# Properties

 Label 13.4.c.a Level $13$ Weight $4$ Character orbit 13.c Analytic conductor $0.767$ Analytic rank $0$ Dimension $2$ Inner twists $2$

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Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [13,4,Mod(3,13)]

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([2]))

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

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

chi := DirichletCharacter("13.3");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 13.c (of order $$3$$, 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: $$0.767024830075$$ 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: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + (4 \zeta_{6} - 4) q^{2} + (2 \zeta_{6} - 2) q^{3} - 8 \zeta_{6} q^{4} + 17 q^{5} - 8 \zeta_{6} q^{6} - 20 \zeta_{6} q^{7} + 23 \zeta_{6} q^{9} +O(q^{10})$$ q + (4*z - 4) * q^2 + (2*z - 2) * q^3 - 8*z * q^4 + 17 * q^5 - 8*z * q^6 - 20*z * q^7 + 23*z * q^9 $$q + (4 \zeta_{6} - 4) q^{2} + (2 \zeta_{6} - 2) q^{3} - 8 \zeta_{6} q^{4} + 17 q^{5} - 8 \zeta_{6} q^{6} - 20 \zeta_{6} q^{7} + 23 \zeta_{6} q^{9} + (68 \zeta_{6} - 68) q^{10} + ( - 32 \zeta_{6} + 32) q^{11} + 16 q^{12} + ( - 13 \zeta_{6} - 39) q^{13} + 80 q^{14} + (34 \zeta_{6} - 34) q^{15} + ( - 64 \zeta_{6} + 64) q^{16} + 13 \zeta_{6} q^{17} - 92 q^{18} - 30 \zeta_{6} q^{19} - 136 \zeta_{6} q^{20} + 40 q^{21} + 128 \zeta_{6} q^{22} + (78 \zeta_{6} - 78) q^{23} + 164 q^{25} + ( - 156 \zeta_{6} + 208) q^{26} - 100 q^{27} + (160 \zeta_{6} - 160) q^{28} + (197 \zeta_{6} - 197) q^{29} - 136 \zeta_{6} q^{30} - 74 q^{31} + 256 \zeta_{6} q^{32} + 64 \zeta_{6} q^{33} - 52 q^{34} - 340 \zeta_{6} q^{35} + ( - 184 \zeta_{6} + 184) q^{36} + ( - 227 \zeta_{6} + 227) q^{37} + 120 q^{38} + ( - 78 \zeta_{6} + 104) q^{39} + ( - 165 \zeta_{6} + 165) q^{41} + (160 \zeta_{6} - 160) q^{42} + 156 \zeta_{6} q^{43} - 256 q^{44} + 391 \zeta_{6} q^{45} - 312 \zeta_{6} q^{46} - 162 q^{47} + 128 \zeta_{6} q^{48} + (57 \zeta_{6} - 57) q^{49} + (656 \zeta_{6} - 656) q^{50} - 26 q^{51} + (416 \zeta_{6} - 104) q^{52} + 93 q^{53} + ( - 400 \zeta_{6} + 400) q^{54} + ( - 544 \zeta_{6} + 544) q^{55} + 60 q^{57} - 788 \zeta_{6} q^{58} + 864 \zeta_{6} q^{59} + 272 q^{60} - 145 \zeta_{6} q^{61} + ( - 296 \zeta_{6} + 296) q^{62} + ( - 460 \zeta_{6} + 460) q^{63} - 512 q^{64} + ( - 221 \zeta_{6} - 663) q^{65} - 256 q^{66} + (862 \zeta_{6} - 862) q^{67} + ( - 104 \zeta_{6} + 104) q^{68} - 156 \zeta_{6} q^{69} + 1360 q^{70} - 654 \zeta_{6} q^{71} + 215 q^{73} + 908 \zeta_{6} q^{74} + (328 \zeta_{6} - 328) q^{75} + (240 \zeta_{6} - 240) q^{76} - 640 q^{77} + (416 \zeta_{6} - 104) q^{78} - 76 q^{79} + ( - 1088 \zeta_{6} + 1088) q^{80} + (421 \zeta_{6} - 421) q^{81} + 660 \zeta_{6} q^{82} + 628 q^{83} - 320 \zeta_{6} q^{84} + 221 \zeta_{6} q^{85} - 624 q^{86} - 394 \zeta_{6} q^{87} + ( - 266 \zeta_{6} + 266) q^{89} - 1564 q^{90} + (1040 \zeta_{6} - 260) q^{91} + 624 q^{92} + ( - 148 \zeta_{6} + 