# Properties

 Label 33.4.e.a Level $33$ Weight $4$ Character orbit 33.e Analytic conductor $1.947$ 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] = [33,4,Mod(4,33)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, base_ring=CyclotomicField(10))

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("33.4");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 33.e (of order $$5$$, degree $$4$$, 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: $$1.94706303019$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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 $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (4 \zeta_{10}^{2} - 2 \zeta_{10} + 4) q^{2} - 3 \zeta_{10}^{3} q^{3} + 12 \zeta_{10}^{2} q^{4} + ( - \zeta_{10}^{3} + 12 \zeta_{10}^{2} - 12 \zeta_{10} + 1) q^{5} + ( - 6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} + 6) q^{6} + (6 \zeta_{10}^{3} - 25 \zeta_{10}^{2} + 6 \zeta_{10}) q^{7} + (8 \zeta_{10}^{3} + 16 \zeta_{10} - 16) q^{8} - 9 \zeta_{10} q^{9} +O(q^{10})$$ q + (4*z^2 - 2*z + 4) * q^2 - 3*z^3 * q^3 + 12*z^2 * q^4 + (-z^3 + 12*z^2 - 12*z + 1) * q^5 + (-6*z^3 - 6*z^2 + 6*z + 6) * q^6 + (6*z^3 - 25*z^2 + 6*z) * q^7 + (8*z^3 + 16*z - 16) * q^8 - 9*z * q^9 $$q + (4 \zeta_{10}^{2} - 2 \zeta_{10} + 4) q^{2} - 3 \zeta_{10}^{3} q^{3} + 12 \zeta_{10}^{2} q^{4} + ( - \zeta_{10}^{3} + 12 \zeta_{10}^{2} - 12 \zeta_{10} + 1) q^{5} + ( - 6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} + 6) q^{6} + (6 \zeta_{10}^{3} - 25 \zeta_{10}^{2} + 6 \zeta_{10}) q^{7} + (8 \zeta_{10}^{3} + 16 \zeta_{10} - 16) q^{8} - 9 \zeta_{10} q^{9} + ( - 26 \zeta_{10}^{3} + 26 \zeta_{10}^{2} - 42) q^{10} + (41 \zeta_{10}^{3} - 25 \zeta_{10}^{2} + 35 \zeta_{10} - 15) q^{11} + 36 q^{12} + ( - 22 \zeta_{10}^{2} - 11 \zeta_{10} - 22) q^{13} + ( - 14 \zeta_{10}^{3} - 88 \zeta_{10} + 88) q^{14} + (33 \zeta_{10}^{3} - 36 \zeta_{10}^{2} + 33 \zeta_{10}) q^{15} + ( - 16 \zeta_{10}^{3} + 16 \zeta_{10}^{2} - 16 \zeta_{10} + 16) q^{16} + ( - 78 \zeta_{10}^{3} + 21 \zeta_{10}^{2} - 21 \zeta_{10} + 78) q^{17} + ( - 36 \zeta_{10}^{3} + 18 \zeta_{10}^{2} - 36 \zeta_{10}) q^{18} + (83 \zeta_{10}^{3} + 45 \zeta_{10} - 45) q^{19} + ( - 132 \zeta_{10}^{2} + 144 \zeta_{10} - 132) q^{20} + ( - 18 \zeta_{10}^{3} + 18 \zeta_{10}^{2} - 57) q^{21} + (172 \zeta_{10}^{3} - 48 \zeta_{10}^{2} - 12 \zeta_{10} - 42) q^{22} + ( - 44 \zeta_{10}^{3} + 44 \zeta_{10}^{2} - 15) q^{23} + (48 \zeta_{10}^{2} - 24 \zeta_{10} + 48) q^{24} + (3 \zeta_{10}^{3} + 143 \zeta_{10} - 143) q^{25} + ( - 88 \zeta_{10}^{3} - 66 \zeta_{10}^{2} - 88 \zeta_{10}) q^{26} + (27 \zeta_{10}^{3} - 27 \zeta_{10}^{2} + 27 \zeta_{10} - 27) q^{27} + ( - 228 \zeta_{10}^{3} + 300 \zeta_{10}^{2} - 300 \zeta_{10} + 228) q^{28} + (192 \zeta_{10}^{3} - 30 \zeta_{10}^{2} + 192 \zeta_{10}) q^{29} + (126 \zeta_{10}^{3} - 78 \zeta_{10} + 78) q^{30} + ( - 77 \zeta_{10}^{2} + 33 \zeta_{10} - 77) q^{31} + ( - 192 \zeta_{10}^{3} + 192 \zeta_{10}^{2} + 96) q^{32} + ( - 60 \zeta_{10}^{3} + 105 \zeta_{10}^{2} + 18 \zeta_{10} + 30) q^{33} + ( - 198 \zeta_{10}^{3} + 198 \zeta_{10}^{2} + 384) q^{34} + (281 \zeta_{10}^{2} - 366 \zeta_{10} + 281) q^{35} - 108 \zeta_{10}^{3} q^{36} + ( - 22 \zeta_{10}^{3} - 245 \zeta_{10}^{2} - 22 \zeta_{10}) q^{37} + (346 \zeta_{10}^{3} - 104 \zeta_{10}^{2} + 104 \zeta_{10} - 346) q^{38} + (99 \zeta_{10}^{3} - 33 \zeta_{10}^{2} + 33 \zeta_{10} - 99) q^{39} + (104 \zeta_{10}^{3} - 272 \zeta_{10}^{2} + 104 \zeta_{10}) q^{40} + ( - 27 \zeta_{10}^{3} + 34 \zeta_{10} - 34) q^{41} + ( - 264 \zeta_{10}^{2} + 222 \zeta_{10} - 264) q^{42} + (84 \zeta_{10}^{3} - 84 \zeta_{10}^{2} - 75) q^{43} + (120 \zeta_{10}^{3} + 120 \zeta_{10}^{2} - 300 \zeta_{10} - 192) q^{44} + ( - 99 \zeta_{10}^{3} + 99 \zeta_{10}^{2} - 9) q^{45} + ( - 148 \zeta_{10}^{2} + 294 \zeta_{10} - 148) q^{46} + ( - 200 \zeta_{10}^{3} - 55 \zeta_{10} + 55) q^{47} - 48 \zeta_{10}^{2} q^{48} + (54 \zeta_{10}^{3} - 318 \zeta_{10}^{2} + 318 \zeta_{10} - 54) q^{49} + (578 \zeta_{10}^{3} - 852 \zeta_{10}^{2} + 852 \zeta_{10} - 578) q^{50} + ( - 171 \zeta_{10}^{3} - 63 \zeta_{10}^{2} - 171 \zeta_{10}) q^{51} + ( - 396 \zeta_{10}^{3} - 264 \zeta_{10} + 264) q^{52} + (143 \zeta_{10}^{2} - 241 \zeta_{10} + 143) q^{53} + (108 \zeta_{10}^{3} - 108 \zeta_{10}^{2} - 54) q^{54} + ( - 51 \zeta_{10}^{3} + 202 \zeta_{10}^{2} - 571 \zeta_{10} + 295) q^{55} + ( - 352 \zeta_{10}^{3} + 352 \zeta_{10}^{2} + 56) q^{56} + (135 \zeta_{10}^{2} + 114 \zeta_{10} + 135) q^{57} + (1092 \zeta_{10}^{3} + 264 \zeta_{10} - 264) q^{58} + ( - 451 \zeta_{10}^{3} + 418 \zeta_{10}^{2} - 451 \zeta_{10}) q^{59} + ( - 36 \zeta_{10}^{3} + 432 \zeta_{10}^{2} - 432 \zeta_{10} + 36) q^{60} + ( - 717 \zeta_{10}^{3} + 438 \zeta_{10}^{2} - 438 \zeta_{10} + 717) q^{61} + ( - 22 \zeta_{10}^{3} - 374 \zeta_{10}^{2} - 22 \zeta_{10}) q^{62} + (171 \zeta_{10}^{3} - 54 \zeta_{10} + 54) q^{63} + 832 \zeta_{10} q^{64} + ( - 99 \zeta_{10}^{3} + 99 \zeta_{10}^{2} + 209) q^{65} + (162 \zeta_{10}^{3} - 36 \zeta_{10}^{2} + 552 \zeta_{10} - 180) q^{66} + (561 \zeta_{10}^{3} - 561 \zeta_{10}^{2} - 243) q^{67} + (684 \zeta_{10}^{2} + 252 \zeta_{10} + 684) q^{68} + (45 \zeta_{10}^{3} - 132 \zeta_{10} + 132) q^{69} + ( - 902 \zeta_{10}^{3} + 1856 \zeta_{10}^{2} - 902 \zeta_{10}) q^{70} + (433 \zeta_{10}^{3} - 708 \zeta_{10}^{2} + 708 \zeta_{10} - 433) q^{71} + ( - 72 \zeta_{10}^{3} - 72 \zeta_{10}^{2} + 72 \zeta_{10} + 72) q^{72} + ( - 318 \zeta_{10}^{3} + 346 \zeta_{10}^{2} - 318 \zeta_{10}) q^{73} + ( - 622 \zeta_{10}^{3} - 1024 \zeta_{10} + 1024) q^{74} + (429 \zeta_{10}^{2} - 420 \zeta_{10} + 429) q^{75} + (540 \zeta_{10}^{3} - 540 \zeta_{10}^{2} - 996) q^{76} + ( - 34 \zeta_{10}^{3} - 496 \zeta_{10}^{2} + 745 \zeta_{10} + 94) q^{77} + (264 \zeta_{10}^{3} - 264 \zeta_{10}^{2} - 462) q^{78} + ( - 702 \zeta_{10}^{2} + 637 \zeta_{10} - 702) q^{79} + ( - 16 \zeta_{10}^{3} + 176 \zeta_{10} - 176) q^{80} + 81 \zeta_{10}^{2} q^{81} + (82 \zeta_{10}^{3} - 258 \zeta_{10}^{2} + 258 \zeta_{10} - 82) q^{82} + (35 \zeta_{10}^{3} + 431 \zeta_{10}^{2} - 431 \zeta_{10} - 35) q^{83} + (216 \zeta_{10}^{3} - 900 \zeta_{10}^{2} + 216 \zeta_{10}) q^{84} + (549 \zeta_{10}^{3} + 174 \zeta_{10} - 174) q^{85} + ( - 132 \zeta_{10}^{2} - 354 \zeta_{10} - 132) q^{86} + ( - 576 \zeta_{10}^{3} + 576 \zeta_{10}^{2} + 486) q^{87} + ( - 240 \zeta_{10}^{3} + 24 \zeta_{10}^{2} - 192 \zeta_{10} - 496) q^{88} + ( - 154 \zeta_{10}^{3} + 154 \zeta_{10}^{2} - 1279) q^{89} + ( - 234 \zeta_{10}^{2} + 612 \zeta_{10} - 234) q^{90} + (495 \zeta_{10}^{3} + 352 \zeta_{10} - 352) q^{91} + (528 \zeta_{10}^{3} - 708 \zeta_{10}^{2} + 528 \zeta_{10}) q^{92} + (132 \zeta_{10}^{3} + 99 \zeta_{10}^{2} - 99 \zeta_{10} - 132) q^{93} + ( - 620 \zeta_{10}^{3} - 70 \zeta_{10}^{2} + 70 \zeta_{10} + 620) q^{94} + ( - 373 \zeta_{10}^{3} - 39 \zeta_{10}^{2} - 373 \zeta_{10}) q^{95} + ( - 288 \zeta_{10}^{3} - 576 \zeta_{10} + 