↑

Properties

Label 7920.2.f.e
Level $7920$
Weight $2$
Character orbit 7920.f
Analytic conductor $63.242$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7920,2,Mod(3761,7920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7920, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7920.3761");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7920.f (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(63.2415184009\)
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: \( x^{12} - 4 x^{11} + 2 x^{10} + 8 x^{9} + 2 x^{8} + 12 x^{7} + 40 x^{6} + 4 x^{5} + 49 x^{4} - 56 x^{3} + \cdots + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 3960)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{5} + (\beta_{11} + \beta_{10} + \cdots - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{5} + (\beta_{11} + \beta_{10} + \cdots - \beta_1) q^{7}+ \cdots + ( - 4 \beta_{11} + 2 \beta_{10} + \cdots + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{11} - 32 q^{17} - 12 q^{25} + 24 q^{29} + 16 q^{35} + 16 q^{37} + 16 q^{41} - 36 q^{49} + 12 q^{55} + 8 q^{65} - 48 q^{67} - 4 q^{77} + 56 q^{83} - 88 q^{91} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 2 x^{10} + 8 x^{9} + 2 x^{8} + 12 x^{7} + 40 x^{6} + 4 x^{5} + 49 x^{4} - 56 x^{3} + \cdots + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 57365257 \nu^{11} + 234079984 \nu^{10} - 135078180 \nu^{9} - 426165507 \nu^{8} + \cdots + 956458910 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 115966669 \nu^{11} + 849598162 \nu^{10} - 1571021987 \nu^{9} - 911824391 \nu^{8} + \cdots + 4905249237 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 196292730 \nu^{11} - 773034542 \nu^{10} + 308733235 \nu^{9} + 1729623155 \nu^{8} + \cdots - 899780200 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 254894142 \nu^{11} - 1388552720 \nu^{10} + 1744677042 \nu^{9} + 2215282039 \nu^{8} + \cdots - 5940581884 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 373180 \nu^{11} + 1289467 \nu^{10} + 6628 \nu^{9} - 3154647 \nu^{8} - 2485934 \nu^{7} + \cdots + 798369 ) / 915349 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 434045 \nu^{11} - 1641264 \nu^{10} + 538704 \nu^{9} + 3498216 \nu^{8} + 1575564 \nu^{7} + \cdots - 2865682 ) / 1000927 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 575585 \nu^{11} + 2289018 \nu^{10} - 1021492 \nu^{9} - 4893056 \nu^{8} - 1247090 \nu^{7} + \cdots + 3721140 ) / 1301563 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 502501593 \nu^{11} - 1727828242 \nu^{10} + 25274663 \nu^{9} + 4007712809 \nu^{8} + \cdots - 3980416628 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 690670795 \nu^{11} + 2575702016 \nu^{10} - 649151894 \nu^{9} - 5859203412 \nu^{8} + \cdots + 1451028663 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 904570864 \nu^{11} + 3324521305 \nu^{10} - 681328203 \nu^{9} - 7627862381 \nu^{8} + \cdots + 2478907030 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1031914316 \nu^{11} - 3715390861 \nu^{10} + 451387854 \nu^{9} + 8894215017 \nu^{8} + \cdots - 3802112492 ) / 1092011357 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + \beta_{10} - \beta_{8} + 2\beta_{7} + \beta_{6} - 2\beta_{5} + \beta_{4} - 2\beta_{3} - 2\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{10} - 4 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - 10 \beta_{5} + 3 \beta_{4} - 5 \beta_{3} + \cdots + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 5 \beta_{11} + 15 \beta_{10} - 6 \beta_{9} - 9 \beta_{8} + 5 \beta_{7} + 2 \beta_{6} + \cdots - 26 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 20 \beta_{11} + 47 \beta_{10} - 27 \beta_{9} - 26 \beta_{8} - 2 \beta_{7} + 4 \beta_{6} - 99 \beta_{5} + \cdots - 38 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 83 \beta_{11} + 87 \beta_{10} - 50 \beta_{9} - 83 \beta_{8} - 54 \beta_{7} + 17 \beta_{6} + \cdots - 236 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 266 \beta_{11} + 93 \beta_{10} - 93 \beta_{9} - 192 \beta_{8} - 242 \beta_{7} + 22 \beta_{6} + \cdots - 952 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 781 \beta_{11} - 329 \beta_{10} - 329 \beta_{8} - 967 \beta_{7} - 1503 \beta_{4} + 2578 \beta_{3} + \cdots - 3216 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1832 \beta_{11} - 2706 \beta_{10} + 874 \beta_{9} - 3086 \beta_{7} - 214 \beta_{6} + 3746 \beta_{5} + \cdots - 9085 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 3079 \beta_{11} - 12323 \beta_{10} + 4918 \beta_{9} + 3079 \beta_{8} - 8652 \beta_{7} - 1247 \beta_{6} + \cdots - 21476 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 44656 \beta_{10} + 20010 \beta_{9} + 17448 \beta_{8} - 19494 \beta_{7} - 4918 \beta_{6} + \cdots - 35797 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7920\mathbb{Z}\right)^\times\).

