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

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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:

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