# Properties

 Label 128.3.c.a Level $128$ Weight $3$ Character orbit 128.c Analytic conductor $3.488$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$128 = 2^{7}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 128.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.48774738381$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{3} + ( -2 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{5} + ( -4 \zeta_{8} + 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{7} + ( -3 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{3} + ( -2 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{5} + ( -4 \zeta_{8} + 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{7} + ( -3 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{9} + ( 6 \zeta_{8} - 10 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{11} + ( 6 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{13} + ( -12 \zeta_{8} - 20 \zeta_{8}^{2} - 12 \zeta_{8}^{3} ) q^{15} + ( -2 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{17} + ( 2 \zeta_{8} + 18 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{19} + 8 q^{21} + ( 4 \zeta_{8} + 28 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{23} + ( 11 + 16 \zeta_{8} - 16 \zeta_{8}^{3} ) q^{25} + ( -4 \zeta_{8} - 20 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{27} + ( -34 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{29} + ( 32 \zeta_{8} + 32 \zeta_{8}^{3} ) q^{31} + ( -4 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{33} + ( -8 \zeta_{8} + 24 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{35} + ( -10 + 28 \zeta_{8} - 28 \zeta_{8}^{3} ) q^{37} + ( 4 \zeta_{8} - 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{39} + ( 2 + 16 \zeta_{8} - 16 \zeta_{8}^{3} ) q^{41} + ( -34 \zeta_{8} - 2 \zeta_{8}^{2} - 34 \zeta_{8}^{3} ) q^{43} + ( 70 + 28 \zeta_{8} - 28 \zeta_{8}^{3} ) q^{45} + ( -8 \zeta_{8} - 24 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{47} + ( 1 + 32 \zeta_{8} - 32 \zeta_{8}^{3} ) q^{49} + ( -20 \zeta_{8} - 36 \zeta_{8}^{2} - 20 \zeta_{8}^{3} ) q^{51} + ( 22 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{53} + ( 28 \zeta_{8} - 28 \zeta_{8}^{2} + 28 \zeta_{8}^{3} ) q^{55} + ( -44 - 40 \zeta_{8} + 40 \zeta_{8}^{3} ) q^{57} + ( 22 \zeta_{8} + 22 \zeta_{8}^{2} + 22 \zeta_{8}^{3} ) q^{59} + ( -74 + 28 \zeta_{8} - 28 \zeta_{8}^{3} ) q^{61} + ( -20 \zeta_{8} + 52 \zeta_{8}^{2} - 20 \zeta_{8}^{3} ) q^{63} + ( 20 - 16 \zeta_{8} + 16 \zeta_{8}^{3} ) q^{65} + ( -6 \zeta_{8} - 54 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{67} + ( -72 - 64 \zeta_{8} + 64 \zeta_{8}^{3} ) q^{69} + ( -20 \zeta_{8} - 12 \zeta_{8}^{2} - 20 \zeta_{8}^{3} ) q^{71} + ( 22 + 24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{73} + ( 54 \zeta_{8} + 86 \zeta_{8}^{2} + 54 \zeta_{8}^{3} ) q^{75} + ( 88 - 64 \zeta_{8} + 64 \zeta_{8}^{3} ) q^{77} + ( 24 \zeta_{8} + 104 \zeta_{8}^{2} + 24 \zeta_{8}^{3} ) q^{79} + ( 29 - 24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{81} + ( 74 \zeta_{8} + 10 \zeta_{8}^{2} + 74 \zeta_{8}^{3} ) q^{83} + ( 