Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5,33,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, 33, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5.2");
S:= CuspForms(chi, 33);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 5 \) |
Weight: | \( k \) | \(=\) | \( 33 \) |
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: | \(32.4333275711\) |
Analytic rank: | \(0\) |
Dimension: | \(30\) |
Relative dimension: | \(15\) 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 | −86676.0 | − | 86676.0i | 3.32299e7 | − | 3.32299e7i | 1.07305e10i | 1.18818e11 | + | 9.57363e10i | −5.76047e12 | −3.06435e13 | − | 3.06435e13i | 5.57804e14 | − | 5.57804e14i | − | 3.55432e14i | −2.00059e15 | − | 1.85967e16i | |||||
2.2 | −73528.8 | − | 73528.8i | −4.29377e6 | + | 4.29377e6i | 6.51799e9i | −1.18376e11 | − | 9.62823e10i | 6.31431e11 | 1.36489e13 | + | 1.36489e13i | 1.63456e14 | − | 1.63456e14i | 1.81615e15i | 1.62449e15 | + | 1.57835e16i | ||||||
2.3 | −70960.9 | − | 70960.9i | −4.83700e7 | + | 4.83700e7i | 5.77593e9i | 1.51118e11 | + | 2.11291e10i | 6.86476e12 | 2.09740e13 | + | 2.09740e13i | 1.05090e14 | − | 1.05090e14i | − | 2.82629e15i | −9.22412e15 | − | 1.22228e16i | |||||
2.4 | −42029.2 | − | 42029.2i | −2.23370e7 | + | 2.23370e7i | − | 7.62067e8i | −7.62027e10 | + | 1.32198e11i | 1.87761e12 | −2.39791e13 | − | 2.39791e13i | −2.12543e14 | + | 2.12543e14i | 8.55140e14i | 8.75889e15 | − | 2.35342e15i | |||||
2.5 | −39535.2 | − | 39535.2i | 4.69567e7 | − | 4.69567e7i | − | 1.16890e9i | −7.61848e10 | + | 1.32208e11i | −3.71289e12 | 2.85172e13 | + | 2.85172e13i | −2.16015e14 | + | 2.16015e14i | − | 2.55684e15i | 8.23885e15 | − | 2.21489e15i | ||||
2.6 | −37971.9 | − | 37971.9i | 3.04306e7 | − | 3.04306e7i | − | 1.41124e9i | 7.42414e10 | − | 1.33309e11i | −2.31101e12 | −9.26812e12 | − | 9.26812e12i | −2.16675e14 | + | 2.16675e14i | 9.78102e11i | −7.88108e15 | + | 2.24291e15i | |||||
2.7 | −7096.95 | − | 7096.95i | −4.70380e7 | + | 4.70380e7i | − | 4.19423e9i | −3.87281e10 | − | 1.47591e11i | 6.67652e11 | −3.04405e13 | − | 3.04405e13i | −6.02475e13 | + | 6.02475e13i | − | 2.57212e15i | −7.72597e14 | + | 1.32230e15i | ||||
2.8 | −5100.58 | − | 5100.58i | −7.18185e6 | + | 7.18185e6i | − | 4.24294e9i | 1.52141e11 | + | 1.16729e10i | 7.32632e10 | 2.17171e13 | + | 2.17171e13i | −4.35482e13 | + | 4.35482e13i | 1.74986e15i | −7.16467e14 | − | 8.35545e14i | |||||
2.9 | 24352.5 | + | 24352.5i | −2.67962e7 | + | 2.67962e7i | − | 3.10888e9i | −1.49880e11 | + | 2.86196e10i | −1.30511e12 | 4.33877e13 | + | 4.33877e13i | 1.80302e14 | − | 1.80302e14i | 4.16950e14i | −4.34691e15 | − | 2.95300e15i | |||||
2.10 | 25524.4 | + | 25524.4i | 3.82280e7 | − | 3.82280e7i | − | 2.99198e9i | −1.26030e11 | − | 8.60210e10i | 1.95149e12 | −1.22489e13 | − | 1.22489e13i | 1.85995e14 | − | 1.85995e14i | − | 1.06974e15i | −1.02119e15 | − | 5.41245e15i | ||||