576) q^{96} + (891 \zeta_{10}^{2} - 282 \zeta_{10} + 891) q^{97} + (744 \zeta_{10}^{3} - 744 \zeta_{10}^{2} + 948) q^{98} + ( - 144 \zeta_{10}^{3} + 54 \zeta_{10}^{2} - 234 \zeta_{10} + 369) q^{99} +O(q^{100})$$ q + (4*z^2 - 2*z + 4) * q^2 - 3*z^3 * q^3 + 12*z^2 * q^4 + (-z^3 + 12*z^2 - 12*z + 1) * q^5 + (-6*z^3 - 6*z^2 + 6*z + 6) * q^6 + (6*z^3 - 25*z^2 + 6*z) * q^7 + (8*z^3 + 16*z - 16) * q^8 - 9*z * q^9 + (-26*z^3 + 26*z^2 - 42) * q^10 + (41*z^3 - 25*z^2 + 35*z - 15) * q^11 + 36 * q^12 + (-22*z^2 - 11*z - 22) * q^13 + (-14*z^3 - 88*z + 88) * q^14 + (33*z^3 - 36*z^2 + 33*z) * q^15 + (-16*z^3 + 16*z^2 - 16*z + 16) * q^16 + (-78*z^3 + 21*z^2 - 21*z + 78) * q^17 + (-36*z^3 + 18*z^2 - 36*z) * q^18 + (83*z^3 + 45*z - 45) * q^19 + (-132*z^2 + 144*z - 132) * q^20 + (-18*z^3 + 18*z^2 - 57) * q^21 + (172*z^3 - 48*z^2 - 12*z - 42) * q^22 + (-44*z^3 + 44*z^2 - 15) * q^23 + (48*z^2 - 24*z + 48) * q^24 + (3*z^3 + 143*z - 143) * q^25 + (-88*z^3 - 66*z^2 - 88*z) * q^26 + (27*z^3 - 27*z^2 + 27*z - 27) * q^27 + (-228*z^3 + 300*z^2 - 300*z + 228) * q^28 + (192*z^3 - 30*z^2 + 192*z) * q^29 + (126*z^3 - 78*z + 78) * q^30 + (-77*z^2 + 33*z - 77) * q^31 + (-192*z^3 + 192*z^2 + 96) * q^32 + (-60*z^3 + 105*z^2 + 18*z + 30) * q^33 + (-198*z^3 + 198*z^2 + 384) * q^34 + (281*z^2 - 366*z + 281) * q^35 - 108*z^3 * q^36 + (-22*z^3 - 245*z^2 - 22*z) * q^37 + (346*z^3 - 104*z^2 + 104*z - 346) * q^38 + (99*z^3 - 33*z^2 + 33*z - 99) * q^39 + (104*z^3 - 272*z^2 + 104*z) * q^40 + (-27*z^3 + 34*z - 34) * q^41 + (-264*z^2 + 222*z - 264) * q^42 + (84*z^3 - 84*z^2 - 75) * q^43 + (120*z^3 + 120*z^2 - 300*z - 192) * q^44 + (-99*z^3 + 99*z^2 - 9) * q^45 + (-148*z^2 + 294*z - 148) * q^46 + (-200*z^3 - 55*z + 55) * q^47 - 48*z^2 * q^48 + (54*z^3 - 318*z^2 + 318*z - 54) * q^49 + (578*z^3 - 852*z^2 + 852*z - 578) * q^50 + (-171*z^3 - 63*z^2 - 171*z) * q^51 + (-396*z^3 - 264*z + 264) * q^52 + (143*z^2 - 241*z + 143) * q^53 + (108*z^3 - 108*z^2 - 54) * q^54 + (-51*z^3 + 202*z^2 - 571*z + 295) * q^55 + (-352*z^3 + 352*z^2 + 56) * q^56 + (135*z^2 + 114*z + 135) * q^57 + (1092*z^3 + 264*z - 264) * q^58 + (-451*z^3 + 418*z^2 - 451*z) * q^59 + (-36*z^3 + 432*z^2 - 432*z + 36) * q^60 + (-717*z^3 + 438*z^2 - 438*z + 717) * q^61 + (-22*z^3 - 374*z^2 - 22*z) * q^62 + (171*z^3 - 54*z + 54) * q^63 + 