Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [51,7,Mod(38,51)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(51, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("51.38");
S:= CuspForms(chi, 7);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 51 = 3 \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 51.f (of order \(4\), degree \(2\), 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: | \(11.7327582646\) |
Analytic rank: | \(0\) |
Dimension: | \(68\) |
Relative dimension: | \(34\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
38.1 | −15.2598 | −17.0979 | − | 20.8965i | 168.862 | 23.5746 | + | 23.5746i | 260.911 | + | 318.876i | −238.814 | − | 238.814i | −1600.18 | −144.325 | + | 714.571i | −359.744 | − | 359.744i | ||||||
38.2 | −14.7512 | 3.59419 | + | 26.7597i | 153.599 | −68.2416 | − | 68.2416i | −53.0188 | − | 394.739i | 124.784 | + | 124.784i | −1321.70 | −703.164 | + | 192.359i | 1006.65 | + | 1006.65i | ||||||
38.3 | −14.0069 | 25.1159 | − | 9.90914i | 132.195 | 154.796 | + | 154.796i | −351.797 | + | 138.797i | 321.666 | + | 321.666i | −955.198 | 532.618 | − | 497.754i | −2168.22 | − | 2168.22i | ||||||
38.4 | −13.3013 | 20.6191 | − | 17.4313i | 112.924 | −135.530 | − | 135.530i | −274.261 | + | 231.859i | −104.927 | − | 104.927i | −650.746 | 121.297 | − | 718.838i | 1802.72 | + | 1802.72i | ||||||
38.5 | −12.6359 | −17.0115 | + | 20.9669i | 95.6654 | 152.781 | + | 152.781i | 214.955 | − | 264.935i | −234.270 | − | 234.270i | −400.120 | −150.221 | − | 713.355i | −1930.52 | − | 1930.52i | ||||||
38.6 | −11.5147 | −26.9464 | + | 1.69997i | 68.5886 | −61.6662 | − | 61.6662i | 310.280 | − | 19.5747i | 157.734 | + | 157.734i | −52.8363 | 723.220 | − | 91.6164i | 710.068 | + | 710.068i | ||||||
38.7 | −10.6271 | 24.8978 | + | 10.4451i | 48.9356 | 5.72766 | + | 5.72766i | −264.592 | − | 111.002i | −324.888 | − | 324.888i | 160.091 | 510.798 | + | 520.121i | −60.8685 | − | 60.8685i | ||||||
38.8 | −9.96367 | −2.68583 | − | 26.8661i | 35.2747 | −16.8097 | − | 16.8097i | 26.7607 | + | 267.685i | 296.612 | + | 296.612i | 286.209 | −714.573 | + | 144.315i | 167.486 | + | 167.486i | ||||||
38.9 | −8.95432 | 15.4594 | + | 22.1361i | 16.1799 | 34.6227 | + | 34.6227i | −138.428 | − | 198.214i | 218.156 | + | 218.156i | 428.197 | −251.014 | + | 684.422i | −310.022 | − | 310.022i | ||||||
38.10 | −7.21375 | −16.6299 | + | 21.2708i | −11.9618 | −104.798 | − | 104.798i | 119.964 | − | 153.442i | −266.693 | − | 266.693i | 547.970 | −175.895 | − | 707.462i | 755.989 | + | 755.989i | ||||||
38.11 | −7.18422 | 8.83916 | − | 25.5121i | −12.3869 | 72.9187 | + | 72.9187i | −63.5025 | + | 183.285i | −433.048 | − | 433.048i | 548.781 | −572.739 | − | 451.012i | −523.864 | − | 523.864i | ||||||
38.12 | −6.18050 | −23.8985 | − | 12.5644i | −25.8014 | 111.665 | + | 111.665i | 147.704 | + | 77.6542i | 7.14097 | + | 7.14097i | 555.018 | 413.272 | + | 600.539i | −690.145 | − | 690.145i | ||||||
38.13 | −4.10952 | 26.3021 | + | 6.09898i | −47.1119 | −147.997 | − | 147.997i | −108.089 | − | 25.0638i | 245.141 | + | 245.141i | 456.616 | 654.605 | + | 320.832i | 608.198 | + | 608.198i | ||||||
38.14 | −3.73857 | −13.3008 | + | 23.4965i | −50.0231 | 94.5565 | + | 94.5565i | 49.7261 | − | 87.8435i | 434.343 | + | 434.343i | 426.283 | −375.175 | − | 625.048i | −353.506 | − | 353.506i | ||||||
38.15 | −3.17896 | 24.3955 | − | 11.5698i | −53.8942 | 27.6632 | + | 27.6632i | −77.5524 | + | 36.7800i | 99.5174 | + | 99.5174i | 374.781 | 461.280 | − | 564.501i | −87.9404 | − | 87.9404i | ||||||
38.16 | −2.39460 | −13.6109 | − | 23.3183i | −58.2659 | −151.447 | − | 151.447i | 32.5926 | + | 55.8380i | −127.576 | − | 127.576i | 292.778 | −358.488 | + | 634.765i | 362.655 | + | 362.655i | ||||||
38.17 | −1.10526 | 2.63593 | + | 26.8710i | −62.7784 | −6.38746 | − | 6.38746i | −2.91338 | − | 29.6994i | −175.878 | − | 175.878i | 140.123 | −715.104 | + | 141.660i | 7.05978 | + | 7.05978i | ||||||
38.18 | 1.10526 | −26.8710 | − | 2.63593i | −62.7784 | 6.38746 | + | 6.38746i | −29.6994 | − | 2.91338i | −175.878 | − | 175.878i | −140.123 | 715.104 | + | 141.660i | 7.05978 | + | 7.05978i | ||||||
38.19 | 2.39460 | 23.3183 | + | 13.6109i | −58.2659 | 151.447 | + | 151.447i | 55.8380 | + | 32.5926i | −127.576 | − | 127.576i | −292.778 | 358.488 | + | 634.765i | 362.655 | + | 362.655i | ||||||
38.20 | 3.17896 | 11.5698 | − | 24.3955i | −53.8942 | −27.6632 | − | 27.6632i | 36.7800 | − | 77.5524i | 99.5174 | + | 99.5174i | −374.781 | −461.280 | − | 564.501i | −87.9404 | − | 87.9404i | ||||||
See all 68 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
17.c | even | 4 | 1 | inner |
51.f | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 51.7.f.a | ✓ | 68 |
3.b | odd | 2 | 1 | inner | 51.7.f.a | ✓ | 68 |
17.c | even | 4 | 1 | inner | 51.7.f.a | ✓ | 68 |
51.f | odd | 4 | 1 | inner | 51.7.f.a | ✓ | 68 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
51.7.f.a | ✓ | 68 | 1.a | even | 1 | 1 | trivial |
51.7.f.a | ✓ | 68 | 3.b | odd | 2 | 1 | inner |
51.7.f.a | ✓ | 68 | 17.c | even | 4 | 1 | inner |
51.7.f.a | ✓ | 68 | 51.f | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(51, [\chi])\).