Properties

 Label 112.4.i.a 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) 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.

 $$f(q)$$ $$=$$ $$q + ( -5 + 5 \zeta_{6} ) q^{3} + 9 \zeta_{6} q^{5} + ( 21 - 14 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -5 + 5 \zeta_{6} ) q^{3} + 9 \zeta_{6} q^{5} + ( 21 - 14 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} + ( -57 + 57 \zeta_{6} ) q^{11} -70 q^{13} -45 q^{15} + ( -51 + 51 \zeta_{6} ) q^{17} + 5 \zeta_{6} q^{19} + ( -35 + 105 \zeta_{6} ) q^{21} + 69 \zeta_{6} q^{23} + ( 44 - 44 \zeta_{6} ) q^{25} -145 q^{27} + 114 q^{29} + ( 23 - 23 \zeta_{6} ) q^{31} -285 \zeta_{6} q^{33} + ( 126 + 63 \zeta_{6} ) q^{35} + 253 \zeta_{6} q^{37} + ( 350 - 350 \zeta_{6} ) q^{39} -42 q^{41} + 124 q^{43} + ( -18 + 18 \zeta_{6} ) q^{45} + 201 \zeta_{6} q^{47} + ( 245 - 392 \zeta_{6} ) q^{49} -255 \zeta_{6} q^{51} + ( 393 - 393 \zeta_{6} ) q^{53} -513 q^{55} -25 q^{57} + ( 219 - 219 \zeta_{6} ) q^{59} + 709 \zeta_{6} q^{61} + ( 28 + 14 \zeta_{6} ) q^{63} -630 \zeta_{6} q^{65} + ( 419 - 419 \zeta_{6} ) q^{67} -345 q^{69} + 96 q^{71} + ( 313 - 313 \zeta_{6} ) q^{73} + 220 \zeta_{6} q^{75} + ( -399 + 1197 \zeta_{6} ) q^{77} + 461 \zeta_{6} q^{79} + ( 671 - 671 \zeta_{6} ) q^{81} + 588 q^{83} -459 q^{85} + ( -570 + 570 \zeta_{6} ) q^{87} + 1017 \zeta_{6} q^{89} + ( -1470 + 980 \zeta_{6} ) q^{91} + 115 \zeta_{6} q^{93} + ( -45 + 45 \zeta_{6} ) q^{95} -1834 q^{97} -114 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 5q^{3} + 9q^{5} + 28q^{7} + 2q^{9} + O(q^{10})$$ $$2q - 5q^{3} + 9q^{5} + 28q^{7} + 2q^{9} - 57q^{11} - 140q^{13} - 90q^{15} - 51q^{17} + 5q^{19} + 35q^{21} + 69q^{23} + 44q^{25} - 290q^{27} + 228q^{29} + 23q^{31} - 285q^{33} + 315q^{35} + 253q^{37} + 350q^{39} - 84q^{41} + 248q^{43} - 18q^{45} + 201q^{47} + 98q^{49} - 255q^{51} + 393q^{53} - 1026q^{55} - 50q^{57} + 219q^{59} + 709q^{61} + 70q^{63} - 630q^{65} + 419q^{67} - 690q^{69} + 192q^{71} + 313q^{73} + 220q^{75} + 399q^{77} + 461q^{79} + 671q^{81} + 1176q^{83} - 918q^{85} - 570q^{87} + 1017q^{89} - 1960q^{91} + 115q^{93} - 45q^{95} - 3668q^{97} - 228q^{99} + O(q^{100})$$

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 −2.50000 4.33013i 0 4.50000 7.79423i 0 14.0000 + 12.1244i 0 1.00000 1.73205i 0
81.1 0 −2.50000 + 4.33013i 0 4.50000 + 7.79423i 0 14.0000 12.1244i 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
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.a 2
4.b odd 2 1 14.4.c.a 2
7.c even 3 1 inner 112.4.i.a 2
7.c even 3 1 784.4.a.p 1
7.d odd 6 1 784.4.a.c 1
8.b even 2 1 448.4.i.e 2
8.d odd 2 1 448.4.i.b 2
12.b even 2 1 126.4.g.d 2
20.d odd 2 1 350.4.e.e 2
20.e even 4 2 350.4.j.b 4
28.d even 2 1 98.4.c.a 2
28.f even 6 1 98.4.a.f 1
28.f even 6 1 98.4.c.a 2
28.g odd 6 1 14.4.c.a 2
28.g odd 6 1 98.4.a.d 1
56.k odd 6 1 448.4.i.b 2
56.p even 6 1 448.4.i.e 2
84.h odd 2 1 882.4.g.u 2
84.j odd 6 1 882.4.a.c 1
84.j odd 6 1 882.4.g.u 2
84.n even 6 1 126.4.g.d 2
84.n even 6 1 882.4.a.f 1
140.p odd 6 1 350.4.e.e 2
140.p odd 6 1 2450.4.a.q 1
140.s even 6 1 2450.4.a.d 1
140.w even 12 2 350.4.j.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a 2 4.b odd 2 1
14.4.c.a 2 28.g odd 6 1
98.4.a.d 1 28.g odd 6 1
98.4.a.f 1 28.f even 6 1
98.4.c.a 2 28.d even 2 1
98.4.c.a 2 28.f even 6 1
112.4.i.a 2 1.a even 1 1 trivial
112.4.i.a 2 7.c even 3 1 inner
126.4.g.d 2 12.b even 2 1
126.4.g.d 2 84.n even 6 1
350.4.e.e 2 20.d odd 2 1
350.4.e.e 2 140.p odd 6 1
350.4.j.b 4 20.e even 4 2
350.4.j.b 4 140.w even 12 2
448.4.i.b 2 8.d odd 2 1
448.4.i.b 2 56.k odd 6 1
448.4.i.e 2 8.b even 2 1
448.4.i.e 2 56.p even 6 1
784.4.a.c 1 7.d odd 6 1
784.4.a.p 1 7.c even 3 1
882.4.a.c 1 84.j odd 6 1
882.4.a.f 1 84.n even 6 1
882.4.g.u 2 84.h odd 2 1
882.4.g.u 2 84.j odd 6 1
2450.4.a.d 1 140.s even 6 1
2450.4.a.q 1 140.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 5 T_{3} + 25$$ acting on $$S_{4}^{\mathrm{new}}(112, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$25 + 5 T + T^{2}$$
$5$ $$81 - 9 T + T^{2}$$
$7$ $$343 - 28 T + T^{2}$$
$11$ $$3249 + 57 T + T^{2}$$
$13$ $$( 70 + T )^{2}$$
$17$ $$2601 + 51 T + T^{2}$$
$19$ $$25 - 5 T + T^{2}$$
$23$ $$4761 - 69 T + T^{2}$$
$29$ $$( -114 + T )^{2}$$
$31$ $$529 - 23 T + T^{2}$$
$37$ $$64009 - 253 T + T^{2}$$
$41$ $$( 42 + T )^{2}$$
$43$ $$( -124 + T )^{2}$$
$47$ $$40401 - 201 T + T^{2}$$
$53$ $$154449 - 393 T + T^{2}$$
$59$ $$47961 - 219 T + T^{2}$$
$61$ $$502681 - 709 T + T^{2}$$
$67$ $$175561 - 419 T + T^{2}$$
$71$ $$( -96 + T )^{2}$$
$73$ $$97969 - 313 T + T^{2}$$
$79$ $$212521 - 461 T + T^{2}$$
$83$ $$( -588 + T )^{2}$$
$89$ $$1034289 - 1017 T + T^{2}$$
$97$ $$( 1834 + T )^{2}$$