Properties

Label 324.3.f.r
Level $324$
Weight $3$
Character orbit 324.f
Analytic conductor $8.828$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 324.f (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.82836056527\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.119023932416481.2
Defining polynomial: \(x^{12} - 2 x^{11} - 3 x^{10} + 11 x^{9} - 5 x^{8} - 14 x^{7} + 29 x^{6} - 28 x^{5} - 20 x^{4} + 88 x^{3} - 48 x^{2} - 64 x + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \beta_{3} - \beta_{6} ) q^{2} + ( 1 + \beta_{2} + \beta_{4} ) q^{4} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{5} + ( -\beta_{4} - \beta_{9} ) q^{7} + ( \beta_{1} + \beta_{2} - \beta_{6} + \beta_{7} - \beta_{9} - \beta_{11} ) q^{8} +O(q^{10})\) \( q + ( \beta_{3} - \beta_{6} ) q^{2} + ( 1 + \beta_{2} + \beta_{4} ) q^{4} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{5} + ( -\beta_{4} - \beta_{9} ) q^{7} + ( \beta_{1} + \beta_{2} - \beta_{6} + \beta_{7} - \beta_{9} - \beta_{11} ) q^{8} + ( 2 + \beta_{1} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{10} + ( -1 - \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{11} + ( -1 + \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{10} ) q^{13} + ( -\beta_{1} - \beta_{3} + 2 \beta_{5} + 2 \beta_{7} - \beta_{10} + \beta_{11} ) q^{14} + ( 3 + 3 \beta_{1} - 3 \beta_{3} + \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - \beta_{9} ) q^{16} + ( 2 + 4 \beta_{6} - \beta_{7} + \beta_{8} ) q^{17} + ( 2 + \beta_{1} - \beta_{2} + 6 \beta_{6} + 3 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{19} + ( -12 - 12 \beta_{1} + \beta_{4} + 4 \beta_{5} ) q^{20} + ( -4 + 7 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - 6 \beta_{5} - 6 \beta_{7} - \beta_{10} + \beta_{11} ) q^{22} + ( -5 - 5 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} - 5 \beta_{4} + 4 \beta_{5} + 4 \beta_{7} + \beta_{11} ) q^{23} + ( 1 + \beta_{1} + 4 \beta_{4} + 3 \beta_{5} - \beta_{8} - \beta_{10} ) q^{25} + ( 10 - \beta_{1} - 4 \beta_{2} + 2 \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{26} + ( -8 + 4 \beta_{8} ) q^{28} + ( -3 - 3 \beta_{1} + 7 \beta_{3} + 5 \beta_{4} + 2 \beta_{5} - 7 \beta_{6} - 3 \beta_{8} - 3 \beta_{10} ) q^{29} + ( 4 + 4 \beta_{2} + 12 \beta_{3} + 4 \beta_{4} + 4 \beta_{10} ) q^{31} + ( -1 - 6 \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - 4 \beta_{10} - \beta_{11} ) q^{32} + ( -16 - 16 \beta_{1} + 3 \beta_{3} - 4 \beta_{4} - 3 \beta_{6} ) q^{34} + ( -2 + \beta_{1} - \beta_{2} - 10 \beta_{6} - \beta_{7} + 