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Properties

Label 1332.2.j.g
Level $1332$
Weight $2$
Character orbit 1332.j
Analytic conductor $10.636$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1332,2,Mod(433,1332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1332, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1332.433");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1332 = 2^{2} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1332.j (of order \(3\), degree \(2\), 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: \(10.6360735492\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 30x^{10} + 672x^{8} + 6550x^{6} + 47634x^{4} + 33060x^{2} + 21025 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_1) q^{5} + ( - \beta_{7} - \beta_{5} - \beta_{4}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_1) q^{5} + ( - \beta_{7} - \beta_{5} - \beta_{4}) q^{7} + ( - \beta_{10} - \beta_{6} - \beta_{3}) q^{11} + (\beta_{9} - \beta_{7} + \cdots + \beta_{2}) q^{13}+ \cdots + ( - \beta_{4} + 3 \beta_{2} + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{7} - 2 q^{13} + 14 q^{19} - 30 q^{25} - 48 q^{31} + 26 q^{37} + 4 q^{43} - 18 q^{49} - 36 q^{55} + 38 q^{61} + 30 q^{67} + 16 q^{73} + 18 q^{79} + 48 q^{85} - 58 q^{91} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 30x^{10} + 672x^{8} + 6550x^{6} + 47634x^{4} + 33060x^{2} + 21025 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -25\nu^{10} - 560\nu^{8} - 12544\nu^{6} - 39695\nu^{4} - 27550\nu^{2} + 7999060 ) / 2584854 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -25\nu^{11} - 560\nu^{9} - 12544\nu^{7} - 39695\nu^{5} - 27550\nu^{3} + 13168768\nu ) / 2584854 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -235\nu^{10} - 5264\nu^{8} - 89193\nu^{6} - 373133\nu^{4} - 258970\nu^{2} + 11575035 ) / 7754562 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4256\nu^{10} + 124055\nu^{8} + 2778832\nu^{6} + 26057920\nu^{4} + 196974529\nu^{2} + 11774000 ) / 124934610 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4256\nu^{11} + 124055\nu^{9} + 2778832\nu^{7} + 26057920\nu^{5} + 196974529\nu^{3} + 11774000\nu ) / 124934610 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -8955\nu^{10} - 344195\nu^{8} - 7709968\nu^{6} - 85733043\nu^{4} - 546512821\nu^{2} - 379302890 ) / 224882298 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 385\nu^{11} + 8624\nu^{9} + 164457\nu^{7} + 611303\nu^{5} + 424270\nu^{3} - 82833081\nu ) / 7754562 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -7787\nu^{10} - 231870\nu^{8} - 5193888\nu^{6} - 50964685\nu^{4} - 368163186\nu^{2} - 255520740 ) / 74960766 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 64471 \nu^{11} - 1903680 \nu^{9} - 42642432 \nu^{7} - 411170945 \nu^{5} + \cdots - 2097855360 \nu ) / 374803830 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 393297 \nu^{11} + 12026560 \nu^{9} + 269394944 \nu^{7} + 2661169605 \nu^{5} + \cdots + 13253269120 \nu ) / 1124411490 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{9} + 10\beta_{5} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{10} + 16\beta_{6} + 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -51\beta_{9} + 9\beta_{7} - 145\beta_{5} - 145 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{11} - 69\beta_{10} - 256\beta_{6} - 256\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 270\beta_{4} - 846\beta_{2} + 2215 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -270\beta_{8} - 1386\beta_{3} + 4177\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 14187\beta_{9} - 6048\beta_{7} + 34840\beta_{5} - 6048\beta_{4} + 14187\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 6048\beta_{11} + 26283\beta_{10} + 6048\beta_{8} + 69262\beta_{6} + 26283\beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -240117\beta_{9} + 121185\beta_{7} - 561205\beta_{5} - 561205 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -121185\beta_{11} - 482487\beta_{10} - 1162624\beta_{6} - 1162624\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1332\mathbb{Z}\right)^\times\).

