# Properties

 Label 13.4.e.b Level $13$ Weight $4$ Character orbit 13.e Analytic conductor $0.767$ 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] = [13,4,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, 4, names="a")

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

chi := DirichletCharacter("13.4");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$0.767024830075$$ 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} + 1) q^{2} + (2 \zeta_{6} - 2) q^{3} - 5 \zeta_{6} q^{4} + ( - 2 \zeta_{6} + 1) q^{5} + (2 \zeta_{6} - 4) q^{6} + (8 \zeta_{6} - 16) q^{7} + ( - 26 \zeta_{6} + 13) q^{8} + 23 \zeta_{6} q^{9} +O(q^{10})$$ q + (z + 1) * q^2 + (2*z - 2) * q^3 - 5*z * q^4 + (-2*z + 1) * q^5 + (2*z - 4) * q^6 + (8*z - 16) * q^7 + (-26*z + 13) * q^8 + 23*z * q^9 $$q + (\zeta_{6} + 1) q^{2} + (2 \zeta_{6} - 2) q^{3} - 5 \zeta_{6} q^{4} + ( - 2 \zeta_{6} + 1) q^{5} + (2 \zeta_{6} - 4) q^{6} + (8 \zeta_{6} - 16) q^{7} + ( - 26 \zeta_{6} + 13) q^{8} + 23 \zeta_{6} q^{9} + ( - 3 \zeta_{6} + 3) q^{10} + (8 \zeta_{6} + 8) q^{11} + 10 q^{12} + (13 \zeta_{6} + 39) q^{13} - 24 q^{14} + (2 \zeta_{6} + 2) q^{15} + (\zeta_{6} - 1) q^{16} - 117 \zeta_{6} q^{17} + (46 \zeta_{6} - 23) q^{18} + (66 \zeta_{6} - 132) q^{19} + (5 \zeta_{6} - 10) q^{20} + ( - 32 \zeta_{6} + 16) q^{21} + 24 \zeta_{6} q^{22} + ( - 78 \zeta_{6} + 78) q^{23} + (26 \zeta_{6} + 26) q^{24} + 122 q^{25} + (65 \zeta_{6} + 26) q^{26} - 100 q^{27} + (40 \zeta_{6} + 40) q^{28} + ( - 141 \zeta_{6} + 141) q^{29} + 6 \zeta_{6} q^{30} + (180 \zeta_{6} - 90) q^{31} + (105 \zeta_{6} - 210) q^{32} + (16 \zeta_{6} - 32) q^{33} + ( - 234 \zeta_{6} + 117) q^{34} + 24 \zeta_{6} q^{35} + ( - 115 \zeta_{6} + 115) q^{36} + ( - 83 \zeta_{6} - 83) q^{37} - 198 q^{38} + (78 \zeta_{6} - 104) q^{39} - 39 q^{40} + (157 \zeta_{6} + 157) q^{41} + ( - 48 \zeta_{6} + 48) q^{42} - 104 \zeta_{6} q^{43} + ( - 80 \zeta_{6} + 40) q^{44} + ( - 23 \zeta_{6} + 46) q^{45} + ( - 78 \zeta_{6} + 156) q^{46} + (348 \zeta_{6} - 174) q^{47} - 2 \zeta_{6} q^{48} + (151 \zeta_{6} - 151) q^{49} + (122 \zeta_{6} + 122) q^{50} + 234 q^{51} + ( - 260 \zeta_{6} + 65) q^{52} + 93 q^{53} + ( - 100 \zeta_{6} - 100) q^{54} + ( - 24 \zeta_{6} + 24) q^{55} + 312 \zeta_{6} q^{56} + ( - 264 \zeta_{6} + 132) q^{57} + ( - 141 \zeta_{6} + 