# Properties

 Label 13.4.e.a 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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} - 2) q^{2} + ( - 7 \zeta_{6} + 7) q^{3} + 4 \zeta_{6} q^{4} + (16 \zeta_{6} - 8) q^{5} + (14 \zeta_{6} - 28) q^{6} + ( - 13 \zeta_{6} + 26) q^{7} + (16 \zeta_{6} - 8) q^{8} - 22 \zeta_{6} q^{9} +O(q^{10})$$ q + (-2*z - 2) * q^2 + (-7*z + 7) * q^3 + 4*z * q^4 + (16*z - 8) * q^5 + (14*z - 28) * q^6 + (-13*z + 26) * q^7 + (16*z - 8) * q^8 - 22*z * q^9 $$q + ( - 2 \zeta_{6} - 2) q^{2} + ( - 7 \zeta_{6} + 7) q^{3} + 4 \zeta_{6} q^{4} + (16 \zeta_{6} - 8) q^{5} + (14 \zeta_{6} - 28) q^{6} + ( - 13 \zeta_{6} + 26) q^{7} + (16 \zeta_{6} - 8) q^{8} - 22 \zeta_{6} q^{9} + ( - 48 \zeta_{6} + 48) q^{10} + ( - 13 \zeta_{6} - 13) q^{11} + 28 q^{12} + (52 \zeta_{6} - 39) q^{13} - 78 q^{14} + (56 \zeta_{6} + 56) q^{15} + ( - 80 \zeta_{6} + 80) q^{16} - 27 \zeta_{6} q^{17} + (88 \zeta_{6} - 44) q^{18} + (51 \zeta_{6} - 102) q^{19} + (32 \zeta_{6} - 64) q^{20} + ( - 182 \zeta_{6} + 91) q^{21} + 78 \zeta_{6} q^{22} + (57 \zeta_{6} - 57) q^{23} + (56 \zeta_{6} + 56) q^{24} - 67 q^{25} + ( - 130 \zeta_{6} + 182) q^{26} + 35 q^{27} + (52 \zeta_{6} + 52) q^{28} + ( - 69 \zeta_{6} + 69) q^{29} - 336 \zeta_{6} q^{30} + ( - 84 \zeta_{6} + 42) q^{31} + (96 \zeta_{6} - 192) q^{32} + (91 \zeta_{6} - 182) q^{33} + (108 \zeta_{6} - 54) q^{34} + 312 \zeta_{6} q^{35} + ( - 88 \zeta_{6} + 88) q^{36} + ( - 23 \zeta_{6} - 23) q^{37} + 306 q^{38} + (273 \zeta_{6} + 91) q^{39} - 192 q^{40} + ( - 227 \zeta_{6} - 227) q^{41} + (546 \zeta_{6} - 546) q^{42} + 85 \zeta_{6} q^{43} + ( - 104 \zeta_{6} + 52) q^{44} + ( - 176 \zeta_{6} + 352) q^{45} + ( - 114 \zeta_{6} + 228) q^{46} + ( - 396 \zeta_{6} + 198) q^{47} - 560 \zeta_{6} q^{48} + ( - 164 \zeta_{6} + 164) q^{49} + (134 \zeta_{6} + 134) q^{50} - 189 q^{51} + (52 \zeta_{6} - 208) q^{52} + 426 q^{53} + ( - 70 \zeta_{6} - 70) q^{54} + ( - 312 \zeta_{6} + 312) q^{55} + 312 \zeta_{6} q^{56} + (714 \zeta_{6} - 357) q^{57} + (138 \zeta_{6} - 276) q^{58} + (11 \zeta_{6} - 22) q^{59} + (448 \zeta_{6} - 224) q^{60} + 17 \zeta_{6} q^{61} + (252 \zeta_{6} - 252) q^{62} + ( - 286 \zeta_{6} - 286) q^{63} - 64 q^{64} + ( - 208 \zeta_{6} - 520) q^{65} + 546 q^{66} + (95 \zeta_{6} + 