Newspace parameters
| Level: | \( N \) | \(=\) | \( 71 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 71.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(22.1793368094\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{17} - x^{16} - 1541 x^{15} + 843 x^{14} + 955403 x^{13} - 21197 x^{12} - 306314257 x^{11} - 193726721 x^{10} + 54415399577 x^{9} + 78353550093 x^{8} + \cdots - 12\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | multiple of \( 2^{14}\cdot 3 \) |
| Twist minimal: | yes |
| 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
\( x^{17} - x^{16} - 1541 x^{15} + 843 x^{14} + 955403 x^{13} - 21197 x^{12} - 306314257 x^{11} - 193726721 x^{10} + 54415399577 x^{9} + 78353550093 x^{8} + \cdots - 12\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 181 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 61\!\cdots\!97 \nu^{16} + \cdots - 10\!\cdots\!00 ) / 10\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 10\!\cdots\!55 \nu^{16} + \cdots + 21\!\cdots\!00 ) / 32\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 10\!\cdots\!47 \nu^{16} + \cdots + 52\!\cdots\!00 ) / 10\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 19\!\cdots\!83 \nu^{16} + \cdots - 34\!\cdots\!00 ) / 15\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 62\!\cdots\!23 \nu^{16} + \cdots + 34\!\cdots\!00 ) / 27\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 10\!\cdots\!85 \nu^{16} + \cdots + 74\!\cdots\!00 ) / 32\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 10\!\cdots\!09 \nu^{16} + \cdots - 29\!\cdots\!00 ) / 16\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 25\!\cdots\!87 \nu^{16} + \cdots + 30\!\cdots\!00 ) / 32\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 93\!\cdots\!77 \nu^{16} + \cdots + 52\!\cdots\!00 ) / 10\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 18\!\cdots\!29 \nu^{16} + \cdots - 44\!\cdots\!00 ) / 13\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 11\!\cdots\!59 \nu^{16} + \cdots + 20\!\cdots\!00 ) / 81\!\cdots\!00 \)
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| \(\beta_{14}\) | \(=\) |
\( ( 77\!\cdots\!51 \nu^{16} + \cdots - 40\!\cdots\!00 ) / 54\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 18\!\cdots\!93 \nu^{16} + \cdots - 19\!\cdots\!00 ) / 10\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 95\!\cdots\!67 \nu^{16} + \cdots + 49\!\cdots\!00 ) / 32\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 181 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{13} - 2\beta_{9} - 2\beta_{6} - 9\beta_{4} + \beta_{3} - \beta_{2} + 298\beta _1 + 107 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 10 \beta_{16} - 14 \beta_{15} - 18 \beta_{14} - 7 \beta_{13} + 9 \beta_{12} + 6 \beta_{11} - 9 \beta_{10} + 6 \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{6} - 6 \beta_{5} - 76 \beta_{4} - 12 \beta_{3} + 407 \beta_{2} + 322 \beta _1 + 54597 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 74 \beta_{16} - 48 \beta_{15} - 132 \beta_{14} + 384 \beta_{13} - 63 \beta_{12} - 34 \beta_{11} + 53 \beta_{10} - 978 \beta_{9} + 14 \beta_{8} - 41 \beta_{7} - 887 \beta_{6} - 66 \beta_{5} - 4889 \beta_{4} + 399 \beta_{3} + \cdots + 36155 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 6744 \beta_{16} - 8788 \beta_{15} - 11000 \beta_{14} - 4766 \beta_{13} + 3960 \beta_{12} + 3308 \beta_{11} - 3900 \beta_{10} + 1900 \beta_{9} + 1820 \beta_{8} - 