# Properties

 Label 7.9.d.a Level 7 Weight 9 Character orbit 7.d Analytic conductor 2.852 Analytic rank 0 Dimension 8 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$7$$ Weight: $$k$$ = $$9$$ Character orbit: $$[\chi]$$ = 7.d (of order $$6$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$2.85165027043$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}\cdot 3\cdot 7^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{2}$$ $$+ ( -7 + 7 \beta_{2} - \beta_{5} + \beta_{6} ) q^{3}$$ $$+ ( -41 + 3 \beta_{1} - 41 \beta_{2} + \beta_{4} - \beta_{6} ) q^{4}$$ $$+ ( -140 + \beta_{1} - 70 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{5}$$ $$+ ( -12 \beta_{1} + 6 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{6}$$ $$+ ( 182 - 21 \beta_{1} + 399 \beta_{2} + 42 \beta_{3} - 7 \beta_{4} - 21 \beta_{5} + 21 \beta_{6} ) q^{7}$$ $$+ ( 818 - 38 \beta_{3} + 14 \beta_{5} - 22 \beta_{6} - 6 \beta_{7} ) q^{8}$$ $$+ ( -93 \beta_{1} - 99 \beta_{2} + 93 \beta_{3} - 15 \beta_{4} + 84 \beta_{5} - 42 \beta_{6} + 15 \beta_{7} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{2}$$ $$+ ( -7 + 7 \beta_{2} - \beta_{5} + \beta_{6} ) q^{3}$$ $$+ ( -41 + 3 \beta_{1} - 41 \beta_{2} + \beta_{4} - \beta_{6} ) q^{4}$$ $$+ ( -140 + \beta_{1} - 70 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{5}$$ $$+ ( -12 \beta_{1} + 6 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{6}$$ $$+ ( 182 - 21 \beta_{1} + 399 \beta_{2} + 42 \beta_{3} - 7 \beta_{4} - 21 \beta_{5} + 21 \beta_{6} ) q^{7}$$ $$+ ( 818 - 38 \beta_{3} + 14 \beta_{5} - 22 \beta_{6} - 6 \beta_{7} ) q^{8}$$ $$+ ( -93 \beta_{1} - 99 \beta_{2} + 93 \beta_{3} - 15 \beta_{4} + 84 \beta_{5} - 42 \beta_{6} + 15 \beta_{7} ) q^{9}$$ $$+ ( 483 + 267 \beta_{1} - 483 \beta_{2} - 534 \beta_{3} + 8 \beta_{4} + 34 \beta_{5} - 26 \beta_{6} - 16 \beta_{7} ) q^{10}$$ $$+ ( 446 + 721 \beta_{1} + 446 \beta_{2} - 9 \beta_{4} - 77 \beta_{5} - 68 \beta_{6} ) q^{11}$$ $$+ ( -6818 - 261 \beta_{1} - 3409 \beta_{2} - 261 \beta_{3} + 5 \beta_{4} + 200 \beta_{6} + 5 \beta_{7} ) q^{12}$$ $$+ ( 5915 - 1602 \beta_{1} + 11830 \beta_{2} + 801 \beta_{3} - 6 \beta_{4} - 98 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{13}$$ $$+ ( -7434 - 154 \beta_{1} - 12404 \beta_{2} + 1960 \beta_{3} + 77 \beta_{4} - 126 \beta_{5} + 112 \beta_{6} - 7 \beta_{7} ) q^{14}$$ $$+ ( 16476 - 447 \beta_{3} - 35 \beta_{5} - 11 \beta_{6} + 81 \beta_{7} ) q^{15}$$ $$+ ( -1062 \beta_{1} + 2146 \beta_{2} + 1062 \beta_{3} + 158 \beta_{4} - 112 \beta_{5} + 56 \beta_{6} - 158 \beta_{7} ) q^{16}$$ $$+ ( -11788 + 2369 \beta_{1} + 11788 \beta_{2} - 4738 \beta_{3} - 81 \beta_{4} - 78 \beta_{5} - 3 \beta_{6} + 162 \beta_{7} ) q^{17}$$ $$+ ( -30486 + 5430 \beta_{1} - 30486 \beta_{2} + 12 \beta_{4} + 294 \beta_{5} + 282 \beta_{6} ) q^{18}$$ $$+ ( -42924 - 2019 \beta_{1} - 21462 \beta_{2} - 2019 \beta_{3} - 157 \beta_{4} - 865 \beta_{6} - 157 \beta_{7} ) q^{19}$$ $$+ ( 62517 - 4590 \beta_{1} + 125034 \beta_{2} + 2295 \beta_{3} - 138 \beta_{4} + 756 \beta_{5} + 69 \beta_{6} + 69 \beta_{7} ) q^{20}$$ $$+ ( 1792 - 378 \beta_{1} - 114919 \beta_{2} + 3633 \beta_{3} - 308 \beta_{4} + 1316 \beta_{5} - 1477 \beta_{6} + 91 \beta_{7} ) q^{21}$$ $$+ ( 213282 - 1776 \beta_{3} + 28 \beta_{5} + 407 \beta_{6} - 463 \beta_{7} ) q^{22}$$ $$+ ( 296 \beta_{1} + 87235 \beta_{2} - 296 \beta_{3} - 522 \beta_{4} - 1778 \beta_{5} + 889 \beta_{6} + 522 \beta_{7} ) q^{23}$$ $$+ ( -74592 - 144 \beta_{1} + 74592 \beta_{2} + 288 \beta_{3} + 220 \beta_{4} - 722 \beta_{5} + 942 \beta_{6} - 440 \beta_{7} ) q^{24}$$ $$+ ( -139438 - 7080 \beta_{1} - 139438 \beta_{2} + 156 \beta_{4} + 952 \beta_{5} + 796 \beta_{6} ) q^{25}$$ $$+ ( -485520 + 6984 \beta_{1} - 242760 \beta_{2} + 6984 \beta_{3} + 908 \beta_{4} - 1886 \beta_{6} + 908 \beta_{7} ) q^{26}$$ $$+ ( 284088 + 15066 \beta_{1} + 568176 \beta_{2} - 7533 \beta_{3} + 1206 \beta_{4} - 1857 \beta_{5} - 603 \beta_{6} - 603 \beta_{7} ) q^{27}$$ $$+ ( -77077 - 1407 \beta_{1} - 525427 \beta_{2} - 19866 \beta_{3} + 455 \beta_{4} - 1274 \beta_{5} + 3437 \beta_{6} - 462 \beta_{7} ) q^{28}$$ $$+ ( 622897 + 11195 \beta_{3} - 1834 \beta_{5} + 2231 \beta_{6} + 1437 \beta_{7} ) q^{29}$$ $$+ ( 1272 \beta_{1} + 137778 \beta_{2} - 1272 \beta_{3} - 99 \beta_{4} + 2128 \beta_{5} - 1064 \beta_{6} + 99 \beta_{7} ) q^{30}$$ $$+ ( -198058 - 27357 \beta_{1} + 198058 \beta_{2} + 54714 \beta_{3} + 475 \beta_{4} + 3569 \beta_{5} - 3094 \beta_{6} - 950 \beta_{7} ) q^{31}$$ $$+ ( -83004 - 46668 \beta_{1} - 83004 \beta_{2} - 780 \beta_{4} - 5908 \beta_{5} - 5128 \beta_{6} ) q^{32}$$ $$+ ( -953190 - 324 \beta_{1} - 476595 \beta_{2} - 324 \beta_{3} - 1948 \beta_{4} + 12412 \beta_{6} - 1948 \beta_{7} ) q^{33}$$ $$+ ( 678237 + 52086 \beta_{1} + 1356474 \beta_{2} - 26043 \beta_{3} - 4096 \beta_{4} + 3246 \beta_{5} + 2048 \beta_{6} + 2048 \beta_{7} ) q^{34}$$ $$+ ( -170303 + 23947 \beta_{1} - 963424 \beta_{2} - 36512 \beta_{3} - 35 \beta_{4} - 6608 \beta_{5} - 924 \beta_{6} + 1036 \beta_{7} ) q^{35}$$ $$+ ( 1658676 - 9036 \beta_{3} + 10332 \beta_{5} - 18156 \beta_{6} - 2508 \beta_{7} ) q^{36}$$ $$+ ( 59724 \beta_{1} - 123185 \beta_{2} - 59724 \beta_{3} + 4082 \beta_{4} + 21028 \beta_{5} - 10514 \beta_{6} - 4082 \beta_{7} ) q^{37}$$ $$+ ( -590674 - 6028 \beta_{1} + 590674 \beta_{2} + 12056 \beta_{3} - 3669 \beta_{4} - 4236 \beta_{5} + 567 \beta_{6} + 7338 \beta_{7} ) q^{38}$$ $$+ ( -712593 - 21021 \beta_{1} - 712593 \beta_{2} + 861 \beta_{4} + 6216 \beta_{5} + 5355 \beta_{6} ) q^{39}$$ $$+ ( -1266972 + 25626 \beta_{1} - 633486 \beta_{2} + 25626 \beta_{3} - 302 \beta_{4} - 3690 \beta_{6} - 302 \beta_{7} ) q^{40}$$ $$+ ( 817719 - 49242 \beta_{1} + 1635438 \beta_{2} + 24621 \beta_{3} + 5722 \beta_{4} - 14602 \beta_{5} - 2861 \beta_{6} - 2861 \beta_{7} ) q^{41}$$ $$+ ( -48195 - 47838 \beta_{1} - 992922 \beta_{2} + 21945 \beta_{3} + 896 \beta_{4} + 1624 \beta_{5} + 6230 \beta_{6} - 532 \beta_{7} ) q^{42}$$ $$+ ( 556054 + 35340 \beta_{3} - 18074 \beta_{5} + 34436 \beta_{6} + 1712 \beta_{7} ) q^{43}$$ $$+ ( -145527 \beta_{1} + 919701 \beta_{2} + 145527 \beta_{3} - 5553 \beta_{4} - 52276 \beta_{5} + 26138 \beta_{6} + 5553 \beta_{7} ) q^{44}$$ $$+ ( 277347 + 28587 \beta_{1} - 277347 \beta_{2} - 57174 \beta_{3} + 4941 \beta_{4} - 4818 \beta_{5} + 9759 \beta_{6} - 9882 \beta_{7} ) q^{45}$$ $$+ ( -56640 + 180906 \beta_{1} - 56640 \beta_{2} + 3937 \beta_{4} + 5530 \beta_{5} + 1593 \beta_{6} ) q^{46}$$ $$+ ( 450688 - 100769 \beta_{1} + 225344 \beta_{2} - 100769 \beta_{3} + 7619 \beta_{4} - 36815 \beta_{6} + 7619 \beta_{7} ) q^{47}$$ $$+ ( -952966 - 111324 \beta_{1} - 1905932 \beta_{2} + 55662 \beta_{3} + 2972 \beta_{4} + 43076 \beta_{5} - 1486 \beta_{6} - 1486 \beta_{7} ) q^{48}$$ $$+ ( -98 + 51744 \beta_{1} + 952560 \beta_{2} + 220353 \beta_{3} - 5096 \beta_{4} + 40670 \beta_{5} - 39935 \beta_{6} - 833 \beta_{7} ) q^{49}$$ $$+ ( -1953614 - 126682 \beta_{3} + 280 \beta_{5} - 4316 \beta_{6} + 3756 \beta_{7} ) q^{50}$$ $$+ ( 103185 \beta_{1} + 87714 \beta_{2} - 103185 \beta_{3} - 10221 \beta_{4} + 4046 \beta_{5} - 2023 \beta_{6} + 10221 \beta_{7} ) q^{51}$$ $$+ ( 927934 + 276858 \beta_{1} - 927934 \beta_{2} - 553716 \beta_{3} + 8870 \beta_{4} + 13188 \beta_{5} - 4318 \beta_{6} - 17740 \beta_{7} ) q^{52}$$ $$+ ( 570365 + 179944 \beta_{1} + 570365 \beta_{2} - 15834 \beta_{4} - 4690 \beta_{5} + 11144 \beta_{6} ) q^{53}$$ $$+ ( 4092228 + 95940 \beta_{1} + 2046114 \beta_{2} + 95940 \beta_{3} - 7485 \beta_{4} - 14070 \beta_{6} - 7485 \beta_{7} ) q^{54}$$ $$+ ( -1414581 - 290868 \beta_{1} - 2829162 \beta_{2} + 145434 \beta_{3} - 24784 \beta_{4} - 43489 \beta_{5} + 12392 \beta_{6} + 12392 \beta_{7} ) q^{55}$$ $$+ ( 1441076 - 201012 \beta_{1} + 4525500 \beta_{2} + 79240 \beta_{3} + 1372 \beta_{4} + 910 \beta_{5} + 32928 \beta_{6} - 4004 \beta_{7} ) q^{56}$$ $$+ ( -5454801 - 14580 \beta_{3} + 24248 \beta_{5} - 33100 \beta_{6} - 15396 \beta_{7} ) q^{57}$$ $$+ ( -317094 \beta_{1} - 2860674 \beta_{2} + 317094 \beta_{3} + 21008 \beta_{4} + 32900 \beta_{5} - 16450 \beta_{6} - 21008 \beta_{7} ) q^{58}$$ $$+ ( 2107595 - 227362 \beta_{1} - 2107595 \beta_{2} + 454724 \beta_{3} - 27050 \beta_{4} + 11059 \beta_{5} - 38109 \beta_{6} + 54100 \beta_{7} ) q^{59}$$ $$+ ( 4419891 + 86751 \beta_{1} + 4419891 \beta_{2} + 16569 \beta_{4} + 12474 \beta_{5} - 4095 \beta_{6} ) q^{60}$$ $$+ ( 9894794 - 294960 \beta_{1} + 4947397 \beta_{2} - 294960 \beta_{3} - 5762 \beta_{4} + 111362 \beta_{6} - 5762 \beta_{7} ) q^{61}$$ $$+ ( -8251838 + 363688 \beta_{1} - 16503676 \beta_{2} - 181844 \beta_{3} + 44726 \beta_{4} - 12812 \beta_{5} - 22363 \beta_{6} - 22363 \beta_{7} ) q^{62}$$ $$+ ( -2378628 + 485205 \beta_{1} + 10263183 \beta_{2} - 218841 \beta_{3} + 20559 \beta_{4} - 163128 \beta_{5} + 44310 \beta_{6} + 12621 \beta_{7} ) q^{63}$$ $$+ ( -14269132 + 244452 \beta_{3} - 13440 \beta_{5} + 596 \beta_{6} + 26284 \beta_{7} ) q^{64}$$ $$+ ( 711795 \beta_{1} - 4230849 \beta_{2} - 711795 \beta_{3} + 24633 \beta_{4} + 76972 \beta_{5} - 38486 \beta_{6} - 24633 \beta_{7} ) q^{65}$$ $$+ ( 38619 - 115917 \beta_{1} - 38619 \beta_{2} + 231834 \beta_{3} + 2348 \beta_{4} - 98848 \beta_{5} + 101196 \beta_{6} - 4696 \beta_{7} ) q^{66}$$ $$+ ( -26777 - 819354 \beta_{1} - 26777 \beta_{2} + 30738 \beta_{4} - 67417 \beta_{5} - 98155 \beta_{6} ) q^{67}$$ $$+ ( 7386162 + 