# Properties

 Label 108.4.h.a Level $108$ Weight $4$ Character orbit 108.h Analytic conductor $6.372$ Analytic rank $0$ Dimension $8$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,4,Mod(35,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 1]))

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

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

chi := DirichletCharacter("108.35");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 108.h (of order $$6$$, degree $$2$$, not 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: $$6.37220628062$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.553553856144.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 11x^{6} + 96x^{4} + 704x^{2} + 4096$$ x^8 + 11*x^6 + 96*x^4 + 704*x^2 + 4096 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{2} + (\beta_{6} - 2 \beta_{4} + \cdots + 2 \beta_{2}) q^{4}+ \cdots + (3 \beta_{7} + 5 \beta_{6} - 2 \beta_{5} + \cdots - 10) q^{8}+O(q^{10})$$ q + b7 * q^2 + (b6 - 2*b4 - 2*b3 + 2*b2) * q^4 + (b6 - 5*b4 + b3 + b1 - 10) * q^5 + (-2*b7 + 3*b6 - 7*b5 - 5*b3 - 3*b1) * q^7 + (3*b7 + 5*b6 - 2*b5 - 20*b4 - 2*b3 + 2*b2 - 6*b1 - 10) * q^8 $$q + \beta_{7} q^{2} + (\beta_{6} - 2 \beta_{4} + \cdots + 2 \beta_{2}) q^{4}+ \cdots + ( - 131 \beta_{7} - 57 \beta_{6} + \cdots - 370) q^{98}+O(q^{100})$$ q + b7 * q^2 + (b6 - 2*b4 - 2*b3 + 2*b2) * q^4 + (b6 - 5*b4 + b3 + b1 - 10) * q^5 + (-2*b7 + 3*b6 - 7*b5 - 5*b3 - 3*b1) * q^7 + (3*b7 + 5*b6 - 2*b5 - 20*b4 - 2*b3 + 2*b2 - 6*b1 - 10) * q^8 + (-6*b7 + 4*b6 - 2*b5 + 2*b3 - 2*b2 + 2*b1 - 10) * q^10 + (-9*b7 + b6 + b5 - 10*b3 + 2*b2 - b1) * q^11 + (-2*b7 - b6 - 55*b4 + b3 - b1) * q^13 + (-10*b7 + 2*b6 - 20*b5 + 2*b4 - 18*b3 - 10*b2 - 8*b1 + 4) * q^14 + (5*b7 + 24*b6 - 14*b5 + 26*b4 - 8*b3 - 2*b1 + 26) * q^16 + (-13*b7 - 13*b6 - 60*b4 - 13*b1 - 30) * q^17 + (35*b7 - 5*b6 + 30*b5 + 40*b3 + 30*b2 + 5*b1) * q^19 + (-18*b7 - 12*b6 - 12*b5 + 20*b4 + 16*b3 - 24*b2 + 12*b1 - 20) * q^20 + (2*b7 - 7*b6 - 60*b4 + 20*b3 - 16*b2 - 4*b1) * q^22 + (5*b7 - 21*b6 + 10*b5 + 26*b3 + 5*b2 + 21*b1) * q^23 + (-11*b7 - 22*b6 - 30*b4 - 11*b3 - 22*b1 - 30) * q^25 + (56*b7 + 54*b6 + 2*b5 + 20*b4 + 2*b3 - 2*b2 - 2*b1 + 10) * q^26 + (-30*b7 - 14*b6 - 44*b5 - 36*b3 - 44*b2 + 4*b1 + 60) * q^28 + (-23*b7 - 53*b4 + 23*b3 + 53) * q^29 + (-29*b7 + 7*b6 - 36*b3 + 43*b2 - 7*b1) * q^31 + (-38*b7 - 