# Properties

 Label 441.6.a.s Level $441$ Weight $6$ Character orbit 441.a Self dual yes Analytic conductor $70.729$ Analytic rank $1$ 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] = [441,6,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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{249})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 62$$ x^2 - x - 62 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 $$\beta = \frac{1}{2}(1 + \sqrt{249})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + (3 \beta + 31) q^{4} + ( - 7 \beta - 13) q^{5} + (5 \beta + 185) q^{8}+O(q^{10})$$ q + (b + 1) * q^2 + (3*b + 31) * q^4 + (-7*b - 13) * q^5 + (5*b + 185) * q^8 $$q + (\beta + 1) q^{2} + (3 \beta + 31) q^{4} + ( - 7 \beta - 13) q^{5} + (5 \beta + 185) q^{8} + ( - 27 \beta - 447) q^{10} + ( - \beta + 569) q^{11} + ( - 9 \beta - 458) q^{13} + (99 \beta - 497) q^{16} + (148 \beta - 236) q^{17} + ( - 27 \beta - 1142) q^{19} + ( - 277 \beta - 1705) q^{20} + (567 \beta + 507) q^{22} + ( - 308 \beta - 644) q^{23} + (231 \beta + 82) q^{25} + ( - 476 \beta - 1016) q^{26} + ( - 45 \beta + 1131) q^{29} + ( - 768 \beta - 1763) q^{31} + ( - 459 \beta - 279) q^{32} + (60 \beta + 8940) q^{34} + (855 \beta - 9982) q^{37} + ( - 1196 \beta - 2816) q^{38} + ( - 1395 \beta - 4575) q^{40} + (846 \beta - 6852) q^{41} + ( - 2043 \beta - 364) q^{43} + (1673 \beta + 17453) q^{44} + ( - 1260 \beta - 19740) q^{46} + ( - 604 \beta - 11278) q^{47} + (544 \beta + 14404) q^{50} + ( - 1680 \beta - 15872) q^{52} + (1751 \beta + 14951) q^{53} + ( - 3963 \beta - 6963) q^{55} + (1041 \beta - 1659) q^{58} + ( - 3917 \beta + 22507) q^{59} + (2544 \beta - 22298) q^{61} + ( - 3299 \beta - 49379) q^{62} + ( - 4365 \beta - 12833) q^{64} + (3386 \beta + 9860) q^{65} + ( - 4461 \beta + 17612) q^{67} + (4324 \beta + 20212) q^{68} + ( - 1404 \beta - 50346) q^{71} + ( - 5247 \beta + 16912) q^{73} + ( - 8272 \beta + 43028) q^{74} + ( - 4344 \beta - 40424) q^{76} + (6834 \beta - 12649) q^{79} + (1499 \beta - 36505) q^{80} + ( - 5160 \beta + 45600) q^{82} + ( - 1899 \beta + 31539) q^{83} + ( - 1308 \beta - 61164) q^{85} + ( - 4450 \beta - 127030) q^{86} + (2655 \beta + 104955) q^{88} + ( - 130 \beta + 14726) q^{89} + ( - 12404 \beta - 77252) q^{92} + ( - 12486 \beta - 48726) q^{94} + (8534 \beta + 26564) q^{95} + (1017 \beta + 4387) q^{97}+O(q^{100})$$ q + (b + 1) * q^2 + (3*b + 31) * q^4 + (-7*b - 13) * q^5 + (5*b + 185) * q^8 + (-27*b - 447) * q^10 + (-b + 569) * q^11 + (-9*b - 458) * q^13 + (99*b - 497) * q^16 + (148*b - 236) * q^17 + (-27*b - 1142) * q^19 + (-277*b - 1705) * q^20 + (567*b + 507) * q^22 + (-308*b - 644) * q^23 + (231*b + 82) * q^25 + (-476*b - 1016) * q^26 + (-45*b + 1131) * q^29 + (-768*b - 1763) * q^31 + (-459*b - 279) * q^32 + (60*b + 8940) * q^34 + (855*b - 9982) * q^37 + (-1196*b - 2816) * q^38 + (-1395*b - 4575) * q^40 + (846*b - 6852) * q^41 + (-2043*b - 364) * q^43 + (1673*b + 17453) * q^44 + (-1260*b - 19740) * q^46 + (-604*b - 11278) * q^47 + (544*b + 14404) * q^50 + (-1680*b - 15872) * q^52 + (1751*b + 14951) * q^53 + (-3963*b - 6963) * q^55 + (1041*b - 1659) * q^58 + (-3917*b + 22507) * q^59 + (2544*b - 22298) * q^61 + (-3299*b - 49379) * q^62 + (-4365*b - 12833) * q^64 + (3386*b + 9860) * q^65 + (-4461*b + 17612) * q^67 + (4324*b + 20212) * q^68 + (-1404*b - 50346) * q^71 + (-5247*b + 16912) * q^73 + (-8272*b + 43028) * q^74 + (-4344*b - 40424) * q^76 + (6834*b - 12649) * q^79 + (1499*b - 36505) * q^80 + (-5160*b + 45600) * q^82 + (-1899*b + 31539) * q^83 + (-1308*b - 61164) * q^85 + (-4450*b - 127030) * q^86 + (2655*b + 104955) * q^88 + (-130*b + 14726) * q^89 + (-12404*b - 77252) * q^92 + (-12486*b - 48726) * q^94 + (8534*b + 26564) * q^95 + (1017*b + 4387) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 65 q^{4} - 33 q^{5} + 375 q^{8}+O(q^{10})$$ 2 * q + 3 * q^2 + 65 * q^4 - 33 * q^5 + 375 * q^8 $$2 q + 3 q^{2} + 65 q^{4} - 33 q^{5} + 375 q^{8} - 921 q^{10} + 1137 q^{11} - 925 q^{13} - 895 q^{16} - 324 q^{17} - 2311 q^{19} - 3687 q^{20} + 1581 q^{22} - 1596 q^{23} + 395 q^{25} - 2508 q^{26} + 2217 q^{29} - 4294 q^{31} - 1017 q^{32} + 17940 q^{34} - 19109 q^{37} - 6828 q^{38} - 10545 q^{40} - 12858 q^{41} - 2771 q^{43} + 36579 q^{44} - 40740 q^{46} - 23160 q^{47} + 29352 q^{50} - 33424 q^{52} + 31653 q^{53} - 17889 q^{55} - 2277 q^{58} + 41097 q^{59} - 42052 q^{61} - 102057 q^{62} - 30031 q^{64} + 23106 q^{65} + 30763 q^{67} + 44748 q^{68} - 102096 q^{71} + 28577 q^{73} + 77784 q^{74} - 85192 q^{76} - 18464 q^{79} - 71511 q^{80} + 86040 q^{82} + 61179 q^{83} - 123636 q^{85} - 258510 q^{86} + 212565 q^{88} + 29322 q^{89} - 166908 q^{92} - 109938 q^{94} + 61662 q^{95} + 9791 q^{97}+O(q^{100})$$ 2 * q + 3 * q^2 + 65 * q^4 - 33 * q^5 + 375 * q^8 - 921 * q^10 + 1137 * q^11 - 925 * q^13 - 895 * q^16 - 324 * q^17 - 2311 * q^19 - 3687 * q^20 + 1581 * q^22 - 1596 * q^23 + 395 * q^25 - 2508 * q^26 + 2217 * q^29 - 4294 * q^31 - 1017 * q^32 + 17940 * q^34 - 19109 * q^37 - 6828 * q^38 - 10545 * q^40 - 12858 * q^41 - 2771 * q^43 + 36579 * q^44 - 40740 * q^46 - 23160 * q^47 + 29352 * q^50 - 33424 * q^52 + 31653 * q^53 - 17889 * q^55 - 2277 * q^58 + 41097 * q^59 - 42052 * q^61 - 102057 * q^62 - 30031 * q^64 + 23106 * q^65 + 30763 * q^67 + 44748 * q^68 - 102096 * q^71 + 28577 * q^73 + 77784 * q^74 - 85192 * q^76 - 18464 * q^79 - 71511 * q^80 + 86040 * q^82 + 61179 * q^83 - 123636 * q^85 - 258510 * q^86 + 212565 * q^88 + 29322 * q^89 - 166908 * q^92 - 109938 * q^94 + 61662 * q^95 + 9791 * 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
 −7.38987 8.38987
−6.38987 0 8.83040 38.7291 0 0 148.051 0 −247.474
1.2 9.38987 0 56.1696 −71.7291 0 0 226.949 0 −673.526
 $$n$$: 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 441.6.a.s 2
3.b odd 2 1 147.6.a.k 2
7.b odd 2 1 441.6.a.t 2
7.d odd 6 2 63.6.e.c 4
21.c even 2 1 147.6.a.i 2
21.g even 6 2 21.6.e.b 4
21.h odd 6 2 147.6.e.l 4
84.j odd 6 2 336.6.q.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.b 4 21.g even 6 2
63.6.e.c 4 7.d odd 6 2
147.6.a.i 2 21.c even 2 1
147.6.a.k 2 3.b odd 2 1
147.6.e.l 4 21.h odd 6 2
336.6.q.e 4 84.j odd 6 2
441.6.a.s 2 1.a even 1 1 trivial
441.6.a.t 2 7.b odd 2 1

