Properties

Label 207.8.a.a
Level $207$
Weight $8$
Character orbit 207.a
Self dual yes
Analytic conductor $64.664$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 207.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.6637002752\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 455 x^{3} - 474 x^{2} + 42284 x + 127016\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 69)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 54 + 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{4} + ( 54 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{5} + ( -96 + 3 \beta_{1} - \beta_{2} - 12 \beta_{3} - 4 \beta_{4} ) q^{7} + ( -284 - 9 \beta_{1} - 11 \beta_{2} + \beta_{4} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 54 + 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{4} + ( 54 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{5} + ( -96 + 3 \beta_{1} - \beta_{2} - 12 \beta_{3} - 4 \beta_{4} ) q^{7} + ( -284 - 9 \beta_{1} - 11 \beta_{2} + \beta_{4} ) q^{8} + ( 296 - 72 \beta_{1} + 8 \beta_{2} - 10 \beta_{3} + 4 \beta_{4} ) q^{10} + ( 220 + 117 \beta_{1} - 4 \beta_{2} + 29 \beta_{3} + 5 \beta_{4} ) q^{11} + ( -172 - 372 \beta_{1} - 13 \beta_{2} + 82 \beta_{3} - 27 \beta_{4} ) q^{13} + ( -1158 + 589 \beta_{1} + 125 \beta_{2} + 23 \beta_{3} + 6 \beta_{4} ) q^{14} + ( -4578 - 108 \beta_{1} + 10 \beta_{2} + 83 \beta_{3} - 59 \beta_{4} ) q^{16} + ( 1126 + 1103 \beta_{1} - 65 \beta_{2} - 76 \beta_{3} - 160 \beta_{4} ) q^{17} + ( -1208 - 598 \beta_{1} + 121 \beta_{2} + 127 \beta_{3} + 125 \beta_{4} ) q^{19} + ( 5416 + 196 \beta_{1} - 40 \beta_{2} + 92 \beta_{3} - 56 \beta_{4} ) q^{20} + ( -19598 - 1461 \beta_{1} - 291 \beta_{2} - 83 \beta_{3} - 130 \beta_{4} ) q^{22} -12167 q^{23} + ( -52391 - 1198 \beta_{1} + 257 \beta_{2} - 280 \beta_{3} - 19 \beta_{4} ) q^{25} + ( 71120 + 316 \beta_{1} - 780 \beta_{2} + 334 \beta_{3} + 138 \beta_{4} ) q^{26} + ( -101182 - 161 \beta_{1} - 233 \beta_{2} - 1325 \beta_{3} - 718 \beta_{4} ) q^{28} + ( 33850 + 3746 \beta_{1} - 286 \beta_{2} - 540 \beta_{3} - 496 \beta_{4} ) q^{29} + ( -39540 + 4906 \beta_{1} - 744 \beta_{2} - 1400 \beta_{3} - 430 \beta_{4} ) q^{31} + ( 56936 + 7289 \beta_{1} + 627 \beta_{2} - 474 \beta_{3} - 531 \beta_{4} ) q^{32} + ( -205950 + 6707 \beta_{1} + 985 \beta_{2} - 421 \beta_{3} - 1426 \beta_{4} ) q^{34} + ( 28046 - 2746 \beta_{1} + 517 \beta_{2} - 1166 \beta_{3} + 109 \beta_{4} ) q^{35} + ( -40902 + 13985 \beta_{1} - 660 \beta_{2} + 1431 \beta_{3} + 177 \beta_{4} ) q^{37} + ( 111664 - 6430 \beta_{1} - 1516 \beta_{2} - 1334 \beta_{3} + 364 \beta_{4} ) q^{38} + ( -69128 + 3920 \beta_{1} - 1848 \beta_{2} + 1396 \beta_{3} - 972 \beta_{4} ) q^{40} + ( -107914 - 8482 \beta_{1} + 2772 \beta_{2} - 422 \beta_{3} + 434 \beta_{4} ) q^{41} + ( -180296 + 25824 \beta_{1} + 5109 \beta_{2} - 1083 \beta_{3} + 2975 \beta_{4} ) q^{43} + ( 246870 + 14067 \beta_{1} + 1763 \beta_{2} + 2633 \beta_{3} + 1792 \beta_{4} ) q^{44} + 12167 \beta_{1} q^{46} + ( -20306 + 9026 \beta_{1} - 99 \beta_{2} + 7350 \beta_{3} - 3311 \beta_{4} ) q^{47} + ( 170843 + 14966 \beta_{1} - 1043 \beta_{2} + 3080 \beta_{3} + 4893 \beta_{4} ) q^{49} + ( 189612 + 64865 \beta_{1} + 2562 \beta_{2} - 2944 \beta_{3} + 378 \beta_{4} ) q^{50} + ( 30968 - 46762 \beta_{1} - 1034 \beta_{2} + 3112 \beta_{3} + 7062 \beta_{4} ) q^{52} + ( 61330 + 10232 \beta_{1} - 2343 \beta_{2} - 8805 \beta_{3} + 5451 \beta_{4} ) q^{53} + ( -188820 + 19000 \beta_{1} - 3110 \beta_{2} + 4700 \beta_{3} - 570 \beta_{4} ) q^{55} + ( 107358 + 95305 \beta_{1} - 1799 \beta_{2} + 1189 \beta_{3} - 382 \beta_{4} ) q^{56} + ( -706228 - 3676 \beta_{1} + 6182 \beta_{2} + 498 \beta_{3} - 3716 \beta_{4} ) q^{58} + ( 634970 + 39800 \beta_{1} - 12851 \beta_{2} + 6800 \beta_{3} + 5535 \beta_{4} ) q^{59} + ( 23102 + 50535 \beta_{1} + 7862 \beta_{2} + 20597 \beta_{3} + 4461 \beta_{4} ) q^{61} + ( -925332 + 80282 \beta_{1} + 15174 \beta_{2} + 9566 \beta_{3} - 536 \beta_{4} ) q^{62} + ( -818206 - 11466 \beta_{1} + 3364 \beta_{2} - 30375 \beta_{3} - 4579 \beta_{4} ) q^{64} + ( -776048 + 13664 \beta_{1} - 3432 \beta_{2} + 6662 \beta_{3} + 5208 \beta_{4} ) q^{65} + ( -96120 - 38948 \beta_{1} - 32053 \beta_{2} + 1711 \beta_{3} - 6131 \beta_{4} ) q^{67} + ( -1493762 + 148751 \beta_{1} + 12971 \beta_{2} - 19571 \beta_{3} + 2560 \beta_{4} ) q^{68} + ( 421372 + 4088 \beta_{1} + 11034 \beta_{2} - 3792 \beta_{3} + 3038 \beta_{4} ) q^{70} + ( -322352 + 18550 \beta_{1} - 7316 \beta_{2} + 14726 \beta_{3} - 2758 \beta_{4} ) q^{71} + ( -11110 - 15000 \beta_{1} - 14012 \beta_{2} + 11284 \beta_{3} + 2604 \beta_{4} ) q^{73} + ( -2436334 - 37087 \beta_{1} - 13827 \beta_{2} - 4259 \beta_{3} - 12662 \beta_{4} ) q^{74} + ( 1370252 - 23908 \beta_{1} - 996 \beta_{2} + 21586 \beta_{3} + 2498 \beta_{4} ) q^{76} + ( -2068324 - 69014 \beta_{1} - 15122 \beta_{2} - 8018 \beta_{3} - 12636 \beta_{4} ) q^{77} + ( -782888 + 122297 \beta_{1} + 21401 \beta_{2} - 7904 \beta_{3} - 8456 \beta_{4} ) q^{79} + ( -1269376 + 39524 \beta_{1} - 6020 \beta_{2} + 10888 \beta_{3} + 4836 \beta_{4} ) q^{80} + ( 1374460 + 136084 \beta_{1} + 1014 \beta_{2} - 38834 \beta_{3} - 2364 \beta_{4} ) q^{82} + ( -3176088 - 72749 \beta_{1} + 178 \beta_{2} + 25355 \beta_{3} - 27507 \beta_{4} ) q^{83} + ( -1702262 + 106802 \beta_{1} - 9729 \beta_{2} + 6172 \beta_{3} + 10887 \beta_{4} ) q^{85} + ( -4947060 + 36996 \beta_{1} + 2746 \beta_{2} - 103720 \beta_{3} - 34328 \beta_{4} ) q^{86} + ( 22934 - 233881 \beta_{1} + 7363 \beta_{2} - 33275 \beta_{3} - 2548 \beta_{4} ) q^{88} + ( -221606 - 375585 \beta_{1} + 12543 \beta_{2} + 33662 \beta_{3} + 37888 \beta_{4} ) q^{89} + ( -5495056 + 131090 \beta_{1} + 43414 \beta_{2} + 30604 \beta_{3} - 22260 \beta_{4} ) q^{91} + ( -657018 - 24334 \beta_{1} - 12167 \beta_{3} - 12167 \beta_{4} ) q^{92} + ( -1432588 + 7068 \beta_{1} - 70090 \beta_{2} - 