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Properties

Label 1441.2.a.f
Level $1441$
Weight $2$
Character orbit 1441.a
Self dual yes
Analytic conductor $11.506$
Analytic rank $0$
Dimension $31$
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: \(31\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 31 q + 6 q^{2} + 4 q^{3} + 38 q^{4} + 8 q^{5} + 7 q^{6} + 4 q^{7} + 24 q^{8} + 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 31 q + 6 q^{2} + 4 q^{3} + 38 q^{4} + 8 q^{5} + 7 q^{6} + 4 q^{7} + 24 q^{8} + 45 q^{9} - 8 q^{10} - 31 q^{11} + 10 q^{12} - 8 q^{13} + 29 q^{14} + 36 q^{15} + 52 q^{16} - q^{17} + 33 q^{18} - 2 q^{19} + 22 q^{20} - 13 q^{21} - 6 q^{22} + 45 q^{23} + 16 q^{24} + 41 q^{25} + 24 q^{26} + 22 q^{27} + 17 q^{28} + 5 q^{29} + 29 q^{30} + 28 q^{31} + 69 q^{32} - 4 q^{33} + 14 q^{34} + 36 q^{35} + 63 q^{36} - 3 q^{37} + 4 q^{38} + 40 q^{39} - 48 q^{40} + 21 q^{41} - 9 q^{42} - 20 q^{43} - 38 q^{44} + 28 q^{45} - 24 q^{46} + 57 q^{47} - 46 q^{48} + 37 q^{49} + 64 q^{50} + 17 q^{51} - 11 q^{52} + 32 q^{53} - 26 q^{54} - 8 q^{55} + 84 q^{56} + 10 q^{57} - 17 q^{58} + 70 q^{59} - 33 q^{60} - 51 q^{61} - 34 q^{62} + 32 q^{63} + 80 q^{64} - q^{65} - 7 q^{66} + 24 q^{67} - 13 q^{68} + 19 q^{69} - 9 q^{70} + 128 q^{71} + 118 q^{72} - 27 q^{73} - 23 q^{74} + 41 q^{75} - 34 q^{76} - 4 q^{77} + 9 q^{78} + 2 q^{79} - 45 q^{80} + 43 q^{81} - 18 q^{82} + 46 q^{83} - 103 q^{84} - 50 q^{85} + 78 q^{86} - 9 q^{87} - 24 q^{88} + 52 q^{89} - 46 q^{90} + 38 q^{91} + 54 q^{92} + 4 q^{93} + 3 q^{94} + 70 q^{95} - 21 q^{96} + 3 q^{97} - 120 q^{98} - 45 q^{99}+O(q^{100}) \) 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 −2.67025 −2.19223 5.13021 1.01111 5.85378 −2.59765 −8.35844 1.80585 −2.69992
1.2 −2.51757 −0.0469654 4.33818 2.78146 0.118239 3.17303 −5.88653 −2.99779 −7.00252
1.3 −2.29335 2.59523 3.25943 0.281199 −5.95177 −4.71795 −2.88831 3.73524 −0.644885
1.4 −2.27484 0.433632 3.17492 −2.29703 −0.986446 −4.86805 −2.67276 −2.81196 5.22539
1.5 −2.10874 −1.73148 2.44680 1.13214 3.65125 3.68428 −0.942192 −0.00197058 −2.38740
1.6 −1.89426 2.98322 1.58823 3.71742 −5.65101 2.24937 0.780003 5.89963 −7.04177
1.7 −1.61641 −1.01358 0.612772 4.09374 1.63835 −3.10004 2.24232 −1.97266 −6.61715
1.8 −1.51851 2.67622 0.305884 −2.61923 −4.06387 2.29369 2.57254 4.16214 3.97734
1.9 −1.31847 −2.88980 −0.261634 −1.34376 3.81011 0.384750 2.98190 5.35092 1.77171
1.10 −1.09944 −1.67977 −0.791230 −0.843613 1.84680 0.723048 3.06879 −0.178383 0.927502
1.11 −0.999239 0.560764 −1.00152 0.738008 −0.560337 −0.683383 2.99924 −2.68554 −0.737446
1.12 −0.749638 0.436865 −1.43804 −3.66899 −0.327491 −2.83791 2.57729 −2.80915 2.75042
1.13 −0.413206 −2.92261 −1.82926 −1.23288 1.20764 5.16907 1.58227 5.54166 0.509432
1.14 −0.187162 3.03513 −1.96497 3.17183 −0.568060 −3.01076 0.742092 6.21199 −0.593646
1.15 0.00234753 2.44529 −1.99999 1.45917 0.00574040 4.30357 −0.00939012 2.97943 0.00342546
1.16 0.262885 0.849379 −1.93089 −1.83441 0.223289 −0.312976 −1.03337 −2.27856 −0.482239
1.17 0.651423 −0.506634 −1.57565 −2.90422 −0.330033 −3.71271 −2.32926 −2.74332 −1.89188
1.18 0.800366 −3.25822 −1.35941 0.350077 −2.60777 −2.72656 −2.68876 7.61603 0.280189
1.19 0.867375 0.595469 −1.24766 3.84366 0.516495 1.26797 −2.81694 −2.64542 3.33390
1.20 1.13757 −1.95057 −0.705942 1.75766 −2.21891 1.09174 −3.07819 0.804735 1.99945
See all 31 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.31
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.f ✓ 31
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1441.2.a.f ✓ 31 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{31} - 6 T_{2}^{30} - 32 T_{2}^{29} + 248 T_{2}^{28} + 368 T_{2}^{27} - 4537 T_{2}^{26} + \cdots - 32 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1441))\). Copy content Toggle raw display