Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1,110,Mod(1,1)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1, base_ring=CyclotomicField(1))
chi = DirichletCharacter(H, H._module([]))
N = Newforms(chi, 110, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1.1");
S:= CuspForms(chi, 110);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 110 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1.a (trivial) |
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: | yes |
| Analytic conductor: | \(75.2394221917\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2 x^{7} + \cdots + 46\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | multiple of \( 2^{118}\cdot 3^{40}\cdot 5^{14}\cdot 7^{6}\cdot 11^{3}\cdot 13 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2 x^{7} + \cdots + 46\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 192\nu - 48 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 18\!\cdots\!09 \nu^{7} + \cdots - 59\!\cdots\!64 ) / 10\!\cdots\!76 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 19\!\cdots\!01 \nu^{7} + \cdots - 20\!\cdots\!36 ) / 15\!\cdots\!68 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 25\!\cdots\!07 \nu^{7} + \cdots + 99\!\cdots\!60 ) / 64\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 37\!\cdots\!77 \nu^{7} + \cdots - 11\!\cdots\!00 ) / 64\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 29\!\cdots\!71 \nu^{7} + \cdots - 92\!\cdots\!20 ) / 19\!\cdots\!20 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 21\!\cdots\!61 \nu^{7} + \cdots + 76\!\cdots\!00 ) / 33\!\cdots\!76 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 48 ) / 192 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 733523\beta_{2} + 1973266568918041\beta _1 + 969131934460330611034670049512448 ) / 36864 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{7} + 79 \beta_{6} + 2011782 \beta_{5} - 23678718391 \beta_{4} + \cdots + 19\!\cdots\!28 ) / 7077888 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 29719859398329 \beta_{7} + \cdots + 12\!\cdots\!36 ) / 10616832 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 69\!\cdots\!93 \beta_{7} + \cdots + 15\!\cdots\!28 ) / 63700992 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 33\!\cdots\!81 \beta_{7} + \cdots + 82\!\cdots\!88 ) / 1327104 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 14\!\cdots\!23 \beta_{7} + \cdots + 33\!\cdots\!64 ) / 21233664 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−4.71735e16 | −1.38575e26 | 1.57631e33 | −1.39304e38 | 6.53705e42 | 1.59904e46 | −4.37425e49 | 9.05875e51 | 6.57147e54 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.56139e16 | 1.00458e26 | 6.19316e32 | −8.07110e36 | −3.57771e42 | −7.18940e45 | 1.05848e48 | −5.23267e49 | 2.87444e53 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.02081e16 | −7.88466e25 | −2.40671e32 | 1.60701e38 | 1.59334e42 | 1.73591e45 | 1.79793e49 | −3.92739e51 | −3.24746e54 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.02754e15 | −9.96254e25 | −6.44926e32 | −2.14431e38 | 2.01995e41 | −1.47165e46 | 2.62357e48 | −2.18949e50 | 4.34769e53 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 3.08871e15 | 1.03355e26 | −6.39497e32 | −4.34627e37 | 3.19235e41 | 1.35697e46 | −3.97991e48 | 5.38156e50 | −1.34244e53 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.87062e16 | 1.16403e26 | 1.75010e32 | 1.23460e38 | 3.34150e42 | −1.99654e46 | −1.36075e49 | 3.40559e51 | 3.54406e54 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.87423e16 | −1.13913e26 | 1.77081e32 | 7.39260e37 | −3.27412e42 | 4.38791e45 | −1.35651e49 | 2.83200e51 | 2.12480e54 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.67743e16 | 3.08304e25 | 1.53880e33 | −1.66073e38 | 1.44207e42 | 8.52831e45 | 4.16178e49 | −9.19366e51 | −7.76792e54 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1.110.a.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.110.a.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace is the entire newspace \(S_{110}^{\mathrm{new}}(\Gamma_0(1))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots + 82\!\cdots\!96 \)
|
| $3$ |
\( T^{8} + \cdots + 46\!\cdots\!96 \)
|
| $5$ |
\( T^{8} + \cdots - 25\!\cdots\!00 \)
|
| $7$ |
\( T^{8} + \cdots - 29\!\cdots\!84 \)
|
| $11$ |
\( T^{8} + \cdots - 13\!\cdots\!84 \)
|
| $13$ |
\( T^{8} + \cdots + 25\!\cdots\!76 \)
|
| $17$ |
\( T^{8} + \cdots + 22\!\cdots\!56 \)
|
| $19$ |
\( T^{8} + \cdots - 12\!\cdots\!00 \)
|
| $23$ |
\( T^{8} + \cdots + 10\!\cdots\!56 \)
|
| $29$ |
\( T^{8} + \cdots + 26\!\cdots\!00 \)
|
| $31$ |
\( T^{8} + \cdots - 13\!\cdots\!64 \)
|
| $37$ |
\( T^{8} + \cdots + 21\!\cdots\!36 \)
|
| $41$ |
\( T^{8} + \cdots + 25\!\cdots\!96 \)
|
| $43$ |
\( T^{8} + \cdots + 73\!\cdots\!16 \)
|
| $47$ |
\( T^{8} + \cdots + 16\!\cdots\!76 \)
|
| $53$ |
\( T^{8} + \cdots - 14\!\cdots\!04 \)
|
| $59$ |
\( T^{8} + \cdots - 14\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots + 63\!\cdots\!16 \)
|
| $67$ |
\( T^{8} + \cdots - 60\!\cdots\!44 \)
|
| $71$ |
\( T^{8} + \cdots - 47\!\cdots\!24 \)
|
| $73$ |
\( T^{8} + \cdots - 55\!\cdots\!44 \)
|
| $79$ |
\( T^{8} + \cdots - 35\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots + 36\!\cdots\!36 \)
|
| $89$ |
\( T^{8} + \cdots - 12\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots - 36\!\cdots\!24 \)
|
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