\(n\) \(667\) \(1037\) \(1297\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
2.08074 + 3.60395i
1.73085 + 2.99792i
0.417943 + 0.723898i
−0.417943 − 0.723898i
−1.73085 − 2.99792i
−2.08074 − 3.60395i
2.08074 − 3.60395i
1.73085 − 2.99792i
0.417943 − 0.723898i
−0.417943 + 0.723898i
−1.73085 + 2.99792i
−2.08074 + 3.60395i
0 0 0 −2.08074 + 3.60395i 0 −1.19478 + 2.06941i 0 0 0
433.2 0 0 0 −1.73085 + 2.99792i 0 2.45090 − 4.24508i 0 0 0
433.3 0 0 0 −0.417943 + 0.723898i 0 −0.256123 + 0.443618i 0 0 0
433.4 0 0 0 0.417943 − 0.723898i 0 −0.256123 + 0.443618i 0 0 0
433.5 0 0 0 1.73085 − 2.99792i 0 2.45090 − 4.24508i 0 0 0
433.6 0 0 0 2.08074 − 3.60395i 0 −1.19478 + 2.06941i 0 0 0
1009.1 0 0 0 −2.08074 − 3.60395i 0 −1.19478 − 2.06941i 0 0 0
1009.2 0 0 0 −1.73085 − 2.99792i 0 2.45090 + 4.24508i 0 0 0
1009.3 0 0 0 −0.417943 − 0.723898i 0 −0.256123 − 0.443618i 0 0 0
1009.4 0 0 0 0.417943 + 0.723898i 0 −0.256123 − 0.443618i 0 0 0
1009.5 0 0 0 1.73085 + 2.99792i 0 2.45090 + 4.24508i 0 0 0
1009.6 0 0 0 2.08074 + 3.60395i 0 −1.19478 − 2.06941i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 433.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
37.c even 3 1 inner
111.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1332.2.j.g ✓ 12
3.b odd 2 1 inner 1332.2.j.g ✓ 12
37.c even 3 1 inner 1332.2.j.g ✓ 12
111.i odd 6 1 inner 1332.2.j.g ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1332.2.j.g ✓ 12 1.a even 1 1 trivial
1332.2.j.g ✓ 12 3.b odd 2 1 inner
1332.2.j.g ✓ 12 37.c even 3 1 inner
1332.2.j.g ✓ 12 111.i odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1332, [\chi])\):

\( T_{5}^{12} + 30T_{5}^{10} + 672T_{5}^{8} + 6550T_{5}^{6} + 47634T_{5}^{4} + 33060T_{5}^{2} + 21025 \) Copy content Toggle raw display
\( T_{7}^{6} - 2T_{7}^{5} + 17T_{7}^{4} + 38T_{7}^{3} + 157T_{7}^{2} + 78T_{7} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 30 T^{10} + \cdots + 21025 \) Copy content Toggle raw display
$7$ \( (T^{6} - 2 T^{5} + 17 T^{4} + \cdots + 36)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 49 T^{4} + \cdots - 580)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + T^{5} + 24 T^{4} + \cdots + 900)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 43 T^{10} + \cdots + 1703025 \) Copy content Toggle raw display
$19$ \( (T^{6} - 7 T^{5} + \cdots + 2500)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 179 T^{4} + \cdots - 187920)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 43 T^{4} + \cdots - 1305)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 12 T^{2} + \cdots - 104)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} - 13 T^{5} + \cdots + 50653)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 48640097025 \) Copy content Toggle raw display
$43$ \( (T^{3} - T^{2} - 83 T - 250)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} - 196 T^{4} + \cdots - 580)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 1785751142400 \) Copy content Toggle raw display
$59$ \( T^{12} + 97 T^{10} + \cdots + 86118400 \) Copy content Toggle raw display
$61$ \( (T^{6} - 19 T^{5} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 15 T^{5} + \cdots + 8100)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 6975590400 \) Copy content Toggle raw display
$73$ \( (T^{3} - 4 T^{2} - 80 T + 8)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} - 9 T^{5} + \cdots + 2916)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 352740966400 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 153726726400 \) Copy content Toggle raw display
$97$ \( (T^{3} - 2 T^{2} + \cdots + 249)^{4} \) Copy content Toggle raw display
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