Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [19,12,Mod(1,19)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(19, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("19.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 19 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 19.a (trivial) |
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: | yes |
| Analytic conductor: | \(14.5985204306\) |
| Analytic rank: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 3 x^{8} - 14682 x^{7} - 159158 x^{6} + 62351712 x^{5} + 1328163744 x^{4} - 57315079008 x^{3} + \cdots + 22065977221632 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 3 x^{8} - 14682 x^{7} - 159158 x^{6} + 62351712 x^{5} + 1328163744 x^{4} - 57315079008 x^{3} + \cdots + 22065977221632 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 10916976827839 \nu^{8} + \cdots - 15\!\cdots\!80 ) / 32\!\cdots\!44 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 11263056385771 \nu^{8} - 238223507973219 \nu^{7} + \cdots + 13\!\cdots\!48 ) / 16\!\cdots\!72 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 11263056385771 \nu^{8} + 238223507973219 \nu^{7} + \cdots - 26\!\cdots\!00 ) / 81\!\cdots\!36 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 11507317049092 \nu^{8} + 412009576030167 \nu^{7} + \cdots - 46\!\cdots\!56 ) / 81\!\cdots\!36 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 20093286956195 \nu^{8} - 384380021874879 \nu^{7} + \cdots + 62\!\cdots\!84 ) / 23\!\cdots\!96 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 35325515929191 \nu^{8} + 388211050428283 \nu^{7} + \cdots - 16\!\cdots\!72 ) / 36\!\cdots\!16 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 8746592937961 \nu^{8} - 101797892610993 \nu^{7} + \cdots + 32\!\cdots\!92 ) / 77\!\cdots\!32 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 20\beta _1 + 3257 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{8} + 12\beta_{7} + 10\beta_{6} - 9\beta_{5} + 28\beta_{4} - 29\beta_{3} - 12\beta_{2} + 6277\beta _1 + 65652 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 229 \beta_{8} + 546 \beta_{7} + 790 \beta_{6} - 1503 \beta_{5} + 8286 \beta_{4} + 12545 \beta_{3} + \cdots + 20378938 \)
|
| \(\nu^{5}\) | \(=\) |
\( 22455 \beta_{8} + 197526 \beta_{7} + 143298 \beta_{6} - 143631 \beta_{5} + 375572 \beta_{4} + \cdots + 801587056 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 1827467 \beta_{8} + 12540630 \beta_{7} + 14215838 \beta_{6} - 18935577 \beta_{5} + \cdots + 147091830780 \)
|
| \(\nu^{7}\) | \(=\) |
\( 324812983 \beta_{8} + 2297517546 \beta_{7} + 1669365506 \beta_{6} - 1741230603 \beta_{5} + \cdots + 7980488205872 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 6592655583 \beta_{8} + 172054428126 \beta_{7} + 174206377278 \beta_{6} - 199977565077 \beta_{5} + \cdots + 11\!\cdots\!84 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−85.4113 | −330.174 | 5247.09 | −1438.22 | 28200.6 | −77005.8 | −273239. | −68132.2 | 122840. | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −70.8864 | 236.948 | 2976.89 | 8874.43 | −16796.4 | 27550.4 | −65845.4 | −121003. | −629077. | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −23.9874 | −160.418 | −1472.61 | −10003.4 | 3848.00 | −19243.4 | 84450.1 | −151413. | 239956. | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 6.12640 | 681.144 | −2010.47 | −2336.97 | 4172.96 | 76450.0 | −24863.8 | 286810. | −14317.2 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 15.0735 | −545.376 | −1820.79 | 8144.87 | −8220.73 | −81138.6 | −58316.2 | 120288. | 122772. | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 28.7044 | −650.711 | −1224.06 | −7939.47 | −18678.3 | 45630.6 | −93922.4 | 246277. | −227898. | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 57.2072 | 659.655 | 1224.67 | 12905.5 | 37737.0 | −44251.2 | −47100.6 | 257998. | 738290. | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 72.0715 | −38.8044 | 3146.31 | 5253.13 | −2796.69 | 45608.0 | 79156.5 | −175641. | 378601. | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 88.1020 | 643.736 | 5713.97 | −11345.9 | 56714.5 | 7320.12 | 322979. | 237249. | −999595. | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(19\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 19.12.a.b | ✓ | 9 |
| 3.b | odd | 2 | 1 | 171.12.a.d | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 19.12.a.b | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 171.12.a.d | 9 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} - 87 T_{2}^{8} - 11322 T_{2}^{7} + 1111298 T_{2}^{6} + 23062032 T_{2}^{5} + \cdots + 139839674073088 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(19))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + \cdots + 139839674073088 \)
|
| $3$ |
\( T^{9} + \cdots + 49\!\cdots\!68 \)
|
| $5$ |
\( T^{9} + \cdots + 14\!\cdots\!00 \)
|
| $7$ |
\( T^{9} + \cdots - 17\!\cdots\!28 \)
|
| $11$ |
\( T^{9} + \cdots + 56\!\cdots\!16 \)
|
| $13$ |
\( T^{9} + \cdots - 33\!\cdots\!48 \)
|
| $17$ |
\( T^{9} + \cdots - 93\!\cdots\!82 \)
|
| $19$ |
\( (T + 2476099)^{9} \)
|
| $23$ |
\( T^{9} + \cdots + 67\!\cdots\!32 \)
|
| $29$ |
\( T^{9} + \cdots - 14\!\cdots\!60 \)
|
| $31$ |
\( T^{9} + \cdots - 92\!\cdots\!44 \)
|
| $37$ |
\( T^{9} + \cdots - 41\!\cdots\!92 \)
|
| $41$ |
\( T^{9} + \cdots - 89\!\cdots\!64 \)
|
| $43$ |
\( T^{9} + \cdots - 69\!\cdots\!96 \)
|
| $47$ |
\( T^{9} + \cdots - 18\!\cdots\!08 \)
|
| $53$ |
\( T^{9} + \cdots - 26\!\cdots\!36 \)
|
| $59$ |
\( T^{9} + \cdots + 54\!\cdots\!20 \)
|
| $61$ |
\( T^{9} + \cdots + 17\!\cdots\!92 \)
|
| $67$ |
\( T^{9} + \cdots + 32\!\cdots\!52 \)
|
| $71$ |
\( T^{9} + \cdots + 12\!\cdots\!84 \)
|
| $73$ |
\( T^{9} + \cdots - 58\!\cdots\!06 \)
|
| $79$ |
\( T^{9} + \cdots + 69\!\cdots\!40 \)
|
| $83$ |
\( T^{9} + \cdots - 54\!\cdots\!56 \)
|
| $89$ |
\( T^{9} + \cdots - 26\!\cdots\!20 \)
|
| $97$ |
\( T^{9} + \cdots + 26\!\cdots\!24 \)
|
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