Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [13,9,Mod(2,13)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(13, base_ring=CyclotomicField(12))
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("13.2");
S:= CuspForms(chi, 9);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 13 \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 13.f (of order \(12\), degree \(4\), 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: | \(5.29592193079\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | −28.0190 | − | 7.50766i | 22.8511 | + | 39.5793i | 506.995 | + | 292.714i | 131.372 | − | 131.372i | −343.117 | − | 1280.53i | −2789.95 | + | 747.565i | −6757.00 | − | 6757.00i | 2236.15 | − | 3873.13i | −4667.19 | + | 2694.60i |
2.2 | −16.0639 | − | 4.30431i | −60.2561 | − | 104.367i | 17.8198 | + | 10.2883i | 169.528 | − | 169.528i | 518.723 | + | 1935.90i | −1635.34 | + | 438.187i | 2768.49 | + | 2768.49i | −3981.10 | + | 6895.47i | −3452.98 | + | 1993.58i |
2.3 | −15.5495 | − | 4.16648i | 11.3712 | + | 19.6955i | 2.72537 | + | 1.57349i | −311.308 | + | 311.308i | −94.7557 | − | 353.633i | 4365.04 | − | 1169.61i | 2878.23 | + | 2878.23i | 3021.89 | − | 5234.07i | 6137.75 | − | 3543.63i |
2.4 | −4.97311 | − | 1.33254i | 73.7308 | + | 127.705i | −198.746 | − | 114.746i | −65.5592 | + | 65.5592i | −196.499 | − | 733.343i | −2444.94 | + | 655.120i | 1767.47 | + | 1767.47i | −7591.95 | + | 13149.6i | 413.394 | − | 238.673i |
2.5 | 4.84258 | + | 1.29756i | 0.725648 | + | 1.25686i | −199.936 | − | 115.433i | 724.141 | − | 724.141i | 1.88315 | + | 7.02801i | 258.924 | − | 69.3786i | −1725.95 | − | 1725.95i | 3279.45 | − | 5680.17i | 4446.33 | − | 2567.09i |
2.6 | 8.65394 | + | 2.31882i | −30.7307 | − | 53.2271i | −152.189 | − | 87.8662i | −612.629 | + | 612.629i | −142.518 | − | 531.883i | −968.610 | + | 259.538i | −2735.08 | − | 2735.08i | 1391.75 | − | 2410.58i | −6722.23 | + | 3881.08i |
2.7 | 22.9875 | + | 6.15949i | 34.8819 | + | 60.4172i | 268.785 | + | 155.183i | −80.1911 | + | 80.1911i | 429.709 | + | 1603.70i | 249.971 | − | 66.9796i | 914.868 | + | 914.868i | 847.009 | − | 1467.06i | −2337.33 | + | 1349.46i |
2.8 | 26.2554 | + | 7.03513i | −76.4565 | − | 132.427i | 418.153 | + | 241.421i | 365.377 | − | 365.377i | −1075.76 | − | 4014.80i | 1567.43 | − | 419.990i | 4359.97 | + | 4359.97i | −8410.69 | + | 14567.7i | 12163.6 | − | 7022.67i |
6.1 | −7.06838 | − | 26.3795i | −12.3990 | + | 21.4757i | −424.216 | + | 244.921i | −289.939 | + | 289.939i | 654.160 | + | 175.282i | 169.108 | − | 631.121i | 4515.77 | + | 4515.77i | 2973.03 | + | 5149.44i | 9697.86 | + | 5599.06i |
6.2 | −4.98932 | − | 18.6204i | 51.4031 | − | 89.0328i | −100.124 | + | 57.8065i | 663.519 | − | 663.519i | −1914.29 | − | 512.934i | −974.220 | + | 3635.84i | −1913.63 | − | 1913.63i | −2004.06 | − | 3471.13i | −15665.5 | − | 9044.49i |
