# Properties

 Label 39.4.e.b Level $39$ Weight $4$ Character orbit 39.e Analytic conductor $2.301$ 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] = [39,4,Mod(16,39)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(39, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("39.16");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$39 = 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 39.e (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: $$2.30107449022$$ 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: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} + (3 \zeta_{6} - 3) q^{3} + 7 \zeta_{6} q^{4} + 7 q^{5} + 3 \zeta_{6} q^{6} + 10 \zeta_{6} q^{7} + 15 q^{8} - 9 \zeta_{6} q^{9} +O(q^{10})$$ q + (-z + 1) * q^2 + (3*z - 3) * q^3 + 7*z * q^4 + 7 * q^5 + 3*z * q^6 + 10*z * q^7 + 15 * q^8 - 9*z * q^9 $$q + ( - \zeta_{6} + 1) q^{2} + (3 \zeta_{6} - 3) q^{3} + 7 \zeta_{6} q^{4} + 7 q^{5} + 3 \zeta_{6} q^{6} + 10 \zeta_{6} q^{7} + 15 q^{8} - 9 \zeta_{6} q^{9} + ( - 7 \zeta_{6} + 7) q^{10} + ( - 22 \zeta_{6} + 22) q^{11} - 21 q^{12} + ( - 13 \zeta_{6} - 39) q^{13} + 10 q^{14} + (21 \zeta_{6} - 21) q^{15} + (41 \zeta_{6} - 41) q^{16} - 37 \zeta_{6} q^{17} - 9 q^{18} - 30 \zeta_{6} q^{19} + 49 \zeta_{6} q^{20} - 30 q^{21} - 22 \zeta_{6} q^{22} + ( - 162 \zeta_{6} + 162) q^{23} + (45 \zeta_{6} - 45) q^{24} - 76 q^{25} + (39 \zeta_{6} - 52) q^{26} + 27 q^{27} + (70 \zeta_{6} - 70) q^{28} + ( - 113 \zeta_{6} + 113) q^{29} + 21 \zeta_{6} q^{30} + 196 q^{31} + 161 \zeta_{6} q^{32} + 66 \zeta_{6} q^{33} - 37 q^{34} + 70 \zeta_{6} q^{35} + ( - 63 \zeta_{6} + 63) q^{36} + (13 \zeta_{6} - 13) q^{37} - 30 q^{38} + ( - 117 \zeta_{6} + 156) q^{39} + 105 q^{40} + (285 \zeta_{6} - 285) q^{41} + (30 \zeta_{6} - 30) q^{42} + 246 \zeta_{6} q^{43} + 154 q^{44} - 63 \zeta_{6} q^{45} - 162 \zeta_{6} q^{46} - 462 q^{47} - 123 \zeta_{6} q^{48} + ( - 243 \zeta_{6} + 243) q^{49} + (76 \zeta_{6} - 76) q^{50} + 111 q^{51} + ( - 364 \zeta_{6} + 91) q^{52} - 537 q^{53} + ( - 27 \zeta_{6} + 27) q^{54} + ( - 154 \zeta_{6} + 154) q^{55} + 150 \zeta_{6} q^{56} + 90 q^{57} - 113 \zeta_{6} q^{58} - 576 \zeta_{6} q^{59} - 147 q^{60} + 635 \zeta_{6} q^{61} + ( - 196 \zeta_{6} + 196) q^{62} + ( - 90 \zeta_{6} + 90) q^{63} - 167 q^{64} + ( - 91 \zeta_{6} - 273) q^{65} + 66 q^{66} + (202 \zeta_{6} - 202) q^{67} + ( - 259 \zeta_{6} + 259) q^{68} + 486 \zeta_{6} q^{69} + 70 q^{70} + 1086 \zeta_{6} q^{71} - 135 \zeta_{6} q^{72} - 805 q^{73} + 13 \zeta_{6} q^{74} + ( - 228 \zeta_{6} + 228) q^{75} + ( - 210 \zeta_{6} + 210) q^{76} + 220 q^{77} + ( - 156 \zeta_{6} + 39) q^{78} + 884 q^{79} + (287 \zeta_{6} - 287) q^{80} + (81 \zeta_{6} - 81) q^{81} + 285 \zeta_{6} q^{82} + 518 q^{83} - 210 \zeta_{6} q^{84} - 259 \zeta_{6} q^{85} + 246 q^{86} + 339 \zeta_{6} q^{87} + ( - 330 \zeta_{6} + 330) q^{88} + (194 \zeta_{6} - 194) q^{89} - 63 q^{90} + ( - 520 \zeta_{6} + 130) q^{91} + 1134 q^{92} + (588 \zeta_{6} - 588) q^{93} + (462 \zeta_{6} - 462) q^{94} - 210 \zeta_{6} q^{95} - 483 q^{96} + 1202 \zeta_{6} q^{97} - 243 \zeta_{6} q^{98} - 198 q^{99} +O(q^{100})$$ q + (-z + 1) * q^2 + (3*z - 3) * q^3 + 7*z * q^4 + 7 * q^5 + 3*z * q^6 + 10*z * q^7 + 15 * q^8 - 9*z * q^9 + (-7*z + 7) * q^10 + (-22*z + 22) * q^11 - 21 * q^12 + (-13*z - 39) * q^13 + 10 * q^14 + (21*z - 21) * q^15 + (41*z - 41) * q^16 - 37*z * q^17 - 9 * q^18 - 30*z * q^19 + 49*z * q^20 - 30 * q^21 - 22*z * q^22 + (-162*z + 162) * q^23 + (45*z - 45) * q^24 - 76 * q^25 + (39*z - 52) * q^26 + 27 * q^27 + (70*z - 70) * q^28 + (-113*z + 113) * q^29 + 21*z * q^30 + 196 * q^31 + 161*z * q^32 + 66*z * q^33 - 37 * q^34 + 70*z * q^35 + (-63*z + 63) * q^36 + (13*z - 13) * q^37 - 30 * q^38 + (-117*z + 156) * q^39 + 105 * q^40 + (285*z - 285) * q^41 + (30*z - 30) * q^42 + 246*z * q^43 + 154 * q^44 - 63*z * q^45 - 162*z * q^46 - 462 * q^47 - 123*z * q^48 + (-243*z + 243) * q^49 + (76*z - 76) * q^50 + 111 * q^51 + (-364*z + 91) * q^52 - 537 * q^53 + (-27*z + 27) * q^54 + (-154*z + 154) * q^55 + 150*z * q^56 + 90 * q^57 - 113*z * q^58 - 576*z * q^59 - 147 * q^60 + 635*z * q^61 + (-196*z + 196) * q^62 + (-90*z + 90) * q^63 - 167 * q^64 + (-91*z - 273) * q^65 + 66 * q^66 + (202*z - 202) * q^67 + (-259*z + 259) * q^68 + 486*z * q^69 + 70 * q^70 + 1086*z * q^71 - 135*z * q^72 - 805 * q^73 + 13*z * q^74 + (-228*z + 228) * q^75 + (-210*z + 210) * q^76 + 220 * q^77 + (-156*z + 39) * q^78 + 884 * q^79 + (287*z - 287) * q^80 + (81*z - 81) * q^81 + 285*z * q^82 + 518 * q^83 - 210*z * q^84 - 259*z * q^85 + 246 * q^86 + 339*z * q^87 + (-330*z + 330) * q^88 + (194*z - 194) * q^89 - 63 * q^90 + (-520*z + 130) * q^91 + 1134 * q^92 + (588*z - 588) * q^93 + (462*z - 462) * q^94 - 210*z * q^95 - 483 * q^96 + 1202*z * q^97 - 243*z * q^98 - 198 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 3 q^{3} + 7 q^{4} + 14 q^{5} + 3 q^{6} + 10 q^{7} + 30 q^{8} - 9 q^{9}+O(q^{10})$$ 2 * q + q^2 - 3 * q^3 + 7 * q^4 + 14 * q^5 + 3 * q^6 + 10 * q^7 + 30 * q^8 - 9 * q^9 $$2 q + q^{2} - 3 q^{3} + 7 q^{4} + 14 q^{5} + 3 q^{6} + 10 q^{7} + 30 q^{8} - 9 q^{9} + 7 q^{10} + 22 q^{11} - 42 q^{12} - 91 q^{13} + 20 q^{14} - 21 q^{15} - 41 q^{16} - 37 q^{17} - 18 q^{18} - 30 q^{19} + 49 q^{20} - 60 q^{21} - 22 q^{22} + 162 q^{23} - 45 q^{24} - 152 q^{25} - 65 q^{26} + 54 q^{27} - 70 q^{28} + 113 q^{29} + 21 q^{30} + 392 q^{31} + 161 q^{32} + 66 q^{33} - 74 q^{34} + 70 q^{35} + 63 q^{36} - 13 q^{37} - 60 q^{38} + 195 q^{39} + 210 q^{40} - 285 q^{41} - 30 q^{42} + 246 q^{43} + 308 q^{44} - 63 q^{45} - 162 q^{46} - 924 q^{47} - 123 q^{48} + 243 q^{49} - 76 q^{50} + 222 q^{51} - 182 q^{52} - 1074 q^{53} + 27 q^{54} + 154 q^{55} + 150 q^{56} + 180 q^{57} - 113 q^{58} - 576 q^{59} - 294 q^{60} + 635 q^{61} + 196 q^{62} + 90 q^{63} - 334 q^{64} - 637 q^{65} + 132 q^{66} - 202 q^{67} + 259 q^{68} + 486 q^{69} + 140 q^{70} + 1086 q^{71} - 135 q^{72} - 1610 q^{73} + 13 q^{74} + 228 q^{75} + 210 q^{76} + 440 q^{77} - 78 q^{78} + 1768 q^{79} - 287 q^{80} - 81 q^{81} + 285 q^{82} + 1036 q^{83} - 210 q^{84} - 259 