Properties

Label 3381.2.a.w
Level $3381$
Weight $2$
Character orbit 3381.a
Self dual yes
Analytic conductor $26.997$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.24197.1
Defining polynomial: \(x^{4} - 6 x^{2} - x + 2\)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{3} ) q^{5} -\beta_{1} q^{6} + ( -1 - \beta_{1} - \beta_{3} ) q^{8} + q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{3} ) q^{5} -\beta_{1} q^{6} + ( -1 - \beta_{1} - \beta_{3} ) q^{8} + q^{9} + ( -1 + \beta_{1} - \beta_{2} ) q^{10} + ( -1 - \beta_{2} + \beta_{3} ) q^{11} + ( 1 + \beta_{2} ) q^{12} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{13} + ( -1 + \beta_{3} ) q^{15} + ( 2 + \beta_{1} ) q^{16} + ( -3 - \beta_{1} + \beta_{2} ) q^{17} -\beta_{1} q^{18} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{19} + ( 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{20} + ( 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{22} - q^{23} + ( -1 - \beta_{1} - \beta_{3} ) q^{24} + ( 1 + \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{25} + ( -1 + 4 \beta_{1} + \beta_{3} ) q^{26} + q^{27} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{29} + ( -1 + \beta_{1} - \beta_{2} ) q^{30} + ( -1 - \beta_{1} + \beta_{2} ) q^{31} + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{32} + ( -1 - \beta_{2} + \beta_{3} ) q^{33} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{34} + ( 1 + \beta_{2} ) q^{36} + ( 5 + 3 \beta_{1} - \beta_{2} ) q^{37} + ( -6 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{38} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{39} + ( -5 + \beta_{3} ) q^{40} + ( -1 - \beta_{2} - \beta_{3} ) q^{41} + ( 2 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{43} + ( -7 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{44} + ( -1 + \beta_{3} ) q^{45} + \beta_{1} q^{46} + ( -2 + \beta_{1} - \beta_{3} ) q^{47} + ( 2 + \beta_{1} ) q^{48} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{50} + ( -3 - \beta_{1} + \beta_{2} ) q^{51} + ( -9 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{52} + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{53} -\beta_{1} q^{54} + ( 5 - 2 \beta_{1} - \beta_{3} ) q^{55} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{57} + ( -2 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{58} + ( 1 - 7 \beta_{1} + 2 \beta_{3} ) q^{59} + ( 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{60} + ( -6 + 2 \beta_{1} + \beta_{2} ) q^{61} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{62} + ( -5 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{64} + ( -3 - 5 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{65} + ( 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{66} + ( 1 + 2 \beta_{1} + 3 \beta_{3} ) q^{67} + ( 3 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{68} - q^{69} + ( -4 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{71} + ( -1 - \beta_{1} - \beta_{3} ) q^{72} + ( -5 + \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{73} + ( -8 - 3 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{74} + ( 1 + \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{75} + ( 7 + 4 \beta_{1} + 3 \beta_{3} ) q^{76} + ( -1 + 4 \beta_{1} + \beta_{3} ) q^{78} + ( -3 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{79} + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{80} + q^{81} + ( 2 + 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{82} + ( -1 - \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{83} + ( 3 + 4 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} ) q^{85} + ( -9 - 4 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{86} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{87} + ( -3 + 5 \beta_{1} + \beta_{2} ) q^{88} + ( \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{89} + ( -1 + \beta_{1} - \beta_{2} ) q^{90} + ( -1 - \beta_{2} ) q^{92} + ( -1 - \beta_{1} + \beta_{2} ) q^{93} + ( -2 + 2 \beta_{1} ) q^{94} + ( -1 + 2 \beta_{2} - \beta_{3} ) q^{95} + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{96} + ( 1 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{97} + ( -1 - \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 4q^{4} - 5q^{5} - 3q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 4q^{4} - 5q^{5} - 3q^{8} + 4q^{9} - 4q^{10} - 5q^{11} + 4q^{12} - 7q^{13} - 5q^{15} + 8q^{16} - 12q^{17} - 3q^{19} + q^{20} - q^{22} - 4q^{23} - 3q^{24} + 7q^{25} - 5q^{26} + 4q^{27} + 6q^{29} - 4q^{30} - 4q^{31} - 6q^{32} - 5q^{33} + 9q^{34} + 4q^{36} + 20q^{37} - 23q^{38} - 7q^{39} - 21q^{40} - 3q^{41} + 9q^{43} - 27q^{44} - 5q^{45} - 7q^{47} + 8q^{48} + 3q^{50} - 12q^{51} - 38q^{52} - 6q^{53} + 21q^{55} - 3q^{57} - 7q^{58} + 2q^{59} + q^{60} - 24q^{61} + 9q^{62} - 21q^{64} - 14q^{65} - q^{66} + q^{67} + 13q^{68} - 4q^{69} - 17q^{71} - 3q^{72} - 16q^{73} - 33q^{74} + 7q^{75} + 25q^{76} - 5q^{78} - 10q^{79} - 6q^{80} + 4q^{81} + 7q^{82} - 8q^{83} + 17q^{85} - 35q^{86} + 6q^{87} - 12q^{88} + 3q^{89} - 4q^{90} - 4q^{92} - 4q^{93} - 8q^{94} - 3q^{95} - 6q^{96} + 2q^{97} - 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 6 x^{2} - x + 2\):

