Properties

Label 5.17.c.a
Level 5
Weight 17
Character orbit 5.c
Analytic conductor 8.116
Analytic rank 0
Dimension 14
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 17 \)
Character orbit: \([\chi]\) = 5.c (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(8.11622719283\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{34}\cdot 3^{10}\cdot 5^{18} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{2} q^{2} \) \( + ( 565 - 3 \beta_{1} - 565 \beta_{4} - \beta_{6} ) q^{3} \) \( + ( 7 \beta_{1} - 7 \beta_{2} + 35670 \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{4} \) \( + ( 13777 - 280 \beta_{1} + 259 \beta_{2} - \beta_{3} + 62039 \beta_{4} - 8 \beta_{6} - 5 \beta_{7} + \beta_{9} ) q^{5} \) \( + ( 316045 - 1762 \beta_{1} - 1762 \beta_{2} - \beta_{3} + 26 \beta_{6} + 27 \beta_{7} + \beta_{8} + \beta_{9} ) q^{6} \) \( + ( -28046 + 2 \beta_{1} + 2764 \beta_{2} - 13 \beta_{3} - 28044 \beta_{4} - 13 \beta_{5} - 180 \beta_{7} - \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + \beta_{12} + \beta_{13} ) q^{7} \) \( + ( -755234 + 39323 \beta_{1} - 7 \beta_{2} + 27 \beta_{3} + 755236 \beta_{4} - 27 \beta_{5} - 321 \beta_{6} + 5 \beta_{7} - 6 \beta_{8} - 17 \beta_{9} - 17 \beta_{10} + \beta_{11} - 6 \beta_{12} + 5 \beta_{13} ) q^{8} \) \( + ( -15 + 7897 \beta_{1} - 7882 \beta_{2} - 16378788 \beta_{4} + 74 \beta_{5} + 478 \beta_{6} - 449 \beta_{7} - 60 \beta_{9} + 31 \beta_{10} + 14 \beta_{12} + 15 \beta_{13} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{2} q^{2} \) \( + ( 565 - 3 \beta_{1} - 565 \beta_{4} - \beta_{6} ) q^{3} \) \( + ( 7 \beta_{1} - 7 \beta_{2} + 35670 \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{4} \) \( + ( 13777 - 280 \beta_{1} + 259 \beta_{2} - \beta_{3} + 62039 \beta_{4} - 8 \beta_{6} - 5 \beta_{7} + \beta_{9} ) q^{5} \) \( + ( 316045 - 1762 \beta_{1} - 1762 \beta_{2} - \beta_{3} + 26 \beta_{6} + 27 \beta_{7} + \beta_{8} + \beta_{9} ) q^{6} \) \( + ( -28046 + 2 \beta_{1} + 2764 \beta_{2} - 13 \beta_{3} - 28044 \beta_{4} - 13 \beta_{5} - 180 \beta_{7} - \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + \beta_{12} + \beta_{13} ) q^{7} \) \( + ( -755234 + 39323 \beta_{1} - 7 \beta_{2} + 27 \beta_{3} + 755236 \beta_{4} - 27 \beta_{5} - 321 \beta_{6} + 5 \beta_{7} - 6 \beta_{8} - 17 \beta_{9} - 17 \beta_{10} + \beta_{11} - 6 \beta_{12} + 5 \beta_{13} ) q^{8} \) \( + ( -15 + 7897 \beta_{1} - 7882 \beta_{2} - 16378788 \beta_{4} + 74 \beta_{5} + 478 \beta_{6} - 449 \beta_{7} - 60 \beta_{9} + 31 \beta_{10} + 14 \beta_{12} + 15 \beta_{13} ) q^{9} \) \( + ( 28486372 + 54584 \beta_{1} - 