2028 \beta_{7} + 1652 \beta_{6} + \cdots + 19079067 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 60732 \beta_{16} - 48472 \beta_{15} - 116788 \beta_{14} + 107475 \beta_{13} - 38102 \beta_{12} - 20232 \beta_{11} + 39626 \beta_{10} - 406110 \beta_{9} + 6048 \beta_{8} - 44510 \beta_{7} + \cdots + 19137267 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 3480150 \beta_{16} - 4378130 \beta_{15} - 5511506 \beta_{14} - 2668585 \beta_{13} + 1426163 \beta_{12} + 1502142 \beta_{11} - 1258555 \beta_{10} + 183638 \beta_{9} + \cdots + 7111493541 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 35981134 \beta_{16} - 32280168 \beta_{15} - 72025600 \beta_{14} + 21229810 \beta_{13} - 17614153 \beta_{12} - 9082682 \beta_{11} + 22869883 \beta_{10} - 164106158 \beta_{9} + \cdots + 12378804367 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 1647390836 \beta_{16} - 2034393880 \beta_{15} - 2603137040 \beta_{14} - 1372646840 \beta_{13} + 487301810 \beta_{12} + 646594828 \beta_{11} + \cdots + 2748005901131 \)
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| \(\nu^{11}\) | \(=\) |
\( - 18972461216 \beta_{16} - 18303169520 \beta_{15} - 38823593336 \beta_{14} - 312135843 \beta_{13} - 7483135720 \beta_{12} - 3674619672 \beta_{11} + \cdots + 7774521265423 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 754044812722 \beta_{16} - 920934222582 \beta_{15} - 1201058686306 \beta_{14} - 671600384843 \beta_{13} + 163134185909 \beta_{12} + \cdots + 10\!\cdots\!45 \)
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| \(\nu^{13}\) | \(=\) |
\( - 9485799283698 \beta_{16} - 9594043142912 \beta_{15} - 19615907273772 \beta_{14} - 3595823865868 \beta_{13} - 3078459363315 \beta_{12} + \cdots + 45\!\cdots\!71 \)
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| \(\nu^{14}\) | \(=\) |
\( - 340425300199504 \beta_{16} - 412559599016636 \beta_{15} - 548537597080696 \beta_{14} - 318491113957730 \beta_{13} + 53818546878684 \beta_{12} + \cdots + 44\!\cdots\!19 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 46\!\cdots\!64 \beta_{16} + \cdots + 25\!\cdots\!59 \)
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| \(\nu^{16}\) | \(=\) |
\( - 15\!\cdots\!86 \beta_{16} + \cdots + 18\!\cdots\!01 \)
|
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 |
|
−20.5769 | −24.8746 | 295.408 | 51.8138 | 511.841 | −964.382 | −3444.74 | −1568.26 | −1066.17 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −20.1512 | 50.2064 | 278.072 | 480.331 | −1011.72 | −492.743 | −3024.14 | 333.680 | −9679.26 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −19.8483 | 14.7419 | 265.954 | −39.2067 | −292.602 | 290.193 | −2738.15 | −1969.68 | 778.186 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −13.0476 | −71.6658 | 42.2389 | −299.002 | 935.065 | −838.975 | 1118.97 | 2948.99 | 3901.25 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −10.6733 | 55.6664 | −14.0807 | −123.787 | −594.144 | 478.240 | 1516.47 | 911.744 | 1321.21 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −9.80500 | −22.7314 | −31.8619 | −426.903 | 222.881 | 1439.06 | 1567.45 | −1670.28 | 4185.78 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −8.56494 | 16.9829 | −54.6417 | 344.458 | −145.458 | −965.607 | 1564.32 | −1898.58 | −2950.26 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −2.93898 | −48.7576 | −119.362 | 213.468 | 143.298 | 152.324 | 726.993 | 190.307 | −627.378 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −2.62329 | −87.0131 | −121.118 | −27.1608 | 228.261 | 757.218 | 653.510 | 5384.28 | 71.2507 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −1.77978 | 72.7305 | −124.832 | −243.222 | −129.444 | 397.740 | 449.987 | 3102.73 | 432.883 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 7.72947 | 35.5389 | −68.2553 | 193.986 | 274.697 | −930.389 | −1516.95 | −923.987 | 1499.41 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 7.77673 | −0.574168 | −67.5225 | 74.6631 | −4.46514 | 1081.19 | −1520.53 | −2186.67 | 580.635 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 10.9708 | 80.3704 | −7.64129 | −472.591 | 881.729 | −1084.98 | −1488.10 | 4272.41 | −5184.70 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 13.8364 | −72.7594 | 63.4464 | 226.462 | −1006.73 | 1016.68 | −893.191 | 3106.93 | 3133.42 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 16.4632 | 9.09105 | 143.038 | −425.151 | 149.668 | 306.368 | 247.579 | −2104.35 | −6999.36 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 16.8807 | −7.07215 | 156.956 | −52.2973 | −119.382 | −940.822 | 488.802 | −2136.98 | −882.813 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 20.3520 | −67.8802 | 286.202 | 94.1391 | −1381.49 | −1417.12 | 3219.72 | 2420.72 | 1915.91 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(71\) | \(-1\) |
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 | 71.8.a.a | ✓ | 17 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 71.8.a.a | ✓ | 17 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{17} + 16 T_{2}^{16} - 1421 T_{2}^{15} - 21712 T_{2}^{14} + 807220 T_{2}^{13} + 11778968 T_{2}^{12} - 233835792 T_{2}^{11} - 3295109632 T_{2}^{10} + 36303285888 T_{2}^{9} + \cdots - 68\!\cdots\!16 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(71))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{17} + 16 T^{16} + \cdots - 68\!\cdots\!16 \)
$3$
\( T^{17} + 68 T^{16} + \cdots + 45\!\cdots\!00 \)
$5$
\( T^{17} + 430 T^{16} + \cdots + 24\!\cdots\!00 \)
$7$
\( T^{17} + 1716 T^{16} + \cdots - 15\!\cdots\!76 \)
$11$
\( T^{17} + 7550 T^{16} + \cdots + 27\!\cdots\!48 \)
$13$
\( T^{17} + 17280 T^{16} + \cdots + 32\!\cdots\!48 \)
$17$
\( T^{17} + 55074 T^{16} + \cdots - 34\!\cdots\!56 \)
$19$
\( T^{17} + 56892 T^{16} + \cdots + 11\!\cdots\!00 \)
$23$
\( T^{17} + 57160 T^{16} + \cdots - 28\!\cdots\!00 \)
$29$
\( T^{17} + 376658 T^{16} + \cdots + 69\!\cdots\!80 \)
$31$
\( T^{17} + 637476 T^{16} + \cdots + 78\!\cdots\!68 \)
$37$
\( T^{17} + 1196566 T^{16} + \cdots + 63\!\cdots\!24 \)
$41$
\( T^{17} + 1620042 T^{16} + \cdots - 64\!\cdots\!36 \)
$43$
\( T^{17} + 1545900 T^{16} + \cdots - 72\!\cdots\!40 \)
$47$
\( T^{17} - 368904 T^{16} + \cdots - 21\!\cdots\!60 \)
$53$
\( T^{17} - 981948 T^{16} + \cdots - 10\!\cdots\!12 \)
$59$
\( T^{17} - 1926882 T^{16} + \cdots + 37\!\cdots\!00 \)
$61$
\( T^{17} + 7380164 T^{16} + \cdots - 18\!\cdots\!60 \)
$67$
\( T^{17} - 1751278 T^{16} + \cdots + 15\!\cdots\!00 \)
$71$
\( (T - 357911)^{17} \)
$73$
\( T^{17} + 5262378 T^{16} + \cdots + 20\!\cdots\!06 \)
$79$
\( T^{17} + 514148 T^{16} + \cdots + 29\!\cdots\!00 \)
$83$
\( T^{17} - 18344544 T^{16} + \cdots + 13\!\cdots\!24 \)
$89$
\( T^{17} + 7940378 T^{16} + \cdots - 12\!\cdots\!10 \)
$97$
\( T^{17} + 41277242 T^{16} + \cdots + 25\!\cdots\!80 \)
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