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.46506
0.509552
−0.700017
−2.27460
−2.46506 1.00000 4.07653 0.653724 −2.46506 0 −5.11879 1.00000 −1.61147
1.2 −0.509552 1.00000 −1.74036 −4.41546 −0.509552 0 1.90591 1.00000 2.24991
1.3 0.700017 1.00000 −1.50998 1.15706 0.700017 0 −2.45704 1.00000 0.809960
1.4 2.27460 1.00000 3.17380 −2.39532 2.27460 0 2.66992 1.00000 −5.44840
\(n\): 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.w 4
7.b odd 2 1 483.2.a.i 4
21.c even 2 1 1449.2.a.p 4
28.d even 2 1 7728.2.a.cd 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.i 4 7.b odd 2 1
1449.2.a.p 4 21.c even 2 1
3381.2.a.w 4 1.a even 1 1 trivial
7728.2.a.cd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\):

\( T_{2}^{4} - 6 T_{2}^{2} + T_{2} + 2 \)
\( T_{5}^{4} + 5 T_{5}^{3} - T_{5}^{2} - 14 T_{5} + 8 \)
\( T_{11}^{4} + 5 T_{11}^{3} - 9 T_{11}^{2} - 65 T_{11} - 68 \)
\( T_{13}^{4} + 7 T_{13}^{3} - 11 T_{13}^{2} - 152 T_{13} - 236 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T - 6 T^{2} + T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( 8 - 14 T - T^{2} + 5 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( -68 - 65 T - 9 T^{2} + 5 T^{3} + T^{4} \)
$13$ \( -236 - 152 T - 11 T^{2} + 7 T^{3} + T^{4} \)
$17$ \( -104 - 10 T + 37 T^{2} + 12 T^{3} + T^{4} \)
$19$ \( 52 - 51 T - 37 T^{2} + 3 T^{3} + T^{4} \)
$23$ \( ( 1 + T )^{4} \)
$29$ \( -436 + 264 T - 31 T^{2} - 6 T^{3} + T^{4} \)
$31$ \( -16 - 46 T - 11 T^{2} + 4 T^{3} + T^{4} \)
$37$ \( -1868 + 280 T + 91 T^{2} - 20 T^{3} + T^{4} \)
$41$ \( -34 - 71 T - 27 T^{2} + 3 T^{3} + T^{4} \)
$43$ \( 272 + 184 T - 39 T^{2} - 9 T^{3} + T^{4} \)
$47$ \( -32 - 28 T + 6 T^{2} + 7 T^{3} + T^{4} \)
$53$ \( -202 - 249 T - 68 T^{2} + 6 T^{3} + T^{4} \)
$59$ \( 17564 + 263 T - 278 T^{2} - 2 T^{3} + T^{4} \)
$61$ \( -1286 + 191 T + 172 T^{2} + 24 T^{3} + T^{4} \)
$67$ \( 4208 - 24 T - 141 T^{2} - T^{3} + T^{4} \)
$71$ \( 64 + 128 T + 79 T^{2} + 17 T^{3} + T^{4} \)
$73$ \( -6656 - 2270 T - 95 T^{2} + 16 T^{3} + T^{4} \)
$79$ \( 128 - 214 T - 17 T^{2} + 10 T^{3} + T^{4} \)
$83$ \( -7256 - 2450 T - 195 T^{2} + 8 T^{3} + T^{4} \)
$89$ \( -92 - 380 T - 113 T^{2} - 3 T^{3} + T^{4} \)
$97$ \( 4 + 356 T - 87 T^{2} - 2 T^{3} + T^{4} \)
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