282) q^{58} + (164 \zeta_{6} - 328) q^{59} + ( - 20 \zeta_{6} + 10) q^{60} - 145 \zeta_{6} q^{61} + (270 \zeta_{6} - 270) q^{62} + ( - 184 \zeta_{6} - 184) q^{63} - 307 q^{64} + ( - 91 \zeta_{6} + 65) q^{65} - 48 q^{66} + ( - 454 \zeta_{6} - 454) q^{67} + (585 \zeta_{6} - 585) q^{68} + 156 \zeta_{6} q^{69} + (48 \zeta_{6} - 24) q^{70} + ( - 610 \zeta_{6} + 1220) q^{71} + ( - 299 \zeta_{6} + 598) q^{72} + (530 \zeta_{6} - 265) q^{73} - 249 \zeta_{6} q^{74} + (244 \zeta_{6} - 244) q^{75} + (330 \zeta_{6} + 330) q^{76} - 192 q^{77} + (52 \zeta_{6} - 182) q^{78} + 1276 q^{79} + (\zeta_{6} + 1) q^{80} + (421 \zeta_{6} - 421) q^{81} + 471 \zeta_{6} q^{82} + ( - 912 \zeta_{6} + 456) q^{83} + (80 \zeta_{6} - 160) q^{84} + (117 \zeta_{6} - 234) q^{85} + ( - 208 \zeta_{6} + 104) q^{86} + 282 \zeta_{6} q^{87} + ( - 312 \zeta_{6} + 312) q^{88} + ( - 564 \zeta_{6} - 564) q^{89} + 69 q^{90} + (208 \zeta_{6} - 728) q^{91} - 390 q^{92} + ( - 180 \zeta_{6} - 180) q^{93} + (522 \zeta_{6} - 522) q^{94} + 198 \zeta_{6} q^{95} + ( - 420 \zeta_{6} + 210) q^{96} + ( - 116 \zeta_{6} + 232) q^{97} + (151 \zeta_{6} - 302) q^{98} + (368 \zeta_{6} - 184) q^{99} +O(q^{100})$$ q + (z + 1) * q^2 + (2*z - 2) * q^3 - 5*z * q^4 + (-2*z + 1) * q^5 + (2*z - 4) * q^6 + (8*z - 16) * q^7 + (-26*z + 13) * q^8 + 23*z * q^9 + (-3*z + 3) * q^10 + (8*z + 8) * q^11 + 10 * q^12 + (13*z + 39) * q^13 - 24 * q^14 + (2*z + 2) * q^15 + (z - 1) * q^16 - 117*z * q^17 + (46*z - 23) * q^18 + (66*z - 132) * q^19 + (5*z - 10) * q^20 + (-32*z + 16) * q^21 + 24*z * q^22 + (-78*z + 78) * q^23 + (26*z + 26) * q^24 + 122 * q^25 + (65*z + 26) * q^26 - 100 * q^27 + (40*z + 40) * q^28 + (-141*z + 141) * q^29 + 6*z * q^30 + (180*z - 90) * q^31 + (105*z - 210) * q^32 + (16*z - 32) * q^33 + (-234*z + 117) * q^34 + 24*z * q^35 + (-115*z + 115) * q^36 + (-83*z - 83) * q^37 - 198 * q^38 + (78*z - 104) * q^39 - 39 * q^40 + (157*z + 157) * q^41 + (-48*z + 48) * q^42 - 104*z * q^43 + (-80*z + 40) * q^44 + (-23*z + 46) * q^45 + (-78*z + 156) * q^46 + (348*z - 174) * q^47 - 2*z * q^48 + (151*z - 151) * q^49 + (122*z + 122) * q^50 + 234 * q^51 + (-260*z + 65) * q^52 + 93 * q^53 + (-100*z - 100) * q^54 + (-24*z + 24) * q^55 + 312*z * q^56 + (-264*z + 132) * q^57 + (-141*z + 282) * q^58 + (164*z - 328) * q^59 + (-20*z + 10) * q^60 - 145*z * q^61 + (270*z - 270) * q^62 + (-184*z - 184) * q^63 - 307 * q^64 + (-91*z + 65) * q^65 - 48 * q^66 + (-454*z - 454) * q^67 + (585*z - 585) * q^68 + 156*z * q^69 + (48*z - 24) * q^70 + (-610*z + 1220) * q^71 + (-299*z + 598) * q^72 + (530*z - 265) * q^73 - 249*z * q^74 + (244*z - 244) * q^75 + (330*z + 330) * q^76 - 192 * q^77 + (52*z - 182) * q^78 + 1276 * q^79 + (z + 1) * q^80 + (421*z - 421) * q^81 + 471*z * q^82 + (-912*z + 456) * q^83 + (80*z - 160) * q^84 + (117*z - 234) * q^85 + (-208*z + 104) * q^86 + 282*z * q^87 + (-312*z + 312) * q^88 + (-564*z - 564) * q^89 + 69 * q^90 + (208*z - 728) * q^91 - 390 * q^92 + (-180*z - 180) * q^93 + (522*z - 522) * q^94 + 198*z * q^95 + (-420*z + 210) * q^96 + (-116*z + 232) * q^97 + (151*z - 302) * q^98 + (368*z - 184) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} - 2 q^{3} - 5 q^{4} - 6 q^{6} - 24 q^{7} + 23 q^{9}+O(q^{10})$$ 2 * q + 3 * q^2 - 2 * q^3 - 5 * q^4 - 6 * q^6 - 24 * q^7 + 23 * q^9 $$2 q + 3 q^{2} - 2 q^{3} - 5 q^{4} - 6 q^{6} - 24 q^{7} + 23 q^{9} + 3 q^{10} + 24 q^{11} + 20 q^{12} + 91 q^{13} - 48 q^{14} + 6 q^{15} - q^{16} - 117 q^{17} - 198 q^{19} - 15 q^{20} + 24 q^{22} + 78 q^{23} + 78 q^{24} + 244 q^{25} + 117 q^{26} - 200 q^{27} + 120 q^{28} + 141 q^{29} + 6 q^{30} - 315 q^{32} - 48 q^{33} + 24 q^{35} + 115 q^{36} - 249 q^{37} - 396 q^{38} - 130 q^{39} - 78 q^{40} + 471 q^{41} + 48 q^{42} - 104 q^{43} + 69 q^{45} + 234 q^{46} - 2 q^{48} - 151 q^{49} + 366 q^{50} + 468 q^{51} - 130 q^{52} + 186 q^{53} - 300 q^{54} + 24 q^{55} + 312 q^{56} + 423 q^{58} - 492 q^{59} - 145 q^{61} - 270 q^{62} - 552 q^{63} - 614 q^{64} + 39 q^{65} - 96 q^{66} - 1362 q^{67} - 585 q^{68} + 156 q^{69} + 1830 q^{71} + 897 q^{72} - 249 q^{74} - 244 q^{75} + 990 q^{76} - 384 q^{77} - 312 q^{78} + 2552 q^{79} + 3 q^{80} - 421 q^{81} + 471 q^{82} - 240 q^{84} - 351 q^{85} + 282 q^{87} + 312 q^{88} - 1692 q^{89} + 138 q^{90} - 1248 q^{91} - 780 q^{92} - 540 q^{93} - 522 q^{94} + 198 q^{95} + 348 q^{97} - 453 q^{98}+O(q^{100})$$ 2 * q + 3 * q^2 - 2 * q^3 - 5 * q^4 - 6 * q^6 - 24 * q^7 + 23 * q^9 + 3 * q^10 + 24 * q^11 + 20 * q^12 + 91 * q^13 - 48 * q^14 + 6 * q^15 - q^16 - 117 * q^17 - 198 * q^19 - 15 * q^20 + 24 * q^22 + 78 * q^23 + 78 * q^24 + 244 * q^25 + 117 * q^26 - 200 * q^27 + 