Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [384,10,Mod(193,384)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(384, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("384.193");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 384 = 2^{7} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 384.d (of order \(2\), degree \(1\), 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: | \(197.773761087\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 8 x^{19} - 300700 x^{18} + 6140664 x^{17} + 35387063979 x^{16} - 1130222504088 x^{15} + \cdots + 12\!\cdots\!52 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{175}\cdot 3^{32} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{20} - 8 x^{19} - 300700 x^{18} + 6140664 x^{17} + 35387063979 x^{16} - 1130222504088 x^{15} + \cdots + 12\!\cdots\!52 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 18\!\cdots\!92 \nu^{19} + \cdots + 47\!\cdots\!08 ) / 23\!\cdots\!20 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 39\!\cdots\!29 \nu^{19} + \cdots - 10\!\cdots\!28 ) / 22\!\cdots\!90 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 23\!\cdots\!12 \nu^{19} + \cdots - 95\!\cdots\!09 ) / 14\!\cdots\!35 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 73\!\cdots\!39 \nu^{19} + \cdots + 21\!\cdots\!42 ) / 31\!\cdots\!30 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 91\!\cdots\!09 \nu^{19} + \cdots - 73\!\cdots\!32 ) / 28\!\cdots\!70 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 11\!\cdots\!59 \nu^{19} + \cdots - 86\!\cdots\!00 ) / 40\!\cdots\!10 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 15\!\cdots\!44 \nu^{19} + \cdots - 38\!\cdots\!36 ) / 25\!\cdots\!40 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 23\!\cdots\!90 \nu^{19} + \cdots + 12\!\cdots\!56 ) / 19\!\cdots\!80 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 27\!\cdots\!64 \nu^{19} + \cdots - 13\!\cdots\!08 ) / 11\!\cdots\!80 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 63\!\cdots\!13 \nu^{19} + \cdots - 16\!\cdots\!20 ) / 17\!\cdots\!20 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 14\!\cdots\!96 \nu^{19} + \cdots + 30\!\cdots\!80 ) / 31\!\cdots\!65 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 14\!\cdots\!68 \nu^{19} + \cdots + 21\!\cdots\!76 ) / 16\!\cdots\!40 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 48\!\cdots\!81 \nu^{19} + \cdots + 10\!\cdots\!60 ) / 11\!\cdots\!80 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 10\!\cdots\!32 \nu^{19} + \cdots - 27\!\cdots\!28 ) / 22\!\cdots\!60 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 15\!\cdots\!80 \nu^{19} + \cdots - 55\!\cdots\!76 ) / 19\!\cdots\!80 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 35\!\cdots\!72 \nu^{19} + \cdots - 88\!\cdots\!12 ) / 22\!\cdots\!60 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 51\!\cdots\!13 \nu^{19} + \cdots + 15\!\cdots\!80 ) / 19\!\cdots\!60 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 69\!\cdots\!92 \nu^{19} + \cdots - 23\!\cdots\!56 ) / 25\!\cdots\!40 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 91\!\cdots\!87 \nu^{19} + \cdots - 22\!\cdots\!40 ) / 11\!\cdots\!80 \)
|
| \(\nu\) | \(=\) |
