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Properties

Label 392.3.k.f
Level $392$
Weight $3$
Character orbit 392.k
Analytic conductor $10.681$
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] = [392,3,Mod(67,392)]
 
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([3, 3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.67");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 392.k (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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

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

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} - 2) q^{2} + ( - \beta_{3} - 4 \beta_{2} - \beta_1) q^{3} + 4 \beta_{2} q^{4} + (2 \beta_{3} - 8) q^{6} + 8 q^{8} + ( - 9 \beta_{2} - 8 \beta_1 - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} - 2) q^{2} + ( - \beta_{3} - 4 \beta_{2} - \beta_1) q^{3} + 4 \beta_{2} q^{4} + (2 \beta_{3} - 8) q^{6} + 8 q^{8} + ( - 9 \beta_{2} - 8 \beta_1 - 9) q^{9} + ( - 12 \beta_{3} - 12 \beta_1) q^{11} + (16 \beta_{2} + 4 \beta_1 + 16) q^{12} + ( - 16 \beta_{2} - 16) q^{16} + ( - \beta_{3} + 24 \beta_{2} - \beta_1) q^{17} + (16 \beta_{3} + 18 \beta_{2} + 16 \beta_1) q^{18} + ( - 12 \beta_{2} - 17 \beta_1 - 12) q^{19} + 24 \beta_{3} q^{22} + ( - 8 \beta_{3} - 32 \beta_{2} - 8 \beta_1) q^{24} + 25 \beta_{2} q^{25} + (32 \beta_{3} - 16) q^{27} + 32 \beta_{2} q^{32} + ( - 24 \beta_{2} - 48 \beta_1 - 24) q^{33} + (2 \beta_{3} + 48) q^{34} + ( - 32 \beta_{3} + 36) q^{36} + (34 \beta_{3} + 24 \beta_{2} + 34 \beta_1) q^{38} + (23 \beta_{3} + 48) q^{41} - 60 \beta_{3} q^{43} + 48 \beta_1 q^{44} + (16 \beta_{3} - 64) q^{48} + 50 q^{50} + (94 \beta_{2} + 20 \beta_1 + 94) q^{51} + (32 \beta_{2} + 64 \beta_1 + 32) q^{54} + (80 \beta_{3} - 82) q^{57} + (41 \beta_{3} - 60 \beta_{2} + 41 \beta_1) q^{59} + 64 q^{64} + (96 \beta_{3} + 48 \beta_{2} + 96 \beta_1) q^{66} - 62 \beta_{2} q^{67} + ( - 96 \beta_{2} + 4 \beta_1 - 96) q^{68} + ( - 72 \beta_{2} - 64 \beta_1 - 72) q^{72} + ( - 71 \beta_{3} + 24 \beta_{2} - 71 \beta_1) q^{73} + (100 \beta_{2} + 25 \beta_1 + 100) q^{75} + ( - 68 \beta_{3} + 48) q^{76} + (72 \beta_{3} + 47 \beta_{2} + 72 \beta_1) q^{81} + ( - 96 \beta_{2} + 46 \beta_1 - 96) q^{82} + (79 \beta_{3} - 36) q^{83} - 120 \beta_1 q^{86} + ( - 96 \beta_{3} - 96 \beta_1) q^{88} + (72 \beta_{2} - 73 \beta_1 + 72) q^{89} + (128 \beta_{2} + 32 \beta_1 + 128) q^{96} + ( - 47 \beta_{3} - 120) q^{97} + (108 \beta_{3} - 192) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 8 q^{3} - 8 q^{4} - 32 q^{6} + 32 q^{8} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 8 q^{3} - 8 q^{4} - 32 q^{6} + 32 q^{8} - 18 q^{9} + 32 q^{12} - 32 q^{16} - 48 q^{17} - 36 q^{18} - 24 q^{19} + 64 q^{24} - 50 q^{25} - 64 q^{27} - 64 q^{32} - 48 q^{33} + 192 q^{34} + 144 q^{36} - 48 q^{38} + 192 q^{41} - 256 q^{48} + 200 q^{50} + 188 q^{51} + 64 q^{54} - 328 q^{57} + 120 q^{59} + 256 q^{64} - 96 q^{66} + 124 q^{67} - 192 q^{68} - 144 q^{72} - 48 q^{73} + 200 q^{75} + 192 q^{76} - 94 q^{81} - 192 q^{82} - 144 q^{83} + 144 q^{89} + 256 q^{96} - 480 q^{97} - 768 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

