Properties

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

Newform invariants

Self dual: No
Analytic conductor: \(32.195682095\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.162447943996702457856.1
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{18} \)
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.

\(f(q)\) \(=\) \( q\) \( + \beta_{13} q^{5} \) \( + \beta_{1} q^{7} \) \(+O(q^{10})\) \( q\) \( + \beta_{13} q^{5} \) \( + \beta_{1} q^{7} \) \( + \beta_{9} q^{11} \) \( -\beta_{5} q^{13} \) \( + ( \beta_{8} + 3 \beta_{10} ) q^{17} \) \( + ( \beta_{4} - \beta_{7} ) q^{19} \) \( -\beta_{12} q^{23} \) \( + \beta_{3} q^{25} \) \( + ( -3 \beta_{11} - \beta_{13} ) q^{29} \) \( -2 \beta_{15} q^{31} \) \( -\beta_{14} q^{35} \) \( + ( -2 \beta_{2} - \beta_{5} ) q^{37} \) \( + ( -3 \beta_{8} - \beta_{10} ) q^{41} \) \( + ( -3 \beta_{4} - \beta_{7} ) q^{43} \) \( -4 \beta_{6} q^{47} \) \(- q^{49}\) \( + ( -\beta_{11} + 3 \beta_{13} ) q^{53} \) \( -2 \beta_{1} q^{55} \) \( + ( -2 \beta_{2} - 2 \beta_{5} ) q^{61} \) \( + ( -2 \beta_{8} + 4 \beta_{10} ) q^{65} \) \( + ( -2 \beta_{4} - \beta_{7} ) q^{67} \) \( + ( -2 \beta_{6} + \beta_{12} ) q^{71} \) \( + ( -5 - \beta_{3} ) q^{73} \) \( + \beta_{11} q^{77} \) \( + ( -3 \beta_{1} + 3 \beta_{15} ) q^{79} \) \( + ( 6 \beta_{9} - 4 \beta_{14} ) q^{83} \) \( + \beta_{2} q^{85} \) \( + ( 3 \beta_{8} + \beta_{10} ) q^{89} \) \( -\beta_{7} q^{91} \) \( + ( -6 \beta_{6} - 2 \beta_{12} ) q^{95} \) \( + ( -7 + \beta_{3} ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(16q \) \(\mathstrut +\mathstrut O(q^{10}) \) \(16q \) \(\mathstrut -\mathstrut 16q^{49} \) \(\mathstrut -\mathstrut 80q^{73} \) \(\mathstrut -\mathstrut 112q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut -\mathstrut \) \(x^{12}\mathstrut -\mathstrut \) \(15\) \(x^{8}\mathstrut -\mathstrut \) \(16\) \(x^{4}\mathstrut +\mathstrut \) \(256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{14} - 15 \nu^{10} - 33 \nu^{6} + 256 \nu^{2} \)\()/576\)
\(\beta_{2}\)\(=\)\((\)\( -13 \nu^{12} + 45 \nu^{8} - 45 \nu^{4} - 32 \)\()/360\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{12} + \nu^{8} + 31 \nu^{4} + 8 \)\()/24\)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{14} + 35 \nu^{10} - 115 \nu^{6} - 112 \nu^{2} \)\()/480\)
\(\beta_{5}\)\(=\)\((\)\( 19 \nu^{12} + 45 \nu^{8} - 45 \nu^{4} - 784 \)\()/360\)
\(\beta_{6}\)\(=\)\((\)\( 4 \nu^{15} - 17 \nu^{13} - 15 \nu^{9} + 255 \nu^{5} + 356 \nu^{3} + 272 \nu \)\()/1440\)
\(\beta_{7}\)\(=\)\((\)\( -3 \nu^{14} - 5 \nu^{10} + 85 \nu^{6} + 328 \nu^{2} \)\()/240\)
\(\beta_{8}\)\(=\)\((\)\( -19 \nu^{15} + 88 \nu^{13} + 195 \nu^{11} + 360 \nu^{9} + 1005 \nu^{7} - 360 \nu^{5} - 416 \nu^{3} - 11968 \nu \)\()/5760\)
