Properties

Label 144.3.m.b
Level $144$
Weight $3$
Character orbit 144.m
Analytic conductor $3.924$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 144.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.92371580679\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{14} + 10 x^{12} + 88 x^{10} - 752 x^{8} + 1408 x^{6} + 2560 x^{4} - 24576 x^{2} + 65536\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{4} q^{2} + ( -1 - \beta_{2} ) q^{4} + \beta_{8} q^{5} + ( \beta_{11} + \beta_{12} ) q^{7} + ( \beta_{4} + \beta_{7} - \beta_{13} ) q^{8} +O(q^{10})\) \( q -\beta_{4} q^{2} + ( -1 - \beta_{2} ) q^{4} + \beta_{8} q^{5} + ( \beta_{11} + \beta_{12} ) q^{7} + ( \beta_{4} + \beta_{7} - \beta_{13} ) q^{8} + ( \beta_{3} - \beta_{6} + \beta_{11} - \beta_{15} ) q^{10} + ( 2 \beta_{1} - \beta_{4} + \beta_{5} + \beta_{9} + \beta_{13} ) q^{11} + ( -1 - \beta_{2} + \beta_{3} - \beta_{6} + \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} ) q^{13} + ( -\beta_{1} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{13} ) q^{14} + ( 3 + \beta_{2} - \beta_{3} + \beta_{6} + 2 \beta_{11} + \beta_{12} - \beta_{14} ) q^{16} + ( -3 \beta_{1} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{10} + 2 \beta_{13} ) q^{17} + ( -4 - 3 \beta_{2} - 2 \beta_{3} - \beta_{11} - \beta_{12} + 2 \beta_{14} - \beta_{15} ) q^{19} + ( -\beta_{1} - \beta_{4} - 2 \beta_{5} + \beta_{7} - 3 \beta_{8} - \beta_{9} - \beta_{13} ) q^{20} + ( 6 - 2 \beta_{2} - 2 \beta_{3} - \beta_{6} + 2 \beta_{12} + \beta_{14} + 2 \beta_{15} ) q^{22} + ( 3 \beta_{1} + 4 \beta_{4} - \beta_{5} - 2 \beta_{7} + 2 \beta_{8} + \beta_{10} + 2 \beta_{13} ) q^{23} + ( 2 - 2 \beta_{2} + 7 \beta_{3} + 4 \beta_{12} - 2 \beta_{14} + 2 \beta_{15} ) q^{25} + ( -\beta_{1} + \beta_{4} + 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{10} - 2 \beta_{13} ) q^{26} + ( 1 + \beta_{2} + 9 \beta_{3} + 2 \beta_{6} - 2 \beta_{11} - \beta_{12} - 2 \beta_{15} ) q^{28} + ( 6 \beta_{1} + 6 \beta_{4} - \beta_{7} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{13} ) q^{29} + ( 3 \beta_{2} - 12 \beta_{3} + 2 \beta_{6} - 4 \beta_{12} + 2 \beta_{14} + \beta_{15} ) q^{31} + ( -3 \beta_{1} - 3 \beta_{4} + 2 \beta_{5} - 3 \beta_{7} - 3 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{13} ) q^{32} + ( -4 - 14 \beta_{3} - 3 \beta_{6} - 2 \beta_{11} - 4 \beta_{12} + \beta_{14} ) q^{34} + ( -13 \beta_{1} + 5 \beta_{4} + 4 \beta_{5} + 4 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{13} ) q^{35} + ( 