# Properties

 Label 4032.2.b.r Level 4032 Weight 2 Character orbit 4032.b 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.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.1956820950$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: 16.0.29960650073923649536.7 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{30}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} + \beta_{8} q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{5} + \beta_{8} q^{7} + ( -\beta_{7} - \beta_{11} ) q^{11} -\beta_{15} q^{13} + ( -\beta_{2} + \beta_{6} ) q^{17} -\beta_{4} q^{19} + \beta_{11} q^{23} + ( -2 + \beta_{12} ) q^{25} + \beta_{14} q^{29} + ( \beta_{4} + \beta_{5} + \beta_{8} ) q^{31} + ( -\beta_{3} - \beta_{11} ) q^{35} + ( -1 + \beta_{12} ) q^{37} + ( -\beta_{2} + \beta_{6} ) q^{41} + ( -\beta_{1} - \beta_{5} + \beta_{8} ) q^{43} + ( \beta_{3} + \beta_{13} ) q^{47} + ( -2 - \beta_{10} + \beta_{12} ) q^{49} + ( -\beta_{9} + \beta_{14} ) q^{53} + ( -3 \beta_{4} - 2 \beta_{5} - 2 \beta_{8} ) q^{55} + ( \beta_{3} - \beta_{13} ) q^{59} + ( -2 \beta_{10} + \beta_{15} ) q^{61} + ( -\beta_{9} - 2 \beta_{14} ) q^{65} + ( \beta_{1} + \beta_{5} - \beta_{8} ) q^{67} + ( 2 \beta_{7} + 3 \beta_{11} ) q^{71} -2 \beta_{15} q^{73} + ( -\beta_{2} + \beta_{9} + \beta_{14} ) q^{77} + ( \beta_{5} - \beta_{8} ) q^{79} + ( -\beta_{3} - \beta_{13} ) q^{83} + ( -5 + 3 \beta_{12} ) q^{85} + ( \beta_{2} + \beta_{6} ) q^{89} + ( \beta_{1} + 3 \beta_{4} + \beta_{5} + \beta_{8} ) q^{91} + ( 4 \beta_{7} + 2 \beta_{11} ) q^{95} -2 \beta_{10} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 32q^{25} - 16q^{37} - 32q^{49} - 80q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 7 x^{12} + 40 x^{8} - 112 x^{4} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{10} + 3 \nu^{6} - 12 \nu^{2}$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{12} - 11 \nu^{8} + 84 \nu^{4} - 192$$$$)/64$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{14} - 3 \nu^{10} - 4 \nu^{6} + 96 \nu^{2}$$$$)/32$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{15} - 2 \nu^{13} + 11 \nu^{11} - 10 \nu^{9} - 20 \nu^{7} + 56 \nu^{5} - 384 \nu$$$$)/128$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{15} + 4 \nu^{14} - 2 \nu^{13} + 9 \nu^{11} - 28 \nu^{10} + 22 \nu^{9} - 52 \nu^{7} + 160 \nu^{6} - 40 \nu^{5} + 32 \nu^{3} - 192 \nu^{2} + 256 \nu$$$$)/256$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{12} - \nu^{8} + 28 \nu^{4} + 