Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [767,2,Mod(1,767)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(767, 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("767.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 767 = 13 \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 767.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: | \(6.12452583503\) |
| Analytic rank: | \(0\) |
| Dimension: | \(17\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{17} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{17} - 3 x^{16} - 20 x^{15} + 62 x^{14} + 156 x^{13} - 505 x^{12} - 603 x^{11} + 2066 x^{10} + \cdots - 7 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 5 \) |
| 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 \(1,\beta_1,\ldots,\beta_{16}\) 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
\( x^{17} - 3 x^{16} - 20 x^{15} + 62 x^{14} + 156 x^{13} - 505 x^{12} - 603 x^{11} + 2066 x^{10} + \cdots - 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 13089 \nu^{16} + 43898 \nu^{15} + 284542 \nu^{14} - 981688 \nu^{13} - 2443788 \nu^{12} + \cdots - 309647 ) / 361112 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 21365 \nu^{16} + 36692 \nu^{15} + 425464 \nu^{14} - 604386 \nu^{13} - 3408870 \nu^{12} + \cdots - 902969 ) / 180556 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 13378 \nu^{16} + 45119 \nu^{15} + 243375 \nu^{14} - 890921 \nu^{13} - 1660569 \nu^{12} + \cdots + 30037 ) / 90278 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 73977 \nu^{16} + 354874 \nu^{15} + 1271326 \nu^{14} - 7491136 \nu^{13} - 7368756 \nu^{12} + \cdots + 2746649 ) / 361112 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 28411 \nu^{16} + 91295 \nu^{15} + 545420 \nu^{14} - 1873693 \nu^{13} - 3958027 \nu^{12} + \cdots + 438422 ) / 45139 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 32258 \nu^{16} - 111453 \nu^{15} - 600792 \nu^{14} + 2275309 \nu^{13} + 4155555 \nu^{12} + \cdots - 754830 ) / 45139 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 266583 \nu^{16} + 873418 \nu^{15} + 5101790 \nu^{14} - 17983292 \nu^{13} - 36798008 \nu^{12} + \cdots + 5219099 ) / 361112 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 135607 \nu^{16} + 425328 \nu^{15} + 2635980 \nu^{14} - 8790694 \nu^{13} - 19419338 \nu^{12} + \cdots + 2655361 ) / 180556 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 73931 \nu^{16} - 252375 \nu^{15} - 1376747 \nu^{14} + 5121963 \nu^{13} + 9566847 \nu^{12} + \cdots - 1458522 ) / 90278 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 38320 \nu^{16} + 134253 \nu^{15} + 713003 \nu^{14} - 2749398 \nu^{13} - 4913640 \nu^{12} + \cdots + 1134263 ) / 45139 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 240117 \nu^{16} - 771910 \nu^{15} - 4556862 \nu^{14} + 15725516 \nu^{13} + 32460404 \nu^{12} + \cdots - 5451153 ) / 180556 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 125661 \nu^{16} - 393147 \nu^{15} - 2420719 \nu^{14} + 8037661 \nu^{13} + 17659813 \nu^{12} + \cdots - 2274644 ) / 90278 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 870015 \nu^{16} + 2881198 \nu^{15} + 16464778 \nu^{14} - 59031808 \nu^{13} - 116538180 \nu^{12} + \cdots + 19880359 ) / 361112 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{16} - \beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} - 2\beta_{8} + \beta_{3} + 8\beta_{2} + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{13} - \beta_{11} + \beta_{10} + 2\beta_{9} + \beta_{8} - \beta_{4} + 7\beta_{3} + \beta_{2} + 27\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 9 \beta_{16} - 9 \beta_{14} + 11 \beta_{13} + \beta_{12} + 9 \beta_{11} + 10 \beta_{10} + 2 \beta_{9} + \cdots + 75 \)
|
| \(\nu^{7}\) | \(=\) |
\( - \beta_{15} + 14 \beta_{13} + 2 \beta_{12} - 13 \beta_{11} + 13 \beta_{10} + 26 \beta_{9} + 11 \beta_{8} + \cdots + 12 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 66 \beta_{16} - 66 \beta_{14} + 97 \beta_{13} + 16 \beta_{12} + 65 \beta_{11} + 79 \beta_{10} + \cdots + 433 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 2 \beta_{16} - 16 \beta_{15} + 144 \beta_{13} + 36 \beta_{12} - 119 \beta_{11} + 122 \beta_{10} + \cdots + 105 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 458 \beta_{16} - 4 \beta_{15} - 456 \beta_{14} + 789 \beta_{13} + 177 \beta_{12} + 438 \beta_{11} + \cdots + 2621 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 37 \beta_{16} - 173 \beta_{15} + \beta_{14} + 1301 \beta_{13} + 422 \beta_{12} - 947 \beta_{11} + \cdots + 824 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 3118 \beta_{16} - 86 \beta_{15} - 3071 \beta_{14} + 6149 \beta_{13} + 1669 \beta_{12} + 2864 \beta_{11} + \cdots + 16372 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 452 \beta_{16} - 1595 \beta_{15} + 19 \beta_{14} + 10935 \beta_{13} + 4103 \beta_{12} - 7018 \beta_{11} + \cdots + 6183 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 21082 \beta_{16} - 1170 \beta_{15} - 20396 \beta_{14} + 46655 \beta_{13} + 14400 \beta_{12} + \cdots + 104461 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 4610 \beta_{16} - 13538 \beta_{15} + 232 \beta_{14} + 87805 \beta_{13} + 36014 \beta_{12} + \cdots + 45512 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 142279 \beta_{16} - 12883 \beta_{15} - 134305 \beta_{14} + 347548 \beta_{13} + 117520 \beta_{12} + \cdots + 676272 \)
