Properties

Label 1764.2.b.m
Level 1764
Weight 2
Character orbit 1764.b
Analytic conductor 14.086
Analytic rank 0
Dimension 12
CM no
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.15911316233388032.1
Defining polynomial: \(x^{12} - 4 x^{11} + 10 x^{10} - 20 x^{9} + 35 x^{8} - 56 x^{7} + 84 x^{6} - 112 x^{5} + 140 x^{4} - 160 x^{3} + 160 x^{2} - 128 x + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 588)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{11} + 10 x^{10} - 20 x^{9} + 35 x^{8} - 56 x^{7} + 84 x^{6} - 112 x^{5} + 140 x^{4} - 160 x^{3} + 160 x^{2} - 128 x + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{11} + 2 \nu^{9} + 3 \nu^{7} + 4 \nu^{6} - 4 \nu^{5} + 4 \nu^{4} - 4 \nu^{3} + 40 \nu^{2} + 32 \)\()/32\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{11} - 3 \nu^{10} + 8 \nu^{9} - 16 \nu^{8} + 23 \nu^{7} - 37 \nu^{6} + 58 \nu^{5} - 70 \nu^{4} + 88 \nu^{3} - 80 \nu^{2} + 80 \nu - 64 \)\()/32\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{11} - 5 \nu^{10} + 10 \nu^{9} - 20 \nu^{8} + 31 \nu^{7} - 51 \nu^{6} + 72 \nu^{5} - 102 \nu^{4} + 116 \nu^{3} - 136 \nu^{2} + 128 \nu - 96 \)\()/32\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{11} - 4 \nu^{10} + 10 \nu^{9} - 20 \nu^{8} + 35 \nu^{7} - 56 \nu^{6} + 84 \nu^{5} - 112 \nu^{4} + 140 \nu^{3} - 160 \nu^{2} + 160 \nu - 128 \)\()/32\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{11} - 5 \nu^{10} + 10 \nu^{9} - 20 \nu^{8} + 34 \nu^{7} - 55 \nu^{6} + 78 \nu^{5} - 90 \nu^{4} + 132 \nu^{3} - 112 \nu^{2} + 112 \nu - 64 \)\()/32\)
\(\beta_{6}\)\(=\)\((\)\( -2 \nu^{11} + 5 \nu^{10} - 12 \nu^{9} + 24 \nu^{8} - 34 \nu^{7} + 63 \nu^{6} - 84 \nu^{5} + 110 \nu^{4} - 128 \nu^{3} + 136 \nu^{2} - 128 \nu + 96 \)\()/32\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{11} - 3 \nu^{10} + 7 \nu^{9} - 12 \nu^{8} + 19 \nu^{7} - 33 \nu^{6} + 47 \nu^{5} - 54 \nu^{4} + 70 \nu^{3} - 72 \nu^{2} + 80 \nu - 48 \)\()/16\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{11} + 3 \nu^{10} - 8 \nu^{9} + 14 \nu^{8} - 23 \nu^{7} + 37 \nu^{6} - 50 \nu^{5} + 64 \nu^{4} - 80 \nu^{3} + 84 \nu^{2} - 80 \nu + 48 \)\()/16\)
\(\beta_{9}\)\(=\)\((\)\( 3 \nu^{11} - 8 \nu^{10} + 18 \nu^{9} - 28 \nu^{8} + 49 \nu^{7} - 76 \nu^{6} + 104 \nu^{5} - 136 \nu^{4} + 164 \nu^{3} - 160 \nu^{2} + 144 \nu - 64 \)\()/32\)
\(\beta_{10}\)\(=\)\((\)\( -4 \nu^{11} + 9 \nu^{10} - 20 \nu^{9} + 36 \nu^{8} - 56 \nu^{7} + 83 \nu^{6} - 120 \nu^{5} + 146 \nu^{4} - 192 \nu^{3} + 192 \nu^{2} - 160 \nu + 96 \)\()/32\)
