# Properties

 Label 441.4.a.w Level $441$ Weight $4$ Character orbit 441.a Self dual yes Analytic conductor $26.020$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,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, 4, names="a")

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

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$26.0198423125$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.6257832.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 19x^{2} + 42$$ x^4 - 19*x^2 + 42 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + (\beta_{3} + \beta_1) q^{5} + (\beta_{3} - \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + 2) * q^4 + (b3 + b1) * q^5 + (b3 - b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + (\beta_{3} + \beta_1) q^{5} + (\beta_{3} - \beta_1) q^{8} + (5 \beta_{2} + 8) q^{10} + (3 \beta_{3} - \beta_1) q^{11} + ( - \beta_{2} + 25) q^{13} + ( - 5 \beta_{2} - 28) q^{16} + ( - 2 \beta_{3} + 26 \beta_1) q^{17} + (11 \beta_{2} + 61) q^{19} + ( - 3 \beta_{3} + 25 \beta_1) q^{20} + (11 \beta_{2} - 16) q^{22} + (2 \beta_{3} + 46 \beta_1) q^{23} + ( - 17 \beta_{2} + 83) q^{25} + ( - \beta_{3} + 20 \beta_1) q^{26} + ( - \beta_{3} - 45 \beta_1) q^{29} + ( - 20 \beta_{2} + 45) q^{31} + ( - 13 \beta_{3} - 45 \beta_1) q^{32} + (18 \beta_{2} + 264) q^{34} + (25 \beta_{2} - 81) q^{37} + (11 \beta_{3} + 116 \beta_1) q^{38} + ( - 27 \beta_{2} + 192) q^{40} + (16 \beta_{3} + 72 \beta_1) q^{41} + ( - 27 \beta_{2} - 223) q^{43} + ( - 13 \beta_{3} + 47 \beta_1) q^{44} + (54 \beta_{2} + 456) q^{46} + ( - 16 \beta_{3} - 44 \beta_1) q^{47} + ( - 17 \beta_{3} - 2 \beta_1) q^{50} + (24 \beta_{2} + 2) q^{52} + (7 \beta_{3} - \beta_1) q^{53} + ( - 71 \beta_{2} + 592) q^{55} + ( - 49 \beta_{2} - 448) q^{58} + ( - 17 \beta_{3} - 73 \beta_1) q^{59} + ( - 46 \beta_{2} + 310) q^{61} + ( - 20 \beta_{3} - 55 \beta_1) q^{62} + ( - 57 \beta_{2} - 200) q^{64} + (30 \beta_{3} + 2 \beta_1) q^{65} + ( - 19 \beta_{2} + 463) q^{67} + (34 \beta_{3} + 146 \beta_1) q^{68} + 36 \beta_{3} q^{71} + (7 \beta_{2} + 441) q^{73} + (25 \beta_{3} + 44 \beta_1) q^{74} + (72 \beta_{2} + 650) q^{76} + (42 \beta_{2} + 23) q^{79} + ( - 3 \beta_{3} - 143 \beta_1) q^{80} + (136 \beta_{2} + 688) q^{82} + ( - 21 \beta_{3} - 189 \beta_1) q^{83} + (174 \beta_{2} - 192) q^{85} + ( - 27 \beta_{3} - 358 \beta_1) q^{86} + ( - 93 \beta_{2} + 624) q^{88} + (22 \beta_{3} - 314 \beta_1) q^{89} + (38 \beta_{3} + 358 \beta_1) q^{92} + ( - 108 \beta_{2} - 408) q^{94} + (6 \beta_{3} + 314 \beta_1) q^{95} + (91 \beta_{2} + 798) q^{97}+O(q^{100})$$ q + b1 * q^2 + (b2 + 2) * q^4 + (b3 + b1) * q^5 + (b3 - b1) * q^8 + (5*b2 + 8) * q^10 + (3*b3 - b1) * q^11 + (-b2 + 25) * q^13 + (-5*b2 - 28) * q^16 + (-2*b3 + 26*b1) * q^17 + (11*b2 + 61) * q^19 + (-3*b3 + 25*b1) * q^20 + (11*b2 - 16) * q^22 + (2*b3 + 46*b1) * q^23 + (-17*b2 + 83) * q^25 + (-b3 + 20*b1) * q^26 + (-b3 - 45*b1) * q^29 + (-20*b2 + 45) * q^31 + (-13*b3 - 45*b1) * q^32 + (18*b2 + 264) * q^34 + (25*b2 - 81) * q^37 + (11*b3 + 116*b1) * q^38 + (-27*b2 + 192) * q^40 + (16*b3 + 72*b1) * q^41 + (-27*b2 - 223) * q^43 + (-13*b3 + 47*b1) * q^44 + (54*b2 + 456) * q^46 + (-16*b3 - 44*b1) * q^47 + (-17*b3 - 2*b1) * q^50 + (24*b2 + 2) * q^52 + (7*b3 - b1) * q^53 + (-71*b2 + 592) * q^55 + (-49*b2 - 448) * q^58 + (-17*b3 - 73*b1) * q^59 + (-46*b2 + 310) * q^61 + (-20*b3 - 55*b1) * q^62 + (-57*b2 - 200) * q^64 + (30*b3 + 2*b1) * q^65 + (-19*b2 + 463) * q^67 + (34*b3 + 146*b1) * q^68 + 36*b3 * q^71 + (7*b2 + 441) * q^73 + (25*b3 + 44*b1) * q^74 + (72*b2 + 650) * q^76 + (42*b2 + 23) * q^79 + (-3*b3 - 143*b1) * q^80 + (136*b2 + 688) * q^82 + (-21*b3 - 189*b1) * q^83 + (174*b2 - 192) * q^85 + (-27*b3 - 358*b1) * q^86 + (-93*b2 + 624) * q^88 + (22*b3 - 314*b1) * q^89 + (38*b3 + 358*b1) * q^92 + (-108*b2 - 408) * q^94 + (6*b3 + 314*b1) * q^95 + (91*b2 + 798) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{4}+O(q^{10})$$ 4 * q + 6 * q^4 $$4 q + 6 q^{4} + 22 q^{10} + 102 q^{13} - 102 q^{16} + 222 q^{19} - 86 q^{22} + 366 q^{25} + 220 q^{31} + 1020 q^{34} - 374 q^{37} + 822 q^{40} - 838 q^{43} + 1716 q^{46} - 40 q^{52} + 2510 q^{55} - 1694 q^{58} + 1332 q^{61} - 686 q^{64} + 1890 q^{67} + 1750 q^{73} + 2456 q^{76} + 8 q^{79} + 2480 q^{82} - 1116 q^{85} + 2682 q^{88} - 1416 q^{94} + 3010 q^{97}+O(q^{100})$$ 4 * q + 6 * q^4 + 22 * q^10 + 102 * q^13 - 102 * q^16 + 222 * q^19 - 86 * q^22 + 366 * q^25 + 220 * q^31 + 1020 * q^34 - 374 * q^37 + 822 * q^40 - 838 * q^43 + 1716 * q^46 - 40 * q^52 + 2510 * q^55 - 1694 * q^58 + 1332 * q^61 - 686 * q^64 + 1890 * q^67 + 1750 * q^73 + 2456 * q^76 + 8 * q^79 + 2480 * q^82 - 1116 * q^85 + 2682 * q^88 - 1416 * q^94 + 3010 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 19x^{2} + 42$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 10$$ v^2 - 10 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 15\nu$$ v^3 - 15*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 10$$ b2 + 10 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 15\beta_1$$ b3 + 15*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
 −4.05539 −1.59805 1.59805 4.05539
−4.05539 0 8.44622 −9.92039 0 0 −1.80961 0 40.2311
1.2 −1.59805 0 −5.44622 18.2917 0 0 21.4878 0 −29.2311
1.3 1.59805 0 −5.44622 −18.2917 0 0 −21.4878 0 −29.2311
1.4 4.05539 0 8.44622 9.92039 0 0 1.80961 0 40.2311
 $$n$$: 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 441.4.a.w 4
3.b odd 2 1 inner 441.4.a.w 4
7.b odd 2 1 441.4.a.v 4
7.c even 3 2 63.4.e.d 8
7.d odd 6 2 441.4.e.x 8
21.c even 2 1 441.4.a.v 4
21.g even 6 2 441.4.e.x 8
21.h odd 6 2 63.4.e.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.e.d 8 7.c even 3 2
63.4.e.d 8 21.h odd 6 2
441.4.a.v 4 7.b odd 2 1
441.4.a.v 4 21.c even 2 1
441.4.a.w 4 1.a even 1 1 trivial
441.4.a.w 4 3.b odd 2 1 inner
441.4.e.x 8 7.d odd 6 2
441.4.e.x 8 21.g even 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2}^{4} - 19T_{2}^{2} + 42$$ T2^4 - 19*T2^2 + 42 $$T_{5}^{4} - 433T_{5}^{2} + 32928$$ T5^4 - 433*T5^2 + 32928 $$T_{13}^{2} - 51T_{13} + 602$$ T13^2 - 51*T13 + 602

