Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [69,6,Mod(4,69)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(69, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("69.4");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 69 = 3 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 69.e (of order \(11\), 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: | \(11.0664835671\) |
Analytic rank: | \(0\) |
Dimension: | \(100\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{11})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{11}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −4.05453 | − | 8.87818i | −8.63544 | + | 2.53559i | −41.4274 | + | 47.8097i | −10.1643 | − | 6.53221i | 57.5241 | + | 66.3863i | 18.4916 | + | 128.612i | 292.757 | + | 85.9613i | 68.1415 | − | 43.7919i | −16.7826 | + | 116.726i |
4.2 | −2.89326 | − | 6.33535i | −8.63544 | + | 2.53559i | −10.8102 | + | 12.4756i | 59.0516 | + | 37.9501i | 41.0484 | + | 47.3724i | −20.5053 | − | 142.618i | −103.530 | − | 30.3990i | 68.1415 | − | 43.7919i | 69.5761 | − | 483.912i |
4.3 | −2.87865 | − | 6.30338i | −8.63544 | + | 2.53559i | −10.4903 | + | 12.1065i | −49.4324 | − | 31.7683i | 40.8412 | + | 47.1333i | −20.2428 | − | 140.792i | −106.255 | − | 31.1992i | 68.1415 | − | 43.7919i | −57.9485 | + | 403.041i |
4.4 | −0.582830 | − | 1.27622i | −8.63544 | + | 2.53559i | 19.6665 | − | 22.6963i | −65.1725 | − | 41.8838i | 8.26897 | + | 9.54290i | 14.8045 | + | 102.968i | −83.5054 | − | 24.5194i | 68.1415 | − | 43.7919i | −15.4685 | + | 107.586i |
4.5 | −0.310130 | − | 0.679089i | −8.63544 | + | 2.53559i | 20.5906 | − | 23.7628i | 14.4659 | + | 9.29667i | 4.40000 | + | 5.07787i | 1.15767 | + | 8.05176i | −45.4448 | − | 13.3438i | 68.1415 | − | 43.7919i | 1.82696 | − | 12.7068i |
4.6 | 2.02749 | + | 4.43958i | −8.63544 | + | 2.53559i | 5.35641 | − | 6.18163i | 66.9970 | + | 43.0564i | −28.7652 | − | 33.1968i | −25.3233 | − | 176.127i | 188.158 | + | 55.2480i | 68.1415 | − | 43.7919i | −55.3165 | + | 384.735i |
4.7 | 2.17624 | + | 4.76530i | −8.63544 | + | 2.53559i | 2.98345 | − | 3.44308i | 20.2644 | + | 13.0232i | −30.8757 | − | 35.6324i | 8.80898 | + | 61.2678i | 183.748 | + | 53.9534i | 68.1415 | − | 43.7919i | −17.9590 | + | 124.908i |
4.8 | 2.60326 | + | 5.70034i | −8.63544 | + | 2.53559i | −4.76135 | + | 5.49489i | −65.3942 | − | 42.0263i | −36.9340 | − | 42.6241i | 8.25660 | + | 57.4259i | 148.692 | + | 43.6598i | 68.1415 | − | 43.7919i | 69.3261 | − | 482.174i |
4.9 | 4.09417 | + | 8.96497i | −8.63544 | + | 2.53559i | −42.6530 | + | 49.2242i | −26.2487 | − | 16.8690i | −58.0864 | − | 67.0353i | −28.2239 | − | 196.302i | −313.318 | − | 91.9985i | 68.1415 | − | 43.7919i | 43.7638 | − | 304.384i |
4.10 | 4.34821 | + | 9.52125i | −8.63544 | + | 2.53559i | −50.7917 | + | 58.6168i | 82.5719 | + | 53.0657i | −61.6907 | − | 71.1949i | 29.0096 | + | 201.766i | −457.577 | − | 134.357i | 68.1415 | − | 43.7919i | −146.212 | + | 1016.93i |
