Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [103,9,Mod(102,103)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(103, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("103.102");
S:= CuspForms(chi, 9);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 103 \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 103.b (of order \(2\), degree \(1\), 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: | \(41.9599968363\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
102.1 | −29.3312 | − | 87.6804i | 604.317 | 634.155i | 2571.77i | 972.142 | −10216.6 | −1126.85 | − | 18600.5i | ||||||||||||||||
102.2 | −29.3312 | 87.6804i | 604.317 | − | 634.155i | − | 2571.77i | 972.142 | −10216.6 | −1126.85 | 18600.5i | ||||||||||||||||
102.3 | −28.8717 | 63.7101i | 577.575 | 899.927i | − | 1839.42i | 2825.20 | −9284.41 | 2502.02 | − | 25982.4i | ||||||||||||||||
102.4 | −28.8717 | − | 63.7101i | 577.575 | − | 899.927i | 1839.42i | 2825.20 | −9284.41 | 2502.02 | 25982.4i | ||||||||||||||||
102.5 | −27.9433 | − | 148.875i | 524.827 | − | 866.759i | 4160.05i | −4361.90 | −7511.92 | −15602.7 | 24220.1i | ||||||||||||||||
102.6 | −27.9433 | 148.875i | 524.827 | 866.759i | − | 4160.05i | −4361.90 | −7511.92 | −15602.7 | − | 24220.1i | ||||||||||||||||
102.7 | −25.5002 | − | 139.257i | 394.260 | 85.7228i | 3551.08i | 3044.06 | −3525.66 | −12831.4 | − | 2185.95i | ||||||||||||||||
102.8 | −25.5002 | 139.257i | 394.260 | − | 85.7228i | − | 3551.08i | 3044.06 | −3525.66 | −12831.4 | 2185.95i | ||||||||||||||||
102.9 | −24.6420 | − | 51.4776i | 351.226 | − | 30.8704i | 1268.51i | −941.911 | −2346.56 | 3911.05 | 760.707i | ||||||||||||||||
102.10 | −24.6420 | 51.4776i | 351.226 | 30.8704i | − | 1268.51i | −941.911 | −2346.56 | 3911.05 | − | 760.707i | ||||||||||||||||
102.11 | −22.6597 | − | 138.463i | 257.460 | 943.340i | 3137.52i | −3425.78 | −33.0865 | −12611.0 | − | 21375.8i | ||||||||||||||||
102.12 | −22.6597 | 138.463i | 257.460 | − | 943.340i | − | 3137.52i | −3425.78 | −33.0865 | −12611.0 | 21375.8i | ||||||||||||||||
102.13 | −21.3916 | 22.4279i | 201.601 | − | 876.654i | − | 479.770i | −2429.16 | 1163.68 | 6057.99 | 18753.0i | ||||||||||||||||
102.14 | −21.3916 | − | 22.4279i | 201.601 | 876.654i | 479.770i | −2429.16 | 1163.68 | 6057.99 | − | 18753.0i | ||||||||||||||||
102.15 | −18.9906 | − | 9.68308i | 104.644 | 820.259i | 183.888i | 3261.63 | 2874.34 | 6467.24 | − | 15577.2i | ||||||||||||||||
102.16 | −18.9906 | 9.68308i | 104.644 | − | 820.259i | − | 183.888i | 3261.63 | 2874.34 | 6467.24 | 15577.2i | ||||||||||||||||
102.17 | −17.6874 | − | 104.607i | 56.8444 | − | 374.832i | 1850.22i | 878.650 | 3522.55 | −4381.57 | 6629.81i | ||||||||||||||||
102.18 | −17.6874 | 104.607i | 56.8444 | 374.832i | − | 1850.22i | 878.650 | 3522.55 | −4381.57 | − | 6629.81i | ||||||||||||||||
102.19 | −15.1473 | − | 80.0038i | −26.5585 | − | 1018.10i | 1211.84i | −1821.42 | 4280.01 | 160.387 | 15421.5i | ||||||||||||||||
102.20 | −15.1473 | 80.0038i | −26.5585 | 1018.10i | − | 1211.84i | −1821.42 | 4280.01 | 160.387 | − | 15421.5i | ||||||||||||||||
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
103.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 103.9.b.b | ✓ | 64 |
103.b | odd | 2 | 1 | inner | 103.9.b.b | ✓ | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
103.9.b.b | ✓ | 64 | 1.a | even | 1 | 1 | trivial |
103.9.b.b | ✓ | 64 | 103.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{32} + T_{2}^{31} - 6015 T_{2}^{30} - 7214 T_{2}^{29} + 16213023 T_{2}^{28} + 21330897 T_{2}^{27} - 25891004273 T_{2}^{26} - 34998546836 T_{2}^{25} + 27290668796296 T_{2}^{24} + \cdots + 12\!\cdots\!00 \)
acting on \(S_{9}^{\mathrm{new}}(103, [\chi])\).