# Properties

 Label 448.2.b.a Level $448$ Weight $2$ Character orbit 448.b 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.57729801055$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} -4 i q^{5} - q^{7} - q^{9} +O(q^{10})$$ $$q + 2 i q^{3} -4 i q^{5} - q^{7} - q^{9} -2 i q^{11} -4 i q^{13} + 8 q^{15} + 2 q^{17} -6 i q^{19} -2 i q^{21} -11 q^{25} + 4 i q^{27} + 8 i q^{29} + 8 q^{31} + 4 q^{33} + 4 i q^{35} -8 i q^{37} + 8 q^{39} + 10 q^{41} -2 i q^{43} + 4 i q^{45} -8 q^{47} + q^{49} + 4 i q^{51} -8 q^{55} + 12 q^{57} + 10 i q^{59} + 4 i q^{61} + q^{63} -16 q^{65} -2 i q^{67} -8 q^{71} -6 q^{73} -22 i q^{75} + 2 i q^{77} -8 q^{79} -11 q^{81} -6 i q^{83} -8 i q^{85} -16 q^{87} + 10 q^{89} + 4 i q^{91} + 16 i q^{93} -24 q^{95} + 2 q^{97} + 2 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{7} - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{7} - 2 q^{9} + 16 q^{15} + 4 q^{17} - 22 q^{25} + 16 q^{31} + 8 q^{33} + 16 q^{39} + 20 q^{41} - 16 q^{47} + 2 q^{49} - 16 q^{55} + 24 q^{57} + 2 q^{63} - 32 q^{65} - 16 q^{71} - 12 q^{73} - 16 q^{79} - 22 q^{81} - 32 q^{87} + 20 q^{89} - 48 q^{95} + 4 q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
225.1
 − 1.00000i 1.00000i
0 2.00000i 0 4.00000i 0 −1.00000 0 −1.00000 0
225.2 0 2.00000i 0 4.00000i 0 −1.00000 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
8.b 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.b.a 2
3.b odd 2 1 4032.2.c.b 2
4.b odd 2 1 448.2.b.b yes 2
7.b odd 2 1 3136.2.b.c 2
8.b even 2 1 inner 448.2.b.a 2
8.d odd 2 1 448.2.b.b yes 2
12.b even 2 1 4032.2.c.f 2
16.e even 4 1 1792.2.a.a 1
16.e even 4 1 1792.2.a.h 1
16.f odd 4 1 1792.2.a.d 1
16.f odd 4 1 1792.2.a.e 1
24.f even 2 1 4032.2.c.f 2
24.h odd 2 1 4032.2.c.b 2
28.d even 2 1 3136.2.b.a 2
56.e even 2 1 3136.2.b.a 2
56.h odd 2 1 3136.2.b.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.2.b.a 2 1.a even 1 1 trivial
448.2.b.a 2 8.b even 2 1 inner
448.2.b.b yes 2 4.b odd 2 1
448.2.b.b yes 2 8.d odd 2 1
1792.2.a.a 1 16.e even 4 1
1792.2.a.d 1 16.f odd 4 1
1792.2.a.e 1 16.f odd 4 1
1792.2.a.h 1 16.e even 4 1
3136.2.b.a 2 28.d even 2 1
3136.2.b.a 2 56.e even 2 1
3136.2.b.c 2 7.b odd 2 1
3136.2.b.c 2 56.h odd 2 1
4032.2.c.b 2 3.b odd 2 1
4032.2.c.b 2 24.h odd 2 1
4032.2.c.f 2 12.b even 2 1
4032.2.c.f 2 24.f even 2 1

## Hecke kernels

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

 $$T_{3}^{2} + 4$$ $$T_{31} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$16 + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$4 + T^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$36 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$64 + T^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$( -10 + T )^{2}$$
$43$ $$4 + T^{2}$$
$47$ $$( 8 + T )^{2}$$
$53$ $$T^{2}$$
$59$ $$100 + T^{2}$$
$61$ $$16 + T^{2}$$
$67$ $$4 + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$( 6 + T )^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$36 + T^{2}$$
$89$ $$( -10 + T )^{2}$$
$97$ $$( -2 + T )^{2}$$