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Properties

Label 392.6.i.o
Level $392$
Weight $6$
Character orbit 392.i
Analytic conductor $62.870$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [392,6,Mod(177,392)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("392.177"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(392, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 392.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,5,0,-81,0,0,0,-390] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(62.8704573667\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 200 x^{8} - 198 x^{7} + 34197 x^{6} - 16185 x^{5} + 1170401 x^{4} + 2020497 x^{3} + \cdots + 13068225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{18}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1 + 1) q^{3} + (\beta_{7} - \beta_{4} + 16 \beta_{3} + \cdots - \beta_1) q^{5} + (\beta_{8} + 2 \beta_{7} + \cdots - 4 \beta_1) q^{9} + ( - \beta_{9} - 2 \beta_{7} + \cdots - 72) q^{11}+ \cdots + (51 \beta_{6} - 234 \beta_{5} + \cdots + 49911) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 5 q^{3} - 81 q^{5} - 390 q^{9} - 361 q^{11} + 684 q^{13} - 2098 q^{15} - 1809 q^{17} - 1277 q^{19} - 911 q^{23} - 3940 q^{25} - 9502 q^{27} + 10884 q^{29} - 2187 q^{31} + 5553 q^{33} + 8181 q^{37}+ \cdots + 499596 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 200 x^{8} - 198 x^{7} + 34197 x^{6} - 16185 x^{5} + 1170401 x^{4} + 2020497 x^{3} + \cdots + 13068225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2025121474 \nu^{9} + 10800435830 \nu^{8} - 43579563150 \nu^{7} + 1409134397898 \nu^{6} + \cdots - 28\!\cdots\!70 ) / 22\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 12994831946063 \nu^{9} - 1220135688085 \nu^{8} + \cdots - 27\!\cdots\!35 ) / 27\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 32866953925838 \nu^{9} - 572138277687598 \nu^{8} + \cdots - 19\!\cdots\!35 ) / 28\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 33695665706284 \nu^{9} - 50643606919436 \nu^{8} + 204346037569980 \nu^{7} + \cdots + 52\!\cdots\!50 ) / 28\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 45766530368302 \nu^{9} + \cdots - 73\!\cdots\!15 ) / 28\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 20\!\cdots\!81 \nu^{9} + \cdots + 43\!\cdots\!00 ) / 68\!\cdots\!83 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 63\!\cdots\!46 \nu^{9} + \cdots + 17\!\cdots\!00 ) / 68\!\cdots\!83 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 96\!\cdots\!87 \nu^{9} + \cdots - 17\!\cdots\!15 ) / 68\!\cdots\!83 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} + 2\beta_{7} + \beta_{5} - 2\beta_{4} + 320\beta_{3} + 2\beta_{2} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{6} + 3\beta_{5} + 52\beta_{4} - 589\beta_{2} + 465 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -50\beta_{9} - 353\beta_{8} - 585\beta_{7} + 50\beta_{6} - 90895\beta_{3} + 1303\beta _1 - 90895 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 700 \beta_{9} - 201 \beta_{8} + 5398 \beta_{7} - 201 \beta_{5} - 5398 \beta_{4} + 92640 \beta_{3} + \cdots - 47492 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -10693\beta_{6} - 58697\beta_{5} + 98936\beta_{4} - 310159\beta_{2} + 14511115 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 244329 \beta_{9} + 20814 \beta_{8} - 1929819 \beta_{7} + 244329 \beta_{6} - 45669810 \beta_{3} + \cdots - 45669810 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2012355 \beta_{9} + 9640298 \beta_{8} + 17649229 \beta_{7} + 9640298 \beta_{5} - 17649229 \beta_{4} + \cdots - 66003142 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -41980957\beta_{6} - 5257104\beta_{5} + 335224051\beta_{4} - 2628681934\beta_{2} + 9823967970 ) / 8 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{3}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
6.16047 + 10.6702i
3.36475 + 5.82792i
−0.319443 − 0.553292i
−2.57166 − 4.45425i
−6.63412 − 11.4906i
6.16047 − 10.6702i
3.36475 − 5.82792i
−0.319443 + 0.553292i
−2.57166 + 4.45425i
−6.63412 + 11.4906i
0 −11.8209 − 20.4745i 0 10.0228 − 17.3600i 0 0 0 −157.969 + 273.611i 0
177.2 0 −6.22951 − 10.7898i 0 −48.8402 + 84.5938i 0 0 0 43.8865 − 76.0136i 0
177.3 0 1.13889 + 1.97261i 0 25.2555 − 43.7438i 0 0 0 118.906 − 205.951i 0
177.4 0 5.64333 + 9.77453i 0 13.0334 − 22.5744i 0 0 0 57.8057 − 100.122i 0
177.5 0 13.7682 + 23.8473i 0 −39.9714 + 69.2326i 0 0 0 −257.629 + 446.226i 0
361.1 0 −11.8209 + 20.4745i 0 10.0228 + 17.3600i 0 0 0 −157.969 − 273.611i 0
361.2 0 −6.22951 + 10.7898i 0 −48.8402 − 84.5938i 0 0 0 43.8865 + 76.0136i 0
361.3 0 1.13889 − 1.97261i 0 25.2555 + 43.7438i 0 0 0 118.906 + 205.951i 0
361.4 0 5.64333 − 9.77453i 0 13.0334 + 22.5744i 0 0 0 57.8057 + 100.122i 0
361.5 0 13.7682 − 23.8473i 0 −39.9714 − 69.2326i 0 0 0 −257.629 − 446.226i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 177.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.6.i.o 10
7.b odd 2 1 56.6.i.b ✓ 10
7.c even 3 1 392.6.a.j 5
7.c even 3 1 inner 392.6.i.o 10
7.d odd 6 1 56.6.i.b ✓ 10
7.d odd 6 1 392.6.a.k 5
21.c even 2 1 504.6.s.b 10
21.g even 6 1 504.6.s.b 10
28.d even 2 1 112.6.i.f 10
28.f even 6 1 112.6.i.f 10
28.f even 6 1 784.6.a.bk 5
28.g odd 6 1 784.6.a.bl 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.i.b ✓ 10 7.b odd 2 1
56.6.i.b ✓ 10 7.d odd 6 1
112.6.i.f 10 28.d even 2 1
112.6.i.f 10 28.f even 6 1
392.6.a.j 5 7.c even 3 1
392.6.a.k 5 7.d odd 6 1
392.6.i.o 10 1.a even 1 1 trivial
392.6.i.o 10 7.c even 3 1 inner
504.6.s.b 10 21.c even 2 1
504.6.s.b 10 21.g even 6 1
784.6.a.bk 5 28.f even 6 1
784.6.a.bl 5 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} - 5 T_{3}^{9} + 815 T_{3}^{8} + 754 T_{3}^{7} + 540053 T_{3}^{6} - 550571 T_{3}^{5} + \cdots + 43481007441 \) acting on \(S_{6}^{\mathrm{new}}(392, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 43481007441 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 42\!\cdots\!01 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 40\!\cdots\!49 \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots - 165262458987360)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 15\!\cdots\!09 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 32\!\cdots\!89 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots + 32\!\cdots\!72)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 74\!\cdots\!41 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 63\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 13\!\cdots\!40)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots + 34\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 36\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 65\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 19\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 64\!\cdots\!21 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 12\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 50\!\cdots\!21 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots + 16\!\cdots\!68)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 40\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 15\!\cdots\!48)^{2} \) Copy content Toggle raw display
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