Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [49,5,Mod(6,49)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(49, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([9]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("49.6");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 49.f (of order \(14\), 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: | \(5.06512819111\) |
Analytic rank: | \(0\) |
Dimension: | \(102\) |
Relative dimension: | \(17\) over \(\Q(\zeta_{14})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{14}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
6.1 | −4.72332 | + | 5.92286i | 3.57449 | − | 7.42250i | −9.21016 | − | 40.3523i | −7.63324 | + | 15.8506i | 27.0790 | + | 56.2301i | 48.8460 | − | 3.88170i | 173.297 | + | 83.4557i | 8.18611 | + | 10.2651i | −57.8265 | − | 120.078i |
6.2 | −4.23981 | + | 5.31655i | −3.45640 | + | 7.17729i | −6.72942 | − | 29.4835i | 10.4167 | − | 21.6306i | −23.5040 | − | 48.8065i | −33.0265 | − | 36.1974i | 87.2550 | + | 42.0198i | 10.9359 | + | 13.7132i | 70.8352 | + | 147.091i |
6.3 | −3.61910 | + | 4.53821i | −4.63704 | + | 9.62891i | −3.93710 | − | 17.2496i | −13.6463 | + | 28.3369i | −26.9161 | − | 55.8918i | −0.846585 | + | 48.9927i | 8.85487 | + | 4.26428i | −20.7111 | − | 25.9709i | −79.2113 | − | 164.484i |
6.4 | −3.02455 | + | 3.79266i | 2.80761 | − | 5.83006i | −1.67606 | − | 7.34331i | −2.45023 | + | 5.08795i | 13.6197 | + | 28.2816i | −31.7325 | + | 37.3369i | −37.0095 | − | 17.8228i | 24.3958 | + | 30.5913i | −11.8860 | − | 24.6816i |
6.5 | −2.51536 | + | 3.15417i | 6.48273 | − | 13.4615i | −0.0613763 | − | 0.268907i | 0.0863095 | − | 0.179223i | 26.1535 | + | 54.3083i | −13.2950 | − | 47.1619i | −57.1543 | − | 27.5241i | −88.6843 | − | 111.207i | 0.348201 | + | 0.723047i |
6.6 | −2.24615 | + | 2.81658i | 0.457590 | − | 0.950195i | 0.672393 | + | 2.94595i | 17.5653 | − | 36.4748i | 1.64849 | + | 3.42312i | 45.9934 | + | 16.9000i | −61.7402 | − | 29.7325i | 49.8092 | + | 62.4588i | 63.2798 | + | 131.402i |
6.7 | −1.53378 | + | 1.92330i | −4.33690 | + | 9.00566i | 2.21373 | + | 9.69900i | −8.18548 | + | 16.9973i | −10.6688 | − | 22.1539i | 24.7221 | − | 42.3063i | −57.5115 | − | 27.6961i | −11.7906 | − | 14.7850i | −20.1362 | − | 41.8133i |
6.8 | −0.190677 | + | 0.239101i | −7.10131 | + | 14.7460i | 3.53952 | + | 15.5077i | 11.9420 | − | 24.7979i | −2.17173 | − | 4.50965i | −32.6258 | + | 36.5590i | −8.79138 | − | 4.23370i | −116.514 | − | 146.104i | 3.65213 | + | 7.58373i |
6.9 | 0.101848 | − | 0.127714i | 0.716533 | − | 1.48790i | 3.55440 | + | 15.5728i | −5.00109 | + | 10.3849i | −0.117047 | − | 0.243050i | −48.8363 | + | 4.00153i | 4.70567 | + | 2.26613i | 48.8023 | + | 61.1961i | 0.816937 | + | 1.69639i |
6.10 | 0.238140 | − | 0.298618i | 3.88212 | − | 8.06131i | 3.52787 | + | 15.4566i | −18.4576 | + | 38.3276i | −1.48276 | − | 3.07899i | 46.2093 | + | 16.3002i | 10.9617 | + | 5.27888i | 0.588851 | + | 0.738396i | 7.04983 | + | 14.6391i |
