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Properties

Label 342.2.j.d
Level $342$
Weight $2$
Character orbit 342.j
Analytic conductor $2.731$
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] = [342,2,Mod(65,342)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(342, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.j (of order \(6\), degree \(2\), 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: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.764411904.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 6x^{6} + 21x^{4} - 54x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + ( - \beta_{6} - \beta_1) q^{3} + ( - \beta_{3} - 1) q^{4} + ( - \beta_{5} + 2 \beta_{3} + 1) q^{5} + ( - \beta_{6} - \beta_{4}) q^{6} + ( - \beta_{7} - \beta_{6} - 2 \beta_{3} + \cdots - 2) q^{7}+ \cdots + ( - 2 \beta_{5} - \beta_{3} + \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + ( - \beta_{6} - \beta_1) q^{3} + ( - \beta_{3} - 1) q^{4} + ( - \beta_{5} + 2 \beta_{3} + 1) q^{5} + ( - \beta_{6} - \beta_{4}) q^{6} + ( - \beta_{7} - \beta_{6} - 2 \beta_{3} + \cdots - 2) q^{7}+ \cdots + (2 \beta_{6} - 2 \beta_{5} + \beta_{4} + \cdots - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{4} - 8 q^{7} - 8 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 4 q^{4} - 8 q^{7} - 8 q^{8} + 12 q^{9} + 12 q^{10} - 16 q^{14} - 4 q^{16} + 12 q^{17} + 8 q^{19} + 12 q^{20} - 8 q^{28} - 24 q^{29} - 24 q^{31} + 4 q^{32} + 24 q^{35} - 12 q^{36} + 16 q^{38} + 24 q^{39} - 24 q^{41} + 12 q^{42} - 8 q^{43} - 12 q^{45} + 24 q^{51} + 12 q^{53} - 16 q^{55} + 8 q^{56} - 12 q^{57} - 12 q^{58} - 8 q^{61} - 24 q^{62} - 24 q^{63} + 8 q^{64} + 12 q^{66} - 24 q^{67} - 12 q^{68} + 24 q^{69} + 24 q^{71} - 12 q^{72} + 4 q^{73} - 24 q^{74} + 8 q^{76} + 12 q^{77} + 24 q^{78} + 24 q^{79} - 12 q^{80} - 12 q^{81} - 12 q^{82} - 24 q^{83} + 12 q^{84} - 4 q^{85} + 8 q^{86} - 48 q^{87} + 12 q^{89} + 24 q^{90} + 48 q^{91} + 60 q^{93} + 24 q^{94} - 24 q^{95} + 24 q^{97} - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 6x^{6} + 21x^{4} - 54x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 9\nu^{4} - 3\nu^{2} + 18 ) / 45 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{6} + 3\nu^{4} - 6\nu^{2} - 9 ) / 45 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4\nu^{7} - 6\nu^{5} + 57\nu^{3} - 27\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - 4\nu^{4} + 18\nu^{2} - 33 ) / 15 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 6\nu^{5} - 21\nu^{3} + 27\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} + 3\nu^{5} - 6\nu^{3} - 9\nu ) / 45 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{3} + \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} + 3\beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{3} + 6\beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{7} + 6\beta_{6} + 12\beta_{4} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3\beta_{5} - 30\beta_{3} + 6\beta_{2} - 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -24\beta_{7} + 9\beta_{6} + 9\beta_{4} - 12\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(-\beta_{3}\) \(-\beta_{3}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
1.69185 + 0.370982i
1.27970 − 1.16721i
−1.27970 + 1.16721i
−1.69185 − 0.370982i
1.69185 − 0.370982i
1.27970 + 1.16721i
−1.27970 − 1.16721i
−1.69185 + 0.370982i
0.500000 + 0.866025i −1.69185 + 0.370982i −0.500000 + 0.866025i − 3.14626i −1.16721 − 1.27970i −2.16721 + 3.75371i −1.00000 2.72474 − 1.25529i 2.72474 − 1.57313i
65.2 0.500000 + 0.866025i −1.27970 − 1.16721i −0.500000 + 0.866025i − 0.317837i 0.370982 − 1.69185i −0.629018 + 1.08949i −1.00000 0.275255 + 2.98735i 0.275255 − 0.158919i
