Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [195,2,Mod(79,195)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(195, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("195.79");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 195 = 3 \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 195.c (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: | \(1.55708283941\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} + 16x^{8} + 90x^{6} + 208x^{4} + 169x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{10} + 16x^{8} + 90x^{6} + 208x^{4} + 169x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 8\nu^{3} + 13\nu ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + 8\nu^{4} + 13\nu^{2} ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{8} + 13\nu^{6} + 53\nu^{4} + 4\nu^{3} + 65\nu^{2} + 20\nu ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} + 11\nu^{5} + 33\nu^{3} + 19\nu ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{8} - 13\nu^{6} - 53\nu^{4} + 4\nu^{3} - 65\nu^{2} + 20\nu ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{8} + 13\nu^{6} + 53\nu^{4} + 69\nu^{2} + 12 ) / 4 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{9} + 15\nu^{7} + 2\nu^{6} + 77\nu^{5} + 20\nu^{4} + 155\nu^{3} + 54\nu^{2} + 96\nu + 32 ) / 8 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -\nu^{9} - 15\nu^{7} + 2\nu^{6} - 77\nu^{5} + 20\nu^{4} - 155\nu^{3} + 54\nu^{2} - 96\nu + 32 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta_{4} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{4} - 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} + \beta_{8} - 7\beta_{7} - 7\beta_{6} + 7\beta_{4} - 2\beta_{3} + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( -8\beta_{6} - 8\beta_{4} + 4\beta_{2} + 27\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -8\beta_{9} - 8\beta_{8} + 43\beta_{7} + 43\beta_{6} - 43\beta_{4} + 20\beta_{3} - 65 \)
|
| \(\nu^{7}\) | \(=\) |
\( 55\beta_{6} + 4\beta_{5} + 55\beta_{4} - 44\beta_{2} - 151\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 51\beta_{9} + 51\beta_{8} - 253\beta_{7} - 257\beta_{6} + 257\beta_{4} - 154\beta_{3} + 351 \)
|
| \(\nu^{9}\) | \(=\) |
\( -4\beta_{9} + 4\beta_{8} - 364\beta_{6} - 60\beta_{5} - 364\beta_{4} + 352\beta_{2} + 865\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/195\mathbb{Z}\right)^\times\).
| \(n\) | \(106\) | \(131\) | \(157\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 79.1 |
|
− | 2.47948i | − | 1.00000i | −4.14785 | −1.72481 | − | 1.42303i | −2.47948 | 0.949959i | 5.32555i | −1.00000 | −3.52839 | + | 4.27665i | ||||||||||||||||||||||||||||||||||||||||||
| 79.2 | − | 2.26036i | 1.00000i | −3.10922 | −2.23266 | − | 0.123438i | 2.26036 | − | 4.96953i | 2.50723i | −1.00000 | −0.279015 | + | 5.04661i | |||||||||||||||||||||||||||||||||||||||||||
| 79.3 | − | 1.77159i | 1.00000i | −1.13853 | 1.51036 | + | 1.64888i | 1.77159 | 0.437634i | − | 1.52618i | −1.00000 | 2.92114 | − | 2.67573i | |||||||||||||||||||||||||||||||||||||||||||
| 79.4 | − | 1.22308i | − | 1.00000i | 0.504085 | −0.638796 | + | 2.14288i | −1.22308 | − | 4.18388i | − | 3.06269i | −1.00000 | 2.62091 | + | 0.781296i | |||||||||||||||||||||||||||||||||||||||||
| 79.5 | − | 0.329386i | − | 1.00000i | 1.89150 | 2.08591 | + | 0.805596i | −0.329386 | 3.70203i | − | 1.28181i | −1.00000 | 0.265352 | − | 0.687069i | ||||||||||||||||||||||||||||||||||||||||||
