Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [175,10,Mod(1,175)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(175, 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("175.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 175 = 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 175.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: | \(90.1312713287\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{10} - 2 x^{9} - 3718 x^{8} + 13493 x^{7} + 4507090 x^{6} - 16532868 x^{5} - 1970350208 x^{4} + \cdots + 455292166912 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{4}\cdot 5^{6} \) |
| 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
\( x^{10} - 2 x^{9} - 3718 x^{8} + 13493 x^{7} + 4507090 x^{6} - 16532868 x^{5} - 1970350208 x^{4} + \cdots + 455292166912 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 3\nu - 745 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 6753464097 \nu^{9} + 210598487008 \nu^{8} - 18245939894918 \nu^{7} - 519178904763767 \nu^{6} + \cdots - 31\!\cdots\!04 ) / 24\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 6753464097 \nu^{9} + 210598487008 \nu^{8} - 18245939894918 \nu^{7} - 519178904763767 \nu^{6} + \cdots - 25\!\cdots\!04 ) / 48\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 85038851213 \nu^{9} + 2589325860832 \nu^{8} - 232167938332622 \nu^{7} + \cdots - 14\!\cdots\!16 ) / 12\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 17105665541 \nu^{9} + 481787218524 \nu^{8} - 47996255081154 \nu^{7} + \cdots - 44\!\cdots\!12 ) / 15\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 97882427571 \nu^{9} - 3143346691444 \nu^{8} + 262795730146874 \nu^{7} + \cdots - 83\!\cdots\!28 ) / 30\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 1830376805529 \nu^{9} + 55860665639456 \nu^{8} + \cdots - 22\!\cdots\!28 ) / 24\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 457193416271 \nu^{9} - 13995111520644 \nu^{8} + \cdots + 79\!\cdots\!72 ) / 30\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 3\beta _1 + 745 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - 5\beta_{3} - 9\beta_{2} + 1292\beta _1 - 2080 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3 \beta_{8} - 2 \beta_{7} + 4 \beta_{6} - 44 \beta_{5} - 19 \beta_{4} - 4 \beta_{3} + 1664 \beta_{2} + \cdots + 959892 \)
|
| \(\nu^{5}\) | \(=\) |
\( 35 \beta_{9} + 106 \beta_{8} - 9 \beta_{7} - 173 \beta_{6} + 750 \beta_{5} + 2244 \beta_{4} + \cdots - 7915385 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 1960 \beta_{9} + 4962 \beta_{8} - 3812 \beta_{7} + 11432 \beta_{6} - 134516 \beta_{5} + \cdots + 1406935496 \)
|
| \(\nu^{7}\) | \(=\) |
\( 138110 \beta_{9} + 339593 \beta_{8} - 18568 \beta_{7} - 562838 \beta_{6} + 2835240 \beta_{5} + \cdots - 20957760014 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 6853371 \beta_{9} + 4980630 \beta_{8} - 6547791 \beta_{7} + 27095141 \beta_{6} - 306232246 \beta_{5} + \cdots + 2221205621513 \)
|
| \(\nu^{9}\) | \(=\) |
\( 366067184 \beta_{9} + 776031952 \beta_{8} - 17484512 \beta_{7} - 1347238696 \beta_{6} + \cdots - 47514903926952 \)
|
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 |
|
−41.8803 | −90.6666 | 1241.96 | 0 | 3797.14 | 2401.00 | −30570.9 | −11462.6 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −32.8310 | 153.262 | 565.874 | 0 | −5031.73 | 2401.00 | −1768.74 | 3806.15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −18.3371 | 40.1240 | −175.752 | 0 | −735.755 | 2401.00 | 12611.4 | −18073.1 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −14.2268 | −156.992 | −309.597 | 0 | 2233.51 | 2401.00 | 11688.7 | 4963.60 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.56944 | −179.608 | −505.398 | 0 | −461.492 | 2401.00 | −2614.14 | 12576.1 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 5.73748 | 271.282 | −479.081 | 0 | 1556.47 | 2401.00 | −5686.31 | 53910.9 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 12.8030 | 59.9558 | −348.084 | 0 | 767.614 | 2401.00 | −11011.6 | −16088.3 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 34.5156 | −270.505 | 679.328 | 0 | −9336.64 | 2401.00 | 5775.42 | 53489.8 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 34.5212 | −75.4776 | 679.716 | 0 | −2605.58 | 2401.00 | 5789.77 | −13986.1 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 39.1284 | 171.626 | 1019.03 | 0 | 6715.46 | 2401.00 | 19839.5 | 9772.48 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
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 | 175.10.a.k | yes | 10 |
| 5.b | even | 2 | 1 | 175.10.a.j | ✓ | 10 | |
| 5.c | odd | 4 | 2 | 175.10.b.i | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 175.10.a.j | ✓ | 10 | 5.b | even | 2 | 1 | |
| 175.10.a.k | yes | 10 | 1.a | even | 1 | 1 | trivial |
| 175.10.b.i | 20 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - 22 T_{2}^{9} - 3502 T_{2}^{8} + 71733 T_{2}^{7} + 3906476 T_{2}^{6} - 67830968 T_{2}^{5} + \cdots + 3156433418240 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(175))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + \cdots + 3156433418240 \)
|
| $3$ |
\( T^{10} + \cdots - 89\!\cdots\!56 \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( (T - 2401)^{10} \)
|
| $11$ |
\( T^{10} + \cdots + 39\!\cdots\!39 \)
|
| $13$ |
\( T^{10} + \cdots - 19\!\cdots\!60 \)
|
| $17$ |
\( T^{10} + \cdots + 10\!\cdots\!96 \)
|
| $19$ |
\( T^{10} + \cdots - 10\!\cdots\!00 \)
|
| $23$ |
\( T^{10} + \cdots + 72\!\cdots\!36 \)
|
| $29$ |
\( T^{10} + \cdots + 18\!\cdots\!00 \)
|
| $31$ |
\( T^{10} + \cdots - 66\!\cdots\!00 \)
|
| $37$ |
\( T^{10} + \cdots - 18\!\cdots\!24 \)
|
| $41$ |
\( T^{10} + \cdots + 41\!\cdots\!16 \)
|
| $43$ |
\( T^{10} + \cdots + 14\!\cdots\!20 \)
|
| $47$ |
\( T^{10} + \cdots - 12\!\cdots\!12 \)
|
| $53$ |
\( T^{10} + \cdots + 84\!\cdots\!68 \)
|
| $59$ |
\( T^{10} + \cdots + 78\!\cdots\!00 \)
|
| $61$ |
\( T^{10} + \cdots + 55\!\cdots\!96 \)
|
| $67$ |
\( T^{10} + \cdots - 24\!\cdots\!57 \)
|
| $71$ |
\( T^{10} + \cdots - 31\!\cdots\!40 \)
|
| $73$ |
\( T^{10} + \cdots - 59\!\cdots\!68 \)
|
| $79$ |
\( T^{10} + \cdots + 26\!\cdots\!00 \)
|
| $83$ |
\( T^{10} + \cdots + 85\!\cdots\!68 \)
|
| $89$ |
\( T^{10} + \cdots - 18\!\cdots\!00 \)
|
| $97$ |
\( T^{10} + \cdots - 45\!\cdots\!28 \)
|
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