# Properties

 Label 576.2.l.a Level $576$ Weight $2$ Character orbit 576.l Analytic conductor $4.599$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 576.l (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.59938315643$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 4 x^{14} + 6 x^{12} - 12 x^{10} + 33 x^{8} - 48 x^{6} + 96 x^{4} - 256 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{18}$$ Twist minimal: no (minimal twist has level 144) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{5} -\beta_{9} q^{7} +O(q^{10})$$ $$q -\beta_{3} q^{5} -\beta_{9} q^{7} + ( \beta_{2} + \beta_{14} ) q^{11} + \beta_{4} q^{13} + ( \beta_{7} - \beta_{8} + \beta_{10} + \beta_{14} ) q^{17} + ( -1 + \beta_{5} + \beta_{6} - \beta_{12} ) q^{19} + ( -\beta_{2} - \beta_{3} - \beta_{8} + \beta_{10} + \beta_{14} + \beta_{15} ) q^{23} + ( -\beta_{1} + \beta_{5} + \beta_{6} - 2 \beta_{11} ) q^{25} + ( \beta_{2} - \beta_{7} - \beta_{10} - \beta_{13} - \beta_{15} ) q^{29} + ( -\beta_{1} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{11} + \beta_{12} ) q^{31} + ( -\beta_{3} - 2 \beta_{7} + \beta_{8} + \beta_{10} + \beta_{13} - \beta_{15} ) q^{35} + ( -\beta_{5} - 2 \beta_{9} + 2 \beta_{11} - \beta_{12} ) q^{37} + ( \beta_{8} + 2 \beta_{13} + \beta_{14} ) q^{41} + ( 2 + 2 \beta_{1} + 2 \beta_{6} + \beta_{9} + \beta_{11} ) q^{43} + ( \beta_{2} - \beta_{3} - \beta_{8} - 3 \beta_{10} - \beta_{13} - \beta_{14} ) q^{47} + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} + \beta_{12} ) q^{49} + ( -\beta_{3} - \beta_{7} + \beta_{10} - \beta_{13} + \beta_{15} ) q^{53} + ( 4 - 2 \beta_{1} - 2 \beta_{5} - 2 \beta_{9} ) q^{55} + ( 2 \beta_{7} + 2 \beta_{10} - \beta_{13} - \beta_{15} ) q^{59} + ( -2 + \beta_{1} + \beta_{4} - 2 \beta_{6} ) q^{61} + ( 2 \beta_{7} - \beta_{8} + \beta_{10} + \beta_{14} + 2 \beta_{15} ) q^{65} + ( 1 - \beta_{5} - \beta_{6} + \beta_{9} - \beta_{11} - \beta_{12} ) q^{67} + ( 4 \beta_{7} + \beta_{15} ) q^{71} + ( -\beta_{1} - \beta_{4} + \beta_{5} - \beta_{12} ) q^{73} + ( -2 \beta_{7} - 2 \beta_{10} - 2 \beta_{14} ) q^{77} + ( \beta_{1} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{11} + \beta_{12} ) q^{79} + ( 3 \beta_{3} + 2 \beta_{7} + \beta_{8} - 3 \beta_{10} + \beta_{13} - \beta_{15} ) q^{83} + ( -2 - \beta_{5} + 2 \beta_{6} ) q^{85} + ( -2 \beta_{2} + 2 \beta_{3} + \beta_{10} ) q^{89} + ( -3 - \beta_{1} - \beta_{4} - 3 \beta_{6} + 2 \beta_{9} + 2 \beta_{11} ) q^{91} + ( -3 \beta_{2} + 3 \beta_{3} - \beta_{8} + 5 \beta_{10} - \beta_{14} ) q^{95} + ( -\beta_{4} + 2 \beta_{9} + \beta_{12} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 16q^{19} + 32q^{43} + 16q^{49} + 64q^{55} - 32q^{61} + 16q^{67} - 32q^{85} - 48q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{14} + 6 x^{12} - 12 x^{10} + 33 x^{8} - 48 x^{6} + 96 x^{4} - 256 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$17 \nu^{14} - 