# Properties

 Label 4032.2.j.f Level 4032 Weight 2 Character orbit 4032.j 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.j (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.1956820950$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: 16.0.11007531417600000000.1 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{18}\cdot 3^{6}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q -\beta_{12} q^{5} + \beta_{8} q^{7} +O(q^{10})$$ $$q -\beta_{12} q^{5} + \beta_{8} q^{7} + ( -\beta_{11} - \beta_{13} ) q^{11} + \beta_{2} q^{13} + \beta_{4} q^{17} + \beta_{10} q^{19} + ( \beta_{3} - \beta_{7} ) q^{23} + ( 4 - \beta_{1} ) q^{25} + \beta_{14} q^{29} -2 \beta_{8} q^{31} + \beta_{11} q^{35} -\beta_{2} q^{37} + \beta_{4} q^{41} + ( -2 \beta_{9} + \beta_{10} ) q^{43} - q^{49} + ( 2 \beta_{12} + 3 \beta_{14} ) q^{53} + ( -12 \beta_{8} + 2 \beta_{15} ) q^{55} -4 \beta_{13} q^{59} + ( 2 \beta_{2} - 2 \beta_{5} ) q^{61} + ( -2 \beta_{4} - 2 \beta_{6} ) q^{65} + \beta_{9} q^{67} + ( -\beta_{3} - \beta_{7} ) q^{71} + ( 1 - \beta_{1} ) q^{73} + ( -\beta_{12} + \beta_{14} ) q^{77} + ( -5 \beta_{8} + \beta_{15} ) q^{79} + ( -2 \beta_{11} - 2 \beta_{13} ) q^{83} + ( -3 \beta_{2} - 3 \beta_{5} ) q^{85} -\beta_{4} q^{89} + \beta_{9} q^{91} + 2 \beta_{3} q^{95} + ( 11 + \beta_{1} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q + 64q^{25} - 16q^{49} + 16q^{73} + 176q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 7 x^{12} + 48 x^{8} - 7 x^{4} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{12} + 161$$$$)/24$$ $$\beta_{2}$$ $$=$$ $$($$$$9 \nu^{12} - 64 \nu^{8} + 416 \nu^{4} - 31$$$$)/24$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{15} + 21 \nu^{13} - 144 \nu^{9} + 1008 \nu^{5} - 377 \nu^{3} - 147 \nu$$$$)/48$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{15} + 21 \nu^{13} - 144 \nu^{9} + 1008 \nu^{5} + 377 \nu^{3} - 147 \nu$$$$)/48$$ $$\beta_{5}$$ $$=$$ $$($$$$-7 \nu^{12} + 48 \nu^{8} - 336 \nu^{4} + 25$$$$)/12$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{15} - 17 \nu^{13} + 120 \nu^{9} - 816 \nu^{5} - 305 \nu^{3} + 119 \nu$$$$)/24$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{15} - 17 \nu^{13} + 120 \nu^{9} - 816 \nu^{5} + 305 \nu^{3} + 119 \nu$$$$)/24$$ $$\beta_{8}$$ $$=$$ $$($$$$-7 \nu^{14} + 48 \nu^{10} - 330 \nu^{6} + \nu^{2}$$$$)/18$$ $$\beta_{9}$$ $$=$$ $$($$$$-11 \nu^{14} + 80 \nu^{10} - 544 \nu^{6} + 157 \nu^{2}$$$$)/24$$ $$\beta_{10}$$ $$=$$ $$($$$$9 \nu^{14} - 64 \nu^{10} + 440 \nu^{6} - 127 \nu^{2}$$$$)/12$$ $$\beta_{11}$$ $$=$$ $$($$$$-37 \nu^{15} - 7 \nu^{13} + 256 \nu^{11} + 48 \nu^{9} - 1760 \nu^{7} - 336 \nu^{5} + 131 \nu^{3} - 47 \nu$$$$)/48$$ $$\beta_{12}$$ $$=$$ $$($$$$37 \nu^{15} - 7 \nu^{13} - 256 \nu^{11} + 48 \nu^{9} + 1760 \nu^{7} - 336 \nu^{5} - 131 \nu^{3} - 47 \nu$$$$)/48$$ $$\beta_{13}$$ $$=$$ $$($$$$15 \nu^{15} + 3 \nu^{13} - 104 \nu^{11} - 20 \nu^{9} + 712 \nu^{7} + 136 \nu^{5} - 53 \nu^{3} + 19 \nu$$$$)/12$$ $$\beta_{14}$$ $$=$$ $$($$$$15 \nu^{15} - 3 \nu^{13} - 104 \nu^{11} + 20 \nu^{9} + 712 \nu^{7} - 136 \nu^{5} - 53 \nu^{3} - 19 \nu$$$$)/12$$ $$\beta_{15}$$ $$=$$ $$($$$$-21 \nu^{14} + 144 \nu^{10} - 984 \nu^{6} + 3 \nu^{2}$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-3 \beta_{12} - 3 \beta_{11} - \beta_{4} - \beta_{3}$$$$)/12$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{15} - 3 \beta_{10} - 3 \beta_{9} - 9 \beta_{8}$$$$)/12$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} + 2 \beta_{4} - 2 \beta_{3}$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{5} - 3 \beta_{2} - \beta_{1} + 7$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$9 \beta_{14} - 9 \beta_{13} - 15 \beta_{12} - 15 \beta_{11} + 3 \beta_{7} + 3 \beta_{6} + 5 \beta_{4} + 5 \beta_{3}$$$$)/12$$ $$\nu^{6}$$ $$=$$ $$($$$$4 \beta_{15} - 27 \beta_{8}$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-24 \beta_{14} - 24 \beta_{13} + 39 \beta_{12} - 39 \beta_{11} - 8 \beta_{7} + 8 \beta_{6} + 13 \beta_{4} - 13 \beta_{3}$$$$)/12$$ $$\nu^{8}$$ $$=$$ $$($$$$-13 \beta_{5} - 21 \beta_{2} + 7 \beta_{1} - 47$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$21 \beta_{7} + 21 \beta_{6} + 34 \beta_{4} + 34 \beta_{3}$$$$)/6$$ $$\nu^{10}$$ $$=$$ $$($$$$55 \beta_{15} + 102 \beta_{10} + 165 \beta_{9} - 369 \beta_{8}$$$$)/12$$ $$\nu^{11}$$ $$=$$ $$($$$$-165 \beta_{14} - 165 \beta_{13} + 267 \beta_{12} - 267 \beta_{11} + 55 \beta_{7} - 55 \beta_{6} - 89 \beta_{4} + 89 \beta_{3}$$$$)/12$$ $$\nu^{12}$$ $$=$$ $$24 \beta_{1} - 161$$ $$\nu^{13}$$ $$=$$ $$($$$$-432 \beta_{14} + 432 \beta_{13} + 699 \beta_{12} + 699 \beta_{11} + 144 \beta_{7} + 144 \beta_{6} + 233 \beta_{4} + 233 \beta_{3}$$$$)/12$$ $$\nu^{14}$$ $$=$$ $$($$$$-377 \beta_{15} + 699 \beta_{10} + 1131 \beta_{9} + 2529 \beta_{8}$$$$)/12$$ $$\nu^{15}$$ $$=$$ $$($$$$377 \beta_{7} - 377 \beta_{6} - 610 \beta_{4} + 610 \beta_{3}$$$$)/6$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2591.1
 −1.56290 − 0.418778i −0.418778 − 1.56290i −0.418778 + 1.56290i −1.56290 + 0.418778i −0.159959 − 0.596975i −0.596975 − 0.159959i −0.596975 + 0.159959i −0.159959 + 0.596975i 0.159959 + 0.596975i 0.596975 + 0.159959i 0.596975 − 0.159959i 0.159959 − 0.596975i 1.56290 + 0.418778i 0.418778 + 1.56290i 0.418778 − 1.56290i 1.56290 − 0.418778i
0 0 0 −3.96336 0 1.00000i 0 0 0
2591.2 0 0 0 −3.96336 0 1.00000i 0 0 0
2591.3 0 0 0 −3.96336 0 1.00000i 0 0 0
2591.4 0 0 0 −3.96336 0 1.00000i 0 0 0
2591.5 0 0 0 −1.51387 0 1.00000i 0 0 0
2591.6 0 0 0 −1.51387 0 1.00000i 0 0 0
2591.7 0 0 0 −1.51387 0 1.00000i 0 0 0
