Newspace parameters
| Level: | \( N \) | \(=\) | \( 5002 = 2 \cdot 41 \cdot 61 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5002.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(39.9411710910\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{20} - 9 x^{19} - x^{18} + 205 x^{17} - 305 x^{16} - 1919 x^{15} + 4028 x^{14} + 9460 x^{13} + \cdots - 45 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 9 x^{19} - x^{18} + 205 x^{17} - 305 x^{16} - 1919 x^{15} + 4028 x^{14} + 9460 x^{13} + \cdots - 45 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 16\!\cdots\!43 \nu^{19} + \cdots - 47\!\cdots\!59 ) / 22\!\cdots\!24 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 38\!\cdots\!09 \nu^{19} + \cdots - 27\!\cdots\!31 ) / 44\!\cdots\!48 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 83\!\cdots\!75 \nu^{19} + \cdots + 34\!\cdots\!29 ) / 44\!\cdots\!48 \)
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| \(\beta_{5}\) | \(=\) |
\( ( - 75\!\cdots\!36 \nu^{19} + \cdots + 64\!\cdots\!49 ) / 22\!\cdots\!24 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 78\!\cdots\!26 \nu^{19} + \cdots + 56\!\cdots\!75 ) / 11\!\cdots\!12 \)
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| \(\beta_{7}\) | \(=\) |
\( ( - 26\!\cdots\!13 \nu^{19} + \cdots - 25\!\cdots\!25 ) / 37\!\cdots\!04 \)
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| \(\beta_{8}\) | \(=\) |
\( ( - 37\!\cdots\!85 \nu^{19} + \cdots - 11\!\cdots\!41 ) / 49\!\cdots\!72 \)
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| \(\beta_{9}\) | \(=\) |
\( ( - 16\!\cdots\!55 \nu^{19} + \cdots - 40\!\cdots\!27 ) / 11\!\cdots\!12 \)
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| \(\beta_{10}\) | \(=\) |
\( ( - 78\!\cdots\!69 \nu^{19} + \cdots - 39\!\cdots\!31 ) / 44\!\cdots\!48 \)
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| \(\beta_{11}\) | \(=\) |
\( ( - 22\!\cdots\!53 \nu^{19} + \cdots - 73\!\cdots\!94 ) / 11\!\cdots\!12 \)
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| \(\beta_{12}\) | \(=\) |
\( ( - 10\!\cdots\!61 \nu^{19} + \cdots - 56\!\cdots\!63 ) / 44\!\cdots\!48 \)
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| \(\beta_{13}\) | \(=\) |
\( ( - 11\!\cdots\!83 \nu^{19} + \cdots - 33\!\cdots\!63 ) / 44\!\cdots\!48 \)
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| \(\beta_{14}\) | \(=\) |
\( ( 40\!\cdots\!81 \nu^{19} + \cdots - 71\!\cdots\!07 ) / 14\!\cdots\!16 \)
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| \(\beta_{15}\) | \(=\) |
\( ( - 10\!\cdots\!65 \nu^{19} + \cdots - 20\!\cdots\!18 ) / 37\!\cdots\!04 \)
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| \(\beta_{16}\) | \(=\) |
\( ( - 38\!\cdots\!57 \nu^{19} + \cdots - 80\!\cdots\!20 ) / 12\!\cdots\!68 \)
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| \(\beta_{17}\) | \(=\) |
\( ( 72\!\cdots\!26 \nu^{19} + \cdots - 32\!\cdots\!37 ) / 22\!\cdots\!24 \)
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| \(\beta_{18}\) | \(=\) |
\( ( 74\!\cdots\!86 \nu^{19} + \cdots - 34\!\cdots\!23 ) / 22\!\cdots\!24 \)
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| \(\beta_{19}\) | \(=\) |
\( ( 15\!\cdots\!17 \nu^{19} + \cdots + 14\!\cdots\!93 ) / 44\!\cdots\!48 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{17} + \beta_{15} + \beta_{5} + \beta _1 + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( - \beta_{19} + 2 \beta_{17} + \beta_{16} + \beta_{15} - \beta_{12} - \beta_{9} + 3 \beta_{5} - \beta_{4} + \cdots + 4 \)
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| \(\nu^{4}\) | \(=\) |
\( - 6 \beta_{19} + 2 \beta_{18} + 9 \beta_{17} + 4 \beta_{16} + 6 \beta_{15} + 2 \beta_{14} - 5 \beta_{12} + \cdots + 32 \)
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| \(\nu^{5}\) | \(=\) |
\( - 26 \beta_{19} + 5 \beta_{18} + 29 \beta_{17} + 17 \beta_{16} + 10 \beta_{15} + 5 \beta_{14} + \cdots + 77 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 124 \beta_{19} + 35 \beta_{18} + 109 \beta_{17} + 72 \beta_{16} + 32 \beta_{15} + 38 \beta_{14} + \cdots + 370 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 497 \beta_{19} + 110 \beta_{18} + 418 \beta_{17} + 275 \beta_{16} + 45 \beta_{15} + 126 \beta_{14} + \cdots + 1227 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 2148 \beta_{19} + 529 \beta_{18} + 1585 \beta_{17} + 1145 \beta_{16} - 10 \beta_{15} + 634 \beta_{14} + \cdots + 5109 \)