6.3 | −2.99242 | − | 11.1679i | −49.8723 | + | 86.3814i | 105.936 | − | 61.1621i | 403.844 | − | 403.844i | 1113.93 | + | 298.478i | 636.017 | − | 2373.65i | −3092.97 | − | 3092.97i | −1694.00 | − | 2934.09i | −5718.55 | − | 3301.61i |
6.4 | −1.72361 | − | 6.43259i | 53.7668 | − | 93.1269i | 183.295 | − | 105.825i | −590.780 | + | 590.780i | −691.720 | − | 185.346i | 973.383 | − | 3632.71i | −2202.16 | − | 2202.16i | −2501.24 | − | 4332.28i | 4818.52 | + | 2781.97i |
6.5 | −0.0710913 | − | 0.265316i | −24.7951 | + | 42.9464i | 221.637 | − | 127.962i | −462.533 | + | 462.533i | 13.1571 | + | 3.52543i | −1022.80 | + | 3817.16i | −99.4285 | − | 99.4285i | 2050.90 | + | 3552.27i | 155.600 | + | 89.8354i |
6.6 | 3.54920 | + | 13.2458i | 29.1358 | − | 50.4648i | 58.8481 | − | 33.9759i | 367.308 | − | 367.308i | 771.855 | + | 206.818i | 60.5481 | − | 225.969i | 3141.23 | + | 3141.23i | 1582.71 | + | 2741.33i | 6168.94 | + | 3561.64i |
6.7 | 5.40711 | + | 20.1796i | −57.1883 | + | 99.0531i | −156.277 | + | 90.2265i | 54.1625 | − | 54.1625i | −2308.07 | − | 618.447i | 319.890 | − | 1193.84i | 1116.02 | + | 1116.02i | −3260.51 | − | 5647.36i | 1385.84 | + | 800.116i |
6.8 | 7.75454 | + | 28.9403i | 32.8317 | − | 56.8661i | −555.707 | + | 320.838i | −552.313 | + | 552.313i | 1900.32 | + | 509.189i | −156.442 | + | 583.848i | −8170.84 | − | 8170.84i | 1124.66 | + | 1947.97i | −20267.1 | − | 11701.2i |
7.1 | −28.0190 | + | 7.50766i | 22.8511 | − | 39.5793i | 506.995 | − | 292.714i | 131.372 | + | 131.372i | −343.117 | + | 1280.53i | −2789.95 | − | 747.565i | −6757.00 | + | 6757.00i | 2236.15 | + | 3873.13i | −4667.19 | − | 2694.60i |
7.2 | −16.0639 | + | 4.30431i | −60.2561 | + | 104.367i | 17.8198 | − | 10.2883i | 169.528 | + | 169.528i | 518.723 | − | 1935.90i | −1635.34 | − | 438.187i | 2768.49 | − | 2768.49i | −3981.10 | − | 6895.47i | −3452.98 | − | 1993.58i |
7.3 | −15.5495 | + | 4.16648i | 11.3712 | − | 19.6955i | 2.72537 | − | 1.57349i | −311.308 | − | 311.308i | −94.7557 | + | 353.633i | 4365.04 | + | 1169.61i | 2878.23 | − | 2878.23i | 3021.89 | + | 5234.07i | 6137.75 | + | 3543.63i |
7.4 | −4.97311 | + | 1.33254i | 73.7308 | − | 127.705i | −198.746 | + | 114.746i | −65.5592 | − | 65.5592i | −196.499 | + | 733.343i | −2444.94 | − | 655.120i | 1767.47 | − | 1767.47i | −7591.95 | − | 13149.6i | 413.394 | + | 238.673i |
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.f | odd | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 13.9.f.a | ✓ | 32 |
13.f | odd | 12 | 1 | inner | 13.9.f.a | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
13.9.f.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
13.9.f.a | ✓ | 32 | 13.f | odd | 12 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(13, [\chi])\).