72 \nu^{12} + 6 \nu^{10} - 36 \nu^{8} + 513 \nu^{6} - 1332 \nu^{4} + 3696 \nu^{2} - 4544$$$$)/1920$$ $$\beta_{2}$$ $$=$$ $$($$$$11 \nu^{15} - 66 \nu^{13} + 138 \nu^{11} - 168 \nu^{9} + 339 \nu^{7} - 486 \nu^{5} + 1008 \nu^{3} - 1952 \nu$$$$)/1920$$ $$\beta_{3}$$ $$=$$ $$($$$$19 \nu^{15} - 54 \nu^{13} + 42 \nu^{11} - 192 \nu^{9} + 651 \nu^{7} - 954 \nu^{5} + 1872 \nu^{3} - 1888 \nu$$$$)/1920$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{14} - 9 \nu^{12} - 30 \nu^{8} + 78 \nu^{6} + 15 \nu^{4} + 192 \nu^{2} - 560$$$$)/96$$ $$\beta_{5}$$ $$=$$ $$($$$$43 \nu^{14} - 48 \nu^{12} - 126 \nu^{10} + 36 \nu^{8} + 27 \nu^{6} + 492 \nu^{4} + 144 \nu^{2} - 1216$$$$)/1920$$ $$\beta_{6}$$ $$=$$ $$($$$$43 \nu^{14} - 108 \nu^{12} + 114 \nu^{10} - 324 \nu^{8} + 747 \nu^{6} - 528 \nu^{4} + 3024 \nu^{2} - 6016$$$$)/1920$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{15} + \nu^{13} - 2 \nu^{11} + 10 \nu^{9} - 13 \nu^{7} + 13 \nu^{5} - 56 \nu^{3} + 80 \nu$$$$)/64$$ $$\beta_{8}$$ $$=$$ $$($$$$61 \nu^{15} - 96 \nu^{13} + 78 \nu^{11} - 228 \nu^{9} + 909 \nu^{7} - 156 \nu^{5} + 5328 \nu^{3} - 2752 \nu$$$$)/3840$$ $$\beta_{9}$$ $$=$$ $$($$$$-\nu^{14} + 3 \nu^{12} - 2 \nu^{10} + 6 \nu^{8} - 21 \nu^{6} + 31 \nu^{4} - 48 \nu^{2} + 144$$$$)/32$$ $$\beta_{10}$$ $$=$$ $$($$$$-89 \nu^{15} + 204 \nu^{13} - 102 \nu^{11} + 732 \nu^{9} - 1401 \nu^{7} + 984 \nu^{5} - 4752 \nu^{3} + 9728 \nu$$$$)/3840$$ $$\beta_{11}$$ $$=$$ $$($$$$32 \nu^{14} - 57 \nu^{12} + 36 \nu^{10} - 246 \nu^{8} + 588 \nu^{6} - 297 \nu^{4} + 1776 \nu^{2} - 3824$$$$)/480$$ $$\beta_{12}$$ $$=$$ $$($$$$-17 \nu^{14} + 30 \nu^{12} - 30 \nu^{10} + 168 \nu^{8} - 201 \nu^{6} + 234 \nu^{4} - 1296 \nu^{2} + 1760$$$$)/192$$ $$\beta_{13}$$ $$=$$ $$($$$$199 \nu^{15} - 504 \nu^{13} + 522 \nu^{11} - 2172 \nu^{9} + 3351 \nu^{7} - 3564 \nu^{5} + 17232 \nu^{3} - 33088 \nu$$$$)/3840$$ $$\beta_{14}$$ $$=$$ $$($$$$-7 \nu^{15} + 18 \nu^{13} - 18 \nu^{11} + 56 \nu^{9} - 143 \nu^{7} + 134 \nu^{5} - 528 \nu^{3} + 1184 \nu$$$$)/128$$ $$\beta_{15}$$ $$=$$ $$($$$$17 \nu^{15} - 32 \nu^{13} + 38 \nu^{11} - 116 \nu^{9} + 225 \nu^{7} - 332 \nu^{5} + 1136 \nu^{3} - 1600 \nu$$$$)/256$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{14} - \beta_{10} + \beta_{8} + \beta_{3} + \beta_{2}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{9} + \beta_{6} + \beta_{1} + 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{14} + 2 \beta_{13} + 3 \beta_{10} + 3 \beta_{8} + 2 \beta_{7} + \beta_{3} - \beta_{2}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{12} + 4 \beta_{9} + 8 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{1} + 2$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-4 \beta_{15} - \beta_{14} - 3 \beta_{10} + 7 \beta_{8} - 4 \beta_{7} - 5 \beta_{3} + 3 \beta_{2}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$2 \beta_{12} + 4 \beta_{11} + \beta_{9} - \beta_{6} - 2 \beta_{5} + \beta_{1} + 