Properties

Label 144.2.l.a
Level 144
Weight 2
Character orbit 144.l
Analytic conductor 1.150
Analytic rank 0
Dimension 16
CM no
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.14984578911\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 4 x^{14} + 6 x^{12} - 12 x^{10} + 33 x^{8} - 48 x^{6} + 96 x^{4} - 256 x^{2} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{10} \)
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.

\(f(q)\) \(=\) \( q + \beta_{11} q^{2} -\beta_{7} q^{4} + \beta_{6} q^{5} + ( -\beta_{4} - \beta_{9} ) q^{7} + ( -\beta_{2} - \beta_{8} + \beta_{12} - \beta_{13} ) q^{8} +O(q^{10})\) \( q + \beta_{11} q^{2} -\beta_{7} q^{4} + \beta_{6} q^{5} + ( -\beta_{4} - \beta_{9} ) q^{7} + ( -\beta_{2} - \beta_{8} + \beta_{12} - \beta_{13} ) q^{8} + ( \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{10} ) q^{10} + ( 2 \beta_{2} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{11} + ( -1 + \beta_{1} - \beta_{3} + \beta_{5} + \beta_{7} + \beta_{9} ) q^{13} + ( -\beta_{6} + \beta_{8} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{14} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{16} + ( -\beta_{2} + \beta_{6} + \beta_{8} - 2 \beta_{11} + \beta_{12} + 2 \beta_{15} ) q^{17} + ( 2 + 2 \beta_{5} + 2 \beta_{7} - 2 \beta_{10} ) q^{19} + ( -\beta_{8} - \beta_{11} + \beta_{12} - 2 \beta_{14} + \beta_{15} ) q^{20} + ( -3 + \beta_{1} + \beta_{5} + 2 \beta_{9} - \beta_{10} ) q^{22} + ( \beta_{2} - \beta_{6} - \beta_{8} - 2 \beta_{12} + \beta_{13} - \beta_{15} ) q^{23} + ( \beta_{3} + \beta_{4} - \beta_{5} - \beta_{9} + 3 \beta_{10} ) q^{25} + ( -\beta_{6} + \beta_{8} + \beta_{11} - \beta_{12} + \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{26} + ( -2 - 2 \beta_{1} - 2 \beta_{4} - 4 \beta_{5} - \beta_{7} - \beta_{9} ) q^{28} + ( -\beta_{2} + \beta_{8} - 2 \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{29} + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{7} - \beta_{10} ) q^{31} + ( -2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{32} + ( 3 + 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{9} + \beta_{10} ) q^{34} + ( -2 \beta_{2} - 2 \beta_{6} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{35} + ( -1 - 2 \beta_{1} - 2 \beta_{3} - \beta_{5} + 2 \beta_{9} - 2 \beta_{10} ) q^{37} + ( 2 \beta_{11} - 4 \beta_{12} + 2 \beta_{13} - 2 \beta_{15} ) q^{38} + ( 5 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{9} - 2 \beta_{10} ) q^{40} + ( \beta_{2} + \beta_{6} - \beta_{8} + 2 \beta_{11} + \beta_{14} + 2 \beta_{15} ) q^{41} + ( -2 - 2 \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - \beta_{9} + \beta_{10} ) q^{43} + ( -\beta_{2} + \beta_{6} - \beta_{8} - 2 \beta_{11} - \beta_{12} + \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{44} + ( 1 + \beta_{1} - 2 \beta_{3} - \beta_{5} + 2 \beta_{7} + 2 \beta_{9} - \beta_{10} ) q^{46} + ( \beta_{2} - \beta_{6} + \beta_{8} + 2 \beta_{11} + \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{47} + ( 3 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{7} ) q^{49} + ( \beta_{2} + 2 \beta_{8} - 2 \beta_{11} + 2 \beta_{12} - 4 \beta_{14} + 2 \beta_{15} ) q^{50} + ( -2 \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{9} + 2 \beta_{10} ) q^{52} + ( -2 \beta_{2} - \beta_{6} + 2 \beta_{11} + \beta_{12} - \beta_{14} - 2 \beta_{15} ) q^{53} + ( -4 + 4 \beta_{1} + 2 \beta_{3} - 4 \beta_{7} + 2 \beta_{10} ) q^{55} + ( 3 \beta_{2} + \beta_{6} + \beta_{8} - 2 \beta_{11} + 3 \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{56} + ( 2 - \beta_{1} + \beta_{3} - \beta_{4} - 5 \beta_{5} + \beta_{7} - 2 \beta_{9} + \beta_{10} ) q^{58} + ( -2 \beta_{8} + 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} ) q^{59} + ( -3 + 2 \beta_{1} + 3 \beta_{5} + 2 \beta_{9} ) q^{61} + ( 2 \beta_{2} + 3 \beta_{6} - \beta_{8} - \beta_{11} + 3 \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{62} + ( -2 + 2 \beta_{1} + 6 \beta_{5} + 2 \beta_{7} - 2 \beta_{10} ) q^{64} + ( -3 \beta_{2} - \beta_{6} - \beta_{8} + 2 \beta_{11} - 2 \beta_{13} ) q^{65} + ( 2 \beta_{1} - \beta_{3} + 3 \beta_{4} + \beta_{9} - \beta_{10} ) q^{67} + ( -2 \beta_{2} + \beta_{8} + \beta_{11} + \beta_{12} + 2 \beta_{13} - 4 \beta_{14} - \beta_{15} ) q^{68} + ( -5 - \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{7} - 2 \beta_{9} - \beta_{10} ) q^{70} + ( \beta_{2} + \beta_{6} + \beta_{8} - 2 \beta_{11} + 4 \beta_{12} + \beta_{13} + \beta_{15} ) q^{71} + ( -2 \beta_{1} - 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{10} ) q^{73} + ( \beta_{2} + 2 \beta_{6} - 2 \beta_{8} - \beta_{11} - 2 \beta_{12} - 4 \beta_{13} + 2 \beta_{14} ) q^{74} + ( 4 \beta_{1} - 4 \beta_{5} - 2 \beta_{7} ) q^{76} + ( 2 \beta_{2} - 2 \beta_{8} - 2 \beta_{13} - 2 \beta_{15} ) q^{77} + ( -2 \beta_{1} - \beta_{3} - 2 \beta_{7} - 2 \beta_{9} + \beta_{10} ) q^{79} + ( \beta_{2} + \beta_{6} - \beta_{8} + 4 \beta_{11} - 5 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{80} + ( 4 + \beta_{1} + 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} ) q^{82} + ( -2 \beta_{2} + 2 \beta_{6} + 3 \beta_{12} - \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{83} + ( -2 - \beta_{1} - \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{10} ) q^{85} + ( 2 \beta_{8} - 4 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 4 \beta_{14} ) q^{86} + ( 2 + 2 \beta_{1} + 2 \beta_{4} + 2 \beta_{7} + 4 \beta_{10} ) q^{88} + ( 