Properties

Label 128.3.d.c
Level 128
Weight 3
Character orbit 128.d
Analytic conductor 3.488
Analytic rank 0
Dimension 4
CM no
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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{3} + 4 \zeta_{8}^{2} q^{5} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{7} - q^{9} +O(q^{10})\) \( q + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{3} + 4 \zeta_{8}^{2} q^{5} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{7} - q^{9} + ( 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{11} -20 \zeta_{8}^{2} q^{13} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{15} -10 q^{17} + ( 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{19} + 32 \zeta_{8}^{2} q^{21} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{23} + 9 q^{25} + ( -20 \zeta_{8} + 20 \zeta_{8}^{3} ) q^{27} -20 \zeta_{8}^{2} q^{29} + 40 q^{33} + ( -32 \zeta_{8} + 32 \zeta_{8}^{3} ) q^{35} + 20 \zeta_{8}^{2} q^{37} + ( -40 \zeta_{8} - 40 \zeta_{8}^{3} ) q^{39} -30 q^{41} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{43} -4 \zeta_{8}^{2} q^{45} + ( -48 \zeta_{8} - 48 \zeta_{8}^{3} ) q^{47} -79 q^{49} + ( -20 \zeta_{8} + 20 \zeta_{8}^{3} ) q^{51} -60 \zeta_{8}^{2} q^{53} + ( 40 \zeta_{8} + 40 \zeta_{8}^{3} ) q^{55} + 40 q^{57} + ( -30 \zeta_{8} + 30 \zeta_{8}^{3} ) q^{59} + 28 \zeta_{8}^{2} q^{61} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{63} + 80 q^{65} + ( 58 \zeta_{8} - 58 \zeta_{8}^{3} ) q^{67} -32 \zeta_{8}^{2} q^{69} + ( 40 \zeta_{8} + 40 \zeta_{8}^{3} ) q^{71} -10 q^{73} + ( 18 \zeta_{8} - 18 \zeta_{8}^{3} ) q^{75} + 160 \zeta_{8}^{2} q^{77} + ( 80 \zeta_{8} + 80 \zeta_{8}^{3} ) q^{79} -71 q^{81} + ( 18 \zeta_{8} - 18 \zeta_{8}^{3} ) q^{83} -40 \zeta_{8}^{2} q^{85} + ( -40 \zeta_{8} - 40 \zeta_{8}^{3} ) q^{87} + 22 q^{89} + ( 160 \zeta_{8} - 160 \zeta_{8}^{3} ) q^{91} + ( 40 \zeta_{8} + 40 \zeta_{8}^{3} ) q^{95} + 150 q^{97} + ( -10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} - 40q^{17} + 36q^{25} + 160q^{33} - 120q^{41} - 316q^{49} + 160q^{57} + 320q^{65} - 40q^{73} - 284q^{81} + 88q^{89} + 600q^{97} + O(q^{100}) \)

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
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
0 −2.82843 0 4.00000i 0 11.3137i 0 −1.00000 0
63.2 0 −2.82843 0 4.00000i 0 11.3137i 0 −1.00000 0
63.3 0 2.82843 0 4.00000i 0 11.3137i 0 −1.00000 0
63.4 0 2.82843 0 4.00000i 0 11.3137i 0 −1.00000 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 inner
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.c 4
3.b odd 2 1 1152.3.b.g 4
4.b odd 2 1 inner 128.3.d.c 4
8.b even 2 1 inner 128.3.d.c 4
8.d odd 2 1 inner 128.3.d.c 4
12.b even 2 1 1152.3.b.g 4
16.e even 4 1 256.3.c.c 2
16.e even 4 1 256.3.c.f 2
16.f odd 4 1 256.3.c.c 2
16.f odd 4 1 256.3.c.f 2
24.f even 2 1 1152.3.b.g 4
24.h odd 2 1 1152.3.b.g 4
48.i odd 4 1 2304.3.g.h 2
48.i odd 4 1 2304.3.g.m 2
48.k even 4 1 2304.3.g.h 2
48.k even 4 1 2304.3.g.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.3.d.c 4 1.a even 1 1 trivial
128.3.d.c 4 4.b odd 2 1 inner
128.3.d.c 4 8.b even 2 1 inner
128.3.d.c 4 8.d odd 2 1 inner
256.3.c.c 2 16.e even 4 1
256.3.c.c 2 16.f odd 4 1
256.3.c.f 2 16.e even 4 1
256.3.c.f 2 16.f odd 4 1
1152.3.b.g 4 3.b odd 2 1
1152.3.b.g 4 12.b even 2 1
1152.3.b.g 4 24.f even 2 1
1152.3.b.g 4 24.h odd 2 1
2304.3.g.h 2 48.i odd 4 1
2304.3.g.h 2 48.k even 4 1
2304.3.g.m 2 48.i odd 4 1
2304.3.g.m 2 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 8 \) acting on \(S_{3}^{\mathrm{new}}(128, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + 10 T^{2} + 81 T^{4} )^{2} \)
$5$ \( ( 1 - 34 T^{2} + 625 T^{4} )^{2} \)
$7$ \( ( 1 + 30 T^{2} + 2401 T^{4} )^{2} \)
$11$ \( ( 1 + 42 T^{2} + 14641 T^{4} )^{2} \)
$13$ \( ( 1 + 62 T^{2} + 28561 T^{4} )^{2} \)
$17$ \( ( 1 + 10 T + 289 T^{2} )^{4} \)
$19$ \( ( 1 + 522 T^{2} + 130321 T^{4} )^{2} \)
$23$ \( ( 1 - 930 T^{2} + 279841 T^{4} )^{2} \)
$29$ \( ( 1 - 1282 T^{2} + 707281 T^{4} )^{2} \)
$31$ \( ( 1 - 31 T )^{4}( 1 + 31 T )^{4} \)
$37$ \( ( 1 - 2338 T^{2} + 1874161 T^{4} )^{2} \)
$41$ \( ( 1 + 30 T + 1681 T^{2} )^{4} \)
$43$ \( ( 1 + 3690 T^{2} + 3418801 T^{4} )^{2} \)
$47$ \( ( 1 + 190 T^{2} + 4879681 T^{4} )^{2} \)
$53$ \( ( 1 - 2018 T^{2} + 7890481 T^{4} )^{2} \)
$59$ \( ( 1 + 5162 T^{2} + 12117361 T^{4} )^{2} \)
$61$ \( ( 1 - 6658 T^{2} + 13845841 T^{4} )^{2} \)
$67$ \( ( 1 + 2250 T^{2} + 20151121 T^{4} )^{2} \)
$71$ \( ( 1 - 6882 T^{2} + 25411681 T^{4} )^{2} \)
$73$ \( ( 1 + 10 T + 5329 T^{2} )^{4} \)
$79$ \( ( 1 + 318 T^{2} + 38950081 T^{4} )^{2} \)
$83$ \( ( 1 + 13130 T^{2} + 47458321 T^{4} )^{2} \)
$89$ \( ( 1 - 22 T + 7921 T^{2} )^{4} \)
$97$ \( ( 1 - 150 T + 9409 T^{2} )^{4} \)
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