Properties

Label 7168.2.a.bf
Level $7168$
Weight $2$
Character orbit 7168.a
Self dual yes
Analytic conductor $57.237$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.9433055232.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 6x^{6} + 32x^{5} + 9x^{4} - 76x^{3} - 4x^{2} + 48x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1792)
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + (\beta_{2} + 1) q^{5} + q^{7} + (\beta_{6} - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} + (\beta_{2} + 1) q^{5} + q^{7} + (\beta_{6} - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{9} + ( - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + 1) q^{11} + ( - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 2) q^{13} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} - 1) q^{15} + (\beta_{7} - 2 \beta_{6} - \beta_{4} + \beta_{2}) q^{17} + ( - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{19} - \beta_{5} q^{21} + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} + 3 \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{23} + ( - \beta_{7} + \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 1) q^{25} + (\beta_{7} - 2 \beta_{6} - \beta_{4} - 3 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{27} + ( - \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{2} + 2 \beta_1 + 1) q^{29} + ( - \beta_{6} - \beta_{5} + 3 \beta_{4} - \beta_1 - 1) q^{31} + ( - 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_1) q^{33} + (\beta_{2} + 1) q^{35} + (2 \beta_{7} + 2 \beta_{4} + \beta_1 + 2) q^{37} + ( - 3 \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{39} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{5} - \beta_{4} - 3 \beta_{3} + \beta_{2} - \beta_1 - 2) q^{41} + (\beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{43} + (3 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 8) q^{45} + (\beta_{6} - \beta_{5} - \beta_{4} - \beta_1 + 3) q^{47} + q^{49} + (\beta_{7} + \beta_{6} + \beta_{5} - 4 \beta_{4} + 3 \beta_{2} + 3 \beta_1 + 5) q^{51} + (2 \beta_{6} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 6) q^{53} + ( - 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + 2) q^{55} + (\beta_{7} + \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_1) q^{57} + ( - \beta_{7} + 3 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{59} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{61} + (\beta_{6} - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{63} + (2 \beta_{7} - \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 5) q^{65} + (2 \beta_{7} + 3 \beta_{6} + \beta_{4} - 3 \beta_{3} - \beta_{2} - \beta_1 + 6) q^{67} + (\beta_{6} - \beta_{5} + 5 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{69} + (\beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{71} + ( - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 3 \beta_1 + 2) q^{73} + ( - \beta_{7} - 4 \beta_{6} - \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + \beta_{2} - 3 \beta_1 - 6) q^{75} + ( - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + 1) q^{77} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} + \beta_{3} + \beta_1 + 2) q^{79} + (\beta_{7} + 4 \beta_{6} + \beta_{5} + 2 \beta_{4} + 5 \beta_{3} + 4 \beta_1 + 5) q^{81} + (2 \beta_{7} - \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{83} + (\beta_{7} - 3 \beta_{6} + \beta_{5} - 6 \beta_{4} + \beta_{2} - 3 \beta_1 + 1) q^{85} + (4 \beta_{7} - 3 \beta_{6} + \beta_{5} - 3 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 