Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [717,2,Mod(1,717)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(717, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("717.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 717 = 3 \cdot 239 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 717.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: | \(5.72527382493\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 3 x^{11} - 15 x^{10} + 47 x^{9} + 75 x^{8} - 256 x^{7} - 134 x^{6} + 571 x^{5} + 23 x^{4} + \cdots - 31 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 3 x^{11} - 15 x^{10} + 47 x^{9} + 75 x^{8} - 256 x^{7} - 134 x^{6} + 571 x^{5} + 23 x^{4} + \cdots - 31 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{11} - 2 \nu^{10} - 13 \nu^{9} + 26 \nu^{8} + 45 \nu^{7} - 103 \nu^{6} + 3 \nu^{5} + 118 \nu^{4} + \cdots - 5 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{11} - 3 \nu^{10} - 12 \nu^{9} + 40 \nu^{8} + 35 \nu^{7} - 162 \nu^{6} + 23 \nu^{5} + 177 \nu^{4} + \cdots - 18 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - \nu^{11} + 4 \nu^{10} + 7 \nu^{9} - 50 \nu^{8} + 35 \nu^{7} + 173 \nu^{6} - 331 \nu^{5} - 68 \nu^{4} + \cdots + 59 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 3 \nu^{11} + 4 \nu^{10} + 49 \nu^{9} - 58 \nu^{8} - 279 \nu^{7} + 291 \nu^{6} + 655 \nu^{5} + \cdots - 95 ) / 4 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 5 \nu^{11} + 14 \nu^{10} + 85 \nu^{9} - 222 \nu^{8} - 533 \nu^{7} + 1215 \nu^{6} + 1509 \nu^{5} + \cdots - 371 ) / 8 \)
|
| \(\beta_{9}\) | \(=\) |
\( \nu^{11} - 3 \nu^{10} - 15 \nu^{9} + 45 \nu^{8} + 78 \nu^{7} - 227 \nu^{6} - 172 \nu^{5} + 437 \nu^{4} + \cdots + 43 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 7 \nu^{11} - 10 \nu^{10} - 127 \nu^{9} + 162 \nu^{8} + 839 \nu^{7} - 933 \nu^{6} - 2423 \nu^{5} + \cdots + 345 ) / 8 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 9 \nu^{11} + 20 \nu^{10} + 151 \nu^{9} - 310 \nu^{8} - 917 \nu^{7} + 1653 \nu^{6} + 2453 \nu^{5} + \cdots - 429 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{11} - 2\beta_{10} - \beta_{9} - \beta_{7} - \beta_{4} + \beta_{3} + 6\beta_{2} + \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + 2\beta_{6} + \beta_{5} - \beta_{4} + 9\beta_{3} + 28\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 12 \beta_{11} - 23 \beta_{10} - 10 \beta_{9} + 3 \beta_{8} - 11 \beta_{7} + \beta_{6} - 12 \beta_{4} + \cdots + 85 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 4 \beta_{11} - 19 \beta_{10} + 10 \beta_{9} + 15 \beta_{8} - 16 \beta_{7} + 23 \beta_{6} + 11 \beta_{5} + \cdots + 12 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 111 \beta_{11} - 208 \beta_{10} - 78 \beta_{9} + 48 \beta_{8} - 98 \beta_{7} + 15 \beta_{6} + \cdots + 509 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 69 \beta_{11} - 229 \beta_{10} + 78 \beta_{9} + 165 \beta_{8} - 176 \beta_{7} + 203 \beta_{6} + \cdots + 123 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 934 \beta_{11} - 1732 \beta_{10} - 563 \beta_{9} + 530 \beta_{8} - 818 \beta_{7} + 164 \beta_{6} + \cdots + 3162 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 817 \beta_{11} - 2308 \beta_{10} + 551 \beta_{9} + 1588 \beta_{8} - 1668 \beta_{7} + 1639 \beta_{6} + \cdots + 1190 \)
|
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 |
|
