Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [21,11,Mod(10,21)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(21, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("21.10");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 21 = 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 21.f (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: | \(13.3425023061\) |
| 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: |
\( x^{12} - x^{11} + 3773 x^{10} + 44516 x^{9} + 11068388 x^{8} + 100480832 x^{7} + 11177140432 x^{6} + \cdots + 30\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3^{7}\cdot 7^{7} \) |
| 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
\( x^{12} - x^{11} + 3773 x^{10} + 44516 x^{9} + 11068388 x^{8} + 100480832 x^{7} + 11177140432 x^{6} + \cdots + 30\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 13\!\cdots\!85 \nu^{11} + \cdots + 55\!\cdots\!60 ) / 54\!\cdots\!60 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 19\!\cdots\!86 \nu^{11} + \cdots - 24\!\cdots\!00 ) / 32\!\cdots\!60 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 40\!\cdots\!73 \nu^{11} + \cdots + 25\!\cdots\!60 ) / 81\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 11\!\cdots\!23 \nu^{11} + \cdots + 12\!\cdots\!20 ) / 10\!\cdots\!20 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 38\!\cdots\!59 \nu^{11} + \cdots - 41\!\cdots\!20 ) / 16\!\cdots\!80 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 51\!\cdots\!55 \nu^{11} + \cdots - 21\!\cdots\!20 ) / 19\!\cdots\!20 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 32\!\cdots\!91 \nu^{11} + \cdots - 74\!\cdots\!20 ) / 33\!\cdots\!20 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 19\!\cdots\!31 \nu^{11} + \cdots - 19\!\cdots\!60 ) / 16\!\cdots\!80 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 41\!\cdots\!91 \nu^{11} + \cdots + 76\!\cdots\!00 ) / 27\!\cdots\!80 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 27\!\cdots\!63 \nu^{11} + \cdots - 54\!\cdots\!00 ) / 81\!\cdots\!40 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{5} + 9\beta_{3} + 1256\beta_{2} - 1256 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2 \beta_{11} + 7 \beta_{10} + 6 \beta_{9} - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 19 \beta_{5} + \cdots - 11720 \)
|
| \(\nu^{4}\) | \(=\) |
\( 49 \beta_{11} + 92 \beta_{10} + 100 \beta_{9} + 49 \beta_{8} - 3041 \beta_{7} + 89 \beta_{6} + 46 \beta_{5} + \cdots - 46 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 4985 \beta_{11} - 11807 \beta_{10} - 18490 \beta_{9} + 9970 \beta_{8} - 85383 \beta_{7} + \cdots + 57905494 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 420370 \beta_{11} - 873419 \beta_{10} - 285046 \beta_{9} + 210185 \beta_{8} - 221078 \beta_{7} + \cdots + 6929260332 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 16872609 \beta_{11} - 41224652 \beta_{10} - 6142124 \beta_{9} - 16872609 \beta_{8} + 296313649 \beta_{7} + \cdots + 20612326 \)
|
| \(\nu^{8}\) | \(=\) |
\( 740063673 \beta_{11} + 1554437295 \beta_{10} + 165376074 \beta_{9} - 1480127346 \beta_{8} + \cdots - 19677039076598 \)
|
| \(\nu^{9}\) | \(=\) |
\( 105427992482 \beta_{11} + 238563843427 \beta_{10} + 158675910438 \beta_{9} - 52713996241 \beta_{8} + \cdots - 772687374238988 \)
|
| \(\nu^{10}\) | \(=\) |
\( 2468471109865 \beta_{11} + 5532921404972 \beta_{10} + 3298579732876 \beta_{9} + 2468471109865 \beta_{8} + \cdots - 2766460702486 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 162499451784257 \beta_{11} - 350726880646247 \beta_{10} - 357292586394394 \beta_{9} + \cdots + 25\!\cdots\!18 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).