567741 \beta_{1} + 3693081 \beta_{2} + 567741 \beta_{3} - 2409 \beta_{4} + 117294 \beta_{6} - 2409 \beta_{7} ) q^{68}$$ $$+ ( -4484424 + 129474 \beta_{1} - 8968848 \beta_{2} - 64737 \beta_{3} - 35614 \beta_{4} + 55144 \beta_{5} + 17807 \beta_{6} + 17807 \beta_{7} ) q^{69}$$ $$+ ( 8138634 - 697158 \beta_{1} + 11497248 \beta_{2} - 476322 \beta_{3} - 26719 \beta_{4} + 28364 \beta_{5} - 15883 \beta_{6} + 16016 \beta_{7} ) q^{70}$$ $$+ ( -10351220 + 104234 \beta_{3} + 20930 \beta_{5} - 21718 \beta_{6} - 20142 \beta_{7} ) q^{71}$$ $$+ ( -503196 \beta_{1} - 3324396 \beta_{2} + 503196 \beta_{3} - 44484 \beta_{4} + 121632 \beta_{5} - 60816 \beta_{6} + 44484 \beta_{7} ) q^{72}$$ $$+ ( 9729867 - 595146 \beta_{1} - 9729867 \beta_{2} + 1190292 \beta_{3} + 39502 \beta_{4} + 156708 \beta_{5} - 117206 \beta_{6} - 79004 \beta_{7} ) q^{73}$$ $$+ ( 18172585 - 840841 \beta_{1} + 18172585 \beta_{2} - 103512 \beta_{4} - 36120 \beta_{5} + 67392 \beta_{6} ) q^{74}$$ $$+ ( 13305404 + 2532 \beta_{1} + 6652702 \beta_{2} + 2532 \beta_{3} + 22608 \beta_{4} - 268102 \beta_{6} + 22608 \beta_{7} ) q^{75}$$ $$+ ( -8371587 + 1874082 \beta_{1} - 16743174 \beta_{2} - 937041 \beta_{3} - 37842 \beta_{4} - 156198 \beta_{5} + 18921 \beta_{6} + 18921 \beta_{7} ) q^{76}$$ $$+ ( -7691761 + 931735 \beta_{1} - 822472 \beta_{2} - 967197 \beta_{3} + 5915 \beta_{4} + 312690 \beta_{5} + 71092 \beta_{6} - 94171 \beta_{7} ) q^{77}$$ $$+ ( -5515314 - 640794 \beta_{3} - 378 \beta_{5} + 966 \beta_{6} - 210 \beta_{7} ) q^{78}$$ $$+ ( 757308 \beta_{1} + 12657023 \beta_{2} - 757308 \beta_{3} - 80346 \beta_{4} - 500458 \beta_{5} + 250229 \beta_{6} + 80346 \beta_{7} ) q^{79}$$ $$+ ( -7799302 + 78446 \beta_{1} + 7799302 \beta_{2} - 156892 \beta_{3} + 2762 \beta_{4} + 189440 \beta_{5} - 186678 \beta_{6} - 5524 \beta_{7} ) q^{80}$$ $$+ ( -24217218 + 68355 \beta_{1} - 24217218 \beta_{2} + 62433 \beta_{4} + 242046 \beta_{5} + 179613 \beta_{6} ) q^{81}$$ $$+ ( -15176952 - 29532 \beta_{1} - 7588476 \beta_{2} - 29532 \beta_{3} + 30640 \beta_{4} - 210646 \beta_{6} + 30640 \beta_{7} ) q^{82}$$ $$+ ( 8989554 - 921852 \beta_{1} + 17979108 \beta_{2} + 460926 \beta_{3} + 137108 \beta_{4} + 347704 \beta_{5} - 68554 \beta_{6} - 68554 \beta_{7} ) q^{83}$$ $$+ ( 16771916 - 263571 \beta_{1} - 4980311 \beta_{2} + 203007 \beta_{3} - 30233 \beta_{4} - 112308 \beta_{5} - 396704 \beta_{6} + 88837 \beta_{7} ) q^{84}$$ $$+ ( 14903901 + 1468062 \beta_{3} - 110642 \beta_{5} + 200644 \beta_{6} + 20640 \beta_{7} ) q^{85}$$ $$+ ( -767396 \beta_{1} - 9773272 \beta_{2} + 767396 \beta_{3} + 94698 \beta_{4} - 24360 \beta_{5} + 12180 \beta_{6} - 94698 \beta_{7} ) q^{86}$$ $$+ ( 2225223 + 163191 \beta_{1} - 2225223 \beta_{2} - 326382 \beta_{3} - 27811 \beta_{4} - 811666 \beta_{5} + 783855 \beta_{6} + 55622 \beta_{7} ) q^{87}$$ $$+ ( 10348212 + 1108308 \beta_{1} + 10348212 \beta_{2} + 122072 \beta_{4} + 18298 \beta_{5} - 103774 \beta_{6} ) q^{88}$$ $$+ ( -387086 - 245408 \beta_{1} - 193543 \beta_{2} - 245408 \beta_{3} - 63324 \beta_{4} + 82020 \beta_{6} - 63324 \beta_{7} ) q^{89}$$ $$+ ( 9489270 - 3235164 \beta_{1} + 