# Properties

 Label 441.6.a.z Level $441$ Weight $6$ Character orbit 441.a Self dual yes Analytic conductor $70.729$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{113})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 59x^{2} + 60x + 674$$ x^4 - 2*x^3 - 59*x^2 + 60*x + 674 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$7$$ Twist minimal: no (minimal twist has level 49) 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 2) q^{2} - 5 \beta_1 q^{4} + ( - 2 \beta_{3} + 8 \beta_{2}) q^{5} + (17 \beta_1 + 76) q^{8}+O(q^{10})$$ q + (-b1 + 2) * q^2 - 5*b1 * q^4 + (-2*b3 + 8*b2) * q^5 + (17*b1 + 76) * q^8 $$q + ( - \beta_1 + 2) q^{2} - 5 \beta_1 q^{4} + ( - 2 \beta_{3} + 8 \beta_{2}) q^{5} + (17 \beta_1 + 76) q^{8} + ( - 38 \beta_{3} + 30 \beta_{2}) q^{10} + (12 \beta_1 + 494) q^{11} + (64 \beta_{3} + 56 \beta_{2}) q^{13} + (135 \beta_1 - 324) q^{16} + ( - 7 \beta_{3} + 152 \beta_{2}) q^{17} + ( - 53 \beta_{3} - 78 \beta_{2}) q^{19} + ( - 170 \beta_{3} + 70 \beta_{2}) q^{20} + ( - 458 \beta_1 + 652) q^{22} + ( - 344 \beta_1 + 1612) q^{23} + ( - 320 \beta_1 + 531) q^{25} + ( - 32 \beta_{3} - 336 \beta_{2}) q^{26} + ( - 784 \beta_1 + 446) q^{29} + (246 \beta_{3} + 772 \beta_{2}) q^{31} + (185 \beta_1 - 6860) q^{32} + ( - 629 \beta_{3} + 353 \beta_{2}) q^{34} + ( - 48 \beta_1 - 2326) q^{37} + (153 \beta_{3} + 215 \beta_{2}) q^{38} + (426 \beta_{3} + 370 \beta_{2}) q^{40} + (1103 \beta_{3} + 224 \beta_{2}) q^{41} + ( - 980 \beta_1 + 4622) q^{43} + ( - 2410 \beta_1 - 1680) q^{44} + ( - 2644 \beta_1 + 12856) q^{46} + (590 \beta_{3} - 1644 \beta_{2}) q^{47} + ( - 1491 \beta_1 + 10022) q^{50} + ( - 800 \beta_{3} - 2240 \beta_{2}) q^{52} + ( - 2592 \beta_1 + 24434) q^{53} + ( - 580 \beta_{3} + 3784 \beta_{2}) q^{55} + ( - 2798 \beta_1 + 22844) q^{58} + (1425 \beta_{3} - 2914 \beta_{2}) q^{59} + ( - 3326 \beta_{3} - 680 \beta_{2}) q^{61} + ( - 2350 \beta_{3} - 178 \beta_{2}) q^{62} + (3095 \beta_1 - 8532) q^{64} + (6496 \beta_1 + 19040) q^{65} + ( - 11632 \beta_1 - 11540) q^{67} + ( - 3075 \beta_{3} + 245 \beta_{2}) q^{68} + (4312 \beta_1 + 40612) q^{71} + (4407 \beta_{3} - 5080 \beta_{2}) q^{73} + (2182 \beta_1 - 3308) q^{74} + (1295 \beta_{3} + 1855 \beta_{2}) q^{76} + (4264 \beta_1 - 20540) q^{79} + (5238 \beta_{3} - 4482 \beta_{2}) q^{80} + (2413 \beta_{3} - 7273 \beta_{2}) q^{82} + (4425 \beta_{3} + 6230 \beta_{2}) q^{83} + ( - 2608 \beta_1 + 66860) q^{85} + ( - 7562 \beta_1 + 36684) q^{86} + (9106 \beta_1 + 43256) q^{88} + ( - 6259 \beta_{3} + 1016 \beta_{2}) q^{89} + ( - 9780 \beta_1 + 48160) q^{92} + (8346 \beta_{3} - 7418 \beta_{2}) q^{94} + ( - 