Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [653,2,Mod(1,653)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(653, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("653.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 653 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 653.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: | \(5.21423125199\) |
| Analytic rank: | \(1\) |
| Dimension: | \(17\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{17} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{17} - 4 x^{16} - 17 x^{15} + 80 x^{14} + 95 x^{13} - 609 x^{12} - 155 x^{11} + 2251 x^{10} + \cdots - 13 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{16}\) 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
\( x^{17} - 4 x^{16} - 17 x^{15} + 80 x^{14} + 95 x^{13} - 609 x^{12} - 155 x^{11} + 2251 x^{10} + \cdots - 13 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 2071773182 \nu^{16} - 383354199 \nu^{15} + 69773196228 \nu^{14} - 18462584534 \nu^{13} + \cdots - 561820394510 ) / 63138701303 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 2470155409 \nu^{16} - 915412877 \nu^{15} + 78211148312 \nu^{14} + 3907792032 \nu^{13} + \cdots - 275196063397 ) / 63138701303 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2722252451 \nu^{16} + 9982901232 \nu^{15} + 49259023042 \nu^{14} - 201084621448 \nu^{13} + \cdots - 164970625673 ) / 63138701303 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 3013643940 \nu^{16} + 5043174400 \nu^{15} + 75335424608 \nu^{14} - 113407422434 \nu^{13} + \cdots - 21615074049 ) / 63138701303 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 4093044327 \nu^{16} + 22471644840 \nu^{15} + 40746271281 \nu^{14} - 411243716466 \nu^{13} + \cdots + 159894068453 ) / 63138701303 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4182667025 \nu^{16} - 9283374907 \nu^{15} - 97712219529 \nu^{14} + 201948658899 \nu^{13} + \cdots - 27571318812 ) / 63138701303 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 9355728539 \nu^{16} + 18559742542 \nu^{15} + 228692284743 \nu^{14} - 412982729209 \nu^{13} + \cdots - 444241181556 ) / 63138701303 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 9585855170 \nu^{16} + 32272935763 \nu^{15} + 182983598712 \nu^{14} - 650402569230 \nu^{13} + \cdots - 343638253007 ) / 63138701303 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 10890187791 \nu^{16} + 36712609970 \nu^{15} + 206115662267 \nu^{14} - 735523015234 \nu^{13} + \cdots - 126326162112 ) / 63138701303 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 11926600754 \nu^{16} + 43428239471 \nu^{15} + 214671800420 \nu^{14} - 859077951034 \nu^{13} + \cdots - 160365843433 ) / 63138701303 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 12832709326 \nu^{16} + 46408970846 \nu^{15} + 231367375052 \nu^{14} - 918903582128 \nu^{13} + \cdots - 69477722690 ) / 63138701303 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 14260899522 \nu^{16} + 43509357113 \nu^{15} + 291727744679 \nu^{14} - 898047763697 \nu^{13} + \cdots - 536411911050 ) / 63138701303 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 15195990709 \nu^{16} - 55301509944 \nu^{15} - 275757645247 \nu^{14} + 1109584691706 \nu^{13} + \cdots + 267698096846 ) / 63138701303 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 15426000628 \nu^{16} + 54799083036 \nu^{15} + 281508286815 \nu^{14} - 1087108807669 \nu^{13} + \cdots - 324200935724 ) / 