Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [218,2,Mod(45,218)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(218, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("218.45");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 218 = 2 \cdot 109 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 218.c (of order \(3\), 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: | \(1.74073876406\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 2x^{9} + 14x^{8} + 6x^{7} + 95x^{6} + 2x^{5} + 231x^{4} + 53x^{3} + 389x^{2} - 76x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{10} - 2x^{9} + 14x^{8} + 6x^{7} + 95x^{6} + 2x^{5} + 231x^{4} + 53x^{3} + 389x^{2} - 76x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 5 \nu^{9} + 212 \nu^{8} - 848 \nu^{7} + 3082 \nu^{6} - 3180 \nu^{5} + 11660 \nu^{4} + \cdots + 19052 ) / 91773 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 128 \nu^{9} + 1163 \nu^{8} - 4652 \nu^{7} + 14380 \nu^{6} - 17445 \nu^{5} + 63965 \nu^{4} + \cdots - 73660 ) / 91773 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 335 \nu^{9} + 2153 \nu^{8} - 8612 \nu^{7} + 20167 \nu^{6} - 32295 \nu^{5} + 118415 \nu^{4} + \cdots + 312404 ) / 91773 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 581 \nu^{9} + 1274 \nu^{8} - 5096 \nu^{7} - 7295 \nu^{6} - 19110 \nu^{5} + 70070 \nu^{4} + \cdots + 546911 ) / 91773 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 4763 \nu^{9} + 9506 \nu^{8} - 65834 \nu^{7} - 31970 \nu^{6} - 440157 \nu^{5} - 22246 \nu^{4} + \cdots - 25456 ) / 367092 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1360 \nu^{9} + 2971 \nu^{8} - 18373 \nu^{7} - 10828 \nu^{6} - 104820 \nu^{5} - 12725 \nu^{4} + \cdots + 87352 ) / 91773 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 3413 \nu^{9} + 6959 \nu^{8} - 49157 \nu^{7} - 14978 \nu^{6} - 341697 \nu^{5} + 13799 \nu^{4} + \cdots + 292388 ) / 91773 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 31745 \nu^{9} - 61166 \nu^{8} + 428210 \nu^{7} + 255350 \nu^{6} + 2844723 \nu^{5} + 306790 \nu^{4} + \cdots - 2023340 ) / 367092 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{8} + \beta_{7} - 4\beta_{6} - 2\beta_{2} + 2\beta _1 - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + 3\beta_{4} - 3\beta_{3} - 8\beta_{2} - 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 3 \beta_{9} - 13 \beta_{8} - 16 \beta_{7} + 35 \beta_{6} - 3 \beta_{5} + 13 \beta_{4} + \cdots - 36 \beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( -16\beta_{9} - 49\beta_{8} - 55\beta_{7} + 96\beta_{6} + 149\beta_{2} - 149\beta _1 + 96 \)
|
| \(\nu^{6}\) | \(=\) |
\( 55\beta_{5} - 188\beta_{4} + 230\beta_{3} + 317\beta_{2} + 431 \)
|
| \(\nu^{7}\) | \(=\) |
\( 230 \beta_{9} + 722 \beta_{8} + 845 \beta_{7} - 1498 \beta_{6} + 230 \beta_{5} - 722 \beta_{4} + \cdots + 2131 \beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 845\beta_{9} + 2746\beta_{8} + 3313\beta_{7} - 5989\beta_{6} - 8049\beta_{2} + 8049\beta _1 - 5989 \)
|
| \(\nu^{9}\) | \(=\) |
\( -3313\beta_{5} + 10517\beta_{4} - 12485\beta_{3} - 18407\beta_{2} - 22257 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/218\mathbb{Z}\right)^\times\).
