Properties

Label 2169.2.a.e
Level $2169$
Weight $2$
Character orbit 2169.a
Self dual yes
Analytic conductor $17.320$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2169 = 3^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2169.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.3195521984\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: 7.7.31056073.1
Defining polynomial: \(x^{7} - 3 x^{6} - 3 x^{5} + 11 x^{4} + x^{3} - 9 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 241)
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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{4} + ( 1 + \beta_{1} + \beta_{2} + \beta_{6} ) q^{5} + ( -2 + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} ) q^{7} + ( 2 + \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} ) q^{8} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{4} + ( 1 + \beta_{1} + \beta_{2} + \beta_{6} ) q^{5} + ( -2 + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} ) q^{7} + ( 2 + \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} ) q^{8} + ( 1 - \beta_{1} - \beta_{4} + \beta_{6} ) q^{10} + ( 3 + \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{11} + ( 1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{13} + ( -1 + \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{14} + ( 3 - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{16} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{17} + ( -3 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{19} + ( 2 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} ) q^{20} + ( 4 - 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} ) q^{22} + ( 2 - \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{23} + ( 1 + \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{25} + ( 3 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{26} + ( -\beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{28} + ( 2 - 3 \beta_{1} - \beta_{2} - 3 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} ) q^{29} + ( -3 + 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{31} + ( 4 + 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{6} ) q^{32} + ( 1 - 2 \beta_{1} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{34} + ( -3 - 2 \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} - 4 \beta_{6} ) q^{35} + ( -2 + \beta_{1} - 3 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} ) q^{37} + ( -7 + 2 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{38} + ( 4 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{40} + ( 2 - 3 \beta_{1} + 3 \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{41} + ( 5 - 5 \beta_{1} - 4 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} ) q^{43} + ( 4 - 4 \beta_{1} + \beta_{2} - 5 \beta_{3} - \beta_{4} - \beta_{5} ) q^{44} + ( -4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{46} + ( 2 + 5 \beta_{1} + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} + 4 \beta_{6} ) q^{47} + ( -\beta_{1} - 4 \beta_{2} + \beta_{3} - 5 \beta_{4} - 4 \beta_{5} ) q^{49} + ( -2 \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{50} + ( 9 - \beta_{1} + \beta_{2} - 6 \beta_{3} + 2 \beta_{4} + \beta_{5} + 4 \beta_{6} ) q^{52} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + 5 \beta_{4} + 4 \beta_{5} + \beta_{6} ) q^{53} + ( 5 + 5 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 6 \beta_{6} ) q^{55} + ( -3 + 2 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{56} + ( 1 - 3 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} ) q^{58} + ( 4 - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{59} + ( 2 + 3 \beta_{1} + 4 \beta_{2} - \beta_{3} + 7 \beta_{4} + 5 \beta_{5} + 2 \beta_{6} ) q^{61} + ( -3 + 3 \beta_{1} - \beta_{2} - \beta_{5} ) q^{62} + ( 3 - 3 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{64} + ( 3 + 2 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{65} + ( 6 + \beta_{2} + 5 \beta_{4} + 4 \beta_{5} + 5 \beta_{6} ) q^{67} + ( 4 - 5 \beta_{1} - 3 \beta_{2} - \beta_{3} - 4 \beta_{4} - 6 \beta_{5} ) q^{68} + ( -3 + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - 5 \beta_{6} ) q^{70} + ( 8 - 4 \beta_{1} - 2 \beta_{2} - \beta_{3} - 4 \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{71} + ( -1 + \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} ) q^{73} + ( -9 + 3 \beta_{1} - \beta_{2} + 7 \beta_{3} - 3 \beta_{4} - 4 \beta_{5} - 6 \beta_{6} ) q^{74} + ( -11 + 3 \beta_{1} - 4 \beta_{2} + 7 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} - 7 \beta_{6} ) q^{76} + ( -4 - 2 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{77} + ( -5 \beta_{1} - 3 \beta_{2} + \beta_{3} - 4 \beta_{5} ) q^{79} + ( 7 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + 5 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} ) q^{80} + ( 4 - 3 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} ) q^{82} + ( 3 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{83} + ( -3 + \beta_{2} - 3 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} ) q^{85} + ( 10 - 7 \beta_{1} - 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} ) q^{86} + ( 4 + \beta_{1} + 6 \beta_{2} - 6 \beta_{3} + \beta_{4} + 5 \beta_{5} + 3 \beta_{6} ) q^{88} + ( -4 + 9 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{89} + ( 1 - 7 \beta_{2} - \beta_{3} - \beta_{5} + 3 \beta_{6} ) q^{91} + ( -2 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{6} ) q^{92} + ( -1 - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + 5 \beta_{6} ) q^{94} + ( 3 - \beta_{1} - 3 \beta_{2} + 5 \beta_{3} - \beta_{4} - 5 \beta_{5} - 3 \beta_{6} ) q^{95} + ( 2 - 4 \beta_{2} + 3 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{97} + ( -6 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 4q^{2} + 2q^{4} + 8q^{5} - 7q^{7} + 6q^{8} + O(q^{10}) \) \( 7q + 4q^{2} + 2q^{4} + 8q^{5} - 7q^{7} + 6q^{8} + 3q^{10} + 18q^{11} - q^{13} + 6q^{14} + 4q^{16} + 2q^{17} - 6q^{19} + 8q^{20} + 10q^{22} + 22q^{23} + 5q^{25} - 8q^{26} + 9q^{28} + 16q^{29} - 18q^{31} + 6q^{32} + 11q^{34} - 7q^{35} + 8q^{37} - 16q^{38} + 14q^{40} + 15q^{41} + 14q^{43} + 4q^{44} + 11q^{46} + 10q^{47} + 6q^{49} + 4q^{50} + 27q^{52} - 15q^{53} + 29q^{55} - 13q^{56} + 17q^{58} + 18q^{59} + 4q^{61} - 13q^{62} + 2q^{64} + 7q^{65} + 18q^{67} + 15q^{68} + 8q^{70} + 50q^{71} - 10q^{74} - 20q^{76} - 17q^{77} - 15q^{79} + 11q^{80} + 45q^{82} + 24q^{83} - 2q^{85} + 23q^{86} + 8q^{88} + 13q^{89} - 12q^{91} + 10q^{92} - 32q^{94} + 41q^{95} + q^{97} - 9q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 3 x^{6} - 3 x^{5} + 11 x^{4} + x^{3} - 9 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 4 \nu^{3} + 5 \nu^{2} + 2 \nu - 1 \)
\(\beta_{3}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 4 \nu^{3} + 4 \nu^{2} + 3 \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{6} - 3 \nu^{5} - 2 \nu^{4} + 9 \nu^{3} - 3 \nu^{2} - 4 \nu + 2 \)
\(\beta_{5}\)\(=\)\( \nu^{6} - 2 \nu^{5} - 5 \nu^{4} + 6 \nu^{3} + 7 \nu^{2} - 3 \nu - 2 \)
\(\beta_{6}\)\(=\)\( -\nu^{6} + 3 \nu^{5} + 3 \nu^{4} - 11 \nu^{3} - \nu^{2} + 8 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(-\beta_{6} - \beta_{5} - \beta_{3} + 2 \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{6} - 2 \beta_{5} + \beta_{4} - 6 \beta_{3} + 8 \beta_{2} + 8 \beta_{1} + 8\)
