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Properties

Label 7360.2.a.cg
Level $7360$
Weight $2$
Character orbit 7360.a
Self dual yes
Analytic conductor $58.770$
Analytic rank $1$
Dimension $4$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.15317.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 5x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 115)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{3} - q^{5} + (\beta_{3} + 1) q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{3} - q^{5} + (\beta_{3} + 1) q^{7} + (\beta_{2} + 2) q^{9} + ( - \beta_{3} + \beta_1 + 1) q^{11} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{13} + (\beta_{2} + 1) q^{15} + \beta_1 q^{17} + (\beta_{3} - \beta_1 - 1) q^{19} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{21} + q^{23} + q^{25} + (\beta_{2} - 3) q^{27} + ( - \beta_{2} + \beta_1 - 5) q^{29} + ( - \beta_{3} + \beta_{2} - 2 \beta_1) q^{31} + (\beta_{3} - 3 \beta_1 - 1) q^{33} + ( - \beta_{3} - 1) q^{35} + (\beta_{3} - 4 \beta_{2} - 1) q^{37} + ( - 3 \beta_{3} + \beta_{2} + \beta_1) q^{39} + ( - \beta_{2} + \beta_1 + 3) q^{41} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{43} + ( - \beta_{2} - 2) q^{45} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{47} + ( - 2 \beta_{2} - \beta_1 + 1) q^{49} + (2 \beta_{3} - 2 \beta_1 - 2) q^{51} + (2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 4) q^{53} + (\beta_{3} - \beta_1 - 1) q^{55} + ( - \beta_{3} + 3 \beta_1 + 1) q^{57} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 6) q^{59}+ \cdots + ( - 2 \beta_{3} + 4 \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 4 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 4 q^{5} + 3 q^{7} + 6 q^{9} + 4 q^{11} + 2 q^{15} - q^{17} - 4 q^{19} - 10 q^{21} + 4 q^{23} + 4 q^{25} - 14 q^{27} - 19 q^{29} + q^{31} - 2 q^{33} - 3 q^{35} + 3 q^{37} + 13 q^{41} - 6 q^{43} - 6 q^{45} - 6 q^{47} + 9 q^{49} - 8 q^{51} - 19 q^{53} - 4 q^{55} + 2 q^{57} + 23 q^{59} + 13 q^{63} - 3 q^{67} - 2 q^{69} + 3 q^{71} - 32 q^{73} - 2 q^{75} - 18 q^{77} - 2 q^{79} - 28 q^{81} - 21 q^{83} + q^{85} + 18 q^{87} - 40 q^{91} + 8 q^{93} + 4 q^{95} - 18 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 1 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + \beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + \beta_{2} + 3\beta _1 + 5 \) 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
2.69353
−1.69353
1.32973
−0.329727
0 −2.56155 0 −1.00000 0 0.819031 0 3.56155 0
1.2 0 −2.56155 0 −1.00000 0 2.74252 0 3.56155 0
1.3 0 1.56155 0 −1.00000 0 −4.06562 0 −0.561553 0
1.4 0 1.56155 0 −1.00000 0 3.50407 0 −0.561553 0
\(n\): e.g. 2-40 or 990-1000
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.cg 4
4.b odd 2 1 7360.2.a.cj 4
8.b even 2 1 1840.2.a.u 4
8.d odd 2 1 115.2.a.c ✓ 4
24.f even 2 1 1035.2.a.o 4
40.e odd 2 1 575.2.a.h 4
40.f even 2 1 9200.2.a.cl 4
40.k even 4 2 575.2.b.e 8
56.e even 2 1 5635.2.a.v 4
120.m even 2 1 5175.2.a.bx 4
184.h even 2 1 2645.2.a.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.a.c ✓ 4 8.d odd 2 1
575.2.a.h 4 40.e odd 2 1
575.2.b.e 8 40.k even 4 2
1035.2.a.o 4 24.f even 2 1
1840.2.a.u 4 8.b even 2 1
2645.2.a.m 4 184.h even 2 1
5175.2.a.bx 4 120.m even 2 1
5635.2.a.v 4 56.e even 2 1
7360.2.a.cg 4 1.a even 1 1 trivial
7360.2.a.cj 4 4.b odd 2 1
9200.2.a.cl 4 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7360))\):

\( T_{3}^{2} + T_{3} - 4 \) Copy content Toggle raw display
\( T_{7}^{4} - 3T_{7}^{3} - 14T_{7}^{2} + 52T_{7} - 32 \) Copy content Toggle raw display
\( T_{11}^{4} - 4T_{11}^{3} - 16T_{11}^{2} + 40T_{11} + 32 \) Copy content Toggle raw display
\( T_{13}^{4} - 41T_{13}^{2} + 212 \) Copy content Toggle raw display
\( T_{17}^{4} + T_{17}^{3} - 18T_{17}^{2} - 24T_{17} + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + T - 4)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 3 T^{3} + \cdots - 32 \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} + \cdots + 32 \) Copy content Toggle raw display
$13$ \( T^{4} - 41T^{2} + 212 \) Copy content Toggle raw display
$17$ \( T^{4} + T^{3} + \cdots + 32 \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} + \cdots + 32 \) Copy content Toggle raw display
$23$ \( (T - 1)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 19 T^{3} + \cdots + 202 \) Copy content Toggle raw display
$31$ \( T^{4} - T^{3} + \cdots + 2144 \) Copy content Toggle raw display
$37$ \( T^{4} - 3 T^{3} + \cdots + 2008 \) Copy content Toggle raw display
$41$ \( T^{4} - 13 T^{3} + \cdots - 94 \) Copy content Toggle raw display
$43$ \( T^{4} + 6 T^{3} + \cdots + 128 \) Copy content Toggle raw display
$47$ \( T^{4} + 6 T^{3} + \cdots - 128 \) Copy content Toggle raw display
$53$ \( T^{4} + 19 T^{3} + \cdots - 8776 \) Copy content Toggle raw display
$59$ \( T^{4} - 23 T^{3} + \cdots - 3136 \) Copy content Toggle raw display
$61$ \( T^{4} - 56 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$67$ \( T^{4} + 3 T^{3} + \cdots + 2032 \) Copy content Toggle raw display
$71$ \( T^{4} - 3 T^{3} + \cdots - 8 \) Copy content Toggle raw display
$73$ \( T^{4} + 32 T^{3} + \cdots + 1684 \) Copy content Toggle raw display
$79$ \( T^{4} + 2 T^{3} + \cdots + 512 \) Copy content Toggle raw display
$83$ \( T^{4} + 21 T^{3} + \cdots - 1216 \) Copy content Toggle raw display
$89$ \( T^{4} - 216 T^{2} + \cdots - 2752 \) Copy content Toggle raw display
$97$ \( T^{4} + 18 T^{3} + \cdots - 1072 \) Copy content Toggle raw display
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