Properties

Label 128.2.e.a
Level 128
Weight 2
Character orbit 128.e
Analytic conductor 1.022
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(1.02208514587\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -1 + i ) q^{3} \) \( + ( 1 + i ) q^{5} \) \( + 2 i q^{7} \) \( + i q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 + i ) q^{3} \) \( + ( 1 + i ) q^{5} \) \( + 2 i q^{7} \) \( + i q^{9} \) \( + ( 1 + i ) q^{11} \) \( + ( 1 - i ) q^{13} \) \( -2 q^{15} \) \( -2 q^{17} \) \( + ( 3 - 3 i ) q^{19} \) \( + ( -2 - 2 i ) q^{21} \) \( -6 i q^{23} \) \( -3 i q^{25} \) \( + ( -4 - 4 i ) q^{27} \) \( + ( -3 + 3 i ) q^{29} \) \( + 8 q^{31} \) \( -2 q^{33} \) \( + ( -2 + 2 i ) q^{35} \) \( + ( -3 - 3 i ) q^{37} \) \( + 2 i q^{39} \) \( + ( 5 + 5 i ) q^{43} \) \( + ( -1 + i ) q^{45} \) \( -8 q^{47} \) \( + 3 q^{49} \) \( + ( 2 - 2 i ) q^{51} \) \( + ( 5 + 5 i ) q^{53} \) \( + 2 i q^{55} \) \( + 6 i q^{57} \) \( + ( -3 - 3 i ) q^{59} \) \( + ( 9 - 9 i ) q^{61} \) \( -2 q^{63} \) \( + 2 q^{65} \) \( + ( -5 + 5 i ) q^{67} \) \( + ( 6 + 6 i ) q^{69} \) \( + 10 i q^{71} \) \( -4 i q^{73} \) \( + ( 3 + 3 i ) q^{75} \) \( + ( -2 + 2 i ) q^{77} \) \( + 5 q^{81} \) \( + ( -1 + i ) q^{83} \) \( + ( -2 - 2 i ) q^{85} \) \( -6 i q^{87} \) \( + 4 i q^{89} \) \( + ( 2 + 2 i ) q^{91} \) \( + ( -8 + 8 i ) q^{93} \) \( + 6 q^{95} \) \( -2 q^{97} \) \( + ( -1 + i ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 2q^{11} \) \(\mathstrut +\mathstrut 2q^{13} \) \(\mathstrut -\mathstrut 4q^{15} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 6q^{19} \) \(\mathstrut -\mathstrut 4q^{21} \) \(\mathstrut -\mathstrut 8q^{27} \) \(\mathstrut -\mathstrut 6q^{29} \) \(\mathstrut +\mathstrut 16q^{31} \) \(\mathstrut -\mathstrut 4q^{33} \) \(\mathstrut -\mathstrut 4q^{35} \) \(\mathstrut -\mathstrut 6q^{37} \) \(\mathstrut +\mathstrut 10q^{43} \) \(\mathstrut -\mathstrut 2q^{45} \) \(\mathstrut -\mathstrut 16q^{47} \) \(\mathstrut +\mathstrut 6q^{49} \) \(\mathstrut +\mathstrut 4q^{51} \) \(\mathstrut +\mathstrut 10q^{53} \) \(\mathstrut -\mathstrut 6q^{59} \) \(\mathstrut +\mathstrut 18q^{61} \) \(\mathstrut -\mathstrut 4q^{63} \) \(\mathstrut +\mathstrut 4q^{65} \) \(\mathstrut -\mathstrut 10q^{67} \) \(\mathstrut +\mathstrut 12q^{69} \) \(\mathstrut +\mathstrut 6q^{75} \) \(\mathstrut -\mathstrut 4q^{77} \) \(\mathstrut +\mathstrut 10q^{81} \) \(\mathstrut -\mathstrut 2q^{83} \) \(\mathstrut -\mathstrut 4q^{85} \) \(\mathstrut +\mathstrut 4q^{91} \) \(\mathstrut -\mathstrut 16q^{93} \) \(\mathstrut +\mathstrut 12q^{95} \) \(\mathstrut -\mathstrut 4q^{97} \) \(\mathstrut -\mathstrut 2q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(5\) \(127\)
\(\chi(n)\) \(i\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
1.00000i
1.00000i
0 −1.00000 1.00000i 0 1.00000 1.00000i 0 2.00000i 0 1.00000i 0
97.1 0 −1.00000 + 1.00000i 0 1.00000 + 1.00000i 0 2.00000i 0 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

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