# Properties

 Label 441.6.a.m Level $441$ Weight $6$ Character orbit 441.a Self dual yes Analytic conductor $70.729$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

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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: $$\Q(\sqrt{37})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 9$$ x^2 - x - 9 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 7) 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 $$\beta = \sqrt{37}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{2} + (2 \beta + 6) q^{4} + ( - 10 \beta - 19) q^{5} + (24 \beta - 48) q^{8}+O(q^{10})$$ q + (-b - 1) * q^2 + (2*b + 6) * q^4 + (-10*b - 19) * q^5 + (24*b - 48) * q^8 $$q + ( - \beta - 1) q^{2} + (2 \beta + 6) q^{4} + ( - 10 \beta - 19) q^{5} + (24 \beta - 48) q^{8} + (29 \beta + 389) q^{10} + ( - 23 \beta - 212) q^{11} + ( - 28 \beta + 462) q^{13} + ( - 40 \beta - 1032) q^{16} + (132 \beta - 1173) q^{17} + ( - 277 \beta + 180) q^{19} + ( - 98 \beta - 854) q^{20} + (235 \beta + 1063) q^{22} + (69 \beta + 6) q^{23} + (380 \beta + 936) q^{25} + ( - 434 \beta + 574) q^{26} + (700 \beta + 3526) q^{29} + (715 \beta - 1774) q^{31} + (304 \beta + 4048) q^{32} + (1041 \beta - 3711) q^{34} + ( - 790 \beta + 5545) q^{37} + (97 \beta + 10069) q^{38} + (24 \beta - 7968) q^{40} + (868 \beta + 1750) q^{41} + (1344 \beta - 6340) q^{43} + ( - 562 \beta - 2974) q^{44} + ( - 75 \beta - 2559) q^{46} + (1635 \beta - 11478) q^{47} + ( - 1316 \beta - 14996) q^{50} + (756 \beta + 700) q^{52} + (1818 \beta - 1521) q^{53} + (2557 \beta + 12538) q^{55} + ( - 4226 \beta - 29426) q^{58} + (531 \beta - 32904) q^{59} + (4154 \beta + 21243) q^{61} + (1059 \beta - 24681) q^{62} + ( - 3072 \beta + 17728) q^{64} + ( - 4088 \beta + 1582) q^{65} + (919 \beta + 21156) q^{67} + ( - 1554 \beta + 2730) q^{68} + (2184 \beta + 1104) q^{71} + (7372 \beta + 25253) q^{73} + ( - 4755 \beta + 23685) q^{74} + ( - 1302 \beta - 19418) q^{76} + ( - 5193 \beta + 4502) q^{79} + (11080 \beta + 34408) q^{80} + ( - 2618 \beta - 33866) q^{82} + ( - 4536 \beta - 52164) q^{83} + (9222 \beta - 26553) q^{85} + (4996 \beta - 43388) q^{86} + ( - 3984 \beta - 10248) q^{88} + (9356 \beta - 13333) q^{89} + (426 \beta + 5142) q^{92} + (9843 \beta - 49017) q^{94} + (3463 \beta + 99070) q^{95} + (196 \beta - 104566) q^{97}+O(q^{100})$$ q + (-b - 1) * q^2 + (2*b + 6) * q^4 + (-10*b - 19) * q^5 + (24*b - 48) * q^8 + (29*b + 389) * q^10 + (-23*b - 212) * q^11 + (-28*b + 462) * q^13 + (-40*b - 1032) * q^16 + (132*b - 1173) * q^17 + (-277*b + 180) * q^19 + (-98*b - 854) * q^20 + (235*b + 1063) * q^22 + (69*b + 6) * q^23 + (380*b + 936) * q^25 + (-434*b + 574) * q^26 + (700*b + 3526) * q^29 + (715*b - 1774) * q^31 + (304*b + 4048) * q^32 + (1041*b - 3711) * q^34 + (-790*b + 5545) * q^37 + (97*b + 10069) * q^38 + (24*b - 7968) * q^40 + (868*b + 1750) * q^41 + (1344*b - 6340) * q^43 + (-562*b - 2974) * q^44 + (-75*b - 2559) * q^46 + (1635*b - 11478) * q^47 + (-1316*b - 14996) * q^50 + (756*b + 700) * q^52 + (1818*b - 1521) * q^53 + (2557*b + 12538) * q^55 + (-4226*b - 29426) * q^58 + (531*b - 32904) * q^59 + (4154*b + 21243) * q^61 + (1059*b - 24681) * q^62 + (-3072*b + 17728) * q^64 + (-4088*b + 1582) * q^65 + (919*b + 21156) * q^67 + (-1554*b + 2730) * q^68 + (2184*b + 1104) * q^71 + (7372*b + 25253) * q^73 + (-4755*b + 23685) * q^74 + (-1302*b - 19418) * q^76 + (-5193*b + 4502) * q^79 + (11080*b + 34408) * q^80 + (-2618*b - 33866) * q^82 + (-4536*b - 52164) * q^83 + (9222*b - 26553) * q^85 + (4996*b - 43388) * q^86 + (-3984*b - 10248) * q^88 + (9356*b - 13333) * q^89 + (426*b + 5142) * q^92 + (9843*b - 49017) * q^94 + (3463*b + 99070) * q^95 + (196*b - 104566) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 12 q^{4} - 38 q^{5} - 96 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 12 * q^4 - 38 * q^5 - 96 * q^8 $$2 q - 2 q^{2} + 12 q^{4} - 38 q^{5} - 96 q^{8} + 778 q^{10} - 424 q^{11} + 924 q^{13} - 2064 q^{16} - 2346 q^{17} + 360 q^{19} - 1708 q^{20} + 2126 q^{22} + 12 q^{23} + 1872 q^{25} + 1148 q^{26} + 7052 q^{29} - 3548 q^{31} + 8096 q^{32} - 7422 q^{34} + 11090 q^{37} + 20138 q^{38} - 15936 q^{40} + 3500 q^{41} - 12680 q^{43} - 5948 q^{44} - 5118 q^{46} - 22956 q^{47} - 29992 q^{50} + 1400 q^{52} - 3042 q^{53} + 25076 q^{55} - 58852 q^{58} - 65808 q^{59} + 42486 q^{61} - 49362 q^{62} + 35456 q^{64} + 3164 q^{65} + 42312 q^{67} + 5460 q^{68} + 2208 q^{71} + 50506 q^{73} + 47370 q^{74} - 38836 q^{76} + 9004 q^{79} + 68816 q^{80} - 67732 q^{82} - 104328 q^{83} - 53106 q^{85} - 86776 q^{86} - 20496 q^{88} - 26666 q^{89} + 10284 q^{92} - 98034 q^{94} + 198140 q^{95} - 209132 q^{97}+O(q^{100})$$ 2 * q - 2 * q^2 + 12 * q^4 - 38 * q^5 - 96 * q^8 + 778 * q^10 - 424 * q^11 + 924 * q^13 - 2064 * q^16 - 2346 * q^17 + 360 * q^19 - 1708 * q^20 + 2126 * q^22 + 12 * q^23 + 1872 * q^25 + 1148 * q^26 + 7052 * q^29 - 3548 * q^31 + 8096 * q^32 - 7422 * q^34 + 11090 * q^37 + 20138 * q^38 - 15936 * q^40 + 3500 * q^41 - 12680 * q^43 - 5948 * q^44 - 5118 * q^46 - 22956 * q^47 - 29992 * q^50 + 1400 * q^52 - 3042 * q^53 + 25076 * q^55 - 58852 * q^58 - 65808 * q^59 + 42486 * q^61 - 49362 * q^62 + 35456 * q^64 + 3164 * q^65 + 42312 * q^67 + 5460 * q^68 + 2208 * q^71 + 50506 * q^73 + 47370 * q^74 - 38836 * q^76 + 9004 * q^79 + 68816 * q^80 - 67732 * q^82 - 104328 * q^83 - 53106 * q^85 - 86776 * q^86 - 20496 * q^88 - 26666 * q^89 + 10284 * q^92 - 98034 * q^94 + 198140 * q^95 - 209132 * 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
 3.54138 −2.54138
−7.08276 0 18.1655 −79.8276 0 0 97.9863 0 565.400
1.2 5.08276 0 −6.16553 41.8276 0 0 −193.986 0 212.600
 $$n$$: 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.m 2
3.b odd 2 1 49.6.a.e 2
7.b odd 2 1 441.6.a.n 2
7.d odd 6 2 63.6.e.d 4
12.b even 2 1 784.6.a.t 2
21.c even 2 1 49.6.a.d 2
21.g even 6 2 7.6.c.a 4
21.h odd 6 2 49.6.c.f 4
84.h odd 2 1 784.6.a.ba 2
84.j odd 6 2 112.6.i.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.c.a 4 21.g even 6 2
49.6.a.d 2 21.c even 2 1
49.6.a.e 2 3.b odd 2 1
49.6.c.f 4 21.h odd 6 2
63.6.e.d 4 7.d odd 6 2
112.6.i.c 4 84.j odd 6 2
441.6.a.m 2 1.a even 1 1 trivial
441.6.a.n 2 7.b odd 2 1
784.6.a.t 2 12.b even 2 1
784.6.a.ba 2 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2}^{2} + 2T_{2} - 36$$ T2^2 + 2*T2 - 36 $$T_{5}^{2} + 38T_{5} - 3339$$ T5^2 + 38*T5 - 3339 $$T_{13}^{2} - 924T_{13} + 184436$$ T13^2 - 924*T13 + 184436