64$$$$)/32$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{15} - 6 \nu^{13} - 3 \nu^{11} + 34 \nu^{9} - 4 \nu^{7} - 152 \nu^{5} - 32 \nu^{3} + 256 \nu$$$$)/128$$ $$\beta_{8}$$ $$=$$ $$($$$$-3 \nu^{15} - 4 \nu^{14} - 2 \nu^{13} + 9 \nu^{11} + 28 \nu^{10} + 22 \nu^{9} - 52 \nu^{7} - 160 \nu^{6} - 40 \nu^{5} + 32 \nu^{3} + 192 \nu^{2} + 256 \nu$$$$)/256$$ $$\beta_{9}$$ $$=$$ $$($$$$\nu^{15} + 6 \nu^{13} - 3 \nu^{11} - 34 \nu^{9} - 4 \nu^{7} + 152 \nu^{5} - 32 \nu^{3} - 256 \nu$$$$)/64$$ $$\beta_{10}$$ $$=$$ $$($$$$\nu^{15} + \nu^{11} + 16 \nu^{9} + 16 \nu^{7} - 48 \nu^{5} - 16 \nu^{3} + 320 \nu$$$$)/64$$ $$\beta_{11}$$ $$=$$ $$($$$$-3 \nu^{15} + 6 \nu^{13} + 33 \nu^{11} - 2 \nu^{9} - 124 \nu^{7} + 56 \nu^{5} + 448 \nu^{3} - 128 \nu$$$$)/256$$ $$\beta_{12}$$ $$=$$ $$($$$$-\nu^{12} + 7 \nu^{8} - 24 \nu^{4} + 56$$$$)/8$$ $$\beta_{13}$$ $$=$$ $$($$$$-\nu^{14} + 5 \nu^{10} - 18 \nu^{6} + 40 \nu^{2}$$$$)/8$$ $$\beta_{14}$$ $$=$$ $$($$$$-\nu^{15} + 15 \nu^{11} - 16 \nu^{9} - 64 \nu^{7} + 48 \nu^{5} + 208 \nu^{3} - 64 \nu$$$$)/64$$ $$\beta_{15}$$ $$=$$ $$($$$$\nu^{15} - \nu^{13} - 5 \nu^{11} + 3 \nu^{9} + 18 \nu^{7} + 4 \nu^{5} - 8 \nu^{3} - 32 \nu$$$$)/32$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{14} - 2 \beta_{11} + \beta_{10} + \beta_{8} - \beta_{7} + \beta_{5} - \beta_{4}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{13} - 2 \beta_{8} + 2 \beta_{5} + 2 \beta_{3}$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{15} + \beta_{14} + 2 \beta_{11} - \beta_{10} - 2 \beta_{9} - \beta_{8} - 3 \beta_{7} - \beta_{5} - 3 \beta_{4}$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{12} + \beta_{6} + 10 \beta_{2} + 14$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$6 \beta_{15} + \beta_{14} - 2 \beta_{11} + 3 \beta_{10} + 2 \beta_{9} + 9 \beta_{8} - 5 \beta_{7} + 9 \beta_{5} + 3 \beta_{4}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$3 \beta_{13} - 14 \beta_{8} + 14 \beta_{5} - 2 \beta_{3} - 4 \beta_{1}$$$$)/8$$ $$\nu^{7}$$ $$=$$ $$($$$$2 \beta_{15} - \beta_{14} - 2 \beta_{11} + 5 \beta_{10} - 10 \beta_{9} - 7 \beta_{8} - 21 \beta_{7} - 7 \beta_{5} + 3 \beta_{4}$$$$)/8$$ $$\nu^{8}$$ $$=$$ $$($$$$14 \beta_{12} + 15 \beta_{6} + 22 \beta_{2} - 62$$$$)/8$$ $$\nu^{9}$$ $$=$$ $$($$$$18 \beta_{15} - 17 \beta_{14} + 34 \beta_{11} + 5 \beta_{10} + 6 \beta_{9} + 23 \beta_{8} + 5 \beta_{7} + 23 \beta_{5} + 13 \beta_{4}$$$$)/8$$ $$\nu^{10}$$ $$=$$ $$($$$$-3 \beta_{13} - 18 \beta_{8} + 18 \beta_{5} - 30 \beta_{3} - 44 \beta_{1}$$$$)/8$$ $$\nu^{11}$$ $$=$$ $$($$$$-18 \beta_{15} + \beta_{14} + 2 \beta_{11} + 43 \beta_{10} - 6 \beta_{9} - 25 \beta_{8} - 11 \beta_{7} - 25 \beta_{5} + 61 \beta_{4}$$$$)/8$$ $$\nu^{12}$$ $$=$$ $$($$$$-14 \beta_{12} + 81 \beta_{6} - 86 \beta_{2} - 322$$$$)/8$$ $$\nu^{13}$$ $$=$$ $$($$$$-50 \beta_{15} - 79 \beta_{14} + 158 \beta_{11} - 5 \beta_{10} + 26 \beta_{9} - 55 \beta_{8} + 27 \beta_{7} - 55 \beta_{5} - 45 \beta_{4}$$$$)/8$$ $$\nu^{14}$$ $$=$$ $$($$$$-93 \beta_{13} + 82 \beta_{8} - 82 \beta_{5} - 34 \beta_{3} - 148 \beta_{1}$$$$)/8$$ $$\nu^{15}$$ $$=$$ $$($$$$18 \beta_{15} + 31 \beta_{14} + 62 \beta_{11} + 117 \beta_{10} + 134 \beta_{9} - 135 \beta_{8} + 299 \beta_{7} - 135 \beta_{5} + 99 \beta_{4}$$$$)/8$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3583.1
 1.32968 + 0.481610i −0.481610 + 1.32968i −1.32968 − 0.481610i 0.481610 − 1.32968i −1.38588 − 0.281691i −0.281691 + 1.38588i 1.38588 + 0.281691i 0.281691 − 1.38588i −0.281691 − 1.38588i −1.38588 + 0.281691i 0.281691 + 1.38588i 1.38588 − 0.281691i −0.481610 − 1.32968i 1.32968 − 0.481610i 0.481610 + 1.32968i −1.32968 + 0.481610i
0 0 0 3.33513i 0 −0.662153 2.56155i 0 0 0
3583.2 0 0 0 3.33513i 0 −0.662153 + 2.56155i 0 0 0
3583.3 0 0 0 3.33513i 0 0.662153 2.56155i 0 0 0
3583.4 0 0 0 3.33513i 0 0.662153 + 2.56155i 0 0 0
3583.5 0 0 0 1.69614i 0 −2.13578 1.56155i 0 0 0
3583.6 0 0 0 1.69614i 0 −2.13578 + 1.56155i 0 0 0
3583.7 0 0 0 1.69614i 0 2.13578 1.56155i 0 0 0
3583.8 0 0 0 1.69614i 0 2.13578 + 1.56155i 0 0 0
3583.9 0 0 0 1.69614i 0 −2.13578 1.56155i 0 0 0
3583.10 0 0 0 1.69614i 0 −2.13578 + 1.56155i 0 0 0
3583.11 0 0 0 1.69614i 0 2.13578 1.56155i 0 0 0
3583.12 0 0 0 1.69614i 0 2.13578 + 1.56155i 0 0 0
3583.13 0 0 0 3.33513i 0 −0.662153 2.56155i 0 0 0
3583.14 0 0 0 3.33513i 0 −0.662153 + 2.56155i 0 0 0
3583.15 0 0 0 3.33513i 0 0.662153 2.56155i 0 0 0
3583.16 0 0 0 3.33513i 0 0.662153 + 2.56155i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3583.16 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.b.r 16
3.b odd 2 1 inner 4032.2.b.r 16
4.b odd 2 1 inner 4032.2.b.r 16
7.b odd 2 1 inner 4032.2.b.r 16
8.b even 2 1 2016.2.b.d 16
8.d odd 2 1 2016.2.b.d 16
12.b even 2 1 inner 4032.2.b.r 16
21.c even 2 1 inner 4032.2.b.r 16
24.f even 2 1 2016.2.b.d 16
24.h odd 2 1 2016.2.b.d 16
28.d even 2 1 inner 4032.2.b.r 16
56.e even 2 1 2016.2.b.d 16
56.h odd 2 1 2016.2.b.d 16
84.h odd 2 1 inner 4032.2.b.r 16
168.e odd 2 1 2016.2.b.d 16
