Properties

Label 7168.2.a.bd
Level $7168$
Weight $2$
Character orbit 7168.a
Self dual yes
Analytic conductor $57.237$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

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: \(8\)
Coefficient field: 8.8.13747093504.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 12x^{6} + 38x^{4} - 20x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 112)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{3} + \beta_{3} q^{5} + q^{7} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{3} + \beta_{3} q^{5} + q^{7} + (\beta_{2} + 1) q^{9} + (\beta_{7} + \beta_{6} - \beta_{3}) q^{11} + (\beta_{7} - \beta_{5} - 2 \beta_{3}) q^{13} + ( - \beta_{4} + \beta_{2} - 1) q^{15} + ( - \beta_{4} - 3) q^{17} + (\beta_{7} - \beta_{6} + \beta_{5}) q^{19} - \beta_{7} q^{21} + ( - \beta_{2} - \beta_1) q^{23} + (2 \beta_{4} - \beta_{2} - 1) q^{25} + ( - \beta_{6} - \beta_{5} + 2 \beta_{3}) q^{27} + ( - \beta_{7} + \beta_{6} + 3 \beta_{5} + \beta_{3}) q^{29} + (2 \beta_{4} + \beta_1 - 2) q^{31} + ( - \beta_{4} - 2 \beta_{2} - \beta_1 - 3) q^{33} + \beta_{3} q^{35} + ( - 2 \beta_{6} + \beta_{5}) q^{37} + (2 \beta_{4} - 3 \beta_{2} + \beta_1 - 2) q^{39} + (\beta_{4} - 4 \beta_{2} + 2 \beta_1 - 1) q^{41} + ( - 2 \beta_{6} - 4 \beta_{5} + 2 \beta_{3}) q^{43} + ( - \beta_{7} - \beta_{5}) q^{45} + (\beta_{4} + 2 \beta_1 + 3) q^{47} + q^{49} + (3 \beta_{7} + \beta_{6} + \beta_{3}) q^{51} + ( - 2 \beta_{7} + \beta_{5}) q^{53} + (\beta_{4} - \beta_1 - 1) q^{55} + (2 \beta_{4} - \beta_{2} - 4) q^{57} + (\beta_{7} + \beta_{6} + 5 \beta_{5} + 2 \beta_{3}) q^{59} + ( - 4 \beta_{5} - \beta_{3}) q^{61} + (\beta_{2} + 1) q^{63} + ( - 2 \beta_{4} + \beta_{2} - 6) q^{65} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{3}) q^{67} + (3 \beta_{7} + 2 \beta_{6} + 5 \beta_{5} - 3 \beta_{3}) q^{69} + ( - 3 \beta_{4} + 2 \beta_{2} - 3 \beta_1 - 7) q^{71} + (2 \beta_{2} - 2 \beta_1 - 6) q^{73} + (3 \beta_{7} - \beta_{6} + \beta_{5} - 4 \beta_{3}) q^{75} + (\beta_{7} + \beta_{6} - \beta_{3}) q^{77} + ( - 3 \beta_{4} - 2 \beta_{2} + \beta_1 - 3) q^{79} + ( - \beta_{2} + 2 \beta_1 - 5) q^{81} + \beta_{7} q^{83} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} - 5 \beta_{3}) q^{85} + ( - 3 \beta_{4} + 2 \beta_{2} - 4 \beta_1 + 3) q^{87} + ( - 3 \beta_{4} + 4 \beta_{2} - 3 \beta_1 - 3) q^{89} + (\beta_{7} - \beta_{5} - 2 \beta_{3}) q^{91} + (\beta_{7} - 3 \beta_{6} - 4 \beta_{5} - \beta_{3}) q^{93} + ( - 2 \beta_{4} - \beta_{2} + \beta_1 - 2) q^{95} + (2 \beta_{2} - 3 \beta_1 - 4) q^{97} + (5 \beta_{7} + \beta_{6} + 6 \beta_{5} - \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{7} + 8 q^{9} - 8 q^{15} - 24 q^{17} - 8 q^{25} - 16 q^{31} - 24 q^{33} - 16 q^{39} - 8 q^{41} + 24 q^{47} + 8 q^{49} - 8 q^{55} - 32 q^{57} + 8 q^{63} - 48 q^{65} - 56 q^{71} - 48 q^{73} - 24 q^{79} - 40 q^{81} + 24 q^{87} - 24 q^{89} - 16 q^{95} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 12x^{6} + 38x^{4} - 20x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 6\nu^{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 11\nu^{5} + 27\nu^{3} + 7\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 11\nu^{4} + 29\nu^{2} - 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 12\nu^{5} - 37\nu^{3} + 14\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 12\nu^{5} - 39\nu^{3} + 28\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} - 35\nu^{5} + 105\nu^{3} - 41\nu ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{7} + 5\beta_{6} + 2\beta_{5} - 7\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{2} + 6\beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 49\beta_{7} + 29\beta_{6} + 24\beta_{5} - 41\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{4} + 22\beta_{2} + 37\beta _1 + 105 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 343\beta_{7} + 177\beta_{6} + 210\beta_{5} - 247\beta_{3} ) / 2 \) 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
2.63251
−0.236253
0.763878
−2.10489
2.10489
−0.763878
0.236253
−2.63251
0 −2.77902 0 −0.887844 0 1.00000 0 4.72294 0
1.2 0 −2.08185 0 −0.500437 0 1.00000 0 1.33411 0
1.3 0 −1.70872 0 3.66816 0 1.00000 0 −0.0802864 0
1.4 0 −1.01155 0 1.22714 0 1.00000 0 −1.97676 0
1.5 0 1.01155 0 −1.22714 0 1.00000 0 −1.97676 0
1.6 0 1.70872 0 −3.66816 0 1.00000 0 −0.0802864 0
1.7 0 2.08185 0 0.500437 0 1.00000 0 1.33411 0
1.8 0 2.77902 0 0.887844 0 1.00000 0 4.72294 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.bd 8
4.b odd 2 1 7168.2.a.bc 8
8.b even 2 1 inner 7168.2.a.bd 8
8.d odd 2 1 7168.2.a.bc 8
32.g even 8 2 448.2.m.c 8
32.g even 8 2 896.2.m.f 8
32.h odd 8 2 112.2.m.c 8
32.h odd 8 2 896.2.m.e 8
224.x even 8 2 784.2.m.g 8
224.be even 24 4 784.2.x.j 16
224.bf odd 24 4 784.2.x.k 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.m.c 8 32.h odd 8 2
448.2.m.c 8 32.g even 8 2
784.2.m.g 8 224.x even 8 2
784.2.x.j 16 224.be even 24 4
784.2.x.k 16 224.bf odd 24 4
896.2.m.e 8 32.h odd 8 2
896.2.m.f 8 32.g even 8 2
7168.2.a.bc 8 4.b odd 2 1
7168.2.a.bc 8 8.d odd 2 1
7168.2.a.bd 8 1.a even 1 1 trivial
7168.2.a.bd 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7168))\):

