Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [245,10,Mod(1,245)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(245, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("245.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 245 = 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 245.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: | \(126.183779860\) |
| Analytic rank: | \(0\) |
| Dimension: | \(13\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{13} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{13} - x^{12} - 5109 x^{11} + 3203 x^{10} + 9635922 x^{9} + 242128 x^{8} - 8405086048 x^{7} + \cdots - 96\!\cdots\!52 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{14}\cdot 3^{3}\cdot 5^{3}\cdot 7^{7} \) |
| Twist minimal: | no (minimal twist has level 35) |
| 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 \(1,\beta_1,\ldots,\beta_{12}\) 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
\( x^{13} - x^{12} - 5109 x^{11} + 3203 x^{10} + 9635922 x^{9} + 242128 x^{8} - 8405086048 x^{7} + \cdots - 96\!\cdots\!52 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 15\!\cdots\!47 \nu^{12} + \cdots - 42\!\cdots\!80 ) / 46\!\cdots\!12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 15\!\cdots\!47 \nu^{12} + \cdots + 60\!\cdots\!48 ) / 46\!\cdots\!12 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 11\!\cdots\!89 \nu^{12} + \cdots + 30\!\cdots\!92 ) / 11\!\cdots\!28 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 68\!\cdots\!19 \nu^{12} + \cdots - 87\!\cdots\!76 ) / 23\!\cdots\!56 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 19\!\cdots\!43 \nu^{12} + \cdots - 14\!\cdots\!72 ) / 15\!\cdots\!04 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 31\!\cdots\!25 \nu^{12} + \cdots - 36\!\cdots\!08 ) / 11\!\cdots\!28 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 55\!\cdots\!27 \nu^{12} + \cdots - 31\!\cdots\!00 ) / 15\!\cdots\!04 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 58\!\cdots\!75 \nu^{12} + \cdots + 32\!\cdots\!88 ) / 15\!\cdots\!04 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 60\!\cdots\!87 \nu^{12} + \cdots + 46\!\cdots\!76 ) / 58\!\cdots\!64 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 62\!\cdots\!43 \nu^{12} + \cdots + 26\!\cdots\!40 ) / 46\!\cdots\!12 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 10\!\cdots\!19 \nu^{12} + \cdots + 38\!\cdots\!80 ) / 46\!\cdots\!12 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 786 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} - 8\beta_{3} - 2\beta_{2} + 1335\beta _1 + 334 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 2 \beta_{12} + 5 \beta_{11} + 3 \beta_{9} + \beta_{8} + 6 \beta_{7} + 2 \beta_{6} - 7 \beta_{5} + \cdots + 1051468 \)
|
| \(\nu^{5}\) | \(=\) |
\( 14 \beta_{12} - 90 \beta_{11} + 102 \beta_{10} + 2 \beta_{9} + 68 \beta_{8} - 140 \beta_{7} + \cdots - 219792 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 4960 \beta_{12} + 16113 \beta_{11} - 1250 \beta_{10} + 7535 \beta_{9} + 4191 \beta_{8} + \cdots + 1675490862 \)
|
| \(\nu^{7}\) | \(=\) |
\( 38956 \beta_{12} - 298852 \beta_{11} + 327268 \beta_{10} - 17980 \beta_{9} + 204948 \beta_{8} + \cdots - 3460687922 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 10078794 \beta_{12} + 39066893 \beta_{11} - 4060376 \beta_{10} + 15408491 \beta_{9} + \cdots + 2892994683596 \)
|
| \(\nu^{9}\) | \(=\) |
\( 80096150 \beta_{12} - 751957078 \beta_{11} + 799945494 \beta_{10} - 92840322 \beta_{9} + \cdots - 12799081766744 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 19599857440 \beta_{12} + 85663414677 \beta_{11} - 9925013066 \beta_{10} + 29962185483 \beta_{9} + \cdots + 52\!\cdots\!54 \)
|
| \(\nu^{11}\) | \(=\) |
\( 156594515668 \beta_{12} - 1729478338264 \beta_{11} + 1777006887972 \beta_{10} - 294213318376 \beta_{9} + \cdots - 35\!\cdots\!34 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 37853573041914 \beta_{12} + 179558986649977 \beta_{11} - 22373125097184 \beta_{10} + \cdots + 97\!\cdots\!96 \)
