Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [50,6,Mod(11,50)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(50, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([8]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("50.11");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 50 = 2 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 50.d (of order \(5\), 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: | \(8.01919099065\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(7\) over \(\Q(\zeta_{5})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | 3.23607 | − | 2.35114i | −9.01407 | + | 27.7425i | 4.94427 | − | 15.2169i | −53.0123 | − | 17.7397i | 36.0563 | + | 110.970i | 66.5258 | −19.7771 | − | 60.8676i | −491.799 | − | 357.313i | −213.260 | + | 67.2326i | ||
11.2 | 3.23607 | − | 2.35114i | −4.67506 | + | 14.3884i | 4.94427 | − | 15.2169i | 27.5965 | + | 48.6152i | 18.7003 | + | 57.5535i | −181.803 | −19.7771 | − | 60.8676i | 11.4223 | + | 8.29879i | 203.605 | + | 92.4389i | ||
11.3 | 3.23607 | − | 2.35114i | −4.00985 | + | 12.3411i | 4.94427 | − | 15.2169i | 32.6779 | − | 45.3558i | 16.0394 | + | 49.3642i | 24.1971 | −19.7771 | − | 60.8676i | 60.3684 | + | 43.8602i | −0.889977 | − | 223.605i | ||
11.4 | 3.23607 | − | 2.35114i | −0.839907 | + | 2.58497i | 4.94427 | − | 15.2169i | −19.5514 | + | 52.3712i | 3.35963 | + | 10.3399i | 200.961 | −19.7771 | − | 60.8676i | 190.615 | + | 138.490i | 59.8626 | + | 215.445i | ||
11.5 | 3.23607 | − | 2.35114i | 1.57970 | − | 4.86182i | 4.94427 | − | 15.2169i | −52.9218 | − | 18.0078i | −6.31881 | − | 19.4473i | −251.763 | −19.7771 | − | 60.8676i | 175.449 | + | 127.471i | −213.597 | + | 66.1524i | ||
11.6 | 3.23607 | − | 2.35114i | 5.47253 | − | 16.8427i | 4.94427 | − | 15.2169i | 53.8479 | − | 15.0136i | −21.8901 | − | 67.3709i | −2.11162 | −19.7771 | − | 60.8676i | −57.1374 | − | 41.5128i | 138.956 | − | 175.189i | ||
11.7 | 3.23607 | − | 2.35114i | 8.20551 | − | 25.2540i | 4.94427 | − | 15.2169i | −55.6327 | − | 5.47754i | −32.8220 | − | 101.016i | 129.715 | −19.7771 | − | 60.8676i | −373.841 | − | 271.611i | −192.910 | + | 113.075i | ||
21.1 | −1.23607 | + | 3.80423i | −23.0554 | − | 16.7507i | −12.9443 | − | 9.40456i | −50.3632 | − | 24.2601i | 92.2216 | − | 67.0029i | 20.3938 | 51.7771 | − | 37.6183i | 175.874 | + | 541.284i | 154.543 | − | 161.606i | ||
21.2 | −1.23607 | + | 3.80423i | −14.1625 | − | 10.2897i | −12.9443 | − | 9.40456i | 55.6267 | + | 5.53762i | 56.6502 | − | 41.1588i | −46.1918 | 51.7771 | − | 37.6183i | 19.6088 | + | 60.3498i | −89.8248 | + | 204.772i | ||
21.3 | −1.23607 | + | 3.80423i | −7.20148 | − | 5.23218i | −12.9443 | − | 9.40456i | −5.94028 | + | 55.5852i | 28.8059 | − | 20.9287i | −16.8909 | 51.7771 | − | 37.6183i | −50.6055 | − | 155.748i | −204.116 | − | 91.3052i | ||
