# Properties

 Label 338.2.c.b Level $338$ Weight $2$ Character orbit 338.c Analytic conductor $2.699$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 338.c (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.69894358832$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{4} -3 q^{5} + \zeta_{6} q^{6} + 3 \zeta_{6} q^{7} + q^{8} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{4} -3 q^{5} + \zeta_{6} q^{6} + 3 \zeta_{6} q^{7} + q^{8} + 2 \zeta_{6} q^{9} + ( 3 - 3 \zeta_{6} ) q^{10} - q^{12} -3 q^{14} + ( -3 + 3 \zeta_{6} ) q^{15} + ( -1 + \zeta_{6} ) q^{16} + 3 \zeta_{6} q^{17} -2 q^{18} + 6 \zeta_{6} q^{19} + 3 \zeta_{6} q^{20} + 3 q^{21} + ( -6 + 6 \zeta_{6} ) q^{23} + ( 1 - \zeta_{6} ) q^{24} + 4 q^{25} + 5 q^{27} + ( 3 - 3 \zeta_{6} ) q^{28} -3 \zeta_{6} q^{30} -\zeta_{6} q^{32} -3 q^{34} -9 \zeta_{6} q^{35} + ( 2 - 2 \zeta_{6} ) q^{36} + ( 3 - 3 \zeta_{6} ) q^{37} -6 q^{38} -3 q^{40} + ( -3 + 3 \zeta_{6} ) q^{42} -\zeta_{6} q^{43} -6 \zeta_{6} q^{45} -6 \zeta_{6} q^{46} -3 q^{47} + \zeta_{6} q^{48} + ( -2 + 2 \zeta_{6} ) q^{49} + ( -4 + 4 \zeta_{6} ) q^{50} + 3 q^{51} -6 q^{53} + ( -5 + 5 \zeta_{6} ) q^{54} + 3 \zeta_{6} q^{56} + 6 q^{57} -6 \zeta_{6} q^{59} + 3 q^{60} + 8 \zeta_{6} q^{61} + ( -6 + 6 \zeta_{6} ) q^{63} + q^{64} + ( 12 - 12 \zeta_{6} ) q^{67} + ( 3 - 3 \zeta_{6} ) q^{68} + 6 \zeta_{6} q^{69} + 9 q^{70} -15 \zeta_{6} q^{71} + 2 \zeta_{6} q^{72} -6 q^{73} + 3 \zeta_{6} q^{74} + ( 4 - 4 \zeta_{6} ) q^{75} + ( 6 - 6 \zeta_{6} ) q^{76} + 10 q^{79} + ( 3 - 3 \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} + 6 q^{83} -3 \zeta_{6} q^{84} -9 \zeta_{6} q^{85} + q^{86} + ( -6 + 6 \zeta_{6} ) q^{89} + 6 q^{90} + 6 q^{92} + ( 3 - 3 \zeta_{6} ) q^{94} -18 \zeta_{6} q^{95} - q^{96} + 12 \zeta_{6} q^{97} -2 \zeta_{6} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + q^{3} - q^{4} - 6q^{5} + q^{6} + 3q^{7} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - q^{2} + q^{3} - q^{4} - 6q^{5} + q^{6} + 3q^{7} + 2q^{8} + 2q^{9} + 3q^{10} - 2q^{12} - 6q^{14} - 3q^{15} - q^{16} + 3q^{17} - 4q^{18} + 6q^{19} + 3q^{20} + 6q^{21} - 6q^{23} + q^{24} + 8q^{25} + 10q^{27} + 3q^{28} - 3q^{30} - q^{32} - 6q^{34} - 9q^{35} + 2q^{36} + 3q^{37} - 12q^{38} - 6q^{40} - 3q^{42} - q^{43} - 6q^{45} - 6q^{46} - 6q^{47} + q^{48} - 2q^{49} - 4q^{50} + 6q^{51} - 12q^{53} - 5q^{54} + 3q^{56} + 12q^{57} - 6q^{59} + 6q^{60} + 8q^{61} - 6q^{63} + 2q^{64} + 12q^{67} + 3q^{68} + 6q^{69} + 18q^{70} - 15q^{71} + 2q^{72} - 12q^{73} + 3q^{74} + 4q^{75} + 6q^{76} + 20q^{79} + 3q^{80} - q^{81} + 12q^{83} - 3q^{84} - 9q^{85} + 2q^{86} - 6q^{89} + 12q^{90} + 12q^{92} + 3q^{94} - 18q^{95} - 2q^{96} + 12q^{97} - 2q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/338\mathbb{Z}\right)^\times$$.

