Properties

Label 1764.2.e.j
Level $1764$
Weight $2$
Character orbit 1764.e
Analytic conductor $14.086$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 8 x^{14} + 49 x^{12} - 104 x^{10} + 160 x^{8} - 104 x^{6} + 49 x^{4} - 8 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{3} q^{2} + ( 1 - \beta_{8} ) q^{4} + \beta_{2} q^{5} + ( -\beta_{3} + \beta_{12} + \beta_{14} ) q^{8} +O(q^{10})\) \( q -\beta_{3} q^{2} + ( 1 - \beta_{8} ) q^{4} + \beta_{2} q^{5} + ( -\beta_{3} + \beta_{12} + \beta_{14} ) q^{8} + ( -\beta_{5} + \beta_{10} + \beta_{11} ) q^{10} + ( \beta_{3} + 2 \beta_{12} + \beta_{14} ) q^{11} -\beta_{10} q^{13} + ( \beta_{4} + \beta_{6} - 2 \beta_{8} ) q^{16} + ( -2 \beta_{2} - \beta_{9} ) q^{17} + ( -\beta_{5} - \beta_{10} + 2 \beta_{15} ) q^{19} + ( -\beta_{1} + \beta_{2} - \beta_{7} ) q^{20} + ( -1 + 2 \beta_{4} + \beta_{6} ) q^{22} + ( \beta_{3} + 2 \beta_{12} + \beta_{14} ) q^{23} + ( 1 + \beta_{4} - \beta_{8} ) q^{25} + \beta_{7} q^{26} + ( -\beta_{3} + 2 \beta_{13} + \beta_{14} ) q^{29} + ( -\beta_{5} + \beta_{10} + 4 \beta_{11} - 2 \beta_{15} ) q^{31} + ( 2 \beta_{12} + 3 \beta_{13} + \beta_{14} ) q^{32} + ( 2 \beta_{5} - 3 \beta_{10} - 3 \beta_{11} + \beta_{15} ) q^{34} + ( \beta_{4} - \beta_{8} ) q^{37} + ( -\beta_{1} + 3 \beta_{2} - 4 \beta_{9} ) q^{38} + ( 3 \beta_{10} - \beta_{11} - \beta_{15} ) q^{40} + ( -3 \beta_{2} + 4 \beta_{9} ) q^{41} + ( -2 - \beta_{4} - 4 \beta_{6} + \beta_{8} ) q^{43} + ( \beta_{3} + \beta_{12} + 5 \beta_{13} - 2 \beta_{14} ) q^{44} + ( -1 + 2 \beta_{4} + \beta_{6} ) q^{46} + ( 2 \beta_{1} + \beta_{2} - 4 \beta_{7} + \beta_{9} ) q^{47} + ( -\beta_{3} + 2 \beta_{12} + 2 \beta_{13} ) q^{50} + ( -\beta_{5} - \beta_{10} + \beta_{11} ) q^{52} + ( -2 \beta_{3} + 3 \beta_{13} + 2 \beta_{14} ) q^{53} + ( 3 \beta_{5} + \beta_{10} - 4 \beta_{11} - 2 \beta_{15} ) q^{55} + ( 1 + 3 \beta_{6} - 2 \beta_{8} ) q^{58} + ( 2 \beta_{1} - 3 \beta_{2} + 4 \beta_{7} + \beta_{9} ) q^{59} + ( 5 \beta_{5} - 2 \beta_{10} ) q^{61} + ( -\beta_{1} + 3 \beta_{2} - 2 \beta_{7} + 4 \beta_{9} ) q^{62} + ( 3 + 2 \beta_{4} + 4 \beta_{6} - \beta_{8} ) q^{64} + ( -\beta_{3} - \beta_{13} + \beta_{14} ) q^{65} + ( \beta_{4} + 3 \beta_{8} ) q^{67} + ( 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{7} - 2 \beta_{9} ) q^{68} + ( 7 \beta_{3} + 2 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} ) q^{71} + ( -3 \beta_{5} + 4 \beta_{10} ) q^{73} + ( 2 \beta_{12} + 2 \beta_{13} ) q^{74} + ( -3 \beta_{5} - 2 \beta_{11} + 3 \beta_{15} ) q^{76} + ( -2 - 2 \beta_{4} - 4 \beta_{6} - 2 \beta_{8} ) q^{79} + ( -4 \beta_{2} - \beta_{7} + 2 \beta_{9} ) q^{80} + ( 3 \beta_{5} + \beta_{10} + \beta_{11} - 4 \beta_{15} ) q^{82} + ( -4 \beta_{1} + 4 \beta_{7} - 2 \beta_{9} ) q^{83} + ( 10 - 2 \beta_{4} + 2 \beta_{8} ) q^{85} + ( 2 \beta_{3} + 2 \beta_{12} - 6 \beta_{13} ) q^{86} + ( 9 + \beta_{4} + 3 \beta_{6} + 3 \beta_{8} ) q^{88} + ( 3 \beta_{2} - 4 \beta_{9} ) q^{89} + ( \beta_{3} + \beta_{12} + 5 \beta_{13} - 2 \beta_{14} ) q^{92} + ( 3 \beta_{5} + 4 \beta_{10} + \beta_{15} ) q^{94} + ( 2 \beta_{3} + 4 \beta_{12} + 2 \beta_{14} ) q^{95} + ( -5 \beta_{5} - 2 \beta_{10} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4} + O(q^{10}) \) \( 16 q + 16 q^{4} - 8 q^{16} - 24 q^{22} + 16 q^{25} - 24 q^{46} - 8 q^{58} + 16 q^{64} + 160 q^{85} + 120 q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 8 x^{14} + 49 x^{12} - 104 x^{10} + 160 x^{8} - 104 x^{6} + 49 x^{4} - 8 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -52 \nu^{14} + 655 \nu^{12} - 4512 \nu^{10} + 17312 \nu^{8} - 33992 \nu^{6} + 38784 \nu^{4} - 17044 \nu^{2} - 305 \)\()/1584\)
\(\beta_{2}\)\(=\)\((\)\( 116 \nu^{14} - 791 \nu^{12} + 4704 \nu^{10} - 6208 \nu^{8} + 9544 \nu^{6} + 480 \nu^{4} + 7316 \nu^{2} - 599 \)\()/1584\)
\(\beta_{3}\)\(=\)\((\)\( -13 \nu^{15} + 227 \nu^{13} - 1568 \nu^{11} + 6968 \nu^{9} - 12392 \nu^{7} + 16208 \nu^{5} - 6637 \nu^{3} + 2451 \nu \)\()/528\)
\(\beta_{4}\)\(=\)\((\)\( 5 \nu^{14} - 32 \nu^{12} + 184 \nu^{10} - 152 \nu^{8} + 112 \nu^{6} + 464 \nu^{4} - 195 \nu^{2} + 88 \)\()/24\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{15} + 13 \nu^{13} - 88 \nu^{11} + 340 \nu^{9} - 624 \nu^{7} + 760 \nu^{5} - 357 \nu^{3} + 101 \nu \)\()/12\)
\(\beta_{6}\)\(=\)\((\)\( 7 \nu^{14} - 56 \nu^{12} + 344 \nu^{10} - 736 \nu^{8} + 1168 \nu^{6} - 824 \nu^{4} + 447 \nu^{2} - 80 \)\()/24\)
\(\beta_{7}\)\(=\)\((\)\( 284 \nu^{14} - 2105 \nu^{12} + 12600 \nu^{10} - 21544 \nu^{8} + 29320 \nu^{6} - 6408 \nu^{4} + 2900 \nu^{2} + 1351 \)\()/792\)
\(\beta_{8}\)\(=\)\((\)\( -5 \nu^{14} + 40 \nu^{12} - 244 \nu^{10} + 512 \nu^{8} - 752 \nu^{6} + 424 \nu^{4} - 129 \nu^{2} + 4 \)\()/12\)
\(\beta_{9}\)\(=\)\((\)\( 1060 \nu^{14} - 8407 \nu^{12} + 51360 \nu^{10} - 106688 \nu^{8} + 162152 \nu^{6} - 98592 \nu^{4} + 44932 \nu^{2} - 4759 \)\()/1584\)