\( ( 2 \beta_{16} - \beta_{15} + 2 \beta_{14} + 81 \beta_{11} + 79 \beta_{7} - 9 \beta_{5} + \cdots + 530686 ) / 1327104 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 192 \beta_{19} - 694 \beta_{16} - 325 \beta_{15} + 4490 \beta_{14} - 1280 \beta_{13} + \cdots + 79820576560 ) / 2654208 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 10368 \beta_{19} + 27216 \beta_{18} + 27216 \beta_{17} + 215168 \beta_{16} - 128968 \beta_{15} + \cdots - 743501056640 ) / 1327104 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 12849408 \beta_{19} - 6912864 \beta_{18} - 6912864 \beta_{17} - 40858082 \beta_{16} + \cdots + 25\!\cdots\!00 ) / 1327104 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2853705600 \beta_{19} + 7327635840 \beta_{18} + 7327272960 \beta_{17} + 50167925846 \beta_{16} + \cdots - 18\!\cdots\!20 ) / 2654208 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 2719601510784 \beta_{19} - 1659649479264 \beta_{18} - 1659511221984 \beta_{17} + \cdots + 36\!\cdots\!48 ) / 2654208 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 161649862732512 \beta_{19} + 372952919493504 \beta_{18} + 372901627312704 \beta_{17} + \cdots - 89\!\cdots\!64 ) / 1327104 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 21\!\cdots\!48 \beta_{19} + \cdots + 21\!\cdots\!98 ) / 221184 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 33\!\cdots\!32 \beta_{19} + \cdots - 16\!\cdots\!08 ) / 2654208 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 22\!\cdots\!40 \beta_{19} + \cdots + 17\!\cdots\!16 ) / 2654208 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 15\!\cdots\!64 \beta_{19} + \cdots - 68\!\cdots\!72 ) / 1327104 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 98\!\cdots\!16 \beta_{19} + \cdots + 63\!\cdots\!12 ) / 1327104 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 29\!\cdots\!56 \beta_{19} + \cdots - 11\!\cdots\!96 ) / 2654208 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 16\!\cdots\!92 \beta_{19} + \cdots + 89\!\cdots\!24 ) / 2654208 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 12\!\cdots\!44 \beta_{19} + \cdots - 46\!\cdots\!16 ) / 1327104 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 11\!\cdots\!28 \beta_{19} + \cdots + 53\!\cdots\!62 ) / 221184 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 22\!\cdots\!40 \beta_{19} + \cdots - 73\!\cdots\!44 ) / 2654208 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 10\!\cdots\!60 \beta_{19} + \cdots + 45\!\cdots\!24 ) / 2654208 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( 95\!\cdots\!40 \beta_{19} + \cdots - 28\!\cdots\!88 ) / 1327104 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(133\) | \(257\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 193.1 |
|
0 | − | 81.0000i | 0 | − | 2358.11i | 0 | −1310.82 | 0 | −6561.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.2 | 0 | − | 81.0000i | 0 | − | 1825.80i | 0 | 10705.0 | 0 | −6561.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.3 | 0 | − | 81.0000i | 0 | − | 1655.59i | 0 | −11370.0 | 0 | −6561.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.4 | 0 | − | 81.0000i | 0 | − | 573.512i | 0 | 2185.20 | 0 | −6561.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.5 | 0 | − | 81.0000i | 0 | − | 471.295i | 0 | −3820.46 | 0 | −6561.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.6 | 0 | − | 81.0000i | 0 | 471.295i | 0 | 3820.46 | 0 | −6561.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.7 | 0 | − | 81.0000i | 0 | 573.512i | 0 | −2185.20 | 0 | −6561.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.8 | 0 | − | 81.0000i | 0 | 1655.59i | 0 | 11370.0 | 0 | −6561.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.9 | 0 | − | 81.0000i | 0 | 1825.80i | 0 | −10705.0 | 0 | −6561.