31*b6 - 76*b5 - 30*b4 - 18*b3 - 38*b2 + 20*b1 - 60) * q^32 + (43*b7 + 60*b6 + 26*b5 + 130*b4 - 26*b1 + 130) * q^34 + (-15*b7 - 70*b6 + 55*b5 + 55*b3 - 55*b2 + 70*b1) * q^35 + (2*b7 - 2*b6 - 4*b3 - 2*b1 - 160) * q^37 + (65*b7 + 70*b6 + 70*b5 + 90*b4 - 20*b3 + 140*b2 - 70*b1 - 90) * q^38 + (-64*b7 - 62*b6 + 140*b4 + 12*b3 - 60*b2 + 48*b1) * q^40 + (-50*b6 + 81*b4 - 50*b3 - 50*b1 + 162) * q^41 + (b7 - 51*b6 + 53*b5 + 52*b3 + 51*b1) * q^43 + (51*b7 + 37*b6 + 14*b5 + 220*b4 + 14*b3 - 14*b2 + 50*b1 + 110) * q^44 + (36*b7 + 26*b6 + 62*b5 - 22*b3 + 62*b2 - 42*b1 - 126) * q^46 + (10*b7 + 87*b6 + 87*b5 - 77*b3 + 174*b2 - 87*b1) * q^47 + (74*b7 + 37*b6 + 94*b4 - 37*b3 + 37*b1) * q^49 + (22*b7 + 41*b6 + 44*b5 + 110*b4 - 22*b3 + 22*b2 - 44*b1 + 220) * q^50 + (-64*b7 - 24*b6 - 104*b5 - 80*b4 + 8*b3 + 120*b1 - 80) * q^52 + (74*b7 + 74*b6 + 20*b4 + 74*b1 + 10) * q^53 + (43*b7 - 26*b6 + 17*b5 + 69*b3 + 17*b2 + 26*b1) * q^55 + (-14*b7 - 60*b6 - 60*b5 - 84*b4 - 56*b3 - 120*b2 + 60*b1 + 84) * q^56 + (106*b7 + 30*b6 + 230*b4 + 46*b3 - 46*b2) * q^58 + (3*b7 - 70*b6 + 6*b5 + 73*b3 + 3*b2 + 70*b1) * q^59 + (-71*b7 - 142*b6 - 41*b4 - 71*b3 - 142*b1 - 41) * q^61 + (36*b7 + 50*b6 - 14*b5 - 260*b4 - 14*b3 + 14*b2 - 158*b1 - 130) * q^62 + (-113*b7 + 23*b6 - 90*b5 - 214*b3 - 90*b2 - 62*b1 - 98) * q^64 + (-37*b7 + 255*b4 + 37*b3 - 255) * q^65 + (-51*b7 - 76*b6 + 25*b3 - 101*b2 + 76*b1) * q^67 + (-34*b7 - 173*b6 - 68*b5 + 70*b4 + 138*b3 - 34*b2 + 172*b1 + 140) * q^68 + (70*b7 - 110*b6 + 250*b5 - 90*b4 + 220*b3 + 190*b1 - 90) * q^70 + (80*b7 + 106*b6 - 26*b5 - 26*b3 + 26*b2 - 106*b1) * q^71 + (-85*b7 + 85*b6 + 170*b3 + 85*b1 + 620) * q^73 + (-158*b7 + 4*b6 + 4*b5 - 20*b4 - 8*b3 + 8*b2 - 4*b1 + 20) * q^74 + (-40*b7 + 115*b6 - 150*b4 + 10*b3 + 270*b2 - 280*b1) * q^76 + (45*b6 - 5*b4 + 45*b3 + 45*b1 - 10) * q^77 + (134*b7 + 341*b6 - 73*b5 - 207*b3 - 341*b1) * q^79 + (-138*b7 - 262*b6 + 124*b5 - 40*b4 + 124*b3 - 124*b2 + 116*b1 - 20) * q^80 + (131*b7 - 31*b6 + 100*b5 - 100*b3 + 100*b2 - 100*b1 + 500) * q^82 + (-30*b7 - 161*b6 - 161*b5 + 131*b3 - 322*b2 + 161*b1) * q^83 + (96*b7 + 48*b6 + 190*b4 - 48*b3 + 48*b1) * q^85 + (104*b7 - b6 + 208*b5 - 100*b4 + 108*b3 + 104*b2 + 4*b1 - 200) * q^86 + (-147*b7 - 248*b6 - 46*b5 - 390*b4 + 56*b3 + 158*b1 - 390) * q^88 + (-46*b7 - 46*b6 - 204*b4 - 46*b1 - 102) * q^89 + (-225*b7 - 100*b6 - 325*b5 - 