Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [345,3,Mod(29,345)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(345, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([11, 11, 18]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("345.29");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 345 = 3 \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 345.p (of order \(22\), degree \(10\), 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: | \(9.40056912043\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{22})\) |
| Coefficient field: | 20.0.3206128490667995866421572265625.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - x^{19} - 3 x^{18} + 7 x^{17} + 5 x^{16} - 33 x^{15} + 13 x^{14} + 119 x^{13} - 171 x^{12} + \cdots + 1048576 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{22}]$ |
$q$-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
\( x^{20} - x^{19} - 3 x^{18} + 7 x^{17} + 5 x^{16} - 33 x^{15} + 13 x^{14} + 119 x^{13} - 171 x^{12} + \cdots + 1048576 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{13} - 4187\nu^{2} ) / 15824 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{12} + 231\nu ) / 3956 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{11} + 1220 ) / 989 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{14} + 3263\nu^{3} ) / 63296 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{14} + 4187\nu^{3} ) / 15824 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3\nu^{13} + 3263\nu^{2} ) / 15824 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -5\nu^{16} - 26537\nu^{5} ) / 1012736 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 5\nu^{15} + 26537\nu^{4} ) / 253184 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 7\nu^{15} - 13485\nu^{4} ) / 253184 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 13\nu^{18} - 133551\nu^{7} ) / 16203776 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 13\nu^{17} - 133551\nu^{6} ) / 4050944 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 171 \nu^{19} - 513 \nu^{18} + 1197 \nu^{17} + 855 \nu^{16} - 5643 \nu^{15} + 2223 \nu^{14} + \cdots + 179306496 ) / 259260416 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 33\nu^{16} - 27403\nu^{5} ) / 1012736 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( -13\nu^{19} + 133551\nu^{8} ) / 16203776 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 305 \nu^{19} - 305 \nu^{18} - 915 \nu^{17} + 2135 \nu^{16} + 1525 \nu^{15} - 10065 \nu^{14} + \cdots - 79953920 ) / 259260416 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( -119\nu^{19} - 23939\nu^{8} ) / 64815104 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 33\nu^{18} - 27403\nu^{7} ) / 4050944 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 171 \nu^{19} + 171 \nu^{18} + 513 \nu^{17} - 1197 \nu^{16} - 855 \nu^{15} + 5643 \nu^{14} + \cdots + 44826624 ) / 64815104 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} - 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{6} + 4\beta_{5} \)
|
| \(\nu^{4}\) | \(=\) |
\( -5\beta_{10} + 7\beta_{9} \)
|
| \(\nu^{5}\) | \(=\) |
\( -5\beta_{14} - 33\beta_{8} \)
|
| \(\nu^{6}\) | \(=\) |
\( - 13 \beta_{17} - 13 \beta_{16} + 13 \beta_{13} - 33 \beta_{12} + 13 \beta_{11} + 13 \beta_{10} + \cdots - 13 \)
|
| \(\nu^{7}\) | \(=\) |
\( 13\beta_{18} - 132\beta_{11} \)
|
| \(\nu^{8}\) | \(=\) |
\( -52\beta_{17} + 119\beta_{15} \)
|
| \(\nu^{9}\) | \(=\) |
\( - 171 \beta_{19} + 171 \beta_{18} + 171 \beta_{17} + 171 \beta_{16} + 171 \beta_{15} - 171 \beta_{14} + \cdots + 171 \)
|
| \(\nu^{10}\) | \(=\) |
\( 305\beta_{19} + 684\beta_{16} \)
|
| \(\nu^{11}\) | \(=\) |
\( -989\beta_{4} + 1220 \)
|
| \(\nu^{12}\) | \(=\) |
\( -3956\beta_{3} + 231\beta_1 \)
|
| \(\nu^{13}\) | \(=\) |
\( 4187\beta_{7} + 3263\beta_{2} \)
|
| \(\nu^{14}\) | \(=\) |
\( -3263\beta_{6} + 16748\beta_{5} \)
|
| \(\nu^{15}\) | \(=\) |
\( 26537\beta_{10} + 13485\beta_{9} \)
|
| \(\nu^{16}\) | \(=\) |
\( 26537\beta_{14} - 27403\beta_{8} \)
|
| \(\nu^{17}\) | \(=\) |
\( - 133551 \beta_{17} - 133551 \beta_{16} + 133551 \beta_{13} - 27403 \beta_{12} + 133551 \beta_{11} + \cdots - 133551 \)
|
| \(\nu^{18}\) | \(=\) |
\( 133551\beta_{18} - 109612\beta_{11} \)
|
| \(\nu^{19}\) | \(=\) |
\( -534204\beta_{17} - 23939\beta_{15} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/345\mathbb{Z}\right)^\times\).
