# Properties

 Label 338.2.c.g Level $338$ Weight $2$ Character orbit 338.c Analytic conductor $2.699$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,2,Mod(191,338)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("338.191");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 338.c (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: $$2.69894358832$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} + ( - 3 \zeta_{6} + 3) q^{3} - \zeta_{6} q^{4} + q^{5} - 3 \zeta_{6} q^{6} + \zeta_{6} q^{7} - q^{8} - 6 \zeta_{6} q^{9} +O(q^{10})$$ q + (-z + 1) * q^2 + (-3*z + 3) * q^3 - z * q^4 + q^5 - 3*z * q^6 + z * q^7 - q^8 - 6*z * q^9 $$q + ( - \zeta_{6} + 1) q^{2} + ( - 3 \zeta_{6} + 3) q^{3} - \zeta_{6} q^{4} + q^{5} - 3 \zeta_{6} q^{6} + \zeta_{6} q^{7} - q^{8} - 6 \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{10} + (2 \zeta_{6} - 2) q^{11} - 3 q^{12} + q^{14} + ( - 3 \zeta_{6} + 3) q^{15} + (\zeta_{6} - 1) q^{16} + 3 \zeta_{6} q^{17} - 6 q^{18} + 6 \zeta_{6} q^{19} - \zeta_{6} q^{20} + 3 q^{21} + 2 \zeta_{6} q^{22} + ( - 4 \zeta_{6} + 4) q^{23} + (3 \zeta_{6} - 3) q^{24} - 4 q^{25} - 9 q^{27} + ( - \zeta_{6} + 1) q^{28} + (2 \zeta_{6} - 2) q^{29} - 3 \zeta_{6} q^{30} - 4 q^{31} + \zeta_{6} q^{32} + 6 \zeta_{6} q^{33} + 3 q^{34} + \zeta_{6} q^{35} + (6 \zeta_{6} - 6) q^{36} + ( - 3 \zeta_{6} + 3) q^{37} + 6 q^{38} - q^{40} + ( - 3 \zeta_{6} + 3) q^{42} + 5 \zeta_{6} q^{43} + 2 q^{44} - 6 \zeta_{6} q^{45} - 4 \zeta_{6} q^{46} - 13 q^{47} + 3 \zeta_{6} q^{48} + ( - 6 \zeta_{6} + 6) q^{49} + (4 \zeta_{6} - 4) q^{50} + 9 q^{51} + 12 q^{53} + (9 \zeta_{6} - 9) q^{54} + (2 \zeta_{6} - 2) q^{55} - \zeta_{6} q^{56} + 18 q^{57} + 2 \zeta_{6} q^{58} - 10 \zeta_{6} q^{59} - 3 q^{60} + 8 \zeta_{6} q^{61} + (4 \zeta_{6} - 4) q^{62} + ( - 6 \zeta_{6} + 6) q^{63} + q^{64} + 6 q^{66} + (2 \zeta_{6} - 2) q^{67} + ( - 3 \zeta_{6} + 3) q^{68} - 12 \zeta_{6} q^{69} + q^{70} - 5 \zeta_{6} q^{71} + 6 \zeta_{6} q^{72} + 10 q^{73} - 3 \zeta_{6} q^{74} + (12 \zeta_{6} - 12) q^{75} + ( - 6 \zeta_{6} + 6) q^{76} - 2 q^{77} - 4 q^{79} + (\zeta_{6} - 1) q^{80} + (9 \zeta_{6} - 9) q^{81} - 3 \zeta_{6} q^{84} + 3 \zeta_{6} q^{85} + 5 q^{86} + 6 \zeta_{6} q^{87} + ( - 2 \zeta_{6} + 2) q^{88} + ( - 6 \zeta_{6} + 6) q^{89} - 6 q^{90} - 4 q^{92} + (12 \zeta_{6} - 12) q^{93} + (13 \zeta_{6} - 13) q^{94} + 6 \zeta_{6} q^{95} + 3 q^{96} + 14 \zeta_{6} q^{97} - 6 \zeta_{6} q^{98} + 12 q^{99} +O(q^{100})$$ q + (-z + 1) * q^2 + (-3*z + 3) * q^3 - z * q^4 + q^5 - 3*z * q^6 + z * q^7 - q^8 - 6*z * q^9 + (-z + 1) * q^10 + (2*z - 2) * q^11 - 3 * q^12 + q^14 + (-3*z + 3) * q^15 + (z - 1) * q^16 + 3*z * q^17 - 6 * q^18 + 6*z * q^19 - z * q^20 + 3 * q^21 + 2*z * q^22 + (-4*z + 4) * q^23 + (3*z - 3) * q^24 - 4 * q^25 - 9 * q^27 + (-z + 1) * q^28 + (2*z - 2) * q^29 - 3*z * q^30 - 4 * q^31 + z * q^32 + 6*z * q^33 + 3 * q^34 + z * q^35 + (6*z - 6) * q^36 + (-3*z + 