# Properties

 Label 17.10.a.b Level $17$ Weight $10$ Character orbit 17.a Self dual yes Analytic conductor $8.756$ Analytic rank $0$ Dimension $7$ 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: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - x^{6} - 2986 x^{5} + 8252 x^{4} + 2252056 x^{3} - 10388768 x^{2} - 243559296 x - 675998208$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{10}\cdot 3^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 12 + 2 \beta_{1} - \beta_{4} ) q^{3} + ( 341 - 3 \beta_{1} + \beta_{3} - \beta_{4} ) q^{4} + ( 198 - 26 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{5} + ( -1669 - 39 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} - \beta_{5} - \beta_{6} ) q^{6} + ( 1358 - 69 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 14 \beta_{4} + 7 \beta_{5} + 6 \beta_{6} ) q^{7} + ( 2467 - 464 \beta_{1} - 10 \beta_{2} + \beta_{3} + 16 \beta_{4} - 17 \beta_{5} - 8 \beta_{6} ) q^{8} + ( 11655 - 36 \beta_{1} - 15 \beta_{2} - 5 \beta_{3} + 30 \beta_{4} + 14 \beta_{5} - 14 \beta_{6} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 12 + 2 \beta_{1} - \beta_{4} ) q^{3} + ( 341 - 3 \beta_{1} + \beta_{3} - \beta_{4} ) q^{4} + ( 198 - 26 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{5} + ( -1669 - 39 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} - \beta_{5} - \beta_{6} ) q^{6} + ( 1358 - 69 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 14 \beta_{4} + 7 \beta_{5} + 6 \beta_{6} ) q^{7} + ( 2467 - 464 \beta_{1} - 10 \beta_{2} + \beta_{3} + 16 \beta_{4} - 17 \beta_{5} - 8 \beta_{6} ) q^{8} + ( 11655 - 36 \beta_{1} - 15 \beta_{2} - 5 \beta_{3} + 30 \beta_{4} + 14 \beta_{5} - 14 \beta_{6} ) q^{9} + ( 22024 + 45 \beta_{1} + 26 \beta_{2} + 46 \beta_{3} + 67 \beta_{4} + 7 \beta_{5} + 28 \beta_{6} ) q^{10} + ( 19292 + 206 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 59 \beta_{4} - 36 \beta_{5} + 28 \beta_{6} ) q^{11} + ( 27990 + 2436 \beta_{1} + 34 \beta_{2} - 80 \beta_{4} + 72 \beta_{5} - 58 \beta_{6} ) q^{12} + ( 23312 + 2444 \beta_{1} + 15 \beta_{2} - 71 \beta_{3} - 306 \beta_{4} - 22 \beta_{5} + 22 \beta_{6} ) q^{13} + ( 63921 - 336 \beta_{1} - 105 \beta_{2} - 52 \beta_{3} - 46 \beta_{4} - 70 \beta_{5} - 55 \beta_{6} ) q^{14} + ( 22332 + 2044 \beta_{1} - 110 \beta_{2} - 210 \beta_{3} - 550 \beta_{4} - 108 \beta_{5} - 112 \beta_{6} ) q^{15} + ( 209555 - 2152 \beta_{1} + 94 \beta_{2} + 281 \beta_{3} + 980 \beta_{4} + 203 \beta_{5} + 336 \beta_{6} ) q^{16} + 83521 q^{17} + ( 22606 - 7580 \beta_{1} + 360 \beta_{2} + 274 \beta_{3} + 1101 \beta_{4} + 89 \beta_{5} + 186 \beta_{6} ) q^{18} + ( 110184 + 8144 \beta_{1} + 100 \beta_{2} + 264 \beta_{3} + 46 \beta_{4} + 572 \beta_{5} - 572 \beta_{6} ) q^{19} + ( -125858 - 35484 \beta_{1} - 924 \beta_{2} - 