Properties

Label 4032.2.h.h
Level 4032
Weight 2
Character orbit 4032.h
Analytic conductor 32.196
Analytic rank 0
Dimension 12
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: \(12\)
Coefficient field: 12.0.653473922154496.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{5} + \beta_{3} q^{7} +O(q^{10})\) \( q -\beta_{2} q^{5} + \beta_{3} q^{7} + ( -\beta_{5} - \beta_{6} ) q^{11} + \beta_{10} q^{13} + ( \beta_{2} + 2 \beta_{4} ) q^{17} + ( \beta_{1} - \beta_{3} + \beta_{7} ) q^{19} + ( -3 \beta_{6} - \beta_{9} ) q^{23} + ( -1 - \beta_{10} ) q^{25} + \beta_{4} q^{29} + ( \beta_{1} - \beta_{3} - \beta_{7} ) q^{31} -\beta_{5} q^{35} + ( 2 \beta_{8} - \beta_{10} ) q^{37} + ( -2 \beta_{4} + \beta_{11} ) q^{41} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{43} + ( -3 \beta_{5} + \beta_{9} ) q^{47} - q^{49} + ( \beta_{2} - \beta_{4} - \beta_{11} ) q^{53} + ( -\beta_{1} - 7 \beta_{3} + \beta_{7} ) q^{55} + ( \beta_{5} - 4 \beta_{6} + \beta_{9} ) q^{59} + ( -5 - \beta_{8} - \beta_{10} ) q^{61} + ( -3 \beta_{2} + 2 \beta_{4} - \beta_{11} ) q^{65} + ( -\beta_{1} - 3 \beta_{3} - 2 \beta_{7} ) q^{67} + ( -5 \beta_{6} + \beta_{9} ) q^{71} + ( -1 - 3 \beta_{8} + 2 \beta_{10} ) q^{73} + ( \beta_{2} - \beta_{4} ) q^{77} + ( 3 \beta_{1} + \beta_{3} ) q^{79} + ( \beta_{5} - 2 \beta_{6} - \beta_{9} ) q^{83} + ( 4 - 2 \beta_{8} + \beta_{10} ) q^{85} + ( 2 \beta_{2} - 2 \beta_{4} - \beta_{11} ) q^{89} -\beta_{7} q^{91} + ( 4 \beta_{5} - 4 \beta_{6} ) q^{95} + ( 1 + 3 \beta_{8} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + O(q^{10}) \) \( 12q - 12q^{25} - 8q^{37} - 12q^{49} - 56q^{61} + 56q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{10} + 13 x^{8} - 28 x^{6} + 52 x^{4} - 64 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{10} - 14 \nu^{8} + 9 \nu^{6} - 22 \nu^{4} - 64 \nu^{2} + 32 \)\()/64\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{9} - 2 \nu^{7} + 9 \nu^{5} - 10 \nu^{3} + 16 \nu \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{10} + 10 \nu^{8} - 27 \nu^{6} + 34 \nu^{4} - 64 \nu^{2} + 32 \)\()/64\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{11} - 4 \nu^{9} + 5 \nu^{7} - 12 \nu^{5} + 12 \nu^{3} - 16 \nu \)\()/32\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{11} + 2 \nu^{9} - 11 \nu^{7} + 26 \nu^{5} - 48 \nu^{3} + 96 \nu \)\()/64\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{11} + 6 \nu^{9} - 19 \nu^{7} + 30 \nu^{5} - 24 \nu^{3} \)\()/64\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{10} + 10 \nu^{8} - 17 \nu^{6} + 50 \nu^{4} - 72 \nu^{2} + 96 \)\()/16\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{10} + 4 \nu^{8} - 13 \nu^{6} + 28 \nu^{4} - 36 \nu^{2} + 40 \)\()/8\)
\(\beta_{9}\)\(=\)\((\)\( -5 \nu^{11} + 38 \nu^{9} - 109 \nu^{7} + 206 \nu^{5} - 320 \nu^{3} + 480 \nu \)\()/64\)
\(\beta_{10}\)\(=\)\((\)\( -\nu^{10} + 2 \nu^{8} - 5 \nu^{6} + 10 \nu^{4} - 12 \nu^{2} + 8 \)\()/4\)
