Properties

Label 7360.2.a.bq
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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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: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
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 \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + q^{5} + ( - \beta + 1) q^{7} + (\beta + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + q^{5} + ( - \beta + 1) q^{7} + (\beta + 2) q^{9} + (\beta - 2) q^{11} + ( - \beta - 3) q^{13} + \beta q^{15} + (\beta - 2) q^{17} + ( - \beta - 3) q^{19} - 5 q^{21} + q^{23} + q^{25} + 5 q^{27} + ( - 2 \beta - 2) q^{29} + ( - 3 \beta + 5) q^{31} + ( - \beta + 5) q^{33} + ( - \beta + 1) q^{35} + 4 q^{37} + ( - 4 \beta - 5) q^{39} + ( - \beta - 4) q^{41} - 4 \beta q^{43} + (\beta + 2) q^{45} + (2 \beta - 10) q^{47} + ( - \beta - 1) q^{49} + ( - \beta + 5) q^{51} - 6 q^{53} + (\beta - 2) q^{55} + ( - 4 \beta - 5) q^{57} + (2 \beta + 8) q^{59} + ( - 3 \beta - 2) q^{61} + ( - 2 \beta - 3) q^{63} + ( - \beta - 3) q^{65} - 4 \beta q^{67} + \beta q^{69} + 3 \beta q^{71} + (6 \beta - 4) q^{73} + \beta q^{75} + (2 \beta - 7) q^{77} + 8 q^{79} + (2 \beta - 6) q^{81} + 6 q^{83} + (\beta - 2) q^{85} + ( - 4 \beta - 10) q^{87} + (4 \beta + 4) q^{89} + (3 \beta + 2) q^{91} + (2 \beta - 15) q^{93} + ( - \beta - 3) q^{95} + ( - 5 \beta + 6) q^{97} + (\beta + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 2 q^{5} + q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 2 q^{5} + q^{7} + 5 q^{9} - 3 q^{11} - 7 q^{13} + q^{15} - 3 q^{17} - 7 q^{19} - 10 q^{21} + 2 q^{23} + 2 q^{25} + 10 q^{27} - 6 q^{29} + 7 q^{31} + 9 q^{33} + q^{35} + 8 q^{37} - 14 q^{39} - 9 q^{41} - 4 q^{43} + 5 q^{45} - 18 q^{47} - 3 q^{49} + 9 q^{51} - 12 q^{53} - 3 q^{55} - 14 q^{57} + 18 q^{59} - 7 q^{61} - 8 q^{63} - 7 q^{65} - 4 q^{67} + q^{69} + 3 q^{71} - 2 q^{73} + q^{75} - 12 q^{77} + 16 q^{79} - 10 q^{81} + 12 q^{83} - 3 q^{85} - 24 q^{87} + 12 q^{89} + 7 q^{91} - 28 q^{93} - 7 q^{95} + 7 q^{97} + 3 q^{99}+O(q^{100}) \) 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
−1.79129
2.79129
0 −1.79129 0 1.00000 0 2.79129 0 0.208712 0
1.2 0 2.79129 0 1.00000 0 −1.79129 0 4.79129 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.bq 2
4.b odd 2 1 7360.2.a.bk 2
8.b even 2 1 230.2.a.a 2
8.d odd 2 1 1840.2.a.n 2
24.h odd 2 1 2070.2.a.x 2
40.e odd 2 1 9200.2.a.bs 2
40.f even 2 1 1150.2.a.o 2
40.i odd 4 2 1150.2.b.g 4
184.e odd 2 1 5290.2.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.a 2 8.b even 2 1
1150.2.a.o 2 40.f even 2 1
1150.2.b.g 4 40.i odd 4 2
1840.2.a.n 2 8.d odd 2 1
2070.2.a.x 2 24.h odd 2 1
5290.2.a.e 2 184.e odd 2 1
7360.2.a.bk 2 4.b odd 2 1
7360.2.a.bq 2 1.a even 1 1 trivial
9200.2.a.bs 2 40.e odd 2 1

Hecke kernels

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

\( T_{3}^{2} - T_{3} - 5 \) Copy content Toggle raw display
\( T_{7}^{2} - T_{7} - 5 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} - 3 \) Copy content Toggle raw display
\( T_{13}^{2} + 7T_{13} + 7 \) Copy content Toggle raw display
\( T_{17}^{2} + 3T_{17} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T - 5 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T - 5 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T - 3 \) Copy content Toggle raw display
$13$ \( T^{2} + 7T + 7 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T - 3 \) Copy content Toggle raw display
$19$ \( T^{2} + 7T + 7 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 6T - 12 \) Copy content Toggle raw display
$31$ \( T^{2} - 7T - 35 \) Copy content Toggle raw display
$37$ \( (T - 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 9T + 15 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T - 80 \) Copy content Toggle raw display
$47$ \( T^{2} + 18T + 60 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 18T + 60 \) Copy content Toggle raw display
$61$ \( T^{2} + 7T - 35 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 80 \) Copy content Toggle raw display
$71$ \( T^{2} - 3T - 45 \) Copy content Toggle raw display
$73$ \( T^{2} + 2T - 188 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 12T - 48 \) Copy content Toggle raw display
$97$ \( T^{2} - 7T - 119 \) Copy content Toggle raw display
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