Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1075,2,Mod(474,1075)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1075, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1075.474");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1075 = 5^{2} \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1075.b (of order \(2\), degree \(1\), not minimal) |
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: | no |
| Analytic conductor: | \(8.58391821729\) |
| 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: |
\( x^{10} + 18x^{8} + 111x^{6} + 255x^{4} + 129x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 215) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{10} + 18x^{8} + 111x^{6} + 255x^{4} + 129x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 6\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} + 6\nu^{2} + 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{9} - 10\nu^{7} - 19\nu^{5} + 29\nu^{3} + 7\nu ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{9} + 14\nu^{7} + 63\nu^{5} + 95\nu^{3} + 25\nu ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{6} + 11\nu^{4} + 31\nu^{2} + 8 \)
|
| \(\beta_{8}\) | \(=\) |
\( \nu^{8} + 12\nu^{6} + 40\nu^{4} + 26\nu^{2} + 4 \)
|
| \(\beta_{9}\) | \(=\) |
\( 2\nu^{7} + 23\nu^{5} + 69\nu^{3} + 21\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} - 6\beta_{2} + 23 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{9} - 2\beta_{6} - 2\beta_{5} - 7\beta_{3} + 37\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{7} - 11\beta_{4} + 35\beta_{2} - 137 \)
|
| \(\nu^{7}\) | \(=\) |
\( -11\beta_{9} + 23\beta_{6} + 23\beta_{5} + 46\beta_{3} - 229\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{8} - 12\beta_{7} + 92\beta_{4} - 206\beta_{2} + 824 \)
|
| \(\nu^{9}\) | \(=\) |
\( 91\beta_{9} - 192\beta_{6} - 196\beta_{5} - 298\beta_{3} + 1420\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1075\mathbb{Z}\right)^\times\).
| \(n\) | \(302\) | \(476\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 474.1 |
|
− | 2.50989i | − | 3.26173i | −4.29955 | 0 | −8.18658 | − | 3.13519i | 5.77162i | −7.63887 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 474.2 | − | 2.48695i | − | 2.94683i | −4.18492 | 0 | −7.32862 | 3.60359i | 5.43378i | −5.68382 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 474.3 | − | 2.20940i | 0.261901i | −2.88146 | 0 | 0.578644 | 0.988801i | 1.94750i | 2.93141 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 474.4 | − | 0.667116i | 3.03868i | 1.55496 | 0 | 2.02715 | − | 4.17800i | − | 2.37157i | −6.23360 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 474.5 | − | 0.434772i | 2.09168i | 1.81097 | 0 | 0.909404 | − | 3.42802i | − | 1.65691i | −1.37512 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 474.6 | 0.434772i | − | 2.09168i | 1.81097 | 0 | 0.909404 | 3.42802i | 1.65691i | −1.37512 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 474.7 | 0.667116i | − | 3.03868i | 1.55496 | 0 | 2.02715 | 4.17800i | 2.37157i | −6.23360 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 474.8 | 2.20940i | − | 0.261901i | −2.88146 | 0 | 0.578644 | − | 0.988801i | − | 1.94750i | 2.93141 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 474.9 | 2.48695i | 2.94683i | −4.18492 | 0 | −7.32862 | − | 3.60359i | − | 5.43378i | −5.68382 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 474.10 | 2.50989i | 3.26173i | −4.29955 | 0 | −8.18658 | 3.13519i | − | 5.77162i | −7.63887 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
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 | 1075.2.b.h | 10 | |
| 5.b | even | 2 | 1 | inner | 1075.2.b.h | 10 | |
| 5.c | odd | 4 | 1 | 215.2.a.c | ✓ | 5 | |
| 5.c | odd | 4 | 1 | 1075.2.a.m | 5 | ||
| 15.e | even | 4 | 1 | 1935.2.a.u | 5 | ||
| 15.e | even | 4 | 1 | 9675.2.a.ch | 5 | ||
| 20.e | even | 4 | 1 | 3440.2.a.w | 5 | ||
| 215.g | even | 4 | 1 | 9245.2.a.l | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 215.2.a.c | ✓ | 5 | 5.c | odd | 4 | 1 | |
| 1075.2.a.m | 5 | 5.c | odd | 4 | 1 | ||
| 1075.2.b.h | 10 | 1.a | even | 1 | 1 | trivial | |
| 1075.2.b.h | 10 | 5.b | even | 2 | 1 | inner | |
| 1935.2.a.u | 5 | 15.e | even | 4 | 1 | ||
| 3440.2.a.w | 5 | 20.e | even | 4 | 1 | ||
| 9245.2.a.l | 5 | 215.g | even | 4 | 1 | ||
| 9675.2.a.ch | 5 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1075, [\chi])\):
|
\( T_{2}^{10} + 18T_{2}^{8} + 111T_{2}^{6} + 255T_{2}^{4} + 129T_{2}^{2} + 16 \)
|
|
\( T_{3}^{10} + 33T_{3}^{8} + 398T_{3}^{6} + 2065T_{3}^{4} + 3872T_{3}^{2} + 256 \)
|
|
\( T_{7}^{10} + 53T_{7}^{8} + 1050T_{7}^{6} + 9385T_{7}^{4} + 34404T_{7}^{2} + 25600 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 18 T^{8} + \cdots + 16 \)
|
| $3$ |
\( T^{10} + 33 T^{8} + \cdots + 256 \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} + 53 T^{8} + \cdots + 25600 \)
|
| $11$ |
\( (T^{5} + 6 T^{4} + T^{3} + \cdots - 12)^{2} \)
|
| $13$ |
\( T^{10} + 125 T^{8} + \cdots + 4000000 \)
|
| $17$ |
\( T^{10} + 101 T^{8} + \cdots + 256 \)
|
| $19$ |
\( (T^{5} - 6 T^{4} + \cdots - 4608)^{2} \)
|
| $23$ |
\( T^{10} + 109 T^{8} + \cdots + 147456 \)
|
| $29$ |
\( (T^{5} + 6 T^{4} + \cdots - 1152)^{2} \)
|
| $31$ |
\( (T^{5} - 6 T^{4} + \cdots + 128)^{2} \)
|
| $37$ |
\( T^{10} + 81 T^{8} + \cdots + 160000 \)
|
| $41$ |
\( (T^{5} - 2 T^{4} - 99 T^{3} + \cdots + 30)^{2} \)
|
| $43$ |
\( (T^{2} + 1)^{5} \)
|
| $47$ |
\( T^{10} + 248 T^{8} + \cdots + 4194304 \)
|
| $53$ |
\( T^{10} + 149 T^{8} + \cdots + 160000 \)
|
| $59$ |
\( (T^{5} - T^{4} - 16 T^{3} + \cdots + 16)^{2} \)
|
| $61$ |
\( (T^{5} - 20 T^{4} + \cdots - 60672)^{2} \)
|
| $67$ |
\( T^{10} + 353 T^{8} + \cdots + 9216 \)
|
| $71$ |
\( (T^{5} - 4 T^{4} + \cdots - 20352)^{2} \)
|
| $73$ |
\( T^{10} + 193 T^{8} + \cdots + 1236544 \)
|
| $79$ |
\( (T^{5} + 41 T^{4} + \cdots + 18688)^{2} \)
|
| $83$ |
\( T^{10} + 245 T^{8} + \cdots + 5760000 \)
|
| $89$ |
\( (T^{5} + 20 T^{4} + \cdots + 2656)^{2} \)
|
| $97$ |
\( T^{10} + 549 T^{8} + \cdots + 1327104 \)
|
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