Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [175,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("175.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 175 = 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(28.0671684673\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 158x^{4} + 131x^{3} + 6470x^{2} + 700x - 36400 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2\cdot 5 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 158x^{4} + 131x^{3} + 6470x^{2} + 700x - 36400 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 86\nu^{4} - 126\nu^{3} - 7591\nu^{2} + 6574\nu + 61968 ) / 4164 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 86\nu^{4} + 126\nu^{3} + 11755\nu^{2} - 14902\nu - 282660 ) / 4164 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 17\nu^{5} - 74\nu^{4} - 634\nu^{3} + 4127\nu^{2} - 56238\nu - 59648 ) / 4164 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -17\nu^{5} + 74\nu^{4} + 2022\nu^{3} - 5515\nu^{2} - 50638\nu + 37440 ) / 1388 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 2\beta _1 + 53 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 3\beta_{4} + \beta_{3} + \beta_{2} + 79\beta _1 + 69 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} + 9\beta_{4} + 92\beta_{3} + 143\beta_{2} + 298\beta _1 + 4192 \)
|
| \(\nu^{5}\) | \(=\) |
\( 46\beta_{5} + 396\beta_{4} + 195\beta_{3} + 417\beta_{2} + 7066\beta _1 + 11463 \)
|
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 |
|
−11.0278 | 13.8765 | 89.6120 | 0 | −153.027 | −49.0000 | −635.333 | −50.4430 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −9.12728 | −30.4362 | 51.3073 | 0 | 277.800 | −49.0000 | −176.223 | 683.363 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −4.77018 | −7.35274 | −9.24535 | 0 | 35.0739 | −49.0000 | 196.748 | −188.937 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.433414 | 7.62979 | −31.8122 | 0 | 3.30686 | −49.0000 | −27.6571 | −184.786 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 5.99461 | −22.9537 | 3.93531 | 0 | −137.599 | −49.0000 | −168.237 | 283.874 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 8.49723 | 23.2364 | 40.2029 | 0 | 197.445 | −49.0000 | 69.7019 | 296.930 | 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 | 175.6.a.i | ✓ | 6 |
| 5.b | even | 2 | 1 | 175.6.a.j | yes | 6 | |
| 5.c | odd | 4 | 2 | 175.6.b.h | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 175.6.a.i | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 175.6.a.j | yes | 6 | 5.b | even | 2 | 1 | |
| 175.6.b.h | 12 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 10T_{2}^{5} - 118T_{2}^{4} - 1053T_{2}^{3} + 3544T_{2}^{2} + 23128T_{2} - 10600 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(175))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 10 T^{5} + \cdots - 10600 \)
|
| $3$ |
\( T^{6} + 16 T^{5} + \cdots - 12637296 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( (T + 49)^{6} \)
|
| $11$ |
\( T^{6} + \cdots - 39052128034249 \)
|
| $13$ |
\( T^{6} + \cdots + 11\!\cdots\!40 \)
|
| $17$ |
\( T^{6} + \cdots + 21\!\cdots\!52 \)
|
| $19$ |
\( T^{6} + \cdots + 14\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots - 82\!\cdots\!36 \)
|
| $29$ |
\( T^{6} + \cdots - 32\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots - 30\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots + 58\!\cdots\!44 \)
|
| $41$ |
\( T^{6} + \cdots + 61\!\cdots\!96 \)
|
| $43$ |
\( T^{6} + \cdots - 23\!\cdots\!20 \)
|
| $47$ |
\( T^{6} + \cdots - 32\!\cdots\!56 \)
|
| $53$ |
\( T^{6} + \cdots + 16\!\cdots\!92 \)
|
| $59$ |
\( T^{6} + \cdots + 26\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 29\!\cdots\!24 \)
|
| $67$ |
\( T^{6} + \cdots - 33\!\cdots\!93 \)
|
| $71$ |
\( T^{6} + \cdots - 28\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots - 25\!\cdots\!76 \)
|
| $79$ |
\( T^{6} + \cdots + 98\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 79\!\cdots\!08 \)
|
| $89$ |
\( T^{6} + \cdots + 48\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 59\!\cdots\!08 \)
|
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