# Properties

 Label 672.2.bc.b Level 672 Weight 2 Character orbit 672.bc 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.bc (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.36594701583$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{5} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{5} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + 5 \zeta_{12} q^{11} + ( -2 + 4 \zeta_{12}^{2} ) q^{13} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{15} + ( 1 - \zeta_{12}^{2} ) q^{17} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{19} + ( -4 - \zeta_{12}^{2} ) q^{21} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{23} + ( 4 - 4 \zeta_{12}^{2} ) q^{25} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( -2 + 4 \zeta_{12}^{2} ) q^{29} -9 \zeta_{12} q^{31} + ( 5 - 10 \zeta_{12}^{2} ) q^{33} + ( 3 \zeta_{12} - \zeta_{12}^{3} ) q^{35} -3 \zeta_{12}^{2} q^{37} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{39} + 8 q^{41} -3 q^{45} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{47} + ( 3 - 8 \zeta_{12}^{2} ) q^{49} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{51} + ( 7 + 7 \zeta_{12}^{2} ) q^{53} + 5 \zeta_{12}^{3} q^{55} + ( -1 - \zeta_{12}^{2} ) q^{57} + ( 5 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{59} + ( -14 + 7 \zeta_{12}^{2} ) q^{61} + ( 3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{63} + ( -4 + 2 \zeta_{12}^{2} ) q^{65} + ( 7 \zeta_{12} - 14 \zeta_{12}^{3} ) q^{67} + ( -5 - 5 \zeta_{12}^{2} ) q^{69} -2 \zeta_{12}^{3} q^{71} + ( 5 + 5 \zeta_{12}^{2} ) q^{73} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{75} + ( 15 - 5 \zeta_{12}^{2} ) q^{77} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{79} -9 \zeta_{12}^{2} q^{81} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{83} + q^{85} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{87} -13 \zeta_{12}^{2} q^{89} + ( 8 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{91} + ( -9 + 18 \zeta_{12}^{2} ) q^{93} + \zeta_{12} q^{95} + ( 2 - 4 \zeta_{12}^{2} ) q^{97} + ( -15 \zeta_{12} + 15 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{5} - 6q^{9} + O(q^{10})$$ $$4q + 2q^{5} - 6q^{9} + 2q^{17} - 18q^{21} + 8q^{25} - 6q^{37} + 32q^{41} - 12q^{45} - 4q^{49} + 42q^{53} - 6q^{57} - 42q^{61} - 12q^{65} - 30q^{69} + 30q^{73} + 50q^{77} - 18q^{81} + 4q^{85} - 26q^{89} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$421$$ $$449$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 −0.866025 + 1.50000i 0 0.500000 0.866025i 0 1.73205 + 2.00000i 0 −1.50000 2.59808i 0
257.2 0 0.866025 1.50000i 0 0.500000 0.866025i 0 −1.73205 2.00000i 0 −1.50000 2.59808i 0
353.1 0 −0.866025 1.50000i 0 0.500000 + 0.866025i 0 1.73205 2.00000i 0 −1.50000 + 2.59808i 0
353.2 0 0.866025 + 1.50000i 0 0.500000 + 0.866025i 0 −1.73205 + 2.00000i 0 −1.50000 + 2.59808i 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
4.b odd 2 1 inner
21.g even 6 1 inner
84.j odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.bc.b yes 4
3.b odd 2 1 672.2.bc.a 4
4.b odd 2 1 inner 672.2.bc.b yes 4
7.d odd 6 1 672.2.bc.a 4
12.b even 2 1 672.2.bc.a 4
21.g even 6 1 inner 672.2.bc.b yes 4
28.f even 6 1 672.2.bc.a 4
84.j odd 6 1 inner 672.2.bc.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.bc.a 4 3.b odd 2 1
672.2.bc.a 4 7.d odd 6 1
672.2.bc.a 4 12.b even 2 1
672.2.bc.a 4 28.f even 6 1
672.2.bc.b yes 4 1.a even 1 1 trivial
672.2.bc.b yes 4 4.b odd 2 1 inner
672.2.bc.b yes 4 21.g even 6 1 inner
672.2.bc.b yes 4 84.j odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(672, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 3 T^{2} + 9 T^{4}$$
$5$ $$( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} )^{2}$$
$7$ $$1 + 2 T^{2} + 49 T^{4}$$
$11$ $$1 - 3 T^{2} - 112 T^{4} - 363 T^{6} + 14641 T^{8}$$
$13$ $$( 1 - 14 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 - T - 16 T^{2} - 17 T^{3} + 289 T^{4} )^{2}$$
$19$ $$( 1 + 11 T^{2} + 361 T^{4} )( 1 + 26 T^{2} + 361 T^{4} )$$
$23$ $$1 + 21 T^{2} - 88 T^{4} + 11109 T^{6} + 279841 T^{8}$$
$29$ $$( 1 - 46 T^{2} + 841 T^{4} )^{2}$$
$31$ $$1 - 19 T^{2} - 600 T^{4} - 18259 T^{6} + 923521 T^{8}$$
$37$ $$( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 8 T + 41 T^{2} )^{4}$$
$43$ $$( 1 + 43 T^{2} )^{4}$$
$47$ $$1 - 91 T^{2} + 6072 T^{4} - 201019 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 - 21 T + 200 T^{2} - 1113 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$1 - 43 T^{2} - 1632 T^{4} - 149683 T^{6} + 12117361 T^{8}$$
$61$ $$( 1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 109 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} )$$
$71$ $$( 1 - 138 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 15 T + 148 T^{2} - 1095 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$1 - 155 T^{2} + 17784 T^{4} - 967355 T^{6} + 38950081 T^{8}$$
$83$ $$( 1 - 26 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 13 T + 80 T^{2} + 1157 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 182 T^{2} + 9409 T^{4} )^{2}$$