Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [38,10,Mod(7,38)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(38, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("38.7");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 38 = 2 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 38.c (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: | \(19.5713617742\) |
| 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: |
\( x^{14} - 3 x^{13} + 93094 x^{12} + 3763497 x^{11} + 6525212431 x^{10} + 244392759522 x^{9} + \cdots + 37\!\cdots\!25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3^{6}\cdot 5^{2} \) |
| 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
\( x^{14} - 3 x^{13} + 93094 x^{12} + 3763497 x^{11} + 6525212431 x^{10} + 244392759522 x^{9} + \cdots + 37\!\cdots\!25 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 28\!\cdots\!82 \nu^{13} + \cdots + 29\!\cdots\!50 ) / 24\!\cdots\!75 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13\!\cdots\!34 \nu^{13} + \cdots - 21\!\cdots\!50 ) / 49\!\cdots\!25 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 11\!\cdots\!68 \nu^{13} + \cdots - 78\!\cdots\!25 ) / 31\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 17\!\cdots\!24 \nu^{13} + \cdots + 39\!\cdots\!75 ) / 21\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 50\!\cdots\!63 \nu^{13} + \cdots - 23\!\cdots\!75 ) / 39\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 62\!\cdots\!63 \nu^{13} + \cdots - 57\!\cdots\!25 ) / 23\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 28\!\cdots\!24 \nu^{13} + \cdots + 43\!\cdots\!25 ) / 31\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 41\!\cdots\!97 \nu^{13} + \cdots + 68\!\cdots\!75 ) / 44\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 61\!\cdots\!28 \nu^{13} + \cdots - 86\!\cdots\!75 ) / 31\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 76\!\cdots\!93 \nu^{13} + \cdots - 66\!\cdots\!75 ) / 35\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 10\!\cdots\!77 \nu^{13} + \cdots - 16\!\cdots\!00 ) / 35\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 15\!\cdots\!31 \nu^{13} + \cdots + 13\!\cdots\!25 ) / 35\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{12} + \beta_{9} + 2\beta_{7} - \beta_{6} - \beta_{4} + 32\beta_{3} - 26583\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( -36\beta_{10} - 52\beta_{8} + 35\beta_{6} - 406\beta_{5} - 71\beta_{4} + 47365\beta_{3} - 47365\beta _1 - 845938 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 11448 \beta_{13} + 52030 \beta_{12} - 15702 \beta_{11} - 11448 \beta_{10} - 45184 \beta_{9} + \cdots - 1258162311 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 3830256 \beta_{13} + 6051722 \beta_{12} - 4180162 \beta_{11} + 1545560 \beta_{9} + \cdots + 79216825858 \beta_{2} \)
|
| \(\nu^{6}\) | \(=\) |
\( 1140296184 \beta_{10} + 1542167838 \beta_{8} + 2009199649 \beta_{6} + 14985660806 \beta_{5} + \cdots + 63436713684549 \)
|
| \(\nu^{7}\) | \(=\) |
\( 303066446004 \beta_{13} - 441616433321 \beta_{12} + 310103988814 \beta_{11} + 303066446004 \beta_{10} + \cdots + 60\!\cdots\!72 \)
|
| \(\nu^{8}\) | \(=\) |
\( 86792913157104 \beta_{13} - 164655104212204 \beta_{12} + 114641442952860 \beta_{11} + \cdots - 33\!\cdots\!05 \beta_{2} \)
|
| \(\nu^{9}\) | \(=\) |
\( - 21\!\cdots\!52 \beta_{10} + \cdots - 43\!\cdots\!40 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 59\!\cdots\!80 \beta_{13} + \cdots - 18\!\cdots\!75 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 14\!\cdots\!56 \beta_{13} + \cdots + 29\!\cdots\!34 \beta_{2} \)
|
| \(\nu^{12}\) | \(=\) |
\( 39\!\cdots\!12 \beta_{10} + \cdots + 10\!\cdots\!95 \)
|
| \(\nu^{13}\) | \(=\) |
\( 95\!\cdots\!40 \beta_{13} + \cdots + 19\!\cdots\!70 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).