| \(n\) | \(115\) |
| \(\chi(n)\) | \(\beta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 45.1 |
|
1.00000 | −1.64978 | + | 2.85751i | 1.00000 | −0.899930 | + | 1.55872i | −1.64978 | + | 2.85751i | 0.430258 | − | 0.745229i | 1.00000 | −3.94357 | − | 6.83047i | −0.899930 | + | 1.55872i | ||||||||||||||||||||||||||||||||||||
| 45.2 | 1.00000 | −0.262161 | + | 0.454076i | 1.00000 | −0.189751 | + | 0.328658i | −0.262161 | + | 0.454076i | 1.23508 | − | 2.13922i | 1.00000 | 1.36254 | + | 2.35999i | −0.189751 | + | 0.328658i | |||||||||||||||||||||||||||||||||||||
| 45.3 | 1.00000 | −0.176633 | + | 0.305938i | 1.00000 | −2.04721 | + | 3.54588i | −0.176633 | + | 0.305938i | −1.41716 | + | 2.45459i | 1.00000 | 1.43760 | + | 2.49000i | −2.04721 | + | 3.54588i | |||||||||||||||||||||||||||||||||||||
| 45.4 | 1.00000 | 1.09691 | − | 1.89990i | 1.00000 | 0.908328 | − | 1.57327i | 1.09691 | − | 1.89990i | −2.27819 | + | 3.94595i | 1.00000 | −0.906417 | − | 1.56996i | 0.908328 | − | 1.57327i | |||||||||||||||||||||||||||||||||||||
| 45.5 | 1.00000 | 1.49167 | − | 2.58365i | 1.00000 | −1.77143 | + | 3.06821i | 1.49167 | − | 2.58365i | 1.53001 | − | 2.65006i | 1.00000 | −2.95015 | − | 5.10982i | −1.77143 | + | 3.06821i | |||||||||||||||||||||||||||||||||||||
| 63.1 | 1.00000 | −1.64978 | − | 2.85751i | 1.00000 | −0.899930 | − | 1.55872i | −1.64978 | − | 2.85751i | 0.430258 | + | 0.745229i | 1.00000 | −3.94357 | + | 6.83047i | −0.899930 | − | 1.55872i | |||||||||||||||||||||||||||||||||||||
| 63.2 | 1.00000 | −0.262161 | − | 0.454076i | 1.00000 | −0.189751 | − | 0.328658i | −0.262161 | − | 0.454076i | 1.23508 | + | 2.13922i | 1.00000 | 1.36254 | − | 2.35999i | −0.189751 | − | 0.328658i | |||||||||||||||||||||||||||||||||||||
| 63.3 | 1.00000 | −0.176633 | − | 0.305938i | 1.00000 | −2.04721 | − | 3.54588i | −0.176633 | − | 0.305938i | −1.41716 | − | 2.45459i | 1.00000 | 1.43760 | − | 2.49000i | −2.04721 | − | 3.54588i | |||||||||||||||||||||||||||||||||||||
| 63.4 | 1.00000 | 1.09691 | + | 1.89990i | 1.00000 | 0.908328 | + | 1.57327i | 1.09691 | + | 1.89990i | −2.27819 | − | 3.94595i | 1.00000 | −0.906417 | + | 1.56996i | 0.908328 | + | 1.57327i | |||||||||||||||||||||||||||||||||||||
| 63.5 | 1.00000 | 1.49167 | + | 2.58365i | 1.00000 | −1.77143 | − | 3.06821i | 1.49167 | + | 2.58365i | 1.53001 | + | 2.65006i | 1.00000 | −2.95015 | + | 5.10982i | −1.77143 | − | 3.06821i | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 109.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 218.2.c.c | ✓ | 10 |
| 3.b | odd | 2 | 1 | 1962.2.f.k | 10 | ||
| 109.c | even | 3 | 1 | inner | 218.2.c.c | ✓ | 10 |
| 327.i | odd | 6 | 1 | 1962.2.f.k | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 218.2.c.c | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 218.2.c.c | ✓ | 10 | 109.c | even | 3 | 1 | inner |
| 1962.2.f.k | 10 | 3.b | odd | 2 | 1 | ||
| 1962.2.f.k | 10 | 327.i | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} - T_{3}^{9} + 13T_{3}^{8} - 12T_{3}^{7} + 139T_{3}^{6} - 106T_{3}^{5} + 352T_{3}^{4} + 300T_{3}^{3} + 241T_{3}^{2} + 68T_{3} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(218, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{10} \)
|
| $3$ |
\( T^{10} - T^{9} + \cdots + 16 \)
|
| $5$ |
\( T^{10} + 8 T^{9} + \cdots + 324 \)
|
| $7$ |
\( T^{10} + T^{9} + \cdots + 7056 \)
|
| $11$ |
\( T^{10} - T^{9} + \cdots + 576 \)
|
| $13$ |
\( T^{10} + 48 T^{8} + \cdots + 9801 \)
|
| $17$ |
\( (T^{5} + 8 T^{4} + \cdots - 363)^{2} \)
|
| $19$ |
\( (T^{5} + 3 T^{4} + \cdots + 2592)^{2} \)
|
| $23$ |
\( (T^{5} - 4 T^{4} + \cdots + 324)^{2} \)
|
| $29$ |
\( T^{10} + \cdots + 402443721 \)
|
| $31$ |
\( T^{10} - 7 T^{9} + \cdots + 944784 \)
|
| $37$ |
\( T^{10} - 4 T^{9} + \cdots + 1 \)
|
| $41$ |
\( (T^{5} + 11 T^{4} + \cdots + 5949)^{2} \)
|
| $43$ |
\( (T^{5} - T^{4} - 97 T^{3} + \cdots - 2412)^{2} \)
|
| $47$ |
\( T^{10} \)
|
| $53$ |
\( T^{10} + 2 T^{9} + \cdots + 38266596 \)
|
| $59$ |
\( T^{10} + \cdots + 241864704 \)
|
| $61$ |
\( T^{10} - 7 T^{9} + \cdots + 293764 \)
|
| $67$ |
\( T^{10} + \cdots + 3332060176 \)
|
| $71$ |
\( (T^{5} - 2 T^{4} + \cdots - 504)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 3731132889 \)
|
| $79$ |
\( T^{10} + \cdots + 45715571344 \)
|
| $83$ |
\( T^{10} + 7 T^{9} + \cdots + 944784 \)
|
| $89$ |
\( T^{10} - 9 T^{9} + \cdots + 1640961 \)
|
| $97$ |
\( T^{10} + \cdots + 308072704 \)
|
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