18978540 \beta_{2} + 1617582 \beta_{3} - 77184 \beta_{4} - 217158 \beta_{5} + 38592 \beta_{6} + 38592 \beta_{7} ) q^{90}$$ $$+ ( -30745001 - 1690500 \beta_{1} - 23541364 \beta_{2} + 3357039 \beta_{3} + 26068 \beta_{4} - 50372 \beta_{5} + 140287 \beta_{6} + 91777 \beta_{7} ) q^{91}$$ $$+ ( 31727391 - 1250997 \beta_{3} - 183526 \beta_{5} + 442727 \beta_{6} - 75675 \beta_{7} ) q^{92}$$ $$+ ( -558084 \beta_{1} + 14423301 \beta_{2} + 558084 \beta_{3} + 147006 \beta_{4} + 212044 \beta_{5} - 106022 \beta_{6} - 147006 \beta_{7} ) q^{93}$$ $$+ ( -28797174 + 1735512 \beta_{1} + 28797174 \beta_{2} - 3471024 \beta_{3} - 99489 \beta_{4} + 363152 \beta_{5} - 462641 \beta_{6} + 198978 \beta_{7} ) q^{94}$$ $$+ ( -43196763 + 2734806 \beta_{1} - 43196763 \beta_{2} - 138948 \beta_{4} - 258125 \beta_{5} - 119177 \beta_{6} ) q^{95}$$ $$+ ( -69409200 - 611832 \beta_{1} - 34704600 \beta_{2} - 611832 \beta_{3} - 48192 \beta_{4} + 304956 \beta_{6} - 48192 \beta_{7} ) q^{96}$$ $$+ ( 48586783 + 999222 \beta_{1} + 97173566 \beta_{2} - 499611 \beta_{3} - 59766 \beta_{4} + 45962 \beta_{5} + 29883 \beta_{6} + 29883 \beta_{7} ) q^{97}$$ $$+ ( 13376412 + 2312555 \beta_{1} - 66662099 \beta_{2} + 256417 \beta_{3} + 131320 \beta_{4} - 2940 \beta_{5} + 215894 \beta_{6} - 203252 \beta_{7} ) q^{98}$$ $$+ ( 84030501 - 2313849 \beta_{3} + 741846 \beta_{5} - 1624701 \beta_{6} + 141009 \beta_{7} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q$$ $$\mathstrut -\mathstrut 4q^{2}$$ $$\mathstrut -\mathstrut 84q^{3}$$ $$\mathstrut -\mathstrut 164q^{4}$$ $$\mathstrut -\mathstrut 840q^{5}$$ $$\mathstrut -\mathstrut 140q^{7}$$ $$\mathstrut +\mathstrut 6544q^{8}$$ $$\mathstrut +\mathstrut 396q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$8q$$ $$\mathstrut -\mathstrut 4q^{2}$$ $$\mathstrut -\mathstrut 84q^{3}$$ $$\mathstrut -\mathstrut 164q^{4}$$ $$\mathstrut -\mathstrut 840q^{5}$$ $$\mathstrut -\mathstrut 140q^{7}$$ $$\mathstrut +\mathstrut 6544q^{8}$$ $$\mathstrut +\mathstrut 396q^{9}$$ $$\mathstrut +\mathstrut 5796q^{10}$$ $$\mathstrut +\mathstrut 1784q^{11}$$ $$\mathstrut -\mathstrut 40908q^{12}$$ $$\mathstrut -\mathstrut 9856q^{14}$$ $$\mathstrut +\mathstrut 131808q^{15}$$ $$\mathstrut -\mathstrut 8584q^{16}$$ $$\mathstrut -\mathstrut 141456q^{17}$$ $$\mathstrut -\mathstrut 121944q^{18}$$ $$\mathstrut -\mathstrut 257544q^{19}$$ $$\mathstrut +\mathstrut 474012q^{21}$$ $$\mathstrut +\mathstrut 1706256q^{22}$$ $$\mathstrut -\mathstrut 348940q^{23}$$ $$\mathstrut -\mathstrut 895104q^{24}$$ $$\mathstrut -\mathstrut 557752q^{25}$$ $$\mathstrut -\mathstrut 2913120q^{26}$$ $$\mathstrut +\mathstrut 1485092q^{28}$$ $$\mathstrut +\mathstrut 4983176q^{29}$$ $$\mathstrut -\mathstrut 551112q^{30}$$ $$\mathstrut -\mathstrut 2376696q^{31}$$ $$\mathstrut -\mathstrut 332016q^{32}$$ $$\mathstrut -\mathstrut 5719140q^{33}$$ $$\mathstrut +\mathstrut 2491272q^{35}$$ $$\mathstrut +\mathstrut 13269408q^{36}$$ $$\mathstrut +\mathstrut 492740q^{37}$$ $$\mathstrut -\mathstrut 7088088q^{38}$$ $$\mathstrut -\mathstrut 2850372q^{39}$$ $$\mathstrut -\mathstrut 7601832q^{40}$$ $$\mathstrut +\mathstrut 3586128q^{42}$$ $$\mathstrut +\mathstrut 4448432q^{43}$$ $$\mathstrut -\mathstrut 3678804q^{44}$$ $$\mathstrut +\mathstrut 3328164q^{45}$$ $$\mathstrut -\mathstrut 226560q^{46}$$ $$\mathstrut +\mathstrut 2704128q^{47}$$ $$\mathstrut -\mathstrut 3811024q^{49}$$ $$\mathstrut -\mathstrut 15628912q^{50}$$ $$\mathstrut -\mathstrut 350856q^{51}$$ $$\mathstrut +\mathstrut 11135208q^{52}$$ $$\mathstrut +\mathstrut 2281460q^{53}$$ $$\mathstrut +\mathstrut 24553368q^{54}$$ $$\mathstrut -\mathstrut 6573392q^{56}$$ $$\mathstrut -\mathstrut 43638408q^{57}$$ $$\mathstrut +\mathstrut 11442696q^{58}$$ $$\mathstrut +\mathstrut 25291140q^{59}$$ $$\mathstrut +\mathstrut 17679564q^{60}$$ $$\mathstrut +\mathstrut 59368764q^{61}$$ $$\mathstrut -\mathstrut 60081756q^{63}$$ $$\mathstrut -\mathstrut 114153056q^{64}$$ $$\mathstrut +\mathstrut 16923396q^{65}$$ $$\mathstrut +\mathstrut 463428q^{66}$$ $$\mathstrut -\mathstrut 107108q^{67}$$ $$\mathstrut +\mathstrut 44316972q^{68}$$ $$\mathstrut +\mathstrut 19120080q^{70}$$ $$\mathstrut -\mathstrut 82809760q^{71}$$ $$\mathstrut +\mathstrut 13297584q^{72}$$ $$\mathstrut +\mathstrut 116758404q^{73}$$ $$\mathstrut +\mathstrut 72690340q^{74}$$ $$\mathstrut +\mathstrut 79832424q^{75}$$ $$\mathstrut -\mathstrut 58244200q^{77}$$ $$\mathstrut -\mathstrut 44122512q^{78}$$ $$\mathstrut -\mathstrut 50628092q^{79}$$ $$\mathstrut -\mathstrut 93591624q^{80}$$ $$\mathstrut -\mathstrut 96868872q^{81}$$ $$\mathstrut -\mathstrut 91061712q^{82}$$ $$\mathstrut +\mathstrut 154096572q^{84}$$ $$\mathstrut +\mathstrut 119231208q^{85}$$ $$\mathstrut +\mathstrut 39093088q^{86}$$ $$\mathstrut +\mathstrut 26702676q^{87}$$ $$\mathstrut +\mathstrut 41392848q^{88}$$ $$\mathstrut -\mathstrut 2322516q^{89}$$ $$\mathstrut -\mathstrut 151794552q^{91}$$ $$\mathstrut +\mathstrut 253819128q^{92}$$ $$\mathstrut -\mathstrut 57693204q^{93}$$ $$\mathstrut -\mathstrut 345566088q^{94}$$ $$\mathstrut -\mathstrut 172787052q^{95}$$ $$\mathstrut -\mathstrut 416455200q^{96}$$ $$\mathstrut +\mathstrut 373659692q^{98}$$ $$\mathstrut +\mathstrut 672244008q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8}\mathstrut +\mathstrut$$ $$592$$ $$x^{6}\mathstrut -\mathstrut$$ $$1176$$ $$x^{5}\mathstrut +\mathstrut$$ $$336397$$ $$x^{4}\mathstrut -\mathstrut$$ $$348096$$ $$x^{3}\mathstrut +\mathstrut$$ $$8673408$$ $$x^{2}\mathstrut +\mathstrut$$ $$8271396$$ $$x\mathstrut +\mathstrut$$ $$197880489$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$40665038848$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$933800395536$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$23107427488768$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$578443118889849$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$14228671705925824$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$327714819020519376$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$339657458304359916$$ $$\nu\mathstrut -\mathstrut$$ $$24226776665406171$$$$)/$$$$7800602713756314075$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$199147024$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$206072832$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$113162941609$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$117098450112$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$66871290607312$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$2782472800212$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$66566402858133$$ $$\nu\mathstrut -\mathstrut$$ $$1716105304424448$$$$)/$$$$1663596228141675$$ $$\beta_{4}$$ $$=$$ $$($$$$4986016178489$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$729234428373612$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$1954798485604859$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$440139351248781243$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$208410678677305067$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$242602731949311036057$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$549685532050544077272$$ $$\nu\mathstrut +\mathstrut$$ $$5840361335857569042933$$$$)/$$$$21841687598517679410$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$142450897959062$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$415028421151641$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$89730500895110567$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$81489430662144594$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$50276218913682852731$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$79052178550067250006$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$2640294308877978734829$$ $$\nu\mathstrut -\mathstrut$$ $$2293927437451668499449$$$$)/$$$$10\!\cdots\!50$$ $$\beta_{6}$$ $$=$$ $$($$$$35983628949817$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$17429303220966$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$19568825417085307$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$37683429688699701$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$11054456035703486131$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$7203424572374123829$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$247240100774204675046$$ $$\nu\mathstrut -\mathstrut$$ $$87312579891615556011$$$$)/$$$$21841687598517679410$$ $$\beta_{7}$$ $$=$$ $$($$$$206519207416061$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$209968840103652$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$112959866406343301$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$204058573291680393$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$64395501685950743543$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$35645454072445999857$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$1227308896964839498488$$ $$\nu\mathstrut +\mathstrut$$ $$29414857890603535700697$$$$)/$$$$10\!