Properties

Label 4032.2.k.g
Level 4032
Weight 2
Character orbit 4032.k
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.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{25} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + \beta_{6} q^{7} +O(q^{10})\) \( q + \beta_{2} q^{5} + \beta_{6} q^{7} -\beta_{3} q^{11} + \beta_{13} q^{13} + ( -\beta_{1} + \beta_{2} ) q^{17} + ( \beta_{7} + \beta_{15} ) q^{19} + ( \beta_{3} - \beta_{9} + \beta_{10} ) q^{23} + ( -1 - \beta_{5} ) q^{25} + \beta_{8} q^{29} + ( \beta_{6} - \beta_{15} ) q^{31} + ( -\beta_{3} - \beta_{9} ) q^{35} -\beta_{5} q^{37} + ( 2 \beta_{1} + \beta_{2} ) q^{41} + ( 2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{43} + ( -\beta_{10} - \beta_{11} ) q^{47} + ( -1 + \beta_{5} - 2 \beta_{12} + \beta_{13} ) q^{49} + ( -\beta_{8} - \beta_{14} ) q^{53} + ( 2 \beta_{6} + \beta_{7} - \beta_{15} ) q^{55} + ( -\beta_{9} + \beta_{11} ) q^{59} + ( -\beta_{12} - 3 \beta_{13} ) q^{61} -\beta_{14} q^{65} + ( \beta_{4} - 3 \beta_{6} + 3 \beta_{7} ) q^{67} + ( -\beta_{3} - 2 \beta_{9} + 2 \beta_{10} ) q^{71} + 3 \beta_{12} q^{73} + ( -\beta_{1} - 3 \beta_{2} - \beta_{8} + \beta_{14} ) q^{77} + ( -3 \beta_{4} + \beta_{6} - \beta_{7} ) q^{79} + ( -2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{83} + ( 4 + \beta_{5} ) q^{85} + ( -2 \beta_{1} - 5 \beta_{2} ) q^{89} + ( \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{15} ) q^{91} + ( 2 \beta_{3} - \beta_{9} + \beta_{10} ) q^{95} + ( -\beta_{12} - 2 \beta_{13} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 16q^{25} - 16q^{49} + 64q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 24 x^{14} + 192 x^{12} + 672 x^{10} + 1092 x^{8} + 880 x^{6} + 352 x^{4} + 64 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{14} - 20 \nu^{12} - 100 \nu^{10} + 4 \nu^{8} + 918 \nu^{6} + 1440 \nu^{4} + 664 \nu^{2} + 72 \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( -7 \nu^{14} - 164 \nu^{12} - 1250 \nu^{10} - 3984 \nu^{8} - 5334 \nu^{6} - 3024 \nu^{4} - 596 \nu^{2} - 16 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{15} - 110 \nu^{13} - 728 \nu^{11} - 1625 \nu^{9} - 118 \nu^{7} + 2276 \nu^{5} + 1648 \nu^{3} + 270 \nu \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -11 \nu^{14} - 261 \nu^{12} - 2040 \nu^{10} - 6818 \nu^{8} - 10038 \nu^{6} - 6666 \nu^{4} - 1856 \nu^{2} - 172 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -6 \nu^{14} - 141 \nu^{12} - 1082 \nu^{10} - 3502 \nu^{8} - 4876 \nu^{6} - 3046 \nu^{4} - 804 \nu^{2} - 68 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( -11 \nu^{15} - 6 \nu^{14} - 260 \nu^{13} - 138 \nu^{12} - 2018 \nu^{11} - 1012 \nu^{10} - 6672 \nu^{9} - 2976 \nu^{8} - 9702 \nu^{7} - 3276 \nu^{6} - 6544 \nu^{5} - 1188 \nu^{4} - 2004 \nu^{3} - 72 \nu^{2} - 272 \nu - 16 \)\()/16\)
\(\beta_{7}\)\(=\)\((\)\( -11 \nu^{15} + 6 \nu^{14} - 260 \nu^{13} + 138 \nu^{12} - 2018 \nu^{11} + 1012 \nu^{10} - 6672 \nu^{9} + 2976 \nu^{8} - 9702 \nu^{7} + 3276 \nu^{6} - 6544 \nu^{5} + 1188 \nu^{4} - 2004 \nu^{3} + 72 \nu^{2} - 272 \nu + 16 \)\()/16\)