95) q^{67} + ( - 108 \zeta_{6} + 108) q^{68} + 399 \zeta_{6} q^{69} + ( - 1248 \zeta_{6} + 624) q^{70} + ( - 337 \zeta_{6} + 674) q^{71} + ( - 176 \zeta_{6} + 352) q^{72} + (1160 \zeta_{6} - 580) q^{73} + 138 \zeta_{6} q^{74} + (469 \zeta_{6} - 469) q^{75} + ( - 204 \zeta_{6} - 204) q^{76} - 507 q^{77} + ( - 1274 \zeta_{6} + 364) q^{78} - 1244 q^{79} + (640 \zeta_{6} + 640) q^{80} + ( - 839 \zeta_{6} + 839) q^{81} + 1362 \zeta_{6} q^{82} + ( - 492 \zeta_{6} + 246) q^{83} + ( - 364 \zeta_{6} + 728) q^{84} + ( - 216 \zeta_{6} + 432) q^{85} + ( - 340 \zeta_{6} + 170) q^{86} - 483 \zeta_{6} q^{87} + ( - 312 \zeta_{6} + 312) q^{88} + (177 \zeta_{6} + 177) q^{89} - 1056 q^{90} + (1183 \zeta_{6} - 338) q^{91} - 228 q^{92} + ( - 294 \zeta_{6} - 294) q^{93} + (1188 \zeta_{6} - 1188) q^{94} - 1224 \zeta_{6} q^{95} + (1344 \zeta_{6} - 672) q^{96} + ( - 713 \zeta_{6} + 1426) q^{97} + (328 \zeta_{6} - 656) q^{98} + (572 \zeta_{6} - 286) q^{99} +O(q^{100})$$ q + (-2*z - 2) * q^2 + (-7*z + 7) * q^3 + 4*z * q^4 + (16*z - 8) * q^5 + (14*z - 28) * q^6 + (-13*z + 26) * q^7 + (16*z - 8) * q^8 - 22*z * q^9 + (-48*z + 48) * q^10 + (-13*z - 13) * q^11 + 28 * q^12 + (52*z - 39) * q^13 - 78 * q^14 + (56*z + 56) * q^15 + (-80*z + 80) * q^16 - 27*z * q^17 + (88*z - 44) * q^18 + (51*z - 102) * q^19 + (32*z - 64) * q^20 + (-182*z + 91) * q^21 + 78*z * q^22 + (57*z - 57) * q^23 + (56*z + 56) * q^24 - 67 * q^25 + (-130*z + 182) * q^26 + 35 * q^27 + (52*z + 52) * q^28 + (-69*z + 69) * q^29 - 336*z * q^30 + (-84*z + 42) * q^31 + (96*z - 192) * q^32 + (91*z - 182) * q^33 + (108*z - 54) * q^34 + 312*z * q^35 + (-88*z + 88) * q^36 + (-23*z - 23) * q^37 + 306 * q^38 + (273*z + 91) * q^39 - 192 * q^40 + (-227*z - 227) * q^41 + (546*z - 546) * q^42 + 85*z * q^43 + (-104*z + 52) * q^44 + (-176*z + 352) * q^45 + (-114*z + 228) * q^46 + (-396*z + 198) * q^47 - 560*z * q^48 + (-164*z + 164) * q^49 + (134*z + 134) * q^50 - 189 * q^51 + (52*z - 208) * q^52 + 426 * q^53 + (-70*z - 70) * q^54 + (-312*z + 312) * q^55 + 312*z * q^56 + (714*z - 357) * q^57 + (138*z - 276) * q^58 + (11*z - 22) * q^59 + (448*z - 224) * q^60 + 17*z * q^61 + (252*z - 252) * q^62 + (-286*z - 286) * q^63 - 64 * q^64 + (-208*z - 520) * q^65 + 546 * q^66 + (95*z + 95) * q^67 + (-108*z + 108) * q^68 + 