2.11 | 32395.0 | + | 32395.0i | 1.63684e7 | − | 1.63684e7i | − | 2.19610e9i | 5.30856e10 | + | 1.43056e11i | 1.06051e12 | −2.79912e13 | − | 2.79912e13i | 2.10278e14 | − | 2.10278e14i | 1.31717e15i | −2.91458e15 | + | 6.35400e15i | |||||
2.12 | 61735.6 | + | 61735.6i | −5.69775e7 | + | 5.69775e7i | 3.32760e9i | 7.47335e10 | + | 1.33034e11i | −7.03508e12 | −1.03601e13 | − | 1.03601e13i | 5.97211e13 | − | 5.97211e13i | − | 4.63985e15i | −3.59920e15 | + | 1.28266e16i | |||||
2.13 | 65410.6 | + | 65410.6i | −1.37699e7 | + | 1.37699e7i | 4.26212e9i | 5.68539e10 | − | 1.41600e11i | −1.80139e12 | −2.41000e12 | − | 2.41000e12i | 2.14856e12 | − | 2.14856e12i | 1.47380e15i | 1.29810e16 | − | 5.54333e15i | ||||||
2.14 | 68938.0 | + | 68938.0i | 5.43018e7 | − | 5.43018e7i | 5.20993e9i | 1.52114e11 | + | 1.20131e10i | 7.48691e12 | 3.40627e13 | + | 3.40627e13i | −6.30755e13 | + | 6.30755e13i | − | 4.04435e15i | 9.65829e15 | + | 1.13146e16i | |||||
2.15 | 84542.4 | + | 84542.4i | 5.85262e6 | − | 5.85262e6i | 9.99987e9i | −1.33071e11 | + | 7.46670e10i | 9.89589e11 | −4.06221e12 | − | 4.06221e12i | −4.82306e14 | + | 4.82306e14i | 1.78451e15i | −1.75627e16 | − | 4.93762e15i | ||||||
3.1 | −86676.0 | + | 86676.0i | 3.32299e7 | + | 3.32299e7i | − | 1.07305e10i | 1.18818e11 | − | 9.57363e10i | −5.76047e12 | −3.06435e13 | + | 3.06435e13i | 5.57804e14 | + | 5.57804e14i | 3.55432e14i | −2.00059e15 | + | 1.85967e16i | |||||
3.2 | −73528.8 | + | 73528.8i | −4.29377e6 | − | 4.29377e6i | − | 6.51799e9i | −1.18376e11 | + | 9.62823e10i | 6.31431e11 | 1.36489e13 | − | 1.36489e13i | 1.63456e14 | + | 1.63456e14i | − | 1.81615e15i | 1.62449e15 | − | 1.57835e16i | ||||
3.3 | −70960.9 | + | 70960.9i | −4.83700e7 | − | 4.83700e7i | − | 5.77593e9i | 1.51118e11 | − | 2.11291e10i | 6.86476e12 | 2.09740e13 | − | 2.09740e13i | 1.05090e14 | + | 1.05090e14i | 2.82629e15i | −9.22412e15 | + | 1.22228e16i | |||||
3.4 | −42029.2 | + | 42029.2i | −2.23370e7 | − | 2.23370e7i | 7.62067e8i | −7.62027e10 | − | 1.32198e11i | 1.87761e12 | −2.39791e13 | + | 2.39791e13i | −2.12543e14 | − | 2.12543e14i | − | 8.55140e14i | 8.75889e15 | + | 2.35342e15i | |||||
3.5 | −39535.2 | + | 39535.2i | 4.69567e7 | + | 4.69567e7i | 1.16890e9i | −7.61848e10 | − | 1.32208e11i | −3.71289e12 | 2.85172e13 | − | 2.85172e13i | −2.16015e14 | − | 2.16015e14i | 2.55684e15i | 8.23885e15 | + | 2.21489e15i | ||||||
See all 30 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 | 5.33.c.a | ✓ | 30 |
5.b | even | 2 | 1 | 25.33.c.b | 30 | ||
5.c | odd | 4 | 1 | inner | 5.33.c.a | ✓ | 30 |
5.c | odd | 4 | 1 | 25.33.c.b | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
5.33.c.a | ✓ | 30 | 1.a | even | 1 | 1 | trivial |
5.33.c.a | ✓ | 30 | 5.c | odd | 4 | 1 | inner |
25.33.c.b | 30 | 5.b | even | 2 | 1 | ||
25.33.c.b | 30 | 5.c | odd | 4 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{33}^{\mathrm{new}}(5, [\chi])\).