# Properties

 Label 128.5.c.b Level 128 Weight 5 Character orbit 128.c Analytic conductor 13.231 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$13.2313552747$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.205520896.4 Defining polynomial: $$x^{8} - 4 x^{7} + 12 x^{6} - 12 x^{5} - 8 x^{4} + 12 x^{3} + 12 x^{2} + 4 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{39}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( 6 + \beta_{2} ) q^{5} + ( \beta_{1} + \beta_{4} ) q^{7} + ( -27 - \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( 6 + \beta_{2} ) q^{5} + ( \beta_{1} + \beta_{4} ) q^{7} + ( -27 - \beta_{2} - \beta_{3} ) q^{9} + ( -2 \beta_{1} + \beta_{4} + \beta_{6} ) q^{11} + ( 30 + \beta_{3} + \beta_{7} ) q^{13} + ( 11 \beta_{1} - \beta_{5} - 2 \beta_{6} ) q^{15} + ( 30 - 3 \beta_{2} + \beta_{3} + 2 \beta_{7} ) q^{17} + ( -10 \beta_{1} - 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{19} + ( -152 - \beta_{2} - 7 \beta_{3} + \beta_{7} ) q^{21} + ( 7 \beta_{1} + 2 \beta_{4} + \beta_{5} - 6 \beta_{6} ) q^{23} + ( 83 + 24 \beta_{2} - 4 \beta_{3} + 2 \beta_{7} ) q^{25} + ( -31 \beta_{1} - 21 \beta_{4} - 2 \beta_{5} + 5 \beta_{6} ) q^{27} + ( 54 - 3 \beta_{2} - 4 \beta_{3} + 4 \beta_{7} ) q^{29} + ( -64 \beta_{1} + 3 \beta_{4} - 7 \beta_{5} - 6 \beta_{6} ) q^{31} + ( 124 + 21 \beta_{2} - 3 \beta_{3} + 4 \beta_{7} ) q^{33} + ( -10 \beta_{1} + 12 \beta_{4} + 10 \beta_{5} + 10 \beta_{6} ) q^{35} + ( 350 + 12 \beta_{2} + 5 \beta_{3} + 5 \beta_{7} ) q^{37} + ( 103 \beta_{1} + 15 \beta_{4} + 8 \beta_{5} - 8 \beta_{6} ) q^{39} + ( -366 - 76 \beta_{2} + 6 \beta_{7} ) q^{41} + ( \beta_{1} + 10 \beta_{4} - 12 \beta_{5} + 6 \beta_{6} ) q^{43} + ( -610 + 18 \beta_{2} - 9 \beta_{3} - \beta_{7} ) q^{45} + ( 242 \beta_{1} - 9 \beta_{4} - 17 \beta_{5} - 2 \beta_{6} ) q^{47} + ( -719 - 36 \beta_{2} - 20 \beta_{3} - 8 \beta_{7} ) q^{49} + ( 81 \beta_{1} + 9 \beta_{4} + 16 \beta_{5} - 7 \beta_{6} ) q^{51} + ( 222 - 2 \beta_{2} + 27 \beta_{3} + 3 \beta_{7} ) q^{53} + ( -175 \beta_{1} + 41 \beta_{4} + 24 \beta_{5} + 24 \beta_{6} ) q^{55} + ( 1076 + 95 \beta_{2} + 23 \beta_{3} - 4 \beta_{7} ) q^{57} + ( -29 \beta_{1} + 2 \beta_{4} - 26 \beta_{5} - 12 \beta_{6} ) q^{59} + ( 1582 + 36 \beta_{2} + 5 \beta_{3} - 19 \beta_{7} ) q^{61} + ( -643 \beta_{1} - 72 \beta_{4} - 15 \beta_{5} + 18 \beta_{6} ) q^{63} + ( 84 + 36 \beta_{2} + 8 \beta_{3} - 14 \beta_{7} ) q^{65} + ( 