Properties

Label 7.10.c.a
Level 7
Weight 10
Character orbit 7.c
Analytic conductor 3.605
Analytic rank 0
Dimension 10
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 7 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 7.c (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(3.60525085315\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{3}\cdot 7^{4} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -\beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{2} \) \( + ( 33 - \beta_{1} - 33 \beta_{3} - \beta_{5} - \beta_{7} ) q^{3} \) \( + ( -191 + 7 \beta_{1} + 191 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{4} \) \( + ( 307 \beta_{3} - \beta_{9} ) q^{5} \) \( + ( -897 + 60 \beta_{2} + 7 \beta_{4} + 11 \beta_{5} - 2 \beta_{6} ) q^{6} \) \( + ( -797 + 26 \beta_{1} - 67 \beta_{2} + 1430 \beta_{3} + 2 \beta_{4} - 9 \beta_{5} + 3 \beta_{6} - 10 \beta_{7} - 10 \beta_{8} - \beta_{9} ) q^{7} \) \( + ( 3462 - 26 \beta_{2} - 10 \beta_{4} - 66 \beta_{5} ) q^{8} \) \( + ( -258 \beta_{1} + 258 \beta_{2} - 7284 \beta_{3} - 86 \beta_{7} + 16 \beta_{8} + 7 \beta_{9} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -\beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{2} \) \( + ( 33 - \beta_{1} - 33 \beta_{3} - \beta_{5} - \beta_{7} ) q^{3} \) \( + ( -191 + 7 \beta_{1} + 191 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{4} \) \( + ( 307 \beta_{3} - \beta_{9} ) q^{5} \) \( + ( -897 + 60 \beta_{2} + 7 \beta_{4} + 11 \beta_{5} - 2 \beta_{6} ) q^{6} \) \( + ( -797 + 26 \beta_{1} - 67 \beta_{2} + 1430 \beta_{3} + 2 \beta_{4} - 9 \beta_{5} + 3 \beta_{6} - 10 \beta_{7} - 10 \beta_{8} - \beta_{9} ) q^{7} \) \( + ( 3462 - 26 \beta_{2} - 10 \beta_{4} - 66 \beta_{5} ) q^{8} \) \( + ( -258 \beta_{1} + 258 \beta_{2} - 7284 \beta_{3} - 86 \beta_{7} + 16 \beta_{8} + 7 \beta_{9} ) q^{9} \) \( + ( 910 - 299 \beta_{1} - 910 \beta_{3} - 14 \beta_{4} + 170 \beta_{5} - 4 \beta_{6} + 170 \beta_{7} + 14 \beta_{8} + 4 \beta_{9} ) q^{10} \) \( + ( 8425 + 3 \beta_{1} - 8425 \beta_{3} - 86 \beta_{4} - 9 \beta_{5} + 7 \beta_{6} - 9 \beta_{7} + 86 \beta_{8} - 7 \beta_{9} ) q^{11} \) \( + ( 2959 \beta_{1} - 2959 \beta_{2} + 28419 \beta_{3} + 341 \beta_{7} - 77 \beta_{8} + 16 \beta_{9} ) q^{12} \) \( + ( -32504 + 1042 \beta_{2} + 28 \beta_{4} + 286 \beta_{5} + 27 \beta_{6} ) q^{13} \) \( + ( 21074 - 3242 \beta_{1} - 1038 \beta_{2} - 47991 \beta_{3} + 50 \beta_{4} - 246 \beta_{5} - 44 \beta_{6} - 789 \beta_{7} + 9 \beta_{8} + 10 \beta_{9} ) q^{14} \) \( + ( 13101 + 6201 \beta_{2} + 86 \beta_{4} - 1055 \beta_{5} - 7 \beta_{6} ) q^{15} \) \( + ( -4560 \beta_{1} + 4560 \beta_{2} + 62344 \beta_{3} - 664 \beta_{7} - 120 \beta_{8} - 112 \beta_{9} ) q^{16} \) \( + ( 68163 - 8798 \beta_{1} - 68163 \beta_{3} + 168 \beta_{4} + 750 \beta_{5} + 65 \beta_{6} + 750 \beta_{7} - 168 \beta_{8} - 65 \beta_{9} ) q^{17} \) \( + ( -208608 + 14898 \beta_{1} + 208608 \beta_{3} + 824 \beta_{4} + 712 \beta_{5} - 112 \beta_{6} + 712 \beta_{7} - 824 \beta_{8} + 112 \beta_{9} ) q^{18} \) \( + ( 19463 \beta_{1} - 19463 \beta_{2} + 4211 \beta_{3} - 1397 \beta_{7} + 826 \beta_{8} - 109 \beta_{9} ) q^{19} \) \( + ( -35363 + 1923 \beta_{2} - 707 \beta_{4} - 1395 \beta_{5} - 128 \beta_{6} ) q^{20} \) \( + ( -25725 - 18130 \beta_{1} - 16758 \beta_{2} - 238875 \beta_{3} - 784 \beta_{4} + 245 \beta_{6} + 2450 \beta_{7} + 588 \beta_{8} ) q^{21} \) \( + ( -22441 + 34660 \beta_{2} - 305 \beta_{4} + 5491 \beta_{5} + 126 \beta_{6} ) q^{22} \) \( + ( -7791 \beta_{1} + 7791 \beta_{2} + 526371 \beta_{3} + 4941 \beta_{7} + 148 \beta_{8} + 728 \beta_{9} ) q^{23} \) \( + ( 1708074 - 38142 \beta_{1} - 1708074 \beta_{3} - 966 \beta_{4} - 7230 \beta_{5} - 432 \beta_{6} - 7230 \beta_{7} + 966 \beta_{8} + 432 \beta_{9} ) q^{24} \) \( + ( -259928 - 1520 \beta_{1} + 259928 \beta_{3} - 2840 \beta_{4} - 2840 \beta_{5} + 714 \beta_{6} - 2840 \beta_{7} + 2840 \beta_{8} - 714 \beta_{9} ) q^{25} \) \( + ( 50734 \beta_{1} - 50734 \beta_{2} + 855644 \beta_{3} + 684 \beta_{7} - 3052 \beta_{8} + 408 \beta_{9} ) q^{26} \) \( + ( -1835079 + 6339 \beta_{2} + 2282 \beta_{4} - 521 \beta_{5} + 131 \beta_{6} ) q^{27} \) \( + ( -2152215 + 33933 \beta_{1} + 11054 \beta_{2} - 152527 \beta_{3} + 3993 \beta_{4} + 10385 \beta_{5} - 496 \beta_{6} + 8455 \beta_{7} - 2591 \beta_{8} - 432 \beta_{9} ) q^{28} \) \( + ( 1530044 - 5930 \beta_{2} + 1268 \beta_{4} + 5538 \beta_{5} - 959 \beta_{6} ) q^{29} \) \( + ( 6210 \beta_{1} - 6210 \beta_{2} + 4115493 \beta_{3} + 1055 \beta_{7} + 485 \beta_{8} - 2254 \beta_{9} ) q^{30} \) \( + ( 3799953 + 90345 \beta_{1} - 3799953 \beta_{3} + 4158 \beta_{4} + 73 \beta_{5} + 1397 \beta_{6} + 73 \beta_{7} - 4158 \beta_{8} - 1397 \beta_{9} ) q^{31} \) \( + ( -1215128 - 81384 \beta_{1} + 1215128 \beta_{3} + 2216 \beta_{4} - 9432 \beta_{5} - 2016 \beta_{6} - 9432 \beta_{7} - 2216 \beta_{8} + 2016 \beta_{9} ) q^{32} \) \( + ( -245136 \beta_{1} + 245136 \beta_{2} + 237315 \beta_{3} - 8312 \beta_{7} + 2296 \beta_{8} - 908 \beta_{9} ) q^{33} \) \( + ( -6302736 + 26341 \beta_{2} + 3892 \beta_{4} + 4116 \beta_{5} + 904 \beta_{6} ) q^{34} \) \( + ( -2273306 + 80213 \beta_{1} + 54341 \beta_{2} - 4262559 \beta_{3} - 8526 \beta_{4} - 33810 \beta_{5} - 833 \beta_{6} - 18375 \beta_{7} - 1078 \beta_{8} + 2499 \beta_{9} ) q^{35} \) \( + ( 7314918 - 388038 \beta_{2} - 6938 \beta_{4} - 37402 \beta_{5} + 3808 \beta_{6} ) q^{36} \) \( + ( 132990 \beta_{1} - 132990 \beta_{2} + 7949757 \beta_{3} - 41494 \beta_{7} + 2868 \beta_{8} + 2030 \beta_{9} ) q^{37} \) \( + ( 13360763 + 221252 \beta_{1} - 13360763 \beta_{3} - 15085 \beta_{4} + 53511 \beta_{5} - 1578 \beta_{6} + 53511 \beta_{7} + 15085 \beta_{8} + 1578 \beta_{9} ) q^{38} \) \( + ( -8898414 + 25270 \beta_{1} + 8898414 \beta_{3} + 8078 \beta_{4} + 74690 \beta_{5} + 371 \beta_{6} + 74690 \beta_{7} - 8078 \beta_{8} - 371 \beta_{9} ) q^{39} \) \( + ( -357926 \beta_{1} + 357926 \beta_{2} + 1028214 \beta_{3} + 62130 \beta_{7} + 13286 \beta_{8} + 1184 \beta_{9} ) q^{40} \) \( + ( -5185700 - 435718 \beta_{2} - 32816 \beta_{4} + 26982 \beta_{5} - 2635 \beta_{6} ) q^{41} \) \( + ( -13136802 + 94374 \beta_{1} + 317667 \beta_{2} - 10695720 \beta_{3} + 1666 \beta_{4} + 22442 \beta_{5} + 6076 \beta_{6} - 92512 \beta_{7} + 27440 \beta_{8} - 5488 \beta_{9} ) q^{42} \) \( + ( 10269764 - 132344 \beta_{2} + 18832 \beta_{4} - 66624 \beta_{5} - 7028 \beta_{6} ) q^{43} \) \( + ( 166623 \beta_{1} - 166623 \beta_{2} + 20018819 \beta_{3} + 48597 \beta_{7} - 26253 \beta_{8} + 7504 \beta_{9} ) q^{44} \) \( + ( 16879164 + 391218 \beta_{1} - 16879164 \beta_{3} + 36148 \beta_{4} - 18730 \beta_{5} - 3209 \beta_{6} - 18730 \beta_{7} - 36148 \beta_{8} + 3209 \beta_{9} ) q^{45} \) \( + ( -2609163 - 582710 \beta_{1} + 2609163 \beta_{3} - 12107 \beta_{4} - 158127 \beta_{5} + 13090 \beta_{6} - 158127 \beta_{7} + 12107 \beta_{8} - 13090 \beta_{9} ) q^{46} \) \( + ( -88975 \beta_{1} + 88975 \beta_{2} + 6414639 \beta_{3} - 113391 \beta_{7} - 31990 \beta_{8} - 129 \beta_{9} ) q^{47} \) \( + ( -18775680 + 769288 \beta_{2} + 45248 \beta_{4} - 90256 \beta_{5} - 2608 \beta_{6} ) q^{48} \) \( + ( -6311697 - 301532 \beta_{1} - 139622 \beta_{2} + 3142748 \beta_{3} + 27328 \beta_{4} - 2534 \beta_{5} - 9037 \beta_{6} + 202244 \beta_{7} - 47264 \beta_{8} - 