35188 \beta_{3} - 39786 \beta_{4} ) q^{94} + ( 1212362 - 66138 \beta_{1} + 11495 \beta_{2} - 1866 \beta_{3} - 4877 \beta_{4} ) q^{95} + ( -3153854 + 490412 \beta_{1} + 4426 \beta_{2} - 782 \beta_{3} - 16134 \beta_{4} ) q^{97} + ( -2349564 - 553197 \beta_{1} - 34650 \beta_{2} + 17220 \beta_{3} + 8554 \beta_{4} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 270q^{4} + 266q^{5} - 496q^{7} - 1422q^{8} + O(q^{10}) \) \( 5q + 270q^{4} + 266q^{5} - 496q^{7} - 1422q^{8} + 1452q^{10} + 1148q^{11} - 642q^{13} - 5756q^{14} - 22606q^{16} + 5798q^{17} - 6036q^{19} + 27376q^{20} - 97896q^{22} - 60835q^{23} - 262477q^{25} + 355992q^{26} - 507124q^{28} + 169162q^{29} - 199640q^{31} + 284794q^{32} - 1027740q^{34} + 137680q^{35} - 202002q^{37} + 554924q^{38} - 340904q^{40} - 541282q^{41} - 909596q^{43} + 1236032q^{44} - 80208q^{47} + 850589q^{49} + 941416q^{50} + 146940q^{52} + 278138q^{53} - 933560q^{55} + 539932q^{56} - 3522712q^{58} + 3177380q^{59} + 147782q^{61} - 4606456q^{62} - 4142622q^{64} - 3877332q^{65} - 464916q^{67} - 7513072q^{68} + 2093200q^{70} - 1576792q^{71} - 38190q^{73} - 12164864q^{74} + 6889436q^{76} - 10332384q^{77} - 3913336q^{79} - 6334776q^{80} + 6799360q^{82} - 15774716q^{83} - 8520740q^{85} - 24874084q^{86} + 53216q^{88} - 1116482q^{89} - 27369552q^{91} - 3285090q^{92} - 7153744q^{94} + 6067832q^{95} - 15738566q^{97} - 11730488q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 455 x^{3} - 474 x^{2} + 42284 x + 127016\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{4} + 142 \nu^{3} + 467 \nu^{2} - 37904 \nu - 76396 \)\()/1552\)
\(\beta_{3}\)\(=\)\((\)\( 11 \nu^{4} - 10 \nu^{3} - 3585 \nu^{2} + 2560 \nu + 117124 \)\()/1552\)
\(\beta_{4}\)\(=\)\((\)\( -11 \nu^{4} + 10 \nu^{3} + 5137 \nu^{2} - 5664 \nu - 399588 \)\()/1552\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + 2 \beta_{1} + 182\)
\(\nu^{3}\)\(=\)\(-\beta_{4} + 11 \beta_{2} + 265 \beta_{1} + 284\)
\(\nu^{4}\)\(=\)\(325 \beta_{4} + 467 \beta_{3} + 10 \beta_{2} + 660 \beta_{1} + 48926\)

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
18.2528
12.4672
−3.24502
−9.64907
−17.8260
−18.2528 0 205.166 55.2224 0 −1258.24 −1408.51 0 −1007.97
1.2 −12.4672 0 27.4323 40.8788 0 1126.70 1253.80 0 −509.647
1.3 3.24502 0 −117.470 −168.083 0 149.817 −796.555 0 −545.434
1.4 9.64907 0 −34.8955 306.939 0 733.069 −1571.79 0 2961.68
1.5 17.8260 0 189.767 31.0428 0 −1247.35 1101.05 0 553.369
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(23\) \(1\)

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 207.8.a.a 5
3.b odd 2 1 69.8.a.a 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.8.a.a 5 3.b odd 2 1
207.8.a.a 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} - 455 T_{2}^{3} + 474 T_{2}^{2} + 42284 T_{2} - 127016 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(207))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -127016 + 42284 T + 474 T^{2} - 455 T^{3} + T^{5} \)
