# Properties

 Label 128.5.d.d Level 128 Weight 5 Character orbit 128.d Analytic conductor 13.231 Analytic rank 0 Dimension 8 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: $$8$$ Coefficient field: 8.0.1871773696.1 Defining polynomial: $$x^{8} + 31 x^{4} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{38}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} -\beta_{2} q^{5} -\beta_{3} q^{7} + ( 55 - \beta_{6} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} -\beta_{2} q^{5} -\beta_{3} q^{7} + ( 55 - \beta_{6} ) q^{9} + ( -\beta_{1} + \beta_{5} ) q^{11} + ( -4 \beta_{2} - \beta_{4} ) q^{13} + ( 7 \beta_{3} + \beta_{7} ) q^{15} + ( -42 + \beta_{6} ) q^{17} + ( -9 \beta_{1} + 5 \beta_{5} ) q^{19} + ( 13 \beta_{2} + 5 \beta_{4} ) q^{21} + ( -15 \beta_{3} + \beta_{7} ) q^{23} + ( -351 + 6 \beta_{6} ) q^{25} + ( -74 \beta_{1} + 9 \beta_{5} ) q^{27} + ( 17 \beta_{2} - 12 \beta_{4} ) q^{29} + 3 \beta_{7} q^{31} + ( 104 - 5 \beta_{6} ) q^{33} + ( -88 \beta_{1} + 10 \beta_{5} ) q^{35} + ( 4 \beta_{2} + 25 \beta_{4} ) q^{37} + ( 37 \beta_{3} + 4 \beta_{7} ) q^{39} + ( 498 - 6 \beta_{6} ) q^{41} + ( 159 \beta_{1} + 10 \beta_{5} ) q^{43} + ( -146 \beta_{2} - 43 \beta_{4} ) q^{45} + ( -42 \beta_{3} - 3 \beta_{7} ) q^{47} + 737 q^{49} + ( 142 \beta_{1} - 9 \beta_{5} ) q^{51} + ( 66 \beta_{2} + 51 \beta_{4} ) q^{53} + 43 \beta_{3} q^{55} + ( 1064 - 29 \beta_{6} ) q^{57} + ( -393 \beta_{1} - 36 \beta_{5} ) q^{59} + ( 68 \beta_{2} - 61 \beta_{4} ) q^{61} + ( -55 \beta_{3} - 13 \beta_{7} ) q^{63} + ( -3696 + 34 \beta_{6} ) q^{65} + ( -33 \beta_{1} - 31 \beta_{5} ) q^{67} + ( 59 \beta_{2} + 67 \beta_{4} ) q^{69} + ( -69 \beta_{3} - 17 \beta_{7} ) q^{71} + ( -2458 + 63 \beta_{6} ) q^{73} + ( 951 \beta_{1} - 54 \beta_{5} ) q^{75} + ( 65 \beta_{2} - 39 \beta_{4} ) q^{77} + 134 \beta_{3} q^{79} + ( 5321 - 29 \beta_{6} ) q^{81} + ( 567 \beta_{1} - 60 \beta_{5} ) q^{83} + ( 133 \beta_{2} + 43 \beta_{4} ) q^{85} + ( -11 \beta_{3} - 17 \beta_{7} ) q^{87} + ( -3354 + 23 \beta_{6} ) q^{89} + ( -456 \beta_{1} + 14 \beta_{5} ) q^{91} + ( -408 \beta_{2} - 24 \beta_{4} ) q^{93} + ( 243 \beta_{3} + 4 \beta_{7} ) q^{95} + ( -2730 - 87 \beta_{6} ) q^{97} + ( -523 \beta_{1} - 36 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 440q^{9} + O(q^{10})$$ $$8q + 440q^{9} - 336q^{17} - 2808q^{25} + 832q^{33} + 3984q^{41} + 5896q^{49} + 8512q^{57} - 29568q^{65} - 19664q^{73} + 