# Properties

 Label 37.4.a.a Level $37$ Weight $4$ Character orbit 37.a Self dual yes Analytic conductor $2.183$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,4,Mod(1,37)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(37, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("37.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$37$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 37.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: $$2.18307067021$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.21208.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 8x^{2} + 13x - 3$$ x^4 - x^3 - 8*x^2 + 13*x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + \beta_{2} - 2) q^{2} + (\beta_{3} - 2 \beta_{2} - 2) q^{3} + (6 \beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{4} + ( - \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 9) q^{5} + ( - 2 \beta_{3} - \beta_{2} - 7) q^{6} + (\beta_{2} - 13 \beta_1 - 5) q^{7} + ( - 20 \beta_{3} + \beta_{2} + 9 \beta_1 - 30) q^{8} + ( - 3 \beta_{3} + 15 \beta_{2} + 3 \beta_1 - 5) q^{9}+O(q^{10})$$ q + (-b3 + b2 - 2) * q^2 + (b3 - 2*b2 - 2) * q^3 + (6*b3 - 3*b2 - b1 + 4) * q^4 + (-b3 + 2*b2 + 4*b1 - 9) * q^5 + (-2*b3 - b2 - 7) * q^6 + (b2 - 13*b1 - 5) * q^7 + (-20*b3 + b2 + 9*b1 - 30) * q^8 + (-3*b3 + 15*b2 + 3*b1 - 5) * q^9 $$q + ( - \beta_{3} + \beta_{2} - 2) q^{2} + (\beta_{3} - 2 \beta_{2} - 2) q^{3} + (6 \beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{4} + ( - \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 9) q^{5} + ( - 2 \beta_{3} - \beta_{2} - 7) q^{6} + (\beta_{2} - 13 \beta_1 - 5) q^{7} + ( - 20 \beta_{3} + \beta_{2} + 9 \beta_1 - 30) q^{8} + ( - 3 \beta_{3} + 15 \beta_{2} + 3 \beta_1 - 5) q^{9} + (13 \beta_{3} - 6 \beta_{2} + 25) q^{10} + (5 \beta_{3} + 9 \beta_{2} + \beta_1 - 4) q^{11} + (7 \beta_{3} + 7 \beta_{2} - 5 \beta_1 + 37) q^{12} + ( - 17 \beta_{3} - 13 \beta_{2} + 19 \beta_1 - 17) q^{13} + (5 \beta_{3} - 18 \beta_{2} + \beta_1 + 26) q^{14} + ( - 8 \beta_{3} - 5 \beta_{2} - 19 \beta_1) q^{15} + (62 \beta_{3} - 17 \beta_{2} - 31 \beta_1 + 122) q^{16} + (8 \beta_{2} - 8 \beta_1 - 14) q^{17} + (17 \beta_{3} - 5 \beta_{2} + 9 \beta_1 + 67) q^{18} + ( - 20 \beta_{3} - 36 \beta_{2} + 40 \beta_1 - 42) q^{19} + ( - 69 \beta_{3} + 22 \beta_{2} - 12 \beta_1 - 61) q^{20} + ( - 4 \beta_{3} + 43 \beta_{2} + 49 \beta_1 + 3) q^{21} + ( - 16 \beta_{3} + 2 \beta_{2} + 19 \beta_1 + 9) q^{22} + (41 \beta_{3} - 9 \beta_{2} - 29 \beta_1 - 49) q^{23} + ( - 49 \beta_{3} + 47 \beta_{2} + 21 \beta_1 - 27) q^{24} + (3 \beta_{3} + 11 \beta_{2} - 69 \beta_1 + 38) q^{25} + (85 \beta_{3} - 15 \beta_{2} - 47 \beta_1 + 61) q^{26} + ( - 20 \beta_{3} - 32 \beta_{2} - 48 \beta_1 - 53) q^{27} + ( - 46 \beta_{3} + 24 \beta_{2} + 96 \beta_1 - 92) q^{28} + (51 \beta_{3} - 15 \beta_{2} + \beta_1 + 9) q^{29} + (32 \beta_{3} - 27 \beta_{2} - 21 \beta_1 + 44) q^{30} + (55 \beta_{3} + 16 \beta_{2} + 10 \beta_1 + 35) q^{31} + ( - 210 \beta_{3} + 145 \beta_{2} + 35 \beta_1 - 334) q^{32} + (10 \beta_{3} - 54 \beta_{2} - 46 \beta_1 - 35) q^{33} + (14 \beta_{3} - 22 \beta_{2} + 8 \beta_1 + 60) q^{34} + (56 \beta_{3} - 98 \beta_{2} + 126 \beta_1 - 152) q^{35} + ( - 111 \beta_{3} - 27 \beta_{2} + 5 \beta_1 - 203) q^{36} + 37 q^{37} + (122 \beta_{3} - 22 \beta_{2} - 76 \beta_1 + 36) q^{38} + ( - 47 \beta_{3} + 72 \beta_{2} + 14 \beta_1 + 57) q^{39} + (233 \beta_{3} - 94 \beta_{2} - 116 \beta_1 + 345) q^{40} + ( - 45 \beta_{3} - 54 \beta_{2} + 24 \beta_1 - 6) q^{41} + (13 \beta_{3} + 48 \beta_{2} + 35 \beta_1 + 94) q^{42} + (10 \beta_{3} + 60 \beta_{2} - 72 \beta_1 - 42) q^{43} + (15 \beta_{3} - 60 \beta_{2} - 38 \beta_1 + 81) q^{44} + (14 \beta_{3} + 41 \beta_{2} + 7 \beta_1 + 246) q^{45} + ( - 115 \beta_{3} - 37 \beta_{2} + 73 \beta_1 - 105) q^{46} + ( - 8 \beta_{3} + 71 \beta_{2} + 125 \beta_1 - 169) q^{47} + (167 \beta_{3} - 111 \beta_{2} - 11 \beta_1 + 123) q^{48} + ( - 170 \beta_{3} + 109 \beta_{2} - 37 \beta_1 + 336) q^{49} + ( - 50 \beta_{3} - 28 \beta_{2} + 17 \beta_1 + 11) q^{50} + ( - 6 \beta_{3} + 4 \beta_{2} + 8 \beta_1 - 28) q^{51} + ( - 265 \beta_{3} + 203 \beta_{2} + 3 \beta_1 - 409) q^{52} + (70 \beta_{3} - 25 \beta_{2} - 15 \beta_1 + 107) q^{53} + (133 \beta_{3} - 121 \beta_{2} - 72 \beta_1 + 158) q^{54} + ( - 69 \beta_{3} + 51 \beta_{2} - 25 \beta_1 + 171) q^{55} + (236 \beta_{3} + 102 \beta_{2} - 76 \beta_1 + 182) q^{56} + ( - 98 \beta_{3} + 200 \beta_{2} + 8 \beta_1 + 256) q^{57} + ( - 213 \beta_{3} + 61 \beta_{2} + 87 \beta_1 - 319) q^{58} + ( - 54 \beta_{3} - 6 \beta_{2} - 114 \beta_1 - 366) q^{59} + ( - 108 \beta_{3} + 95 \beta_{2} + 189 \beta_1 - 308) q^{60} + (5 \beta_{3} - 202 \beta_{2} + 12 \beta_1 + 41) q^{61} + ( - 255 \beta_{3} + 100 \beta_{2} + 126 \beta_1 - 307) q^{62} + (42 \beta_{3} - 434 \beta_{2} + 38 \beta_1 - 188) q^{63} + (678 \beta_{3} - 373 \beta_{2} - 27 \beta_1 + 1142) q^{64} + (158 \beta_{3} - 25 \beta_{2} - 231 \beta_1 + 246) q^{65} + ( - 5 \beta_{3} - 71 \beta_{2} - 34 \beta_1 - 96) q^{66} + ( - 69 \beta_{3} + 90 \beta_{2} + 128 \beta_1 - 73) q^{67} + ( - 116 \beta_{3} + 18 \beta_{2} + 70 \beta_1 - 152) q^{68} + ( - 17 \beta_{3} + 198 \beta_{2} + 20 \beta_1 + 325) q^{69} + ( - 72 \beta_{3} + 30 \beta_{2} + 14 \beta_1 - 396) q^{70} + (62 \beta_{3} + 163 \beta_{2} - 267 \beta_1 + 139) q^{71} + (511 \beta_{3} - 269 \beta_{2} - 321 \beta_1 + 339) q^{72} + (217 \beta_{3} - \beta_{2} + 3 \beta_1 - 188) q^{73} + ( - 37 \beta_{3} + 37 \beta_{2} - 74) q^{74} + (52 \beta_{3} + 62 \beta_{2} + 234 \beta_1 - 141) q^{75} + ( - 364 \beta_{3} + 370 \beta_{2} - 98 \beta_1 - 336) q^{76} + ( - 26 \beta_{3} - 341 \beta_{2} + 213 \beta_1 - 237) q^{77} + (131 \beta_{3} + 24 \beta_{2} - 22 \beta_1 + 323) q^{78} + ( - 267 \beta_{3} + 111 \beta_{2} + 27 \beta_1 - 97) q^{79} + ( - 725 \beta_{3} + 286 \beta_{2} + 468 \beta_1 - 1533) q^{80} + ( - 24 \beta_{3} + 57 \beta_{2} + 267 \beta_1 + 385) q^{81} + (186 \beta_{3} - 27 \beta_{2} - 144 \beta_1 + 51) q^{82} + ( - 114 \beta_{3} - 35 \beta_{2} - 189 \beta_1 - 757) q^{83} + ( - 114 \beta_{3} - 202 \beta_{2} - 318 \beta_1 - 168) q^{84} + (38 \beta_{3} - 60 \beta_{2} + 56 \beta_1 + 86) q^{85} + (2 \beta_{3} - 104 \beta_{2} + 80 \beta_1 + 286) q^{86} + (45 \beta_{3} + 18 \beta_{2} - 112 \beta_1 + 291) q^{87} + ( - 13 \beta_{3} + 42 \beta_{2} - 182 \beta_1 - 451) q^{88} + (188 \beta_{3} - 182 \beta_{2} + 362 \beta_1 + 58) q^{89} + ( - 302 \beta_{3} + 267 \beta_{2} + 69 \beta_1 - 446) q^{90} + (362 \beta_{3} + 372 \beta_{2} - 112 \beta_1 - 342) q^{91} + (237 \beta_{3} - 75 \beta_{2} - 35 \beta_1 + 993) q^{92} + (106 \beta_{3} - 251 \beta_{2} - 253 \beta_1 + 38) q^{93} + (201 \beta_{3} - 52 \beta_{2} + 55 \beta_1 + 466) q^{94} + (158 \beta_{3} - 12 \beta_{2} - 616 \beta_1 + 542) q^{95} + ( - 399 \beta_{3} - 97 \beta_{2} + 55 \beta_1 - 1187) q^{96} + ( - 80 \beta_{3} - 74 \beta_{2} - 122 \beta_1 - 604) q^{97} + (344 \beta_{3} + 129 \beta_{2} - 231 \beta_1 + 542) q^{98} + ( - 214 \beta_{3} + 279 \beta_{2} + 289 \beta_1 + 596) q^{99}+O(q^{100})$$ q + (-b3 + b2 - 2) * q^2 + (b3 - 2*b2 - 2) * q^3 + (6*b3 - 3*b2 - b1 + 4) * q^4 + (-b3 + 2*b2 + 4*b1 - 9) * q^5 + (-2*b3 - b2 - 7) * q^6 + (b2 - 13*b1 - 5) * q^7 + (-20*b3 + b2 + 9*b1 - 30) * q^8 + (-3*b3 + 15*b2 + 3*b1 - 5) * q^9 + (13*b3 - 6*b2 + 25) * q^10 + (5*b3 + 9*b2 + b1 - 4) * q^11 + (7*b3 + 7*b2 - 5*b1 + 37) * q^12 + (-17*b3 - 13*b2 + 19*b1 - 17) * q^13 + (5*b3 - 18*b2 + b1 + 26) * q^14 + (-8*b3 - 5*b2 - 19*b1) * q^15 + (62*b3 - 17*b2 - 31*b1 + 122) * q^16 + (8*b2 - 8*b1 - 14) * q^17 + (17*b3 - 5*b2 + 9*b1 + 67) * q^18 + (-20*b3 - 36*b2 + 40*b1 - 42) * q^19 + (-69*b3 + 22*b2 - 12*b1 - 61) * q^20 + (-4*b3 + 43*b2 + 49*b1 + 3) * q^21 + (-16*b3 + 2*b2 + 19*b1 + 9) * q^22 + (41*b3 - 9*b2 - 29*b1 - 49) * q^23 + (-49*b3 + 47*b2 + 21*b1 - 27) * q^24 + (3*b3 + 11*b2 - 69*b1 + 38) * q^25 + (85*b3 - 15*b2 - 47*b1 + 61) * q^26 + (-20*b3 - 32*b2 - 48*b1 - 53) * q^27 + (-46*b3 + 24*b2 + 96*b1 - 92) * q^28 + (51*b3 - 15*b2 + b1 + 9) * q^29 + (32*b3 - 27*b2 - 21*b1 + 44) * q^30 + (55*b3 + 16*b2 + 10*b1 + 35) * q^31 + (-210*b3 + 145*b2 + 35*b1 - 334) * q^32 + (10*b3 - 54*b2 - 46*b1 - 35) * q^33 + (14*b3 - 22*b2 + 8*b1 + 60) * q^34 + (56*b3 - 98*b2 + 126*b1 - 152) * q^35 + (-111*b3 - 27*b2 + 5*b1 - 203) * q^36 + 37 * q^37 + (122*b3 - 22*b2 - 76*b1 + 36) * q^38 + (-47*b3 + 72*b2 + 14*b1 + 57) * q^39 + (233*b3 - 94*b2 - 116*b1 + 345) * q^40 + (-45*b3 - 54*b2 + 24*b1 - 6) * q^41 + (13*b3 + 48*b2 + 35*b1 + 94) * q^42 + (10*b3 + 60*b2 - 72*b1 - 42) * q^43 + (15*b3 - 60*b2 - 38*b1 + 81) * q^44 + (14*b3 + 41*b2 + 7*b1 + 246) * q^45 + (-115*b3 - 37*b2 + 73*b1 - 105) * q^46 + (-8*b3 + 71*b2 + 125*b1 - 169) * q^47 + (167*b3 - 111*b2 - 11*b1 + 123) * q^48 + (-170*b3 + 109*b2 - 37*b1 + 336) * q^49 + (-50*b3 - 28*b2 + 17*b1 + 11) * q^50 + (-6*b3 + 4*b2 + 8*b1 - 28) * q^51 + (-265*b3 + 203*b2 + 3*b1 - 409) * q^52 + (70*b3 - 25*b2 - 15*b1 + 107) * q^53 + (133*b3 - 121*b2 - 72*b1 + 158) * q^54 + (-69*b3 + 51*b2 - 25*b1 + 171) * q^55 + (236*b3 + 102*b2 - 76*b1 + 182) * q^56 + (-98*b3 + 200*b2 + 8*b1 + 256) * q^57 + (-213*b3 + 61*b2 + 87*b1 - 319) * q^58 + (-54*b3 - 6*b2 - 114*b1 - 366) * q^59 + (-108*b3 + 95*b2 + 189*b1 - 308) * q^60 + (5*b3 - 202*b2 + 12*b1 + 41) * q^61 + (-255*b3 + 100*b2 + 126*b1 - 307) * q^62 + (42*b3 - 434*b2 + 38*b1 - 188) * q^63 + (678*b3 - 373*b2 - 27*b1 + 1142) * q^64 + (158*b3 - 25*b2 - 231*b1 + 246) * q^65 + (-5*b3 - 71*b2 - 34*b1 - 96) * q^66 + (-69*b3 + 90*b2 + 128*b1 - 73) * q^67 + (-116*b3 + 18*b2 + 70*b1 - 152) * q^68 + (-17*b3 + 198*b2 + 20*b1 + 325) * q^69 + (-72*b3 + 30*b2 + 14*b1 - 396) * q^70 + (62*b3 + 163*b2 - 267*b1 + 139) * q^71 + (511*b3 - 269*b2 - 321*b1 + 339) * q^72 + (217*b3 - b2 + 3*b1 - 188) * q^73 + (-37*b3 + 37*b2 - 74) * q^74 + (52*b3 + 62*b2 + 234*b1 - 141) * q^75 + (-364*b3 + 370*b2 - 98*b1 - 336) * q^76 + (-26*b3 - 341*b2 + 213*b1 - 237) * q^77 + (131*b3 + 24*b2 - 22*b1 + 323) * q^78 + (-267*b3 + 111*b2 + 27*b1 - 97) * q^79 + (-725*b3 + 286*b2 + 468*b1 - 1533) * q^80 + (-24*b3 + 57*b2 + 267*b1 + 385) * q^81 + (186*b3 - 27*b2 - 144*b1 + 51) * q^82 + (-114*b3 - 35*b2 - 189*b1 - 757) * q^83 + (-114*b3 - 202*b2 - 318*b1 - 168) * q^84 + (38*b3 - 60*b2 + 56*b1 + 86) * q^85 + (2*b3 - 104*b2 + 80*b1 + 286) * q^86 + (45*b3 + 18*b2 - 112*b1 + 291) * q^87 + (-13*b3 + 42*b2 - 182*b1 - 451) * q^88 + (188*b3 - 182*b2 + 362*b1 + 58) * q^89 + (-302*b3 + 267*b2 + 69*b1 - 446) * q^90 + (362*b3 + 372*b2 - 112*b1 - 342) * q^91 + (237*b3 - 75*b2 - 35*b1 + 993) * q^92 + (106*b3 - 251*b2 - 253*b1 + 38) * q^93 + (201*b3 - 52*b2 + 55*b1 + 466) * q^94 + (158*b3 - 12*b2 - 616*b1 + 542) * q^95 + (-399*b3 - 97*b2 + 55*b1 - 1187) * q^96 + (-80*b3 - 74*b2 - 122*b1 - 604) * q^97 + (344*b3 + 129*b2 - 231*b1 + 542) * q^98 + (-214*b3 + 279*b2 + 289*b1 + 596) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{2} - 11 q^{3} + 6 q^{4} - 29 q^{5} - 27 q^{6} - 32 q^{7} - 90 q^{8} + q^{9}+O(q^{10})$$ 4 * q - 6 * q^2 - 11 * q^3 + 6 * q^4 - 29 * q^5 - 27 * q^6 - 32 * q^7 - 90 * q^8 + q^9 $$4 q - 6 q^{2} - 11 q^{3} + 6 q^{4} - 29 q^{5} - 27 q^{6} - 32 q^{7} - 90 q^{8} + q^{9} + 81 q^{10} - 11 q^{11} + 143 q^{12} - 45 q^{13} + 82 q^{14} - 16 q^{15} + 378 q^{16} - 56 q^{17} + 255 q^{18} - 144 q^{19} - 165 q^{20} + 108 q^{21} + 73 q^{22} - 275 q^{23} + 9 q^{24} + 91 q^{25} + 97 q^{26} - 272 q^{27} - 202 q^{28} - 29 q^{29} + 96 q^{30} + 111 q^{31} - 946 q^{32} - 250 q^{33} + 212 q^{34} - 636 q^{35} - 723 q^{36} + 148 q^{37} - 76 q^{38} + 361 q^{39} + 937 q^{40} - 9 q^{41} + 446 q^{42} - 190 q^{43} + 211 q^{44} + 1018 q^{45} - 269 q^{46} - 472 q^{47} + 203 q^{48} + 1586 q^{49} + 83 q^{50} - 94 q^{51} - 1165 q^{52} + 318 q^{53} + 306 q^{54} + 779 q^{55} + 518 q^{56} + 1330 q^{57} - 915 