# Properties

 Label 84.6.i.c Level $84$ Weight $6$ Character orbit 84.i Analytic conductor $13.472$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} + 703x^{6} + 2770x^{5} + 427565x^{4} + 718170x^{3} + 42175732x^{2} - 40929504x + 3559792896$$ x^8 - 2*x^7 + 703*x^6 + 2770*x^5 + 427565*x^4 + 718170*x^3 + 42175732*x^2 - 40929504*x + 3559792896 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{6}\cdot 3^{3}\cdot 7$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 9 \beta_1 q^{3} + ( - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{6} + 30 \beta_1 + 10) q^{7} + ( - 81 \beta_1 - 81) q^{9}+O(q^{10})$$ q - 9*b1 * q^3 + (-b3 - b2) * q^5 + (-b6 + 30*b1 + 10) * q^7 + (-81*b1 - 81) * q^9 $$q - 9 \beta_1 q^{3} + ( - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{6} + 30 \beta_1 + 10) q^{7} + ( - 81 \beta_1 - 81) q^{9} + (\beta_{7} + \beta_{5} + 3 \beta_{4} + \cdots + 2) q^{11}+ \cdots + ( - 162 \beta_{7} + 243 \beta_{6} + \cdots + 9153) q^{99}+O(q^{100})$$ q - 9*b1 * q^3 + (-b3 - b2) * q^5 + (-b6 + 30*b1 + 10) * q^7 + (-81*b1 - 81) * q^9 + (b7 + b5 + 3*b4 + 2*b2 + 115*b1 + 2) * q^11 + (-4*b7 + b6 + 3*b5 - 2*b4 - b3 + 3*b2 - 2*b1 - 154) * q^13 - 9*b2 * q^15 + (-2*b7 - 3*b6 + b5 + 3*b4 + 6*b3 - b2 - 56*b1 + 2) * q^17 + (-2*b7 + 6*b6 + b5 + 9*b4 - 5*b3 - b2 + 86*b1 + 90) * q^19 + (9*b4 + 9*b2 + 180*b1 + 270) * q^21 + (-b7 + 3*b6 - b5 + 6*b4 - 30*b3 - 28*b2 - 538*b1 - 536) * q^23 + (-5*b7 - 10*b6 + 5*b5 + 15*b4 - 20*b3 + 1366*b1 + 10) * q^25 - 729 * q^27 + (-13*b7 - 18*b6 + 14*b5 - 15*b4 - 16*b3 - 40*b2 + 2*b1 - 1388) * q^29 + (-3*b7 - 10*b6 + 7*b5 + 21*b4 + 75*b3 + 4*b2 - 206*b1 + 14) * q^31 + (-9*b7 + 27*b6 + 18*b5 + 27*b4 + 9*b3 + 27*b2 + 1017*b1 + 1035) * q^33 + (-14*b7 + 12*b6 + 28*b5 + 3*b4 + 35*b3 + 108*b2 - 2498*b1 - 289) * q^35 + (-11*b7 + 33*b6 + 22*b5 + 33*b4 - 119*b3 - 97*b2 - 1006*b1 - 984) * q^37 + (-27*b7 - 18*b6 - 9*b5 - 27*b4 - 54*b3 - 36*b2 + 1368*b1 - 18) * q^39 + (50*b7 - 90*b6 - 22*b5 - 6*b4 - 34*b3 + 28*b2 + 56*b1 - 2012) * q^41 + (25*b7 + 15*b6 - 23*b5 + 21*b4 + 19*b3 - 28*b2 + 4*b1 - 3614) * q^43 + 81*b3 * q^45 + (25*b7 - 75*b6 - 35*b5 - 90*b4 - 146*b3 - 196*b2 + 10480*b1 + 10430) * q^47 + (14*b7 - 31*b6 - 63*b5 - 62*b4 + 77*b3 - 27*b2 - 8927*b1 + 739) * q^49 + (-9*b7 + 27*b6 - 9*b5 + 54*b4 + 90*b3 + 108*b2 - 522*b1 - 504) * q^51 + (78*b7 + 90*b6 - 12*b5 - 36*b4 - 45*b3 + 66*b2 - 5580*b1 - 24) * q^53 + (-99*b7 + 206*b6 + 38*b5 + 23*b4 + 84*b3 + 328*b2 - 122*b1 + 832) * q^55 + (-9*b7 + 81*b6 - 9*b5 + 27*b4 + 45*b3 - 18*b2 - 36*b1 + 