# Properties

 Label 13.4.b.a Level $13$ Weight $4$ Character orbit 13.b Analytic conductor $0.767$ 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] = [13,4,Mod(12,13)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(13, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("13.12");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 13.b (of order $$2$$, degree $$1$$, 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: $$0.767024830075$$ 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: $$3$$ Twist minimal: yes 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 $$\beta = 3i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - q^{3} - q^{4} - 3 \beta q^{5} - \beta q^{6} - 5 \beta q^{7} + 7 \beta q^{8} - 26 q^{9} +O(q^{10})$$ q + b * q^2 - q^3 - q^4 - 3*b * q^5 - b * q^6 - 5*b * q^7 + 7*b * q^8 - 26 * q^9 $$q + \beta q^{2} - q^{3} - q^{4} - 3 \beta q^{5} - \beta q^{6} - 5 \beta q^{7} + 7 \beta q^{8} - 26 q^{9} + 27 q^{10} + 16 \beta q^{11} + q^{12} + ( - 13 \beta + 26) q^{13} + 45 q^{14} + 3 \beta q^{15} - 71 q^{16} - 45 q^{17} - 26 \beta q^{18} + 2 \beta q^{19} + 3 \beta q^{20} + 5 \beta q^{21} - 144 q^{22} + 162 q^{23} - 7 \beta q^{24} + 44 q^{25} + (26 \beta + 117) q^{26} + 53 q^{27} + 5 \beta q^{28} - 144 q^{29} - 27 q^{30} + 88 \beta q^{31} - 15 \beta q^{32} - 16 \beta q^{33} - 45 \beta q^{34} - 135 q^{35} + 26 q^{36} - 101 \beta q^{37} - 18 q^{38} + (13 \beta - 26) q^{39} + 189 q^{40} - 64 \beta q^{41} - 45 q^{42} - 97 q^{43} - 16 \beta q^{44} + 78 \beta q^{45} + 162 \beta q^{46} - 37 \beta q^{47} + 71 q^{48} + 118 q^{49} + 44 \beta q^{50} + 45 q^{51} + (13 \beta - 26) q^{52} - 414 q^{53} + 53 \beta q^{54} + 432 q^{55} + 315 q^{56} - 2 \beta q^{57} - 144 \beta q^{58} - 174 \beta q^{59} - 3 \beta q^{60} + 376 q^{61} - 792 q^{62} + 130 \beta q^{63} - 433 q^{64} + ( - 78 \beta - 351) q^{65} + 144 q^{66} - 12 \beta q^{67} + 45 q^{68} - 162 q^{69} - 135 \beta q^{70} + 119 \beta q^{71} - 182 \beta q^{72} + 366 \beta q^{73} + 909 q^{74} - 44 q^{75} - 2 \beta q^{76} + 720 q^{77} + ( - 26 \beta - 117) q^{78} - 830 q^{79} + 213 \beta q^{80} + 649 q^{81} + 576 q^{82} - 146 \beta q^{83} - 5 \beta q^{84} + 135 \beta q^{85} - 97 \beta q^{86} + 144 q^{87} - 1008 q^{88} + 146 \beta q^{89} - 702 q^{90} + ( - 130 \beta - 585) q^{91} - 162 q^{92} - 88 \beta q^{93} + 333 q^{94} + 54 q^{95} + 15 \beta q^{96} - 284 \beta q^{97} + 118 \beta q^{98} - 416 \beta q^{99} +O(q^{100})$$ q + b * q^2 - q^3 - q^4 - 3*b * q^5 - b * q^6 - 5*b * q^7 + 7*b * q^8 - 26 * q^9 + 27 * q^10 + 16*b * q^11 + q^12 + (-13*b + 26) * q^13 + 45 * q^14 + 3*b * q^15 - 71 * q^16 - 45 * q^17 - 26*b * q^18 + 2*b * q^19 + 3*b * q^20 + 5*b * q^21 - 144 * q^22 + 162 * q^23 - 7*b * q^24 + 44 * q^25 + (26*b + 117) * q^26 + 53 * q^27 + 5*b * q^28 - 144 * q^29 - 27 * q^30 + 88*b * q^31 - 15*b * q^32 - 16*b * q^33 - 45*b * q^34 - 135 * q^35 + 26 * q^36 - 101*b * q^37 - 18 * q^38 + (13*b - 26) * q^39 + 189 * q^40 - 64*b * q^41 - 45 * q^42 - 97 * q^43 - 16*b * q^44 + 78*b * q^45 + 162*b * q^46 - 37*b * q^47 + 71 * q^48 + 118 * q^49 + 44*b * q^50 + 45 * q^51 + (13*b - 26) * q^52 - 414 * q^53 + 53*b * q^54 + 432 * q^55 + 315 * q^56 - 2*b * q^57 - 144*b * q^58 - 174*b * q^59 - 3*b * q^60 + 376 * q^61 - 792 * q^62 + 130*b * q^63 - 433 * q^64 + (-78*b - 351) * q^65 + 144 * q^66 - 12*b * q^67 + 45 * q^68 - 162 * q^69 - 135*b * q^70 + 119*b * q^71 - 182*b * q^72 + 366*b * q^73 + 909 * q^74 - 44 * q^75 - 2*b * q^76 + 720 * q^77 + (-26*b - 117) * q^78 - 830 * q^79 + 213*b * q^80 + 649 * q^81 + 576 * q^82 - 146*b * q^83 - 5*b * q^84 + 135*b * q^85 - 97*b * q^86 + 144 * q^87 - 1008 * q^88 + 146*b * q^89 - 702 * q^90 + (-130*b - 585) * q^91 - 162 * q^92 - 88*b * q^93 + 333 * q^94 + 54 * q^95 + 15*b * q^96 - 284*b * q^97 + 118*b * q^98 - 416*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{4} - 52 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^4 - 52 * q^9 $$2 q - 2 q^{3} - 2 q^{4} - 52 q^{9} + 54 q^{10} + 2 q^{12} + 52 q^{13} + 90 q^{14} - 142 q^{16} - 90 q^{17} - 288 q^{22} + 324 q^{23} + 88 q^{25} + 234 q^{26} + 106 q^{27} - 288 q^{29} - 54 q^{30} - 270 q^{35} + 52 q^{36} - 36 q^{38} - 52 q^{39} + 378 q^{40} - 90 q^{42} - 194 q^{43} + 142 q^{48} + 236 q^{49} + 90 q^{51} - 52 q^{52} - 828 q^{53} + 864 q^{55} + 630 q^{56} + 752 q^{61} - 1584 q^{62} - 866 q^{64} - 702 q^{65} + 288 q^{66} + 90 q^{68} - 324 q^{69} + 1818 q^{74} - 88 q^{75} + 1440 q^{77} - 234 q^{78} - 1660 q^{79} + 1298 q^{81} + 1152 q^{82} + 288 q^{87} - 2016 q^{88} - 1404 q^{90} - 1170 q^{91} - 324 q^{92} + 666 q^{94} + 108 q^{95}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^4 - 52 * q^9 + 54 * q^10 + 2 * q^12 + 52 * q^13 + 90 * q^14 - 142 * q^16 - 90 * q^17 - 288 * q^22 + 324 * q^23 + 88 * q^25 + 234 * q^26 + 106 * q^27 - 288 * q^29 - 54 * q^30 - 270 * q^35 + 52 * q^36 - 36 * q^38 - 52 * q^39 + 378 * q^40 - 90 * q^42 - 194 * q^43 + 142 * q^48 + 236 * q^49 + 90 * q^51 - 52 * q^52 - 828 * q^53 + 864 * q^55 + 630 * q^56 + 752 * q^61 - 1584 * q^62 - 866 * q^64 - 702 * q^65 + 288 * q^66 + 90 * q^68 - 324 * q^69 + 1818 * q^74 - 88 * q^75 + 1440 * q^77 - 234 * q^78 - 1660 * q^79 + 1298 * q^81 + 1152 * q^82 + 288 * q^87 - 2016 * q^88 - 1404 * q^90 - 1170 * q^91 - 324 * q^92 + 666 * q^94 + 108 * q^95

