Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [30,10,Mod(17,30)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(30, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("30.17");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 30 = 2 \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 30.e (of order \(4\), degree \(2\), 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: | \(15.4510750849\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Relative dimension: | \(18\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | −11.3137 | + | 11.3137i | −138.928 | + | 19.5470i | − | 256.000i | 442.457 | + | 1325.65i | 1350.64 | − | 1792.94i | 4229.25 | + | 4229.25i | 2896.31 | + | 2896.31i | 18918.8 | − | 5431.23i | −20003.9 | − | 9992.22i | |
17.2 | −11.3137 | + | 11.3137i | −125.652 | + | 62.4065i | − | 256.000i | −1151.16 | − | 792.434i | 715.542 | − | 2127.64i | −1314.95 | − | 1314.95i | 2896.31 | + | 2896.31i | 11893.9 | − | 15683.0i | 21989.3 | − | 4058.54i | |
17.3 | −11.3137 | + | 11.3137i | −89.2589 | − | 108.240i | − | 256.000i | 1142.31 | − | 805.147i | 2234.44 | + | 214.744i | 2602.86 | + | 2602.86i | 2896.31 | + | 2896.31i | −3748.70 | + | 19322.7i | −3814.52 | + | 22032.9i | |
17.4 | −11.3137 | + | 11.3137i | −65.1884 | − | 124.232i | − | 256.000i | −1015.91 | + | 959.718i | 2143.04 | + | 667.997i | −7930.01 | − | 7930.01i | 2896.31 | + | 2896.31i | −11184.0 | + | 16196.9i | 635.693 | − | 22351.6i | |
17.5 | −11.3137 | + | 11.3137i | 43.1765 | + | 133.487i | − | 256.000i | −445.093 | + | 1324.77i | −1998.72 | − | 1021.75i | −3280.20 | − | 3280.20i | 2896.31 | + | 2896.31i | −15954.6 | + | 11527.0i | −9952.42 | − | 20023.7i | |
17.6 | −11.3137 | + | 11.3137i | 73.5417 | − | 119.476i | − | 256.000i | −1371.94 | + | 266.282i | 519.692 | + | 2183.75i | 8083.49 | + | 8083.49i | 2896.31 | + | 2896.31i | −8866.24 | − | 17573.0i | 12509.1 | − | 18534.4i | |
17.7 | −11.3137 | + | 11.3137i | 85.7313 | + | 111.055i | − | 256.000i | −520.827 | − | 1296.87i | −2226.38 | − | 286.501i | 5788.99 | + | 5788.99i | 2896.31 | + | 2896.31i | −4983.28 | + | 19041.7i | 20564.9 | + | 8779.89i | |
17.8 | −11.3137 | + | 11.3137i | 100.077 | − | 98.3235i | − | 256.000i | 257.681 | − | 1373.58i | −19.8436 | + | 2244.65i | −6195.32 | − | 6195.32i | 2896.31 | + | 2896.31i | 347.984 | − | 19679.9i | 12625.0 | + | 18455.6i | |
17.9 | −11.3137 | + | 11.3137i | 140.295 | − | 0.428453i | − | 256.000i | 1192.41 | + | 728.894i | −1582.41 | + | 1592.11i | 454.888 | + | 454.888i | 2896.31 | + | 2896.31i | 19682.6 | − | 120.220i | −21737.1 | + | 5244.06i | |
17.10 | 11.3137 | − | 11.3137i | −133.487 | − | 43.1765i | − | 256.000i | 445.093 | − | 1324.77i | −1998.72 | + | 1021.75i | −3280.20 | − | 3280.20i | −2896.31 | − | 2896.31i | 15954.6 | + | 11527.0i | −9952.42 | − | 20023.7i | |
