# Properties

 Label 672.2.k.b Level 672 Weight 2 Character orbit 672.k Analytic conductor 5.366 Analytic rank 0 Dimension 8 CM discriminant -84 Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$672 = 2^{5} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 672.k (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.36594701583$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.49787136.1 Defining polynomial: $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + \beta_{5} q^{5} -\beta_{2} q^{7} -3 q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + \beta_{5} q^{5} -\beta_{2} q^{7} -3 q^{9} + \beta_{3} q^{11} + ( \beta_{3} + \beta_{6} ) q^{15} -\beta_{7} q^{17} + 2 \beta_{2} q^{19} + \beta_{4} q^{21} + ( \beta_{3} - \beta_{6} ) q^{23} + ( 5 - 2 \beta_{4} ) q^{25} + 3 \beta_{1} q^{27} + 2 \beta_{1} q^{31} + ( -2 \beta_{5} + \beta_{7} ) q^{33} + ( \beta_{3} + 2 \beta_{6} ) q^{35} + 2 \beta_{4} q^{37} + ( 2 \beta_{5} + \beta_{7} ) q^{41} -3 \beta_{5} q^{45} -7 q^{49} + ( \beta_{3} - 2 \beta_{6} ) q^{51} + ( -6 \beta_{1} - 2 \beta_{2} ) q^{55} -2 \beta_{4} q^{57} + 3 \beta_{2} q^{63} + ( -\beta_{5} + 2 \beta_{7} ) q^{69} + ( \beta_{3} + 3 \beta_{6} ) q^{71} + ( -5 \beta_{1} - 6 \beta_{2} ) q^{75} + ( -\beta_{5} - 2 \beta_{7} ) q^{77} + 9 q^{81} + ( -2 + 2 \beta_{4} ) q^{85} + ( 4 \beta_{5} - \beta_{7} ) q^{89} + 6 q^{93} + ( -2 \beta_{3} - 4 \beta_{6} ) q^{95} -3 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 24q^{9} + O(q^{10})$$ $$8q - 24q^{9} + 40q^{25} - 56q^{49} + 72q^{81} - 16q^{85} + 48q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$3 \nu^{6} + 5 \nu^{4} + 15 \nu^{2} + 26$$$$)/10$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{6} - 9$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 3 \nu^{5} + 5 \nu^{3} + 20 \nu$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} + 3 \nu^{4} + \nu^{2} + 6$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 3 \nu^{5} + 5 \nu^{3} + 4 \nu$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{7} - 5 \nu^{5} + 5 \nu^{3} - 16 \nu$$$$)/10$$ $$\beta_{7}$$ $$=$$ $$($$$$-3 \nu^{7} - \nu^{5} - 7 \nu^{3} - 20 \nu$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{5} + \beta_{3}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{4} + \beta_{2} + 3 \beta_{1} - 3$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} + 5 \beta_{6} + 3 \beta_{5}$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{4} + 3 \beta_{2} - \beta_{1} - 1$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{7} - 5 \beta_{6} + 7 \beta_{5} + 2 \beta_{3}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-5 \beta_{2} - 9$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-5 \beta_{7} - 5 \beta_{6} + 2 \beta_{5} - 7 \beta_{3}$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
545.1
 1.09445 + 0.895644i 0.228425 + 1.39564i −0.228425 − 1.39564i −1.09445 − 0.895644i 1.09445 − 0.895644i 0.228425 − 1.39564i −0.228425 + 1.39564i −1.09445 + 0.895644i
0 1.73205i 0 −4.37780 0 2.64575i 0 −3.00000 0
545.2 0 1.73205i 0 −0.913701 0 2.64575i 0 −3.00000 0
545.3 0 1.73205i 0 0.913701 0 2.64575i 0 −3.00000 0
545.4 0 1.73205i 0 4.37780 0 2.64575i 0 −3.00000 0
545.5 0 1.73205i 0 −4.37780 0 2.64575i 0 −3.00000 0
545.6 0 1.73205i 0 −0.913701 0 2.64575i 0 −3.00000 0
545.7 0 1.73205i 0 0.913701 0 2.64575i 0 −3.00000 0
545.8 0 1.73205i 0 4.37780 0 2.64575i 0 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 545.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
84.h odd 2 1 CM by $$\Q(\sqrt{-21})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d 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.k.b 8
3.b odd 2 1 inner 672.2.k.b 8
4.b odd 2 1 inner 672.2.k.b 8
7.b odd 2 1 inner 672.2.k.b 8
8.b even 2 1 1344.2.k.g 8
8.d odd 2 1 1344.2.k.g 8
12.b even 2 1 inner 672.2.k.b 8
21.c even 2 1 inner 672.2.k.b 8
24.f even 2 1 1344.2.k.g 8
24.h odd 2 1 1344.2.k.g 8
28.d even 2 1 inner 672.2.k.b 8
56.e even 2 1 1344.2.k.g 8
56.h odd 2 1 1344.2.k.g 8
84.h odd 2 1 CM 672.2.k.b 8
168.e odd 2 1 1344.2.k.g 8
168.i even 2 1 1344.2.k.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.k.b 8 1.a even 1 1 trivial
672.2.k.b 8 3.b odd 2 1 inner
672.2.k.b 8 4.b odd 2 1 inner
672.2.k.b 8 7.b odd 2 1 inner
672.2.k.b 8 12.b even 2 1 inner
672.2.k.b 8 21.c even 2 1 inner
672.2.k.b 8 28.d even 2 1 inner
672.2.k.b 8 84.h odd 2 1 CM
1344.2.k.g 8 8.b even 2 1
1344.2.k.g 8 8.d odd 2 1
1344.2.k.g 8 24.f even 2 1
1344.2.k.g 8 24.h odd 2 1
1344.2.k.g 8 56.e even 2 1
1344.2.k.g 8 56.h odd 2 1
1344.2.k.g 8 168.e odd 2 1
1344.2.k.g 8 168.i even 2 1