9 + 7 \beta_{2} + 7 \beta_{3} - \beta_{6} - \beta_{11} + \beta_{12} - 3 \beta_{14} - \beta_{15} ) q^{37} + ( 2 \beta_{1} - 2 \beta_{5} + 6 \beta_{7} - 4 \beta_{10} - 2 \beta_{13} ) q^{38} + ( -16 + 8 \beta_{3} + 4 \beta_{6} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} + 2 \beta_{15} ) q^{40} + ( 13 \beta_{1} - 2 \beta_{4} + 3 \beta_{5} + \beta_{7} + \beta_{8} + 2 \beta_{9} + 5 \beta_{10} ) q^{41} + ( 7 \beta_{2} - 2 \beta_{3} + 2 \beta_{6} - \beta_{11} + 3 \beta_{12} + 4 \beta_{14} + \beta_{15} ) q^{43} + ( 6 \beta_{1} - 2 \beta_{4} - 6 \beta_{5} - 2 \beta_{7} + 4 \beta_{8} + 2 \beta_{10} - 2 \beta_{13} ) q^{44} + ( -16 + 4 \beta_{2} + 6 \beta_{3} - 4 \beta_{6} - 2 \beta_{11} + 2 \beta_{14} - 2 \beta_{15} ) q^{46} + ( \beta_{1} - 14 \beta_{4} - 5 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} ) q^{47} + ( 7 + 4 \beta_{2} - 2 \beta_{6} - 2 \beta_{11} - 10 \beta_{12} - 4 \beta_{14} ) q^{49} + ( -5 \beta_{1} + 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} + 6 \beta_{8} - 2 \beta_{9} + 6 \beta_{10} - 2 \beta_{13} ) q^{50} + ( -11 + \beta_{2} - 23 \beta_{3} + 6 \beta_{6} + \beta_{12} - 2 \beta_{14} + 4 \beta_{15} ) q^{52} + ( -8 \beta_{1} - 22 \beta_{4} - 2 \beta_{5} - \beta_{8} - 2 \beta_{9} - 8 \beta_{10} + 2 \beta_{13} ) q^{53} + ( -16 - 4 \beta_{2} + 2 \beta_{6} - 4 \beta_{11} + 8 \beta_{12} + 6 \beta_{14} ) q^{55} + ( -13 \beta_{1} - 7 \beta_{4} + 8 \beta_{5} + 5 \beta_{7} + \beta_{8} - \beta_{9} - 4 \beta_{10} + 3 \beta_{13} ) q^{56} + ( 20 + 4 \beta_{2} + 27 \beta_{3} - 2 \beta_{6} - \beta_{11} + 4 \beta_{12} + 3 \beta_{14} + 3 \beta_{15} ) q^{58} + ( 12 \beta_{1} - 6 \beta_{4} + 6 \beta_{5} - 4 \beta_{7} - 2 \beta_{9} - 4 \beta_{10} - 2 \beta_{13} ) q^{59} + ( -1 - 11 \beta_{2} + 5 \beta_{3} + \beta_{6} - 5 \beta_{11} - 3 \beta_{12} - 3 \beta_{14} + 5 \beta_{15} ) q^{61} + ( 9 \beta_{1} - 2 \beta_{4} - \beta_{5} - 5 \beta_{7} - 3 \beta_{8} + 5 \beta_{9} - 7 \beta_{10} + 3 \beta_{13} ) q^{62} + ( 30 - 2 \beta_{2} - 10 \beta_{3} + 4 \beta_{6} - 6 \beta_{11} + 2 \beta_{14} - 2 \beta_{15} ) q^{64} + ( -27 \beta_{1} + 16 \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} + 9 \beta_{10} - 6 \beta_{13} ) q^{65} + ( -16 - 4 \beta_{2} - 20 \beta_{3} + 4 \beta_{6} + 4 \beta_{11} - 4 \beta_{12} + 4 \beta_{15} ) q^{67} + ( 22 \beta_{1} + 6 \beta_{4} - 6 \beta_{5} + 2 \beta_{7} + 4 \beta_{9} + 6 \beta_{10} + 2 \beta_{13} ) q^{68} + ( 42 + 6 \beta_{2} - 28 \beta_{3} - 5 \beta_{6} + 2 \beta_{11} - 10 \beta_{12} - \beta_{14} - 4 \beta_{15} ) q^{70} + ( 22 \beta_{1} + 24 \beta_{4} - 10 \beta_{5} + 8 \beta_{7} - 8 \beta_{8} + 2 \beta_{10} - 4 \beta_{13} ) q^{71} + ( -2 + 2 \beta_{2} + 12 \beta_{3} - 4 \beta_{6} + 4 \beta_{12} - 2 \beta_{14} - 10 \beta_{15} ) q^{73} + ( -5 \beta_{1} - 7 \beta_{4} + 6 \beta_{5} - 4 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 8 \beta_{10} + 8 \beta_{13} ) q^{74} + ( 20 + 50 \beta_{3} + 2 \beta_{6} + 8 \beta_{11} + 2 \beta_{12} - 6 \beta_{14} + 4 \beta_{15} ) q^{76} + ( 6 \beta_{1} + 30 \beta_{4} - 8 \beta_{5} + 2 \beta_{7} - 6 \beta_{9} - 10 \beta_{10} - 6 \beta_{13} ) q^{77} + ( -8 - 3 \beta_{2} + 8 \beta_{3} - 4 \beta_{6} - 8 \beta_{12} + 4 \beta_{14} - 5 \beta_{15} ) q^{79} + ( -12 \beta_{1} + 18 \beta_{4} + 12 \beta_{5} - 2 \beta_{7} + 4 \beta_{8} + 4 \beta_{9} + 2 \beta_{13} ) q^{80} + ( -16 - 4 \beta_{2} - 44 \beta_{3} - \beta_{6} + 4 \beta_{11} + 16 \beta_{12} + \beta_{14} + 2 \beta_{15} ) q^{82} + ( -37 \beta_{1} + 9 \beta_{4} + 8 \beta_{5} - 16 \beta_{8} + 3 \beta_{9} - 5 \beta_{10} - 3 \beta_{13} ) q^{83} + ( 12 - 2 \beta_{2} + 12 \beta_{3} - 2 \beta_{6} + 6 \beta_{11} + 10 \beta_{12} - 2 \beta_{14} + 6 \beta_{15} ) q^{85} + ( -4 \beta_{4} - 4 \beta_{5} - 8 \beta_{7} + 6 \beta_{8} - 2 \beta_{9} - 10 \beta_{10} + 8 \beta_{13} ) q^{86} + ( -62 - 2 \beta_{2} + 30 \beta_{3} - 2 \beta_{6} + 4 \beta_{11} + 2 \beta_{12} + 2 \beta_{14} - 8 \beta_{15} ) q^{88} + ( 16 \beta_{1} - 24 \beta_{4} - 2 \beta_{7} - 2 \beta_{8} - 8 \beta_{9} + 8 \beta_{10} ) q^{89} + ( 16 + 5 \beta_{2} - 18 \beta_{3} - 2 \beta_{6} + 5 \beta_{11} + \beta_{12} - 5 \beta_{15} ) q^{91} + ( 2 \beta_{1} + 14 \beta_{4} - 8 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 4 \beta_{10} + 6 \beta_{13} ) q^{92} + ( -76 - 16 \beta_{2} + 18 \beta_{3} - 2 \beta_{6} + 6 \beta_{11} - 4 \beta_{12} + 2 \beta_{15} ) q^{94} + ( -7 \beta_{1} - 38 \beta_{4} - 13 \beta_{5} - 10 \beta_{7} - 10 \beta_{8} - 6 \beta_{9} + 3 \beta_{10} ) q^{95} + ( 12 \beta_{11} + 4 \beta_{12} - 4 \beta_{14} ) q^{97} + ( 6 \beta_{1} - \beta_{4} + 6 \beta_{5} - 2 \beta_{7} - 6 \beta_{8} + 10 \beta_{9} + 14 \beta_{10} + 6 \beta_{13} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 12q^{4} + O(q^{10}) \) \( 16q - 12q^{4} + 16q^{10} + 32q^{16} - 32q^{19} + 104q^{22} - 24q^{34} + 96q^{37} - 312q^{40} - 32q^{43} - 224q^{46} + 112q^{49} - 264q^{52} - 256q^{55} + 312q^{58} - 32q^{61} + 456q^{64} - 256q^{67} + 744q^{70} + 264q^{76} - 280q^{82} + 160q^{85} - 912q^{88} + 