Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [196,9,Mod(97,196)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(196, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("196.97");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 196 = 2^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 196.b (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: | \(79.8462075720\) |
| 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: |
\( x^{10} - x^{9} + 1442 x^{8} + 59551 x^{7} + 2229058 x^{6} + 41253567 x^{5} + 582209889 x^{4} + \cdots + 63214027776 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{17}\cdot 3^{6}\cdot 7^{12} \) |
| Twist minimal: | no (minimal twist has level 28) |
| 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
\( x^{10} - x^{9} + 1442 x^{8} + 59551 x^{7} + 2229058 x^{6} + 41253567 x^{5} + 582209889 x^{4} + \cdots + 63214027776 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 10\!\cdots\!92 \nu^{9} + \cdots + 26\!\cdots\!16 ) / 32\!\cdots\!36 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 26\!\cdots\!79 \nu^{9} + \cdots - 60\!\cdots\!40 ) / 73\!\cdots\!32 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 32\!\cdots\!17 \nu^{9} + \cdots - 95\!\cdots\!20 ) / 11\!\cdots\!48 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 38\!\cdots\!49 \nu^{9} + \cdots + 74\!\cdots\!88 ) / 55\!\cdots\!24 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 53\!\cdots\!25 \nu^{9} + \cdots - 24\!\cdots\!88 ) / 47\!\cdots\!88 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 18\!\cdots\!37 \nu^{9} + \cdots + 16\!\cdots\!32 ) / 14\!\cdots\!64 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 38\!\cdots\!79 \nu^{9} + \cdots - 35\!\cdots\!04 ) / 28\!\cdots\!28 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 75\!\cdots\!69 \nu^{9} + \cdots - 16\!\cdots\!64 ) / 47\!\cdots\!88 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 48\!\cdots\!97 \nu^{9} + \cdots - 12\!\cdots\!16 ) / 11\!\cdots\!48 \)
|
| \(\nu\) | \(=\) |
\( ( 4\beta_{9} + 4\beta_{8} - 3\beta_{6} - 3\beta_{5} + \beta_{4} - 25\beta_{3} - 266\beta_{2} + 25\beta _1 + 234 ) / 2352 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 76 \beta_{9} - 132 \beta_{8} - 24 \beta_{7} + 141 \beta_{6} + 53 \beta_{5} + 103 \beta_{4} + \cdots - 678030 ) / 2352 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -7004\beta_{8} - 856\beta_{7} + 5995\beta_{6} + 4219\beta_{5} - 21515986 ) / 1176 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 225164 \beta_{9} - 315716 \beta_{8} - 41752 \beta_{7} + 297421 \beta_{6} + 163893 \beta_{5} + \cdots - 1176236302 ) / 2352 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 11154540 \beta_{9} + 14970268 \beta_{8} + 1972568 \beta_{7} - 13577867 \beta_{6} + \cdots + 52219110290 ) / 2352 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 698399364\beta_{8} + 91367448\beta_{7} - 642710637\beta_{6} - 377461909\beta_{5} + 2495533372878 ) / 1176 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 24154257452 \beta_{9} + 32749022428 \beta_{8} + 4308453464 \beta_{7} - 29974318763 \beta_{6} + \cdots + 115997281639250 ) / 2352 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 1127789661964 \beta_{9} - 1533208744900 \beta_{8} - 201080627480 \beta_{7} + 1406153730317 \beta_{6} + \cdots - 54\!\cdots\!26 ) / 2352 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 71815001881628 \beta_{8} - 9433022196568 \beta_{7} + 65814058542283 \beta_{6} + \cdots - 25\!\cdots\!94 ) / 1176 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/196\mathbb{Z}\right)^\times\).
