LMFDB - Embedded newform 9702.2.a.ch.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9702,2,Mod(1,9702)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9702.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9702, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9702.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,2,-4,0,0,-2,0,4,-2,0,2,0,0,2,-4,0,4,-4,0,2,4,0,2,-2,0, 0,6,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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

Self dual:	yes
Analytic conductor:	$77.4708600410$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	9702.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -0.585786 q^{5} -1.00000 q^{8} +0.585786 q^{10} -1.00000 q^{11} +3.82843 q^{13} +1.00000 q^{16} +3.65685 q^{17} +0.585786 q^{19} -0.585786 q^{20} +1.00000 q^{22} +6.24264 q^{23} -4.65685 q^{25} -3.82843 q^{26} -2.65685 q^{29} +4.00000 q^{31} -1.00000 q^{32} -3.65685 q^{34} -9.41421 q^{37} -0.585786 q^{38} +0.585786 q^{40} -5.41421 q^{41} -5.65685 q^{43} -1.00000 q^{44} -6.24264 q^{46} -10.4853 q^{47} +4.65685 q^{50} +3.82843 q^{52} -7.89949 q^{53} +0.585786 q^{55} +2.65685 q^{58} -5.58579 q^{59} -11.8284 q^{61} -4.00000 q^{62} +1.00000 q^{64} -2.24264 q^{65} +2.75736 q^{67} +3.65685 q^{68} +11.0711 q^{71} +9.41421 q^{73} +9.41421 q^{74} +0.585786 q^{76} -13.2426 q^{79} -0.585786 q^{80} +5.41421 q^{82} -12.1421 q^{83} -2.14214 q^{85} +5.65685 q^{86} +1.00000 q^{88} +12.4853 q^{89} +6.24264 q^{92} +10.4853 q^{94} -0.343146 q^{95} +3.82843 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} + 4 q^{10} - 2 q^{11} + 2 q^{13} + 2 q^{16} - 4 q^{17} + 4 q^{19} - 4 q^{20} + 2 q^{22} + 4 q^{23} + 2 q^{25} - 2 q^{26} + 6 q^{29} + 8 q^{31} - 2 q^{32} + 4 q^{34}+ \cdots + 2 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^8 + 4 * q^10 - 2 * q^11 + 2 * q^13 + 2 * q^16 - 4 * q^17 + 4 * q^19 - 4 * q^20 + 2 * q^22 + 4 * q^23 + 2 * q^25 - 2 * q^26 + 6 * q^29 + 8 * q^31 - 2 * q^32 + 4 * q^34 - 16 * q^37 - 4 * q^38 + 4 * q^40 - 8 * q^41 - 2 * q^44 - 4 * q^46 - 4 * q^47 - 2 * q^50 + 2 * q^52 + 4 * q^53 + 4 * q^55 - 6 * q^58 - 14 * q^59 - 18 * q^61 - 8 * q^62 + 2 * q^64 + 4 * q^65 + 14 * q^67 - 4 * q^68 + 8 * q^71 + 16 * q^73 + 16 * q^74 + 4 * q^76 - 18 * q^79 - 4 * q^80 + 8 * q^82 + 4 * q^83 + 24 * q^85 + 2 * q^88 + 8 * q^89 + 4 * q^92 + 4 * q^94 - 12 * q^95 + 2 * q^97

