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

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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:

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 \(S_{9}^{\mathrm{new}}(7, [\chi])\).