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Properties

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

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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: \( x^{16} - 8x^{14} + 32x^{12} + 128x^{10} + 223x^{8} + 128x^{6} + 32x^{4} - 8x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 1980)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + ( - \beta_{10} - \beta_{7}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} + ( - \beta_{10} - \beta_{7}) q^{7} + (\beta_{9} - \beta_{8}) q^{11} + (\beta_{15} + \beta_{10}) q^{13} + ( - \beta_{13} - 2 \beta_{12}) q^{17} + (\beta_{15} - \beta_{14} + \beta_{7}) q^{19} + (\beta_{9} - \beta_{6} + 2 \beta_{3} + \beta_1) q^{23} - q^{25} + (\beta_{13} - 2 \beta_{12} - \beta_{11}) q^{29} + ( - \beta_{5} + \beta_{4} + 2) q^{31} - \beta_{11} q^{35} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{37} + (\beta_{13} + \beta_{12} + \cdots + \beta_{8}) q^{41}+ \cdots + (\beta_{5} + 3 \beta_{4} - \beta_{2} + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{25} + 32 q^{31} + 16 q^{49} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8x^{14} + 32x^{12} + 128x^{10} + 223x^{8} + 128x^{6} + 32x^{4} - 8x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 8\nu^{14} - 35\nu^{12} + 8\nu^{10} + 2080\nu^{8} + 5000\nu^{6} + 5286\nu^{4} + 1136\nu^{2} - 96 ) / 155 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16\nu^{14} - 132\nu^{12} + 543\nu^{10} + 1928\nu^{8} + 3025\nu^{6} + 1024\nu^{4} - 239\nu^{2} - 192 ) / 155 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 584 \nu^{14} + 4479 \nu^{12} - 17084 \nu^{10} - 81440 \nu^{8} - 152700 \nu^{6} - 111808 \nu^{4} + \cdots + 2608 ) / 4991 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3768 \nu^{14} + 30608 \nu^{12} - 122260 \nu^{10} - 486758 \nu^{8} - 689324 \nu^{6} - 233504 \nu^{4} + \cdots - 38621 ) / 24955 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4906 \nu^{14} - 40276 \nu^{12} + 164165 \nu^{10} + 604706 \nu^{8} + 917963 \nu^{6} + 310808 \nu^{4} + \cdots - 69973 ) / 24955 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 176\nu^{14} - 1416\nu^{12} + 5711\nu^{10} + 22144\nu^{8} + 38783\nu^{6} + 22280\nu^{4} + 7649\nu^{2} - 636 ) / 805 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3656 \nu^{15} + 25638 \nu^{13} - 88387 \nu^{11} - 580390 \nu^{9} - 1293949 \nu^{7} + \cdots + 17617 \nu ) / 24955 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 835 \nu^{15} - 6397 \nu^{13} + 24164 \nu^{11} + 118371 \nu^{9} + 212392 \nu^{7} + 134697 \nu^{5} + \cdots - 11024 \nu ) / 3565 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 74\nu^{14} - 595\nu^{12} + 2399\nu^{10} + 9320\nu^{8} + 16335\nu^{6} + 9758\nu^{4} + 3223\nu^{2} - 268 ) / 155 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 523 \nu^{15} + 4189 \nu^{13} - 16760 \nu^{11} - 66977 \nu^{9} - 114892 \nu^{7} - 66397 \nu^{5} + \cdots + 268 \nu ) / 1085 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13330 \nu^{15} - 107524 \nu^{13} + 433599 \nu^{11} + 1677976 \nu^{9} + 2861235 \nu^{7} + \cdots - 168005 \nu ) / 24955 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2249 \nu^{15} - 18181 \nu^{13} + 73544 \nu^{11} + 281261 \nu^{9} + 479740 \nu^{7} + 252757 \nu^{5} + \cdots - 27940 \nu ) / 3565 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 16290 \nu^{15} - 129670 \nu^{13} + 514821 \nu^{11} + 2116320 \nu^{9} + 3672987 \nu^{7} + \cdots - 205281 \nu ) / 24955 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 811 \nu^{15} - 6319 \nu^{13} + 24620 \nu^{11} + 109029 \nu^{9} + 203368 \nu^{7} + 142955 \nu^{5} + \cdots - 1184 \nu ) / 1085 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 23717 \nu^{15} - 187133 \nu^{13} + 738331 \nu^{11} + 3117733 \nu^{9} + 5625763 \nu^{7} + \cdots - 24499 \nu ) / 24955 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{8} + \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{9} + 5\beta_{6} + 3\beta_{5} + \beta_{4} - 4\beta_{3} - 5\beta_{2} - \beta _1 + 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8\beta_{15} - 9\beta_{14} - 4\beta_{13} + 3\beta_{12} + 3\beta_{10} + 3\beta_{8} - 4\beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -16\beta_{9} + 20\beta_{6} - 31\beta_{3} - 7\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 63 \beta_{15} - 68 \beta_{14} + 27 \beta_{13} - 27 \beta_{12} + 9 \beta_{11} + 27 \beta_{10} + \cdots - 27 \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -88\beta_{9} + 115\beta_{6} - 88\beta_{5} - 36\beta_{4} - 160\beta_{3} + 115\beta_{2} - 36\beta _1 - 160 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 203 \beta_{15} - 214 \beta_{14} + 487 \beta_{13} - 521 \beta_{12} + 203 \beta_{11} + 93 \beta_{10} + \cdots - 81 \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -956\beta_{5} - 397\beta_{4} + 1240\beta_{2} - 1759 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1556 \beta_{15} + 1657 \beta_{14} + 3752 \beta_{13} - 4007 \beta_{12} + 1556 \beta_{11} + \cdots + 640 \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 10413 \beta_{9} - 13525 \beta_{6} - 10413 \beta_{5} - 4311 \beta_{4} + 19116 \beta_{3} + 13525 \beta_{2} + \cdots - 19116 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 28889 \beta_{15} + 30842 \beta_{14} + 11969 \beta_{13} - 12771 \beta_{12} + 4951 \beta_{11} + \cdots + 11969 \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 28340\beta_{9} - 36800\beta_{6} + 52049\beta_{3} + 11740\beta_1 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 222409 \beta_{15} + 237429 \beta_{14} - 92129 \beta_{13} + 98340 \beta_{12} - 38151 \beta_{11} + \cdots + 92129 \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 617107 \beta_{9} - 801365 \beta_{6} + 617107 \beta_{5} + 255609 \beta_{4} + 1133276 \beta_{3} + \cdots + 1133276 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 709236 \beta_{15} + 757107 \beta_{14} - 1712232 \beta_{13} + 1827841 \beta_{12} + \cdots + 293760 \beta_{7} ) / 4 \) 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
−0.470061 + 1.13483i
−0.332975 − 0.137923i
−0.311548 + 0.752144i
−2.56342 − 1.06180i
2.56342 + 1.06180i
0.311548 − 0.752144i
0.332975 + 0.137923i
0.470061 − 1.13483i
0.470061 + 1.13483i
0.332975 − 0.137923i
0.311548 + 0.752144i
2.56342 − 1.06180i
−2.56342 + 1.06180i
−0.311548 − 0.752144i
−0.332975 + 0.137923i
−0.470061 − 1.13483i
0 0 0 − 1.00000i 0 − 3.97514i 0 0 0
3761.2 0 0 0 − 1.00000i 0 − 2.23786i 0 0 0
3761.3 0 0 0 − 1.00000i 0 − 1.36202i 0 0 0
3761.4 0 0 0 − 1.00000i 0 − 1.15547i 0 0 0
3761.5 0 0 0 − 1.00000i 0 1.15547i 0 0 0
3761.6 0 0 0 − 1.00000i 0 1.36202i 0 0 0
3761.7 0 0 0 − 1.00000i 0 2.23786i 0 0 0
3761.8 0 0 0 − 1.00000i 0 3.97514i 0 0 0
3761.9 0 0 0 1.00000i 0 − 3.97514i 0 0 0
3761.10 0 0 0 1.00000i 0 − 2.23786i 0 0 0
3761.11 0 0 0 1.00000i 0 − 1.36202i 0 0 0
3761.12 0 0 0 1.00000i 0 − 1.15547i 0 0 0