| \(n\) | \(8\) | \(10\) |
| \(\chi(n)\) | \(1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 10.1 |
|
−27.0722 | − | 46.8904i | −121.500 | − | 70.1481i | −953.808 | + | 1652.04i | −1899.28 | + | 1096.55i | 7596.25i | 16751.6 | − | 1363.71i | 47842.8 | 9841.50 | + | 17046.0i | 102836. | + | 59372.1i | ||||||||||||||||||||||||||||||||||||||||
| 10.2 | −13.5243 | − | 23.4248i | −121.500 | − | 70.1481i | 146.185 | − | 253.199i | −1211.43 | + | 699.418i | 3794.82i | −12945.1 | − | 10719.1i | −35606.0 | 9841.50 | + | 17046.0i | 32767.5 | + | 18918.3i | |||||||||||||||||||||||||||||||||||||||||
| 10.3 | −4.98749 | − | 8.63859i | −121.500 | − | 70.1481i | 462.250 | − | 800.640i | 2618.43 | − | 1511.75i | 1399.45i | 13450.1 | + | 10078.2i | −19436.3 | 9841.50 | + | 17046.0i | −26118.8 | − | 15079.7i | |||||||||||||||||||||||||||||||||||||||||
| 10.4 | 11.2102 | + | 19.4166i | −121.500 | − | 70.1481i | 260.663 | − | 451.481i | −354.602 | + | 204.730i | − | 3145.50i | −12291.7 | + | 11462.5i | 34646.8 | 9841.50 | + | 17046.0i | −7950.32 | − | 4590.12i | ||||||||||||||||||||||||||||||||||||||||
| 10.5 | 16.0682 | + | 27.8309i | −121.500 | − | 70.1481i | −4.37327 | + | 7.57472i | −4433.83 | + | 2559.87i | − | 4508.61i | 10835.6 | − | 12847.8i | 32626.6 | 9841.50 | + | 17046.0i | −142487. | − | 82265.1i | ||||||||||||||||||||||||||||||||||||||||
| 10.6 | 23.8056 | + | 41.2326i | −121.500 | − | 70.1481i | −621.416 | + | 1076.32i | 4637.22 | − | 2677.30i | − | 6679.68i | −5755.42 | − | 15790.8i | −10418.9 | 9841.50 | + | 17046.0i | 220784. | + | 127470.i | ||||||||||||||||||||||||||||||||||||||||
| 19.1 | −27.0722 | + | 46.8904i | −121.500 | + | 70.1481i | −953.808 | − | 1652.04i | −1899.28 | − | 1096.55i | − | 7596.25i | 16751.6 | + | 1363.71i | 47842.8 | 9841.50 | − | 17046.0i | 102836. | − | 59372.1i | ||||||||||||||||||||||||||||||||||||||||
| 19.2 | −13.5243 | + | 23.4248i | −121.500 | + | 70.1481i | 146.185 | + | 253.199i | −1211.43 | − | 699.418i | − | 3794.82i | −12945.1 | + | 10719.1i | −35606.0 | 9841.50 | − | 17046.0i | 32767.5 | − | 18918.3i | ||||||||||||||||||||||||||||||||||||||||
| 19.3 | −4.98749 | + | 8.63859i | −121.500 | + | 70.1481i | 462.250 | + | 800.640i | 2618.43 | + | 1511.75i | − | 1399.45i | 13450.1 | − | 10078.2i | −19436.3 | 9841.50 | − | 17046.0i | −26118.8 | + | 15079.7i | ||||||||||||||||||||||||||||||||||||||||
| 19.4 | 11.2102 | − | 19.4166i | −121.500 | + | 70.1481i | 260.663 | + | 451.481i | −354.602 | − | 204.730i | 3145.50i | −12291.7 | − | 11462.5i | 34646.8 | 9841.50 | − | 17046.0i | −7950.32 | + | 4590.12i | |||||||||||||||||||||||||||||||||||||||||