\(\beta_{11}\)\(=\)\((\)\( 3 \nu^{11} - 11 \nu^{10} + 24 \nu^{9} - 44 \nu^{8} + 77 \nu^{7} - 117 \nu^{6} + 166 \nu^{5} - 210 \nu^{4} + 248 \nu^{3} - 280 \nu^{2} + 240 \nu - 128 \)\()/32\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{9} + \beta_{7} + \beta_{3} - \beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{10} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} - 1\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{11} + \beta_{9} - 2 \beta_{8} - \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \beta_{3} + \beta_{2} - \beta_{1} + 1\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{10} - 2 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_{1} - 1\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(2 \beta_{11} + \beta_{9} + 6 \beta_{8} + 3 \beta_{7} - 2 \beta_{5} + 2 \beta_{4} - 5 \beta_{3} + 5 \beta_{2} - \beta_{1} + 1\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-3 \beta_{10} + 4 \beta_{8} + 4 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} + \beta_{3} + 5 \beta_{2} + \beta_{1} + 3\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(6 \beta_{11} - 5 \beta_{9} - 2 \beta_{8} - 3 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - \beta_{3} - 5 \beta_{2} + 5 \beta_{1} - 9\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(5 \beta_{10} + 14 \beta_{9} + 6 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - \beta_{5} + 9 \beta_{4} - 11 \beta_{3} - 3 \beta_{2} - 3 \beta_{1} - 7\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-10 \beta_{11} + 15 \beta_{9} - 14 \beta_{8} + 5 \beta_{7} + 12 \beta_{6} - 14 \beta_{5} + 2 \beta_{4} - 7 \beta_{3} + 11 \beta_{2} - 7 \beta_{1} - 1\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(-\beta_{10} - 4 \beta_{9} - 16 \beta_{8} - 12 \beta_{7} - 16 \beta_{6} - 13 \beta_{5} + 21 \beta_{4} - 37 \beta_{3} + 7 \beta_{2} + 3 \beta_{1} + 13\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(2 \beta_{11} - 24 \beta_{10} + \beta_{9} + 42 \beta_{8} - \beta_{7} - 46 \beta_{6} - 10 \beta_{5} + 36 \beta_{4} - 39 \beta_{3} - 39 \beta_{2} + 15 \beta_{1} + 5\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
1.34902 0.424442i
1.34902 + 0.424442i
1.22594 0.705031i
1.22594 + 0.705031i
0.639847 1.26119i
0.639847 + 1.26119i
0.250649 1.39182i
0.250649 + 1.39182i
−0.476589 1.33149i
−0.476589 + 1.33149i
−0.988865 1.01101i
−0.988865 + 1.01101i
−1.34902 0.424442i 0 1.63970 + 1.14516i 0.127929i 0 0 −1.72593 2.24080i 0 −0.0542984 + 0.172579i
1567.2 −1.34902 + 0.424442i 0 1.63970 1.14516i 0.127929i 0 0 −1.72593 + 2.24080i 0 −0.0542984 0.172579i
1567.3 −1.22594 0.705031i 0 1.00586 + 1.72865i 3.64758i 0 0 −0.0143727 2.82839i 0 −2.57166 + 4.47172i
1567.4 −1.22594 + 0.705031i 0 1.00586 1.72865i 3.64758i 0 0 −0.0143727 + 2.82839i 0 −2.57166 4.47172i
