Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [169,10,Mod(1,169)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(169, 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("169.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 169 = 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 169.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: | \(87.0410563117\) |
| Analytic rank: | \(1\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 3 x^{8} - 3184 x^{7} + 4328 x^{6} + 3323368 x^{5} - 2832720 x^{4} - 1268725952 x^{3} + \cdots + 390142272000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3\cdot 13^{2} \) |
| Twist minimal: | no (minimal twist has level 13) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 3 x^{8} - 3184 x^{7} + 4328 x^{6} + 3323368 x^{5} - 2832720 x^{4} - 1268725952 x^{3} + \cdots + 390142272000 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 708 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 60199 \nu^{8} - 23080813 \nu^{7} - 543953414 \nu^{6} + 83217824628 \nu^{5} + \cdots + 10\!\cdots\!20 ) / 417575199150080 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 31848991 \nu^{8} + 959555337 \nu^{7} - 116006648294 \nu^{6} - 2042879267332 \nu^{5} + \cdots + 21\!\cdots\!00 ) / 730756598512640 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 136245217 \nu^{8} + 7231100859 \nu^{7} - 384065441318 \nu^{6} - 20404537289644 \nu^{5} + \cdots + 80\!\cdots\!00 ) / 29\!\cdots\!60 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2326141 \nu^{8} + 73562703 \nu^{7} - 8347836590 \nu^{6} - 162190499836 \nu^{5} + \cdots - 43\!\cdots\!92 ) / 41757519915008 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 295739873 \nu^{8} + 3080314837 \nu^{7} + 1144549491446 \nu^{6} - 9922631049620 \nu^{5} + \cdots - 41\!\cdots\!40 ) / 584605278810112 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 862743299 \nu^{8} - 13119464927 \nu^{7} - 2428839028066 \nu^{6} + 28600740670332 \nu^{5} + \cdots + 45\!\cdots\!00 ) / 14\!\cdots\!80 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 708 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 21\beta_{3} + 9\beta_{2} + 1111\beta _1 + 1373 \)
|
| \(\nu^{4}\) | \(=\) |
\( -44\beta_{6} + 19\beta_{5} + 37\beta_{4} + 327\beta_{3} + 1507\beta_{2} + 7017\beta _1 + 780567 \)
|
| \(\nu^{5}\) | \(=\) |
\( 48 \beta_{8} + 112 \beta_{7} - 500 \beta_{6} + 1929 \beta_{5} - 577 \beta_{4} + 59573 \beta_{3} + \cdots + 4825349 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2096 \beta_{8} + 3280 \beta_{7} - 103676 \beta_{6} + 47251 \beta_{5} + 95109 \beta_{4} + \cdots + 986992551 \)
|
| \(\nu^{7}\) | \(=\) |
\( 103888 \beta_{8} + 300208 \beta_{7} - 1582868 \beta_{6} + 3287801 \beta_{5} + 958607 \beta_{4} + \cdots + 12016457973 \)
|
| \(\nu^{8}\) | \(=\) |
\( 7583344 \beta_{8} + 10086288 \beta_{7} - 186733116 \beta_{6} + 94197483 \beta_{5} + 182981037 \beta_{4} + \cdots + 1364356391391 \)
|
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 |
|
−36.5476 | −184.214 | 823.724 | 529.393 | 6732.57 | −8675.83 | −11392.7 | 14251.8 | −19348.0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −34.5507 | 57.7993 | 681.751 | −2017.22 | −1997.01 | 11757.7 | −5865.01 | −16342.2 | 69696.2 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −28.9303 | 190.754 | 324.962 | 1752.49 | −5518.56 | −4404.48 | 5411.06 | 16704.0 | −50700.2 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −9.18744 | −148.535 | −427.591 | 1229.19 | 1364.65 | 8333.07 | 8632.44 | 2379.53 | −11293.1 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −6.38428 | −5.82434 | −471.241 | −1063.81 | 37.1842 | −6475.20 | 6277.28 | −19649.1 | 6791.63 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 12.0491 | 193.813 | −366.820 | −26.3134 | 2335.27 | 2696.19 | −10589.0 | 17880.6 | −317.053 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 22.2574 | −256.527 | −16.6063 | −2488.59 | −5709.64 | 167.896 | −11765.4 | 46123.2 | −55389.7 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 27.1723 | −80.4157 | 226.333 | 1952.16 | −2185.08 | −1080.19 | −7762.23 | −13216.3 | 53044.7 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 39.1215 | 72.1497 | 1018.49 | −1007.31 | 2822.60 | −380.109 | 19814.6 | −14477.4 | −39407.4 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(13\) | \( +1 \) |
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 | 169.10.a.c | 9 | |
| 13.b | even | 2 | 1 | 169.10.a.d | 9 | ||
| 13.c | even | 3 | 2 | 13.10.c.a | ✓ | 18 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.10.c.a | ✓ | 18 | 13.c | even | 3 | 2 | |
| 169.10.a.c | 9 | 1.a | even | 1 | 1 | trivial | |
| 169.10.a.d | 9 | 13.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} + 15 T_{2}^{8} - 3088 T_{2}^{7} - 39912 T_{2}^{6} + 3108520 T_{2}^{5} + 29769792 T_{2}^{4} + \cdots + 610864349184 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(169))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + \cdots + 610864349184 \)
|
| $3$ |
\( T^{9} + \cdots + 50\!\cdots\!40 \)
|
| $5$ |
\( T^{9} + \cdots + 31\!\cdots\!00 \)
|
| $7$ |
\( T^{9} + \cdots + 45\!\cdots\!20 \)
|
| $11$ |
\( T^{9} + \cdots + 53\!\cdots\!48 \)
|
| $13$ |
\( T^{9} \)
|
| $17$ |
\( T^{9} + \cdots - 44\!\cdots\!43 \)
|
| $19$ |
\( T^{9} + \cdots + 24\!\cdots\!60 \)
|
| $23$ |
\( T^{9} + \cdots - 32\!\cdots\!24 \)
|
| $29$ |
\( T^{9} + \cdots - 43\!\cdots\!61 \)
|
| $31$ |
\( T^{9} + \cdots - 33\!\cdots\!00 \)
|
| $37$ |
\( T^{9} + \cdots + 96\!\cdots\!43 \)
|
| $41$ |
\( T^{9} + \cdots - 20\!\cdots\!75 \)
|
| $43$ |
\( T^{9} + \cdots - 34\!\cdots\!16 \)
|
| $47$ |
\( T^{9} + \cdots + 79\!\cdots\!00 \)
|
| $53$ |
\( T^{9} + \cdots - 17\!\cdots\!00 \)
|
| $59$ |
\( T^{9} + \cdots - 65\!\cdots\!24 \)
|
| $61$ |
\( T^{9} + \cdots - 47\!\cdots\!85 \)
|
| $67$ |
\( T^{9} + \cdots - 24\!\cdots\!44 \)
|
| $71$ |
\( T^{9} + \cdots - 48\!\cdots\!76 \)
|
| $73$ |
\( T^{9} + \cdots + 12\!\cdots\!00 \)
|
| $79$ |
\( T^{9} + \cdots - 35\!\cdots\!00 \)
|
| $83$ |
\( T^{9} + \cdots - 95\!\cdots\!00 \)
|
| $89$ |
\( T^{9} + \cdots + 73\!\cdots\!92 \)
|
| $97$ |
\( T^{9} + \cdots + 16\!\cdots\!76 \)
|
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