Properties

Label 576.2.l.a
Level 576
Weight 2
Character orbit 576.l
Analytic conductor 4.599
Analytic rank 0
Dimension 16
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 576.l (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(4.59938315643\)
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: \( 2^{18} \)
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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{5} -\beta_{9} q^{7} +O(q^{10})\) \( q -\beta_{3} q^{5} -\beta_{9} q^{7} + ( \beta_{2} + \beta_{14} ) q^{11} + \beta_{4} q^{13} + ( \beta_{7} - \beta_{8} + \beta_{10} + \beta_{14} ) q^{17} + ( -1 + \beta_{5} + \beta_{6} - \beta_{12} ) q^{19} + ( -\beta_{2} - \beta_{3} - \beta_{8} + \beta_{10} + \beta_{14} + \beta_{15} ) q^{23} + ( -\beta_{1} + \beta_{5} + \beta_{6} - 2 \beta_{11} ) q^{25} + ( \beta_{2} - \beta_{7} - \beta_{10} - \beta_{13} - \beta_{15} ) q^{29} + ( -\beta_{1} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{11} + \beta_{12} ) q^{31} + ( -\beta_{3} - 2 \beta_{7} + \beta_{8} + \beta_{10} + \beta_{13} - \beta_{15} ) q^{35} + ( -\beta_{5} - 2 \beta_{9} + 2 \beta_{11} - \beta_{12} ) q^{37} + ( \beta_{8} + 2 \beta_{13} + \beta_{14} ) q^{41} + ( 2 + 2 \beta_{1} + 2 \beta_{6} + \beta_{9} + \beta_{11} ) q^{43} + ( \beta_{2} - \beta_{3} - \beta_{8} - 3 \beta_{10} - \beta_{13} - \beta_{14} ) q^{47} + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} + \beta_{12} ) q^{49} + ( -\beta_{3} - \beta_{7} + \beta_{10} - \beta_{13} + \beta_{15} ) q^{53} + ( 4 - 2 \beta_{1} - 2 \beta_{5} - 2 \beta_{9} ) q^{55} + ( 2 \beta_{7} + 2 \beta_{10} - \beta_{13} - \beta_{15} ) q^{59} + ( -2 + \beta_{1} + \beta_{4} - 2 \beta_{6} ) q^{61} + ( 2 \beta_{7} - \beta_{8} + \beta_{10} + \beta_{14} + 2 \beta_{15} ) q^{65} + ( 1 - \beta_{5} - \beta_{6} + \beta_{9} - \beta_{11} - \beta_{12} ) q^{67} + ( 4 \beta_{7} + \beta_{15} ) q^{71} + ( -\beta_{1} - \beta_{4} + \beta_{5} - \beta_{12} ) q^{73} + ( -2 \beta_{7} - 2 \beta_{10} - 2 \beta_{14} ) q^{77} + ( \beta_{1} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{11} + \beta_{12} ) q^{79} + ( 3 \beta_{3} + 2 \beta_{7} + \beta_{8} - 3 \beta_{10} + \beta_{13} - \beta_{15} ) q^{83} + ( -2 - \beta_{5} + 2 \beta_{6} ) q^{85} + ( -2 \beta_{2} + 2 \beta_{3} + \beta_{10} ) q^{89} + ( -3 - \beta_{1} - \beta_{4} - 3 \beta_{6} + 2 \beta_{9} + 2 \beta_{11} ) q^{91} + ( -3 \beta_{2} + 3 \beta_{3} - \beta_{8} + 5 \beta_{10} - \beta_{14} ) q^{95} + ( -\beta_{4} + 2 \beta_{9} + \beta_{12} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 16q^{19} + 32q^{43} + 16q^{49} + 64q^{55} - 32q^{61} + 16q^{67} - 32q^{85} - 48q^{91} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 17 \nu^{14} - 72 \nu^{12} + 6 \nu^{10} - 36 \nu^{8} + 513 \nu^{6} - 1332 \nu^{4} + 3696 \nu^{2} - 4544 \)\()/1920\)
