Properties

Label 2151.2.a.d
Level $2151$
Weight $2$
Character orbit 2151.a
Self dual yes
Analytic conductor $17.176$
Analytic rank $1$
Dimension $3$
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: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 239)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( 1 - \beta_{2} ) q^{5} - q^{7} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( 1 - \beta_{2} ) q^{5} - q^{7} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{8} + ( -1 + \beta_{1} - \beta_{2} ) q^{10} -\beta_{2} q^{11} + ( -2 + \beta_{2} ) q^{13} -\beta_{1} q^{14} + ( -3 + \beta_{1} - 3 \beta_{2} ) q^{16} + ( 1 + \beta_{2} ) q^{17} + ( -2 - 3 \beta_{1} + \beta_{2} ) q^{19} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{20} + ( -1 - \beta_{2} ) q^{22} + ( -1 + \beta_{1} + \beta_{2} ) q^{23} + ( -3 + \beta_{1} - 3 \beta_{2} ) q^{25} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{26} -\beta_{2} q^{28} + ( 1 - \beta_{1} + 6 \beta_{2} ) q^{29} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{31} + ( -3 + \beta_{1} - 4 \beta_{2} ) q^{32} + ( 1 + \beta_{1} + \beta_{2} ) q^{34} + ( -1 + \beta_{2} ) q^{35} -3 \beta_{1} q^{37} + ( -5 - 2 \beta_{1} - 2 \beta_{2} ) q^{38} + ( 2 - 3 \beta_{1} + 3 \beta_{2} ) q^{40} + ( 1 - 3 \beta_{1} - 4 \beta_{2} ) q^{41} + ( 2 - \beta_{1} ) q^{43} + ( -1 - \beta_{1} + \beta_{2} ) q^{44} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{46} + ( -3 + \beta_{1} - 6 \beta_{2} ) q^{47} -6 q^{49} + ( -1 - 3 \beta_{1} - 2 \beta_{2} ) q^{50} + ( 1 + \beta_{1} - 3 \beta_{2} ) q^{52} + ( -3 - 3 \beta_{1} + 2 \beta_{2} ) q^{53} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{55} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{56} + ( 4 + \beta_{1} + 5 \beta_{2} ) q^{58} + ( -1 - \beta_{1} + 3 \beta_{2} ) q^{59} + ( -8 + 6 \beta_{1} - 8 \beta_{2} ) q^{61} + ( 2 - 4 \beta_{1} ) q^{62} + ( 4 - 5 \beta_{1} + 3 \beta_{2} ) q^{64} + ( -3 - \beta_{1} + 4 \beta_{2} ) q^{65} + ( 4 + \beta_{1} + 5 \beta_{2} ) q^{67} + ( 1 + \beta_{1} ) q^{68} + ( 1 - \beta_{1} + \beta_{2} ) q^{70} + ( 3 - 4 \beta_{1} + 5 \beta_{2} ) q^{71} + ( -7 + 5 \beta_{1} ) q^{73} + ( -6 - 3 \beta_{2} ) q^{74} + ( -2 + \beta_{1} - 6 \beta_{2} ) q^{76} + \beta_{2} q^{77} + ( 2 - 2 \beta_{1} + 6 \beta_{2} ) q^{79} + ( -1 + 4 \beta_{1} - 4 \beta_{2} ) q^{80} + ( -10 + \beta_{1} - 7 \beta_{2} ) q^{82} + ( 4 - 6 \beta_{1} + 6 \beta_{2} ) q^{83} + ( -\beta_{1} + \beta_{2} ) q^{85} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{86} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{88} + ( 7 + 6 \beta_{1} - 2 \beta_{2} ) q^{89} + ( 2 - \beta_{2} ) q^{91} + ( 2 + \beta_{1} - \beta_{2} ) q^{92} + ( -4 - 3 \beta_{1} - 5 \beta_{2} ) q^{94} + ( -4 \beta_{1} + 7 \beta_{2} ) q^{95} + ( -4 - 4 \beta_{1} + 7 \beta_{2} ) q^{97} -6 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{2} - q^{4} + 4q^{5} - 3q^{7} + O(q^{10}) \) \( 3q + q^{2} - q^{4} + 4q^{5} - 3q^{7} - q^{10} + q^{11} - 7q^{13} - q^{14} - 5q^{16} + 2q^{17} - 10q^{19} - 6q^{20} - 2q^{22} - 3q^{23} - 5q^{25} + q^{28} - 4q^{29} - 8q^{31} - 4q^{32} + 3q^{34} - 4q^{35} - 3q^{37} - 15q^{38} + 4q^{41} + 5q^{43} - 5q^{44} + 6q^{46} - 2q^{47} - 18q^{49} - 4q^{50} + 7q^{52} - 14q^{53} + 6q^{55} + 8q^{58} - 7q^{59} - 10q^{61} + 2q^{62} + 4q^{64} - 14q^{65} + 8q^{67} + 4q^{68} + q^{70} - 16q^{73} - 15q^{74} + q^{76} - q^{77} - 2q^{79} + 5q^{80} - 22q^{82} - 2q^{85} - 3q^{86} + 29q^{89} + 7q^{91} + 8q^{92} - 10q^{94} - 11q^{95} - 23q^{97} - 6q^{98} + O(q^{100}) \)

