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Properties

Label 392.3.g.i
Level $392$
Weight $3$
Character orbit 392.g
Analytic conductor $10.681$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.15582448.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 13x^{4} - 21x^{3} + 20x^{2} - 10x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + (\beta_{3} + \beta_{2} + \beta_1) q^{3} + (\beta_{4} + \beta_1 + 1) q^{4} + ( - \beta_{5} + \beta_{4} + \cdots - \beta_{2}) q^{5}+ \cdots + (\beta_{3} + \beta_{2} - 4 \beta_1 + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + (\beta_{3} + \beta_{2} + \beta_1) q^{3} + (\beta_{4} + \beta_1 + 1) q^{4} + ( - \beta_{5} + \beta_{4} + \cdots - \beta_{2}) q^{5}+ \cdots + ( - 21 \beta_{3} - 21 \beta_{2} + \cdots + 24) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} - 6 q^{3} + 4 q^{4} + 28 q^{6} + 4 q^{8} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} - 6 q^{3} + 4 q^{4} + 28 q^{6} + 4 q^{8} + 40 q^{9} - 6 q^{10} - 30 q^{11} + 32 q^{12} - 16 q^{16} + 30 q^{17} + 16 q^{18} + 78 q^{19} - 24 q^{20} + 12 q^{22} - 76 q^{24} + 92 q^{25} - 128 q^{26} - 78 q^{27} + 16 q^{30} - 112 q^{32} - 78 q^{33} - 38 q^{34} - 124 q^{36} + 80 q^{38} + 44 q^{40} + 116 q^{41} - 100 q^{43} - 132 q^{44} + 156 q^{46} - 88 q^{48} + 24 q^{50} - 10 q^{51} + 132 q^{52} - 36 q^{54} + 166 q^{57} - 4 q^{58} - 110 q^{59} - 84 q^{60} + 48 q^{62} - 80 q^{64} + 32 q^{65} - 138 q^{66} - 434 q^{67} + 96 q^{68} + 328 q^{72} + 102 q^{73} + 34 q^{74} - 60 q^{75} + 84 q^{76} + 360 q^{78} - 256 q^{80} + 82 q^{81} - 24 q^{82} + 268 q^{83} - 240 q^{86} + 204 q^{88} + 214 q^{89} - 220 q^{90} + 80 q^{92} - 16 q^{94} + 48 q^{96} + 76 q^{97} + 252 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 13x^{4} - 21x^{3} + 20x^{2} - 10x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{4} - 2\nu^{3} + 10\nu^{2} - 9\nu + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{5} - 2\nu^{4} + 11\nu^{3} - 10\nu^{2} + 10\nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{5} + 3\nu^{4} - 13\nu^{3} + 21\nu^{2} - 20\nu + 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -6\nu^{5} + 15\nu^{4} - 70\nu^{3} + 90\nu^{2} - 69\nu + 20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -6\nu^{5} + 15\nu^{4} - 70\nu^{3} + 90\nu^{2} - 71\nu + 21 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{4} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + \beta_{4} + 2\beta_{3} + 2\beta_{2} - 2\beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9\beta_{5} - 8\beta_{4} + 6\beta_{2} - 3\beta _1 - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 19\beta_{5} - 17\beta_{4} - 20\beta_{3} - 8\beta_{2} + 16\beta _1 + 37 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -61\beta_{5} + 54\beta_{4} - 20\beta_{3} - 60\beta_{2} + 45\beta _1 + 106 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
0.500000 + 2.94141i
0.500000 − 2.94141i
0.500000 + 0.759064i
0.500000 − 0.759064i
0.500000 + 0.148124i
0.500000 − 0.148124i
−1.88766 − 0.660851i −1.64878 3.12655 + 2.49493i 4.56111i 3.11234 + 1.08960i 0 −4.25310 − 6.77577i −6.28154 3.01422 − 8.60985i
99.2 −1.88766 + 0.660851i −1.64878 3.12655 − 2.49493i − 4.56111i 3.11234 − 1.08960i 0 −4.25310 + 6.77577i −6.28154 3.01422 + 8.60985i
99.3 −0.789608 − 1.83753i −5.33225 −2.75304 + 2.90186i − 2.15693i 4.21039 + 9.79818i 0 7.50608 + 2.76746i 19.4329 −3.96343 + 1.70313i
