# Properties

 Label 441.4.a.r Level $441$ Weight $4$ Character orbit 441.a Self dual yes Analytic conductor $26.020$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 14$$ x^2 - x - 14 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) 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 $$\beta = \frac{1}{2}(1 + \sqrt{57})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + (3 \beta + 7) q^{4} + (2 \beta + 2) q^{5} + (5 \beta + 41) q^{8}+O(q^{10})$$ q + (b + 1) * q^2 + (3*b + 7) * q^4 + (2*b + 2) * q^5 + (5*b + 41) * q^8 $$q + (\beta + 1) q^{2} + (3 \beta + 7) q^{4} + (2 \beta + 2) q^{5} + (5 \beta + 41) q^{8} + (6 \beta + 30) q^{10} + ( - 10 \beta + 8) q^{11} + (12 \beta - 14) q^{13} + (27 \beta + 55) q^{16} + ( - 2 \beta - 2) q^{17} + (24 \beta - 44) q^{19} + (26 \beta + 98) q^{20} + ( - 12 \beta - 132) q^{22} + (34 \beta - 20) q^{23} + (12 \beta - 65) q^{25} + (10 \beta + 154) q^{26} + ( - 24 \beta + 138) q^{29} + ( - 72 \beta + 16) q^{31} + (69 \beta + 105) q^{32} + ( - 6 \beta - 30) q^{34} + ( - 36 \beta - 106) q^{37} + (4 \beta + 292) q^{38} + (102 \beta + 222) q^{40} + ( - 30 \beta - 210) q^{41} + ( - 48 \beta + 212) q^{43} + ( - 76 \beta - 364) q^{44} + (48 \beta + 456) q^{46} + (68 \beta - 40) q^{47} + ( - 41 \beta + 103) q^{50} + (78 \beta + 406) q^{52} + ( - 4 \beta + 554) q^{53} + ( - 24 \beta - 264) q^{55} + (90 \beta - 198) q^{58} + ( - 116 \beta + 460) q^{59} + ( - 72 \beta + 250) q^{61} + ( - 128 \beta - 992) q^{62} + (27 \beta + 631) q^{64} + (20 \beta + 308) q^{65} + (108 \beta + 20) q^{67} + ( - 26 \beta - 98) q^{68} + (30 \beta - 492) q^{71} + ( - 12 \beta - 530) q^{73} + ( - 178 \beta - 610) q^{74} + (108 \beta + 700) q^{76} + ( - 108 \beta - 232) q^{79} + (218 \beta + 866) q^{80} + ( - 270 \beta - 630) q^{82} + (96 \beta + 924) q^{83} + ( - 12 \beta - 60) q^{85} + (116 \beta - 460) q^{86} + ( - 420 \beta - 372) q^{88} + ( - 142 \beta + 254) q^{89} + (280 \beta + 1288) q^{92} + (96 \beta + 912) q^{94} + (8 \beta + 584) q^{95} + ( - 276 \beta - 266) q^{97}+O(q^{100})$$ q + (b + 1) * q^2 + (3*b + 7) * q^4 + (2*b + 2) * q^5 + (5*b + 41) * q^8 + (6*b + 30) * q^10 + (-10*b + 8) * q^11 + (12*b - 14) * q^13 + (27*b + 55) * q^16 + (-2*b - 2) * q^17 + (24*b - 44) * q^19 + (26*b + 98) * q^20 + (-12*b - 132) * q^22 + (34*b - 20) * q^23 + (12*b - 65) * q^25 + (10*b + 154) * q^26 + (-24*b + 138) * q^29 + (-72*b + 16) * q^31 + (69*b + 105) * q^32 + (-6*b - 30) * q^34 + (-36*b - 106) * q^37 + (4*b + 292) * q^38 + (102*b + 222) * q^40 + (-30*b - 210) * q^41 + (-48*b + 212) * q^43 + (-76*b - 364) * q^44 + (48*b + 456) * q^46 + (68*b - 40) * q^47 + (-41*b + 103) * q^50 + (78*b + 406) * q^52 + (-4*b + 554) * q^53 + (-24*b - 264) * q^55 + (90*b - 198) * q^58 + (-116*b + 460) * q^59 + (-72*b + 250) * q^61 + (-128*b - 992) * q^62 + (27*b + 631) * q^64 + (20*b + 308) * q^65 + (108*b + 20) * q^67 + (-26*b - 98) * q^68 + (30*b - 492) * q^71 + (-12*b - 530) * q^73 + (-178*b - 610) * q^74 + (108*b + 700) * q^76 + (-108*b - 232) * q^79 + (218*b + 866) * q^80 + (-270*b - 630) * q^82 + (96*b + 924) * q^83 + (-12*b - 60) * q^85 + (116*b - 460) * q^86 + (-420*b - 372) * q^88 + (-142*b + 254) * q^89 + (280*b + 1288) * q^92 + (96*b + 912) * q^94 + (8*b + 584) * q^95 + (-276*b - 266) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 17 q^{4} + 6 q^{5} + 87 q^{8}+O(q^{10})$$ 2 * q + 3 * q^2 + 17 * q^4 + 6 * q^5 + 87 * q^8 $$2 q + 3 q^{2} + 17 q^{4} + 6 q^{5} + 87 q^{8} + 66 q^{10} + 6 q^{11} - 16 q^{13} + 137 q^{16} - 6 q^{17} - 64 q^{19} + 222 q^{20} - 276 q^{22} - 6 q^{23} - 118 q^{25} + 318 q^{26} + 252 q^{29} - 40 q^{31} + 279 q^{32} - 66 q^{34} - 248 q^{37} + 588 q^{38} + 546 q^{40} - 450 q^{41} + 376 q^{43} - 804 q^{44} + 960 q^{46} - 12 q^{47} + 165 q^{50} + 890 q^{52} + 1104 q^{53} - 552 q^{55} - 306 q^{58} + 804 q^{59} + 428 q^{61} - 2112 q^{62} + 1289 q^{64} + 636 q^{65} + 148 q^{67} - 222 q^{68} - 954 q^{71} - 1072 q^{73} - 1398 q^{74} + 1508 q^{76} - 572 q^{79} + 1950 q^{80} - 1530 q^{82} + 1944 q^{83} - 132 q^{85} - 804 q^{86} - 1164 q^{88} + 366 q^{89} + 2856 q^{92} + 1920 q^{94} + 1176 q^{95} - 808 q^{97}+O(q^{100})$$ 2 * q + 3 * q^2 + 17 * q^4 + 6 * q^5 + 87 * q^8 + 66 * q^10 + 6 * q^11 - 16 * q^13 + 137 * q^16 - 6 * q^17 - 64 * q^19 + 222 * q^20 - 276 * q^22 - 6 * q^23 - 118 * q^25 + 318 * q^26 + 252 * q^29 - 40 * q^31 + 279 * q^32 - 66 * q^34 - 248 * q^37 + 588 * q^38 + 546 * q^40 - 450 * q^41 + 376 * q^43 - 804 * q^44 + 960 * q^46 - 12 * q^47 + 165 * q^50 + 890 * q^52 + 1104 * q^53 - 552 * q^55 - 306 * q^58 + 804 * q^59 + 428 * q^61 - 2112 * q^62 + 1289 * q^64 + 636 * q^65 + 148 * q^67 - 222 * q^68 - 954 * q^71 - 1072 * q^73 - 1398 * q^74 + 1508 * q^76 - 572 * q^79 + 1950 * q^80 - 1530 * q^82 + 1944 * q^83 - 132 * q^85 - 804 * q^86 - 1164 * q^88 + 366 * q^89 + 2856 * q^92 + 1920 * q^94 + 1176 * q^95 - 808 * q^97