774) * q^57 + (43*b7 + 81*b6 - 38*b5 - 114*b4 + 104*b3 + 5*b2 - 8243*b1 - 76) * q^59 + (20*b7 - 60*b6 - 42*b5 - 58*b4 + 260*b3 + 220*b2 + 20984*b1 + 20944) * q^61 + (81*b6 + 81*b4 + 81*b2 - 810*b1 + 1620) * q^63 + (121*b7 - 363*b6 - 77*b5 - 528*b4 + 160*b3 - 82*b2 - 23078*b1 - 23320) * q^65 + (17*b7 + 13*b6 + 4*b5 + 12*b4 + 434*b3 + 21*b2 + 20000*b1 + 8) * q^67 + (9*b7 + 54*b6 - 18*b5 + 27*b4 + 36*b3 - 243*b2 - 18*b1 - 4842) * q^69 + (91*b7 - 114*b6 - 50*b5 + 9*b4 - 32*b3 + 391*b2 + 82*b1 + 11300) * q^71 + (11*b7 + 46*b6 - 35*b5 - 105*b4 - 652*b3 - 24*b2 + 5612*b1 - 70) * q^73 + (-45*b7 + 135*b6 + 225*b4 - 45*b3 + 45*b2 + 12204*b1 + 12294) * q^75 + (189*b7 - 231*b6 - 63*b5 - 228*b4 - 623*b3 - 333*b2 - 33204*b1 + 762) * q^77 + (-47*b7 + 141*b6 - 64*b5 + 299*b4 - 158*b3 - 64*b2 - 18860*b1 - 18766) * q^79 + 6561*b1 * q^81 + (57*b7 - 153*b6 - 15*b5 - 27*b4 - 69*b3 - 700*b2 + 84*b1 - 4263) * q^83 + (-132*b7 + 98*b6 + 86*b5 - 40*b4 + 6*b3 + 316*b2 - 92*b1 + 34678) * q^85 + (-126*b7 - 135*b6 + 9*b5 + 27*b4 + 81*b3 - 117*b2 + 12510*b1 + 18) * q^87 + (80*b7 - 240*b6 + 176*b5 - 576*b4 + 242*b3 + 82*b2 + 7268*b1 + 7108) * q^89 + (-196*b7 + 294*b6 + 217*b5 + 383*b4 + 707*b3 - 877*b2 - 39702*b1 - 21284) * q^91 + (-63*b7 + 189*b6 + 36*b5 + 279*b4 + 828*b3 + 954*b2 - 1980*b1 - 1854) * q^93 + (-204*b7 - 177*b6 - 27*b5 - 81*b4 - 922*b3 - 231*b2 + 36132*b1 - 54) * q^95 + (282*b7 - 343*b6 - 157*b5 + 32*b4 - 93*b3 + 103*b2 + 250*b1 - 53841) * q^97 + (-162*b7 + 243*b6 + 81*b5 + 81*b3 + 81*b2 - 162*b1 + 9153) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 36 q^{3} - 42 q^{7} - 324 q^{9}+O(q^{10})$$ 8 * q + 36 * q^3 - 42 * q^7 - 324 * q^9 $$8 q + 36 q^{3} - 42 q^{7} - 324 q^{9} - 462 q^{11} - 1204 q^{13} + 228 q^{17} + 358 q^{19} + 1404 q^{21} - 2148 q^{23} - 5454 q^{25} - 5832 q^{27} - 11064 q^{29} + 830 q^{31} + 4158 q^{33} + 7692 q^{35} - 3914 q^{37} - 5418 q^{39} - 16632 q^{41} - 29036 q^{43} + 41700 q^{47} + 41876 q^{49} - 2052 q^{51} + 22164 q^{53} + 7784 q^{55} + 6444 q^{57} + 32886 q^{59} + 83732 q^{61} + 16038 q^{63} - 93192 q^{65} - 80034 q^{67} - 38664 q^{69} + 89544 q^{71} - 22470 q^{73} + 49086 q^{75} + 138732 q^{77} - 75286 q^{79} - 26244 q^{81} - 34836 q^{83} + 278504 q^{85} - 49788 q^{87} + 28944 q^{89} - 12058 q^{91} - 7470 q^{93} - 144120 q^{95} - 433356 q^{97} + 74844 q^{99}+O(q^{100})$$ 8 * q + 36 * q^3 - 42 * q^7 - 324 * q^9 - 462 * q^11 - 1204 * q^13 + 228 * q^17 + 358 * q^19 + 1404 * q^21 - 2148 * q^23 - 5454 * q^25 - 5832 * q^27 - 11064 * q^29 + 830 * q^31 + 4158 * q^33 + 7692 * q^35 - 3914 * q^37 - 5418 * q^39 - 16632 * q^41 - 29036 * q^43 + 41700 * q^47 + 41876 * q^49 - 2052 * q^51 + 22164 * q^53 + 7784 * q^55 + 6444 * q^57 + 32886 * q^59 + 83732 * q^61 + 16038 * q^63 - 93192 * q^65 - 80034 * q^67 - 38664 * q^69 + 89544 * q^71 - 22470 * q^73 + 49086 * q^75 + 138732 * q^77 - 75286 * q^79 - 26244 * q^81 - 34836 * q^83 + 278504 * q^85 - 49788 * q^87 + 28944 * q^89 - 12058 * q^91 - 7470 * q^93 - 144120 * q^95 - 433356 * q^97 + 74844 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 703x^{6} + 2770x^{5} + 427565x^{4} + 718170x^{3} + 42175732x^{2} - 40929504x + 3559792896$$ :

 $$\beta_{1}$$ $$=$$ $$( - 202509581 \nu^{7} - 15579673694 \nu^{6} - 97000119635 \nu^{5} - 10465172425490 \nu^{4} + \cdots - 66\!\cdots\!28 ) / 59\!\cdots\!80$$ (-202509581*v^7 - 15579673694*v^6 - 97000119635*v^5 - 10465172425490*v^4 - 122564145295465*v^3 - 6949782824559210*v^2 - 9533960765338532*v - 665045423124812928) / 593800842443904480 $$\beta_{2}$$ $$=$$ $$( - 4074390615 \nu^{7} - 75239346984 \nu^{6} - 2524519237925 \nu^{5} - 9170584770500 \nu^{4} + \cdots + 91\!\cdots\!92 ) / 13\!\cdots\!20$$ (-4074390615*v^7 - 75239346984*v^6 - 2524519237925*v^5 - 9170584770500*v^4 - 2358643147423695*v^3 - 253101611314600*v^2 - 20272476670754880*v + 9157021379768685092) / 137203982534387020 $$\beta_{3}$$ $$=$$ $$( - 1106077550275 \nu^{7} - 64933012777282 \nu^{6} - 404277402169405 \nu^{5} + \cdots - 27\!\cdots\!84 ) / 36\!\cdots\!80$$ (-1106077550275*v^7 - 64933012777282*v^6 - 404277402169405*v^5 - 58866216604990510*v^4 - 510823228317805895*v^3 - 28965326605025760630*v^2 - 92202170495235626620*v - 2771778395709729293184) / 36221851389078173280 $$\beta_{4}$$ $$=$$ $$( 2298609090161 \nu^{7} - 10640696920522 \nu^{6} - 380580964066225 \nu^{5} + \cdots - 12\!\cdots\!04 ) / 36\!\cdots\!80$$ (2298609090161*v^7 - 10640696920522*v^6 - 380580964066225*v^5 + 3350711258275130*v^4 + 34630317119624605*v^3 - 8879646131762419950*v^2 - 424202705317524470188*v - 1207000672719081592704) / 36221851389078173280 $$\beta_{5}$$ $$=$$ $$( - 3837039574679 \nu^{7} + 46767932099422 \nu^{6} + \cdots - 18\!\cdots\!96 ) / 18\!\cdots\!40$$ (-3837039574679*v^7 + 46767932099422*v^6 - 3165735808019705*v^5 + 13958280574520530*v^4 - 2019877844122019515*v^3 + 8695553339011612650*v^2 - 419787840498489820988*v - 189862036976927202096) / 18110925694539086640 $$\beta_{6}$$ $$=$$ $$( - 608647346745 \nu^{7} + 5497197755738 \nu^{6} - 365756430948295 \nu^{5} + \cdots + 77\!\cdots\!76 ) / 17\!