# Properties

 Label 112.2.p.c Level $112$ Weight $2$ Character orbit 112.p Analytic conductor $0.894$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7x^{2} + 49$$ x^4 + 7*x^2 + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + ( - \beta_{2} - 2) q^{5} - \beta_{3} q^{7} + 4 \beta_{2} q^{9}+O(q^{10})$$ q + b1 * q^3 + (-b2 - 2) * q^5 - b3 * q^7 + 4*b2 * q^9 $$q + \beta_1 q^{3} + ( - \beta_{2} - 2) q^{5} - \beta_{3} q^{7} + 4 \beta_{2} q^{9} + (2 \beta_{3} + \beta_1) q^{11} + ( - 4 \beta_{2} - 2) q^{13} + ( - \beta_{3} - 2 \beta_1) q^{15} + ( - 3 \beta_{2} + 3) q^{17} + ( - \beta_{3} - \beta_1) q^{19} + (7 \beta_{2} + 7) q^{21} + (\beta_{3} - \beta_1) q^{23} + ( - 2 \beta_{2} - 2) q^{25} + \beta_{3} q^{27} - \beta_1 q^{31} + ( - 7 \beta_{2} - 14) q^{33} + (\beta_{3} - \beta_1) q^{35} + 7 \beta_{2} q^{37} + ( - 4 \beta_{3} - 2 \beta_1) q^{39} + (4 \beta_{2} + 2) q^{41} + (2 \beta_{3} + 4 \beta_1) q^{43} + ( - 4 \beta_{2} + 4) q^{45} + (3 \beta_{3} + 3 \beta_1) q^{47} + 7 q^{49} + ( - 3 \beta_{3} + 3 \beta_1) q^{51} + ( - 3 \beta_{2} - 3) q^{53} - 3 \beta_{3} q^{55} + 7 q^{57} - 3 \beta_1 q^{59} + (\beta_{2} + 2) q^{61} + (4 \beta_{3} + 4 \beta_1) q^{63} + 6 \beta_{2} q^{65} + ( - 2 \beta_{3} - \beta_1) q^{67} + ( - 14 \beta_{2} - 7) q^{69} + (2 \beta_{3} + 4 \beta_1) q^{71} + (3 \beta_{2} - 3) q^{73} + ( - 2 \beta_{3} - 2 \beta_1) q^{75} + (7 \beta_{2} - 7) q^{77} + (\beta_{3} - \beta_1) q^{79} + (5 \beta_{2} + 5) q^{81} - 9 q^{85} + (\beta_{2} + 2) q^{89} + ( - 2 \beta_{3} - 4 \beta_1) q^{91} - 7 \beta_{2} q^{93} + (2 \beta_{3} + \beta_1) q^{95} + (4 \beta_{2} + 2) q^{97} + ( - 4 \beta_{3} - 8 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^3 + (-b2 - 2) * q^5 - b3 * q^7 + 4*b2 * q^9 + (2*b3 + b1) * q^11 + (-4*b2 - 2) * q^13 + (-b3 - 2*b1) * q^15 + (-3*b2 + 3) * q^17 + (-b3 - b1) * q^19 + (7*b2 + 7) * q^21 + (b3 - b1) * q^23 + (-2*b2 - 2) * q^25 + b3 * q^27 - b1 * q^31 + (-7*b2 - 14) * q^33 + (b3 - b1) * q^35 + 7*b2 * q^37 + (-4*b3 - 2*b1) * q^39 + (4*b2 + 2) * q^41 + (2*b3 + 4*b1) * q^43 + (-4*b2 + 4) * q^45 + (3*b3 + 3*b1) * q^47 + 7 * q^49 + (-3*b3 + 3*b1) * q^51 + (-3*b2 - 3) * q^53 - 3*b3 * q^55 + 7 * q^57 - 3*b1 * q^59 + (b2 + 2) * q^61 + (4*b3 + 4*b1) * q^63 + 6*b2 * q^65 + (-2*b3 - b1) * q^67 + (-14*b2 - 7) * q^69 + (2*b3 + 4*b1) * q^71 + (3*b2 - 3) * q^73 + (-2*b3 - 2*b1) * q^75 + (7*b2 - 7) * q^77 + (b3 - b1) * q^79 + (5*b2 + 5) * q^81 - 9 * q^85 + (b2 + 2) * q^89 + (-2*b3 - 4*b1) * q^91 - 7*b2 * q^93 + (2*b3 + b1) * q^95 + (4*b2 + 2) * q^97 + (-4*b3 - 8*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{5} - 8 q^{9}+O(q^{10})$$ 4 * q - 6 * q^5 - 8 * q^9 $$4 q - 6 q^{5} - 8 q^{9} + 18 q^{17} + 14 q^{21} - 4 q^{25} - 42 q^{33} - 14 q^{37} + 24 q^{45} + 28 q^{49} - 6 q^{53} + 28 q^{57} + 6 q^{61} - 12 q^{65} - 18 q^{73} - 42 q^{77} + 10 q^{81} - 36 q^{85} + 6 q^{89} + 14 q^{93}+O(q^{100})$$ 4 * q - 6 * q^5 - 8 * q^9 + 18 * q^17 + 14 * q^21 - 4 * q^25 - 42 * q^33 - 14 * q^37 + 24 * q^45 + 28 * q^49 - 6 * q^53 + 28 * q^57 + 6 * q^61 - 12 * q^65 - 18 * q^73 - 42 * q^77 + 10 * q^81 - 36 * q^85 + 6 * q^89 + 14 * q^93