q^{58} - 1530 q^{59} - 840 q^{60} - 31 q^{61} - 747 q^{62} - 1190 q^{63} + 3490 q^{64} + 570 q^{65} - 484 q^{66} - 5 q^{67} - 404 q^{68} + 1535 q^{69} - 1468 q^{70} + 390 q^{71} + 255 q^{72} - 967 q^{73} - 222 q^{74} - 320 q^{75} - 708 q^{76} - 1050 q^{77} + 1163 q^{78} + 17 q^{79} - 4653 q^{80} + 1888 q^{81} - 153 q^{82} - 3138 q^{83} - 1078 q^{84} + 302 q^{85} + 1118 q^{86} + 1025 q^{87} - 1931 q^{88} + 224 q^{89} - 1146 q^{90} - 1470 q^{91} + 3625 q^{92} - 458 q^{93} + 1666 q^{94} + 1382 q^{95} - 4391 q^{96} - 2532 q^{97} + 1722 q^{98} + 3166 q^{99}+O(q^{100})$$ 4 * q - 6 * q^2 - 11 * q^3 + 6 * q^4 - 29 * q^5 - 27 * q^6 - 32 * q^7 - 90 * q^8 + q^9 + 81 * q^10 - 11 * q^11 + 143 * q^12 - 45 * q^13 + 82 * q^14 - 16 * q^15 + 378 * q^16 - 56 * q^17 + 255 * q^18 - 144 * q^19 - 165 * q^20 + 108 * q^21 + 73 * q^22 - 275 * q^23 + 9 * q^24 + 91 * q^25 + 97 * q^26 - 272 * q^27 - 202 * q^28 - 29 * q^29 + 96 * q^30 + 111 * q^31 - 946 * q^32 - 250 * q^33 + 212 * q^34 - 636 * q^35 - 723 * q^36 + 148 * q^37 - 76 * q^38 + 361 * q^39 + 937 * q^40 - 9 * q^41 + 446 * q^42 - 190 * q^43 + 211 * q^44 + 1018 * q^45 - 269 * q^46 - 472 * q^47 + 203 * q^48 + 1586 * q^49 + 83 * q^50 - 94 * q^51 - 1165 * q^52 + 318 * q^53 + 306 * q^54 + 779 * q^55 + 518 * q^56 + 1330 * q^57 - 915 * q^58 - 1530 * q^59 - 840 * q^60 - 31 * q^61 - 747 * q^62 - 1190 * q^63 + 3490 * q^64 + 570 * q^65 - 484 * q^66 - 5 * q^67 - 404 * q^68 + 1535 * q^69 - 1468 * q^70 + 390 * q^71 + 255 * q^72 - 967 * q^73 - 222 * q^74 - 320 * q^75 - 708 * q^76 - 1050 * q^77 + 1163 * q^78 + 17 * q^79 - 4653 * q^80 + 1888 * q^81 - 153 * q^82 - 3138 * q^83 - 1078 * q^84 + 302 * q^85 + 1118 * q^86 + 1025 * q^87 - 1931 * q^88 + 224 * q^89 - 1146 * q^90 - 1470 * q^91 + 3625 * q^92 - 458 * q^93 + 1666 * q^94 + 1382 * q^95 - 4391 * q^96 - 2532 * q^97 + 1722 * q^98 + 3166 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 8x^{2} + 13x - 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{3} + \nu^{2} - 6\nu + 1$$ v^3 + v^2 - 6*v + 1 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 7\nu + 5$$ v^3 - 7*v + 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} - \beta _1 + 4$$ -b3 + b2 - b1 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 7\beta _1 - 5$$ b3 + 7*b1 - 5

## 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
 0.280359 1.54469 −3.07664 2.25158