254 \beta_{3} + 1152 \beta_{4} - 970 \beta_{5} - 720 \beta_{6} ) q^{20} + ( -488282 - 1724 \beta_{1} + 965 \beta_{2} + 83 \beta_{3} - 2050 \beta_{4} - 842 \beta_{5} + 842 \beta_{6} ) q^{21} + ( -170451 - 27903 \beta_{1} - 447 \beta_{2} - 106 \beta_{3} + 1455 \beta_{4} - 173 \beta_{5} + 401 \beta_{6} ) q^{22} + ( 190922 + 26063 \beta_{1} - 203 \beta_{2} - 141 \beta_{3} + 246 \beta_{4} + 1815 \beta_{5} + 702 \beta_{6} ) q^{23} + ( -1214348 + 17172 \beta_{1} - 332 \beta_{2} - 2296 \beta_{3} - 3972 \beta_{4} + 892 \beta_{5} - 1404 \beta_{6} ) q^{24} + ( 144635 + 43904 \beta_{1} + 52 \beta_{2} + 1472 \beta_{3} - 2056 \beta_{4} + 296 \beta_{5} + 924 \beta_{6} ) q^{25} + ( -2055310 + 13901 \beta_{1} + 1296 \beta_{2} - 2506 \beta_{3} - 2065 \beta_{4} + 803 \beta_{5} + 198 \beta_{6} ) q^{26} + ( -651412 + 12466 \beta_{1} - 2168 \beta_{2} + 1132 \beta_{3} - 6288 \beta_{4} - 4786 \beta_{5} - 2072 \beta_{6} ) q^{27} + ( -476978 - 6878 \beta_{1} + 738 \beta_{2} + 4576 \beta_{3} + 1102 \beta_{4} - 2078 \beta_{5} - 94 \beta_{6} ) q^{28} + ( 135550 + 40114 \beta_{1} + 869 \beta_{2} + 1525 \beta_{3} + 10978 \beta_{4} + 1840 \beta_{5} + 904 \beta_{6} ) q^{29} + ( -1806858 + 97584 \beta_{1} + 6730 \beta_{2} + 1560 \beta_{3} - 140 \beta_{4} + 4692 \beta_{5} + 3638 \beta_{6} ) q^{30} + ( 507130 + 6735 \beta_{1} - 3979 \beta_{2} + 411 \beta_{3} + 6272 \beta_{4} - 2957 \beta_{5} - 5722 \beta_{6} ) q^{31} + ( 718807 - 181628 \beta_{1} - 7234 \beta_{2} - 3379 \beta_{3} + 13832 \beta_{4} - 1021 \beta_{5} + 1472 \beta_{6} ) q^{32} + ( 1697098 + 108892 \beta_{1} - 249 \beta_{2} - 6223 \beta_{3} + 9402 \beta_{4} + 9226 \beta_{5} + 2170 \beta_{6} ) q^{33} -83521 \beta_{1} q^{34} + ( -48804 - 157146 \beta_{1} + 4516 \beta_{2} - 5184 \beta_{3} + 15072 \beta_{4} - 2286 \beta_{5} + 5696 \beta_{6} ) q^{35} + ( 647685 - 127385 \beta_{1} - 4600 \beta_{2} + 1745 \beta_{3} - 43527 \beta_{4} - 11522 \beta_{5} - 1428 \beta_{6} ) q^{36} + ( 2617230 - 45102 \beta_{1} + 6105 \beta_{2} + 5441 \beta_{3} - 4266 \beta_{4} - 5648 \beta_{5} - 7352 \beta_{6} ) q^{37} + ( -7040126 - 84622 \beta_{1} + 7170 \beta_{2} - 12188 \beta_{3} - 36294 \beta_{4} - 46 \beta_{5} - 13254 \beta_{6} ) q^{38} + ( 12339088 - 56830 \beta_{1} - 9932 \beta_{2} - 1588 \beta_{3} - 196 \beta_{4} + 7678 \beta_{5} + 3780 \beta_{6} ) q^{39} + ( 18125498 + 192340 \beta_{1} + 2332 \beta_{2} + 38742 \beta_{3} + 6324 \beta_{4} + 14694 \beta_{5} + 13536 \beta_{6} ) q^{40} + ( 1492278 - 165676 \beta_{1} + 14290 \beta_{2} - 4594 \beta_{3} - 804 \beta_{4} - 5456 \beta_{5} + 7756 \beta_{6} ) q^{41} + ( 2017466 + 365477 \beta_{1} - 18524 \beta_{2} - 13286 \beta_{3} - 85847 \beta_{4} - 1675 \beta_{5} - 11354 \beta_{6} ) q^{42} + ( 3136628 + 42186 \beta_{1} + 11144 \beta_{2} - 16112 \beta_{3} + 26088 \beta_{4} + 