# Properties

 Label 672.2.q.k Level $672$ Weight $2$ Character orbit 672.q Analytic conductor $5.366$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.1156923.1 Defining polynomial: $$x^{6} - 3 x^{5} + 12 x^{4} - 19 x^{3} + 27 x^{2} - 18 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + ( -\beta_{1} + \beta_{4} ) q^{5} + ( -1 - \beta_{1} + \beta_{4} - \beta_{5} ) q^{7} + ( -1 - \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + ( -\beta_{1} + \beta_{4} ) q^{5} + ( -1 - \beta_{1} + \beta_{4} - \beta_{5} ) q^{7} + ( -1 - \beta_{2} ) q^{9} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{11} + ( 2 \beta_{1} - \beta_{3} + \beta_{5} ) q^{13} + \beta_{1} q^{15} + ( 2 \beta_{2} - 2 \beta_{4} ) q^{17} + ( -1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{19} + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{21} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{23} + ( -1 - \beta_{1} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{25} + q^{27} + ( 4 - \beta_{1} ) q^{29} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{31} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{33} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{35} + ( -1 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{37} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{39} + ( -4 + 2 \beta_{3} - 2 \beta_{5} ) q^{41} + ( 4 + \beta_{3} - \beta_{5} ) q^{43} -\beta_{4} q^{45} + ( -4 - 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{4} ) q^{47} + ( -1 + 2 \beta_{1} - 5 \beta_{2} - \beta_{3} - \beta_{5} ) q^{49} + ( -2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} ) q^{51} + ( 2 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} ) q^{53} + ( -4 + 3 \beta_{1} ) q^{55} + ( -2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{57} + ( -1 - \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{59} + ( 6 + 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{61} + ( 1 - \beta_{3} ) q^{63} + ( -8 - 6 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{65} + ( 1 + \beta_{1} + 4 \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{67} + ( -2 - 2 \beta_{1} ) q^{69} -2 \beta_{1} q^{71} + ( -1 - \beta_{1} + 10 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{73} + ( -1 + \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{75} + ( -8 + \beta_{1} - 12 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{77} + ( 9 - 2 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{79} + \beta_{2} q^{81} + ( 5 - 4 \beta_{1} + \beta_{3} - \beta_{5} ) q^{83} + ( 14 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} ) q^{85} + ( 4 \beta_{2} + \beta_{4} ) q^{87} + ( 4 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} ) q^{89} + ( -7 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} ) q^{91} + ( -1 + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{93} + ( 2 + 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{95} + ( 1 - \beta_{3} + \beta_{5} ) q^{97} + ( 1 - \beta_{3} + \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{3} - 3 q^{7} - 3 q^{9} + O(q^{10})$$ $$6 q - 3 q^{3} - 3 q^{7} - 3 q^{9} - 6 q^{13} - 6 q^{17} - 3 q^{19} + 6 q^{23} - 3 q^{25} + 6 q^{27} + 24 q^{29} - 3 q^{31} - 12 q^{35} - 3 q^{37} + 3 q^{39} - 12 q^{41} + 30 q^{43} - 12 q^{47} + 9 q^{49} - 6 q^{51} - 6 q^{53} - 24 q^{55} + 6 q^{57} - 12 q^{59} + 18 q^{61} + 3 q^{63} - 24 q^{65} - 9 q^{67} - 12 q^{69} - 33 q^{73} - 3 q^{75} - 12 q^{77} + 27 q^{79} - 3 q^{81} + 36 q^{83} + 72 q^{85} - 12 q^{87} + 12 q^{89} - 51 q^{91} - 3 q^{93} + 24 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} + 12 x^{4} - 19 x^{3} + 27 x^{2} - 18 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 3$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{5} - 5 \nu^{4} + 22 \nu^{3} - 28 \nu^{2} + 43 \nu - 18$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$-3 \nu^{5} + 7 \nu^{4} - 31 \nu^{3} + 37 \nu^{2} - 56 \nu + 22$$ $$\beta_{4}$$ $$=$$ $$($$$$-6 \nu^{5} + 15 \nu^{4} - 64 \nu^{3} + 82 \nu^{2} - 121 \nu + 50$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$-3 \nu^{5} + 8 \nu^{4} - 33 \nu^{3} + 44 \nu^{2} - 62 \nu + 24$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} - 2 \beta_{4} + \beta_{3} + 3 \beta_{1} - 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{5} + 8 \beta_{4} - 3 \beta_{3} + 6 \beta_{2} - \beta_{1} - 5$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-5 \beta_{5} + 18 \beta_{4} - 9 \beta_{3} + 12 \beta_{2} - 17 \beta_{1} + 27$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$13 \beta_{5} - 28 \beta_{4} + 3 \beta_{3} - 34 \beta_{2} - 11 \beta_{1} + 49$$$$)/2$$

