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Properties

Label 392.3.c.c
Level $392$
Weight $3$
Character orbit 392.c
Analytic conductor $10.681$
Analytic rank $0$
Dimension $8$
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] = [392,3,Mod(97,392)]
 
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, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.97");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 392.c (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: \(10.6812263629\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.126303473664.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 9x^{6} + 2x^{5} + 92x^{4} + 14x^{3} - 441x^{2} - 686x + 2401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + \beta_{7} q^{5} + (\beta_1 - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} + \beta_{7} q^{5} + (\beta_1 - 5) q^{9} + (\beta_{4} + \beta_1 - 1) q^{11} + (2 \beta_{7} - \beta_{6} + \cdots + \beta_{2}) q^{13}+ \cdots + ( - 12 \beta_{4} - 8 \beta_{3} + \cdots + 91) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 40 q^{9} - 8 q^{11} + 8 q^{15} + 96 q^{23} - 48 q^{25} - 104 q^{29} - 136 q^{37} + 104 q^{39} + 160 q^{43} + 360 q^{51} + 56 q^{53} - 8 q^{57} - 440 q^{65} + 112 q^{67} - 208 q^{71} + 320 q^{79} - 192 q^{81} - 88 q^{85} - 536 q^{93} - 432 q^{95} + 728 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 9x^{6} + 2x^{5} + 92x^{4} + 14x^{3} - 441x^{2} - 686x + 2401 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{7} - 20\nu^{6} - 48\nu^{5} + 230\nu^{4} + 1032\nu^{3} - 1344\nu^{2} - 11515\nu + 1372 ) / 2744 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -99\nu^{7} - 75\nu^{6} + 800\nu^{5} + 1818\nu^{4} - 4950\nu^{3} - 16800\nu^{2} + 10731\nu + 100499 ) / 10976 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -13\nu^{7} + 12\nu^{6} + 96\nu^{5} + 198\nu^{4} - 440\nu^{3} - 1568\nu^{2} + 1029\nu + 9604 ) / 1372 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -55\nu^{7} - 44\nu^{6} + 656\nu^{5} + 1570\nu^{4} - 3016\nu^{3} - 9744\nu^{2} + 3087\nu + 61740 ) / 5488 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\nu^{7} - 5\nu^{6} - 112\nu^{5} - 26\nu^{4} + 710\nu^{3} + 976\nu^{2} - 1603\nu - 7987 ) / 784 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 263\nu^{7} + 223\nu^{6} - 1856\nu^{5} - 4402\nu^{4} + 10014\nu^{3} + 32704\nu^{2} - 18767\nu - 214375 ) / 10976 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -435\nu^{7} - 187\nu^{6} + 3040\nu^{5} + 7418\nu^{4} - 21750\nu^{3} - 54432\nu^{2} + 30331\nu + 391363 ) / 10976 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + 2\beta_{2} - 2\beta _1 + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - 2\beta_{6} + 2\beta_{4} - \beta_{3} - 11\beta_{2} - \beta _1 + 11 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{7} - 9\beta_{6} + \beta_{5} + 8\beta_{3} + 28 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8\beta_{7} + 3\beta_{6} + 15\beta_{5} + 18\beta_{4} + 6\beta_{3} - 31\beta_{2} - 7\beta _1 - 31 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -69\beta_{7} - 57\beta_{6} - 63\beta_{5} + 120\beta_{4} + 3\beta_{3} - 82\beta_{2} + 6\beta _1 + 82 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 15\beta_{7} + 65\beta_{6} + 7\beta_{5} + 112\beta_{3} + 22 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -57\beta_{7} + 923\beta_{6} + 29\beta_{5} + 952\beta_{4} - 447\beta_{3} + 1262\beta_{2} - 86\beta _1 + 1262 ) / 8 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
2.18070 + 1.49818i
2.64247 − 0.131782i
−2.38781 + 1.13946i
−1.43536 − 2.22255i
−1.43536 + 2.22255i
