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Properties

Label 269.4.b.a
Level $269$
Weight $4$
Character orbit 269.b
Analytic conductor $15.872$
Analytic rank $0$
Dimension $66$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [269,4,Mod(268,269)] 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("269.268"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(269, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 269 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 269.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.8715137915\)
Analytic rank: \(0\)
Dimension: \(66\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 66 q - 258 q^{4} - 34 q^{5} + 48 q^{6} - 564 q^{9} + 22 q^{11} + 118 q^{13} + 60 q^{14} + 1030 q^{16} + 144 q^{20} - 64 q^{21} - 80 q^{23} - 778 q^{24} + 1676 q^{25} - 1146 q^{30} + 308 q^{34} + 2030 q^{36}+ \cdots - 3982 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
268.1 − 5.46293i 9.84378i −21.8436 −17.0489 53.7759 − 10.8168i 75.6265i −69.9000 93.1370i
268.2 − 5.43117i 1.93468i −21.4976 −1.46854 10.5076 4.84948i 73.3079i 23.2570 7.97591i
268.3 − 5.20185i − 7.70369i −19.0592 −10.1744 −40.0734 − 6.15414i 57.5285i −32.3469 52.9259i
268.4 − 5.10783i − 5.66809i −18.0900 3.10801 −28.9517 23.9191i 51.5379i −5.12722 − 15.8752i
268.5 − 5.09712i 0.677318i −17.9807 15.4092 3.45237 − 7.81953i 50.8727i 26.5412 − 78.5428i
268.6 − 4.95781i 7.62155i −16.5798 15.1989 37.7862 12.6208i 42.5372i −31.0881 − 75.3533i
268.7 − 4.85932i 1.08269i −15.6130 −4.38506 5.26114 − 35.2002i 36.9940i 25.8278 21.3084i
268.8 − 4.73556i − 8.07694i −14.4255 20.1267 −38.2489 − 21.3096i 30.4286i −38.2370 − 95.3113i
268.9 − 4.45843i 0.747483i −11.8776 −16.4504 3.33261 20.0331i 17.2882i 26.4413 73.3432i
268.10 − 4.12762i − 3.54781i −9.03728 −21.5186 −14.6440 − 4.84095i 4.28152i 14.4131 88.8207i
268.11 − 4.07649i 5.35145i −8.61779 −7.46062 21.8151 − 7.87834i 2.51843i −1.63798 30.4132i
268.12 − 4.04411i 7.14699i −8.35486 −6.15143 28.9033 27.7589i 1.43509i −24.0795 24.8771i
268.13 − 3.88278i − 3.34083i −7.07596 10.6926 −12.9717 26.6615i − 3.58783i 15.8389 − 41.5171i
268.14 − 3.52574i − 2.12667i −4.43082 3.51546 −7.49808 − 22.1988i − 12.5840i 22.4773 − 12.3946i
268.15 − 3.49361i 8.75650i −4.20529 10.1781 30.5918 − 29.7936i − 13.2572i −49.6762 − 35.5583i
268.16 − 3.48326i − 3.82589i −4.13307 5.71821 −13.3265 9.78714i − 13.4695i 12.3626 − 19.9180i
268.17 − 3.30572i 4.74020i −2.92775 4.68152 15.6697 20.5036i − 16.7674i 4.53054 − 15.4758i
268.18 − 3.22535i − 8.16338i −2.40290 −2.36759 −26.3298 − 13.0246i − 18.0526i −39.6408 7.63633i
268.19 − 2.73656i − 9.85676i 0.511250 −13.5631 −26.9736 33.9604i − 23.2915i −70.1558 37.1162i
268.20 − 2.62939i 2.67922i 1.08630 17.6598 7.04471 − 5.55539i − 23.8914i 19.8218 − 46.4345i
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 268.66
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 269.4.b.a ✓ 66
269.b even 2 1 inner 269.4.b.a ✓ 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
269.4.b.a ✓ 66 1.a even 1 1 trivial
269.4.b.a ✓ 66 269.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(269, [\chi])\).