6.11 | 1.25074 | − | 1.56837i | 7.29580 | − | 15.1499i | 2.66488 | + | 11.6756i | 13.2800 | − | 27.5763i | −14.6355 | − | 30.3910i | 1.44197 | + | 48.9788i | 50.5626 | + | 24.3497i | −125.788 | − | 157.733i | −26.6401 | − | 55.3187i |
6.12 | 1.54183 | − | 1.93339i | 0.771738 | − | 1.60253i | 2.19957 | + | 9.63692i | 14.3069 | − | 29.7086i | −1.90843 | − | 3.96290i | −14.7592 | − | 46.7244i | 57.6714 | + | 27.7731i | 48.5302 | + | 60.8549i | −35.3796 | − | 73.4666i |
6.13 | 2.38635 | − | 2.99238i | −3.28786 | + | 6.82732i | 0.300624 | + | 1.31712i | 0.646610 | − | 1.34270i | 12.5840 | + | 26.1309i | 43.0574 | + | 23.3893i | 59.8326 | + | 28.8139i | 14.7005 | + | 18.4338i | −2.47484 | − | 5.13905i |
6.14 | 3.33676 | − | 4.18417i | −4.73280 | + | 9.82775i | −2.81295 | − | 12.3243i | −18.3794 | + | 38.1652i | 25.3288 | + | 52.5957i | −48.9209 | − | 2.78244i | 16.1949 | + | 7.79907i | −23.6827 | − | 29.6972i | 98.3620 | + | 204.251i |
6.15 | 3.56270 | − | 4.46749i | 4.66338 | − | 9.68361i | −3.70526 | − | 16.2338i | −6.40559 | + | 13.3014i | −26.6472 | − | 55.3334i | 12.2647 | − | 47.4402i | −3.35304 | − | 1.61474i | −21.5224 | − | 26.9883i | 36.6024 | + | 76.0057i |
6.16 | 4.29229 | − | 5.38236i | 2.34281 | − | 4.86490i | −6.98570 | − | 30.6064i | 3.49106 | − | 7.24926i | −16.1286 | − | 33.4914i | −15.3426 | + | 46.5360i | −95.4785 | − | 45.9800i | 32.3242 | + | 40.5332i | −24.0335 | − | 49.9061i |
6.17 | 4.60462 | − | 5.77401i | −5.81900 | + | 12.0833i | −8.57632 | − | 37.5753i | 15.6615 | − | 32.5215i | 42.9747 | + | 89.2379i | 8.91373 | − | 48.1824i | −149.989 | − | 72.2309i | −61.6423 | − | 77.2970i | −115.664 | − | 240.179i |
13.1 | −1.74700 | − | 7.65410i | −7.57880 | + | 6.04389i | −41.1178 | + | 19.8013i | 5.79433 | − | 4.62082i | 59.5007 | + | 47.4502i | 15.8681 | − | 46.3595i | 145.074 | + | 181.917i | 2.88536 | − | 12.6416i | −45.4909 | − | 36.2778i |
13.2 | −1.44765 | − | 6.34256i | 9.73419 | − | 7.76276i | −23.7169 | + | 11.4215i | −17.9952 | + | 14.3507i | −63.3275 | − | 50.5020i | −38.0892 | − | 30.8256i | 41.8754 | + | 52.5101i | 16.4699 | − | 72.1592i | 117.071 | + | 93.3611i |
13.3 | −1.26018 | − | 5.52122i | −2.76841 | + | 2.20774i | −14.4803 | + | 6.97334i | −15.1005 | + | 12.0422i | 15.6781 | + | 12.5029i | 3.85271 | + | 48.8483i | 0.253896 | + | 0.318375i | −15.2342 | + | 66.7453i | 85.5170 | + | 68.1975i |
See next 80 embeddings (of 102 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
49.f | odd | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 49.5.f.a | ✓ | 102 |
49.f | odd | 14 | 1 | inner | 49.5.f.a | ✓ | 102 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
49.5.f.a | ✓ | 102 | 1.a | even | 1 | 1 | trivial |
49.5.f.a | ✓ | 102 | 49.f | odd | 14 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(49, [\chi])\).