# Properties

 Label 84.6.i.b Level $84$ Weight $6$ Character orbit 84.i Analytic conductor $13.472$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,6,Mod(25,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("84.25");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$84 = 2^{2} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 84.i (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: $$13.4722408643$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7081})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 1771x^{2} + 1770x + 3132900$$ x^4 - x^3 + 1771*x^2 + 1770*x + 3132900 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.

 $$f(q)$$ $$=$$ $$q + (9 \beta_{2} - 9) q^{3} + ( - 24 \beta_{2} + \beta_1) q^{5} + (\beta_{3} + 59 \beta_{2} - 2 \beta_1 - 73) q^{7} - 81 \beta_{2} q^{9}+O(q^{10})$$ q + (9*b2 - 9) * q^3 + (-24*b2 + b1) * q^5 + (b3 + 59*b2 - 2*b1 - 73) * q^7 - 81*b2 * q^9 $$q + (9 \beta_{2} - 9) q^{3} + ( - 24 \beta_{2} + \beta_1) q^{5} + (\beta_{3} + 59 \beta_{2} - 2 \beta_1 - 73) q^{7} - 81 \beta_{2} q^{9} + (\beta_{3} - 204 \beta_{2} + \cdots + 203) q^{11}+ \cdots + ( - 81 \beta_{3} - 16443) q^{99}+O(q^{100})$$ q + (9*b2 - 9) * q^3 + (-24*b2 + b1) * q^5 + (b3 + 59*b2 - 2*b1 - 73) * q^7 - 81*b2 * q^9 + (b3 - 204*b2 + b1 + 203) * q^11 + (5*b3 + 222) * q^13 + (9*b3 + 207) * q^15 + (4*b3 - 936*b2 + 4*b1 + 932) * q^17 + (757*b2 - 51*b1) * q^19 + (-27*b3 - 648*b2 - 9*b1 + 144) * q^21 + (72*b2 - 100*b1) * q^23 + (-47*b3 - 779*b2 - 47*b1 + 826) * q^25 + 729 * q^27 + (109*b3 + 329) * q^29 + (76*b3 - 5623*b2 + 76*b1 + 5547) * q^31 + (1836*b2 - 9*b1) * q^33 + (105*b3 - 4998*b2 + 56*b1 + 4851) * q^35 + (1567*b2 - 21*b1) * q^37 + (-45*b3 + 2043*b2 - 45*b1 - 1998) * q^39 + (6*b3 - 3924) * q^41 + (-21*b3 - 6304) * q^43 + (-81*b3 + 1944*b2 - 81*b1 - 1863) * q^45 + (-4770*b2 - 36*b1) * q^47 + (-377*b3 + 9145*b2 - 58*b1 - 3230) * q^49 + (8424*b2 - 36*b1) * q^51 + (231*b3 + 6582*b2 + 231*b1 - 6813) * q^53 + (-227*b3 - 6439) * q^55 + (-459*b3 - 6354) * q^57 + (659*b3 - 24090*b2 + 659*b1 + 23431) * q^59 + (31606*b2 + 440*b1) * q^61 + (162*b3 + 1053*b2 + 243*b1 + 4617) * q^63 + (-14298*b2 + 342*b1) * q^65 + (-205*b3 + 22373*b2 - 205*b1 - 22168) * q^67 + (-900*b3 + 252) * q^69 + (500*b3 - 63170) * q^71 + (-751*b3 + 3395*b2 - 751*b1 - 2644) * q^73 + (7011*b2 + 423*b1) * q^75 + (596*b3 + 12918*b2 + 131*b1 + 2062) * q^77 + (-9611*b2 + 1634*b1) * q^79 + (6561*b2 - 6561) * q^81 + (1931*b3 + 18697) * q^83 + (-1028*b3 - 28516) * q^85 + (-981*b3 + 3942*b2 - 981*b1 - 2961) * q^87 + (-41532*b2 - 18*b1) * q^89 + (-138*b3 + 31093*b2 - 739*b1 - 7356) * q^91 + (50607*b2 - 684*b1) * q^93 + (1930*b3 - 108438*b2 + 1930*b1 + 106508) * q^95 + (1135*b3 + 91825) * q^97 + (-81*b3 - 16443) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 18 q^{3} - 47 q^{5} - 174 q^{7} - 162 q^{9}+O(q^{10})$$ 4 * q - 18 * q^3 - 47 * q^5 - 174 * q^7 - 162 * q^9 $$4 q - 18 q^{3} - 47 q^{5} - 174 q^{7} - 162 q^{9} + 407 q^{11} + 898 q^{13} + 846 q^{15} + 1868 q^{17} + 1463 q^{19} - 783 q^{21} + 44 q^{23} + 1605 q^{25} + 2916 q^{27} + 1534 q^{29} + 11170 q^{31} + 3663 q^{33} + 9674 q^{35} + 3113 q^{37} - 4041 q^{39} - 15684 q^{41} - 25258 q^{43} - 3807 q^{45} - 9576 q^{47} + 4558 q^{49} + 16812 q^{51} - 13395 q^{53} - 26210 q^{55} - 26334 q^{57} + 47521 q^{59} + 63652 q^{61} + 21141 q^{63} - 28254 q^{65} - 44541 q^{67} - 792 q^{69} - 251680 q^{71} - 6039 q^{73} + 14445 q^{75} + 35407 q^{77} - 17588 q^{79} - 13122 q^{81} + 78650 q^{83} - 116120 q^{85} - 6903 q^{87} - 83082 q^{89} + 31747 q^{91} + 100530 q^{93} + 214946 q^{95} + 369570 q^{97} - 65934 q^{99}+O(q^{100})$$ 4 * q - 18 * q^3 - 47 * q^5 - 174 * q^7 - 162 * q^9 + 407 * q^11 + 898 * q^13 + 846 * q^15 + 1868 * q^17 + 1463 * q^19 - 783 * q^21 + 44 * q^23 + 1605 * q^25 + 2916 * q^27 + 1534 * q^29 + 11170 * q^31 + 3663 * q^33 + 9674 * q^35 + 3113 * q^37 - 4041 * q^39 - 15684 * q^41 - 25258 * q^43 - 3807 * q^45 - 9576 * q^47 + 4558 * q^49 + 16812 * q^51 - 13395 * q^53 - 26210 * q^55 - 26334 * q^57 + 47521 * q^59 + 63652 * q^61 + 21141 * q^63 - 28254 * q^65 - 44541 * q^67 - 792 * q^69 - 251680 * q^71 - 6039 * q^73 + 14445 * q^75 + 35407 * q^77 - 17588 * q^79 - 13122 * q^81 + 78650 * q^83 - 116120 * q^85 - 6903 * q^87 - 83082 * q^89 + 31747 * q^91 + 100530 * q^93 + 214946 * q^95 + 369570 * q^97 - 65934 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 1771x^{2} + 1770x + 3132900$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 1771\nu^{2} - 1771\nu + 3132900 ) / 3134670$$ (-v^3 + 1771*v^2 - 1771*v + 3132900) / 3134670 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 3541 ) / 1771$$ (v^3 + 3541) / 1771
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 1770\beta_{2} + \beta _1 - 1771$$ b3 + 1770*b2 + b1 - 1771 $$\nu^{3}$$ $$=$$ $$1771\beta_{3} - 3541$$ 1771*b3 - 3541

