# Properties

 Label 126.4.g.b 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) 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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + (6 \zeta_{6} - 6) q^{5} + ( - 21 \zeta_{6} + 7) q^{7} + 8 q^{8} +O(q^{10})$$ q + (2*z - 2) * q^2 - 4*z * q^4 + (6*z - 6) * q^5 + (-21*z + 7) * q^7 + 8 * q^8 $$q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + (6 \zeta_{6} - 6) q^{5} + ( - 21 \zeta_{6} + 7) q^{7} + 8 q^{8} - 12 \zeta_{6} q^{10} - 30 \zeta_{6} q^{11} + 53 q^{13} + (14 \zeta_{6} + 28) q^{14} + (16 \zeta_{6} - 16) q^{16} - 84 \zeta_{6} q^{17} + ( - 97 \zeta_{6} + 97) q^{19} + 24 q^{20} + 60 q^{22} + ( - 84 \zeta_{6} + 84) q^{23} + 89 \zeta_{6} q^{25} + (106 \zeta_{6} - 106) q^{26} + (56 \zeta_{6} - 84) q^{28} + 180 q^{29} - 179 \zeta_{6} q^{31} - 32 \zeta_{6} q^{32} + 168 q^{34} + (42 \zeta_{6} + 84) q^{35} + ( - 145 \zeta_{6} + 145) q^{37} + 194 \zeta_{6} q^{38} + (48 \zeta_{6} - 48) q^{40} - 126 q^{41} - 325 q^{43} + (120 \zeta_{6} - 120) q^{44} + 168 \zeta_{6} q^{46} + (366 \zeta_{6} - 366) q^{47} + (147 \zeta_{6} - 392) q^{49} - 178 q^{50} - 212 \zeta_{6} q^{52} - 768 \zeta_{6} q^{53} + 180 q^{55} + ( - 168 \zeta_{6} + 56) q^{56} + (360 \zeta_{6} - 360) q^{58} - 264 \zeta_{6} q^{59} + (818 \zeta_{6} - 818) q^{61} + 358 q^{62} + 64 q^{64} + (318 \zeta_{6} - 318) q^{65} + 523 \zeta_{6} q^{67} + (336 \zeta_{6} - 336) q^{68} + (168 \zeta_{6} - 252) q^{70} + 342 q^{71} + 43 \zeta_{6} q^{73} + 290 \zeta_{6} q^{74} - 388 q^{76} + (420 \zeta_{6} - 630) q^{77} + ( - 1171 \zeta_{6} + 1171) q^{79} - 96 \zeta_{6} q^{80} + ( - 252 \zeta_{6} + 252) q^{82} + 810 q^{83} + 504 q^{85} + ( - 650 \zeta_{6} + 650) q^{86} - 240 \zeta_{6} q^{88} + (600 \zeta_{6} - 600) q^{89} + ( - 1113 \zeta_{6} + 371) q^{91} - 336 q^{92} - 732 \zeta_{6} q^{94} + 582 \zeta_{6} q^{95} + 386 q^{97} + ( - 784 \zeta_{6} + 490) q^{98} +O(q^{100})$$ q + (2*z - 2) * q^2 - 4*z * q^4 + (6*z - 6) * q^5 + (-21*z + 7) * q^7 + 8 * q^8 - 12*z * q^10 - 30*z * q^11 + 53 * q^13 + (14*z + 28) * q^14 + (16*z - 16) * q^16 - 84*z * q^17 + (-97*z + 97) * q^19 + 24 * q^20 + 60 * q^22 + (-84*z + 84) * q^23 + 89*z * q^25 + (106*z - 106) * q^26 + (56*z - 84) * q^28 + 180 * q^29 - 179*z * q^31 - 32*z * q^32 + 168 * q^34 + (42*z + 84) * q^35 + (-145*z + 145) * q^37 + 194*z * q^38 + (48*z - 48) * q^40 - 126 * q^41 - 325 * q^43 + (120*z - 120) * q^44 + 168*z * q^46 + (366*z - 366) * q^47 + (147*z - 392) * q^49 - 178 * q^50 - 212*z * q^52 - 768*z * q^53 + 180 * q^55 + (-168*z + 56) * q^56 + (360*z - 360) * q^58 - 264*z * q^59 + (818*z - 818) * q^61 + 358 * q^62 + 64 * q^64 + (318*z - 318) * q^65 + 523*z * q^67 + (336*z - 336) * q^68 + (168*z - 252) * q^70 + 342 * q^71 + 43*z * q^73 + 290*z * q^74 - 388 * q^76 + (420*z - 630) * q^77 + (-1171*z + 1171) * q^79 - 96*z * q^80 + (-252*z + 252) * q^82 + 810 * q^83 + 504 * q^85 + (-650*z + 650) * q^86 - 240*z * q^88 + (600*z - 600) * q^89 + (-1113*z + 371) * q^91 - 336 * q^92 - 732*z * q^94 + 582*z * q^95 + 386 * q^97 + (-784*z + 490) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{4} - 6 q^{5} - 7 q^{7} + 16 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^4 - 6 * q^5 - 7 * q^7 + 16 * q^8 $$2 q - 2 q^{2} - 4 q^{4} - 6 q^{5} - 7 q^{7} + 16 q^{8} - 12 q^{10} - 30 q^{11} + 106 q^{13} + 70 q^{14} - 16 q^{16} - 84 q^{17} + 97 q^{19} + 48 q^{20} + 120 q^{22} + 84 q^{23} + 89 q^{25} - 