# 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,4,Mod(65,112)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("112.65");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 112.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$6.60821392064$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 $$T_{3}^{2} + 5T_{3} + 25$$ acting on $$S_{4}^{\mathrm{new}}(112, [\chi])$$.

## Hecke characteristic polynomials

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