↑

Properties

Label 392.2.b.f
Level $392$
Weight $2$
Character orbit 392.b
Analytic conductor $3.130$
Analytic rank $0$
Dimension $6$
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] = [392,2,Mod(197,392)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(392, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.197");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 392.b (of order \(2\), degree \(1\), 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: \(3.13013575923\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.1142512.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - x^{4} + 5x^{3} - 2x^{2} - 4x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} - \beta_{3} q^{3} - \beta_{2} q^{4} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{5} + (\beta_{5} - \beta_{3} + \beta_1 + 1) q^{6} + (\beta_{5} + \beta_{4} + \beta_{3} + \cdots + 1) q^{8}+ \cdots + ( - \beta_{5} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} - \beta_{3} q^{3} - \beta_{2} q^{4} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{5} + (\beta_{5} - \beta_{3} + \beta_1 + 1) q^{6} + (\beta_{5} + \beta_{4} + \beta_{3} + \cdots + 1) q^{8}+ \cdots + ( - \beta_{5} + 2 \beta_{4} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} + 4 q^{6} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{2} + 4 q^{6} + 2 q^{8} - 8 q^{10} - 2 q^{12} - 10 q^{15} - 8 q^{16} - 2 q^{17} - 6 q^{18} - 4 q^{20} + 6 q^{22} - 2 q^{23} + 18 q^{24} + 4 q^{25} - 2 q^{26} - 14 q^{30} + 10 q^{31} + 12 q^{32} - 14 q^{33} - 16 q^{34} - 16 q^{36} + 18 q^{38} - 4 q^{39} + 10 q^{40} + 4 q^{41} + 30 q^{44} + 4 q^{46} + 30 q^{47} + 28 q^{48} - 8 q^{50} - 32 q^{52} + 2 q^{54} - 2 q^{55} - 2 q^{57} + 22 q^{58} - 6 q^{60} + 14 q^{62} + 12 q^{64} - 8 q^{65} - 38 q^{66} + 4 q^{68} + 16 q^{71} - 20 q^{72} - 10 q^{73} - 18 q^{74} - 26 q^{76} + 26 q^{78} + 22 q^{79} + 36 q^{80} - 22 q^{81} - 26 q^{82} - 40 q^{86} - 20 q^{87} + 14 q^{88} - 10 q^{89} - 26 q^{90} - 10 q^{92} + 42 q^{94} + 34 q^{95} + 16 q^{96} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - x^{4} + 5x^{3} - 2x^{2} - 4x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + \nu^{4} + \nu^{3} - \nu^{2} + 2\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - \nu^{4} - \nu^{3} + 5\nu^{2} + 2\nu - 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - \nu^{4} + 3\nu^{3} - 3\nu^{2} - 4\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} + \nu^{4} + \nu^{3} - \nu^{2} + 4\nu + 4 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{5} - 5\beta_{4} - 3\beta_{3} + 3\beta_{2} + \beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 5\beta _1 - 5 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
1.17445 − 0.787829i
1.17445 + 0.787829i
−1.37241 + 0.341295i
−1.37241 − 0.341295i
0.697966 − 1.22998i
0.697966 + 1.22998i
−0.933099 − 1.06270i 1.57566i −0.258652 + 1.98320i − 0.549738i 1.67445 − 1.47024i 0 2.34889 − 1.57566i 0.517304 −0.584205 + 0.512960i
197.2 −0.933099 + 1.06270i − 1.57566i −0.258652 − 1.98320i 0.549738i 1.67445 + 1.47024i 0 2.34889 + 1.57566i 0.517304 −0.584205 − 0.512960i
197.3 0.605378 − 1.27809i − 0.682591i −1.26704 − 1.54746i − 3.23877i −0.872413 − 0.413225i 0 −2.74483 + 0.682591i 2.53407 −4.13945 − 1.96068i
197.4 0.605378 + 1.27809i 0.682591i −1.26704 + 1.54746i 3.23877i −0.872413 + 0.413225i 0 −2.74483 − 0.682591i 2.53407 −4.13945 + 1.96068i
197.5 1.32772 − 0.486987i 2.45995i 1.52569 − 1.29317i 1.48598i 1.19797 + 3.26613i 0 1.39593 − 2.45995i −3.05137 0.723653 + 1.97297i
197.6 1.32772 + 0.486987i − 2.45995i 1.52569 + 1.29317i − 1.48598i 1.19797 − 3.26613i 0 1.39593 + 2.45995i −3.05137 0.723653 − 1.97297i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.2.b.f 6
4.b odd 2 1 1568.2.b.e 6
7.b odd 2 1 392.2.b.e 6
7.c even 3 2 392.2.p.g 12
7.d odd 6 2 56.2.p.a ✓ 12
8.b even 2 1 inner 392.2.b.f 6
8.d odd 2 1 1568.2.b.e 6
21.g even 6 2 504.2.cj.c 12
28.d even 2 1 1568.2.b.f 6
28.f even 6 2 224.2.t.a 12
28.g odd 6 2 1568.2.t.g 12
56.e even 2 1 1568.2.b.f 6
56.h odd 2 1 392.2.b.e 6
56.j odd 6 2 56.2.p.a ✓ 12
56.k odd 6 2 1568.2.t.g 12
56.m even 6 2 224.2.t.a 12
56.p even 6 2 392.2.p.g 12
84.j odd 6 2 2016.2.cr.c 12
168.ba even 6 2 504.2.cj.c 12
168.be odd 6 2 2016.2.cr.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.p.a ✓ 12 7.d odd 6 2
56.2.p.a ✓ 12 56.j odd 6 2
224.2.t.a 12 28.f even 6 2
224.2.t.a 12 56.m even 6 2
392.2.b.e 6 7.b odd 2 1
392.2.b.e 6 56.h odd 2 1
392.2.b.f 6 1.a even 1 1 trivial
392.2.b.f 6 8.b even 2 1 inner
392.2.p.g 12 7.c even 3 2
392.2.p.g 12 56.p even 6 2
504.2.cj.c 12 21.g even 6 2
504.2.cj.c 12 168.ba even 6 2
1568.2.b.e 6 4.b odd 2 1
1568.2.b.e 6 8.d odd 2 1
1568.2.b.f 6 28.d even 2 1
1568.2.b.f 6 56.e even 2 1
1568.2.t.g 12 28.g odd 6 2
1568.2.t.g 12 56.k odd 6 2
2016.2.cr.c 12 84.j odd 6 2
2016.2.cr.c 12 168.be odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(392, [\chi])\):

