# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$128 = 2^{7}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 128.d (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.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{3} + 4 \zeta_{8}^{2} q^{5} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{7} - q^{9} +O(q^{10})$$ $$q + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{3} + 4 \zeta_{8}^{2} q^{5} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{7} - q^{9} + ( 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{11} -20 \zeta_{8}^{2} q^{13} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{15} -10 q^{17} + ( 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{19} + 32 \zeta_{8}^{2} q^{21} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{23} + 9 q^{25} + ( -20 \zeta_{8} + 20 \zeta_{8}^{3} ) q^{27} -20 \zeta_{8}^{2} q^{29} + 40 q^{33} + ( -32 \zeta_{8} + 32 \zeta_{8}^{3} ) q^{35} + 20 \zeta_{8}^{2} q^{37} + ( -40 \zeta_{8} - 40 \zeta_{8}^{3} ) q^{39} -30 q^{41} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{43} -4 \zeta_{8}^{2} q^{45} + ( -48 \zeta_{8} - 48 \zeta_{8}^{3} ) q^{47} -79 q^{49} + ( -20 \zeta_{8} + 20 \zeta_{8}^{3} ) q^{51} -60 \zeta_{8}^{2} q^{53} + ( 40 \zeta_{8} + 40 \zeta_{8}^{3} ) q^{55} + 40 q^{57} + ( -30 \zeta_{8} + 30 \zeta_{8}^{3} ) q^{59} + 28 \zeta_{8}^{2} q^{61} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{63} + 80 q^{65} + ( 58 \zeta_{8} - 58 \zeta_{8}^{3} ) q^{67} -32 \zeta_{8}^{2} q^{69} + ( 40 \zeta_{8} + 40 \zeta_{8}^{3} ) q^{71} -10 q^{73} + ( 18 \zeta_{8} - 18 \zeta_{8}^{3} ) q^{75} + 160 \zeta_{8}^{2} q^{77} + ( 80 \zeta_{8} + 80 \zeta_{8}^{3} ) q^{79} -71 q^{81} + ( 18 \zeta_{8} - 18 \zeta_{8}^{3} ) q^{83} -40 \zeta_{8}^{2} q^{85} + ( -40 \zeta_{8} - 40 \zeta_{8}^{3} ) q^{87} + 22 q^{89} + ( 160 \zeta_{8} - 160 \zeta_{8}^{3} ) q^{91} + ( 40 \zeta_{8} + 40 \zeta_{8}^{3} ) q^{95} + 150 q^{97} + ( -10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{9} + O(q^{10})$$ $$4q - 4q^{9} - 40q^{17} + 36q^{25} + 160q^{33} - 120q^{41} - 316q^{49} + 160q^{57} + 320q^{65} - 40q^{73} - 284q^{81} + 88q^{89} + 600q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
63.1
 −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i
0 −2.82843 0 4.00000i 0 11.3137i 0 −1.00000 0
63.2 0 −2.82843 0 4.00000i 0 11.3137i 0 −1.00000 0
63.3 0 2.82843 0 4.00000i 0 11.3137i 0 −1.00000 0
63.4 0 2.82843 0 4.00000i 0 11.3137i 0 −1.00000 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
8.b even 2 1 inner
8.d 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.d.c 4
3.b odd 2 1 1152.3.b.g 4
4.b odd 2 1 inner 128.3.d.c 4
8.b even 2 1 inner 128.3.d.c 4
8.d odd 2 1 inner 128.3.d.c 4
12.b even 2 1 1152.3.b.g 4
16.e even 4 1 256.3.c.c 2
16.e even 4 1 256.3.c.f 2
16.f odd 4 1 256.3.c.c 2
16.f odd 4 1 256.3.c.f 2
24.f even 2 1 1152.3.b.g 4
24.h odd 2 1 1152.3.b.g 4
48.i odd 4 1 2304.3.g.h 2
48.i odd 4 1 2304.3.g.m 2
48.k even 4 1 2304.3.g.h 2
48.k even 4 1 2304.3.g.m 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 8$$ acting on $$S_{3}^{\mathrm{new}}(128, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + 10 T^{2} + 81 T^{4} )^{2}$$
$5$ $$( 1 - 34 T^{2} + 625 T^{4} )^{2}$$
$7$ $$( 1 + 30 T^{2} + 2401 T^{4} )^{2}$$
$11$ $$( 1 + 42 T^{2} + 14641 T^{4} )^{2}$$
$13$ $$( 1 + 62 T^{2} + 28561 T^{4} )^{2}$$
$17$ $$( 1 + 10 T + 289 T^{2} )^{4}$$
$19$ $$( 1 + 522 T^{2} + 130321 T^{4} )^{2}$$
$23$ $$( 1 - 930 T^{2} + 279841 T^{4} )^{2}$$
$29$ $$( 1 - 1282 T^{2} + 707281 T^{4} )^{2}$$
$31$ $$( 1 - 31 T )^{4}( 1 + 31 T )^{4}$$
$37$ $$( 1 - 2338 T^{2} + 1874161 T^{4} )^{2}$$
$41$ $$( 1 + 30 T + 1681 T^{2} )^{4}$$
$43$ $$( 1 + 3690 T^{2} + 3418801 T^{4} )^{2}$$
$47$ $$( 1 + 190 T^{2} + 4879681 T^{4} )^{2}$$
$53$ $$( 1 - 2018 T^{2} + 7890481 T^{4} )^{2}$$
$59$ $$( 1 + 5162 T^{2} + 12117361 T^{4} )^{2}$$
$61$ $$( 1 - 6658 T^{2} + 13845841 T^{4} )^{2}$$
$67$ $$( 1 + 2250 T^{2} + 20151121 T^{4} )^{2}$$
$71$ $$( 1 - 6882 T^{2} + 25411681 T^{4} )^{2}$$
$73$ $$( 1 + 10 T + 5329 T^{2} )^{4}$$
$79$ $$( 1 + 318 T^{2} + 38950081 T^{4} )^{2}$$
$83$ $$( 1 + 13130 T^{2} + 47458321 T^{4} )^{2}$$
$89$ $$( 1 - 22 T + 7921 T^{2} )^{4}$$
$97$ $$( 1 - 150 T + 9409 T^{2} )^{4}$$