1334 \beta_{5} + 15140 \beta_{6} ) q^{43} + ( 13844554 - 98624 \beta_{1} - 8218 \beta_{2} + 34744 \beta_{3} + 96052 \beta_{4} + 13996 \beta_{5} + 2506 \beta_{6} ) q^{44} + ( 15495510 + 362986 \beta_{1} - 19267 \beta_{2} + 8053 \beta_{3} - 31434 \beta_{4} - 2656 \beta_{5} - 14704 \beta_{6} ) q^{45} + ( -21571583 - 269466 \beta_{1} - 3973 \beta_{2} - 37640 \beta_{3} + 98152 \beta_{4} - 25784 \beta_{5} - 5519 \beta_{6} ) q^{46} + ( 8171072 - 451462 \beta_{1} + 1104 \beta_{2} + 31664 \beta_{3} + 93354 \beta_{4} - 12878 \beta_{5} - 26028 \beta_{6} ) q^{47} + ( -29006728 + 1521632 \beta_{1} + 53224 \beta_{2} - 7232 \beta_{3} - 79280 \beta_{4} + 6608 \beta_{5} + 31576 \beta_{6} ) q^{48} + ( 3837823 + 373028 \beta_{1} - 19379 \beta_{2} - 48605 \beta_{3} + 25702 \beta_{4} + 20590 \beta_{5} + 6086 \beta_{6} ) q^{49} + ( -37099512 - 1181463 \beta_{1} - 28152 \beta_{2} - 53552 \beta_{3} + 122776 \beta_{4} - 41768 \beta_{5} - 9752 \beta_{6} ) q^{50} + ( 1002252 + 167042 \beta_{1} - 83521 \beta_{4} ) q^{51} + ( -22656430 + 2467632 \beta_{1} + 21064 \beta_{2} - 2278 \beta_{3} - 68140 \beta_{4} + 48170 \beta_{5} - 31436 \beta_{6} ) q^{52} + ( 17409274 - 446976 \beta_{1} + 21210 \beta_{2} + 74926 \beta_{3} - 132416 \beta_{4} + 27436 \beta_{5} + 4764 \beta_{6} ) q^{53} + ( -13318218 - 35786 \beta_{1} + 40190 \beta_{2} + 66084 \beta_{3} + 147402 \beta_{4} + 34578 \beta_{5} + 64222 \beta_{6} ) q^{54} + ( 5991932 - 1239616 \beta_{1} - 28710 \beta_{2} - 41770 \beta_{3} - 70230 \beta_{4} + 592 \beta_{5} + 23568 \beta_{6} ) q^{55} + ( -27769176 - 1839768 \beta_{1} - 18836 \beta_{2} + 25988 \beta_{3} + 38912 \beta_{4} - 10996 \beta_{5} - 1700 \beta_{6} ) q^{56} + ( 21851348 - 987844 \beta_{1} - 15170 \beta_{2} + 150994 \beta_{3} - 465068 \beta_{4} - 111616 \beta_{5} - 45884 \beta_{6} ) q^{57} + ( -33698520 - 636397 \beta_{1} - 68690 \beta_{2} - 75302 \beta_{3} + 101601 \beta_{4} - 37459 \beta_{5} - 29916 \beta_{6} ) q^{58} + ( 4338396 - 1749298 \beta_{1} - 12644 \beta_{2} - 21988 \beta_{3} - 277836 \beta_{4} - 9518 \beta_{5} + 36004 \beta_{6} ) q^{59} + ( -89857732 + 514276 \beta_{1} - 58420 \beta_{2} - 161720 \beta_{3} - 232580 \beta_{4} - 19492 \beta_{5} - 125268 \beta_{6} ) q^{60} + ( -7146054 + 586486 \beta_{1} + 65505 \beta_{2} - 60195 \beta_{3} + 41082 \beta_{4} + 58500 \beta_{5} + 11900 \beta_{6} ) q^{61} + ( -10483545 + 151632 \beta_{1} + 92097 \beta_{2} + 107652 \beta_{3} + 207158 \beta_{4} + 84286 \beta_{5} + 81471 \beta_{6} ) q^{62} + ( 374174 - 2267759 \beta_{1} + 81943 \beta_{2} - 14431 \beta_{3} + 1011660 \beta_{4} + 138109 \beta_{5} + 13034 \beta_{6} ) q^{63} + ( 45327171 + 452052 \beta_{1} - 23146 \beta_{2} + 175985 \beta_{3} + 386328 \beta_{4} - 99569 \beta_{5} + 51328 \beta_{6} ) q^{64} + ( -17747372 + 1293780 \beta_{1} - 44968 \beta_{2} - 189528 \beta_{3} - 460056 \beta_{4} + 28764 \beta_{5} - 3024 \beta_{6} ) q^{65} + ( -89407334 + 1882163 \beta_{1} + 45224 \beta_{2} - 165226 \beta_{3} + 191415 \beta_{4} - 8357 \beta_{5} - 9154 \beta_{6} ) q^{66} + ( 43085328 + 1297892 \beta_{1} + 26640 \beta_{2} + 3484 \beta_{3} + 610060 \beta_{4} - 70028 \beta_{5} - 2772 \beta_{6} ) q^{67} + ( 28480661 - 250563 \beta_{1} + 83521 \beta_{3} - 83521 \beta_{4} ) q^{68} + ( 53398266 + 4220872 \beta_{1} - 56291 \beta_{2} + 76047 \beta_{3} - 346414 \beta_{4} - 212646 \beta_{5} + 29750 \beta_{6} ) q^{69} + ( 137522490 + 2674638 \beta_{1} - 113126 \beta_{2} + 79444 \beta_{3} - 405422 \beta_{4} + 64762 \beta_{5} + 7938 \beta_{6} ) q^{70} + ( 93694086 - 1768777 \beta_{1} - 112733 \beta_{2} - 19755 \beta_{3} + 594476 \beta_{4} + 91999 \beta_{5} - 102202 \beta_{6} ) q^{71} + ( 93461663 - 22020 \beta_{1} - 54514 \beta_{2} + 152773 \beta_{3} - 434292 \beta_{4} - 24097 \beta_{5} + 48528 \beta_{6} ) q^{72} + ( 43711854 + 444796 \beta_{1} + 70496 \beta_{2} + 8708 \beta_{3} - 149704 \beta_{4} - 71740 \beta_{5} - 120844 \beta_{6} ) q^{73} + ( 35763068 - 2911773 \beta_{1} + 7370 \beta_{2} - 6726 \beta_{3} - 1238047 \beta_{4} + 112525 \beta_{5} - 183400 \beta_{6} ) q^{74} + ( 130611068 + 7020026 \beta_{1} - 53840 \beta_{2} + 1480 \beta_{3} + 860365 \beta_{4} + 200828 \beta_{5} + 19672 \beta_{6} ) q^{75} + ( 15864176 + 13477332 \beta_{1} + 391116 \beta_{2} - 101892 \beta_{3} - 2520748 \beta_{4} + 115264 \beta_{5} + 84612 \beta_{6} ) q^{76} + ( -14744882 - 1848760 \beta_{1} + 105915 \beta_{2} - 62607 \beta_{3} - 1088242 \beta_{4} - 27290 \beta_{5} + 327890 \beta_{6} ) q^{77} + ( 48531346 - 14699542 \beta_{1} + 49530 \beta_{2} + 169916 \beta_{3} + 1474726 \beta_{4} - 170114 \beta_{5} + 196714 \beta_{6} ) q^{78} + ( 136620946 + 2802391 \beta_{1} - 11463 \beta_{2} + 200759 \beta_{3} + 449114 \beta_{4} - 119953 \beta_{5} - 38250 \beta_{6} ) q^{79} + ( -94987878 - 25325304 \beta_{1} - 251884 \beta_{2} - 338754 \beta_{3} + 1250832 \beta_{4} - 435870 \beta_{5} + 12640 \beta_{6} ) q^{80} + ( -53782541 + 5953220 \beta_{1} - 200845 \beta_{2} - 288499 \beta_{3} + 1021074 \beta_{4} + 345802 \beta_{5} - 109102 \beta_{6} ) q^{81} + ( 148866360 + 1777172 \beta_{1} - 186116 \beta_{2} - 75188 \beta_{3} - 1671666 \beta_{4} + 105494 \beta_{5} - 193920 \beta_{6} ) q^{82} + ( -215742436 - 1909818 \beta_{1} + 129096 \beta_{2} + 177744 \beta_{3} + 965128 \beta_{4} - 401926 \beta_{5} - 259028 \beta_{6} ) q^{83} + ( -68478052 + 4090718 \beta_{1} + 253840 \beta_{2} + 9172 \beta_{3} + 2171686 \beta_{4} + 613550 \beta_{5} - 62300 \beta_{6} ) q^{84} + ( 16537158 - 2171546 \beta_{1} - 83521 \beta_{2} - 83521 \beta_{3} - 167042 \beta_{4} ) q^{85} + ( -24438530 + 5849150 \beta_{1} - 200194 \beta_{2} - 280452 \beta_{3} - 