Properties

Label 7360.2.a.bs
Level $7360$
Weight $2$
Character orbit 7360.a
Self dual yes
Analytic conductor $58.770$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) 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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + (\beta - 1) q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + q^{5} + (\beta - 1) q^{7} - 2 q^{9} + (\beta - 1) q^{11} - \beta q^{13} + q^{15} + ( - 2 \beta + 2) q^{17} + (\beta - 1) q^{19} + (\beta - 1) q^{21} + q^{23} + q^{25} - 5 q^{27} + ( - 4 \beta - 1) q^{29} + (\beta + 2) q^{31} + (\beta - 1) q^{33} + (\beta - 1) q^{35} + ( - \beta + 1) q^{37} - \beta q^{39} + ( - 2 \beta - 1) q^{41} + (\beta - 5) q^{43} - 2 q^{45} + ( - 4 \beta - 1) q^{47} + ( - 2 \beta - 3) q^{49} + ( - 2 \beta + 2) q^{51} + (2 \beta + 4) q^{53} + (\beta - 1) q^{55} + (\beta - 1) q^{57} - 6 \beta q^{59} + (5 \beta + 1) q^{61} + ( - 2 \beta + 2) q^{63} - \beta q^{65} + (5 \beta - 3) q^{67} + q^{69} + 7 \beta q^{71} + ( - \beta - 14) q^{73} + q^{75} + ( - 2 \beta + 4) q^{77} + ( - 5 \beta - 7) q^{79} + q^{81} - 4 q^{83} + ( - 2 \beta + 2) q^{85} + ( - 4 \beta - 1) q^{87} + (5 \beta - 9) q^{89} + (\beta - 3) q^{91} + (\beta + 2) q^{93} + (\beta - 1) q^{95} + (5 \beta + 1) q^{97} + ( - 2 \beta + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} - 4 q^{9} - 2 q^{11} + 2 q^{15} + 4 q^{17} - 2 q^{19} - 2 q^{21} + 2 q^{23} + 2 q^{25} - 10 q^{27} - 2 q^{29} + 4 q^{31} - 2 q^{33} - 2 q^{35} + 2 q^{37} - 2 q^{41} - 10 q^{43} - 4 q^{45} - 2 q^{47} - 6 q^{49} + 4 q^{51} + 8 q^{53} - 2 q^{55} - 2 q^{57} + 2 q^{61} + 4 q^{63} - 6 q^{67} + 2 q^{69} - 28 q^{73} + 2 q^{75} + 8 q^{77} - 14 q^{79} + 2 q^{81} - 8 q^{83} + 4 q^{85} - 2 q^{87} - 18 q^{89} - 6 q^{91} + 4 q^{93} - 2 q^{95} + 2 q^{97} + 4 q^{99}+O(q^{100}) \) 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
−1.73205
1.73205
0 1.00000 0 1.00000 0 −2.73205 0 −2.00000 0
1.2 0 1.00000 0 1.00000 0 0.732051 0 −2.00000 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.bs 2
4.b odd 2 1 7360.2.a.bg 2
8.b even 2 1 3680.2.a.l 2
8.d odd 2 1 3680.2.a.n yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3680.2.a.l 2 8.b even 2 1
3680.2.a.n yes 2 8.d odd 2 1
7360.2.a.bg 2 4.b odd 2 1
7360.2.a.bs 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7360))\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 2 \) Copy content Toggle raw display
\( T_{13}^{2} - 3 \) Copy content Toggle raw display
\( T_{17}^{2} - 4T_{17} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$13$ \( T^{2} - 3 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 8 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 2T - 47 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T + 1 \) Copy content Toggle raw display
$37$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$41$ \( T^{2} + 2T - 11 \) Copy content Toggle raw display
$43$ \( T^{2} + 10T + 22 \) Copy content Toggle raw display
$47$ \( T^{2} + 2T - 47 \) Copy content Toggle raw display
$53$ \( T^{2} - 8T + 4 \) Copy content Toggle raw display
$59$ \( T^{2} - 108 \) Copy content Toggle raw display
$61$ \( T^{2} - 2T - 74 \) Copy content Toggle raw display
$67$ \( T^{2} + 6T - 66 \) Copy content Toggle raw display
$71$ \( T^{2} - 147 \) Copy content Toggle raw display
$73$ \( T^{2} + 28T + 193 \) Copy content Toggle raw display
$79$ \( T^{2} + 14T - 26 \) Copy content Toggle raw display
$83$ \( (T + 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 18T + 6 \) Copy content Toggle raw display
$97$ \( T^{2} - 2T - 74 \) Copy content Toggle raw display
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