Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [19,9,Mod(2,19)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(19, base_ring=CyclotomicField(18))
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("19.2");
S:= CuspForms(chi, 9);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 19 \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 19.f (of order \(18\), degree \(6\), 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: | \(7.74019359116\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{18})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | −27.0408 | − | 4.76802i | 43.2321 | − | 51.5220i | 467.907 | + | 170.304i | −872.668 | + | 317.625i | −1414.68 | + | 1187.06i | 536.781 | + | 929.732i | −5753.07 | − | 3321.53i | 353.804 | + | 2006.52i | 25112.0 | − | 4427.93i |
2.2 | −25.8887 | − | 4.56487i | −45.3255 | + | 54.0168i | 408.823 | + | 148.799i | 502.442 | − | 182.874i | 1420.00 | − | 1191.52i | 126.399 | + | 218.929i | −4076.51 | − | 2353.57i | 275.889 | + | 1564.64i | −13842.3 | + | 2440.78i |
2.3 | −17.6477 | − | 3.11176i | 88.1568 | − | 105.061i | 61.1966 | + | 22.2738i | 955.890 | − | 347.916i | −1882.69 | + | 1579.76i | 678.261 | + | 1174.78i | 2962.23 | + | 1710.24i | −2126.93 | − | 12062.4i | −17951.9 | + | 3165.40i |
2.4 | −13.6035 | − | 2.39867i | 15.5877 | − | 18.5766i | −61.2592 | − | 22.2965i | −36.1678 | + | 13.1640i | −256.606 | + | 215.318i | −1384.17 | − | 2397.46i | 3842.32 | + | 2218.36i | 1037.19 | + | 5882.19i | 523.585 | − | 92.3221i |
2.5 | −10.0012 | − | 1.76348i | −70.0946 | + | 83.5355i | −143.647 | − | 52.2832i | −433.854 | + | 157.910i | 848.344 | − | 711.845i | 171.924 | + | 297.780i | 3595.94 | + | 2076.12i | −925.619 | − | 5249.44i | 4617.53 | − | 814.196i |
2.6 | −1.18052 | − | 0.208158i | 63.0914 | − | 75.1894i | −239.211 | − | 87.0657i | −753.457 | + | 274.236i | −90.1321 | + | 75.6298i | 994.028 | + | 1721.71i | 530.033 | + | 306.015i | −533.617 | − | 3026.29i | 946.557 | − | 166.904i |
2.7 | 3.54293 | + | 0.624714i | −19.5543 | + | 23.3039i | −228.399 | − | 83.1305i | 773.341 | − | 281.473i | −83.8379 | + | 70.3484i | 2158.31 | + | 3738.30i | −1554.86 | − | 897.701i | 978.604 | + | 5549.94i | 2915.73 | − | 514.122i |
2.8 | 7.58647 | + | 1.33770i | 27.9671 | − | 33.3299i | −184.796 | − | 67.2603i | 295.009 | − | 107.375i | 256.757 | − | 215.445i | −1865.56 | − | 3231.24i | −3019.86 | − | 1743.52i | 810.583 | + | 4597.05i | 2381.71 | − | 419.960i |
2.9 | 14.3082 | + | 2.52292i | −93.7652 | + | 111.745i | −42.2029 | − | 15.3606i | 404.530 | − | 147.237i | −1623.53 | + | 1362.30i | −1320.16 | − | 2286.59i | −3786.18 | − | 2185.95i | −2555.73 | − | 14494.2i | 6159.55 | − | 1086.10i |
2.10 | 18.3368 | + | 3.23327i | −30.1192 | + | 35.8947i | 85.2225 | + | 31.0185i | −993.378 | + | 361.560i | −668.347 | + | 560.810i | 465.610 | + | 806.461i | −2665.61 | − | 1538.99i | 758.045 | + | 4299.09i | −19384.4 | + | 3417.99i |
