Properties

Label 3381.2.a.v
Level $3381$
Weight $2$
Character orbit 3381.a
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9974209234\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
Defining polynomial: \(x^{3} - 6 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} + ( -1 + \beta_{1} ) q^{5} -\beta_{1} q^{6} + ( 1 + 2 \beta_{1} ) q^{8} + q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} + ( -1 + \beta_{1} ) q^{5} -\beta_{1} q^{6} + ( 1 + 2 \beta_{1} ) q^{8} + q^{9} + ( 4 - \beta_{1} + \beta_{2} ) q^{10} + ( 2 - \beta_{1} + \beta_{2} ) q^{11} + ( -2 - \beta_{2} ) q^{12} + ( -3 + \beta_{2} ) q^{13} + ( 1 - \beta_{1} ) q^{15} + ( 4 + \beta_{1} ) q^{16} + ( -2 + \beta_{1} + \beta_{2} ) q^{17} + \beta_{1} q^{18} + ( 2 - \beta_{1} + \beta_{2} ) q^{19} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{20} + ( -3 + 4 \beta_{1} - \beta_{2} ) q^{22} - q^{23} + ( -1 - 2 \beta_{1} ) q^{24} + ( -2 \beta_{1} + \beta_{2} ) q^{25} + ( 1 - \beta_{1} ) q^{26} - q^{27} + ( 2 + 3 \beta_{1} - \beta_{2} ) q^{29} + ( -4 + \beta_{1} - \beta_{2} ) q^{30} + ( 6 - \beta_{1} + \beta_{2} ) q^{31} + ( 2 + \beta_{2} ) q^{32} + ( -2 + \beta_{1} - \beta_{2} ) q^{33} + ( 5 + \beta_{2} ) q^{34} + ( 2 + \beta_{2} ) q^{36} + ( 2 - 3 \beta_{1} + \beta_{2} ) q^{37} + ( -3 + 4 \beta_{1} - \beta_{2} ) q^{38} + ( 3 - \beta_{2} ) q^{39} + ( 7 - \beta_{1} + 2 \beta_{2} ) q^{40} + ( 2 - \beta_{1} + 3 \beta_{2} ) q^{41} + ( -3 - 2 \beta_{1} + \beta_{2} ) q^{43} + ( 11 - 3 \beta_{1} + 2 \beta_{2} ) q^{44} + ( -1 + \beta_{1} ) q^{45} -\beta_{1} q^{46} + ( -6 - 2 \beta_{2} ) q^{47} + ( -4 - \beta_{1} ) q^{48} + ( -7 + 2 \beta_{1} - 2 \beta_{2} ) q^{50} + ( 2 - \beta_{1} - \beta_{2} ) q^{51} + ( 2 + \beta_{1} - 3 \beta_{2} ) q^{52} + ( 1 - \beta_{1} ) q^{53} -\beta_{1} q^{54} + ( -5 + 5 \beta_{1} - 2 \beta_{2} ) q^{55} + ( -2 + \beta_{1} - \beta_{2} ) q^{57} + ( 11 + 3 \beta_{2} ) q^{58} + ( -1 + \beta_{1} + 2 \beta_{2} ) q^{59} + ( 1 - 4 \beta_{1} + \beta_{2} ) q^{60} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{61} + ( -3 + 8 \beta_{1} - \beta_{2} ) q^{62} + ( -7 + 2 \beta_{1} ) q^{64} + ( 4 - \beta_{1} - \beta_{2} ) q^{65} + ( 3 - 4 \beta_{1} + \beta_{2} ) q^{66} + ( -7 + 3 \beta_{1} + 2 \beta_{2} ) q^{67} + ( 5 + 5 \beta_{1} - 2 \beta_{2} ) q^{68} + q^{69} + ( 3 + 2 \beta_{1} + 3 \beta_{2} ) q^{71} + ( 1 + 2 \beta_{1} ) q^{72} + ( -4 - 3 \beta_{1} - \beta_{2} ) q^{73} + ( -11 + 4 \beta_{1} - 3 \beta_{2} ) q^{74} + ( 2 \beta_{1} - \beta_{2} ) q^{75} + ( 11 - 3 \beta_{1} + 2 \beta_{2} ) q^{76} + ( -1 + \beta_{1} ) q^{78} + ( 4 + 3 \beta_{1} - \beta_{2} ) q^{79} + ( 3 \beta_{1} + \beta_{2} ) q^{80} + q^{81} + ( -1 + 8 \beta_{1} - \beta_{2} ) q^{82} + ( -4 + \beta_{1} + \beta_{2} ) q^{83} + ( 7 - \beta_{1} ) q^{85} + ( -7 - \beta_{1} - 2 \beta_{2} ) q^{86} + ( -2 - 3 \beta_{1} + \beta_{2} ) q^{87} + ( -4 + 7 \beta_{1} - \beta_{2} ) q^{88} + ( -5 - \beta_{2} ) q^{89} + ( 4 - \beta_{1} + \beta_{2} ) q^{90} + ( -2 - \beta_{2} ) q^{92} + ( -6 + \beta_{1} - \beta_{2} ) q^{93} + ( -2 - 10 \beta_{1} ) q^{94} + ( -5 + 5 \beta_{1} - 2 \beta_{2} ) q^{95} + ( -2 - \beta_{2} ) q^{96} + ( 4 - 3 \beta_{1} - \beta_{2} ) q^{97} + ( 2 - \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + 6q^{4} - 3q^{5} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + 6q^{4} - 3q^{5} + 3q^{8} + 3q^{9} + 12q^{10} + 6q^{11} - 6q^{12} - 9q^{13} + 3q^{15} + 12q^{16} - 6q^{17} + 6q^{19} - 3q^{20} - 9q^{22} - 3q^{23} - 3q^{24} + 3q^{26} - 3q^{27} + 6q^{29} - 12q^{30} + 18q^{31} + 6q^{32} - 6q^{33} + 15q^{34} + 6q^{36} + 6q^{37} - 9q^{38} + 9q^{39} + 21q^{40} + 6q^{41} - 9q^{43} + 33q^{44} - 3q^{45} - 18q^{47} - 12q^{48} - 21q^{50} + 6q^{51} + 6q^{52} + 3q^{53} - 15q^{55} - 6q^{57} + 33q^{58} - 3q^{59} + 3q^{60} + 3q^{61} - 9q^{62} - 21q^{64} + 12q^{65} + 9q^{66} - 21q^{67} + 15q^{68} + 3q^{69} + 9q^{71} + 3q^{72} - 12q^{73} - 33q^{74} + 33q^{76} - 3q^{78} + 12q^{79} + 3q^{81} - 3q^{82} - 12q^{83} + 21q^{85} - 21q^{86} - 6q^{87} - 12q^{88} - 15q^{89} + 12q^{90} - 6q^{92} - 18q^{93} - 6q^{94} - 15q^{95} - 6q^{96} + 12q^{97} + 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 6 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)