\cdots\!50$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$296$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$\beta_{1}$$ $$\nu^{3}$$ $$=$$ $$-$$$$3$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$25$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$14$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$544$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$441$$ $$\nu^{4}$$ $$=$$ $$-$$$$592$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$592$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$161165$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$1180$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$161165$$ $$\nu^{5}$$ $$=$$ $$2364$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$8288$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$16576$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$2364$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$308569$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$435120$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$308569$$ $$\beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-$$$$338161$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$321697$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$8232$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$1004365$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$91505156$$ $$\nu^{7}$$ $$=$$ $$3004175$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$4709558$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$1705383$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$346152513$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$175714240$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$346152513$$

## Character Values

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

 $$n$$ $$3$$ $$\chi(n)$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 −12.1698 + 21.0787i −2.23583 + 3.87257i 2.77916 − 4.81365i 11.6264 − 20.1376i −12.1698 − 21.0787i −2.23583 − 3.87257i 2.77916 + 4.81365i 11.6264 + 20.1376i
−12.6698 21.9447i −7.68317 4.43588i −193.046 + 334.366i −538.352 + 310.818i 224.806i 207.497 2392.02i 3296.47 −3241.15 5613.83i 13641.6 + 7875.98i
3.2 −2.73583 4.73860i 58.8837 + 33.9965i 113.030 195.774i 586.541 338.640i 372.035i −168.652 + 2395.07i −2637.67 −968.977 1678.32i −3209.35 1852.92i
3.3 2.27916 + 3.94762i −124.416 71.8315i 117.611 203.708i −163.006 + 94.1113i 654.862i −2345.65 512.580i 2239.15 7039.03 + 12192.0i −743.031 428.989i
3.4 11.1264 + 19.2716i 31.2153 + 18.0222i −119.595 + 207.145i −305.183 + 176.198i 802.090i 2236.80 872.649i 374.058 −2630.90 4556.86i −6791.21 3920.91i
5.1 −12.6698 + 21.9447i −7.68317 + 4.43588i −193.046 334.366i −538.352 310.818i 224.806i 207.497 + 2392.02i 3296.47 −3241.15 + 5613.83i 13641.6 7875.98i
5.2 −2.73583 + 4.73860i 58.8837 33.9965i 113.030 + 195.774i 586.541 + 338.640i 372.035i −168.652 2395.07i −2637.67 −968.977 + 1678.32i −3209.35 + 1852.92i
5.3 2.27916 3.94762i −124.416 + 71.8315i 117.611 + 203.708i −163.006 94.1113i 654.862i −2345.65 + 512.580i 2239.15 7039.03 12192.0i −743.031 + 428.989i
5.4 11.1264 19.2716i 31.2153 18.0222i −119.595 207.145i −305.183 176.198i 802.090i 2236.80 + 872.649i 374.058 −2630.90 + 4556.86i −6791.21 + 3920.91i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.4 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.d Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{9}^{\mathrm{new}}(7, [\chi])$$.