3) q^{87} + (2 \beta_{4} + \beta_{3} + \beta_1 + 2) q^{89} + ( - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 2) q^{91} + ( - 5 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} - \beta_{2} + \cdots + 1) q^{93}+ \cdots + ( - 3 \beta_{7} + 2 \beta_{6} - \beta_{5} + 3 \beta_{4} + \beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5} + 8 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{5} + 8 q^{7} + 12 q^{9} + 12 q^{11} + 20 q^{13} + 4 q^{17} + 4 q^{19} + 8 q^{23} + 12 q^{25} - 12 q^{27} + 8 q^{29} - 4 q^{31} + 8 q^{33} + 8 q^{35} + 8 q^{37} - 16 q^{39} - 12 q^{41} - 4 q^{43} + 52 q^{45} + 20 q^{47} + 8 q^{49} + 32 q^{51} + 40 q^{53} + 24 q^{55} - 4 q^{57} + 4 q^{59} - 8 q^{61} + 12 q^{63} + 36 q^{65} + 28 q^{67} + 4 q^{69} - 16 q^{71} + 16 q^{73} - 28 q^{75} + 12 q^{77} + 20 q^{81} - 8 q^{83} + 16 q^{85} + 20 q^{87} + 16 q^{89} + 20 q^{91} + 16 q^{93} - 40 q^{95} - 36 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 6x^{6} + 32x^{5} + 9x^{4} - 76x^{3} - 4x^{2} + 48x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 6\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 4\nu^{6} - 2\nu^{5} + 20\nu^{4} - 15\nu^{3} - 8\nu^{2} + 32\nu - 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 3\nu^{5} - 6\nu^{4} + 17\nu^{3} + 9\nu^{2} - 18\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 3\nu^{5} + 6\nu^{4} - 17\nu^{3} - 7\nu^{2} + 16\nu - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 4\nu^{6} - 4\nu^{5} + 26\nu^{4} - \nu^{3} - 42\nu^{2} + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} + 4\nu^{5} - 26\nu^{4} + 5\nu^{3} + 38\nu^{2} - 16\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} + 6\nu^{5} - 30\nu^{4} - 13\nu^{3} + 62\nu^{2} + 20\nu - 24 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} - \beta_{5} + \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{4} + 2\beta_{3} + \beta_{2} + \beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{7} + 7\beta_{6} - 3\beta_{5} + 2\beta_{4} + 2\beta_{3} + 5\beta_{2} + 5\beta _1 + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -9\beta_{7} + 13\beta_{6} - 5\beta_{5} + 14\beta_{4} + 14\beta_{3} + 9\beta_{2} + 13\beta _1 + 43 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -29\beta_{7} + 55\beta_{6} - 7\beta_{5} + 22\beta_{4} + 22\beta_{3} + 33\beta_{2} + 41\beta _1 + 99 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -65\beta_{7} + 133\beta_{6} - 9\beta_{5} + 98\beta_{4} + 102\beta_{3} + 77\beta_{2} + 125\beta _1 + 323 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -189\beta_{7} + 463\beta_{6} + 29\beta_{5} + 202\beta_{4} + 218\beta_{3} + 253\beta_{2} + 373\beta _1 + 859 ) / 2 \) 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
3.02908
−1.38887
0.177920
−2.02908
−1.21236
2.21236
2.38887
0.822080
0 −3.42180 0 3.59402 0 1.00000 0 8.70871 0
1.2 0 −2.09974 0 2.59647 0 1.00000 0 1.40890 0
1.3 0 −1.67251 0 −2.65618 0 1.00000 0 −0.202696 0
1.4 0 −0.242103 0 −0.379610 0 1.00000 0 −2.94139 0
1.5 0 0.463900 0 −1.98268 0 1.00000 0 −2.78480 0
1.6 0 1.78579 0 4.18248 0 1.00000 0 0.189043 0
1.7 0 2.29954 0 1.65321 0 1.00000 0 2.28788 0
1.8 0 2.88693 0 0.992279 0 1.00000 0 5.33435 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

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 7168.2.a.bf 8
4.b odd 2 1 7168.2.a.be 8
8.b even 2 1 7168.2.a.bb 8
8.d odd 2 1 7168.2.a.ba 8
32.g even 8 2 1792.2.m.e 16
32.g even 8 2 1792.2.m.h yes 16
32.h odd 8 2 1792.2.m.f yes 16
32.h odd 8 2 1792.2.m.g yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1792.2.m.e 16 32.g even 8 2
