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Properties

Label 7254.2.a.bs
Level $7254$
Weight $2$
Character orbit 7254.a
Self dual yes
Analytic conductor $57.923$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7254,2,Mod(1,7254)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7254, 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("7254.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7254 = 2 \cdot 3^{2} \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7254.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: \(57.9234816262\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2x^{6} - 15x^{5} + 35x^{4} + 33x^{3} - 107x^{2} + 60x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + (\beta_1 + 1) q^{5} + (\beta_{4} + 1) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + (\beta_1 + 1) q^{5} + (\beta_{4} + 1) q^{7} + q^{8} + (\beta_1 + 1) q^{10} + ( - \beta_{2} + 2) q^{11} + q^{13} + (\beta_{4} + 1) q^{14} + q^{16} + ( - \beta_1 + 2) q^{17} + (\beta_{3} - \beta_1 + 1) q^{19} + (\beta_1 + 1) q^{20} + ( - \beta_{2} + 2) q^{22} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots + 1) q^{23}+ \cdots + ( - \beta_{6} + 2 \beta_{5} + \beta_{3} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 7 q^{2} + 7 q^{4} + 9 q^{5} + 4 q^{7} + 7 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 7 q^{2} + 7 q^{4} + 9 q^{5} + 4 q^{7} + 7 q^{8} + 9 q^{10} + 11 q^{11} + 7 q^{13} + 4 q^{14} + 7 q^{16} + 12 q^{17} + 7 q^{19} + 9 q^{20} + 11 q^{22} + 8 q^{23} + 10 q^{25} + 7 q^{26} + 4 q^{28} + 3 q^{29} + 7 q^{31} + 7 q^{32} + 12 q^{34} + 3 q^{35} + 7 q^{37} + 7 q^{38} + 9 q^{40} + 2 q^{41} + 11 q^{43} + 11 q^{44} + 8 q^{46} - 4 q^{47} + 13 q^{49} + 10 q^{50} + 7 q^{52} + 8 q^{55} + 4 q^{56} + 3 q^{58} + 7 q^{59} - 3 q^{61} + 7 q^{62} + 7 q^{64} + 9 q^{65} + 11 q^{67} + 12 q^{68} + 3 q^{70} + 28 q^{71} + 8 q^{73} + 7 q^{74} + 7 q^{76} + 6 q^{77} + 12 q^{79} + 9 q^{80} + 2 q^{82} + 25 q^{83} - 18 q^{85} + 11 q^{86} + 11 q^{88} + 12 q^{89} + 4 q^{91} + 8 q^{92} - 4 q^{94} - 12 q^{95} + 13 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 2x^{6} - 15x^{5} + 35x^{4} + 33x^{3} - 107x^{2} + 60x - 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + \nu^{5} - 12\nu^{4} - \nu^{3} + 24\nu^{2} - 32\nu + 15 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{5} + \nu^{4} - 13\nu^{3} - 4\nu^{2} + 34\nu - 11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{6} + \nu^{5} + 30\nu^{4} - 28\nu^{3} - 90\nu^{2} + 100\nu - 15 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{6} + \nu^{5} + 15\nu^{4} - 21\nu^{3} - 42\nu^{2} + 69\nu - 17 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{6} + \nu^{5} + 15\nu^{4} - 21\nu^{3} - 41\nu^{2} + 70\nu - 22 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} - \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 9\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10\beta_{6} - 13\beta_{5} + 4\beta_{4} + 2\beta_{3} - \beta_{2} - 15\beta _1 + 46 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{6} + 22\beta_{5} - 17\beta_{4} - 14\beta_{3} + 14\beta_{2} + 94\beta _1 - 67 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 102\beta_{6} - 153\beta_{5} + 64\beta_{4} + 37\beta_{3} - 22\beta_{2} - 209\beta _1 + 480 \) 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
