Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [74,4,Mod(7,74)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(74, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([16]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("74.7");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 74 = 2 \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 74.f (of order \(9\), 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: | \(4.36614134042\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{9})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 | 1.87939 | − | 0.684040i | −4.40012 | − | 1.60151i | 3.06418 | − | 2.57115i | −0.0856491 | − | 0.485740i | −9.36501 | −4.85295 | − | 27.5224i | 4.00000 | − | 6.92820i | −3.88702 | − | 3.26159i | −0.493234 | − | 0.854306i | ||
7.2 | 1.87939 | − | 0.684040i | −1.71848 | − | 0.625476i | 3.06418 | − | 2.57115i | −3.13691 | − | 17.7903i | −3.65754 | 1.67925 | + | 9.52347i | 4.00000 | − | 6.92820i | −18.1212 | − | 15.2055i | −18.0648 | − | 31.2891i | ||
7.3 | 1.87939 | − | 0.684040i | 2.73860 | + | 0.996769i | 3.06418 | − | 2.57115i | 3.22893 | + | 18.3122i | 5.82871 | 2.30707 | + | 13.0840i | 4.00000 | − | 6.92820i | −14.1768 | − | 11.8958i | 18.5947 | + | 32.2069i | ||
7.4 | 1.87939 | − | 0.684040i | 6.85178 | + | 2.49384i | 3.06418 | − | 2.57115i | −0.811775 | − | 4.60380i | 14.5830 | −0.602158 | − | 3.41501i | 4.00000 | − | 6.92820i | 20.0444 | + | 16.8192i | −4.67483 | − | 8.09704i | ||
9.1 | −0.347296 | − | 1.96962i | −1.43352 | + | 8.12991i | −3.75877 | + | 1.36808i | −2.21967 | − | 1.86252i | 16.5106 | −21.7866 | − | 18.2811i | 4.00000 | + | 6.92820i | −38.6687 | − | 14.0742i | −2.89757 | + | 5.01874i | ||
9.2 | −0.347296 | − | 1.96962i | −0.666426 | + | 3.77949i | −3.75877 | + | 1.36808i | −4.75553 | − | 3.99036i | 7.67559 | 19.2298 | + | 16.1357i | 4.00000 | + | 6.92820i | 11.5313 | + | 4.19704i | −6.20790 | + | 10.7524i | ||
9.3 | −0.347296 | − | 1.96962i | −0.0459350 | + | 0.260510i | −3.75877 | + | 1.36808i | 12.6348 | + | 10.6018i | 0.529058 | 0.748056 | + | 0.627693i | 4.00000 | + | 6.92820i | 25.3059 | + | 9.21061i | 16.4935 | − | 28.5676i | ||
9.4 | −0.347296 | − | 1.96962i | 1.09285 | − | 6.19786i | −3.75877 | + | 1.36808i | −4.09539 | − | 3.43644i | −12.5869 | −9.53718 | − | 8.00265i | 4.00000 | + | 6.92820i | −11.8474 | − | 4.31210i | −5.34616 | + | 9.25982i | ||
33.1 | −0.347296 | + | 1.96962i | −1.43352 | − | 8.12991i | −3.75877 | − | 1.36808i | −2.21967 | + | 1.86252i | 16.5106 | −21.7866 | + | 18.2811i | 4.00000 | − | 6.92820i | −38.6687 | + | 14.0742i | −2.89757 | − | 5.01874i | ||
33.2 | −0.347296 | + | 1.96962i | −0.666426 | − | 3.77949i | −3.75877 | − | 1.36808i | −4.75553 | + | 3.99036i | 7.67559 | 19.2298 | − | 16.1357i | 4.00000 | − | 6.92820i | 11.5313 | − | 4.19704i | −6.20790 | − | 10.7524i | ||
