Properties

Label 21.7.f.b
Level $21$
Weight $7$
Character orbit 21.f
Analytic conductor $4.831$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 21.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.83113575602\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - x^{7} + 212 x^{6} - 787 x^{5} + 38792 x^{4} - 92833 x^{3} + 1563109 x^{2} + 3107772 x + 38787984\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} - \beta_{2} ) q^{2} + ( 9 + 9 \beta_{2} ) q^{3} + ( -43 + 43 \beta_{2} - \beta_{5} + \beta_{6} ) q^{4} + ( -6 + 3 \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{5} + ( 9 - 18 \beta_{1} - 18 \beta_{2} + 9 \beta_{3} ) q^{6} + ( 77 + 5 \beta_{1} + 35 \beta_{2} - 10 \beta_{3} - \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{7} + ( -61 + 2 \beta_{1} + 14 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 4 \beta_{7} ) q^{8} + 243 \beta_{2} q^{9} +O(q^{10})\) \( q + ( -\beta_{1} - \beta_{2} ) q^{2} + ( 9 + 9 \beta_{2} ) q^{3} + ( -43 + 43 \beta_{2} - \beta_{5} + \beta_{6} ) q^{4} + ( -6 + 3 \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{5} + ( 9 - 18 \beta_{1} - 18 \beta_{2} + 9 \beta_{3} ) q^{6} + ( 77 + 5 \beta_{1} + 35 \beta_{2} - 10 \beta_{3} - \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{7} + ( -61 + 2 \beta_{1} + 14 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 4 \beta_{7} ) q^{8} + 243 \beta_{2} q^{9} + ( 9 + 48 \beta_{1} + 9 \beta_{2} + 34 \beta_{3} + 3 \beta_{5} + 3 \beta_{6} + 14 \beta_{7} ) q^{10} + ( -265 - \beta_{1} + 265 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 9 \beta_{5} + 9 \beta_{6} - \beta_{7} ) q^{11} + ( -774 + 387 \beta_{2} - 18 \beta_{5} + 9 \beta_{6} ) q^{12} + ( 864 - 11 \beta_{1} - 1728 \beta_{2} + 14 \beta_{3} - 17 \beta_{4} + 5 \beta_{5} - 10 \beta_{6} ) q^{13} + ( 511 - 159 \beta_{1} + 546 \beta_{2} - 53 \beta_{3} + 22 \beta_{4} + 32 \beta_{5} - 19 \beta_{6} - 2 \beta_{7} ) q^{14} + ( -81 - 9 \beta_{1} - 9 \beta_{4} - 27 \beta_{5} - 18 \beta_{7} ) q^{15} + ( 184 \beta_{1} + 933 \beta_{2} + 10 \beta_{4} + 5 \beta_{6} - 10 \beta_{7} ) q^{16} + ( 626 - 124 \beta_{1} + 626 \beta_{2} - 70 \beta_{3} + 24 \beta_{5} + 24 \beta_{6} - 54 \beta_{7} ) q^{17} + ( 243 - 243 \beta_{1} - 243 \beta_{2} + 243 \beta_{3} ) q^{18} + ( -3908 + 360 \beta_{1} + 1954 \beta_{2} - 720 \beta_{3} - 27 \beta_{4} + 2 \beta_{5} - \beta_{6} - 27 \beta_{7} ) q^{19} + ( 3945 - 114 \beta_{1} - 7890 \beta_{2} + 24 \beta_{3} + 66 \beta_{4} + 45 \beta_{5} - 90 \beta_{6} ) q^{20} + ( 378 - 9 \beta_{1} + 1323 \beta_{2} - 108 \beta_{3} - 36 \beta_{4} + 45 \beta_{5} - 9 \beta_{6} + 9 \beta_{7} ) q^{21} + ( -169 + 26 \beta_{1} + 570 \beta_{3} + 26 \beta_{4} - 11 \beta_{5} + 52 \beta_{7} ) q^{22} + ( 1302 \beta_{1} - 4144 \beta_{2} + 