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Properties

Label 392.3.g.m
Level $392$
Weight $3$
Character orbit 392.g
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(99,392)]
 
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([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.99");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 392.g (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: 8.0.292213762624.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 2x^{6} - 2x^{5} + 24x^{4} - 8x^{3} - 32x^{2} - 64x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + ( - \beta_{3} - \beta_{2} + 1) q^{3} + (\beta_{4} + 1) q^{4} + ( - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{5} + ( - \beta_{6} + \beta_{2} + \beta_1 + 3) q^{6} + (\beta_{6} - \beta_{5} - \beta_{3} + \cdots + 1) q^{8}+ \cdots + ( - \beta_{7} - \beta_{6} - \beta_{4} + \cdots + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + ( - \beta_{3} - \beta_{2} + 1) q^{3} + (\beta_{4} + 1) q^{4} + ( - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{5} + ( - \beta_{6} + \beta_{2} + \beta_1 + 3) q^{6} + (\beta_{6} - \beta_{5} - \beta_{3} + \cdots + 1) q^{8}+ \cdots + ( - 7 \beta_{7} - 7 \beta_{6} + \cdots + 32) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 8 q^{3} + 5 q^{4} + 22 q^{6} + 13 q^{8} + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} + 8 q^{3} + 5 q^{4} + 22 q^{6} + 13 q^{8} + 48 q^{9} - 16 q^{10} - 32 q^{11} - 30 q^{12} - 71 q^{16} + 80 q^{17} - 29 q^{18} - 56 q^{19} + 108 q^{20} + 66 q^{22} - 22 q^{24} - 16 q^{25} - 24 q^{26} + 32 q^{27} + 96 q^{30} - 19 q^{32} - 32 q^{33} - 74 q^{34} - 33 q^{36} + 14 q^{38} - 84 q^{40} - 128 q^{41} + 50 q^{44} - 152 q^{46} - 134 q^{48} + 33 q^{50} - 368 q^{51} - 132 q^{52} + 228 q^{54} + 56 q^{57} + 24 q^{58} - 104 q^{59} + 192 q^{60} - 120 q^{62} - 55 q^{64} - 72 q^{65} + 276 q^{66} + 304 q^{67} + 190 q^{68} - 209 q^{72} + 112 q^{73} + 8 q^{74} - 72 q^{75} - 70 q^{76} - 304 q^{78} - 124 q^{80} + 48 q^{81} - 450 q^{82} - 72 q^{83} + 210 q^{86} - 486 q^{88} + 512 q^{89} + 184 q^{90} - 472 q^{92} - 472 q^{94} - 558 q^{96} - 64 q^{97} + 256 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 2x^{6} - 2x^{5} + 24x^{4} - 8x^{3} - 32x^{2} - 64x + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + \nu^{5} - 2\nu^{3} + 4\nu^{2} + 16\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 3\nu^{6} + 6\nu^{5} + 10\nu^{4} - 16\nu^{3} - 24\nu^{2} + 128 ) / 64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - \nu^{6} - 2\nu^{5} - 2\nu^{4} + 24\nu^{3} - 8\nu^{2} - 32\nu - 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 3\nu^{6} + 6\nu^{5} + 10\nu^{4} - 16\nu^{3} - 88\nu^{2} + 64\nu + 128 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - \nu^{6} - 2\nu^{5} - 2\nu^{4} + 24\nu^{3} + 56\nu^{2} + 160\nu - 128 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + \nu^{6} - 6\nu^{5} + 26\nu^{4} + 8\nu^{3} + 24\nu^{2} - 64\nu + 192 ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + \nu^{6} - 2\nu^{5} - 2\nu^{4} + 8\nu^{3} + 16\nu^{2} - 16\nu - 32 ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 3\beta_{4} - \beta_{3} + 3\beta_{2} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{7} + \beta_{5} + \beta_{4} + 11\beta_{3} + 3\beta_{2} + 4\beta _1 + 9 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{7} + 8\beta_{6} + \beta_{5} + \beta_{4} + 3\beta_{3} + 3\beta_{2} + 4\beta _1 - 31 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{7} - 8\beta_{6} - 3\beta_{5} - 11\beta_{4} + 31\beta_{3} + 39\beta_{2} + 28\beta _1 - 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4\beta_{7} + 8\beta_{6} - 15\beta_{5} + 9\beta_{4} + 11\beta_{3} - 29\beta_{2} + 44\beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -92\beta_{7} + 8\beta_{6} - 3\beta_{5} - 27\beta_{4} + 31\beta_{3} - 25\beta_{2} + 12\beta _1 + 13 ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
−1.67467 + 1.09337i
−1.67467 − 1.09337i
−1.05468 + 1.69931i
−1.05468 − 1.69931i
1.37098 + 1.45617i
1.37098 − 1.45617i
1.85837 + 0.739226i
1.85837 − 0.739226i
