# Properties

 Label 13.2.e.a Level $13$ Weight $2$ Character orbit 13.e Analytic conductor $0.104$ 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] = [13,2,Mod(4,13)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(13, base_ring=CyclotomicField(6))

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("13.4");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 13.e (of order $$6$$, degree $$2$$, 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: $$0.103805522628$$ 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: yes 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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} - 1) q^{2} + (2 \zeta_{6} - 2) q^{3} + \zeta_{6} q^{4} + ( - 2 \zeta_{6} + 1) q^{5} + ( - 2 \zeta_{6} + 4) q^{6} + (2 \zeta_{6} - 1) q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + (-z - 1) * q^2 + (2*z - 2) * q^3 + z * q^4 + (-2*z + 1) * q^5 + (-2*z + 4) * q^6 + (2*z - 1) * q^8 - z * q^9 $$q + ( - \zeta_{6} - 1) q^{2} + (2 \zeta_{6} - 2) q^{3} + \zeta_{6} q^{4} + ( - 2 \zeta_{6} + 1) q^{5} + ( - 2 \zeta_{6} + 4) q^{6} + (2 \zeta_{6} - 1) q^{8} - \zeta_{6} q^{9} + (3 \zeta_{6} - 3) q^{10} - 2 q^{12} + ( - 3 \zeta_{6} - 1) q^{13} + (2 \zeta_{6} + 2) q^{15} + ( - 5 \zeta_{6} + 5) q^{16} + 3 \zeta_{6} q^{17} + (2 \zeta_{6} - 1) q^{18} + (2 \zeta_{6} - 4) q^{19} + ( - \zeta_{6} + 2) q^{20} + ( - 6 \zeta_{6} + 6) q^{23} + ( - 2 \zeta_{6} - 2) q^{24} + 2 q^{25} + (7 \zeta_{6} - 2) q^{26} - 4 q^{27} + (3 \zeta_{6} - 3) q^{29} - 6 \zeta_{6} q^{30} + (4 \zeta_{6} - 2) q^{31} + (3 \zeta_{6} - 6) q^{32} + ( - 6 \zeta_{6} + 3) q^{34} + ( - \zeta_{6} + 1) q^{36} + (5 \zeta_{6} + 5) q^{37} + 6 q^{38} + ( - 2 \zeta_{6} + 8) q^{39} + 3 q^{40} + ( - 3 \zeta_{6} - 3) q^{41} - 8 \zeta_{6} q^{43} + (\zeta_{6} - 2) q^{45} + (6 \zeta_{6} - 12) q^{46} + ( - 4 \zeta_{6} + 2) q^{47} + 10 \zeta_{6} q^{48} + (7 \zeta_{6} - 7) q^{49} + ( - 2 \zeta_{6} - 2) q^{50} - 6 q^{51} + ( - 4 \zeta_{6} + 3) q^{52} - 3 q^{53} + (4 \zeta_{6} + 4) q^{54} + ( - 8 \zeta_{6} + 4) q^{57} + ( - 3 \zeta_{6} + 6) q^{58} + ( - 4 \zeta_{6} + 8) q^{59} + (4 \zeta_{6} - 2) q^{60} - \zeta_{6} q^{61} + ( - 6 \zeta_{6} + 6) q^{62} - q^{64} + (5 \zeta_{6} - 7) q^{65} + (2 \zeta_{6} + 2) q^{67} + (3 \zeta_{6} - 3) q^{68} + 12 \zeta_{6} q^{69} + ( - 2 \zeta_{6} + 