Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2303,1,Mod(281,2303)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2303, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([4, 7]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2303.281");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2303 = 7^{2} \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2303.j (of order \(14\), degree \(6\), 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: | \(1.14934672409\) |
| Analytic rank: | \(0\) |
| Dimension: | \(24\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{14})\) |
| Coefficient field: | \(\Q(\zeta_{35})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{24} - x^{23} + x^{19} - x^{18} + x^{17} - x^{16} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} + \cdots + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{35}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{35} - \cdots)\) |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2303\mathbb{Z}\right)^\times\).
| \(n\) | \(99\) | \(2257\) |
| \(\chi(n)\) | \(-1\) | \(-\zeta_{70}^{15}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 281.1 |
|
−0.137526 | − | 0.602539i | 0.590905 | + | 0.740971i | 0.556829 | − | 0.268155i | 0 | 0.365199 | − | 0.457945i | 0.858449 | + | 0.512899i | −0.623490 | − | 0.781831i | 0.0226513 | − | 0.0992418i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 281.2 | −0.137526 | − | 0.602539i | 1.22694 | + | 1.53853i | 0.556829 | − | 0.268155i | 0 | 0.758291 | − | 0.950866i | −0.995974 | + | 0.0896393i | −0.623490 | − | 0.781831i | −0.639185 | + | 2.80045i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 281.3 | 0.360046 | + | 1.57747i | −0.861741 | − | 1.08059i | −1.45780 | + | 0.702039i | 0 | 1.39433 | − | 1.74843i | −0.393025 | − | 0.919528i | −0.623490 | − | 0.781831i | −0.202554 | + | 0.887448i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 281.4 | 0.360046 | + | 1.57747i | 0.167386 | + | 0.209896i | −1.45780 | + | 0.702039i | 0 | −0.270836 | + | 0.339618i | 0.753071 | − | 0.657939i | −0.623490 | − | 0.781831i | 0.206483 | − | 0.904661i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 610.1 | −0.556829 | + | 0.268155i | 0.0199667 | − | 0.0874800i | −0.385338 | + | 0.483198i | 0 | 0.0123401 | + | 0.0540656i | −0.691063 | + | 0.722795i | 0.222521 | − | 0.974928i | 0.893715 | + | 0.430390i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 610.2 | −0.556829 | + | 0.268155i | 0.429004 | − | 1.87959i | −0.385338 | + | 0.483198i | 0 | 0.265139 | + | 1.16165i | 0.134233 | − | 0.990950i | 0.222521 | − | 0.974928i | −2.44784 | − | 1.17882i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 610.3 | 1.45780 | − | 0.702039i | −0.416664 | + | 1.82552i | 1.00883 | − | 1.26503i | 0 | 0.674176 | + | 2.95376i | 0.983930 | − | 0.178557i | 0.222521 | − | 0.974928i | −2.25796 | − | 1.08737i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 610.4 | 1.45780 | − | 0.702039i | 0.245172 | − | 1.07417i | 1.00883 | − | 1.26503i | 0 | −0.396697 | − | 1.73804i | 0.473869 | + | 0.880596i | 0.222521 | − | 0.974928i | −0.192762 | − | 0.0928294i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 939.1 | −1.00883 | − | 1.26503i | −1.54687 | + | 0.744934i | −0.360046 | + | 1.57747i | 0 | 2.50289 | + | 1.20533i | −0.963963 | + | 0.266037i | 0.900969 | − | 0.433884i | 1.21439 | − | 1.52280i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 939.2 | −1.00883 | − | 1.26503i | 1.79468 | − | 0.864274i | −0.360046 | + | 1.57747i | 0 | −2.90386 | − | 1.39842i | −0.0448648 | + | 0.998993i | 0.900969 | − | 0.433884i | 1.85043 | − | 2.32037i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 939.3 | 0.385338 | + | 0.483198i | −1.35699 | + | 0.653491i | 0.137526 | − | 0.602539i | 0 | −0.838665 | − | 0.403880i | 0.936235 | + | 0.351375i | 0.900969 | − | 0.433884i | 0.790876 | − | 0.991727i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 939.4 | 0.385338 | + | 0.483198i | 0.708207 | − | 0.341054i | 0.137526 | − | 0.602539i | 0 | 0.437696 | + | 0.210783i | −0.550897 | − | 0.834573i | 0.900969 | − | 