Properties

 Label 21.6.e.a Level $21$ Weight $6$ Character orbit 21.e Analytic conductor $3.368$ 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] = [21,6,Mod(4,21)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(21, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("21.4");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 21.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: $$3.36806021607$$ 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: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{2} + 9 \zeta_{6} q^{3} + 28 \zeta_{6} q^{4} + (11 \zeta_{6} - 11) q^{5} + 18 q^{6} + (7 \zeta_{6} + 126) q^{7} + 120 q^{8} + (81 \zeta_{6} - 81) q^{9}+O(q^{10})$$ q + (-2*z + 2) * q^2 + 9*z * q^3 + 28*z * q^4 + (11*z - 11) * q^5 + 18 * q^6 + (7*z + 126) * q^7 + 120 * q^8 + (81*z - 81) * q^9 $$q + ( - 2 \zeta_{6} + 2) q^{2} + 9 \zeta_{6} q^{3} + 28 \zeta_{6} q^{4} + (11 \zeta_{6} - 11) q^{5} + 18 q^{6} + (7 \zeta_{6} + 126) q^{7} + 120 q^{8} + (81 \zeta_{6} - 81) q^{9} + 22 \zeta_{6} q^{10} - 269 \zeta_{6} q^{11} + (252 \zeta_{6} - 252) q^{12} - 308 q^{13} + ( - 252 \zeta_{6} + 266) q^{14} - 99 q^{15} + (656 \zeta_{6} - 656) q^{16} - 1896 \zeta_{6} q^{17} + 162 \zeta_{6} q^{18} + ( - 164 \zeta_{6} + 164) q^{19} - 308 q^{20} + (1197 \zeta_{6} - 63) q^{21} - 538 q^{22} + ( - 3264 \zeta_{6} + 3264) q^{23} + 1080 \zeta_{6} q^{24} + 3004 \zeta_{6} q^{25} + (616 \zeta_{6} - 616) q^{26} - 729 q^{27} + (3724 \zeta_{6} - 196) q^{28} + 2417 q^{29} + (198 \zeta_{6} - 198) q^{30} - 2841 \zeta_{6} q^{31} + 5152 \zeta_{6} q^{32} + ( - 2421 \zeta_{6} + 2421) q^{33} - 3792 q^{34} + (1386 \zeta_{6} - 1463) q^{35} - 2268 q^{36} + ( - 11328 \zeta_{6} + 11328) q^{37} - 328 \zeta_{6} q^{38} - 2772 \zeta_{6} q^{39} + (1320 \zeta_{6} - 1320) q^{40} - 16856 q^{41} + (126 \zeta_{6} + 2268) q^{42} - 7894 q^{43} + ( - 7532 \zeta_{6} + 7532) q^{44} - 891 \zeta_{6} q^{45} - 6528 \zeta_{6} q^{46} + (21102 \zeta_{6} - 21102) q^{47} - 5904 q^{48} + (1813 \zeta_{6} + 15827) q^{49} + 6008 q^{50} + ( - 17064 \zeta_{6} + 17064) q^{51} - 8624 \zeta_{6} q^{52} + 29691 \zeta_{6} q^{53} + (1458 \zeta_{6} - 1458) q^{54} + 2959 q^{55} + (840 \zeta_{6} + 15120) q^{56} + 1476 q^{57} + ( - 4834 \zeta_{6} + 4834) q^{58} + 8163 \zeta_{6} q^{59} - 2772 \zeta_{6} q^{60} + (15166 \zeta_{6} - 15166) q^{61} - 5682 q^{62} + (10206 \zeta_{6} - 10773) q^{63} - 10688 q^{64} + ( - 3388 \zeta_{6} + 3388) q^{65} - 4842 \zeta_{6} q^{66} + 32078 \zeta_{6} q^{67} + ( - 53088 \zeta_{6} + 53088) q^{68} + 29376 q^{69} + (2926 \zeta_{6} - 154) q^{70} - 38274 q^{71} + (9720 \zeta_{6} - 9720) q^{72} - 34866 \zeta_{6} q^{73} - 22656 \zeta_{6} q^{74} + (27036 \zeta_{6} - 27036) q^{75} + 4592 q^{76} + ( - 35777 \zeta_{6} + 1883) q^{77} - 5544 q^{78} + (13529 \zeta_{6} - 13529) q^{79} - 