13.1 | −6.62196 | − | 7.64215i | 7.57128 | + | 4.86577i | −9.99801 | + | 69.5377i | 22.3300 | − | 48.8959i | −12.9518 | − | 90.0818i | −79.5712 | − | 23.3642i | 325.407 | − | 209.127i | 33.6486 | + | 73.6802i | −521.539 | + | 153.138i |
13.2 | −5.44390 | − | 6.28260i | 7.57128 | + | 4.86577i | −5.28090 | + | 36.7294i | −18.8225 | + | 41.2154i | −10.6477 | − | 74.0561i | −66.5270 | − | 19.5341i | 35.7163 | − | 22.9535i | 33.6486 | + | 73.6802i | 361.408 | − | 106.119i |
13.3 | −3.21781 | − | 3.71355i | 7.57128 | + | 4.86577i | 1.11793 | − | 7.77535i | 37.1130 | − | 81.2661i | −6.29366 | − | 43.7734i | 196.204 | + | 57.6107i | −164.749 | + | 105.878i | 33.6486 | + | 73.6802i | −421.208 | + | 123.678i |
13.4 | −2.79827 | − | 3.22937i | 7.57128 | + | 4.86577i | 1.95553 | − | 13.6010i | −19.4524 | + | 42.5948i | −5.47309 | − | 38.0662i | 36.6143 | + | 10.7509i | −164.426 | + | 105.670i | 33.6486 | + | 73.6802i | 191.988 | − | 56.3726i |
13.5 | −0.220291 | − | 0.254230i | 7.57128 | + | 4.86577i | 4.53797 | − | 31.5623i | 21.3588 | − | 46.7692i | −0.430865 | − | 2.99673i | −219.257 | − | 64.3798i | −18.0795 | + | 11.6190i | 33.6486 | + | 73.6802i | −16.5953 | + | 4.87282i |
13.6 | 2.18855 | + | 2.52572i | 7.57128 | + | 4.86577i | 2.96456 | − | 20.6189i | −1.08863 | + | 2.38376i | 4.28056 | + | 29.7719i | 60.1265 | + | 17.6547i | 148.533 | − | 95.4564i | 33.6486 | + | 73.6802i | −8.40324 | + | 2.46741i |
13.7 | 4.37852 | + | 5.05308i | 7.57128 | + | 4.86577i | −1.80812 | + | 12.5757i | 41.9916 | − | 91.9487i | 8.56389 | + | 59.5631i | −7.37026 | − | 2.16411i | 108.530 | − | 69.7479i | 33.6486 | + | 73.6802i | 648.485 | − | 190.412i |
13.8 | 4.58877 | + | 5.29572i | 7.57128 | + | 4.86577i | −2.43379 | + | 16.9274i | −42.3452 | + | 92.7231i | 8.97511 | + | 62.4233i | −216.718 | − | 63.6341i | 87.8249 | − | 56.4417i | 33.6486 | + | 73.6802i | −685.348 | + | 201.236i |
13.9 | 5.18515 | + | 5.98398i | 7.57128 | + | 4.86577i | −4.36819 | + | 30.3814i | −11.2004 | + | 24.5255i | 10.1416 | + | 70.5362i | 164.984 | + | 48.4438i | 8.70029 | − | 5.59133i | 33.6486 | + | 73.6802i | −204.836 | + | 60.1453i |
13.10 | 7.28389 | + | 8.40605i | 7.57128 | + | 4.86577i | −13.0526 | + | 90.7832i | 7.34378 | − | 16.0806i | 14.2465 | + | 99.0863i | −9.51310 | − | 2.79330i | −558.775 | + | 359.103i | 33.6486 | + | 73.6802i | 188.666 | − | 55.3973i |
See all 100 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.c | even | 11 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 69.6.e.b | ✓ | 100 |
23.c | even | 11 | 1 | inner | 69.6.e.b | ✓ | 100 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
69.6.e.b | ✓ | 100 | 1.a | even | 1 | 1 | trivial |
69.6.e.b | ✓ | 100 | 23.c | even | 11 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{100} - 12 T_{2}^{99} + 306 T_{2}^{98} - 1564 T_{2}^{97} + 39671 T_{2}^{96} + \cdots + 59\!\cdots\!44 \) acting on \(S_{6}^{\mathrm{new}}(69, [\chi])\).