## 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
 −3.27492 4.27492
−2.27492 0 −2.82475 −4.54983 0 0 24.6254 0 10.3505
1.2 5.27492 0 19.8248 10.5498 0 0 62.3746 0 55.6495
 $$n$$: 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.a.r 2
3.b odd 2 1 147.4.a.i 2
7.b odd 2 1 63.4.a.e 2
7.c even 3 2 441.4.e.p 4
7.d odd 6 2 441.4.e.q 4
12.b even 2 1 2352.4.a.bz 2
21.c even 2 1 21.4.a.c 2
21.g even 6 2 147.4.e.l 4
21.h odd 6 2 147.4.e.m 4
28.d even 2 1 1008.4.a.ba 2
35.c odd 2 1 1575.4.a.p 2
84.h odd 2 1 336.4.a.m 2
105.g even 2 1 525.4.a.n 2
105.k odd 4 2 525.4.d.g 4
168.e odd 2 1 1344.4.a.bo 2
168.i even 2 1 1344.4.a.bg 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.c 2 21.c even 2 1
63.4.a.e 2 7.b odd 2 1
147.4.a.i 2 3.b odd 2 1
147.4.e.l 4 21.g even 6 2
147.4.e.m 4 21.h odd 6 2
336.4.a.m 2 84.h odd 2 1
441.4.a.r 2 1.a even 1 1 trivial
441.4.e.p 4 7.c even 3 2
441.4.e.q 4 7.d odd 6 2
525.4.a.n 2 105.g even 2 1
525.4.d.g 4 105.k odd 4 2
1008.4.a.ba 2 28.d even 2 1
1344.4.a.bg 2 168.i even 2 1
1344.4.a.bo 2 168.e odd 2 1
1575.4.a.p 2 35.c odd 2 1
2352.4.a.bz 2 12.b even 2 1

## Hecke kernels

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

 $$T_{2}^{2} - 3T_{2} - 12$$ T2^2 - 3*T2 - 12 $$T_{5}^{2} - 6T_{5} - 48$$ T5^2 - 6*T5 - 48 $$T_{13}^{2} + 16T_{13} - 1988$$ T13^2 + 16*T13 - 1988

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3T - 12$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 6T - 48$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 6T - 1416$$
$13$ $$T^{2} + 16T - 1988$$
$17$ $$T^{2} + 6T - 48$$
$19$ $$T^{2} + 64T - 7184$$
$23$ $$T^{2} + 6T - 16464$$
$29$ $$T^{2} - 252T + 7668$$
$31$ $$T^{2} + 40T - 73472$$
$37$ $$T^{2} + 248T - 3092$$
$41$ $$T^{2} + 450T + 37800$$
$43$ $$T^{2} - 376T + 2512$$
$47$ $$T^{2} + 12T - 65856$$
$53$ $$T^{2} - 1104 T + 304476$$
$59$ $$T^{2} - 804T - 30144$$
$61$ $$T^{2} - 428T - 28076$$
$67$ $$T^{2} - 148T - 160736$$
$71$ $$T^{2} + 954T + 214704$$
$73$ $$T^{2} + 1072 T + 285244$$
$79$ $$T^{2} + 572T - 84416$$
$83$ $$T^{2} - 1944 T + 813456$$
$89$ $$T^{2} - 366T - 253848$$
$97$ $$T^{2} + 808T - 922292$$