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

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(13.2313552747\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1871773696.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{38} \)
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.

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

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

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

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:

Inner twists

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

Hecke kernels

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