# Properties

 Label 552.2.m.b Level $552$ Weight $2$ Character orbit 552.m Analytic conductor $4.408$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$552 = 2^{3} \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 552.m (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.40774219157$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 2 x^{14} - 2 x^{12} + 8 x^{10} - 8 x^{8} + 32 x^{6} - 32 x^{4} - 128 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{7} q^{3} -\beta_{5} q^{5} -\beta_{1} q^{7} + ( \beta_{2} + \beta_{4} + \beta_{6} ) q^{9} +O(q^{10})$$ $$q -\beta_{7} q^{3} -\beta_{5} q^{5} -\beta_{1} q^{7} + ( \beta_{2} + \beta_{4} + \beta_{6} ) q^{9} + \beta_{14} q^{11} + ( 1 - \beta_{6} + \beta_{7} + \beta_{11} ) q^{13} + ( -\beta_{10} + \beta_{15} ) q^{15} + \beta_{5} q^{17} + \beta_{10} q^{19} + ( \beta_{5} + \beta_{13} ) q^{21} + ( -\beta_{2} - \beta_{3} - \beta_{4} - \beta_{8} + \beta_{14} ) q^{23} + ( 2 + \beta_{2} - \beta_{3} - \beta_{11} ) q^{25} + ( 1 - \beta_{4} - \beta_{8} + \beta_{11} ) q^{27} + ( -\beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{29} + ( 4 - \beta_{6} + \beta_{7} ) q^{31} + ( \beta_{1} - \beta_{10} - \beta_{12} + \beta_{14} ) q^{33} + ( \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{35} + ( \beta_{1} - \beta_{9} - \beta_{10} ) q^{37} + ( -3 - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{11} ) q^{39} + ( -\beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} ) q^{41} + ( -\beta_{1} + \beta_{9} + \beta_{12} - \beta_{13} ) q^{43} + ( -\beta_{5} + \beta_{9} + \beta_{13} - \beta_{14} ) q^{45} + ( \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{47} + ( -2 + \beta_{2} - \beta_{3} - \beta_{11} ) q^{49} + ( \beta_{10} - \beta_{15} ) q^{51} + ( \beta_{1} + \beta_{5} - \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{53} + ( 2 - 2 \beta_{11} ) q^{55} + ( -\beta_{5} - \beta_{9} + \beta_{14} ) q^{57} + ( \beta_{2} + \beta_{3} - 3 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} ) q^{59} + ( -3 \beta_{1} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} ) q^{61} + ( \beta_{1} - \beta_{9} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{63} + ( -\beta_{1} + \beta_{10} + \beta_{12} + \beta_{13} - 3 \beta_{14} - 2 \beta_{15} ) q^{65} + ( -\beta_{1} + 2 \beta_{10} ) q^{67} + ( \beta_{1} - \beta_{2} + 2 \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{10} - \beta_{12} + \beta_{14} ) q^{69} + ( 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 3 \beta_{6} + 3 \beta_{7} ) q^{71} + ( 4 - \beta_{2} + \beta_{3} - 3 \beta_{6} + 3 \beta_{7} ) q^{73} + ( -2 - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{75} + ( 2 \beta_{4} + 4 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} ) q^{77} + ( \beta_{1} - \beta_{9} + \beta_{12} - \beta_{13} ) q^{79} + ( \beta_{3} + \beta_{4} - \beta_{7} + 2 \beta_{8} + 2 \beta_{11} ) q^{81} + ( -2 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} ) q^{83} + ( -7 - \beta_{2} + \beta_{3} + \beta_{11} ) q^{85} + ( -3 - 2 \beta_{3} + \beta_{4} + 