Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [819,2,Mod(235,819)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(819, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("819.235");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 819.j (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: | \(6.53974792554\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} + 18 x^{18} + 214 x^{16} + 1450 x^{14} + 7087 x^{12} + 20465 x^{10} + 42361 x^{8} + 50535 x^{6} + \cdots + 3969 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{3} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} + 18 x^{18} + 214 x^{16} + 1450 x^{14} + 7087 x^{12} + 20465 x^{10} + 42361 x^{8} + 50535 x^{6} + \cdots + 3969 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 63483510895 \nu^{18} - 1071726339501 \nu^{16} - 12404989720237 \nu^{14} + \cdots + 41\!\cdots\!09 ) / 11\!\cdots\!33 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 972440103969 \nu^{18} + 17059537295177 \nu^{16} + 200600097872859 \nu^{14} + \cdots + 40\!\cdots\!22 ) / 81\!\cdots\!31 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 972440103969 \nu^{19} - 17059537295177 \nu^{17} - 200600097872859 \nu^{15} + \cdots - 12\!\cdots\!53 \nu ) / 81\!\cdots\!31 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 15906241212796 \nu^{19} + 183416141110176 \nu^{17} + \cdots - 39\!\cdots\!72 \nu ) / 12\!\cdots\!65 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 34329053073829 \nu^{19} - 643881681068694 \nu^{17} + \cdots - 67\!\cdots\!57 \nu ) / 24\!\cdots\!30 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 36362255678023 \nu^{18} + 518705271356931 \nu^{16} + \cdots - 10\!\cdots\!74 ) / 24\!\cdots\!30 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 14981283821149 \nu^{18} + 228809603427294 \nu^{16} + \cdots - 20\!\cdots\!63 ) / 81\!\cdots\!10 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 45939561634201 \nu^{18} + 742171850598621 \nu^{16} + \cdots - 95\!\cdots\!12 ) / 24\!\cdots\!30 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 23730211055842 \nu^{19} - 481739104565112 \nu^{17} + \cdots - 69\!\cdots\!06 \nu ) / 12\!\cdots\!65 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 7070517816554 \nu^{19} + 119757399916923 \nu^{17} + \cdots - 44\!\cdots\!17 \nu ) / 34\!\cdots\!90 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 4167081224956 \nu^{18} + 71907524691147 \nu^{16} + 832313317051339 \nu^{14} + \cdots + 80\!\cdots\!17 ) / 11\!\cdots\!30 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 96954046827562 \nu^{18} + \cdots - 10\!\cdots\!93 ) / 24\!\cdots\!30 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 3445375839611 \nu^{18} - 60736064804201 \nu^{16} - 715565463449777 \nu^{14} + \cdots - 44\!\cdots\!51 ) / 81\!\cdots\!31 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 127729036819714 \nu^{18} + \cdots - 11\!\cdots\!57 ) / 24\!\cdots\!30 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 9268731387422 \nu^{19} + 158882523648984 \nu^{17} + \cdots - 42\!\cdots\!41 \nu ) / 17\!\cdots\!95 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 10220984050847 \nu^{19} - 174958418741499 \nu^{17} + \cdots + 13\!