4 \beta_{4} + 54 \beta_{6} - 4 \beta_{7} ) q^{23} + ( -549 + 180 \beta_{1} - 549 \beta_{2} + 126 \beta_{3} - 27 \beta_{5} - 27 \beta_{6} + 54 \beta_{7} ) q^{24} + ( 5198 - 1399 \beta_{1} - 5198 \beta_{2} + 1398 \beta_{3} + 2 \beta_{4} + 81 \beta_{5} - 81 \beta_{6} + \beta_{7} ) q^{25} + ( -2732 + 1093 \beta_{1} + 1366 \beta_{2} - 2186 \beta_{3} + 106 \beta_{4} + 206 \beta_{5} - 103 \beta_{6} + 106 \beta_{7} ) q^{26} + ( -2187 + 4374 \beta_{2} ) q^{27} + ( -7952 - 1174 \beta_{1} + 15659 \beta_{2} - 312 \beta_{3} - 76 \beta_{4} - 290 \beta_{5} + 191 \beta_{6} - 86 \beta_{7} ) q^{28} + ( 3949 - 61 \beta_{1} + 466 \beta_{3} - 61 \beta_{4} + 183 \beta_{5} - 122 \beta_{7} ) q^{29} + ( 1170 \beta_{1} + 243 \beta_{2} - 126 \beta_{4} + 81 \beta_{6} + 126 \beta_{7} ) q^{30} + ( 3401 + 78 \beta_{1} + 3401 \beta_{2} + 122 \beta_{3} - 282 \beta_{5} - 282 \beta_{6} - 44 \beta_{7} ) q^{31} + ( 17073 - 1978 \beta_{1} - 17073 \beta_{2} + 1920 \beta_{3} + 116 \beta_{4} - 183 \beta_{5} + 183 \beta_{6} + 58 \beta_{7} ) q^{32} + ( -4770 - 18 \beta_{1} + 2385 \beta_{2} + 36 \beta_{3} - 27 \beta_{4} - 162 \beta_{5} + 81 \beta_{6} - 27 \beta_{7} ) q^{33} + ( -10238 - 3216 \beta_{1} + 20476 \beta_{2} + 1752 \beta_{3} - 288 \beta_{4} - 466 \beta_{5} + 932 \beta_{6} ) q^{34} + ( -34419 - 1529 \beta_{1} + 9786 \beta_{2} - 2136 \beta_{3} + 9 \beta_{4} - 293 \beta_{5} - 38 \beta_{6} + 94 \beta_{7} ) q^{35} + ( -10449 - 243 \beta_{5} ) q^{36} + ( 2756 \beta_{1} - 11970 \beta_{2} + 71 \beta_{4} - 475 \beta_{6} - 71 \beta_{7} ) q^{37} + ( 41396 + 1661 \beta_{1} + 41396 \beta_{2} + 1451 \beta_{3} + 525 \beta_{5} + 525 \beta_{6} + 210 \beta_{7} ) q^{38} + ( 23328 - 225 \beta_{1} - 23328 \beta_{2} + 378 \beta_{3} - 306 \beta_{4} + 135 \beta_{5} - 135 \beta_{6} - 153 \beta_{7} ) q^{39} + ( -14298 + 3056 \beta_{1} + 7149 \beta_{2} - 6112 \beta_{3} + 98 \beta_{4} - 186 \beta_{5} + 93 \beta_{6} + 98 \beta_{7} ) q^{40} + ( 6362 + 424 \beta_{1} - 12724 \beta_{2} - 340 \beta_{3} + 256 \beta_{4} + 436 \beta_{5} - 872 \beta_{6} ) q^{41} + ( -315 - 3141 \beta_{1} + 14427 \beta_{2} + 738 \beta_{3} + 414 \beta_{4} + 459 \beta_{5} - 54 \beta_{6} + 180 \beta_{7} ) q^{42} + ( -57678 + 257 \beta_{1} + 6060 \beta_{3} + 257 \beta_{4} + 1051 \beta_{5} + 514 \beta_{7} ) q^{43} + ( 1064 \beta_{1} - 47319 \beta_{2} + 122 \beta_{4} - 495 \beta_{6} - 122 \beta_{7} ) q^{44} + ( -729 - 243 \beta_{1} - 729 \beta_{2} - 243 \beta_{5} - 243 \beta_{6} - 243 \beta_{7} ) q^{45} + ( 132004 + 3440 \beta_{1} - 132004 \beta_{2} - 3516 \beta_{3} + 152 \beta_{4} + 1076 \beta_{5} - 1076 \beta_{6} + 76 \beta_{7} ) q^{46} + ( 13376 - 1702 \beta_{1} - 6688 \beta_{2} + 3404 \beta_{3} - 770 \beta_{4} - 112 \beta_{5} + 56 \beta_{6} - 770 \beta_{7} ) q^{47} + ( -8397 + 3402 \beta_{1} + 16794 \beta_{2} - 1836 \beta_{3} + 270 \beta_{4} - 45 \beta_{5} + 90 \beta_{6} ) q^{48} + ( -20356 + 9179 \beta_{1} + 45535 \beta_{2} - 2762 \beta_{3} - 139 \beta_{4} + 1187 \beta_{5} - 586 \beta_{6} - 240 \beta_{7} ) q^{49} + ( -150284 - 170 \beta_{1} - 6623 \beta_{3} - 170 \beta_{4} - 1173 \beta_{5} - 340 \beta_{7} ) q^{50} + ( -2862 \beta_{1} + 16902 \beta_{2} + 486 \beta_{4} + 648 \beta_{6} - 486 \beta_{7} ) q^{51} + ( 61172 - 4182 \beta_{1} + 61172 \beta_{2} - 3804 \beta_{3} + 446 \beta_{5} + 446 \beta_{6} - 378 \beta_{7} ) q^{52} + ( -3055 - 1947 \beta_{1} + 3055 \beta_{2} + 2096 \beta_{3} - 298 \beta_{4} - 2463 \beta_{5} + 2463 \beta_{6} - 149 \beta_{7} ) q^{53} + ( 4374 - 2187 \beta_{1} - 2187 \beta_{2} + 4374 \beta_{3} ) q^{54} + ( 44475 - 3319 \beta_{1} - 88950 \beta_{2} + 1474 \beta_{3} + 371 \beta_{4} + 345 \beta_{5} - 690 \beta_{6} ) q^{55} + ( -44681 - 2652 \beta_{1} + 116501 \beta_{2} + 11212 \beta_{3} - 24 \beta_{4} - 509 \beta_{5} - 293 \beta_{6} + 34 \beta_{7} ) q^{56} + ( -52758 - 243 \beta_{1} - 9720 \beta_{3} - 243 \beta_{4} + 27 \beta_{5} - 486 \beta_{7} ) q^{57} + ( -10376 \beta_{1} - 37607 \beta_{2} - 122 \beta_{4} + 1181 \beta_{6} + 122 \beta_{7} ) q^{58} + ( 151147 - 757 \beta_{1} + 151147 \beta_{2} - 1460 \beta_{3} - 377 \beta_{5} - 377 \beta_{6} + 703 \beta_{7} ) q^{59} + ( 106515 - 1242 \beta_{1} - 106515 \beta_{2} + 648 \beta_{3} + 1188 \beta_{4} + 1215 \beta_{5} - 1215 \beta_{6} + 594 \beta_{7} ) q^{60} + ( -67056 - 2144 \beta_{1} + 33528 \beta_{2} + 4288 \beta_{3} + 964 \beta_{4} + 104 \beta_{5} - 52 \beta_{6} + 964 \beta_{7} ) q^{61} + ( 25589 + 14874 \beta_{1} - 51178 \beta_{2} - 6415 \beta_{3} - 2044 \beta_{4} + 704 \beta_{5} - 1408 \beta_{6} ) q^{62} + ( -8505 - 1458 \beta_{1} + 27216 \beta_{2} - 486 \beta_{3} - 729 \beta_{4} + 486 \beta_{5} + 243 \beta_{6} - 243 \beta_{7} ) q^{63} + ( -176205 - 738 \beta_{1} + 5004 \beta_{3} - 738 \beta_{4} - 3193 \beta_{5} - 1476 \beta_{7} ) q^{64} + ( -17582 \beta_{1} - 93648 \beta_{2} - 1586 \beta_{4} + 1584 \beta_{6} + 1586 \beta_{7} ) q^{65} + ( -1521 + 5832 \beta_{1} - 1521 \beta_{2} + 5130 \beta_{3} - 99 \beta_{5} - 99 \beta_{6} + 702 \beta_{7} ) q^{66} + ( 101116 + 18853 \beta_{1} - 101116 \beta_{2} - 19020 \beta_{3} + 334 \beta_{4} - 1851 \beta_{5} + 1851 \beta_{6} + 167 \beta_{7} ) q^{67} + ( -281604 - 22668 \beta_{1} + 140802 \beta_{2} + 45336 \beta_{3} + 1644 \beta_{4} + 228 \beta_{5} - 114 \beta_{6} + 1644 \beta_{7} ) q^{68} + ( 37296 + 23472 \beta_{1} - 74592 \beta_{2} - 11790 \beta_{3} + 108 \beta_{4} - 486 \beta_{5} + 972 \beta_{6} ) q^{69} + ( -156282 + 33680 \beta_{1} + 400659 \beta_{2} + 9766 \beta_{3} - 58 \beta_{4} - 1044 \beta_{5} + 1683 \beta_{6} + 438 \beta_{7} ) q^{70} + ( 28372 + 1706 \beta_{1} - 7094 \beta_{3} + 1706 \beta_{4} + 1332 \beta_{5} + 3412 \beta_{7} ) q^{71} + ( 4374 \beta_{1} - 14823 \beta_{2} - 486 \beta_{4} - 729 \beta_{6} + 486 \beta_{7} ) q^{72} + ( 77984 - 10023 \beta_{1} + 77984 \beta_{2} - 10218 \beta_{3} - 887 \beta_{5} - 887 \beta_{6} + 195 \beta_{7} ) q^{73} + ( 304420 + 32159 \beta_{1} - 304420 \beta_{2} - 30641 \beta_{3} - 3036 \beta_{4} + 3045 \beta_{5} - 3045 \beta_{6} - 1518 \beta_{7} ) q^{74} + ( 93564 - 12582 \beta_{1} - 46782 \beta_{2} + 25164 \beta_{3} + 27 \beta_{4} + 1458 \beta_{5} - 729 \beta_{6} + 27 \beta_{7} ) q^{75} + ( 58806 - 71982 \beta_{1} - 117612 \beta_{2} + 34440 \beta_{3} + 3102 \beta_{4} + 1200 \beta_{5} - 2400 \beta_{6} ) q^{76} + ( -107856 - 6826 \beta_{1} + 140273 \beta_{2} - 7488 \beta_{3} - 305 \beta_{4} - 2326 \beta_{5} + 1427 \beta_{6} - 293 \beta_{7} ) q^{77} + ( -36882 + 954 \beta_{1} - 29511 \beta_{3} + 954 \beta_{4} + 2781 \beta_{5} + 1908 \beta_{7} ) q^{78} + ( -51686 \beta_{1} - 127061 \beta_{2} + 484 \beta_{4} + 822 \beta_{6} - 484 \beta_{7} ) q^{79} + ( 70377 + 1754 \beta_{1} + 70377 \beta_{2} + 6204 \beta_{3} - 691 \beta_{5} - 691 \beta_{6} - 4450 \beta_{7} ) q^{80} + ( -59049 + 59049 \beta_{2} ) q^{81} + ( 70684 + 20938 \beta_{1} - 35342 \beta_{2} - 41876 \beta_{3} - 4664 \beta_{4} + 224 \beta_{5} - 112 \beta_{6} - 4664 \beta_{7} ) q^{82} + ( -73913 + 49367 \beta_{1} + 147826 \beta_{2} - 25496 \beta_{3} + 1625 \beta_{4} - 4843 \beta_{5} + 9686 \beta_{6} ) q^{83} + ( -212499 - 24624 \beta_{1} + 210294 \beta_{2} + 7668 \beta_{3} - 594 \beta_{4} - 4329 \beta_{5} + 828 \beta_{6} - 1458 \beta_{7} ) q^{84} + ( 225666 - 1618 \beta_{1} + 77016 \beta_{3} - 1618 \beta_{4} + 1962 \beta_{5} - 3236 \beta_{7} ) q^{85} + ( 21721 \beta_{1} - 578108 \beta_{2} + 4158 \beta_{4} - 7533 \beta_{6} - 4158 \beta_{7} ) q^{86} + ( 35541 + 2547 \beta_{1} + 35541 \beta_{2} + 4194 \beta_{3} + 1647 \beta_{5} + 1647 \beta_{6} - 1647 \beta_{7} ) q^{87} + ( 83477 + 28310 \beta_{1} - 83477 \beta_{2} - 28008 \beta_{3} - 604 \beta_{4} - 107 \beta_{5} + 107 \beta_{6} - 302 \beta_{7} ) q^{88} + ( -296972 - 3980 \beta_{1} + 148486 \beta_{2} + 7960 \beta_{3} + 78 \beta_{4} - 252 \beta_{5} + 126 \beta_{6} + 78 \beta_{7} ) q^{89} + ( -2187 + 19926 \beta_{1} + 4374 \beta_{2} - 8262 \beta_{3} - 3402 \beta_{4} - 729 \beta_{5} + 1458 \beta_{6} ) q^{90} + ( -420476 + 47800 \beta_{1} + 96628 \beta_{2} - 22996 \beta_{3} + 5581 \beta_{4} + 3690 \beta_{5} - 5505 \beta_{6} + 3657 \beta_{7} ) q^{91} + ( 8028 - 3016 \beta_{1} - 88428 \beta_{3} - 3016 \beta_{4} + 8832 \beta_{5} - 6032 \beta_{7} ) q^{92} + ( 2502 \beta_{1} + 91827 \beta_{2} + 396 \beta_{4} - 7614 \beta_{6} - 396 \beta_{7} ) q^{93} + ( -153920 + 390 \beta_{1} - 