661882 \beta_{4} + 122782 \beta_{5} - 23466 \beta_{6} ) q^{86} + ( -231308132 + 4749564 \beta_{1} + 208606 \beta_{2} + 335594 \beta_{3} - 21678 \beta_{4} - 162020 \beta_{5} + 264456 \beta_{6} ) q^{87} + ( 164829420 - 19838428 \beta_{1} - 454052 \beta_{2} + 123336 \beta_{3} + 1993020 \beta_{4} - 592356 \beta_{5} - 271876 \beta_{6} ) q^{88} + ( -280993836 - 5328572 \beta_{1} + 228211 \beta_{2} + 51165 \beta_{3} - 1024942 \beta_{4} + 452218 \beta_{5} + 169898 \beta_{6} ) q^{89} + ( -322979028 - 20425697 \beta_{1} + 366482 \beta_{2} + 45122 \beta_{3} + 1929669 \beta_{4} - 50687 \beta_{5} + 276312 \beta_{6} ) q^{90} + ( -151624560 - 1391840 \beta_{1} + 526030 \beta_{2} - 405626 \beta_{3} - 1253802 \beta_{4} + 192080 \beta_{5} + 756220 \beta_{6} ) q^{91} + ( 123433086 + 33968878 \beta_{1} + 343082 \beta_{2} + 691016 \beta_{3} - 528358 \beta_{4} + 58142 \beta_{5} + 287010 \beta_{6} ) q^{92} + ( -115833522 + 6473568 \beta_{1} - 749761 \beta_{2} + 89157 \beta_{3} - 2601490 \beta_{4} - 270378 \beta_{5} - 634598 \beta_{6} ) q^{93} + ( 364294644 - 18202972 \beta_{1} - 295580 \beta_{2} + 562520 \beta_{3} - 865428 \beta_{4} + 100780 \beta_{5} - 219980 \beta_{6} ) q^{94} + ( -464516808 + 3295152 \beta_{1} - 336936 \beta_{2} - 374776 \beta_{3} - 287932 \beta_{4} - 282616 \beta_{5} - 972664 \beta_{6} ) q^{95} + ( -638951248 + 29321584 \beta_{1} - 387696 \beta_{2} - 1462208 \beta_{3} - 2150896 \beta_{4} - 580816 \beta_{5} - 372400 \beta_{6} ) q^{96} + ( 289042294 - 13616496 \beta_{1} + 129756 \beta_{2} + 524064 \beta_{3} + 834488 \beta_{4} + 960136 \beta_{5} - 23180 \beta_{6} ) q^{97} + ( -312344506 + 21090920 \beta_{1} + 648720 \beta_{2} - 95470 \beta_{3} + 2274381 \beta_{4} + 391129 \beta_{5} + 668946 \beta_{6} ) q^{98} + ( -370196824 + 211100 \beta_{1} - 315246 \beta_{2} + 569038 \beta_{3} - 3700869 \beta_{4} - 932830 \beta_{5} - 107596 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q - q^{2} + 88q^{3} + 2389q^{4} + 1362q^{5} - 11720q^{6} + 9388q^{7} + 16821q^{8} + 81419q^{9} + O(q^{10})$$ $$7q - q^{2} + 88q^{3} + 2389q^{4} + 1362q^{5} - 11720q^{6} + 9388q^{7} + 16821q^{8} + 81419q^{9} + 154226q^{10} + 135536q^{11} + 198160q^{12} + 166122q^{13} + 447252q^{14} + 159048q^{15} + 1463585q^{16} + 584647q^{17} + 149027q^{18} + 777172q^{19} - 917162q^{20} - 3412104q^{21} - 1222520q^{22} + 1357764q^{23} - 8487360q^{24} + 1065785q^{25} - 14379966q^{26} - 4519064q^{27} - 3328892q^{28} + 967002q^{29} - 12558992q^{30} + 3546740q^{31} + 4825461q^{32} + 11928016q^{33} - 83521q^{34} - 530736q^{35} + 4535009q^{36} + 18296498q^{37} - 49363020q^{38} + 86306872q^{39} + 127155062q^{40} + 10285686q^{41} + 14620416q^{42} + 21913204q^{43} + 96696624q^{44} + 108916410q^{45} - 151509484q^{46} + 56639800q^{47} - 201398496q^{48} + 27010351q^{49} - 