# Properties

 Label 24.4.d.a Level $24$ Weight $4$ Character orbit 24.d Analytic conductor $1.416$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [24,4,Mod(13,24)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(24, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("24.13");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$24 = 2^{3} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 24.d (of order $$2$$, degree $$1$$, 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: $$1.41604584014$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.8248384.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + x^{4} - 12x^{3} + 4x^{2} + 64$$ x^6 + x^4 - 12*x^3 + 4*x^2 + 64 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{5}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + \beta_1 q^{3} + ( - \beta_{5} + 3) q^{4} + ( - \beta_{4} - \beta_{3} + \cdots + 2 \beta_1) q^{5}+ \cdots - 9 q^{9}+O(q^{10})$$ q + b3 * q^2 + b1 * q^3 + (-b5 + 3) * q^4 + (-b4 - b3 - b2 + 2*b1) * q^5 + (b2 - 1) * q^6 + (b5 + 2*b4 - 3*b3 - b2 + 6) * q^7 + (-b5 - b4 + 4*b3 - 4*b1 - 14) * q^8 - 9 * q^9 $$q + \beta_{3} q^{2} + \beta_1 q^{3} + ( - \beta_{5} + 3) q^{4} + ( - \beta_{4} - \beta_{3} + \cdots + 2 \beta_1) q^{5}+ \cdots + ( - 36 \beta_{5} + 36 \beta_{3} - 36 \beta_{2}) q^{99}+O(q^{100})$$ q + b3 * q^2 + b1 * q^3 + (-b5 + 3) * q^4 + (-b4 - b3 - b2 + 2*b1) * q^5 + (b2 - 1) * q^6 + (b5 + 2*b4 - 3*b3 - b2 + 6) * q^7 + (-b5 - b4 + 4*b3 - 4*b1 - 14) * q^8 - 9 * q^9 + (2*b5 + 2*b4 + 2*b2 - 8*b1 + 10) * q^10 + (4*b5 - 4*b3 + 4*b2) * q^11 + (-3*b4 + 4*b1 - 3) * q^12 + (-2*b5 - 2*b4 - 4*b2 - 4*b1) * q^13 + (2*b5 + 6*b4 + 2*b3 + 8*b1 - 16) * q^14 + (3*b5 + 9*b3 - b2 - 14) * q^15 + (-4*b5 - 2*b4 - 12*b3 - 4*b2 - 8*b1 + 14) * q^16 + (-8*b5 - 4*b4 - 12*b3 + 4*b2 + 14) * q^17 - 9*b3 * q^18 + (-4*b5 + 8*b4 + 12*b3 + 4*b2 - 4*b1) * q^19 + (-2*b4 + 8*b3 - 8*b2 + 24*b1 + 6) * q^20 + (-6*b5 + 3*b4 + 9*b3 - 3*b2 + 2*b1) * q^21 + (8*b5 - 8*b4 + 32*b1 + 32) * q^22 + (-2*b5 - 4*b4 + 6*b3 + 2*b2 + 52) * q^23 + (3*b5 - 3*b4 + 4*b2 - 12*b1 + 32) * q^24 + (8*b5 - 8*b4 + 48*b3 - 39) * q^25 + (8*b4 - 4*b2 - 32*b1 + 12) * q^26 - 9*b1 * q^27 + (-6*b5 + 8*b4 - 24*b3 + 8*b2 + 32*b1 - 46) * q^28 + (4*b5 + 3*b4 - b3 + 7*b2 - 22*b1) * q^29 + (-6*b5 + 6*b4 - 18*b3 + 8*b1 + 72) * q^30 + (-9*b5 + 6*b4 - 45*b3 + b2 - 86) * q^31 + (10*b5 + 6*b4 + 16*b3 - 8*b2 - 40*b1 - 48) * q^32 + (12*b4 - 36*b3 - 4*b2 + 16) * q^33 + (8*b5 - 24*b4 + 30*b3 - 32*b1 - 112) * q^34 + (4*b5 - 16*b4 - 20*b3 - 12*b2 + 4*b1) * q^35 + (9*b5 - 27) * q^36 + (22*b5 - 8*b4 - 30*b3 + 14*b2 + 40*b1) * q^37 + (-24*b5 - 8*b4 - 4*b2 + 32*b1 - 124) * q^38 + (6*b5 - 6*b4 + 36*b3 + 36) * q^39 + (-6*b5 + 22*b4 + 24*b2 - 40*b1 + 48) * q^40 + (8*b5 + 28*b4 - 60*b3 - 12*b2 + 66) * q^41 + (-18*b5 + 6*b4 + 2*b2 - 24*b1 - 86) * q^42 + (-20*b5 - 8*b4 + 12*b3 - 28*b2 + 36*b1) * q^43 + (16*b5 + 32*b3 + 32*b2 + 176) * q^44 + (9*b4 + 9*b3 + 9*b2 - 18*b1) * q^45 + (-4*b5 - 12*b4 + 60*b3 - 16*b1 + 32) * q^46 + (6*b5 - 12*b4 + 54*b3 + 2*b2 - 92) * q^47 + (6*b5 - 12*b4 + 36*b3 - 12*b2 + 16*b1 + 78) * q^48 + (24*b5 + 72*b3 - 8*b2 + 77) * q^49 + (-32*b5 - 39*b3 + 352) * q^50 + (12*b5 - 24*b4 - 36*b3 - 12*b2 + 30*b1) * q^51 + (-8*b5 + 20*b4 - 32*b2 + 16*b1 - 52) * q^52 + (-20*b5 + 21*b4 + 41*b3 + b2 + 102*b1) * q^53 + (-9*b2 + 9) * q^54 + (-32*b5 + 8*b4 - 120*b3 + 8*b2 + 224) * q^55 + (10*b5 - 22*b4 - 40*b3 + 32*b2 + 40*b1 - 308) * q^56 + (-24*b5 - 12*b4 - 36*b3 + 12*b2 - 12) * q^57 + (2*b5 - 14*b4 - 22*b2 + 56*b1 + 18) * q^58 + (-16*b5 + 16*b4 + 32*b3 - 36*b1) * q^59 + (6*b5 + 72*b3 + 8*b2 - 218) * q^60 + (18*b5 - 12*b4 - 30*b3 + 6*b2 - 160*b1) * q^61 + (30*b5 - 6*b4 - 82*b3 - 8*b1 - 336) * q^62 + (-9*b5 - 18*b4 + 27*b3 + 9*b2 - 54) * q^63 + (-12*b5 + 40*b4 - 72*b3 - 40*b2 + 32*b1 + 180) * q^64 + (8*b5 - 20*b4 + 84*b3 + 4*b2 - 328) * q^65 + (24*b5 + 24*b4 + 32*b1 - 240) * q^66 + (48*b4 + 48*b3 + 48*b2 - 116*b1) * q^67 + (2*b5 - 16*b4 - 96*b3 - 32*b2 - 64*b1 + 522) * q^68 + (12*b5 - 6*b4 - 18*b3 + 6*b2 + 60*b1) * q^69 + (40*b5 + 24*b4 + 4*b2 - 96*b1 + 220) * q^70 + (42*b5 + 12*b4 + 90*b3 - 18*b2 - 324) * q^71 + (9*b5 + 9*b4 - 36*b3 + 36*b1 + 126) * q^72 + (-40*b5 - 32*b4 - 24*b3 + 24*b2 + 170) * q^73 + (60*b5 - 28*b4 + 40*b2 + 112*b1 + 232) * q^74 + (24*b5 + 24*b4 + 48*b2 - 39*b1) * q^75 + (-16*b5 - 20*b4 - 96*b3 + 32*b2 - 144*b1 - 260) * q^76 + (-40*b5 + 40*b3 - 40*b2 - 256*b1) * q^77 + (-24*b5 + 36*b3 + 264) * q^78 + (19*b5 + 14*b4 + 15*b3 - 11*b2 - 14) * q^79 + (-28*b5 - 56*b4 + 56*b3 - 40*b2 + 160*b1 - 380) * q^80 + 81 * q^81 + (40*b5 + 72*b4 + 18*b3 + 96*b1 - 368) * q^82 + (-4*b5 + 16*b4 + 20*b3 + 12*b2 + 112*b1) * q^83 + (-24*b5 - 18*b4 - 72*b3 - 24*b2 - 40*b1 - 306) * q^84 + (-56*b5 - 14*b4 + 42*b3 - 70*b2 + 316*b1) * q^85 + (-24*b5 + 56*b4 + 36*b2 - 224*b1 - 100) * q^86 + (-9*b5 + 12*b4 - 63*b3 - b2 + 202) * q^87 + (-16*b5 - 80*b4 + 192*b3 + 192*b1 + 224) * q^88 + (-16*b5 - 8*b4 - 24*b3 + 8*b2 - 26) * q^89 + (-18*b5 - 18*b4 - 18*b2 + 72*b1 - 90) * q^90 + (76*b5 - 56*b4 - 132*b3 + 20*b2 + 120*b1) * q^91 + (-52*b5 - 16*b4 + 48*b3 - 16*b2 - 64*b1 + 284) * q^92 + (-18*b5 - 27*b4 - 9*b3 - 45*b2 - 82*b1) * q^93 + (-36*b5 - 12*b4 - 84*b3 - 16*b1 + 384) * q^94 + (4*b5 + 56*b4 - 156*b3 - 20*b2 + 920) * q^95 + (-18*b5 + 30*b4 + 72*b3 + 16*b2 - 72*b1 + 356) * q^96 + (-32*b5 + 8*b4 - 120*b3 + 8*b2 - 354) * q^97 + (-48*b5 + 48*b4 + 45*b3 + 64*b1 + 576) * q^98 + (-36*b5 + 36*b3 - 36*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{2} + 16 q^{4} - 6 q^{6} + 28 q^{7} - 76 q^{8} - 54 q^{9}+O(q^{10})$$ 6 * q + 2 * q^2 + 16 * q^4 - 6 * q^6 + 28 * q^7 - 76 * q^8 - 54 * q^9 $$6 q + 2 q^{2} + 16 q^{4} - 6 q^{6} + 28 q^{7} - 76 q^{8} - 54 q^{9} + 60 q^{10} - 12 q^{12} - 100 q^{14} - 60 q^{15} + 56 q^{16} + 52 q^{17} - 18 q^{18} + 56 q^{20} + 224 q^{22} + 328 q^{23} + 204 q^{24} - 106 q^{25} + 56 q^{26} - 352 q^{28} + 372 q^{30} - 636 q^{31} - 248 q^{32} - 548 q^{34} - 144 q^{36} - 776 q^{38} + 312 q^{39} + 232 q^{40} + 236 q^{41} - 564 q^{42} + 1152 q^{44} + 328 q^{46} - 408 q^{47} + 576 q^{48} + 654 q^{49} + 1970 q^{50} - 368 q^{52} + 54 q^{54} + 1024 q^{55} - 1864 q^{56} - 168 q^{57} + 140 q^{58} - 1152 q^{60} - 2108 q^{62} - 252 q^{63} + 832 q^{64} - 1744 q^{65} - 1440 q^{66} + 2976 q^{68} + 1352 q^{70} - 1704 q^{71} + 684 q^{72} + 956 q^{73} + 1568 q^{74} - 1744 q^{76} + 1608 q^{78} - 44 q^{79} - 2112 q^{80} + 486 q^{81} - 2236 q^{82} - 1992 q^{84} - 760 q^{86} + 1044 q^{87} + 1856 q^{88} - 220 q^{89} - 540 q^{90} + 1728 q^{92} + 2088 q^{94} + 5104 q^{95} + 2184 q^{96} - 2444 q^{97} + 3354 q^{98}+O(q^{100})$$ 6 * q + 2 * q^2 + 16 * q^4 - 6 * q^6 + 28 * q^7 - 76 * q^8 - 54 * q^9 + 60 * q^10 - 12 * q^12 - 100 * q^14 - 60 * q^15 + 56 * q^16 + 52 * q^17 - 18 * q^18 + 56 * q^20 + 224 * q^22 + 328 * q^23 + 204 * q^24 - 106 * q^25 + 56 * q^26 - 352 * q^28 + 372 * q^30 - 636 * q^31 - 248 * q^32 - 548 * q^34 - 144 * q^36 - 776 * q^38 + 312 * q^39 + 232 * q^40 + 236 * q^41 - 564 * q^42 + 1152 * q^44 + 328 * q^46 - 408 * q^47 + 576 * q^48 + 654 * q^49 + 1970 * q^50 - 368 * q^52 + 54 * q^54 + 1024 * q^55 - 1864 * q^56 - 168 * q^57 + 140 * q^58 - 1152 * q^60 - 2108 * q^62 - 252 * q^63 + 832 * q^64 - 1744 * q^65 - 1440 * q^66 + 2976 * q^68 + 1352 * q^70 - 1704 * q^71 + 684 * q^72 + 956 * q^73 + 1568 * q^74 - 1744 * q^76 + 1608 * q^78 - 44 * q^79 - 2112 * q^80 + 486 * q^81 - 2236 * q^82 - 1992 * q^84 - 760 * q^86 + 1044 * q^87 + 1856 * q^88 - 220 * q^89 - 540 * q^90 + 1728 * q^92 + 2088 * q^94 + 5104 * q^95 + 2184 * q^96 - 2444 * q^97 + 3354 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + x^{4} - 12x^{3} + 4x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$( -3\nu^{5} - 6\nu^{4} + 9\nu^{3} + 6\nu^{2} + 24\nu - 96 ) / 32$$ (-3*v^5 - 6*v^4 + 9*v^3 + 6*v^2 + 24*v - 96) / 32 $$\beta_{2}$$ $$=$$ $$( -3\nu^{4} - 6\nu^{3} + 9\nu^{2} + 6\nu + 32 ) / 8$$ (-3*v^4 - 6*v^3 + 9*v^2 + 6*v + 32) / 8 $$\beta_{3}$$ $$=$$ $$( -\nu^{5} + \nu^{4} + \nu^{3} + 9\nu^{2} - 6\nu - 8 ) / 8$$ (-v^5 + v^4 + v^3 + 9*v^2 - 6*v - 8) / 8 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} + 2\nu^{4} + 3\nu^{3} + 6\nu^{2} - 40\nu - 24 ) / 8$$ (-v^5 + 2*v^4 + 3*v^3 + 6*v^2 - 40*v - 24) / 8 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} + 4\nu^{4} - 9\nu^{3} - 28\nu + 56 ) / 8$$ (-v^5 + 4*v^4 - 9*v^3 - 28*v + 56) / 8
 $$\nu$$ $$=$$ $$( -3\beta_{4} + 3\beta_{3} - \beta_{2} - 2 ) / 12$$ (-3*b4 + 3*b3 - b2 - 2) / 12 $$\nu^{2}$$ $$=$$ $$( -3\beta_{5} + 9\beta_{3} + 3\beta_{2} - 8\beta _1 - 6 ) / 12$$ (-3*b5 + 9*b3 + 3*b2 - 8*b1 - 6) / 12 $$\nu^{3}$$ $$=$$ $$( -6\beta_{5} + 3\beta_{4} + 3\beta_{3} - 5\beta_{2} + 74 ) / 12$$ (-6*b5 + 3*b4 + 3*b3 - 5*b2 + 74) / 12 $$\nu^{4}$$ $$=$$ $$( \beta_{5} - 4\beta_{4} + 9\beta_{3} - 5\beta_{2} - 8\beta _1 - 14 ) / 4$$ (b5 - 4*b4 + 9*b3 - 5*b2 - 8*b1 - 14) / 4 $$\nu^{5}$$ $$=$$ $$( -30\beta_{5} + 9\beta_{4} - 3\beta_{3} + 13\beta_{2} - 96\beta _1 - 106 ) / 12$$ (-30*b5 + 9*b4 - 3*b3 + 13*b2 - 96*b1 - 106) / 12

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/24\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$13$$ $$17$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 −1.24181 + 1.56777i −1.24181 − 1.56777i −0.641412 + 1.89436i −0.641412 − 1.89436i 1.88322 − 0.673417i 1.88322 + 0.673417i
−2.80958 0.325969i 3.00000i 7.78749 + 1.83167i 18.5422i 0.977907 8.42874i 9.32669 −21.2825 7.68472i −9.00000 6.04419 52.0958i
13.2 −2.80958 + 0.325969i 3.00000i 7.78749 1.83167i 18.5422i 0.977907 + 8.42874i 9.32669 −21.2825 + 7.68472i −9.00000 6.04419 + 52.0958i
13.3 1.25295 2.53577i 3.00000i −4.86025 6.35436i 9.15486i −7.60731 3.75884i 27.4175 −22.2028 + 4.36281i −9.00000 23.2146 + 11.4705i
13.4 1.25295 + 2.53577i 3.00000i −4.86025 + 6.35436i 9.15486i −7.60731 + 3.75884i 27.4175 −22.2028 4.36281i −9.00000 23.2146 11.4705i
