Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [12,21,Mod(7,12)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 21, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("12.7");
S:= CuspForms(chi, 21);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 12 = 2^{2} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 21 \) |
| Character orbit: | \([\chi]\) | \(=\) | 12.d (of order \(2\), degree \(1\), 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: | \(30.4216518123\) |
| 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: |
\( x^{20} - 8 x^{19} - 1800529 x^{18} + 10471600 x^{17} + 1401323871170 x^{16} - 8255926364216 x^{15} + \cdots + 17\!\cdots\!88 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{174}\cdot 3^{82}\cdot 5^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{20} - 8 x^{19} - 1800529 x^{18} + 10471600 x^{17} + 1401323871170 x^{16} - 8255926364216 x^{15} + \cdots + 17\!\cdots\!88 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 28\!\cdots\!53 \nu^{19} + \cdots + 13\!\cdots\!96 ) / 11\!\cdots\!40 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 45\!\cdots\!87 \nu^{19} + \cdots - 20\!\cdots\!84 ) / 55\!\cdots\!20 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 19\!\cdots\!37 \nu^{19} + \cdots + 91\!\cdots\!12 ) / 27\!\cdots\!60 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 90\!\cdots\!68 \nu^{19} + \cdots + 41\!\cdots\!80 ) / 11\!\cdots\!80 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 16\!\cdots\!99 \nu^{19} + \cdots - 86\!\cdots\!76 ) / 11\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 31\!\cdots\!55 \nu^{19} + \cdots - 16\!\cdots\!44 ) / 12\!\cdots\!40 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 14\!\cdots\!17 \nu^{19} + \cdots + 66\!\cdots\!04 ) / 45\!\cdots\!60 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 16\!\cdots\!16 \nu^{19} + \cdots + 17\!\cdots\!64 ) / 43\!\cdots\!40 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 31\!\cdots\!05 \nu^{19} + \cdots - 35\!\cdots\!36 ) / 62\!\cdots\!20 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 23\!\cdots\!85 \nu^{19} + \cdots + 16\!\cdots\!24 ) / 20\!\cdots\!40 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 67\!\cdots\!08 \nu^{19} + \cdots - 20\!\cdots\!20 ) / 35\!\cdots\!40 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 51\!\cdots\!70 \nu^{19} + \cdots + 24\!\cdots\!48 ) / 25\!\cdots\!80 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 80\!\cdots\!38 \nu^{19} + \cdots + 21\!\cdots\!20 ) / 25\!\cdots\!80 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 11\!\cdots\!81 \nu^{19} + \cdots - 59\!\cdots\!76 ) / 12\!\cdots\!40 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 87\!\cdots\!78 \nu^{19} + \cdots - 41\!\cdots\!80 ) / 83\!\cdots\!60 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 11\!\cdots\!72 \nu^{19} + \cdots + 56\!\cdots\!36 ) / 83\!\cdots\!60 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 42\!\cdots\!36 \nu^{19} + \cdots + 19\!\cdots\!44 ) / 25\!\cdots\!80 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 45\!\cdots\!88 \nu^{19} + \cdots - 20\!\cdots\!16 ) / 25\!\cdots\!80 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 41\!\cdots\!67 \nu^{19} + \cdots + 19\!\cdots\!28 ) / 17\!\cdots\!20 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{19} + 9 \beta_{18} + 12 \beta_{17} + 3 \beta_{15} + 3 \beta_{14} - 15 \beta_{13} + \cdots + 837932855 ) / 2579890176 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 26968 \beta_{19} + 11736 \beta_{18} - 3384 \beta_{17} - 2592 \beta_{16} + \cdots + 929045568739400 ) / 5159780352 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 880964 \beta_{19} - 233052 \beta_{18} + 688676 \beta_{17} - 2508624 \beta_{16} + \cdots + 491593836890236 ) / 859963392 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 4618748563 \beta_{19} + 1291489011 \beta_{18} - 299669784 \beta_{17} - 1340154720 \beta_{16} + \cdots + 11\!\cdots\!37 ) / 2579890176 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2489752913992 \beta_{19} - 1386065948328 \beta_{18} + 719148588120 \beta_{17} + \cdots + 52\!\cdots\!40 ) / 5159780352 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 252080348387418 \beta_{19} + 47811515662410 \beta_{18} - 12631288723820 \beta_{17} + \cdots + 52\!