Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [23,13,Mod(5,23)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(23, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 13, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("23.5");
S:= CuspForms(chi, 13);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 23 \) |
Weight: | \( k \) | \(=\) | \( 13 \) |
Character orbit: | \([\chi]\) | \(=\) | 23.d (of order \(22\), degree \(10\), 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: | \(21.0218577974\) |
Analytic rank: | \(0\) |
Dimension: | \(230\) |
Relative dimension: | \(23\) over \(\Q(\zeta_{22})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −120.139 | − | 35.2760i | −90.7690 | + | 104.753i | 9743.22 | + | 6261.59i | −23272.0 | − | 3346.01i | 14600.2 | − | 9382.96i | −140828. | + | 64314.1i | −613803. | − | 708366.i | 72897.8 | + | 507015.i | 2.67784e6 | + | 1.22293e6i |
5.2 | −107.972 | − | 31.7035i | −294.207 | + | 339.533i | 7207.11 | + | 4631.73i | 15513.5 | + | 2230.51i | 42530.6 | − | 27332.7i | 176390. | − | 80554.6i | −329484. | − | 380244.i | 46907.1 | + | 326246.i | −1.60431e6 | − | 732666.i |
5.3 | −102.129 | − | 29.9877i | 821.786 | − | 948.392i | 6085.27 | + | 3910.76i | 12183.7 | + | 1751.75i | −112368. | + | 72214.7i | −67347.8 | + | 30756.7i | −218700. | − | 252393.i | −148483. | − | 1.03272e6i | −1.19178e6 | − | 544267.i |
5.4 | −90.4098 | − | 26.5467i | −761.907 | + | 879.288i | 4023.43 | + | 2585.70i | 13366.6 | + | 1921.82i | 92226.1 | − | 59270.1i | −116992. | + | 53428.4i | −42370.4 | − | 48898.1i | −117012. | − | 813840.i | −1.15745e6 | − | 528590.i |
5.5 | −87.7954 | − | 25.7791i | 198.298 | − | 228.848i | 3597.70 | + | 2312.10i | 1515.00 | + | 217.824i | −23309.1 | + | 14979.9i | −31197.4 | + | 14247.4i | −10821.5 | − | 12488.7i | 62582.6 | + | 435272.i | −127395. | − | 58179.2i |
5.6 | −77.3817 | − | 22.7213i | 495.427 | − | 571.753i | 2025.90 | + | 1301.96i | −13357.6 | − | 1920.53i | −51328.0 | + | 32986.5i | 130792. | − | 59730.6i | 89139.4 | + | 102872.i | −5821.98 | − | 40492.8i | 989994. | + | 452115.i |
5.7 | −72.3634 | − | 21.2478i | −762.412 | + | 879.870i | 1339.21 | + | 860.659i | −30802.3 | − | 4428.71i | 73866.0 | − | 47470.8i | 143265. | − | 65427.1i | 123673. | + | 142726.i | −117268. | − | 815615.i | 2.13486e6 | + | 974958.i |
5.8 | −54.9700 | − | 16.1406i | −407.479 | + | 470.255i | −684.594 | − | 439.962i | 1588.57 | + | 228.402i | 29989.3 | − | 19273.0i | −79757.6 | + | 36424.1i | 184202. | + | 212581.i | 20530.6 | + | 142794.i | −83637.1 | − | 38195.8i |
5.9 | −35.7422 | − | 10.4949i | 347.101 | − | 400.576i | −2278.41 | − | 1464.25i | 28946.6 | + | 4161.89i | −16610.1 | + | 10674.7i | −39526.9 | + | 18051.3i | 165987. | + | 191560.i | 35650.0 | + | 247951.i | −990936. | − | 452545.i |
5.10 | −32.4770 | − | 9.53610i | 448.373 | − | 517.450i | −2481.96 | − | 1595.06i | −21831.2 | − | 3138.85i | −19496.3 | + | 12529.5i | −104777. | + | 47850.0i | 156187. | + | 180249.i | 8915.82 | + | 62010.9i | 679079. | + | 310125.i |
