Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [35,6,Mod(3,35)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(35, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([9, 2]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("35.3");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 35 = 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 35.k (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: | \(5.61343369345\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(18\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −9.95978 | − | 2.66872i | −4.43456 | − | 16.5500i | 64.3624 | + | 37.1597i | 55.6892 | + | 4.86925i | 176.669i | 127.528 | − | 23.3159i | −308.553 | − | 308.553i | −43.7930 | + | 25.2839i | −541.658 | − | 197.115i | ||
3.2 | −8.82949 | − | 2.36586i | −1.01039 | − | 3.77082i | 44.6498 | + | 25.7786i | −51.5659 | + | 21.5861i | 35.6848i | −68.0178 | + | 110.366i | −126.411 | − | 126.411i | 197.246 | − | 113.880i | 506.370 | − | 68.5968i | ||
3.3 | −8.64179 | − | 2.31556i | 7.51818 | + | 28.0582i | 41.6059 | + | 24.0212i | −15.7494 | + | 53.6373i | − | 259.882i | 119.105 | − | 51.1954i | −101.488 | − | 101.488i | −520.297 | + | 300.393i | 260.303 | − | 427.054i | |
3.4 | −8.12393 | − | 2.17680i | 3.17140 | + | 11.8358i | 33.5470 | + | 19.3683i | 32.4351 | − | 45.5298i | − | 103.057i | −127.425 | + | 23.8695i | −40.0639 | − | 40.0639i | 80.4154 | − | 46.4279i | −362.610 | + | 299.276i | |
3.5 | −5.77451 | − | 1.54727i | −5.02978 | − | 18.7714i | 3.23807 | + | 1.86950i | −46.7278 | − | 30.6841i | 116.178i | 36.5834 | − | 124.373i | 119.466 | + | 119.466i | −116.622 | + | 67.3320i | 222.353 | + | 249.486i | ||
3.6 | −4.18974 | − | 1.12264i | 0.900875 | + | 3.36211i | −11.4192 | − | 6.59287i | 36.1647 | + | 42.6276i | − | 15.0977i | −63.7723 | − | 112.872i | 138.589 | + | 138.589i | 199.952 | − | 115.442i | −103.665 | − | 219.199i | |
3.7 | −3.55905 | − | 0.953643i | 3.79917 | + | 14.1787i | −15.9554 | − | 9.21188i | −11.6079 | − | 54.6832i | − | 54.0857i | 120.817 | + | 47.0133i | 131.374 | + | 131.374i | 23.8423 | − | 13.7654i | −10.8354 | + | 205.690i | |
3.8 | −2.46827 | − | 0.661371i | −7.06878 | − | 26.3810i | −22.0579 | − | 12.7351i | 47.7429 | − | 29.0794i | 69.7907i | −83.3911 | + | 99.2619i | 103.843 | + | 103.843i | −435.548 | + | 251.464i | −137.075 | + | 40.2001i | ||
3.9 | −1.58619 | − | 0.425019i | −2.61466 | − | 9.75805i | −25.3775 | − | 14.6517i | −7.86751 | + | 55.3453i | 16.5894i | 65.3428 | + | 111.970i | 71.1838 | + | 71.1838i | 122.061 | − | 70.4720i | 36.0022 | − | 84.4444i | ||
3.10 | 0.911102 | + | 0.244129i | 4.40748 | + | 16.4489i | −26.9423 | − | 15.5551i | −55.1076 | + | 9.38874i | 16.0626i | −68.6887 | − | 109.949i | −42.0929 | − | 42.0929i | −40.6973 | + | 23.4966i | −52.5007 | − | 4.89927i | ||