3 \beta_{8} - \beta_{9} - \beta_{11} ) q^{35} + ( 11 + 5 \beta_{2} - 21 \beta_{6} + 9 \beta_{7} - 4 \beta_{8} ) q^{37} + ( 24 + 24 \beta_{1} - 3 \beta_{3} - 4 \beta_{4} + 6 \beta_{5} + 3 \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{38} + ( 1 - 10 \beta_{1} + \beta_{2} - 9 \beta_{3} + \beta_{4} - 3 \beta_{5} - 3 \beta_{7} - 4 \beta_{10} - \beta_{11} ) q^{40} + ( 6 - 13 \beta_{1} + 6 \beta_{2} - 10 \beta_{3} + 6 \beta_{4} + 7 \beta_{5} + 7 \beta_{7} + \beta_{10} ) q^{41} + ( -3 - 3 \beta_{1} - 6 \beta_{3} - \beta_{4} + 3 \beta_{5} + 6 \beta_{6} - 3 \beta_{8} + 5 \beta_{9} - 3 \beta_{10} ) q^{43} + ( -28 - 4 \beta_{1} + 4 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} + 4 \beta_{9} + 4 \beta_{11} ) q^{44} + ( 16 - 5 \beta_{1} + 3 \beta_{6} - 6 \beta_{7} + 5 \beta_{8} + 5 \beta_{9} + 5 \beta_{11} ) q^{46} + ( 2 + 2 \beta_{1} - 8 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} + 8 \beta_{6} - 2 \beta_{8} - 6 \beta_{9} - 2 \beta_{10} ) q^{47} + ( -2 - 10 \beta_{1} - 2 \beta_{2} - 18 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 3 \beta_{7} + 5 \beta_{10} ) q^{49} + ( 4 + 12 \beta_{1} + 4 \beta_{2} + 4 \beta_{4} - 4 \beta_{10} - 4 \beta_{11} ) q^{50} + ( 16 + 16 \beta_{1} + 12 \beta_{3} + \beta_{4} - 12 \beta_{6} + 4 \beta_{9} ) q^{52} + ( -16 + 4 \beta_{2} + 4 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{53} + ( 1 - 5 \beta_{1} - 5 \beta_{2} + 24 \beta_{6} + 6 \beta_{7} - 6 \beta_{8} + 5 \beta_{9} + 5 \beta_{11} ) q^{55} + ( -52 - 52 \beta_{1} - 8 \beta_{3} + 4 \beta_{5} + 8 \beta_{6} + 4 \beta_{8} + 4 \beta_{10} ) q^{56} + ( 12 + 43 \beta_{1} + 12 \beta_{2} - 6 \beta_{3} + 12 \beta_{4} - 5 \beta_{10} - 5 \beta_{11} ) q^{58} + ( 6 + 2 \beta_{1} + 6 \beta_{2} + 16 \beta_{3} + 6 \beta_{4} + 4 \beta_{5} + 4 \beta_{7} + 4 \beta_{10} - 6 \beta_{11} ) q^{59} + ( -13 - 13 \beta_{1} + 15 \beta_{3} - 11 \beta_{4} - 12 \beta_{5} - 15 \beta_{6} - \beta_{8} - \beta_{10} ) q^{61} + ( 64 + 4 \beta_{1} + 16 \beta_{2} - 4 \beta_{6} + 8 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} - 4 \beta_{11} ) q^{62} + ( -46 - \beta_{1} + \beta_{2} - 3 \beta_{6} - 9 \beta_{7} + 4 \beta_{8} + \beta_{9} + \beta_{11} ) q^{64} + ( 11 + 11 \beta_{1} + 14 \beta_{3} - 6 \beta_{4} - 8 \beta_{5} - 14 \beta_{6} - 2 \beta_{8} - 2 \beta_{10} ) q^{65} + ( -7 - 10 \beta_{1} - 7 \beta_{2} + 18 \beta_{3} - 7 \beta_{4} + 9 \beta_{5} + 9 \beta_{7} + 3 \beta_{10} + \beta_{11} ) q^{67} + ( -1 - 8 \beta_{1} - \beta_{2} - 12 \beta_{3} - \beta_{4} - 4 \beta_{5} - 4 \beta_{7} + 4 \beta_{11} ) q^{68} + ( -40 - 40 \beta_{1} - 3 \beta_{3} + 12 \beta_{4} + 6 \beta_{5} + 3 \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{70} + ( -7 - 5 \beta_{1} - \beta_{2} - 12 \beta_{6} - 6 \beta_{7} + 2 \beta_{8} + 5 \beta_{9} + 5 \beta_{11} ) q^{71} + ( -22 - 10 \beta_{2} - 18 \beta_{6} - 3 \beta_{7} - 7 \beta_{8} ) q^{73} + ( 74 + 74 \beta_{1} + 2 \beta_{3} + 16 \beta_{4} - 2 \beta_{6} + 5 \beta_{8} - 5 \beta_{9} + 5 \beta_{10} ) q^{74} + ( -8 - 40 \beta_{1} - 8 \beta_{2} + 36 \beta_{3} - 8 \beta_{4} - 12 \beta_{5} - 12 \beta_{7} - 4 \beta_{10} + 4 \beta_{11} ) q^{76} + ( -14 + 25 \beta_{1} - 14 \beta_{2} - 6 \beta_{3} - 14 \beta_{4} - 9 \beta_{5} - 9 \beta_{7} + 5 \beta_{10} ) q^{77} + ( 6 + 6 \beta_{1} + 3 \beta_{4} - 6 \beta_{5} + 2 \beta_{8} - 5 \beta_{9} + 2 \beta_{10} ) q^{79} + ( -70 + \beta_{1} - 9 \beta_{2} - 13 \beta_{6} - 3 \beta_{7} - \beta_{9} - \beta_{11} ) q^{80} + ( 24 + 6 \beta_{1} - 4 \beta_{2} - 12 \beta_{6} + 6 \beta_{8} - 6 \beta_{9} - 6 \beta_{11} ) q^{82} + ( 8 + 8 \beta_{1} - 20 \beta_{3} + 14 \beta_{4} - 8 \beta_{5} + 20 \beta_{6} - 4 \beta_{8} + 10 \beta_{9} - 4 \beta_{10} ) q^{83} + ( 3 + 13 \beta_{1} + 3 \beta_{2} - 15 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} + 6 \beta_{7} + 3 \beta_{10} ) q^{85} + ( -12 + 23 \beta_{1} - 12 \beta_{2} + 11 \beta_{3} - 12 \beta_{4} - 22 \beta_{5} - 22 \beta_{7} - \beta_{10} + \beta_{11} ) q^{86} + ( 60 + 60 \beta_{1} - 24 \beta_{3} - 8 \beta_{4} - 12 \beta_{5} + 24 \beta_{6} + 4 \beta_{8} + 4 \beta_{10} ) q^{88} + ( 35 - 12 \beta_{2} + 28 \beta_{6} - 16 \beta_{7} + 4 \beta_{8} ) q^{89} + ( 12 + 5 \beta_{1} + 15 \beta_{2} + 18 \beta_{6} - 3 \beta_{7} - 7 \beta_{8} - 5 \beta_{9} - 5 \beta_{11} ) q^{91} + ( -88 - 88 \beta_{1} + 32 \beta_{3} - 8 \beta_{4} - 16 \beta_{5} - 32 \beta_{6} + 4 \beta_{8} + 4 \beta_{10} ) q^{92} + ( -8 + 26 \beta_{1} - 8 \beta_{2} - 6 \beta_{3} - 8 \beta_{4} + 12 \beta_{5} + 12 \beta_{7} - 6 \beta_{10} + 6 \beta_{11} ) q^{94} + ( 5 + \beta_{1} + 5 \beta_{2} + 5 \beta_{4} - 12 \beta_{5} - 12 \beta_{7} + 4 \beta_{10} + 11 \beta_{11} ) q^{95} + ( 39 + 39 \beta_{1} - 18 \beta_{3} + 14 \beta_{4} + 15 \beta_{5} + 18 \beta_{6} + \beta_{8} + \beta_{10} ) q^{97} + ( 56 - 2 \beta_{1} - 20 \beta_{2} - 5 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{2} + 3 q^{4} + 2 q^{5} - 14 q^{8} + O(q^{10}) \) \( 12 q + q^{2} + 3 q^{4} + 2 q^{5} - 14 q^{8} + 18 q^{10} - 6 q^{13} + 15 q^{16} + 20 q^{17} - 67 q^{20} - 48 q^{22} + 146 q^{26} - 96 q^{28} - 22 q^{29} + 31 q^{32} - 81 q^{34} + 108 q^{37} + 168 