\(\beta_{8}\)\(=\)\((\)\( 11 \nu^{15} + 258 \nu^{13} + 1971 \nu^{11} + 6311 \nu^{9} + 8530 \nu^{7} + 4916 \nu^{5} + 1046 \nu^{3} + 38 \nu \)\()/8\)
\(\beta_{9}\)\(=\)\((\)\( -11 \nu^{15} - 15 \nu^{14} - 261 \nu^{13} - 352 \nu^{12} - 2040 \nu^{11} - 2692 \nu^{10} - 6817 \nu^{9} - 8638 \nu^{8} - 10018 \nu^{7} - 11726 \nu^{6} - 6554 \nu^{5} - 6808 \nu^{4} - 1640 \nu^{3} - 1496 \nu^{2} - 82 \nu - 76 \)\()/8\)
\(\beta_{10}\)\(=\)\((\)\( 11 \nu^{15} - 15 \nu^{14} + 261 \nu^{13} - 352 \nu^{12} + 2040 \nu^{11} - 2692 \nu^{10} + 6817 \nu^{9} - 8638 \nu^{8} + 10018 \nu^{7} - 11726 \nu^{6} + 6554 \nu^{5} - 6808 \nu^{4} + 1640 \nu^{3} - 1496 \nu^{2} + 82 \nu - 76 \)\()/8\)
\(\beta_{11}\)\(=\)\((\)\( -11 \nu^{15} - 27 \nu^{14} - 261 \nu^{13} - 636 \nu^{12} - 2040 \nu^{11} - 4904 \nu^{10} - 6817 \nu^{9} - 16030 \nu^{8} - 10018 \nu^{7} - 22886 \nu^{6} - 6554 \nu^{5} - 15248 \nu^{4} - 1640 \nu^{3} - 4576 \nu^{2} - 82 \nu - 428 \)\()/8\)
\(\beta_{12}\)\(=\)\((\)\( -19 \nu^{15} - 448 \nu^{13} - 3460 \nu^{11} - 11326 \nu^{9} - 16094 \nu^{7} - 10328 \nu^{5} - 2904 \nu^{3} - 316 \nu \)\()/8\)
\(\beta_{13}\)\(=\)\((\)\( -19 \nu^{15} - 454 \nu^{13} - 3598 \nu^{11} - 12338 \nu^{9} - 19070 \nu^{7} - 13604 \nu^{5} - 4092 \nu^{3} - 372 \nu \)\()/8\)
\(\beta_{14}\)\(=\)\((\)\( 11 \nu^{15} + 261 \nu^{13} + 2042 \nu^{11} + 6862 \nu^{9} + 10334 \nu^{7} + 7410 \nu^{5} + 2412 \nu^{3} + 252 \nu \)\()/4\)
\(\beta_{15}\)\(=\)\((\)\( 141 \nu^{15} - 6 \nu^{14} + 3324 \nu^{13} - 138 \nu^{12} + 25666 \nu^{11} - 1012 \nu^{10} + 84024 \nu^{9} - 2976 \nu^{8} + 119626 \nu^{7} - 3276 \nu^{6} + 77296 \nu^{5} - 1188 \nu^{4} + 21364 \nu^{3} - 72 \nu^{2} + 1888 \nu - 16 \)\()/16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{13} - \beta_{12} + \beta_{10} - \beta_{9} - 2 \beta_{8}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{11} + 2 \beta_{10} + \beta_{9} + 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{2} + \beta_{1} - 12\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{15} + 3 \beta_{14} - 11 \beta_{13} + 7 \beta_{12} - 8 \beta_{10} + 8 \beta_{9} + 16 \beta_{8} - 6 \beta_{7} - 3 \beta_{6} - 4 \beta_{3}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-6 \beta_{11} - 10 \beta_{10} - 4 \beta_{9} - 14 \beta_{7} + 14 \beta_{6} + 13 \beta_{5} - 4 \beta_{4} + 12 \beta_{2} - 10 \beta_{1} + 48\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(50 \beta_{15} - 55 \beta_{14} + 126 \beta_{13} - 64 \beta_{12} + 83 \beta_{10} - 83 \beta_{9} - 188 \beta_{8} + 80 \beta_{7} + 30 \beta_{6} + 58 \beta_{3}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(72 \beta_{11} + 109 \beta_{10} + 37 \beta_{9} + 181 \beta_{7} - 181 \beta_{6} - 161 \beta_{5} + 67 \beta_{4} - 134 \beta_{2} + 140 \beta_{1} - 504\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-336 \beta_{15} + 378 \beta_{14} - 752 \beta_{13} + 339 \beta_{12} - 478 \beta_{10} + 478 \beta_{9} + 1142 \beta_{8} - 497 \beta_{7} - 161 \beta_{6} - 372 \beta_{3}\)\()/2\)
