Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1775,4,Mod(1,1775)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1775, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1775.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1775 = 5^{2} \cdot 71 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1775.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: | \(104.728390260\) |
| Analytic rank: | \(0\) |
| Dimension: | \(19\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{19} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{19} - 9 x^{18} - 79 x^{17} + 878 x^{16} + 2001 x^{15} - 34381 x^{14} - 4553 x^{13} + 689017 x^{12} + \cdots - 9007680 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | no (minimal twist has level 355) |
| 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 \(1,\beta_1,\ldots,\beta_{18}\) 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
\( x^{19} - 9 x^{18} - 79 x^{17} + 878 x^{16} + 2001 x^{15} - 34381 x^{14} - 4553 x^{13} + 689017 x^{12} + \cdots - 9007680 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 60\!\cdots\!17 \nu^{18} + \cdots - 17\!\cdots\!24 ) / 17\!\cdots\!68 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 20\!\cdots\!78 \nu^{18} + \cdots + 37\!\cdots\!08 ) / 21\!\cdots\!96 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 69\!\cdots\!79 \nu^{18} + \cdots + 29\!\cdots\!32 ) / 42\!\cdots\!92 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 23\!\cdots\!87 \nu^{18} + \cdots - 12\!\cdots\!96 ) / 85\!\cdots\!84 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 12\!\cdots\!53 \nu^{18} + \cdots - 95\!\cdots\!96 ) / 42\!\cdots\!92 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 58\!\cdots\!91 \nu^{18} + \cdots + 40\!\cdots\!88 ) / 17\!\cdots\!68 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 15\!\cdots\!15 \nu^{18} + \cdots - 81\!\cdots\!08 ) / 42\!\cdots\!92 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 48\!\cdots\!85 \nu^{18} + \cdots - 30\!\cdots\!32 ) / 99\!\cdots\!44 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 23\!\cdots\!41 \nu^{18} + \cdots + 30\!\cdots\!52 ) / 42\!\cdots\!92 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 98\!\cdots\!01 \nu^{18} + \cdots + 60\!\cdots\!80 ) / 17\!\cdots\!68 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 50\!\cdots\!05 \nu^{18} + \cdots + 14\!\cdots\!64 ) / 85\!\cdots\!84 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 52\!\cdots\!83 \nu^{18} + \cdots - 18\!\cdots\!12 ) / 85\!\cdots\!84 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 28\!\cdots\!67 \nu^{18} + \cdots + 67\!\cdots\!88 ) / 42\!\cdots\!92 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 13\!\cdots\!45 \nu^{18} + \cdots + 50\!\cdots\!24 ) / 17\!\cdots\!68 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 16\!\cdots\!41 \nu^{18} + \cdots - 54\!\cdots\!36 ) / 21\!\cdots\!96 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 15\!\cdots\!77 \nu^{18} + \cdots + 56\!\cdots\!76 ) / 17\!\cdots\!68 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{16} - \beta_{10} + 2\beta_{9} - 2\beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} + 22\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{18} + \beta_{17} + \beta_{16} + 2 \beta_{14} + 2 \beta_{12} + \beta_{9} - \beta_{8} + \beta_{7} + \cdots + 270 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 2 \beta_{18} + 38 \beta_{16} - 6 \beta_{15} + 4 \beta_{14} + 5 \beta_{13} + 5 \beta_{12} + \beta_{11} + \cdots + 106 \)
|
| \(\nu^{6}\) | \(=\) |
\( 26 \beta_{18} + 41 \beta_{17} + 53 \beta_{16} + \beta_{15} + 88 \beta_{14} + 98 \beta_{12} + 3 \beta_{11} + \cdots + 6377 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 107 \beta_{18} + \beta_{17} + 1223 \beta_{16} - 317 \beta_{15} + 234 \beta_{14} + 250 \beta_{13} + \cdots + 4892 \)
|
| \(\nu^{8}\) | \(=\) |
\( 501 \beta_{18} + 1402 \beta_{17} + 2173 \beta_{16} - 9 \beta_{15} + 3052 \beta_{14} + 67 \beta_{13} + \cdots + 161087 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 3956 \beta_{18} + 125 \beta_{17} + 37406 \beta_{16} - 11830 \beta_{15} + 9922 \beta_{14} + \cdots + 184566 \)
|
| \(\nu^{10}\) | \(=\) |
\( 7181 \beta_{18} + 45229 \beta_{17} + 79758 \beta_{16} - 2635 \beta_{15} + 97802 \beta_{14} + \cdots + 4251788 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 128294 \beta_{18} + 10871 \beta_{17} + 1120059 \beta_{16} - 386403 \beta_{15} + 367550 \beta_{14} + \cdots + 6467310 \)
|
| \(\nu^{12}\) | \(=\) |