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 36$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 38T - 3339$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 424T + 25371$$
$13$ $$T^{2} - 924T + 184436$$
$17$ $$T^{2} + 2346 T + 731241$$
$19$ $$T^{2} - 360 T - 2806573$$
$23$ $$T^{2} - 12T - 176121$$
$29$ $$T^{2} - 7052 T - 5697324$$
$31$ $$T^{2} + 3548 T - 15768249$$
$37$ $$T^{2} - 11090 T + 7655325$$
$41$ $$T^{2} - 3500 T - 24814188$$
$43$ $$T^{2} + 12680 T - 26638832$$
$47$ $$T^{2} + 22956 T + 32835159$$
$53$ $$T^{2} + 3042 T - 119976147$$
$59$ $$T^{2} + \cdots + 1072240659$$
$61$ $$T^{2} - 42486 T - 187196443$$
$67$ $$T^{2} - 42312 T + 416327579$$
$71$ $$T^{2} - 2208 T - 175265856$$
$73$ $$T^{2} + \cdots - 1373102199$$
$79$ $$T^{2} - 9004 T - 977520209$$
$83$ $$T^{2} + \cdots + 1959796944$$
$89$ $$T^{2} + \cdots - 3061016343$$
$97$ $$T^{2} + \cdots + 10932626964$$
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