# Properties

 Label 405.4.e.r Level $405$ Weight $4$ Character orbit 405.e Analytic conductor $23.896$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(136,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("405.136");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 405.e (of order $$3$$, 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: $$23.8957735523$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.95327307.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} + 20x^{4} - 35x^{3} + 85x^{2} - 68x + 16$$ x^6 - 3*x^5 + 20*x^4 - 35*x^3 + 85*x^2 - 68*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{3}$$ Twist minimal: no (minimal twist has level 135) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + (\beta_{5} - \beta_{4} - 7 \beta_{3} - \beta_1) q^{4} + 5 \beta_{3} q^{5} + (2 \beta_{4} + 14 \beta_{3} - 14) q^{7} + (\beta_{2} + 8 \beta_1 + 9) q^{8}+O(q^{10})$$ q + b4 * q^2 + (b5 - b4 - 7*b3 - b1) * q^4 + 5*b3 * q^5 + (2*b4 + 14*b3 - 14) * q^7 + (b2 + 8*b1 + 9) * q^8 $$q + \beta_{4} q^{2} + (\beta_{5} - \beta_{4} - 7 \beta_{3} - \beta_1) q^{4} + 5 \beta_{3} q^{5} + (2 \beta_{4} + 14 \beta_{3} - 14) q^{7} + (\beta_{2} + 8 \beta_1 + 9) q^{8} - 5 \beta_1 q^{10} + (4 \beta_{5} + 2 \beta_{4} - 12 \beta_{3} + 4 \beta_{2} + 12) q^{11} + ( - 4 \beta_{5} + 10 \beta_{4} - 14 \beta_{3} + 10 \beta_1) q^{13} + (2 \beta_{5} - 16 \beta_{4} - 30 \beta_{3} - 16 \beta_1) q^{14} + (17 \beta_{4} + 58 \beta_{3} - 58) q^{16} + (8 \beta_{2} + 2 \beta_1 + 3) q^{17} + ( - 4 \beta_{2} + 14 \beta_1 + 59) q^{19} + (5 \beta_{5} - 5 \beta_{4} - 35 \beta_{3} + 5 \beta_{2} + 35) q^{20} + (2 \beta_{5} + 42 \beta_{4} - 54 \beta_{3} + 42 \beta_1) q^{22} + ( - 4 \beta_{5} + 32 \beta_{4} - 39 \beta_{3} + 32 \beta_1) q^{23} + (25 \beta_{3} - 25) q^{25} + ( - 10 \beta_{2} - 28 \beta_1 - 126) q^{26} + (16 \beta_{2} + 46 \beta_1 + 116) q^{28} + ( - 8 \beta_{5} - 42 \beta_{4} - 42 \beta_{3} - 8 \beta_{2} + 42) q^{29} + (8 \beta_{5} + 30 \beta_{4} - 83 \beta_{3} + 30 \beta_1) q^{31} + (9 \beta_{5} - 11 \beta_{4} - 183 \beta_{3} - 11 \beta_1) q^{32} + ( - 2 \beta_{5} + 69 \beta_{4} - 18 \beta_{3} - 2 \beta_{2} + 18) q^{34} + ( - 10 \beta_1 - 70) q^{35} + ( - 8 \beta_{2} - 64 \beta_1 + 50) q^{37} + ( - 14 \beta_{5} + 41 \beta_{4} + 234 \beta_{3} - 14 \beta_{2} + \cdots - 234) q^{38}+ \cdots + (60 \beta_{2} + 5 \beta_1 + 876) q^{98}+O(q^{100})$$ q + b4 * q^2 + (b5 - b4 - 7*b3 - b1) * q^4 + 5*b3 * q^5 + (2*b4 + 14*b3 - 14) * q^7 + (b2 + 8*b1 + 9) * q^8 - 5*b1 * q^10 + (4*b5 + 2*b4 - 12*b3 + 4*b2 + 12) * q^11 + (-4*b5 + 10*b4 - 14*b3 + 10*b1) * q^13 + (2*b5 - 16*b4 - 30*b3 - 16*b1) * q^14 + (17*b4 + 58*b3 - 58) * q^16 + (8*b2 + 2*b1 + 3) * q^17 + (-4*b2 + 14*b1 + 59) * q^19 + (5*b5 - 5*b4 - 35*b3 + 5*b2 + 35) * q^20 + (2*b5 + 42*b4 - 54*b3 + 42*b1) * q^22 + (-4*b5 + 32*b4 - 39*b3 + 32*b1) * q^23 + (25*b3 - 25) * q^25 + (-10*b2 - 28*b1 - 126) * q^26 + (16*b2 + 46*b1 + 116) * q^28 + (-8*b5 - 42*b4 - 42*b3 - 8*b2 + 42) * q^29 + (8*b5 + 30*b4 - 83*b3 + 30*b1) * q^31 + (9*b5 - 11*b4 - 183*b3 - 11*b1) * q^32 + (-2*b5 + 69*b4 - 18*b3 - 2*b2 + 18) * q^34 + (-10*b1 - 70) * q^35 + (-8*b2 - 64*b1 + 50) * q^37 + (-14*b5 + 41*b4 + 234*b3 - 14*b2 - 234) * q^38 + (-5*b5 + 40*b4 + 45*b3 + 40*b1) * q^40 + (-16*b5 + 42*b4 - 132*b3 + 42*b1) * q^41 + (8*b5 + 78*b4 - 16*b3 + 8*b2 + 16) * q^43 + (-10*b2 + 12*b1 - 546) * q^44 + (-32*b2 - 25*b1 - 456) * q^46 + (4*b5 - 28*b4 + 168*b3 + 4*b2 - 168) * q^47 + (4*b5 - 60*b4 + 87*b3 - 60*b1) * q^49 + (-25*b4 - 25*b1) * q^50 + (-4*b5 - 154*b4 - 472*b3 - 4*b2 + 472) * q^52 + (-12*b2 - 14*b1 - 165) * q^53 + (20*b2 - 10*b1 + 60) * q^55 + (-30*b5 + 162*b4 + 354*b3 - 30*b2 - 354) * q^56 + (-42*b5 + 20*b4 + 678*b3 + 20*b1) * q^58 + (12*b5 - 82*b4 + 78*b3 - 82*b1) * q^59 + (-16*b5 + 60*b4 + 173*b3 - 16*b2 - 173) * q^61 + (-30*b2 + 117*b1 - 498) * q^62 + (11*b2 + 130*b1 - 353) * q^64 + (-20*b5 + 50*b4 - 70*b3 - 20*b2 + 70) * q^65 + (24*b5 + 52*b4 - 302*b3 + 52*b1) * q^67 + (5*b5 - 51*b4 - 999*b3 - 51*b1) * q^68 + (10*b5 - 80*b4 - 150*b3 + 10*b2 + 150) * q^70 + (60*b2 - 34*b1 + 192) * q^71 + (60*b2 + 22*b1 + 404) * q^73 + (64*b5 - 78*b4 - 912*b3 + 64*b2 + 912) * q^74 + (73*b5 - 275*b4 - 59*b3 - 275*b1) * q^76 + (-52*b5 + 56*b4 + 60*b3 + 56*b1) * q^77 + (12*b5 - 22*b4 + 221*b3 + 12*b2 - 221) * q^79 + (-85*b1 - 290) * q^80 + (-42*b2 - 38*b1 - 534) * q^82 + (60*b5 - 120*b4 + 489*b3 + 60*b2 - 489) * q^83 + (-40*b5 + 10*b4 + 15*b3 + 10*b1) * q^85 + (78*b5 + 2*b4 - 1218*b3 + 2*b1) * q^86 + (4*b5 - 278*b4 - 192*b3 + 4*b2 + 192) * q^88 + (48*b2 + 66*b1 - 756) * q^89 + (-76*b2 - 196*b1 - 56) * q^91 + (-7*b5 - 481*b4 - 495*b3 - 7*b2 + 495) * q^92 + (-28*b5 - 108*b4 + 396*b3 - 108*b1) * q^94 + (20*b5 + 70*b4 + 295*b3 + 70*b1) * q^95 + (-8*b5 + 64*b4 + 440*b3 - 8*b2 - 440) * q^97 + (60*b2 + 5*b1 + 876) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - q^{2} - 23 q^{4} + 15 q^{5} - 44 q^{7} + 72 q^{8}+O(q^{10})$$ 6 * q - q^2 - 23 * q^4 + 15 * q^5 - 44 * q^7 + 72 * q^8 $$6 q - q^{2} - 23 q^{4} + 15 q^{5} - 44 q^{7} + 72 q^{8} - 10 q^{10} + 38 q^{11} - 28 q^{13} - 108 q^{14} - 191 q^{16} + 38 q^{17} + 374 q^{19} + 115 q^{20} - 122 q^{22} - 81 q^{23} - 75 q^{25} - 832 q^{26} + 820 q^{28} + 160 q^{29} - 227 q^{31} - 569 q^{32} - 17 q^{34} - 440 q^{35} + 156 q^{37} - 757 q^{38} + 180 q^{40} - 338 q^{41} - 22 q^{43} - 3272 q^{44} - 2850 q^{46} - 472 q^{47} + 197 q^{49} - 25 q^{50} + 1566 q^{52} - 1042 q^{53} + 380 q^{55} - 1254 q^{56} + 2096 q^{58} + 140 q^{59} - 595 q^{61} - 2814 q^{62} - 1836 q^{64} + 140 q^{65} - 878 q^{67} - 3053 q^{68} + 540 q^{70} + 1204 q^{71} + 2588 q^{73} + 2878 q^{74} - 525 q^{76} + 288 q^{77} - 629 q^{79} - 1910 q^{80} - 3364 q^{82} - 1287 q^{83} + 95 q^{85} - 3730 q^{86} + 858 q^{88} - 4308 q^{89} - 880 q^{91} + 1959 q^{92} + 1108 q^{94} + 935 q^{95} - 1392 q^{97} + 5386 q^{98}+O(q^{100})$$ 6 * q - q^2 - 23 * q^4 + 15 * q^5 - 44 * q^7 + 72 * q^8 - 10 * q^10 + 38 * q^11 - 28 * q^13 - 108 * q^14 - 191 * q^16 + 38 * q^17 + 374 * q^19 + 115 * q^20 - 122 * q^22 - 81 * q^23 - 75 * q^25 - 832 * q^26 + 820 * q^28 + 160 * q^29 - 227 * q^31 - 569 * q^32 - 17 * q^34 - 440 * q^35 + 156 * q^37 - 757 * q^38 + 180 * q^40 - 338 * q^41 - 22 * q^43 - 3272 * q^44 - 2850 * q^46 - 472 * q^47 + 197 * q^49 - 25 * q^50 + 1566 * q^52 - 1042 * q^53 + 380 * q^55 - 1254 * q^56 + 2096 * q^58 + 140 * q^59 - 595 * q^61 - 2814 * q^62 - 1836 * q^64 + 140 * q^65 - 878 * q^67 - 3053 * q^68 + 540 * q^70 + 1204 * q^71 + 2588 * q^73 + 2878 * q^74 - 525 * q^76 + 288 * q^77 - 629 * q^79 - 1910 * q^80 - 3364 * q^82 - 1287 * q^83 + 95 * q^85 - 3730 * q^86 + 858 * q^88 - 4308 * q^89 - 880 * q^91 + 1959 * q^92 + 1108 * q^94 + 935 * q^95 - 1392 * q^97 + 5386 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} + 20x^{4} - 35x^{3} + 85x^{2} - 68x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{4} - 2\nu^{3} + 14\nu^{2} - 13\nu + 24 ) / 4$$ (v^4 - 2*v^3 + 14*v^2 - 13*v + 24) / 4 $$\beta_{2}$$ $$=$$ $$( -\nu^{4} + 2\nu^{3} - 8\nu^{2} + 7\nu + 12 ) / 2$$ (-v^4 + 2*v^3 - 8*v^2 + 7*v + 12) / 2 $$\beta_{3}$$ $$=$$ $$( 2\nu^{5} - 5\nu^{4} + 38\nu^{3} - 52\nu^{2} + 149\nu - 64 ) / 4$$ (2*v^5 - 5*v^4 + 38*v^3 - 52*v^2 + 149*v - 64) / 4 $$\beta_{4}$$ $$=$$ $$( 11\nu^{5} - 28\nu^{4} + 206\nu^{3} - 287\nu^{2} + 786\nu - 356 ) / 4$$ (11*v^5 - 28*v^4 + 206*v^3 - 287*v^2 + 786*v - 356) / 4 $$\beta_{5}$$ $$=$$ $$( -7\nu^{5} + 18\nu^{4} - 132\nu^{3} + 183\nu^{2} - 514\nu + 220 ) / 2$$ (-7*v^5 + 18*v^4 - 132*v^3 + 183*v^2 - 514*v + 220) / 2
 $$\nu$$ $$=$$ $$( -2\beta_{5} - 2\beta_{4} - 3\beta_{3} - \beta_{2} - \beta _1 + 6 ) / 9$$ (-2*b5 - 2*b4 - 3*b3 - b2 - b1 + 6) / 9 $$\nu^{2}$$ $$=$$ $$( -2\beta_{5} - 2\beta_{4} - 3\beta_{3} + 2\beta_{2} + 5\beta _1 - 48 ) / 9$$ (-2*b5 - 2*b4 - 3*b3 + 2*b2 + 5*b1 - 48) / 9 $$\nu^{3}$$ $$=$$ $$( 17\beta_{5} + 8\beta_{4} + 75\beta_{3} + 13\beta_{2} + 13\beta _1 - 114 ) / 9$$ (17*b5 + 8*b4 + 75*b3 + 13*b2 + 13*b1 - 114) / 9 $$\nu^{4}$$ $$=$$ $$( 12\beta_{5} + 6\beta_{4} + 51\beta_{3} - 5\beta_{2} - 7\beta _1 + 102 ) / 3$$ (12*b5 + 6*b4 + 51*b3 - 5*b2 - 7*b1 + 102) / 3 $$\nu^{5}$$ $$=$$ $$( -136\beta_{5} - 10\beta_{4} - 879\beta_{3} - 158\beta_{2} - 95\beta _1 + 1524 ) / 9$$ (-136*b5 - 10*b4 - 879*b3 - 158*b2 - 95*b1 + 1524) / 9

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$$.

 $$n$$ $$82$$ $$326$$ $$\chi(n)$$ $$1$$ $$-\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
136.1
 0.5 + 0.0280269i 0.5 − 3.26212i 0.5 + 2.36807i 0.5 − 0.0280269i 0.5 + 3.26212i 0.5 − 2.36807i
−2.60034 + 4.50391i 0 −9.52349 16.4952i 2.50000 + 4.33013i 0 −12.2007 + 21.1322i 57.4517 0 −26.0034
136.2 −0.129356 + 0.224051i 0 3.96653 + 6.87024i 2.50000 + 4.33013i 0 −7.25871 + 12.5725i −4.12208 0 −1.29356
136.3 2.22969 3.86194i 0 −5.94305 10.2937i 2.50000 + 4.33013i 0 −2.54062 + 4.40048i −17.3296 0 22.2969
271.1 −2.60034 4.50391i 0 −9.52349 + 16.4952i 2.50000 4.33013i 0 −12.2007 21.1322i 57.4517 0 −26.0034
271.2 −0.129356 0.224051i 0 3.96653 6.87024i 2.50000 4.33013i 0 −7.25871 12.5725i −4.12208 0 −1.29356
271.3 2.22969 + 3.86194i 0 −5.94305 + 10.2937i 2.50000 4.33013i 0 −2.54062 4.40048i −17.3296 0 22.2969
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 271.3 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 405.4.e.r 6
