Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [123,2,Mod(10,123)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(123, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("123.10");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 123 = 3 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 123.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: | \(0.982159944862\) |
| 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: |
\( x^{12} - 2 x^{11} + 2 x^{10} + 3 x^{9} + 14 x^{8} - 26 x^{7} + 113 x^{6} - 52 x^{5} + 70 x^{4} + \cdots + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{12} - 2 x^{11} + 2 x^{10} + 3 x^{9} + 14 x^{8} - 26 x^{7} + 113 x^{6} - 52 x^{5} + 70 x^{4} + \cdots + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 23609499 \nu^{11} + 69932271 \nu^{10} - 101864069 \nu^{9} + 963675 \nu^{8} + \cdots + 800876661 ) / 1289475791 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 60838904 \nu^{11} - 122804370 \nu^{10} + 316728779 \nu^{9} - 532709523 \nu^{8} + \cdots - 982025859 ) / 1289475791 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 101597624 \nu^{11} - 111106411 \nu^{10} + 57702032 \nu^{9} + 438840162 \nu^{8} + \cdots + 615512499 ) / 1289475791 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 153218003 \nu^{11} + 38063585 \nu^{10} - 291158324 \nu^{9} + 939791443 \nu^{8} + \cdots + 1574825085 ) / 1289475791 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 208077435 \nu^{11} - 300166272 \nu^{10} + 261481814 \nu^{9} + 751666829 \nu^{8} + \cdots - 309675059 ) / 1289475791 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 252120492 \nu^{11} - 512853545 \nu^{10} + 454566979 \nu^{9} + 862778631 \nu^{8} + \cdots - 746017229 ) / 1289475791 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 344499591 \nu^{11} - 597594330 \nu^{10} + 480137434 \nu^{9} + 1269860551 \nu^{8} + \cdots - 153218003 ) / 1289475791 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 746017229 \nu^{11} - 1239913966 \nu^{10} + 979180913 \nu^{9} + 2692618666 \nu^{8} + \cdots - 1786158146 ) / 1289475791 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 832063482 \nu^{11} - 1286218825 \nu^{10} + 1091739599 \nu^{9} + 2893369856 \nu^{8} + \cdots - 1786404404 ) / 1289475791 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 982025859 \nu^{11} - 2024890622 \nu^{10} + 1841247348 \nu^{9} + 3262806356 \nu^{8} + \cdots - 3707641437 ) / 1289475791 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 1427087835 \nu^{11} + 2421408099 \nu^{10} - 1943558448 \nu^{9} - 5231083053 \nu^{8} + \cdots + 812336886 ) / 1289475791 \)
|
| \(\nu\) | \(=\) |
\( -\beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{9} + \beta_{8} - \beta_{7} + 2\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{9} + \beta_{6} + 5\beta_{4} + 5\beta_{2} - 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{11} - 6\beta_{10} - 6\beta_{9} + 10\beta_{8} + 5\beta_{7} + 6\beta_{5} - 6\beta_{4} - \beta_{3} - 2\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 2 \beta_{11} - 7 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + 26 \beta_{7} - 9 \beta_{6} - 28 \beta_{4} + \cdots - 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -26\beta_{11} - 26\beta_{10} + 21\beta_{7} - 21\beta_{6} - 33\beta_{5} - 68\beta_{4} + 9\beta_{3} - 47\beta_{2} - 33 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 26 \beta_{11} + 21 \beta_{9} - 21 \beta_{8} + 63 \beta_{7} - 141 \beta_{6} - 33 \beta_{5} - 204 \beta_{4} + \cdots - 21 \)
|
| \(\nu^{8}\) | \(=\) |
\( 141 \beta_{10} + 204 \beta_{9} - 298 \beta_{8} - 146 \beta_{6} - 298 \beta_{5} - 49 \beta_{4} + \cdots - 115 \)
|
| \(\nu^{9}\) | \(=\) |
\( 146 \beta_{11} + 310 \beta_{10} + 310 \beta_{9} - 270 \beta_{8} - 405 \beta_{7} - 169 \beta_{5} + \cdots + 789 \beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( 789 \beta_{11} + 1194 \beta_{10} + 789 \beta_{9} - 1030 \beta_{8} - 1149 \beta_{7} + 863 \beta_{6} + \cdots + 643 \)
|
| \(\nu^{11}\) | \(=\) |
\( 1149 \beta_{11} + 1149 \beta_{10} - 4519 \beta_{7} + 4519 \beta_{6} + 1223 \beta_{5} + 7773 \beta_{4} + \cdots + 1223 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/123\mathbb{Z}\right)^\times\).
