# 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 $4$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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