# 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

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

## 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(448))$$:

 $$T_{3} - 8$$ T3 - 8 $$T_{5} - 14$$ T5 - 14

## Hecke characteristic polynomials

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