\(\nu^{5}\)\(=\)\(-6 \beta_{6} - 8 \beta_{5} + 2 \beta_{4} - 11 \beta_{3} + 20 \beta_{2} + 25 \beta_{1} + 11\)
\(\nu^{6}\)\(=\)\(-11 \beta_{6} - 19 \beta_{5} + 9 \beta_{4} - 39 \beta_{3} + 61 \beta_{2} + 62 \beta_{1} + 44\)

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
2.73684
1.48734
1.27758
0.369356
−0.356270
−0.911223
−1.60363
−1.73684 0 1.01662 2.63180 0 −2.01025 1.70797 0 −4.57103
1.2 −0.487343 0 −1.76250 −0.961999 0 −4.61392 1.83363 0 0.468824
1.3 −0.277577 0 −1.92295 1.23324 0 1.36627 1.08892 0 −0.342320
1.4 0.630644 0 −1.60229 3.89634 0 −3.68231 −2.27176 0 2.45721
1.5 1.35627 0 −0.160532 −2.74184 0 −0.283608 −2.93026 0 −3.71867
1.6 1.91122 0 1.65278 2.25110 0 3.52970 −0.663624 0 4.30235
1.7 2.60363 0 4.77887 1.69135 0 −1.30586 7.23513 0 4.40364
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(241\) \(1\)

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 2169.2.a.e 7
3.b odd 2 1 241.2.a.a 7
12.b even 2 1 3856.2.a.j 7
15.d odd 2 1 6025.2.a.f 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.2.a.a 7 3.b odd 2 1
2169.2.a.e 7 1.a even 1 1 trivial
3856.2.a.j 7 12.b even 2 1
6025.2.a.f 7 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - 4 T_{2}^{6} + 14 T_{2}^{4} - 10 T_{2}^{3} - 6 T_{2}^{2} + 3 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2169))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T - 6 T^{2} - 10 T^{3} + 14 T^{4} - 4 T^{6} + T^{7} \)
$3$ \( T^{7} \)
$5$ \( -127 + 137 T + 93 T^{2} - 165 T^{3} + 50 T^{4} + 12 T^{5} - 8 T^{6} + T^{7} \)
$7$ \( 61 + 260 T + 127 T^{2} - 138 T^{3} - 98 T^{4} - 3 T^{5} + 7 T^{6} + T^{7} \)
$11$ \( -1069 - 1281 T + 1559 T^{2} - 137 T^{3} - 283 T^{4} + 117 T^{5} - 18 T^{6} + T^{7} \)
$13$ \( -1 + 13 T + 860 T^{2} + 533 T^{3} - 62 T^{4} - 48 T^{5} + T^{6} + T^{7} \)
$17$ \( 1039 - 633 T - 1382 T^{2} + 967 T^{3} + 86 T^{4} - 65 T^{5} - 2 T^{6} + T^{7} \)
$19$ \( -5983 - 6849 T + 3337 T^{2} + 1067 T^{3} - 276 T^{4} - 56 T^{5} + 6 T^{6} + T^{7} \)
$23$ \( -1369 - 532 T + 1693 T^{2} - 75 T^{3} - 463 T^{4} + 168 T^{5} - 22 T^{6} + T^{7} \)
$29$ \( 10769 - 23012 T + 17808 T^{2} - 6173 T^{3} + 839 T^{4} + 25 T^{5} - 16 T^{6} + T^{7} \)
$31$ \( -617 - 3770 T - 3600 T^{2} - 1006 T^{3} + 109 T^{4} + 104 T^{5} + 18 T^{6} + T^{7} \)
$37$ \( -78167 + 88791 T - 27243 T^{2} - 828 T^{3} + 1330 T^{4} - 119 T^{5} - 8 T^{6} + T^{7} \)
$41$ \( -101009 - 157642 T - 58348 T^{2} + 5058 T^{3} + 1974 T^{4} - 122 T^{5} - 15 T^{6} + T^{7} \)
$43$ \( 296569 + 67463 T - 74048 T^{2} - 49 T^{3} + 2460 T^{4} - 141 T^{5} - 14 T^{6} + T^{7} \)
$47$ \( 7793 - 48600 T - 18600 T^{2} + 3904 T^{3} + 997 T^{4} - 116 T^{5} - 10 T^{6} + T^{7} \)
$53$ \( 230663 + 281202 T + 84407 T^{2} - 35 T^{3} - 2311 T^{4} - 123 T^{5} + 15 T^{6} + T^{7} \)
$59$ \( 2076763 - 114293 T - 152610 T^{2} + 5036 T^{3} + 3030 T^{4} - 130 T^{5} - 18 T^{6} + T^{7} \)
$61$ \( 23149 + 291498 T - 136478 T^{2} + 13144 T^{3} + 1693 T^{4} - 254 T^{5} - 4 T^{6} + T^{7} \)
$67$ \( 2288147 + 4468 T - 189536 T^{2} + 4503 T^{3} + 3579 T^{4} - 157 T^{5} - 18 T^{6} + T^{7} \)
$71$ \( 255937 - 122885 T - 37139 T^{2} + 34990 T^{3} - 8586 T^{4} + 955 T^{5} - 50 T^{6} + T^{7} \)
$73$ \( 11879 - 5297 T - 192681 T^{2} + 37009 T^{3} + 1068 T^{4} - 378 T^{5} + T^{7} \)
$79$ \( 52709 + 79692 T + 33855 T^{2} + 711 T^{3} - 1557 T^{4} - 85 T^{5} + 15 T^{6} + T^{7} \)
$83$ \( -4333 + 2523 T + 1594 T^{2} - 1238 T^{3} + 14 T^{4} + 124 T^{5} - 24 T^{6} + T^{7} \)
$89$ \( 89477 + 26169 T - 119043 T^{2} + 12156 T^{3} + 4667 T^{4} - 363 T^{5} - 13 T^{6} + T^{7} \)
$97$ \( 40121 - 136236 T - 43003 T^{2} + 15061 T^{3} + 939 T^{4} - 283 T^{5} - T^{6} + T^{7} \)
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