# Properties

 Label 126.4.g.f Level $126$ Weight $4$ Character orbit 126.g Analytic conductor $7.434$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{193})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 49x^{2} + 48x + 2304$$ x^4 - x^3 + 49*x^2 + 48*x + 2304 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta_{2} q^{2} + (4 \beta_{2} - 4) q^{4} + ( - 3 \beta_{2} - \beta_1) q^{5} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{7} - 8 q^{8}+O(q^{10})$$ q + 2*b2 * q^2 + (4*b2 - 4) * q^4 + (-3*b2 - b1) * q^5 + (b3 - b2 - 2*b1 + 2) * q^7 - 8 * q^8 $$q + 2 \beta_{2} q^{2} + (4 \beta_{2} - 4) q^{4} + ( - 3 \beta_{2} - \beta_1) q^{5} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{7} - 8 q^{8} + ( - 2 \beta_{3} - 6 \beta_{2} - 2 \beta_1 + 8) q^{10} + ( - 7 \beta_{3} - 9 \beta_{2} - 7 \beta_1 + 16) q^{11} + (5 \beta_{3} - 27) q^{13} + ( - 4 \beta_{3} + 4 \beta_{2} - 6 \beta_1 + 6) q^{14} - 16 \beta_{2} q^{16} + ( - 10 \beta_{3} + 54 \beta_{2} - 10 \beta_1 - 44) q^{17} + (58 \beta_{2} + 3 \beta_1) q^{19} + ( - 4 \beta_{3} + 16) q^{20} + ( - 14 \beta_{3} + 32) q^{22} + ( - 54 \beta_{2} - 14 \beta_1) q^{23} + (7 \beta_{3} - 68 \beta_{2} + 7 \beta_1 + 61) q^{25} + ( - 44 \beta_{2} - 10 \beta_1) q^{26} + ( - 12 \beta_{3} + 12 \beta_{2} - 4 \beta_1 + 4) q^{28} + ( - 7 \beta_{3} + 40) q^{29} + (28 \beta_{3} + 35 \beta_{2} + 28 \beta_1 - 63) q^{31} + ( - 32 \beta_{2} + 32) q^{32} + ( - 20 \beta_{3} - 88) q^{34} + (9 \beta_{3} + 138 \beta_{2} + 10 \beta_1 - 108) q^{35} + ( - 134 \beta_{2} - 21 \beta_1) q^{37} + (6 \beta_{3} + 116 \beta_{2} + 6 \beta_1 - 122) q^{38} + (24 \beta_{2} + 8 \beta_1) q^{40} + 168 q^{41} + (21 \beta_{3} + 143) q^{43} + (36 \beta_{2} + 28 \beta_1) q^{44} + ( - 28 \beta_{3} - 108 \beta_{2} - 28 \beta_1 + 136) q^{46} + ( - 348 \beta_{2} + 24 \beta_1) q^{47} + (13 \beta_{3} + 379 \beta_{2} + 2 \beta_1 - 149) q^{49} + (14 \beta_{3} + 122) q^{50} + ( - 20 \beta_{3} - 88 \beta_{2} - 20 \beta_1 + 108) q^{52} + (63 \beta_{3} - 219 \beta_{2} + 63 \beta_1 + 156) q^{53} + (37 \beta_{3} - 400) q^{55} + ( - 8 \beta_{3} + 8 \beta_{2} + 16 \beta_1 - 16) q^{56} + (66 \beta_{2} + 14 \beta_1) q^{58} + (61 \beta_{3} + 351 \beta_{2} + 61 \beta_1 - 412) q^{59} + (220 \beta_{2} - 34 \beta_1) q^{61} + (56 \beta_{3} - 126) q^{62} + 64 q^{64} + (306 \beta_{2} + 42 \beta_1) q^{65} + (35 \beta_{3} - 538 \beta_{2} + 35 \beta_1 + 503) q^{67} + ( - 216 \beta_{2} + 40 \beta_1) q^{68} + (20 \beta_{3} + 78 \beta_{2} + 2 \beta_1 - 296) q^{70} + (28 \beta_{3} + 812) q^{71} + ( - 97 \beta_{3} - 46 \beta_{2} - 97 \beta_1 + 143) q^{73} + ( - 