0.433884i | −0.238251 | + | 0.298758i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1268.1 | −1.00883 | + | 1.26503i | −1.54687 | − | 0.744934i | −0.360046 | − | 1.57747i | 0 | 2.50289 | − | 1.20533i | −0.963963 | − | 0.266037i | 0.900969 | + | 0.433884i | 1.21439 | + | 1.52280i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1268.2 | −1.00883 | + | 1.26503i | 1.79468 | + | 0.864274i | −0.360046 | − | 1.57747i | 0 | −2.90386 | + | 1.39842i | −0.0448648 | − | 0.998993i | 0.900969 | + | 0.433884i | 1.85043 | + | 2.32037i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1268.3 | 0.385338 | − | 0.483198i | −1.35699 | − | 0.653491i | 0.137526 | + | 0.602539i | 0 | −0.838665 | + | 0.403880i | 0.936235 | − | 0.351375i | 0.900969 | + | 0.433884i | 0.790876 | + | 0.991727i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1268.4 | 0.385338 | − | 0.483198i | 0.708207 | + | 0.341054i | 0.137526 | + | 0.602539i | 0 | 0.437696 | − | 0.210783i | −0.550897 | + | 0.834573i | 0.900969 | + | 0.433884i | −0.238251 | − | 0.298758i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1597.1 | −0.556829 | − | 0.268155i | 0.0199667 | + | 0.0874800i | −0.385338 | − | 0.483198i | 0 | 0.0123401 | − | 0.0540656i | −0.691063 | − | 0.722795i | 0.222521 | + | 0.974928i | 0.893715 | − | 0.430390i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1597.2 | −0.556829 | − | 0.268155i | 0.429004 | + | 1.87959i | −0.385338 | − | 0.483198i | 0 | 0.265139 | − | 1.16165i | 0.134233 | + | 0.990950i | 0.222521 | + | 0.974928i | −2.44784 | + | 1.17882i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1597.3 | 1.45780 | + | 0.702039i | −0.416664 | − | 1.82552i | 1.00883 | + | 1.26503i | 0 | 0.674176 | − | 2.95376i | 0.983930 | + | 0.178557i | 0.222521 | + | 0.974928i | −2.25796 | + | 1.08737i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1597.4 | 1.45780 | + | 0.702039i | 0.245172 | + | 1.07417i | 1.00883 | + | 1.26503i | 0 | −0.396697 | + | 1.73804i | 0.473869 | − | 0.880596i | 0.222521 | + | 0.974928i | −0.192762 | + | 0.0928294i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| See all 24 embeddings | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 47.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-47}) \) |
| 49.e | even | 7 | 1 | inner |
| 2303.j | odd | 14 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2303.1.j.b | ✓ | 24 |
| 47.b | odd | 2 | 1 | CM | 2303.1.j.b | ✓ | 24 |
| 49.e | even | 7 | 1 | inner | 2303.1.j.b | ✓ | 24 |
| 2303.j | odd | 14 | 1 | inner | 2303.1.j.b | ✓ | 24 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2303.1.j.b | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
| 2303.1.j.b | ✓ | 24 | 47.b | odd | 2 | 1 | CM |
| 2303.1.j.b | ✓ | 24 | 49.e | even | 7 | 1 | inner |
| 2303.1.j.b | ✓ | 24 | 2303.j | odd | 14 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - T_{2}^{11} + 2 T_{2}^{10} - 3 T_{2}^{9} + 5 T_{2}^{8} - 8 T_{2}^{7} + 13 T_{2}^{6} + \cdots + 1 \)
acting on \(S_{1}^{\mathrm{new}}(2303, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{12} - T^{11} + 2 T^{10} + \cdots + 1)^{2} \)
|
| $3$ |
\( T^{24} - 2 T^{23} + \cdots + 1 \)
|
| $5$ |
\( T^{24} \)
|
| $7$ |
\( T^{24} - T^{23} + \cdots + 1 \)
|
| $11$ |
\( T^{24} \)
|
| $13$ |
\( T^{24} \)
|
| $17$ |
\( T^{24} - 2 T^{23} + \cdots + 1 \)
|
| $19$ |
\( T^{24} \)
|
| $23$ |
\( T^{24} \)
|
| $29$ |
\( T^{24} \)
|
| $31$ |
\( T^{24} \)
|
| $37$ |
\( T^{24} - 2 T^{23} + \cdots + 1 \)
|
| $41$ |
\( T^{24} \)
|
| $43$ |
\( T^{24} \)
|
| $47$ |
\( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{4} \)
|
| $53$ |
\( T^{24} + 5 T^{23} + \cdots + 1 \)
|
| $59$ |
\( T^{24} + 5 T^{23} + \cdots + 1 \)
|
| $61$ |
\( T^{24} - 2 T^{23} + \cdots + 1 \)
|
| $67$ |
\( T^{24} \)
|
| $71$ |
\( T^{24} + 5 T^{23} + \cdots + 1 \)
|
| $73$ |
\( T^{24} \)
|
| $79$ |
\( (T^{12} - T^{11} - 12 T^{10} + \cdots + 1)^{2} \)
|
| $83$ |
\( (T^{6} + 2 T^{5} + 4 T^{4} + \cdots + 1)^{4} \)
|
| $89$ |
\( (T^{12} - T^{11} + 2 T^{10} + \cdots + 1)^{2} \)
|
| $97$ |
\( (T^{2} + T - 1)^{12} \)
|
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