Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [495,2,Mod(188,495)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(495, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("495.188");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 495 = 3^{2} \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 495.l (of order \(4\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(3.95259490005\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} + 34 x^{18} + 489 x^{16} + 3880 x^{14} + 18578 x^{12} + 55156 x^{10} + 100314 x^{8} + 106120 x^{6} + \cdots + 625 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{20} + 34 x^{18} + 489 x^{16} + 3880 x^{14} + 18578 x^{12} + 55156 x^{10} + 100314 x^{8} + 106120 x^{6} + \cdots + 625 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2 \nu^{18} + 13 \nu^{16} - 767 \nu^{14} - 14900 \nu^{12} - 117714 \nu^{10} - 485013 \nu^{8} + \cdots - 17950 ) / 3700 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2 \nu^{19} + 13 \nu^{17} - 767 \nu^{15} - 15270 \nu^{13} - 126224 \nu^{11} - 560493 \nu^{9} + \cdots - 106010 \nu ) / 11100 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 121 \nu^{19} + 30 \nu^{18} - 3839 \nu^{17} + 750 \nu^{16} - 50444 \nu^{15} + 6810 \nu^{14} + \cdots - 19500 ) / 44400 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 121 \nu^{19} - 30 \nu^{18} - 3839 \nu^{17} - 750 \nu^{16} - 50444 \nu^{15} - 6810 \nu^{14} + \cdots + 19500 ) / 44400 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 52 \nu^{19} - 510 \nu^{18} + 1818 \nu^{17} - 16265 \nu^{16} + 26678 \nu^{15} - 215115 \nu^{14} + \cdots - 103250 ) / 37000 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 583 \nu^{19} - 525 \nu^{18} - 18497 \nu^{17} - 13125 \nu^{16} - 244562 \nu^{15} - 110850 \nu^{14} + \cdots + 2020125 ) / 222000 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 51 \nu^{19} + 1045 \nu^{18} - 2459 \nu^{17} + 33155 \nu^{16} - 46764 \nu^{15} + 436830 \nu^{14} + \cdots + 592625 ) / 74000 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 583 \nu^{19} - 3345 \nu^{18} - 18497 \nu^{17} - 103605 \nu^{16} - 244562 \nu^{15} + \cdots - 808875 ) / 222000 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 583 \nu^{19} + 3345 \nu^{18} - 18497 \nu^{17} + 103605 \nu^{16} - 244562 \nu^{15} + \cdots + 142875 ) / 222000 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 179 \nu^{19} - 57 \nu^{18} + 5881 \nu^{17} - 1203 \nu^{16} + 80926 \nu^{15} - 6168 \nu^{14} + \cdots - 37875 ) / 44400 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 17 \nu^{19} + 553 \nu^{17} + 7513 \nu^{15} + 55335 \nu^{13} + 240001 \nu^{11} + 623127 \nu^{9} + \cdots + 25750 \nu ) / 3000 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 459 \nu^{19} - 25 \nu^{18} + 14731 \nu^{17} - 625 \nu^{16} + 197026 \nu^{15} - 4750 \nu^{14} + \cdots - 34625 ) / 74000 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 26 \nu^{19} - 859 \nu^{17} - 11914 \nu^{15} - 90180 \nu^{13} - 405478 \nu^{11} - 1103856 \nu^{9} + \cdots - 66625 \nu ) / 3000 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 1771 \nu^{19} - 3135 \nu^{18} + 56189 \nu^{17} - 99465 \nu^{16} + 741344 \nu^{15} - 1310490 \nu^{14} + \cdots - 1777875 ) / 222000 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 751 \nu^{19} - 1040 \nu^{18} - 23659 \nu^{17} - 32660 \nu^{16} - 309264 \nu^{15} - 424410 \nu^{14} + \cdots - 267500 ) / 74000 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 751 \nu^{19} + 1040 \nu^{18} - 23659 \nu^{17} + 32660 \nu^{16} - 309264 \nu^{15} + 