Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [76,7,Mod(23,76)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(76, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([9, 2]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("76.23");
S:= CuspForms(chi, 7);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 76 = 2^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 76.l (of order \(18\), 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: | \(17.4841103551\) |
Analytic rank: | \(0\) |
Dimension: | \(348\) |
Relative dimension: | \(58\) over \(\Q(\zeta_{18})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 | −7.99410 | + | 0.307308i | −27.6808 | + | 4.88087i | 63.8111 | − | 4.91330i | −34.1068 | + | 28.6190i | 219.783 | − | 47.5247i | −83.9559 | − | 48.4720i | −508.602 | + | 58.8871i | 57.3665 | − | 20.8797i | 263.858 | − | 239.264i |
23.2 | −7.91695 | − | 1.14977i | 3.83980 | − | 0.677060i | 61.3561 | + | 18.2054i | −157.228 | + | 131.930i | −31.1780 | + | 0.945358i | 546.642 | + | 315.604i | −464.821 | − | 214.676i | −670.750 | + | 244.133i | 1396.45 | − | 863.704i |
23.3 | −7.84835 | + | 1.55031i | −22.9636 | + | 4.04910i | 59.1931 | − | 24.3348i | −26.2597 | + | 22.0345i | 173.949 | − | 67.3795i | −137.300 | − | 79.2704i | −426.841 | + | 282.756i | −174.105 | + | 63.3690i | 171.935 | − | 213.645i |
23.4 | −7.76312 | − | 1.93236i | 51.9029 | − | 9.15188i | 56.5319 | + | 30.0023i | −46.4533 | + | 38.9790i | −420.613 | − | 29.2482i | 69.3277 | + | 40.0264i | −380.888 | − | 342.152i | 1925.12 | − | 700.686i | 435.944 | − | 212.834i |
23.5 | −7.72475 | − | 2.08045i | 16.9114 | − | 2.98194i | 55.3435 | + | 32.1418i | 113.326 | − | 95.0915i | −136.840 | − | 12.1485i | −331.831 | − | 191.583i | −360.645 | − | 363.427i | −407.932 | + | 148.475i | −1073.25 | + | 498.790i |
23.6 | −7.71608 | + | 2.11236i | 26.4348 | − | 4.66117i | 55.0759 | − | 32.5982i | 6.31656 | − | 5.30022i | −194.127 | + | 91.8057i | −142.890 | − | 82.4974i | −356.111 | + | 367.871i | −7.96412 | + | 2.89870i | −37.5431 | + | 54.2398i |
23.7 | −7.31367 | + | 3.24195i | 23.5223 | − | 4.14762i | 42.9795 | − | 47.4211i | 104.114 | − | 87.3621i | −158.588 | + | 106.593i | 538.554 | + | 310.935i | −160.601 | + | 486.160i | −148.939 | + | 54.2093i | −478.233 | + | 976.470i |
23.8 | −7.25479 | − | 3.37166i | −16.9114 | + | 2.98194i | 41.2638 | + | 48.9213i | 113.326 | − | 95.0915i | 132.743 | + | 35.3862i | 331.831 | + | 191.583i | −134.414 | − | 494.041i | −407.932 | + | 148.475i | −1142.77 | + | 307.773i |
23.9 | −7.25120 | + | 3.37936i | −38.7503 | + | 6.83273i | 41.1598 | − | 49.0089i | 178.137 | − | 149.475i | 257.896 | − | 180.497i | −72.9496 | − | 42.1175i | −132.840 | + | 494.467i | 769.865 | − | 280.208i | −786.579 | + | 1685.86i |
23.10 | −7.18899 | − | 3.50976i | −51.9029 | + | 9.15188i | 39.3632 | + | 50.4632i | −46.4533 | + | 38.9790i | 405.250 | + | 116.374i | −69.3277 | − | 40.0264i | −105.868 | − | 500.935i | 1925.12 | − | 700.686i | 470.759 | − | 117.180i |