\cdots\!80$$ (-608647346745*v^7 + 5497197755738*v^6 - 365756430948295*v^5 + 694293952024790*v^4 - 189468012630439685*v^3 + 1121773665776127390*v^2 + 104411600727777100*v + 77846717154246757376) / 1724850066146579680 $$\beta_{7}$$ $$=$$ $$( 15200991659713 \nu^{7} + 29669373236422 \nu^{6} + \cdots + 12\!\cdots\!44 ) / 36\!\cdots\!80$$ (15200991659713*v^7 + 29669373236422*v^6 + 9127156669914655*v^5 + 94254291918546250*v^4 + 5586821641990408205*v^3 + 32379920546328895170*v^2 - 269208956785846512524*v + 1294165410554261202144) / 36221851389078173280
 $$\nu$$ $$=$$ $$( -16\beta_{7} - 15\beta_{6} - \beta_{5} - 3\beta_{4} - 28\beta_{3} - 17\beta_{2} - 118\beta _1 - 2 ) / 252$$ (-16*b7 - 15*b6 - b5 - 3*b4 - 28*b3 - 17*b2 - 118*b1 - 2) / 252 $$\nu^{2}$$ $$=$$ $$( -2\beta_{7} + 6\beta_{6} - 29\beta_{5} + 39\beta_{4} + 217\beta_{3} + 221\beta_{2} - 22070\beta _1 - 22066 ) / 63$$ (-2*b7 + 6*b6 - 29*b5 + 39*b4 + 217*b3 + 221*b2 - 22070*b1 - 22066) / 63 $$\nu^{3}$$ $$=$$ $$( 1033 \beta_{7} + 1968 \beta_{6} - 1220 \beta_{5} + 1407 \beta_{4} + 1594 \beta_{3} + 1289 \beta_{2} + \cdots - 56104 ) / 36$$ (1033*b7 + 1968*b6 - 1220*b5 + 1407*b4 + 1594*b3 + 1289*b2 - 374*b1 - 56104) / 36 $$\nu^{4}$$ $$=$$ $$( 3187 \beta_{7} + 3396 \beta_{6} - 209 \beta_{5} - 627 \beta_{4} - 12936 \beta_{3} + 2978 \beta_{2} + \cdots - 418 ) / 7$$ (3187*b7 + 3396*b6 - 209*b5 - 627*b4 - 12936*b3 + 2978*b2 + 1320066*b1 - 418) / 7 $$\nu^{5}$$ $$=$$ $$( 995191 \beta_{7} - 2985573 \beta_{6} + 4334173 \beta_{5} - 9310128 \beta_{4} - 2044322 \beta_{3} + \cdots + 421240394 ) / 252$$ (995191*b7 - 2985573*b6 + 4334173*b5 - 9310128*b4 - 2044322*b3 - 4034704*b2 + 423230776*b1 + 421240394) / 252 $$\nu^{6}$$ $$=$$ $$( - 2915935 \beta_{7} - 4285635 \beta_{6} + 3189875 \beta_{5} - 3463815 \beta_{4} - 3737755 \beta_{3} + \cdots + 1038478372 ) / 9$$ (-2915935*b7 - 4285635*b6 + 3189875*b5 - 3463815*b4 - 3737755*b3 - 16605650*b2 + 547880*b1 + 1038478372) / 9 $$\nu^{7}$$ $$=$$ $$( - 3473034028 \beta_{7} - 4111074465 \beta_{6} + 638040437 \beta_{5} + 1914121311 \beta_{4} + \cdots + 1276080874 ) / 252$$ (-3473034028*b7 - 4111074465*b6 + 638040437*b5 + 1914121311*b4 - 2126412064*b3 - 2834993591*b2 - 361895969854*b1 + 1276080874) / 252

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 4.59067 − 7.95128i −5.49618 + 9.51967i 13.1471 − 22.7714i −11.2416 + 19.4709i 4.59067 + 7.95128i −5.49618 − 9.51967i 13.1471 + 22.7714i −11.2416 − 19.4709i
0 4.50000 7.79423i 0 −39.3359 68.1317i 0 −100.606 + 81.7641i 0 −40.5000 70.1481i 0