 $$n$$ $$171$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i −3.00000 0.500000 + 0.866025i 1.50000 + 2.59808i 1.00000 1.00000 + 1.73205i 1.50000 2.59808i
315.1 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −3.00000 0.500000 0.866025i 1.50000 2.59808i 1.00000 1.00000 1.73205i 1.50000 + 2.59808i
 $$n$$: 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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.2.c.b 2
13.b even 2 1 338.2.c.f 2
13.c even 3 1 338.2.a.d 1
13.c even 3 1 inner 338.2.c.b 2
13.d odd 4 2 338.2.e.c 4
13.e even 6 1 338.2.a.b 1
13.e even 6 1 338.2.c.f 2
13.f odd 12 2 26.2.b.a 2
13.f odd 12 2 338.2.e.c 4
39.h odd 6 1 3042.2.a.j 1
39.i odd 6 1 3042.2.a.g 1
39.k even 12 2 234.2.b.b 2
52.i odd 6 1 2704.2.a.k 1
52.j odd 6 1 2704.2.a.j 1
52.l even 12 2 208.2.f.a 2
65.l even 6 1 8450.2.a.u 1
65.n even 6 1 8450.2.a.h 1
65.o even 12 2 650.2.c.d 2
65.s odd 12 2 650.2.d.b 2
65.t even 12 2 650.2.c.a 2
91.w even 12 2 1274.2.n.c 4
91.x odd 12 2 1274.2.n.d 4
91.ba even 12 2 1274.2.n.c 4
91.bc even 12 2 1274.2.d.c 2
91.bd odd 12 2 1274.2.n.d 4
104.u even 12 2 832.2.f.b 2
104.x odd 12 2 832.2.f.d 2
156.v odd 12 2 1872.2.c.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.b.a 2 13.f odd 12 2
208.2.f.a 2 52.l even 12 2
234.2.b.b 2 39.k even 12 2
338.2.a.b 1 13.e even 6 1
338.2.a.d 1 13.c even 3 1
338.2.c.b 2 1.a even 1 1 trivial
338.2.c.b 2 13.c even 3 1 inner
338.2.c.f 2 13.b even 2 1
338.2.c.f 2 13.e even 6 1
338.2.e.c 4 13.d odd 4 2
338.2.e.c 4 13.f odd 12 2
650.2.c.a 2 65.t even 12 2
650.2.c.d 2 65.o even 12 2
650.2.d.b 2 65.s odd 12 2
832.2.f.b 2 104.u even 12 2
832.2.f.d 2 104.x odd 12 2
1274.2.d.c 2 91.bc even 12 2
1274.2.n.c 4 91.w even 12 2
1274.2.n.c 4 91.ba even 12 2
1274.2.n.d 4 91.x odd 12 2
1274.2.n.d 4 91.bd odd 12 2
1872.2.c.f 2 156.v odd 12 2
2704.2.a.j 1 52.j odd 6 1
2704.2.a.k 1 52.i odd 6 1
3042.2.a.g 1 39.i odd 6 1
3042.2.a.j 1 39.h odd 6 1
8450.2.a.h 1 65.n even 6 1
8450.2.a.u 1 65.l even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(338, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ $$T_{5} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$( 3 + T )^{2}$$
$7$ $$9 - 3 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$9 - 3 T + T^{2}$$
$19$ $$36 - 6 T + T^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$9 - 3 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$1 + T + T^{2}$$
$47$ $$( 3 + T )^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$36 + 6 T + T^{2}$$
$61$ $$64 - 8 T + T^{2}$$
$67$ $$144 - 12 T + T^{2}$$
$71$ $$225 + 15 T + T^{2}$$
$73$ $$( 6 + T )^{2}$$
$79$ $$( -10 + T )^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$36 + 6 T + T^{2}$$
$97$ $$144 - 12 T + T^{2}$$