Properties

 Label 126.4.g.c Level $126$ Weight $4$ Character orbit 126.g Analytic conductor $7.434$ 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] = [126,4,Mod(37,126)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(126, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([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("126.37");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 126.g (of order $$3$$, degree $$2$$, 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: $$7.43424066072$$ 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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + ( - 7 \zeta_{6} + 7) q^{5} + (18 \zeta_{6} - 19) q^{7} + 8 q^{8} +O(q^{10})$$ q + (2*z - 2) * q^2 - 4*z * q^4 + (-7*z + 7) * q^5 + (18*z - 19) * q^7 + 8 * q^8 $$q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + ( - 7 \zeta_{6} + 7) q^{5} + (18 \zeta_{6} - 19) q^{7} + 8 q^{8} + 14 \zeta_{6} q^{10} + 35 \zeta_{6} q^{11} + 66 q^{13} + ( - 38 \zeta_{6} + 2) q^{14} + (16 \zeta_{6} - 16) q^{16} + 59 \zeta_{6} q^{17} + (137 \zeta_{6} - 137) q^{19} - 28 q^{20} - 70 q^{22} + (7 \zeta_{6} - 7) q^{23} + 76 \zeta_{6} q^{25} + (132 \zeta_{6} - 132) q^{26} + (4 \zeta_{6} + 72) q^{28} - 106 q^{29} - 75 \zeta_{6} q^{31} - 32 \zeta_{6} q^{32} - 118 q^{34} + (133 \zeta_{6} - 7) q^{35} + (11 \zeta_{6} - 11) q^{37} - 274 \zeta_{6} q^{38} + ( - 56 \zeta_{6} + 56) q^{40} + 498 q^{41} + 260 q^{43} + ( - 140 \zeta_{6} + 140) q^{44} - 14 \zeta_{6} q^{46} + (171 \zeta_{6} - 171) q^{47} + ( - 360 \zeta_{6} + 37) q^{49} - 152 q^{50} - 264 \zeta_{6} q^{52} - 417 \zeta_{6} q^{53} + 245 q^{55} + (144 \zeta_{6} - 152) q^{56} + ( - 212 \zeta_{6} + 212) q^{58} - 17 \zeta_{6} q^{59} + (51 \zeta_{6} - 51) q^{61} + 150 q^{62} + 64 q^{64} + ( - 462 \zeta_{6} + 462) q^{65} - 439 \zeta_{6} q^{67} + ( - 236 \zeta_{6} + 236) q^{68} + ( - 14 \zeta_{6} - 252) q^{70} + 784 q^{71} - 295 \zeta_{6} q^{73} - 22 \zeta_{6} q^{74} + 548 q^{76} + ( - 35 \zeta_{6} - 630) q^{77} + ( - 495 \zeta_{6} + 495) q^{79} + 112 \zeta_{6} q^{80} + (996 \zeta_{6} - 996) q^{82} - 932 q^{83} + 413 q^{85} + (520 \zeta_{6} - 520) q^{86} + 280 \zeta_{6} q^{88} + (873 \zeta_{6} - 873) q^{89} + (1188 \zeta_{6} - 1254) q^{91} + 28 q^{92} - 342 \zeta_{6} q^{94} + 959 \zeta_{6} q^{95} - 290 q^{97} + (74 \zeta_{6} + 646) q^{98} +O(q^{100})$$ q + (2*z - 2) * q^2 - 4*z * q^4 + (-7*z + 7) * q^5 + (18*z - 19) * q^7 + 8 * q^8 + 14*z * q^10 + 35*z * q^11 + 66 * q^13 + (-38*z + 2) * q^14 + (16*z - 16) * q^16 + 59*z * q^17 + (137*z - 137) * q^19 - 28 * q^20 - 70 * q^22 + (7*z - 7) * q^23 + 76*z * q^25 + (132*z - 132) * q^26 + (4*z + 72) * q^28 - 