Properties

Label 1045.1.k.b
Level $1045$
Weight $1$
Character orbit 1045.k
Analytic conductor $0.522$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -19
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: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.287375.1

$q$-expansion

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

\(f(q)\) \(=\) \( q - i q^{4} + q^{5} + (i - 1) q^{7} - i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{4} + q^{5} + (i - 1) q^{7} - i q^{9} - i q^{11} - q^{16} + ( - i + 1) q^{17} + q^{19} - i q^{20} + (i - 1) q^{23} + q^{25} + (i + 1) q^{28} + (i - 1) q^{35} - q^{36} + ( - i - 1) q^{43} - q^{44} - i q^{45} + (i + 1) q^{47} - i q^{49} - i q^{55} + i q^{61} + (i + 1) q^{63} + i q^{64} + ( - i - 1) q^{68} + ( - i - 1) q^{73} - i q^{76} + (i + 1) q^{77} - q^{80} - q^{81} + (i + 1) q^{83} + ( - i + 1) q^{85} + (i + 1) q^{92} + q^{95} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 2 q^{7} - 2 q^{16} + 2 q^{17} + 2 q^{19} - 2 q^{23} + 2 q^{25} + 2 q^{28} - 2 q^{35} - 2 q^{36} - 2 q^{43} - 2 q^{44} + 2 q^{47} + 2 q^{63} - 2 q^{68} - 2 q^{73} + 2 q^{77} - 2 q^{80} - 2 q^{81} + 2 q^{83} + 2 q^{85} + 2 q^{92} + 2 q^{95} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
0 0 1.00000i 1.00000 0 −1.00000 1.00000i 0 1.00000i 0
417.1 0 0 1.00000i 1.00000 0 −1.00000 + 1.00000i 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
55.e even 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.b 2
5.c odd 4 1 1045.1.k.c yes 2
11.b odd 2 1 1045.1.k.c yes 2
19.b odd 2 1 CM 1045.1.k.b 2
55.e even 4 1 inner 1045.1.k.b 2
95.g even 4 1 1045.1.k.c yes 2
209.d even 2 1 1045.1.k.c yes 2
1045.k odd 4 1 inner 1045.1.k.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.1.k.b 2 1.a even 1 1 trivial
1045.1.k.b 2 19.b odd 2 1 CM
1045.1.k.b 2 55.e even 4 1 inner
1045.1.k.b 2 1045.k odd 4 1 inner
1045.1.k.c yes 2 5.c odd 4 1
1045.1.k.c yes 2 11.b odd 2 1
1045.1.k.c yes 2 95.g even 4 1
1045.1.k.c yes 2 209.d even 2 1

Hecke kernels

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

\( T_{2} \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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