Newspace parameters
| Level: | \( N \) | \(=\) | \( 9065 = 5 \cdot 7^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9065.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(72.3843894323\) |
| Analytic rank: | \(0\) |
| Dimension: | \(15\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{15} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{15} - x^{14} - 26 x^{13} + 24 x^{12} + 266 x^{11} - 222 x^{10} - 1368 x^{9} + 998 x^{8} + 3770 x^{7} + \cdots - 158 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 1295) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{14}\) 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
\( x^{15} - x^{14} - 26 x^{13} + 24 x^{12} + 266 x^{11} - 222 x^{10} - 1368 x^{9} + 998 x^{8} + 3770 x^{7} + \cdots - 158 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 13885 \nu^{14} - 92473 \nu^{13} - 245409 \nu^{12} + 2219889 \nu^{11} + 984050 \nu^{10} + \cdots - 5632185 ) / 501919 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 17739 \nu^{14} + 1011 \nu^{13} - 456233 \nu^{12} - 62904 \nu^{11} + 4559762 \nu^{10} + \cdots + 3326114 ) / 45629 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 230988 \nu^{14} - 43885 \nu^{13} + 6018929 \nu^{12} + 1502450 \nu^{11} - 61212587 \nu^{10} + \cdots - 20523869 ) / 501919 \)
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| \(\beta_{6}\) | \(=\) |
\( ( - 239608 \nu^{14} + 13885 \nu^{13} + 6151220 \nu^{12} + 155219 \nu^{11} - 61360620 \nu^{10} + \cdots - 34735474 ) / 501919 \)
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| \(\beta_{7}\) | \(=\) |
\( ( - 260988 \nu^{14} - 79232 \nu^{13} + 6800753 \nu^{12} + 2450110 \nu^{11} - 69062089 \nu^{10} + \cdots - 43132127 ) / 501919 \)
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| \(\beta_{8}\) | \(=\) |
\( ( - 270767 \nu^{14} - 31745 \nu^{13} + 7049177 \nu^{12} + 1285716 \nu^{11} - 71564182 \nu^{10} + \cdots - 34293396 ) / 501919 \)
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| \(\beta_{9}\) | \(=\) |
\( ( 271358 \nu^{14} + 144821 \nu^{13} - 7179418 \nu^{12} - 4086020 \nu^{11} + 74327491 \nu^{10} + \cdots + 47143506 ) / 501919 \)
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| \(\beta_{10}\) | \(=\) |
\( ( 356335 \nu^{14} + 112755 \nu^{13} - 9250237 \nu^{12} - 3528268 \nu^{11} + 93562495 \nu^{10} + \cdots + 70982304 ) / 501919 \)
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| \(\beta_{11}\) | \(=\) |
\( ( - 49606 \nu^{14} - 13947 \nu^{13} + 1292261 \nu^{12} + 442628 \nu^{11} - 13126296 \nu^{10} + \cdots - 8215608 ) / 45629 \)
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| \(\beta_{12}\) | \(=\) |
\( ( 610987 \nu^{14} + 16680 \nu^{13} - 15762999 \nu^{12} - 1646435 \nu^{11} + 158261225 \nu^{10} + \cdots + 75027581 ) / 501919 \)
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| \(\beta_{13}\) | \(=\) |
\( ( - 707761 \nu^{14} - 22589 \nu^{13} + 18286613 \nu^{12} + 1987011 \nu^{11} - 183915222 \nu^{10} + \cdots - 90790934 ) / 501919 \)
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| \(\beta_{14}\) | \(=\) |
\( ( - 742461 \nu^{14} - 45070 \nu^{13} + 19170846 \nu^{12} + 2614680 \nu^{11} - 192616367 \nu^{10} + \cdots - 103800913 ) / 501919 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{12} - \beta_{10} + \beta_{8} - \beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} + 5\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( -\beta_{14} + \beta_{13} + \beta_{11} + \beta_{9} - \beta_{6} + 8\beta_{2} + 23 \)
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| \(\nu^{5}\) | \(=\) |
\( 10 \beta_{12} - \beta_{11} - 10 \beta_{10} - \beta_{9} + 9 \beta_{8} - 9 \beta_{7} + 2 \beta_{6} + \cdots + 1 \)
