Properties

Label 4032.2.h.g
Level 4032
Weight 2
Character orbit 4032.h
Analytic conductor 32.196
Analytic rank 0
Dimension 8
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.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.5473632256.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 2016)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{5} + \beta_{6} ) q^{5} -\beta_{1} q^{7} +O(q^{10})\) \( q + ( \beta_{5} + \beta_{6} ) q^{5} -\beta_{1} q^{7} + ( -\beta_{4} - 2 \beta_{7} ) q^{11} + ( 1 - \beta_{2} ) q^{13} + ( -\beta_{5} - 3 \beta_{6} ) q^{17} + 4 \beta_{1} q^{19} + ( \beta_{4} + 2 \beta_{7} ) q^{23} + ( -4 - \beta_{2} ) q^{25} + ( -2 \beta_{5} - 3 \beta_{6} ) q^{29} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{31} + ( \beta_{4} + \beta_{7} ) q^{35} + ( 7 - \beta_{2} ) q^{37} + ( -\beta_{5} - 5 \beta_{6} ) q^{41} + 4 \beta_{1} q^{43} + ( -2 \beta_{4} + 4 \beta_{7} ) q^{47} - q^{49} + ( -2 \beta_{5} - 3 \beta_{6} ) q^{53} + ( -10 \beta_{1} + 2 \beta_{3} ) q^{55} + ( -2 \beta_{4} - 4 \beta_{7} ) q^{59} + 2 q^{61} -8 \beta_{6} q^{65} + ( \beta_{1} + \beta_{3} ) q^{67} + ( 3 \beta_{4} + 2 \beta_{7} ) q^{71} + ( -3 - \beta_{2} ) q^{73} + ( \beta_{5} + 2 \beta_{6} ) q^{77} + ( -\beta_{1} + 3 \beta_{3} ) q^{79} + 4 \beta_{4} q^{83} + ( 11 + 3 \beta_{2} ) q^{85} + ( -\beta_{5} - \beta_{6} ) q^{89} + ( -\beta_{1} - \beta_{3} ) q^{91} + ( -4 \beta_{4} - 4 \beta_{7} ) q^{95} + ( -3 - \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{13} - 32q^{25} + 56q^{37} - 8q^{49} + 16q^{61} - 24q^{73} + 88q^{85} - 24q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 49 x^{4} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 65 \nu^{2} \)\()/144\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{4} + 49 \)\()/9\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} - 33 \nu^{2} \)\()/16\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - 65 \nu^{3} + 144 \nu \)\()/144\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + 65 \nu^{3} + 144 \nu \)\()/144\)
\(\beta_{6}\)\(=\)\((\)\( 5 \nu^{7} + 16 \nu^{5} + 181 \nu^{3} + 464 \nu \)\()/576\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{7} + 16 \nu^{5} - 181 \nu^{3} + 464 \nu \)\()/576\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 9 \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(4 \beta_{7} - 4 \beta_{6} + 5 \beta_{5} - 5 \beta_{4}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(9 \beta_{2} - 49\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(36 \beta_{7} + 36 \beta_{6} - 29 \beta_{5} - 29 \beta_{4}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-65 \beta_{3} - 297 \beta_{1}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-260 \beta_{7} + 260 \beta_{6} - 181 \beta_{5} + 181 \beta_{4}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
575.1
−1.10418 1.10418i
1.10418 1.10418i
−1.81129 1.81129i
1.81129 1.81129i
1.81129 + 1.81129i
−1.81129 + 1.81129i
1.10418 + 1.10418i
−1.10418 + 1.10418i
0 0 0 3.62258i 0 1.00000i 0 0 0
575.2 0 0 0 3.62258i 0 1.00000i 0 0 0
575.3 0 0 0 2.20837i 0 1.00000i 0 0 0
575.4 0 0 0 2.20837i 0 1.00000i 0 0 0
575.5 0 0 0 2.20837i 0 1.00000i 0 0 0
575.6 0 0 0 2.20837i 0 1.00000i 0 0 0
575.7 0 0 0 3.62258i 0 1.00000i 0 0 0
575.8 0 0 0 3.62258i 0 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 575.8
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
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.h.g 8
3.b odd 2 1 inner 4032.2.h.g 8
4.b odd 2 1 inner 4032.2.h.g 8
8.b even 2 1 2016.2.h.e 8
8.d odd 2 1 2016.2.h.e 8
12.b even 2 1 inner 4032.2.h.g 8
24.f even 2 1 2016.2.h.e 8
24.h odd 2 1 2016.2.h.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2016.2.h.e 8 8.b even 2 1
2016.2.h.e 8 8.d odd 2 1
2016.2.h.e 8 24.f even 2 1
2016.2.h.e 8 24.h odd 2 1
4032.2.h.g 8 1.a even 1 1 trivial
4032.2.h.g 8 3.b odd 2 1 inner
4032.2.h.g 8 4.b odd 2 1 inner
4032.2.h.g 8 12.b even 2 1 inner

Hecke kernels

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

\( T_{5}^{4} + 18 T_{5}^{2} + 64 \)
\( T_{11}^{4} - 26 T_{11}^{2} + 16 \)
\( T_{13}^{2} - 2 T_{13} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 2 T^{2} + 34 T^{4} - 50 T^{6} + 625 T^{8} )^{2} \)
$7$ \( ( 1 + T^{2} )^{4} \)
$11$ \( ( 1 + 18 T^{2} + 170 T^{4} + 2178 T^{6} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - 2 T + 10 T^{2} - 26 T^{3} + 169 T^{4} )^{4} \)
$17$ \( ( 1 - 26 T^{2} + 322 T^{4} - 7514 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 22 T^{2} + 361 T^{4} )^{4} \)
$23$ \( ( 1 + 66 T^{2} + 1994 T^{4} + 34914 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 32 T^{2} + 850 T^{4} - 26912 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 + 20 T^{2} + 934 T^{4} + 19220 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 - 14 T + 106 T^{2} - 518 T^{3} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 - 66 T^{2} + 3074 T^{4} - 110946 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 70 T^{2} + 1849 T^{4} )^{4} \)
$47$ \( ( 1 + 20 T^{2} - 2282 T^{4} + 44180 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 - 128 T^{2} + 8626 T^{4} - 359552 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 + 132 T^{2} + 8870 T^{4} + 459492 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 2 T + 61 T^{2} )^{8} \)
$67$ \( ( 1 - 232 T^{2} + 22366 T^{4} - 1041448 T^{6} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 + 130 T^{2} + 14154 T^{4} + 655330 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 + 6 T + 138 T^{2} + 438 T^{3} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 - 8 T^{2} + 11886 T^{4} - 49928 T^{6} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 + 44 T^{2} + 9910 T^{4} + 303116 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 338 T^{2} + 44386 T^{4} - 2677298 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 + 6 T + 186 T^{2} + 582 T^{3} + 9409 T^{4} )^{4} \)
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