Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [177,8,Mod(1,177)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(177, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("177.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 177 = 3 \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 177.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: | \(55.2921495107\) |
| 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: |
\( x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} + \cdots - 58\!\cdots\!76 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | multiple of \( 2^{10}\cdot 3^{5} \) |
| 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
\( x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} + \cdots - 58\!\cdots\!76 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 193 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 59\!\cdots\!07 \nu^{16} + \cdots + 21\!\cdots\!32 ) / 86\!\cdots\!08 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 48\!\cdots\!11 \nu^{16} + \cdots + 12\!\cdots\!20 ) / 13\!\cdots\!72 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 35\!\cdots\!73 \nu^{16} + \cdots - 84\!\cdots\!00 ) / 86\!\cdots\!08 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 45\!\cdots\!93 \nu^{16} + \cdots + 11\!\cdots\!36 ) / 86\!\cdots\!08 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 13\!\cdots\!07 \nu^{16} + \cdots - 31\!\cdots\!48 ) / 21\!\cdots\!52 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 70\!\cdots\!35 \nu^{16} + \cdots + 16\!\cdots\!92 ) / 10\!\cdots\!76 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 56\!\cdots\!87 \nu^{16} + \cdots - 13\!\cdots\!76 ) / 86\!\cdots\!08 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 31\!\cdots\!73 \nu^{16} + \cdots - 77\!\cdots\!04 ) / 43\!\cdots\!04 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 66\!\cdots\!61 \nu^{16} + \cdots - 15\!\cdots\!32 ) / 86\!\cdots\!08 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 31\!\cdots\!53 \nu^{16} + \cdots + 75\!\cdots\!48 ) / 21\!\cdots\!52 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 66\!\cdots\!77 \nu^{16} + \cdots - 15\!\cdots\!20 ) / 43\!\cdots\!04 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 10\!\cdots\!89 \nu^{16} + \cdots - 25\!\cdots\!20 ) / 54\!\cdots\!88 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 21\!\cdots\!51 \nu^{16} + \cdots + 52\!\cdots\!68 ) / 86\!\cdots\!08 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 37\!\cdots\!17 \nu^{16} + \cdots - 91\!\cdots\!84 ) / 86\!\cdots\!08 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 193 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{9} - \beta_{8} - \beta_{3} + 3\beta_{2} + 336\beta _1 + 253 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{16} - \beta_{15} + 2 \beta_{14} + 4 \beta_{13} + 4 \beta_{12} + 6 \beta_{11} + 3 \beta_{10} + \cdots + 64899 \)
|
| \(\nu^{5}\) | \(=\) |
\( 148 \beta_{16} - 49 \beta_{15} - 132 \beta_{14} + \beta_{13} - 10 \beta_{12} - 147 \beta_{11} + \cdots + 234684 \)
|
| \(\nu^{6}\) | \(=\) |
\( 766 \beta_{16} - 1912 \beta_{15} + 1532 \beta_{14} + 5300 \beta_{13} + 5146 \beta_{12} + 3504 \beta_{11} + \cdots + 25899709 \)
|
| \(\nu^{7}\) | \(=\) |
\( 114021 \beta_{16} - 57379 \beta_{15} - 98702 \beta_{14} + 27464 \beta_{13} + 12614 \beta_{12} + \cdots + 168642875 \)
|
| \(\nu^{8}\) | \(=\) |
\( 635630 \beta_{16} - 1725907 \beta_{15} + 716880 \beta_{14} + 3990095 \beta_{13} + 3826244 \beta_{12} + \cdots + 11512527046 \)
|
| \(\nu^{9}\) | \(=\) |
\( 66441380 \beta_{16} - 46530816 \beta_{15} - 54081184 \beta_{14} + 34859488 \beta_{13} + \cdots + 111819940985 \)
|
| \(\nu^{10}\) | \(=\) |
\( 501465529 \beta_{16} - 1237972625 \beta_{15} + 233253826 \beta_{14} + 2511200180 \beta_{13} + \cdots + 5535112643107 \)
|
| \(\nu^{11}\) | \(=\) |
\( 35736675116 \beta_{16} - 32314549633 \beta_{15} - 26860437140 \beta_{14} + 29741506937 \beta_{13} + \cdots + 71002791721804 \)
|
| \(\nu^{12}\) | \(=\) |
\( 361105418406 \beta_{16} - 804046573904 \beta_{15} + 30850932748 \beta_{14} + 1473767444980 \beta_{13} + \cdots + 28\!\cdots\!65 \)
|
| \(\nu^{13}\) | \(=\) |
\( 18880479302309 \beta_{16} - 20707039000419 \beta_{15} - 12995109669646 \beta_{14} + \cdots + 43\!\cdots\!91 \)
|
| \(\nu^{14}\) | \(=\) |
\( 241898320413574 \beta_{16} - 496953590726531 \beta_{15} - 31391010068768 \beta_{14} + \cdots + 14\!\cdots\!94 \)
|
| \(\nu^{15}\) | \(=\) |
