# Properties

 Label 112.4.i.c Level $112$ Weight $4$ Character orbit 112.i 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.i (of order $$3$$, degree $$2$$, not 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: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 7 - 7 \zeta_{6} ) q^{3} -7 \zeta_{6} q^{5} + ( -7 - 14 \zeta_{6} ) q^{7} -22 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 7 - 7 \zeta_{6} ) q^{3} -7 \zeta_{6} q^{5} + ( -7 - 14 \zeta_{6} ) q^{7} -22 \zeta_{6} q^{9} + ( -5 + 5 \zeta_{6} ) q^{11} -14 q^{13} -49 q^{15} + ( 21 - 21 \zeta_{6} ) q^{17} + 49 \zeta_{6} q^{19} + ( -147 + 49 \zeta_{6} ) q^{21} -159 \zeta_{6} q^{23} + ( 76 - 76 \zeta_{6} ) q^{25} + 35 q^{27} + 58 q^{29} + ( 147 - 147 \zeta_{6} ) q^{31} + 35 \zeta_{6} q^{33} + ( -98 + 147 \zeta_{6} ) q^{35} -219 \zeta_{6} q^{37} + ( -98 + 98 \zeta_{6} ) q^{39} + 350 q^{41} + 124 q^{43} + ( -154 + 154 \zeta_{6} ) q^{45} + 525 \zeta_{6} q^{47} + ( -147 + 392 \zeta_{6} ) q^{49} -147 \zeta_{6} q^{51} + ( -303 + 303 \zeta_{6} ) q^{53} + 35 q^{55} + 343 q^{57} + ( -105 + 105 \zeta_{6} ) q^{59} + 413 \zeta_{6} q^{61} + ( -308 + 462 \zeta_{6} ) q^{63} + 98 \zeta_{6} q^{65} + ( 415 - 415 \zeta_{6} ) q^{67} -1113 q^{69} + 432 q^{71} + ( 1113 - 1113 \zeta_{6} ) q^{73} -532 \zeta_{6} q^{75} + ( 105 - 35 \zeta_{6} ) q^{77} -103 \zeta_{6} q^{79} + ( 839 - 839 \zeta_{6} ) q^{81} -1092 q^{83} -147 q^{85} + ( 406 - 406 \zeta_{6} ) q^{87} + 329 \zeta_{6} q^{89} + ( 98 + 196 \zeta_{6} ) q^{91} -1029 \zeta_{6} q^{93} + ( 343 - 343 \zeta_{6} ) q^{95} -882 q^{97} + 110 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 7q^{3} - 7q^{5} - 28q^{7} - 22q^{9} + O(q^{10})$$ $$2q + 7q^{3} - 7q^{5} - 28q^{7} - 22q^{9} - 5q^{11} - 28q^{13} - 98q^{15} + 21q^{17} + 49q^{19} - 245q^{21} - 159q^{23} + 76q^{25} + 70q^{27} + 116q^{29} + 147q^{31} + 35q^{33} - 49q^{35} - 219q^{37} - 98q^{39} + 700q^{41} + 248q^{43} - 154q^{45} + 525q^{47} + 98q^{49} - 147q^{51} - 303q^{53} + 70q^{55} + 686q^{57} - 105q^{59} + 413q^{61} - 154q^{63} + 98q^{65} + 415q^{67} - 2226q^{69} + 864q^{71} + 1113q^{73} - 532q^{75} + 175q^{77} - 103q^{79} + 839q^{81} - 2184q^{83} - 294q^{85} + 406q^{87} + 329q^{89} + 392q^{91} - 1029q^{93} + 343q^{95} - 1764q^{97} + 220q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 3.50000 + 6.06218i 0 −3.50000 + 6.06218i 0 −14.0000 + 12.1244i 0 −11.0000 + 19.0526i 0
