# Properties

 Label 1152.3.m.e Level $1152$ Weight $3$ Character orbit 1152.m Analytic conductor $31.390$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1152.m (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$31.3897264543$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 6 x^{14} + 10 x^{12} + 88 x^{10} - 752 x^{8} + 1408 x^{6} + 2560 x^{4} - 24576 x^{2} + 65536$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{28}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{5} + \beta_{8} q^{7} +O(q^{10})$$ $$q + \beta_{3} q^{5} + \beta_{8} q^{7} + ( -\beta_{4} - \beta_{14} ) q^{11} -\beta_{15} q^{13} + ( -\beta_{10} + \beta_{11} + \beta_{14} ) q^{17} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{12} ) q^{19} + ( -\beta_{3} - \beta_{4} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{14} ) q^{23} + ( -5 \beta_{1} + \beta_{2} + \beta_{6} + 2 \beta_{13} ) q^{25} + ( -\beta_{4} + \beta_{5} - \beta_{9} - 2 \beta_{14} ) q^{29} + ( 8 \beta_{1} - \beta_{2} - \beta_{6} - \beta_{12} - \beta_{13} - \beta_{15} ) q^{31} + ( 3 \beta_{3} + \beta_{5} - \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{35} + ( -6 + 6 \beta_{1} - 2 \beta_{2} + 2 \beta_{8} + \beta_{12} + 2 \beta_{13} ) q^{37} + ( -2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{7} - \beta_{11} + \beta_{14} ) q^{41} + ( 2 + 2 \beta_{1} + \beta_{6} - 2 \beta_{8} + 2 \beta_{13} + 3 \beta_{15} ) q^{43} + ( -3 \beta_{3} + 3 \beta_{4} + \beta_{5} - 3 \beta_{7} - \beta_{11} + \beta_{14} ) q^{47} + ( 7 + \beta_{2} - \beta_{6} - 6 \beta_{8} - 2 \beta_{12} + 2 \beta_{15} ) q^{49} + ( -3 \beta_{3} + \beta_{5} + 2 \beta_{7} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{53} + ( -16 - \beta_{2} + \beta_{6} + 2 \beta_{8} + 3 \beta_{12} - 3 \beta_{15} ) q^{55} + ( 6 \beta_{4} - \beta_{5} - 3 \beta_{7} + \beta_{9} - 3 \beta_{10} + 2 \beta_{14} ) q^{59} + ( 2 + 2 \beta_{1} - 2 \beta_{6} + 6 \beta_{8} - 6 \beta_{13} - \beta_{15} ) q^{61} + ( -2 \beta_{3} - 2 \beta_{4} - 2 \beta_{9} - 3 \beta_{10} - 3 \beta_{11} - 3 \beta_{14} ) q^{65} + ( 16 - 16 \beta_{1} + 2 \beta_{2} - 2 \beta_{8} + 2 \beta_{12} - 2 \beta_{13} ) q^{67} + ( 6 \beta_{3} + 6 \beta_{4} + 4 \beta_{9} - 2 \beta_{11} - 2 \beta_{14} ) q^{71} + ( -6 \beta_{1} + \beta_{2} + \beta_{6} + 2 \beta_{12} - 6 \beta_{13} + 2 \beta_{15} ) q^{73} + ( 4 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} - 3 \beta_{9} - 2 \beta_{10} + 6 \beta_{14} ) q^{77} + ( -8 \beta_{1} - 2 \beta_{2} - 2 \beta_{6} + 2 \beta_{12} - \beta_{13} + 2 \beta_{15} ) q^{79} + ( -13 \beta_{3} + 4 \beta_{5} + 