4) q^{71} + ( - \zeta_{6} + 2) q^{72} + (2 \zeta_{6} - 1) q^{73} - 15 \zeta_{6} q^{74} + (4 \zeta_{6} - 4) q^{75} + ( - 2 \zeta_{6} - 2) q^{76} + ( - 4 \zeta_{6} - 10) q^{78} + 4 q^{79} + ( - 5 \zeta_{6} - 5) q^{80} + ( - 11 \zeta_{6} + 11) q^{81} + 9 \zeta_{6} q^{82} + (16 \zeta_{6} - 8) q^{83} + ( - 3 \zeta_{6} + 6) q^{85} + (16 \zeta_{6} - 8) q^{86} - 6 \zeta_{6} q^{87} + ( - 4 \zeta_{6} - 4) q^{89} + 3 q^{90} + 6 q^{92} + ( - 4 \zeta_{6} - 4) q^{93} + (6 \zeta_{6} - 6) q^{94} + 6 \zeta_{6} q^{95} + ( - 12 \zeta_{6} + 6) q^{96} + ( - 4 \zeta_{6} + 8) q^{97} + ( - 7 \zeta_{6} + 14) q^{98} +O(q^{100})$$ q + (-z - 1) * q^2 + (2*z - 2) * q^3 + z * q^4 + (-2*z + 1) * q^5 + (-2*z + 4) * q^6 + (2*z - 1) * q^8 - z * q^9 + (3*z - 3) * q^10 - 2 * q^12 + (-3*z - 1) * q^13 + (2*z + 2) * q^15 + (-5*z + 5) * q^16 + 3*z * q^17 + (2*z - 1) * q^18 + (2*z - 4) * q^19 + (-z + 2) * q^20 + (-6*z + 6) * q^23 + (-2*z - 2) * q^24 + 2 * q^25 + (7*z - 2) * q^26 - 4 * q^27 + (3*z - 3) * q^29 - 6*z * q^30 + (4*z - 2) * q^31 + (3*z - 6) * q^32 + (-6*z + 3) * q^34 + (-z + 1) * q^36 + (5*z + 5) * q^37 + 6 * q^38 + (-2*z + 8) * q^39 + 3 * q^40 + (-3*z - 3) * q^41 - 8*z * q^43 + (z - 2) * q^45 + (6*z - 12) * q^46 + (-4*z + 2) * q^47 + 10*z * q^48 + (7*z - 7) * q^49 + (-2*z - 2) * q^50 - 6 * q^51 + (-4*z + 3) * q^52 - 3 * q^53 + (4*z + 4) * q^54 + (-8*z + 4) * q^57 + (-3*z + 6) * q^58 + (-4*z + 8) * q^59 + (4*z - 2) * q^60 - z * q^61 + (-6*z + 6) * q^62 - q^64 + (5*z - 7) * q^65 + (2*z + 2) * q^67 + (3*z - 3) * q^68 + 12*z * q^69 + (-2*z + 4) * q^71 + (-z + 2) * q^72 + (2*z - 1) * q^73 - 15*z * q^74 + (4*z - 4) * q^75 + (-2*z - 2) * q^76 + (-4*z - 10) * q^78 + 4 * q^79 + (-5*z - 5) * q^80 + (-11*z + 11) * q^81 + 9*z * q^82 + (16*z - 8) * q^83 + (-3*z + 6) * q^85 + (16*z - 8) * q^86 - 6*z * q^87 + (-4*z - 4) * q^89 + 3 * q^90 + 6 * q^92 + (-4*z - 4) * q^93 + (6*z - 6) * q^94 + 6*z * q^95 + (-12*z + 6) * q^96 + (-4*z + 8) * q^97 + (-7*z + 14) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} - 2 q^{3} + q^{4} + 6 q^{6} - q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 - 2 * q^3 + q^4 + 6 * q^6 - q^9 $$2 q - 3 q^{2} - 2 q^{3} + q^{4} + 6 q^{6} - q^{9} - 3 q^{10} - 4 q^{12} - 5 q^{13} + 6 q^{15} + 5 q^{16} + 3 q^{17} - 6 q^{19} + 3 q^{20} + 6 q^{23} - 6 q^{24} + 4 q^{25} + 3 q^{26} - 8 q^{27} - 