Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2365,2,Mod(1,2365)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2365, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2365.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2365 = 5 \cdot 11 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2365.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: | \(18.8846200780\) |
| 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: |
\( x^{20} - 6 x^{19} - 13 x^{18} + 136 x^{17} - 19 x^{16} - 1235 x^{15} + 1169 x^{14} + 5712 x^{13} + \cdots - 36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 6 x^{19} - 13 x^{18} + 136 x^{17} - 19 x^{16} - 1235 x^{15} + 1169 x^{14} + 5712 x^{13} + \cdots - 36 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 6\nu^{2} + 4\nu + 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 15137361 \nu^{19} - 82625972 \nu^{18} - 199147229 \nu^{17} + 1500989060 \nu^{16} + \cdots + 27175129134 ) / 8191241446 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 229513885 \nu^{19} + 1055602910 \nu^{18} + 4520479944 \nu^{17} - 25333241704 \nu^{16} + \cdots - 12193416570 ) / 24573724338 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 83054263 \nu^{19} + 81400354 \nu^{18} - 3786618911 \nu^{17} + 296014053 \nu^{16} + \cdots + 24540607040 ) / 8191241446 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 321480400 \nu^{19} + 1536799439 \nu^{18} + 5880646656 \nu^{17} - 36174601969 \nu^{16} + \cdots - 8262499860 ) / 24573724338 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 349931870 \nu^{19} - 2592893611 \nu^{18} - 1978858744 \nu^{17} + 56078776940 \nu^{16} + \cdots - 116002797616 ) / 16382482892 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 562977658 \nu^{19} - 3811703240 \nu^{18} - 5124227835 \nu^{17} + 84537277288 \nu^{16} + \cdots - 155694891420 ) / 24573724338 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 2092008820 \nu^{19} + 7810396121 \nu^{18} + 49255397490 \nu^{17} - 194109587236 \nu^{16} + \cdots - 109543151784 ) / 49147448676 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 2490302882 \nu^{19} - 11554764211 \nu^{18} - 48100257150 \nu^{17} + 273207049736 \nu^{16} + \cdots + 5008205100 ) / 49147448676 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 2738912872 \nu^{19} + 15362892245 \nu^{18} + 40781278350 \nu^{17} - 351652211212 \nu^{16} + \cdots + 98990971284 ) / 49147448676 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 3080099149 \nu^{19} + 15131833412 \nu^{18} + 53628837492 \nu^{17} - 346795477384 \nu^{16} + \cdots + 321831458076 ) / 49147448676 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 1226978386 \nu^{19} + 7341301889 \nu^{18} + 16103133532 \nu^{17} - 166348015086 \nu^{16} + \cdots + 101408007784 ) / 16382482892 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 1360121996 \nu^{19} + 7315161397 \nu^{18} + 21088730146 \nu^{17} - 166854449952 \nu^{16} + \cdots + 25097715400 ) / 16382482892 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 4082047433 \nu^{19} - 18602570347 \nu^{18} - 80185267290 \nu^{17} + 441922405328 \nu^{16} + \cdots + 182385018756 ) / 49147448676 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 5066250521 \nu^{19} - 26315455693 \nu^{18} - 84463827120 \nu^{17} + 608824803566 \nu^{16} + \cdots + 50097758364 ) / 49147448676 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 1886389303 \nu^{19} + 9748756747 \nu^{18} + 32164413974 \nu^{17} - 227521621100 \nu^{16} + \cdots + 32159375240 ) / 16382482892 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 7\beta_{2} + \beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{15} - \beta_{13} + \beta_{10} + \beta_{8} - \beta_{6} + \beta_{4} + 9\beta_{3} + 10\beta_{2} + 28\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{18} + \beta_{16} + \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} + 2 \beta_{10} - \beta_{9} + \cdots + 75 \)
