# Properties

 Label 112.4.p.b Level $112$ Weight $4$ Character orbit 112.p Analytic conductor $6.608$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$6.60821392064$$ 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: 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{3} + ( -6 + 3 \zeta_{6} ) q^{5} + ( 21 - 14 \zeta_{6} ) q^{7} + 26 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{3} + ( -6 + 3 \zeta_{6} ) q^{5} + ( 21 - 14 \zeta_{6} ) q^{7} + 26 \zeta_{6} q^{9} + ( 15 + 15 \zeta_{6} ) q^{11} + ( -40 + 80 \zeta_{6} ) q^{13} + ( 3 - 6 \zeta_{6} ) q^{15} + ( 21 + 21 \zeta_{6} ) q^{17} + 17 \zeta_{6} q^{19} + ( -7 + 21 \zeta_{6} ) q^{21} + ( 162 - 81 \zeta_{6} ) q^{23} + ( -98 + 98 \zeta_{6} ) q^{25} -53 q^{27} + 90 q^{29} + ( -17 + 17 \zeta_{6} ) q^{31} + ( -30 + 15 \zeta_{6} ) q^{33} + ( -84 + 105 \zeta_{6} ) q^{35} -199 \zeta_{6} q^{37} + ( -40 - 40 \zeta_{6} ) q^{39} + ( 108 - 216 \zeta_{6} ) q^{41} + ( -146 + 292 \zeta_{6} ) q^{43} + ( -78 - 78 \zeta_{6} ) q^{45} -567 \zeta_{6} q^{47} + ( 245 - 392 \zeta_{6} ) q^{49} + ( -42 + 21 \zeta_{6} ) q^{51} + ( 333 - 333 \zeta_{6} ) q^{53} -135 q^{55} -17 q^{57} + ( -801 + 801 \zeta_{6} ) q^{59} + ( -414 + 207 \zeta_{6} ) q^{61} + ( 364 + 182 \zeta_{6} ) q^{63} -360 \zeta_{6} q^{65} + ( -125 - 125 \zeta_{6} ) q^{67} + ( -81 + 162 \zeta_{6} ) q^{69} + ( 282 - 564 \zeta_{6} ) q^{71} + ( 233 + 233 \zeta_{6} ) q^{73} -98 \zeta_{6} q^{75} + ( 525 - 105 \zeta_{6} ) q^{77} + ( 1378 - 689 \zeta_{6} ) q^{79} + ( -649 + 649 \zeta_{6} ) q^{81} -468 q^{83} -189 q^{85} + ( -90 + 90 \zeta_{6} ) q^{87} + ( -222 + 111 \zeta_{6} ) q^{89} + ( 280 + 1120 \zeta_{6} ) q^{91} -17 \zeta_{6} q^{93} + ( -51 - 51 \zeta_{6} ) q^{95} + ( -804 + 1608 \zeta_{6} ) q^{97} + ( -390 + 780 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 9 q^{5} + 28 q^{7} + 26 q^{9} + O(q^{10})$$ $$2 q - q^{3} - 9 q^{5} + 28 q^{7} + 26 q^{9} + 45 q^{11} + 63 q^{17} + 17 q^{19} + 7 q^{21} + 243 q^{23} - 98 q^{25} - 106 q^{27} + 180 q^{29} - 17 q^{31} - 45 q^{33} - 63 q^{35} - 199 q^{37} - 120 q^{39} - 234 q^{45} - 567 q^{47} + 98 q^{49} - 63 q^{51} + 333 q^{53} - 270 q^{55} - 34 q^{57} - 801 q^{59} - 621 q^{61} + 910 q^{63} - 360 q^{65} - 375 q^{67} + 699 q^{73} - 98 q^{75} + 945 q^{77} + 2067 q^{79} - 649 q^{81} - 936 q^{83} - 378 q^{85} - 90 q^{87} - 333 q^{89} + 1680 q^{91} - 17 q^{93} - 153 q^{95} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −0.500000 + 0.866025i 0 −4.50000 + 2.59808i 0 14.0000 12.1244i 0 13.0000 + 22.5167i 0
47.1 0 −0.500000 0.866025i 0 −4.50000 2.59808i 0 14.0000 + 12.1244i 0 13.0000 22.5167i 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 112.4.p.b 2
4.b odd 2 1 112.4.p.c yes 2
7.c even 3 1 784.4.f.c 2
7.d odd 6 1 112.4.p.c yes 2
7.d odd 6 1 784.4.f.b 2
8.b even 2 1 448.4.p.c 2
8.d odd 2 1 448.4.p.b 2
28.f even 6 1 inner 112.4.p.b 2
28.f even 6 1 784.4.f.c 2
28.g odd 6 1 784.4.f.b 2
56.j odd 6 1 448.4.p.b 2
56.m even 6 1 448.4.p.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.4.p.b 2 1.a even 1 1 trivial
112.4.p.b 2 28.f even 6 1 inner
112.4.p.c yes 2 4.b odd 2 1
112.4.p.c yes 2 7.d odd 6 1
448.4.p.b 2 8.d odd 2 1
448.4.p.b 2 56.j odd 6 1
448.4.p.c 2 8.b even 2 1
448.4.p.c 2 56.m even 6 1
784.4.f.b 2 7.d odd 6 1
784.4.f.b 2 28.g odd 6 1
784.4.f.c 2 7.c even 3 1
784.4.f.c 2 28.f 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 $$S_{4}^{\mathrm{new}}(112, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$27 + 9 T + T^{2}$$
$7$ $$343 - 28 T + T^{2}$$
$11$ $$675 - 45 T + T^{2}$$
$13$ $$4800 + T^{2}$$
$17$ $$1323 - 63 T + T^{2}$$
$19$ $$289 - 17 T + T^{2}$$
$23$ $$19683 - 243 T + T^{2}$$
$29$ $$( -90 + T )^{2}$$
$31$ $$289 + 17 T + T^{2}$$
$37$ $$39601 + 199 T + T^{2}$$
$41$ $$34992 + T^{2}$$
$43$ $$63948 + T^{2}$$
$47$ $$321489 + 567 T + T^{2}$$
$53$ $$110889 - 333 T + T^{2}$$
$59$ $$641601 + 801 T + T^{2}$$
$61$ $$128547 + 621 T + T^{2}$$
$67$ $$46875 + 375 T + T^{2}$$
$71$ $$238572 + T^{2}$$
$73$ $$162867 - 699 T + T^{2}$$
$79$ $$1424163 - 2067 T + T^{2}$$
$83$ $$( 468 + T )^{2}$$
$89$ $$36963 + 333 T + T^{2}$$
$97$ $$1939248 + T^{2}$$