Newspace parameters
| Level: | \( N \) | \(=\) | \( 2523 = 3 \cdot 29^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2523.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(20.1462564300\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
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| Defining polynomial: |
\( x^{12} - x^{11} - 17 x^{10} + 13 x^{9} + 105 x^{8} - 49 x^{7} - 292 x^{6} + 44 x^{5} + 355 x^{4} + \cdots + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - x^{11} - 17 x^{10} + 13 x^{9} + 105 x^{8} - 49 x^{7} - 292 x^{6} + 44 x^{5} + 355 x^{4} + \cdots + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( - 485 \nu^{11} + 834 \nu^{10} + 7634 \nu^{9} - 12224 \nu^{8} - 41950 \nu^{7} + 59596 \nu^{6} + \cdots - 15412 ) / 1916 \)
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| \(\beta_{3}\) | \(=\) |
\( ( - 292 \nu^{11} + 352 \nu^{10} + 4767 \nu^{9} - 4933 \nu^{8} - 27691 \nu^{7} + 22347 \nu^{6} + \cdots - 6004 ) / 958 \)
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| \(\beta_{4}\) | \(=\) |
\( ( - 1381 \nu^{11} + 2275 \nu^{10} + 21835 \nu^{9} - 32643 \nu^{8} - 121795 \nu^{7} + 154447 \nu^{6} + \cdots - 45200 ) / 3832 \)
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| \(\beta_{5}\) | \(=\) |
\( ( - 1561 \nu^{11} + 2387 \nu^{10} + 24767 \nu^{9} - 33407 \nu^{8} - 138727 \nu^{7} + 150867 \nu^{6} + \cdots - 32392 ) / 3832 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 921 \nu^{11} - 1563 \nu^{10} - 14555 \nu^{9} + 22175 \nu^{8} + 81079 \nu^{7} - 102071 \nu^{6} + \cdots + 19600 ) / 1916 \)
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| \(\beta_{7}\) | \(=\) |
\( ( - 1031 \nu^{11} + 1525 \nu^{10} + 16879 \nu^{9} - 21471 \nu^{8} - 98771 \nu^{7} + 97435 \nu^{6} + \cdots - 23056 ) / 1916 \)
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| \(\beta_{8}\) | \(=\) |
\( ( - 1225 \nu^{11} + 2146 \nu^{10} + 19262 \nu^{9} - 30480 \nu^{8} - 106450 \nu^{7} + 141104 \nu^{6} + \cdots - 29604 ) / 1916 \)
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| \(\beta_{9}\) | \(=\) |
\( ( 1505 \nu^{11} - 2267 \nu^{10} - 24089 \nu^{9} + 32041 \nu^{8} + 136461 \nu^{7} - 146765 \nu^{6} + \cdots + 25860 ) / 1916 \)
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| \(\beta_{10}\) | \(=\) |
\( ( 4049 \nu^{11} - 7171 \nu^{10} - 64059 \nu^{9} + 102171 \nu^{8} + 358331 \nu^{7} - 475855 \nu^{6} + \cdots + 101712 ) / 3832 \)
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| \(\beta_{11}\) | \(=\) |
\( ( - 3807 \nu^{11} + 6584 \nu^{10} + 60000 \nu^{9} - 93086 \nu^{8} - 333108 \nu^{7} + 426754 \nu^{6} + \cdots - 81080 ) / 1916 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{9} - \beta_{6} + \beta_{3} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{11} + \beta_{9} - 2\beta_{8} + \beta_{6} + \beta_{3} + \beta_{2} + 5\beta _1 + 2 \)
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| \(\nu^{4}\) | \(=\) |
\( -\beta_{10} + 7\beta_{9} - \beta_{7} - 7\beta_{6} - \beta_{5} - 2\beta_{4} + 9\beta_{3} - \beta _1 + 16 \)
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| \(\nu^{5}\) | \(=\) |
\( 9 \beta_{11} + \beta_{10} + 7 \beta_{9} - 16 \beta_{8} + 10 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} + \cdots + 15 \)
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| \(\nu^{6}\) | \(=\) |
\( - \beta_{11} - 11 \beta_{10} + 47 \beta_{9} + 3 \beta_{8} - 10 \beta_{7} - 44 \beta_{6} - 8 \beta_{5} + \cdots + 96 \)
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| \(\nu^{7}\) | \(=\) |
\( 68 \beta_{11} + 13 \beta_{10} + 44 \beta_{9} - 107 \beta_{8} + \beta_{7} + 85 \beta_{6} - 35 \beta_{5} + \cdots + 97 \)
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| \(\nu^{8}\) | \(=\) |
\( - 12 \beta_{11} - 91 \beta_{10} + 316 \beta_{9} + 43 \beta_{8} - 79 \beta_{7} - 274 \beta_{6} + \cdots + 601 \)
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| \(\nu^{9}\) | \(=\) |
\( 488 \beta_{11} + 117 \beta_{10} + 275 \beta_{9} - 693 \beta_{8} + 17 \beta_{7} + 672 \beta_{6} + \cdots + 611 \)
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| \(\nu^{10}\) | \(=\) |
\( - 106 \beta_{11} - 687 \beta_{10} + 2136 \beta_{9} + 430 \beta_{8} - 577 \beta_{7} - 1726 \beta_{6} + \cdots + 3857 \)
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| \(\nu^{11}\) | \(=\) |
\( 3430 \beta_{11} + 917 \beta_{10} + 1743 \beta_{9} - 4490 \beta_{8} + 188 \beta_{7} + 5084 \beta_{6} + \cdots + 3844 \)
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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 |
|
