Properties

Label 2151.2.a.f
Level $2151$
Weight $2$
Character orbit 2151.a
Self dual yes
Analytic conductor $17.176$
Analytic rank $1$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - x^{6} - 10 x^{5} + 8 x^{4} + 22 x^{3} - 5 x^{2} - 7 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( 1 + \beta_{6} ) q^{4} -\beta_{6} q^{5} + ( 1 + \beta_{1} + \beta_{4} ) q^{7} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{8} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( 1 + \beta_{6} ) q^{4} -\beta_{6} q^{5} + ( 1 + \beta_{1} + \beta_{4} ) q^{7} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{8} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{10} + ( -2 - \beta_{1} - \beta_{2} ) q^{11} + ( -1 + \beta_{3} ) q^{13} + ( -1 - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{14} + ( -1 - \beta_{4} - \beta_{5} + \beta_{6} ) q^{16} + ( -2 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{17} + ( -1 - \beta_{1} - \beta_{5} - \beta_{6} ) q^{19} + ( -4 + \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{20} + ( -3 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} ) q^{22} + ( -3 - \beta_{3} - \beta_{6} ) q^{23} + ( -1 - \beta_{4} - \beta_{5} + \beta_{6} ) q^{25} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{6} ) q^{26} + ( 1 + \beta_{1} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{28} + ( -2 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{29} + ( -1 - \beta_{1} - \beta_{4} + \beta_{5} ) q^{31} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{32} + ( -4 + \beta_{1} + \beta_{3} + 2 \beta_{4} - 3 \beta_{6} ) q^{34} + ( -\beta_{4} + \beta_{5} + \beta_{6} ) q^{35} + ( 2 - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{37} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{38} + ( -1 - 2 \beta_{1} - 4 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{40} + ( -4 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{41} + ( 1 - \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{43} + ( -2 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{44} + ( 1 + \beta_{1} - 4 \beta_{2} - \beta_{4} + \beta_{5} ) q^{46} + ( -2 + 2 \beta_{1} + 2 \beta_{3} + 3 \beta_{6} ) q^{47} + ( 1 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{49} + ( 1 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} ) q^{50} + ( -3 - \beta_{1} - \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{52} + ( -\beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{53} + ( -2 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{55} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{56} + ( 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} - 5 \beta_{5} + 4 \beta_{6} ) q^{58} + ( 1 + 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{59} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - 3 \beta_{6} ) q^{61} + ( 1 - \beta_{1} + 2 \beta_{4} - 2 \beta_{6} ) q^{62} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{64} + ( 2 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{65} + ( 2 - 2 \beta_{1} - 2 \beta_{5} + 2 \beta_{6} ) q^{67} + ( 1 - 4 \beta_{1} - 8 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} ) q^{68} + ( 1 + \beta_{1} + 2 \beta_{2} + 3 \beta_{4} - \beta_{5} - \beta_{6} ) q^{70} + ( 4 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{71} + ( -2 