# Properties

 Label 4032.2.j.e 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: 16.0.162447943996702457856.1 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{18}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{13} q^{5} + \beta_{1} q^{7} +O(q^{10})$$ $$q + \beta_{13} q^{5} + \beta_{1} q^{7} + \beta_{9} q^{11} -\beta_{5} q^{13} + ( \beta_{8} + 3 \beta_{10} ) q^{17} + ( \beta_{4} - \beta_{7} ) q^{19} -\beta_{12} q^{23} + \beta_{3} q^{25} + ( -3 \beta_{11} - \beta_{13} ) q^{29} -2 \beta_{15} q^{31} -\beta_{14} q^{35} + ( -2 \beta_{2} - \beta_{5} ) q^{37} + ( -3 \beta_{8} - \beta_{10} ) q^{41} + ( -3 \beta_{4} - \beta_{7} ) q^{43} -4 \beta_{6} q^{47} - q^{49} + ( -\beta_{11} + 3 \beta_{13} ) q^{53} -2 \beta_{1} q^{55} + ( -2 \beta_{2} - 2 \beta_{5} ) q^{61} + ( -2 \beta_{8} + 4 \beta_{10} ) q^{65} + ( -2 \beta_{4} - \beta_{7} ) q^{67} + ( -2 \beta_{6} + \beta_{12} ) q^{71} + ( -5 - \beta_{3} ) q^{73} + \beta_{11} q^{77} + ( -3 \beta_{1} + 3 \beta_{15} ) q^{79} + ( 6 \beta_{9} - 4 \beta_{14} ) q^{83} + \beta_{2} q^{85} + ( 3 \beta_{8} + \beta_{10} ) q^{89} -\beta_{7} q^{91} + ( -6 \beta_{6} - 2 \beta_{12} ) q^{95} + ( -7 + \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 16q^{49} - 80q^{73} - 112q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - x^{12} - 15 x^{8} - 16 x^{4} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{14} - 15 \nu^{10} - 33 \nu^{6} + 256 \nu^{2}$$$$)/576$$ $$\beta_{2}$$ $$=$$ $$($$$$-13 \nu^{12} + 45 \nu^{8} - 45 \nu^{4} - 32$$$$)/360$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{12} + \nu^{8} + 31 \nu^{4} + 8$$$$)/24$$ $$\beta_{4}$$ $$=$$ $$($$$$-3 \nu^{14} + 35 \nu^{10} - 115 \nu^{6} - 112 \nu^{2}$$$$)/480$$ $$\beta_{5}$$ $$=$$ $$($$$$19 \nu^{12} + 45 \nu^{8} - 45 \nu^{4} - 784$$$$)/360$$ $$\beta_{6}$$ $$=$$ $$($$$$4 \nu^{15} - 17 \nu^{13} - 15 \nu^{9} + 255 \nu^{5} + 356 \nu^{3} + 272 \nu$$$$)/1440$$ $$\beta_{7}$$ $$=$$ $$($$$$-3 \nu^{14} - 5 \nu^{10} + 85 \nu^{6} + 328 \nu^{2}$$$$)/240$$ $$\beta_{8}$$ $$=$$ $$($$$$-19 \nu^{15} + 88 \nu^{13} + 195 \nu^{11} + 360 \nu^{9} + 1005 \nu^{7} - 360 \nu^{5} - 416 \nu^{3} - 11968 \nu$$$$)/5760$$ $$\beta_{9}$$ $$=$$ $$($$$$5 \nu^{15} - 8 \nu^{13} - 21 \nu^{11} + 72 \nu^{9} + 69 \nu^{7} - 72 \nu^{5} - 224 \nu^{3} - 64 \nu$$$$)/1152$$ $$\beta_{10}$$ $$=$$ $$($$$$-4 \nu^{15} - 17 \nu^{13} - 15 \nu^{9} + 255 \nu^{5} - 356 \nu^{3} + 272 \nu$$$$)/1440$$ $$\beta_{11}$$ $$=$$ $$($$$$-5 \nu^{15} - 8 \nu^{13} + 21 \nu^{11} + 72 \nu^{9} - 69 \nu^{7} - 72 \nu^{5} + 224 \nu^{3} - 64 \nu$$$$)/1152$$ $$\beta_{12}$$ $$=$$ $$($$$$-19 \nu^{15} - 88 \nu^{13} + 195 \nu^{11} - 360 \nu^{9} + 1005 \nu^{7} + 360 \nu^{5} - 416 \nu^{3} + 11968 \nu$$$$)/5760$$ $$\beta_{13}$$ $$=$$ $$($$$$-11 \nu^{15} + 4 \nu^{13} - 21 \nu^{11} + 60 \nu^{9} + 69 \nu^{7} + 132 \nu^{5} + 656 \nu^{3} + 128 \nu$$$$)/1152$$ $$\beta_{14}$$ $$=$$ $$($$$$11 \nu^{15} + 4 \nu^{13} + 21 \nu^{11} + 60 \nu^{9} - 69 \nu^{7} + 132 \nu^{5} - 656 \nu^{3} + 128 \nu$$$$)/1152$$ $$\beta_{15}$$ $$=$$ $$($$$$7 \nu^{14} + 9 \nu^{10} - 57 \nu^{6} + 32 \nu^{2}$$$$)/192$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{14} + \beta_{13} + \beta_{12} - \beta_{10} - \beta_{8} - \beta_{6}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{15} + 2 \beta_{7} + \beta_{4} + 3 \beta_{1}$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{14} + \beta_{13} + \beta_{11} - 5 \beta_{10} - \beta_{9} + 5 \beta_{6}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{5} + 3 \beta_{3} - 2 \beta_{2} + 1$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$6 \beta_{14} + 6 \beta_{13} - \beta_{12} - 5 \beta_{11} + 6 \beta_{10} - 5 \beta_{9} + \beta_{8} + 6 \beta_{6}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$5 \beta_{7} - 5 \beta_{4} - 18 \beta_{1}$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-3 \beta_{14} + 3 \beta_{13} + 7 \beta_{12} - 10 \beta_{11} - 3 \beta_{10} + 10 \beta_{9} + 7 \beta_{8} + 3 \beta_{6}$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$14 \beta_{5} + 3 \beta_{3} + 17 \beta_{2} + 31$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$17 \beta_{14} + 17 \beta_{13} + 17 \beta_{11} - 5 \beta_{10} + 17 \beta_{9} - 5 \beta_{6}$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$11 \beta_{15} + 23 \beta_{7} + 34 \beta_{4} - 57 \beta_{1}$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$22 \beta_{14} - 22 \beta_{13} + 23 \beta_{12} + 45 \beta_{11} - 22 \beta_{10} - 45 \beta_{9} + 23 \beta_{8} + 22 \beta_{6}$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$45 \beta_{5} - 45 \beta_{2} + 94$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$91 \beta_{14} + 91 \beta_{13} + \beta_{12} - 90 \beta_{11} - 91 \beta_{10} - 90 \beta_{9} - \beta_{8} - 91 \beta_{6}$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$91 \beta_{15} + 2 \beta_{7} - 89 \beta_{4} - 87 \beta_{1}$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$($$$$89 \beta_{14} - 89 \beta_{13} - 89 \beta_{11} - 275 \beta_{10} + 89 \beta_{9} + 275 \beta_{6}$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2591.1
 −1.40721 − 0.140577i −0.140577 − 1.40721i −0.140577 + 1.40721i −1.40721 + 0.140577i 0.825348 − 1.14839i −1.14839 + 0.825348i −1.14839 − 0.825348i 0.825348 + 1.14839i −0.825348 + 1.14839i 1.14839 − 0.825348i 1.14839 + 0.825348i −0.825348 − 1.14839i 1.40721 + 0.140577i 0.140577 + 1.40721i 0.140577 − 1.40721i 1.40721 − 0.140577i
