Properties

 Label 36.4.e.a Level $36$ Weight $4$ Character orbit 36.e Analytic conductor $2.124$ 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] = [36,4,Mod(13,36)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(36, base_ring=CyclotomicField(6))

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("36.13");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$36 = 2^{2} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 36.e (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: $$2.12406876021$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.6831243.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 13x^{4} + 49x^{2} + 48$$ x^6 + 13*x^4 + 49*x^2 + 48 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{3}$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 1) q^{3} + (\beta_{5} - \beta_{3} - 2 \beta_{2} + \cdots + 1) q^{5}+ \cdots + (\beta_{5} + 3 \beta_{4} + \beta_{2} + \cdots + 7) q^{9}+O(q^{10})$$ q + (-b2 - 1) * q^3 + (b5 - b3 - 2*b2 - 3*b1 + 1) * q^5 + (-2*b4 + b3 - 3*b2 - 3*b1 - 1) * q^7 + (b5 + 3*b4 + b2 - b1 + 7) * q^9 $$q + ( - \beta_{2} - 1) q^{3} + (\beta_{5} - \beta_{3} - 2 \beta_{2} + \cdots + 1) q^{5}+ \cdots + ( - 168 \beta_{5} + 9 \beta_{4} + \cdots - 15) q^{99}+O(q^{100})$$ q + (-b2 - 1) * q^3 + (b5 - b3 - 2*b2 - 3*b1 + 1) * q^5 + (-2*b4 + b3 - 3*b2 - 3*b1 - 1) * q^7 + (b5 + 3*b4 + b2 - b1 + 7) * q^9 + (-5*b5 + b4 - 3*b3 + 4*b2 + 20*b1 + 3) * q^11 + (3*b5 - 4*b4 + 5*b3 + 6*b2 - 3*b1 + 5) * q^13 + (-3*b5 + 3*b4 + 3*b3 + 3*b2 - 18*b1 - 18) * q^15 + (8*b5 + b4 - 7*b3 + 5*b2 - b1 - 38) * q^17 + (-12*b5 - 7*b4 + 5*b3 + 9*b2 + 7*b1 + 12) * q^19 + (b5 + 9*b4 + 5*b2 + 80*b1 - 58) * q^21 + (-b5 + 3*b4 - 5*b3 - 7*b2 - 72*b1 + 68) * q^23 + (9*b5 - 3*b4 + 6*b3 - 9*b2 - 7*b1 - 6) * q^25 + (-3*b5 - 6*b4 + 3*b3 - 9*b2 - 72*b1 + 141) * q^27 + (-5*b5 - 5*b4 - 5*b2 + 152*b1) * q^29 + (9*b5 + 3*b4 - 15*b3 - 27*b2 - 28*b1 + 4) * q^31 + (18*b5 + 3*b4 - 15*b3 - 21*b2 - 156*b1 - 42) * q^33 + (-13*b5 + b4 + 14*b3 - 16*b2 - b1 - 185) * q^35 + (12*b5 + 10*b4 - 2*b3 - 18*b2 - 10*b1 - 26) * q^37 + (-7*b5 - 27*b4 + 9*b3 - 3*b2 + 268*b1 - 138) * q^39 + (10*b5 - 12*b4 + 14*b3 + 16*b2 - 297*b1 + 301) * q^41 + (-27*b5 + 27*b4 - 27*b3 + 54*b2 + 70*b1 + 