Properties

 Label 441.6.a.c 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 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)

 $$f(q)$$ $$=$$ $$q - 5 q^{2} - 7 q^{4} + 94 q^{5} + 195 q^{8}+O(q^{10})$$ q - 5 * q^2 - 7 * q^4 + 94 * q^5 + 195 * q^8 $$q - 5 q^{2} - 7 q^{4} + 94 q^{5} + 195 q^{8} - 470 q^{10} - 52 q^{11} + 770 q^{13} - 751 q^{16} - 2022 q^{17} - 1732 q^{19} - 658 q^{20} + 260 q^{22} + 576 q^{23} + 5711 q^{25} - 3850 q^{26} - 5518 q^{29} - 6336 q^{31} - 2485 q^{32} + 10110 q^{34} - 7338 q^{37} + 8660 q^{38} + 18330 q^{40} - 3262 q^{41} + 5420 q^{43} + 364 q^{44} - 2880 q^{46} + 864 q^{47} - 28555 q^{50} - 5390 q^{52} - 4182 q^{53} - 4888 q^{55} + 27590 q^{58} - 11220 q^{59} + 45602 q^{61} + 31680 q^{62} + 36457 q^{64} + 72380 q^{65} + 1396 q^{67} + 14154 q^{68} - 18720 q^{71} - 46362 q^{73} + 36690 q^{74} + 12124 q^{76} + 97424 q^{79} - 70594 q^{80} + 16310 q^{82} - 81228 q^{83} - 190068 q^{85} - 27100 q^{86} - 10140 q^{88} - 3182 q^{89} - 4032 q^{92} - 4320 q^{94} - 162808 q^{95} - 4914 q^{97}+O(q^{100})$$ q - 5 * q^2 - 7 * q^4 + 94 * q^5 + 195 * q^8 - 470 * q^10 - 52 * q^11 + 770 * q^13 - 751 * q^16 - 2022 * q^17 - 1732 * q^19 - 658 * q^20 + 260 * q^22 + 576 * q^23 + 5711 * q^25 - 3850 * q^26 - 5518 * q^29 - 6336 * q^31 - 2485 * q^32 + 10110 * q^34 - 7338 * q^37 + 8660 * q^38 + 18330 * q^40 - 3262 * q^41 + 5420 * q^43 + 364 * q^44 - 2880 * q^46 + 864 * q^47 - 28555 * q^50 - 5390 * q^52 - 4182 * q^53 - 4888 * q^55 + 27590 * q^58 - 11220 * q^59 + 45602 * q^61 + 31680 * q^62 + 36457 * q^64 + 72380 * q^65 + 1396 * q^67 + 14154 * q^68 - 18720 * q^71 - 46362 * q^73 + 36690 * q^74 + 12124 * q^76 + 97424 * q^79 - 70594 * q^80 + 16310 * q^82 - 81228 * q^83 - 190068 * q^85 - 27100 * q^86 - 10140 * q^88 - 3182 * q^89 - 4032 * q^92 - 4320 * q^94 - 162808 * q^95 - 4914 * 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
−5.00000 0 −7.00000 94.0000 0 0 195.000 0 −470.000
 $$n$$: 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.c 1
3.b odd 2 1 147.6.a.f 1
7.b odd 2 1 63.6.a.b 1
21.c even 2 1 21.6.a.c 1
21.g even 6 2 147.6.e.c 2
21.h odd 6 2 147.6.e.d 2
28.d even 2 1 1008.6.a.a 1
84.h odd 2 1 336.6.a.i 1
105.g even 2 1 525.6.a.b 1
105.k odd 4 2 525.6.d.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.c 1 21.c even 2 1
63.6.a.b 1 7.b odd 2 1
147.6.a.f 1 3.b odd 2 1
147.6.e.c 2 21.g even 6 2
147.6.e.d 2 21.h odd 6 2
336.6.a.i 1 84.h odd 2 1
441.6.a.c 1 1.a even 1 1 trivial
525.6.a.b 1 105.g even 2 1
525.6.d.c 2 105.k odd 4 2
1008.6.a.a 1 28.d even 2 1

Hecke kernels

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

 $$T_{2} + 5$$ T2 + 5 $$T_{5} - 94$$ T5 - 94 $$T_{13} - 770$$ T13 - 770

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 5$$
$3$ $$T$$
$5$ $$T - 94$$
$7$ $$T$$
$11$ $$T + 52$$
$13$ $$T - 770$$
$17$ $$T + 2022$$
$19$ $$T + 1732$$
$23$ $$T - 576$$
$29$ $$T + 5518$$
$31$ $$T + 6336$$
$37$ $$T + 7338$$
$41$ $$T + 3262$$
$43$ $$T - 5420$$
$47$ $$T - 864$$
$53$ $$T + 4182$$
$59$ $$T + 11220$$
$61$ $$T - 45602$$
$67$ $$T - 1396$$
$71$ $$T + 18720$$
$73$ $$T + 46362$$
$79$ $$T - 97424$$
$83$ $$T + 81228$$
$89$ $$T + 3182$$
$97$ $$T + 4914$$