17.11 | 11.3137 | − | 11.3137i | −111.055 | − | 85.7313i | − | 256.000i | 520.827 | + | 1296.87i | −2226.38 | + | 286.501i | 5788.99 | + | 5788.99i | −2896.31 | − | 2896.31i | 4983.28 | + | 19041.7i | 20564.9 | + | 8779.89i | |
17.12 | 11.3137 | − | 11.3137i | −62.4065 | + | 125.652i | − | 256.000i | 1151.16 | + | 792.434i | 715.542 | + | 2127.64i | −1314.95 | − | 1314.95i | −2896.31 | − | 2896.31i | −11893.9 | − | 15683.0i | 21989.3 | − | 4058.54i | |
17.13 | 11.3137 | − | 11.3137i | −19.5470 | + | 138.928i | − | 256.000i | −442.457 | − | 1325.65i | 1350.64 | + | 1792.94i | 4229.25 | + | 4229.25i | −2896.31 | − | 2896.31i | −18918.8 | − | 5431.23i | −20003.9 | − | 9992.22i | |
17.14 | 11.3137 | − | 11.3137i | 0.428453 | − | 140.295i | − | 256.000i | −1192.41 | − | 728.894i | −1582.41 | − | 1592.11i | 454.888 | + | 454.888i | −2896.31 | − | 2896.31i | −19682.6 | − | 120.220i | −21737.1 | + | 5244.06i | |
17.15 | 11.3137 | − | 11.3137i | 98.3235 | − | 100.077i | − | 256.000i | −257.681 | + | 1373.58i | −19.8436 | − | 2244.65i | −6195.32 | − | 6195.32i | −2896.31 | − | 2896.31i | −347.984 | − | 19679.9i | 12625.0 | + | 18455.6i | |
17.16 | 11.3137 | − | 11.3137i | 108.240 | + | 89.2589i | − | 256.000i | −1142.31 | + | 805.147i | 2234.44 | − | 214.744i | 2602.86 | + | 2602.86i | −2896.31 | − | 2896.31i | 3748.70 | + | 19322.7i | −3814.52 | + | 22032.9i | |
17.17 | 11.3137 | − | 11.3137i | 119.476 | − | 73.5417i | − | 256.000i | 1371.94 | − | 266.282i | 519.692 | − | 2183.75i | 8083.49 | + | 8083.49i | −2896.31 | − | 2896.31i | 8866.24 | − | 17573.0i | 12509.1 | − | 18534.4i | |
17.18 | 11.3137 | − | 11.3137i | 124.232 | + | 65.1884i | − | 256.000i | 1015.91 | − | 959.718i | 2143.04 | − | 667.997i | −7930.01 | − | 7930.01i | −2896.31 | − | 2896.31i | 11184.0 | + | 16196.9i | 635.693 | − | 22351.6i | |
23.1 | −11.3137 | − | 11.3137i | −138.928 | − | 19.5470i | 256.000i | 442.457 | − | 1325.65i | 1350.64 | + | 1792.94i | 4229.25 | − | 4229.25i | 2896.31 | − | 2896.31i | 18918.8 | + | 5431.23i | −20003.9 | + | 9992.22i | ||
23.2 | −11.3137 | − | 11.3137i | −125.652 | − | 62.4065i | 256.000i | −1151.16 | + | 792.434i | 715.542 | + | 2127.64i | −1314.95 | + | 1314.95i | 2896.31 | − | 2896.31i | 11893.9 | + | 15683.0i | 21989.3 | + | 4058.54i | ||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
15.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 30.10.e.a | ✓ | 36 |
3.b | odd | 2 | 1 | inner | 30.10.e.a | ✓ | 36 |
5.c | odd | 4 | 1 | inner | 30.10.e.a | ✓ | 36 |
15.e | even | 4 | 1 | inner | 30.10.e.a | ✓ | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
30.10.e.a | ✓ | 36 | 1.a | even | 1 | 1 | trivial |
30.10.e.a | ✓ | 36 | 3.b | odd | 2 | 1 | inner |
30.10.e.a | ✓ | 36 | 5.c | odd | 4 | 1 | inner |
30.10.e.a | ✓ | 36 | 15.e | even | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(30, [\chi])\).