832*z * q^64 + (-99*z^3 + 99*z^2 + 209) * q^65 + (162*z^3 - 36*z^2 + 552*z - 180) * q^66 + (561*z^3 - 561*z^2 - 243) * q^67 + (684*z^2 + 252*z + 684) * q^68 + (45*z^3 - 132*z + 132) * q^69 + (-902*z^3 + 1856*z^2 - 902*z) * q^70 + (433*z^3 - 708*z^2 + 708*z - 433) * q^71 + (-72*z^3 - 72*z^2 + 72*z + 72) * q^72 + (-318*z^3 + 346*z^2 - 318*z) * q^73 + (-622*z^3 - 1024*z + 1024) * q^74 + (429*z^2 - 420*z + 429) * q^75 + (540*z^3 - 540*z^2 - 996) * q^76 + (-34*z^3 - 496*z^2 + 745*z + 94) * q^77 + (264*z^3 - 264*z^2 - 462) * q^78 + (-702*z^2 + 637*z - 702) * q^79 + (-16*z^3 + 176*z - 176) * q^80 + 81*z^2 * q^81 + (82*z^3 - 258*z^2 + 258*z - 82) * q^82 + (35*z^3 + 431*z^2 - 431*z - 35) * q^83 + (216*z^3 - 900*z^2 + 216*z) * q^84 + (549*z^3 + 174*z - 174) * q^85 + (-132*z^2 - 354*z - 132) * q^86 + (-576*z^3 + 576*z^2 + 486) * q^87 + (-240*z^3 + 24*z^2 - 192*z - 496) * q^88 + (-154*z^3 + 154*z^2 - 1279) * q^89 + (-234*z^2 + 612*z - 234) * q^90 + (495*z^3 + 352*z - 352) * q^91 + (528*z^3 - 708*z^2 + 528*z) * q^92 + (132*z^3 + 99*z^2 - 99*z - 132) * q^93 + (-620*z^3 - 70*z^2 + 70*z + 620) * q^94 + (-373*z^3 - 39*z^2 - 373*z) * q^95 + (-288*z^3 - 576*z + 576) * q^96 + (891*z^2 - 282*z + 891) * q^97 + (744*z^3 - 744*z^2 + 948) * q^98 + (-144*z^3 + 54*z^2 - 234*z + 369) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 10 q^{2} - 3 q^{3} - 12 q^{4} - 21 q^{5} + 30 q^{6} + 37 q^{7} - 40 q^{8} - 9 q^{9}+O(q^{10})$$ 4 * q + 10 * q^2 - 3 * q^3 - 12 * q^4 - 21 * q^5 + 30 * q^6 + 37 * q^7 - 40 * q^8 - 9 * q^9 $$4 q + 10 q^{2} - 3 q^{3} - 12 q^{4} - 21 q^{5} + 30 q^{6} + 37 q^{7} - 40 q^{8} - 9 q^{9} - 220 q^{10} + 41 q^{11} + 144 q^{12} - 77 q^{13} + 250 q^{14} + 102 q^{15} + 16 q^{16} + 192 q^{17} - 90 q^{18} - 52 q^{19} - 252 q^{20} - 264 q^{21} + 40 q^{22} - 148 q^{23} + 120 q^{24} - 426 q^{25} - 110 q^{26} - 27 q^{27} + 84 q^{28} + 414 q^{29} + 360 q^{30} - 198 q^{31} - 27 q^{33} + 1140 q^{34} + 477 q^{35} - 108 q^{36} + 201 q^{37} - 830 q^{38} - 231 q^{39} + 480 q^{40} - 129 q^{41} - 570 q^{42} - 132 q^{43} - 1068 q^{44} - 234 q^{45} - 150 q^{46} - 35 q^{47} + 48 q^{48} + 474 q^{49} - 30 q^{50} - 279 q^{51} + 396 q^{52} + 188 q^{53} + 356 q^{55} - 480 q^{56} + 519 q^{57} + 300 q^{58} - 1320 q^{59} - 756 q^{60} + 1275 q^{61} + 330 q^{62} + 333 q^{63} + 832 