Properties

 Label 441.6.a.x Level $441$ Weight $6$ Character orbit 441.a Self dual yes Analytic conductor $70.729$ 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] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

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: yes Analytic conductor: $$70.7292645375$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.358541904.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 111x^{2} + 756$$ x^4 - 111*x^2 + 756 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 63) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} + 24) q^{4} + (\beta_{2} - 3 \beta_1) q^{5} + (3 \beta_{2} + 32 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b3 + 24) * q^4 + (b2 - 3*b1) * q^5 + (3*b2 + 32*b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{3} + 24) q^{4} + (\beta_{2} - 3 \beta_1) q^{5} + (3 \beta_{2} + 32 \beta_1) q^{8} + (2 \beta_{3} - 140) q^{10} + (\beta_{2} + 45 \beta_1) q^{11} + (16 \beta_{3} - 210) q^{13} + (15 \beta_{3} + 1108) q^{16} + (31 \beta_{2} + 131 \beta_1) q^{17} + ( - 16 \beta_{3} - 1708) q^{19} + ( - 26 \beta_{2} + 36 \beta_1) q^{20} + (50 \beta_{3} + 2548) q^{22} + (47 \beta_{2} - 237 \beta_1) q^{23} + ( - 80 \beta_{3} + 711) q^{25} + (48 \beta_{2} + 430 \beta_1) q^{26} + (32 \beta_{2} - 496 \beta_1) q^{29} + (48 \beta_{3} + 1204) q^{31} + ( - 51 \beta_{2} + 684 \beta_1) q^{32} + (286 \beta_{3} + 8204) q^{34} + (240 \beta_{3} + 1634) q^{37} + ( - 48 \beta_{2} - 2348 \beta_1) q^{38} + ( - 158 \beta_{3} + 5768) q^{40} + (29 \beta_{2} + 809 \beta_1) q^{41} + (128 \beta_{3} + 2700) q^{43} + (118 \beta_{2} + 3108 \beta_1) q^{44} + ( - 2 \beta_{3} - 11956) q^{46} + ( - 114 \beta_{2} + 2582 \beta_1) q^{47} + ( - 240 \beta_{2} - 2489 \beta_1) q^{50} + (158 \beta_{3} + 32144) q^{52} + (14 \beta_{2} + 1510 \beta_1) q^{53} + (16 \beta_{3} - 2884) q^{55} + ( - 336 \beta_{3} - 26880) q^{58} + ( - 174 \beta_{2} + 1642 \beta_1) q^{59} + (240 \beta_{3} - 3206) q^{61} + (144 \beta_{2} + 3124 \beta_1) q^{62} + ( - 51 \beta_{3} + 1420) q^{64} + ( - 1010 \beta_{2} + 2358 \beta_1) q^{65} + ( - 96 \beta_{3} + 50828) q^{67} + ( - 134 \beta_{2} + 15452 \beta_1) q^{68} + (717 \beta_{2} - 1287 \beta_1) q^{71} + ( - 1120 \beta_{3} - 4942) q^{73} + (720 \beta_{2} + 11234 \beta_1) q^{74} + ( - 2076 \beta_{3} - 78176) q^{76} + ( - 672 \beta_{3} + 8264) q^{79} + (358 \beta_{2} - 1704 \beta_1) q^{80} + (954 \beta_{3} + 46116) q^{82} + (112 \beta_{2} - 9296 \beta_1) q^{83} + ( - 2032 \beta_{3} + 87556) q^{85} + (384 \beta_{2} + 7820 \beta_1) q^{86} + (2098 \beta_{3} + 95816) q^{88} + (113 \beta_{2} + 1453 \beta_1) q^{89} + ( - 1510 \beta_{2} - 4452 \beta_1) q^{92} + (2012 \beta_{3} + 141400) q^{94} + ( - 908 \beta_{2} + 3396 \beta_1) q^{95} + ( - 224 \beta_{3} - 109326) q^{97}+O(q^{100})$$ q + b1 * q^2 + (b3 + 24) * q^4 + (b2 - 3*b1) * q^5 + (3*b2 + 32*b1) * q^8 + (2*b3 - 140) * q^10 + (b2 + 45*b1) * q^11 + (16*b3 - 210) * q^13 + (15*b3 + 1108) * q^16 + (31*b2 + 131*b1) * q^17 + (-16*b3 - 1708) * q^19 + (-26*b2 + 36*b1) * q^20 + (50*b3 + 2548) * q^22 + (47*b2 - 237*b1) * q^23 + (-80*b3 + 711) * q^25 + (48*b2 + 430*b1) * q^26 + (32*b2 - 496*b1) * q^29 + (48*b3 + 1204) * q^31 + (-51*b2 + 684*b1) * q^32 + (286*b3 + 8204) * q^34 + (240*b3 + 1634) * q^37 + (-48*b2 - 2348*b1) * q^38 + (-158*b3 + 5768) * q^40 + (29*b2 + 809*b1) * q^41 + (128*b3 + 2700) * q^43 + (118*b2 + 3108*b1) * q^44 + (-2*b3 - 11956) * q^46 + (-114*b2 + 2582*b1) * q^47 + (-240*b2 - 2489*b1) * q^50 + (158*b3 + 32144) * q^52 + (14*b2 + 1510*b1) * q^53 + (16*b3 - 2884) * q^55 + (-336*b3 - 26880) * q^58 + (-174*b2 + 1642*b1) * q^59 + (240*b3 - 3206) * q^61 + (144*b2 + 3124*b1) * q^62 + (-51*b3 + 1420) * q^64 + (-1010*b2 + 2358*b1) * q^65 + (-96*b3 + 50828) * q^67 + (-134*b2 + 15452*b1) * q^68 + (717*b2 - 1287*b1) * q^71 + (-1120*b3 - 4942) * q^73 + (720*b2 + 11234*b1) * q^74 + (-2076*b3 - 78176) * q^76 + (-672*b3 + 8264) * q^79 + (358*b2 - 1704*b1) * q^80 + (954*b3 + 46116) * q^82 + (112*b2 - 9296*b1) * q^83 + (-2032*b3 + 87556) * q^85 + (384*b2 + 7820*b1) * q^86 + (2098*b3 + 95816) * q^88 + (113*b2 + 1453*b1) * q^89 + (-1510*b2 - 4452*b1) * q^92 + (2012*b3 + 141400) * q^94 + (-908*b2 + 3396*b1) * q^95 + (-224*b3 - 109326) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 94 q^{4}+O(q^{10})$$ 4 * q + 94 * q^4 $$4 q + 94 q^{4} - 564 q^{10} - 872 q^{13} + 4402 q^{16} - 6800 q^{19} + 10092 q^{22} + 3004 q^{25} + 4720 q^{31} + 32244 q^{34} + 6056 q^{37} + 23388 q^{40} + 10544 q^{43} - 47820 q^{46} + 128260 q^{52} - 11568 q^{55} - 106848 q^{58} - 13304 q^{61} + 5782 q^{64} + 203504 q^{67} - 17528 q^{73} - 308552 q^{76} + 34400 q^{79} + 182556 q^{82} + 354288 q^{85} + 379068 q^{88} + 561576 q^{94} - 436856 q^{97}+O(q^{100})$$ 4 * q + 94 * q^4 - 564 * q^10 - 872 * q^13 + 4402 * q^16 - 6800 * q^19 + 10092 * q^22 + 3004 * q^25 + 4720 * q^31 + 32244 * q^34 + 6056 * q^37 + 23388 * q^40 + 10544 * q^43 - 47820 * q^46 + 128260 * q^52 - 11568 * q^55 - 106848 * q^58 - 13304 * q^61 + 5782 * q^64 + 203504 * q^67 - 17528 * q^73 - 308552 * q^76 + 34400 * q^79 + 182556 * q^82 + 354288 * q^85 + 379068 * q^88 + 561576 * q^94 - 436856 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 111x^{2} + 756$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 96\nu ) / 3$$ (v^3 - 96*v) / 3 $$\beta_{3}$$ $$=$$ $$\nu^{2} - 56$$ v^2 - 56
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 56$$ b3 + 56 $$\nu^{3}$$ $$=$$ $$3\beta_{2} + 96\beta_1$$ 3*b2 + 96*b1

