# Properties

 Label 672.2.k.c Level 672 Weight 2 Character orbit 672.k Analytic conductor 5.366 Analytic rank 0 Dimension 8 CM no Inner twists 8

# Learn more about

## 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: 8.0.40960000.1 Defining polynomial: $$x^{8} + 7 x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{4} + \beta_{5} ) q^{3} + \beta_{7} q^{5} + ( -\beta_{3} - \beta_{4} ) q^{7} + ( 2 - \beta_{6} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{4} + \beta_{5} ) q^{3} + \beta_{7} q^{5} + ( -\beta_{3} - \beta_{4} ) q^{7} + ( 2 - \beta_{6} ) q^{9} + ( -2 \beta_{2} - \beta_{7} ) q^{13} + ( \beta_{1} - \beta_{3} ) q^{15} -2 \beta_{7} q^{17} -3 \beta_{4} q^{19} + ( -1 + \beta_{2} - \beta_{6} + 3 \beta_{7} ) q^{21} -6 \beta_{1} q^{23} -3 q^{25} + ( -4 \beta_{4} + \beta_{5} ) q^{27} + 2 \beta_{6} q^{29} -4 \beta_{4} q^{31} + ( 2 \beta_{1} - \beta_{4} + 2 \beta_{5} ) q^{35} -2 q^{37} + ( -5 \beta_{1} - \beta_{3} ) q^{39} -6 \beta_{7} q^{41} -4 \beta_{3} q^{43} + ( 2 \beta_{2} + 3 \beta_{7} ) q^{45} + ( -2 \beta_{4} + 4 \beta_{5} ) q^{47} + ( 3 + 4 \beta_{2} + 2 \beta_{7} ) q^{49} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{51} + 6 \beta_{6} q^{53} + ( -3 - 3 \beta_{6} ) q^{57} + ( \beta_{4} - 2 \beta_{5} ) q^{59} + ( -2 \beta_{2} - \beta_{7} ) q^{61} + ( 5 \beta_{1} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{63} + 2 \beta_{6} q^{65} -6 \beta_{2} q^{69} + 8 \beta_{1} q^{71} + ( 8 \beta_{2} + 4 \beta_{7} ) q^{73} + ( 3 \beta_{4} - 3 \beta_{5} ) q^{75} + 6 \beta_{3} q^{79} + ( -1 - 4 \beta_{6} ) q^{81} + ( -\beta_{4} + 2 \beta_{5} ) q^{83} -4 q^{85} + ( 4 \beta_{4} + 2 \beta_{5} ) q^{87} -8 \beta_{7} q^{89} + ( -2 \beta_{3} + 5 \beta_{4} ) q^{91} + ( -4 - 4 \beta_{6} ) q^{93} + 6 \beta_{1} q^{95} + ( -12 \beta_{2} - 6 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 16q^{9} + O(q^{10})$$ $$8q + 16q^{9} - 8q^{21} - 24q^{25} - 16q^{37} + 24q^{49} - 24q^{57} - 8q^{81} - 32q^{85} - 32q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 7 x^{4} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} + 8 \nu^{2}$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + \nu^{5} + 8 \nu^{3} + 8 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{4} + 7$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{7} + \nu^{5} + 13 \nu^{3} + 5 \nu$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{5} - 8 \nu^{3} + 8 \nu$$$$)/3$$ $$\beta_{6}$$ $$=$$ $$\nu^{6} + 6 \nu^{2}$$ $$\beta_{7}$$ $$=$$ $$($$$$2 \nu^{7} - \nu^{5} + 13 \nu^{3} - 5 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{5} - \beta_{4} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{6} + 3 \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} - 2 \beta_{5} - \beta_{4} + 2 \beta_{2}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{3} - 7$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-8 \beta_{7} - 5 \beta_{5} + 8 \beta_{4} - 5 \beta_{2}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$4 \beta_{6} - 9 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$($$$$8 \beta_{7} + 13 \beta_{5} + 8 \beta_{4} - 13 \beta_{2}$$$$)/2$$

## Character values

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

 $$n$$ $$127$$ $$421$$ $$449$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$-1$$