q^{38} + 81 q^{40} + 92 q^{41} - 336 q^{44} + 240 q^{46} + 66 q^{49} - 48 q^{50} + 117 q^{52} - 232 q^{53} - 312 q^{56} - 201 q^{58} - 54 q^{61} + 624 q^{62} - 510 q^{64} + 82 q^{65} + 53 q^{68} - 264 q^{70} - 156 q^{73} + 383 q^{74} + 192 q^{76} - 168 q^{77} - 754 q^{80} + 300 q^{82} - 66 q^{85} - 144 q^{86} + 336 q^{88} + 500 q^{89} - 504 q^{92} - 216 q^{94} + 204 q^{97} + 814 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 2 x^{11} - 3 x^{10} + 11 x^{9} - 5 x^{8} - 14 x^{7} + 29 x^{6} - 28 x^{5} - 20 x^{4} + 88 x^{3} - 48 x^{2} - 64 x + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{10} - 5 \nu^{9} + \nu^{8} + 12 \nu^{7} - 16 \nu^{6} + 7 \nu^{5} + 13 \nu^{4} - 51 \nu^{3} + 38 \nu^{2} + 40 \nu - 64 \)\()/24\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{11} - 3 \nu^{9} + 5 \nu^{8} - 3 \nu^{7} - 4 \nu^{6} + 21 \nu^{5} - 18 \nu^{4} - 8 \nu^{3} + 16 \nu^{2} - 32 \nu + 24 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{11} - 2 \nu^{10} - 11 \nu^{9} + 13 \nu^{8} + 3 \nu^{7} - 16 \nu^{6} + 37 \nu^{5} - 32 \nu^{4} - 78 \nu^{3} + 92 \nu^{2} + 40 \nu - 64 \)\()/24\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{10} + \nu^{9} + 4 \nu^{8} - 3 \nu^{7} - \nu^{6} + \nu^{5} - 14 \nu^{4} + 3 \nu^{3} + 32 \nu^{2} - 8 \nu - 40 \)\()/6\)
\(\beta_{5}\)\(=\)\((\)\( 9 \nu^{11} - 10 \nu^{10} - 13 \nu^{9} + 35 \nu^{8} - 63 \nu^{7} + 40 \nu^{6} + 131 \nu^{5} - 244 \nu^{4} + 114 \nu^{3} + 52 \nu^{2} - 208 \nu + 400 \)\()/48\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{11} + \nu^{10} - 7 \nu^{9} + 2 \nu^{8} + 14 \nu^{7} - 19 \nu^{6} + 13 \nu^{5} + 19 \nu^{4} - 78 \nu^{3} + 48 \nu^{2} + 72 \nu - 80 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{11} - 4 \nu^{10} - \nu^{9} + 13 \nu^{8} - 13 \nu^{7} - 2 \nu^{6} + 19 \nu^{5} - 42 \nu^{4} + 18 \nu^{3} + 64 \nu^{2} - 48 \nu - 40 \)\()/8\)
\(\beta_{8}\)\(=\)\((\)\( \nu^{11} - 2 \nu^{10} + 5 \nu^{8} - 10 \nu^{7} + 3 \nu^{6} + 10 \nu^{5} - 26 \nu^{4} + 23 \nu^{3} + 12 \nu^{2} - 48 \nu + 16 \)\()/4\)
\(\beta_{9}\)\(=\)\((\)\( 3 \nu^{11} + 22 \nu^{10} - 65 \nu^{9} + 13 \nu^{8} + 165 \nu^{7} - 262 \nu^{6} + 79 \nu^{5} + 280 \nu^{4} - 780 \nu^{3} + 440 \nu^{2} + 736 \nu - 928 \)\()/48\)
\(\beta_{10}\)\(=\)\((\)\( 15 \nu^{11} - 8 \nu^{10} - 89 \nu^{9} + 139 \nu^{8} + 87 \nu^{7} - 280 \nu^{6} + 343 \nu^{5} - 206 \nu^{4} - 708 \nu^{3} + 1304 \nu^{2} + 256 \nu - 1120 \)\()/48\)