\(\beta_{11}\)\(=\)\((\)\( 3 \nu^{11} - 10 \nu^{9} + 27 \nu^{7} - 50 \nu^{5} + 96 \nu^{3} - 48 \nu \)\()/16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11} + \beta_{9} - 2 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} - \beta_{2}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{8} - \beta_{7} - 2 \beta_{3} - 2 \beta_{1} + 3\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{11} + \beta_{9} + 10 \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{2}\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{10} + 2 \beta_{8} - 11 \beta_{3} - \beta_{1} - 6\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{11} - 3 \beta_{9} + 14 \beta_{6} - \beta_{5} + 6 \beta_{4} + 15 \beta_{2}\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(4 \beta_{10} - 5 \beta_{8} + 5 \beta_{7} - 14 \beta_{3} + 2 \beta_{1} - 7\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-3 \beta_{11} - 9 \beta_{9} - 26 \beta_{6} + 9 \beta_{5} - 30 \beta_{4} + 15 \beta_{2}\)\()/8\)
\(\nu^{8}\)\(=\)\((\)\(-\beta_{10} - 10 \beta_{8} + 8 \beta_{7} + 19 \beta_{3} - 7 \beta_{1} - 2\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-\beta_{11} + 3 \beta_{9} - 46 \beta_{6} - 31 \beta_{5} - 102 \beta_{4} - 15 \beta_{2}\)\()/8\)
\(\nu^{10}\)\(=\)\((\)\(-28 \beta_{10} + 13 \beta_{8} + 3 \beta_{7} + 22 \beta_{3} - 10 \beta_{1} - 33\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(3 \beta_{11} + 25 \beta_{9} - 38 \beta_{6} - 121 \beta_{5} + 62 \beta_{4} + 17 \beta_{2}\)\()/8\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
575.1
1.35489 + 0.405301i
−1.35489 + 0.405301i
0.892524 1.09700i
−0.892524 1.09700i
−1.16947 0.795191i
1.16947 0.795191i
1.16947 + 0.795191i
−1.16947 + 0.795191i
−0.892524 + 1.09700i
0.892524 + 1.09700i
−1.35489 0.405301i
1.35489 0.405301i
0 0 0 3.31339i 0 1.00000i 0 0 0
575.2 0 0 0 3.31339i 0 1.00000i 0 0 0
575.3 0 0 0 2.56483i 0 1.00000i 0 0 0
575.4 0 0 0 2.56483i 0 1.00000i 0 0 0
575.5 0 0 0 0.665647i 0 1.00000i 0 0 0
575.6 0 0 0 0.665647i 0 1.00000i 0 0 0
575.7 0 0 0 0.665647i 0 1.00000i 0 0 0
575.8 0 0 0 0.665647i 0 1.00000i 0 0 0
575.9 0 0 0 2.56483i 0 1.00000i 0 0 0
575.10 0 0 0 2.56483i 0 1.00000i 0 0 0
575.11 0 0 0 3.31339i 0 1.00000i 0 0 0
575.12 0 0 0 3.31339i 0 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 575.12
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.h 12
3.b odd 2 1 inner 4032.2.h.h 12
4.b odd 2 1 inner 4032.2.h.h 12
8.b even 2 1 252.2.e.a 12
8.d odd 2 1 252.2.e.a 12
12.b even 2 1 inner 4032.2.h.h 12
24.f even 2 1 252.2.e.a 12
24.h odd 2 1 252.2.e.a 12
56.e even 2 1 1764.2.e.g 12
56.h odd 2 1 1764.2.e.g 12
168.e odd 2 1 1764.2.e.g 12
168.i even 2 1 1764.2.e.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.e.a 12 8.b even 2 1
252.2.e.a 12 8.d odd 2 1
252.2.e.a 12 24.f even 2 1
252.2.e.a 12 24.h odd 2 1
