Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [15,11,Mod(7,15)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15, base_ring=CyclotomicField(4))
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("15.7");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 15 = 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 15.f (of order \(4\), 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: | \(9.53035879011\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 4 x^{19} + 8 x^{18} - 49316 x^{17} + 18276332 x^{16} - 230627572 x^{15} + 1992333560 x^{14} + \cdots + 14\!\cdots\!04 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{21}\cdot 3^{34}\cdot 5^{14} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{20} - 4 x^{19} + 8 x^{18} - 49316 x^{17} + 18276332 x^{16} - 230627572 x^{15} + 1992333560 x^{14} + \cdots + 14\!\cdots\!04 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 47\!\cdots\!94 \nu^{19} + \cdots + 75\!\cdots\!44 ) / 31\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 53\!\cdots\!77 \nu^{19} + \cdots + 25\!\cdots\!48 ) / 33\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 53\!\cdots\!28 \nu^{19} + \cdots - 66\!\cdots\!72 ) / 31\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 69\!\cdots\!59 \nu^{19} + \cdots + 34\!\cdots\!84 ) / 21\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 16\!\cdots\!02 \nu^{19} + \cdots + 16\!\cdots\!52 ) / 15\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 14\!\cdots\!82 \nu^{19} + \cdots + 22\!\cdots\!32 ) / 11\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 23\!\cdots\!88 \nu^{19} + \cdots - 20\!\cdots\!12 ) / 11\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 77\!\cdots\!07 \nu^{19} + \cdots + 37\!\cdots\!68 ) / 31\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 26\!\cdots\!99 \nu^{19} + \cdots - 12\!\cdots\!76 ) / 15\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 51\!\cdots\!57 \nu^{19} + \cdots - 55\!\cdots\!64 ) / 20\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 36\!\cdots\!16 \nu^{19} + \cdots - 74\!\cdots\!84 ) / 12\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 86\!\cdots\!09 \nu^{19} + \cdots + 67\!\cdots\!16 ) / 25\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 11\!\cdots\!59 \nu^{19} + \cdots + 40\!\cdots\!16 ) / 25\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 40\!\cdots\!78 \nu^{19} + \cdots - 75\!\cdots\!72 ) / 63\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 59\!\cdots\!69 \nu^{19} + \cdots + 62\!\cdots\!56 ) / 63\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 27\!\cdots\!37 \nu^{19} + \cdots - 12\!\cdots\!12 ) / 25\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 14\!\cdots\!98 \nu^{19} + \cdots - 46\!\cdots\!52 ) / 12\!\cdots\!00 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 29\!\cdots\!51 \nu^{19} + \cdots + 36\!\cdots\!24 ) / 25\!\cdots\!00 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 10\!\cdots\!59 \nu^{19} + \cdots + 54\!\cdots\!16 ) / 25\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{6} + 81\beta_1 ) / 81 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -81\beta_{9} - 6\beta_{8} - 164\beta_{7} - 164\beta_{6} + 438\beta_{3} - 117756\beta_{2} - 438\beta_1 ) / 81 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 11 \beta_{19} + 9 \beta_{18} + 2 \beta_{17} + 9 \beta_{16} + 9 \beta_{15} - 3 \beta_{14} + \cdots + 71030 ) / 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 1917 \beta_{19} - 3834 \beta_{18} - 507 \beta_{17} + 2169 \beta_{15} - 3033 \beta_{14} + \cdots - 293795235 ) / 81 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 13422 \beta_{19} + 416664 \beta_{18} + 921942 \beta_{17} - 416664 \beta_{16} - 25632 \beta_{15} + \cdots + 3321567852 ) / 81 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 2532484 \beta_{19} - 2615180 \beta_{17} + 2339136 \beta_{16} + 2314992 \beta_{15} + 1329096 \beta_{14} + \cdots - 2581360 ) / 9 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 1558330731 \beta_{19} - 1766753289 \beta_{18} + 280052478 \beta_{17} - 1766753289 \beta_{16} + \cdots - 14663812861566 ) / 81 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 47156659461 \beta_{19} + 94313318922 \beta_{18} + 11335635891 \beta_{17} - 49893993657 \beta_{15} + \cdots + 30\!