Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [300,9,Mod(157,300)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(300, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("300.157");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 300.k (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: | \(122.213583018\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 30270 x^{6} - 12 x^{5} + 343617491 x^{4} - 181632 x^{3} - 1733702051118 x^{2} + \cdots + 32\!\cdots\!50 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{16}\cdot 3^{4}\cdot 5^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 30270 x^{6} - 12 x^{5} + 343617491 x^{4} - 181632 x^{3} - 1733702051118 x^{2} + \cdots + 32\!\cdots\!50 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 12\!\cdots\!78 \nu^{7} + \cdots + 46\!\cdots\!75 ) / 96\!\cdots\!25 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 332873137307952 \nu^{7} - 92368156032 \nu^{6} + \cdots - 48\!\cdots\!00 ) / 19\!\cdots\!45 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13093074 \nu^{7} - 39274026 \nu^{6} - 297252088426 \nu^{5} + 1485913118511 \nu^{4} + \cdots + 51\!\cdots\!75 ) / 52\!\cdots\!25 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 71\!\cdots\!06 \nu^{7} - 599108130614224 \nu^{6} + \cdots + 78\!\cdots\!25 ) / 96\!\cdots\!25 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 314939472 \nu^{7} - 14267311131528 \nu^{6} - 9536045734368 \nu^{5} + \cdots + 61\!\cdots\!00 ) / 10\!\cdots\!85 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 1885277952 \nu^{7} - 2380634007168 \nu^{6} - 57067836016248 \nu^{5} + \cdots + 10\!\cdots\!00 ) / 10\!\cdots\!85 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2304 \nu^{7} - 2906016 \nu^{6} - 69742656 \nu^{5} + 65973797136 \nu^{4} + 659794970880 \nu^{3} + \cdots + 12\!\cdots\!00 ) / 6414335095 \)
|
| \(\nu\) | \(=\) |
\( ( -120\beta_{3} + \beta_{2} + 240\beta_1 ) / 240 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - 2\beta_{6} - 240\beta_{4} + 720\beta_{3} + 1816200 ) / 240 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -9\beta_{7} + 3\beta_{5} + 720\beta_{4} - 2724360\beta_{3} + 7567\beta_{2} + 5448600\beta _1 + 1080 ) / 240 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 15135 \beta_{7} - 30268 \beta_{6} - 12 \beta_{5} - 10897440 \beta_{4} + 32691600 \beta_{3} + \cdots + 13742275080 ) / 240 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 227010 \beta_{7} - 30 \beta_{6} + 75675 \beta_{5} + 54486000 \beta_{4} - 34360228320 \beta_{3} + \cdots + 81730800 ) / 240 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 171793466 \beta_{7} - 343511472 \beta_{6} - 454020 \beta_{5} - 206161395840 \beta_{4} + \cdots + 103966948731600 ) / 240 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 3606870834 \beta_{7} - 1589175 \beta_{6} + 1202555018 \beta_{5} + 1442939103720 \beta_{4} + \cdots + 2164694701680 ) / 240 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(151\) | \(277\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 157.1 |
|
0 | −33.0681 | + | 33.0681i | 0 | 0 | 0 | −745.870 | − | 745.870i | 0 | − | 2187.00i | 0 | |||||||||||||||||||||||||||||||||||||
| 157.2 | 0 | −33.0681 | + | 33.0681i | 0 | 0 | 0 | −745.870 | − | 745.870i | 0 | − | 2187.00i | 0 | ||||||||||||||||||||||||||||||||||||||
| 157.3 | 0 | 33.0681 | − | 33.0681i | 0 | 0 | 0 | 745.870 | + | 745.870i | 0 | − | 2187.00i | 0 | ||||||||||||||||||||||||||||||||||||||
| 157.4 | 0 | 33.0681 | − | 33.0681i | 0 | 0 | 0 | 745.870 | + | 745.870i | 0 | − | 2187.00i | 0 | ||||||||||||||||||||||||||||||||||||||
| 193.1 | 0 | −33.0681 | − | 33.0681i | 0 | 0 | 0 | −745.870 | + | 745.870i | 0 | 2187.00i | 0 | |||||||||||||||||||||||||||||||||||||||
| 193.2 | 0 | −33.0681 | − | 33.0681i | 0 | 0 | 0 | −745.870 | + | 745.870i | 0 | 2187.00i | 0 | |||||||||||||||||||||||||||||||||||||||
| 193.3 | 0 | 33.0681 | + | 33.0681i | 0 | 0 | 0 | 745.870 | − | 745.870i | 0 | 2187.00i | 0 | |||||||||||||||||||||||||||||||||||||||
| 193.4 | 0 | 33.0681 | + | 33.0681i | 0 | 0 | 0 | 745.870 | − | 745.870i | 0 | 2187.00i | 0 | |||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 5.c | odd | 4 | 2 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 300.9.k.b | ✓ | 8 |
| 5.b | even | 2 | 1 | inner | 300.9.k.b | ✓ | 8 |
| 5.c | odd | 4 | 2 | inner | 300.9.k.b | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 300.9.k.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 300.9.k.b | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 300.9.k.b | ✓ | 8 | 5.c | odd | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 1237974445449 \)
acting on \(S_{9}^{\mathrm{new}}(300, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} + 4782969)^{2} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( (T^{4} + 1237974445449)^{2} \)
|
| $11$ |
\( (T^{2} + 10068 T - 410561244)^{4} \)
|
| $13$ |
\( T^{8} + \cdots + 18\!\cdots\!25 \)
|
| $17$ |
\( T^{8} + \cdots + 15\!\cdots\!56 \)
|
| $19$ |
\( (T^{4} + \cdots + 32\!\cdots\!25)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 89\!\cdots\!96 \)
|
| $29$ |
\( (T^{4} + \cdots + 27\!\cdots\!76)^{2} \)
|
| $31$ |
\( (T^{2} + 1404262 T + 351755563561)^{4} \)
|
| $37$ |
\( T^{8} + \cdots + 72\!\cdots\!76 \)
|
| $41$ |
\( (T^{2} - 2146200 T - 72906231600)^{4} \)
|
| $43$ |
\( T^{8} + \cdots + 76\!\cdots\!01 \)
|
| $47$ |
\( T^{8} + \cdots + 57\!\cdots\!76 \)
|
| $53$ |
\( T^{8} + \cdots + 88\!\cdots\!56 \)
|
| $59$ |
\( (T^{4} + \cdots + 53\!\cdots\!56)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots - 329983936225751)^{4} \)
|
| $67$ |
\( T^{8} + \cdots + 12\!\cdots\!21 \)
|
| $71$ |
\( (T^{2} + \cdots - 230556872804400)^{4} \)
|
| $73$ |
\( T^{8} + \cdots + 24\!\cdots\!36 \)
|
| $79$ |
\( (T^{4} + \cdots + 67\!\cdots\!76)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 87\!\cdots\!56 \)
|
| $89$ |
\( (T^{4} + \cdots + 16\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 55\!\cdots\!61 \)
|
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