# Properties

 Label 448.2.i.i Level $448$ Weight $2$ Character orbit 448.i 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.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.57729801055$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ 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 224) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + ( - \zeta_{12}^{2} + 1) q^{5} + (3 \zeta_{12}^{3} - 2 \zeta_{12}) q^{7}+O(q^{10})$$ q + (-z^3 - z) * q^3 + (-z^2 + 1) * q^5 + (3*z^3 - 2*z) * q^7 $$q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + ( - \zeta_{12}^{2} + 1) q^{5} + (3 \zeta_{12}^{3} - 2 \zeta_{12}) q^{7} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{11} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{15} - 5 \zeta_{12}^{2} q^{17} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{19} + (\zeta_{12}^{2} + 4) q^{21} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{23} + 4 \zeta_{12}^{2} q^{25} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} - 8 q^{29} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{31} + (9 \zeta_{12}^{2} - 9) q^{33} + (2 \zeta_{12}^{3} + \zeta_{12}) q^{35} + (5 \zeta_{12}^{2} - 5) q^{37} + 4 q^{41} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}) q^{43} + ( - 10 \zeta_{12}^{3} + 5 \zeta_{12}) q^{47} + ( - 8 \zeta_{12}^{2} + 3) q^{49} + (10 \zeta_{12}^{3} - 5 \zeta_{12}) q^{51} - \zeta_{12}^{2} q^{53} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{55} + 3 q^{57} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{59} + ( - 11 \zeta_{12}^{2} + 11) q^{61} + (7 \zeta_{12}^{3} + 7 \zeta_{12}) q^{67} + 3 q^{69} + ( - 8 \zeta_{12}^{3} + 16 \zeta_{12}) q^{71} - 15 \zeta_{12}^{2} q^{73} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{75} + (3 \zeta_{12}^{2} + 12) q^{77} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{79} + 9 \zeta_{12}^{2} q^{81} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}) q^{83} - 5 q^{85} + (8 \zeta_{12}^{3} + 8 \zeta_{12}) q^{87} + (7 \zeta_{12}^{2} - 7) q^{89} + (15 \zeta_{12}^{2} - 15) q^{93} + (\zeta_{12}^{3} + \zeta_{12}) q^{95} + 12 q^{97} +O(q^{100})$$ q + (-z^3 - z) * q^3 + (-z^2 + 1) * q^5 + (3*z^3 - 2*z) * q^7 + (-3*z^3 - 3*z) * q^11 + (z^3 - 2*z) * q^15 - 5*z^2 * q^17 + (2*z^3 - z) * q^19 + (z^2 + 4) * q^21 + (2*z^3 - z) * q^23 + 4*z^2 * q^25 + (3*z^3 - 6*z) * q^27 - 8 * q^29 + (-5*z^3 - 5*z) * q^31 + (9*z^2 - 9) * q^33 + (2*z^3 + z) * q^35 + (5*z^2 - 5) * q^37 + 4 * q^41 + (-4*z^3 + 8*z) * q^43 + (-10*z^3 + 5*z) * q^47 + (-8*z^2 + 3) * q^49 + (10*z^3 - 5*z) * q^51 - z^2 * q^53 + (3*z^3 - 6*z) * q^55 + 3 * q^57 + (-z^3 - z) * q^59 + (-11*z^2 + 11) * q^61 + (7*z^3 + 7*z) * q^67 + 3 * q^69 + (-8*z^3 + 16*z) * q^71 - 15*z^2 * q^73 + (-8*z^3 + 4*z) * q^75 + (3*z^2 + 12) * q^77 + (2*z^3 - z) * q^79 + 9*z^2 * q^81 + (-4*z^3 + 8*z) * q^83 - 5 * q^85 + (8*z^3 + 8*z) * q^87 + (7*z^2 - 7) * q^89 + (15*z^2 - 15) * q^93 + (z^3 + z) * q^95 + 12 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{5}+O(q^{10})$$ 4 * q + 2 * q^5 $$4 q + 2 q^{5} - 10 q^{17} + 18 q^{21} + 8 q^{25} - 32 q^{29} - 18 q^{33} - 10 q^{37} + 16 q^{41} - 4 q^{49} - 2 q^{53} + 12 q^{57} + 22 q^{61} + 12 q^{69} - 30 q^{73} + 54 q^{77} + 18 q^{81} - 20 q^{85} - 14 q^{89} - 30 q^{93} + 48 q^{97}+O(q^{100})$$ 4 * q + 2 * q^5 - 10 * q^17 + 18 * q^21 + 8 * q^25 - 32 * q^29 - 18 * q^33 - 10 * q^37 + 16 * q^41 - 4 * q^49 - 2 * q^53 + 12 * q^57 + 22 * q^61 + 12 * q^69 - 30 * q^73 + 54 * q^77 + 18 * q^81 - 20 * q^85 - 14 * q^89 - 30 * q^93 + 48 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 −0.866025 1.50000i 0 0.500000 0.866025i 0 −1.73205 + 2.00000i 0 0 0
65.2 0 0.866025 + 1.50000i 0 0.500000 0.866025i 0 1.73205 2.00000i 0 0 0
193.1 0 −0.866025 + 1.50000i 0 0.500000 + 0.866025i 0 −1.73205 2.00000i 0 0 0
193.2 0 0.866025 1.50000i 0 0.500000 + 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.c even 3 1 inner
28.g odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.2.i.i 4
4.b odd 2 1 inner 448.2.i.i 4
7.c even 3 1 inner 448.2.i.i 4
7.c even 3 1 3136.2.a.bh 2
7.d odd 6 1 3136.2.a.bu 2
8.b even 2 1 224.2.i.b 4
8.d odd 2 1 224.2.i.b 4
24.f even 2 1 2016.2.s.r 4
24.h odd 2 1 2016.2.s.r 4
28.f even 6 1 3136.2.a.bu 2
28.g odd 6 1 inner 448.2.i.i 4
28.g odd 6 1 3136.2.a.bh 2
56.e even 2 1 1568.2.i.u 4
56.h odd 2 1 1568.2.i.u 4
56.j odd 6 1 1568.2.a.n 2
56.j odd 6 1 1568.2.i.u 4
56.k odd 6 1 224.2.i.b 4
56.k odd 6 1 1568.2.a.s 2
56.m even 6 1 1568.2.a.n 2
56.m even 6 1 1568.2.i.u 4
56.p even 6 1 224.2.i.b 4
56.p even 6 1 1568.2.a.s 2
168.s odd 6 1 2016.2.s.r 4
168.v even 6 1 2016.2.s.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.b 4 8.b even 2 1
224.2.i.b 4 8.d odd 2 1
224.2.i.b 4 56.k odd 6 1
224.2.i.b 4 56.p even 6 1
448.2.i.i 4 1.a even 1 1 trivial
448.2.i.i 4 4.b odd 2 1 inner
448.2.i.i 4 7.c even 3 1 inner
448.2.i.i 4 28.g odd 6 1 inner
1568.2.a.n 2 56.j odd 6 1
1568.2.a.n 2 56.m even 6 1
1568.2.a.s 2 56.k odd 6 1
1568.2.a.s 2 56.p even 6 1
1568.2.i.u 4 56.e even 2 1
1568.2.i.u 4 56.h odd 2 1
1568.2.i.u 4 56.j odd 6 1
1568.2.i.u 4 56.m even 6 1
2016.2.s.r 4 24.f even 2 1
2016.2.s.r 4 24.h odd 2 1
2016.2.s.r 4 168.s odd 6 1
2016.2.s.r 4 168.v even 6 1
3136.2.a.bh 2 7.c even 3 1
3136.2.a.bh 2 28.g odd 6 1
3136.2.a.bu 2 7.d odd 6 1
3136.2.a.bu 2 28.f even 6 1

