# Properties

 Label 448.2.p.a Level $448$ Weight $2$ Character orbit 448.p Analytic conductor $3.577$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

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## 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: $$1$$ 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 112) 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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{3} + ( -2 + \zeta_{6} ) q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{3} + ( -2 + \zeta_{6} ) q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} + ( -1 - \zeta_{6} ) q^{11} + ( 1 - 2 \zeta_{6} ) q^{15} + ( -3 - 3 \zeta_{6} ) q^{17} -7 \zeta_{6} q^{19} + ( 3 - \zeta_{6} ) q^{21} + ( -10 + 5 \zeta_{6} ) q^{23} + ( -2 + 2 \zeta_{6} ) q^{25} -5 q^{27} + 6 q^{29} + ( 5 - 5 \zeta_{6} ) q^{31} + ( 2 - \zeta_{6} ) q^{33} + ( 4 + \zeta_{6} ) q^{35} -5 \zeta_{6} q^{37} + ( 4 - 8 \zeta_{6} ) q^{41} + ( -2 + 4 \zeta_{6} ) q^{43} + ( -2 - 2 \zeta_{6} ) q^{45} + 3 \zeta_{6} q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + ( 6 - 3 \zeta_{6} ) q^{51} + ( -9 + 9 \zeta_{6} ) q^{53} + 3 q^{55} + 7 q^{57} + ( -9 + 9 \zeta_{6} ) q^{59} + ( -10 + 5 \zeta_{6} ) q^{61} + ( 4 - 6 \zeta_{6} ) q^{63} + ( 3 + 3 \zeta_{6} ) q^{67} + ( 5 - 10 \zeta_{6} ) q^{69} + ( -2 + 4 \zeta_{6} ) q^{71} + ( 1 + \zeta_{6} ) q^{73} -2 \zeta_{6} q^{75} + ( -1 + 5 \zeta_{6} ) q^{77} + ( 6 - 3 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} + 12 q^{83} + 9 q^{85} + ( -6 + 6 \zeta_{6} ) q^{87} + ( -14 + 7 \zeta_{6} ) q^{89} + 5 \zeta_{6} q^{93} + ( 7 + 7 \zeta_{6} ) q^{95} + ( 4 - 8 \zeta_{6} ) q^{97} + ( 2 - 4 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} - 3q^{5} - 4q^{7} + 2q^{9} + O(q^{10})$$ $$2q - q^{3} - 3q^{5} - 4q^{7} + 2q^{9} - 3q^{11} - 9q^{17} - 7q^{19} + 5q^{21} - 15q^{23} - 2q^{25} - 10q^{27} + 12q^{29} + 5q^{31} + 3q^{33} + 9q^{35} - 5q^{37} - 6q^{45} + 3q^{47} + 2q^{49} + 9q^{51} - 9q^{53} + 6q^{55} + 14q^{57} - 9q^{59} - 15q^{61} + 2q^{63} + 9q^{67} + 3q^{73} - 2q^{75} + 3q^{77} + 9q^{79} - q^{81} + 24q^{83} + 18q^{85} - 6q^{87} - 21q^{89} + 5q^{93} + 21q^{95} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
255.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −0.500000 + 0.866025i 0 −1.50000 + 0.866025i 0 −2.00000 1.73205i 0 1.00000 + 1.73205i 0
383.1 0 −0.500000 0.866025i 0 −1.50000 0.866025i 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
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.a 2
4.b odd 2 1 448.2.p.b 2
7.c even 3 1 3136.2.f.b 2
7.d odd 6 1 448.2.p.b 2
7.d odd 6 1 3136.2.f.a 2
8.b even 2 1 112.2.p.b yes 2
8.d odd 2 1 112.2.p.a 2
24.f even 2 1 1008.2.cs.f 2
24.h odd 2 1 1008.2.cs.c 2
28.f even 6 1 inner 448.2.p.a 2
28.f even 6 1 3136.2.f.b 2
28.g odd 6 1 3136.2.f.a 2
56.e even 2 1 784.2.p.d 2
56.h odd 2 1 784.2.p.c 2
56.j odd 6 1 112.2.p.a 2
56.j odd 6 1 784.2.f.b 2
56.k odd 6 1 784.2.f.b 2
56.k odd 6 1 784.2.p.c 2
56.m even 6 1 112.2.p.b yes 2
56.m even 6 1 784.2.f.a 2
56.p even 6 1 784.2.f.a 2
56.p even 6 1 784.2.p.d 2
168.s odd 6 1 7056.2.b.b 2
168.v even 6 1 7056.2.b.m 2
168.ba even 6 1 1008.2.cs.f 2
168.ba even 6 1 7056.2.b.m 2
168.be odd 6 1 1008.2.cs.c 2
168.be odd 6 1 7056.2.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.a 2 8.d odd 2 1
112.2.p.a 2 56.j odd 6 1
112.2.p.b yes 2 8.b even 2 1
112.2.p.b yes 2 56.m even 6 1
448.2.p.a 2 1.a even 1 1 trivial
448.2.p.a 2 28.f even 6 1 inner
448.2.p.b 2 4.b odd 2 1
448.2.p.b 2 7.d odd 6 1
784.2.f.a 2 56.m even 6 1
784.2.f.a 2 56.p even 6 1
784.2.f.b 2 56.j odd 6 1
784.2.f.b 2 56.k odd 6 1
784.2.p.c 2 56.h odd 2 1
784.2.p.c 2 56.k odd 6 1
784.2.p.d 2 56.e even 2 1
784.2.p.d 2 56.p even 6 1
1008.2.cs.c 2 24.h odd 2 1
1008.2.cs.c 2 168.be odd 6 1
1008.2.cs.f 2 24.f even 2 1
1008.2.cs.f 2 168.ba even 6 1
3136.2.f.a 2 7.d odd 6 1
3136.2.f.a 2 28.g odd 6 1
3136.2.f.b 2 7.c even 3 1
3136.2.f.b 2 28.f even 6 1
7056.2.b.b 2 168.s odd 6 1
7056.2.b.b 2 168.be odd 6 1
7056.2.b.m 2 168.v even 6 1
7056.2.b.m 2 168.ba even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(448, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$3 + 3 T + T^{2}$$
$7$ $$7 + 4 T + T^{2}$$
$11$ $$3 + 3 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$27 + 9 T + T^{2}$$
$19$ $$49 + 7 T + T^{2}$$
$23$ $$75 + 15 T + T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$25 - 5 T + T^{2}$$
$37$ $$25 + 5 T + T^{2}$$
$41$ $$48 + T^{2}$$
$43$ $$12 + T^{2}$$
$47$ $$9 - 3 T + T^{2}$$
$53$ $$81 + 9 T + T^{2}$$
$59$ $$81 + 9 T + T^{2}$$
$61$ $$75 + 15 T + T^{2}$$
$67$ $$27 - 9 T + T^{2}$$
$71$ $$12 + T^{2}$$
$73$ $$3 - 3 T + T^{2}$$
$79$ $$27 - 9 T + T^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$147 + 21 T + T^{2}$$
$97$ $$48 + T^{2}$$
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