Properties

Label 3381.2.a.r
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: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.61803
−0.618034
−1.61803 1.00000 0.618034 0.618034 −1.61803 0 2.23607 1.00000 −1.00000
1.2 0.618034 1.00000 −1.61803 −1.61803 0.618034 0 −2.23607 1.00000 −1.00000
\(n\): 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.r 2
7.b odd 2 1 483.2.a.d 2
21.c even 2 1 1449.2.a.h 2
28.d even 2 1 7728.2.a.bn 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.d 2 7.b odd 2 1
1449.2.a.h 2 21.c even 2 1
3381.2.a.r 2 1.a even 1 1 trivial
7728.2.a.bn 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\):

\( T_{2}^{2} + T_{2} - 1 \)
\( T_{5}^{2} + T_{5} - 1 \)
\( T_{11}^{2} - 5 \)
\( T_{13}^{2} - 7 T_{13} + 11 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -1 + T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -5 + T^{2} \)
$13$ \( 11 - 7 T + T^{2} \)
$17$ \( -45 + T^{2} \)
$19$ \( -19 + 2 T + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( 31 + 12 T + T^{2} \)
$31$ \( -45 + T^{2} \)
$37$ \( ( 11 + T )^{2} \)
$41$ \( -11 + 6 T + T^{2} \)
$43$ \( -1 + T + T^{2} \)
$47$ \( 20 - 10 T + T^{2} \)
$53$ \( 25 + 15 T + T^{2} \)
$59$ \( 109 + 21 T + T^{2} \)
$61$ \( -9 - 3 T + T^{2} \)
$67$ \( -31 - T + T^{2} \)
$71$ \( 29 - 11 T + T^{2} \)
$73$ \( -9 - 12 T + T^{2} \)
$79$ \( 5 + 10 T + T^{2} \)
$83$ \( -121 + 4 T + T^{2} \)
$89$ \( 109 + 21 T + T^{2} \)
$97$ \( -171 - 6 T + T^{2} \)
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