|
| \(\nu^{7}\) | \(=\) |
\( 2 \beta_{18} + \beta_{17} + \beta_{16} + 12 \beta_{15} - \beta_{14} - 11 \beta_{13} - 2 \beta_{12} + \cdots + 20 \)
|
| \(\nu^{8}\) | \(=\) |
\( - \beta_{19} + 15 \beta_{18} + \beta_{17} + 12 \beta_{16} + 16 \beta_{15} - 13 \beta_{14} - 13 \beta_{13} + \cdots + 436 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 4 \beta_{19} + 31 \beta_{18} + 14 \beta_{17} + 14 \beta_{16} + 110 \beta_{15} - 17 \beta_{14} + \cdots + 238 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 24 \beta_{19} + 152 \beta_{18} + 16 \beta_{17} + 101 \beta_{16} + 179 \beta_{15} - 124 \beta_{14} + \cdots + 2685 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 84 \beta_{19} + 328 \beta_{18} + 133 \beta_{17} + 129 \beta_{16} + 921 \beta_{15} - 195 \beta_{14} + \cdots + 2328 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 349 \beta_{19} + 1317 \beta_{18} + 172 \beta_{17} + 727 \beta_{16} + 1722 \beta_{15} - 1054 \beta_{14} + \cdots + 17309 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 1137 \beta_{19} + 2961 \beta_{18} + 1084 \beta_{17} + 970 \beta_{16} + 7396 \beta_{15} - 1894 \beta_{14} + \cdots + 20712 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 4059 \beta_{19} + 10552 \beta_{18} + 1569 \beta_{17} + 4732 \beta_{16} + 15264 \beta_{15} + \cdots + 115946 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 12656 \beta_{19} + 24582 \beta_{18} + 8213 \beta_{17} + 6292 \beta_{16} + 58045 \beta_{15} + \cdots + 174957 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 41693 \beta_{19} + 80942 \beta_{18} + 13171 \beta_{17} + 28150 \beta_{16} + 128688 \beta_{15} + \cdots + 802204 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 126270 \beta_{19} + 194239 \beta_{18} + 60052 \beta_{17} + 34754 \beta_{16} + 449197 \beta_{15} + \cdots + 1433425 \)
|
| \(\nu^{18}\) | \(=\) |
\( - 396358 \beta_{19} + 605193 \beta_{18} + 105712 \beta_{17} + 149458 \beta_{16} + 1049791 \beta_{15} + \cdots + 5700937 \)
|
| \(\nu^{19}\) | \(=\) |
\( - 1176246 \beta_{19} + 1488120 \beta_{18} + 432625 \beta_{17} + 143608 \beta_{16} + 3444637 \beta_{15} + \cdots + 11525135 \)
|
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 |
|
−2.37119 | −2.92709 | 3.62252 | 1.00000 | 6.94068 | 0.666008 | −3.84731 | 5.56787 | −2.37119 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.35054 | 0.967763 | 3.52503 | 1.00000 | −2.27476 | −3.44547 | −3.58465 | −2.06344 | −2.35054 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.19850 | −0.116781 | 2.83339 | 1.00000 | 0.256743 | 3.77570 | −1.83220 | −2.98636 | −2.19850 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.82167 | 2.81505 | 1.31850 | 1.00000 | −5.12810 | 0.159714 | 1.24148 | 4.92449 | −1.82167 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.37721 | −2.44735 | −0.103282 | 1.00000 | 3.37053 | −2.87853 | 2.89667 | 2.98954 | −1.37721 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.13416 | −0.414449 | −0.713688 | 1.00000 | 0.470050 | 1.51087 | 3.07775 | −2.82823 | −1.13416 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.731326 | 2.73782 | −1.46516 | 1.00000 | −2.00224 | 3.95302 | 2.53416 | 4.49565 | −0.731326 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.310156 | −1.02109 | −1.90380 | 1.00000 | 0.316696 | −3.19924 | 1.21079 | −1.95738 | −0.310156 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.167505 | −0.640984 | −1.97194 | 1.00000 | 0.107368 | 2.18875 | 0.665321 | −2.58914 | −0.167505 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.200229 | 0.783630 | −1.95991 | 1.00000 | 0.156906 | −3.60565 | −0.792889 | −2.38592 | 0.200229 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0.551185 | 1.43586 | −1.69620 | 1.00000 | 0.791423 | 3.88921 | −2.03729 | −0.938313 | 0.551185 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0.803598 | 3.12612 | −1.35423 | 1.00000 | 2.51215 | −0.0444752 | −2.69545 | 6.77265 | 0.803598 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.36327 | −2.56305 | −0.141496 | 1.00000 | −3.49412 | 3.81154 | −2.91944 | 3.56921 | 1.36327 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 1.44603 | −1.77521 | 0.0910038 | 1.00000 | −2.56700 | −3.09193 | −2.76047 | 0.151360 | 1.44603 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 1.68047 | 1.22689 | 0.823977 | 1.00000 | 2.06174 | −1.29978 | −1.97627 | −1.49475 | 1.68047 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.14696 | 2.25608 | 2.60946 | 1.00000 | 4.84372 | 2.63008 | 1.30848 | 2.08988 | 2.14696 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 2.32868 | −0.773496 | 3.42275 | 1.00000 | −1.80122 | 3.55721 | 3.31313 | −2.40170 | 2.32868 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 2.50737 | 2.89465 | 4.28692 | 1.00000 | 7.25798 | −2.30286 | 5.73417 | 5.37901 | 2.50737 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.19 | 2.64828 | 1.78498 | 5.01340 | 1.00000 | 4.72714 | −0.915010 | 7.98034 | 0.186163 | 2.64828 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.20 | 2.78617 | −2.34934 | 5.76275 | 1.00000 | −6.54567 | −2.35915 | 10.4837 | 2.51941 | 2.78617 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(11\) | \(-1\) |
| \(43\) | \(-1\) |
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 | 2365.2.a.o | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2365.2.a.o | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2365))\):
|
\( T_{2}^{20} - 6 T_{2}^{19} - 13 T_{2}^{18} + 136 T_{2}^{17} - 19 T_{2}^{16} - 1235 T_{2}^{15} + 1169 T_{2}^{14} + \cdots - 36 \)
|
|
\( T_{3}^{20} - 5 T_{3}^{19} - 27 T_{3}^{18} + 162 T_{3}^{17} + 255 T_{3}^{16} - 2137 T_{3}^{15} + \cdots + 704 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} - 6 T^{19} + \cdots - 36 \)
|
| $3$ |
\( T^{20} - 5 T^{19} + \cdots + 704 \)
|
| $5$ |
\( (T - 1)^{20} \)
|
| $7$ |
\( T^{20} - 3 T^{19} + \cdots + 74016 \)
|
| $11$ |
\( (T - 1)^{20} \)
|
| $13$ |
\( T^{20} - 3 T^{19} + \cdots - 5530112 \)
|
| $17$ |
\( T^{20} - 6 T^{19} + \cdots - 22891392 \)
|
| $19$ |
\( T^{20} + \cdots - 18096312960 \)
|
| $23$ |
\( T^{20} + \cdots - 7263738432 \)
|
| $29$ |
\( T^{20} + \cdots - 289294085450 \)
|
| $31$ |
\( T^{20} + \cdots - 54577253376 \)
|
| $37$ |
\( T^{20} + \cdots - 658179584 \)
|
| $41$ |
\( T^{20} + \cdots - 122032705536 \)
|
| $43$ |
\( (T - 1)^{20} \)
|
| $47$ |
\( T^{20} + \cdots - 8534065984 \)
|
| $53$ |
\( T^{20} + \cdots + 97522795008 \)
|
| $59$ |
\( T^{20} + \cdots - 582946099200 \)
|
| $61$ |
\( T^{20} + \cdots + 32904706815684 \)
|
| $67$ |
\( T^{20} + \cdots + 619966094336 \)
|
| $71$ |
\( T^{20} + \cdots + 533645703936 \)
|
| $73$ |
\( T^{20} + \cdots - 75955259049472 \)
|
| $79$ |
\( T^{20} + \cdots + 10\!\cdots\!00 \)
|
| $83$ |
\( T^{20} + \cdots - 16049712393344 \)
|
| $89$ |
\( T^{20} + \cdots + 797327855104000 \)
|
| $97$ |
\( T^{20} + \cdots + 335524826829248 \)
|
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