5000 \beta_1 - 29556) q^{95} + ( - 5873 \beta_{3} + 7448 \beta_{2}) q^{97}+O(q^{100})$$ q + (-b1 + 2) * q^2 - 5*b1 * q^4 + (-2*b3 + 8*b2) * q^5 + (17*b1 + 76) * q^8 + (-38*b3 + 30*b2) * q^10 + (12*b1 + 494) * q^11 + (64*b3 + 56*b2) * q^13 + (135*b1 - 324) * q^16 + (-7*b3 + 152*b2) * q^17 + (-53*b3 - 78*b2) * q^19 + (-170*b3 + 70*b2) * q^20 + (-458*b1 + 652) * q^22 + (-344*b1 + 1612) * q^23 + (-320*b1 + 531) * q^25 + (-32*b3 - 336*b2) * q^26 + (-784*b1 + 446) * q^29 + (246*b3 + 772*b2) * q^31 + (185*b1 - 6860) * q^32 + (-629*b3 + 353*b2) * q^34 + (-48*b1 - 2326) * q^37 + (153*b3 + 215*b2) * q^38 + (426*b3 + 370*b2) * q^40 + (1103*b3 + 224*b2) * q^41 + (-980*b1 + 4622) * q^43 + (-2410*b1 - 1680) * q^44 + (-2644*b1 + 12856) * q^46 + (590*b3 - 1644*b2) * q^47 + (-1491*b1 + 10022) * q^50 + (-800*b3 - 2240*b2) * q^52 + (-2592*b1 + 24434) * q^53 + (-580*b3 + 3784*b2) * q^55 + (-2798*b1 + 22844) * q^58 + (1425*b3 - 2914*b2) * q^59 + (-3326*b3 - 680*b2) * q^61 + (-2350*b3 - 178*b2) * q^62 + (3095*b1 - 8532) * q^64 + (6496*b1 + 19040) * q^65 + (-11632*b1 - 11540) * q^67 + (-3075*b3 + 245*b2) * q^68 + (4312*b1 + 40612) * q^71 + (4407*b3 - 5080*b2) * q^73 + (2182*b1 - 3308) * q^74 + (1295*b3 + 1855*b2) * q^76 + (4264*b1 - 20540) * q^79 + (5238*b3 - 4482*b2) * q^80 + (2413*b3 - 7273*b2) * q^82 + (4425*b3 + 6230*b2) * q^83 + (-2608*b1 + 66860) * q^85 + (-7562*b1 + 36684) * q^86 + (9106*b1 + 43256) * q^88 + (-6259*b3 + 1016*b2) * q^89 + (-9780*b1 + 48160) * q^92 + (8346*b3 - 7418*b2) * q^94 + (-5000*b1 - 29556) * q^95 + (-5873*b3 + 7448*b2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 10 q^{2} + 10 q^{4} + 270 q^{8}+O(q^{10})$$ 4 * q + 10 * q^2 + 10 * q^4 + 270 * q^8 $$4 q + 10 q^{2} + 10 q^{4} + 270 q^{8} + 1952 q^{11} - 1566 q^{16} + 3524 q^{22} + 7136 q^{23} + 2764 q^{25} + 3352 q^{29} - 27810 q^{32} - 9208 q^{37} + 20448 q^{43} - 1900 q^{44} + 56712 q^{46} + 43070 q^{50} + 102920 q^{53} + 96972 q^{58} - 40318 q^{64} + 63168 q^{65} - 22896 q^{67} + 153824 q^{71} - 17596 q^{74} - 90688 q^{79} + 272656 q^{85} + 161860 q^{86} + 154812 q^{88} + 212200 q^{92} - 108224 q^{95}+O(q^{100})$$ 4 * q + 10 * q^2 + 10 * q^4 + 270 * q^8 + 1952 * q^11 - 1566 * q^16 + 3524 * q^22 + 7136 * q^23 + 2764 * q^25 + 3352 * q^29 - 27810 * q^32 - 9208 * q^37 + 20448 * q^43 - 1900 * q^44 + 56712 * q^46 + 43070 * q^50 + 102920 * q^53 + 96972 * q^58 - 40318 * q^64 + 63168 * q^65 - 22896 * q^67 + 153824 * q^71 - 17596 * q^74 - 90688 * q^79 + 272656 * q^85 + 161860 * q^86 + 154812 * q^88 + 212200 * q^92 - 108224 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 59x^{2} + 60x + 674$$ :

