# Properties

 Label 128.4.b.d Level 128 Weight 4 Character orbit 128.b Analytic conductor 7.552 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$128 = 2^{7}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 128.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.55224448073$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 8 i q^{3} + 12 i q^{5} + 32 q^{7} -37 q^{9} +O(q^{10})$$ $$q + 8 i q^{3} + 12 i q^{5} + 32 q^{7} -37 q^{9} + 8 i q^{11} + 20 i q^{13} -96 q^{15} -98 q^{17} -88 i q^{19} + 256 i q^{21} -32 q^{23} -19 q^{25} -80 i q^{27} -172 i q^{29} + 256 q^{31} -64 q^{33} + 384 i q^{35} + 92 i q^{37} -160 q^{39} -102 q^{41} + 296 i q^{43} -444 i q^{45} + 320 q^{47} + 681 q^{49} -784 i q^{51} + 76 i q^{53} -96 q^{55} + 704 q^{57} -408 i q^{59} -636 i q^{61} -1184 q^{63} -240 q^{65} + 552 i q^{67} -256 i q^{69} + 416 q^{71} -138 q^{73} -152 i q^{75} + 256 i q^{77} + 64 q^{79} -359 q^{81} + 392 i q^{83} -1176 i q^{85} + 1376 q^{87} + 582 q^{89} + 640 i q^{91} + 2048 i q^{93} + 1056 q^{95} + 238 q^{97} -296 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 64q^{7} - 74q^{9} + O(q^{10})$$ $$2q + 64q^{7} - 74q^{9} - 192q^{15} - 196q^{17} - 64q^{23} - 38q^{25} + 512q^{31} - 128q^{33} - 320q^{39} - 204q^{41} + 640q^{47} + 1362q^{49} - 192q^{55} + 1408q^{57} - 2368q^{63} - 480q^{65} + 832q^{71} - 276q^{73} + 128q^{79} - 718q^{81} + 2752q^{87} + 1164q^{89} + 2112q^{95} + 476q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 − 1.00000i 1.00000i
0 8.00000i 0 12.0000i 0 32.0000 0 −37.0000 0
65.2 0 8.00000i 0 12.0000i 0 32.0000 0 −37.0000 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
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.4.b.d yes 2
3.b odd 2 1 1152.4.d.h 2
4.b odd 2 1 128.4.b.a 2
8.b even 2 1 inner 128.4.b.d yes 2
8.d odd 2 1 128.4.b.a 2
12.b even 2 1 1152.4.d.a 2
16.e even 4 1 256.4.a.c 1
16.e even 4 1 256.4.a.f 1
16.f odd 4 1 256.4.a.b 1
16.f odd 4 1 256.4.a.g 1
24.f even 2 1 1152.4.d.a 2
24.h odd 2 1 1152.4.d.h 2
48.i odd 4 1 2304.4.a.c 1
48.i odd 4 1 2304.4.a.m 1
48.k even 4 1 2304.4.a.d 1
48.k even 4 1 2304.4.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.4.b.a 2 4.b odd 2 1
128.4.b.a 2 8.d odd 2 1
128.4.b.d yes 2 1.a even 1 1 trivial
128.4.b.d yes 2 8.b even 2 1 inner
256.4.a.b 1 16.f odd 4 1
256.4.a.c 1 16.e even 4 1
256.4.a.f 1 16.e even 4 1
256.4.a.g 1 16.f odd 4 1
1152.4.d.a 2 12.b even 2 1
1152.4.d.a 2 24.f even 2 1
1152.4.d.h 2 3.b odd 2 1
1152.4.d.h 2 24.h odd 2 1
2304.4.a.c 1 48.i odd 4 1
2304.4.a.d 1 48.k even 4 1
2304.4.a.m 1 48.i odd 4 1
2304.4.a.n 1 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(128, [\chi])$$:

 $$T_{3}^{2} + 64$$ $$T_{7} - 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 10 T^{2} + 729 T^{4}$$
$5$ $$1 - 106 T^{2} + 15625 T^{4}$$
$7$ $$( 1 - 32 T + 343 T^{2} )^{2}$$
$11$ $$1 - 2598 T^{2} + 1771561 T^{4}$$
$13$ $$1 - 3994 T^{2} + 4826809 T^{4}$$
$17$ $$( 1 + 98 T + 4913 T^{2} )^{2}$$
$19$ $$1 - 5974 T^{2} + 47045881 T^{4}$$
$23$ $$( 1 + 32 T + 12167 T^{2} )^{2}$$
$29$ $$1 - 19194 T^{2} + 594823321 T^{4}$$
$31$ $$( 1 - 256 T + 29791 T^{2} )^{2}$$
$37$ $$1 - 92842 T^{2} + 2565726409 T^{4}$$
$41$ $$( 1 + 102 T + 68921 T^{2} )^{2}$$
$43$ $$1 - 71398 T^{2} + 6321363049 T^{4}$$
$47$ $$( 1 - 320 T + 103823 T^{2} )^{2}$$
$53$ $$1 - 291978 T^{2} + 22164361129 T^{4}$$
$59$ $$1 - 244294 T^{2} + 42180533641 T^{4}$$
$61$ $$1 - 49466 T^{2} + 51520374361 T^{4}$$
$67$ $$1 - 296822 T^{2} + 90458382169 T^{4}$$
$71$ $$( 1 - 416 T + 357911 T^{2} )^{2}$$
$73$ $$( 1 + 138 T + 389017 T^{2} )^{2}$$
$79$ $$( 1 - 64 T + 493039 T^{2} )^{2}$$
$83$ $$1 - 989910 T^{2} + 326940373369 T^{4}$$
$89$ $$( 1 - 582 T + 704969 T^{2} )^{2}$$
$97$ $$( 1 - 238 T + 912673 T^{2} )^{2}$$