Newspace parameters
| Level: | \( N \) | \(=\) | \( 1859 = 11 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1859.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(109.684550701\) |
| Analytic rank: | \(0\) |
| Dimension: | \(17\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{17} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{17} - 93 x^{15} - 7 x^{14} + 3449 x^{13} + 406 x^{12} - 65242 x^{11} - 7942 x^{10} + 669163 x^{9} + \cdots - 2210688 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{7} \) |
| Twist minimal: | no (minimal twist has level 143) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
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
\( x^{17} - 93 x^{15} - 7 x^{14} + 3449 x^{13} + 406 x^{12} - 65242 x^{11} - 7942 x^{10} + 669163 x^{9} + \cdots - 2210688 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 19\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 41310869995 \nu^{16} + 278727206397 \nu^{15} - 4357449870192 \nu^{14} + \cdots + 19\!\cdots\!24 ) / 12\!\cdots\!76 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 22792255353 \nu^{16} - 173935419925 \nu^{15} - 1445507747716 \nu^{14} + \cdots - 92\!\cdots\!40 ) / 424284949467392 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 124885018663 \nu^{16} + 68376766059 \nu^{15} + 11092500475884 \nu^{14} + \cdots - 31\!\cdots\!20 ) / 12\!\cdots\!76 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 294827222141 \nu^{16} + 219464620161 \nu^{15} + 25860702212988 \nu^{14} + \cdots - 63\!\cdots\!08 ) / 25\!\cdots\!52 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 177480472921 \nu^{16} - 398292066701 \nu^{15} - 14709832798652 \nu^{14} + \cdots + 13\!\cdots\!48 ) / 848569898934784 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 183146367977 \nu^{16} - 411803762979 \nu^{15} + 17660657765996 \nu^{14} + \cdots - 68\!\cdots\!80 ) / 848569898934784 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 278509815379 \nu^{16} + 240283192845 \nu^{15} - 25773301577160 \nu^{14} + \cdots + 10\!\cdots\!16 ) / 12\!\cdots\!76 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 719004694813 \nu^{16} - 1831736510361 \nu^{15} - 59159334330924 \nu^{14} + \cdots + 60\!\cdots\!52 ) / 25\!\cdots\!52 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 483443617781 \nu^{16} - 896950422627 \nu^{15} + 47322682896696 \nu^{14} + \cdots - 21\!\cdots\!52 ) / 12\!\cdots\!76 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 508877270167 \nu^{16} - 1725555260937 \nu^{15} + 52228351735548 \nu^{14} + \cdots - 29\!\cdots\!36 ) / 12\!\cdots\!76 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 675317839955 \nu^{16} + 243395485401 \nu^{15} - 61841470872860 \nu^{14} + \cdots + 20\!\cdots\!52 ) / 848569898934784 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 1065195885323 \nu^{16} + 1645100590137 \nu^{15} - 102145188288576 \nu^{14} + \cdots + 42\!\cdots\!80 ) / 12\!\cdots\!76 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 545813418179 \nu^{16} + 175570637925 \nu^{15} - 49939980916872 \nu^{14} + \cdots + 18\!\cdots\!36 ) / 636427424201088 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 11 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 19\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{14} + \beta_{12} - \beta_{10} - \beta_{8} - \beta_{6} - \beta_{5} - \beta_{4} + 26\beta_{2} + 4\beta _1 + 207 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 2 \beta_{16} + 2 \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + 4 \beta_{9} + \cdots + 63 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{15} + 36 \beta_{14} + \beta_{13} + 40 \beta_{12} + 8 \beta_{11} - 37 \beta_{10} - 8 \beta_{9} + \cdots + 4479 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 88 \beta_{16} + 16 \beta_{15} + 77 \beta_{14} - 28 \beta_{13} + 43 \beta_{12} - 44 \beta_{11} + \cdots + 2488 \)
|
| \(\nu^{8}\) | \(=\) |
\( 2 \beta_{16} + 70 \beta_{15} + 1024 \beta_{14} + 65 \beta_{13} + 1243 \beta_{12} + 411 \beta_{11} + \cdots + 103475 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 2892 \beta_{16} + 887 \beta_{15} + 2231 \beta_{14} - 603 \beta_{13} + 1393 \beta_{12} - 1450 \beta_{11} + \cdots + 82167 \)
|
| \(\nu^{10}\) | \(=\) |
\( 154 \beta_{16} + 2932 \beta_{15} + 27330 \beta_{14} + 2713 \beta_{13} + 35465 \beta_{12} + \cdots + 2487570 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 85404 \beta_{16} + 33111 \beta_{15} + 59472 \beta_{14} - 11917 \beta_{13} + 40444 \beta_{12} + \cdots + 2465723 \)
|
| \(\nu^{12}\) | \(=\) |
\( 7496 \beta_{16} + 99188 \beta_{15} + 714934 \beta_{14} + 93336 \beta_{13} + 973360 \beta_{12} + \cdots + 61381628 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 2395480 \beta_{16} + 1052466 \beta_{15} + 1542287 \beta_{14} - 225820 \beta_{13} + 1112321 \beta_{12} + \cdots + 69948387 \)
|
| \(\nu^{14}\) | \(=\) |
\( 295642 \beta_{16} + 3011076 \beta_{15} + 18586881 \beta_{14} + 2905893 \beta_{13} + 26170874 \beta_{12} + \cdots + 1541335383 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 65313734 \beta_{16} + 30847469 \beta_{15} + 39648325 \beta_{14} - 4120830 \beta_{13} + \cdots + 1916892151 \)
|
| \(\nu^{16}\) | \(=\) |
\( 10358698 \beta_{16} + 86056501 \beta_{15} + 482470073 \beta_{14} + 85537486 \beta_{13} + \cdots + 39163943496 \)
|
