Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1140,2,Mod(151,1140)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1140, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1140.151");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1140 = 2^{2} \cdot 3 \cdot 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1140.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: | \(9.10294583043\) |
| Analytic rank: | \(0\) |
| Dimension: | \(18\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{18} - x^{17} - x^{16} + 3 x^{15} - x^{14} - x^{13} - x^{12} - x^{11} + 6 x^{10} - 4 x^{9} + \cdots + 512 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{11} \) |
| 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 \(1,\beta_1,\ldots,\beta_{17}\) 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
\( x^{18} - x^{17} - x^{16} + 3 x^{15} - x^{14} - x^{13} - x^{12} - x^{11} + 6 x^{10} - 4 x^{9} + \cdots + 512 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{2} \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3 \nu^{17} + 7 \nu^{16} - \nu^{15} - 5 \nu^{14} - 9 \nu^{13} - \nu^{12} - 17 \nu^{11} + 15 \nu^{10} + \cdots + 768 ) / 1024 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - \nu^{17} - \nu^{16} + 3 \nu^{15} - \nu^{14} - 5 \nu^{13} + 3 \nu^{12} + 3 \nu^{11} + 3 \nu^{10} + \cdots + 512 ) / 256 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{17} - \nu^{16} - \nu^{15} + 3 \nu^{14} - \nu^{13} - \nu^{12} - \nu^{11} - \nu^{10} + 6 \nu^{9} + \cdots - 256 ) / 256 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 3 \nu^{17} + 9 \nu^{16} + 17 \nu^{15} - 11 \nu^{14} - 7 \nu^{13} - 15 \nu^{12} + \nu^{11} + \cdots + 2304 ) / 1024 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( \nu^{17} - 2 \nu^{16} + 8 \nu^{14} + 8 \nu^{13} + 12 \nu^{12} - 4 \nu^{11} + 12 \nu^{10} + 19 \nu^{9} + \cdots - 384 ) / 256 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( -\nu^{16} - 4\nu^{11} + 4\nu^{10} + 4\nu^{9} - 3\nu^{8} - 20\nu^{6} - 12\nu^{4} + 32\nu^{3} + 64\nu^{2} - 128\nu ) / 64 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 3 \nu^{17} + 5 \nu^{16} + \nu^{15} - 11 \nu^{14} - 7 \nu^{13} + 9 \nu^{12} + 25 \nu^{11} + 25 \nu^{10} + \cdots + 1536 ) / 512 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - \nu^{17} - 4 \nu^{12} + 4 \nu^{11} + 4 \nu^{10} - 3 \nu^{9} - 20 \nu^{7} - 12 \nu^{5} + \cdots - 128 \nu^{2} ) / 128 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 9 \nu^{17} + \nu^{16} - 47 \nu^{15} + 21 \nu^{14} + 25 \nu^{13} + \nu^{12} + 17 \nu^{11} + \cdots - 6400 ) / 1024 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 9 \nu^{17} + 17 \nu^{16} - 31 \nu^{15} - 27 \nu^{14} - 23 \nu^{13} + 17 \nu^{12} + 33 \nu^{11} + \cdots + 768 ) / 1024 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 3 \nu^{17} + 4 \nu^{16} - 2 \nu^{15} - 10 \nu^{14} - 2 \nu^{13} + 2 \nu^{12} + 18 \nu^{11} + \cdots + 640 ) / 256 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 3 \nu^{17} - 6 \nu^{16} - 8 \nu^{15} + 8 \nu^{14} + 8 \nu^{13} + 4 \nu^{12} - 12 \nu^{11} + \cdots - 1664 ) / 256 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 17 \nu^{17} + 3 \nu^{16} + 27 \nu^{15} + 7 \nu^{14} - 13 \nu^{13} - 37 \nu^{12} + 11 \nu^{11} + \cdots + 1792 ) / 1024 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{3} + \beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2\beta_{4} + \beta_{2} ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2\beta_{17} + 2\beta_{16} + 2\beta_{11} - 2\beta_{7} + 2\beta_{5} + 2\beta_{4} + \beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 2 \beta_{17} + 2 \beta_{16} + 2 \beta_{15} - 2 \beta_{13} - 2 \beta_{11} - 2 \beta_{10} - 4 \beta_{8} + \cdots + 2 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 2 \beta_{17} - 2 \beta_{14} + 2 \beta_{13} - 4 \beta_{12} + 2 \beta_{11} - 2 \beta_{9} - 2 \beta_{8} + \cdots + 4 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 2 \beta_{17} + 2 \beta_{16} - 2 \beta_{15} + 4 \beta_{14} - 6 \beta_{13} + 2 \beta_{11} - 2 \beta_{10} + \cdots - 2 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 4 \beta_{17} - 2 \beta_{16} + 4 \beta_{15} + 2 \beta_{14} + 2 \beta_{13} - 4 \beta_{12} - 4 \beta_{11} + \cdots + \beta_1 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 4 \beta_{17} - 8 \beta_{16} - 4 \beta_{15} + 8 \beta_{12} + 8 \beta_{11} + 8 \beta_{10} + 4 \beta_{9} + \cdots - 20 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 4 \beta_{16} + 8 \beta_{15} - 4 \beta_{14} + 4 \beta_{13} + 16 \beta_{12} + 8 \beta_{11} - 8 \beta_{10} + \cdots - 8 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 8 \beta_{17} - 8 \beta_{16} - 8 \beta_{15} + 8 \beta_{14} + 8 \beta_{13} - 16 \beta_{12} + 16 \beta_{9} + \cdots - 40 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 2 \beta_{17} + 10 \beta_{16} + 16 \beta_{15} - 24 \beta_{14} + 8 \beta_{13} + 2 \beta_{11} + 8 \beta_{9} + \cdots + 80 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 6 \beta_{17} - 22 \beta_{16} - 30 \beta_{15} + 14 \beta_{13} + 6 \beta_{11} - 18 \beta_{10} + \cdots + 98 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 6 \beta_{17} - 8 \beta_{16} + 24 \beta_{15} - 34 \beta_{14} - 22 \beta_{13} - 36 \beta_{12} + \cdots - 132 ) / 2 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 46 \beta_{17} - 102 \beta_{16} - 66 \beta_{15} + 12 \beta_{14} + 50 \beta_{13} - 48 \beta_{12} + \cdots - 82 ) / 2 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 60 \beta_{17} - 2 \beta_{16} + 36 \beta_{15} - 14 \beta_{14} - 62 \beta_{13} - 4 \beta_{12} + \cdots - 32 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1140\mathbb{Z}\right)^\times\).
| \(n\) | \(457\) | \(571\) | \(761\) | \(781\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 151.1 |
|
−1.39086 | − | 0.255925i | −1.00000 | 1.86901 | + | 0.711913i | −1.00000 | 1.39086 | + | 0.255925i | − | 2.31417i | −2.41734 | − | 1.46850i | 1.00000 | 1.39086 | + | 0.255925i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.2 | −1.39086 | + | 0.255925i | −1.00000 | 1.86901 | − | 0.711913i | −1.00000 | 1.39086 | − | 0.255925i | 2.31417i | −2.41734 | + | 1.46850i | 1.00000 | 1.39086 | − | 0.255925i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.3 | −1.26146 | − | 0.639314i | −1.00000 | 1.18256 | + | 1.61294i | −1.00000 | 1.26146 | + | 0.639314i | 4.53677i | −0.460573 | − | 2.79068i | 1.00000 | 1.26146 | + | 0.639314i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.4 | −1.26146 | + | 0.639314i | −1.00000 | 1.18256 | − | 1.61294i | −1.00000 | 1.26146 | − | 0.639314i | − | 4.53677i | −0.460573 | + | 2.79068i | 1.00000 | 1.26146 | − | 0.639314i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.5 | −0.891738 | − | 1.09763i | −1.00000 | −0.409605 | + | 1.95761i | −1.00000 | 0.891738 | + | 1.09763i | − | 1.17043i | 2.51400 | − | 1.29608i | 1.00000 | 0.891738 | + | 1.09763i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.6 | −0.891738 | + | 1.09763i | −1.00000 | −0.409605 | − | 1.95761i | −1.00000 | 0.891738 | − | 1.09763i | 1.17043i | 2.51400 | + | 1.29608i | 1.00000 | 0.891738 | − | 1.09763i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.7 | −0.829037 | − | 1.14573i | −1.00000 | −0.625394 | + | 1.89971i | −1.00000 | 0.829037 | + | 1.14573i | − | 3.64704i | 2.69503 | − | 0.858394i | 1.00000 | 0.829037 | + | 1.14573i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.8 | −0.829037 | + | 1.14573i | −1.00000 | −0.625394 | − | 1.89971i | −1.00000 | 0.829037 | − | 1.14573i | 3.64704i | 2.69503 | + | 0.858394i | 1.00000 | 0.829037 | − | 1.14573i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.9 | −0.184079 | − | 1.40218i | −1.00000 | −1.93223 | + | 0.516225i | −1.00000 | 0.184079 | + | 1.40218i | 5.09617i | 1.07953 | + | 2.61431i | 1.00000 | 0.184079 | + | 1.40218i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.10 | −0.184079 | + | 1.40218i | −1.00000 | −1.93223 | − | 0.516225i | −1.00000 | 0.184079 | − | 1.40218i | − | 5.09617i | 1.07953 | − | 2.61431i | 1.00000 | 0.184079 | − | 1.40218i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.11 | 0.427068 | − | 1.34819i | −1.00000 | −1.63523 | − | 1.15154i | −1.00000 | −0.427068 | + | 1.34819i | − | 3.16188i | −2.25084 | + | 1.71281i | 1.00000 | −0.427068 | + | 1.34819i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.12 | 0.427068 | + | 1.34819i | −1.00000 | −1.63523 | + | 1.15154i | −1.00000 | −0.427068 | − | 1.34819i | 3.16188i | −2.25084 | − | 1.71281i | 1.00000 | −0.427068 | − | 1.34819i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.13 | 0.916572 | − | 1.07698i | −1.00000 | −0.319793 | − | 1.97427i | −1.00000 | −0.916572 | + | 1.07698i | − | 0.0984762i | −2.41937 | − | 1.46515i | 1.00000 | −0.916572 | + | 1.07698i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.14 | 0.916572 | + | 1.07698i | −1.00000 | −0.319793 | + | 1.97427i | −1.00000 | −0.916572 | − | 1.07698i | 0.0984762i | −2.41937 | + | 1.46515i | 1.00000 | −0.916572 | − | 1.07698i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.15 | 1.31376 | − | 0.523494i | −1.00000 | 1.45191 | − | 1.37549i | −1.00000 | −1.31376 | + | 0.523494i | 1.57687i | 1.18740 | − | 2.56712i | 1.00000 | −1.31376 | + | 0.523494i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.16 | 1.31376 | + | 0.523494i | −1.00000 | 1.45191 | + | 1.37549i | −1.00000 | −1.31376 | − | 0.523494i | − | 1.57687i | 1.18740 | + | 2.56712i | 1.00000 | −1.31376 | − | 0.523494i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.17 | 1.39978 | − | 0.201522i | −1.00000 | 1.91878 | − | 0.564173i | −1.00000 | −1.39978 | + | 0.201522i | − | 2.21964i | 2.57218 | − | 1.17640i | 1.00000 | −1.39978 | + | 0.201522i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.18 | 1.39978 | + | 0.201522i | −1.00000 | 1.91878 | + | 0.564173i | −1.00000 | −1.39978 | − | 0.201522i | 2.21964i | 2.57218 | + | 1.17640i | 1.00000 | −1.39978 | − | 0.201522i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 76.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1140.2.b.c | ✓ | 18 |
| 4.b | odd | 2 | 1 | 1140.2.b.d | yes | 18 | |
| 19.b | odd | 2 | 1 | 1140.2.b.d | yes | 18 | |
| 76.d | even | 2 | 1 | inner | 1140.2.b.c | ✓ | 18 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1140.2.b.c | ✓ | 18 | 1.a | even | 1 | 1 | trivial |
| 1140.2.b.c | ✓ | 18 | 76.d | even | 2 | 1 | inner |
| 1140.2.b.d | yes | 18 | 4.b | odd | 2 | 1 | |
| 1140.2.b.d | yes | 18 | 19.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1140, [\chi])\):
|
\( T_{7}^{18} + 84 T_{7}^{16} + 2810 T_{7}^{14} + 48432 T_{7}^{12} + 466476 T_{7}^{10} + 2550176 T_{7}^{8} + \cdots + 61952 \)
|
|
\( T_{31}^{9} - 8 T_{31}^{8} - 150 T_{31}^{7} + 1328 T_{31}^{6} + 4204 T_{31}^{5} - 52144 T_{31}^{4} + \cdots + 667904 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} + T^{17} + \cdots + 512 \)
|
| $3$ |
\( (T + 1)^{18} \)
|
| $5$ |
\( (T + 1)^{18} \)
|
| $7$ |
\( T^{18} + 84 T^{16} + \cdots + 61952 \)
|
| $11$ |
\( T^{18} + 108 T^{16} + \cdots + 373248 \)
|
| $13$ |
\( T^{18} + \cdots + 1015021568 \)
|
| $17$ |
\( (T^{9} + 10 T^{8} + \cdots + 7616)^{2} \)
|
| $19$ |
\( T^{18} + \cdots + 322687697779 \)
|
| $23$ |
\( T^{18} + 148 T^{16} + \cdots + 524288 \)
|
| $29$ |
\( T^{18} + 112 T^{16} + \cdots + 9469952 \)
|
| $31$ |
\( (T^{9} - 8 T^{8} + \cdots + 667904)^{2} \)
|
| $37$ |
\( T^{18} + \cdots + 17557880832 \)
|
| $41$ |
\( T^{18} + \cdots + 4471437467648 \)
|
| $43$ |
\( T^{18} + \cdots + 8396605297152 \)
|
| $47$ |
\( T^{18} + \cdots + 1917665280000 \)
|
| $53$ |
\( T^{18} + \cdots + 87275077632 \)
|
| $59$ |
\( (T^{9} + 16 T^{8} + \cdots + 4022272)^{2} \)
|
| $61$ |
\( (T^{9} + 2 T^{8} + \cdots + 54022208)^{2} \)
|
| $67$ |
\( (T^{9} - 360 T^{7} + \cdots - 11870208)^{2} \)
|
| $71$ |
\( (T^{9} + 8 T^{8} + \cdots - 14680064)^{2} \)
|
| $73$ |
\( (T^{9} - 10 T^{8} + \cdots + 8135936)^{2} \)
|
| $79$ |
\( (T^{9} + 8 T^{8} + \cdots - 1048576)^{2} \)
|
| $83$ |
\( T^{18} + \cdots + 40\!\cdots\!08 \)
|
| $89$ |
\( T^{18} + \cdots + 18\!\cdots\!08 \)
|
| $97$ |
\( T^{18} + \cdots + 361459901530112 \)
|
| show more | |
| show less |