\cdots\!66 \nu ) / 17\!\cdots\!95 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 11984094316579 \nu^{19} + 206865110727243 \nu^{17} + \cdots - 80\!\cdots\!57 \nu ) / 17\!\cdots\!95 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 104584051769077 \nu^{19} + \cdots + 17\!\cdots\!61 \nu ) / 12\!\cdots\!65 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{14} + 4\beta_{3} + \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{19} + \beta_{17} + \beta_{16} - \beta_{10} - 6\beta_{4} - 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{15} - 8\beta_{14} + \beta_{9} - \beta_{8} - 24\beta_{3} - 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{19} + 10\beta_{10} - \beta_{5} + 40\beta_{4} \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{15} + 9\beta_{13} + 12\beta_{12} - 10\beta_{9} - 10\beta_{7} - 58\beta_{2} + 159 \)
|
| \(\nu^{7}\) | \(=\) |
\( 12\beta_{18} - 58\beta_{17} - 81\beta_{16} + 4\beta_{11} + 275\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 81 \beta_{15} + 414 \beta_{14} - 81 \beta_{13} - 109 \beta_{12} + 19 \beta_{9} + 109 \beta_{8} + \cdots + 414 \beta_{2} \)
|
| \(\nu^{9}\) | \(=\) |
\( - 414 \beta_{19} - 109 \beta_{18} + 414 \beta_{17} + 623 \beta_{16} - 66 \beta_{11} - 623 \beta_{10} + \cdots - 1912 \beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( 391\beta_{15} - 2949\beta_{14} + 232\beta_{13} + 391\beta_{9} - 907\beta_{8} + 232\beta_{7} - 7473\beta_{3} - 7473 \)
|
| \(\nu^{11}\) | \(=\) |
\( 2949\beta_{19} + 4711\beta_{10} - 748\beta_{6} - 907\beta_{5} + 13371\beta_{4} \)
|
| \(\nu^{12}\) | \(=\) |
\( 2351\beta_{15} + 2360\beta_{13} + 7273\beta_{12} - 4711\beta_{9} - 4711\beta_{7} - 21031\beta_{2} + 51829 \)
|
| \(\nu^{13}\) | \(=\) |
\( 7273\beta_{18} - 21031\beta_{17} - 35366\beta_{16} + 7264\beta_{11} + 93891\beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 35366 \beta_{15} + 150288 \beta_{14} - 35366 \beta_{13} - 57176 \beta_{12} + 21590 \beta_{9} + \cdots + 150288 \beta_{2} \)
|
| \(\nu^{15}\) | \(=\) |
\( - 150288 \beta_{19} - 57176 \beta_{18} + 150288 \beta_{17} + 264420 \beta_{16} - 64990 \beta_{11} + \cdots - 661603 \beta_1 \)
|
| \(\nu^{16}\) | \(=\) |
\( 77484 \beta_{15} - 1076311 \beta_{14} + 186936 \beta_{13} + 77484 \beta_{9} - 443762 \beta_{8} + \cdots - 2524246 \)
|
| \(\nu^{17}\) | \(=\) |
\( 1076311\beta_{19} + 1971429\beta_{10} - 553214\beta_{6} - 443762\beta_{5} + 4676868\beta_{4} \)
|
| \(\nu^{18}\) | \(=\) |
\( 1557784 \beta_{15} + 413645 \beta_{13} + 3412167 \beta_{12} - 1971429 \beta_{9} - 1971429 \beta_{7} + \cdots + 17710496 \)
|
| \(\nu^{19}\) | \(=\) |
\( 3412167\beta_{18} - 7724608\beta_{17} - 14665988\beta_{16} + 4556306\beta_{11} + 33159712\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/819\mathbb{Z}\right)^\times\).
| \(n\) | \(92\) | \(379\) | \(703\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 235.1 |
|
