# Properties

 Label 672.2.q.g Level 672 Weight 2 Character orbit 672.q Analytic conductor 5.366 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$672 = 2^{5} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 672.q (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.36594701583$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + ( 3 - \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + ( 3 - \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( -1 + \zeta_{6} ) q^{11} - q^{15} + ( 8 - 8 \zeta_{6} ) q^{17} + 4 \zeta_{6} q^{19} + ( 2 - 3 \zeta_{6} ) q^{21} -4 \zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} - q^{27} -5 q^{29} + ( -7 + 7 \zeta_{6} ) q^{31} + \zeta_{6} q^{33} + ( -1 - 2 \zeta_{6} ) q^{35} -8 \zeta_{6} q^{37} + 4 q^{41} + 10 q^{43} + ( -1 + \zeta_{6} ) q^{45} -6 \zeta_{6} q^{47} + ( 8 - 5 \zeta_{6} ) q^{49} -8 \zeta_{6} q^{51} + ( 1 - \zeta_{6} ) q^{53} + q^{55} + 4 q^{57} + ( -9 + 9 \zeta_{6} ) q^{59} + 2 \zeta_{6} q^{61} + ( -1 - 2 \zeta_{6} ) q^{63} + ( -2 + 2 \zeta_{6} ) q^{67} -4 q^{69} + 6 q^{71} + ( -2 + 2 \zeta_{6} ) q^{73} -4 \zeta_{6} q^{75} + ( -2 + 3 \zeta_{6} ) q^{77} + 9 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -3 q^{83} -8 q^{85} + ( -5 + 5 \zeta_{6} ) q^{87} + 6 \zeta_{6} q^{89} + 7 \zeta_{6} q^{93} + ( 4 - 4 \zeta_{6} ) q^{95} - q^{97} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} - q^{5} + 5q^{7} - q^{9} + O(q^{10})$$ $$2q + q^{3} - q^{5} + 5q^{7} - q^{9} - q^{11} - 2q^{15} + 8q^{17} + 4q^{19} + q^{21} - 4q^{23} + 4q^{25} - 2q^{27} - 10q^{29} - 7q^{31} + q^{33} - 4q^{35} - 8q^{37} + 8q^{41} + 20q^{43} - q^{45} - 6q^{47} + 11q^{49} - 8q^{51} + q^{53} + 2q^{55} + 8q^{57} - 9q^{59} + 2q^{61} - 4q^{63} - 2q^{67} - 8q^{69} + 12q^{71} - 2q^{73} - 4q^{75} - q^{77} + 9q^{79} - q^{81} - 6q^{83} - 16q^{85} - 5q^{87} + 6q^{89} + 7q^{93} + 4q^{95} - 2q^{97} + 2q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$421$$ $$449$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 0.866025i 0 −0.500000 0.866025i 0 2.50000 0.866025i 0 −0.500000 0.866025i 0
289.1 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 2.50000 + 0.866025i 0 −0.500000 + 0.866025i 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.q.g yes 2
3.b odd 2 1 2016.2.s.j 2
4.b odd 2 1 672.2.q.b 2
7.c even 3 1 inner 672.2.q.g yes 2
7.c even 3 1 4704.2.a.l 1
7.d odd 6 1 4704.2.a.w 1
8.b even 2 1 1344.2.q.h 2
8.d odd 2 1 1344.2.q.r 2
12.b even 2 1 2016.2.s.i 2
21.h odd 6 1 2016.2.s.j 2
28.f even 6 1 4704.2.a.f 1
28.g odd 6 1 672.2.q.b 2
28.g odd 6 1 4704.2.a.bc 1
56.j odd 6 1 9408.2.a.bb 1
56.k odd 6 1 1344.2.q.r 2
56.k odd 6 1 9408.2.a.o 1
56.m even 6 1 9408.2.a.cp 1
56.p even 6 1 1344.2.q.h 2
56.p even 6 1 9408.2.a.cg 1
84.n even 6 1 2016.2.s.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.b 2 4.b odd 2 1
672.2.q.b 2 28.g odd 6 1
672.2.q.g yes 2 1.a even 1 1 trivial
672.2.q.g yes 2 7.c even 3 1 inner
1344.2.q.h 2 8.b even 2 1
1344.2.q.h 2 56.p even 6 1
1344.2.q.r 2 8.d odd 2 1
1344.2.q.r 2 56.k odd 6 1
2016.2.s.i 2 12.b even 2 1
2016.2.s.i 2 84.n even 6 1
2016.2.s.j 2 3.b odd 2 1
2016.2.s.j 2 21.h odd 6 1
4704.2.a.f 1 28.f even 6 1
4704.2.a.l 1 7.c even 3 1
4704.2.a.w 1 7.d odd 6 1
4704.2.a.bc 1 28.g odd 6 1
9408.2.a.o 1 56.k odd 6 1
9408.2.a.bb 1 56.j odd 6 1
9408.2.a.cg 1 56.p even 6 1
9408.2.a.cp 1 56.m even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(672, [\chi])$$:

 $$T_{5}^{2} + T_{5} + 1$$ $$T_{11}^{2} + T_{11} + 1$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T + T^{2}$$
$5$ $$1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4}$$
$7$ $$1 - 5 T + 7 T^{2}$$
$11$ $$1 + T - 10 T^{2} + 11 T^{3} + 121 T^{4}$$
$13$ $$( 1 + 13 T^{2} )^{2}$$
$17$ $$1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4}$$
$19$ $$1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4}$$
$23$ $$1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 5 T + 29 T^{2} )^{2}$$
$31$ $$( 1 - 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} )$$
$37$ $$1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4}$$
$41$ $$( 1 - 4 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - 10 T + 43 T^{2} )^{2}$$
$47$ $$1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4}$$
$53$ $$1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4}$$
$59$ $$1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4}$$
$61$ $$1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4}$$
$67$ $$1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 6 T + 71 T^{2} )^{2}$$
$73$ $$1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4}$$
$79$ $$1 - 9 T + 2 T^{2} - 711 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 3 T + 83 T^{2} )^{2}$$
$89$ $$1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + T + 97 T^{2} )^{2}$$