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: | \(\mathbb{Q}[x]/(x^{17} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{17} - 4 x^{16} - 99 x^{15} + 375 x^{14} + 3949 x^{13} - 13998 x^{12} - 81750 x^{11} + 267574 x^{10} + \cdots + 2596992 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3\cdot 5 \) |
| 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
\( x^{17} - 4 x^{16} - 99 x^{15} + 375 x^{14} + 3949 x^{13} - 13998 x^{12} - 81750 x^{11} + 267574 x^{10} + \cdots + 2596992 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 13 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 20\nu + 10 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 32\!\cdots\!77 \nu^{16} + \cdots - 40\!\cdots\!80 ) / 19\!\cdots\!24 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 38\!\cdots\!51 \nu^{16} + \cdots + 16\!\cdots\!60 ) / 19\!\cdots\!24 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 83\!\cdots\!63 \nu^{16} + \cdots + 94\!\cdots\!32 ) / 19\!\cdots\!24 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 24\!\cdots\!43 \nu^{16} + \cdots + 18\!\cdots\!00 ) / 33\!\cdots\!04 \)
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| \(\beta_{8}\) | \(=\) |
\( ( - 79\!\cdots\!35 \nu^{16} + \cdots - 31\!\cdots\!24 ) / 99\!\cdots\!12 \)
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| \(\beta_{9}\) | \(=\) |
\( ( 15\!\cdots\!27 \nu^{16} + \cdots + 44\!\cdots\!40 ) / 16\!\cdots\!52 \)
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| \(\beta_{10}\) | \(=\) |
\( ( - 40\!\cdots\!89 \nu^{16} + \cdots - 46\!\cdots\!32 ) / 16\!\cdots\!52 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 48\!\cdots\!93 \nu^{16} + \cdots - 71\!\cdots\!56 ) / 19\!\cdots\!24 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 14\!\cdots\!39 \nu^{16} + \cdots - 51\!\cdots\!52 ) / 49\!\cdots\!56 \)
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| \(\beta_{13}\) | \(=\) |
\( ( - 35\!\cdots\!11 \nu^{16} + \cdots + 65\!\cdots\!92 ) / 99\!\cdots\!12 \)
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| \(\beta_{14}\) | \(=\) |
\( ( - 11\!\cdots\!67 \nu^{16} + \cdots - 57\!\cdots\!84 ) / 30\!\cdots\!64 \)
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| \(\beta_{15}\) | \(=\) |
\( ( - 57\!\cdots\!55 \nu^{16} + \cdots + 42\!\cdots\!68 ) / 99\!\cdots\!12 \)
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| \(\beta_{16}\) | \(=\) |
\( ( 69\!\cdots\!81 \nu^{16} + \cdots - 21\!\cdots\!96 ) / 99\!\cdots\!12 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 13 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 20\beta _1 + 3 \)
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| \(\nu^{4}\) | \(=\) |
\( - \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} - 2 \beta_{8} + \beta_{6} - \beta_{5} + \cdots + 270 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{12} + \beta_{11} - 4 \beta_{9} - 6 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \cdots + 141 \)
|
| \(\nu^{6}\) | \(=\) |
\( - \beta_{16} - \beta_{15} - 38 \beta_{14} + 45 \beta_{13} - 42 \beta_{12} - 38 \beta_{11} + 51 \beta_{10} + \cdots + 6530 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 4 \beta_{16} + 24 \beta_{15} - 13 \beta_{14} + 5 \beta_{13} + 63 \beta_{12} + 75 \beta_{11} + \cdots + 5057 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 76 \beta_{16} - 28 \beta_{15} - 1170 \beta_{14} + 1574 \beta_{13} - 1413 \beta_{12} - 1173 \beta_{11} + \cdots + 169491 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 345 \beta_{16} + 1579 \beta_{15} - 723 \beta_{14} + 250 \beta_{13} + 2655 \beta_{12} + 3491 \beta_{11} + \cdots + 163801 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 3718 \beta_{16} - 190 \beta_{15} - 34032 \beta_{14} + 50710 \beta_{13} - 44323 \beta_{12} + \cdots + 4568062 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 18009 \beta_{16} + 70427 \beta_{15} - 27372 \beta_{14} + 9395 \beta_{13} + 94656 \beta_{12} + \cdots + 5080318 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 153116 \beta_{16} + 18360 \beta_{15} - 970618 \beta_{14} + 1577410 \beta_{13} - 1347192 \beta_{12} + \cdots + 125942716 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 764378 \beta_{16} + 2660158 \beta_{15} - 879847 \beta_{14} + 342061 \beta_{13} + 3096891 \beta_{12} + \cdots + 154334378 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 5757958 \beta_{16} + 1212598 \beta_{15} - 27452667 \beta_{14} + 48227045 \beta_{13} + \cdots + 3524491166 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 29143851 \beta_{16} + 91765041 \beta_{15} - 25797947 \beta_{14} + 12824784 \beta_{13} + \cdots + 4638375275 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 204737235 \beta_{16} + 52327109 \beta_{15} - 772903021 \beta_{14} + 1460641142 \beta_{13} + \cdots + 99659560772 \)
|
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 |
|
