# Properties

 Label 9.4.c.a Level $9$ Weight $4$ Character orbit 9.c Analytic conductor $0.531$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9,4,Mod(4,9)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2]))

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

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

chi := DirichletCharacter("9.4");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 9.c (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: $$0.531017190052$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - \beta_1) q^{2} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{3} + (3 \beta_{3} - 3 \beta_{2} + \beta_1 - 4) q^{4} + ( - \beta_{3} + \beta_{2} + 8 \beta_1 - 7) q^{5} + ( - 3 \beta_{3} + 3 \beta_{2} + \cdots + 12) q^{6}+ \cdots + (3 \beta_{3} - 6 \beta_{2} + 21 \beta_1 - 3) q^{9}+O(q^{10})$$ q + (-b3 - b1) * q^2 + (b3 + b2 + b1 - 1) * q^3 + (3*b3 - 3*b2 + b1 - 4) * q^4 + (-b3 + b2 + 8*b1 - 7) * q^5 + (-3*b3 + 3*b2 - 15*b1 + 12) * q^6 + (-3*b3 - 2*b1) * q^7 + (-b2 + 16) * q^8 + (3*b3 - 6*b2 + 21*b1 - 3) * q^9 $$q + ( - \beta_{3} - \beta_1) q^{2} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{3} + (3 \beta_{3} - 3 \beta_{2} + \beta_1 - 4) q^{4} + ( - \beta_{3} + \beta_{2} + 8 \beta_1 - 7) q^{5} + ( - 3 \beta_{3} + 3 \beta_{2} + \cdots + 12) q^{6}+ \cdots + (279 \beta_{3} - 81 \beta_{2} + \cdots + 504) q^{99}+O(q^{100})$$ q + (-b3 - b1) * q^2 + (b3 + b2 + b1 - 1) * q^3 + (3*b3 - 3*b2 + b1 - 4) * q^4 + (-b3 + b2 + 8*b1 - 7) * q^5 + (-3*b3 + 3*b2 - 15*b1 + 12) * q^6 + (-3*b3 - 2*b1) * q^7 + (-b2 + 16) * q^8 + (3*b3 - 6*b2 + 21*b1 - 3) * q^9 + (6*b2 + 6) * q^10 + (8*b3 - 37*b1) * q^11 + (-2*b3 - 5*b2 + 22*b1 - 52) * q^12 + (-15*b3 + 15*b2 + 2*b1 + 13) * q^13 + (8*b3 - 8*b2 + 26*b1 - 34) * q^14 + (9*b3 - 15*b2 - 24*b1 + 9) * q^15 + (9*b3 - b1) * q^16 + (9*b2 + 54) * q^17 + (-18*b3 + 27*b2 + 72) * q^18 + (-27*b2 - 52) * q^19 + (-20*b3 + 16*b1) * q^20 + (-7*b3 + 8*b2 - 46*b1 + 34) * q^21 + (21*b3 - 21*b2 - 27*b1 + 6) * q^22 + (-19*b3 + 19*b2 + 26*b1 - 7) * q^23 + (15*b3 + 18*b2 + 9*b1 - 24) * q^24 + (15*b3 + 53*b1) * q^25 + (-28*b2 - 146) * q^26 + (36*b3 - 18*b2 - 18*b1 - 117) * q^27 + (18*b2 + 92) * q^28 + (-b3 + 26*b1) * q^29 + (12*b3 - 6*b2 + 48*b1 + 42) * q^30 + (3*b3 - 3*b2 + 20*b1 - 23) * q^31 + (-9*b3 + 9*b2 - 207*b1 + 216) * q^32 + (-66*b3 + 21*b2 + 165*b1 - 6) * q^33 + (-63*b3 - 117*b1) * q^34 + (19*b2 + 11) * q^35 + (-15*b3 - 60*b2 - 141*b1 - 12) * q^36 + (54*b2 + 2) * q^37 + (79*b3 + 241*b1) * q^38 + (17*b3 + 11*b2 - 124*b1 + 253) * q^39 + (-24*b3 + 24*b2 + 144*b1 - 120) * q^40 + (98*b3 - 98*b2 + 17*b1 - 115) * q^41 + (18*b3 - 60*b2 + 12*b1 - 162) * q^42 + (6*b3 - 47*b1) * q^43 + (79*b2 - 76) * q^44 + (-45*b3 + 45*b2 + 72*b1 - 171) * q^45 + (-12*b2 - 138) * q^46 + (-91*b3 - 154*b1) * q^47 + (7*b3 - 17*b2 + 145*b1 - 88) * q^48 + (21*b3 - 21*b2 - 267*b1 + 246) * q^49 + (-83*b3 + 83*b2 - 173*b1 + 256) * q^50 + (63*b3 + 36*b2 + 117*b1 + 18) * q^51 + (54*b3 + 358*b1) * q^52 + (-162*b2 - 54) * q^53 + (81*b3 + 54*b2 - 27*b1 + 324) * q^54 + (-93*b2 + 267) * q^55 + (-46*b3 - 10*b1) * q^56 + (-79*b3 + 2*b2 - 241*b1 - 164) * q^57 + (-24*b3 + 24*b2 - 18*b1 + 42) * q^58 + (-136*b3 + 136*b2 + 467*b1 - 331) * q^59 + (12*b3 + 24*b2 - 336*b1 + 168) * q^60 + (105*b3 - 272*b1) * q^61 + (26*b2 + 70) * q^62 + (-57*b3 + 78*b2 + 24*b1 + 192) * q^63 + (-153*b2 - 440) * q^64 + (107*b3 - 136*b1) * q^65 + (-48*b3 + 33*b2 + 222*b1 - 330) * q^66 + (66*b3 - 66*b2 + 461*b1 - 527) * q^67 + (171*b3 - 171*b2 + 261*b1 - 432) * q^68 + (45*b3 - 33*b2 - 204*b1 + 297) * q^69 + (-30*b3 - 144*b1) * q^70 + (144*b2 + 756) * q^71 + (27*b3 - 99*b2 + 333*b1) * q^72 + (243*b2 - 106) * q^73 + (-56*b3 - 380*b1) * q^74 + (121*b3 - 83*b2 + 187*b1 - 256) * q^75 + (-183*b3 + 183*b2 - 673*b1 + 856) * q^76 + (71*b3 - 71*b2 - 118*b1 + 47) * q^77 + (-174*b3 - 90*b2 - 342*b1 - 78) * q^78 + (-309*b3 + 556*b1) * q^79 + (-64*b2 + 16) * q^80 + (-135*b3 - 135*b2 + 351*b1 - 351) * q^81 + (213*b2 + 1014) * q^82 + (107*b3 - 460*b1) * q^83 + (110*b3 + 56*b2 + 218*b1 + 52) * q^84 + (18*b3 - 18*b2 + 288*b1 - 306) * q^85 + (35*b3 - 35*b2 - b1 - 34) * q^86 + (51*b3 - 24*b2 - 42*b1 - 42) * q^87 + (165*b3 - 693*b1) * q^88 + (72*b2 - 162) * q^89 + (144*b3 - 18*b2 + 144*b1 - 306) * q^90 + (-69*b2 - 425) * q^91 + (-2*b3 + 430*b1) * q^92 + (17*b3 - 43*b2 - 16*b1 - 71) * q^93 + (336*b3 - 336*b2 + 882*b1 - 1218) * q^94 + (-164*b3 + 164*b2 + 16*b1 + 148) * q^95 + (-198*b3 + 423*b2 + 342*b1 + 360) * q^96 + (-102*b3 - 317*b1) * q^97 + (-225*b2 - 324) * q^98 + (279*b3 - 81*b2 - 1080*b1 + 504) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{2} - 3 q^{3} - 5 q^{4} - 15 q^{5} + 9 q^{6} - 7 q^{7} + 66 q^{8} + 45 q^{9}+O(q^{10})$$ 4 * q - 3 * q^2 - 3 * q^3 - 5 * q^4 - 15 * q^5 + 9 * q^6 - 7 * q^7 + 66 * q^8 + 45 * q^9 $$4 q - 3 q^{2} - 3 q^{3} - 5 q^{4} - 15 q^{5} + 9 q^{6} - 7 q^{7} + 66 q^{8} + 45 q^{9} + 12 q^{10} - 66 q^{11} - 156 q^{12} + 11 q^{13} - 60 q^{14} + 27 q^{15} + 7 q^{16} + 198 q^{17} + 216 q^{18} - 154 q^{19} + 12 q^{20} + 21 q^{21} + 33 q^{22} - 33 q^{23} - 99 q^{24} + 121 q^{25} - 528 q^{26} - 432 q^{27} + 332 q^{28} + 51 q^{29} + 288 q^{30} - 43 q^{31} + 423 q^{32} + 198 q^{33} - 297 q^{34} + 6 q^{35} - 225 q^{36} - 100 q^{37} + 561 q^{38} + 759 q^{39} - 264 q^{40} - 132 q^{41} - 486 q^{42} - 88 q^{43} - 462 q^{44} - 675 q^{45} - 528 q^{46} - 399 q^{47} - 21 q^{48} + 513 q^{49} + 429 q^{50} + 297 q^{51} + 