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Properties

Label 169.8.a.i
Level $169$
Weight $8$
Character orbit 169.a
Self dual yes
Analytic conductor $52.793$
Analytic rank $0$
Dimension $21$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [169,8,Mod(1,169)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(169, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("169.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 169.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [21,31] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(52.7930693068\)
Analytic rank: \(0\)
Dimension: \(21\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q + 31 q^{2} - 26 q^{3} + 1409 q^{4} + 680 q^{5} + 1470 q^{6} + 2929 q^{7} + 4716 q^{8} + 15465 q^{9} - 5167 q^{10} + 14824 q^{11} + 21795 q^{12} - 179 q^{14} + 36398 q^{15} + 113205 q^{16} + 45016 q^{17}+ \cdots + 37605493 q^{99}+O(q^{100}) \) 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 −20.8456 28.0386 306.538 273.942 −584.480 1317.98 −3721.73 −1400.84 −5710.48
1.2 −20.4423 −14.6497 289.887 478.639 299.473 −1197.76 −3309.33 −1972.39 −9784.47
1.3 −16.7260 −13.7721 151.758 0.583029 230.352 −274.333 −397.369 −1997.33 −9.75172
1.4 −15.0231 −85.3884 97.6921 −439.379 1282.79 536.518 455.317 5104.17 6600.82
1.5 −13.2906 38.7978 48.6390 −274.097 −515.644 774.181 1054.75 −681.732 3642.90
1.6 −8.99747 87.5208 −47.0455 190.033 −787.466 44.6495 1574.97 5472.89 −1709.81
1.7 −8.58294 18.1259 −54.3331 −37.6136 −155.574 −711.860 1564.95 −1858.45 322.835
1.8 −6.93821 −64.5145 −79.8613 −33.5113 447.615 160.658 1442.18 1975.12 232.508
1.9 −2.90053 −90.5117 −119.587 240.795 262.532 1391.07 718.132 6005.38 −698.433
1.10 1.25436 8.41428 −126.427 −307.776 10.5545 83.6974 −319.143 −2116.20 −386.063
1.11 2.91821 44.2582 −119.484 197.474 129.155 −303.075 −722.210 −228.212 576.269
1.12 3.19657 −57.0706 −117.782 431.499 −182.430 225.139 −785.660 1070.05 1379.32
1.13 5.77104 82.1608 −94.6950 −17.6329 474.154 888.558 −1285.18 4563.40 −101.761
1.14 10.5911 −6.51000 −15.8290 397.099 −68.9480 −1187.35 −1523.30 −2144.62 4205.71
1.15 11.8381 −26.1495 12.1398 −422.131 −309.560 −1215.99 −1371.56 −1503.20 −4997.21
1.16 13.1620 −21.4559 45.2387 −344.184 −282.403 −168.957 −1089.31 −1726.64 −4530.16
1.17 15.3948 −46.3939 108.998 391.457 −714.223 1742.39 −292.525 −34.6060 6026.39
1.18 17.7683 58.1116 187.714 15.5333 1032.55 1688.45 1061.02 1189.96 276.002
1.19 19.6237 −72.4635 257.091 −413.760 −1422.00 −1005.88 2533.24 3063.96 −8119.51
1.20 21.5931 28.3750 338.261 473.706 612.705 −99.1966 4540.19 −1381.86 10228.8
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
Significant digits:
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Atkin-Lehner signs

\( p \) Sign
\(13\) \( +1 \)

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 169.8.a.i yes 21
13.b even 2 1 169.8.a.h ✓ 21
13.d odd 4 2 169.8.b.f 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.8.a.h ✓ 21 13.b even 2 1
169.8.a.i yes 21 1.a even 1 1 trivial
169.8.b.f 42 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{21} - 31 T_{2}^{20} - 1568 T_{2}^{19} + 54321 T_{2}^{18} + 947492 T_{2}^{17} + \cdots + 61\!\cdots\!68 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(169))\). Copy content Toggle raw display