Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [14,15,Mod(13,14)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 15, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("14.13");
S:= CuspForms(chi, 15);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 14 = 2 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 15 \) |
| Character orbit: | \([\chi]\) | \(=\) | 14.b (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: | \(17.4060555413\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 93828x^{6} + 1769902470x^{4} + 614765459916x^{2} + 53884819834839 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{35}\cdot 3^{2}\cdot 7^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 93828x^{6} + 1769902470x^{4} + 614765459916x^{2} + 53884819834839 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 35456\nu^{6} + 3320523456\nu^{4} + 62168970525696\nu^{2} + 10847181370643136 ) / 24658345539 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 1306394098 \nu^{7} + 122346235552590 \nu^{5} + \cdots + 39\!\cdots\!90 \nu ) / 634220866711593 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 56280279646 \nu^{7} - 572879163948 \nu^{6} + \cdots - 17\!\cdots\!12 ) / 57\!\cdots\!37 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 70428485086 \nu^{7} + \cdots + 21\!\cdots\!42 \nu ) / 57\!\cdots\!37 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 56280279646 \nu^{7} - 98250006349908 \nu^{6} + \cdots - 30\!\cdots\!28 ) / 57\!\cdots\!37 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 260626328 \nu^{7} - 24408141246888 \nu^{5} + \cdots - 79\!\cdots\!12 \nu ) / 8687957078241 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3271041706 \nu^{7} + 306339177148662 \nu^{5} + \cdots + 10\!\cdots\!94 \nu ) / 78191613704169 \)
|
| \(\nu\) | \(=\) |
\( ( 10\beta_{7} + 9\beta_{6} + 2\beta_{4} - 84\beta_{2} ) / 3584 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -102\beta_{6} - 10\beta_{5} - 102\beta_{4} + 214\beta_{3} + 102\beta_{2} - 104771\beta _1 - 21017472 ) / 896 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -624066\beta_{7} - 344637\beta_{6} - 139458\beta_{4} + 8490492\beta_{2} ) / 3584 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 1390761 \beta_{6} + 221991 \beta_{5} + 1390761 \beta_{4} - 3003513 \beta_{3} - 1390761 \beta_{2} + \cdots + 294777764064 ) / 224 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 41320423710\beta_{7} + 16383032271\beta_{6} + 7898675922\beta_{4} - 647902257408\beta_{2} ) / 3584 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 342141896862 \beta_{6} - 65625439554 \beta_{5} - 342141896862 \beta_{4} + 749909233278 \beta_{3} + \cdots - 73\!\cdots\!80 ) / 896 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 27\!\cdots\!22 \beta_{7} - 932765950539729 \beta_{6} - 495810941491206 \beta_{4} + 45\!\cdots\!68 \beta_{2} ) / 3584 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 13.1 |
|
−90.5097 | − | 2707.42i | 8192.00 | 15052.3i | 245048.i | −651367. | + | 503928.i | −741455. | −2.54716e6 | − | 1.36238e6i | ||||||||||||||||||||||||||||||||||||||
| 13.2 | −90.5097 | − | 1665.69i | 8192.00 | − | 92849.1i | 150761.i | 608394. | − | 555049.i | −741455. | 2.00845e6 | 8.40374e6i | |||||||||||||||||||||||||||||||||||||||
| 13.3 | −90.5097 | 1665.69i | 8192.00 | 92849.1i | − | 150761.i | 608394. | + | 555049.i | −741455. | 2.00845e6 | − | 8.40374e6i | |||||||||||||||||||||||||||||||||||||||
| 13.4 | −90.5097 | 2707.42i | 8192.00 | − | 15052.3i | − | 245048.i | −651367. | − | 503928.i | −741455. | −2.54716e6 | 1.36238e6i | |||||||||||||||||||||||||||||||||||||||
| 13.5 | 90.5097 | − | 1924.91i | 8192.00 | − | 119634.i | − | 174223.i | −768476. | + | 296087.i | 741455. | 1.07769e6 | − | 1.08280e7i | |||||||||||||||||||||||||||||||||||||
| 13.6 | 90.5097 | − | 1775.85i | 8192.00 | 35197.4i | − | 160732.i | 668957. | + | 480332.i | 741455. | 1.62932e6 | 3.18570e6i | |||||||||||||||||||||||||||||||||||||||
| 13.7 | 90.5097 | 1775.85i | 8192.00 | − | 35197.4i | 160732.i | 668957. | − | 480332.i | 741455. | 1.62932e6 | − | 3.18570e6i | |||||||||||||||||||||||||||||||||||||||
| 13.8 | 90.5097 | 1924.91i | 8192.00 | 119634.i | 174223.i | −768476. | − | 296087.i | 741455. | 1.07769e6 | 1.08280e7i | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 14.15.b.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 126.15.c.a | 8 | ||
| 4.b | odd | 2 | 1 | 112.15.c.c | 8 | ||
| 7.b | odd | 2 | 1 | inner | 14.15.b.a | ✓ | 8 |
| 7.c | even | 3 | 2 | 98.15.d.a | 16 | ||
| 7.d | odd | 6 | 2 | 98.15.d.a | 16 | ||
| 21.c | even | 2 | 1 | 126.15.c.a | 8 | ||
| 28.d | even | 2 | 1 | 112.15.c.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.15.b.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 14.15.b.a | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 98.15.d.a | 16 | 7.c | even | 3 | 2 | ||
| 98.15.d.a | 16 | 7.d | odd | 6 | 2 | ||
| 112.15.c.c | 8 | 4.b | odd | 2 | 1 | ||
| 112.15.c.c | 8 | 28.d | even | 2 | 1 | ||
| 126.15.c.a | 8 | 3.b | odd | 2 | 1 | ||
| 126.15.c.a | 8 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{15}^{\mathrm{new}}(14, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 8192)^{4} \)
|
| $3$ |
\( T^{8} + \cdots + 23\!\cdots\!24 \)
|
| $5$ |
\( T^{8} + \cdots + 34\!\cdots\!00 \)
|
| $7$ |
\( T^{8} + \cdots + 21\!\cdots\!01 \)
|
| $11$ |
\( (T^{4} + \cdots + 33\!\cdots\!24)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 55\!\cdots\!24 \)
|
| $17$ |
\( T^{8} + \cdots + 37\!\cdots\!04 \)
|
| $19$ |
\( T^{8} + \cdots + 10\!\cdots\!84 \)
|
| $23$ |
\( (T^{4} + \cdots + 20\!\cdots\!76)^{2} \)
|
| $29$ |
\( (T^{4} + \cdots - 34\!\cdots\!24)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 98\!\cdots\!64 \)
|
| $37$ |
\( (T^{4} + \cdots - 67\!\cdots\!24)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 11\!\cdots\!04 \)
|
| $43$ |
\( (T^{4} + \cdots - 10\!\cdots\!24)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 40\!\cdots\!64 \)
|
| $53$ |
\( (T^{4} + \cdots + 64\!\cdots\!84)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 46\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots + 57\!\cdots\!84 \)
|
| $67$ |
\( (T^{4} + \cdots + 25\!\cdots\!04)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots - 22\!\cdots\!56)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 14\!\cdots\!04 \)
|
| $79$ |
\( (T^{4} + \cdots - 45\!\cdots\!44)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 80\!\cdots\!64 \)
|
| $89$ |
\( T^{8} + \cdots + 57\!\cdots\!84 \)
|
| $97$ |
\( T^{8} + \cdots + 23\!\cdots\!04 \)
|
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