Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [14,11,Mod(3,14)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("14.3");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 14 = 2 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 14.d (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: | \(8.89500153743\) |
| 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: |
\( x^{12} + 7168 x^{10} - 191104 x^{9} + 39872585 x^{8} - 837614684 x^{7} + 83400850488 x^{6} + \cdots + 16\!\cdots\!56 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{24}\cdot 3^{4}\cdot 7^{4} \) |
| 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
\( x^{12} + 7168 x^{10} - 191104 x^{9} + 39872585 x^{8} - 837614684 x^{7} + 83400850488 x^{6} + \cdots + 16\!\cdots\!56 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 59\!\cdots\!91 \nu^{11} + \cdots - 20\!\cdots\!92 ) / 74\!\cdots\!80 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 28\!\cdots\!47 \nu^{11} + \cdots - 24\!\cdots\!60 ) / 30\!\cdots\!15 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 36\!\cdots\!01 \nu^{11} + \cdots - 54\!\cdots\!20 ) / 25\!\cdots\!90 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 70\!\cdots\!13 \nu^{11} + \cdots + 11\!\cdots\!36 ) / 40\!\cdots\!20 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 37\!\cdots\!01 \nu^{11} + \cdots - 31\!\cdots\!12 ) / 13\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 13\!\cdots\!72 \nu^{11} + \cdots + 13\!\cdots\!16 ) / 44\!\cdots\!80 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 11\!\cdots\!76 \nu^{11} + \cdots + 96\!\cdots\!64 ) / 13\!\cdots\!40 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 55\!\cdots\!52 \nu^{11} + \cdots - 93\!\cdots\!64 ) / 20\!\cdots\!10 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 19\!\cdots\!74 \nu^{11} + \cdots + 15\!\cdots\!20 ) / 60\!\cdots\!30 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 45\!\cdots\!17 \nu^{11} + \cdots + 49\!\cdots\!28 ) / 12\!\cdots\!60 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 17\!\cdots\!19 \nu^{11} + \cdots - 11\!\cdots\!00 ) / 40\!\cdots\!20 \)
|
| \(\nu\) | \(=\) |
\( ( 2 \beta_{11} + \beta_{10} - 2 \beta_{7} - 17 \beta_{6} - 34 \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 1 ) / 168 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 64 \beta_{11} + 64 \beta_{10} - 20 \beta_{9} - 10 \beta_{8} - 423 \beta_{7} - 591 \beta_{6} + \cdots - 359 ) / 336 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 4429 \beta_{11} - 8858 \beta_{10} + 856 \beta_{9} - 856 \beta_{8} - 3899 \beta_{7} + 131434 \beta_{6} + \cdots + 8030267 ) / 168 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 995464 \beta_{11} + 497732 \beta_{10} + 190930 \beta_{9} + 381860 \beta_{8} + 4750574 \beta_{7} + \cdots - 3177371684 ) / 336 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 23896209 \beta_{11} + 23896209 \beta_{10} - 15725860 \beta_{9} - 7862930 \beta_{8} + \cdots - 26101033 ) / 168 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 3372257800 \beta_{11} - 6744515600 \beta_{10} + 1417094874 \beta_{9} - 