826 \beta_{9} ) q^{49} \) \( + ( -52660 + 676184 \beta_{2} + 44 \beta_{4} + 290460 \beta_{5} - 2856 \beta_{6} ) q^{50} \) \( + ( -486345 \beta_{1} + 486345 \beta_{2} + 8644815 \beta_{3} - 37389 \beta_{7} + 57234 \beta_{8} - 22953 \beta_{9} ) q^{51} \) \( + ( 21387550 - 1415726 \beta_{1} - 21387550 \beta_{3} - 25634 \beta_{4} - 130050 \beta_{5} + 10720 \beta_{6} - 130050 \beta_{7} + 25634 \beta_{8} - 10720 \beta_{9} ) q^{52} \) \( + ( -5847531 + 1630882 \beta_{1} + 5847531 \beta_{3} - 11844 \beta_{4} + 153342 \beta_{5} - 28672 \beta_{6} + 153342 \beta_{7} + 11844 \beta_{8} + 28672 \beta_{9} ) q^{53} \) \( + ( 2529708 \beta_{1} - 2529708 \beta_{2} + 11291151 \beta_{3} + 90677 \beta_{7} + 1799 \beta_{8} - 6130 \beta_{9} ) q^{54} \) \( + ( -14307503 - 495623 \beta_{2} + 41692 \beta_{4} + 91565 \beta_{5} + 19856 \beta_{6} ) q^{55} \) \( + ( 9077142 + 623692 \beta_{1} - 2086366 \beta_{2} + 6271140 \beta_{3} - 52730 \beta_{4} + 106014 \beta_{5} - 7408 \beta_{6} + 130860 \beta_{7} - 31036 \beta_{8} + 27296 \beta_{9} ) q^{56} \) \( + ( -12832221 + 1674308 \beta_{2} - 98600 \beta_{4} - 112292 \beta_{5} + 41650 \beta_{6} ) q^{57} \) \( + ( -1024314 \beta_{1} + 1024314 \beta_{2} - 9635948 \beta_{3} + 274612 \beta_{7} - 10068 \beta_{8} + 12376 \beta_{9} ) q^{58} \) \( + ( 10188249 - 1888789 \beta_{1} - 10188249 \beta_{3} - 107072 \beta_{4} - 244461 \beta_{5} - 2652 \beta_{6} - 244461 \beta_{7} + 107072 \beta_{8} + 2652 \beta_{9} ) q^{59} \) \( + ( 26869857 - 820029 \beta_{1} - 26869857 \beta_{3} - 1519 \beta_{4} - 149975 \beta_{5} - 9520 \beta_{6} - 149975 \beta_{7} + 1519 \beta_{8} + 9520 \beta_{9} ) q^{60} \) \( + ( 307914 \beta_{1} - 307914 \beta_{2} - 22604563 \beta_{3} - 296018 \beta_{7} + 75684 \beta_{8} + 23626 \beta_{9} ) q^{61} \) \( + ( 45842373 + 2282626 \beta_{2} - 97867 \beta_{4} - 57327 \beta_{5} - 13758 \beta_{6} ) q^{62} \) \( + ( 35103588 - 3933990 \beta_{1} + 1330818 \beta_{2} - 6105438 \beta_{3} + 32680 \beta_{4} + 10244 \beta_{5} + 35020 \beta_{6} - 98034 \beta_{7} + 122718 \beta_{8} - 40749 \beta_{9} ) q^{63} \) \( + ( -19537232 + 904752 \beta_{2} + 111408 \beta_{4} + 64368 \beta_{5} - 72576 \beta_{6} ) q^{64} \) \( + ( 1411242 \beta_{1} - 1411242 \beta_{2} - 67114040 \beta_{3} - 56490 \beta_{7} - 110432 \beta_{8} + 27853 \beta_{9} ) q^{65} \) \( + ( -168072516 + 1398237 \beta_{1} + 168072516 \beta_{3} + 275408 \beta_{4} + 595120 \beta_{5} - 15664 \beta_{6} + 595120 \beta_{7} - 275408 \beta_{8} + 15664 \beta_{9} ) q^{66} \) \( + ( -26836615 + 3520839 \beta_{1} + 26836615 \beta_{3} + 45732 \beta_{4} + 133127 \beta_{5} + 106498 \beta_{6} + 133127 \beta_{7} - 45732 \beta_{8} - 106498 \beta_{9} ) q^{67} \) \( + ( 3144119 \beta_{1} - 3144119 \beta_{2} + 7892559 \beta_{3} + 488529 \beta_{7} - 138033 \beta_{8} - 36448 \beta_{9} ) q^{68} \) \( + ( 134567397 - 5336292 \beta_{2} - 49728 \beta_{4} - 146436 \beta_{5} - 56583 \beta_{6} ) q^{69} \) \( + ( 37265529 + 3414712 \beta_{1} + 702660 \beta_{2} + 63096173 \beta_{3} + 68257 \beta_{4} - 374115 \beta_{5} - 28910 \beta_{6} - 853825 \beta_{7} - 13867 \beta_{8} - 18326 \beta_{9} ) q^{70} \) \( + ( -118841568 + 618504 \beta_{2} + 247788 \beta_{4} - 577176 \beta_{5} + 2478 \beta_{6} ) q^{71} \) \( + ( -3874644 \beta_{1} + 3874644 \beta_{2} - 192184524 \beta_{3} - 1384836 \beta_{7} + 116436 \beta_{8} - 18816 \beta_{9} ) q^{72} \) \( + ( -173198403 + 2667688 \beta_{1} + 173198403 \beta_{3} - 81536 \beta_{4} + 681424 \beta_{5} - 22698 \beta_{6} + 681424 \beta_{7} + 81536 \beta_{8} + 22698 \beta_{9} ) q^{73} \) \( + ( 120976554 - 6502975 \beta_{1} - 120976554 \beta_{3} + 135790 \beta_{4} + 77382 \beta_{5} - 69132 \beta_{6} + 77382 \beta_{7} - 135790 \beta_{8} + 69132 \beta_{9} ) q^{74} \) \( + ( -11219744 \beta_{1} + 11219744 \beta_{2} - 38065836 \beta_{3} + 496520 \beta_{7} + 151564 \beta_{8} - 13718 \beta_{9} ) q^{75} \) \( + ( 107515751 + 5990733 \beta_{2} - 142569 \beta_{4} + 429807 \beta_{5} + 87696 \beta_{6} ) q^{76} \) \( + ( 190811013 - 2401520 \beta_{1} + 3572614 \beta_{2} - 983785 \beta_{3} - 261140 \beta_{4} - 606342 \beta_{5} + 9180 \beta_{6} - 510240 \beta_{7} + 81568 \beta_{8} + 70363 \beta_{9} ) q^{77} \) \( + ( 57522990 - 13203372 \beta_{2} - 454370 \beta_{4} - 1104922 \beta_{5} + 131740 \beta_{6} ) q^{78} \) \( + ( 11509413 \beta_{1} - 11509413 \beta_{2} - 100369637 \beta_{3} + 1186521 \beta_{7} - 190092 \beta_{8} - 7728 \beta_{9} ) q^{79} \) \( + ( -252792184 + 3541920 \beta_{1} + 252792184 \beta_{3} - 400120 \beta_{4} - 527880 \beta_{5} + 90032 \beta_{6} - 527880 \beta_{7} + 400120 \beta_{8} - 90032 \beta_{9} ) q^{80} \) \( + ( 109718097 + 1207626 \beta_{1} - 109718097 \beta_{3} - 243152 \beta_{4} + 386246 \beta_{5} - 179333 \beta_{6} + 386246 \beta_{7} + 243152 \beta_{8} + 179333 \beta_{9} ) q^{81} \) \( + ( -5363238 \beta_{1} + 5363238 \beta_{2} - 271025884 \beta_{3} - 1325932 \beta_{7} + 212268 \beta_{8} + 130136 \beta_{9} ) q^{82} \) \( + ( -11965020 - 6939344 \beta_{2} + 779044 \beta_{4} + 977208 \beta_{5} + 48406 \beta_{6} ) q^{83} \) \( + ( -118515222 + 18467071 \beta_{1} - 6083497 \beta_{2} + 331413999 \beta_{3} + 201194 \beta_{4} + 1853866 \beta_{5} - 26656 \beta_{6} + 2793441 \beta_{7} - 526897 \beta_{8} + 43904 \beta_{9} ) q^{84} \) \( + ( -117971535 - 6540962 \beta_{2} - 501212 \beta_{4} + 1989570 \beta_{5} - 110432 \beta_{6} ) q^{85} \) \( + ( -6522284 \beta_{1} + 6522284 \beta_{2} - 137616464 \beta_{3} + 1318944 \beta_{7} + 686976 \beta_{8} - 142800 \beta_{9} ) q^{86} \) \( + ( -70609326 + 8113446 \beta_{1} + 70609326 \beta_{3} + 114898 \beta_{4} - 1964398 \beta_{5} + 41041 \beta_{6} - 1964398 \beta_{7} - 114898 \beta_{8} - 41041 \beta_{9} ) q^{87} \) \( + ( 185530650 - 11927534 \beta_{1} - 185530650 \beta_{3} - 430550 \beta_{4} - 403278 \beta_{5} + 139216 \beta_{6} - 403278 \beta_{7} + 430550 \beta_{8} - 139216 \beta_{9} ) q^{88} \) \( + ( -5558276 \beta_{1} + 5558276 \beta_{2} + 330210049 \beta_{3} + 654420 \beta_{7} - 985432 \beta_{8} - 144296 \beta_{9} ) q^{89} \) \( + ( 195712764 + 5212662 \beta_{2} - 125468 \beta_{4} - 2520700 \beta_{5} - 96920 \beta_{6} ) q^{90} \) \( + ( 109658962 - 10990700 \beta_{1} + 14868854 \beta_{2} - 81478180 \beta_{3} + 502250 \beta_{4} + 412678 \beta_{5} - 76881 \beta_{6} + 162092 \beta_{7} - 89768 \beta_{8} - 77812 \beta_{9} ) q^{91} \) \( + ( -135172605 + 10684041 \beta_{2} + 1387667 \beta_{4} + 2452731 \beta_{5} + 28336 \beta_{6} ) q^{92} \) \( + ( 1356734 \beta_{1} - 1356734 \beta_{2} - 85029801 \beta_{3} - 1890534 \beta_{7} - 591932 \beta_{8} + 215992 \beta_{9} ) q^{93} \) \( + ( -48863379 - 13086278 \beta_{1} + 48863379 \beta_{3} + 863485 \beta_{4} - 322695 \beta_{5} - 291278 \beta_{6} - 322695 \beta_{7} - 863485 \beta_{8} + 291278 \beta_{9} ) q^{94} \) \( + ( -241561379 + 9505677 \beta_{1} + 241561379 \beta_{3} + 684952 \beta_{4} - 4148415 \beta_{5} + 326914 \beta_{6} - 4148415 \beta_{7} - 684952 \beta_{8} - 326914 \beta_{9} ) q^{95} \) \( + ( 13074456 \beta_{1} - 13074456 \beta_{2} - 283520904 \beta_{3} - 1174520 \beta_{7} + 439096 \beta_{8} - 39392 \beta_{9} ) q^{96} \) \( + ( 90081768 - 6851474 \beta_{2} - 1237712 \beta_{4} - 1405558 \beta_{5} - 123697 \beta_{6} ) q^{97} \) \( + ( -185733352 - 8106357 \beta_{1} - 13252771 \beta_{2} - 93269792 \beta_{3} - 