# Properties

 Label 21.6.e.b Level $21$ Weight $6$ Character orbit 21.e Analytic conductor $3.368$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-83})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 20x^{2} - 21x + 441$$ x^4 - x^3 - 20*x^2 - 21*x + 441 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + \beta_1 + 1) q^{2} - 9 \beta_1 q^{3} + ( - 3 \beta_{3} + 3 \beta_{2} + 31 \beta_1) q^{4} + ( - 7 \beta_{3} + 13 \beta_1 + 13) q^{5} + ( - 9 \beta_{2} + 9) q^{6} + (14 \beta_{3} - 7 \beta_{2} + 7 \beta_1 - 84) q^{7} + (5 \beta_{2} - 185) q^{8} + ( - 81 \beta_1 - 81) q^{9}+O(q^{10})$$ q + (-b3 + b1 + 1) * q^2 - 9*b1 * q^3 + (-3*b3 + 3*b2 + 31*b1) * q^4 + (-7*b3 + 13*b1 + 13) * q^5 + (-9*b2 + 9) * q^6 + (14*b3 - 7*b2 + 7*b1 - 84) * q^7 + (5*b2 - 185) * q^8 + (-81*b1 - 81) * q^9 $$q + ( - \beta_{3} + \beta_1 + 1) q^{2} - 9 \beta_1 q^{3} + ( - 3 \beta_{3} + 3 \beta_{2} + 31 \beta_1) q^{4} + ( - 7 \beta_{3} + 13 \beta_1 + 13) q^{5} + ( - 9 \beta_{2} + 9) q^{6} + (14 \beta_{3} - 7 \beta_{2} + 7 \beta_1 - 84) q^{7} + (5 \beta_{2} - 185) q^{8} + ( - 81 \beta_1 - 81) q^{9} + ( - 27 \beta_{3} + 27 \beta_{2} + 447 \beta_1) q^{10} + ( - \beta_{3} + \beta_{2} - 569 \beta_1) q^{11} + ( - 27 \beta_{3} + 279 \beta_1 + 279) q^{12} + ( - 9 \beta_{2} + 458) q^{13} + (98 \beta_{3} - 21 \beta_{2} - 518 \beta_1 + 343) q^{14} + ( - 63 \beta_{2} + 117) q^{15} + (99 \beta_{3} + 497 \beta_1 + 497) q^{16} + ( - 148 \beta_{3} + 148 \beta_{2} - 236 \beta_1) q^{17} + (81 \beta_{3} - 81 \beta_{2} - 81 \beta_1) q^{18} + (27 \beta_{3} - 1142 \beta_1 - 1142) q^{19} + (277 \beta_{2} - 1705) q^{20} + (63 \beta_{3} + 63 \beta_{2} + 819 \beta_1 + 63) q^{21} + ( - 567 \beta_{2} + 507) q^{22} + (308 \beta_{3} - 644 \beta_1 - 644) q^{23} + ( - 45 \beta_{3} + 45 \beta_{2} + 1665 \beta_1) q^{24} + ( - 231 \beta_{3} + 231 \beta_{2} + 82 \beta_1) q^{25} + ( - 476 \beta_{3} + 1016 \beta_1 + 1016) q^{26} - 729 q^{27} + (35 \beta_{3} - 490 \beta_{2} - 1519 \beta_1 + 2387) q^{28} + ( - 45 \beta_{2} - 1131) q^{29} + ( - 243 \beta_{3} + 4023 \beta_1 + 4023) q^{30} + ( - 768 \beta_{3} + 768 \beta_{2} + 1763 \beta_1) q^{31} + ( - 459 \beta_{3} + 459 \beta_{2} + 279 \beta_1) q^{32} + ( - 9 \beta_{3} - 5121 \beta_1 - 5121) q^{33} + (60 \beta_{2} - 8940) q^{34} + (728 \beta_{3} - 231 \beta_{2} - 4130 \beta_1 + 1855) q^{35} + ( - 243 \beta_{2} + 2511) q^{36} + (855 \beta_{3} + 9982 \beta_1 + 9982) q^{37} + (1196 \beta_{3} - 1196 \beta_{2} - 2816 \beta_1) q^{38} + (81 \beta_{3} - 81 \beta_{2} - 4122 \beta_1) q^{39} + (1395 \beta_{3} - 4575 \beta_1 - 4575) q^{40} + ( - 846 \beta_{2} - 6852) q^{41} + (189 \beta_{3} + 693 \beta_{2} - 7749 \beta_1 - 4662) q^{42} + (2043 \beta_{2} - 364) q^{43} + ( - 1673 \beta_{3} + 17453 \beta_1 + 17453) q^{44} + (567 \beta_{3} - 567 \beta_{2} - 1053 \beta_1) q^{45} + (1260 \beta_{3} - 1260 \beta_{2} - 19740 \beta_1) q^{46} + ( - 604 \beta_{3} + 11278 \beta_1 + 11278) q^{47} + (891 \beta_{2} + 4473) q^{48} + ( - 2450 \beta_{3} + 1225 \beta_{2} - 1225 \beta_1 - 2107) q^{49} + (544 \beta_{2} - 14404) q^{50} + ( - 1332 \beta_{3} - 2124 \beta_1 - 2124) q^{51} + ( - 1680 \beta_{3} + 1680 \beta_{2} + 15872 \beta_1) q^{52} + (1751 \beta_{3} - 1751 \beta_{2} - 14951 \beta_1) q^{53} + (729 \beta_{3} - 729 \beta_1 - 729) q^{54} + ( - 3963 \beta_{2} + 6963) q^{55} + ( - 2625 \beta_{3} + 875 \beta_{2} + 3045 \beta_1 + 17710) q^{56} + (243 \beta_{2} - 10278) q^{57} + (1041 \beta_{3} + 1659 \beta_1 + 1659) q^{58} + (3917 \beta_{3} - 3917 \beta_{2} + 22507 \beta_1) q^{59} + ( - 2493 \beta_{3} + 2493 \beta_{2} + 15345 \beta_1) q^{60} + ( - 2544 \beta_{3} - 22298 \beta_1 - 22298) q^{61} + (3299 \beta_{2} - 49379) q^{62} + ( - 567 \beta_{3} + 1134 \beta_{2} + 6804 \beta_1 + 7371) q^{63} + (4365 \beta_{2} - 12833) q^{64} + ( - 3386 \beta_{3} + 9860 \beta_1 + 9860) q^{65} + (5103 \beta_{3} - 5103 \beta_{2} - 4563 \beta_1) q^{66} + (4461 \beta_{3} - 4461 \beta_{2} + 17612 \beta_1) q^{67} + (4324 \beta_{3} - 20212 \beta_1 - 20212) q^{68} + (2772 \beta_{2} - 5796) q^{69} + ( - 861 \beta_{3} - 5586 \beta_{2} - 28959 \beta_1 + 20307) q^{70} + ( - 1404 \beta_{2} + 50346) q^{71} + ( - 405 \beta_{3} + 14985 \beta_1 + 14985) q^{72} + ( - 5247 \beta_{3} + 5247 \beta_{2} - 16912 \beta_1) q^{73} + ( - 8272 \beta_{3} + 8272 \beta_{2} - 43028 \beta_1) q^{74} + ( - 2079 \beta_{3} + 738 \beta_1 + 738) q^{75} + ( - 4344 \beta_{2} + 40424) q^{76} + (4067 \beta_{3} + 3892 \beta_{2} + 52213 \beta_1 + 4851) q^{77} + ( - 4284 \beta_{2} + 9144) q^{78} + (6834 \beta_{3} + 12649 \beta_1 + 12649) q^{79} + ( - 1499 \beta_{3} + 1499 \beta_{2} - 36505 \beta_1) q^{80} + 6561 \beta_1 q^{81} + (5160 \beta_{3} + 45600 \beta_1 + 45600) q^{82} + (1899 \beta_{2} + 31539) q^{83} + (4410 \beta_{3} - 4095 \beta_{2} - 35154 \beta_1 - 13671) q^{84} + (1308 \beta_{2} - 61164) q^{85} + (4450 \beta_{3} - 127030 \beta_1 - 127030) q^{86} + (405 \beta_{3} - 405 \beta_{2} + 10179 \beta_1) q^{87} + ( - 2655 \beta_{3} + 2655 \beta_{2} + 104955 \beta_1) q^{88} + ( - 130 \beta_{3} - 14726 \beta_1 - 14726) q^{89} + ( - 2187 \beta_{2} + 36207) q^{90} + (6475 \beta_{3} - 2450 \beta_{2} - 4606 \beta_1 - 42378) q^{91} + ( - 12404 \beta_{2} + 77252) q^{92} + ( - 6912 \beta_{3} + 15867 \beta_1 + 15867) q^{93} + ( - 12486 \beta_{3} + 12486 \beta_{2} + 48726 \beta_1) q^{94} + (8534 \beta_{3} - 8534 \beta_{2} - 26564 \beta_1) q^{95} + ( - 4131 \beta_{3} + 2511 \beta_1 + 2511) q^{96} + (1017 \beta_{2} - 4387) q^{97} + ( - 343 \beta_{3} + 3675 \beta_{2} + 73843 \beta_1 - 76832) q^{98} + ( - 81 \beta_{2} - 46089) q^{99}+O(q^{100})$$ q + (-b3 + b1 + 1) * q^2 - 9*b1 * q^3 + (-3*b3 + 3*b2 + 31*b1) * q^4 + (-7*b3 + 13*b1 + 13) * q^5 + (-9*b2 + 9) * q^6 + (14*b3 - 7*b2 + 7*b1 - 84) * q^7 + (5*b2 - 185) * q^8 + (-81*b1 - 81) * q^9 + (-27*b3 + 27*b2 + 447*b1) * q^10 + (-b3 + b2 - 569*b1) * q^11 + (-27*b3 + 279*b1 + 279) * q^12 + (-9*b2 + 458) * q^13 + (98*b3 - 21*b2 - 518*b1 + 343) * q^14 + (-63*b2 + 117) * q^15 + (99*b3 + 497*b1 + 497) * q^16 + (-148*b3 + 148*b2 - 236*b1) * q^17 + (81*b3 - 81*b2 - 81*b1) * q^18 + (27*b3 - 1142*b1 - 1142) * q^19 + (277*b2 - 1705) * q^20 + (63*b3 + 63*b2 + 819*b1 + 63) * q^21 + (-567*b2 + 507) * q^22 + (308*b3 - 644*b1 - 644) * q^23 + (-45*b3 + 45*b2 + 1665*b1) * q^24 + (-231*b3 + 231*b2 + 82*b1) * q^25 + (-476*b3 + 1016*b1 + 1016) * q^26 - 729 * q^27 + (35*b3 - 490*b2 - 1519*b1 + 2387) * q^28 + (-45*b2 - 1131) * q^29 + (-243*b3 + 4023*b1 + 4023) * q^30 + (-768*b3 + 768*b2 + 1763*b1) * q^31 + (-459*b3 + 459*b2 + 279*b1) * q^32 + (-9*b3 - 5121*b1 - 5121) * q^33 + (60*b2 - 8940) * q^34 + (728*b3 - 231*b2 - 4130*b1 + 1855) * q^35 + (-243*b2 + 2511) * q^36 + (855*b3 + 9982*b1 + 9982) * q^37 + (1196*b3 - 1196*b2 - 2816*b1) * q^38 + (81*b3 - 81*b2 - 4122*b1) * q^39 + (1395*b3 - 4575*b1 - 4575) * q^40 + (-846*b2 - 6852) * q^41 + (189*b3 + 693*b2 - 7749*b1 - 4662) * q^42 + (2043*b2 - 364) * q^43 + (-1673*b3 + 17453*b1 + 17453) * q^44 + (567*b3 - 567*b2 - 1053*b1) * q^45 + (1260*b3 - 1260*b2 - 19740*b1) * q^46 + (-604*b3 + 11278*b1 + 11278) * q^47 + (891*b2 + 4473) * q^48 + (-2450*b3 + 1225*b2 - 1225*b1 - 2107) * q^49 + (544*b2 - 14404) * q^50 + (-1332*b3 - 2124*b1 - 2124) * q^51 + (-1680*b3 + 1680*b2 + 15872*b1) * q^52 + (1751*b3 - 1751*b2 - 14951*b1) * q^53 + (729*b3 - 729*b1 - 729) * q^54 + (-3963*b2 + 6963) * q^55 + (-2625*b3 + 875*b2 + 3045*b1 + 17710) * q^56 + (243*b2 - 10278) * q^57 + (1041*b3 + 1659*b1 + 1659) * q^58 + (3917*b3 - 3917*b2 + 22507*b1) * q^59 + (-2493*b3 + 2493*b2 + 15345*b1) * q^60 + (-2544*b3 - 22298*b1 - 22298) * q^61 + (3299*b2 - 49379) * q^62 + (-567*b3 + 1134*b2 + 6804*b1 + 7371) * q^63 + (4365*b2 - 12833) * q^64 + (-3386*b3 + 9860*b1 + 