\( T_{3}^{6} + 9T_{3}^{4} + 19T_{3}^{2} + 7 \) Copy content Toggle raw display
\( T_{17}^{3} + T_{17}^{2} - 17T_{17} - 21 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 2 T^{5} + \cdots + 8 \) Copy content Toggle raw display
$3$ \( T^{6} + 9 T^{4} + \cdots + 7 \) Copy content Toggle raw display
$5$ \( T^{6} + 13 T^{4} + \cdots + 7 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 37 T^{4} + \cdots + 847 \) Copy content Toggle raw display
$13$ \( T^{6} + 32 T^{4} + \cdots + 1008 \) Copy content Toggle raw display
$17$ \( (T^{3} + T^{2} - 17 T - 21)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 61 T^{4} + \cdots + 3087 \) Copy content Toggle raw display
$23$ \( (T^{3} + T^{2} - 7 T - 9)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + 72 T^{4} + \cdots + 112 \) Copy content Toggle raw display
$31$ \( (T^{3} - 5 T^{2} + 3 T + 7)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 61 T^{4} + \cdots + 63 \) Copy content Toggle raw display
$41$ \( (T^{3} - 2 T^{2} - 40 T + 84)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 164 T^{4} + \cdots + 4032 \) Copy content Toggle raw display
$47$ \( (T^{3} - 15 T^{2} + \cdots + 189)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + 157 T^{4} + \cdots + 26047 \) Copy content Toggle raw display
$59$ \( T^{6} + 177 T^{4} + \cdots + 26047 \) Copy content Toggle raw display
$61$ \( T^{6} + 293 T^{4} + \cdots + 642663 \) Copy content Toggle raw display
$67$ \( T^{6} + 41 T^{4} + \cdots + 63 \) Copy content Toggle raw display
$71$ \( (T^{3} - 8 T^{2} + \cdots + 432)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 5 T^{2} + \cdots - 441)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 11 T^{2} + 21 T - 9)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 52 T^{4} + \cdots + 448 \) Copy content Toggle raw display
$89$ \( (T^{3} + 5 T^{2} + \cdots + 231)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 10 T^{2} + \cdots + 28)^{2} \) Copy content Toggle raw display
show more
show less