Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [29,9,Mod(2,29)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(29, base_ring=CyclotomicField(28))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("29.2");
S:= CuspForms(chi, 9);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 29 \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 29.f (of order \(28\), degree \(12\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(11.8139796918\) |
Analytic rank: | \(0\) |
Dimension: | \(228\) |
Relative dimension: | \(19\) over \(\Q(\zeta_{28})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{28}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −31.1717 | − | 3.51221i | 2.26708 | + | 1.42450i | 709.757 | + | 161.997i | 40.2894 | − | 32.1297i | −65.6655 | − | 52.3665i | 235.726 | + | 1032.78i | −13975.6 | − | 4890.26i | −2843.60 | − | 5904.80i | −1368.74 | + | 860.034i |
2.2 | −24.5490 | − | 2.76601i | −135.176 | − | 84.9370i | 345.422 | + | 78.8403i | −399.043 | + | 318.226i | 3083.51 | + | 2459.02i | 40.6060 | + | 177.907i | −2292.28 | − | 802.105i | 8211.67 | + | 17051.7i | 10676.3 | − | 6708.39i |
2.3 | −22.9178 | − | 2.58222i | 95.4072 | + | 59.9483i | 268.977 | + | 61.3923i | 748.172 | − | 596.647i | −2031.73 | − | 1620.25i | −79.6660 | − | 349.039i | −433.075 | − | 151.540i | 2662.02 | + | 5527.74i | −18687.1 | + | 11741.9i |
2.4 | −21.5736 | − | 2.43076i | −17.4548 | − | 10.9676i | 209.928 | + | 47.9147i | −119.794 | + | 95.5326i | 349.902 | + | 279.037i | −857.743 | − | 3758.02i | 833.466 | + | 291.642i | −2662.33 | − | 5528.38i | 2816.60 | − | 1769.79i |
2.5 | −21.4136 | − | 2.41273i | 91.7346 | + | 57.6407i | 203.138 | + | 46.3649i | −731.114 | + | 583.044i | −1825.29 | − | 1455.62i | 246.757 | + | 1081.11i | 968.946 | + | 339.049i | 2246.08 | + | 4664.03i | 17062.5 | − | 10721.1i |
2.6 | −18.5588 | − | 2.09108i | −58.4072 | − | 36.6997i | 90.4766 | + | 20.6507i | 495.290 | − | 394.980i | 1007.23 | + | 803.238i | 641.791 | + | 2811.87i | 2876.87 | + | 1006.66i | −782.176 | − | 1624.21i | −10017.9 | + | 6294.69i |
2.7 | −7.14136 | − | 0.804638i | −31.1173 | − | 19.5523i | −199.230 | − | 45.4729i | −619.444 | + | 493.990i | 206.487 | + | 164.668i | 232.502 | + | 1018.66i | 3122.70 | + | 1092.68i | −2260.72 | − | 4694.43i | 4821.16 | − | 3029.33i |
2.8 | −5.51607 | − | 0.621512i | 67.5754 | + | 42.4604i | −219.541 | − | 50.1088i | 264.780 | − | 211.155i | −346.361 | − | 276.213i | 606.153 | + | 2655.73i | 2521.16 | + | 882.193i | −83.1675 | − | 172.699i | −1591.78 | + | 1000.18i |
2.9 | −4.14353 | − | 0.466864i | −78.5126 | − | 49.3327i | −232.631 | − | 53.0964i | 799.859 | − | 637.866i | 302.287 | + | 241.066i | −699.825 | − | 3066.14i | 1946.68 | + | 681.171i | 883.797 | + | 1835.22i | −3612.03 | + | 2269.59i |
2.10 | −3.38682 | − | 0.381603i | 91.1302 | + | 57.2609i | −238.257 | − | 54.3805i | 116.314 | − | 92.7574i | −286.791 | − | 228.708i | −926.307 | − | 4058.42i | 1609.73 | + | 563.269i | 2179.19 | + | 4525.14i | −429.332 | + | 269.767i |