288q^{91} - 1104q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 6 x^{14} + 10 x^{12} + 88 x^{10} - 752 x^{8} + 1408 x^{6} + 2560 x^{4} - 24576 x^{2} + 65536\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 1 \)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{14} + 14 \nu^{12} + 110 \nu^{10} - 472 \nu^{8} + 944 \nu^{6} + 5504 \nu^{4} - 24064 \nu^{2} - 20480 \)\()/12288\)
\(\beta_{4}\)\(=\)\((\)\( 5 \nu^{15} - 14 \nu^{13} - 110 \nu^{11} + 472 \nu^{9} - 944 \nu^{7} - 5504 \nu^{5} + 24064 \nu^{3} + 20480 \nu \)\()/12288\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{15} + 46 \nu^{13} - 122 \nu^{11} - 968 \nu^{9} + 4528 \nu^{7} - 8960 \nu^{5} - 46592 \nu^{3} + 229376 \nu \)\()/12288\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{14} - 6 \nu^{12} + 10 \nu^{10} + 88 \nu^{8} - 752 \nu^{6} + 1408 \nu^{4} + 2560 \nu^{2} - 22528 \)\()/2048\)
\(\beta_{7}\)\(=\)\((\)\( -17 \nu^{15} - 34 \nu^{13} + 326 \nu^{11} - 424 \nu^{9} - 1360 \nu^{7} + 26624 \nu^{5} + 9728 \nu^{3} - 167936 \nu \)\()/24576\)
\(\beta_{8}\)\(=\)\((\)\( -11 \nu^{15} + 14 \nu^{13} + 170 \nu^{11} - 784 \nu^{9} + 1328 \nu^{7} + 9536 \nu^{5} - 30208 \nu^{3} - 26624 \nu \)\()/12288\)
\(\beta_{9}\)\(=\)\((\)\( -11 \nu^{15} + 14 \nu^{13} + 170 \nu^{11} - 784 \nu^{9} + 1328 \nu^{7} + 9536 \nu^{5} - 17920 \nu^{3} - 38912 \nu \)\()/12288\)
\(\beta_{10}\)\(=\)\((\)\( \nu^{15} - 16 \nu^{13} + 38 \nu^{11} + 308 \nu^{9} - 1696 \nu^{7} + 3296 \nu^{5} + 13568 \nu^{3} - 72704 \nu \)\()/3072\)
\(\beta_{11}\)\(=\)\((\)\( -5 \nu^{14} + 38 \nu^{12} - 34 \nu^{10} - 1000 \nu^{8} + 4592 \nu^{6} - 1792 \nu^{4} - 45568 \nu^{2} + 182272 \)\()/6144\)
\(\beta_{12}\)\(=\)\((\)\( -11 \nu^{14} + 146 \nu^{12} - 142 \nu^{10} - 2344 \nu^{8} + 11600 \nu^{6} - 16768 \nu^{4} - 119296 \nu^{2} + 348160 \)\()/12288\)
\(\beta_{13}\)\(=\)\((\)\( -13 \nu^{15} + 86 \nu^{13} + 14 \nu^{11} - 1704 \nu^{9} + 7280 \nu^{7} - 2304 \nu^{5} - 76288 \nu^{3} + 176128 \nu \)\()/8192\)
\(\beta_{14}\)\(=\)\((\)\( -5 \nu^{14} + 62 \nu^{12} - 82 \nu^{10} - 1336 \nu^{8} + 6128 \nu^{6} - 8320 \nu^{4} - 62464 \nu^{2} + 246784 \)\()/3072\)
\(\beta_{15}\)\(=\)\((\)\( 47 \nu^{14} - 122 \nu^{12} - 554 \nu^{10} + 4072 \nu^{8} - 10640 \nu^{6} - 33152 \nu^{4} + 162304 \nu^{2} - 77824 \)\()/12288\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{9} - \beta_{8} + \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{14} + \beta_{12} + 2 \beta_{11} + \beta_{6} - \beta_{3} + \beta_{2} + 3\)