\(\nu^{8}\)\(=\)\(-438 \beta_{11} - 636 \beta_{10} - 198 \beta_{9} - 1130 \beta_{7} + 1130 \beta_{6} + 990 \beta_{5} - 450 \beta_{4} + 784 \beta_{2} - 896 \beta_{1} + 2910\)
\(\nu^{9}\)\(=\)\((\)\(4248 \beta_{15} - 4812 \beta_{14} + 9127 \beta_{13} - 3907 \beta_{12} + 5723 \beta_{10} - 5723 \beta_{9} - 13942 \beta_{8} + 6108 \beta_{7} + 1860 \beta_{6} + 4632 \beta_{3}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(10707 \beta_{11} + 15294 \beta_{10} + 4587 \beta_{9} + 27862 \beta_{7} - 27862 \beta_{6} - 24294 \beta_{5} + 11376 \beta_{4} - 18846 \beta_{2} + 22287 \beta_{1} - 69716\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-52613 \beta_{15} + 59741 \beta_{14} - 111425 \beta_{13} + 46733 \beta_{12} - 69508 \beta_{10} + 69508 \beta_{9} + 170488 \beta_{8} - 74866 \beta_{7} - 22253 \beta_{6} - 57084 \beta_{3}\)\()/2\)
\(\nu^{12}\)\(=\)\(-65522 \beta_{11} - 93010 \beta_{10} - 27488 \beta_{9} - 170986 \beta_{7} + 170986 \beta_{6} + 148879 \beta_{5} - 70448 \beta_{4} + 114548 \beta_{2} - 137214 \beta_{1} + 423384\)
\(\nu^{13}\)\(=\)\((\)\(646906 \beta_{15} - 735189 \beta_{14} + 1363030 \beta_{13} - 567236 \beta_{12} + 848621 \beta_{10} - 848621 \beta_{9} - 2086356 \beta_{8} + 916968 \beta_{7} + 270062 \beta_{6} + 700742 \beta_{3}\)\()/2\)
\(\nu^{14}\)\(=\)\(802232 \beta_{11} + 1136119 \beta_{10} + 333887 \beta_{9} + 2095431 \beta_{7} - 2095431 \beta_{6} - 1823791 \beta_{5} + 866233 \beta_{4} - 1398722 \beta_{2} + 1683568 \beta_{1} - 5168936\)
\(\nu^{15}\)\(=\)\(-3966632 \beta_{15} + 4509412 \beta_{14} - 8342408 \beta_{13} + 3461601 \beta_{12} - 5190196 \beta_{10} + 5190196 \beta_{9} + 12770386 \beta_{8} - 5614479 \beta_{7} - 1647847 \beta_{6} - 4294488 \beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3905.1
0.724535i
0.724535i
1.05636i
1.05636i
2.08509i
2.08509i
0.886177i
0.886177i
0.528036i
0.528036i
3.49930i
3.49930i
0.357857i
0.357857i
2.13875i
2.13875i
0 0 0 −2.61313 0 −1.25928 2.32685i 0 0 0
3905.2 0 0 0 −2.61313 0 −1.25928 + 2.32685i 0 0 0
3905.3 0 0 0 −2.61313 0 1.25928 2.32685i 0 0 0
3905.4 0 0 0 −2.61313 0 1.25928 + 2.32685i 0 0 0
3905.5 0 0 0 −1.08239 0 −2.10100 1.60804i 0 0 0
3905.6 0 0 0 −1.08239 0 −2.10100 + 1.60804i 0 0 0
3905.7 0 0 0 −1.08239 0 2.10100 1.60804i 0 0 0
3905.8 0 0 0 −1.08239 0 2.10100 + 1.60804i 0 0 0
3905.9 0 0 0 1.08239 0 −2.10100 1.60804i 0 0 0
3905.10 0 0 0 1.08239 0 −2.10100 + 1.60804i 0 0 0
3905.11 0 0 0 1.08239 0 2.10100 1.60804i 0 0 0
3905.12 0 0 0 1.08239 0 2.10100 + 1.60804i 0 0 0
3905.13 0 0 0 2.61313 0 −1.25928 2.32685i 0 0 0
3905.14 0 0 0 2.61313 0 −1.25928 + 2.32685i 0 0 0
3905.15 0 0 0 2.61313 0 1.25928 2.32685i 0 0 0
3905.16 0 0 0 2.61313 0 1.25928 + 2.32685i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3905.16
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
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.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.k.g 16