## Character values

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

 $$n$$ $$127$$ $$421$$ $$449$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1 - \beta_{2}$$

## 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 + 0.0585812i 0.5 + 1.51496i 0.5 − 2.43956i 0.5 − 0.0585812i 0.5 − 1.51496i 0.5 + 2.43956i
0 −0.500000 + 0.866025i 0 −1.37328 2.37860i 0 −2.64510 0.0585812i 0 −0.500000 0.866025i 0
193.2 0 −0.500000 + 0.866025i 0 −0.227452 0.393958i 0 2.16908 1.51496i 0 −0.500000 0.866025i 0
193.3 0 −0.500000 + 0.866025i 0 1.60074 + 2.77256i 0 −1.02398 + 2.43956i 0 −0.500000 0.866025i 0
289.1 0 −0.500000 0.866025i 0 −1.37328 + 2.37860i 0 −2.64510 + 0.0585812i 0 −0.500000 + 0.866025i 0
289.2 0 −0.500000 0.866025i 0 −0.227452 + 0.393958i 0 2.16908 + 1.51496i 0 −0.500000 + 0.866025i 0
289.3 0 −0.500000 0.866025i 0 1.60074 2.77256i 0 −1.02398 2.43956i 0 −0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.3 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.k 6
3.b odd 2 1 2016.2.s.u 6
4.b odd 2 1 672.2.q.l yes 6
7.c even 3 1 inner 672.2.q.k 6
7.c even 3 1 4704.2.a.bu 3
7.d odd 6 1 4704.2.a.bt 3
8.b even 2 1 1344.2.q.z 6
8.d odd 2 1 1344.2.q.y 6
12.b even 2 1 2016.2.s.v 6
21.h odd 6 1 2016.2.s.u 6
28.f even 6 1 4704.2.a.bv 3
28.g odd 6 1 672.2.q.l yes 6
28.g odd 6 1 4704.2.a.bs 3
56.j odd 6 1 9408.2.a.ei 3
56.k odd 6 1 1344.2.q.y 6
56.k odd 6 1 9408.2.a.ej 3
56.m even 6 1 9408.2.a.eg 3
56.p even 6 1 1344.2.q.z 6
56.p even 6 1 9408.2.a.eh 3
84.n even 6 1 2016.2.s.v 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.k 6 1.a even 1 1 trivial
672.2.q.k 6 7.c even 3 1 inner
672.2.q.l yes 6 4.b odd 2 1
672.2.q.l yes 6 28.g odd 6 1
1344.2.q.y 6 8.d odd 2 1
1344.2.q.y 6 56.k odd 6 1
1344.2.q.z 6 8.b even 2 1
1344.2.q.z 6 56.p even 6 1
2016.2.s.u 6 3.b odd 2 1
2016.2.s.u 6 21.h odd 6 1
2016.2.s.v 6 12.b even 2 1
2016.2.s.v 6 84.n even 6 1
4704.2.a.bs 3 28.g odd 6 1
4704.2.a.bt 3 7.d odd 6 1
4704.2.a.bu 3 7.c even 3 1
4704.2.a.bv 3 28.f even 6 1
9408.2.a.eg 3 56.m even 6 1
9408.2.a.eh 3 56.p even 6 1
9408.2.a.ei 3 56.j odd 6 1
9408.2.a.ej 3 56.k odd 6 1

