Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [186,4,Mod(97,186)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(186, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 6]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("186.97");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 186 = 2 \cdot 3 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 186.f (of order \(5\), 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: | \(10.9743552611\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 3 x^{15} + 146 x^{14} - 489 x^{13} + 9589 x^{12} - 36007 x^{11} + 235459 x^{10} + \cdots + 8212890625 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{4}\cdot 11^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 3 x^{15} + 146 x^{14} - 489 x^{13} + 9589 x^{12} - 36007 x^{11} + 235459 x^{10} + \cdots + 8212890625 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 11\!\cdots\!58 \nu^{15} + \cdots + 76\!\cdots\!75 ) / 27\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 12\!\cdots\!56 \nu^{15} + \cdots - 19\!\cdots\!75 ) / 29\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 19\!\cdots\!46 \nu^{15} + \cdots - 37\!\cdots\!25 ) / 27\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 21\!\cdots\!34 \nu^{15} + \cdots + 16\!\cdots\!75 ) / 27\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 22\!\cdots\!23 \nu^{15} + \cdots - 19\!\cdots\!00 ) / 86\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 94\!\cdots\!86 \nu^{15} + \cdots - 70\!\cdots\!50 ) / 29\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 11\!\cdots\!24 \nu^{15} + \cdots - 65\!\cdots\!25 ) / 29\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 11\!\cdots\!89 \nu^{15} + \cdots + 53\!\cdots\!50 ) / 29\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 17\!\cdots\!17 \nu^{15} + \cdots - 12\!\cdots\!00 ) / 29\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 59\!\cdots\!08 \nu^{15} + \cdots + 11\!\cdots\!75 ) / 86\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 12\!\cdots\!37 \nu^{15} + \cdots + 10\!\cdots\!25 ) / 17\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 12\!\cdots\!06 \nu^{15} + \cdots + 10\!\cdots\!25 ) / 17\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 23\!\cdots\!96 \nu^{15} + \cdots + 24\!\cdots\!50 ) / 29\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 13\!\cdots\!11 \nu^{15} + \cdots + 11\!\cdots\!25 ) / 86\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 16\!\cdots\!96 \nu^{15} + \cdots + 22\!\cdots\!00 ) / 86\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( 5 \beta_{14} + 9 \beta_{12} + 2 \beta_{11} - 2 \beta_{10} + 4 \beta_{9} - 2 \beta_{8} + 5 \beta_{7} + \cdots + 11 ) / 33 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 6 \beta_{15} - 2 \beta_{14} - 7 \beta_{13} - 6 \beta_{12} + 6 \beta_{11} - 28 \beta_{9} + 13 \beta_{8} + \cdots - 149 ) / 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 346 \beta_{15} + 180 \beta_{13} - 361 \beta_{12} + 152 \beta_{11} + 118 \beta_{10} - 34 \beta_{8} + \cdots + 236 ) / 33 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 40 \beta_{15} + 449 \beta_{14} + 1530 \beta_{13} + 449 \beta_{12} - 1417 \beta_{11} + 632 \beta_{10} + \cdots - 40 ) / 11 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 35341 \beta_{15} - 35341 \beta_{14} - 46095 \beta_{13} + 5850 \beta_{10} - 46095 \beta_{9} + \cdots - 70200 ) / 33 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 44329 \beta_{14} - 54567 \beta_{12} + 133217 \beta_{11} - 133217 \beta_{10} - 134560 \beta_{9} + \cdots + 1900162 ) / 11 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 3501594 \beta_{15} + 6045379 \beta_{14} + 4397857 \beta_{13} + 3501594 \beta_{12} - 274469 \beta_{11} + \cdots + 12306918 ) / 33 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 1737903 \beta_{15} - 10441646 \beta_{13} + 4257723 \beta_{12} - 6750730 \beta_{11} + 12391290 \beta_{10} + \cdots - 331543907 ) / 11 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 231609284 \beta_{15} - 574219275 \beta_{14} - 232038932 \beta_{13} - 574219275 \beta_{12} + \cdots - 231609284 ) / 33 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 422204470 \beta_{15} + 422204470 \beta_{14} + 1403492922 \beta_{13} - 667010636 \beta_{10} + \cdots + 31540806655 ) / 11 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 3021312001 \beta_{14} + 4982796176 \beta_{12} - 47730899 \beta_{11} + 47730899 \beta_{10} + \cdots - 8730824756 ) / 3 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 42522160266 \beta_{15} - 66338794138 \beta_{14} - 138897208646 \beta_{13} - 42522160266 \beta_{12} + \cdots - 1832491883181 ) / 11 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 2037572558919 \beta_{15} + 2367046485374 \beta_{13} - 3209341235909 \beta_{12} + \cdots + 17662515973450 ) / 33 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 2533879131809 \beta_{15} + 6842947016448 \beta_{14} + 8667863464577 \beta_{13} + \cdots + 2533879131809 ) / 11 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 309266239715097 \beta_{15} - 309266239715097 \beta_{14} - 387387773169533 \beta_{13} + \cdots - 20\!\cdots\!20 ) / 33 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/186\mathbb{Z}\right)^\times\).
