Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [130,3,Mod(27,130)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(130, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("130.27");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 130 = 2 \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 130.i (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: | \(3.54224343668\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 4 x^{11} + 8 x^{10} + 116 x^{9} + 872 x^{8} - 112 x^{7} + 200 x^{6} + 2320 x^{5} + 11324 x^{4} + \cdots + 16900 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2\cdot 5 \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 4 x^{11} + 8 x^{10} + 116 x^{9} + 872 x^{8} - 112 x^{7} + 200 x^{6} + 2320 x^{5} + 11324 x^{4} + \cdots + 16900 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3955747697 \nu^{11} - 68999771737 \nu^{10} + 246932784305 \nu^{9} + 21110340042 \nu^{8} + \cdots + 5562749767900 ) / 946423916434750 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 21546039449 \nu^{11} - 315826023579 \nu^{10} + 1600713381860 \nu^{9} + \cdots - 11\!\cdots\!00 ) / 436811038354500 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 53176780949 \nu^{11} - 215286802729 \nu^{10} + 437756392810 \nu^{9} + \cdots + 66852136079300 ) / 946423916434750 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 109370795189 \nu^{11} + 418608466316 \nu^{10} - 5559330630945 \nu^{9} + \cdots + 10\!\cdots\!00 ) / 11\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 741543837839 \nu^{11} - 9318826175019 \nu^{10} + 32244489070660 \nu^{9} + \cdots + 17\!\cdots\!00 ) / 56\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1165858030894 \nu^{11} + 2806374583749 \nu^{10} + 1275161668040 \nu^{9} + \cdots - 66\!\cdots\!00 ) / 56\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 1246064286666 \nu^{11} - 8517504488436 \nu^{10} + 28873582744565 \nu^{9} + \cdots + 61\!\cdots\!00 ) / 56\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 2839172476297 \nu^{11} + 10414239537062 \nu^{10} - 14783081978255 \nu^{9} + \cdots - 26\!\cdots\!00 ) / 56\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 3561601788788 \nu^{11} - 16577911161373 \nu^{10} + 37148504228920 \nu^{9} + \cdots + 20\!\cdots\!00 ) / 56\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 7734562402683 \nu^{11} + 37737431484418 \nu^{10} - 91836310238095 \nu^{9} + \cdots - 35\!\cdots\!00 ) / 56\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{11} - 2\beta_{10} + \beta_{6} - \beta_{5} + \beta_{4} - 3\beta_{3} - 12\beta_{2} + 2\beta _1 - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( - 5 \beta_{11} - 17 \beta_{10} + 6 \beta_{8} - \beta_{7} + \beta_{6} + 3 \beta_{5} + 36 \beta_{4} + \cdots - 23 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 35 \beta_{11} - 129 \beta_{10} - 15 \beta_{9} + 9 \beta_{8} - 32 \beta_{7} + 47 \beta_{5} + \cdots - 386 \)
|
| \(\nu^{5}\) | \(=\) |