−5.36576 | −0.836231 | 20.7914 | 1.92861 | 4.48702 | 17.6723 | −68.6355 | −26.3007 | −10.3485 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.61453 | 3.68758 | 13.2939 | 19.7910 | −17.0165 | −18.7869 | −24.4287 | −13.4017 | −91.3259 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.18927 | −8.20774 | 9.54998 | −18.8743 | 34.3844 | −14.8122 | −6.49330 | 40.3670 | 79.0697 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −3.21160 | −6.73941 | 2.31437 | 9.24473 | 21.6443 | 8.56428 | 18.2600 | 18.4196 | −29.6904 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −3.17483 | 2.79441 | 2.07954 | −8.68483 | −8.87177 | −6.73544 | 18.7965 | −19.1913 | 27.5728 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.47248 | 6.49330 | −1.88686 | −10.7576 | −16.0545 | 24.9760 | 24.4450 | 15.1629 | 26.5979 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.07101 | −5.99193 | −6.85294 | 0.973634 | 6.41740 | −28.3191 | 15.9076 | 8.90321 | −1.04277 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.549217 | −0.561969 | −7.69836 | 16.1600 | 0.308643 | 31.1335 | 8.62181 | −26.6842 | −8.87537 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.114461 | 5.80602 | −7.98690 | 11.2238 | −0.664561 | −15.9656 | 1.82987 | 6.70989 | −1.28469 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.75097 | 3.47555 | −4.93410 | −8.23074 | 6.08559 | −17.9650 | −22.6472 | −14.9205 | −14.4118 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.02345 | −7.47211 | −3.90564 | −6.22433 | −15.1195 | 28.5679 | −24.0905 | 28.8324 | −12.5946 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.74484 | −4.19208 | −0.465832 | 5.57379 | −11.5066 | −6.88148 | −23.2374 | −9.42645 | 15.2992 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 2.79502 | 10.1456 | −0.187860 | 5.45712 | 28.3572 | 7.73594 | −22.8852 | 75.9332 | 15.2528 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 3.98991 | 1.67407 | 7.91939 | −20.8270 | 6.67941 | 9.96761 | −0.321630 | −24.1975 | −83.0979 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 4.70666 | −10.1964 | 14.1527 | 19.7628 | −47.9909 | −2.86520 | 28.9586 | 76.9660 | 93.0167 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 5.30056 | −2.59422 | 20.0959 | −4.49343 | −13.7508 | −33.2644 | 64.1151 | −20.2700 | −23.8177 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 5.45173 | 6.71552 | 21.7213 | 3.97680 | 36.6112 | 10.9778 | 74.8051 | 18.0982 | 21.6804 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.i | 17 | |
| 13.b | even | 2 | 1 | 1859.4.a.f | 17 | ||
| 13.c | even | 3 | 2 | 143.4.e.a | ✓ | 34 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 143.4.e.a | ✓ | 34 | 13.c | even | 3 | 2 | |
| 1859.4.a.f | 17 | 13.b | even | 2 | 1 | ||
| 1859.4.a.i | 17 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{17} - 4 T_{2}^{16} - 99 T_{2}^{15} + 375 T_{2}^{14} + 3949 T_{2}^{13} - 13998 T_{2}^{12} + \cdots + 2596992 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1859))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{17} - 4 T^{16} + \cdots + 2596992 \)
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| $3$ |
\( T^{17} + \cdots + 19875034496 \)
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| $5$ |
\( T^{17} + \cdots + 12\!\cdots\!88 \)
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| $7$ |
\( T^{17} + \cdots + 28\!\cdots\!00 \)
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| $11$ |
\( (T + 11)^{17} \)
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| $13$ |
\( T^{17} \)
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| $17$ |
\( T^{17} + \cdots + 14\!\cdots\!36 \)
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| $19$ |
\( T^{17} + \cdots + 13\!\cdots\!00 \)
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| $23$ |
\( T^{17} + \cdots + 12\!\cdots\!64 \)
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| $29$ |
\( T^{17} + \cdots - 40\!\cdots\!57 \)
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| $31$ |
\( T^{17} + \cdots + 59\!\cdots\!24 \)
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| $37$ |
\( T^{17} + \cdots + 64\!\cdots\!74 \)
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| $41$ |
\( T^{17} + \cdots - 87\!\cdots\!50 \)
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| $43$ |
\( T^{17} + \cdots - 89\!\cdots\!56 \)
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| $47$ |
\( T^{17} + \cdots - 74\!\cdots\!68 \)
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| $53$ |
\( T^{17} + \cdots - 44\!\cdots\!54 \)
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| $59$ |
\( T^{17} + \cdots + 68\!\cdots\!32 \)
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| $61$ |
\( T^{17} + \cdots + 13\!\cdots\!50 \)
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| $67$ |
\( T^{17} + \cdots + 28\!\cdots\!28 \)
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| $71$ |
\( T^{17} + \cdots - 30\!\cdots\!48 \)
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| $73$ |
\( T^{17} + \cdots + 18\!\cdots\!00 \)
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| $79$ |
\( T^{17} + \cdots - 30\!\cdots\!88 \)
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| $83$ |
\( T^{17} + \cdots + 81\!\cdots\!36 \)
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| $89$ |
\( T^{17} + \cdots + 12\!\cdots\!04 \)
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| $97$ |
\( T^{17} + \cdots - 49\!\cdots\!76 \)
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