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Properties

Label 392.2.b.c
Level $392$
Weight $2$
Character orbit 392.b
Analytic conductor $3.130$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(4\)
Coefficient field: 4.0.2312.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} - \beta_1) q^{3} + \beta_{2} q^{4} + (\beta_{3} + \beta_1) q^{5} + ( - \beta_{3} + \beta_{2} - 2) q^{6} + ( - \beta_{3} - 2) q^{8} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} - \beta_1) q^{3} + \beta_{2} q^{4} + (\beta_{3} + \beta_1) q^{5} + ( - \beta_{3} + \beta_{2} - 2) q^{6} + ( - \beta_{3} - 2) q^{8} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{9} + ( - \beta_{3} - \beta_{2} + 2) q^{10} + (\beta_{3} + \beta_{2}) q^{11} + (2 \beta_1 - 4) q^{12} + ( - \beta_{3} - \beta_1) q^{13} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{15} + (\beta_{3} + 2 \beta_1 - 2) q^{16} - 2 q^{17} + ( - 2 \beta_{2} + 3 \beta_1 - 4) q^{18} + (\beta_{2} - \beta_1) q^{19} + (2 \beta_{3} - 2 \beta_1) q^{20} - 2 \beta_{3} q^{22} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{23} + ( - 2 \beta_{2} + 4 \beta_1) q^{24} + (\beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{25} + (\beta_{3} + \beta_{2} - 2) q^{26} + (2 \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{27} + (2 \beta_{2} - 2 \beta_1) q^{29} + (2 \beta_{2} - 2 \beta_1 + 4) q^{30} + (2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 4) q^{31} + ( - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{32} - 4 q^{33} + 2 \beta_1 q^{34} + (2 \beta_{3} - 3 \beta_{2} + 4 \beta_1 + 4) q^{36} + ( - 2 \beta_{2} + 2 \beta_1) q^{37} + ( - \beta_{3} + \beta_{2} - 2) q^{38} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{39} + ( - 2 \beta_{3} + 2 \beta_{2} + 4) q^{40} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{41}+ \cdots + (3 \beta_{3} - \beta_{2} + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + q^{4} - 6 q^{6} - 7 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + q^{4} - 6 q^{6} - 7 q^{8} - 8 q^{9} + 8 q^{10} - 14 q^{12} + 4 q^{15} - 7 q^{16} - 8 q^{17} - 15 q^{18} - 4 q^{20} + 2 q^{22} + 4 q^{23} + 2 q^{24} - 8 q^{25} - 8 q^{26} + 16 q^{30} + 8 q^{31} + 9 q^{32} - 16 q^{33} + 2 q^{34} + 15 q^{36} - 6 q^{38} - 4 q^{39} + 20 q^{40} + 16 q^{41} - 18 q^{44} + 16 q^{46} + 10 q^{48} + 19 q^{50} + 4 q^{52} + 28 q^{54} - 24 q^{55} - 20 q^{57} - 12 q^{58} - 16 q^{60} + 32 q^{62} + q^{64} + 28 q^{65} + 4 q^{66} - 2 q^{68} - 32 q^{71} + 31 q^{72} + 24 q^{73} + 12 q^{74} - 14 q^{76} - 16 q^{78} - 36 q^{80} + 24 q^{81} - 38 q^{82} + 2 q^{86} - 40 q^{87} + 22 q^{88} + 32 q^{89} - 16 q^{90} - 16 q^{92} + 4 q^{95} + 42 q^{96} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2 \) 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
1.28078 + 0.599676i
1.28078 − 0.599676i
−0.780776 + 1.17915i
−0.780776 − 1.17915i
−1.28078 − 0.599676i 0.936426i 1.28078 + 1.53610i 3.33513i 0.561553 − 1.19935i 0 −0.719224 − 2.73546i 2.12311 2.00000 − 4.27156i
197.2 −1.28078 + 0.599676i − 0.936426i 1.28078 − 1.53610i − 3.33513i 0.561553 + 1.19935i 0 −0.719224 + 2.73546i 2.12311 2.00000 + 4.27156i
197.3 0.780776 − 1.17915i − 3.02045i −0.780776 − 1.84130i 1.69614i −3.56155 − 2.35829i 0 −2.78078 − 0.516994i −6.12311 2.00000 + 1.32431i
197.4 0.780776 + 1.17915i 3.02045i −0.780776 + 1.84130i − 1.69614i −3.56155 + 2.35829i 0 −2.78078 + 0.516994i −6.12311 2.00000 − 1.32431i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.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.c 4
4.b odd 2 1 1568.2.b.d 4
7.b odd 2 1 56.2.b.b ✓ 4
7.c even 3 2 392.2.p.e 8
7.d odd 6 2 392.2.p.f 8
8.b even 2 1 inner 392.2.b.c 4
8.d odd 2 1 1568.2.b.d 4
21.c even 2 1 504.2.c.d 4
28.d even 2 1 224.2.b.b 4
28.f even 6 2 1568.2.t.d 8
28.g odd 6 2 1568.2.t.e 8
56.e even 2 1 224.2.b.b 4
56.h odd 2 1 56.2.b.b ✓ 4
56.j odd 6 2 392.2.p.f 8
56.k odd 6 2 1568.2.t.e 8
56.m even 6 2 1568.2.t.d 8
56.p even 6 2 392.2.p.e 8
84.h odd 2 1 2016.2.c.c 4
112.j even 4 2 1792.2.a.v 4
112.l odd 4 2 1792.2.a.x 4
168.e odd 2 1 2016.2.c.c 4
168.i even 2 1 504.2.c.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.b.b ✓ 4 7.b odd 2 1
56.2.b.b ✓ 4 56.h odd 2 1
224.2.b.b 4 28.d even 2 1
224.2.b.b 4 56.e even 2 1
392.2.b.c 4 1.a even 1 1 trivial
392.2.b.c 4 8.b even 2 1 inner
392.2.p.e 8 7.c even 3 2
392.2.p.e 8 56.p even 6 2
392.2.p.f 8 7.d odd 6 2
392.2.p.f 8 56.j odd 6 2
504.2.c.d 4 21.c even 2 1
504.2.c.d 4 168.i even 2 1
1568.2.b.d 4 4.b odd 2 1
1568.2.b.d 4 8.d odd 2 1
1568.2.t.d 8 28.f even 6 2
1568.2.t.d 8 56.m even 6 2
1568.2.t.e 8 28.g odd 6 2
1568.2.t.e 8 56.k odd 6 2
1792.2.a.v 4 112.j even 4 2
1792.2.a.x 4 112.l odd 4 2
2016.2.c.c 4 84.h odd 2 1
2016.2.c.c 4 168.e odd 2 1