\(\beta_{11}\)\(=\)\((\)\( -33 \nu^{11} + 22 \nu^{10} + 133 \nu^{9} - 167 \nu^{8} - 141 \nu^{7} + 332 \nu^{6} - 371 \nu^{5} + 136 \nu^{4} + 1110 \nu^{3} - 1324 \nu^{2} - 992 \nu + 1568 \)\()/48\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + 3 \beta_{5} - 3 \beta_{3} - 2 \beta_{2} - 5 \beta_{1} - 2\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{11} + 2 \beta_{10} - \beta_{9} + \beta_{8} + 3 \beta_{6} - 2 \beta_{4} - 4 \beta_{2} - 5 \beta_{1} + 6\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + 3 \beta_{5} - 4 \beta_{4} - 12 \beta_{3} - 4 \beta_{2} + 6 \beta_{1} - 8\)\()/12\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{9} + 2 \beta_{7} + 3 \beta_{6} - \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - \beta_{1} + 5\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{10} - 2 \beta_{9} - 3 \beta_{8} + 4 \beta_{7} + 16 \beta_{6} + 9 \beta_{5} + 4 \beta_{4} - 30 \beta_{3} + 6 \beta_{2} - \beta_{1} - 19\)\()/12\)
\(\nu^{6}\)\(=\)\((\)\(2 \beta_{10} - 9 \beta_{9} - 5 \beta_{8} + 3 \beta_{7} + 3 \beta_{6} - 9 \beta_{5} - 8 \beta_{4} + 15 \beta_{3} + 2 \beta_{2} - 5 \beta_{1} - 20\)\()/12\)
\(\nu^{7}\)\(=\)\((\)\(-9 \beta_{11} - 11 \beta_{9} - \beta_{8} - 14 \beta_{7} - 11 \beta_{6} - 6 \beta_{5} + 6 \beta_{4} - 24 \beta_{3} + 16 \beta_{2} + 43 \beta_{1} - 2\)\()/12\)
\(\nu^{8}\)\(=\)\((\)\(-3 \beta_{11} - \beta_{10} - \beta_{9} - 5 \beta_{8} + 3 \beta_{7} - 11 \beta_{6} - 9 \beta_{5} - 4 \beta_{4} + 10 \beta_{3} + 8 \beta_{2} - 8 \beta_{1} + 10\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-4 \beta_{11} + 2 \beta_{10} + 5 \beta_{9} - 6 \beta_{8} - 26 \beta_{7} + 13 \beta_{6} + 3 \beta_{5} + 46 \beta_{4} - 45 \beta_{3} + 36 \beta_{2} - 133 \beta_{1} - 3\)\()/12\)
\(\nu^{10}\)\(=\)\((\)\(8 \beta_{11} + 27 \beta_{10} + 16 \beta_{9} - 9 \beta_{8} - 18 \beta_{7} - 54 \beta_{6} - 21 \beta_{5} - 36 \beta_{4} + 108 \beta_{3} - 18 \beta_{2} - 207 \beta_{1} - 191\)\()/12\)
\(\nu^{11}\)\(=\)\((\)\(-18 \beta_{11} + 26 \beta_{10} - 11 \beta_{9} + 57 \beta_{8} - 89 \beta_{7} + \beta_{6} - 33 \beta_{5} - 8 \beta_{4} - 21 \beta_{3} - 18 \beta_{2} - 235 \beta_{1} - 88\)\()/12\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
−1.36237 0.379393i
0.0389494 + 1.41368i
1.38685 + 0.276848i
1.25131 0.658947i
1.08837 + 0.903022i
−1.40311 + 0.176844i
−1.36237 + 0.379393i
0.0389494 1.41368i
1.38685 0.276848i
1.25131 + 0.658947i
1.08837 0.903022i
−1.40311 0.176844i
−1.91597 + 0.573621i 0 3.34192 2.19809i −2.18740 3.78869i 0 1.84918 + 1.06762i −5.14216 + 6.12848i 0 6.36427 + 6.00429i
55.2 −1.62609 1.16440i 0 1.28836 + 3.78684i 3.61903 + 6.26834i 0 −5.95847 3.44013i 2.31438 7.65792i 0 1.41395 14.4069i