3.b odd 2 1 405.4.e.t 6
9.c even 3 1 135.4.a.g yes 3
9.c even 3 1 inner 405.4.e.r 6
9.d odd 6 1 135.4.a.f 3
9.d odd 6 1 405.4.e.t 6
36.f odd 6 1 2160.4.a.be 3
36.h even 6 1 2160.4.a.bm 3
45.h odd 6 1 675.4.a.r 3
45.j even 6 1 675.4.a.q 3
45.k odd 12 2 675.4.b.k 6
45.l even 12 2 675.4.b.l 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.f 3 9.d odd 6 1
135.4.a.g yes 3 9.c even 3 1
405.4.e.r 6 1.a even 1 1 trivial
405.4.e.r 6 9.c even 3 1 inner
405.4.e.t 6 3.b odd 2 1
405.4.e.t 6 9.d odd 6 1
675.4.a.q 3 45.j even 6 1
675.4.a.r 3 45.h odd 6 1
675.4.b.k 6 45.k odd 12 2
675.4.b.l 6 45.l even 12 2
2160.4.a.be 3 36.f odd 6 1
2160.4.a.bm 3 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(405, [\chi])$$:

 $$T_{2}^{6} + T_{2}^{5} + 24T_{2}^{4} - 11T_{2}^{3} + 535T_{2}^{2} + 138T_{2} + 36$$ T2^6 + T2^5 + 24*T2^4 - 11*T2^3 + 535*T2^2 + 138*T2 + 36 $$T_{7}^{6} + 44T_{7}^{5} + 1384T_{7}^{4} + 20688T_{7}^{3} + 225504T_{7}^{2} + 993600T_{7} + 3240000$$ T7^6 + 44*T7^5 + 1384*T7^4 + 20688*T7^3 + 225504*T7^2 + 993600*T7 + 3240000

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + T^{5} + 24 T^{4} - 11 T^{3} + \cdots + 36$$
$3$ $$T^{6}$$
$5$ $$(T^{2} - 5 T + 25)^{3}$$
$7$ $$T^{6} + 44 T^{5} + 1384 T^{4} + \cdots + 3240000$$
$11$ $$T^{6} - 38 T^{5} + \cdots + 6935558400$$
$13$ $$T^{6} + 28 T^{5} + \cdots + 10024014400$$
$17$ $$(T^{3} - 19 T^{2} - 11477 T + 553887)^{2}$$
$19$ $$(T^{3} - 187 T^{2} + 3587 T + 525871)^{2}$$
$23$ $$T^{6} + 81 T^{5} + \cdots + 4177858328361$$
$29$ $$T^{6} - 160 T^{5} + \cdots + 62295660417600$$
$31$ $$T^{6} + 227 T^{5} + \cdots + 60674035041$$
$37$ $$(T^{3} - 78 T^{2} - 99924 T + 13637080)^{2}$$
$41$ $$T^{6} + \cdots + 146812964889600$$
$43$ $$T^{6} + \cdots + 340939975993600$$
$47$ $$T^{6} + 472 T^{5} + \cdots + 80202240000$$
$53$ $$(T^{3} + 521 T^{2} + 61387 T - 939789)^{2}$$
$59$ $$T^{6} - 140 T^{5} + \cdots + 11\!\cdots\!00$$
$61$ $$T^{6} + 595 T^{5} + \cdots + 3177687716449$$
$67$ $$T^{6} + \cdots + 127577025000000$$
$71$ $$(T^{3} - 602 T^{2} - 583652 T + 280550880)^{2}$$
$73$ $$(T^{3} - 1294 T^{2} - 95908 T + 404091280)^{2}$$
$79$ $$T^{6} + 629 T^{5} + \cdots + 4041318151809$$
$83$ $$T^{6} + 1287 T^{5} + \cdots + 11\!\cdots\!21$$
$89$ $$(T^{3} + 2154 T^{2} + 1057572 T + 74325600)^{2}$$
$97$ $$T^{6} + 1392 T^{5} + \cdots + 40\!\cdots\!00$$