Hecke kernels

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

\( T_{3}^{4} + 10T_{3}^{2} + 8 \) Copy content Toggle raw display
\( T_{17} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{4} + 10T^{2} + 8 \) Copy content Toggle raw display
$5$ \( T^{4} + 14T^{2} + 32 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 20T^{2} + 32 \) Copy content Toggle raw display
$13$ \( T^{4} + 14T^{2} + 32 \) Copy content Toggle raw display
$17$ \( (T + 2)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 10T^{2} + 8 \) Copy content Toggle raw display
$23$ \( (T^{2} - 2 T - 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 40T^{2} + 128 \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 64)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 40T^{2} + 128 \) Copy content Toggle raw display
$41$ \( (T^{2} - 8 T - 52)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 20T^{2} + 32 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 80T^{2} + 512 \) Copy content Toggle raw display
$59$ \( T^{4} + 58T^{2} + 8 \) Copy content Toggle raw display
$61$ \( T^{4} + 14T^{2} + 32 \) Copy content Toggle raw display
$67$ \( T^{4} + 380 T^{2} + 32768 \) Copy content Toggle raw display
$71$ \( (T + 8)^{4} \) Copy content Toggle raw display
$73$ \( (T - 6)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 122T^{2} + 2888 \) Copy content Toggle raw display
$89$ \( (T^{2} - 16 T - 4)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T - 52)^{2} \) Copy content Toggle raw display
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