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Properties

Label 7168.2.a.bk
Level $7168$
Weight $2$
Character orbit 7168.a
Self dual yes
Analytic conductor $57.237$
Analytic rank $0$
Dimension $12$
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] = [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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 14 x^{10} + 60 x^{9} + 71 x^{8} - 312 x^{7} - 164 x^{6} + 648 x^{5} + 167 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} - \beta_{5} q^{5} - q^{7} + (\beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} - \beta_{5} q^{5} - q^{7} + (\beta_{2} - \beta_1 + 1) q^{9} + ( - \beta_{7} - \beta_{3} + 1) q^{11} + ( - \beta_{11} - \beta_1) q^{13} + (\beta_{10} - \beta_{5} + \beta_{3}) q^{15} + (\beta_{11} - \beta_{7} + \cdots - \beta_{4}) q^{17}+ \cdots + ( - 2 \beta_{11} - 2 \beta_{10} + \cdots + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{3} - 12 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 8 q^{3} - 12 q^{7} + 12 q^{9} + 16 q^{11} + 8 q^{19} - 8 q^{21} + 12 q^{25} + 32 q^{27} - 24 q^{29} + 16 q^{33} - 8 q^{37} + 32 q^{39} + 16 q^{41} + 16 q^{43} + 12 q^{49} + 16 q^{51} - 24 q^{53} + 16 q^{57} + 40 q^{59} - 12 q^{63} - 8 q^{65} + 32 q^{67} - 8 q^{73} + 40 q^{75} - 16 q^{77} + 28 q^{81} + 56 q^{83} - 8 q^{89} + 32 q^{93} + 32 q^{95} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 14 x^{10} + 60 x^{9} + 71 x^{8} - 312 x^{7} - 164 x^{6} + 648 x^{5} + 167 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} - 6\nu^{9} - \nu^{8} + 56\nu^{7} - 42\nu^{6} - 164\nu^{5} + 106\nu^{4} + 200\nu^{3} - 31\nu^{2} - 70\nu - 1 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - 5 \nu^{10} - 7 \nu^{9} + 59 \nu^{8} - 2 \nu^{7} - 238 \nu^{6} + 78 \nu^{5} + 402 \nu^{4} + \cdots + 59 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{10} + 5 \nu^{9} + 8 \nu^{8} - 60 \nu^{7} - 14 \nu^{6} + 242 \nu^{5} + 4 \nu^{4} - 364 \nu^{3} + \cdots + 4 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{11} + 3 \nu^{10} + 15 \nu^{9} - 37 \nu^{8} - 94 \nu^{7} + 146 \nu^{6} + 306 \nu^{5} - 174 \nu^{4} + \cdots + 27 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} - 3 \nu^{10} - 17 \nu^{9} + 47 \nu^{8} + 102 \nu^{7} - 242 \nu^{6} - 254 \nu^{5} + 458 \nu^{4} + \cdots + 31 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{11} - 4 \nu^{10} - 13 \nu^{9} + 54 \nu^{8} + 70 \nu^{7} - 256 \nu^{6} - 206 \nu^{5} + 484 \nu^{4} + \cdots + 22 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{11} + 3 \nu^{10} + 17 \nu^{9} - 43 \nu^{8} - 118 \nu^{7} + 210 \nu^{6} + 406 \nu^{5} + \cdots - 35 ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2 \nu^{11} + 9 \nu^{10} + 22 \nu^{9} - 119 \nu^{8} - 84 \nu^{7} + 526 \nu^{6} + 172 \nu^{5} + \cdots + 1 ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} - 4 \nu^{10} - 14 \nu^{9} + 61 \nu^{8} + 66 \nu^{7} - 316 \nu^{6} - 120 \nu^{5} + 630 \nu^{4} + \cdots + 45 ) / 8 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{5} + \beta_{4} + \beta_{2} + 7\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{11} + 2 \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - 2 \beta_{5} + 3 \beta_{4} + \beta_{3} + \cdots + 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{11} + 14 \beta_{9} + 5 \beta_{8} + 3 \beta_{7} + 2 \beta_{6} - 12 \beta_{5} + 14 \beta_{4} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 17 \beta_{11} + 36 \beta_{9} + 23 \beta_{8} + 15 \beta_{7} + 13 \beta_{6} - 30 \beta_{5} + 45 \beta_{4} + \cdots + 149 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 55 \beta_{11} + 2 \beta_{10} + 163 \beta_{9} + 95 \beta_{8} + 47 \beta_{7} + 34 \beta_{6} + \cdots + 314 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 226 \beta_{11} + 8 \beta_{10} + 480 \beta_{9} + 358 \beta_{8} + 182 \beta_{7} + 148 \beta_{6} + \cdots + 1305 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 754 \beta_{11} + 48 \beta_{10} + 1840 \beta_{9} + 1338 \beta_{8} + 590 \beta_{7} + 438 \beta_{6} + \cdots + 3340 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2770 \beta_{11} + 184 \beta_{10} + 5788 \beta_{9} + 4746 \beta_{8} + 2106 \beta_{7} + 1640 \beta_{6} + \cdots + 12291 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 9314 \beta_{11} + 788 \beta_{10} + 20601 \beta_{9} + 16846 \beta_{8} + 6950 \beta_{7} + 5138 \beta_{6} + \cdots + 35134 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
3.30856
2.73903
2.68373
1.68305
0.822055
0.571033
−0.0108380
−0.513747
−1.16999
−1.50510
−2.29773
−2.31007
0 −2.30856 0 −0.653822 0 −1.00000 0 2.32947 0
1.2 0 −1.73903 0 2.27204 0 −1.00000 0 0.0242347 0
1.3 0 −1.68373 0 0.524564 0 −1.00000 0 −0.165048 0
1.4 0 −0.683046 0 −3.22341 0 −1.00000 0 −2.53345 0
1.5 0 0.177945 0 1.87282 0 −1.00000 0 −2.96834 0
1.6 0 0.428967 0 −3.38043 0 −1.00000 0 −2.81599 0
1.7 0 1.01084 0 −0.303370 0 −1.00000 0 −1.97821 0
1.8 0 1.51375 0 4.20887 0 −1.00000 0 −0.708572 0
1.9 0 2.16999 0 −2.23683 0 −1.00000 0 1.70884 0
1.10 0 2.50510 0 2.54951 0 −1.00000 0 3.27553 0
1.11 0 3.29773 0 −3.07015 0 −1.00000 0 7.87499 0
1.12 0 3.31007 0 1.44021 0 −1.00000 0 7.95654 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.bk 12
4.b odd 2 1 7168.2.a.bh 12
8.b even 2 1 7168.2.a.bg 12
8.d odd 2 1 7168.2.a.bl 12
32.g even 8 2 3584.2.m.bk ✓ 24
32.g even 8 2 3584.2.m.bn yes 24
32.h odd 8 2 3584.2.m.bl yes 24
32.h odd 8 2 3584.2.m.bm yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.m.bk ✓ 24 32.g even 8 2
3584.2.m.bl yes 24 32.h odd 8 2
3584.2.m.bm yes 24 32.h odd 8 2
3584.2.m.bn yes 24 32.g even 8 2
7168.2.a.bg 12 8.b even 2 1
7168.2.a.bh 12 4.b odd 2 1
7168.2.a.bk 12 1.a even 1 1 trivial
7168.2.a.bl 12 8.d odd 2 1