261150303q^{50} + 7349848q^{51} - 156226378q^{52} + 121813562q^{53} - 93375344q^{54} + 40793128q^{55} - 196175436q^{56} + 153612960q^{57} - 236833910q^{58} + 29222388q^{59} - 628643488q^{60} - 49915846q^{61} - 73506556q^{62} - 2185356q^{63} + 317922057q^{64} - 122633668q^{65} - 624886144q^{66} + 301863420q^{67} + 199531669q^{68} + 379683432q^{69} + 966315960q^{70} + 652473940q^{71} + 655760385q^{72} + 306656342q^{73} + 249173874q^{74} + 919071912q^{75} + 128694700q^{76} - 102442536q^{77} + 323434416q^{78} + 959147884q^{79} - 692173602q^{80} - 374486977q^{81} + 1046441254q^{82} - 1512945268q^{83} - 481790592q^{84} + 113755602q^{85} - 164953236q^{86} - 1612550856q^{87} + 1132038848q^{88} - 1971327114q^{89} - 2284664662q^{90} - 1061062864q^{91} + 901186756q^{92} - 798598936q^{93} + 2534831232q^{94} - 3249631512q^{95} - 4442036640q^{96} + 2006526254q^{97} - 2170640009q^{98} - 2579159272q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - x^{6} - 2986 x^{5} + 8252 x^{4} + 2252056 x^{3} - 10388768 x^{2} - 243559296 x - 675998208$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$8711 \nu^{6} + 479085 \nu^{5} - 21966986 \nu^{4} - 962897524 \nu^{3} + 9962276152 \nu^{2} + 249666517824 \nu + 1595734267008$$$$)/ 3478080000$$ $$\beta_{3}$$ $$=$$ $$($$$$-41053 \nu^{6} + 292545 \nu^{5} + 124232878 \nu^{4} - 1085670148 \nu^{3} - 87332865896 \nu^{2} + 980540221248 \nu + 1729601433216$$$$)/ 6260544000$$ $$\beta_{4}$$ $$=$$ $$($$$$-41053 \nu^{6} + 292545 \nu^{5} + 124232878 \nu^{4} - 1085670148 \nu^{3} - 93593409896 \nu^{2} + 961758589248 \nu + 7069845465216$$$$)/ 6260544000$$ $$\beta_{5}$$ $$=$$ $$($$$$-843359 \nu^{6} + 2902635 \nu^{5} + 2465651834 \nu^{4} - 12878937644 \nu^{3} - 1752000948088 \nu^{2} + 13251504106944 \nu + 113709524082048$$$$)/ 31302720000$$ $$\beta_{6}$$ $$=$$ $$($$$$419317 \nu^{6} - 2816505 \nu^{5} - 1224135742 \nu^{4} + 10192644772 \nu^{3} + 873469755944 \nu^{2} - 8853358980672 \nu - 59384105783424$$$$)/ 10434240000$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{4} + \beta_{3} - 3 \beta_{1} + 853$$ $$\nu^{3}$$ $$=$$ $$8 \beta_{6} + 17 \beta_{5} - 16 \beta_{4} - \beta_{3} + 10 \beta_{2} + 1488 \beta_{1} - 2467$$ $$\nu^{4}$$ $$=$$ $$336 \beta_{6} + 203 \beta_{5} - 556 \beta_{4} + 1817 \beta_{3} + 94 \beta_{2} - 6760 \beta_{1} + 1257619$$ $$\nu^{5}$$ $$=$$ $$14912 \beta_{6} + 35837 \beta_{5} - 46600 \beta_{4} + 1331 \beta_{3} + 27714 \beta_{2} + 2442620 \beta_{1} - 5771223$$ $$\nu^{6}$$ $$=$$ $$911488 \beta_{6} + 420111 \beta_{5} + 535832 \beta_{4} + 3254641 \beta_{3} + 217494 \beta_{2} - 12134956 \beta_{1} + 2057396547$$

## 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
 42.3973 28.6400 16.8116 −4.12962 −5.44491 −34.1532 −43.1213
−42.3973 109.740 1285.53 −2498.37 −4652.67 2872.61 −32795.8 −7640.20 105924.