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{73} + ( 3 + 4 \beta_{1} + 4 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{74} + ( -4 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{76} + ( -4 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{77} + ( 1 + \beta_{3} - \beta_{4} + 3 \beta_{6} ) q^{79} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 5 \beta_{6} ) q^{80} + ( 3 - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 5 \beta_{6} ) q^{82} + ( -5 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} ) q^{83} + ( -3 + 3 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} ) q^{85} + ( 4 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + 6 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{86} + ( -1 - \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} - 5 \beta_{6} ) q^{88} + ( -5 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{89} + ( -4 + \beta_{1} + 3 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{91} + ( -5 + \beta_{1} + 2 \beta_{4} - 4 \beta_{6} ) q^{92} + ( -2 + \beta_{1} - \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{94} + ( 3 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{6} ) q^{95} + ( -2 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{97} + ( -3 - \beta_{1} - \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 10q^{4} - 3q^{5} + 3q^{7} + O(q^{10}) \) \( 7q + 10q^{4} - 3q^{5} + 3q^{7} - 11q^{11} - 5q^{13} - 6q^{14} - 4q^{16} - 11q^{17} - 8q^{19} - 34q^{20} - 19q^{22} - 26q^{23} - 4q^{25} - 6q^{26} - 2q^{28} - 6q^{29} - 2q^{31} + 5q^{32} - 40q^{34} + 5q^{35} + 12q^{37} + 9q^{38} - 5q^{40} - 26q^{41} + 10q^{43} - 3q^{44} + 6q^{46} - 7q^{47} + 2q^{49} + 5q^{50} - 22q^{52} - 2q^{53} - 8q^{55} + 9q^{56} - 6q^{58} - 6q^{59} - 8q^{61} + 2q^{62} - 18q^{64} + 17q^{65} + 24q^{67} + 9q^{68} - 3q^{70} + 25q^{71} - 16q^{73} + 9q^{74} - 32q^{76} - 24q^{77} + 19q^{79} - 18q^{80} + 39q^{82} - 37q^{83} - 20q^{85} + q^{86} - 21q^{88} - 29q^{89} - 19q^{91} - 52q^{92} - 22q^{94} + 24q^{95} - 12q^{97} - 23q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - x^{6} - 10 x^{5} + 8 x^{4} + 22 x^{3} - 5 x^{2} - 7 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{4} + \nu^{3} - 7 \nu^{2} - 5 \nu + 3 \)
\(\beta_{2}\)\(=\)\( \nu^{6} - \nu^{5} - 10 \nu^{4} + 7 \nu^{3} + 21 \nu^{2} + 2 \nu - 3 \)
\(\beta_{3}\)\(=\)\( -\nu^{6} + \nu^{5} + 10 \nu^{4} - 7 \nu^{3} - 21 \nu^{2} + 3 \)
\(\beta_{4}\)\(=\)\( \nu^{6} - \nu^{5} - 10 \nu^{4} + 7 \nu^{3} + 22 \nu^{2} + \nu - 6 \)
\(\beta_{5}\)\(=\)\( \nu^{6} - \nu^{5} - 10 \nu^{4} + 8 \nu^{3} + 22 \nu^{2} - 4 \nu - 6 \)
\(\beta_{6}\)\(=\)\( 2 \nu^{6} - \nu^{5} - 21 \nu^{4} + 6 \nu^{3} + 51 \nu^{2} + 11 \nu - 12 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{4} + \beta_{3} - \beta_{2} + 6\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{5} - 2 \beta_{4} + 5 \beta_{3} + 5 \beta_{2}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-2 \beta_{5} + 16 \beta_{4} + 7 \beta_{3} - 7 \beta_{2} + 2 \beta_{1} + 36\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(2 \beta_{6} + 14 \beta_{5} - 18 \beta_{4} + 31 \beta_{3} + 31 \beta_{2} + 2 \beta_{1} - 6\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(2 \beta_{6} - 20 \beta_{5} + 114 \beta_{4} + 43 \beta_{3} - 53 \beta_{2} + 22 \beta_{1} + 234\)\()/2\)