148) q^{93} + ( - 648 \zeta_{6} + 648) q^{94} - 510 \zeta_{6} q^{95} - 512 q^{96} - 238 \zeta_{6} q^{97} - 228 \zeta_{6} q^{98} + 736 q^{99} +O(q^{100})$$ q + (4*z - 4) * q^2 + (2*z - 2) * q^3 - 8*z * q^4 + 17 * q^5 - 8*z * q^6 - 20*z * q^7 + 23*z * q^9 + (68*z - 68) * q^10 + (-32*z + 32) * q^11 + 16 * q^12 + (-13*z - 39) * q^13 + 80 * q^14 + (34*z - 34) * q^15 + (-64*z + 64) * q^16 + 13*z * q^17 - 92 * q^18 - 30*z * q^19 - 136*z * q^20 + 40 * q^21 + 128*z * q^22 + (78*z - 78) * q^23 + 164 * q^25 + (-156*z + 208) * q^26 - 100 * q^27 + (160*z - 160) * q^28 + (197*z - 197) * q^29 - 136*z * q^30 - 74 * q^31 + 256*z * q^32 + 64*z * q^33 - 52 * q^34 - 340*z * q^35 + (-184*z + 184) * q^36 + (-227*z + 227) * q^37 + 120 * q^38 + (-78*z + 104) * q^39 + (-165*z + 165) * q^41 + (160*z - 160) * q^42 + 156*z * q^43 - 256 * q^44 + 391*z * q^45 - 312*z * q^46 - 162 * q^47 + 128*z * q^48 + (57*z - 57) * q^49 + (656*z - 656) * q^50 - 26 * q^51 + (416*z - 104) * q^52 + 93 * q^53 + (-400*z + 400) * q^54 + (-544*z + 544) * q^55 + 60 * q^57 - 788*z * q^58 + 864*z * q^59 + 272 * q^60 - 145*z * q^61 + (-296*z + 296) * q^62 + (-460*z + 460) * q^63 - 512 * q^64 + (-221*z - 663) * q^65 - 256 * q^66 + (862*z - 862) * q^67 + (-104*z + 104) * q^68 - 156*z * q^69 + 1360 * q^70 - 654*z * q^71 + 215 * q^73 + 908*z * q^74 + (328*z - 328) * q^75 + (240*z - 240) * q^76 - 640 * q^77 + (416*z - 104) * q^78 - 76 * q^79 + (-1088*z + 1088) * q^80 + (421*z - 421) * q^81 + 660*z * q^82 + 628 * q^83 - 320*z * q^84 + 221*z * q^85 - 624 * q^86 - 394*z * q^87 + (-266*z + 266) * q^89 - 1564 * q^90 + (1040*z - 260) * q^91 + 624 * q^92 + (-148*z + 148) * q^93 + (-648*z + 648) * q^94 - 510*z * q^95 - 512 * q^96 - 238*z * q^97 - 228*z * q^98 + 736 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} - 2 q^{3} - 8 q^{4} + 34 q^{5} - 8 q^{6} - 20 q^{7} + 23 q^{9}+O(q^{10})$$ 2 * q - 4 * q^2 - 2 * q^3 - 8 * q^4 + 34 * q^5 - 8 * q^6 - 20 * q^7 + 23 * q^9 $$2 q - 4 q^{2} - 2 q^{3} - 8 q^{4} + 34 q^{5} - 8 q^{6} - 20 q^{7} + 23 q^{9} - 68 q^{10} + 32 q^{11} + 32 q^{12} - 91 q^{13} + 160 q^{14} - 34 q^{15} + 64 q^{16} + 13 q^{17} - 184 q^{18} - 30 q^{19} - 136 q^{20} + 80 q^{21} + 128 q^{22} - 78 q^{23} + 328 q^{25} + 260 q^{26} - 200 q^{27} - 160 q^{28} - 197 q^{29} - 136 q^{30} - 148 q^{31} + 256 q^{32} + 64 q^{33} - 104 q^{34} - 340 q^{35} + 184 q^{36} + 227 q^{37} + 240 q^{38} + 130 q^{39} + 165 q^{41} - 160 q^{42} + 156 q^{43} - 512 q^{44} + 391 q^{45} - 312 q^{46} - 324 q^{47} + 128 q^{48} - 57 q^{49} - 656 q^{50} - 52 q^{51} + 208 q^{52} + 186 q^{53} + 400 q^{54} + 544 q^{55} + 120 q^{57} - 788 q^{58} + 864 q^{59} + 544 q^{60} - 145 q^{61} + 296 q^{62} + 460 q^{63} - 1024 q^{64} - 1547 q^{65} - 512 q^{66} - 862 q^{67} + 104 q^{68} - 156 q^{69} + 2720 q^{70} - 654 q^{71} + 430 q^{73} + 908 q^{74} - 328 q^{75} - 240 q^{76} - 1280 q^{77} + 208 q^{78} - 152 q^{79} + 1088 q^{80} - 421 q^{81} + 660 q^{82} + 1256 q^{83} - 320 q^{84} + 221 q^{85} - 1248 q^{86} - 394 q^{87} + 266 q^{89} - 3128 