2591.8 0 0 0 −1.51387 0 1.00000i 0 0 0
2591.9 0 0 0 1.51387 0 1.00000i 0 0 0
2591.10 0 0 0 1.51387 0 1.00000i 0 0 0
2591.11 0 0 0 1.51387 0 1.00000i 0 0 0
2591.12 0 0 0 1.51387 0 1.00000i 0 0 0
2591.13 0 0 0 3.96336 0 1.00000i 0 0 0
2591.14 0 0 0 3.96336 0 1.00000i 0 0 0
2591.15 0 0 0 3.96336 0 1.00000i 0 0 0
2591.16 0 0 0 3.96336 0 1.00000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2591.16 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
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.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.j.f 16
3.b odd 2 1 inner 4032.2.j.f 16
4.b odd 2 1 inner 4032.2.j.f 16
8.b even 2 1 inner 4032.2.j.f 16
8.d odd 2 1 inner 4032.2.j.f 16
12.b even 2 1 inner 4032.2.j.f 16
24.f even 2 1 inner 4032.2.j.f 16
24.h odd 2 1 inner 4032.2.j.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4032.2.j.f 16 1.a even 1 1 trivial
4032.2.j.f 16 3.b odd 2 1 inner
4032.2.j.f 16 4.b odd 2 1 inner
4032.2.j.f 16 8.b even 2 1 inner
4032.2.j.f 16 8.d odd 2 1 inner
4032.2.j.f 16 12.b even 2 1 inner
4032.2.j.f 16 24.f even 2 1 inner
4032.2.j.f 16 24.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 $$S_{2}^{\mathrm{new}}(4032, [\chi])$$:

 $$T_{5}^{4} - 18 T_{5}^{2} + 36$$ $$T_{19}^{2} - 12$$ $$T_{43}^{2} - 60$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 2 T^{2} + 6 T^{4} + 50 T^{6} + 625 T^{8} )^{4}$$
$7$ $$( 1 + T^{2} )^{8}$$
$11$ $$( 1 - 2 T^{2} - 162 T^{4} - 242 T^{6} + 14641 T^{8} )^{4}$$
$13$ $$( 1 - 16 T^{2} + 222 T^{4} - 2704 T^{6} + 28561 T^{8} )^{4}$$
$17$ $$( 1 - 14 T^{2} + 222 T^{4} - 4046 T^{6} + 83521 T^{8} )^{4}$$
$19$ $$( 1 + 26 T^{2} + 361 T^{4} )^{8}$$
$23$ $$( 1 + 38 T^{2} + 1014 T^{4} + 20102 T^{6} + 279841 T^{8} )^{4}$$
$29$ $$( 1 + 52 T^{2} + 841 T^{4} )^{8}$$
$31$ $$( 1 - 58 T^{2} + 961 T^{4} )^{8}$$
$37$ $$( 1 - 112 T^{2} + 5694 T^{4} - 153328 T^{6} + 1874161 T^{8} )^{4}$$
$41$ $$( 1 - 110 T^{2} + 5982 T^{4} - 184910 T^{6} + 2825761 T^{8} )^{4}$$
$43$ $$( 1 + 26 T^{2} + 1849 T^{4} )^{8}$$
$47$ $$( 1 + 47 T^{2} )^{16}$$
$53$ $$( 1 + 104 T^{2} + 5442 T^{4} + 292136 T^{6} + 7890481 T^{8} )^{4}$$
$59$ $$( 1 - 22 T^{2} + 3481 T^{4} )^{8}$$
$61$ $$( 1 - 100 T^{2} + 7062 T^{4} - 372100 T^{6} + 13845841 T^{8} )^{4}$$
$67$ $$( 1 + 232 T^{2} + 22254 T^{4} + 1041448 T^{6} + 20151121 T^{8} )^{4}$$
$71$ $$( 1 + 158 T^{2} + 12678 T^{4} + 796478 T^{6} + 25411681 T^{8} )^{4}$$
$73$ $$( 1 - 2 T + 102 T^{2} - 146 T^{3} + 5329 T^{4} )^{8}$$
$79$ $$( 1 - 176 T^{2} + 15726 T^{4} - 1098416 T^{6} + 38950081 T^{8} )^{4}$$
$83$ $$( 1 - 164 T^{2} + 14022 T^{4} - 1129796 T^{6} + 47458321 T^{8} )^{4}$$
$89$ $$( 1 - 302 T^{2} + 38238 T^{4} - 2392142 T^{6} + 62742241 T^{8} )^{4}$$
$97$ $$( 1 - 22 T + 270 T^{2} - 2134 T^{3} + 9409 T^{4} )^{8}$$