9860) * q^65 + (5103*b3 - 5103*b2 - 4563*b1) * q^66 + (4461*b3 - 4461*b2 + 17612*b1) * q^67 + (4324*b3 - 20212*b1 - 20212) * q^68 + (2772*b2 - 5796) * q^69 + (-861*b3 - 5586*b2 - 28959*b1 + 20307) * q^70 + (-1404*b2 + 50346) * q^71 + (-405*b3 + 14985*b1 + 14985) * q^72 + (-5247*b3 + 5247*b2 - 16912*b1) * q^73 + (-8272*b3 + 8272*b2 - 43028*b1) * q^74 + (-2079*b3 + 738*b1 + 738) * q^75 + (-4344*b2 + 40424) * q^76 + (4067*b3 + 3892*b2 + 52213*b1 + 4851) * q^77 + (-4284*b2 + 9144) * q^78 + (6834*b3 + 12649*b1 + 12649) * q^79 + (-1499*b3 + 1499*b2 - 36505*b1) * q^80 + 6561*b1 * q^81 + (5160*b3 + 45600*b1 + 45600) * q^82 + (1899*b2 + 31539) * q^83 + (4410*b3 - 4095*b2 - 35154*b1 - 13671) * q^84 + (1308*b2 - 61164) * q^85 + (4450*b3 - 127030*b1 - 127030) * q^86 + (405*b3 - 405*b2 + 10179*b1) * q^87 + (-2655*b3 + 2655*b2 + 104955*b1) * q^88 + (-130*b3 - 14726*b1 - 14726) * q^89 + (-2187*b2 + 36207) * q^90 + (6475*b3 - 2450*b2 - 4606*b1 - 42378) * q^91 + (-12404*b2 + 77252) * q^92 + (-6912*b3 + 15867*b1 + 15867) * q^93 + (-12486*b3 + 12486*b2 + 48726*b1) * q^94 + (8534*b3 - 8534*b2 - 26564*b1) * q^95 + (-4131*b3 + 2511*b1 + 2511) * q^96 + (1017*b2 - 4387) * q^97 + (-343*b3 + 3675*b2 + 73843*b1 - 76832) * q^98 + (-81*b2 - 46089) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 3 q^{2} + 18 q^{3} - 65 q^{4} + 33 q^{5} + 54 q^{6} - 350 q^{7} - 750 q^{8} - 162 q^{9}+O(q^{10})$$ 4 * q + 3 * q^2 + 18 * q^3 - 65 * q^4 + 33 * q^5 + 54 * q^6 - 350 * q^7 - 750 * q^8 - 162 * q^9 $$4 q + 3 q^{2} + 18 q^{3} - 65 q^{4} + 33 q^{5} + 54 q^{6} - 350 q^{7} - 750 q^{8} - 162 q^{9} - 921 q^{10} + 1137 q^{11} + 585 q^{12} + 1850 q^{13} + 2352 q^{14} + 594 q^{15} + 895 q^{16} + 324 q^{17} + 243 q^{18} - 2311 q^{19} - 7374 q^{20} - 1575 q^{21} + 3162 q^{22} - 1596 q^{23} - 3375 q^{24} - 395 q^{25} + 2508 q^{26} - 2916 q^{27} + 13531 q^{28} - 4434 q^{29} + 8289 q^{30} - 4294 q^{31} - 1017 q^{32} - 10233 q^{33} - 35880 q^{34} + 15414 q^{35} + 10530 q^{36} + 19109 q^{37} + 6828 q^{38} + 8325 q^{39} - 10545 q^{40} - 25716 q^{41} - 4725 q^{42} - 5542 q^{43} + 36579 q^{44} + 2673 q^{45} + 40740 q^{46} + 23160 q^{47} + 16110 q^{48} - 5978 q^{49} - 58704 q^{50} - 2916 q^{51} - 33424 q^{52} + 31653 q^{53} - 2187 q^{54} + 35778 q^{55} + 65625 q^{56} - 41598 q^{57} + 2277 q^{58} - 41097 q^{59} - 33183 q^{60} - 42052 q^{61} - 204114 q^{62} + 14175 q^{63} - 60062 q^{64} + 23106 q^{65} + 14229 q^{66} - 30763 q^{67} - 44748 q^{68} - 28728 q^{69} + 151179 q^{70} + 204192 q^{71} + 30375 q^{72} + 28577 q^{73} + 77784 q^{74} + 3555 q^{75} + 170384 