174 \beta_{1} + 15 \beta_{4} + 12 \beta_{5} - 45 \beta_{6} ) q^{67} + ( -552 - 107 \beta_{2} - 29 \beta_{3} - 21 \beta_{7} ) q^{69} + ( 221 \beta_{1} - 30 \beta_{4} + 31 \beta_{5} + 22 \beta_{6} ) q^{71} + ( 70 - 21 \beta_{2} + 27 \beta_{3} - 24 \beta_{7} ) q^{73} + ( -131 \beta_{1} - 96 \beta_{4} - 26 \beta_{5} - 46 \beta_{6} ) q^{75} + ( -3912 - 119 \beta_{2} + 15 \beta_{3} - 9 \beta_{7} ) q^{77} + ( 514 \beta_{1} - 58 \beta_{4} - 4 \beta_{5} + 48 \beta_{6} ) q^{79} + ( 1837 + 17 \beta_{2} + 89 \beta_{3} + 4 \beta_{7} ) q^{81} + ( -111 \beta_{1} + 110 \beta_{4} + 14 \beta_{5} - 8 \beta_{6} ) q^{83} + ( -1804 + 51 \beta_{2} + 15 \beta_{3} - 33 \beta_{7} ) q^{85} + ( -309 \beta_{1} - 108 \beta_{4} + 11 \beta_{5} - 2 \beta_{6} ) q^{87} + ( -2874 - 181 \beta_{2} - 53 \beta_{3} - 16 \beta_{7} ) q^{89} + ( 378 \beta_{1} + 128 \beta_{4} - 14 \beta_{5} - 42 \beta_{6} ) q^{91} + ( 7040 - 148 \beta_{2} + 68 \beta_{3} + 20 \beta_{7} ) q^{93} + ( -661 \beta_{1} + 127 \beta_{4} - 4 \beta_{5} + 24 \beta_{6} ) q^{95} + ( -466 - 183 \beta_{2} + 13 \beta_{3} - 14 \beta_{7} ) q^{97} + ( -201 \beta_{1} - 6 \beta_{4} - 10 \beta_{5} + 28 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 48q^{5} - 216q^{9} + O(q^{10})$$ $$8q + 48q^{5} - 216q^{9} + 240q^{13} + 240q^{17} - 1216q^{21} + 664q^{25} + 432q^{29} + 992q^{33} + 2800q^{37} - 2928q^{41} - 4880q^{45} - 5752q^{49} + 1776q^{53} + 8608q^{57} + 12656q^{61} + 672q^{65} - 4416q^{69} + 560q^{73} - 31296q^{77} + 14696q^{81} - 14432q^{85} - 22992q^{89} + 56320q^{93} - 3728q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 12 x^{6} - 12 x^{5} - 8 x^{4} + 12 x^{3} + 12 x^{2} + 4 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$152 \nu^{7} - 638 \nu^{6} + 1996 \nu^{5} - 2362 \nu^{4} - 268 \nu^{3} + 1570 \nu^{2} + 1528 \nu + 344$$$$)/51$$ $$\beta_{2}$$ $$=$$ $$($$$$144 \nu^{7} - 676 \nu^{6} + 2120 \nu^{5} - 2864 \nu^{4} - 168 \nu^{3} + 2740 \nu^{2} + 832 \nu - 1120$$$$)/51$$ $$\beta_{3}$$ $$=$$ $$($$$$-176 \nu^{7} + 524 \nu^{6} - 1624 \nu^{5} + 720 \nu^{4} + 1656 \nu^{3} + 580 \nu^{2} + 192 \nu - 2016$$$$)/51$$ $$\beta_{4}$$ $$=$$ $$($$$$-824 \nu^{7} + 3226 \nu^{6} - 9396 \nu^{5} + 8406 \nu^{4} + 9620 \nu^{3} - 10662 \nu^{2} - 12120 \nu - 2824$$$$)/51$$ $$\beta_{5}$$ $$=$$ $$($$$$1288 \nu^{7} - 3742 \nu^{6} + 8732 \nu^{5} + 6430 \nu^{4} - 42620 \nu^{3} + 24674 \nu^{2} + 32360 \nu + 7768$$$$)/51$$ $$\beta_{6}$$ $$=$$ $$($$$$1488 \nu^{7} - 7008 \nu^{6} + 22768 \nu^{5} - 33720 \nu^{4} + 11184 \nu^{3} + 11200 \nu^{2} + 8144 \nu + 1664$$$$)/51$$ $$\beta_{7}$$ $$=$$ $$($$$$-4672 \nu^{7} + 20784 \nu^{6} - 64672 \nu^{5} + 81920 \nu^{4} + 10528 \nu^{3} - 73968 \nu^{2} - 22400 \nu - 896$$$$)/51$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{7} - 8 \beta_{6} - 5 \beta_{5} - 13 \beta_{4} + 6 \beta_{3} + 26 \beta_{2} + 94 \beta_{1} + 512$$$$)/1024$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} - 6 \beta_{6} - 3 \beta_{5} - 5 \beta_{4} + 5 \beta_{3} - 21 \beta_{2} + 52 \beta_{1} - 256$$$$)/256$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{7} + 7 \beta_{6} + 6 \beta_{5} + 19 \beta_{4} + 13 \beta_{3} - 85 \beta_{2} - 13 \beta_{1} - 1408$$$$)/256$$ $$\nu^{4}$$ $$=$$ $$($$$$60 \beta_{6} + 43 \beta_{5} + 111 \beta_{4} - 350 \beta_{1}$$$$)/256$$ $$\nu^{5}$$ $$=$$ $$($$$$27 \beta_{7} + 67 \beta_{6} + 49 \beta_{5} + 130 \beta_{4} - 101 \beta_{3} + 749 \beta_{2} - 363 \beta_{1} + 12928$$$$)/256$$ $$\nu^{6}$$ $$=$$ $$($$$$61 \beta_{7} - 390 \beta_{6} - 254 \beta_{5} - 592 \beta_{4} - 233 \beta_{3} + 1689 \beta_{2} + 2766 \beta_{1} + 28928$$$$)/256$$ $$\nu^{7}$$ $$=$$ $$($$$$-394 \beta_{7} - 5960 \beta_{6} - 3949 \beta_{5} - 9397 \beta_{4} + 1218 \beta_{3} - 10978 \beta_{2} + 40910 \beta_{1} - 200192$$$$)/1024$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 1.55050 − 0.642239i 1.12964 − 2.72719i −0.550501 − 0.228025i −0.129640 − 0.312979i −0.129640 + 0.312979i −0.550501 + 0.228025i 1.12964 + 2.72719i 1.55050 + 0.642239i
0 16.7135i 0 10.1891 0 80.4325i 0 −198.341 0
127.2 0 7.95199i 0 46.0525 0 57.5735i 0 17.7658 0
127.3 0 7.05664i 0 −9.50277 0 6.49140i 0 31.2038 0
127.4 0 6.29514i 0 −22.7388 0 51.5147i 0 41.3713 0
127.5 0 6.29514i 0 −22.7388 0 51.5147i 0 41.3713 0
127.6 0 7.05664i 0 −9.50277 0 6.49140i 0 31.2038 0
127.7 0 7.95199i 0 46.0525 0 57.5735i 0 17.7658 0
127.8 0 16.7135i 0 10.1891 0 80.4325i 0 −198.341 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 127.8 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

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 24 T_{5}^{3} - 1128 T_{5}^{2} + 2976 T_{5} + 101392$$ acting on $$S_{5}^{\mathrm{new}}(128, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 216 T^{2} + 24028 T^{4} - 1277928 T^{6} + 70075206 T^{8} - 8384485608 T^{10} + 1034326612188 T^{12} - 61004779879896 T^{14} + 1853020188851841 T^{16}$$
