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Properties

Label 7623.2.a.da
Level $7623$
Weight $2$
Character orbit 7623.a
Self dual yes
Analytic conductor $60.870$
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] = [7623,2,Mod(1,7623)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7623, 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("7623.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7623.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: \(60.8699614608\)
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: \( x^{12} - 22x^{10} + 181x^{8} - 692x^{6} + 1240x^{4} - 936x^{2} + 244 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} - \beta_{8} q^{5} + q^{7} + (\beta_{3} + 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} - \beta_{8} q^{5} + q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + ( - \beta_{6} - \beta_{4} + 2) q^{10} + ( - \beta_{5} + \beta_{4} + \beta_{2} + 2) q^{13} + \beta_1 q^{14} + (\beta_{10} - \beta_{6} - 2 \beta_{5} + \cdots + 3) q^{16}+ \cdots + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 20 q^{4} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 20 q^{4} + 12 q^{7} + 20 q^{10} + 20 q^{13} + 28 q^{16} + 12 q^{19} + 32 q^{25} + 20 q^{28} - 16 q^{31} - 24 q^{34} + 4 q^{37} + 48 q^{40} + 16 q^{43} + 24 q^{46} + 12 q^{49} + 96 q^{52} + 20 q^{58} + 44 q^{61} + 76 q^{64} + 20 q^{70} + 52 q^{73} - 8 q^{79} - 68 q^{82} + 72 q^{85} + 20 q^{91} + 20 q^{94} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 22x^{10} + 181x^{8} - 692x^{6} + 1240x^{4} - 936x^{2} + 244 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 6\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} - 17\nu^{6} + 94\nu^{4} - 186\nu^{2} + 92 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{10} - 21\nu^{8} + 160\nu^{6} - 536\nu^{4} + 748\nu^{2} - 308 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{10} - 21\nu^{8} + 164\nu^{6} - 588\nu^{4} + 924\nu^{2} - 412 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{11} + 21\nu^{9} - 160\nu^{7} + 536\nu^{5} - 748\nu^{3} + 308\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{11} - 21\nu^{9} + 162\nu^{7} - 562\nu^{5} + 840\nu^{3} - 384\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{11} + 22\nu^{9} - 178\nu^{7} + 643\nu^{5} - 980\nu^{3} + 438\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3\nu^{10} - 63\nu^{8} + 484\nu^{6} - 1652\nu^{4} + 2356\nu^{2} - 956 ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3\nu^{11} - 63\nu^{9} + 482\nu^{7} - 1626\nu^{5} + 2264\nu^{3} - 880\nu ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} - \beta_{6} - 2\beta_{5} + 8\beta_{2} + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} - \beta_{8} + 2\beta_{7} + 9\beta_{3} + 39\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13\beta_{10} - 11\beta_{6} - 28\beta_{5} + 60\beta_{2} + 149 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13\beta_{11} - 9\beta_{8} + 30\beta_{7} + 71\beta_{3} + 269\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 127\beta_{10} - 93\beta_{6} - 288\beta_{5} + 4\beta_{4} + 454\beta_{2} + 1023 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 127\beta_{11} + 4\beta_{9} - 55\beta_{8} + 318\beta_{7} + 547\beta_{3} + 1931\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1123\beta_{10} - 729\beta_{6} - 2632\beta_{5} + 84\beta_{4} + 3474\beta_{2} + 7287 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1123\beta_{11} + 84\beta_{9} - 251\beta_{8} + 2942\beta_{7} + 4203\beta_{3} + 14235\beta_1 \) 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
−2.79401
−2.41714
−2.09771
−1.60257
−0.870105
−0.790737
0.790737
0.870105
1.60257
2.09771
2.41714
2.79401
−2.79401 0 5.80650 −1.62657 0 1.00000 −10.6354 0 4.54465
1.2 −2.41714 0 3.84255 −4.01054 0 1.00000 −4.45369 0 9.69401
1.3 −2.09771 0 2.40037 2.53213 0 1.00000 −0.839859 0 −5.31167
1.4 −1.60257 0 0.568238 1.76461 0 1.00000 2.29450 0 −2.82792
1.5 −0.870105 0 −1.24292 −0.709862 0 1.00000 2.82168 0 0.617654
1.6 −0.790737 0 −1.37474 −4.15216 0 1.00000 2.66853 0 3.28327
1.7 0.790737 0 −1.37474 4.15216 0 1.00000 −2.66853 0 3.28327
1.8 0.870105 0 −1.24292 0.709862 0 1.00000 −2.82168 0 0.617654
1.9 1.60257 0 0.568238 −1.76461 0 1.00000 −2.29450 0 −2.82792
1.10 2.09771 0 2.40037 −2.53213 0 1.00000 0.839859 0 −5.31167
1.11 2.41714 0 3.84255 4.01054 0 1.00000 4.45369 0 9.69401
1.12 2.79401 0 5.80650 1.62657 0 1.00000 10.6354 0 4.54465
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(11\) \(1\)

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 7623.2.a.da yes 12
3.b odd 2 1 inner 7623.2.a.da yes 12
11.b odd 2 1 7623.2.a.cz ✓ 12
33.d even 2 1 7623.2.a.cz ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7623.2.a.cz ✓ 12 11.b odd 2 1
7623.2.a.cz ✓ 12 33.d even 2 1
7623.2.a.da yes 12 1.a even 1 1 trivial
7623.2.a.da yes 12 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{12} - 22T_{2}^{10} + 181T_{2}^{8} - 692T_{2}^{6} + 1240T_{2}^{4} - 936T_{2}^{2} + 244 \) Copy content Toggle raw display
\( T_{5}^{12} - 46T_{5}^{10} + 751T_{5}^{8} - 5300T_{5}^{6} + 16771T_{5}^{4} - 21846T_{5}^{2} + 7381 \) Copy content Toggle raw display
\( T_{13}^{6} - 10T_{13}^{5} + 5T_{13}^{4} + 150T_{13}^{3} - 174T_{13}^{2} - 260T_{13} - 50 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 22 T^{10} + \cdots + 244 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 46 T^{10} + \cdots + 7381 \) Copy content Toggle raw display
$7$ \( (T - 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( (T^{6} - 10 T^{5} + \cdots - 50)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} - 114 T^{10} + \cdots + 325069 \) Copy content Toggle raw display
$19$ \( (T^{6} - 6 T^{5} - 8 T^{4} + \cdots - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 108 T^{10} + \cdots + 999424 \) Copy content Toggle raw display
$29$ \( T^{12} - 190 T^{10} + \cdots + 18452500 \) Copy content Toggle raw display
$31$ \( (T^{6} + 8 T^{5} + \cdots + 352)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 2 T^{5} + \cdots - 28748)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 161673424 \) Copy content Toggle raw display
$43$ \( (T^{6} - 8 T^{5} + \cdots + 6100)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 232 T^{10} + \cdots + 74887504 \) Copy content Toggle raw display
$53$ \( T^{12} - 254 T^{10} + \cdots + 249856 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 3646813264 \) Copy content Toggle raw display
$61$ \( (T^{6} - 22 T^{5} + \cdots + 36928)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 206 T^{4} + \cdots - 41456)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 1072030752064 \) Copy content Toggle raw display
$73$ \( (T^{6} - 26 T^{5} + \cdots + 46432)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 4 T^{5} + \cdots + 443344)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 790976775424 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 1406025661 \) Copy content Toggle raw display
$97$ \( (T^{6} + 16 T^{5} + \cdots - 92066)^{2} \) Copy content Toggle raw display
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