5.11 | −15.4298 | − | 4.53059i | −161.338 | + | 186.194i | −3228.22 | − | 2074.65i | 3924.46 | + | 564.252i | 3332.97 | − | 2141.97i | 124602. | − | 56904.0i | 83546.0 | + | 96417.2i | 66993.7 | + | 465951.i | −57997.1 | − | 26486.4i |
5.12 | −0.705443 | − | 0.207137i | 890.238 | − | 1027.39i | −3445.32 | − | 2214.17i | −536.567 | − | 77.1467i | −840.822 | + | 540.363i | 114910. | − | 52477.8i | 3943.94 | + | 4551.55i | −187373. | − | 1.30321e6i | 362.538 | + | 165.565i |
5.13 | 8.53737 | + | 2.50680i | −912.924 | + | 1053.57i | −3379.17 | − | 2171.66i | 21701.0 | + | 3120.14i | −10435.0 | + | 6706.20i | 94155.6 | − | 42999.4i | −47271.9 | − | 54554.7i | −200948. | − | 1.39763e6i | 177448. | + | 81037.8i |
5.14 | 16.1031 | + | 4.72830i | −479.913 | + | 553.849i | −3208.82 | − | 2062.18i | −13434.2 | − | 1931.55i | −10346.9 | + | 6649.52i | −82122.5 | + | 37504.1i | −86938.4 | − | 100332.i | −800.303 | − | 5566.23i | −207200. | − | 94625.2i |
5.15 | 35.2606 | + | 10.3534i | 563.490 | − | 650.302i | −2309.66 | − | 1484.33i | 2550.09 | + | 366.648i | 26601.9 | − | 17096.0i | −174986. | + | 79913.5i | −164645. | − | 190010.i | −29740.0 | − | 206846.i | 86121.7 | + | 39330.5i |
5.16 | 47.4597 | + | 13.9354i | 72.2874 | − | 83.4241i | −1387.55 | − | 891.721i | −16609.8 | − | 2388.12i | 4593.29 | − | 2951.93i | 138459. | − | 63232.2i | −186102. | − | 214773.i | 73897.8 | + | 513971.i | −755015. | − | 344804.i |
5.17 | 53.6967 | + | 15.7668i | 228.961 | − | 264.235i | −811.028 | − | 521.216i | 16209.3 | + | 2330.55i | 16460.6 | − | 10578.6i | 57943.8 | − | 26462.0i | −185443. | − | 214013.i | 58234.9 | + | 405033.i | 833643. | + | 380712.i |
5.18 | 68.8275 | + | 20.2096i | −392.302 | + | 452.741i | 883.025 | + | 567.486i | 18893.6 | + | 2716.49i | −36150.9 | + | 23232.8i | −145477. | + | 66436.9i | −143103. | − | 165150.i | 24558.6 | + | 170809.i | 1.24550e6 | + | 568801.i |
5.19 | 85.4052 | + | 25.0772i | 424.703 | − | 490.133i | 3219.40 | + | 2068.98i | −25609.1 | − | 3682.03i | 48563.0 | − | 31209.5i | −11513.2 | + | 5257.91i | −15685.2 | − | 18101.7i | 15774.0 | + | 109710.i | −2.09481e6 | − | 956670.i |
5.20 | 87.7164 | + | 25.7559i | −741.301 | + | 855.507i | 3585.04 | + | 2303.96i | −13018.4 | − | 1871.76i | −87058.6 | + | 55949.2i | 29354.1 | − | 13405.6i | 9910.44 | + | 11437.3i | −106733. | − | 742344.i | −1.09372e6 | − | 499484.i |
See next 80 embeddings (of 230 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.d | odd | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 23.13.d.a | ✓ | 230 |
23.d | odd | 22 | 1 | inner | 23.13.d.a | ✓ | 230 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
23.13.d.a | ✓ | 230 | 1.a | even | 1 | 1 | trivial |
23.13.d.a | ✓ | 230 | 23.d | odd | 22 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{13}^{\mathrm{new}}(23, [\chi])\).