\( 24577 \beta_{18} + 1418606 \beta_{17} + 2752839 \beta_{16} - 157914 \beta_{15} + 3032080 \beta_{14} + \cdots + 115847950 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 3926311 \beta_{18} + 676818 \beta_{17} + 33217369 \beta_{16} - 11820510 \beta_{15} + 12667802 \beta_{14} + \cdots + 218909775 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 3654669 \beta_{18} + 43851711 \beta_{17} + 91508359 \beta_{16} - 6872339 \beta_{15} + \cdots + 3232780483 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 116800611 \beta_{18} + 34303019 \beta_{17} + 981094787 \beta_{16} - 348481464 \beta_{15} + \cdots + 7260950811 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 207500592 \beta_{18} + 1346414449 \beta_{17} + 2968560460 \beta_{16} - 257202035 \beta_{15} + \cdots + 91867260750 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 3423926490 \beta_{18} + 1528499997 \beta_{17} + 28941500551 \beta_{16} - 10038983480 \beta_{15} + \cdots + 237451303503 \)
|
| \(\nu^{18}\) | \(=\) |
\( - 8347743500 \beta_{18} + 41253611739 \beta_{17} + 94716870014 \beta_{16} - 8802317644 \beta_{15} + \cdots + 2647339653317 \)
|
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 |
|
−5.55572 | 9.77036 | 22.8660 | 0 | −54.2814 | 28.6578 | −82.5916 | 68.4599 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −5.49865 | −3.08291 | 22.2352 | 0 | 16.9518 | −13.5280 | −78.2743 | −17.4957 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.60196 | −9.86031 | 13.1781 | 0 | 45.3768 | 14.8949 | −23.8293 | 70.2256 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −4.43810 | −7.36264 | 11.6967 | 0 | 32.6761 | −31.7612 | −16.4063 | 27.2085 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −4.33225 | 4.03934 | 10.7684 | 0 | −17.4995 | −13.8724 | −11.9935 | −10.6837 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.43043 | 3.89023 | −2.09301 | 0 | −9.45492 | −22.3557 | 24.5304 | −11.8661 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −2.39662 | 5.67406 | −2.25622 | 0 | −13.5985 | 29.5196 | 24.5802 | 5.19493 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −2.03120 | −6.41558 | −3.87421 | 0 | 13.0313 | −11.6120 | 24.1189 | 14.1596 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.99176 | 7.65305 | −4.03290 | 0 | −15.2430 | 13.1951 | 23.9666 | 31.5692 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −0.371025 | 3.72119 | −7.86234 | 0 | −1.38066 | −21.7374 | 5.88533 | −13.1527 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −0.267791 | −0.962650 | −7.92829 | 0 | 0.257789 | 26.1528 | 4.26545 | −26.0733 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0.656965 | −2.12236 | −7.56840 | 0 | −1.39432 | 20.6566 | −10.2279 | −22.4956 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 0.705683 | −8.20008 | −7.50201 | 0 | −5.78666 | −5.75838 | −10.9395 | 40.2413 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.44227 | 2.89055 | −2.03532 | 0 | 7.05949 | 21.1158 | −24.5090 | −18.6447 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 3.25185 | −0.957679 | 2.57455 | 0 | −3.11423 | −17.6553 | −17.6428 | −26.0829 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 3.85991 | 8.26010 | 6.89888 | 0 | 31.8832 | −31.1291 | −4.25021 | 41.2293 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 4.07087 | −9.75612 | 8.57200 | 0 | −39.7159 | −12.9895 | 2.32855 | 68.1819 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 4.76325 | 9.95894 | 14.6885 | 0 | 47.4369 | 17.2158 | 31.8591 | 72.1805 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.19 | 5.16472 | −3.13750 | 18.6743 | 0 | −16.2043 | 28.9907 | 55.1298 | −17.1561 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(71\) | \(1\) |
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 | 1775.4.a.f | 19 | |
| 5.b | even | 2 | 1 | 355.4.a.c | ✓ | 19 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 355.4.a.c | ✓ | 19 | 5.b | even | 2 | 1 | |
| 1775.4.a.f | 19 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{19} + 9 T_{2}^{18} - 79 T_{2}^{17} - 878 T_{2}^{16} + 2001 T_{2}^{15} + 34381 T_{2}^{14} + \cdots + 9007680 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1775))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{19} + 9 T^{18} + \cdots + 9007680 \)
|
| $3$ |
\( T^{19} + \cdots - 4159981775872 \)
|
| $5$ |
\( T^{19} \)
|
| $7$ |
\( T^{19} + \cdots - 13\!\cdots\!36 \)
|
| $11$ |
\( T^{19} + \cdots - 14\!\cdots\!00 \)
|
| $13$ |
\( T^{19} + \cdots - 33\!\cdots\!00 \)
|
| $17$ |
\( T^{19} + \cdots + 83\!\cdots\!52 \)
|
| $19$ |
\( T^{19} + \cdots - 39\!\cdots\!16 \)
|
| $23$ |
\( T^{19} + \cdots - 36\!\cdots\!80 \)
|
| $29$ |
\( T^{19} + \cdots + 51\!\cdots\!32 \)
|
| $31$ |
\( T^{19} + \cdots - 12\!\cdots\!64 \)
|
| $37$ |
\( T^{19} + \cdots + 14\!\cdots\!44 \)
|
| $41$ |
\( T^{19} + \cdots - 12\!\cdots\!28 \)
|
| $43$ |
\( T^{19} + \cdots - 97\!\cdots\!68 \)
|
| $47$ |
\( T^{19} + \cdots - 58\!\cdots\!32 \)
|
| $53$ |
\( T^{19} + \cdots + 82\!\cdots\!88 \)
|
| $59$ |
\( T^{19} + \cdots - 35\!\cdots\!24 \)
|
| $61$ |
\( T^{19} + \cdots + 20\!\cdots\!28 \)
|
| $67$ |
\( T^{19} + \cdots + 16\!\cdots\!68 \)
|
| $71$ |
\( (T + 71)^{19} \)
|
| $73$ |
\( T^{19} + \cdots - 85\!\cdots\!08 \)
|
| $79$ |
\( T^{19} + \cdots + 88\!\cdots\!96 \)
|
| $83$ |
\( T^{19} + \cdots + 82\!\cdots\!96 \)
|
| $89$ |
\( T^{19} + \cdots - 50\!\cdots\!60 \)
|
| $97$ |
\( T^{19} + \cdots + 51\!\cdots\!28 \)
|
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