\( 120 \beta_{11} - 87 \beta_{10} - 240 \beta_{9} - 18 \beta_{8} - 87 \beta_{7} - 87 \beta_{6} + \cdots - 1091 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2240 \beta_{11} + 4606 \beta_{10} - 690 \beta_{9} - 966 \beta_{8} - 1274 \beta_{6} + 1412 \beta_{5} + \cdots + 2102 \)
|
| \(\nu^{7}\) | \(=\) |
\( 16382 \beta_{11} + 49040 \beta_{10} + 1104 \beta_{9} - 10380 \beta_{8} + 5044 \beta_{7} - 5044 \beta_{6} + \cdots + 80532 \)
|
| \(\nu^{8}\) | \(=\) |
\( 63632 \beta_{11} + 279660 \beta_{10} + 51102 \beta_{9} - 38514 \beta_{8} + 57338 \beta_{7} + \cdots + 758288 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 241146 \beta_{11} + 261208 \beta_{10} + 482292 \beta_{9} + 57396 \beta_{8} + 261208 \beta_{7} + \cdots + 3058888 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 5283478 \beta_{11} - 10818986 \beta_{10} + 1969698 \beta_{9} + 2566782 \beta_{8} + 2716696 \beta_{6} + \cdots - 4984936 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 40503600 \beta_{11} - 119536578 \beta_{10} - 2865324 \beta_{9} + 23131968 \beta_{8} + \cdots - 200983054 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/130\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(41\) |
| \(\chi(n)\) | \(-\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 27.1 |
|
−1.00000 | − | 1.00000i | −3.94518 | + | 3.94518i | 2.00000i | −3.79773 | + | 3.25227i | 7.89036 | −2.14293 | − | 2.14293i | 2.00000 | − | 2.00000i | − | 22.1289i | 7.05000 | + | 0.545467i | |||||||||||||||||||||||||||||||||||||||||
| 27.2 | −1.00000 | − | 1.00000i | −0.375483 | + | 0.375483i | 2.00000i | 0.689291 | + | 4.95226i | 0.750965 | −8.24785 | − | 8.24785i | 2.00000 | − | 2.00000i | 8.71803i | 4.26297 | − | 5.64155i | |||||||||||||||||||||||||||||||||||||||||||
| 27.3 | −1.00000 | − | 1.00000i | 0.254871 | − | 0.254871i | 2.00000i | −4.94461 | + | 0.742193i | −0.509741 | 5.32789 | + | 5.32789i | 2.00000 | − | 2.00000i | 8.87008i | 5.68680 | + | 4.20242i | |||||||||||||||||||||||||||||||||||||||||||
| 27.4 | −1.00000 | − | 1.00000i | 1.95768 | − | 1.95768i | 2.00000i | 3.22545 | + | 3.82054i | −3.91537 | 7.25829 | + | 7.25829i | 2.00000 | − | 2.00000i | 1.33494i | 0.595091 | − | 7.04598i | |||||||||||||||||||||||||||||||||||||||||||
| 27.5 | −1.00000 | − | 1.00000i | 2.12061 | − | 2.12061i | 2.00000i | 3.80483 | − | 3.24396i | −4.24122 | −6.17905 | − | 6.17905i | 2.00000 | − | 2.00000i | 0.00600950i | −7.04879 | − | 0.560865i | |||||||||||||||||||||||||||||||||||||||||||
| 27.6 | −1.00000 | − | 1.00000i | 3.98749 | − | 3.98749i | 2.00000i | −4.97722 | + | 0.476708i | −7.97499 | −4.01634 | − | 4.01634i | 2.00000 | − | 2.00000i | − | 22.8002i | 5.45393 | + | 4.50052i | ||||||||||||||||||||||||||||||||||||||||||
| 53.1 | −1.00000 | + | 1.00000i | −3.94518 | − | 3.94518i | − | 2.00000i | −3.79773 | − | 3.25227i | 7.89036 | −2.14293 | + | 2.14293i | 2.00000 | + | 2.00000i | 22.1289i | 7.05000 | − | 0.545467i | ||||||||||||||||||||||||||||||||||||||||||
| 53.2 | −1.00000 | + | 1.00000i | −0.375483 | − | 0.375483i | − | 2.00000i | 0.689291 | − | 4.95226i | 0.750965 | −8.24785 | + | 8.24785i | 2.00000 | + | 2.00000i | − | 8.71803i | 4.26297 | + | 5.64155i | |||||||||||||||||||||||||||||||||||||||||
