Properties

Label 4032.2.b.r
Level 4032
Weight 2
Character orbit 4032.b
Analytic conductor 32.196
Analytic rank 0
Dimension 16
CM No
Inner twists 8

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(32.195682095\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.29960650073923649536.7
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{30} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)  \(=\)  \( q\) \( - \beta_{2} q^{5} \) \( + \beta_{8} q^{7} \) \(+O(q^{10})\) \( q\) \( - \beta_{2} q^{5} \) \( + \beta_{8} q^{7} \) \( + ( - \beta_{7} - \beta_{11} ) q^{11} \) \( - \beta_{15} q^{13} \) \( + ( - \beta_{2} + \beta_{6} ) q^{17} \) \( - \beta_{4} q^{19} \) \( + \beta_{11} q^{23} \) \( + ( -2 + \beta_{12} ) q^{25} \) \( + \beta_{14} q^{29} \) \( + ( \beta_{4} + \beta_{5} + \beta_{8} ) q^{31} \) \( + ( - \beta_{3} - \beta_{11} ) q^{35} \) \( + ( -1 + \beta_{12} ) q^{37} \) \( + ( - \beta_{2} + \beta_{6} ) q^{41} \) \( + ( - \beta_{1} - \beta_{5} + \beta_{8} ) q^{43} \) \( + ( \beta_{3} + \beta_{13} ) q^{47} \) \( + ( -2 - \beta_{10} + \beta_{12} ) q^{49} \) \( + ( - \beta_{9} + \beta_{14} ) q^{53} \) \( + ( -3 \beta_{4} - 2 \beta_{5} - 2 \beta_{8} ) q^{55} \) \( + ( \beta_{3} - \beta_{13} ) q^{59} \) \( + ( -2 \beta_{10} + \beta_{15} ) q^{61} \) \( + ( - \beta_{9} - 2 \beta_{14} ) q^{65} \) \( + ( \beta_{1} + \beta_{5} - \beta_{8} ) q^{67} \) \( + ( 2 \beta_{7} + 3 \beta_{11} ) q^{71} \) \( -2 \beta_{15} q^{73} \) \( + ( - \beta_{2} + \beta_{9} + \beta_{14} ) q^{77} \) \( + ( \beta_{5} - \beta_{8} ) q^{79} \) \( + ( - \beta_{3} - \beta_{13} ) q^{83} \) \( + ( -5 + 3 \beta_{12} ) q^{85} \) \( + ( \beta_{2} + \beta_{6} ) q^{89} \) \( + ( \beta_{1} + 3 \beta_{4} + \beta_{5} + \beta_{8} ) q^{91} \) \( + ( 4 \beta_{7} + 2 \beta_{11} ) q^{95} \) \( -2 \beta_{10} q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)  \(=\)  \(16q \) \(\mathstrut +\mathstrut O(q^{10}) \) \(16q \) \(\mathstrut -\mathstrut 32q^{25} \) \(\mathstrut -\mathstrut 16q^{37} \) \(\mathstrut -\mathstrut 32q^{49} \) \(\mathstrut -\mathstrut 80q^{85} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut -\mathstrut \) \(7\) \(x^{12}\mathstrut +\mathstrut \) \(40\) \(x^{8}\mathstrut -\mathstrut \) \(112\) \(x^{4}\mathstrut +\mathstrut \) \(256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{10} + 3 \nu^{6} - 12 \nu^{2} \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{12} - 11 \nu^{8} + 84 \nu^{4} - 192 \)\()/64\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{14} - 3 \nu^{10} - 4 \nu^{6} + 96 \nu^{2} \)\()/32\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{15} - 2 \nu^{13} + 11 \nu^{11} - 10 \nu^{9} - 20 \nu^{7} + 56 \nu^{5} - 384 \nu \)\()/128\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{15} + 4 \nu^{14} - 2 \nu^{13} + 9 \nu^{11} - 28 \nu^{10} + 22 \nu^{9} - 52 \nu^{7} + 160 \nu^{6} - 40 \nu^{5} + 32 \nu^{3} - 192 \nu^{2} + 256 \nu \)\()/256\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{12} - \nu^{8} + 28 \nu^{4} + 64 \)\()/32\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{15} - 6 \nu^{13} - 3 \nu^{11} + 34 \nu^{9} - 4 \nu^{7} - 152 \nu^{5} - 32 \nu^{3} + 256 \nu \)\()/128\)
