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Properties

Label 2744.2.a.h
Level $2744$
Weight $2$
Character orbit 2744.a
Self dual yes
Analytic conductor $21.911$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2744,2,Mod(1,2744)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2744, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2744.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2744 = 2^{3} \cdot 7^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2744.a (trivial)

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: yes
Analytic conductor: \(21.9109503146\)
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: \( x^{12} - 29x^{10} + 304x^{8} - 1393x^{6} + 2574x^{4} - 1164x^{2} + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{10} q^{5} + (\beta_{3} + \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{10} q^{5} + (\beta_{3} + \beta_{2} + 2) q^{9} + ( - \beta_{7} - \beta_{6} + \beta_{3} + 2) q^{11} + ( - \beta_{10} - 2 \beta_{8} + \beta_{5}) q^{13} + (\beta_{6} - 2 \beta_{4} + \beta_{2} + 3) q^{15} + (\beta_{10} + \beta_{9} + 2 \beta_{8} + \cdots + \beta_1) q^{17}+ \cdots + ( - \beta_{7} - 7 \beta_{6} - 2 \beta_{4} + \cdots + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 22 q^{9} + 18 q^{11} + 28 q^{15} + 20 q^{23} + 14 q^{25} - 18 q^{29} - 18 q^{37} + 36 q^{39} + 10 q^{43} + 48 q^{51} - 38 q^{53} + 12 q^{57} + 8 q^{65} + 42 q^{67} + 56 q^{71} + 56 q^{79} + 52 q^{81} + 8 q^{85} - 48 q^{93} + 84 q^{95} + 90 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 29x^{10} + 304x^{8} - 1393x^{6} + 2574x^{4} - 1164x^{2} + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -9\nu^{10} + 680\nu^{8} - 13379\nu^{6} + 97733\nu^{4} - 226709\nu^{2} + 1782 ) / 18452 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9\nu^{10} - 680\nu^{8} + 13379\nu^{6} - 97733\nu^{4} + 245161\nu^{2} - 94042 ) / 18452 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -37\nu^{10} + 2283\nu^{8} - 36038\nu^{6} + 196769\nu^{4} - 310296\nu^{2} + 35004 ) / 73808 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -37\nu^{11} + 2283\nu^{9} - 36038\nu^{7} + 196769\nu^{5} - 310296\nu^{3} + 35004\nu ) / 73808 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 30\nu^{10} - 729\nu^{8} + 6155\nu^{6} - 24394\nu^{4} + 52983\nu^{2} - 33618 ) / 18452 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1395\nu^{10} - 36205\nu^{8} + 325418\nu^{6} - 1198903\nu^{4} + 1543416\nu^{2} - 174724 ) / 73808 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -2299\nu^{11} + 65553\nu^{9} - 670806\nu^{7} + 2962367\nu^{5} - 5110644\nu^{3} + 1866780\nu ) / 73808 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2419\nu^{11} - 68469\nu^{9} + 695426\nu^{7} - 3059943\nu^{5} + 5322576\nu^{3} - 1927444\nu ) / 73808 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3009\nu^{11} - 87419\nu^{9} + 919498\nu^{7} - 4234717\nu^{5} + 7845348\nu^{3} - 3428164\nu ) / 73808 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1643\nu^{11} - 47767\nu^{9} + 502388\nu^{7} - 2313319\nu^{5} + 4326658\nu^{3} - 2087480\nu ) / 36904 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{10} - 2\beta_{9} - 2\beta_{8} + \beta_{5} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - 11\beta_{6} + 3\beta_{4} + 11\beta_{3} + 10\beta_{2} + 36 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{11} - 9\beta_{10} - 35\beta_{9} - 36\beta_{8} + 16\beta_{5} + 80\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 17\beta_{7} - 186\beta_{6} + 63\beta_{4} + 124\beta_{3} + 98\beta_{2} + 294 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 72\beta_{11} - 63\beta_{10} - 477\beta_{9} - 485\beta_{8} + 