| \(n\) | \(116\) | \(166\) | \(277\) |
| \(\chi(n)\) | \(-1\) | \(\beta_{5}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 29.1 |
|
−0.0546904 | + | 0.119755i | −2.87848 | − | 0.845198i | 2.60809 | + | 3.00990i | −4.20627 | + | 2.70320i | 0.258642 | − | 0.298489i | 0 | −1.00837 | + | 0.296084i | 7.57128 | + | 4.86577i | −0.0936805 | − | 0.651562i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 29.2 | 0.851866 | − | 1.86533i | −2.87848 | − | 0.845198i | −0.134324 | − | 0.155018i | −4.20627 | + | 2.70320i | −4.02865 | + | 4.64931i | 0 | 7.46671 | − | 2.19242i | 7.57128 | + | 4.86577i | 1.45918 | + | 10.1488i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.1 | −0.820304 | − | 0.946682i | 2.52376 | + | 1.62192i | 0.345952 | − | 2.40615i | −2.07708 | + | 4.54816i | −0.534807 | − | 3.71967i | 0 | −6.77679 | + | 4.35518i | 3.73874 | + | 8.18669i | 6.00949 | − | 1.76455i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.2 | 1.92211 | + | 2.21824i | 2.52376 | + | 1.62192i | −0.656795 | + | 4.56811i | −2.07708 | + | 4.54816i | 1.25314 | + | 8.71581i | 0 | −1.51877 | + | 0.976053i | 3.73874 | + | 8.18669i | −14.0813 | + | 4.13463i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 104.1 | −3.10527 | + | 1.99564i | −0.426945 | + | 2.96946i | 3.99850 | − | 8.75549i | 4.79746 | − | 1.40866i | −4.60020 | − | 10.0730i | 0 | 2.95508 | + | 20.5530i | −8.63544 | − | 2.53559i | −12.0863 | + | 13.9483i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 104.2 | 3.34472 | − | 2.14952i | −0.426945 | + | 2.96946i | 4.90505 | − | 10.7406i | 4.79746 | − | 1.40866i | 4.95492 | + | 10.8498i | 0 | −4.41774 | − | 30.7261i | −8.63544 | − | 2.53559i | 13.0182 | − | 15.0238i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 119.1 | −0.0546904 | − | 0.119755i | −2.87848 | + | 0.845198i | 2.60809 | − | 3.00990i | −4.20627 | − | 2.70320i | 0.258642 | + | 0.298489i | 0 | −1.00837 | − | 0.296084i | 7.57128 | − | 4.86577i | −0.0936805 | + | 0.651562i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 119.2 | 0.851866 | + | 1.86533i | −2.87848 | + | 0.845198i | −0.134324 | + | 0.155018i | −4.20627 | − | 2.70320i | −4.02865 | − | 4.64931i | 0 | 7.46671 | + | 2.19242i | 7.57128 | − | 4.86577i | 1.45918 | − | 10.1488i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 164.1 | −0.442254 | − | 3.07594i | 1.24625 | − | 2.72890i | −5.42787 | + | 1.59377i | 3.27430 | − | 3.77875i | −8.94509 | − | 2.62651i | 0 | 2.13910 | + | 4.68398i | −5.89375 | − | 6.80175i | −13.0713 | − | 8.40041i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 164.2 | 0.560493 | + | 3.89832i | 1.24625 | − | 2.72890i | −11.0448 | + | 3.24303i | 3.27430 | − | 3.77875i | 11.3366 | + | 3.32873i | 0 | −12.2886 | − | 26.9083i | −5.89375 | − | 6.80175i | 16.5660 | + | 10.6463i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 179.1 | −3.43678 | − | 1.00913i | −1.96458 | + | 2.26725i | 7.42807 | + | 4.77373i | 0.711574 | − | 4.94911i | 9.03977 | − | 5.80951i | 0 | −11.3288 | − | 13.0741i | −1.28083 | − | 8.90839i | −7.43981 | + | 16.2909i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 179.2 | 2.18011 | + | 0.640137i | −1.96458 | + | 2.26725i | 0.978077 | + | 0.628572i | 0.711574 | − | 4.94911i | −5.73435 | + | 3.68524i | 0 | −4.22181 | − | 4.87223i | −1.28083 | − | 8.90839i | 4.71942 | − | 10.3341i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 209.1 | −3.10527 | − | 1.99564i | −0.426945 | − | 2.96946i | 3.99850 | + | 8.75549i | 4.79746 | + | 1.40866i | −4.60020 | + | 10.0730i | 0 | 2.95508 | − | 20.5530i | −8.63544 | + | 2.53559i | −12.0863 | − | 13.9483i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 209.2 | 3.34472 | + | 2.14952i | −0.426945 | − | 2.96946i | 4.90505 | + | 10.7406i | 4.79746 | + | 1.40866i | 4.95492 | − | 10.8498i | 0 | −4.41774 | + | 30.7261i | −8.63544 | + | 2.53559i | 13.0182 | + | 