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: |
\( x^{20} - 3 x^{19} - 436 x^{18} + 1032 x^{17} + 79722 x^{16} - 137625 x^{15} - 7955280 x^{14} + \cdots + 80139762021616 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 5\cdot 11^{8} \) |
| 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
\( x^{20} - 3 x^{19} - 436 x^{18} + 1032 x^{17} + 79722 x^{16} - 137625 x^{15} - 7955280 x^{14} + \cdots + 80139762021616 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 44 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 33\!\cdots\!91 \nu^{19} + \cdots - 91\!\cdots\!16 ) / 66\!\cdots\!60 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 11\!\cdots\!97 \nu^{19} + \cdots - 22\!\cdots\!12 ) / 12\!\cdots\!20 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 23\!\cdots\!99 \nu^{19} + \cdots - 26\!\cdots\!96 ) / 13\!\cdots\!20 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 68\!\cdots\!61 \nu^{19} + \cdots - 60\!\cdots\!56 ) / 33\!\cdots\!80 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 38\!\cdots\!37 \nu^{19} + \cdots + 88\!\cdots\!32 ) / 12\!\cdots\!20 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 14\!\cdots\!13 \nu^{19} + \cdots + 27\!\cdots\!88 ) / 44\!\cdots\!40 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 22\!\cdots\!36 \nu^{19} + \cdots - 12\!\cdots\!76 ) / 41\!\cdots\!60 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 18\!\cdots\!93 \nu^{19} + \cdots + 46\!\cdots\!88 ) / 13\!\cdots\!20 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 10\!\cdots\!17 \nu^{19} + \cdots - 22\!\cdots\!32 ) / 66\!\cdots\!60 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 21\!\cdots\!51 \nu^{19} + \cdots + 51\!\cdots\!36 ) / 13\!\cdots\!20 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 22\!\cdots\!79 \nu^{19} + \cdots - 46\!\cdots\!36 ) / 13\!\cdots\!72 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 26\!\cdots\!19 \nu^{19} + \cdots + 61\!\cdots\!04 ) / 13\!\cdots\!20 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 28\!\cdots\!81 \nu^{19} + \cdots - 69\!\cdots\!76 ) / 13\!\cdots\!20 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 22\!\cdots\!79 \nu^{19} + \cdots + 36\!\cdots\!64 ) / 66\!\cdots\!60 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 53\!\cdots\!47 \nu^{19} + \cdots + 12\!\cdots\!52 ) / 13\!\cdots\!20 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 19\!\cdots\!89 \nu^{19} + \cdots - 51\!\cdots\!24 ) / 41\!\cdots\!60 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 14\!\cdots\!63 \nu^{19} + \cdots + 30\!\cdots\!08 ) / 30\!\cdots\!88 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 44 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{12} + \beta_{10} + \beta_{7} + \beta_{4} + \beta_{3} + \beta_{2} + 73\beta _1 + 32 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{15} + 2 \beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} - 2 \beta_{9} - \beta_{8} + 4 \beta_{7} + \cdots + 3235 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 3 \beta_{19} + 6 \beta_{18} + 2 \beta_{16} - 9 \beta_{15} + 7 \beta_{14} + 3 \beta_{13} - 126 \beta_{12} + \cdots + 4456 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 61 \beta_{19} + 52 \beta_{18} + 16 \beta_{17} + 40 \beta_{16} + 99 \beta_{15} + 345 \beta_{14} + \cdots + 278502 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 592 \beta_{19} + 1446 \beta_{18} - 132 \beta_{17} + 224 \beta_{16} - 1925 \beta_{15} + 1410 \beta_{14} + \cdots + 524677 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 13436 \beta_{19} + 11564 \beta_{18} + 3160 \beta_{17} + 9032 \beta_{16} + 5714 \beta_{15} + \cdots + 25599852 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 87004 \beta_{19} + 235008 \beta_{18} - 26464 \beta_{17} + 18528 \beta_{16} - 288465 \beta_{15} + \cdots + 58438297 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 2054309 \beta_{19} + 1832474 \beta_{18} + 463392 \beta_{17} + 1383526 \beta_{16} - 2842 \beta_{15} + \cdots + 2436032367 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 11539774 \beta_{19} + 32636570 \beta_{18} - 3583776 \beta_{17} + 1366890 \beta_{16} + \cdots + 6362725497 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 271575108 \beta_{19} + 253905572 \beta_{18} + 60875140 \beta_{17} + 180656874 \beta_{16} + \cdots + 237068131781 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 1454884697 \beta_{19} + 4182431214 \beta_{18} - 406788188 \beta_{17} + 95948840 \beta_{16} + \cdots + 687241069259 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 33342049256 \beta_{19} + 32804771242 \beta_{18} + 7560877620 \beta_{17} + 21704440984 \beta_{16} + \cdots + 23459616118208 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 177778368331 \beta_{19} + 511447348986 \beta_{18} - 41317194632 \beta_{17} + 6711433594 \beta_{16} + \cdots + 74150322302858 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 3923113674559 \beta_{19} + 4063815496656 \beta_{18} + 908031506784 \beta_{17} + 2482937401780 \beta_{16} + \cdots + 23\!\cdots\!15 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 21255554557483 \beta_{19} + 60708015231800 \beta_{18} - 3829896076292 \beta_{17} + \cdots + 80\!\cdots\!03 \)
|
| \(\nu^{18}\) | \(=\) |
\( - 449610493190765 \beta_{19} + 489670124139272 \beta_{18} + 106672111798256 \beta_{17} + \cdots + 23\!\cdots\!23 \)
|
| \(\nu^{19}\) | \(=\) |
\( - 25\!\cdots\!09 \beta_{19} + \cdots + 86\!\cdots\!64 \)
|
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 |
|
−9.46090 | 6.31148 | 57.5086 | 25.0000 | −59.7123 | 132.083 | −241.334 | −203.165 | −236.522 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −8.79354 | −22.5193 | 45.3263 | 25.0000 | 198.024 | −38.0979 | −117.185 | 264.118 | −219.838 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −8.11748 | −11.1242 | 33.8935 | 25.0000 | 90.3005 | 93.8217 | −15.3706 | −119.252 | −202.937 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −7.89249 | 20.7410 | 30.2914 | 25.0000 | −163.698 | 181.733 | 13.4853 | 187.189 | −197.312 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −4.90749 | 11.7032 | −7.91656 | 25.0000 | −57.4334 | −217.582 | 195.890 | −106.035 | −122.687 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −4.51773 | 22.9237 | −11.5902 | 25.0000 | −103.563 | 56.4114 | 196.928 | 282.498 | −112.943 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −3.82252 | −23.1708 | −17.3883 | 25.0000 | 88.5706 | −62.4720 | 188.788 | 293.884 | −95.5630 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −2.74430 | −12.8364 | −24.4688 | 25.0000 | 35.2269 | −161.649 | 154.967 | −78.2269 | −68.6075 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.628180 | −4.58825 | −31.6054 | 25.0000 | 2.88225 | 116.292 | 39.9557 | −221.948 | −15.7045 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.111181 | 4.60961 | −31.9876 | 25.0000 | 0.512503 | 61.3924 | −7.11423 | −221.751 | 2.77953 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.91082 | 18.7218 | −23.5272 | 25.0000 | 54.4957 | −208.489 | −161.629 | 107.505 | 72.7704 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 3.66765 | 24.7362 | −18.5483 | 25.0000 | 90.7237 | 169.967 | −185.394 | 368.878 | 91.6913 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 4.45406 | −2.47240 | −12.1614 | 25.0000 | −11.0122 | −82.4631 | −196.697 | −236.887 | 111.351 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 4.62368 | −27.1017 | −10.6216 | 25.0000 | −125.310 | 225.648 | −197.069 | 491.501 | 115.592 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 6.47490 | −29.5208 | 9.92436 | 25.0000 | −191.144 | −16.3316 | −142.938 | 628.477 | 161.873 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 6.90598 | −1.02322 | 15.6925 | 25.0000 | −7.06633 | −3.68045 | −112.619 | −241.953 | 172.649 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 7.97082 | 27.8505 | 31.5340 | 25.0000 | 221.991 | 220.809 | −3.71413 | 532.648 | 199.271 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 9.70460 | −7.56086 | 62.1793 | 25.0000 | −73.3751 | 199.276 | 292.879 | −185.833 | 242.615 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.19 | 9.94562 | 23.0590 | 66.9153 | 25.0000 | 229.336 | −80.2641 | 347.254 | 288.718 | 248.640 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.20 | 11.1153 | 3.26137 | 91.5500 | 25.0000 | 36.2511 | −27.4041 | 661.917 | −232.363 | 277.883 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.q | 20 | |
| 11.b | odd | 2 | 1 | 605.6.a.n | 20 | ||
| 11.c | even | 5 | 2 | 55.6.g.a | ✓ | 40 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 55.6.g.a | ✓ | 40 | 11.c | even | 5 | 2 | |
| 605.6.a.n | 20 | 11.b | odd | 2 | 1 | ||
| 605.6.a.q | 20 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{20} - 17 T_{2}^{19} - 303 T_{2}^{18} + 6189 T_{2}^{17} + 32496 T_{2}^{16} - 926379 T_{2}^{15} + \cdots - 7278582132736 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(605))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + \cdots - 7278582132736 \)
|
| $3$ |
\( T^{20} + \cdots + 82\!\cdots\!80 \)
|
| $5$ |
\( (T - 25)^{20} \)
|
| $7$ |
\( T^{20} + \cdots + 29\!\cdots\!80 \)
|
| $11$ |
\( T^{20} \)
|
| $13$ |
\( T^{20} + \cdots + 39\!\cdots\!00 \)
|
| $17$ |
\( T^{20} + \cdots - 24\!\cdots\!96 \)
|
| $19$ |
\( T^{20} + \cdots + 37\!\cdots\!25 \)
|
| $23$ |
\( T^{20} + \cdots + 82\!\cdots\!36 \)
|
| $29$ |
\( T^{20} + \cdots + 30\!\cdots\!00 \)
|
| $31$ |
\( T^{20} + \cdots - 50\!\cdots\!64 \)
|
| $37$ |
\( T^{20} + \cdots + 24\!\cdots\!16 \)
|
| $41$ |
\( T^{20} + \cdots + 58\!\cdots\!81 \)
|
| $43$ |
\( T^{20} + \cdots + 14\!\cdots\!36 \)
|
| $47$ |
\( T^{20} + \cdots - 56\!\cdots\!84 \)
|
| $53$ |
\( T^{20} + \cdots + 33\!\cdots\!64 \)
|
| $59$ |
\( T^{20} + \cdots + 26\!\cdots\!75 \)
|
| $61$ |
\( T^{20} + \cdots - 56\!\cdots\!20 \)
|
| $67$ |
\( T^{20} + \cdots + 37\!\cdots\!84 \)
|
| $71$ |
\( T^{20} + \cdots - 22\!\cdots\!84 \)
|
| $73$ |
\( T^{20} + \cdots + 60\!\cdots\!56 \)
|
| $79$ |
\( T^{20} + \cdots + 72\!\cdots\!00 \)
|
| $83$ |
\( T^{20} + \cdots - 18\!\cdots\!24 \)
|
| $89$ |
\( T^{20} + \cdots - 15\!\cdots\!75 \)
|
| $97$ |
\( T^{20} + \cdots + 63\!\cdots\!00 \)
|
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