\( T_{3}^{8} - 16T_{3}^{6} + 84T_{3}^{4} - 168T_{3}^{2} + 100 \) Copy content Toggle raw display
\( T_{5}^{8} - 16T_{5}^{6} + 36T_{5}^{4} - 24T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{8} - 48T_{11}^{6} + 608T_{11}^{4} - 1280T_{11}^{2} + 256 \) Copy content Toggle raw display
\( T_{13}^{8} - 56T_{13}^{6} + 1124T_{13}^{4} - 9528T_{13}^{2} + 28900 \) Copy content Toggle raw display
\( T_{17}^{4} + 12T_{17}^{3} + 40T_{17}^{2} + 40T_{17} + 8 \) Copy content Toggle raw display
\( T_{23}^{4} - 40T_{23}^{2} + 128T_{23} - 108 \) Copy content Toggle raw display
\( T_{31}^{4} + 8T_{31}^{3} - 32T_{31}^{2} - 184T_{31} + 488 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 16 T^{6} + 84 T^{4} + \cdots + 100 \) Copy content Toggle raw display
$5$ \( T^{8} - 16 T^{6} + 36 T^{4} - 24 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( (T - 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 48 T^{6} + 608 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{8} - 56 T^{6} + 1124 T^{4} + \cdots + 28900 \) Copy content Toggle raw display
$17$ \( (T^{4} + 12 T^{3} + 40 T^{2} + 40 T + 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 72 T^{6} + 1572 T^{4} + \cdots + 39204 \) Copy content Toggle raw display
$23$ \( (T^{4} - 40 T^{2} + 128 T - 108)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 104 T^{6} + 632 T^{4} + \cdots + 144 \) Copy content Toggle raw display
$31$ \( (T^{4} + 8 T^{3} - 32 T^{2} - 184 T + 488)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 184 T^{6} + 10296 T^{4} + \cdots + 435600 \) Copy content Toggle raw display
$41$ \( (T^{4} + 4 T^{3} - 136 T^{2} - 536 T + 2280)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 288 T^{6} + \cdots + 10863616 \) Copy content Toggle raw display
$47$ \( (T^{4} - 12 T^{3} - 8 T^{2} + 232 T - 376)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 72 T^{6} + 1688 T^{4} + \cdots + 4624 \) Copy content Toggle raw display
$59$ \( T^{8} - 248 T^{6} + 21668 T^{4} + \cdots + 9771876 \) Copy content Toggle raw display
$61$ \( T^{8} - 112 T^{6} + 4164 T^{4} + \cdots + 161604 \) Copy content Toggle raw display
$67$ \( T^{8} - 288 T^{6} + 19328 T^{4} + \cdots + 495616 \) Copy content Toggle raw display
$71$ \( (T^{4} + 28 T^{3} + 144 T^{2} + \cdots - 12080)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 24 T^{3} + 152 T^{2} + 224 T - 176)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 12 T^{3} - 160 T^{2} - 1712 T - 2160)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 16 T^{6} + 84 T^{4} + \cdots + 100 \) Copy content Toggle raw display
$89$ \( (T^{4} + 12 T^{3} - 144 T^{2} - 1808 T - 1136)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 16 T^{3} - 24 T^{2} - 648 T + 712)^{2} \) Copy content Toggle raw display
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