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Properties

Label 392.4.i.o
Level $392$
Weight $4$
Character orbit 392.i
Analytic conductor $23.129$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [392,4,Mod(177,392)]
 
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([0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.177");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 392.i (of order \(3\), 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: \(23.1287487223\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.54095201243136.19
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 102x^{6} + 320x^{5} + 4283x^{4} - 9104x^{3} - 85298x^{2} + 89904x + 714364 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{3} + (10 \beta_{6} + \beta_{5}) q^{5} + ( - 3 \beta_{7} + 34 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{3} + (10 \beta_{6} + \beta_{5}) q^{5} + ( - 3 \beta_{7} + 34 \beta_1) q^{9} + (\beta_{7} - \beta_{3} + 29 \beta_1 + 29) q^{11} + (12 \beta_{6} + 9 \beta_{4} - 12 \beta_{2}) q^{13} + ( - 8 \beta_{3} + 28) q^{15} - 65 \beta_{2} q^{17} + (11 \beta_{6} + 17 \beta_{5}) q^{19} + ( - 8 \beta_{7} - 20 \beta_1) q^{23} + ( - 19 \beta_{7} + 19 \beta_{3} + \cdots - 112) q^{25}+ \cdots + ( - 53 \beta_{3} - 647) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 136 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 136 q^{9} + 116 q^{11} + 224 q^{15} + 80 q^{23} - 448 q^{25} + 72 q^{29} - 436 q^{37} - 2232 q^{39} + 744 q^{43} + 780 q^{51} - 976 q^{53} + 7624 q^{57} - 780 q^{65} + 1176 q^{67} - 4816 q^{71} - 56 q^{79} - 2104 q^{81} + 9880 q^{85} - 2376 q^{93} - 4032 q^{95} - 5176 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 102x^{6} + 320x^{5} + 4283x^{4} - 9104x^{3} - 85298x^{2} + 89904x + 714364 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{6} + 6\nu^{5} + 269\nu^{4} - 548\nu^{3} - 10445\nu^{2} + 10720\nu + 126196 ) / 26418 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2276 \nu^{7} - 7966 \nu^{6} - 249872 \nu^{5} + 644595 \nu^{4} + 11918032 \nu^{3} + \cdots + 95317544 ) / 181452033 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4552 \nu^{7} + 15932 \nu^{6} + 499744 \nu^{5} - 1289190 \nu^{4} - 23836064 \nu^{3} + \cdots - 372087121 ) / 181452033 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11543 \nu^{7} + 722003 \nu^{6} - 3076121 \nu^{5} - 55106985 \nu^{4} + 136362466 \nu^{3} + \cdots - 12298890268 ) / 362904066 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13819 \nu^{7} - 741511 \nu^{6} + 3408415 \nu^{5} + 58157643 \nu^{4} - 155808374 \nu^{3} + \cdots + 19017784100 ) / 362904066 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 27638 \nu^{7} - 96733 \nu^{6} - 2077565 \nu^{5} + 5435745 \nu^{4} + 60215911 \nu^{3} + \cdots + 313025636 ) / 362904066 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 101120 \nu^{7} - 353920 \nu^{6} - 10463726 \nu^{5} + 27044115 \nu^{4} + 352197748 \nu^{3} + \cdots + 1978335548 ) / 362904066 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{5} - 4\beta_{4} + \beta_{3} + 4\beta _1 + 57 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6\beta_{7} + 4\beta_{6} - 6\beta_{5} - 6\beta_{4} + 29\beta_{3} + 164\beta_{2} + 6\beta _1 + 85 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -12\beta_{7} - 232\beta_{5} - 216\beta_{4} + 57\beta_{3} + 224\beta_{2} + 676\beta _1 + 1673 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -560\beta_{7} + 1112\beta_{6} - 570\beta_{5} - 530\beta_{4} + 849\beta_{3} + 7000\beta_{2} + 1680\beta _1 + 4041 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1650 \beta_{7} + 2240 \beta_{6} - 10380 \beta_{5} - 8108 \beta_{4} + 2405 \beta_{3} + 16912 \beta_{2} + \cdots + 47725 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 32438 \beta_{7} + 108976 \beta_{6} - 34342 \beta_{5} - 26530 \beta_{4} + 22793 \beta_{3} + \cdots + 152993 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
