Properties

Label 288.2.p.b
Level $288$
Weight $2$
Character orbit 288.p
Analytic conductor $2.300$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 288.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.29969157821\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 3 x^{15} + 7 x^{14} - 12 x^{13} + 16 x^{12} - 12 x^{11} - 8 x^{10} + 36 x^{9} - 68 x^{8} + 72 x^{7} - 32 x^{6} - 96 x^{5} + 256 x^{4} - 384 x^{3} + 448 x^{2} - 384 x + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{3} - \beta_{8} ) q^{3} -\beta_{9} q^{5} -\beta_{4} q^{7} + ( -2 - \beta_{1} - \beta_{2} - \beta_{5} + \beta_{7} - \beta_{10} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{3} - \beta_{8} ) q^{3} -\beta_{9} q^{5} -\beta_{4} q^{7} + ( -2 - \beta_{1} - \beta_{2} - \beta_{5} + \beta_{7} - \beta_{10} ) q^{9} + ( -2 - \beta_{2} - \beta_{3} - \beta_{10} ) q^{11} -\beta_{13} q^{13} + ( -\beta_{6} - \beta_{11} + \beta_{14} ) q^{15} + ( -\beta_{1} + \beta_{3} + \beta_{5} - 2 \beta_{7} - \beta_{8} ) q^{17} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{7} + 2 \beta_{10} ) q^{19} + ( -2 \beta_{4} - \beta_{6} + \beta_{9} - 2 \beta_{11} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{21} + ( -\beta_{9} + \beta_{12} - \beta_{14} - \beta_{15} ) q^{23} + ( \beta_{1} + \beta_{5} + 2 \beta_{7} - \beta_{8} + \beta_{10} ) q^{25} + ( -\beta_{1} - 2 \beta_{3} + \beta_{8} - 3 \beta_{10} ) q^{27} + ( \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} ) q^{29} + ( \beta_{4} + 2 \beta_{6} - \beta_{9} + \beta_{11} - \beta_{13} ) q^{31} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{8} ) q^{33} + ( 3 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - 3 \beta_{7} ) q^{35} + ( \beta_{4} + \beta_{6} + \beta_{9} + 2 \beta_{11} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{37} + ( -\beta_{6} + \beta_{9} - \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{39} + ( -2 + \beta_{1} + \beta_{3} - \beta_{5} - \beta_{7} - \beta_{10} ) q^{41} + ( 2 \beta_{2} - 2 \beta_{3} - \beta_{5} + 4 \beta_{7} + \beta_{8} + 2 \beta_{10} ) q^{43} + ( \beta_{4} - \beta_{11} - \beta_{12} + 2 \beta_{15} ) q^{45} + ( \beta_{4} + 2 \beta_{11} - \beta_{14} + \beta_{15} ) q^{47} + ( 2 + 2 \beta_{1} + \beta_{2} - \beta_{5} - 2 \beta_{7} - \beta_{8} + 2 \beta_{10} ) q^{49} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{7} + \beta_{10} ) q^{51} + ( 2 \beta_{4} + \beta_{6} - \beta_{9} + \beta_{11} - \beta_{12} - \beta_{14} ) q^{53} + ( \beta_{4} + \beta_{6} + \beta_{9} + \beta_{11} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{55} + ( 2 + \beta_{1} - 2 \beta_{2} + \beta_{5} + 2 \beta_{7} + \beta_{8} + \beta_{10} ) q^{57} + ( 3 \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} + \beta_{8} - 2 \beta_{10} ) q^{59} + ( -2 \beta_{4} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{61} + ( -\beta_{4} - \beta_{6} + 2 \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{63} + ( 2 + 4 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 4 \beta_{5} + 4 \beta_{8} ) q^{65} + ( 5 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} - 3 \beta_{7} - \beta_{8} + 3 \beta_{10} ) q^{67} + ( 2 \beta_{4} - \beta_{6} + \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{15} ) q^{69} + ( \beta_{4} + \beta_{6} - \beta_{9} + \beta_{13} + \beta_{14} ) q^{71} + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} + 3 \beta_{8} ) q^{73} + ( -7 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{5} + 5 \beta_{7} + \beta_{8} + \beta_{10} ) q^{75} + ( \beta_{4} + \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{77} + ( -\beta_{4} - \beta_{6} + 2 \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{79} + ( 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{7} - 3 \beta_{8} - 3 \beta_{10} ) q^{81} + ( -2 - 5 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} - 2 \beta_{8} + 5 \beta_{10} ) q^{83} + ( \beta_{4} + 2 \beta_{6} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{85} + ( \beta_{4} + 2 \beta_{6} + \beta_{9} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{87} + ( -1 + 2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 4 \beta_{5} + \beta_{7} + \beta_{8} ) q^{89} + ( 4 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 4 \beta_{5} - 4 \beta_{7} + \beta_{8} - 8 \beta_{10} ) q^{91} + ( \beta_{4} + \beta_{6} - \beta_{9} - 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{93} + ( -\beta_{4} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{95} + ( -3 \beta_{1} + \beta_{2} - \beta_{3} - 5 \beta_{5} + 6 \beta_{7} + 5 \beta_{8} + 3 \beta_{10} ) q^{97} + ( 5 + 4 \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{5} - \beta_{7} + \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 6q^{3} - 6q^{9} + O(q^{10}) \) \( 16q + 6q^{3} - 6q^{9} - 12q^{11} + 4q^{19} - 14q^{25} + 36q^{27} + 12q^{33} - 36q^{41} - 8q^{43} + 10q^{49} - 18q^{51} + 18q^{57} - 12q^{59} - 6q^{65} + 16q^{67} - 4q^{73} - 78q^{75} - 6q^{81} - 54q^{83} + 36q^{91} + 8q^{97} + 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 3 x^{15} + 7 x^{14} - 12 x^{13} + 16 x^{12} - 12 x^{11} - 8 x^{10} + 36 x^{9} - 68 x^{8} + 72 x^{7} - 32 x^{6} - 96 x^{5} + 256 x^{4} - 384 x^{3} + 448 x^{2} - 384 x + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{15} - 13 \nu^{14} + 11 \nu^{13} + 4 \nu^{12} - 38 \nu^{11} + 60 \nu^{10} - 104 \nu^{9} + 68 \nu^{8} + 148 \nu^{7} - 344 \nu^{6} + 