\(n\) \(991\) \(3521\) \(5941\) \(6337\) \(6481\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3761.1
0.568642 − 1.37282i
−0.445186 + 1.07477i
−1.48663 − 0.615783i
0.169437 − 0.409058i
2.84980 + 1.18043i
0.343932 + 0.142462i
0.343932 − 0.142462i
2.84980 − 1.18043i
0.169437 + 0.409058i
−1.48663 + 0.615783i
−0.445186 − 1.07477i
0.568642 + 1.37282i
0 0 0 − 1.00000i 0 − 2.40128i 0 0 0
3761.2 0 0 0 − 1.00000i 0 − 1.51092i 0 0 0
3761.3 0 0 0 − 1.00000i 0 − 0.490014i 0 0 0
3761.4 0 0 0 − 1.00000i 0 3.66956i 0 0 0
3761.5 0 0 0 − 1.00000i 0 4.12597i 0 0 0
3761.6 0 0 0 − 1.00000i 0 4.60668i 0 0 0
3761.7 0 0 0 1.00000i 0 − 4.60668i 0 0 0
3761.8 0 0 0 1.00000i 0 − 4.12597i 0 0 0
3761.9 0 0 0 1.00000i 0 − 3.66956i 0 0 0
3761.10 0 0 0 1.00000i 0 0.490014i 0 0 0
3761.11 0 0 0 1.00000i 0 1.51092i 0 0 0
3761.12 0 0 0 1.00000i 0 2.40128i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3761.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7920.2.f.e 12
3.b odd 2 1 7920.2.f.d 12
4.b odd 2 1 3960.2.f.a ✓ 12
11.b odd 2 1 7920.2.f.d 12
12.b even 2 1 3960.2.f.d yes 12
33.d even 2 1 inner 7920.2.f.e 12
44.c even 2 1 3960.2.f.d yes 12
132.d odd 2 1 3960.2.f.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3960.2.f.a ✓ 12 4.b odd 2 1
3960.2.f.a ✓ 12 132.d odd 2 1
3960.2.f.d yes 12 12.b even 2 1
3960.2.f.d yes 12 44.c even 2 1
7920.2.f.d 12 3.b odd 2 1
7920.2.f.d 12 11.b odd 2 1
7920.2.f.e 12 1.a even 1 1 trivial
7920.2.f.e 12 33.d 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 \(S_{2}^{\mathrm{new}}(7920, [\chi])\):

\( T_{7}^{12} + 60T_{7}^{10} + 1320T_{7}^{8} + 12912T_{7}^{6} + 53716T_{7}^{4} + 76208T_{7}^{2} + 15376 \) Copy content Toggle raw display
\( T_{17}^{6} + 16T_{17}^{5} + 44T_{17}^{4} - 512T_{17}^{3} - 3758T_{17}^{2} - 8680T_{17} - 6664 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} + 60 T^{10} + \cdots + 15376 \) Copy content Toggle raw display
$11$ \( T^{12} - 4 T^{11} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( T^{12} + 68 T^{10} + \cdots + 10000 \) Copy content Toggle raw display
$17$ \( (T^{6} + 16 T^{5} + \cdots - 6664)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 176 T^{10} + \cdots + 62980096 \) Copy content Toggle raw display
$23$ \( T^{12} + 160 T^{10} + \cdots + 16777216 \) Copy content Toggle raw display
$29$ \( (T^{6} - 12 T^{5} + \cdots + 64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 68 T^{4} + \cdots - 4064)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 8 T^{5} + \cdots - 20704)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 8 T^{5} + \cdots + 57632)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 176 T^{10} + \cdots + 430336 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 695271424 \) Copy content Toggle raw display
$53$ \( T^{12} + 272 T^{10} + \cdots + 65536 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 1683080696896 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 17213440000 \) Copy content Toggle raw display
$67$ \( (T^{6} + 24 T^{5} + \cdots - 18688)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 1349533696 \) Copy content Toggle raw display
$73$ \( T^{12} + 360 T^{10} + \cdots + 24840256 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 13389266944 \) Copy content Toggle raw display
$83$ \( (T^{6} - 28 T^{5} + \cdots - 187904)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 2595291136 \) Copy content Toggle raw display
$97$ \( (T^{6} - 4 T^{5} + \cdots + 164800)^{2} \) Copy content Toggle raw display
show more
show less