# Properties

 Label 448.2.p.d Level $448$ Weight $2$ Character orbit 448.p Analytic conductor $3.577$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$448 = 2^{6} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 448.p (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.57729801055$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 28) 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + (\zeta_{12}^{2} + 1) q^{5} + ( - \zeta_{12}^{3} - 2 \zeta_{12}) q^{7}+O(q^{10})$$ q + (-z^3 - z) * q^3 + (z^2 + 1) * q^5 + (-z^3 - 2*z) * q^7 $$q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + (\zeta_{12}^{2} + 1) q^{5} + ( - \zeta_{12}^{3} - 2 \zeta_{12}) q^{7} + (\zeta_{12}^{3} - \zeta_{12}) q^{11} + ( - 4 \zeta_{12}^{2} + 2) q^{13} - 3 \zeta_{12}^{3} q^{15} + (\zeta_{12}^{2} - 2) q^{17} + ( - 6 \zeta_{12}^{3} + 3 \zeta_{12}) q^{19} + (5 \zeta_{12}^{2} - 4) q^{21} + \zeta_{12} q^{23} - 2 \zeta_{12}^{2} q^{25} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} - 4 q^{29} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{31} + (\zeta_{12}^{2} + 1) q^{33} + ( - 4 \zeta_{12}^{3} - \zeta_{12}) q^{35} + ( - 3 \zeta_{12}^{2} + 3) q^{37} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{39} + ( - 4 \zeta_{12}^{2} + 2) q^{41} + 2 \zeta_{12}^{3} q^{43} + ( - 10 \zeta_{12}^{3} + 5 \zeta_{12}) q^{47} + (8 \zeta_{12}^{2} - 5) q^{49} + 3 \zeta_{12} q^{51} - \zeta_{12}^{2} q^{53} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{55} - 9 q^{57} + (3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{59} + (3 \zeta_{12}^{2} + 3) q^{61} + ( - 6 \zeta_{12}^{2} + 6) q^{65} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{67} + ( - 2 \zeta_{12}^{2} + 1) q^{69} + 14 \zeta_{12}^{3} q^{71} + ( - 5 \zeta_{12}^{2} + 10) q^{73} + (4 \zeta_{12}^{3} - 2 \zeta_{12}) q^{75} + (\zeta_{12}^{2} + 2) q^{77} + 9 \zeta_{12} q^{79} + 9 \zeta_{12}^{2} q^{81} + ( - 8 \zeta_{12}^{3} + 16 \zeta_{12}) q^{83} - 3 q^{85} + (4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{87} + (9 \zeta_{12}^{2} + 9) q^{89} + (10 \zeta_{12}^{3} - 8 \zeta_{12}) q^{91} + (3 \zeta_{12}^{2} - 3) q^{93} + ( - 9 \zeta_{12}^{3} + 9 \zeta_{12}) q^{95} + ( - 20 \zeta_{12}^{2} + 10) q^{97} +O(q^{100})$$ q + (-z^3 - z) * q^3 + (z^2 + 1) * q^5 + (-z^3 - 2*z) * q^7 + (z^3 - z) * q^11 + (-4*z^2 + 2) * q^13 - 3*z^3 * q^15 + (z^2 - 2) * q^17 + (-6*z^3 + 3*z) * q^19 + (5*z^2 - 4) * q^21 + z * q^23 - 2*z^2 * q^25 + (3*z^3 - 