3.11 | 1.93900 | + | 0.519555i | 6.66647 | + | 24.8796i | −24.2230 | − | 13.9852i | 54.2832 | + | 13.3542i | 51.7053i | −57.1121 | + | 116.384i | −85.1248 | − | 85.1248i | −364.109 | + | 210.219i | 98.3171 | + | 54.0970i | ||
3.12 | 3.92292 | + | 1.05114i | −1.12570 | − | 4.20116i | −13.4284 | − | 7.75291i | 47.6558 | − | 29.2220i | − | 17.6641i | 109.646 | − | 69.1724i | −136.426 | − | 136.426i | 194.062 | − | 112.042i | 217.666 | − | 64.5422i | |
3.13 | 4.04554 | + | 1.08400i | −2.28884 | − | 8.54208i | −12.5215 | − | 7.22927i | −38.9420 | − | 40.1063i | − | 37.0384i | −111.439 | + | 66.2439i | −137.589 | − | 137.589i | 142.716 | − | 82.3971i | −114.066 | − | 204.465i | |
3.14 | 5.18499 | + | 1.38931i | −6.95065 | − | 25.9402i | −2.75889 | − | 1.59285i | −15.4213 | + | 53.7325i | − | 144.156i | −8.69466 | − | 129.350i | −133.553 | − | 133.553i | −414.137 | + | 239.102i | −154.611 | + | 257.177i | |
3.15 | 7.58665 | + | 2.03284i | 3.15524 | + | 11.7755i | 25.7121 | + | 14.8449i | −36.2352 | + | 42.5677i | 95.7510i | 105.610 | + | 75.1901i | −12.8308 | − | 12.8308i | 81.7366 | − | 47.1907i | −361.437 | + | 249.286i | ||
3.16 | 8.99064 | + | 2.40904i | 6.51729 | + | 24.3229i | 47.3154 | + | 27.3176i | −2.18478 | − | 55.8590i | 234.379i | −13.7803 | − | 128.907i | 148.975 | + | 148.975i | −338.682 | + | 195.538i | 114.924 | − | 507.471i | ||
3.17 | 9.05026 | + | 2.42501i | −0.494625 | − | 1.84597i | 48.3138 | + | 27.8940i | 47.5992 | + | 29.3141i | − | 17.9059i | −129.513 | + | 5.78206i | 157.601 | + | 157.601i | 207.281 | − | 119.674i | 359.698 | + | 380.729i | |
3.18 | 10.1356 | + | 2.71583i | −5.75210 | − | 21.4671i | 67.6421 | + | 39.0532i | −25.1607 | − | 49.9193i | − | 233.204i | 112.890 | + | 63.7413i | 342.099 | + | 342.099i | −217.307 | + | 125.463i | −119.446 | − | 574.295i | |
12.1 | −9.95978 | + | 2.66872i | −4.43456 | + | 16.5500i | 64.3624 | − | 37.1597i | 55.6892 | − | 4.86925i | − | 176.669i | 127.528 | + | 23.3159i | −308.553 | + | 308.553i | −43.7930 | − | 25.2839i | −541.658 | + | 197.115i | |
12.2 | −8.82949 | + | 2.36586i | −1.01039 | + | 3.77082i | 44.6498 | − | 25.7786i | −51.5659 | − | 21.5861i | − | 35.6848i | −68.0178 | − | 110.366i | −126.411 | + | 126.411i | 197.246 | + | 113.880i | 506.370 | + | 68.5968i | |
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
7.d | odd | 6 | 1 | inner |
35.k | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 35.6.k.a | ✓ | 72 |
5.c | odd | 4 | 1 | inner | 35.6.k.a | ✓ | 72 |
7.d | odd | 6 | 1 | inner | 35.6.k.a | ✓ | 72 |
35.k | even | 12 | 1 | inner | 35.6.k.a | ✓ | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
35.6.k.a | ✓ | 72 | 1.a | even | 1 | 1 | trivial |
35.6.k.a | ✓ | 72 | 5.c | odd | 4 | 1 | inner |
35.6.k.a | ✓ | 72 | 7.d | odd | 6 | 1 | inner |
35.6.k.a | ✓ | 72 | 35.k | even | 12 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(35, [\chi])\).