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
12.1
 − 1.00000i 1.00000i
3.00000i −1.00000 −1.00000 9.00000i 3.00000i 15.0000i 21.0000i −26.0000 27.0000
12.2 3.00000i −1.00000 −1.00000 9.00000i 3.00000i 15.0000i 21.0000i −26.0000 27.0000
 $$n$$: 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 13.4.b.a 2
3.b odd 2 1 117.4.b.a 2
4.b odd 2 1 208.4.f.b 2
5.b even 2 1 325.4.c.b 2
5.c odd 4 1 325.4.d.a 2
5.c odd 4 1 325.4.d.b 2
8.b even 2 1 832.4.f.e 2
8.d odd 2 1 832.4.f.c 2
13.b even 2 1 inner 13.4.b.a 2
13.c even 3 2 169.4.e.d 4
13.d odd 4 1 169.4.a.b 1
13.d odd 4 1 169.4.a.c 1
13.e even 6 2 169.4.e.d 4
13.f odd 12 2 169.4.c.b 2
13.f odd 12 2 169.4.c.c 2
39.d odd 2 1 117.4.b.a 2
39.f even 4 1 1521.4.a.d 1
39.f even 4 1 1521.4.a.i 1
52.b odd 2 1 208.4.f.b 2
65.d even 2 1 325.4.c.b 2
65.h odd 4 1 325.4.d.a 2
65.h odd 4 1 325.4.d.b 2
104.e even 2 1 832.4.f.e 2
104.h odd 2 1 832.4.f.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.b.a 2 1.a even 1 1 trivial
13.4.b.a 2 13.b even 2 1 inner
117.4.b.a 2 3.b odd 2 1
117.4.b.a 2 39.d odd 2 1
169.4.a.b 1 13.d odd 4 1
169.4.a.c 1 13.d odd 4 1
169.4.c.b 2 13.f odd 12 2
169.4.c.c 2 13.f odd 12 2
169.4.e.d 4 13.c even 3 2
169.4.e.d 4 13.e even 6 2
208.4.f.b 2 4.b odd 2 1
208.4.f.b 2 52.b odd 2 1
325.4.c.b 2 5.b even 2 1
325.4.c.b 2 65.d even 2 1
325.4.d.a 2 5.c odd 4 1
325.4.d.a 2 65.h odd 4 1
325.4.d.b 2 5.c odd 4 1
325.4.d.b 2 65.h odd 4 1
832.4.f.c 2 8.d odd 2 1
832.4.f.c 2 104.h odd 2 1
832.4.f.e 2 8.b even 2 1
832.4.f.e 2 104.e even 2 1
1521.4.a.d 1 39.f even 4 1
1521.4.a.i 1 39.f even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(13, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 9$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 81$$
$7$ $$T^{2} + 225$$
$11$ $$T^{2} + 2304$$
$13$ $$T^{2} - 52T + 2197$$
$17$ $$(T + 45)^{2}$$
$19$ $$T^{2} + 36$$
$23$ $$(T - 162)^{2}$$
$29$ $$(T + 144)^{2}$$
$31$ $$T^{2} + 69696$$
$37$ $$T^{2} + 91809$$
$41$ $$T^{2} + 36864$$
$43$ $$(T + 97)^{2}$$
$47$ $$T^{2} + 12321$$
$53$ $$(T + 414)^{2}$$
$59$ $$T^{2} + 272484$$
$61$ $$(T - 376)^{2}$$
$67$ $$T^{2} + 1296$$
$71$ $$T^{2} + 127449$$
$73$ $$T^{2} + 1205604$$
$79$ $$(T + 830)^{2}$$
$83$ $$T^{2} + 191844$$
$89$ $$T^{2} + 191844$$
$97$ $$T^{2} + 725904$$