q^{64} + 638 q^{65} + 30 q^{66} + 150 q^{67} + 2304 q^{68} + 441 q^{69} - 3660 q^{70} + 117 q^{71} + 360 q^{72} - 982 q^{73} + 2450 q^{74} + 867 q^{75} - 2904 q^{76} + 1583 q^{77} - 1320 q^{78} - 1469 q^{79} - 544 q^{80} - 81 q^{81} + 270 q^{82} - 967 q^{83} + 1332 q^{84} + 27 q^{85} - 750 q^{86} + 792 q^{87} - 2440 q^{88} - 5424 q^{89} - 90 q^{90} - 561 q^{91} + 1764 q^{92} - 594 q^{93} + 2000 q^{94} - 707 q^{95} + 1440 q^{96} + 2391 q^{97} + 5280 q^{98} + 1044 q^{99}+O(q^{100})$$ 4 * q + 10 * q^2 - 3 * q^3 - 12 * q^4 - 21 * q^5 + 30 * q^6 + 37 * q^7 - 40 * q^8 - 9 * q^9 - 220 * q^10 + 41 * q^11 + 144 * q^12 - 77 * q^13 + 250 * q^14 + 102 * q^15 + 16 * q^16 + 192 * q^17 - 90 * q^18 - 52 * q^19 - 252 * q^20 - 264 * q^21 + 40 * q^22 - 148 * q^23 + 120 * q^24 - 426 * q^25 - 110 * q^26 - 27 * q^27 + 84 * q^28 + 414 * q^29 + 360 * q^30 - 198 * q^31 - 27 * q^33 + 1140 * q^34 + 477 * q^35 - 108 * q^36 + 201 * q^37 - 830 * q^38 - 231 * q^39 + 480 * q^40 - 129 * q^41 - 570 * q^42 - 132 * q^43 - 1068 * q^44 - 234 * q^45 - 150 * q^46 - 35 * q^47 + 48 * q^48 + 474 * q^49 - 30 * q^50 - 279 * q^51 + 396 * q^52 + 188 * q^53 + 356 * q^55 - 480 * q^56 + 519 * q^57 + 300 * q^58 - 1320 * q^59 - 756 * q^60 + 1275 * q^61 + 330 * q^62 + 333 * q^63 + 832 * q^64 + 638 * q^65 + 30 * q^66 + 150 * q^67 + 2304 * q^68 + 441 * q^69 - 3660 * q^70 + 117 * q^71 + 360 * q^72 - 982 * q^73 + 2450 * q^74 + 867 * q^75 - 2904 * q^76 + 1583 * q^77 - 1320 * q^78 - 1469 * q^79 - 544 * q^80 - 81 * q^81 + 270 * q^82 - 967 * q^83 + 1332 * q^84 + 27 * q^85 - 750 * q^86 + 792 * q^87 - 2440 * q^88 - 5424 * q^89 - 90 * q^90 - 561 * q^91 + 1764 * q^92 - 594 * q^93 + 2000 * q^94 - 707 * q^95 + 1440 * q^96 + 2391 * q^97 + 5280 * q^98 + 1044 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$\zeta_{10}^{2}$$ $$1$$

## 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
 0.809017 + 0.587785i −0.309017 − 0.951057i 0.809017 − 0.587785i −0.309017 + 0.951057i
3.61803 + 2.62866i 0.927051 2.85317i 3.70820 + 11.4127i −4.69098 + 3.40820i 10.8541 7.88597i −4.72542 14.5434i −5.52786 + 17.0130i −7.28115 5.29007i −25.9311
16.1 1.38197 + 4.25325i −2.42705 1.76336i −9.70820 + 7.05342i −5.80902 + 17.8783i 4.14590 12.7598i 23.2254 16.8743i −14.4721 10.5146i 2.78115 + 8.55951i −84.0689
