# Properties

 Label 112.2.p.b Level $112$ Weight $2$ Character orbit 112.p Analytic conductor $0.894$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 112.p (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.894324502638$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{3} + ( - \zeta_{6} + 2) q^{5} + ( - 2 \zeta_{6} - 1) q^{7} + 2 \zeta_{6} q^{9}+O(q^{10})$$ q + (-z + 1) * q^3 + (-z + 2) * q^5 + (-2*z - 1) * q^7 + 2*z * q^9 $$q + ( - \zeta_{6} + 1) q^{3} + ( - \zeta_{6} + 2) q^{5} + ( - 2 \zeta_{6} - 1) q^{7} + 2 \zeta_{6} q^{9} + (\zeta_{6} + 1) q^{11} + ( - 2 \zeta_{6} + 1) q^{15} + ( - 3 \zeta_{6} - 3) q^{17} + 7 \zeta_{6} q^{19} + (\zeta_{6} - 3) q^{21} + (5 \zeta_{6} - 10) q^{23} + (2 \zeta_{6} - 2) q^{25} + 5 q^{27} - 6 q^{29} + ( - 5 \zeta_{6} + 5) q^{31} + ( - \zeta_{6} + 2) q^{33} + ( - \zeta_{6} - 4) q^{35} + 5 \zeta_{6} q^{37} + ( - 8 \zeta_{6} + 4) q^{41} + ( - 4 \zeta_{6} + 2) q^{43} + (2 \zeta_{6} + 2) q^{45} + 3 \zeta_{6} q^{47} + (8 \zeta_{6} - 3) q^{49} + (3 \zeta_{6} - 6) q^{51} + ( - 9 \zeta_{6} + 9) q^{53} + 3 q^{55} + 7 q^{57} + ( - 9 \zeta_{6} + 9) q^{59} + ( - 5 \zeta_{6} + 10) q^{61} + ( - 6 \zeta_{6} + 4) q^{63} + ( - 3 \zeta_{6} - 3) q^{67} + (10 \zeta_{6} - 5) q^{69} + (4 \zeta_{6} - 2) q^{71} + (\zeta_{6} + 1) q^{73} + 2 \zeta_{6} q^{75} + ( - 5 \zeta_{6} + 1) q^{77} + ( - 3 \zeta_{6} + 6) q^{79} + (\zeta_{6} - 1) q^{81} - 12 q^{83} - 9 q^{85} + (6 \zeta_{6} - 6) q^{87} + (7 \zeta_{6} - 14) q^{89} - 5 \zeta_{6} q^{93} + (7 \zeta_{6} + 7) q^{95} + ( - 8 \zeta_{6} + 4) q^{97} + (4 \zeta_{6} - 2) q^{99} +O(q^{100})$$ q + (-z + 1) * q^3 + (-z + 2) * q^5 + (-2*z - 1) * q^7 + 2*z * q^9 + (z + 1) * q^11 + (-2*z + 1) * q^15 + (-3*z - 3) * q^17 + 7*z * q^19 + (z - 3) * q^21 + (5*z - 10) * q^23 + (2*z - 2) * q^25 + 5 * q^27 - 6 * q^29 + (-5*z + 5) * q^31 + (-z + 2) * q^33 + (-z - 4) * q^35 + 5*z * q^37 + (-8*z + 4) * q^41 + (-4*z + 2) * q^43 + (2*z + 2) * q^45 + 3*z * q^47 + (8*z - 3) * q^49 + (3*z - 6) * q^51 + (-9*z + 9) * q^53 + 3 * q^55 + 7 * q^57 + (-9*z + 9) * q^59 + (-5*z + 10) * q^61 + (-6*z + 4) * q^63 + (-3*z - 3) * q^67 + (10*z - 5) * q^69 + (4*z - 2) * q^71 + (z + 1) * q^73 + 2*z * q^75 + (-5*z + 1) * q^77 + (-3*z + 6) * q^79 + (z - 1) * q^81 - 12 * q^83 - 9 * q^85 + (6*z - 6) * q^87 + (7*z - 14) * q^89 - 5*z * q^93 + (7*z + 7) * q^95 + (-8*z + 4) * q^97 + (4*z - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 3 q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + q^3 + 3 * q^5 - 4 * q^7 + 2 * q^9 $$2 q + q^{3} + 3 q^{5} - 4 q^{7} + 2 q^{9} + 3 q^{11} - 9 q^{17} + 7 q^{19} - 5 q^{21} - 15 q^{23} - 2 q^{25} + 10 q^{27} - 12 q^{29} + 5 q^{31} + 3 q^{33} - 9 q^{35} + 5 q^{37} + 6 q^{45} + 3 q^{47} + 2 q^{49} - 9 q^{51} + 9 q^{53} + 6 q^{55} + 14 q^{57} + 9 q^{59} + 15 q^{61} + 2 q^{63} - 9 q^{67} + 3 q^{73} + 2 q^{75} - 3 q^{77} + 9 q^{79} - q^{81} - 24 q^{83} - 18 q^{85} - 6 q^{87} - 21 q^{89} - 5 q^{93} + 21 q^{95}+O(q^{100})$$ 2 * q + q^3 + 3 * q^5 - 4 * q^7 + 2 * q^9 + 3 * q^11 - 9 * q^17 + 7 * q^19 - 5 * q^21 - 15 * q^23 - 2 * q^25 + 10 * q^27 - 12 * q^29 + 5 * q^31 + 3 * q^33 - 9 * q^35 + 5 * q^37 + 6 * q^45 + 3 * q^47 + 2 * q^49 - 9 * q^51 + 9 * q^53 + 6 * q^55 + 14 * q^57 + 9 * q^59 + 15 * q^61 + 2 * q^63 - 9 * q^67 + 3 * q^73 + 2 * q^75 - 3 * q^77 + 9 * q^79 - q^81 - 24 * q^83 - 18 * q^85 - 6 * q^87 - 21 * q^89 - 5 * q^93 + 21 * q^95