| \(n\) | \(83\) | \(88\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{4} - \beta_{5} - \beta_{8}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 10.1 |
|
−1.49960 | − | 1.08952i | −1.00000 | 0.443698 | + | 1.36556i | 1.15855 | + | 3.56564i | 1.49960 | + | 1.08952i | 1.77509 | − | 1.28968i | −0.323149 | + | 0.994549i | 1.00000 | 2.14749 | − | 6.60928i | ||||||||||||||||||||||||||||||||||||||||
| 10.2 | 0.254476 | + | 0.184888i | −1.00000 | −0.587459 | − | 1.80801i | −0.477560 | − | 1.46978i | −0.254476 | − | 0.184888i | 2.55033 | − | 1.85292i | 0.379188 | − | 1.16702i | 1.00000 | 0.150216 | − | 0.462318i | |||||||||||||||||||||||||||||||||||||||||
| 10.3 | 2.24512 | + | 1.63117i | −1.00000 | 1.76179 | + | 5.42225i | −0.990005 | − | 3.04692i | −2.24512 | − | 1.63117i | −0.0893533 | + | 0.0649190i | −3.17407 | + | 9.76879i | 1.00000 | 2.74738 | − | 8.45557i | |||||||||||||||||||||||||||||||||||||||||
| 16.1 | −0.0717988 | + | 0.220974i | −1.00000 | 1.57436 | + | 1.14384i | −0.217758 | − | 0.158211i | 0.0717988 | − | 0.220974i | 0.917757 | + | 2.82457i | −0.741740 | + | 0.538905i | 1.00000 | 0.0505953 | − | 0.0367596i | |||||||||||||||||||||||||||||||||||||||||
| 16.2 | 0.353737 | − | 1.08869i | −1.00000 | 0.557919 | + | 0.405352i | 2.52896 | + | 1.83739i | −0.353737 | + | 1.08869i | −0.982466 | − | 3.02372i | 2.49085 | − | 1.80971i | 1.00000 | 2.89494 | − | 2.10329i | |||||||||||||||||||||||||||||||||||||||||
| 16.3 | 0.718062 | − | 2.20997i | −1.00000 | −2.75031 | − | 1.99822i | −1.50218 | − | 1.09140i | −0.718062 | + | 2.20997i | −0.171359 | − | 0.527389i | −2.63107 | + | 1.91159i | 1.00000 | −3.49062 | + | 2.53608i | |||||||||||||||||||||||||||||||||||||||||
| 37.1 | −1.49960 | + | 1.08952i | −1.00000 | 0.443698 | − | 1.36556i | 1.15855 | − | 3.56564i | 1.49960 | − | 1.08952i | 1.77509 | + | 1.28968i | −0.323149 | − | 0.994549i | 1.00000 | 2.14749 | + | 6.60928i | |||||||||||||||||||||||||||||||||||||||||
| 37.2 | 0.254476 | − | 0.184888i | −1.00000 | −0.587459 | + | 1.80801i | −0.477560 | + | 1.46978i | −0.254476 | + | 0.184888i | 2.55033 | + | 1.85292i | 0.379188 | + | 1.16702i | 1.00000 | 0.150216 | + | 0.462318i | |||||||||||||||||||||||||||||||||||||||||
| 37.3 | 2.24512 | − | 1.63117i | −1.00000 | 1.76179 | − | 5.42225i | −0.990005 | + | 3.04692i | −2.24512 | + | 1.63117i | −0.0893533 | − | 0.0649190i | −3.17407 | − | 9.76879i | 1.00000 | 2.74738 | + | 8.45557i | |||||||||||||||||||||||||||||||||||||||||
| 100.1 | −0.0717988 | − | 0.220974i | −1.00000 | 1.57436 | − | 1.14384i | −0.217758 | + | 0.158211i | 0.0717988 | + | 0.220974i | 0.917757 | − | 2.82457i | −0.741740 | − | 0.538905i | 1.00000 | 0.0505953 | + | 0.0367596i | |||||||||||||||||||||||||||||||||||||||||
