# Properties

 Label 336.6.q.j Level $336$ Weight $6$ Character orbit 336.q Analytic conductor $53.889$ 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] = [336,6,Mod(193,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 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("336.193");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 336.q (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: $$53.8889634572$$ 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: $$x^{8} - x^{7} + 98x^{6} + 83x^{5} + 9122x^{4} - 91x^{3} + 28567x^{2} + 2058x + 86436$$ x^8 - x^7 + 98*x^6 + 83*x^5 + 9122*x^4 - 91*x^3 + 28567*x^2 + 2058*x + 86436 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{8}\cdot 3\cdot 7^{2}$$ Twist minimal: no (minimal twist has level 21) 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.

 $$f(q)$$ $$=$$ $$q + ( - 9 \beta_1 + 9) q^{3} + ( - \beta_{6} + \beta_{4}) q^{5} + (\beta_{7} - \beta_{4} + \beta_{3} + \cdots - 9) q^{7}+ \cdots - 81 \beta_1 q^{9}+O(q^{10})$$ q + (-9*b1 + 9) * q^3 + (-b6 + b4) * q^5 + (b7 - b4 + b3 - b2 - 45*b1 - 9) * q^7 - 81*b1 * q^9 $$q + ( - 9 \beta_1 + 9) q^{3} + ( - \beta_{6} + \beta_{4}) q^{5} + (\beta_{7} - \beta_{4} + \beta_{3} + \cdots - 9) q^{7}+ \cdots + (243 \beta_{7} - 486 \beta_{6} + \cdots - 8019) q^{99}+O(q^{100})$$ q + (-9*b1 + 9) * q^3 + (-b6 + b4) * q^5 + (b7 - b4 + b3 - b2 - 45*b1 - 9) * q^7 - 81*b1 * q^9 + (6*b4 + 3*b3 - b2 - 102*b1 + 102) * q^11 + (5*b7 + 4*b6 - 13*b5 + 13*b2 + 118) * q^13 - 9*b6 * q^15 + (2*b4 + 6*b3 + 11*b2 + 66*b1 - 66) * q^17 + (-7*b7 - 8*b6 - 22*b5 + 8*b4 - 7*b3 + 131*b1 - 7) * q^19 + (9*b7 + 9*b6 - 9*b5 - 9*b4 + 90*b1 - 486) * q^21 + (6*b7 + 14*b6 + 31*b5 - 14*b4 + 6*b3 + 1722*b1 + 6) * q^23 + (70*b4 + 25*b3 - 5*b2 + 691*b1 - 691) * q^25 - 729 * q^27 + (54*b7 - 49*b6 + 17*b5 - 17*b2 + 162) * q^29 + (59*b4 + 35*b3 - 19*b2 + 1585*b1 - 1585) * q^31 + (27*b7 - 54*b6 - 9*b5 + 54*b4 + 27*b3 - 918*b1 + 27) * q^33 + (21*b7 - 50*b6 - 27*b5 - 56*b4 - 48*b3 + 34*b2 - 1728*b1 + 5001) * q^35 + (-43*b7 + 52*b6 + 137*b5 - 52*b4 - 43*b3 - 3791*b1 - 43) * q^37 + (36*b4 - 45*b3 + 117*b2 - 1017*b1 + 1017) * q^39 + (102*b7 + 40*b6 - 20*b5 + 20*b2 + 1128) * q^41 + (63*b7 - 30*b6 + 78*b5 - 78*b2 - 7268) * q^43 - 81*b4 * q^45 + (-42*b7 + 14*b6 - 23*b5 - 14*b4 - 42*b3 - 3744*b1 - 42) * q^47 + (-89*b7 + 150*b6 - 171*b5 - 72*b4 - 22*b3 + 183*b2 + 5241*b1 - 10702) * q^49 + (54*b7 - 18*b6 + 99*b5 + 18*b4 + 54*b3 + 594*b1 + 54) * q^51 + (113*b4 - 126*b3 - 76*b2 + 3486*b1 - 3486) * q^53 + (-10*b7 + 325*b6 - 13*b5 + 13*b2 - 18286) * q^55 + (-63*b7 - 72*b6 - 198*b5 + 198*b2 + 1116) * q^57 + (46*b4 + 117*b3 - 158*b2 - 8766*b1 + 8766) * q^59 + (-146*b7 - 94*b6 + 196*b5 + 94*b4 - 146*b3 + 1414*b1 - 146) * q^61 + (81*b6 - 81*b5 - 81*b3 + 81*b2 + 4455*b1 - 3645) * q^63 + (-288*b7 + 42*b6 + 341*b5 - 42*b4 - 288*b3 - 16572*b1 - 