↑

Properties

Label 9200.2.a.l.1.1
Level $9200$
Weight $2$
Character 9200.1
Self dual yes
Analytic conductor $73.462$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9200,2,Mod(1,9200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9200.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9200.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-1,0,0,0,-4,0,-2,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(73.4623698596\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 184)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9200.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -4.00000 q^{7} -2.00000 q^{9} +2.00000 q^{11} -7.00000 q^{13} +4.00000 q^{17} +6.00000 q^{19} +4.00000 q^{21} -1.00000 q^{23} +5.00000 q^{27} +5.00000 q^{29} -3.00000 q^{31} -2.00000 q^{33} -2.00000 q^{37} +7.00000 q^{39} -9.00000 q^{41} +8.00000 q^{43} -1.00000 q^{47} +9.00000 q^{49} -4.00000 q^{51} +6.00000 q^{53} -6.00000 q^{57} +8.00000 q^{59} -10.0000 q^{61} +8.00000 q^{63} +2.00000 q^{67} +1.00000 q^{69} +13.0000 q^{71} +3.00000 q^{73} -8.00000 q^{77} -6.00000 q^{79} +1.00000 q^{81} -5.00000 q^{87} -4.00000 q^{89} +28.0000 q^{91} +3.00000 q^{93} +8.00000 q^{97} -4.00000 q^{99} +O(q^{100})\)

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 \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) −7.00000 −1.94145 −0.970725 − 0.240192i \(-0.922790\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 − 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 − 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 7.00000 1.12090
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 − 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.00000 −0.145865 −0.0729325 − 0.997337i \(-0.523236\pi\)
−0.0729325 + 0.997337i \(0.523236\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 − 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 8.00000 1.00791
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 13.0000 1.54282 0.771408 − 0.636341i \(-0.219553\pi\)
0.771408 + 0.636341i \(0.219553\pi\)
\(72\) 0 0
\(73\) 3.00000 0.351123 0.175562 − 0.984468i \(-0.443826\pi\)
0.175562 + 0.984468i \(0.443826\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.00000 −0.911685
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 − 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −5.00000 −0.536056
\(88\) 0 0
\(89\) −4.00000 −0.423999 −0.212000 − 0.977270i \(-0.567998\pi\)
−0.212000 + 0.977270i \(0.567998\pi\)
\(90\) 0 0
\(91\) 28.0000 2.93520
\(92\) 0 0
\(93\) 3.00000 0.311086
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
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 9200.2.a.l.1.1 1
4.3 odd 2 4600.2.a.k.1.1 1
5.4 even 2 368.2.a.f.1.1 1
15.14 odd 2 3312.2.a.o.1.1 1
20.3 even 4 4600.2.e.f.4049.2 2
20.7 even 4 4600.2.e.f.4049.1 2
20.19 odd 2 184.2.a.b.1.1 ✓ 1
40.19 odd 2 1472.2.a.k.1.1 1
40.29 even 2 1472.2.a.d.1.1 1
60.59 even 2 1656.2.a.g.1.1 1
115.114 odd 2 8464.2.a.m.1.1 1
140.139 even 2 9016.2.a.j.1.1 1
460.459 even 2 4232.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.a.b.1.1 ✓ 1 20.19 odd 2
368.2.a.f.1.1 1 5.4 even 2
1472.2.a.d.1.1 1 40.29 even 2
1472.2.a.k.1.1 1 40.19 odd 2
1656.2.a.g.1.1 1 60.59 even 2
3312.2.a.o.1.1 1 15.14 odd 2
4232.2.a.e.1.1 1 460.459 even 2
4600.2.a.k.1.1 1 4.3 odd 2
4600.2.e.f.4049.1 2 20.7 even 4
4600.2.e.f.4049.2 2 20.3 even 4
8464.2.a.m.1.1 1 115.114 odd 2
9016.2.a.j.1.1 1 140.139 even 2
9200.2.a.l.1.1 1 1.1 even 1 trivial