Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [605,6,Mod(1,605)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(605, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("605.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 605 = 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 605.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: | \(97.0322109869\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - x^{19} - 523 x^{18} + 521 x^{17} + 115018 x^{16} - 115347 x^{15} - 13821739 x^{14} + \cdots - 32708279373824 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 11^{8} \) |
| Twist minimal: | no (minimal twist has level 55) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - x^{19} - 523 x^{18} + 521 x^{17} + 115018 x^{16} - 115347 x^{15} - 13821739 x^{14} + \cdots - 32708279373824 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 52 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 17\!\cdots\!77 \nu^{19} + \cdots - 27\!\cdots\!12 ) / 30\!\cdots\!44 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 19\!\cdots\!37 \nu^{19} + \cdots + 66\!\cdots\!72 ) / 19\!\cdots\!76 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 93\!\cdots\!31 \nu^{19} + \cdots - 11\!\cdots\!36 ) / 39\!\cdots\!52 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 30\!\cdots\!17 \nu^{19} + \cdots - 85\!\cdots\!28 ) / 11\!\cdots\!56 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 41\!\cdots\!31 \nu^{19} + \cdots - 65\!\cdots\!08 ) / 98\!\cdots\!88 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 50\!\cdots\!33 \nu^{19} + \cdots - 28\!\cdots\!28 ) / 11\!\cdots\!56 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 64\!\cdots\!77 \nu^{19} + \cdots + 12\!\cdots\!12 ) / 10\!\cdots\!96 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 71\!\cdots\!39 \nu^{19} + \cdots - 50\!\cdots\!68 ) / 11\!\cdots\!56 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 94\!\cdots\!03 \nu^{19} + \cdots - 75\!\cdots\!16 ) / 14\!\cdots\!32 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 28\!\cdots\!35 \nu^{19} + \cdots + 86\!\cdots\!00 ) / 29\!\cdots\!64 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 14\!\cdots\!31 \nu^{19} + \cdots + 15\!\cdots\!36 ) / 11\!\cdots\!56 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 37\!\cdots\!85 \nu^{19} + \cdots - 30\!\cdots\!56 ) / 29\!\cdots\!64 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 72\!\cdots\!51 \nu^{19} + \cdots - 66\!\cdots\!12 ) / 39\!\cdots\!52 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 28\!\cdots\!07 \nu^{19} + \cdots - 42\!\cdots\!24 ) / 11\!\cdots\!56 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 68\!\cdots\!59 \nu^{19} + \cdots + 94\!\cdots\!48 ) / 22\!\cdots\!52 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 80\!\cdots\!03 \nu^{19} + \cdots + 11\!\cdots\!12 ) / 17\!\cdots\!16 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 28\!\cdots\!09 \nu^{19} + \cdots + 20\!\cdots\!92 ) / 49\!\cdots\!44 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 52 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + 2\beta_{4} - \beta_{3} + 83\beta _1 - 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{19} - 2 \beta_{18} + \beta_{17} - \beta_{16} - \beta_{15} - 3 \beta_{11} + \beta_{10} + \cdots + 4310 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{19} - 8 \beta_{18} + 2 \beta_{17} + 5 \beta_{16} - 5 \beta_{15} + 9 \beta_{14} + 3 \beta_{13} + \cdots - 157 \)
|
| \(\nu^{6}\) | \(=\) |
