Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [600,2,Mod(109,600)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(600, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 5, 0, 7]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("600.109");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 600.be (of order \(10\), 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: | \(4.79102412128\) |
Analytic rank: | \(0\) |
Dimension: | \(120\) |
Relative dimension: | \(30\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
109.1 | −1.41414 | − | 0.0145696i | 0.309017 | + | 0.951057i | 1.99958 | + | 0.0412067i | 0.432940 | + | 2.19376i | −0.423136 | − | 1.34943i | − | 0.643164i | −2.82708 | − | 0.0874049i | −0.809017 | + | 0.587785i | −0.580275 | − | 3.10858i | |
109.2 | −1.40750 | − | 0.137619i | 0.309017 | + | 0.951057i | 1.96212 | + | 0.387399i | 1.96621 | + | 1.06491i | −0.304058 | − | 1.38114i | 1.46554i | −2.70838 | − | 0.815291i | −0.809017 | + | 0.587785i | −2.62089 | − | 1.76944i | ||
109.3 | −1.33241 | + | 0.474014i | 0.309017 | + | 0.951057i | 1.55062 | − | 1.26316i | 1.59403 | − | 1.56814i | −0.862551 | − | 1.12072i | − | 1.85305i | −1.46730 | + | 2.41806i | −0.809017 | + | 0.587785i | −1.38058 | + | 2.84499i | |
109.4 | −1.33227 | − | 0.474398i | 0.309017 | + | 0.951057i | 1.54989 | + | 1.26405i | −2.08356 | − | 0.811661i | 0.0394846 | − | 1.41366i | − | 4.66350i | −1.46521 | − | 2.41933i | −0.809017 | + | 0.587785i | 2.39081 | + | 2.06979i | |
109.5 | −1.31688 | − | 0.515578i | 0.309017 | + | 0.951057i | 1.46836 | + | 1.35791i | 1.16726 | − | 1.90723i | 0.0834045 | − | 1.41175i | − | 1.91925i | −1.23355 | − | 2.54526i | −0.809017 | + | 0.587785i | −2.52046 | + | 1.90978i | |
109.6 | −1.31138 | + | 0.529425i | 0.309017 | + | 0.951057i | 1.43942 | − | 1.38855i | −1.06976 | − | 1.96357i | −0.908750 | − | 1.08359i | 3.44588i | −1.15249 | + | 2.58298i | −0.809017 | + | 0.587785i | 2.44242 | + | 2.00863i | ||
109.7 | −1.30099 | − | 0.554466i | 0.309017 | + | 0.951057i | 1.38514 | + | 1.44271i | −2.13436 | + | 0.666722i | 0.125301 | − | 1.40865i | 5.07654i | −1.00211 | − | 2.64495i | −0.809017 | + | 0.587785i | 3.14645 | + | 0.316032i | ||
109.8 | −1.15683 | + | 0.813479i | 0.309017 | + | 0.951057i | 0.676504 | − | 1.88211i | −2.09775 | + | 0.774239i | −1.13114 | − | 0.848831i | − | 2.82006i | 0.748458 | + | 2.72760i | −0.809017 | + | 0.587785i | 1.79691 | − | 2.60214i | |
109.9 | −0.846671 | + | 1.13276i | 0.309017 | + | 0.951057i | −0.566296 | − | 1.91815i | −0.00633452 | + | 2.23606i | −1.33896 | − | 0.455190i | 4.12547i | 2.65228 | + | 0.982566i | −0.809017 | + | 0.587785i | −2.52756 | − | 1.90038i | ||
109.10 | −0.822953 | − | 1.15011i | 0.309017 | + | 0.951057i | −0.645497 | + | 1.89297i | −1.36330 | − | 1.77240i | 0.839511 | − | 1.13808i | 0.996218i | 2.70833 | − | 0.815433i | −0.809017 | + | 0.587785i | −0.916527 | + | 3.02655i | ||
109.11 | −0.571475 | − | 1.29361i | 0.309017 | + | 0.951057i | −1.34683 | + | 1.47853i | −1.14312 | + | 1.92179i | 1.05370 | − | 0.943251i | − | 0.646785i | 2.68231 | + | 0.897331i | −0.809017 | + | 0.587785i | 3.13930 | + | 0.380487i | |