2.11 | 5.20158 | + | 0.586077i | −110.135 | − | 69.2024i | −222.869 | − | 50.8683i | −222.021 | + | 177.056i | −532.318 | − | 424.509i | −45.3512 | − | 198.696i | −2394.29 | − | 837.798i | 4494.03 | + | 9331.94i | −1258.63 | + | 790.849i |
2.12 | 9.08550 | + | 1.02369i | 24.7393 | + | 15.5448i | −168.083 | − | 38.3639i | −38.0474 | + | 30.3418i | 208.856 | + | 166.557i | −85.6032 | − | 375.052i | −3697.10 | − | 1293.67i | −2476.32 | − | 5142.12i | −376.740 | + | 236.721i |
2.13 | 14.2835 | + | 1.60937i | 128.283 | + | 80.6053i | −48.1524 | − | 10.9905i | −467.828 | + | 373.081i | 1702.60 | + | 1357.78i | 469.147 | + | 2055.47i | −4143.33 | − | 1449.81i | 7112.49 | + | 14769.2i | −7282.66 | + | 4576.00i |
2.14 | 14.6914 | + | 1.65532i | 1.69586 | + | 1.06558i | −36.4848 | − | 8.32741i | 784.225 | − | 625.399i | 23.1506 | + | 18.4620i | 313.531 | + | 1373.67i | −4094.63 | − | 1432.77i | −2844.97 | − | 5907.64i | 12556.6 | − | 7889.83i |
2.15 | 18.1222 | + | 2.04189i | 3.92609 | + | 2.46693i | 74.6647 | + | 17.0417i | −848.114 | + | 676.349i | 66.1123 | + | 52.7228i | −652.501 | − | 2858.79i | −3088.37 | − | 1080.67i | −2837.38 | − | 5891.89i | −16750.8 | + | 10525.2i |
2.16 | 19.6004 | + | 2.20843i | −83.4552 | − | 52.4384i | 129.715 | + | 29.6067i | 14.4052 | − | 11.4877i | −1519.95 | − | 1212.12i | 772.012 | + | 3382.40i | −2289.00 | − | 800.955i | 1368.27 | + | 2841.25i | 307.716 | − | 193.351i |
2.17 | 25.7386 | + | 2.90004i | 82.6049 | + | 51.9041i | 404.482 | + | 92.3203i | 418.219 | − | 333.519i | 1975.61 | + | 1575.49i | −324.689 | − | 1422.56i | 3884.38 | + | 1359.20i | 1282.82 | + | 2663.80i | 11731.6 | − | 7371.44i |
2.18 | 26.7386 | + | 3.01272i | −87.9248 | − | 55.2468i | 456.297 | + | 104.147i | 197.846 | − | 157.777i | −2184.55 | − | 1742.12i | −752.858 | − | 3298.49i | 5385.14 | + | 1884.34i | 1831.85 | + | 3803.87i | 5765.47 | − | 3622.69i |
2.19 | 29.9840 | + | 3.37838i | 11.3862 | + | 7.15440i | 638.043 | + | 145.629i | −500.927 | + | 399.476i | 317.232 | + | 252.984i | 838.383 | + | 3673.20i | 11348.1 | + | 3970.86i | −2768.25 | − | 5748.33i | −16369.4 | + | 10285.5i |
3.1 | −25.9849 | − | 16.3274i | 93.5645 | − | 32.7396i | 297.558 | + | 617.886i | −652.275 | + | 148.877i | −2965.82 | − | 676.929i | 2496.53 | + | 1202.27i | 1476.82 | − | 13107.1i | 2552.84 | − | 2035.82i | 19380.1 | + | 6781.39i |
See next 80 embeddings (of 228 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
29.f | odd | 28 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 29.9.f.a | ✓ | 228 |
29.f | odd | 28 | 1 | inner | 29.9.f.a | ✓ | 228 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
29.9.f.a | ✓ | 228 | 1.a | even | 1 | 1 | trivial |
29.9.f.a | ✓ | 228 | 29.f | odd | 28 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(29, [\chi])\).