# Properties

 Label 39.4.e.a Level $39$ Weight $4$ Character orbit 39.e Analytic conductor $2.301$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$39 = 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 39.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.30107449022$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + (3 \zeta_{6} - 3) q^{2} + (3 \zeta_{6} - 3) q^{3} - \zeta_{6} q^{4} - 9 q^{5} - 9 \zeta_{6} q^{6} - 2 \zeta_{6} q^{7} - 21 q^{8} - 9 \zeta_{6} q^{9} +O(q^{10})$$ q + (3*z - 3) * q^2 + (3*z - 3) * q^3 - z * q^4 - 9 * q^5 - 9*z * q^6 - 2*z * q^7 - 21 * q^8 - 9*z * q^9 $$q + (3 \zeta_{6} - 3) q^{2} + (3 \zeta_{6} - 3) q^{3} - \zeta_{6} q^{4} - 9 q^{5} - 9 \zeta_{6} q^{6} - 2 \zeta_{6} q^{7} - 21 q^{8} - 9 \zeta_{6} q^{9} + ( - 27 \zeta_{6} + 27) q^{10} + (30 \zeta_{6} - 30) q^{11} + 3 q^{12} + (39 \zeta_{6} + 13) q^{13} + 6 q^{14} + ( - 27 \zeta_{6} + 27) q^{15} + ( - 71 \zeta_{6} + 71) q^{16} + 111 \zeta_{6} q^{17} + 27 q^{18} + 46 \zeta_{6} q^{19} + 9 \zeta_{6} q^{20} + 6 q^{21} - 90 \zeta_{6} q^{22} + ( - 6 \zeta_{6} + 6) q^{23} + ( - 63 \zeta_{6} + 63) q^{24} - 44 q^{25} + (39 \zeta_{6} - 156) q^{26} + 27 q^{27} + (2 \zeta_{6} - 2) q^{28} + ( - 105 \zeta_{6} + 105) q^{29} + 81 \zeta_{6} q^{30} - 100 q^{31} + 45 \zeta_{6} q^{32} - 90 \zeta_{6} q^{33} - 333 q^{34} + 18 \zeta_{6} q^{35} + (9 \zeta_{6} - 9) q^{36} + (17 \zeta_{6} - 17) q^{37} - 138 q^{38} + (39 \zeta_{6} - 156) q^{39} + 189 q^{40} + ( - 231 \zeta_{6} + 231) q^{41} + (18 \zeta_{6} - 18) q^{42} + 514 \zeta_{6} q^{43} + 30 q^{44} + 81 \zeta_{6} q^{45} + 18 \zeta_{6} q^{46} - 162 q^{47} + 213 \zeta_{6} q^{48} + ( - 339 \zeta_{6} + 339) q^{49} + ( - 132 \zeta_{6} + 132) q^{50} - 333 q^{51} + ( - 52 \zeta_{6} + 39) q^{52} + 639 q^{53} + (81 \zeta_{6} - 81) q^{54} + ( - 270 \zeta_{6} + 270) q^{55} + 42 \zeta_{6} q^{56} - 138 q^{57} + 315 \zeta_{6} q^{58} - 600 \zeta_{6} q^{59} - 27 q^{60} - 233 \zeta_{6} q^{61} + ( - 300 \zeta_{6} + 300) q^{62} + (18 \zeta_{6} - 18) q^{63} + 433 q^{64} + ( - 351 \zeta_{6} - 117) q^{65} + 270 q^{66} + (926 \zeta_{6} - 926) q^{67} + ( - 111 \zeta_{6} + 111) q^{68} + 18 \zeta_{6} q^{69} - 54 q^{70} + 930 \zeta_{6} q^{71} + 189 \zeta_{6} q^{72} - 253 q^{73} - 51 \zeta_{6} q^{74} + ( - 132 \zeta_{6} + 132) q^{75} + ( - 46 \zeta_{6} + 46) q^{76} + 60 q^{77} + ( - 468 \zeta_{6} + 351) q^{78} - 1324 q^{79} + (639 \zeta_{6} - 639) q^{80} + (81 \zeta_{6} - 81) q^{81} + 693 \zeta_{6} q^{82} + 810 q^{83} - 6 \zeta_{6} q^{84} - 999 \zeta_{6} q^{85} - 1542 q^{86} + 315 \zeta_{6} q^{87} + ( - 630 \zeta_{6} + 630) q^{88} + (498 \zeta_{6} - 498) q^{89} - 243 q^{90} + ( - 104 \zeta_{6} + 78) q^{91} - 6 q^{92} + ( - 300 \zeta_{6} + 300) q^{93} + ( - 486 \zeta_{6} + 486) q^{94} - 414 \zeta_{6} q^{95} - 135 q^{96} - 1358 \zeta_{6} q^{97} + 1017 \zeta_{6} q^{98} + 270 q^{99} +O(q^{100})$$ q + (3*z - 3) * q^2 + (3*z - 3) * q^3 - z * q^4 - 9 * q^5 - 9*z * q^6 - 2*z * q^7 - 21 * q^8 - 9*z * q^9 + (-27*z + 27) * q^10 + (30*z - 30) * q^11 + 3 * q^12 + (39*z + 13) * q^13 + 6 * q^14 + (-27*z + 27) * q^15 + (-71*z + 71) * q^16 + 111*z * q^17 + 27 * q^18 + 46*z * q^19 + 9*z * q^20 + 6 * q^21 - 90*z * q^22 + (-6*z + 6) * q^23 + (-63*z + 63) * q^24 - 44 * q^25 + (39*z - 156) * q^26 + 27 * q^27 + (2*z - 2) * q^28 + (-105*z + 105) * q^29 + 81*z * q^30 - 100 * q^31 + 45*z * q^32 - 90*z * q^33 - 333 * q^34 + 18*z * q^35 + (9*z - 9) * q^36 + (17*z - 17) * q^37 - 138 * q^38 + (39*z - 156) * q^39 + 189 * q^40 + (-231*z + 231) * q^41 + (18*z - 18) * q^42 + 514*z * q^43 + 30 * q^44 + 81*z * q^45 + 18*z * q^46 - 162 * q^47 + 213*z * q^48 + (-339*z + 339) * q^49 + (-132*z + 132) * q^50 - 333 * q^51 + (-52*z + 39) * q^52 + 639 * q^53 + (81*z - 81) * q^54 + (-270*z + 270) * q^55 + 42*z * q^56 - 138 * q^57 + 315*z * q^58 - 600*z * q^59 - 27 * q^60 - 233*z * q^61 + (-300*z + 300) * q^62 + (18*z - 18) * q^63 + 433 * q^64 + (-351*z - 117) * q^65 + 270 * q^66 + (926*z - 926) * q^67 + (-111*z + 111) * q^68 + 18*z * q^69 - 54 * q^70 + 930*z * q^71 + 189*z * q^72 - 253 * q^73 - 51*z * q^74 + (-132*z + 132) * q^75 + (-46*z + 46) * q^76 + 60 * q^77 + (-468*z + 351) * q^78 - 1324 * q^79 + (639*z - 639) * q^80 + (81*z - 81) * q^81 + 693*z * q^82 + 810 * q^83 - 6*z * q^84 - 999*z * q^85 - 1542 * q^86 + 315*z * q^87 + (-630*z + 630) * q^88 + (498*z - 498) * q^89 - 243 * q^90 + (-104*z + 78) * q^91 - 6 * q^92 + (-300*z + 300) * q^93 + (-486*z + 486) * q^94 - 414*z * q^95 - 135 * q^96 - 1358*z * q^97 + 1017*z * q^98 + 270 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} - 3 q^{3} - q^{4} - 18 q^{5} - 9 q^{6} - 2 q^{7} - 42 q^{8} - 9 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 - 3 * q^3 - q^4 - 18 * q^5 - 9 * q^6 - 2 * q^7 - 42 * q^8 - 9 * q^9 $$2 q - 3 q^{2} - 3 q^{3} - q^{4} - 18 q^{5} - 9 q^{6} - 2 q^{7} - 42 q^{8} - 9 q^{9} + 27 q^{10} - 30 q^{11} + 6 q^{12} + 65 q^{13} + 12 q^{14} + 27 q^{15} + 71 q^{16} + 111 q^{17} + 54 q^{18} + 46 q^{19} + 9 q^{20} + 12 q^{21} - 90 q^{22} + 6 q^{23} + 63 q^{24} - 88 q^{25} - 273 q^{26} + 54 q^{27} - 2 q^{28} + 105 q^{29} + 81 q^{30} - 200 q^{31} + 45 q^{32} - 90 q^{33} - 666 q^{34} + 18 q^{35} - 9 q^{36} - 17 q^{37} - 276 q^{38} - 273 q^{39} + 378 q^{40} + 231 q^{41} - 18 q^{42} + 514 q^{43} + 60 q^{44} + 81 q^{45} + 18 q^{46} - 324 q^{47} + 213 q^{48} + 339 q^{49} + 132 q^{50} - 666 q^{51} + 26 q^{52} + 1278 q^{53} - 81 q^{54} + 270 q^{55} + 42 q^{56} - 276 q^{57} + 315 q^{58} - 600 q^{59} - 54 q^{60} - 233 q^{61} + 300 q^{62} - 18 q^{63} + 866 q^{64} - 585 q^{65} + 540 q^{66} - 926 q^{67} + 111 q^{68} + 18 q^{69} - 108 q^{70} + 930 q^{71} + 189 q^{72} - 506 q^{73} - 51 q^{74} + 132 q^{75} + 46 q^{76} + 120 q^{77} + 234 q^{78} - 2648 q^{79} - 639 q^{80} - 81 q^{81} + 693 q^{82} + 1620 q^{83} - 6 q^{84} - 999 q^{85} - 3084 q^{86} + 315 q^{87} + 630 q^{88} - 498 q^{89} - 486 q^{90} + 52 