# Properties

 Label 17.10.a.a Level $17$ Weight $10$ Character orbit 17.a Self dual yes Analytic conductor $8.756$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$17$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 17.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.75560921479$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ Defining polynomial: $$x^{5} - 2 x^{4} - 1596 x^{3} + 5754 x^{2} + 488987 x - 2711704$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}\cdot 3$$ 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -7 + \beta_{1} ) q^{2} + ( -48 + 2 \beta_{1} + \beta_{4} ) q^{3} + ( 177 - 17 \beta_{1} + \beta_{3} - 3 \beta_{4} ) q^{4} + ( 309 - 26 \beta_{1} + \beta_{2} - 6 \beta_{3} - 6 \beta_{4} ) q^{5} + ( 1574 - 154 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} - 12 \beta_{4} ) q^{6} + ( -2625 - 44 \beta_{1} + 13 \beta_{2} + 12 \beta_{3} + 13 \beta_{4} ) q^{7} + ( -8547 + 179 \beta_{1} + 28 \beta_{2} - 9 \beta_{3} + 67 \beta_{4} ) q^{8} + ( 2404 - 606 \beta_{1} - 73 \beta_{2} + 50 \beta_{3} - 14 \beta_{4} ) q^{9} +O(q^{10})$$ $$q + ( -7 + \beta_{1} ) q^{2} + ( -48 + 2 \beta_{1} + \beta_{4} ) q^{3} + ( 177 - 17 \beta_{1} + \beta_{3} - 3 \beta_{4} ) q^{4} + ( 309 - 26 \beta_{1} + \beta_{2} - 6 \beta_{3} - 6 \beta_{4} ) q^{5} + ( 1574 - 154 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} - 12 \beta_{4} ) q^{6} + ( -2625 - 44 \beta_{1} + 13 \beta_{2} + 12 \beta_{3} + 13 \beta_{4} ) q^{7} + ( -8547 + 179 \beta_{1} + 28 \beta_{2} - 9 \beta_{3} + 67 \beta_{4} ) q^{8} + ( 2404 - 606 \beta_{1} - 73 \beta_{2} + 50 \beta_{3} - 14 \beta_{4} ) q^{9} + ( -18048 + 560 \beta_{1} - 104 \beta_{3} + 104 \beta_{4} ) q^{10} + ( -13582 - 138 \beta_{1} - 54 \beta_{2} + 48 \beta_{3} - 139 \beta_{4} ) q^{11} + ( -82450 + 3538 \beta_{1} + 304 \beta_{2} - 266 \beta_{3} + 190 \beta_{4} ) q^{12} + ( -31799 - 150 \beta_{1} - 37 \beta_{2} + 198 \beta_{3} - 546 \beta_{4} ) q^{13} + ( -16212 - 2388 \beta_{1} - 368 \beta_{2} + 384 \beta_{3} - 256 \beta_{4} ) q^{14} + ( -139040 + 4156 \beta_{1} + 548 \beta_{2} + 56 \beta_{3} - 354 \beta_{4} ) q^{15} + ( 73093 - 8421 \beta_{1} - 1244 \beta_{2} + 179 \beta_{3} - \beta_{4} ) q^{16} -83521 q^{17} + ( -387255 + 14625 \beta_{1} + 2064 \beta_{2} - 1248 \beta_{3} + 3408 \beta_{4} ) q^{18} + ( -75406 + 4968 \beta_{1} + 234 \beta_{2} - 1832 \beta_{3} + 858 \beta_{4} ) q^{19} + ( 336512 - 28224 \beta_{1} - 1760 \beta_{2} + 2384 \beta_{3} + 976 \beta_{4} ) q^{20} + ( 361375 - 11086 \beta_{1} - 685 \beta_{2} - 754 \beta_{3} - 3310 \beta_{4} ) q^{21} + ( 23518 + 4366 \beta_{1} + 2600 \beta_{2} - 716 \beta_{3} + 2340 \beta_{4} ) q^{22} + ( 329681 - 6084 \beta_{1} + 167 \beta_{2} + 4900 \beta_{3} - 2325 \beta_{4} ) q^{23} + ( 