106 q^{26} - 112 q^{28} + 360 q^{29} - 179 q^{31} - 32 q^{32} + 336 q^{34} + 210 q^{35} + 145 q^{37} + 194 q^{38} - 48 q^{40} - 252 q^{41} - 650 q^{43} - 120 q^{44} + 168 q^{46} - 366 q^{47} - 637 q^{49} - 356 q^{50} - 212 q^{52} - 768 q^{53} + 360 q^{55} - 56 q^{56} - 360 q^{58} - 264 q^{59} - 818 q^{61} + 716 q^{62} + 128 q^{64} - 318 q^{65} + 523 q^{67} - 336 q^{68} - 336 q^{70} + 684 q^{71} + 43 q^{73} + 290 q^{74} - 776 q^{76} - 840 q^{77} + 1171 q^{79} - 96 q^{80} + 252 q^{82} + 1620 q^{83} + 1008 q^{85} + 650 q^{86} - 240 q^{88} - 600 q^{89} - 371 q^{91} - 672 q^{92} - 732 q^{94} + 582 q^{95} + 772 q^{97} + 196 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^4 - 6 * q^5 - 7 * q^7 + 16 * q^8 - 12 * q^10 - 30 * q^11 + 106 * q^13 + 70 * q^14 - 16 * q^16 - 84 * q^17 + 97 * q^19 + 48 * q^20 + 120 * q^22 + 84 * q^23 + 89 * q^25 - 106 * q^26 - 112 * q^28 + 360 * q^29 - 179 * q^31 - 32 * q^32 + 336 * q^34 + 210 * q^35 + 145 * q^37 + 194 * q^38 - 48 * q^40 - 252 * q^41 - 650 * q^43 - 120 * q^44 + 168 * q^46 - 366 * q^47 - 637 * q^49 - 356 * q^50 - 212 * q^52 - 768 * q^53 + 360 * q^55 - 56 * q^56 - 360 * q^58 - 264 * q^59 - 818 * q^61 + 716 * q^62 + 128 * q^64 - 318 * q^65 + 523 * q^67 - 336 * q^68 - 336 * q^70 + 684 * q^71 + 43 * q^73 + 290 * q^74 - 776 * q^76 - 840 * q^77 + 1171 * q^79 - 96 * q^80 + 252 * q^82 + 1620 * q^83 + 1008 * q^85 + 650 * q^86 - 240 * q^88 - 600 * q^89 - 371 * q^91 - 672 * q^92 - 732 * q^94 + 582 * q^95 + 772 * q^97 + 196 * 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 −3.00000 + 5.19615i 0 −3.50000 18.1865i 8.00000 0 −6.00000 10.3923i
109.1 −1.00000 1.73205i 0 −2.00000 + 3.46410i −3.00000 5.19615i 0 −3.50000 + 18.1865i 8.00000 0 −6.00000 + 10.3923i
 $$n$$: 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.b 2
3.b odd 2 1 42.4.e.a 2
7.b odd 2 1 882.4.g.g 2
7.c even 3 1 inner 126.4.g.b 2
7.c even 3 1 882.4.a.o 1
7.d odd 6 1 882.4.a.l 1
7.d odd 6 1 882.4.g.g 2
12.b even 2 1 336.4.q.f 2
21.c even 2 1 294.4.e.i 2
21.g even 6 1 294.4.a.c 1
21.g even 6 1 294.4.e.i 2
21.h odd 6 1 42.4.e.a 2
21.h odd 6 1 294.4.a.d 1
84.j odd 6 1 2352.4.a.bf 1
84.n even 6 1 336.4.q.f 2
84.n even 6 1 2352.4.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.a 2 3.b odd 2 1
42.4.e.a 2 21.h odd 6 1
126.4.g.b 2 1.a even 1 1 trivial
126.4.g.b 2 7.c even 3 1 inner
294.4.a.c 1 21.g even 6 1
294.4.a.d 1 21.h odd 6 1
294.4.e.i 2 21.c even 2 1
294.4.e.i 2 21.g even 6 1
336.4.q.f 2 12.b even 2 1
336.4.q.f 2 84.n even 6 1
882.4.a.l 1 7.d odd 6 1
882.4.a.o 1 7.c even 3 1
882.4.g.g 2 7.b odd 2 1
882.4.g.g 2 7.d odd 6 1
2352.4.a.f 1 84.n even 6 1
2352.4.a.bf 1 84.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 6T_{5} + 36$$ acting on $$S_{4}^{\mathrm{new}}(126, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 6T + 36$$
$7$ $$T^{2} + 7T + 343$$
$11$ $$T^{2} + 30T + 900$$
$13$ $$(T - 53)^{2}$$
$17$ $$T^{2} + 84T + 7056$$
$19$ $$T^{2} - 97T + 9409$$
$23$ $$T^{2} - 84T + 7056$$
$29$ $$(T - 180)^{2}$$
$31$ $$T^{2} + 179T + 32041$$
$37$ $$T^{2} - 145T + 21025$$
$41$ $$(T + 126)^{2}$$
$43$ $$(T + 325)^{2}$$
$47$ $$T^{2} + 366T + 133956$$
$53$ $$T^{2} + 768T + 589824$$
$59$ $$T^{2} + 264T + 69696$$
$61$ $$T^{2} + 818T + 669124$$
$67$ $$T^{2} - 523T + 273529$$
$71$ $$(T - 342)^{2}$$
$73$ $$T^{2} - 43T + 1849$$
$79$ $$T^{2} - 1171 T + 1371241$$
$83$ $$(T - 810)^{2}$$
$89$ $$T^{2} + 600T + 360000$$
$97$ $$(T - 386)^{2}$$