Properties

Label 7280.2.a.cf
Level $7280$
Weight $2$
Character orbit 7280.a
Self dual yes
Analytic conductor $58.131$
Analytic rank $1$
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] = [7280,2,Mod(1,7280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7280, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("7280.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7280 = 2^{4} \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7280.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: \(58.1310926715\)
Analytic rank: \(1\)
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: \( x^{7} - 3x^{6} - 11x^{5} + 24x^{4} + 33x^{3} - 41x^{2} - 31x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 3640)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} + q^{5} - q^{7} + (\beta_{2} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} + q^{5} - q^{7} + (\beta_{2} - \beta_1 + 2) q^{9} + ( - \beta_{3} - \beta_1 - 1) q^{11} + q^{13} + (\beta_1 - 1) q^{15} - \beta_{6} q^{17} + (\beta_{5} - \beta_1 - 1) q^{19} + ( - \beta_1 + 1) q^{21} + ( - \beta_{5} - \beta_{2} - 2) q^{23} + q^{25} + (\beta_{6} - \beta_{4} + 2 \beta_{3} - \beta_{2} + 4 \beta_1 - 2) q^{27} + (\beta_{6} + \beta_{4} - \beta_1 + 1) q^{29} + ( - \beta_{6} + \beta_{4} - \beta_{3} - 2 \beta_1 - 2) q^{31} + ( - \beta_{6} - \beta_{5} + \beta_{3} - 2 \beta_{2} - 2) q^{33} - q^{35} + (2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 + 1) q^{37} + (\beta_1 - 1) q^{39} + (\beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} + 2) q^{41} + (\beta_{6} - \beta_{5} + 2 \beta_{3} + \beta_1 - 1) q^{43} + (\beta_{2} - \beta_1 + 2) q^{45} + (\beta_{4} + \beta_{3} + \beta_{2} - 2) q^{47} + q^{49} + (\beta_{6} - \beta_{3}) q^{51} + (\beta_{6} - \beta_{4} - \beta_{3} + \beta_1 + 1) q^{53} + ( - \beta_{3} - \beta_1 - 1) q^{55} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{57} + ( - \beta_{6} + \beta_{5} + \beta_{3} - \beta_1 - 1) q^{59} + (\beta_{4} + \beta_{3} - \beta_{2}) q^{61} + ( - \beta_{2} + \beta_1 - 2) q^{63} + q^{65} + ( - \beta_{6} - \beta_{5} - \beta_{4} + 3 \beta_1 - 3) q^{67} + ( - \beta_{6} + \beta_{5} - 3 \beta_{3} - 6 \beta_1) q^{69} + ( - \beta_{6} + \beta_{3} + \beta_{2} - 2 \beta_1 - 2) q^{71} + ( - 2 \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} - 4) q^{73} + (\beta_1 - 1) q^{75} + (\beta_{3} + \beta_1 + 1) q^{77} + (\beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{2} + \beta_1 - 1) q^{79} + (\beta_{5} + \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 6 \beta_1 + 7) q^{81} + ( - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} - 4) q^{83} - \beta_{6} q^{85} + ( - \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + \beta_1 - 3) q^{87} + ( - 2 \beta_{4} + \beta_{3} + 4 \beta_1) q^{89} - q^{91} + ( - 3 \beta_{2} - \beta_1 - 3) q^{93} + (\beta_{5} - \beta_1 - 1) q^{95} + ( - \beta_{6} + \beta_{5} - \beta_{3} - 4 \beta_1) q^{97} + (2 \beta_{5} + \beta_{4} - 4 \beta_{3} + \beta_{2} - 9 \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 4 q^{3} + 7 q^{5} - 7 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 4 q^{3} + 7 q^{5} - 7 q^{7} + 11 q^{9} - 7 q^{11} + 7 q^{13} - 4 q^{15} + 3 q^{17} - 13 q^{19} + 4 q^{21} - 11 q^{23} + 7 q^{25} - 13 q^{27} + 3 q^{29} - 12 q^{31} - 11 q^{33} - 7 q^{35} - 4 q^{37} - 4 q^{39} + 6 q^{41} - 10 q^{43} + 11 q^{45} - 15 q^{47} + 7 q^{49} + 8 q^{53} - 7 q^{55} - 18 q^{57} - 13 q^{59} - q^{61} - 11 q^{63} + 7 q^{65} - 8 q^{67} - 9 q^{69} - 20 q^{71} - 17 q^{73} - 4 q^{75} + 7 q^{77} - 6 q^{79} + 39 q^{81} - 30 q^{83} + 3 q^{85} - 21 q^{87} + 5 q^{89} - 7 q^{91} - 24 q^{93} - 13 q^{95} - 9 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 11x^{5} + 24x^{4} + 33x^{3} - 41x^{2} - 31x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 2\nu^{5} - 13\nu^{4} + 15\nu^{3} + 36\nu^{2} - 33\nu - 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 6\nu^{5} + 5\nu^{4} - 59\nu^{3} - 4\nu^{2} + 125\nu + 24 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} - 3\nu^{5} - 10\nu^{4} + 22\nu^{3} + 22\nu^{2} - 33\nu - 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{6} + 10\nu^{5} + 31\nu^{4} - 85\nu^{3} - 84\nu^{2} + 159\nu + 60 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{4} + 2\beta_{3} + 2\beta_{2} + 10\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{6} + \beta_{5} - 3\beta_{4} + 5\beta_{3} + 14\beta_{2} + 23\beta _1 + 38 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 19\beta_{6} + 2\beta_{5} - 16\beta_{4} + 33\beta_{3} + 42\beta_{2} + 125\beta _1 + 96 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 75\beta_{6} + 17\beta_{5} - 56\beta_{4} + 105\beta_{3} + 200\beta_{2} + 396\beta _1 + 479 \) 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
−2.44487
−1.59001
−0.436282
−0.173825
1.70423
2.04214
3.89861
0 −3.44487 0 1.00000 0 −1.00000 0 8.86710 0
1.2 0 −2.59001 0 1.00000 0 −1.00000 0 3.70817 0
1.3 0 −1.43628 0 1.00000 0 −1.00000 0 −0.937093 0
1.4 0 −1.17382 0 1.00000 0 −1.00000 0 −1.62214 0
1.5 0 0.704230 0 1.00000 0 −1.00000 0 −2.50406 0
1.6 0 1.04214 0 1.00000 0 −1.00000 0 −1.91394 0
1.7 0 2.89861 0 1.00000 0 −1.00000 0 5.40195 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
\(2\) \(1\)
\(5\) \(-1\)
\(7\) \(1\)
\(13\) \(-1\)