2 \beta_{2} - 2 \beta_{6} + 2 \beta_{8} - \beta_{14} ) q^{89} + ( 2 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{9} + 2 \beta_{10} ) q^{91} + ( 3 \beta_{2} - \beta_{6} + \beta_{8} + 4 \beta_{11} - 3 \beta_{12} + \beta_{13} + 5 \beta_{14} + \beta_{15} ) q^{92} + ( 5 - \beta_{1} - 2 \beta_{4} + \beta_{5} - 2 \beta_{7} - 3 \beta_{10} ) q^{94} + ( -4 \beta_{2} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{11} + 2 \beta_{13} + 6 \beta_{14} ) q^{95} + ( 2 + \beta_{3} - \beta_{4} - 3 \beta_{9} - \beta_{10} ) q^{97} + ( -2 \beta_{2} - 4 \beta_{8} + \beta_{11} - 4 \beta_{14} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 8q^{10} - 16q^{16} + 16q^{19} - 40q^{22} - 24q^{28} + 24q^{34} + 72q^{40} - 32q^{43} + 40q^{46} + 16q^{49} + 24q^{52} - 64q^{55} + 24q^{58} - 32q^{61} - 48q^{64} - 16q^{67} - 72q^{70} + 80q^{82} - 32q^{85} + 48q^{88} + 48q^{91} + 72q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 4 x^{14} + 6 x^{12} - 12 x^{10} + 33 x^{8} - 48 x^{6} + 96 x^{4} - 256 x^{2} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{14} + 21 \nu^{12} - 18 \nu^{10} + 18 \nu^{8} - 189 \nu^{6} + 81 \nu^{4} + 72 \nu^{2} + 1072 \)\()/480\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{15} - 6 \nu^{11} - 12 \nu^{9} - 81 \nu^{7} - 276 \nu^{5} + 336 \nu^{3} + 64 \nu \)\()/1536\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{14} + 2 \nu^{12} - 6 \nu^{10} + 16 \nu^{8} + 27 \nu^{6} + 62 \nu^{4} - 256 \nu^{2} - 96 \)\()/320\)
\(\beta_{4}\)\(=\)\((\)\( -29 \nu^{14} + 24 \nu^{12} + 18 \nu^{10} + 372 \nu^{8} - 141 \nu^{6} - 876 \nu^{4} - 1872 \nu^{2} + 1088 \)\()/1920\)
\(\beta_{5}\)\(=\)\((\)\( 43 \nu^{14} - 108 \nu^{12} + 114 \nu^{10} - 324 \nu^{8} + 747 \nu^{6} - 528 \nu^{4} + 3024 \nu^{2} - 6016 \)\()/1920\)
\(\beta_{6}\)\(=\)\((\)\( 11 \nu^{15} - 66 \nu^{13} + 138 \nu^{11} - 168 \nu^{9} + 339 \nu^{7} - 486 \nu^{5} + 1008 \nu^{3} - 1952 \nu \)\()/1920\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{14} - \nu^{12} + 2 \nu^{10} - 10 \nu^{8} + 13 \nu^{6} - 13 \nu^{4} + 56 \nu^{2} - 80 \)\()/32\)
\(\beta_{8}\)\(=\)\((\)\( 27 \nu^{15} - 72 \nu^{13} + 66 \nu^{11} - 236 \nu^{9} + 1003 \nu^{7} - 812 \nu^{5} + 1936 \nu^{3} - 2624 \nu \)\()/2560\)
\(\beta_{9}\)\(=\)\((\)\( 89 \nu^{14} - 204 \nu^{12} + 102 \nu^{10} - 732 \nu^{8} + 1401 \nu^{6} - 984 \nu^{4} + 4752 \nu^{2} - 9728 \)\()/1920\)
\(\beta_{10}\)\(=\)\((\)\( 11 \nu^{14} - 24 \nu^{12} + 18 \nu^{10} - 108 \nu^{8} + 219 \nu^{6} - 156 \nu^{4} + 864 \nu^{2} - 1472 \)\()/192\)
\(\beta_{11}\)\(=\)\((\)\( -39 \nu^{15} + 64 \nu^{13} - 42 \nu^{11} + 172 \nu^{9} - 471 \nu^{7} + 564 \nu^{5} - 1552 \nu^{3} + 4288 \nu \)\()/2560\)
