Properties

Label 2151.2.a.g
Level $2151$
Weight $2$
Character orbit 2151.a
Self dual yes
Analytic conductor $17.176$
Analytic rank $0$
Dimension $8$
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: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.2585660609.1
Defining polynomial: \(x^{8} - 3 x^{7} - 4 x^{6} + 15 x^{5} + x^{4} - 19 x^{3} + 6 x^{2} + 3 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{2} ) q^{2} + ( 1 + \beta_{2} + \beta_{3} ) q^{4} + ( 2 - \beta_{5} ) q^{5} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} ) q^{7} + ( 1 + 2 \beta_{3} + \beta_{5} - \beta_{7} ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta_{2} ) q^{2} + ( 1 + \beta_{2} + \beta_{3} ) q^{4} + ( 2 - \beta_{5} ) q^{5} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} ) q^{7} + ( 1 + 2 \beta_{3} + \beta_{5} - \beta_{7} ) q^{8} + ( 1 + 2 \beta_{2} - \beta_{6} ) q^{10} + ( 2 - \beta_{1} - \beta_{6} + \beta_{7} ) q^{11} + ( 2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{13} + ( 1 - \beta_{1} - \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{14} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{16} + ( 1 - \beta_{1} - 2 \beta_{4} + \beta_{6} ) q^{17} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{7} ) q^{19} + ( 2 + \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{6} ) q^{20} + ( 3 - \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{22} + ( 4 + \beta_{1} + 3 \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{23} + ( 2 - \beta_{2} + \beta_{4} - 3 \beta_{5} ) q^{25} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{26} + ( 1 - 2 \beta_{3} + 2 \beta_{4} + \beta_{7} ) q^{28} + ( 3 + \beta_{2} - 3 \beta_{3} - \beta_{5} + \beta_{6} ) q^{29} + ( -2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} - \beta_{7} ) q^{31} + ( 1 + 3 \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{32} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + 4 \beta_{5} + \beta_{6} - \beta_{7} ) q^{34} + ( 3 - \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{7} ) q^{35} + ( \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{37} + ( -4 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{38} + ( 3 + 2 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{6} - 3 \beta_{7} ) q^{40} + ( 5 + \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{41} + ( -4 - \beta_{1} + 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{43} + ( 4 + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{44} + ( 5 + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{46} + ( \beta_{1} + \beta_{3} - 3 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{47} + ( 2 + 3 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{49} + ( -2 + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{50} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} + \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{52} + ( -2 - \beta_{1} - 5 \beta_{4} - 3 \beta_{7} ) q^{53} + ( 7 + 2 \beta_{2} - \beta_{3} + \beta_{4} - 5 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{55} + ( 1 + \beta_{1} - \beta_{2} - 4 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} ) q^{56} + ( 3 - \beta_{1} - 2 \beta_{3} - \beta_{5} + 3 \beta_{7} ) q^{58} + ( 1 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{59} + ( 2 + 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{61} + ( -4 - \beta_{1} - \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{62} + ( 1 + 3 \beta_{1} - \beta_{2} + 4 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - \beta_{6} - 5 \beta_{7} ) q^{64} + ( \beta_{1} + 4 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{65} + ( -6 - \beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{67} + ( 1 + 4 \beta_{3} + \beta_{4} + 4 \beta_{5} + 2 \beta_{6} ) q^{68} + ( 6 + \beta_{2} + \beta_{4} - 5 \beta_{5} - 2 \beta_{6} + 6 \beta_{7} ) q^{70} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + 3 \beta_{7} ) q^{71} + ( -3 - 2 \beta_{2} - \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{73} + ( -2 - 2 \beta_{2} - \beta_{3} + \beta_{5} ) q^{74} + ( 4 - 3 \beta_{1} - \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{6} + 4 \beta_{7} ) q^{76} + ( 5 + \beta_{1} + 6 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} + \beta_{6} ) q^{77} + ( 1 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{79} + ( -4 + 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} + 5 \beta_{5} - 5 \beta_{7} ) q^{80} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} + 3 \beta_{4} + \beta_{6} + 3 \beta_{7} ) q^{82} + ( 3 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} - 3 \beta_{7} ) q^{83} + ( -1 - 6 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{85} + ( -2 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{86} + ( -\beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{7} ) q^{88} + ( 8 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{89} + ( -7 - 4 \beta_{1} + \beta_{2} - \beta_{3} + 4 \beta_{5} - 3 \beta_{6} ) q^{91} + ( 5 - 5 \beta_{1} + 7 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{92} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{4} - 5 \beta_{7} ) q^{94} + ( -4 - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + 4 \beta_{7} ) q^{95} + ( -4 - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{97} + ( -3 + 2 \beta_{1} - 4 \beta_{3} + 3 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 5q^{2} + 7q^{4} + 13q^{5} - 7q^{7} + 15q^{8} + O(q^{10}) \) \( 8q + 5q^{2} + 7q^{4} + 13q^{5} - 7q^{7} + 15q^{8} + 4q^{10} + 19q^{11} - 3q^{13} + 8q^{14} + 9q^{16} + 13q^{17} - 6q^{19} + 18q^{20} + 3q^{22} + 18q^{23} + 7q^{25} - 2q^{28} + 10q^{29} - 2q^{31} + 20q^{32} - 4q^{34} + 7q^{35} - 8q^{37} - 9q^{38} + 29q^{40} + 22q^{41} - 24q^{43} + 7q^{44} + 30q^{46} + 17q^{47} + 15q^{49} - 24q^{50} + 22q^{52} + 32q^{55} - 19q^{56} + 18q^{58} + 24q^{59} + 10q^{61} - 30q^{62} + 33q^{64} + 17q^{65} - 48q^{67} + 21q^{68} + 31q^{70} + 17q^{71} + 2q^{73} - 9q^{74} + 10q^{76} - 10q^{77} + 17q^{79} - 8q^{80} - 17q^{82} + 37q^{83} + 28q^{85} + q^{86} + 15q^{88} + 41q^{89} - 39q^{91} + 38q^{92} + 2q^{94} - 16q^{95} - 20q^{97} - 46q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} - 4 x^{6} + 15 x^{5} + x^{4} - 19 x^{3} + 6 x^{2} + 3 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu^{7} + 3 \nu^{6} + 4 \nu^{5} - 15 \nu^{4} - \nu^{3} + 18 \nu^{2} - 5 \nu - 1 \)
\(\beta_{2}\)\(=\)\( \nu^{7} - 2 \nu^{6} - 6 \nu^{5} + 9 \nu^{4} + 10 \nu^{3} - 9 \nu^{2} - 4 \nu \)
\(\beta_{3}\)\(=\)\( -\nu^{7} + 3 \nu^{6} + 5 \nu^{5} - 16 \nu^{4} - 8 \nu^{3} + 22 \nu^{2} + 5 \nu - 4 \)
\(\beta_{4}\)\(=\)\( 2 \nu^{7} - 5 \nu^{6} - 10 \nu^{5} + 24 \nu^{4} + 12 \nu^{3} - 29 \nu^{2} - 2 \nu + 4 \)
\(\beta_{5}\)\(=\)\( 2 \nu^{7} - 5 \nu^{6} - 11 \nu^{5} + 26 \nu^{4} + 16 \nu^{3} - 35 \nu^{2} - 4 \nu + 7 \)
\(\beta_{6}\)\(=\)\( -3 \nu^{7} + 8 \nu^{6} + 14 \nu^{5} - 38 \nu^{4} - 15 \nu^{3} + 45 \nu^{2} - 7 \)
\(\beta_{7}\)\(=\)\( -3 \nu^{7} + 7 \nu^{6} + 16 \nu^{5} - 33 \nu^{4} - 22 \nu^{3} + 38 \nu^{2} + 4 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} - \beta_{4} - \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} - \beta_{1} + 5\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - \beta_{4} - 2 \beta_{3} - 5 \beta_{2} + 4\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-9 \beta_{7} + 8 \beta_{6} - 6 \beta_{5} + 3 \beta_{4} - 6 \beta_{3} - 7 \beta_{2} - 4 \beta_{1} + 22\)\()/2\)
\(\nu^{5}\)\(=\)\(-15 \beta_{7} + 9 \beta_{6} - 8 \beta_{5} + 3 \beta_{4} - 7 \beta_{3} - 16 \beta_{2} - \beta_{1} + 18\)
\(\nu^{6}\)\(=\)\((\)\(-69 \beta_{7} + 57 \beta_{6} - 41 \beta_{5} + 30 \beta_{4} - 37 \beta_{3} - 68 \beta_{2} - 17 \beta_{1} + 125\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-200 \beta_{7} + 139 \beta_{6} - 113 \beta_{5} + 75 \beta_{4} - 93 \beta_{3} - 217 \beta_{2} - 19 \beta_{1} + 273\)\()/2\)