−2.50712 | −1.00000 | 4.28565 | 0.414699 | 2.50712 | 3.85993 | −5.73041 | 1.00000 | −1.03970 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.27963 | −1.00000 | 3.19672 | −2.95582 | 2.27963 | 0.183248 | −2.72808 | 1.00000 | 6.73817 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.42188 | −1.00000 | 0.0217417 | 3.65064 | 1.42188 | 2.24006 | 2.81285 | 1.00000 | −5.19077 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.40336 | −1.00000 | −0.0305942 | −0.939648 | 1.40336 | 3.20775 | 2.84964 | 1.00000 | 1.31866 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.496586 | −1.00000 | −1.75340 | 1.64136 | 0.496586 | −3.34365 | 1.86389 | 1.00000 | −0.815074 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.411990 | −1.00000 | −1.83026 | −3.83303 | −0.411990 | −3.43834 | −1.57803 | 1.00000 | −1.57917 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.650058 | −1.00000 | −1.57742 | 3.22288 | −0.650058 | 3.97632 | −2.32553 | 1.00000 | 2.09506 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.719502 | −1.00000 | −1.48232 | −1.41968 | −0.719502 | −0.0123010 | −2.50553 | 1.00000 | −1.02146 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.86392 | −1.00000 | 1.47419 | −3.80019 | −1.86392 | 4.40195 | −0.980069 | 1.00000 | −7.08324 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.33213 | −1.00000 | 3.43882 | 3.08269 | −2.33213 | 0.766491 | 3.35552 | 1.00000 | 7.18923 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.34937 | −1.00000 | 3.51954 | 1.00390 | −2.34937 | −3.03107 | 3.56996 | 1.00000 | 2.35853 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.78161 | −1.00000 | 5.73734 | −1.06781 | −2.78161 | 2.18960 | 10.3958 | 1.00000 | −2.97024 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(239\) | \(-1\) |
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 | 717.2.a.g | ✓ | 12 |
| 3.b | odd | 2 | 1 | 2151.2.a.h | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 717.2.a.g | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 2151.2.a.h | 12 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(717))\):
|
\( T_{2}^{12} - 3 T_{2}^{11} - 15 T_{2}^{10} + 47 T_{2}^{9} + 75 T_{2}^{8} - 256 T_{2}^{7} - 134 T_{2}^{6} + \cdots - 31 \)
|
|
\( T_{5}^{12} + T_{5}^{11} - 39 T_{5}^{10} - 31 T_{5}^{9} + 546 T_{5}^{8} + 339 T_{5}^{7} - 3247 T_{5}^{6} + \cdots + 1520 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - 3 T^{11} + \cdots - 31 \)
|
| $3$ |
\( (T + 1)^{12} \)
|
| $5$ |
\( T^{12} + T^{11} + \cdots + 1520 \)
|
| $7$ |
\( T^{12} - 11 T^{11} + \cdots + 64 \)
|
| $11$ |
\( T^{12} - 15 T^{11} + \cdots - 15536 \)
|
| $13$ |
\( T^{12} - 7 T^{11} + \cdots + 1280 \)
|
| $17$ |
\( T^{12} + 3 T^{11} + \cdots - 2880080 \)
|
| $19$ |
\( T^{12} - 10 T^{11} + \cdots - 4568000 \)
|
| $23$ |
\( T^{12} - 20 T^{11} + \cdots - 520192 \)
|
| $29$ |
\( T^{12} - 2 T^{11} + \cdots - 27648560 \)
|
| $31$ |
\( T^{12} + \cdots + 622739200 \)
|
| $37$ |
\( T^{12} + \cdots - 6555065600 \)
|
| $41$ |
\( T^{12} + \cdots - 605102912 \)
|
| $43$ |
\( T^{12} + \cdots - 142624448 \)
|
| $47$ |
\( T^{12} + \cdots + 2011774976 \)
|
| $53$ |
\( T^{12} + \cdots + 649713088 \)
|
| $59$ |
\( T^{12} + \cdots + 13769595904 \)
|
| $61$ |
\( T^{12} + \cdots + 15680924560 \)
|
| $67$ |
\( T^{12} + \cdots - 862161920 \)
|
| $71$ |
\( T^{12} + 7 T^{11} + \cdots + 55275520 \)
|
| $73$ |
\( T^{12} + \cdots + 1080955648 \)
|
| $79$ |
\( T^{12} - 15 T^{11} + \cdots + 35608000 \)
|
| $83$ |
\( T^{12} + \cdots + 112487344 \)
|
| $89$ |
\( T^{12} + \cdots - 8473904128 \)
|
| $97$ |
\( T^{12} + \cdots - 108705900800 \)
|
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