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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

Character values

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

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

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 \(S_{2}^{\mathrm{new}}(342, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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