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Properties

Label 342.2.g.f
Level $342$
Weight $2$
Character orbit 342.g
Analytic conductor $2.731$
Analytic rank $0$
Dimension $4$
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(163,342)]
 
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([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.163");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.g (of order \(3\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{2} + \beta_{2} q^{4} + (\beta_{2} + \beta_1 + 1) q^{5} + (\beta_{3} + 1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{2} + \beta_{2} q^{4} + (\beta_{2} + \beta_1 + 1) q^{5} + (\beta_{3} + 1) q^{7} - q^{8} + (\beta_{3} + \beta_{2} + \beta_1) q^{10} + (\beta_{3} + 2) q^{11} + 2 \beta_{2} q^{13} + (\beta_{2} - \beta_1 + 1) q^{14} + ( - \beta_{2} - 1) q^{16} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 4) q^{19} + (\beta_{3} - 1) q^{20} + (2 \beta_{2} - \beta_1 + 2) q^{22} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{23} + (2 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{25} - 2 q^{26} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{28} + (\beta_{3} + \beta_{2} + \beta_1) q^{29} + ( - \beta_{3} - 3) q^{31} - \beta_{2} q^{32} + ( - 6 \beta_{2} - 6) q^{35} + ( - \beta_{3} + 3) q^{37} + ( - \beta_{3} + 4 \beta_{2} + 2) q^{38} + ( - \beta_{2} - \beta_1 - 1) q^{40} + (5 \beta_{2} + 2 \beta_1 + 5) q^{41} + ( - 6 \beta_{2} + 2 \beta_1 - 6) q^{43} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{44} + ( - \beta_{3} + 1) q^{46} + ( - \beta_{3} - 7 \beta_{2} - \beta_1) q^{47} + (2 \beta_{3} + 1) q^{49} + (2 \beta_{3} - 3) q^{50} + ( - 2 \beta_{2} - 2) q^{52} + ( - 4 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{53} + ( - 5 \beta_{2} + \beta_1 - 5) q^{55} + ( - \beta_{3} - 1) q^{56} + (\beta_{3} - 1) q^{58} + 3 \beta_1 q^{59} + ( - 3 \beta_{3} - 7 \beta_{2} - 3 \beta_1) q^{61} + ( - 3 \beta_{2} + \beta_1 - 3) q^{62} + q^{64} + (2 \beta_{3} - 2) q^{65} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{67} - 6 \beta_{2} q^{70} + ( - 8 \beta_{2} - 2 \beta_1 - 8) q^{71} + ( - 7 \beta_{2} - 2 \beta_1 - 7) q^{73} + (3 \beta_{2} + \beta_1 + 3) q^{74} + (2 \beta_{2} + \beta_1 - 2) q^{76} + (3 \beta_{3} + 9) q^{77} + (4 \beta_{2} + 4) q^{79} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{80} + (2 \beta_{3} + 5 \beta_{2} + 2 \beta_1) q^{82} - 3 \beta_{3} q^{83} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1) q^{86} + ( - \beta_{3} - 2) q^{88} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{91} + (\beta_{2} + \beta_1 + 1) q^{92} + ( - \beta_{3} + 7) q^{94} + (\beta_{3} + 4 \beta_{2} + 4 \beta_1 + 9) q^{95} + (9 \beta_{2} + 2 \beta_1 + 9) q^{97} + (\beta_{2} - 2 \beta_1 + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 4 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 4 q^{7} - 4 q^{8} - 2 q^{10} + 8 q^{11} - 4 q^{13} + 2 q^{14} - 2 q^{16} + 12 q^{19} - 4 q^{20} + 4 q^{22} + 2 q^{23} - 6 q^{25} - 8 q^{26} - 2 q^{28} - 2 q^{29} - 12 q^{31} + 2 q^{32} - 12 q^{35} + 12 q^{37} - 2 q^{40} + 10 q^{41} - 12 q^{43} - 4 q^{44} + 4 q^{46} + 14 q^{47} + 4 q^{49} - 12 q^{50} - 4 q^{52} - 4 q^{53} - 10 q^{55} - 4 q^{56} - 4 q^{58} + 14 q^{61} - 6 q^{62} + 4 q^{64} - 8 q^{65} + 4 q^{67} + 12 q^{70} - 16 q^{71} - 14 q^{73} + 6 q^{74} - 12 q^{76} + 36 q^{77} + 8 q^{79} + 2 q^{80} - 10 q^{82} + 12 q^{86} - 8 q^{88} - 4 q^{91} + 2 q^{92} + 28 q^{94} + 28 q^{95} + 18 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 7 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{3} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
