Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [684,6,Mod(505,684)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(684, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 4]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("684.505");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 684.k (of order \(3\), 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: | \(109.702532752\) |
| Analytic rank: | \(0\) |
| Dimension: | \(18\) |
| Relative dimension: | \(9\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{18} - 7 x^{17} + 17616 x^{16} - 456301 x^{15} + 216301789 x^{14} - 6076762674 x^{13} + \cdots + 55\!\cdots\!41 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{16}\cdot 3^{7}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 76) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) 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
\( x^{18} - 7 x^{17} + 17616 x^{16} - 456301 x^{15} + 216301789 x^{14} - 6076762674 x^{13} + \cdots + 55\!\cdots\!41 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 14\!\cdots\!75 \nu^{17} + \cdots - 55\!\cdots\!83 ) / 31\!\cdots\!64 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 14\!\cdots\!75 \nu^{17} + \cdots + 55\!\cdots\!83 ) / 31\!\cdots\!64 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 42\!\cdots\!41 \nu^{17} + \cdots + 12\!\cdots\!35 ) / 20\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 36\!\cdots\!35 \nu^{17} + \cdots - 27\!\cdots\!79 ) / 10\!\cdots\!76 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 14\!\cdots\!03 \nu^{17} + \cdots - 50\!\cdots\!95 ) / 40\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 50\!\cdots\!61 \nu^{17} + \cdots - 77\!\cdots\!65 ) / 20\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 34\!\cdots\!11 \nu^{17} + \cdots + 14\!\cdots\!85 ) / 44\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 33\!\cdots\!69 \nu^{17} + \cdots + 19\!\cdots\!85 ) / 40\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 17\!\cdots\!59 \nu^{17} + \cdots - 24\!\cdots\!79 ) / 20\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 80\!\cdots\!81 \nu^{17} + \cdots + 23\!\cdots\!61 ) / 54\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 31\!\cdots\!17 \nu^{17} + \cdots - 48\!\cdots\!05 ) / 20\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 51\!\cdots\!95 \nu^{17} + \cdots - 41\!\cdots\!29 ) / 20\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 81\!\cdots\!55 \nu^{17} + \cdots - 54\!\cdots\!99 ) / 20\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 31\!\cdots\!15 \nu^{17} + \cdots - 33\!\cdots\!13 ) / 20\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 91\!\cdots\!65 \nu^{17} + \cdots + 86\!\cdots\!31 ) / 44\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 59\!\cdots\!13 \nu^{17} + \cdots + 87\!\cdots\!53 ) / 18\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 16\!\cdots\!45 \nu^{17} + \cdots + 92\!\cdots\!09 ) / 44\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_{2} + \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{16} + \beta_{14} + 10 \beta_{13} - 2 \beta_{12} + 4 \beta_{10} + 11 \beta_{9} - \beta_{8} + \cdots - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( - 45 \beta_{17} - 23 \beta_{16} - 90 \beta_{15} + 162 \beta_{14} + 180 \beta_{13} + 56 \beta_{12} + \cdots + 59110 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 5330 \beta_{17} + 2477 \beta_{16} - 2935 \beta_{15} + 3217 \beta_{14} + 660 \beta_{13} + \cdots - 26366473 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 401865 \beta_{17} + 618262 \beta_{16} + 361890 \beta_{15} - 1450833 \beta_{14} - 1737650 \beta_{13} + \cdots - 132423474 \)
|
| \(\nu^{6}\) | \(=\) |
\( 31315085 \beta_{17} - 76163096 \beta_{16} + 69508015 \beta_{15} - 93640341 \beta_{14} + \cdots + 224305429259 \)
|
| \(\nu^{7}\) | \(=\) |
\( 6309274755 \beta_{17} - 3586945138 \beta_{16} + 3519764325 \beta_{15} - 3665815433 \beta_{14} + \cdots - 3897627431452 \)
|
| \(\nu^{8}\) | \(=\) |
\( 393961777125 \beta_{17} + 341214452599 \beta_{16} - 318154178895 \beta_{15} + 320234613514 \beta_{14} + \cdots - 37467138222115 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 22127547800730 \beta_{17} - 14750449644371 \beta_{16} - 53881807088895 \beta_{15} + \cdots + 58\!\cdots\!34 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 68\!\cdots\!05 \beta_{17} + \cdots - 18\!\cdots\!95 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 29\!\cdots\!30 \beta_{17} + \cdots - 21\!\cdots\!78 \)
|
| \(\nu^{12}\) | \(=\) |
\( 27\!\cdots\!10 \beta_{17} + \cdots + 18\!\cdots\!73 \)
|
| \(\nu^{13}\) | \(=\) |
\( 41\!\cdots\!10 \beta_{17} + \cdots - 47\!\cdots\!12 \)
|
| \(\nu^{14}\) | \(=\) |
\( 34\!\cdots\!30 \beta_{17} + \cdots - 83\!\cdots\!81 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 11\!\cdots\!35 \beta_{17} + \cdots + 79\!\cdots\!86 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 55\!\cdots\!60 \beta_{17} + \cdots - 16\!\cdots\!97 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 23\!\cdots\!35 \beta_{17} + \cdots - 29\!\cdots\!46 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).
