Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [72,10,Mod(37,72)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(72, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("72.37");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 72 = 2^{3} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 72.d (of order \(2\), degree \(1\), 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: | \(37.0825802038\) |
| 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: |
\( x^{8} - x^{7} + 59x^{6} - 313x^{5} - 315x^{4} - 92091x^{3} + 1261649x^{2} - 16074123x + 251007534 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{28}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 8) |
| 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
\( x^{8} - x^{7} + 59x^{6} - 313x^{5} - 315x^{4} - 92091x^{3} + 1261649x^{2} - 16074123x + 251007534 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 4\nu^{2} + 2\nu + 58 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - \nu^{7} - 318 \nu^{6} + 2947 \nu^{5} + 70006 \nu^{4} + 49989 \nu^{3} - 742730 \nu^{2} + \cdots - 529423278 ) / 8912896 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 55 \nu^{7} - 2258 \nu^{6} - 22875 \nu^{5} - 188326 \nu^{4} - 3365165 \nu^{3} + 4878490 \nu^{2} + \cdots + 120340126 ) / 4456448 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 545 \nu^{7} + 770 \nu^{6} - 30237 \nu^{5} + 273462 \nu^{4} + 17893 \nu^{3} + 49386870 \nu^{2} + \cdots + 7650083474 ) / 8912896 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 11 \nu^{7} + 22 \nu^{6} + 1377 \nu^{5} - 4078 \nu^{4} - 123529 \nu^{3} + 501650 \nu^{2} + \cdots + 79996934 ) / 131072 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 703 \nu^{7} + 10306 \nu^{6} - 39357 \nu^{5} + 642294 \nu^{4} - 14156923 \nu^{3} + \cdots + 4806907090 ) / 8912896 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} - \beta _1 - 58 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{7} - 2\beta_{6} + \beta_{5} - 4\beta_{4} - 15\beta_{3} - 2\beta_{2} - 57\beta _1 + 774 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 14\beta_{7} - 10\beta_{6} + 33\beta_{5} - 12\beta_{4} + 657\beta_{3} + 4\beta_{2} + 301\beta _1 + 9346 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -16\beta_{7} + 88\beta_{6} - 128\beta_{5} - 72\beta_{4} + 608\beta_{3} + 99\beta_{2} + 574\beta _1 + 97102 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 1066 \beta_{7} + 410 \beta_{6} + 1467 \beta_{5} - 2972 \beta_{4} - 29877 \beta_{3} - 469 \beta_{2} + \cdots - 3751528 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 13522 \beta_{7} - 23454 \beta_{6} - 81689 \beta_{5} + 1428 \beta_{4} + 109815 \beta_{3} + \cdots + 74348202 ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(55\) | \(65\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
−18.9692 | − | 12.3357i | 0 | 207.662 | + | 467.996i | − | 506.862i | 0 | 300.249 | 1833.85 | − | 11439.2i | 0 | −6252.48 | + | 9614.77i | |||||||||||||||||||||||||||||||||
| 37.2 | −18.9692 | + | 12.3357i | 0 | 207.662 | − | 467.996i | 506.862i | 0 | 300.249 | 1833.85 | + | 11439.2i | 0 | −6252.48 | − | 9614.77i | |||||||||||||||||||||||||||||||||||
| 37.3 | −2.86961 | − | 22.4447i | 0 | −495.531 | + | 128.815i | 1417.55i | 0 | 5087.57 | 4313.20 | + | 10752.4i | 0 | 31816.6 | − | 4067.83i | |||||||||||||||||||||||||||||||||||
