Properties

Label 2151.2.a.e
Level $2151$
Weight $2$
Character orbit 2151.a
Self dual yes
Analytic conductor $17.176$
Analytic rank $1$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.1767625.1
Defining polynomial: \(x^{6} - 7 x^{4} - x^{3} + 11 x^{2} + x - 1\)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{5} q^{2} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{4} + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{5} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{7} + ( \beta_{2} - \beta_{3} ) q^{8} +O(q^{10})\) \( q -\beta_{5} q^{2} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{4} + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{5} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{7} + ( \beta_{2} - \beta_{3} ) q^{8} + ( -2 - \beta_{3} + 2 \beta_{4} ) q^{10} + ( 2 + \beta_{1} - \beta_{2} - \beta_{4} ) q^{11} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{13} + ( \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{14} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{16} + ( -2 + 2 \beta_{1} + \beta_{4} - \beta_{5} ) q^{17} + ( -4 + \beta_{1} + \beta_{2} - \beta_{5} ) q^{19} + ( 2 + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{20} + ( -3 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{22} + ( 1 + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{23} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{25} + ( -1 - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{26} + ( -4 + 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{28} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{29} + ( -3 - \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{31} + ( 1 - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{5} ) q^{32} + ( 2 - \beta_{1} - \beta_{2} + 3 \beta_{3} + 3 \beta_{5} ) q^{34} + ( 2 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{35} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{37} + ( 3 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + 4 \beta_{5} ) q^{38} + ( -1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{40} + ( -3 - \beta_{1} - \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{41} + ( \beta_{2} + 3 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{43} + ( \beta_{1} + \beta_{2} - \beta_{3} - 5 \beta_{4} + 2 \beta_{5} ) q^{44} + ( -5 - \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{46} + ( \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{47} + ( 1 + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{49} + ( -3 + \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{50} + ( -1 - 3 \beta_{1} + 2 \beta_{4} + 4 \beta_{5} ) q^{52} + ( -1 - 3 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{53} + ( -3 \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} + 5 \beta_{5} ) q^{55} + ( -1 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{56} + ( -2 - \beta_{2} + \beta_{4} + \beta_{5} ) q^{58} + ( 3 - 3 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{59} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{61} + ( 7 - 3 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} + \beta_{4} + 5 \beta_{5} ) q^{62} + ( -6 + 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{64} + ( 3 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} ) q^{65} + ( -8 + 2 \beta_{1} - \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{67} + ( -5 - 3 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} ) q^{68} + ( 3 - 2 \beta_{1} - \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{70} + ( 4 + 2 \beta_{1} + 7 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{71} + ( -4 - 4 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{73} + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{74} + ( -3 - 2 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{76} + ( -2 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{77} + ( -2 + \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} ) q^{79} + ( -1 + \beta_{1} - 3 \beta_{3} - 2 \beta_{5} ) q^{80} + ( 6 - 2 \beta_{1} + \beta_{2} + 6 \beta_{3} - \beta_{4} + 6 \beta_{5} ) q^{82} + ( 3 + \beta_{1} + 3 \beta_{2} + 7 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} ) q^{83} + ( -5 - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} ) q^{85} + ( -9 - 2 \beta_{2} - 4 \beta_{3} + 5 \beta_{4} - \beta_{5} ) q^{86} + ( -5 + 2 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{88} + ( -4 + 3 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{89} + ( -4 + 3 \beta_{2} - \beta_{4} ) q^{91} + ( 3 + 2 \beta_{1} - \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{92} + ( 12 - \beta_{1} + 2 \beta_{2} + 7 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{94} + ( -3 + 3 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} - 5 \beta_{5} ) q^{95} + ( -6 - 3 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{97} + ( 1 - 4 \beta_{1} + \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{2} + 4q^{4} - 5q^{5} - 9q^{7} + 3q^{8} + O(q^{10}) \) \( 6q + 2q^{2} + 4q^{4} - 5q^{5} - 9q^{7} + 3q^{8} - 11q^{10} + 13q^{11} - q^{13} - 4q^{16} - 11q^{17} - 22q^{19} + q^{20} - 2q^{22} + 12q^{23} - q^{25} - 12q^{26} - 16q^{28} - 18q^{31} - 7q^{32} - 3q^{34} + 9q^{35} - 8q^{37} + 5q^{38} - 11q^{40} - 10q^{41} - 14q^{43} + 4q^{44} - 18q^{46} + 9q^{47} + 5q^{49} - 4q^{50} - 16q^{52} + 8q^{53} - 20q^{55} - 11q^{56} - 15q^{58} + 10q^{59} - 12q^{61} + 13q^{62} - 31q^{64} + 11q^{65} - 36q^{67} - 22q^{68} + q^{70} + 3q^{71} - 32q^{73} - 9q^{74} - 4q^{76} - 6q^{77} - q^{79} + 7q^{80} + 7q^{82} + 7q^{83} - 14q^{85} - 45q^{86} - 15q^{88} - 17q^{89} - 23q^{91} + 12q^{92} + 50q^{94} - 28q^{97} - 13q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 7 x^{4} - x^{3} + 11 x^{2} + x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - \nu^{4} - 4 \nu^{3} + 3 \nu^{2} - 1 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} - \nu^{4} - 6 \nu^{3} + 3 \nu^{2} + 8 \nu - 1 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} - \nu^{4} - 6 \nu^{3} + 5 \nu^{2} + 8 \nu - 5 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} + \nu^{4} - 8 \nu^{3} - 5 \nu^{2} + 14 \nu + 1 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} - \beta_{3} + 2\)
\(\nu^{3}\)\(=\)\(-\beta_{3} + \beta_{2} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{5} + 4 \beta_{4} - 6 \beta_{3} + \beta_{2} + \beta_{1} + 7\)
\(\nu^{5}\)\(=\)\(\beta_{5} + \beta_{4} - 7 \beta_{3} + 7 \beta_{2} + 17 \beta_{1} + 2\)

