Newspace parameters
| Level: | \( N \) | \(=\) | \( 5819 = 11 \cdot 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5819.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(46.4649489362\) |
| Analytic rank: | \(0\) |
| Dimension: | \(18\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
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| Defining polynomial: |
\( x^{18} - 26 x^{16} - 2 x^{15} + 275 x^{14} + 48 x^{13} - 1521 x^{12} - 434 x^{11} + 4680 x^{10} + \cdots + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\ldots,\beta_{17}\) 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
\( x^{18} - 26 x^{16} - 2 x^{15} + 275 x^{14} + 48 x^{13} - 1521 x^{12} - 434 x^{11} + 4680 x^{10} + \cdots + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( ( - 328565 \nu^{17} + 2987085 \nu^{16} + 6798802 \nu^{15} - 74806734 \nu^{14} - 49662910 \nu^{13} + \cdots + 80782880 ) / 15441383 \)
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| \(\beta_{4}\) | \(=\) |
\( ( - 984466 \nu^{17} - 380825 \nu^{16} + 23362110 \nu^{15} + 8866189 \nu^{14} - 220021595 \nu^{13} + \cdots - 1708753 ) / 15441383 \)
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| \(\beta_{5}\) | \(=\) |
\( ( - 1708753 \nu^{17} + 984466 \nu^{16} + 44808403 \nu^{15} - 19944604 \nu^{14} - 478773264 \nu^{13} + \cdots - 4156670 ) / 15441383 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 1708753 \nu^{17} - 984466 \nu^{16} - 44808403 \nu^{15} + 19944604 \nu^{14} + 478773264 \nu^{13} + \cdots + 4156670 ) / 15441383 \)
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| \(\beta_{7}\) | \(=\) |
\( ( 2438491 \nu^{17} + 792919 \nu^{16} - 63051704 \nu^{15} - 22546430 \nu^{14} + 660638610 \nu^{13} + \cdots - 30337775 ) / 15441383 \)
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| \(\beta_{8}\) | \(=\) |
\( ( 3356951 \nu^{17} + 2126298 \nu^{16} - 87007460 \nu^{15} - 59091504 \nu^{14} + 912564442 \nu^{13} + \cdots - 5197969 ) / 15441383 \)
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| \(\beta_{9}\) | \(=\) |
\( ( 4031929 \nu^{17} + 447948 \nu^{16} - 106536316 \nu^{15} - 19863881 \nu^{14} + 1149371424 \nu^{13} + \cdots + 91405616 ) / 15441383 \)
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| \(\beta_{10}\) | \(=\) |
\( ( 4686441 \nu^{17} + 176628 \nu^{16} - 124918594 \nu^{15} - 14492291 \nu^{14} + 1358800014 \nu^{13} + \cdots + 98659328 ) / 15441383 \)
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| \(\beta_{11}\) | \(=\) |
\( ( - 7253712 \nu^{17} + 654512 \nu^{16} + 188325192 \nu^{15} - 3874854 \nu^{14} - 1989399210 \nu^{13} + \cdots - 35457028 ) / 15441383 \)
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| \(\beta_{12}\) | \(=\) |
\( ( - 8765642 \nu^{17} + 1231060 \nu^{16} + 229002906 \nu^{15} - 15576181 \nu^{14} - 2439736042 \nu^{13} + \cdots - 84822899 ) / 15441383 \)
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| \(\beta_{13}\) | \(=\) |
\( ( 9563456 \nu^{17} - 385571 \nu^{16} - 247977222 \nu^{15} - 7939320 \nu^{14} + 2617183371 \nu^{13} + \cdots + 110208007 ) / 15441383 \)
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| \(\beta_{14}\) | \(=\) |
\( ( 11002028 \nu^{17} - 2007514 \nu^{16} - 284424442 \nu^{15} + 31402566 \nu^{14} + 2989568132 \nu^{13} + \cdots + 89594943 ) / 15441383 \)
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| \(\beta_{15}\) | \(=\) |
\( ( 13293155 \nu^{17} - 1834850 \nu^{16} - 348268130 \nu^{15} + 20000541 \nu^{14} + 3720004421 \nu^{13} + \cdots + 122423289 ) / 15441383 \)
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| \(\beta_{16}\) | \(=\) |
\( ( - 14272726 \nu^{17} + 1950452 \nu^{16} + 370325372 \nu^{15} - 24703438 \nu^{14} + \cdots - 96308443 ) / 15441383 \)
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| \(\beta_{17}\) | \(=\) |
\( ( 27105813 \nu^{17} - 4885552 \nu^{16} - 704855985 \nu^{15} + 69618017 \nu^{14} + 7461970829 \nu^{13} + \cdots + 194187329 ) / 15441383 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 5\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( -\beta_{16} - \beta_{15} + \beta_{14} + \beta_{12} - \beta_{8} + \beta_{7} + 2\beta_{4} + \beta_{3} + 6\beta_{2} + \beta _1 + 16 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{16} - \beta_{13} - 2 \beta_{12} + \beta_{8} - \beta_{7} + 10 \beta_{6} + 8 \beta_{5} - 2 \beta_{4} + \cdots - 3 \)
