Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [37,4,Mod(36,37)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(37, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("37.36");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 37 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 37.b (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: | \(2.18307067021\) |
| 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: |
\( x^{8} - 4x^{7} + 8x^{6} + 58x^{5} + 197x^{4} + 26x^{3} + 2x^{2} + 28x + 196 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{5}\cdot 7 \) |
| Twist minimal: | yes |
| 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
\( x^{8} - 4x^{7} + 8x^{6} + 58x^{5} + 197x^{4} + 26x^{3} + 2x^{2} + 28x + 196 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 14585 \nu^{7} - 398747 \nu^{6} + 1478205 \nu^{5} - 1813577 \nu^{4} - 17599612 \nu^{3} + \cdots - 685412 ) / 2793216 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 18131 \nu^{7} + 44297 \nu^{6} - 62271 \nu^{5} - 1177789 \nu^{4} - 5224748 \nu^{3} + \cdots - 804916 ) / 2793216 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 12445 \nu^{7} + 48263 \nu^{6} - 88337 \nu^{5} - 735795 \nu^{4} - 2595316 \nu^{3} - 145702 \nu^{2} + \cdots + 638388 ) / 698304 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 54955 \nu^{7} - 254737 \nu^{6} + 635895 \nu^{5} + 2705381 \nu^{4} + 9009676 \nu^{3} + \cdots + 2022356 ) / 2793216 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 41609 \nu^{7} + 152779 \nu^{6} - 196621 \nu^{5} - 3002535 \nu^{4} - 7694276 \nu^{3} + \cdots - 15235548 ) / 1396608 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 118039 \nu^{7} - 587029 \nu^{6} + 1393299 \nu^{5} + 5757881 \nu^{4} + 18933052 \nu^{3} + \cdots + 4318244 ) / 2793216 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1205\nu^{7} - 4489\nu^{6} + 6436\nu^{5} + 81594\nu^{4} + 230213\nu^{3} + 12911\nu^{2} - 226730\nu + 918 ) / 21822 \)
|
| \(\nu\) | \(=\) |
\( ( 3\beta_{7} + 2\beta_{6} + 3\beta_{5} - 2\beta_{4} + 9\beta_{3} - 6\beta_{2} + 21 ) / 36 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{6} - 14\beta_{4} - 27\beta_{2} + 3\beta_1 ) / 18 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -87\beta_{7} + 34\beta_{6} - 99\beta_{5} - 142\beta_{4} - 105\beta_{3} - 192\beta_{2} + 18\beta _1 - 981 ) / 36 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -51\beta_{7} - 70\beta_{5} - 41\beta_{3} - 724 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 2469 \beta_{7} - 878 \beta_{6} - 3093 \beta_{5} + 4010 \beta_{4} - 2427 \beta_{3} + 5802 \beta_{2} + \cdots - 31443 ) / 36 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -3194\beta_{6} + 14942\beta_{4} + 22509\beta_{2} - 2469\beta_1 ) / 18 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 72465 \beta_{7} - 25270 \beta_{6} + 92877 \beta_{5} + 116962 \beta_{4} + 68031 \beta_{3} + \cdots + 947571 ) / 36 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 36.1 |
|
− | 5.39603i | 2.81527 | −21.1172 | − | 6.77552i | − | 15.1913i | 16.2590 | 70.7808i | −19.0743 | −36.5609 | |||||||||||||||||||||||||||||||||||||||
| 36.2 | − | 3.73687i | −7.61586 | −5.96420 | 2.08874i | 28.4595i | −23.3854 | − | 7.60752i | 31.0013 | 7.80537 | |||||||||||||||||||||||||||||||||||||||||
