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Properties

Label 392.3.o.d
Level $392$
Weight $3$
Character orbit 392.o
Analytic conductor $10.681$
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] = [392,3,Mod(129,392)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(392, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.129");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 392.o (of order \(6\), degree \(2\), 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: \(10.6812263629\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} - 68 x^{14} + 568 x^{13} + 2134 x^{12} - 16640 x^{11} - 41092 x^{10} + 246584 x^{9} + \cdots + 24404548 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( - \beta_{10} - \beta_{9} - \beta_{5}) q^{5} + (\beta_{15} - \beta_{6} - 8 \beta_{4} + \cdots + 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + ( - \beta_{10} - \beta_{9} - \beta_{5}) q^{5} + (\beta_{15} - \beta_{6} - 8 \beta_{4} + \cdots + 8) q^{9}+ \cdots + (3 \beta_{14} + 5 \beta_{6} + \cdots - 110) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 64 q^{9} + 24 q^{11} + 80 q^{15} + 136 q^{23} + 80 q^{25} + 128 q^{29} + 64 q^{37} + 376 q^{39} + 272 q^{43} + 408 q^{51} + 104 q^{53} - 480 q^{57} - 224 q^{65} + 16 q^{67} + 704 q^{71} + 112 q^{79} - 584 q^{81} + 400 q^{85} - 784 q^{93} - 120 q^{95} - 1760 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} - 68 x^{14} + 568 x^{13} + 2134 x^{12} - 16640 x^{11} - 41092 x^{10} + 246584 x^{9} + \cdots + 24404548 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 11\!\cdots\!72 \nu^{15} + \cdots - 14\!\cdots\!76 ) / 24\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 69\!\cdots\!27 \nu^{15} + \cdots - 44\!\cdots\!32 ) / 10\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10\!\cdots\!10 \nu^{15} + \cdots - 64\!\cdots\!32 ) / 88\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 31947656064 \nu^{15} + 319927501932 \nu^{14} + 1617746278720 \nu^{13} + \cdots - 34\!\cdots\!88 ) / 25\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 22\!\cdots\!02 \nu^{15} + \cdots - 96\!\cdots\!36 ) / 10\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 28\!\cdots\!20 \nu^{15} + \cdots + 37\!\cdots\!62 ) / 10\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 28\!\cdots\!83 \nu^{15} + \cdots + 47\!\cdots\!76 ) / 10\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 35\!\cdots\!64 \nu^{15} + \cdots - 46\!\cdots\!76 ) / 10\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 37\!\cdots\!94 \nu^{15} + \cdots - 39\!\cdots\!92 ) / 10\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 39\!\cdots\!84 \nu^{15} + \cdots - 83\!\cdots\!36 ) / 10\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 43\!\cdots\!58 \nu^{15} + \cdots + 51\!\cdots\!80 ) / 10\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 48\!\cdots\!36 \nu^{15} + \cdots + 61\!\cdots\!56 ) / 10\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 61\!\cdots\!98 \nu^{15} + \cdots + 82\!\cdots\!28 ) / 10\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 12\!\cdots\!38 \nu^{15} + \cdots - 16\!\cdots\!78 ) / 10\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 32\!\cdots\!04 \nu^{15} + \cdots - 47\!\cdots\!60 ) / 21\!\cdots\!90 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} - 2\beta_{11} - 6\beta_{8} - 4\beta_{6} - 9\beta_{3} + 7 ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{14} + 4\beta_{11} + 12\beta_{8} - 6\beta_{6} - 28\beta_{4} - 45\beta_{3} + 28\beta_{2} + 14\beta _1 + 189 ) / 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 18 \beta_{15} + 27 \beta_{14} - 42 \beta_{13} - 6 \beta_{12} - 28 \beta_{11} - 16 \beta_{9} + \cdots + 385 ) / 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12 \beta_{15} + 187 \beta_{14} - 224 \beta_{13} + 24 \beta_{12} + 208 \beta_{11} - 56 \beta_{10} + \cdots + 4207 ) / 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 660 \beta_{15} + 937 \beta_{14} - 2310 \beta_{13} - 10 \beta_{12} - 50 \beta_{11} + 140 \beta_{10} + \cdots + 13657 ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1590 \beta_{15} + 6163 \beta_{14} - 12516 \beta_{13} + 1710 \beta_{12} + 7828 \beta_{11} + \cdots + 105889 ) / 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 23310 \beta_{15} + 33237 \beta_{14} - 94276 \beta_{13} + 7322 \beta_{12} + 20620 \beta_{11} + \cdots + 473739 ) / 14 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 99456 \beta_{15} + 199855 \beta_{14} - 536256 \beta_{13} + 85036 \beta_{12} + 262112 \beta_{11} + \cdots + 3024567 ) / 14 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 879120 \beta_{15} + 1139501 \beta_{14} - 3493938 \beta_{13} + 476064 \beta_{12} + 1175146 \beta_{11} + \cdots + 16197433 ) / 14 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4821150 \beta_{15} + 6515535 \beta_{14} - 20521620 \beta_{13} + 3691740 \beta_{12} + 8696980 \beta_{11} + \cdots + 94252081 ) / 14 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 34269510 \beta_{15} + 38004973 \beta_{14} - 124768952 \beta_{13} + 22401544 \beta_{12} + \cdots + 543409643 ) / 14 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 206180304 \beta_{15} + 212901139 \beta_{14} - 741657504 \beta_{13} + 149814786 \beta_{12} + \cdots + 3051658015 ) / 14 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1331667792 \beta_{15} + 1242142525 \beta_{14} - 4371316950 \beta_{13} + 928110638 \beta_{12} + \cdots + 17872944733 ) / 14 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 8200902710 \beta_{15} + 6913528543 \beta_{14} - 25952396260 \beta_{13} + 5827920098 \beta_{12} + \cdots + 99299044481 ) / 14 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 50720967126 \beta_{15} + 39885133861 \beta_{14} - 150909948280 \beta_{13} + 36005568642 \beta_{12} + \cdots + 576287897395 ) / 14 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(1\) \(1\) \(1 - \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
0.601058 − 0.923880i
4.50657 − 0.382683i
−1.01527 + 0.923880i
−2.09236 + 0.382683i
−3.41801 − 0.382683i
−4.21568 − 0.923880i
5.83222 + 0.382683i
3.80147 + 0.923880i
0.601058 + 0.923880i
4.50657 + 0.382683i
−1.01527 − 0.923880i
−2.09236 − 0.382683i
−3.41801 + 0.382683i
−4.21568 + 0.923880i
5.83222 − 0.382683i
3.80147 − 0.923880i
0 −5.12268 − 2.95758i 0 4.65947 − 2.69015i 0 0 0 12.9946 + 22.5073i 0
129.2 0 −4.08924 − 2.36093i 0 −6.94645 + 4.01053i 0 0 0 6.64795 + 11.5146i 0
129.3 0 −2.58510 − 1.49251i 0 −1.46680 + 0.846859i 0 0 0 −0.0448447 − 0.0776732i 0
129.4 0 −1.16338 − 0.671680i 0 −5.73452 + 3.31082i 0 0 0 −3.59769 − 6.23139i 0
129.5 0 1.16338 + 0.671680i 0 5.73452 − 3.31082i 0 0 0 −3.59769 − 6.23139i 0
129.6 0 2.58510 + 1.49251i 0 1.46680 − 0.846859i 0 0 0 −0.0448447 − 0.0776732i 0
129.7 0 4.08924 + 2.36093i 0 6.94645 − 4.01053i 0 0 0 6.64795 + 11.5146i 0
129.8 0 5.12268 + 2.95758i 0 −4.65947 + 2.69015i 0 0 0 12.9946 + 22.5073i 0
313.1 0 −5.12268 + 2.95758i 0 4.65947 + 2.69015i 0 0 0 12.9946 − 22.5073i 0
313.2 0 −4.08924 + 2.36093i 0 −6.94645 − 4.01053i 0 0 0 6.64795 − 11.5146i 0
313.3 0 −2.58510 + 1.49251i 0 −1.46680 − 0.846859i 0 0 0 −0.0448447 + 0.0776732i 0
313.4 0 −1.16338 + 0.671680i 0 −5.73452 − 3.31082i 0 0 0 −3.59769 + 6.23139i 0
313.5 0 1.16338 − 0.671680i 0 5.73452 + 3.31082i 0 0 0 −3.59769 + 6.23139i 0
313.6 0 2.58510 − 1.49251i 0 1.46680 + 0.846859i 0 0 0 −0.0448447 + 0.0776732i 0
313.7 0 4.08924 − 2.36093i 0 6.94645 + 4.01053i 0 0 0 6.64795 − 11.5146i 0
313.8 0 5.12268 − 2.95758i 0 −4.65947 − 2.69015i 0 0 0 12.9946 − 22.5073i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.3.o.d 16
4.b odd 2 1 784.3.s.k 16
7.b odd 2 1 inner 392.3.o.d 16
7.c even 3 1 392.3.c.b ✓ 8
7.c even 3 1 inner 392.3.o.d 16
7.d odd 6 1 392.3.c.b ✓ 8
7.d odd 6 1 inner 392.3.o.d 16
21.g even 6 1 3528.3.f.e 8
21.h odd 6 1 3528.3.f.e 8
28.d even 2 1 784.3.s.k 16
28.f even 6 1 784.3.c.g 8
28.f even 6 1 784.3.s.k 16
28.g odd 6 1 784.3.c.g 8
28.g odd 6 1 784.3.s.k 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.3.c.b ✓ 8 7.c even 3 1
392.3.c.b ✓ 8 7.d odd 6 1
392.3.o.d 16 1.a even 1 1 trivial
392.3.o.d 16 7.b odd 2 1 inner
392.3.o.d 16 7.c even 3 1 inner
392.3.o.d 16 7.d odd 6 1 inner
784.3.c.g 8 28.f even 6 1
784.3.c.g 8 28.g odd 6 1
784.3.s.k 16 4.b odd 2 1
784.3.s.k 16 28.d even 2 1
784.3.s.k 16 28.f even 6 1
784.3.s.k 16 28.g odd 6 1
3528.3.f.e 8 21.g even 6 1
3528.3.f.e 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 68 T_{3}^{14} + 3214 T_{3}^{12} - 77320 T_{3}^{10} + 1344516 T_{3}^{8} - 11378816 T_{3}^{6} + \cdots + 157351936 \) acting on \(S_{3}^{\mathrm{new}}(392, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 157351936 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 54875873536 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 12 T^{7} + \cdots + 242861056)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 620 T^{6} + \cdots + 49056016)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 82\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 89\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( (T^{8} - 68 T^{7} + \cdots + 463096582144)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 32 T^{3} + \cdots - 14704)^{4} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( (T^{8} - 32 T^{7} + \cdots + 1183744)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 5992 T^{6} + \cdots + 307783067524)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 68 T^{3} + \cdots - 103904)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 41\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 7518651744256)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 49\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 3577121603584)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 176 T^{3} + \cdots - 3110912)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 86\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 921096277786624)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 6466157322496)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 62\!\cdots\!96)^{2} \) Copy content Toggle raw display
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