\(\nu^{5}\)\(=\)\(\beta_{13} + 2 \beta_{10} - \beta_{9} - 3 \beta_{8} + 3 \beta_{7} + 2 \beta_{5} - \beta_{4} + \beta_{1}\)
\(\nu^{6}\)\(=\)\(2 \beta_{15} - 2 \beta_{14} + 6 \beta_{11} - 4 \beta_{6} + 10 \beta_{3} + 2 \beta_{2} - 30\)
\(\nu^{7}\)\(=\)\(-2 \beta_{13} - 12 \beta_{10} - 4 \beta_{9} - 12 \beta_{8} + 10 \beta_{7} - 8 \beta_{5} - 18 \beta_{4} - 32 \beta_{1}\)
\(\nu^{8}\)\(=\)\(4 \beta_{15} - 14 \beta_{14} + 14 \beta_{12} + 16 \beta_{11} - 14 \beta_{6} + 14 \beta_{3} - 38 \beta_{2} + 110\)
\(\nu^{9}\)\(=\)\(10 \beta_{13} - 20 \beta_{10} - 54 \beta_{9} + 14 \beta_{8} + 14 \beta_{7} - 44 \beta_{5} - 18 \beta_{4} + 118 \beta_{1}\)
\(\nu^{10}\)\(=\)\(36 \beta_{15} + 24 \beta_{14} - 36 \beta_{12} - 44 \beta_{11} - 68 \beta_{6} + 328 \beta_{3} + 64 \beta_{2} + 488\)
\(\nu^{11}\)\(=\)\(-72 \beta_{13} - 176 \beta_{10} + 108 \beta_{9} - 92 \beta_{8} - 88 \beta_{7} - 168 \beta_{5} - 288 \beta_{4} + 540 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-164 \beta_{14} + 396 \beta_{12} + 40 \beta_{11} + 196 \beta_{6} - 60 \beta_{3} + 444 \beta_{2} + 3268\)
\(\nu^{13}\)\(=\)\(396 \beta_{13} + 680 \beta_{10} + 404 \beta_{9} - 484 \beta_{8} - 316 \beta_{7} + 104 \beta_{5} + 308 \beta_{4} + 3196 \beta_{1}\)
\(\nu^{14}\)\(=\)\(792 \beta_{15} - 88 \beta_{14} + 96 \beta_{12} + 968 \beta_{11} + 720 \beta_{6} + 4056 \beta_{3} + 2904 \beta_{2} - 1768\)
\(\nu^{15}\)\(=\)\(-696 \beta_{13} - 144 \beta_{10} + 1936 \beta_{9} - 5456 \beta_{8} + 1048 \beta_{7} + 1440 \beta_{5} - 5816 \beta_{4} - 4160 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.136762 1.99532i
1.64663 1.13516i
−1.66730 1.10459i
1.99750 0.0999235i
−1.99750 + 0.0999235i
1.66730 + 1.10459i
−1.64663 + 1.13516i
−0.136762 + 1.99532i
0.136762 + 1.99532i
1.64663 + 1.13516i
−1.66730 + 1.10459i
1.99750 + 0.0999235i
−1.99750 0.0999235i
1.66730 1.10459i
−1.64663 1.13516i
−0.136762 1.99532i
−1.99532 0.136762i 0 3.96259 + 0.545766i −0.227650 + 0.227650i 0 3.90219 −7.83199 1.63091i 0 0.485368 0.423100i
19.2 −1.13516 1.64663i 0 −1.42280 + 3.73840i 2.41234 2.41234i 0 −11.8718 7.77089 1.90087i 0 −6.71063 1.23383i
19.3 −1.10459 + 1.66730i 0 −1.55976 3.68336i −4.23991 + 4.23991i 0 −0.262225 7.86415 + 1.46802i 0 −2.38583 11.7525i
19.4 −0.0999235 1.99750i 0 −3.98003 + 0.399195i −6.01265 + 6.01265i 0 8.23187 1.19509 + 7.91023i 0 12.6111 + 11.4095i
19.5 0.0999235 + 1.99750i 0 −3.98003 + 0.399195i 6.01265 6.01265i 0 8.23187 −1.19509 7.91023i 0 12.6111 + 11.4095i
19.6 1.10459 1.66730i 0 −1.55976 3.68336i 4.23991 4.23991i 0 −0.262225 −7.86415 1.46802i 0 −2.38583 11.7525i
