# Properties

 Label 128.3.f.b Level $128$ Weight $3$ Character orbit 128.f Analytic conductor $3.488$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$128 = 2^{7}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 128.f (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.48774738381$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(i)$$ Coefficient field: 6.0.399424.1 Defining polynomial: $$x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 16) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{3} + ( 1 + \beta_{1} + \beta_{3} + \beta_{5} ) q^{5} + ( \beta_{2} + \beta_{3} ) q^{7} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{9} +O(q^{10})$$ $$q -\beta_{5} q^{3} + ( 1 + \beta_{1} + \beta_{3} + \beta_{5} ) q^{5} + ( \beta_{2} + \beta_{3} ) q^{7} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{9} + ( 4 - 4 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{11} + ( 1 - \beta_{1} - \beta_{2} + 3 \beta_{4} ) q^{13} + ( -8 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{15} + ( 2 + \beta_{2} + \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{17} + ( -4 - 4 \beta_{1} + 2 \beta_{3} + \beta_{5} ) q^{19} + ( 2 + 2 \beta_{1} - 4 \beta_{5} ) q^{21} + ( 8 - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{23} + ( \beta_{1} - 4 \beta_{4} + 4 \beta_{5} ) q^{25} + ( -12 + 12 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{27} + ( 1 - \beta_{1} + \beta_{2} - 7 \beta_{4} ) q^{29} + ( 24 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{31} + ( -6 - \beta_{2} - \beta_{3} - 7 \beta_{4} - 7 \beta_{5} ) q^{33} + ( 16 + 16 \beta_{1} - 4 \beta_{3} + 2 \beta_{5} ) q^{35} + ( -7 - 7 \beta_{1} - 5 \beta_{3} + 7 \beta_{5} ) q^{37} + ( -32 - 3 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{39} + ( -8 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} ) q^{41} + ( 16 - 16 \beta_{1} - 8 \beta_{2} - \beta_{4} ) q^{43} + ( -9 + 9 \beta_{1} + 5 \beta_{2} + \beta_{4} ) q^{45} + ( -24 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 10 \beta_{4} + 10 \beta_{5} ) q^{47} + ( -13 - 6 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{49} + ( -28 - 28 \beta_{1} - 6 \beta_{3} ) q^{51} + ( -15 - 15 \beta_{1} - \beta_{3} - 5 \beta_{5} ) q^{53} + ( 48 + \beta_{2} + \beta_{3} + 8 \beta_{4} + 8 \beta_{5} ) q^{55} + ( -6 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{57} + ( -32 + 32 \beta_{1} + 4 \beta_{2} + 3 \beta_{4} ) q^{59} + ( -7 + 7 \beta_{1} - 5 \beta_{2} - \beta_{4} ) q^{61} + ( 40 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} ) q^{63} + ( 10 + 6 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} ) q^{65} + ( 44 + 44 \beta_{1} + 14 \beta_{3} + 5 \beta_{5} ) q^{67} + ( 18 + 18 \beta_{1} + 4 \beta_{3} - 8 \beta_{5} ) q^{69} + ( -56 - \beta_{2} - \beta_{3} - 18 \beta_{4} - 18 \beta_{5} ) q^{71} + ( 24 \beta_{1} - 11 \beta_{2} + 11 \beta_{3} - 13 \beta_{4} + 13 \beta_{5} ) q^{73} + ( 40 - 40 \beta_{1} + 8 \beta_{2} - 7 \beta_{4} ) q^{75} + ( 34 - 34 \beta_{1} - 4 \beta_{2} ) q^{77} + ( -64 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + 20 \beta_{4} - 20 \beta_{5} ) q^{79} + ( 25 + 11 \beta_{2} + 11 