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Properties

Label 392.2.e.a
Level $392$
Weight $2$
Character orbit 392.e
Analytic conductor $3.130$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [392,2,Mod(195,392)]
 
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([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.195");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 392.e (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: \(4\)
Coefficient field: 4.0.2048.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - \beta_{3} q^{3} + 2 q^{4} - \beta_1 q^{6} - 2 \beta_{2} q^{8} + (\beta_{2} - 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} - \beta_{3} q^{3} + 2 q^{4} - \beta_1 q^{6} - 2 \beta_{2} q^{8} + (\beta_{2} - 7) q^{9} + ( - 3 \beta_{2} - 2) q^{11} - 2 \beta_{3} q^{12} + 4 q^{16} + \beta_{3} q^{17} + (7 \beta_{2} - 2) q^{18} + ( - \beta_{3} + \beta_1) q^{19} + (2 \beta_{2} + 6) q^{22} - 2 \beta_1 q^{24} - 5 q^{25} + (4 \beta_{3} + \beta_1) q^{27} - 4 \beta_{2} q^{32} + (2 \beta_{3} - 3 \beta_1) q^{33} + \beta_1 q^{34} + (2 \beta_{2} - 14) q^{36} + (2 \beta_{3} - \beta_1) q^{38} + ( - \beta_{3} - \beta_1) q^{41} + ( - 5 \beta_{2} + 6) q^{43} + ( - 6 \beta_{2} - 4) q^{44} - 4 \beta_{3} q^{48} + 5 \beta_{2} q^{50} + ( - \beta_{2} + 10) q^{51} + (2 \beta_{3} + 4 \beta_1) q^{54} + ( - 9 \beta_{2} - 8) q^{57} + ( - \beta_{3} + 3 \beta_1) q^{59} + 8 q^{64} + ( - 6 \beta_{3} + 2 \beta_1) q^{66} + 6 \beta_{2} q^{67} + 2 \beta_{3} q^{68} + (14 \beta_{2} - 4) q^{72} + (3 \beta_{3} + \beta_1) q^{73} + 5 \beta_{3} q^{75} + ( - 2 \beta_{3} + 2 \beta_1) q^{76} + ( - 11 \beta_{2} + 21) q^{81} + ( - 2 \beta_{3} - \beta_1) q^{82} + ( - \beta_{3} - 3 \beta_1) q^{83} + ( - 6 \beta_{2} + 10) q^{86} + (4 \beta_{2} + 12) q^{88} + ( - 5 \beta_{3} + \beta_1) q^{89} - 4 \beta_1 q^{96} + ( - \beta_{3} + 4 \beta_1) q^{97} + (19 \beta_{2} + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} - 28 q^{9} - 8 q^{11} + 16 q^{16} - 8 q^{18} + 24 q^{22} - 20 q^{25} - 56 q^{36} + 24 q^{43} - 16 q^{44} + 40 q^{51} - 32 q^{57} + 32 q^{64} - 16 q^{72} + 84 q^{81} + 40 q^{86} + 48 q^{88} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\nu^{3} - 5\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6\beta_{3} - 5\beta_1 ) / 7 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
195.1
− 0.765367i
0.765367i
1.84776i
− 1.84776i
−1.41421 − 2.93015i 2.00000 0 4.14386i 0 −2.82843 −5.58579 0
195.2 −1.41421 2.93015i 2.00000 0 − 4.14386i 0 −2.82843 −5.58579 0
195.3 1.41421 − 3.37849i 2.00000 0 − 4.77791i 0 2.82843 −8.41421 0
195.4 1.41421 3.37849i 2.00000 0 4.77791i 0 2.82843 −8.41421 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
7.b odd 2 1 inner
56.e 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.e.a ✓ 4
4.b odd 2 1 1568.2.e.a 4
7.b odd 2 1 inner 392.2.e.a ✓ 4
7.c even 3 2 392.2.m.d 8
7.d odd 6 2 392.2.m.d 8
8.b even 2 1 1568.2.e.a 4
8.d odd 2 1 CM 392.2.e.a ✓ 4
28.d even 2 1 1568.2.e.a 4
28.f even 6 2 1568.2.q.f 8
28.g odd 6 2 1568.2.q.f 8
56.e even 2 1 inner 392.2.e.a ✓ 4
56.h odd 2 1 1568.2.e.a 4
56.j odd 6 2 1568.2.q.f 8
56.k odd 6 2 392.2.m.d 8
56.m even 6 2 392.2.m.d 8
56.p even 6 2 1568.2.q.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.e.a ✓ 4 1.a even 1 1 trivial
392.2.e.a ✓ 4 7.b odd 2 1 inner
392.2.e.a ✓ 4 8.d odd 2 1 CM
392.2.e.a ✓ 4 56.e even 2 1 inner
392.2.m.d 8 7.c even 3 2
392.2.m.d 8 7.d odd 6 2
392.2.m.d 8 56.k odd 6 2
392.2.m.d 8 56.m even 6 2
1568.2.e.a 4 4.b odd 2 1
1568.2.e.a 4 8.b even 2 1
1568.2.e.a 4 28.d even 2 1
1568.2.e.a 4 56.h odd 2 1
1568.2.q.f 8 28.f even 6 2
1568.2.q.f 8 28.g odd 6 2
1568.2.q.f 8 56.j odd 6 2
1568.2.q.f 8 56.p even 6 2

Hecke kernels

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

\( T_{3}^{4} + 20T_{3}^{2} + 98 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 20T^{2} + 98 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 4 T - 14)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 20T^{2} + 98 \) Copy content Toggle raw display
$19$ \( T^{4} + 52T^{2} + 98 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 68T^{2} + 98 \) Copy content Toggle raw display
$43$ \( (T^{2} - 12 T - 14)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 356 T^{2} + 28322 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 244T^{2} + 4802 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 404 T^{2} + 28322 \) Copy content Toggle raw display
$89$ \( T^{4} + 500 T^{2} + 51842 \) Copy content Toggle raw display
$97$ \( T^{4} + 628 T^{2} + 94178 \) Copy content Toggle raw display
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