Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [21,12,Mod(4,21)]
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, 4]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("21.4");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 21 = 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 21.e (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: | \(16.1352067918\) |
| Analytic rank: | \(0\) |
| Dimension: | \(14\) |
| Relative dimension: | \(7\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{14} - 2 x^{13} + 11519 x^{12} + 118106 x^{11} + 96026178 x^{10} + 984766908 x^{9} + \cdots + 24\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{7}\cdot 7^{8} \) |
| 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 \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} - 2 x^{13} + 11519 x^{12} + 118106 x^{11} + 96026178 x^{10} + 984766908 x^{9} + \cdots + 24\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 73\!\cdots\!79 \nu^{13} + \cdots - 10\!\cdots\!30 ) / 10\!\cdots\!30 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 28\!\cdots\!46 \nu^{13} + \cdots + 10\!\cdots\!30 ) / 66\!\cdots\!41 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 64\!\cdots\!12 \nu^{13} + \cdots - 21\!\cdots\!38 ) / 66\!\cdots\!41 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 11\!\cdots\!06 \nu^{13} + \cdots - 12\!\cdots\!00 ) / 54\!\cdots\!65 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 71\!\cdots\!79 \nu^{13} + \cdots + 14\!\cdots\!60 ) / 17\!\cdots\!20 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 28\!\cdots\!89 \nu^{13} + \cdots + 11\!\cdots\!84 ) / 53\!\cdots\!92 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 58\!\cdots\!17 \nu^{13} + \cdots - 14\!\cdots\!40 ) / 41\!\cdots\!60 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 10\!\cdots\!47 \nu^{13} + \cdots + 82\!\cdots\!40 ) / 48\!\cdots\!20 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 56\!\cdots\!34 \nu^{13} + \cdots - 18\!\cdots\!00 ) / 17\!\cdots\!20 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 96\!\cdots\!24 \nu^{13} + \cdots - 14\!\cdots\!60 ) / 29\!\cdots\!20 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 45\!\cdots\!62 \nu^{13} + \cdots - 97\!\cdots\!80 ) / 88\!\cdots\!60 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 98\!\cdots\!37 \nu^{13} + \cdots + 54\!\cdots\!60 ) / 88\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 9\beta_{3} + 3288\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{12} - 3\beta_{8} - 3\beta_{7} - 2\beta_{6} - 11\beta_{4} + 5203\beta_{3} - 5203\beta _1 - 28756 \)
|
| \(\nu^{4}\) | \(=\) |
\( 23 \beta_{13} + 174 \beta_{10} - 132 \beta_{9} - 23 \beta_{8} + 132 \beta_{7} - 7804 \beta_{5} + \cdots - 17095119 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8816 \beta_{13} + 8816 \beta_{12} + 5184 \beta_{11} + 16416 \beta_{10} + 47808 \beta_{9} + 13824 \beta_{8} + \cdots + 8816 \)
|
| \(\nu^{6}\) | \(=\) |
\( 268498 \beta_{12} + 71520 \beta_{11} + 2242454 \beta_{8} - 1359242 \beta_{7} + 1902436 \beta_{6} + \cdots + 105888453472 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 66672639 \beta_{13} - 29989920 \beta_{11} - 128158782 \beta_{10} - 396156732 \beta_{9} + \cdots + 3091627954833 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 2513822469 \beta_{13} - 2513822469 \beta_{12} - 2013502272 \beta_{11} - 16013839050 \beta_{10} + \cdots - 2513822469 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 488789730742 \beta_{12} - 256206816000 \beta_{11} - 1756780248354 \beta_{8} - 2430221957346 \beta_{7} + \cdots - 28\!\cdots\!30 \)
|
| \(\nu^{10}\) | \(=\) |
\( 21948270232448 \beta_{13} + 10166702661312 \beta_{11} + 124810535425152 \beta_{10} + \cdots - 50\!\cdots\!40 \)
|
| \(\nu^{11}\) | \(=\) |
\( 35\!\cdots\!25 \beta_{13} + \cdots + 35\!\cdots\!25 \)
|
| \(\nu^{12}\) | \(=\) |
\( 18\!\cdots\!55 \beta_{12} + \cdots + 36\!\cdots\!98 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 26\!\cdots\!68 \beta_{13} + \cdots + 20\!\cdots\!34 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).
