# Properties

 Label 441.6.a.i Level $441$ Weight $6$ Character orbit 441.a Self dual yes Analytic conductor $70.729$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3) 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)

 $$f(q)$$ $$=$$ $$q + 6 q^{2} + 4 q^{4} + 6 q^{5} - 168 q^{8}+O(q^{10})$$ q + 6 * q^2 + 4 * q^4 + 6 * q^5 - 168 * q^8 $$q + 6 q^{2} + 4 q^{4} + 6 q^{5} - 168 q^{8} + 36 q^{10} + 564 q^{11} - 638 q^{13} - 1136 q^{16} + 882 q^{17} + 556 q^{19} + 24 q^{20} + 3384 q^{22} + 840 q^{23} - 3089 q^{25} - 3828 q^{26} - 4638 q^{29} - 4400 q^{31} - 1440 q^{32} + 5292 q^{34} - 2410 q^{37} + 3336 q^{38} - 1008 q^{40} - 6870 q^{41} + 9644 q^{43} + 2256 q^{44} + 5040 q^{46} - 18672 q^{47} - 18534 q^{50} - 2552 q^{52} - 33750 q^{53} + 3384 q^{55} - 27828 q^{58} - 18084 q^{59} - 39758 q^{61} - 26400 q^{62} + 27712 q^{64} - 3828 q^{65} - 23068 q^{67} + 3528 q^{68} + 4248 q^{71} + 41110 q^{73} - 14460 q^{74} + 2224 q^{76} + 21920 q^{79} - 6816 q^{80} - 41220 q^{82} + 82452 q^{83} + 5292 q^{85} + 57864 q^{86} - 94752 q^{88} - 94086 q^{89} + 3360 q^{92} - 112032 q^{94} + 3336 q^{95} - 49442 q^{97}+O(q^{100})$$ q + 6 * q^2 + 4 * q^4 + 6 * q^5 - 168 * q^8 + 36 * q^10 + 564 * q^11 - 638 * q^13 - 1136 * q^16 + 882 * q^17 + 556 * q^19 + 24 * q^20 + 3384 * q^22 + 840 * q^23 - 3089 * q^25 - 3828 * q^26 - 4638 * q^29 - 4400 * q^31 - 1440 * q^32 + 5292 * q^34 - 2410 * q^37 + 3336 * q^38 - 1008 * q^40 - 6870 * q^41 + 9644 * q^43 + 2256 * q^44 + 5040 * q^46 - 18672 * q^47 - 18534 * q^50 - 2552 * q^52 - 33750 * q^53 + 3384 * q^55 - 27828 * q^58 - 18084 * q^59 - 39758 * q^61 - 26400 * q^62 + 27712 * q^64 - 3828 * q^65 - 23068 * q^67 + 3528 * q^68 + 4248 * q^71 + 41110 * q^73 - 14460 * q^74 + 2224 * q^76 + 21920 * q^79 - 6816 * q^80 - 41220 * q^82 + 82452 * q^83 + 5292 * q^85 + 57864 * q^86 - 94752 * q^88 - 94086 * q^89 + 3360 * q^92 - 112032 * q^94 + 3336 * q^95 - 49442 * 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
 0
6.00000 0 4.00000 6.00000 0 0 −168.000 0 36.0000
 $$n$$: 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.i 1
3.b odd 2 1 147.6.a.a 1
7.b odd 2 1 9.6.a.a 1
21.c even 2 1 3.6.a.a 1
21.g even 6 2 147.6.e.h 2
21.h odd 6 2 147.6.e.k 2
28.d even 2 1 144.6.a.f 1
35.c odd 2 1 225.6.a.a 1
35.f even 4 2 225.6.b.b 2
56.e even 2 1 576.6.a.t 1
56.h odd 2 1 576.6.a.s 1
63.l odd 6 2 81.6.c.a 2
63.o even 6 2 81.6.c.c 2
77.b even 2 1 1089.6.a.b 1
84.h odd 2 1 48.6.a.a 1
105.g even 2 1 75.6.a.e 1
105.k odd 4 2 75.6.b.b 2
168.e odd 2 1 192.6.a.l 1
168.i even 2 1 192.6.a.d 1
231.h odd 2 1 363.6.a.d 1
273.g even 2 1 507.6.a.b 1
336.v odd 4 2 768.6.d.h 2
336.y even 4 2 768.6.d.k 2
357.c even 2 1 867.6.a.a 1
399.h odd 2 1 1083.6.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.6.a.a 1 21.c even 2 1
9.6.a.a 1 7.b odd 2 1
48.6.a.a 1 84.h odd 2 1
75.6.a.e 1 105.g even 2 1
75.6.b.b 2 105.k odd 4 2
81.6.c.a 2 63.l odd 6 2
81.6.c.c 2 63.o even 6 2
144.6.a.f 1 28.d even 2 1
147.6.a.a 1 3.b odd 2 1
147.6.e.h 2 21.g even 6 2
147.6.e.k 2 21.h odd 6 2
192.6.a.d 1 168.i even 2 1
192.6.a.l 1 168.e odd 2 1
225.6.a.a 1 35.c odd 2 1
225.6.b.b 2 35.f even 4 2
363.6.a.d 1 231.h odd 2 1
441.6.a.i 1 1.a even 1 1 trivial
507.6.a.b 1 273.g even 2 1
576.6.a.s 1 56.h odd 2 1
576.6.a.t 1 56.e even 2 1
768.6.d.h 2 336.v odd 4 2
768.6.d.k 2 336.y even 4 2
867.6.a.a 1 357.c even 2 1
1083.6.a.c 1 399.h odd 2 1
1089.6.a.b 1 77.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 $$S_{6}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2} - 6$$ T2 - 6 $$T_{5} - 6$$ T5 - 6 $$T_{13} + 638$$ T13 + 638

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 6$$
$3$ $$T$$
$5$ $$T - 6$$
$7$ $$T$$
$11$ $$T - 564$$
$13$ $$T + 638$$
$17$ $$T - 882$$
$19$ $$T - 556$$
$23$ $$T - 840$$
$29$ $$T + 4638$$
$31$ $$T + 4400$$
$37$ $$T + 2410$$
$41$ $$T + 6870$$
$43$ $$T - 9644$$
$47$ $$T + 18672$$
$53$ $$T + 33750$$
$59$ $$T + 18084$$
$61$ $$T + 39758$$
$67$ $$T + 23068$$
$71$ $$T - 4248$$
$73$ $$T - 41110$$
$79$ $$T - 21920$$
$83$ $$T - 82452$$
$89$ $$T + 94086$$
$97$ $$T + 49442$$