Properties

Label 1045.1.k.a
Level $1045$
Weight $1$
Character orbit 1045.k
Analytic conductor $0.522$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
RM discriminant 209
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1045.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.521522938201\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.26125.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( -1 - i ) q^{2} + i q^{4} -i q^{5} -i q^{9} +O(q^{10})\) \( q + ( -1 - i ) q^{2} + i q^{4} -i q^{5} -i q^{9} + ( -1 + i ) q^{10} + q^{11} + ( 1 - i ) q^{13} + q^{16} + ( -1 + i ) q^{18} + i q^{19} + q^{20} + ( -1 - i ) q^{22} + ( -1 + i ) q^{23} - q^{25} -2 q^{26} -2 i q^{29} + ( -1 - i ) q^{32} + q^{36} + ( 1 - i ) q^{38} + i q^{44} - q^{45} + 2 q^{46} + ( -1 - i ) q^{47} + i q^{49} + ( 1 + i ) q^{50} + ( 1 + i ) q^{52} -i q^{55} + ( -2 + 2 i ) q^{58} + i q^{64} + ( -1 - i ) q^{65} - q^{76} -i q^{80} - q^{81} + ( 1 + i ) q^{90} + ( -1 - i ) q^{92} + 2 i q^{94} + q^{95} + ( 1 - i ) q^{98} -i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + O(q^{10}) \) \( 2 q - 2 q^{2} - 2 q^{10} + 2 q^{11} + 2 q^{13} + 2 q^{16} - 2 q^{18} + 2 q^{20} - 2 q^{22} - 2 q^{23} - 2 q^{25} - 4 q^{26} - 2 q^{32} + 2 q^{36} + 2 q^{38} - 2 q^{45} + 4 q^{46} - 2 q^{47} + 2 q^{50} + 2 q^{52} - 4 q^{58} - 2 q^{65} - 2 q^{76} - 2 q^{81} + 2 q^{90} - 2 q^{92} + 2 q^{95} + 2 q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times\).

\(n\) \(496\) \(761\) \(837\)
\(\chi(n)\) \(-1\) \(-1\) \(i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
208.1
1.00000i
1.00000i
−1.00000 + 1.00000i 0 1.00000i 1.00000i 0 0 0 1.00000i −1.00000 1.00000i
417.1 −1.00000 1.00000i 0 1.00000i 1.00000i 0 0 0 1.00000i −1.00000 + 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
209.d even 2 1 RM by \(\Q(\sqrt{209}) \)
5.c odd 4 1 inner
1045.k odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.1.k.a 2
5.c odd 4 1 inner 1045.1.k.a 2
11.b odd 2 1 1045.1.k.d yes 2
19.b odd 2 1 1045.1.k.d yes 2
55.e even 4 1 1045.1.k.d yes 2
95.g even 4 1 1045.1.k.d yes 2
209.d even 2 1 RM 1045.1.k.a 2
1045.k odd 4 1 inner 1045.1.k.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.1.k.a 2 1.a even 1 1 trivial
1045.1.k.a 2 5.c odd 4 1 inner
1045.1.k.a 2 209.d even 2 1 RM
1045.1.k.a 2 1045.k odd 4 1 inner
1045.1.k.d yes 2 11.b odd 2 1
1045.1.k.d yes 2 19.b odd 2 1
1045.1.k.d yes 2 55.e even 4 1
1045.1.k.d yes 2 95.g even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1045, [\chi])\):

\( T_{2}^{2} + 2 T_{2} + 2 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( 2 - 2 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 1 + T^{2} \)
$23$ \( 2 + 2 T + T^{2} \)
$29$ \( 4 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( 2 + 2 T + T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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