Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [624,3,Mod(385,624)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(624, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("624.385");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 624 = 2^{4} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 624.ba (of order \(4\), degree \(2\), 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: | \(17.0027684961\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.1579585536.10 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} - 4x^{6} + 28x^{5} - 38x^{4} + 8x^{3} + 200x^{2} - 352x + 484 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | no (minimal twist has level 156) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{8} - 2x^{7} - 4x^{6} + 28x^{5} - 38x^{4} + 8x^{3} + 200x^{2} - 352x + 484 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{7} - 46\nu^{6} + 150\nu^{5} - 27\nu^{4} - 720\nu^{3} + 1768\nu^{2} - 2792\nu + 1881 ) / 935 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 118\nu^{7} + 607\nu^{6} - 490\nu^{5} + 384\nu^{4} + 8710\nu^{3} - 19856\nu^{2} + 35024\nu + 91398 ) / 68510 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3448 \nu^{7} - 21163 \nu^{6} - 45670 \nu^{5} + 200494 \nu^{4} - 126360 \nu^{3} - 635936 \nu^{2} + \cdots - 2421122 ) / 753610 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 3448 \nu^{7} + 21163 \nu^{6} + 45670 \nu^{5} - 200494 \nu^{4} + 126360 \nu^{3} + 635936 \nu^{2} + \cdots + 2421122 ) / 753610 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 86\nu^{7} - 227\nu^{6} + 162\nu^{5} + 758\nu^{4} - 1794\nu^{3} + 8124\nu^{2} - 8408\nu + 13662 ) / 8866 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3777 \nu^{7} - 11052 \nu^{6} - 45875 \nu^{5} + 222466 \nu^{4} - 199615 \nu^{3} - 521764 \nu^{2} + \cdots - 3038343 ) / 376805 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -1123\nu^{7} + 3803\nu^{6} - 3465\nu^{5} - 12944\nu^{4} + 51805\nu^{3} - 107134\nu^{2} + 75416\nu - 40743 ) / 34255 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + 4\beta_{5} + \beta_{4} - \beta_{3} - 4\beta_{2} + 2 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{7} + 6\beta_{5} + \beta_{4} + 2\beta_{3} + \beta_{2} + \beta _1 - 7 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{7} + 5\beta_{6} + 13\beta_{5} - 3\beta_{4} - 7\beta_{3} + \beta_{2} + 4\beta _1 + 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( 5\beta_{7} + 2\beta_{6} + 7\beta_{5} + \beta_{4} + 2\beta_{3} + 34\beta_{2} + 12\beta _1 - 55 \)
|
| \(\nu^{6}\) | \(=\) |
\( 22\beta_{6} - 4\beta_{5} - 22\beta_{4} - 70\beta_{3} + 22\beta_{2} \)
|
| \(\nu^{7}\) | \(=\) |
\( -59\beta_{7} - 37\beta_{6} - 99\beta_{5} - 11\beta_{4} + 11\beta_{3} + 195\beta_{2} - 37\beta _1 - 331 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).
| \(n\) | \(79\) | \(145\) | \(209\) | \(469\) |
| \(\chi(n)\) | \(1\) | \(\beta_{5}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 385.1 |
|
0 | −1.73205 | 0 | −0.848026 | − | 0.848026i | 0 | −5.42810 | + | 5.42810i | 0 | 3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||
| 385.2 | 0 | −1.73205 | 0 | 5.58008 | + | 5.58008i | 0 | 7.42810 | − | 7.42810i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
| 385.3 | 0 | 1.73205 | 0 | −0.658261 | − | 0.658261i | 0 | −1.58447 | + | 1.58447i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
| 385.4 | 0 | 1.73205 | 0 | 1.92621 | + | 1.92621i | 0 | 3.58447 | − | 3.58447i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
| 577.1 | 0 | −1.73205 | 0 | −0.848026 | + | 0.848026i | 0 | −5.42810 | − | 5.42810i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
| 577.2 | 0 | −1.73205 | 0 | 5.58008 | − | 5.58008i | 0 | 7.42810 | + | 7.42810i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
| 577.3 | 0 | 1.73205 | 0 | −0.658261 | + | 0.658261i | 0 | −1.58447 | − | 1.58447i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
| 577.4 | 0 | 1.73205 | 0 | 1.92621 | − | 1.92621i | 0 | 3.58447 | + | 3.58447i | 0 | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.d | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 624.3.ba.d | 8 | |
| 4.b | odd | 2 | 1 | 156.3.j.a | ✓ | 8 | |
| 12.b | even | 2 | 1 | 468.3.m.d | 8 | ||
| 13.d | odd | 4 | 1 | inner | 624.3.ba.d | 8 | |
| 52.f | even | 4 | 1 | 156.3.j.a | ✓ | 8 | |
| 156.l | odd | 4 | 1 | 468.3.m.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 156.3.j.a | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 156.3.j.a | ✓ | 8 | 52.f | even | 4 | 1 | |
| 468.3.m.d | 8 | 12.b | even | 2 | 1 | ||
| 468.3.m.d | 8 | 156.l | odd | 4 | 1 | ||
| 624.3.ba.d | 8 | 1.a | even | 1 | 1 | trivial | |
| 624.3.ba.d | 8 | 13.d | odd | 4 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 12T_{5}^{7} + 72T_{5}^{6} - 48T_{5}^{5} - 48T_{5}^{4} + 288T_{5}^{3} + 1152T_{5}^{2} + 1152T_{5} + 576 \)
acting on \(S_{3}^{\mathrm{new}}(624, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{2} - 3)^{4} \)
|
| $5$ |
\( T^{8} - 12 T^{7} + \cdots + 576 \)
|
| $7$ |
\( T^{8} - 8 T^{7} + \cdots + 839056 \)
|
| $11$ |
\( T^{8} + 12 T^{7} + \cdots + 90782784 \)
|
| $13$ |
\( T^{8} - 24 T^{7} + \cdots + 815730721 \)
|
| $17$ |
\( T^{8} + 1296 T^{6} + \cdots + 427993344 \)
|
| $19$ |
\( T^{8} + 88 T^{7} + \cdots + 2972176 \)
|
| $23$ |
\( T^{8} + \cdots + 8422834176 \)
|
| $29$ |
\( (T^{4} - 12 T^{3} + \cdots - 75504)^{2} \)
|
| $31$ |
\( T^{8} - 16 T^{7} + \cdots + 10061584 \)
|
| $37$ |
\( T^{8} + \cdots + 149561639824 \)
|
| $41$ |
\( T^{8} + \cdots + 28199813184 \)
|
| $43$ |
\( T^{8} + \cdots + 619986161664 \)
|
| $47$ |
\( T^{8} + \cdots + 4080820170816 \)
|
| $53$ |
\( (T^{4} + 36 T^{3} + \cdots - 976944)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 4746925417536 \)
|
| $61$ |
\( (T^{4} + 96 T^{3} + \cdots - 249024)^{2} \)
|
| $67$ |
\( T^{8} + 16 T^{7} + \cdots + 929274256 \)
|
| $71$ |
\( T^{8} + \cdots + 448998131059776 \)
|
| $73$ |
\( T^{8} + \cdots + 601874565729424 \)
|
| $79$ |
\( (T^{4} + 24 T^{3} + \cdots + 613248)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 31\!\cdots\!84 \)
|
| $89$ |
\( T^{8} + \cdots + 1057162337856 \)
|
| $97$ |
\( T^{8} + \cdots + 63914204772496 \)
|
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