Newspace parameters
| Level: | \( N \) | \(=\) | \( 7105 = 5 \cdot 7^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7105.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(56.7337106361\) |
| Analytic rank: | \(1\) |
| Dimension: | \(17\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{17} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{17} - 6 x^{16} - 7 x^{15} + 101 x^{14} - 66 x^{13} - 640 x^{12} + 855 x^{11} + 1899 x^{10} + \cdots + 71 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1015) |
| 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
\( x^{17} - 6 x^{16} - 7 x^{15} + 101 x^{14} - 66 x^{13} - 640 x^{12} + 855 x^{11} + 1899 x^{10} + \cdots + 71 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( ( - 2680 \nu^{16} + 2635 \nu^{15} + 94919 \nu^{14} - 167508 \nu^{13} - 1067161 \nu^{12} + \cdots + 1411185 ) / 135123 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 7724 \nu^{16} + 65072 \nu^{15} - 2664 \nu^{14} - 1112137 \nu^{13} + 1490361 \nu^{12} + \cdots + 1108909 ) / 45041 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 36793 \nu^{16} - 128358 \nu^{15} - 617301 \nu^{14} + 2325823 \nu^{13} + 3874657 \nu^{12} + \cdots - 2197949 ) / 135123 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 59248 \nu^{16} + 298920 \nu^{15} + 711021 \nu^{14} - 5367946 \nu^{13} - 1290070 \nu^{12} + \cdots + 5617793 ) / 135123 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 100598 \nu^{16} + 496715 \nu^{15} + 1210684 \nu^{14} - 8820324 \nu^{13} - 2328698 \nu^{12} + \cdots + 8813550 ) / 135123 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 137491 \nu^{16} + 454755 \nu^{15} + 2348826 \nu^{14} - 8080870 \nu^{13} - 15464815 \nu^{12} + \cdots + 1915118 ) / 135123 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 139443 \nu^{16} + 596503 \nu^{15} + 1952627 \nu^{14} - 10558184 \nu^{13} - 8103249 \nu^{12} + \cdots + 7414798 ) / 135123 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 147122 \nu^{16} + 553550 \nu^{15} + 2321386 \nu^{14} - 9867549 \nu^{13} - 13164989 \nu^{12} + \cdots + 4984881 ) / 135123 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 152683 \nu^{16} + 756072 \nu^{15} + 1801065 \nu^{14} - 13318453 \nu^{13} - 2944654 \nu^{12} + \cdots + 10711262 ) / 135123 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 181719 \nu^{16} - 635044 \nu^{15} - 2984672 \nu^{14} + 11219294 \nu^{13} + 18396354 \nu^{12} + \cdots - 1446100 ) / 135123 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 190811 \nu^{16} + 831206 \nu^{15} + 2644720 \nu^{14} - 14762025 \nu^{13} - 10605701 \nu^{12} + \cdots + 10351488 ) / 135123 \)
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| \(\beta_{14}\) | \(=\) |
\( ( 70803 \nu^{16} - 342129 \nu^{15} - 874814 \nu^{14} + 6068992 \nu^{13} + 2043310 \nu^{12} + \cdots - 4956902 ) / 45041 \)
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| \(\beta_{15}\) | \(=\) |
\( ( 259145 \nu^{16} - 936543 \nu^{15} - 4203348 \nu^{14} + 16703207 \nu^{13} + 25196135 \nu^{12} + \cdots - 7338292 ) / 135123 \)
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| \(\beta_{16}\) | \(=\) |
\( ( - 402495 \nu^{16} + 1628566 \nu^{15} + 5963384 \nu^{14} - 28922096 \nu^{13} - 29177493 \nu^{12} + \cdots + 16301554 ) / 135123 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( -\beta_{15} + \beta_{12} - \beta_{7} + \beta_{4} + \beta_{2} + 4\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( -\beta_{15} + \beta_{13} + \beta_{12} - \beta_{11} - \beta_{9} - \beta_{6} - \beta_{5} + 2\beta_{4} + 8\beta_{2} + 14 \)
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| \(\nu^{5}\) | \(=\) |
\( - 3 \beta_{16} - 10 \beta_{15} - \beta_{14} + 2 \beta_{13} + 9 \beta_{12} + 2 \beta_{10} + \cdots + 19 \beta_1 \)
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| \(\nu^{6}\) | \(=\) |
\( - 2 \beta_{16} - 13 \beta_{15} - \beta_{14} + 11 \beta_{13} + 12 \beta_{12} - 10 \beta_{11} + 2 \beta_{10} + \cdots + 75 \)
