Properties

Label 2151.2.a.a
Level $2151$
Weight $2$
Character orbit 2151.a
Self dual yes
Analytic conductor $17.176$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1758214748\)
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 717)
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta ) q^{2} + 3 \beta q^{4} + ( -1 + 2 \beta ) q^{5} + ( -1 - 4 \beta ) q^{8} +O(q^{10})\) \( q + ( -1 - \beta ) q^{2} + 3 \beta q^{4} + ( -1 + 2 \beta ) q^{5} + ( -1 - 4 \beta ) q^{8} + ( -1 - 3 \beta ) q^{10} -3 q^{11} + ( 2 - 2 \beta ) q^{13} + ( 5 + 3 \beta ) q^{16} + ( 3 - 2 \beta ) q^{17} + ( 6 + 3 \beta ) q^{20} + ( 3 + 3 \beta ) q^{22} + 2 \beta q^{23} + 2 \beta q^{26} + ( -3 + 2 \beta ) q^{29} + ( 5 - 6 \beta ) q^{31} + ( -6 - 3 \beta ) q^{32} + ( -1 + \beta ) q^{34} + ( 6 - 6 \beta ) q^{37} + ( -7 - 6 \beta ) q^{40} + ( 2 - 2 \beta ) q^{41} + ( -6 + 4 \beta ) q^{43} -9 \beta q^{44} + ( -2 - 4 \beta ) q^{46} + 4 \beta q^{47} -7 q^{49} -6 q^{52} + ( -8 - 2 \beta ) q^{53} + ( 3 - 6 \beta ) q^{55} + ( 1 - \beta ) q^{58} + ( 6 - 6 \beta ) q^{59} -7 q^{61} + ( 1 + 7 \beta ) q^{62} + ( -1 + 6 \beta ) q^{64} + ( -6 + 2 \beta ) q^{65} + ( 1 - 6 \beta ) q^{67} + ( -6 + 3 \beta ) q^{68} -12 q^{71} + ( -2 + 6 \beta ) q^{73} + 6 \beta q^{74} + ( -6 + 2 \beta ) q^{79} + ( 1 + 13 \beta ) q^{80} + 2 \beta q^{82} + ( 11 + 4 \beta ) q^{83} + ( -7 + 4 \beta ) q^{85} + ( 2 - 2 \beta ) q^{86} + ( 3 + 12 \beta ) q^{88} -4 \beta q^{89} + ( 6 + 6 \beta ) q^{92} + ( -4 - 8 \beta ) q^{94} + ( -6 + 10 \beta ) q^{97} + ( 7 + 7 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{2} + 3q^{4} - 6q^{8} + O(q^{10}) \) \( 2q - 3q^{2} + 3q^{4} - 6q^{8} - 5q^{10} - 6q^{11} + 2q^{13} + 13q^{16} + 4q^{17} + 15q^{20} + 9q^{22} + 2q^{23} + 2q^{26} - 4q^{29} + 4q^{31} - 15q^{32} - q^{34} + 6q^{37} - 20q^{40} + 2q^{41} - 8q^{43} - 9q^{44} - 8q^{46} + 4q^{47} - 14q^{49} - 12q^{52} - 18q^{53} + q^{58} + 6q^{59} - 14q^{61} + 9q^{62} + 4q^{64} - 10q^{65} - 4q^{67} - 9q^{68} - 24q^{71} + 2q^{73} + 6q^{74} - 10q^{79} + 15q^{80} + 2q^{82} + 26q^{83} - 10q^{85} + 2q^{86} + 18q^{88} - 4q^{89} + 18q^{92} - 16q^{94} - 2q^{97} + 21q^{98} + O(q^{100}) \)

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
−2.61803 0 4.85410 2.23607 0 0 −7.47214 0 −5.85410
1.2 −0.381966 0 −1.85410 −2.23607 0 0 1.47214 0 0.854102
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(239\) \(-1\)

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 2151.2.a.a 2
3.b odd 2 1 717.2.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
717.2.a.c 2 3.b odd 2 1
2151.2.a.a 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2151))\):

\( T_{2}^{2} + 3 T_{2} + 1 \)
\( T_{5}^{2} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( -5 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 3 + T )^{2} \)
$13$ \( -4 - 2 T + T^{2} \)
$17$ \( -1 - 4 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( -4 - 2 T + T^{2} \)
$29$ \( -1 + 4 T + T^{2} \)
$31$ \( -41 - 4 T + T^{2} \)
$37$ \( -36 - 6 T + T^{2} \)
$41$ \( -4 - 2 T + T^{2} \)
$43$ \( -4 + 8 T + T^{2} \)
$47$ \( -16 - 4 T + T^{2} \)
$53$ \( 76 + 18 T + T^{2} \)
$59$ \( -36 - 6 T + T^{2} \)
$61$ \( ( 7 + T )^{2} \)
$67$ \( -41 + 4 T + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( -44 - 2 T + T^{2} \)
$79$ \( 20 + 10 T + T^{2} \)
$83$ \( 149 - 26 T + T^{2} \)
$89$ \( -16 + 4 T + T^{2} \)
$97$ \( -124 + 2 T + T^{2} \)
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