Hecke kernels

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

\( T_{3}^{12} - 8 T_{3}^{11} + 8 T_{3}^{10} + 80 T_{3}^{9} - 184 T_{3}^{8} - 208 T_{3}^{7} + 816 T_{3}^{6} + \cdots + 32 \) Copy content Toggle raw display
\( T_{5}^{12} - 36 T_{5}^{10} + 460 T_{5}^{8} - 80 T_{5}^{7} - 2608 T_{5}^{6} + 1024 T_{5}^{5} + 6136 T_{5}^{4} + \cdots + 512 \) Copy content Toggle raw display
\( T_{11}^{12} - 16 T_{11}^{11} + 56 T_{11}^{10} + 352 T_{11}^{9} - 2808 T_{11}^{8} + 3328 T_{11}^{7} + \cdots + 8192 \) Copy content Toggle raw display
\( T_{13}^{12} - 84 T_{13}^{10} + 48 T_{13}^{9} + 2604 T_{13}^{8} - 3056 T_{13}^{7} - 35408 T_{13}^{6} + \cdots - 397312 \) Copy content Toggle raw display
\( T_{17}^{12} - 128 T_{17}^{10} + 5792 T_{17}^{8} - 896 T_{17}^{7} - 112000 T_{17}^{6} + 61440 T_{17}^{5} + \cdots + 526336 \) Copy content Toggle raw display
\( T_{23}^{12} - 112 T_{23}^{10} + 192 T_{23}^{9} + 4304 T_{23}^{8} - 13184 T_{23}^{7} - 60160 T_{23}^{6} + \cdots + 2048 \) Copy content Toggle raw display
\( T_{31}^{12} - 176 T_{31}^{10} + 9632 T_{31}^{8} - 2176 T_{31}^{7} - 209280 T_{31}^{6} + 76288 T_{31}^{5} + \cdots - 96256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 8 T^{11} + \cdots + 32 \) Copy content Toggle raw display
$5$ \( T^{12} - 36 T^{10} + \cdots + 512 \) Copy content Toggle raw display
$7$ \( (T + 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 16 T^{11} + \cdots + 8192 \) Copy content Toggle raw display
$13$ \( T^{12} - 84 T^{10} + \cdots - 397312 \) Copy content Toggle raw display
$17$ \( T^{12} - 128 T^{10} + \cdots + 526336 \) Copy content Toggle raw display
$19$ \( T^{12} - 8 T^{11} + \cdots + 8224 \) Copy content Toggle raw display
$23$ \( T^{12} - 112 T^{10} + \cdots + 2048 \) Copy content Toggle raw display
$29$ \( T^{12} + 24 T^{11} + \cdots - 192448 \) Copy content Toggle raw display
$31$ \( T^{12} - 176 T^{10} + \cdots - 96256 \) Copy content Toggle raw display
$37$ \( T^{12} + 8 T^{11} + \cdots + 417856 \) Copy content Toggle raw display
$41$ \( T^{12} - 16 T^{11} + \cdots - 526336 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 219922432 \) Copy content Toggle raw display
$47$ \( T^{12} - 272 T^{10} + \cdots - 6191104 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 2680538048 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 108119072 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 1712357888 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 418906112 \) Copy content Toggle raw display
$71$ \( T^{12} - 448 T^{10} + \cdots + 32505856 \) Copy content Toggle raw display
$73$ \( T^{12} + 8 T^{11} + \cdots - 2625536 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 8560443392 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 104617958944 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 20967452672 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 1771931648 \) Copy content Toggle raw display
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