6*z) * q^27 - 4 * q^29 + (-z^3 - z) * q^31 + (z^2 + 1) * q^33 + (-4*z^3 - z) * q^35 + (-3*z^2 + 3) * q^37 + (6*z^3 - 6*z) * q^39 + (-4*z^2 + 2) * q^41 + 2*z^3 * q^43 + (-10*z^3 + 5*z) * q^47 + (8*z^2 - 5) * q^49 + 3*z * q^51 - z^2 * q^53 + (z^3 - 2*z) * q^55 - 9 * q^57 + (3*z^3 + 3*z) * q^59 + (3*z^2 + 3) * q^61 + (-6*z^2 + 6) * q^65 + (3*z^3 - 3*z) * q^67 + (-2*z^2 + 1) * q^69 + 14*z^3 * q^71 + (-5*z^2 + 10) * q^73 + (4*z^3 - 2*z) * q^75 + (z^2 + 2) * q^77 + 9*z * q^79 + 9*z^2 * q^81 + (-8*z^3 + 16*z) * q^83 - 3 * q^85 + (4*z^3 + 4*z) * q^87 + (9*z^2 + 9) * q^89 + (10*z^3 - 8*z) * q^91 + (3*z^2 - 3) * q^93 + (-9*z^3 + 9*z) * q^95 + (-20*z^2 + 10) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{5}+O(q^{10})$$ 4 * q + 6 * q^5 $$4 q + 6 q^{5} - 6 q^{17} - 6 q^{21} - 4 q^{25} - 16 q^{29} + 6 q^{33} + 6 q^{37} - 4 q^{49} - 2 q^{53} - 36 q^{57} + 18 q^{61} + 12 q^{65} + 30 q^{73} + 10 q^{77} + 18 q^{81} - 12 q^{85} + 54 q^{89} - 6 q^{93}+O(q^{100})$$ 4 * q + 6 * q^5 - 6 * q^17 - 6 * q^21 - 4 * q^25 - 16 * q^29 + 6 * q^33 + 6 * q^37 - 4 * q^49 - 2 * q^53 - 36 * q^57 + 18 * q^61 + 12 * q^65 + 30 * q^73 + 10 * q^77 + 18 * q^81 - 12 * q^85 + 54 * q^89 - 6 * q^93

## Character values

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

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$-1$$ $$1 - \zeta_{12}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
255.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 −0.866025 + 1.50000i 0 1.50000 0.866025i 0 −1.73205 + 2.00000i 0 0 0
255.2 0 0.866025 1.50000i 0 1.50000 0.866025i 0 1.73205 2.00000i 0 0 0
383.1 0 −0.866025 1.50000i 0 1.50000 + 0.866025i 0 −1.73205 2.00000i 0 0 0
383.2 0 0.866025 + 1.50000i 0 1.50000 + 0.866025i 0 1.73205 + 2.00000i 0 0 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
7.d odd 6 1 inner
28.f 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.p.d 4
4.b odd 2 1 inner 448.2.p.d 4
7.c even 3 1 3136.2.f.e 4
7.d odd 6 1 inner 448.2.p.d 4
7.d odd 6 1 3136.2.f.e 4
8.b even 2 1 28.2.f.a 4
8.d odd 2 1 28.2.f.a 4
24.f even 2 1 252.2.bf.e 4
24.h odd 2 1 252.2.bf.e 4
28.f even 6 1 inner 448.2.p.d 4
28.f even 6 1 3136.2.f.e 4
28.g odd 6 1 3136.2.f.e 4
40.e odd 2 1 700.2.p.a 4
40.f even 2 1 700.2.p.a 4
40.i odd 4 1 700.2.t.a 4
40.i odd 4 1 700.2.t.b 4
40.k even 4 1 700.2.t.a 4
40.k even 4 1 700.2.t.b 4
56.e even 2 1 196.2.f.a 4
56.h odd 2 1 196.2.f.a 4
56.j odd 6 1 28.2.f.a 4
56.j odd 6 1 196.2.d.b 4