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
−1.24698
0.445042
1.80194
−1.24698 0 −0.445042 1.44504 0 −1.00000 3.04892 0 −1.80194
1.2 0.445042 0 −1.80194 2.80194 0 −1.00000 −1.69202 0 1.24698
1.3 1.80194 0 1.24698 −0.246980 0 −1.00000 −1.35690 0 −0.445042
\(n\): 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.d 3
3.b odd 2 1 239.2.a.a 3
12.b even 2 1 3824.2.a.e 3
15.d odd 2 1 5975.2.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
239.2.a.a 3 3.b odd 2 1
2151.2.a.d 3 1.a even 1 1 trivial
3824.2.a.e 3 12.b even 2 1
5975.2.a.d 3 15.d odd 2 1

Hecke kernels

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

\( T_{2}^{3} - T_{2}^{2} - 2 T_{2} + 1 \)
\( T_{5}^{3} - 4 T_{5}^{2} + 3 T_{5} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T - T^{2} + T^{3} \)
$3$ \( T^{3} \)
$5$ \( 1 + 3 T - 4 T^{2} + T^{3} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( 1 - 2 T - T^{2} + T^{3} \)
$13$ \( 7 + 14 T + 7 T^{2} + T^{3} \)
$17$ \( 1 - T - 2 T^{2} + T^{3} \)
$19$ \( -41 + 17 T + 10 T^{2} + T^{3} \)
$23$ \( -13 - 4 T + 3 T^{2} + T^{3} \)
$29$ \( -29 - 67 T + 4 T^{2} + T^{3} \)
$31$ \( -8 + 12 T + 8 T^{2} + T^{3} \)
$37$ \( -27 - 18 T + 3 T^{2} + T^{3} \)
$41$ \( 421 - 81 T - 4 T^{2} + T^{3} \)
$43$ \( -1 + 6 T - 5 T^{2} + T^{3} \)
$47$ \( -113 - 71 T + 2 T^{2} + T^{3} \)
$53$ \( 7 + 49 T + 14 T^{2} + T^{3} \)
$59$ \( -7 + 7 T^{2} + T^{3} \)
$61$ \( -776 - 88 T + 10 T^{2} + T^{3} \)
$67$ \( 29 - 51 T - 8 T^{2} + T^{3} \)
$71$ \( 91 - 49 T + T^{3} \)
$73$ \( -127 + 27 T + 16 T^{2} + T^{3} \)
$79$ \( 104 - 64 T + 2 T^{2} + T^{3} \)
$83$ \( 56 - 84 T + T^{3} \)
$89$ \( -83 + 215 T - 29 T^{2} + T^{3} \)
$97$ \( 97 + 90 T + 23 T^{2} + T^{3} \)
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