Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2763,2,Mod(1,2763)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2763, 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("2763.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2763 = 3^{2} \cdot 307 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2763.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: | \(22.0626660785\) |
| Analytic rank: | \(0\) |
| Dimension: | \(14\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{14} - 3 x^{13} - 20 x^{12} + 61 x^{11} + 150 x^{10} - 467 x^{9} - 521 x^{8} + 1662 x^{7} + 833 x^{6} + \cdots + 28 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 921) |
| 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 \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} - 3 x^{13} - 20 x^{12} + 61 x^{11} + 150 x^{10} - 467 x^{9} - 521 x^{8} + 1662 x^{7} + 833 x^{6} + \cdots + 28 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2597 \nu^{13} - 2891 \nu^{12} - 46127 \nu^{11} + 41140 \nu^{10} + 268505 \nu^{9} - 159405 \nu^{8} + \cdots + 58268 ) / 125724 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1819 \nu^{13} - 3211 \nu^{12} - 34729 \nu^{11} + 51956 \nu^{10} + 242191 \nu^{9} - 269157 \nu^{8} + \cdots + 198052 ) / 62862 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 4820 \nu^{13} + 10703 \nu^{12} + 111257 \nu^{11} - 223465 \nu^{10} - 1017722 \nu^{9} + \cdots + 1429348 ) / 125724 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 6935 \nu^{13} - 21953 \nu^{12} - 136829 \nu^{11} + 446560 \nu^{10} + 1002155 \nu^{9} + \cdots - 813256 ) / 125724 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 1242 \nu^{13} - 2371 \nu^{12} - 25360 \nu^{11} + 44518 \nu^{10} + 193399 \nu^{9} - 305845 \nu^{8} + \cdots - 74588 ) / 20954 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 8083 \nu^{13} + 23824 \nu^{12} + 160354 \nu^{11} - 484655 \nu^{10} - 1162021 \nu^{9} + \cdots + 241904 ) / 125724 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 1493 \nu^{13} - 6604 \nu^{12} - 25913 \nu^{11} + 133113 \nu^{10} + 151308 \nu^{9} - 1000433 \nu^{8} + \cdots - 89628 ) / 20954 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 3560 \nu^{13} + 8905 \nu^{12} + 68827 \nu^{11} - 175265 \nu^{10} - 476842 \nu^{9} + 1284889 \nu^{8} + \cdots + 19304 ) / 41908 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 6053 \nu^{13} + 18599 \nu^{12} + 117605 \nu^{11} - 370912 \nu^{10} - 854033 \nu^{9} + \cdots + 689530 ) / 62862 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 8299 \nu^{13} + 16948 \nu^{12} + 176608 \nu^{11} - 340253 \nu^{10} - 1434349 \nu^{9} + \cdots + 866186 ) / 62862 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{11} + \beta_{9} - \beta_{4} + 6\beta_{2} + \beta _1 + 23 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{13} - \beta_{11} - \beta_{10} - \beta_{8} + 2\beta_{7} - \beta_{6} + \beta_{5} + 9\beta_{3} + 30\beta _1 + 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{13} + \beta_{12} - 12 \beta_{11} + 12 \beta_{9} + 2 \beta_{7} - 2 \beta_{6} - 10 \beta_{4} + \cdots + 142 \)
|
| \(\nu^{7}\) | \(=\) |
\( 15 \beta_{13} - \beta_{12} - 15 \beta_{11} - 15 \beta_{10} + 2 \beta_{9} - 11 \beta_{8} + 30 \beta_{7} + \cdots + 70 \)
|
| \(\nu^{8}\) | \(=\) |
\( 12 \beta_{13} + 17 \beta_{12} - 107 \beta_{11} + \beta_{10} + 110 \beta_{9} - 3 \beta_{8} + 33 \beta_{7} + \cdots + 914 \)
|
| \(\nu^{9}\) | \(=\) |
\( 153 \beta_{13} - 18 \beta_{12} - 157 \beta_{11} - 156 \beta_{10} + 39 \beta_{9} - 94 \beta_{8} + \cdots + 535 \)
|
| \(\nu^{10}\) | \(=\) |
\( 102 \beta_{13} + 188 \beta_{12} - 860 \beta_{11} + 13 \beta_{10} + 915 \beta_{9} - 59 \beta_{8} + \cdots + 6060 \)
|
| \(\nu^{11}\) | \(=\) |
\( 1335 \beta_{13} - 212 \beta_{12} - 1423 \beta_{11} - 1398 \beta_{10} + 495 \beta_{9} - 754 \beta_{8} + \cdots + 4117 \)
|
| \(\nu^{12}\) | \(=\) |
\( 755 \beta_{13} + 1730 \beta_{12} - 6603 \beta_{11} + 88 \beta_{10} + 7271 \beta_{9} - 768 \beta_{8} + \cdots + 41074 \)
