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Properties

Label 2011.2.a.b
Level $2011$
Weight $2$
Character orbit 2011.a
Self dual yes
Analytic conductor $16.058$
Analytic rank $0$
Dimension $90$
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: \(0\)
Dimension: \(90\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 90 q + 11 q^{2} + 9 q^{3} + 95 q^{4} + 47 q^{5} + 20 q^{6} + 4 q^{7} + 33 q^{8} + 109 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 90 q + 11 q^{2} + 9 q^{3} + 95 q^{4} + 47 q^{5} + 20 q^{6} + 4 q^{7} + 33 q^{8} + 109 q^{9} + 19 q^{10} + 24 q^{11} + 14 q^{12} + 36 q^{13} + 43 q^{14} + 4 q^{15} + 93 q^{16} + 55 q^{17} + 18 q^{18} + 15 q^{19} + 76 q^{20} + 65 q^{21} - 3 q^{22} + 30 q^{23} + 46 q^{24} + 107 q^{25} + 38 q^{26} + 21 q^{27} + 2 q^{28} + 149 q^{29} + q^{30} + 33 q^{31} + 67 q^{32} + 13 q^{33} + 15 q^{34} + 34 q^{35} + 103 q^{36} + 23 q^{37} + 38 q^{38} + 32 q^{39} + 43 q^{40} + 144 q^{41} - 20 q^{42} - 5 q^{43} + 37 q^{44} + 103 q^{45} + 8 q^{46} + 28 q^{47} + 12 q^{48} + 114 q^{49} + 67 q^{50} + 11 q^{51} + 59 q^{52} + 59 q^{53} + 38 q^{54} + 3 q^{55} + 106 q^{56} + 2 q^{57} - 5 q^{58} + 86 q^{59} - 28 q^{60} + 113 q^{61} + 12 q^{62} - 29 q^{63} + 71 q^{64} + 51 q^{65} + 15 q^{66} - 14 q^{67} + 96 q^{68} + 116 q^{69} - 24 q^{70} + 47 q^{71} + 13 q^{72} + 22 q^{73} + 57 q^{74} + 7 q^{75} + 2 q^{76} + 100 q^{77} - 34 q^{78} + 18 q^{79} + 100 q^{80} + 154 q^{81} - 4 q^{82} + 24 q^{83} + 35 q^{84} + 30 q^{85} - q^{86} + 49 q^{87} - 74 q^{88} + 97 q^{89} + 22 q^{90} - 25 q^{91} + 23 q^{92} + 32 q^{93} + 21 q^{94} + 56 q^{95} + 29 q^{96} + 26 q^{97} + 15 q^{98} - 11 q^{99}+O(q^{100}) \) 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 −2.73560 −2.44822 5.48349 −1.66786 6.69734 −3.51512 −9.52943 2.99378 4.56259
1.2 −2.71918 0.710368 5.39393 1.00988 −1.93162 −1.70425 −9.22870 −2.49538 −2.74606
1.3 −2.64310 1.72816 4.98599 0.778079 −4.56770 −3.99781 −7.89228 −0.0134675 −2.05654
1.4 −2.57947 −2.56013 4.65368 2.86545 6.60379 0.908168 −6.84510 3.55428 −7.39135
1.5 −2.47357 2.87390 4.11857 −1.72812 −7.10880 2.21112 −5.24043 5.25930 4.27462
1.6 −2.38950 2.69179 3.70972 2.78252 −6.43204 −1.06979 −4.08539 4.24574 −6.64884
1.7 −2.36376 −1.29958 3.58735 1.46524 3.07189 4.02276 −3.75210 −1.31109 −3.46348
1.8 −2.32550 −0.161403 3.40797 0.999797 0.375342 −4.86701 −3.27423 −2.97395 −2.32503
1.9 −2.32243 1.62400 3.39368 3.22643 −3.77162 3.90862 −3.23673 −0.362630 −7.49316
1.10 −2.30849 −0.566470 3.32914 −0.195371 1.30769 0.321334 −3.06830 −2.67911 0.451012
1.11 −2.25469 −2.58869 3.08364 −1.87674 5.83670 −0.960990 −2.44328 3.70130 4.23147
1.12 −2.24510 −3.30517 3.04046 3.11080 7.42044 −4.52544 −2.33594 7.92418 −6.98405
1.13 −2.19848 −0.0830286 2.83330 −1.81066 0.182537 1.42772 −1.83200 −2.99311 3.98069
1.14 −2.00555 0.462381 2.02225 −2.40107 −0.927330 1.12009 −0.0446211 −2.78620 4.81548
1.15 −1.94922 −2.03830 1.79948 3.33364 3.97310 2.50256 0.390866 1.15465 −6.49802
1.16 −1.87387 −0.640540 1.51138 −2.41353 1.20029 −1.29099 0.915613 −2.58971 4.52263
1.17 −1.86200 −0.824474 1.46703 4.18296 1.53517 2.51013 0.992388 −2.32024 −7.78865
1.18 −1.82168 2.92306 1.31851 −3.21312 −5.32488 −3.48423 1.24146 5.54431 5.85327
1.19 −1.78972 3.22143 1.20310 2.22731 −5.76546 4.02762 1.42623 7.37762 −3.98627
1.20 −1.77648 −3.05691 1.15589 −3.10407 5.43054 −0.458013 1.49955 6.34468 5.51432
See all 90 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.90
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.b ✓ 90
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2011.2.a.b ✓ 90 1.a even 1 1 trivial