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Properties

Label 900.2.r.g
Level $900$
Weight $2$
Character orbit 900.r
Analytic conductor $7.187$
Analytic rank $0$
Dimension $48$
Inner twists $8$

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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(551,900)] 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("900.551"); 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, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,0,2,0,0,0,0,12,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(12)] == 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 2 q^{4} + 12 q^{9} + 42 q^{14} + 30 q^{16} - 12 q^{21} + 6 q^{24} + 4 q^{34} + 96 q^{36} + 96 q^{41} + 4 q^{46} - 32 q^{49} + 30 q^{54} + 6 q^{56} + 8 q^{61} + 20 q^{64} + 36 q^{66} + 96 q^{69} - 72 q^{74}+ \cdots + 54 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
551.1 −1.41352 + 0.0443404i 0.461412 − 1.66946i 1.99607 − 0.125352i 0 −0.578190 + 2.38027i −1.15604 − 0.667441i −2.81592 + 0.265693i −2.57420 − 1.54062i 0
551.2 −1.41248 + 0.0700780i −1.67915 + 0.424796i 1.99018 − 0.197967i 0 2.34199 − 0.717686i 2.78632 + 1.60868i −2.79721 + 0.419091i 2.63910 − 1.42659i 0
551.3 −1.36866 − 0.356046i −1.14213 + 1.30213i 1.74646 + 0.974612i 0 2.02680 − 1.37553i −2.87089 − 1.65751i −2.04331 − 1.95573i −0.391092 − 2.97440i 0
551.4 −1.34329 + 0.442228i 1.72840 + 0.112475i 1.60887 − 1.18808i 0 −2.37148 + 0.613258i −0.993605 − 0.573658i −1.63578 + 2.30743i 2.97470 + 0.388803i 0
551.5 −1.20167 + 0.745653i 0.589100 + 1.62879i 0.888004 − 1.79205i 0 −1.92241 − 1.51800i −2.50234 − 1.44473i 0.269163 + 2.81559i −2.30592 + 1.91904i 0
551.6 −1.05892 + 0.937387i −1.44177 − 0.959839i 0.242609 − 1.98523i 0 2.42646 − 0.335110i −0.953419 − 0.550457i 1.60403 + 2.32961i 1.15742 + 2.76774i 0
551.7 −0.992675 − 1.00727i 1.14213 − 1.30213i −0.0291929 + 1.99979i 0 −2.44536 + 0.142161i 2.87089 + 1.65751i 2.04331 − 1.95573i −0.391092 − 2.97440i 0
551.8 −0.668359 − 1.24631i −0.461412 + 1.66946i −1.10659 + 1.66597i 0 2.38906 − 0.540736i 1.15604 + 0.667441i 2.81592 + 0.265693i −2.57420 − 1.54062i 0
551.9 −0.645549 − 1.25828i 1.67915 − 0.424796i −1.16653 + 1.62456i 0 −1.61849 − 1.83861i −2.78632 − 1.60868i 2.79721 + 0.419091i 2.63910 − 1.42659i 0
551.10 −0.288666 − 1.38444i −1.72840 − 0.112475i −1.83334 + 0.799280i 0 0.343213 + 2.42533i 0.993605 + 0.573658i 1.63578 + 2.30743i 2.97470 + 0.388803i 0
551.11 −0.282343 + 1.38574i −1.44177 − 0.959839i −1.84056 − 0.782509i 0 1.73716 − 1.72692i −0.953419 − 0.550457i 1.60403 − 2.32961i 1.15742 + 2.76774i 0
551.12 −0.0449209 + 1.41350i 0.589100 + 1.62879i −1.99596 − 0.126991i 0 −2.32876 + 0.759526i −2.50234 − 1.44473i 0.269163 − 2.81559i −2.30592 + 1.91904i 0
