Properties

Label 1024.2.e.m
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: \(x^{4} + 1\)
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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(-\zeta_{8}^{2}\) \(1\)

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

Hecke kernels

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

\( T_{3}^{4} + 16 \)
\( T_{5}^{4} + 16 \)
\( T_{47} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 4 T + 8 T^{2} - 12 T^{3} + 9 T^{4} )( 1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4} ) \)
$5$ \( ( 1 - 8 T^{2} + 25 T^{4} )( 1 + 8 T^{2} + 25 T^{4} ) \)
$7$ \( ( 1 + 2 T^{2} + 49 T^{4} )^{2} \)
$11$ \( 1 + 82 T^{4} + 14641 T^{8} \)
$13$ \( 1 + 146 T^{4} + 28561 T^{8} \)
$17$ \( ( 1 - 2 T + 17 T^{2} )^{4} \)
$19$ \( ( 1 - 12 T + 72 T^{2} - 228 T^{3} + 361 T^{4} )( 1 + 12 T + 72 T^{2} + 228 T^{3} + 361 T^{4} ) \)
$23$ \( ( 1 - 30 T^{2} + 529 T^{4} )^{2} \)
$29$ \( 1 - 1198 T^{4} + 707281 T^{8} \)
$31$ \( ( 1 + 31 T^{2} )^{4} \)
$37$ \( 1 - 2062 T^{4} + 1874161 T^{8} \)
$41$ \( ( 1 - 46 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( 1 - 1198 T^{4} + 3418801 T^{8} \)
$47$ \( ( 1 + 8 T + 47 T^{2} )^{4} \)
$53$ \( 1 - 718 T^{4} + 7890481 T^{8} \)
$59$ \( 1 - 878 T^{4} + 12117361 T^{8} \)
$61$ \( 1 + 6482 T^{4} + 13845841 T^{8} \)
$67$ \( 1 - 7822 T^{4} + 20151121 T^{8} \)
$71$ \( ( 1 + 2 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + 50 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 + 8 T + 79 T^{2} )^{4} \)
$83$ \( 1 + 3122 T^{4} + 47458321 T^{8} \)
$89$ \( ( 1 - 174 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 2 T + 97 T^{2} )^{4} \)
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