Properties

Label 4032.2.h.h
Level 4032
Weight 2
Character orbit 4032.h
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.h (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(32.195682095\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.653473922154496.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{13} \)
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.

\(f(q)\)  \(=\)  \( q\) \( - \beta_{2} q^{5} \) \( + \beta_{3} q^{7} \) \(+O(q^{10})\) \( q\) \( - \beta_{2} q^{5} \) \( + \beta_{3} q^{7} \) \( + ( - \beta_{5} - \beta_{6} ) q^{11} \) \( + \beta_{10} q^{13} \) \( + ( \beta_{2} + 2 \beta_{4} ) q^{17} \) \( + ( \beta_{1} - \beta_{3} + \beta_{7} ) q^{19} \) \( + ( -3 \beta_{6} - \beta_{9} ) q^{23} \) \( + ( -1 - \beta_{10} ) q^{25} \) \( + \beta_{4} q^{29} \) \( + ( \beta_{1} - \beta_{3} - \beta_{7} ) q^{31} \) \( - \beta_{5} q^{35} \) \( + ( 2 \beta_{8} - \beta_{10} ) q^{37} \) \( + ( -2 \beta_{4} + \beta_{11} ) q^{41} \) \( + ( 2 \beta_{1} - 2 \beta_{3} ) q^{43} \) \( + ( -3 \beta_{5} + \beta_{9} ) q^{47} \) \(- q^{49}\) \( + ( \beta_{2} - \beta_{4} - \beta_{11} ) q^{53} \) \( + ( - \beta_{1} - 7 \beta_{3} + \beta_{7} ) q^{55} \) \( + ( \beta_{5} - 4 \beta_{6} + \beta_{9} ) q^{59} \) \( + ( -5 - \beta_{8} - \beta_{10} ) q^{61} \) \( + ( -3 \beta_{2} + 2 \beta_{4} - \beta_{11} ) q^{65} \) \( + ( - \beta_{1} - 3 \beta_{3} - 2 \beta_{7} ) q^{67} \) \( + ( -5 \beta_{6} + \beta_{9} ) q^{71} \) \( + ( -1 - 3 \beta_{8} + 2 \beta_{10} ) q^{73} \) \( + ( \beta_{2} - \beta_{4} ) q^{77} \) \( + ( 3 \beta_{1} + \beta_{3} ) q^{79} \) \( + ( \beta_{5} - 2 \beta_{6} - \beta_{9} ) q^{83} \) \( + ( 4 - 2 \beta_{8} + \beta_{10} ) q^{85} \) \( + ( 2 \beta_{2} - 2 \beta_{4} - \beta_{11} ) q^{89} \) \( - \beta_{7} q^{91} \) \( + ( 4 \beta_{5} - 4 \beta_{6} ) q^{95} \) \( + ( 1 + 3 \beta_{8} ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)  \(=\)  \(12q \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut -\mathstrut 12q^{25} \) \(\mathstrut -\mathstrut 8q^{37} \) \(\mathstrut -\mathstrut 12q^{49} \) \(\mathstrut -\mathstrut 56q^{61} \) \(\mathstrut +\mathstrut 56q^{85} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(4\) \(x^{10}\mathstrut +\mathstrut \) \(13\) \(x^{8}\mathstrut -\mathstrut \) \(28\) \(x^{6}\mathstrut +\mathstrut \) \(52\) \(x^{4}\mathstrut -\mathstrut \) \(64\) \(x^{2}\mathstrut +\mathstrut \) \(64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{10} - 14 \nu^{8} + 9 \nu^{6} - 22 \nu^{4} - 64 \nu^{2} + 32 \)\()/64\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{9} - 2 \nu^{7} + 9 \nu^{5} - 10 \nu^{3} + 16 \nu \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{10} + 10 \nu^{8} - 27 \nu^{6} + 34 \nu^{4} - 64 \nu^{2} + 32 \)\()/64\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{11} - 4 \nu^{9} + 5 \nu^{7} - 12 \nu^{5} + 12 \nu^{3} - 16 \nu \)\()/32\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{11} + 2 \nu^{9} - 11 \nu^{7} + 26 \nu^{5} - 48 \nu^{3} + 96 \nu \)\()/64\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{11} + 6 \nu^{9} - 19 \nu^{7} + 30 \nu^{5} - 24 \nu^{3} \)\()/64\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{10} + 10 \nu^{8} - 17 \nu^{6} + 50 \nu^{4} - 72 \nu^{2} + 96 \)\()/16\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{10} + 4 \nu^{8} - 13 \nu^{6} + 28 \nu^{4} - 36 \nu^{2} + 40 \)\()/8\)
\(\beta_{9}\)\(=\)\((\)\( -5 \nu^{11} + 38 \nu^{9} - 109 \nu^{7} + 206 \nu^{5} - 320 \nu^{3} + 480 \nu \)\()/64\)
\(\beta_{10}\)\(=\)\((\)\( -\nu^{10} + 2 \nu^{8} - 5 \nu^{6} + 10 \nu^{4} - 12 \nu^{2} + 8 \)\()/4\)
