# Properties

 Label 882.4.g.z Level $882$ Weight $4$ Character orbit 882.g Analytic conductor $52.040$ 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] = [882,4,Mod(361,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("882.361");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 882.g (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: $$52.0396846251$$ 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: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) 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.

 $$f(q)$$ $$=$$ $$q - 2 \beta_{2} q^{2} + (4 \beta_{2} - 4) q^{4} + ( - 3 \beta_{2} - \beta_1) q^{5} + 8 q^{8}+O(q^{10})$$ q - 2*b2 * q^2 + (4*b2 - 4) * q^4 + (-3*b2 - b1) * q^5 + 8 * q^8 $$q - 2 \beta_{2} q^{2} + (4 \beta_{2} - 4) q^{4} + ( - 3 \beta_{2} - \beta_1) q^{5} + 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} - 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} + (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} + ( - 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} + ( - 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} + (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} + ( - 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} + ( - 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} + (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}+O(q^{100})$$ q - 2*b2 * q^2 + (4*b2 - 4) * q^4 + (-3*b2 - b1) * q^5 + 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 - 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 + (7*b3 - 40) * q^29 + (-28*b3 - 35*b2 - 28*b1 + 63) * q^31 + (32*b2 - 32) * q^32 + (20*b3 + 88) * q^34 + (-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 + (-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 + (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 + (-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 + (-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 + (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 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} - 8 q^{4} - 7 q^{5} + 32 q^{8}+O(q^{10})$$ 4 * q - 4 * q^2 - 8 * q^4 - 7 * q^5 + 32 * q^8 $$4 q - 4 q^{2} - 8 q^{4} - 7 q^{5} + 32 q^{8} - 14 q^{10} - 25 q^{11} + 98 q^{13} - 32 q^{16} - 98 q^{17} - 119 q^{19} + 56 q^{20} + 100 q^{22} + 122 q^{23} + 129 q^{25} - 98 q^{26} - 146 q^{29} + 98 q^{31} - 64 q^{32} + 392 q^{34} - 289 q^{37} - 238 q^{38} - 56 q^{40} + 672 q^{41} + 614 q^{43} - 100 q^{44} + 244 q^{46} - 672 q^{47} - 516 q^{50} - 196 q^{52} - 375 q^{53} + 1526 q^{55} + 146 q^{58} - 763 q^{59} - 406 q^{61} - 392 q^{62} + 256 q^{64} - 654 q^{65} + 1041 q^{67} - 392 q^{68} - 3304 q^{71} - 189 q^{73} - 578 q^{74} + 952 q^{76} - 524 q^{79} - 112 q^{80} - 672 q^{82} + 574 q^{83} - 1244 q^{85} - 614 q^{86} - 200 q^{88} - 2394 q^{89} - 976 q^{92} - 1344 q^{94} - 706 q^{95} + 126 q^{97}+O(q^{100})$$ 4 * q - 4 * q^2 - 8 * q^4 - 7 * q^5 + 32 * q^8 - 14 * q^10 - 25 * q^11 + 98 * q^13 - 32 * q^16 - 98 * q^17 - 119 * q^19 + 56 * q^20 + 100 * q^22 + 122 * q^23 + 129 * q^25 - 98 * q^26 - 146 * q^29 + 98 * q^31 - 64 * q^32 + 392 * q^34 - 289 * q^37 - 238 * q^38 - 56 * q^40 + 672 * q^41 + 614 * q^43 - 100 * q^44 + 244 * q^46 - 672 * q^47 - 516 * q^50 - 196 * q^52 - 375 * q^53 + 1526 * q^55 + 146 * q^58 - 763 * q^59 - 406 * q^61 - 392 * q^62 + 256 * q^64 - 654 * q^65 + 1041 * q^67 - 392 * q^68 - 3304 * q^71 - 189 * q^73 - 578 * q^74 + 952 * q^76 - 524 * q^79 - 112 * q^80 - 672 * q^82 + 574 * q^83 - 1244 * q^85 - 614 * q^86 - 200 * q^88 - 2394 * q^89 - 976 * q^92 - 1344 * q^94 - 706 * q^95 + 126 * q^97

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 $$\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$785$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 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 0 8.00000 0 −10.4462 + 18.0934i
361.2 −1.00000 1.73205i 0 −2.00000 + 3.46410i 1.72311 + 2.98452i 0 0 8.00000 0 3.44622 5.96903i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −5.22311 + 9.04669i 0 0 8.00000 0 −10.4462 18.0934i
667.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i 1.72311 2.98452i 0 0 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 882.4.g.z 4
3.b odd 2 1 882.4.g.bj 4
7.b odd 2 1 126.4.g.e 4
7.c even 3 1 882.4.a.bh 2
7.c even 3 1 inner 882.4.g.z 4
7.d odd 6 1 126.4.g.e 4
7.d odd 6 1 882.4.a.bd 2
21.c even 2 1 126.4.g.f yes 4
21.g even 6 1 126.4.g.f yes 4
21.g even 6 1 882.4.a.ba 2
21.h odd 6 1 882.4.a.u 2
21.h odd 6 1 882.4.g.bj 4

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{4} + 7T_{5}^{3} + 85T_{5}^{2} - 252T_{5} + 1296$$ T5^4 + 7*T5^3 + 85*T5^2 - 252*T5 + 1296 $$T_{11}^{4} + 25T_{11}^{3} + 2833T_{11}^{2} - 55200T_{11} + 4875264$$ T11^4 + 25*T11^3 + 2833*T11^2 - 55200*T11 + 4875264 $$T_{13}^{2} - 49T_{13} - 606$$ T13^2 - 49*T13 - 606

## Hecke characteristic polynomials

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