Properties

Label 2151.2.a.b
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.

\(f(q)\) \(=\) \( q -\beta q^{2} + ( -1 + \beta ) q^{4} -3 q^{5} + ( 2 - 2 \beta ) q^{7} + ( -1 + 2 \beta ) q^{8} +O(q^{10})\) \( q -\beta q^{2} + ( -1 + \beta ) q^{4} -3 q^{5} + ( 2 - 2 \beta ) q^{7} + ( -1 + 2 \beta ) q^{8} + 3 \beta q^{10} + ( 1 - 2 \beta ) q^{11} + 2 q^{14} -3 \beta q^{16} -3 q^{17} + ( 2 + 4 \beta ) q^{19} + ( 3 - 3 \beta ) q^{20} + ( 2 + \beta ) q^{22} + ( 2 + 2 \beta ) q^{23} + 4 q^{25} + ( -4 + 2 \beta ) q^{28} + 3 q^{29} + ( -1 - 2 \beta ) q^{31} + ( 5 - \beta ) q^{32} + 3 \beta q^{34} + ( -6 + 6 \beta ) q^{35} + ( -8 + 2 \beta ) q^{37} + ( -4 - 6 \beta ) q^{38} + ( 3 - 6 \beta ) q^{40} + 4 \beta q^{41} + 6 q^{43} + ( -3 + \beta ) q^{44} + ( -2 - 4 \beta ) q^{46} + ( -6 + 4 \beta ) q^{47} + ( 1 - 4 \beta ) q^{49} -4 \beta q^{50} + 8 \beta q^{53} + ( -3 + 6 \beta ) q^{55} + ( -6 + 2 \beta ) q^{56} -3 \beta q^{58} + ( -10 + 2 \beta ) q^{59} + q^{61} + ( 2 + 3 \beta ) q^{62} + ( 1 + 2 \beta ) q^{64} + ( -5 - 6 \beta ) q^{67} + ( 3 - 3 \beta ) q^{68} -6 q^{70} + ( -4 + 4 \beta ) q^{71} + ( -2 + 8 \beta ) q^{73} + ( -2 + 6 \beta ) q^{74} + ( 2 + 2 \beta ) q^{76} + ( 6 - 2 \beta ) q^{77} -6 q^{79} + 9 \beta q^{80} + ( -4 - 4 \beta ) q^{82} + ( -5 - 2 \beta ) q^{83} + 9 q^{85} -6 \beta q^{86} -5 q^{88} + 2 \beta q^{92} + ( -4 + 2 \beta ) q^{94} + ( -6 - 12 \beta ) q^{95} + ( -6 + 6 \beta ) q^{97} + ( 4 + 3 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} - 6q^{5} + 2q^{7} + O(q^{10}) \) \( 2q - q^{2} - q^{4} - 6q^{5} + 2q^{7} + 3q^{10} + 4q^{14} - 3q^{16} - 6q^{17} + 8q^{19} + 3q^{20} + 5q^{22} + 6q^{23} + 8q^{25} - 6q^{28} + 6q^{29} - 4q^{31} + 9q^{32} + 3q^{34} - 6q^{35} - 14q^{37} - 14q^{38} + 4q^{41} + 12q^{43} - 5q^{44} - 8q^{46} - 8q^{47} - 2q^{49} - 4q^{50} + 8q^{53} - 10q^{56} - 3q^{58} - 18q^{59} + 2q^{61} + 7q^{62} + 4q^{64} - 16q^{67} + 3q^{68} - 12q^{70} - 4q^{71} + 4q^{73} + 2q^{74} + 6q^{76} + 10q^{77} - 12q^{79} + 9q^{80} - 12q^{82} - 12q^{83} + 18q^{85} - 6q^{86} - 10q^{88} + 2q^{92} - 6q^{94} - 24q^{95} - 6q^{97} + 11q^{98} + O(q^{100}) \)

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
−1.61803 0 0.618034 −3.00000 0 −1.23607 2.23607 0 4.85410
1.2 0.618034 0 −1.61803 −3.00000 0 3.23607 −2.23607 0 −1.85410
\(n\): 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.b 2
3.b odd 2 1 717.2.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
717.2.a.b 2 3.b odd 2 1
2151.2.a.b 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2151))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 3 + T )^{2} \)
$7$ \( -4 - 2 T + T^{2} \)
$11$ \( -5 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( 3 + T )^{2} \)
$19$ \( -4 - 8 T + T^{2} \)
$23$ \( 4 - 6 T + T^{2} \)
$29$ \( ( -3 + T )^{2} \)
$31$ \( -1 + 4 T + T^{2} \)
$37$ \( 44 + 14 T + T^{2} \)
$41$ \( -16 - 4 T + T^{2} \)
$43$ \( ( -6 + T )^{2} \)
$47$ \( -4 + 8 T + T^{2} \)
$53$ \( -64 - 8 T + T^{2} \)
$59$ \( 76 + 18 T + T^{2} \)
$61$ \( ( -1 + T )^{2} \)
$67$ \( 19 + 16 T + T^{2} \)
$71$ \( -16 + 4 T + T^{2} \)
$73$ \( -76 - 4 T + T^{2} \)
$79$ \( ( 6 + T )^{2} \)
$83$ \( 31 + 12 T + T^{2} \)
$89$ \( T^{2} \)
$97$ \( -36 + 6 T + T^{2} \)
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