Properties

Label 342.2.n.c
Level $342$
Weight $2$
Character orbit 342.n
Analytic conductor $2.731$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [342,2,Mod(293,342)]
 
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([5, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.293");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.n (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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( - 2 \zeta_{6} + 1) q^{3} + q^{4} + ( - \zeta_{6} + 2) q^{5} + ( - 2 \zeta_{6} + 1) q^{6} + 5 \zeta_{6} q^{7} + q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + ( - 2 \zeta_{6} + 1) q^{3} + q^{4} + ( - \zeta_{6} + 2) q^{5} + ( - 2 \zeta_{6} + 1) q^{6} + 5 \zeta_{6} q^{7} + q^{8} - 3 q^{9} + ( - \zeta_{6} + 2) q^{10} + ( - \zeta_{6} + 2) q^{11} + ( - 2 \zeta_{6} + 1) q^{12} + 5 \zeta_{6} q^{14} - 3 \zeta_{6} q^{15} + q^{16} + ( - 3 \zeta_{6} - 3) q^{17} - 3 q^{18} + ( - 2 \zeta_{6} - 3) q^{19} + ( - \zeta_{6} + 2) q^{20} + ( - 5 \zeta_{6} + 10) q^{21} + ( - \zeta_{6} + 2) q^{22} + ( - 4 \zeta_{6} + 2) q^{23} + ( - 2 \zeta_{6} + 1) q^{24} + (2 \zeta_{6} - 2) q^{25} + (6 \zeta_{6} - 3) q^{27} + 5 \zeta_{6} q^{28} + (3 \zeta_{6} - 3) q^{29} - 3 \zeta_{6} q^{30} + ( - 5 \zeta_{6} - 5) q^{31} + q^{32} - 3 \zeta_{6} q^{33} + ( - 3 \zeta_{6} - 3) q^{34} + (5 \zeta_{6} + 5) q^{35} - 3 q^{36} + ( - 2 \zeta_{6} - 3) q^{38} + ( - \zeta_{6} + 2) q^{40} + 9 \zeta_{6} q^{41} + ( - 5 \zeta_{6} + 10) q^{42} + 8 q^{43} + ( - \zeta_{6} + 2) q^{44} + (3 \zeta_{6} - 6) q^{45} + ( - 4 \zeta_{6} + 2) q^{46} + (3 \zeta_{6} + 3) q^{47} + ( - 2 \zeta_{6} + 1) q^{48} + (18 \zeta_{6} - 18) q^{49} + (2 \zeta_{6} - 2) q^{50} + (9 \zeta_{6} - 9) q^{51} - 3 \zeta_{6} q^{53} + (6 \zeta_{6} - 3) q^{54} + ( - 3 \zeta_{6} + 3) q^{55} + 5 \zeta_{6} q^{56} + (8 \zeta_{6} - 7) q^{57} + (3 \zeta_{6} - 3) q^{58} - 3 \zeta_{6} q^{59} - 3 \zeta_{6} q^{60} + (7 \zeta_{6} - 7) q^{61} + ( - 5 \zeta_{6} - 5) q^{62} - 15 \zeta_{6} q^{63} + q^{64} - 3 \zeta_{6} q^{66} + (4 \zeta_{6} - 2) q^{67} + ( - 3 \zeta_{6} - 3) q^{68} - 6 q^{69} + (5 \zeta_{6} + 5) q^{70} + ( - 15 \zeta_{6} + 15) q^{71} - 3 q^{72} + (11 \zeta_{6} - 11) q^{73} + (2 \zeta_{6} + 2) q^{75} + ( - 2 \zeta_{6} - 3) q^{76} + (5 \zeta_{6} + 5) q^{77} + ( - 12 \zeta_{6} + 6) q^{79} + ( - \zeta_{6} + 2) q^{80} + 9 q^{81} + 9 \zeta_{6} q^{82} + (3 \zeta_{6} - 6) q^{83} + ( - 5 \zeta_{6} + 10) q^{84} - 9 q^{85} + 8 q^{86} + (3 \zeta_{6} + 3) q^{87} + ( - \zeta_{6} + 2) q^{88} - 15 \zeta_{6} q^{89} + (3 \zeta_{6} - 6) q^{90} + ( - 4 \zeta_{6} + 2) q^{92} + (15 \zeta_{6} - 15) q^{93} + (3 \zeta_{6} + 3) q^{94} + (\zeta_{6} - 8) q^{95} + ( - 2 \zeta_{6} + 1) q^{96} + ( - 8 \zeta_{6} + 4) q^{97} + (18 \zeta_{6} - 18) q^{98} + (3 \zeta_{6} - 6) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 3 q^{5} + 5 q^{7} + 2 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 3 q^{5} + 5 q^{7} + 2 q^{8} - 6 q^{9} + 3 q^{10} + 3 q^{11} + 5 q^{14} - 3 q^{15} + 2 q^{16} - 9 q^{17} - 6 q^{18} - 8 q^{19} + 3 q^{20} + 15 q^{21} + 3 q^{22} - 2 q^{25} + 5 q^{28} - 3 q^{29} - 3 q^{30} - 15 q^{31} + 2 q^{32} - 3 q^{33} - 9 q^{34} + 15 q^{35} - 6 q^{36} - 8 q^{38} + 3 q^{40} + 9 q^{41} + 15 q^{42} + 16 q^{43} + 3 q^{44} - 9 q^{45} + 9 q^{47} - 18 q^{49} - 2 q^{50} - 9 q^{51} - 3 q^{53} + 3 q^{55} + 5 q^{56} - 6 q^{57} - 3 q^{58} - 3 q^{59} - 3 q^{60} - 7 q^{61} - 15 q^{62} - 15 q^{63} + 2 q^{64} - 3 q^{66} - 9 q^{68} - 12 q^{69} + 15 q^{70} + 15 q^{71} - 6 q^{72} - 11 q^{73} + 6 q^{75} - 8 q^{76} + 15 q^{77} + 3 q^{80} + 18 q^{81} + 9 q^{82} - 9 q^{83} + 15 q^{84} - 18 q^{85} + 16 q^{86} + 9 q^{87} + 3 q^{88} - 15 q^{89} - 9 q^{90} - 15 q^{93} + 9 q^{94} - 15 q^{95} - 18 q^{98} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1 - \zeta_{6}\) \(\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1
0.500000 + 0.866025i
0.500000 0.866025i
1.00000 1.73205i 1.00000 1.50000 0.866025i 1.73205i 2.50000 + 4.33013i 1.00000 −3.00000 1.50000 0.866025i
335.1 1.00000 1.73205i 1.00000 1.50000 + 0.866025i 1.73205i 2.50000 4.33013i 1.00000 −3.00000 1.50000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 3T_{5} + 3 \) acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$7$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$19$ \( T^{2} + 8T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} + 12 \) Copy content Toggle raw display
$29$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$31$ \( T^{2} + 15T + 75 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$43$ \( (T - 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$61$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} + 12 \) Copy content Toggle raw display
$71$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$73$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$79$ \( T^{2} + 108 \) Copy content Toggle raw display
$83$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$89$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$97$ \( T^{2} + 48 \) Copy content Toggle raw display
show more
show less