Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [26,9,Mod(7,26)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(26, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([11]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("26.7");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 26 = 2 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 26.f (of order \(12\), degree \(4\), 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: | \(10.5918438616\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} + 29976 x^{18} + 369207906 x^{16} + 2417155536656 x^{14} + \cdots + 14\!\cdots\!36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3^{8}\cdot 13^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} + 29976 x^{18} + 369207906 x^{16} + 2417155536656 x^{14} + \cdots + 14\!\cdots\!36 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 79\!\cdots\!25 \nu^{19} + \cdots + 22\!\cdots\!44 ) / 23\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 31\!\cdots\!45 \nu^{19} + \cdots + 23\!\cdots\!28 \nu ) / 45\!\cdots\!20 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 79\!\cdots\!25 \nu^{19} + \cdots - 41\!\cdots\!24 ) / 11\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 79\!\cdots\!25 \nu^{19} + \cdots + 41\!\cdots\!24 ) / 11\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 93\!\cdots\!91 \nu^{19} + \cdots - 19\!\cdots\!00 ) / 39\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 34\!\cdots\!01 \nu^{19} + \cdots - 35\!\cdots\!00 ) / 41\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 32\!\cdots\!89 \nu^{19} + \cdots - 11\!\cdots\!00 ) / 23\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 23\!\cdots\!37 \nu^{19} + \cdots + 33\!\cdots\!64 ) / 23\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 16\!\cdots\!44 \nu^{19} + \cdots - 43\!\cdots\!76 ) / 35\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 40\!\cdots\!99 \nu^{19} + \cdots - 18\!\cdots\!44 ) / 70\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 40\!\cdots\!99 \nu^{19} + \cdots - 18\!\cdots\!44 ) / 70\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 89\!\cdots\!23 \nu^{19} + \cdots - 86\!\cdots\!72 ) / 47\!\cdots\!20 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 31\!\cdots\!49 \nu^{19} + \cdots - 41\!\cdots\!80 ) / 14\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 31\!\cdots\!99 \nu^{19} + \cdots - 41\!\cdots\!80 ) / 14\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 21\!\cdots\!63 \nu^{19} + \cdots + 37\!\cdots\!28 ) / 89\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 49\!\cdots\!39 \nu^{19} + \cdots - 30\!\cdots\!12 ) / 14\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 58\!\cdots\!35 \nu^{19} + \cdots - 15\!\cdots\!52 ) / 14\!\cdots\!00 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 33\!\cdots\!55 \nu^{19} + \cdots - 15\!\cdots\!56 ) / 70\!\cdots\!00 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 38\!\cdots\!38 \nu^{19} + \cdots + 10\!\cdots\!16 ) / 35\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} + 2\beta_{2} ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{19} + 2 \beta_{17} + 2 \beta_{15} - \beta_{14} - 2 \beta_{13} + \beta_{11} + 2 \beta_{10} + \cdots - 8988 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 36 \beta_{19} + 11 \beta_{18} + 11 \beta_{17} - 11 \beta_{16} + \beta_{14} - 48 \beta_{13} + \cdots - 69626 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 6946 \beta_{19} - 684 \beta_{18} - 12590 \beta_{17} - 684 \beta_{16} - 13892 \beta_{15} + \cdots + 47972184 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 375642 \beta_{19} - 124259 \beta_{18} - 29966 \beta_{17} + 124259 \beta_{16} + 110531 \beta_{14} + \cdots + 805718021 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 45704251 \beta_{19} + 12501657 \beta_{18} + 82553807 \beta_{17} + 12501657 \beta_{16} + \cdots - 294625601421 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 3338233497 \beta_{19} + 1207810682 \beta_{18} - 244583227 \beta_{17} - 1207810682 \beta_{16} + \cdots - 7444681862282 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 305332473100 \beta_{19} - 151265245968 \beta_{18} - 575905887644 \beta_{17} - 151265245968 \beta_{16} + \cdots + 19\!\cdots\!46 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 27900717919668 \beta_{19} - 10964338395581 \beta_{18} + 4974423861382 \beta_{17} + 10964338395581 \beta_{16} + \cdots + 63\!