$3$ \( T^{5} \)
$5$ \( 3615360000 - 260643200 T + 5761760 T^{2} - 28696 T^{3} - 266 T^{4} + T^{5} \)
$7$ \( -194204974150720 + 1423528608720 T - 510098608 T^{2} - 2361144 T^{3} + 496 T^{4} + T^{5} \)
$11$ \( -154691619430400 + 8190801003520 T + 8448479456 T^{2} - 17900584 T^{3} - 1148 T^{4} + T^{5} \)
$13$ \( 16949964907397805984 + 8797382529202512 T - 46113337584 T^{2} - 206597592 T^{3} + 642 T^{4} + T^{5} \)
$17$ \( -\)\(11\!\cdots\!80\)\( + 269929356591517120 T + 11180076292112 T^{2} - 1285951984 T^{3} - 5798 T^{4} + T^{5} \)
$19$ \( -\)\(14\!\cdots\!36\)\( + 185171606250158832 T + 3874831298480 T^{2} - 976972672 T^{3} + 6036 T^{4} + T^{5} \)
$23$ \( ( 12167 + T )^{5} \)
$29$ \( -\)\(24\!\cdots\!88\)\( - 20455752789771350832 T + 1428794454126960 T^{2} - 5923396536 T^{3} - 169162 T^{4} + T^{5} \)
$31$ \( -\)\(23\!\cdots\!28\)\( - \)\(10\!\cdots\!88\)\( T - 13426584591784320 T^{2} - 43994925216 T^{3} + 199640 T^{4} + T^{5} \)
$37$ \( \)\(82\!\cdots\!20\)\( + \)\(80\!\cdots\!60\)\( T - 12266792719102368 T^{2} - 105154566272 T^{3} + 202002 T^{4} + T^{5} \)
$41$ \( -\)\(16\!\cdots\!52\)\( - \)\(70\!\cdots\!52\)\( T - 66617897199338800 T^{2} - 103444637496 T^{3} + 541282 T^{4} + T^{5} \)
$43$ \( \)\(66\!\cdots\!00\)\( + \)\(89\!\cdots\!12\)\( T - 636892817243404752 T^{2} - 720482936448 T^{3} + 909596 T^{4} + T^{5} \)
$47$ \( \)\(24\!\cdots\!00\)\( + \)\(44\!\cdots\!60\)\( T - 143091146720317440 T^{2} - 1392905639296 T^{3} + 80208 T^{4} + T^{5} \)
$53$ \( -\)\(10\!\cdots\!60\)\( + \)\(79\!\cdots\!40\)\( T + 499432731442902464 T^{2} - 3057292616632 T^{3} - 278138 T^{4} + T^{5} \)
$59$ \( -\)\(75\!\cdots\!64\)\( - \)\(53\!\cdots\!24\)\( T + 13664970599888805632 T^{2} - 3306712956160 T^{3} - 3177380 T^{4} + T^{5} \)
$61$ \( -\)\(14\!\cdots\!84\)\( + \)\(22\!\cdots\!08\)\( T - 7818212974425020704 T^{2} - 9781093585984 T^{3} - 147782 T^{4} + T^{5} \)
$67$ \( -\)\(79\!\cdots\!40\)\( + \)\(18\!\cdots\!32\)\( T - 290932021540136880 T^{2} - 27027287614400 T^{3} + 464916 T^{4} + T^{5} \)
$71$ \( -\)\(49\!\cdots\!52\)\( + \)\(15\!\cdots\!48\)\( T + 179997306641763584 T^{2} - 3496700575968 T^{3} + 1576792 T^{4} + T^{5} \)
$73$ \( \)\(11\!\cdots\!64\)\( + \)\(36\!\cdots\!64\)\( T - 1042990056947080848 T^{2} - 7488464098136 T^{3} + 38190 T^{4} + T^{5} \)
$79$ \( -\)\(36\!\cdots\!20\)\( + \)\(60\!\cdots\!88\)\( T - 7813983134821600528 T^{2} - 20007982009272 T^{3} + 3913336 T^{4} + T^{5} \)
$83$ \( -\)\(18\!\cdots\!20\)\( - \)\(12\!\cdots\!40\)\( T - \)\(15\!\cdots\!08\)\( T^{2} + 58536185513432 T^{3} + 15774716 T^{4} + T^{5} \)
$89$ \( \)\(25\!\cdots\!96\)\( + \)\(95\!\cdots\!32\)\( T - \)\(40\!\cdots\!64\)\( T^{2} - 117042154036144 T^{3} + 1116482 T^{4} + T^{5} \)
$97$ \( \)\(11\!\cdots\!20\)\( + \)\(21\!\cdots\!28\)\( T - \)\(62\!\cdots\!88\)\( T^{2} - 22269450612824 T^{3} + 15738566 T^{4} + T^{5} \)
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