| 100.2 | 0.353737 | + | 1.08869i | −1.00000 | 0.557919 | − | 0.405352i | 2.52896 | − | 1.83739i | −0.353737 | − | 1.08869i | −0.982466 | + | 3.02372i | 2.49085 | + | 1.80971i | 1.00000 | 2.89494 | + | 2.10329i | |||||||||||||||||||||||||||||||||||||||||
| 100.3 | 0.718062 | + | 2.20997i | −1.00000 | −2.75031 | + | 1.99822i | −1.50218 | + | 1.09140i | −0.718062 | − | 2.20997i | −0.171359 | + | 0.527389i | −2.63107 | − | 1.91159i | 1.00000 | −3.49062 | − | 2.53608i | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 41.d | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 123.2.g.b | ✓ | 12 |
| 3.b | odd | 2 | 1 | 369.2.h.c | 12 | ||
| 41.d | even | 5 | 1 | inner | 123.2.g.b | ✓ | 12 |
| 41.d | even | 5 | 1 | 5043.2.a.u | 6 | ||
| 41.f | even | 10 | 1 | 5043.2.a.v | 6 | ||
| 123.k | odd | 10 | 1 | 369.2.h.c | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 123.2.g.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 123.2.g.b | ✓ | 12 | 41.d | even | 5 | 1 | inner |
| 369.2.h.c | 12 | 3.b | odd | 2 | 1 | ||
| 369.2.h.c | 12 | 123.k | odd | 10 | 1 | ||
| 5043.2.a.u | 6 | 41.d | even | 5 | 1 | ||
| 5043.2.a.v | 6 | 41.f | even | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 4 T_{2}^{11} + 10 T_{2}^{10} - 8 T_{2}^{9} + 10 T_{2}^{8} + 2 T_{2}^{7} + 143 T_{2}^{6} + \cdots + 1 \)
acting on \(S_{2}^{\mathrm{new}}(123, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - 4 T^{11} + \cdots + 1 \)
|
| $3$ |
\( (T + 1)^{12} \)
|
| $5$ |
\( T^{12} - T^{11} + \cdots + 841 \)
|
| $7$ |
\( T^{12} - 8 T^{11} + \cdots + 16 \)
|
| $11$ |
\( T^{12} - 9 T^{11} + \cdots + 1635841 \)
|
| $13$ |
\( T^{12} - 2 T^{11} + \cdots + 1936 \)
|
| $17$ |
\( T^{12} + 2 T^{11} + \cdots + 3748096 \)
|
| $19$ |
\( T^{12} + 2 T^{11} + \cdots + 160801 \)
|
| $23$ |
\( T^{12} - 15 T^{11} + \cdots + 7623121 \)
|
| $29$ |
\( T^{12} + 14 T^{11} + \cdots + 3916441 \)
|
| $31$ |
\( T^{12} + 18 T^{11} + \cdots + 6806881 \)
|
| $37$ |
\( T^{12} - 15 T^{11} + \cdots + 2430481 \)
|
| $41$ |
\( T^{12} + \cdots + 4750104241 \)
|
| $43$ |
\( T^{12} + 119 T^{10} + \cdots + 30976 \)
|
| $47$ |
\( T^{12} - T^{11} + \cdots + 34562641 \)
|
| $53$ |
\( T^{12} + 9 T^{11} + \cdots + 25411681 \)
|
| $59$ |
\( T^{12} + \cdots + 147889921 \)
|
| $61$ |
\( T^{12} - 13 T^{11} + \cdots + 361201 \)
|
| $67$ |
\( T^{12} + \cdots + 3917633281 \)
|
| $71$ |
\( T^{12} + \cdots + 128551648681 \)
|
| $73$ |
\( (T^{6} + 17 T^{5} + \cdots - 70064)^{2} \)
|
| $79$ |
\( (T^{6} - 40 T^{5} + \cdots - 4096)^{2} \)
|
| $83$ |
\( (T^{6} + 3 T^{5} + \cdots + 15616)^{2} \)
|
| $89$ |
\( T^{12} + \cdots + 30261397073296 \)
|
| $97$ |
\( T^{12} + \cdots + 7665177601 \)
|
| show more | |
| show less |