# Properties

 Label 336.6.q.e Level $336$ Weight $6$ Character orbit 336.q Analytic conductor $53.889$ 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] = [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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-83})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 20x^{2} - 21x + 441$$ x^4 - x^3 - 20*x^2 - 21*x + 441 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$3$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 9 \beta_1 q^{3} + (7 \beta_{3} + 20 \beta_1 + 20) q^{5} + (14 \beta_{3} - 7 \beta_{2} + 7 \beta_1 + 91) q^{7} + ( - 81 \beta_1 - 81) q^{9}+O(q^{10})$$ q + 9*b1 * q^3 + (7*b3 + 20*b1 + 20) * q^5 + (14*b3 - 7*b2 + 7*b1 + 91) * q^7 + (-81*b1 - 81) * q^9 $$q + 9 \beta_1 q^{3} + (7 \beta_{3} + 20 \beta_1 + 20) q^{5} + (14 \beta_{3} - 7 \beta_{2} + 7 \beta_1 + 91) q^{7} + ( - 81 \beta_1 - 81) q^{9} + ( - \beta_{3} + \beta_{2} + 568 \beta_1) q^{11} + (9 \beta_{2} + 467) q^{13} + ( - 63 \beta_{2} - 180) q^{15} + (148 \beta_{3} - 148 \beta_{2} - 88 \beta_1) q^{17} + (27 \beta_{3} + 1169 \beta_1 + 1169) q^{19} + ( - 63 \beta_{3} - 63 \beta_{2} + 756 \beta_1 - 63) q^{21} + (308 \beta_{3} + 952 \beta_1 + 952) q^{23} + (231 \beta_{3} - 231 \beta_{2} + 313 \beta_1) q^{25} + 729 q^{27} + (45 \beta_{2} - 1086) q^{29} + ( - 768 \beta_{3} + 768 \beta_{2} - 2531 \beta_1) q^{31} + (9 \beta_{3} - 5112 \beta_1 - 5112) q^{33} + (728 \beta_{3} - 231 \beta_{2} + 4858 \beta_1 - 1358) q^{35} + ( - 855 \beta_{3} + 9127 \beta_1 + 9127) q^{37} + (81 \beta_{3} - 81 \beta_{2} + 4203 \beta_1) q^{39} + (846 \beta_{2} - 6006) q^{41} + (2043 \beta_{2} + 2407) q^{43} + ( - 567 \beta_{3} + 567 \beta_{2} - 1620 \beta_1) q^{45} + ( - 604 \beta_{3} - 11882 \beta_1 - 11882) q^{47} + (2450 \beta_{3} - 1225 \beta_{2} + 1225 \beta_1 - 882) q^{49} + ( - 1332 \beta_{3} + 792 \beta_1 + 792) q^{51} + ( - 1751 \beta_{3} + 1751 \beta_{2} - 16702 \beta_1) q^{53} + ( - 3963 \beta_{2} - 10926) q^{55} + ( - 243 \beta_{2} - 10521) q^{57} + (3917 \beta_{3} - 3917 \beta_{2} - 18590 \beta_1) q^{59} + (2544 \beta_{3} - 19754 \beta_1 - 19754) q^{61} + ( - 567 \beta_{3} + 1134 \beta_{2} - 7371 \beta_1 - 6804) q^{63} + (3386 \beta_{3} + 13246 \beta_1 + 13246) q^{65} + (4461 \beta_{3} - 4461 \beta_{2} - 13151 \beta_1) q^{67} + ( - 2772 \beta_{2} - 8568) q^{69} + ( - 1404 \beta_{2} - 51750) q^{71} + (5247 \beta_{3} - 5247 \beta_{2} - 11665 \beta_1) q^{73} + ( - 2079 \beta_{3} - 2817 \beta_1 - 2817) q^{75} + ( - 4067 \beta_{3} - 3892 \beta_{2} + 48146 \beta_1 - 3108) q^{77} + (6834 \beta_{3} - 5815 \beta_1 - 5815) q^{79} + 6561 \beta_1 q^{81} + (1899 \beta_{2} - 29640) q^{83} + ( - 1308 \beta_{2} - 62472) q^{85} + (405 \beta_{3} - 405 \beta_{2} - 9774 \beta_1) q^{87} + (130 \beta_{3} - 14596 \beta_1 - 14596) q^{89} + (6475 \beta_{3} - 2450 \beta_{2} + 11081 \beta_1 + 46403) q^{91} + (6912 \beta_{3} + 22779 \beta_1 + 22779) q^{93} + (8534 \beta_{3} - 8534 \beta_{2} + 35098 \beta_1) q^{95} + ( - 1017 \beta_{2} - 5404) q^{97} + ( - 81 \beta_{2} + 46008) q^{99}+O(q^{100})$$ q + 9*b1 * q^3 + (7*b3 + 20*b1 + 20) * q^5 + (14*b3 - 7*b2 + 7*b1 + 91) * q^7 + (-81*b1 - 81) * q^9 + (-b3 + b2 + 568*b1) * q^11 + (9*b2 + 467) * q^13 + (-63*b2 - 180) * q^15 + (148*b3 - 148*b2 - 88*b1) * q^17 + (27*b3 + 1169*b1 + 1169) * q^19 + (-63*b3 - 63*b2 + 756*b1 - 63) * q^21 + (308*b3 + 952*b1 + 952) * q^23 + (231*b3 - 231*b2 + 313*b1) * q^25 + 729 * q^27 + (45*b2 - 1086) * q^29 + (-768*b3 + 768*b2 - 2531*b1) * q^31 + (9*b3 - 5112*b1 - 5112) * q^33 + (728*b3 - 231*b2 + 4858*b1 - 1358) * q^35 + (-855*b3 + 9127*b1 + 9127) * q^37 + (81*b3 - 81*b2 + 4203*b1) * q^39 + (846*b2 - 6006) * q^41 + (2043*b2 + 2407) * q^43 + (-567*b3 + 567*b2 - 1620*b1) * q^45 + (-604*b3 - 11882*b1 - 11882) * q^47 + (2450*b3 - 1225*b2 + 1225*b1 - 882) * q^49 + (-1332*b3 + 792*b1 + 792) * q^51 + (-1751*b3 + 1751*b2 - 16702*b1) * q^53 + (-3963*b2 - 10926) * q^55 + (-243*b2 - 10521) * q^57 + (3917*b3 - 3917*b2 - 18590*b1) * q^59 + (2544*b3 - 19754*b1 - 19754) * q^61 + (-567*b3 + 1134*b2 - 7371*b1 - 6804) * q^63 + (3386*b3 + 13246*b1 + 13246) * q^65 + (4461*b3 - 4461*b2 - 13151*b1) * q^67 + (-2772*b2 - 8568) * q^69 + (-1404*b2 - 51750) * q^71 + (5247*b3 - 5247*b2 - 11665*b1) * q^73 + (-2079*b3 - 2817*b1 - 2817) * q^75 + (-4067*b3 - 3892*b2 + 48146*b1 - 3108) * q^77 + (6834*b3 - 5815*b1 - 5815) * q^79 + 6561*b1 * q^81 + (1899*b2 - 29640) * q^83 + (-1308*b2 - 62472) * q^85 + (405*b3 - 405*b2 - 9774*b1) * q^87 + (130*b3 - 14596*b1 - 14596) * q^89 + (6475*b3 - 2450*b2 + 11081*b1 + 46403) * q^91 + (6912*b3 + 22779*b1 + 22779) * q^93 + (8534*b3 - 8534*b2 + 35098*b1) * q^95 + (-1017*b2 - 5404) * q^97 + (-81*b2 + 46008) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 18 q^{3} + 33 q^{5} + 350 q^{7} - 162 q^{9}+O(q^{10})$$ 4 * q - 18 * q^3 + 33 * q^5 + 350 * q^7 - 162 * q^9 $$4 q - 18 q^{3} + 33 q^{5} + 350 q^{7} - 162 q^{9} - 1137 q^{11} + 1850 q^{13} - 594 q^{15} + 324 q^{17} + 2311 q^{19} - 1575 q^{21} + 1596 q^{23} - 395 q^{25} + 2916 q^{27} - 4434 q^{29} + 4294 q^{31} - 10233 q^{33} - 15414 q^{35} + 19109 q^{37} - 8325 q^{39} - 25716 q^{41} + 5542 q^{43} + 2673 q^{45} - 23160 q^{47} - 5978 q^{49} + 2916 q^{51} + 31653 q^{53} - 35778 q^{55} - 41598 q^{57} + 41097 q^{59} - 42052 q^{61} - 14175 q^{63} + 23106 q^{65} + 30763 q^{67} - 28728 q^{69} - 204192 q^{71} + 28577 q^{73} - 3555 q^{75} - 96873 q^{77} - 18464 q^{79} - 13122 q^{81} - 122358 q^{83} - 247272 q^{85} + 19953 q^{87} - 29322 q^{89} + 161875 q^{91} + 38646 q^{93} - 61662 q^{95} - 19582 q^{97} + 184194 q^{99}+O(q^{100})$$ 4 * q - 18 * q^3 + 33 * q^5 + 350 * q^7 - 162 * q^9 - 1137 * q^11 + 1850 * q^13 - 594 * q^15 + 324 * q^17 + 2311 * q^19 - 1575 * q^21 + 1596 * q^23 - 395 * q^25 + 2916 * q^27 - 4434 * q^29 + 4294 * q^31 - 10233 * q^33 - 15414 * q^35 + 19109 * q^37 - 8325 * q^39 - 25716 * q^41 + 5542 * q^43 + 2673 * q^45 - 23160 * q^47 - 5978 * q^49 + 2916 * q^51 + 31653 * q^53 - 35778 * q^55 - 41598 * q^57 + 41097 * q^59 - 42052 * q^61 - 14175 * q^63 + 23106 * q^65 + 30763 * q^67 - 28728 * q^69 - 204192 * q^71 + 28577 * q^73 - 3555 * q^75 - 96873 * q^77 - 18464 * q^79 - 13122 * q^81 - 122358 * q^83 - 247272 * q^85 + 19953 * q^87 - 29322 * q^89 + 161875 * q^91 + 38646 * q^93 - 61662 * q^95 - 19582 * q^97 + 184194 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 20x^{2} - 21x + 441$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 20\nu^{2} - 20\nu - 441 ) / 420$$ (v^3 + 20*v^2 - 20*v - 441) / 420 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 41\nu ) / 21$$ (-v^3 + v^2 + 41*v) / 21 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 20\nu - 41 ) / 20$$ (v^3 + 20*v - 41) / 20
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3$$ (b3 + b2 - b1 + 1) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 61\beta _1 + 62 ) / 3$$ (-b3 + 2*b2 + 61*b1 + 62) / 3 $$\nu^{3}$$ $$=$$ $$( 40\beta_{3} - 20\beta_{2} + 20\beta _1 + 103 ) / 3$$ (40*b3 - 20*b2 + 20*b1 + 103) / 3