\cdots\!59 ) / 81 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 61221448742 \beta_{19} - 260185352976 \beta_{18} - 395057340258 \beta_{17} + 260185352976 \beta_{16} + \cdots - 22\!\cdots\!08 ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 313839976508808 \beta_{19} + 427914435625320 \beta_{17} - 401941184329872 \beta_{16} + \cdots + 502041471433440 ) / 81 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 18\!\cdots\!39 \beta_{19} + \cdots + 25\!\cdots\!38 ) / 81 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 93\!\cdots\!49 \beta_{19} + \cdots - 42\!\cdots\!43 ) / 9 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 42\!\cdots\!14 \beta_{19} + \cdots + 10\!\cdots\!96 ) / 81 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 39\!\cdots\!08 \beta_{19} + \cdots - 87\!\cdots\!60 ) / 81 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 23\!\cdots\!99 \beta_{19} + \cdots - 47\!\cdots\!02 ) / 9 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 13\!\cdots\!77 \beta_{19} + \cdots + 52\!\cdots\!27 ) / 81 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 79\!\cdots\!78 \beta_{19} + \cdots - 17\!\cdots\!12 ) / 81 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 18\!\cdots\!48 \beta_{19} + \cdots + 53\!\cdots\!80 ) / 3 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 26\!\cdots\!79 \beta_{19} + \cdots + 70\!\cdots\!66 ) / 81 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) | \(11\) |
| \(\chi(n)\) | \(\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−41.2250 | − | 41.2250i | −99.2043 | + | 99.2043i | 2375.00i | 953.956 | − | 2975.83i | 8179.40 | 16792.7 | + | 16792.7i | 55695.2 | − | 55695.2i | − | 19683.0i | −162006. | + | 83352.0i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.2 | −32.7993 | − | 32.7993i | 99.2043 | − | 99.2043i | 1127.59i | −1119.22 | − | 2917.70i | −6507.67 | −16059.1 | − | 16059.1i | 3397.64 | − | 3397.64i | − | 19683.0i | −58989.1 | + | 132408.i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.3 | −28.0945 | − | 28.0945i | 99.2043 | − | 99.2043i | 554.607i | 2550.39 | + | 1805.86i | −5574.20 | 20119.5 | + | 20119.5i | −13187.4 | + | 13187.4i | − | 19683.0i | −20917.3 | − | 122387.i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.4 | −19.3738 | − | 19.3738i | −99.2043 | + | 99.2043i | − | 273.309i | −2998.53 | + | 880.025i | 3843.94 | 3127.99 | + | 3127.99i | −25133.9 | + | 25133.9i | − | 19683.0i | 75142.5 | + | 41043.6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.5 | −12.5555 | − | 12.5555i | −99.2043 | + | 99.2043i | − | 708.721i | 3105.25 | + | 350.813i | 2491.11 | −8580.16 | − | 8580.16i | −21755.1 | + | 21755.1i | − | 19683.0i | −34583.2 | − | 43392.4i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.6 | −2.85456 | − | 2.85456i | 99.2043 | − | 99.2043i | − | 1007.70i | −3074.01 | + | 562.191i | −566.369 | 5862.00 | + | 5862.00i | −5799.62 | + | 5799.62i | − | 19683.0i | 10379.8 | + | 7170.15i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.7 | 11.6866 | + | 11.6866i | 99.2043 | − | 99.2043i | − | 750.847i | 2257.04 | + | 2161.34i | 2318.72 | −21416.5 | − | 21416.5i | 20741.9 | − | 20741.9i | − | 19683.0i | 1118.39 | + | 51635.8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.8 | 20.8144 | + | 20.8144i | −99.2043 | + | 99.2043i | − | 157.526i | −1515.02 | − | 2733.19i | −4129.75 | −6932.43 | − | 6932.43i | 24592.7 | − | 24592.7i | − | 19683.0i | 25355.6 | − | 88423.8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.9 | 29.9381 | + | 29.9381i | 99.2043 | − | 99.2043i | 768.579i | 2095.22 | − | 2318.55i | 5939.98 | 15970.0 | + | 15970.0i | 7646.81 | − | 7646.81i | − | 19683.0i | 132140. | − | 6686.35i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.10 | 42.4637 | + | 42.4637i | −99.2043 | + | 99.2043i | 2582.33i | 3082.93 | + | 511.056i | −8425.16 | −3582.02 | − | 3582.02i | −66172.3 | + | 66172.3i | − | 19683.0i | 109211. | + | 152614.i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.1 | −41.2250 | + | 41.2250i | −99.2043 | − | 99.2043i | − | 2375.00i | 953.956 | + | 2975.83i | 8179.40 | 16792.7 | − | 16792.7i | 55695.2 | + | 55695.2i | 19683.0i | −162006. | − | 