−1.35395 | − | 2.34511i | 0 | −2.66636 | + | 4.61827i | 2.03914 | + | 3.53190i | 0 | −0.0214054 | − | 2.64566i | 9.02468 | 0 | 5.52180 | − | 9.56403i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 235.2 | −1.27547 | − | 2.20918i | 0 | −2.25366 | + | 3.90345i | −0.856194 | − | 1.48297i | 0 | 0.787089 | + | 2.52596i | 6.39604 | 0 | −2.18410 | + | 3.78298i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 235.3 | −0.785666 | − | 1.36081i | 0 | −0.234542 | + | 0.406239i | −0.670183 | − | 1.16079i | 0 | −1.91002 | − | 1.83080i | −2.40558 | 0 | −1.05308 | + | 1.82399i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 235.4 | −0.563413 | − | 0.975859i | 0 | 0.365133 | − | 0.632428i | −1.71089 | − | 2.96335i | 0 | −2.13253 | + | 1.56598i | −3.07653 | 0 | −1.92787 | + | 3.33917i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 235.5 | −0.324476 | − | 0.562008i | 0 | 0.789431 | − | 1.36733i | 0.991232 | + | 1.71686i | 0 | 2.27687 | − | 1.34754i | −2.32251 | 0 | 0.643261 | − | 1.11416i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 235.6 | 0.324476 | + | 0.562008i | 0 | 0.789431 | − | 1.36733i | −0.991232 | − | 1.71686i | 0 | 2.27687 | − | 1.34754i | 2.32251 | 0 | 0.643261 | − | 1.11416i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 235.7 | 0.563413 | + | 0.975859i | 0 | 0.365133 | − | 0.632428i | 1.71089 | + | 2.96335i | 0 | −2.13253 | + | 1.56598i | 3.07653 | 0 | −1.92787 | + | 3.33917i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 235.8 | 0.785666 | + | 1.36081i | 0 | −0.234542 | + | 0.406239i | 0.670183 | + | 1.16079i | 0 | −1.91002 | − | 1.83080i | 2.40558 | 0 | −1.05308 | + | 1.82399i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 235.9 | 1.27547 | + | 2.20918i | 0 | −2.25366 | + | 3.90345i | 0.856194 | + | 1.48297i | 0 | 0.787089 | + | 2.52596i | −6.39604 | 0 | −2.18410 | + | 3.78298i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 235.10 | 1.35395 | + | 2.34511i | 0 | −2.66636 | + | 4.61827i | −2.03914 | − | 3.53190i | 0 | −0.0214054 | − | 2.64566i | −9.02468 | 0 | 5.52180 | − | 9.56403i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 352.1 | −1.35395 | + | 2.34511i | 0 | −2.66636 | − | 4.61827i | 2.03914 | − | 3.53190i | 0 | −0.0214054 | + | 2.64566i | 9.02468 | 0 | 5.52180 | + | 9.56403i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 352.2 | −1.27547 | + | 2.20918i | 0 | −2.25366 | − | 3.90345i | −0.856194 | + | 1.48297i | 0 | 0.787089 | − | 2.52596i | 6.39604 | 0 | −2.18410 | − | 3.78298i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 352.3 | −0.785666 | + | 1.36081i | 0 | −0.234542 | − | 0.406239i | −0.670183 | + | 1.16079i | 0 | −1.91002 | + | 1.83080i | −2.40558 | 0 | −1.05308 | − | 1.82399i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 352.4 | −0.563413 | + | 0.975859i | 0 | 0.365133 | + | 0.632428i | −1.71089 | + | 2.96335i | 0 | −2.13253 | − | 1.56598i | −3.07653 | 0 | −1.92787 | − | 3.33917i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 352.5 | −0.324476 | + | 0.562008i | 0 | 0.789431 | + | 1.36733i | 0.991232 | − | 1.71686i | 0 | 2.27687 | + | 1.34754i | −2.32251 | 0 | 0.643261 | + | 1.11416i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 352.6 | 0.324476 | − | 0.562008i | 0 | 0.789431 | + | 1.36733i | −0.991232 | + | 1.71686i | 0 | 2.27687 | + | 1.34754i | 2.32251 | 0 | 0.643261 | + | 1.11416i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 352.7 | 0.563413 | − | 0.975859i | 0 | 0.365133 | + | 0.632428i | 1.71089 | − | 2.96335i | 0 | −2.13253 | − | 1.56598i | 3.07653 | 0 | −1.92787 | − | 3.33917i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 352.8 | 0.785666 | − | 1.36081i | 0 | −0.234542 | − | 0.406239i | 0.670183 | − | 1.16079i | 0 | −1.91002 | + | 1.83080i | 2.40558 | 0 | −1.05308 | − | 1.82399i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 352.9 | 1.27547 | − | 2.20918i | 0 | −2.25366 | − | 3.90345i | 0.856194 | − | 1.48297i | 0 | 0.787089 | − | 2.52596i | −6.39604 | 0 | −2.18410 | − | 3.78298i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 352.10 | 1.35395 | − | 2.34511i | 0 | −2.66636 | − | 4.61827i | −2.03914 | + | 3.53190i | 0 | −0.0214054 | + | 2.64566i | −9.02468 | 0 | 5.52180 | + | 9.56403i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 21.h | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 819.2.j.j | ✓ | 20 |
| 3.b | odd | 2 | 1 | inner | 819.2.j.j | ✓ | 20 |
| 7.c | even | 3 | 1 | inner | 819.2.j.j | ✓ | 20 |
| 7.c | even | 3 | 1 | 5733.2.a.by | 10 | ||
| 7.d | odd | 6 | 1 | 5733.2.a.bz | 10 | ||
| 21.g | even | 6 | 1 | 5733.2.a.bz | 10 | ||
| 21.h | odd | 6 | 1 | inner | 819.2.j.j | ✓ | 20 |
| 21.h | odd | 6 | 1 | 5733.2.a.by | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 819.2.j.j | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 819.2.j.j | ✓ | 20 | 3.b | odd | 2 | 1 | inner |
| 819.2.j.j | ✓ | 20 | 7.c | even | 3 | 1 | inner |
| 819.2.j.j | ✓ | 20 | 21.h | odd | 6 | 1 | inner |
| 5733.2.a.by | 10 | 7.c | even | 3 | 1 | ||
| 5733.2.a.by | 10 | 21.h | odd | 6 | 1 | ||
| 5733.2.a.bz | 10 | 7.d | odd | 6 | 1 | ||
| 5733.2.a.bz | 10 | 21.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{20} + 18 T_{2}^{18} + 214 T_{2}^{16} + 1450 T_{2}^{14} + 7087 T_{2}^{12} + 20465 T_{2}^{10} + \cdots + 3969 \)
acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + 18 T^{18} + \cdots + 3969 \)
|
| $3$ |
\( T^{20} \)
|
| $5$ |
\( T^{20} + 37 T^{18} + \cdots + 16257024 \)
|
| $7$ |
\( (T^{10} + 2 T^{9} + \cdots + 16807)^{2} \)
|
| $11$ |
\( T^{20} + 79 T^{18} + \cdots + 3969 \)
|
| $13$ |
\( (T - 1)^{20} \)
|
| $17$ |
\( T^{20} + 81 T^{18} + \cdots + 3969 \)
|
| $19$ |
\( (T^{10} + 3 T^{9} + \cdots + 2401)^{2} \)
|
| $23$ |
\( T^{20} + \cdots + 69676549173504 \)
|
| $29$ |
\( (T^{10} - 146 T^{8} + \cdots - 642663)^{2} \)
|
| $31$ |
\( (T^{10} + 6 T^{9} + \cdots + 784)^{2} \)
|
| $37$ |
\( (T^{10} + 13 T^{9} + \cdots + 64)^{2} \)
|
| $41$ |
\( (T^{10} - 228 T^{8} + \cdots - 197568)^{2} \)
|
| $43$ |
\( (T^{5} - 20 T^{4} + \cdots + 3748)^{4} \)
|
| $47$ |
\( T^{20} + \cdots + 10160640000 \)
|
| $53$ |
\( T^{20} + 167 T^{18} + \cdots + 9529569 \)
|
| $59$ |
\( T^{20} + \cdots + 21\!\cdots\!09 \)
|
| $61$ |
\( (T^{10} - 11 T^{9} + \cdots + 82369)^{2} \)
|
| $67$ |
\( (T^{10} + 13 T^{9} + \cdots + 218951209)^{2} \)
|
| $71$ |
\( (T^{10} - 215 T^{8} + \cdots - 22743)^{2} \)
|
| $73$ |
\( (T^{10} + 12 T^{9} + \cdots + 132496)^{2} \)
|
| $79$ |
\( (T^{10} + 14 T^{9} + \cdots + 13104400)^{2} \)
|
| $83$ |
\( (T^{10} - 368 T^{8} + \cdots - 326592)^{2} \)
|
| $89$ |
\( T^{20} + \cdots + 45\!\cdots\!64 \)
|
| $97$ |
\( (T^{5} - 9 T^{4} + \cdots - 178556)^{4} \)
|
| show more | |
| show less |