Properties

Label 1859.2.a.l
Level $1859$
Weight $2$
Character orbit 1859.a
Self dual yes
Analytic conductor $14.844$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1859.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.8441897358\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.28561300.1
Defining polynomial: \(x^{6} - 9 x^{4} - x^{3} + 22 x^{2} + 4 x - 12\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 143)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + ( -\beta_{2} - \beta_{4} ) q^{3} + ( 1 + \beta_{2} + \beta_{4} + \beta_{5} ) q^{4} + ( 1 + \beta_{1} ) q^{5} + ( 1 + \beta_{2} + \beta_{3} + \beta_{5} ) q^{6} + ( -\beta_{1} + \beta_{5} ) q^{7} + ( -\beta_{2} - \beta_{3} ) q^{8} + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} + ( -\beta_{2} - \beta_{4} ) q^{3} + ( 1 + \beta_{2} + \beta_{4} + \beta_{5} ) q^{4} + ( 1 + \beta_{1} ) q^{5} + ( 1 + \beta_{2} + \beta_{3} + \beta_{5} ) q^{6} + ( -\beta_{1} + \beta_{5} ) q^{7} + ( -\beta_{2} - \beta_{3} ) q^{8} + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{9} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{10} + q^{11} + ( -3 - 2 \beta_{1} - 2 \beta_{2} - \beta_{4} ) q^{12} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} ) q^{14} + ( -2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{15} + ( 2 + 2 \beta_{1} + \beta_{4} - \beta_{5} ) q^{16} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{17} + ( -2 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{4} + \beta_{5} ) q^{18} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{19} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{20} + ( -\beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{21} -\beta_{2} q^{22} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{23} + ( 2 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{24} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{25} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{5} ) q^{27} + ( 3 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{28} + ( 2 + 3 \beta_{4} - 2 \beta_{5} ) q^{29} + ( 2 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{30} + ( 1 + \beta_{3} + \beta_{4} - \beta_{5} ) q^{31} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{32} + ( -\beta_{2} - \beta_{4} ) q^{33} + ( 3 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{34} + ( -2 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{35} + ( 1 + \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + \beta_{4} + \beta_{5} ) q^{36} + ( 4 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{37} + ( -4 + \beta_{1} - 2 \beta_{2} - \beta_{5} ) q^{38} + ( -\beta_{1} - 4 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{40} + ( 4 + \beta_{2} + \beta_{3} - \beta_{5} ) q^{41} + ( -3 \beta_{1} - 4 \beta_{2} + \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{42} + ( -1 - 2 \beta_{1} - \beta_{4} - \beta_{5} ) q^{43} + ( 1 + \beta_{2} + \beta_{4} + \beta_{5} ) q^{44} + ( 4 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{45} + ( 4 + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{46} + ( 2 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{47} + ( -2 - 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{48} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{49} + ( -3 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{50} + ( -2 + \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{51} + ( 2 + 4 \beta_{1} - 4 \beta_{2} - \beta_{4} - \beta_{5} ) q^{53} + ( 5 + 2 \beta_{1} + 3 \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{54} + ( 1 + \beta_{1} ) q^{55} + ( 3 + \beta_{1} + 2 \beta_{2} + 3 \beta_{4} ) q^{56} + ( 