106 * q^29 - 75*z * q^31 - 32*z * q^32 - 118 * q^34 + (133*z - 7) * q^35 + (11*z - 11) * q^37 - 274*z * q^38 + (-56*z + 56) * q^40 + 498 * q^41 + 260 * q^43 + (-140*z + 140) * q^44 - 14*z * q^46 + (171*z - 171) * q^47 + (-360*z + 37) * q^49 - 152 * q^50 - 264*z * q^52 - 417*z * q^53 + 245 * q^55 + (144*z - 152) * q^56 + (-212*z + 212) * q^58 - 17*z * q^59 + (51*z - 51) * q^61 + 150 * q^62 + 64 * q^64 + (-462*z + 462) * q^65 - 439*z * q^67 + (-236*z + 236) * q^68 + (-14*z - 252) * q^70 + 784 * q^71 - 295*z * q^73 - 22*z * q^74 + 548 * q^76 + (-35*z - 630) * q^77 + (-495*z + 495) * q^79 + 112*z * q^80 + (996*z - 996) * q^82 - 932 * q^83 + 413 * q^85 + (520*z - 520) * q^86 + 280*z * q^88 + (873*z - 873) * q^89 + (1188*z - 1254) * q^91 + 28 * q^92 - 342*z * q^94 + 959*z * q^95 - 290 * q^97 + (74*z + 646) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{4} + 7 q^{5} - 20 q^{7} + 16 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^4 + 7 * q^5 - 20 * q^7 + 16 * q^8 $$2 q - 2 q^{2} - 4 q^{4} + 7 q^{5} - 20 q^{7} + 16 q^{8} + 14 q^{10} + 35 q^{11} + 132 q^{13} - 34 q^{14} - 16 q^{16} + 59 q^{17} - 137 q^{19} - 56 q^{20} - 140 q^{22} - 7 q^{23} + 76 q^{25} - 132 q^{26} + 148 q^{28} - 212 q^{29} - 75 q^{31} - 32 q^{32} - 236 q^{34} + 119 q^{35} - 11 q^{37} - 274 q^{38} + 56 q^{40} + 996 q^{41} + 520 q^{43} + 140 q^{44} - 14 q^{46} - 171 q^{47} - 286 q^{49} - 304 q^{50} - 264 q^{52} - 417 q^{53} + 490 q^{55} - 160 q^{56} + 212 q^{58} - 17 q^{59} - 51 q^{61} + 300 q^{62} + 128 q^{64} + 462 q^{65} - 439 q^{67} + 236 q^{68} - 518 q^{70} + 1568 q^{71} - 295 q^{73} - 22 q^{74} + 1096 q^{76} - 1295 q^{77} + 495 q^{79} + 112 q^{80} - 996 q^{82} - 1864 q^{83} + 826 q^{85} - 520 q^{86} + 280 q^{88} - 873 q^{89} - 1320 q^{91} + 56 q^{92} - 342 q^{94} + 959 q^{95} - 580 q^{97} + 1366 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^4 + 7 * q^5 - 20 * q^7 + 16 * q^8 + 14 * q^10 + 35 * q^11 + 132 * q^13 - 34 * q^14 - 16 * q^16 + 59 * q^17 - 137 * q^19 - 56 * q^20 - 140 * q^22 - 7 * q^23 + 76 * q^25 - 132 * q^26 + 148 * q^28 - 212 * q^29 - 75 * q^31 - 32 * q^32 - 236 * q^34 + 119 * q^35 - 11 * q^37 - 274 * q^38 + 56 * q^40 + 996 * q^41 + 520 * q^43 + 140 * q^44 - 14 * q^46 - 171 * q^47 - 286 * q^49 - 304 * q^50 - 264 * q^52 - 417 * q^53 + 490 * q^55 - 160 * q^56 + 212 * q^58 - 17 * q^59 - 51 * q^61 + 300 * q^62 + 128 * q^64 + 462 * q^65 - 439 * q^67 + 236 * q^68 - 518 * q^70 + 1568 * q^71 - 295 * q^73 - 22 * q^74 + 1096 * q^76 - 1295 * q^77 + 495 * q^79 + 112 * q^80 - 996 * q^82 - 1864 * q^83 + 826 * q^85 - 520 * q^86 + 280 * q^88 - 873 * q^89 - 1320 * q^91 + 56 * q^92 - 342 * q^94 + 959 * q^95 - 580 * q^97 + 1366 * q^98

