Properties

Label 128.4.a.f
Level 128
Weight 4
Character orbit 128.a
Self dual Yes
Analytic conductor 7.552
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 128 = 2^{7} \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 128.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(7.55224448073\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -2 + \beta ) q^{3} \) \( + ( 2 + 2 \beta ) q^{5} \) \( + ( 4 + 2 \beta ) q^{7} \) \( + ( 25 - 4 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -2 + \beta ) q^{3} \) \( + ( 2 + 2 \beta ) q^{5} \) \( + ( 4 + 2 \beta ) q^{7} \) \( + ( 25 - 4 \beta ) q^{9} \) \( + ( -46 - \beta ) q^{11} \) \( + ( 50 - 6 \beta ) q^{13} \) \( + ( 92 - 2 \beta ) q^{15} \) \( + ( 46 + 12 \beta ) q^{17} \) \( + ( -2 - 7 \beta ) q^{19} \) \( + 88 q^{21} \) \( + ( -4 - 18 \beta ) q^{23} \) \( + ( 71 + 8 \beta ) q^{25} \) \( + ( -188 + 6 \beta ) q^{27} \) \( + ( 42 + 10 \beta ) q^{29} \) \( + ( 192 + 16 \beta ) q^{31} \) \( + ( 44 - 44 \beta ) q^{33} \) \( + ( 200 + 12 \beta ) q^{35} \) \( + ( -86 - 14 \beta ) q^{37} \) \( + ( -388 + 62 \beta ) q^{39} \) \( + ( -150 + 8 \beta ) q^{41} \) \( + ( -150 - 5 \beta ) q^{43} \) \( + ( -334 + 42 \beta ) q^{45} \) \( + ( 8 - 44 \beta ) q^{47} \) \( + ( -135 + 16 \beta ) q^{49} \) \( + ( 484 + 22 \beta ) q^{51} \) \( + ( -6 - 14 \beta ) q^{53} \) \( + ( -188 - 94 \beta ) q^{55} \) \( + ( -332 + 12 \beta ) q^{57} \) \( + ( 322 - 33 \beta ) q^{59} \) \( + ( 146 - 70 \beta ) q^{61} \) \( + ( -284 + 34 \beta ) q^{63} \) \( + ( -476 + 88 \beta ) q^{65} \) \( + ( 86 - 83 \beta ) q^{67} \) \( + ( -856 + 32 \beta ) q^{69} \) \( + ( -204 + 42 \beta ) q^{71} \) \( + ( 206 - 52 \beta ) q^{73} \) \( + ( 242 + 55 \beta ) q^{75} \) \( + ( -280 - 96 \beta ) q^{77} \) \( + ( 200 + 36 \beta ) q^{79} \) \( + ( -11 - 92 \beta ) q^{81} \) \( + ( -474 + 13 \beta ) q^{83} \) \( + ( 1244 + 116 \beta ) q^{85} \) \( + ( 396 + 22 \beta ) q^{87} \) \( + ( 286 - 116 \beta ) q^{89} \) \( + ( -376 + 76 \beta ) q^{91} \) \( + ( 384 + 160 \beta ) q^{93} \) \( + ( -676 - 18 \beta ) q^{95} \) \( + ( 1102 + 92 \beta ) q^{97} \) \( + ( -958 + 159 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 50q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 50q^{9} \) \(\mathstrut -\mathstrut 92q^{11} \) \(\mathstrut +\mathstrut 100q^{13} \) \(\mathstrut +\mathstrut 184q^{15} \) \(\mathstrut +\mathstrut 92q^{17} \) \(\mathstrut -\mathstrut 4q^{19} \) \(\mathstrut +\mathstrut 176q^{21} \) \(\mathstrut -\mathstrut 8q^{23} \) \(\mathstrut +\mathstrut 142q^{25} \) \(\mathstrut -\mathstrut 376q^{27} \) \(\mathstrut +\mathstrut 84q^{29} \) \(\mathstrut +\mathstrut 384q^{31} \) \(\mathstrut +\mathstrut 88q^{33} \) \(\mathstrut +\mathstrut 400q^{35} \) \(\mathstrut -\mathstrut 172q^{37} \) \(\mathstrut -\mathstrut 776q^{39} \) \(\mathstrut -\mathstrut 300q^{41} \) \(\mathstrut -\mathstrut 300q^{43} \) \(\mathstrut -\mathstrut 668q^{45} \) \(\mathstrut +\mathstrut 16q^{47} \) \(\mathstrut -\mathstrut 270q^{49} \) \(\mathstrut +\mathstrut 968q^{51} \) \(\mathstrut -\mathstrut 12q^{53} \) \(\mathstrut -\mathstrut 376q^{55} \) \(\mathstrut -\mathstrut 664q^{57} \) \(\mathstrut +\mathstrut 644q^{59} \) \(\mathstrut +\mathstrut 292q^{61} \) \(\mathstrut -\mathstrut 568q^{63} \) \(\mathstrut -\mathstrut 952q^{65} \) \(\mathstrut +\mathstrut 172q^{67} \) \(\mathstrut -\mathstrut 1712q^{69} \) \(\mathstrut -\mathstrut 408q^{71} \) \(\mathstrut +\mathstrut 412q^{73} \) \(\mathstrut +\mathstrut 484q^{75} \) \(\mathstrut -\mathstrut 560q^{77} \) \(\mathstrut +\mathstrut 400q^{79} \) \(\mathstrut -\mathstrut 22q^{81} \) \(\mathstrut -\mathstrut 948q^{83} \) \(\mathstrut +\mathstrut 2488q^{85} \) \(\mathstrut +\mathstrut 792q^{87} \) \(\mathstrut +\mathstrut 572q^{89} \) \(\mathstrut -\mathstrut 752q^{91} \) \(\mathstrut +\mathstrut 768q^{93} \) \(\mathstrut -\mathstrut 1352q^{95} \) \(\mathstrut +\mathstrut 2204q^{97} \) \(\mathstrut -\mathstrut 1916q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.73205
1.73205
0 −8.92820 0 −11.8564 0 −9.85641 0 52.7128 0
1.2 0 4.92820 0 15.8564 0 17.8564 0 −2.71281 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(128))\):

\(T_{3}^{2} \) \(\mathstrut +\mathstrut 4 T_{3} \) \(\mathstrut -\mathstrut 44 \)
\(T_{5}^{2} \) \(\mathstrut -\mathstrut 4 T_{5} \) \(\mathstrut -\mathstrut 188 \)