\(\beta_{8}\)\(=\)\((\)\( -3 \nu^{15} - 4 \nu^{14} - 2 \nu^{13} + 9 \nu^{11} + 28 \nu^{10} + 22 \nu^{9} - 52 \nu^{7} - 160 \nu^{6} - 40 \nu^{5} + 32 \nu^{3} + 192 \nu^{2} + 256 \nu \)\()/256\)
\(\beta_{9}\)\(=\)\((\)\( \nu^{15} + 6 \nu^{13} - 3 \nu^{11} - 34 \nu^{9} - 4 \nu^{7} + 152 \nu^{5} - 32 \nu^{3} - 256 \nu \)\()/64\)
\(\beta_{10}\)\(=\)\((\)\( \nu^{15} + \nu^{11} + 16 \nu^{9} + 16 \nu^{7} - 48 \nu^{5} - 16 \nu^{3} + 320 \nu \)\()/64\)
\(\beta_{11}\)\(=\)\((\)\( -3 \nu^{15} + 6 \nu^{13} + 33 \nu^{11} - 2 \nu^{9} - 124 \nu^{7} + 56 \nu^{5} + 448 \nu^{3} - 128 \nu \)\()/256\)
\(\beta_{12}\)\(=\)\((\)\( -\nu^{12} + 7 \nu^{8} - 24 \nu^{4} + 56 \)\()/8\)
\(\beta_{13}\)\(=\)\((\)\( -\nu^{14} + 5 \nu^{10} - 18 \nu^{6} + 40 \nu^{2} \)\()/8\)
\(\beta_{14}\)\(=\)\((\)\( -\nu^{15} + 15 \nu^{11} - 16 \nu^{9} - 64 \nu^{7} + 48 \nu^{5} + 208 \nu^{3} - 64 \nu \)\()/64\)
\(\beta_{15}\)\(=\)\((\)\( \nu^{15} - \nu^{13} - 5 \nu^{11} + 3 \nu^{9} + 18 \nu^{7} + 4 \nu^{5} - 8 \nu^{3} - 32 \nu \)\()/32\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{14}\mathstrut -\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{13}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(2\) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(3\) \(\beta_{4}\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(10\) \(\beta_{2}\mathstrut +\mathstrut \) \(14\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(6\) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(3\) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(9\) \(\beta_{8}\mathstrut -\mathstrut \) \(5\) \(\beta_{7}\mathstrut +\mathstrut \) \(9\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(3\) \(\beta_{13}\mathstrut -\mathstrut \) \(14\) \(\beta_{8}\mathstrut +\mathstrut \) \(14\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{1}\)\()/8\)
\(\nu^{7}\)\(=\)\((\)\(2\) \(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(5\) \(\beta_{10}\mathstrut -\mathstrut \) \(10\) \(\beta_{9}\mathstrut -\mathstrut \) \(7\) \(\beta_{8}\mathstrut -\mathstrut \) \(21\) \(\beta_{7}\mathstrut -\mathstrut \) \(7\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\)\()/8\)
\(\nu^{8}\)\(=\)\((\)\(14\) \(\beta_{12}\mathstrut +\mathstrut \) \(15\) \(\beta_{6}\mathstrut +\mathstrut \) \(22\) \(\beta_{2}\mathstrut -\mathstrut \) \(62\)\()/8\)
\(\nu^{9}\)\(=\)\((\)\(18\) \(\beta_{15}\mathstrut -\mathstrut \) \(17\) \(\beta_{14}\mathstrut +\mathstrut \) \(34\) \(\beta_{11}\mathstrut +\mathstrut \) \(5\) \(\beta_{10}\mathstrut +\mathstrut \) \(6\) \(\beta_{9}\mathstrut +\mathstrut \) \(23\) \(\beta_{8}\mathstrut +\mathstrut \) \(5\) \(\beta_{7}\mathstrut +\mathstrut \) \(23\) \(\beta_{5}\mathstrut +\mathstrut \) \(13\) \(\beta_{4}\)\()/8\)
\(\nu^{10}\)\(=\)\((\)\(-\)\(3\) \(\beta_{13}\mathstrut -\mathstrut \) \(18\) \(\beta_{8}\mathstrut +\mathstrut \) \(18\) \(\beta_{5}\mathstrut -\mathstrut \) \(30\) \(\beta_{3}\mathstrut -\mathstrut \) \(44\) \(\beta_{1}\)\()/8\)