1.2 −28.6400 243.971 308.250 1776.79 −6987.32 −9598.61 5835.40 39838.7 −50887.2
1.3 −16.8116 −116.887 −229.369 −1103.40 1965.06 −5164.29 12463.6 −6020.47 18549.9
1.4 4.12962 −254.074 −494.946 151.544 −1049.23 9407.97 −4158.31 44870.8 625.818
1.5 5.44491 106.475 −482.353 1303.94 579.746 9199.27 −5414.17 −8346.12 7099.84
1.6 34.1532 169.801 654.438 195.287 5799.26 −356.628 4864.71 9149.54 6669.66
1.7 43.1213 −171.025 1347.45 1536.21 −7374.84 3027.69 36025.5 9566.70 66243.5
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 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.b 7
3.b odd 2 1 153.10.a.f 7
4.b odd 2 1 272.10.a.g 7
17.b even 2 1 289.10.a.b 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.10.a.b 7 1.a even 1 1 trivial
153.10.a.f 7 3.b odd 2 1
272.10.a.g 7 4.b odd 2 1
289.10.a.b 7 17.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{7} + T_{2}^{6} - 2986 T_{2}^{5} - 8252 T_{2}^{4} + 2252056 T_{2}^{3} + 10388768 T_{2}^{2} - 243559296 T_{2} + 675998208$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(17))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$675998208 - 243559296 T + 10388768 T^{2} + 2252056 T^{3} - 8252 T^{4} - 2986 T^{5} + T^{6} + T^{7}$$
$3$ $$2458538542080000 - 24964560910080 T - 298088384256 T^{2} + 3088987488 T^{3} + 9882840 T^{4} - 105728 T^{5} - 88 T^{6} + T^{7}$$
$5$ $$-$$$$29\!\cdots\!00$$$$+ 3598666811889720000 T - 11836046160700000 T^{2} + 3254663233200 T^{3} + 11440345400 T^{4} - 6441308 T^{5} - 1362 T^{6} + T^{7}$$
$7$ $$13\!\cdots\!04$$$$+$$$$29\!\cdots\!24$$$$T - 21870118739669049472 T^{2} + 1394754860681056 T^{3} + 1101314051384 T^{4} - 110675528 T^{5} - 9388 T^{6} + T^{7}$$
$11$ $$-$$$$12\!\cdots\!00$$$$+$$$$89\!\cdots\!80$$$$T +$$$$10\!\cdots\!12$$$$T^{2} - 20473108288822205856 T^{3} + 420218473383624 T^{4} + 2210349776 T^{5} - 135536 T^{6} + T^{7}$$
$13$ $$51\!\cdots\!32$$$$+$$$$71\!\cdots\!52$$$$T -$$$$18\!\cdots\!76$$$$T^{2} -$$$$11\!\cdots\!64$$$$T^{3} + 5366197083279160 T^{4} - 23816588620 T^{5} - 166122 T^{6} + T^{7}$$
$17$ $$( -83521 + T )^{7}$$
$19$ $$35\!\cdots\!00$$$$+$$$$53\!\cdots\!40$$$$T -$$$$82\!\cdots\!08$$$$T^{2} +$$$$82\!\cdots\!84$$$$T^{3} + 1573658158893097216 T^{4} - 1884318745136 T^{5} - 777172 T^{6} + T^{7}$$
$23$ $$31\!\cdots\!72$$$$-$$$$29\!\cdots\!76$$$$T -$$$$11\!\cdots\!64$$$$T^{2} +$$$$89\!\cdots\!52$$$$T^{3} + 7952165690620827416 T^{4} - 6093544607192 T^{5} - 1357764 T^{6} + T^{7}$$
$29$ $$34\!\cdots\!60$$$$-$$$$54\!\cdots\!64$$$$T +$$$$17\!\cdots\!32$$$$T^{2} +$$$$16\!\cdots\!00$$$$T^{3} - 9055408693964232360 T^{4} - 26389618394492 T^{5} - 967002 T^{6} + T^{7}$$
$31$ $$15\!\cdots\!00$$$$+$$$$25\!\cdots\!76$$$$T -$$$$57\!\cdots\!84$$$$T^{2} +$$$$15\!\cdots\!28$$$$T^{3} +$$$$27\!\cdots\!20$$$$T^{4} - 87320100382184 T^{5} - 3546740 T^{6} + T^{7}$$