| \(n\) | \(21\) |
| \(\chi(n)\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
8.00000 | + | 13.8564i | −113.259 | − | 196.170i | −128.000 | + | 221.703i | 563.006 | + | 975.155i | 1812.14 | − | 3138.71i | 3883.37 | −4096.00 | −15813.5 | + | 27389.8i | −9008.10 | + | 15602.5i | ||||||||||||||||||||||||||||||||||||||||||||||
| 7.2 | 8.00000 | + | 13.8564i | −85.2191 | − | 147.604i | −128.000 | + | 221.703i | −867.647 | − | 1502.81i | 1363.51 | − | 2361.66i | −2874.94 | −4096.00 | −4683.10 | + | 8111.37i | 13882.3 | − | 24044.9i | |||||||||||||||||||||||||||||||||||||||||||||||
| 7.3 | 8.00000 | + | 13.8564i | 6.53417 | + | 11.3175i | −128.000 | + | 221.703i | 80.6193 | + | 139.637i | −104.547 | + | 181.080i | −6270.17 | −4096.00 | 9756.11 | − | 16898.1i | −1289.91 | + | 2234.19i | |||||||||||||||||||||||||||||||||||||||||||||||
| 7.4 | 8.00000 | + | 13.8564i | 9.29294 | + | 16.0959i | −128.000 | + | 221.703i | 830.940 | + | 1439.23i | −148.687 | + | 257.534i | 10512.6 | −4096.00 | 9668.78 | − | 16746.8i | −13295.0 | + | 23027.7i | |||||||||||||||||||||||||||||||||||||||||||||||
| 7.5 | 8.00000 | + | 13.8564i | 37.7605 | + | 65.4031i | −128.000 | + | 221.703i | −802.961 | − | 1390.77i | −604.168 | + | 1046.45i | 1586.11 | −4096.00 | 6989.79 | − | 12106.7i | 12847.4 | − | 22252.3i | |||||||||||||||||||||||||||||||||||||||||||||||
| 7.6 | 8.00000 | + | 13.8564i | 110.618 | + | 191.597i | −128.000 | + | 221.703i | 1160.30 | + | 2009.69i | −1769.90 | + | 3065.55i | −7717.55 | −4096.00 | −14631.4 | + | 25342.3i | −18564.7 | + | 32155.1i | |||||||||||||||||||||||||||||||||||||||||||||||
| 7.7 | 8.00000 | + | 13.8564i | 116.772 | + | 202.254i | −128.000 | + | 221.703i | −509.754 | − | 882.920i | −1868.34 | + | 3236.07i | 2726.62 | −4096.00 | −17429.7 | + | 30189.1i | 8156.07 | − | 14126.7i | |||||||||||||||||||||||||||||||||||||||||||||||
| 11.1 | 8.00000 | − | 13.8564i | −113.259 | + | 196.170i | −128.000 | − | 221.703i | 563.006 | − | 975.155i | 1812.14 | + | 3138.71i | 3883.37 | −4096.00 | −15813.5 | − | 27389.8i | −9008.10 | − | 15602.5i | |||||||||||||||||||||||||||||||||||||||||||||||
| 11.2 | 8.00000 | − | 13.8564i | −85.2191 | + | 147.604i | −128.000 | − | 221.703i | −867.647 | + | 1502.81i | 1363.51 | + | 2361.66i | −2874.94 | −4096.00 | −4683.10 | − | 8111.37i | 13882.3 | + | 24044.9i | |||||||||||||||||||||||||||||||||||||||||||||||
| 11.3 | 8.00000 | − | 13.8564i | 6.53417 | − | 11.3175i | −128.000 | − | 221.703i | 80.6193 | − | 139.637i | −104.547 | − | 181.080i | −6270.17 | −4096.00 | 9756.11 | + | 16898.1i | −1289.91 | − | 2234.19i | |||||||||||||||||||||||||||||||||||||||||||||||
