Properties

Label 3042.2.a.bd
Level $3042$
Weight $2$
Character orbit 3042.a
Self dual yes
Analytic conductor $24.290$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1014)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + (\beta_{2} + \beta_1 + 1) q^{5} + ( - 2 \beta_{2} + \beta_1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + (\beta_{2} + \beta_1 + 1) q^{5} + ( - 2 \beta_{2} + \beta_1) q^{7} - q^{8} + ( - \beta_{2} - \beta_1 - 1) q^{10} + ( - \beta_{2} - 2 \beta_1) q^{11} + (2 \beta_{2} - \beta_1) q^{14} + q^{16} + ( - 4 \beta_{2} + 2 \beta_1 - 6) q^{17} + (4 \beta_{2} - 4 \beta_1 + 4) q^{19} + (\beta_{2} + \beta_1 + 1) q^{20} + (\beta_{2} + 2 \beta_1) q^{22} + (2 \beta_{2} - 2 \beta_1 - 4) q^{23} + (4 \beta_{2} + 3 \beta_1 + 1) q^{25} + ( - 2 \beta_{2} + \beta_1) q^{28} + (3 \beta_{2} + 2 \beta_1 - 4) q^{29} + (5 \beta_{2} - 4 \beta_1) q^{31} - q^{32} + (4 \beta_{2} - 2 \beta_1 + 6) q^{34} + ( - \beta_1 - 1) q^{35} + ( - 2 \beta_{2} + 4 \beta_1 - 6) q^{37} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{38} + ( - \beta_{2} - \beta_1 - 1) q^{40} + ( - 2 \beta_{2} + 4) q^{41} + (2 \beta_{2} - 6 \beta_1) q^{43} + ( - \beta_{2} - 2 \beta_1) q^{44} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{46} - 4 \beta_1 q^{47} + ( - 7 \beta_{2} + 4 \beta_1 - 5) q^{49} + ( - 4 \beta_{2} - 3 \beta_1 - 1) q^{50} + (3 \beta_{2} - 9 \beta_1 - 1) q^{53} + ( - 5 \beta_{2} - 3 \beta_1 - 8) q^{55} + (2 \beta_{2} - \beta_1) q^{56} + ( - 3 \beta_{2} - 2 \beta_1 + 4) q^{58} + ( - 2 \beta_{2} + \beta_1 + 2) q^{59} + (2 \beta_1 - 4) q^{61} + ( - 5 \beta_{2} + 4 \beta_1) q^{62} + q^{64} + ( - 10 \beta_{2} + 8 \beta_1 - 8) q^{67} + ( - 4 \beta_{2} + 2 \beta_1 - 6) q^{68} + (\beta_1 + 1) q^{70} + (6 \beta_{2} - 6 \beta_1 + 2) q^{71} + (\beta_{2} + 3 \beta_1 + 1) q^{73} + (2 \beta_{2} - 4 \beta_1 + 6) q^{74} + (4 \beta_{2} - 4 \beta_1 + 4) q^{76} + ( - \beta_{2} + 2 \beta_1 + 1) q^{77} + (6 \beta_{2} - 11 \beta_1 + 4) q^{79} + (\beta_{2} + \beta_1 + 1) q^{80} + (2 \beta_{2} - 4) q^{82} + ( - 2 \beta_{2} + 11 \beta_1 - 2) q^{83} + ( - 6 \beta_{2} - 8 \beta_1 - 8) q^{85} + ( - 2 \beta_{2} + 6 \beta_1) q^{86} + (\beta_{2} + 2 \beta_1) q^{88} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{89} + (2 \beta_{2} - 2 \beta_1 - 4) q^{92} + 4 \beta_1 q^{94} + 4 \beta_1 q^{95} + (3 \beta_{2} + \beta_1 + 3) q^{97} + (7 \beta_{2} - 4 \beta_1 + 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{10} - q^{11} - 3 q^{14} + 3 q^{16} - 12 q^{17} + 4 q^{19} + 3 q^{20} + q^{22} - 16 q^{23} + 2 q^{25} + 3 q^{28} - 13 q^{29} - 9 q^{31} - 3 q^{32} + 12 q^{34} - 4 q^{35} - 12 q^{37} - 4 q^{38} - 3 q^{40} + 14 q^{41} - 8 q^{43} - q^{44} + 16 q^{46} - 4 q^{47} - 4 q^{49} - 2 q^{50} - 15 q^{53} - 22 q^{55} - 3 q^{56} + 13 q^{58} + 9 q^{59} - 10 q^{61} + 9 q^{62} + 3 q^{64} - 6 q^{67} - 12 q^{68} + 4 q^{70} - 6 q^{71} + 5 q^{73} + 12 q^{74} + 4 q^{76} + 6 q^{77} - 5 q^{79} + 3 q^{80} - 14 q^{82} + 7 q^{83} - 26 q^{85} + 8 q^{86} + q^{88} + 10 q^{89} - 16 q^{92} + 4 q^{94} + 4 q^{95} + 7 q^{97} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display