$37$ $$26\!\cdots\!04$$$$-$$$$37\!\cdots\!60$$$$T -$$$$30\!\cdots\!32$$$$T^{2} +$$$$27\!\cdots\!60$$$$T^{3} +$$$$28\!\cdots\!76$$$$T^{4} - 244849889723324 T^{5} - 18296498 T^{6} + T^{7}$$
$41$ $$11\!\cdots\!40$$$$-$$$$21\!\cdots\!92$$$$T -$$$$85\!\cdots\!60$$$$T^{2} +$$$$15\!\cdots\!24$$$$T^{3} +$$$$15\!\cdots\!12$$$$T^{4} - 908777787122252 T^{5} - 10285686 T^{6} + T^{7}$$
$43$ $$32\!\cdots\!64$$$$-$$$$66\!\cdots\!80$$$$T -$$$$13\!\cdots\!08$$$$T^{2} +$$$$14\!\cdots\!64$$$$T^{3} +$$$$12\!\cdots\!56$$$$T^{4} - 890347107965552 T^{5} - 21913204 T^{6} + T^{7}$$
$47$ $$-$$$$18\!\cdots\!00$$$$+$$$$13\!\cdots\!80$$$$T -$$$$63\!\cdots\!64$$$$T^{2} -$$$$26\!\cdots\!88$$$$T^{3} +$$$$20\!\cdots\!76$$$$T^{4} - 2369048503609792 T^{5} - 56639800 T^{6} + T^{7}$$
$53$ $$-$$$$68\!\cdots\!00$$$$-$$$$64\!\cdots\!76$$$$T -$$$$12\!\cdots\!64$$$$T^{2} +$$$$31\!\cdots\!80$$$$T^{3} +$$$$81\!\cdots\!16$$$$T^{4} - 7137963194574220 T^{5} - 121813562 T^{6} + T^{7}$$
$59$ $$53\!\cdots\!60$$$$-$$$$33\!\cdots\!36$$$$T -$$$$24\!\cdots\!08$$$$T^{2} +$$$$15\!\cdots\!40$$$$T^{3} +$$$$46\!\cdots\!60$$$$T^{4} - 21897186066653360 T^{5} - 29222388 T^{6} + T^{7}$$
$61$ $$-$$$$16\!\cdots\!00$$$$+$$$$26\!\cdots\!60$$$$T -$$$$13\!\cdots\!28$$$$T^{2} +$$$$21\!\cdots\!28$$$$T^{3} +$$$$57\!\cdots\!00$$$$T^{4} - 30822482868012572 T^{5} + 49915846 T^{6} + T^{7}$$
$67$ $$18\!\cdots\!16$$$$+$$$$60\!\cdots\!88$$$$T -$$$$36\!\cdots\!72$$$$T^{2} -$$$$28\!\cdots\!64$$$$T^{3} +$$$$71\!\cdots\!68$$$$T^{4} - 16130575396626608 T^{5} - 301863420 T^{6} + T^{7}$$
$71$ $$-$$$$21\!\cdots\!16$$$$+$$$$61\!\cdots\!28$$$$T -$$$$11\!\cdots\!12$$$$T^{2} -$$$$18\!\cdots\!64$$$$T^{3} +$$$$14\!\cdots\!84$$$$T^{4} + 74845948151756744 T^{5} - 652473940 T^{6} + T^{7}$$
$73$ $$86\!\cdots\!04$$$$-$$$$10\!\cdots\!20$$$$T -$$$$41\!\cdots\!32$$$$T^{2} +$$$$31\!\cdots\!04$$$$T^{3} +$$$$13\!\cdots\!60$$$$T^{4} - 43284425854107020 T^{5} - 306656342 T^{6} + T^{7}$$
$79$ $$-$$$$18\!\cdots\!00$$$$+$$$$44\!\cdots\!84$$$$T +$$$$46\!\cdots\!68$$$$T^{2} -$$$$78\!\cdots\!16$$$$T^{3} -$$$$11\!\cdots\!52$$$$T^{4} + 266761687131878824 T^{5} - 959147884 T^{6} + T^{7}$$
$83$ $$-$$$$61\!\cdots\!12$$$$-$$$$43\!\cdots\!48$$$$T -$$$$53\!\cdots\!68$$$$T^{2} -$$$$24\!\cdots\!48$$$$T^{3} -$$$$33\!\cdots\!88$$$$T^{4} + 442229020241797520 T^{5} + 1512945268 T^{6} + T^{7}$$
$89$ $$-$$$$45\!\cdots\!00$$$$-$$$$23\!\cdots\!00$$$$T -$$$$36\!\cdots\!80$$$$T^{2} -$$$$17\!\cdots\!56$$$$T^{3} -$$$$74\!\cdots\!60$$$$T^{4} + 1038612277666203316 T^{5} + 1971327114 T^{6} + T^{7}$$
$97$ $$15\!\cdots\!00$$$$+$$$$19\!\cdots\!00$$$$T -$$$$11\!\cdots\!20$$$$T^{2} -$$$$55\!\cdots\!76$$$$T^{3} +$$$$31\!\cdots\!16$$$$T^{4} - 711563132961471020 T^{5} - 2006526254 T^{6} + T^{7}$$