424410 \nu^{14} + \cdots + 267500 ) / 74000 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 1811 \nu^{19} - 2115 \nu^{18} - 56449 \nu^{17} - 65085 \nu^{16} - 726004 \nu^{15} + \cdots - 678750 ) / 111000 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 1811 \nu^{19} - 2115 \nu^{18} + 56449 \nu^{17} - 65085 \nu^{16} + 726004 \nu^{15} - 824760 \nu^{14} + \cdots - 678750 ) / 111000 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{16} - 2\beta_{13} + \beta_{11} + \beta_{10} + 2\beta_{9} + \beta_{6} + 2\beta_{4} + \beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta _1 - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 3 \beta_{19} + 3 \beta_{18} + 3 \beta_{17} + 13 \beta_{16} + 9 \beta_{15} - 3 \beta_{14} + \cdots - 8 \beta_1 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{15} - \beta_{14} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{5} + \beta_{4} + \beta_{2} - 7\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 30 \beta_{19} - 30 \beta_{18} - 30 \beta_{17} - 92 \beta_{16} - 96 \beta_{15} + 36 \beta_{14} + \cdots + 49 \beta_1 ) / 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2 \beta_{19} + 2 \beta_{18} + 10 \beta_{15} + 10 \beta_{14} - 2 \beta_{11} + 12 \beta_{10} - 15 \beta_{9} + \cdots - 87 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 237 \beta_{19} + 237 \beta_{18} + 243 \beta_{17} + 649 \beta_{16} + 789 \beta_{15} - 327 \beta_{14} + \cdots - 302 \beta_1 ) / 6 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 29 \beta_{19} - 29 \beta_{18} + \beta_{17} - \beta_{16} - 78 \beta_{15} - 78 \beta_{14} + 28 \beta_{11} + \cdots + 543 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 1734 \beta_{19} - 1734 \beta_{18} - 1830 \beta_{17} - 4568 \beta_{16} - 5952 \beta_{15} + \cdots + 1891 \beta_1 ) / 6 \)
|
| \(\nu^{10}\) | \(=\) |
\( 293 \beta_{19} + 293 \beta_{18} - 21 \beta_{17} + 21 \beta_{16} + 562 \beta_{15} + 562 \beta_{14} + \cdots - 3529 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 12303 \beta_{19} + 12303 \beta_{18} + 13383 \beta_{17} + 32161 \beta_{16} + 43371 \beta_{15} + \cdots - 11996 \beta_1 ) / 6 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 2569 \beta_{19} - 2569 \beta_{18} + 289 \beta_{17} - 289 \beta_{16} - 3925 \beta_{15} - 3925 \beta_{14} + \cdots + 23473 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 86196 \beta_{19} - 86196 \beta_{18} - 96612 \beta_{17} - 226658 \beta_{16} - 311244 \beta_{15} + \cdots + 76879 \beta_1 ) / 6 \)
|
| \(\nu^{14}\) | \(=\) |
\( 20995 \beta_{19} + 20995 \beta_{18} - 3273 \beta_{17} + 3273 \beta_{16} + 27068 \beta_{15} + \cdots - 158309 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 601257 \beta_{19} + 601257 \beta_{18} + 693075 \beta_{17} + 1598953 \beta_{16} + 2218347 \beta_{15} + \cdots - 496898 \beta_1 ) / 6 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 164936 \beta_{19} - 164936 \beta_{18} + 33088 \beta_{17} - 33088 \beta_{16} - 185856 \beta_{15} + \cdots + 1077097 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 4192584 \beta_{19} - 4192584 \beta_{18} - 4955040 \beta_{17} - 11288210 \beta_{16} + \cdots + 3236353 \beta_1 ) / 6 \)
|
| \(\nu^{18}\) | \(=\) |
\( 1264756 \beta_{19} + 1264756 \beta_{18} - 310708 \beta_{17} + 310708 \beta_{16} + 1275664 \beta_{15} + \cdots - 7371907 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 29282907 \beta_{19} + 29282907 \beta_{18} + 35351091 \beta_{17} + 79734349 \beta_{16} + \cdots - 21233672 \beta_1 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).
