# Properties

 Label 1008.3.f.a Level 1008 Weight 3 Character orbit 1008.f Self dual yes Analytic conductor 27.466 Analytic rank 0 Dimension 1 CM discriminant -7 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1008.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$27.4660106475$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 7q^{7} + O(q^{10})$$ $$q + 7q^{7} - 6q^{11} + 18q^{23} + 25q^{25} + 54q^{29} - 38q^{37} - 58q^{43} + 49q^{49} + 6q^{53} + 118q^{67} + 114q^{71} - 42q^{77} + 94q^{79} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
433.1
 0
0 0 0 0 0 7.00000 0 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 $$\Q(\sqrt{-7})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.3.f.a 1
3.b odd 2 1 112.3.c.a 1
4.b odd 2 1 63.3.d.a 1
7.b odd 2 1 CM 1008.3.f.a 1
12.b even 2 1 7.3.b.a 1
21.c even 2 1 112.3.c.a 1
21.g even 6 2 784.3.s.a 2
21.h odd 6 2 784.3.s.a 2
24.f even 2 1 448.3.c.a 1
24.h odd 2 1 448.3.c.b 1
28.d even 2 1 63.3.d.a 1
28.f even 6 2 441.3.m.a 2
28.g odd 6 2 441.3.m.a 2
60.h even 2 1 175.3.d.a 1
60.l odd 4 2 175.3.c.a 2
84.h odd 2 1 7.3.b.a 1
84.j odd 6 2 49.3.d.a 2
84.n even 6 2 49.3.d.a 2
168.e odd 2 1 448.3.c.a 1
168.i even 2 1 448.3.c.b 1
420.o odd 2 1 175.3.d.a 1
420.w even 4 2 175.3.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.3.b.a 1 12.b even 2 1
7.3.b.a 1 84.h odd 2 1
49.3.d.a 2 84.j odd 6 2
49.3.d.a 2 84.n even 6 2
63.3.d.a 1 4.b odd 2 1
63.3.d.a 1 28.d even 2 1
112.3.c.a 1 3.b odd 2 1
112.3.c.a 1 21.c even 2 1
175.3.c.a 2 60.l odd 4 2
175.3.c.a 2 420.w even 4 2
175.3.d.a 1 60.h even 2 1
175.3.d.a 1 420.o odd 2 1
441.3.m.a 2 28.f even 6 2
441.3.m.a 2 28.g odd 6 2
448.3.c.a 1 24.f even 2 1
448.3.c.a 1 168.e odd 2 1
448.3.c.b 1 24.h odd 2 1
448.3.c.b 1 168.i even 2 1
784.3.s.a 2 21.g even 6 2
784.3.s.a 2 21.h odd 6 2
1008.3.f.a 1 1.a even 1 1 trivial
1008.3.f.a 1 7.b odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1008, [\chi])$$:

 $$T_{5}$$ $$T_{11} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 5 T )( 1 + 5 T )$$
$7$ $$1 - 7 T$$
$11$ $$1 + 6 T + 121 T^{2}$$
$13$ $$( 1 - 13 T )( 1 + 13 T )$$
$17$ $$( 1 - 17 T )( 1 + 17 T )$$
$19$ $$( 1 - 19 T )( 1 + 19 T )$$
$23$ $$1 - 18 T + 529 T^{2}$$
$29$ $$1 - 54 T + 841 T^{2}$$
$31$ $$( 1 - 31 T )( 1 + 31 T )$$
$37$ $$1 + 38 T + 1369 T^{2}$$
$41$ $$( 1 - 41 T )( 1 + 41 T )$$
$43$ $$1 + 58 T + 1849 T^{2}$$
$47$ $$( 1 - 47 T )( 1 + 47 T )$$
$53$ $$1 - 6 T + 2809 T^{2}$$
$59$ $$( 1 - 59 T )( 1 + 59 T )$$
$61$ $$( 1 - 61 T )( 1 + 61 T )$$
$67$ $$1 - 118 T + 4489 T^{2}$$
$71$ $$1 - 114 T + 5041 T^{2}$$
$73$ $$( 1 - 73 T )( 1 + 73 T )$$
$79$ $$1 - 94 T + 6241 T^{2}$$
$83$ $$( 1 - 83 T )( 1 + 83 T )$$
$89$ $$( 1 - 89 T )( 1 + 89 T )$$
$97$ $$( 1 - 97 T )( 1 + 97 T )$$