19.7 1.13516 + 1.64663i 0 −1.42280 + 3.73840i −2.41234 + 2.41234i 0 −11.8718 −7.77089 + 1.90087i 0 −6.71063 1.23383i
19.8 1.99532 + 0.136762i 0 3.96259 + 0.545766i 0.227650 0.227650i 0 3.90219 7.83199 + 1.63091i 0 0.485368 0.423100i
91.1 −1.99532 + 0.136762i 0 3.96259 0.545766i −0.227650 0.227650i 0 3.90219 −7.83199 + 1.63091i 0 0.485368 + 0.423100i
91.2 −1.13516 + 1.64663i 0 −1.42280 3.73840i 2.41234 + 2.41234i 0 −11.8718 7.77089 + 1.90087i 0 −6.71063 + 1.23383i
91.3 −1.10459 1.66730i 0 −1.55976 + 3.68336i −4.23991 4.23991i 0 −0.262225 7.86415 1.46802i 0 −2.38583 + 11.7525i
91.4 −0.0999235 + 1.99750i 0 −3.98003 0.399195i −6.01265 6.01265i 0 8.23187 1.19509 7.91023i 0 12.6111 11.4095i
91.5 0.0999235 1.99750i 0 −3.98003 0.399195i 6.01265 + 6.01265i 0 8.23187 −1.19509 + 7.91023i 0 12.6111 11.4095i
91.6 1.10459 + 1.66730i 0 −1.55976 + 3.68336i 4.23991 + 4.23991i 0 −0.262225 −7.86415 + 1.46802i 0 −2.38583 + 11.7525i
91.7 1.13516 1.64663i 0 −1.42280 3.73840i −2.41234 2.41234i 0 −11.8718 −7.77089 1.90087i 0 −6.71063 + 1.23383i
91.8 1.99532 0.136762i 0 3.96259 0.545766i 0.227650 + 0.227650i 0 3.90219 7.83199 1.63091i 0 0.485368 + 0.423100i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.m.b 16
3.b odd 2 1 inner 144.3.m.b 16
4.b odd 2 1 576.3.m.b 16
8.b even 2 1 1152.3.m.e 16
8.d odd 2 1 1152.3.m.d 16
12.b even 2 1 576.3.m.b 16
16.e even 4 1 576.3.m.b 16
16.e even 4 1 1152.3.m.d 16
16.f odd 4 1 inner 144.3.m.b 16
16.f odd 4 1 1152.3.m.e 16
24.f even 2 1 1152.3.m.d 16
24.h odd 2 1 1152.3.m.e 16
48.i odd 4 1 576.3.m.b 16
48.i odd 4 1 1152.3.m.d 16
48.k even 4 1 inner 144.3.m.b 16
48.k even 4 1 1152.3.m.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.m.b 16 1.a even 1 1 trivial
144.3.m.b 16 3.b odd 2 1 inner
144.3.m.b 16 16.f odd 4 1 inner
144.3.m.b 16 48.k even 4 1 inner
576.3.m.b 16 4.b odd 2 1
576.3.m.b 16 12.b even 2 1
576.3.m.b 16 16.e even 4 1
576.3.m.b 16 48.i odd 4 1
1152.3.m.d 16 8.d odd 2 1
1152.3.m.d 16 16.e even 4 1
1152.3.m.d 16 24.f even 2 1
1152.3.m.d 16 48.i odd 4 1
1152.3.m.e 16 8.b even 2 1
1152.3.m.e 16 16.f odd 4 1
1152.3.m.e 16 24.h odd 2 1
1152.3.m.e 16 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 6656 T_{5}^{12} + 7641216 T_{5}^{8} + 915505152 T_{5}^{4} + 9834496 \) acting on \(S_{3}^{\mathrm{new}}(144, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 65536 + 24576 T^{2} + 2560 T^{4} - 1408 T^{6} - 752 T^{8} - 88 T^{10} + 10 T^{12} + 6 T^{14} + T^{16} \)
