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 \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(13.2313552747\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.205520896.4
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{39} \)
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.

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

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

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 yes

Hecke kernels

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