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
0.551277
2.65398
0.315996
−0.417244
1.66273
−1.35076
−1.74142
1.32544
−1.77982 0 1.16777 3.42938 0 −4.73145 1.48122 0 −6.10369
1.2 −1.04938 0 −0.898808 3.87926 0 2.44573 3.04194 0 −4.07080
1.3 −0.777975 0 −1.39476 −0.966236 0 −3.17897 2.64103 0 0.751707
1.4 0.711649 0 −1.49356 −0.309852 0 −4.19834 −2.48618 0 −0.220506
1.5 0.846183 0 −1.28397 1.31908 0 0.998785 −2.77884 0 1.11619
1.6 1.92536 0 1.70701 3.65027 0 3.22894 −0.564105 0 7.02808
1.7 2.37665 0 3.64845 −0.0257644 0 0.278803 3.91777 0 −0.0612329
1.8 2.74734 0 5.54786 2.02387 0 −1.84350 9.74717 0 5.56025
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.g 8
3.b odd 2 1 717.2.a.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
717.2.a.f 8 3.b odd 2 1
2151.2.a.g 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2151))\):

\(T_{2}^{8} - \cdots\)
\(T_{5}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -11 + 12 T + 31 T^{2} - 33 T^{3} - 22 T^{4} + 25 T^{5} + T^{6} - 5 T^{7} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( -1 - 41 T - 81 T^{2} + 151 T^{3} + 20 T^{4} - 113 T^{5} + 61 T^{6} - 13 T^{7} + T^{8} \)
$7$ \( 256 - 1024 T + 176 T^{2} + 744 T^{3} - 13 T^{4} - 135 T^{5} - 11 T^{6} + 7 T^{7} + T^{8} \)
$11$ \( 125 - 650 T - 530 T^{2} + 827 T^{3} + 84 T^{4} - 334 T^{5} + 128 T^{6} - 19 T^{7} + T^{8} \)
$13$ \( 8912 - 12520 T - 7752 T^{2} + 2616 T^{3} + 1191 T^{4} - 161 T^{5} - 61 T^{6} + 3 T^{7} + T^{8} \)
$17$ \( -76013 + 22981 T + 34869 T^{2} - 10577 T^{3} - 2176 T^{4} + 889 T^{5} - 25 T^{6} - 13 T^{7} + T^{8} \)
$19$ \( -69776 - 48328 T + 13540 T^{2} + 13974 T^{3} + 1237 T^{4} - 583 T^{5} - 82 T^{6} + 6 T^{7} + T^{8} \)
$23$ \( -32528 - 40656 T + 50048 T^{2} - 5690 T^{3} - 4681 T^{4} + 987 T^{5} + 30 T^{6} - 18 T^{7} + T^{8} \)
$29$ \( -85537 + 65198 T + 5679 T^{2} - 11892 T^{3} + 712 T^{4} + 640 T^{5} - 55 T^{6} - 10 T^{7} + T^{8} \)
$31$ \( -88763 - 88535 T + 10934 T^{2} + 21768 T^{3} + 3614 T^{4} - 483 T^{5} - 122 T^{6} + 2 T^{7} + T^{8} \)
$37$ \( -16 - 216 T + 444 T^{2} + 354 T^{3} - 209 T^{4} - 165 T^{5} - 10 T^{6} + 8 T^{7} + T^{8} \)
$41$ \( 258608 - 393512 T + 120472 T^{2} + 20956 T^{3} - 14155 T^{4} + 1686 T^{5} + 63 T^{6} - 22 T^{7} + T^{8} \)
$43$ \( 63184 + 86688 T + 26912 T^{2} - 10340 T^{3} - 6981 T^{4} - 833 T^{5} + 116 T^{6} + 24 T^{7} + T^{8} \)
$47$ \( -1377328 - 285336 T + 451140 T^{2} - 72914 T^{3} - 9107 T^{4} + 2721 T^{5} - 79 T^{6} - 17 T^{7} + T^{8} \)
$53$ \( -23116976 - 13776144 T - 1691448 T^{2} + 238730 T^{3} + 47495 T^{4} - 987 T^{5} - 382 T^{6} + T^{8} \)
$59$ \( 1041776 + 2124456 T + 519960 T^{2} - 170680 T^{3} - 22661 T^{4} + 5558 T^{5} - 97 T^{6} - 24 T^{7} + T^{8} \)
$61$ \( 945611 + 205041 T - 289570 T^{2} - 58852 T^{3} + 13200 T^{4} + 1685 T^{5} - 210 T^{6} - 10 T^{7} + T^{8} \)
$67$ \( 1011184 - 671624 T - 806072 T^{2} - 188196 T^{3} + 2389 T^{4} + 6092 T^{5} + 850 T^{6} + 48 T^{7} + T^{8} \)
$71$ \( -891136 - 548160 T + 657536 T^{2} - 128144 T^{3} - 5787 T^{4} + 3115 T^{5} - 117 T^{6} - 17 T^{7} + T^{8} \)
$73$ \( -495952 + 848936 T - 335620 T^{2} - 45450 T^{3} + 22795 T^{4} + 789 T^{5} - 326 T^{6} - 2 T^{7} + T^{8} \)
$79$ \( -2003312 - 674024 T + 2099628 T^{2} - 488942 T^{3} + 6039 T^{4} + 5788 T^{5} - 278 T^{6} - 17 T^{7} + T^{8} \)
$83$ \( -20237 - 51842 T + 25576 T^{2} + 82225 T^{3} - 20246 T^{4} - 218 T^{5} + 410 T^{6} - 37 T^{7} + T^{8} \)
$89$ \( -2989312 - 1983744 T + 246656 T^{2} + 174516 T^{3} - 31477 T^{4} - 856 T^{5} + 534 T^{6} - 41 T^{7} + T^{8} \)
$97$ \( 340592 + 291368 T + 14408 T^{2} - 50712 T^{3} - 17485 T^{4} - 1914 T^{5} + 35 T^{6} + 20 T^{7} + T^{8} \)
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