Properties

Label 342.2.s.a
Level $342$
Weight $2$
Character orbit 342.s
Analytic conductor $2.731$
Analytic rank $0$
Dimension $4$
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.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
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

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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( -\beta_{1} + \beta_{3} ) q^{5} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{7} + q^{8} +O(q^{10})\) \( q -\beta_{2} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( -\beta_{1} + \beta_{3} ) q^{5} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{7} + q^{8} + \beta_{1} q^{10} + ( 2 - 4 \beta_{2} + 2 \beta_{3} ) q^{11} + ( 1 + 3 \beta_{1} + \beta_{2} ) q^{13} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{14} -\beta_{2} q^{16} + ( 4 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{17} + ( 5 - 2 \beta_{2} ) q^{19} -\beta_{3} q^{20} + ( -4 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{22} + 4 \beta_{1} q^{23} + ( -3 + 3 \beta_{2} ) q^{25} + ( 1 - 2 \beta_{2} - 3 \beta_{3} ) q^{26} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{28} + ( \beta_{1} - 2 \beta_{3} ) q^{29} + ( 3 - 6 \beta_{2} + 3 \beta_{3} ) q^{31} + ( -1 + \beta_{2} ) q^{32} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{34} + ( -4 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{35} + ( -1 + 2 \beta_{2} - 3 \beta_{3} ) q^{37} + ( -2 - 3 \beta_{2} ) q^{38} + ( -\beta_{1} + \beta_{3} ) q^{40} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{43} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{44} -4 \beta_{3} q^{46} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{47} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{49} + 3 q^{50} + ( -2 - 3 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{52} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{53} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{55} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{56} + ( -2 \beta_{1} + \beta_{3} ) q^{58} + ( \beta_{1} + 12 \beta_{2} + \beta_{3} ) q^{59} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{61} + ( -6 + 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{62} + q^{64} + ( -6 - 2 \beta_{1} + \beta_{3} ) q^{65} + ( -1 - 6 \beta_{1} - \beta_{2} ) q^{67} + ( -2 + 4 \beta_{2} - \beta_{3} ) q^{68} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{70} + 6 \beta_{2} q^{71} + ( -2 \beta_{1} + 7 \beta_{2} - 2 \beta_{3} ) q^{73} + ( 2 - 3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{74} + ( -3 + 5 \beta_{2} ) q^{76} + ( -6 + 12 \beta_{2} - 8 \beta_{3} ) q^{77} + ( 6 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{79} + \beta_{1} q^{80} + ( 6 - 12 \beta_{2} - \beta_{3} ) q^{83} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{85} + ( 1 - 2 \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{86} + ( 2 - 4 \beta_{2} + 2 \beta_{3} ) q^{88} + ( -5 \beta_{1} + 10 \beta_{3} ) q^{89} + ( 5 + 5 \beta_{2} ) q^{91} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{92} + ( -2 + 4 \beta_{2} - \beta_{3} ) q^{94} + ( -3 \beta_{1} + 5 \beta_{3} ) q^{95} + ( -12 - 6 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} ) q^{97} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - 2q^{4} - 4q^{7} + 4q^{8} + O(q^{10}) \) \( 4q - 2q^{2} - 2q^{4} - 4q^{7} + 4q^{8} + 6q^{13} + 2q^{14} - 2q^{16} + 12q^{17} + 16q^{19} - 12q^{22} - 6q^{25} + 2q^{28} - 2q^{32} - 12q^{34} - 12q^{35} - 14q^{38} + 2q^{43} + 12q^{44} - 12q^{47} + 12q^{50} - 6q^{52} - 8q^{55} - 4q^{56} + 24q^{59} + 2q^{61} - 18q^{62} + 4q^{64} - 24q^{65} - 6q^{67} + 12q^{70} + 12q^{71} + 14q^{73} + 6q^{74} - 2q^{76} + 18q^{79} + 4q^{85} + 2q^{86} + 30q^{91} - 36q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 0.707107i
−1.22474 + 0.707107i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −1.22474 + 0.707107i 0 1.44949 1.00000 0 1.22474 + 0.707107i
107.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.22474 0.707107i 0 −3.44949 1.00000 0 −1.22474 0.707107i
179.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −1.22474 0.707107i 0 1.44949 1.00000 0 1.22474 0.707107i
179.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 1.22474 + 0.707107i 0 −3.44949 1.00000 0 −1.22474 + 0.707107i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.f 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.s.a 4
3.b odd 2 1 342.2.s.b yes 4
4.b odd 2 1 2736.2.dc.b 4
12.b even 2 1 2736.2.dc.a 4
19.d odd 6 1 342.2.s.b yes 4
57.f even 6 1 inner 342.2.s.a 4
76.f even 6 1 2736.2.dc.a 4
228.n odd 6 1 2736.2.dc.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.s.a 4 1.a even 1 1 trivial
342.2.s.a 4 57.f even 6 1 inner
342.2.s.b yes 4 3.b odd 2 1
342.2.s.b yes 4 19.d odd 6 1
2736.2.dc.a 4 12.b even 2 1
2736.2.dc.a 4 76.f even 6 1
2736.2.dc.b 4 4.b odd 2 1
2736.2.dc.b 4 228.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{4} - 12 T_{17}^{3} + 58 T_{17}^{2} - 120 T_{17} + 100 \) acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 4 - 2 T^{2} + T^{4} \)
$7$ \( ( -5 + 2 T + T^{2} )^{2} \)
$11$ \( 16 + 40 T^{2} + T^{4} \)
$13$ \( 225 + 90 T - 3 T^{2} - 6 T^{3} + T^{4} \)
$17$ \( 100 - 120 T + 58 T^{2} - 12 T^{3} + T^{4} \)
$19$ \( ( 19 - 8 T + T^{2} )^{2} \)
$23$ \( 1024 - 32 T^{2} + T^{4} \)
$29$ \( 36 + 6 T^{2} + T^{4} \)
$31$ \( 81 + 90 T^{2} + T^{4} \)
$37$ \( 225 + 42 T^{2} + T^{4} \)
$41$ \( T^{4} \)
$43$ \( 529 + 46 T + 27 T^{2} - 2 T^{3} + T^{4} \)
$47$ \( 100 + 120 T + 58 T^{2} + 12 T^{3} + T^{4} \)
$53$ \( 576 + 24 T^{2} + T^{4} \)
$59$ \( 19044 - 3312 T + 438 T^{2} - 24 T^{3} + T^{4} \)
$61$ \( 25 + 10 T + 9 T^{2} - 2 T^{3} + T^{4} \)
$67$ \( 4761 - 414 T - 57 T^{2} + 6 T^{3} + T^{4} \)
$71$ \( ( 36 - 6 T + T^{2} )^{2} \)
$73$ \( 625 - 350 T + 171 T^{2} - 14 T^{3} + T^{4} \)
$79$ \( 81 - 162 T + 117 T^{2} - 18 T^{3} + T^{4} \)
$83$ \( 11236 + 220 T^{2} + T^{4} \)
$89$ \( 22500 + 150 T^{2} + T^{4} \)
$97$ \( 1296 + 1296 T + 468 T^{2} + 36 T^{3} + T^{4} \)
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