Properties

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

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Newspace parameters

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

Self dual: no
Analytic conductor: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + q^{7} + q^{8} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + q^{7} + q^{8} + 2 q^{11} + 3 \zeta_{6} q^{13} + ( -1 + \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( 4 - 4 \zeta_{6} ) q^{17} + ( 3 + 2 \zeta_{6} ) q^{19} + ( -2 + 2 \zeta_{6} ) q^{22} + 4 \zeta_{6} q^{23} + 5 \zeta_{6} q^{25} -3 q^{26} -\zeta_{6} q^{28} -3 q^{31} -\zeta_{6} q^{32} + 4 \zeta_{6} q^{34} -5 q^{37} + ( -5 + 3 \zeta_{6} ) q^{38} + ( 4 - 4 \zeta_{6} ) q^{41} + ( 9 - 9 \zeta_{6} ) q^{43} -2 \zeta_{6} q^{44} -4 q^{46} + 10 \zeta_{6} q^{47} -6 q^{49} -5 q^{50} + ( 3 - 3 \zeta_{6} ) q^{52} -4 \zeta_{6} q^{53} + q^{56} + ( -14 + 14 \zeta_{6} ) q^{59} -11 \zeta_{6} q^{61} + ( 3 - 3 \zeta_{6} ) q^{62} + q^{64} -3 \zeta_{6} q^{67} -4 q^{68} + ( 14 - 14 \zeta_{6} ) q^{71} + ( 11 - 11 \zeta_{6} ) q^{73} + ( 5 - 5 \zeta_{6} ) q^{74} + ( 2 - 5 \zeta_{6} ) q^{76} + 2 q^{77} + ( -1 + \zeta_{6} ) q^{79} + 4 \zeta_{6} q^{82} -8 q^{83} + 9 \zeta_{6} q^{86} + 2 q^{88} -14 \zeta_{6} q^{89} + 3 \zeta_{6} q^{91} + ( 4 - 4 \zeta_{6} ) q^{92} -10 q^{94} + ( -2 + 2 \zeta_{6} ) q^{97} + ( 6 - 6 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} + 2q^{7} + 2q^{8} + O(q^{10}) \) \( 2q - q^{2} - q^{4} + 2q^{7} + 2q^{8} + 4q^{11} + 3q^{13} - q^{14} - q^{16} + 4q^{17} + 8q^{19} - 2q^{22} + 4q^{23} + 5q^{25} - 6q^{26} - q^{28} - 6q^{31} - q^{32} + 4q^{34} - 10q^{37} - 7q^{38} + 4q^{41} + 9q^{43} - 2q^{44} - 8q^{46} + 10q^{47} - 12q^{49} - 10q^{50} + 3q^{52} - 4q^{53} + 2q^{56} - 14q^{59} - 11q^{61} + 3q^{62} + 2q^{64} - 3q^{67} - 8q^{68} + 14q^{71} + 11q^{73} + 5q^{74} - q^{76} + 4q^{77} - q^{79} + 4q^{82} - 16q^{83} + 9q^{86} + 4q^{88} - 14q^{89} + 3q^{91} + 4q^{92} - 20q^{94} - 2q^{97} + 6q^{98} + O(q^{100}) \)

Character values

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

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 1.00000 1.00000 0 0
235.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 1.00000 1.00000 0 0
\(n\): 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.c 2
3.b odd 2 1 114.2.e.b 2
4.b odd 2 1 2736.2.s.k 2
12.b even 2 1 912.2.q.b 2
19.c even 3 1 inner 342.2.g.c 2
19.c even 3 1 6498.2.a.u 1
19.d odd 6 1 6498.2.a.g 1
57.f even 6 1 2166.2.a.h 1
57.h odd 6 1 114.2.e.b 2
57.h odd 6 1 2166.2.a.b 1
76.g odd 6 1 2736.2.s.k 2
228.m even 6 1 912.2.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.e.b 2 3.b odd 2 1
114.2.e.b 2 57.h odd 6 1
342.2.g.c 2 1.a even 1 1 trivial
342.2.g.c 2 19.c even 3 1 inner
912.2.q.b 2 12.b even 2 1
912.2.q.b 2 228.m even 6 1
2166.2.a.b 1 57.h odd 6 1
2166.2.a.h 1 57.f even 6 1
2736.2.s.k 2 4.b odd 2 1
2736.2.s.k 2 76.g odd 6 1
6498.2.a.g 1 19.d odd 6 1
6498.2.a.u 1 19.c even 3 1

Hecke kernels

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

\( T_{5} \)
\( T_{7} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( 9 - 3 T + T^{2} \)
$17$ \( 16 - 4 T + T^{2} \)
$19$ \( 19 - 8 T + T^{2} \)
$23$ \( 16 - 4 T + T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( 3 + T )^{2} \)
$37$ \( ( 5 + T )^{2} \)
$41$ \( 16 - 4 T + T^{2} \)
$43$ \( 81 - 9 T + T^{2} \)
$47$ \( 100 - 10 T + T^{2} \)
$53$ \( 16 + 4 T + T^{2} \)
$59$ \( 196 + 14 T + T^{2} \)
$61$ \( 121 + 11 T + T^{2} \)
$67$ \( 9 + 3 T + T^{2} \)
$71$ \( 196 - 14 T + T^{2} \)
$73$ \( 121 - 11 T + T^{2} \)
$79$ \( 1 + T + T^{2} \)
$83$ \( ( 8 + T )^{2} \)
$89$ \( 196 + 14 T + T^{2} \)
$97$ \( 4 + 2 T + T^{2} \)
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