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

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(1.14984578911\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{10} \)
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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
16.f Odd 1 yes
48.k Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(144, [\chi])\).