Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [37,7,Mod(2,37)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(37, base_ring=CyclotomicField(36))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("37.2");
S:= CuspForms(chi, 7);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 37 \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 37.i (of order \(36\), degree \(12\), 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: | \(8.51200109393\) |
Analytic rank: | \(0\) |
Dimension: | \(216\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{36})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{36}]$ |
$q$-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 | −14.8980 | − | 1.30340i | 11.6276 | − | 13.8572i | 157.222 | + | 27.7226i | 4.78254 | − | 10.2562i | −191.288 | + | 191.288i | −248.810 | + | 90.5593i | −1381.66 | − | 370.215i | 69.7682 | + | 395.675i | −84.6179 | + | 146.563i |
2.2 | −12.6746 | − | 1.10888i | −25.8114 | + | 30.7608i | 96.3874 | + | 16.9957i | 50.4999 | − | 108.297i | 361.258 | − | 361.258i | 84.5834 | − | 30.7859i | −416.297 | − | 111.547i | −153.410 | − | 870.029i | −760.153 | + | 1316.62i |
2.3 | −11.5362 | − | 1.00928i | −6.04724 | + | 7.20682i | 69.0368 | + | 12.1730i | −53.1787 | + | 114.042i | 77.0357 | − | 77.0357i | 358.686 | − | 130.551i | −68.2523 | − | 18.2882i | 111.220 | + | 630.762i | 728.579 | − | 1261.94i |
2.4 | −8.95954 | − | 0.783858i | 21.0804 | − | 25.1226i | 16.6313 | + | 2.93254i | 74.3976 | − | 159.546i | −208.563 | + | 208.563i | 147.602 | − | 53.7227i | 409.278 | + | 109.666i | −60.1738 | − | 341.262i | −791.630 | + | 1371.14i |
2.5 | −7.75264 | − | 0.678269i | 26.0231 | − | 31.0131i | −3.38424 | − | 0.596733i | −73.2086 | + | 156.996i | −222.783 | + | 222.783i | −148.093 | + | 53.9015i | 506.925 | + | 135.830i | −158.022 | − | 896.190i | 674.046 | − | 1167.48i |
2.6 | −7.18053 | − | 0.628215i | −17.1726 | + | 20.4655i | −11.8624 | − | 2.09166i | −35.1352 | + | 75.3476i | 136.165 | − | 136.165i | −623.699 | + | 227.008i | 529.455 | + | 141.867i | 2.65032 | + | 15.0307i | 299.623 | − | 518.963i |
2.7 | −3.71967 | − | 0.325429i | −9.90022 | + | 11.7986i | −49.2976 | − | 8.69250i | 68.4388 | − | 146.767i | 40.6652 | − | 40.6652i | −17.2823 | + | 6.29023i | 411.368 | + | 110.226i | 85.3964 | + | 484.307i | −302.332 | + | 523.655i |
2.8 | −1.08129 | − | 0.0946003i | 0.982487 | − | 1.17088i | −61.8675 | − | 10.9089i | −20.8024 | + | 44.6109i | −1.17312 | + | 1.17312i | 425.722 | − | 154.950i | 132.964 | + | 35.6276i | 126.184 | + | 715.624i | 26.7136 | − | 46.2693i |
2.9 | −0.744857 | − | 0.0651666i | −32.3086 | + | 38.5039i | −62.4771 | − | 11.0164i | −73.6496 | + | 157.942i | 26.5745 | − | 26.5745i | 281.089 | − | 102.308i | 92.0410 | + | 24.6623i | −312.115 | − | 1770.09i | 65.1510 | − | 112.845i |
