# Properties

 Label 112.3.s.a Level $112$ Weight $3$ Character orbit 112.s Analytic conductor $3.052$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 112.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.05177896084$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 28) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \zeta_{6} ) q^{3} + ( 2 - \zeta_{6} ) q^{5} + 7 q^{7} -6 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 - \zeta_{6} ) q^{3} + ( 2 - \zeta_{6} ) q^{5} + 7 q^{7} -6 \zeta_{6} q^{9} + ( 15 - 15 \zeta_{6} ) q^{11} + ( 8 - 16 \zeta_{6} ) q^{13} -3 q^{15} + ( 17 + 17 \zeta_{6} ) q^{17} + ( -18 + 9 \zeta_{6} ) q^{19} + ( -7 - 7 \zeta_{6} ) q^{21} -9 \zeta_{6} q^{23} + ( -22 + 22 \zeta_{6} ) q^{25} + ( -15 + 30 \zeta_{6} ) q^{27} -6 q^{29} + ( 7 + 7 \zeta_{6} ) q^{31} + ( -30 + 15 \zeta_{6} ) q^{33} + ( 14 - 7 \zeta_{6} ) q^{35} -31 \zeta_{6} q^{37} + ( -24 + 24 \zeta_{6} ) q^{39} + ( -32 + 64 \zeta_{6} ) q^{41} -10 q^{43} + ( -6 - 6 \zeta_{6} ) q^{45} + ( -50 + 25 \zeta_{6} ) q^{47} + 49 q^{49} -51 \zeta_{6} q^{51} + ( 57 - 57 \zeta_{6} ) q^{53} + ( 15 - 30 \zeta_{6} ) q^{55} + 27 q^{57} + ( 47 + 47 \zeta_{6} ) q^{59} + ( -94 + 47 \zeta_{6} ) q^{61} -42 \zeta_{6} q^{63} -24 \zeta_{6} q^{65} + ( -49 + 49 \zeta_{6} ) q^{67} + ( -9 + 18 \zeta_{6} ) q^{69} + 126 q^{71} + ( -15 - 15 \zeta_{6} ) q^{73} + ( 44 - 22 \zeta_{6} ) q^{75} + ( 105 - 105 \zeta_{6} ) q^{77} -73 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( 8 - 16 \zeta_{6} ) q^{83} + 51 q^{85} + ( 6 + 6 \zeta_{6} ) q^{87} + ( 66 - 33 \zeta_{6} ) q^{89} + ( 56 - 112 \zeta_{6} ) q^{91} -21 \zeta_{6} q^{93} + ( -27 + 27 \zeta_{6} ) q^{95} + ( 16 - 32 \zeta_{6} ) q^{97} -90 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} + 3 q^{5} + 14 q^{7} - 6 q^{9} + O(q^{10})$$ $$2 q - 3 q^{3} + 3 q^{5} + 14 q^{7} - 6 q^{9} + 15 q^{11} - 6 q^{15} + 51 q^{17} - 27 q^{19} - 21 q^{21} - 9 q^{23} - 22 q^{25} - 12 q^{29} + 21 q^{31} - 45 q^{33} + 21 q^{35} - 31 q^{37} - 24 q^{39} - 20 q^{43} - 18 q^{45} - 75 q^{47} + 98 q^{49} - 51 q^{51} + 57 q^{53} + 54 q^{57} + 141 q^{59} - 141 q^{61} - 42 q^{63} - 24 q^{65} - 49 q^{67} + 252 q^{71} - 45 q^{73} + 66 q^{75} + 105 q^{77} - 73 q^{79} - 9 q^{81} + 102 q^{85} + 18 q^{87} + 99 q^{89} - 21 q^{93} - 27 q^{95} - 180 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$$.

