Newspace parameters
| Level: | \( N \) | \(=\) | \( 10 = 2 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 10.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(27.9477344287\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{1179649}) \) |
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| Defining polynomial: |
\( x^{2} - x - 294912 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3\cdot 5 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
Embedding invariants
| Embedding label | 1.2 | ||
| Root | \(-542.558\) of defining polynomial | ||
| Character | \(\chi\) | \(=\) | 10.1 |
$q$-expansion
Coefficient data
For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.
Currently showing only \(a_p\); display all \(a_n\)
Currently showing all \(a_n\); display only \(a_p\)
| \(n\) | \(a_n\) | \(a_n / n^{(k-1)/2}\) | \( \alpha_n \) | \( \theta_n \) | ||||||
|---|---|---|---|---|---|---|---|---|---|---|
| \(p\) | \(a_p\) | \(a_p / p^{(k-1)/2}\) | \( \alpha_p\) | \( \theta_p \) | ||||||
| \(2\) | 1024.00 | 0.707107 | ||||||||
| \(3\) | 145820. | 1.42575 | 0.712876 | − | 0.701290i | \(-0.247392\pi\) | ||||
| 0.712876 | + | 0.701290i | \(0.247392\pi\) | |||||||
| \(4\) | 1.04858e6 | 0.500000 | ||||||||
| \(5\) | −9.76562e6 | −0.447214 | ||||||||
| \(6\) | 1.49320e8 | 1.00816 | ||||||||
| \(7\) | −2.36011e8 | −0.315793 | −0.157896 | − | 0.987456i | \(-0.550471\pi\) | ||||
| −0.157896 | + | 0.987456i | \(0.550471\pi\) | |||||||
| \(8\) | 1.07374e9 | 0.353553 | ||||||||
| \(9\) | 1.08031e10 | 1.03277 | ||||||||
| \(10\) | −1.00000e10 | −0.316228 | ||||||||
| \(11\) | 1.49202e11 | 1.73441 | 0.867206 | − | 0.497949i | \(-0.165913\pi\) | ||||
| 0.867206 | + | 0.497949i | \(0.165913\pi\) | |||||||
| \(12\) | 1.52903e11 | 0.712876 | ||||||||
| \(13\) | 4.89350e11 | 0.984497 | 0.492248 | − | 0.870455i | \(-0.336175\pi\) | ||||
| 0.492248 | + | 0.870455i | \(0.336175\pi\) | |||||||
| \(14\) | −2.41675e11 | −0.223299 | ||||||||
| \(15\) | −1.42402e12 | −0.637615 | ||||||||
| \(16\) | 1.09951e12 | 0.250000 | ||||||||
| \(17\) | 5.46288e12 | 0.657216 | 0.328608 | − | 0.944466i | \(-0.393421\pi\) | ||||
| 0.328608 | + | 0.944466i | \(0.393421\pi\) | |||||||
| \(18\) | 1.10624e13 | 0.730277 | ||||||||
| \(19\) | −1.12049e13 | −0.419270 | −0.209635 | − | 0.977780i | \(-0.567228\pi\) | ||||
| −0.209635 | + | 0.977780i | \(0.567228\pi\) | |||||||
| \(20\) | −1.02400e13 | −0.223607 | ||||||||
| \(21\) | −3.44151e13 | −0.450242 | ||||||||
| \(22\) | 1.52783e14 | 1.22641 | ||||||||
| \(23\) | 1.78791e13 | 0.0899916 | 0.0449958 | − | 0.998987i | \(-0.485673\pi\) | ||||
| 0.0449958 | + | 0.998987i | \(0.485673\pi\) | |||||||
| \(24\) | 1.56573e14 | 0.504079 | ||||||||
| \(25\) | 9.53674e13 | 0.200000 | ||||||||
| \(26\) | 5.01094e14 | 0.696144 | ||||||||