288) * q^65 + (136*b4 - 329*b3 + 544*b2 + 1663*b1 - 1663) * q^67 + (54*b7 + 126*b6 + 279*b5 - 279*b2 + 15552) * q^69 + (78*b7 + 102*b6 + 393*b5 - 393*b2 - 22278) * q^71 + (118*b4 + 25*b3 - 365*b2 + 14897*b1 - 14897) * q^73 + (225*b7 - 630*b6 - 45*b5 + 630*b4 + 225*b3 + 6219*b1 + 225) * q^75 + (348*b7 - 715*b6 - 433*b5 + 177*b4 + 6*b3 + 8*b2 + 17136*b1 - 17850) * q^77 + (-345*b7 - 441*b6 - 3*b5 + 441*b4 - 345*b3 - 10843*b1 - 345) * q^79 + (6561*b1 - 6561) * q^81 + (567*b7 - 438*b6 - 978*b5 + 978*b2 + 52395) * q^83 + (282*b7 + 90*b6 + 210*b5 - 210*b2 + 9222) * q^85 + (-441*b4 - 486*b3 - 153*b2 - 972*b1 + 972) * q^87 + (480*b7 - 1102*b6 + 1224*b5 + 1102*b4 + 480*b3 + 19140*b1 + 480) * q^89 + (317*b7 - 114*b6 + 1332*b5 - 240*b4 + 193*b3 - 270*b2 + 71403*b1 - 48585) * q^91 + (315*b7 - 531*b6 - 171*b5 + 531*b4 + 315*b3 + 14265*b1 + 315) * q^93 + (918*b4 + 624*b3 + 395*b2 + 55032*b1 - 55032) * q^95 + (97*b7 - 766*b6 + 85*b5 - 85*b2 - 47109) * q^97 + (243*b7 - 486*b6 - 81*b5 + 81*b2 - 8019) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 36 q^{3} - 258 q^{7} - 324 q^{9}+O(q^{10})$$ 8 * q + 36 * q^3 - 258 * q^7 - 324 * q^9 $$8 q + 36 q^{3} - 258 q^{7} - 324 q^{9} + 402 q^{11} + 924 q^{13} - 276 q^{17} + 510 q^{19} - 3564 q^{21} + 6900 q^{23} - 2814 q^{25} - 5832 q^{27} + 1080 q^{29} - 6410 q^{31} - 3618 q^{33} + 33108 q^{35} - 15250 q^{37} + 4158 q^{39} + 8616 q^{41} - 58396 q^{43} - 15060 q^{47} - 64252 q^{49} + 2484 q^{51} - 13692 q^{53} - 146248 q^{55} + 9180 q^{57} + 34830 q^{59} + 5364 q^{61} - 11178 q^{63} - 66864 q^{65} - 5994 q^{67} + 124200 q^{69} - 178536 q^{71} - 59638 q^{73} + 25326 q^{75} - 75660 q^{77} - 44062 q^{79} - 26244 q^{81} + 416892 q^{83} + 72648 q^{85} + 4860 q^{87} + 77520 q^{89} - 104722 q^{91} + 57690 q^{93} - 221376 q^{95} - 377260 q^{97} - 65124 q^{99}+O(q^{100})$$ 8 * q + 36 * q^3 - 258 * q^7 - 324 * q^9 + 402 * q^11 + 924 * q^13 - 276 * q^17 + 510 * q^19 - 3564 * q^21 + 6900 * q^23 - 2814 * q^25 - 5832 * q^27 + 1080 * q^29 - 6410 * q^31 - 3618 * q^33 + 33108 * q^35 - 15250 * q^37 + 4158 * q^39 + 8616 * q^41 - 58396 * q^43 - 15060 * q^47 - 64252 * q^49 + 2484 * q^51 - 13692 * q^53 - 146248 * q^55 + 9180 * q^57 + 34830 * q^59 + 5364 * q^61 - 11178 * q^63 - 66864 * q^65 - 5994 * q^67 + 124200 * q^69 - 178536 * q^71 - 59638 * q^73 + 25326 * q^75 - 75660 * q^77 - 44062 * q^79 - 26244 * q^81 + 416892 * q^83 + 72648 * q^85 + 4860 * q^87 + 77520 * q^89 - 104722 * q^91 + 57690 * q^93 - 221376 * q^95 - 377260 * q^97 - 65124 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 98x^{6} + 83x^{5} + 9122x^{4} - 91x^{3} + 28567x^{2} + 2058x + 86436$$ :

