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Properties

Label 2011.2.a.a
Level $2011$
Weight $2$
Character orbit 2011.a
Self dual yes
Analytic conductor $16.058$
Analytic rank $1$
Dimension $77$
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] = [2011,2,Mod(1,2011)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2011, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2011.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2011 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2011.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: \(16.0579158465\)
Analytic rank: \(1\)
Dimension: \(77\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 77 q - 13 q^{2} - 13 q^{3} + 67 q^{4} - 47 q^{5} - 20 q^{6} - 8 q^{7} - 33 q^{8} + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 77 q - 13 q^{2} - 13 q^{3} + 67 q^{4} - 47 q^{5} - 20 q^{6} - 8 q^{7} - 33 q^{8} + 52 q^{9} - 21 q^{10} - 34 q^{11} - 36 q^{12} - 34 q^{13} - 49 q^{14} - 12 q^{15} + 47 q^{16} - 59 q^{17} - 24 q^{18} - 31 q^{19} - 82 q^{20} - 71 q^{21} - 3 q^{22} - 28 q^{23} - 50 q^{24} + 68 q^{25} - 54 q^{26} - 43 q^{27} - 2 q^{28} - 151 q^{29} + q^{30} - 37 q^{31} - 59 q^{32} - 35 q^{33} - q^{34} - 58 q^{35} + 19 q^{36} - 29 q^{37} - 22 q^{38} - 40 q^{39} - 41 q^{40} - 142 q^{41} + 16 q^{42} - 23 q^{43} - 89 q^{44} - 119 q^{45} - 6 q^{46} - 36 q^{47} - 46 q^{48} + 45 q^{49} - 29 q^{50} - 53 q^{51} - 11 q^{52} - 69 q^{53} - 50 q^{54} - 13 q^{55} - 122 q^{56} - 14 q^{57} + 31 q^{58} - 92 q^{59} + 20 q^{60} - 115 q^{61} - 66 q^{62} - 25 q^{63} + 37 q^{64} - 57 q^{65} - 17 q^{66} - 108 q^{68} - 160 q^{69} + 40 q^{70} - 67 q^{71} - 35 q^{72} - 36 q^{73} - 55 q^{74} - 51 q^{75} - 56 q^{76} - 116 q^{77} + 22 q^{78} - 42 q^{79} - 114 q^{80} + 37 q^{81} + 18 q^{82} - 42 q^{83} - 77 q^{84} - 18 q^{85} - 33 q^{86} - 7 q^{87} - 2 q^{88} - 93 q^{89} - 34 q^{90} - 37 q^{91} - 55 q^{92} - 8 q^{93} - 35 q^{94} - 64 q^{95} - 83 q^{96} - 16 q^{97} - 57 q^{98} - 65 q^{99}+O(q^{100}) \) 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 −2.76376 0.184789 5.63836 0.567323 −0.510713 3.98319 −10.0556 −2.96585 −1.56794
1.2 −2.74022 2.07071 5.50879 −2.32761 −5.67420 1.82697 −9.61486 1.28784 6.37817
1.3 −2.70121 −0.800757 5.29656 4.14776 2.16302 −0.544039 −8.90472 −2.35879 −11.2040
1.4 −2.59047 2.90849 4.71055 −0.332972 −7.53436 −0.345794 −7.02159 5.45930 0.862554
1.5 −2.56905 −3.29856 4.60003 −1.98223 8.47417 3.09967 −6.67961 7.88048 5.09244
1.6 −2.56536 −1.45727 4.58106 −3.76790 3.73843 4.67717 −6.62135 −0.876353 9.66602
1.7 −2.55772 −0.969080 4.54191 −3.30788 2.47863 −1.46662 −6.50148 −2.06088 8.46061
1.8 −2.55236 −1.71417 4.51454 0.273588 4.37517 −0.964162 −6.41801 −0.0616276 −0.698295
1.9 −2.30867 1.99206 3.32996 −3.84947 −4.59902 −0.0467895 −3.07045 0.968319 8.88716
1.10 −2.28290 1.00065 3.21164 0.355171 −2.28438 2.08053 −2.76607 −1.99870 −0.810822
1.11 −2.27120 3.19440 3.15836 0.518665 −7.25513 −3.96582 −2.63086 7.20420 −1.17799
1.12 −2.23267 −0.954482 2.98483 2.41155 2.13105 −3.40709 −2.19880 −2.08896 −5.38421
1.13 −2.20702 0.857753 2.87092 −2.40942 −1.89308 −4.04007 −1.92213 −2.26426 5.31763
1.14 −2.10114 −2.11084 2.41479 −0.00654949 4.43518 3.82034 −0.871525 1.45567 0.0137614
1.15 −2.04796 1.69073 2.19412 3.45749 −3.46253 −0.456303 −0.397551 −0.141445 −7.08078
1.16 −1.91229 −2.23317 1.65686 1.03328 4.27047 −0.947039 0.656182 1.98705 −1.97593
1.17 −1.88739 0.958132 1.56223 −3.60051 −1.80837 5.25796 0.826234 −2.08198 6.79555
1.18 −1.74900 −2.40312 1.05899 −4.35016 4.20305 −4.48930 1.64582 2.77499 7.60842
1.19 −1.69280 −3.42746 0.865569 2.32653 5.80201 3.69630 1.92036 8.74749 −3.93835
1.20 −1.67179 0.416908 0.794887 1.68997 −0.696984 −1.05186 2.01470 −2.82619 −2.82529
See all 77 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.77
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2011\) \(1\)

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 2011.2.a.a ✓ 77
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2011.2.a.a ✓ 77 1.a even 1 1 trivial