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| \(\nu^{7}\) | \(=\) |
\( - 36 \beta_{16} - 82 \beta_{15} - 12 \beta_{14} + 25 \beta_{13} + 71 \beta_{12} - \beta_{11} + 24 \beta_{10} + \cdots + 2 \)
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| \(\nu^{8}\) | \(=\) |
\( - 33 \beta_{16} - 127 \beta_{15} - 12 \beta_{14} + 96 \beta_{13} + 113 \beta_{12} - 79 \beta_{11} + \cdots + 433 \)
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| \(\nu^{9}\) | \(=\) |
\( - 323 \beta_{16} - 635 \beta_{15} - 103 \beta_{14} + 234 \beta_{13} + 540 \beta_{12} - 23 \beta_{11} + \cdots + 46 \)
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| \(\nu^{10}\) | \(=\) |
\( - 378 \beta_{16} - 1119 \beta_{15} - 105 \beta_{14} + 781 \beta_{13} + 973 \beta_{12} - 583 \beta_{11} + \cdots + 2631 \)
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| \(\nu^{11}\) | \(=\) |
\( - 2631 \beta_{16} - 4822 \beta_{15} - 775 \beta_{14} + 1982 \beta_{13} + 4052 \beta_{12} - 313 \beta_{11} + \cdots + 639 \)
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| \(\nu^{12}\) | \(=\) |
\( - 3727 \beta_{16} - 9379 \beta_{15} - 816 \beta_{14} + 6173 \beta_{13} + 8011 \beta_{12} - 4216 \beta_{11} + \cdots + 16613 \)
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| \(\nu^{13}\) | \(=\) |
\( - 20619 \beta_{16} - 36397 \beta_{15} - 5448 \beta_{14} + 16062 \beta_{13} + 30281 \beta_{12} + \cdots + 7093 \)
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| \(\nu^{14}\) | \(=\) |
\( - 33946 \beta_{16} - 76448 \beta_{15} - 5956 \beta_{14} + 48135 \beta_{13} + 64313 \beta_{12} + \cdots + 108119 \)
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| \(\nu^{15}\) | \(=\) |
\( - 159112 \beta_{16} - 274601 \beta_{15} - 36751 \beta_{14} + 127348 \beta_{13} + 226367 \beta_{12} + \cdots + 69534 \)
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| \(\nu^{16}\) | \(=\) |
\( - 294836 \beta_{16} - 612625 \beta_{15} - 41654 \beta_{14} + 372860 \beta_{13} + 508557 \beta_{12} + \cdots + 721086 \)
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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 |
|
−2.77892 | −0.635503 | 5.72242 | −1.00000 | 1.76602 | 0 | −10.3443 | −2.59614 | 2.77892 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.64280 | 3.33770 | 4.98441 | −1.00000 | −8.82088 | 0 | −7.88721 | 8.14023 | 2.64280 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.11360 | −2.37828 | 2.46729 | −1.00000 | 5.02673 | 0 | −0.987663 | 2.65623 | 2.11360 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.07006 | −1.28200 | 2.28515 | −1.00000 | 2.65382 | 0 | −0.590283 | −1.35648 | 2.07006 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.04611 | −0.740192 | 2.18655 | −1.00000 | 1.51451 | 0 | −0.381698 | −2.45212 | 2.04611 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.54043 | 1.30845 | 0.372935 | −1.00000 | −2.01558 | 0 | 2.50639 | −1.28796 | 1.54043 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.05626 | 2.11431 | −0.884310 | −1.00000 | −2.23327 | 0 | 3.04659 | 1.47030 | 1.05626 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.859555 | −0.574420 | −1.26117 | −1.00000 | 0.493746 | 0 | 2.80315 | −2.67004 | 0.859555 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.806778 | 3.14968 | −1.34911 | −1.00000 | −2.54109 | 0 | 2.70199 | 6.92047 | 0.806778 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −0.167300 | 0.381104 | −1.97201 | −1.00000 | −0.0637585 | 0 | 0.664516 | −2.85476 | 0.167300 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0.613079 | −0.184450 | −1.62413 | −1.00000 | −0.113083 | 0 | −2.22188 | −2.96598 | −0.613079 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0.697926 | 2.33494 | −1.51290 | −1.00000 | 1.62961 | 0 | −2.45174 | 2.45194 | −0.697926 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.00910 | −1.95675 | −0.981708 | −1.00000 | −1.97456 | 0 | −3.00886 | 0.828862 | −1.00910 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 1.48067 | 1.26684 | 0.192376 | −1.00000 | 1.87577 | 0 | −2.67649 | −1.39511 | −1.48067 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 1.78176 | −2.41987 | 1.17466 | −1.00000 | −4.31162 | 0 | −1.47056 | 2.85577 | −1.78176 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.05246 | 2.30280 | 2.21260 | −1.00000 | 4.72642 | 0 | 0.436351 | 2.30290 | −2.05246 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 2.44682 | 0.975640 | 3.98695 | −1.00000 | 2.38722 | 0 | 4.86171 | −2.04813 | −2.44682 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(7\) | \( -1 \) |