56.k odd 6 1 196.2.d.b 4
56.k odd 6 1 196.2.f.a 4
56.m even 6 1 28.2.f.a 4
56.m even 6 1 196.2.d.b 4
56.p even 6 1 196.2.d.b 4
56.p even 6 1 196.2.f.a 4
168.s odd 6 1 1764.2.b.a 4
168.v even 6 1 1764.2.b.a 4
168.ba even 6 1 252.2.bf.e 4
168.ba even 6 1 1764.2.b.a 4
168.be odd 6 1 252.2.bf.e 4
168.be odd 6 1 1764.2.b.a 4
280.ba even 6 1 700.2.p.a 4
280.bk odd 6 1 700.2.p.a 4
280.bp odd 12 1 700.2.t.a 4
280.bp odd 12 1 700.2.t.b 4
280.bv even 12 1 700.2.t.a 4
280.bv even 12 1 700.2.t.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.f.a 4 8.b even 2 1
28.2.f.a 4 8.d odd 2 1
28.2.f.a 4 56.j odd 6 1
28.2.f.a 4 56.m even 6 1
196.2.d.b 4 56.j odd 6 1
196.2.d.b 4 56.k odd 6 1
196.2.d.b 4 56.m even 6 1
196.2.d.b 4 56.p even 6 1
196.2.f.a 4 56.e even 2 1
196.2.f.a 4 56.h odd 2 1
196.2.f.a 4 56.k odd 6 1
196.2.f.a 4 56.p even 6 1
252.2.bf.e 4 24.f even 2 1
252.2.bf.e 4 24.h odd 2 1
252.2.bf.e 4 168.ba even 6 1
252.2.bf.e 4 168.be odd 6 1
448.2.p.d 4 1.a even 1 1 trivial
448.2.p.d 4 4.b odd 2 1 inner
448.2.p.d 4 7.d odd 6 1 inner
448.2.p.d 4 28.f even 6 1 inner
700.2.p.a 4 40.e odd 2 1
700.2.p.a 4 40.f even 2 1
700.2.p.a 4 280.ba even 6 1
700.2.p.a 4 280.bk odd 6 1
700.2.t.a 4 40.i odd 4 1
700.2.t.a 4 40.k even 4 1
700.2.t.a 4 280.bp odd 12 1
700.2.t.a 4 280.bv even 12 1
700.2.t.b 4 40.i odd 4 1
700.2.t.b 4 40.k even 4 1
700.2.t.b 4 280.bp odd 12 1
700.2.t.b 4 280.bv even 12 1
1764.2.b.a 4 168.s odd 6 1
1764.2.b.a 4 168.v even 6 1
1764.2.b.a 4 168.ba even 6 1
1764.2.b.a 4 168.be odd 6 1
3136.2.f.e 4 7.c even 3 1
3136.2.f.e 4 7.d odd 6 1
3136.2.f.e 4 28.f even 6 1
3136.2.f.e 4 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 3T_{3}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(448, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$(T^{2} - 3 T + 3)^{2}$$
$7$ $$T^{4} + 2T^{2} + 49$$
$11$ $$T^{4} - T^{2} + 1$$
$13$ $$(T^{2} + 12)^{2}$$
$17$ $$(T^{2} + 3 T + 3)^{2}$$
$19$ $$T^{4} + 27T^{2} + 729$$
$23$ $$T^{4} - T^{2} + 1$$
$29$ $$(T + 4)^{4}$$
$31$ $$T^{4} + 3T^{2} + 9$$
$37$ $$(T^{2} - 3 T + 9)^{2}$$
$41$ $$(T^{2} + 12)^{2}$$
$43$ $$(T^{2} + 4)^{2}$$
$47$ $$T^{4} + 75T^{2} + 5625$$
$53$ $$(T^{2} + T + 1)^{2}$$
$59$ $$T^{4} + 27T^{2} + 729$$
$61$ $$(T^{2} - 9 T + 27)^{2}$$
$67$ $$T^{4} - 9T^{2} + 81$$
$71$ $$(T^{2} + 196)^{2}$$
$73$ $$(T^{2} - 15 T + 75)^{2}$$
$79$ $$T^{4} - 81T^{2} + 6561$$
$83$ $$(T^{2} - 192)^{2}$$
$89$ $$(T^{2} - 27 T + 243)^{2}$$
$97$ $$(T^{2} + 300)^{2}$$