42 \beta_{3} - 268 \beta_{2} - 42 \beta_1 + 310) q^{74} + (12 \beta_{3} - 244) q^{76} + (34 \beta_{3} + 309 \beta_{2} - 5 \beta_1 - 1024) q^{77} + ( - 311 \beta_{2} + 98 \beta_1) q^{79} + (16 \beta_{3} + 48 \beta_{2} + 16 \beta_1 - 64) q^{80} + 336 \beta_{2} q^{82} + ( - 89 \beta_{3} + 188) q^{83} + ( - 14 \beta_{3} - 304) q^{85} + (328 \beta_{2} - 42 \beta_1) q^{86} + (56 \beta_{3} + 72 \beta_{2} + 56 \beta_1 - 128) q^{88} + ( - 1170 \beta_{2} - 54 \beta_1) q^{89} + ( - 12 \beta_{3} + 502 \beta_{2} + 59 \beta_1 + 186) q^{91} + ( - 56 \beta_{3} + 272) q^{92} + (48 \beta_{3} - 696 \beta_{2} + 48 \beta_1 + 648) q^{94} + ( - 70 \beta_{3} - 318 \beta_{2} - 70 \beta_1 + 388) q^{95} + ( - 155 \beta_{3} + 46) q^{97} + (4 \beta_{3} + 486 \beta_{2} - 22 \beta_1 - 762) q^{98}+O(q^{100})$$ q + 2*b2 * q^2 + (4*b2 - 4) * q^4 + (-3*b2 - b1) * q^5 + (b3 - b2 - 2*b1 + 2) * q^7 - 8 * q^8 + (-2*b3 - 6*b2 - 2*b1 + 8) * q^10 + (-7*b3 - 9*b2 - 7*b1 + 16) * q^11 + (5*b3 - 27) * q^13 + (-4*b3 + 4*b2 - 6*b1 + 6) * q^14 - 16*b2 * q^16 + (-10*b3 + 54*b2 - 10*b1 - 44) * q^17 + (58*b2 + 3*b1) * q^19 + (-4*b3 + 16) * q^20 + (-14*b3 + 32) * q^22 + (-54*b2 - 14*b1) * q^23 + (7*b3 - 68*b2 + 7*b1 + 61) * q^25 + (-44*b2 - 10*b1) * q^26 + (-12*b3 + 12*b2 - 4*b1 + 4) * q^28 + (-7*b3 + 40) * q^29 + (28*b3 + 35*b2 + 28*b1 - 63) * q^31 + (-32*b2 + 32) * q^32 + (-20*b3 - 88) * q^34 + (9*b3 + 138*b2 + 10*b1 - 108) * q^35 + (-134*b2 - 21*b1) * q^37 + (6*b3 + 116*b2 + 6*b1 - 122) * q^38 + (24*b2 + 8*b1) * q^40 + 168 * q^41 + (21*b3 + 143) * q^43 + (36*b2 + 28*b1) * q^44 + (-28*b3 - 108*b2 - 28*b1 + 136) * q^46 + (-348*b2 + 24*b1) * q^47 + (13*b3 + 379*b2 + 2*b1 - 149) * q^49 + (14*b3 + 122) * q^50 + (-20*b3 - 88*b2 - 20*b1 + 108) * q^52 + (63*b3 - 219*b2 + 63*b1 + 156) * q^53 + (37*b3 - 400) * q^55 + (-8*b3 + 8*b2 + 16*b1 - 16) * q^56 + (66*b2 + 14*b1) * q^58 + (61*b3 + 351*b2 + 61*b1 - 412) * q^59 + (220*b2 - 34*b1) * q^61 + (56*b3 - 126) * q^62 + 64 * q^64 + (306*b2 + 42*b1) * q^65 + (35*b3 - 538*b2 + 35*b1 + 503) * q^67 + (-216*b2 + 40*b1) * q^68 + (20*b3 + 78*b2 + 2*b1 - 296) * q^70 + (28*b3 + 812) * q^71 + (-97*b3 - 46*b2 - 97*b1 + 143) * q^73 + (-42*b3 - 268*b2 - 42*b1 + 310) * q^74 + (12*b3 - 244) * q^76 + (34*b3 + 309*b2 - 5*b1 - 1024) * q^77 + (-311*b2 + 98*b1) * q^79 + (16*b3 + 48*b2 + 16*b1 - 64) * q^80 + 336*b2 * q^82 + (-89*b3 + 188) * q^83 + (-14*b3 - 304) * q^85 + (328*b2 - 42*b1) * q^86 + (56*b3 + 72*b2 + 56*b1 - 128) * q^88 + (-1170*b2 - 54*b1) * q^89 + (-12*b3 + 502*b2 + 59*b1 + 186) * q^91 + (-56*b3 + 272) * q^92 + (48*b3 - 696*b2 + 48*b1 + 