 $$\beta_{1}$$ $$=$$ $$( 8905 \nu^{7} - 366059 \nu^{6} + 1191820 \nu^{5} - 33153695 \nu^{4} + 47979268 \nu^{3} + \cdots - 307508418 ) / 9888988410$$ (8905*v^7 - 366059*v^6 + 1191820*v^5 - 33153695*v^4 + 47979268*v^3 - 3262309295*v^2 + 141627885*v - 307508418) / 9888988410 $$\beta_{2}$$ $$=$$ $$( 8905 \nu^{7} - 366059 \nu^{6} + 1191820 \nu^{5} - 33153695 \nu^{4} + 47979268 \nu^{3} + \cdots - 10196496828 ) / 4944494205$$ (8905*v^7 - 366059*v^6 + 1191820*v^5 - 33153695*v^4 + 47979268*v^3 - 3262309295*v^2 + 39697581525*v - 10196496828) / 4944494205 $$\beta_{3}$$ $$=$$ $$( 65603 \nu^{7} - 25662916 \nu^{6} + 83553680 \nu^{5} - 2627557456 \nu^{4} + 3363632432 \nu^{3} + \cdots - 714835153872 ) / 9888988410$$ (65603*v^7 - 25662916*v^6 + 83553680*v^5 - 2627557456*v^4 + 3363632432*v^3 - 228707310580*v^2 + 464946681171*v - 714835153872) / 9888988410 $$\beta_{4}$$ $$=$$ $$( 731039 \nu^{7} - 7994302 \nu^{6} + 26027960 \nu^{5} - 628584958 \nu^{4} + 1047811304 \nu^{3} + \cdots - 222679608984 ) / 9888988410$$ (731039*v^7 - 7994302*v^6 + 26027960*v^5 - 628584958*v^4 + 1047811304*v^3 - 71245033510*v^2 - 451136527737*v - 222679608984) / 9888988410 $$\beta_{5}$$ $$=$$ $$( 1437521 \nu^{7} - 1642579 \nu^{6} + 136763060 \nu^{5} + 99854873 \nu^{4} + 13093975028 \nu^{3} + \cdots + 2771341902 ) / 4944494205$$ (1437521*v^7 - 1642579*v^6 + 136763060*v^5 + 99854873*v^4 + 13093975028*v^3 - 2811264295*v^2 + 41000921157*v + 2771341902) / 4944494205 $$\beta_{6}$$ $$=$$ $$( 484709 \nu^{7} - 433105 \nu^{6} + 45997385 \nu^{5} + 45127907 \nu^{4} + 4285048727 \nu^{3} + \cdots - 5171325642 ) / 282542526$$ (484709*v^7 - 433105*v^6 + 45997385*v^5 + 45127907*v^4 + 4285048727*v^3 + 153033125*v^2 + 442204518*v - 5171325642) / 282542526 $$\beta_{7}$$ $$=$$ $$( 2434409 \nu^{7} - 2482177 \nu^{6} + 231017885 \nu^{5} + 226651007 \nu^{4} + 21466747139 \nu^{3} + \cdots + 102356221716 ) / 1412712630$$ (2434409*v^7 - 2482177*v^6 + 231017885*v^5 + 226651007*v^4 + 21466747139*v^3 + 768595625*v^2 + 2220933918*v + 102356221716) / 1412712630
 $$\nu$$ $$=$$ $$( \beta_{2} - 2\beta _1 + 2 ) / 8$$ (b2 - 2*b1 + 2) / 8 $$\nu^{2}$$ $$=$$ $$( \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} - 98\beta _1 + 1 ) / 2$$ (b7 - b6 + b4 + b3 - 98*b1 + 1) / 2 $$\nu^{3}$$ $$=$$ $$( -16\beta_{6} + 95\beta_{5} - 95\beta_{2} - 542 ) / 8$$ (-16*b6 + 95*b5 - 95*b2 - 542) / 8 $$\nu^{4}$$ $$=$$ $$( -101\beta_{4} - 97\beta_{3} - 22\beta_{2} + 9050\beta _1 - 9050 ) / 2$$ (-101*b4 - 97*b3 - 22*b2 + 9050*b1 - 9050) / 2 $$\nu^{5}$$ $$=$$ $$( -360\beta_{7} + 1928\beta_{6} - 9009\beta_{5} - 1928\beta_{4} - 360\beta_{3} + 85442\beta _1 - 360 ) / 8$$ (-360*b7 + 1928*b6 - 9009*b5 - 1928*b4 - 360*b3 + 85442*b1 - 360) / 8 $$\nu^{6}$$ $$=$$ $$( -9205\beta_{7} + 9957\beta_{6} - 4220\beta_{5} + 4220\beta_{2} + 860245 ) / 2$$ (-9205*b7 + 9957*b6 - 4220*b5 + 4220*b2 + 860245) / 2 $$\nu^{7}$$ $$=$$ $$( 219312\beta_{4} + 69024\beta_{3} + 862207\beta_{2} - 11352926\beta _1 + 11352926 ) / 8$$ (219312*b4 + 69024*b3 + 862207*b2 - 11352926*b1 + 11352926) / 8

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$$.

