# Properties

 Label 441.6.a.r 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: $$\Q(\sqrt{193})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 48$$ x^2 - x - 48 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 147) 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 $$\beta = \frac{1}{2}(1 + \sqrt{193})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + (3 \beta + 17) q^{4} + 36 q^{5} + ( - 9 \beta + 129) q^{8}+O(q^{10})$$ q + (b + 1) * q^2 + (3*b + 17) * q^4 + 36 * q^5 + (-9*b + 129) * q^8 $$q + (\beta + 1) q^{2} + (3 \beta + 17) q^{4} + 36 q^{5} + ( - 9 \beta + 129) q^{8} + (36 \beta + 36) q^{10} + ( - 8 \beta - 236) q^{11} + ( - 72 \beta - 612) q^{13} + (15 \beta - 847) q^{16} + ( - 216 \beta + 576) q^{17} + (288 \beta - 36) q^{19} + (108 \beta + 612) q^{20} + ( - 252 \beta - 620) q^{22} + (56 \beta + 224) q^{23} - 1829 q^{25} + ( - 756 \beta - 4068) q^{26} + ( - 640 \beta - 2866) q^{29} + (576 \beta - 5256) q^{31} + ( - 529 \beta - 4255) q^{32} + (144 \beta - 9792) q^{34} + ( - 1056 \beta + 6090) q^{37} + (540 \beta + 13788) q^{38} + ( - 324 \beta + 4644) q^{40} + (216 \beta + 10368) q^{41} + ( - 2400 \beta - 1932) q^{43} + ( - 868 \beta - 5164) q^{44} + (336 \beta + 2912) q^{46} + (3456 \beta + 2232) q^{47} + ( - 1829 \beta - 1829) q^{50} + ( - 3276 \beta - 20772) q^{52} + (1184 \beta - 1702) q^{53} + ( - 288 \beta - 8496) q^{55} + ( - 4146 \beta - 33586) q^{58} + (864 \beta + 14436) q^{59} + ( - 4680 \beta + 10980) q^{61} + ( - 4104 \beta + 22392) q^{62} + ( - 5793 \beta - 2543) q^{64} + ( - 2592 \beta - 22032) q^{65} + (6480 \beta - 13580) q^{67} + ( - 2592 \beta - 21312) q^{68} + ( - 2552 \beta + 47416) q^{71} + (1872 \beta - 29232) q^{73} + (3978 \beta - 44598) q^{74} + (5652 \beta + 40860) q^{76} + ( - 6480 \beta - 24808) q^{79} + (540 \beta - 30492) q^{80} + (10800 \beta + 20736) q^{82} + (9504 \beta - 40428) q^{83} + ( - 7776 \beta + 20736) q^{85} + ( - 6732 \beta - 117132) q^{86} + (1164 \beta - 26988) q^{88} + ( - 3672 \beta + 63432) q^{89} + (1792 \beta + 11872) q^{92} + (9144 \beta + 168120) q^{94} + (10368 \beta - 1296) q^{95} + ( - 19584 \beta - 8136) q^{97}+O(q^{100})$$ q + (b + 1) * q^2 + (3*b + 17) * q^4 + 36 * q^5 + (-9*b + 129) * q^8 + (36*b + 36) * q^10 + (-8*b - 236) * q^11 + (-72*b - 612) * q^13 + (15*b - 847) * q^16 + (-216*b + 576) * q^17 + (288*b - 36) * q^19 + (108*b + 612) * q^20 + (-252*b - 620) * q^22 + (56*b + 224) * q^23 - 1829 * q^25 + (-756*b - 4068) * q^26 + (-640*b - 2866) * q^29 + (576*b - 5256) * q^31 + (-529*b - 4255) * q^32 + (144*b - 9792) * q^34 + (-1056*b + 6090) * q^37 + (540*b + 13788) * q^38 + (-324*b + 4644) * q^40 + (216*b + 10368) * q^41 + (-2400*b - 1932) * q^43 + (-868*b - 5164) * q^44 + (336*b + 2912) * q^46 + (3456*b + 2232) * q^47 + (-1829*b - 1829) * q^50 + (-3276*b - 20772) * q^52 + (1184*b - 1702) * q^53 + (-288*b - 8496) * q^55 + (-4146*b - 33586) * q^58 + (864*b + 14436) * q^59 + (-4680*b + 10980) * q^61 + (-4104*b + 22392) * q^62 + (-5793*b - 2543) * q^64 + (-2592*b - 22032) * q^65 + (6480*b - 13580) * q^67 + (-2592*b - 21312) * q^68 + (-2552*b + 47416) * q^71 + (1872*b - 29232) * q^73 + (3978*b - 44598) * q^74 + (5652*b + 40860) * q^76 + (-6480*b - 24808) * q^79 + (540*b - 30492) * q^80 + (10800*b + 20736) * q^82 + (9504*b - 40428) * q^83 + (-7776*b + 20736) * q^85 + (-6732*b - 117132) * q^86 + (1164*b - 26988) * q^88 + (-3672*b + 63432) * q^89 + (1792*b + 11872) * q^92 + (9144*b + 168120) * q^94 + (10368*b - 1296) * q^95 + (-19584*b - 8136) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 37 q^{4} + 72 q^{5} + 249 q^{8}+O(q^{10})$$ 2 * q + 3 * q^2 + 37 * q^4 + 72 * q^5 + 249 * q^8 $$2 q + 3 q^{2} + 37 q^{4} + 72 q^{5} + 249 q^{8} + 108 q^{10} - 480 q^{11} - 1296 q^{13} - 1679 q^{16} + 936 q^{17} + 216 q^{19} + 1332 q^{20} - 1492 q^{22} + 504 q^{23} - 3658 q^{25} - 8892 q^{26} - 6372 q^{29} - 9936 q^{31} - 9039 q^{32} - 19440 q^{34} + 11124 q^{37} + 28116 q^{38} + 8964 q^{40} + 20952 q^{41} - 6264 q^{43} - 11196 q^{44} + 6160 q^{46} + 7920 q^{47} - 5487 q^{50} - 44820 q^{52} - 2220 q^{53} - 17280 q^{55} - 71318 q^{58} + 29736 q^{59} + 17280 q^{61} + 40680 q^{62} - 10879 q^{64} - 46656 q^{65} - 20680 q^{67} - 45216 q^{68} + 92280 q^{71} - 56592 q^{73} - 85218 q^{74} + 87372 q^{76} - 56096 q^{79} - 60444 q^{80} + 52272 q^{82} - 71352 q^{83} + 33696 q^{85} - 240996 q^{86} - 52812 q^{88} + 123192 q^{89} + 25536 q^{92} + 345384 q^{94} + 7776 q^{95} - 35856 q^{97}+O(q^{100})$$ 2 * q + 3 * q^2 + 37 * q^4 + 72 * q^5 + 249 * q^8 + 108 * q^10 - 480 * q^11 - 1296 * q^13 - 1679 * q^16 + 936 * q^17 + 216 * q^19 + 1332 * q^20 - 1492 * q^22 + 504 * q^23 - 3658 * q^25 - 8892 * q^26 - 6372 * q^29 - 9936 * q^31 - 9039 * q^32 - 19440 * q^34 + 11124 * q^37 + 28116 * q^38 + 8964 * q^40 + 20952 * q^41 - 6264 * q^43 - 11196 * q^44 + 6160 * q^46 + 7920 * q^47 - 5487 * q^50 - 44820 * q^52 - 2220 * q^53 - 17280 * q^55 - 71318 * q^58 + 29736 * q^59 + 17280 * q^61 + 40680 * q^62 - 10879 * q^64 - 46656 * q^65 - 20680 * q^67 - 45216 * q^68 + 92280 * q^71 - 56592 * q^73 - 85218 * q^74 + 87372 * q^76 - 56096 * q^79 - 60444 * q^80 + 52272 * q^82 - 71352 * q^83 + 33696 * q^85 - 240996 * q^86 - 52812 * q^88 + 123192 * q^89 + 25536 * q^92 + 345384 * q^94 + 7776 * q^95 - 35856 * 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
 −6.44622 7.44622
−5.44622 0 −2.33867 36.0000 0 0 187.016 0 −196.064
1.2 8.44622 0 39.3387 36.0000 0 0 61.9840 0 304.064
 $$n$$: 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.r 2
3.b odd 2 1 147.6.a.j yes 2
7.b odd 2 1 441.6.a.q 2
21.c even 2 1 147.6.a.h 2
21.g even 6 2 147.6.e.n 4
21.h odd 6 2 147.6.e.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.6.a.h 2 21.c even 2 1
147.6.a.j yes 2 3.b odd 2 1
147.6.e.m 4 21.h odd 6 2
147.6.e.n 4 21.g even 6 2
441.6.a.q 2 7.b odd 2 1
441.6.a.r 2 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{2}^{2} - 3T_{2} - 46$$ T2^2 - 3*T2 - 46 $$T_{5} - 36$$ T5 - 36 $$T_{13}^{2} + 1296T_{13} + 169776$$ T13^2 + 1296*T13 + 169776

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3T - 46$$
$3$ $$T^{2}$$
$5$ $$(T - 36)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 480T + 54512$$
$13$ $$T^{2} + 1296 T + 169776$$
$17$ $$T^{2} - 936 T - 2032128$$
$19$ $$T^{2} - 216 T - 3990384$$
$23$ $$T^{2} - 504T - 87808$$
$29$ $$T^{2} + 6372 T - 9612604$$
$31$ $$T^{2} + 9936 T + 8672832$$
$37$ $$T^{2} - 11124 T - 22869468$$
$41$ $$T^{2} - 20952 T + 107495424$$
$43$ $$T^{2} + 6264 T - 268110576$$
$47$ $$T^{2} - 7920 T - 560613312$$
$53$ $$T^{2} + 2220 T - 66407452$$
$59$ $$T^{2} - 29736 T + 185038992$$
$61$ $$T^{2} - 17280 T - 982141200$$
$67$ $$T^{2} + 20680 T - 1919121200$$
$71$ $$T^{2} - 92280 T + 1814661632$$
$73$ $$T^{2} + 56592 T + 631577088$$
$79$ $$T^{2} + 56096 T - 1239346496$$
$83$ $$T^{2} + 71352 T - 3085453296$$
$89$ $$T^{2} - 123192 T + 3143484288$$
$97$ $$T^{2} + 35856 T - 18184056768$$