7$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-8 \beta_{15} - 7 \beta_{14} - 6 \beta_{13} - \beta_{10} + 7 \beta_{8} - 14 \beta_{7} + 9 \beta_{3} + 3 \beta_{2}$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$9 \beta_{12} + 4 \beta_{11} + 8 \beta_{9} + 32 \beta_{6} + 5 \beta_{5} - 7 \beta_{4} + 11 \beta_{1} + 2$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$8 \beta_{15} - 13 \beta_{14} - 12 \beta_{13} + 13 \beta_{10} + 7 \beta_{8} + 20 \beta_{7} - 9 \beta_{3} - 5 \beta_{2}$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$4 \beta_{12} + 5 \beta_{9} + 41 \beta_{6} - 8 \beta_{5} - 8 \beta_{4} - 7 \beta_{1} + 1$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-15 \beta_{14} - 14 \beta_{13} + 43 \beta_{10} + 11 \beta_{8} - 46 \beta_{7} - 31 \beta_{3} + 55 \beta_{2}$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$9 \beta_{12} + 72 \beta_{11} + 44 \beta_{9} - 56 \beta_{6} - 31 \beta_{5} - 39 \beta_{4} + 15 \beta_{1} - 94$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$-12 \beta_{15} - 49 \beta_{14} - 24 \beta_{13} + 133 \beta_{10} - \beta_{8} - 180 \beta_{7} - 5 \beta_{3} - 53 \beta_{2}$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$6 \beta_{12} + 36 \beta_{11} + 9 \beta_{9} + 27 \beta_{6} + 42 \beta_{5} - 48 \beta_{4} - 15 \beta_{1} - 13$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$120 \beta_{15} - 47 \beta_{14} - 150 \beta_{13} - 97 \beta_{10} - 41 \beta_{8} - 126 \beta_{7} - 191 \beta_{3} - 77 \beta_{2}$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$325$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1
 −0.517174 − 1.31626i 0.944649 − 1.05244i 1.36166 − 0.381939i 1.40927 − 0.118126i −1.40927 + 0.118126i −1.36166 + 0.381939i −0.944649 + 1.05244i 0.517174 + 1.31626i −0.517174 + 1.31626i 0.944649 + 1.05244i 1.36166 + 0.381939i 1.40927 + 0.118126i −1.40927 − 0.118126i −1.36166 − 0.381939i −0.944649 − 1.05244i 0.517174 − 1.31626i
0 0 0 −2.63251 + 2.63251i 0 0.207188 0 0 0
143.2 0 0 0 −2.10489 + 2.10489i 0 4.40731 0 0 0
143.3 0 0 0 −0.763878 + 0.763878i 0 −1.33620 0 0 0
143.4 0 0 0 −0.236253 + 0.236253i 0 −3.27830 0 0 0
143.5 0 0 0 0.236253 0.236253i 0 −3.27830 0 0 0
143.6 0 0 0 0.763878 0.763878i 0 −1.33620 0 0 0
143.7 0 0 0 2.10489 2.10489i 0 4.40731 0 0 0
143.8 0 0 0 2.63251 2.63251i 0 0.207188 0 0 0
431.1 0 0 0 −2.63251 2.63251i 0 0.207188 0 0 0
431.2 0 0 0 −2.10489 2.10489i 0 4.40731 0 0 0
431.3 0 0 0 −0.763878 0.763878i 0 −1.33620 0 0 0
431.4 0 0 0 −0.236253 0.236253i 0 −3.27830 0 0 0
431.5 0 0 0 0.236253 + 0.236253i 0 −3.27830 0 0 0
431.6 0 0 0 0.763878 + 0.763878i 0 −1.33620 0 0 0
431.7 0 0 0 2.10489 + 2.10489i 0 4.40731 0 0 0
431.8 0 0 0 2.63251 + 2.63251i 0 0.207188 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 431.8 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
16.f odd 4 1 inner
48.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.l.a 16