2.10 | 1.35176 | + | 0.118264i | 12.8836 | − | 15.3541i | −61.2144 | − | 10.7938i | 3.06542 | − | 6.57381i | 19.2315 | − | 19.2315i | −365.776 | + | 133.132i | −165.355 | − | 44.3068i | 56.8286 | + | 322.291i | 4.92117 | − | 8.52371i |
2.11 | 5.23865 | + | 0.458323i | 29.7705 | − | 35.4790i | −35.7943 | − | 6.31149i | −1.99257 | + | 4.27309i | 172.218 | − | 172.218i | 376.658 | − | 137.092i | −509.708 | − | 136.576i | −245.893 | − | 1394.53i | −12.3969 | + | 21.4720i |
2.12 | 5.48890 | + | 0.480216i | −25.0783 | + | 29.8871i | −33.1303 | − | 5.84176i | 69.6929 | − | 149.457i | −152.004 | + | 152.004i | 37.2020 | − | 13.5404i | −519.659 | − | 139.242i | −137.731 | − | 781.110i | 454.309 | − | 786.886i |
2.13 | 8.35059 | + | 0.730582i | 1.76500 | − | 2.10345i | 6.17099 | + | 1.08811i | −100.792 | + | 216.150i | 16.2756 | − | 16.2756i | −148.724 | + | 54.1313i | −467.463 | − | 125.256i | 125.280 | + | 710.500i | −999.591 | + | 1731.34i |
2.14 | 8.51484 | + | 0.744952i | −15.5833 | + | 18.5715i | 8.91992 | + | 1.57282i | −14.3618 | + | 30.7990i | −146.524 | + | 146.524i | −88.0064 | + | 32.0317i | −453.612 | − | 121.545i | 24.5295 | + | 139.114i | −145.232 | + | 251.550i |
2.15 | 9.97417 | + | 0.872627i | 16.2625 | − | 19.3809i | 35.6949 | + | 6.29397i | 95.3651 | − | 204.511i | 179.118 | − | 179.118i | −390.942 | + | 142.291i | −268.416 | − | 71.9219i | 15.4390 | + | 87.5587i | 1129.65 | − | 1956.61i |
2.16 | 12.8119 | + | 1.12090i | −2.07304 | + | 2.47056i | 99.8612 | + | 17.6082i | 27.6431 | − | 59.2807i | −29.3289 | + | 29.3289i | 497.914 | − | 181.226i | 464.628 | + | 124.497i | 124.783 | + | 707.682i | 420.608 | − | 728.515i |
2.17 | 14.3464 | + | 1.25514i | 22.3262 | − | 26.6073i | 141.215 | + | 24.9000i | −37.3574 | + | 80.1132i | 353.696 | − | 353.696i | −152.507 | + | 55.5081i | 1104.40 | + | 295.923i | −82.9011 | − | 470.155i | −636.496 | + | 1102.44i |
2.18 | 15.1491 | + | 1.32538i | −28.9017 | + | 34.4437i | 164.711 | + | 29.0431i | −13.1400 | + | 28.1787i | −483.486 | + | 483.486i | −429.463 | + | 156.312i | 1516.66 | + | 406.387i | −224.471 | − | 1273.04i | −236.406 | + | 409.468i |
5.1 | −6.52282 | + | 13.9882i | 4.44480 | − | 12.2120i | −111.985 | − | 133.458i | −123.914 | + | 176.967i | 141.831 | + | 141.831i | −94.3880 | − | 535.301i | 1643.16 | − | 440.285i | 429.070 | + | 360.033i | −1667.19 | − | 2887.66i |
5.2 | −5.53836 | + | 11.8770i | 6.06356 | − | 16.6595i | −69.2525 | − | 82.5319i | 60.5521 | − | 86.4774i | 164.283 | + | 164.283i | 72.0577 | + | 408.660i | 553.647 | − | 148.349i | 317.674 | + | 266.560i | 691.737 | + | 1198.12i |
See next 80 embeddings (of 216 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
37.i | odd | 36 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 37.7.i.a | ✓ | 216 |
37.i | odd | 36 | 1 | inner | 37.7.i.a | ✓ | 216 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
37.7.i.a | ✓ | 216 | 1.a | even | 1 | 1 | trivial |
37.7.i.a | ✓ | 216 | 37.i | odd | 36 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(37, [\chi])\).