| \(n\) | \(125\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{1} - \beta_{3} + \beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 97.1 |
|
0.618034 | + | 1.90211i | −0.927051 | + | 2.85317i | −3.23607 | + | 2.35114i | −21.1256 | −6.00000 | 24.4941 | − | 17.7960i | −6.47214 | − | 4.70228i | −7.28115 | − | 5.29007i | −13.0563 | − | 40.1833i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.2 | 0.618034 | + | 1.90211i | −0.927051 | + | 2.85317i | −3.23607 | + | 2.35114i | −5.01840 | −6.00000 | −14.0322 | + | 10.1950i | −6.47214 | − | 4.70228i | −7.28115 | − | 5.29007i | −3.10154 | − | 9.54556i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.3 | 0.618034 | + | 1.90211i | −0.927051 | + | 2.85317i | −3.23607 | + | 2.35114i | 6.04356 | −6.00000 | −1.08488 | + | 0.788211i | −6.47214 | − | 4.70228i | −7.28115 | − | 5.29007i | 3.73513 | + | 11.4955i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.4 | 0.618034 | + | 1.90211i | −0.927051 | + | 2.85317i | −3.23607 | + | 2.35114i | 12.4824 | −6.00000 | 23.5845 | − | 17.1352i | −6.47214 | − | 4.70228i | −7.28115 | − | 5.29007i | 7.71456 | + | 23.7430i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.1 | −1.61803 | + | 1.17557i | 2.42705 | + | 1.76336i | 1.23607 | − | 3.80423i | −15.0956 | −6.00000 | 6.46176 | − | 19.8873i | 2.47214 | + | 7.60845i | 2.78115 | + | 8.55951i | 24.4252 | − | 17.7460i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.2 | −1.61803 | + | 1.17557i | 2.42705 | + | 1.76336i | 1.23607 | − | 3.80423i | −8.76806 | −6.00000 | −6.43450 | + | 19.8034i | 2.47214 | + | 7.60845i | 2.78115 | + | 8.55951i | 14.1870 | − | 10.3075i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.3 | −1.61803 | + | 1.17557i | 2.42705 | + | 1.76336i | 1.23607 | − | 3.80423i | 2.62099 | −6.00000 | 6.74371 | − | 20.7550i | 2.47214 | + | 7.60845i | 2.78115 | + | 8.55951i | −4.24085 | + | 3.08116i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 109.4 | −1.61803 | + | 1.17557i | 2.42705 | + | 1.76336i | 1.23607 | − | 3.80423i | 15.8607 | −6.00000 | −6.23247 | + | 19.1816i | 2.47214 | + | 7.60845i | 2.78115 | + | 8.55951i | −25.6632 | + | 18.6454i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 157.1 | −1.61803 | − | 1.17557i | 2.42705 | − | 1.76336i | 1.23607 | + | 3.80423i | −15.0956 | −6.00000 | 6.46176 | + | 19.8873i | 2.47214 | − | 7.60845i | 2.78115 | − | 8.55951i | 24.4252 | + | 17.7460i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 157.2 | −1.61803 | − | 1.17557i | 2.42705 | − | 1.76336i | 1.23607 | + | 3.80423i | −8.76806 | −6.00000 | −6.43450 | − | 19.8034i | 2.47214 | − | 7.60845i | 2.78115 | − | 8.55951i | 14.1870 | + | 10.3075i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 157.3 | −1.61803 | − | 1.17557i | 2.42705 | − | 1.76336i | 1.23607 | + | 3.80423i | 2.62099 | −6.00000 | 6.74371 | + | 20.7550i | 2.47214 | − | 7.60845i | 2.78115 | − | 8.55951i | −4.24085 | − | 3.08116i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 157.4 | −1.61803 | − | 1.17557i | 2.42705 | − | 1.76336i | 1.23607 | + | 