Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [985,2,Mod(1,985)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(985, 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("985.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 985 = 5 \cdot 197 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 985.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: | \(7.86526459910\) |
| 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: |
\( x^{17} - 6 x^{16} - 8 x^{15} + 106 x^{14} - 60 x^{13} - 698 x^{12} + 877 x^{11} + 2076 x^{10} - 3556 x^{9} + \cdots + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{17} - 6 x^{16} - 8 x^{15} + 106 x^{14} - 60 x^{13} - 698 x^{12} + 877 x^{11} + 2076 x^{10} - 3556 x^{9} + \cdots + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2191 \nu^{16} + 28927 \nu^{15} - 60841 \nu^{14} - 425083 \nu^{13} + 1562505 \nu^{12} + \cdots - 270722 ) / 245192 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 54273 \nu^{16} - 92866 \nu^{15} - 2278479 \nu^{14} + 5511036 \nu^{13} + 28342853 \nu^{12} + \cdots - 24548096 ) / 5639416 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 35046 \nu^{16} + 110817 \nu^{15} + 764848 \nu^{14} - 2404159 \nu^{13} - 6491506 \nu^{12} + \cdots - 1062386 ) / 2819708 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 100298 \nu^{16} + 181737 \nu^{15} + 3360696 \nu^{14} - 7555731 \nu^{13} - 37921246 \nu^{12} + \cdots + 48232934 ) / 5639416 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 241552 \nu^{16} + 806883 \nu^{15} + 5112434 \nu^{14} - 17850929 \nu^{13} - 42261732 \nu^{12} + \cdots + 2082482 ) / 5639416 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 246041 \nu^{16} + 1704318 \nu^{15} + 66483 \nu^{14} - 25609808 \nu^{13} + 46293035 \nu^{12} + \cdots - 15791344 ) / 5639416 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 350679 \nu^{16} - 1997896 \nu^{15} - 3275133 \nu^{14} + 35199734 \nu^{13} - 11047357 \nu^{12} + \cdots + 5393812 ) / 5639416 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 522455 \nu^{16} + 3194212 \nu^{15} + 3528121 \nu^{14} - 54341730 \nu^{13} + 40686141 \nu^{12} + \cdots - 16088228 ) / 5639416 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 596404 \nu^{16} - 3091147 \nu^{15} - 6671366 \nu^{14} + 54710157 \nu^{13} + 1577112 \nu^{12} + \cdots + 11922902 ) / 5639416 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 34331 \nu^{16} - 211845 \nu^{15} - 220207 \nu^{14} + 3614653 \nu^{13} - 2959421 \nu^{12} + \cdots - 823298 ) / 245192 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 984739 \nu^{16} + 4852934 \nu^{15} + 12396845 \nu^{14} - 88461580 \nu^{13} - 24382447 \nu^{12} + \cdots + 16024912 ) / 5639416 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 988531 \nu^{16} + 5841395 \nu^{15} + 7686195 \nu^{14} - 100325551 \nu^{13} + 59472133 \nu^{12} + \cdots + 19076038 ) / 5639416 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 1136939 \nu^{16} - 7098868 \nu^{15} - 7090713 \nu^{14} + 121011790 \nu^{13} - 101282793 \nu^{12} + \cdots - 25816036 ) / 5639416 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{15} + \beta_{13} + \beta_{10} + \beta_{8} - 2\beta_{6} + \beta_{4} + 2\beta_{3} + 7\beta_{2} + \beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2 \beta_{15} - \beta_{14} + 2 \beta_{13} - \beta_{12} + \beta_{10} - \beta_{9} + 3 \beta_{8} + \beta_{7} + \cdots + 12 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 2 \beta_{16} + 9 \beta_{15} - \beta_{14} + 13 \beta_{13} + 2 \beta_{11} + 9 \beta_{10} + 13 \beta_{8} + \cdots + 99 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 2 \beta_{16} + 23 \beta_{15} - 12 \beta_{14} + 27 \beta_{13} - 10 \beta_{12} + \beta_{11} + \cdots + 112 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 29 \beta_{16} + 69 \beta_{15} - 16 \beta_{14} + 128 \beta_{13} - 2 \beta_{12} + 26 \beta_{11} + \cdots + 655 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 38 \beta_{16} + 201 \beta_{15} - 109 \beta_{14} + 279 \beta_{13} - 79 \beta_{12} + 17 \beta_{11} + \cdots + 954 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 299 \beta_{16} + 516 \beta_{15} - 175 \beta_{14} + 1136 \beta_{13} - 36 \beta_{12} + 243 \beta_{11} + \cdots + 4501 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 475 \beta_{16} + 1605 \beta_{15} - 897 \beta_{14} + 2602 \beta_{13} - 583 \beta_{12} + 203 \beta_{11} + \cdots + 7769 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 2703 \beta_{16} + 3878 \beta_{15} - 1639 \beta_{14} + 9589 \beta_{13} - 428 \beta_{12} + 2005 \beta_{11} + \cdots + 31689 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 4948 \beta_{16} + 12367 \beta_{15} - 7054 \beta_{14} + 22957 \beta_{13} - 4204 \beta_{12} + \cdots + 61666 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 22922 \beta_{16} + 29449 \beta_{15} - 14175 \beta_{14} + 78742 \beta_{13} - 4271 \beta_{12} + \cdots + 226889 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 46643 \beta_{16} + 93975 \beta_{15} - 54224 \beta_{14} + 195520 \beta_{13} - 30145 \beta_{12} + \cdots + 481705 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 187753 \beta_{16} + 225590 \beta_{15} - 117146 \beta_{14} + 635904 \beta_{13} - 38843 \beta_{12} + \cdots + 1644546 \)
