Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [209,4,Mod(1,209)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(209, 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("209.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 209 = 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 209.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: | \(12.3313991912\) |
| Analytic rank: | \(0\) |
| Dimension: | \(13\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{13} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{13} - 2 x^{12} - 91 x^{11} + 176 x^{10} + 3117 x^{9} - 5786 x^{8} - 49725 x^{7} + 87196 x^{6} + \cdots - 86016 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{12}\) 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
\( x^{13} - 2 x^{12} - 91 x^{11} + 176 x^{10} + 3117 x^{9} - 5786 x^{8} - 49725 x^{7} + 87196 x^{6} + \cdots - 86016 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 14 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 179351 \nu^{12} - 6111956 \nu^{11} + 22627657 \nu^{10} + 624540444 \nu^{9} + \cdots + 665042042368 ) / 142316773888 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 632365 \nu^{12} - 12143320 \nu^{11} + 16549089 \nu^{10} + 1022191756 \nu^{9} + \cdots + 4814040741376 ) / 142316773888 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1490925 \nu^{12} - 3374106 \nu^{11} - 117105595 \nu^{10} + 266779852 \nu^{9} + \cdots - 863686138368 ) / 142316773888 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 1017601 \nu^{12} - 276553 \nu^{11} - 80542386 \nu^{10} - 32232160 \nu^{9} + \cdots - 637257679104 ) / 71158386944 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1920205 \nu^{12} - 11132819 \nu^{11} - 167383848 \nu^{10} + 911265656 \nu^{9} + \cdots + 1253807937024 ) / 71158386944 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 139235 \nu^{12} - 445304 \nu^{11} - 12795239 \nu^{10} + 39284284 \nu^{9} + 442283335 \nu^{8} + \cdots + 4926684672 ) / 3309692416 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 10489149 \nu^{12} - 29313602 \nu^{11} + 912896051 \nu^{10} + 2404111664 \nu^{9} + \cdots - 463328483328 ) / 142316773888 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 7125793 \nu^{12} + 3639059 \nu^{11} - 649932012 \nu^{10} - 291399086 \nu^{9} + \cdots + 456899993088 ) / 71158386944 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 16394151 \nu^{12} + 20435252 \nu^{11} + 1462858283 \nu^{10} - 1790208568 \nu^{9} + \cdots - 802773986816 ) / 142316773888 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 17020531 \nu^{12} + 12986530 \nu^{11} - 1507070505 \nu^{10} - 1046131624 \nu^{9} + \cdots + 548980252160 ) / 142316773888 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 14 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{12} + \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} + 23\beta _1 - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2 \beta_{12} - 4 \beta_{10} - 2 \beta_{9} - 2 \beta_{6} + 6 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + \cdots + 310 \)
|
| \(\nu^{5}\) | \(=\) |
\( 39\beta_{12} + 39\beta_{9} - 39\beta_{8} - 8\beta_{7} - 39\beta_{6} + 63\beta_{5} + 8\beta_{4} + 595\beta _1 - 62 \)
|
| \(\nu^{6}\) | \(=\) |
\( 126 \beta_{12} + 8 \beta_{11} - 212 \beta_{10} - 78 \beta_{9} + 24 \beta_{8} - 32 \beta_{7} + \cdots + 7782 \)
|
| \(\nu^{7}\) | \(=\) |
\( 1275 \beta_{12} + 72 \beta_{11} + 64 \beta_{10} + 1283 \beta_{9} - 1251 \beta_{8} - 416 \beta_{7} + \cdots - 1566 \)
|
| \(\nu^{8}\) | \(=\) |
\( 5526 \beta_{12} + 480 \beta_{11} - 8460 \beta_{10} - 2454 \beta_{9} + 1392 \beta_{8} - 2016 \beta_{7} + \cdots + 208294 \)
|
| \(\nu^{9}\) | \(=\) |
\( 40243 \beta_{12} + 4864 \beta_{11} + 3776 \beta_{10} + 40307 \beta_{9} - 37843 \beta_{8} - 16536 \beta_{7} + \cdots - 32310 \)
|
| \(\nu^{10}\) | \(=\) |
\( 210454 \beta_{12} + 19096 \beta_{11} - 304948 \beta_{10} - 73094 \beta_{9} + 55848 \beta_{8} + \cdots + 5793542 \)
|
| \(\nu^{11}\) | \(=\) |
\( 1263587 \beta_{12} + 225368 \beta_{11} + 150720 \beta_{10} + 1243259 \beta_{9} - 1122331 \beta_{8} + \cdots - 421022 \)