q^{76} - 96873 q^{77} + 45144 q^{78} + 18464 q^{79} + 71511 q^{80} - 13122 q^{81} + 86040 q^{82} + 122358 q^{83} + 19404 q^{84} - 247272 q^{85} - 258510 q^{86} - 19953 q^{87} - 212565 q^{88} - 29322 q^{89} + 149202 q^{90} - 161875 q^{91} + 333816 q^{92} + 38646 q^{93} - 109938 q^{94} + 61662 q^{95} + 9153 q^{96} - 19582 q^{97} - 462021 q^{98} - 184194 q^{99}+O(q^{100})$$ 4 * q + 3 * q^2 + 18 * q^3 - 65 * q^4 + 33 * q^5 + 54 * q^6 - 350 * q^7 - 750 * q^8 - 162 * q^9 - 921 * q^10 + 1137 * q^11 + 585 * q^12 + 1850 * q^13 + 2352 * q^14 + 594 * q^15 + 895 * q^16 + 324 * q^17 + 243 * q^18 - 2311 * q^19 - 7374 * q^20 - 1575 * q^21 + 3162 * q^22 - 1596 * q^23 - 3375 * q^24 - 395 * q^25 + 2508 * q^26 - 2916 * q^27 + 13531 * q^28 - 4434 * q^29 + 8289 * q^30 - 4294 * q^31 - 1017 * q^32 - 10233 * q^33 - 35880 * q^34 + 15414 * q^35 + 10530 * q^36 + 19109 * q^37 + 6828 * q^38 + 8325 * q^39 - 10545 * q^40 - 25716 * q^41 - 4725 * q^42 - 5542 * q^43 + 36579 * q^44 + 2673 * q^45 + 40740 * q^46 + 23160 * q^47 + 16110 * q^48 - 5978 * q^49 - 58704 * q^50 - 2916 * q^51 - 33424 * q^52 + 31653 * q^53 - 2187 * q^54 + 35778 * q^55 + 65625 * q^56 - 41598 * q^57 + 2277 * q^58 - 41097 * q^59 - 33183 * q^60 - 42052 * q^61 - 204114 * q^62 + 14175 * q^63 - 60062 * q^64 + 23106 * q^65 + 14229 * q^66 - 30763 * q^67 - 44748 * q^68 - 28728 * q^69 + 151179 * q^70 + 204192 * q^71 + 30375 * q^72 + 28577 * q^73 + 77784 * q^74 + 3555 * q^75 + 170384 * q^76 - 96873 * q^77 + 45144 * q^78 + 18464 * q^79 + 71511 * q^80 - 13122 * q^81 + 86040 * q^82 + 122358 * q^83 + 19404 * q^84 - 247272 * q^85 - 258510 * q^86 - 19953 * q^87 - 212565 * q^88 - 29322 * q^89 + 149202 * q^90 - 161875 * q^91 + 333816 * q^92 + 38646 * q^93 - 109938 * q^94 + 61662 * q^95 + 9153 * q^96 - 19582 * q^97 - 462021 * q^98 - 184194 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 20x^{2} - 21x + 441$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 20\nu^{2} - 20\nu - 441 ) / 420$$ (v^3 + 20*v^2 - 20*v - 441) / 420 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 41\nu ) / 21$$ (-v^3 + v^2 + 41*v) / 21 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 20\nu - 41 ) / 20$$ (v^3 + 20*v - 41) / 20
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3$$ (b3 + b2 - b1 + 1) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 61\beta _1 + 62 ) / 3$$ (-b3 + 2*b2 + 61*b1 + 62) / 3 $$\nu^{3}$$ $$=$$ $$( 40\beta_{3} - 20\beta_{2} + 20\beta _1 + 103 ) / 3$$ (40*b3 - 20*b2 + 20*b1 + 103) / 3

## Character values

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