| 36.3 | − | 2.49424i | 5.45287 | 1.77875 | 5.37972i | − | 13.6008i | −8.18671 | − | 24.3906i | 2.73384 | 13.4183 | ||||||||||||||||||||||||||||||||||||||||
| 36.4 | − | 0.835079i | −3.65228 | 7.30264 | − | 18.7560i | 3.04994i | 17.3131 | − | 12.7789i | −13.6608 | −15.6628 | ||||||||||||||||||||||||||||||||||||||||
| 36.5 | 0.835079i | −3.65228 | 7.30264 | 18.7560i | − | 3.04994i | 17.3131 | 12.7789i | −13.6608 | −15.6628 | ||||||||||||||||||||||||||||||||||||||||||
| 36.6 | 2.49424i | 5.45287 | 1.77875 | − | 5.37972i | 13.6008i | −8.18671 | 24.3906i | 2.73384 | 13.4183 | ||||||||||||||||||||||||||||||||||||||||||
| 36.7 | 3.73687i | −7.61586 | −5.96420 | − | 2.08874i | − | 28.4595i | −23.3854 | 7.60752i | 31.0013 | 7.80537 | |||||||||||||||||||||||||||||||||||||||||
| 36.8 | 5.39603i | 2.81527 | −21.1172 | 6.77552i | 15.1913i | 16.2590 | − | 70.7808i | −19.0743 | −36.5609 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 37.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 37.4.b.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 333.4.c.d | 8 | ||
| 4.b | odd | 2 | 1 | 592.4.g.c | 8 | ||
| 37.b | even | 2 | 1 | inner | 37.4.b.a | ✓ | 8 |
| 37.d | odd | 4 | 1 | 1369.4.a.a | 4 | ||
| 37.d | odd | 4 | 1 | 1369.4.a.b | 4 | ||
| 111.d | odd | 2 | 1 | 333.4.c.d | 8 | ||
| 148.b | odd | 2 | 1 | 592.4.g.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 37.4.b.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 37.4.b.a | ✓ | 8 | 37.b | even | 2 | 1 | inner |
| 333.4.c.d | 8 | 3.b | odd | 2 | 1 | ||
| 333.4.c.d | 8 | 111.d | odd | 2 | 1 | ||
| 592.4.g.c | 8 | 4.b | odd | 2 | 1 | ||
| 592.4.g.c | 8 | 148.b | odd | 2 | 1 | ||
| 1369.4.a.a | 4 | 37.d | odd | 4 | 1 | ||
| 1369.4.a.b | 4 | 37.d | odd | 4 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(37, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 50 T^{6} + \cdots + 1764 \)
|
| $3$ |
\( (T^{4} + 3 T^{3} + \cdots + 427)^{2} \)
|
| $5$ |
\( T^{8} + 431 T^{6} + \cdots + 2039184 \)
|
| $7$ |
\( (T^{4} - 2 T^{3} + \cdots + 53892)^{2} \)
|
| $11$ |
\( (T^{4} - 45 T^{3} + \cdots + 19197)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 19901912856336 \)
|
| $17$ |
\( T^{8} + \cdots + 7376047628544 \)
|
| $19$ |
\( T^{8} + \cdots + 3733922416896 \)
|
| $23$ |
\( T^{8} + \cdots + 76\!\cdots\!64 \)
|
| $29$ |
\( T^{8} + \cdots + 17\!\cdots\!56 \)
|
| $31$ |
\( T^{8} + \cdots + 48\!\cdots\!04 \)
|
| $37$ |
\( T^{8} + \cdots + 65\!\cdots\!81 \)
|
| $41$ |
\( (T^{4} - 345 T^{3} + \cdots - 99666693)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 12\!\cdots\!96 \)
|
| $47$ |
\( (T^{4} + 696 T^{3} + \cdots - 2544743628)^{2} \)
|
| $53$ |
\( (T^{4} - 384 T^{3} + \cdots - 7068105036)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 33\!\cdots\!76 \)
|
| $61$ |
\( T^{8} + \cdots + 19\!\cdots\!56 \)
|
| $67$ |
\( (T^{4} - 551 T^{3} + \cdots - 18172915056)^{2} \)
|
| $71$ |
\( (T^{4} + 6 T^{3} + \cdots + 46886442732)^{2} \)
|
| $73$ |
\( (T^{4} + 721 T^{3} + \cdots + 35735285943)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 12\!\cdots\!16 \)
|
| $83$ |
\( (T^{4} - 846 T^{3} + \cdots + 56854414512)^{2} \)
|
| $89$ |
\( T^{8} + \cdots + 28\!\cdots\!44 \)
|
| $97$ |
\( T^{8} + \cdots + 46\!\cdots\!96 \)
|
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