| \(n\) | \(8\) | \(10\) |
| \(\chi(n)\) | \(1\) | \(-1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4.1 |
|
−38.1650 | − | 66.1038i | −121.500 | + | 210.444i | −1889.14 | + | 3272.09i | 2577.44 | + | 4464.27i | 18548.2 | −25394.3 | + | 36502.8i | 132073. | −29524.5 | − | 51137.9i | 196737. | − | 340758.i | ||||||||||||||||||||||||||||||||||||||||||||||
| 4.2 | −29.3173 | − | 50.7791i | −121.500 | + | 210.444i | −695.013 | + | 1203.80i | −3860.97 | − | 6687.40i | 14248.2 | −3025.67 | − | 44364.1i | −38580.1 | −29524.5 | − | 51137.9i | −226387. | + | 392114.i | |||||||||||||||||||||||||||||||||||||||||||||||
| 4.3 | −11.1356 | − | 19.2874i | −121.500 | + | 210.444i | 775.998 | − | 1344.07i | 6671.24 | + | 11554.9i | 5411.89 | −19453.6 | − | 39986.1i | −80176.1 | −29524.5 | − | 51137.9i | 148576. | − | 257341.i | |||||||||||||||||||||||||||||||||||||||||||||||
| 4.4 | −10.7443 | − | 18.6097i | −121.500 | + | 210.444i | 793.120 | − | 1373.72i | −387.534 | − | 671.228i | 5221.73 | 29537.7 | + | 33239.3i | −78094.8 | −29524.5 | − | 51137.9i | −8327.56 | + | 14423.8i | |||||||||||||||||||||||||||||||||||||||||||||||
| 4.5 | 22.0968 | + | 38.2727i | −121.500 | + | 210.444i | 47.4666 | − | 82.2146i | −1604.38 | − | 2778.87i | −10739.0 | 26454.9 | − | 35741.6i | 94703.8 | −29524.5 | − | 51137.9i | 70903.3 | − | 122808.i | |||||||||||||||||||||||||||||||||||||||||||||||
| 4.6 | 27.4471 | + | 47.5397i | −121.500 | + | 210.444i | −482.683 | + | 836.031i | 3859.11 | + | 6684.18i | −13339.3 | −28528.7 | + | 34109.2i | 59430.3 | −29524.5 | − | 51137.9i | −211843. | + | 366922.i | |||||||||||||||||||||||||||||||||||||||||||||||
| 4.7 | 44.3184 | + | 76.7618i | −121.500 | + | 210.444i | −2904.25 | + | 5030.31i | −3645.90 | − | 6314.89i | −21538.8 | 25019.2 | + | 36760.9i | −333319. | −29524.5 | − | 51137.9i | 323162. | − | 559732.i | |||||||||||||||||||||||||||||||||||||||||||||||
| 16.1 | −38.1650 | + | 66.1038i | −121.500 | − | 210.444i | −1889.14 | − | 3272.09i | 2577.44 | − | 4464.27i | 18548.2 | −25394.3 | − | 36502.8i | 132073. | −29524.5 | + | 51137.9i | 196737. | + | 340758.i | |||||||||||||||||||||||||||||||||||||||||||||||
| 16.2 | −29.3173 | + | 50.7791i | −121.500 | − | 210.444i | −695.013 | − | 1203.80i | −3860.97 | + | 6687.40i | 14248.2 | −3025.67 | + | 44364.1i | −38580.1 | −29524.5 | + | 51137.9i | −226387. | − | 392114.i | |||||||||||||||||||||||||||||||||||||||||||||||
| 16.3 | −11.1356 | + | 19.2874i | −121.500 | − | 210.444i | 775.998 | + | 1344.07i | 6671.24 | − | 11554.9i | 5411.89 | −19453.6 | + | 39986.1i | −80176.1 | −29524.5 | + | 51137.9i | 148576. | + | 257341.i | |||||||||||||||||||||||||||||||||||||||||||||||
| 16.4 | −10.7443 | + | 18.6097i | −121.500 | − | 210.444i | 793.120 | + | 1373.72i | −387.534 | + | 671.228i | 5221.73 | 29537.7 | − | 33239.3i | −78094.8 | −29524.5 | + | 51137.9i | −8327.56 | − | 14423.8i | |||||||||||||||||||||||||||||||||||||||||||||||