q^{91} - 12 q^{92} + 300 q^{93} + 486 q^{94} - 414 q^{95} - 270 q^{96} - 1358 q^{97} + 1017 q^{98} + 540 q^{99}+O(q^{100})$$ 2 * q - 3 * q^2 - 3 * q^3 - q^4 - 18 * q^5 - 9 * q^6 - 2 * q^7 - 42 * q^8 - 9 * q^9 + 27 * q^10 - 30 * q^11 + 6 * q^12 + 65 * q^13 + 12 * q^14 + 27 * q^15 + 71 * q^16 + 111 * q^17 + 54 * q^18 + 46 * q^19 + 9 * q^20 + 12 * q^21 - 90 * q^22 + 6 * q^23 + 63 * q^24 - 88 * q^25 - 273 * q^26 + 54 * q^27 - 2 * q^28 + 105 * q^29 + 81 * q^30 - 200 * q^31 + 45 * q^32 - 90 * q^33 - 666 * q^34 + 18 * q^35 - 9 * q^36 - 17 * q^37 - 276 * q^38 - 273 * q^39 + 378 * q^40 + 231 * q^41 - 18 * q^42 + 514 * q^43 + 60 * q^44 + 81 * q^45 + 18 * q^46 - 324 * q^47 + 213 * q^48 + 339 * q^49 + 132 * q^50 - 666 * q^51 + 26 * q^52 + 1278 * q^53 - 81 * q^54 + 270 * q^55 + 42 * q^56 - 276 * q^57 + 315 * q^58 - 600 * q^59 - 54 * q^60 - 233 * q^61 + 300 * q^62 - 18 * q^63 + 866 * q^64 - 585 * q^65 + 540 * q^66 - 926 * q^67 + 111 * q^68 + 18 * q^69 - 108 * q^70 + 930 * q^71 + 189 * q^72 - 506 * q^73 - 51 * q^74 + 132 * q^75 + 46 * q^76 + 120 * q^77 + 234 * q^78 - 2648 * q^79 - 639 * q^80 - 81 * q^81 + 693 * q^82 + 1620 * q^83 - 6 * q^84 - 999 * q^85 - 3084 * q^86 + 315 * q^87 + 630 * q^88 - 498 * q^89 - 486 * q^90 + 52 * q^91 - 12 * q^92 + 300 * q^93 + 486 * q^94 - 414 * q^95 - 270 * q^96 - 1358 * q^97 + 1017 * q^98 + 540 * 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.

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
−1.50000 2.59808i −1.50000 2.59808i −0.500000 + 0.866025i −9.00000 −4.50000 + 7.79423i −1.00000 + 1.73205i −21.0000 −4.50000 + 7.79423i 13.5000 + 23.3827i
22.1 −1.50000 + 2.59808i −1.50000 + 2.59808i −0.500000 0.866025i −9.00000 −4.50000 7.79423i −1.00000 1.73205i −21.0000 −4.50000 7.79423i 13.5000 23.3827i
 $$n$$: 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.a 2
3.b odd 2 1 117.4.g.b 2
4.b odd 2 1 624.4.q.b 2
13.c even 3 1 inner 39.4.e.a 2
13.c even 3 1 507.4.a.e 1
13.e even 6 1 507.4.a.a 1
13.f odd 12 2 507.4.b.c 2
39.h odd 6 1 1521.4.a.j 1
39.i odd 6 1 117.4.g.b 2
39.i odd 6 1 1521.4.a.c 1
52.j odd 6 1 624.4.q.b 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3T_{2} + 9$$ acting on $$S_{4}^{\mathrm{new}}(39, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T + 9$$
$3$ $$T^{2} + 3T + 9$$
$5$ $$(T + 9)^{2}$$
$7$ $$T^{2} + 2T + 4$$
$11$ $$T^{2} + 30T + 900$$
$13$ $$T^{2} - 65T + 2197$$
$17$ $$T^{2} - 111T + 12321$$
$19$ $$T^{2} - 46T + 2116$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} - 105T + 11025$$
$31$ $$(T + 100)^{2}$$
$37$ $$T^{2} + 17T + 289$$
$41$ $$T^{2} - 231T + 53361$$
$43$ $$T^{2} - 514T + 264196$$
$47$ $$(T + 162)^{2}$$
$53$ $$(T - 639)^{2}$$
$59$ $$T^{2} + 600T + 360000$$
$61$ $$T^{2} + 233T + 54289$$
$67$ $$T^{2} + 926T + 857476$$
$71$ $$T^{2} - 930T + 864900$$
$73$ $$(T + 253)^{2}$$
$79$ $$(T + 1324)^{2}$$
$83$ $$(T - 810)^{2}$$
$89$ $$T^{2} + 498T + 248004$$
$97$ $$T^{2} + 1358 T + 1844164$$