1965510 - 80934 \beta_{1} - 5784 \beta_{2} + 3618 \beta_{3} - 11766 \beta_{4} ) q^{24} + ( 642933 + 31460 \beta_{1} - 2518 \beta_{2} - 4932 \beta_{3} - 1648 \beta_{4} ) q^{25} + ( 134226 + 34018 \beta_{1} + 6048 \beta_{2} + 864 \beta_{3} + 4144 \beta_{4} ) q^{26} + ( -643184 + 117456 \beta_{1} + 6956 \beta_{2} - 5152 \beta_{3} + 472 \beta_{4} ) q^{27} + ( 6524 + 88548 \beta_{1} + 5760 \beta_{2} - 10292 \beta_{3} + 9372 \beta_{4} ) q^{28} + ( 721211 + 24298 \beta_{1} - 8977 \beta_{2} + 2494 \beta_{3} + 5182 \beta_{4} ) q^{29} + ( 3462160 - 150272 \beta_{1} - 10096 \beta_{2} + 14096 \beta_{3} - 22512 \beta_{4} ) q^{30} + ( -1493895 + 87056 \beta_{1} - 10641 \beta_{2} + 11180 \beta_{3} + 34533 \beta_{4} ) q^{31} + ( -1145387 + 92331 \beta_{1} + 16244 \beta_{2} - 23701 \beta_{3} + 17975 \beta_{4} ) q^{32} + ( -2290557 + 72690 \beta_{1} + 9963 \beta_{2} - 8766 \beta_{3} - 7002 \beta_{4} ) q^{33} + ( 584647 - 83521 \beta_{1} ) q^{34} + ( -5284272 - 77964 \beta_{1} + 9164 \beta_{2} + 41232 \beta_{3} + 38410 \beta_{4} ) q^{35} + ( 10198225 - 642225 \beta_{1} - 44416 \beta_{2} + 15521 \beta_{3} - 100067 \beta_{4} ) q^{36} + ( -6287685 + 22746 \beta_{1} + 15899 \beta_{2} - 5938 \beta_{3} - 12574 \beta_{4} ) q^{37} + ( 3848092 - 358532 \beta_{1} - 19808 \beta_{2} - 14752 \beta_{3} - 21536 \beta_{4} ) q^{38} + ( -8522696 + 135080 \beta_{1} + 32468 \beta_{2} - 37048 \beta_{3} - 36940 \beta_{4} ) q^{39} + ( -10976352 + 468960 \beta_{1} + 43968 \beta_{2} + 28672 \beta_{3} + 59520 \beta_{4} ) q^{40} + ( -1421988 - 387124 \beta_{1} - 17594 \beta_{2} - 65172 \beta_{3} - 41192 \beta_{4} ) q^{41} + ( -9158968 + 698216 \beta_{1} + 39904 \beta_{2} - 40592 \beta_{3} + 69696 \beta_{4} ) q^{42} + ( -11492294 + 231296 \beta_{1} - 64242 \beta_{2} + 13184 \beta_{3} + 123526 \beta_{4} ) q^{43} + ( 8739126 - 235702 \beta_{1} - 56336 \beta_{2} + 21246 \beta_{3} - 11738 \beta_{4} ) q^{44} + ( 2788251 - 697662 \beta_{1} - 43881 \beta_{2} + 104862 \beta_{3} - 146550 \beta_{4} ) q^{45} + ( -6834352 + 1010368 \beta_{1} + 34192 \beta_{2} + 60872 \beta_{3} + 18728 \beta_{4} ) q^{46} + ( -3131634 - 723308 \beta_{1} + 2726 \beta_{2} + 102088 \beta_{3} - 22048 \beta_{4} ) q^{47} + ( -21467242 + 2338474 \beta_{1} + 91768 \beta_{2} - 21734 \beta_{3} + 336130 \beta_{4} ) q^{48} + ( -2354982 + 213486 \beta_{1} + 16821 \beta_{2} - 209650 \beta_{3} - 102102 \beta_{4} ) q^{49} + ( 17202051 + 68571 \beta_{1} + 53888 \beta_{2} - 86208 \beta_{3} - 19232 \beta_{4} ) q^{50} + ( 4009008 - 167042 \beta_{1} - 83521 \beta_{4} ) q^{51} + ( 34854146 - 465666 \beta_{1} - 155904 \beta_{2} + 61890 \beta_{3} + 17850 \beta_{4} ) q^{52} + ( -16615966 + 152616 \beta_{1} + 258816 \beta_{2} - 164784 \beta_{3} + 55220 \beta_{4} ) q^{53} + ( 