Coefficient data

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

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display 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	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−0.585786	−0.261972	−0.130986	−	0.991384i	$-0.541814\pi$
$5$	−0.585786	−0.261972	−0.130986	+	0.991384i	$0.541814\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0.585786	0.185242
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.82843	1.06181	0.530907	−	0.847430i	$-0.321851\pi$
$13$	3.82843	1.06181	0.530907	+	0.847430i	$0.321851\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	3.65685	0.886917	0.443459	−	0.896295i	$-0.353751\pi$
$17$	3.65685	0.886917	0.443459	+	0.896295i	$0.353751\pi$
$18$	0	0
$19$	0.585786	0.134389	0.0671943	−	0.997740i	$-0.478595\pi$
$19$	0.585786	0.134389	0.0671943	+	0.997740i	$0.478595\pi$
$20$	−0.585786	−0.130986
$21$	0	0
$22$	1.00000	0.213201
$23$	6.24264	1.30168	0.650840	−	0.759215i	$-0.274417\pi$
$23$	6.24264	1.30168	0.650840	+	0.759215i	$0.274417\pi$
$24$	0	0
$25$	−4.65685	−0.931371
$26$	−3.82843	−0.750816
$27$	0	0
$28$	0	0
$29$	−2.65685	−0.493365	−0.246683	−	0.969096i	$-0.579341\pi$
$29$	−2.65685	−0.493365	−0.246683	+	0.969096i	$0.579341\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.65685	−0.627145
$35$	0	0
$36$	0	0
$37$	−9.41421	−1.54769	−0.773844	−	0.633377i	$-0.781668\pi$
$37$	−9.41421	−1.54769	−0.773844	+	0.633377i	$0.781668\pi$
$38$	−0.585786	−0.0950271
$39$	0	0
$40$	0.585786	0.0926210
$41$	−5.41421	−0.845558	−0.422779	−	0.906233i	$-0.638945\pi$
$41$	−5.41421	−0.845558	−0.422779	+	0.906233i	$0.638945\pi$
$42$	0	0
$43$	−5.65685	−0.862662	−0.431331	−	0.902194i	$-0.641956\pi$
$43$	−5.65685	−0.862662	−0.431331	+	0.902194i	$0.641956\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−6.24264	−0.920427
$47$	−10.4853	−1.52944	−0.764718	−	0.644365i	$-0.777122\pi$
$47$	−10.4853	−1.52944	−0.764718	+	0.644365i	$0.777122\pi$
$48$	0	0
$49$	0	0
$50$	4.65685	0.658579
$51$	0	0
$52$	3.82843	0.530907
$53$	−7.89949	−1.08508	−0.542540	−	0.840030i	$-0.682537\pi$
$53$	−7.89949	−1.08508	−0.542540	+	0.840030i	$0.682537\pi$
$54$	0	0
$55$	0.585786	0.0789874
$56$	0	0
$57$	0	0
$58$	2.65685	0.348862
$59$	−5.58579	−0.727207	−0.363604	−	0.931554i	$-0.618454\pi$
$59$	−5.58579	−0.727207	−0.363604	+	0.931554i	$0.618454\pi$
$60$	0	0
$61$	−11.8284	−1.51447	−0.757237	−	0.653140i	$-0.773451\pi$
$61$	−11.8284	−1.51447	−0.757237	+	0.653140i	$0.773451\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.24264	−0.278165
$66$	0	0
$67$	2.75736	0.336865	0.168433	−	0.985713i	$-0.446129\pi$
$67$	2.75736	0.336865	0.168433	+	0.985713i	$0.446129\pi$
$68$	3.65685	0.443459
$69$	0	0
$70$	0	0
$71$	11.0711	1.31389	0.656947	−	0.753937i	$-0.271848\pi$
$71$	11.0711	1.31389	0.656947	+	0.753937i	$0.271848\pi$
$72$	0	0
$73$	9.41421	1.10185	0.550925	−	0.834555i	$-0.314275\pi$
$73$	9.41421	1.10185	0.550925	+	0.834555i	$0.314275\pi$
$74$	9.41421	1.09438
$75$	0	0
$76$	0.585786	0.0671943
$77$	0	0
$78$	0	0
$79$	−13.2426	−1.48991	−0.744957	−	0.667113i	$-0.767530\pi$
$79$	−13.2426	−1.48991	−0.744957	+	0.667113i	$0.767530\pi$
$80$	−0.585786	−0.0654929
$81$	0	0
$82$	5.41421	0.597900
$83$	−12.1421	−1.33277	−0.666386	−	0.745607i	$-0.732160\pi$
$83$	−12.1421	−1.33277	−0.666386	+	0.745607i	$0.732160\pi$
$84$	0	0
$85$	−2.14214	−0.232347
$86$	5.65685	0.609994
$87$	0	0
$88$	1.00000	0.106600
$89$	12.4853	1.32344	0.661719	−	0.749752i	$-0.269827\pi$
$89$	12.4853	1.32344	0.661719	+	0.749752i	$0.269827\pi$
$90$	0	0
$91$	0	0
$92$	6.24264	0.650840
$93$	0	0
$94$	10.4853	1.08147
$95$	−0.343146	−0.0352060
$96$	0	0
$97$	3.82843	0.388718	0.194359	−	0.980930i	$-0.437737\pi$
$97$	3.82843	0.388718	0.194359	+	0.980930i	$0.437737\pi$
$98$	0	0
$99$	0	0

