# Properties

 Label 448.2.q.a Level $448$ Weight $2$ Character orbit 448.q Analytic conductor $3.577$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$448 = 2^{6} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 448.q (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.57729801055$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.12960000.1 Defining polynomial: $$x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{2} + \beta_{7} ) q^{3} + \beta_{6} q^{5} + ( 2 \beta_{1} - \beta_{5} ) q^{7} -2 \beta_{4} q^{9} +O(q^{10})$$ $$q + ( -\beta_{2} + \beta_{7} ) q^{3} + \beta_{6} q^{5} + ( 2 \beta_{1} - \beta_{5} ) q^{7} -2 \beta_{4} q^{9} + ( -\beta_{2} - \beta_{7} ) q^{11} -5 \beta_{5} q^{15} + ( 3 - 3 \beta_{4} ) q^{17} + 3 \beta_{2} q^{19} + ( -3 \beta_{3} + \beta_{6} ) q^{21} + 3 \beta_{1} q^{23} -\beta_{7} q^{27} + ( 2 \beta_{3} - 4 \beta_{6} ) q^{29} + ( \beta_{1} - \beta_{5} ) q^{31} + ( 10 + 5 \beta_{4} ) q^{33} + ( 3 \beta_{2} - 2 \beta_{7} ) q^{35} + ( 6 \beta_{3} - 3 \beta_{6} ) q^{37} + ( -6 - 12 \beta_{4} ) q^{41} + ( -2 \beta_{3} + 2 \beta_{6} ) q^{45} + ( 3 \beta_{1} + 6 \beta_{5} ) q^{47} + ( 3 + 8 \beta_{4} ) q^{49} + ( -3 \beta_{2} + 6 \beta_{7} ) q^{51} + ( -\beta_{3} - \beta_{6} ) q^{53} + ( -10 \beta_{1} - 5 \beta_{5} ) q^{55} -15 q^{57} + ( -\beta_{2} + \beta_{7} ) q^{59} + 3 \beta_{6} q^{61} + ( 6 \beta_{1} + 4 \beta_{5} ) q^{63} + ( -3 \beta_{2} - 3 \beta_{7} ) q^{67} -3 \beta_{3} q^{69} -6 \beta_{5} q^{71} + ( -3 + 3 \beta_{4} ) q^{73} + ( -\beta_{3} + 5 \beta_{6} ) q^{77} -13 \beta_{1} q^{79} + ( 11 + 11 \beta_{4} ) q^{81} -2 \beta_{7} q^{83} + ( -3 \beta_{3} + 6 \beta_{6} ) q^{85} + ( -10 \beta_{1} + 10 \beta_{5} ) q^{87} + ( -6 - 3 \beta_{4} ) q^{89} + ( -2 \beta_{3} + \beta_{6} ) q^{93} + ( 15 \beta_{1} + 15 \beta_{5} ) q^{95} + ( -2 - 4 \beta_{4} ) q^{97} + ( -4 \beta_{2} + 2 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{9} + O(q^{10})$$ $$8q + 8q^{9} + 36q^{17} + 60q^{33} - 8q^{49} - 120q^{57} - 36q^{73} + 44q^{81} - 36q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 13 \nu$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 29 \nu$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 9$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{6} - 8 \nu^{4} + 24 \nu^{2} - 9$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} + 8 \nu^{5} - 20 \nu^{3} + \nu$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$-7 \nu^{6} + 24 \nu^{4} - 56 \nu^{2} + 21$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$3 \nu^{7} - 8 \nu^{5} + 22 \nu^{3} - \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + 3 \beta_{4} - \beta_{3} + 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + 2 \beta_{5}$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{6} + 7 \beta_{4}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{7} + 11 \beta_{5} - 5 \beta_{2} + 11 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$4 \beta_{3} - 9$$ $$\nu^{7}$$ $$=$$ $$($$$$-13 \beta_{2} + 29 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$-1$$ $$-\beta_{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 1.40126 − 0.809017i 0.535233 − 0.309017i −1.40126 + 0.809017i −0.535233 + 0.309017i 1.40126 + 0.809017i 0.535233 + 0.309017i −1.40126 − 0.809017i −0.535233 − 0.309017i