| \(29\) | \( -1 \) |
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 | 7105.2.a.ba | 17 | |
| 7.b | odd | 2 | 1 | 7105.2.a.z | 17 | ||
| 7.d | odd | 6 | 2 | 1015.2.i.c | ✓ | 34 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1015.2.i.c | ✓ | 34 | 7.d | odd | 6 | 2 | |
| 7105.2.a.z | 17 | 7.b | odd | 2 | 1 | ||
| 7105.2.a.ba | 17 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7105))\):
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\( T_{2}^{17} + 6 T_{2}^{16} - 7 T_{2}^{15} - 101 T_{2}^{14} - 66 T_{2}^{13} + 640 T_{2}^{12} + 855 T_{2}^{11} + \cdots - 71 \)
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\( T_{3}^{17} - 7 T_{3}^{16} - 5 T_{3}^{15} + 126 T_{3}^{14} - 122 T_{3}^{13} - 842 T_{3}^{12} + 1339 T_{3}^{11} + \cdots - 53 \)
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\( T_{17}^{17} - 10 T_{17}^{16} - 100 T_{17}^{15} + 1239 T_{17}^{14} + 3216 T_{17}^{13} - 61485 T_{17}^{12} + \cdots - 1709299327 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{17} + 6 T^{16} + \cdots - 71 \)
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| $3$ |
\( T^{17} - 7 T^{16} + \cdots - 53 \)
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| $5$ |
\( (T + 1)^{17} \)
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| $7$ |
\( T^{17} \)
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| $11$ |
\( T^{17} + 17 T^{16} + \cdots - 767233 \)
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| $13$ |
\( T^{17} - T^{16} + \cdots - 510995 \)
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| $17$ |
\( T^{17} + \cdots - 1709299327 \)
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| $19$ |
\( T^{17} + \cdots + 289832845 \)
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| $23$ |
\( T^{17} + 33 T^{16} + \cdots + 32039705 \)
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| $29$ |
\( (T - 1)^{17} \)
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| $31$ |
\( T^{17} + \cdots + 143537475385 \)
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| $37$ |
\( T^{17} + \cdots + 137735368245 \)
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| $41$ |
\( T^{17} + \cdots - 305747778159 \)
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| $43$ |
\( T^{17} + \cdots + 635753133 \)
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| $47$ |
\( T^{17} - 18 T^{16} + \cdots + 82348151 \)
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| $53$ |
\( T^{17} + \cdots + 2903542098461 \)
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| $59$ |
\( T^{17} + \cdots + 634129387425 \)
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| $61$ |
\( T^{17} + \cdots - 102946247333 \)
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| $67$ |
\( T^{17} + \cdots + 853762351131 \)
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| $71$ |
\( T^{17} + \cdots + 97088231499709 \)
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| $73$ |
\( T^{17} + \cdots + 129077800774155 \)
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| $79$ |
\( T^{17} + \cdots - 20\!\cdots\!27 \)
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| $83$ |
\( T^{17} + \cdots - 4587660101581 \)
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| $89$ |
\( T^{17} + \cdots + 173002933715 \)
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| $97$ |
\( T^{17} + \cdots + 85948078849 \)
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