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9702.2.a.ch.1.2		2
3.2	odd	2		1078.2.a.x.1.2		2
7.3	odd	6		1386.2.k.t.793.2		4
7.5	odd	6		1386.2.k.t.991.2		4
7.6	odd	2		9702.2.a.cx.1.1		2
12.11	even	2		8624.2.a.bh.1.1		2
21.2	odd	6		1078.2.e.m.67.1		4
21.5	even	6		154.2.e.e.67.2	yes	4
21.11	odd	6		1078.2.e.m.177.1		4
21.17	even	6		154.2.e.e.23.2	✓	4
21.20	even	2		1078.2.a.t.1.1		2
84.47	odd	6		1232.2.q.f.529.1		4
84.59	odd	6		1232.2.q.f.177.1		4
84.83	odd	2		8624.2.a.cc.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.e.e.23.2	✓	4	21.17	even	6
154.2.e.e.67.2	yes	4	21.5	even	6
1078.2.a.t.1.1		2	21.20	even	2
1078.2.a.x.1.2		2	3.2	odd	2
1078.2.e.m.67.1		4	21.2	odd	6
1078.2.e.m.177.1		4	21.11	odd	6
1232.2.q.f.177.1		4	84.59	odd	6
1232.2.q.f.529.1		4	84.47	odd	6
1386.2.k.t.793.2		4	7.3	odd	6
1386.2.k.t.991.2		4	7.5	odd	6
8624.2.a.bh.1.1		2	12.11	even	2
8624.2.a.cc.1.2		2	84.83	odd	2
9702.2.a.ch.1.2		2	1.1	even	1	trivial
9702.2.a.cx.1.1		2	7.6	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9702.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.585786 q^{5} -1.00000 q^{8} +0.585786 q^{10} -1.00000 q^{11} +3.82843 q^{13} +1.00000 q^{16} +3.65685 q^{17} +0.585786 q^{19} -0.585786 q^{20} +1.00000 q^{22} +6.24264 q^{23} -4.65685 q^{25} -3.82843 q^{26} -2.65685 q^{29} +4.00000 q^{31} -1.00000 q^{32} -3.65685 q^{34} -9.41421 q^{37} -0.585786 q^{38} +0.585786 q^{40} -5.41421 q^{41} -5.65685 q^{43} -1.00000 q^{44} -6.24264 q^{46} -10.4853 q^{47} +4.65685 q^{50} +3.82843 q^{52} -7.89949 q^{53} +0.585786 q^{55} +2.65685 q^{58} -5.58579 q^{59} -11.8284 q^{61} -4.00000 q^{62} +1.00000 q^{64} -2.24264 q^{65} +2.75736 q^{67} +3.65685 q^{68} +11.0711 q^{71} +9.41421 q^{73} +9.41421 q^{74} +0.585786 q^{76} -13.2426 q^{79} -0.585786 q^{80} +5.41421 q^{82} -12.1421 q^{83} -2.14214 q^{85} +5.65685 q^{86} +1.00000 q^{88} +12.4853 q^{89} +6.24264 q^{92} +10.4853 q^{94} -0.343146 q^{95} +3.82843 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} + 4 q^{10} - 2 q^{11} + 2 q^{13} + 2 q^{16} - 4 q^{17} + 4 q^{19} - 4 q^{20} + 2 q^{22} + 4 q^{23} + 2 q^{25} - 2 q^{26} + 6 q^{29} + 8 q^{31} - 2 q^{32} + 4 q^{34}+ \cdots + 2 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^8 + 4 * q^10 - 2 * q^11 + 2 * q^13 + 2 * q^16 - 4 * q^17 + 4 * q^19 - 4 * q^20 + 2 * q^22 + 4 * q^23 + 2 * q^25 - 2 * q^26 + 6 * q^29 + 8 * q^31 - 2 * q^32 + 4 * q^34 - 16 * q^37 - 4 * q^38 + 4 * q^40 - 8 * q^41 - 2 * q^44 - 4 * q^46 - 4 * q^47 - 2 * q^50 + 2 * q^52 + 4 * q^53 + 4 * q^55 - 6 * q^58 - 14 * q^59 - 18 * q^61 - 8 * q^62 + 2 * q^64 + 4 * q^65 + 14 * q^67 - 4 * q^68 + 8 * q^71 + 16 * q^73 + 16 * q^74 + 4 * q^76 - 18 * q^79 - 4 * q^80 + 8 * q^82 + 4 * q^83 + 24 * q^85 + 2 * q^88 + 8 * q^89 + 4 * q^92 + 4 * q^94 - 12 * q^95 + 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more