# Properties

 Label 338.2.e.d Level $338$ Weight $2$ Character orbit 338.e Analytic conductor $2.699$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,2,Mod(23,338)]

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([5]))

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

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

chi := DirichletCharacter("338.23");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 338.e (of order $$6$$, 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field 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

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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 + 0.500000i 1.50000 + 2.59808i 0.500000 0.866025i 1.00000i −2.59808 1.50000i −0.866025 0.500000i 1.00000i −3.00000 + 5.19615i −0.500000 0.866025i
23.2 0.866025 0.500000i 1.50000 + 2.59808i 0.500000 0.866025i 1.00000i 2.59808 + 1.50000i 0.866025 + 0.500000i 1.00000i −3.00000 + 5.19615i −0.500000 0.866025i
147.1 −0.866025 0.500000i 1.50000 2.59808i 0.500000 + 0.866025i 1.00000i −2.59808 + 1.50000i −0.866025 + 0.500000i 1.00000i −3.00000 5.19615i −0.500000 + 0.866025i
147.2 0.866025 + 0.500000i 1.50000 2.59808i 0.500000 + 0.866025i 1.00000i 2.59808 1.50000i 0.866025 0.500000i 1.00000i −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.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.d 4
13.b even 2 1 inner 338.2.e.d 4
13.c even 3 1 338.2.b.a 2
13.c even 3 1 inner 338.2.e.d 4
13.d odd 4 1 338.2.c.c 2
13.d odd 4 1 338.2.c.g 2
13.e even 6 1 338.2.b.a 2
13.e even 6 1 inner 338.2.e.d 4
13.f odd 12 1 26.2.a.b 1
13.f odd 12 1 338.2.a.a 1
13.f odd 12 1 338.2.c.c 2
13.f odd 12 1 338.2.c.g 2
39.h odd 6 1 3042.2.b.f 2
39.i odd 6 1 3042.2.b.f 2
39.k even 12 1 234.2.a.b 1
39.k even 12 1 3042.2.a.l 1
52.i odd 6 1 2704.2.f.j 2
52.j odd 6 1 2704.2.f.j 2
52.l even 12 1 208.2.a.d 1
52.l even 12 1 2704.2.a.n 1
65.o even 12 1 650.2.b.a 2
65.s odd 12 1 650.2.a.g 1
65.s odd 12 1 8450.2.a.y 1
65.t even 12 1 650.2.b.a 2
91.w even 12 1 1274.2.f.a 2
91.x odd 12 1 1274.2.f.l 2
91.ba even 12 1 1274.2.f.a 2
91.bc even 12 1 1274.2.a.o 1
91.bd odd 12 1 1274.2.f.l 2
104.u even 12 1 832.2.a.a 1
104.x odd 12 1 832.2.a.j 1
117.w odd 12 1 2106.2.e.h 2
117.x even 12 1 2106.2.e.t 2
117.bb odd 12 1 2106.2.e.h 2
117.bc even 12 1 2106.2.e.t 2
143.o even 12 1 3146.2.a.a 1
156.v odd 12 1 1872.2.a.m 1
195.bc odd 12 1 5850.2.e.v 2
195.bh even 12 1 5850.2.a.bn 1
195.bn odd 12 1 5850.2.e.v 2
208.be odd 12 1 3328.2.b.g 2
208.bf even 12 1 3328.2.b.k 2
208.bk even 12 1 3328.2.b.k 2
208.bl odd 12 1 3328.2.b.g 2
221.w odd 12 1 7514.2.a.i 1
247.bd even 12 1 9386.2.a.f 1
260.bc even 12 1 5200.2.a.c 1
312.bo even 12 1 7488.2.a.w 1
312.bq odd 12 1 7488.2.a.v 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 3T_{3} + 9$$ acting on $$S_{2}^{\mathrm{new}}(338, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$(T^{2} - 3 T + 9)^{2}$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$T^{4} - T^{2} + 1$$
$11$ $$T^{4} - 4T^{2} + 16$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 3 T + 9)^{2}$$
$19$ $$T^{4} - 36T^{2} + 1296$$
$23$ $$(T^{2} + 4 T + 16)^{2}$$
$29$ $$(T^{2} + 2 T + 4)^{2}$$
$31$ $$(T^{2} + 16)^{2}$$
$37$ $$T^{4} - 9T^{2} + 81$$
$41$ $$T^{4}$$
$43$ $$(T^{2} + 5 T + 25)^{2}$$
$47$ $$(T^{2} + 169)^{2}$$
$53$ $$(T - 12)^{4}$$
$59$ $$T^{4} - 100 T^{2} + 10000$$
$61$ $$(T^{2} - 8 T + 64)^{2}$$
$67$ $$T^{4} - 4T^{2} + 16$$
$71$ $$T^{4} - 25T^{2} + 625$$
$73$ $$(T^{2} + 100)^{2}$$
$79$ $$(T + 4)^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4} - 36T^{2} + 1296$$
$97$ $$T^{4} - 196 T^{2} + 38416$$