−0.00413948	+	0.999991i	$0.501318\pi$
$734$	20.5579	0.758807
$735$	−6.58642	−0.242944
$736$	2.04892	0.0755241
$737$	6.41981	0.236477
$738$	5.74094	0.211327
$739$	−2.39075	−0.0879451	−0.0439725	−	0.999033i	$-0.514001\pi$
$739$	−2.39075	−0.0879451	−0.0439725	+	0.999033i	$0.514001\pi$
$740$	6.07069	0.223163
$741$	0	0
$742$	−1.08469	−0.0398202
$743$	−25.5144	−0.936033	−0.468016	−	0.883720i	$-0.655031\pi$
$743$	−25.5144	−0.936033	−0.468016	+	0.883720i	$0.655031\pi$
$744$	−2.08815	−0.0765551
$745$	−22.6504	−0.829846
$746$	−1.31037	−0.0479761
$747$	11.2959	0.413295
$748$	4.13706	0.151266
$749$	−9.43296	−0.344673
$750$	−1.00000	−0.0365148
$751$	18.2755	0.666881	0.333440	−	0.942771i	$-0.391790\pi$
$751$	18.2755	0.666881	0.333440	+	0.942771i	$0.391790\pi$
$752$	7.00000	0.255264
$753$	24.5972	0.896371
$754$	0	0
$755$	10.2959	0.374706
$756$	0.643104	0.0233895
$757$	−20.2301	−0.735276	−0.367638	−	0.929969i	$-0.619833\pi$
$757$	−20.2301	−0.735276	−0.367638	+	0.929969i	$0.619833\pi$
$758$	−28.1551	−1.02264
$759$	−9.02715	−0.327665
$760$	−2.75302	−0.0998625
$761$	−0.217440	−0.00788218	−0.00394109	−	0.999992i	$-0.501254\pi$
$761$	−0.217440	−0.00788218	−0.00394109	+	0.999992i	$0.501254\pi$
$762$	−4.85086	−0.175728
$763$	5.24698	0.189953
$764$	0.753020	0.0272433
$765$	0.939001	0.0339497
$766$	18.6679	0.674497
$767$	0	0
$768$	1.00000	0.0360844
$769$	0.124982	0.00450696	0.00225348	−	0.999997i	$-0.499283\pi$
$769$	0.124982	0.00450696	0.00225348	+	0.999997i	$0.499283\pi$
$770$	−2.83340	−0.102109
$771$	−26.9463	−0.970447
$772$	−7.84117	−0.282210
$773$	11.7028	0.420920	0.210460	−	0.977603i	$-0.432504\pi$
$773$	11.7028	0.420920	0.210460	+	0.977603i	$0.432504\pi$
$774$	−4.69202	−0.168651
$775$	2.08815	0.0750084
$776$	5.05861	0.181593
$777$	3.90408	0.140058
$778$	29.6752	1.06391
$779$	−15.8049	−0.566270
$780$	0	0
$781$	−31.0030	−1.10937
$782$	1.92394	0.0687998
$783$	−0.149145	−0.00533000
$784$	−6.58642	−0.235229
$785$	0.994623	0.0354996
$786$	−1.52781	−0.0544952
$787$	−50.6926	−1.80700	−0.903498	−	0.428592i	$-0.859010\pi$
$787$	−50.6926	−1.80700	−0.903498	+	0.428592i	$0.859010\pi$
$788$	10.2010	0.363397
$789$	−24.0465	−0.856079
$790$	−0.929312	−0.0330635
$791$	10.8325	0.385161
$792$	−4.40581	−0.156554
$793$	0	0
$794$	28.7536	1.02043
$795$	1.68664	0.0598191
$796$	1.65279	0.0585817
$797$	45.7977	1.62224	0.811120	−	0.584880i	$-0.198859\pi$
$797$	45.7977	1.62224	0.811120	+	0.584880i	$0.198859\pi$
$798$	−1.77048	−0.0626743
$799$	6.57301	0.232536
$800$	−1.00000	−0.0353553
$801$	−7.47650	−0.264169
$802$	−18.6993	−0.660296
$803$	65.4728	2.31048
$804$	1.45712	0.0513888
$805$	−1.31767	−0.0464417
$806$	0	0
$807$	−7.17523	−0.252580
$808$	6.81163	0.239632
$809$	22.5579	0.793095	0.396548	−	0.918014i	$-0.370208\pi$
$809$	22.5579	0.793095	0.396548	+	0.918014i	$0.370208\pi$