125*b3 - 325*b2 + 100*b1) * q^91 + (-28*b7 + 72*b6 + 72*b5 - 160*b4 + 104*b3 + 144*b2 - 72*b1 + 160) * q^92 + (174*b7 + 184*b6 - 462*b4 + 154*b3 + 194*b2 - 348*b1) * q^94 + (-290*b7 + 200*b6 - 580*b5 - 490*b3 - 290*b2 - 200*b1) * q^95 + (92*b7 + 184*b6 - 5*b4 + 92*b3 + 184*b1 - 5) * q^97 + (-131*b7 - 57*b6 - 74*b5 - 740*b4 - 74*b3 + 74*b2 + 74*b1 - 370) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 3 q^{2} + 11 q^{4} - 66 q^{5}+O(q^{10})$$ 8 * q + 3 * q^2 + 11 * q^4 - 66 * q^5 $$8 q + 3 q^{2} + 11 q^{4} - 66 q^{5} - 116 q^{10} + 214 q^{13} + 42 q^{14} + 71 q^{16} - 306 q^{20} + 207 q^{22} - 54 q^{25} + 540 q^{28} + 498 q^{29} - 327 q^{32} + 469 q^{34} - 1256 q^{37} - 1035 q^{38} - 602 q^{40} + 1272 q^{41} - 912 q^{46} - 154 q^{49} + 1329 q^{50} - 464 q^{52} + 1314 q^{56} - 830 q^{58} + 262 q^{61} - 550 q^{64} - 3282 q^{65} + 843 q^{68} - 480 q^{70} + 3940 q^{73} - 222 q^{74} + 105 q^{76} - 330 q^{77} + 4786 q^{82} - 472 q^{85} - 1209 q^{86} - 1425 q^{88} + 1308 q^{92} + 1356 q^{94} - 572 q^{97}+O(q^{100})$$ 8 * q + 3 * q^2 + 11 * q^4 - 66 * q^5 - 116 * q^10 + 214 * q^13 + 42 * q^14 + 71 * q^16 - 306 * q^20 + 207 * q^22 - 54 * q^25 + 540 * q^28 + 498 * q^29 - 327 * q^32 + 469 * q^34 - 1256 * q^37 - 1035 * q^38 - 602 * q^40 + 1272 * q^41 - 912 * q^46 - 154 * q^49 + 1329 * q^50 - 464 * q^52 + 1314 * q^56 - 830 * q^58 + 262 * q^61 - 550 * q^64 - 3282 * q^65 + 843 * q^68 - 480 * q^70 + 3940 * q^73 - 222 * q^74 + 105 * q^76 - 330 * q^77 + 4786 * q^82 - 472 * q^85 - 1209 * q^86 - 1425 * q^88 + 1308 * q^92 + 1356 * q^94 - 572 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 11x^{6} + 96x^{4} + 704x^{2} + 4096$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{6} - 5\nu^{5} - \nu^{4} - 15\nu^{3} + 28\nu^{2} - 40\nu + 128 ) / 320$$ (v^6 - 5*v^5 - v^4 - 15*v^3 + 28*v^2 - 40*v + 128) / 320 $$\beta_{3}$$ $$=$$ $$( -\nu^{6} - 19\nu^{4} - 88\nu^{2} - 160\nu - 768 ) / 320$$ (-v^6 - 19*v^4 - 88*v^2 - 160*v - 768) / 320 $$\beta_{4}$$ $$=$$ $$( -3\nu^{7} - 17\nu^{5} + 16\nu^{3} - 704\nu - 2560 ) / 5120$$ (-3*v^7 - 17*v^5 + 16*v^3 - 704*v - 2560) / 5120 $$\beta_{5}$$ $$=$$ $$( -5\nu^{7} + 8\nu^{6} + 25\nu^{5} + 152\nu^{4} - 240\nu^{3} + 1984\nu^{2} - 320\nu + 8704 ) / 5120$$ (-5*v^7 + 8*v^6 + 25*v^5 + 152*v^4 - 240*v^3 + 1984*v^2 - 320*v + 8704) / 5120 $$\beta_{6}$$ $$=$$ $$( 5\nu^{7} - 24\nu^{6} + 