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 \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.162447943996702457856.1
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: yes
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.

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

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.j.e 16
3.b odd 2 1 inner 4032.2.j.e 16
4.b odd 2 1 inner 4032.2.j.e 16
8.b even 2 1 inner 4032.2.j.e 16
8.d odd 2 1 inner 4032.2.j.e 16
12.b even 2 1 inner 4032.2.j.e 16
24.f even 2 1 inner 4032.2.j.e 16
24.h odd 2 1 inner 4032.2.j.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4032.2.j.e 16 1.a even 1 1 trivial
4032.2.j.e 16 3.b odd 2 1 inner
4032.2.j.e 16 4.b odd 2 1 inner
4032.2.j.e 16 8.b even 2 1 inner
4032.2.j.e 16 8.d odd 2 1 inner
4032.2.j.e 16 12.b even 2 1 inner
4032.2.j.e 16 24.f even 2 1 inner
4032.2.j.e 16 24.h odd 2 1 inner

Hecke kernels

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( ( 1 + 10 T^{2} + 54 T^{4} + 250 T^{6} + 625 T^{8} )^{4} \)
$7$ \( ( 1 + T^{2} )^{8} \)
$11$ \( ( 1 - 34 T^{2} + 510 T^{4} - 4114 T^{6} + 14641 T^{8} )^{4} \)
$13$ \( ( 1 - 32 T^{2} + 510 T^{4} - 5408 T^{6} + 28561 T^{8} )^{4} \)
$17$ \( ( 1 - 22 T^{2} + 174 T^{4} - 6358 T^{6} + 83521 T^{8} )^{4} \)
$19$ \( ( 1 + 10 T^{2} + 361 T^{4} )^{8} \)
$23$ \( ( 1 + 70 T^{2} + 2262 T^{4} + 37030 T^{6} + 279841 T^{8} )^{4} \)
$29$ \( ( 1 - 8 T^{2} + 354 T^{4} - 6728 T^{6} + 707281 T^{8} )^{4} \)
$31$ \( ( 1 + 22 T^{2} + 961 T^{4} )^{8} \)
$37$ \( ( 1 - 80 T^{2} + 3582 T^{4} - 109520 T^{6} + 1874161 T^{8} )^{4} \)
$41$ \( ( 1 + 26 T^{2} + 3342 T^{4} + 43706 T^{6} + 2825761 T^{8} )^{4} \)
$43$ \( ( 1 + 20 T^{2} - 1578 T^{4} + 36980 T^{6} + 3418801 T^{8} )^{4} \)
$47$ \( ( 1 + 62 T^{2} + 2209 T^{4} )^{8} \)
$53$ \( ( 1 + 136 T^{2} + 8898 T^{4} + 382024 T^{6} + 7890481 T^{8} )^{4} \)
$59$ \( ( 1 - 59 T^{2} )^{16} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )^{8}( 1 + 14 T + 61 T^{2} )^{8} \)
$67$ \( ( 1 + 200 T^{2} + 18222 T^{4} + 897800 T^{6} + 20151121 T^{8} )^{4} \)
$71$ \( ( 1 + 254 T^{2} + 26022 T^{4} + 1280414 T^{6} + 25411681 T^{8} )^{4} \)
$73$ \( ( 1 + 10 T + 150 T^{2} + 730 T^{3} + 5329 T^{4} )^{8} \)
$79$ \( ( 1 + 80 T^{2} + 7278 T^{4} + 499280 T^{6} + 38950081 T^{8} )^{4} \)
$83$ \( ( 1 - 4 T^{2} + 5382 T^{4} - 27556 T^{6} + 47458321 T^{8} )^{4} \)
$89$ \( ( 1 - 166 T^{2} + 22542 T^{4} - 1314886 T^{6} + 62742241 T^{8} )^{4} \)
$97$ \( ( 1 + 14 T + 222 T^{2} + 1358 T^{3} + 9409 T^{4} )^{8} \)
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