## 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
 −1.14412 − 1.14412i −0.437016 + 0.437016i −1.14412 + 1.14412i −0.437016 − 0.437016i 0.437016 + 0.437016i 1.14412 − 1.14412i 0.437016 − 0.437016i 1.14412 + 1.14412i
0 −1.58114 0.707107i 0 −1.41421 0 2.23607 1.41421i 0 2.00000 + 2.23607i 0
545.2 0 −1.58114 0.707107i 0 1.41421 0 −2.23607 1.41421i 0 2.00000 + 2.23607i 0
545.3 0 −1.58114 + 0.707107i 0 −1.41421 0 2.23607 + 1.41421i 0 2.00000 2.23607i 0
545.4 0 −1.58114 + 0.707107i 0 1.41421 0 −2.23607 + 1.41421i 0 2.00000 2.23607i 0
545.5 0 1.58114 0.707107i 0 −1.41421 0 −2.23607 1.41421i 0 2.00000 2.23607i 0
545.6 0 1.58114 0.707107i 0 1.41421 0 2.23607 1.41421i 0 2.00000 2.23607i 0
545.7 0 1.58114 + 0.707107i 0 −1.41421 0 −2.23607 + 1.41421i 0 2.00000 + 2.23607i 0
545.8 0 1.58114 + 0.707107i 0 1.41421 0 2.23607 + 1.41421i 0 2.00000 + 2.23607i 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
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
84.h odd 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.c 8
3.b odd 2 1 inner 672.2.k.c 8
4.b odd 2 1 inner 672.2.k.c 8
7.b odd 2 1 inner 672.2.k.c 8
8.b even 2 1 1344.2.k.h 8
8.d odd 2 1 1344.2.k.h 8
12.b even 2 1 inner 672.2.k.c 8
21.c even 2 1 inner 672.2.k.c 8
24.f even 2 1 1344.2.k.h 8
24.h odd 2 1 1344.2.k.h 8
28.d even 2 1 inner 672.2.k.c 8
56.e even 2 1 1344.2.k.h 8
56.h odd 2 1 1344.2.k.h 8
84.h odd 2 1 inner 672.2.k.c 8
168.e odd 2 1 1344.2.k.h 8
168.i even 2 1 1344.2.k.h 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.k.c 8 1.a even 1 1 trivial
672.2.k.c 8 3.b odd 2 1 inner
672.2.k.c 8 4.b odd 2 1 inner
672.2.k.c 8 7.b odd 2 1 inner
672.2.k.c 8 12.b even 2 1 inner
672.2.k.c 8 21.c even 2 1 inner
672.2.k.c 8 28.d even 2 1 inner
672.2.k.c 8 84.h odd 2 1 inner
1344.2.k.h 8 8.b even 2 1
1344.2.k.h 8 8.d odd 2 1
1344.2.k.h 8 24.f even 2 1
1344.2.k.h 8 24.h odd 2 1
1344.2.k.h 8 56.e even 2 1
1344.2.k.h 8 56.h odd 2 1
1344.2.k.h 8 168.e odd 2 1
1344.2.k.h 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 $$S_{2}^{\mathrm{new}}(672, [\chi])$$:

 $$T_{5}^{2} - 2$$ $$T_{43}^{2} - 80$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 4 T^{2} + 9 T^{4} )^{2}$$
$5$ $$( 1 + 8 T^{2} + 25 T^{4} )^{4}$$
$7$ $$( 1 - 6 T^{2} + 49 T^{4} )^{2}$$
$11$ $$( 1 - 11 T^{2} )^{8}$$
$13$ $$( 1 - 16 T^{2} + 169 T^{4} )^{4}$$
$17$ $$( 1 + 26 T^{2} + 289 T^{4} )^{4}$$
$19$ $$( 1 - 20 T^{2} + 361 T^{4} )^{4}$$
$23$ $$( 1 - 10 T^{2} + 529 T^{4} )^{4}$$
$29$ $$( 1 - 38 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 - 30 T^{2} + 961 T^{4} )^{4}$$
$37$ $$( 1 + 2 T + 37 T^{2} )^{8}$$
$41$ $$( 1 + 10 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$( 1 + 6 T^{2} + 1849 T^{4} )^{4}$$
$47$ $$( 1 + 54 T^{2} + 2209 T^{4} )^{4}$$
$53$ $$( 1 + 74 T^{2} + 2809 T^{4} )^{4}$$
$59$ $$( 1 + 108 T^{2} + 3481 T^{4} )^{4}$$
$61$ $$( 1 - 112 T^{2} + 3721 T^{4} )^{4}$$
$67$ $$( 1 + 67 T^{2} )^{8}$$
$71$ $$( 1 - 78 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$( 1 + 14 T^{2} + 5329 T^{4} )^{4}$$
$79$ $$( 1 - 22 T^{2} + 6241 T^{4} )^{4}$$
$83$ $$( 1 + 156 T^{2} + 6889 T^{4} )^{4}$$
$89$ $$( 1 + 50 T^{2} + 7921 T^{4} )^{4}$$
$97$ $$( 1 + 166 T^{2} + 9409 T^{4} )^{4}$$
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