Properties

Label 1040.2.f.a
Level $1040$
Weight $2$
Character orbit 1040.f
Analytic conductor $8.304$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.30444181021\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 i q^{3} + ( -1 - 2 i ) q^{5} - q^{9} +O(q^{10})\) \( q + 2 i q^{3} + ( -1 - 2 i ) q^{5} - q^{9} + 2 i q^{11} + ( 3 + 2 i ) q^{13} + ( 4 - 2 i ) q^{15} -6 i q^{19} + 6 i q^{23} + ( -3 + 4 i ) q^{25} + 4 i q^{27} + 6 q^{29} + 6 i q^{31} -4 q^{33} + 6 q^{37} + ( -4 + 6 i ) q^{39} + 8 i q^{41} -6 i q^{43} + ( 1 + 2 i ) q^{45} + 8 q^{47} -7 q^{49} + 12 i q^{53} + ( 4 - 2 i ) q^{55} + 12 q^{57} + 2 i q^{59} + 6 q^{61} + ( 1 - 8 i ) q^{65} -12 q^{67} -12 q^{69} -2 i q^{71} -6 q^{73} + ( -8 - 6 i ) q^{75} -11 q^{81} + 4 q^{83} + 12 i q^{87} + 8 i q^{89} -12 q^{93} + ( -12 + 6 i ) q^{95} -6 q^{97} -2 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{5} - 2 q^{9} + 6 q^{13} + 8 q^{15} - 6 q^{25} + 12 q^{29} - 8 q^{33} + 12 q^{37} - 8 q^{39} + 2 q^{45} + 16 q^{47} - 14 q^{49} + 8 q^{55} + 24 q^{57} + 12 q^{61} + 2 q^{65} - 24 q^{67} - 24 q^{69} - 12 q^{73} - 16 q^{75} - 22 q^{81} + 8 q^{83} - 24 q^{93} - 24 q^{95} - 12 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
1.00000i
1.00000i
0 2.00000i 0 −1.00000 + 2.00000i 0 0 0 −1.00000 0
129.2 0 2.00000i 0 −1.00000 2.00000i 0 0 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
65.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.f.a 2
4.b odd 2 1 65.2.d.a 2
5.b even 2 1 1040.2.f.b 2
12.b even 2 1 585.2.h.c 2
13.b even 2 1 1040.2.f.b 2
20.d odd 2 1 65.2.d.b yes 2
20.e even 4 1 325.2.c.b 2
20.e even 4 1 325.2.c.e 2
52.b odd 2 1 65.2.d.b yes 2
52.f even 4 1 845.2.b.a 2
52.f even 4 1 845.2.b.b 2
52.i odd 6 2 845.2.l.a 4
52.j odd 6 2 845.2.l.b 4
52.l even 12 2 845.2.n.a 4
52.l even 12 2 845.2.n.b 4
60.h even 2 1 585.2.h.b 2
65.d even 2 1 inner 1040.2.f.a 2
156.h even 2 1 585.2.h.b 2
260.g odd 2 1 65.2.d.a 2
260.l odd 4 1 4225.2.a.k 1
260.l odd 4 1 4225.2.a.m 1
260.p even 4 1 325.2.c.b 2
260.p even 4 1 325.2.c.e 2
260.s odd 4 1 4225.2.a.e 1
260.s odd 4 1 4225.2.a.h 1
260.u even 4 1 845.2.b.a 2
260.u even 4 1 845.2.b.b 2
260.v odd 6 2 845.2.l.a 4
260.w odd 6 2 845.2.l.b 4
260.bc even 12 2 845.2.n.a 4
260.bc even 12 2 845.2.n.b 4
780.d even 2 1 585.2.h.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.d.a 2 4.b odd 2 1
65.2.d.a 2 260.g odd 2 1
65.2.d.b yes 2 20.d odd 2 1
65.2.d.b yes 2 52.b odd 2 1
325.2.c.b 2 20.e even 4 1
325.2.c.b 2 260.p even 4 1
325.2.c.e 2 20.e even 4 1
325.2.c.e 2 260.p even 4 1
585.2.h.b 2 60.h even 2 1
585.2.h.b 2 156.h even 2 1
585.2.h.c 2 12.b even 2 1
585.2.h.c 2 780.d even 2 1
845.2.b.a 2 52.f even 4 1
845.2.b.a 2 260.u even 4 1
845.2.b.b 2 52.f even 4 1
845.2.b.b 2 260.u even 4 1
845.2.l.a 4 52.i odd 6 2
845.2.l.a 4 260.v odd 6 2
845.2.l.b 4 52.j odd 6 2
845.2.l.b 4 260.w odd 6 2
845.2.n.a 4 52.l even 12 2
845.2.n.a 4 260.bc even 12 2
845.2.n.b 4 52.l even 12 2
845.2.n.b 4 260.bc even 12 2
1040.2.f.a 2 1.a even 1 1 trivial
1040.2.f.a 2 65.d even 2 1 inner
1040.2.f.b 2 5.b even 2 1
1040.2.f.b 2 13.b even 2 1
4225.2.a.e 1 260.s odd 4 1
4225.2.a.h 1 260.s odd 4 1
4225.2.a.k 1 260.l odd 4 1
4225.2.a.m 1 260.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1040, [\chi])\):

\( T_{3}^{2} + 4 \)
\( T_{7} \)
\( T_{37} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( 5 + 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 4 + T^{2} \)
$13$ \( 13 - 6 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 36 + T^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( 36 + T^{2} \)
$37$ \( ( -6 + T )^{2} \)
$41$ \( 64 + T^{2} \)
$43$ \( 36 + T^{2} \)
$47$ \( ( -8 + T )^{2} \)
$53$ \( 144 + T^{2} \)
$59$ \( 4 + T^{2} \)
$61$ \( ( -6 + T )^{2} \)
$67$ \( ( 12 + T )^{2} \)
$71$ \( 4 + T^{2} \)
$73$ \( ( 6 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( ( -4 + T )^{2} \)
$89$ \( 64 + T^{2} \)
$97$ \( ( 6 + T )^{2} \)
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