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
−2.36147
−0.167449
2.52892
−2.36147 −1.00000 3.57653 −3.36147 2.36147 0 −3.72294 1.00000 7.93800
1.2 −0.167449 −1.00000 −1.97196 −1.16745 0.167449 0 0.665102 1.00000 0.195488
1.3 2.52892 −1.00000 4.39543 1.52892 −2.52892 0 6.05784 1.00000 3.86651
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(23\) \(1\)

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 3381.2.a.v 3
7.b odd 2 1 483.2.a.h 3
21.c even 2 1 1449.2.a.l 3
28.d even 2 1 7728.2.a.bt 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.h 3 7.b odd 2 1
1449.2.a.l 3 21.c even 2 1
3381.2.a.v 3 1.a even 1 1 trivial
7728.2.a.bt 3 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\):

\( T_{2}^{3} - 6 T_{2} - 1 \)
\( T_{5}^{3} + 3 T_{5}^{2} - 3 T_{5} - 6 \)
\( T_{11}^{3} - 6 T_{11}^{2} - 3 T_{11} + 20 \)
\( T_{13}^{3} + 9 T_{13}^{2} + 15 T_{13} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - 6 T + T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( -6 - 3 T + 3 T^{2} + T^{3} \)
$7$ \( T^{3} \)
$11$ \( 20 - 3 T - 6 T^{2} + T^{3} \)
$13$ \( 6 + 15 T + 9 T^{2} + T^{3} \)
$17$ \( -50 - 9 T + 6 T^{2} + T^{3} \)
$19$ \( 20 - 3 T - 6 T^{2} + T^{3} \)
$23$ \( ( 1 + T )^{3} \)
$29$ \( 262 - 45 T - 6 T^{2} + T^{3} \)
$31$ \( -128 + 93 T - 18 T^{2} + T^{3} \)
$37$ \( -50 - 45 T - 6 T^{2} + T^{3} \)
$41$ \( 590 - 93 T - 6 T^{2} + T^{3} \)
$43$ \( -124 - 3 T + 9 T^{2} + T^{3} \)
$47$ \( -192 + 60 T + 18 T^{2} + T^{3} \)
$53$ \( 6 - 3 T - 3 T^{2} + T^{3} \)
$59$ \( -12 - 57 T + 3 T^{2} + T^{3} \)
$61$ \( 90 - 27 T - 3 T^{2} + T^{3} \)
$67$ \( -908 + 27 T + 21 T^{2} + T^{3} \)
$71$ \( 424 - 123 T - 9 T^{2} + T^{3} \)
$73$ \( 10 - 27 T + 12 T^{2} + T^{3} \)
$79$ \( 320 - 9 T - 12 T^{2} + T^{3} \)
$83$ \( -36 + 27 T + 12 T^{2} + T^{3} \)
$89$ \( 50 + 63 T + 15 T^{2} + T^{3} \)
$97$ \( 482 - 27 T - 12 T^{2} + T^{3} \)
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