Properties

 Label 44.6.a.b Level $44$ Weight $6$ Character orbit 44.a Self dual yes Analytic conductor $7.057$ Analytic rank $0$ 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] = [44,6,Mod(1,44)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(44, 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("44.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$44 = 2^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 44.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: $$7.05688807177$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{31})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 31$$ x^2 - 31 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes 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 $$\beta = 4\sqrt{31}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 3) q^{3} + (2 \beta - 11) q^{5} + ( - 3 \beta + 134) q^{7} + ( - 6 \beta + 262) q^{9}+O(q^{10})$$ q + (b - 3) * q^3 + (2*b - 11) * q^5 + (-3*b + 134) * q^7 + (-6*b + 262) * q^9 $$q + (\beta - 3) q^{3} + (2 \beta - 11) q^{5} + ( - 3 \beta + 134) q^{7} + ( - 6 \beta + 262) q^{9} - 121 q^{11} + ( - 5 \beta + 616) q^{13} + ( - 17 \beta + 1025) q^{15} + (47 \beta - 62) q^{17} + ( - 31 \beta + 972) q^{19} + (143 \beta - 1890) q^{21} + ( - 59 \beta + 1673) q^{23} + ( - 44 \beta - 1020) q^{25} + (37 \beta - 3033) q^{27} + ( - 78 \beta - 3288) q^{29} + ( - 371 \beta + 1249) q^{31} + ( - 121 \beta + 363) q^{33} + (301 \beta - 4450) q^{35} + (94 \beta - 7337) q^{37} + (631 \beta - 4328) q^{39} + ( - 477 \beta - 3252) q^{41} + (644 \beta + 5814) q^{43} + (590 \beta - 8834) q^{45} + ( - 522 \beta + 18408) q^{47} + ( - 804 \beta + 5613) q^{49} + ( - 203 \beta + 23498) q^{51} + ( - 1150 \beta - 1646) q^{53} + ( - 242 \beta + 1331) q^{55} + (1065 \beta - 18292) q^{57} + (585 \beta + 6063) q^{59} + ( - 562 \beta + 36564) q^{61} + ( - 1590 \beta + 44036) q^{63} + (1287 \beta - 11736) q^{65} + (1733 \beta + 14667) q^{67} + (1850 \beta - 34283) q^{69} + ( - 951 \beta - 23061) q^{71} + (799 \beta + 4120) q^{73} + ( - 888 \beta - 18764) q^{75} + (363 \beta - 16214) q^{77} + ( - 852 \beta - 7390) q^{79} + ( - 1686 \beta - 36215) q^{81} + (802 \beta - 37282) q^{83} + ( - 641 \beta + 47306) q^{85} + ( - 3054 \beta - 28824) q^{87} + ( - 650 \beta - 16849) q^{89} + ( - 2518 \beta + 89984) q^{91} + (2362 \beta - 187763) q^{93} + (2285 \beta - 41444) q^{95} + (3188 \beta + 63081) q^{97} + (726 \beta - 31702) q^{99}+O(q^{100})$$ q + (b - 3) * q^3 + (2*b - 11) * q^5 + (-3*b + 134) * q^7 + (-6*b + 262) * q^9 - 121 * q^11 + (-5*b + 616) * q^13 + (-17*b + 1025) * q^15 + (47*b - 62) * q^17 + (-31*b + 972) * q^19 + (143*b - 1890) * q^21 + (-59*b + 1673) * q^23 + (-44*b - 1020) * q^25 + (37*b - 3033) * q^27 + (-78*b - 3288) * q^29 + (-371*b + 1249) * q^31 + (-121*b + 363) * q^33 + (301*b - 4450) * q^35 + (94*b - 7337) * q^37 + (631*b - 4328) * q^39 + (-477*b - 3252) * q^41 + (644*b + 5814) * q^43 + (590*b - 8834) * q^45 + (-522*b + 18408) * q^47 + (-804*b + 5613) * q^49 + (-203*b + 23498) * q^51 + (-1150*b - 1646) * q^53 + (-242*b + 1331) * q^55 + (1065*b - 18292) * q^57 + (585*b + 6063) * q^59 + (-562*b + 36564) * q^61 + (-1590*b + 44036) * q^63 + (1287*b - 11736) * q^65 + (1733*b + 14667) * q^67 + (1850*b - 34283) * q^69 + (-951*b - 23061) * q^71 + (799*b + 4120) * q^73 + (-888*b - 18764) * q^75 + (363*b - 16214) * q^77 + (-852*b - 7390) * q^79 + (-1686*b - 36215) * q^81 + (802*b - 37282) * q^83 + (-641*b + 47306) * q^85 + (-3054*b - 28824) * q^87 + (-650*b - 16849) * q^89 + (-2518*b + 89984) * q^91 + (2362*b - 187763) * q^93 + (2285*b - 41444) * q^95 + (3188*b + 63081) * q^97 + (726*b - 31702) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} - 22 q^{5} + 268 q^{7} + 524 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 - 22 * q^5 + 268 * q^7 + 524 * q^9 $$2 q - 6 q^{3} - 22 q^{5} + 268 q^{7} + 524 q^{9} - 242 q^{11} + 1232 q^{13} + 2050 q^{15} - 124 q^{17} + 1944 q^{19} - 3780 q^{21} + 3346 q^{23} - 2040 q^{25} - 6066 q^{27} - 6576 q^{29} + 2498 q^{31} + 726 q^{33} - 8900 q^{35} - 14674 q^{37} - 8656 q^{39} - 6504 q^{41} + 11628 q^{43} - 17668 q^{45} + 36816 q^{47} + 11226 q^{49} + 46996 q^{51} - 3292 q^{53} + 2662 q^{55} - 36584 q^{57} + 12126 q^{59} + 73128 q^{61} + 88072 q^{63} - 23472 q^{65} + 29334 q^{67} - 68566 q^{69} - 46122 q^{71} + 8240 q^{73} - 37528 q^{75} - 32428 q^{77} - 14780 q^{79} - 72430 q^{81} - 74564 q^{83} + 94612 q^{85} - 57648 q^{87} - 33698 q^{89} + 179968 q^{91} - 375526 q^{93} - 82888 q^{95} + 126162 q^{97} - 63404 q^{99}+O(q^{100})$$ 2 * q - 6 * q^3 - 22 * q^5 + 268 * q^7 + 524 * q^9 - 242 * q^11 + 1232 * q^13 + 2050 * q^15 - 124 * q^17 + 1944 * q^19 - 3780 * q^21 + 3346 * q^23 - 2040 * q^25 - 6066 * q^27 - 6576 * q^29 + 2498 * q^31 + 726 * q^33 - 8900 * q^35 - 14674 * q^37 - 8656 * q^39 - 6504 * q^41 + 11628 * q^43 - 17668 * q^45 + 36816 * q^47 + 11226 * q^49 + 46996 * q^51 - 3292 * q^53 + 2662 * q^55 - 36584 * q^57 + 12126 * q^59 + 73128 * q^61 + 88072 * q^63 - 23472 * q^65 + 29334 * q^67 - 68566 * q^69 - 46122 * q^71 + 8240 * q^73 - 37528 * q^75 - 32428 * q^77 - 14780 * q^79 - 72430 * q^81 - 74564 * q^83 + 94612 * q^85 - 57648 * q^87 - 33698 * q^89 + 179968 * q^91 - 375526 * q^93 - 82888 * q^95 + 126162 * q^97 - 63404 * q^99

