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Properties

Label 900.2.bj.d
Level $900$
Weight $2$
Character orbit 900.bj
Analytic conductor $7.187$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,2,Mod(127,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.bj (of order \(20\), degree \(8\), 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: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(12\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 100)
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 10 q^{2} - 10 q^{4} + 20 q^{5} + 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 10 q^{2} - 10 q^{4} + 20 q^{5} + 10 q^{8} - 10 q^{10} - 20 q^{13} + 10 q^{14} - 14 q^{16} + 20 q^{17} - 10 q^{20} - 10 q^{22} - 20 q^{25} + 12 q^{26} - 10 q^{28} + 20 q^{29} + 50 q^{32} - 60 q^{34} + 40 q^{37} - 20 q^{38} + 40 q^{40} + 28 q^{41} - 60 q^{44} - 6 q^{46} - 80 q^{50} + 80 q^{52} + 40 q^{53} + 6 q^{56} + 60 q^{58} + 12 q^{61} - 40 q^{62} + 20 q^{64} + 100 q^{65} + 10 q^{68} - 10 q^{70} - 20 q^{73} + 20 q^{77} + 10 q^{80} - 50 q^{82} + 100 q^{85} + 6 q^{86} - 130 q^{88} - 160 q^{89} + 110 q^{92} - 170 q^{94} + 180 q^{97} + 130 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1 −1.08354 + 0.908815i 0 0.348110 − 1.96947i 1.19833 − 1.88786i 0 −2.85931 + 2.85931i 1.41269 + 2.45037i 0 0.417281 + 3.13463i
127.2 −0.882822 + 1.10482i 0 −0.441252 − 1.95072i −1.77085 + 1.36532i 0 0.797631 − 0.797631i 2.54474 + 1.23463i 0 0.0549088 − 3.16180i
127.3 −0.749667 − 1.19917i 0 −0.875999 + 1.79795i 1.19833 − 1.88786i 0 2.85931 − 2.85931i 2.81275 − 0.297395i 0 −3.16220 − 0.0217285i
127.4 −0.498205 − 1.32355i 0 −1.50358 + 1.31880i −1.77085 + 1.36532i 0 −0.797631 + 0.797631i 2.49460 + 1.33304i 0 2.68932 + 1.66360i
127.5 0.124137 + 1.40875i 0 −1.96918 + 0.349758i 2.06099 + 0.867367i 0 2.36078 − 2.36078i −0.737172 − 2.73067i 0 −0.966062 + 3.01110i
127.6 0.153683 + 1.40584i 0 −1.95276 + 0.432108i 0.577818 − 2.16012i 0 −1.01154 + 1.01154i −0.907581 − 2.67886i 0 3.12558 + 0.480343i
127.7 0.553391 − 1.30144i 0 −1.38752 − 1.44041i 2.06099 + 0.867367i 0 −2.36078 + 2.36078i −2.64246 + 1.00867i 0 2.26936 − 2.20227i
127.8 0.580589 − 1.28954i 0 −1.32583 − 1.49739i 0.577818 − 2.16012i 0 1.01154 − 1.01154i −2.70071 + 0.840347i 0 −2.45009 − 1.99926i
127.9 1.07329 + 0.920891i 0 0.303921 + 1.97677i −1.79614 − 1.33187i 0 1.40168 − 1.40168i −1.49420 + 2.40154i 0 −0.701275 − 3.08354i
127.10 1.30533 − 0.544153i 0 1.40780 − 1.42060i −1.79614 − 1.33187i 0 −1.40168 + 1.40168i 1.06462 − 2.62042i 0 −3.06930 − 0.761166i
127.11 1.35816 + 0.394222i 0 1.68918 + 1.07083i 0.420835 + 2.19611i 0 2.60816 − 2.60816i 1.87202 + 2.12027i 0 −0.294196 + 3.14856i
127.12 1.41350 + 0.0447659i 0 1.99599 + 0.126554i 0.420835 + 2.19611i 0 −2.60816 + 2.60816i 2.81568 + 0.268237i 0 0.496541 + 3.12305i