15.0238i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.1 | −3.43678 | + | 1.00913i | −1.96458 | − | 2.26725i | 7.42807 | − | 4.77373i | 0.711574 | + | 4.94911i | 9.03977 | + | 5.80951i | 0 | −11.3288 | + | 13.0741i | −1.28083 | + | 8.90839i | −7.43981 | − | 16.2909i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.2 | 2.18011 | − | 0.640137i | −1.96458 | − | 2.26725i | 0.978077 | − | 0.628572i | 0.711574 | + | 4.94911i | −5.73435 | − | 3.68524i | 0 | −4.22181 | + | 4.87223i | −1.28083 | + | 8.90839i | 4.71942 | + | 10.3341i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 269.1 | −0.820304 | + | 0.946682i | 2.52376 | − | 1.62192i | 0.345952 | + | 2.40615i | −2.07708 | − | 4.54816i | −0.534807 | + | 3.71967i | 0 | −6.77679 | − | 4.35518i | 3.73874 | − | 8.18669i | 6.00949 | + | 1.76455i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 269.2 | 1.92211 | − | 2.21824i | 2.52376 | − | 1.62192i | −0.656795 | − | 4.56811i | −2.07708 | − | 4.54816i | 1.25314 | − | 8.71581i | 0 | −1.51877 | − | 0.976053i | 3.73874 | − | 8.18669i | −14.0813 | − | 4.13463i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 284.1 | −0.442254 | + | 3.07594i | 1.24625 | + | 2.72890i | −5.42787 | − | 1.59377i | 3.27430 | + | 3.77875i | −8.94509 | + | 2.62651i | 0 | 2.13910 | − | 4.68398i | −5.89375 | + | 6.80175i | −13.0713 | + | 8.40041i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 284.2 | 0.560493 | − | 3.89832i | 1.24625 | + | 2.72890i | −11.0448 | − | 3.24303i | 3.27430 | + | 3.77875i | 11.3366 | − | 3.32873i | 0 | −12.2886 | + | 26.9083i | −5.89375 | + | 6.80175i | 16.5660 | − | 10.6463i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 15.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-15}) \) |
| 23.c | even | 11 | 1 | inner |
| 345.p | odd | 22 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 345.3.p.b | yes | 20 |
| 3.b | odd | 2 | 1 | 345.3.p.a | ✓ | 20 | |
| 5.b | even | 2 | 1 | 345.3.p.a | ✓ | 20 | |
| 15.d | odd | 2 | 1 | CM | 345.3.p.b | yes | 20 |
| 23.c | even | 11 | 1 | inner | 345.3.p.b | yes | 20 |
| 69.h | odd | 22 | 1 | 345.3.p.a | ✓ | 20 | |
| 115.j | even | 22 | 1 | 345.3.p.a | ✓ | 20 | |
| 345.p | odd | 22 | 1 | inner | 345.3.p.b | yes | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 345.3.p.a | ✓ | 20 | 3.b | odd | 2 | 1 | |
| 345.3.p.a | ✓ | 20 | 5.b | even | 2 | 1 | |
| 345.3.p.a | ✓ | 20 | 69.h | odd | 22 | 1 | |
| 345.3.p.a | ✓ | 20 | 115.j | even | 22 | 1 | |
| 345.3.p.b | yes | 20 | 1.a | even | 1 | 1 | trivial |
| 345.3.p.b | yes | 20 | 15.d | odd | 2 | 1 | CM |
| 345.3.p.b | yes | 20 | 23.c | even | 11 | 1 | inner |
| 345.3.p.b | yes | 20 | 345.p | odd | 22 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{20} - 2 T_{2}^{19} + 3 T_{2}^{18} + 40 T_{2}^{17} - 83 T_{2}^{16} + 126 T_{2}^{15} + \cdots + 2105401 \)
acting on \(S_{3}^{\mathrm{new}}(345, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} - 2 T^{19} + \cdots + 2105401 \)
|
| $3$ |
\( (T^{10} + 3 T^{9} + \cdots + 59049)^{2} \)
|
| $5$ |
\( (T^{10} - 5 T^{9} + \cdots + 9765625)^{2} \)
|
| $7$ |
\( T^{20} \)
|
| $11$ |
\( T^{20} \)
|
| $13$ |
\( T^{20} \)
|
| $17$ |
\( T^{20} + \cdots + 23\!\cdots\!01 \)
|
| $19$ |
\( T^{20} + \cdots + 26\!\cdots\!01 \)
|
| $23$ |
\( T^{20} + \cdots + 17\!\cdots\!01 \)
|
| $29$ |
\( T^{20} \)
|
| $31$ |
\( T^{20} + \cdots + 77\!\cdots\!01 \)
|
| $37$ |
\( T^{20} \)
|
| $41$ |
\( T^{20} \)
|
| $43$ |
\( T^{20} \)
|
| $47$ |
\( (T^{10} + \cdots - 35\!\cdots\!51)^{2} \)
|
| $53$ |
\( T^{20} + \cdots + 32\!\cdots\!01 \)
|
| $59$ |
\( T^{20} \)
|
| $61$ |
\( T^{20} + \cdots + 49\!\cdots\!01 \)
|
| $67$ |
\( T^{20} \)
|
| $71$ |
\( T^{20} \)
|
| $73$ |
\( T^{20} \)
|
| $79$ |
\( T^{20} + \cdots + 18\!\cdots\!01 \)
|
| $83$ |
\( T^{20} + \cdots + 65\!\cdots\!01 \)
|
| $89$ |
\( T^{20} \)
|
| $97$ |
\( T^{20} \)
|
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