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Properties

Label 630.2.bv.d
Level $630$
Weight $2$
Character orbit 630.bv
Analytic conductor $5.031$
Analytic rank $0$
Dimension $32$
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] = [630,2,Mod(73,630)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(630, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 9, 2])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("630.73"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.bv (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{7} + 12 q^{10} + 16 q^{16} + 8 q^{22} + 20 q^{28} + 48 q^{31} + 32 q^{37} + 80 q^{43} + 8 q^{46} - 28 q^{58} + 48 q^{61} + 16 q^{67} - 60 q^{70} - 24 q^{73} + 48 q^{82} - 144 q^{85} + 4 q^{88}+ \cdots - 32 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 −0.965926 − 0.258819i 0 0.866025 + 0.500000i −2.10096 − 0.765474i 0 −2.01472 + 1.71490i −0.707107 − 0.707107i 0 1.83126 + 1.28316i
73.2 −0.965926 − 0.258819i 0 0.866025 + 0.500000i −1.78539 + 1.34625i 0 1.03034 − 2.43688i −0.707107 − 0.707107i 0 2.07299 − 0.838283i
73.3 −0.965926 − 0.258819i 0 0.866025 + 0.500000i 0.316731 − 2.21352i 0 1.51871 + 2.16646i −0.707107 − 0.707107i 0 −0.878841 + 2.05612i
73.4 −0.965926 − 0.258819i 0 0.866025 + 0.500000i 1.89659 + 1.18446i 0 2.56375 + 0.653601i −0.707107 − 0.707107i 0 −1.52540 − 1.63497i
73.5 0.965926 + 0.258819i 0 0.866025 + 0.500000i −1.89659 − 1.18446i 0 2.56375 + 0.653601i 0.707107 + 0.707107i 0 −1.52540 − 1.63497i
73.6 0.965926 + 0.258819i 0 0.866025 + 0.500000i −0.316731 + 2.21352i 0 1.51871 + 2.16646i 0.707107 + 0.707107i 0 −0.878841 + 2.05612i
73.7 0.965926 + 0.258819i 0 0.866025 + 0.500000i 1.78539 − 1.34625i 0 1.03034 − 2.43688i 0.707107 + 0.707107i 0 2.07299 − 0.838283i
73.8 0.965926 + 0.258819i 0 0.866025 + 0.500000i 2.10096 + 0.765474i 0 −2.01472 + 1.71490i 0.707107 + 0.707107i 0 1.83126 + 1.28316i
397.1 −0.965926 + 0.258819i 0 0.866025 − 0.500000i −2.10096 + 0.765474i 0 −2.01472 − 1.71490i −0.707107 + 0.707107i 0 1.83126 − 1.28316i
397.2 −0.965926 + 0.258819i 0 0.866025 − 0.500000i −1.78539 − 1.34625i 0 1.03034 + 2.43688i −0.707107 + 0.707107i 0 2.07299 + 0.838283i
397.3 −0.965926 + 0.258819i 0 0.866025 − 0.500000i 0.316731 + 2.21352i 0 1.51871 − 2.16646i −0.707107 + 0.707107i 0 −0.878841 − 2.05612i
397.4 −0.965926 + 0.258819i 0 0.866025 − 0.500000i 1.89659 − 1.18446i 0 2.56375 − 0.653601i −0.707107 + 0.707107i 0 −1.52540 + 1.63497i
397.5 0.965926 − 0.258819i 0 0.866025 − 0.500000i −1.89659 + 1.18446i 0 2.56375 − 0.653601i 0.707107 − 0.707107i 0 −1.52540 + 1.63497i
397.6 0.965926 − 0.258819i 0 0.866025 − 0.500000i −0.316731 − 2.21352i 0 1.51871 − 2.16646i 0.707107 − 0.707107i 0 −0.878841 − 2.05612i
397.7 0.965926 − 0.258819i 0 0.866025 − 0.500000i 1.78539 + 1.34625i 0 1.03034 + 2.43688i 0.707107 − 0.707107i 0 2.07299 + 0.838283i
397.8 0.965926 − 0.258819i 0 0.866025 − 0.500000i 2.10096 − 0.765474i 0 −2.01472 − 1.71490i 0.707107 − 0.707107i 0 1.83126 − 1.28316i
523.1 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i −2.07533 + 0.832464i 0 −2.16646 − 1.51871i 0.707107 + 0.707107i 0 1.34123 + 1.78916i
523.2 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i 0.0774781 − 2.23473i 0 −0.653601 − 2.56375i 0.707107 + 0.707107i 0 −2.17863 + 0.503551i
523.3 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i 0.387562 + 2.20223i 0 −1.71490 + 2.01472i 0.707107 + 0.707107i 0 2.02688 − 0.944334i
523.4 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i 2.05858 + 0.873069i 0 2.43688 − 1.03034i 0.707107 + 0.707107i 0 0.310520 − 2.21440i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
7.d odd 6 1 inner
15.e even 4 1 inner
21.g even 6 1 inner
35.k even 12 1 inner
105.w odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.bv.d ✓ 32
3.b odd 2 1 inner 630.2.bv.d ✓ 32
5.c odd 4 1 inner 630.2.bv.d ✓ 32
7.d odd 6 1 inner 630.2.bv.d ✓ 32
15.e even 4 1 inner 630.2.bv.d ✓ 32
21.g even 6 1 inner 630.2.bv.d ✓ 32
35.k even 12 1 inner 630.2.bv.d ✓ 32
105.w odd 12 1 inner 630.2.bv.d ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.bv.d ✓ 32 1.a even 1 1 trivial
630.2.bv.d ✓ 32 3.b odd 2 1 inner
630.2.bv.d ✓ 32 5.c odd 4 1 inner
630.2.bv.d ✓ 32 7.d odd 6 1 inner
630.2.bv.d ✓ 32 15.e even 4 1 inner
630.2.bv.d ✓ 32 21.g even 6 1 inner
630.2.bv.d ✓ 32 35.k even 12 1 inner
630.2.bv.d ✓ 32 105.w odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\):

\( T_{11}^{16} + 74 T_{11}^{14} + 3946 T_{11}^{12} + 98192 T_{11}^{10} + 1774255 T_{11}^{8} + \cdots + 112550881 \) Copy content Toggle raw display
\( T_{13}^{16} + 2754T_{13}^{12} + 1807361T_{13}^{8} + 48864T_{13}^{4} + 256 \) Copy content Toggle raw display