↑

Properties

Label 7920.2.f.g
Level $7920$
Weight $2$
Character orbit 7920.f
Analytic conductor $63.242$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4x^{14} + 8x^{12} - 20x^{10} + 49x^{8} - 80x^{6} + 128x^{4} - 256x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 495)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{5} + (\beta_{14} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{5} + (\beta_{14} + \beta_1) q^{7} + ( - \beta_{7} + \beta_{5} + \beta_{4}) q^{11} + \beta_1 q^{13} + (\beta_{9} - \beta_{8} - \beta_{6}) q^{17} + ( - \beta_{11} - \beta_1) q^{19} + (\beta_{12} + \beta_{4}) q^{23} - q^{25} + (2 \beta_{9} - \beta_{8} + \cdots + \beta_{6}) q^{29}+ \cdots + ( - \beta_{13} + 3 \beta_{3} - \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{25} - 32 q^{37} - 16 q^{49} + 16 q^{55} + 32 q^{67} - 64 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4x^{14} + 8x^{12} - 20x^{10} + 49x^{8} - 80x^{6} + 128x^{4} - 256x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{15} + 3\nu^{13} - 12\nu^{9} + 5\nu^{7} + 23\nu^{5} + 4\nu^{3} - 160\nu ) / 160 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{14} - 28\nu^{12} + 56\nu^{10} - 100\nu^{8} + 341\nu^{6} - 600\nu^{4} + 176\nu^{2} - 1152 ) / 320 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{14} - 12\nu^{12} + 24\nu^{10} - 100\nu^{8} + 149\nu^{6} - 200\nu^{4} + 464\nu^{2} - 768 ) / 320 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19\nu^{14} - 12\nu^{12} + 40\nu^{10} - 188\nu^{8} + 355\nu^{6} - 496\nu^{4} + 976\nu^{2} - 1920 ) / 1088 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -53\nu^{14} + 114\nu^{12} - 176\nu^{10} + 596\nu^{8} - 1341\nu^{6} + 1006\nu^{4} - 2608\nu^{2} + 6272 ) / 2720 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -61\nu^{15} + 26\nu^{13} + 72\nu^{11} + 532\nu^{9} + 27\nu^{7} - 138\nu^{5} - 1752\nu^{3} - 4544\nu ) / 5440 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 141\nu^{15} - 336\nu^{13} + 168\nu^{11} - 1252\nu^{9} + 4653\nu^{7} - 5932\nu^{5} + 4752\nu^{3} - 28736\nu ) / 10880 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 151 \nu^{15} - 396 \nu^{13} + 1048 \nu^{11} - 3212 \nu^{9} + 3623 \nu^{7} - 7392 \nu^{5} + \cdots - 27456 \nu ) / 10880 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 183\nu^{15} - 588\nu^{13} + 1144\nu^{11} - 2956\nu^{9} + 4679\nu^{7} - 5536\nu^{5} + 7296\nu^{3} - 19008\nu ) / 10880 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -13\nu^{15} + 36\nu^{13} - 40\nu^{11} + 196\nu^{9} - 445\nu^{7} + 256\nu^{5} - 1152\nu^{3} + 2240\nu ) / 640 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 7\nu^{15} - 14\nu^{13} + 40\nu^{11} - 124\nu^{9} + 255\nu^{7} - 354\nu^{5} + 648\nu^{3} - 960\nu ) / 320 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 477\nu^{14} - 1196\nu^{12} + 2264\nu^{10} - 6724\nu^{8} + 15469\nu^{6} - 20104\nu^{4} + 47952\nu^{2} - 78208 ) / 5440 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 15\nu^{14} - 32\nu^{12} + 24\nu^{10} - 140\nu^{8} + 239\nu^{6} - 340\nu^{4} + 784\nu^{2} - 1408 ) / 160 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 31\nu^{15} - 72\nu^{13} + 120\nu^{11} - 332\nu^{9} + 735\nu^{7} - 892\nu^{5} + 2064\nu^{3} - 3520\nu ) / 640 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -269\nu^{14} + 492\nu^{12} - 1368\nu^{10} + 2948\nu^{8} - 4253\nu^{6} + 7688\nu^{4} - 13904\nu^{2} + 21056 ) / 2720 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{12} + 2\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{14} - \beta_{10} - 3\beta_{9} + \beta_{8} - 2\beta_{7} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{13} - 10\beta_{5} - 4\beta_{4} - 2\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{14} - 2\beta_{11} - 5\beta_{10} + \beta_{9} - 3\beta_{8} - 6\beta_{7} + 2\beta_{6} + 3\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{15} - 2\beta_{13} + \beta_{12} - 14\beta_{5} + 5\beta_{4} - 5\beta_{3} + \beta_{2} + 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 9\beta_{11} - 6\beta_{10} - 2\beta_{9} - 13\beta_{8} - 5\beta_{7} + 3\beta_{6} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -\beta_{15} + 6\beta_{12} - 40\beta_{3} - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 20\beta_{14} - 3\beta_{11} + 8\beta_{10} - 16\beta_{9} - 13\beta_{8} - 9\beta_{7} + 15\beta_{6} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -12\beta_{15} - 12\beta_{13} + 3\beta_{12} - 18\beta_{5} - 19\beta_{4} - 19\beta_{3} - 3\beta_{2} - 18 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 27\beta_{14} + 4\beta_{11} + 21\beta_{10} + 15\beta_{9} - 17\beta_{8} - 50\beta_{7} + 44\beta_{6} + 25\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( -23\beta_{13} + 10\beta_{5} + 140\beta_{4} - 6\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 11\beta_{14} + 62\beta_{11} + 33\beta_{10} - 45\beta_{9} - 29\beta_{8} - 30\beta_{7} + 18\beta_{6} + 85\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -22\beta_{15} + 22\beta_{13} - 17\beta_{12} - 58\beta_{5} + 211\beta_{4} - 211\beta_{3} - 17\beta_{2} + 58 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 180\beta_{14} - 61\beta_{11} + 146\beta_{10} - 58\beta_{9} - 99\beta_{8} - 7\beta_{7} - 95\beta_{6} + 153\beta_1 ) / 4 \) 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
