# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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