221\beta_{5} + 852\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 229\beta_{7} - 2482\beta_{6} + 987\beta_{4} + 1401\beta_{3} + 987\beta_{2} + 2597 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 573\beta_{11} - 388\beta_{10} - 5927\beta_{9} - 5971\beta_{8} + 2846\beta_{5} + 9282\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2890\beta_{7} - 30481\beta_{6} + 13498\beta_{4} + 15782\beta_{3} + 10243\beta_{2} + 24351 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4704\beta_{11} - 2055\beta_{10} - 70474\beta_{9} - 70715\beta_{8} + 35060\beta_{5} + 102178\beta_1 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.35261
−2.74923
−2.46901
−1.85982
−0.799876
−0.0835475
0.0835475
0.799876
1.85982
2.46901
2.74923
3.35261
0 −3.35261 0 −0.758778 0 0 0 8.24000 0
1.2 0 −2.74923 0 −3.64717 0 0 0 4.55826 0
1.3 0 −2.46901 0 −2.10686 0 0 0 3.09602 0
1.4 0 −1.85982 0 1.80700 0 0 0 0.458937 0
1.5 0 −0.799876 0 0.913902 0 0 0 −2.36020 0
1.6 0 −0.0835475 0 −3.81878 0 0 0 −2.99302 0
1.7 0 0.0835475 0 3.81878 0 0 0 −2.99302 0
1.8 0 0.799876 0 −0.913902 0 0 0 −2.36020 0
1.9 0 1.85982 0 −1.80700 0 0 0 0.458937 0
1.10 0 2.46901 0 2.10686 0 0 0 3.09602 0
1.11 0 2.74923 0 3.64717 0 0 0 4.55826 0
1.12 0 3.35261 0 0.758778 0 0 0 8.24000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(7\) \( -1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2744.2.a.h ✓ 12
4.b odd 2 1 5488.2.a.w 12
7.b odd 2 1 inner 2744.2.a.h ✓ 12
28.d even 2 1 5488.2.a.w 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2744.2.a.h ✓ 12 1.a even 1 1 trivial
2744.2.a.h ✓ 12 7.b odd 2 1 inner
5488.2.a.w 12 4.b odd 2 1
5488.2.a.w 12 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 29T_{3}^{10} + 304T_{3}^{8} - 1393T_{3}^{6} + 2574T_{3}^{4} - 1164T_{3}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2744))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 29 T^{10} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( T^{12} - 37 T^{10} + \cdots + 1352 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} - 9 T^{5} + \cdots + 728)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} - 129 T^{10} + \cdots + 724808 \) Copy content Toggle raw display
$17$ \( T^{12} - 108 T^{10} + \cdots + 86528 \) Copy content Toggle raw display
$19$ \( T^{12} - 162 T^{10} + \cdots + 73544192 \) Copy content Toggle raw display
$23$ \( (T^{6} - 10 T^{5} + \cdots + 2639)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 9 T^{5} + \cdots + 9304)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 5776835072 \) Copy content Toggle raw display
$37$ \( (T^{6} + 9 T^{5} + \cdots - 28904)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 276 T^{10} + \cdots + 512 \) Copy content Toggle raw display
$43$ \( (T^{6} - 5 T^{5} + \cdots - 5384)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 74 T^{10} + \cdots + 86528 \) Copy content Toggle raw display
$53$ \( (T^{6} + 19 T^{5} + \cdots + 24408)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 42838986632 \) Copy content Toggle raw display
$61$ \( T^{12} - 193 T^{10} + \cdots + 3623432 \) Copy content Toggle raw display
$67$ \( (T^{6} - 21 T^{5} + \cdots + 5144)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 28 T^{5} + \cdots + 323743)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 97067704832 \) Copy content Toggle raw display
$79$ \( (T^{6} - 28 T^{5} + \cdots + 41957)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 5051733128 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 45858455552 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 796164608 \) Copy content Toggle raw display
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