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\beta_{1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 −0.874091 + 1.51397i −4.61193 + 7.98809i 5.09061 − 8.81720i 0.895402 − 1.55088i −0.874091 − 1.51397i −4.61193 − 7.98809i 5.09061 + 8.81720i 0.895402 + 1.55088i
0 4.50000 7.79423i 0 −29.1836 50.5475i 0 21.4366 + 127.857i 0 −40.5000 70.1481i 0
193.2 0 4.50000 7.79423i 0 −11.8764 20.5705i 0 −30.6840 125.958i 0 −40.5000 70.1481i 0
193.3 0 4.50000 7.79423i 0 −11.0358 19.1146i 0 −126.882 26.6059i 0 −40.5000 70.1481i 0
193.4 0 4.50000 7.79423i 0 52.0958 + 90.2327i 0 7.12980 129.446i 0 −40.5000 70.1481i 0
289.1 0 4.50000 + 7.79423i 0 −29.1836 + 50.5475i 0 21.4366 127.857i 0 −40.5000 + 70.1481i 0
289.2 0 4.50000 + 7.79423i 0 −11.8764 + 20.5705i 0 −30.6840 + 125.958i 0 −40.5000 + 70.1481i 0
289.3 0 4.50000 + 7.79423i 0 −11.0358 + 19.1146i 0 −126.882 + 26.6059i 0 −40.5000 + 70.1481i 0
289.4 0 4.50000 + 7.79423i 0 52.0958 90.2327i 0 7.12980 + 129.446i 0 −40.5000 + 70.1481i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 193.4 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 336.6.q.j 8
4.b odd 2 1 21.6.e.c 8
7.c even 3 1 inner 336.6.q.j 8
12.b even 2 1 63.6.e.e 8
28.d even 2 1 147.6.e.o 8
28.f even 6 1 147.6.a.l 4
28.f even 6 1 147.6.e.o 8
28.g odd 6 1 21.6.e.c 8
28.g odd 6 1 147.6.a.m 4
84.j odd 6 1 441.6.a.v 4
84.n even 6 1 63.6.e.e 8
84.n even 6 1 441.6.a.w 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.c 8 4.b odd 2 1
21.6.e.c 8 28.g odd 6 1
63.6.e.e 8 12.b even 2 1
63.6.e.e 8 84.n even 6 1
147.6.a.l 4 28.f even 6 1
147.6.a.m 4 28.g odd 6 1
147.6.e.o 8 28.d even 2 1
147.6.e.o 8 28.f even 6 1
336.6.q.j 8 1.a even 1 1 trivial
336.6.q.j 8 7.c even 3 1 inner
441.6.a.v 4 84.j odd 6 1
441.6.a.w 4 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 7657 T_{5}^{6} + 605400 T_{5}^{5} + 61817893 T_{5}^{4} + 2317773900 T_{5}^{3} + \cdots + 10164899803536$$ acting on $$S_{6}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{2} - 9 T + 81)^{4}$$
$5$ $$T^{8} + \cdots + 10164899803536$$
$7$ $$T^{8} + \cdots + 79\!\cdots\!01$$
$11$ $$T^{8} + \cdots + 28\!\cdots\!76$$
$13$ $$(T^{4} - 462 T^{3} + \cdots + 149501563456)^{2}$$
$17$ $$T^{8} + \cdots + 25\!\cdots\!24$$
$19$ $$T^{8} + \cdots + 54\!\cdots\!76$$
$23$ $$T^{8} + \cdots + 90\!\cdots\!76$$
$29$ $$(T^{4} + \cdots + 408027025117872)^{2}$$
$31$ $$T^{8} + \cdots + 75\!\cdots\!09$$
$37$ $$T^{8} + \cdots + 25\!\cdots\!36$$
$41$ $$(T^{4} + \cdots - 18\!\cdots\!52)^{2}$$
$43$ $$(T^{4} + \cdots - 991662745581932)^{2}$$
$47$ $$T^{8} + \cdots + 73\!\cdots\!36$$
$53$ $$T^{8} + \cdots + 72\!\cdots\!84$$
$59$ $$T^{8} + \cdots + 66\!\cdots\!96$$
$61$ $$T^{8} + \cdots + 32\!\cdots\!76$$
$67$ $$T^{8} + \cdots + 30\!\cdots\!56$$
$71$ $$(T^{4} + \cdots + 21\!\cdots\!68)^{2}$$
$73$ $$T^{8} + \cdots + 14\!\cdots\!00$$
$79$ $$T^{8} + \cdots + 27\!\cdots\!81$$
$83$ $$(T^{4} + \cdots - 41\!\cdots\!32)^{2}$$
$89$ $$T^{8} + \cdots + 22\!\cdots\!24$$
$97$ $$(T^{4} + \cdots - 11\!\cdots\!44)^{2}$$