7216 \zeta_{6} q^{80} - 6561 \zeta_{6} q^{81} + (33712 \zeta_{6} - 33712) q^{82} - 68103 q^{83} + (31752 \zeta_{6} - 33516) q^{84} + 20856 q^{85} + (15788 \zeta_{6} - 15788) q^{86} + 21753 \zeta_{6} q^{87} - 32280 \zeta_{6} q^{88} + ( - 114922 \zeta_{6} + 114922) q^{89} - 1782 q^{90} + ( - 2156 \zeta_{6} - 38808) q^{91} + 91392 q^{92} + ( - 25569 \zeta_{6} + 25569) q^{93} + 42204 \zeta_{6} q^{94} + 1804 \zeta_{6} q^{95} + (46368 \zeta_{6} - 46368) q^{96} + 154959 q^{97} + ( - 31654 \zeta_{6} + 35280) q^{98} + 21789 q^{99} +O(q^{100})$$ q + (-2*z + 2) * q^2 + 9*z * q^3 + 28*z * q^4 + (11*z - 11) * q^5 + 18 * q^6 + (7*z + 126) * q^7 + 120 * q^8 + (81*z - 81) * q^9 + 22*z * q^10 - 269*z * q^11 + (252*z - 252) * q^12 - 308 * q^13 + (-252*z + 266) * q^14 - 99 * q^15 + (656*z - 656) * q^16 - 1896*z * q^17 + 162*z * q^18 + (-164*z + 164) * q^19 - 308 * q^20 + (1197*z - 63) * q^21 - 538 * q^22 + (-3264*z + 3264) * q^23 + 1080*z * q^24 + 3004*z * q^25 + (616*z - 616) * q^26 - 729 * q^27 + (3724*z - 196) * q^28 + 2417 * q^29 + (198*z - 198) * q^30 - 2841*z * q^31 + 5152*z * q^32 + (-2421*z + 2421) * q^33 - 3792 * q^34 + (1386*z - 1463) * q^35 - 2268 * q^36 + (-11328*z + 11328) * q^37 - 328*z * q^38 - 2772*z * q^39 + (1320*z - 1320) * q^40 - 16856 * q^41 + (126*z + 2268) * q^42 - 7894 * q^43 + (-7532*z + 7532) * q^44 - 891*z * q^45 - 6528*z * q^46 + (21102*z - 21102) * q^47 - 5904 * q^48 + (1813*z + 15827) * q^49 + 6008 * q^50 + (-17064*z + 17064) * q^51 - 8624*z * q^52 + 29691*z * q^53 + (1458*z - 1458) * q^54 + 2959 * q^55 + (840*z + 15120) * q^56 + 1476 * q^57 + (-4834*z + 4834) * q^58 + 8163*z * q^59 - 2772*z * q^60 + (15166*z - 15166) * q^61 - 5682 * q^62 + (10206*z - 10773) * q^63 - 10688 * q^64 + (-3388*z + 3388) * q^65 - 4842*z * q^66 + 32078*z * q^67 + (-53088*z + 53088) * q^68 + 29376 * q^69 + (2926*z - 154) * q^70 - 38274 * q^71 + (9720*z - 9720) * q^72 - 34866*z * q^73 - 22656*z * q^74 + (27036*z - 27036) * q^75 + 4592 * q^76 + (-35777*z + 1883) * q^77 - 5544 * q^78 + (13529*z - 13529) * q^79 - 7216*z * q^80 - 6561*z * q^81 + (33712*z - 33712) * q^82 - 68103 * q^83 + (31752*z - 33516) * q^84 + 20856 * q^85 + (15788*z - 15788) * q^86 + 21753*z * q^87 - 32280*z * q^88 + (-114922*z + 114922) * q^89 - 1782 * q^90 + (-2156*z - 38808) * q^91 + 91392 * q^92 + (-25569*z + 25569) * q^93 + 42204*z * q^94 + 1804*z * q^95 + (46368*z - 46368) * q^96 + 154959 * q^97 + (-31654*z + 35280) * q^98 + 21789 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 9 q^{3} + 28 q^{4} - 11 q^{5} + 36 q^{6} + 259 q^{7} + 240 q^{8} - 81 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 9 * q^3 + 28 * q^4 - 11 * q^5 + 36 * q^6 + 259 * q^7 + 240 * q^8 - 81 * q^9 $$2 q + 2 q^{2} + 9 q^{3} + 28 q^{4} - 11 q^{5} + 36 q^{6} + 259 q^{7} + 240 q^{8} - 81 