\(\nu^{11}\)\(=\)\((\)\(-\)\(18\) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(43\) \(\beta_{10}\mathstrut -\mathstrut \) \(6\) \(\beta_{9}\mathstrut -\mathstrut \) \(25\) \(\beta_{8}\mathstrut -\mathstrut \) \(11\) \(\beta_{7}\mathstrut -\mathstrut \) \(25\) \(\beta_{5}\mathstrut +\mathstrut \) \(61\) \(\beta_{4}\)\()/8\)
\(\nu^{12}\)\(=\)\((\)\(-\)\(14\) \(\beta_{12}\mathstrut +\mathstrut \) \(81\) \(\beta_{6}\mathstrut -\mathstrut \) \(86\) \(\beta_{2}\mathstrut -\mathstrut \) \(322\)\()/8\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(50\) \(\beta_{15}\mathstrut -\mathstrut \) \(79\) \(\beta_{14}\mathstrut +\mathstrut \) \(158\) \(\beta_{11}\mathstrut -\mathstrut \) \(5\) \(\beta_{10}\mathstrut +\mathstrut \) \(26\) \(\beta_{9}\mathstrut -\mathstrut \) \(55\) \(\beta_{8}\mathstrut +\mathstrut \) \(27\) \(\beta_{7}\mathstrut -\mathstrut \) \(55\) \(\beta_{5}\mathstrut -\mathstrut \) \(45\) \(\beta_{4}\)\()/8\)
\(\nu^{14}\)\(=\)\((\)\(-\)\(93\) \(\beta_{13}\mathstrut +\mathstrut \) \(82\) \(\beta_{8}\mathstrut -\mathstrut \) \(82\) \(\beta_{5}\mathstrut -\mathstrut \) \(34\) \(\beta_{3}\mathstrut -\mathstrut \) \(148\) \(\beta_{1}\)\()/8\)
\(\nu^{15}\)\(=\)\((\)\(18\) \(\beta_{15}\mathstrut +\mathstrut \) \(31\) \(\beta_{14}\mathstrut +\mathstrut \) \(62\) \(\beta_{11}\mathstrut +\mathstrut \) \(117\) \(\beta_{10}\mathstrut +\mathstrut \) \(134\) \(\beta_{9}\mathstrut -\mathstrut \) \(135\) \(\beta_{8}\mathstrut +\mathstrut \) \(299\) \(\beta_{7}\mathstrut -\mathstrut \) \(135\) \(\beta_{5}\mathstrut +\mathstrut \) \(99\) \(\beta_{4}\)\()/8\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3583.1
1.32968 + 0.481610i
−0.481610 + 1.32968i
−1.32968 0.481610i
0.481610 1.32968i
−1.38588 0.281691i
−0.281691 + 1.38588i
1.38588 + 0.281691i
0.281691 1.38588i
−0.281691 1.38588i
−1.38588 + 0.281691i
0.281691 + 1.38588i
1.38588 0.281691i
−0.481610 1.32968i
1.32968 0.481610i
0.481610 + 1.32968i
−1.32968 + 0.481610i
0 0 0 3.33513i 0 −0.662153 2.56155i 0 0 0
3583.2 0 0 0 3.33513i 0 −0.662153 + 2.56155i 0 0 0
3583.3 0 0 0 3.33513i 0 0.662153 2.56155i 0 0 0
3583.4 0 0 0 3.33513i 0 0.662153 + 2.56155i 0 0 0
3583.5 0 0 0 1.69614i 0 −2.13578 1.56155i 0 0 0
3583.6 0 0 0 1.69614i 0 −2.13578 + 1.56155i 0 0 0
3583.7 0 0 0 1.69614i 0 2.13578 1.56155i 0 0 0
3583.8 0 0 0 1.69614i 0 2.13578 + 1.56155i 0 0 0
3583.9 0 0 0 1.69614i 0 −2.13578 1.56155i 0 0 0
3583.10 0 0 0 1.69614i 0 −2.13578 + 1.56155i 0 0 0
3583.11 0 0 0 1.69614i 0 2.13578 1.56155i 0 0 0
3583.12 0 0 0 1.69614i 0 2.13578 + 1.56155i 0 0 0
3583.13 0 0 0 3.33513i 0 −0.662153 2.56155i 0 0 0
3583.14 0 0 0 3.33513i 0 −0.662153 + 2.56155i 0 0 0
3583.15 0 0 0 3.33513i 0 0.662153 2.56155i 0 0 0
3583.16 0 0 0 3.33513i 0 0.662153 + 2.56155i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3583.16
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
7.b Odd 1 no
12.b Even 1 yes
21.c Even 1 no
28.d Even 1 no
84.h Odd 1 no

Hecke kernels

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

\(T_{5}^{4} \) \(\mathstrut +\mathstrut 14 T_{5}^{2} \) \(\mathstrut +\mathstrut 32 \)
\(T_{11}^{4} \) \(\mathstrut +\mathstrut 26 T_{11}^{2} \) \(\mathstrut +\mathstrut 16 \)
\(T_{19}^{4} \) \(\mathstrut -\mathstrut 28 T_{19}^{2} \) \(\mathstrut +\mathstrut 128 \)