Properties

Label 1008.1.u.b
Level $1008$
Weight $1$
Character orbit 1008.u
Analytic conductor $0.503$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1008.u (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.503057532734\)
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: no (minimal twist has level 112)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.14336.1
Artin image: $C_2\times C_4\wr C_2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{16} + \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} -i q^{7} -i q^{8} +O(q^{10})\) \( q + i q^{2} - q^{4} -i q^{7} -i q^{8} + ( 1 - i ) q^{11} + q^{14} + q^{16} + ( 1 + i ) q^{22} -i q^{25} + i q^{28} + ( 1 + i ) q^{29} + i q^{32} + ( -1 + i ) q^{37} + ( 1 - i ) q^{43} + ( -1 + i ) q^{44} - q^{49} + q^{50} + ( -1 + i ) q^{53} - q^{56} + ( -1 + i ) q^{58} - q^{64} + ( 1 + i ) q^{67} -2 i q^{71} + ( -1 - i ) q^{74} + ( -1 - i ) q^{77} + ( 1 + i ) q^{86} + ( -1 - i ) q^{88} -i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + O(q^{10}) \) \( 2 q - 2 q^{4} + 2 q^{11} + 2 q^{14} + 2 q^{16} + 2 q^{22} + 2 q^{29} - 2 q^{37} + 2 q^{43} - 2 q^{44} - 2 q^{49} + 2 q^{50} - 2 q^{53} - 2 q^{56} - 2 q^{58} - 2 q^{64} + 2 q^{67} - 2 q^{74} - 2 q^{77} + 2 q^{86} - 2 q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(-1\) \(-i\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
1.00000i
1.00000i
1.00000i 0 −1.00000 0 0 1.00000i 1.00000i 0 0
685.1 1.00000i 0 −1.00000 0 0 1.00000i 1.00000i 0 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
16.e even 4 1 inner
112.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.1.u.b 2
3.b odd 2 1 112.1.l.a 2
7.b odd 2 1 CM 1008.1.u.b 2
12.b even 2 1 448.1.l.a 2
15.d odd 2 1 2800.1.z.a 2
15.e even 4 1 2800.1.bf.a 2
15.e even 4 1 2800.1.bf.b 2
16.e even 4 1 inner 1008.1.u.b 2
21.c even 2 1 112.1.l.a 2
21.g even 6 2 784.1.y.a 4
21.h odd 6 2 784.1.y.a 4
24.f even 2 1 896.1.l.a 2
24.h odd 2 1 896.1.l.b 2
48.i odd 4 1 112.1.l.a 2
48.i odd 4 1 896.1.l.b 2
48.k even 4 1 448.1.l.a 2
48.k even 4 1 896.1.l.a 2
84.h odd 2 1 448.1.l.a 2
84.j odd 6 2 3136.1.bc.a 4
84.n even 6 2 3136.1.bc.a 4
105.g even 2 1 2800.1.z.a 2
105.k odd 4 1 2800.1.bf.a 2
105.k odd 4 1 2800.1.bf.b 2
112.l odd 4 1 inner 1008.1.u.b 2
168.e odd 2 1 896.1.l.a 2
168.i even 2 1 896.1.l.b 2
240.bb even 4 1 2800.1.bf.a 2
240.bf even 4 1 2800.1.bf.b 2
240.bm odd 4 1 2800.1.z.a 2
336.v odd 4 1 448.1.l.a 2
336.v odd 4 1 896.1.l.a 2
336.y even 4 1 112.1.l.a 2
336.y even 4 1 896.1.l.b 2
336.bo even 12 2 784.1.y.a 4
336.br odd 12 2 3136.1.bc.a 4
336.bt odd 12 2 784.1.y.a 4
336.bu even 12 2 3136.1.bc.a 4
1680.br odd 4 1 2800.1.bf.b 2
1680.bx even 4 1 2800.1.z.a 2
1680.cn odd 4 1 2800.1.bf.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.1.l.a 2 3.b odd 2 1
112.1.l.a 2 21.c even 2 1
112.1.l.a 2 48.i odd 4 1
112.1.l.a 2 336.y even 4 1
448.1.l.a 2 12.b even 2 1
448.1.l.a 2 48.k even 4 1
448.1.l.a 2 84.h odd 2 1
448.1.l.a 2 336.v odd 4 1
784.1.y.a 4 21.g even 6 2
784.1.y.a 4 21.h odd 6 2
784.1.y.a 4 336.bo even 12 2
784.1.y.a 4 336.bt odd 12 2
896.1.l.a 2 24.f even 2 1
896.1.l.a 2 48.k even 4 1
896.1.l.a 2 168.e odd 2 1
896.1.l.a 2 336.v odd 4 1
896.1.l.b 2 24.h odd 2 1
896.1.l.b 2 48.i odd 4 1
896.1.l.b 2 168.i even 2 1
896.1.l.b 2 336.y even 4 1
1008.1.u.b 2 1.a even 1 1 trivial
1008.1.u.b 2 7.b odd 2 1 CM
1008.1.u.b 2 16.e even 4 1 inner
1008.1.u.b 2 112.l odd 4 1 inner
2800.1.z.a 2 15.d odd 2 1
2800.1.z.a 2 105.g even 2 1
2800.1.z.a 2 240.bm odd 4 1
2800.1.z.a 2 1680.bx even 4 1
2800.1.bf.a 2 15.e even 4 1
2800.1.bf.a 2 105.k odd 4 1
2800.1.bf.a 2 240.bb even 4 1
2800.1.bf.a 2 1680.cn odd 4 1
2800.1.bf.b 2 15.e even 4 1
2800.1.bf.b 2 105.k odd 4 1
2800.1.bf.b 2 240.bf even 4 1
2800.1.bf.b 2 1680.br odd 4 1
3136.1.bc.a 4 84.j odd 6 2
3136.1.bc.a 4 84.n even 6 2
3136.1.bc.a 4 336.br odd 12 2
3136.1.bc.a 4 336.bu even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1008, [\chi])\):

\( T_{11}^{2} - 2 T_{11} + 2 \)
\( T_{23} \)

Hecke characteristic polynomials

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