Properties

Label 7360.2.a.cv
Level $7360$
Weight $2$
Character orbit 7360.a
Self dual yes
Analytic conductor $58.770$
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] = [7360,2,Mod(1,7360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7360, 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("7360.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7360.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: \(58.7698958877\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.255601784.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 10x^{4} + 5x^{3} + 25x^{2} - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3680)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} - q^{5} + ( - \beta_{3} - 1) q^{7} + (\beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} - q^{5} + ( - \beta_{3} - 1) q^{7} + (\beta_{2} - \beta_1 + 1) q^{9} + ( - \beta_{5} + 1) q^{11} + ( - \beta_{4} + \beta_{2} + \beta_1 - 1) q^{13} + (\beta_1 - 1) q^{15} + ( - 2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{17} + ( - \beta_{5} + 2 \beta_{4} - \beta_{2} - 2 \beta_1 + 2) q^{19} + (\beta_{4} - 2 \beta_{3} - \beta_{2}) q^{21} - q^{23} + q^{25} + (\beta_{5} - \beta_{4} + 3 \beta_{2} + \beta_1 + 2) q^{27} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{29} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 - 2) q^{31} + ( - \beta_{5} - \beta_{2} - 2 \beta_1) q^{33} + (\beta_{3} + 1) q^{35} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1) q^{37} + ( - \beta_{5} + 2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{39} + (\beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{41} + (2 \beta_{5} - \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 3) q^{43} + ( - \beta_{2} + \beta_1 - 1) q^{45} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{47} + (2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{49} + ( - \beta_{5} - \beta_{4} - \beta_{2} + 1) q^{51} + ( - \beta_{5} - \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{53} + (\beta_{5} - 1) q^{55} + (2 \beta_{5} - \beta_{4} - 4 \beta_{3} - \beta_{2} + 2) q^{57} + (\beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{59} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 1) q^{61} + (\beta_{5} + 2 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + 2) q^{63} + (\beta_{4} - \beta_{2} - \beta_1 + 1) q^{65} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 + 4) q^{67} + (\beta_1 - 1) q^{69} + (3 \beta_{5} - \beta_{4} - 3 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{71} + ( - 3 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1) q^{73} + ( - \beta_1 + 1) q^{75} + (2 \beta_{5} - \beta_{2} + 3) q^{77} + ( - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 1) q^{79} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 2) q^{81} + ( - 3 \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 5) q^{83} + (2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{85} + (3 \beta_{5} - \beta_{4} - 4 \beta_{3} - \beta_1 - 3) q^{87} + ( - \beta_{4} - 2 \beta_{3} - 4 \beta_1 - 1) q^{89} + (\beta_{5} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 1) q^{91} + ( - 3 \beta_{5} + 4 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{93} + (\beta_{5} - 2 \beta_{4} + \beta_{2} + 2 \beta_1 - 2) q^{95} + ( - \beta_{5} + 2 \beta_{3} - 4 \beta_1 + 1) q^{97} + (\beta_{5} + \beta_{4} - \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 5 q^{3} - 6 q^{5} - 4 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 5 q^{3} - 6 q^{5} - 4 q^{7} + 7 q^{9} + 7 q^{11} - q^{13} - 5 q^{15} + 5 q^{19} - 6 q^{23} + 6 q^{25} + 20 q^{27} + 3 q^{29} - 12 q^{31} - 3 q^{33} + 4 q^{35} - 7 q^{37} - 8 q^{39} + 8 q^{41} + 28 q^{43} - 7 q^{45} - 8 q^{47} + 6 q^{49} + 7 q^{51} + 5 q^{53} - 7 q^{55} + 18 q^{57} + 9 q^{59} + 3 q^{61} + 5 q^{63} + q^{65} + 27 q^{67} - 5 q^{69} + 2 q^{71} - 2 q^{73} + 5 q^{75} + 14 q^{77} - 4 q^{79} + 14 q^{81} + 23 q^{83} - 12 q^{87} - 4 q^{89} - q^{91} + 4 q^{93} - 5 q^{95} - q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 10x^{4} + 5x^{3} + 25x^{2} - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 2\nu^{3} - 6\nu^{2} + 9\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - \nu^{4} - 8\nu^{3} + 5\nu^{2} + 13\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - \nu^{4} - 10\nu^{3} + 5\nu^{2} + 23\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} + 5\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{5} + 2\beta_{4} + 2\beta_{3} + 6\beta_{2} + 7\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -10\beta_{5} + 12\beta_{4} + 2\beta_{3} + \beta_{2} + 29\beta _1 + 23 \) 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
2.82746
2.15791
0.397065
−0.436047
−1.64604
−2.30036
0 −1.82746 0 −1.00000 0 −1.09205 0 0.339621 0
1.2 0 −1.15791 0 −1.00000 0 0.465759 0 −1.65924 0
1.3 0 0.602935 0 −1.00000 0 −4.26364 0 −2.63647 0
1.4 0 1.43605 0 −1.00000 0 −0.568362 0 −0.937769 0
1.5 0 2.64604 0 −1.00000 0 4.40512 0 4.00151 0
1.6 0 3.30036 0 −1.00000 0 −2.94683 0 7.89234 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
\(2\) \(-1\)
\(5\) \(1\)
\(23\) \(1\)