$810$	−1.00000	−0.0351364
$811$	19.2728	0.676759	0.338380	−	0.941010i	$-0.390121\pi$
$811$	19.2728	0.676759	0.338380	+	0.941010i	$0.390121\pi$
$812$	−0.0959157	−0.00336598
$813$	10.9825	0.385175
$814$	−26.7463	−0.937458
$815$	21.6015	0.756667
$816$	0.939001	0.0328716
$817$	12.9172	0.451917
$818$	−10.6233	−0.371433
$819$	0	0
$820$	−5.74094	−0.200482
$821$	11.5036	0.401480	0.200740	−	0.979645i	$-0.435665\pi$
$821$	11.5036	0.401480	0.200740	+	0.979645i	$0.435665\pi$
$822$	6.41119	0.223616
$823$	−25.2610	−0.880542	−0.440271	−	0.897865i	$-0.645118\pi$
$823$	−25.2610	−0.880542	−0.440271	+	0.897865i	$0.645118\pi$
$824$	12.0858	0.421027
$825$	4.40581	0.153391
$826$	−5.05238	−0.175795
$827$	−7.09677	−0.246779	−0.123389	−	0.992358i	$-0.539376\pi$
$827$	−7.09677	−0.246779	−0.123389	+	0.992358i	$0.539376\pi$
$828$	−2.04892	−0.0712048
$829$	10.1226	0.351572	0.175786	−	0.984428i	$-0.443753\pi$
$829$	10.1226	0.351572	0.175786	+	0.984428i	$0.443753\pi$
$830$	−11.2959	−0.392086
$831$	16.2107	0.562344
$832$	0	0
$833$	−6.18465	−0.214286
$834$	8.78017	0.304032
$835$	7.96615	0.275680
$836$	12.1293	0.419500
$837$	2.08815	0.0721769
$838$	−19.4198	−0.670846
$839$	−43.0844	−1.48744	−0.743720	−	0.668492i	$-0.766940\pi$
$839$	−43.0844	−1.48744	−0.743720	+	0.668492i	$0.766940\pi$
$840$	−0.643104	−0.0221892
$841$	−28.9778	−0.999233
$842$	−20.5719	−0.708956
$843$	−12.9584	−0.446310
$844$	11.7453	0.404288
$845$	0	0
$846$	−7.00000	−0.240665
$847$	5.40927	0.185865
$848$	1.68664	0.0579196
$849$	21.5415	0.739304
$850$	−0.939001	−0.0322075
$851$	−12.4383	−0.426381
$852$	−7.03684	−0.241078
$853$	43.7837	1.49913	0.749563	−	0.661933i	$-0.230263\pi$
$853$	43.7837	1.49913	0.749563	+	0.661933i	$0.230263\pi$
$854$	7.76617	0.265753
$855$	2.75302	0.0941513
$856$	14.6679	0.501337
$857$	−16.3067	−0.557025	−0.278512	−	0.960433i	$-0.589841\pi$
$857$	−16.3067	−0.557025	−0.278512	+	0.960433i	$0.589841\pi$
$858$	0	0
$859$	−31.8629	−1.08715	−0.543575	−	0.839361i	$-0.682930\pi$
$859$	−31.8629	−1.08715	−0.543575	+	0.839361i	$0.682930\pi$
$860$	4.69202	0.159997
$861$	−3.69202	−0.125824
$862$	−33.8079	−1.15150
$863$	−0.511418	−0.0174089	−0.00870443	−	0.999962i	$-0.502771\pi$
$863$	−0.511418	−0.0174089	−0.00870443	+	0.999962i	$0.502771\pi$
$864$	−1.00000	−0.0340207
$865$	4.11960	0.140071
$866$	−27.2838	−0.927142
$867$	−16.1183	−0.547405
$868$	1.34290	0.0455808
$869$	4.09438	0.138892
$870$	0.149145	0.00505649
$871$	0	0
$872$	−8.15883	−0.276293
$873$	−5.05861	−0.171208
$874$	5.64071	0.190800
$875$	0.643104	0.0217409
$876$	14.8605	0.502091
$877$	3.83638	0.129545	0.0647727	−	0.997900i	$-0.479368\pi$
$877$	3.83638	0.129545	0.0647727	+	0.997900i	$0.479368\pi$
$878$	1.56943	0.0529658
$879$	9.57971	0.323116
$880$	4.40581	0.148520