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| \(\nu^{9}\) | \(=\) |
\( - 8584 \beta_{19} + 1860 \beta_{18} + 6332 \beta_{17} + 4424 \beta_{16} - 735 \beta_{15} + 2364 \beta_{14} + \cdots + 18948 \)
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| \(\nu^{10}\) | \(=\) |
\( - 35489 \beta_{19} + 7823 \beta_{18} + 24648 \beta_{17} + 18006 \beta_{16} - 5306 \beta_{15} + \cdots + 75765 \)
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| \(\nu^{11}\) | \(=\) |
\( - 141892 \beta_{19} + 28957 \beta_{18} + 98937 \beta_{17} + 70580 \beta_{16} - 28377 \beta_{15} + \cdots + 292109 \)
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| \(\nu^{12}\) | \(=\) |
\( - 574753 \beta_{19} + 115534 \beta_{18} + 390836 \beta_{17} + 283333 \beta_{16} - 139873 \beta_{15} + \cdots + 1155331 \)
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| \(\nu^{13}\) | \(=\) |
\( - 2295172 \beta_{19} + 437756 \beta_{18} + 1565129 \beta_{17} + 1118564 \beta_{16} - 638577 \beta_{15} + \cdots + 4516982 \)
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| \(\nu^{14}\) | \(=\) |
\( - 9204987 \beta_{19} + 1711644 \beta_{18} + 6217035 \beta_{17} + 4459116 \beta_{16} - 2841341 \beta_{15} + \cdots + 17819346 \)
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| \(\nu^{15}\) | \(=\) |
\( - 36686758 \beta_{19} + 6558949 \beta_{18} + 24821626 \beta_{17} + 17650968 \beta_{16} + \cdots + 70041765 \)
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| \(\nu^{16}\) | \(=\) |
\( - 146367546 \beta_{19} + 25476008 \beta_{18} + 98701609 \beta_{17} + 70126009 \beta_{16} + \cdots + 276223946 \)
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| \(\nu^{17}\) | \(=\) |
\( - 582239163 \beta_{19} + 98238183 \beta_{18} + 393065887 \beta_{17} + 277776491 \beta_{16} + \cdots + 1088165791 \)
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| \(\nu^{18}\) | \(=\) |
\( - 2315937299 \beta_{19} + 381191485 \beta_{18} + 1562003724 \beta_{17} + 1101676243 \beta_{16} + \cdots + 4292349140 \)
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| \(\nu^{19}\) | \(=\) |
\( - 9197992431 \beta_{19} + 1476119021 \beta_{18} + 6208084098 \beta_{17} + 4363604443 \beta_{16} + \cdots + 16927036492 \)
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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 |
|
−1.00000 | −2.95254 | 1.00000 | 0.567950 | 2.95254 | −0.536131 | −1.00000 | 5.71747 | −0.567950 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.49728 | 1.00000 | −1.18967 | 2.49728 | 1.90395 | −1.00000 | 3.23642 | 1.18967 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −2.40221 | 1.00000 | 2.77621 | 2.40221 | −1.26994 | −1.00000 | 2.77063 | −2.77621 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −1.83747 | 1.00000 | 3.44644 | 1.83747 | 3.34177 | −1.00000 | 0.376288 | −3.44644 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | −1.31508 | 1.00000 | −3.45676 | 1.31508 | 1.60623 | −1.00000 | −1.27057 | 3.45676 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | −0.553960 | 1.00000 | −2.58493 | 0.553960 | 2.53063 | −1.00000 | −2.69313 | 2.58493 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | −0.320175 | 1.00000 | 0.801058 | 0.320175 | −1.32458 | −1.00000 | −2.89749 | −0.801058 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | −0.0709782 | 1.00000 | 3.26241 | 0.0709782 | −1.37639 | −1.00000 | −2.99496 | −3.26241 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.00000 | 0.176460 | 1.00000 | 3.59938 | −0.176460 | 2.42930 | −1.00000 | −2.96886 | −3.59938 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −1.00000 | 0.375367 | 1.00000 | 1.22435 | −0.375367 | 5.01127 | −1.00000 | −2.85910 | −1.22435 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −1.00000 | 0.532632 | 1.00000 | −0.662168 | −0.532632 | −1.94148 | −1.00000 | −2.71630 | 0.662168 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | −1.00000 | 1.01391 | 1.00000 | −2.19014 | −1.01391 | −1.21109 | −1.00000 | −1.97198 | 2.19014 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | −1.00000 | 1.48671 | 1.00000 | −2.46211 | −1.48671 | 2.44897 | −1.00000 | −0.789685 | 2.46211 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | −1.00000 | 1.97601 | 1.00000 | 2.99900 | −1.97601 | −4.16691 | −1.00000 | 0.904619 | −2.99900 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | −1.00000 | 2.30243 | 1.00000 | −2.87294 | −2.30243 | −4.36976 | −1.00000 | 2.30120 | 2.87294 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | −1.00000 | 2.63542 | 1.00000 | 1.99693 | −2.63542 | 4.42425 | −1.00000 | 3.94542 | −1.99693 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | −1.00000 | 2.88262 | 1.00000 | 0.0528086 | −2.88262 | 2.97329 | −1.00000 | 5.30951 | −0.0528086 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | −1.00000 | 3.01613 | 1.00000 | 4.03782 | −3.01613 | 1.17454 | −1.00000 | 6.09704 | −4.03782 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.19 | −1.00000 | 3.13601 | 1.00000 | 1.31161 | −3.13601 | 0.413239 | −1.00000 | 6.83458 | −1.31161 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.20 | −1.00000 | 3.41598 | 1.00000 | −1.65726 | −3.41598 | −0.0611560 | −1.00000 | 8.66891 | 1.65726 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(41\) | \( -1 \) |
| \(61\) | \( +1 \) |
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 | 5002.2.a.k | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5002.2.a.k | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5002))\):
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\( T_{3}^{20} - 11 T_{3}^{19} + 18 T_{3}^{18} + 212 T_{3}^{17} - 849 T_{3}^{16} - 885 T_{3}^{15} + \cdots + 32 \)
|
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\( T_{7}^{20} - 12 T_{7}^{19} + 3 T_{7}^{18} + 486 T_{7}^{17} - 1586 T_{7}^{16} - 5389 T_{7}^{15} + \cdots - 16000 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{20} \)
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| $3$ |
\( T^{20} - 11 T^{19} + \cdots + 32 \)
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| $5$ |
\( T^{20} - 9 T^{19} + \cdots + 18944 \)
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| $7$ |
\( T^{20} - 12 T^{19} + \cdots - 16000 \)
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| $11$ |
\( T^{20} - 18 T^{19} + \cdots - 135672 \)
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| $13$ |
\( T^{20} - 8 T^{19} + \cdots - 141696 \)
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| $17$ |
\( T^{20} - 25 T^{19} + \cdots + 5299200 \)
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| $19$ |
\( T^{20} - 18 T^{19} + \cdots + 10688160 \)
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| $23$ |
\( T^{20} + \cdots + 1616473600 \)
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| $29$ |
\( T^{20} + \cdots - 285585868800 \)
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| $31$ |
\( T^{20} - T^{19} + \cdots + 16542720 \)
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| $37$ |
\( T^{20} + \cdots - 49096259616 \)
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| $41$ |
\( (T - 1)^{20} \)
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| $43$ |
\( T^{20} + \cdots - 1254737078784 \)
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| $47$ |
\( T^{20} + \cdots - 291141817128 \)
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| $53$ |
\( T^{20} + \cdots - 947442087936 \)
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| $59$ |
\( T^{20} + \cdots - 677485775523840 \)
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| $61$ |
\( (T + 1)^{20} \)
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| $67$ |
\( T^{20} + \cdots - 66\!\cdots\!04 \)
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| $71$ |
\( T^{20} + \cdots + 73478473279520 \)
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| $73$ |
\( T^{20} + \cdots - 358993207274976 \)
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| $79$ |
\( T^{20} + \cdots + 1410236544160 \)
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| $83$ |
\( T^{20} + \cdots - 43275345931392 \)
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| $89$ |
\( T^{20} + \cdots + 19\!\cdots\!04 \)
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| $97$ |
\( T^{20} + \cdots - 10\!\cdots\!96 \)
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