\cdots\!93 ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 21\!\cdots\!95 \beta_{19} + \cdots - 13\!\cdots\!13 ) / 3 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 22\!\cdots\!73 \beta_{19} + \cdots - 51\!\cdots\!18 ) / 3 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 14\!\cdots\!52 \beta_{19} + \cdots + 97\!\cdots\!90 ) / 3 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 18\!\cdots\!64 \beta_{19} + \cdots + 41\!\cdots\!13 ) / 3 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 10\!\cdots\!43 \beta_{19} + \cdots - 72\!\cdots\!17 ) / 3 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 14\!\cdots\!73 \beta_{19} + \cdots - 33\!\cdots\!34 ) / 3 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 81\!\cdots\!48 \beta_{19} + \cdots + 54\!\cdots\!98 ) / 3 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 11\!\cdots\!00 \beta_{19} + \cdots + 26\!\cdots\!69 ) / 3 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( 62\!\cdots\!71 \beta_{19} + \cdots - 42\!\cdots\!21 ) / 3 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 89\!\cdots\!93 \beta_{19} + \cdots - 20\!\cdots\!74 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/26\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) |
| \(\chi(n)\) | \(\beta_{1} + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−10.9282 | + | 2.92820i | −56.5107 | + | 97.8794i | 110.851 | − | 64.0000i | −635.686 | − | 635.686i | 330.950 | − | 1235.12i | −1071.14 | − | 287.011i | −1024.00 | + | 1024.00i | −3106.42 | − | 5380.47i | 8808.33 | + | 5085.49i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.2 | −10.9282 | + | 2.92820i | −41.0220 | + | 71.0521i | 110.851 | − | 64.0000i | 424.520 | + | 424.520i | 240.241 | − | 896.593i | 2644.56 | + | 708.608i | −1024.00 | + | 1024.00i | −85.1037 | − | 147.404i | −5882.32 | − | 3396.16i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.3 | −10.9282 | + | 2.92820i | 6.04717 | − | 10.4740i | 110.851 | − | 64.0000i | 123.967 | + | 123.967i | −35.4147 | + | 132.169i | −1762.46 | − | 472.250i | −1024.00 | + | 1024.00i | 3207.36 | + | 5555.32i | −1717.74 | − | 991.736i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.4 | −10.9282 | + | 2.92820i | 44.2774 | − | 76.6907i | 110.851 | − | 64.0000i | −815.818 | − | 815.818i | −259.306 | + | 967.744i | 1265.85 | + | 339.184i | −1024.00 | + | 1024.00i | −640.471 | − | 1109.33i | 11304.3 | + | 6526.54i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.5 | −10.9282 | + | 2.92820i | 70.5908 | − | 122.267i | 110.851 | − | 64.0000i | 614.411 | + | 614.411i | −413.408 | + | 1542.86i | 2457.17 | + | 658.398i | −1024.00 | + | 1024.00i | −6685.62 | − | 11579.8i | −8513.53 | − | 4915.29i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.1 | 2.92820 | − | 10.9282i | −65.6810 | − | 113.763i | −110.851 | − | 64.0000i | −8.81831 | − | 8.81831i | −1435.55 | + | 384.654i | −1049.82 | − | 3917.97i | −1024.00 | + | 1024.00i | −5347.47 | + | 9262.10i | −122.190 | + | 70.5465i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.2 | 2.92820 | − | 10.9282i | −27.6837 | − | 47.9496i | −110.851 | − | 64.0000i | −604.733 | − | 604.733i | −605.067 | + | 162.127i | 877.488 | + | 3274.83i | −1024.00 | + | 1024.00i | 1747.72 | − | 3027.15i | −8379.42 | + | 4837.86i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.3 | 2.92820 | − | 10.9282i | −16.1706 | − | 28.0084i | −110.851 | − | 64.0000i | 721.790 | + | 721.790i | −353.432 | + | 94.7018i | 557.837 | + | 2081.88i | −1024.00 | + | 1024.00i | 2757.52 | − | 4776.17i | 10001.4 | − | 5774.32i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.4 | 2.92820 | − | 10.9282i | 13.1587 | + | 22.7916i | −110.851 | − | 64.0000i | 142.545 | + | 142.545i | 287.603 | − | 77.0629i | −678.918 | − | 2533.76i | −1024.00 | + | 1024.00i | 2934.20 | − | 5082.18i | 1975.16 | − | 1140.36i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.5 | 2.92820 | − | 10.9282i | 72.9939 | + | 126.429i | −110.851 | − | 64.0000i | 163.823 | + | 163.823i | 1595.38 | − | 427.482i | 181.428 | + | 677.099i | −1024.00 | + | 1024.00i | −7375.71 | + | 12775.1i | 2269.99 | − | 1310.58i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 15.1 | −10.9282 | − | 2.92820i | −56.5107 | − | 97.8794i | 110.851 | + | 64.0000i | −635.686 | + | 635.686i | 330.950 | + | 1235.12i | −1071.14 | + | 287.011i | −1024.00 | − | 1024.00i | −3106.42 | + | 5380.47i | 8808.33 | − | 5085.49i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 