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Properties

Label 3.39.b.a
Level $3$
Weight $39$
Character orbit 3.b
Analytic conductor $27.439$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3,39,Mod(2,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 39, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.2");
 
S:= CuspForms(chi, 39);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 39 \)
Character orbit: \([\chi]\) \(=\) 3.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(27.4390407101\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 17353504902 x^{10} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{75}\cdot 3^{91}\cdot 5^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 97 \beta_1 - 9561867) q^{3} + (\beta_{3} - \beta_{2} - 141606210704) q^{4} + (\beta_{4} - 340 \beta_{2} - 1705115 \beta_1) q^{5} + ( - \beta_{7} + 6 \beta_{4} + \cdots - 40300933723344) q^{6}+ \cdots + (\beta_{9} + 2 \beta_{8} + \cdots - 35\!\cdots\!95) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 97 \beta_1 - 9561867) q^{3} + (\beta_{3} - \beta_{2} - 141606210704) q^{4} + (\beta_{4} - 340 \beta_{2} - 1705115 \beta_1) q^{5} + ( - \beta_{7} + 6 \beta_{4} + \cdots - 40300933723344) q^{6}+ \cdots + (53\!\cdots\!54 \beta_{11} + \cdots - 28\!\cdots\!20) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 114742404 q^{3} - 1699274528448 q^{4} - 483611204680128 q^{6} + 81\!\cdots\!48 q^{7}+ \cdots - 42\!\cdots\!40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 114742404 q^{3} - 1699274528448 q^{4} - 483611204680128 q^{6} + 81\!\cdots\!48 q^{7}+ \cdots - 34\!\cdots\!40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 17353504902 x^{10} + \cdots + 27\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 12\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 64\!\cdots\!25 \nu^{11} + \cdots + 74\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 64\!\cdots\!25 \nu^{11} + \cdots + 96\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 12\!\cdots\!73 \nu^{11} + \cdots + 25\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 54\!\cdots\!91 \nu^{11} + \cdots - 36\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 38\!\cdots\!31 \nu^{11} + \cdots - 18\!\cdots\!00 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 19\!\cdots\!49 \nu^{11} + \cdots - 86\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 75\!\cdots\!71 \nu^{11} + \cdots + 45\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25\!\cdots\!11 \nu^{11} + \cdots - 24\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10\!\cdots\!23 \nu^{11} + \cdots + 14\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 24\!\cdots\!09 \nu^{11} + \cdots + 24\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - \beta_{2} - 416484117648 ) / 144 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{8} - 20\beta_{7} - 9\beta_{5} + 1334\beta_{4} - 714\beta_{3} - 45304936\beta_{2} - 656647616971\beta_1 ) / 1728 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 799 \beta_{11} + 6669 \beta_{10} - 6392 \beta_{9} + 12539 \beta_{8} + 47088136 \beta_{7} + \cdots + 13\!\cdots\!44 ) / 10368 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 6705600 \beta_{11} - 61826900 \beta_{10} - 67056000 \beta_{9} - 35393950167 \beta_{8} + \cdots + 15\!\cdots\!13 \beta_1 ) / 7776 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 15258505739191 \beta_{11} - 98697773101653 \beta_{10} + 122068045913528 \beta_{9} + \cdots - 10\!\cdots\!64 ) / 15552 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 99\!\cdots\!00 \beta_{11} + \cdots - 13\!\cdots\!63 \beta_1 ) / 11664 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 69\!\cdots\!31 \beta_{11} + \cdots + 31\!\cdots\!28 ) / 7776 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 37\!\cdots\!00 \beta_{11} + \cdots + 41\!\cdots\!71 \beta_1 ) / 5832 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 27\!\cdots\!75 \beta_{11} + \cdots - 95\!\cdots\!56 ) / 3888 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 12\!\cdots\!00 \beta_{11} + \cdots - 12\!\cdots\!07 \beta_1 ) / 2916 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
− 81459.4i
− 65373.8i
− 54008.5i
− 53338.0i
− 18842.7i
− 18089.7i
18089.7i
18842.7i
53338.0i
54008.5i
65373.8i
81459.4i
− 977513.i −4.43499e8 − 1.07432e9i −6.80653e11 1.94075e13i −1.05016e15 + 4.33526e14i −1.85443e15 3.96650e17i −9.57469e17 + 9.52919e17i 1.89711e19
2.2 − 784486.i 1.15997e9 + 7.30136e7i −3.40540e11 − 3.07374e13i 5.72781e13 − 9.09976e14i 1.07708e16 5.15107e16i 1.34019e18 + 1.69386e17i −2.41131e19
2.3 − 648102.i −9.89523e8 + 6.09669e8i −1.45158e11 − 1.81665e13i 3.95128e14 + 6.41312e14i −1.77541e16 − 8.40715e16i 6.07458e17 − 1.20656e18i −1.17737e19
2.4 − 640056.i 2.88152e8 + 1.12598e9i −1.34793e11 3.15820e13i 7.20687e14 − 1.84433e14i 9.50531e15 − 8.96619e16i −1.18479e18 + 6.48904e17i 2.02142e19
2.5 − 226112.i −8.54601e8 − 7.87723e8i 2.23751e11 − 6.89065e12i −1.78114e14 + 1.93236e14i 1.19822e16 − 1.12746e17i 1.09835e17 + 1.34638e18i −1.55806e18
2.6 − 217076.i 7.82134e8 − 8.59720e8i 2.27756e11 1.16100e13i −1.86625e14 − 1.69783e14i −8.59590e15 − 1.09110e17i −1.27385e17 − 1.34483e18i 2.52026e18
2.7 217076.i 7.82134e8 + 8.59720e8i 2.27756e11 − 1.16100e13i −1.86625e14 + 1.69783e14i −8.59590e15 1.09110e17i −1.27385e17 + 1.34483e18i 2.52026e18
2.8 226112.i −8.54601e8 + 7.87723e8i 2.23751e11 6.89065e12i −1.78114e14 − 1.93236e14i 1.19822e16 1.12746e17i 1.09835e17 − 1.34638e18i −1.55806e18
2.9 640056.i 2.88152e8 − 1.12598e9i −1.34793e11 − 3.15820e13i 7.20687e14 + 1.84433e14i 9.50531e15 8.96619e16i −1.18479e18 − 6.48904e17i 2.02142e19
2.10 648102.i −9.89523e8 − 6.09669e8i −1.45158e11 1.81665e13i 3.95128e14 − 6.41312e14i −1.77541e16 8.40715e16i 6.07458e17 + 1.20656e18i −1.17737e19
2.11 784486.i 1.15997e9 − 7.30136e7i −3.40540e11 3.07374e13i 5.72781e13 + 9.09976e14i 1.07708e16 − 5.15107e16i 1.34019e18 − 1.69386e17i −2.41131e19
2.12 977513.i −4.43499e8 + 1.07432e9i −6.80653e11 − 1.94075e13i −1.05016e15 − 4.33526e14i −1.85443e15 − 3.96650e17i −9.57469e17 − 9.52919e17i 1.89711e19
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.39.b.a ✓ 12
3.b odd 2 1 inner 3.39.b.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.39.b.a ✓ 12 1.a even 1 1 trivial
3.39.b.a ✓ 12 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{39}^{\mathrm{new}}(3, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 60\!\cdots\!61 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots - 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 12\!\cdots\!96)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 92\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 53\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 28\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 66\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 10\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
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