−2.61265 | 1.00000 | 4.82594 | −2.90921 | −2.61265 | −2.92312 | −7.38321 | 1.00000 | 7.60076 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.32830 | 1.00000 | 3.42098 | 0.126501 | −2.32830 | 4.62264 | −3.30846 | 1.00000 | −0.294531 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.21330 | 1.00000 | 2.89869 | 3.59353 | −2.21330 | 4.20833 | −1.98908 | 1.00000 | −7.95355 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.40064 | 1.00000 | −0.0382014 | 2.49985 | −1.40064 | 2.29755 | 2.85479 | 1.00000 | −3.50140 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.742506 | 1.00000 | −1.44869 | −0.0547619 | −0.742506 | 1.46183 | 2.56067 | 1.00000 | 0.0406610 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.282500 | 1.00000 | −1.92019 | −2.99747 | −0.282500 | 2.93733 | 1.10745 | 1.00000 | 0.846784 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.668708 | 1.00000 | −1.55283 | 3.36024 | 0.668708 | −4.86588 | −2.37581 | 1.00000 | 2.24702 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.726454 | 1.00000 | −1.47226 | −1.17772 | 0.726454 | 0.775671 | −2.52244 | 1.00000 | −0.855558 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.06848 | 1.00000 | −0.858356 | 3.93653 | 1.06848 | 2.58610 | −3.05409 | 1.00000 | 4.20609 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.46888 | 1.00000 | 0.157605 | −3.40129 | 1.46888 | −1.63441 | −2.70625 | 1.00000 | −4.99608 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.01690 | 1.00000 | 2.06787 | 3.13689 | 2.01690 | 3.05231 | 0.136879 | 1.00000 | 6.32677 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.63048 | 1.00000 | 4.91944 | 0.886923 | 2.63048 | −3.51834 | 7.67955 | 1.00000 | 2.33304 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(29\) | \( +1 \) |
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 | 2523.2.a.t | ✓ | 12 |
| 3.b | odd | 2 | 1 | 7569.2.a.bs | 12 | ||
| 29.b | even | 2 | 1 | 2523.2.a.u | yes | 12 | |
| 87.d | odd | 2 | 1 | 7569.2.a.bo | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2523.2.a.t | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 2523.2.a.u | yes | 12 | 29.b | even | 2 | 1 | |
| 7569.2.a.bo | 12 | 87.d | odd | 2 | 1 | ||
| 7569.2.a.bs | 12 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2523))\):
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\( T_{2}^{12} + T_{2}^{11} - 17 T_{2}^{10} - 13 T_{2}^{9} + 105 T_{2}^{8} + 49 T_{2}^{7} - 292 T_{2}^{6} + \cdots + 16 \)
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\( T_{5}^{12} - 7 T_{5}^{11} - 19 T_{5}^{10} + 214 T_{5}^{9} - 26 T_{5}^{8} - 2258 T_{5}^{7} + 2294 T_{5}^{6} + \cdots - 80 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + T^{11} + \cdots + 16 \)
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| $3$ |
\( (T - 1)^{12} \)
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| $5$ |
\( T^{12} - 7 T^{11} + \cdots - 80 \)
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| $7$ |
\( T^{12} - 9 T^{11} + \cdots + 96111 \)
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| $11$ |
\( T^{12} + 12 T^{11} + \cdots - 3098864 \)
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| $13$ |
\( T^{12} - 19 T^{11} + \cdots - 1355625 \)
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| $17$ |
\( T^{12} - 7 T^{11} + \cdots - 3600 \)
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| $19$ |
\( T^{12} - 16 T^{11} + \cdots + 605191 \)
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| $23$ |
\( T^{12} - 29 T^{11} + \cdots - 1265904 \)
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| $29$ |
\( T^{12} \)
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| $31$ |
\( T^{12} - 8 T^{11} + \cdots - 412619 \)
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| $37$ |
\( T^{12} + \cdots - 107348609 \)
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| $41$ |
\( T^{12} - 193 T^{10} + \cdots + 919120 \)
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| $43$ |
\( T^{12} + 3 T^{11} + \cdots + 418491 \)
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| $47$ |
\( T^{12} + \cdots + 386568976 \)
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| $53$ |
\( T^{12} + \cdots - 400419584 \)
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| $59$ |
\( T^{12} + 16 T^{11} + \cdots - 2116080 \)
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| $61$ |
\( T^{12} + \cdots + 54907178661 \)
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| $67$ |
\( T^{12} + \cdots + 825555631 \)
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| $71$ |
\( T^{12} + \cdots - 1261879664 \)
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| $73$ |
\( T^{12} - 216 T^{10} + \cdots + 106111 \)
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| $79$ |
\( T^{12} + \cdots + 8602913331 \)
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| $83$ |
\( T^{12} + \cdots - 203096304 \)
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| $89$ |
\( T^{12} + \cdots - 201310544 \)
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| $97$ |
\( T^{12} + \cdots + 13491351961 \)
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