# Properties

 Label 126.4.g.a 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 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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + (15 \zeta_{6} - 15) q^{5} + (7 \zeta_{6} + 14) q^{7} + 8 q^{8} +O(q^{10})$$ q + (2*z - 2) * q^2 - 4*z * q^4 + (15*z - 15) * q^5 + (7*z + 14) * q^7 + 8 * q^8 $$q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + (15 \zeta_{6} - 15) q^{5} + (7 \zeta_{6} + 14) q^{7} + 8 q^{8} - 30 \zeta_{6} q^{10} - 9 \zeta_{6} q^{11} - 88 q^{13} + (28 \zeta_{6} - 42) q^{14} + (16 \zeta_{6} - 16) q^{16} - 84 \zeta_{6} q^{17} + (104 \zeta_{6} - 104) q^{19} + 60 q^{20} + 18 q^{22} + (84 \zeta_{6} - 84) q^{23} - 100 \zeta_{6} q^{25} + ( - 176 \zeta_{6} + 176) q^{26} + ( - 84 \zeta_{6} + 28) q^{28} - 51 q^{29} - 185 \zeta_{6} q^{31} - 32 \zeta_{6} q^{32} + 168 q^{34} + (210 \zeta_{6} - 315) q^{35} + (44 \zeta_{6} - 44) q^{37} - 208 \zeta_{6} q^{38} + (120 \zeta_{6} - 120) q^{40} + 168 q^{41} + 326 q^{43} + (36 \zeta_{6} - 36) q^{44} - 168 \zeta_{6} q^{46} + (138 \zeta_{6} - 138) q^{47} + (245 \zeta_{6} + 147) q^{49} + 200 q^{50} + 352 \zeta_{6} q^{52} + 639 \zeta_{6} q^{53} + 135 q^{55} + (56 \zeta_{6} + 112) q^{56} + ( - 102 \zeta_{6} + 102) q^{58} + 159 \zeta_{6} q^{59} + (722 \zeta_{6} - 722) q^{61} + 370 q^{62} + 64 q^{64} + ( - 1320 \zeta_{6} + 1320) q^{65} + 166 \zeta_{6} q^{67} + (336 \zeta_{6} - 336) q^{68} + ( - 630 \zeta_{6} + 210) q^{70} - 1086 q^{71} - 218 \zeta_{6} q^{73} - 88 \zeta_{6} q^{74} + 416 q^{76} + ( - 189 \zeta_{6} + 63) q^{77} + ( - 583 \zeta_{6} + 583) q^{79} - 240 \zeta_{6} q^{80} + (336 \zeta_{6} - 336) q^{82} + 597 q^{83} + 1260 q^{85} + (652 \zeta_{6} - 652) q^{86} - 72 \zeta_{6} q^{88} + (1038 \zeta_{6} - 1038) q^{89} + ( - 616 \zeta_{6} - 1232) q^{91} + 336 q^{92} - 276 \zeta_{6} q^{94} - 1560 \zeta_{6} q^{95} - 169 q^{97} + (294 \zeta_{6} - 784) q^{98} +O(q^{100})$$ q + (2*z - 2) * q^2 - 4*z * q^4 + (15*z - 15) * q^5 + (7*z + 14) * q^7 + 8 * q^8 - 30*z * q^10 - 9*z * q^11 - 88 * q^13 + (28*z - 42) * q^14 + (16*z - 16) * q^16 - 84*z * q^17 + (104*z - 104) * q^19 + 60 * q^20 + 18 * q^22 + (84*z - 84) * q^23 - 100*z * q^25 + (-176*z + 176) * q^26 + (-84*z + 28) * q^28 - 51 * q^29 - 185*z * q^31 - 32*z * q^32 + 168 * q^34 + (210*z - 315) * q^35 + (44*z - 44) * q^37 - 208*z * q^38 + (120*z - 120) * q^40 + 168 * q^41 + 326 * q^43 + (36*z - 36) * q^44 - 168*z * q^46 + (138*z - 138) * q^47 + (245*z + 147) * q^49 + 200 * q^50 + 352*z * q^52 + 639*z * q^53 + 135 * q^55 + (56*z + 112) * q^56 + (-102*z + 102) * q^58 + 159*z * q^59 + (722*z - 722) * q^61 + 370 * q^62 + 64 * q^64 + (-1320*z + 1320) * q^65 + 166*z * q^67 + (336*z - 336) * q^68 + (-630*z + 210) * q^70 - 1086 * q^71 - 218*z * q^73 - 88*z * q^74 + 416 * q^76 + (-189*z + 63) * q^77 + (-583*z + 583) * q^79 - 240*z * q^80 + (336*z - 336) * q^82 + 597 * q^83 + 1260 * q^85 + (652*z - 652) * q^86 - 72*z * q^88 + (1038*z - 1038) * q^89 + (-616*z - 1232) * q^91 + 336 * q^92 - 276*z * q^94 - 1560*z * q^95 - 169 * q^97 + (294*z - 784) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{4} - 15 q^{5} + 35 q^{7} + 16 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^4 - 15 * q^5 + 35 * q^7 + 16 * q^8 $$2 q - 2 q^{2} - 4 q^{4} - 15 q^{5} + 35 q^{7} + 16 q^{8} - 30 q^{10} - 9 q^{11} - 176 q^{13} - 56 q^{14} - 16 q^{16} - 84 q^{17} - 104 q^{19} + 120 q^{20} + 36 q^{22} - 84 q^{23} - 100 q^{25} + 176 q^{26} - 28 q^{28} - 102 q^{29} - 185 q^{31} - 32 q^{32} + 336 q^{34} - 420 q^{35} - 44 q^{37} - 208 q^{38} - 120 q^{40} + 336 q^{41} + 652 q^{43} - 36 q^{44} - 168 q^{46} - 138 q^{47} + 539 q^{49} + 400 q^{50} + 352 q^{52} + 639 q^{53} + 270 q^{55} + 280 q^{56} + 102 q^{58} + 159 q^{59} - 722 q^{61} + 740 q^{62} + 128 q^{64} + 1320 q^{65} + 166 q^{67} - 336 q^{68} - 210 q^{70} - 2172 q^{71} - 218 q^{73} - 88 q^{74} + 832 q^{76} - 63 q^{77} + 583 q^{79} - 240 q^{80} - 336 q^{82} + 1194 q^{83} + 2520 q^{85} - 652 q^{86} - 72 q^{88} - 1038 q^{89} - 3080 q^{91} + 672 q^{92} - 276 q^{94} - 1560 q^{95} - 338 q^{97} - 1274 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^4 - 15 * q^5 + 35 * q^7 + 16 * q^8 - 30 * q^10 - 9 * q^11 - 176 * q^13 - 56 * q^14 - 16 * q^16 - 84 * q^17 - 104 * q^19 + 120 * q^20 + 36 * q^22 - 84 * q^23 - 100 * q^25 + 176 * q^26 - 28 * q^28 - 102 * q^29 - 185 * q^31 - 32 * q^32 + 336 * q^34 - 420 * q^35 - 44 * q^37 - 208 * q^38 - 120 * q^40 + 336 * q^41 + 652 * q^43 - 36 * q^44 - 168 * q^46 - 138 * q^47 + 539 * q^49 + 400 * q^50 + 352 * q^52 + 639 * q^53 + 270 * q^55 + 280 * q^56 + 102 * q^58 + 159 * q^59 - 722 * q^61 + 740 * q^62 + 128 * q^64 + 1320 * q^65 + 166 * q^67 - 336 * q^68 - 210 * q^70 - 2172 * q^71 - 218 * q^73 - 88 * q^74 + 832 * q^76 - 63 * q^77 + 583 * q^79 - 240 * q^80 - 336 * q^82 + 1194 * q^83 + 2520 * q^85 - 652 * q^86 - 72 * q^88 - 1038 * q^89 - 3080 * q^91 + 672 * q^92 - 276 * q^94 - 1560 * q^95 - 338 * q^97 - 1274 * 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 −7.50000 + 12.9904i 0 17.5000 + 6.06218i 8.00000 0 −15.0000 25.9808i
109.1 −1.00000 1.73205i 0 −2.00000 + 3.46410i −7.50000 12.9904i 0 17.5000 6.06218i 8.00000 0 −15.0000 + 25.9808i
 $$n$$: 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.a 2
3.b odd 2 1 42.4.e.b 2
7.b odd 2 1 882.4.g.l 2
7.c even 3 1 inner 126.4.g.a 2
7.c even 3 1 882.4.a.r 1
7.d odd 6 1 882.4.a.h 1
7.d odd 6 1 882.4.g.l 2
12.b even 2 1 336.4.q.d 2
21.c even 2 1 294.4.e.e 2
21.g even 6 1 294.4.a.g 1
21.g even 6 1 294.4.e.e 2
21.h odd 6 1 42.4.e.b 2
21.h odd 6 1 294.4.a.a 1
84.j odd 6 1 2352.4.a.q 1
84.n even 6 1 336.4.q.d 2
84.n even 6 1 2352.4.a.u 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 15T_{5} + 225$$ acting on $$S_{4}^{\mathrm{new}}(126, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 15T + 225$$
$7$ $$T^{2} - 35T + 343$$
$11$ $$T^{2} + 9T + 81$$
$13$ $$(T + 88)^{2}$$
$17$ $$T^{2} + 84T + 7056$$
$19$ $$T^{2} + 104T + 10816$$
$23$ $$T^{2} + 84T + 7056$$
$29$ $$(T + 51)^{2}$$
$31$ $$T^{2} + 185T + 34225$$
$37$ $$T^{2} + 44T + 1936$$
$41$ $$(T - 168)^{2}$$
$43$ $$(T - 326)^{2}$$
$47$ $$T^{2} + 138T + 19044$$
$53$ $$T^{2} - 639T + 408321$$
$59$ $$T^{2} - 159T + 25281$$
$61$ $$T^{2} + 722T + 521284$$
$67$ $$T^{2} - 166T + 27556$$
$71$ $$(T + 1086)^{2}$$
$73$ $$T^{2} + 218T + 47524$$
$79$ $$T^{2} - 583T + 339889$$
$83$ $$(T - 597)^{2}$$
$89$ $$T^{2} + 1038 T + 1077444$$
$97$ $$(T + 169)^{2}$$