168.i even 2 1 2016.2.b.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2016.2.b.d 16 8.b even 2 1
2016.2.b.d 16 8.d odd 2 1
2016.2.b.d 16 24.f even 2 1
2016.2.b.d 16 24.h odd 2 1
2016.2.b.d 16 56.e even 2 1
2016.2.b.d 16 56.h odd 2 1
2016.2.b.d 16 168.e odd 2 1
2016.2.b.d 16 168.i even 2 1
4032.2.b.r 16 1.a even 1 1 trivial
4032.2.b.r 16 3.b odd 2 1 inner
4032.2.b.r 16 4.b odd 2 1 inner
4032.2.b.r 16 7.b odd 2 1 inner
4032.2.b.r 16 12.b even 2 1 inner
4032.2.b.r 16 21.c even 2 1 inner
4032.2.b.r 16 28.d even 2 1 inner
4032.2.b.r 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 $$S_{2}^{\mathrm{new}}(4032, [\chi])$$:

 $$T_{5}^{4} + 14 T_{5}^{2} + 32$$ $$T_{11}^{4} + 26 T_{11}^{2} + 16$$ $$T_{19}^{4} - 28 T_{19}^{2} + 128$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 6 T^{2} + 42 T^{4} - 150 T^{6} + 625 T^{8} )^{4}$$
$7$ $$( 1 + 8 T^{2} + 46 T^{4} + 392 T^{6} + 2401 T^{8} )^{2}$$
$11$ $$( 1 - 18 T^{2} + 170 T^{4} - 2178 T^{6} + 14641 T^{8} )^{4}$$
$13$ $$( 1 - 12 T^{2} + 102 T^{4} - 2028 T^{6} + 28561 T^{8} )^{4}$$
$17$ $$( 1 - 22 T^{2} + 682 T^{4} - 6358 T^{6} + 83521 T^{8} )^{4}$$
$19$ $$( 1 + 48 T^{2} + 1230 T^{4} + 17328 T^{6} + 130321 T^{8} )^{4}$$
$23$ $$( 1 - 74 T^{2} + 2410 T^{4} - 39146 T^{6} + 279841 T^{8} )^{4}$$
$29$ $$( 1 + 44 T^{2} + 1894 T^{4} + 37004 T^{6} + 707281 T^{8} )^{4}$$
$31$ $$( 1 + 84 T^{2} + 3414 T^{4} + 80724 T^{6} + 923521 T^{8} )^{4}$$
$37$ $$( 1 + 2 T + 58 T^{2} + 74 T^{3} + 1369 T^{4} )^{8}$$
$41$ $$( 1 - 118 T^{2} + 6826 T^{4} - 198358 T^{6} + 2825761 T^{8} )^{4}$$
$43$ $$( 1 - 88 T^{2} + 3934 T^{4} - 162712 T^{6} + 3418801 T^{8} )^{4}$$
$47$ $$( 1 + 28 T^{2} + 262 T^{4} + 61852 T^{6} + 4879681 T^{8} )^{4}$$
$53$ $$( 1 + 108 T^{2} + 6086 T^{4} + 303372 T^{6} + 7890481 T^{8} )^{4}$$
$59$ $$( 1 + 12 T^{2} + 2646 T^{4} + 41772 T^{6} + 12117361 T^{8} )^{4}$$
$61$ $$( 1 + 52 T^{2} + 7846 T^{4} + 193492 T^{6} + 13845841 T^{8} )^{4}$$
$67$ $$( 1 - 184 T^{2} + 15742 T^{4} - 825976 T^{6} + 20151121 T^{8} )^{4}$$
$71$ $$( 1 - 106 T^{2} + 9066 T^{4} - 534346 T^{6} + 25411681 T^{8} )^{4}$$
$73$ $$( 1 - 132 T^{2} + 10662 T^{4} - 703428 T^{6} + 28398241 T^{8} )^{4}$$
$79$ $$( 1 - 280 T^{2} + 32014 T^{4} - 1747480 T^{6} + 38950081 T^{8} )^{4}$$
$83$ $$( 1 + 172 T^{2} + 16822 T^{4} + 1184908 T^{6} + 47458321 T^{8} )^{4}$$
$89$ $$( 1 - 294 T^{2} + 36618 T^{4} - 2328774 T^{6} + 62742241 T^{8} )^{4}$$
$97$ $$( 1 - 4 T - 74 T^{2} - 388 T^{3} + 9409 T^{4} )^{4}( 1 + 4 T - 74 T^{2} + 388 T^{3} + 9409 T^{4} )^{4}$$