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| \(\nu^{6}\) | \(=\) |
\( - 15 \beta_{14} + 13 \beta_{13} - 2 \beta_{12} + 14 \beta_{11} + 3 \beta_{10} + 13 \beta_{9} + \cdots + 150 \)
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| \(\nu^{7}\) | \(=\) |
\( \beta_{14} + 83 \beta_{12} - 17 \beta_{11} - 85 \beta_{10} - 15 \beta_{9} + 70 \beta_{8} - 68 \beta_{7} + \cdots + 17 \)
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| \(\nu^{8}\) | \(=\) |
\( - 159 \beta_{14} + 126 \beta_{13} - 35 \beta_{12} + 144 \beta_{11} + 47 \beta_{10} + 128 \beta_{9} + \cdots + 1034 \)
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| \(\nu^{9}\) | \(=\) |
\( 21 \beta_{14} - 3 \beta_{13} + 658 \beta_{12} - 199 \beta_{11} - 693 \beta_{10} - 164 \beta_{9} + \cdots + 190 \)
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| \(\nu^{10}\) | \(=\) |
\( - 1474 \beta_{14} + 1100 \beta_{13} - 414 \beta_{12} + 1316 \beta_{11} + 521 \beta_{10} + 1136 \beta_{9} + \cdots + 7343 \)
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| \(\nu^{11}\) | \(=\) |
\( 284 \beta_{14} - 63 \beta_{13} + 5147 \beta_{12} - 1993 \beta_{11} - 5563 \beta_{10} - 1587 \beta_{9} + \cdots + 1779 \)
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| \(\nu^{12}\) | \(=\) |
\( - 12790 \beta_{14} + 9154 \beta_{13} - 4169 \beta_{12} + 11337 \beta_{11} + 5043 \beta_{10} + \cdots + 53162 \)
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| \(\nu^{13}\) | \(=\) |
\( 3162 \beta_{14} - 851 \beta_{13} + 40093 \beta_{12} - 18385 \beta_{11} - 44300 \beta_{10} - 14420 \beta_{9} + \cdots + 15099 \)
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| \(\nu^{14}\) | \(=\) |
\( - 106947 \beta_{14} + 74286 \beta_{13} - 38635 \beta_{12} + 94428 \beta_{11} + 45568 \beta_{10} + \cdots + 390290 \)
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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 |
|
−2.78012 | −2.49546 | 5.72907 | 1.00000 | 6.93769 | 0 | −10.3673 | 3.22733 | −2.78012 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.71485 | 1.40009 | 5.37043 | 1.00000 | −3.80103 | 0 | −9.15021 | −1.03976 | −2.71485 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.26763 | 2.33889 | 3.14213 | 1.00000 | −5.30373 | 0 | −2.58991 | 2.47041 | −2.26763 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.76603 | −1.25337 | 1.11887 | 1.00000 | 2.21348 | 0 | 1.55611 | −1.42907 | −1.76603 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.44547 | 3.38423 | 0.0893801 | 1.00000 | −4.89180 | 0 | 2.76174 | 8.45300 | −1.44547 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.11780 | −2.63196 | −0.750532 | 1.00000 | 2.94200 | 0 | 3.07453 | 3.92722 | −1.11780 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.11140 | 0.528459 | −0.764787 | 1.00000 | −0.587330 | 0 | 3.07279 | −2.72073 | −1.11140 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.217863 | 1.49352 | −1.95254 | 1.00000 | 0.325383 | 0 | −0.861110 | −0.769386 | 0.217863 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.347681 | 0.0759576 | −1.87912 | 1.00000 | 0.0264090 | 0 | −1.34870 | −2.99423 | 0.347681 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.19283 | 0.0386931 | −0.577162 | 1.00000 | 0.0461542 | 0 | −3.07411 | −2.99850 | 1.19283 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.28924 | −3.37140 | −0.337867 | 1.00000 | −4.34654 | 0 | −3.01407 | 8.36634 | 1.28924 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.54002 | 3.41172 | 0.371666 | 1.00000 | 5.25412 | 0 | −2.50767 | 8.63984 | 1.54002 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 2.37325 | −3.18420 | 3.63232 | 1.00000 | −7.55690 | 0 | 3.87391 | 7.13912 | 2.37325 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.43860 | 2.20221 | 3.94679 | 1.00000 | 5.37032 | 0 | 4.74745 | 1.84974 | 2.43860 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.80381 | −0.937381 | 5.86135 | 1.00000 | −2.62824 | 0 | 10.8265 | −2.12132 | 2.80381 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(7\) | \( -1 \) |