33.3 | −0.347296 | + | 1.96962i | −0.0459350 | − | 0.260510i | −3.75877 | − | 1.36808i | 12.6348 | − | 10.6018i | 0.529058 | 0.748056 | − | 0.627693i | 4.00000 | − | 6.92820i | 25.3059 | − | 9.21061i | 16.4935 | + | 28.5676i | ||
33.4 | −0.347296 | + | 1.96962i | 1.09285 | + | 6.19786i | −3.75877 | − | 1.36808i | −4.09539 | + | 3.43644i | −12.5869 | −9.53718 | + | 8.00265i | 4.00000 | − | 6.92820i | −11.8474 | + | 4.31210i | −5.34616 | − | 9.25982i | ||
49.1 | −1.53209 | + | 1.28558i | −6.58177 | − | 5.52276i | 0.694593 | − | 3.93923i | −2.85390 | + | 1.03873i | 17.1838 | −0.561982 | + | 0.204545i | 4.00000 | + | 6.92820i | 8.13033 | + | 46.1094i | 3.03706 | − | 5.26034i | ||
49.2 | −1.53209 | + | 1.28558i | −1.34104 | − | 1.12526i | 0.694593 | − | 3.93923i | 6.03617 | − | 2.19699i | 3.50120 | −28.8243 | + | 10.4912i | 4.00000 | + | 6.92820i | −4.15634 | − | 23.5718i | −6.42356 | + | 11.1259i | ||
49.3 | −1.53209 | + | 1.28558i | 3.31240 | + | 2.77944i | 0.694593 | − | 3.93923i | 10.2218 | − | 3.72042i | −8.64807 | 12.8394 | − | 4.67315i | 4.00000 | + | 6.92820i | −1.44175 | − | 8.17656i | −10.8778 | + | 18.8409i | ||
49.4 | −1.53209 | + | 1.28558i | 5.19166 | + | 4.35632i | 0.694593 | − | 3.93923i | −18.6628 | + | 6.79271i | −13.5545 | −8.13840 | + | 2.96214i | 4.00000 | + | 6.92820i | 3.28731 | + | 18.6432i | 19.8606 | − | 34.3995i | ||
53.1 | 1.87939 | + | 0.684040i | −4.40012 | + | 1.60151i | 3.06418 | + | 2.57115i | −0.0856491 | + | 0.485740i | −9.36501 | −4.85295 | + | 27.5224i | 4.00000 | + | 6.92820i | −3.88702 | + | 3.26159i | −0.493234 | + | 0.854306i | ||
53.2 | 1.87939 | + | 0.684040i | −1.71848 | + | 0.625476i | 3.06418 | + | 2.57115i | −3.13691 | + | 17.7903i | −3.65754 | 1.67925 | − | 9.52347i | 4.00000 | + | 6.92820i | −18.1212 | + | 15.2055i | −18.0648 | + | 31.2891i | ||
53.3 | 1.87939 | + | 0.684040i | 2.73860 | − | 0.996769i | 3.06418 | + | 2.57115i | 3.22893 | − | 18.3122i | 5.82871 | 2.30707 | − | 13.0840i | 4.00000 | + | 6.92820i | −14.1768 | + | 11.8958i | 18.5947 | − | 32.2069i | ||
53.4 | 1.87939 | + | 0.684040i | 6.85178 | − | 2.49384i | 3.06418 | + | 2.57115i | −0.811775 | + | 4.60380i | 14.5830 | −0.602158 | + | 3.41501i | 4.00000 | + | 6.92820i | 20.0444 | − | 16.8192i | −4.67483 | + | 8.09704i | ||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
37.f | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 74.4.f.a | ✓ | 24 |
37.f | even | 9 | 1 | inner | 74.4.f.a | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
74.4.f.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
74.4.f.a | ✓ | 24 | 37.f | even | 9 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{24} - 6 T_{3}^{23} + 42 T_{3}^{22} - 487 T_{3}^{21} + 3579 T_{3}^{20} - 22725 T_{3}^{19} + 334902 T_{3}^{18} - 2355801 T_{3}^{17} + 9595461 T_{3}^{16} - 1079640 T_{3}^{15} - 201808932 T_{3}^{14} + \cdots + 17897307931441 \)
acting on \(S_{4}^{\mathrm{new}}(74, [\chi])\).