Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [37,10,Mod(1,37)]
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([0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("37.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 37 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 37.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: | \(19.0563259381\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{13} - 6 x^{12} - 4637 x^{11} + 28852 x^{10} + 8006690 x^{9} - 52024972 x^{8} - 6415977160 x^{7} + \cdots - 99\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{4} \) |
| 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
\( x^{13} - 6 x^{12} - 4637 x^{11} + 28852 x^{10} + 8006690 x^{9} - 52024972 x^{8} - 6415977160 x^{7} + \cdots - 99\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 29\!\cdots\!07 \nu^{12} + \cdots - 45\!\cdots\!60 ) / 17\!\cdots\!80 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 48\!\cdots\!29 \nu^{12} + \cdots + 43\!\cdots\!00 ) / 22\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 95\!\cdots\!01 \nu^{12} + \cdots - 34\!\cdots\!00 ) / 38\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 95\!\cdots\!01 \nu^{12} + \cdots - 66\!\cdots\!00 ) / 38\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 11\!\cdots\!31 \nu^{12} + \cdots - 13\!\cdots\!00 ) / 38\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 89\!\cdots\!97 \nu^{12} + \cdots - 30\!\cdots\!00 ) / 26\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 15\!\cdots\!29 \nu^{12} + \cdots + 62\!\cdots\!00 ) / 13\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 13\!\cdots\!73 \nu^{12} + \cdots + 86\!\cdots\!00 ) / 89\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 29\!\cdots\!65 \nu^{12} + \cdots + 70\!\cdots\!60 ) / 17\!\cdots\!80 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 24\!\cdots\!83 \nu^{12} + \cdots + 22\!\cdots\!00 ) / 89\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 14\!\cdots\!17 \nu^{12} + \cdots + 47\!\cdots\!00 ) / 26\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + 716 \)
|
| \(\nu^{3}\) | \(=\) |
\( -4\beta_{11} - \beta_{10} + 2\beta_{9} + \beta_{8} + \beta_{7} - 39\beta_{5} - 3\beta_{4} + 1193\beta _1 - 780 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 33 \beta_{12} + 4 \beta_{11} + 38 \beta_{10} + 51 \beta_{9} + 2 \beta_{8} - 53 \beta_{7} + \cdots + 842060 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 95 \beta_{12} - 8248 \beta_{11} - 2286 \beta_{10} + 4897 \beta_{9} + 1818 \beta_{8} + 857 \beta_{7} + \cdots - 1099204 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 87801 \beta_{12} + 23888 \beta_{11} + 82984 \beta_{10} + 90707 \beta_{9} - 10708 \beta_{8} + \cdots + 1161780604 \)
|
| \(\nu^{7}\) | \(=\) |
\( 77019 \beta_{12} - 14041560 \beta_{11} - 4533532 \beta_{10} + 9403063 \beta_{9} + 3218392 \beta_{8} + \cdots - 1834152692 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 179166405 \beta_{12} + 65427560 \beta_{11} + 146968080 \beta_{10} + 136600335 \beta_{9} + \cdots + 1720483979356 \)
|
| \(\nu^{9}\) | \(=\) |
\( 780822343 \beta_{12} - 22736645448 \beta_{11} - 8586838112 \beta_{10} + 16487608923 \beta_{9} + \cdots - 3661488436788 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 333503309421 \beta_{12} + 145590909016 \beta_{11} + 246156622536 \beta_{10} + 200220734599 \beta_{9} + \cdots + 26\!\cdots\!76 \)
|
| \(\nu^{11}\) | \(=\) |
\( 2448672829367 \beta_{12} - 36293262572104 \beta_{11} - 15790210549896 \beta_{10} + 27823504604155 \beta_{9} + \cdots - 77\!\cdots\!64 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 594920210706789 \beta_{12} + 296669226450296 \beta_{11} + 405858436830440 \beta_{10} + \cdots + 40\!\cdots\!00 \)
|