27) * q^43 + (-15*b5 - 9*b3 + 30*b2 - 3*b1 + 273) * q^45 + (60*b5 + 18*b4 + 21*b3 - 3*b2 + 153*b1 - 21) * q^47 + (-57*b5 + 22*b4 + 13*b3 + 48*b2 + 110*b1 - 40) * q^49 + (-21*b5 - 39*b4 + 24*b3 + 45*b2 - 147*b1 - 168) * q^51 + (-44*b5 - 22*b4 + 22*b3 + 22*b2 + 22*b1 - 346) * q^53 + (81*b5 + 33*b4 - 48*b3 - 18*b2 - 33*b1 - 69) * q^55 + (15*b5 + 9*b4 - 36*b3 - 17*b2 + 237*b1 - 242) * q^57 + (-64*b5 + 15*b4 + 34*b3 + 83*b2 - 102*b1 + 200) * q^59 + (9*b5 - 15*b4 + 12*b3 - 27*b2 - 146*b1 - 12) * q^61 + (74*b5 - 18*b4 + 3*b3 - 31*b2 - 311*b1 + 419) * q^63 + (-31*b5 - 43*b4 + 6*b3 - 49*b2 + 364*b1 - 6) * q^65 + (18*b5 + 3*b4 - 24*b3 - 45*b2 + 50*b1 - 92) * q^67 + (-69*b5 + 24*b4 - 3*b3 + 6*b2 - 153*b1 + 81) * q^69 + (92*b5 + 52*b4 - 40*b3 - 64*b2 - 52*b1 - 32) * q^71 + (-72*b5 - 21*b4 + 51*b3 - 9*b2 + 21*b1 + 146) * q^73 + (-b5 + 27*b3 + 10*b2 + 298*b1 + 10) * q^75 + (89*b5 - 12*b4 - 65*b3 - 142*b2 - 453*b1 + 299) * q^77 + (-18*b5 - 28*b4 + 5*b3 - 33*b2 + 179*b1 - 5) * q^79 + (-57*b5 + 36*b4 - 9*b3 - 66*b2 + 282*b1 - 237) * q^81 + (-48*b5 + 18*b4 - 33*b3 + 51*b2 - 171*b1 + 33) * q^83 + (90*b5 - 60*b4 + 30*b3 - 426*b1 + 366) * q^85 + (162*b5 + 30*b4 - 15*b3 - 147*b2 - 57*b1 - 357) * q^87 + (-28*b5 - 44*b4 - 16*b3 + 104*b2 + 44*b1 - 110) * q^89 + (-117*b5 - 75*b4 + 42*b3 + 108*b2 + 75*b1 + 31) * q^91 + (-25*b5 + 54*b4 + 27*b3 + 36*b2 - 407*b1 - 99) * q^93 + (-4*b5 + 24*b4 - 44*b3 - 64*b2 + 708*b1 - 748) * q^95 + (126*b5 + 14*b4 + 56*b3 - 42*b2 - 25*b1 - 56) * q^97 + (-168*b5 + 9*b4 + 54*b3 + 210*b2 - 210*b1 - 15) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{3} + 6 q^{5} - 6 q^{7} + 39 q^{9}+O(q^{10})$$ 6 * q - 3 * q^3 + 6 * q^5 - 6 * q^7 + 39 * q^9 $$6 q - 3 q^{3} + 6 q^{5} - 6 q^{7} + 39 q^{9} + 51 q^{11} + 12 q^{13} - 180 q^{15} - 222 q^{17} + 30 q^{19} - 120 q^{21} + 210 q^{23} - 3 q^{25} + 648 q^{27} + 456 q^{29} + 48 q^{31} - 603 q^{33} - 1104 q^{35} - 96 q^{37} - 36 q^{39} + 897 q^{41} + 129 q^{43} + 1494 q^{45} + 522 q^{47} - 225 q^{49} - 1647 q^{51} - 2208 q^{53} - 216 q^{55} - 645 q^{57} + 453 q^{59} - 402 q^{61} + 1896 q^{63} + 1110 q^{65} - 213 q^{67} - 198 q^{69} + 120 q^{71} + 750 q^{73} + 921 q^{75} + 1128 q^{77} + 552 q^{79} - 549 q^{81} - 612 q^{83} + 