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Properties

Label 9200.2.a.cu.1.5
Level $9200$
Weight $2$
Character 9200.1
Self dual yes
Analytic conductor $73.462$
Analytic rank $1$
Dimension $5$
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] = [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 = [5,0,0,0,0,0,-2,0,13,0,1] 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: \(5\)
Coefficient field: 5.5.13955077.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 14x^{3} - x^{2} + 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(3.30649\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q+3.30649 q^{3} -2.55040 q^{7} +7.93288 q^{9} +2.72314 q^{11} -7.12637 q^{13} -0.924010 q^{17} -7.51623 q^{19} -8.43286 q^{21} -1.00000 q^{23} +16.3105 q^{27} -2.38248 q^{29} -0.866248 q^{31} +9.00402 q^{33} -0.352855 q^{37} -23.5633 q^{39} +4.34066 q^{41} -13.3239 q^{47} -0.495474 q^{49} -3.05523 q^{51} -3.99262 q^{53} -24.8523 q^{57} +3.84064 q^{59} -9.14262 q^{61} -20.2320 q^{63} -3.15933 q^{67} -3.30649 q^{69} +6.07883 q^{71} +11.3239 q^{73} -6.94508 q^{77} +12.0593 q^{79} +30.1319 q^{81} -6.35285 q^{83} -7.87765 q^{87} -9.71377 q^{89} +18.1751 q^{91} -2.86424 q^{93} -8.76465 q^{97} +21.6023 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{7} + 13 q^{9} + q^{11} - 4 q^{13} - 4 q^{17} - 7 q^{19} + 6 q^{21} - 5 q^{23} + 3 q^{27} + 4 q^{29} - 19 q^{31} - 17 q^{33} - 15 q^{37} - 19 q^{39} + 25 q^{41} - 11 q^{47} + 25 q^{49} - 19 q^{51}+ \cdots + 65 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 3.30649 1.90900 0.954501 − 0.298206i \(-0.0963883\pi\)
0.954501 + 0.298206i \(0.0963883\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.55040 −0.963960 −0.481980 − 0.876182i \(-0.660082\pi\)
−0.481980 + 0.876182i \(0.660082\pi\)
\(8\) 0 0
\(9\) 7.93288 2.64429
\(10\) 0 0
\(11\) 2.72314 0.821056 0.410528 − 0.911848i \(-0.365344\pi\)
0.410528 + 0.911848i \(0.365344\pi\)
\(12\) 0 0
\(13\) −7.12637 −1.97650 −0.988250 − 0.152845i \(-0.951156\pi\)
−0.988250 + 0.152845i \(0.951156\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.924010 −0.224105 −0.112053 − 0.993702i \(-0.535743\pi\)
−0.112053 + 0.993702i \(0.535743\pi\)
\(18\) 0 0
\(19\) −7.51623 −1.72434 −0.862171 − 0.506617i \(-0.830896\pi\)
−0.862171 + 0.506617i \(0.830896\pi\)
\(20\) 0 0
\(21\) −8.43286 −1.84020
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 16.3105 3.13896
\(28\) 0 0
\(29\) −2.38248 −0.442415 −0.221208 − 0.975227i \(-0.571000\pi\)
−0.221208 + 0.975227i \(0.571000\pi\)
\(30\) 0 0
\(31\) −0.866248 −0.155583 −0.0777913 − 0.996970i \(-0.524787\pi\)
−0.0777913 + 0.996970i \(0.524787\pi\)
\(32\) 0 0
\(33\) 9.00402 1.56740
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.352855 −0.0580089 −0.0290045 − 0.999579i \(-0.509234\pi\)
−0.0290045 + 0.999579i \(0.509234\pi\)
\(38\) 0 0
\(39\) −23.5633 −3.77315
\(40\) 0 0
\(41\) 4.34066 0.677896 0.338948 − 0.940805i \(-0.389929\pi\)
0.338948 + 0.940805i \(0.389929\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −13.3239 −1.94349 −0.971746 − 0.236027i \(-0.924155\pi\)
−0.971746 + 0.236027i \(0.924155\pi\)
\(48\) 0 0
\(49\) −0.495474 −0.0707819
\(50\) 0 0
\(51\) −3.05523 −0.427818
\(52\) 0 0
\(53\) −3.99262 −0.548429 −0.274214 − 0.961669i \(-0.588418\pi\)
−0.274214 + 0.961669i \(0.588418\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −24.8523 −3.29177
\(58\) 0 0
\(59\) 3.84064 0.500009 0.250004 − 0.968245i \(-0.419568\pi\)
0.250004 + 0.968245i \(0.419568\pi\)
\(60\) 0 0
\(61\) −9.14262 −1.17059 −0.585296 − 0.810820i \(-0.699022\pi\)
−0.585296 + 0.810820i \(0.699022\pi\)
\(62\) 0 0
\(63\) −20.2320 −2.54899
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.15933 −0.385974 −0.192987 − 0.981201i \(-0.561817\pi\)
−0.192987 + 0.981201i \(0.561817\pi\)
\(68\) 0 0
\(69\) −3.30649 −0.398055
\(70\) 0 0
\(71\) 6.07883 0.721424 0.360712 − 0.932677i \(-0.382534\pi\)
0.360712 + 0.932677i \(0.382534\pi\)
\(72\) 0 0
\(73\) 11.3239 1.32536 0.662682 − 0.748901i \(-0.269418\pi\)
0.662682 + 0.748901i \(0.269418\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.94508 −0.791465
\(78\) 0 0
\(79\) 12.0593 1.35677 0.678386 − 0.734706i \(-0.262680\pi\)
0.678386 + 0.734706i \(0.262680\pi\)
\(80\) 0 0
\(81\) 30.1319 3.34799
\(82\) 0 0
\(83\) −6.35285 −0.697316 −0.348658 − 0.937250i \(-0.613363\pi\)
−0.348658 + 0.937250i \(0.613363\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −7.87765 −0.844572
\(88\) 0 0
\(89\) −9.71377 −1.02966 −0.514829 − 0.857293i \(-0.672145\pi\)
−0.514829 + 0.857293i \(0.672145\pi\)
\(90\) 0 0
\(91\) 18.1751 1.90527
\(92\) 0 0
\(93\) −2.86424 −0.297008
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.76465 −0.889916 −0.444958 − 0.895552i \(-0.646781\pi\)
−0.444958 + 0.895552i \(0.646781\pi\)
\(98\) 0 0
\(99\) 21.6023 2.17111
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.cu.1.5 5
4.3 odd 2 4600.2.a.be.1.1 5
5.4 even 2 1840.2.a.v.1.1 5
20.3 even 4 4600.2.e.u.4049.2 10
20.7 even 4 4600.2.e.u.4049.9 10
20.19 odd 2 920.2.a.j.1.5 ✓ 5
40.19 odd 2 7360.2.a.co.1.1 5
40.29 even 2 7360.2.a.cp.1.5 5
60.59 even 2 8280.2.a.bs.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.j.1.5 ✓ 5 20.19 odd 2
1840.2.a.v.1.1 5 5.4 even 2
4600.2.a.be.1.1 5 4.3 odd 2
4600.2.e.u.4049.2 10 20.3 even 4
4600.2.e.u.4049.9 10 20.7 even 4
7360.2.a.co.1.1 5 40.19 odd 2
7360.2.a.cp.1.5 5 40.29 even 2
8280.2.a.bs.1.2 5 60.59 even 2
9200.2.a.cu.1.5 5 1.1 even 1 trivial