Properties

Label 342.3.d.a
Level $342$
Weight $3$
Character orbit 342.d
Analytic conductor $9.319$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 342.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.31882504112\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} -2 q^{4} + q^{5} + 5 q^{7} + 2 \beta q^{8} +O(q^{10})\) \( q -\beta q^{2} -2 q^{4} + q^{5} + 5 q^{7} + 2 \beta q^{8} -\beta q^{10} -5 q^{11} -12 \beta q^{13} -5 \beta q^{14} + 4 q^{16} + 25 q^{17} + 19 q^{19} -2 q^{20} + 5 \beta q^{22} + 10 q^{23} -24 q^{25} -24 q^{26} -10 q^{28} -30 \beta q^{29} -30 \beta q^{31} -4 \beta q^{32} -25 \beta q^{34} + 5 q^{35} -18 \beta q^{37} -19 \beta q^{38} + 2 \beta q^{40} + 30 \beta q^{41} + 5 q^{43} + 10 q^{44} -10 \beta q^{46} -5 q^{47} -24 q^{49} + 24 \beta q^{50} + 24 \beta q^{52} + 18 \beta q^{53} -5 q^{55} + 10 \beta q^{56} -60 q^{58} -60 \beta q^{59} + 95 q^{61} -60 q^{62} -8 q^{64} -12 \beta q^{65} + 78 \beta q^{67} -50 q^{68} -5 \beta q^{70} -25 q^{73} -36 q^{74} -38 q^{76} -25 q^{77} + 30 \beta q^{79} + 4 q^{80} + 60 q^{82} + 130 q^{83} + 25 q^{85} -5 \beta q^{86} -10 \beta q^{88} + 90 \beta q^{89} -60 \beta q^{91} -20 q^{92} + 5 \beta q^{94} + 19 q^{95} + 12 \beta q^{97} + 24 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} + 2q^{5} + 10q^{7} + O(q^{10}) \) \( 2q - 4q^{4} + 2q^{5} + 10q^{7} - 10q^{11} + 8q^{16} + 50q^{17} + 38q^{19} - 4q^{20} + 20q^{23} - 48q^{25} - 48q^{26} - 20q^{28} + 10q^{35} + 10q^{43} + 20q^{44} - 10q^{47} - 48q^{49} - 10q^{55} - 120q^{58} + 190q^{61} - 120q^{62} - 16q^{64} - 100q^{68} - 50q^{73} - 72q^{74} - 76q^{76} - 50q^{77} + 8q^{80} + 120q^{82} + 260q^{83} + 50q^{85} - 40q^{92} + 38q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
1.41421i
1.41421i
1.41421i 0 −2.00000 1.00000 0 5.00000 2.82843i 0 1.41421i
37.2 1.41421i 0 −2.00000 1.00000 0 5.00000 2.82843i 0 1.41421i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.3.d.a 2
3.b odd 2 1 38.3.b.a 2
4.b odd 2 1 2736.3.o.h 2
12.b even 2 1 304.3.e.c 2
15.d odd 2 1 950.3.c.a 2
15.e even 4 2 950.3.d.a 4
19.b odd 2 1 inner 342.3.d.a 2
24.f even 2 1 1216.3.e.i 2
24.h odd 2 1 1216.3.e.j 2
57.d even 2 1 38.3.b.a 2
76.d even 2 1 2736.3.o.h 2
228.b odd 2 1 304.3.e.c 2
285.b even 2 1 950.3.c.a 2
285.j odd 4 2 950.3.d.a 4
456.l odd 2 1 1216.3.e.i 2
456.p even 2 1 1216.3.e.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.3.b.a 2 3.b odd 2 1
38.3.b.a 2 57.d even 2 1
304.3.e.c 2 12.b even 2 1
304.3.e.c 2 228.b odd 2 1
342.3.d.a 2 1.a even 1 1 trivial
342.3.d.a 2 19.b odd 2 1 inner
950.3.c.a 2 15.d odd 2 1
950.3.c.a 2 285.b even 2 1
950.3.d.a 4 15.e even 4 2
950.3.d.a 4 285.j odd 4 2
1216.3.e.i 2 24.f even 2 1
1216.3.e.i 2 456.l odd 2 1
1216.3.e.j 2 24.h odd 2 1
1216.3.e.j 2 456.p even 2 1
2736.3.o.h 2 4.b odd 2 1
2736.3.o.h 2 76.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 1 \) acting on \(S_{3}^{\mathrm{new}}(342, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( ( -5 + T )^{2} \)
$11$ \( ( 5 + T )^{2} \)
$13$ \( 288 + T^{2} \)
$17$ \( ( -25 + T )^{2} \)
$19$ \( ( -19 + T )^{2} \)
$23$ \( ( -10 + T )^{2} \)
$29$ \( 1800 + T^{2} \)
$31$ \( 1800 + T^{2} \)
$37$ \( 648 + T^{2} \)
$41$ \( 1800 + T^{2} \)
$43$ \( ( -5 + T )^{2} \)
$47$ \( ( 5 + T )^{2} \)
$53$ \( 648 + T^{2} \)
$59$ \( 7200 + T^{2} \)
$61$ \( ( -95 + T )^{2} \)
$67$ \( 12168 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 25 + T )^{2} \)
$79$ \( 1800 + T^{2} \)
$83$ \( ( -130 + T )^{2} \)
$89$ \( 16200 + T^{2} \)
$97$ \( 288 + T^{2} \)
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