−5.64104 2.22256 23.8213 −12.1011 −12.5375 −9.22618 −89.2487 −22.0602 68.2629
1.2 −2.06923 0.265551 −3.71830 −5.08678 −0.549486 −27.2773 24.2478 −26.9295 10.5257
1.3 0.389055 −4.19207 −7.84864 −19.1145 −1.63095 34.7993 −6.16599 −9.42651 −7.43658
1.4 1.32121 −9.29603 −6.25440 7.30237 −12.2820 −30.2958 −18.8331 59.4162 9.64798
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$37$$ $$-1$$

## 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 37.4.a.a 4
3.b odd 2 1 333.4.a.e 4
4.b odd 2 1 592.4.a.f 4
5.b even 2 1 925.4.a.a 4
7.b odd 2 1 1813.4.a.b 4
8.b even 2 1 2368.4.a.l 4
8.d odd 2 1 2368.4.a.g 4
37.b even 2 1 1369.4.a.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.4.a.a 4 1.a even 1 1 trivial
333.4.a.e 4 3.b odd 2 1
592.4.a.f 4 4.b odd 2 1
925.4.a.a 4 5.b even 2 1
1369.4.a.c 4 37.b even 2 1
1813.4.a.b 4 7.b odd 2 1
2368.4.a.g 4 8.d odd 2 1
2368.4.a.l 4 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 6T_{2}^{3} - T_{2}^{2} - 16T_{2} + 6$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(37))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 6 T^{3} - T^{2} - 16 T + 6$$
$3$ $$T^{4} + 11 T^{3} + 6 T^{2} - 89 T + 23$$
$5$ $$T^{4} + 29 T^{3} + 125 T^{2} + \cdots - 8592$$
$7$ $$T^{4} + 32 T^{3} - 967 T^{2} + \cdots - 265324$$
$11$ $$T^{4} + 11 T^{3} - 1432 T^{2} + \cdots + 169317$$
$13$ $$T^{4} + 45 T^{3} - 4635 T^{2} + \cdots - 4630592$$
$17$ $$T^{4} + 56 T^{3} + 344 T^{2} + \cdots - 1776$$
$19$ $$T^{4} + 144 T^{3} + \cdots - 115380208$$
$23$ $$T^{4} + 275 T^{3} + \cdots - 93600684$$
$29$ $$T^{4} + 29 T^{3} - 24515 T^{2} + \cdots - 20921868$$
$31$ $$T^{4} - 111 T^{3} + \cdots + 443578864$$
$37$ $$(T - 37)^{4}$$
$41$ $$T^{4} + 9 T^{3} - 55962 T^{2} + \cdots + 239475447$$
$43$ $$T^{4} + 190 T^{3} + \cdots + 118152128$$
$47$ $$T^{4} + 472 T^{3} + \cdots - 4884453612$$
$53$ $$T^{4} - 318 T^{3} - 4891 T^{2} + \cdots + 1515396$$
$59$ $$T^{4} + 1530 T^{3} + \cdots - 19839940416$$
$61$ $$T^{4} + 31 T^{3} + \cdots - 1974966496$$
$67$ $$T^{4} + 5 T^{3} + \cdots + 3584880464$$
$71$ $$T^{4} - 390 T^{3} + \cdots - 21095220732$$
$73$ $$T^{4} + 967 T^{3} + \cdots - 10092864749$$
$79$ $$T^{4} - 17 T^{3} + \cdots - 29390715884$$
$83$ $$T^{4} + 3138 T^{3} + \cdots - 146330370096$$
$89$ $$T^{4} - 224 T^{3} + \cdots - 212866753728$$
$97$ $$T^{4} + 2532 T^{3} + \cdots + 3402009472$$