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 |
|
−41.5553 | 233.991 | 1214.84 | 1451.63 | −9723.56 | −11035.4 | −29206.7 | 35068.8 | −60322.9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −40.3784 | 50.3948 | 1118.42 | −1136.25 | −2034.86 | 800.534 | −24486.1 | −17143.4 | 45879.8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −27.8517 | −234.213 | 263.715 | −396.481 | 6523.22 | −10779.8 | 6915.15 | 35172.6 | 11042.7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −22.6648 | −201.289 | 1.69319 | −1574.64 | 4562.17 | 8238.87 | 11566.0 | 20834.2 | 35688.8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −22.0873 | 104.113 | −24.1499 | 861.677 | −2299.57 | −5593.71 | 11842.1 | −8843.54 | −19032.1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −12.3415 | −116.116 | −359.688 | 44.3181 | 1433.04 | 8404.17 | 10757.9 | −6200.11 | −546.952 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −3.19145 | 133.736 | −501.815 | 16.5967 | −426.813 | 4316.78 | 3235.54 | −1797.60 | −52.9675 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.674428 | −256.621 | −511.545 | 2353.95 | −173.072 | 2771.95 | −690.308 | 46171.4 | 1587.57 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 18.4888 | 161.312 | −170.163 | −1979.14 | 2982.47 | 3622.73 | −12612.4 | 6338.52 | −36592.1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 20.7923 | 116.492 | −79.6812 | −29.0102 | 2422.13 | −10914.4 | −12302.4 | −6112.64 | −603.188 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 24.7693 | −84.2227 | 101.518 | 1373.33 | −2086.14 | −3156.66 | −10167.4 | −12589.5 | 34016.4 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 34.3196 | −129.318 | 665.837 | −1076.40 | −4438.14 | 9336.44 | 5279.62 | −2959.93 | −36941.6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 39.0259 | −29.2595 | 1011.02 | −2068.58 | −1141.88 | −8587.50 | 19474.9 | −18826.9 | −80728.4 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(37\) | \(1\) |
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 | 37.10.a.a | ✓ | 13 |
| 3.b | odd | 2 | 1 | 333.10.a.c | 13 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 37.10.a.a | ✓ | 13 | 1.a | even | 1 | 1 | trivial |
| 333.10.a.c | 13 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{13} + 32 T_{2}^{12} - 4181 T_{2}^{11} - 126994 T_{2}^{10} + 6431510 T_{2}^{9} + \cdots + 79\!\cdots\!76 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(37))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{13} + \cdots + 79\!\cdots\!76 \)
|
| $3$ |
\( T^{13} + \cdots + 13\!\cdots\!12 \)
|
| $5$ |
\( T^{13} + \cdots + 26\!\cdots\!00 \)
|
| $7$ |
\( T^{13} + \cdots - 44\!\cdots\!20 \)
|
| $11$ |
\( T^{13} + \cdots + 62\!\cdots\!00 \)
|
| $13$ |
\( T^{13} + \cdots - 21\!\cdots\!04 \)
|
| $17$ |
\( T^{13} + \cdots + 13\!\cdots\!00 \)
|
| $19$ |
\( T^{13} + \cdots - 84\!\cdots\!00 \)
|
| $23$ |
\( T^{13} + \cdots - 30\!\cdots\!84 \)
|
| $29$ |
\( T^{13} + \cdots - 38\!\cdots\!84 \)
|
| $31$ |
\( T^{13} + \cdots + 25\!\cdots\!80 \)
|
| $37$ |
\( (T + 1874161)^{13} \)
|
| $41$ |
\( T^{13} + \cdots + 18\!\cdots\!10 \)
|
| $43$ |
\( T^{13} + \cdots + 85\!\cdots\!20 \)
|
| $47$ |
\( T^{13} + \cdots + 66\!\cdots\!80 \)
|
| $53$ |
\( T^{13} + \cdots + 15\!\cdots\!52 \)
|
| $59$ |
\( T^{13} + \cdots - 11\!\cdots\!00 \)
|
| $61$ |
\( T^{13} + \cdots + 92\!\cdots\!56 \)
|
| $67$ |
\( T^{13} + \cdots - 20\!\cdots\!28 \)
|
| $71$ |
\( T^{13} + \cdots + 40\!\cdots\!00 \)
|
| $73$ |
\( T^{13} + \cdots + 13\!\cdots\!02 \)
|
| $79$ |
\( T^{13} + \cdots - 37\!\cdots\!36 \)
|
| $83$ |
\( T^{13} + \cdots - 50\!\cdots\!00 \)
|
| $89$ |
\( T^{13} + \cdots + 15\!\cdots\!84 \)
|
| $97$ |
\( T^{13} + \cdots - 13\!\cdots\!00 \)
|
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