Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1100,6,Mod(749,1100)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1100, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1100.749");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1100.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: | \(176.422201794\) |
| 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: |
\( x^{10} + 1622x^{8} + 834457x^{6} + 142814980x^{4} + 7536339200x^{2} + 102851055616 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 5^{4} \) |
| Twist minimal: | no (minimal twist has level 220) |
| 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
\( x^{10} + 1622x^{8} + 834457x^{6} + 142814980x^{4} + 7536339200x^{2} + 102851055616 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{8} - 1150\nu^{6} - 361857\nu^{4} + 7974524\nu^{2} + 11923811072 ) / 37065600 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{8} + 1150\nu^{6} + 361857\nu^{4} + 29091076\nu^{2} + 92856448 ) / 7413120 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 523\nu^{8} + 917350\nu^{6} + 470788311\nu^{4} + 62742981148\nu^{2} + 1437134748544 ) / 1964476800 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -89\nu^{8} - 140180\nu^{6} - 69962343\nu^{4} - 11252548604\nu^{2} - 360898841792 ) / 65482560 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -191\nu^{9} - 149450\nu^{7} + 25023513\nu^{5} + 30746832484\nu^{3} + 6197142977152\nu ) / 2971771545600 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 191\nu^{9} + 149450\nu^{7} - 25023513\nu^{5} - 30746832484\nu^{3} - 3225371431552\nu ) / 1188708618240 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 7903919 \nu^{9} - 14101529450 \nu^{7} - 7104734436183 \nu^{5} + \cdots + 42\!\cdots\!68 \nu ) / 315007783833600 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 2078879 \nu^{9} - 3308763050 \nu^{7} - 1611893475303 \nu^{5} - 231250877339804 \nu^{3} - 74\!\cdots\!12 \nu ) / 78751945958400 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 1768013 \nu^{9} + 2764610750 \nu^{7} + 1353872201541 \nu^{5} + 208727640039188 \nu^{3} + 70\!\cdots\!64 \nu ) / 24231367987200 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{6} + 5\beta_{5} ) / 5 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 5\beta _1 - 1621 ) / 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -19\beta_{9} - 66\beta_{8} + 13\beta_{7} - 1362\beta_{6} - 2942\beta_{5} ) / 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -81\beta_{4} - 291\beta_{3} - 977\beta_{2} - 3676\beta _1 + 961251 ) / 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 17375\beta_{9} + 57810\beta_{8} - 9725\beta_{7} + 864142\beta_{6} + 1937870\beta_{5} ) / 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 72189\beta_{4} + 290439\beta_{3} + 658905\beta_{2} + 2523864\beta _1 - 634780691 ) / 5 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -13070823\beta_{9} - 42786882\beta_{8} + 6520101\beta_{7} - 598816718\beta_{6} - 1307441534\beta_{5} ) / 5 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( -53706933\beta_{4} - 228704463\beta_{3} - 396231937\beta_{2} - 1717712648\beta _1 + 428854743499 ) / 5 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 9445173919 \beta_{9} + 30428365746 \beta_{8} - 4283109613 \beta_{7} + 427403807262 \beta_{6} + 887744127662 \beta_{5} ) / 5 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(177\) | \(551\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 749.1 |
|
0 | − | 29.7722i | 0 | 0 | 0 | 54.8255i | 0 | −643.385 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 749.2 | 0 | − | 22.3640i | 0 | 0 | 0 | − | 139.452i | 0 | −257.147 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 749.3 | 0 | − | 11.5771i | 0 | 0 | 0 | − | 76.8335i | 0 | 108.972 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 749.4 | 0 | − | 9.57436i | 0 | 0 | 0 | 205.693i | 0 | 151.332 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 749.5 | 0 | − | 8.58904i | 0 | 0 | 0 | 218.249i | 0 | 169.228 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 749.6 | 0 | 8.58904i | 0 | 0 | 0 | − | 218.249i | 0 | 169.228 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 749.7 | 0 | 9.57436i | 0 | 0 | 0 | − | 205.693i | 0 | 151.332 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 749.8 | 0 | 11.5771i | 0 | 0 | 0 | 76.8335i | 0 | 108.972 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 749.9 | 0 | 22.3640i | 0 | 0 | 0 | 139.452i | 0 | −257.147 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 749.10 | 0 | 29.7722i | 0 | 0 | 0 | − | 54.8255i | 0 | −643.385 | 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 | 1100.6.b.g | 10 | |
| 5.b | even | 2 | 1 | inner | 1100.6.b.g | 10 | |
| 5.c | odd | 4 | 1 | 220.6.a.e | ✓ | 5 | |
| 5.c | odd | 4 | 1 | 1100.6.a.g | 5 | ||
| 20.e | even | 4 | 1 | 880.6.a.p | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 220.6.a.e | ✓ | 5 | 5.c | odd | 4 | 1 | |
| 880.6.a.p | 5 | 20.e | even | 4 | 1 | ||
| 1100.6.a.g | 5 | 5.c | odd | 4 | 1 | ||
| 1100.6.b.g | 10 | 1.a | even | 1 | 1 | trivial | |
| 1100.6.b.g | 10 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} + 1686T_{3}^{8} + 887481T_{3}^{6} + 173788324T_{3}^{4} + 14084763264T_{3}^{2} + 401813996544 \)
acting on \(S_{6}^{\mathrm{new}}(1100, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( T^{10} + \cdots + 401813996544 \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} + \cdots + 69\!\cdots\!00 \)
|
| $11$ |
\( (T - 121)^{10} \)
|
| $13$ |
\( T^{10} + \cdots + 23\!\cdots\!00 \)
|
| $17$ |
\( T^{10} + \cdots + 53\!\cdots\!96 \)
|
| $19$ |
\( (T^{5} + \cdots - 696671609378624)^{2} \)
|
| $23$ |
\( T^{10} + \cdots + 78\!\cdots\!56 \)
|
| $29$ |
\( (T^{5} + \cdots - 56\!\cdots\!08)^{2} \)
|
| $31$ |
\( (T^{5} + \cdots + 58\!\cdots\!60)^{2} \)
|
| $37$ |
\( T^{10} + \cdots + 13\!\cdots\!16 \)
|
| $41$ |
\( (T^{5} + \cdots - 17\!\cdots\!60)^{2} \)
|
| $43$ |
\( T^{10} + \cdots + 39\!\cdots\!76 \)
|
| $47$ |
\( T^{10} + \cdots + 91\!\cdots\!00 \)
|
| $53$ |
\( T^{10} + \cdots + 13\!\cdots\!00 \)
|
| $59$ |
\( (T^{5} + \cdots - 80\!\cdots\!04)^{2} \)
|
| $61$ |
\( (T^{5} + \cdots + 81\!\cdots\!24)^{2} \)
|
| $67$ |
\( T^{10} + \cdots + 59\!\cdots\!04 \)
|
| $71$ |
\( (T^{5} + \cdots - 13\!\cdots\!96)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 14\!\cdots\!16 \)
|
| $79$ |
\( (T^{5} + \cdots - 27\!\cdots\!12)^{2} \)
|
| $83$ |
\( T^{10} + \cdots + 21\!\cdots\!24 \)
|
| $89$ |
\( (T^{5} + \cdots - 53\!\cdots\!64)^{2} \)
|
| $97$ |
\( T^{10} + \cdots + 31\!\cdots\!56 \)
|
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