3) * q^37 + 6 * q^38 - q^40 + (-3*z + 3) * q^42 + 5*z * q^43 + 2 * q^44 - 6*z * q^45 - 4*z * q^46 - 13 * q^47 + 3*z * q^48 + (-6*z + 6) * q^49 + (4*z - 4) * q^50 + 9 * q^51 + 12 * q^53 + (9*z - 9) * q^54 + (2*z - 2) * q^55 - z * q^56 + 18 * q^57 + 2*z * q^58 - 10*z * q^59 - 3 * q^60 + 8*z * q^61 + (4*z - 4) * q^62 + (-6*z + 6) * q^63 + q^64 + 6 * q^66 + (2*z - 2) * q^67 + (-3*z + 3) * q^68 - 12*z * q^69 + q^70 - 5*z * q^71 + 6*z * q^72 + 10 * q^73 - 3*z * q^74 + (12*z - 12) * q^75 + (-6*z + 6) * q^76 - 2 * q^77 - 4 * q^79 + (z - 1) * q^80 + (9*z - 9) * q^81 - 3*z * q^84 + 3*z * q^85 + 5 * q^86 + 6*z * q^87 + (-2*z + 2) * q^88 + (-6*z + 6) * q^89 - 6 * q^90 - 4 * q^92 + (12*z - 12) * q^93 + (13*z - 13) * q^94 + 6*z * q^95 + 3 * q^96 + 14*z * q^97 - 6*z * q^98 + 12 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 3 q^{3} - q^{4} + 2 q^{5} - 3 q^{6} + q^{7} - 2 q^{8} - 6 q^{9}+O(q^{10})$$ 2 * q + q^2 + 3 * q^3 - q^4 + 2 * q^5 - 3 * q^6 + q^7 - 2 * q^8 - 6 * q^9 $$2 q + q^{2} + 3 q^{3} - q^{4} + 2 q^{5} - 3 q^{6} + q^{7} - 2 q^{8} - 6 q^{9} + q^{10} - 2 q^{11} - 6 q^{12} + 2 q^{14} + 3 q^{15} - q^{16} + 3 q^{17} - 12 q^{18} + 6 q^{19} - q^{20} + 6 q^{21} + 2 q^{22} + 4 q^{23} - 3 q^{24} - 8 q^{25} - 18 q^{27} + q^{28} - 2 q^{29} - 3 q^{30} - 8 q^{31} + q^{32} + 6 q^{33} + 6 q^{34} + q^{35} - 6 q^{36} + 3 q^{37} + 12 q^{38} - 2 q^{40} + 3 q^{42} + 5 q^{43} + 4 q^{44} - 6 q^{45} - 4 q^{46} - 26 q^{47} + 3 q^{48} + 6 q^{49} - 4 q^{50} + 18 q^{51} + 24 q^{53} - 9 q^{54} - 2 q^{55} - q^{56} + 36 q^{57} + 2 q^{58} - 10 q^{59} - 6 q^{60} + 8 q^{61} - 4 q^{62} + 6 q^{63} + 2 q^{64} + 12 q^{66} - 2 q^{67} + 3 q^{68} - 12 q^{69} + 2 q^{70} - 5 q^{71} + 6 q^{72} + 20 q^{73} - 3 q^{74} - 12 q^{75} + 6 q^{76} - 4 q^{77} - 8 q^{79} - q^{80} - 9 q^{81} - 3 q^{84} + 3 q^{85} + 10 q^{86} + 6 q^{87} + 2 q^{88} + 6 q^{89} - 12 q^{90} - 8 q^{92} - 12 q^{93} - 13 q^{94} + 6 q^{95} + 6 q^{96} + 14 q^{97} - 6 q^{98} + 24 q^{99}+O(q^{100})$$ 2 * q + q^2 + 3 * q^3 - q^4 + 2 * q^5 - 3 * q^6 + q^7 - 2 * q^8 - 6 * q^9 + q^10 - 2 * q^11 - 6 * q^12 + 2 * q^14 + 3 * q^15 - q^16 + 3 * q^17 - 12 * q^18 + 6 * q^19 - q^20 + 6 * q^21 + 2 * q^22 + 4 * q^23 - 3 * q^24 - 8 * q^25 - 18 * q^27 + q^28 - 2 * q^29 - 3 * q^30 - 8 * q^31 + q^32 + 6 * q^33 + 6 * q^34 + q^35 - 6 * q^36 + 3 * q^37 + 12 * q^38 - 2 * q^40 + 3 * q^42 + 5 * q^43 + 4 * q^44 - 6 * q^45 - 4 * q^46 - 26 * q^47 + 3 * q^48 + 6 * q^49 - 4 * q^50 + 18 * q^51 + 24 * q^53 - 9 * q^54 - 2 * q^55 - q^56 + 36 * q^57 + 2 * q^58 - 10 * q^59 - 6 * q^60 + 8 * q^61 - 4 * q^62 + 6 * q^63 + 2 * q^64 + 12 * q^66 - 2 * q^67 + 3 * q^68 - 12 * q^69 + 2 * q^70 - 5 * q^71 + 6 * q^72 + 20 * q^73 - 3 * q^74 - 12 * q^75 + 6 * q^76 - 4 * q^77 - 8 * q^79 - q^80 - 9 * q^81 - 3 * q^84 + 3 * q^85 + 10 * q^86 + 6 * q^87 + 2 * q^88 + 6 * q^89 - 12 * q^90 - 8 * q^92 - 12 * q^93 - 13 * q^94 + 6 * q^95 + 6 * q^96 + 14 * q^97 - 6 * q^98 + 24 * q^99