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−5.11519 | −3.58195 | 18.1651 | 2.16776 | 18.3223 | −20.9807 | −51.9966 | −14.1696 | −11.0885 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.19083 | 4.70544 | 9.56305 | −14.4259 | −19.7197 | 34.7012 | −6.55048 | −4.85885 | 60.4566 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.15571 | 7.17608 | 9.26990 | 11.1721 | −29.8217 | 7.78423 | −5.27734 | 24.4962 | −46.4282 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −4.13824 | −5.47866 | 9.12507 | 3.23279 | 22.6720 | −3.07367 | −4.65582 | 3.01573 | −13.3781 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.78419 | −8.94641 | −0.248293 | 17.0433 | 24.9085 | 30.9353 | 22.9648 | 53.0382 | −47.4517 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.88904 | 1.56640 | −4.43152 | −2.20557 | −2.95900 | 1.80472 | 23.4837 | −24.5464 | 4.16640 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.41378 | 5.76511 | −6.00123 | 4.58681 | −8.15059 | −36.4239 | 19.7947 | 6.23646 | −6.48474 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.991815 | −5.36099 | −7.01630 | −18.5970 | 5.31711 | 13.4049 | 14.8934 | 1.74021 | 18.4447 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.631226 | 9.24271 | −7.60155 | 19.0524 | 5.83424 | 25.4436 | −9.84810 | 58.4278 | 12.0263 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.664163 | −0.846704 | −7.55889 | 8.92433 | −0.562349 | −3.53056 | −10.3336 | −26.2831 | 5.92721 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.06843 | −9.85108 | −6.85846 | −2.35929 | −10.5252 | −12.2506 | −15.8752 | 70.0437 | −2.52073 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.71387 | 7.58218 | −5.06264 | −12.0244 | 12.9949 | −1.14865 | −22.3877 | 30.4895 | −20.6082 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 3.12700 | −0.537604 | 1.77811 | 0.516265 | −1.68109 | 34.6419 | −19.4558 | −26.7110 | 1.61436 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 3.17505 | −0.600459 | 2.08094 | −12.4679 | −1.90649 | −10.6254 | −18.7933 | −26.6394 | −39.5863 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 4.11212 | −4.01551 | 8.90957 | 12.3424 | −16.5123 | −29.6370 | 3.74025 | −10.8757 | 50.7537 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 5.08131 | −7.39360 | 17.8197 | −8.39690 | −37.5692 | 14.1925 | 49.8972 | 27.6653 | −42.6673 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 5.10562 | 4.57504 | 18.0674 | 15.4387 | 23.3584 | 16.7622 | 51.4001 | −6.06900 | 78.8244 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \( +1 \) |
| \(13\) | \( +1 \) |
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 | 1859.4.a.h | 17 | |
| 13.b | even | 2 | 1 | 1859.4.a.g | 17 | ||
| 13.e | even | 6 | 2 | 143.4.e.b | ✓ | 34 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 143.4.e.b | ✓ | 34 | 13.e | even | 6 | 2 | |
| 1859.4.a.g | 17 | 13.b | even | 2 | 1 | ||
| 1859.4.a.h | 17 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{17} - 93 T_{2}^{15} - 7 T_{2}^{14} + 3449 T_{2}^{13} + 406 T_{2}^{12} - 65242 T_{2}^{11} + \cdots - 2210688 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1859))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{17} - 93 T^{15} + \cdots - 2210688 \)
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| $3$ |
\( T^{17} + \cdots - 7355831456 \)
|
| $5$ |
\( T^{17} + \cdots + 179912814666900 \)
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| $7$ |
\( T^{17} + \cdots - 15\!\cdots\!24 \)
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| $11$ |
\( (T + 11)^{17} \)
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| $13$ |
\( T^{17} \)
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| $17$ |
\( T^{17} + \cdots + 20\!\cdots\!00 \)
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| $19$ |
\( T^{17} + \cdots - 20\!\cdots\!36 \)
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| $23$ |
\( T^{17} + \cdots + 38\!\cdots\!76 \)
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| $29$ |
\( T^{17} + \cdots + 17\!\cdots\!77 \)
|
| $31$ |
\( T^{17} + \cdots - 11\!\cdots\!16 \)
|
| $37$ |
\( T^{17} + \cdots - 60\!\cdots\!98 \)
|
| $41$ |
\( T^{17} + \cdots - 56\!\cdots\!50 \)
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| $43$ |
\( T^{17} + \cdots - 12\!\cdots\!00 \)
|
| $47$ |
\( T^{17} + \cdots + 24\!\cdots\!72 \)
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| $53$ |
\( T^{17} + \cdots - 60\!\cdots\!50 \)
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| $59$ |
\( T^{17} + \cdots - 23\!\cdots\!00 \)
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| $61$ |
\( T^{17} + \cdots - 28\!\cdots\!14 \)
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| $67$ |
\( T^{17} + \cdots + 56\!\cdots\!48 \)
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| $71$ |
\( T^{17} + \cdots + 60\!\cdots\!84 \)
|
| $73$ |
\( T^{17} + \cdots + 66\!\cdots\!32 \)
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| $79$ |
\( T^{17} + \cdots - 82\!\cdots\!00 \)
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| $83$ |
\( T^{17} + \cdots + 65\!\cdots\!56 \)
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| $89$ |
\( T^{17} + \cdots - 19\!\cdots\!36 \)
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| $97$ |
\( T^{17} + \cdots + 46\!\cdots\!84 \)
|
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