# Properties

 Label 4032.2.p.j Level 4032 Weight 2 Character orbit 4032.p Analytic conductor 32.196 Analytic rank 0 Dimension 12 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4032 = 2^{6} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4032.p (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.1956820950$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{14}$$ Twist minimal: no (minimal twist has level 1344) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{5} + \beta_{10} q^{7} +O(q^{10})$$ $$q + \beta_{2} q^{5} + \beta_{10} q^{7} + \beta_{11} q^{11} -2 q^{13} + ( \beta_{3} - \beta_{5} ) q^{17} + ( \beta_{9} + \beta_{10} ) q^{19} + ( \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{23} + ( 3 + 2 \beta_{2} + \beta_{4} ) q^{25} + ( \beta_{1} - 2 \beta_{3} ) q^{29} + ( -\beta_{7} - 2 \beta_{11} ) q^{31} + ( \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{35} + ( \beta_{1} + \beta_{3} ) q^{37} + ( -2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{41} + ( -\beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{43} + ( 2 \beta_{7} + 2 \beta_{11} ) q^{47} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{49} + ( \beta_{1} + 2 \beta_{5} ) q^{53} + ( \beta_{7} - \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{55} + 2 \beta_{6} q^{59} + ( 6 + 2 \beta_{2} + 2 \beta_{4} ) q^{61} -2 \beta_{2} q^{65} + ( 2 \beta_{7} + 3 \beta_{9} - 3 \beta_{10} ) q^{67} + ( 3 \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{71} + ( 2 \beta_{3} + 2 \beta_{5} ) q^{73} + ( -\beta_{1} + \beta_{2} + 2 \beta_{5} ) q^{77} + ( -2 \beta_{6} + 2 \beta_{8} - \beta_{9} - \beta_{10} ) q^{79} + ( 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{83} + ( \beta_{1} + \beta_{3} + 4 \beta_{5} ) q^{85} + ( 2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{89} -2 \beta_{10} q^{91} + ( 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{95} + ( 2 \beta_{1} + 2 \beta_{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + O(q^{10})$$ $$12q - 24q^{13} + 36q^{25} - 12q^{49} + 72q^{61} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 15 x^{10} + 90 x^{8} - 247 x^{6} + 270 x^{4} + 21 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$8 \nu^{10} - 332 \nu^{8} + 3624 \nu^{6} - 15496 \nu^{4} + 23800 \nu^{2} + 126$$$$)/2947$$ $$\beta_{2}$$ $$=$$ $$($$$$-27 \nu^{10} + 489 \nu^{8} - 2969 \nu^{6} + 6410 \nu^{4} + 928 \nu^{2} - 7056$$$$)/2947$$ $$\beta_{3}$$ $$=$$ $$($$$$-54 \nu^{10} + 557 \nu^{8} - 1728 \nu^{6} - 1073 \nu^{4} + 6908 \nu^{2} + 623$$$$)/2947$$ $$\beta_{4}$$ $$=$$ $$($$$$58 \nu^{10} - 723 \nu^{8} + 3540 \nu^{6} - 6675 \nu^{4} - 902 \nu^{2} + 14175$$$$)/2947$$ $$\beta_{5}$$ $$=$$ $$($$$$-149 \nu^{10} + 2184 \nu^{8} - 12767 \nu^{6} + 34329 \nu^{4} - 38694 \nu^{2} + 1337$$$$)/2947$$ $$\beta_{6}$$ $$=$$ $$($$$$-346 \nu^{11} + 5939 \nu^{9} - 40121 \nu^{7} + 121218 \nu^{5} - 123779 \nu^{3} - 65863 \nu$$$$)/20629$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{11} - 15 \nu^{9} + 97 \nu^{7} - 317 \nu^{5} + 501 \nu^{3} - 161 \nu$$$$)/49$$ $$\beta_{8}$$ $$=$$ $$($$$$501 \nu^{11} - 7109 \nu^{9} + 40029 \nu^{7} - 98967 \nu^{5} + 88545 \nu^{3} + 45465 \nu$$$$)/20629$$ $$\beta_{9}$$ $$=$$ $$($$$$-151 \nu^{11} + 2267 \nu^{9} - 13673 \nu^{7} + 38203 \nu^{5} - 41697 \nu^{3} - 14903 \nu$$$$)/5894$$ $$\beta_{10}$$ $$=$$ $$($$$$-1137 \nu^{11} + 16663 \nu^{9} - 94903 \nu^{7} + 232931 \nu^{5} - 169503 \nu^{3} - 135051 \nu$$$$)/41258$$ $$\beta_{11}$$ $$=$$ $$($$$$-834 \nu^{11} + 13140 \nu^{9} - 83523 \nu^{7} + 252681 \nu^{5} - 346259 \nu^{3} + 118006 \nu$$$$)/20629$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{10} - 2 \beta_{9} + \beta_{8} - \beta_{7}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{4} - 2 \beta_{3} + \beta_{1} + 10$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{11} + 10 \beta_{10} - 4 \beta_{9} + 8 \beta_{8} - 5 \beta_{7}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$4 \beta_{5} - 8 \beta_{4} - 16 \beta_{3} - 4 \beta_{2} + 11 \beta_{1} + 30$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-8 \beta_{11} + 42 \beta_{10} + 8 \beta_{9} + 51 \beta_{8} - 18 \beta_{7} - 10 \beta_{6}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$17 \beta_{5} - 9 \beta_{4} - 46 \beta_{3} - 9 \beta_{2} + 41 \beta_{1} + 22$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-10 \beta_{11} + 148 \beta_{10} + 160 \beta_{9} + 265 \beta_{8} - 15 \beta_{7} - 98 \beta_{6}$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$208 \beta_{5} + 60 \beta_{4} - 444 \beta_{3} + 8 \beta_{2} + 469 \beta_{1} - 290$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$168 \beta_{11} + 312 \beta_{10} + 1218 \beta_{9} + 1168 \beta_{8} + 405 \beta_{7} - 594 \beta_{6}$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$978 \beta_{5} + 1098 \beta_{4} - 1792 \beta_{3} + 738 \beta_{2} + 2123 \beta_{1} - 3670$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$1956 \beta_{11} - 1050 \beta_{10} + 6968 \beta_{9} + 4135 \beta_{8} + 4364 \beta_{7} - 2574 \beta_{6}$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1567.1
 0.385124 + 0.500000i 0.385124 − 0.500000i −0.385124 + 0.500000i −0.385124 − 0.500000i −2.23871 − 0.500000i −2.23871 + 0.500000i 2.23871 − 0.500000i 2.23871 + 0.500000i −1.75780 − 0.500000i −1.75780 + 0.500000i 1.75780 − 0.500000i 1.75780 + 0.500000i
0 0 0 −2.76873 0 −0.480901 2.60168i 0 0 0
1567.2 0 0 0 −2.76873 0 −0.480901 + 2.60168i 0 0 0
1567.3 0 0 0 −2.76873 0 0.480901 2.60168i 0 0 0
1567.4 0 0 0 −2.76873 0 0.480901 + 2.60168i 0 0 0
1567.5 0 0 0 −1.11575 0 −1.37268 2.26180i 0 0 0
1567.6 0 0 0 −1.11575 0 −1.37268 + 2.26180i 0 0 0
1567.7 0 0 0 −1.11575 0 1.37268 2.26180i 0 0 0
1567.8 0 0 0 −1.11575 0 1.37268 + 2.26180i 0 0 0
1567.9 0 0 0 3.88448 0 −2.62383 0.339877i 0 0 0
1567.10 0 0 0 3.88448 0 −2.62383 + 0.339877i 0 0 0
1567.11 0 0 0 3.88448 0 2.62383 0.339877i 0 0 0
1567.12 0 0 0 3.88448 0 2.62383 + 0.339877i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1567.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
56.e even 2 1 inner
56.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.p.j 12
3.b odd 2 1 1344.2.p.c 12
4.b odd 2 1 inner 4032.2.p.j 12
7.b odd 2 1 4032.2.p.k 12
8.b even 2 1 4032.2.p.k 12
8.d odd 2 1 4032.2.p.k 12
12.b even 2 1 1344.2.p.c 12
21.c even 2 1 1344.2.p.d yes 12
24.f even 2 1 1344.2.p.d yes 12
24.h odd 2 1 1344.2.p.d yes 12
28.d even 2 1 4032.2.p.k 12
56.e even 2 1 inner 4032.2.p.j 12
56.h odd 2 1 inner 4032.2.p.j 12
84.h odd 2 1 1344.2.p.d yes 12
168.e odd 2 1 1344.2.p.c 12