| 19.5 | 16.0682 | − | 27.8309i | −121.500 | + | 70.1481i | −4.37327 | − | 7.57472i | −4433.83 | − | 2559.87i | 4508.61i | 10835.6 | + | 12847.8i | 32626.6 | 9841.50 | − | 17046.0i | −142487. | + | 82265.1i | |||||||||||||||||||||||||||||||||||||||||
| 19.6 | 23.8056 | − | 41.2326i | −121.500 | + | 70.1481i | −621.416 | − | 1076.32i | 4637.22 | + | 2677.30i | 6679.68i | −5755.42 | + | 15790.8i | −10418.9 | 9841.50 | − | 17046.0i | 220784. | − | 127470.i | |||||||||||||||||||||||||||||||||||||||||
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 | 21.11.f.a | ✓ | 12 |
| 3.b | odd | 2 | 1 | 63.11.m.b | 12 | ||
| 7.c | even | 3 | 1 | 147.11.d.a | 12 | ||
| 7.d | odd | 6 | 1 | inner | 21.11.f.a | ✓ | 12 |
| 7.d | odd | 6 | 1 | 147.11.d.a | 12 | ||
| 21.g | even | 6 | 1 | 63.11.m.b | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.11.f.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 21.11.f.a | ✓ | 12 | 7.d | odd | 6 | 1 | inner |
| 63.11.m.b | 12 | 3.b | odd | 2 | 1 | ||
| 63.11.m.b | 12 | 21.g | even | 6 | 1 | ||
| 147.11.d.a | 12 | 7.c | even | 3 | 1 | ||
| 147.11.d.a | 12 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 11 T_{2}^{11} + 3843 T_{2}^{10} - 59914 T_{2}^{9} + 11481368 T_{2}^{8} + \cdots + 25\!\cdots\!00 \)
acting on \(S_{11}^{\mathrm{new}}(21, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots + 25\!\cdots\!00 \)
|
| $3$ |
\( (T^{2} + 243 T + 19683)^{6} \)
|
| $5$ |
\( T^{12} + \cdots + 10\!\cdots\!00 \)
|
| $7$ |
\( T^{12} + \cdots + 50\!\cdots\!01 \)
|
| $11$ |
\( T^{12} + \cdots + 26\!\cdots\!00 \)
|
| $13$ |
\( T^{12} + \cdots + 22\!\cdots\!00 \)
|
| $17$ |
\( T^{12} + \cdots + 73\!\cdots\!00 \)
|
| $19$ |
\( T^{12} + \cdots + 62\!\cdots\!24 \)
|
| $23$ |
\( T^{12} + \cdots + 61\!\cdots\!00 \)
|
| $29$ |
\( (T^{6} + \cdots - 47\!\cdots\!00)^{2} \)
|
| $31$ |
\( T^{12} + \cdots + 28\!\cdots\!49 \)
|
| $37$ |
\( T^{12} + \cdots + 16\!\cdots\!00 \)
|
| $41$ |
\( T^{12} + \cdots + 14\!\cdots\!84 \)
|
| $43$ |
\( (T^{6} + \cdots - 26\!\cdots\!24)^{2} \)
|
| $47$ |
\( T^{12} + \cdots + 26\!\cdots\!00 \)
|
| $53$ |
\( T^{12} + \cdots + 35\!\cdots\!00 \)
|
| $59$ |
\( T^{12} + \cdots + 39\!\cdots\!64 \)
|
| $61$ |
\( T^{12} + \cdots + 95\!\cdots\!00 \)
|
| $67$ |
\( T^{12} + \cdots + 86\!\cdots\!00 \)
|
| $71$ |
\( (T^{6} + \cdots + 36\!\cdots\!44)^{2} \)
|
| $73$ |
\( T^{12} + \cdots + 10\!\cdots\!64 \)
|
| $79$ |
\( T^{12} + \cdots + 42\!\cdots\!25 \)
|
| $83$ |
\( T^{12} + \cdots + 78\!\cdots\!04 \)
|
| $89$ |
\( T^{12} + \cdots + 17\!\cdots\!44 \)
|
| $97$ |
\( T^{12} + \cdots + 28\!\cdots\!00 \)
|
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