−2.38781 − 1.13946i
2.64247 + 0.131782i
2.18070 − 1.49818i
0 − 4.95492i 0 − 2.82578i 0 0 0 −15.5512 0
97.2 0 − 4.55428i 0 7.15816i 0 0 0 −11.7414 0
97.3 0 − 2.50543i 0 − 7.72476i 0 0 0 2.72281 0
97.4 0 − 2.10479i 0 2.25918i 0 0 0 4.56987 0
97.5 0 2.10479i 0 − 2.25918i 0 0 0 4.56987 0
97.6 0 2.50543i 0 7.72476i 0 0 0 2.72281 0
97.7 0 4.55428i 0 − 7.15816i 0 0 0 −11.7414 0
97.8 0 4.95492i 0 2.82578i 0 0 0 −15.5512 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.3.c.c 8
3.b odd 2 1 3528.3.f.d 8
4.b odd 2 1 784.3.c.h 8
7.b odd 2 1 inner 392.3.c.c 8
7.c even 3 1 56.3.o.a ✓ 8
7.c even 3 1 392.3.o.c 8
7.d odd 6 1 56.3.o.a ✓ 8
7.d odd 6 1 392.3.o.c 8
21.c even 2 1 3528.3.f.d 8
21.g even 6 1 504.3.by.b 8
21.h odd 6 1 504.3.by.b 8
28.d even 2 1 784.3.c.h 8
28.f even 6 1 112.3.s.c 8
28.f even 6 1 784.3.s.i 8
28.g odd 6 1 112.3.s.c 8
28.g odd 6 1 784.3.s.i 8
56.j odd 6 1 448.3.s.e 8
56.k odd 6 1 448.3.s.f 8
56.m even 6 1 448.3.s.f 8
56.p even 6 1 448.3.s.e 8
84.j odd 6 1 1008.3.cg.o 8
84.n even 6 1 1008.3.cg.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.o.a ✓ 8 7.c even 3 1
56.3.o.a ✓ 8 7.d odd 6 1
112.3.s.c 8 28.f even 6 1
112.3.s.c 8 28.g odd 6 1
392.3.c.c 8 1.a even 1 1 trivial
392.3.c.c 8 7.b odd 2 1 inner
392.3.o.c 8 7.c even 3 1
392.3.o.c 8 7.d odd 6 1
448.3.s.e 8 56.j odd 6 1
448.3.s.e 8 56.p even 6 1
448.3.s.f 8 56.k odd 6 1
448.3.s.f 8 56.m even 6 1
504.3.by.b 8 21.g even 6 1
504.3.by.b 8 21.h odd 6 1
784.3.c.h 8 4.b odd 2 1
784.3.c.h 8 28.d even 2 1
784.3.s.i 8 28.f even 6 1
784.3.s.i 8 28.g odd 6 1
1008.3.cg.o 8 84.j odd 6 1
1008.3.cg.o 8 84.n even 6 1
3528.3.f.d 8 3.b odd 2 1
3528.3.f.d 8 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 56T_{3}^{6} + 1022T_{3}^{4} + 6712T_{3}^{2} + 14161 \) acting on \(S_{3}^{\mathrm{new}}(392, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 56 T^{6} + \cdots + 14161 \) Copy content Toggle raw display
$5$ \( T^{8} + 124 T^{6} + \cdots + 124609 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 4 T^{3} + \cdots - 161)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 712 T^{6} + \cdots + 205520896 \) Copy content Toggle raw display
$17$ \( T^{8} + 908 T^{6} + \cdots + 711875761 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 1818937201 \) Copy content Toggle raw display
$23$ \( (T^{4} - 48 T^{3} + \cdots + 6223)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 52 T^{3} + \cdots - 1025504)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 222339597841 \) Copy content Toggle raw display
$37$ \( (T^{4} + 68 T^{3} + \cdots - 1620191)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 2179615227904 \) Copy content Toggle raw display
$43$ \( (T^{4} - 80 T^{3} + \cdots - 3066512)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 162997105441 \) Copy content Toggle raw display
$53$ \( (T^{4} - 28 T^{3} + \cdots + 4756897)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 6191643969 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 15423708853809 \) Copy content Toggle raw display
$67$ \( (T^{4} - 56 T^{3} + \cdots - 5899241)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 104 T^{3} + \cdots - 1612688)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 27500867992129 \) Copy content Toggle raw display
$79$ \( (T^{4} - 160 T^{3} + \cdots - 55457)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 546575022555136 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 16081439367889 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 7518651744256 \) Copy content Toggle raw display
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