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

Character values

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

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(-1\) \(-1\) \(-1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−0.707107 − 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 − 1.22474i
−1.00000 − 1.73205i 1.29289 − 2.23936i −2.00000 + 3.46410i 0 −5.17157 0 8.00000 1.15685 + 2.00373i 0
67.2 −1.00000 − 1.73205i 2.70711 − 4.68885i −2.00000 + 3.46410i 0 −10.8284 0 8.00000 −10.1569 − 17.5922i 0
275.1 −1.00000 + 1.73205i 1.29289 + 2.23936i −2.00000 − 3.46410i 0 −5.17157 0 8.00000 1.15685 − 2.00373i 0
275.2 −1.00000 + 1.73205i 2.70711 + 4.68885i −2.00000 − 3.46410i 0 −10.8284 0 8.00000 −10.1569 + 17.5922i 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.c even 3 1 inner
56.k 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.k.f 4
7.b odd 2 1 392.3.k.e 4
7.c even 3 1 392.3.g.f ✓ 2
7.c even 3 1 inner 392.3.k.f 4
7.d odd 6 1 392.3.g.g yes 2
7.d odd 6 1 392.3.k.e 4
8.d odd 2 1 CM 392.3.k.f 4
28.f even 6 1 1568.3.g.b 2
28.g odd 6 1 1568.3.g.g 2
56.e even 2 1 392.3.k.e 4
56.j odd 6 1 1568.3.g.b 2
56.k odd 6 1 392.3.g.f ✓ 2
56.k odd 6 1 inner 392.3.k.f 4
56.m even 6 1 392.3.g.g yes 2
56.m even 6 1 392.3.k.e 4
56.p even 6 1 1568.3.g.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.3.g.f ✓ 2 7.c even 3 1
392.3.g.f ✓ 2 56.k odd 6 1
392.3.g.g yes 2 7.d odd 6 1
392.3.g.g yes 2 56.m even 6 1
392.3.k.e 4 7.b odd 2 1
392.3.k.e 4 7.d odd 6 1
392.3.k.e 4 56.e even 2 1
392.3.k.e 4 56.m even 6 1
392.3.k.f 4 1.a even 1 1 trivial
392.3.k.f 4 7.c even 3 1 inner
392.3.k.f 4 8.d odd 2 1 CM
392.3.k.f 4 56.k odd 6 1 inner
1568.3.g.b 2 28.f even 6 1
1568.3.g.b 2 56.j odd 6 1
1568.3.g.g 2 28.g odd 6 1
1568.3.g.g 2 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(392, [\chi])\):

\( T_{3}^{4} - 8T_{3}^{3} + 50T_{3}^{2} - 112T_{3} + 196 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 8 T^{3} + \cdots + 196 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 288 T^{2} + 82944 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 48 T^{3} + \cdots + 329476 \) Copy content Toggle raw display
$19$ \( T^{4} + 24 T^{3} + \cdots + 188356 \) 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
$41$ \( (T^{2} - 96 T + 1246)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 7200)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 120 T^{3} + \cdots + 56644 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 62 T + 3844)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 48 T^{3} + \cdots + 90364036 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 72 T - 11186)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 144 T^{3} + \cdots + 29964676 \) Copy content Toggle raw display
$97$ \( (T^{2} + 240 T + 9982)^{2} \) Copy content Toggle raw display
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