# Properties

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

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 28) 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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} + 3 \zeta_{6} q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} + 3 \zeta_{6} q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} + ( -3 + 3 \zeta_{6} ) q^{11} -2 q^{13} + 3 q^{15} + ( -3 + 3 \zeta_{6} ) q^{17} -\zeta_{6} q^{19} + ( -1 + 3 \zeta_{6} ) q^{21} -3 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + 5 q^{27} + 6 q^{29} + ( 7 - 7 \zeta_{6} ) q^{31} + 3 \zeta_{6} q^{33} + ( -6 - 3 \zeta_{6} ) q^{35} -\zeta_{6} q^{37} + ( -2 + 2 \zeta_{6} ) q^{39} + 6 q^{41} + 4 q^{43} + ( -6 + 6 \zeta_{6} ) q^{45} + 9 \zeta_{6} q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + 3 \zeta_{6} q^{51} + ( 3 - 3 \zeta_{6} ) q^{53} -9 q^{55} - q^{57} + ( 9 - 9 \zeta_{6} ) q^{59} -\zeta_{6} q^{61} + ( -4 - 2 \zeta_{6} ) q^{63} -6 \zeta_{6} q^{65} + ( -7 + 7 \zeta_{6} ) q^{67} -3 q^{69} + ( 1 - \zeta_{6} ) q^{73} + 4 \zeta_{6} q^{75} + ( 3 - 9 \zeta_{6} ) q^{77} + 13 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -12 q^{83} -9 q^{85} + ( 6 - 6 \zeta_{6} ) q^{87} -15 \zeta_{6} q^{89} + ( 6 - 4 \zeta_{6} ) q^{91} -7 \zeta_{6} q^{93} + ( 3 - 3 \zeta_{6} ) q^{95} -10 q^{97} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} + 3q^{5} - 4q^{7} + 2q^{9} + O(q^{10})$$ $$2q + q^{3} + 3q^{5} - 4q^{7} + 2q^{9} - 3q^{11} - 4q^{13} + 6q^{15} - 3q^{17} - q^{19} + q^{21} - 3q^{23} - 4q^{25} + 10q^{27} + 12q^{29} + 7q^{31} + 3q^{33} - 15q^{35} - q^{37} - 2q^{39} + 12q^{41} + 8q^{43} - 6q^{45} + 9q^{47} + 2q^{49} + 3q^{51} + 3q^{53} - 18q^{55} - 2q^{57} + 9q^{59} - q^{61} - 10q^{63} - 6q^{65} - 7q^{67} - 6q^{69} + q^{73} + 4q^{75} - 3q^{77} + 13q^{79} - q^{81} - 24q^{83} - 18q^{85} + 6q^{87} - 15q^{89} + 8q^{91} - 7q^{93} + 3q^{95} - 20q^{97} - 12q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 + 0.866025i 0 1.50000 2.59808i 0 −2.00000 1.73205i 0 1.00000 1.73205i 0
193.1 0 0.500000 0.866025i 0 1.50000 + 2.59808i 0 −2.00000 + 1.73205i 0 1.00000 + 1.73205i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.2.i.e 2
4.b odd 2 1 448.2.i.c 2
7.c even 3 1 inner 448.2.i.e 2
7.c even 3 1 3136.2.a.h 1
7.d odd 6 1 3136.2.a.v 1
8.b even 2 1 28.2.e.a 2
8.d odd 2 1 112.2.i.b 2
24.f even 2 1 1008.2.s.p 2
24.h odd 2 1 252.2.k.c 2
28.f even 6 1 3136.2.a.k 1
28.g odd 6 1 448.2.i.c 2
28.g odd 6 1 3136.2.a.s 1
40.f even 2 1 700.2.i.c 2
40.i odd 4 2 700.2.r.b 4
56.e even 2 1 784.2.i.d 2
56.h odd 2 1 196.2.e.a 2
56.j odd 6 1 196.2.a.a 1
56.j odd 6 1 196.2.e.a 2
56.k odd 6 1 112.2.i.b 2
56.k odd 6 1 784.2.a.d 1
56.m even 6 1 784.2.a.g 1
56.m even 6 1 784.2.i.d 2
