Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [627,4,Mod(1,627)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(627, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("627.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 627 = 3 \cdot 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 627.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: | \(36.9941975736\) |
| 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: |
\( x^{10} - 63x^{8} - 11x^{7} + 1335x^{6} + 424x^{5} - 10579x^{4} - 4725x^{3} + 21298x^{2} + 7896x - 216 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| Twist minimal: | yes |
| 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
\( x^{10} - 63x^{8} - 11x^{7} + 1335x^{6} + 424x^{5} - 10579x^{4} - 4725x^{3} + 21298x^{2} + 7896x - 216 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 353 \nu^{9} - 9729 \nu^{8} - 48022 \nu^{7} + 543763 \nu^{6} + 1557316 \nu^{5} - 8637956 \nu^{4} + \cdots - 16261016 ) / 2577760 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1519 \nu^{9} - 42113 \nu^{8} + 77026 \nu^{7} + 1515811 \nu^{6} + 473332 \nu^{5} + \cdots + 163634088 ) / 10311040 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 17939 \nu^{9} + 9453 \nu^{8} - 1018266 \nu^{7} - 656311 \nu^{6} + 18048588 \nu^{5} + \cdots + 256280952 ) / 10311040 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3697 \nu^{9} - 6961 \nu^{8} - 216318 \nu^{7} + 327587 \nu^{6} + 4220724 \nu^{5} - 4642364 \nu^{4} + \cdots - 5852184 ) / 1288880 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 37823 \nu^{9} - 1839 \nu^{8} - 2474562 \nu^{7} - 273587 \nu^{6} + 54422956 \nu^{5} + \cdots + 197129944 ) / 10311040 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 65987 \nu^{9} + 18611 \nu^{8} + 4013018 \nu^{7} - 172057 \nu^{6} - 81959484 \nu^{5} + \cdots - 246289656 ) / 10311040 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 67719 \nu^{9} - 7927 \nu^{8} - 4270546 \nu^{7} - 66331 \nu^{6} + 90885708 \nu^{5} + \cdots + 254221592 ) / 10311040 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} + \beta_{7} + \beta_{6} - \beta_{3} + 21\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{9} + 3\beta_{8} + \beta_{6} + 2\beta_{5} + \beta_{4} - \beta_{3} + 24\beta_{2} + 7\beta _1 + 267 \)
|
| \(\nu^{5}\) | \(=\) |
\( 12\beta_{9} + 42\beta_{8} + 26\beta_{7} + 35\beta_{6} + 4\beta_{4} - 38\beta_{3} + 4\beta_{2} + 493\beta _1 + 118 \)
|
| \(\nu^{6}\) | \(=\) |
\( 102 \beta_{9} + 153 \beta_{8} + \beta_{7} + 65 \beta_{6} + 76 \beta_{5} + 34 \beta_{4} - 59 \beta_{3} + \cdots + 6038 \)
|
| \(\nu^{7}\) | \(=\) |
\( 570 \beta_{9} + 1384 \beta_{8} + 593 \beta_{7} + 1066 \beta_{6} + 42 \beta_{5} + 189 \beta_{4} + \cdots + 4301 \)
|
| \(\nu^{8}\) | \(=\) |
\( 3766 \beta_{9} + 5704 \beta_{8} + 173 \beta_{7} + 2619 \beta_{6} + 2362 \beta_{5} + 957 \beta_{4} + \cdots + 144330 \)
|
| \(\nu^{9}\) | \(=\) |
\( 20216 \beta_{9} + 42080 \beta_{8} + 13351 \beta_{7} + 31291 \beta_{6} + 2682 \beta_{5} + 6537 \beta_{4} + \cdots + 152653 \)
|
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 |
|
