Properties

Label 6960.2.a.bx
Level $6960$
Weight $2$
Character orbit 6960.a
Self dual yes
Analytic conductor $55.576$
Analytic rank $0$
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] = [6960,2,Mod(1,6960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6960.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6960 = 2^{4} \cdot 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6960.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: \(55.5758798068\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} - 2 \beta q^{7} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{5} - 2 \beta q^{7} + q^{9} + ( - \beta + 4) q^{11} - 2 q^{13} - q^{15} + ( - 2 \beta - 2) q^{17} - 2 \beta q^{19} + 2 \beta q^{21} + ( - \beta - 4) q^{23} + q^{25} - q^{27} + q^{29} - 4 q^{31} + (\beta - 4) q^{33} - 2 \beta q^{35} + ( - 3 \beta + 6) q^{37} + 2 q^{39} + ( - 3 \beta + 6) q^{41} - 3 \beta q^{43} + q^{45} + (2 \beta + 8) q^{47} + (4 \beta + 9) q^{49} + (2 \beta + 2) q^{51} + (\beta - 6) q^{53} + ( - \beta + 4) q^{55} + 2 \beta q^{57} - 12 q^{59} + ( - 2 \beta + 6) q^{61} - 2 \beta q^{63} - 2 q^{65} + (6 \beta - 4) q^{67} + (\beta + 4) q^{69} + ( - 2 \beta + 8) q^{71} + (3 \beta - 6) q^{73} - q^{75} + ( - 6 \beta + 8) q^{77} + 12 q^{79} + q^{81} + \beta q^{83} + ( - 2 \beta - 2) q^{85} - q^{87} + (4 \beta + 2) q^{89} + 4 \beta q^{91} + 4 q^{93} - 2 \beta q^{95} + ( - 3 \beta + 2) q^{97} + ( - \beta + 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 7 q^{11} - 4 q^{13} - 2 q^{15} - 6 q^{17} - 2 q^{19} + 2 q^{21} - 9 q^{23} + 2 q^{25} - 2 q^{27} + 2 q^{29} - 8 q^{31} - 7 q^{33} - 2 q^{35} + 9 q^{37} + 4 q^{39} + 9 q^{41} - 3 q^{43} + 2 q^{45} + 18 q^{47} + 22 q^{49} + 6 q^{51} - 11 q^{53} + 7 q^{55} + 2 q^{57} - 24 q^{59} + 10 q^{61} - 2 q^{63} - 4 q^{65} - 2 q^{67} + 9 q^{69} + 14 q^{71} - 9 q^{73} - 2 q^{75} + 10 q^{77} + 24 q^{79} + 2 q^{81} + q^{83} - 6 q^{85} - 2 q^{87} + 8 q^{89} + 4 q^{91} + 8 q^{93} - 2 q^{95} + q^{97} + 7 q^{99}+O(q^{100}) \) 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
2.56155
−1.56155
0 −1.00000 0 1.00000 0 −5.12311 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 3.12311 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(5\) \( -1 \)
\(29\) \( -1 \)

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 6960.2.a.bx 2
4.b odd 2 1 435.2.a.h 2
12.b even 2 1 1305.2.a.i 2
20.d odd 2 1 2175.2.a.m 2
20.e even 4 2 2175.2.c.h 4
60.h even 2 1 6525.2.a.bc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.h 2 4.b odd 2 1
1305.2.a.i 2 12.b even 2 1
2175.2.a.m 2 20.d odd 2 1
2175.2.c.h 4 20.e even 4 2
6525.2.a.bc 2 60.h even 2 1
6960.2.a.bx 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6960))\):

\( T_{7}^{2} + 2T_{7} - 16 \) Copy content Toggle raw display
\( T_{11}^{2} - 7T_{11} + 8 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{17}^{2} + 6T_{17} - 8 \) 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
$7$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$11$ \( T^{2} - 7T + 8 \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$23$ \( T^{2} + 9T + 16 \) Copy content Toggle raw display
$29$ \( (T - 1)^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 9T - 18 \) Copy content Toggle raw display
$41$ \( T^{2} - 9T - 18 \) Copy content Toggle raw display
$43$ \( T^{2} + 3T - 36 \) Copy content Toggle raw display
$47$ \( T^{2} - 18T + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 11T + 26 \) Copy content Toggle raw display
$59$ \( (T + 12)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 10T + 8 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T - 152 \) Copy content Toggle raw display
$71$ \( T^{2} - 14T + 32 \) Copy content Toggle raw display
$73$ \( T^{2} + 9T - 18 \) Copy content Toggle raw display
$79$ \( (T - 12)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$89$ \( T^{2} - 8T - 52 \) Copy content Toggle raw display
$97$ \( T^{2} - T - 38 \) Copy content Toggle raw display
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