Properties

Label 7360.2.a.bc
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: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{3} - q^{5} + ( - \beta + 2) q^{7} + (3 \beta + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{3} - q^{5} + ( - \beta + 2) q^{7} + (3 \beta + 1) q^{9} + (\beta + 3) q^{11} + (\beta - 2) q^{13} + (\beta + 1) q^{15} + ( - 3 \beta + 3) q^{17} + (3 \beta - 2) q^{19} + q^{21} - q^{23} + q^{25} + ( - 4 \beta - 7) q^{27} - 2 \beta q^{29} + (3 \beta - 4) q^{31} + ( - 5 \beta - 6) q^{33} + (\beta - 2) q^{35} - 8 q^{37} - q^{39} + ( - 3 \beta - 3) q^{41} + ( - 4 \beta + 4) q^{43} + ( - 3 \beta - 1) q^{45} + 2 \beta q^{47} - 3 \beta q^{49} + (3 \beta + 6) q^{51} + ( - 4 \beta + 6) q^{53} + ( - \beta - 3) q^{55} + ( - 4 \beta - 7) q^{57} + (2 \beta + 6) q^{59} + (5 \beta - 5) q^{61} + (2 \beta - 7) q^{63} + ( - \beta + 2) q^{65} + 4 q^{67} + (\beta + 1) q^{69} + (\beta - 15) q^{71} + (6 \beta + 2) q^{73} + ( - \beta - 1) q^{75} + ( - 2 \beta + 3) q^{77} + (8 \beta - 4) q^{79} + (6 \beta + 16) q^{81} + (4 \beta - 6) q^{83} + (3 \beta - 3) q^{85} + (4 \beta + 6) q^{87} + (3 \beta - 7) q^{91} + ( - 2 \beta - 5) q^{93} + ( - 3 \beta + 2) q^{95} + ( - \beta + 5) q^{97} + (13 \beta + 12) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - 2 q^{5} + 3 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} - 2 q^{5} + 3 q^{7} + 5 q^{9} + 7 q^{11} - 3 q^{13} + 3 q^{15} + 3 q^{17} - q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} - 18 q^{27} - 2 q^{29} - 5 q^{31} - 17 q^{33} - 3 q^{35} - 16 q^{37} - 2 q^{39} - 9 q^{41} + 4 q^{43} - 5 q^{45} + 2 q^{47} - 3 q^{49} + 15 q^{51} + 8 q^{53} - 7 q^{55} - 18 q^{57} + 14 q^{59} - 5 q^{61} - 12 q^{63} + 3 q^{65} + 8 q^{67} + 3 q^{69} - 29 q^{71} + 10 q^{73} - 3 q^{75} + 4 q^{77} + 38 q^{81} - 8 q^{83} - 3 q^{85} + 16 q^{87} - 11 q^{91} - 12 q^{93} + q^{95} + 9 q^{97} + 37 q^{99}+O(q^{100}) \) 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
2.30278
−1.30278
0 −3.30278 0 −1.00000 0 −0.302776 0 7.90833 0
1.2 0 0.302776 0 −1.00000 0 3.30278 0 −2.90833 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.bc 2
4.b odd 2 1 7360.2.a.bu 2
8.b even 2 1 230.2.a.b 2
8.d odd 2 1 1840.2.a.j 2
24.h odd 2 1 2070.2.a.w 2
40.e odd 2 1 9200.2.a.ca 2
40.f even 2 1 1150.2.a.m 2
40.i odd 4 2 1150.2.b.f 4
184.e odd 2 1 5290.2.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.b 2 8.b even 2 1
1150.2.a.m 2 40.f even 2 1
1150.2.b.f 4 40.i odd 4 2
1840.2.a.j 2 8.d odd 2 1
2070.2.a.w 2 24.h odd 2 1
5290.2.a.j 2 184.e odd 2 1
7360.2.a.bc 2 1.a even 1 1 trivial
7360.2.a.bu 2 4.b odd 2 1
9200.2.a.ca 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7360))\):

\( T_{3}^{2} + 3T_{3} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 3T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 7T_{11} + 9 \) Copy content Toggle raw display
\( T_{13}^{2} + 3T_{13} - 1 \) Copy content Toggle raw display
\( T_{17}^{2} - 3T_{17} - 27 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T - 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 3T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 7T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} + 3T - 1 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T - 27 \) Copy content Toggle raw display
$19$ \( T^{2} + T - 29 \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$31$ \( T^{2} + 5T - 23 \) Copy content Toggle raw display
$37$ \( (T + 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 9T - 9 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 48 \) Copy content Toggle raw display
$47$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$53$ \( T^{2} - 8T - 36 \) Copy content Toggle raw display
$59$ \( T^{2} - 14T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} + 5T - 75 \) Copy content Toggle raw display
$67$ \( (T - 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 29T + 207 \) Copy content Toggle raw display
$73$ \( T^{2} - 10T - 92 \) Copy content Toggle raw display
$79$ \( T^{2} - 208 \) Copy content Toggle raw display
$83$ \( T^{2} + 8T - 36 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 9T + 17 \) Copy content Toggle raw display
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