0 0 0 −3.09557 0 1.00000i 0 0 0
2591.2 0 0 0 −3.09557 0 1.00000i 0 0 0
2591.3 0 0 0 −3.09557 0 1.00000i 0 0 0
2591.4 0 0 0 −3.09557 0 1.00000i 0 0 0
2591.5 0 0 0 −0.646084 0 1.00000i 0 0 0
2591.6 0 0 0 −0.646084 0 1.00000i 0 0 0
2591.7 0 0 0 −0.646084 0 1.00000i 0 0 0
2591.8 0 0 0 −0.646084 0 1.00000i 0 0 0
2591.9 0 0 0 0.646084 0 1.00000i 0 0 0
2591.10 0 0 0 0.646084 0 1.00000i 0 0 0
2591.11 0 0 0 0.646084 0 1.00000i 0 0 0
2591.12 0 0 0 0.646084 0 1.00000i 0 0 0
2591.13 0 0 0 3.09557 0 1.00000i 0 0 0
2591.14 0 0 0 3.09557 0 1.00000i 0 0 0
2591.15 0 0 0 3.09557 0 1.00000i 0 0 0
2591.16 0 0 0 3.09557 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.e 16
3.b odd 2 1 inner 4032.2.j.e 16
4.b odd 2 1 inner 4032.2.j.e 16
8.b even 2 1 inner 4032.2.j.e 16
8.d odd 2 1 inner 4032.2.j.e 16
12.b even 2 1 inner 4032.2.j.e 16
24.f even 2 1 inner 4032.2.j.e 16
24.h odd 2 1 inner 4032.2.j.e 16

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

 $$T_{5}^{4} - 10 T_{5}^{2} + 4$$ $$T_{19}^{2} - 28$$ $$T_{43}^{4} - 152 T_{43}^{2} + 400$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 10 T^{2} + 54 T^{4} + 250 T^{6} + 625 T^{8} )^{4}$$
$7$ $$( 1 + T^{2} )^{8}$$
$11$ $$( 1 - 34 T^{2} + 510 T^{4} - 4114 T^{6} + 14641 T^{8} )^{4}$$
$13$ $$( 1 - 32 T^{2} + 510 T^{4} - 5408 T^{6} + 28561 T^{8} )^{4}$$
$17$ $$( 1 - 22 T^{2} + 174 T^{4} - 6358 T^{6} + 83521 T^{8} )^{4}$$
$19$ $$( 1 + 10 T^{2} + 361 T^{4} )^{8}$$
$23$ $$( 1 + 70 T^{2} + 2262 T^{4} + 37030 T^{6} + 279841 T^{8} )^{4}$$
$29$ $$( 1 - 8 T^{2} + 354 T^{4} - 6728 T^{6} + 707281 T^{8} )^{4}$$
$31$ $$( 1 + 22 T^{2} + 961 T^{4} )^{8}$$
$37$ $$( 1 - 80 T^{2} + 3582 T^{4} - 109520 T^{6} + 1874161 T^{8} )^{4}$$
$41$ $$( 1 + 26 T^{2} + 3342 T^{4} + 43706 T^{6} + 2825761 T^{8} )^{4}$$
$43$ $$( 1 + 20 T^{2} - 1578 T^{4} + 36980 T^{6} + 3418801 T^{8} )^{4}$$
$47$ $$( 1 + 62 T^{2} + 2209 T^{4} )^{8}$$
$53$ $$( 1 + 136 T^{2} + 8898 T^{4} + 382024 T^{6} + 7890481 T^{8} )^{4}$$
$59$ $$( 1 - 59 T^{2} )^{16}$$
$61$ $$( 1 - 14 T + 61 T^{2} )^{8}( 1 + 14 T + 61 T^{2} )^{8}$$
$67$ $$( 1 + 200 T^{2} + 18222 T^{4} + 897800 T^{6} + 20151121 T^{8} )^{4}$$
$71$ $$( 1 + 254 T^{2} + 26022 T^{4} + 1280414 T^{6} + 25411681 T^{8} )^{4}$$
$73$ $$( 1 + 10 T + 150 T^{2} + 730 T^{3} + 5329 T^{4} )^{8}$$
$79$ $$( 1 + 80 T^{2} + 7278 T^{4} + 499280 T^{6} + 38950081 T^{8} )^{4}$$
$83$ $$( 1 - 4 T^{2} + 5382 T^{4} - 27556 T^{6} + 47458321 T^{8} )^{4}$$
$89$ $$( 1 - 166 T^{2} + 22542 T^{4} - 1314886 T^{6} + 62742241 T^{8} )^{4}$$
$97$ $$( 1 + 14 T + 222 T^{2} + 1358 T^{3} + 9409 T^{4} )^{8}$$