# Properties

 Label 1008.4.k.b Level $1008$ Weight $4$ Character orbit 1008.k Analytic conductor $59.474$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$59.4739252858$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{111})$$ Defining polynomial: $$x^{4} + 112 x^{2} + 3025$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{5} + ( 11 + \beta_{2} ) q^{7} +O(q^{10})$$ $$q + \beta_{3} q^{5} + ( 11 + \beta_{2} ) q^{7} -11 \beta_{1} q^{11} + 2 \beta_{2} q^{13} -3 \beta_{3} q^{17} + 6 \beta_{2} q^{19} + 55 \beta_{1} q^{23} + 319 q^{25} + 89 \beta_{1} q^{29} + 16 \beta_{2} q^{31} + ( 222 \beta_{1} + 11 \beta_{3} ) q^{35} -184 q^{37} -5 \beta_{3} q^{41} + 190 q^{43} -2 \beta_{3} q^{47} + ( -101 + 22 \beta_{2} ) q^{49} -253 \beta_{1} q^{53} -22 \beta_{2} q^{55} -4 \beta_{3} q^{59} -44 \beta_{2} q^{61} + 444 \beta_{1} q^{65} -296 q^{67} -233 \beta_{1} q^{71} + 54 \beta_{2} q^{73} + ( -121 \beta_{1} + 11 \beta_{3} ) q^{77} -836 q^{79} + 58 \beta_{3} q^{83} -1332 q^{85} + 33 \beta_{3} q^{89} + ( -444 + 22 \beta_{2} ) q^{91} + 1332 \beta_{1} q^{95} -38 \beta_{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 44q^{7} + O(q^{10})$$ $$4q + 44q^{7} + 1276q^{25} - 736q^{37} + 760q^{43} - 404q^{49} - 1184q^{67} - 3344q^{79} - 5328q^{85} - 1776q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 112 x^{2} + 3025$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 57 \nu$$$$)/55$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 167 \nu$$$$)/55$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} + 112$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 112$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-57 \beta_{2} + 167 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1
 − 8.15694i 8.15694i − 6.74273i 6.74273i
0 0 0 −21.0713 0 11.0000 14.8997i 0 0 0
881.2 0 0 0 −21.0713 0 11.0000 + 14.8997i 0 0 0
881.3 0 0 0 21.0713 0 11.0000 14.8997i 0 0 0
881.4 0 0 0 21.0713 0 11.0000 + 14.8997i 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
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.4.k.b 4
3.b odd 2 1 inner 1008.4.k.b 4
4.b odd 2 1 63.4.c.b 4
7.b odd 2 1 inner 1008.4.k.b 4
12.b even 2 1 63.4.c.b 4
21.c even 2 1 inner 1008.4.k.b 4
28.d even 2 1 63.4.c.b 4
28.f even 6 2 441.4.p.a 8
28.g odd 6 2 441.4.p.a 8
84.h odd 2 1 63.4.c.b 4
84.j odd 6 2 441.4.p.a 8
84.n even 6 2 441.4.p.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.c.b 4 4.b odd 2 1
63.4.c.b 4 12.b even 2 1
63.4.c.b 4 28.d even 2 1
63.4.c.b 4 84.h odd 2 1
441.4.p.a 8 28.f even 6 2
441.4.p.a 8 28.g odd 6 2
441.4.p.a 8 84.j odd 6 2
441.4.p.a 8 84.n even 6 2
1008.4.k.b 4 1.a even 1 1 trivial
1008.4.k.b 4 3.b odd 2 1 inner
1008.4.k.b 4 7.b odd 2 1 inner
1008.4.k.b 4 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 444$$ acting on $$S_{4}^{\mathrm{new}}(1008, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -444 + T^{2} )^{2}$$
$7$ $$( 343 - 22 T + T^{2} )^{2}$$
$11$ $$( 242 + T^{2} )^{2}$$
$13$ $$( 888 + T^{2} )^{2}$$
$17$ $$( -3996 + T^{2} )^{2}$$
$19$ $$( 7992 + T^{2} )^{2}$$
$23$ $$( 6050 + T^{2} )^{2}$$
$29$ $$( 15842 + T^{2} )^{2}$$
$31$ $$( 56832 + T^{2} )^{2}$$
$37$ $$( 184 + T )^{4}$$
$41$ $$( -11100 + T^{2} )^{2}$$
$43$ $$( -190 + T )^{4}$$
$47$ $$( -1776 + T^{2} )^{2}$$
$53$ $$( 128018 + T^{2} )^{2}$$
$59$ $$( -7104 + T^{2} )^{2}$$
$61$ $$( 429792 + T^{2} )^{2}$$
$67$ $$( 296 + T )^{4}$$
$71$ $$( 108578 + T^{2} )^{2}$$
$73$ $$( 647352 + T^{2} )^{2}$$
$79$ $$( 836 + T )^{4}$$
$83$ $$( -1493616 + T^{2} )^{2}$$
$89$ $$( -483516 + T^{2} )^{2}$$
$97$ $$( 320568 + T^{2} )^{2}$$