648) * q^94 + (-70*b3 - 318*b2 - 70*b1 + 388) * q^95 + (-155*b3 + 46) * q^97 + (4*b3 + 486*b2 - 22*b1 - 762) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} - 8 q^{4} - 7 q^{5} + 6 q^{7} - 32 q^{8}+O(q^{10})$$ 4 * q + 4 * q^2 - 8 * q^4 - 7 * q^5 + 6 * q^7 - 32 * q^8 $$4 q + 4 q^{2} - 8 q^{4} - 7 q^{5} + 6 q^{7} - 32 q^{8} + 14 q^{10} + 25 q^{11} - 98 q^{13} + 18 q^{14} - 32 q^{16} - 98 q^{17} + 119 q^{19} + 56 q^{20} + 100 q^{22} - 122 q^{23} + 129 q^{25} - 98 q^{26} + 12 q^{28} + 146 q^{29} - 98 q^{31} + 64 q^{32} - 392 q^{34} - 128 q^{35} - 289 q^{37} - 238 q^{38} + 56 q^{40} + 672 q^{41} + 614 q^{43} + 100 q^{44} + 244 q^{46} - 672 q^{47} + 190 q^{49} + 516 q^{50} + 196 q^{52} + 375 q^{53} - 1526 q^{55} - 48 q^{56} + 146 q^{58} - 763 q^{59} + 406 q^{61} - 392 q^{62} + 256 q^{64} + 654 q^{65} + 1041 q^{67} - 392 q^{68} - 986 q^{70} + 3304 q^{71} + 189 q^{73} + 578 q^{74} - 952 q^{76} - 3415 q^{77} - 524 q^{79} - 112 q^{80} + 672 q^{82} + 574 q^{83} - 1244 q^{85} + 614 q^{86} - 200 q^{88} - 2394 q^{89} + 1783 q^{91} + 976 q^{92} + 1344 q^{94} + 706 q^{95} - 126 q^{97} - 2090 q^{98}+O(q^{100})$$ 4 * q + 4 * q^2 - 8 * q^4 - 7 * q^5 + 6 * q^7 - 32 * q^8 + 14 * q^10 + 25 * q^11 - 98 * q^13 + 18 * q^14 - 32 * q^16 - 98 * q^17 + 119 * q^19 + 56 * q^20 + 100 * q^22 - 122 * q^23 + 129 * q^25 - 98 * q^26 + 12 * q^28 + 146 * q^29 - 98 * q^31 + 64 * q^32 - 392 * q^34 - 128 * q^35 - 289 * q^37 - 238 * q^38 + 56 * q^40 + 672 * q^41 + 614 * q^43 + 100 * q^44 + 244 * q^46 - 672 * q^47 + 190 * q^49 + 516 * q^50 + 196 * q^52 + 375 * q^53 - 1526 * q^55 - 48 * q^56 + 146 * q^58 - 763 * q^59 + 406 * q^61 - 392 * q^62 + 256 * q^64 + 654 * q^65 + 1041 * q^67 - 392 * q^68 - 986 * q^70 + 3304 * q^71 + 189 * q^73 + 578 * q^74 - 952 * q^76 - 3415 * q^77 - 524 * q^79 - 112 * q^80 + 672 * q^82 + 574 * q^83 - 1244 * q^85 + 614 * q^86 - 200 * q^88 - 2394 * q^89 + 1783 * q^91 + 976 * q^92 + 1344 * q^94 + 706 * q^95 - 126 * q^97 - 2090 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 49x^{2} + 48x + 2304$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 49\nu^{2} - 49\nu + 2304 ) / 2352$$ (-v^3 + 49*v^2 - 49*v + 2304) / 2352 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 97 ) / 49$$ (v^3 + 97) / 49
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 48\beta_{2} + \beta _1 - 49$$ b3 + 48*b2 + b1 - 49 $$\nu^{3}$$ $$=$$ $$49\beta_{3} - 97$$ 49*b3 - 97

## Character values

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

 $$n$$ $$29$$ $$73$$ $$\chi(n)$$ $$1$$ $$-\beta_{2}$$