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.10 | 0 | − | 81.0000i | 0 | 2358.11i | 0 | 1310.82 | 0 | −6561.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.11 | 0 | 81.0000i | 0 | − | 2358.11i | 0 | 1310.82 | 0 | −6561.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.12 | 0 | 81.0000i | 0 | − | 1825.80i | 0 | −10705.0 | 0 | −6561.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.13 | 0 | 81.0000i | 0 | − | 1655.59i | 0 | 11370.0 | 0 | −6561.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.14 | 0 | 81.0000i | 0 | − | 573.512i | 0 | −2185.20 | 0 | −6561.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.15 | 0 | 81.0000i | 0 | − | 471.295i | 0 | 3820.46 | 0 | −6561.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.16 | 0 | 81.0000i | 0 | 471.295i | 0 | −3820.46 | 0 | −6561.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.17 | 0 | 81.0000i | 0 | 573.512i | 0 | 2185.20 | 0 | −6561.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.18 | 0 | 81.0000i | 0 | 1655.59i | 0 | −11370.0 | 0 | −6561.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.19 | 0 | 81.0000i | 0 | 1825.80i | 0 | 10705.0 | 0 | −6561.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.20 | 0 | 81.0000i | 0 | 2358.11i | 0 | −1310.82 | 0 | −6561.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
| 8.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 384.10.d.f | ✓ | 20 |
| 4.b | odd | 2 | 1 | inner | 384.10.d.f | ✓ | 20 |
| 8.b | even | 2 | 1 | inner | 384.10.d.f | ✓ | 20 |
| 8.d | odd | 2 | 1 | inner | 384.10.d.f | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 384.10.d.f | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 384.10.d.f | ✓ | 20 | 4.b | odd | 2 | 1 | inner |
| 384.10.d.f | ✓ | 20 | 8.b | even | 2 | 1 | inner |
| 384.10.d.f | ✓ | 20 | 8.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(384, [\chi])\):
|
\( T_{5}^{10} + 12186280 T_{5}^{8} + 49400434717312 T_{5}^{6} + \cdots + 37\!\cdots\!00 \)
|
|
\( T_{7}^{10} - 264961960 T_{7}^{8} + \cdots - 17\!\cdots\!00 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( (T^{2} + 6561)^{10} \)
|
| $5$ |
\( (T^{10} + \cdots + 37\!\cdots\!00)^{2} \)
|
| $7$ |
\( (T^{10} + \cdots - 17\!\cdots\!00)^{2} \)
|
| $11$ |
\( (T^{10} + \cdots + 25\!\cdots\!00)^{2} \)
|
| $13$ |
\( (T^{10} + \cdots + 21\!\cdots\!68)^{2} \)
|
| $17$ |
\( (T^{5} + \cdots + 23\!\cdots\!36)^{4} \)
|
| $19$ |
\( (T^{10} + \cdots + 44\!\cdots\!36)^{2} \)
|
| $23$ |
\( (T^{10} + \cdots - 45\!\cdots\!88)^{2} \)
|
| $29$ |
\( (T^{10} + \cdots + 10\!\cdots\!32)^{2} \)
|
| $31$ |
\( (T^{10} + \cdots - 14\!\cdots\!92)^{2} \)
|
| $37$ |
\( (T^{10} + \cdots + 24\!\cdots\!00)^{2} \)
|
| $41$ |
\( (T^{5} + \cdots - 30\!\cdots\!00)^{4} \)
|
| $43$ |
\( (T^{10} + \cdots + 66\!\cdots\!36)^{2} \)
|
| $47$ |
\( (T^{10} + \cdots - 17\!\cdots\!72)^{2} \)
|
| $53$ |
\( (T^{10} + \cdots + 36\!\cdots\!48)^{2} \)
|
| $59$ |
\( (T^{10} + \cdots + 21\!\cdots\!76)^{2} \)
|
| $61$ |
\( (T^{10} + \cdots + 16\!\cdots\!00)^{2} \)
|
| $67$ |
\( (T^{10} + \cdots + 36\!\cdots\!36)^{2} \)
|
| $71$ |
\( (T^{10} + \cdots - 86\!\cdots\!52)^{2} \)
|
| $73$ |
\( (T^{5} + \cdots - 10\!\cdots\!04)^{4} \)
|
| $79$ |
\( (T^{10} + \cdots - 18\!\cdots\!12)^{2} \)
|
| $83$ |
\( (T^{10} + \cdots + 57\!\cdots\!56)^{2} \)
|
| $89$ |
\( (T^{5} + \cdots + 12\!\cdots\!84)^{4} \)
|
| $97$ |
\( (T^{5} + \cdots - 23\!\cdots\!04)^{4} \)
|
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