−3.51854
−2.05802
0.253507
0.507118
1.24718
2.75272
2.81603
1.00000 0 1.00000 −2.51854 0 1.80690 1.00000 0 −2.51854
1.2 1.00000 0 1.00000 −1.05802 0 −1.87771 1.00000 0 −1.05802
1.3 1.00000 0 1.00000 1.25351 0 2.41168 1.00000 0 1.25351
1.4 1.00000 0 1.00000 1.50712 0 4.63287 1.00000 0 1.50712
1.5 1.00000 0 1.00000 2.24718 0 −4.50574 1.00000 0 2.24718
1.6 1.00000 0 1.00000 3.75272 0 2.56245 1.00000 0 3.75272
1.7 1.00000 0 1.00000 3.81603 0 −1.03044 1.00000 0 3.81603
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(13\) \( -1 \)
\(31\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7254.2.a.bs yes 7
3.b odd 2 1 7254.2.a.bn ✓ 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7254.2.a.bn ✓ 7 3.b odd 2 1
7254.2.a.bs yes 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7254))\):

\( T_{5}^{7} - 9T_{5}^{6} + 18T_{5}^{5} + 45T_{5}^{4} - 182T_{5}^{3} + 103T_{5}^{2} + 177T_{5} - 162 \) Copy content Toggle raw display
\( T_{7}^{7} - 4T_{7}^{6} - 23T_{7}^{5} + 101T_{7}^{4} + 49T_{7}^{3} - 435T_{7}^{2} + 68T_{7} + 451 \) Copy content Toggle raw display
\( T_{11}^{7} - 11T_{11}^{6} + 4T_{11}^{5} + 231T_{11}^{4} - 196T_{11}^{3} - 1531T_{11}^{2} - 87T_{11} + 702 \) Copy content Toggle raw display
\( T_{17}^{7} - 12T_{17}^{6} + 45T_{17}^{5} - 45T_{17}^{4} - 47T_{17}^{3} + 77T_{17}^{2} + 12T_{17} - 27 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{7} \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( T^{7} - 9 T^{6} + \cdots - 162 \) Copy content Toggle raw display
$7$ \( T^{7} - 4 T^{6} + \cdots + 451 \) Copy content Toggle raw display
$11$ \( T^{7} - 11 T^{6} + \cdots + 702 \) Copy content Toggle raw display
$13$ \( (T - 1)^{7} \) Copy content Toggle raw display
$17$ \( T^{7} - 12 T^{6} + \cdots - 27 \) Copy content Toggle raw display
$19$ \( T^{7} - 7 T^{6} + \cdots + 484 \) Copy content Toggle raw display
$23$ \( T^{7} - 8 T^{6} + \cdots + 6336 \) Copy content Toggle raw display
$29$ \( T^{7} - 3 T^{6} + \cdots + 4392 \) Copy content Toggle raw display
$31$ \( (T - 1)^{7} \) Copy content Toggle raw display
$37$ \( T^{7} - 7 T^{6} + \cdots - 24784 \) Copy content Toggle raw display
$41$ \( T^{7} - 2 T^{6} + \cdots + 20709 \) Copy content Toggle raw display
$43$ \( T^{7} - 11 T^{6} + \cdots - 98758 \) Copy content Toggle raw display
$47$ \( T^{7} + 4 T^{6} + \cdots - 144 \) Copy content Toggle raw display
$53$ \( T^{7} - 132 T^{5} + \cdots + 15552 \) Copy content Toggle raw display
$59$ \( T^{7} - 7 T^{6} + \cdots - 41364 \) Copy content Toggle raw display
$61$ \( T^{7} + 3 T^{6} + \cdots - 85532 \) Copy content Toggle raw display
$67$ \( T^{7} - 11 T^{6} + \cdots - 24142 \) Copy content Toggle raw display
$71$ \( T^{7} - 28 T^{6} + \cdots - 2523096 \) Copy content Toggle raw display
$73$ \( T^{7} - 8 T^{6} + \cdots + 124063 \) Copy content Toggle raw display
$79$ \( T^{7} - 12 T^{6} + \cdots - 5419 \) Copy content Toggle raw display
$83$ \( T^{7} - 25 T^{6} + \cdots - 8078832 \) Copy content Toggle raw display
$89$ \( T^{7} - 12 T^{6} + \cdots + 718992 \) Copy content Toggle raw display
$97$ \( T^{7} - 368 T^{5} + \cdots + 574424 \) Copy content Toggle raw display
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