78104580 - 2364012 \beta_{1} - 191328 \beta_{2} + 171480 \beta_{3} - 497928 \beta_{4} ) q^{54} + ( 1354508 - 394556 \beta_{1} - 121312 \beta_{2} + 129312 \beta_{3} + 110278 \beta_{4} ) q^{55} + ( 64080268 - 1399244 \beta_{1} - 65968 \beta_{2} - 129724 \beta_{3} - 296940 \beta_{4} ) q^{56} + ( 27808062 - 1439252 \beta_{1} - 87834 \beta_{2} + 183132 \beta_{3} - 211960 \beta_{4} ) q^{57} + ( 12857952 + 327856 \beta_{1} + 183968 \beta_{2} - 92008 \beta_{3} + 88520 \beta_{4} ) q^{58} + ( -7188478 - 1081736 \beta_{1} - 101866 \beta_{2} + 28696 \beta_{3} + 838738 \beta_{4} ) q^{59} + ( -46939264 + 6076160 \beta_{1} + 198208 \beta_{2} - 208352 \beta_{3} + 961056 \beta_{4} ) q^{60} + ( -15322337 - 601494 \beta_{1} + 252183 \beta_{2} + 190702 \beta_{3} - 679618 \beta_{4} ) q^{61} + ( 66638204 - 4277044 \beta_{1} + 23840 \beta_{2} + 121104 \beta_{3} - 256624 \beta_{4} ) q^{62} + ( -39986065 + 4362816 \beta_{1} + 37441 \beta_{2} - 351572 \beta_{3} + 85811 \beta_{4} ) q^{63} + ( 26922421 - 1487477 \beta_{1} + 8468 \beta_{2} - 2789 \beta_{3} - 694297 \beta_{4} ) q^{64} + ( -7317776 - 2449816 \beta_{1} - 380232 \beta_{2} + 393496 \beta_{3} + 683460 \beta_{4} ) q^{65} + ( 60821316 - 3258828 \beta_{1} - 218160 \beta_{2} + 115296 \beta_{3} - 377712 \beta_{4} ) q^{66} + ( -61385498 + 1506284 \beta_{1} - 104058 \beta_{2} - 346368 \beta_{3} - 351296 \beta_{4} ) q^{67} + ( -14783217 + 1419857 \beta_{1} - 83521 \beta_{3} + 250563 \beta_{4} ) q^{68} + ( -67446455 + 2471206 \beta_{1} + 163157 \beta_{2} - 478126 \beta_{3} + 523578 \beta_{4} ) q^{69} + ( -23188592 - 4344416 \beta_{1} - 362288 \beta_{2} + 741056 \beta_{3} - 280640 \beta_{4} ) q^{70} + ( -96618793 + 3613408 \beta_{1} + 572997 \beta_{2} - 81388 \beta_{3} - 671613 \beta_{4} ) q^{71} + ( -267640899 + 19472787 \beta_{1} + 871836 \beta_{2} - 785577 \beta_{3} + 1728291 \beta_{4} ) q^{72} + ( -58202062 - 413264 \beta_{1} - 551952 \beta_{2} + 652200 \beta_{3} - 1040600 \beta_{4} ) q^{73} + ( 54751612 - 6087540 \beta_{1} - 304736 \beta_{2} + 200648 \beta_{3} - 330696 \beta_{4} ) q^{74} + ( -30423556 - 1292378 \beta_{1} - 92588 \beta_{2} + 462088 \beta_{3} + 542743 \beta_{4} ) q^{75} + ( -208411364 + 5651492 \beta_{1} + 468864 \beta_{2} - 26692 \beta_{3} + 1230796 \beta_{4} ) q^{76} + ( -30391837 + 4369154 \beta_{1} - 318833 \beta_{2} - 241802 \beta_{3} + 84822 \beta_{4} ) q^{77} + ( 142222076 - 10174996 \beta_{1} - 631904 \beta_{2} + 126952 \beta_{3} - 823800 \beta_{4} ) q^{78} + ( -164909903 - 468556 \beta_{1} + 671111 \beta_{2} - 1041436 \beta_{3} + 399047 \beta_{4} ) q^{79} + ( 183928544 - 4263904 \beta_{1} - 515584 \beta_{2} + 560224 \beta_{3} - 3288352 \beta_{4} ) q^{80} + ( 183259756 - 12049086 \beta_{1} + 346643 \beta_{2} - 262678 \beta_{3} - 2418818 \beta_{4} ) q^{81} + ( -221363594 + 545318 \beta_{1} + 491104 \beta_{2} - 1698608 \beta_{3} + 1925936 \beta_{4} ) q^{82} + ( 41276982 - 6790888 \beta_{1} + 96290 \beta_{2} + 987152 \beta_{3} + 1443942 \beta_{4} ) q^{83} + ( 315652680 - 20309000 \beta_{1} - 1326912 \beta_{2} + 1373640 \beta_{3} - 1614808 \beta_{4} ) q^{84} + ( -25807989 + 2171546 \beta_{1} - 83521 \beta_{2} + 501126 \beta_{3} + 501126 \beta_{4} ) q^{85} + ( 242346664 - 22716488 \beta_{1} + 606336 \beta_{2} - 493432 \beta_{3} - 48088 \beta_{4} ) q^{86} + ( 29721792 + 4296804 \beta_{1} - 317700 \beta_{2} + 314808 \beta_{3} + 2494482 \beta_{4} ) q^{87} + ( -208194130 + 12204402 \beta_{1} + 199752 \beta_{2} - 609190 \beta_{3} + 776354 \beta_{4} ) q^{88} + ( 66779497 + 20899250 \beta_{1} + 460683 \beta_{2} + 806402 \beta_{3} + 1206630 \beta_{4} ) q^{89} + ( -460075968 + 31781232 \beta_{1} + 2644992 \beta_{2} - 312552 \beta_{3} + 3727944 \beta_{4} ) q^{90} + ( 34422318 + 12123668 \beta_{1} - 679922 \beta_{2} - 1247592 \beta_{3} + 374208 \beta_{4} ) q^{91} + ( 505380000 - 10506368 \beta_{1} - 812448 \beta_{2} + 6688 \beta_{3} - 2827040 \beta_{4} ) q^{92} + ( 703328359 - 13352182 \beta_{1} - 2485549 \beta_{2} + 1206470 \beta_{3} + 2059406 \beta_{4} ) q^{93} + ( -455043760 + 14752608 \beta_{1} + 519312 \beta_{2} + 710896 \beta_{3} + 2038064 \beta_{4} ) q^{94} + ( 290829908 + 24844352 \beta_{1} + 687740 \beta_{2} - 2235320 \beta_{3} - 886660 \beta_{4} ) q^{95} + ( 599610166 - 35015990 \beta_{1} - 2017000 \beta_{2} + 2505866 \beta_{3} - 4983438 \beta_{4} ) q^{96} + ( 125658398 + 33525784 \beta_{1} + 236504 \beta_{2} - 1535904 \beta_{3} - 5543956 \beta_{4} ) q^{97} + ( 180869073 - 13890359 \beta_{1} - 425488 \beta_{2} - 2623040 \beta_{3} + 21392 \beta_{4} ) q^{98} + ( 413521590 - 19380798 \beta_{1} + 707514 \beta_{2} - 516216 \beta_{3} - 2435865 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q - 33q^{2} - 236q^{3} + 853q^{4} + 1480q^{5} + 7578q^{6} - 13202q^{7} - 42423q^{8} + 10981q^{9} + O(q^{10})$$ $$5q - 33q^{2} - 236q^{3} + 853q^{4} + 1480q^{5} + 7578q^{6} - 13202q^{7} - 42423q^{8} + 10981q^{9} - 89328q^{10} - 68036q^{11} - 406010q^{12} - 158862q^{13} - 84700q^{14} - 687324q^{15} + 350225q^{16} - 417605q^{17} - 1911585q^{18} - 370992q^{19} + 1632640q^{20} + 1783880q^{21} + 122290q^{22} + 1645870q^{23} + 9678702q^{24} + 3270239q^{25} + 734846q^{26} - 2998268q^{27} + 183372q^{28} + 3668616q^{29} + 17048544q^{30} - 7262362q^{31} - 5605919q^{32} - 11334900q^{33} + 2756193q^{34} - 26503988q^{35} + 49782133q^{36} - 31420708q^{37} + 18513700q^{38} - 42449884q^{39} - 53930464q^{40} - 7996938q^{41} - 44519496q^{42} - 56908268q^{43} + 43323054q^{44} + 12799536q^{45} - 32063472q^{46} - 16903336q^{47} - 