Properties

Label 128.3.f.a
Level 128
Weight 3
Character orbit 128.f
Analytic conductor 3.488
Analytic rank 0
Dimension 6
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 128 = 2^{7} \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 128.f (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(3.48774738381\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.399424.1
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{6} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{4} q^{3} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{4} ) q^{5} \) \( + ( -\beta_{2} - \beta_{3} ) q^{7} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{4} q^{3} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{4} ) q^{5} \) \( + ( -\beta_{2} - \beta_{3} ) q^{7} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{9} \) \( + ( -4 - 4 \beta_{1} - 2 \beta_{3} - \beta_{5} ) q^{11} \) \( + ( 1 + \beta_{1} - \beta_{3} + 3 \beta_{5} ) q^{13} \) \( + ( -8 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{15} \) \( + ( 2 + \beta_{2} + \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{17} \) \( + ( 4 - 4 \beta_{1} - 2 \beta_{2} - \beta_{4} ) q^{19} \) \( + ( 2 - 2 \beta_{1} - 4 \beta_{4} ) q^{21} \) \( + ( -8 + \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{23} \) \( + ( -\beta_{1} + 4 \beta_{4} - 4 \beta_{5} ) q^{25} \) \( + ( 12 + 12 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{27} \) \( + ( 1 + \beta_{1} + \beta_{3} - 7 \beta_{5} ) q^{29} \) \( + ( 24 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{31} \) \( + ( -6 - \beta_{2} - \beta_{3} - 7 \beta_{4} - 7 \beta_{5} ) q^{33} \) \( + ( -16 + 16 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} ) q^{35} \) \( + ( -7 + 7 \beta_{1} - 5 \beta_{2} + 7 \beta_{4} ) q^{37} \) \( + ( 32 + 3 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} ) q^{39} \) \( + ( 8 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} ) q^{41} \) \( + ( -16 - 16 \beta_{1} + 8 \beta_{3} + \beta_{5} ) q^{43} \) \( + ( -9 - 9 \beta_{1} + 5 \beta_{3} + \beta_{5} ) q^{45} \) \( + ( -24 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 10 \beta_{4} + 10 \beta_{5} ) q^{47} \) \( + ( -13 - 6 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{49} \) \( + ( 28 - 28 \beta_{1} + 6 \beta_{2} ) q^{51} \) \( + ( -15 + 15 \beta_{1} - \beta_{2} - 5 \beta_{4} ) q^{53} \) \( + ( -48 - \beta_{2} - \beta_{3} - 8 \beta_{4} - 8 \beta_{5} ) q^{55} \) \( + ( 6 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{57} \) \( + ( 32 + 32 \beta_{1} - 4 \beta_{3} - 3 \beta_{5} ) q^{59} \) \( + ( -7 - 7 \beta_{1} - 5 \beta_{3} - \beta_{5} ) q^{61} \) \( + ( 40 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} ) q^{63} \) \( + ( 10 + 6 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} ) q^{65} \) \( + ( -44 + 44 \beta_{1} - 14 \beta_{2} - 5 \beta_{4} ) q^{67} \) \( + ( 18 - 18 \beta_{1} + 4 \beta_{2} - 8 \beta_{4} ) q^{69} \) \( + ( 56 + \beta_{2} + \beta_{3} + 18 \beta_{4} + 18 \beta_{5} ) q^{71} \) \( + ( -24 \beta_{1} + 11 \beta_{2} - 11 \beta_{3} + 13 \beta_{4} - 13 \beta_{5} ) q^{73} \) \( + ( -40 - 40 \beta_{1} - 8 \beta_{3} + 7 \beta_{5} ) q^{75} \) \( + ( 34 + 34 \beta_{1} - 4 \beta_{3} ) q^{77} \) \( + ( -64 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + 20 \beta_{4} - 20 \beta_{5} ) q^{79} \) \( + ( 25 + 11 \beta_{2} + 11 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} ) q^{81} \) \( + ( 56 - 56 \beta_{1} + 9 \beta_{4} ) q^{83} \) \( + ( 42 - 42 \beta_{1} + 4 \beta_{2} + 16 \beta_{4} ) q^{85} \) \( + ( -72 - 7 \beta_{2} - 7 \beta_{3} + 10 \beta_{4} + 10 \beta_{5} ) q^{87} \) \( + ( -8 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} - 13 \beta_{4} + 13 \beta_{5} ) q^{89} \) \( + ( 24 + 24 \beta_{1} - 8 \beta_{3} - 14 \beta_{5} ) q^{91} \) \( + ( 16 + 16 \beta_{1} + 4 \beta_{3} + 28 \beta_{5} ) q^{93} \) \( + ( 32 \beta_{1} + 7 \beta_{2} - 7 \beta_{3} ) q^{95} \) \( + ( 2 - 11 \beta_{2} - 11 \beta_{3} + 15 \beta_{4} + 15 \beta_{5} ) q^{97} \) \( + ( -32 + 32 \beta_{1} + 4 \beta_{2} + 13 \beta_{4} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut -\mathstrut 18q^{11} \) \(\mathstrut +\mathstrut 2q^{13} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 30q^{19} \) \(\mathstrut +\mathstrut 20q^{21} \) \(\mathstrut -\mathstrut 60q^{23} \) \(\mathstrut +\mathstrut 64q^{27} \) \(\mathstrut +\mathstrut 18q^{29} \) \(\mathstrut -\mathstrut 4q^{33} \) \(\mathstrut -\mathstrut 100q^{35} \) \(\mathstrut -\mathstrut 46q^{37} \) \(\mathstrut +\mathstrut 196q^{39} \) \(\mathstrut -\mathstrut 114q^{43} \) \(\mathstrut -\mathstrut 66q^{45} \) \(\mathstrut -\mathstrut 46q^{49} \) \(\mathstrut +\mathstrut 156q^{51} \) \(\mathstrut -\mathstrut 78q^{53} \) \(\mathstrut -\mathstrut 252q^{55} \) \(\mathstrut +\mathstrut 206q^{59} \) \(\mathstrut -\mathstrut 30q^{61} \) \(\mathstrut +\mathstrut 12q^{65} \) \(\mathstrut -\mathstrut 226q^{67} \) \(\mathstrut +\mathstrut 116q^{69} \) \(\mathstrut +\mathstrut 260q^{71} \) \(\mathstrut -\mathstrut 238q^{75} \) \(\mathstrut +\mathstrut 212q^{77} \) \(\mathstrut +\mathstrut 86q^{81} \) \(\mathstrut +\mathstrut 318q^{83} \) \(\mathstrut +\mathstrut 212q^{85} \) \(\mathstrut -\mathstrut 444q^{87} \) \(\mathstrut +\mathstrut 188q^{91} \) \(\mathstrut +\mathstrut 32q^{93} \) \(\mathstrut -\mathstrut 4q^{97} \) \(\mathstrut -\mathstrut 226q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{5} - 3 \nu^{3} + 4 \nu^{2} - 2 \nu + 8 \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 4 \nu^{4} - 5 \nu^{3} + 8 \nu^{2} - 14 \nu + 4 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{5} + 4 \nu^{4} - 9 \nu^{3} + 8 \nu^{2} + 2 \nu + 12 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{5} - 4 \nu^{4} + 9 \nu^{3} - 8 \nu^{2} + 14 \nu - 20 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -5 \nu^{5} + 4 \nu^{4} - 7 \nu^{3} + 16 \nu^{2} - 10 \nu + 20 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(6\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(10\) \(\beta_{1}\mathstrut +\mathstrut \) \(8\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(10\)\()/4\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(-\beta_{1}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.264658 1.38923i
−0.671462 + 1.24464i
1.40680 + 0.144584i
0.264658 + 1.38923i
−0.671462 1.24464i
1.40680 0.144584i
0 −3.24914 3.24914i 0 0.0586332 + 0.0586332i 0 −4.61555 0 12.1138i 0
31.2 0 0.146365 + 0.146365i 0 −3.68585 3.68585i 0 9.66442 0 8.95715i 0
31.3 0 2.10278 + 2.10278i 0 4.62721 + 4.62721i 0 −3.04888 0 0.156674i 0
95.1 0 −3.24914 + 3.24914i 0 0.0586332 0.0586332i 0 −4.61555 0 12.1138i 0
95.2 0 0.146365 0.146365i 0 −3.68585 + 3.68585i 0 9.66442 0 8.95715i 0
95.3 0 2.10278 2.10278i 0 4.62721 4.62721i 0 −3.04888 0 0.156674i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 95.3
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
16.f Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{6} \) \(\mathstrut +\mathstrut 2 T_{3}^{5} \) \(\mathstrut +\mathstrut 2 T_{3}^{4} \) \(\mathstrut -\mathstrut 32 T_{3}^{3} \) \(\mathstrut +\mathstrut 196 T_{3}^{2} \) \(\mathstrut -\mathstrut 56 T_{3} \) \(\mathstrut +\mathstrut 8 \) acting on \(S_{3}^{\mathrm{new}}(128, [\chi])\).