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Properties

Label 4232.2.a.bb.1.8
Level $4232$
Weight $2$
Character 4232.1
Self dual yes
Analytic conductor $33.793$
Analytic rank $0$
Dimension $15$
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] = [4232,2,Mod(1,4232)] 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("4232.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4232, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4232 = 2^{3} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4232.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [15,0,1,0,0,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(33.7926901354\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 30 x^{13} + 28 x^{12} + 354 x^{11} - 302 x^{10} - 2111 x^{9} + 1596 x^{8} + 6777 x^{7} + \cdots - 419 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 1.8
Root \(-0.159078\) of defining polynomial
Character \(\chi\) \(=\) 4232.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.159078 q^{3} -0.0374616 q^{5} -3.17851 q^{7} -2.97469 q^{9} -2.40638 q^{11} -6.46974 q^{13} +0.00595930 q^{15} +1.41265 q^{17} -3.79359 q^{19} +0.505629 q^{21} -4.99860 q^{25} +0.950440 q^{27} -1.34251 q^{29} +5.26684 q^{31} +0.382801 q^{33} +0.119072 q^{35} +2.86429 q^{37} +1.02919 q^{39} +1.53077 q^{41} +3.15025 q^{43} +0.111437 q^{45} -7.00603 q^{47} +3.10291 q^{49} -0.224721 q^{51} +12.8957 q^{53} +0.0901467 q^{55} +0.603475 q^{57} -0.932170 q^{59} -4.33766 q^{61} +9.45509 q^{63} +0.242367 q^{65} +10.5734 q^{67} +12.6991 q^{71} -12.6489 q^{73} +0.795165 q^{75} +7.64869 q^{77} -7.50885 q^{79} +8.77289 q^{81} +12.5406 q^{83} -0.0529202 q^{85} +0.213563 q^{87} +7.79190 q^{89} +20.5641 q^{91} -0.837836 q^{93} +0.142114 q^{95} -2.28969 q^{97} +7.15824 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} + q^{21} + 23 q^{25} + q^{27} - 2 q^{29} + 20 q^{31} + 18 q^{33} - 18 q^{35} + 24 q^{37} - 19 q^{39} + 9 q^{41} + 48 q^{43} + 4 q^{45}+ \cdots + 63 q^{99}+O(q^{100}) \) 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
\(3\) −0.159078 −0.0918435 −0.0459217 − 0.998945i \(-0.514622\pi\)
−0.0459217 + 0.998945i \(0.514622\pi\)
\(4\) 0 0
\(5\) −0.0374616 −0.0167533 −0.00837667 − 0.999965i \(-0.502666\pi\)
−0.00837667 + 0.999965i \(0.502666\pi\)
\(6\) 0 0
\(7\) −3.17851 −1.20136 −0.600682 − 0.799488i \(-0.705104\pi\)
−0.600682 + 0.799488i \(0.705104\pi\)
\(8\) 0 0
\(9\) −2.97469 −0.991565
\(10\) 0 0
\(11\) −2.40638 −0.725550 −0.362775 − 0.931877i \(-0.618171\pi\)
−0.362775 + 0.931877i \(0.618171\pi\)
\(12\) 0 0
\(13\) −6.46974 −1.79438 −0.897191 − 0.441642i \(-0.854396\pi\)
−0.897191 + 0.441642i \(0.854396\pi\)
\(14\) 0 0
\(15\) 0.00595930 0.00153868
\(16\) 0 0
\(17\) 1.41265 0.342618 0.171309 − 0.985217i \(-0.445200\pi\)
0.171309 + 0.985217i \(0.445200\pi\)
\(18\) 0 0
\(19\) −3.79359 −0.870310 −0.435155 − 0.900356i \(-0.643306\pi\)
−0.435155 + 0.900356i \(0.643306\pi\)
\(20\) 0 0
\(21\) 0.505629 0.110337
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) −4.99860 −0.999719
\(26\) 0 0
\(27\) 0.950440 0.182912
\(28\) 0 0
