Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5,35,Mod(2,5)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 35, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5.2");
S:= CuspForms(chi, 35);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 5 \) |
Weight: | \( k \) | \(=\) | \( 35 \) |
Character orbit: | \([\chi]\) | \(=\) | 5.c (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: | \(36.6128270213\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −169510. | − | 169510.i | −198644. | + | 198644.i | 4.02874e10i | −3.95922e11 | + | 6.52168e11i | 6.73444e10 | 1.10969e14 | + | 1.10969e14i | 3.91696e15 | − | 3.91696e15i | 1.66771e16i | 1.77662e17 | − | 4.34363e16i | ||||||
2.2 | −151262. | − | 151262.i | −1.00065e8 | + | 1.00065e8i | 2.85804e10i | 1.61018e11 | − | 7.45755e11i | 3.02721e13 | −1.52632e14 | − | 1.52632e14i | 1.72447e15 | − | 1.72447e15i | − | 3.34887e15i | −1.37160e17 | + | 8.84484e16i | |||||
2.3 | −134834. | − | 134834.i | 1.19387e8 | − | 1.19387e8i | 1.91805e10i | 6.75906e11 | − | 3.53875e11i | −3.21948e13 | 2.58783e14 | + | 2.58783e14i | 2.69752e14 | − | 2.69752e14i | − | 1.18293e16i | −1.38849e17 | − | 4.34207e16i | |||||
2.4 | −115430. | − | 115430.i | 1.39345e8 | − | 1.39345e8i | 9.46824e9i | −7.62275e11 | − | 3.18319e10i | −3.21691e13 | −2.97793e14 | − | 2.97793e14i | −8.90153e14 | + | 8.90153e14i | − | 2.21567e16i | 8.43150e16 | + | 9.16637e16i | |||||
2.5 | −85294.4 | − | 85294.4i | −1.68126e8 | + | 1.68126e8i | − | 2.62960e9i | −5.92285e11 | + | 4.80911e11i | 2.86804e13 | 2.04951e14 | + | 2.04951e14i | −1.68964e15 | + | 1.68964e15i | − | 3.98554e16i | 9.15376e16 | + | 9.49962e15i | ||||
2.6 | −83255.7 | − | 83255.7i | −2.65055e7 | + | 2.65055e7i | − | 3.31684e9i | 5.91968e11 | + | 4.81301e11i | 4.41347e12 | −1.23465e14 | − | 1.23465e14i | −1.70647e15 | + | 1.70647e15i | 1.52721e16i | −9.21369e15 | − | 8.93558e16i | |||||
2.7 | −39637.5 | − | 39637.5i | 937048. | − | 937048.i | − | 1.40376e10i | −4.84629e11 | − | 5.89246e11i | −7.42844e10 | 1.00289e14 | + | 1.00289e14i | −1.23738e15 | + | 1.23738e15i | 1.66754e16i | −4.14676e15 | + | 4.25657e16i | |||||
2.8 | 8626.63 | + | 8626.63i | 9.89110e7 | − | 9.89110e7i | − | 1.70310e10i | −8.06672e10 | + | 7.58663e11i | 1.70654e12 | 1.10848e14 | + | 1.10848e14i | 2.95125e14 | − | 2.95125e14i | − | 2.88960e15i | −7.24059e15 | + | 5.84882e15i | ||||
2.9 | 26483.0 | + | 26483.0i | −1.33847e8 | + | 1.33847e8i | − | 1.57772e10i | 6.94260e11 | − | 3.16354e11i | −7.08935e12 | 1.17241e13 | + | 1.17241e13i | 8.72802e14 | − | 8.72802e14i | − | 1.91529e16i | 2.67641e16 | + | 1.00081e16i | ||||
2.10 | 36635.8 | + | 36635.8i | 1.35857e8 | − | 1.35857e8i | − | 1.44955e10i | 6.50279e11 | − | 3.99016e11i | 9.95447e12 | −1.47937e14 | − | 1.47937e14i | 1.16045e15 | − | 1.16045e15i | − | 2.02371e16i | 3.84418e16 | + | 9.20523e15i | ||||