15.2 | −10.9282 | − | 2.92820i | −41.0220 | − | 71.0521i | 110.851 | + | 64.0000i | 424.520 | − | 424.520i | 240.241 | + | 896.593i | 2644.56 | − | 708.608i | −1024.00 | − | 1024.00i | −85.1037 | + | 147.404i | −5882.32 | + | 3396.16i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 15.3 | −10.9282 | − | 2.92820i | 6.04717 | + | 10.4740i | 110.851 | + | 64.0000i | 123.967 | − | 123.967i | −35.4147 | − | 132.169i | −1762.46 | + | 472.250i | −1024.00 | − | 1024.00i | 3207.36 | − | 5555.32i | −1717.74 | + | 991.736i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 15.4 | −10.9282 | − | 2.92820i | 44.2774 | + | 76.6907i | 110.851 | + | 64.0000i | −815.818 | + | 815.818i | −259.306 | − | 967.744i | 1265.85 | − | 339.184i | −1024.00 | − | 1024.00i | −640.471 | + | 1109.33i | 11304.3 | − | 6526.54i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 15.5 | −10.9282 | − | 2.92820i | 70.5908 | + | 122.267i | 110.851 | + | 64.0000i | 614.411 | − | 614.411i | −413.408 | − | 1542.86i | 2457.17 | − | 658.398i | −1024.00 | − | 1024.00i | −6685.62 | + | 11579.8i | −8513.53 | + | 4915.29i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.1 | 2.92820 | + | 10.9282i | −65.6810 | + | 113.763i | −110.851 | + | 64.0000i | −8.81831 | + | 8.81831i | −1435.55 | − | 384.654i | −1049.82 | + | 3917.97i | −1024.00 | − | 1024.00i | −5347.47 | − | 9262.10i | −122.190 | − | 70.5465i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.2 | 2.92820 | + | 10.9282i | −27.6837 | + | 47.9496i | −110.851 | + | 64.0000i | −604.733 | + | 604.733i | −605.067 | − | 162.127i | 877.488 | − | 3274.83i | −1024.00 | − | 1024.00i | 1747.72 | + | 3027.15i | −8379.42 | − | 4837.86i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.3 | 2.92820 | + | 10.9282i | −16.1706 | + | 28.0084i | −110.851 | + | 64.0000i | 721.790 | − | 721.790i | −353.432 | − | 94.7018i | 557.837 | − | 2081.88i | −1024.00 | − | 1024.00i | 2757.52 | + | 4776.17i | 10001.4 | + | 5774.32i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.4 | 2.92820 | + | 10.9282i | 13.1587 | − | 22.7916i | −110.851 | + | 64.0000i | 142.545 | − | 142.545i | 287.603 | + | 77.0629i | −678.918 | + | 2533.76i | −1024.00 | − | 1024.00i | 2934.20 | + | 5082.18i | 1975.16 | + | 1140.36i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.5 | 2.92820 | + | 10.9282i | 72.9939 | − | 126.429i | −110.851 | + | 64.0000i | 163.823 | − | 163.823i | 1595.38 | + | 427.482i | 181.428 | − | 677.099i | −1024.00 | − | 1024.00i | −7375.71 | − | 12775.1i | 2269.99 | + | 1310.58i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.f | odd | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 26.9.f.a | ✓ | 20 |
| 13.f | odd | 12 | 1 | inner | 26.9.f.a | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 26.9.f.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 26.9.f.a | ✓ | 20 | 13.f | odd | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{20} + 45399 T_{3}^{18} + 1449036 T_{3}^{17} + 1388097252 T_{3}^{16} + 52578853326 T_{3}^{15} + \cdots + 16\!\cdots\!04 \)
acting on \(S_{9}^{\mathrm{new}}(26, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 16 T^{3} + \cdots + 16384)^{5} \)
|
| $3$ |
\( T^{20} + \cdots + 16\!\cdots\!04 \)
|
| $5$ |
\( T^{20} + \cdots + 23\!\cdots\!00 \)
|
| $7$ |
\( T^{20} + \cdots + 10\!\cdots\!84 \)
|
| $11$ |
\( T^{20} + \cdots + 87\!\cdots\!24 \)
|
| $13$ |
\( T^{20} + \cdots + 13\!\cdots\!01 \)
|
| $17$ |
\( T^{20} + \cdots + 31\!\cdots\!81 \)
|
| $19$ |
\( T^{20} + \cdots + 23\!\cdots\!84 \)
|
| $23$ |
\( T^{20} + \cdots + 37\!\cdots\!44 \)
|
| $29$ |
\( T^{20} + \cdots + 49\!\cdots\!21 \)
|
| $31$ |
\( T^{20} + \cdots + 40\!\cdots\!56 \)
|
| $37$ |
\( T^{20} + \cdots + 67\!\cdots\!29 \)
|
| $41$ |
\( T^{20} + \cdots + 12\!\cdots\!41 \)
|
| $43$ |
\( T^{20} + \cdots + 38\!\cdots\!96 \)
|
| $47$ |
\( T^{20} + \cdots + 26\!\cdots\!96 \)
|
| $53$ |
\( (T^{10} + \cdots - 90\!\cdots\!96)^{2} \)
|
| $59$ |
\( T^{20} + \cdots + 41\!\cdots\!16 \)
|
| $61$ |
\( T^{20} + \cdots + 24\!\cdots\!21 \)
|
| $67$ |
\( T^{20} + \cdots + 31\!\cdots\!76 \)
|
| $71$ |
\( T^{20} + \cdots + 11\!\cdots\!84 \)
|
| $73$ |
\( T^{20} + \cdots + 13\!\cdots\!56 \)
|
| $79$ |
\( (T^{10} + \cdots - 32\!\cdots\!64)^{2} \)
|
| $83$ |
\( T^{20} + \cdots + 27\!\cdots\!24 \)
|
| $89$ |
\( T^{20} + \cdots + 79\!\cdots\!16 \)
|
| $97$ |
\( T^{20} + \cdots + 47\!\cdots\!56 \)
|
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