| 11.4 | 8.00000 | − | 13.8564i | 9.29294 | − | 16.0959i | −128.000 | − | 221.703i | 830.940 | − | 1439.23i | −148.687 | − | 257.534i | 10512.6 | −4096.00 | 9668.78 | + | 16746.8i | −13295.0 | − | 23027.7i | |||||||||||||||||||||||||||||||||||||||||||||||
| 11.5 | 8.00000 | − | 13.8564i | 37.7605 | − | 65.4031i | −128.000 | − | 221.703i | −802.961 | + | 1390.77i | −604.168 | − | 1046.45i | 1586.11 | −4096.00 | 6989.79 | + | 12106.7i | 12847.4 | + | 22252.3i | |||||||||||||||||||||||||||||||||||||||||||||||
| 11.6 | 8.00000 | − | 13.8564i | 110.618 | − | 191.597i | −128.000 | − | 221.703i | 1160.30 | − | 2009.69i | −1769.90 | − | 3065.55i | −7717.55 | −4096.00 | −14631.4 | − | 25342.3i | −18564.7 | − | 32155.1i | |||||||||||||||||||||||||||||||||||||||||||||||
| 11.7 | 8.00000 | − | 13.8564i | 116.772 | − | 202.254i | −128.000 | − | 221.703i | −509.754 | + | 882.920i | −1868.34 | − | 3236.07i | 2726.62 | −4096.00 | −17429.7 | − | 30189.1i | 8156.07 | + | 14126.7i | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 38.10.c.a | ✓ | 14 |
| 19.c | even | 3 | 1 | inner | 38.10.c.a | ✓ | 14 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.10.c.a | ✓ | 14 | 1.a | even | 1 | 1 | trivial |
| 38.10.c.a | ✓ | 14 | 19.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{14} - 165 T_{3}^{13} + 108646 T_{3}^{12} - 11474337 T_{3}^{11} + 7242607471 T_{3}^{10} + \cdots + 13\!\cdots\!25 \)
acting on \(S_{10}^{\mathrm{new}}(38, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 16 T + 256)^{7} \)
|
| $3$ |
\( T^{14} + \cdots + 13\!\cdots\!25 \)
|
| $5$ |
\( T^{14} + \cdots + 39\!\cdots\!00 \)
|
| $7$ |
\( (T^{7} + \cdots + 24\!\cdots\!96)^{2} \)
|
| $11$ |
\( (T^{7} + \cdots + 10\!\cdots\!52)^{2} \)
|
| $13$ |
\( T^{14} + \cdots + 62\!\cdots\!96 \)
|
| $17$ |
\( T^{14} + \cdots + 59\!\cdots\!84 \)
|
| $19$ |
\( T^{14} + \cdots + 36\!\cdots\!59 \)
|
| $23$ |
\( T^{14} + \cdots + 74\!\cdots\!84 \)
|
| $29$ |
\( T^{14} + \cdots + 16\!\cdots\!00 \)
|
| $31$ |
\( (T^{7} + \cdots - 39\!\cdots\!32)^{2} \)
|
| $37$ |
\( (T^{7} + \cdots + 58\!\cdots\!00)^{2} \)
|
| $41$ |
\( T^{14} + \cdots + 25\!\cdots\!25 \)
|
| $43$ |
\( T^{14} + \cdots + 17\!\cdots\!00 \)
|
| $47$ |
\( T^{14} + \cdots + 27\!\cdots\!00 \)
|
| $53$ |
\( T^{14} + \cdots + 31\!\cdots\!16 \)
|
| $59$ |
\( T^{14} + \cdots + 77\!\cdots\!25 \)
|
| $61$ |
\( T^{14} + \cdots + 77\!\cdots\!96 \)
|
| $67$ |
\( T^{14} + \cdots + 20\!\cdots\!25 \)
|
| $71$ |
\( T^{14} + \cdots + 37\!\cdots\!36 \)
|
| $73$ |
\( T^{14} + \cdots + 13\!\cdots\!25 \)
|
| $79$ |
\( T^{14} + \cdots + 39\!\cdots\!16 \)
|
| $83$ |
\( (T^{7} + \cdots - 53\!\cdots\!64)^{2} \)
|
| $89$ |
\( T^{14} + \cdots + 16\!\cdots\!56 \)
|
| $97$ |
\( T^{14} + \cdots + 12\!\cdots\!25 \)
|
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