Properties

 Label 338.2.e.c Level $338$ Weight $2$ Character orbit 338.e Analytic conductor $2.699$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 338.e (of order $$6$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$2.69894358832$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

Character values

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

 $$n$$ $$171$$ $$\chi(n)$$ $$\zeta_{12}^{2}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 3.00000i −0.866025 0.500000i 2.59808 + 1.50000i 1.00000i 1.00000 1.73205i −1.50000 2.59808i
23.2 0.866025 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 3.00000i 0.866025 + 0.500000i −2.59808 1.50000i 1.00000i 1.00000 1.73205i −1.50000 2.59808i
147.1 −0.866025 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 3.00000i −0.866025 + 0.500000i 2.59808 1.50000i 1.00000i 1.00000 + 1.73205i −1.50000 + 2.59808i
147.2 0.866025 + 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 3.00000i 0.866025 0.500000i −2.59808 + 1.50000i 1.00000i 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.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

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

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(338, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$( 1 - T + T^{2} )^{2}$$
$5$ $$( 9 + T^{2} )^{2}$$
$7$ $$81 - 9 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$( 9 + 3 T + T^{2} )^{2}$$
$19$ $$1296 - 36 T^{2} + T^{4}$$
$23$ $$( 36 - 6 T + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$81 - 9 T^{2} + T^{4}$$
$41$ $$T^{4}$$
$43$ $$( 1 - T + T^{2} )^{2}$$
$47$ $$( 9 + T^{2} )^{2}$$
$53$ $$( 6 + T )^{4}$$
$59$ $$1296 - 36 T^{2} + T^{4}$$
$61$ $$( 64 - 8 T + T^{2} )^{2}$$
$67$ $$20736 - 144 T^{2} + T^{4}$$
$71$ $$50625 - 225 T^{2} + T^{4}$$
$73$ $$( 36 + T^{2} )^{2}$$
$79$ $$( -10 + T )^{4}$$
$83$ $$( 36 + T^{2} )^{2}$$
$89$ $$1296 - 36 T^{2} + T^{4}$$
$97$ $$20736 - 144 T^{2} + T^{4}$$