153920 \beta_{2} - 6106 \beta_{3} + 2750 \beta_{5} + 2750 \beta_{6} + 6496 \beta_{7} ) q^{94} + ( 265014 - 78386 \beta_{1} - 265014 \beta_{2} + 72450 \beta_{3} + 11872 \beta_{4} + 9798 \beta_{5} - 9798 \beta_{6} + 5936 \beta_{7} ) q^{95} + ( 307314 - 17280 \beta_{1} - 153657 \beta_{2} + 34560 \beta_{3} + 1566 \beta_{4} - 3294 \beta_{5} + 1647 \beta_{6} + 1566 \beta_{7} ) q^{96} + ( 135633 - 217 \beta_{1} - 271266 \beta_{2} + 3574 \beta_{3} - 6931 \beta_{4} + 1627 \beta_{5} - 3254 \beta_{6} ) q^{97} + ( 1054403 - 42716 \beta_{1} - 676361 \beta_{2} + 15593 \beta_{3} - 778 \beta_{4} + 11266 \beta_{5} - 2669 \beta_{6} - 2434 \beta_{7} ) q^{98} + ( -64395 - 243 \beta_{1} + 486 \beta_{3} - 243 \beta_{4} - 2187 \beta_{5} - 486 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 5q^{2} + 108q^{3} - 173q^{4} - 42q^{5} + 748q^{7} - 454q^{8} + 972q^{9} + O(q^{10}) \) \( 8q - 5q^{2} + 108q^{3} - 173q^{4} - 42q^{5} + 748q^{7} - 454q^{8} + 972q^{9} + 261q^{10} - 1070q^{11} - 4671q^{12} + 6070q^{14} - 756q^{15} + 3911q^{16} + 7212q^{17} + 1215q^{18} - 24606q^{19} + 8154q^{21} - 78q^{22} - 15224q^{23} - 6129q^{24} + 22274q^{25} - 19044q^{26} - 3415q^{28} + 32524q^{29} + 2349q^{30} + 40200q^{31} + 70203q^{32} - 28890q^{33} - 242436q^{35} - 84078q^{36} - 45670q^{37} + 503310q^{38} + 93366q^{39} - 94941q^{40} + 55161q^{42} - 445660q^{43} - 188829q^{44} - 10206q^{45} + 525804q^{46} + 82884q^{47} + 24116q^{49} - 1218884q^{50} + 64908q^{51} + 722856q^{52} - 13034q^{53} + 32805q^{54} + 127061q^{56} - 442908q^{57} - 159501q^{58} + 1810362q^{59} + 429705q^{60} - 392856q^{61} + 38394q^{63} - 1410446q^{64} - 389004q^{65} - 1053q^{66} + 384094q^{67} - 1616346q^{68} + 406005q^{70} + 225688q^{71} - 55161q^{72} + 903078q^{73} + 1185530q^{74} + 601398q^{75} - 327674q^{77} - 342792q^{78} - 559592q^{79} + 847713q^{80} - 236196q^{81} + 347634q^{82} - 879444q^{84} + 1953576q^{85} - 2302402q^{86} + 439074q^{87} + 304887q^{88} - 1770036q^{89} - 2960718q^{91} - 113064q^{92} + 361800q^{93} - 1837620q^{94} + 1160112q^{95} + 1895481q^{96} + 5732467q^{98} - 520020q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} + 212 x^{6} - 787 x^{5} + 38792 x^{4} - 92833 x^{3} + 1563109 x^{2} + 3107772 x + 38787984\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-1585013359 \nu^{7} + 18232571539 \nu^{6} - 303603349712 \nu^{5} + 4245938445433 \nu^{4} - 65749585575908 \nu^{3} + 780366646056751 \nu^{2} - 2109023702500351 \nu + 22182278913510768\)\()/ 23802631772961636 \)
\(\beta_{3}\)\(=\)\((\)\(-8019055 \nu^{7} - 15616321 \nu^{6} - 1444380025 \nu^{5} + 2053828205 \nu^{4} - 305021724904 \nu^{3} - 177516832405 \nu^{2} - 1592248206780 \nu - 29614389599556\)\()/ 11465622241311 \)