102794498q^{48} - 11784059q^{49} + 85921093q^{50} + 19710956q^{51} + 173619082q^{52} - 83362982q^{53} + 386329164q^{54} + 6363364q^{55} + 317409372q^{56} + 136615904q^{57} + 64577488q^{58} - 37946604q^{59} - 223158912q^{60} - 77685452q^{61} + 324855300q^{62} - 191945278q^{63} + 131623105q^{64} - 40321288q^{65} + 298037676q^{66} - 304503600q^{67} - 71243413q^{68} - 333409272q^{69} - 122787392q^{70} - 476602922q^{71} - 1301701911q^{72} - 289980486q^{73} + 262289012q^{74} - 153685772q^{75} - 1031276084q^{76} - 143385648q^{77} + 691646196q^{78} - 828240610q^{79} + 912750944q^{80} + 891328609q^{81} - 1109615654q^{82} + 194681148q^{83} + 1541719592q^{84} - 123611080q^{85} + 1164707144q^{86} + 158149884q^{87} - 1017979978q^{88} + 376848106q^{89} - 2240087472q^{90} + 194543664q^{91} + 2506713088q^{92} + 3494835920q^{93} - 2244811104q^{94} + 1498679864q^{95} + 2935047582q^{96} + 692035246q^{97} + 871744055q^{98} + 2027106408q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 1596 x^{3} + 5754 x^{2} + 488987 x - 2711704$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{4} + 207 \nu^{3} + 6301 \nu^{2} - 209023 \nu - 509736$$$$)/8096$$ $$\beta_{3}$$ $$=$$ $$($$$$21 \nu^{4} + 345 \nu^{3} - 24845 \nu^{2} - 323145 \nu + 3450824$$$$)/16192$$ $$\beta_{4}$$ $$=$$ $$($$$$7 \nu^{4} + 115 \nu^{3} - 13679 \nu^{2} - 123907 \nu + 4604568$$$$)/16192$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-3 \beta_{4} + \beta_{3} - 3 \beta_{1} + 640$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{4} + 12 \beta_{3} + 28 \beta_{2} + 993 \beta_{1} - 1932$$ $$\nu^{4}$$ $$=$$ $$-3615 \beta_{4} + 1757 \beta_{3} - 460 \beta_{2} - 4475 \beta_{1} + 624596$$

## 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
 −35.0613 −21.1654 5.77274 18.8209 33.6330
−42.0613 −256.773 1257.15 1407.25 10800.2 −5138.64 −31342.2 46249.6 −59190.7
1.2 −28.1654 −3.02373 281.287 762.851 85.1644 5573.11 6498.11 −19673.9 −21486.0
1.3 −1.22726 177.437 −510.494 −1620.18 −217.762 −1834.42 1254.87 11801.0 1988.39
1.4 11.8209 −67.6654 −372.266 2390.67 −799.867 −11355.8 −10452.8 −15104.4 28259.9
1.5 26.6330 −85.9747 197.318 −1460.58 −2289.77 −446.232 −8380.94 −12291.3 −38899.6
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$17$$ $$1$$

## 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 17.10.a.a 5
3.b odd 2 1 153.10.a.c 5
4.b odd 2 1 272.10.a.f 5
17.b even 2 1 289.10.a.a 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.10.a.a 5 1.a even 1 1 trivial
153.10.a.c 5 3.b odd 2 1
272.10.a.f 5 4.b odd 2 1
289.10.a.a 5 17.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{5} + 33 T_{2}^{4} - 1162 T_{2}^{3} - 24920 T_{2}^{2} + 344192 T_{2} + 457728$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(17))$$.