83352.0i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.2 | −32.7993 | + | 32.7993i | 99.2043 | + | 99.2043i | − | 1127.59i | −1119.22 | + | 2917.70i | −6507.67 | −16059.1 | + | 16059.1i | 3397.64 | + | 3397.64i | 19683.0i | −58989.1 | − | 132408.i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.3 | −28.0945 | + | 28.0945i | 99.2043 | + | 99.2043i | − | 554.607i | 2550.39 | − | 1805.86i | −5574.20 | 20119.5 | − | 20119.5i | −13187.4 | − | 13187.4i | 19683.0i | −20917.3 | + | 122387.i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.4 | −19.3738 | + | 19.3738i | −99.2043 | − | 99.2043i | 273.309i | −2998.53 | − | 880.025i | 3843.94 | 3127.99 | − | 3127.99i | −25133.9 | − | 25133.9i | 19683.0i | 75142.5 | − | 41043.6i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.5 | −12.5555 | + | 12.5555i | −99.2043 | − | 99.2043i | 708.721i | 3105.25 | − | 350.813i | 2491.11 | −8580.16 | + | 8580.16i | −21755.1 | − | 21755.1i | 19683.0i | −34583.2 | + | 43392.4i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.6 | −2.85456 | + | 2.85456i | 99.2043 | + | 99.2043i | 1007.70i | −3074.01 | − | 562.191i | −566.369 | 5862.00 | − | 5862.00i | −5799.62 | − | 5799.62i | 19683.0i | 10379.8 | − | 7170.15i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.7 | 11.6866 | − | 11.6866i | 99.2043 | + | 99.2043i | 750.847i | 2257.04 | − | 2161.34i | 2318.72 | −21416.5 | + | 21416.5i | 20741.9 | + | 20741.9i | 19683.0i | 1118.39 | − | 51635.8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.8 | 20.8144 | − | 20.8144i | −99.2043 | − | 99.2043i | 157.526i | −1515.02 | + | 2733.19i | −4129.75 | −6932.43 | + | 6932.43i | 24592.7 | + | 24592.7i | 19683.0i | 25355.6 | + | 88423.8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.9 | 29.9381 | − | 29.9381i | 99.2043 | + | 99.2043i | − | 768.579i | 2095.22 | + | 2318.55i | 5939.98 | 15970.0 | − | 15970.0i | 7646.81 | + | 7646.81i | 19683.0i | 132140. | + | 6686.35i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.10 | 42.4637 | − | 42.4637i | −99.2043 | − | 99.2043i | − | 2582.33i | 3082.93 | − | 511.056i | −8425.16 | −3582.02 | + | 3582.02i | −66172.3 | − | 66172.3i | 19683.0i | 109211. | − | 152614.i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 15.11.f.a | ✓ | 20 |
| 3.b | odd | 2 | 1 | 45.11.g.c | 20 | ||
| 5.b | even | 2 | 1 | 75.11.f.d | 20 | ||
| 5.c | odd | 4 | 1 | inner | 15.11.f.a | ✓ | 20 |
| 5.c | odd | 4 | 1 | 75.11.f.d | 20 | ||
| 15.e | even | 4 | 1 | 45.11.g.c | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.11.f.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 15.11.f.a | ✓ | 20 | 5.c | odd | 4 | 1 | inner |
| 45.11.g.c | 20 | 3.b | odd | 2 | 1 | ||
| 45.11.g.c | 20 | 15.e | even | 4 | 1 | ||
| 75.11.f.d | 20 | 5.b | even | 2 | 1 | ||
| 75.11.f.d | 20 | 5.c | odd | 4 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(15, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + \cdots + 68\!\cdots\!76 \)
|
| $3$ |
\( (T^{4} + 387420489)^{5} \)
|
| $5$ |
\( T^{20} + \cdots + 78\!\cdots\!25 \)
|
| $7$ |
\( T^{20} + \cdots + 53\!\cdots\!00 \)
|
| $11$ |
\( (T^{10} + \cdots + 18\!\cdots\!32)^{2} \)
|
| $13$ |
\( T^{20} + \cdots + 19\!\cdots\!76 \)
|
| $17$ |
\( T^{20} + \cdots + 21\!\cdots\!24 \)
|
| $19$ |
\( T^{20} + \cdots + 16\!\cdots\!00 \)
|
| $23$ |
\( T^{20} + \cdots + 11\!\cdots\!00 \)
|
| $29$ |
\( T^{20} + \cdots + 17\!\cdots\!00 \)
|
| $31$ |
\( (T^{10} + \cdots - 37\!\cdots\!00)^{2} \)
|
| $37$ |
\( T^{20} + \cdots + 10\!\cdots\!00 \)
|
| $41$ |
\( (T^{10} + \cdots + 24\!\cdots\!00)^{2} \)
|
| $43$ |
\( T^{20} + \cdots + 98\!\cdots\!24 \)
|
| $47$ |
\( T^{20} + \cdots + 71\!\cdots\!24 \)
|
| $53$ |
\( T^{20} + \cdots + 50\!\cdots\!00 \)
|
| $59$ |
\( T^{20} + \cdots + 30\!\cdots\!00 \)
|
| $61$ |
\( (T^{10} + \cdots - 79\!\cdots\!24)^{2} \)
|
| $67$ |
\( T^{20} + \cdots + 16\!\cdots\!24 \)
|
| $71$ |
\( (T^{10} + \cdots + 23\!\cdots\!24)^{2} \)
|
| $73$ |
\( T^{20} + \cdots + 61\!\cdots\!00 \)
|
| $79$ |
\( T^{20} + \cdots + 31\!\cdots\!00 \)
|
| $83$ |
\( T^{20} + \cdots + 68\!\cdots\!76 \)
|
| $89$ |
\( T^{20} + \cdots + 70\!\cdots\!00 \)
|
| $97$ |
\( T^{20} + \cdots + 23\!\cdots\!24 \)
|
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