# Properties

 Label 128.5.d.c Level 128 Weight 5 Character orbit 128.d Analytic conductor 13.231 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$128 = 2^{7}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 128.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.2313552747$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{11}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} -\beta_{1} q^{5} + \beta_{3} q^{7} + 15 q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} -\beta_{1} q^{5} + \beta_{3} q^{7} + 15 q^{9} + 11 \beta_{2} q^{11} -27 \beta_{1} q^{13} + \beta_{3} q^{15} -162 q^{17} -45 \beta_{2} q^{19} -96 \beta_{1} q^{21} -9 \beta_{3} q^{23} + 561 q^{25} + 66 \beta_{2} q^{27} -163 \beta_{1} q^{29} -8 \beta_{3} q^{31} -1056 q^{33} + 64 \beta_{2} q^{35} -189 \beta_{1} q^{37} + 27 \beta_{3} q^{39} + 1890 q^{41} -297 \beta_{2} q^{43} -15 \beta_{1} q^{45} + 18 \beta_{3} q^{47} -3743 q^{49} + 162 \beta_{2} q^{51} + 247 \beta_{1} q^{53} -11 \beta_{3} q^{55} + 4320 q^{57} + 231 \beta_{2} q^{59} + 297 \beta_{1} q^{61} + 15 \beta_{3} q^{63} -1728 q^{65} + 171 \beta_{2} q^{67} + 864 \beta_{1} q^{69} -99 \beta_{3} q^{71} + 2750 q^{73} -561 \beta_{2} q^{75} + 1056 \beta_{1} q^{77} -102 \beta_{3} q^{79} -7551 q^{81} -953 \beta_{2} q^{83} + 162 \beta_{1} q^{85} + 163 \beta_{3} q^{87} + 2430 q^{89} + 1728 \beta_{2} q^{91} + 768 \beta_{1} q^{93} + 45 \beta_{3} q^{95} + 7454 q^{97} + 165 \beta_{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 60q^{9} + O(q^{10})$$ $$4q + 60q^{9} - 648q^{17} + 2244q^{25} - 4224q^{33} + 7560q^{41} - 14972q^{49} + 17280q^{57} - 6912q^{65} + 11000q^{73} - 30204q^{81} + 9720q^{89} + 29816q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$8 \nu^{2}$$$$/3$$ $$\beta_{2}$$ $$=$$ $$($$$$-4 \nu^{3} + 12 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$32 \nu^{3} + 96 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 8 \beta_{2}$$$$)/64$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{1}$$$$/8$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{3} - 24 \beta_{2}$$$$)/64$$

## 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
 1.22474 + 1.22474i 1.22474 − 1.22474i −1.22474 − 1.22474i −1.22474 + 1.22474i
0 −9.79796 0 8.00000i 0 78.3837i 0 15.0000 0
63.2 0 −9.79796 0 8.00000i 0 78.3837i 0 15.0000 0
63.3 0 9.79796 0 8.00000i 0 78.3837i 0 15.0000 0
63.4 0 9.79796 0 8.00000i 0 78.3837i 0 15.0000 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
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 128.5.d.c 4
3.b odd 2 1 1152.5.b.i 4
4.b odd 2 1 inner 128.5.d.c 4
8.b even 2 1 inner 128.5.d.c 4
8.d odd 2 1 inner 128.5.d.c 4
12.b even 2 1 1152.5.b.i 4
16.e even 4 1 256.5.c.c 2
16.e even 4 1 256.5.c.f 2
16.f odd 4 1 256.5.c.c 2
16.f odd 4 1 256.5.c.f 2
24.f even 2 1 1152.5.b.i 4
24.h odd 2 1 1152.5.b.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.5.d.c 4 1.a even 1 1 trivial
128.5.d.c 4 4.b odd 2 1 inner
128.5.d.c 4 8.b even 2 1 inner
128.5.d.c 4 8.d odd 2 1 inner
256.5.c.c 2 16.e even 4 1
256.5.c.c 2 16.f odd 4 1
256.5.c.f 2 16.e even 4 1
256.5.c.f 2 16.f odd 4 1
1152.5.b.i 4 3.b odd 2 1
1152.5.b.i 4 12.b even 2 1
1152.5.b.i 4 24.f even 2 1
1152.5.b.i 4 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 96$$ acting on $$S_{5}^{\mathrm{new}}(128, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + 66 T^{2} + 6561 T^{4} )^{2}$$
$5$ $$( 1 - 1186 T^{2} + 390625 T^{4} )^{2}$$
$7$ $$( 1 + 1342 T^{2} + 5764801 T^{4} )^{2}$$
$11$ $$( 1 + 17666 T^{2} + 214358881 T^{4} )^{2}$$
$13$ $$( 1 - 10466 T^{2} + 815730721 T^{4} )^{2}$$
$17$ $$( 1 + 162 T + 83521 T^{2} )^{4}$$
$19$ $$( 1 + 66242 T^{2} + 16983563041 T^{4} )^{2}$$
$23$ $$( 1 - 62018 T^{2} + 78310985281 T^{4} )^{2}$$
$29$ $$( 1 + 285854 T^{2} + 500246412961 T^{4} )^{2}$$
$31$ $$( 1 - 1453826 T^{2} + 852891037441 T^{4} )^{2}$$
$37$ $$( 1 - 1462178 T^{2} + 3512479453921 T^{4} )^{2}$$
$41$ $$( 1 - 1890 T + 2825761 T^{2} )^{4}$$
$43$ $$( 1 - 1630462 T^{2} + 11688200277601 T^{4} )^{2}$$
$47$ $$( 1 - 7768706 T^{2} + 23811286661761 T^{4} )^{2}$$
$53$ $$( 1 - 11876386 T^{2} + 62259690411361 T^{4} )^{2}$$
$59$ $$( 1 + 19112066 T^{2} + 146830437604321 T^{4} )^{2}$$
$61$ $$( 1 - 22046306 T^{2} + 191707312997281 T^{4} )^{2}$$
$67$ $$( 1 + 37495106 T^{2} + 406067677556641 T^{4} )^{2}$$
$71$ $$( 1 + 9393982 T^{2} + 645753531245761 T^{4} )^{2}$$
$73$ $$( 1 - 2750 T + 28398241 T^{2} )^{4}$$
$79$ $$( 1 - 13977986 T^{2} + 1517108809906561 T^{4} )^{2}$$
$83$ $$( 1 + 7728578 T^{2} + 2252292232139041 T^{4} )^{2}$$
$89$ $$( 1 - 2430 T + 62742241 T^{2} )^{4}$$
$97$ $$( 1 - 7454 T + 88529281 T^{2} )^{4}$$