1567.5 −0.639847 1.26119i 0 −1.18119 + 1.61393i 3.10455i 0 0 2.79126 + 0.457034i 0 −3.91542 + 1.98644i
1567.6 −0.639847 + 1.26119i 0 −1.18119 1.61393i 3.10455i 0 0 2.79126 0.457034i 0 −3.91542 1.98644i
1567.7 −0.250649 1.39182i 0 −1.87435 + 0.697718i 3.39209i 0 0 1.44090 + 2.43388i 0 4.72119 0.850222i
1567.8 −0.250649 + 1.39182i 0 −1.87435 0.697718i 3.39209i 0 0 1.44090 2.43388i 0 4.72119 + 0.850222i
1567.9 0.476589 1.33149i 0 −1.54572 1.26915i 0.509876i 0 0 −2.42653 + 1.45325i 0 0.678894 + 0.243001i
1567.10 0.476589 + 1.33149i 0 −1.54572 + 1.26915i 0.509876i 0 0 −2.42653 1.45325i 0 0.678894 0.243001i
1567.11 0.988865 1.01101i 0 −0.0442929 1.99951i 1.12886i 0 0 −2.06533 1.93246i 0 1.14129 + 1.11629i
1567.12 0.988865 + 1.01101i 0 −0.0442929 + 1.99951i 1.12886i 0 0 −2.06533 + 1.93246i 0 1.14129 1.11629i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1567.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.b.m 12
3.b odd 2 1 588.2.b.d yes 12
4.b odd 2 1 1764.2.b.l 12
7.b odd 2 1 1764.2.b.l 12
12.b even 2 1 588.2.b.c 12
21.c even 2 1 588.2.b.c 12
21.g even 6 2 588.2.o.f 24
21.h odd 6 2 588.2.o.e 24
28.d even 2 1 inner 1764.2.b.m 12
84.h odd 2 1 588.2.b.d yes 12
84.j odd 6 2 588.2.o.e 24
84.n even 6 2 588.2.o.f 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.b.c 12 12.b even 2 1
588.2.b.c 12 21.c even 2 1
588.2.b.d yes 12 3.b odd 2 1
588.2.b.d yes 12 84.h odd 2 1
588.2.o.e 24 21.h odd 6 2
588.2.o.e 24 84.j odd 6 2
588.2.o.f 24 21.g even 6 2
588.2.o.f 24 84.n even 6 2
1764.2.b.l 12 4.b odd 2 1
1764.2.b.l 12 7.b odd 2 1
1764.2.b.m 12 1.a even 1 1 trivial
1764.2.b.m 12 28.d even 2 1 inner

Hecke kernels

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

\( T_{5}^{12} + 36 T_{5}^{10} + 446 T_{5}^{8} + 2096 T_{5}^{6} + 2428 T_{5}^{4} + 528 T_{5}^{2} + 8 \)
\( T_{11}^{12} + 104 T_{11}^{10} + 4040 T_{11}^{8} + 72672 T_{11}^{6} + 603920 T_{11}^{4} + 1962496 T_{11}^{2} + 1968128 \)
\( T_{19}^{6} - 72 T_{19}^{4} - 32 T_{19}^{3} + 1288 T_{19}^{2} + 1280 T_{19} - 256 \)
\( T_{29}^{6} + 16 T_{29}^{5} + 26 T_{29}^{4} - 592 T_{29}^{3} - 2268 T_{29}^{2} + 1440 T_{29} + 9544 \)
\( T_{53}^{6} - 16 T_{53}^{5} + 24 T_{53}^{4} + 288 T_{53}^{3} + 144 T_{53}^{2} - 384 T_{53} - 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T + 10 T^{2} + 20 T^{3} + 35 T^{4} + 56 T^{5} + 84 T^{6} + 112 T^{7} + 140 T^{8} + 160 T^{9} + 160 T^{10} + 128 T^{11} + 64 T^{12} \)
$3$ 1
$5$ \( 1 - 24 T^{2} + 296 T^{4} - 2744 T^{6} + 21123 T^{8} - 134032 T^{10} + 718928 T^{12} - 3350800 T^{14} + 13201875 T^{16} - 42875000 T^{18} + 115625000 T^{20} - 234375000 T^{22} + 244140625 T^{24} \)