168.i even 2 1 1344.2.p.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.2.p.c 12 3.b odd 2 1
1344.2.p.c 12 12.b even 2 1
1344.2.p.c 12 168.e odd 2 1
1344.2.p.c 12 168.i even 2 1
1344.2.p.d yes 12 21.c even 2 1
1344.2.p.d yes 12 24.f even 2 1
1344.2.p.d yes 12 24.h odd 2 1
1344.2.p.d yes 12 84.h odd 2 1
4032.2.p.j 12 1.a even 1 1 trivial
4032.2.p.j 12 4.b odd 2 1 inner
4032.2.p.j 12 56.e even 2 1 inner
4032.2.p.j 12 56.h odd 2 1 inner
4032.2.p.k 12 7.b odd 2 1
4032.2.p.k 12 8.b even 2 1
4032.2.p.k 12 8.d odd 2 1
4032.2.p.k 12 28.d even 2 1

## Hecke kernels

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

 $$T_{5}^{3} - 12 T_{5} - 12$$ $$T_{11}^{6} - 36 T_{11}^{4} + 96 T_{11}^{2} - 48$$ $$T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 3 T^{2} - 12 T^{3} + 15 T^{4} + 125 T^{6} )^{4}$$
$7$ $$1 + 6 T^{2} - 33 T^{4} - 556 T^{6} - 1617 T^{8} + 14406 T^{10} + 117649 T^{12}$$
$11$ $$( 1 + 30 T^{2} + 327 T^{4} + 2548 T^{6} + 39567 T^{8} + 439230 T^{10} + 1771561 T^{12} )^{2}$$
$13$ $$( 1 + 2 T + 13 T^{2} )^{12}$$
$17$ $$( 1 - 30 T^{2} + 831 T^{4} - 12628 T^{6} + 240159 T^{8} - 2505630 T^{10} + 24137569 T^{12} )^{2}$$
$19$ $$( 1 - 66 T^{2} + 2343 T^{4} - 54844 T^{6} + 845823 T^{8} - 8601186 T^{10} + 47045881 T^{12} )^{2}$$
$23$ $$( 1 - 78 T^{2} + 3567 T^{4} - 98836 T^{6} + 1886943 T^{8} - 21827598 T^{10} + 148035889 T^{12} )^{2}$$
$29$ $$( 1 - 42 T^{2} + 1575 T^{4} - 60076 T^{6} + 1324575 T^{8} - 29705802 T^{10} + 594823321 T^{12} )^{2}$$
$31$ $$( 1 + 54 T^{2} + 3471 T^{4} + 101684 T^{6} + 3335631 T^{8} + 49870134 T^{10} + 887503681 T^{12} )^{2}$$
$37$ $$( 1 - 126 T^{2} + 7863 T^{4} - 337412 T^{6} + 10764447 T^{8} - 236144286 T^{10} + 2565726409 T^{12} )^{2}$$
$41$ $$( 1 - 126 T^{2} + 6639 T^{4} - 258580 T^{6} + 11160159 T^{8} - 356045886 T^{10} + 4750104241 T^{12} )^{2}$$
$43$ $$( 1 + 78 T^{2} + 4503 T^{4} + 248420 T^{6} + 8326047 T^{8} + 266666478 T^{10} + 6321363049 T^{12} )^{2}$$
$47$ $$( 1 + 90 T^{2} + 7791 T^{4} + 391852 T^{6} + 17210319 T^{8} + 439171290 T^{10} + 10779215329 T^{12} )^{2}$$
$53$ $$( 1 - 186 T^{2} + 19575 T^{4} - 1258444 T^{6} + 54986175 T^{8} - 1467629466 T^{10} + 22164361129 T^{12} )^{2}$$
$59$ $$( 1 - 258 T^{2} + 31863 T^{4} - 2365180 T^{6} + 110915103 T^{8} - 3126279138 T^{10} + 42180533641 T^{12} )^{2}$$
$61$ $$( 1 - 18 T + 195 T^{2} - 1484 T^{3} + 11895 T^{4} - 66978 T^{5} + 226981 T^{6} )^{4}$$
$67$ $$( 1 + 78 T^{2} + 3399 T^{4} + 64676 T^{6} + 15258111 T^{8} + 1571787438 T^{10} + 90458382169 T^{12} )^{2}$$
$71$ $$( 1 - 222 T^{2} + 23727 T^{4} - 1839796 T^{6} + 119607807 T^{8} - 5641393182 T^{10} + 128100283921 T^{12} )^{2}$$
$73$ $$( 1 - 150 T^{2} + 16575 T^{4} - 1350452 T^{6} + 88328175 T^{8} - 4259736150 T^{10} + 151334226289 T^{12} )^{2}$$
$79$ $$( 1 - 234 T^{2} + 22767 T^{4} - 1649932 T^{6} + 142088847 T^{8} - 9114318954 T^{10} + 243087455521 T^{12} )^{2}$$
$83$ $$( 1 - 258 T^{2} + 42087 T^{4} - 4123708 T^{6} + 289937343 T^{8} - 12244246818 T^{10} + 326940373369 T^{12} )^{2}$$
$89$ $$( 1 - 222 T^{2} + 32463 T^{4} - 3429460 T^{6} + 257139423 T^{8} - 13928777502 T^{10} + 496981290961 T^{12} )^{2}$$
$97$ $$( 1 - 390 T^{2} + 77391 T^{4} - 9349652 T^{6} + 728171919 T^{8} - 34526419590 T^{10} + 832972004929 T^{12} )^{2}$$