56.p even 6 1 28.2.e.a 2
56.p even 6 1 196.2.a.b 1
72.j odd 6 1 2268.2.i.h 2
72.j odd 6 1 2268.2.l.a 2
72.n even 6 1 2268.2.i.a 2
72.n even 6 1 2268.2.l.h 2
168.i even 2 1 1764.2.k.b 2
168.s odd 6 1 252.2.k.c 2
168.s odd 6 1 1764.2.a.a 1
168.v even 6 1 1008.2.s.p 2
168.v even 6 1 7056.2.a.f 1
168.ba even 6 1 1764.2.a.j 1
168.ba even 6 1 1764.2.k.b 2
168.be odd 6 1 7056.2.a.bw 1
280.bf even 6 1 700.2.i.c 2
280.bf even 6 1 4900.2.a.g 1
280.bk odd 6 1 4900.2.a.n 1
280.bt odd 12 2 700.2.r.b 4
280.bt odd 12 2 4900.2.e.i 2
280.bv even 12 2 4900.2.e.h 2
504.w even 6 1 2268.2.i.a 2
504.bi odd 6 1 2268.2.l.a 2
504.cq even 6 1 2268.2.l.h 2
504.db odd 6 1 2268.2.i.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.e.a 2 8.b even 2 1
28.2.e.a 2 56.p even 6 1
112.2.i.b 2 8.d odd 2 1
112.2.i.b 2 56.k odd 6 1
196.2.a.a 1 56.j odd 6 1
196.2.a.b 1 56.p even 6 1
196.2.e.a 2 56.h odd 2 1
196.2.e.a 2 56.j odd 6 1
252.2.k.c 2 24.h odd 2 1
252.2.k.c 2 168.s odd 6 1
448.2.i.c 2 4.b odd 2 1
448.2.i.c 2 28.g odd 6 1
448.2.i.e 2 1.a even 1 1 trivial
448.2.i.e 2 7.c even 3 1 inner
700.2.i.c 2 40.f even 2 1
700.2.i.c 2 280.bf even 6 1
700.2.r.b 4 40.i odd 4 2
700.2.r.b 4 280.bt odd 12 2
784.2.a.d 1 56.k odd 6 1
784.2.a.g 1 56.m even 6 1
784.2.i.d 2 56.e even 2 1
784.2.i.d 2 56.m even 6 1
1008.2.s.p 2 24.f even 2 1
1008.2.s.p 2 168.v even 6 1
1764.2.a.a 1 168.s odd 6 1
1764.2.a.j 1 168.ba even 6 1
1764.2.k.b 2 168.i even 2 1
1764.2.k.b 2 168.ba even 6 1
2268.2.i.a 2 72.n even 6 1
2268.2.i.a 2 504.w even 6 1
2268.2.i.h 2 72.j odd 6 1
2268.2.i.h 2 504.db odd 6 1
2268.2.l.a 2 72.j odd 6 1
2268.2.l.a 2 504.bi odd 6 1
2268.2.l.h 2 72.n even 6 1
2268.2.l.h 2 504.cq even 6 1
3136.2.a.h 1 7.c even 3 1
3136.2.a.k 1 28.f even 6 1
3136.2.a.s 1 28.g odd 6 1
3136.2.a.v 1 7.d odd 6 1
4900.2.a.g 1 280.bf even 6 1
4900.2.a.n 1 280.bk odd 6 1
4900.2.e.h 2 280.bv even 12 2
4900.2.e.i 2 280.bt odd 12 2
7056.2.a.f 1 168.v even 6 1
7056.2.a.bw 1 168.be odd 6 1

## Hecke kernels

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

 $$T_{3}^{2} - T_{3} + 1$$ $$T_{5}^{2} - 3 T_{5} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$9 - 3 T + T^{2}$$
$7$ $$7 + 4 T + T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$9 + 3 T + T^{2}$$
$19$ $$1 + T + T^{2}$$
$23$ $$9 + 3 T + T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$49 - 7 T + T^{2}$$
$37$ $$1 + T + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$81 - 9 T + T^{2}$$
$53$ $$9 - 3 T + T^{2}$$
$59$ $$81 - 9 T + T^{2}$$
$61$ $$1 + T + T^{2}$$
$67$ $$49 + 7 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$1 - T + T^{2}$$
$79$ $$169 - 13 T + T^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$225 + 15 T + T^{2}$$
$97$ $$( 10 + T )^{2}$$