| \(37\) | \( -1 \) |
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 | 9065.2.a.o | 15 | |
| 7.b | odd | 2 | 1 | 1295.2.a.j | ✓ | 15 | |
| 35.c | odd | 2 | 1 | 6475.2.a.u | 15 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1295.2.a.j | ✓ | 15 | 7.b | odd | 2 | 1 | |
| 6475.2.a.u | 15 | 35.c | odd | 2 | 1 | ||
| 9065.2.a.o | 15 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9065))\):
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\( T_{2}^{15} + T_{2}^{14} - 26 T_{2}^{13} - 24 T_{2}^{12} + 266 T_{2}^{11} + 222 T_{2}^{10} - 1368 T_{2}^{9} + \cdots + 158 \)
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\( T_{3}^{15} - T_{3}^{14} - 37 T_{3}^{13} + 37 T_{3}^{12} + 522 T_{3}^{11} - 532 T_{3}^{10} - 3497 T_{3}^{9} + \cdots - 16 \)
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\( T_{11}^{15} - 17 T_{11}^{14} + 11 T_{11}^{13} + 1325 T_{11}^{12} - 6534 T_{11}^{11} - 27490 T_{11}^{10} + \cdots - 83533568 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{15} + T^{14} + \cdots + 158 \)
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| $3$ |
\( T^{15} - T^{14} + \cdots - 16 \)
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| $5$ |
\( (T - 1)^{15} \)
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| $7$ |
\( T^{15} \)
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| $11$ |
\( T^{15} - 17 T^{14} + \cdots - 83533568 \)
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| $13$ |
\( T^{15} + \cdots - 115659776 \)
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| $17$ |
\( T^{15} - 7 T^{14} + \cdots - 6575104 \)
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| $19$ |
\( T^{15} + \cdots - 115847552 \)
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| $23$ |
\( T^{15} + \cdots + 532537216 \)
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| $29$ |
\( T^{15} + \cdots + 127761408 \)
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| $31$ |
\( T^{15} + \cdots + 756503552 \)
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| $37$ |
\( (T - 1)^{15} \)
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| $41$ |
\( T^{15} + \cdots + 198148096 \)
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| $43$ |
\( T^{15} + \cdots + 214502062592 \)
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| $47$ |
\( T^{15} + \cdots + 4284762304 \)
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| $53$ |
\( T^{15} + \cdots - 1364707328 \)
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| $59$ |
\( T^{15} + \cdots + 1694452410496 \)
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| $61$ |
\( T^{15} + \cdots - 4722102304 \)
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| $67$ |
\( T^{15} + \cdots + 133074509824 \)
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| $71$ |
\( T^{15} + \cdots + 131794019584 \)
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| $73$ |
\( T^{15} + \cdots - 99835887104 \)
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| $79$ |
\( T^{15} + \cdots + 159723487232 \)
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| $83$ |
\( T^{15} - 6 T^{14} + \cdots + 62968832 \)
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| $89$ |
\( T^{15} + \cdots - 413093256704 \)
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| $97$ |
\( T^{15} + \cdots + 12\!\cdots\!12 \)
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