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 0.866025i 0 1.50000 0.866025i 0 −2.00000 1.73205i 0 1.00000 + 1.73205i 0
47.1 0 0.500000 + 0.866025i 0 1.50000 + 0.866025i 0 −2.00000 + 1.73205i 0 1.00000 1.73205i 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
28.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.2.p.b yes 2
3.b odd 2 1 1008.2.cs.c 2
4.b odd 2 1 112.2.p.a 2
7.b odd 2 1 784.2.p.c 2
7.c even 3 1 784.2.f.a 2
7.c even 3 1 784.2.p.d 2
7.d odd 6 1 112.2.p.a 2
7.d odd 6 1 784.2.f.b 2
8.b even 2 1 448.2.p.a 2
8.d odd 2 1 448.2.p.b 2
12.b even 2 1 1008.2.cs.f 2
21.g even 6 1 1008.2.cs.f 2
21.g even 6 1 7056.2.b.m 2
21.h odd 6 1 7056.2.b.b 2
28.d even 2 1 784.2.p.d 2
28.f even 6 1 inner 112.2.p.b yes 2
28.f even 6 1 784.2.f.a 2
28.g odd 6 1 784.2.f.b 2
28.g odd 6 1 784.2.p.c 2
56.j odd 6 1 448.2.p.b 2
56.j odd 6 1 3136.2.f.a 2
56.k odd 6 1 3136.2.f.a 2
56.m even 6 1 448.2.p.a 2
56.m even 6 1 3136.2.f.b 2
56.p even 6 1 3136.2.f.b 2
84.j odd 6 1 1008.2.cs.c 2
84.j odd 6 1 7056.2.b.b 2
84.n even 6 1 7056.2.b.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.a 2 4.b odd 2 1
112.2.p.a 2 7.d odd 6 1
112.2.p.b yes 2 1.a even 1 1 trivial
112.2.p.b yes 2 28.f even 6 1 inner
448.2.p.a 2 8.b even 2 1
448.2.p.a 2 56.m even 6 1
448.2.p.b 2 8.d odd 2 1
448.2.p.b 2 56.j odd 6 1
784.2.f.a 2 7.c even 3 1
784.2.f.a 2 28.f even 6 1
784.2.f.b 2 7.d odd 6 1
784.2.f.b 2 28.g odd 6 1
784.2.p.c 2 7.b odd 2 1
784.2.p.c 2 28.g odd 6 1
784.2.p.d 2 7.c even 3 1
784.2.p.d 2 28.d even 2 1
1008.2.cs.c 2 3.b odd 2 1
1008.2.cs.c 2 84.j odd 6 1
1008.2.cs.f 2 12.b even 2 1
1008.2.cs.f 2 21.g even 6 1
3136.2.f.a 2 56.j odd 6 1
3136.2.f.a 2 56.k odd 6 1
3136.2.f.b 2 56.m even 6 1
3136.2.f.b 2 56.p even 6 1
7056.2.b.b 2 21.h odd 6 1
7056.2.b.b 2 84.j odd 6 1
7056.2.b.m 2 21.g even 6 1
7056.2.b.m 2 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(112, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} - 3T + 3$$
$7$ $$T^{2} + 4T + 7$$
$11$ $$T^{2} - 3T + 3$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 9T + 27$$
$19$ $$T^{2} - 7T + 49$$
$23$ $$T^{2} + 15T + 75$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} - 5T + 25$$
$37$ $$T^{2} - 5T + 25$$
$41$ $$T^{2} + 48$$
$43$ $$T^{2} + 12$$
$47$ $$T^{2} - 3T + 9$$
$53$ $$T^{2} - 9T + 81$$
$59$ $$T^{2} - 9T + 81$$
$61$ $$T^{2} - 15T + 75$$
$67$ $$T^{2} + 9T + 27$$
$71$ $$T^{2} + 12$$
$73$ $$T^{2} - 3T + 3$$
$79$ $$T^{2} - 9T + 27$$
$83$ $$(T + 12)^{2}$$
$89$ $$T^{2} + 21T + 147$$
$97$ $$T^{2} + 48$$