23.11 | −6.80379 | − | 4.20814i | −3.83980 | + | 0.677060i | 28.5831 | + | 57.2626i | −157.228 | + | 131.930i | 28.9744 | + | 11.5518i | −546.642 | − | 315.604i | 46.4952 | − | 509.884i | −670.750 | + | 244.133i | 1624.92 | − | 235.986i |
23.12 | −6.50920 | + | 4.65084i | 33.9040 | − | 5.97820i | 20.7395 | − | 60.5465i | −158.311 | + | 132.839i | −192.885 | + | 196.595i | −237.498 | − | 137.119i | 146.594 | + | 490.565i | 428.710 | − | 156.038i | 412.668 | − | 1600.95i |
23.13 | −6.18379 | + | 5.07550i | −9.73276 | + | 1.71615i | 12.4786 | − | 62.7717i | −96.0367 | + | 80.5844i | 51.4751 | − | 60.0109i | −51.8068 | − | 29.9106i | 241.432 | + | 451.502i | −593.255 | + | 215.927i | 184.865 | − | 985.751i |
23.14 | −5.92630 | − | 5.37392i | 27.6808 | − | 4.88087i | 6.24202 | + | 63.6949i | −34.1068 | + | 28.6190i | −190.274 | − | 119.829i | 83.9559 | + | 48.4720i | 305.299 | − | 411.019i | 57.3665 | − | 20.8797i | 355.923 | + | 13.6824i |
23.15 | −5.75352 | + | 5.55851i | −45.0914 | + | 7.95083i | 2.20600 | − | 63.9620i | −116.601 | + | 97.8402i | 215.240 | − | 296.386i | 418.318 | + | 241.516i | 342.841 | + | 380.269i | 1284.98 | − | 467.695i | 127.023 | − | 1211.06i |
23.16 | −5.01566 | − | 6.23243i | 22.9636 | − | 4.04910i | −13.6863 | + | 62.5195i | −26.2597 | + | 22.0345i | −140.413 | − | 122.810i | 137.300 | + | 79.2704i | 458.294 | − | 228.277i | −174.105 | + | 63.3690i | 269.038 | + | 53.1441i |
23.17 | −4.61729 | + | 6.53304i | 1.04918 | − | 0.184999i | −21.3612 | − | 60.3299i | 59.4109 | − | 49.8517i | −3.63577 | + | 7.70853i | −459.233 | − | 265.139i | 492.769 | + | 139.007i | −683.969 | + | 248.944i | 51.3653 | + | 618.313i |
23.18 | −4.55307 | − | 6.57796i | −26.4348 | + | 4.66117i | −22.5392 | + | 59.8998i | 6.31656 | − | 5.30022i | 151.020 | + | 152.664i | 142.890 | + | 82.4974i | 496.641 | − | 124.466i | −7.96412 | + | 2.89870i | −63.6244 | − | 17.4178i |
23.19 | −4.53942 | + | 6.58739i | 50.5024 | − | 8.90493i | −22.7874 | − | 59.8058i | 146.923 | − | 123.283i | −170.591 | + | 373.102i | −251.055 | − | 144.947i | 497.406 | + | 121.374i | 1786.16 | − | 650.108i | 145.169 | + | 1527.48i |
23.20 | −4.45512 | + | 6.64469i | −11.1633 | + | 1.96839i | −24.3038 | − | 59.2058i | 71.7659 | − | 60.2188i | 36.6545 | − | 82.9459i | 245.952 | + | 142.001i | 501.680 | + | 102.278i | −564.292 | + | 205.385i | 80.4088 | + | 745.144i |
See next 80 embeddings (of 348 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
19.e | even | 9 | 1 | inner |
76.l | odd | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 76.7.l.a | ✓ | 348 |
4.b | odd | 2 | 1 | inner | 76.7.l.a | ✓ | 348 |
19.e | even | 9 | 1 | inner | 76.7.l.a | ✓ | 348 |
76.l | odd | 18 | 1 | inner | 76.7.l.a | ✓ | 348 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
76.7.l.a | ✓ | 348 | 1.a | even | 1 | 1 | trivial |
76.7.l.a | ✓ | 348 | 4.b | odd | 2 | 1 | inner |
76.7.l.a | ✓ | 348 | 19.e | even | 9 | 1 | inner |
76.7.l.a | ✓ | 348 | 76.l | odd | 18 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(76, [\chi])\).