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Properties

Label 528.6.d.c
Level $528$
Weight $6$
Character orbit 528.d
Analytic conductor $84.683$
Analytic rank $0$
Dimension $34$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [528,6,Mod(287,528)] 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("528.287"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(528, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 528.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [34,0,-31] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(84.6826568613\)
Analytic rank: \(0\)
Dimension: \(34\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 34 q - 31 q^{3} - 31 q^{9} - 4114 q^{11} - 328 q^{13} + 2065 q^{15} + 1414 q^{21} + 2606 q^{23} - 27056 q^{25} + 4088 q^{27} + 3751 q^{33} - 21052 q^{35} - 13586 q^{37} - 32736 q^{39} + 16663 q^{45} + 55648 q^{47}+ \cdots + 3751 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1 0 −15.4473 − 2.09270i 0 − 108.895i 0 − 70.1061i 0 234.241 + 64.6533i 0
287.2 0 −15.4473 + 2.09270i 0 108.895i 0 70.1061i 0 234.241 − 64.6533i 0
287.3 0 −15.4402 − 2.14451i 0 − 0.769475i 0 36.9856i 0 233.802 + 66.2236i 0
287.4 0 −15.4402 + 2.14451i 0 0.769475i 0 − 36.9856i 0 233.802 − 66.2236i 0
287.5 0 −15.2456 − 3.25132i 0 18.1984i 0 250.737i 0 221.858 + 99.1367i 0
287.6 0 −15.2456 + 3.25132i 0 − 18.1984i 0 − 250.737i 0 221.858 − 99.1367i 0
287.7 0 −12.0595 − 9.87767i 0 3.93760i 0 − 28.5888i 0 47.8633 + 238.240i 0
287.8 0 −12.0595 + 9.87767i 0 − 3.93760i 0 28.5888i 0 47.8633 − 238.240i 0
287.9 0 −10.0282 − 11.9346i 0 98.7017i 0 − 25.8652i 0 −41.8699 + 239.366i 0
287.10 0 −10.0282 + 11.9346i 0 − 98.7017i 0 25.8652i 0 −41.8699 − 239.366i 0
287.11 0 −8.23797 − 13.2339i 0 − 71.1572i 0 66.5719i 0 −107.272 + 218.041i 0
287.12 0 −8.23797 + 13.2339i 0 71.1572i 0 − 66.5719i 0 −107.272 − 218.041i 0
287.13 0 −8.10177 − 13.3177i 0 − 39.0806i 0 − 227.976i 0 −111.723 + 215.794i 0
287.14 0 −8.10177 + 13.3177i 0 39.0806i 0 227.976i 0 −111.723 − 215.794i 0
287.15 0 −4.86222 − 14.8108i 0 − 57.1666i 0 148.891i 0 −195.718 + 144.027i 0
287.16 0 −4.86222 + 14.8108i 0 57.1666i 0 − 148.891i 0 −195.718 − 144.027i 0
287.17 0 −2.06606 − 15.4509i 0 85.8379i 0 204.540i 0 −234.463 + 63.8452i 0
287.18 0 −2.06606 + 15.4509i 0 − 85.8379i 0 − 204.540i 0 −234.463 − 63.8452i 0
287.19 0 1.29049 − 15.5349i 0 48.7276i 0 − 126.772i 0 −239.669 − 40.0955i 0
287.20 0 1.29049 + 15.5349i 0 − 48.7276i 0 126.772i 0 −239.669 + 40.0955i 0
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 287.34
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.6.d.c ✓ 34
3.b odd 2 1 528.6.d.d yes 34
4.b odd 2 1 528.6.d.d yes 34
12.b even 2 1 inner 528.6.d.c ✓ 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
528.6.d.c ✓ 34 1.a even 1 1 trivial
528.6.d.c ✓ 34 12.b even 2 1 inner
528.6.d.d yes 34 3.b odd 2 1
528.6.d.d yes 34 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(528, [\chi])\):

\( T_{5}^{34} + 66653 T_{5}^{32} + 1980220725 T_{5}^{30} + 34719695961993 T_{5}^{28} + \cdots + 15\!\cdots\!28 \) Copy content Toggle raw display
\( T_{23}^{17} - 1303 T_{23}^{16} - 59709975 T_{23}^{15} + 54261762849 T_{23}^{14} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display