Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1150,4,Mod(599,1150)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1150, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1150.599");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1150 = 2 \cdot 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1150.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: | \(67.8521965066\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 136x^{6} + 5308x^{4} + 58833x^{2} + 116964 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 230) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 136x^{6} + 5308x^{4} + 58833x^{2} + 116964 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} + 136\nu^{5} + 4966\nu^{3} + 35577\nu ) / 37962 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 68\nu^{2} + 342 ) / 111 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} - 79\nu^{5} + 1019\nu^{3} + 89367\nu ) / 6327 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 17\nu^{7} + 2141\nu^{5} + 72794\nu^{3} + 584289\nu ) / 18981 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2\nu^{4} + 247\nu^{2} + 4458 ) / 111 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{6} + 118\nu^{4} + 3409\nu^{2} + 15102 ) / 333 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - 2\beta_{3} - 34 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{5} + 3\beta_{4} - 84\beta_{2} - 56\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -68\beta_{6} + 247\beta_{3} + 1970 \)
|
| \(\nu^{5}\) | \(=\) |
\( -315\beta_{5} - 204\beta_{4} + 9486\beta_{2} + 3688\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 333\beta_{7} + 4615\beta_{6} - 22328\beta_{3} - 131656 \)
|
| \(\nu^{7}\) | \(=\) |
\( 27942\beta_{5} + 12846\beta_{4} - 834990\beta_{2} - 259049\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).
| \(n\) | \(51\) | \(277\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 599.1 |
|
− | 2.00000i | − | 7.73081i | −4.00000 | 0 | −15.4616 | 23.5622i | 8.00000i | −32.7654 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 599.2 | − | 2.00000i | − | 0.589969i | −4.00000 | 0 | −1.17994 | − | 18.5077i | 8.00000i | 26.6519 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 599.3 | − | 2.00000i | 4.74869i | −4.00000 | 0 | 9.49738 | 29.3684i | 8.00000i | 4.44993 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 599.4 | − | 2.00000i | 7.57209i | −4.00000 | 0 | 15.1442 | − | 35.4229i | 8.00000i | −30.3365 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 599.5 | 2.00000i | − | 7.57209i | −4.00000 | 0 | 15.1442 | 35.4229i | − | 8.00000i | −30.3365 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 599.6 | 2.00000i | − | 4.74869i | −4.00000 | 0 | 9.49738 | − | 29.3684i | − | 8.00000i | 4.44993 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 599.7 | 2.00000i | 0.589969i | −4.00000 | 0 | −1.17994 | 18.5077i | − | 8.00000i | 26.6519 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 599.8 | 2.00000i | 7.73081i | −4.00000 | 0 | −15.4616 | − | 23.5622i | − | 8.00000i | −32.7654 | 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 | 1150.4.b.n | 8 | |
| 5.b | even | 2 | 1 | inner | 1150.4.b.n | 8 | |
| 5.c | odd | 4 | 1 | 230.4.a.h | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 1150.4.a.p | 4 | ||
| 15.e | even | 4 | 1 | 2070.4.a.bj | 4 | ||
| 20.e | even | 4 | 1 | 1840.4.a.m | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 230.4.a.h | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 1150.4.a.p | 4 | 5.c | odd | 4 | 1 | ||
| 1150.4.b.n | 8 | 1.a | even | 1 | 1 | trivial | |
| 1150.4.b.n | 8 | 5.b | even | 2 | 1 | inner | |
| 1840.4.a.m | 4 | 20.e | even | 4 | 1 | ||
| 2070.4.a.bj | 4 | 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 \(S_{4}^{\mathrm{new}}(1150, [\chi])\):
|
\( T_{3}^{8} + 140T_{3}^{6} + 6116T_{3}^{4} + 79385T_{3}^{2} + 26896 \)
|
|
\( T_{7}^{8} + 3015T_{7}^{6} + 3173141T_{7}^{4} + 1374197220T_{7}^{2} + 205811024896 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{4} \)
|
| $3$ |
\( T^{8} + 140 T^{6} + \cdots + 26896 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + \cdots + 205811024896 \)
|
| $11$ |
\( (T^{4} + 39 T^{3} + \cdots - 977536)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 9357493236004 \)
|
| $17$ |
\( T^{8} + \cdots + 1098689697856 \)
|
| $19$ |
\( (T^{4} + 53 T^{3} + \cdots - 78336)^{2} \)
|
| $23$ |
\( (T^{2} + 529)^{4} \)
|
| $29$ |
\( (T^{4} + 161 T^{3} + \cdots + 1597064)^{2} \)
|
| $31$ |
\( (T^{4} - 388 T^{3} + \cdots - 397027152)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 62742621208576 \)
|
| $41$ |
\( (T^{4} - 484 T^{3} + \cdots + 1506099394)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 29\!\cdots\!04 \)
|
| $47$ |
\( T^{8} + \cdots + 17\!\cdots\!44 \)
|
| $53$ |
\( T^{8} + \cdots + 10\!\cdots\!24 \)
|
| $59$ |
\( (T^{4} - 94 T^{3} + \cdots + 11673423616)^{2} \)
|
| $61$ |
\( (T^{4} - 1153 T^{3} + \cdots - 42772329400)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 10\!\cdots\!64 \)
|
| $71$ |
\( (T^{4} - 200 T^{3} + \cdots + 274201266224)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 22\!\cdots\!36 \)
|
| $79$ |
\( (T^{4} - 908 T^{3} + \cdots + 145785325568)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 31\!\cdots\!84 \)
|
| $89$ |
\( (T^{4} - 1784 T^{3} + \cdots - 969417760000)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 43\!\cdots\!76 \)
|
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