42568q^{81} - 26832q^{89} - 21840q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 31 x^{4} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-4 \nu^{7} - 18 \nu^{5} - 286 \nu^{3} + 414 \nu$$$$)/189$$ $$\beta_{2}$$ $$=$$ $$($$$$-44 \nu^{6} - 752 \nu^{2}$$$$)/63$$ $$\beta_{3}$$ $$=$$ $$($$$$80 \nu^{7} + 72 \nu^{5} + 2696 \nu^{3} + 4392 \nu$$$$)/189$$ $$\beta_{4}$$ $$=$$ $$($$$$12 \nu^{6} + 368 \nu^{2}$$$$)/7$$ $$\beta_{5}$$ $$=$$ $$($$$$-272 \nu^{7} + 504 \nu^{5} - 7352 \nu^{3} + 12600 \nu$$$$)/189$$ $$\beta_{6}$$ $$=$$ $$($$$$64 \nu^{4} + 992$$$$)/7$$ $$\beta_{7}$$ $$=$$ $$($$$$-1024 \nu^{7} - 2304 \nu^{5} - 24832 \nu^{3} - 43776 \nu$$$$)/189$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + 4 \beta_{5} + 32 \beta_{3} + 112 \beta_{1}$$$$)/1024$$ $$\nu^{2}$$ $$=$$ $$($$$$11 \beta_{4} + 27 \beta_{2}$$$$)/256$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{7} + 4 \beta_{5} + 64 \beta_{3} - 272 \beta_{1}$$$$)/512$$ $$\nu^{4}$$ $$=$$ $$($$$$7 \beta_{6} - 992$$$$)/64$$ $$\nu^{5}$$ $$=$$ $$($$$$-61 \beta_{7} + 92 \beta_{5} - 608 \beta_{3} - 2800 \beta_{1}$$$$)/1024$$ $$\nu^{6}$$ $$=$$ $$($$$$-47 \beta_{4} - 207 \beta_{2}$$$$)/64$$ $$\nu^{7}$$ $$=$$ $$($$$$-337 \beta_{7} - 572 \beta_{5} - 3104 \beta_{3} + 14704 \beta_{1}$$$$)/1024$$

## 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
 1.62831 + 1.62831i 1.62831 − 1.62831i 0.921201 − 0.921201i 0.921201 + 0.921201i −0.921201 + 0.921201i −0.921201 − 0.921201i −1.62831 − 1.62831i −1.62831 + 1.62831i
0 −15.8549 0 40.8444i 0 40.7922i 0 170.378 0
63.2 0 −15.8549 0 40.8444i 0 40.7922i 0 170.378 0
63.3 0 −4.54118 0 16.8444i 0 40.7922i 0 −60.3776 0
63.4 0 −4.54118 0 16.8444i 0 40.7922i 0 −60.3776 0
63.5 0 4.54118 0 16.8444i 0 40.7922i 0 −60.3776 0
63.6 0 4.54118 0 16.8444i 0 40.7922i 0 −60.3776 0
63.7 0 15.8549 0 40.8444i 0 40.7922i 0 170.378 0
63.8 0 15.8549 0 40.8444i 0 40.7922i 0 170.378 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 63.8 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.d 8
3.b odd 2 1 1152.5.b.l 8
4.b odd 2 1 inner 128.5.d.d 8
8.b even 2 1 inner 128.5.d.d 8
8.d odd 2 1 inner 128.5.d.d 8
12.b even 2 1 1152.5.b.l 8
16.e even 4 1 256.5.c.g 4
16.e even 4 1 256.5.c.k 4
16.f odd 4 1 256.5.c.g 4
16.f odd 4 1 256.5.c.k 4
24.f even 2 1 1152.5.b.l 8
