Properties

Label 4032.2.j.f
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: 16.0.11007531417600000000.1
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{6} \)
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.

\(f(q)\) \(=\) \( q -\beta_{12} q^{5} + \beta_{8} q^{7} +O(q^{10})\) \( q -\beta_{12} q^{5} + \beta_{8} q^{7} + ( -\beta_{11} - \beta_{13} ) q^{11} + \beta_{2} q^{13} + \beta_{4} q^{17} + \beta_{10} q^{19} + ( \beta_{3} - \beta_{7} ) q^{23} + ( 4 - \beta_{1} ) q^{25} + \beta_{14} q^{29} -2 \beta_{8} q^{31} + \beta_{11} q^{35} -\beta_{2} q^{37} + \beta_{4} q^{41} + ( -2 \beta_{9} + \beta_{10} ) q^{43} - q^{49} + ( 2 \beta_{12} + 3 \beta_{14} ) q^{53} + ( -12 \beta_{8} + 2 \beta_{15} ) q^{55} -4 \beta_{13} q^{59} + ( 2 \beta_{2} - 2 \beta_{5} ) q^{61} + ( -2 \beta_{4} - 2 \beta_{6} ) q^{65} + \beta_{9} q^{67} + ( -\beta_{3} - \beta_{7} ) q^{71} + ( 1 - \beta_{1} ) q^{73} + ( -\beta_{12} + \beta_{14} ) q^{77} + ( -5 \beta_{8} + \beta_{15} ) q^{79} + ( -2 \beta_{11} - 2 \beta_{13} ) q^{83} + ( -3 \beta_{2} - 3 \beta_{5} ) q^{85} -\beta_{4} q^{89} + \beta_{9} q^{91} + 2 \beta_{3} q^{95} + ( 11 + \beta_{1} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 64q^{25} - 16q^{49} + 16q^{73} + 176q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 7 x^{12} + 48 x^{8} - 7 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{12} + 161 \)\()/24\)
\(\beta_{2}\)\(=\)\((\)\( 9 \nu^{12} - 64 \nu^{8} + 416 \nu^{4} - 31 \)\()/24\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{15} + 21 \nu^{13} - 144 \nu^{9} + 1008 \nu^{5} - 377 \nu^{3} - 147 \nu \)\()/48\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{15} + 21 \nu^{13} - 144 \nu^{9} + 1008 \nu^{5} + 377 \nu^{3} - 147 \nu \)\()/48\)
\(\beta_{5}\)\(=\)\((\)\( -7 \nu^{12} + 48 \nu^{8} - 336 \nu^{4} + 25 \)\()/12\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{15} - 17 \nu^{13} + 120 \nu^{9} - 816 \nu^{5} - 305 \nu^{3} + 119 \nu \)\()/24\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{15} - 17 \nu^{13} + 120 \nu^{9} - 816 \nu^{5} + 305 \nu^{3} + 119 \nu \)\()/24\)
\(\beta_{8}\)\(=\)\((\)\( -7 \nu^{14} + 48 \nu^{10} - 330 \nu^{6} + \nu^{2} \)\()/18\)
\(\beta_{9}\)\(=\)\((\)\( -11 \nu^{14} + 80 \nu^{10} - 544 \nu^{6} + 157 \nu^{2} \)\()/24\)
\(\beta_{10}\)\(=\)\((\)\( 9 \nu^{14} - 64 \nu^{10} + 440 \nu^{6} - 127 \nu^{2} \)\()/12\)
\(\beta_{11}\)\(=\)\((\)\( -37 \nu^{15} - 7 \nu^{13} + 256 \nu^{11} + 48 \nu^{9} - 1760 \nu^{7} - 336 \nu^{5} + 131 \nu^{3} - 47 \nu \)\()/48\)
\(\beta_{12}\)\(=\)\((\)\( 37 \nu^{15} - 7 \nu^{13} - 256 \nu^{11} + 48 \nu^{9} + 1760 \nu^{7} - 336 \nu^{5} - 131 \nu^{3} - 47 \nu \)\()/48\)
\(\beta_{13}\)\(=\)\((\)\( 15 \nu^{15} + 3 \nu^{13} - 104 \nu^{11} - 20 \nu^{9} + 712 \nu^{7} + 136 \nu^{5} - 53 \nu^{3} + 19 \nu \)\()/12\)
\(\beta_{14}\)\(=\)\((\)\( 15 \nu^{15} - 3 \nu^{13} - 104 \nu^{11} + 20 \nu^{9} + 712 \nu^{7} - 136 \nu^{5} - 53 \nu^{3} - 19 \nu \)\()/12\)
\(\beta_{15}\)\(=\)\((\)\( -21 \nu^{14} + 144 \nu^{10} - 984 \nu^{6} + 3 \nu^{2} \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-3 \beta_{12} - 3 \beta_{11} - \beta_{4} - \beta_{3}\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{15} - 3 \beta_{10} - 3 \beta_{9} - 9 \beta_{8}\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} + \beta_{6} + 2 \beta_{4} - 2 \beta_{3}\)\()/6\)
\(\nu^{4}\)\(=\)\((\)\(-2 \beta_{5} - 3 \beta_{2} - \beta_{1} + 7\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(9 \beta_{14} - 9 \beta_{13} - 15 \beta_{12} - 15 \beta_{11} + 3 \beta_{7} + 3 \beta_{6} + 5 \beta_{4} + 5 \beta_{3}\)\()/12\)
\(\nu^{6}\)\(=\)\((\)\(4 \beta_{15} - 27 \beta_{8}\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-24 \beta_{14} - 24 \beta_{13} + 39 \beta_{12} - 39 \beta_{11} - 8 \beta_{7} + 8 \beta_{6} + 13 \beta_{4} - 13 \beta_{3}\)\()/12\)
\(\nu^{8}\)\(=\)\((\)\(-13 \beta_{5} - 21 \beta_{2} + 7 \beta_{1} - 47\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(21 \beta_{7} + 21 \beta_{6} + 34 \beta_{4} + 34 \beta_{3}\)\()/6\)
\(\nu^{10}\)\(=\)\((\)\(55 \beta_{15} + 102 \beta_{10} + 165 \beta_{9} - 369 \beta_{8}\)\()/12\)
\(\nu^{11}\)\(=\)\((\)\(-165 \beta_{14} - 165 \beta_{13} + 267 \beta_{12} - 267 \beta_{11} + 55 \beta_{7} - 55 \beta_{6} - 89 \beta_{4} + 89 \beta_{3}\)\()/12\)
\(\nu^{12}\)\(=\)\(24 \beta_{1} - 161\)
\(\nu^{13}\)\(=\)\((\)\(-432 \beta_{14} + 432 \beta_{13} + 699 \beta_{12} + 699 \beta_{11} + 144 \beta_{7} + 144 \beta_{6} + 233 \beta_{4} + 233 \beta_{3}\)\()/12\)
\(\nu^{14}\)\(=\)\((\)\(-377 \beta_{15} + 699 \beta_{10} + 1131 \beta_{9} + 2529 \beta_{8}\)\()/12\)
\(\nu^{15}\)\(=\)\((\)\(377 \beta_{7} - 377 \beta_{6} - 610 \beta_{4} + 610 \beta_{3}\)\()/6\)