781704 \beta_{4} - 3896424 \beta_{5} + 306656 \beta_{6} - 1487556 \beta_{7} + 1446564 \beta_{8} - 330232 \beta_{9} ) q^{98} \) \( + ( -200734674 + 25930614 \beta_{2} + 219734 \beta_{4} + 1130938 \beta_{5} - 309295 \beta_{6} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(10q \) \(\mathstrut -\mathstrut 18q^{2} \) \(\mathstrut +\mathstrut 161q^{3} \) \(\mathstrut -\mathstrut 940q^{4} \) \(\mathstrut +\mathstrut 1533q^{5} \) \(\mathstrut -\mathstrut 8708q^{6} \) \(\mathstrut -\mathstrut 1036q^{7} \) \(\mathstrut +\mathstrut 34272q^{8} \) \(\mathstrut -\mathstrut 35734q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut -\mathstrut 18q^{2} \) \(\mathstrut +\mathstrut 161q^{3} \) \(\mathstrut -\mathstrut 940q^{4} \) \(\mathstrut +\mathstrut 1533q^{5} \) \(\mathstrut -\mathstrut 8708q^{6} \) \(\mathstrut -\mathstrut 1036q^{7} \) \(\mathstrut +\mathstrut 34272q^{8} \) \(\mathstrut -\mathstrut 35734q^{9} \) \(\mathstrut +\mathstrut 4298q^{10} \) \(\mathstrut +\mathstrut 42213q^{11} \) \(\mathstrut +\mathstrut 135604q^{12} \) \(\mathstrut -\mathstrut 319676q^{13} \) \(\mathstrut -\mathstrut 39522q^{14} \) \(\mathstrut +\mathstrut 151394q^{15} \) \(\mathstrut +\mathstrut 322064q^{16} \) \(\mathstrut +\mathstrut 324681q^{17} \) \(\mathstrut -\mathstrut 1012868q^{18} \) \(\mathstrut -\mathstrut 16121q^{19} \) \(\mathstrut -\mathstrut 350616q^{20} \) \(\mathstrut -\mathstrut 1557857q^{21} \) \(\mathstrut -\mathstrut 62692q^{22} \) \(\mathstrut +\mathstrut 2638863q^{23} \) \(\mathstrut +\mathstrut 8449728q^{24} \) \(\mathstrut -\mathstrut 1304092q^{25} \) \(\mathstrut +\mathstrut 4179252q^{26} \) \(\mathstrut -\mathstrut 18331558q^{27} \) \(\mathstrut -\mathstrut 22156316q^{28} \) \(\mathstrut +\mathstrut 15292500q^{29} \) \(\mathstrut +\mathstrut 20557942q^{30} \) \(\mathstrut +\mathstrut 19179237q^{31} \) \(\mathstrut -\mathstrut 6263520q^{32} \) \(\mathstrut +\mathstrut 1689359q^{33} \) \(\mathstrut -\mathstrut 62909700q^{34} \) \(\mathstrut -\mathstrut 43746759q^{35} \) \(\mathstrut +\mathstrut 71476528q^{36} \) \(\mathstrut +\mathstrut 39566985q^{37} \) \(\mathstrut +\mathstrut 67365270q^{38} \) \(\mathstrut -\mathstrut 44299486q^{39} \) \(\mathstrut +\mathstrut 5721744q^{40} \) \(\mathstrut -\mathstrut 53436852q^{41} \) \(\mathstrut -\mathstrut 183129856q^{42} \) \(\mathstrut +\mathstrut 101835992q^{43} \) \(\mathstrut +\mathstrut 99704916q^{44} \) \(\mathstrut +\mathstrut 85098230q^{45} \) \(\mathstrut -\mathstrut 14489202q^{46} \) \(\mathstrut +\mathstrut 32509659q^{47} \) \(\mathstrut -\mathstrut 185141600q^{48} \) \(\mathstrut -\mathstrut 49024598q^{49} \) \(\mathstrut +\mathstrut 3328464q^{50} \) \(\mathstrut +\mathstrut 44168403q^{51} \) \(\mathstrut +\mathstrut 103893272q^{52} \) \(\mathstrut -\mathstrut 25714707q^{53} \) \(\mathstrut +\mathstrut 51200926q^{54} \) \(\mathstrut -\mathstrut 144695222q^{55} \) \(\mathstrut +\mathstrut 115352832q^{56} \) \(\mathstrut -\mathstrut 121710346q^{57} \) \(\mathstrut -\mathstrut 46645516q^{58} \) \(\mathstrut +\mathstrut 46776513q^{59} \) \(\mathstrut +\mathstrut 132391756q^{60} \) \(\mathstrut -\mathstrut 113075039q^{61} \) \(\mathstrut +\mathstrut 467465628q^{62} \) \(\mathstrut +\mathstrut 318071530q^{63} \) \(\mathstrut -\mathstrut 192008960q^{64} \) \(\mathstrut -\mathstrut 338113566q^{65} \) \(\mathstrut -\mathstrut 836682602q^{66} \) \(\mathstrut -\mathstrut 126707879q^{67} \) \(\mathstrut +\mathstrut 32262636q^{68} \) \(\mathstrut +\mathstrut 1323616182q^{69} \) \(\mathstrut +\mathstrut 697712470q^{70} \) \(\mathstrut -\mathstrut 1188736032q^{71} \) \(\mathstrut -\mathstrut 