63138701303 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 31730587167 \nu^{16} - 120125205204 \nu^{15} - 553660946969 \nu^{14} + 2373077582374 \nu^{13} + \cdots + 350943157043 ) / 63138701303 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{15} - \beta_{14} + \beta_{12} - \beta_{10} + \beta_{7} + \beta_{3} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{14} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 4\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( - \beta_{16} - 10 \beta_{15} - 7 \beta_{14} - \beta_{13} + 8 \beta_{12} - 6 \beta_{10} + 7 \beta_{7} + \cdots + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{16} + 3 \beta_{15} - 10 \beta_{14} - 2 \beta_{13} - 4 \beta_{12} - 6 \beta_{10} - 8 \beta_{9} + \cdots + 16 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 11 \beta_{16} - 82 \beta_{15} - 48 \beta_{14} - 13 \beta_{13} + 58 \beta_{12} + \beta_{11} + \cdots + 86 \)
|
| \(\nu^{7}\) | \(=\) |
\( 15 \beta_{16} + 35 \beta_{15} - 85 \beta_{14} - 30 \beta_{13} - 50 \beta_{12} + 6 \beta_{11} + \cdots + 109 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 92 \beta_{16} - 629 \beta_{15} - 339 \beta_{14} - 132 \beta_{13} + 408 \beta_{12} + 19 \beta_{11} + \cdots + 521 \)
|
| \(\nu^{9}\) | \(=\) |
\( 163 \beta_{16} + 315 \beta_{15} - 681 \beta_{14} - 325 \beta_{13} - 468 \beta_{12} + 101 \beta_{11} + \cdots + 704 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 691 \beta_{16} - 4686 \beta_{15} - 2445 \beta_{14} - 1212 \beta_{13} + 2838 \beta_{12} + 240 \beta_{11} + \cdots + 3251 \)
|
| \(\nu^{11}\) | \(=\) |
\( 1545 \beta_{16} + 2605 \beta_{15} - 5296 \beta_{14} - 3072 \beta_{13} - 3960 \beta_{12} + 1150 \beta_{11} + \cdots + 4432 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 4900 \beta_{16} - 34413 \beta_{15} - 17856 \beta_{14} - 10502 \beta_{13} + 19676 \beta_{12} + \cdots + 20671 \)
|
| \(\nu^{13}\) | \(=\) |
\( 13587 \beta_{16} + 20720 \beta_{15} - 40500 \beta_{14} - 26979 \beta_{13} - 31987 \beta_{12} + \cdots + 27468 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 33531 \beta_{16} - 250868 \beta_{15} - 131347 \beta_{14} - 87626 \beta_{13} + 136444 \beta_{12} + \cdots + 133198 \)
|
| \(\nu^{15}\) | \(=\) |
\( 114077 \beta_{16} + 161116 \beta_{15} - 306645 \beta_{14} - 226577 \beta_{13} - 251955 \beta_{12} + \cdots + 168186 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 223917 \beta_{16} - 1822068 \beta_{15} - 970224 \beta_{14} - 712271 \beta_{13} + 947754 \beta_{12} + \cdots + 867146 \)
|
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 |
|
−2.73523 | −2.56890 | 5.48147 | 4.41900 | 7.02651 | −0.424780 | −9.52262 | 3.59922 | −12.0870 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.55380 | 1.80381 | 4.52188 | 0.391007 | −4.60656 | −1.95624 | −6.44038 | 0.253717 | −0.998553 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.43550 | −2.17976 | 3.93167 | −1.87318 | 5.30881 | −2.97577 | −4.70458 | 1.75136 | 4.56212 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.09202 | −0.00533424 | 2.37656 | −0.350114 | 0.0111594 | 2.45921 | −0.787774 | −2.99997 | 0.732446 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.89228 | 2.37042 | 1.58073 | −0.281698 | −4.48550 | −4.77153 | 0.793374 | 2.61887 | 0.533052 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.02021 | −3.27019 | −0.959165 | −3.68519 | 3.33629 | 1.16202 | 3.01898 | 7.69413 | 3.75968 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.874235 | −0.207560 | −1.23571 | −0.886243 | 0.181457 | 0.993525 | 2.82877 | −2.95692 | 0.774784 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.626990 | 2.27486 | −1.60688 | −3.19845 | −1.42631 | 1.33297 | 2.26148 | 2.17497 | 2.00540 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.461150 | 0.371086 | −1.78734 | 3.62010 | −0.171126 | −3.24444 | 1.74653 | −2.86230 | −1.66941 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.100074 | −0.322837 | −1.98999 | 0.328957 | −0.0323077 | 2.85306 | −0.399295 | −2.89578 | 0.0329201 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0.637509 | 1.17207 | −1.59358 | −0.257769 | 0.747203 | −2.66587 | −2.29094 | −1.62626 | −0.164330 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0.750464 | −3.26431 | −1.43680 | 1.26240 | −2.44974 | 3.42149 | −2.57920 | 7.65570 | 0.947388 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 0.936766 | 1.80873 | −1.12247 | −2.76772 | 1.69435 | −0.739094 | −2.92502 | 0.271497 | −2.59271 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 1.51313 | −1.86619 | 0.289547 | 1.48787 | −2.82378 | −1.31812 | −2.58813 | 0.482674 | 2.25133 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 1.87380 | −2.39826 | 1.51114 | −2.38611 | −4.49386 | 4.20064 | −0.916034 | 2.75163 | −4.47111 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.18167 | −0.413638 | 2.75970 | −4.05570 | −0.902423 | −2.97213 | 1.65742 | −2.82890 | −8.84822 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 2.69801 | −3.30399 | 5.27924 | −1.76716 | −8.91419 | −4.35496 | 8.84741 | 7.91635 | −4.76780 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(653\) | \(1\) |
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 | 653.2.a.b | ✓ | 17 |
| 3.b | odd | 2 | 1 | 5877.2.a.d | 17 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 653.2.a.b | ✓ | 17 | 1.a | even | 1 | 1 | trivial |
| 5877.2.a.d | 17 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{17} + 4 T_{2}^{16} - 17 T_{2}^{15} - 80 T_{2}^{14} + 95 T_{2}^{13} + 609 T_{2}^{12} - 155 T_{2}^{11} + \cdots + 13 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(653))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{17} + 4 T^{16} + \cdots + 13 \)
|
| $3$ |
\( T^{17} + 10 T^{16} + \cdots + 1 \)
|
| $5$ |
\( T^{17} + 10 T^{16} + \cdots + 91 \)
|
| $7$ |
\( T^{17} + 9 T^{16} + \cdots - 199701 \)
|
| $11$ |
\( T^{17} + 3 T^{16} + \cdots - 19447 \)
|
| $13$ |
\( T^{17} + 37 T^{16} + \cdots + 31562864 \)
|
| $17$ |
\( T^{17} + 21 T^{16} + \cdots + 12220031 \)
|
| $19$ |
\( T^{17} + 36 T^{16} + \cdots + 120368 \)
|
| $23$ |
\( T^{17} + \cdots - 13215704719 \)
|
| $29$ |
\( T^{17} + \cdots - 39740622047 \)
|
| $31$ |
\( T^{17} + \cdots - 213173247737 \)
|
| $37$ |
\( T^{17} + \cdots - 115818363013 \)
|
| $41$ |
\( T^{17} + \cdots + 21973784823 \)
|
| $43$ |
\( T^{17} + \cdots - 33364915253 \)
|
| $47$ |
\( T^{17} - 29 T^{16} + \cdots - 1565739 \)
|
| $53$ |
\( T^{17} + \cdots - 105203574329 \)
|
| $59$ |
\( T^{17} + \cdots - 72065761349 \)
|
| $61$ |
\( T^{17} + \cdots + 1322760141 \)
|
| $67$ |
\( T^{17} + \cdots + 3501182559253 \)
|
| $71$ |
\( T^{17} + \cdots + 16\!\cdots\!09 \)
|
| $73$ |
\( T^{17} + \cdots + 5653871088257 \)
|
| $79$ |
\( T^{17} + \cdots + 1138201693407 \)
|
| $83$ |
\( T^{17} + \cdots + 110811504229616 \)
|
| $89$ |
\( T^{17} + \cdots - 1664866098864 \)
|
| $97$ |
\( T^{17} + \cdots + 137254649953 \)
|
| show more | |
| show less |