3 q^{29} - 6 q^{30} - 9 q^{32} + q^{36} + 15 q^{37} + 12 q^{38} + 14 q^{39} + 6 q^{40} - 9 q^{41} - 8 q^{43} - 3 q^{45} - 18 q^{46} + 10 q^{48} - 7 q^{49} - 6 q^{50} - 12 q^{51} + 2 q^{52} - 6 q^{53} + 12 q^{54} + 9 q^{58} + 12 q^{59} - q^{61} + 6 q^{62} - 2 q^{64} - 9 q^{65} + 6 q^{67} - 3 q^{68} + 12 q^{69} + 6 q^{71} + 3 q^{72} - 15 q^{74} - 4 q^{75} - 6 q^{76} - 24 q^{78} + 8 q^{79} - 15 q^{80} + 11 q^{81} + 9 q^{82} + 9 q^{85} - 6 q^{87} - 12 q^{89} + 6 q^{90} + 12 q^{92} - 12 q^{93} - 6 q^{94} + 6 q^{95} + 12 q^{97} + 21 q^{98}+O(q^{100})$$ 2 * q - 3 * q^2 - 2 * q^3 + q^4 + 6 * q^6 - q^9 - 3 * q^10 - 4 * q^12 - 5 * q^13 + 6 * q^15 + 5 * q^16 + 3 * q^17 - 6 * q^19 + 3 * q^20 + 6 * q^23 - 6 * q^24 + 4 * q^25 + 3 * q^26 - 8 * q^27 - 3 * q^29 - 6 * q^30 - 9 * q^32 + q^36 + 15 * q^37 + 12 * q^38 + 14 * q^39 + 6 * q^40 - 9 * q^41 - 8 * q^43 - 3 * q^45 - 18 * q^46 + 10 * q^48 - 7 * q^49 - 6 * q^50 - 12 * q^51 + 2 * q^52 - 6 * q^53 + 12 * q^54 + 9 * q^58 + 12 * q^59 - q^61 + 6 * q^62 - 2 * q^64 - 9 * q^65 + 6 * q^67 - 3 * q^68 + 12 * q^69 + 6 * q^71 + 3 * q^72 - 15 * q^74 - 4 * q^75 - 6 * q^76 - 24 * q^78 + 8 * q^79 - 15 * q^80 + 11 * q^81 + 9 * q^82 + 9 * q^85 - 6 * q^87 - 12 * q^89 + 6 * q^90 + 12 * q^92 - 12 * q^93 - 6 * q^94 + 6 * q^95 + 12 * q^97 + 21 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/13\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.50000 0.866025i −1.00000 + 1.73205i 0.500000 + 0.866025i 1.73205i 3.00000 1.73205i 0 1.73205i −0.500000 0.866025i −1.50000 + 2.59808i
10.1 −1.50000 + 0.866025i −1.00000 1.73205i 0.500000 0.866025i 1.73205i 3.00000 + 1.73205i 0 1.73205i −0.500000 + 0.866025i −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.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.2.e.a 2
3.b odd 2 1 117.2.q.c 2
4.b odd 2 1 208.2.w.b 2
5.b even 2 1 325.2.n.a 2
5.c odd 4 2 325.2.m.a 4
7.b odd 2 1 637.2.q.a 2
7.c even 3 1 637.2.k.a 2
7.c even 3 1 637.2.u.c 2
7.d odd 6 1 637.2.k.c 2
7.d odd 6 1 637.2.u.b 2
8.b even 2 1 832.2.w.d 2
8.d odd 2 1 832.2.w.a 2
12.b even 2 1 1872.2.by.d 2
13.b even 2 1 169.2.e.a 2
13.c even 3 1 169.2.b.a 2
13.c even 3 1 169.2.e.a 2
13.d odd 4 2 169.2.c.a 4
13.e even 6 1 inner 13.2.e.a 2
13.e even 6 1 169.2.b.a 2
13.f odd 12 2 169.2.a.a 2
13.f odd 12 2 169.2.c.a 4
39.h odd 6 1 117.2.q.c 2