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| \(\nu^{6}\) | \(=\) |
\( \beta_{17} - 11 \beta_{16} - 12 \beta_{15} + 9 \beta_{14} - \beta_{13} + 10 \beta_{12} - 2 \beta_{11} + \cdots + 95 \)
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| \(\nu^{7}\) | \(=\) |
\( 14 \beta_{16} + \beta_{14} - 12 \beta_{13} - 24 \beta_{12} - 2 \beta_{11} - \beta_{10} + \beta_{9} + \cdots - 40 \)
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| \(\nu^{8}\) | \(=\) |
\( 11 \beta_{17} - 97 \beta_{16} - 105 \beta_{15} + 65 \beta_{14} - 10 \beta_{13} + 82 \beta_{12} + \cdots + 595 \)
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| \(\nu^{9}\) | \(=\) |
\( - \beta_{17} + 147 \beta_{16} + 8 \beta_{15} + 17 \beta_{14} - 106 \beta_{13} - 217 \beta_{12} + \cdots - 398 \)
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| \(\nu^{10}\) | \(=\) |
\( 89 \beta_{17} - 793 \beta_{16} - 827 \beta_{15} + 435 \beta_{14} - 66 \beta_{13} + 638 \beta_{12} + \cdots + 3869 \)
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| \(\nu^{11}\) | \(=\) |
\( - 19 \beta_{17} + 1365 \beta_{16} + 164 \beta_{15} + 185 \beta_{14} - 842 \beta_{13} - 1781 \beta_{12} + \cdots - 3538 \)
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| \(\nu^{12}\) | \(=\) |
\( 650 \beta_{17} - 6249 \beta_{16} - 6225 \beta_{15} + 2815 \beta_{14} - 329 \beta_{13} + 4878 \beta_{12} + \cdots + 25897 \)
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| \(\nu^{13}\) | \(=\) |
\( - 235 \beta_{17} + 11826 \beta_{16} + 2162 \beta_{15} + 1654 \beta_{14} - 6380 \beta_{13} + \cdots - 29710 \)
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| \(\nu^{14}\) | \(=\) |
\( 4568 \beta_{17} - 48236 \beta_{16} - 45840 \beta_{15} + 17942 \beta_{14} - 983 \beta_{13} + \cdots + 177341 \)
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| \(\nu^{15}\) | \(=\) |
\( - 2411 \beta_{17} + 98099 \beta_{16} + 23415 \beta_{15} + 13265 \beta_{14} - 47213 \beta_{13} + \cdots - 241403 \)
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| \(\nu^{16}\) | \(=\) |
\( 31720 \beta_{17} - 367747 \beta_{16} - 333872 \beta_{15} + 113682 \beta_{14} + 3684 \beta_{13} + \cdots + 1236559 \)
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| \(\nu^{17}\) | \(=\) |
\( - 22349 \beta_{17} + 790691 \beta_{16} + 227257 \beta_{15} + 99427 \beta_{14} - 345054 \beta_{13} + \cdots - 1921101 \)
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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 |
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−2.73796 | 0.200998 | 5.49642 | −3.20428 | −0.550325 | −0.207965 | −9.57305 | −2.95960 | 8.77318 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.33159 | 2.16091 | 3.43632 | −0.751052 | −5.03835 | −3.40928 | −3.34890 | 1.66952 | 1.75115 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.92566 | −2.85748 | 1.70816 | 2.48248 | 5.50252 | 1.99527 | 0.561991 | 5.16518 | −4.78041 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.82277 | 3.06090 | 1.32247 | 2.03363 | −5.57929 | 4.14508 | 1.23497 | 6.36909 | −3.70683 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.56027 | 0.376498 | 0.434433 | −1.07262 | −0.587437 | −1.19333 | 2.44270 | −2.85825 | 1.67358 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.30660 | −2.02698 | −0.292802 | 0.604117 | 2.64845 | 4.26811 | 2.99577 | 1.10866 | −0.789338 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.693371 | 2.51104 | −1.51924 | 4.17000 | −1.74108 | −2.28893 | 2.44014 | 3.30530 | −2.89136 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.542399 | 1.05221 | −1.70580 | −1.58565 | −0.570719 | −3.07693 | 2.01002 | −1.89285 | 0.860056 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.169979 | 0.193934 | −1.97111 | 2.49566 | −0.0329647 | 1.42197 | 0.675003 | −2.96239 | −0.424208 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −0.110336 | −2.45131 | −1.98783 | −1.83126 | 0.270467 | 1.29692 | 0.439999 | 3.00894 | 0.202053 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0.0580425 | 2.73846 | −1.99663 | −2.50598 | 0.158947 | 3.34211 | −0.231975 | 4.49914 | −0.145454 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0.974917 | 1.12173 | −1.04954 | −0.457847 | 1.09360 | −4.56202 | −2.97304 | −1.74171 | −0.446362 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.01567 | −1.32280 | −0.968420 | 3.69433 | −1.34353 | 0.310386 | −3.01493 | −1.25020 | 3.75221 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 1.98706 | −3.40891 | 1.94840 | −0.261295 | −6.77370 | −2.94073 | −0.102524 | 8.62066 | −0.519209 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.00682 | −1.31486 | 2.02733 | 1.49710 | −2.63870 | 0.0253913 | 0.0548547 | −1.27113 | 3.00441 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.26251 | 1.80997 | 3.11895 | 1.61936 | 4.09508 | 3.02973 | 2.53162 | 0.275996 | 3.66383 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 2.36446 | 2.21586 | 3.59066 | 4.15771 | 5.23930 | 1.17054 | 3.76104 | 1.91003 | 9.83074 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 2.53145 | −0.0601502 | 4.40823 | −3.08441 | −0.152267 | 4.67368 | 6.09630 | −2.99638 | −7.80801 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \( -1 \) |
| \(23\) | \( +1 \) |
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 | 5819.2.a.m | yes | 18 |
| 23.b | odd | 2 | 1 | 5819.2.a.l | ✓ | 18 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5819.2.a.l | ✓ | 18 | 23.b | odd | 2 | 1 | |
| 5819.2.a.m | yes | 18 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5819))\):
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\( T_{2}^{18} - 26 T_{2}^{16} - 2 T_{2}^{15} + 275 T_{2}^{14} + 48 T_{2}^{13} - 1521 T_{2}^{12} - 434 T_{2}^{11} + \cdots + 1 \)
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\( T_{5}^{18} - 8 T_{5}^{17} - 20 T_{5}^{16} + 280 T_{5}^{15} - 23 T_{5}^{14} - 3880 T_{5}^{13} + \cdots - 8192 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} - 26 T^{16} + \cdots + 1 \)
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| $3$ |
\( T^{18} - 4 T^{17} + \cdots - 16 \)
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| $5$ |
\( T^{18} - 8 T^{17} + \cdots - 8192 \)
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| $7$ |
\( T^{18} - 8 T^{17} + \cdots - 2272 \)
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| $11$ |
\( (T - 1)^{18} \)
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| $13$ |
\( T^{18} - 4 T^{17} + \cdots + 53518076 \)
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| $17$ |
\( T^{18} - 8 T^{17} + \cdots - 712072 \)
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| $19$ |
\( T^{18} + \cdots + 6036527168 \)
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| $23$ |
\( T^{18} \)
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| $29$ |
\( T^{18} + \cdots - 207306514696 \)
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| $31$ |
\( T^{18} + \cdots - 33347852897168 \)
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| $37$ |
\( T^{18} + \cdots + 151151247608 \)
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| $41$ |
\( T^{18} + \cdots - 576752708 \)
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| $43$ |
\( T^{18} + \cdots - 1251666440768 \)
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| $47$ |
\( T^{18} + \cdots - 39102918747136 \)
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| $53$ |
\( T^{18} + \cdots + 242919386508344 \)
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| $59$ |
\( T^{18} + \cdots - 17963460657152 \)
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| $61$ |
\( T^{18} + \cdots - 8621466201352 \)
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| $67$ |
\( T^{18} + \cdots + 11353501349888 \)
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| $71$ |
\( T^{18} + \cdots + 2812021378576 \)
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| $73$ |
\( T^{18} + \cdots + 61809600133832 \)
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| $79$ |
\( T^{18} + \cdots - 16\!\cdots\!92 \)
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| $83$ |
\( T^{18} + \cdots + 685990631699168 \)
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| $89$ |
\( T^{18} + \cdots - 25\!\cdots\!04 \)
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| $97$ |
\( T^{18} + \cdots + 163661738744 \)
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