$3$ \( T^{16} \)
$5$ \( 9834496 + 915505152 T^{4} + 7641216 T^{8} + 6656 T^{12} + T^{16} \)
$7$ \( ( 100 + 352 T - 112 T^{2} + T^{4} )^{4} \)
$11$ \( 1228250152960000 + 13103358017536 T^{4} + 2539489280 T^{8} + 110592 T^{12} + T^{16} \)
$13$ \( ( 35760400 - 26024960 T + 9469952 T^{2} - 1584128 T^{3} + 144456 T^{4} - 4352 T^{5} + T^{8} )^{2} \)
$17$ \( ( 472105984 - 42764800 T^{2} + 382592 T^{4} - 1104 T^{6} + T^{8} )^{2} \)
$19$ \( ( 6629867776 + 78167040 T + 460800 T^{2} - 2070784 T^{3} + 477152 T^{4} - 13760 T^{5} + 128 T^{6} + 16 T^{7} + T^{8} )^{2} \)
$23$ \( ( 467943424 - 37695488 T^{2} + 502272 T^{4} - 1632 T^{6} + T^{8} )^{2} \)
$29$ \( 2847880354408960000 + 1898534369613070336 T^{4} + 4853968416896 T^{8} + 3912960 T^{12} + T^{16} \)
$31$ \( ( 11588953104 + 1680413440 T^{2} + 4834568 T^{4} + 4032 T^{6} + T^{8} )^{2} \)
$37$ \( ( 137744899600 + 40629438080 T + 5992059392 T^{2} - 265498816 T^{3} + 5955464 T^{4} + 14752 T^{5} + 1152 T^{6} - 48 T^{7} + T^{8} )^{2} \)
$41$ \( ( 198844646400 + 5017549312 T^{2} + 12842112 T^{4} + 7248 T^{6} + T^{8} )^{2} \)
$43$ \( ( 21036601600 - 22398817280 T + 11924621312 T^{2} + 610465536 T^{3} + 15454944 T^{4} + 90944 T^{5} + 128 T^{6} + 16 T^{7} + T^{8} )^{2} \)
$47$ \( ( 15982596734976 + 46512123904 T^{2} + 40460800 T^{4} + 11360 T^{6} + T^{8} )^{2} \)
$53$ \( \)\(18\!\cdots\!00\)\( + \)\(80\!\cdots\!16\)\( T^{4} + 1104026742079616 T^{8} + 58906368 T^{12} + T^{16} \)
$59$ \( \)\(11\!\cdots\!00\)\( + \)\(58\!\cdots\!96\)\( T^{4} + 4866440880979968 T^{8} + 140066816 T^{12} + T^{16} \)
$61$ \( ( 51506974970896 - 743864697728 T + 5371453952 T^{2} + 899677248 T^{3} + 81451272 T^{4} - 52960 T^{5} + 128 T^{6} + 16 T^{7} + T^{8} )^{2} \)
$67$ \( ( 7615833702400 + 1119061278720 T + 82216747008 T^{2} + 1261568000 T^{3} + 10536960 T^{4} + 118784 T^{5} + 8192 T^{6} + 128 T^{7} + T^{8} )^{2} \)
$71$ \( ( 5121887017369600 - 2814020681728 T^{2} + 525148160 T^{4} - 39552 T^{6} + T^{8} )^{2} \)
$73$ \( ( 534697177190400 + 469936807936 T^{2} + 148044928 T^{4} + 20096 T^{6} + T^{8} )^{2} \)
$79$ \( ( 56939504400 + 93640045696 T^{2} + 65683656 T^{4} + 14624 T^{6} + T^{8} )^{2} \)
$83$ \( \)\(60\!\cdots\!36\)\( + \)\(18\!\cdots\!20\)\( T^{4} + 269780128958810112 T^{8} + 1008676864 T^{12} + T^{16} \)
$89$ \( ( 236165588582400 + 800601505792 T^{2} + 558270464 T^{4} + 45760 T^{6} + T^{8} )^{2} \)
$97$ \( ( 29866240 - 77824 T - 12384 T^{2} + T^{4} )^{4} \)
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