# Properties

 Label 448.2.m.a Level $448$ Weight $2$ Character orbit 448.m Analytic conductor $3.577$ Analytic rank $0$ 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.m (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.57729801055$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 112) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (2 i + 2) q^{5} + i q^{7} + 3 i q^{9}+O(q^{10})$$ q + (2*i + 2) * q^5 + i * q^7 + 3*i * q^9 $$q + (2 i + 2) q^{5} + i q^{7} + 3 i q^{9} + ( - i - 1) q^{11} - 2 q^{17} + (2 i - 2) q^{19} + 6 i q^{23} + 3 i q^{25} + ( - 7 i + 7) q^{29} + 8 q^{31} + (2 i - 2) q^{35} + ( - 5 i - 5) q^{37} + 10 i q^{41} + (i + 1) q^{43} + (6 i - 6) q^{45} + 12 q^{47} - q^{49} + ( - i - 1) q^{53} - 4 i q^{55} + ( - 8 i - 8) q^{59} + ( - 6 i + 6) q^{61} - 3 q^{63} + (3 i - 3) q^{67} - 6 i q^{73} + ( - i + 1) q^{77} - 10 q^{79} - 9 q^{81} + ( - 10 i + 10) q^{83} + ( - 4 i - 4) q^{85} - 14 i q^{89} - 8 q^{95} - 2 q^{97} + ( - 3 i + 3) q^{99} +O(q^{100})$$ q + (2*i + 2) * q^5 + i * q^7 + 3*i * q^9 + (-i - 1) * q^11 - 2 * q^17 + (2*i - 2) * q^19 + 6*i * q^23 + 3*i * q^25 + (-7*i + 7) * q^29 + 8 * q^31 + (2*i - 2) * q^35 + (-5*i - 5) * q^37 + 10*i * q^41 + (i + 1) * q^43 + (6*i - 6) * q^45 + 12 * q^47 - q^49 + (-i - 1) * q^53 - 4*i * q^55 + (-8*i - 8) * q^59 + (-6*i + 6) * q^61 - 3 * q^63 + (3*i - 3) * q^67 - 6*i * q^73 + (-i + 1) * q^77 - 10 * q^79 - 9 * q^81 + (-10*i + 10) * q^83 + (-4*i - 4) * q^85 - 14*i * q^89 - 8 * q^95 - 2 * q^97 + (-3*i + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5}+O(q^{10})$$ 2 * q + 4 * q^5 $$2 q + 4 q^{5} - 2 q^{11} - 4 q^{17} - 4 q^{19} + 14 q^{29} + 16 q^{31} - 4 q^{35} - 10 q^{37} + 2 q^{43} - 12 q^{45} + 24 q^{47} - 2 q^{49} - 2 q^{53} - 16 q^{59} + 12 q^{61} - 6 q^{63} - 6 q^{67} + 2 q^{77} - 20 q^{79} - 18 q^{81} + 20 q^{83} - 8 q^{85} - 16 q^{95} - 4 q^{97} + 6 q^{99}+O(q^{100})$$ 2 * q + 4 * q^5 - 2 * q^11 - 4 * q^17 - 4 * q^19 + 14 * q^29 + 16 * q^31 - 4 * q^35 - 10 * q^37 + 2 * q^43 - 12 * q^45 + 24 * q^47 - 2 * q^49 - 2 * q^53 - 16 * q^59 + 12 * q^61 - 6 * q^63 - 6 * q^67 + 2 * q^77 - 20 * q^79 - 18 * q^81 + 20 * q^83 - 8 * q^85 - 16 * q^95 - 4 * q^97 + 6 * q^99

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
113.1
 1.00000i − 1.00000i
0 0 0 2.00000 + 2.00000i 0 1.00000i 0 3.00000i 0
337.1 0 0 0 2.00000 2.00000i 0 1.00000i 0 3.00000i 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
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.2.m.a 2
4.b odd 2 1 112.2.m.b 2
8.b even 2 1 896.2.m.c 2
8.d odd 2 1 896.2.m.b 2
16.e even 4 1 inner 448.2.m.a 2
16.e even 4 1 896.2.m.c 2
16.f odd 4 1 112.2.m.b 2
16.f odd 4 1 896.2.m.b 2
28.d even 2 1 784.2.m.a 2
28.f even 6 2 784.2.x.e 4
28.g odd 6 2 784.2.x.d 4
32.g even 8 2 7168.2.a.k 2
32.h odd 8 2 7168.2.a.b 2
112.j even 4 1 784.2.m.a 2
112.u odd 12 2 784.2.x.d 4
112.v even 12 2 784.2.x.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.m.b 2 4.b odd 2 1
112.2.m.b 2 16.f odd 4 1
448.2.m.a 2 1.a even 1 1 trivial
448.2.m.a 2 16.e even 4 1 inner
784.2.m.a 2 28.d even 2 1
784.2.m.a 2 112.j even 4 1
784.2.x.d 4 28.g odd 6 2
784.2.x.d 4 112.u odd 12 2
784.2.x.e 4 28.f even 6 2
784.2.x.e 4 112.v even 12 2
896.2.m.b 2 8.d odd 2 1
896.2.m.b 2 16.f odd 4 1
896.2.m.c 2 8.b even 2 1
896.2.m.c 2 16.e even 4 1
7168.2.a.b 2 32.h odd 8 2
7168.2.a.k 2 32.g even 8 2

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 4T + 8$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2} + 2T + 2$$
$13$ $$T^{2}$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} + 4T + 8$$
$23$ $$T^{2} + 36$$
$29$ $$T^{2} - 14T + 98$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} + 10T + 50$$
$41$ $$T^{2} + 100$$
$43$ $$T^{2} - 2T + 2$$
$47$ $$(T - 12)^{2}$$
$53$ $$T^{2} + 2T + 2$$
$59$ $$T^{2} + 16T + 128$$
$61$ $$T^{2} - 12T + 72$$
$67$ $$T^{2} + 6T + 18$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 36$$
$79$ $$(T + 10)^{2}$$
$83$ $$T^{2} - 20T + 200$$
$89$ $$T^{2} + 196$$
$97$ $$(T + 2)^{2}$$
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