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Properties

Label 4232.2.a.y.1.15
Level $4232$
Weight $2$
Character 4232.1
Self dual yes
Analytic conductor $33.793$
Analytic rank $1$
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,-10,0,-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: \(1\)
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: \( x^{15} - x^{14} - 30 x^{13} + 26 x^{12} + 338 x^{11} - 238 x^{10} - 1773 x^{9} + 894 x^{8} + 4319 x^{7} + \cdots + 11 \) 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 1.15
Root \(-3.23165\) 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)
 
\(f(q)\) \(=\) \(q+3.23165 q^{3} -2.81977 q^{5} -2.60337 q^{7} +7.44356 q^{9} -3.84347 q^{11} -0.602670 q^{13} -9.11252 q^{15} +5.45338 q^{17} +2.16793 q^{19} -8.41317 q^{21} +2.95113 q^{25} +14.3600 q^{27} -7.30531 q^{29} -4.94985 q^{31} -12.4207 q^{33} +7.34091 q^{35} -1.76675 q^{37} -1.94762 q^{39} -8.76626 q^{41} -2.93669 q^{43} -20.9892 q^{45} +6.55118 q^{47} -0.222481 q^{49} +17.6234 q^{51} +0.742047 q^{53} +10.8377 q^{55} +7.00599 q^{57} -6.36945 q^{59} -6.96226 q^{61} -19.3783 q^{63} +1.69939 q^{65} -9.39883 q^{67} -10.7646 q^{71} -12.7408 q^{73} +9.53700 q^{75} +10.0060 q^{77} +1.27846 q^{79} +24.0759 q^{81} +4.65683 q^{83} -15.3773 q^{85} -23.6082 q^{87} -13.4445 q^{89} +1.56897 q^{91} -15.9962 q^{93} -6.11307 q^{95} +7.86437 q^{97} -28.6091 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - q^{3} - 10 q^{5} - 10 q^{7} + 16 q^{9} + 9 q^{11} - 12 q^{15} - 20 q^{17} + 3 q^{19} - 21 q^{21} + 11 q^{25} - 7 q^{27} - 4 q^{29} - 14 q^{31} - 34 q^{33} + 18 q^{35} - 26 q^{37} + 33 q^{39} - 11 q^{41}+ \cdots + 29 q^{99}+O(q^{100}) \) 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
\(3\) 3.23165 1.86579 0.932897 − 0.360144i \(-0.117272\pi\)
0.932897 + 0.360144i \(0.117272\pi\)
\(4\) 0 0
\(5\) −2.81977 −1.26104 −0.630521 − 0.776172i \(-0.717159\pi\)
−0.630521 + 0.776172i \(0.717159\pi\)
\(6\) 0 0
\(7\) −2.60337 −0.983980 −0.491990 − 0.870601i \(-0.663730\pi\)
−0.491990 + 0.870601i \(0.663730\pi\)
\(8\) 0 0
\(9\) 7.44356 2.48119
\(10\) 0 0
\(11\) −3.84347 −1.15885 −0.579425 − 0.815026i \(-0.696723\pi\)
−0.579425 + 0.815026i \(0.696723\pi\)
\(12\) 0 0
\(13\) −0.602670 −0.167151 −0.0835753 − 0.996501i \(-0.526634\pi\)
−0.0835753 + 0.996501i \(0.526634\pi\)
\(14\) 0 0
\(15\) −9.11252 −2.35284
\(16\) 0 0
\(17\) 5.45338 1.32264 0.661320 − 0.750104i \(-0.269997\pi\)
0.661320 + 0.750104i \(0.269997\pi\)
\(18\) 0 0
\(19\) 2.16793 0.497357 0.248679 − 0.968586i \(-0.420004\pi\)
0.248679 + 0.968586i \(0.420004\pi\)
\(20\) 0 0
\(21\) −8.41317 −1.83590
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) 2.95113 0.590225
\(26\) 0 0
\(27\) 14.3600 2.76359
\(28\) 0 0
\(29\) −7.30531 −1.35656 −0.678281 − 0.734802i \(-0.737275\pi\)
−0.678281 + 0.734802i \(0.737275\pi\)
\(30\) 0 0
\(31\) −4.94985 −0.889019 −0.444510 − 0.895774i \(-0.646622\pi\)