1417094874 \beta_{8} + \cdots + 16460698858519 ) / 336 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 286644095922 \beta_{11} + 143322047961 \beta_{10} + 54869325196 \beta_{9} + 109738650392 \beta_{8} + \cdots - 532355126776537 ) / 168 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 21921601335180 \beta_{11} + 21921601335180 \beta_{10} - 18485762966292 \beta_{9} + \cdots - 55527029516471 ) / 336 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 893649686832565 \beta_{11} + \cdots + 35\!\cdots\!64 ) / 168 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 28\!\cdots\!16 \beta_{11} + \cdots - 59\!\cdots\!64 ) / 336 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 56\!\cdots\!77 \beta_{11} + \cdots - 12\!\cdots\!66 ) / 168 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−11.3137 | − | 19.5959i | −304.740 | − | 175.942i | −256.000 | + | 443.405i | −3227.42 | + | 1863.35i | 7962.22i | 4145.11 | − | 16287.8i | 11585.2 | 32386.5 | + | 56095.1i | 73028.2 | + | 42162.9i | ||||||||||||||||||||||||||||||||||||||||
| 3.2 | −11.3137 | − | 19.5959i | 43.3342 | + | 25.0190i | −256.000 | + | 443.405i | 1336.07 | − | 771.378i | − | 1132.23i | 14974.8 | + | 7630.84i | 11585.2 | −28272.6 | − | 48969.6i | −30231.7 | − | 17454.3i | ||||||||||||||||||||||||||||||||||||||||
| 3.3 | −11.3137 | − | 19.5959i | 181.381 | + | 104.720i | −256.000 | + | 443.405i | −1221.89 | + | 705.456i | − | 4739.09i | −16732.6 | − | 1579.95i | 11585.2 | −7591.90 | − | 13149.6i | 27648.1 | + | 15962.6i | ||||||||||||||||||||||||||||||||||||||||
| 3.4 | 11.3137 | + | 19.5959i | −102.202 | − | 59.0063i | −256.000 | + | 443.405i | 32.1973 | − | 18.5891i | − | 2670.32i | −11036.6 | − | 12675.5i | −11585.2 | −22561.0 | − | 39076.8i | 728.542 | + | 420.624i | ||||||||||||||||||||||||||||||||||||||||
| 3.5 | 11.3137 | + | 19.5959i | 173.391 | + | 100.107i | −256.000 | + | 443.405i | 4505.30 | − | 2601.14i | 4530.33i | 7983.44 | + | 14789.9i | −11585.2 | −9481.60 | − | 16422.6i | 101943. | + | 58857.1i | |||||||||||||||||||||||||||||||||||||||||
| 3.6 | 11.3137 | + | 19.5959i | 251.837 | + | 145.398i | −256.000 | + | 443.405i | −4757.26 | + | 2746.61i | 6579.96i | 15953.8 | − | 5286.95i | −11585.2 | 12756.6 | + | 22095.0i | −107645. | − | 62148.6i | |||||||||||||||||||||||||||||||||||||||||
| 5.1 | −11.3137 | + | 19.5959i | −304.740 | + | 175.942i | −256.000 | − | 443.405i | −3227.42 | − | 1863.35i | − | 7962.22i | 4145.11 | + | 16287.8i | 11585.2 | 32386.5 | − | 56095.1i | 73028.2 | − | 42162.9i | ||||||||||||||||||||||||||||||||||||||||
| 5.2 | −11.3137 | + | 19.5959i | 43.3342 | − | 25.0190i | −256.000 | − | 443.405i | 1336.07 | + | 771.378i | 1132.23i | 14974.8 | − | 7630.84i | 11585.2 | −28272.6 | + | 48969.6i | −30231.7 | + | 17454.3i | |||||||||||||||||||||||||||||||||||||||||
| 5.3 | −11.3137 | + | 19.5959i | 181.381 | − | 104.720i | −256.000 | − | 443.405i | −1221.89 | − | 705.456i | 4739.09i | −16732.6 | + | 1579.95i | 11585.2 | −7591.90 | + | 13149.6i | 27648.1 | − | 15962.6i | |||||||||||||||||||||||||||||||||||||||||