\(\beta_{12}\)\(=\)\((\)\( -\nu^{15} + \nu^{13} - 2 \nu^{11} + 10 \nu^{9} - 13 \nu^{7} + 13 \nu^{5} - 56 \nu^{3} + 80 \nu \)\()/64\)
\(\beta_{13}\)\(=\)\((\)\( -71 \nu^{15} + 216 \nu^{13} - 138 \nu^{11} + 828 \nu^{9} - 2199 \nu^{7} + 1356 \nu^{5} - 5328 \nu^{3} + 14912 \nu \)\()/3840\)
\(\beta_{14}\)\(=\)\((\)\( 89 \nu^{15} - 204 \nu^{13} + 102 \nu^{11} - 732 \nu^{9} + 1401 \nu^{7} - 984 \nu^{5} + 4752 \nu^{3} - 9728 \nu \)\()/3840\)
\(\beta_{15}\)\(=\)\((\)\( 149 \nu^{15} - 264 \nu^{13} + 222 \nu^{11} - 1332 \nu^{9} + 2181 \nu^{7} - 1764 \nu^{5} + 11952 \nu^{3} - 18368 \nu \)\()/3840\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{14} + \beta_{13} + \beta_{11} + \beta_{8}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{10} - \beta_{9} - \beta_{7} + \beta_{5} + \beta_{1} + 1\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{15} - \beta_{14} + \beta_{13} + 2 \beta_{12} + \beta_{11} + \beta_{8}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-2 \beta_{9} - \beta_{7} + 3 \beta_{5} - 3 \beta_{4} + \beta_{3} - \beta_{1} + 1\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(2 \beta_{15} + \beta_{14} + \beta_{13} + 2 \beta_{12} + 3 \beta_{11} - \beta_{8} - 8 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(5 \beta_{10} - 5 \beta_{9} - \beta_{7} - 3 \beta_{5} + 4 \beta_{3} - 3 \beta_{1} + 9\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-3 \beta_{14} - \beta_{13} + \beta_{11} + 7 \beta_{8} - 2 \beta_{6} - 6 \beta_{2}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(4 \beta_{10} - 6 \beta_{9} - 9 \beta_{7} + 15 \beta_{5} + 5 \beta_{4} + 9 \beta_{3} + 7 \beta_{1} + 9\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(8 \beta_{15} - 13 \beta_{14} + 7 \beta_{13} + 20 \beta_{12} - 21 \beta_{11} + 9 \beta_{8} + 2 \beta_{6} - 10 \beta_{2}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(-15 \beta_{10} - 17 \beta_{9} + 19 \beta_{7} + 45 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} - 11 \beta_{1} + 13\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(6 \beta_{15} - 29 \beta_{14} + 5 \beta_{13} - 10 \beta_{12} - 15 \beta_{11} - 3 \beta_{8} + 28 \beta_{6} - 28 \beta_{2}\)\()/2\)
\(\nu^{12}\)\(=\)\((\)\(24 \beta_{10} - 42 \beta_{9} + 23 \beta_{7} - 13 \beta_{5} + 5 \beta_{4} + 25 \beta_{3} + 7 \beta_{1} - 23\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(6 \beta_{15} - 91 \beta_{14} - 19 \beta_{13} - 66 \beta_{12} - 37 \beta_{11} + 3 \beta_{8} - 20 \beta_{6} - 20 \beta_{2}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(-27 \beta_{10} + 27 \beta_{9} + 15 \beta_{7} + 69 \beta_{5} + 24 \beta_{4} + 84 \beta_{3} + 69 \beta_{1} + 41\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(-12 \beta_{15} + 25 \beta_{14} + 91 \beta_{13} - 60 \beta_{12} - 167 \beta_{11} + 19 \beta_{8} - 30 \beta_{6} - 90 \beta_{2}\)\()/2\)