## Hecke kernels

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

 $$T_{5}^{6} + 9 T_{5}^{4} + 8 T_{5}^{3} + 81 T_{5}^{2} + 36 T_{5} + 16$$ $$T_{11}^{6} + 27 T_{11}^{4} + 76 T_{11}^{3} + 729 T_{11}^{2} + 1026 T_{11} + 1444$$ $$T_{13}^{3} + 3 T_{13}^{2} - 36 T_{13} - 112$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$( 1 + T + T^{2} )^{3}$$
$5$ $$16 + 36 T + 81 T^{2} + 8 T^{3} + 9 T^{4} + T^{6}$$
$7$ $$343 + 147 T - 5 T^{3} + 3 T^{5} + T^{6}$$
$11$ $$1444 + 1026 T + 729 T^{2} + 76 T^{3} + 27 T^{4} + T^{6}$$
$13$ $$( -112 - 36 T + 3 T^{2} + T^{3} )^{2}$$
$17$ $$9216 + 2304 T + 1152 T^{2} + 48 T^{3} + 60 T^{4} + 6 T^{5} + T^{6}$$
$19$ $$12544 + 4032 T + 1632 T^{2} + 116 T^{3} + 45 T^{4} + 3 T^{5} + T^{6}$$
$23$ $$9216 - 2304 T + 1152 T^{2} - 48 T^{3} + 60 T^{4} - 6 T^{5} + T^{6}$$
$29$ $$( -32 + 39 T - 12 T^{2} + T^{3} )^{2}$$
$31$ $$2209 + 987 T + 582 T^{2} + 31 T^{3} + 30 T^{4} + 3 T^{5} + T^{6}$$
$37$ $$135424 + 30912 T + 8160 T^{2} + 484 T^{3} + 93 T^{4} + 3 T^{5} + T^{6}$$
$41$ $$( -512 - 96 T + 6 T^{2} + T^{3} )^{2}$$
$43$ $$( -28 + 48 T - 15 T^{2} + T^{3} )^{2}$$
$47$ $$12544 - 1344 T + 1488 T^{2} + 368 T^{3} + 132 T^{4} + 12 T^{5} + T^{6}$$
$53$ $$27556 - 13446 T + 5565 T^{2} - 818 T^{3} + 117 T^{4} + 6 T^{5} + T^{6}$$
$59$ $$6724 - 1722 T + 1425 T^{2} + 416 T^{3} + 123 T^{4} + 12 T^{5} + T^{6}$$
$61$ $$304704 + 6624 T + 10080 T^{2} - 1320 T^{3} + 312 T^{4} - 18 T^{5} + T^{6}$$
$67$ $$26896 + 1968 T + 1620 T^{2} + 220 T^{3} + 93 T^{4} + 9 T^{5} + T^{6}$$
$71$ $$( -32 - 36 T + T^{3} )^{2}$$
$73$ $$992016 + 334656 T + 80028 T^{2} + 9096 T^{3} + 753 T^{4} + 33 T^{5} + T^{6}$$
$79$ $$124609 - 68835 T + 28494 T^{2} - 4559 T^{3} + 534 T^{4} - 27 T^{5} + T^{6}$$
$83$ $$( 948 - 15 T - 18 T^{2} + T^{3} )^{2}$$
$89$ $$12544 + 1344 T + 1488 T^{2} - 368 T^{3} + 132 T^{4} - 12 T^{5} + T^{6}$$
$97$ $$( 38 - 27 T + T^{3} )^{2}$$