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} ) q^{81} + ( -56 - 56 \beta_{1} - 9 \beta_{5} ) q^{83} + ( 42 + 42 \beta_{1} + 4 \beta_{3} + 16 \beta_{5} ) q^{85} + ( 72 + 7 \beta_{2} + 7 \beta_{3} - 10 \beta_{4} - 10 \beta_{5} ) q^{87} + ( 8 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} + 13 \beta_{4} - 13 \beta_{5} ) q^{89} + ( -24 + 24 \beta_{1} + 8 \beta_{2} + 14 \beta_{4} ) q^{91} + ( 16 - 16 \beta_{1} + 4 \beta_{2} + 28 \beta_{4} ) q^{93} + ( 32 \beta_{1} + 7 \beta_{2} - 7 \beta_{3} ) q^{95} + ( 2 - 11 \beta_{2} - 11 \beta_{3} + 15 \beta_{4} + 15 \beta_{5} ) q^{97} + ( 32 + 32 \beta_{1} - 4 \beta_{3} - 13 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 2q^{3} + 2q^{5} - 4q^{7} + O(q^{10})$$ $$6q + 2q^{3} + 2q^{5} - 4q^{7} + 18q^{11} + 2q^{13} - 4q^{17} - 30q^{19} + 20q^{21} + 60q^{23} - 64q^{27} + 18q^{29} - 4q^{33} + 100q^{35} - 46q^{37} - 196q^{39} + 114q^{43} - 66q^{45} - 46q^{49} - 156q^{51} - 78q^{53} + 252q^{55} - 206q^{59} - 30q^{61} + 12q^{65} + 226q^{67} + 116q^{69} - 260q^{71} + 238q^{75} + 212q^{77} + 86q^{81} - 318q^{83} + 212q^{85} + 444q^{87} - 188q^{91} + 32q^{93} - 4q^{97} + 226q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} - 3 \nu^{3} + 4 \nu^{2} - 2 \nu + 8$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + 4 \nu^{4} - 5 \nu^{3} + 8 \nu^{2} - 14 \nu + 4$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{5} + 4 \nu^{4} - 9 \nu^{3} + 8 \nu^{2} + 2 \nu + 12$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{5} - 4 \nu^{4} + 9 \nu^{3} - 8 \nu^{2} + 14 \nu - 20$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-5 \nu^{5} + 4 \nu^{4} - 7 \nu^{3} + 16 \nu^{2} - 10 \nu + 20$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} + 2$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + 2 \beta_{4} + \beta_{2} + 2 \beta_{1}$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{5} + \beta_{4} - \beta_{3} - 4 \beta_{1} + 6$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{5} + 2 \beta_{3} + \beta_{2} - 10 \beta_{1} + 8$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-2 \beta_{5} + 3 \beta_{4} + \beta_{3} + 4 \beta_{2} + 4 \beta_{1} + 10$$$$)/4$$

## Character values

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

 $$n$$ $$5$$ $$127$$ $$\chi(n)$$ $$\beta_{1}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 1.40680 − 0.144584i −0.671462 − 1.24464i 0.264658 + 1.38923i 1.40680 + 0.144584i −0.671462 + 1.24464i 0.264658 − 1.38923i
0 −2.10278 2.10278i 0 4.62721 + 4.62721i 0 3.04888 0 0.156674i 0
31.2 0 −0.146365 0.146365i 0 −3.68585 3.68585i 0 −9.66442 0 8.95715i 0
31.3 0 3.24914 + 3.24914i 0 0.0586332 + 0.0586332i 0 4.61555 0 12.1138i 0
95.1 0 −2.10278 + 2.10278i 0 4.62721 4.62721i 0 3.04888 0 0.156674i 0
95.2 0 −0.146365 + 0.146365i 0 −3.68585 + 3.68585i 0 −9.66442 0 8.95715i 0
95.3 0 3.24914 3.24914i 0 0.0586332 0.0586332i 0 4.61555 0 12.1138i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 95.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.3.f.b 6
3.b odd 2 1 1152.3.m.a 6
4.b odd 2 1 128.3.f.a 6
8.b even 2 1 16.3.f.a 6
8.d odd 2 1 64.3.f.a 6
12.b even 2 1 1152.3.m.b 6
16.e even 4 1 64.3.f.a 6
16.e even 4 1 128.3.f.a 6