| 53.3 | −1.00000 | + | 1.00000i | 0.254871 | + | 0.254871i | − | 2.00000i | −4.94461 | − | 0.742193i | −0.509741 | 5.32789 | − | 5.32789i | 2.00000 | + | 2.00000i | − | 8.87008i | 5.68680 | − | 4.20242i | |||||||||||||||||||||||||||||||||||||||||
| 53.4 | −1.00000 | + | 1.00000i | 1.95768 | + | 1.95768i | − | 2.00000i | 3.22545 | − | 3.82054i | −3.91537 | 7.25829 | − | 7.25829i | 2.00000 | + | 2.00000i | − | 1.33494i | 0.595091 | + | 7.04598i | |||||||||||||||||||||||||||||||||||||||||
| 53.5 | −1.00000 | + | 1.00000i | 2.12061 | + | 2.12061i | − | 2.00000i | 3.80483 | + | 3.24396i | −4.24122 | −6.17905 | + | 6.17905i | 2.00000 | + | 2.00000i | − | 0.00600950i | −7.04879 | + | 0.560865i | |||||||||||||||||||||||||||||||||||||||||
| 53.6 | −1.00000 | + | 1.00000i | 3.98749 | + | 3.98749i | − | 2.00000i | −4.97722 | − | 0.476708i | −7.97499 | −4.01634 | + | 4.01634i | 2.00000 | + | 2.00000i | 22.8002i | 5.45393 | − | 4.50052i | ||||||||||||||||||||||||||||||||||||||||||
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 | 130.3.i.a | ✓ | 12 |
| 5.b | even | 2 | 1 | 650.3.i.d | 12 | ||
| 5.c | odd | 4 | 1 | inner | 130.3.i.a | ✓ | 12 |
| 5.c | odd | 4 | 1 | 650.3.i.d | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 130.3.i.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 130.3.i.a | ✓ | 12 | 5.c | odd | 4 | 1 | inner |
| 650.3.i.d | 12 | 5.b | even | 2 | 1 | ||
| 650.3.i.d | 12 | 5.c | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} - 8 T_{3}^{11} + 32 T_{3}^{10} - 60 T_{3}^{9} + 1028 T_{3}^{8} - 7744 T_{3}^{7} + 30856 T_{3}^{6} + \cdots + 2500 \)
acting on \(S_{3}^{\mathrm{new}}(130, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 2)^{6} \)
|
| $3$ |
\( T^{12} - 8 T^{11} + \cdots + 2500 \)
|
| $5$ |
\( T^{12} + \cdots + 244140625 \)
|
| $7$ |
\( T^{12} + \cdots + 18414490000 \)
|
| $11$ |
\( (T^{6} - 8 T^{5} + \cdots - 187450)^{2} \)
|
| $13$ |
\( (T^{4} + 169)^{3} \)
|
| $17$ |
\( T^{12} + \cdots + 32546203485184 \)
|
| $19$ |
\( T^{12} + \cdots + 84617148742564 \)
|
| $23$ |
\( T^{12} + \cdots + 124018074244 \)
|
| $29$ |
\( T^{12} + \cdots + 26\!\cdots\!56 \)
|
| $31$ |
\( (T^{6} - 8 T^{5} + \cdots - 21345130)^{2} \)
|
| $37$ |
\( T^{12} + \cdots + 35\!\cdots\!76 \)
|
| $41$ |
\( (T^{6} + 100 T^{5} + \cdots + 295775000)^{2} \)
|
| $43$ |
\( T^{12} + \cdots + 79\!\cdots\!00 \)
|
| $47$ |
\( T^{12} + \cdots + 44\!\cdots\!84 \)
|
| $53$ |
\( T^{12} + \cdots + 16\!\cdots\!76 \)
|
| $59$ |
\( T^{12} + \cdots + 13\!\cdots\!00 \)
|
| $61$ |
\( (T^{6} - 92 T^{5} + \cdots - 1850306396)^{2} \)
|
| $67$ |
\( T^{12} + \cdots + 24\!\cdots\!00 \)
|
| $71$ |
\( (T^{6} + 72 T^{5} + \cdots - 22493746250)^{2} \)
|
| $73$ |
\( T^{12} + \cdots + 18\!\cdots\!00 \)
|
| $79$ |
\( T^{12} + \cdots + 27\!\cdots\!44 \)
|
| $83$ |
\( T^{12} + \cdots + 16\!\cdots\!00 \)
|
| $89$ |
\( T^{12} + \cdots + 18\!\cdots\!00 \)
|
| $97$ |
\( T^{12} + \cdots + 35\!\cdots\!24 \)
|
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