\(\beta_{4}\)\(=\)\((\)\(27322708193 \nu^{7} + 1000103862583 \nu^{6} + 22563530233273 \nu^{5} + 219959885092027 \nu^{4} + 2723907868150765 \nu^{3} + 19609334703803473 \nu^{2} + 313020438506182988 \nu + 699314583065439780\)\()/ 11901315886480818 \)
\(\beta_{5}\)\(=\)\((\)\(-39673486 \nu^{7} + 224426993 \nu^{6} - 7145928130 \nu^{5} + 10161113066 \nu^{4} - 1531993215028 \nu^{3} - 878247070906 \nu^{2} - 7877491417656 \nu - 963541759742958\)\()/ 11465622241311 \)
\(\beta_{6}\)\(=\)\((\)\(59472187739 \nu^{7} - 700951590437 \nu^{6} + 11672037067696 \nu^{5} - 218751249599021 \nu^{4} + 2527744179224764 \nu^{3} - 30001211870054633 \nu^{2} + 130709559191228591 \nu - 852798172253616144\)\()/ 11901315886480818 \)
\(\beta_{7}\)\(=\)\((\)\(226304020214 \nu^{7} + 436984015237 \nu^{6} + 31940434175641 \nu^{5} - 171439510285889 \nu^{4} + 4754608612331077 \nu^{3} - 4492583429983241 \nu^{2} - 114813811933007845 \nu + 14206534053079428\)\()/ 11901315886480818 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - \beta_{5} + 2 \beta_{3} + 106 \beta_{2} - 2 \beta_{1} - 106\)
\(\nu^{3}\)\(=\)\(-4 \beta_{7} + 6 \beta_{5} - 2 \beta_{4} - 145 \beta_{3} - 2 \beta_{1} + 252\)
\(\nu^{4}\)\(=\)\(-2 \beta_{7} - 205 \beta_{6} + 2 \beta_{4} - 15886 \beta_{2} + 772 \beta_{1}\)
\(\nu^{5}\)\(=\)\(424 \beta_{7} + 1560 \beta_{6} - 1560 \beta_{5} + 848 \beta_{4} + 24589 \beta_{3} + 90180 \beta_{2} - 25013 \beta_{1} - 90180\)
\(\nu^{6}\)\(=\)\(-304 \beta_{7} + 38461 \beta_{5} - 152 \beta_{4} - 199046 \beta_{3} - 152 \beta_{1} + 2727346\)
\(\nu^{7}\)\(=\)\(75858 \beta_{7} - 355626 \beta_{6} - 75858 \beta_{4} - 22658292 \beta_{2} + 4625113 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
5.73828 + 9.93899i
4.15432 + 7.19549i
−2.30325 3.98935i
−7.08935 12.2791i
5.73828 9.93899i
4.15432 7.19549i
−2.30325 + 3.98935i
−7.08935 + 12.2791i
−6.23828 10.8050i 13.5000 + 7.79423i −45.8323 + 79.3839i −175.367 + 101.248i 194.490i 284.280 + 191.921i 345.159 121.500 + 210.444i 2187.98 + 1263.23i
10.2 −4.65432 8.06151i 13.5000 + 7.79423i −11.3253 + 19.6160i 151.343 87.3778i 145.107i −271.614 209.463i −384.906 121.500 + 210.444i −1408.79 813.368i
10.3 1.80325 + 3.12332i 13.5000 + 7.79423i 25.4966 44.1614i 71.9311 41.5295i 56.2198i 77.0894 + 334.225i 414.723 121.500 + 210.444i 259.420 + 149.776i
10.4 6.58935 + 11.4131i 13.5000 + 7.79423i −54.8390 + 94.9839i −68.9069 + 39.7834i 205.435i 284.244 191.975i −601.976 121.500 + 210.444i −908.103 524.293i
19.1 −6.23828 + 10.8050i 13.5000 7.79423i −45.8323 79.3839i −175.367 101.248i 194.490i 284.280 191.921i 345.159 121.500 210.444i 2187.98 1263.23i
19.2 −4.65432 + 8.06151i 13.5000 7.79423i −11.3253 19.6160i 151.343 + 87.3778i 145.107i −271.614 + 209.463i −384.906 121.500 210.444i −1408.79 + 813.368i