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
−1.56290 0.418778i
−0.418778 1.56290i
−0.418778 + 1.56290i
−1.56290 + 0.418778i
−0.159959 0.596975i
−0.596975 0.159959i
−0.596975 + 0.159959i
−0.159959 + 0.596975i
0.159959 + 0.596975i
0.596975 + 0.159959i
0.596975 0.159959i
0.159959 0.596975i
1.56290 + 0.418778i
0.418778 + 1.56290i
0.418778 1.56290i
1.56290 0.418778i
0 0 0 −3.96336 0 1.00000i 0 0 0
2591.2 0 0 0 −3.96336 0 1.00000i 0 0 0
2591.3 0 0 0 −3.96336 0 1.00000i 0 0 0
2591.4 0 0 0 −3.96336 0 1.00000i 0 0 0
2591.5 0 0 0 −1.51387 0 1.00000i 0 0 0
2591.6 0 0 0 −1.51387 0 1.00000i 0 0 0
2591.7 0 0 0 −1.51387 0 1.00000i 0 0 0
2591.8 0 0 0 −1.51387 0 1.00000i 0 0 0
2591.9 0 0 0 1.51387 0 1.00000i 0 0 0
2591.10 0 0 0 1.51387 0 1.00000i 0 0 0
2591.11 0 0 0 1.51387 0 1.00000i 0 0 0
2591.12 0 0 0 1.51387 0 1.00000i 0 0 0
2591.13 0 0 0 3.96336 0 1.00000i 0 0 0
2591.14 0 0 0 3.96336 0 1.00000i 0 0 0
2591.15 0 0 0 3.96336 0 1.00000i 0 0 0
2591.16 0 0 0 3.96336 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 \(S_{2}^{\mathrm{new}}(4032, [\chi])\):

\( T_{5}^{4} - 18 T_{5}^{2} + 36 \)
\( T_{19}^{2} - 12 \)
\( T_{43}^{2} - 60 \)