# Properties

 Label 39.4.a.b Level $39$ Weight $4$ Character orbit 39.a Self dual yes Analytic conductor $2.301$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [39,4,Mod(1,39)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(39, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("39.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$39 = 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 39.a (trivial)

## 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: yes Analytic conductor: $$2.30107449022$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{14})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 14$$ x^2 - 14 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \sqrt{14}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} - 3 q^{3} + (2 \beta + 7) q^{4} + ( - 2 \beta + 12) q^{5} + ( - 3 \beta - 3) q^{6} - 2 \beta q^{7} + (\beta + 27) q^{8} + 9 q^{9} +O(q^{10})$$ q + (b + 1) * q^2 - 3 * q^3 + (2*b + 7) * q^4 + (-2*b + 12) * q^5 + (-3*b - 3) * q^6 - 2*b * q^7 + (b + 27) * q^8 + 9 * q^9 $$q + (\beta + 1) q^{2} - 3 q^{3} + (2 \beta + 7) q^{4} + ( - 2 \beta + 12) q^{5} + ( - 3 \beta - 3) q^{6} - 2 \beta q^{7} + (\beta + 27) q^{8} + 9 q^{9} + (10 \beta - 16) q^{10} + ( - 12 \beta - 22) q^{11} + ( - 6 \beta - 21) q^{12} - 13 q^{13} + ( - 2 \beta - 28) q^{14} + (6 \beta - 36) q^{15} + (12 \beta - 15) q^{16} + (4 \beta + 82) q^{17} + (9 \beta + 9) q^{18} + (2 \beta + 24) q^{19} + (10 \beta + 28) q^{20} + 6 \beta q^{21} + ( - 34 \beta - 190) q^{22} + (48 \beta + 4) q^{23} + ( - 3 \beta - 81) q^{24} + ( - 48 \beta + 75) q^{25} + ( - 13 \beta - 13) q^{26} - 27 q^{27} + ( - 14 \beta - 56) q^{28} + ( - 24 \beta + 202) q^{29} + ( - 30 \beta + 48) q^{30} + ( - 26 \beta + 20) q^{31} + ( - 11 \beta - 63) q^{32} + (36 \beta + 66) q^{33} + (86 \beta + 138) q^{34} + ( - 24 \beta + 56) q^{35} + (18 \beta + 63) q^{36} + (28 \beta - 50) q^{37} + (26 \beta + 52) q^{38} + 39 q^{39} + ( - 42 \beta + 296) q^{40} + (94 \beta + 100) q^{41} + (6 \beta + 84) q^{42} + (52 \beta - 308) q^{43} + ( - 128 \beta - 490) q^{44} + ( - 18 \beta + 108) q^{45} + (52 \beta + 676) q^{46} + (32 \beta - 162) q^{47} + ( - 36 \beta + 45) q^{48} - 287 q^{49} + (27 \beta - 597) q^{50} + ( - 12 \beta - 246) q^{51} + ( - 26 \beta - 91) q^{52} + ( - 120 \beta - 82) q^{53} + ( - 27 \beta - 27) q^{54} + ( - 100 \beta + 72) q^{55} + ( - 54 \beta - 28) q^{56} + ( - 6 \beta - 72) q^{57} + (178 \beta - 134) q^{58} + (40 \beta + 70) q^{59} + ( - 30 \beta - 84) q^{60} + (136 \beta + 