# Properties

 Label 448.2.f.c Level 448 Weight 2 Character orbit 448.f 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.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.57729801055$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 112) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 q^{3} + ( -2 + 4 \zeta_{6} ) q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} + q^{9} +O(q^{10})$$ $$q + 2 q^{3} + ( -2 + 4 \zeta_{6} ) q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} + q^{9} + ( -2 + 4 \zeta_{6} ) q^{11} + ( 2 - 4 \zeta_{6} ) q^{13} + ( -4 + 8 \zeta_{6} ) q^{15} + 2 q^{19} + ( 6 - 4 \zeta_{6} ) q^{21} + ( -2 + 4 \zeta_{6} ) q^{23} -7 q^{25} -4 q^{27} -6 q^{29} + 8 q^{31} + ( -4 + 8 \zeta_{6} ) q^{33} + ( 2 + 8 \zeta_{6} ) q^{35} + 2 q^{37} + ( 4 - 8 \zeta_{6} ) q^{39} + ( 4 - 8 \zeta_{6} ) q^{41} + ( 6 - 12 \zeta_{6} ) q^{43} + ( -2 + 4 \zeta_{6} ) q^{45} + ( 5 - 8 \zeta_{6} ) q^{49} -6 q^{53} -12 q^{55} + 4 q^{57} + 6 q^{59} + ( 2 - 4 \zeta_{6} ) q^{61} + ( 3 - 2 \zeta_{6} ) q^{63} + 12 q^{65} + ( -2 + 4 \zeta_{6} ) q^{67} + ( -4 + 8 \zeta_{6} ) q^{69} + ( 2 - 4 \zeta_{6} ) q^{71} + ( -4 + 8 \zeta_{6} ) q^{73} -14 q^{75} + ( 2 + 8 \zeta_{6} ) q^{77} + ( 2 - 4 \zeta_{6} ) q^{79} -11 q^{81} -6 q^{83} -12 q^{87} + ( 4 - 8 \zeta_{6} ) q^{89} + ( -2 - 8 \zeta_{6} ) q^{91} + 16 q^{93} + ( -4 + 8 \zeta_{6} ) q^{95} + ( -8 + 16 \zeta_{6} ) q^{97} + ( -2 + 4 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{3} + 4q^{7} + 2q^{9} + O(q^{10})$$ $$2q + 4q^{3} + 4q^{7} + 2q^{9} + 4q^{19} + 8q^{21} - 14q^{25} - 8q^{27} - 12q^{29} + 16q^{31} + 12q^{35} + 4q^{37} + 2q^{49} - 12q^{53} - 24q^{55} + 8q^{57} + 12q^{59} + 4q^{63} + 24q^{65} - 28q^{75} + 12q^{77} - 22q^{81} - 12q^{83} - 24q^{87} - 12q^{91} + 32q^{93} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
447.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 2.00000 0 3.46410i 0 2.00000 + 1.73205i 0 1.00000 0
447.2 0 2.00000 0 3.46410i 0 2.00000 1.73205i 0 1.00000 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
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.2.f.c 2
3.b odd 2 1 4032.2.b.h 2
4.b odd 2 1 448.2.f.a 2
7.b odd 2 1 448.2.f.a 2
8.b even 2 1 112.2.f.a 2
8.d odd 2 1 112.2.f.b yes 2
12.b even 2 1 4032.2.b.b 2
16.e even 4 2 1792.2.e.a 4
16.f odd 4 2 1792.2.e.c 4
21.c even 2 1 4032.2.b.b 2
24.f even 2 1 1008.2.b.b 2
24.h odd 2 1 1008.2.b.g 2
28.d even 2 1 inner 448.2.f.c 2
40.e odd 2 1 2800.2.k.b 2
40.f even 2 1 2800.2.k.e 2
40.i odd 4 2 2800.2.e.c 4
40.k even 4 2 2800.2.e.b 4
56.e even 2 1 112.2.f.a 2
56.h odd 2 1 112.2.f.b yes 2
56.j odd 6 1 784.2.p.a 2
56.j odd 6 1 784.2.p.b 2
56.k odd 6 1 784.2.p.a 2
56.k odd 6 1 784.2.p.b 2
56.m even 6 1 784.2.p.e 2
56.m even 6 1 784.2.p.f 2
56.p even 6 1 784.2.p.e 2
56.p even 6 1 784.2.p.f 2
84.h odd 2 1 4032.2.b.h 2
112.j even 4 2 1792.2.e.a 4
112.l odd 4 2 1792.2.e.c 4
168.e odd 2 1 1008.2.b.g 2
168.i even 2 1 1008.2.b.b 2
280.c odd 2 1 2800.2.k.b 2
280.n even 2 1 2800.2.k.e 2
280.s even 4 2 2800.2.e.b 4
280.y odd 4 2 2800.2.e.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.f.a 2 8.b even 2 1
112.2.f.a 2 56.e even 2 1
112.2.f.b yes 2 8.d odd 2 1
112.2.f.b yes 2 56.h odd 2 1
448.2.f.a 2 4.b odd 2 1
448.2.f.a 2 7.b odd 2 1
448.2.f.c 2 1.a even 1 1 trivial
448.2.f.c 2 28.d even 2 1 inner
784.2.p.a 2 56.j odd 6 1
784.2.p.a 2 56.k odd 6 1
784.2.p.b 2 56.j odd 6 1
784.2.p.b 2 56.k odd 6 1
784.2.p.e 2 56.m even 6 1
784.2.p.e 2 56.p even 6 1
784.2.p.f 2 56.m even 6 1
784.2.p.f 2 56.p even 6 1
1008.2.b.b 2 24.f even 2 1
1008.2.b.b 2 168.i even 2 1
1008.2.b.g 2 24.h odd 2 1
1008.2.b.g 2 168.e odd 2 1
1792.2.e.a 4 16.e even 4 2
1792.2.e.a 4 112.j even 4 2
1792.2.e.c 4 16.f odd 4 2
1792.2.e.c 4 112.l odd 4 2
2800.2.e.b 4 40.k even 4 2
2800.2.e.b 4 280.s even 4 2
2800.2.e.c 4 40.i odd 4 2
2800.2.e.c 4 280.y odd 4 2
2800.2.k.b 2 40.e odd 2 1
2800.2.k.b 2 280.c odd 2 1
2800.2.k.e 2 40.f even 2 1
2800.2.k.e 2 280.n even 2 1
4032.2.b.b 2 12.b even 2 1
4032.2.b.b 2 21.c even 2 1
4032.2.b.h 2 3.b odd 2 1
4032.2.b.h 2 84.h odd 2 1

## Hecke kernels

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