# Properties

 Label 336.4.q.d Level $336$ Weight $4$ Character orbit 336.q Analytic conductor $19.825$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,4,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, 4, names="a")

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

chi := DirichletCharacter("336.193");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$19.8246417619$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (3 \zeta_{6} - 3) q^{3} + 15 \zeta_{6} q^{5} + (7 \zeta_{6} - 21) q^{7} - 9 \zeta_{6} q^{9} +O(q^{10})$$ q + (3*z - 3) * q^3 + 15*z * q^5 + (7*z - 21) * q^7 - 9*z * q^9 $$q + (3 \zeta_{6} - 3) q^{3} + 15 \zeta_{6} q^{5} + (7 \zeta_{6} - 21) q^{7} - 9 \zeta_{6} q^{9} + (9 \zeta_{6} - 9) q^{11} - 88 q^{13} - 45 q^{15} + ( - 84 \zeta_{6} + 84) q^{17} + 104 \zeta_{6} q^{19} + ( - 63 \zeta_{6} + 42) q^{21} - 84 \zeta_{6} q^{23} + (100 \zeta_{6} - 100) q^{25} + 27 q^{27} + 51 q^{29} + ( - 185 \zeta_{6} + 185) q^{31} - 27 \zeta_{6} q^{33} + ( - 210 \zeta_{6} - 105) q^{35} - 44 \zeta_{6} q^{37} + ( - 264 \zeta_{6} + 264) q^{39} - 168 q^{41} - 326 q^{43} + ( - 135 \zeta_{6} + 135) q^{45} - 138 \zeta_{6} q^{47} + ( - 245 \zeta_{6} + 392) q^{49} + 252 \zeta_{6} q^{51} + (639 \zeta_{6} - 639) q^{53} - 135 q^{55} - 312 q^{57} + ( - 159 \zeta_{6} + 159) q^{59} - 722 \zeta_{6} q^{61} + (126 \zeta_{6} + 63) q^{63} - 1320 \zeta_{6} q^{65} + (166 \zeta_{6} - 166) q^{67} + 252 q^{69} - 1086 q^{71} + (218 \zeta_{6} - 218) q^{73} - 300 \zeta_{6} q^{75} + ( - 189 \zeta_{6} + 126) q^{77} - 583 \zeta_{6} q^{79} + (81 \zeta_{6} - 81) q^{81} + 597 q^{83} + 1260 q^{85} + (153 \zeta_{6} - 153) q^{87} + 1038 \zeta_{6} q^{89} + ( - 616 \zeta_{6} + 1848) q^{91} + 555 \zeta_{6} q^{93} + (1560 \zeta_{6} - 1560) q^{95} - 169 q^{97} + 81 q^{99} +O(q^{100})$$ q + (3*z - 3) * q^3 + 15*z * q^5 + (7*z - 21) * q^7 - 9*z * q^9 + (9*z - 9) * q^11 - 88 * q^13 - 45 * q^15 + (-84*z + 84) * q^17 + 104*z * q^19 + (-63*z + 42) * q^21 - 84*z * q^23 + (100*z - 100) * q^25 + 27 * q^27 + 51 * q^29 + (-185*z + 185) * q^31 - 27*z * q^33 + (-210*z - 105) * q^35 - 44*z * q^37 + (-264*z + 264) * q^39 - 168 * q^41 - 326 * q^43 + (-135*z + 135) * q^45 - 138*z * q^47 + (-245*z + 392) * q^49 + 252*z * q^51 + (639*z - 639) * q^53 - 135 * q^55 - 312 * q^57 + (-159*z + 159) * q^59 - 722*z * q^61 + (126*z + 63) * q^63 - 1320*z * q^65 + (166*z - 166) * q^67 + 252 * q^69 - 1086 * q^71 + (218*z - 218) * q^73 - 300*z * q^75 + (-189*z + 126) * q^77 - 583*z * q^79 + (81*z - 81) * q^81 + 597 * q^83 + 1260 * q^85 + (153*z - 153) * q^87 + 1038*z * q^89 + (-616*z + 1848) * q^91 + 555*z * q^93 + (1560*z - 1560) * q^95 - 169 * q^97 + 81 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} + 15 q^{5} - 35 q^{7} - 9 q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 + 15 * q^5 - 35 * q^7 - 9 * q^9 $$2 q - 3 q^{3} + 15 q^{5} - 35 q^{7} - 9 q^{9} - 9 q^{11} - 176 q^{13} - 90 q^{15} + 84 q^{17} + 104 q^{19} + 21 q^{21} - 84 q^{23} - 100 q^{25} + 54 q^{27} + 102 q^{29} + 185 q^{31} - 27 q^{33} - 420 q^{35} - 44 q^{37} + 264 q^{39} - 336 q^{41} - 652 q^{43} + 135 q^{45} - 138 q^{47} + 539 q^{49} + 252 q^{51} - 639 q^{53} - 270 q^{55} - 624 q^{57} + 159 q^{59} - 722 q^{61} + 252 q^{63} - 1320 q^{65} - 166 q^{67} + 504 q^{69} - 2172 q^{71} - 218 q^{73} - 300 q^{75} + 63 q^{77} - 583 q^{79} - 81 q^{81} + 1194 q^{83} + 2520 q^{85} - 153 q^{87} + 1038 q^{89} + 3080 q^{91} + 555 q^{93} - 1560 q^{95} - 338 q^{97} + 162 q^{99}+O(q^{100})$$ 2 * q - 3 * q^3 + 15 * q^5 - 35 * q^7 - 9 * q^9 - 9 * q^11 - 176 * q^13 - 90 * q^15 + 84 * q^17 + 104 * q^19 + 21 * q^21 - 84 * q^23 - 100 * q^25 + 54 * q^27 + 102 * q^29 + 185 * q^31 - 27 * q^33 - 420 * q^35 - 44 * q^37 + 264 * q^39 - 336 * q^41 - 652 * q^43 + 135 * q^45 - 138 * q^47 + 539 * q^49 + 252 * q^51 - 639 * q^53 - 270 * q^55 - 624 * q^57 + 159 * q^59 - 722 * q^61 + 252 * q^63 - 1320 * q^65 - 166 * q^67 + 504 * q^69 - 2172 * q^71 - 218 * q^73 - 300 * q^75 + 63 * q^77 - 583 * q^79 - 81 * q^81 + 1194 * q^83 + 2520 * q^85 - 153 * q^87 + 1038 * q^89 + 3080 * q^91 + 555 * q^93 - 1560 * q^95 - 338 * q^97 + 162 * q^99