21.4 | −1.23607 | + | 3.80423i | 4.87185 | + | 3.53960i | −12.9443 | − | 9.40456i | −25.7617 | − | 49.6118i | −19.4874 | + | 14.1584i | 192.464 | 51.7771 | − | 37.6183i | −63.8850 | − | 196.618i | 220.578 | − | 36.6799i | ||
21.5 | −1.23607 | + | 3.80423i | 6.10314 | + | 4.43419i | −12.9443 | − | 9.40456i | 17.4261 | − | 53.1162i | −24.4126 | + | 17.7368i | −227.988 | 51.7771 | − | 37.6183i | −57.5049 | − | 176.982i | 180.526 | + | 131.948i | ||
21.6 | −1.23607 | + | 3.80423i | 17.4616 | + | 12.6866i | −12.9443 | − | 9.40456i | 53.8689 | + | 14.9378i | −69.8465 | + | 50.7465i | 157.525 | 51.7771 | − | 37.6183i | 68.8673 | + | 211.952i | −123.412 | + | 186.465i | ||
21.7 | −1.23607 | + | 3.80423i | 22.7640 | + | 16.5390i | −12.9443 | − | 9.40456i | −50.3607 | + | 24.2652i | −91.0559 | + | 66.1560i | −183.034 | 51.7771 | − | 37.6183i | 169.569 | + | 521.880i | −30.0609 | − | 221.577i | ||
31.1 | −1.23607 | − | 3.80423i | −23.0554 | + | 16.7507i | −12.9443 | + | 9.40456i | −50.3632 | + | 24.2601i | 92.2216 | + | 67.0029i | 20.3938 | 51.7771 | + | 37.6183i | 175.874 | − | 541.284i | 154.543 | + | 161.606i | ||
31.2 | −1.23607 | − | 3.80423i | −14.1625 | + | 10.2897i | −12.9443 | + | 9.40456i | 55.6267 | − | 5.53762i | 56.6502 | + | 41.1588i | −46.1918 | 51.7771 | + | 37.6183i | 19.6088 | − | 60.3498i | −89.8248 | − | 204.772i | ||
31.3 | −1.23607 | − | 3.80423i | −7.20148 | + | 5.23218i | −12.9443 | + | 9.40456i | −5.94028 | − | 55.5852i | 28.8059 | + | 20.9287i | −16.8909 | 51.7771 | + | 37.6183i | −50.6055 | + | 155.748i | −204.116 | + | 91.3052i | ||
31.4 | −1.23607 | − | 3.80423i | 4.87185 | − | 3.53960i | −12.9443 | + | 9.40456i | −25.7617 | + | 49.6118i | −19.4874 | − | 14.1584i | 192.464 | 51.7771 | + | 37.6183i | −63.8850 | + | 196.618i | 220.578 | + | 36.6799i | ||
31.5 | −1.23607 | − | 3.80423i | 6.10314 | − | 4.43419i | −12.9443 | + | 9.40456i | 17.4261 | + | 53.1162i | −24.4126 | − | 17.7368i | −227.988 | 51.7771 | + | 37.6183i | −57.5049 | + | 176.982i | 180.526 | − | 131.948i | ||
31.6 | −1.23607 | − | 3.80423i | 17.4616 | − | 12.6866i | −12.9443 | + | 9.40456i | 53.8689 | − | 14.9378i | −69.8465 | − | 50.7465i | 157.525 | 51.7771 | + | 37.6183i | 68.8673 | − | 211.952i | −123.412 | − | 186.465i | ||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
25.d | even | 5 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 50.6.d.b | ✓ | 28 |
25.d | even | 5 | 1 | inner | 50.6.d.b | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
50.6.d.b | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
50.6.d.b | ✓ | 28 | 25.d | even | 5 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} - 7 T_{3}^{27} + 1098 T_{3}^{26} - 10074 T_{3}^{25} + 1054175 T_{3}^{24} + \cdots + 21\!\cdots\!76 \) acting on \(S_{6}^{\mathrm{new}}(50, [\chi])\).