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$43$$ $$73$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 −20.7872 − 36.0044i 21.2872 + 36.8705i −20.7872 + 36.0044i 21.2872 − 36.8705i
0 −4.50000 + 7.79423i 0 −32.7872 56.7890i 0 40.6487 + 123.104i 0 −40.5000 70.1481i 0
25.2 0 −4.50000 + 7.79423i 0 9.28717 + 16.0858i 0 −127.649 22.6454i 0 −40.5000 70.1481i 0
37.1 0 −4.50000 7.79423i 0 −32.7872 + 56.7890i 0 40.6487 123.104i 0 −40.5000 + 70.1481i 0
37.2 0 −4.50000 7.79423i 0 9.28717 16.0858i 0 −127.649 + 22.6454i 0 −40.5000 + 70.1481i 0
 $$n$$: 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 84.6.i.b 4
3.b odd 2 1 252.6.k.e 4
4.b odd 2 1 336.6.q.g 4
7.b odd 2 1 588.6.i.m 4
7.c even 3 1 inner 84.6.i.b 4
7.c even 3 1 588.6.a.l 2
7.d odd 6 1 588.6.a.h 2
7.d odd 6 1 588.6.i.m 4
21.h odd 6 1 252.6.k.e 4
28.g odd 6 1 336.6.q.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.6.i.b 4 1.a even 1 1 trivial
84.6.i.b 4 7.c even 3 1 inner
252.6.k.e 4 3.b odd 2 1
252.6.k.e 4 21.h odd 6 1
336.6.q.g 4 4.b odd 2 1
336.6.q.g 4 28.g odd 6 1
588.6.a.h 2 7.d odd 6 1
588.6.a.l 2 7.c even 3 1
588.6.i.m 4 7.b odd 2 1
588.6.i.m 4 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 47T_{5}^{3} + 3427T_{5}^{2} - 57246T_{5} + 1483524$$ acting on $$S_{6}^{\mathrm{new}}(84, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 9 T + 81)^{2}$$
$5$ $$T^{4} + 47 T^{3} + \cdots + 1483524$$
$7$ $$T^{4} + 174 T^{3} + \cdots + 282475249$$
$11$ $$T^{4} + \cdots + 1571488164$$
$13$ $$(T^{2} - 449 T + 6144)^{2}$$
$17$ $$T^{4} + \cdots + 712390017024$$
$19$ $$T^{4} + \cdots + 16559430371584$$
$23$ $$T^{4} + \cdots + 313361370464256$$
$29$ $$(T^{2} - 767 T - 20885268)^{2}$$
$31$ $$T^{4} + \cdots + 439626033842121$$
$37$ $$T^{4} + \cdots + 2696203408144$$
$41$ $$(T^{2} + 7842 T + 15310512)^{2}$$
$43$ $$(T^{2} + 12629 T + 39092230)^{2}$$
$47$ $$T^{4} + \cdots + 425625782490000$$
$53$ $$T^{4} + \cdots + 24\!\cdots\!16$$
$59$ $$T^{4} + \cdots + 41\!\cdots\!00$$
$61$ $$T^{4} + \cdots + 44\!\cdots\!76$$
$67$ $$T^{4} + \cdots + 17\!\cdots\!96$$
$71$ $$(T^{2} + 125840 T + 3516363900)^{2}$$
$73$ $$T^{4} + \cdots + 97\!\cdots\!00$$
$79$ $$T^{4} + \cdots + 21\!\cdots\!29$$
$83$ $$(T^{2} - 39325 T - 6214225254)^{2}$$
$89$ $$T^{4} + \cdots + 29\!\cdots\!00$$
$97$ $$(T^{2} - 184785 T + 6255893750)^{2}$$