Properties

Label 289.10.a.b
Level $289$
Weight $10$
Character orbit 289.a
Self dual yes
Analytic conductor $148.845$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 289.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(148.845356651\)
Analytic rank: \(1\)
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: no (minimal twist has level 17)
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} + ( 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} + ( -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} + ( -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} + ( 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} + ( -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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - q^{2} - 88q^{3} + 2389q^{4} - 1362q^{5} + 11720q^{6} - 9388q^{7} + 16821q^{8} + 81419q^{9} + O(q^{10}) \) \( 7q - q^{2} - 88q^{3} + 2389q^{4} - 1362q^{5} + 11720q^{6} - 9388q^{7} + 16821q^{8} + 81419q^{9} - 154226q^{10} - 135536q^{11} - 198160q^{12} + 166122q^{13} - 447252q^{14} + 159048q^{15} + 1463585q^{16} + 149027q^{18} + 777172q^{19} + 917162q^{20} - 3412104q^{21} + 1222520q^{22} - 1357764q^{23} + 8487360q^{24} + 1065785q^{25} - 14379966q^{26} + 4519064q^{27} + 3328892q^{28} - 967002q^{29} - 12558992q^{30} - 3546740q^{31} + 4825461q^{32} + 11928016q^{33} - 530736q^{35} + 4535009q^{36} - 18296498q^{37} - 49363020q^{38} - 86306872q^{39} - 127155062q^{40} - 10285686q^{41} + 14620416q^{42} + 21913204q^{43} - 96696624q^{44} - 108916410q^{45} + 151509484q^{46} + 56639800q^{47} + 201398496q^{48} + 27010351q^{49} - 261150303q^{50} - 156226378q^{52} + 121813562q^{53} + 93375344q^{54} + 40793128q^{55} + 196175436q^{56} - 153612960q^{57} + 236833910q^{58} + 29222388q^{59} - 628643488q^{60} + 49915846q^{61} + 73506556q^{62} + 2185356q^{63} + 317922057q^{64} + 122633668q^{65} - 624886144q^{66} + 301863420q^{67} + 379683432q^{69} + 966315960q^{70} - 652473940q^{71} + 655760385q^{72} - 306656342q^{73} - 249173874q^{74} - 919071912q^{75} + 128694700q^{76} - 102442536q^{77} - 323434416q^{78} - 959147884q^{79} + 692173602q^{80} - 374486977q^{81} - 1046441254q^{82} - 1512945268q^{83} - 481790592q^{84} - 164953236q^{86} - 1612550856q^{87} - 1132038848q^{88} - 1971327114q^{89} + 2284664662q^{90} + 1061062864q^{91} - 901186756q^{92} - 798598936q^{93} + 2534831232q^{94} + 3249631512q^{95} + 4442036640q^{96} - 2006526254q^{97} - 2170640009q^{98} + 2579159272q^{99} + O(q^{100}) \)

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:

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 289.10.a.b 7
17.b even 2 1 17.10.a.b 7
51.c odd 2 1 153.10.a.f 7
68.d odd 2 1 272.10.a.g 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.10.a.b 7 17.b even 2 1
153.10.a.f 7 51.c odd 2 1
272.10.a.g 7 68.d odd 2 1
289.10.a.b 7 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 \(S_{10}^{\mathrm{new}}(\Gamma_0(289))\):

\( T_{2}^{7} + T_{2}^{6} - 2986 T_{2}^{5} - 8252 T_{2}^{4} + 2252056 T_{2}^{3} + 10388768 T_{2}^{2} - 243559296 T_{2} + 675998208 \)
\( T_{3}^{7} + 88 T_{3}^{6} - 105728 T_{3}^{5} - 9882840 T_{3}^{4} + 3088987488 T_{3}^{3} + 298088384256 T_{3}^{2} - \)\(24\!\cdots\!80\)\( T_{3} - \)\(24\!\cdots\!00\)\( \)

Hecke characteristic polynomials

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