\(\beta_{10}\)\(=\)\((\)\( -41 \nu^{15} + 329 \nu^{13} - 2016 \nu^{11} + 4312 \nu^{9} - 6664 \nu^{7} + 4608 \nu^{5} - 2345 \nu^{3} + 665 \nu \)\()/144\)
\(\beta_{11}\)\(=\)\((\)\( -47 \nu^{15} + 383 \nu^{13} - 2352 \nu^{11} + 5176 \nu^{9} - 7912 \nu^{7} + 5328 \nu^{5} - 2015 \nu^{3} + 287 \nu \)\()/48\)
\(\beta_{12}\)\(=\)\( \nu^{15} - 8 \nu^{13} + 49 \nu^{11} - 104 \nu^{9} + 160 \nu^{7} - 104 \nu^{5} + 49 \nu^{3} - 9 \nu \)
\(\beta_{13}\)\(=\)\((\)\( -183 \nu^{15} + 1449 \nu^{13} - 8832 \nu^{11} + 18184 \nu^{9} - 27048 \nu^{7} + 15456 \nu^{5} - 5943 \nu^{3} + 345 \nu \)\()/176\)
\(\beta_{14}\)\(=\)\((\)\( -689 \nu^{15} + 5519 \nu^{13} - 33824 \nu^{11} + 72040 \nu^{9} - 111176 \nu^{7} + 72656 \nu^{5} - 33377 \nu^{3} + 3711 \nu \)\()/528\)
\(\beta_{15}\)\(=\)\((\)\( 97 \nu^{15} - 745 \nu^{13} + 4512 \nu^{11} - 8624 \nu^{9} + 12632 \nu^{7} - 5808 \nu^{5} + 2569 \nu^{3} + 263 \nu \)\()/72\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{15} - 2 \beta_{12} - 2 \beta_{10} - \beta_{5}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(4 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} - 4 \beta_{6} + \beta_{2} + \beta_{1} + 2\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(4 \beta_{15} + 3 \beta_{14} + 4 \beta_{13} - 2 \beta_{11} - 2 \beta_{10} - \beta_{5} - 3 \beta_{3}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(18 \beta_{9} + 20 \beta_{8} + 12 \beta_{7} - 22 \beta_{6} - 6 \beta_{4} - 7 \beta_{2} - 5 \beta_{1} - 28\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(21 \beta_{15} + 12 \beta_{14} + 42 \beta_{13} + 28 \beta_{12} - 14 \beta_{11} + 36 \beta_{10} + 21 \beta_{5} - 54 \beta_{3}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-13 \beta_{9} + 24 \beta_{8} + 68 \beta_{7} - 24 \beta_{4} - 21 \beta_{2} - 26 \beta_{1} - 88\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-115 \beta_{15} - 108 \beta_{14} - 42 \beta_{13} + 148 \beta_{12} + 82 \beta_{11} + 308 \beta_{10} + 157 \beta_{5} - 138 \beta_{3}\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(-718 \beta_{9} - 388 \beta_{8} + 380 \beta_{7} + 662 \beta_{6} - 90 \beta_{4} + 49 \beta_{2} - 141 \beta_{1} - 148\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-636 \beta_{15} - 459 \beta_{14} - 788 \beta_{13} + 462 \beta_{11} + 318 \beta_{10} + 87 \beta_{5} + 459 \beta_{3}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(-3212 \beta_{9} - 3682 \beta_{8} - 2114 \beta_{7} + 3668 \beta_{6} + 1008 \beta_{4} + 1625 \beta_{2} + 777 \beta_{1} + 4478\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(-3527 \beta_{15} - 1728 \beta_{14} - 7440 \beta_{13} - 4478 \beta_{12} + 2576 \beta_{11} - 5858 \beta_{10} - 3913 \beta_{5} + 9456 \beta_{3}\)\()/4\)