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 7$$ (v^2) / 7 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 7$$ (v^3) / 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$7\beta_{2}$$ 7*b2 $$\nu^{3}$$ $$=$$ $$7\beta_{3}$$ 7*b3

## Character values

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

 $$n$$ $$15$$ $$17$$ $$85$$ $$\chi(n)$$ $$-1$$ $$-\beta_{2}$$ $$1$$

## 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
 −1.32288 + 2.29129i 1.32288 − 2.29129i −1.32288 − 2.29129i 1.32288 + 2.29129i
0 −1.32288 + 2.29129i 0 −1.50000 + 0.866025i 0 −2.64575 0 −2.00000 3.46410i 0
31.2 0 1.32288 2.29129i 0 −1.50000 + 0.866025i 0 2.64575 0 −2.00000 3.46410i 0
47.1 0 −1.32288 2.29129i 0 −1.50000 0.866025i 0 −2.64575 0 −2.00000 + 3.46410i 0
47.2 0 1.32288 + 2.29129i 0 −1.50000 0.866025i 0 2.64575 0 −2.00000 + 3.46410i 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
4.b odd 2 1 inner
7.d odd 6 1 inner
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.c 4
3.b odd 2 1 1008.2.cs.q 4
4.b odd 2 1 inner 112.2.p.c 4
7.b odd 2 1 784.2.p.g 4
7.c even 3 1 784.2.f.d 4
7.c even 3 1 784.2.p.g 4
7.d odd 6 1 inner 112.2.p.c 4
7.d odd 6 1 784.2.f.d 4
8.b even 2 1 448.2.p.c 4
8.d odd 2 1 448.2.p.c 4
12.b even 2 1 1008.2.cs.q 4
21.g even 6 1 1008.2.cs.q 4
21.g even 6 1 7056.2.b.s 4
21.h odd 6 1 7056.2.b.s 4
28.d even 2 1 784.2.p.g 4
28.f even 6 1 inner 112.2.p.c 4
28.f even 6 1 784.2.f.d 4
28.g odd 6 1 784.2.f.d 4
28.g odd 6 1 784.2.p.g 4
56.j odd 6 1 448.2.p.c 4
56.j odd 6 1 3136.2.f.f 4
56.k odd 6 1 3136.2.f.f 4
56.m even 6 1 448.2.p.c 4
56.m even 6 1 3136.2.f.f 4
56.p even 6 1 3136.2.f.f 4
84.j odd 6 1 1008.2.cs.q 4
84.j odd 6 1 7056.2.b.s 4
84.n even 6 1 7056.2.b.s 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 7T_{3}^{2} + 49$$ acting on $$S_{2}^{\mathrm{new}}(112, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 7T^{2} + 49$$
$5$ $$(T^{2} + 3 T + 3)^{2}$$
$7$ $$(T^{2} - 7)^{2}$$
$11$ $$T^{4} - 21T^{2} + 441$$
$13$ $$(T^{2} + 12)^{2}$$
$17$ $$(T^{2} - 9 T + 27)^{2}$$
$19$ $$T^{4} + 7T^{2} + 49$$
$23$ $$T^{4} - 21T^{2} + 441$$
$29$ $$T^{4}$$
$31$ $$T^{4} + 7T^{2} + 49$$
$37$ $$(T^{2} + 7 T + 49)^{2}$$
$41$ $$(T^{2} + 12)^{2}$$
$43$ $$(T^{2} + 84)^{2}$$
$47$ $$T^{4} + 63T^{2} + 3969$$
$53$ $$(T^{2} + 3 T + 9)^{2}$$
$59$ $$T^{4} + 63T^{2} + 3969$$
$61$ $$(T^{2} - 3 T + 3)^{2}$$
$67$ $$T^{4} - 21T^{2} + 441$$
$71$ $$(T^{2} + 84)^{2}$$
$73$ $$(T^{2} + 9 T + 27)^{2}$$
$79$ $$T^{4} - 21T^{2} + 441$$
$83$ $$T^{4}$$
$89$ $$(T^{2} - 3 T + 3)^{2}$$
$97$ $$(T^{2} + 12)^{2}$$