## 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
 3.72311 − 6.44862i −3.22311 + 5.58259i 3.72311 + 6.44862i −3.22311 − 5.58259i
1.00000 1.73205i 0 −2.00000 3.46410i −5.22311 + 9.04669i 0 −12.3924 + 13.7633i −8.00000 0 10.4462 + 18.0934i
37.2 1.00000 1.73205i 0 −2.00000 3.46410i 1.72311 2.98452i 0 15.3924 10.2992i −8.00000 0 −3.44622 5.96903i
109.1 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −5.22311 9.04669i 0 −12.3924 13.7633i −8.00000 0 10.4462 18.0934i
109.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 1.72311 + 2.98452i 0 15.3924 + 10.2992i −8.00000 0 −3.44622 + 5.96903i
 $$n$$: 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.f yes 4
3.b odd 2 1 126.4.g.e 4
7.b odd 2 1 882.4.g.bj 4
7.c even 3 1 inner 126.4.g.f yes 4
7.c even 3 1 882.4.a.ba 2
7.d odd 6 1 882.4.a.u 2
7.d odd 6 1 882.4.g.bj 4
21.c even 2 1 882.4.g.z 4
21.g even 6 1 882.4.a.bh 2
21.g even 6 1 882.4.g.z 4
21.h odd 6 1 126.4.g.e 4
21.h odd 6 1 882.4.a.bd 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.g.e 4 3.b odd 2 1
126.4.g.e 4 21.h odd 6 1
126.4.g.f yes 4 1.a even 1 1 trivial
126.4.g.f yes 4 7.c even 3 1 inner
882.4.a.u 2 7.d odd 6 1
882.4.a.ba 2 7.c even 3 1
882.4.a.bd 2 21.h odd 6 1
882.4.a.bh 2 21.g even 6 1
882.4.g.z 4 21.c even 2 1
882.4.g.z 4 21.g even 6 1
882.4.g.bj 4 7.b odd 2 1
882.4.g.bj 4 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 7T_{5}^{3} + 85T_{5}^{2} - 252T_{5} + 1296$$ acting on $$S_{4}^{\mathrm{new}}(126, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2 T + 4)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 7 T^{3} + 85 T^{2} + \cdots + 1296$$
$7$ $$T^{4} - 6 T^{3} - 77 T^{2} + \cdots + 117649$$
$11$ $$T^{4} - 25 T^{3} + 2833 T^{2} + \cdots + 4875264$$
$13$ $$(T^{2} + 49 T - 606)^{2}$$
$17$ $$T^{4} + 98 T^{3} + 12028 T^{2} + \cdots + 5875776$$
$19$ $$T^{4} - 119 T^{3} + 11055 T^{2} + \cdots + 9647236$$
$23$ $$T^{4} + 122 T^{3} + \cdots + 32901696$$
$29$ $$(T^{2} - 73 T - 1032)^{2}$$
$31$ $$T^{4} + 98 T^{3} + \cdots + 1255072329$$
$37$ $$T^{4} + 289 T^{3} + 83919 T^{2} + \cdots + 158404$$
$41$ $$(T - 168)^{4}$$
$43$ $$(T^{2} - 307 T + 2284)^{2}$$
$47$ $$T^{4} + 672 T^{3} + \cdots + 7242690816$$
$53$ $$T^{4} - 375 T^{3} + \cdots + 24444697104$$
$59$ $$T^{4} + 763 T^{3} + \cdots + 1155728016$$
$61$ $$T^{4} - 406 T^{3} + \cdots + 212226624$$
$67$ $$T^{4} - 1041 T^{3} + \cdots + 44865170596$$
$71$ $$(T^{2} - 1652 T + 644448)^{2}$$
$73$ $$T^{4} - 189 T^{3} + \cdots + 198073062916$$
$79$ $$T^{4} + 524 T^{3} + \cdots + 155826773001$$
$83$ $$(T^{2} - 287 T - 361596)^{2}$$
$89$ $$T^{4} + 2394 T^{3} + \cdots + 1669553420544$$
$97$ $$(T^{2} + 63 T - 1158214)^{2}$$