# Properties

 Label 128.3.d.a Level 128 Weight 3 Character orbit 128.d Analytic conductor 3.488 Analytic rank 0 Dimension 2 CM discriminant -4 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 8 i q^{5} -9 q^{9} +O(q^{10})$$ $$q + 8 i q^{5} -9 q^{9} + 24 i q^{13} + 30 q^{17} -39 q^{25} -40 i q^{29} -24 i q^{37} + 18 q^{41} -72 i q^{45} + 49 q^{49} -56 i q^{53} + 120 i q^{61} -192 q^{65} + 110 q^{73} + 81 q^{81} + 240 i q^{85} + 78 q^{89} -130 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 18q^{9} + O(q^{10})$$ $$2q - 18q^{9} + 60q^{17} - 78q^{25} + 36q^{41} + 98q^{49} - 384q^{65} + 220q^{73} + 162q^{81} + 156q^{89} - 260q^{97} + O(q^{100})$$

## 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
 − 1.00000i 1.00000i
0 0 0 8.00000i 0 0 0 −9.00000 0
63.2 0 0 0 8.00000i 0 0 0 −9.00000 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 CM by $$\Q(\sqrt{-1})$$
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.a 2
3.b odd 2 1 1152.3.b.a 2
4.b odd 2 1 CM 128.3.d.a 2
8.b even 2 1 inner 128.3.d.a 2
8.d odd 2 1 inner 128.3.d.a 2
12.b even 2 1 1152.3.b.a 2
16.e even 4 1 256.3.c.a 1
16.e even 4 1 256.3.c.b 1
16.f odd 4 1 256.3.c.a 1
16.f odd 4 1 256.3.c.b 1
24.f even 2 1 1152.3.b.a 2
24.h odd 2 1 1152.3.b.a 2
48.i odd 4 1 2304.3.g.a 1
48.i odd 4 1 2304.3.g.f 1
48.k even 4 1 2304.3.g.a 1
48.k even 4 1 2304.3.g.f 1

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + 9 T^{2} )^{2}$$
$5$ $$( 1 - 6 T + 25 T^{2} )( 1 + 6 T + 25 T^{2} )$$
$7$ $$( 1 - 7 T )^{2}( 1 + 7 T )^{2}$$
$11$ $$( 1 + 121 T^{2} )^{2}$$
$13$ $$( 1 - 10 T + 169 T^{2} )( 1 + 10 T + 169 T^{2} )$$
$17$ $$( 1 - 30 T + 289 T^{2} )^{2}$$
$19$ $$( 1 + 361 T^{2} )^{2}$$
$23$ $$( 1 - 23 T )^{2}( 1 + 23 T )^{2}$$
$29$ $$( 1 - 42 T + 841 T^{2} )( 1 + 42 T + 841 T^{2} )$$
$31$ $$( 1 - 31 T )^{2}( 1 + 31 T )^{2}$$
$37$ $$( 1 - 70 T + 1369 T^{2} )( 1 + 70 T + 1369 T^{2} )$$
$41$ $$( 1 - 18 T + 1681 T^{2} )^{2}$$
$43$ $$( 1 + 1849 T^{2} )^{2}$$
$47$ $$( 1 - 47 T )^{2}( 1 + 47 T )^{2}$$
$53$ $$( 1 - 90 T + 2809 T^{2} )( 1 + 90 T + 2809 T^{2} )$$
$59$ $$( 1 + 3481 T^{2} )^{2}$$
$61$ $$( 1 - 22 T + 3721 T^{2} )( 1 + 22 T + 3721 T^{2} )$$
$67$ $$( 1 + 4489 T^{2} )^{2}$$
$71$ $$( 1 - 71 T )^{2}( 1 + 71 T )^{2}$$
$73$ $$( 1 - 110 T + 5329 T^{2} )^{2}$$
$79$ $$( 1 - 79 T )^{2}( 1 + 79 T )^{2}$$
$83$ $$( 1 + 6889 T^{2} )^{2}$$
$89$ $$( 1 - 78 T + 7921 T^{2} )^{2}$$
$97$ $$( 1 + 130 T + 9409 T^{2} )^{2}$$