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
 −5.56776 5.56776
0 −25.2711 0 −55.5421 0 200.813 0 395.626 0
1.2 0 19.2711 0 33.5421 0 67.1868 0 128.374 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$11$$ $$+1$$

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 44.6.a.b 2
3.b odd 2 1 396.6.a.f 2
4.b odd 2 1 176.6.a.g 2
5.b even 2 1 1100.6.a.b 2
5.c odd 4 2 1100.6.b.c 4
8.b even 2 1 704.6.a.n 2
8.d odd 2 1 704.6.a.m 2
11.b odd 2 1 484.6.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.6.a.b 2 1.a even 1 1 trivial
176.6.a.g 2 4.b odd 2 1
396.6.a.f 2 3.b odd 2 1
484.6.a.d 2 11.b odd 2 1
704.6.a.m 2 8.d odd 2 1
704.6.a.n 2 8.b even 2 1
1100.6.a.b 2 5.b even 2 1
1100.6.b.c 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 6T_{3} - 487$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(44))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 6T - 487$$
$5$ $$T^{2} + 22T - 1863$$
$7$ $$T^{2} - 268T + 13492$$
$11$ $$(T + 121)^{2}$$
$13$ $$T^{2} - 1232 T + 367056$$
$17$ $$T^{2} + 124 T - 1091820$$
$19$ $$T^{2} - 1944 T + 468128$$
$23$ $$T^{2} - 3346 T + 1072353$$
$29$ $$T^{2} + 6576 T + 7793280$$
$31$ $$T^{2} - 2498 T - 66709935$$
$37$ $$T^{2} + 14674 T + 49448913$$
$41$ $$T^{2} + 6504 T - 102278880$$
$43$ $$T^{2} - 11628 T - 171906460$$
$47$ $$T^{2} - 36816 T + 203702400$$
$53$ $$T^{2} + 3292 T - 653250684$$
$59$ $$T^{2} - 12126 T - 132983631$$
$61$ $$T^{2} + \cdots + 1180267472$$
$67$ $$T^{2} + \cdots - 1274510455$$
$71$ $$T^{2} + 46122 T + 83226825$$
$73$ $$T^{2} - 8240 T - 299672496$$
$79$ $$T^{2} + 14780 T - 305436284$$
$83$ $$T^{2} + \cdots + 1070918340$$
$89$ $$T^{2} + 33698 T + 74328801$$
$97$ $$T^{2} + \cdots - 1061806063$$