| \(27\) | 4.99812e13 | 0.0467183 | ||||||||
| \(28\) | −2.47475e14 | −0.157896 | ||||||||
| \(29\) | −2.82120e15 | −1.24525 | −0.622623 | − | 0.782522i | \(-0.713933\pi\) | ||||
| −0.622623 | + | 0.782522i | \(0.713933\pi\) | |||||||
| \(30\) | −1.45820e15 | −0.450862 | ||||||||
| \(31\) | 4.24797e15 | 0.930858 | 0.465429 | − | 0.885085i | \(-0.345900\pi\) | ||||
| 0.465429 | + | 0.885085i | \(0.345900\pi\) | |||||||
| \(32\) | 1.12590e15 | 0.176777 | ||||||||
| \(33\) | 2.17567e16 | 2.47284 | ||||||||
| \(34\) | 5.59399e15 | 0.464722 | ||||||||
| \(35\) | 2.30479e15 | 0.141227 | ||||||||
| \(36\) | 1.13279e16 | 0.516384 | ||||||||
| \(37\) | 5.74457e16 | 1.96399 | 0.981995 | − | 0.188909i | \(-0.0604951\pi\) | ||||
| 0.981995 | + | 0.188909i | \(0.0604951\pi\) | |||||||
| \(38\) | −1.14738e16 | −0.296469 | ||||||||
| \(39\) | 7.13570e16 | 1.40365 | ||||||||
| \(40\) | −1.04858e16 | −0.158114 | ||||||||
| \(41\) | −3.51283e16 | −0.408721 | −0.204361 | − | 0.978896i | \(-0.565512\pi\) | ||||
| −0.204361 | + | 0.978896i | \(0.565512\pi\) | |||||||
| \(42\) | −3.52410e16 | −0.318369 | ||||||||
| \(43\) | −2.62144e17 | −1.84978 | −0.924891 | − | 0.380233i | \(-0.875844\pi\) | ||||
| −0.924891 | + | 0.380233i | \(0.875844\pi\) | |||||||
| \(44\) | 1.56450e17 | 0.867206 | ||||||||
| \(45\) | −1.05499e17 | −0.461868 | ||||||||
| \(46\) | 1.83081e16 | 0.0636337 | ||||||||
| \(47\) | 1.08406e17 | 0.300626 | 0.150313 | − | 0.988638i | \(-0.451972\pi\) | ||||
| 0.150313 | + | 0.988638i | \(0.451972\pi\) | |||||||
| \(48\) | 1.60331e17 | 0.356438 | ||||||||
| \(49\) | −5.02845e17 | −0.900275 | ||||||||
| \(50\) | 9.76562e16 | 0.141421 | ||||||||
| \(51\) | 7.96597e17 | 0.937027 | ||||||||
| \(52\) | 5.13120e17 | 0.492248 | ||||||||
| \(53\) | −2.47953e18 | −1.94748 | −0.973740 | − | 0.227662i | \(-0.926892\pi\) | ||||
| −0.973740 | + | 0.227662i | \(0.926892\pi\) | |||||||
| \(54\) | 5.11807e16 | 0.0330348 | ||||||||
| \(55\) | −1.45705e18 | −0.775653 | ||||||||
| \(56\) | −2.53415e17 | −0.111650 | ||||||||
| \(57\) | −1.63389e18 | −0.597775 | ||||||||
| \(58\) | −2.88891e18 | −0.880521 | ||||||||
| \(59\) | 2.14259e17 | 0.0545748 | 0.0272874 | − | 0.999628i | \(-0.491313\pi\) | ||||
| 0.0272874 | + | 0.999628i | \(0.491313\pi\) | |||||||
| \(60\) | −1.49320e18 | −0.318808 | ||||||||
| \(61\) | 3.64272e18 | 0.653826 | 0.326913 | − | 0.945054i | \(-0.393992\pi\) | ||||
| 0.326913 | + | 0.945054i | \(0.393992\pi\) | |||||||
| \(62\) | 4.34992e18 | 0.658216 | ||||||||
| \(63\) | −2.54965e18 | −0.326141 | ||||||||
| \(64\) | 1.15292e18 | 0.125000 | ||||||||
| \(65\) | −4.77881e18 | −0.440280 | ||||||||
| \(66\) | 2.22788e19 | 1.74856 | ||||||||
| \(67\) | −1.13372e19 | −0.759835 | −0.379918 | − | 0.925020i | \(-0.624048\pi\) | ||||
| −0.379918 | + | 0.925020i | \(0.624048\pi\) | |||||||