770 q^{52} + 108 q^{53} + 1215 q^{54} + 1254 q^{55} - 66 q^{56} - 1221 q^{57} + 60 q^{58} - 798 q^{59} - 36 q^{60} - 439 q^{61} + 228 q^{62} + 603 q^{63} - 1454 q^{64} - 165 q^{65} - 990 q^{66} - 988 q^{67} - 693 q^{68} + 891 q^{69} - 318 q^{70} + 2736 q^{71} + 891 q^{72} - 910 q^{73} - 816 q^{74} - 363 q^{75} + 1529 q^{76} + 165 q^{77} - 990 q^{78} + 803 q^{79} + 192 q^{80} - 567 q^{81} + 3630 q^{82} - 813 q^{83} + 642 q^{84} - 594 q^{85} - 33 q^{86} - 153 q^{87} - 1221 q^{88} - 792 q^{89} - 756 q^{90} - 1562 q^{91} + 858 q^{92} - 213 q^{93} - 2100 q^{94} + 132 q^{95} + 1080 q^{96} - 736 q^{97} - 846 q^{98} + 297 q^{99}+O(q^{100})$$ 4 * q - 3 * q^2 - 3 * q^3 - 5 * q^4 - 15 * q^5 + 9 * q^6 - 7 * q^7 + 66 * q^8 + 45 * q^9 + 12 * q^10 - 66 * q^11 - 156 * q^12 + 11 * q^13 - 60 * q^14 + 27 * q^15 + 7 * q^16 + 198 * q^17 + 216 * q^18 - 154 * q^19 + 12 * q^20 + 21 * q^21 + 33 * q^22 - 33 * q^23 - 99 * q^24 + 121 * q^25 - 528 * q^26 - 432 * q^27 + 332 * q^28 + 51 * q^29 + 288 * q^30 - 43 * q^31 + 423 * q^32 + 198 * q^33 - 297 * q^34 + 6 * q^35 - 225 * q^36 - 100 * q^37 + 561 * q^38 + 759 * q^39 - 264 * q^40 - 132 * q^41 - 486 * q^42 - 88 * q^43 - 462 * q^44 - 675 * q^45 - 528 * q^46 - 399 * q^47 - 21 * q^48 + 513 * q^49 + 429 * q^50 + 297 * q^51 + 770 * q^52 + 108 * q^53 + 1215 * q^54 + 1254 * q^55 - 66 * q^56 - 1221 * q^57 + 60 * q^58 - 798 * q^59 - 36 * q^60 - 439 * q^61 + 228 * q^62 + 603 * q^63 - 1454 * q^64 - 165 * q^65 - 990 * q^66 - 988 * q^67 - 693 * q^68 + 891 * q^69 - 318 * q^70 + 2736 * q^71 + 891 * q^72 - 910 * q^73 - 816 * q^74 - 363 * q^75 + 1529 * q^76 + 165 * q^77 - 990 * q^78 + 803 * q^79 + 192 * q^80 - 567 * q^81 + 3630 * q^82 - 813 * q^83 + 642 * q^84 - 594 * q^85 - 33 * q^86 - 153 * q^87 - 1221 * q^88 - 792 * q^89 - 756 * q^90 - 1562 * q^91 + 858 * q^92 - 213 * q^93 - 2100 * q^94 + 132 * q^95 + 1080 * q^96 - 736 * q^97 - 846 * q^98 + 297 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6$$ (v^3 + 2*v^2 - 2*v - 3) / 6 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 5\nu ) / 3$$ (-v^3 + v^2 + 5*v) / 3 $$\beta_{3}$$ $$=$$ $$( 2\nu^{3} + \nu^{2} + 2\nu - 9 ) / 3$$ (2*v^3 + v^2 + 2*v - 9) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3$$ (b3 + b2 - 2*b1 + 2) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 8\beta _1 + 1 ) / 3$$ (-b3 + 2*b2 + 8*b1 + 1) / 3 $$\nu^{3}$$ $$=$$ $$( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 11 ) / 3$$ (4*b3 - 2*b2 - 2*b1 + 11) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/9\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1 + \beta_{1}$$

## 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
 1.68614 + 0.396143i −1.18614 − 1.26217i 1.68614 − 0.396143i −1.18614 + 1.26217i