|
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 |
|
−43.3172 | 196.236 | 1364.38 | 625.000 | −8500.37 | 0 | −36922.6 | 18825.4 | −27073.2 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −36.4811 | −102.189 | 818.871 | 625.000 | 3727.97 | 0 | −11195.0 | −9240.37 | −22800.7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −34.1284 | −128.345 | 652.750 | 625.000 | 4380.23 | 0 | −4803.59 | −3210.47 | −21330.3 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −21.3241 | 190.064 | −57.2810 | 625.000 | −4052.95 | 0 | 12139.4 | 16441.3 | −13327.6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −15.1795 | 45.9146 | −281.582 | 625.000 | −696.962 | 0 | 12046.2 | −17574.9 | −9487.21 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −9.52028 | −175.497 | −421.364 | 625.000 | 1670.78 | 0 | 8885.89 | 11116.2 | −5950.18 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −2.56137 | 177.556 | −505.439 | 625.000 | −454.786 | 0 | 2606.03 | 11843.1 | −1600.85 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 13.9930 | −255.495 | −316.195 | 625.000 | −3575.14 | 0 | −11589.0 | 45594.6 | 8745.64 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 15.4429 | −46.3250 | −273.516 | 625.000 | −715.393 | 0 | −12130.7 | −17537.0 | 9651.82 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 24.0875 | 256.792 | 68.2094 | 625.000 | 6185.48 | 0 | −10689.8 | 46259.1 | 15054.7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 26.9956 | 40.9716 | 216.763 | 625.000 | 1106.05 | 0 | −7970.10 | −18004.3 | 16872.3 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 35.9424 | −97.4212 | 779.854 | 625.000 | −3501.55 | 0 | 9627.32 | −10192.1 | 22464.0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 45.0506 | 165.739 | 1517.55 | 625.000 | 7466.64 | 0 | 45300.8 | 7786.42 | 28156.6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(7\) | \( +1 \) |
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 | 245.10.a.m | 13 | |
| 7.b | odd | 2 | 1 | 245.10.a.l | 13 | ||
| 7.c | even | 3 | 2 | 35.10.e.b | ✓ | 26 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.10.e.b | ✓ | 26 | 7.c | even | 3 | 2 | |
| 245.10.a.l | 13 | 7.b | odd | 2 | 1 | ||
| 245.10.a.m | 13 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(245))\):
|
\( T_{2}^{13} + T_{2}^{12} - 5109 T_{2}^{11} - 3203 T_{2}^{10} + 9635922 T_{2}^{9} - 242128 T_{2}^{8} + \cdots + 96\!\cdots\!52 \)
|
|
\( T_{3}^{13} - 268 T_{3}^{12} - 133081 T_{3}^{11} + 36954174 T_{3}^{10} + 6155444118 T_{3}^{9} + \cdots - 14\!\cdots\!28 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{13} + \cdots + 96\!\cdots\!52 \)
|
| $3$ |
\( T^{13} + \cdots - 14\!\cdots\!28 \)
|
| $5$ |
\( (T - 625)^{13} \)
|
| $7$ |
\( T^{13} \)
|
| $11$ |
\( T^{13} + \cdots + 29\!\cdots\!00 \)
|
| $13$ |
\( T^{13} + \cdots + 17\!\cdots\!28 \)
|
| $17$ |
\( T^{13} + \cdots - 12\!\cdots\!04 \)
|
| $19$ |
\( T^{13} + \cdots + 19\!\cdots\!72 \)
|
| $23$ |
\( T^{13} + \cdots - 12\!\cdots\!81 \)
|
| $29$ |
\( T^{13} + \cdots - 32\!\cdots\!00 \)
|
| $31$ |
\( T^{13} + \cdots - 33\!\cdots\!48 \)
|
| $37$ |
\( T^{13} + \cdots - 18\!\cdots\!00 \)
|
| $41$ |
\( T^{13} + \cdots + 38\!\cdots\!25 \)
|
| $43$ |
\( T^{13} + \cdots - 56\!\cdots\!48 \)
|
| $47$ |
\( T^{13} + \cdots + 75\!\cdots\!00 \)
|
| $53$ |
\( T^{13} + \cdots - 18\!\cdots\!68 \)
|
| $59$ |
\( T^{13} + \cdots - 13\!\cdots\!00 \)
|
| $61$ |
\( T^{13} + \cdots + 26\!\cdots\!48 \)
|
| $67$ |
\( T^{13} + \cdots + 23\!\cdots\!00 \)
|
| $71$ |
\( T^{13} + \cdots - 17\!\cdots\!68 \)
|
| $73$ |
\( T^{13} + \cdots + 18\!\cdots\!72 \)
|
| $79$ |
\( T^{13} + \cdots + 24\!\cdots\!36 \)
|
| $83$ |
\( T^{13} + \cdots + 71\!\cdots\!04 \)
|
| $89$ |
\( T^{13} + \cdots - 15\!\cdots\!52 \)
|
| $97$ |
\( T^{13} + \cdots + 26\!\cdots\!16 \)
|
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