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 19T^{2} + 42$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 433 T^{2} + 32928$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 3937 T^{2} + 688128$$
$13$ $$(T^{2} - 51 T + 602)^{2}$$
$17$ $$T^{4} - 15396 T^{2} + \cdots + 58084992$$
$19$ $$(T^{2} - 111 T - 2758)^{2}$$
$23$ $$T^{4} - 40452 T^{2} + \cdots + 44731008$$
$29$ $$T^{4} - 38185 T^{2} + \cdots + 96018048$$
$31$ $$(T^{2} - 110 T - 16275)^{2}$$
$37$ $$(T^{2} + 187 T - 21414)^{2}$$
$41$ $$T^{4} - 190144 T^{2} + \cdots + 6145155072$$
$43$ $$(T^{2} + 419 T + 8716)^{2}$$
$47$ $$T^{4} - 135600 T^{2} + \cdots + 4556708352$$
$53$ $$T^{4} - 21201 T^{2} + \cdots + 27149472$$
$59$ $$T^{4} - 205665 T^{2} + \cdots + 7681740192$$
$61$ $$(T^{2} - 666 T + 8792)^{2}$$
$67$ $$(T^{2} - 945 T + 205838)^{2}$$
$71$ $$T^{4} - 557280 T^{2} + \cdots + 22856214528$$
$73$ $$(T^{2} - 875 T + 189042)^{2}$$
$79$ $$(T^{2} - 4 T - 85109)^{2}$$
$83$ $$T^{4} - 804825 T^{2} + \cdots + 10585989792$$
$89$ $$T^{4} - 2191972 T^{2} + \cdots + 1155567856128$$
$97$ $$(T^{2} - 1505 T + 166698)^{2}$$