| 16.5 | 22.0968 | − | 38.2727i | −121.500 | − | 210.444i | 47.4666 | + | 82.2146i | −1604.38 | + | 2778.87i | −10739.0 | 26454.9 | + | 35741.6i | 94703.8 | −29524.5 | + | 51137.9i | 70903.3 | + | 122808.i | |||||||||||||||||||||||||||||||||||||||||||||||
| 16.6 | 27.4471 | − | 47.5397i | −121.500 | − | 210.444i | −482.683 | − | 836.031i | 3859.11 | − | 6684.18i | −13339.3 | −28528.7 | − | 34109.2i | 59430.3 | −29524.5 | + | 51137.9i | −211843. | − | 366922.i | |||||||||||||||||||||||||||||||||||||||||||||||
| 16.7 | 44.3184 | − | 76.7618i | −121.500 | − | 210.444i | −2904.25 | − | 5030.31i | −3645.90 | + | 6314.89i | −21538.8 | 25019.2 | − | 36760.9i | −333319. | −29524.5 | + | 51137.9i | 323162. | + | 559732.i | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 21.12.e.a | ✓ | 14 |
| 3.b | odd | 2 | 1 | 63.12.e.c | 14 | ||
| 7.c | even | 3 | 1 | inner | 21.12.e.a | ✓ | 14 |
| 7.c | even | 3 | 1 | 147.12.a.j | 7 | ||
| 7.d | odd | 6 | 1 | 147.12.a.i | 7 | ||
| 21.h | odd | 6 | 1 | 63.12.e.c | 14 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.12.e.a | ✓ | 14 | 1.a | even | 1 | 1 | trivial |
| 21.12.e.a | ✓ | 14 | 7.c | even | 3 | 1 | inner |
| 63.12.e.c | 14 | 3.b | odd | 2 | 1 | ||
| 63.12.e.c | 14 | 21.h | odd | 6 | 1 | ||
| 147.12.a.i | 7 | 7.d | odd | 6 | 1 | ||
| 147.12.a.j | 7 | 7.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{14} - 9 T_{2}^{13} + 11563 T_{2}^{12} + 83394 T_{2}^{11} + 95377024 T_{2}^{10} + \cdots + 21\!\cdots\!00 \)
acting on \(S_{12}^{\mathrm{new}}(21, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} + \cdots + 21\!\cdots\!00 \)
|
| $3$ |
\( (T^{2} + 243 T + 59049)^{7} \)
|
| $5$ |
\( T^{14} + \cdots + 55\!\cdots\!00 \)
|
| $7$ |
\( T^{14} + \cdots + 11\!\cdots\!07 \)
|
| $11$ |
\( T^{14} + \cdots + 38\!\cdots\!00 \)
|
| $13$ |
\( (T^{7} + \cdots - 28\!\cdots\!12)^{2} \)
|
| $17$ |
\( T^{14} + \cdots + 84\!\cdots\!00 \)
|
| $19$ |
\( T^{14} + \cdots + 48\!\cdots\!00 \)
|
| $23$ |
\( T^{14} + \cdots + 29\!\cdots\!00 \)
|
| $29$ |
\( (T^{7} + \cdots + 44\!\cdots\!16)^{2} \)
|
| $31$ |
\( T^{14} + \cdots + 33\!\cdots\!25 \)
|
| $37$ |
\( T^{14} + \cdots + 60\!\cdots\!04 \)
|
| $41$ |
\( (T^{7} + \cdots - 15\!\cdots\!00)^{2} \)
|
| $43$ |
\( (T^{7} + \cdots + 24\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{14} + \cdots + 61\!\cdots\!00 \)
|
| $53$ |
\( T^{14} + \cdots + 39\!\cdots\!00 \)
|
| $59$ |
\( T^{14} + \cdots + 46\!\cdots\!00 \)
|
| $61$ |
\( T^{14} + \cdots + 60\!\cdots\!00 \)
|
| $67$ |
\( T^{14} + \cdots + 88\!\cdots\!00 \)
|
| $71$ |
\( (T^{7} + \cdots + 22\!\cdots\!56)^{2} \)
|
| $73$ |
\( T^{14} + \cdots + 56\!\cdots\!36 \)
|
| $79$ |
\( T^{14} + \cdots + 28\!\cdots\!09 \)
|
| $83$ |
\( (T^{7} + \cdots - 53\!\cdots\!64)^{2} \)
|
| $89$ |
\( T^{14} + \cdots + 60\!\cdots\!44 \)
|
| $97$ |
\( (T^{7} + \cdots - 69\!\cdots\!00)^{2} \)
|
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