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Properties

Label 8800.2.a.bb.1.1
Level $8800$
Weight $2$
Character 8800.1
Self dual yes
Analytic conductor $70.268$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8800,2,Mod(1,8800)] 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("8800.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8800 = 2^{5} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8800.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,3,0,0,0,0,0,6,0,1,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(70.2683537787\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 352)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8800.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +6.00000 q^{9} +1.00000 q^{11} +6.00000 q^{13} +4.00000 q^{17} -6.00000 q^{19} +3.00000 q^{23} +9.00000 q^{27} -4.00000 q^{29} +9.00000 q^{31} +3.00000 q^{33} -7.00000 q^{37} +18.0000 q^{39} -2.00000 q^{41} +6.00000 q^{43} +12.0000 q^{47} -7.00000 q^{49} +12.0000 q^{51} -2.00000 q^{53} -18.0000 q^{57} -9.00000 q^{59} +8.00000 q^{61} -15.0000 q^{67} +9.00000 q^{69} +3.00000 q^{71} +6.00000 q^{73} +6.00000 q^{79} +9.00000 q^{81} -6.00000 q^{83} -12.0000 q^{87} -5.00000 q^{89} +27.0000 q^{93} +3.00000 q^{97} +6.00000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 1.73205 0.866025 − 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 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 \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 0.625543 0.312772 − 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 − 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 9.00000 1.61645 0.808224 − 0.588875i \(-0.200429\pi\)
0.808224 + 0.588875i \(0.200429\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 − 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 0 0
\(39\) 18.0000 2.88231
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 12.0000 1.68034
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −18.0000 −2.38416
\(58\) 0 0
\(59\) −9.00000 −1.17170 −0.585850 − 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −15.0000 −1.83254 −0.916271 − 0.400559i \(-0.868816\pi\)
−0.916271 + 0.400559i \(0.868816\pi\)
\(68\) 0 0
\(69\) 9.00000 1.08347
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 − 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 − 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −12.0000 −1.28654
\(88\) 0 0
\(89\) −5.00000 −0.529999 −0.264999 − 0.964249i \(-0.585372\pi\)
−0.264999 + 0.964249i \(0.585372\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 27.0000 2.79977
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.00000 0.304604 0.152302 − 0.988334i \(-0.451331\pi\)
0.152302 + 0.988334i \(0.451331\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
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 8800.2.a.bb.1.1 1
4.3 odd 2 8800.2.a.a.1.1 1
5.4 even 2 352.2.a.a.1.1 ✓ 1
15.14 odd 2 3168.2.a.h.1.1 1
20.19 odd 2 352.2.a.f.1.1 yes 1
40.19 odd 2 704.2.a.a.1.1 1
40.29 even 2 704.2.a.k.1.1 1
55.54 odd 2 3872.2.a.a.1.1 1
60.59 even 2 3168.2.a.i.1.1 1
80.19 odd 4 2816.2.c.h.1409.2 2
80.29 even 4 2816.2.c.g.1409.1 2
80.59 odd 4 2816.2.c.h.1409.1 2
80.69 even 4 2816.2.c.g.1409.2 2
120.29 odd 2 6336.2.a.bt.1.1 1
120.59 even 2 6336.2.a.bs.1.1 1
220.219 even 2 3872.2.a.m.1.1 1
440.109 odd 2 7744.2.a.bj.1.1 1
440.219 even 2 7744.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
352.2.a.a.1.1 ✓ 1 5.4 even 2
352.2.a.f.1.1 yes 1 20.19 odd 2
704.2.a.a.1.1 1 40.19 odd 2
704.2.a.k.1.1 1 40.29 even 2
2816.2.c.g.1409.1 2 80.29 even 4
2816.2.c.g.1409.2 2 80.69 even 4
2816.2.c.h.1409.1 2 80.59 odd 4
2816.2.c.h.1409.2 2 80.19 odd 4
3168.2.a.h.1.1 1 15.14 odd 2
3168.2.a.i.1.1 1 60.59 even 2
3872.2.a.a.1.1 1 55.54 odd 2
3872.2.a.m.1.1 1 220.219 even 2
6336.2.a.bs.1.1 1 120.59 even 2
6336.2.a.bt.1.1 1 120.29 odd 2
7744.2.a.a.1.1 1 440.219 even 2
7744.2.a.bj.1.1 1 440.109 odd 2
8800.2.a.a.1.1 1 4.3 odd 2
8800.2.a.bb.1.1 1 1.1 even 1 trivial