\( 134 \beta_{19} - 361 \beta_{18} + 170 \beta_{17} - 140 \beta_{16} - 147 \beta_{15} + 29 \beta_{14} + \cdots + 401873 \)
|
| \(\nu^{7}\) | \(=\) |
\( 14 \beta_{19} - 1616 \beta_{18} + 536 \beta_{17} + 1130 \beta_{16} - 892 \beta_{15} + 1670 \beta_{14} + \cdots + 7452 \)
|
| \(\nu^{8}\) | \(=\) |
\( 13892 \beta_{19} - 49832 \beta_{18} + 22680 \beta_{17} - 15070 \beta_{16} - 17058 \beta_{15} + \cdots + 39541608 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 12481 \beta_{19} - 249102 \beta_{18} + 93275 \beta_{17} + 181555 \beta_{16} - 125929 \beta_{15} + \cdots + 5447998 \)
|
| \(\nu^{10}\) | \(=\) |
\( 1337413 \beta_{19} - 6321964 \beta_{18} + 2797532 \beta_{17} - 1442413 \beta_{16} - 1864295 \beta_{15} + \cdots + 4011058321 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 2692252 \beta_{19} - 34661771 \beta_{18} + 13649034 \beta_{17} + 25644280 \beta_{16} + \cdots + 1351708001 \)
|
| \(\nu^{12}\) | \(=\) |
\( 125800466 \beta_{19} - 774466126 \beta_{18} + 333510036 \beta_{17} - 125448632 \beta_{16} + \cdots + 415378775286 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 405101796 \beta_{19} - 4579359616 \beta_{18} + 1827374616 \beta_{17} + 3397046160 \beta_{16} + \cdots + 253404682644 \)
|
| \(\nu^{14}\) | \(=\) |
\( 11781256913 \beta_{19} - 93275343778 \beta_{18} + 39078781481 \beta_{17} - 9572480593 \beta_{16} + \cdots + 43701921820630 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 52873075427 \beta_{19} - 587133723344 \beta_{18} + 232550480486 \beta_{17} + 433532994865 \beta_{16} + \cdots + 41223854994291 \)
|
| \(\nu^{16}\) | \(=\) |
\( 1107156456910 \beta_{19} - 11130736807385 \beta_{18} + 4536158683702 \beta_{17} - 547555567636 \beta_{16} + \cdots + 46\!\cdots\!89 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 6397086175894 \beta_{19} - 73874705672452 \beta_{18} + 28704694377240 \beta_{17} + \cdots + 61\!\cdots\!48 \)
|
| \(\nu^{18}\) | \(=\) |
\( 104749406552836 \beta_{19} + \cdots + 50\!\cdots\!40 \)
|
| \(\nu^{19}\) | \(=\) |
\( - 738711912072257 \beta_{19} + \cdots + 86\!\cdots\!62 \)
|
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 |
|
−10.8658 | −20.9346 | 86.0646 | −25.0000 | 227.471 | 150.113 | −587.453 | 195.259 | 271.644 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −10.1332 | −21.6453 | 70.6821 | −25.0000 | 219.336 | −136.893 | −391.974 | 225.517 | 253.330 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −8.99127 | 10.3783 | 48.8429 | −25.0000 | −93.3145 | −132.461 | −151.440 | −135.290 | 224.782 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −8.98229 | 23.5092 | 48.6815 | −25.0000 | −211.166 | 232.955 | −149.838 | 309.680 | 224.557 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −8.03981 | 5.09012 | 32.6385 | −25.0000 | −40.9236 | −87.3874 | −5.13363 | −217.091 | 200.995 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −5.77720 | 22.4828 | 1.37599 | −25.0000 | −129.888 | 56.5871 | 176.921 | 262.477 | 144.430 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −5.27089 | −17.5243 | −4.21770 | −25.0000 | 92.3688 | −132.134 | 190.900 | 64.1015 | 131.772 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −3.12060 | −6.47958 | −22.2618 | −25.0000 | 20.2202 | 46.8054 | 169.330 | −201.015 | 78.0151 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.84684 | −21.5804 | −28.5892 | −25.0000 | 39.8555 | 85.4988 | 111.898 | 222.716 | 46.1709 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −1.08038 | 9.44854 | −30.8328 | −25.0000 | −10.2080 | 16.3713 | 67.8833 | −153.725 | 27.0095 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −0.238351 | −19.5755 | −31.9432 | −25.0000 | 4.66584 | 208.743 | 15.2409 | 140.200 | 5.95878 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.05592 | 22.6862 | −30.8850 | −25.0000 | 23.9549 | −12.2465 | −66.4016 | 271.665 | −26.3980 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 4.73032 | 8.08839 | −9.62406 | −25.0000 | 38.2607 | −159.979 | −196.895 | −177.578 | −118.258 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 5.37088 | −0.268292 | −3.15366 | −25.0000 | −1.44096 | −221.735 | −188.806 | −242.928 | −134.272 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 5.73355 | 3.90196 | 0.873636 | −25.0000 | 22.3721 | 214.940 | −178.465 | −227.775 | −143.339 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 6.92807 | −8.76999 | 15.9982 | −25.0000 | −60.7591 | 97.5351 | −110.862 | −166.087 | −173.202 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 9.34075 | −29.7904 | 55.2496 | −25.0000 | −278.265 | −104.862 | 217.169 | 644.469 | −233.519 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 9.60667 | 21.6000 | 60.2880 | −25.0000 | 207.504 | 51.1649 | 271.754 | 223.560 | −240.167 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.19 | 10.1575 | 30.4037 | 71.1747 | −25.0000 | 308.825 | −10.6636 | 397.917 | 681.384 | −253.937 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.20 | 10.4229 | −11.0208 | 76.6377 | −25.0000 | −114.870 | 4.64894 | 465.256 | −121.541 | −260.573 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(11\) | \(-1\) |
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 | 605.6.a.o | 20 | |
| 11.b | odd | 2 | 1 | 605.6.a.p | 20 | ||
| 11.d | odd | 10 | 2 | 55.6.g.b | ✓ | 40 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 55.6.g.b | ✓ | 40 | 11.d | odd | 10 | 2 | |
| 605.6.a.o | 20 | 1.a | even | 1 | 1 | trivial | |
| 605.6.a.p | 20 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{20} + T_{2}^{19} - 523 T_{2}^{18} - 521 T_{2}^{17} + 115018 T_{2}^{16} + 115347 T_{2}^{15} + \cdots - 32708279373824 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(605))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + \cdots - 32708279373824 \)
|
| $3$ |
\( T^{20} + \cdots + 20\!\cdots\!04 \)
|
| $5$ |
\( (T + 25)^{20} \)
|
| $7$ |
\( T^{20} + \cdots - 13\!\cdots\!64 \)
|
| $11$ |
\( T^{20} \)
|
| $13$ |
\( T^{20} + \cdots + 18\!\cdots\!00 \)
|
| $17$ |
\( T^{20} + \cdots - 17\!\cdots\!96 \)
|
| $19$ |
\( T^{20} + \cdots + 49\!\cdots\!25 \)
|
| $23$ |
\( T^{20} + \cdots + 43\!\cdots\!24 \)
|
| $29$ |
\( T^{20} + \cdots + 10\!\cdots\!00 \)
|
| $31$ |
\( T^{20} + \cdots + 47\!\cdots\!00 \)
|
| $37$ |
\( T^{20} + \cdots + 16\!\cdots\!84 \)
|
| $41$ |
\( T^{20} + \cdots - 35\!\cdots\!59 \)
|
| $43$ |
\( T^{20} + \cdots - 10\!\cdots\!24 \)
|
| $47$ |
\( T^{20} + \cdots + 74\!\cdots\!04 \)
|
| $53$ |
\( T^{20} + \cdots - 39\!\cdots\!04 \)
|
| $59$ |
\( T^{20} + \cdots - 32\!\cdots\!25 \)
|
| $61$ |
\( T^{20} + \cdots + 58\!\cdots\!36 \)
|
| $67$ |
\( T^{20} + \cdots - 27\!\cdots\!44 \)
|
| $71$ |
\( T^{20} + \cdots - 23\!\cdots\!16 \)
|
| $73$ |
\( T^{20} + \cdots + 21\!\cdots\!44 \)
|
| $79$ |
\( T^{20} + \cdots + 83\!\cdots\!00 \)
|
| $83$ |
\( T^{20} + \cdots + 44\!\cdots\!36 \)
|
| $89$ |
\( T^{20} + \cdots + 52\!\cdots\!25 \)
|
| $97$ |
\( T^{20} + \cdots + 12\!\cdots\!64 \)
|
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