120 * q^28 + 141 * q^29 + 6 * q^30 - 315 * q^32 - 48 * q^33 + 24 * q^35 + 115 * q^36 - 249 * q^37 - 396 * q^38 - 130 * q^39 - 78 * q^40 + 471 * q^41 + 48 * q^42 - 104 * q^43 + 69 * q^45 + 234 * q^46 - 2 * q^48 - 151 * q^49 + 366 * q^50 + 468 * q^51 - 130 * q^52 + 186 * q^53 - 300 * q^54 + 24 * q^55 + 312 * q^56 + 423 * q^58 - 492 * q^59 - 145 * q^61 - 270 * q^62 - 552 * q^63 - 614 * q^64 + 39 * q^65 - 96 * q^66 - 1362 * q^67 - 585 * q^68 + 156 * q^69 + 1830 * q^71 + 897 * q^72 - 249 * q^74 - 244 * q^75 + 990 * q^76 - 384 * q^77 - 312 * q^78 + 2552 * q^79 + 3 * q^80 - 421 * q^81 + 471 * q^82 - 240 * q^84 - 351 * q^85 + 282 * q^87 + 312 * q^88 - 1692 * q^89 + 138 * q^90 - 1248 * q^91 - 780 * q^92 - 540 * q^93 - 522 * q^94 + 198 * q^95 + 348 * q^97 - 453 * 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 −2.50000 4.33013i 1.73205i −3.00000 + 1.73205i −12.0000 + 6.92820i 22.5167i 11.5000 + 19.9186i 1.50000 2.59808i
10.1 1.50000 0.866025i −1.00000 1.73205i −2.50000 + 4.33013i 1.73205i −3.00000 1.73205i −12.0000 6.92820i 22.5167i 11.5000 19.9186i 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 13.4.e.b 2
3.b odd 2 1 117.4.q.a 2
4.b odd 2 1 208.4.w.b 2
13.b even 2 1 169.4.e.a 2
13.c even 3 1 169.4.b.d 2
13.c even 3 1 169.4.e.a 2
13.d odd 4 2 169.4.c.h 4
13.e even 6 1 inner 13.4.e.b 2
13.e even 6 1 169.4.b.d 2
13.f odd 12 2 169.4.a.i 2
13.f odd 12 2 169.4.c.h 4
39.h odd 6 1 117.4.q.a 2
39.k even 12 2 1521.4.a.o 2
52.i odd 6 1 208.4.w.b 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 3T_{2} + 3$$ acting on $$S_{4}^{\mathrm{new}}(13, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3T + 3$$
$3$ $$T^{2} + 2T + 4$$
$5$ $$T^{2} + 3$$
$7$ $$T^{2} + 24T + 192$$
$11$ $$T^{2} - 24T + 192$$
$13$ $$T^{2} - 91T + 2197$$
$17$ $$T^{2} + 117T + 13689$$
$19$ $$T^{2} + 198T + 13068$$
$23$ $$T^{2} - 78T + 6084$$
$29$ $$T^{2} - 141T + 19881$$
$31$ $$T^{2} + 24300$$
$37$ $$T^{2} + 249T + 20667$$
$41$ $$T^{2} - 471T + 73947$$
$43$ $$T^{2} + 104T + 10816$$
$47$ $$T^{2} + 90828$$
$53$ $$(T - 93)^{2}$$
$59$ $$T^{2} + 492T + 80688$$
$61$ $$T^{2} + 145T + 21025$$
$67$ $$T^{2} + 1362 T + 618348$$
$71$ $$T^{2} - 1830 T + 1116300$$
$73$ $$T^{2} + 210675$$
$79$ $$(T - 1276)^{2}$$
$83$ $$T^{2} + 623808$$
$89$ $$T^{2} + 1692 T + 954288$$
$97$ $$T^{2} - 348T + 40368$$