2.11 | 21.4759 | + | 3.78678i | 79.9115 | − | 95.2348i | 206.314 | + | 75.0920i | 64.9468 | − | 23.6387i | 2076.80 | − | 1742.65i | 256.522 | + | 444.310i | −688.296 | − | 397.388i | −1544.51 | − | 8759.37i | 1484.31 | − | 261.723i |
2.12 | 28.1724 | + | 4.96755i | −19.0758 | + | 22.7336i | 528.446 | + | 192.339i | 353.774 | − | 128.763i | −650.341 | + | 545.701i | 44.1972 | + | 76.5518i | 7589.89 | + | 4382.03i | 986.373 | + | 5594.00i | 10606.3 | − | 1870.18i |
3.1 | −19.5948 | + | 23.3522i | −44.0282 | − | 120.966i | −116.914 | − | 663.050i | −107.009 | + | 606.875i | 3687.55 | + | 1342.16i | −1489.58 | − | 2580.03i | 11016.2 | + | 6360.18i | −7668.38 | + | 6434.54i | −12075.0 | − | 14390.5i |
3.2 | −16.5036 | + | 19.6683i | 23.6465 | + | 64.9681i | −70.0169 | − | 397.085i | 121.428 | − | 688.654i | −1668.06 | − | 607.125i | −892.850 | − | 1546.46i | 3273.28 | + | 1889.83i | 1364.32 | − | 1144.80i | 11540.6 | + | 13753.6i |
3.3 | −14.7997 | + | 17.6376i | 25.5124 | + | 70.0948i | −47.6002 | − | 269.954i | −204.279 | + | 1158.52i | −1613.88 | − | 587.406i | 1812.74 | + | 3139.75i | 361.273 | + | 208.581i | 763.617 | − | 640.751i | −17410.3 | − | 20748.8i |
3.4 | −10.9746 | + | 13.0790i | −26.6938 | − | 73.3406i | −6.16460 | − | 34.9612i | 90.6436 | − | 514.065i | 1252.17 | + | 455.754i | 1689.23 | + | 2925.83i | −3260.30 | − | 1882.34i | 359.732 | − | 301.851i | 5728.67 | + | 6827.17i |
3.5 | −6.34622 | + | 7.56314i | −20.1988 | − | 55.4957i | 27.5275 | + | 156.116i | −27.4521 | + | 155.689i | 547.908 | + | 199.422i | −956.918 | − | 1657.43i | −3544.29 | − | 2046.29i | 2354.23 | − | 1975.44i | −1003.28 | − | 1195.66i |
3.6 | −2.04200 | + | 2.43356i | 36.3630 | + | 99.9066i | 42.7015 | + | 242.172i | −2.12030 | + | 12.0248i | −317.382 | − | 115.518i | −453.880 | − | 786.144i | −1380.84 | − | 797.228i | −3633.05 | + | 3048.49i | −24.9335 | − | 29.7146i |
3.7 | 6.68795 | − | 7.97039i | −14.7756 | − | 40.5957i | 25.6555 | + | 145.500i | −158.021 | + | 896.184i | −422.383 | − | 153.735i | 332.389 | + | 575.714i | 3638.00 | + | 2100.40i | 3596.32 | − | 3017.67i | 6086.09 | + | 7253.12i |
3.8 | 6.72219 | − | 8.01120i | 9.24008 | + | 25.3869i | 25.4625 | + | 144.405i | 179.746 | − | 1019.39i | 265.493 | + | 96.6316i | 1532.60 | + | 2654.54i | 3646.56 | + | 2105.34i | 4466.90 | − | 3748.18i | −6958.26 | − | 8292.54i |
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.f | odd | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 19.9.f.a | ✓ | 72 |
19.f | odd | 18 | 1 | inner | 19.9.f.a | ✓ | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
19.9.f.a | ✓ | 72 | 1.a | even | 1 | 1 | trivial |
19.9.f.a | ✓ | 72 | 19.f | odd | 18 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(19, [\chi])\).