 $$n$$ $$15$$ $$17$$ $$85$$ $$\chi(n)$$ $$1$$ $$\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −1.50000 0.866025i 0 1.50000 0.866025i 0 7.00000 0 −3.00000 5.19615i 0
33.1 0 −1.50000 + 0.866025i 0 1.50000 + 0.866025i 0 7.00000 0 −3.00000 + 5.19615i 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
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.3.s.a 2
3.b odd 2 1 1008.3.cg.c 2
4.b odd 2 1 28.3.h.a 2
7.b odd 2 1 784.3.s.b 2
7.c even 3 1 784.3.c.a 2
7.c even 3 1 784.3.s.b 2
7.d odd 6 1 inner 112.3.s.a 2
7.d odd 6 1 784.3.c.a 2
8.b even 2 1 448.3.s.b 2
8.d odd 2 1 448.3.s.a 2
12.b even 2 1 252.3.z.a 2
20.d odd 2 1 700.3.s.a 2
20.e even 4 2 700.3.o.a 4
21.g even 6 1 1008.3.cg.c 2
28.d even 2 1 196.3.h.a 2
28.f even 6 1 28.3.h.a 2
28.f even 6 1 196.3.b.a 2
28.g odd 6 1 196.3.b.a 2
28.g odd 6 1 196.3.h.a 2
56.j odd 6 1 448.3.s.b 2
56.m even 6 1 448.3.s.a 2
84.h odd 2 1 1764.3.z.f 2
84.j odd 6 1 252.3.z.a 2
84.j odd 6 1 1764.3.d.a 2
84.n even 6 1 1764.3.d.a 2
84.n even 6 1 1764.3.z.f 2
140.s even 6 1 700.3.s.a 2
140.x odd 12 2 700.3.o.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.3.h.a 2 4.b odd 2 1
28.3.h.a 2 28.f even 6 1
112.3.s.a 2 1.a even 1 1 trivial
112.3.s.a 2 7.d odd 6 1 inner
196.3.b.a 2 28.f even 6 1
196.3.b.a 2 28.g odd 6 1
196.3.h.a 2 28.d even 2 1
196.3.h.a 2 28.g odd 6 1
252.3.z.a 2 12.b even 2 1
252.3.z.a 2 84.j odd 6 1
448.3.s.a 2 8.d odd 2 1
448.3.s.a 2 56.m even 6 1
448.3.s.b 2 8.b even 2 1
448.3.s.b 2 56.j odd 6 1
700.3.o.a 4 20.e even 4 2
700.3.o.a 4 140.x odd 12 2
700.3.s.a 2 20.d odd 2 1
700.3.s.a 2 140.s even 6 1
784.3.c.a 2 7.c even 3 1
784.3.c.a 2 7.d odd 6 1
784.3.s.b 2 7.b odd 2 1
784.3.s.b 2 7.c even 3 1
1008.3.cg.c 2 3.b odd 2 1
1008.3.cg.c 2 21.g even 6 1
1764.3.d.a 2 84.j odd 6 1
1764.3.d.a 2 84.n even 6 1
1764.3.z.f 2 84.h odd 2 1
1764.3.z.f 2 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 3 T_{3} + 3$$ acting on $$S_{3}^{\mathrm{new}}(112, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 + 3 T + T^{2}$$
$5$ $$3 - 3 T + T^{2}$$
$7$ $$( -7 + T )^{2}$$
$11$ $$225 - 15 T + T^{2}$$
$13$ $$192 + T^{2}$$
$17$ $$867 - 51 T + T^{2}$$
$19$ $$243 + 27 T + T^{2}$$
$23$ $$81 + 9 T + T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$147 - 21 T + T^{2}$$
$37$ $$961 + 31 T + T^{2}$$
$41$ $$3072 + T^{2}$$
$43$ $$( 10 + T )^{2}$$
$47$ $$1875 + 75 T + T^{2}$$
$53$ $$3249 - 57 T + T^{2}$$
$59$ $$6627 - 141 T + T^{2}$$
$61$ $$6627 + 141 T + T^{2}$$
$67$ $$2401 + 49 T + T^{2}$$
$71$ $$( -126 + T )^{2}$$
$73$ $$675 + 45 T + T^{2}$$
$79$ $$5329 + 73 T + T^{2}$$
$83$ $$192 + T^{2}$$
$89$ $$3267 - 99 T + T^{2}$$
$97$ $$768 + T^{2}$$