$881$	−0.499336	−0.0168231	−0.00841153	−	0.999965i	$-0.502678\pi$
$881$	−0.499336	−0.0168231	−0.00841153	+	0.999965i	$0.502678\pi$
$882$	6.58642	0.221776
$883$	−38.8993	−1.30907	−0.654533	−	0.756034i	$-0.727135\pi$
$883$	−38.8993	−1.30907	−0.654533	+	0.756034i	$0.727135\pi$
$884$	0	0
$885$	7.85623	0.264084
$886$	−38.0489	−1.27828
$887$	−8.32245	−0.279441	−0.139720	−	0.990191i	$-0.544620\pi$
$887$	−8.32245	−0.279441	−0.139720	+	0.990191i	$0.544620\pi$
$888$	−6.07069	−0.203719
$889$	3.11960	0.104628
$890$	7.47650	0.250613
$891$	4.40581	0.147600
$892$	−9.73855	−0.326071
$893$	19.2711	0.644884
$894$	22.6504	0.757543
$895$	−8.78017	−0.293489
$896$	−0.643104	−0.0214846
$897$	0	0
$898$	24.2373	0.808809
$899$	−0.311436	−0.0103870
$900$	1.00000	0.0333333
$901$	1.58376	0.0527627
$902$	25.2935	0.842182
$903$	3.01746	0.100415
$904$	−16.8442	−0.560228
$905$	19.4741	0.647341
$906$	−10.2959	−0.342058
$907$	27.4566	0.911683	0.455842	−	0.890061i	$-0.349338\pi$
$907$	27.4566	0.911683	0.455842	+	0.890061i	$0.349338\pi$
$908$	−25.2500	−0.837949
$909$	−6.81163	−0.225927
$910$	0	0
$911$	−29.7224	−0.984748	−0.492374	−	0.870384i	$-0.663871\pi$
$911$	−29.7224	−0.984748	−0.492374	+	0.870384i	$0.663871\pi$
$912$	2.75302	0.0911616
$913$	49.7676	1.64707
$914$	−9.75063	−0.322522
$915$	−12.0761	−0.399222
$916$	23.7845	0.785861
$917$	0.982542	0.0324464
$918$	−0.939001	−0.0309917
$919$	−2.02954	−0.0669483	−0.0334742	−	0.999440i	$-0.510657\pi$
$919$	−2.02954	−0.0669483	−0.0334742	+	0.999440i	$0.510657\pi$
$920$	2.04892	0.0675508
$921$	19.0140	0.626533
$922$	−31.0358	−1.02211
$923$	0	0
$924$	2.83340	0.0932119
$925$	6.07069	0.199603
$926$	−33.1618	−1.08976
$927$	−12.0858	−0.396948
$928$	0.149145	0.00489592
$929$	0.815938	0.0267701	0.0133850	−	0.999910i	$-0.495739\pi$
$929$	0.815938	0.0267701	0.0133850	+	0.999910i	$0.495739\pi$
$930$	−2.08815	−0.0684730
$931$	−18.1325	−0.594270
$932$	0.603875	0.0197806
$933$	−3.74764	−0.122692
$934$	16.1153	0.527308
$935$	4.13706	0.135296
$936$	0	0
$937$	−29.5864	−0.966546	−0.483273	−	0.875470i	$-0.660552\pi$
$937$	−29.5864	−0.966546	−0.483273	+	0.875470i	$0.660552\pi$
$938$	−0.937082	−0.0305968
$939$	25.1196	0.819747
$940$	7.00000	0.228315
$941$	30.2573	0.986358	0.493179	−	0.869928i	$-0.335835\pi$
$941$	30.2573	0.986358	0.493179	+	0.869928i	$0.335835\pi$
$942$	−0.994623	−0.0324066
$943$	11.7627	0.383047
$944$	7.85623	0.255699
$945$	0.643104	0.0209202
$946$	−20.6722	−0.672110
$947$	7.12152	0.231418	0.115709	−	0.993283i	$-0.463086\pi$
$947$	7.12152	0.231418	0.115709	+	0.993283i	$0.463086\pi$
$948$	0.929312	0.0301827
$949$	0	0
$950$	−2.75302	−0.0893198
$951$	12.6679	0.410783
$952$	−0.603875	−0.0195717
$953$	−38.3279	−1.24156	−0.620782	−	0.783983i	$-0.713185\pi$
$953$	−38.3279	−1.24156	−0.620782	+	0.783983i	$0.713185\pi$
