Properties

Label 1024.2.e.i
Level $1024$
Weight $2$
Character orbit 1024.e
Analytic conductor $8.177$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.e (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.17668116698\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{3} q^{5} + 4 \beta_{2} q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{3} q^{5} + 4 \beta_{2} q^{7} + \beta_{2} q^{9} + \beta_{3} q^{11} - \beta_1 q^{13} - 4 q^{15} + 2 q^{17} - \beta_1 q^{19} + 4 \beta_{3} q^{21} + 4 \beta_{2} q^{23} + \beta_{2} q^{25} - 2 \beta_{3} q^{27} - 3 \beta_1 q^{29} - 4 q^{33} - 4 \beta_1 q^{35} + 5 \beta_{3} q^{37} - 4 \beta_{2} q^{39} - 6 \beta_{2} q^{41} - 3 \beta_{3} q^{43} - \beta_1 q^{45} + 8 q^{47} - 9 q^{49} + 2 \beta_1 q^{51} + 3 \beta_{3} q^{53} - 4 \beta_{2} q^{55} - 4 \beta_{2} q^{57} + 7 \beta_{3} q^{59} + \beta_1 q^{61} - 4 q^{63} + 4 q^{65} + 5 \beta_1 q^{67} + 4 \beta_{3} q^{69} - 12 \beta_{2} q^{71} + 14 \beta_{2} q^{73} + \beta_{3} q^{75} - 4 \beta_1 q^{77} + 8 q^{79} + 11 q^{81} + 3 \beta_1 q^{83} + 2 \beta_{3} q^{85} - 12 \beta_{2} q^{87} + 2 \beta_{2} q^{89} - 4 \beta_{3} q^{91} + 4 q^{95} - 2 q^{97} - \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{15} + 8 q^{17} - 16 q^{33} + 32 q^{47} - 36 q^{49} - 16 q^{63} + 16 q^{65} + 32 q^{79} + 44 q^{81} + 16 q^{95} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{8} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{8}^{3} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(-\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
0 −1.41421 1.41421i 0 1.41421 1.41421i 0 4.00000i 0 1.00000i 0
257.2 0 1.41421 + 1.41421i 0 −1.41421 + 1.41421i 0 4.00000i 0 1.00000i 0
769.1 0 −1.41421 + 1.41421i 0 1.41421 + 1.41421i 0 4.00000i 0 1.00000i 0
769.2 0 1.41421 1.41421i 0 −1.41421 1.41421i 0 4.00000i 0 1.00000i 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
8.b even 2 1 inner
16.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1024.2.e.i 4
4.b odd 2 1 1024.2.e.m 4
8.b even 2 1 inner 1024.2.e.i 4
8.d odd 2 1 1024.2.e.m 4
16.e even 4 2 inner 1024.2.e.i 4
16.f odd 4 2 1024.2.e.m 4
32.g even 8 1 128.2.a.a 1
32.g even 8 1 128.2.a.d yes 1
32.g even 8 2 256.2.b.c 2
32.h odd 8 1 128.2.a.b yes 1
32.h odd 8 1 128.2.a.c yes 1
32.h odd 8 2 256.2.b.a 2
96.o even 8 1 1152.2.a.h 1
96.o even 8 1 1152.2.a.r 1
96.o even 8 2 2304.2.d.b 2
96.p odd 8 1 1152.2.a.c 1
96.p odd 8 1 1152.2.a.m 1
96.p odd 8 2 2304.2.d.r 2
160.u even 8 1 3200.2.c.f 2
160.u even 8 1 3200.2.c.k 2
160.v odd 8 1 3200.2.c.e 2
160.v odd 8 1 3200.2.c.l 2
160.y odd 8 1 3200.2.a.e 1
160.y odd 8 1 3200.2.a.u 1
160.z even 8 1 3200.2.a.h 1
160.z even 8 1 3200.2.a.x 1
160.ba even 8 1 3200.2.c.f 2
160.ba even 8 1 3200.2.c.k 2
160.bb odd 8 1 3200.2.c.e 2
160.bb odd 8 1 3200.2.c.l 2
224.v odd 8 1 6272.2.a.a 1
224.v odd 8 1 6272.2.a.h 1
224.x even 8 1 6272.2.a.b 1
224.x even 8 1 6272.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.a.a 1 32.g even 8 1
128.2.a.b yes 1 32.h odd 8 1
128.2.a.c yes 1 32.h odd 8 1
128.2.a.d yes 1 32.g even 8 1
256.2.b.a 2 32.h odd 8 2
256.2.b.c 2 32.g even 8 2
1024.2.e.i 4 1.a even 1 1 trivial
1024.2.e.i 4 8.b even 2 1 inner
1024.2.e.i 4 16.e even 4 2 inner
1024.2.e.m 4 4.b odd 2 1
1024.2.e.m 4 8.d odd 2 1
1024.2.e.m 4 16.f odd 4 2
1152.2.a.c 1 96.p odd 8 1
1152.2.a.h 1 96.o even 8 1
1152.2.a.m 1 96.p odd 8 1
1152.2.a.r 1 96.o even 8 1
2304.2.d.b 2 96.o even 8 2
2304.2.d.r 2 96.p odd 8 2
3200.2.a.e 1 160.y odd 8 1
3200.2.a.h 1 160.z even 8 1
3200.2.a.u 1 160.y odd 8 1
3200.2.a.x 1 160.z even 8 1
3200.2.c.e 2 160.v odd 8 1
3200.2.c.e 2 160.bb odd 8 1
3200.2.c.f 2 160.u even 8 1
3200.2.c.f 2 160.ba even 8 1
3200.2.c.k 2 160.u even 8 1
3200.2.c.k 2 160.ba even 8 1
3200.2.c.l 2 160.v odd 8 1
3200.2.c.l 2 160.bb odd 8 1
6272.2.a.a 1 224.v odd 8 1
6272.2.a.b 1 224.x even 8 1
6272.2.a.g 1 224.x even 8 1
6272.2.a.h 1 224.v odd 8 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1024, [\chi])\):

\( T_{3}^{4} + 16 \) Copy content Toggle raw display
\( T_{5}^{4} + 16 \) Copy content Toggle raw display
\( T_{47} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} + 16 \) Copy content Toggle raw display
$7$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 16 \) Copy content Toggle raw display
$13$ \( T^{4} + 16 \) Copy content Toggle raw display
$17$ \( (T - 2)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 16 \) Copy content Toggle raw display
$23$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 1296 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 10000 \) Copy content Toggle raw display
$41$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 1296 \) Copy content Toggle raw display
$47$ \( (T - 8)^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 1296 \) Copy content Toggle raw display
$59$ \( T^{4} + 38416 \) Copy content Toggle raw display
$61$ \( T^{4} + 16 \) Copy content Toggle raw display
$67$ \( T^{4} + 10000 \) Copy content Toggle raw display
$71$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 1296 \) Copy content Toggle raw display
$89$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$97$ \( (T + 2)^{4} \) Copy content Toggle raw display
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