\(\beta_{2}\)\(=\)\((\)\( 11 \nu^{15} - 66 \nu^{13} + 138 \nu^{11} - 168 \nu^{9} + 339 \nu^{7} - 486 \nu^{5} + 1008 \nu^{3} - 1952 \nu \)\()/1920\)
\(\beta_{3}\)\(=\)\((\)\( 19 \nu^{15} - 54 \nu^{13} + 42 \nu^{11} - 192 \nu^{9} + 651 \nu^{7} - 954 \nu^{5} + 1872 \nu^{3} - 1888 \nu \)\()/1920\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{14} - 9 \nu^{12} - 30 \nu^{8} + 78 \nu^{6} + 15 \nu^{4} + 192 \nu^{2} - 560 \)\()/96\)
\(\beta_{5}\)\(=\)\((\)\( 43 \nu^{14} - 48 \nu^{12} - 126 \nu^{10} + 36 \nu^{8} + 27 \nu^{6} + 492 \nu^{4} + 144 \nu^{2} - 1216 \)\()/1920\)
\(\beta_{6}\)\(=\)\((\)\( 43 \nu^{14} - 108 \nu^{12} + 114 \nu^{10} - 324 \nu^{8} + 747 \nu^{6} - 528 \nu^{4} + 3024 \nu^{2} - 6016 \)\()/1920\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{15} + \nu^{13} - 2 \nu^{11} + 10 \nu^{9} - 13 \nu^{7} + 13 \nu^{5} - 56 \nu^{3} + 80 \nu \)\()/64\)
\(\beta_{8}\)\(=\)\((\)\( 61 \nu^{15} - 96 \nu^{13} + 78 \nu^{11} - 228 \nu^{9} + 909 \nu^{7} - 156 \nu^{5} + 5328 \nu^{3} - 2752 \nu \)\()/3840\)
\(\beta_{9}\)\(=\)\((\)\( -\nu^{14} + 3 \nu^{12} - 2 \nu^{10} + 6 \nu^{8} - 21 \nu^{6} + 31 \nu^{4} - 48 \nu^{2} + 144 \)\()/32\)
\(\beta_{10}\)\(=\)\((\)\( -89 \nu^{15} + 204 \nu^{13} - 102 \nu^{11} + 732 \nu^{9} - 1401 \nu^{7} + 984 \nu^{5} - 4752 \nu^{3} + 9728 \nu \)\()/3840\)
\(\beta_{11}\)\(=\)\((\)\( 32 \nu^{14} - 57 \nu^{12} + 36 \nu^{10} - 246 \nu^{8} + 588 \nu^{6} - 297 \nu^{4} + 1776 \nu^{2} - 3824 \)\()/480\)
\(\beta_{12}\)\(=\)\((\)\( -17 \nu^{14} + 30 \nu^{12} - 30 \nu^{10} + 168 \nu^{8} - 201 \nu^{6} + 234 \nu^{4} - 1296 \nu^{2} + 1760 \)\()/192\)
\(\beta_{13}\)\(=\)\((\)\( 199 \nu^{15} - 504 \nu^{13} + 522 \nu^{11} - 2172 \nu^{9} + 3351 \nu^{7} - 3564 \nu^{5} + 17232 \nu^{3} - 33088 \nu \)\()/3840\)
\(\beta_{14}\)\(=\)\((\)\( -7 \nu^{15} + 18 \nu^{13} - 18 \nu^{11} + 56 \nu^{9} - 143 \nu^{7} + 134 \nu^{5} - 528 \nu^{3} + 1184 \nu \)\()/128\)
\(\beta_{15}\)\(=\)\((\)\( 17 \nu^{15} - 32 \nu^{13} + 38 \nu^{11} - 116 \nu^{9} + 225 \nu^{7} - 332 \nu^{5} + 1136 \nu^{3} - 1600 \nu \)\()/256\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{14} - \beta_{10} + \beta_{8} + \beta_{3} + \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{9} + \beta_{6} + \beta_{1} + 1\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{14} + 2 \beta_{13} + 3 \beta_{10} + 3 \beta_{8} + 2 \beta_{7} + \beta_{3} - \beta_{2}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{12} + 4 \beta_{9} + 8 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{1} + 2\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-4 \beta_{15} - \beta_{14} - 3 \beta_{10} + 7 \beta_{8} - 4 \beta_{7} - 5 \beta_{3} + 3 \beta_{2}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(2 \beta_{12} + 4 \beta_{11} + \beta_{9} - \beta_{6} - 2 \beta_{5} + \beta_{1} + 