↑

Properties

Label 900.2.o.c
Level $900$
Weight $2$
Character orbit 900.o
Analytic conductor $7.187$
Analytic rank $0$
Dimension $48$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [900,2,Mod(299,900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("900.299"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 1, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,0,0,0,6,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{6} + 12 q^{8} - 4 q^{9} + 28 q^{12} + 30 q^{14} - 18 q^{18} - 4 q^{21} - 42 q^{22} + 28 q^{24} + 12 q^{29} - 48 q^{33} + 6 q^{34} + 42 q^{36} - 6 q^{38} - 60 q^{41} + 16 q^{42} - 12 q^{46} - 74 q^{48}+ \cdots + 84 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
299.1 −1.41203 − 0.0785043i −1.67842 + 0.427686i 1.98767 + 0.221701i 0 2.40356 − 0.472143i 2.34956 − 4.06955i −2.78926 − 0.469091i 2.63417 − 1.43567i 0
299.2 −1.40670 + 0.145547i 1.13944 + 1.30449i 1.95763 − 0.409483i 0 −1.79271 − 1.66919i −1.45253 + 2.51585i −2.69421 + 0.860949i −0.403372 + 2.97276i 0
299.3 −1.30771 + 0.538420i 1.68556 + 0.398623i 1.42021 − 1.40819i 0 −2.41884 + 0.386254i −0.514892 + 0.891819i −1.09902 + 2.60618i 2.68220 + 1.34380i 0
299.4 −1.25064 − 0.660227i 0.425108 + 1.67907i 1.12820 + 1.65141i 0 0.576911 − 2.38058i 1.08216 − 1.87435i −0.320666 − 2.81019i −2.63857 + 1.42758i 0
299.5 −1.03621 − 0.962433i 1.43605 − 0.968388i 0.147445 + 1.99456i 0 −2.42005 − 0.378648i −1.09783 + 1.90150i 1.76684 − 2.20868i 1.12445 − 2.78130i 0
299.6 −1.01770 + 0.981980i −0.585384 − 1.63013i 0.0714305 − 1.99872i 0 2.19650 + 1.08415i −0.836639 + 1.44910i 1.89001 + 2.10425i −2.31465 + 1.90850i 0
299.7 −0.901786 − 1.08940i −1.07843 + 1.35535i −0.373566 + 1.96480i 0 2.44903 − 0.0473955i −1.92400 + 3.33246i 2.47732 − 1.36487i −0.673959 − 2.92332i 0
299.8 −0.769686 + 1.18642i 0.296046 − 1.70656i −0.815168 − 1.82634i 0 1.79683 + 1.66475i 2.05246 − 3.55496i 2.79422 + 0.438576i −2.82471 − 1.01044i 0
299.9 −0.642624 + 1.25978i −0.296046 + 1.70656i −1.17407 − 1.61912i 0 −1.95964 − 1.46963i −2.05246 + 3.55496i 2.79422 − 0.438576i −2.82471 − 1.01044i 0
299.10 −0.435401 − 1.34552i −1.64850 − 0.531460i −1.62085 + 1.17168i 0 0.00266887 + 2.44949i −2.08506 + 3.61144i 2.28224 + 1.67074i 2.43510 + 1.75222i 0
299.11 −0.426910 − 1.34824i 0.936861 − 1.45681i −1.63550 + 1.15115i 0 −2.36408 − 0.641186i −0.425874 + 0.737635i 2.25024 + 1.71360i −1.24458 − 2.72965i 0
299.12 −0.341569 + 1.37234i 0.585384 + 1.63013i −1.76666 − 0.937501i 0 −2.43705 + 0.246546i 0.836639 − 1.44910i 1.89001 − 2.10425i −2.31465 + 1.90850i 0
299.13 0.187569 + 1.40172i −1.68556 − 0.398623i −1.92964 + 0.525838i 0 0.242600 − 2.43745i 0.514892 − 0.891819i −1.09902 − 2.60618i 2.68220 + 1.34380i 0
299.14 0.228922 − 1.39556i −0.654027 + 1.60382i −1.89519 − 0.638951i 0 2.08851 + 1.27989i 0.604565 − 1.04714i −1.32555 + 2.49858i −2.14450 − 2.09789i 0