440 \nu^{5} - 240 \nu^{4} - 32 \nu^{3} + 608 \nu^{2} - 1152 \nu + 384 \)\()/896\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{15} - \nu^{14} - 25 \nu^{13} + 38 \nu^{12} - 46 \nu^{11} + 24 \nu^{10} - 8 \nu^{9} - 68 \nu^{8} + 244 \nu^{7} - 272 \nu^{6} + 8 \nu^{5} + 128 \nu^{4} - 416 \nu^{3} + 1184 \nu^{2} - 1088 \nu + 512 \)\()/896\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{15} - 25 \nu^{14} + 33 \nu^{13} - 44 \nu^{12} + 40 \nu^{11} + 40 \nu^{10} - 144 \nu^{9} + 316 \nu^{8} - 172 \nu^{7} - 24 \nu^{6} + 480 \nu^{5} - 720 \nu^{4} + 1024 \nu^{3} - 1088 \nu^{2} + 128 \nu - 640 \)\()/896\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{15} - 29 \nu^{14} + 59 \nu^{13} - 102 \nu^{12} + 66 \nu^{11} + 24 \nu^{10} - 176 \nu^{9} + 492 \nu^{8} - 540 \nu^{7} + 288 \nu^{6} + 456 \nu^{5} - 1216 \nu^{4} + 2496 \nu^{3} - 2400 \nu^{2} + 1600 \nu - 384 \)\()/896\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{15} + 11 \nu^{14} - 19 \nu^{13} + 16 \nu^{12} + 2 \nu^{11} - 54 \nu^{10} + 116 \nu^{9} - 92 \nu^{8} + 4 \nu^{7} + 192 \nu^{6} - 424 \nu^{5} + 440 \nu^{4} - 352 \nu^{3} - 256 \nu^{2} + 544 \nu - 704 \)\()/448\)
\(\beta_{6}\)\(=\)\((\)\( 6 \nu^{15} - 8 \nu^{14} + 3 \nu^{13} + 17 \nu^{12} - 39 \nu^{11} + 80 \nu^{10} - 92 \nu^{9} + 16 \nu^{8} + 272 \nu^{7} - 300 \nu^{6} + 260 \nu^{5} - 152 \nu^{4} - 304 \nu^{3} + 1072 \nu^{2} - 1312 \nu + 64 \)\()/448\)
\(\beta_{7}\)\(=\)\((\)\( -13 \nu^{15} + 15 \nu^{14} - 31 \nu^{13} + 4 \nu^{12} + 32 \nu^{11} - 136 \nu^{10} + 176 \nu^{9} - 100 \nu^{8} - 76 \nu^{7} + 552 \nu^{6} - 736 \nu^{5} + 880 \nu^{4} - 256 \nu^{3} - 512 \nu^{2} + 1536 \nu - 1408 \)\()/896\)
\(\beta_{8}\)\(=\)\((\)\( -3 \nu^{15} + 18 \nu^{14} - 47 \nu^{13} + 100 \nu^{12} - 117 \nu^{11} + 100 \nu^{10} + 60 \nu^{9} - 316 \nu^{8} + 592 \nu^{7} - 648 \nu^{6} + 52 \nu^{5} + 888 \nu^{4} - 2144 \nu^{3} + 2768 \nu^{2} - 3040 \nu + 1536 \)\()/448\)
\(\beta_{9}\)\(=\)\((\)\( 6 \nu^{15} - 15 \nu^{14} + 24 \nu^{13} - 32 \nu^{12} + 45 \nu^{11} + 24 \nu^{10} - 64 \nu^{9} + 128 \nu^{8} - 92 \nu^{7} + 176 \nu^{6} - 20 \nu^{5} - 600 \nu^{4} + 592 \nu^{3} - 944 \nu^{2} + 928 \nu - 1728 \)\()/448\)
\(\beta_{10}\)\(=\)\((\)\( 5 \nu^{15} - 16 \nu^{14} + 48 \nu^{13} - 71 \nu^{12} + 76 \nu^{11} - 22 \nu^{10} - 100 \nu^{9} + 284 \nu^{8} - 408 \nu^{7} + 212 \nu^{6} + 184 \nu^{5} - 920 \nu^{4} + 1520 \nu^{3} - 1888 \nu^{2} + 1184 \nu - 768 \)\()/448\)
\(\beta_{11}\)\(=\)\((\)\( -19 \nu^{15} + 37 \nu^{14} - 13 \nu^{13} - 48 \nu^{12} + 232 \nu^{11} - 384 \nu^{10} + 464 \nu^{9} - 60 \nu^{8} - 740 \nu^{7} + 1608 \nu^{6} - 1696 \nu^{5} + 752 \nu^{4} + 1280 \nu^{3} - 4608 \nu^{2} + 5760 \nu - 4608 \)\()/896\)