24.h odd 2 1 1152.5.b.l 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 272 T_{3}^{2} + 5184$$ acting on $$S_{5}^{\mathrm{new}}(128, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + 52 T^{2} + 486 T^{4} + 341172 T^{6} + 43046721 T^{8} )^{2}$$
$5$ $$( 1 - 548 T^{2} + 377094 T^{4} - 214062500 T^{6} + 152587890625 T^{8} )^{2}$$
$7$ $$( 1 - 3138 T^{2} + 5764801 T^{4} )^{4}$$
$11$ $$( 1 + 45876 T^{2} + 934622054 T^{4} + 9833928024756 T^{6} + 45949729863572161 T^{8} )^{2}$$
$13$ $$( 1 - 79268 T^{2} + 2902795398 T^{4} - 64661342792228 T^{6} + 665416609183179841 T^{8} )^{2}$$
$17$ $$( 1 + 84 T + 155494 T^{2} + 7015764 T^{3} + 6975757441 T^{4} )^{4}$$
$19$ $$( 1 + 191412 T^{2} + 35457974246 T^{4} + 3250857768803892 T^{6} +$$$$28\!\cdots\!81$$$$T^{8} )^{2}$$
$23$ $$( 1 - 108420 T^{2} - 36732732538 T^{4} - 8490477024166020 T^{6} +$$$$61\!\cdots\!61$$$$T^{8} )^{2}$$
$29$ $$( 1 - 1076900 T^{2} + 796704441222 T^{4} - 538715362117700900 T^{6} +$$$$25\!\cdots\!21$$$$T^{8} )^{2}$$
$31$ $$( 1 - 667394 T^{2} + 852891037441 T^{4} )^{4}$$
$37$ $$( 1 - 3128612 T^{2} + 6805143118086 T^{4} - 10989185369290687652 T^{6} +$$$$12\!\cdots\!41$$$$T^{8} )^{2}$$
$41$ $$( 1 - 996 T + 5420294 T^{2} - 2814457956 T^{3} + 7984925229121 T^{4} )^{4}$$
$43$ $$( 1 + 5340852 T^{2} + 29705752269926 T^{4} + 62424947829025856052 T^{6} +$$$$13\!\cdots\!01$$$$T^{8} )^{2}$$
$47$ $$( 1 - 11288836 T^{2} + 65631563659014 T^{4} -$$$$26\!\cdots\!96$$$$T^{6} +$$$$56\!\cdots\!21$$$$T^{8} )^{2}$$
$53$ $$( 1 - 7465252 T^{2} + 129570259041798 T^{4} -$$$$46\!\cdots\!72$$$$T^{6} +$$$$38\!\cdots\!21$$$$T^{8} )^{2}$$
$59$ $$( 1 - 11608652 T^{2} + 321512309200230 T^{4} -$$$$17\!\cdots\!92$$$$T^{6} +$$$$21\!\cdots\!41$$$$T^{8} )^{2}$$
$61$ $$( 1 - 16591268 T^{2} + 144825482457990 T^{4} -$$$$31\!\cdots\!08$$$$T^{6} +$$$$36\!\cdots\!61$$$$T^{8} )^{2}$$
$67$ $$( 1 + 68122548 T^{2} + 1934597002567910 T^{4} +$$$$27\!\cdots\!68$$$$T^{6} +$$$$16\!\cdots\!81$$$$T^{8} )^{2}$$
$71$ $$( 1 - 10042500 T^{2} + 116338596303494 T^{4} -$$$$64\!\cdots\!00$$$$T^{6} +$$$$41\!\cdots\!21$$$$T^{8} )^{2}$$
$73$ $$( 1 + 4916 T + 10002918 T^{2} + 139605752756 T^{3} + 806460091894081 T^{4} )^{4}$$
$79$ $$( 1 - 48021378 T^{2} + 1517108809906561 T^{4} )^{4}$$
$83$ $$( 1 + 61584436 T^{2} + 3095005172414694 T^{4} +$$$$13\!\cdots\!76$$$$T^{6} +$$$$50\!\cdots\!81$$$$T^{8} )^{2}$$
$89$ $$( 1 + 6708 T + 129691750 T^{2} + 420874952628 T^{3} + 3936588805702081 T^{4} )^{4}$$
$97$ $$( 1 + 5460 T + 83752934 T^{2} + 483369874260 T^{3} + 7837433594376961 T^{4} )^{4}$$