1.41088 − 0.0971138i
0.608209 + 1.27675i
0.928970 + 1.06631i
1.33286 − 0.472728i
−1.33286 + 0.472728i
−0.928970 − 1.06631i
−0.608209 − 1.27675i
−1.41088 + 0.0971138i
−1.41088 − 0.0971138i
−0.608209 + 1.27675i
−0.928970 + 1.06631i
−1.33286 − 0.472728i
1.33286 + 0.472728i
0.928970 − 1.06631i
0.608209 − 1.27675i
1.41088 + 0.0971138i
0 0 0 − 1.00000i 0 − 4.95437i 0 0 0
3761.2 0 0 0 − 1.00000i 0 − 2.16187i 0 0 0
3761.3 0 0 0 − 1.00000i 0 − 1.66371i 0 0 0
3761.4 0 0 0 − 1.00000i 0 − 0.112236i 0 0 0
3761.5 0 0 0 − 1.00000i 0 0.112236i 0 0 0
3761.6 0 0 0 − 1.00000i 0 1.66371i 0 0 0
3761.7 0 0 0 − 1.00000i 0 2.16187i 0 0 0
3761.8 0 0 0 − 1.00000i 0 4.95437i 0 0 0
3761.9 0 0 0 1.00000i 0 − 4.95437i 0 0 0
3761.10 0 0 0 1.00000i 0 − 2.16187i 0 0 0
3761.11 0 0 0 1.00000i 0 − 1.66371i 0 0 0
3761.12 0 0 0 1.00000i 0 − 0.112236i 0 0 0
3761.13 0 0 0 1.00000i 0 0.112236i 0 0 0
3761.14 0 0 0 1.00000i 0 1.66371i 0 0 0
3761.15 0 0 0 1.00000i 0 2.16187i 0 0 0
3761.16 0 0 0 1.00000i 0 4.95437i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3761.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
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.g 16
3.b odd 2 1 inner 7920.2.f.g 16
4.b odd 2 1 495.2.f.a ✓ 16
11.b odd 2 1 inner 7920.2.f.g 16
12.b even 2 1 495.2.f.a ✓ 16
20.d odd 2 1 2475.2.f.h 16
20.e even 4 1 2475.2.d.b 16
20.e even 4 1 2475.2.d.c 16
33.d even 2 1 inner 7920.2.f.g 16
44.c even 2 1 495.2.f.a ✓ 16
60.h even 2 1 2475.2.f.h 16
60.l odd 4 1 2475.2.d.b 16
60.l odd 4 1 2475.2.d.c 16
132.d odd 2 1 495.2.f.a ✓ 16
220.g even 2 1 2475.2.f.h 16
220.i odd 4 1 2475.2.d.b 16
220.i odd 4 1 2475.2.d.c 16
660.g odd 2 1 2475.2.f.h 16
660.q even 4 1 2475.2.d.b 16
660.q even 4 1 2475.2.d.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.f.a ✓ 16 4.b odd 2 1
495.2.f.a ✓ 16 12.b even 2 1
495.2.f.a ✓ 16 44.c even 2 1
495.2.f.a ✓ 16 132.d odd 2 1
2475.2.d.b 16 20.e even 4 1
2475.2.d.b 16 60.l odd 4 1
2475.2.d.b 16 220.i odd 4 1
2475.2.d.b 16 660.q even 4 1
2475.2.d.c 16 20.e even 4 1
2475.2.d.c 16 60.l odd 4 1
2475.2.d.c 16 220.i odd 4 1
2475.2.d.c 16 660.q even 4 1
2475.2.f.h 16 20.d odd 2 1
2475.2.f.h 16 60.h even 2 1
2475.2.f.h 16 220.g even 2 1
2475.2.f.h 16 660.g odd 2 1
7920.2.f.g 16 1.a even 1 1 trivial
7920.2.f.g 16 3.b odd 2 1 inner
7920.2.f.g 16 11.b odd 2 1 inner
7920.2.f.g 16 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 \(S_{2}^{\mathrm{new}}(7920, [\chi])\):

\( T_{7}^{8} + 32T_{7}^{6} + 196T_{7}^{4} + 320T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{17}^{8} - 80T_{17}^{6} + 1924T_{17}^{4} - 12960T_{17}^{2} + 1156 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} + 32 T^{6} + 196 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 20 T^{6} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 32 T^{6} + 196 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 80 T^{6} + \cdots + 1156)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 80 T^{6} + \cdots + 73984)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 112 T^{6} + \cdots + 295936)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 208 T^{6} + \cdots + 4804864)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 108 T^{2} + \cdots + 2372)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 8 T^{3} + \cdots - 496)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 176 T^{6} + \cdots + 246016)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 208 T^{6} + \cdots + 228484)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 256 T^{6} + \cdots + 2560000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 320 T^{6} + \cdots + 135424)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 152 T^{6} + \cdots + 8464)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 448 T^{6} + \cdots + 80856064)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 8 T^{3} + \cdots - 736)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 248 T^{6} + \cdots + 4624)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 304 T^{6} + \cdots + 334084)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 304 T^{6} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 176 T^{6} + \cdots + 2116)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 352 T^{6} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 160 T^{2} + \cdots - 1264)^{4} \) Copy content Toggle raw display
show more
show less