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(\beta_{5}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1
0.944649 1.05244i
−1.36166 + 0.381939i
−1.40927 + 0.118126i
0.517174 + 1.31626i
−0.517174 1.31626i
1.40927 0.118126i
1.36166 0.381939i
−0.944649 + 1.05244i
0.944649 + 1.05244i
−1.36166 0.381939i
−1.40927 0.118126i
0.517174 1.31626i
−0.517174 + 1.31626i
1.40927 + 0.118126i
1.36166 + 0.381939i
−0.944649 1.05244i
−1.32068 + 0.505776i 0 1.48838 1.33594i −2.10489 2.10489i 0 −4.40731 −1.28999 + 2.51713i 0 3.84448 + 1.71528i
35.2 −1.12697 + 0.854358i 0 0.540143 1.92568i 0.763878 + 0.763878i 0 1.33620 1.03649 + 2.63167i 0 −1.51350 0.208245i
35.3 −0.957325 1.04093i 0 −0.167056 + 1.99301i 0.236253 + 0.236253i 0 3.27830 2.23450 1.73407i 0 0.0197510 0.472092i
35.4 −0.263185 + 1.38951i 0 −1.86147 0.731395i 2.63251 + 2.63251i 0 −0.207188 1.50619 2.39403i 0 −4.35074 + 2.96506i
35.5 0.263185 1.38951i 0 −1.86147 0.731395i −2.63251 2.63251i 0 −0.207188 −1.50619 + 2.39403i 0 −4.35074 + 2.96506i
35.6 0.957325 + 1.04093i 0 −0.167056 + 1.99301i −0.236253 0.236253i 0 3.27830 −2.23450 + 1.73407i 0 0.0197510 0.472092i
35.7 1.12697 0.854358i 0 0.540143 1.92568i −0.763878 0.763878i 0 1.33620 −1.03649 2.63167i 0 −1.51350 0.208245i
35.8 1.32068 0.505776i 0 1.48838 1.33594i 2.10489 + 2.10489i 0 −4.40731 1.28999 2.51713i 0 3.84448 + 1.71528i
107.1 −1.32068 0.505776i 0 1.48838 + 1.33594i −2.10489 + 2.10489i 0 −4.40731 −1.28999 2.51713i 0 3.84448 1.71528i
107.2 −1.12697 0.854358i 0 0.540143 + 1.92568i 0.763878 0.763878i 0 1.33620 1.03649 2.63167i 0 −1.51350 + 0.208245i
107.3 −0.957325 + 1.04093i 0 −0.167056 1.99301i 0.236253 0.236253i 0 3.27830 2.23450 + 1.73407i 0 0.0197510 + 0.472092i
107.4 −0.263185 1.38951i 0 −1.86147 + 0.731395i 2.63251 2.63251i 0 −0.207188 1.50619 + 2.39403i 0 −4.35074 2.96506i
107.5 0.263185 + 1.38951i 0 −1.86147 + 0.731395i −2.63251 + 2.63251i 0 −0.207188 −1.50619 2.39403i 0 −4.35074 2.96506i
107.6 0.957325 1.04093i 0 −0.167056 1.99301i −0.236253 + 0.236253i 0 3.27830 −2.23450 1.73407i 0 0.0197510 + 0.472092i
107.7 1.12697 + 0.854358i 0 0.540143 + 1.92568i −0.763878 + 0.763878i 0 1.33620 −1.03649 + 2.63167i 0 −1.51350 + 0.208245i
107.8 1.32068 + 0.505776i 0 1.48838 + 1.33594i 2.10489 2.10489i 0 −4.40731 1.28999 + 2.51713i 0 3.84448 1.71528i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.8
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 144.2.l.a 16
3.b odd 2 1 inner 144.2.l.a 16
4.b odd 2 1 576.2.l.a 16
8.b even 2 1 1152.2.l.a 16
8.d odd 2 1 1152.2.l.b 16
12.b even 2 1 576.2.l.a 16
16.e even 4 1 576.2.l.a 16
16.e even 4 1 1152.2.l.b 16
