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Properties

Label 900.2.bj.f
Level $900$
Weight $2$
Character orbit 900.bj
Analytic conductor $7.187$
Analytic rank $0$
Dimension $240$
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: \(240\)
Relative dimension: \(30\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 300)
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.

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 12 q^{8} + 8 q^{10} + 4 q^{13} - 20 q^{17} + 20 q^{20} - 12 q^{22} + 20 q^{25} + 4 q^{28} + 20 q^{32} - 4 q^{37} + 76 q^{38} - 92 q^{40} + 140 q^{44} + 164 q^{50} - 172 q^{52} + 4 q^{53} - 120 q^{58} + 44 q^{62} - 60 q^{64} + 20 q^{65} - 16 q^{68} - 44 q^{70} - 44 q^{73} + 48 q^{77} + 4 q^{80} + 24 q^{82} - 64 q^{85} + 60 q^{88} + 260 q^{89} - 144 q^{92} + 40 q^{94} - 180 q^{97} - 256 q^{98}+O(q^{100}) \) 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 −1.41129 + 0.0909451i 0 1.98346 − 0.256699i 1.83309 − 1.28054i 0 1.46352 − 1.46352i −2.77588 + 0.542662i 0 −2.47056 + 1.97391i
127.2 −1.36363 + 0.374857i 0 1.71896 − 1.02233i −2.16584 + 0.556007i 0 −3.13412 + 3.13412i −1.96080 + 2.03845i 0 2.74498 − 1.57007i
127.3 −1.36070 + 0.385348i 0 1.70301 − 1.04869i 0.0577937 + 2.23532i 0 −0.0211854 + 0.0211854i −1.91318 + 2.08320i 0 −0.940016 − 3.01933i
127.4 −1.33300 + 0.472333i 0 1.55380 − 1.25924i −1.01218 − 1.99386i 0 0.189838 − 0.189838i −1.47645 + 2.41249i 0 2.29101 + 2.17974i
127.5 −1.31411 − 0.522605i 0 1.45377 + 1.37352i 1.83309 − 1.28054i 0 −1.46352 + 1.46352i −1.19260 − 2.56470i 0 −3.07810 + 0.724779i
127.6 −1.18105 − 0.777895i 0 0.789760 + 1.83747i −2.16584 + 0.556007i 0 3.13412 − 3.13412i 0.496609 − 2.78449i 0 2.99048 + 1.02812i
127.7 −1.17502 − 0.786967i 0 0.761365 + 1.84941i 0.0577937 + 2.23532i 0 0.0211854 − 0.0211854i 0.560803 − 2.77227i 0 1.69122 − 2.67204i
127.8 −1.12180 − 0.861136i 0 0.516889 + 1.93205i −1.01218 − 1.99386i 0 −0.189838 + 0.189838i 1.08391 − 2.61250i 0 −0.581516 + 3.10835i
127.9 −0.872257 + 1.11318i 0 −0.478336 − 1.94196i 2.16662 − 0.552943i 0 2.97520 − 2.97520i 2.57898 + 1.16141i 0 −1.27433 + 2.89415i
127.10 −0.729889 + 1.21131i 0 −0.934525 − 1.76824i −1.31656 − 1.80739i 0 1.89964 − 1.89964i 2.82398 + 0.158622i 0 3.15025 − 0.275559i
127.11 −0.510631 + 1.31881i 0 −1.47851 − 1.34685i 1.27726 + 1.83538i 0 −3.58459 + 3.58459i 2.53121 − 1.26213i 0 −3.07272 + 0.747264i
127.12 −0.485574 − 1.32824i 0 −1.52844 + 1.28992i 2.16662 − 0.552943i 0 −2.97520 + 2.97520i 2.45549 + 1.40378i 0 −1.78650 − 2.60930i
127.13 −0.319851 − 1.37757i 0 −1.79539 + 0.881234i −1.31656 − 1.80739i 0 −1.89964 + 1.89964i 1.78822 + 2.19141i 0 −2.06871 + 2.39175i
127.14 −0.262182 + 1.38970i 0 −1.86252 − 0.728708i −0.583536 + 2.15858i 0 −0.889008 + 0.889008i 1.50100 − 2.39729i 0 −2.84679 − 1.37688i
127.15 −0.0781043 − 1.41206i 0 −1.98780 + 0.220575i 1.27726 + 1.83538i 0 3.58459 − 3.58459i 0.466720 + 2.78965i 0 2.49189 − 1.94691i
127.16 0.0475977 + 1.41341i 0 −1.99547 + 0.134550i −2.23568 + 0.0415918i 0 1.15642 − 1.15642i −0.285155 − 2.81402i 0 −0.165200 − 3.15796i
127.17 0.180090 − 1.40270i 0 −1.93514 − 0.505225i −0.583536 + 2.15858i 0 0.889008 − 0.889008i −1.05718 + 2.62343i 0 2.92276 + 1.20727i
127.18 0.482036 − 1.32953i 0 −1.53528 − 1.28176i −2.23568 + 0.0415918i 0 −1.15642 + 1.15642i −2.44420 + 1.42334i 0 −1.02238 + 2.99245i
127.19 0.510574 + 1.31883i 0 −1.47863 + 1.34672i −0.968294 − 2.01554i 0 −2.67499 + 2.67499i −2.53105 − 1.26246i 0 2.16377 − 2.30610i
127.20 0.753043 + 1.19705i 0 −0.865853 + 1.80286i −1.25978 + 1.84742i 0 1.55959 − 1.55959i −2.81013 + 0.321160i 0 −3.16012 − 0.116836i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.30
Significant digits:
Format:

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.f 240
3.b odd 2 1 300.2.w.a ✓ 240
4.b odd 2 1 inner 900.2.bj.f 240
12.b even 2 1 300.2.w.a ✓ 240
25.f odd 20 1 inner 900.2.bj.f 240
75.l even 20 1 300.2.w.a ✓ 240
100.l even 20 1 inner 900.2.bj.f 240
300.u odd 20 1 300.2.w.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.w.a ✓ 240 3.b odd 2 1
300.2.w.a ✓ 240 12.b even 2 1
300.2.w.a ✓ 240 75.l even 20 1
300.2.w.a ✓ 240 300.u odd 20 1
900.2.bj.f 240 1.a even 1 1 trivial
900.2.bj.f 240 4.b odd 2 1 inner
900.2.bj.f 240 25.f odd 20 1 inner
900.2.bj.f 240 100.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 \(S_{2}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{240} + 7300 T_{7}^{236} + 25284330 T_{7}^{232} + 55350374200 T_{7}^{228} + 86068531713275 T_{7}^{224} + \cdots + 52\!\cdots\!96 \) Copy content Toggle raw display
\( T_{13}^{120} - 2 T_{13}^{119} + 2 T_{13}^{118} - 8 T_{13}^{117} - 2336 T_{13}^{116} + \cdots + 32\!\cdots\!36 \) Copy content Toggle raw display
\( T_{17}^{120} + 10 T_{17}^{119} + 90 T_{17}^{118} + 40 T_{17}^{117} - 6870 T_{17}^{116} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display