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
−1.24698
0.445042
1.80194
−1.00000 0 1.00000 −0.692021 0 −0.356896 −1.00000 0 0.692021
1.2 −1.00000 0 1.00000 −0.356896 0 4.04892 −1.00000 0 0.356896
1.3 −1.00000 0 1.00000 4.04892 0 −0.692021 −1.00000 0 −4.04892
\(n\): 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.bd 3
3.b odd 2 1 1014.2.a.o yes 3
12.b even 2 1 8112.2.a.bz 3
13.b even 2 1 3042.2.a.be 3
13.d odd 4 2 3042.2.b.r 6
39.d odd 2 1 1014.2.a.m 3
39.f even 4 2 1014.2.b.g 6
39.h odd 6 2 1014.2.e.m 6
39.i odd 6 2 1014.2.e.k 6
39.k even 12 4 1014.2.i.g 12
156.h even 2 1 8112.2.a.ce 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1014.2.a.m 3 39.d odd 2 1
1014.2.a.o yes 3 3.b odd 2 1
1014.2.b.g 6 39.f even 4 2
1014.2.e.k 6 39.i odd 6 2
1014.2.e.m 6 39.h odd 6 2
1014.2.i.g 12 39.k even 12 4
3042.2.a.bd 3 1.a even 1 1 trivial
3042.2.a.be 3 13.b even 2 1
3042.2.b.r 6 13.d odd 4 2
8112.2.a.bz 3 12.b even 2 1
8112.2.a.ce 3 156.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3042))\):

\( T_{5}^{3} - 3T_{5}^{2} - 4T_{5} - 1 \) Copy content Toggle raw display
\( T_{7}^{3} - 3T_{7}^{2} - 4T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{3} + T_{11}^{2} - 16T_{11} + 13 \) Copy content Toggle raw display
\( T_{17}^{3} + 12T_{17}^{2} + 20T_{17} - 104 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 3 T^{2} - 4 T - 1 \) Copy content Toggle raw display
$7$ \( T^{3} - 3 T^{2} - 4 T - 1 \) Copy content Toggle raw display
$11$ \( T^{3} + T^{2} - 16 T + 13 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 12 T^{2} + 20 T - 104 \) Copy content Toggle raw display
$19$ \( T^{3} - 4 T^{2} - 32 T + 64 \) Copy content Toggle raw display
$23$ \( T^{3} + 16 T^{2} + 76 T + 104 \) Copy content Toggle raw display
$29$ \( T^{3} + 13 T^{2} + 12 T - 223 \) Copy content Toggle raw display
$31$ \( T^{3} + 9 T^{2} - 22 T - 29 \) Copy content Toggle raw display
$37$ \( T^{3} + 12 T^{2} + 20 T + 8 \) Copy content Toggle raw display
$41$ \( T^{3} - 14 T^{2} + 56 T - 56 \) Copy content Toggle raw display
$43$ \( T^{3} + 8 T^{2} - 44 T - 344 \) Copy content Toggle raw display
$47$ \( T^{3} + 4 T^{2} - 32 T - 64 \) Copy content Toggle raw display
$53$ \( T^{3} + 15 T^{2} - 72 T - 1247 \) Copy content Toggle raw display
$59$ \( T^{3} - 9 T^{2} + 20 T - 13 \) Copy content Toggle raw display
$61$ \( T^{3} + 10 T^{2} + 24 T + 8 \) Copy content Toggle raw display
$67$ \( T^{3} + 6 T^{2} - 184 T - 1112 \) Copy content Toggle raw display
$71$ \( T^{3} + 6 T^{2} - 72 T - 104 \) Copy content Toggle raw display
$73$ \( T^{3} - 5 T^{2} - 22 T + 13 \) Copy content Toggle raw display
$79$ \( T^{3} + 5 T^{2} - 204 T - 1469 \) Copy content Toggle raw display
$83$ \( T^{3} - 7 T^{2} - 224 T + 1477 \) Copy content Toggle raw display
$89$ \( T^{3} - 10 T^{2} + 24 T - 8 \) Copy content Toggle raw display
$97$ \( T^{3} - 7 T^{2} - 14 T + 7 \) Copy content Toggle raw display
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