Properties

Label 4032.2.p.j
Level 4032
Weight 2
Character orbit 4032.p
Analytic conductor 32.196
Analytic rank 0
Dimension 12
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4032.p (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(32.195682095\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{14} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{2} q^{5} \) \( + \beta_{10} q^{7} \) \(+O(q^{10})\) \( q\) \( + \beta_{2} q^{5} \) \( + \beta_{10} q^{7} \) \( + \beta_{11} q^{11} \) \( -2 q^{13} \) \( + ( \beta_{3} - \beta_{5} ) q^{17} \) \( + ( \beta_{9} + \beta_{10} ) q^{19} \) \( + ( \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{23} \) \( + ( 3 + 2 \beta_{2} + \beta_{4} ) q^{25} \) \( + ( \beta_{1} - 2 \beta_{3} ) q^{29} \) \( + ( - \beta_{7} - 2 \beta_{11} ) q^{31} \) \( + ( \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{35} \) \( + ( \beta_{1} + \beta_{3} ) q^{37} \) \( + ( -2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{41} \) \( + ( - \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{43} \) \( + ( 2 \beta_{7} + 2 \beta_{11} ) q^{47} \) \( + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{49} \) \( + ( \beta_{1} + 2 \beta_{5} ) q^{53} \) \( + ( \beta_{7} - \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{55} \) \( + 2 \beta_{6} q^{59} \) \( + ( 6 + 2 \beta_{2} + 2 \beta_{4} ) q^{61} \) \( -2 \beta_{2} q^{65} \) \( + ( 2 \beta_{7} + 3 \beta_{9} - 3 \beta_{10} ) q^{67} \) \( + ( 3 \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{71} \) \( + ( 2 \beta_{3} + 2 \beta_{5} ) q^{73} \) \( + ( - \beta_{1} + \beta_{2} + 2 \beta_{5} ) q^{77} \) \( + ( -2 \beta_{6} + 2 \beta_{8} - \beta_{9} - \beta_{10} ) q^{79} \) \( + ( 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{83} \) \( + ( \beta_{1} + \beta_{3} + 4 \beta_{5} ) q^{85} \) \( + ( 2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{89} \) \( -2 \beta_{10} q^{91} \) \( + ( 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{95} \) \( + ( 2 \beta_{1} + 2 \beta_{5} ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut -\mathstrut 24q^{13} \) \(\mathstrut +\mathstrut 36q^{25} \) \(\mathstrut -\mathstrut 12q^{49} \) \(\mathstrut +\mathstrut 72q^{61} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(15\) \(x^{10}\mathstrut +\mathstrut \) \(90\) \(x^{8}\mathstrut -\mathstrut \) \(247\) \(x^{6}\mathstrut +\mathstrut \) \(270\) \(x^{4}\mathstrut +\mathstrut \) \(21\) \(x^{2}\mathstrut +\mathstrut \) \(49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 8 \nu^{10} - 332 \nu^{8} + 3624 \nu^{6} - 15496 \nu^{4} + 23800 \nu^{2} + 126 \)\()/2947\)
\(\beta_{2}\)\(=\)\((\)\( -27 \nu^{10} + 489 \nu^{8} - 2969 \nu^{6} + 6410 \nu^{4} + 928 \nu^{2} - 7056 \)\()/2947\)
\(\beta_{3}\)\(=\)\((\)\( -54 \nu^{10} + 557 \nu^{8} - 1728 \nu^{6} - 1073 \nu^{4} + 6908 \nu^{2} + 623 \)\()/2947\)
\(\beta_{4}\)\(=\)\((\)\( 58 \nu^{10} - 723 \nu^{8} + 3540 \nu^{6} - 6675 \nu^{4} - 902 \nu^{2} + 14175 \)\()/2947\)
\(\beta_{5}\)\(=\)\((\)\( -149 \nu^{10} + 2184 \nu^{8} - 12767 \nu^{6} + 34329 \nu^{4} - 38694 \nu^{2} + 1337 \)\()/2947\)
\(\beta_{6}\)\(=\)\((\)\( -346 \nu^{11} + 5939 \nu^{9} - 40121 \nu^{7} + 121218 \nu^{5} - 123779 \nu^{3} - 65863 \nu \)\()/20629\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{11} - 15 \nu^{9} + 97 \nu^{7} - 317 \nu^{5} + 501 \nu^{3} - 161 \nu \)\()/49\)
\(\beta_{8}\)\(=\)\((\)\( 501 \nu^{11} - 7109 \nu^{9} + 40029 \nu^{7} - 98967 \nu^{5} + 88545 \nu^{3} + 45465 \nu \)\()/20629\)
\(\beta_{9}\)\(=\)\((\)\( -151 \nu^{11} + 2267 \nu^{9} - 13673 \nu^{7} + 38203 \nu^{5} - 41697 \nu^{3} - 14903 \nu \)\()/5894\)
\(\beta_{10}\)\(=\)\((\)\( -1137 \nu^{11} + 16663 \nu^{9} - 94903 \nu^{7} + 232931 \nu^{5} - 169503 \nu^{3} - 135051 \nu \)\()/41258\)