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
−1.19620
−2.67022
2.51519
0.326576
−0.727328
2.29537
0.456608
−2.42067 0 3.85962 −2.85962 0 −2.49298 −4.50152 0 6.92219
1.2 −2.03375 0 2.13614 −1.13614 0 4.00620 −0.276881 0 2.31063
1.3 −1.42661 0 0.0352187 0.964781 0 2.45740 2.80298 0 −1.37637
1.4 0.0204073 0 −1.99958 2.99958 0 −1.53276 −0.0816207 0 0.0612134
1.5 1.51425 0 0.292958 0.707042 0 3.59927 −2.58489 0 1.07064
1.6 1.83393 0 1.36331 −0.363312 0 −2.69764 −1.16764 0 −0.666290
1.7 2.51244 0 4.31233 −3.31233 0 −0.339486 5.80958 0 −8.32202
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.f 7
3.b odd 2 1 717.2.a.e 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
717.2.a.e 7 3.b odd 2 1
2151.2.a.f 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2151))\):

\( T_{2}^{7} - 12 T_{2}^{5} + 44 T_{2}^{3} - T_{2}^{2} - 49 T_{2} + 1 \)
\( T_{5}^{7} + 3 T_{5}^{6} - 11 T_{5}^{5} - 31 T_{5}^{4} + 19 T_{5}^{3} + 38 T_{5}^{2} - 12 T_{5} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 49 T - T^{2} + 44 T^{3} - 12 T^{5} + T^{7} \)
$3$ \( T^{7} \)
$5$ \( -8 - 12 T + 38 T^{2} + 19 T^{3} - 31 T^{4} - 11 T^{5} + 3 T^{6} + T^{7} \)
$7$ \( -124 - 426 T - 130 T^{2} + 161 T^{3} + 43 T^{4} - 21 T^{5} - 3 T^{6} + T^{7} \)
$11$ \( -592 + 524 T + 312 T^{2} - 221 T^{3} - 88 T^{4} + 22 T^{5} + 11 T^{6} + T^{7} \)
$13$ \( 152 + 588 T + 446 T^{2} - 135 T^{3} - 169 T^{4} - 25 T^{5} + 5 T^{6} + T^{7} \)
$17$ \( 6376 + 5108 T - 1686 T^{2} - 2441 T^{3} - 645 T^{4} - 21 T^{5} + 11 T^{6} + T^{7} \)
$19$ \( -620 + 446 T + 1534 T^{2} - 297 T^{3} - 341 T^{4} - 30 T^{5} + 8 T^{6} + T^{7} \)
$23$ \( -512 - 1008 T + 608 T^{2} + 2077 T^{3} + 1129 T^{4} + 254 T^{5} + 26 T^{6} + T^{7} \)
$29$ \( -73880 - 6748 T + 31314 T^{2} + 4661 T^{3} - 886 T^{4} - 139 T^{5} + 6 T^{6} + T^{7} \)
$31$ \( -128 + 208 T + 976 T^{2} + 361 T^{3} - 91 T^{4} - 40 T^{5} + 2 T^{6} + T^{7} \)
$37$ \( 11608 + 8252 T - 7158 T^{2} - 617 T^{3} + 789 T^{4} - 52 T^{5} - 12 T^{6} + T^{7} \)
$41$ \( -9208 - 34622 T - 33454 T^{2} - 9841 T^{3} - 534 T^{4} + 175 T^{5} + 26 T^{6} + T^{7} \)
$43$ \( 2564 - 4986 T - 6314 T^{2} + 2467 T^{3} + 1439 T^{4} - 158 T^{5} - 10 T^{6} + T^{7} \)
$47$ \( -464896 - 28160 T + 51848 T^{2} + 4959 T^{3} - 1453 T^{4} - 173 T^{5} + 7 T^{6} + T^{7} \)
$53$ \( -928 - 958 T + 2248 T^{2} + 1595 T^{3} - 89 T^{4} - 96 T^{5} + 2 T^{6} + T^{7} \)
$59$ \( 24320 - 91304 T - 1180 T^{2} + 8143 T^{3} - 324 T^{4} - 171 T^{5} + 6 T^{6} + T^{7} \)
$61$ \( -47728 - 10704 T + 14742 T^{2} + 3861 T^{3} - 703 T^{4} - 156 T^{5} + 8 T^{6} + T^{7} \)
$67$ \( -3733504 + 1798784 T - 194432 T^{2} - 22992 T^{3} + 4776 T^{4} - 56 T^{5} - 24 T^{6} + T^{7} \)
$71$ \( 37696 - 62204 T + 34024 T^{2} - 7267 T^{3} + 135 T^{4} + 179 T^{5} - 25 T^{6} + T^{7} \)
$73$ \( 3496 + 8092 T - 354 T^{2} - 3809 T^{3} - 1115 T^{4} - 26 T^{5} + 16 T^{6} + T^{7} \)
$79$ \( 115900 + 8098 T - 20640 T^{2} - 1177 T^{3} + 1086 T^{4} + 12 T^{5} - 19 T^{6} + T^{7} \)
$83$ \( 69584 - 31484 T - 21056 T^{2} + 2733 T^{3} + 2704 T^{4} + 496 T^{5} + 37 T^{6} + T^{7} \)
$89$ \( 3800 - 13834 T + 13882 T^{2} - 3661 T^{3} - 682 T^{4} + 184 T^{5} + 29 T^{6} + T^{7} \)
$97$ \( -335144 + 85996 T + 110574 T^{2} + 11815 T^{3} - 2322 T^{4} - 231 T^{5} + 12 T^{6} + T^{7} \)
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