55\nu^{5} - 136\nu^{4} + 480\nu^{3} + 128\nu^{2} + 3520\nu - 5632 ) / 5120$$ (5*v^7 - 24*v^6 + 55*v^5 - 136*v^4 + 480*v^3 + 128*v^2 + 3520*v - 5632) / 5120 $$\beta_{7}$$ $$=$$ $$( 5\nu^{7} + 24\nu^{6} + 55\nu^{5} + 136\nu^{4} + 480\nu^{3} - 128\nu^{2} + 3520\nu + 5632 ) / 5120$$ (5*v^7 + 24*v^6 + 55*v^5 + 136*v^4 + 480*v^3 - 128*v^2 + 3520*v + 5632) / 5120
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{6} + 2\beta_{5} + 2\beta_{2} - \beta _1 - 2$$ 2*b6 + 2*b5 + 2*b2 - b1 - 2 $$\nu^{3}$$ $$=$$ $$4\beta_{7} + 6\beta_{6} - 2\beta_{5} + 20\beta_{4} - 2\beta_{3} + 2\beta_{2} - 5\beta _1 + 10$$ 4*b7 + 6*b6 - 2*b5 + 20*b4 - 2*b3 + 2*b2 - 5*b1 + 10 $$\nu^{4}$$ $$=$$ $$-8\beta_{7} - 6\beta_{6} - 14\beta_{5} - 24\beta_{3} - 14\beta_{2} - 5\beta _1 - 26$$ -8*b7 - 6*b6 - 14*b5 - 24*b3 - 14*b2 - 5*b1 - 26 $$\nu^{5}$$ $$=$$ $$20\beta_{7} - 18\beta_{6} + 38\beta_{5} - 60\beta_{4} + 38\beta_{3} - 38\beta_{2} + 7\beta _1 - 30$$ 20*b7 - 18*b6 + 38*b5 - 60*b4 + 38*b3 - 38*b2 + 7*b1 - 30 $$\nu^{6}$$ $$=$$ $$152\beta_{7} - 62\beta_{6} + 90\beta_{5} + 136\beta_{3} + 90\beta_{2} + 23\beta _1 - 98$$ 152*b7 - 62*b6 + 90*b5 + 136*b3 + 90*b2 + 23*b1 - 98 $$\nu^{7}$$ $$=$$ $$-92\beta_{7} + 134\beta_{6} - 226\beta_{5} - 1260\beta_{4} - 226\beta_{3} + 226\beta_{2} - 301\beta _1 - 630$$ -92*b7 + 134*b6 - 226*b5 - 1260*b4 - 226*b3 + 226*b2 - 301*b1 - 630

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$55$$ $$\chi(n)$$ $$-\beta_{4}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
35.1
 2.14417 − 1.84460i −2.14417 − 1.84460i 0.807795 + 2.71062i −0.807795 + 2.71062i 2.14417 + 1.84460i −2.14417 + 1.84460i 0.807795 − 2.71062i −0.807795 − 2.71062i
−2.66955 + 0.934608i 0 6.25302 4.98997i −4.30507 + 2.48553i 0 3.07102 + 1.77306i −12.0291 + 19.1651i 0 9.16960 10.6588i
35.2 −0.525382 2.77920i 0 −7.44795 + 2.92028i −4.30507 + 2.48553i 0 −3.07102 1.77306i 12.0291 + 19.1651i 0 9.16960 + 10.6588i
35.3 1.94357 + 2.05488i 0 −0.445079 + 7.98761i −12.1949 + 7.04075i 0 −21.1499 12.2109i −17.2786 + 14.6099i 0 −38.1696 11.3750i
35.4 2.75136 + 0.655739i 0 7.14001 + 3.60836i −12.1949 + 7.04075i 0 21.1499 + 12.2109i 17.2786 + 14.6099i 0 −38.1696 + 11.3750i
71.1 −2.66955 0.934608i 0 6.25302 + 4.98997i −4.30507 2.48553i 0 3.07102 1.77306i −12.0291 19.1651i 0 9.16960 + 10.6588i
71.2 −0.525382 + 2.77920i 0 −7.44795 2.92028i −4.30507 2.48553i 0 −3.07102 + 1.77306i 12.0291 19.1651i 0 9.16960 10.6588i