q^{9} + 22 q^{10} - 269 q^{11} - 252 q^{12} - 616 q^{13} + 280 q^{14} - 198 q^{15} - 656 q^{16} - 1896 q^{17} + 162 q^{18} + 164 q^{19} - 616 q^{20} + 1071 q^{21} - 1076 q^{22} + 3264 q^{23} + 1080 q^{24} + 3004 q^{25} - 616 q^{26} - 1458 q^{27} + 3332 q^{28} + 4834 q^{29} - 198 q^{30} - 2841 q^{31} + 5152 q^{32} + 2421 q^{33} - 7584 q^{34} - 1540 q^{35} - 4536 q^{36} + 11328 q^{37} - 328 q^{38} - 2772 q^{39} - 1320 q^{40} - 33712 q^{41} + 4662 q^{42} - 15788 q^{43} + 7532 q^{44} - 891 q^{45} - 6528 q^{46} - 21102 q^{47} - 11808 q^{48} + 33467 q^{49} + 12016 q^{50} + 17064 q^{51} - 8624 q^{52} + 29691 q^{53} - 1458 q^{54} + 5918 q^{55} + 31080 q^{56} + 2952 q^{57} + 4834 q^{58} + 8163 q^{59} - 2772 q^{60} - 15166 q^{61} - 11364 q^{62} - 11340 q^{63} - 21376 q^{64} + 3388 q^{65} - 4842 q^{66} + 32078 q^{67} + 53088 q^{68} + 58752 q^{69} + 2618 q^{70} - 76548 q^{71} - 9720 q^{72} - 34866 q^{73} - 22656 q^{74} - 27036 q^{75} + 9184 q^{76} - 32011 q^{77} - 11088 q^{78} - 13529 q^{79} - 7216 q^{80} - 6561 q^{81} - 33712 q^{82} - 136206 q^{83} - 35280 q^{84} + 41712 q^{85} - 15788 q^{86} + 21753 q^{87} - 32280 q^{88} + 114922 q^{89} - 3564 q^{90} - 79772 q^{91} + 182784 q^{92} + 25569 q^{93} + 42204 q^{94} + 1804 q^{95} - 46368 q^{96} + 309918 q^{97} + 38906 q^{98} + 43578 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 9 * q^3 + 28 * q^4 - 11 * q^5 + 36 * q^6 + 259 * q^7 + 240 * q^8 - 81 * q^9 + 22 * q^10 - 269 * q^11 - 252 * q^12 - 616 * q^13 + 280 * q^14 - 198 * q^15 - 656 * q^16 - 1896 * q^17 + 162 * q^18 + 164 * q^19 - 616 * q^20 + 1071 * q^21 - 1076 * q^22 + 3264 * q^23 + 1080 * q^24 + 3004 * q^25 - 616 * q^26 - 1458 * q^27 + 3332 * q^28 + 4834 * q^29 - 198 * q^30 - 2841 * q^31 + 5152 * q^32 + 2421 * q^33 - 7584 * q^34 - 1540 * q^35 - 4536 * q^36 + 11328 * q^37 - 328 * q^38 - 2772 * q^39 - 1320 * q^40 - 33712 * q^41 + 4662 * q^42 - 15788 * q^43 + 7532 * q^44 - 891 * q^45 - 6528 * q^46 - 21102 * q^47 - 11808 * q^48 + 33467 * q^49 + 12016 * q^50 + 17064 * q^51 - 8624 * q^52 + 29691 * q^53 - 1458 * q^54 + 5918 * q^55 + 31080 * q^56 + 2952 * q^57 + 4834 * q^58 + 8163 * q^59 - 2772 * q^60 - 15166 * q^61 - 11364 * q^62 - 11340 * q^63 - 21376 * q^64 + 3388 * q^65 - 4842 * q^66 + 32078 * q^67 + 53088 * q^68 + 58752 * q^69 + 2618 * q^70 - 76548 * q^71 - 9720 * q^72 - 34866 * q^73 - 22656 * q^74 - 27036 * q^75 + 9184 * q^76 - 32011 * q^77 - 11088 * q^78 - 13529 * q^79 - 7216 * q^80 - 6561 * q^81 - 33712 * q^82 - 136206 * q^83 - 35280 * q^84 + 41712 * q^85 - 15788 * q^86 + 21753 * q^87 - 32280 * q^88 + 114922 * q^89 - 3564 * q^90 - 79772 * q^91 + 182784 * q^92 + 25569 * q^93 + 42204 * q^94 + 1804 * q^95 - 46368 * q^96 + 309918 * q^97 + 38906 * q^98 + 43578 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/21\mathbb{Z}\right)^\times$$.