4 \beta_{9} + 3 \beta_{11} ) q^{83} + ( -10 + 10 \beta_{1} - 2 \beta_{2} - 6 \beta_{8} - 6 \beta_{13} ) q^{85} + ( 6 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} + 2 \beta_{7} + 4 \beta_{11} - 4 \beta_{14} ) q^{89} + ( -18 - 18 \beta_{1} + \beta_{6} - 4 \beta_{8} + 4 \beta_{13} - \beta_{15} ) q^{91} + ( 13 \beta_{3} - 13 \beta_{4} + 3 \beta_{5} - 5 \beta_{7} + 3 \beta_{11} - 3 \beta_{14} ) q^{95} + ( 8 \beta_{8} - 2 \beta_{12} + 2 \beta_{15} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q + 32q^{19} - 96q^{37} + 32q^{43} + 112q^{49} - 256q^{55} + 32q^{61} + 256q^{67} - 160q^{85} - 288q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 6 x^{14} + 10 x^{12} + 88 x^{10} - 752 x^{8} + 1408 x^{6} + 2560 x^{4} - 24576 x^{2} + 65536$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-5 \nu^{14} + 14 \nu^{12} + 110 \nu^{10} - 472 \nu^{8} + 944 \nu^{6} + 5504 \nu^{4} - 24064 \nu^{2} - 20480$$$$)/12288$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{14} - 22 \nu^{12} - 214 \nu^{10} - 712 \nu^{8} + 4880 \nu^{6} - 13696 \nu^{4} + 22016 \nu^{2} + 409600$$$$)/12288$$ $$\beta_{3}$$ $$=$$ $$($$$$-17 \nu^{15} - 34 \nu^{13} + 326 \nu^{11} - 424 \nu^{9} - 1360 \nu^{7} + 26624 \nu^{5} + 9728 \nu^{3} - 167936 \nu$$$$)/24576$$ $$\beta_{4}$$ $$=$$ $$($$$$11 \nu^{15} - 14 \nu^{13} - 170 \nu^{11} + 784 \nu^{9} - 1328 \nu^{7} - 9536 \nu^{5} + 30208 \nu^{3} + 26624 \nu$$$$)/12288$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{15} + 6 \nu^{13} - 62 \nu^{11} - 48 \nu^{9} + 816 \nu^{7} - 4032 \nu^{5} - 1024 \nu^{3} + 51200 \nu$$$$)/1024$$ $$\beta_{6}$$ $$=$$ $$($$$$13 \nu^{14} - 94 \nu^{12} - 94 \nu^{10} + 344 \nu^{8} - 4144 \nu^{6} + 3200 \nu^{4} + 3584 \nu^{2} + 188416$$$$)/12288$$ $$\beta_{7}$$ $$=$$ $$($$$$-3 \nu^{15} - 6 \nu^{13} + 114 \nu^{11} + 8 \nu^{9} - 880 \nu^{7} + 6656 \nu^{5} - 512 \nu^{3} - 94208 \nu$$$$)/2048$$ $$\beta_{8}$$ $$=$$ $$($$$$-7 \nu^{14} + 74 \nu^{12} - 70 \nu^{10} - 1448 \nu^{8} + 6928 \nu^{6} - 6784 \nu^{4} - 70144 \nu^{2} + 237568$$$$)/4096$$ $$\beta_{9}$$ $$=$$ $$($$$$11 \nu^{15} - 74 \nu^{13} - 98 \nu^{11} + 1912 \nu^{9} - 6416 \nu^{7} - 2048 \nu^{5} + 94720 \nu^{3} - 151552 \nu$$$$)/6144$$ $$\beta_{10}$$ $$=$$ $$($$$$-7 \nu^{15} + 46 \nu^{13} + 34 \nu^{11} - 1088 \nu^{9} + 4336 \nu^{7} - 1088 \nu^{5} - 51200 \nu^{3} + 137216 \nu$$$$)/3072$$ $$\beta_{11}$$ $$=$$ $$($$$$-7 \nu^{15} + 55 \nu^{13} + 4 \nu^{11} - 1142 \nu^{9} + 4600 \nu^{7} - 2672 \nu^{5} - 51968 \nu^{3} + 135680 \nu$$$$)/3072$$ $$\beta_{12}$$ $$=$$ $$($$$$37 \nu^{14} + 146 \nu^{12} - 1006 \nu^{10} - 616 \nu^{8} + 10832 \nu^{6} - 76672 \nu^{4} - 64000 \nu^{2} + 778240$$$$)/12288$$ $$\beta_{13}$$ $$=$$ $$($$$$-47 \nu^{14} + 122 \nu^{12} + 554 \nu^{10} - 4072 \nu^{8} + 10640 \nu^{6} + 33152 \nu^{4} - 150016 \nu^{2} + 65536$$$$)/12288$$ $$\beta_{14}$$ $$=$$ $$($$$$-77 \nu^{15} + 470 \nu^{13} + 398 \nu^{11} - 10408 \nu^{9} + 41072 \nu^{7} - 256 \nu^{5} - 506368 \nu^{3} + 987136 \nu$$$$)/24576$$ $$\beta_{15}$$ $$=$$ $$($$$$79 \nu^{14} - 490 \nu^{12} - 202 \nu^{10} + 10760 \nu^{8} - 43024 \nu^{6} + 16256 \nu^{4} + 492032 \nu^{2} - 1163264$$$$)/12288$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{10} + \beta_{9} - \beta_{7} - \beta_{5}$$$$)/16$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{15} + 2 \beta_{13} + \beta_{12} - \beta_{6} + \beta_{2} + 2 \beta_{1} + 6$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{14} + 4 \beta_{11} + \beta_{10} + 2 \beta_{7} + \beta_{5} + 4 \beta_{4} + 4 \beta_{3}$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{15} + 3 \beta_{13} + 5 \beta_{8} - \beta_{2} - 2 \beta_{1} + 8$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$8 \beta_{14} - 5 \beta_{10} - \beta_{9} - 7 \beta_{7} - 7 \beta_{5} + 32 \beta_{4} + 24 \beta_{3}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$3 \beta_{15} - 12 \beta_{13} - 7 \beta_{12} + 14 \beta_{8} - 5 \beta_{6} + 7 \beta_{2} + 38 \beta_{1} - 114$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$4 \beta_{14} - 12 \beta_{11} + 8 \beta_{10} - 7 \beta_{9} + 17 \beta_{7} + 14 \beta_{5} + 52 \beta_{4} + 28 \beta_{3}$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$-8 \beta_{15} - 39 \beta_{13} - 23 \beta_{12} + 17 \beta_{8} - 5 \beta_{6} - 8 \beta_{2} - 16 \beta_{1} + 238$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$128 \beta_{14} - 88 \beta_{11} + 17 \beta_{10} + 79 \beta_{9} - 35 \beta_{7} - 41 \beta_{5} + 72 \beta_{4} - 32 \beta_{3}$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$-23 \beta_{15} - 90 \beta_{13} + 5 \beta_{12} - 60 \beta_{8} - 49 \beta_{6} + 69 \beta_{2} + 658 \beta_{1} + 1014$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-180 \beta_{14} + 36 \beta_{11} + 229 \beta_{10} + 54 \beta_{9} + 132 \beta_{7} + 37 \beta_{5} + 4 \beta_{4} - 140 \beta_{3}$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$101 \beta_{15} + 209 \beta_{13} + 108 \beta_{12} + 47 \beta_{8} - 92 \beta_{6} - 79 \beta_{2} - 22 \beta_{1} + 2888$$ $$\nu^{13}$$ $$=$$ $$($$$$-8 \beta_{14} + 800 \beta_{11} - 163 \beta_{10} + 497 \beta_{9} - 201 \beta_{7} - 513 \beta_{5} + 960 \beta_{4} + 168 \beta_{3}$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$773 \beta_{15} + 492 \beta_{13} + 511 \beta_{12} + 706 \beta_{8} - 195 \beta_{6} + 369 \beta_{2} + 5514 \beta_{1} - 2878$$ $$\nu^{15}$$ $$=$$ $$-1316 \beta_{14} + 620 \beta_{11} + 568 \beta_{10} - 1145 \beta_{9} + 735 \beta_{7} + 162 \beta_{5} + 4140 \beta_{4} + 1668 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$641$$ $$901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
415.1
 −1.99750 + 0.0999235i 1.66730 + 1.10459i 1.64663 − 1.13516i −0.136762 + 1.99532i 0.136762 − 1.99532i −1.64663 + 1.13516i −1.66730 − 1.10459i 1.99750 − 0.0999235i −1.99750 − 0.0999235i 1.66730 − 1.10459i 1.64663 + 1.13516i −0.136762 − 1.99532i 0.136762 + 1.99532i −1.64663 − 1.13516i −1.66730 + 1.10459i 1.99750 + 0.0999235i
0 0 0 −6.01265 6.01265i 0 8.23187 0 0 0
415.2 0 0 0 −4.23991 4.23991i 0 −0.262225 0 0 0
415.3 0 0 0 −2.41234 2.41234i 0 −11.8718 0 0 0
415.4 0 0 0 −0.227650 0.227650i 0 3.90219 0 0 0
415.5 0 0 0 0.227650 + 0.227650i 0 3.90219 0 0 0
415.6 0 0 0 2.41234 + 2.41234i 0 −11.8718 0 0 0
415.7 0 0 0 4.23991 + 4.23991i 0 −0.262225 0 0 0
415.8 0 0 0 6.01265 + 6.01265i 0 8.23187 0 0 0
991.1 0 0 0 −6.01265 + 6.01265i 0 8.23187 0 0 0
991.2 0 0 0 −4.23991 + 4.23991i 0 −0.262225 0 0 0
991.3 0 0 0 −2.41234 + 2.41234i 0 −11.8718 0 0 0
991.4 0 0 0 −0.227650 + 0.227650i 0 3.90219 0 0 0
991.5 0 0 0 0.227650 0.227650i 0 3.90219 0 0 0
991.6 0 0 0 2.41234 2.41234i 0 −11.8718 0 0 0
991.7 0 0 0 4.23991 4.23991i 0 −0.262225 0 0 0
991.8 0 0 0 6.01265 6.01265i 0 8.23187 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 991.8 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 1152.3.m.e 16
3.b odd 2 1 inner 1152.3.m.e 16
4.b odd 2 1 1152.3.m.d 16
8.b even 2 1 144.3.m.b 16
8.d odd 2 1 576.3.m.b 16
12.b even 2 1 1152.3.m.d 16
16.e even 4 1 576.3.m.b 16
16.e even 4 1 1152.3.m.d 16
16.f odd 4 1 144.3.m.b 16
16.f odd 4 1 inner 1152.3.m.e 16
24.f even 2 1 576.3.m.b 16
24.h odd 2 1 144.3.m.b 16
48.i odd 4 1 576.3.m.b 16
48.i odd 4 1 1152.3.m.d 16
48.k even 4 1 144.3.m.b 16
48.k even 4 1 inner 1152.3.m.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.m.b 16 8.b even 2 1
144.3.m.b 16 16.f odd 4 1
144.3.m.b 16 24.h odd 2 1
144.3.m.b 16 48.k even 4 1
576.3.m.b 16 8.d odd 2 1
576.3.m.b 16 16.e even 4 1
576.3.m.b 16 24.f even 2 1
576.3.m.b 16 48.i odd 4 1
1152.3.m.d 16 4.b odd 2 1
1152.3.m.d 16 12.b even 2 1
1152.3.m.d 16 16.e even 4 1
1152.3.m.d 16 48.i odd 4 1
1152.3.m.e 16 1.a even 1 1 trivial
1152.3.m.e 16 3.b odd 2 1 inner
1152.3.m.e 16 16.f odd 4 1 inner
1152.3.m.e 16 48.k even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1152, [\chi])$$:

