Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1210,4,Mod(1,1210)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1210, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1210.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1210 = 2 \cdot 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1210.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: | \(71.3923111069\) |
| 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: |
\( x^{8} - 3x^{7} - 154x^{6} + 466x^{5} + 5711x^{4} - 12645x^{3} - 46521x^{2} + 74790x + 4455 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 5\cdot 11^{2} \) |
| Twist minimal: | no (minimal twist has level 110) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{7} - 154x^{6} + 466x^{5} + 5711x^{4} - 12645x^{3} - 46521x^{2} + 74790x + 4455 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 5141 \nu^{7} - 37788 \nu^{6} + 954776 \nu^{5} + 6885520 \nu^{4} - 49727731 \nu^{3} + \cdots + 1572912945 ) / 120381930 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 224 \nu^{7} - 3904 \nu^{6} - 41861 \nu^{5} + 517296 \nu^{4} + 1522413 \nu^{3} - 13779638 \nu^{2} + \cdots + 43785417 ) / 1337577 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 36209 \nu^{7} + 121518 \nu^{6} + 6115844 \nu^{5} - 18619640 \nu^{4} - 279535399 \nu^{3} + \cdots - 3815725725 ) / 120381930 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 21373 \nu^{7} - 9664 \nu^{6} + 3558268 \nu^{5} + 228000 \nu^{4} - 159108813 \nu^{3} + \cdots - 274379445 ) / 40127310 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 145357 \nu^{7} + 367764 \nu^{6} + 22046812 \nu^{5} - 56923810 \nu^{4} - 792879467 \nu^{3} + \cdots - 7116425235 ) / 120381930 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 8361 \nu^{7} - 18332 \nu^{6} - 1277866 \nu^{5} + 2779905 \nu^{4} + 46811511 \nu^{3} + \cdots + 47145390 ) / 6687885 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{2} - \beta _1 + 39 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} + 3\beta_{5} - 6\beta_{4} - 2\beta_{3} - 2\beta_{2} + 75\beta _1 - 26 \)
|
| \(\nu^{4}\) | \(=\) |
\( 89\beta_{7} + 92\beta_{6} - 18\beta_{5} + 12\beta_{4} + 2\beta_{3} + 152\beta_{2} - 160\beta _1 + 3017 \)
|
| \(\nu^{5}\) | \(=\) |
\( 27\beta_{7} - 21\beta_{6} + 411\beta_{5} - 630\beta_{4} - 263\beta_{3} - 336\beta_{2} + 6764\beta _1 - 5591 \)
|
| \(\nu^{6}\) | \(=\) |
\( 7997 \beta_{7} + 8441 \beta_{6} - 3018 \beta_{5} + 2493 \beta_{4} + 298 \beta_{3} + 16694 \beta_{2} + \cdots + 273124 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 6391 \beta_{7} - 14551 \beta_{6} + 45387 \beta_{5} - 61218 \beta_{4} - 29010 \beta_{3} + \cdots - 871641 \)
|
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 |
|
−2.00000 | −9.39956 | 4.00000 | 5.00000 | 18.7991 | −24.7661 | −8.00000 | 61.3518 | −10.0000 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −4.75698 | 4.00000 | 5.00000 | 9.51396 | −5.27257 | −8.00000 | −4.37115 | −10.0000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | −2.19411 | 4.00000 | 5.00000 | 4.38823 | 32.8447 | −8.00000 | −22.1859 | −10.0000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 0.942460 | 4.00000 | 5.00000 | −1.88492 | 31.4118 | −8.00000 | −26.1118 | −10.0000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 2.49944 | 4.00000 | 5.00000 | −4.99888 | −17.8076 | −8.00000 | −20.7528 | −10.0000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.00000 | 4.65134 | 4.00000 | 5.00000 | −9.30268 | −35.3231 | −8.00000 | −5.36503 | −10.0000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −2.00000 | 8.91779 | 4.00000 | 5.00000 | −17.8356 | 13.1674 | −8.00000 | 52.5269 | −10.0000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −2.00000 | 10.3396 | 4.00000 | 5.00000 | −20.6793 | −10.2544 | −8.00000 | 79.9079 | −10.0000 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(11\) | \( -1 \) |
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 | 1210.4.a.bi | 8 | |
| 11.b | odd | 2 | 1 | 1210.4.a.bj | 8 | ||
| 11.d | odd | 10 | 2 | 110.4.g.d | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 110.4.g.d | ✓ | 16 | 11.d | odd | 10 | 2 | |
| 1210.4.a.bi | 8 | 1.a | even | 1 | 1 | trivial | |
| 1210.4.a.bj | 8 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1210))\):
|
\( T_{3}^{8} - 11T_{3}^{7} - 105T_{3}^{6} + 1271T_{3}^{5} + 1246T_{3}^{4} - 27910T_{3}^{3} + 18801T_{3}^{2} + 110278T_{3} - 99116 \)
|
|
\( T_{7}^{8} + 16 T_{7}^{7} - 2147 T_{7}^{6} - 36575 T_{7}^{5} + 1232421 T_{7}^{4} + 24254160 T_{7}^{3} + \cdots - 11442319344 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{8} \)
|
| $3$ |
\( T^{8} - 11 T^{7} + \cdots - 99116 \)
|
| $5$ |
\( (T - 5)^{8} \)
|
| $7$ |
\( T^{8} + \cdots - 11442319344 \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} + \cdots - 68679308738480 \)
|
| $17$ |
\( T^{8} + \cdots - 262433469242780 \)
|
| $19$ |
\( T^{8} + \cdots - 6590822813225 \)
|
| $23$ |
\( T^{8} + \cdots + 43\!\cdots\!44 \)
|
| $29$ |
\( T^{8} + \cdots - 390205858748400 \)
|
| $31$ |
\( T^{8} + \cdots - 269616652052400 \)
|
| $37$ |
\( T^{8} + \cdots - 10\!\cdots\!80 \)
|
| $41$ |
\( T^{8} + \cdots + 16\!\cdots\!89 \)
|
| $43$ |
\( T^{8} + \cdots + 17\!\cdots\!20 \)
|
| $47$ |
\( T^{8} + \cdots + 28\!\cdots\!80 \)
|
| $53$ |
\( T^{8} + \cdots + 18\!\cdots\!36 \)
|
| $59$ |
\( T^{8} + \cdots - 33\!\cdots\!75 \)
|
| $61$ |
\( T^{8} + \cdots + 21\!\cdots\!00 \)
|
| $67$ |
\( T^{8} + \cdots + 10\!\cdots\!84 \)
|
| $71$ |
\( T^{8} + \cdots + 36\!\cdots\!84 \)
|
| $73$ |
\( T^{8} + \cdots + 44\!\cdots\!20 \)
|
| $79$ |
\( T^{8} + \cdots + 46\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots + 40\!\cdots\!20 \)
|
| $89$ |
\( T^{8} + \cdots + 47\!\cdots\!25 \)
|
| $97$ |
\( T^{8} + \cdots - 25\!\cdots\!24 \)
|
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