16.f odd 4 1 inner 144.2.l.a 16
16.f odd 4 1 1152.2.l.a 16
24.f even 2 1 1152.2.l.b 16
24.h odd 2 1 1152.2.l.a 16
48.i odd 4 1 576.2.l.a 16
48.i odd 4 1 1152.2.l.b 16
48.k even 4 1 inner 144.2.l.a 16
48.k even 4 1 1152.2.l.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.l.a 16 1.a even 1 1 trivial
144.2.l.a 16 3.b odd 2 1 inner
144.2.l.a 16 16.f odd 4 1 inner
144.2.l.a 16 48.k even 4 1 inner
576.2.l.a 16 4.b odd 2 1
576.2.l.a 16 12.b even 2 1
576.2.l.a 16 16.e even 4 1
576.2.l.a 16 48.i odd 4 1
1152.2.l.a 16 8.b even 2 1
1152.2.l.a 16 16.f odd 4 1
1152.2.l.a 16 24.h odd 2 1
1152.2.l.a 16 48.k even 4 1
1152.2.l.b 16 8.d odd 2 1
1152.2.l.b 16 16.e even 4 1
1152.2.l.b 16 24.f even 2 1
1152.2.l.b 16 48.i odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(144, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T^{4} + 8 T^{6} + 4 T^{8} + 32 T^{10} + 64 T^{12} + 256 T^{16} \)
$3$ 1
$5$ \( 1 - 8 T^{4} - 804 T^{8} + 2376 T^{12} + 502406 T^{16} + 1485000 T^{20} - 314062500 T^{24} - 1953125000 T^{28} + 152587890625 T^{32} \)
$7$ \( ( 1 + 12 T^{2} + 16 T^{3} + 74 T^{4} + 112 T^{5} + 588 T^{6} + 2401 T^{8} )^{4} \)
$11$ \( 1 - 184 T^{4} + 8796 T^{8} + 2616696 T^{12} - 490804602 T^{16} + 38311046136 T^{20} + 1885500717276 T^{24} - 577470821316664 T^{28} + 45949729863572161 T^{32} \)
$13$ \( ( 1 + 64 T^{3} - 36 T^{4} - 704 T^{5} + 2048 T^{6} + 384 T^{7} - 43930 T^{8} + 4992 T^{9} + 346112 T^{10} - 1546688 T^{11} - 1028196 T^{12} + 23762752 T^{13} + 815730721 T^{16} )^{2} \)
$17$ \( ( 1 - 64 T^{2} + 1956 T^{4} - 40128 T^{6} + 697542 T^{8} - 11596992 T^{10} + 163367076 T^{12} - 1544804416 T^{14} + 6975757441 T^{16} )^{2} \)
$19$ \( ( 1 - 8 T + 32 T^{2} - 184 T^{3} + 388 T^{4} + 1992 T^{5} - 11424 T^{6} + 82616 T^{7} - 538074 T^{8} + 1569704 T^{9} - 4124064 T^{10} + 13663128 T^{11} + 50564548 T^{12} - 455602216 T^{13} + 1505468192 T^{14} - 7150973912 T^{15} + 16983563041 T^{16} )^{2} \)
$23$ \( ( 1 - 88 T^{2} + 3004 T^{4} - 46440 T^{6} + 546118 T^{8} - 24566760 T^{10} + 840642364 T^{12} - 13027158232 T^{14} + 78310985281 T^{16} )^{2} \)
$29$ \( 1 - 1672 T^{4} + 1133916 T^{8} - 595679544 T^{12} + 410245939974 T^{16} - 421312823559864 T^{20} + 567237411599085276 T^{24} - \)\(59\!\cdots\!52\)\( T^{28} + \)\(25\!\cdots\!21\)\( T^{32} \)
$31$ \( ( 1 - 56 T^{2} + 3492 T^{4} - 137064 T^{6} + 4924550 T^{8} - 131718504 T^{10} + 3224935332 T^{12} - 49700206136 T^{14} + 852891037441 T^{16} )^{2} \)
$37$ \( ( 1 - 512 T^{3} + 700 T^{4} + 9728 T^{5} + 131072 T^{6} - 437248 T^{7} - 3354522 T^{8} - 16178176 T^{9} + 179437568 T^{10} + 492752384 T^{11} + 1311912700 T^{12} - 35504105984 T^{13} + 3512479453921 T^{16} )^{2} \)