99.4 −0.789608 + 1.83753i −5.33225 −2.75304 − 2.90186i 2.15693i 4.21039 − 9.79818i 0 7.50608 − 2.76746i 19.4329 −3.96343 − 1.70313i
99.5 1.67727 − 1.08938i 3.98103 1.62649 − 3.65439i − 1.88252i 6.67727 − 4.33687i 0 −1.25297 − 7.90127i 6.84860 −2.05079 − 3.15750i
99.6 1.67727 + 1.08938i 3.98103 1.62649 + 3.65439i 1.88252i 6.67727 + 4.33687i 0 −1.25297 + 7.90127i 6.84860 −2.05079 + 3.15750i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.6
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.i 6
4.b odd 2 1 1568.3.g.l 6
7.b odd 2 1 392.3.g.j 6
7.c even 3 2 392.3.k.l 12
7.d odd 6 2 56.3.k.d ✓ 12
8.b even 2 1 1568.3.g.l 6
8.d odd 2 1 inner 392.3.g.i 6
28.d even 2 1 1568.3.g.j 6
28.f even 6 2 224.3.o.d 12
56.e even 2 1 392.3.g.j 6
56.h odd 2 1 1568.3.g.j 6
56.j odd 6 2 224.3.o.d 12
56.k odd 6 2 392.3.k.l 12
56.m even 6 2 56.3.k.d ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.k.d ✓ 12 7.d odd 6 2
56.3.k.d ✓ 12 56.m even 6 2
224.3.o.d 12 28.f even 6 2
224.3.o.d 12 56.j odd 6 2
392.3.g.i 6 1.a even 1 1 trivial
392.3.g.i 6 8.d odd 2 1 inner
392.3.g.j 6 7.b odd 2 1
392.3.g.j 6 56.e even 2 1
392.3.k.l 12 7.c even 3 2
392.3.k.l 12 56.k odd 6 2
1568.3.g.j 6 28.d even 2 1
1568.3.g.j 6 56.h odd 2 1
1568.3.g.l 6 4.b odd 2 1
1568.3.g.l 6 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + 3T_{3}^{2} - 19T_{3} - 35 \) acting on \(S_{3}^{\mathrm{new}}(392, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 2 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( (T^{3} + 3 T^{2} - 19 T - 35)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} + 29 T^{4} + \cdots + 343 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( (T^{3} + 15 T^{2} + \cdots - 513)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 928 T^{4} + \cdots + 20420848 \) Copy content Toggle raw display
$17$ \( (T^{3} - 15 T^{2} + \cdots + 35)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 39 T^{2} + \cdots - 539)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 2481 T^{4} + \cdots + 103165447 \) Copy content Toggle raw display
$29$ \( T^{6} + 1384 T^{4} + \cdots + 35753200 \) Copy content Toggle raw display
$31$ \( T^{6} + 2205 T^{4} + \cdots + 306307575 \) Copy content Toggle raw display
$37$ \( T^{6} + 2429 T^{4} + \cdots + 64563583 \) Copy content Toggle raw display
$41$ \( (T^{3} - 58 T^{2} + \cdots + 106036)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 50 T^{2} + \cdots - 77000)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 5967253663 \) Copy content Toggle raw display
$53$ \( T^{6} + 6293 T^{4} + \cdots + 91073143 \) Copy content Toggle raw display
$59$ \( (T^{3} + 55 T^{2} + \cdots - 6559)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 4507234375 \) Copy content Toggle raw display
$67$ \( (T^{3} + 217 T^{2} + \cdots + 368695)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 1372000000 \) Copy content Toggle raw display
$73$ \( (T^{3} - 51 T^{2} + \cdots - 46305)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 41185599175 \) Copy content Toggle raw display
$83$ \( (T^{3} - 134 T^{2} + \cdots + 185080)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 107 T^{2} + \cdots + 908663)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 38 T^{2} + \cdots + 117740)^{2} \) Copy content Toggle raw display
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