3 \beta_{6} - \beta_{8} - \beta_{11} ) q^{87} + ( \beta_{1} - \beta_{5} - \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{89} + ( -2 \beta_{1} - \beta_{9} + 2 \beta_{10} + \beta_{12} - \beta_{13} ) q^{91} + ( -3 - \beta_{2} - \beta_{4} - \beta_{6} - 4 \beta_{7} ) q^{93} + ( \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} ) q^{95} + ( 2 \beta_{1} + 2 \beta_{10} - \beta_{12} + \beta_{13} ) q^{97} + ( -2 \beta_{1} + \beta_{9} - \beta_{12} - \beta_{13} - \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 4q^{3} + 4q^{9} + O(q^{10})$$ $$16q + 4q^{3} + 4q^{9} + 8q^{13} + 32q^{25} + 16q^{27} + 56q^{31} - 44q^{39} - 32q^{49} + 32q^{55} + 4q^{69} + 40q^{73} - 20q^{75} + 4q^{81} - 112q^{85} - 36q^{87} - 36q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{14} - 2 x^{12} + 8 x^{10} - 8 x^{8} + 32 x^{6} - 32 x^{4} - 128 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{15} - 2 \nu^{11} - 12 \nu^{9} + 8 \nu^{7} - 128 \nu$$$$)/128$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{14} + 2 \nu^{10} - 4 \nu^{8} - 8 \nu^{6} + 32 \nu^{4} + 64 \nu^{2}$$$$)/64$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{14} - 2 \nu^{12} + 2 \nu^{10} + 8 \nu^{8} + 16 \nu^{4} - 64 \nu^{2}$$$$)/64$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{12} - 2 \nu^{8} + 4 \nu^{6} + 8 \nu^{4} - 32 \nu^{2}$$$$)/32$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{15} - 4 \nu^{13} + 2 \nu^{11} + 4 \nu^{9} + 8 \nu^{7} + 32 \nu^{5} - 128 \nu^{3} - 128 \nu$$$$)/128$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{14} + 2 \nu^{10} + 4 \nu^{8} - 16 \nu^{4} + 64$$$$)/64$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{14} - 6 \nu^{10} + 4 \nu^{8} - 8 \nu^{6} + 16 \nu^{4} + 64 \nu^{2} - 192$$$$)/64$$ $$\beta_{8}$$ $$=$$ $$($$$$\nu^{12} - 4 \nu^{10} + 2 \nu^{8} + 4 \nu^{6} + 8 \nu^{4} + 32 \nu^{2} - 96$$$$)/32$$ $$\beta_{9}$$ $$=$$ $$($$$$\nu^{15} - 4 \nu^{13} - 6 \nu^{11} + 4 \nu^{9} - 8 \nu^{7} + 64 \nu^{5} - 64 \nu^{3} - 128 \nu$$$$)/64$$ $$\beta_{10}$$ $$=$$ $$($$$$\nu^{15} - 6 \nu^{11} + 4 \nu^{9} + 8 \nu^{7} + 16 \nu^{5} + 96 \nu^{3} - 192 \nu$$$$)/64$$ $$\beta_{11}$$ $$=$$ $$($$$$-\nu^{14} + \nu^{12} + 2 \nu^{10} - 2 \nu^{8} + 12 \nu^{6} - 40 \nu^{4} + 32 \nu^{2} + 96$$$$)/32$$ $$\beta_{12}$$ $$=$$ $$($$$$-3 \nu^{15} + 2 \nu^{13} + 2 \nu^{11} - 16 \nu^{9} + 32 \nu^{7} - 144 \nu^{5} - 64 \nu^{3} + 384 \nu$$$$)/128$$ $$\beta_{13}$$ $$=$$ $$($$$$\nu^{15} + \nu^{13} - 2 \nu^{11} - 2 \nu^{9} + 12 \nu^{7} + 40 \nu^{5} + 64 \nu$$$$)/64$$ $$\beta_{14}$$ $$=$$ $$($$$$\nu^{15} + 4 \nu^{13} - 14 \nu^{11} + 4 \nu^{9} + 8 \nu^{7} + 64 \nu^{3} - 256 \nu$$$$)/128$$ $$\beta_{15}$$ $$=$$ $$($$$$-3 \nu^{15} + 2 \nu^{13} + 2 \nu^{11} + 16 \nu^{9} + 32 \nu^{7} - 80 \nu^{5} + 64 \nu^{3} + 128 \nu$$$$)/128$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{14} + \beta_{13} + \beta_{12} + \beta_{9} - 2 \beta_{5} - 2 \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{11} + 2 \beta_{7} + 2 \beta_{6} + \beta_{4} + 2 \beta_{2} + 