2 + 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{57} + ( 4 - 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{58} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{59} + ( -9 - 6 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{60} + ( -4 - 2 \beta_{1} + \beta_{2} + \beta_{5} ) q^{61} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{62} + ( -2 - 3 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{63} + ( -5 + \beta_{3} ) q^{64} + ( 1 + \beta_{2} + \beta_{3} + \beta_{5} ) q^{66} + ( 3 - \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{67} + ( 4 - \beta_{1} - 3 \beta_{3} + \beta_{4} ) q^{68} + ( 2 + 2 \beta_{1} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{69} + ( 5 + \beta_{1} + 3 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{70} + ( -2 - 4 \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{71} + ( -6 - 3 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} ) q^{72} + ( 2 - \beta_{1} - \beta_{4} ) q^{73} + ( -6 + \beta_{1} - 6 \beta_{2} - \beta_{4} - \beta_{5} ) q^{74} + ( -1 - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{75} + ( 1 - 3 \beta_{1} + 5 \beta_{2} + \beta_{3} + 4 \beta_{4} + \beta_{5} ) q^{76} + ( -\beta_{1} + \beta_{5} ) q^{77} + ( 2 - \beta_{1} - \beta_{3} - 2 \beta_{4} - 5 \beta_{5} ) q^{79} + ( 7 + 3 \beta_{1} + 3 \beta_{2} + \beta_{4} + \beta_{5} ) q^{80} + ( 4 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{81} + ( -5 - 2 \beta_{1} - 5 \beta_{2} - 3 \beta_{4} - 2 \beta_{5} ) q^{82} + ( 1 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{83} + ( 5 + 3 \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{84} + ( -2 - 2 \beta_{2} + \beta_{4} ) q^{85} + ( -3 + 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{5} ) q^{86} + ( -6 - \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{87} + ( -\beta_{2} - \beta_{3} ) q^{88} + ( 1 + \beta_{1} - 4 \beta_{2} - \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{89} + ( -10 - 4 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} ) q^{90} + ( 4 - 2 \beta_{2} - \beta_{3} + 6 \beta_{4} ) q^{92} + ( -3 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{93} + ( 2 - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{94} + ( 5 + 5 \beta_{1} - 2 \beta_{2} - 3 \beta_{4} - 2 \beta_{5} ) q^{95} + ( -2 + 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{4} - \beta_{5} ) q^{96} + ( 2 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{5} ) q^{97} + ( -\beta_{1} + 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{98} + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{3} + 8 q^{4} + 6 q^{5} + 12 q^{6} + 3 q^{7} - 3 q^{8} + 7 q^{9} + O(q^{10}) \) \( 6 q + q^{3} + 8 q^{4} + 6 q^{5} + 12 q^{6} + 3 q^{7} - 3 q^{8} + 7 q^{9} - 3 q^{10} + 6 q^{11} - 17 q^{12} + 12 q^{14} - 4 q^{15} + 8 q^{16} + 2 q^{17} - 6 q^{18} + 10 q^{19} + 15 q^{20} + 12 q^{21} + 3 q^{23} + 14 q^{24} - 6 q^{25} + 10 q^{27} + 16 q^{28} + 3 q^{29} + 19 q^{30} + 5 q^{31} - q^{32} + q^{33} - 5 q^{34} - 13 q^{35} + 20 q^{36} + 25 q^{37} - 27 q^{38} - 8 q^{40} + 24 q^{41} + 13 q^{42} - 8 q^{43} + 8 q^{44} + 27 q^{45} + 18 q^{46} + 10 q^{47} - 28 q^{48} - q^{49} - 26 q^{50} - 17 q^{51} + 10 q^{53} + 47 q^{54} + 6 q^{55} + 15 q^{56} + 6 q^{58} - 4 q^{59} - 61 q^{60} - 21 q^{61} - 5 q^{62} + 6 q^{63} - 27 q^{64} + 12 q^{66} + 21 q^{67} + 14 q^{68} + 5 q^{69} + 31 q^{70} - 3 q^{71} - 50 q^{72} + 13 q^{73} - 38 q^{74} - 23 q^{75} + 8 q^{76} + 3 q^{77} - 4 q^{79} + 44 q^{80} + 34 q^{81} - 33 q^{82} + 8 q^{83} + 47 q^{84} - 13 q^{85} - 11 q^{86} - 51 q^{87} - 3 q^{88} - 9 q^{89} - 70 q^{90} + 15 q^{92} - 21 q^{93} + 10 q^{94} + 27 q^{95} - 19 q^{96} + 15 q^{97} + 21 q^{98} + 7 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 9 x^{4} - x^{3} + 22 x^{2} + 4 x - 12\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 4 \nu \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} - 7 \nu^{3} - \nu^{2} + 10 \nu + 2 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{5} + 2 \nu^{4} + 7 \nu^{3} - 13 \nu^{2} - 10 \nu + 16 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{5} + \beta_{4} + 7 \beta_{2} + 12\)
