# Properties

 Label 112.4.a.f Level $112$ Weight $4$ Character orbit 112.a Self dual yes Analytic conductor $6.608$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 112.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.60821392064$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{3} + 16q^{5} + 7q^{7} - 23q^{9} + O(q^{10})$$ $$q + 2q^{3} + 16q^{5} + 7q^{7} - 23q^{9} + 8q^{11} + 28q^{13} + 32q^{15} + 54q^{17} + 110q^{19} + 14q^{21} - 48q^{23} + 131q^{25} - 100q^{27} - 110q^{29} - 12q^{31} + 16q^{33} + 112q^{35} - 246q^{37} + 56q^{39} + 182q^{41} - 128q^{43} - 368q^{45} - 324q^{47} + 49q^{49} + 108q^{51} - 162q^{53} + 128q^{55} + 220q^{57} - 810q^{59} - 488q^{61} - 161q^{63} + 448q^{65} - 244q^{67} - 96q^{69} + 768q^{71} - 702q^{73} + 262q^{75} + 56q^{77} - 440q^{79} + 421q^{81} + 1302q^{83} + 864q^{85} - 220q^{87} + 730q^{89} + 196q^{91} - 24q^{93} + 1760q^{95} + 294q^{97} - 184q^{99} + O(q^{100})$$

## 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 2.00000 0 16.0000 0 7.00000 0 −23.0000 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 112.4.a.f 1
3.b odd 2 1 1008.4.a.c 1
4.b odd 2 1 7.4.a.a 1
7.b odd 2 1 784.4.a.g 1
8.b even 2 1 448.4.a.e 1
8.d odd 2 1 448.4.a.i 1
12.b even 2 1 63.4.a.b 1
20.d odd 2 1 175.4.a.b 1
20.e even 4 2 175.4.b.b 2
28.d even 2 1 49.4.a.b 1
28.f even 6 2 49.4.c.b 2
28.g odd 6 2 49.4.c.c 2
44.c even 2 1 847.4.a.b 1
52.b odd 2 1 1183.4.a.b 1
60.h even 2 1 1575.4.a.e 1
68.d odd 2 1 2023.4.a.a 1
84.h odd 2 1 441.4.a.i 1
84.j odd 6 2 441.4.e.e 2
84.n even 6 2 441.4.e.h 2
140.c even 2 1 1225.4.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 4.b odd 2 1
49.4.a.b 1 28.d even 2 1
49.4.c.b 2 28.f even 6 2
49.4.c.c 2 28.g odd 6 2
63.4.a.b 1 12.b even 2 1
112.4.a.f 1 1.a even 1 1 trivial
175.4.a.b 1 20.d odd 2 1
175.4.b.b 2 20.e even 4 2
441.4.a.i 1 84.h odd 2 1
441.4.e.e 2 84.j odd 6 2
441.4.e.h 2 84.n even 6 2
448.4.a.e 1 8.b even 2 1
448.4.a.i 1 8.d odd 2 1
784.4.a.g 1 7.b odd 2 1
847.4.a.b 1 44.c even 2 1
1008.4.a.c 1 3.b odd 2 1
1183.4.a.b 1 52.b odd 2 1
1225.4.a.j 1 140.c even 2 1
1575.4.a.e 1 60.h even 2 1
2023.4.a.a 1 68.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(112))$$:

 $$T_{3} - 2$$ $$T_{5} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-2 + T$$
$5$ $$-16 + T$$
$7$ $$-7 + T$$
$11$ $$-8 + T$$
$13$ $$-28 + T$$
$17$ $$-54 + T$$
$19$ $$-110 + T$$
$23$ $$48 + T$$
$29$ $$110 + T$$
$31$ $$12 + T$$
$37$ $$246 + T$$
$41$ $$-182 + T$$
$43$ $$128 + T$$
$47$ $$324 + T$$
$53$ $$162 + T$$
$59$ $$810 + T$$
$61$ $$488 + T$$
$67$ $$244 + T$$
$71$ $$-768 + T$$
$73$ $$702 + T$$
$79$ $$440 + T$$
$83$ $$-1302 + T$$
$89$ $$-730 + T$$
$97$ $$-294 + T$$