25.1 3.61803 2.62866i 0.927051 + 2.85317i 3.70820 11.4127i −4.69098 3.40820i 10.8541 + 7.88597i −4.72542 + 14.5434i −5.52786 17.0130i −7.28115 + 5.29007i −25.9311
31.1 1.38197 4.25325i −2.42705 + 1.76336i −9.70820 7.05342i −5.80902 17.8783i 4.14590 + 12.7598i 23.2254 + 16.8743i −14.4721 + 10.5146i 2.78115 8.55951i −84.0689
 $$n$$: 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
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.4.e.a 4
3.b odd 2 1 99.4.f.a 4
11.c even 5 1 inner 33.4.e.a 4
11.c even 5 1 363.4.a.n 2
11.d odd 10 1 363.4.a.o 2
33.f even 10 1 1089.4.a.q 2
33.h odd 10 1 99.4.f.a 4
33.h odd 10 1 1089.4.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.e.a 4 1.a even 1 1 trivial
33.4.e.a 4 11.c even 5 1 inner
99.4.f.a 4 3.b odd 2 1
99.4.f.a 4 33.h odd 10 1
363.4.a.n 2 11.c even 5 1
363.4.a.o 2 11.d odd 10 1
1089.4.a.p 2 33.h odd 10 1
1089.4.a.q 2 33.f even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 10T_{2}^{3} + 60T_{2}^{2} - 200T_{2} + 400$$ acting on $$S_{4}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 10 T^{3} + 60 T^{2} + \cdots + 400$$
$3$ $$T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81$$
$5$ $$T^{4} + 21 T^{3} + 496 T^{2} + \cdots + 11881$$
$7$ $$T^{4} - 37 T^{3} + 619 T^{2} + \cdots + 192721$$
$11$ $$T^{4} - 41 T^{3} + 1881 T^{2} + \cdots + 1771561$$
$13$ $$T^{4} + 77 T^{3} + 2299 T^{2} + \cdots + 14641$$
$17$ $$T^{4} - 192 T^{3} + 14634 T^{2} + \cdots + 2595321$$
$19$ $$T^{4} + 52 T^{3} + 11254 T^{2} + \cdots + 1274641$$
$23$ $$(T^{2} + 74 T - 1051)^{2}$$
$29$ $$T^{4} - 414 T^{3} + \cdots + 1740892176$$
$31$ $$T^{4} + 198 T^{3} + \cdots + 54479161$$
$37$ $$T^{4} - 201 T^{3} + \cdots + 4216034761$$
$41$ $$T^{4} + 129 T^{3} + 6271 T^{2} + \cdots + 241081$$
$43$ $$(T^{2} + 66 T - 7731)^{2}$$
$47$ $$T^{4} + 35 T^{3} + \cdots + 674700625$$
$53$ $$T^{4} - 188 T^{3} + \cdots + 10042561$$
$59$ $$T^{4} + 1320 T^{3} + \cdots + 47173668025$$
$61$ $$T^{4} - 1275 T^{3} + \cdots + 55792802025$$
$67$ $$(T^{2} - 75 T - 391995)^{2}$$
$71$ $$T^{4} - 117 T^{3} + \cdots + 53332821721$$
$73$ $$T^{4} + 982 T^{3} + \cdots + 8360542096$$
$79$ $$T^{4} + 1469 T^{3} + \cdots + 285379255681$$
$83$ $$T^{4} + 967 T^{3} + \cdots + 53935882081$$
$89$ $$(T^{2} + 2712 T + 1809091)^{2}$$
$97$ $$T^{4} - 2391 T^{3} + \cdots + 932420053161$$