## Character values

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

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

## 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
 −3.69493 − 2.71062i 4.19493 + 1.84460i −3.69493 + 2.71062i 4.19493 − 1.84460i
0 −4.50000 + 7.79423i 0 −19.3645 33.5404i 0 87.5000 95.6596i 0 −40.5000 70.1481i 0
193.2 0 −4.50000 + 7.79423i 0 35.8645 + 62.1192i 0 87.5000 + 95.6596i 0 −40.5000 70.1481i 0
289.1 0 −4.50000 7.79423i 0 −19.3645 + 33.5404i 0 87.5000 + 95.6596i 0 −40.5000 + 70.1481i 0
289.2 0 −4.50000 7.79423i 0 35.8645 62.1192i 0 87.5000 95.6596i 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 336.6.q.e 4
4.b odd 2 1 21.6.e.b 4
7.c even 3 1 inner 336.6.q.e 4
12.b even 2 1 63.6.e.c 4
28.d even 2 1 147.6.e.l 4
28.f even 6 1 147.6.a.k 2
28.f even 6 1 147.6.e.l 4
28.g odd 6 1 21.6.e.b 4
28.g odd 6 1 147.6.a.i 2
84.j odd 6 1 441.6.a.s 2
84.n even 6 1 63.6.e.c 4
84.n even 6 1 441.6.a.t 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 33T_{5}^{3} + 3867T_{5}^{2} + 91674T_{5} + 7717284$$ acting on $$S_{6}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 9 T + 81)^{2}$$
$5$ $$T^{4} - 33 T^{3} + 3867 T^{2} + \cdots + 7717284$$
$7$ $$(T^{2} - 175 T + 16807)^{2}$$
$11$ $$T^{4} + 1137 T^{3} + \cdots + 104412996900$$
$13$ $$(T^{2} - 925 T + 208864)^{2}$$
$17$ $$T^{4} - 324 T^{3} + \cdots + 1788317798400$$
$19$ $$T^{4} - 2311 T^{3} + \cdots + 1663584040000$$
$23$ $$T^{4} - 1596 T^{3} + \cdots + 27756881510400$$
$29$ $$(T^{2} + 2217 T + 1102716)^{2}$$
$31$ $$T^{4} - 4294 T^{3} + \cdots + 10\!\cdots\!25$$
$37$ $$T^{4} - 19109 T^{3} + \cdots + 20\!\cdots\!96$$
$41$ $$(T^{2} + 12858 T - 3221280)^{2}$$
$43$ $$(T^{2} - 2771 T - 257902490)^{2}$$
$47$ $$T^{4} + 23160 T^{3} + \cdots + 12\!\cdots\!16$$
$53$ $$T^{4} - 31653 T^{3} + \cdots + 35\!\cdots\!00$$
$59$ $$T^{4} - 41097 T^{3} + \cdots + 28\!\cdots\!44$$
$61$ $$T^{4} + 42052 T^{3} + \cdots + 15\!\cdots\!00$$
$67$ $$T^{4} - 30763 T^{3} + \cdots + 10\!\cdots\!00$$
$71$ $$(T^{2} + 102096 T + 2483190108)^{2}$$
$73$ $$T^{4} - 28577 T^{3} + \cdots + 22\!\cdots\!84$$
$79$ $$T^{4} + 18464 T^{3} + \cdots + 79\!\cdots\!69$$
$83$ $$(T^{2} + 61179 T + 711231498)^{2}$$
$89$ $$T^{4} + 29322 T^{3} + \cdots + 45\!\cdots\!16$$
$97$ $$(T^{2} + 9791 T - 40418570)^{2}$$