1188 q^{85} - 1386 q^{87} - 924 q^{89} - 264 q^{91} - 1998 q^{93} - 2184 q^{95} + 93 q^{97} - 1854 q^{99}+O(q^{100})$$ 6 * q - 3 * q^3 + 6 * q^5 - 6 * q^7 + 39 * q^9 + 51 * q^11 + 12 * q^13 - 180 * q^15 - 222 * q^17 + 30 * q^19 - 120 * q^21 + 210 * q^23 - 3 * q^25 + 648 * q^27 + 456 * q^29 + 48 * q^31 - 603 * q^33 - 1104 * q^35 - 96 * q^37 - 36 * q^39 + 897 * q^41 + 129 * q^43 + 1494 * q^45 + 522 * q^47 - 225 * q^49 - 1647 * q^51 - 2208 * q^53 - 216 * q^55 - 645 * q^57 + 453 * q^59 - 402 * q^61 + 1896 * q^63 + 1110 * q^65 - 213 * q^67 - 198 * q^69 + 120 * q^71 + 750 * q^73 + 921 * q^75 + 1128 * q^77 + 552 * q^79 - 549 * q^81 - 612 * q^83 + 1188 * q^85 - 1386 * q^87 - 924 * q^89 - 264 * q^91 - 1998 * q^93 - 2184 * q^95 + 93 * q^97 - 1854 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 13x^{4} + 49x^{2} + 48$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{5} + 9\nu^{3} + 17\nu + 4 ) / 8$$ (v^5 + 9*v^3 + 17*v + 4) / 8 $$\beta_{2}$$ $$=$$ $$( -\nu^{5} - 4\nu^{4} - 5\nu^{3} - 20\nu^{2} + 15\nu + 4 ) / 8$$ (-v^5 - 4*v^4 - 5*v^3 - 20*v^2 + 15*v + 4) / 8 $$\beta_{3}$$ $$=$$ $$( -\nu^{5} - 9\nu^{3} - 12\nu^{2} - 5\nu - 52 ) / 4$$ (-v^5 - 9*v^3 - 12*v^2 - 5*v - 52) / 4 $$\beta_{4}$$ $$=$$ $$( 3\nu^{4} + 3\nu^{3} + 21\nu^{2} + 18\nu + 20 ) / 2$$ (3*v^4 + 3*v^3 + 21*v^2 + 18*v + 20) / 2 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} + 4\nu^{4} - 13\nu^{3} + 44\nu^{2} - 25\nu + 100 ) / 8$$ (-v^5 + 4*v^4 - 13*v^3 + 44*v^2 - 25*v + 100) / 8
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{3} + \beta_{2} + 4\beta _1 - 2 ) / 6$$ (b5 + b3 + b2 + 4*b1 - 2) / 6 $$\nu^{2}$$ $$=$$ $$( \beta_{5} - \beta_{3} + \beta_{2} - 26 ) / 6$$ (b5 - b3 + b2 - 26) / 6 $$\nu^{3}$$ $$=$$ $$( -4\beta_{5} + \beta_{4} - 3\beta_{3} - \beta_{2} - 11\beta _1 + 7 ) / 3$$ (-4*b5 + b4 - 3*b3 - b2 - 11*b1 + 7) / 3 $$\nu^{4}$$ $$=$$ $$( -5\beta_{5} + 2\beta_{4} + 7\beta_{3} - 11\beta_{2} - 2\beta _1 + 140 ) / 6$$ (-5*b5 + 2*b4 + 7*b3 - 11*b2 - 2*b1 + 140) / 6 $$\nu^{5}$$ $$=$$ $$( 55\beta_{5} - 18\beta_{4} + 37\beta_{3} + \beta_{2} + 178\beta _1 - 116 ) / 6$$ (55*b5 - 18*b4 + 37*b3 + b2 + 178*b1 - 116) / 6

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/36\mathbb{Z}\right)^\times$$.