## Hecke kernels

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

 $$T_{3}^{4} + 3T_{3}^{2} + 9$$ T3^4 + 3*T3^2 + 9 $$T_{5}^{2} - T_{5} + 1$$ T5^2 - T5 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$(T^{2} - T + 1)^{2}$$
$7$ $$T^{4} + 2T^{2} + 49$$
$11$ $$T^{4} + 27T^{2} + 729$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 5 T + 25)^{2}$$
$19$ $$T^{4} + 3T^{2} + 9$$
$23$ $$T^{4} + 3T^{2} + 9$$
$29$ $$(T + 8)^{4}$$
$31$ $$T^{4} + 75T^{2} + 5625$$
$37$ $$(T^{2} + 5 T + 25)^{2}$$
$41$ $$(T - 4)^{4}$$
$43$ $$(T^{2} - 48)^{2}$$
$47$ $$T^{4} + 75T^{2} + 5625$$
$53$ $$(T^{2} + T + 1)^{2}$$
$59$ $$T^{4} + 3T^{2} + 9$$
$61$ $$(T^{2} - 11 T + 121)^{2}$$
$67$ $$T^{4} + 147 T^{2} + 21609$$
$71$ $$(T^{2} - 192)^{2}$$
$73$ $$(T^{2} + 15 T + 225)^{2}$$
$79$ $$T^{4} + 3T^{2} + 9$$
$83$ $$(T^{2} - 48)^{2}$$
$89$ $$(T^{2} + 7 T + 49)^{2}$$
$97$ $$(T - 12)^{4}$$