\(\beta_{11}\)\(=\)\((\)\( -834 \nu^{11} + 13140 \nu^{9} - 83523 \nu^{7} + 252681 \nu^{5} - 346259 \nu^{3} + 118006 \nu \)\()/20629\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(10\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(10\) \(\beta_{10}\mathstrut -\mathstrut \) \(4\) \(\beta_{9}\mathstrut +\mathstrut \) \(8\) \(\beta_{8}\mathstrut -\mathstrut \) \(5\) \(\beta_{7}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(4\) \(\beta_{5}\mathstrut -\mathstrut \) \(8\) \(\beta_{4}\mathstrut -\mathstrut \) \(16\) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(30\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(8\) \(\beta_{11}\mathstrut +\mathstrut \) \(42\) \(\beta_{10}\mathstrut +\mathstrut \) \(8\) \(\beta_{9}\mathstrut +\mathstrut \) \(51\) \(\beta_{8}\mathstrut -\mathstrut \) \(18\) \(\beta_{7}\mathstrut -\mathstrut \) \(10\) \(\beta_{6}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(17\) \(\beta_{5}\mathstrut -\mathstrut \) \(9\) \(\beta_{4}\mathstrut -\mathstrut \) \(46\) \(\beta_{3}\mathstrut -\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(41\) \(\beta_{1}\mathstrut +\mathstrut \) \(22\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(10\) \(\beta_{11}\mathstrut +\mathstrut \) \(148\) \(\beta_{10}\mathstrut +\mathstrut \) \(160\) \(\beta_{9}\mathstrut +\mathstrut \) \(265\) \(\beta_{8}\mathstrut -\mathstrut \) \(15\) \(\beta_{7}\mathstrut -\mathstrut \) \(98\) \(\beta_{6}\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(208\) \(\beta_{5}\mathstrut +\mathstrut \) \(60\) \(\beta_{4}\mathstrut -\mathstrut \) \(444\) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(469\) \(\beta_{1}\mathstrut -\mathstrut \) \(290\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(168\) \(\beta_{11}\mathstrut +\mathstrut \) \(312\) \(\beta_{10}\mathstrut +\mathstrut \) \(1218\) \(\beta_{9}\mathstrut +\mathstrut \) \(1168\) \(\beta_{8}\mathstrut +\mathstrut \) \(405\) \(\beta_{7}\mathstrut -\mathstrut \) \(594\) \(\beta_{6}\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(978\) \(\beta_{5}\mathstrut +\mathstrut \) \(1098\) \(\beta_{4}\mathstrut -\mathstrut \) \(1792\) \(\beta_{3}\mathstrut +\mathstrut \) \(738\) \(\beta_{2}\mathstrut +\mathstrut \) \(2123\) \(\beta_{1}\mathstrut -\mathstrut \) \(3670\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(1956\) \(\beta_{11}\mathstrut -\mathstrut \) \(1050\) \(\beta_{10}\mathstrut +\mathstrut \) \(6968\) \(\beta_{9}\mathstrut +\mathstrut \) \(4135\) \(\beta_{8}\mathstrut +\mathstrut \) \(4364\) \(\beta_{7}\mathstrut -\mathstrut \) \(2574\) \(\beta_{6}\)\()/4\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
0.385124 + 0.500000i
0.385124 0.500000i
−0.385124 + 0.500000i
−0.385124 0.500000i
−2.23871 0.500000i
−2.23871 + 0.500000i
2.23871 0.500000i
2.23871 + 0.500000i
−1.75780 0.500000i
−1.75780 + 0.500000i
1.75780 0.500000i
1.75780 + 0.500000i
0 0 0 −2.76873 0 −0.480901 2.60168i 0 0 0
1567.2 0 0 0 −2.76873 0 −0.480901 + 2.60168i 0 0 0
1567.3 0 0 0 −2.76873 0 0.480901 2.60168i 0 0 0
1567.4 0 0 0 −2.76873 0 0.480901 + 2.60168i 0 0 0
1567.5 0 0 0 −1.11575 0 −1.37268 2.26180i 0 0 0
1567.6 0 0 0 −1.11575 0 −1.37268 + 2.26180i 0 0 0
1567.7 0 0 0 −1.11575 0 1.37268 2.26180i 0 0 0
1567.8 0 0 0 −1.11575 0 1.37268 + 2.26180i 0 0 0
1567.9 0 0 0 3.88448 0 −2.62383 0.339877i 0 0 0
1567.10 0 0 0 3.88448 0 −2.62383 + 0.339877i 0 0 0
1567.11 0 0 0 3.88448 0 2.62383 0.339877i 0 0 0
1567.12 0 0 0 3.88448 0 2.62383 + 0.339877i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1567.12
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 yes
56.e Even 1 no
56.h Odd 1 no

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4032, \chi)\):

\(T_{5}^{3} \) \(\mathstrut -\mathstrut 12 T_{5} \) \(\mathstrut -\mathstrut 12 \)
\(T_{11}^{6} \) \(\mathstrut -\mathstrut 36 T_{11}^{4} \) \(\mathstrut +\mathstrut 96 T_{11}^{2} \) \(\mathstrut -\mathstrut 48 \)
\(T_{13} \) \(\mathstrut +\mathstrut 2 \)