Properties

Label 3381.2.a.p
Level $3381$
Weight $2$
Character orbit 3381.a
Self dual yes
Analytic conductor $26.997$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} - q^{3} + ( 1 + \beta ) q^{4} + ( 3 - \beta ) q^{5} + \beta q^{6} -3 q^{8} + q^{9} +O(q^{10})\) \( q -\beta q^{2} - q^{3} + ( 1 + \beta ) q^{4} + ( 3 - \beta ) q^{5} + \beta q^{6} -3 q^{8} + q^{9} + ( 3 - 2 \beta ) q^{10} -5 q^{11} + ( -1 - \beta ) q^{12} -\beta q^{13} + ( -3 + \beta ) q^{15} + ( -2 + \beta ) q^{16} + ( 1 + 2 \beta ) q^{17} -\beta q^{18} + ( -3 + 2 \beta ) q^{19} + \beta q^{20} + 5 \beta q^{22} + q^{23} + 3 q^{24} + ( 7 - 5 \beta ) q^{25} + ( 3 + \beta ) q^{26} - q^{27} + ( -3 + 4 \beta ) q^{29} + ( -3 + 2 \beta ) q^{30} -3 q^{31} + ( 3 + \beta ) q^{32} + 5 q^{33} + ( -6 - 3 \beta ) q^{34} + ( 1 + \beta ) q^{36} -9 q^{37} + ( -6 + \beta ) q^{38} + \beta q^{39} + ( -9 + 3 \beta ) q^{40} + ( 3 + 4 \beta ) q^{41} + ( -6 + 5 \beta ) q^{43} + ( -5 - 5 \beta ) q^{44} + ( 3 - \beta ) q^{45} -\beta q^{46} + ( 4 + 2 \beta ) q^{47} + ( 2 - \beta ) q^{48} + ( 15 - 2 \beta ) q^{50} + ( -1 - 2 \beta ) q^{51} + ( -3 - 2 \beta ) q^{52} + ( -1 - 5 \beta ) q^{53} + \beta q^{54} + ( -15 + 5 \beta ) q^{55} + ( 3 - 2 \beta ) q^{57} + ( -12 - \beta ) q^{58} + ( 3 - 3 \beta ) q^{59} -\beta q^{60} + ( 8 - 3 \beta ) q^{61} + 3 \beta q^{62} + ( 1 - 6 \beta ) q^{64} + ( 3 - 2 \beta ) q^{65} -5 \beta q^{66} + ( -5 - 3 \beta ) q^{67} + ( 7 + 5 \beta ) q^{68} - q^{69} + ( -6 + 3 \beta ) q^{71} -3 q^{72} + ( -7 + 4 \beta ) q^{73} + 9 \beta q^{74} + ( -7 + 5 \beta ) q^{75} + ( 3 + \beta ) q^{76} + ( -3 - \beta ) q^{78} - q^{79} + ( -9 + 4 \beta ) q^{80} + q^{81} + ( -12 - 7 \beta ) q^{82} + ( -1 - 2 \beta ) q^{83} + ( -3 + 3 \beta ) q^{85} + ( -15 + \beta ) q^{86} + ( 3 - 4 \beta ) q^{87} + 15 q^{88} + ( -4 - 3 \beta ) q^{89} + ( 3 - 2 \beta ) q^{90} + ( 1 + \beta ) q^{92} + 3 q^{93} + ( -6 - 6 \beta ) q^{94} + ( -15 + 7 \beta ) q^{95} + ( -3 - \beta ) q^{96} + ( 13 + 2 \beta ) q^{97} -5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - 2q^{3} + 3q^{4} + 5q^{5} + q^{6} - 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - q^{2} - 2q^{3} + 3q^{4} + 5q^{5} + q^{6} - 6q^{8} + 2q^{9} + 4q^{10} - 10q^{11} - 3q^{12} - q^{13} - 5q^{15} - 3q^{16} + 4q^{17} - q^{18} - 4q^{19} + q^{20} + 5q^{22} + 2q^{23} + 6q^{24} + 9q^{25} + 7q^{26} - 2q^{27} - 2q^{29} - 4q^{30} - 6q^{31} + 7q^{32} + 10q^{33} - 15q^{34} + 3q^{36} - 18q^{37} - 11q^{38} + q^{39} - 15q^{40} + 10q^{41} - 7q^{43} - 15q^{44} + 5q^{45} - q^{46} + 10q^{47} + 3q^{48} + 28q^{50} - 4q^{51} - 8q^{52} - 7q^{53} + q^{54} - 25q^{55} + 4q^{57} - 25q^{58} + 3q^{59} - q^{60} + 13q^{61} + 3q^{62} - 4q^{64} + 4q^{65} - 5q^{66} - 13q^{67} + 19q^{68} - 2q^{69} - 9q^{71} - 6q^{72} - 10q^{73} + 9q^{74} - 9q^{75} + 7q^{76} - 7q^{78} - 2q^{79} - 14q^{80} + 2q^{81} - 31q^{82} - 4q^{83} - 3q^{85} - 29q^{86} + 2q^{87} + 30q^{88} - 11q^{89} + 4q^{90} + 3q^{92} + 6q^{93} - 18q^{94} - 23q^{95} - 7q^{96} + 28q^{97} - 10q^{99} + O(q^{100}) \)

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
2.30278
−1.30278
−2.30278 −1.00000 3.30278 0.697224 2.30278 0 −3.00000 1.00000 −1.60555
1.2 1.30278 −1.00000 −0.302776 4.30278 −1.30278 0 −3.00000 1.00000 5.60555
\(n\): 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.p 2
7.b odd 2 1 483.2.a.f 2
21.c even 2 1 1449.2.a.j 2
28.d even 2 1 7728.2.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.f 2 7.b odd 2 1
1449.2.a.j 2 21.c even 2 1
3381.2.a.p 2 1.a even 1 1 trivial
7728.2.a.x 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\):

\( T_{2}^{2} + T_{2} - 3 \)
\( T_{5}^{2} - 5 T_{5} + 3 \)
\( T_{11} + 5 \)
\( T_{13}^{2} + T_{13} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 + T + T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( 3 - 5 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 5 + T )^{2} \)
$13$ \( -3 + T + T^{2} \)
$17$ \( -9 - 4 T + T^{2} \)
$19$ \( -9 + 4 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( -51 + 2 T + T^{2} \)
$31$ \( ( 3 + T )^{2} \)
$37$ \( ( 9 + T )^{2} \)
$41$ \( -27 - 10 T + T^{2} \)
$43$ \( -69 + 7 T + T^{2} \)
$47$ \( 12 - 10 T + T^{2} \)
$53$ \( -69 + 7 T + T^{2} \)
$59$ \( -27 - 3 T + T^{2} \)
$61$ \( 13 - 13 T + T^{2} \)
$67$ \( 13 + 13 T + T^{2} \)
$71$ \( -9 + 9 T + T^{2} \)
$73$ \( -27 + 10 T + T^{2} \)
$79$ \( ( 1 + T )^{2} \)
$83$ \( -9 + 4 T + T^{2} \)
$89$ \( 1 + 11 T + T^{2} \)
$97$ \( 183 - 28 T + T^{2} \)
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