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
0.267500
2.30642
−1.94590
1.39213
−0.360520
−1.65963
−2.12029 0 2.49562 2.47082 0 −2.45271 −1.05086 0 −5.23885
1.2 −1.05161 0 −0.894124 −2.87285 0 −4.11383 3.04348 0 3.02110
1.3 −0.104133 0 −1.98916 0.431998 0 2.76657 0.415402 0 −0.0449852
1.4 0.899709 0 −1.19052 −1.67380 0 −1.75765 −2.87054 0 −1.50593
1.5 2.15574 0 2.64721 −3.41325 0 0.201367 1.39522 0 −7.35809
1.6 2.22058 0 2.93097 0.0570810 0 −3.64374 2.06730 0 0.126753
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.e 6
3.b odd 2 1 717.2.a.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
717.2.a.d 6 3.b odd 2 1
2151.2.a.e 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2151))\):

\( T_{2}^{6} - 2 T_{2}^{5} - 6 T_{2}^{4} + 11 T_{2}^{3} + 7 T_{2}^{2} - 9 T_{2} - 1 \)
\( T_{5}^{6} + 5 T_{5}^{5} - 2 T_{5}^{4} - 34 T_{5}^{3} - 24 T_{5}^{2} + 19 T_{5} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - 9 T + 7 T^{2} + 11 T^{3} - 6 T^{4} - 2 T^{5} + T^{6} \)
$3$ \( T^{6} \)
$5$ \( -1 + 19 T - 24 T^{2} - 34 T^{3} - 2 T^{4} + 5 T^{5} + T^{6} \)
$7$ \( 36 - 138 T - 193 T^{2} - 51 T^{3} + 17 T^{4} + 9 T^{5} + T^{6} \)
$11$ \( -521 + 1110 T - 721 T^{2} + 99 T^{3} + 41 T^{4} - 13 T^{5} + T^{6} \)
$13$ \( -44 + 78 T + 167 T^{2} - 59 T^{3} - 43 T^{4} + T^{5} + T^{6} \)
$17$ \( 1459 + 655 T - 408 T^{2} - 176 T^{3} + 14 T^{4} + 11 T^{5} + T^{6} \)
$19$ \( 436 + 1328 T + 1453 T^{2} + 735 T^{3} + 184 T^{4} + 22 T^{5} + T^{6} \)
$23$ \( 5584 + 140 T - 2291 T^{2} + 701 T^{3} - 24 T^{4} - 12 T^{5} + T^{6} \)
$29$ \( -1 - 6 T + 36 T^{2} - 24 T^{3} - 22 T^{4} + T^{6} \)
$31$ \( -5869 - 8533 T - 3931 T^{2} - 505 T^{3} + 63 T^{4} + 18 T^{5} + T^{6} \)
$37$ \( 36 + 60 T - 13 T^{2} - 47 T^{3} - 4 T^{4} + 8 T^{5} + T^{6} \)
$41$ \( -484 + 902 T + 501 T^{2} - 338 T^{3} - 43 T^{4} + 10 T^{5} + T^{6} \)
$43$ \( -10924 - 13648 T - 5963 T^{2} - 1001 T^{3} - 8 T^{4} + 14 T^{5} + T^{6} \)
$47$ \( -2396 + 5644 T - 4157 T^{2} + 1135 T^{3} - 79 T^{4} - 9 T^{5} + T^{6} \)
$53$ \( 4 - 98 T + 335 T^{2} + 175 T^{3} - 60 T^{4} - 8 T^{5} + T^{6} \)
$59$ \( 2000 - 2225 T^{2} + 1000 T^{3} - 75 T^{4} - 10 T^{5} + T^{6} \)
$61$ \( 2719 + 2075 T - 571 T^{2} - 441 T^{3} - 19 T^{4} + 12 T^{5} + T^{6} \)
$67$ \( -61424 - 36664 T - 3440 T^{2} + 1700 T^{3} + 435 T^{4} + 36 T^{5} + T^{6} \)
$71$ \( -27584 - 69304 T + 19451 T^{2} + 903 T^{3} - 283 T^{4} - 3 T^{5} + T^{6} \)
$73$ \( 8236 - 4086 T - 3943 T^{2} + 449 T^{3} + 302 T^{4} + 32 T^{5} + T^{6} \)
$79$ \( -37484 - 81018 T + 26057 T^{2} + 148 T^{3} - 334 T^{4} + T^{5} + T^{6} \)
$83$ \( 140741 - 349864 T + 40967 T^{2} + 3091 T^{3} - 413 T^{4} - 7 T^{5} + T^{6} \)
$89$ \( 5744 + 17980 T + 10199 T^{2} - 2406 T^{3} - 174 T^{4} + 17 T^{5} + T^{6} \)
$97$ \( 1192624 + 151580 T - 32531 T^{2} - 5364 T^{3} + T^{4} + 28 T^{5} + T^{6} \)
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