# Properties

 Label 338.2.e.a 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: $$x^{4} - x^{2} + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + (\zeta_{12}^{2} - 1) q^{3} + \zeta_{12}^{2} q^{4} - 3 \zeta_{12}^{3} q^{5} + (\zeta_{12}^{3} - \zeta_{12}) q^{6} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + 2 \zeta_{12}^{2} q^{9} +O(q^{10})$$ q + z * q^2 + (z^2 - 1) * q^3 + z^2 * q^4 - 3*z^3 * q^5 + (z^3 - z) * q^6 + (-z^3 + z) * q^7 + z^3 * q^8 + 2*z^2 * q^9 $$q + \zeta_{12} q^{2} + (\zeta_{12}^{2} - 1) q^{3} + \zeta_{12}^{2} q^{4} - 3 \zeta_{12}^{3} q^{5} + (\zeta_{12}^{3} - \zeta_{12}) q^{6} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + 2 \zeta_{12}^{2} q^{9} + ( - 3 \zeta_{12}^{2} + 3) q^{10} + 6 \zeta_{12} q^{11} - q^{12} + q^{14} + 3 \zeta_{12} q^{15} + (\zeta_{12}^{2} - 1) q^{16} - 3 \zeta_{12}^{2} q^{17} + 2 \zeta_{12}^{3} q^{18} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{19} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{20} + \zeta_{12}^{3} q^{21} + 6 \zeta_{12}^{2} q^{22} - \zeta_{12} q^{24} - 4 q^{25} - 5 q^{27} + \zeta_{12} q^{28} + (6 \zeta_{12}^{2} - 6) q^{29} + 3 \zeta_{12}^{2} q^{30} - 4 \zeta_{12}^{3} q^{31} + (\zeta_{12}^{3} - \zeta_{12}) q^{32} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{33} - 3 \zeta_{12}^{3} q^{34} - 3 \zeta_{12}^{2} q^{35} + (2 \zeta_{12}^{2} - 2) q^{36} - 7 \zeta_{12} q^{37} + 2 q^{38} + 3 q^{40} + (\zeta_{12}^{2} - 1) q^{42} - \zeta_{12}^{2} q^{43} + 6 \zeta_{12}^{3} q^{44} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{45} - 3 \zeta_{12}^{3} q^{47} - \zeta_{12}^{2} q^{48} + (6 \zeta_{12}^{2} - 6) q^{49} - 4 \zeta_{12} q^{50} + 3 q^{51} - 5 \zeta_{12} q^{54} + ( - 18 \zeta_{12}^{2} + 18) q^{55} + \zeta_{12}^{2} q^{56} + 2 \zeta_{12}^{3} q^{57} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{58} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{59} + 3 \zeta_{12}^{3} q^{60} - 8 \zeta_{12}^{2} q^{61} + ( - 4 \zeta_{12}^{2} + 4) q^{62} + 2 \zeta_{12} q^{63} - q^{64} - 6 q^{66} - 14 \zeta_{12} q^{67} + ( - 3 \zeta_{12}^{2} + 3) q^{68} - 3 \zeta_{12}^{3} q^{70} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{71} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{72} - 2 \zeta_{12}^{3} q^{73} - 7 \zeta_{12}^{2} q^{74} + ( - 4 \zeta_{12}^{2} + 4) q^{75} + 2 \zeta_{12} q^{76} + 6 q^{77} + 8 q^{79} + 3 \zeta_{12} q^{80} + (\zeta_{12}^{2} - 1) q^{81} + 12 \zeta_{12}^{3} q^{83} + (\zeta_{12}^{3} - \zeta_{12}) q^{84} + (9 \zeta_{12}^{3} - 9 \zeta_{12}) q^{85} - \zeta_{12}^{3} q^{86} - 6 \zeta_{12}^{2} q^{87} + (6 \zeta_{12}^{2} - 6) q^{88} - 6 \zeta_{12} q^{89} + 6 q^{90} + 4 \zeta_{12} q^{93} + ( - 3 \zeta_{12}^{2} + 3) q^{94} - 6 \zeta_{12}^{2} q^{95} - \zeta_{12}^{3} q^{96} + (10 \zeta_{12}^{3} - 10 \zeta_{12}) q^{97} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{98} + 12 \zeta_{12}^{3} q^{99} +O(q^{100})$$ q + z * q^2 + (z^2 - 1) * q^3 + z^2 * q^4 - 3*z^3 * q^5 + (z^3 - z) * q^6 + (-z^3 + z) * q^7 + z^3 * q^8 + 2*z^2 * q^9 + (-3*z^2 + 3) * q^10 + 6*z * q^11 - q^12 + q^14 + 