Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [600,2,Mod(121,600)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(600, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 0, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("600.121");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 600.y (of order \(5\), degree \(4\), 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: | \(4.79102412128\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 8 x^{15} + 64 x^{14} - 308 x^{13} + 1299 x^{12} - 4154 x^{11} + 11123 x^{10} - 23924 x^{9} + \cdots + 176 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 5 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 8 x^{15} + 64 x^{14} - 308 x^{13} + 1299 x^{12} - 4154 x^{11} + 11123 x^{10} - 23924 x^{9} + \cdots + 176 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 470632 \nu^{15} - 11412117 \nu^{14} + 81647677 \nu^{13} - 564747969 \nu^{12} + 2406351257 \nu^{11} + \cdots - 2989759752 ) / 36193424 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 576682 \nu^{15} - 1370105 \nu^{14} + 14637241 \nu^{13} + 2815755 \nu^{12} - 32466589 \nu^{11} + \cdots + 507641048 ) / 18096712 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 576682 \nu^{15} + 7280125 \nu^{14} - 56007381 \nu^{13} + 330810643 \nu^{12} + \cdots + 1269939128 ) / 18096712 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1541280 \nu^{15} + 7500147 \nu^{14} - 62519103 \nu^{13} + 185864031 \nu^{12} + \cdots - 1604537400 ) / 36193424 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 1541280 \nu^{15} + 15619053 \nu^{14} - 119351445 \nu^{13} + 645653331 \nu^{12} + \cdots + 1442484304 ) / 36193424 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 1816627 \nu^{15} + 12842362 \nu^{14} - 103619478 \nu^{13} + 457243524 \nu^{12} + \cdots - 183591296 ) / 36193424 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 4823098 \nu^{15} + 34898927 \nu^{14} - 280355205 \nu^{13} + 1259553795 \nu^{12} + \cdots + 72477560 ) / 36193424 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 4823098 \nu^{15} + 37447543 \nu^{14} - 298195517 \nu^{13} + 1403771103 \nu^{12} + \cdots + 1353421192 ) / 36193424 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 6841419 \nu^{15} + 44058048 \nu^{14} - 360059920 \nu^{13} + 1482492314 \nu^{12} + \cdots - 1398827200 ) / 36193424 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 6841419 \nu^{15} - 58563237 \nu^{14} + 461596243 \nu^{13} - 2301849923 \nu^{12} + \cdots - 4214902088 ) / 36193424 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 4082148 \nu^{15} + 29978956 \nu^{14} - 240831032 \nu^{13} + 1093993553 \nu^{12} + \cdots + 601521308 ) / 18096712 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 15198346 \nu^{15} + 106650709 \nu^{14} - 862032301 \nu^{13} + 3793794913 \nu^{12} + \cdots + 1089418984 ) / 36193424 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 15198346 \nu^{15} + 121324481 \nu^{14} - 964748705 \nu^{13} + 4622657911 \nu^{12} + \cdots + 6083973040 ) / 36193424 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 21012657 \nu^{15} + 151719842 \nu^{14} - 1221166782 \nu^{13} + 5483888396 \nu^{12} + \cdots + 2782403936 ) / 36193424 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 22358652 \nu^{15} - 166159823 \nu^{14} + 1334206735 \nu^{13} - 6111112253 \nu^{12} + \cdots - 4055310928 ) / 36193424 \)
|
| \(\nu\) | \(=\) |