314) q^{61} + ( - 6 \beta - 344) q^{62} - 18 \beta q^{63} + ( - 170 \beta - 97) q^{64} + (26 \beta - 156) q^{65} + (102 \beta + 570) q^{66} + ( - 170 \beta - 236) q^{67} + (192 \beta + 686) q^{68} + ( - 144 \beta - 12) q^{69} + (32 \beta - 280) q^{70} + ( - 84 \beta + 214) q^{71} + (9 \beta + 243) q^{72} + (76 \beta - 450) q^{73} + ( - 22 \beta + 342) q^{74} + (144 \beta - 225) q^{75} + (62 \beta + 224) q^{76} + (44 \beta + 336) q^{77} + (39 \beta + 39) q^{78} + ( - 88 \beta - 216) q^{79} + (174 \beta - 516) q^{80} + 81 q^{81} + (194 \beta + 1416) q^{82} + (64 \beta - 694) q^{83} + (42 \beta + 168) q^{84} + ( - 116 \beta + 872) q^{85} + ( - 256 \beta + 420) q^{86} + (72 \beta - 606) q^{87} + ( - 346 \beta - 762) q^{88} + ( - 190 \beta + 480) q^{89} + (90 \beta - 144) q^{90} + 26 \beta q^{91} + (344 \beta + 1372) q^{92} + (78 \beta - 60) q^{93} + ( - 130 \beta + 286) q^{94} + ( - 24 \beta + 232) q^{95} + (33 \beta + 189) q^{96} + ( - 220 \beta - 266) q^{97} + ( - 287 \beta - 287) q^{98} + ( - 108 \beta - 198) q^{99} +O(q^{100})$$ q + (b + 1) * q^2 - 3 * q^3 + (2*b + 7) * q^4 + (-2*b + 12) * q^5 + (-3*b - 3) * q^6 - 2*b * q^7 + (b + 27) * q^8 + 9 * q^9 + (10*b - 16) * q^10 + (-12*b - 22) * q^11 + (-6*b - 21) * q^12 - 13 * q^13 + (-2*b - 28) * q^14 + (6*b - 36) * q^15 + (12*b - 15) * q^16 + (4*b + 82) * q^17 + (9*b + 9) * q^18 + (2*b + 24) * q^19 + (10*b + 28) * q^20 + 6*b * q^21 + (-34*b - 190) * q^22 + (48*b + 4) * q^23 + (-3*b - 81) * q^24 + (-48*b + 75) * q^25 + (-13*b - 13) * q^26 - 27 * q^27 + (-14*b - 56) * q^28 + (-24*b + 202) * q^29 + (-30*b + 48) * q^30 + (-26*b + 20) * q^31 + (-11*b - 63) * q^32 + (36*b + 66) * q^33 + (86*b + 138) * q^34 + (-24*b + 56) * q^35 + (18*b + 63) * q^36 + (28*b - 50) * q^37 + (26*b + 52) * q^38 + 39 * q^39 + (-42*b + 296) * q^40 + (94*b + 100) * q^41 + (6*b + 84) * q^42 + (52*b - 308) * q^43 + (-128*b - 490) * q^44 + (-18*b + 108) * q^45 + (52*b + 676) * q^46 + (32*b - 162) * q^47 + (-36*b + 45) * q^48 - 287 * q^49 + (27*b - 597) * q^50 + (-12*b - 246) * q^51 + (-26*b - 91) * q^52 + (-120*b - 82) * q^53 + (-27*b - 27) * q^54 + (-100*b + 72) * q^55 + (-54*b - 28) * q^56 + (-6*b - 72) * q^57 + (178*b - 134) * q^58 + (40*b + 70) * q^59 + (-30*b - 84) * q^60 + (136*b + 314) * q^61 + (-6*b - 344) * q^62 - 18*b * q^63 + (-170*b - 97) * q^64 + (26*b - 156) * q^65 + (102*b + 570) * q^66 + (-170*b - 236) * q^67 + (192*b + 686) * q^68 + (-144*b - 12) * q^69 + (32*b - 280) * q^70 + (-84*b + 214) * q^71 + (9*b + 243) * q^72 + (76*b - 450) * q^73 + (-22*b + 342) * q^74 + (144*b - 225) * q^75 + (62*b + 224) * q^76 + (44*b + 336) * q^77 + (39*b + 39) * q^78 + (-88*b - 216) * q^79 + (174*b - 516) * q^80 + 81 * q^81 + (194*b + 1416) * q^82 + (64*b - 694) * q^83 + (42*b + 168) * q^84 + (-116*b + 872) * q^85 + (-256*b + 420) * q^86 + (72*b - 606) * q^87 + (-346*b - 762) * q^88 + (-190*b + 480) * q^89 + (90*b - 144) * q^90 + 26*b * q^91 + (344*b + 1372) * q^92 + (78*b - 60) * q^93 + (-130*b + 286) * q^94 + (-24*b + 232) * q^95 + (33*b + 189) * q^96 + (-220*b - 266) * q^97 + (-287*b - 287) * q^98 + (-108*b - 198) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 6 q^{3} + 14 q^{4} + 24 q^{5} - 6 q^{6} + 54 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 6 * q^3 + 14 * q^4 + 24 * q^5 - 6 * q^6 + 54 * q^8 + 18 * q^9 $$2 q + 2 q^{2} - 6 q^{3} + 14 q^{4} + 24 q^{5} - 6 q^{6} + 54 q^{8} + 18 q^{9} - 32 q^{10} - 44 q^{11} - 42 q^{12} - 26 q^{13} - 56 q^{14} - 72 q^{15} - 30 q^{16} + 164 q^{17} + 18 q^{18} + 48 q^{19} + 56 q^{20} - 380 q^{22} + 8 q^{23} - 162 q^{24} + 150 q^{25} - 26 q^{26} - 54 q^{27} - 112 q^{28} + 404 q^{29} + 96 q^{30} + 40 q^{31} - 126 q^{32} + 132 q^{33} + 276 q^{34} + 112 q^{35} + 126 q^{36} - 100 q^{37} + 104 q^{38} + 78 q^{39} + 592 q^{40} + 200 q^{41} + 168 q^{42} - 616 q^{43} - 980 q^{44} + 216 q^{45} + 1352 q^{46} - 324 q^{47} + 90 q^{48} - 574 q^{49} - 1194 q^{50} - 492 q^{51} - 182 q^{52} - 164 q^{53} - 54 q^{54} + 144 q^{55} - 56 q^{56} - 144 q^{57} - 268 q^{58} + 140 q^{59} - 168 q^{60} + 628 q^{61} - 688 q^{62} - 194 q^{64} - 312 q^{65} + 1140 q^{66} - 472 q^{67} + 1372 q^{68} - 24 q^{69} - 560 q^{70} + 428 q^{71} + 486 q^{72} - 900 q^{73} + 684 q^{74} - 450 q^{75} + 448 q^{76} + 672 q^{77} + 78 q^{78} - 432 q^{79} - 1032 q^{80} + 162 q^{81} + 2832 q^{82} - 1388 q^{83} + 336 q^{84} + 1744 q^{85} + 840 q^{86} - 1212 q^{87} - 1524 q^{88} + 960 q^{89} - 288 q^{90} + 2744 q^{92} - 120 q^{93} + 572 q^{94} + 464 q^{95} + 378 q^{96} - 532 q^{97} - 574 q^{98} - 396 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 6 * q^3 + 14 * q^4 + 24 * q^5 - 6 * q^6 + 54 * q^8 + 18 * q^9 - 32 * q^10 - 44 * q^11 - 42 * q^12 - 26 * q^13 - 56 * q^14 - 72 * q^15 - 30 * q^16 + 164 * q^17 + 18 * q^18 + 48 * q^19 + 56 * q^20 - 380 * q^22 + 8 * q^23 - 162 * q^24 + 150 * q^25 - 26 * q^26 - 54 * q^27 - 112 * q^28 + 404 * q^29 + 96 * q^30 + 40 * q^31 - 126 * q^32 + 132 * q^33 + 276 * q^34 + 112 * q^35 + 126 * q^36 - 100 * q^37 + 104 * q^38 + 78 * q^39 + 592 * q^40 + 200 * q^41 + 168 * q^42 - 616 * q^43 - 980 * q^44 + 216 * q^45 + 1352 * q^46 - 324 * q^47 + 90 * q^48 - 574 * q^49 - 1194 * q^50 - 492 * q^51 - 182 * q^52 - 164 * q^53 - 54 * q^54 + 144 * q^55 - 56 * q^56 - 144 * q^57 - 268 * q^58 + 140 * q^59 - 168 * q^60 + 628 * q^61 - 688 * q^62 - 194 * q^64 - 312 * q^65 + 1140 * q^66 - 472 * q^67 + 1372 * q^68 - 24 * q^69 - 560 * q^70 + 428 * q^71 + 486 * q^72 - 900 * q^73 + 684 * q^74 - 450 * q^75 + 448 * q^76 + 672 * q^77 + 78 * q^78 - 432 * q^79 - 1032 * q^80 + 162 * q^81 + 2832 * q^82 - 1388 * q^83 + 336 * q^84 + 1744 * q^85 + 840 * q^86 - 1212 * q^87 - 1524 * q^88 + 960 * q^89 - 288 * q^90 + 2744 * q^92 - 120 * q^93 + 572 * q^94 + 464 * q^95 + 378 * q^96 - 532 * q^97 - 574 * q^98 - 396 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −3.74166 3.74166
−2.74166 −3.00000 −0.483315 19.4833 8.22497 7.48331 23.2583 9.00000 −53.4166
1.2 4.74166 −3.00000 14.4833 4.51669 −14.2250 −7.48331 30.7417 9.00000 21.4166
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$13$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.4.a.b 2
3.b odd 2 1 117.4.a.c 2
4.b odd 2 1 624.4.a.r 2
5.b even 2 1 975.4.a.j 2
7.b odd 2 1 1911.4.a.h 2
8.b even 2 1 2496.4.a.bc 2
8.d odd 2 1 2496.4.a.s 2
12.b even 2 1 1872.4.a.t 2
13.b even 2 1 507.4.a.f 2
13.d odd 4 2 507.4.b.f 4
39.d odd 2 1 1521.4.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.b 2 1.a even 1 1 trivial
117.4.a.c 2 3.b odd 2 1
507.4.a.f 2 13.b even 2 1
507.4.b.f 4 13.d odd 4 2
624.4.a.r 2 4.b odd 2 1
975.4.a.j 2 5.b even 2 1
1521.4.a.s 2 39.d odd 2 1
1872.4.a.t 2 12.b even 2 1
1911.4.a.h 2 7.b odd 2 1
2496.4.a.s 2 8.d odd 2 1
2496.4.a.bc 2 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2T_{2} - 13$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(39))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 13$$
$3$ $$(T + 3)^{2}$$
$5$ $$T^{2} - 24T + 88$$
$7$ $$T^{2} - 56$$
$11$ $$T^{2} + 44T - 1532$$
$13$ $$(T + 13)^{2}$$
$17$ $$T^{2} - 164T + 6500$$
$19$ $$T^{2} - 48T + 520$$
$23$ $$T^{2} - 8T - 32240$$
$29$ $$T^{2} - 404T + 32740$$
$31$ $$T^{2} - 40T - 9064$$
$37$ $$T^{2} + 100T - 8476$$
$41$ $$T^{2} - 200T - 113704$$
$43$ $$T^{2} + 616T + 57008$$
$47$ $$T^{2} + 324T + 11908$$
$53$ $$T^{2} + 164T - 194876$$
$59$ $$T^{2} - 140T - 17500$$
$61$ $$T^{2} - 628T - 160348$$
$67$ $$T^{2} + 472T - 348904$$
$71$ $$T^{2} - 428T - 52988$$
$73$ $$T^{2} + 900T + 121636$$
$79$ $$T^{2} + 432T - 61760$$
$83$ $$T^{2} + 1388 T + 424292$$
$89$ $$T^{2} - 960T - 275000$$
$97$ $$T^{2} + 532T - 606844$$