\(\beta_{11}\)\(=\)\((\)\( 3 \nu^{11} - 10 \nu^{9} + 27 \nu^{7} - 50 \nu^{5} + 96 \nu^{3} - 48 \nu \)\()/16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(3\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{2}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(3\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(10\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(11\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(6\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{11}\mathstrut -\mathstrut \) \(3\) \(\beta_{9}\mathstrut +\mathstrut \) \(14\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(6\) \(\beta_{4}\mathstrut +\mathstrut \) \(15\) \(\beta_{2}\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(4\) \(\beta_{10}\mathstrut -\mathstrut \) \(5\) \(\beta_{8}\mathstrut +\mathstrut \) \(5\) \(\beta_{7}\mathstrut -\mathstrut \) \(14\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut -\mathstrut \) \(7\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(3\) \(\beta_{11}\mathstrut -\mathstrut \) \(9\) \(\beta_{9}\mathstrut -\mathstrut \) \(26\) \(\beta_{6}\mathstrut +\mathstrut \) \(9\) \(\beta_{5}\mathstrut -\mathstrut \) \(30\) \(\beta_{4}\mathstrut +\mathstrut \) \(15\) \(\beta_{2}\)\()/8\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(\beta_{10}\mathstrut -\mathstrut \) \(10\) \(\beta_{8}\mathstrut +\mathstrut \) \(8\) \(\beta_{7}\mathstrut +\mathstrut \) \(19\) \(\beta_{3}\mathstrut -\mathstrut \) \(7\) \(\beta_{1}\mathstrut -\mathstrut \) \(2\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(\beta_{11}\mathstrut +\mathstrut \) \(3\) \(\beta_{9}\mathstrut -\mathstrut \) \(46\) \(\beta_{6}\mathstrut -\mathstrut \) \(31\) \(\beta_{5}\mathstrut -\mathstrut \) \(102\) \(\beta_{4}\mathstrut -\mathstrut \) \(15\) \(\beta_{2}\)\()/8\)
\(\nu^{10}\)\(=\)\((\)\(-\)\(28\) \(\beta_{10}\mathstrut +\mathstrut \) \(13\) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(22\) \(\beta_{3}\mathstrut -\mathstrut \) \(10\) \(\beta_{1}\mathstrut -\mathstrut \) \(33\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(3\) \(\beta_{11}\mathstrut +\mathstrut \) \(25\) \(\beta_{9}\mathstrut -\mathstrut \) \(38\) \(\beta_{6}\mathstrut -\mathstrut \) \(121\) \(\beta_{5}\mathstrut +\mathstrut \) \(62\) \(\beta_{4}\mathstrut +\mathstrut \) \(17\) \(\beta_{2}\)\()/8\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
575.1
1.35489 + 0.405301i
−1.35489 + 0.405301i
0.892524 1.09700i
−0.892524 1.09700i
−1.16947 0.795191i
1.16947 0.795191i
1.16947 + 0.795191i
−1.16947 + 0.795191i
−0.892524 + 1.09700i
0.892524 + 1.09700i
−1.35489 0.405301i
1.35489 0.405301i
0 0 0 3.31339i 0 1.00000i 0 0 0
575.2 0 0 0 3.31339i 0 1.00000i 0 0 0
575.3 0 0 0 2.56483i 0 1.00000i 0 0 0
575.4 0 0 0 2.56483i 0 1.00000i 0 0 0
575.5 0 0 0 0.665647i 0 1.00000i 0 0 0
575.6 0 0 0 0.665647i 0 1.00000i 0 0 0
575.7 0 0 0 0.665647i 0 1.00000i 0 0 0
575.8 0 0 0 0.665647i 0 1.00000i 0 0 0
575.9 0 0 0 2.56483i 0 1.00000i 0 0 0
575.10 0 0 0 2.56483i 0 1.00000i 0 0 0
575.11 0 0 0 3.31339i 0 1.00000i 0 0 0
575.12 0 0 0 3.31339i 0 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 575.12
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
4.b Odd 1 yes
12.b Even 1 yes

Hecke kernels

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

\(T_{5}^{6} \) \(\mathstrut +\mathstrut 18 T_{5}^{4} \) \(\mathstrut +\mathstrut 80 T_{5}^{2} \) \(\mathstrut +\mathstrut 32 \)
\(T_{11}^{6} \) \(\mathstrut -\mathstrut 28 T_{11}^{4} \) \(\mathstrut +\mathstrut 132 T_{11}^{2} \) \(\mathstrut -\mathstrut 128 \)
\(T_{13}^{3} \) \(\mathstrut -\mathstrut 28 T_{13} \) \(\mathstrut +\mathstrut 16 \)