# Properties

 Label 336.2.q.d Level $336$ Weight $2$ Character orbit 336.q Analytic conductor $2.683$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 336.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.68297350792$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} -3 \zeta_{6} q^{5} + ( -3 + \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} -3 \zeta_{6} q^{5} + ( -3 + \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( 3 - 3 \zeta_{6} ) q^{11} -4 q^{13} -3 q^{15} -4 \zeta_{6} q^{19} + ( -2 + 3 \zeta_{6} ) q^{21} + ( -4 + 4 \zeta_{6} ) q^{25} - q^{27} + 9 q^{29} + ( -1 + \zeta_{6} ) q^{31} -3 \zeta_{6} q^{33} + ( 3 + 6 \zeta_{6} ) q^{35} -8 \zeta_{6} q^{37} + ( -4 + 4 \zeta_{6} ) q^{39} + 10 q^{43} + ( -3 + 3 \zeta_{6} ) q^{45} -6 \zeta_{6} q^{47} + ( 8 - 5 \zeta_{6} ) q^{49} + ( 3 - 3 \zeta_{6} ) q^{53} -9 q^{55} -4 q^{57} + ( 3 - 3 \zeta_{6} ) q^{59} + 10 \zeta_{6} q^{61} + ( 1 + 2 \zeta_{6} ) q^{63} + 12 \zeta_{6} q^{65} + ( -10 + 10 \zeta_{6} ) q^{67} + 6 q^{71} + ( -2 + 2 \zeta_{6} ) q^{73} + 4 \zeta_{6} q^{75} + ( -6 + 9 \zeta_{6} ) q^{77} -\zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 9 q^{83} + ( 9 - 9 \zeta_{6} ) q^{87} -6 \zeta_{6} q^{89} + ( 12 - 4 \zeta_{6} ) q^{91} + \zeta_{6} q^{93} + ( -12 + 12 \zeta_{6} ) q^{95} - q^{97} -3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} - 3q^{5} - 5q^{7} - q^{9} + O(q^{10})$$ $$2q + q^{3} - 3q^{5} - 5q^{7} - q^{9} + 3q^{11} - 8q^{13} - 6q^{15} - 4q^{19} - q^{21} - 4q^{25} - 2q^{27} + 18q^{29} - q^{31} - 3q^{33} + 12q^{35} - 8q^{37} - 4q^{39} + 20q^{43} - 3q^{45} - 6q^{47} + 11q^{49} + 3q^{53} - 18q^{55} - 8q^{57} + 3q^{59} + 10q^{61} + 4q^{63} + 12q^{65} - 10q^{67} + 12q^{71} - 2q^{73} + 4q^{75} - 3q^{77} - q^{79} - q^{81} + 18q^{83} + 9q^{87} - 6q^{89} + 20q^{91} + q^{93} - 12q^{95} - 2q^{97} - 6q^{99} + O(q^{100})$$