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/126\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$\chi(n)$$ $$1$$ $$-1 + \zeta_{6}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 + 1.73205i 0 −2.00000 3.46410i 3.50000 6.06218i 0 −10.0000 + 15.5885i 8.00000 0 7.00000 + 12.1244i
109.1 −1.00000 1.73205i 0 −2.00000 + 3.46410i 3.50000 + 6.06218i 0 −10.0000 15.5885i 8.00000 0 7.00000 12.1244i
 $$n$$: 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 126.4.g.c 2
3.b odd 2 1 14.4.c.b 2
7.b odd 2 1 882.4.g.d 2
7.c even 3 1 inner 126.4.g.c 2
7.c even 3 1 882.4.a.k 1
7.d odd 6 1 882.4.a.p 1
7.d odd 6 1 882.4.g.d 2
12.b even 2 1 112.4.i.b 2
15.d odd 2 1 350.4.e.b 2
15.e even 4 2 350.4.j.d 4
21.c even 2 1 98.4.c.e 2
21.g even 6 1 98.4.a.c 1
21.g even 6 1 98.4.c.e 2
21.h odd 6 1 14.4.c.b 2
21.h odd 6 1 98.4.a.b 1
24.f even 2 1 448.4.i.d 2
24.h odd 2 1 448.4.i.c 2
84.j odd 6 1 784.4.a.j 1
84.n even 6 1 112.4.i.b 2
84.n even 6 1 784.4.a.l 1
105.o odd 6 1 350.4.e.b 2
105.o odd 6 1 2450.4.a.bh 1
105.p even 6 1 2450.4.a.bf 1
105.x even 12 2 350.4.j.d 4
168.s odd 6 1 448.4.i.c 2
168.v even 6 1 448.4.i.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.b 2 3.b odd 2 1
14.4.c.b 2 21.h odd 6 1
98.4.a.b 1 21.h odd 6 1
98.4.a.c 1 21.g even 6 1
98.4.c.e 2 21.c even 2 1
98.4.c.e 2 21.g even 6 1
112.4.i.b 2 12.b even 2 1
112.4.i.b 2 84.n even 6 1
126.4.g.c 2 1.a even 1 1 trivial
126.4.g.c 2 7.c even 3 1 inner
350.4.e.b 2 15.d odd 2 1
350.4.e.b 2 105.o odd 6 1
350.4.j.d 4 15.e even 4 2
350.4.j.d 4 105.x even 12 2
448.4.i.c 2 24.h odd 2 1
448.4.i.c 2 168.s odd 6 1
448.4.i.d 2 24.f even 2 1
448.4.i.d 2 168.v even 6 1
784.4.a.j 1 84.j odd 6 1
784.4.a.l 1 84.n even 6 1
882.4.a.k 1 7.c even 3 1
882.4.a.p 1 7.d odd 6 1
882.4.g.d 2 7.b odd 2 1
882.4.g.d 2 7.d odd 6 1
2450.4.a.bf 1 105.p even 6 1
2450.4.a.bh 1 105.o odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 7T_{5} + 49$$ acting on $$S_{4}^{\mathrm{new}}(126, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 7T + 49$$
$7$ $$T^{2} + 20T + 343$$
$11$ $$T^{2} - 35T + 1225$$
$13$ $$(T - 66)^{2}$$
$17$ $$T^{2} - 59T + 3481$$
$19$ $$T^{2} + 137T + 18769$$
$23$ $$T^{2} + 7T + 49$$
$29$ $$(T + 106)^{2}$$
$31$ $$T^{2} + 75T + 5625$$
$37$ $$T^{2} + 11T + 121$$
$41$ $$(T - 498)^{2}$$
$43$ $$(T - 260)^{2}$$
$47$ $$T^{2} + 171T + 29241$$
$53$ $$T^{2} + 417T + 173889$$
$59$ $$T^{2} + 17T + 289$$
$61$ $$T^{2} + 51T + 2601$$
$67$ $$T^{2} + 439T + 192721$$
$71$ $$(T - 784)^{2}$$
$73$ $$T^{2} + 295T + 87025$$
$79$ $$T^{2} - 495T + 245025$$
$83$ $$(T + 932)^{2}$$
$89$ $$T^{2} + 873T + 762129$$
$97$ $$(T + 290)^{2}$$