|
| \(\nu^{13}\) | \(=\) |
\( 10719 \beta_{13} - 2066 \beta_{12} - 11980 \beta_{11} - 11602 \beta_{10} + 5228 \beta_{9} - 5976 \beta_{8} + \cdots + 31988 \)
|
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 |
|
−2.60225 | 0 | 4.77172 | −0.875556 | 0 | −0.346190 | −7.21270 | 0 | 2.27842 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.36813 | 0 | 3.60805 | −1.87697 | 0 | 0.0865868 | −3.80809 | 0 | 4.44490 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.93653 | 0 | 1.75015 | −2.69450 | 0 | −1.79761 | 0.483849 | 0 | 5.21798 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.75176 | 0 | 1.06868 | 0.563260 | 0 | 3.94174 | 1.63146 | 0 | −0.986699 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.781842 | 0 | −1.38872 | 2.85654 | 0 | −0.597454 | 2.64945 | 0 | −2.23336 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.636824 | 0 | −1.59446 | −1.20484 | 0 | −0.712261 | 2.28904 | 0 | 0.767273 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.144275 | 0 | −1.97918 | 3.64245 | 0 | 5.06827 | −0.574098 | 0 | 0.525515 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.277950 | 0 | −1.92274 | −0.827490 | 0 | −4.74203 | −1.09033 | 0 | −0.230001 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.11126 | 0 | −0.765091 | −0.685781 | 0 | 2.60866 | −3.07275 | 0 | −0.762084 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.62353 | 0 | 0.635856 | −4.22119 | 0 | −1.37888 | −2.21473 | 0 | −6.85324 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.20190 | 0 | 2.84837 | −0.0952912 | 0 | 4.15061 | 1.86803 | 0 | −0.209822 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.31203 | 0 | 3.34548 | 2.76750 | 0 | −3.76591 | 3.11078 | 0 | 6.39854 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 2.64253 | 0 | 4.98295 | 3.25754 | 0 | 2.71142 | 7.88252 | 0 | 8.60813 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.76386 | 0 | 5.63895 | −3.60566 | 0 | 0.773038 | 10.0576 | 0 | −9.96555 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(307\) | \( +1 \) |
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 | 2763.2.a.s | 14 | |
| 3.b | odd | 2 | 1 | 921.2.a.h | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 921.2.a.h | ✓ | 14 | 3.b | odd | 2 | 1 | |
| 2763.2.a.s | 14 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2763))\):
|
\( T_{2}^{14} - 3 T_{2}^{13} - 20 T_{2}^{12} + 61 T_{2}^{11} + 150 T_{2}^{10} - 467 T_{2}^{9} - 521 T_{2}^{8} + \cdots + 28 \)
|
|
\( T_{5}^{14} + 3 T_{5}^{13} - 38 T_{5}^{12} - 115 T_{5}^{11} + 497 T_{5}^{10} + 1641 T_{5}^{9} + \cdots - 232 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} - 3 T^{13} + \cdots + 28 \)
|
| $3$ |
\( T^{14} \)
|
| $5$ |
\( T^{14} + 3 T^{13} + \cdots - 232 \)
|
| $7$ |
\( T^{14} - 6 T^{13} + \cdots - 256 \)
|
| $11$ |
\( T^{14} + 18 T^{13} + \cdots - 262528 \)
|
| $13$ |
\( T^{14} - 6 T^{13} + \cdots - 1024 \)
|
| $17$ |
\( T^{14} - 2 T^{13} + \cdots - 2198336 \)
|
| $19$ |
\( T^{14} + \cdots - 420324608 \)
|
| $23$ |
\( T^{14} - 2 T^{13} + \cdots - 237568 \)
|
| $29$ |
\( T^{14} + 24 T^{13} + \cdots - 3872288 \)
|
| $31$ |
\( T^{14} + \cdots - 113494048 \)
|
| $37$ |
\( T^{14} + \cdots - 150416672 \)
|
| $41$ |
\( T^{14} - 291 T^{12} + \cdots + 9524224 \)
|
| $43$ |
\( T^{14} + \cdots - 1143486256 \)
|
| $47$ |
\( T^{14} + 7 T^{13} + \cdots - 75235328 \)
|
| $53$ |
\( T^{14} + \cdots + 30088548864 \)
|
| $59$ |
\( T^{14} + \cdots + 213569512192 \)
|
| $61$ |
\( T^{14} + \cdots - 445571928064 \)
|
| $67$ |
\( T^{14} + \cdots + 80989584896 \)
|
| $71$ |
\( T^{14} + \cdots - 8502212704 \)
|
| $73$ |
\( T^{14} - 13 T^{13} + \cdots + 3514624 \)
|
| $79$ |
\( T^{14} - 29 T^{13} + \cdots + 334336 \)
|
| $83$ |
\( T^{14} + \cdots + 13762537488 \)
|
| $89$ |
\( T^{14} + \cdots - 5341069929056 \)
|
| $97$ |
\( T^{14} + \cdots + 146925952 \)
|
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