1$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{14} + \beta_{10} - \beta_{5}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{11} + 2 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + \beta_{4} + 2 \beta_{2} + 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$2 \beta_{15} - \beta_{14} + 3 \beta_{13} - 3 \beta_{12} - 2 \beta_{10} + \beta_{9} - 2 \beta_{5} + 4 \beta_{1}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$3 \beta_{11} + 4 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + 3 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 1$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$2 \beta_{15} - \beta_{14} + 3 \beta_{13} + \beta_{12} + 2 \beta_{10} - \beta_{9} + 4 \beta_{5} + 2 \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$2 \beta_{8} + 4 \beta_{6} - 2 \beta_{4} + 4 \beta_{3} + 2$$ $$\nu^{9}$$ $$=$$ $$($$$$6 \beta_{15} + 3 \beta_{14} - \beta_{13} - 3 \beta_{12} - 2 \beta_{10} + \beta_{9} + 2 \beta_{5} - 8 \beta_{1}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$\beta_{11} - 6 \beta_{7} + 2 \beta_{6} - 3 \beta_{4} + 4 \beta_{3} + 6 \beta_{2} - 23$$ $$\nu^{11}$$ $$=$$ $$-4 \beta_{14} - 4 \beta_{12} - 6 \beta_{9} + 6 \beta_{5} + 2 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$-6 \beta_{11} + 12 \beta_{8} - 28 \beta_{7} - 20 \beta_{6} - 26 \beta_{4} - 4 \beta_{2} - 10$$ $$\nu^{13}$$ $$=$$ $$2 \beta_{15} + 17 \beta_{14} + 5 \beta_{13} - 9 \beta_{12} - 14 \beta_{10} - 9 \beta_{9} + 2 \beta_{5} + 4 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$10 \beta_{11} - 8 \beta_{8} + 4 \beta_{7} - 60 \beta_{6} - 22 \beta_{4} + 24 \beta_{3} - 4 \beta_{2} + 18$$ $$\nu^{15}$$ $$=$$ $$-28 \beta_{15} + 2 \beta_{14} + 2 \beta_{13} + 14 \beta_{12} + 20 \beta_{10} - 14 \beta_{9} + 24 \beta_{5} - 44 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/552\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$185$$ $$277$$ $$415$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
137.1
 −1.32734 − 0.488022i 1.32734 + 0.488022i −1.32734 + 0.488022i 1.32734 − 0.488022i 0.883519 + 1.10426i −0.883519 − 1.10426i 0.883519 − 1.10426i −0.883519 + 1.10426i −0.108061 + 1.41008i 0.108061 − 1.41008i −0.108061 − 1.41008i 0.108061 + 1.41008i −1.39495 + 0.232632i 1.39495 − 0.232632i −1.39495 − 0.232632i 1.39495 + 0.232632i
0 −1.47398 0.909606i 0 −3.63073 0 1.67864i 0 1.34523 + 2.68148i 0
137.2 0 −1.47398 0.909606i 0 3.63073 0 1.67864i 0 1.34523 + 2.68148i 0
137.3 0 −1.47398 + 0.909606i 0 −3.63073 0 1.67864i 0 1.34523 2.68148i 0
137.4 0 −1.47398 + 0.909606i 0 3.63073 0 1.67864i 0 1.34523 2.68148i 0
137.5 0 −0.356193 1.69503i 0 −0.441484 0 3.97556i 0 −2.74625 + 1.20752i 0
137.6 0 −0.356193 1.69503i 0 0.441484 0 3.97556i 0 −2.74625 + 1.20752i 0
137.7 0 −0.356193 + 1.69503i 0 −0.441484 0 3.97556i 0 −2.74625 1.20752i 0
137.8 0 −0.356193 + 1.69503i 0 0.441484 0 3.97556i 0 −2.74625 1.20752i 0
137.9 0 1.10238 1.33595i 0 −3.03628 0 2.60404i 0 −0.569517 2.94545i 0
137.10 0 1.10238 1.33595i 0 3.03628 0 2.60404i 0 −0.569517 2.94545i 0
137.11 0 1.10238 + 1.33595i 0 −3.03628 0 2.60404i 0 −0.569517 + 2.94545i 0