950557728q^{72} \) \(\mathstrut -\mathstrut 859257651q^{73} \) \(\mathstrut +\mathstrut 591757530q^{74} \) \(\mathstrut -\mathstrut 169061732q^{75} \) \(\mathstrut +\mathstrut 1101475592q^{76} \) \(\mathstrut +\mathstrut 1911891891q^{77} \) \(\mathstrut +\mathstrut 519432424q^{78} \) \(\mathstrut -\mathstrut 527065417q^{79} \) \(\mathstrut -\mathstrut 1257352656q^{80} \) \(\mathstrut +\mathstrut 551662715q^{81} \) \(\mathstrut -\mathstrut 1341703076q^{82} \) \(\mathstrut -\mathstrut 144863208q^{83} \) \(\mathstrut +\mathstrut 486452204q^{84} \) \(\mathstrut -\mathstrut 1197360222q^{85} \) \(\mathstrut -\mathstrut 678648216q^{86} \) \(\mathstrut -\mathstrut 340781350q^{87} \) \(\mathstrut +\mathstrut 903700608q^{88} \) \(\mathstrut +\mathstrut 1661554797q^{89} \) \(\mathstrut +\mathstrut 1967758744q^{90} \) \(\mathstrut +\mathstrut 726641384q^{91} \) \(\mathstrut -\mathstrut 1301840952q^{92} \) \(\mathstrut -\mathstrut 423057489q^{93} \) \(\mathstrut -\mathstrut 272580882q^{94} \) \(\mathstrut -\mathstrut 1197123495q^{95} \) \(\mathstrut -\mathstrut 1441922272q^{96} \) \(\mathstrut +\mathstrut 869770188q^{97} \) \(\mathstrut -\mathstrut 2404833858q^{98} \) \(\mathstrut -\mathstrut 1900777180q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(x^{9}\mathstrut +\mathstrut \) \(430\) \(x^{8}\mathstrut +\mathstrut \) \(61\) \(x^{7}\mathstrut +\mathstrut \) \(146753\) \(x^{6}\mathstrut +\mathstrut \) \(23608\) \(x^{5}\mathstrut +\mathstrut \) \(16136944\) \(x^{4}\mathstrut +\mathstrut \) \(30575648\) \(x^{3}\mathstrut +\mathstrut \) \(1399072384\) \(x^{2}\mathstrut +\mathstrut \) \(1034227200\) \(x\mathstrut +\mathstrut \) \(761760000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\(174976544647\) \(\nu^{9}\mathstrut +\mathstrut \) \(543280294119\) \(\nu^{8}\mathstrut -\mathstrut \) \(2080323777608\) \(\nu^{7}\mathstrut +\mathstrut \) \(306747495137243\) \(\nu^{6}\mathstrut -\mathstrut \) \(527825025841115\) \(\nu^{5}\mathstrut +\mathstrut \) \(79251473775272906\) \(\nu^{4}\mathstrut -\mathstrut \) \(8455568604050729964\) \(\nu^{3}\mathstrut +\mathstrut \) \(8633477355188858864\) \(\nu^{2}\mathstrut +\mathstrut \) \(6390233594342311200\) \(\nu\mathstrut -\mathstrut \) \(728499442105846008000\)\()/\)\(49\!\cdots\!80\)
\(\beta_{3}\)\(=\)\((\)\(439915122044593\) \(\nu^{9}\mathstrut -\mathstrut \) \(359425911506973\) \(\nu^{8}\mathstrut +\mathstrut \) \(189413411414469730\) \(\nu^{7}\mathstrut +\mathstrut \) \(25877873507020493\) \(\nu^{6}\mathstrut +\mathstrut \) \(64699967753173288309\) \(\nu^{5}\mathstrut +\mathstrut \) \(10142716689341838644\) \(\nu^{4}\mathstrut +\mathstrut \) \(7135341367123388280552\) \(\nu^{3}\mathstrut +\mathstrut \) \(9561128363649180087824\) \(\nu^{2}\mathstrut +\mathstrut \) \(619444498139966557897152\) \(\nu\mathstrut +\mathstrut \) \(457911692363235156681600\)\()/\)\(45\!\cdots\!00\)
\(\beta_{4}\)\(=\)\((\)\(9108600924689\) \(\nu^{9}\mathstrut +\mathstrut \) \(744234087958953\) \(\nu^{8}\mathstrut -\mathstrut \) \(2849814149431096\) \(\nu^{7}\mathstrut +\mathstrut \) \(263155948009015381\) \(\nu^{6}\mathstrut -\mathstrut \) \(723062074883078005\) \(\nu^{5}\mathstrut +\mathstrut \) \(108565778922995107222\) \(\nu^{4}\mathstrut -\mathstrut \) \(451992977525138313948\) \(\nu^{3}\mathstrut +\mathstrut \) \(11826911844414978154768\) \(\nu^{2}\mathstrut +\mathstrut \) \(8753915285372555954400\) \(\nu\mathstrut +\mathstrut \) \(1498083923068056865537920\)\()/\)\(23\!