\(\nu^{12}\)\(=\)\(1076 \beta_{9} - 2112 \beta_{8} - 5872 \beta_{7} + 2112 \beta_{4} + 1860 \beta_{2} + 2152 \beta_{1} + 7327\)
\(\nu^{13}\)\(=\)\((\)\(19575 \beta_{15} + 18624 \beta_{14} + 7440 \beta_{13} - 24830 \beta_{12} - 14320 \beta_{11} - 52062 \beta_{10} - 27015 \beta_{5} + 24336 \beta_{3}\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(122988 \beta_{9} + 66782 \beta_{8} - 65214 \beta_{7} - 112972 \beta_{6} + 15888 \beta_{4} - 9065 \beta_{2} + 23879 \beta_{1} + 24830\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(108668 \beta_{15} + 78213 \beta_{14} + 135436 \beta_{13} - 79534 \beta_{11} - 54334 \beta_{10} - 14567 \beta_{5} - 78213 \beta_{3}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1079.1
2.04058 1.17813i
0.367543 + 0.212201i
0.367543 0.212201i
2.04058 + 1.17813i
−0.670418 + 0.387066i
−1.11871 0.645885i
−1.11871 + 0.645885i
−0.670418 0.387066i
1.11871 0.645885i
0.670418 + 0.387066i
0.670418 0.387066i
1.11871 + 0.645885i
−0.367543 + 0.212201i
−2.04058 1.17813i
−2.04058 + 1.17813i
−0.367543 0.212201i
−1.39033 0.258819i 0 1.86603 + 0.719687i 0.732051i 0 0 −2.40812 1.48356i 0 −0.189469 + 1.01779i
1079.2 −1.39033 0.258819i 0 1.86603 + 0.719687i 0.732051i 0 0 −2.40812 1.48356i 0 0.189469 1.01779i
1079.3 −1.39033 + 0.258819i 0 1.86603 0.719687i 0.732051i 0 0 −2.40812 + 1.48356i 0 0.189469 + 1.01779i
1079.4 −1.39033 + 0.258819i 0 1.86603 0.719687i 0.732051i 0 0 −2.40812 + 1.48356i 0 −0.189469 1.01779i
1079.5 −1.03295 0.965926i 0 0.133975 + 1.99551i 2.73205i 0 0 1.78912 2.19067i 0 −2.63896 + 2.82207i
1079.6 −1.03295 0.965926i 0 0.133975 + 1.99551i 2.73205i 0 0 1.78912 2.19067i 0 2.63896 2.82207i
1079.7 −1.03295 + 0.965926i 0 0.133975 1.99551i 2.73205i 0 0 1.78912 + 2.19067i 0 2.63896 + 2.82207i
1079.8 −1.03295 + 0.965926i 0 0.133975 1.99551i 2.73205i 0 0 1.78912 + 2.19067i 0 −2.63896 2.82207i
1079.9 1.03295 0.965926i 0 0.133975 1.99551i 2.73205i 0 0 −1.78912 2.19067i 0 −2.63896 2.82207i
1079.10 1.03295 0.965926i 0 0.133975 1.99551i 2.73205i 0 0 −1.78912 2.19067i 0 2.63896 + 2.82207i
1079.11 1.03295 + 0.965926i 0 0.133975 + 1.99551i 2.73205i 0 0 −1.78912 + 2.19067i 0 2.63896 2.82207i