−1.32288 + 2.29129i
1.32288 − 2.29129i
−1.32288 − 2.29129i
1.32288 + 2.29129i
0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.822876 + 1.42526i 0 3.64575 −1.00000 0 0.822876 + 1.42526i
163.2 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 1.82288 − 3.15731i 0 −1.64575 −1.00000 0 −1.82288 − 3.15731i
235.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.822876 − 1.42526i 0 3.64575 −1.00000 0 0.822876 − 1.42526i
235.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.82288 + 3.15731i 0 −1.64575 −1.00000 0 −1.82288 + 3.15731i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.2.g.f 4
3.b odd 2 1 38.2.c.b ✓ 4
4.b odd 2 1 2736.2.s.v 4
12.b even 2 1 304.2.i.e 4
15.d odd 2 1 950.2.e.k 4
15.e even 4 2 950.2.j.g 8
19.c even 3 1 inner 342.2.g.f 4
19.c even 3 1 6498.2.a.ba 2
19.d odd 6 1 6498.2.a.bg 2
24.f even 2 1 1216.2.i.k 4
24.h odd 2 1 1216.2.i.l 4
57.d even 2 1 722.2.c.j 4
57.f even 6 1 722.2.a.g 2
57.f even 6 1 722.2.c.j 4
57.h odd 6 1 38.2.c.b ✓ 4
57.h odd 6 1 722.2.a.j 2
57.j even 18 6 722.2.e.o 12
57.l odd 18 6 722.2.e.n 12
76.g odd 6 1 2736.2.s.v 4
228.m even 6 1 304.2.i.e 4
228.m even 6 1 5776.2.a.ba 2
228.n odd 6 1 5776.2.a.z 2
285.n odd 6 1 950.2.e.k 4
285.v even 12 2 950.2.j.g 8
456.u even 6 1 1216.2.i.k 4
456.x odd 6 1 1216.2.i.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.c.b ✓ 4 3.b odd 2 1
38.2.c.b ✓ 4 57.h odd 6 1
304.2.i.e 4 12.b even 2 1
304.2.i.e 4 228.m even 6 1
342.2.g.f 4 1.a even 1 1 trivial
342.2.g.f 4 19.c even 3 1 inner
722.2.a.g 2 57.f even 6 1
722.2.a.j 2 57.h odd 6 1
722.2.c.j 4 57.d even 2 1
722.2.c.j 4 57.f even 6 1
722.2.e.n 12 57.l odd 18 6
722.2.e.o 12 57.j even 18 6
950.2.e.k 4 15.d odd 2 1
950.2.e.k 4 285.n odd 6 1
950.2.j.g 8 15.e even 4 2
950.2.j.g 8 285.v even 12 2
1216.2.i.k 4 24.f even 2 1
1216.2.i.k 4 456.u even 6 1
1216.2.i.l 4 24.h odd 2 1
1216.2.i.l 4 456.x odd 6 1
2736.2.s.v 4 4.b odd 2 1
2736.2.s.v 4 76.g odd 6 1
5776.2.a.z 2 228.n odd 6 1
5776.2.a.ba 2 228.m even 6 1
6498.2.a.ba 2 19.c even 3 1
6498.2.a.bg 2 19.d odd 6 1

Hecke kernels

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

\( T_{5}^{4} - 2T_{5}^{3} + 10T_{5}^{2} + 12T_{5} + 36 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 2 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T - 6)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 4 T - 3)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 12 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$29$ \( T^{4} + 2 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$31$ \( (T^{2} + 6 T + 2)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 6 T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 10 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$43$ \( T^{4} + 12 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$47$ \( T^{4} - 14 T^{3} + \cdots + 1764 \) Copy content Toggle raw display
$53$ \( T^{4} + 4 T^{3} + \cdots + 11664 \) Copy content Toggle raw display
$59$ \( T^{4} + 63T^{2} + 3969 \) Copy content Toggle raw display
$61$ \( T^{4} - 14 T^{3} + \cdots + 196 \) Copy content Toggle raw display
$67$ \( T^{4} - 4 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$71$ \( T^{4} + 16 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$73$ \( T^{4} + 14 T^{3} + \cdots + 441 \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 63)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 18 T^{3} + \cdots + 2809 \) Copy content Toggle raw display
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