# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,2,Mod(337,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

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

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

chi := DirichletCharacter("338.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 338.b (of order $$2$$, degree $$1$$, 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: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ 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: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$171$$ $$\chi(n)$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 1.00000i 1.00000i
1.00000i 1.00000 −1.00000 3.00000i 1.00000i 1.00000i 1.00000i −2.00000 −3.00000
337.2 1.00000i 1.00000 −1.00000 3.00000i 1.00000i 1.00000i 1.00000i −2.00000 −3.00000
 $$n$$: 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

## Twists

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

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

## Hecke kernels

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

## Hecke characteristic polynomials

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