# Properties

 Label 1441.2.a.c Level $1441$ Weight $2$ Character orbit 1441.a Self dual yes Analytic conductor $11.506$ Analytic rank $1$ Dimension $23$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1441,2,Mod(1,1441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1441.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1441 = 11 \cdot 131$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1441.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: $$11.5064429313$$ Analytic rank: $$1$$ Dimension: $$23$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$23 q - 7 q^{2} - 3 q^{3} + 15 q^{4} - 9 q^{5} - 11 q^{6} - 12 q^{7} - 21 q^{8} + 12 q^{9}+O(q^{10})$$ 23 * q - 7 * q^2 - 3 * q^3 + 15 * q^4 - 9 * q^5 - 11 * q^6 - 12 * q^7 - 21 * q^8 + 12 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$23 q - 7 q^{2} - 3 q^{3} + 15 q^{4} - 9 q^{5} - 11 q^{6} - 12 q^{7} - 21 q^{8} + 12 q^{9} - 2 q^{10} + 23 q^{11} + 8 q^{12} - 24 q^{13} - 13 q^{14} - 27 q^{15} + 7 q^{16} - 7 q^{17} - 14 q^{18} - 18 q^{19} - 4 q^{20} - 29 q^{21} - 7 q^{22} - 26 q^{23} - 4 q^{24} + 18 q^{25} + 8 q^{26} - 3 q^{27} - 11 q^{28} - 45 q^{29} + 19 q^{30} - 23 q^{31} - 34 q^{32} - 3 q^{33} - 2 q^{34} - 18 q^{35} - 6 q^{36} - 2 q^{37} - 8 q^{38} - 40 q^{39} - 24 q^{40} - 23 q^{41} + 59 q^{42} - 14 q^{43} + 15 q^{44} - 18 q^{45} - 12 q^{46} - 55 q^{47} + 10 q^{48} + 11 q^{49} - 41 q^{50} - 21 q^{51} - 37 q^{52} - 10 q^{53} - 68 q^{54} - 9 q^{55} + 2 q^{56} - 18 q^{57} + 27 q^{58} - 75 q^{59} - 63 q^{60} - 55 q^{61} + 14 q^{62} - 16 q^{63} + 19 q^{64} - 25 q^{65} - 11 q^{66} + 17 q^{67} + 41 q^{68} - 22 q^{69} + 27 q^{70} - 105 q^{71} - 11 q^{72} - 3 q^{73} - 39 q^{74} + 25 q^{75} - 30 q^{76} - 12 q^{77} + 25 q^{78} - 48 q^{79} - 37 q^{80} + 3 q^{81} + 36 q^{82} + 4 q^{83} - 111 q^{84} - 30 q^{85} + 22 q^{86} + 5 q^{87} - 21 q^{88} - 39 q^{89} + 100 q^{90} - 22 q^{91} - 30 q^{92} - 5 q^{93} + 11 q^{94} - 88 q^{95} + 13 q^{96} + 24 q^{97} - 91 q^{98} + 12 q^{99}+O(q^{100})$$ 23 * q - 7 * q^2 - 3 * q^3 + 15 * q^4 - 9 * q^5 - 11 * q^6 - 12 * q^7 - 21 * q^8 + 12 * q^9 - 2 * q^10 + 23 * q^11 + 8 * q^12 - 24 * q^13 - 13 * q^14 - 27 * q^15 + 7 * q^16 - 7 * q^17 - 14 * q^18 - 18 * q^19 - 4 * q^20 - 29 * q^21 - 7 * q^22 - 26 * q^23 - 4 * q^24 + 18 * q^25 + 8 * q^26 - 3 * q^27 - 11 * q^28 - 45 * q^29 + 19 * q^30 - 23 * q^31 - 34 * q^32 - 3 * q^33 - 2 * q^34 - 18 * q^35 - 6 * q^36 - 2 * q^37 - 8 * q^38 - 40 * q^39 - 24 * q^40 - 23 * q^41 + 59 * q^42 - 14 * q^43 + 15 * q^44 - 18 * q^45 - 12 * q^46 - 55 * q^47 + 10 * q^48 + 11 * q^49 - 41 * q^50 - 21 * q^51 - 37 * q^52 - 10 * q^53 - 68 * q^54 - 9 * q^55 + 2 * q^56 - 18 * q^57 + 27 * q^58 - 75 * q^59 - 63 * q^60 - 55 * q^61 + 14 * q^62 - 16 * q^63 + 19 * q^64 - 25 * q^65 - 11 * q^66 + 