| \(n\) | \(46\) | \(56\) | \(397\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-\beta_{12}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 188.1 |
|
−1.89557 | − | 1.89557i | 0 | 5.18637i | 2.04940 | + | 0.894408i | 0 | −2.22091 | + | 2.22091i | 6.03998 | − | 6.03998i | 0 | −2.18937 | − | 5.58019i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 188.2 | −1.85768 | − | 1.85768i | 0 | 4.90196i | 1.13954 | − | 1.92391i | 0 | 1.46709 | − | 1.46709i | 5.39091 | − | 5.39091i | 0 | −5.69092 | + | 1.45712i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 188.3 | −1.45192 | − | 1.45192i | 0 | 2.21616i | −1.97309 | + | 1.05210i | 0 | 1.16900 | − | 1.16900i | 0.313853 | − | 0.313853i | 0 | 4.39234 | + | 1.33722i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 188.4 | −0.506593 | − | 0.506593i | 0 | − | 1.48673i | −1.73564 | + | 1.40980i | 0 | −0.499440 | + | 0.499440i | −1.76635 | + | 1.76635i | 0 | 1.59346 | + | 0.165069i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 188.5 | −0.172296 | − | 0.172296i | 0 | − | 1.94063i | −1.26816 | − | 1.84168i | 0 | 1.85562 | − | 1.85562i | −0.678954 | + | 0.678954i | 0 | −0.0988147 | + | 0.535812i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 188.6 | 0.632341 | + | 0.632341i | 0 | − | 1.20029i | 0.106291 | − | 2.23354i | 0 | −0.865046 | + | 0.865046i | 2.02367 | − | 2.02367i | 0 | 1.47957 | − | 1.34515i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 188.7 | 1.06895 | + | 1.06895i | 0 | 0.285287i | 1.39342 | + | 1.74882i | 0 | 2.04096 | − | 2.04096i | 1.83293 | − | 1.83293i | 0 | −0.379901 | + | 3.35889i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 188.8 | 1.17559 | + | 1.17559i | 0 | 0.764004i | −2.23321 | − | 0.113046i | 0 | 3.01828 | − | 3.01828i | 1.45302 | − | 1.45302i | 0 | −2.49243 | − | 2.75822i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 188.9 | 1.26345 | + | 1.26345i | 0 | 1.19263i | 0.601700 | + | 2.15359i | 0 | −3.12088 | + | 3.12088i | 1.02007 | − | 1.02007i | 0 | −1.96074 | + | 3.48118i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 188.10 | 1.74374 | + | 1.74374i | 0 | 4.08124i | 1.91975 | − | 1.14654i | 0 | −2.84467 | + | 2.84467i | −3.62913 | + | 3.62913i | 0 | 5.34681 | + | 1.34827i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.1 | −1.89557 | + | 1.89557i | 0 | − | 5.18637i | 2.04940 | − | 0.894408i | 0 | −2.22091 | − | 2.22091i | 6.03998 | + | 6.03998i | 0 | −2.18937 | + | 5.58019i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.2 | −1.85768 | + | 1.85768i | 0 | − | 4.90196i | 1.13954 | + | 1.92391i | 0 | 1.46709 | + | 1.46709i | 5.39091 | + | 5.39091i | 0 | −5.69092 | − | 1.45712i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.3 | −1.45192 | + | 1.45192i | 0 | − | 2.21616i | −1.97309 | − | 1.05210i | 0 | 1.16900 | + | 1.16900i | 0.313853 | + | 0.313853i | 0 | 4.39234 | − | 1.33722i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.4 | −0.506593 | + | 0.506593i | 0 | 1.48673i | −1.73564 | − | 1.40980i | 0 | −0.499440 | − | 0.499440i | −1.76635 | − | 1.76635i | 0 | 1.59346 | − | 0.165069i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.5 | −0.172296 | + | 0.172296i | 0 | 1.94063i | −1.26816 | + | 1.84168i | 0 | 1.85562 | + | 1.85562i | −0.678954 | − | 0.678954i | 0 | −0.0988147 | − | 0.535812i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.6 | 0.632341 | − | 0.632341i | 0 | 1.20029i | 0.106291 | + | 2.23354i | 0 | −0.865046 | − | 0.865046i | 2.02367 | + | 2.02367i | 0 | 1.47957 | + | 1.34515i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.7 | 1.06895 | − | 1.06895i | 0 | − | 0.285287i | 1.39342 | − | 1.74882i | 0 | 2.04096 | + | 2.04096i | 1.83293 | + | 1.83293i | 0 | −0.379901 | − | 3.35889i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.8 | 1.17559 | − | 1.17559i | 0 | − | 0.764004i | −2.23321 | + | 0.113046i | 0 | 3.01828 | + | 3.01828i | 1.45302 | + | 1.45302i | 0 | −2.49243 | + | 2.75822i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.9 | 1.26345 | − | 1.26345i | 0 | − | 1.19263i | 0.601700 | − | 2.15359i | 0 | −3.12088 | − | 3.12088i | 1.02007 | + | 1.02007i | 0 | −1.96074 | − | 3.48118i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.10 | 1.74374 | − | 1.74374i | 0 | − | 4.08124i | 1.91975 | + | 1.14654i | 0 | −2.84467 | − | 2.84467i | −3.62913 | − | 3.62913i | 0 | 5.34681 | − | 1.34827i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 15.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 495.2.l.b | yes | 20 |
| 3.b | odd | 2 | 1 | 495.2.l.a | ✓ | 20 | |
| 5.c | odd | 4 | 1 | 495.2.l.a | ✓ | 20 | |
| 15.e | even | 4 | 1 | inner | 495.2.l.b | yes | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 495.2.l.a | ✓ | 20 | 3.b | odd | 2 | 1 | |
| 495.2.l.a | ✓ | 20 | 5.c | odd | 4 | 1 | |
| 495.2.l.b | yes | 20 | 1.a | even | 1 | 1 | trivial |
| 495.2.l.b | yes | 20 | 15.e | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{20} - 8 T_{2}^{17} + 89 T_{2}^{16} - 48 T_{2}^{15} + 32 T_{2}^{14} - 440 T_{2}^{13} + 2642 T_{2}^{12} + \cdots + 625 \)
acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} - 8 T^{17} + \cdots + 625 \)
|
| $3$ |
\( T^{20} \)
|
| $5$ |
\( T^{20} + 16 T^{17} + \cdots + 9765625 \)
|
| $7$ |
\( T^{20} - 24 T^{17} + \cdots + 28558336 \)
|
| $11$ |
\( (T^{2} + 1)^{10} \)
|
| $13$ |
\( T^{20} - 4 T^{19} + \cdots + 5184 \)
|
| $17$ |
\( T^{20} + \cdots + 70960435456 \)
|
| $19$ |
\( T^{20} + 160 T^{18} + \cdots + 95883264 \)
|
| $23$ |
\( T^{20} + \cdots + 16122904576 \)
|
| $29$ |
\( (T^{10} + 8 T^{9} + \cdots - 874528)^{2} \)
|
| $31$ |
\( (T^{10} - 16 T^{9} + \cdots - 160256)^{2} \)
|
| $37$ |
\( T^{20} + \cdots + 3805183673344 \)
|
| $41$ |
\( T^{20} + \cdots + 2893579264 \)
|
| $43$ |
\( T^{20} + \cdots + 96586694656 \)
|
| $47$ |
\( T^{20} + \cdots + 6850489614336 \)
|
| $53$ |
\( T^{20} + \cdots + 86935343104 \)
|
| $59$ |
\( (T^{10} - 8 T^{9} + \cdots + 1192832)^{2} \)
|
| $61$ |
\( (T^{10} + 24 T^{9} + \cdots + 22720000)^{2} \)
|
| $67$ |
\( T^{20} + \cdots + 31\!\cdots\!24 \)
|
| $71$ |
\( T^{20} + \cdots + 251015098023936 \)
|
| $73$ |
\( T^{20} + \cdots + 50\!\cdots\!36 \)
|
| $79$ |
\( T^{20} + \cdots + 15\!\cdots\!96 \)
|
| $83$ |
\( T^{20} + \cdots + 355975462105344 \)
|
| $89$ |
\( (T^{10} - 514 T^{8} + \cdots - 1792678016)^{2} \)
|
| $97$ |
\( T^{20} + \cdots + 54\!\cdots\!44 \)
|
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