Properties

Label 4032.2.v.c
Level 4032
Weight 2
Character orbit 4032.v
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.v (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(32.195682095\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.653473922154496.1
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{12} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(f(q)\)  \(=\)  \( q\) \( + ( \beta_{3} - \beta_{5} ) q^{5} \) \(+ q^{7}\) \(+O(q^{10})\) \( q\) \( + ( \beta_{3} - \beta_{5} ) q^{5} \) \(+ q^{7}\) \( + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{11} \) \( + ( 1 - \beta_{2} - \beta_{8} ) q^{13} \) \( + ( - \beta_{1} - 3 \beta_{3} - \beta_{7} - \beta_{10} ) q^{17} \) \( - \beta_{10} q^{23} \) \( + \beta_{2} q^{25} \) \( + ( \beta_{1} - \beta_{6} - \beta_{10} ) q^{29} \) \( + ( 2 \beta_{2} + \beta_{4} + \beta_{8} ) q^{31} \) \( + ( \beta_{3} - \beta_{5} ) q^{35} \) \( + ( -1 - \beta_{2} ) q^{37} \) \( + ( \beta_{1} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{41} \) \( + ( 2 - 2 \beta_{2} + 2 \beta_{8} + \beta_{11} ) q^{43} \) \( + ( 2 \beta_{5} - 2 \beta_{6} ) q^{47} \) \(+ q^{49}\) \( + ( 3 \beta_{3} - 3 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{10} ) q^{53} \) \( + ( -4 + \beta_{4} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{55} \) \( + ( 2 \beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{10} ) q^{59} \) \( + ( -3 + 3 \beta_{2} - \beta_{8} ) q^{61} \) \( + ( 2 \beta_{3} + 2 \beta_{10} ) q^{65} \) \( + ( -4 - 4 \beta_{2} + \beta_{4} ) q^{67} \) \( + ( - \beta_{1} + 2 \beta_{3} - \beta_{7} - 3 \beta_{10} ) q^{71} \) \( + ( -4 \beta_{2} + \beta_{9} - \beta_{11} ) q^{73} \) \( + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{77} \) \( + ( -4 \beta_{2} - 2 \beta_{4} - 2 \beta_{8} ) q^{79} \) \( + ( -5 \beta_{3} + 5 \beta_{5} - \beta_{6} + \beta_{10} ) q^{83} \) \( + ( 6 + 6 \beta_{2} - 2 \beta_{9} ) q^{85} \) \( + ( \beta_{1} - 7 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{89} \) \( + ( 1 - \beta_{2} - \beta_{8} ) q^{91} \) \( + ( -6 + 3 \beta_{4} - 3 \beta_{8} - \beta_{9} - \beta_{11} ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)  \(=\)  \(12q \) \(\mathstrut +\mathstrut 12q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut +\mathstrut 12q^{7} \) \(\mathstrut +\mathstrut 16q^{13} \) \(\mathstrut -\mathstrut 12q^{37} \) \(\mathstrut +\mathstrut 20q^{43} \) \(\mathstrut +\mathstrut 12q^{49} \) \(\mathstrut -\mathstrut 32q^{55} \) \(\mathstrut -\mathstrut 32q^{61} \) \(\mathstrut -\mathstrut 44q^{67} \) \(\mathstrut +\mathstrut 64q^{85} \) \(\mathstrut +\mathstrut 16q^{91} \) \(\mathstrut -\mathstrut 56q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 3 \nu^{11} + 6 \nu^{9} - 5 \nu^{7} + 46 \nu^{5} - 32 \nu^{3} + 32 \nu \)\()/64\)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{10} + 10 \nu^{8} - 27 \nu^{6} + 34 \nu^{4} - 64 \nu^{2} + 32 \)\()/64\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{11} - 4 \nu^{9} + 5 \nu^{7} - 12 \nu^{5} + 12 \nu^{3} - 16 \nu \)\()/32\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{10} + 18 \nu^{8} - 55 \nu^{6} + 138 \nu^{4} - 256 \nu^{2} + 352 \)\()/64\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{11} + 6 \nu^{9} - 19 \nu^{7} + 30 \nu^{5} - 24 \nu^{3} \)\()/64\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{11} - 18 \nu^{9} + 49 \nu^{7} - 90 \nu^{5} + 136 \nu^{3} - 192 \nu \)\()/64\)
\(\beta_{7}\)\(=\)\((\)\( -3 \nu^{11} + 10 \nu^{9} - 27 \nu^{7} + 98 \nu^{5} - 128 \nu^{3} + 224 \nu \)\()/64\)
\(\beta_{8}\)\(=\)\((\)\( -7 \nu^{10} + 18 \nu^{8} - 31 \nu^{6} + 74 \nu^{4} - 160 \nu^{2} + 96 \)\()/64\)
\(\beta_{9}\)\(=\)\((\)\( -11 \nu^{10} + 10 \nu^{8} - 67 \nu^{6} + 98 \nu^{4} - 160 \nu^{2} + 96 \)\()/64\)
\(\beta_{10}\)\(=\)\((\)\( 3 \nu^{11} - 12 \nu^{9} + 31 \nu^{7} - 68 \nu^{5} + 116 \nu^{3} - 80 \nu \)\()/32\)
\(\beta_{11}\)\(=\)\((\)\( -13 \nu^{10} + 54 \nu^{8} - 117 \nu^{6} + 286 \nu^{4} - 320 \nu^{2} + 416 \)\()/64\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{11}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(2\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(3\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(5\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(10\) \(\beta_{2}\mathstrut -\mathstrut \) \(8\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{10}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut +\mathstrut \) \(7\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(\beta_{11}\mathstrut -\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(8\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(16\) \(\beta_{2}\mathstrut -\mathstrut \) \(2\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(3\) \(\beta_{10}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(9\) \(\beta_{6}\mathstrut -\mathstrut \) \(13\) \(\beta_{5}\mathstrut -\mathstrut \) \(15\) \(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(13\) \(\beta_{9}\mathstrut +\mathstrut \) \(5\) \(\beta_{8}\mathstrut -\mathstrut \) \(4\) \(\beta_{4}\mathstrut +\mathstrut \) \(26\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(\beta_{10}\mathstrut -\mathstrut \) \(11\) \(\beta_{7}\mathstrut -\mathstrut \) \(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(23\) \(\beta_{5}\mathstrut -\mathstrut \) \(51\) \(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(-\)\(\beta_{11}\mathstrut -\mathstrut \) \(14\) \(\beta_{9}\mathstrut -\mathstrut \) \(24\) \(\beta_{8}\mathstrut +\mathstrut \) \(17\) \(\beta_{4}\mathstrut +\mathstrut \) \(32\) \(\beta_{2}\mathstrut -\mathstrut \) \(46\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(3\) \(\beta_{10}\mathstrut -\mathstrut \) \(19\) \(\beta_{7}\mathstrut -\mathstrut \) \(25\) \(\beta_{6}\mathstrut -\mathstrut \) \(19\) \(\beta_{5}\mathstrut +\mathstrut \) \(31\) \(\beta_{3}\mathstrut +\mathstrut \) \(29\) \(\beta_{1}\)\()/4\)

