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Properties

Label 990.2.ba.i
Level $990$
Weight $2$
Character orbit 990.ba
Analytic conductor $7.905$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [990,2,Mod(289,990)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(990, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 5, 8])) 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("990.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 990.ba (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,8,2,0,0] 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: \(7.90518980011\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 330)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{4} + 2 q^{5} + 4 q^{10} - 10 q^{14} - 8 q^{16} + 20 q^{19} - 2 q^{20} + 4 q^{25} + 2 q^{26} - 2 q^{29} - 24 q^{31} - 28 q^{34} - 44 q^{35} - 4 q^{40} + 34 q^{41} + 20 q^{44} - 38 q^{46} + 14 q^{49}+ \cdots + 60 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1 −0.587785 − 0.809017i 0 −0.309017 + 0.951057i −2.23339 + 0.109316i 0 1.51700 + 0.492904i 0.951057 − 0.309017i 0 1.40119 + 1.74260i
289.2 −0.587785 − 0.809017i 0 −0.309017 + 0.951057i 0.256687 + 2.22129i 0 3.54705 + 1.15250i 0.951057 − 0.309017i 0 1.64618 − 1.51330i
289.3 −0.587785 − 0.809017i 0 −0.309017 + 0.951057i 0.450764 − 2.19016i 0 −2.88658 − 0.937907i 0.951057 − 0.309017i 0 −2.03683 + 0.922670i
289.4 −0.587785 − 0.809017i 0 −0.309017 + 0.951057i 1.38390 + 1.75636i 0 −3.49179 − 1.13455i 0.951057 − 0.309017i 0 0.607489 − 2.15197i
289.5 0.587785 + 0.809017i 0 −0.309017 + 0.951057i −1.94367 + 1.10550i 0 2.88658 + 0.937907i −0.951057 + 0.309017i 0 −2.03683 − 0.922670i
289.6 0.587785 + 0.809017i 0 −0.309017 + 0.951057i −0.586191 − 2.15786i 0 −1.51700 − 0.492904i −0.951057 + 0.309017i 0 1.40119 − 1.74260i
289.7 0.587785 + 0.809017i 0 −0.309017 + 0.951057i 2.09805 + 0.773425i 0 3.49179 + 1.13455i −0.951057 + 0.309017i 0 0.607489 + 2.15197i
289.8 0.587785 + 0.809017i 0 −0.309017 + 0.951057i 2.19189 − 0.442291i 0 −3.54705 − 1.15250i −0.951057 + 0.309017i 0 1.64618 + 1.51330i
379.1 −0.951057 + 0.309017i 0 0.809017 − 0.587785i −2.14525 + 0.630779i 0 −0.890727 − 1.22598i −0.587785 + 0.809017i 0 1.84534 − 1.26283i
379.2 −0.951057 + 0.309017i 0 0.809017 − 0.587785i −0.178263 − 2.22895i 0 2.46794 + 3.39683i −0.587785 + 0.809017i 0 0.858322 + 2.06477i
379.3 −0.951057 + 0.309017i 0 0.809017 − 0.587785i 0.434075 + 2.19353i 0 2.37317 + 3.26639i −0.587785 + 0.809017i 0 −1.09067 − 1.95204i
379.4 −0.951057 + 0.309017i 0 0.809017 − 0.587785i 2.16821 + 0.546681i 0 −1.82376 − 2.51019i −0.587785 + 0.809017i 0 −2.23103 + 0.150090i
379.5 0.951057 − 0.309017i 0 0.809017 − 0.587785i −2.07545 − 0.832169i 0 1.82376 + 2.51019i 0.587785 − 0.809017i 0 −2.23103 − 0.150090i
379.6 0.951057 − 0.309017i 0 0.809017 − 0.587785i −1.64050 + 1.51946i 0 −2.37317 − 3.26639i 0.587785 − 0.809017i 0 −1.09067 + 1.95204i
379.7 0.951057 − 0.309017i 0 0.809017 − 0.587785i 1.36479 + 1.77126i 0 0.890727 + 1.22598i 0.587785 − 0.809017i 0 1.84534 + 1.26283i
379.8 0.951057 − 0.309017i 0 0.809017 − 0.587785i 1.45436 − 1.69848i 0 −2.46794 − 3.39683i 0.587785 − 0.809017i 0 0.858322 − 2.06477i
559.1 −0.951057 − 0.309017i 0 0.809017 + 0.587785i −2.14525 − 0.630779i 0 −0.890727 + 1.22598i −0.587785 − 0.809017i 0 1.84534 + 1.26283i
559.2 −0.951057 − 0.309017i 0 0.809017 + 0.587785i −0.178263 + 2.22895i 0 2.46794 − 3.39683i −0.587785 − 0.809017i 0 0.858322 − 2.06477i
559.3 −0.951057 − 0.309017i 0 0.809017 + 0.587785i 0.434075 − 2.19353i 0 2.37317 − 3.26639i −0.587785 − 0.809017i 0 −1.09067 + 1.95204i
559.4 −0.951057 − 0.309017i 0 0.809017 + 0.587785i 2.16821 − 0.546681i 0 −1.82376 + 2.51019i −0.587785 − 0.809017i 0 −2.23103 − 0.150090i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 990.2.ba.i 32
3.b odd 2 1 330.2.s.c ✓ 32
5.b even 2 1 inner 990.2.ba.i 32
11.c even 5 1 inner 990.2.ba.i 32
15.d odd 2 1 330.2.s.c ✓ 32
33.h odd 10 1 330.2.s.c ✓ 32
55.j even 10 1 inner 990.2.ba.i 32
165.o odd 10 1 330.2.s.c ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.s.c ✓ 32 3.b odd 2 1
330.2.s.c ✓ 32 15.d odd 2 1
330.2.s.c ✓ 32 33.h odd 10 1
330.2.s.c ✓ 32 165.o odd 10 1
990.2.ba.i 32 1.a even 1 1 trivial
990.2.ba.i 32 5.b even 2 1 inner
990.2.ba.i 32 11.c even 5 1 inner
990.2.ba.i 32 55.j even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(990, [\chi])\):

\( T_{7}^{32} - 35 T_{7}^{30} + 1014 T_{7}^{28} - 25754 T_{7}^{26} + 552757 T_{7}^{24} + \cdots + 779563031470336 \) Copy content Toggle raw display
\( T_{13}^{32} - 103 T_{13}^{30} + 5800 T_{13}^{28} - 243581 T_{13}^{26} + 10462051 T_{13}^{24} + \cdots + 26\!\cdots\!36 \) Copy content Toggle raw display
\( T_{29}^{16} + T_{29}^{15} + 109 T_{29}^{14} + 177 T_{29}^{13} + 7399 T_{29}^{12} - 2516 T_{29}^{11} + \cdots + 118810000 \) Copy content Toggle raw display