Properties

Label 7448.2.a.bp
Level $7448$
Weight $2$
Character orbit 7448.a
Self dual yes
Analytic conductor $59.473$
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] = [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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 14x^{5} + 13x^{4} + 50x^{3} - 53x^{2} - 25x + 28 \) 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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{5} q^{5} + (\beta_{5} + \beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{5} q^{5} + (\beta_{5} + \beta_{4} + 1) q^{9} + ( - \beta_{3} + \beta_1) q^{11} + ( - \beta_{2} + 1) q^{13} + (\beta_{5} + \beta_{2} + \beta_1 - 1) q^{15} + ( - \beta_{6} - \beta_{3} + 1) q^{17} + q^{19} + (\beta_{6} - \beta_{4} - \beta_{2}) q^{23} + ( - \beta_{6} - \beta_{2} + \beta_1 + 1) q^{25} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{27} + ( - \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2}) q^{29} + ( - \beta_{4} + \beta_{2} - 1) q^{31} + (\beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 3) q^{33} + ( - \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_1 - 1) q^{37} + ( - \beta_{6} - 2 \beta_{5} + \beta_{3} + \beta_{2}) q^{39} + (\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{41} + (\beta_{5} + \beta_{3} + 2 \beta_1) q^{43} + (\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 + 3) q^{45} + (2 \beta_{6} - \beta_{4} - \beta_{3} + \beta_1 + 2) q^{47} + ( - \beta_{6} + \beta_{4} + \beta_{3} + 1) q^{51} + (2 \beta_{6} + \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{53} + (\beta_{6} + \beta_{5} + \beta_{4} + \beta_1 - 1) q^{55} + \beta_1 q^{57} + (2 \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{59} + ( - 2 \beta_{6} + \beta_{4} + \beta_{3} - \beta_1 + 2) q^{61} + (\beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_1 - 1) q^{65} + (\beta_{6} - 2 \beta_{5} - \beta_{2} + 2 \beta_1) q^{67} + ( - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{69} + (\beta_{6} + 2 \beta_{5} - \beta_{3} + 3 \beta_1 - 1) q^{71} + (\beta_{6} - 2 \beta_{5} + 2 \beta_{3} + 7) q^{73} + ( - 2 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 + 6) q^{75} + ( - \beta_{5} + \beta_{3}) q^{79} + (\beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 3 \beta_1 - 1) q^{81} + ( - \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1 - 3) q^{83} + (2 \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3} - 2) q^{85} + ( - 2 \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1) q^{87} + (2 \beta_{6} + \beta_{4} + \beta_1 + 6) q^{89} + (\beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{2} - 2 \beta_1) q^{93} + \beta_{5} q^{95} + ( - 2 \beta_{6} + \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{97} + (\beta_{6} + 3 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{3} + q^{5} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + q^{3} + q^{5} + 8 q^{9} + 3 q^{11} + 6 q^{13} - 4 q^{15} + 10 q^{17} + 7 q^{19} - 2 q^{23} + 8 q^{25} - 2 q^{27} + 3 q^{29} - 6 q^{31} + 21 q^{33} - 7 q^{37} - 2 q^{39} + 9 q^{41} + q^{43} + 24 q^{45} + 15 q^{47} + 6 q^{51} + 5 q^{53} - 6 q^{55} + q^{57} - 9 q^{59} + 13 q^{61} - 6 q^{65} - 2 q^{67} - 20 q^{69} - q^{71} + 42 q^{73} + 40 q^{75} - 3 q^{79} - q^{81} - 12 q^{83} - 16 q^{85} - 2 q^{87} + 41 q^{89} - 2 q^{93} + q^{95} + 5 q^{97} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 14x^{5} + 13x^{4} + 50x^{3} - 53x^{2} - 25x + 28 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 12\nu^{2} - \nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 9\nu^{3} + 14\nu^{2} + 13\nu - 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 13\nu^{4} + 2\nu^{3} - 39\nu^{2} + 16 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - 13\nu^{4} - 2\nu^{3} + 41\nu^{2} - 24 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 15\nu^{4} - 57\nu^{2} + 8\nu + 34 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 7\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 10\beta_{5} + 9\beta_{4} - \beta_{3} + \beta_{2} + 3\beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{6} + 15\beta_{5} + 13\beta_{4} - 9\beta_{3} + 11\beta_{2} + 56\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13\beta_{6} + 93\beta_{5} + 78\beta_{4} - 15\beta_{3} + 15\beta_{2} + 53\beta _1 + 196 \) 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
−2.62810
−2.52291
−0.779807
0.824576
1.13946
1.84812
3.11867
0 −2.62810 0 2.40717 0 0 0 3.90693 0
1.2 0 −2.52291 0 0.138247 0 0 0 3.36506 0
1.3 0 −0.779807 0 −1.35094 0 0 0 −2.39190 0
1.4 0 0.824576 0 −1.46996 0 0 0 −2.32007 0
1.5 0 1.13946 0 3.27400 0 0 0 −1.70164 0
1.6 0 1.84812 0 −4.19961 0 0 0 0.415553 0
1.7 0 3.11867 0 2.20110 0 0 0 6.72607 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

