Properties

Label 7448.2.a.bn
Level $7448$
Weight $2$
Character orbit 7448.a
Self dual yes
Analytic conductor $59.473$
Analytic rank $0$
Dimension $6$
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] = [7448,2,Mod(1,7448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7448, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("7448.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7448.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: \(59.4725794254\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.98211824.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 6x^{4} + 15x^{3} + 8x^{2} - 9x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} - \beta_{3} q^{5} + (\beta_{5} - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} - \beta_{3} q^{5} + (\beta_{5} - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{9} + (\beta_{5} + \beta_{2} - \beta_1 + 1) q^{11} + (\beta_{5} - \beta_{4} + 2 \beta_{2} - \beta_1 + 2) q^{13} + ( - \beta_{5} + \beta_{4} - \beta_{3} - 1) q^{15} + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{17} - q^{19} + ( - \beta_{4} - 2 \beta_{3}) q^{23} + ( - \beta_{5} + \beta_{3} + \beta_{2} + 2) q^{25} + (\beta_{5} - 2 \beta_{4} - \beta_1 + 1) q^{27} + ( - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{29} + (2 \beta_{5} + \beta_{3} + \beta_{2} + 2) q^{31} + ( - \beta_{4} - \beta_{3} - \beta_{2} - 3 \beta_1 + 4) q^{33} + (\beta_{5} + \beta_{3} + \beta_1 - 2) q^{37} + ( - 2 \beta_{4} - \beta_{3} - 5 \beta_1 + 6) q^{39} + (3 \beta_{5} - 2 \beta_1 - 4) q^{41} + (\beta_{5} - \beta_{3} + \beta_{2}) q^{43} + ( - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_1 - 1) q^{45} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} + 5) q^{47} + ( - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + 3 \beta_1) q^{51} + (\beta_{4} - 3 \beta_{2} - 2 \beta_1 - 3) q^{53} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{55} + (\beta_1 - 1) q^{57} + ( - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 4) q^{59} + (\beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 3) q^{61} + ( - 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{2} + 3 \beta_1 + 2) q^{65} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + 3 \beta_{2} + \beta_1 + 2) q^{67} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{69} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 + 3) q^{71} + (2 \beta_{5} - \beta_{4} - \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 2) q^{73} + ( - \beta_{4} + 2 \beta_{2} + 5) q^{75} + (3 \beta_{5} - \beta_{3} - \beta_{2} + 2 \beta_1 - 6) q^{79} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{81} + ( - \beta_{4} - \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{83} + ( - \beta_{4} - \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 10) q^{85} + (3 \beta_{5} - \beta_{4} + 3 \beta_{2} + 5) q^{87} + (\beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} + 4 \beta_1 - 3) q^{89} + ( - \beta_{4} - 4 \beta_{2} - 6 \beta_1 + 2) q^{93} + \beta_{3} q^{95} + ( - 2 \beta_{5} + \beta_{3} - \beta_{2} + 3 \beta_1 - 3) q^{97} + (\beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} + q^{5} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} + q^{5} + 3 q^{9} + 3 q^{11} + 6 q^{13} - 8 q^{15} - 2 q^{17} - 6 q^{19} + 2 q^{23} + 5 q^{25} + 6 q^{27} - q^{29} + 14 q^{31} + 19 q^{33} - 7 q^{37} + 22 q^{39} - 21 q^{41} + q^{43} - 4 q^{45} + 23 q^{47} + 2 q^{51} - 15 q^{53} + 2 q^{55} - 3 q^{57} + 25 q^{59} + 21 q^{61} + 6 q^{65} + 2 q^{67} - 20 q^{69} + 13 q^{71} - 2 q^{73} + 24 q^{75} - 17 q^{79} - 2 q^{81} + 4 q^{83} + 40 q^{85} + 30 q^{87} - q^{89} + 6 q^{93} - q^{95} - 13 q^{97} + 41 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} - 6x^{4} + 15x^{3} + 8x^{2} - 9x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 3\nu^{4} - 5\nu^{3} + 13\nu^{2} + 3\nu - 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 9\nu^{3} + 10\nu^{2} + 19\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} + 7\nu^{3} - 17\nu^{2} - 13\nu + 11 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{5} + 3\nu^{4} + 6\nu^{3} - 14\nu^{2} - 9\nu + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} - \beta_{4} + 3\beta_{2} + 7\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 11\beta_{5} - 7\beta_{4} + 2\beta_{3} + 13\beta_{2} + 15\beta _1 + 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 30\beta_{5} - 13\beta_{4} + 6\beta_{3} + 43\beta_{2} + 64\beta _1 + 41 \) Copy content Toggle raw display