| 37.4 | −2.86961 | + | 22.4447i | 0 | −495.531 | − | 128.815i | − | 1417.55i | 0 | 5087.57 | 4313.20 | − | 10752.4i | 0 | 31816.6 | + | 4067.83i | ||||||||||||||||||||||||||||||||||
| 37.5 | 9.36065 | − | 20.6004i | 0 | −336.757 | − | 385.667i | 292.339i | 0 | −9955.46 | −11097.2 | + | 3327.24i | 0 | 6022.31 | + | 2736.48i | |||||||||||||||||||||||||||||||||||
| 37.6 | 9.36065 | + | 20.6004i | 0 | −336.757 | + | 385.667i | − | 292.339i | 0 | −9955.46 | −11097.2 | − | 3327.24i | 0 | 6022.31 | − | 2736.48i | ||||||||||||||||||||||||||||||||||
| 37.7 | 21.4782 | − | 7.11952i | 0 | 410.625 | − | 305.829i | − | 2583.09i | 0 | 6967.65 | 6642.12 | − | 9492.10i | 0 | −18390.4 | − | 55480.1i | ||||||||||||||||||||||||||||||||||
| 37.8 | 21.4782 | + | 7.11952i | 0 | 410.625 | + | 305.829i | 2583.09i | 0 | 6967.65 | 6642.12 | + | 9492.10i | 0 | −18390.4 | + | 55480.1i | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 72.10.d.b | 8 | |
| 3.b | odd | 2 | 1 | 8.10.b.a | ✓ | 8 | |
| 4.b | odd | 2 | 1 | 288.10.d.b | 8 | ||
| 8.b | even | 2 | 1 | inner | 72.10.d.b | 8 | |
| 8.d | odd | 2 | 1 | 288.10.d.b | 8 | ||
| 12.b | even | 2 | 1 | 32.10.b.a | 8 | ||
| 24.f | even | 2 | 1 | 32.10.b.a | 8 | ||
| 24.h | odd | 2 | 1 | 8.10.b.a | ✓ | 8 | |
| 48.i | odd | 4 | 2 | 256.10.a.p | 8 | ||
| 48.k | even | 4 | 2 | 256.10.a.s | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.10.b.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 8.10.b.a | ✓ | 8 | 24.h | odd | 2 | 1 | |
| 32.10.b.a | 8 | 12.b | even | 2 | 1 | ||
| 32.10.b.a | 8 | 24.f | even | 2 | 1 | ||
| 72.10.d.b | 8 | 1.a | even | 1 | 1 | trivial | |
| 72.10.d.b | 8 | 8.b | even | 2 | 1 | inner | |
| 256.10.a.p | 8 | 48.i | odd | 4 | 2 | ||
| 256.10.a.s | 8 | 48.k | even | 4 | 2 | ||
| 288.10.d.b | 8 | 4.b | odd | 2 | 1 | ||
| 288.10.d.b | 8 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 9024192T_{5}^{6} + 16402176226816T_{5}^{4} + 4781063725538918400T_{5}^{2} + 294381261264939950080000 \)
acting on \(S_{10}^{\mathrm{new}}(72, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots + 68719476736 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + \cdots + 29\!\cdots\!00 \)
|
| $7$ |
\( (T^{4} + \cdots - 105959154151424)^{2} \)
|
| $11$ |
\( T^{8} + \cdots + 18\!\cdots\!00 \)
|
| $13$ |
\( T^{8} + \cdots + 18\!\cdots\!00 \)
|
| $17$ |
\( (T^{4} + \cdots + 49\!\cdots\!48)^{2} \)
|
| $19$ |
\( T^{8} + \cdots + 48\!\cdots\!72 \)
|
| $23$ |
\( (T^{4} + \cdots - 42\!\cdots\!76)^{2} \)
|
| $29$ |
\( T^{8} + \cdots + 46\!\cdots\!00 \)
|
| $31$ |
\( (T^{4} + \cdots + 67\!\cdots\!56)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 17\!\cdots\!48 \)
|
| $41$ |
\( (T^{4} + \cdots + 74\!\cdots\!80)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 94\!\cdots\!08 \)
|
| $47$ |
\( (T^{4} + \cdots + 54\!\cdots\!12)^{2} \)
|
| $53$ |
\( T^{8} + \cdots + 11\!\cdots\!88 \)
|
| $59$ |
\( T^{8} + \cdots + 77\!\cdots\!08 \)
|
| $61$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
|
| $67$ |
\( T^{8} + \cdots + 13\!\cdots\!68 \)
|
| $71$ |
\( (T^{4} + \cdots + 60\!\cdots\!52)^{2} \)
|
| $73$ |
\( (T^{4} + \cdots - 27\!\cdots\!12)^{2} \)
|
| $79$ |
\( (T^{4} + \cdots + 14\!\cdots\!40)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 37\!\cdots\!68 \)
|
| $89$ |
\( (T^{4} + \cdots + 17\!\cdots\!20)^{2} \)
|
| $97$ |
\( (T^{4} + \cdots + 41\!\cdots\!72)^{2} \)
|
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