\(29\) −1.34251 −0.249298 −0.124649 − 0.992201i \(-0.539780\pi\)
−0.124649 + 0.992201i \(0.539780\pi\)
\(30\) 0 0
\(31\) 5.26684 0.945952 0.472976 − 0.881075i \(-0.343180\pi\)
0.472976 + 0.881075i \(0.343180\pi\)
\(32\) 0 0
\(33\) 0.382801 0.0666370
\(34\) 0 0
\(35\) 0.119072 0.0201268
\(36\) 0 0
\(37\) 2.86429 0.470886 0.235443 − 0.971888i \(-0.424346\pi\)
0.235443 + 0.971888i \(0.424346\pi\)
\(38\) 0 0
\(39\) 1.02919 0.164802
\(40\) 0 0
\(41\) 1.53077 0.239066 0.119533 − 0.992830i \(-0.461860\pi\)
0.119533 + 0.992830i \(0.461860\pi\)
\(42\) 0 0
\(43\) 3.15025 0.480408 0.240204 − 0.970722i \(-0.422786\pi\)
0.240204 + 0.970722i \(0.422786\pi\)
\(44\) 0 0
\(45\) 0.111437 0.0166120
\(46\) 0 0
\(47\) −7.00603 −1.02193 −0.510967 − 0.859600i \(-0.670713\pi\)
−0.510967 + 0.859600i \(0.670713\pi\)
\(48\) 0 0
\(49\) 3.10291 0.443274
\(50\) 0 0
\(51\) −0.224721 −0.0314673
\(52\) 0 0
\(53\) 12.8957 1.77136 0.885681 − 0.464294i \(-0.153692\pi\)
0.885681 + 0.464294i \(0.153692\pi\)
\(54\) 0 0
\(55\) 0.0901467 0.0121554
\(56\) 0 0
\(57\) 0.603475 0.0799323
\(58\) 0 0
\(59\) −0.932170 −0.121358 −0.0606791 − 0.998157i \(-0.519327\pi\)
−0.0606791 + 0.998157i \(0.519327\pi\)
\(60\) 0 0
\(61\) −4.33766 −0.555380 −0.277690 − 0.960671i \(-0.589569\pi\)
−0.277690 + 0.960671i \(0.589569\pi\)
\(62\) 0 0
\(63\) 9.45509 1.19123
\(64\) 0 0
\(65\) 0.242367 0.0300619
\(66\) 0 0
\(67\) 10.5734 1.29175 0.645874 − 0.763444i \(-0.276493\pi\)
0.645874 + 0.763444i \(0.276493\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.6991 1.50710 0.753551 − 0.657390i \(-0.228340\pi\)
0.753551 + 0.657390i \(0.228340\pi\)
\(72\) 0 0
\(73\) −12.6489 −1.48044 −0.740218 − 0.672367i \(-0.765278\pi\)
−0.740218 + 0.672367i \(0.765278\pi\)
\(74\) 0 0
\(75\) 0.795165 0.0918177
\(76\) 0 0
\(77\) 7.64869 0.871649
\(78\) 0 0
\(79\) −7.50885 −0.844812 −0.422406 − 0.906407i \(-0.638814\pi\)
−0.422406 + 0.906407i \(0.638814\pi\)
\(80\) 0 0
\(81\) 8.77289 0.974765
\(82\) 0 0
\(83\) 12.5406 1.37651 0.688255 − 0.725469i \(-0.258377\pi\)
0.688255 + 0.725469i \(0.258377\pi\)
\(84\) 0 0
\(85\) −0.0529202 −0.00574000
\(86\) 0 0
\(87\) 0.213563 0.0228964
\(88\) 0 0
\(89\) 7.79190 0.825940 0.412970 − 0.910745i \(-0.364491\pi\)
0.412970 + 0.910745i \(0.364491\pi\)
\(90\) 0 0
\(91\) 20.5641 2.15571
\(92\) 0 0
\(93\) −0.837836 −0.0868795
\(94\) 0 0
\(95\) 0.142114 0.0145806
\(96\) 0 0
\(97\) −2.28969 −0.232482 −0.116241 − 0.993221i \(-0.537085\pi\)
−0.116241 + 0.993221i \(0.537085\pi\)
\(98\) 0 0
\(99\) 7.15824 0.719430
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 4232.2.a.bb.1.8 15
4.3 odd 2 8464.2.a.cg.1.8 15
23.9 even 11 184.2.i.b.81.2 yes 30
23.18 even 11 184.2.i.b.25.2 ✓ 30
23.22 odd 2 4232.2.a.ba.1.8 15
92.55 odd 22 368.2.m.e.81.2 30
92.87 odd 22 368.2.m.e.209.2 30
92.91 even 2 8464.2.a.ch.1.8 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.i.b.25.2 ✓ 30 23.18 even 11
184.2.i.b.81.2 yes 30 23.9 even 11
368.2.m.e.81.2 30 92.55 odd 22
368.2.m.e.209.2 30 92.87 odd 22
4232.2.a.ba.1.8 15 23.22 odd 2
4232.2.a.bb.1.8 15 1.1 even 1 trivial
8464.2.a.cg.1.8 15 4.3 odd 2
8464.2.a.ch.1.8 15 92.91 even 2