Character Values

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

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1583.1
1.35489 + 0.405301i
−1.16947 0.795191i
−0.892524 + 1.09700i
0.892524 1.09700i
1.16947 + 0.795191i
−1.35489 0.405301i
1.35489 0.405301i
−1.16947 + 0.795191i
−0.892524 1.09700i
0.892524 + 1.09700i
1.16947 0.795191i
−1.35489 + 0.405301i
0 0 0 −1.41421 1.41421i 0 1.00000 0 0 0
1583.2 0 0 0 −1.41421 1.41421i 0 1.00000 0 0 0
1583.3 0 0 0 −1.41421 1.41421i 0 1.00000 0 0 0
1583.4 0 0 0 1.41421 + 1.41421i 0 1.00000 0 0 0
1583.5 0 0 0 1.41421 + 1.41421i 0 1.00000 0 0 0
1583.6 0 0 0 1.41421 + 1.41421i 0 1.00000 0 0 0
3599.1 0 0 0 −1.41421 + 1.41421i 0 1.00000 0 0 0
3599.2 0 0 0 −1.41421 + 1.41421i 0 1.00000 0 0 0
3599.3 0 0 0 −1.41421 + 1.41421i 0 1.00000 0 0 0
3599.4 0 0 0 1.41421 1.41421i 0 1.00000 0 0 0
3599.5 0 0 0 1.41421 1.41421i 0 1.00000 0 0 0
3599.6 0 0 0 1.41421 1.41421i 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3599.6
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 no
48.k Even 1 no

Hecke kernels

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

\(T_{5}^{4} \) \(\mathstrut +\mathstrut 16 \)
\(T_{11}^{12} \) \(\mathstrut +\mathstrut 1056 T_{11}^{8} \) \(\mathstrut +\mathstrut 53504 T_{11}^{4} \) \(\mathstrut +\mathstrut 65536 \)