\(\beta_{12}\)\(=\)\((\)\( -13 \nu^{15} + 29 \nu^{14} - 59 \nu^{13} + 116 \nu^{12} - 150 \nu^{11} + 88 \nu^{10} + 120 \nu^{9} - 324 \nu^{8} + 652 \nu^{7} - 680 \nu^{6} + 104 \nu^{5} + 1216 \nu^{4} - 2720 \nu^{3} + 3968 \nu^{2} - 4288 \nu + 3072 \)\()/448\)
\(\beta_{13}\)\(=\)\((\)\( -16 \nu^{15} + 40 \nu^{14} - 99 \nu^{13} + 153 \nu^{12} - 113 \nu^{11} - 36 \nu^{10} + 376 \nu^{9} - 640 \nu^{8} + 768 \nu^{7} - 124 \nu^{6} - 964 \nu^{5} + 2440 \nu^{4} - 3072 \nu^{3} + 3152 \nu^{2} - 2400 \nu - 320 \)\()/448\)
\(\beta_{14}\)\(=\)\((\)\( -37 \nu^{15} + 103 \nu^{14} - 183 \nu^{13} + 244 \nu^{12} - 232 \nu^{11} - 64 \nu^{10} + 712 \nu^{9} - 1172 \nu^{8} + 1300 \nu^{7} - 600 \nu^{6} - 1216 \nu^{5} + 4176 \nu^{4} - 6432 \nu^{3} + 6848 \nu^{2} - 3968 \nu + 2816 \)\()/896\)
\(\beta_{15}\)\(=\)\((\)\( -57 \nu^{15} + 167 \nu^{14} - 375 \nu^{13} + 528 \nu^{12} - 536 \nu^{11} + 24 \nu^{10} + 1112 \nu^{9} - 2308 \nu^{8} + 2932 \nu^{7} - 1448 \nu^{6} - 1952 \nu^{5} + 7856 \nu^{4} - 12512 \nu^{3} + 14400 \nu^{2} - 12288 \nu + 5888 \)\()/896\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{15} + \beta_{14} - 2 \beta_{10}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} - 2 \beta_{8} - 2 \beta_{5} - \beta_{4} + 2 \beta_{2} - 4 \beta_{1} - 4\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{13} - \beta_{12} + \beta_{11} - 2 \beta_{9} - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_{1} - 2\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{15} + \beta_{14} + 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - 4 \beta_{9} + 2 \beta_{8} + 4 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 4 \beta_{3} - 4 \beta_{2} - 6 \beta_{1}\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} + 3 \beta_{11} - 10 \beta_{10} - 2 \beta_{9} - 6 \beta_{8} - 10 \beta_{7} - \beta_{4} + 6 \beta_{3} - 6 \beta_{2} + 4 \beta_{1} - 10\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(2 \beta_{15} - 6 \beta_{14} - \beta_{13} + 5 \beta_{12} + 3 \beta_{11} - 12 \beta_{10} + 2 \beta_{9} - 8 \beta_{8} - 6 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{3} - 10 \beta_{2} + 2 \beta_{1} - 2\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(3 \beta_{15} + \beta_{14} - 4 \beta_{13} - 4 \beta_{12} + 8 \beta_{11} - 2 \beta_{10} + 2 \beta_{8} + 14 \beta_{6} + 2 \beta_{5} + 6 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 18 \beta_{1} + 28\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(-5 \beta_{15} + 5 \beta_{14} - \beta_{13} + 5 \beta_{12} - \beta_{11} - 2 \beta_{10} - 6 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 12 \beta_{6} + 16 \beta_{5} - \beta_{4} + 18 \beta_{3} + 2 \beta_{2} - 16 \beta_{1} + 2\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(12 \beta_{14} + 9 \beta_{13} + 3 \beta_{12} - 3 \beta_{11} + 6 \beta_{9} - 28 \beta_{8} - 50 \beta_{7} - 6 \beta_{6} + 18 \beta_{5} + 3 \beta_{4} + 18 \beta_{3} + 10 \beta_{2} + 2 \beta_{1} + 10\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(-\beta_{15} + 5 \beta_{14} + 6 \beta_{13} - 6 \beta_{12} - 6 \beta_{11} - 2 \beta_{10} + 12 \beta_{9} + 18 \beta_{8} - 4 \beta_{7} - 6 \beta_{6} - 18 \beta_{5} + 28 \beta_{4} + 8 \beta_{3} - 8 \beta_{2} - 58 \beta_{1}\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(-3 \beta_{15} + 11 \beta_{14} + 7 \beta_{13} - 11 \beta_{12} + 7 \beta_{11} + 26 \beta_{10} + 2 \beta_{9} + 26 \beta_{8} + 26 \beta_{7} - 76 \beta_{5} - 21 \beta_{4} + 50 \beta_{3} + 26 \beta_{2} - 84 \beta_{1} + 34\)\()/4\)
\(\nu^{12}\)\(=\)\((\)\(-38 \beta_{15} - 30 \beta_{14} + 19 \beta_{13} + 49 \beta_{12} + 39 \beta_{11} - 108 \beta_{10} - 14 \beta_{9} - 8 \beta_{8} - 54 \beta_{7} - 14 \beta_{6} + 2 \beta_{5} - 19 \beta_{4} - 2 \beta_{3} - 10 \beta_{2} + 2 \beta_{1} - 74\)\()/4\)
\(\nu^{13}\)\(=\)\((\)\(-41 \beta_{15} + 5 \beta_{14} + 36 \beta_{13} + 36 \beta_{12} + 16 \beta_{11} - 2 \beta_{10} - 6 \beta_{8} + 22 \beta_{6} - 6 \beta_{5} - 10 \beta_{4} - 108 \beta_{3} - 108 \beta_{2} + 130 \beta_{1} + 44\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(7 \beta_{15} - 7 \beta_{14} - 77 \beta_{13} - 7 \beta_{12} + 91 \beta_{11} - 34 \beta_{10} - 86 \beta_{9} + 102 \beta_{8} + 34 \beta_{7} + 172 \beta_{6} + 8 \beta_{5} + 91 \beta_{4} - 94 \beta_{3} - 102 \beta_{2} - 96 \beta_{1} - 190\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(12 \beta_{14} + 29 \beta_{13} - 17 \beta_{12} - 79 \beta_{11} - 130 \beta_{9} - 156 \beta_{8} - 218 \beta_{7} + 130 \beta_{6} + 202 \beta_{5} - 129 \beta_{4} + 202 \beta_{3} - 46 \beta_{2} - 326 \beta_{1} - 62\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(1 + \beta_{1}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−0.533474 + 1.30973i
0.867527 1.11687i
1.12063 0.862658i
−0.186766 + 1.40183i
−0.409484 1.35363i
−1.37702 + 0.322193i
0.608741 + 1.27649i
1.40985 0.111062i
−0.533474 1.30973i
0.867527 + 1.11687i
1.12063 + 0.862658i
−0.186766 1.40183i
−0.409484 + 1.35363i
−1.37702 0.322193i
0.608741 1.27649i
1.40985 + 0.111062i
0 −0.925606 1.46399i 0 −0.895377 1.55084i 0 −2.08793 1.20546i 0 −1.28651 + 2.71015i 0
47.2 0 −0.925606 1.46399i 0 0.895377 + 1.55084i 0 2.08793 + 1.20546i 0 −1.28651 + 2.71015i 0
47.3 0 −0.418594 + 1.68071i 0 −1.60936 2.78750i 0 −1.82223 1.05206i 0 −2.64956 1.40707i 0
47.4 0 −0.418594 + 1.68071i 0 1.60936 + 2.78750i 0 1.82223 + 1.05206i 0 −2.64956 1.40707i 0
47.5 0 1.12774 1.31461i 0 −0.565188 0.978934i 0 −3.71499 2.14485i 0 −0.456412 2.96508i 0
