Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [84,9,Mod(61,84)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(84, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 5]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("84.61");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 84 = 2^{2} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 84.m (of order \(6\), 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: | \(34.2198032451\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 3 x^{11} + 148097 x^{10} + 46071824 x^{9} + 21578502553 x^{8} + 3561445462121 x^{7} + \cdots + 45\!\cdots\!96 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{20}\cdot 3^{10}\cdot 7^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 3 x^{11} + 148097 x^{10} + 46071824 x^{9} + 21578502553 x^{8} + 3561445462121 x^{7} + \cdots + 45\!\cdots\!96 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 15\!\cdots\!43 \nu^{11} + \cdots - 14\!\cdots\!20 ) / 14\!\cdots\!72 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 34\!\cdots\!83 \nu^{11} + \cdots + 41\!\cdots\!04 ) / 20\!\cdots\!24 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 42\!\cdots\!13 \nu^{11} + \cdots + 17\!\cdots\!08 ) / 20\!\cdots\!24 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 49\!\cdots\!23 \nu^{11} + \cdots - 86\!\cdots\!88 ) / 20\!\cdots\!24 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 92\!\cdots\!02 \nu^{11} + \cdots + 10\!\cdots\!68 ) / 20\!\cdots\!24 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 15\!\cdots\!68 \nu^{11} + \cdots + 78\!\cdots\!92 ) / 28\!\cdots\!32 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 18\!\cdots\!39 \nu^{11} + \cdots - 44\!\cdots\!40 ) / 20\!\cdots\!24 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 13\!\cdots\!45 \nu^{11} + \cdots - 20\!\cdots\!48 ) / 50\!\cdots\!56 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 14\!\cdots\!29 \nu^{11} + \cdots - 79\!\cdots\!00 ) / 50\!\cdots\!56 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 17\!\cdots\!71 \nu^{11} + \cdots + 25\!\cdots\!72 ) / 20\!\cdots\!24 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 25\!\cdots\!43 \nu^{11} + \cdots - 99\!\cdots\!36 ) / 20\!\cdots\!24 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{9} + 2\beta_{8} + 11\beta_{5} - 2\beta_{4} - 13\beta_{3} + 5\beta_{2} + 89\beta _1 - 4 ) / 168 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 49 \beta_{11} - 98 \beta_{10} + 379 \beta_{9} + \beta_{8} - 357 \beta_{6} + 460 \beta_{5} + \cdots - 8295551 ) / 168 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 35084 \beta_{11} - 35084 \beta_{10} + 789666 \beta_{9} - 376181 \beta_{8} - 275352 \beta_{7} + \cdots - 5863038373 ) / 504 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 53732714 \beta_{11} + 26866357 \beta_{10} + 42409788 \beta_{9} - 238180922 \beta_{8} + \cdots + 602856833 ) / 504 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 8924207908 \beta_{11} + 17848415816 \beta_{10} - 91153716975 \beta_{9} - 55734876919 \beta_{8} + \cdots + 13\!\cdots\!03 ) / 504 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 4764138914563 \beta_{11} + 4764138914563 \beta_{10} - 56310552509697 \beta_{9} + \cdots + 66\!\cdots\!34 ) / 504 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 12\!\cdots\!96 \beta_{11} - 637399972567148 \beta_{10} + \cdots - 18\!\cdots\!84 ) / 168 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 29\!\cdots\!67 \beta_{11} + \cdots - 42\!\cdots\!61 ) / 168 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 38\!\cdots\!56 \beta_{11} + \cdots - 56\!\cdots\!67 ) / 504 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 11\!\cdots\!10 \beta_{11} + \cdots + 16\!\cdots\!09 ) / 168 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 76\!\cdots\!72 \beta_{11} + \cdots + 11\!\cdots\!37 ) / 504 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(43\) | \(73\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1 - \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 61.1 |