|
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 |
|
−2.54781 | 0.381465 | 4.49134 | −0.451716 | −0.971901 | 2.04678 | −6.34746 | −2.85448 | 1.15089 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.10683 | −3.01509 | 2.43871 | −2.60961 | 6.35226 | 3.49346 | −0.924295 | 6.09075 | 5.49799 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.89234 | 3.38592 | 1.58094 | 1.86483 | −6.40731 | −0.0315346 | 0.793008 | 8.46448 | −3.52888 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.86759 | 0.991483 | 1.48789 | 4.00796 | −1.85168 | −4.05930 | 0.956406 | −2.01696 | −7.48523 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.22478 | −0.889382 | −0.499903 | −2.74194 | 1.08930 | −2.44726 | 3.06184 | −2.20900 | 3.35828 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.849499 | −0.868536 | −1.27835 | 2.40799 | 0.737820 | 1.94310 | 2.78496 | −2.24565 | −2.04559 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.319332 | 2.41747 | −1.89803 | 1.08914 | −0.771974 | 5.16999 | 1.24476 | 2.84414 | −0.347796 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.304218 | 2.49154 | −1.90745 | −2.69888 | −0.757972 | 0.112577 | 1.18872 | 3.20777 | 0.821047 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.121254 | −2.62119 | −1.98530 | 1.67358 | −0.317829 | −4.77864 | −0.483232 | 3.87065 | 0.202928 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.386740 | −1.21158 | −1.85043 | −3.99175 | −0.468567 | −2.31135 | −1.48912 | −1.53208 | −1.54377 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.12805 | −2.17590 | −0.727508 | −1.72355 | −2.45452 | 4.08782 | −3.07676 | 1.73453 | −1.94425 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.28826 | 1.94768 | −0.340386 | 1.61929 | 2.50911 | −0.107919 | −3.01503 | 0.793441 | 2.08607 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.87382 | 2.86813 | 1.51122 | 2.74620 | 5.37438 | −1.68299 | −0.915895 | 5.22619 | 5.14589 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 1.91345 | −0.0416765 | 1.66129 | 3.05767 | −0.0797459 | 4.76401 | −0.648103 | −2.99826 | 5.85069 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.29602 | −2.58214 | 3.27171 | 3.60890 | −5.92865 | 1.35555 | 2.91987 | 3.66746 | 8.28611 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.46648 | 2.32610 | 4.08353 | −3.92562 | 5.73728 | 3.57863 | 5.13900 | 2.41073 | −9.68246 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 2.63832 | 1.59571 | 4.96072 | 0.0674986 | 4.20999 | −2.13293 | 7.81133 | −0.453707 | 0.178083 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(13\) | \(1\) |
| \(59\) | \(-1\) |
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 | 767.2.a.e | ✓ | 17 |
| 3.b | odd | 2 | 1 | 6903.2.a.v | 17 | ||
| 13.b | even | 2 | 1 | 9971.2.a.f | 17 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 767.2.a.e | ✓ | 17 | 1.a | even | 1 | 1 | trivial |
| 6903.2.a.v | 17 | 3.b | odd | 2 | 1 | ||
| 9971.2.a.f | 17 | 13.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{17} - 3 T_{2}^{16} - 20 T_{2}^{15} + 62 T_{2}^{14} + 156 T_{2}^{13} - 505 T_{2}^{12} - 603 T_{2}^{11} + \cdots - 7 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(767))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{17} - 3 T^{16} + \cdots - 7 \)
|
| $3$ |
\( T^{17} - 5 T^{16} + \cdots - 277 \)
|
| $5$ |
\( T^{17} - 4 T^{16} + \cdots + 25600 \)
|
| $7$ |
\( T^{17} - 9 T^{16} + \cdots - 1024 \)
|
| $11$ |
\( T^{17} - 7 T^{16} + \cdots - 1759678 \)
|
| $13$ |
\( (T + 1)^{17} \)
|
| $17$ |
\( T^{17} - 9 T^{16} + \cdots - 376492 \)
|
| $19$ |
\( T^{17} - 8 T^{16} + \cdots - 637952 \)
|
| $23$ |
\( T^{17} - 8 T^{16} + \cdots + 628736 \)
|
| $29$ |
\( T^{17} + \cdots + 34635299125 \)
|
| $31$ |
\( T^{17} + \cdots - 1240455662 \)
|
| $37$ |
\( T^{17} + \cdots - 24052339000 \)
|
| $41$ |
\( T^{17} + \cdots - 63678211072 \)
|
| $43$ |
\( T^{17} + \cdots + 776731664384 \)
|
| $47$ |
\( T^{17} + \cdots - 143189622416 \)
|
| $53$ |
\( T^{17} + \cdots + 454335190381 \)
|
| $59$ |
\( (T - 1)^{17} \)
|
| $61$ |
\( T^{17} + \cdots + 24953376679936 \)
|
| $67$ |
\( T^{17} + \cdots - 200522322400 \)
|
| $71$ |
\( T^{17} + \cdots - 88363028992000 \)
|
| $73$ |
\( T^{17} + \cdots + 22876047744442 \)
|
| $79$ |
\( T^{17} + \cdots + 48182561477 \)
|
| $83$ |
\( T^{17} + \cdots + 11\!\cdots\!82 \)
|
| $89$ |
\( T^{17} + \cdots + 7106135439250 \)
|
| $97$ |
\( T^{17} + \cdots + 86\!\cdots\!72 \)
|
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