0 −1.93649 1.11803i 0 −1.11803 1.93649i 0 −1.73205 + 2.00000i 0 1.00000 + 1.73205i 0
31.2 0 −1.93649 1.11803i 0 1.11803 + 1.93649i 0 1.73205 2.00000i 0 1.00000 + 1.73205i 0
31.3 0 1.93649 + 1.11803i 0 −1.11803 1.93649i 0 1.73205 2.00000i 0 1.00000 + 1.73205i 0
31.4 0 1.93649 + 1.11803i 0 1.11803 + 1.93649i 0 −1.73205 + 2.00000i 0 1.00000 + 1.73205i 0
159.1 0 −1.93649 + 1.11803i 0 −1.11803 + 1.93649i 0 −1.73205 2.00000i 0 1.00000 1.73205i 0
159.2 0 −1.93649 + 1.11803i 0 1.11803 1.93649i 0 1.73205 + 2.00000i 0 1.00000 1.73205i 0
159.3 0 1.93649 1.11803i 0 −1.11803 + 1.93649i 0 1.73205 + 2.00000i 0 1.00000 1.73205i 0
159.4 0 1.93649 1.11803i 0 1.11803 1.93649i 0 −1.73205 2.00000i 0 1.00000 1.73205i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 159.4 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
7.d odd 6 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
28.f even 6 1 inner
56.j odd 6 1 inner
56.m even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.2.q.a 8
4.b odd 2 1 inner 448.2.q.a 8
7.c even 3 1 3136.2.e.a 8
7.d odd 6 1 inner 448.2.q.a 8
7.d odd 6 1 3136.2.e.a 8
8.b even 2 1 inner 448.2.q.a 8
8.d odd 2 1 inner 448.2.q.a 8
28.f even 6 1 inner 448.2.q.a 8
28.f even 6 1 3136.2.e.a 8
28.g odd 6 1 3136.2.e.a 8
56.j odd 6 1 inner 448.2.q.a 8
56.j odd 6 1 3136.2.e.a 8
56.k odd 6 1 3136.2.e.a 8
56.m even 6 1 inner 448.2.q.a 8
56.m even 6 1 3136.2.e.a 8
56.p even 6 1 3136.2.e.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.2.q.a 8 1.a even 1 1 trivial
448.2.q.a 8 4.b odd 2 1 inner
448.2.q.a 8 7.d odd 6 1 inner
448.2.q.a 8 8.b even 2 1 inner
448.2.q.a 8 8.d odd 2 1 inner
448.2.q.a 8 28.f even 6 1 inner
448.2.q.a 8 56.j odd 6 1 inner
448.2.q.a 8 56.m even 6 1 inner
3136.2.e.a 8 7.c even 3 1
3136.2.e.a 8 7.d odd 6 1
3136.2.e.a 8 28.f even 6 1
3136.2.e.a 8 28.g odd 6 1
3136.2.e.a 8 56.j odd 6 1
3136.2.e.a 8 56.k odd 6 1
3136.2.e.a 8 56.m even 6 1
3136.2.e.a 8 56.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(448, [\chi])$$:

 $$T_{3}^{4} - 5 T_{3}^{2} + 25$$ $$T_{5}^{4} + 5 T_{5}^{2} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 25 - 5 T^{2} + T^{4} )^{2}$$
$5$ $$( 25 + 5 T^{2} + T^{4} )^{2}$$
$7$ $$( 49 + 2 T^{2} + T^{4} )^{2}$$
$11$ $$( 225 + 15 T^{2} + T^{4} )^{2}$$
$13$ $$T^{8}$$
$17$ $$( 27 - 9 T + T^{2} )^{4}$$
$19$ $$( 2025 - 45 T^{2} + T^{4} )^{2}$$
$23$ $$( 81 - 9 T^{2} + T^{4} )^{2}$$
$29$ $$( 60 + T^{2} )^{4}$$
$31$ $$( 9 + 3 T^{2} + T^{4} )^{2}$$
$37$ $$( 18225 - 135 T^{2} + T^{4} )^{2}$$
$41$ $$( 108 + T^{2} )^{4}$$
$43$ $$T^{8}$$
$47$ $$( 729 + 27 T^{2} + T^{4} )^{2}$$
$53$ $$( 225 - 15 T^{2} + T^{4} )^{2}$$
$59$ $$( 25 - 5 T^{2} + T^{4} )^{2}$$
$61$ $$( 2025 + 45 T^{2} + T^{4} )^{2}$$
$67$ $$( 18225 + 135 T^{2} + T^{4} )^{2}$$
$71$ $$( 36 + T^{2} )^{4}$$
$73$ $$( 27 + 9 T + T^{2} )^{4}$$
$79$ $$( 28561 - 169 T^{2} + T^{4} )^{2}$$
$83$ $$( 20 + T^{2} )^{4}$$
$89$ $$( 27 + 9 T + T^{2} )^{4}$$
$97$ $$( 12 + T^{2} )^{4}$$