 $$n$$ $$8$$ $$10$$ $$\chi(n)$$ $$1$$ $$-1 + \zeta_{6}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.00000 + 1.73205i 4.50000 7.79423i 14.0000 24.2487i −5.50000 9.52628i 18.0000 129.500 6.06218i 120.000 −40.5000 70.1481i 11.0000 19.0526i
16.1 1.00000 1.73205i 4.50000 + 7.79423i 14.0000 + 24.2487i −5.50000 + 9.52628i 18.0000 129.500 + 6.06218i 120.000 −40.5000 + 70.1481i 11.0000 + 19.0526i
 $$n$$: 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
7.c even 3 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.6.e.a 2
3.b odd 2 1 63.6.e.a 2
4.b odd 2 1 336.6.q.b 2
7.b odd 2 1 147.6.e.g 2
7.c even 3 1 inner 21.6.e.a 2
7.c even 3 1 147.6.a.c 1
7.d odd 6 1 147.6.a.d 1
7.d odd 6 1 147.6.e.g 2
21.g even 6 1 441.6.a.h 1
21.h odd 6 1 63.6.e.a 2
21.h odd 6 1 441.6.a.g 1
28.g odd 6 1 336.6.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.a 2 1.a even 1 1 trivial
21.6.e.a 2 7.c even 3 1 inner
63.6.e.a 2 3.b odd 2 1
63.6.e.a 2 21.h odd 6 1
147.6.a.c 1 7.c even 3 1
147.6.a.d 1 7.d odd 6 1
147.6.e.g 2 7.b odd 2 1
147.6.e.g 2 7.d odd 6 1
336.6.q.b 2 4.b odd 2 1
336.6.q.b 2 28.g odd 6 1
441.6.a.g 1 21.h odd 6 1
441.6.a.h 1 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2T_{2} + 4$$ acting on $$S_{6}^{\mathrm{new}}(21, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2} - 9T + 81$$
$5$ $$T^{2} + 11T + 121$$
$7$ $$T^{2} - 259T + 16807$$
$11$ $$T^{2} + 269T + 72361$$
$13$ $$(T + 308)^{2}$$
$17$ $$T^{2} + 1896 T + 3594816$$
$19$ $$T^{2} - 164T + 26896$$
$23$ $$T^{2} - 3264 T + 10653696$$
$29$ $$(T - 2417)^{2}$$
$31$ $$T^{2} + 2841 T + 8071281$$
$37$ $$T^{2} - 11328 T + 128323584$$
$41$ $$(T + 16856)^{2}$$
$43$ $$(T + 7894)^{2}$$
$47$ $$T^{2} + 21102 T + 445294404$$
$53$ $$T^{2} - 29691 T + 881555481$$
$59$ $$T^{2} - 8163 T + 66634569$$
$61$ $$T^{2} + 15166 T + 230007556$$
$67$ $$T^{2} - 32078 T + 1028998084$$
$71$ $$(T + 38274)^{2}$$
$73$ $$T^{2} + 34866 T + 1215637956$$
$79$ $$T^{2} + 13529 T + 183033841$$
$83$ $$(T + 68103)^{2}$$
$89$ $$T^{2} - 114922 T + 13207066084$$
$97$ $$(T - 154959)^{2}$$