17 * q^67 + 41 * q^68 - 22 * q^69 + 27 * q^70 - 105 * q^71 - 11 * q^72 - 3 * q^73 - 39 * q^74 + 25 * q^75 - 30 * q^76 - 12 * q^77 + 25 * q^78 - 48 * q^79 - 37 * q^80 + 3 * q^81 + 36 * q^82 + 4 * q^83 - 111 * q^84 - 30 * q^85 + 22 * q^86 + 5 * q^87 - 21 * q^88 - 39 * q^89 + 100 * q^90 - 22 * q^91 - 30 * q^92 - 5 * q^93 + 11 * q^94 - 88 * q^95 + 13 * q^96 + 24 * q^97 - 91 * q^98 + 12 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −2.78830 −0.518611 5.77463 −0.346547 1.44605 3.25187 −10.5248 −2.73104 0.966278
1.2 −2.56442 0.292938 4.57625 3.66360 −0.751215 −5.05392 −6.60659 −2.91419 −9.39502
1.3 −2.50019 3.16294 4.25093 −4.26480 −7.90794 −1.67608 −5.62776 7.00418 10.6628
1.4 −2.24881 −1.99348 3.05713 −0.623302 4.48295 4.70119 −2.37729 0.973955 1.40169
1.5 −1.84752 2.72852 1.41334 1.47029 −5.04100 −2.80791 1.08387 4.44483 −2.71640
1.6 −1.75710 −2.59102 1.08739 2.84782 4.55267 2.84815 1.60354 3.71338 −5.00390
1.7 −1.59929 1.84862 0.557714 −2.20532 −2.95647 0.234218 2.30663 0.417399 3.52693
1.8 −1.52508 −0.420875 0.325868 −3.40099 0.641868 −3.60038 2.55318 −2.82286 5.18678
1.9 −1.24808 −0.917360 −0.442292 1.92098 1.14494 0.0608487 3.04818 −2.15845 −2.39755
1.10 −1.00403 1.28366 −0.991926 0.810675 −1.28883 −2.46332 3.00398 −1.35222 −0.813941
1.11 −0.363851 −2.50967 −1.86761 −1.53260 0.913144 −1.87155 1.40723 3.29843 0.557639
1.12 −0.288878 2.13203 −1.91655 −3.62334 −0.615896 3.72928 1.13141 1.54555 1.04670
1.13 −0.169669 −1.85024 −1.97121 −3.73136 0.313927 1.69497 0.673790 0.423370 0.633095
1.14 0.338964 0.415229 −1.88510 2.48639 0.140748 1.37068 −1.31691 −2.82758 0.842796
1.15 0.495822 −0.133739 −1.75416 −0.493567 −0.0663105 1.67893 −1.86140 −2.98211 −0.244721
1.16 0.665447 −3.11635 −1.55718 −0.365961 −2.07377 −1.41982 −2.36711 6.71165 −0.243528
1.17 0.714474 2.60520 −1.48953 −0.266226 1.86135 −4.25119 −2.49318 3.78706 −0.190211
1.18 1.32496 0.816970 −0.244480 −1.13056 1.08245 0.353069 −2.97385 −2.33256 −1.49794
1.19 1.34408 −2.56743 −0.193462 4.05031 −3.45081 −3.27147 −2.94818 3.59168 5.44393
1.20 1.75422 −0.900924 1.07730 1.37756 −1.58042 −1.58298 −1.61862 −2.18834 2.41656
See all 23 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.23 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$11$$ $$-1$$
$$131$$ $$-1$$

## 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 1441.2.a.c 23

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1441.2.a.c 23 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{23} + 7 T_{2}^{22} - 6 T_{2}^{21} - 140 T_{2}^{20} - 138 T_{2}^{19} + 1131 T_{2}^{18} + 2001 T_{2}^{17} - 4693 T_{2}^{16} - 11448 T_{2}^{15} + 10339 T_{2}^{14} + 35570 T_{2}^{13} - 10374 T_{2}^{12} - 64708 T_{2}^{11} + \cdots + 16$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1441))$$.