19.3 1.80325 3.12332i 13.5000 7.79423i 25.4966 + 44.1614i 71.9311 + 41.5295i 56.2198i 77.0894 334.225i 414.723 121.500 210.444i 259.420 149.776i
19.4 6.58935 11.4131i 13.5000 7.79423i −54.8390 94.9839i −68.9069 39.7834i 205.435i 284.244 + 191.975i −601.976 121.500 210.444i −908.103 + 524.293i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.7.f.b 8
3.b odd 2 1 63.7.m.c 8
4.b odd 2 1 336.7.bh.b 8
7.b odd 2 1 147.7.f.a 8
7.c even 3 1 147.7.d.a 8
7.c even 3 1 147.7.f.a 8
7.d odd 6 1 inner 21.7.f.b 8
7.d odd 6 1 147.7.d.a 8
21.g even 6 1 63.7.m.c 8
21.g even 6 1 441.7.d.d 8
21.h odd 6 1 441.7.d.d 8
28.f even 6 1 336.7.bh.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.f.b 8 1.a even 1 1 trivial
21.7.f.b 8 7.d odd 6 1 inner
63.7.m.c 8 3.b odd 2 1
63.7.m.c 8 21.g even 6 1
147.7.d.a 8 7.c even 3 1
147.7.d.a 8 7.d odd 6 1
147.7.f.a 8 7.b odd 2 1
147.7.f.a 8 7.c even 3 1
336.7.bh.b 8 4.b odd 2 1
336.7.bh.b 8 28.f even 6 1
441.7.d.d 8 21.g even 6 1
441.7.d.d 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{8} + \cdots\) acting on \(S_{7}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 30470400 - 5045280 T + 1950436 T^{2} + 129428 T^{3} + 39854 T^{4} + 818 T^{5} + 227 T^{6} + 5 T^{7} + T^{8} \)
$3$ \( ( 243 - 27 T + T^{2} )^{4} \)
$5$ \( 54693259182810000 - 24205120650000 T - 9840546578700 T^{2} + 4356625500 T^{3} + 1536505749 T^{4} - 1767906 T^{5} - 41505 T^{6} + 42 T^{7} + T^{8} \)
$7$ \( \)\(19\!\cdots\!01\)\( - 1218053371237015852 T + 3705229535984494 T^{2} - 5061633633224 T^{3} + 8234236703 T^{4} - 43023176 T^{5} + 267694 T^{6} - 748 T^{7} + T^{8} \)
$11$ \( \)\(34\!\cdots\!00\)\( - 18488408967989050800 T + 113509429921169980 T^{2} - 43785002211340 T^{3} + 345504620141 T^{4} + 343064750 T^{5} + 1409111 T^{6} + 1070 T^{7} + T^{8} \)
$13$ \( \)\(34\!\cdots\!00\)\( + 53257929335845404768 T^{2} + 105787465152777 T^{4} + 22315686 T^{6} + T^{8} \)
$17$ \( \)\(22\!\cdots\!00\)\( + \)\(46\!\cdots\!60\)\( T - \)\(83\!\cdots\!72\)\( T^{2} - 23263548118111082880 T^{3} + 3514480347514032 T^{4} + 547654326480 T^{5} - 58598892 T^{6} - 7212 T^{7} + T^{8} \)
$19$ \( \)\(12\!\cdots\!96\)\( + \)\(11\!\cdots\!92\)\( T + \)\(36\!\cdots\!68\)\( T^{2} + 27399467456974423980 T^{3} - 6359170662393867 T^{4} - 670975723710 T^{5} + 174549627 T^{6} + 24606 T^{7} + T^{8} \)
$23$ \( \)\(63\!\cdots\!00\)\( - \)\(71\!\cdots\!00\)\( T + \)\(15\!\cdots\!00\)\( T^{2} + 51980122560355404800 T^{3} + 101423010754761920 T^{4} + 1256314418240 T^{5} + 520738856 T^{6} + 15224 T^{7} + T^{8} \)
$29$ \( ( -39242852020022400 + 9430137809600 T - 442580539 T^{2} - 16262 T^{3} + T^{4} )^{2} \)
$31$ \( \)\(42\!\cdots\!21\)\( + \)\(16\!\cdots\!80\)\( T - \)\(92\!\cdots\!06\)\( T^{2} - \)\(43\!\cdots\!80\)\( T^{3} + 2063880899710463727 T^{4} + 70137742150800 T^{5} - 1206039954 T^{6} - 40200 T^{7} + T^{8} \)