 $$\beta_{1}$$ $$=$$ $$( -2\nu^{3} + 3\nu^{2} + 172\nu - 139 ) / 105$$ (-2*v^3 + 3*v^2 + 172*v - 139) / 105 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 51\nu^{2} - 86\nu - 1558 ) / 105$$ (v^3 + 51*v^2 - 86*v - 1558) / 105 $$\beta_{3}$$ $$=$$ $$( 2\nu^{3} - 3\nu^{2} - 67\nu + 34 ) / 15$$ (2*v^3 - 3*v^2 - 67*v + 34) / 15
 $$\nu$$ $$=$$ $$( \beta_{3} + 7\beta _1 + 7 ) / 7$$ (b3 + 7*b1 + 7) / 7 $$\nu^{2}$$ $$=$$ $$2\beta_{2} + \beta _1 + 31$$ 2*b2 + b1 + 31 $$\nu^{3}$$ $$=$$ $$( 86\beta_{3} + 21\beta_{2} + 245\beta _1 + 441 ) / 7$$ (86*b3 + 21*b2 + 245*b1 + 441) / 7

## 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
 4.40086 7.22929 −3.40086 −6.22929
−2.81507 0 −24.0754 −45.9910 0 0 157.856 0 129.468
1.2 −2.81507 0 −24.0754 45.9910 0 0 157.856 0 −129.468
1.3 7.81507 0 29.0754 −74.2753 0 0 −22.8562 0 −580.467
1.4 7.81507 0 29.0754 74.2753 0 0 −22.8562 0 580.467
 $$n$$: 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.6.a.z 4
3.b odd 2 1 49.6.a.g 4
7.b odd 2 1 inner 441.6.a.z 4
12.b even 2 1 784.6.a.bf 4
21.c even 2 1 49.6.a.g 4
21.g even 6 2 49.6.c.h 8
21.h odd 6 2 49.6.c.h 8
84.h odd 2 1 784.6.a.bf 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.6.a.g 4 3.b odd 2 1
49.6.a.g 4 21.c even 2 1
49.6.c.h 8 21.g even 6 2
49.6.c.h 8 21.h odd 6 2
441.6.a.z 4 1.a even 1 1 trivial
441.6.a.z 4 7.b odd 2 1 inner
784.6.a.bf 4 12.b even 2 1
784.6.a.bf 4 84.h odd 2 1

## Hecke kernels

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

 $$T_{2}^{2} - 5T_{2} - 22$$ T2^2 - 5*T2 - 22 $$T_{5}^{4} - 7632T_{5}^{2} + 11669056$$ T5^4 - 7632*T5^2 + 11669056 $$T_{13}^{4} - 1260672T_{13}^{2} + 76158337024$$ T13^4 - 1260672*T13^2 + 76158337024

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 5 T - 22)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 7632 T^{2} + \cdots + 11669056$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 976 T + 234076)^{2}$$
$13$ $$T^{4} - 1260672 T^{2} + \cdots + 76158337024$$
$17$ $$T^{4} - 2613668 T^{2} + \cdots + 1700202150724$$
$19$ $$T^{4} - 1359892 T^{2} + \cdots + 56942116$$
$23$ $$(T^{2} - 3568 T - 160336)^{2}$$
$29$ $$(T^{2} - 1676 T - 16661788)^{2}$$
$31$ $$T^{4} + \cdots + 614334295349824$$
$37$ $$(T^{2} + 4604 T + 5234116)^{2}$$
$41$ $$T^{4} - 251093444 T^{2} + \cdots + 14\!\cdots\!56$$
$43$ $$(T^{2} - 10224 T - 998756)^{2}$$
$47$ $$T^{4} - 349180624 T^{2} + \cdots + 17\!\cdots\!36$$
$53$ $$(T^{2} - 51460 T + 472236292)^{2}$$
$59$ $$T^{4} - 1249753044 T^{2} + \cdots + 11\!\cdots\!76$$
$61$ $$T^{4} - 2284246736 T^{2} + \cdots + 11\!\cdots\!24$$
$67$ $$(T^{2} + 11448 T - 3789557552)^{2}$$
$71$ $$(T^{2} - 76912 T + 953601968)^{2}$$
$73$ $$T^{4} - 6121721124 T^{2} + \cdots + 20\!\cdots\!44$$
$79$ $$(T^{2} + 45344 T + 386672)^{2}$$
$83$ $$T^{4} - 9034370100 T^{2} + \cdots + 17\!\cdots\!00$$
$89$ $$T^{4} - 7617937028 T^{2} + \cdots + 13\!\cdots\!96$$
$97$ $$T^{4} - 11859566628 T^{2} + \cdots + 11\!\cdots\!44$$