−2.18614 3.78651i 3.55842 + 3.78651i −5.55842 + 9.62747i −2.31386 + 4.00772i 6.55842 21.7518i −6.05842 10.4935i 13.6277 −1.67527 + 26.9480i 20.2337
4.2 0.686141 + 1.18843i −5.05842 1.18843i 3.05842 5.29734i −5.18614 + 8.98266i −2.05842 6.82701i 2.55842 + 4.43132i 19.3723 24.1753 + 12.0232i −14.2337
7.1 −2.18614 + 3.78651i 3.55842 3.78651i −5.55842 9.62747i −2.31386 4.00772i 6.55842 + 21.7518i −6.05842 + 10.4935i 13.6277 −1.67527 26.9480i 20.2337
7.2 0.686141 1.18843i −5.05842 + 1.18843i 3.05842 + 5.29734i −5.18614 8.98266i −2.05842 + 6.82701i 2.55842 4.43132i 19.3723 24.1753 12.0232i −14.2337
 $$n$$: 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.4.c.a 4
3.b odd 2 1 27.4.c.a 4
4.b odd 2 1 144.4.i.c 4
5.b even 2 1 225.4.e.b 4
5.c odd 4 2 225.4.k.b 8
9.c even 3 1 inner 9.4.c.a 4
9.c even 3 1 81.4.a.d 2
9.d odd 6 1 27.4.c.a 4
9.d odd 6 1 81.4.a.a 2
12.b even 2 1 432.4.i.c 4
36.f odd 6 1 144.4.i.c 4
36.f odd 6 1 1296.4.a.u 2
36.h even 6 1 432.4.i.c 4
36.h even 6 1 1296.4.a.i 2
45.h odd 6 1 2025.4.a.n 2
45.j even 6 1 225.4.e.b 4
45.j even 6 1 2025.4.a.g 2
45.k odd 12 2 225.4.k.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.c.a 4 1.a even 1 1 trivial
9.4.c.a 4 9.c even 3 1 inner
27.4.c.a 4 3.b odd 2 1
27.4.c.a 4 9.d odd 6 1
81.4.a.a 2 9.d odd 6 1
81.4.a.d 2 9.c even 3 1
144.4.i.c 4 4.b odd 2 1
144.4.i.c 4 36.f odd 6 1
225.4.e.b 4 5.b even 2 1
225.4.e.b 4 45.j even 6 1
225.4.k.b 8 5.c odd 4 2
225.4.k.b 8 45.k odd 12 2
432.4.i.c 4 12.b even 2 1
432.4.i.c 4 36.h even 6 1
1296.4.a.i 2 36.h even 6 1
1296.4.a.u 2 36.f odd 6 1
2025.4.a.g 2 45.j even 6 1
2025.4.a.n 2 45.h odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(9, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3 T^{3} + \cdots + 36$$
$3$ $$T^{4} + 3 T^{3} + \cdots + 729$$
$5$ $$T^{4} + 15 T^{3} + \cdots + 2304$$
$7$ $$T^{4} + 7 T^{3} + \cdots + 3844$$
$11$ $$T^{4} + 66 T^{3} + \cdots + 314721$$
$13$ $$T^{4} - 11 T^{3} + \cdots + 3334276$$
$17$ $$(T^{2} - 99 T + 1782)^{2}$$
$19$ $$(T^{2} + 77 T - 4532)^{2}$$
$23$ $$T^{4} + 33 T^{3} + \cdots + 7322436$$
$29$ $$T^{4} - 51 T^{3} + \cdots + 412164$$
$31$ $$T^{4} + 43 T^{3} + \cdots + 150544$$
$37$ $$(T^{2} + 50 T - 23432)^{2}$$
$41$ $$T^{4} + \cdots + 5606565129$$
$43$ $$T^{4} + 88 T^{3} + \cdots + 2686321$$
$47$ $$T^{4} + 399 T^{3} + \cdots + 813276324$$
$53$ $$(T^{2} - 54 T - 215784)^{2}$$
$59$ $$T^{4} + 798 T^{3} + \cdots + 43678881$$
$61$ $$T^{4} + \cdots + 1829786176$$
$67$ $$T^{4} + \cdots + 43305193801$$
$71$ $$(T^{2} - 1368 T + 296784)^{2}$$
$73$ $$(T^{2} + 455 T - 435398)^{2}$$
$79$ $$T^{4} + \cdots + 392522298256$$
$83$ $$T^{4} + \cdots + 5010940944$$
$89$ $$(T^{2} + 396 T - 3564)^{2}$$
$97$ $$T^{4} + \cdots + 2459267281$$