$37$ \( \)\(25\!\cdots\!00\)\( - \)\(12\!\cdots\!00\)\( T + \)\(21\!\cdots\!00\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + 10133981768848670525 T^{4} - 93188930684650 T^{5} + 5175060655 T^{6} + 45670 T^{7} + T^{8} \)
$41$ \( \)\(53\!\cdots\!16\)\( + \)\(87\!\cdots\!08\)\( T^{2} + 50479416076831845984 T^{4} + 11994986352 T^{6} + T^{8} \)
$43$ \( ( -\)\(14\!\cdots\!84\)\( - 2845100102192540 T - 1577056827 T^{2} + 222830 T^{3} + T^{4} )^{2} \)
$47$ \( \)\(30\!\cdots\!00\)\( + \)\(12\!\cdots\!00\)\( T + \)\(18\!\cdots\!00\)\( T^{2} + \)\(87\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!20\)\( T^{4} + 1025758292062320 T^{5} - 10085910828 T^{6} - 82884 T^{7} + T^{8} \)
$53$ \( \)\(23\!\cdots\!00\)\( - \)\(38\!\cdots\!00\)\( T + \)\(25\!\cdots\!64\)\( T^{2} + \)\(27\!\cdots\!44\)\( T^{3} + \)\(20\!\cdots\!21\)\( T^{4} + 946587441999554 T^{5} + 50746432799 T^{6} + 13034 T^{7} + T^{8} \)
$59$ \( \)\(13\!\cdots\!36\)\( - \)\(41\!\cdots\!92\)\( T + \)\(55\!\cdots\!32\)\( T^{2} - \)\(43\!\cdots\!28\)\( T^{3} + \)\(21\!\cdots\!13\)\( T^{4} - 715051889313496398 T^{5} + 1487447487327 T^{6} - 1810362 T^{7} + T^{8} \)
$61$ \( \)\(78\!\cdots\!00\)\( + \)\(11\!\cdots\!40\)\( T + \)\(61\!\cdots\!72\)\( T^{2} + \)\(15\!\cdots\!00\)\( T^{3} - \)\(51\!\cdots\!48\)\( T^{4} - 1477092202992000 T^{5} + 47685396912 T^{6} + 392856 T^{7} + T^{8} \)
$67$ \( \)\(93\!\cdots\!00\)\( + \)\(14\!\cdots\!40\)\( T + \)\(25\!\cdots\!04\)\( T^{2} + \)\(80\!\cdots\!20\)\( T^{3} + \)\(67\!\cdots\!57\)\( T^{4} - 12184102069585486 T^{5} + 192425428651 T^{6} - 384094 T^{7} + T^{8} \)
$71$ \( ( -\)\(25\!\cdots\!12\)\( + 44439430271275520 T - 187952640388 T^{2} - 112844 T^{3} + T^{4} )^{2} \)
$73$ \( \)\(12\!\cdots\!24\)\( + \)\(24\!\cdots\!32\)\( T + \)\(16\!\cdots\!20\)\( T^{2} + \)\(44\!\cdots\!04\)\( T^{3} - \)\(63\!\cdots\!51\)\( T^{4} - 18164126696429538 T^{5} + 291963532599 T^{6} - 903078 T^{7} + T^{8} \)
$79$ \( \)\(58\!\cdots\!25\)\( + \)\(18\!\cdots\!80\)\( T + \)\(57\!\cdots\!34\)\( T^{2} + \)\(12\!\cdots\!12\)\( T^{3} + \)\(36\!\cdots\!07\)\( T^{4} + 233175179655084368 T^{5} + 779488395718 T^{6} + 559592 T^{7} + T^{8} \)
$83$ \( \)\(36\!\cdots\!24\)\( + \)\(24\!\cdots\!96\)\( T^{2} + \)\(12\!\cdots\!25\)\( T^{4} + 2038066317246 T^{6} + T^{8} \)
$89$ \( \)\(13\!\cdots\!44\)\( + \)\(38\!\cdots\!40\)\( T + \)\(51\!\cdots\!48\)\( T^{2} + \)\(40\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!44\)\( T^{4} + 673833907778767344 T^{5} + 1425031860636 T^{6} + 1770036 T^{7} + T^{8} \)
$97$ \( \)\(22\!\cdots\!00\)\( + \)\(33\!\cdots\!12\)\( T^{2} + \)\(14\!\cdots\!01\)\( T^{4} + 2348711138742 T^{6} + T^{8} \)
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