68 + 24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{85} + ( -76 \zeta_{8} - 84 \zeta_{8}^{2} - 76 \zeta_{8}^{3} ) q^{87} + ( 54 - 40 \zeta_{8} + 40 \zeta_{8}^{3} ) q^{89} + ( -40 \zeta_{8} + 56 \zeta_{8}^{2} - 40 \zeta_{8}^{3} ) q^{91} + ( -128 - 64 \zeta_{8} + 64 \zeta_{8}^{3} ) q^{93} + ( -76 \zeta_{8} - 52 \zeta_{8}^{2} - 76 \zeta_{8}^{3} ) q^{95} + ( -82 - 40 \zeta_{8} + 40 \zeta_{8}^{3} ) q^{97} + ( 62 \zeta_{8} - 66 \zeta_{8}^{2} + 62 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{5} - 12q^{9} + O(q^{10})$$ $$4q - 8q^{5} - 12q^{9} + 24q^{13} - 8q^{17} + 32q^{21} + 44q^{25} - 136q^{29} - 16q^{33} - 40q^{37} + 8q^{41} + 280q^{45} + 4q^{49} + 88q^{53} - 176q^{57} - 296q^{61} + 80q^{65} - 288q^{69} + 88q^{73} + 352q^{77} + 116q^{81} + 272q^{85} + 216q^{89} - 512q^{93} - 328q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/128\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
0 4.82843i 0 −7.65685 0 1.65685i 0 −14.3137 0
127.2 0 0.828427i 0 3.65685 0 9.65685i 0 8.31371 0
127.3 0 0.828427i 0 3.65685 0 9.65685i 0 8.31371 0
127.4 0 4.82843i 0 −7.65685 0 1.65685i 0 −14.3137 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.3.c.a 4
3.b odd 2 1 1152.3.g.b 4
4.b odd 2 1 inner 128.3.c.a 4
8.b even 2 1 128.3.c.b yes 4
8.d odd 2 1 128.3.c.b yes 4
12.b even 2 1 1152.3.g.b 4
16.e even 4 1 256.3.d.d 4
16.e even 4 1 256.3.d.e 4
16.f odd 4 1 256.3.d.d 4
16.f odd 4 1 256.3.d.e 4
24.f even 2 1 1152.3.g.a 4
24.h odd 2 1 1152.3.g.a 4
48.i odd 4 1 2304.3.b.j 4
48.i odd 4 1 2304.3.b.p 4
48.k even 4 1 2304.3.b.j 4
48.k even 4 1 2304.3.b.p 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.3.c.a 4 1.a even 1 1 trivial
128.3.c.a 4 4.b odd 2 1 inner
128.3.c.b yes 4 8.b even 2 1
128.3.c.b yes 4 8.d odd 2 1
256.3.d.d 4 16.e even 4 1
256.3.d.d 4 16.f odd 4 1
256.3.d.e 4 16.e even 4 1
256.3.d.e 4 16.f odd 4 1
1152.3.g.a 4 24.f even 2 1
1152.3.g.a 4 24.h odd 2 1
1152.3.g.b 4 3.b odd 2 1
1152.3.g.b 4 12.b even 2 1
2304.3.b.j 4 48.i odd 4 1
2304.3.b.j 4 48.k even 4 1
2304.3.b.p 4 48.i odd 4 1
2304.3.b.p 4 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 4 T_{5} - 28$$ acting on $$S_{3}^{\mathrm{new}}(128, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$16 + 24 T^{2} + T^{4}$$
$5$ $$( -28 + 4 T + T^{2} )^{2}$$
$7$ $$256 + 96 T^{2} + T^{4}$$
$11$ $$784 + 344 T^{2} + T^{4}$$
$13$ $$( 4 - 12 T + T^{2} )^{2}$$
$17$ $$( -124 + 4 T + T^{2} )^{2}$$
$19$ $$99856 + 664 T^{2} + T^{4}$$
$23$ $$565504 + 1632 T^{2} + T^{4}$$
$29$ $$( 1124 + 68 T + T^{2} )^{2}$$
$31$ $$( 2048 + T^{2} )^{2}$$
$37$ $$( -1468 + 20 T + T^{2} )^{2}$$
$41$ $$( -508 - 4 T + T^{2} )^{2}$$
$43$ $$5326864 + 4632 T^{2} + T^{4}$$
$47$ $$200704 + 1408 T^{2} + T^{4}$$
$53$ $$( 452 - 44 T + T^{2} )^{2}$$
$59$ $$234256 + 2904 T^{2} + T^{4}$$
$61$ $$( 3908 + 148 T + T^{2} )^{2}$$
$67$ $$8088336 + 5976 T^{2} + T^{4}$$
$71$ $$430336 + 1888 T^{2} + T^{4}$$
$73$ $$( -668 - 44 T + T^{2} )^{2}$$
$79$ $$93392896 + 23936 T^{2} + T^{4}$$
$83$ $$117765904 + 22104 T^{2} + T^{4}$$
$89$ $$( -284 - 108 T + T^{2} )^{2}$$
$97$ $$( 3524 + 164 T + T^{2} )^{2}$$