Properties

Label 3381.2.a.o
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.

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

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
1.61803
−0.618034
−1.61803 −1.00000 0.618034 −1.38197 1.61803 0 2.23607 1.00000 2.23607
1.2 0.618034 −1.00000 −1.61803 −3.61803 −0.618034 0 −2.23607 1.00000 −2.23607
\(n\): 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.o 2
7.b odd 2 1 483.2.a.e 2
21.c even 2 1 1449.2.a.g 2
28.d even 2 1 7728.2.a.be 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.e 2 7.b odd 2 1
1449.2.a.g 2 21.c even 2 1
3381.2.a.o 2 1.a even 1 1 trivial
7728.2.a.be 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T + T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( 5 + 5 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( -1 - T + T^{2} \)
$17$ \( -19 - 2 T + T^{2} \)
$19$ \( -1 + 4 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( 11 - 8 T + T^{2} \)
$31$ \( -41 - 4 T + T^{2} \)
$37$ \( -1 + 4 T + T^{2} \)
$41$ \( -19 + 2 T + T^{2} \)
$43$ \( -11 + T + T^{2} \)
$47$ \( -20 + 10 T + T^{2} \)
$53$ \( -1 - 11 T + T^{2} \)
$59$ \( 5 - 5 T + T^{2} \)
$61$ \( 9 - 9 T + T^{2} \)
$67$ \( -25 + 5 T + T^{2} \)
$71$ \( -55 + 5 T + T^{2} \)
$73$ \( 11 + 8 T + T^{2} \)
$79$ \( 139 + 24 T + T^{2} \)
$83$ \( 1 - 18 T + T^{2} \)
$89$ \( -89 + 7 T + T^{2} \)
$97$ \( -45 + T^{2} \)
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