 $$n$$ $$19$$ $$29$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{1}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 1.23396i − 2.13353i 2.63162i − 1.23396i 2.13353i − 2.63162i
0 −4.14739 3.13036i 0 6.92194 11.9892i 0 −15.3540 26.5939i 0 7.40171 + 25.9656i 0
13.2 0 −2.51979 + 4.54430i 0 −6.37096 + 11.0348i 0 7.02674 + 12.1707i 0 −14.3013 22.9014i 0
13.3 0 5.16718 0.547914i 0 2.44901 4.24182i 0 5.32725 + 9.22708i 0 26.3996 5.66234i 0
25.1 0 −4.14739 + 3.13036i 0 6.92194 + 11.9892i 0 −15.3540 + 26.5939i 0 7.40171 25.9656i 0
25.2 0 −2.51979 4.54430i 0 −6.37096 11.0348i 0 7.02674 12.1707i 0 −14.3013 + 22.9014i 0
25.3 0 5.16718 + 0.547914i 0 2.44901 + 4.24182i 0 5.32725 9.22708i 0 26.3996 + 5.66234i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 13.3 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 36.4.e.a 6
3.b odd 2 1 108.4.e.a 6
4.b odd 2 1 144.4.i.d 6
9.c even 3 1 inner 36.4.e.a 6
9.c even 3 1 324.4.a.c 3
9.d odd 6 1 108.4.e.a 6
9.d odd 6 1 324.4.a.d 3
12.b even 2 1 432.4.i.d 6
36.f odd 6 1 144.4.i.d 6
36.f odd 6 1 1296.4.a.v 3
36.h even 6 1 432.4.i.d 6
36.h even 6 1 1296.4.a.w 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.4.e.a 6 1.a even 1 1 trivial
36.4.e.a 6 9.c even 3 1 inner
108.4.e.a 6 3.b odd 2 1
108.4.e.a 6 9.d odd 6 1
144.4.i.d 6 4.b odd 2 1
144.4.i.d 6 36.f odd 6 1
324.4.a.c 3 9.c even 3 1
324.4.a.d 3 9.d odd 6 1
432.4.i.d 6 12.b even 2 1
432.4.i.d 6 36.h even 6 1
1296.4.a.v 3 36.f odd 6 1
1296.4.a.w 3 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(36, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 3 T^{5} + \cdots + 19683$$
$5$ $$T^{6} - 6 T^{5} + \cdots + 746496$$
$7$ $$T^{6} + 6 T^{5} + \cdots + 21141604$$
$11$ $$T^{6} + \cdots + 4386545361$$
$13$ $$T^{6} + \cdots + 4155865156$$
$17$ $$(T^{3} + 111 T^{2} + \cdots - 577476)^{2}$$
$19$ $$(T^{3} - 15 T^{2} + \cdots - 216368)^{2}$$
$23$ $$T^{6} - 210 T^{5} + \cdots + 5391684$$
$29$ $$T^{6} + \cdots + 8292430196964$$
$31$ $$T^{6} + \cdots + 9331729724944$$
$37$ $$(T^{3} + 48 T^{2} + \cdots - 682352)^{2}$$
$41$ $$T^{6} + \cdots + 139158426750849$$
$43$ $$T^{6} + \cdots + 2031049672201$$
$47$ $$T^{6} + \cdots + 41\!\cdots\!24$$
$53$ $$(T^{3} + 1104 T^{2} + \cdots + 11853648)^{2}$$
$59$ $$T^{6} + \cdots + 93\!\cdots\!69$$
$61$ $$T^{6} + \cdots + 1463461189696$$
$67$ $$T^{6} + \cdots + 9579414973969$$
$71$ $$(T^{3} - 60 T^{2} + \cdots + 113211648)^{2}$$
$73$ $$(T^{3} - 375 T^{2} + \cdots + 158369284)^{2}$$
$79$ $$T^{6} + \cdots + 318578661907984$$
$83$ $$T^{6} + \cdots + 12101965948944$$
$89$ $$(T^{3} + 462 T^{2} + \cdots + 170122248)^{2}$$
$97$ $$T^{6} + \cdots + 74\!\cdots\!29$$