$5$ $$( 1 - 24 T + 1372 T^{2} - 42024 T^{3} + 1035142 T^{4} - 26265000 T^{5} + 535937500 T^{6} - 5859375000 T^{7} + 152587890625 T^{8} )^{2}$$
$7$ $$1 - 6728 T^{2} + 29560476 T^{4} - 97383797752 T^{6} + 248996961551942 T^{8} - 561398214664527352 T^{10} +$$$$98\!\cdots\!76$$$$T^{12} -$$$$12\!\cdots\!28$$$$T^{14} +$$$$11\!\cdots\!01$$$$T^{16}$$
$11$ $$1 - 64600 T^{2} + 2083873756 T^{4} - 45907087190632 T^{6} + 765101105580719686 T^{8} -$$$$98\!\cdots\!92$$$$T^{10} +$$$$95\!\cdots\!16$$$$T^{12} -$$$$63\!\cdots\!00$$$$T^{14} +$$$$21\!\cdots\!21$$$$T^{16}$$
$13$ $$( 1 - 120 T + 68124 T^{2} - 3870408 T^{3} + 2251470470 T^{4} - 110542722888 T^{5} + 55570839637404 T^{6} - 2795770214697720 T^{7} + 665416609183179841 T^{8} )^{2}$$
$17$ $$( 1 - 120 T + 168220 T^{2} - 16065864 T^{3} + 15632527174 T^{4} - 1341837027144 T^{5} + 1173461916725020 T^{6} - 69914668467571320 T^{7} + 48661191875666868481 T^{8} )^{2}$$
$19$ $$1 - 550872 T^{2} + 164466729436 T^{4} - 33320633036248296 T^{6} +$$$$49\!\cdots\!46$$$$T^{8} -$$$$56\!\cdots\!36$$$$T^{10} +$$$$47\!\cdots\!16$$$$T^{12} -$$$$26\!\cdots\!12$$$$T^{14} +$$$$83\!\cdots\!61$$$$T^{16}$$
$23$ $$1 - 1025096 T^{2} + 455029229724 T^{4} - 118004258074639864 T^{6} +$$$$27\!\cdots\!14$$$$T^{8} -$$$$92\!\cdots\!84$$$$T^{10} +$$$$27\!\cdots\!64$$$$T^{12} -$$$$49\!\cdots\!36$$$$T^{14} +$$$$37\!\cdots\!21$$$$T^{16}$$
$29$ $$( 1 - 216 T + 1846876 T^{2} - 423831144 T^{3} + 1803449019142 T^{4} - 299767715359464 T^{5} + 923893094183759836 T^{6} - 76423993172381312856 T^{7} +$$$$25\!\cdots\!21$$$$T^{8} )^{2}$$
$31$ $$1 - 2366472 T^{2} + 1444435485724 T^{4} + 1509721701480657864 T^{6} -$$$$28\!\cdots\!50$$$$T^{8} +$$$$12\!\cdots\!24$$$$T^{10} +$$$$10\!\cdots\!44$$$$T^{12} -$$$$14\!\cdots\!12$$$$T^{14} +$$$$52\!\cdots\!61$$$$T^{16}$$
$37$ $$( 1 - 1400 T + 6773148 T^{2} - 6215850184 T^{3} + 17798225143046 T^{4} - 11649503996695624 T^{5} + 23790543188366113308 T^{6} -$$$$92\!\cdots\!00$$$$T^{7} +$$$$12\!\cdots\!41$$$$T^{8} )^{2}$$
$41$ $$( 1 + 1464 T + 3213340 T^{2} + 1762829064 T^{3} - 832430784314 T^{4} + 4981333618717704 T^{5} + 25658279635743674140 T^{6} +$$$$33\!\cdots\!84$$$$T^{7} +$$$$63\!\cdots\!41$$$$T^{8} )^{2}$$
$43$ $$1 - 14239448 T^{2} + 102258927107292 T^{4} -$$$$53\!\cdots\!24$$$$T^{6} +$$$$21\!\cdots\!14$$$$T^{8} -$$$$62\!\cdots\!24$$$$T^{10} +$$$$13\!\cdots\!92$$$$T^{12} -$$$$22\!\cdots\!48$$$$T^{14} +$$$$18\!\cdots\!01$$$$T^{16}$$