47.6 0 1.12774 1.31461i 0 0.565188 + 0.978934i 0 3.71499 + 2.14485i 0 −0.456412 2.96508i 0
47.7 0 1.71646 + 0.231865i 0 −1.74322 3.01934i 0 1.80802 + 1.04386i 0 2.89248 + 0.795973i 0
47.8 0 1.71646 + 0.231865i 0 1.74322 + 3.01934i 0 −1.80802 1.04386i 0 2.89248 + 0.795973i 0
239.1 0 −0.925606 + 1.46399i 0 −0.895377 + 1.55084i 0 −2.08793 + 1.20546i 0 −1.28651 2.71015i 0
239.2 0 −0.925606 + 1.46399i 0 0.895377 1.55084i 0 2.08793 1.20546i 0 −1.28651 2.71015i 0
239.3 0 −0.418594 1.68071i 0 −1.60936 + 2.78750i 0 −1.82223 + 1.05206i 0 −2.64956 + 1.40707i 0
239.4 0 −0.418594 1.68071i 0 1.60936 2.78750i 0 1.82223 1.05206i 0 −2.64956 + 1.40707i 0
239.5 0 1.12774 + 1.31461i 0 −0.565188 + 0.978934i 0 −3.71499 + 2.14485i 0 −0.456412 + 2.96508i 0
239.6 0 1.12774 + 1.31461i 0 0.565188 0.978934i 0 3.71499 2.14485i 0 −0.456412 + 2.96508i 0
239.7 0 1.71646 0.231865i 0 −1.74322 + 3.01934i 0 1.80802 1.04386i 0 2.89248 0.795973i 0
239.8 0 1.71646 0.231865i 0 1.74322 3.01934i 0 −1.80802 + 1.04386i 0 2.89248 0.795973i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
9.d odd 6 1 inner
72.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.p.b 16
3.b odd 2 1 864.2.p.b 16
4.b odd 2 1 72.2.l.b 16
8.b even 2 1 72.2.l.b 16
8.d odd 2 1 inner 288.2.p.b 16
9.c even 3 1 864.2.p.b 16
9.c even 3 1 2592.2.f.b 16
9.d odd 6 1 inner 288.2.p.b 16
9.d odd 6 1 2592.2.f.b 16
12.b even 2 1 216.2.l.b 16
24.f even 2 1 864.2.p.b 16
24.h odd 2 1 216.2.l.b 16
36.f odd 6 1 216.2.l.b 16
36.f odd 6 1 648.2.f.b 16
36.h even 6 1 72.2.l.b 16
36.h even 6 1 648.2.f.b 16
72.j odd 6 1 72.2.l.b 16
72.j odd 6 1 648.2.f.b 16
72.l even 6 1 inner 288.2.p.b 16
72.l even 6 1 2592.2.f.b 16
72.n even 6 1 216.2.l.b 16
72.n even 6 1 648.2.f.b 16
72.p odd 6 1 864.2.p.b 16
72.p odd 6 1 2592.2.f.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.l.b 16 4.b odd 2 1
72.2.l.b 16 8.b even 2 1
72.2.l.b 16 36.h even 6 1
72.2.l.b 16 72.j odd 6 1
216.2.l.b 16 12.b even 2 1
216.2.l.b 16 24.h odd 2 1
216.2.l.b 16 36.f odd 6 1
216.2.l.b 16 72.n even 6 1
288.2.p.b 16 1.a even 1 1 trivial
288.2.p.b 16 8.d odd 2 1 inner
288.2.p.b 16 9.d odd 6 1 inner
288.2.p.b 16 72.l even 6 1 inner
648.2.f.b 16 36.f odd 6 1
648.2.f.b 16 36.h even 6 1
648.2.f.b 16 72.j odd 6 1
648.2.f.b 16 72.n even 6 1
864.2.p.b 16 3.b odd 2 1
864.2.p.b 16 9.c even 3 1
864.2.p.b 16 24.f even 2 1
864.2.p.b 16 72.p odd 6 1
2592.2.f.b 16 9.c even 3 1
2592.2.f.b 16 9.d odd 6 1
2592.2.f.b 16 72.l even 6 1
2592.2.f.b 16 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{16} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(288, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( ( 81 - 81 T + 54 T^{2} - 45 T^{3} + 30 T^{4} - 15 T^{5} + 6 T^{6} - 3 T^{7} + T^{8} )^{2} \)
