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Properties

Label 1441.2.a.e
Level $1441$
Weight $2$
Character orbit 1441.a
Self dual yes
Analytic conductor $11.506$
Analytic rank $0$
Dimension $28$
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] = [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: \(0\)
Dimension: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 7 q^{2} - 2 q^{3} + 27 q^{4} + 3 q^{5} - q^{6} + 7 q^{7} + 21 q^{8} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 7 q^{2} - 2 q^{3} + 27 q^{4} + 3 q^{5} - q^{6} + 7 q^{7} + 21 q^{8} + 28 q^{9} + 14 q^{10} + 28 q^{11} - 6 q^{12} + 3 q^{13} - q^{14} + 19 q^{15} + 29 q^{16} + 9 q^{17} - 2 q^{18} + 20 q^{19} + 6 q^{20} + 6 q^{21} + 7 q^{22} + 24 q^{23} + 20 q^{24} + 23 q^{25} + 10 q^{26} - 20 q^{27} - 3 q^{28} + 43 q^{29} + 11 q^{30} + 3 q^{31} + 44 q^{32} - 2 q^{33} - 28 q^{34} + 32 q^{35} + 24 q^{36} - 4 q^{37} + 24 q^{38} + 37 q^{39} + 22 q^{40} + 32 q^{41} - 27 q^{42} + 25 q^{43} + 27 q^{44} - 36 q^{45} + 10 q^{46} + 19 q^{47} + 42 q^{48} + 17 q^{49} + q^{50} + 39 q^{51} - 19 q^{52} + 5 q^{53} + 6 q^{54} + 3 q^{55} + 8 q^{56} + 2 q^{57} + 21 q^{58} + 44 q^{59} + 65 q^{60} + 28 q^{61} + 60 q^{62} - 8 q^{63} + 5 q^{64} + 33 q^{65} - q^{66} + 7 q^{67} + 13 q^{68} - 22 q^{69} + 9 q^{70} + 117 q^{71} - 17 q^{72} + 7 q^{73} + 41 q^{74} - 40 q^{75} + 34 q^{76} + 7 q^{77} - 97 q^{78} + 48 q^{79} + 41 q^{80} + 40 q^{81} + 2 q^{82} + 22 q^{83} + 27 q^{84} + 30 q^{85} + 24 q^{86} + 37 q^{87} + 21 q^{88} - 6 q^{89} + 4 q^{90} - 33 q^{91} + 18 q^{92} + 5 q^{93} - 43 q^{94} + 64 q^{95} + 55 q^{96} - 50 q^{97} + 97 q^{98} + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 −2.46758 −2.74867 4.08895 0.488919 6.78255 −1.50214 −5.15465 4.55516 −1.20645
1.2 −2.36893 0.351154 3.61185 −1.64180 −0.831861 −1.35814 −3.81837 −2.87669 3.88932
1.3 −2.33619 2.29063 3.45776 −0.314509 −5.35135 0.543204 −3.40560 2.24701 0.734752
1.4 −2.09032 1.02488 2.36944 3.67569 −2.14232 2.35755 −0.772245 −1.94963 −7.68337
1.5 −1.79989 −0.0643329 1.23961 −3.25644 0.115792 2.45508 1.36862 −2.99586 5.86125
1.6 −1.77267 −3.40600 1.14234 −3.60328 6.03770 0.417068 1.52034 8.60085 6.38741
1.7 −1.13972 3.40925 −0.701031 −0.104086 −3.88560 1.00506 3.07843 8.62297 0.118629
1.8 −1.03351 −2.03030 −0.931861 −0.515805 2.09833 −3.45818 3.03010 1.12211 0.533089
1.9 −1.01237 −1.33231 −0.975116 3.44940 1.34878 1.50282 3.01190 −1.22496 −3.49205
1.10 −0.660505 2.12062 −1.56373 2.37769 −1.40068 3.00733 2.35386 1.49703 −1.57048
1.11 −0.574950 1.45629 −1.66943 −2.31674 −0.837294 −3.08132 2.10974 −0.879222 1.33201
1.12 −0.452030 −0.254542 −1.79567 −0.679800 0.115061 3.95437 1.71576 −2.93521 0.307290
1.13 0.00299522 0.0189616 −1.99999 1.18959 5.67941e−5 0 −5.08173 −0.0119809 −2.99964 0.00356309
1.14 0.235116 −2.49285 −1.94472 2.71737 −0.586108 3.95783 −0.927465 3.21431 0.638897
1.15 0.544027 2.32406 −1.70404 −4.20016 1.26435 −0.904548 −2.01509 2.40124 −2.28500
1.16 0.635564 −1.42688 −1.59606 2.67263 −0.906871 −2.11184 −2.28553 −0.964025 1.69862
1.17 0.851882 2.55735 −1.27430 2.24117 2.17856 −0.305679 −2.78931 3.54005 1.90921
1.18 1.06795 −2.13569 −0.859479 −3.41004 −2.28082 1.64444 −3.05379 1.56119 −3.64175
1.19 1.13327 0.404748 −0.715692 −1.18320 0.458690 4.77646 −3.07762 −2.83618 −1.34089
1.20 1.67300 −1.49184 0.798939 −1.12664 −2.49585 −0.438386 −2.00938 −0.774411 −1.88488
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.28
Significant digits:
Format:

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.e ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1441.2.a.e ✓ 28 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} - 7 T_{2}^{27} - 17 T_{2}^{26} + 217 T_{2}^{25} - 45 T_{2}^{24} - 2862 T_{2}^{23} + 3378 T_{2}^{22} + \cdots + 6 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1441))\). Copy content Toggle raw display