Properties

Label 7168.2.a.v
Level $7168$
Weight $2$
Character orbit 7168.a
Self dual yes
Analytic conductor $57.237$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7168,2,Mod(1,7168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7168, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("7168.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7168 = 2^{10} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7168.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: \(57.2367681689\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 3584)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( - \beta_{2} - \beta_1) q^{5} + q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + ( - \beta_{2} - \beta_1) q^{5} + q^{7} + 3 q^{9} - 2 \beta_1 q^{11} + (\beta_{2} + 3 \beta_1) q^{13} + ( - \beta_{3} - 6) q^{15} + \beta_{3} q^{17} - \beta_{2} q^{19} + \beta_{2} q^{21} + ( - \beta_{3} - 6) q^{23} + (2 \beta_{3} + 3) q^{25} + ( - 2 \beta_{2} + \beta_1) q^{29} + ( - \beta_{3} - 4) q^{31} - 2 \beta_{3} q^{33} + ( - \beta_{2} - \beta_1) q^{35} + ( - 2 \beta_{2} + 3 \beta_1) q^{37} + (3 \beta_{3} + 6) q^{39} + \beta_{3} q^{41} + (2 \beta_{2} + 4 \beta_1) q^{43} + ( - 3 \beta_{2} - 3 \beta_1) q^{45} + ( - \beta_{3} - 8) q^{47} + q^{49} + 6 \beta_1 q^{51} + \beta_1 q^{53} + (2 \beta_{3} + 4) q^{55} - 6 q^{57} + (\beta_{2} - 4 \beta_1) q^{59} + ( - \beta_{2} + 7 \beta_1) q^{61} + 3 q^{63} + ( - 4 \beta_{3} - 12) q^{65} + (4 \beta_{2} - 4 \beta_1) q^{67} + ( - 6 \beta_{2} - 6 \beta_1) q^{69} - 4 q^{71} - 6 q^{73} + (3 \beta_{2} + 12 \beta_1) q^{75} - 2 \beta_1 q^{77} - 4 q^{79} - 9 q^{81} + (\beta_{2} - 4 \beta_1) q^{83} + ( - 2 \beta_{2} - 6 \beta_1) q^{85} + (\beta_{3} - 12) q^{87} + (2 \beta_{3} + 2) q^{89} + (\beta_{2} + 3 \beta_1) q^{91} + ( - 4 \beta_{2} - 6 \beta_1) q^{93} + (\beta_{3} + 6) q^{95} + ( - 3 \beta_{3} + 8) q^{97} - 6 \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{7} + 12 q^{9} - 24 q^{15} - 24 q^{23} + 12 q^{25} - 16 q^{31} + 24 q^{39} - 32 q^{47} + 4 q^{49} + 16 q^{55} - 24 q^{57} + 12 q^{63} - 48 q^{65} - 16 q^{71} - 24 q^{73} - 16 q^{79} - 36 q^{81} - 48 q^{87} + 8 q^{89} + 24 q^{95} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{24} + \zeta_{24}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{2} + 5\beta_1 ) / 2 \) 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
−0.517638
−1.93185
1.93185
0.517638
0 −2.44949 0 1.03528 0 1.00000 0 3.00000 0
1.2 0 −2.44949 0 3.86370 0 1.00000 0 3.00000 0
1.3 0 2.44949 0 −3.86370 0 1.00000 0 3.00000 0
1.4 0 2.44949 0 −1.03528 0 1.00000 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7168.2.a.v 4
4.b odd 2 1 7168.2.a.u 4
8.b even 2 1 inner 7168.2.a.v 4
8.d odd 2 1 7168.2.a.u 4
32.g even 8 2 3584.2.m.bd yes 4
32.g even 8 2 3584.2.m.be yes 4
32.h odd 8 2 3584.2.m.bc 4
32.h odd 8 2 3584.2.m.bf yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.m.bc 4 32.h odd 8 2
3584.2.m.bd yes 4 32.g even 8 2
3584.2.m.be yes 4 32.g even 8 2
3584.2.m.bf yes 4 32.h odd 8 2
7168.2.a.u 4 4.b odd 2 1
7168.2.a.u 4 8.d odd 2 1
7168.2.a.v 4 1.a even 1 1 trivial
7168.2.a.v 4 8.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7168))\):

\( T_{3}^{2} - 6 \) Copy content Toggle raw display
\( T_{5}^{4} - 16T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} - 8 \) Copy content Toggle raw display
\( T_{13}^{4} - 48T_{13}^{2} + 144 \) Copy content Toggle raw display
\( T_{17}^{2} - 12 \) Copy content Toggle raw display
\( T_{23}^{2} + 12T_{23} + 24 \) Copy content Toggle raw display
\( T_{31}^{2} + 8T_{31} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 16T^{2} + 16 \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 48T^{2} + 144 \) Copy content Toggle raw display
$17$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 52T^{2} + 484 \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 84T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 112T^{2} + 64 \) Copy content Toggle raw display
$47$ \( (T^{2} + 16 T + 52)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 76T^{2} + 676 \) Copy content Toggle raw display
$61$ \( T^{4} - 208T^{2} + 8464 \) Copy content Toggle raw display
$67$ \( T^{4} - 256T^{2} + 4096 \) Copy content Toggle raw display
$71$ \( (T + 4)^{4} \) Copy content Toggle raw display
$73$ \( (T + 6)^{4} \) Copy content Toggle raw display
$79$ \( (T + 4)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} - 76T^{2} + 676 \) Copy content Toggle raw display
$89$ \( (T^{2} - 4 T - 44)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 16 T - 44)^{2} \) Copy content Toggle raw display
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