\cdots\!40\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(44987322634457\) \(\nu^{9}\mathstrut -\mathstrut \) \(60129676436289\) \(\nu^{8}\mathstrut +\mathstrut \) \(230247989821048\) \(\nu^{7}\mathstrut -\mathstrut \) \(51400958387300173\) \(\nu^{6}\mathstrut +\mathstrut \) \(58419104028555565\) \(\nu^{5}\mathstrut -\mathstrut \) \(8771467558811176486\) \(\nu^{4}\mathstrut +\mathstrut \) \(1145664200820139449324\) \(\nu^{3}\mathstrut -\mathstrut \) \(955543953106873546384\) \(\nu^{2}\mathstrut -\mathstrut \) \(707264155426816927200\) \(\nu\mathstrut +\mathstrut \) \(261687922576606706402880\)\()/\)\(69\!\cdots\!20\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(10228081391315\) \(\nu^{9}\mathstrut +\mathstrut \) \(1952450458321161\) \(\nu^{8}\mathstrut -\mathstrut \) \(7476304878007352\) \(\nu^{7}\mathstrut +\mathstrut \) \(897435986095250741\) \(\nu^{6}\mathstrut -\mathstrut \) \(1896907038176377685\) \(\nu^{5}\mathstrut +\mathstrut \) \(284815366892851129814\) \(\nu^{4}\mathstrut -\mathstrut \) \(297921422945101274472\) \(\nu^{3}\mathstrut +\mathstrut \) \(31027145658538503980816\) \(\nu^{2}\mathstrut +\mathstrut \) \(22965336024722535832800\) \(\nu\mathstrut +\mathstrut \) \(1379362333342473784891584\)\()/\)\(69\!\cdots\!72\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(24593260288252157\) \(\nu^{9}\mathstrut +\mathstrut \) \(42692492469767637\) \(\nu^{8}\mathstrut -\mathstrut \) \(13545670542188346350\) \(\nu^{7}\mathstrut +\mathstrut \) \(9874684454720909303\) \(\nu^{6}\mathstrut -\mathstrut \) \(5152118620797417711701\) \(\nu^{5}\mathstrut +\mathstrut \) \(2305464026693842938944\) \(\nu^{4}\mathstrut -\mathstrut \) \(830194281200809808698368\) \(\nu^{3}\mathstrut -\mathstrut \) \(538575932743903852034896\) \(\nu^{2}\mathstrut -\mathstrut \) \(81614206561156272103139328\) \(\nu\mathstrut -\mathstrut \) \(60375780584089979592902400\)\()/\)\(15\!\cdots\!00\)
\(\beta_{8}\)\(=\)\((\)\(2067492150283470103\) \(\nu^{9}\mathstrut -\mathstrut \) \(1639595498403371523\) \(\nu^{8}\mathstrut +\mathstrut \) \(883784357122680443350\) \(\nu^{7}\mathstrut +\mathstrut \) \(146171516473387214363\) \(\nu^{6}\mathstrut +\mathstrut \) \(300834508490249091273979\) \(\nu^{5}\mathstrut +\mathstrut \) \(53897632103545345431524\) \(\nu^{4}\mathstrut +\mathstrut \) \(32599014210284667922809672\) \(\nu^{3}\mathstrut +\mathstrut \) \(57713122284011650146364784\) \(\nu^{2}\mathstrut +\mathstrut \) \(2809343727409862629926750912\) \(\nu\mathstrut +\mathstrut \) \(2076654230999456261004489600\)\()/\)\(31\!\cdots\!00\)
\(\beta_{9}\)\(=\)\((\)\(1042602787806930287\) \(\nu^{9}\mathstrut -\mathstrut \) \(1349683788498539907\) \(\nu^{8}\mathstrut +\mathstrut \) \(446731469300652376670\) \(\nu^{7}\mathstrut +\mathstrut \) \(69678488713722111187\) \(\nu^{6}\mathstrut +\mathstrut \) \(152247759899824475413931\) \(\nu^{5}\mathstrut +\mathstrut \) \(26156328254249575048396\) \(\nu^{4}\mathstrut +\mathstrut \) \(16592811419754117068838168\) \(\nu^{3}\mathstrut +\mathstrut \) \(40035573671756716045423216\) \(\nu^{2}\mathstrut +\mathstrut \) \(1433445544800945125082504768\) \(\nu\mathstrut +\mathstrut \) \(1059612788993006122246454400\)\()/\)\(53\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(687\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(27\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(507\) \(\beta_{2}\mathstrut -\mathstrut \) \(375\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(14\) \(\beta_{9}\mathstrut -\mathstrut \) \(169\) \(\beta_{8}\mathstrut +\mathstrut \) \(155\) \(\beta_{7}\mathstrut -\mathstrut \) \(14\) \(\beta_{6}\mathstrut +\mathstrut \) \(155\) \(\beta_{5}\mathstrut +\mathstrut \) \(169\) \(\beta_{4}\mathstrut +\mathstrut \) \(87639\) \(\beta_{3}\mathstrut -\mathstrut \) \(134\) \(\beta_{1}\mathstrut -\mathstrut \) \(87639\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(28\) \(\beta_{9}\mathstrut -\mathstrut \) \(583\) \(\beta_{8}\mathstrut -\mathstrut \) \(11457\) \(\beta_{7}\mathstrut +\mathstrut \) \(99345\) \(\beta_{3}\mathstrut +\mathstrut \) \(142643\) \(\beta_{2}\mathstrut -\mathstrut \) \(142643\) \(\beta_{1}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(12040\) \(\beta_{6}\mathstrut -\mathstrut \) \(89029\) \(\beta_{5}\mathstrut -\mathstrut \) \(107929\) \(\beta_{4}\mathstrut +\mathstrut \) \(71655\) \(\beta_{2}\mathstrut +\mathstrut \) \(49481343\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(9450\) \(\beta_{9}\mathstrut +\mathstrut \) \(115386\) \(\beta_{8}\mathstrut +\mathstrut \) \(1949460\) \(\beta_{7}\mathstrut +\mathstrut \) \(9450\) \(\beta_{6}\mathstrut +\mathstrut \) \(1949460\) \(\beta_{5}\mathstrut -\mathstrut \) \(115386\) \(\beta_{4}\mathstrut -\mathstrut \) \(13417998\) \(\beta_{3}\mathstrut +\mathstrut \) \(21095143\) \(\beta_{1}\mathstrut +\mathstrut \) \(13417998\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(4129692\) \(\beta_{9}\mathstrut +\mathstrut \) \(33731905\) \(\beta_{8}\mathstrut -\mathstrut \) \(25531489\) \(\beta_{7}\mathstrut -\mathstrut \) \(14657685447\) \(\beta_{3}\mathstrut -\mathstrut \) \(22648273\) \(\beta_{2}\mathstrut +\mathstrut \) \(22648273\) \(\beta_{1}\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(8200416\) \(\beta_{6}\mathstrut -\mathstrut \) \(1242725823\) \(\beta_{5}\mathstrut +\mathstrut \) \(81699181\) \(\beta_{4}\mathstrut -\mathstrut \) \(12756581151\) \(\beta_{2}\mathstrut -\mathstrut \) \(8505431979\)\()/4\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
−8.71912 15.1020i
−5.11725 8.86334i
−0.371984 0.644295i
5.89912 + 10.2176i
8.80924 + 15.2580i
−8.71912 + 15.1020i
−5.11725 + 8.86334i
−0.371984 + 0.644295i
5.89912 10.2176i
8.80924 15.2580i
−19.4382 + 33.6680i 113.728 + 196.982i −499.691 865.489i −162.760 + 281.909i −8842.67 5234.95 3598.46i 18947.7 −16026.5 + 27758.7i −6327.54 10959.6i
2.2 −12.2345 + 21.1908i −79.7348 138.105i −43.3662 75.1124i 1014.15 1756.56i 3902.06 −4235.51 4734.35i −10405.9 −2873.78 + 4977.54i 24815.2 + 42981.2i
2.3 −2.74397 + 4.75269i −1.70307 2.94981i 240.941 + 417.323i −828.924 + 1435.74i 18.6927 2822.68 + 5690.88i −5454.36 9835.70 17035.9i −4549.08 7879.24i
2.4 9.79824 16.9710i 104.977 + 181.826i 63.9892 + 110.832i 983.791 1703.98i 4114.37 −5768.52 + 2660.41i 12541.3 −12198.9 + 21129.2i −19278.8 33391.9i
2.5 15.6185 27.0520i −56.7670 98.3234i −231.874 401.617i −239.755 + 415.269i −3546.46 1428.40 6189.77i 1507.26 3396.51 5882.92i 7489.23 + 12971.7i
4.1 −19.4382 33.6680i 113.728 196.982i −499.691 + 865.489i −162.760 281.909i −8842.67 5234.95 + 3598.46i 18947.7 −16026.5 27758.7i −6327.54 + 10959.6i
4.2 −12.2345 21.1908i −79.7348 + 138.105i −43.3662 + 75.1124i 1014.15 + 1756.56i 3902.06 −4235.51 + 4734.35i −10405.9 −2873.78 4977.54i 24815.2 42981.2i
4.3 −2.74397 4.75269i −1.70307 + 2.94981i 240.941 417.323i −828.924 1435.74i 18.6927 2822.68 5690.88i −5454.36 9835.70 + 17035.9i −4549.08 + 7879.24i
4.4 9.79824 + 16.9710i 104.977 181.826i 63.9892 110.832i 983.791 + 1703.98i 4114.37 −5768.52 2660.41i 12541.3 −12198.9 21129.2i −19278.8 + 33391.9i
4.5 15.6185 + 27.0520i −56.7670 + 98.3234i −231.874 + 401.617i −239.755 415.269i −3546.46 1428.40 + 6189.77i 1507.26 3396.51 + 5882.92i 7489.23 12971.7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.5
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.c Even 1 yes

Hecke kernels

There are no other newforms in \(S_{10}^{\mathrm{new}}(7, [\chi])\).