299.15 0.422205 − 1.34972i −1.63836 − 0.561951i −1.64349 − 1.13972i 0 −1.45020 + 1.97406i −0.468677 + 0.811773i −2.23218 + 1.73705i 2.36842 + 1.84135i 0
299.16 0.577305 + 1.29101i −1.13944 − 1.30449i −1.33344 + 1.49062i 0 1.02631 − 2.22411i 1.45253 − 2.51585i −2.69421 − 0.860949i −0.403372 + 2.97276i 0
299.17 0.774003 + 1.18360i 1.67842 − 0.427686i −0.801838 + 1.83223i 0 1.80531 + 1.65555i −2.34956 + 4.06955i −2.78926 + 0.469091i 2.63417 − 1.43567i 0
299.18 0.957789 − 1.04050i 1.63836 + 0.561951i −0.165280 − 1.99316i 0 2.15391 − 1.16648i 0.468677 − 0.811773i −2.23218 − 1.73705i 2.36842 + 1.84135i 0
299.19 1.09413 − 0.896034i 0.654027 − 1.60382i 0.394247 − 1.96076i 0 −0.721488 − 2.34082i −0.604565 + 1.04714i −1.32555 − 2.49858i −2.14450 − 2.09789i 0
299.20 1.19709 + 0.752973i −0.425108 − 1.67907i 0.866065 + 1.80276i 0 0.755401 − 2.33010i −1.08216 + 1.87435i −0.320666 + 2.81019i −2.63857 + 1.42758i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 299.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
45.h odd 6 1 inner
180.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.o.c 48
4.b odd 2 1 inner 900.2.o.c 48
5.b even 2 1 900.2.o.b 48
5.c odd 4 1 180.2.q.a ✓ 48
5.c odd 4 1 900.2.r.f 48
9.d odd 6 1 900.2.o.b 48
15.e even 4 1 540.2.q.a 48
20.d odd 2 1 900.2.o.b 48
20.e even 4 1 180.2.q.a ✓ 48
20.e even 4 1 900.2.r.f 48
36.h even 6 1 900.2.o.b 48
45.h odd 6 1 inner 900.2.o.c 48
45.k odd 12 1 540.2.q.a 48
45.k odd 12 1 1620.2.e.b 48
45.l even 12 1 180.2.q.a ✓ 48
45.l even 12 1 900.2.r.f 48
45.l even 12 1 1620.2.e.b 48
60.l odd 4 1 540.2.q.a 48
180.n even 6 1 inner 900.2.o.c 48
180.v odd 12 1 180.2.q.a ✓ 48
180.v odd 12 1 900.2.r.f 48
180.v odd 12 1 1620.2.e.b 48
180.x even 12 1 540.2.q.a 48
180.x even 12 1 1620.2.e.b 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.q.a ✓ 48 5.c odd 4 1
180.2.q.a ✓ 48 20.e even 4 1
180.2.q.a ✓ 48 45.l even 12 1
180.2.q.a ✓ 48 180.v odd 12 1
540.2.q.a 48 15.e even 4 1
540.2.q.a 48 45.k odd 12 1
540.2.q.a 48 60.l odd 4 1
540.2.q.a 48 180.x even 12 1
900.2.o.b 48 5.b even 2 1
900.2.o.b 48 9.d odd 6 1
900.2.o.b 48 20.d odd 2 1
900.2.o.b 48 36.h even 6 1
900.2.o.c 48 1.a even 1 1 trivial
900.2.o.c 48 4.b odd 2 1 inner
900.2.o.c 48 45.h odd 6 1 inner
900.2.o.c 48 180.n even 6 1 inner
900.2.r.f 48 5.c odd 4 1
900.2.r.f 48 20.e even 4 1
900.2.r.f 48 45.l even 12 1
900.2.r.f 48 180.v odd 12 1
1620.2.e.b 48 45.k odd 12 1
1620.2.e.b 48 45.l even 12 1
1620.2.e.b 48 180.v odd 12 1
1620.2.e.b 48 180.x even 12 1

Hecke kernels

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

\( T_{7}^{48} + 96 T_{7}^{46} + 5319 T_{7}^{44} + 199112 T_{7}^{42} + 5590407 T_{7}^{40} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
\( T_{13}^{24} - 84 T_{13}^{22} + 4740 T_{13}^{20} - 3672 T_{13}^{19} - 146040 T_{13}^{18} + \cdots + 82591744 \) Copy content Toggle raw display