| \(68\) | 5.72825e18 | 0.328608 | ||||||||
| \(69\) | 2.60712e18 | 0.128306 | ||||||||
| \(70\) | 2.36011e18 | 0.0998625 | ||||||||
| \(71\) | 2.44087e19 | 0.889882 | 0.444941 | − | 0.895560i | \(-0.353225\pi\) | ||||
| 0.444941 | + | 0.895560i | \(0.353225\pi\) | |||||||
| \(72\) | 1.15998e19 | 0.365138 | ||||||||
| \(73\) | −4.12244e19 | −1.12270 | −0.561350 | − | 0.827578i | \(-0.689718\pi\) | ||||
| −0.561350 | + | 0.827578i | \(0.689718\pi\) | |||||||
| \(74\) | 5.88244e19 | 1.38875 | ||||||||
| \(75\) | 1.39065e19 | 0.285150 | ||||||||
| \(76\) | −1.17492e19 | −0.209635 | ||||||||
| \(77\) | −3.52133e19 | −0.547715 | ||||||||
| \(78\) | 7.30695e19 | 0.992529 | ||||||||
| \(79\) | 6.83832e19 | 0.812578 | 0.406289 | − | 0.913745i | \(-0.366823\pi\) | ||||
| 0.406289 | + | 0.913745i | \(0.366823\pi\) | |||||||
| \(80\) | −1.07374e19 | −0.111803 | ||||||||
| \(81\) | −1.05716e20 | −0.966159 | ||||||||
| \(82\) | −3.59714e19 | −0.289009 | ||||||||
| \(83\) | 1.39311e20 | 0.985522 | 0.492761 | − | 0.870165i | \(-0.335988\pi\) | ||||
| 0.492761 | + | 0.870165i | \(0.335988\pi\) | |||||||
| \(84\) | −3.60868e19 | −0.225121 | ||||||||
| \(85\) | −5.33484e19 | −0.293916 | ||||||||
| \(86\) | −2.68435e20 | −1.30799 | ||||||||
| \(87\) | −4.11387e20 | −1.77541 | ||||||||
| \(88\) | 1.60205e20 | 0.613207 | ||||||||
| \(89\) | −4.04817e20 | −1.37614 | −0.688072 | − | 0.725643i | \(-0.741542\pi\) | ||||
| −0.688072 | + | 0.725643i | \(0.741542\pi\) | |||||||
| \(90\) | −1.08031e20 | −0.326590 | ||||||||
| \(91\) | −1.15492e20 | −0.310897 | ||||||||
| \(92\) | 1.87475e19 | 0.0449958 | ||||||||
| \(93\) | 6.19438e20 | 1.32717 | ||||||||
| \(94\) | 1.11008e20 | 0.212575 | ||||||||
| \(95\) | 1.09423e20 | 0.187503 | ||||||||
| \(96\) | 1.64179e20 | 0.252040 | ||||||||
| \(97\) | −9.18963e20 | −1.26530 | −0.632652 | − | 0.774436i | \(-0.718034\pi\) | ||||
| −0.632652 | + | 0.774436i | \(0.718034\pi\) | |||||||
| \(98\) | −5.14913e20 | −0.636590 | ||||||||
| \(99\) | 1.61185e21 | 1.79124 | ||||||||
Currently showing only \(a_p\); display all \(a_n\)
Currently showing all \(a_n\); display only \(a_p\)
Twists
| By twisting character | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Type | Twist | Min | Dim | |
| 1.1 | even | 1 | trivial | 10.22.a.d.1.2 | ✓ | 2 | |
| 4.3 | odd | 2 | 80.22.a.c.1.1 | 2 | |||
| 5.2 | odd | 4 | 50.22.b.e.49.3 | 4 | |||
| 5.3 | odd | 4 | 50.22.b.e.49.2 | 4 | |||
| 5.4 | even | 2 | 50.22.a.d.1.1 | 2 | |||
| By twisted newform | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Type | |
| 10.22.a.d.1.2 | ✓ | 2 | 1.1 | even | 1 | trivial | |
| 50.22.a.d.1.1 | 2 | 5.4 | even | 2 | |||
| 50.22.b.e.49.2 | 4 | 5.3 | odd | 4 | |||
| 50.22.b.e.49.3 | 4 | 5.2 | odd | 4 | |||
| 80.22.a.c.1.1 | 2 | 4.3 | odd | 2 | |||