3*z * q^15 + (z^2 - 1) * q^16 - 3*z^2 * q^17 + 2*z^3 * q^18 + (-2*z^3 + 2*z) * q^19 + (-3*z^3 + 3*z) * q^20 + z^3 * q^21 + 6*z^2 * q^22 - z * q^24 - 4 * q^25 - 5 * q^27 + z * q^28 + (6*z^2 - 6) * q^29 + 3*z^2 * q^30 - 4*z^3 * q^31 + (z^3 - z) * q^32 + (6*z^3 - 6*z) * q^33 - 3*z^3 * q^34 - 3*z^2 * q^35 + (2*z^2 - 2) * q^36 - 7*z * q^37 + 2 * q^38 + 3 * q^40 + (z^2 - 1) * q^42 - z^2 * q^43 + 6*z^3 * q^44 + (-6*z^3 + 6*z) * q^45 - 3*z^3 * q^47 - z^2 * q^48 + (6*z^2 - 6) * q^49 - 4*z * q^50 + 3 * q^51 - 5*z * q^54 + (-18*z^2 + 18) * q^55 + z^2 * q^56 + 2*z^3 * q^57 + (6*z^3 - 6*z) * q^58 + (-6*z^3 + 6*z) * q^59 + 3*z^3 * q^60 - 8*z^2 * q^61 + (-4*z^2 + 4) * q^62 + 2*z * q^63 - q^64 - 6 * q^66 - 14*z * q^67 + (-3*z^2 + 3) * q^68 - 3*z^3 * q^70 + (3*z^3 - 3*z) * q^71 + (2*z^3 - 2*z) * q^72 - 2*z^3 * q^73 - 7*z^2 * q^74 + (-4*z^2 + 4) * q^75 + 2*z * q^76 + 6 * q^77 + 8 * q^79 + 3*z * q^80 + (z^2 - 1) * q^81 + 12*z^3 * q^83 + (z^3 - z) * q^84 + (9*z^3 - 9*z) * q^85 - z^3 * q^86 - 6*z^2 * q^87 + (6*z^2 - 6) * q^88 - 6*z * q^89 + 6 * q^90 + 4*z * q^93 + (-3*z^2 + 3) * q^94 - 6*z^2 * q^95 - z^3 * q^96 + (10*z^3 - 10*z) * q^97 + (6*z^3 - 6*z) * q^98 + 12*z^3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} + 2 q^{4} + 4 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 + 2 * q^4 + 4 * q^9 $$4 q - 2 q^{3} + 2 q^{4} + 4 q^{9} + 6 q^{10} - 4 q^{12} + 4 q^{14} - 2 q^{16} - 6 q^{17} + 12 q^{22} - 16 q^{25} - 20 q^{27} - 12 q^{29} + 6 q^{30} - 6 q^{35} - 4 q^{36} + 8 q^{38} + 12 q^{40} - 2 q^{42} - 2 q^{43} - 2 q^{48} - 12 q^{49} + 12 q^{51} + 36 q^{55} + 2 q^{56} - 16 q^{61} + 8 q^{62} - 4 q^{64} - 24 q^{66} + 6 q^{68} - 14 q^{74} + 8 q^{75} + 24 q^{77} + 32 q^{79} - 2 q^{81} - 12 q^{87} - 12 q^{88} + 24 q^{90} + 6 q^{94} - 12 q^{95}+O(q^{100})$$ 4 * q - 2 * q^3 + 2 * q^4 + 4 * q^9 + 6 * q^10 - 4 * q^12 + 4 * q^14 - 2 * q^16 - 6 * q^17 + 12 * q^22 - 16 * q^25 - 20 * q^27 - 12 * q^29 + 6 * q^30 - 6 * q^35 - 4 * q^36 + 8 * q^38 + 12 * q^40 - 2 * q^42 - 2 * q^43 - 2 * q^48 - 12 * q^49 + 12 * q^51 + 36 * q^55 + 2 * q^56 - 16 * q^61 + 8 * q^62 - 4 * q^64 - 24 * q^66 + 6 * q^68 - 14 * q^74 + 8 * q^75 + 24 * q^77 + 32 * q^79 - 2 * q^81 - 12 * q^87 - 12 * q^88 + 24 * q^90 + 6 * q^94 - 12 * q^95

## 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 −0.866025 0.500000i 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 0.866025 + 0.500000i 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 −0.866025 + 0.500000i 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 0.866025 0.500000i 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.a 4
13.b even 2 1 inner 338.2.e.a 4
13.c even 3 1 338.2.b.c 2
13.c even 3 1 inner 338.2.e.a 4
13.d odd 4 1 338.2.c.a 2
13.d odd 4 1 338.2.c.d 2
13.e even 6 1 338.2.b.c 2
13.e even 6 1 inner 338.2.e.a 4
13.f odd 12 1 26.2.a.a 1
13.f odd 12 1 338.2.a.f 1
13.f odd 12 1 338.2.c.a 2
13.f odd 12 1 338.2.c.d 2
39.h odd 6 1 3042.2.b.a 2
39.i odd 6 1 3042.2.b.a 2
39.k even 12 1 234.2.a.e 1
39.k even 12 1 3042.2.a.a 1
52.i odd 6 1 2704.2.f.d 2
52.j odd 6 1 2704.2.f.d 2
