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Properties

Label 669.6.a.d
Level $669$
Weight $6$
Character orbit 669.a
Self dual yes
Analytic conductor $107.297$
Analytic rank $0$
Dimension $51$
CM no
Inner twists $1$

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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] = [669,6,Mod(1,669)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(669, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("669.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: \( N \) \(=\) \( 669 = 3 \cdot 223 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 669.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [51] 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: yes
Analytic conductor: \(107.296775455\)
Analytic rank: \(0\)
Dimension: \(51\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 51 q + 18 q^{2} + 459 q^{3} + 940 q^{4} + 97 q^{5} + 162 q^{6} + 794 q^{7} + 411 q^{8} + 4131 q^{9} + 1256 q^{10} + 2349 q^{11} + 8460 q^{12} + 2954 q^{13} + 1816 q^{14} + 873 q^{15} + 18580 q^{16} + 1567 q^{17}+ \cdots + 190269 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −11.2372 9.00000 94.2747 45.8963 −101.135 −182.875 −699.794 81.0000 −515.746
1.2 −11.1501 9.00000 92.3238 −38.4136 −100.351 177.847 −672.614 81.0000 428.314
1.3 −10.5386 9.00000 79.0626 −28.4789 −94.8476 −77.5234 −495.976 81.0000 300.129
1.4 −10.2333 9.00000 72.7206 75.0887 −92.0998 161.649 −416.707 81.0000 −768.406
1.5 −9.81248 9.00000 64.2847 −96.0863 −88.3123 217.807 −316.793 81.0000 942.844
1.6 −9.27536 9.00000 54.0323 9.00538 −83.4782 −223.458 −204.358 81.0000 −83.5282
1.7 −8.23985 9.00000 35.8952 71.4029 −74.1587 −149.227 −32.0957 81.0000 −588.350
1.8 −7.99466 9.00000 31.9146 24.3219 −71.9520 8.88342 0.682408 81.0000 −194.445
1.9 −7.98446 9.00000 31.7515 −109.109 −71.8601 33.0673 1.98387 81.0000 871.175
1.10 −7.84052 9.00000 29.4738 53.8508 −70.5647 209.823 19.8068 81.0000 −422.218
1.11 −7.79860 9.00000 28.8182 −46.0998 −70.1874 161.880 24.8134 81.0000 359.514
1.12 −6.71396 9.00000 13.0773 65.8642 −60.4256 −82.2935 127.047 81.0000 −442.210
1.13 −6.68652 9.00000 12.7095 −80.2857 −60.1787 −121.262 128.986 81.0000 536.832
1.14 −6.24059 9.00000 6.94497 93.3979 −56.1653 218.351 156.358 81.0000 −582.858
1.15 −4.77660 9.00000 −9.18411 −97.7541 −42.9894 5.34001 196.720 81.0000 466.932
1.16 −4.70327 9.00000 −9.87929 −33.0996 −42.3294 −135.345 196.969 81.0000 155.676
1.17 −4.46581 9.00000 −12.0565 14.1739 −40.1923 50.8168 196.748 81.0000 −63.2979
1.18 −4.33601 9.00000 −13.1991 −45.5590 −39.0241 −59.1993 195.983 81.0000 197.544
1.19 −3.13600 9.00000 −22.1655 −69.7481 −28.2240 144.343 169.863 81.0000 218.730
1.20 −2.89052 9.00000 −23.6449 92.0484 −26.0147 −29.5222 160.843 81.0000 −266.068
See all 51 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.51
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(223\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 669.6.a.d ✓ 51
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
669.6.a.d ✓ 51 1.a even 1 1 trivial