71.3 1.94357 2.05488i 0 −0.445079 7.98761i −12.1949 7.04075i 0 −21.1499 + 12.2109i −17.2786 14.6099i 0 −38.1696 + 11.3750i
71.4 2.75136 0.655739i 0 7.14001 3.60836i −12.1949 7.04075i 0 21.1499 12.2109i 17.2786 14.6099i 0 −38.1696 11.3750i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 35.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.4.h.a 8
3.b odd 2 1 36.4.h.a 8
4.b odd 2 1 inner 108.4.h.a 8
9.c even 3 1 36.4.h.a 8
9.c even 3 1 324.4.b.b 8
9.d odd 6 1 inner 108.4.h.a 8
9.d odd 6 1 324.4.b.b 8
12.b even 2 1 36.4.h.a 8
36.f odd 6 1 36.4.h.a 8
36.f odd 6 1 324.4.b.b 8
36.h even 6 1 inner 108.4.h.a 8
36.h even 6 1 324.4.b.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.4.h.a 8 3.b odd 2 1
36.4.h.a 8 9.c even 3 1
36.4.h.a 8 12.b even 2 1
36.4.h.a 8 36.f odd 6 1
108.4.h.a 8 1.a even 1 1 trivial
108.4.h.a 8 4.b odd 2 1 inner
108.4.h.a 8 9.d odd 6 1 inner
108.4.h.a 8 36.h even 6 1 inner
324.4.b.b 8 9.c even 3 1
324.4.b.b 8 9.d odd 6 1
324.4.b.b 8 36.f odd 6 1
324.4.b.b 8 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 33T_{5}^{3} + 433T_{5}^{2} + 2310T_{5} + 4900$$ acting on $$S_{4}^{\mathrm{new}}(108, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 3 T^{7} + \cdots + 4096$$
$3$ $$T^{8}$$
$5$ $$(T^{4} + 33 T^{3} + \cdots + 4900)^{2}$$
$7$ $$T^{8} - 609 T^{6} + \cdots + 56250000$$
$11$ $$T^{8} + \cdots + 44383955625$$
$13$ $$(T^{4} - 107 T^{3} + \cdots + 7840000)^{2}$$
$17$ $$(T^{4} + 10327 T^{2} + 3422500)^{2}$$
$19$ $$(T^{4} + 27675 T^{2} + 149107500)^{2}$$
$23$ $$T^{8} + \cdots + 732895782914304$$
$29$ $$(T^{4} - 249 T^{3} + \cdots + 33756100)^{2}$$
$31$ $$T^{8} + \cdots + 75\!\cdots\!00$$
$37$ $$(T^{2} + 314 T + 24400)^{4}$$
$41$ $$(T^{4} - 636 T^{3} + \cdots + 330039889)^{2}$$
$43$ $$T^{8} + \cdots + 94\!\cdots\!25$$
$47$ $$T^{8} + \cdots + 94\!\cdots\!24$$
$53$ $$(T^{4} + 231628 T^{2} + 12418873600)^{2}$$
$59$ $$T^{8} + \cdots + 22\!\cdots\!25$$
$61$ $$(T^{4} - 131 T^{3} + \cdots + 95797678144)^{2}$$
$67$ $$T^{8} + \cdots + 44\!\cdots\!29$$
$71$ $$(T^{4} - 171216 T^{2} + 7279642800)^{2}$$
$73$ $$(T^{2} - 985 T - 207200)^{4}$$
$79$ $$T^{8} + \cdots + 34\!\cdots\!00$$
$83$ $$T^{8} + \cdots + 11\!\cdots\!64$$
$89$ $$(T^{4} + 125260 T^{2} + 634233856)^{2}$$
$97$ $$(T^{4} + 286 T^{3} + \cdots + 256476409225)^{2}$$