## Hecke kernels

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

 $$T_{2}^{2} - 3T_{2} - 60$$ T2^2 - 3*T2 - 60 $$T_{5}^{2} + 33T_{5} - 2778$$ T5^2 + 33*T5 - 2778 $$T_{13}^{2} + 925T_{13} + 208864$$ T13^2 + 925*T13 + 208864

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3T - 60$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 33T - 2778$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 1137 T + 323130$$
$13$ $$T^{2} + 925T + 208864$$
$17$ $$T^{2} + 324 T - 1337280$$
$19$ $$T^{2} + 2311 T + 1289800$$
$23$ $$T^{2} + 1596 T - 5268480$$
$29$ $$T^{2} - 2217 T + 1102716$$
$31$ $$T^{2} + 4294 T - 32106935$$
$37$ $$T^{2} + 19109 T + 45782164$$
$41$ $$T^{2} + 12858 T - 3221280$$
$43$ $$T^{2} + 2771 T - 257902490$$
$47$ $$T^{2} + 23160 T + 111386604$$
$53$ $$T^{2} - 31653 T + 59619540$$
$59$ $$T^{2} - 41097 T - 532853988$$
$61$ $$T^{2} + 42052 T + 39214660$$
$67$ $$T^{2} - 30763 T - 1002216890$$
$71$ $$T^{2} + 102096 T + 2483190108$$
$73$ $$T^{2} - 28577 T - 1509644078$$
$79$ $$T^{2} + 18464 T - 2822066537$$
$83$ $$T^{2} - 61179 T + 711231498$$
$89$ $$T^{2} - 29322 T + 213892896$$
$97$ $$T^{2} - 9791 T - 40418570$$