3.80423i | 15.8607 | −6.00000 | −6.23247 | − | 19.1816i | 2.47214 | − | 7.60845i | 2.78115 | − | 8.55951i | −25.6632 | − | 18.6454i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.1 | 0.618034 | − | 1.90211i | −0.927051 | − | 2.85317i | −3.23607 | − | 2.35114i | −21.1256 | −6.00000 | 24.4941 | + | 17.7960i | −6.47214 | + | 4.70228i | −7.28115 | + | 5.29007i | −13.0563 | + | 40.1833i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.2 | 0.618034 | − | 1.90211i | −0.927051 | − | 2.85317i | −3.23607 | − | 2.35114i | −5.01840 | −6.00000 | −14.0322 | − | 10.1950i | −6.47214 | + | 4.70228i | −7.28115 | + | 5.29007i | −3.10154 | + | 9.54556i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.3 | 0.618034 | − | 1.90211i | −0.927051 | − | 2.85317i | −3.23607 | − | 2.35114i | 6.04356 | −6.00000 | −1.08488 | − | 0.788211i | −6.47214 | + | 4.70228i | −7.28115 | + | 5.29007i | 3.73513 | − | 11.4955i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.4 | 0.618034 | − | 1.90211i | −0.927051 | − | 2.85317i | −3.23607 | − | 2.35114i | 12.4824 | −6.00000 | 23.5845 | + | 17.1352i | −6.47214 | + | 4.70228i | −7.28115 | + | 5.29007i | 7.71456 | − | 23.7430i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.d | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 186.4.f.c | ✓ | 16 |
| 31.d | even | 5 | 1 | inner | 186.4.f.c | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 186.4.f.c | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 186.4.f.c | ✓ | 16 | 31.d | even | 5 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 13 T_{5}^{7} - 529 T_{5}^{6} - 5111 T_{5}^{5} + 83371 T_{5}^{4} + 523230 T_{5}^{3} + \cdots + 44005680 \)
acting on \(S_{4}^{\mathrm{new}}(186, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{4} \)
|
| $3$ |
\( (T^{4} - 3 T^{3} + 9 T^{2} + \cdots + 81)^{4} \)
|
| $5$ |
\( (T^{8} + 13 T^{7} + \cdots + 44005680)^{2} \)
|
| $7$ |
\( T^{16} + \cdots + 15\!\cdots\!01 \)
|
| $11$ |
\( T^{16} + \cdots + 35\!\cdots\!96 \)
|
| $13$ |
\( T^{16} + \cdots + 48\!\cdots\!01 \)
|
| $17$ |
\( T^{16} + \cdots + 47\!\cdots\!00 \)
|
| $19$ |
\( T^{16} + \cdots + 73\!\cdots\!25 \)
|
| $23$ |
\( T^{16} + \cdots + 33\!\cdots\!00 \)
|
| $29$ |
\( T^{16} + \cdots + 18\!\cdots\!36 \)
|
| $31$ |
\( T^{16} + \cdots + 62\!\cdots\!21 \)
|
| $37$ |
\( (T^{8} + \cdots - 29\!\cdots\!20)^{2} \)
|
| $41$ |
\( T^{16} + \cdots + 35\!\cdots\!00 \)
|
| $43$ |
\( T^{16} + \cdots + 12\!\cdots\!25 \)
|
| $47$ |
\( T^{16} + \cdots + 23\!\cdots\!00 \)
|
| $53$ |
\( T^{16} + \cdots + 89\!\cdots\!00 \)
|
| $59$ |
\( T^{16} + \cdots + 14\!\cdots\!56 \)
|
| $61$ |
\( (T^{8} + \cdots + 40\!\cdots\!95)^{2} \)
|
| $67$ |
\( (T^{8} + \cdots - 98\!\cdots\!04)^{2} \)
|
| $71$ |
\( T^{16} + \cdots + 14\!\cdots\!00 \)
|
| $73$ |
\( T^{16} + \cdots + 10\!\cdots\!81 \)
|
| $79$ |
\( T^{16} + \cdots + 34\!\cdots\!81 \)
|
| $83$ |
\( T^{16} + \cdots + 12\!\cdots\!00 \)
|
| $89$ |
\( T^{16} + \cdots + 54\!\cdots\!96 \)
|
| $97$ |
\( T^{16} + \cdots + 21\!\cdots\!25 \)
|
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