Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 512)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + i ) q^{3} + ( 2 + 2 i ) q^{5} + 4 i q^{7} + i q^{9} +O(q^{10})\) \( q + ( -1 + i ) q^{3} + ( 2 + 2 i ) q^{5} + 4 i q^{7} + i q^{9} + ( -1 - i ) q^{11} + ( -2 + 2 i ) q^{13} -4 q^{15} + 4 q^{17} + ( -5 + 5 i ) q^{19} + ( -4 - 4 i ) q^{21} -4 i q^{23} + 3 i q^{25} + ( -4 - 4 i ) q^{27} + ( 6 - 6 i ) q^{29} + 8 q^{31} + 2 q^{33} + ( -8 + 8 i ) q^{35} + ( -2 - 2 i ) q^{37} -4 i q^{39} -2 i q^{41} + ( -3 - 3 i ) q^{43} + ( -2 + 2 i ) q^{45} -9 q^{49} + ( -4 + 4 i ) q^{51} + ( 2 + 2 i ) q^{53} -4 i q^{55} -10 i q^{57} + ( 3 + 3 i ) q^{59} + ( -6 + 6 i ) q^{61} -4 q^{63} -8 q^{65} + ( -3 + 3 i ) q^{67} + ( 4 + 4 i ) q^{69} -4 i q^{71} + 4 i q^{73} + ( -3 - 3 i ) q^{75} + ( 4 - 4 i ) q^{77} + 8 q^{79} + 5 q^{81} + ( -7 + 7 i ) q^{83} + ( 8 + 8 i ) q^{85} + 12 i q^{87} + 12 i q^{89} + ( -8 - 8 i ) q^{91} + ( -8 + 8 i ) q^{93} -20 q^{95} -4 q^{97} + ( 1 - i ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 4q^{5} + O(q^{10}) \) \( 2q - 2q^{3} + 4q^{5} - 2q^{11} - 4q^{13} - 8q^{15} + 8q^{17} - 10q^{19} - 8q^{21} - 8q^{27} + 12q^{29} + 16q^{31} + 4q^{33} - 16q^{35} - 4q^{37} - 6q^{43} - 4q^{45} - 18q^{49} - 8q^{51} + 4q^{53} + 6q^{59} - 12q^{61} - 8q^{63} - 16q^{65} - 6q^{67} + 8q^{69} - 6q^{75} + 8q^{77} + 16q^{79} + 10q^{81} - 14q^{83} + 16q^{85} - 16q^{91} - 16q^{93} - 40q^{95} - 8q^{97} + 2q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(i\) \(1\)

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
1.00000i
1.00000i
0 −1.00000 1.00000i 0 2.00000 2.00000i 0 4.00000i 0 1.00000i 0
769.1 0 −1.00000 + 1.00000i 0 2.00000 + 2.00000i 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
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1024.2.e.c 2
4.b odd 2 1 1024.2.e.e 2
8.b even 2 1 1024.2.e.d 2
8.d odd 2 1 1024.2.e.b 2
16.e even 4 1 inner 1024.2.e.c 2
16.e even 4 1 1024.2.e.d 2
16.f odd 4 1 1024.2.e.b 2
16.f odd 4 1 1024.2.e.e 2
32.g even 8 2 512.2.a.d yes 2
32.g even 8 2 512.2.b.a 2
32.h odd 8 2 512.2.a.c 2
32.h odd 8 2 512.2.b.b 2
96.o even 8 2 4608.2.a.f 2
96.o even 8 2 4608.2.d.b 2
96.p odd 8 2 4608.2.a.m 2
96.p odd 8 2 4608.2.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.a.c 2 32.h odd 8 2
512.2.a.d yes 2 32.g even 8 2
512.2.b.a 2 32.g even 8 2
512.2.b.b 2 32.h odd 8 2
1024.2.e.b 2 8.d odd 2 1
1024.2.e.b 2 16.f odd 4 1
1024.2.e.c 2 1.a even 1 1 trivial
1024.2.e.c 2 16.e even 4 1 inner
1024.2.e.d 2 8.b even 2 1
1024.2.e.d 2 16.e even 4 1
1024.2.e.e 2 4.b odd 2 1
1024.2.e.e 2 16.f odd 4 1
4608.2.a.f 2 96.o even 8 2
4608.2.a.m 2 96.p odd 8 2
4608.2.d.a 2 96.p odd 8 2
4608.2.d.b 2 96.o even 8 2

Hecke kernels

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

\( T_{3}^{2} + 2 T_{3} + 2 \)
\( T_{5}^{2} - 4 T_{5} + 8 \)
\( T_{47} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 2 + 2 T + T^{2} \)
$5$ \( 8 - 4 T + T^{2} \)
$7$ \( 16 + T^{2} \)
$11$ \( 2 + 2 T + T^{2} \)
$13$ \( 8 + 4 T + T^{2} \)
$17$ \( ( -4 + T )^{2} \)
$19$ \( 50 + 10 T + T^{2} \)
$23$ \( 16 + T^{2} \)
$29$ \( 72 - 12 T + T^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( 8 + 4 T + T^{2} \)
$41$ \( 4 + T^{2} \)
$43$ \( 18 + 6 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 8 - 4 T + T^{2} \)
$59$ \( 18 - 6 T + T^{2} \)
$61$ \( 72 + 12 T + T^{2} \)
$67$ \( 18 + 6 T + T^{2} \)
$71$ \( 16 + T^{2} \)
$73$ \( 16 + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 98 + 14 T + T^{2} \)
$89$ \( 144 + T^{2} \)
$97$ \( ( 4 + T )^{2} \)
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