|
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 |
|
−2.37113 | 3.22611 | 3.62225 | −1.00000 | −7.64953 | −3.33938 | −3.84656 | 7.40781 | 2.37113 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.21690 | 0.781418 | 2.91464 | −1.00000 | −1.73232 | −1.11516 | −2.02767 | −2.38939 | 2.21690 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.06409 | −1.71472 | 2.26049 | −1.00000 | 3.53934 | 3.04480 | −0.537666 | −0.0597382 | 2.06409 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.51786 | −0.903960 | 0.303910 | −1.00000 | 1.37209 | −1.33147 | 2.57443 | −2.18286 | 1.51786 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.695153 | −2.78934 | −1.51676 | −1.00000 | 1.93902 | 0.663651 | 2.44469 | 4.78042 | 0.695153 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.589248 | 0.610291 | −1.65279 | −1.00000 | −0.359613 | 3.82240 | 2.15240 | −2.62754 | 0.589248 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.369022 | 3.16939 | −1.86382 | −1.00000 | −1.16957 | 3.62949 | 1.42584 | 7.04503 | 0.369022 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.0270965 | 0.151367 | −1.99927 | −1.00000 | 0.00410152 | −1.42890 | −0.108366 | −2.97709 | −0.0270965 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.290121 | −0.916709 | −1.91583 | −1.00000 | −0.265957 | −4.90454 | −1.13606 | −2.15964 | −0.290121 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.704734 | 1.91747 | −1.50335 | −1.00000 | 1.35131 | 0.223493 | −2.46893 | 0.676708 | −0.704734 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.17487 | −1.75288 | −0.619690 | −1.00000 | −2.05940 | −0.641186 | −3.07778 | 0.0725852 | −1.17487 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.68026 | −2.94614 | 0.823263 | −1.00000 | −4.95027 | 3.59992 | −1.97722 | 5.67973 | −1.68026 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.78623 | 2.62867 | 1.19063 | −1.00000 | 4.69542 | 2.07657 | −1.44572 | 3.90991 | −1.78623 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.22883 | 3.02604 | 2.96767 | −1.00000 | 6.74452 | 1.28446 | 2.15678 | 6.15690 | −2.22883 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.55893 | −1.95773 | 4.54812 | −1.00000 | −5.00969 | −0.330398 | 6.52046 | 0.832702 | −2.55893 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.61726 | 1.86075 | 4.85003 | −1.00000 | 4.87007 | −2.88349 | 7.45926 | 0.462402 | −2.61726 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 2.75509 | 0.609959 | 5.59050 | −1.00000 | 1.68049 | 4.62975 | 9.89214 | −2.62795 | −2.75509 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(197\) | \( -1 \) |
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 | 985.2.a.g | ✓ | 17 |
| 3.b | odd | 2 | 1 | 8865.2.a.z | 17 | ||
| 5.b | even | 2 | 1 | 4925.2.a.l | 17 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 985.2.a.g | ✓ | 17 | 1.a | even | 1 | 1 | trivial |
| 4925.2.a.l | 17 | 5.b | even | 2 | 1 | ||
| 8865.2.a.z | 17 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{17} - 6 T_{2}^{16} - 8 T_{2}^{15} + 106 T_{2}^{14} - 60 T_{2}^{13} - 698 T_{2}^{12} + 877 T_{2}^{11} + \cdots + 2 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(985))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{17} - 6 T^{16} + \cdots + 2 \)
|
| $3$ |
\( T^{17} - 5 T^{16} + \cdots + 512 \)
|
| $5$ |
\( (T + 1)^{17} \)
|
| $7$ |
\( T^{17} - 7 T^{16} + \cdots - 5912 \)
|
| $11$ |
\( T^{17} - 7 T^{16} + \cdots + 41153536 \)
|
| $13$ |
\( T^{17} - 3 T^{16} + \cdots - 310096 \)
|
| $17$ |
\( T^{17} + \cdots - 349688704 \)
|
| $19$ |
\( T^{17} + 23 T^{16} + \cdots + 1915904 \)
|
| $23$ |
\( T^{17} - 49 T^{16} + \cdots + 40562168 \)
|
| $29$ |
\( T^{17} + \cdots + 5029285904 \)
|
| $31$ |
\( T^{17} + \cdots + 14161566256 \)
|
| $37$ |
\( T^{17} + \cdots + 376364288 \)
|
| $41$ |
\( T^{17} + \cdots - 4430614816 \)
|
| $43$ |
\( T^{17} + \cdots - 278500548904 \)
|
| $47$ |
\( T^{17} + \cdots - 62369461376 \)
|
| $53$ |
\( T^{17} + \cdots - 17917124608 \)
|
| $59$ |
\( T^{17} + \cdots + 2605268451328 \)
|
| $61$ |
\( T^{17} + \cdots + 3030799356 \)
|
| $67$ |
\( T^{17} + \cdots - 275592380032 \)
|
| $71$ |
\( T^{17} + \cdots - 2285257277264 \)
|
| $73$ |
\( T^{17} + \cdots + 25494443456 \)
|
| $79$ |
\( T^{17} + \cdots - 17\!\cdots\!16 \)
|
| $83$ |
\( T^{17} + \cdots + 574612307072 \)
|
| $89$ |
\( T^{17} + \cdots - 1850800446976 \)
|
| $97$ |
\( T^{17} + \cdots + 3084621824 \)
|
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