\(\nu^{5}\)\(=\)\(2 \beta_{4} + 7 \beta_{3} + \beta_{2} + 18 \beta_{1} + 1\)

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
2.35258
−2.23039
1.80787
−1.58147
−1.08549
0.736891
−2.53464 −2.99024 4.42441 3.35258 7.57919 −1.91841 −6.14501 5.94153 −8.49760
1.2 −1.97462 −0.571492 1.89914 −1.23039 1.12848 2.55803 0.199164 −2.67340 2.42955
1.3 −0.268396 2.35117 −1.92796 2.80787 −0.631044 −2.38467 1.05425 2.52800 −0.753621
1.4 0.498953 −0.240597 −1.75105 −0.581470 −0.120047 −1.41017 −1.87160 −2.94211 −0.290126
1.5 1.82172 3.11527 1.31865 −0.0854874 5.67515 4.51942 −1.24122 6.70494 −0.155734
1.6 2.45699 −0.664116 4.03681 1.73689 −1.63173 1.63580 5.00442 −2.55895 4.26753
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(13\) \(1\)

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 1859.2.a.l 6
13.b even 2 1 1859.2.a.k 6
13.e even 6 2 143.2.e.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.2.e.c 12 13.e even 6 2
1859.2.a.k 6 13.b even 2 1
1859.2.a.l 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1859))\):

\( T_{2}^{6} - 10 T_{2}^{4} + T_{2}^{3} + 24 T_{2}^{2} - 5 T_{2} - 3 \)
\( T_{7}^{6} - 3 T_{7}^{5} - 16 T_{7}^{4} + 27 T_{7}^{3} + 82 T_{7}^{2} - 52 T_{7} - 122 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 - 5 T + 24 T^{2} + T^{3} - 10 T^{4} + T^{6} \)
$3$ \( 2 + 14 T + 26 T^{2} + 7 T^{3} - 12 T^{4} - T^{5} + T^{6} \)
$5$ \( -1 - 13 T - 14 T^{2} + 15 T^{3} + 6 T^{4} - 6 T^{5} + T^{6} \)
$7$ \( -122 - 52 T + 82 T^{2} + 27 T^{3} - 16 T^{4} - 3 T^{5} + T^{6} \)
$11$ \( ( -1 + T )^{6} \)
$13$ \( T^{6} \)
$17$ \( -139 + 191 T + 260 T^{2} + 13 T^{3} - 34 T^{4} - 2 T^{5} + T^{6} \)
$19$ \( -302 - 1034 T + 42 T^{2} + 211 T^{3} - 9 T^{4} - 10 T^{5} + T^{6} \)
$23$ \( 592 + 1736 T + 904 T^{2} + 15 T^{3} - 56 T^{4} - 3 T^{5} + T^{6} \)
$29$ \( -19891 - 528 T + 3700 T^{2} + 197 T^{3} - 126 T^{4} - 3 T^{5} + T^{6} \)
$31$ \( 120 + 380 T + 350 T^{2} + 57 T^{3} - 31 T^{4} - 5 T^{5} + T^{6} \)
$37$ \( -5171 + 9344 T - 4290 T^{2} + 81 T^{3} + 176 T^{4} - 25 T^{5} + T^{6} \)
$41$ \( -6427 + 6715 T - 1642 T^{2} - 315 T^{3} + 182 T^{4} - 24 T^{5} + T^{6} \)
$43$ \( 1448 + 2268 T + 546 T^{2} - 309 T^{3} - 43 T^{4} + 8 T^{5} + T^{6} \)
$47$ \( -2 + 164 T^{2} + 183 T^{3} - 15 T^{4} - 10 T^{5} + T^{6} \)
$53$ \( 121 + 1177 T - 4650 T^{2} + 1997 T^{3} - 156 T^{4} - 10 T^{5} + T^{6} \)
$59$ \( -18398 + 12496 T + 5792 T^{2} - 469 T^{3} - 149 T^{4} + 4 T^{5} + T^{6} \)
$61$ \( -655 - 3250 T - 1010 T^{2} + 243 T^{3} + 144 T^{4} + 21 T^{5} + T^{6} \)
$67$ \( 2978 - 3266 T - 2142 T^{2} + 391 T^{3} + 94 T^{4} - 21 T^{5} + T^{6} \)
$71$ \( -101256 + 25820 T + 15530 T^{2} - 911 T^{3} - 272 T^{4} + 3 T^{5} + T^{6} \)
$73$ \( 296 + 196 T - 254 T^{2} - 5 T^{3} + 49 T^{4} - 13 T^{5} + T^{6} \)
$79$ \( -1572016 + 36648 T + 48776 T^{2} - 1034 T^{3} - 408 T^{4} + 4 T^{5} + T^{6} \)
$83$ \( -12114 + 11794 T + 7360 T^{2} + 397 T^{3} - 177 T^{4} - 8 T^{5} + T^{6} \)
$89$ \( 648364 + 288808 T + 17206 T^{2} - 3699 T^{3} - 328 T^{4} + 9 T^{5} + T^{6} \)
$97$ \( -7628 - 15440 T + 9290 T^{2} + 3091 T^{3} - 267 T^{4} - 15 T^{5} + T^{6} \)
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