55.3 0.0560770 1.99921i 0 −3.99371 0.224220i −0.931627 1.61363i 0 9.80189 + 5.65913i −0.672219 + 7.97171i 0 −3.27823 + 1.77203i
55.4 0.461217 + 1.94609i 0 −3.57456 + 1.79514i −2.18740 3.78869i 0 −1.84918 1.06762i −5.14216 6.12848i 0 6.36427 6.00429i
55.5 1.70333 1.04817i 0 1.80268 3.57076i −0.931627 1.61363i 0 −9.80189 5.65913i −0.672219 7.97171i 0 −3.27823 1.77203i
55.6 1.82144 + 0.826041i 0 2.63531 + 3.00917i 3.61903 + 6.26834i 0 5.95847 + 3.44013i 2.31438 + 7.65792i 0 1.41395 + 14.4069i
271.1 −1.91597 0.573621i 0 3.34192 + 2.19809i −2.18740 + 3.78869i 0 1.84918 1.06762i −5.14216 6.12848i 0 6.36427 6.00429i
271.2 −1.62609 + 1.16440i 0 1.28836 3.78684i 3.61903 6.26834i 0 −5.95847 + 3.44013i 2.31438 + 7.65792i 0 1.41395 + 14.4069i
271.3 0.0560770 + 1.99921i 0 −3.99371 + 0.224220i −0.931627 + 1.61363i 0 9.80189 5.65913i −0.672219 7.97171i 0 −3.27823 1.77203i
271.4 0.461217 1.94609i 0 −3.57456 1.79514i −2.18740 + 3.78869i 0 −1.84918 + 1.06762i −5.14216 + 6.12848i 0 6.36427 + 6.00429i
271.5 1.70333 + 1.04817i 0 1.80268 + 3.57076i −0.931627 + 1.61363i 0 −9.80189 + 5.65913i −0.672219 + 7.97171i 0 −3.27823 + 1.77203i
271.6 1.82144 0.826041i 0 2.63531 3.00917i 3.61903 6.26834i 0 5.95847 3.44013i 2.31438 7.65792i 0 1.41395 14.4069i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 271.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.3.f.r 12
3.b odd 2 1 324.3.f.q 12
4.b odd 2 1 inner 324.3.f.r 12
9.c even 3 1 324.3.d.e 6
9.c even 3 1 inner 324.3.f.r 12
9.d odd 6 1 324.3.d.f yes 6
9.d odd 6 1 324.3.f.q 12
12.b even 2 1 324.3.f.q 12
36.f odd 6 1 324.3.d.e 6
36.f odd 6 1 inner 324.3.f.r 12
36.h even 6 1 324.3.d.f yes 6
36.h even 6 1 324.3.f.q 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.3.d.e 6 9.c even 3 1
324.3.d.e 6 36.f odd 6 1
324.3.d.f yes 6 9.d odd 6 1
324.3.d.f yes 6 36.h even 6 1
324.3.f.q 12 3.b odd 2 1
324.3.f.q 12 9.d odd 6 1
324.3.f.q 12 12.b even 2 1
324.3.f.q 12 36.h even 6 1
324.3.f.r 12 1.a even 1 1 trivial
324.3.f.r 12 4.b odd 2 1 inner
324.3.f.r 12 9.c even 3 1 inner
324.3.f.r 12 36.f odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{6} - T_{5}^{5} + 38 T_{5}^{4} + 155 T_{5}^{3} + 1310 T_{5}^{2} + 2183 T_{5} + 3481 \)
\( T_{7}^{12} - 180 T_{7}^{10} + 25536 T_{7}^{8} - 1180224 T_{7}^{6} + 42137856 T_{7}^{4} - 189775872 T_{7}^{2} + 764411904 \)
\( T_{11}^{12} - 516 T_{11}^{10} + 198336 T_{11}^{8} - 31507776 T_{11}^{6} + 3700078848 T_{11}^{4} - 120182538240 T_{11}^{2} + \)\(31\!\cdots\!84\)\( \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4096 - 1024 T - 256 T^{2} + 384 T^{3} - 128 T^{4} - 32 T^{5} + 112 T^{6} - 8 T^{7} - 8 T^{8} + 6 T^{9} - T^{10} - T^{11} + T^{12} \)