## 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.

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 0.500000 0.866025i 0 −1.50000 2.59808i 0 −2.50000 + 0.866025i 0 −0.500000 0.866025i 0
289.1 0 0.500000 + 0.866025i 0 −1.50000 + 2.59808i 0 −2.50000 0.866025i 0 −0.500000 + 0.866025i 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 336.2.q.d 2
3.b odd 2 1 1008.2.s.n 2
4.b odd 2 1 42.2.e.b 2
7.b odd 2 1 2352.2.q.m 2
7.c even 3 1 inner 336.2.q.d 2
7.c even 3 1 2352.2.a.m 1
7.d odd 6 1 2352.2.a.n 1
7.d odd 6 1 2352.2.q.m 2
8.b even 2 1 1344.2.q.j 2
8.d odd 2 1 1344.2.q.v 2
12.b even 2 1 126.2.g.b 2
20.d odd 2 1 1050.2.i.e 2
20.e even 4 2 1050.2.o.b 4
21.g even 6 1 7056.2.a.bz 1
21.h odd 6 1 1008.2.s.n 2
21.h odd 6 1 7056.2.a.g 1
28.d even 2 1 294.2.e.f 2
28.f even 6 1 294.2.a.a 1
28.f even 6 1 294.2.e.f 2
28.g odd 6 1 42.2.e.b 2
28.g odd 6 1 294.2.a.d 1
36.f odd 6 1 1134.2.e.a 2
36.f odd 6 1 1134.2.h.p 2
36.h even 6 1 1134.2.e.p 2
36.h even 6 1 1134.2.h.a 2
56.j odd 6 1 9408.2.a.bm 1
56.k odd 6 1 1344.2.q.v 2
56.k odd 6 1 9408.2.a.d 1
56.m even 6 1 9408.2.a.db 1
56.p even 6 1 1344.2.q.j 2
56.p even 6 1 9408.2.a.bu 1
84.h odd 2 1 882.2.g.b 2
84.j odd 6 1 882.2.a.k 1
84.j odd 6 1 882.2.g.b 2
84.n even 6 1 126.2.g.b 2
84.n even 6 1 882.2.a.g 1
140.p odd 6 1 1050.2.i.e 2
140.p odd 6 1 7350.2.a.ce 1
140.s even 6 1 7350.2.a.cw 1
140.w even 12 2 1050.2.o.b 4
252.o even 6 1 1134.2.e.p 2
252.u odd 6 1 1134.2.h.p 2
252.bb even 6 1 1134.2.h.a 2
252.bl odd 6 1 1134.2.e.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.b 2 4.b odd 2 1
42.2.e.b 2 28.g odd 6 1
126.2.g.b 2 12.b even 2 1
126.2.g.b 2 84.n even 6 1
294.2.a.a 1 28.f even 6 1
294.2.a.d 1 28.g odd 6 1
294.2.e.f 2 28.d even 2 1
294.2.e.f 2 28.f even 6 1
336.2.q.d 2 1.a even 1 1 trivial
336.2.q.d 2 7.c even 3 1 inner
882.2.a.g 1 84.n even 6 1
882.2.a.k 1 84.j odd 6 1
882.2.g.b 2 84.h odd 2 1
882.2.g.b 2 84.j odd 6 1
1008.2.s.n 2 3.b odd 2 1
1008.2.s.n 2 21.h odd 6 1
1050.2.i.e 2 20.d odd 2 1
1050.2.i.e 2 140.p odd 6 1
1050.2.o.b 4 20.e even 4 2
1050.2.o.b 4 140.w even 12 2
1134.2.e.a 2 36.f odd 6 1
1134.2.e.a 2 252.bl odd 6 1
1134.2.e.p 2 36.h even 6 1
1134.2.e.p 2 252.o even 6 1
1134.2.h.a 2 36.h even 6 1
1134.2.h.a 2 252.bb even 6 1
1134.2.h.p 2 36.f odd 6 1
1134.2.h.p 2 252.u odd 6 1
1344.2.q.j 2 8.b even 2 1
1344.2.q.j 2 56.p even 6 1
1344.2.q.v 2 8.d odd 2 1
1344.2.q.v 2 56.k odd 6 1
2352.2.a.m 1 7.c even 3 1
2352.2.a.n 1 7.d odd 6 1
2352.2.q.m 2 7.b odd 2 1
2352.2.q.m 2 7.d odd 6 1
7056.2.a.g 1 21.h odd 6 1
7056.2.a.bz 1 21.g even 6 1
7350.2.a.ce 1 140.p odd 6 1
7350.2.a.cw 1 140.s even 6 1
9408.2.a.d 1 56.k odd 6 1
9408.2.a.bm 1 56.j odd 6 1
9408.2.a.bu 1 56.p even 6 1
9408.2.a.db 1 56.m even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(336, [\chi])$$:

 $$T_{5}^{2} + 3 T_{5} + 9$$ $$T_{11}^{2} - 3 T_{11} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T + T^{2}$$
$5$ $$1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}$$
$7$ $$1 + 5 T + 7 T^{2}$$
$11$ $$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$
$13$ $$( 1 + 4 T + 13 T^{2} )^{2}$$
$17$ $$1 - 17 T^{2} + 289 T^{4}$$
$19$ $$1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4}$$
$23$ $$1 - 23 T^{2} + 529 T^{4}$$
$29$ $$( 1 - 9 T + 29 T^{2} )^{2}$$
$31$ $$1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4}$$
$37$ $$1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 - 10 T + 43 T^{2} )^{2}$$
$47$ $$1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4}$$
$53$ $$1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4}$$
$59$ $$1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4}$$
$61$ $$1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4}$$
$67$ $$1 + 10 T + 33 T^{2} + 670 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 6 T + 71 T^{2} )^{2}$$
$73$ $$1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4}$$
$79$ $$1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4}$$
$83$ $$( 1 - 9 T + 83 T^{2} )^{2}$$
$89$ $$1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + T + 97 T^{2} )^{2}$$