| 5.4 | 11.3137 | − | 19.5959i | −102.202 | + | 59.0063i | −256.000 | − | 443.405i | 32.1973 | + | 18.5891i | 2670.32i | −11036.6 | + | 12675.5i | −11585.2 | −22561.0 | + | 39076.8i | 728.542 | − | 420.624i | |||||||||||||||||||||||||||||||||||||||||
| 5.5 | 11.3137 | − | 19.5959i | 173.391 | − | 100.107i | −256.000 | − | 443.405i | 4505.30 | + | 2601.14i | − | 4530.33i | 7983.44 | − | 14789.9i | −11585.2 | −9481.60 | + | 16422.6i | 101943. | − | 58857.1i | ||||||||||||||||||||||||||||||||||||||||
| 5.6 | 11.3137 | − | 19.5959i | 251.837 | − | 145.398i | −256.000 | − | 443.405i | −4757.26 | − | 2746.61i | − | 6579.96i | 15953.8 | + | 5286.95i | −11585.2 | 12756.6 | − | 22095.0i | −107645. | + | 62148.6i | ||||||||||||||||||||||||||||||||||||||||
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 | 14.11.d.a | ✓ | 12 |
| 3.b | odd | 2 | 1 | 126.11.n.b | 12 | ||
| 4.b | odd | 2 | 1 | 112.11.s.a | 12 | ||
| 7.b | odd | 2 | 1 | 98.11.d.a | 12 | ||
| 7.c | even | 3 | 1 | 98.11.b.c | 12 | ||
| 7.c | even | 3 | 1 | 98.11.d.a | 12 | ||
| 7.d | odd | 6 | 1 | inner | 14.11.d.a | ✓ | 12 |
| 7.d | odd | 6 | 1 | 98.11.b.c | 12 | ||
| 21.g | even | 6 | 1 | 126.11.n.b | 12 | ||
| 28.f | even | 6 | 1 | 112.11.s.a | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.11.d.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 14.11.d.a | ✓ | 12 | 7.d | odd | 6 | 1 | inner |
| 98.11.b.c | 12 | 7.c | even | 3 | 1 | ||
| 98.11.b.c | 12 | 7.d | odd | 6 | 1 | ||
| 98.11.d.a | 12 | 7.b | odd | 2 | 1 | ||
| 98.11.d.a | 12 | 7.c | even | 3 | 1 | ||
| 112.11.s.a | 12 | 4.b | odd | 2 | 1 | ||
| 112.11.s.a | 12 | 28.f | even | 6 | 1 | ||
| 126.11.n.b | 12 | 3.b | odd | 2 | 1 | ||
| 126.11.n.b | 12 | 21.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(14, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 512 T^{2} + 262144)^{3} \)
|
| $3$ |
\( T^{12} + \cdots + 64\!\cdots\!29 \)
|
| $5$ |
\( T^{12} + \cdots + 74\!\cdots\!25 \)
|
| $7$ |
\( T^{12} + \cdots + 50\!\cdots\!01 \)
|
| $11$ |
\( T^{12} + \cdots + 29\!\cdots\!09 \)
|
| $13$ |
\( T^{12} + \cdots + 32\!\cdots\!16 \)
|
| $17$ |
\( T^{12} + \cdots + 63\!\cdots\!41 \)
|
| $19$ |
\( T^{12} + \cdots + 30\!\cdots\!25 \)
|
| $23$ |
\( T^{12} + \cdots + 17\!\cdots\!09 \)
|
| $29$ |
\( (T^{6} + \cdots + 13\!\cdots\!88)^{2} \)
|
| $31$ |
\( T^{12} + \cdots + 34\!\cdots\!69 \)
|
| $37$ |
\( T^{12} + \cdots + 38\!\cdots\!61 \)
|
| $41$ |
\( T^{12} + \cdots + 52\!\cdots\!24 \)
|
| $43$ |
\( (T^{6} + \cdots + 12\!\cdots\!72)^{2} \)
|
| $47$ |
\( T^{12} + \cdots + 17\!\cdots\!69 \)
|
| $53$ |
\( T^{12} + \cdots + 82\!\cdots\!09 \)
|
| $59$ |
\( T^{12} + \cdots + 14\!\cdots\!29 \)
|
| $61$ |
\( T^{12} + \cdots + 36\!\cdots\!09 \)
|
| $67$ |
\( T^{12} + \cdots + 23\!\cdots\!81 \)
|
| $71$ |
\( (T^{6} + \cdots - 12\!\cdots\!24)^{2} \)
|
| $73$ |
\( T^{12} + \cdots + 20\!\cdots\!81 \)
|
| $79$ |
\( T^{12} + \cdots + 28\!\cdots\!25 \)
|
| $83$ |
\( T^{12} + \cdots + 14\!\cdots\!56 \)
|
| $89$ |
\( T^{12} + \cdots + 18\!\cdots\!01 \)
|
| $97$ |
\( T^{12} + \cdots + 26\!\cdots\!76 \)
|
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