$7$ 1
$11$ \( 1 - 28 T^{2} + 586 T^{4} - 9388 T^{6} + 131503 T^{8} - 1628696 T^{10} + 19283052 T^{12} - 197072216 T^{14} + 1925335423 T^{16} - 16631414668 T^{18} + 125614304266 T^{20} - 726247888828 T^{22} + 3138428376721 T^{24} \)
$13$ \( 1 - 64 T^{2} + 2288 T^{4} - 59968 T^{6} + 1236915 T^{8} - 20973696 T^{10} + 297955296 T^{12} - 3544554624 T^{14} + 35327529315 T^{16} - 289454082112 T^{18} + 1866391889648 T^{20} - 8822943478336 T^{22} + 23298085122481 T^{24} \)
$17$ \( 1 - 88 T^{2} + 4536 T^{4} - 165688 T^{6} + 4700147 T^{8} - 107164048 T^{10} + 2007265136 T^{12} - 30970409872 T^{14} + 392560977587 T^{16} - 3999305532472 T^{18} + 31642035752376 T^{20} - 177407463239512 T^{22} + 582622237229761 T^{24} \)
$19$ \( ( 1 + 42 T^{2} - 32 T^{3} + 1231 T^{4} - 544 T^{5} + 29916 T^{6} - 10336 T^{7} + 444391 T^{8} - 219488 T^{9} + 5473482 T^{10} + 47045881 T^{12} )^{2} \)
$23$ \( 1 - 60 T^{2} + 3354 T^{4} - 123436 T^{6} + 4153279 T^{8} - 110540568 T^{10} + 2811666956 T^{12} - 58475960472 T^{14} + 1162257748639 T^{16} - 18272957994604 T^{18} + 262655044632474 T^{20} - 2485590672818940 T^{22} + 21914624432020321 T^{24} \)
$29$ \( ( 1 + 16 T + 200 T^{2} + 1728 T^{3} + 13363 T^{4} + 84496 T^{5} + 496976 T^{6} + 2450384 T^{7} + 11238283 T^{8} + 42144192 T^{9} + 141456200 T^{10} + 328178384 T^{11} + 594823321 T^{12} )^{2} \)
$31$ \( ( 1 - 8 T + 130 T^{2} - 824 T^{3} + 8247 T^{4} - 42224 T^{5} + 314764 T^{6} - 1308944 T^{7} + 7925367 T^{8} - 24547784 T^{9} + 120057730 T^{10} - 229033208 T^{11} + 887503681 T^{12} )^{2} \)
$37$ \( ( 1 - 16 T + 200 T^{2} - 1680 T^{3} + 13131 T^{4} - 82528 T^{5} + 535824 T^{6} - 3053536 T^{7} + 17976339 T^{8} - 85097040 T^{9} + 374832200 T^{10} - 1109503312 T^{11} + 2565726409 T^{12} )^{2} \)
$41$ \( 1 - 264 T^{2} + 32344 T^{4} - 2436136 T^{6} + 127693075 T^{8} - 5269183920 T^{10} + 207449944752 T^{12} - 8857498169520 T^{14} + 360830111305075 T^{16} - 11571899945252776 T^{18} + 258264421610689624 T^{20} - 3543582057880233864 T^{22} + 22563490300366186081 T^{24} \)
$43$ \( 1 - 276 T^{2} + 36338 T^{4} - 3157924 T^{6} + 212530239 T^{8} - 11922137512 T^{10} + 561568704636 T^{12} - 22044032259688 T^{14} + 726598593623439 T^{16} - 19962384085150276 T^{18} + 424725821687465138 T^{20} - 5964769118466452724 T^{22} + 39959630797262576401 T^{24} \)
$47$ \( ( 1 + 90 T^{2} + 160 T^{3} + 6183 T^{4} + 13280 T^{5} + 328892 T^{6} + 624160 T^{7} + 13658247 T^{8} + 16611680 T^{9} + 439171290 T^{10} + 10779215329 T^{12} )^{2} \)
$53$ \( ( 1 - 16 T + 342 T^{2} - 3952 T^{3} + 47367 T^{4} - 404032 T^{5} + 3397044 T^{6} - 21413696 T^{7} + 133053903 T^{8} - 588361904 T^{9} + 2698544502 T^{10} - 6691127888 T^{11} + 22164361129 T^{12} )^{2} \)