\( ( 2 \beta_{15} - 2 \beta_{14} + \beta_{13} + 4 \beta_{12} + \beta_{10} + 4 \beta_{8} + \beta_{7} + \cdots + 1 ) / 5 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2 \beta_{15} - 2 \beta_{14} + \beta_{13} + 9 \beta_{12} - 5 \beta_{11} + \beta_{10} - 5 \beta_{9} + \cdots - 14 ) / 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( - 4 \beta_{15} + 4 \beta_{14} - 2 \beta_{13} - 5 \beta_{12} - 5 \beta_{11} - 3 \beta_{10} + \cdots - 2 \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 52 \beta_{15} + 32 \beta_{14} - 6 \beta_{13} - 124 \beta_{12} + 5 \beta_{11} - 21 \beta_{10} + \cdots + 134 ) / 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 143 \beta_{15} - 193 \beta_{14} + 134 \beta_{13} + 116 \beta_{12} + 285 \beta_{11} + 164 \beta_{10} + \cdots + 94 ) / 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( 165 \beta_{15} - 79 \beta_{14} + 21 \beta_{13} + 291 \beta_{12} + 72 \beta_{11} + 58 \beta_{10} + \cdots - 247 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 353 \beta_{15} + 2033 \beta_{14} - 1754 \beta_{13} + 364 \beta_{12} - 2740 \beta_{11} - 1924 \beta_{10} + \cdots - 1839 ) / 5 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 10348 \beta_{15} + 5088 \beta_{14} - 2344 \beta_{13} - 15471 \beta_{12} - 7960 \beta_{11} + \cdots + 10781 ) / 5 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 2313 \beta_{15} - 4479 \beta_{14} + 4271 \beta_{13} - 4178 \beta_{12} + 4375 \beta_{11} + 4354 \beta_{10} + \cdots + 5030 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 108883 \beta_{15} - 70003 \beta_{14} + 45139 \beta_{13} + 148336 \beta_{12} + 125610 \beta_{11} + \cdots - 92491 ) / 5 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 298918 \beta_{15} + 245252 \beta_{14} - 238296 \beta_{13} + 391076 \beta_{12} - 105180 \beta_{11} + \cdots - 305021 ) / 5 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 185588 \beta_{15} + 200150 \beta_{14} - 151361 \beta_{13} - 248252 \beta_{12} - 339177 \beta_{11} + \cdots + 160800 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 4966173 \beta_{15} - 2514937 \beta_{14} + 2373106 \beta_{13} - 5695461 \beta_{12} - 802625 \beta_{11} + \cdots + 3588226 ) / 5 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 4996052 \beta_{15} - 14298182 \beta_{14} + 11399701 \beta_{13} + 8054549 \beta_{12} + 20387300 \beta_{11} + \cdots - 7178859 ) / 5 \)
|
| \(\nu^{15}\) | \(=\) |
\( 13650085 \beta_{15} + 4416279 \beta_{14} - 3945871 \beta_{13} + 14570277 \beta_{12} + 6924893 \beta_{11} + \cdots - 8591742 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).
| \(n\) | \(151\) | \(301\) | \(401\) | \(577\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1 - \beta_{7} + \beta_{8} + \beta_{11}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 121.1 |
|
0 | −0.809017 | − | 0.587785i | 0 | −2.21049 | − | 0.337273i | 0 | 0.973940 | 0 | 0.309017 | + | 0.951057i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.2 | 0 | −0.809017 | − | 0.587785i | 0 | 0.0720149 | − | 2.23491i | 0 | −1.39848 | 0 | 0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.3 | 0 | −0.809017 | − | 0.587785i | 0 | 1.86268 | + | 1.23711i | 0 | −4.24423 | 0 | 0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.4 | 0 | −0.809017 | − | 0.587785i | 0 | 2.20285 | + | 0.384014i | 0 | 4.05073 | 0 | 0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 241.1 | 0 | 0.309017 | + | 0.951057i | 0 | −2.09044 | − | 0.793762i | 0 | 0.829064 | 0 | −0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 241.2 | 0 | 0.309017 | + | 0.951057i | 0 | −1.73178 | + | 1.41455i | 0 | 0.398624 | 0 | −0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 241.3 | 0 | 0.309017 | + | 0.951057i | 0 | 0.391650 | − | 2.20150i | 0 | −3.02595 | 0 | −0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 241.4 | 0 | 0.309017 | + | 0.951057i | 0 | 2.00352 | + | 0.992933i | 0 | 3.41630 | 0 | −0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 361.1 | 0 | 0.309017 | − | 0.951057i | 0 | −2.09044 | + | 0.793762i | 0 | 0.829064 | 0 | −0.809017 | − | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 361.2 | 0 | 0.309017 | − | 0.951057i | 0 | −1.73178 | − | 1.41455i | 0 | 0.398624 | 0 | −0.809017 | − | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 361.3 | 0 | 0.309017 | − | 0.951057i | 0 | 0.391650 | + | 2.20150i | 0 | −3.02595 | 0 | −0.809017 | − | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 361.4 | 0 | 0.309017 | − | 0.951057i | 0 | 2.00352 | − | 0.992933i | 0 | 3.41630 | 0 | −0.809017 | − | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 481.1 | 0 | −0.809017 | + | 0.587785i | 0 | −2.21049 | + | 0.337273i | 0 | 0.973940 | 0 | 0.309017 | − | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 481.2 | 0 | −0.809017 | + | 0.587785i | 0 | 0.0720149 | + | 2.23491i | 0 | −1.39848 | 0 | 0.309017 | − | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 481.3 | 0 | −0.809017 | + | 0.587785i | 0 | 1.86268 | − | 1.23711i | 0 | −4.24423 | 0 | 0.309017 | − | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 481.4 | 0 | −0.809017 | + | 0.587785i | 0 | 2.20285 | − | 0.384014i | 0 | 4.05073 | 0 | 0.309017 | − | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 25.d | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 600.2.y.e | ✓ | 16 |
| 25.d | even | 5 | 1 | inner | 600.2.y.e | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 600.2.y.e | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 600.2.y.e | ✓ | 16 | 25.d | even | 5 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} - T_{7}^{7} - 29T_{7}^{6} + 29T_{7}^{5} + 216T_{7}^{4} - 200T_{7}^{3} - 255T_{7}^{2} + 320T_{7} - 80 \)
acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \)
|
| $5$ |
\( T^{16} - T^{15} + \cdots + 390625 \)
|
| $7$ |
\( (T^{8} - T^{7} - 29 T^{6} + \cdots - 80)^{2} \)
|
| $11$ |
\( T^{16} + 3 T^{15} + \cdots + 774400 \)
|
| $13$ |
\( T^{16} + 2 T^{15} + \cdots + 4393216 \)
|
| $17$ |
\( T^{16} + \cdots + 286421776 \)
|
| $19$ |
\( T^{16} + \cdots + 13330087936 \)
|
| $23$ |
\( T^{16} + \cdots + 12131700736 \)
|
| $29$ |
\( T^{16} + \cdots + 1168465873936 \)
|
| $31$ |
\( T^{16} + \cdots + 198246400 \)
|
| $37$ |
\( T^{16} + \cdots + 508231936 \)
|
| $41$ |
\( T^{16} + \cdots + 16937340250000 \)
|
| $43$ |
\( (T^{8} - 18 T^{7} + \cdots + 576)^{2} \)
|
| $47$ |
\( T^{16} + 11 T^{15} + \cdots + 8294400 \)
|
| $53$ |
\( T^{16} - 6 T^{15} + \cdots + 10857025 \)
|
| $59$ |
\( T^{16} + \cdots + 83359238400 \)
|
| $61$ |
\( T^{16} + \cdots + 9012700477456 \)
|
| $67$ |
\( T^{16} + \cdots + 1494286336 \)
|
| $71$ |
\( T^{16} + \cdots + 91693647462400 \)
|
| $73$ |
\( T^{16} + \cdots + 5682144400 \)
|
| $79$ |
\( T^{16} + \cdots + 34073651005696 \)
|
| $83$ |
\( T^{16} + \cdots + 510847549696 \)
|
| $89$ |
\( T^{16} + \cdots + 12796674635536 \)
|
| $97$ |
\( T^{16} + \cdots + 3839879312721 \)
|
| show more | |
| show less |