3.b odd 2 1 inner 4032.2.k.g 16
4.b odd 2 1 inner 4032.2.k.g 16
7.b odd 2 1 inner 4032.2.k.g 16
8.b even 2 1 2016.2.k.a 16
8.d odd 2 1 2016.2.k.a 16
12.b even 2 1 inner 4032.2.k.g 16
21.c even 2 1 inner 4032.2.k.g 16
24.f even 2 1 2016.2.k.a 16
24.h odd 2 1 2016.2.k.a 16
28.d even 2 1 inner 4032.2.k.g 16
56.e even 2 1 2016.2.k.a 16
56.h odd 2 1 2016.2.k.a 16
84.h odd 2 1 inner 4032.2.k.g 16
168.e odd 2 1 2016.2.k.a 16
168.i even 2 1 2016.2.k.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2016.2.k.a 16 8.b even 2 1
2016.2.k.a 16 8.d odd 2 1
2016.2.k.a 16 24.f even 2 1
2016.2.k.a 16 24.h odd 2 1
2016.2.k.a 16 56.e even 2 1
2016.2.k.a 16 56.h odd 2 1
2016.2.k.a 16 168.e odd 2 1
2016.2.k.a 16 168.i even 2 1
4032.2.k.g 16 1.a even 1 1 trivial
4032.2.k.g 16 3.b odd 2 1 inner
4032.2.k.g 16 4.b odd 2 1 inner
4032.2.k.g 16 7.b odd 2 1 inner
4032.2.k.g 16 12.b even 2 1 inner
4032.2.k.g 16 21.c even 2 1 inner
4032.2.k.g 16 28.d even 2 1 inner
4032.2.k.g 16 84.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 \(S_{2}^{\mathrm{new}}(4032, [\chi])\):

\( T_{5}^{4} - 8 T_{5}^{2} + 8 \)
\( T_{43}^{4} - 152 T_{43}^{2} + 5488 \)
\( T_{67}^{4} - 208 T_{67}^{2} + 448 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 + 12 T^{2} + 78 T^{4} + 300 T^{6} + 625 T^{8} )^{4} \)
$7$ \( ( 1 + 4 T^{2} + 70 T^{4} + 196 T^{6} + 2401 T^{8} )^{2} \)
$11$ \( ( 1 - 24 T^{2} + 314 T^{4} - 2904 T^{6} + 14641 T^{8} )^{4} \)
$13$ \( ( 1 - 36 T^{2} + 630 T^{4} - 6084 T^{6} + 28561 T^{8} )^{4} \)
$17$ \( ( 1 + 28 T^{2} + 382 T^{4} + 8092 T^{6} + 83521 T^{8} )^{4} \)
$19$ \( ( 1 - 12 T^{2} - 42 T^{4} - 4332 T^{6} + 130321 T^{8} )^{4} \)
$23$ \( ( 1 - 40 T^{2} + 810 T^{4} - 21160 T^{6} + 279841 T^{8} )^{4} \)
$29$ \( ( 1 - 56 T^{2} + 841 T^{4} )^{8} \)
$31$ \( ( 1 - 60 T^{2} + 2694 T^{4} - 57660 T^{6} + 923521 T^{8} )^{4} \)
$37$ \( ( 1 + 66 T^{2} + 1369 T^{4} )^{8} \)
$41$ \( ( 1 + 28 T^{2} + 3166 T^{4} + 47068 T^{6} + 2825761 T^{8} )^{4} \)
$43$ \( ( 1 + 20 T^{2} + 3510 T^{4} + 36980 T^{6} + 3418801 T^{8} )^{4} \)
$47$ \( ( 1 + 60 T^{2} + 2118 T^{4} + 132540 T^{6} + 4879681 T^{8} )^{4} \)
$53$ \( ( 1 - 176 T^{2} + 13234 T^{4} - 494384 T^{6} + 7890481 T^{8} )^{4} \)
$59$ \( ( 1 + 108 T^{2} + 9366 T^{4} + 375948 T^{6} + 12117361 T^{8} )^{4} \)
$61$ \( ( 1 - 84 T^{2} + 9078 T^{4} - 312564 T^{6} + 13845841 T^{8} )^{4} \)
$67$ \( ( 1 + 60 T^{2} - 490 T^{4} + 269340 T^{6} + 20151121 T^{8} )^{4} \)
$71$ \( ( 1 - 40 T^{2} + 10410 T^{4} - 201640 T^{6} + 25411681 T^{8} )^{4} \)
$73$ \( ( 1 - 148 T^{2} + 13542 T^{4} - 788692 T^{6} + 28398241 T^{8} )^{4} \)
$79$ \( ( 1 - 20 T^{2} + 6310 T^{4} - 124820 T^{6} + 38950081 T^{8} )^{4} \)
$83$ \( ( 1 + 76 T^{2} - 266 T^{4} + 523564 T^{6} + 47458321 T^{8} )^{4} \)
$89$ \( ( 1 + 28 T^{2} - 3170 T^{4} + 221788 T^{6} + 62742241 T^{8} )^{4} \)
$97$ \( ( 1 - 308 T^{2} + 42502 T^{4} - 2897972 T^{6} + 88529281 T^{8} )^{4} \)
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