## Hecke kernels

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

 $$T_{5}^{4} - 20 T_{5}^{2} + 16$$ $$T_{43}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + 3 T^{2} )^{4}$$
$5$ $$( 1 - 34 T^{4} + 625 T^{8} )^{2}$$
$7$ $$( 1 + 7 T^{2} )^{4}$$
$11$ $$( 1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4} )^{2}( 1 + 2 T + 2 T^{2} + 22 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 13 T^{2} )^{8}$$
$17$ $$( 1 - 178 T^{4} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 10 T^{2} + 361 T^{4} )^{4}$$
$23$ $$( 1 - 10 T + 50 T^{2} - 230 T^{3} + 529 T^{4} )^{2}( 1 + 10 T + 50 T^{2} + 230 T^{3} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 29 T^{2} )^{8}$$
$31$ $$( 1 - 50 T^{2} + 961 T^{4} )^{4}$$
$37$ $$( 1 - 10 T^{2} + 1369 T^{4} )^{4}$$
$41$ $$( 1 + 1262 T^{4} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 + 43 T^{2} )^{8}$$
$47$ $$( 1 + 47 T^{2} )^{8}$$
$53$ $$( 1 - 53 T^{2} )^{8}$$
$59$ $$( 1 + 59 T^{2} )^{8}$$
$61$ $$( 1 - 61 T^{2} )^{8}$$
$67$ $$( 1 + 67 T^{2} )^{8}$$
$71$ $$( 1 - 22 T + 242 T^{2} - 1562 T^{3} + 5041 T^{4} )^{2}( 1 + 22 T + 242 T^{2} + 1562 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 73 T^{2} )^{8}$$
$79$ $$( 1 + 79 T^{2} )^{8}$$
$83$ $$( 1 + 83 T^{2} )^{8}$$
$89$ $$( 1 + 13742 T^{4} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 97 T^{2} )^{8}$$