Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [300,7,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, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("300.157");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| 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: | \(69.0162250860\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.691798081536.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} + 26x^{6} - 64x^{5} + 229x^{4} - 356x^{3} + 164x^{2} + 4x + 985 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{16}\cdot 3^{8}\cdot 5^{4} \) |
| 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
\( x^{8} - 4x^{7} + 26x^{6} - 64x^{5} + 229x^{4} - 356x^{3} + 164x^{2} + 4x + 985 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{6} + 6\nu^{5} - 55\nu^{4} + 100\nu^{3} - 443\nu^{2} + 394\nu - 140 ) / 975 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -174\nu^{7} + 609\nu^{6} - 1341\nu^{5} + 1830\nu^{4} + 15609\nu^{3} - 24939\nu^{2} + 325776\nu - 158685 ) / 32825 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 16\nu^{6} - 48\nu^{5} + 440\nu^{4} - 800\nu^{3} + 6144\nu^{2} - 5752\nu + 14120 ) / 65 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -684\nu^{7} + 2394\nu^{6} - 17496\nu^{5} + 37755\nu^{4} - 162756\nu^{3} + 207576\nu^{2} - 62019\nu - 2385 ) / 32825 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 152\nu^{6} - 456\nu^{5} + 2880\nu^{4} - 5000\nu^{3} + 19368\nu^{2} - 16944\nu - 33560 ) / 195 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 10944 \nu^{7} - 38304 \nu^{6} + 279936 \nu^{5} - 604080 \nu^{4} + 2604096 \nu^{3} - 3321216 \nu^{2} + \cdots - 2325240 ) / 6565 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 192\nu^{7} - 672\nu^{6} + 4656\nu^{5} - 9960\nu^{4} + 32928\nu^{3} - 39768\nu^{2} - 55056\nu + 33840 ) / 101 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + 80\beta_{4} + 360 ) / 720 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} + 80\beta_{4} + 18\beta_{3} + 2160\beta _1 - 3240 ) / 720 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -9\beta_{7} - 4\beta_{6} - 1080\beta_{4} + 27\beta_{3} - 240\beta_{2} + 3240\beta _1 - 5040 ) / 720 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -18\beta_{7} - 9\beta_{6} - 108\beta_{5} - 2240\beta_{4} - 144\beta_{3} - 480\beta_{2} - 58320\beta _1 + 4680 ) / 720 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 120 \beta_{7} - 34 \beta_{6} - 270 \beta_{5} + 5480 \beta_{4} - 405 \beta_{3} + 10800 \beta_{2} + \cdots + 20160 ) / 720 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 405 \beta_{7} - 79 \beta_{6} + 2160 \beta_{5} + 22080 \beta_{4} + 108 \beta_{3} + 33600 \beta_{2} + \cdots + 417960 ) / 720 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 504 \beta_{7} + 1261 \beta_{6} + 8505 \beta_{5} + 52920 \beta_{4} + 1827 \beta_{3} - 128040 \beta_{2} + \cdots + 1386360 ) / 720 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(151\) | \(277\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 157.1 |
|
0 | −11.0227 | + | 11.0227i | 0 | 0 | 0 | −191.496 | − | 191.496i | 0 | − | 243.000i | 0 | |||||||||||||||||||||||||||||||||||||
| 157.2 | 0 | −11.0227 | + | 11.0227i | 0 | 0 | 0 | 235.587 | + | 235.587i | 0 | − | 243.000i | 0 | ||||||||||||||||||||||||||||||||||||||
| 157.3 | 0 | 11.0227 | − | 11.0227i | 0 | 0 | 0 | −235.587 | − | 235.587i | 0 | − | 243.000i | 0 | ||||||||||||||||||||||||||||||||||||||
| 157.4 | 0 | 11.0227 | − | 11.0227i | 0 | 0 | 0 | 191.496 | + | 191.496i | 0 | − | 243.000i | 0 | ||||||||||||||||||||||||||||||||||||||
| 193.1 | 0 | −11.0227 | − | 11.0227i | 0 | 0 | 0 | −191.496 | + | 191.496i | 0 | 243.000i | 0 | |||||||||||||||||||||||||||||||||||||||
| 193.2 | 0 | −11.0227 | − | 11.0227i | 0 | 0 | 0 | 235.587 | − | 235.587i | 0 | 243.000i | 0 | |||||||||||||||||||||||||||||||||||||||
| 193.3 | 0 | 11.0227 | + | 11.0227i | 0 | 0 | 0 | −235.587 | + | 235.587i | 0 | 243.000i | 0 | |||||||||||||||||||||||||||||||||||||||
| 193.4 | 0 | 11.0227 | + | 11.0227i | 0 | 0 | 0 | 191.496 | − | 191.496i | 0 | 243.000i | 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 | 300.7.k.c | ✓ | 8 |
| 5.b | even | 2 | 1 | inner | 300.7.k.c | ✓ | 8 |
| 5.c | odd | 4 | 2 | inner | 300.7.k.c | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 300.7.k.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 300.7.k.c | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 300.7.k.c | ✓ | 8 | 5.c | odd | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} + 17700526368T_{7}^{4} + 66277378691949056256 \)
acting on \(S_{7}^{\mathrm{new}}(300, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} + 59049)^{2} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + \cdots + 66\!\cdots\!56 \)
|
| $11$ |
\( (T^{2} - 1452 T - 1935324)^{4} \)
|
| $13$ |
\( T^{8} + \cdots + 22\!\cdots\!00 \)
|
| $17$ |
\( T^{8} + \cdots + 84\!\cdots\!76 \)
|
| $19$ |
\( (T^{4} + \cdots + 74\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 16\!\cdots\!76 \)
|
| $29$ |
\( (T^{4} + \cdots + 9905270685696)^{2} \)
|
| $31$ |
\( (T^{2} - 58028 T + 487226596)^{4} \)
|
| $37$ |
\( T^{8} + \cdots + 21\!\cdots\!36 \)
|
| $41$ |
\( (T^{2} - 74940 T - 1442533500)^{4} \)
|
| $43$ |
\( T^{8} + \cdots + 35\!\cdots\!56 \)
|
| $47$ |
\( T^{8} + \cdots + 12\!\cdots\!76 \)
|
| $53$ |
\( T^{8} + \cdots + 32\!\cdots\!96 \)
|
| $59$ |
\( (T^{4} + \cdots + 19\!\cdots\!76)^{2} \)
|
| $61$ |
\( (T^{2} - 32164 T - 1159711676)^{4} \)
|
| $67$ |
\( T^{8} + \cdots + 27\!\cdots\!96 \)
|
| $71$ |
\( (T^{2} + 8400 T - 130243320000)^{4} \)
|
| $73$ |
\( T^{8} + \cdots + 45\!\cdots\!36 \)
|
| $79$ |
\( (T^{4} + \cdots + 10\!\cdots\!16)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 11\!\cdots\!76 \)
|
| $89$ |
\( (T^{4} + \cdots + 10\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 11\!\cdots\!96 \)
|
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