q^{90} + 520 q^{91} + 1248 q^{92} + 148 q^{93} + 648 q^{94} - 510 q^{95} - 1024 q^{96} - 238 q^{97} - 228 q^{98} + 1472 q^{99}+O(q^{100})$$ 2 * q - 4 * q^2 - 2 * q^3 - 8 * q^4 + 34 * q^5 - 8 * q^6 - 20 * q^7 + 23 * q^9 - 68 * q^10 + 32 * q^11 + 32 * q^12 - 91 * q^13 + 160 * q^14 - 34 * q^15 + 64 * q^16 + 13 * q^17 - 184 * q^18 - 30 * q^19 - 136 * q^20 + 80 * q^21 + 128 * q^22 - 78 * q^23 + 328 * q^25 + 260 * q^26 - 200 * q^27 - 160 * q^28 - 197 * q^29 - 136 * q^30 - 148 * q^31 + 256 * q^32 + 64 * q^33 - 104 * q^34 - 340 * q^35 + 184 * q^36 + 227 * q^37 + 240 * q^38 + 130 * q^39 + 165 * q^41 - 160 * q^42 + 156 * q^43 - 512 * q^44 + 391 * q^45 - 312 * q^46 - 324 * q^47 + 128 * q^48 - 57 * q^49 - 656 * q^50 - 52 * q^51 + 208 * q^52 + 186 * q^53 + 400 * q^54 + 544 * q^55 + 120 * q^57 - 788 * q^58 + 864 * q^59 + 544 * q^60 - 145 * q^61 + 296 * q^62 + 460 * q^63 - 1024 * q^64 - 1547 * q^65 - 512 * q^66 - 862 * q^67 + 104 * q^68 - 156 * q^69 + 2720 * q^70 - 654 * q^71 + 430 * q^73 + 908 * q^74 - 328 * q^75 - 240 * q^76 - 1280 * q^77 + 208 * q^78 - 152 * q^79 + 1088 * q^80 - 421 * q^81 + 660 * q^82 + 1256 * q^83 - 320 * q^84 + 221 * q^85 - 1248 * q^86 - 394 * q^87 + 266 * q^89 - 3128 * q^90 + 520 * q^91 + 1248 * q^92 + 148 * q^93 + 648 * q^94 - 510 * q^95 - 1024 * q^96 - 238 * q^97 - 228 * q^98 + 1472 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 0.5 − 0.866025i 0.5 + 0.866025i
−2.00000 3.46410i −1.00000 1.73205i −4.00000 + 6.92820i 17.0000 −4.00000 + 6.92820i −10.0000 + 17.3205i 0 11.5000 19.9186i −34.0000 58.8897i
9.1 −2.00000 + 3.46410i −1.00000 + 1.73205i −4.00000 6.92820i 17.0000 −4.00000 6.92820i −10.0000 17.3205i 0 11.5000 + 19.9186i −34.0000 + 58.8897i
 $$n$$: 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 13.4.c.a 2
3.b odd 2 1 117.4.g.c 2
4.b odd 2 1 208.4.i.b 2
13.b even 2 1 169.4.c.d 2
13.c even 3 1 inner 13.4.c.a 2
13.c even 3 1 169.4.a.d 1
13.d odd 4 2 169.4.e.c 4
13.e even 6 1 169.4.a.a 1
13.e even 6 1 169.4.c.d 2
13.f odd 12 2 169.4.b.c 2
13.f odd 12 2 169.4.e.c 4
39.h odd 6 1 1521.4.a.k 1
39.i odd 6 1 117.4.g.c 2
39.i odd 6 1 1521.4.a.b 1
52.j odd 6 1 208.4.i.b 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 4T_{2} + 16$$ acting on $$S_{4}^{\mathrm{new}}(13, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4T + 16$$
$3$ $$T^{2} + 2T + 4$$
$5$ $$(T - 17)^{2}$$
$7$ $$T^{2} + 20T + 400$$
$11$ $$T^{2} - 32T + 1024$$
$13$ $$T^{2} + 91T + 2197$$
$17$ $$T^{2} - 13T + 169$$
$19$ $$T^{2} + 30T + 900$$
$23$ $$T^{2} + 78T + 6084$$
$29$ $$T^{2} + 197T + 38809$$
$31$ $$(T + 74)^{2}$$
$37$ $$T^{2} - 227T + 51529$$
$41$ $$T^{2} - 165T + 27225$$
$43$ $$T^{2} - 156T + 24336$$
$47$ $$(T + 162)^{2}$$
$53$ $$(T - 93)^{2}$$
$59$ $$T^{2} - 864T + 746496$$
$61$ $$T^{2} + 145T + 21025$$
$67$ $$T^{2} + 862T + 743044$$
$71$ $$T^{2} + 654T + 427716$$
$73$ $$(T - 215)^{2}$$
$79$ $$(T + 76)^{2}$$
$83$ $$(T - 628)^{2}$$
$89$ $$T^{2} - 266T + 70756$$
$97$ $$T^{2} + 238T + 56644$$
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