## Character values

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

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0 −1.50000 + 2.59808i 0 7.50000 + 12.9904i 0 −17.5000 + 6.06218i 0 −4.50000 7.79423i 0
289.1 0 −1.50000 2.59808i 0 7.50000 12.9904i 0 −17.5000 6.06218i 0 −4.50000 + 7.79423i 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 336.4.q.d 2
4.b odd 2 1 42.4.e.b 2
7.c even 3 1 inner 336.4.q.d 2
7.c even 3 1 2352.4.a.u 1
7.d odd 6 1 2352.4.a.q 1
12.b even 2 1 126.4.g.a 2
28.d even 2 1 294.4.e.e 2
28.f even 6 1 294.4.a.g 1
28.f even 6 1 294.4.e.e 2
28.g odd 6 1 42.4.e.b 2
28.g odd 6 1 294.4.a.a 1
84.h odd 2 1 882.4.g.l 2
84.j odd 6 1 882.4.a.h 1
84.j odd 6 1 882.4.g.l 2
84.n even 6 1 126.4.g.a 2
84.n even 6 1 882.4.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.b 2 4.b odd 2 1
42.4.e.b 2 28.g odd 6 1
126.4.g.a 2 12.b even 2 1
126.4.g.a 2 84.n even 6 1
294.4.a.a 1 28.g odd 6 1
294.4.a.g 1 28.f even 6 1
294.4.e.e 2 28.d even 2 1
294.4.e.e 2 28.f even 6 1
336.4.q.d 2 1.a even 1 1 trivial
336.4.q.d 2 7.c even 3 1 inner
882.4.a.h 1 84.j odd 6 1
882.4.a.r 1 84.n even 6 1
882.4.g.l 2 84.h odd 2 1
882.4.g.l 2 84.j odd 6 1
2352.4.a.q 1 7.d odd 6 1
2352.4.a.u 1 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 15T_{5} + 225$$ acting on $$S_{4}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3T + 9$$
$5$ $$T^{2} - 15T + 225$$
$7$ $$T^{2} + 35T + 343$$
$11$ $$T^{2} + 9T + 81$$
$13$ $$(T + 88)^{2}$$
$17$ $$T^{2} - 84T + 7056$$
$19$ $$T^{2} - 104T + 10816$$
$23$ $$T^{2} + 84T + 7056$$
$29$ $$(T - 51)^{2}$$
$31$ $$T^{2} - 185T + 34225$$
$37$ $$T^{2} + 44T + 1936$$
$41$ $$(T + 168)^{2}$$
$43$ $$(T + 326)^{2}$$
$47$ $$T^{2} + 138T + 19044$$
$53$ $$T^{2} + 639T + 408321$$
$59$ $$T^{2} - 159T + 25281$$
$61$ $$T^{2} + 722T + 521284$$
$67$ $$T^{2} + 166T + 27556$$
$71$ $$(T + 1086)^{2}$$
$73$ $$T^{2} + 218T + 47524$$
$79$ $$T^{2} + 583T + 339889$$
$83$ $$(T - 597)^{2}$$
$89$ $$T^{2} - 1038 T + 1077444$$
$97$ $$(T + 169)^{2}$$