$5$ \( 266256 + 339012 T^{2} + 312453 T^{4} + 123903 T^{6} + 35106 T^{8} + 4923 T^{10} + 498 T^{12} + 27 T^{14} + T^{16} \)
$7$ \( 4260096 - 2904048 T^{2} + 1279953 T^{4} - 340749 T^{6} + 66426 T^{8} - 8373 T^{10} + 750 T^{12} - 33 T^{14} + T^{16} \)
$11$ \( ( 1 - 12 T + 44 T^{2} + 48 T^{3} - 9 T^{4} - 24 T^{5} + 8 T^{6} + 6 T^{7} + T^{8} )^{2} \)
$13$ \( 266256 - 1772460 T^{2} + 11382813 T^{4} - 2713221 T^{6} + 454938 T^{8} - 39129 T^{10} + 2442 T^{12} - 57 T^{14} + T^{16} \)
$17$ \( ( 784 + 992 T^{2} + 360 T^{4} + 35 T^{6} + T^{8} )^{2} \)
$19$ \( ( 16 + 8 T - 12 T^{2} - T^{3} + T^{4} )^{4} \)
$23$ \( 639280656 + 454125924 T^{2} + 255771909 T^{4} + 42464691 T^{6} + 5182026 T^{8} + 225735 T^{10} + 7158 T^{12} + 99 T^{14} + T^{16} \)
$29$ \( 17449353216 + 10074829824 T^{2} + 5081053545 T^{4} + 389228679 T^{6} + 20607630 T^{8} + 599547 T^{10} + 12654 T^{12} + 135 T^{14} + T^{16} \)
$31$ \( 279189651456 - 45038923776 T^{2} + 4696155729 T^{4} - 290875401 T^{6} + 13147422 T^{8} - 398493 T^{10} + 8826 T^{12} - 117 T^{14} + T^{16} \)
$37$ \( ( 74304 + 40176 T^{2} + 4896 T^{4} + 156 T^{6} + T^{8} )^{2} \)
$41$ \( ( 7921 + 8010 T + 920 T^{2} - 1800 T^{3} - 51 T^{4} + 360 T^{5} + 128 T^{6} + 18 T^{7} + T^{8} )^{2} \)
$43$ \( ( 6889 - 11786 T + 15184 T^{2} - 7856 T^{3} + 3115 T^{4} - 524 T^{5} + 76 T^{6} + 4 T^{7} + T^{8} )^{2} \)
$47$ \( 266256 + 1701252 T^{2} + 10094661 T^{4} + 4840839 T^{6} + 1892526 T^{8} + 160239 T^{10} + 10818 T^{12} + 111 T^{14} + T^{16} \)
$53$ \( ( 297216 - 331344 T^{2} + 15408 T^{4} - 228 T^{6} + T^{8} )^{2} \)
$59$ \( ( 528529 - 165756 T - 33562 T^{2} + 15960 T^{3} + 3717 T^{4} - 420 T^{5} - 58 T^{6} + 6 T^{7} + T^{8} )^{2} \)
$61$ \( 1192149524736 - 215024661360 T^{2} + 26804642049 T^{4} - 1747852317 T^{6} + 82050270 T^{8} - 1679649 T^{10} + 24750 T^{12} - 189 T^{14} + T^{16} \)
$67$ \( ( 582169 + 209062 T + 125434 T^{2} - 5876 T^{3} + 5785 T^{4} - 20 T^{5} + 130 T^{6} - 8 T^{7} + T^{8} )^{2} \)
$71$ \( ( 74304 - 28944 T^{2} + 3744 T^{4} - 168 T^{6} + T^{8} )^{2} \)
$73$ \( ( 172 - 224 T - 78 T^{2} + T^{3} + T^{4} )^{4} \)
$79$ \( 74509345296 - 36110680524 T^{2} + 13340118429 T^{4} - 1880575641 T^{6} + 199135626 T^{8} - 3530925 T^{10} + 46758 T^{12} - 249 T^{14} + T^{16} \)
$83$ \( ( 432964 - 487578 T + 162629 T^{2} + 22971 T^{3} - 6366 T^{4} - 837 T^{5} + 212 T^{6} + 27 T^{7} + T^{8} )^{2} \)
$89$ \( ( 891136 + 689456 T^{2} + 23808 T^{4} + 272 T^{6} + T^{8} )^{2} \)
$97$ \( ( 1018081 + 904064 T + 978382 T^{2} - 147832 T^{3} + 32851 T^{4} - 1096 T^{5} + 190 T^{6} - 4 T^{7} + T^{8} )^{2} \)
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