| \(n\) | \(325\) | \(343\) | \(533\) |
| \(\chi(n)\) | \(-1 + \beta_{2}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 505.1 |
|
0 | 0 | 0 | −50.5928 | + | 87.6293i | 0 | 95.5451 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 505.2 | 0 | 0 | 0 | −35.6401 | + | 61.7304i | 0 | 252.315 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 505.3 | 0 | 0 | 0 | −31.6056 | + | 54.7425i | 0 | −80.2775 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 505.4 | 0 | 0 | 0 | −12.6095 | + | 21.8402i | 0 | −40.7176 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 505.5 | 0 | 0 | 0 | 15.4645 | − | 26.7852i | 0 | 132.225 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 505.6 | 0 | 0 | 0 | 18.0963 | − | 31.3437i | 0 | −208.885 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 505.7 | 0 | 0 | 0 | 18.6530 | − | 32.3080i | 0 | 15.8772 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 505.8 | 0 | 0 | 0 | 25.2470 | − | 43.7291i | 0 | −187.942 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 505.9 | 0 | 0 | 0 | 47.4871 | − | 82.2500i | 0 | 189.860 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.1 | 0 | 0 | 0 | −50.5928 | − | 87.6293i | 0 | 95.5451 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.2 | 0 | 0 | 0 | −35.6401 | − | 61.7304i | 0 | 252.315 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.3 | 0 | 0 | 0 | −31.6056 | − | 54.7425i | 0 | −80.2775 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.4 | 0 | 0 | 0 | −12.6095 | − | 21.8402i | 0 | −40.7176 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.5 | 0 | 0 | 0 | 15.4645 | + | 26.7852i | 0 | 132.225 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.6 | 0 | 0 | 0 | 18.0963 | + | 31.3437i | 0 | −208.885 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.7 | 0 | 0 | 0 | 18.6530 | + | 32.3080i | 0 | 15.8772 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.8 | 0 | 0 | 0 | 25.2470 | + | 43.7291i | 0 | −187.942 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.9 | 0 | 0 | 0 | 47.4871 | + | 82.2500i | 0 | 189.860 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 684.6.k.f | 18 | |
| 3.b | odd | 2 | 1 | 76.6.e.a | ✓ | 18 | |
| 12.b | even | 2 | 1 | 304.6.i.d | 18 | ||
| 19.c | even | 3 | 1 | inner | 684.6.k.f | 18 | |
| 57.h | odd | 6 | 1 | 76.6.e.a | ✓ | 18 | |
| 228.m | even | 6 | 1 | 304.6.i.d | 18 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 76.6.e.a | ✓ | 18 | 3.b | odd | 2 | 1 | |
| 76.6.e.a | ✓ | 18 | 57.h | odd | 6 | 1 | |
| 304.6.i.d | 18 | 12.b | even | 2 | 1 | ||
| 304.6.i.d | 18 | 228.m | even | 6 | 1 | ||
| 684.6.k.f | 18 | 1.a | even | 1 | 1 | trivial | |
| 684.6.k.f | 18 | 19.c | even | 3 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(684, [\chi])\):
|
\( T_{5}^{18} + 11 T_{5}^{17} + 17650 T_{5}^{16} - 174581 T_{5}^{15} + 211569494 T_{5}^{14} + \cdots + 53\!\cdots\!96 \)
|
|
\( T_{7}^{9} - 168 T_{7}^{8} - 92704 T_{7}^{7} + 13902400 T_{7}^{6} + 2542736868 T_{7}^{5} + \cdots - 12\!\cdots\!00 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} \)
|
| $3$ |
\( T^{18} \)
|
| $5$ |
\( T^{18} + \cdots + 53\!\cdots\!96 \)
|
| $7$ |
\( (T^{9} + \cdots - 12\!\cdots\!00)^{2} \)
|
| $11$ |
\( (T^{9} + \cdots - 15\!\cdots\!96)^{2} \)
|
| $13$ |
\( T^{18} + \cdots + 60\!\cdots\!64 \)
|
| $17$ |
\( T^{18} + \cdots + 16\!\cdots\!84 \)
|
| $19$ |
\( T^{18} + \cdots + 34\!\cdots\!99 \)
|
| $23$ |
\( T^{18} + \cdots + 75\!\cdots\!00 \)
|
| $29$ |
\( T^{18} + \cdots + 96\!\cdots\!24 \)
|
| $31$ |
\( (T^{9} + \cdots - 91\!\cdots\!60)^{2} \)
|
| $37$ |
\( (T^{9} + \cdots - 70\!\cdots\!00)^{2} \)
|
| $41$ |
\( T^{18} + \cdots + 33\!\cdots\!25 \)
|
| $43$ |
\( T^{18} + \cdots + 51\!\cdots\!00 \)
|
| $47$ |
\( T^{18} + \cdots + 16\!\cdots\!16 \)
|
| $53$ |
\( T^{18} + \cdots + 50\!\cdots\!64 \)
|
| $59$ |
\( T^{18} + \cdots + 21\!\cdots\!89 \)
|
| $61$ |
\( T^{18} + \cdots + 21\!\cdots\!56 \)
|
| $67$ |
\( T^{18} + \cdots + 92\!\cdots\!81 \)
|
| $71$ |
\( T^{18} + \cdots + 17\!\cdots\!00 \)
|
| $73$ |
\( T^{18} + \cdots + 43\!\cdots\!25 \)
|
| $79$ |
\( T^{18} + \cdots + 12\!\cdots\!44 \)
|
| $83$ |
\( (T^{9} + \cdots - 24\!\cdots\!00)^{2} \)
|
| $89$ |
\( T^{18} + \cdots + 86\!\cdots\!44 \)
|
| $97$ |
\( T^{18} + \cdots + 93\!\cdots\!25 \)
|
| show more | |
| show less |