1792.2.m.f yes 16 32.h odd 8 2
1792.2.m.g yes 16 32.h odd 8 2
1792.2.m.h yes 16 32.g even 8 2
7168.2.a.ba 8 8.d odd 2 1
7168.2.a.bb 8 8.b even 2 1
7168.2.a.be 8 4.b odd 2 1
7168.2.a.bf 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7168))\):

\( T_{3}^{8} - 18T_{3}^{6} + 4T_{3}^{5} + 94T_{3}^{4} - 24T_{3}^{3} - 152T_{3}^{2} + 32T_{3} + 16 \) Copy content Toggle raw display
\( T_{5}^{8} - 8T_{5}^{7} + 6T_{5}^{6} + 84T_{5}^{5} - 162T_{5}^{4} - 176T_{5}^{3} + 512T_{5}^{2} - 128T_{5} - 128 \) Copy content Toggle raw display
\( T_{11}^{8} - 12T_{11}^{7} + 32T_{11}^{6} + 88T_{11}^{5} - 492T_{11}^{4} + 640T_{11}^{3} - 192T_{11}^{2} - 128T_{11} + 64 \) Copy content Toggle raw display
\( T_{13}^{8} - 20T_{13}^{7} + 138T_{13}^{6} - 284T_{13}^{5} - 882T_{13}^{4} + 4960T_{13}^{3} - 6784T_{13}^{2} + 4096 \) Copy content Toggle raw display
\( T_{17}^{8} - 4T_{17}^{7} - 100T_{17}^{6} + 272T_{17}^{5} + 3528T_{17}^{4} - 4992T_{17}^{3} - 46272T_{17}^{2} + 15872T_{17} + 116608 \) Copy content Toggle raw display
\( T_{23}^{8} - 8T_{23}^{7} - 84T_{23}^{6} + 688T_{23}^{5} + 1428T_{23}^{4} - 15680T_{23}^{3} + 12800T_{23}^{2} + 56064T_{23} - 70592 \) Copy content Toggle raw display
\( T_{31}^{8} + 4 T_{31}^{7} - 132 T_{31}^{6} - 576 T_{31}^{5} + 5304 T_{31}^{4} + 25792 T_{31}^{3} - 59584 T_{31}^{2} - 366848 T_{31} - 251648 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 18 T^{6} + 4 T^{5} + 94 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{8} - 8 T^{7} + 6 T^{6} + 84 T^{5} + \cdots - 128 \) Copy content Toggle raw display
$7$ \( (T - 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 12 T^{7} + 32 T^{6} + 88 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{8} - 20 T^{7} + 138 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{7} - 100 T^{6} + \cdots + 116608 \) Copy content Toggle raw display
$19$ \( T^{8} - 4 T^{7} - 46 T^{6} + 140 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{8} - 8 T^{7} - 84 T^{6} + \cdots - 70592 \) Copy content Toggle raw display
$29$ \( T^{8} - 8 T^{7} - 96 T^{6} + \cdots + 91792 \) Copy content Toggle raw display
$31$ \( T^{8} + 4 T^{7} - 132 T^{6} + \cdots - 251648 \) Copy content Toggle raw display
$37$ \( T^{8} - 8 T^{7} - 168 T^{6} + \cdots + 695056 \) Copy content Toggle raw display
$41$ \( T^{8} + 12 T^{7} - 76 T^{6} + \cdots - 9344 \) Copy content Toggle raw display
$43$ \( T^{8} + 4 T^{7} - 120 T^{6} + \cdots + 45952 \) Copy content Toggle raw display
$47$ \( T^{8} - 20 T^{7} + 68 T^{6} + \cdots - 18176 \) Copy content Toggle raw display
$53$ \( T^{8} - 40 T^{7} + 520 T^{6} + \cdots + 365584 \) Copy content Toggle raw display
$59$ \( T^{8} - 4 T^{7} - 206 T^{6} + \cdots + 407824 \) Copy content Toggle raw display
$61$ \( T^{8} + 8 T^{7} - 122 T^{6} + \cdots - 98432 \) Copy content Toggle raw display
$67$ \( T^{8} - 28 T^{7} + 16 T^{6} + \cdots - 51056384 \) Copy content Toggle raw display
$71$ \( T^{8} + 16 T^{7} - 16 T^{6} + \cdots - 8192 \) Copy content Toggle raw display
$73$ \( T^{8} - 16 T^{7} - 168 T^{6} + \cdots + 7618816 \) Copy content Toggle raw display
$79$ \( T^{8} - 288 T^{6} - 704 T^{5} + \cdots - 4822784 \) Copy content Toggle raw display
$83$ \( T^{8} + 8 T^{7} - 418 T^{6} + \cdots - 20041712 \) Copy content Toggle raw display
$89$ \( (T^{4} - 8 T^{3} - 16 T^{2} + 32 T + 16)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 36 T^{7} + 116 T^{6} + \cdots - 2571392 \) Copy content Toggle raw display
show more
show less