Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [110,4,Mod(31,110)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(110, 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("110.31");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 110 = 2 \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 110.g (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: | \(6.49021010063\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - x^{11} + 80 x^{10} - 71 x^{9} + 3603 x^{8} - 191 x^{7} + 110280 x^{6} + 142285 x^{5} + \cdots + 2085136 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - x^{11} + 80 x^{10} - 71 x^{9} + 3603 x^{8} - 191 x^{7} + 110280 x^{6} + 142285 x^{5} + \cdots + 2085136 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 48\!\cdots\!35 \nu^{11} + \cdots + 23\!\cdots\!92 ) / 12\!\cdots\!12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 87\!\cdots\!97 \nu^{11} + \cdots - 14\!\cdots\!80 ) / 17\!\cdots\!96 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 52\!\cdots\!83 \nu^{11} + \cdots + 32\!\cdots\!98 ) / 43\!\cdots\!49 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 20\!\cdots\!11 \nu^{11} + \cdots + 32\!\cdots\!32 ) / 12\!\cdots\!12 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 38\!\cdots\!65 \nu^{11} + \cdots - 18\!\cdots\!32 ) / 18\!\cdots\!68 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 37\!\cdots\!71 \nu^{11} + \cdots + 58\!\cdots\!68 ) / 17\!\cdots\!96 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 29\!\cdots\!33 \nu^{11} + \cdots + 17\!\cdots\!64 ) / 12\!\cdots\!12 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 45\!\cdots\!43 \nu^{11} + \cdots + 27\!\cdots\!00 ) / 18\!\cdots\!68 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 33\!\cdots\!33 \nu^{11} + \cdots + 16\!\cdots\!60 ) / 93\!\cdots\!34 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 37\!\cdots\!40 \nu^{11} + \cdots - 71\!\cdots\!60 ) / 93\!\cdots\!34 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{10} - 3\beta_{9} + 6\beta_{8} + 3\beta_{6} - 34\beta_{5} - 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{11} - 61\beta_{7} + 9\beta_{4} - 9\beta_{3} + 12\beta_{2} - 12 \)
|
| \(\nu^{4}\) | \(=\) |
\( -27\beta_{10} + 186\beta_{9} - 1822\beta_{8} + 660\beta_{5} - 18\beta_{4} + 9\beta_{3} + 660\beta_{2} - 9\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 213 \beta_{11} + 213 \beta_{10} - 996 \beta_{8} + 3811 \beta_{7} - 2803 \beta_{4} + 3811 \beta_{3} + \cdots + 546 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 3024 \beta_{11} - 3024 \beta_{9} + 783 \beta_{7} - 8622 \beta_{6} - 56286 \beta_{5} - 783 \beta_{4} + \cdots + 56286 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 675 \beta_{11} - 14481 \beta_{10} + 14481 \beta_{9} + 67536 \beta_{8} - 86076 \beta_{7} + \cdots - 67536 \)
|
| \(\nu^{8}\) | \(=\) |
\( 741027 \beta_{11} + 258903 \beta_{10} + 5821360 \beta_{8} + 76464 \beta_{7} + 258903 \beta_{6} + \cdots - 10143112 \)
|
| \(\nu^{9}\) | \(=\) |
\( 111888 \beta_{10} - 970419 \beta_{9} + 2936304 \beta_{8} + 4171494 \beta_{5} + 8973007 \beta_{4} + \cdots - 6616197 \beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 47738031 \beta_{11} - 47738031 \beta_{10} + 19960479 \beta_{9} - 314604246 \beta_{8} + \cdots + 658936114 \)
|
| \(\nu^{11}\) | \(=\) |
\( 12371238 \beta_{11} + 12371238 \beta_{9} + 482105151 \beta_{7} + 52375686 \beta_{6} - 242945736 \beta_{5} + \cdots + 242945736 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/110\mathbb{Z}\right)^\times\).