 $$T_{5}^{16} + 6656 T_{5}^{12} + 7641216 T_{5}^{8} + 915505152 T_{5}^{4} + 9834496$$ $$T_{7}^{4} - 112 T_{7}^{2} + 352 T_{7} + 100$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$9834496 + 915505152 T^{4} + 7641216 T^{8} + 6656 T^{12} + T^{16}$$
$7$ $$( 100 + 352 T - 112 T^{2} + T^{4} )^{4}$$
$11$ $$1228250152960000 + 13103358017536 T^{4} + 2539489280 T^{8} + 110592 T^{12} + T^{16}$$
$13$ $$( 35760400 + 26024960 T + 9469952 T^{2} + 1584128 T^{3} + 144456 T^{4} + 4352 T^{5} + T^{8} )^{2}$$
$17$ $$( 472105984 - 42764800 T^{2} + 382592 T^{4} - 1104 T^{6} + T^{8} )^{2}$$
$19$ $$( 6629867776 - 78167040 T + 460800 T^{2} + 2070784 T^{3} + 477152 T^{4} + 13760 T^{5} + 128 T^{6} - 16 T^{7} + T^{8} )^{2}$$
$23$ $$( 467943424 - 37695488 T^{2} + 502272 T^{4} - 1632 T^{6} + T^{8} )^{2}$$
$29$ $$2847880354408960000 + 1898534369613070336 T^{4} + 4853968416896 T^{8} + 3912960 T^{12} + T^{16}$$
$31$ $$( 11588953104 + 1680413440 T^{2} + 4834568 T^{4} + 4032 T^{6} + T^{8} )^{2}$$
$37$ $$( 137744899600 - 40629438080 T + 5992059392 T^{2} + 265498816 T^{3} + 5955464 T^{4} - 14752 T^{5} + 1152 T^{6} + 48 T^{7} + T^{8} )^{2}$$
$41$ $$( 198844646400 + 5017549312 T^{2} + 12842112 T^{4} + 7248 T^{6} + T^{8} )^{2}$$
$43$ $$( 21036601600 + 22398817280 T + 11924621312 T^{2} - 610465536 T^{3} + 15454944 T^{4} - 90944 T^{5} + 128 T^{6} - 16 T^{7} + T^{8} )^{2}$$
$47$ $$( 15982596734976 + 46512123904 T^{2} + 40460800 T^{4} + 11360 T^{6} + T^{8} )^{2}$$
$53$ $$18\!\cdots\!00$$$$+$$$$80\!\cdots\!16$$$$T^{4} + 1104026742079616 T^{8} + 58906368 T^{12} + T^{16}$$
$59$ $$11\!\cdots\!00$$$$+$$$$58\!\cdots\!96$$$$T^{4} + 4866440880979968 T^{8} + 140066816 T^{12} + T^{16}$$
$61$ $$( 51506974970896 + 743864697728 T + 5371453952 T^{2} - 899677248 T^{3} + 81451272 T^{4} + 52960 T^{5} + 128 T^{6} - 16 T^{7} + T^{8} )^{2}$$
$67$ $$( 7615833702400 - 1119061278720 T + 82216747008 T^{2} - 1261568000 T^{3} + 10536960 T^{4} - 118784 T^{5} + 8192 T^{6} - 128 T^{7} + T^{8} )^{2}$$
$71$ $$( 5121887017369600 - 2814020681728 T^{2} + 525148160 T^{4} - 39552 T^{6} + T^{8} )^{2}$$
$73$ $$( 534697177190400 + 469936807936 T^{2} + 148044928 T^{4} + 20096 T^{6} + T^{8} )^{2}$$
$79$ $$( 56939504400 + 93640045696 T^{2} + 65683656 T^{4} + 14624 T^{6} + T^{8} )^{2}$$
$83$ $$60\!\cdots\!36$$$$+$$$$18\!\cdots\!20$$$$T^{4} + 269780128958810112 T^{8} + 1008676864 T^{12} + T^{16}$$
$89$ $$( 236165588582400 + 800601505792 T^{2} + 558270464 T^{4} + 45760 T^{6} + T^{8} )^{2}$$
$97$ $$( 29866240 - 77824 T - 12384 T^{2} + T^{4} )^{4}$$