1079.12 1.03295 + 0.965926i 0 0.133975 + 1.99551i 2.73205i 0 0 −1.78912 + 2.19067i 0 −2.63896 + 2.82207i
1079.13 1.39033 0.258819i 0 1.86603 0.719687i 0.732051i 0 0 2.40812 1.48356i 0 −0.189469 1.01779i
1079.14 1.39033 0.258819i 0 1.86603 0.719687i 0.732051i 0 0 2.40812 1.48356i 0 0.189469 + 1.01779i
1079.15 1.39033 + 0.258819i 0 1.86603 + 0.719687i 0.732051i 0 0 2.40812 + 1.48356i 0 0.189469 1.01779i
1079.16 1.39033 + 0.258819i 0 1.86603 + 0.719687i 0.732051i 0 0 2.40812 + 1.48356i 0 −0.189469 + 1.01779i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1079.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.e.j 16
3.b odd 2 1 inner 1764.2.e.j 16
4.b odd 2 1 inner 1764.2.e.j 16
7.b odd 2 1 inner 1764.2.e.j 16
12.b even 2 1 inner 1764.2.e.j 16
21.c even 2 1 inner 1764.2.e.j 16
28.d even 2 1 inner 1764.2.e.j 16
84.h odd 2 1 inner 1764.2.e.j 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.2.e.j 16 1.a even 1 1 trivial
1764.2.e.j 16 3.b odd 2 1 inner
1764.2.e.j 16 4.b odd 2 1 inner
1764.2.e.j 16 7.b odd 2 1 inner
1764.2.e.j 16 12.b even 2 1 inner
1764.2.e.j 16 21.c even 2 1 inner
1764.2.e.j 16 28.d even 2 1 inner
1764.2.e.j 16 84.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{4} + 8 T_{5}^{2} + 4 \)
\( T_{13}^{2} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 16 - 16 T^{2} + 9 T^{4} - 4 T^{6} + T^{8} )^{2} \)
$3$ \( T^{16} \)
$5$ \( ( 4 + 8 T^{2} + T^{4} )^{4} \)
$7$ \( T^{16} \)
$11$ \( ( 132 - 36 T^{2} + T^{4} )^{4} \)
$13$ \( ( -2 + T^{2} )^{8} \)
$17$ \( ( 676 + 56 T^{2} + T^{4} )^{4} \)
$19$ \( ( 528 + 48 T^{2} + T^{4} )^{4} \)
$23$ \( ( 132 - 36 T^{2} + T^{4} )^{4} \)
$29$ \( ( 4 + 28 T^{2} + T^{4} )^{4} \)
$31$ \( ( 4752 + 144 T^{2} + T^{4} )^{4} \)
$37$ \( ( -12 + T^{2} )^{8} \)
$41$ \( ( 2116 + 104 T^{2} + T^{4} )^{4} \)
$43$ \( ( 2112 + 96 T^{2} + T^{4} )^{4} \)
$47$ \( ( 528 - 120 T^{2} + T^{4} )^{4} \)
$53$ \( ( 36 + 84 T^{2} + T^{4} )^{4} \)
$59$ \( ( 4752 - 216 T^{2} + T^{4} )^{4} \)
$61$ \( ( 20164 - 316 T^{2} + T^{4} )^{4} \)
$67$ \( ( 528 + 72 T^{2} + T^{4} )^{4} \)
$71$ \( ( 15972 - 276 T^{2} + T^{4} )^{4} \)
$73$ \( ( 484 - 172 T^{2} + T^{4} )^{4} \)
$79$ \( ( 4752 + 216 T^{2} + T^{4} )^{4} \)
$83$ \( ( 19008 - 288 T^{2} + T^{4} )^{4} \)
$89$ \( ( 2116 + 104 T^{2} + T^{4} )^{4} \)
$97$ \( ( 20164 - 316 T^{2} + T^{4} )^{4} \)
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