2.11 | 67956.3 | + | 67956.3i | −7.73822e7 | + | 7.73822e7i | − | 7.94376e9i | −6.92309e11 | + | 3.20600e11i | −1.05172e13 | −2.59708e14 | − | 2.59708e14i | 1.70731e15 | − | 1.70731e15i | 4.70118e15i | −6.88336e16 | − | 2.52599e16i | |||||
2.12 | 109228. | + | 109228.i | 1.53588e7 | − | 1.53588e7i | 6.68164e9i | 1.05031e11 | − | 7.55675e11i | 3.35523e12 | 5.89338e13 | + | 5.89338e13i | 1.14670e15 | − | 1.14670e15i | 1.62054e16i | 9.40132e16 | − | 7.10686e16i | ||||||
2.13 | 117370. | + | 117370.i | −6.15595e7 | + | 6.15595e7i | 1.03717e10i | 3.42977e11 | + | 6.81501e11i | −1.44505e13 | 2.43299e14 | + | 2.43299e14i | 7.99081e14 | − | 7.99081e14i | 9.09803e15i | −3.97325e16 | + | 1.20243e17i | ||||||
2.14 | 136964. | + | 136964.i | 1.30224e8 | − | 1.30224e8i | 2.03383e10i | −7.62514e11 | + | 2.54690e10i | 3.56719e13 | 1.04083e14 | + | 1.04083e14i | −4.32593e14 | + | 4.32593e14i | − | 1.72393e16i | −1.07925e17 | − | 1.00949e17i | |||||
2.15 | 167195. | + | 167195.i | 4.67349e7 | − | 4.67349e7i | 3.87285e10i | 6.83351e11 | + | 3.39275e11i | 1.56277e13 | −2.57154e14 | − | 2.57154e14i | −3.60283e15 | + | 3.60283e15i | 1.23089e16i | 5.75278e16 | + | 1.70978e17i | ||||||
2.16 | 174299. | + | 174299.i | −1.58564e8 | + | 1.58564e8i | 4.35807e10i | −4.99549e11 | − | 5.76652e11i | −5.52754e13 | 3.74774e13 | + | 3.74774e13i | −4.60166e15 | + | 4.60166e15i | − | 3.36082e16i | 1.34389e16 | − | 1.87581e17i | |||||
3.1 | −169510. | + | 169510.i | −198644. | − | 198644.i | − | 4.02874e10i | −3.95922e11 | − | 6.52168e11i | 6.73444e10 | 1.10969e14 | − | 1.10969e14i | 3.91696e15 | + | 3.91696e15i | − | 1.66771e16i | 1.77662e17 | + | 4.34363e16i | ||||
3.2 | −151262. | + | 151262.i | −1.00065e8 | − | 1.00065e8i | − | 2.85804e10i | 1.61018e11 | + | 7.45755e11i | 3.02721e13 | −1.52632e14 | + | 1.52632e14i | 1.72447e15 | + | 1.72447e15i | 3.34887e15i | −1.37160e17 | − | 8.84484e16i | |||||
3.3 | −134834. | + | 134834.i | 1.19387e8 | + | 1.19387e8i | − | 1.91805e10i | 6.75906e11 | + | 3.53875e11i | −3.21948e13 | 2.58783e14 | − | 2.58783e14i | 2.69752e14 | + | 2.69752e14i | 1.18293e16i | −1.38849e17 | + | 4.34207e16i | |||||
3.4 | −115430. | + | 115430.i | 1.39345e8 | + | 1.39345e8i | − | 9.46824e9i | −7.62275e11 | + | 3.18319e10i | −3.21691e13 | −2.97793e14 | + | 2.97793e14i | −8.90153e14 | − | 8.90153e14i | 2.21567e16i | 8.43150e16 | − | 9.16637e16i | |||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 5.35.c.a | ✓ | 32 |
5.b | even | 2 | 1 | 25.35.c.b | 32 | ||
5.c | odd | 4 | 1 | inner | 5.35.c.a | ✓ | 32 |
5.c | odd | 4 | 1 | 25.35.c.b | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
5.35.c.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
5.35.c.a | ✓ | 32 | 5.c | odd | 4 | 1 | inner |
25.35.c.b | 32 | 5.b | even | 2 | 1 | ||
25.35.c.b | 32 | 5.c | odd | 4 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{35}^{\mathrm{new}}(5, [\chi])\).