−5.37532 | −3.00000 | 20.8941 | −2.13401 | 16.1260 | −36.7735 | −69.3098 | 9.00000 | 11.4710 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.25249 | −3.00000 | 10.0837 | 11.4796 | 12.7575 | 8.02199 | −8.86095 | 9.00000 | −48.8167 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.92833 | −3.00000 | 7.43177 | −7.76216 | 11.7850 | 18.1402 | 2.23220 | 9.00000 | 30.4923 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.64270 | −3.00000 | −5.30154 | −17.0227 | 4.92810 | −23.3126 | 21.8504 | 9.00000 | 27.9632 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.0255987 | −3.00000 | −7.99934 | 19.2106 | 0.0767961 | −4.59698 | 0.409562 | 9.00000 | −0.491766 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.392328 | −3.00000 | −7.84608 | −15.0745 | −1.17698 | 8.17399 | −6.21687 | 9.00000 | −5.91416 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.88234 | −3.00000 | −4.45678 | −0.0442548 | −5.64703 | 6.46402 | −23.4479 | 9.00000 | −0.0833027 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 3.40034 | −3.00000 | 3.56234 | 6.34288 | −10.2010 | −5.18031 | −15.0896 | 9.00000 | 21.5680 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 4.64037 | −3.00000 | 13.5330 | −16.1277 | −13.9211 | 20.8330 | 25.6751 | 9.00000 | −74.8384 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 4.90906 | −3.00000 | 16.0989 | 0.132381 | −14.7272 | −29.7697 | 39.7578 | 9.00000 | 0.649867 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(11\) | \( -1 \) |
| \(19\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 627.4.a.c | ✓ | 10 |
| 3.b | odd | 2 | 1 | 1881.4.a.d | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 627.4.a.c | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 1881.4.a.d | 10 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - 63 T_{2}^{8} + 11 T_{2}^{7} + 1335 T_{2}^{6} - 424 T_{2}^{5} - 10579 T_{2}^{4} + 4725 T_{2}^{3} + \cdots - 216 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(627))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 63 T^{8} + \cdots - 216 \)
|
| $3$ |
\( (T + 3)^{10} \)
|
| $5$ |
\( T^{10} + 21 T^{9} + \cdots + 561776 \)
|
| $7$ |
\( T^{10} + \cdots - 97351091856 \)
|
| $11$ |
\( (T - 11)^{10} \)
|
| $13$ |
\( T^{10} + \cdots + 9703633884672 \)
|
| $17$ |
\( T^{10} + \cdots + 68\!\cdots\!28 \)
|
| $19$ |
\( (T + 19)^{10} \)
|
| $23$ |
\( T^{10} + \cdots + 20\!\cdots\!84 \)
|
| $29$ |
\( T^{10} + \cdots - 56\!\cdots\!00 \)
|
| $31$ |
\( T^{10} + \cdots - 33\!\cdots\!64 \)
|
| $37$ |
\( T^{10} + \cdots + 56\!\cdots\!44 \)
|
| $41$ |
\( T^{10} + \cdots - 17\!\cdots\!88 \)
|
| $43$ |
\( T^{10} + \cdots - 16\!\cdots\!76 \)
|
| $47$ |
\( T^{10} + \cdots - 23\!\cdots\!32 \)
|
| $53$ |
\( T^{10} + \cdots - 21\!\cdots\!92 \)
|
| $59$ |
\( T^{10} + \cdots + 80\!\cdots\!00 \)
|
| $61$ |
\( T^{10} + \cdots - 56\!\cdots\!68 \)
|
| $67$ |
\( T^{10} + \cdots + 86\!\cdots\!32 \)
|
| $71$ |
\( T^{10} + \cdots - 93\!\cdots\!84 \)
|
| $73$ |
\( T^{10} + \cdots - 18\!\cdots\!44 \)
|
| $79$ |
\( T^{10} + \cdots - 19\!\cdots\!20 \)
|
| $83$ |
\( T^{10} + \cdots - 11\!\cdots\!64 \)
|
| $89$ |
\( T^{10} + \cdots - 53\!\cdots\!40 \)
|
| $97$ |
\( T^{10} + \cdots - 50\!\cdots\!64 \)
|
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