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Properties

Label 6011.2.a.e
Level $6011$
Weight $2$
Character orbit 6011.a
Self dual yes
Analytic conductor $47.998$
Analytic rank $1$
Dimension $221$
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] = [6011,2,Mod(1,6011)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6011, 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("6011.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6011 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6011.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: \(47.9980766550\)
Analytic rank: \(1\)
Dimension: \(221\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 221 q - 15 q^{2} - 17 q^{3} + 189 q^{4} - 32 q^{5} - 33 q^{6} - 40 q^{7} - 39 q^{8} + 176 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 221 q - 15 q^{2} - 17 q^{3} + 189 q^{4} - 32 q^{5} - 33 q^{6} - 40 q^{7} - 39 q^{8} + 176 q^{9} - 61 q^{10} - 50 q^{11} - 43 q^{12} - 87 q^{13} - 41 q^{14} - 62 q^{15} + 129 q^{16} - 29 q^{17} - 61 q^{18} - 107 q^{19} - 59 q^{20} - 163 q^{21} - 70 q^{22} - 31 q^{23} - 98 q^{24} + 119 q^{25} - 23 q^{26} - 41 q^{27} - 112 q^{28} - 152 q^{29} - 66 q^{30} - 117 q^{31} - 93 q^{32} - 60 q^{33} - 80 q^{34} - 21 q^{35} + 92 q^{36} - 231 q^{37} + 2 q^{38} - 81 q^{39} - 143 q^{40} - 81 q^{41} - 6 q^{42} - 126 q^{43} - 115 q^{44} - 156 q^{45} - 205 q^{46} - 4 q^{47} - 55 q^{48} + 103 q^{49} - 61 q^{50} - 106 q^{51} - 164 q^{52} - 87 q^{53} - 110 q^{54} - 62 q^{55} - 73 q^{56} - 136 q^{57} - 128 q^{58} - 76 q^{59} - 148 q^{60} - 345 q^{61} + 5 q^{62} - 74 q^{63} - 25 q^{64} - 110 q^{65} - 34 q^{66} - 104 q^{67} - 48 q^{68} - 133 q^{69} - 92 q^{70} - 39 q^{71} - 177 q^{72} - 175 q^{73} - 44 q^{74} - 23 q^{75} - 268 q^{76} - 81 q^{77} - 19 q^{78} - 272 q^{79} - 60 q^{80} + 77 q^{81} - 13 q^{82} - 40 q^{83} - 221 q^{84} - 376 q^{85} - 82 q^{86} - 3 q^{87} - 234 q^{88} - 92 q^{89} - 91 q^{90} - 205 q^{91} - 11 q^{92} - 125 q^{93} - 126 q^{94} - 56 q^{95} - 148 q^{96} - 133 q^{97} - 4 q^{98} - 195 q^{99}+O(q^{100}) \) 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 −2.80701 −2.40971 5.87930 2.03862 6.76409 2.90054 −10.8892 2.80672 −5.72241
1.2 −2.77821 2.41290 5.71844 −0.0351443 −6.70355 −1.23918 −10.3306 2.82211 0.0976383
1.3 −2.73985 2.92308 5.50679 −0.916253 −8.00881 0.545086 −9.60807 5.54441 2.51040
1.4 −2.72000 −0.0742815 5.39842 2.94538 0.202046 −4.01910 −9.24371 −2.99448 −8.01145
1.5 −2.69018 0.438844 5.23705 −1.01798 −1.18057 1.02718 −8.70824 −2.80742 2.73853
1.6 −2.66798 −1.54490 5.11814 −2.74500 4.12178 −3.36744 −8.31916 −0.613269 7.32363
1.7 −2.66460 −1.59742 5.10010 2.00952 4.25649 1.11567 −8.26053 −0.448248 −5.35457
1.8 −2.63998 0.698590 4.96949 2.68847 −1.84426 −1.30377 −7.83938 −2.51197 −7.09752
1.9 −2.59248 0.192728 4.72097 1.84783 −0.499645 −2.86648 −7.05406 −2.96286 −4.79047
1.10 −2.55964 0.828942 4.55177 −3.75172 −2.12180 −2.60223 −6.53163 −2.31285 9.60307
1.11 −2.55631 2.23851 4.53471 2.90049 −5.72232 1.63127 −6.47949 2.01093 −7.41455
1.12 −2.53639 −0.500202 4.43325 0.151225 1.26871 0.853062 −6.17167 −2.74980 −0.383566
1.13 −2.53178 −2.36548 4.40990 −0.538488 5.98887 −3.13679 −6.10134 2.59549 1.36333
1.14 −2.52486 3.20374 4.37490 −3.72596 −8.08898 −4.85825 −5.99630 7.26393 9.40752
1.15 −2.51880 2.78779 4.34434 4.21455 −7.02187 −4.02027 −5.90490 4.77177 −10.6156
1.16 −2.51760 1.78803 4.33833 0.754203 −4.50154 4.46405 −5.88698 0.197034 −1.89878
1.17 −2.48261 −2.01031 4.16334 −2.06004 4.99080 −1.12812 −5.37073 1.04134 5.11427
1.18 −2.45556 −3.18088 4.02979 3.67099 7.81084 4.15705 −4.98426 7.11798 −9.01435
1.19 −2.44176 −0.0989876 3.96219 −1.83713 0.241704 2.86274 −4.79120 −2.99020 4.48584
1.20 −2.43824 −2.91124 3.94502 −1.87258 7.09830 1.17191 −4.74244 5.47531 4.56580
See next 80 embeddings (of 221 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.221
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(6011\) \(1\)

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 6011.2.a.e ✓ 221
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6011.2.a.e ✓ 221 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{221} + 15 T_{2}^{220} - 203 T_{2}^{219} - 4137 T_{2}^{218} + 16123 T_{2}^{217} + \cdots + 21232262853324 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6011))\). Copy content Toggle raw display