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −10.1838 −2.69991 2.69991 10.1838
−10.1838 0 71.7105 4.37743 0 0 −404.405 0 −44.5790
1.2 −2.69991 0 −24.7105 87.9366 0 0 153.113 0 −237.421
1.3 2.69991 0 −24.7105 −87.9366 0 0 −153.113 0 −237.421
1.4 10.1838 0 71.7105 −4.37743 0 0 404.405 0 −44.5790
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$+1$$
$$7$$ $$-1$$

Inner twists

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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.6.a.x 4
3.b odd 2 1 inner 441.6.a.x 4
7.b odd 2 1 63.6.a.h 4
21.c even 2 1 63.6.a.h 4
28.d even 2 1 1008.6.a.by 4
84.h odd 2 1 1008.6.a.by 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.6.a.h 4 7.b odd 2 1
63.6.a.h 4 21.c even 2 1
441.6.a.x 4 1.a even 1 1 trivial
441.6.a.x 4 3.b odd 2 1 inner
1008.6.a.by 4 28.d even 2 1
1008.6.a.by 4 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2}^{4} - 111T_{2}^{2} + 756$$ T2^4 - 111*T2^2 + 756 $$T_{5}^{4} - 7752T_{5}^{2} + 148176$$ T5^4 - 7752*T5^2 + 148176 $$T_{13}^{2} + 436T_{13} - 547484$$ T13^2 + 436*T13 - 547484

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 111T^{2} + 756$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 7752 T^{2} + 148176$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 236424 T^{2} + 407299536$$
$13$ $$(T^{2} + 436 T - 547484)^{2}$$
$17$ $$T^{4} + \cdots + 20712534557904$$
$19$ $$(T^{2} + 3400 T + 2294992)^{2}$$
$23$ $$T^{4} + \cdots + 27015936275664$$
$29$ $$T^{4} + \cdots + 269206953394176$$
$31$ $$(T^{2} - 2360 T - 3962672)^{2}$$
$37$ $$(T^{2} - 3028 T - 131584604)^{2}$$
$41$ $$T^{4} + \cdots + 1390182638544$$
$43$ $$(T^{2} - 5272 T - 31132016)^{2}$$
$47$ $$T^{4} + \cdots + 14\!\cdots\!56$$
$53$ $$T^{4} + \cdots + 21\!\cdots\!56$$
$59$ $$T^{4} + \cdots + 49\!\cdots\!96$$
$61$ $$(T^{2} + 6652 T - 122814524)^{2}$$
$67$ $$(T^{2} - 101752 T + 2566947088)^{2}$$
$71$ $$T^{4} + \cdots + 11\!\cdots\!64$$
$73$ $$(T^{2} + 8764 T - 2896337276)^{2}$$
$79$ $$(T^{2} - 17200 T - 975634112)^{2}$$
$83$ $$T^{4} + \cdots + 97\!\cdots\!56$$
$89$ $$T^{4} + \cdots + 81\!\cdots\!64$$
$97$ $$(T^{2} + 218428 T + 11811076228)^{2}$$