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Properties

Label 1000.2.bb.a
Level $1000$
Weight $2$
Character orbit 1000.bb
Analytic conductor $7.985$
Analytic rank $0$
Dimension $2960$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1000,2,Mod(21,1000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1000, base_ring=CyclotomicField(50))
 
chi = DirichletCharacter(H, H._module([0, 25, 46]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1000.21");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1000.bb (of order \(50\), degree \(20\), 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: \(7.98504020213\)
Analytic rank: \(0\)
Dimension: \(2960\)
Relative dimension: \(148\) over \(\Q(\zeta_{50})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{50}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 2960 q - 20 q^{2} - 20 q^{4} - 20 q^{6} - 30 q^{7} - 5 q^{8} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 2960 q - 20 q^{2} - 20 q^{4} - 20 q^{6} - 30 q^{7} - 5 q^{8} - 40 q^{9} - 20 q^{10} - 20 q^{12} - 10 q^{14} - 40 q^{15} - 40 q^{16} - 40 q^{17} - 15 q^{18} - 20 q^{20} - 45 q^{22} - 40 q^{23} - 15 q^{24} - 40 q^{25} - 15 q^{26} - 60 q^{30} - 40 q^{31} - 15 q^{32} - 40 q^{33} - 20 q^{34} - 220 q^{36} - 40 q^{39} + 5 q^{40} - 40 q^{41} - 40 q^{42} + 15 q^{44} - 20 q^{46} - 40 q^{47} + 45 q^{48} - 710 q^{49} - 100 q^{50} - 25 q^{52} - 100 q^{54} - 40 q^{55} + 15 q^{56} - 80 q^{57} + 185 q^{58} + 65 q^{60} - 45 q^{62} + 30 q^{63} - 5 q^{64} + 10 q^{65} + 55 q^{66} - 40 q^{68} + 15 q^{70} - 40 q^{71} - 110 q^{72} - 40 q^{73} - 15 q^{74} - 15 q^{76} - 100 q^{78} - 40 q^{79} + 360 q^{80} - 40 q^{81} - 55 q^{82} - 25 q^{84} - 80 q^{86} - 40 q^{87} - 25 q^{88} - 90 q^{89} - 120 q^{90} - 25 q^{92} - 100 q^{94} - 140 q^{95} + 15 q^{96} - 45 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 −1.41416 + 0.0123216i −2.71790 + 0.518467i 1.99970 − 0.0348495i −0.975747 + 2.01194i 3.83715 − 0.766684i 0.925343 + 0.672301i −2.82746 + 0.0739222i 4.32883 − 1.71390i 1.35507 − 2.85723i
21.2 −1.41118 + 0.0925426i 0.0569975 − 0.0108729i 1.98287 − 0.261189i 0.833519 − 2.07491i −0.0794277 + 0.0206183i 3.68819 + 2.67963i −2.77402 + 0.552085i −2.78620 + 1.10313i −0.984231 + 3.00521i
21.3 −1.41083 + 0.0977404i 2.43374 − 0.464261i 1.98089 − 0.275791i −2.22805 − 0.189190i −3.38822 + 0.892869i 1.13675 + 0.825897i −2.76775 + 0.582708i 2.91823 − 1.15541i 3.16190 + 0.0491448i
21.4 −1.40958 + 0.114420i 1.18091 − 0.225271i 1.97382 − 0.322568i −1.48178 − 1.67461i −1.63881 + 0.452657i −0.253590 − 0.184244i −2.74534 + 0.680529i −1.44552 + 0.572324i 2.28029 + 2.19095i
21.5 −1.40702 − 0.142453i 0.741369 − 0.141424i 1.95941 + 0.400868i −0.0863295 + 2.23440i −1.06327 + 0.0933758i 0.237005 + 0.172194i −2.69983 − 0.843154i −2.25970 + 0.894680i 0.439764 − 3.13155i
21.6 −1.40583 + 0.153780i −2.99114 + 0.570590i 1.95270 − 0.432376i 1.87416 − 1.21965i 4.11728 − 1.26213i −1.85801 − 1.34992i −2.67867 + 0.908133i 5.83201 − 2.30906i −2.44718 + 2.00282i
21.7 −1.39891 − 0.207494i −1.33498 + 0.254661i 1.91389 + 0.580531i −0.0194023 − 2.23598i 1.92035 − 0.0792470i −1.15129 − 0.836463i −2.55690 − 1.20923i −1.07201 + 0.424440i −0.436811 + 3.13196i