1764.2.e.g 12 56.e even 2 1
1764.2.e.g 12 56.h odd 2 1
1764.2.e.g 12 168.e odd 2 1
1764.2.e.g 12 168.i even 2 1
4032.2.h.h 12 1.a even 1 1 trivial
4032.2.h.h 12 3.b odd 2 1 inner
4032.2.h.h 12 4.b odd 2 1 inner
4032.2.h.h 12 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 \(S_{2}^{\mathrm{new}}(4032, [\chi])\):

\( T_{5}^{6} + 18 T_{5}^{4} + 80 T_{5}^{2} + 32 \)
\( T_{11}^{6} - 28 T_{11}^{4} + 132 T_{11}^{2} - 128 \)
\( T_{13}^{3} - 28 T_{13} + 16 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( ( 1 - 12 T^{2} + 95 T^{4} - 568 T^{6} + 2375 T^{8} - 7500 T^{10} + 15625 T^{12} )^{2} \)
$7$ \( ( 1 + T^{2} )^{6} \)
$11$ \( ( 1 + 38 T^{2} + 715 T^{4} + 9068 T^{6} + 86515 T^{8} + 556358 T^{10} + 1771561 T^{12} )^{2} \)
$13$ \( ( 1 + 11 T^{2} + 16 T^{3} + 143 T^{4} + 2197 T^{6} )^{4} \)
$17$ \( ( 1 - 68 T^{2} + 2167 T^{4} - 44168 T^{6} + 626263 T^{8} - 5679428 T^{10} + 24137569 T^{12} )^{2} \)
$19$ \( ( 1 - 50 T^{2} + 1639 T^{4} - 35804 T^{6} + 591679 T^{8} - 6516050 T^{10} + 47045881 T^{12} )^{2} \)
$23$ \( ( 1 - 2 T^{2} + 979 T^{4} + 3772 T^{6} + 517891 T^{8} - 559682 T^{10} + 148035889 T^{12} )^{2} \)
$29$ \( ( 1 - 56 T^{2} + 841 T^{4} )^{6} \)
$31$ \( ( 1 - 58 T^{2} + 3727 T^{4} - 113644 T^{6} + 3581647 T^{8} - 53564218 T^{10} + 887503681 T^{12} )^{2} \)
$37$ \( ( 1 + 2 T + 47 T^{2} + 276 T^{3} + 1739 T^{4} + 2738 T^{5} + 50653 T^{6} )^{4} \)
$41$ \( ( 1 - 132 T^{2} + 10823 T^{4} - 527752 T^{6} + 18193463 T^{8} - 373000452 T^{10} + 4750104241 T^{12} )^{2} \)
$43$ \( ( 1 - 98 T^{2} + 4567 T^{4} - 172988 T^{6} + 8444383 T^{8} - 335042498 T^{10} + 6321363049 T^{12} )^{2} \)
$47$ \( ( 1 + 10 T^{2} + 4591 T^{4} + 70732 T^{6} + 10141519 T^{8} + 48796810 T^{10} + 10779215329 T^{12} )^{2} \)
$53$ \( ( 1 - 184 T^{2} + 16171 T^{4} - 975856 T^{6} + 45424339 T^{8} - 1451848504 T^{10} + 22164361129 T^{12} )^{2} \)
$59$ \( ( 1 + 146 T^{2} + 7799 T^{4} + 306396 T^{6} + 27148319 T^{8} + 1769134706 T^{10} + 42180533641 T^{12} )^{2} \)
$61$ \( ( 1 + 14 T + 187 T^{2} + 1700 T^{3} + 11407 T^{4} + 52094 T^{5} + 226981 T^{6} )^{4} \)
$67$ \( ( 1 - 186 T^{2} + 13175 T^{4} - 680684 T^{6} + 59142575 T^{8} - 3748108506 T^{10} + 90458382169 T^{12} )^{2} \)
$71$ \( ( 1 + 158 T^{2} + 5683 T^{4} - 107140 T^{6} + 28648003 T^{8} + 4015045598 T^{10} + 128100283921 T^{12} )^{2} \)
$73$ \( ( 1 + 47 T^{2} - 352 T^{3} + 3431 T^{4} + 389017 T^{6} )^{4} \)
$79$ \( ( 1 - 162 T^{2} + 19359 T^{4} - 2006332 T^{6} + 120819519 T^{8} - 6309913122 T^{10} + 243087455521 T^{12} )^{2} \)
$83$ \( ( 1 + 370 T^{2} + 65191 T^{4} + 6834652 T^{6} + 449100799 T^{8} + 17559578770 T^{10} + 326940373369 T^{12} )^{2} \)
$89$ \( ( 1 - 308 T^{2} + 49895 T^{4} - 5241384 T^{6} + 395218295 T^{8} - 19324610228 T^{10} + 496981290961 T^{12} )^{2} \)
$97$ \( ( 1 + 135 T^{2} + 128 T^{3} + 13095 T^{4} + 912673 T^{6} )^{4} \)
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