109.12 | −0.447604 | + | 1.34151i | 0.309017 | + | 0.951057i | −1.59930 | − | 1.20093i | 2.18469 | − | 0.476576i | −1.41417 | − | 0.0111470i | − | 1.14258i | 2.32691 | − | 1.60794i | −0.809017 | + | 0.587785i | −0.338544 | + | 3.14410i | |
109.13 | −0.387062 | + | 1.36021i | 0.309017 | + | 0.951057i | −1.70037 | − | 1.05297i | −2.15299 | − | 0.603864i | −1.41325 | + | 0.0522115i | 0.170277i | 2.09042 | − | 1.90530i | −0.809017 | + | 0.587785i | 1.65472 | − | 2.69479i | ||
109.14 | −0.164617 | − | 1.40460i | 0.309017 | + | 0.951057i | −1.94580 | + | 0.462442i | 1.43821 | − | 1.71218i | 1.28498 | − | 0.590605i | 3.38230i | 0.969859 | + | 2.65695i | −0.809017 | + | 0.587785i | −2.64168 | − | 1.73826i | ||
109.15 | −0.142266 | + | 1.40704i | 0.309017 | + | 0.951057i | −1.95952 | − | 0.400348i | −0.665214 | + | 2.13483i | −1.38214 | + | 0.299496i | − | 2.27192i | 0.842080 | − | 2.70017i | −0.809017 | + | 0.587785i | −2.90915 | − | 1.23970i | |
109.16 | −0.0755657 | − | 1.41219i | 0.309017 | + | 0.951057i | −1.98858 | + | 0.213427i | 2.18080 | + | 0.494088i | 1.31972 | − | 0.508259i | − | 3.46121i | 0.451668 | + | 2.79213i | −0.809017 | + | 0.587785i | 0.532954 | − | 3.11704i | |
109.17 | 0.495441 | + | 1.32459i | 0.309017 | + | 0.951057i | −1.50908 | + | 1.31251i | 0.137607 | − | 2.23183i | −1.10666 | + | 0.880513i | − | 5.06961i | −2.48620 | − | 1.34863i | −0.809017 | + | 0.587785i | 3.02443 | − | 0.923468i | |
109.18 | 0.522022 | − | 1.31434i | 0.309017 | + | 0.951057i | −1.45499 | − | 1.37223i | −1.71415 | − | 1.43586i | 1.41133 | + | 0.0903187i | − | 1.87161i | −2.56311 | + | 1.19601i | −0.809017 | + | 0.587785i | −2.78204 | + | 1.50342i | |
109.19 | 0.539646 | + | 1.30720i | 0.309017 | + | 0.951057i | −1.41757 | + | 1.41085i | 2.17022 | + | 0.538631i | −1.07647 | + | 0.917182i | 2.20134i | −2.60926 | − | 1.09169i | −0.809017 | + | 0.587785i | 0.467051 | + | 3.12760i | ||
109.20 | 0.696059 | − | 1.23106i | 0.309017 | + | 0.951057i | −1.03100 | − | 1.71378i | 0.311583 | + | 2.21425i | 1.38590 | + | 0.281574i | 1.39209i | −2.82740 | + | 0.0763343i | −0.809017 | + | 0.587785i | 2.94275 | + | 1.15767i | ||
See next 80 embeddings (of 120 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
200.o | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 600.2.be.a | ✓ | 120 |
8.b | even | 2 | 1 | 600.2.be.b | yes | 120 | |
25.e | even | 10 | 1 | 600.2.be.b | yes | 120 | |
200.o | even | 10 | 1 | inner | 600.2.be.a | ✓ | 120 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
600.2.be.a | ✓ | 120 | 1.a | even | 1 | 1 | trivial |
600.2.be.a | ✓ | 120 | 200.o | even | 10 | 1 | inner |
600.2.be.b | yes | 120 | 8.b | even | 2 | 1 | |
600.2.be.b | yes | 120 | 25.e | even | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{120} - 186 T_{11}^{118} + 18969 T_{11}^{116} - 1407040 T_{11}^{114} - 68880 T_{11}^{113} + \cdots + 20\!\cdots\!00 \) acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\).