3.b odd 2 1 inner 576.2.l.a 16
4.b odd 2 1 144.2.l.a 16
8.b even 2 1 1152.2.l.b 16
8.d odd 2 1 1152.2.l.a 16
12.b even 2 1 144.2.l.a 16
16.e even 4 1 144.2.l.a 16
16.e even 4 1 1152.2.l.a 16
16.f odd 4 1 inner 576.2.l.a 16
16.f odd 4 1 1152.2.l.b 16
24.f even 2 1 1152.2.l.a 16
24.h odd 2 1 1152.2.l.b 16
48.i odd 4 1 144.2.l.a 16
48.i odd 4 1 1152.2.l.a 16
48.k even 4 1 inner 576.2.l.a 16
48.k even 4 1 1152.2.l.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.l.a 16 4.b odd 2 1
144.2.l.a 16 12.b even 2 1
144.2.l.a 16 16.e even 4 1
144.2.l.a 16 48.i odd 4 1
576.2.l.a 16 1.a even 1 1 trivial
576.2.l.a 16 3.b odd 2 1 inner
576.2.l.a 16 16.f odd 4 1 inner
576.2.l.a 16 48.k even 4 1 inner
1152.2.l.a 16 8.d odd 2 1
1152.2.l.a 16 16.e even 4 1
1152.2.l.a 16 24.f even 2 1
1152.2.l.a 16 48.i odd 4 1
1152.2.l.b 16 8.b even 2 1
1152.2.l.b 16 16.f odd 4 1
1152.2.l.b 16 24.h odd 2 1
1152.2.l.b 16 48.k even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(576, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$256 + 20736 T^{4} + 15456 T^{8} + 272 T^{12} + T^{16}$$
$7$ $$( 4 - 16 T - 16 T^{2} + T^{4} )^{4}$$
$11$ $$65536 + 7585792 T^{4} + 181760 T^{8} + 960 T^{12} + T^{16}$$
$13$ $$( 400 - 1280 T + 2048 T^{2} + 1792 T^{3} + 744 T^{4} + 64 T^{5} + T^{8} )^{2}$$
$17$ $$( 1936 + 5024 T^{2} + 1208 T^{4} + 72 T^{6} + T^{8} )^{2}$$
$19$ $$( 30976 + 50688 T + 41472 T^{2} + 10624 T^{3} + 1376 T^{4} + 32 T^{5} + 32 T^{6} + 8 T^{7} + T^{8} )^{2}$$
$23$ $$( 6400 + 5632 T^{2} + 1440 T^{4} + 96 T^{6} + T^{8} )^{2}$$
$29$ $$16062013696 + 1768163584 T^{4} + 7056992 T^{8} + 5520 T^{12} + T^{16}$$
$31$ $$( 1648656 + 288256 T^{2} + 12296 T^{4} + 192 T^{6} + T^{8} )^{2}$$
$37$ $$( 35344 - 96256 T + 131072 T^{2} - 47104 T^{3} + 8840 T^{4} - 512 T^{5} + T^{8} )^{2}$$
$41$ $$( 144 - 30752 T^{2} + 7032 T^{4} - 168 T^{6} + T^{8} )^{2}$$
$43$ $$( 4129024 - 2080768 T + 524288 T^{2} - 65280 T^{3} + 5088 T^{4} - 512 T^{5} + 128 T^{6} - 16 T^{7} + T^{8} )^{2}$$
$47$ $$( 665856 - 129536 T^{2} + 7840 T^{4} - 160 T^{6} + T^{8} )^{2}$$
$53$ $$22663495936 + 1216180480 T^{4} + 6358112 T^{8} + 5904 T^{12} + T^{16}$$
$59$ $$2186423566336 + 18575523840 T^{4} + 34824192 T^{8} + 15104 T^{12} + T^{16}$$
$61$ $$( 258064 + 178816 T + 61952 T^{2} + 6720 T^{3} + 1032 T^{4} + 416 T^{5} + 128 T^{6} + 16 T^{7} + T^{8} )^{2}$$
$67$ $$( 7573504 - 4755456 T + 1492992 T^{2} - 243200 T^{3} + 21888 T^{4} - 704 T^{5} + 32 T^{6} - 8 T^{7} + T^{8} )^{2}$$
$71$ $$( 73984 + 78848 T^{2} + 7712 T^{4} + 192 T^{6} + T^{8} )^{2}$$
$73$ $$( 20736 + 90880 T^{2} + 8032 T^{4} + 176 T^{6} + T^{8} )^{2}$$
$79$ $$( 3825936 + 415360 T^{2} + 15432 T^{4} + 224 T^{6} + T^{8} )^{2}$$
$83$ $$102776124276736 + 231440760832 T^{4} + 167437824 T^{8} + 40384 T^{12} + T^{16}$$
$89$ $$( 104976 - 60704 T^{2} + 8600 T^{4} - 200 T^{6} + T^{8} )^{2}$$
$97$ $$( -176 - 256 T - 72 T^{2} + T^{4} )^{4}$$