65.3 0.500000 + 0.866025i 1.27970 + 1.16721i −0.500000 + 0.866025i − 0.317837i −0.370982 + 1.69185i −1.37098 + 2.37461i −1.00000 0.275255 + 2.98735i 0.275255 − 0.158919i
65.4 0.500000 + 0.866025i 1.69185 − 0.370982i −0.500000 + 0.866025i − 3.14626i 1.16721 + 1.27970i 0.167207 − 0.289611i −1.00000 2.72474 − 1.25529i 2.72474 − 1.57313i
221.1 0.500000 − 0.866025i −1.69185 − 0.370982i −0.500000 − 0.866025i 3.14626i −1.16721 + 1.27970i −2.16721 − 3.75371i −1.00000 2.72474 + 1.25529i 2.72474 + 1.57313i
221.2 0.500000 − 0.866025i −1.27970 + 1.16721i −0.500000 − 0.866025i 0.317837i 0.370982 + 1.69185i −0.629018 − 1.08949i −1.00000 0.275255 − 2.98735i 0.275255 + 0.158919i
221.3 0.500000 − 0.866025i 1.27970 − 1.16721i −0.500000 − 0.866025i 0.317837i −0.370982 − 1.69185i −1.37098 − 2.37461i −1.00000 0.275255 − 2.98735i 0.275255 + 0.158919i
221.4 0.500000 − 0.866025i 1.69185 + 0.370982i −0.500000 − 0.866025i 3.14626i 1.16721 − 1.27970i 0.167207 + 0.289611i −1.00000 2.72474 + 1.25529i 2.72474 + 1.57313i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.k even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.2.j.d ✓ 8
3.b odd 2 1 1026.2.j.d 8
9.c even 3 1 1026.2.n.d 8
9.d odd 6 1 342.2.n.d yes 8
19.d odd 6 1 342.2.n.d yes 8
57.f even 6 1 1026.2.n.d 8
171.k even 6 1 inner 342.2.j.d ✓ 8
171.s odd 6 1 1026.2.j.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.j.d ✓ 8 1.a even 1 1 trivial
342.2.j.d ✓ 8 171.k even 6 1 inner
342.2.n.d yes 8 9.d odd 6 1
342.2.n.d yes 8 19.d odd 6 1
1026.2.j.d 8 3.b odd 2 1
1026.2.j.d 8 171.s odd 6 1
1026.2.n.d 8 9.c even 3 1
1026.2.n.d 8 57.f even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\):

\( T_{5}^{4} + 10T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{8} + 8T_{7}^{7} + 46T_{7}^{6} + 128T_{7}^{5} + 265T_{7}^{4} + 224T_{7}^{3} + 154T_{7}^{2} - 40T_{7} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} - 6 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} + 10 T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 8 T^{7} + \cdots + 25 \) Copy content Toggle raw display
$11$ \( T^{8} - 22 T^{6} + \cdots + 361 \) Copy content Toggle raw display
$13$ \( T^{8} - 36 T^{6} + \cdots + 90000 \) Copy content Toggle raw display
$17$ \( T^{8} - 12 T^{7} + \cdots + 36481 \) Copy content Toggle raw display
$19$ \( T^{8} - 8 T^{7} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( T^{8} - 40 T^{6} + \cdots + 1600 \) Copy content Toggle raw display
$29$ \( (T^{4} + 12 T^{3} + \cdots - 1719)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 24 T^{7} + \cdots + 13995081 \) Copy content Toggle raw display
$37$ \( T^{8} + 312 T^{6} + \cdots + 19184400 \) Copy content Toggle raw display
$41$ \( (T^{4} + 12 T^{3} + \cdots - 303)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 8 T^{7} + \cdots + 400 \) Copy content Toggle raw display
$47$ \( T^{8} + 188 T^{6} + \cdots + 44521 \) Copy content Toggle raw display
$53$ \( T^{8} - 12 T^{7} + \cdots + 5377761 \) Copy content Toggle raw display
$59$ \( (T^{4} - 66 T^{2} + \cdots - 285)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 4 T^{3} + \cdots - 863)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 24 T^{7} + \cdots + 230400 \) Copy content Toggle raw display
$71$ \( T^{8} - 24 T^{7} + \cdots + 1320201 \) Copy content Toggle raw display
$73$ \( T^{8} - 4 T^{7} + \cdots + 10336225 \) Copy content Toggle raw display
$79$ \( T^{8} - 24 T^{7} + \cdots + 32400 \) Copy content Toggle raw display
$83$ \( T^{8} + 24 T^{7} + \cdots + 841 \) Copy content Toggle raw display
$89$ \( T^{8} - 12 T^{7} + \cdots + 2331729 \) Copy content Toggle raw display
$97$ \( T^{8} - 24 T^{7} + \cdots + 2396304 \) Copy content Toggle raw display
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