−0.444510 + 0.895774i \(0.646622\pi\)
\(32\) 0 0
\(33\) −12.4207 −2.16217
\(34\) 0 0
\(35\) 7.34091 1.24084
\(36\) 0 0
\(37\) −1.76675 −0.290453 −0.145226 − 0.989398i \(-0.546391\pi\)
−0.145226 + 0.989398i \(0.546391\pi\)
\(38\) 0 0
\(39\) −1.94762 −0.311868
\(40\) 0 0
\(41\) −8.76626 −1.36906 −0.684530 − 0.728985i \(-0.739993\pi\)
−0.684530 + 0.728985i \(0.739993\pi\)
\(42\) 0 0
\(43\) −2.93669 −0.447841 −0.223920 − 0.974607i \(-0.571886\pi\)
−0.223920 + 0.974607i \(0.571886\pi\)
\(44\) 0 0
\(45\) −20.9892 −3.12888
\(46\) 0 0
\(47\) 6.55118 0.955588 0.477794 − 0.878472i \(-0.341436\pi\)
0.477794 + 0.878472i \(0.341436\pi\)
\(48\) 0 0
\(49\) −0.222481 −0.0317831
\(50\) 0 0
\(51\) 17.6234 2.46777
\(52\) 0 0
\(53\) 0.742047 0.101928 0.0509640 − 0.998700i \(-0.483771\pi\)
0.0509640 + 0.998700i \(0.483771\pi\)
\(54\) 0 0
\(55\) 10.8377 1.46136
\(56\) 0 0
\(57\) 7.00599 0.927966
\(58\) 0 0
\(59\) −6.36945 −0.829231 −0.414616 − 0.909997i \(-0.636084\pi\)
−0.414616 + 0.909997i \(0.636084\pi\)
\(60\) 0 0
\(61\) −6.96226 −0.891426 −0.445713 − 0.895176i \(-0.647050\pi\)
−0.445713 + 0.895176i \(0.647050\pi\)
\(62\) 0 0
\(63\) −19.3783 −2.44144
\(64\) 0 0
\(65\) 1.69939 0.210784
\(66\) 0 0
\(67\) −9.39883 −1.14825 −0.574125 − 0.818768i \(-0.694658\pi\)
−0.574125 + 0.818768i \(0.694658\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.7646 −1.27752 −0.638761 − 0.769405i \(-0.720553\pi\)
−0.638761 + 0.769405i \(0.720553\pi\)
\(72\) 0 0
\(73\) −12.7408 −1.49120 −0.745601 − 0.666393i \(-0.767838\pi\)
−0.745601 + 0.666393i \(0.767838\pi\)
\(74\) 0 0
\(75\) 9.53700 1.10124
\(76\) 0 0
\(77\) 10.0060 1.14028
\(78\) 0 0
\(79\) 1.27846 0.143838 0.0719191 − 0.997410i \(-0.477088\pi\)
0.0719191 + 0.997410i \(0.477088\pi\)
\(80\) 0 0
\(81\) 24.0759 2.67510
\(82\) 0 0
\(83\) 4.65683 0.511153 0.255577 − 0.966789i \(-0.417735\pi\)
0.255577 + 0.966789i \(0.417735\pi\)
\(84\) 0 0
\(85\) −15.3773 −1.66790
\(86\) 0 0
\(87\) −23.6082 −2.53107
\(88\) 0 0
\(89\) −13.4445 −1.42511 −0.712555 − 0.701616i \(-0.752462\pi\)
−0.712555 + 0.701616i \(0.752462\pi\)
\(90\) 0 0
\(91\) 1.56897 0.164473
\(92\) 0 0
\(93\) −15.9962 −1.65873
\(94\) 0 0
\(95\) −6.11307 −0.627188
\(96\) 0 0
\(97\) 7.86437 0.798506 0.399253 − 0.916841i \(-0.369270\pi\)
0.399253 + 0.916841i \(0.369270\pi\)
\(98\) 0 0
\(99\) −28.6091 −2.87532
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.y.1.15 15
4.3 odd 2 8464.2.a.ci.1.1 15
23.9 even 11 184.2.i.a.81.1 yes 30
23.18 even 11 184.2.i.a.25.1 ✓ 30
23.22 odd 2 4232.2.a.z.1.15 15
92.55 odd 22 368.2.m.f.81.3 30
92.87 odd 22 368.2.m.f.209.3 30
92.91 even 2 8464.2.a.cj.1.1 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.i.a.25.1 ✓ 30 23.18 even 11
184.2.i.a.81.1 yes 30 23.9 even 11
368.2.m.f.81.3 30 92.55 odd 22
368.2.m.f.209.3 30 92.87 odd 22
4232.2.a.y.1.15 15 1.1 even 1 trivial
4232.2.a.z.1.15 15 23.22 odd 2
8464.2.a.ci.1.1 15 4.3 odd 2
8464.2.a.cj.1.1 15 92.91 even 2