3761.13 0 0 0 1.00000i 0 1.15547i 0 0 0
3761.14 0 0 0 1.00000i 0 1.36202i 0 0 0
3761.15 0 0 0 1.00000i 0 2.23786i 0 0 0
3761.16 0 0 0 1.00000i 0 3.97514i 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.h 16
3.b odd 2 1 inner 7920.2.f.h 16
4.b odd 2 1 1980.2.d.a ✓ 16
11.b odd 2 1 inner 7920.2.f.h 16
12.b even 2 1 1980.2.d.a ✓ 16
20.d odd 2 1 9900.2.d.c 16
20.e even 4 1 9900.2.n.c 16
20.e even 4 1 9900.2.n.d 16
33.d even 2 1 inner 7920.2.f.h 16
44.c even 2 1 1980.2.d.a ✓ 16
60.h even 2 1 9900.2.d.c 16
60.l odd 4 1 9900.2.n.c 16
60.l odd 4 1 9900.2.n.d 16
132.d odd 2 1 1980.2.d.a ✓ 16
220.g even 2 1 9900.2.d.c 16
220.i odd 4 1 9900.2.n.c 16
220.i odd 4 1 9900.2.n.d 16
660.g odd 2 1 9900.2.d.c 16
660.q even 4 1 9900.2.n.c 16
660.q even 4 1 9900.2.n.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1980.2.d.a ✓ 16 4.b odd 2 1
1980.2.d.a ✓ 16 12.b even 2 1
1980.2.d.a ✓ 16 44.c even 2 1
1980.2.d.a ✓ 16 132.d odd 2 1
7920.2.f.h 16 1.a even 1 1 trivial
7920.2.f.h 16 3.b odd 2 1 inner
7920.2.f.h 16 11.b odd 2 1 inner
7920.2.f.h 16 33.d even 2 1 inner
9900.2.d.c 16 20.d odd 2 1
9900.2.d.c 16 60.h even 2 1
9900.2.d.c 16 220.g even 2 1
9900.2.d.c 16 660.g odd 2 1
9900.2.n.c 16 20.e even 4 1
9900.2.n.c 16 60.l odd 4 1
9900.2.n.c 16 220.i odd 4 1
9900.2.n.c 16 660.q even 4 1
9900.2.n.d 16 20.e even 4 1
9900.2.n.d 16 60.l odd 4 1
9900.2.n.d 16 220.i odd 4 1
9900.2.n.d 16 660.q even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(7920, [\chi])\):

\( T_{7}^{8} + 24T_{7}^{6} + 148T_{7}^{4} + 304T_{7}^{2} + 196 \) Copy content Toggle raw display
\( T_{17}^{8} - 72T_{17}^{6} + 500T_{17}^{4} - 144T_{17}^{2} + 4 \) 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
$7$ \( (T^{8} + 24 T^{6} + \cdots + 196)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 12 T^{6} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 56 T^{6} + \cdots + 1156)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 72 T^{6} + 500 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 80 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 80 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 144 T^{6} + \cdots + 295936)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 8 T^{3} - 28 T^{2} + \cdots - 28)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 32 T^{2} + \cdots + 112)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 112 T^{6} + \cdots + 50176)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 184 T^{6} + \cdots + 1085764)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 256 T^{6} + \cdots + 2560000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 64 T^{6} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 248 T^{6} + \cdots + 4892944)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 80 T^{2} + 32)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 152 T^{2} + \cdots - 1600)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 248 T^{6} + \cdots + 2890000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 440 T^{6} + \cdots + 27227524)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 496 T^{6} + \cdots + 23348224)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 296 T^{6} + \cdots + 24964)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 160 T^{6} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 8 T^{3} + \cdots + 6128)^{4} \) Copy content Toggle raw display
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