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Properties

Label 6011.2.a.f
Level $6011$
Weight $2$
Character orbit 6011.a
Self dual yes
Analytic conductor $47.998$
Analytic rank $0$
Dimension $275$
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: \(0\)
Dimension: \(275\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 275 q + 16 q^{2} + 9 q^{3} + 316 q^{4} + 36 q^{5} + 30 q^{6} + 41 q^{7} + 36 q^{8} + 322 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 275 q + 16 q^{2} + 9 q^{3} + 316 q^{4} + 36 q^{5} + 30 q^{6} + 41 q^{7} + 36 q^{8} + 322 q^{9} + 44 q^{10} + 42 q^{11} + 26 q^{12} + 97 q^{13} + 24 q^{14} + 46 q^{15} + 386 q^{16} + 35 q^{17} + 47 q^{18} + 101 q^{19} + 60 q^{20} + 187 q^{21} + 72 q^{22} + 35 q^{23} + 73 q^{24} + 373 q^{25} + 21 q^{26} + 27 q^{27} + 97 q^{28} + 162 q^{29} + 13 q^{30} + 113 q^{31} + 58 q^{32} + 16 q^{33} + 52 q^{34} + 23 q^{35} + 426 q^{36} + 257 q^{37} + 8 q^{38} + 87 q^{39} + 126 q^{40} + 77 q^{41} - 7 q^{42} + 107 q^{43} + 133 q^{44} + 140 q^{45} + 207 q^{46} + 24 q^{47} + 4 q^{48} + 418 q^{49} + 65 q^{50} + 94 q^{51} + 142 q^{52} + 81 q^{53} + 79 q^{54} + 26 q^{55} + 62 q^{56} + 112 q^{57} + 44 q^{58} + 30 q^{59} + 83 q^{60} + 347 q^{61} + 5 q^{62} + 97 q^{63} + 508 q^{64} + 94 q^{65} + 4 q^{66} + 98 q^{67} + 28 q^{68} + 91 q^{69} + 17 q^{70} + 58 q^{71} + 99 q^{72} + 157 q^{73} + 80 q^{74} + 83 q^{75} + 264 q^{76} + 61 q^{77} + 5 q^{78} + 282 q^{79} + 49 q^{80} + 403 q^{81} + 46 q^{82} + 43 q^{83} + 318 q^{84} + 396 q^{85} + 57 q^{86} + 31 q^{87} + 180 q^{88} + 98 q^{89} + 67 q^{90} + 195 q^{91} + 97 q^{92} + 83 q^{93} + 96 q^{94} + 28 q^{95} + 127 q^{96} + 167 q^{97} + 24 q^{98} + 133 q^{99}+O(q^{100}) \) 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 −2.81994 −1.65347 5.95204 −2.82087 4.66269 −3.44538 −11.1445 −0.266030 7.95467
1.2 −2.79179 0.836540 5.79408 −1.93912 −2.33544 −2.85701 −10.5923 −2.30020 5.41362
1.3 −2.79091 −0.525145 5.78920 2.75280 1.46563 0.824362 −10.5753 −2.72422 −7.68283
1.4 −2.76662 0.922557 5.65417 0.0715024 −2.55236 5.08559 −10.1097 −2.14889 −0.197820
1.5 −2.75181 −3.27979 5.57245 0.268703 9.02535 −2.53711 −9.83069 7.75702 −0.739418
1.6 −2.71378 −2.39167 5.36460 −2.12248 6.49047 2.98262 −9.13079 2.72010 5.75996
1.7 −2.70543 2.79292 5.31937 2.92654 −7.55605 −0.481988 −8.98033 4.80038 −7.91756
1.8 −2.69174 1.47984 5.24548 −2.06553 −3.98336 0.925808 −8.73601 −0.810060 5.55987
1.9 −2.68457 2.63672 5.20694 0.421516 −7.07847 −3.33953 −8.60926 3.95229 −1.13159
1.10 −2.68065 −2.13914 5.18588 0.0996967 5.73429 3.53853 −8.54022 1.57593 −0.267252
1.11 −2.63588 1.89916 4.94785 −3.80955 −5.00596 −0.400599 −7.77017 0.606822 10.0415
1.12 −2.63570 3.36960 4.94691 2.50931 −8.88125 4.49964 −7.76717 8.35420 −6.61379
1.13 −2.61945 −2.32553 4.86151 4.35353 6.09159 −2.02403 −7.49557 2.40807 −11.4039
1.14 −2.61605 −0.908380 4.84374 −1.38921 2.37637 2.01973 −7.43937 −2.17484 3.63425
1.15 −2.61438 −2.85061 4.83497 −3.39585 7.45256 2.58882 −7.41168 5.12595 8.87805
1.16 −2.60468 −1.25355 4.78433 −3.42467 3.26509 −1.07098 −7.25228 −1.42861 8.92017
1.17 −2.59860 2.92349 4.75274 −3.28865 −7.59699 2.97734 −7.15327 5.54680 8.54591
1.18 −2.58380 −1.84401 4.67602 2.13817 4.76456 −0.0377922 −6.91430 0.400387 −5.52460
1.19 −2.53734 2.44850 4.43808 2.44812 −6.21267 1.86592 −6.18624 2.99515 −6.21172
1.20 −2.52281 0.402729 4.36459 0.845770 −1.01601 −3.10596 −5.96541 −2.83781 −2.13372
See next 80 embeddings (of 275 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.275
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.f ✓ 275
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6011.2.a.f ✓ 275 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{275} - 16 T_{2}^{274} - 305 T_{2}^{273} + 6212 T_{2}^{272} + 40464 T_{2}^{271} + \cdots - 12\!\cdots\!60 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6011))\). Copy content Toggle raw display