$3$ \( T^{12} \)
$5$ \( ( 3481 + 2183 T + 1310 T^{2} + 155 T^{3} + 38 T^{4} - T^{5} + T^{6} )^{2} \)
$7$ \( 764411904 - 189775872 T^{2} + 42137856 T^{4} - 1180224 T^{6} + 25536 T^{8} - 180 T^{10} + T^{12} \)
$11$ \( 3131031158784 - 120182538240 T^{2} + 3700078848 T^{4} - 31507776 T^{6} + 198336 T^{8} - 516 T^{10} + T^{12} \)
$13$ \( ( 982081 + 163515 T + 30198 T^{2} + 1487 T^{3} + 174 T^{4} + 3 T^{5} + T^{6} )^{2} \)
$17$ \( ( 233 - 157 T - 5 T^{2} + T^{3} )^{4} \)
$19$ \( ( 20155392 + 352080 T^{2} + 1476 T^{4} + T^{6} )^{2} \)
$23$ \( 180723314174459904 - 960346451607552 T^{2} + 3883957754112 T^{4} - 5628649536 T^{6} + 5966400 T^{8} - 2868 T^{10} + T^{12} \)
$29$ \( ( 1057485361 + 60517859 T + 3821030 T^{2} + 44567 T^{3} + 1982 T^{4} + 11 T^{5} + T^{6} )^{2} \)
$31$ \( 50096498540544 - 50792283242496 T^{2} + 51459680894976 T^{4} - 38565052416 T^{6} + 21725184 T^{8} - 5376 T^{10} + T^{12} \)
$37$ \( ( 144607 - 4029 T - 27 T^{2} + T^{3} )^{4} \)
$41$ \( ( 482944576 - 22239712 T + 2035040 T^{2} + 2600 T^{3} + 3128 T^{4} - 46 T^{5} + T^{6} )^{2} \)
$43$ \( 41813954908913664 - 1691827124502528 T^{2} + 67149745793280 T^{4} - 52310511936 T^{6} + 32328768 T^{8} - 6372 T^{10} + T^{12} \)
$47$ \( 552971608642289664 - 6749838309851136 T^{2} + 76288145817600 T^{4} - 73016709120 T^{6} + 58294272 T^{8} - 8208 T^{10} + T^{12} \)
$53$ \( ( -5128 + 140 T + 58 T^{2} + T^{3} )^{4} \)
$59$ \( \)\(24\!\cdots\!64\)\( - 587886057807151104 T^{2} + 1204413851566080 T^{4} - 472275947520 T^{6} + 141745152 T^{8} - 13392 T^{10} + T^{12} \)
$61$ \( ( 10239618481 + 487841811 T + 25974198 T^{2} + 72215 T^{3} + 5550 T^{4} + 27 T^{5} + T^{6} )^{2} \)
$67$ \( \)\(16\!\cdots\!84\)\( - 612813715067879424 T^{2} + 2075124782999808 T^{4} - 646435812672 T^{6} + 152651712 T^{8} - 14148 T^{10} + T^{12} \)
$71$ \( ( 808455168 + 16256592 T^{2} + 8532 T^{4} + T^{6} )^{2} \)
$73$ \( ( -447851 - 8901 T + 39 T^{2} + T^{3} )^{4} \)
$79$ \( 12824703626379264 - 1389911479418880 T^{2} + 149756122130688 T^{4} - 95063874624 T^{6} + 48006336 T^{8} - 7764 T^{10} + T^{12} \)
$83$ \( \)\(81\!\cdots\!84\)\( - \)\(30\!\cdots\!20\)\( T^{2} + 85047969115865088 T^{4} - 10108229308416 T^{6} + 876435456 T^{8} - 34896 T^{10} + T^{12} \)
$89$ \( ( 107777 - 4477 T - 125 T^{2} + T^{3} )^{4} \)
$97$ \( ( 136361809984 - 1661724000 T + 57915744 T^{2} - 279544 T^{3} + 14904 T^{4} - 102 T^{5} + T^{6} )^{2} \)
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