$59$ \( ( 1 - 8 T + 210 T^{2} - 1624 T^{3} + 18415 T^{4} - 148976 T^{5} + 1121964 T^{6} - 8789584 T^{7} + 64102615 T^{8} - 333535496 T^{9} + 2544645810 T^{10} - 5719394392 T^{11} + 42180533641 T^{12} )^{2} \)
$61$ \( 1 - 544 T^{2} + 142480 T^{4} - 23858720 T^{6} + 2849681139 T^{8} - 255863393088 T^{10} + 17709328352032 T^{12} - 952067685680448 T^{14} + 39456231951292899 T^{16} - 1229210186174277920 T^{18} + 27314457955852596880 T^{20} - \)\(38\!\cdots\!44\)\( T^{22} + \)\(26\!\cdots\!21\)\( T^{24} \)
$67$ \( 1 - 356 T^{2} + 55762 T^{4} - 4560756 T^{6} + 122644383 T^{8} + 13948776888 T^{10} - 1693776977860 T^{12} + 62616059450232 T^{14} + 2471421801803343 T^{16} - 412558609227559764 T^{18} + 22643145835913415442 T^{20} - \)\(64\!\cdots\!44\)\( T^{22} + \)\(81\!\cdots\!61\)\( T^{24} \)
$71$ \( 1 - 556 T^{2} + 151546 T^{4} - 27058972 T^{6} + 3535695487 T^{8} - 356059001528 T^{10} + 28330933942092 T^{12} - 1794893426702648 T^{14} + 89847965828783647 T^{16} - 3466261995810389212 T^{18} + 97861364646170096506 T^{20} - \)\(18\!\cdots\!56\)\( T^{22} + \)\(16\!\cdots\!41\)\( T^{24} \)
$73$ \( 1 - 480 T^{2} + 105792 T^{4} - 14378720 T^{6} + 1390885347 T^{8} - 108814494400 T^{10} + 7928879090304 T^{12} - 579872440657600 T^{14} + 39498697287474627 T^{16} - 2175992466226170080 T^{18} + 85317026041658617152 T^{20} - \)\(20\!\cdots\!20\)\( T^{22} + \)\(22\!\cdots\!21\)\( T^{24} \)
$79$ \( 1 - 516 T^{2} + 134210 T^{4} - 23318900 T^{6} + 3036334383 T^{8} - 315920772616 T^{10} + 27245615350940 T^{12} - 1971661541896456 T^{14} + 118265470160935023 T^{16} - 5668532066548646900 T^{18} + \)\(20\!\cdots\!10\)\( T^{20} - \)\(48\!\cdots\!16\)\( T^{22} + \)\(59\!\cdots\!41\)\( T^{24} \)
$83$ \( ( 1 - 8 T + 290 T^{2} - 1848 T^{3} + 43207 T^{4} - 256016 T^{5} + 4402748 T^{6} - 21249328 T^{7} + 297653023 T^{8} - 1056662376 T^{9} + 13762913090 T^{10} - 31512325144 T^{11} + 326940373369 T^{12} )^{2} \)
$89$ \( 1 - 696 T^{2} + 241496 T^{4} - 54982424 T^{6} + 9125837523 T^{8} - 1161685318736 T^{10} + 116205471399600 T^{12} - 9201709409707856 T^{14} + 572575497194909043 T^{16} - 27325236059685069464 T^{18} + \)\(95\!\cdots\!76\)\( T^{20} - \)\(21\!\cdots\!96\)\( T^{22} + \)\(24\!\cdots\!21\)\( T^{24} \)
$97$ \( 1 - 672 T^{2} + 218848 T^{4} - 46502304 T^{6} + 7344144003 T^{8} - 928722590784 T^{10} + 97980461927872 T^{12} - 8738350856686656 T^{14} + 650171788146051843 T^{16} - 38735117396697856416 T^{18} + \)\(17\!\cdots\!28\)\( T^{20} - \)\(49\!\cdots\!28\)\( T^{22} + \)\(69\!\cdots\!41\)\( T^{24} \)
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