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
3.28578
2.15897
0.511631
0.129411
−1.04143
−2.04436
0 −2.28578 0 1.50233 0 0 0 2.22478 0
1.2 0 −1.15897 0 1.74190 0 0 0 −1.65679 0
1.3 0 0.488369 0 −3.51566 0 0 0 −2.76150 0
1.4 0 0.870589 0 0.696873 0 0 0 −2.24208 0
1.5 0 2.04143 0 3.17676 0 0 0 1.16742 0
1.6 0 3.04436 0 −2.60221 0 0 0 6.26815 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(19\) \(1\)

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 7448.2.a.bn yes 6
7.b odd 2 1 7448.2.a.bm 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7448.2.a.bm 6 7.b odd 2 1
7448.2.a.bn yes 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7448))\):

\( T_{3}^{6} - 3T_{3}^{5} - 6T_{3}^{4} + 19T_{3}^{3} + 2T_{3}^{2} - 19T_{3} + 7 \) Copy content Toggle raw display
\( T_{5}^{6} - T_{5}^{5} - 17T_{5}^{4} + 24T_{5}^{3} + 59T_{5}^{2} - 123T_{5} + 53 \) Copy content Toggle raw display
\( T_{11}^{6} - 3T_{11}^{5} - 28T_{11}^{4} + 121T_{11}^{3} + 4T_{11}^{2} - 539T_{11} + 575 \) Copy content Toggle raw display
\( T_{13}^{6} - 6T_{13}^{5} - 40T_{13}^{4} + 194T_{13}^{3} + 639T_{13}^{2} - 1544T_{13} - 3920 \) Copy content Toggle raw display
\( T_{17}^{6} + 2T_{17}^{5} - 55T_{17}^{4} - 138T_{17}^{3} + 91T_{17}^{2} + 322T_{17} + 148 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 3 T^{5} - 6 T^{4} + 19 T^{3} + \cdots + 7 \) Copy content Toggle raw display
$5$ \( T^{6} - T^{5} - 17 T^{4} + 24 T^{3} + \cdots + 53 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 3 T^{5} - 28 T^{4} + 121 T^{3} + \cdots + 575 \) Copy content Toggle raw display
$13$ \( T^{6} - 6 T^{5} - 40 T^{4} + \cdots - 3920 \) Copy content Toggle raw display
$17$ \( T^{6} + 2 T^{5} - 55 T^{4} - 138 T^{3} + \cdots + 148 \) Copy content Toggle raw display
$19$ \( (T + 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 2 T^{5} - 78 T^{4} + \cdots - 6860 \) Copy content Toggle raw display
$29$ \( T^{6} + T^{5} - 90 T^{4} + 147 T^{3} + \cdots + 9521 \) Copy content Toggle raw display
$31$ \( T^{6} - 14 T^{5} - 3 T^{4} + \cdots + 3148 \) Copy content Toggle raw display
$37$ \( T^{6} + 7 T^{5} - 27 T^{4} - 66 T^{3} + \cdots - 7 \) Copy content Toggle raw display
$41$ \( T^{6} + 21 T^{5} - 6 T^{4} + \cdots - 43133 \) Copy content Toggle raw display
$43$ \( T^{6} - T^{5} - 51 T^{4} + 52 T^{3} + \cdots - 1472 \) Copy content Toggle raw display
$47$ \( T^{6} - 23 T^{5} + 159 T^{4} + \cdots - 4855 \) Copy content Toggle raw display
$53$ \( T^{6} + 15 T^{5} - 36 T^{4} + \cdots + 4549 \) Copy content Toggle raw display
$59$ \( T^{6} - 25 T^{5} + 221 T^{4} + \cdots - 79 \) Copy content Toggle raw display
$61$ \( T^{6} - 21 T^{5} + 11 T^{4} + \cdots - 5867 \) Copy content Toggle raw display
$67$ \( T^{6} - 2 T^{5} - 253 T^{4} + \cdots + 4396 \) Copy content Toggle raw display
$71$ \( T^{6} - 13 T^{5} + T^{4} + 398 T^{3} + \cdots + 1813 \) Copy content Toggle raw display
$73$ \( T^{6} + 2 T^{5} - 245 T^{4} + \cdots - 71932 \) Copy content Toggle raw display
$79$ \( T^{6} + 17 T^{5} - 245 T^{4} + \cdots - 35696 \) Copy content Toggle raw display
$83$ \( T^{6} - 4 T^{5} - 133 T^{4} + \cdots - 32732 \) Copy content Toggle raw display
$89$ \( T^{6} + T^{5} - 379 T^{4} + \cdots - 360640 \) Copy content Toggle raw display
$97$ \( T^{6} + 13 T^{5} - 69 T^{4} + \cdots + 74851 \) Copy content Toggle raw display
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