25.2 0 4.50000 7.79423i 0 −23.0577 39.9371i 0 112.271 + 64.8240i 0 −40.5000 70.1481i 0
25.3 0 4.50000 7.79423i 0 15.9808 + 27.6796i 0 85.7043 97.2716i 0 −40.5000 70.1481i 0
25.4 0 4.50000 7.79423i 0 46.4128 + 80.3893i 0 −118.369 + 52.8745i 0 −40.5000 70.1481i 0
37.1 0 4.50000 + 7.79423i 0 −39.3359 + 68.1317i 0 −100.606 81.7641i 0 −40.5000 + 70.1481i 0
37.2 0 4.50000 + 7.79423i 0 −23.0577 + 39.9371i 0 112.271 64.8240i 0 −40.5000 + 70.1481i 0
37.3 0 4.50000 + 7.79423i 0 15.9808 27.6796i 0 85.7043 + 97.2716i 0 −40.5000 + 70.1481i 0
37.4 0 4.50000 + 7.79423i 0 46.4128 80.3893i 0 −118.369 52.8745i 0 −40.5000 + 70.1481i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 25.4 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.c 8
3.b odd 2 1 252.6.k.f 8
4.b odd 2 1 336.6.q.i 8
7.b odd 2 1 588.6.i.o 8
7.c even 3 1 inner 84.6.i.c 8
7.c even 3 1 588.6.a.n 4
7.d odd 6 1 588.6.a.p 4
7.d odd 6 1 588.6.i.o 8
21.h odd 6 1 252.6.k.f 8
28.g odd 6 1 336.6.q.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.6.i.c 8 1.a even 1 1 trivial
84.6.i.c 8 7.c even 3 1 inner
252.6.k.f 8 3.b odd 2 1
252.6.k.f 8 21.h odd 6 1
336.6.q.i 8 4.b odd 2 1
336.6.q.i 8 28.g odd 6 1
588.6.a.n 4 7.c even 3 1
588.6.a.p 4 7.d odd 6 1
588.6.i.o 8 7.b odd 2 1
588.6.i.o 8 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 8977 T_{5}^{6} + 165000 T_{5}^{5} + 69822853 T_{5}^{4} + 740602500 T_{5}^{3} + \cdots + 115856721032976$$ acting on $$S_{6}^{\mathrm{new}}(84, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{2} - 9 T + 81)^{4}$$
$5$ $$T^{8} + \cdots + 115856721032976$$
$7$ $$T^{8} + \cdots + 79\!\cdots\!01$$
$11$ $$T^{8} + \cdots + 25\!\cdots\!24$$
$13$ $$(T^{4} + 602 T^{3} + \cdots + 755795447424)^{2}$$
$17$ $$T^{8} + \cdots + 23\!\cdots\!44$$
$19$ $$T^{8} + \cdots + 74\!\cdots\!76$$
$23$ $$T^{8} + \cdots + 18\!\cdots\!64$$
$29$ $$(T^{4} + \cdots - 333378202056336)^{2}$$
$31$ $$T^{8} + \cdots + 10\!\cdots\!49$$
$37$ $$T^{8} + \cdots + 16\!\cdots\!16$$
$41$ $$(T^{4} + \cdots - 15\!\cdots\!88)^{2}$$
$43$ $$(T^{4} + \cdots - 359515701753932)^{2}$$
$47$ $$T^{8} + \cdots + 58\!\cdots\!76$$
$53$ $$T^{8} + \cdots + 88\!\cdots\!64$$
$59$ $$T^{8} + \cdots + 16\!\cdots\!84$$
$61$ $$T^{8} + \cdots + 38\!\cdots\!56$$
$67$ $$T^{8} + \cdots + 32\!\cdots\!16$$
$71$ $$(T^{4} + \cdots + 41\!\cdots\!92)^{2}$$
$73$ $$T^{8} + \cdots + 20\!\cdots\!00$$
$79$ $$T^{8} + \cdots + 34\!\cdots\!61$$
$83$ $$(T^{4} + \cdots + 53\!\cdots\!08)^{2}$$
$89$ $$T^{8} + \cdots + 12\!\cdots\!84$$
$97$ $$(T^{4} + \cdots - 37\!\cdots\!72)^{2}$$