\(\beta_{9}\)\(=\)\((\)\( 5 \nu^{15} - 8 \nu^{13} - 21 \nu^{11} + 72 \nu^{9} + 69 \nu^{7} - 72 \nu^{5} - 224 \nu^{3} - 64 \nu \)\()/1152\)
\(\beta_{10}\)\(=\)\((\)\( -4 \nu^{15} - 17 \nu^{13} - 15 \nu^{9} + 255 \nu^{5} - 356 \nu^{3} + 272 \nu \)\()/1440\)
\(\beta_{11}\)\(=\)\((\)\( -5 \nu^{15} - 8 \nu^{13} + 21 \nu^{11} + 72 \nu^{9} - 69 \nu^{7} - 72 \nu^{5} + 224 \nu^{3} - 64 \nu \)\()/1152\)
\(\beta_{12}\)\(=\)\((\)\( -19 \nu^{15} - 88 \nu^{13} + 195 \nu^{11} - 360 \nu^{9} + 1005 \nu^{7} + 360 \nu^{5} - 416 \nu^{3} + 11968 \nu \)\()/5760\)
\(\beta_{13}\)\(=\)\((\)\( -11 \nu^{15} + 4 \nu^{13} - 21 \nu^{11} + 60 \nu^{9} + 69 \nu^{7} + 132 \nu^{5} + 656 \nu^{3} + 128 \nu \)\()/1152\)
\(\beta_{14}\)\(=\)\((\)\( 11 \nu^{15} + 4 \nu^{13} + 21 \nu^{11} + 60 \nu^{9} - 69 \nu^{7} + 132 \nu^{5} - 656 \nu^{3} + 128 \nu \)\()/1152\)
\(\beta_{15}\)\(=\)\((\)\( 7 \nu^{14} + 9 \nu^{10} - 57 \nu^{6} + 32 \nu^{2} \)\()/192\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{6}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{15}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(5\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(5\) \(\beta_{6}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(1\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(6\) \(\beta_{14}\mathstrut +\mathstrut \) \(6\) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(5\) \(\beta_{11}\mathstrut +\mathstrut \) \(6\) \(\beta_{10}\mathstrut -\mathstrut \) \(5\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(6\) \(\beta_{6}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(5\) \(\beta_{7}\mathstrut -\mathstrut \) \(5\) \(\beta_{4}\mathstrut -\mathstrut \) \(18\) \(\beta_{1}\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(3\) \(\beta_{14}\mathstrut +\mathstrut \) \(3\) \(\beta_{13}\mathstrut +\mathstrut \) \(7\) \(\beta_{12}\mathstrut -\mathstrut \) \(10\) \(\beta_{11}\mathstrut -\mathstrut \) \(3\) \(\beta_{10}\mathstrut +\mathstrut \) \(10\) \(\beta_{9}\mathstrut +\mathstrut \) \(7\) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(14\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(17\) \(\beta_{2}\mathstrut +\mathstrut \) \(31\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(17\) \(\beta_{14}\mathstrut +\mathstrut \) \(17\) \(\beta_{13}\mathstrut +\mathstrut \) \(17\) \(\beta_{11}\mathstrut -\mathstrut \) \(5\) \(\beta_{10}\mathstrut +\mathstrut \) \(17\) \(\beta_{9}\mathstrut -\mathstrut \) \(5\) \(\beta_{6}\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(11\) \(\beta_{15}\mathstrut +\mathstrut \) \(23\) \(\beta_{7}\mathstrut +\mathstrut \) \(34\) \(\beta_{4}\mathstrut -\mathstrut \) \(57\) \(\beta_{1}\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(22\) \(\beta_{14}\mathstrut -\mathstrut \) \(22\) \(\beta_{13}\mathstrut +\mathstrut \) \(23\) \(\beta_{12}\mathstrut +\mathstrut \) \(45\) \(\beta_{11}\mathstrut -\mathstrut \) \(22\) \(\beta_{10}\mathstrut -\mathstrut \) \(45\) \(\beta_{9}\mathstrut +\mathstrut \) \(23\) \(\beta_{8}\mathstrut +\mathstrut \) \(22\) \(\beta_{6}\)\()/4\)