$954$	−1.68664	−0.0546071
$955$	0.753020	0.0243672
$956$	−28.5579	−0.923630
$957$	−0.657105	−0.0212412
$958$	30.6474	0.990173
$959$	−4.12306	−0.133141
$960$	1.00000	0.0322749
$961$	−26.6396	−0.859343
$962$	0	0
$963$	−14.6679	−0.472665
$964$	6.31096	0.203262
$965$	−7.84117	−0.252416
$966$	1.31767	0.0423952
$967$	−50.6015	−1.62723	−0.813617	−	0.581401i	$-0.802505\pi$
$967$	−50.6015	−1.62723	−0.813617	+	0.581401i	$0.802505\pi$
$968$	−8.41119	−0.270346
$969$	2.58509	0.0830450
$970$	5.05861	0.162422
$971$	−27.3026	−0.876182	−0.438091	−	0.898931i	$-0.644345\pi$
$971$	−27.3026	−0.876182	−0.438091	+	0.898931i	$0.644345\pi$
$972$	1.00000	0.0320750
$973$	−5.64656	−0.181020
$974$	−17.6829	−0.566597
$975$	0	0
$976$	−12.0761	−0.386545
$977$	−46.5491	−1.48924	−0.744619	−	0.667490i	$-0.767369\pi$
$977$	−46.5491	−1.48924	−0.744619	+	0.667490i	$0.767369\pi$
$978$	−21.6015	−0.690739
$979$	−32.9401	−1.05277
$980$	−6.58642	−0.210395
$981$	8.15883	0.260492
$982$	28.5435	0.910859
$983$	1.10321	0.0351870	0.0175935	−	0.999845i	$-0.494400\pi$
$983$	1.10321	0.0351870	0.0175935	+	0.999845i	$0.494400\pi$
$984$	5.74094	0.183014
$985$	10.2010	0.325032
$986$	0.140047	0.00446001
$987$	4.50173	0.143292
$988$	0	0
$989$	−9.61356	−0.305694
$990$	−4.40581	−0.140026
$991$	−51.4752	−1.63516	−0.817581	−	0.575813i	$-0.804686\pi$
$991$	−51.4752	−1.63516	−0.817581	+	0.575813i	$0.804686\pi$
$992$	−2.08815	−0.0662987
$993$	−6.43057	−0.204068
$994$	4.52542	0.143538
$995$	1.65279	0.0523971
$996$	11.2959	0.357924
$997$	−28.9715	−0.917537	−0.458769	−	0.888556i	$-0.651709\pi$
$997$	−28.9715	−0.917537	−0.458769	+	0.888556i	$0.651709\pi$
$998$	27.9409	0.884454
$999$	6.07069	0.192068

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.bq.1.2	✓	3
13.5	odd	4		5070.2.b.y.1351.5		6
13.8	odd	4		5070.2.b.y.1351.2		6
13.12	even	2		5070.2.a.bv.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bq.1.2	✓	3	1.1	even	1	trivial
5070.2.a.bv.1.2	yes	3	13.12	even	2
5070.2.b.y.1351.2		6	13.8	odd	4
5070.2.b.y.1351.5		6	13.5	odd	4

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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +0.643104 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +0.643104 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.40581 q^{11} +1.00000 q^{12} -0.643104 q^{14} +1.00000 q^{15} +1.00000 q^{16} +0.939001 q^{17} -1.00000 q^{18} +2.75302 q^{19} +1.00000 q^{20} +0.643104 q^{21} -4.40581 q^{22} -2.04892 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +0.643104 q^{28} -0.149145 q^{29} -1.00000 q^{30} +2.08815 q^{31} -1.00000 q^{32} +4.40581 q^{33} -0.939001 q^{34} +0.643104 q^{35} +1.00000 q^{36} +6.07069 q^{37} -2.75302 q^{38} -1.00000 q^{40} -5.74094 q^{41} -0.643104 q^{42} +4.69202 q^{43} +4.40581 q^{44} +1.00000 q^{45} +2.04892 