\( 10\!\cdots\!48 \beta_{16} + \cdots + 26\!\cdots\!45 \)
|
| \(\nu^{16}\) | \(=\) |
\( 15\!\cdots\!53 \beta_{16} + \cdots + 81\!\cdots\!79 \)
|
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 |
|
−22.3182 | 27.0000 | 370.104 | −84.4166 | −602.593 | 1532.13 | −5403.34 | 729.000 | 1884.03 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −21.6388 | 27.0000 | 340.238 | −399.107 | −584.248 | −1780.57 | −4592.58 | 729.000 | 8636.21 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −17.9892 | 27.0000 | 195.612 | −98.9095 | −485.709 | 159.201 | −1216.28 | 729.000 | 1779.30 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −17.8998 | 27.0000 | 192.403 | 255.355 | −483.295 | −1072.82 | −1152.80 | 729.000 | −4570.81 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −13.3335 | 27.0000 | 49.7814 | 152.093 | −360.004 | −1328.47 | 1042.93 | 729.000 | −2027.93 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −12.1391 | 27.0000 | 19.3578 | 236.334 | −327.756 | 1426.66 | 1318.82 | 729.000 | −2868.89 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −6.85375 | 27.0000 | −81.0262 | −190.727 | −185.051 | −799.288 | 1432.61 | 729.000 | 1307.20 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −6.01497 | 27.0000 | −91.8201 | −385.807 | −162.404 | 847.649 | 1322.21 | 729.000 | 2320.62 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.396855 | 27.0000 | −127.843 | 247.233 | 10.7151 | −652.929 | −101.532 | 729.000 | 98.1157 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.413031 | 27.0000 | −127.829 | 231.152 | 11.1518 | 302.150 | −105.665 | 729.000 | 95.4729 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.11298 | 27.0000 | −123.535 | −537.118 | 57.0506 | 1259.15 | −531.490 | 729.000 | −1134.92 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 5.01814 | 27.0000 | −102.818 | −77.5687 | 135.490 | −216.829 | −1158.28 | 729.000 | −389.251 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 10.7630 | 27.0000 | −12.1577 | 451.863 | 290.601 | −504.672 | −1508.52 | 729.000 | 4863.40 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 13.6681 | 27.0000 | 58.8158 | −207.404 | 369.038 | 882.793 | −945.614 | 729.000 | −2834.81 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 15.5255 | 27.0000 | 113.041 | −141.845 | 419.189 | 105.285 | −232.240 | 729.000 | −2202.21 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 16.2619 | 27.0000 | 136.450 | −30.6671 | 439.072 | −1356.62 | 137.421 | 729.000 | −498.706 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 22.0278 | 27.0000 | 357.226 | −492.460 | 594.752 | −1209.81 | 5049.35 | 729.000 | −10847.8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(59\) | \(1\) |
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 | 177.8.a.b | ✓ | 17 |
| 3.b | odd | 2 | 1 | 531.8.a.d | 17 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 177.8.a.b | ✓ | 17 | 1.a | even | 1 | 1 | trivial |
| 531.8.a.d | 17 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{17} + 32 T_{2}^{16} - 1159 T_{2}^{15} - 43065 T_{2}^{14} + 456514 T_{2}^{13} + \cdots - 14\!\cdots\!16 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(177))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{17} + \cdots - 14\!\cdots\!16 \)
|
| $3$ |
\( (T - 27)^{17} \)
|
| $5$ |
\( T^{17} + \cdots + 10\!\cdots\!00 \)
|
| $7$ |
\( T^{17} + \cdots + 24\!\cdots\!92 \)
|
| $11$ |
\( T^{17} + \cdots + 63\!\cdots\!60 \)
|
| $13$ |
\( T^{17} + \cdots + 12\!\cdots\!88 \)
|
| $17$ |
\( T^{17} + \cdots + 39\!\cdots\!68 \)
|
| $19$ |
\( T^{17} + \cdots + 27\!\cdots\!00 \)
|
| $23$ |
\( T^{17} + \cdots - 15\!\cdots\!72 \)
|
| $29$ |
\( T^{17} + \cdots - 15\!\cdots\!00 \)
|
| $31$ |
\( T^{17} + \cdots + 83\!\cdots\!92 \)
|
| $37$ |
\( T^{17} + \cdots - 36\!\cdots\!24 \)
|
| $41$ |
\( T^{17} + \cdots - 57\!\cdots\!00 \)
|
| $43$ |
\( T^{17} + \cdots + 42\!\cdots\!92 \)
|
| $47$ |
\( T^{17} + \cdots + 19\!\cdots\!80 \)
|
| $53$ |
\( T^{17} + \cdots + 92\!\cdots\!84 \)
|
| $59$ |
\( (T + 205379)^{17} \)
|
| $61$ |
\( T^{17} + \cdots + 46\!\cdots\!96 \)
|
| $67$ |
\( T^{17} + \cdots + 41\!\cdots\!12 \)
|
| $71$ |
\( T^{17} + \cdots - 88\!\cdots\!16 \)
|
| $73$ |
\( T^{17} + \cdots - 48\!\cdots\!00 \)
|
| $79$ |
\( T^{17} + \cdots - 80\!\cdots\!28 \)
|
| $83$ |
\( T^{17} + \cdots + 56\!\cdots\!08 \)
|
| $89$ |
\( T^{17} + \cdots - 36\!\cdots\!88 \)
|
| $97$ |
\( T^{17} + \cdots + 80\!\cdots\!88 \)
|
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