| 79.6 | 0.329386i | 1.00000i | 1.89150 | 2.08591 | − | 0.805596i | −0.329386 | − | 3.70203i | 1.28181i | −1.00000 | 0.265352 | + | 0.687069i | ||||||||||||||||||||||||||||||||||||||||||||
| 79.7 | 1.22308i | 1.00000i | 0.504085 | −0.638796 | − | 2.14288i | −1.22308 | 4.18388i | 3.06269i | −1.00000 | 2.62091 | − | 0.781296i | |||||||||||||||||||||||||||||||||||||||||||||
| 79.8 | 1.77159i | − | 1.00000i | −1.13853 | 1.51036 | − | 1.64888i | 1.77159 | − | 0.437634i | 1.52618i | −1.00000 | 2.92114 | + | 2.67573i | |||||||||||||||||||||||||||||||||||||||||||
| 79.9 | 2.26036i | − | 1.00000i | −3.10922 | −2.23266 | + | 0.123438i | 2.26036 | 4.96953i | − | 2.50723i | −1.00000 | −0.279015 | − | 5.04661i | |||||||||||||||||||||||||||||||||||||||||||
| 79.10 | 2.47948i | 1.00000i | −4.14785 | −1.72481 | + | 1.42303i | −2.47948 | − | 0.949959i | − | 5.32555i | −1.00000 | −3.52839 | − | 4.27665i | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 195.2.c.b | ✓ | 10 |
| 3.b | odd | 2 | 1 | 585.2.c.c | 10 | ||
| 4.b | odd | 2 | 1 | 3120.2.l.p | 10 | ||
| 5.b | even | 2 | 1 | inner | 195.2.c.b | ✓ | 10 |
| 5.c | odd | 4 | 1 | 975.2.a.r | 5 | ||
| 5.c | odd | 4 | 1 | 975.2.a.s | 5 | ||
| 15.d | odd | 2 | 1 | 585.2.c.c | 10 | ||
| 15.e | even | 4 | 1 | 2925.2.a.bl | 5 | ||
| 15.e | even | 4 | 1 | 2925.2.a.bm | 5 | ||
| 20.d | odd | 2 | 1 | 3120.2.l.p | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 195.2.c.b | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 195.2.c.b | ✓ | 10 | 5.b | even | 2 | 1 | inner |
| 585.2.c.c | 10 | 3.b | odd | 2 | 1 | ||
| 585.2.c.c | 10 | 15.d | odd | 2 | 1 | ||
| 975.2.a.r | 5 | 5.c | odd | 4 | 1 | ||
| 975.2.a.s | 5 | 5.c | odd | 4 | 1 | ||
| 2925.2.a.bl | 5 | 15.e | even | 4 | 1 | ||
| 2925.2.a.bm | 5 | 15.e | even | 4 | 1 | ||
| 3120.2.l.p | 10 | 4.b | odd | 2 | 1 | ||
| 3120.2.l.p | 10 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 16T_{2}^{8} + 90T_{2}^{6} + 208T_{2}^{4} + 169T_{2}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(195, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 16 T^{8} + \cdots + 16 \)
|
| $3$ |
\( (T^{2} + 1)^{5} \)
|
| $5$ |
\( T^{10} + 2 T^{9} + \cdots + 3125 \)
|
| $7$ |
\( T^{10} + 57 T^{8} + \cdots + 1024 \)
|
| $11$ |
\( (T^{5} - 5 T^{4} + \cdots - 452)^{2} \)
|
| $13$ |
\( (T^{2} + 1)^{5} \)
|
| $17$ |
\( T^{10} + 73 T^{8} + \cdots + 6400 \)
|
| $19$ |
\( (T^{5} + 8 T^{4} + \cdots + 1280)^{2} \)
|
| $23$ |
\( T^{10} + 161 T^{8} + \cdots + 102400 \)
|
| $29$ |
\( (T^{5} + 8 T^{4} + \cdots + 640)^{2} \)
|
| $31$ |
\( (T^{5} - 12 T^{4} + \cdots + 128)^{2} \)
|
| $37$ |
\( T^{10} + 209 T^{8} + \cdots + 350464 \)
|
| $41$ |
\( (T^{5} - 5 T^{4} + \cdots - 196)^{2} \)
|
| $43$ |
\( T^{10} + 128 T^{8} + \cdots + 200704 \)
|
| $47$ |
\( T^{10} + 204 T^{8} + \cdots + 64 \)
|
| $53$ |
\( T^{10} + 217 T^{8} + \cdots + 256 \)
|
| $59$ |
\( (T^{5} + 8 T^{4} + \cdots + 400)^{2} \)
|
| $61$ |
\( (T^{5} - 13 T^{4} + \cdots - 256)^{2} \)
|
| $67$ |
\( T^{10} + 320 T^{8} + \cdots + 15745024 \)
|
| $71$ |
\( (T^{5} - 5 T^{4} + \cdots - 28868)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 223323136 \)
|
| $79$ |
\( (T^{5} + T^{4} - 248 T^{3} + \cdots + 400)^{2} \)
|
| $83$ |
\( T^{10} + \cdots + 3685946944 \)
|
| $89$ |
\( (T^{5} + 19 T^{4} + \cdots - 4900)^{2} \)
|
| $97$ |
\( T^{10} + \cdots + 1260534016 \)
|
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