# Properties

 Label 672.2.q.l 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 $$3$$ and degree $$2$$)

## 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 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 3q^{3} + 3q^{7} - 3q^{9} + O(q^{10})$$ $$6q + 3q^{3} + 3q^{7} - 3q^{9} - 6q^{13} - 6q^{17} + 3q^{19} - 6q^{23} - 3q^{25} - 6q^{27} + 24q^{29} + 3q^{31} + 12q^{35} - 3q^{37} - 3q^{39} - 12q^{41} - 30q^{43} + 12q^{47} + 9q^{49} + 6q^{51} - 6q^{53} + 24q^{55} + 6q^{57} + 12q^{59} + 18q^{61} - 3q^{63} - 24q^{65} + 9q^{67} - 12q^{69} - 33q^{73} + 3q^{75} - 12q^{77} - 27q^{79} - 3q^{81} - 36q^{83} + 72q^{85} + 12q^{87} + 12q^{89} + 51q^{91} - 3q^{93} - 24q^{95} + O(q^{100})$$

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.c Even 1 yes

## Hecke kernels

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

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