Properties

Label 128.3.d.a
Level 128
Weight 3
Character orbit 128.d
Analytic conductor 3.488
Analytic rank 0
Dimension 2
CM discriminant -4
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 128.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.48774738381\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 8 i q^{5} -9 q^{9} +O(q^{10})\) \( q + 8 i q^{5} -9 q^{9} + 24 i q^{13} + 30 q^{17} -39 q^{25} -40 i q^{29} -24 i q^{37} + 18 q^{41} -72 i q^{45} + 49 q^{49} -56 i q^{53} + 120 i q^{61} -192 q^{65} + 110 q^{73} + 81 q^{81} + 240 i q^{85} + 78 q^{89} -130 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 18q^{9} + O(q^{10}) \) \( 2q - 18q^{9} + 60q^{17} - 78q^{25} + 36q^{41} + 98q^{49} - 384q^{65} + 220q^{73} + 162q^{81} + 156q^{89} - 260q^{97} + O(q^{100}) \)

Character values

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

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

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
1.00000i
1.00000i
0 0 0 8.00000i 0 0 0 −9.00000 0
63.2 0 0 0 8.00000i 0 0 0 −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.3.d.a 2
3.b odd 2 1 1152.3.b.a 2
4.b odd 2 1 CM 128.3.d.a 2
8.b even 2 1 inner 128.3.d.a 2
8.d odd 2 1 inner 128.3.d.a 2
12.b even 2 1 1152.3.b.a 2
16.e even 4 1 256.3.c.a 1
16.e even 4 1 256.3.c.b 1
16.f odd 4 1 256.3.c.a 1
16.f odd 4 1 256.3.c.b 1
24.f even 2 1 1152.3.b.a 2
24.h odd 2 1 1152.3.b.a 2
48.i odd 4 1 2304.3.g.a 1
48.i odd 4 1 2304.3.g.f 1
48.k even 4 1 2304.3.g.a 1
48.k even 4 1 2304.3.g.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.3.d.a 2 1.a even 1 1 trivial
128.3.d.a 2 4.b odd 2 1 CM
128.3.d.a 2 8.b even 2 1 inner
128.3.d.a 2 8.d odd 2 1 inner
256.3.c.a 1 16.e even 4 1
256.3.c.a 1 16.f odd 4 1
256.3.c.b 1 16.e even 4 1
256.3.c.b 1 16.f odd 4 1
1152.3.b.a 2 3.b odd 2 1
1152.3.b.a 2 12.b even 2 1
1152.3.b.a 2 24.f even 2 1
1152.3.b.a 2 24.h odd 2 1
2304.3.g.a 1 48.i odd 4 1
2304.3.g.a 1 48.k even 4 1
2304.3.g.f 1 48.i odd 4 1
2304.3.g.f 1 48.k even 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + 9 T^{2} )^{2} \)
$5$ \( ( 1 - 6 T + 25 T^{2} )( 1 + 6 T + 25 T^{2} ) \)
$7$ \( ( 1 - 7 T )^{2}( 1 + 7 T )^{2} \)
$11$ \( ( 1 + 121 T^{2} )^{2} \)
$13$ \( ( 1 - 10 T + 169 T^{2} )( 1 + 10 T + 169 T^{2} ) \)
$17$ \( ( 1 - 30 T + 289 T^{2} )^{2} \)
$19$ \( ( 1 + 361 T^{2} )^{2} \)
$23$ \( ( 1 - 23 T )^{2}( 1 + 23 T )^{2} \)
$29$ \( ( 1 - 42 T + 841 T^{2} )( 1 + 42 T + 841 T^{2} ) \)
$31$ \( ( 1 - 31 T )^{2}( 1 + 31 T )^{2} \)
$37$ \( ( 1 - 70 T + 1369 T^{2} )( 1 + 70 T + 1369 T^{2} ) \)
$41$ \( ( 1 - 18 T + 1681 T^{2} )^{2} \)
$43$ \( ( 1 + 1849 T^{2} )^{2} \)
$47$ \( ( 1 - 47 T )^{2}( 1 + 47 T )^{2} \)
$53$ \( ( 1 - 90 T + 2809 T^{2} )( 1 + 90 T + 2809 T^{2} ) \)
$59$ \( ( 1 + 3481 T^{2} )^{2} \)
$61$ \( ( 1 - 22 T + 3721 T^{2} )( 1 + 22 T + 3721 T^{2} ) \)
$67$ \( ( 1 + 4489 T^{2} )^{2} \)
$71$ \( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \)
$73$ \( ( 1 - 110 T + 5329 T^{2} )^{2} \)
$79$ \( ( 1 - 79 T )^{2}( 1 + 79 T )^{2} \)
$83$ \( ( 1 + 6889 T^{2} )^{2} \)
$89$ \( ( 1 - 78 T + 7921 T^{2} )^{2} \)
$97$ \( ( 1 + 130 T + 9409 T^{2} )^{2} \)
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