7\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-8 \beta_{15} - 7 \beta_{14} - 6 \beta_{13} - \beta_{10} + 7 \beta_{8} - 14 \beta_{7} + 9 \beta_{3} + 3 \beta_{2}\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(9 \beta_{12} + 4 \beta_{11} + 8 \beta_{9} + 32 \beta_{6} + 5 \beta_{5} - 7 \beta_{4} + 11 \beta_{1} + 2\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(8 \beta_{15} - 13 \beta_{14} - 12 \beta_{13} + 13 \beta_{10} + 7 \beta_{8} + 20 \beta_{7} - 9 \beta_{3} - 5 \beta_{2}\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(4 \beta_{12} + 5 \beta_{9} + 41 \beta_{6} - 8 \beta_{5} - 8 \beta_{4} - 7 \beta_{1} + 1\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-15 \beta_{14} - 14 \beta_{13} + 43 \beta_{10} + 11 \beta_{8} - 46 \beta_{7} - 31 \beta_{3} + 55 \beta_{2}\)\()/4\)
\(\nu^{12}\)\(=\)\((\)\(9 \beta_{12} + 72 \beta_{11} + 44 \beta_{9} - 56 \beta_{6} - 31 \beta_{5} - 39 \beta_{4} + 15 \beta_{1} - 94\)\()/4\)
\(\nu^{13}\)\(=\)\((\)\(-12 \beta_{15} - 49 \beta_{14} - 24 \beta_{13} + 133 \beta_{10} - \beta_{8} - 180 \beta_{7} - 5 \beta_{3} - 53 \beta_{2}\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(6 \beta_{12} + 36 \beta_{11} + 9 \beta_{9} + 27 \beta_{6} + 42 \beta_{5} - 48 \beta_{4} - 15 \beta_{1} - 13\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(120 \beta_{15} - 47 \beta_{14} - 150 \beta_{13} - 97 \beta_{10} - 41 \beta_{8} - 126 \beta_{7} - 191 \beta_{3} - 77 \beta_{2}\)\()/4\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(-1\) \(-1\) \(-\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
−0.517174 1.31626i
0.944649 1.05244i
1.36166 0.381939i
1.40927 0.118126i
−1.40927 + 0.118126i
−1.36166 + 0.381939i
−0.944649 + 1.05244i
0.517174 + 1.31626i
−0.517174 + 1.31626i
0.944649 + 1.05244i
1.36166 + 0.381939i
1.40927 + 0.118126i
−1.40927 0.118126i
−1.36166 0.381939i
−0.944649 1.05244i
0.517174 1.31626i
0 0 0 −2.63251 + 2.63251i 0 0.207188 0 0 0
143.2 0 0 0 −2.10489 + 2.10489i 0 4.40731 0 0 0
143.3 0 0 0 −0.763878 + 0.763878i 0 −1.33620 0 0 0
143.4 0 0 0 −0.236253 + 0.236253i 0 −3.27830 0 0 0
143.5 0 0 0 0.236253 0.236253i 0 −3.27830 0 0 0
143.6 0 0 0 0.763878 0.763878i 0 −1.33620 0 0 0
143.7 0 0 0 2.10489 2.10489i 0 4.40731 0 0 0
143.8 0 0 0 2.63251 2.63251i 0 0.207188 0 0 0
431.1 0 0 0 −2.63251 2.63251i 0 0.207188 0 0 0
431.2 0 0 0 −2.10489 2.10489i 0 4.40731 0 0 0
431.3 0 0 0 −0.763878 0.763878i 0 −1.33620 0 0 0
431.4 0 0 0 −0.236253 0.236253i 0 −3.27830 0 0 0
431.5 0 0 0 0.236253 + 0.236253i 0 −3.27830 0 0 0
431.6 0 0 0 0.763878 + 0.763878i 0 −1.33620 0 0 0
431.7 0 0 0 2.10489 + 2.10489i 0 4.40731 0 0 0
431.8 0 0 0 2.63251 + 2.63251i 0 0.207188 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 431.8
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 \(S_{2}^{\mathrm{new}}(576, [\chi])\).