551.13 0.0449209 − 1.41350i −0.589100 − 1.62879i −1.99596 − 0.126991i 0 −2.32876 + 0.759526i 2.50234 + 1.44473i −0.269163 + 2.81559i −2.30592 + 1.91904i 0
551.14 0.282343 − 1.38574i 1.44177 + 0.959839i −1.84056 − 0.782509i 0 1.73716 − 1.72692i 0.953419 + 0.550457i −1.60403 + 2.32961i 1.15742 + 2.76774i 0
551.15 0.288666 + 1.38444i 1.72840 + 0.112475i −1.83334 + 0.799280i 0 0.343213 + 2.42533i −0.993605 − 0.573658i −1.63578 − 2.30743i 2.97470 + 0.388803i 0
551.16 0.645549 + 1.25828i −1.67915 + 0.424796i −1.16653 + 1.62456i 0 −1.61849 − 1.83861i 2.78632 + 1.60868i −2.79721 − 0.419091i 2.63910 − 1.42659i 0
551.17 0.668359 + 1.24631i 0.461412 − 1.66946i −1.10659 + 1.66597i 0 2.38906 − 0.540736i −1.15604 − 0.667441i −2.81592 − 0.265693i −2.57420 − 1.54062i 0
551.18 0.992675 + 1.00727i −1.14213 + 1.30213i −0.0291929 + 1.99979i 0 −2.44536 + 0.142161i −2.87089 − 1.65751i −2.04331 + 1.95573i −0.391092 − 2.97440i 0
551.19 1.05892 − 0.937387i 1.44177 + 0.959839i 0.242609 − 1.98523i 0 2.42646 − 0.335110i 0.953419 + 0.550457i −1.60403 − 2.32961i 1.15742 + 2.76774i 0
551.20 1.20167 − 0.745653i −0.589100 − 1.62879i 0.888004 − 1.79205i 0 −1.92241 − 1.51800i 2.50234 + 1.44473i −0.269163 − 2.81559i −2.30592 + 1.91904i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 551.24
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
5.b even 2 1 inner
9.d odd 6 1 inner
20.d odd 2 1 inner
36.h even 6 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.r.g 48
4.b odd 2 1 inner 900.2.r.g 48
5.b even 2 1 inner 900.2.r.g 48
5.c odd 4 2 180.2.n.d ✓ 48
9.d odd 6 1 inner 900.2.r.g 48
15.e even 4 2 540.2.n.d 48
20.d odd 2 1 inner 900.2.r.g 48
20.e even 4 2 180.2.n.d ✓ 48
36.h even 6 1 inner 900.2.r.g 48
45.h odd 6 1 inner 900.2.r.g 48
45.k odd 12 2 540.2.n.d 48
45.l even 12 2 180.2.n.d ✓ 48
60.l odd 4 2 540.2.n.d 48
180.n even 6 1 inner 900.2.r.g 48
180.v odd 12 2 180.2.n.d ✓ 48
180.x even 12 2 540.2.n.d 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.n.d ✓ 48 5.c odd 4 2
180.2.n.d ✓ 48 20.e even 4 2
180.2.n.d ✓ 48 45.l even 12 2
180.2.n.d ✓ 48 180.v odd 12 2
540.2.n.d 48 15.e even 4 2
540.2.n.d 48 45.k odd 12 2
540.2.n.d 48 60.l odd 4 2
540.2.n.d 48 180.x even 12 2
900.2.r.g 48 1.a even 1 1 trivial
900.2.r.g 48 4.b odd 2 1 inner
900.2.r.g 48 5.b even 2 1 inner
900.2.r.g 48 9.d odd 6 1 inner
900.2.r.g 48 20.d odd 2 1 inner
900.2.r.g 48 36.h even 6 1 inner
900.2.r.g 48 45.h odd 6 1 inner
900.2.r.g 48 180.n even 6 1 inner

Hecke kernels

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

\( T_{7}^{24} - 34 T_{7}^{22} + 730 T_{7}^{20} - 9700 T_{7}^{18} + 94189 T_{7}^{16} - 620404 T_{7}^{14} + \cdots + 7290000 \) Copy content Toggle raw display
\( T_{13}^{24} + 93 T_{13}^{22} + 5247 T_{13}^{20} + 193314 T_{13}^{18} + 5286408 T_{13}^{16} + \cdots + 12990084638976 \) Copy content Toggle raw display