q^{85} + 492 q^{86} + 339 q^{87} + 330 q^{88} - 194 q^{89} - 126 q^{90} - 260 q^{91} + 2268 q^{92} - 588 q^{93} - 462 q^{94} - 210 q^{95} - 966 q^{96} + 1202 q^{97} - 243 q^{98} - 396 q^{99}+O(q^{100})$$ 2 * q + q^2 - 3 * q^3 + 7 * q^4 + 14 * q^5 + 3 * q^6 + 10 * q^7 + 30 * q^8 - 9 * q^9 + 7 * q^10 + 22 * q^11 - 42 * q^12 - 91 * q^13 + 20 * q^14 - 21 * q^15 - 41 * q^16 - 37 * q^17 - 18 * q^18 - 30 * q^19 + 49 * q^20 - 60 * q^21 - 22 * q^22 + 162 * q^23 - 45 * q^24 - 152 * q^25 - 65 * q^26 + 54 * q^27 - 70 * q^28 + 113 * q^29 + 21 * q^30 + 392 * q^31 + 161 * q^32 + 66 * q^33 - 74 * q^34 + 70 * q^35 + 63 * q^36 - 13 * q^37 - 60 * q^38 + 195 * q^39 + 210 * q^40 - 285 * q^41 - 30 * q^42 + 246 * q^43 + 308 * q^44 - 63 * q^45 - 162 * q^46 - 924 * q^47 - 123 * q^48 + 243 * q^49 - 76 * q^50 + 222 * q^51 - 182 * q^52 - 1074 * q^53 + 27 * q^54 + 154 * q^55 + 150 * q^56 + 180 * q^57 - 113 * q^58 - 576 * q^59 - 294 * q^60 + 635 * q^61 + 196 * q^62 + 90 * q^63 - 334 * q^64 - 637 * q^65 + 132 * q^66 - 202 * q^67 + 259 * q^68 + 486 * q^69 + 140 * q^70 + 1086 * q^71 - 135 * q^72 - 1610 * q^73 + 13 * q^74 + 228 * q^75 + 210 * q^76 + 440 * q^77 - 78 * q^78 + 1768 * q^79 - 287 * q^80 - 81 * q^81 + 285 * q^82 + 1036 * q^83 - 210 * q^84 - 259 * q^85 + 492 * q^86 + 339 * q^87 + 330 * q^88 - 194 * q^89 - 126 * q^90 - 260 * q^91 + 2268 * q^92 - 588 * q^93 - 462 * q^94 - 210 * q^95 - 966 * q^96 + 1202 * q^97 - 243 * q^98 - 396 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/39\mathbb{Z}\right)^\times$$.

 $$n$$ $$14$$ $$28$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 + 0.866025i −1.50000 2.59808i 3.50000 6.06218i 7.00000 1.50000 2.59808i 5.00000 8.66025i 15.0000 −4.50000 + 7.79423i 3.50000 + 6.06218i
22.1 0.500000 0.866025i −1.50000 + 2.59808i 3.50000 + 6.06218i 7.00000 1.50000 + 2.59808i 5.00000 + 8.66025i 15.0000 −4.50000 7.79423i 3.50000 6.06218i
 $$n$$: 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 39.4.e.b 2
3.b odd 2 1 117.4.g.a 2
4.b odd 2 1 624.4.q.c 2
13.c even 3 1 inner 39.4.e.b 2
13.c even 3 1 507.4.a.b 1
13.e even 6 1 507.4.a.d 1
13.f odd 12 2 507.4.b.d 2
39.h odd 6 1 1521.4.a.e 1
39.i odd 6 1 117.4.g.a 2
39.i odd 6 1 1521.4.a.h 1
52.j odd 6 1 624.4.q.c 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} + 1$$ acting on $$S_{4}^{\mathrm{new}}(39, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} + 3T + 9$$
$5$ $$(T - 7)^{2}$$
$7$ $$T^{2} - 10T + 100$$
$11$ $$T^{2} - 22T + 484$$
$13$ $$T^{2} + 91T + 2197$$
$17$ $$T^{2} + 37T + 1369$$
$19$ $$T^{2} + 30T + 900$$
$23$ $$T^{2} - 162T + 26244$$
$29$ $$T^{2} - 113T + 12769$$
$31$ $$(T - 196)^{2}$$
$37$ $$T^{2} + 13T + 169$$
$41$ $$T^{2} + 285T + 81225$$
$43$ $$T^{2} - 246T + 60516$$
$47$ $$(T + 462)^{2}$$
$53$ $$(T + 537)^{2}$$
$59$ $$T^{2} + 576T + 331776$$
$61$ $$T^{2} - 635T + 403225$$
$67$ $$T^{2} + 202T + 40804$$
$71$ $$T^{2} - 1086 T + 1179396$$
$73$ $$(T + 805)^{2}$$
$79$ $$(T - 884)^{2}$$
$83$ $$(T - 518)^{2}$$
$89$ $$T^{2} + 194T + 37636$$
$97$ $$T^{2} - 1202 T + 1444804$$