q^{46} +7.00000 q^{47} +1.00000 q^{48} -6.58642 q^{49} -1.00000 q^{50} +0.939001 q^{51} +1.68664 q^{53} -1.00000 q^{54} +4.40581 q^{55} -0.643104 q^{56} +2.75302 q^{57} +0.149145 q^{58} +7.85623 q^{59} +1.00000 q^{60} -12.0761 q^{61} -2.08815 q^{62} +0.643104 q^{63} +1.00000 q^{64} -4.40581 q^{66} +1.45712 q^{67} +0.939001 q^{68} -2.04892 q^{69} -0.643104 q^{70} -7.03684 q^{71} -1.00000 q^{72} +14.8605 q^{73} -6.07069 q^{74} +1.00000 q^{75} +2.75302 q^{76} +2.83340 q^{77} +0.929312 q^{79} +1.00000 q^{80} +1.00000 q^{81} +5.74094 q^{82} +11.2959 q^{83} +0.643104 q^{84} +0.939001 q^{85} -4.69202 q^{86} -0.149145 q^{87} -4.40581 q^{88} -7.47650 q^{89} -1.00000 q^{90} -2.04892 q^{92} +2.08815 q^{93} -7.00000 q^{94} +2.75302 q^{95} -1.00000 q^{96} -5.05861 q^{97} +6.58642 q^{98} +4.40581 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 6 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{10} + 3 q^{12} - 6 q^{14} + 3 q^{15} + 3 q^{16} - 7 q^{17} - 3 q^{18} + 13 q^{19} + 3 q^{20} + 6 q^{21} + 3 q^{23} - 3 q^{24} + 3 q^{25} + 3 q^{27} + 6 q^{28} - 14 q^{29} - 3 q^{30} + 10 q^{31} - 3 q^{32} + 7 q^{34} + 6 q^{35} + 3 q^{36} + 6 q^{37} - 13 q^{38} - 3 q^{40} - 3 q^{41} - 6 q^{42} + 9 q^{43} + 3 q^{45} - 3 q^{46} + 21 q^{47} + 3 q^{48} + 5 q^{49} - 3 q^{50} - 7 q^{51} + 3 q^{53} - 3 q^{54} - 6 q^{56} + 13 q^{57} + 14 q^{58} + 7 q^{59} + 3 q^{60} - 21 q^{61} - 10 q^{62} + 6 q^{63} + 3 q^{64} + 23 q^{67} - 7 q^{68} + 3 q^{69} - 6 q^{70} + 7 q^{71} - 3 q^{72} + 9 q^{73} - 6 q^{74} + 3 q^{75} + 13 q^{76} - 21 q^{77} + 15 q^{79} + 3 q^{80} + 3 q^{81} + 3 q^{82} + 20 q^{83} + 6 q^{84} - 7 q^{85} - 9 q^{86} - 14 q^{87} + 3 q^{89} - 3 q^{90} + 3 q^{92} + 10 q^{93} - 21 q^{94} + 13 q^{95} - 3 q^{96} + 16 q^{97} - 5 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 6 * q^7 - 3 * q^8 + 3 * q^9 - 3 * q^10 + 3 * q^12 - 6 * q^14 + 3 * q^15 + 3 * q^16 - 7 * q^17 - 3 * q^18 + 13 * q^19 + 3 * q^20 + 6 * q^21 + 3 * q^23 - 3 * q^24 + 3 * q^25 + 3 * q^27 + 6 * q^28 - 14 * q^29 - 3 * q^30 + 10 * q^31 - 3 * q^32 + 7 * q^34 + 6 * q^35 + 3 * q^36 + 6 * q^37 - 13 * q^38 - 3 * q^40 - 3 * q^41 - 6 * q^42 + 9 * q^43 + 3 * q^45 - 3 * q^46 + 21 * q^47 + 3 * q^48 + 5 * q^49 - 3 * q^50 - 7 * q^51 + 3 * q^53 - 3 * q^54 - 6 * q^56 + 13 * q^57 + 14 * q^58 + 7 * q^59 + 3 * q^60 - 21 * q^61 - 10 * q^62 + 6 * q^63 + 3 * q^64 + 23 * q^67 - 7 * q^68 + 3 * q^69 - 6 * q^70 + 7 * q^71 - 3 * q^72 + 9 * q^73 - 6 * q^74 + 3 * q^75 + 13 * q^76 - 21 * q^77 + 15 * q^79 + 3 * q^80 + 3 * q^81 + 3 * q^82 + 20 * q^83 + 6 * q^84 - 7 * q^85 - 9 * q^86 - 14 * q^87 + 3 * q^89 - 3 * q^90 + 3 * q^92 + 10 * q^93 - 21 * q^94 + 13 * q^95 - 3 * q^96 + 16 * q^97 - 5 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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