|
| \(\nu^{12}\) | \(=\) |
\( 7470214 \beta_{12} + 645520 \beta_{11} - 10444476 \beta_{10} - 2139558 \beta_{9} + 1929504 \beta_{8} + \cdots + 165422118 \)
|
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 |
|
−5.62918 | 10.0382 | 23.6877 | −2.17343 | −56.5070 | 1.58350 | −88.3091 | 73.7660 | 12.2346 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.82823 | −1.78306 | 15.3118 | −12.0513 | 8.60905 | 32.7048 | −35.3032 | −23.8207 | 58.1866 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.48413 | −7.61802 | 12.1074 | 0.924111 | 34.1602 | −24.8241 | −18.4180 | 31.0342 | −4.14383 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −3.21121 | −2.40039 | 2.31189 | 14.8396 | 7.70816 | 20.3294 | 18.2657 | −21.2381 | −47.6532 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.98386 | 3.69270 | 0.903438 | −11.2324 | −11.0185 | −31.8562 | 21.1752 | −13.3640 | 33.5161 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.33472 | 8.27114 | −6.21852 | 12.3033 | −11.0397 | 31.4819 | 18.9777 | 41.4117 | −16.4215 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.224790 | −4.32964 | −7.94947 | −17.8280 | 0.973257 | −19.8188 | 3.58527 | −8.25425 | 4.00756 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.264443 | −4.63433 | −7.93007 | 7.68704 | −1.22552 | −7.09559 | −4.21260 | −5.52296 | 2.03279 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.57872 | 8.66678 | −1.35022 | 16.9434 | 22.3492 | −6.89893 | −24.1116 | 48.1130 | 43.6921 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 3.02090 | −8.02896 | 1.12583 | −20.1089 | −24.2547 | 33.1653 | −20.7662 | 37.4642 | −60.7468 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 4.33968 | 2.40523 | 10.8328 | 6.68433 | 10.4379 | 24.5435 | 12.2935 | −21.2149 | 29.0079 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 4.99897 | 8.31216 | 16.9897 | −8.69474 | 41.5522 | −3.84543 | 44.9392 | 42.0920 | −43.4648 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 5.49342 | −1.59182 | 22.1777 | 20.7070 | −8.74456 | −10.4694 | 77.8840 | −24.4661 | 113.752 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \( +1 \) |
| \(19\) | \( +1 \) |
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 | 209.4.a.c | ✓ | 13 |
| 3.b | odd | 2 | 1 | 1881.4.a.k | 13 | ||
| 11.b | odd | 2 | 1 | 2299.4.a.k | 13 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 209.4.a.c | ✓ | 13 | 1.a | even | 1 | 1 | trivial |
| 1881.4.a.k | 13 | 3.b | odd | 2 | 1 | ||
| 2299.4.a.k | 13 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{13} + 2 T_{2}^{12} - 91 T_{2}^{11} - 176 T_{2}^{10} + 3117 T_{2}^{9} + 5786 T_{2}^{8} + \cdots + 86016 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(209))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{13} + 2 T^{12} + \cdots + 86016 \)
|
| $3$ |
\( T^{13} + \cdots + 444198064 \)
|
| $5$ |
\( T^{13} + \cdots - 2789379165656 \)
|
| $7$ |
\( T^{13} + \cdots + 833325521817216 \)
|
| $11$ |
\( (T + 11)^{13} \)
|
| $13$ |
\( T^{13} + \cdots + 49\!\cdots\!88 \)
|
| $17$ |
\( T^{13} + \cdots + 90\!\cdots\!88 \)
|
| $19$ |
\( (T + 19)^{13} \)
|
| $23$ |
\( T^{13} + \cdots - 27\!\cdots\!72 \)
|
| $29$ |
\( T^{13} + \cdots - 10\!\cdots\!08 \)
|
| $31$ |
\( T^{13} + \cdots + 19\!\cdots\!88 \)
|
| $37$ |
\( T^{13} + \cdots - 29\!\cdots\!84 \)
|
| $41$ |
\( T^{13} + \cdots + 69\!\cdots\!40 \)
|
| $43$ |
\( T^{13} + \cdots - 13\!\cdots\!28 \)
|
| $47$ |
\( T^{13} + \cdots - 96\!\cdots\!00 \)
|
| $53$ |
\( T^{13} + \cdots - 93\!\cdots\!12 \)
|
| $59$ |
\( T^{13} + \cdots + 89\!\cdots\!68 \)
|
| $61$ |
\( T^{13} + \cdots + 16\!\cdots\!52 \)
|
| $67$ |
\( T^{13} + \cdots + 41\!\cdots\!36 \)
|
| $71$ |
\( T^{13} + \cdots + 18\!\cdots\!56 \)
|
| $73$ |
\( T^{13} + \cdots + 37\!\cdots\!24 \)
|
| $79$ |
\( T^{13} + \cdots - 71\!\cdots\!36 \)
|
| $83$ |
\( T^{13} + \cdots - 24\!\cdots\!72 \)
|
| $89$ |
\( T^{13} + \cdots - 26\!\cdots\!76 \)
|
| $97$ |
\( T^{13} + \cdots - 88\!\cdots\!40 \)
|
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