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Properties

Label 2252.3.d.b
Level $2252$
Weight $3$
Character orbit 2252.d
Analytic conductor $61.363$
Analytic rank $0$
Dimension $76$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2252,3,Mod(1125,2252)] 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("2252.1125"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2252, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 2252 = 2^{2} \cdot 563 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2252.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [76] 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: \(61.3625555339\)
Analytic rank: \(0\)
Dimension: \(76\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 76 q + 4 q^{3} - 8 q^{7} + 128 q^{9} + 2 q^{11} - 6 q^{13} + 22 q^{17} + 12 q^{19} - 6 q^{21} + 24 q^{23} - 912 q^{25} + 22 q^{27} - 52 q^{33} - 70 q^{39} - 28 q^{47} - 340 q^{49} + 314 q^{51} - 98 q^{57}+ \cdots + 348 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1125.1 0 −5.29590 0 6.45884i 0 0.456803 0 19.0466 0
1125.2 0 −5.29590 0 − 6.45884i 0 0.456803 0 19.0466 0
1125.3 0 −5.15375 0 8.57109i 0 −5.01378 0 17.5611 0
1125.4 0 −5.15375 0 − 8.57109i 0 −5.01378 0 17.5611 0
1125.5 0 −5.03693 0 7.68565i 0 9.37074 0 16.3707 0
1125.6 0 −5.03693 0 − 7.68565i 0 9.37074 0 16.3707 0
1125.7 0 −4.37758 0 − 7.43046i 0 −11.0035 0 10.1632 0
1125.8 0 −4.37758 0 7.43046i 0 −11.0035 0 10.1632 0
1125.9 0 −4.25003 0 − 0.993321i 0 4.26516 0 9.06277 0
1125.10 0 −4.25003 0 0.993321i 0 4.26516 0 9.06277 0
1125.11 0 −4.15996 0 6.80727i 0 6.97969 0 8.30523 0
1125.12 0 −4.15996 0 − 6.80727i 0 6.97969 0 8.30523 0
1125.13 0 −4.03034 0 4.25476i 0 −6.68184 0 7.24360 0
1125.14 0 −4.03034 0 − 4.25476i 0 −6.68184 0 7.24360 0
1125.15 0 −3.46678 0 − 1.78369i 0 −7.09205 0 3.01854 0
1125.16 0 −3.46678 0 1.78369i 0 −7.09205 0 3.01854 0
1125.17 0 −2.60138 0 − 4.72318i 0 1.65696 0 −2.23282 0
1125.18 0 −2.60138 0 4.72318i 0 1.65696 0 −2.23282 0
1125.19 0 −2.46129 0 − 6.41237i 0 −2.35196 0 −2.94204 0
1125.20 0 −2.46129 0 6.41237i 0 −2.35196 0 −2.94204 0
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1125.76
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
563.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2252.3.d.b ✓ 76
563.b odd 2 1 inner 2252.3.d.b ✓ 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2252.3.d.b ✓ 76 1.a even 1 1 trivial
2252.3.d.b ✓ 76 563.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{38} - 2 T_{3}^{37} - 201 T_{3}^{36} + 389 T_{3}^{35} + 18318 T_{3}^{34} - 34246 T_{3}^{33} + \cdots - 175110945344480 \) acting on \(S_{3}^{\mathrm{new}}(2252, [\chi])\). Copy content Toggle raw display