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Properties

Label 501.2.e.b
Level $501$
Weight $2$
Character orbit 501.e
Analytic conductor $4.001$
Analytic rank $0$
Dimension $1148$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [501,2,Mod(4,501)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(501, base_ring=CyclotomicField(166))
 
chi = DirichletCharacter(H, H._module([0, 80]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("501.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 501 = 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 501.e (of order \(83\), degree \(82\), minimal)

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: no
Analytic conductor: \(4.00050514127\)
Analytic rank: \(0\)
Dimension: \(1148\)
Relative dimension: \(14\) over \(\Q(\zeta_{83})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{83}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 1148 q + 2 q^{2} + 14 q^{3} - 12 q^{4} + 4 q^{5} - 2 q^{6} + 6 q^{8} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1148 q + 2 q^{2} + 14 q^{3} - 12 q^{4} + 4 q^{5} - 2 q^{6} + 6 q^{8} - 14 q^{9} + 6 q^{10} - 2 q^{11} + 12 q^{12} + 2 q^{13} - 6 q^{14} - 4 q^{15} - 16 q^{16} + 2 q^{17} - 81 q^{18} - 8 q^{19} + 6 q^{20} - 10 q^{22} - 77 q^{23} - 6 q^{24} - 8 q^{25} - 12 q^{26} + 14 q^{27} + 2 q^{28} + 10 q^{29} - 6 q^{30} - 12 q^{31} + 24 q^{32} + 2 q^{33} - 14 q^{34} - 4 q^{35} - 12 q^{36} - 154 q^{37} - 2 q^{39} - 32 q^{40} - 16 q^{41} - 160 q^{42} - 8 q^{43} - 2 q^{44} + 4 q^{45} - 2 q^{47} + 16 q^{48} - 2 q^{49} + 12 q^{50} - 2 q^{51} - 16 q^{52} - 2 q^{53} - 2 q^{54} - 2 q^{56} + 8 q^{57} - 2 q^{58} - 40 q^{59} - 6 q^{60} - 182 q^{61} + 4 q^{62} + 8 q^{64} + 8 q^{65} - 156 q^{66} + 8 q^{67} + 6 q^{68} - 6 q^{69} + 20 q^{70} - 10 q^{71} + 6 q^{72} + 8 q^{73} + 2 q^{74} + 8 q^{75} - 24 q^{77} + 12 q^{78} - 2 q^{79} - 832 q^{80} - 14 q^{81} + 16 q^{82} - 8 q^{83} - 2 q^{84} - 642 q^{85} + 50 q^{86} - 10 q^{87} + 8 q^{88} - 20 q^{89} + 6 q^{90} - 24 q^{91} - 4 q^{92} + 12 q^{93} + 40 q^{94} + 14 q^{95} - 24 q^{96} + 6 q^{97} + 38 q^{98} - 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −1.81080 − 2.14883i 0.584719 + 0.811236i −0.999479 + 5.81113i −0.0689719 − 1.21351i 0.684402 − 2.72545i −1.67237 + 0.739536i 9.44764 − 5.53436i −0.316208 + 0.948690i −2.48273 + 2.34562i
4.2 −1.32671 − 1.57438i 0.584719 + 0.811236i −0.379495 + 2.20645i −0.0889489 − 1.56499i 0.501440 − 1.99685i 2.70747 − 1.19727i 0.424305 − 0.248555i −0.316208 + 0.948690i −2.34588 + 2.21633i
4.3 −1.19417 − 1.41710i 0.584719 + 0.811236i −0.243108 + 1.41347i 0.214840 + 3.77995i 0.451346 − 1.79736i −0.958561 + 0.423884i −0.904681 + 0.529955i −0.316208 + 0.948690i 5.10001 − 4.81837i
4.4 −0.607051 − 0.720374i 0.584719 + 0.811236i 0.188583 − 1.09645i −0.0123069 − 0.216530i 0.229439 − 0.913678i −3.27375 + 1.44768i −2.53003 + 1.48207i −0.316208 + 0.948690i −0.148512 + 0.140311i
4.5 −0.587776 − 0.697500i 0.584719 + 0.811236i 0.197984 − 1.15111i 0.160199 + 2.81858i 0.222153 − 0.884666i 2.75337 − 1.21756i −2.49335 + 1.46058i −0.316208 + 0.948690i 1.87180 − 1.76843i
4.6 −0.409638 − 0.486108i 0.584719 + 0.811236i 0.270512 − 1.57280i −0.0147512 − 0.259535i 0.154825 − 0.616549i −3.53683 + 1.56402i −1.97238 + 1.15541i −0.316208 + 0.948690i −0.120119 + 0.113486i