\cdots\!70 ) / 429981696 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 53\!\cdots\!61 \beta_{19} + \cdots + 18\!\cdots\!31 ) / 2579890176 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 10\!\cdots\!52 \beta_{19} + \cdots + 18\!\cdots\!52 ) / 5159780352 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 72\!\cdots\!52 \beta_{19} + \cdots + 33\!\cdots\!16 ) / 859963392 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 17\!\cdots\!15 \beta_{19} + \cdots + 30\!\cdots\!21 ) / 2579890176 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 17\!\cdots\!08 \beta_{19} + \cdots + 94\!\cdots\!60 ) / 5159780352 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 49\!\cdots\!66 \beta_{19} + \cdots + 82\!\cdots\!78 ) / 214990848 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 32\!\cdots\!85 \beta_{19} + \cdots + 20\!\cdots\!47 ) / 2579890176 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 42\!\cdots\!08 \beta_{19} + \cdots + 66\!\cdots\!68 ) / 5159780352 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 41\!\cdots\!64 \beta_{19} + \cdots + 28\!\cdots\!36 ) / 859963392 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 75\!\cdots\!55 \beta_{19} + \cdots + 11\!\cdots\!33 ) / 2579890176 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 92\!\cdots\!16 \beta_{19} + \cdots + 68\!\cdots\!72 ) / 5159780352 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( 45\!\cdots\!58 \beta_{19} + \cdots + 65\!\cdots\!66 ) / 429981696 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( 17\!\cdots\!13 \beta_{19} + \cdots + 13\!\cdots\!79 ) / 2579890176 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(7\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−1009.00 | − | 174.621i | 34092.0i | 987591. | + | 352385.i | 5.90557e6 | 5.95317e6 | − | 3.43988e7i | − | 2.05952e8i | −9.34947e8 | − | 5.28011e8i | −1.16226e9 | −5.95872e9 | − | 1.03123e9i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.2 | −1009.00 | + | 174.621i | − | 34092.0i | 987591. | − | 352385.i | 5.90557e6 | 5.95317e6 | + | 3.43988e7i | 2.05952e8i | −9.34947e8 | + | 5.28011e8i | −1.16226e9 | −5.95872e9 | + | 1.03123e9i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.3 | −831.424 | − | 597.755i | − | 34092.0i | 333954. | + | 993975.i | −5.52569e6 | −2.03786e7 | + | 2.83449e7i | − | 2.40780e8i | 3.16496e8 | − | 1.02604e9i | −1.16226e9 | 4.59419e9 | + | 3.30301e9i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.4 | −831.424 | + | 597.755i | 34092.0i | 333954. | − | 993975.i | −5.52569e6 | −2.03786e7 | − | 2.83449e7i | 2.40780e8i | 3.16496e8 | + | 1.02604e9i | −1.16226e9 | 4.59419e9 | − | 3.30301e9i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.5 | −718.951 | − | 729.168i | 34092.0i | −14796.1 | + | 1.04847e6i | −6.68527e6 | 2.48588e7 | − | 2.45104e7i | 4.90135e8i | 7.75150e8 | − | 7.43010e8i | −1.16226e9 | 4.80638e9 | + | 4.87468e9i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.6 | −718.951 | + | 729.168i | − | 34092.0i | −14796.1 | − | 1.04847e6i | −6.68527e6 | 2.48588e7 | + | 2.45104e7i | − | 4.90135e8i | 7.75150e8 | + | 7.43010e8i | −1.16226e9 | 4.80638e9 | − | 4.87468e9i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.7 | −146.341 | − | 1013.49i | − | 34092.0i | −1.00574e6 | + | 296630.i | 1.07961e7 | −3.45518e7 | + | 4.98905e6i | − | 1.91267e8i | 4.47813e8 | + | 9.75902e8i | −1.16226e9 | −1.57992e9 | − | 1.09418e10i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.8 | −146.341 | + | 1013.49i | 34092.0i | −1.00574e6 | − | 296630.i | 1.07961e7 | −3.45518e7 | − | 4.98905e6i | 1.91267e8i | 4.47813e8 | − | 9.75902e8i | −1.16226e9 | −1.57992e9 | + | 1.09418e10i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.9 | −128.521 | − | 1015.90i | − | 34092.0i | −1.01554e6 | + | 261130.i | −1.38143e7 | −3.46341e7 | + | 4.38153e6i | 4.88171e8i | 3.95801e8 | + | 9.98130e8i | −1.16226e9 | 1.77543e9 | + | 1.40340e10i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.10 | −128.521 | + | 1015.90i | 34092.0i | −1.01554e6 | − | 261130.i | −1.38143e7 | −3.46341e7 | − | 4.38153e6i | − | 4.88171e8i | 3.95801e8 | − | 9.98130e8i | −1.16226e9 | 1.77543e9 | − | 1.40340e10i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.11 | 171.720 | − | 1009.50i | 34092.0i | −989600. | − | 346703.i | 1.74429e7 | 3.44158e7 | + | 5.85428e6i | 3.10877e8i | −5.19931e8 | + | 9.39464e8i | −1.16226e9 | 2.99530e9 | − | 1.76086e10i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.12 | 171.720 | + | 1009.50i | − | 34092.0i | −989600. | + | 346703.i | 1.74429e7 | 3.44158e7 | − | 5.85428e6i | − | 3.10877e8i | −5.19931e8 | − | 9.39464e8i | −1.16226e9 | 2.99530e9 | + | 1.76086e10i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.13 | 578.030 | − | 845.256i | 34092.0i | −380338. | − | 977166.i | −1.01328e7 | 2.88164e7 | + | 1.97062e7i | − | 2.56091e7i | −1.04580e9 | − | 2.43348e8i | −1.16226e9 | −5.85703e9 | + | 8.56477e9i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.14 | 578.030 | + | 845.256i | − | 34092.0i | −380338. | + | 977166.i | −1.01328e7 | 2.88164e7 | − | 1.97062e7i | 2.56091e7i | −1.04580e9 | + | 2.43348e8i | −1.16226e9 | −5.85703e9 | − | 8.56477e9i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.15 | 734.722 | − | 713.274i | − | 34092.0i | 31057.2 | − | 1.04812e6i | 6.80556e6 | −2.43169e7 | − | 2.50481e7i | − | 4.46066e7i | −7.24775e8 | − | 7.92226e8i | −1.16226e9 | 5.00020e9 | − | 4.85423e9i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.16 | 734.722 | + | 713.274i | 34092.0i | 31057.2 | + | 1.04812e6i | 6.80556e6 | −2.43169e7 | + | 2.50481e7i | 4.46066e7i | −7.24775e8 | + | 7.92226e8i | −1.16226e9 | 5.00020e9 | + | 4.85423e9i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.17 | 981.162 | − | 293.082i | 34092.0i | 876782. | − | 575122.i | 1.08427e7 | 9.99175e6 | + | 3.34497e7i | − | 4.06904e8i | 6.91707e8 | − | 8.21257e8i | −1.16226e9 | 1.06384e10 | − | 3.17779e9i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.18 | 981.162 | + | 293.082i | − | 34092.0i | 876782. | + | 575122.i | 1.08427e7 | 9.99175e6 | − | 3.34497e7i | 4.06904e8i | 6.91707e8 | + | 8.21257e8i | −1.16226e9 | 1.06384e10 | + | 3.17779e9i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.19 | 995.603 | − | 239.481i | − | 34092.0i | 933874. | − | 476855.i | −1.48963e7 | −8.16437e6 | − | 3.39420e7i | − | 1.28787e8i | 8.15570e8 | − | 6.98403e8i | −1.16226e9 | −1.48308e10 | + | 3.56738e9i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.20 | 995.603 | + | 239.481i | 34092.0i | 933874. | + | 476855.i | −1.48963e7 | −8.16437e6 | + | 3.39420e7i | 1.28787e8i | 8.15570e8 | + | 6.98403e8i | −1.16226e9 | −1.48308e10 | − | 3.56738e9i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 12.21.d.a | ✓ | 20 |
| 3.b | odd | 2 | 1 | 36.21.d.c | 20 | ||
| 4.b | odd | 2 | 1 | inner | 12.21.d.a | ✓ | 20 |
| 12.b | even | 2 | 1 | 36.21.d.c | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 12.21.d.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 12.21.d.a | ✓ | 20 | 4.b | odd | 2 | 1 | inner |
| 36.21.d.c | 20 | 3.b | odd | 2 | 1 | ||
| 36.21.d.c | 20 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{21}^{\mathrm{new}}(12, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + \cdots + 16\!\cdots\!76 \)
|
| $3$ |
\( (T^{2} + 1162261467)^{10} \)
|
| $5$ |
\( (T^{10} + \cdots - 63\!\cdots\!00)^{2} \)
|
| $7$ |
\( T^{20} + \cdots + 17\!\cdots\!64 \)
|
| $11$ |
\( T^{20} + \cdots + 73\!\cdots\!96 \)
|
| $13$ |
\( (T^{10} + \cdots - 65\!\cdots\!36)^{2} \)
|
| $17$ |
\( (T^{10} + \cdots + 21\!\cdots\!00)^{2} \)
|
| $19$ |
\( T^{20} + \cdots + 11\!\cdots\!44 \)
|
| $23$ |
\( T^{20} + \cdots + 74\!\cdots\!44 \)
|
| $29$ |
\( (T^{10} + \cdots + 11\!\cdots\!96)^{2} \)
|
| $31$ |
\( T^{20} + \cdots + 19\!\cdots\!16 \)
|
| $37$ |
\( (T^{10} + \cdots - 93\!\cdots\!84)^{2} \)
|
| $41$ |
\( (T^{10} + \cdots - 54\!\cdots\!32)^{2} \)
|
| $43$ |
\( T^{20} + \cdots + 16\!\cdots\!44 \)
|
| $47$ |
\( T^{20} + \cdots + 27\!\cdots\!00 \)
|
| $53$ |
\( (T^{10} + \cdots + 77\!\cdots\!16)^{2} \)
|
| $59$ |
\( T^{20} + \cdots + 89\!\cdots\!64 \)
|
| $61$ |
\( (T^{10} + \cdots - 24\!\cdots\!36)^{2} \)
|
| $67$ |
\( T^{20} + \cdots + 25\!\cdots\!64 \)
|
| $71$ |
\( T^{20} + \cdots + 26\!\cdots\!00 \)
|
| $73$ |
\( (T^{10} + \cdots + 71\!\cdots\!52)^{2} \)
|
| $79$ |
\( T^{20} + \cdots + 55\!\cdots\!36 \)
|
| $83$ |
\( T^{20} + \cdots + 21\!\cdots\!16 \)
|
| $89$ |
\( (T^{10} + \cdots + 26\!\cdots\!52)^{2} \)
|
| $97$ |
\( (T^{10} + \cdots + 66\!\cdots\!48)^{2} \)
|
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