# Properties

 Label 672.2.j.a Level 672 Weight 2 Character orbit 672.j Analytic conductor 5.366 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$672 = 2^{5} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 672.j (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.36594701583$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 168) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \zeta_{8} - \zeta_{8}^{3} ) q^{3} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{5} -\zeta_{8}^{2} q^{7} + ( -1 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \zeta_{8} - \zeta_{8}^{3} ) q^{3} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{5} -\zeta_{8}^{2} q^{7} + ( -1 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{11} -4 \zeta_{8}^{2} q^{13} + ( 2 \zeta_{8} + 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{15} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{17} + 2 q^{19} + ( -\zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{21} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{23} + 3 q^{25} + ( 5 - \zeta_{8} - \zeta_{8}^{3} ) q^{27} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{29} + ( 8 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{33} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{35} + 8 \zeta_{8}^{2} q^{37} + ( -4 \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{39} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{41} + 10 q^{43} + ( 2 \zeta_{8} - 8 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{45} - q^{49} + ( 4 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{51} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{53} -16 \zeta_{8}^{2} q^{55} + ( -2 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{57} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{59} + 12 \zeta_{8}^{2} q^{61} + ( 2 \zeta_{8} + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{63} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{65} -6 q^{67} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{69} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{71} + 10 q^{73} + ( -3 - 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{75} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{77} + 8 \zeta_{8}^{2} q^{79} + ( -7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{81} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{83} -8 \zeta_{8}^{2} q^{85} + ( -4 \zeta_{8} - 8 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{87} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{89} -4 q^{91} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{95} -2 q^{97} + ( -16 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{3} - 4q^{9} + 8q^{19} + 12q^{25} + 20q^{27} + 32q^{33} + 40q^{43} - 4q^{49} + 16q^{51} - 8q^{57} - 24q^{67} + 40q^{73} - 12q^{75} - 28q^{81} - 16q^{91} - 8q^{97} - 64q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$421$$ $$449$$ $$577$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
239.1
 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i
0 −1.00000 1.41421i 0 −2.82843 0 1.00000i 0 −1.00000 + 2.82843i 0
239.2 0 −1.00000 1.41421i 0 2.82843 0 1.00000i 0 −1.00000 + 2.82843i 0
239.3 0 −1.00000 + 1.41421i 0 −2.82843 0 1.00000i 0 −1.00000 2.82843i 0
239.4 0 −1.00000 + 1.41421i 0 2.82843 0 1.00000i 0 −1.00000 2.82843i 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
8.d odd 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.j.a 4
3.b odd 2 1 inner 672.2.j.a 4
4.b odd 2 1 168.2.j.b 4
8.b even 2 1 168.2.j.b 4
8.d odd 2 1 inner 672.2.j.a 4
12.b even 2 1 168.2.j.b 4
24.f even 2 1 inner 672.2.j.a 4
24.h odd 2 1 168.2.j.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.j.b 4 4.b odd 2 1
168.2.j.b 4 8.b even 2 1
168.2.j.b 4 12.b even 2 1
168.2.j.b 4 24.h odd 2 1
672.2.j.a 4 1.a even 1 1 trivial
672.2.j.a 4 3.b odd 2 1 inner
672.2.j.a 4 8.d odd 2 1 inner
672.2.j.a 4 24.f even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 8$$ acting on $$S_{2}^{\mathrm{new}}(672, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + 2 T + 3 T^{2} )^{2}$$
$5$ $$( 1 + 2 T^{2} + 25 T^{4} )^{2}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( 1 + 10 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2}$$
$17$ $$( 1 - 26 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 2 T + 19 T^{2} )^{4}$$
$23$ $$( 1 + 38 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 + 26 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 31 T^{2} )^{4}$$
$37$ $$( 1 - 10 T^{2} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 74 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 10 T + 43 T^{2} )^{4}$$
$47$ $$( 1 + 47 T^{2} )^{4}$$
$53$ $$( 1 + 74 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 - 46 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 10 T + 61 T^{2} )^{2}( 1 + 10 T + 61 T^{2} )^{2}$$
$67$ $$( 1 + 6 T + 67 T^{2} )^{4}$$
$71$ $$( 1 + 70 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 10 T + 73 T^{2} )^{4}$$
$79$ $$( 1 - 94 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 18 T + 83 T^{2} )^{2}( 1 + 18 T + 83 T^{2} )^{2}$$
$89$ $$( 1 - 170 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 2 T + 97 T^{2} )^{4}$$