137.12 0 1.10238 + 1.33595i 0 3.03628 0 2.60404i 0 −0.569517 + 2.94545i 0
137.13 0 1.72779 0.121372i 0 −2.32463 0 3.25516i 0 2.97054 0.419412i 0
137.14 0 1.72779 0.121372i 0 2.32463 0 3.25516i 0 2.97054 0.419412i 0
137.15 0 1.72779 + 0.121372i 0 −2.32463 0 3.25516i 0 2.97054 + 0.419412i 0
137.16 0 1.72779 + 0.121372i 0 2.32463 0 3.25516i 0 2.97054 + 0.419412i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 137.16 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
23.b odd 2 1 inner
69.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 552.2.m.b 16
3.b odd 2 1 inner 552.2.m.b 16
4.b odd 2 1 1104.2.m.e 16
12.b even 2 1 1104.2.m.e 16
23.b odd 2 1 inner 552.2.m.b 16
69.c even 2 1 inner 552.2.m.b 16
92.b even 2 1 1104.2.m.e 16
276.h odd 2 1 1104.2.m.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.m.b 16 1.a even 1 1 trivial
552.2.m.b 16 3.b odd 2 1 inner
552.2.m.b 16 23.b odd 2 1 inner
552.2.m.b 16 69.c even 2 1 inner
1104.2.m.e 16 4.b odd 2 1
1104.2.m.e 16 12.b even 2 1
1104.2.m.e 16 92.b even 2 1
1104.2.m.e 16 276.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 28 T_{5}^{6} + 248 T_{5}^{4} - 704 T_{5}^{2} + 128$$ acting on $$S_{2}^{\mathrm{new}}(552, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$( 81 - 54 T + 9 T^{2} - 6 T^{3} + 4 T^{4} - 2 T^{5} + T^{6} - 2 T^{7} + T^{8} )^{2}$$
$5$ $$( 128 - 704 T^{2} + 248 T^{4} - 28 T^{6} + T^{8} )^{2}$$
$7$ $$( 3200 + 2112 T^{2} + 440 T^{4} + 36 T^{6} + T^{8} )^{2}$$
$11$ $$( 2048 - 3328 T^{2} + 720 T^{4} - 48 T^{6} + T^{8} )^{2}$$
$13$ $$( -140 + 144 T - 35 T^{2} - 2 T^{3} + T^{4} )^{4}$$
$17$ $$( 128 - 704 T^{2} + 248 T^{4} - 28 T^{6} + T^{8} )^{2}$$
$19$ $$( 32768 + 12288 T^{2} + 1512 T^{4} + 68 T^{6} + T^{8} )^{2}$$
$23$ $$78310985281 - 1184287112 T^{2} - 45893924 T^{4} + 2475720 T^{6} + 158726 T^{8} + 4680 T^{10} - 164 T^{12} - 8 T^{14} + T^{16}$$
$29$ $$( 6400 + 5664 T^{2} + 1321 T^{4} + 70 T^{6} + T^{8} )^{2}$$
$31$ $$( 32 - 88 T + 61 T^{2} - 14 T^{3} + T^{4} )^{4}$$
$37$ $$( 100352 + 72448 T^{2} + 7888 T^{4} + 176 T^{6} + T^{8} )^{2}$$
$41$ $$( 1024 + 7936 T^{2} + 1913 T^{4} + 86 T^{6} + T^{8} )^{2}$$
$43$ $$( 32768 + 86016 T^{2} + 11432 T^{4} + 228 T^{6} + T^{8} )^{2}$$
$47$ $$( 400 + 5384 T^{2} + 1489 T^{4} + 102 T^{6} + T^{8} )^{2}$$
$53$ $$( 3276800 - 1456128 T^{2} + 39272 T^{4} - 348 T^{6} + T^{8} )^{2}$$
$59$ $$( 313600 + 100864 T^{2} + 8544 T^{4} + 224 T^{6} + T^{8} )^{2}$$
$61$ $$( 147095552 + 6041600 T^{2} + 85888 T^{4} + 496 T^{6} + T^{8} )^{2}$$
$67$ $$( 28155008 + 1549376 T^{2} + 31928 T^{4} + 292 T^{6} + T^{8} )^{2}$$
$71$ $$( 1817104 + 942872 T^{2} + 29937 T^{4} + 310 T^{6} + T^{8} )^{2}$$
$73$ $$( -508 + 756 T - 75 T^{2} - 10 T^{3} + T^{4} )^{4}$$
$79$ $$( 9193472 + 2460672 T^{2} + 55720 T^{4} + 420 T^{6} + T^{8} )^{2}$$
$83$ $$( 232589312 - 10616064 T^{2} + 132304 T^{4} - 624 T^{6} + T^{8} )^{2}$$
$89$ $$( 147095552 - 6959104 T^{2} + 103016 T^{4} - 572 T^{6} + T^{8} )^{2}$$
$97$ $$( 125960192 + 6469632 T^{2} + 99664 T^{4} + 576 T^{6} + T^{8} )^{2}$$