4.7 −0.198412 − 0.235452i 0.584719 + 0.811236i 0.322940 − 1.87763i −0.221371 − 3.89485i 0.0749912 − 0.298632i 1.05975 − 0.468630i −1.03752 + 0.607770i −0.316208 + 0.948690i −0.873126 + 0.824909i
4.8 0.265125 + 0.314618i 0.584719 + 0.811236i 0.310317 − 1.80423i 0.0735235 + 1.29359i −0.100206 + 0.399042i 2.46568 − 1.09034i 1.35993 − 0.796635i −0.316208 + 0.948690i −0.387493 + 0.366095i
4.9 0.423791 + 0.502903i 0.584719 + 0.811236i 0.265697 − 1.54481i −0.0858791 − 1.51098i −0.160175 + 0.637852i 0.217133 − 0.0960182i 2.02441 − 1.18588i −0.316208 + 0.948690i 0.723480 − 0.683527i
4.10 0.994427 + 1.18006i 0.584719 + 0.811236i −0.0646555 + 0.375918i 0.153454 + 2.69991i −0.375850 + 1.49672i −2.63227 + 1.16401i 2.15519 − 1.26250i −0.316208 + 0.948690i −3.03347 + 2.86595i
4.11 1.00223 + 1.18932i 0.584719 + 0.811236i −0.0710157 + 0.412897i 0.0352992 + 0.621061i −0.378799 + 1.50847i 2.28648 − 1.01110i 2.12175 − 1.24291i −0.316208 + 0.948690i −0.703265 + 0.664429i
4.12 1.30274 + 1.54593i 0.584719 + 0.811236i −0.353761 + 2.05682i −0.204304 − 3.59457i −0.492378 + 1.96076i 2.80547 − 1.24060i −0.151804 + 0.0889257i −0.316208 + 0.948690i 5.29080 − 4.99863i
4.13 1.60433 + 1.90382i 0.584719 + 0.811236i −0.711651 + 4.13765i −0.0968791 − 1.70451i −0.606366 + 2.41469i −3.61447 + 1.59835i −4.72265 + 2.76649i −0.316208 + 0.948690i 3.08966 − 2.91904i
4.14 1.73949 + 2.06421i 0.584719 + 0.811236i −0.896140 + 5.21030i 0.0108187 + 0.190347i −0.657451 + 2.61812i 1.39291 − 0.615958i −7.65561 + 4.48460i −0.316208 + 0.948690i −0.374099 + 0.353440i
7.1 −0.530992 − 2.51371i 0.843109 − 0.537743i −4.20765 + 1.86066i −1.43525 + 1.84022i −1.79941 − 1.83380i −1.32188 + 0.525686i 3.90690 + 5.42041i 0.421665 − 0.906752i 5.38789 + 2.63066i
7.2 −0.524468 − 2.48283i 0.843109 − 0.537743i −4.06022 + 1.79546i 0.989641 − 1.26888i −1.77730 − 1.81126i 2.75094 − 1.09399i 3.61969 + 5.02194i 0.421665 − 0.906752i −3.66945 − 1.79162i
7.3 −0.433814 − 2.05367i 0.843109 − 0.537743i −2.20023 + 0.972959i 2.13663 − 2.73951i −1.47010 − 1.49819i −2.92892 + 1.16477i 0.497988 + 0.690905i 0.421665 − 0.906752i −6.55295 − 3.19950i
7.4 −0.316846 − 1.49995i 0.843109 − 0.537743i −0.320313 + 0.141645i −1.85634 + 2.38012i −1.07372 − 1.09424i 3.66875 − 1.45899i −1.47885 − 2.05175i 0.421665 − 0.906752i 4.15823 + 2.03027i
7.5 −0.244073 − 1.15544i 0.843109 − 0.537743i 0.553666 − 0.244836i −2.06424 + 2.64669i −0.827111 − 0.842914i −4.75555 + 1.89119i −1.79906 − 2.49601i 0.421665 − 0.906752i 3.56192 + 1.73912i
7.6 −0.188647 − 0.893055i 0.843109 − 0.537743i 1.06718 − 0.471916i 1.43426 − 1.83895i −0.639284 − 0.651499i 2.65536 − 1.05599i −1.69019 − 2.34495i 0.421665 − 0.906752i −1.91286 − 0.933958i
See next 80 embeddings (of 1148 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
167.c even 83 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 501.2.e.b ✓ 1148
167.c even 83 1 inner 501.2.e.b ✓ 1148
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
501.2.e.b ✓ 1148 1.a even 1 1 trivial
501.2.e.b ✓ 1148 167.c even 83 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{1148} - 2 T_{2}^{1147} + 22 T_{2}^{1146} - 46 T_{2}^{1145} + 300 T_{2}^{1144} + \cdots + 73\!\cdots\!49 \) acting on \(S_{2}^{\mathrm{new}}(501, [\chi])\). Copy content Toggle raw display