5.10797 − 1.22474i
−4.10797 + 1.22474i
−5.52218 − 1.22474i
6.52218 + 1.22474i
5.10797 + 1.22474i
−4.10797 − 1.22474i
−5.52218 + 1.22474i
6.52218 − 1.22474i
0 −4.81898 − 8.34673i 0 2.95919 − 5.12547i 0 0 0 −32.9452 + 57.0628i 0
177.2 0 −2.69766 − 4.67249i 0 −10.4758 + 18.1447i 0 0 0 −1.05478 + 1.82693i 0
177.3 0 2.69766 + 4.67249i 0 10.4758 − 18.1447i 0 0 0 −1.05478 + 1.82693i 0
177.4 0 4.81898 + 8.34673i 0 −2.95919 + 5.12547i 0 0 0 −32.9452 + 57.0628i 0
361.1 0 −4.81898 + 8.34673i 0 2.95919 + 5.12547i 0 0 0 −32.9452 − 57.0628i 0
361.2 0 −2.69766 + 4.67249i 0 −10.4758 − 18.1447i 0 0 0 −1.05478 − 1.82693i 0
361.3 0 2.69766 − 4.67249i 0 10.4758 + 18.1447i 0 0 0 −1.05478 − 1.82693i 0
361.4 0 4.81898 − 8.34673i 0 −2.95919 − 5.12547i 0 0 0 −32.9452 − 57.0628i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 177.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.4.i.o 8
7.b odd 2 1 inner 392.4.i.o 8
7.c even 3 1 392.4.a.n ✓ 4
7.c even 3 1 inner 392.4.i.o 8
7.d odd 6 1 392.4.a.n ✓ 4
7.d odd 6 1 inner 392.4.i.o 8
28.f even 6 1 784.4.a.bh 4
28.g odd 6 1 784.4.a.bh 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.4.a.n ✓ 4 7.c even 3 1
392.4.a.n ✓ 4 7.d odd 6 1
392.4.i.o 8 1.a even 1 1 trivial
392.4.i.o 8 7.b odd 2 1 inner
392.4.i.o 8 7.c even 3 1 inner
392.4.i.o 8 7.d odd 6 1 inner
784.4.a.bh 4 28.f even 6 1
784.4.a.bh 4 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(392, [\chi])\):

\( T_{3}^{8} + 122T_{3}^{6} + 12180T_{3}^{4} + 329888T_{3}^{2} + 7311616 \) Copy content Toggle raw display
\( T_{5}^{8} + 474T_{5}^{6} + 209300T_{5}^{4} + 7288224T_{5}^{2} + 236421376 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 122 T^{6} + \cdots + 7311616 \) Copy content Toggle raw display
$5$ \( T^{8} + 474 T^{6} + \cdots + 236421376 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 58 T^{3} + \cdots + 529984)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 10242 T^{2} + 16257024)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 8450 T^{2} + 71402500)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 70\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T^{4} - 40 T^{3} + \cdots + 46676224)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 18 T - 5456)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( (T^{4} + 218 T^{3} + \cdots + 835903744)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 162324 T^{2} + 4262261796)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 186 T + 8536)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 41\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{4} + 488 T^{3} + \cdots + 525509776)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 58\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{4} - 588 T^{3} + \cdots + 1007681536)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 1204 T + 163072)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{4} + 28 T^{3} + \cdots + 5805220864)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 2714 T^{2} + 1547536)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 38\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( (T^{4} - 308708 T^{2} + 9932514244)^{2} \) Copy content Toggle raw display
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