# Properties

 Label 112.4.a.a 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 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 112.4.a.a 1
3.b odd 2 1 1008.4.a.s 1
4.b odd 2 1 14.4.a.a 1
7.b odd 2 1 784.4.a.s 1
8.b even 2 1 448.4.a.o 1
8.d odd 2 1 448.4.a.b 1
12.b even 2 1 126.4.a.h 1
20.d odd 2 1 350.4.a.l 1
20.e even 4 2 350.4.c.b 2
28.d even 2 1 98.4.a.a 1
28.f even 6 2 98.4.c.f 2
28.g odd 6 2 98.4.c.d 2
44.c even 2 1 1694.4.a.g 1
52.b odd 2 1 2366.4.a.h 1
84.h odd 2 1 882.4.a.i 1
84.j odd 6 2 882.4.g.k 2
84.n even 6 2 882.4.g.b 2
140.c 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 4.b odd 2 1
98.4.a.a 1 28.d even 2 1
98.4.c.d 2 28.g odd 6 2
98.4.c.f 2 28.f even 6 2
112.4.a.a 1 1.a even 1 1 trivial
126.4.a.h 1 12.b even 2 1
350.4.a.l 1 20.d odd 2 1
350.4.c.b 2 20.e even 4 2
448.4.a.b 1 8.d odd 2 1
448.4.a.o 1 8.b even 2 1
784.4.a.s 1 7.b odd 2 1
882.4.a.i 1 84.h odd 2 1
882.4.g.b 2 84.n even 6 2
882.4.g.k 2 84.j odd 6 2
1008.4.a.s 1 3.b odd 2 1
1694.4.a.g 1 44.c even 2 1
2366.4.a.h 1 52.b odd 2 1
2450.4.a.bo 1 140.c even 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

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