# Properties

 Label 448.4.a.o Level $448$ Weight $4$ Character orbit 448.a Self dual yes Analytic conductor $26.433$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$448 = 2^{6} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 448.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.4328556826$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 8q^{3} + 14q^{5} + 7q^{7} + 37q^{9} + O(q^{10})$$ $$q + 8q^{3} + 14q^{5} + 7q^{7} + 37q^{9} - 28q^{11} - 18q^{13} + 112q^{15} + 74q^{17} + 80q^{19} + 56q^{21} + 112q^{23} + 71q^{25} + 80q^{27} - 190q^{29} - 72q^{31} - 224q^{33} + 98q^{35} + 346q^{37} - 144q^{39} + 162q^{41} - 412q^{43} + 518q^{45} - 24q^{47} + 49q^{49} + 592q^{51} - 318q^{53} - 392q^{55} + 640q^{57} - 200q^{59} + 198q^{61} + 259q^{63} - 252q^{65} - 716q^{67} + 896q^{69} - 392q^{71} + 538q^{73} + 568q^{75} - 196q^{77} - 240q^{79} - 359q^{81} - 1072q^{83} + 1036q^{85} - 1520q^{87} + 810q^{89} - 126q^{91} - 576q^{93} + 1120q^{95} + 1354q^{97} - 1036q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 8.00000 0 14.0000 0 7.00000 0 37.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.4.a.o 1
4.b odd 2 1 448.4.a.b 1
8.b even 2 1 112.4.a.a 1
8.d odd 2 1 14.4.a.a 1
24.f even 2 1 126.4.a.h 1
24.h odd 2 1 1008.4.a.s 1
40.e odd 2 1 350.4.a.l 1
40.k even 4 2 350.4.c.b 2
56.e even 2 1 98.4.a.a 1
56.h odd 2 1 784.4.a.s 1
56.k odd 6 2 98.4.c.d 2
56.m even 6 2 98.4.c.f 2
88.g even 2 1 1694.4.a.g 1
104.h odd 2 1 2366.4.a.h 1
168.e odd 2 1 882.4.a.i 1
168.v even 6 2 882.4.g.b 2
168.be odd 6 2 882.4.g.k 2
280.n even 2 1 2450.4.a.bo 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.a 1 8.d odd 2 1
98.4.a.a 1 56.e even 2 1
98.4.c.d 2 56.k odd 6 2
98.4.c.f 2 56.m even 6 2
112.4.a.a 1 8.b even 2 1
126.4.a.h 1 24.f even 2 1
350.4.a.l 1 40.e odd 2 1
350.4.c.b 2 40.k even 4 2
448.4.a.b 1 4.b odd 2 1
448.4.a.o 1 1.a even 1 1 trivial
784.4.a.s 1 56.h odd 2 1
882.4.a.i 1 168.e odd 2 1
882.4.g.b 2 168.v even 6 2
882.4.g.k 2 168.be odd 6 2
1008.4.a.s 1 24.h odd 2 1
1694.4.a.g 1 88.g even 2 1
2366.4.a.h 1 104.h odd 2 1
2450.4.a.bo 1 280.n even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(448))$$:

 $$T_{3} - 8$$ $$T_{5} - 14$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-8 + T$$
$5$ $$-14 + T$$
$7$ $$-7 + T$$
$11$ $$28 + T$$
$13$ $$18 + T$$
$17$ $$-74 + T$$
$19$ $$-80 + T$$
$23$ $$-112 + T$$
$29$ $$190 + T$$
$31$ $$72 + T$$
$37$ $$-346 + T$$
$41$ $$-162 + T$$
$43$ $$412 + T$$
$47$ $$24 + T$$
$53$ $$318 + T$$
$59$ $$200 + T$$
$61$ $$-198 + T$$
$67$ $$716 + T$$
$71$ $$392 + T$$
$73$ $$-538 + T$$
$79$ $$240 + T$$
$83$ $$1072 + T$$
$89$ $$-810 + T$$
$97$ $$-1354 + T$$