Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [275,6,Mod(1,275)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(275, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("275.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 275 = 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 275.a (trivial) |
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: | yes |
| Analytic conductor: | \(44.1055504486\) |
| Analytic rank: | \(1\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 218x^{8} + 15721x^{6} - 420544x^{4} + 3412272x^{2} - 473344 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 55) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 218x^{8} + 15721x^{6} - 420544x^{4} + 3412272x^{2} - 473344 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 44 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -21\nu^{8} + 5004\nu^{6} - 349531\nu^{4} + 6254180\nu^{2} + 3127936 ) / 418064 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -7\nu^{8} + 1668\nu^{6} - 151349\nu^{4} + 5603432\nu^{2} - 40206336 ) / 418064 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -131\nu^{8} + 23750\nu^{6} - 1257183\nu^{4} + 17115580\nu^{2} + 6413024 ) / 836128 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -64\nu^{9} + 18983\nu^{7} - 1883946\nu^{5} + 67957671\nu^{3} - 562343956\nu ) / 17976752 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -647\nu^{9} + 139240\nu^{7} - 9741143\nu^{5} + 242032302\nu^{3} - 1669880504\nu ) / 35953504 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -5003\nu^{9} + 1080162\nu^{7} - 77436123\nu^{5} + 2076159428\nu^{3} - 16790640320\nu ) / 71907008 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -199\nu^{9} + 45102\nu^{7} - 3449775\nu^{5} + 102191672\nu^{3} - 992688944\nu ) / 2479552 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 44 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} - \beta_{8} - 3\beta_{6} + 73\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -12\beta_{4} + 4\beta_{3} + 101\beta_{2} + 3260 \)
|
| \(\nu^{5}\) | \(=\) |
\( 117\beta_{9} - 165\beta_{8} + 184\beta_{7} - 343\beta_{6} + 6129\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -112\beta_{5} - 1484\beta_{4} + 844\beta_{3} + 9557\beta_{2} + 272332 \)
|
| \(\nu^{7}\) | \(=\) |
\( 12221\beta_{9} - 20701\beta_{8} + 31824\beta_{7} - 31799\beta_{6} + 542249\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( -26688\beta_{5} - 153884\beta_{4} + 114628\beta_{3} + 894037\beta_{2} + 23885244 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1242613\beta_{9} - 2344901\beta_{8} + 4022952\beta_{7} - 2801511\beta_{6} + 49146465\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−9.80621 | 14.6031 | 64.1617 | 0 | −143.201 | 13.1652 | −315.384 | −29.7493 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −8.62547 | −6.81457 | 42.3987 | 0 | 58.7788 | 215.387 | −89.6937 | −196.562 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −5.75907 | −3.98878 | 1.16691 | 0 | 22.9717 | −184.588 | 177.570 | −227.090 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −3.75919 | 21.7683 | −17.8685 | 0 | −81.8312 | −59.6389 | 187.465 | 230.858 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.375714 | −16.7195 | −31.8588 | 0 | 6.28176 | −33.7013 | 23.9927 | 36.5426 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.375714 | 16.7195 | −31.8588 | 0 | 6.28176 | 33.7013 | −23.9927 | 36.5426 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 3.75919 | −21.7683 | −17.8685 | 0 | −81.8312 | 59.6389 | −187.465 | 230.858 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 5.75907 | 3.98878 | 1.16691 | 0 | 22.9717 | 184.588 | −177.570 | −227.090 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 8.62547 | 6.81457 | 42.3987 | 0 | 58.7788 | −215.387 | 89.6937 | −196.562 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 9.80621 | −14.6031 | 64.1617 | 0 | −143.201 | −13.1652 | 315.384 | −29.7493 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(11\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 275.6.a.k | 10 | |
| 5.b | even | 2 | 1 | inner | 275.6.a.k | 10 | |
| 5.c | odd | 4 | 2 | 55.6.b.a | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 55.6.b.a | ✓ | 10 | 5.c | odd | 4 | 2 | |
| 275.6.a.k | 10 | 1.a | even | 1 | 1 | trivial | |
| 275.6.a.k | 10 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - 218T_{2}^{8} + 15721T_{2}^{6} - 420544T_{2}^{4} + 3412272T_{2}^{2} - 473344 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(275))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 218 T^{8} + \cdots - 473344 \)
|
| $3$ |
\( T^{10} + \cdots - 20871003024 \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} + \cdots - 11\!\cdots\!84 \)
|
| $11$ |
\( (T - 121)^{10} \)
|
| $13$ |
\( T^{10} + \cdots - 84\!\cdots\!64 \)
|
| $17$ |
\( T^{10} + \cdots - 39\!\cdots\!76 \)
|
| $19$ |
\( (T^{5} + \cdots - 32083441000000)^{2} \)
|
| $23$ |
\( T^{10} + \cdots - 19\!\cdots\!64 \)
|
| $29$ |
\( (T^{5} + \cdots + 35\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{5} + \cdots - 21\!\cdots\!00)^{2} \)
|
| $37$ |
\( T^{10} + \cdots - 70\!\cdots\!76 \)
|
| $41$ |
\( (T^{5} + \cdots - 83\!\cdots\!68)^{2} \)
|
| $43$ |
\( T^{10} + \cdots - 45\!\cdots\!76 \)
|
| $47$ |
\( T^{10} + \cdots - 11\!\cdots\!76 \)
|
| $53$ |
\( T^{10} + \cdots - 17\!\cdots\!76 \)
|
| $59$ |
\( (T^{5} + \cdots + 67\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{5} + \cdots - 33\!\cdots\!28)^{2} \)
|
| $67$ |
\( T^{10} + \cdots - 15\!\cdots\!64 \)
|
| $71$ |
\( (T^{5} + \cdots - 26\!\cdots\!56)^{2} \)
|
| $73$ |
\( T^{10} + \cdots - 22\!\cdots\!84 \)
|
| $79$ |
\( (T^{5} + \cdots - 49\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{10} + \cdots - 36\!\cdots\!96 \)
|
| $89$ |
\( (T^{5} + \cdots + 74\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{10} + \cdots - 82\!\cdots\!36 \)
|
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