| \(n\) | \(99\) | \(101\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 97.1 |
|
0 | − | 98.1976i | 0 | 1098.72i | 0 | 0 | 0 | −3081.76 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 97.2 | 0 | − | 96.9080i | 0 | 388.547i | 0 | 0 | 0 | −2830.16 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 97.3 | 0 | − | 93.3293i | 0 | − | 377.943i | 0 | 0 | 0 | −2149.36 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 97.4 | 0 | − | 77.8607i | 0 | − | 711.392i | 0 | 0 | 0 | 498.714 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 97.5 | 0 | − | 30.0072i | 0 | − | 560.381i | 0 | 0 | 0 | 5660.57 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 97.6 | 0 | 30.0072i | 0 | 560.381i | 0 | 0 | 0 | 5660.57 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 97.7 | 0 | 77.8607i | 0 | 711.392i | 0 | 0 | 0 | 498.714 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 97.8 | 0 | 93.3293i | 0 | 377.943i | 0 | 0 | 0 | −2149.36 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 97.9 | 0 | 96.9080i | 0 | − | 388.547i | 0 | 0 | 0 | −2830.16 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 97.10 | 0 | 98.1976i | 0 | − | 1098.72i | 0 | 0 | 0 | −3081.76 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 196.9.b.a | 10 | |
| 7.b | odd | 2 | 1 | inner | 196.9.b.a | 10 | |
| 7.c | even | 3 | 1 | 28.9.h.a | ✓ | 10 | |
| 7.c | even | 3 | 1 | 196.9.h.a | 10 | ||
| 7.d | odd | 6 | 1 | 28.9.h.a | ✓ | 10 | |
| 7.d | odd | 6 | 1 | 196.9.h.a | 10 | ||
| 21.g | even | 6 | 1 | 252.9.z.c | 10 | ||
| 21.h | odd | 6 | 1 | 252.9.z.c | 10 | ||
| 28.f | even | 6 | 1 | 112.9.s.b | 10 | ||
| 28.g | odd | 6 | 1 | 112.9.s.b | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 28.9.h.a | ✓ | 10 | 7.c | even | 3 | 1 | |
| 28.9.h.a | ✓ | 10 | 7.d | odd | 6 | 1 | |
| 112.9.s.b | 10 | 28.f | even | 6 | 1 | ||
| 112.9.s.b | 10 | 28.g | odd | 6 | 1 | ||
| 196.9.b.a | 10 | 1.a | even | 1 | 1 | trivial | |
| 196.9.b.a | 10 | 7.b | odd | 2 | 1 | inner | |
| 196.9.h.a | 10 | 7.c | even | 3 | 1 | ||
| 196.9.h.a | 10 | 7.d | odd | 6 | 1 | ||
| 252.9.z.c | 10 | 21.g | even | 6 | 1 | ||
| 252.9.z.c | 10 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} + 34707 T_{3}^{8} + 454983318 T_{3}^{6} + 2725114654098 T_{3}^{4} + \cdots + 43\!\cdots\!27 \)
acting on \(S_{9}^{\mathrm{new}}(196, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( T^{10} + \cdots + 43\!\cdots\!27 \)
|
| $5$ |
\( T^{10} + \cdots + 41\!\cdots\!75 \)
|
| $7$ |
\( T^{10} \)
|
| $11$ |
\( (T^{5} + \cdots - 20\!\cdots\!25)^{2} \)
|
| $13$ |
\( T^{10} + \cdots + 73\!\cdots\!92 \)
|
| $17$ |
\( T^{10} + \cdots + 25\!\cdots\!43 \)
|
| $19$ |
\( T^{10} + \cdots + 38\!\cdots\!47 \)
|
| $23$ |
\( (T^{5} + \cdots + 17\!\cdots\!71)^{2} \)
|
| $29$ |
\( (T^{5} + \cdots + 35\!\cdots\!08)^{2} \)
|
| $31$ |
\( T^{10} + \cdots + 18\!\cdots\!07 \)
|
| $37$ |
\( (T^{5} + \cdots + 28\!\cdots\!29)^{2} \)
|
| $41$ |
\( T^{10} + \cdots + 31\!\cdots\!28 \)
|
| $43$ |
\( (T^{5} + \cdots - 17\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{10} + \cdots + 30\!\cdots\!75 \)
|
| $53$ |
\( (T^{5} + \cdots + 13\!\cdots\!29)^{2} \)
|
| $59$ |
\( T^{10} + \cdots + 16\!\cdots\!83 \)
|
| $61$ |
\( T^{10} + \cdots + 62\!\cdots\!03 \)
|
| $67$ |
\( (T^{5} + \cdots + 42\!\cdots\!59)^{2} \)
|
| $71$ |
\( (T^{5} + \cdots - 10\!\cdots\!12)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 56\!\cdots\!75 \)
|
| $79$ |
\( (T^{5} + \cdots + 34\!\cdots\!11)^{2} \)
|
| $83$ |
\( T^{10} + \cdots + 56\!\cdots\!68 \)
|
| $89$ |
\( T^{10} + \cdots + 32\!\cdots\!23 \)
|
| $97$ |
\( T^{10} + \cdots + 24\!\cdots\!88 \)
|
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