\(\nu^{12}\)\(=\)\((\)\(45\) \(\beta_{5}\mathstrut -\mathstrut \) \(45\) \(\beta_{2}\mathstrut +\mathstrut \) \(94\)\()/4\)
\(\nu^{13}\)\(=\)\((\)\(91\) \(\beta_{14}\mathstrut +\mathstrut \) \(91\) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(90\) \(\beta_{11}\mathstrut -\mathstrut \) \(91\) \(\beta_{10}\mathstrut -\mathstrut \) \(90\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(91\) \(\beta_{6}\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(91\) \(\beta_{15}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(89\) \(\beta_{4}\mathstrut -\mathstrut \) \(87\) \(\beta_{1}\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(89\) \(\beta_{14}\mathstrut -\mathstrut \) \(89\) \(\beta_{13}\mathstrut -\mathstrut \) \(89\) \(\beta_{11}\mathstrut -\mathstrut \) \(275\) \(\beta_{10}\mathstrut +\mathstrut \) \(89\) \(\beta_{9}\mathstrut +\mathstrut \) \(275\) \(\beta_{6}\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2591.1
−1.40721 0.140577i
−0.140577 1.40721i
−0.140577 + 1.40721i
−1.40721 + 0.140577i
0.825348 1.14839i
−1.14839 + 0.825348i
−1.14839 0.825348i
0.825348 + 1.14839i
−0.825348 + 1.14839i
1.14839 0.825348i
1.14839 + 0.825348i
−0.825348 1.14839i
1.40721 + 0.140577i
0.140577 + 1.40721i
0.140577 1.40721i
1.40721 0.140577i
0 0 0 −3.09557 0 1.00000i 0 0 0
2591.2 0 0 0 −3.09557 0 1.00000i 0 0 0
2591.3 0 0 0 −3.09557 0 1.00000i 0 0 0
2591.4 0 0 0 −3.09557 0 1.00000i 0 0 0
2591.5 0 0 0 −0.646084 0 1.00000i 0 0 0
2591.6 0 0 0 −0.646084 0 1.00000i 0 0 0
2591.7 0 0 0 −0.646084 0 1.00000i 0 0 0
2591.8 0 0 0 −0.646084 0 1.00000i 0 0 0
2591.9 0 0 0 0.646084 0 1.00000i 0 0 0
2591.10 0 0 0 0.646084 0 1.00000i 0 0 0
2591.11 0 0 0 0.646084 0 1.00000i 0 0 0
2591.12 0 0 0 0.646084 0 1.00000i 0 0 0
2591.13 0 0 0 3.09557 0 1.00000i 0 0 0
2591.14 0 0 0 3.09557 0 1.00000i 0 0 0
2591.15 0 0 0 3.09557 0 1.00000i 0 0 0
2591.16 0 0 0 3.09557 0 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2591.16
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
8.b Even 1 yes
8.d Odd 1 yes
12.b Even 1 yes
24.f Even 1 yes
24.h Odd 1 yes

Hecke kernels

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

\(T_{5}^{4} \) \(\mathstrut -\mathstrut 10 T_{5}^{2} \) \(\mathstrut +\mathstrut 4 \)
\(T_{19}^{2} \) \(\mathstrut -\mathstrut 28 \)
\(T_{43}^{4} \) \(\mathstrut -\mathstrut 152 T_{43}^{2} \) \(\mathstrut +\mathstrut 400 \)