Properties

Label 3381.2.a.x
Level $3381$
Weight $2$
Character orbit 3381.a
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.15317.1
Defining polynomial: \(x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 2\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 5 \beta_{1} + 2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.69353
−0.329727
1.32973
2.69353
−1.69353 1.00000 0.868028 −3.51256 −1.69353 0 1.91702 1.00000 5.94860
1.2 −0.329727 1.00000 −1.89128 2.73589 −0.329727 0 1.28306 1.00000 −0.902098
1.3 1.32973 1.00000 −0.231826 −3.17434 1.32973 0 −2.96772 1.00000 −4.22101
1.4 2.69353 1.00000 5.25508 −1.04900 2.69353 0 8.76763 1.00000 −2.82550
\(n\): 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.x 4
7.b odd 2 1 483.2.a.j 4
21.c even 2 1 1449.2.a.o 4
28.d even 2 1 7728.2.a.ce 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.j 4 7.b odd 2 1
1449.2.a.o 4 21.c even 2 1
3381.2.a.x 4 1.a even 1 1 trivial
7728.2.a.ce 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\):

\( T_{2}^{4} - 2 T_{2}^{3} - 4 T_{2}^{2} + 5 T_{2} + 2 \)
\( T_{5}^{4} + 5 T_{5}^{3} - 3 T_{5}^{2} - 38 T_{5} - 32 \)
\( T_{11}^{4} - T_{11}^{3} - 17 T_{11}^{2} - 19 T_{11} + 4 \)
\( T_{13}^{4} + 7 T_{13}^{3} - 17 T_{13}^{2} - 164 T_{13} - 188 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + 5 T - 4 T^{2} - 2 T^{3} + T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( -32 - 38 T - 3 T^{2} + 5 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 4 - 19 T - 17 T^{2} - T^{3} + T^{4} \)
$13$ \( -188 - 164 T - 17 T^{2} + 7 T^{3} + T^{4} \)
$17$ \( -64 - 98 T - 29 T^{2} + 2 T^{3} + T^{4} \)
$19$ \( 4 - 19 T - 17 T^{2} - T^{3} + T^{4} \)
$23$ \( ( -1 + T )^{4} \)
$29$ \( ( -38 - T + T^{2} )^{2} \)
$31$ \( 16 - 10 T - 17 T^{2} + 6 T^{3} + T^{4} \)
$37$ \( -188 + 72 T + 55 T^{2} - 16 T^{3} + T^{4} \)
$41$ \( -134 + 183 T - 71 T^{2} + 5 T^{3} + T^{4} \)
$43$ \( -16 + 128 T - 3 T^{2} - 9 T^{3} + T^{4} \)
$47$ \( -1696 + 212 T + 102 T^{2} - 21 T^{3} + T^{4} \)
$53$ \( 3578 + 577 T - 104 T^{2} - 10 T^{3} + T^{4} \)
$59$ \( 1556 - 1027 T + 248 T^{2} - 26 T^{3} + T^{4} \)
$61$ \( 1114 - 929 T - 180 T^{2} + 2 T^{3} + T^{4} \)
$67$ \( 11008 + 1396 T - 281 T^{2} - 5 T^{3} + T^{4} \)
$71$ \( 32 + 116 T + 83 T^{2} + 19 T^{3} + T^{4} \)
$73$ \( 608 - 226 T - 61 T^{2} + 10 T^{3} + T^{4} \)
$79$ \( 8992 - 350 T - 221 T^{2} + 6 T^{3} + T^{4} \)
$83$ \( 8152 + 130 T - 197 T^{2} - 2 T^{3} + T^{4} \)
$89$ \( 11188 + 2312 T - 175 T^{2} - 17 T^{3} + T^{4} \)
$97$ \( 1684 + 1392 T + 343 T^{2} + 32 T^{3} + T^{4} \)
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