13.5 2.55664 1.20980i 3.00000i 5.07277 6.18604i 0.612661i 3.62940 + 7.66991i −22.7441 5.48534 21.9525i −9.00000 0.741198 + 1.56635i
13.6 2.55664 + 1.20980i 3.00000i 5.07277 + 6.18604i 0.612661i 3.62940 7.66991i −22.7441 5.48534 + 21.9525i −9.00000 0.741198 1.56635i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 13.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.4.d.a 6
3.b odd 2 1 72.4.d.d 6
4.b odd 2 1 96.4.d.a 6
8.b even 2 1 inner 24.4.d.a 6
8.d odd 2 1 96.4.d.a 6
12.b even 2 1 288.4.d.d 6
16.e even 4 1 768.4.a.r 3
16.e even 4 1 768.4.a.s 3
16.f odd 4 1 768.4.a.q 3
16.f odd 4 1 768.4.a.t 3
24.f even 2 1 288.4.d.d 6
24.h odd 2 1 72.4.d.d 6
48.i odd 4 1 2304.4.a.bt 3
48.i odd 4 1 2304.4.a.bv 3
48.k even 4 1 2304.4.a.bu 3
48.k even 4 1 2304.4.a.bw 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.d.a 6 1.a even 1 1 trivial
24.4.d.a 6 8.b even 2 1 inner
72.4.d.d 6 3.b odd 2 1
72.4.d.d 6 24.h odd 2 1
96.4.d.a 6 4.b odd 2 1
96.4.d.a 6 8.d odd 2 1
288.4.d.d 6 12.b even 2 1
288.4.d.d 6 24.f even 2 1
768.4.a.q 3 16.f odd 4 1
768.4.a.r 3 16.e even 4 1
768.4.a.s 3 16.e even 4 1
768.4.a.t 3 16.f odd 4 1
2304.4.a.bt 3 48.i odd 4 1
2304.4.a.bu 3 48.k even 4 1
2304.4.a.bv 3 48.i odd 4 1
2304.4.a.bw 3 48.k even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(24, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 2 T^{5} + \cdots + 512$$
$3$ $$(T^{2} + 9)^{3}$$
$5$ $$T^{6} + 428 T^{4} + \cdots + 10816$$
$7$ $$(T^{3} - 14 T^{2} + \cdots + 5816)^{2}$$
$11$ $$T^{6} + \cdots + 2415919104$$
$13$ $$T^{6} + \cdots + 3121680384$$
$17$ $$(T^{3} - 26 T^{2} + \cdots + 477576)^{2}$$
$19$ $$T^{6} + \cdots + 75488661504$$
$23$ $$(T^{3} - 164 T^{2} + \cdots - 45504)^{2}$$
$29$ $$T^{6} + \cdots + 3766031424$$
$31$ $$(T^{3} + 318 T^{2} + \cdots - 3749624)^{2}$$
$37$ $$T^{6} + \cdots + 6879707136$$
$41$ $$(T^{3} - 118 T^{2} + \cdots + 19985976)^{2}$$
$43$ $$T^{6} + \cdots + 73984219582464$$
$47$ $$(T^{3} + 204 T^{2} + \cdots - 1964736)^{2}$$
$53$ $$T^{6} + \cdots + 427051482970176$$
$59$ $$T^{6} + \cdots + 72651484205056$$
$61$ $$T^{6} + \cdots + 10\!\cdots\!56$$
$67$ $$T^{6} + \cdots + 10\!\cdots\!84$$
$71$ $$(T^{3} + 852 T^{2} + \cdots - 85084992)^{2}$$
$73$ $$(T^{3} - 478 T^{2} + \cdots + 120833304)^{2}$$
$79$ $$(T^{3} + 22 T^{2} + \cdots - 7902616)^{2}$$
$83$ $$T^{6} + \cdots + 14\!\cdots\!96$$
$89$ $$(T^{3} + 110 T^{2} + \cdots + 1423656)^{2}$$
$97$ $$(T^{3} + 1222 T^{2} + \cdots - 74802424)^{2}$$