52.l even 12 1 208.2.a.a 1
52.l even 12 1 2704.2.a.f 1
65.o even 12 1 650.2.b.d 2
65.s odd 12 1 650.2.a.j 1
65.s odd 12 1 8450.2.a.c 1
65.t even 12 1 650.2.b.d 2
91.w even 12 1 1274.2.f.r 2
91.x odd 12 1 1274.2.f.p 2
91.ba even 12 1 1274.2.f.r 2
91.bc even 12 1 1274.2.a.d 1
91.bd odd 12 1 1274.2.f.p 2
104.u even 12 1 832.2.a.i 1
104.x odd 12 1 832.2.a.d 1
117.w odd 12 1 2106.2.e.ba 2
117.x even 12 1 2106.2.e.b 2
117.bb odd 12 1 2106.2.e.ba 2
117.bc even 12 1 2106.2.e.b 2
143.o even 12 1 3146.2.a.n 1
156.v odd 12 1 1872.2.a.q 1
195.bc odd 12 1 5850.2.e.a 2
195.bh even 12 1 5850.2.a.p 1
195.bn odd 12 1 5850.2.e.a 2
208.be odd 12 1 3328.2.b.m 2
208.bf even 12 1 3328.2.b.j 2
208.bk even 12 1 3328.2.b.j 2
208.bl odd 12 1 3328.2.b.m 2
221.w odd 12 1 7514.2.a.c 1
247.bd even 12 1 9386.2.a.j 1
260.bc even 12 1 5200.2.a.x 1
312.bo even 12 1 7488.2.a.g 1
312.bq odd 12 1 7488.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.a 1 13.f odd 12 1
208.2.a.a 1 52.l even 12 1
234.2.a.e 1 39.k even 12 1
338.2.a.f 1 13.f odd 12 1
338.2.b.c 2 13.c even 3 1
338.2.b.c 2 13.e even 6 1
338.2.c.a 2 13.d odd 4 1
338.2.c.a 2 13.f odd 12 1
338.2.c.d 2 13.d odd 4 1
338.2.c.d 2 13.f odd 12 1
338.2.e.a 4 1.a even 1 1 trivial
338.2.e.a 4 13.b even 2 1 inner
338.2.e.a 4 13.c even 3 1 inner
338.2.e.a 4 13.e even 6 1 inner
650.2.a.j 1 65.s odd 12 1
650.2.b.d 2 65.o even 12 1
650.2.b.d 2 65.t even 12 1
832.2.a.d 1 104.x odd 12 1
832.2.a.i 1 104.u even 12 1
1274.2.a.d 1 91.bc even 12 1
1274.2.f.p 2 91.x odd 12 1
1274.2.f.p 2 91.bd odd 12 1
1274.2.f.r 2 91.w even 12 1
1274.2.f.r 2 91.ba even 12 1
1872.2.a.q 1 156.v odd 12 1
2106.2.e.b 2 117.x even 12 1
2106.2.e.b 2 117.bc even 12 1
2106.2.e.ba 2 117.w odd 12 1
2106.2.e.ba 2 117.bb odd 12 1
2704.2.a.f 1 52.l even 12 1
2704.2.f.d 2 52.i odd 6 1
2704.2.f.d 2 52.j odd 6 1
3042.2.a.a 1 39.k even 12 1
3042.2.b.a 2 39.h odd 6 1
3042.2.b.a 2 39.i odd 6 1
3146.2.a.n 1 143.o even 12 1
3328.2.b.j 2 208.bf even 12 1
3328.2.b.j 2 208.bk even 12 1
3328.2.b.m 2 208.be odd 12 1
3328.2.b.m 2 208.bl odd 12 1
5200.2.a.x 1 260.bc even 12 1
5850.2.a.p 1 195.bh even 12 1
5850.2.e.a 2 195.bc odd 12 1
5850.2.e.a 2 195.bn odd 12 1
7488.2.a.g 1 312.bo even 12 1
7488.2.a.h 1 312.bq odd 12 1
7514.2.a.c 1 221.w odd 12 1
8450.2.a.c 1 65.s odd 12 1
9386.2.a.j 1 247.bd even 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

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$(T^{2} + 9)^{2}$$
$7$ $$T^{4} - T^{2} + 1$$
$11$ $$T^{4} - 36T^{2} + 1296$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 3 T + 9)^{2}$$
$19$ $$T^{4} - 4T^{2} + 16$$
$23$ $$T^{4}$$
$29$ $$(T^{2} + 6 T + 36)^{2}$$
$31$ $$(T^{2} + 16)^{2}$$
$37$ $$T^{4} - 49T^{2} + 2401$$
$41$ $$T^{4}$$
$43$ $$(T^{2} + T + 1)^{2}$$
$47$ $$(T^{2} + 9)^{2}$$
$53$ $$T^{4}$$
$59$ $$T^{4} - 36T^{2} + 1296$$
$61$ $$(T^{2} + 8 T + 64)^{2}$$
$67$ $$T^{4} - 196 T^{2} + 38416$$
$71$ $$T^{4} - 9T^{2} + 81$$
$73$ $$(T^{2} + 4)^{2}$$
$79$ $$(T - 8)^{4}$$
$83$ $$(T^{2} + 144)^{2}$$
$89$ $$T^{4} - 36T^{2} + 1296$$
$97$ $$T^{4} - 100 T^{2} + 10000$$