 $$n$$ $$8$$ $$10$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{1}$$

## 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
 4.19493 + 1.84460i −3.69493 − 2.71062i 4.19493 − 1.84460i −3.69493 + 2.71062i
−3.19493 5.53379i 4.50000 7.79423i −4.41520 + 7.64735i −19.3645 33.5404i −57.5088 −87.5000 + 95.6596i −148.051 −40.5000 70.1481i −123.737 + 214.318i
4.2 4.69493 + 8.13186i 4.50000 7.79423i −28.0848 + 48.6443i 35.8645 + 62.1192i 84.5088 −87.5000 95.6596i −226.949 −40.5000 70.1481i −336.763 + 583.291i
16.1 −3.19493 + 5.53379i 4.50000 + 7.79423i −4.41520 7.64735i −19.3645 + 33.5404i −57.5088 −87.5000 95.6596i −148.051 −40.5000 + 70.1481i −123.737 214.318i
16.2 4.69493 8.13186i 4.50000 + 7.79423i −28.0848 48.6443i 35.8645 62.1192i 84.5088 −87.5000 + 95.6596i −226.949 −40.5000 + 70.1481i −336.763 583.291i
 $$n$$: 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.b 4
3.b odd 2 1 63.6.e.c 4
4.b odd 2 1 336.6.q.e 4
7.b odd 2 1 147.6.e.l 4
7.c even 3 1 inner 21.6.e.b 4
7.c even 3 1 147.6.a.i 2
7.d odd 6 1 147.6.a.k 2
7.d odd 6 1 147.6.e.l 4
21.g even 6 1 441.6.a.s 2
21.h odd 6 1 63.6.e.c 4
21.h odd 6 1 441.6.a.t 2
28.g odd 6 1 336.6.q.e 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 3T_{2}^{3} + 69T_{2}^{2} + 180T_{2} + 3600$$ acting on $$S_{6}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 3 T^{3} + 69 T^{2} + \cdots + 3600$$
$3$ $$(T^{2} - 9 T + 81)^{2}$$
$5$ $$T^{4} - 33 T^{3} + 3867 T^{2} + \cdots + 7717284$$
$7$ $$(T^{2} + 175 T + 16807)^{2}$$
$11$ $$T^{4} - 1137 T^{3} + \cdots + 104412996900$$
$13$ $$(T^{2} - 925 T + 208864)^{2}$$
$17$ $$T^{4} - 324 T^{3} + \cdots + 1788317798400$$
$19$ $$T^{4} + 2311 T^{3} + \cdots + 1663584040000$$
$23$ $$T^{4} + 1596 T^{3} + \cdots + 27756881510400$$
$29$ $$(T^{2} + 2217 T + 1102716)^{2}$$
$31$ $$T^{4} + 4294 T^{3} + \cdots + 10\!\cdots\!25$$
$37$ $$T^{4} - 19109 T^{3} + \cdots + 20\!\cdots\!96$$
$41$ $$(T^{2} + 12858 T - 3221280)^{2}$$
$43$ $$(T^{2} + 2771 T - 257902490)^{2}$$
$47$ $$T^{4} - 23160 T^{3} + \cdots + 12\!\cdots\!16$$
$53$ $$T^{4} - 31653 T^{3} + \cdots + 35\!\cdots\!00$$
$59$ $$T^{4} + 41097 T^{3} + \cdots + 28\!\cdots\!44$$
$61$ $$T^{4} + 42052 T^{3} + \cdots + 15\!\cdots\!00$$
$67$ $$T^{4} + 30763 T^{3} + \cdots + 10\!\cdots\!00$$
$71$ $$(T^{2} - 102096 T + 2483190108)^{2}$$
$73$ $$T^{4} - 28577 T^{3} + \cdots + 22\!\cdots\!84$$
$79$ $$T^{4} - 18464 T^{3} + \cdots + 79\!\cdots\!69$$
$83$ $$(T^{2} - 61179 T + 711231498)^{2}$$
$89$ $$T^{4} + 29322 T^{3} + \cdots + 45\!\cdots\!16$$
$97$ $$(T^{2} + 9791 T - 40418570)^{2}$$