# Properties

 Label 112.3.d.a Level $112$ Weight $3$ Character orbit 112.d 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.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.05177896084$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-7})$$ Defining polynomial: $$x^{2} - x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 \beta q^{3} + 8 q^{5} + \beta q^{7} -19 q^{9} +O(q^{10})$$ $$q -2 \beta q^{3} + 8 q^{5} + \beta q^{7} -19 q^{9} -4 \beta q^{11} -4 q^{13} -16 \beta q^{15} -2 q^{17} + 10 \beta q^{19} + 14 q^{21} + 8 \beta q^{23} + 39 q^{25} + 20 \beta q^{27} + 14 q^{29} -12 \beta q^{31} -56 q^{33} + 8 \beta q^{35} + 14 q^{37} + 8 \beta q^{39} + 46 q^{41} + 4 \beta q^{43} -152 q^{45} + 12 \beta q^{47} -7 q^{49} + 4 \beta q^{51} -22 q^{53} -32 \beta q^{55} + 140 q^{57} + 34 \beta q^{59} + 48 q^{61} -19 \beta q^{63} -32 q^{65} -24 \beta q^{67} + 112 q^{69} + 32 \beta q^{71} -110 q^{73} -78 \beta q^{75} + 28 q^{77} -48 \beta q^{79} + 109 q^{81} -14 \beta q^{83} -16 q^{85} -28 \beta q^{87} -134 q^{89} -4 \beta q^{91} -168 q^{93} + 80 \beta q^{95} -178 q^{97} + 76 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 16q^{5} - 38q^{9} + O(q^{10})$$ $$2q + 16q^{5} - 38q^{9} - 8q^{13} - 4q^{17} + 28q^{21} + 78q^{25} + 28q^{29} - 112q^{33} + 28q^{37} + 92q^{41} - 304q^{45} - 14q^{49} - 44q^{53} + 280q^{57} + 96q^{61} - 64q^{65} + 224q^{69} - 220q^{73} + 56q^{77} + 218q^{81} - 32q^{85} - 268q^{89} - 336q^{93} - 356q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$15$$ $$17$$ $$85$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
15.1
 0.5 + 1.32288i 0.5 − 1.32288i
0 5.29150i 0 8.00000 0 2.64575i 0 −19.0000 0
15.2 0 5.29150i 0 8.00000 0 2.64575i 0 −19.0000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.3.d.a 2
3.b odd 2 1 1008.3.m.a 2
4.b odd 2 1 inner 112.3.d.a 2
7.b odd 2 1 784.3.d.d 2
7.c even 3 2 784.3.r.k 4
7.d odd 6 2 784.3.r.m 4
8.b even 2 1 448.3.d.a 2
8.d odd 2 1 448.3.d.a 2
12.b even 2 1 1008.3.m.a 2
16.e even 4 2 1792.3.g.b 4
16.f odd 4 2 1792.3.g.b 4
28.d even 2 1 784.3.d.d 2
28.f even 6 2 784.3.r.m 4
28.g odd 6 2 784.3.r.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.3.d.a 2 1.a even 1 1 trivial
112.3.d.a 2 4.b odd 2 1 inner
448.3.d.a 2 8.b even 2 1
448.3.d.a 2 8.d odd 2 1
784.3.d.d 2 7.b odd 2 1
784.3.d.d 2 28.d even 2 1
784.3.r.k 4 7.c even 3 2
784.3.r.k 4 28.g odd 6 2
784.3.r.m 4 7.d odd 6 2
784.3.r.m 4 28.f even 6 2
1008.3.m.a 2 3.b odd 2 1
1008.3.m.a 2 12.b even 2 1
1792.3.g.b 4 16.e even 4 2
1792.3.g.b 4 16.f odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 28$$ acting on $$S_{3}^{\mathrm{new}}(112, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$28 + T^{2}$$
$5$ $$( -8 + T )^{2}$$
$7$ $$7 + T^{2}$$
$11$ $$112 + T^{2}$$
$13$ $$( 4 + T )^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$700 + T^{2}$$
$23$ $$448 + T^{2}$$
$29$ $$( -14 + T )^{2}$$
$31$ $$1008 + T^{2}$$
$37$ $$( -14 + T )^{2}$$
$41$ $$( -46 + T )^{2}$$
$43$ $$112 + T^{2}$$
$47$ $$1008 + T^{2}$$
$53$ $$( 22 + T )^{2}$$
$59$ $$8092 + T^{2}$$
$61$ $$( -48 + T )^{2}$$
$67$ $$4032 + T^{2}$$
$71$ $$7168 + T^{2}$$
$73$ $$( 110 + T )^{2}$$
$79$ $$16128 + T^{2}$$
$83$ $$1372 + T^{2}$$
$89$ $$( 134 + T )^{2}$$
$97$ $$( 178 + T )^{2}$$