| \(n\) | \(67\) | \(101\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
−0.618034 | + | 1.90211i | −5.73804 | + | 4.16893i | −3.23607 | − | 2.35114i | −1.54508 | − | 4.75528i | −4.38347 | − | 13.4909i | −9.28009 | − | 6.74238i | 6.47214 | − | 4.70228i | 7.20166 | − | 22.1644i | 10.0000 | ||||||||||||||||||||||||||||||||||||||
| 31.2 | −0.618034 | + | 1.90211i | −0.396392 | + | 0.287995i | −3.23607 | − | 2.35114i | −1.54508 | − | 4.75528i | −0.302816 | − | 0.931973i | 19.1679 | + | 13.9263i | 6.47214 | − | 4.70228i | −8.26927 | + | 25.4502i | 10.0000 | |||||||||||||||||||||||||||||||||||||||
| 31.3 | −0.618034 | + | 1.90211i | 5.32541 | − | 3.86914i | −3.23607 | − | 2.35114i | −1.54508 | − | 4.75528i | 4.06825 | + | 12.5208i | 7.43847 | + | 5.40436i | 6.47214 | − | 4.70228i | 5.04633 | − | 15.5310i | 10.0000 | |||||||||||||||||||||||||||||||||||||||
| 71.1 | −0.618034 | − | 1.90211i | −5.73804 | − | 4.16893i | −3.23607 | + | 2.35114i | −1.54508 | + | 4.75528i | −4.38347 | + | 13.4909i | −9.28009 | + | 6.74238i | 6.47214 | + | 4.70228i | 7.20166 | + | 22.1644i | 10.0000 | |||||||||||||||||||||||||||||||||||||||
| 71.2 | −0.618034 | − | 1.90211i | −0.396392 | − | 0.287995i | −3.23607 | + | 2.35114i | −1.54508 | + | 4.75528i | −0.302816 | + | 0.931973i | 19.1679 | − | 13.9263i | 6.47214 | + | 4.70228i | −8.26927 | − | 25.4502i | 10.0000 | |||||||||||||||||||||||||||||||||||||||
| 71.3 | −0.618034 | − | 1.90211i | 5.32541 | + | 3.86914i | −3.23607 | + | 2.35114i | −1.54508 | + | 4.75528i | 4.06825 | − | 12.5208i | 7.43847 | − | 5.40436i | 6.47214 | + | 4.70228i | 5.04633 | + | 15.5310i | 10.0000 | |||||||||||||||||||||||||||||||||||||||
| 81.1 | 1.61803 | + | 1.17557i | −2.51400 | + | 7.73730i | 1.23607 | + | 3.80423i | 4.04508 | − | 2.93893i | −13.1635 | + | 9.56383i | 7.65943 | + | 23.5733i | −2.47214 | + | 7.60845i | −31.7021 | − | 23.0330i | 10.0000 | |||||||||||||||||||||||||||||||||||||||
| 81.2 | 1.61803 | + | 1.17557i | 0.292839 | − | 0.901266i | 1.23607 | + | 3.80423i | 4.04508 | − | 2.93893i | 1.53333 | − | 1.11403i | 2.73261 | + | 8.41010i | −2.47214 | + | 7.60845i | 21.1169 | + | 15.3423i | 10.0000 | |||||||||||||||||||||||||||||||||||||||
| 81.3 | 1.61803 | + | 1.17557i | 2.53018 | − | 7.78709i | 1.23607 | + | 3.80423i | 4.04508 | − | 2.93893i | 13.2482 | − | 9.62537i | −8.71828 | − | 26.8321i | −2.47214 | + | 7.60845i | −32.3935 | − | 23.5353i | 10.0000 | |||||||||||||||||||||||||||||||||||||||
| 91.1 | 1.61803 | − | 1.17557i | −2.51400 | − | 7.73730i | 1.23607 | − | 3.80423i | 4.04508 | + | 2.93893i | −13.1635 | − | 9.56383i | 7.65943 | − | 23.5733i | −2.47214 | − | 7.60845i | −31.7021 | + | 23.0330i | 10.0000 | |||||||||||||||||||||||||||||||||||||||