$47$ $$1 + 1700088 T^{2} + 22769502102556 T^{4} -$$$$10\!\cdots\!76$$$$T^{6} +$$$$23\!\cdots\!86$$$$T^{8} -$$$$23\!\cdots\!36$$$$T^{10} +$$$$12\!\cdots\!76$$$$T^{12} +$$$$22\!\cdots\!28$$$$T^{14} +$$$$32\!\cdots\!41$$$$T^{16}$$
$53$ $$( 1 - 888 T + 17878684 T^{2} - 38865956040 T^{3} + 153671441281030 T^{4} - 306671087680455240 T^{5} +$$$$11\!\cdots\!24$$$$T^{6} -$$$$43\!\cdots\!08$$$$T^{7} +$$$$38\!\cdots\!21$$$$T^{8} )^{2}$$
$59$ $$1 - 59081816 T^{2} + 1713928392943068 T^{4} -$$$$32\!\cdots\!48$$$$T^{6} +$$$$45\!\cdots\!06$$$$T^{8} -$$$$47\!\cdots\!08$$$$T^{10} +$$$$36\!\cdots\!88$$$$T^{12} -$$$$18\!\cdots\!76$$$$T^{14} +$$$$46\!\cdots\!81$$$$T^{16}$$
$61$ $$( 1 - 6328 T + 54475164 T^{2} - 210175360136 T^{3} + 1063754656519814 T^{4} - 2910054618560794376 T^{5} +$$$$10\!\cdots\!84$$$$T^{6} -$$$$16\!\cdots\!88$$$$T^{7} +$$$$36\!\cdots\!61$$$$T^{8} )^{2}$$
$67$ $$1 - 74176088 T^{2} + 3133751785162972 T^{4} -$$$$92\!\cdots\!92$$$$T^{6} +$$$$20\!\cdots\!38$$$$T^{8} -$$$$37\!\cdots\!72$$$$T^{10} +$$$$51\!\cdots\!32$$$$T^{12} -$$$$49\!\cdots\!48$$$$T^{14} +$$$$27\!\cdots\!61$$$$T^{16}$$
$71$ $$1 - 116180040 T^{2} + 5397632108208796 T^{4} -$$$$13\!\cdots\!28$$$$T^{6} +$$$$28\!\cdots\!78$$$$T^{8} -$$$$87\!\cdots\!08$$$$T^{10} +$$$$22\!\cdots\!16$$$$T^{12} -$$$$31\!\cdots\!40$$$$T^{14} +$$$$17\!\cdots\!41$$$$T^{16}$$
$73$ $$( 1 - 280 T + 72747100 T^{2} - 19192386472 T^{3} + 2931543571401286 T^{4} - 545030016396995752 T^{5} +$$$$58\!\cdots\!00$$$$T^{6} -$$$$64\!\cdots\!80$$$$T^{7} +$$$$65\!\cdots\!61$$$$T^{8} )^{2}$$
$79$ $$1 - 111270664 T^{2} + 6811470751623196 T^{4} -$$$$27\!\cdots\!16$$$$T^{6} +$$$$10\!\cdots\!62$$$$T^{8} -$$$$41\!\cdots\!76$$$$T^{10} +$$$$15\!\cdots\!16$$$$T^{12} -$$$$38\!\cdots\!84$$$$T^{14} +$$$$52\!\cdots\!41$$$$T^{16}$$
$83$ $$1 - 238640216 T^{2} + 29479046032067292 T^{4} -$$$$23\!\cdots\!92$$$$T^{6} +$$$$13\!\cdots\!54$$$$T^{8} -$$$$52\!\cdots\!72$$$$T^{10} +$$$$14\!\cdots\!52$$$$T^{12} -$$$$27\!\cdots\!36$$$$T^{14} +$$$$25\!\cdots\!61$$$$T^{16}$$
$89$ $$( 1 + 11496 T + 204574044 T^{2} + 1764445531224 T^{3} + 18548660202283334 T^{4} +$$$$11\!\cdots\!84$$$$T^{5} +$$$$80\!\cdots\!64$$$$T^{6} +$$$$28\!\cdots\!16$$$$T^{7} +$$$$15\!\cdots\!61$$$$T^{8} )^{2}$$
$97$ $$( 1 + 1864 T + 295260828 T^{2} + 647805539576 T^{3} + 36499931947241798 T^{4} + 57349758646480324856 T^{5} +$$$$23\!\cdots\!08$$$$T^{6} +$$$$12\!\cdots\!24$$$$T^{7} +$$$$61\!\cdots\!21$$$$T^{8} )^{2}$$