−1.67467 − 1.09337i −4.56747 1.60906 + 3.66209i − 5.73252i 7.64902 + 4.99396i 0 1.30939 − 7.89212i 11.8618 −6.26779 + 9.60010i
99.2 −1.67467 + 1.09337i −4.56747 1.60906 − 3.66209i 5.73252i 7.64902 − 4.99396i 0 1.30939 + 7.89212i 11.8618 −6.26779 − 9.60010i
99.3 −1.05468 − 1.69931i 3.44128 −1.77532 + 3.58445i − 4.88287i −3.62943 − 5.84780i 0 7.96347 − 0.763618i 2.84239 −8.29751 + 5.14984i
99.4 −1.05468 + 1.69931i 3.44128 −1.77532 − 3.58445i 4.88287i −3.62943 + 5.84780i 0 7.96347 + 0.763618i 2.84239 −8.29751 − 5.14984i
99.5 1.37098 − 1.45617i 5.22363 −0.240837 − 3.99274i 6.26788i 7.16148 − 7.60647i 0 −6.14428 − 5.12327i 18.2863 9.12707 + 8.59313i
99.6 1.37098 + 1.45617i 5.22363 −0.240837 + 3.99274i − 6.26788i 7.16148 + 7.60647i 0 −6.14428 + 5.12327i 18.2863 9.12707 − 8.59313i
99.7 1.85837 − 0.739226i −0.0974366 2.90709 − 2.74751i − 3.46547i −0.181073 + 0.0720276i 0 3.37142 − 7.25490i −8.99051 −2.56177 − 6.44013i
99.8 1.85837 + 0.739226i −0.0974366 2.90709 + 2.74751i 3.46547i −0.181073 − 0.0720276i 0 3.37142 + 7.25490i −8.99051 −2.56177 + 6.44013i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d 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.g.m 8
4.b odd 2 1 1568.3.g.m 8
7.b odd 2 1 56.3.g.b ✓ 8
7.c even 3 2 392.3.k.n 16
7.d odd 6 2 392.3.k.o 16
8.b even 2 1 1568.3.g.m 8
8.d odd 2 1 inner 392.3.g.m 8
21.c even 2 1 504.3.g.b 8
28.d even 2 1 224.3.g.b 8
56.e even 2 1 56.3.g.b ✓ 8
56.h odd 2 1 224.3.g.b 8
56.k odd 6 2 392.3.k.n 16
56.m even 6 2 392.3.k.o 16
84.h odd 2 1 2016.3.g.b 8
112.j even 4 2 1792.3.d.j 16
112.l odd 4 2 1792.3.d.j 16
168.e odd 2 1 504.3.g.b 8
168.i even 2 1 2016.3.g.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.g.b ✓ 8 7.b odd 2 1
56.3.g.b ✓ 8 56.e even 2 1
224.3.g.b 8 28.d even 2 1
224.3.g.b 8 56.h odd 2 1
392.3.g.m 8 1.a even 1 1 trivial
392.3.g.m 8 8.d odd 2 1 inner
392.3.k.n 16 7.c even 3 2
392.3.k.n 16 56.k odd 6 2
392.3.k.o 16 7.d odd 6 2
392.3.k.o 16 56.m even 6 2
504.3.g.b 8 21.c even 2 1
504.3.g.b 8 168.e odd 2 1
1568.3.g.m 8 4.b odd 2 1
1568.3.g.m 8 8.b even 2 1
1792.3.d.j 16 112.j even 4 2
1792.3.d.j 16 112.l odd 4 2
2016.3.g.b 8 84.h odd 2 1
2016.3.g.b 8 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 4T_{3}^{3} - 22T_{3}^{2} + 80T_{3} + 8 \) acting on \(S_{3}^{\mathrm{new}}(392, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{4} - 4 T^{3} - 22 T^{2} + \cdots + 8)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 108 T^{6} + \cdots + 369664 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 16 T^{3} + \cdots - 864)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 908 T^{6} + \cdots + 133448704 \) Copy content Toggle raw display
$17$ \( (T^{4} - 40 T^{3} + \cdots - 752)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 28 T^{3} + \cdots + 1288)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 15607005184 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 13389266944 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 61917364224 \) Copy content Toggle raw display
$37$ \( T^{8} + 7440 T^{6} + \cdots + 554696704 \) Copy content Toggle raw display
$41$ \( (T^{4} + 64 T^{3} + \cdots + 837776)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 2716 T^{2} + \cdots - 217952)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 7542537191424 \) Copy content Toggle raw display
$53$ \( T^{8} + 3552 T^{6} + \cdots + 16777216 \) Copy content Toggle raw display
$59$ \( (T^{4} + 52 T^{3} + \cdots - 11252856)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 3223777158144 \) Copy content Toggle raw display
$67$ \( (T^{4} - 152 T^{3} + \cdots - 791808)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 221437256269824 \) Copy content Toggle raw display
$73$ \( (T^{4} - 56 T^{3} + \cdots - 1726704)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 89369947930624 \) Copy content Toggle raw display
$83$ \( (T^{4} + 36 T^{3} + \cdots + 3180872)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 256 T^{3} + \cdots - 618736)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 32 T^{3} + \cdots + 9539216)^{2} \) Copy content Toggle raw display
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