81.1 0 3.50000 6.06218i 0 −3.50000 6.06218i 0 −14.0000 12.1244i 0 −11.0000 19.0526i 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.4.i.c 2
4.b odd 2 1 7.4.c.a 2
7.c even 3 1 inner 112.4.i.c 2
7.c even 3 1 784.4.a.b 1
7.d odd 6 1 784.4.a.r 1
8.b even 2 1 448.4.i.a 2
8.d odd 2 1 448.4.i.f 2
12.b even 2 1 63.4.e.b 2
20.d odd 2 1 175.4.e.a 2
20.e even 4 2 175.4.k.a 4
28.d even 2 1 49.4.c.a 2
28.f even 6 1 49.4.a.c 1
28.f even 6 1 49.4.c.a 2
28.g odd 6 1 7.4.c.a 2
28.g odd 6 1 49.4.a.d 1
56.k odd 6 1 448.4.i.f 2
56.p even 6 1 448.4.i.a 2
84.h odd 2 1 441.4.e.k 2
84.j odd 6 1 441.4.a.e 1
84.j odd 6 1 441.4.e.k 2
84.n even 6 1 63.4.e.b 2
84.n even 6 1 441.4.a.d 1
140.p odd 6 1 175.4.e.a 2
140.p odd 6 1 1225.4.a.c 1
140.s even 6 1 1225.4.a.d 1
140.w even 12 2 175.4.k.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 4.b odd 2 1
7.4.c.a 2 28.g odd 6 1
49.4.a.c 1 28.f even 6 1
49.4.a.d 1 28.g odd 6 1
49.4.c.a 2 28.d even 2 1
49.4.c.a 2 28.f even 6 1
63.4.e.b 2 12.b even 2 1
63.4.e.b 2 84.n even 6 1
112.4.i.c 2 1.a even 1 1 trivial
112.4.i.c 2 7.c even 3 1 inner
175.4.e.a 2 20.d odd 2 1
175.4.e.a 2 140.p odd 6 1
175.4.k.a 4 20.e even 4 2
175.4.k.a 4 140.w even 12 2
441.4.a.d 1 84.n even 6 1
441.4.a.e 1 84.j odd 6 1
441.4.e.k 2 84.h odd 2 1
441.4.e.k 2 84.j odd 6 1
448.4.i.a 2 8.b even 2 1
448.4.i.a 2 56.p even 6 1
448.4.i.f 2 8.d odd 2 1
448.4.i.f 2 56.k odd 6 1
784.4.a.b 1 7.c even 3 1
784.4.a.r 1 7.d odd 6 1
1225.4.a.c 1 140.p odd 6 1
1225.4.a.d 1 140.s even 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$49 - 7 T + T^{2}$$
$5$ $$49 + 7 T + T^{2}$$
$7$ $$343 + 28 T + T^{2}$$
$11$ $$25 + 5 T + T^{2}$$
$13$ $$( 14 + T )^{2}$$
$17$ $$441 - 21 T + T^{2}$$
$19$ $$2401 - 49 T + T^{2}$$
$23$ $$25281 + 159 T + T^{2}$$
$29$ $$( -58 + T )^{2}$$
$31$ $$21609 - 147 T + T^{2}$$
$37$ $$47961 + 219 T + T^{2}$$
$41$ $$( -350 + T )^{2}$$
$43$ $$( -124 + T )^{2}$$
$47$ $$275625 - 525 T + T^{2}$$
$53$ $$91809 + 303 T + T^{2}$$
$59$ $$11025 + 105 T + T^{2}$$
$61$ $$170569 - 413 T + T^{2}$$
$67$ $$172225 - 415 T + T^{2}$$
$71$ $$( -432 + T )^{2}$$
$73$ $$1238769 - 1113 T + T^{2}$$
$79$ $$10609 + 103 T + T^{2}$$
$83$ $$( 1092 + T )^{2}$$
$89$ $$108241 - 329 T + T^{2}$$
$97$ $$( 882 + T )^{2}$$