Properties

Label 3042.2.a.y
Level $3042$
Weight $2$
Character orbit 3042.a
Self dual yes
Analytic conductor $24.290$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3042.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.2904922949\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( 2 + \beta ) q^{5} + ( 1 + \beta ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + ( 2 + \beta ) q^{5} + ( 1 + \beta ) q^{7} + q^{8} + ( 2 + \beta ) q^{10} + ( 3 - \beta ) q^{11} + ( 1 + \beta ) q^{14} + q^{16} + ( 4 + \beta ) q^{17} + ( -3 - \beta ) q^{19} + ( 2 + \beta ) q^{20} + ( 3 - \beta ) q^{22} + ( 1 - 3 \beta ) q^{23} + ( 2 + 4 \beta ) q^{25} + ( 1 + \beta ) q^{28} + ( 1 + 2 \beta ) q^{29} + ( 2 - 2 \beta ) q^{31} + q^{32} + ( 4 + \beta ) q^{34} + ( 5 + 3 \beta ) q^{35} + ( -7 + 2 \beta ) q^{37} + ( -3 - \beta ) q^{38} + ( 2 + \beta ) q^{40} + ( 1 - 6 \beta ) q^{41} + ( -1 - 5 \beta ) q^{43} + ( 3 - \beta ) q^{44} + ( 1 - 3 \beta ) q^{46} + ( -3 + 3 \beta ) q^{47} + ( -3 + 2 \beta ) q^{49} + ( 2 + 4 \beta ) q^{50} + ( 3 + 2 \beta ) q^{53} + ( 3 + \beta ) q^{55} + ( 1 + \beta ) q^{56} + ( 1 + 2 \beta ) q^{58} + 8 q^{59} + ( -4 - 3 \beta ) q^{61} + ( 2 - 2 \beta ) q^{62} + q^{64} + ( -1 - 7 \beta ) q^{67} + ( 4 + \beta ) q^{68} + ( 5 + 3 \beta ) q^{70} + ( 3 + \beta ) q^{71} + ( 8 - \beta ) q^{73} + ( -7 + 2 \beta ) q^{74} + ( -3 - \beta ) q^{76} + 2 \beta q^{77} + ( -6 + 2 \beta ) q^{79} + ( 2 + \beta ) q^{80} + ( 1 - 6 \beta ) q^{82} + ( 5 - 3 \beta ) q^{83} + ( 11 + 6 \beta ) q^{85} + ( -1 - 5 \beta ) q^{86} + ( 3 - \beta ) q^{88} + ( -6 - 2 \beta ) q^{89} + ( 1 - 3 \beta ) q^{92} + ( -3 + 3 \beta ) q^{94} + ( -9 - 5 \beta ) q^{95} + 6 q^{97} + ( -3 + 2 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 2 q^{7} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 2 q^{7} + 2 q^{8} + 4 q^{10} + 6 q^{11} + 2 q^{14} + 2 q^{16} + 8 q^{17} - 6 q^{19} + 4 q^{20} + 6 q^{22} + 2 q^{23} + 4 q^{25} + 2 q^{28} + 2 q^{29} + 4 q^{31} + 2 q^{32} + 8 q^{34} + 10 q^{35} - 14 q^{37} - 6 q^{38} + 4 q^{40} + 2 q^{41} - 2 q^{43} + 6 q^{44} + 2 q^{46} - 6 q^{47} - 6 q^{49} + 4 q^{50} + 6 q^{53} + 6 q^{55} + 2 q^{56} + 2 q^{58} + 16 q^{59} - 8 q^{61} + 4 q^{62} + 2 q^{64} - 2 q^{67} + 8 q^{68} + 10 q^{70} + 6 q^{71} + 16 q^{73} - 14 q^{74} - 6 q^{76} - 12 q^{79} + 4 q^{80} + 2 q^{82} + 10 q^{83} + 22 q^{85} - 2 q^{86} + 6 q^{88} - 12 q^{89} + 2 q^{92} - 6 q^{94} - 18 q^{95} + 12 q^{97} - 6 q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.73205
1.73205
1.00000 0 1.00000 0.267949 0 −0.732051 1.00000 0 0.267949
1.2 1.00000 0 1.00000 3.73205 0 2.73205 1.00000 0 3.73205
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3042.2.a.y 2
3.b odd 2 1 1014.2.a.i 2
12.b even 2 1 8112.2.a.bj 2
13.b even 2 1 3042.2.a.p 2
13.d odd 4 2 3042.2.b.i 4
13.f odd 12 2 234.2.l.c 4
39.d odd 2 1 1014.2.a.k 2
39.f even 4 2 1014.2.b.e 4
39.h odd 6 2 1014.2.e.g 4
39.i odd 6 2 1014.2.e.i 4
39.k even 12 2 78.2.i.a 4
39.k even 12 2 1014.2.i.a 4
52.l even 12 2 1872.2.by.h 4
156.h even 2 1 8112.2.a.bp 2
156.v odd 12 2 624.2.bv.e 4
195.bc odd 12 2 1950.2.y.b 4
195.bh even 12 2 1950.2.bc.d 4
195.bn odd 12 2 1950.2.y.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.i.a 4 39.k even 12 2
234.2.l.c 4 13.f odd 12 2
624.2.bv.e 4 156.v odd 12 2
1014.2.a.i 2 3.b odd 2 1
1014.2.a.k 2 39.d odd 2 1
1014.2.b.e 4 39.f even 4 2
1014.2.e.g 4 39.h odd 6 2
1014.2.e.i 4 39.i odd 6 2
1014.2.i.a 4 39.k even 12 2
1872.2.by.h 4 52.l even 12 2
1950.2.y.b 4 195.bc odd 12 2
1950.2.y.g 4 195.bn odd 12 2
1950.2.bc.d 4 195.bh even 12 2
3042.2.a.p 2 13.b even 2 1
3042.2.a.y 2 1.a even 1 1 trivial
3042.2.b.i 4 13.d odd 4 2
8112.2.a.bj 2 12.b even 2 1
8112.2.a.bp 2 156.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3042))\):

\( T_{5}^{2} - 4 T_{5} + 1 \)
\( T_{7}^{2} - 2 T_{7} - 2 \)
\( T_{11}^{2} - 6 T_{11} + 6 \)
\( T_{17}^{2} - 8 T_{17} + 13 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 - 4 T + T^{2} \)
$7$ \( -2 - 2 T + T^{2} \)
$11$ \( 6 - 6 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 13 - 8 T + T^{2} \)
$19$ \( 6 + 6 T + T^{2} \)
$23$ \( -26 - 2 T + T^{2} \)
$29$ \( -11 - 2 T + T^{2} \)
$31$ \( -8 - 4 T + T^{2} \)
$37$ \( 37 + 14 T + T^{2} \)
$41$ \( -107 - 2 T + T^{2} \)
$43$ \( -74 + 2 T + T^{2} \)
$47$ \( -18 + 6 T + T^{2} \)
$53$ \( -3 - 6 T + T^{2} \)
$59$ \( ( -8 + T )^{2} \)
$61$ \( -11 + 8 T + T^{2} \)
$67$ \( -146 + 2 T + T^{2} \)
$71$ \( 6 - 6 T + T^{2} \)
$73$ \( 61 - 16 T + T^{2} \)
$79$ \( 24 + 12 T + T^{2} \)
$83$ \( -2 - 10 T + T^{2} \)
$89$ \( 24 + 12 T + T^{2} \)
$97$ \( ( -6 + T )^{2} \)
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