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 7280.2.a.cf 7
4.b odd 2 1 3640.2.a.ba 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3640.2.a.ba 7 4.b odd 2 1
7280.2.a.cf 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7280))\):

\( T_{3}^{7} + 4T_{3}^{6} - 8T_{3}^{5} - 41T_{3}^{4} - 6T_{3}^{3} + 68T_{3}^{2} + 16T_{3} - 32 \) Copy content Toggle raw display
\( T_{11}^{7} + 7T_{11}^{6} - 26T_{11}^{5} - 220T_{11}^{4} + 80T_{11}^{3} + 1664T_{11}^{2} + 1280T_{11} - 1024 \) Copy content Toggle raw display
\( T_{17}^{7} - 3T_{17}^{6} - 47T_{17}^{5} + 238T_{17}^{4} - 160T_{17}^{3} - 464T_{17}^{2} + 176T_{17} + 32 \) Copy content Toggle raw display
\( T_{19}^{7} + 13T_{19}^{6} - 15T_{19}^{5} - 712T_{19}^{4} - 1776T_{19}^{3} + 4968T_{19}^{2} + 10880T_{19} - 17728 \) Copy content Toggle raw display
\( T_{23}^{7} + 11T_{23}^{6} - 58T_{23}^{5} - 776T_{23}^{4} + 1112T_{23}^{3} + 18032T_{23}^{2} - 7872T_{23} - 134528 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( T^{7} + 4 T^{6} - 8 T^{5} - 41 T^{4} + \cdots - 32 \) Copy content Toggle raw display
$5$ \( (T - 1)^{7} \) Copy content Toggle raw display
$7$ \( (T + 1)^{7} \) Copy content Toggle raw display
$11$ \( T^{7} + 7 T^{6} - 26 T^{5} + \cdots - 1024 \) Copy content Toggle raw display
$13$ \( (T - 1)^{7} \) Copy content Toggle raw display
$17$ \( T^{7} - 3 T^{6} - 47 T^{5} + 238 T^{4} + \cdots + 32 \) Copy content Toggle raw display
$19$ \( T^{7} + 13 T^{6} - 15 T^{5} + \cdots - 17728 \) Copy content Toggle raw display
$23$ \( T^{7} + 11 T^{6} - 58 T^{5} + \cdots - 134528 \) Copy content Toggle raw display
$29$ \( T^{7} - 3 T^{6} - 155 T^{5} + \cdots + 24224 \) Copy content Toggle raw display
$31$ \( T^{7} + 12 T^{6} - 40 T^{5} + \cdots + 3008 \) Copy content Toggle raw display
$37$ \( T^{7} + 4 T^{6} - 234 T^{5} + \cdots - 281104 \) Copy content Toggle raw display
$41$ \( T^{7} - 6 T^{6} - 136 T^{5} + \cdots - 76624 \) Copy content Toggle raw display
$43$ \( T^{7} + 10 T^{6} - 204 T^{5} + \cdots - 1688576 \) Copy content Toggle raw display
$47$ \( T^{7} + 15 T^{6} - 96 T^{5} + \cdots - 131072 \) Copy content Toggle raw display
$53$ \( T^{7} - 8 T^{6} - 164 T^{5} + \cdots - 77824 \) Copy content Toggle raw display
$59$ \( T^{7} + 13 T^{6} - 171 T^{5} + \cdots - 148864 \) Copy content Toggle raw display
$61$ \( T^{7} + T^{6} - 180 T^{5} + \cdots - 136768 \) Copy content Toggle raw display
$67$ \( T^{7} + 8 T^{6} - 276 T^{5} + \cdots - 107776 \) Copy content Toggle raw display
$71$ \( T^{7} + 20 T^{6} - 76 T^{5} + \cdots - 184832 \) Copy content Toggle raw display
$73$ \( T^{7} + 17 T^{6} - 128 T^{5} + \cdots + 32000 \) Copy content Toggle raw display
$79$ \( T^{7} + 6 T^{6} - 318 T^{5} + \cdots - 29696 \) Copy content Toggle raw display
$83$ \( T^{7} + 30 T^{6} + 60 T^{5} + \cdots - 1110016 \) Copy content Toggle raw display
$89$ \( T^{7} - 5 T^{6} - 379 T^{5} + \cdots - 2801056 \) Copy content Toggle raw display
$97$ \( T^{7} + 9 T^{6} - 278 T^{5} + \cdots + 158528 \) Copy content Toggle raw display
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