16.f odd 4 1 16.3.f.a 6
16.f odd 4 1 inner 128.3.f.b 6
24.f even 2 1 576.3.m.a 6
24.h odd 2 1 144.3.m.a 6
32.g even 8 2 1024.3.c.j 12
32.g even 8 2 1024.3.d.k 12
32.h odd 8 2 1024.3.c.j 12
32.h odd 8 2 1024.3.d.k 12
40.f even 2 1 400.3.r.c 6
40.i odd 4 1 400.3.k.c 6
40.i odd 4 1 400.3.k.d 6
48.i odd 4 1 576.3.m.a 6
48.i odd 4 1 1152.3.m.b 6
48.k even 4 1 144.3.m.a 6
48.k even 4 1 1152.3.m.a 6
80.j even 4 1 400.3.k.c 6
80.k odd 4 1 400.3.r.c 6
80.s even 4 1 400.3.k.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.3.f.a 6 8.b even 2 1
16.3.f.a 6 16.f odd 4 1
64.3.f.a 6 8.d odd 2 1
64.3.f.a 6 16.e even 4 1
128.3.f.a 6 4.b odd 2 1
128.3.f.a 6 16.e even 4 1
128.3.f.b 6 1.a even 1 1 trivial
128.3.f.b 6 16.f odd 4 1 inner
144.3.m.a 6 24.h odd 2 1
144.3.m.a 6 48.k even 4 1
400.3.k.c 6 40.i odd 4 1
400.3.k.c 6 80.j even 4 1
400.3.k.d 6 40.i odd 4 1
400.3.k.d 6 80.s even 4 1
400.3.r.c 6 40.f even 2 1
400.3.r.c 6 80.k odd 4 1
576.3.m.a 6 24.f even 2 1
576.3.m.a 6 48.i odd 4 1
1024.3.c.j 12 32.g even 8 2
1024.3.c.j 12 32.h odd 8 2
1024.3.d.k 12 32.g even 8 2
1024.3.d.k 12 32.h odd 8 2
1152.3.m.a 6 3.b odd 2 1
1152.3.m.a 6 48.k even 4 1
1152.3.m.b 6 12.b even 2 1
1152.3.m.b 6 48.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} - 2 T_{3}^{5} + 2 T_{3}^{4} + 32 T_{3}^{3} + 196 T_{3}^{2} + 56 T_{3} + 8$$ acting on $$S_{3}^{\mathrm{new}}(128, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$8 + 56 T + 196 T^{2} + 32 T^{3} + 2 T^{4} - 2 T^{5} + T^{6}$$
$5$ $$8 - 136 T + 1156 T^{2} + 64 T^{3} + 2 T^{4} - 2 T^{5} + T^{6}$$
$7$ $$( 136 - 60 T + 2 T^{2} + T^{3} )^{2}$$
$11$ $$587528 - 67208 T + 3844 T^{2} + 32 T^{3} + 162 T^{4} - 18 T^{5} + T^{6}$$
$13$ $$1286408 - 311176 T + 37636 T^{2} - 1216 T^{3} + 2 T^{4} - 2 T^{5} + T^{6}$$
$17$ $$( -1544 - 260 T + 2 T^{2} + T^{3} )^{2}$$
$19$ $$13448 - 5576 T + 1156 T^{2} + 1184 T^{3} + 450 T^{4} + 30 T^{5} + T^{6}$$
$23$ $$( 968 + 164 T - 30 T^{2} + T^{3} )^{2}$$
$29$ $$19046792 + 4752440 T + 592900 T^{2} + 20032 T^{3} + 162 T^{4} - 18 T^{5} + T^{6}$$
$31$ $$16777216 + 659456 T^{2} + 1920 T^{4} + T^{6}$$
$37$ $$42632 - 439752 T + 2268036 T^{2} - 69568 T^{3} + 1058 T^{4} + 46 T^{5} + T^{6}$$
$41$ $$67108864 + 6230016 T^{2} + 4992 T^{4} + T^{6}$$
$43$ $$42632 - 80008 T + 75076 T^{2} - 30944 T^{3} + 6498 T^{4} - 114 T^{5} + T^{6}$$
$47$ $$6056574976 + 15044608 T^{2} + 8576 T^{4} + T^{6}$$
$53$ $$783752 + 838840 T + 448900 T^{2} + 51008 T^{3} + 3042 T^{4} + 78 T^{5} + T^{6}$$
$59$ $$8410007432 + 853113976 T + 43270084 T^{2} + 1225376 T^{3} + 21218 T^{4} + 206 T^{5} + T^{6}$$
$61$ $$151449608 + 10059512 T + 334084 T^{2} + 64 T^{3} + 450 T^{4} + 30 T^{5} + T^{6}$$
$67$ $$87233303432 + 1203788344 T + 8305924 T^{2} - 1069024 T^{3} + 25538 T^{4} - 226 T^{5} + T^{6}$$
$71$ $$( -391864 - 3548 T + 130 T^{2} + T^{3} )^{2}$$
$73$ $$7310934016 + 79362304 T^{2} + 18848 T^{4} + T^{6}$$
$79$ $$1550483193856 + 433127424 T^{2} + 37376 T^{4} + T^{6}$$
$83$ $$105636303368 + 7200782904 T + 245423556 T^{2} + 4522144 T^{3} + 50562 T^{4} + 318 T^{5} + T^{6}$$
$89$ $$25681985536 + 41113856 T^{2} + 16288 T^{4} + T^{6}$$
$97$ $$( 519928 - 17540 T + 2 T^{2} + T^{3} )^{2}$$