| 91.2 | 1.61803 | − | 1.17557i | 0.292839 | + | 0.901266i | 1.23607 | − | 3.80423i | 4.04508 | + | 2.93893i | 1.53333 | + | 1.11403i | 2.73261 | − | 8.41010i | −2.47214 | − | 7.60845i | 21.1169 | − | 15.3423i | 10.0000 | |||||||||||||||||||||||||||||||||||||||
| 91.3 | 1.61803 | − | 1.17557i | 2.53018 | + | 7.78709i | 1.23607 | − | 3.80423i | 4.04508 | + | 2.93893i | 13.2482 | + | 9.62537i | −8.71828 | + | 26.8321i | −2.47214 | − | 7.60845i | −32.3935 | + | 23.5353i | 10.0000 | |||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 110.4.g.c | ✓ | 12 |
| 11.c | even | 5 | 1 | inner | 110.4.g.c | ✓ | 12 |
| 11.c | even | 5 | 1 | 1210.4.a.bd | 6 | ||
| 11.d | odd | 10 | 1 | 1210.4.a.bg | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 110.4.g.c | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 110.4.g.c | ✓ | 12 | 11.c | even | 5 | 1 | inner |
| 1210.4.a.bd | 6 | 11.c | even | 5 | 1 | ||
| 1210.4.a.bg | 6 | 11.d | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + T_{3}^{11} + 80 T_{3}^{10} + 71 T_{3}^{9} + 3603 T_{3}^{8} + 191 T_{3}^{7} + 110280 T_{3}^{6} + \cdots + 2085136 \)
acting on \(S_{4}^{\mathrm{new}}(110, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 16)^{3} \)
|
| $3$ |
\( T^{12} + T^{11} + \cdots + 2085136 \)
|
| $5$ |
\( (T^{4} - 5 T^{3} + \cdots + 625)^{3} \)
|
| $7$ |
\( T^{12} + \cdots + 238773729813025 \)
|
| $11$ |
\( T^{12} + \cdots + 55\!\cdots\!81 \)
|
| $13$ |
\( T^{12} + \cdots + 81\!\cdots\!61 \)
|
| $17$ |
\( T^{12} + \cdots + 10\!\cdots\!56 \)
|
| $19$ |
\( T^{12} + \cdots + 16\!\cdots\!61 \)
|
| $23$ |
\( (T^{6} - 236 T^{5} + \cdots + 35478920461)^{2} \)
|
| $29$ |
\( T^{12} + \cdots + 38\!\cdots\!00 \)
|
| $31$ |
\( T^{12} + \cdots + 23\!\cdots\!16 \)
|
| $37$ |
\( T^{12} + \cdots + 99\!\cdots\!61 \)
|
| $41$ |
\( T^{12} + \cdots + 15\!\cdots\!01 \)
|
| $43$ |
\( (T^{6} + \cdots + 78741334432420)^{2} \)
|
| $47$ |
\( T^{12} + \cdots + 68\!\cdots\!21 \)
|
| $53$ |
\( T^{12} + \cdots + 14\!\cdots\!25 \)
|
| $59$ |
\( T^{12} + \cdots + 98\!\cdots\!21 \)
|
| $61$ |
\( T^{12} + \cdots + 58\!\cdots\!00 \)
|
| $67$ |
\( (T^{6} + \cdots + 13\!\cdots\!44)^{2} \)
|
| $71$ |
\( T^{12} + \cdots + 12\!\cdots\!00 \)
|
| $73$ |
\( T^{12} + \cdots + 21\!\cdots\!76 \)
|
| $79$ |
\( T^{12} + \cdots + 17\!\cdots\!76 \)
|
| $83$ |
\( T^{12} + \cdots + 51\!\cdots\!56 \)
|
| $89$ |
\( (T^{6} + \cdots + 27\!\cdots\!19)^{2} \)
|
| $97$ |
\( T^{12} + \cdots + 22\!\cdots\!36 \)
|
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