Properties

Label 392.3.k.b
Level $392$
Weight $3$
Character orbit 392.k
Analytic conductor $10.681$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 8)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{2} - 2 \zeta_{6} q^{3} - 4 \zeta_{6} q^{4} - 4 q^{6} - 8 q^{8} + ( - 5 \zeta_{6} + 5) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 2) q^{2} - 2 \zeta_{6} q^{3} - 4 \zeta_{6} q^{4} - 4 q^{6} - 8 q^{8} + ( - 5 \zeta_{6} + 5) q^{9} - 14 \zeta_{6} q^{11} + (8 \zeta_{6} - 8) q^{12} + (16 \zeta_{6} - 16) q^{16} + 2 \zeta_{6} q^{17} - 10 \zeta_{6} q^{18} + (34 \zeta_{6} - 34) q^{19} - 28 q^{22} + 16 \zeta_{6} q^{24} - 25 \zeta_{6} q^{25} - 28 q^{27} + 32 \zeta_{6} q^{32} + (28 \zeta_{6} - 28) q^{33} + 4 q^{34} - 20 q^{36} + 68 \zeta_{6} q^{38} + 46 q^{41} + 14 q^{43} + (56 \zeta_{6} - 56) q^{44} + 32 q^{48} - 50 q^{50} + ( - 4 \zeta_{6} + 4) q^{51} + (56 \zeta_{6} - 56) q^{54} + 68 q^{57} - 82 \zeta_{6} q^{59} + 64 q^{64} + 56 \zeta_{6} q^{66} - 62 \zeta_{6} q^{67} + ( - 8 \zeta_{6} + 8) q^{68} + (40 \zeta_{6} - 40) q^{72} - 142 \zeta_{6} q^{73} + (50 \zeta_{6} - 50) q^{75} + 136 q^{76} + 11 \zeta_{6} q^{81} + ( - 92 \zeta_{6} + 92) q^{82} - 158 q^{83} + ( - 28 \zeta_{6} + 28) q^{86} + 112 \zeta_{6} q^{88} + ( - 146 \zeta_{6} + 146) q^{89} + ( - 64 \zeta_{6} + 64) q^{96} + 94 q^{97} - 70 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} - 4 q^{4} - 8 q^{6} - 16 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} - 4 q^{4} - 8 q^{6} - 16 q^{8} + 5 q^{9} - 14 q^{11} - 8 q^{12} - 16 q^{16} + 2 q^{17} - 10 q^{18} - 34 q^{19} - 56 q^{22} + 16 q^{24} - 25 q^{25} - 56 q^{27} + 32 q^{32} - 28 q^{33} + 8 q^{34} - 40 q^{36} + 68 q^{38} + 92 q^{41} + 28 q^{43} - 56 q^{44} + 64 q^{48} - 100 q^{50} + 4 q^{51} - 56 q^{54} + 136 q^{57} - 82 q^{59} + 128 q^{64} + 56 q^{66} - 62 q^{67} + 8 q^{68} - 40 q^{72} - 142 q^{73} - 50 q^{75} + 272 q^{76} + 11 q^{81} + 92 q^{82} - 316 q^{83} + 28 q^{86} + 112 q^{88} + 146 q^{89} + 64 q^{96} + 188 q^{97} - 140 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(-1\) \(-1\) \(-1 + \zeta_{6}\)

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
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 + 1.73205i −1.00000 + 1.73205i −2.00000 + 3.46410i 0 −4.00000 0 −8.00000 2.50000 + 4.33013i 0
275.1 1.00000 1.73205i −1.00000 1.73205i −2.00000 3.46410i 0 −4.00000 0 −8.00000 2.50000 4.33013i 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.b 2
7.b odd 2 1 392.3.k.d 2
7.c even 3 1 392.3.g.a 1
7.c even 3 1 inner 392.3.k.b 2
7.d odd 6 1 8.3.d.a 1
7.d odd 6 1 392.3.k.d 2
8.d odd 2 1 CM 392.3.k.b 2
21.g even 6 1 72.3.b.a 1
28.f even 6 1 32.3.d.a 1
28.g odd 6 1 1568.3.g.a 1
35.i odd 6 1 200.3.g.a 1
35.k even 12 2 200.3.e.a 2
56.e even 2 1 392.3.k.d 2
56.j odd 6 1 32.3.d.a 1
56.k odd 6 1 392.3.g.a 1
56.k odd 6 1 inner 392.3.k.b 2
56.m even 6 1 8.3.d.a 1
56.m even 6 1 392.3.k.d 2
56.p even 6 1 1568.3.g.a 1
84.j odd 6 1 288.3.b.a 1
112.v even 12 2 256.3.c.e 2
112.x odd 12 2 256.3.c.e 2
140.s even 6 1 800.3.g.a 1
140.x odd 12 2 800.3.e.a 2
168.ba even 6 1 288.3.b.a 1
168.be odd 6 1 72.3.b.a 1
280.ba even 6 1 200.3.g.a 1
280.bk odd 6 1 800.3.g.a 1
280.bp odd 12 2 200.3.e.a 2
280.bv even 12 2 800.3.e.a 2
336.bo even 12 2 2304.3.g.j 2
336.br odd 12 2 2304.3.g.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.3.d.a 1 7.d odd 6 1
8.3.d.a 1 56.m even 6 1
32.3.d.a 1 28.f even 6 1
32.3.d.a 1 56.j odd 6 1
72.3.b.a 1 21.g even 6 1
72.3.b.a 1 168.be odd 6 1
200.3.e.a 2 35.k even 12 2
200.3.e.a 2 280.bp odd 12 2
200.3.g.a 1 35.i odd 6 1
200.3.g.a 1 280.ba even 6 1
256.3.c.e 2 112.v even 12 2
256.3.c.e 2 112.x odd 12 2
288.3.b.a 1 84.j odd 6 1
288.3.b.a 1 168.ba even 6 1
392.3.g.a 1 7.c even 3 1
392.3.g.a 1 56.k odd 6 1
392.3.k.b 2 1.a even 1 1 trivial
392.3.k.b 2 7.c even 3 1 inner
392.3.k.b 2 8.d odd 2 1 CM
392.3.k.b 2 56.k odd 6 1 inner
392.3.k.d 2 7.b odd 2 1
392.3.k.d 2 7.d odd 6 1
392.3.k.d 2 56.e even 2 1
392.3.k.d 2 56.m even 6 1
800.3.e.a 2 140.x odd 12 2
800.3.e.a 2 280.bv even 12 2
800.3.g.a 1 140.s even 6 1
800.3.g.a 1 280.bk odd 6 1
1568.3.g.a 1 28.g odd 6 1
1568.3.g.a 1 56.p even 6 1
2304.3.g.j 2 336.bo even 12 2
2304.3.g.j 2 336.br odd 12 2

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 34T + 1156 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( (T - 46)^{2} \) Copy content Toggle raw display
$43$ \( (T - 14)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 82T + 6724 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 62T + 3844 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 142T + 20164 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( (T + 158)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 146T + 21316 \) Copy content Toggle raw display
$97$ \( (T - 94)^{2} \) Copy content Toggle raw display
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