163.1 −1.08354 − 0.908815i 0 0.348110 + 1.96947i 1.19833 + 1.88786i 0 −2.85931 − 2.85931i 1.41269 − 2.45037i 0 0.417281 − 3.13463i
163.2 −0.882822 − 1.10482i 0 −0.441252 + 1.95072i −1.77085 − 1.36532i 0 0.797631 + 0.797631i 2.54474 − 1.23463i 0 0.0549088 + 3.16180i
163.3 −0.749667 + 1.19917i 0 −0.875999 − 1.79795i 1.19833 + 1.88786i 0 2.85931 + 2.85931i 2.81275 + 0.297395i 0 −3.16220 + 0.0217285i
163.4 −0.498205 + 1.32355i 0 −1.50358 − 1.31880i −1.77085 − 1.36532i 0 −0.797631 − 0.797631i 2.49460 − 1.33304i 0 2.68932 − 1.66360i
163.5 0.124137 − 1.40875i 0 −1.96918 − 0.349758i 2.06099 − 0.867367i 0 2.36078 + 2.36078i −0.737172 + 2.73067i 0 −0.966062 − 3.01110i
163.6 0.153683 − 1.40584i 0 −1.95276 − 0.432108i 0.577818 + 2.16012i 0 −1.01154 − 1.01154i −0.907581 + 2.67886i 0 3.12558 − 0.480343i
163.7 0.553391 + 1.30144i 0 −1.38752 + 1.44041i 2.06099 − 0.867367i 0 −2.36078 − 2.36078i −2.64246 − 1.00867i 0 2.26936 + 2.20227i
163.8 0.580589 + 1.28954i 0 −1.32583 + 1.49739i 0.577818 + 2.16012i 0 1.01154 + 1.01154i −2.70071 − 0.840347i 0 −2.45009 + 1.99926i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
25.f odd 20 1 inner
100.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.bj.d 96
3.b odd 2 1 100.2.l.b ✓ 96
4.b odd 2 1 inner 900.2.bj.d 96
12.b even 2 1 100.2.l.b ✓ 96
15.d odd 2 1 500.2.l.f 96
15.e even 4 1 500.2.l.d 96
15.e even 4 1 500.2.l.e 96
25.f odd 20 1 inner 900.2.bj.d 96
60.h even 2 1 500.2.l.f 96
60.l odd 4 1 500.2.l.d 96
60.l odd 4 1 500.2.l.e 96
75.h odd 10 1 500.2.l.d 96
75.j odd 10 1 500.2.l.e 96
75.l even 20 1 100.2.l.b ✓ 96
75.l even 20 1 500.2.l.f 96
100.l even 20 1 inner 900.2.bj.d 96
300.n even 10 1 500.2.l.e 96
300.r even 10 1 500.2.l.d 96
300.u odd 20 1 100.2.l.b ✓ 96
300.u odd 20 1 500.2.l.f 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.2.l.b ✓ 96 3.b odd 2 1
100.2.l.b ✓ 96 12.b even 2 1
100.2.l.b ✓ 96 75.l even 20 1
100.2.l.b ✓ 96 300.u odd 20 1
500.2.l.d 96 15.e even 4 1
500.2.l.d 96 60.l odd 4 1
500.2.l.d 96 75.h odd 10 1
500.2.l.d 96 300.r even 10 1
500.2.l.e 96 15.e even 4 1
500.2.l.e 96 60.l odd 4 1
500.2.l.e 96 75.j odd 10 1
500.2.l.e 96 300.n even 10 1
500.2.l.f 96 15.d odd 2 1
500.2.l.f 96 60.h even 2 1
500.2.l.f 96 75.l even 20 1
500.2.l.f 96 300.u odd 20 1
900.2.bj.d 96 1.a even 1 1 trivial
900.2.bj.d 96 4.b odd 2 1 inner
900.2.bj.d 96 25.f odd 20 1 inner
900.2.bj.d 96 100.l even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{96} + 2332 T_{7}^{92} + 2432078 T_{7}^{88} + 1507630344 T_{7}^{84} + 622128956825 T_{7}^{80} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
\( T_{13}^{48} + 10 T_{13}^{47} + 55 T_{13}^{46} + 190 T_{13}^{45} - 218 T_{13}^{44} + \cdots + 57\!\cdots\!25 \) Copy content Toggle raw display
\( T_{17}^{48} - 10 T_{17}^{47} + 15 T_{17}^{46} + 480 T_{17}^{45} - 4203 T_{17}^{44} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display