399*z * q^69 + (-1248*z + 624) * q^70 + (-337*z + 674) * q^71 + (-176*z + 352) * q^72 + (1160*z - 580) * q^73 + 138*z * q^74 + (469*z - 469) * q^75 + (-204*z - 204) * q^76 - 507 * q^77 + (-1274*z + 364) * q^78 - 1244 * q^79 + (640*z + 640) * q^80 + (-839*z + 839) * q^81 + 1362*z * q^82 + (-492*z + 246) * q^83 + (-364*z + 728) * q^84 + (-216*z + 432) * q^85 + (-340*z + 170) * q^86 - 483*z * q^87 + (-312*z + 312) * q^88 + (177*z + 177) * q^89 - 1056 * q^90 + (1183*z - 338) * q^91 - 228 * q^92 + (-294*z - 294) * q^93 + (1188*z - 1188) * q^94 - 1224*z * q^95 + (1344*z - 672) * q^96 + (-713*z + 1426) * q^97 + (328*z - 656) * q^98 + (572*z - 286) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{2} + 7 q^{3} + 4 q^{4} - 42 q^{6} + 39 q^{7} - 22 q^{9}+O(q^{10})$$ 2 * q - 6 * q^2 + 7 * q^3 + 4 * q^4 - 42 * q^6 + 39 * q^7 - 22 * q^9 $$2 q - 6 q^{2} + 7 q^{3} + 4 q^{4} - 42 q^{6} + 39 q^{7} - 22 q^{9} + 48 q^{10} - 39 q^{11} + 56 q^{12} - 26 q^{13} - 156 q^{14} + 168 q^{15} + 80 q^{16} - 27 q^{17} - 153 q^{19} - 96 q^{20} + 78 q^{22} - 57 q^{23} + 168 q^{24} - 134 q^{25} + 234 q^{26} + 70 q^{27} + 156 q^{28} + 69 q^{29} - 336 q^{30} - 288 q^{32} - 273 q^{33} + 312 q^{35} + 88 q^{36} - 69 q^{37} + 612 q^{38} + 455 q^{39} - 384 q^{40} - 681 q^{41} - 546 q^{42} + 85 q^{43} + 528 q^{45} + 342 q^{46} - 560 q^{48} + 164 q^{49} + 402 q^{50} - 378 q^{51} - 364 q^{52} + 852 q^{53} - 210 q^{54} + 312 q^{55} + 312 q^{56} - 414 q^{58} - 33 q^{59} + 17 q^{61} - 252 q^{62} - 858 q^{63} - 128 q^{64} - 1248 q^{65} + 1092 q^{66} + 285 q^{67} + 108 q^{68} + 399 q^{69} + 1011 q^{71} + 528 q^{72} + 138 q^{74} - 469 q^{75} - 612 q^{76} - 1014 q^{77} - 546 q^{78} - 2488 q^{79} + 1920 q^{80} + 839 q^{81} + 1362 q^{82} + 1092 q^{84} + 648 q^{85} - 483 q^{87} + 312 q^{88} + 531 q^{89} - 2112 q^{90} + 507 q^{91} - 456 q^{92} - 882 q^{93} - 1188 q^{94} - 1224 q^{95} + 2139 q^{97} - 984 q^{98}+O(q^{100})$$ 2 * q - 6 * q^2 + 7 * q^3 + 4 * q^4 - 42 * q^6 + 39 * q^7 - 22 * q^9 + 48 * q^10 - 39 * q^11 + 56 * q^12 - 26 * q^13 - 156 * q^14 + 168 * q^15 + 80 * q^16 - 27 * q^17 - 153 * q^19 - 96 * q^20 + 78 * q^22 - 57 * q^23 + 168 * q^24 - 134 * q^25 + 234 * q^26 + 70 * q^27 + 156 * q^28 + 69 * q^29 - 336 * q^30 - 288 * q^32 - 273 * q^33 + 312 * q^35 + 88 * q^36 - 69 * q^37 + 612 * q^38 + 455 * q^39 - 384 * q^40 - 681 * q^41 - 546 * q^42 + 85 * q^43 + 528 * q^45 + 342 * q^46 - 560 * q^48 + 164 * q^49 + 402 * q^50 - 378 * q^51 - 364 * q^52 + 852 * q^53 - 210 * q^54 + 312 * q^55 + 312 * q^56 - 414 * q^58 - 33 * q^59 + 17 * q^61 - 252 * q^62 - 858 * q^63 - 128 * q^64 - 1248 * q^65 + 1092 * q^66 + 285 * q^67 + 108 * q^68 + 399 * q^69 + 1011 * q^71 + 528 * q^72 + 138 * q^74 - 469 * q^75 - 612 * q^76 - 1014 * q^77 - 546 * q^78 - 2488 * q^79 + 1920 * q^80 + 839 * q^81 + 1362 * q^82 + 1092 * q^84 + 648 * q^85 - 483 * q^87 + 312 * q^88 + 531 * q^89 - 2112 * q^90 + 507 * q^91 - 456 * q^92 - 882 * q^93 - 1188 * q^94 - 1224 * q^95 + 2139 * q^97 - 984 * 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
−3.00000 1.73205i 3.50000 6.06218i 2.00000 + 3.46410i 13.8564i −21.0000 + 12.1244i 19.5000 11.2583i 13.8564i −11.0000 19.0526i 24.0000 41.5692i
10.1 −3.00000 + 1.73205i 3.50000 + 6.06218i 2.00000 3.46410i 13.8564i −21.0000 12.1244i 19.5000 + 11.2583i 13.8564i −11.0000 + 19.0526i 24.0000 + 41.5692i
 $$n$$: 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.a 2
3.b odd 2 1 117.4.q.c 2
4.b odd 2 1 208.4.w.a 2
13.b even 2 1 169.4.e.b 2
13.c even 3 1 169.4.b.b 2
13.c even 3 1 169.4.e.b 2
13.d odd 4 2 169.4.c.i 4
13.e even 6 1 inner 13.4.e.a 2
13.e even 6 1 169.4.b.b 2
13.f odd 12 2 169.4.a.h 2
13.f odd 12 2 169.4.c.i 4
39.h odd 6 1 117.4.q.c 2
39.k even 12 2 1521.4.a.q 2
52.i odd 6 1 208.4.w.a 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 6T_{2} + 12$$ acting on $$S_{4}^{\mathrm{new}}(13, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 6T + 12$$
$3$ $$T^{2} - 7T + 49$$
$5$ $$T^{2} + 192$$
$7$ $$T^{2} - 39T + 507$$
$11$ $$T^{2} + 39T + 507$$
$13$ $$T^{2} + 26T + 2197$$
$17$ $$T^{2} + 27T + 729$$
$19$ $$T^{2} + 153T + 7803$$
$23$ $$T^{2} + 57T + 3249$$
$29$ $$T^{2} - 69T + 4761$$
$31$ $$T^{2} + 5292$$
$37$ $$T^{2} + 69T + 1587$$
$41$ $$T^{2} + 681T + 154587$$
$43$ $$T^{2} - 85T + 7225$$
$47$ $$T^{2} + 117612$$
$53$ $$(T - 426)^{2}$$
$59$ $$T^{2} + 33T + 363$$
$61$ $$T^{2} - 17T + 289$$
$67$ $$T^{2} - 285T + 27075$$
$71$ $$T^{2} - 1011 T + 340707$$
$73$ $$T^{2} + 1009200$$
$79$ $$(T + 1244)^{2}$$
$83$ $$T^{2} + 181548$$
$89$ $$T^{2} - 531T + 93987$$
$97$ $$T^{2} - 2139 T + 1525107$$