|
0 | 40.5000 | − | 23.3827i | 0 | −939.615 | − | 542.487i | 0 | 1451.57 | + | 1912.52i | 0 | 1093.50 | − | 1894.00i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 61.2 | 0 | 40.5000 | − | 23.3827i | 0 | −225.043 | − | 129.928i | 0 | 597.275 | − | 2325.52i | 0 | 1093.50 | − | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 61.3 | 0 | 40.5000 | − | 23.3827i | 0 | −203.366 | − | 117.413i | 0 | 1130.34 | − | 2118.29i | 0 | 1093.50 | − | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 61.4 | 0 | 40.5000 | − | 23.3827i | 0 | −133.903 | − | 77.3091i | 0 | −2234.88 | + | 877.558i | 0 | 1093.50 | − | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 61.5 | 0 | 40.5000 | − | 23.3827i | 0 | 632.551 | + | 365.204i | 0 | 984.437 | + | 2189.91i | 0 | 1093.50 | − | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 61.6 | 0 | 40.5000 | − | 23.3827i | 0 | 1011.88 | + | 584.207i | 0 | −1829.74 | − | 1554.62i | 0 | 1093.50 | − | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 73.1 | 0 | 40.5000 | + | 23.3827i | 0 | −939.615 | + | 542.487i | 0 | 1451.57 | − | 1912.52i | 0 | 1093.50 | + | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 73.2 | 0 | 40.5000 | + | 23.3827i | 0 | −225.043 | + | 129.928i | 0 | 597.275 | + | 2325.52i | 0 | 1093.50 | + | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 73.3 | 0 | 40.5000 | + | 23.3827i | 0 | −203.366 | + | 117.413i | 0 | 1130.34 | + | 2118.29i | 0 | 1093.50 | + | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 73.4 | 0 | 40.5000 | + | 23.3827i | 0 | −133.903 | + | 77.3091i | 0 | −2234.88 | − | 877.558i | 0 | 1093.50 | + | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 73.5 | 0 | 40.5000 | + | 23.3827i | 0 | 632.551 | − | 365.204i | 0 | 984.437 | − | 2189.91i | 0 | 1093.50 | + | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 73.6 | 0 | 40.5000 | + | 23.3827i | 0 | 1011.88 | − | 584.207i | 0 | −1829.74 | + | 1554.62i | 0 | 1093.50 | + | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 84.9.m.b | ✓ | 12 |
| 3.b | odd | 2 | 1 | 252.9.z.d | 12 | ||
| 7.d | odd | 6 | 1 | inner | 84.9.m.b | ✓ | 12 |
| 21.g | even | 6 | 1 | 252.9.z.d | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 84.9.m.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 84.9.m.b | ✓ | 12 | 7.d | odd | 6 | 1 | inner |
| 252.9.z.d | 12 | 3.b | odd | 2 | 1 | ||
| 252.9.z.d | 12 | 21.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{12} - 285 T_{5}^{11} - 1570602 T_{5}^{10} + 455337945 T_{5}^{9} + 2118409775334 T_{5}^{8} + \cdots + 76\!\cdots\!00 \)
acting on \(S_{9}^{\mathrm{new}}(84, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( (T^{2} - 81 T + 2187)^{6} \)
|
| $5$ |
\( T^{12} + \cdots + 76\!\cdots\!00 \)
|
| $7$ |
\( T^{12} + \cdots + 36\!\cdots\!01 \)
|
| $11$ |
\( T^{12} + \cdots + 16\!\cdots\!00 \)
|
| $13$ |
\( T^{12} + \cdots + 21\!\cdots\!64 \)
|
| $17$ |
\( T^{12} + \cdots + 10\!\cdots\!00 \)
|
| $19$ |
\( T^{12} + \cdots + 83\!\cdots\!96 \)
|
| $23$ |
\( T^{12} + \cdots + 63\!\cdots\!64 \)
|
| $29$ |
\( (T^{6} + \cdots + 23\!\cdots\!00)^{2} \)
|
| $31$ |
\( T^{12} + \cdots + 15\!\cdots\!81 \)
|
| $37$ |
\( T^{12} + \cdots + 40\!\cdots\!84 \)
|
| $41$ |
\( T^{12} + \cdots + 15\!\cdots\!64 \)
|
| $43$ |
\( (T^{6} + \cdots + 51\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{12} + \cdots + 18\!\cdots\!00 \)
|
| $53$ |
\( T^{12} + \cdots + 11\!\cdots\!56 \)
|
| $59$ |
\( T^{12} + \cdots + 26\!\cdots\!24 \)
|
| $61$ |
\( T^{12} + \cdots + 93\!\cdots\!00 \)
|
| $67$ |
\( T^{12} + \cdots + 65\!\cdots\!44 \)
|
| $71$ |
\( (T^{6} + \cdots - 52\!\cdots\!20)^{2} \)
|
| $73$ |
\( T^{12} + \cdots + 94\!\cdots\!00 \)
|
| $79$ |
\( T^{12} + \cdots + 47\!\cdots\!25 \)
|
| $83$ |
\( T^{12} + \cdots + 78\!\cdots\!04 \)
|
| $89$ |
\( T^{12} + \cdots + 11\!\cdots\!96 \)
|
| $97$ |
\( T^{12} + \cdots + 53\!\cdots\!00 \)
|
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