$41$ \( ( 1 + 160 T^{2} + 12772 T^{4} + 745952 T^{6} + 34631942 T^{8} + 1253945312 T^{10} + 36090619492 T^{12} + 760016678560 T^{14} + 7984925229121 T^{16} )^{2} \)
$43$ \( ( 1 + 16 T + 128 T^{2} + 1200 T^{3} + 6980 T^{4} - 80 T^{5} - 174720 T^{6} - 2568048 T^{7} - 25827098 T^{8} - 110426064 T^{9} - 323057280 T^{10} - 6360560 T^{11} + 23863230980 T^{12} + 176410131600 T^{13} + 809134470272 T^{14} + 4349097777712 T^{15} + 11688200277601 T^{16} )^{2} \)
$47$ \( ( 1 + 216 T^{2} + 24572 T^{4} + 1856872 T^{6} + 101744902 T^{8} + 4101830248 T^{10} + 119903521532 T^{12} + 2328310511064 T^{14} + 23811286661761 T^{16} )^{2} \)
$53$ \( 1 + 2936 T^{4} + 12654300 T^{8} + 22427423688 T^{12} + 91065624180102 T^{16} + 176963160489113928 T^{20} + \)\(78\!\cdots\!00\)\( T^{24} + \)\(14\!\cdots\!76\)\( T^{28} + \)\(38\!\cdots\!21\)\( T^{32} \)
$59$ \( 1 - 2360 T^{4} + 15662172 T^{8} - 90024948744 T^{12} + 175698504275846 T^{16} - 1090864802937544584 T^{20} + \)\(22\!\cdots\!12\)\( T^{24} - \)\(41\!\cdots\!60\)\( T^{28} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( ( 1 + 16 T + 128 T^{2} + 1392 T^{3} + 16892 T^{4} + 124816 T^{5} + 803712 T^{6} + 7357296 T^{7} + 67536550 T^{8} + 448795056 T^{9} + 2990612352 T^{10} + 28330860496 T^{11} + 233883946172 T^{12} + 1175678050992 T^{13} + 6594607918208 T^{14} + 50283885376336 T^{15} + 191707312997281 T^{16} )^{2} \)
$67$ \( ( 1 + 8 T + 32 T^{2} + 1240 T^{3} + 5540 T^{4} - 65000 T^{5} + 71520 T^{6} - 2339576 T^{7} - 86245658 T^{8} - 156751592 T^{9} + 321053280 T^{10} - 19549595000 T^{11} + 111637210340 T^{12} + 1674155132680 T^{13} + 2894668229408 T^{14} + 48485692842584 T^{15} + 406067677556641 T^{16} )^{2} \)
$71$ \( ( 1 - 376 T^{2} + 67068 T^{4} - 7636296 T^{6} + 626574150 T^{8} - 38494568136 T^{10} + 1704310621308 T^{12} - 48165706754296 T^{14} + 645753531245761 T^{16} )^{2} \)
$73$ \( ( 1 - 408 T^{2} + 80156 T^{4} - 9970856 T^{6} + 862104454 T^{8} - 53134691624 T^{10} + 2276289405596 T^{12} - 61744364325912 T^{14} + 806460091894081 T^{16} )^{2} \)
$79$ \( ( 1 - 408 T^{2} + 84004 T^{4} - 11101576 T^{6} + 1033756678 T^{8} - 69284935816 T^{10} + 3271962604324 T^{12} - 99179681852568 T^{14} + 1517108809906561 T^{16} )^{2} \)
$83$ \( 1 - 5432 T^{4} + 103273180 T^{8} - 984945914120 T^{12} + 5200099763862790 T^{16} - 46743879359945392520 T^{20} + \)\(23\!\cdots\!80\)\( T^{24} - \)\(58\!\cdots\!52\)\( T^{28} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( ( 1 + 512 T^{2} + 123588 T^{4} + 18716160 T^{6} + 1970104134 T^{8} + 148250703360 T^{10} + 7754188080708 T^{12} + 254454420972032 T^{14} + 3936588805702081 T^{16} )^{2} \)
$97$ \( ( 1 + 316 T^{2} - 256 T^{3} + 42310 T^{4} - 24832 T^{5} + 2973244 T^{6} + 88529281 T^{8} )^{4} \)
show more
show less