# Properties

 Label 672.2.c.a Level 672 Weight 2 Character orbit 672.c Analytic conductor 5.366 Analytic rank 0 Dimension 4 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.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.36594701583$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{12}^{3} q^{3} + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + q^{7} - q^{9} +O(q^{10})$$ $$q -\zeta_{12}^{3} q^{3} + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + q^{7} - q^{9} + ( -1 + 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{11} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{13} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{15} + ( 1 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{17} + ( 2 - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{19} -\zeta_{12}^{3} q^{21} + ( 5 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{23} + ( 1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{25} + \zeta_{12}^{3} q^{27} + ( 2 - 4 \zeta_{12}^{2} ) q^{29} -2 q^{31} + ( -3 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{33} + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{35} + ( 4 - 8 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{37} + ( -2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{39} + ( -1 - 10 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{41} + ( 2 - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{43} + ( -1 + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{45} + q^{49} + ( -3 + 6 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{51} + ( -6 + 12 \zeta_{12}^{2} ) q^{53} + ( 6 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{55} + ( 4 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{57} + ( 4 - 8 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{59} + ( -4 + 8 \zeta_{12}^{2} ) q^{61} - q^{63} -4 q^{65} + 6 \zeta_{12}^{3} q^{67} + ( -1 + 2 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{69} + ( -9 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{71} + ( 12 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{73} + ( 2 - 4 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{75} + ( -1 + 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{77} + ( 2 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{79} + q^{81} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{83} + ( 4 - 8 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{85} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{87} + ( 5 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{89} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{91} + 2 \zeta_{12}^{3} q^{93} + ( -10 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{95} + ( -8 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{97} + ( 1 - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{7} - 4q^{9} + O(q^{10})$$ $$4q + 4q^{7} - 4q^{9} + 4q^{15} + 4q^{17} + 20q^{23} + 4q^{25} - 8q^{31} - 12q^{33} - 8q^{39} - 4q^{41} + 4q^{49} + 24q^{55} + 16q^{57} - 4q^{63} - 16q^{65} - 36q^{71} + 48q^{73} + 8q^{79} + 4q^{81} + 20q^{89} - 40q^{95} - 32q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i
0 1.00000i 0 0.732051i 0 1.00000 0 −1.00000 0
337.2 0 1.00000i 0 2.73205i 0 1.00000 0 −1.00000 0
337.3 0 1.00000i 0 2.73205i 0 1.00000 0 −1.00000 0
337.4 0 1.00000i 0 0.732051i 0 1.00000 0 −1.00000 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
8.b 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.c.a 4
3.b odd 2 1 2016.2.c.d 4
4.b odd 2 1 168.2.c.a 4
7.b odd 2 1 4704.2.c.b 4
8.b even 2 1 inner 672.2.c.a 4
8.d odd 2 1 168.2.c.a 4
12.b even 2 1 504.2.c.b 4
16.e even 4 1 5376.2.a.o 2
16.e even 4 1 5376.2.a.bc 2
16.f odd 4 1 5376.2.a.u 2
16.f odd 4 1 5376.2.a.y 2
24.f even 2 1 504.2.c.b 4
24.h odd 2 1 2016.2.c.d 4
28.d even 2 1 1176.2.c.b 4
56.e even 2 1 1176.2.c.b 4
56.h odd 2 1 4704.2.c.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.c.a 4 4.b odd 2 1
168.2.c.a 4 8.d odd 2 1
504.2.c.b 4 12.b even 2 1
504.2.c.b 4 24.f even 2 1
672.2.c.a 4 1.a even 1 1 trivial
672.2.c.a 4 8.b even 2 1 inner
1176.2.c.b 4 28.d even 2 1
1176.2.c.b 4 56.e even 2 1
2016.2.c.d 4 3.b odd 2 1
2016.2.c.d 4 24.h odd 2 1
4704.2.c.b 4 7.b odd 2 1
4704.2.c.b 4 56.h odd 2 1
5376.2.a.o 2 16.e even 4 1
5376.2.a.u 2 16.f odd 4 1
5376.2.a.y 2 16.f odd 4 1
5376.2.a.bc 2 16.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 8 T_{5}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(672, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$1 - 12 T^{2} + 74 T^{4} - 300 T^{6} + 625 T^{8}$$
$7$ $$( 1 - T )^{4}$$
$11$ $$1 - 20 T^{2} + 234 T^{4} - 2420 T^{6} + 14641 T^{8}$$
$13$ $$1 - 20 T^{2} + 246 T^{4} - 3380 T^{6} + 28561 T^{8}$$
$17$ $$( 1 - 2 T + 8 T^{2} - 34 T^{3} + 289 T^{4} )^{2}$$
$19$ $$1 - 20 T^{2} + 54 T^{4} - 7220 T^{6} + 130321 T^{8}$$
$23$ $$( 1 - 10 T + 68 T^{2} - 230 T^{3} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 46 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 + 2 T + 31 T^{2} )^{4}$$
$37$ $$1 - 44 T^{2} + 2454 T^{4} - 60236 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 + 2 T + 8 T^{2} + 82 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 - 116 T^{2} + 6294 T^{4} - 214484 T^{6} + 3418801 T^{8}$$
$47$ $$( 1 + 47 T^{2} )^{4}$$
$53$ $$( 1 + 2 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$1 - 12 T^{2} - 5290 T^{4} - 41772 T^{6} + 12117361 T^{8}$$
$61$ $$( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2}$$
$67$ $$( 1 - 98 T^{2} + 4489 T^{4} )^{2}$$
$71$ $$( 1 + 18 T + 220 T^{2} + 1278 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 24 T + 278 T^{2} - 1752 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 4 T + 54 T^{2} - 316 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$1 - 300 T^{2} + 36086 T^{4} - 2066700 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 - 10 T + 200 T^{2} - 890 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 16 T + 246 T^{2} + 1552 T^{3} + 9409 T^{4} )^{2}$$