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 7360.2.a.cv 6
4.b odd 2 1 7360.2.a.cu 6
8.b even 2 1 3680.2.a.bc 6
8.d odd 2 1 3680.2.a.bd yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3680.2.a.bc 6 8.b even 2 1
3680.2.a.bd yes 6 8.d odd 2 1
7360.2.a.cu 6 4.b odd 2 1
7360.2.a.cv 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7360))\):

\( T_{3}^{6} - 5T_{3}^{5} + 25T_{3}^{3} - 15T_{3}^{2} - 26T_{3} + 16 \) Copy content Toggle raw display
\( T_{7}^{6} + 4T_{7}^{5} - 16T_{7}^{4} - 79T_{7}^{3} - 64T_{7}^{2} + 14T_{7} + 16 \) Copy content Toggle raw display
\( T_{11}^{6} - 7T_{11}^{5} - 3T_{11}^{4} + 92T_{11}^{3} - 142T_{11}^{2} - 4T_{11} + 32 \) Copy content Toggle raw display
\( T_{13}^{6} + T_{13}^{5} - 30T_{13}^{4} - 23T_{13}^{3} + 187T_{13}^{2} + 22T_{13} - 16 \) Copy content Toggle raw display
\( T_{17}^{6} - 54T_{17}^{4} - 109T_{17}^{3} + 114T_{17}^{2} + 376T_{17} + 208 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 5 T^{5} + 25 T^{3} - 15 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T + 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 4 T^{5} - 16 T^{4} - 79 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{6} - 7 T^{5} - 3 T^{4} + 92 T^{3} + \cdots + 32 \) Copy content Toggle raw display
$13$ \( T^{6} + T^{5} - 30 T^{4} - 23 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$17$ \( T^{6} - 54 T^{4} - 109 T^{3} + \cdots + 208 \) Copy content Toggle raw display
$19$ \( T^{6} - 5 T^{5} - 71 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( (T + 1)^{6} \) Copy content Toggle raw display
$29$ \( T^{6} - 3 T^{5} - 101 T^{4} + \cdots + 1784 \) Copy content Toggle raw display
$31$ \( T^{6} + 12 T^{5} - 19 T^{4} + \cdots + 1408 \) Copy content Toggle raw display
$37$ \( T^{6} + 7 T^{5} - 16 T^{4} - 62 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$41$ \( T^{6} - 8 T^{5} - 49 T^{4} + \cdots + 1294 \) Copy content Toggle raw display
$43$ \( T^{6} - 28 T^{5} + 206 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$47$ \( T^{6} + 8 T^{5} - 159 T^{4} + \cdots + 150016 \) Copy content Toggle raw display
$53$ \( T^{6} - 5 T^{5} - 82 T^{4} + 128 T^{3} + \cdots + 416 \) Copy content Toggle raw display
$59$ \( T^{6} - 9 T^{5} - 106 T^{4} + \cdots + 31744 \) Copy content Toggle raw display
$61$ \( T^{6} - 3 T^{5} - 273 T^{4} + \cdots - 271784 \) Copy content Toggle raw display
$67$ \( T^{6} - 27 T^{5} + 210 T^{4} + \cdots - 8192 \) Copy content Toggle raw display
$71$ \( T^{6} - 2 T^{5} - 195 T^{4} + \cdots - 2416 \) Copy content Toggle raw display
$73$ \( T^{6} + 2 T^{5} - 317 T^{4} + \cdots + 53056 \) Copy content Toggle raw display
$79$ \( T^{6} + 4 T^{5} - 242 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$83$ \( T^{6} - 23 T^{5} + 92 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$89$ \( T^{6} + 4 T^{5} - 318 T^{4} + \cdots - 201344 \) Copy content Toggle raw display
$97$ \( T^{6} + T^{5} - 181 T^{4} + \cdots - 27112 \) Copy content Toggle raw display
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