# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{10}^{\mathrm{new}}(7, [\chi])$$.