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 1.50000 2.59808i −0.500000 0.866025i 1.00000 −1.50000 2.59808i 0.500000 + 0.866025i −1.00000 −3.00000 5.19615i 0.500000 0.866025i
315.1 0.500000 + 0.866025i 1.50000 + 2.59808i −0.500000 + 0.866025i 1.00000 −1.50000 + 2.59808i 0.500000 0.866025i −1.00000 −3.00000 + 5.19615i 0.500000 + 0.866025i
 $$n$$: 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.g 2
13.b even 2 1 338.2.c.c 2
13.c even 3 1 338.2.a.a 1
13.c even 3 1 inner 338.2.c.g 2
13.d odd 4 2 338.2.e.d 4
13.e even 6 1 26.2.a.b 1
13.e even 6 1 338.2.c.c 2
13.f odd 12 2 338.2.b.a 2
13.f odd 12 2 338.2.e.d 4
39.h odd 6 1 234.2.a.b 1
39.i odd 6 1 3042.2.a.l 1
39.k even 12 2 3042.2.b.f 2
52.i odd 6 1 208.2.a.d 1
52.j odd 6 1 2704.2.a.n 1
52.l even 12 2 2704.2.f.j 2
65.l even 6 1 650.2.a.g 1
65.n even 6 1 8450.2.a.y 1
65.r odd 12 2 650.2.b.a 2
91.k even 6 1 1274.2.f.l 2
91.l odd 6 1 1274.2.f.a 2
91.p odd 6 1 1274.2.f.a 2
91.t odd 6 1 1274.2.a.o 1
91.u even 6 1 1274.2.f.l 2
104.p odd 6 1 832.2.a.a 1
104.s even 6 1 832.2.a.j 1
117.l even 6 1 2106.2.e.h 2
117.m odd 6 1 2106.2.e.t 2
117.r even 6 1 2106.2.e.h 2
117.v odd 6 1 2106.2.e.t 2
143.i odd 6 1 3146.2.a.a 1
156.r even 6 1 1872.2.a.m 1
195.y odd 6 1 5850.2.a.bn 1
195.bf even 12 2 5850.2.e.v 2
208.bh even 12 2 3328.2.b.g 2
208.bi odd 12 2 3328.2.b.k 2
221.n even 6 1 7514.2.a.i 1
247.m odd 6 1 9386.2.a.f 1
260.w odd 6 1 5200.2.a.c 1
312.ba even 6 1 7488.2.a.v 1
312.bg odd 6 1 7488.2.a.w 1

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

## Hecke kernels

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

 $$T_{3}^{2} - 3T_{3} + 9$$ T3^2 - 3*T3 + 9 $$T_{5} - 1$$ T5 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} - 3T + 9$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - T + 1$$
$11$ $$T^{2} + 2T + 4$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2} - 6T + 36$$
$23$ $$T^{2} - 4T + 16$$
$29$ $$T^{2} + 2T + 4$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} - 3T + 9$$
$41$ $$T^{2}$$
$43$ $$T^{2} - 5T + 25$$
$47$ $$(T + 13)^{2}$$
$53$ $$(T - 12)^{2}$$
$59$ $$T^{2} + 10T + 100$$
$61$ $$T^{2} - 8T + 64$$
$67$ $$T^{2} + 2T + 4$$
$71$ $$T^{2} + 5T + 25$$
$73$ $$(T - 10)^{2}$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$T^{2} - 14T + 196$$