21.8 −1.38895 − 0.266116i 0.731499 − 0.139541i 1.85837 + 0.739242i −2.01278 + 0.974028i −1.05315 0.000847880i −2.01926 − 1.46708i −2.38445 − 1.52131i −2.27371 + 0.900225i 3.05485 − 0.817246i
21.9 −1.38765 − 0.272812i 3.10516 − 0.592341i 1.85115 + 0.757136i 1.80042 + 1.32608i −4.47048 − 0.0251633i −4.05933 − 2.94928i −2.36219 − 1.55566i 6.50183 − 2.57426i −2.13658 − 2.33132i
21.10 −1.38727 + 0.274751i 1.03223 − 0.196909i 1.84902 − 0.762307i 1.79845 − 1.32875i −1.37788 + 0.556772i −3.57265 − 2.59568i −2.35565 + 1.56555i −1.76260 + 0.697863i −2.12986 + 2.33745i
21.11 −1.37594 + 0.326796i −2.03622 + 0.388430i 1.78641 − 0.899303i −2.07951 − 0.821962i 2.67478 − 1.19989i 1.70182 + 1.23645i −2.16410 + 1.82118i 1.20600 − 0.477489i 3.12990 + 0.451391i
21.12 −1.37149 − 0.344990i −1.83890 + 0.350790i 1.76196 + 0.946301i 2.08413 − 0.810194i 2.64306 + 0.153300i 3.20941 + 2.33177i −2.09005 − 1.90570i 0.469187 − 0.185764i −3.13787 + 0.392168i
21.13 −1.37090 − 0.347318i −1.10389 + 0.210577i 1.75874 + 0.952278i 1.52030 + 1.63972i 1.58646 + 0.0947190i 0.190184 + 0.138177i −2.08031 − 1.91632i −1.61511 + 0.639467i −1.51468 − 2.77592i
21.14 −1.36975 − 0.351839i 3.04832 − 0.581498i 1.75242 + 0.963861i −0.197801 − 2.22730i −4.38002 − 0.276011i 0.606646 + 0.440754i −2.06125 − 1.93681i 6.16479 − 2.44081i −0.512713 + 3.12044i
21.15 −1.36318 + 0.376489i 2.03622 − 0.388430i 1.71651 − 1.02644i 2.07951 + 0.821962i −2.62950 + 1.29612i 1.70182 + 1.23645i −1.95347 + 2.04547i 1.20600 − 0.477489i −3.14421 − 0.337567i
21.16 −1.34803 + 0.427555i −1.03223 + 0.196909i 1.63439 − 1.15272i −1.79845 + 1.32875i 1.30729 − 0.706775i −3.57265 − 2.59568i −1.71037 + 2.25270i −1.76260 + 0.697863i 1.85627 − 2.56013i
21.17 −1.31693 − 0.515452i −1.40915 + 0.268809i 1.46862 + 1.35763i −1.72661 − 1.42085i 1.99431 + 0.372344i −2.96235 − 2.15227i −1.23428 − 2.54491i −0.875893 + 0.346791i 1.54145 + 2.76115i
21.18 −1.30602 + 0.542504i 2.99114 − 0.570590i 1.41138 − 1.41704i −1.87416 + 1.21965i −3.59694 + 2.36791i −1.85801 − 1.34992i −1.07454 + 2.61637i 5.83201 − 2.30906i 1.78602 − 2.60962i
21.19 −1.29666 − 0.564510i 1.98817 − 0.379263i 1.36266 + 1.46395i 2.23515 + 0.0640677i −2.79207 − 0.630563i 2.70188 + 1.96303i −0.940489 − 2.66749i 1.01963 − 0.403702i −2.86206 − 1.34484i
21.20 −1.29034 + 0.578802i −1.18091 + 0.225271i 1.32998 − 1.49371i 1.48178 + 1.67461i 1.39340 − 0.974191i −0.253590 − 0.184244i −0.851568 + 2.69719i −1.44552 + 0.572324i −2.88127 − 1.30317i
See next 80 embeddings (of 2960 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.148
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
125.g even 25 1 inner
1000.bb even 50 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1000.2.bb.a ✓ 2960
8.b even 2 1 inner 1000.2.bb.a ✓ 2960
125.g even 25 1 inner 1000.2.bb.a ✓ 2960
1000.bb even 50 1 inner 1000.2.bb.a ✓ 2960
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1000.2.bb.a ✓ 2960 1.a even 1 1 trivial
1000.2.bb.a ✓ 2960 8.b even 2 1 inner
1000.2.bb.a ✓ 2960 125.g even 25 1 inner
1000.2.bb.a ✓ 2960 1000.bb even 50 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1000, [\chi])\).