39.h odd 6 1 1521.2.b.a 2
39.i odd 6 1 1521.2.b.a 2
39.k even 12 2 1521.2.a.k 2
52.i odd 6 1 208.2.w.b 2
52.i odd 6 1 2704.2.f.b 2
52.j odd 6 1 2704.2.f.b 2
52.l even 12 2 2704.2.a.o 2
65.l even 6 1 325.2.n.a 2
65.r odd 12 2 325.2.m.a 4
65.s odd 12 2 4225.2.a.v 2
91.k even 6 1 637.2.u.c 2
91.l odd 6 1 637.2.u.b 2
91.p odd 6 1 637.2.k.c 2
91.t odd 6 1 637.2.q.a 2
91.u even 6 1 637.2.k.a 2
91.bc even 12 2 8281.2.a.q 2
104.p odd 6 1 832.2.w.a 2
104.s even 6 1 832.2.w.d 2
156.r even 6 1 1872.2.by.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.2.e.a 2 1.a even 1 1 trivial
13.2.e.a 2 13.e even 6 1 inner
117.2.q.c 2 3.b odd 2 1
117.2.q.c 2 39.h odd 6 1
169.2.a.a 2 13.f odd 12 2
169.2.b.a 2 13.c even 3 1
169.2.b.a 2 13.e even 6 1
169.2.c.a 4 13.d odd 4 2
169.2.c.a 4 13.f odd 12 2
169.2.e.a 2 13.b even 2 1
169.2.e.a 2 13.c even 3 1
208.2.w.b 2 4.b odd 2 1
208.2.w.b 2 52.i odd 6 1
325.2.m.a 4 5.c odd 4 2
325.2.m.a 4 65.r odd 12 2
325.2.n.a 2 5.b even 2 1
325.2.n.a 2 65.l even 6 1
637.2.k.a 2 7.c even 3 1
637.2.k.a 2 91.u even 6 1
637.2.k.c 2 7.d odd 6 1
637.2.k.c 2 91.p odd 6 1
637.2.q.a 2 7.b odd 2 1
637.2.q.a 2 91.t odd 6 1
637.2.u.b 2 7.d odd 6 1
637.2.u.b 2 91.l odd 6 1
637.2.u.c 2 7.c even 3 1
637.2.u.c 2 91.k even 6 1
832.2.w.a 2 8.d odd 2 1
832.2.w.a 2 104.p odd 6 1
832.2.w.d 2 8.b even 2 1
832.2.w.d 2 104.s even 6 1
1521.2.a.k 2 39.k even 12 2
1521.2.b.a 2 39.h odd 6 1
1521.2.b.a 2 39.i odd 6 1
1872.2.by.d 2 12.b even 2 1
1872.2.by.d 2 156.r even 6 1
2704.2.a.o 2 52.l even 12 2
2704.2.f.b 2 52.i odd 6 1
2704.2.f.b 2 52.j odd 6 1
4225.2.a.v 2 65.s odd 12 2
8281.2.a.q 2 91.bc even 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(13, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T + 3$$
$3$ $$T^{2} + 2T + 4$$
$5$ $$T^{2} + 3$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 5T + 13$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2} + 6T + 12$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$T^{2} + 12$$
$37$ $$T^{2} - 15T + 75$$
$41$ $$T^{2} + 9T + 27$$
$43$ $$T^{2} + 8T + 64$$
$47$ $$T^{2} + 12$$
$53$ $$(T + 3)^{2}$$
$59$ $$T^{2} - 12T + 48$$
$61$ $$T^{2} + T + 1$$
$67$ $$T^{2} - 6T + 12$$
$71$ $$T^{2} - 6T + 12$$
$73$ $$T^{2} + 3$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2} + 192$$
$89$ $$T^{2} + 12T + 48$$
$97$ $$T^{2} - 12T + 48$$