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Properties

Label 900.2.w.c
Level $900$
Weight $2$
Character orbit 900.w
Analytic conductor $7.187$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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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(109,900)] 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("900.109"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.w (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{5} + 6 q^{11} - 10 q^{17} + 10 q^{19} - 40 q^{23} - 4 q^{25} - 4 q^{29} + 6 q^{31} + 6 q^{35} + 10 q^{41} + 40 q^{47} - 56 q^{49} + 60 q^{53} - 62 q^{55} + 36 q^{59} - 12 q^{61} + 20 q^{67}+ \cdots - 40 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1 0 0 0 −2.23394 + 0.0974182i 0 − 1.31873i 0 0 0
109.2 0 0 0 −0.971442 + 2.01403i 0 − 1.04684i 0 0 0
109.3 0 0 0 −0.913250 − 2.04107i 0 − 4.62675i 0 0 0
109.4 0 0 0 1.64247 − 1.51733i 0 3.78808i 0 0 0
109.5 0 0 0 1.98828 + 1.02311i 0 − 3.54704i 0 0 0
109.6 0 0 0 2.10592 − 0.751722i 0 0.595901i 0 0 0
289.1 0 0 0 −2.23394 − 0.0974182i 0 1.31873i 0 0 0
289.2 0 0 0 −0.971442 − 2.01403i 0 1.04684i 0 0 0
289.3 0 0 0 −0.913250 + 2.04107i 0 4.62675i 0 0 0
289.4 0 0 0 1.64247 + 1.51733i 0 − 3.78808i 0 0 0
289.5 0 0 0 1.98828 − 1.02311i 0 3.54704i 0 0 0
289.6 0 0 0 2.10592 + 0.751722i 0 − 0.595901i 0 0 0
469.1 0 0 0 −1.99921 − 1.00158i 0 − 3.80992i 0 0 0
469.2 0 0 0 −1.28878 − 1.82730i 0 2.44380i 0 0 0
469.3 0 0 0 −0.892889 + 2.05006i 0 − 4.13266i 0 0 0
469.4 0 0 0 0.900274 + 2.04683i 0 0.957526i 0 0 0
469.5 0 0 0 0.921600 + 2.03732i 0 4.41540i 0 0 0
469.6 0 0 0 1.74098 − 1.40321i 0 1.57893i 0 0 0
829.1 0 0 0 −1.99921 + 1.00158i 0 3.80992i 0 0 0
829.2 0 0 0 −1.28878 + 1.82730i 0 − 2.44380i 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.w.c 24
3.b odd 2 1 300.2.o.a ✓ 24
15.d odd 2 1 1500.2.o.c 24
15.e even 4 1 1500.2.m.c 24
15.e even 4 1 1500.2.m.d 24
25.e even 10 1 inner 900.2.w.c 24
75.h odd 10 1 300.2.o.a ✓ 24
75.h odd 10 1 7500.2.d.g 24
75.j odd 10 1 1500.2.o.c 24
75.j odd 10 1 7500.2.d.g 24
75.l even 20 1 1500.2.m.c 24
75.l even 20 1 1500.2.m.d 24
75.l even 20 1 7500.2.a.m 12
75.l even 20 1 7500.2.a.n 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.o.a ✓ 24 3.b odd 2 1
300.2.o.a ✓ 24 75.h odd 10 1
900.2.w.c 24 1.a even 1 1 trivial
900.2.w.c 24 25.e even 10 1 inner
1500.2.m.c 24 15.e even 4 1
1500.2.m.c 24 75.l even 20 1
1500.2.m.d 24 15.e even 4 1
1500.2.m.d 24 75.l even 20 1
1500.2.o.c 24 15.d odd 2 1
1500.2.o.c 24 75.j odd 10 1
7500.2.a.m 12 75.l even 20 1
7500.2.a.n 12 75.l even 20 1
7500.2.d.g 24 75.h odd 10 1
7500.2.d.g 24 75.j odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 112 T_{7}^{22} + 5396 T_{7}^{20} + 146190 T_{7}^{18} + 2444070 T_{7}^{16} + \cdots + 172554496 \) acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\). Copy content Toggle raw display