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

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

\( T_{3}^{7} - T_{3}^{6} - 14T_{3}^{5} + 13T_{3}^{4} + 50T_{3}^{3} - 53T_{3}^{2} - 25T_{3} + 28 \) Copy content Toggle raw display
\( T_{5}^{7} - T_{5}^{6} - 21T_{5}^{5} + 28T_{5}^{4} + 91T_{5}^{3} - 83T_{5}^{2} - 135T_{5} + 20 \) Copy content Toggle raw display
\( T_{11}^{7} - 3T_{11}^{6} - 32T_{11}^{5} + 101T_{11}^{4} + 128T_{11}^{3} - 327T_{11}^{2} + 19T_{11} + 100 \) Copy content Toggle raw display
\( T_{13}^{7} - 6T_{13}^{6} - 24T_{13}^{5} + 102T_{13}^{4} + 271T_{13}^{3} - 100T_{13}^{2} - 272T_{13} + 128 \) Copy content Toggle raw display
\( T_{17}^{7} - 10T_{17}^{6} - 7T_{17}^{5} + 258T_{17}^{4} - 341T_{17}^{3} - 882T_{17}^{2} + 1772T_{17} - 640 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( T^{7} - T^{6} - 14 T^{5} + 13 T^{4} + \cdots + 28 \) Copy content Toggle raw display
$5$ \( T^{7} - T^{6} - 21 T^{5} + 28 T^{4} + \cdots + 20 \) Copy content Toggle raw display
$7$ \( T^{7} \) Copy content Toggle raw display
$11$ \( T^{7} - 3 T^{6} - 32 T^{5} + 101 T^{4} + \cdots + 100 \) Copy content Toggle raw display
$13$ \( T^{7} - 6 T^{6} - 24 T^{5} + 102 T^{4} + \cdots + 128 \) Copy content Toggle raw display
$17$ \( T^{7} - 10 T^{6} - 7 T^{5} + 258 T^{4} + \cdots - 640 \) Copy content Toggle raw display
$19$ \( (T - 1)^{7} \) Copy content Toggle raw display
$23$ \( T^{7} + 2 T^{6} - 78 T^{5} + \cdots - 20480 \) Copy content Toggle raw display
$29$ \( T^{7} - 3 T^{6} - 96 T^{5} + \cdots + 2630 \) Copy content Toggle raw display
$31$ \( T^{7} + 6 T^{6} - 75 T^{5} - 276 T^{4} + \cdots - 800 \) Copy content Toggle raw display
$37$ \( T^{7} + 7 T^{6} - 133 T^{5} + \cdots + 51278 \) Copy content Toggle raw display
$41$ \( T^{7} - 9 T^{6} - 86 T^{5} + \cdots - 20228 \) Copy content Toggle raw display
$43$ \( T^{7} - T^{6} - 119 T^{5} + 40 T^{4} + \cdots + 1280 \) Copy content Toggle raw display
$47$ \( T^{7} - 15 T^{6} - 105 T^{5} + \cdots + 368216 \) Copy content Toggle raw display
$53$ \( T^{7} - 5 T^{6} - 214 T^{5} + \cdots + 650 \) Copy content Toggle raw display
$59$ \( T^{7} + 9 T^{6} - 227 T^{5} + \cdots - 1337012 \) Copy content Toggle raw display
$61$ \( T^{7} - 13 T^{6} - 129 T^{5} + \cdots - 144140 \) Copy content Toggle raw display
$67$ \( T^{7} + 2 T^{6} - 173 T^{5} + \cdots - 56000 \) Copy content Toggle raw display
$71$ \( T^{7} + T^{6} - 235 T^{5} + \cdots + 15700 \) Copy content Toggle raw display
$73$ \( T^{7} - 42 T^{6} + 587 T^{5} + \cdots - 37216 \) Copy content Toggle raw display
$79$ \( T^{7} + 3 T^{6} - 45 T^{5} - 96 T^{4} + \cdots - 448 \) Copy content Toggle raw display
$83$ \( T^{7} + 12 T^{6} - 245 T^{5} + \cdots - 19760 \) Copy content Toggle raw display
$89$ \( T^{7} - 41 T^{6} + 533 T^{5} + \cdots + 98048 \) Copy content Toggle raw display
$97$ \( T^{7} - 5 T^{6} - 293 T^{5} + \cdots - 1035860 \) Copy content Toggle raw display
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