99609 \beta_{2} - 554 \beta_{3} - 26153671 \beta_{4} + 531 \beta_{5} - 542 \beta_{6} - 2207 \beta_{7} + 20 \beta_{8} - 11 \beta_{9} + 14 \beta_{10} + 5 \beta_{11} - 15 \beta_{12} + 35 \beta_{13} ) q^{10} \) \( + ( -38555658 + 171956 \beta_{1} + 171726 \beta_{2} + 1098 \beta_{3} + 80 \beta_{4} + 6400 \beta_{6} + 6608 \beta_{7} - 12 \beta_{8} + 303 \beta_{9} - 205 \beta_{10} + 5 \beta_{11} + 75 \beta_{13} ) q^{11} \) \( + ( 140961718 + 219 \beta_{1} - 143999 \beta_{2} - 4666 \beta_{3} + 140961952 \beta_{4} - 4666 \beta_{5} - 85 \beta_{6} - 38149 \beta_{7} - 2 \beta_{8} + 289 \beta_{9} + 511 \beta_{10} + 15 \beta_{11} + 2 \beta_{12} + 117 \beta_{13} ) q^{12} \) \( + ( -48653379 - 156377 \beta_{1} - 102 \beta_{2} + 4427 \beta_{3} + 48653361 \beta_{4} - 4427 \beta_{5} - 21178 \beta_{6} + 345 \beta_{7} + 114 \beta_{8} - 282 \beta_{9} - 567 \beta_{10} - 9 \beta_{11} + 114 \beta_{12} + 120 \beta_{13} ) q^{13} \) \( + ( 30 + 1047876 \beta_{1} - 1048006 \beta_{2} - 276880241 \beta_{4} + 7747 \beta_{5} - 90039 \beta_{6} + 89864 \beta_{7} - 230 \beta_{9} + 205 \beta_{10} + 50 \beta_{11} - 345 \beta_{12} + 20 \beta_{13} ) q^{14} \) \( + ( 251152244 - 1594929 \beta_{1} + 496261 \beta_{2} - 9649 \beta_{3} - 505382342 \beta_{4} + 19799 \beta_{5} + 69003 \beta_{6} - 208023 \beta_{7} - 485 \beta_{8} - 56 \beta_{9} - 359 \beta_{10} - 90 \beta_{11} + 395 \beta_{12} - 255 \beta_{13} ) q^{15} \) \( + ( -1639811898 + 1855974 \beta_{1} + 1858844 \beta_{2} + 28642 \beta_{3} - 970 \beta_{4} + 474342 \beta_{6} + 471468 \beta_{7} - 44 \beta_{8} - 3904 \beta_{9} + 2770 \beta_{10} - 20 \beta_{11} - 950 \beta_{13} ) q^{16} \) \( + ( 936561521 - 3587 \beta_{1} + 5628943 \beta_{2} - 2022 \beta_{3} + 936557659 \beta_{4} - 2022 \beta_{5} + 925 \beta_{6} - 267483 \beta_{7} + 456 \beta_{8} - 4237 \beta_{9} - 8093 \beta_{10} - 275 \beta_{11} - 456 \beta_{12} - 1931 \beta_{13} ) q^{17} \) \( + ( -804247557 - 19299161 \beta_{1} + 2694 \beta_{2} - 2239 \beta_{3} + 804247713 \beta_{4} + 2239 \beta_{5} - 1114875 \beta_{6} - 6275 \beta_{7} - 653 \beta_{8} + 8209 \beta_{9} + 11819 \beta_{10} + 78 \beta_{11} - 653 \beta_{12} - 2850 \beta_{13} ) q^{18} \) \( + ( 2020 + 5624087 \beta_{1} - 5623957 \beta_{2} + 910093816 \beta_{4} - 28686 \beta_{5} - 842245 \beta_{6} + 839617 \beta_{7} + 15605 \beta_{9} - 8677 \beta_{10} - 1075 \beta_{11} + 3692 \beta_{12} - 3095 \beta_{13} ) q^{19} \) \( + ( -1429134394 - 39684334 \beta_{1} - 50066490 \beta_{2} + 90815 \beta_{3} + 9204342042 \beta_{4} - 154476 \beta_{5} + 1619725 \beta_{6} - 1665333 \beta_{7} + 4950 \beta_{8} + 1535 \beta_{9} + 4051 \beta_{10} + 925 \beta_{11} - 4650 \beta_{12} - 2275 \beta_{13} ) q^{20} \) \( + ( 9533321393 + 49817587 \beta_{1} + 49811542 \beta_{2} - 277604 \beta_{3} + 1395 \beta_{4} + 1533142 \beta_{6} + 1542627 \beta_{7} + 1580 \beta_{8} + 8090 \beta_{9} - 10695 \beta_{10} - 930 \beta_{11} + 2325 \beta_{13} ) q^{21} \) \( + ( -17502536797 + 17854 \beta_{1} + 123123386 \beta_{2} + 338629 \beta_{3} - 17502516633 \beta_{4} + 338629 \beta_{5} + 1635 \beta_{6} - 814971 \beta_{7} - 7097 \beta_{8} + 13909 \beta_{9} + 40161 \beta_{10} + 2310 \beta_{11} + 7097 \beta_{12} + 10082 \beta_{13} ) q^{22} \) \( + ( 16887812190 - 31090554 \beta_{1} - 16160 \beta_{2} - 316765 \beta_{3} - 16887815680 \beta_{4} + 316765 \beta_{5} - 1829324 \beta_{6} + 35500 \beta_{7} - 2055 \beta_{8} - 67985 \beta_{9} - 71310 \beta_{10} - 1745 \beta_{11} - 2055 \beta_{12} + 19650 \beta_{13} ) q^{23} \) \( + ( -16860 + 376938688 \beta_{1} - 376942828 \beta_{2} - 5458344252 \beta_{4} - 320644 \beta_{5} + 1770136 \beta_{6} - 1733084 \beta_{7} - 140940 \beta_{9} + 61888 \beta_{10} + 10500 \beta_{11} - 21808 \beta_{12} + 27360 \beta_{13} ) q^{24} \) \( + ( 1000225160 - 325283145 \beta_{1} - 103198090 \beta_{2} + 121410 \beta_{3} - 8991992755 \beta_{4} - 239230 \beta_{5} - 1700770 \beta_{6} + 2302685 \beta_{7} - 25450 \beta_{8} - 9010 \beta_{9} - 25845 \beta_{10} - 8550 \beta_{11} + 31900 \beta_{12} + 34525 \beta_{13} ) q^{25} \) \( + ( 15867659676 + 307955429 \beta_{1} + 307871419 \beta_{2} - 76818 \beta_{3} + 37510 \beta_{4} - 12020962 \beta_{6} - 11976074 \beta_{7} - 10602 \beta_{8} + 125178 \beta_{9} - 12710 \beta_{10} + 14260 \beta_{11} + 23250 \beta_{13} ) q^{26} \) \( + ( 43360242102 + 8226 \beta_{1} + 551969994 \beta_{2} - 1178364 \beta_{3} + 43360236738 \beta_{4} - 1178364 \beta_{5} - 79140 \beta_{6} + 9128412 \beta_{7} + 54642 \beta_{8} + 100956 \beta_{9} + 3144 \beta_{10} - 13590 \beta_{11} - 54642 \beta_{12} - 2682 \beta_{13} ) q^{27} \) \( + ( -106604788976 - 793770005 \beta_{1} + 847 \beta_{2} + 1294958 \beta_{3} + 106604832334 \beta_{4} - 1294958 \beta_{5} + 31749417 \beta_{6} - 16905 \beta_{7} + 49826 \beta_{8} + 269157 \beta_{9} + 61957 \beta_{10} + 21679 \beta_{11} + 49826 \beta_{12} - 44205 \beta_{13} ) q^{28} \) \( + ( 20880 + 697792115 \beta_{1} - 697688145 \beta_{2} - 43237322276 \beta_{4} + 1471510 \beta_{5} + 17883591 \beta_{6} - 18081727 \beta_{7} + 520495 \beta_{9} - 72659 \beta_{10} - 62425 \beta_{11} + 72444 \beta_{12} - 83305 \beta_{13} ) q^{29} \) \( + ( 161227965329 - 1472256280 \beta_{1} - 1039416042 \beta_{2} - 2784887 \beta_{3} - 47836548372 \beta_{4} + 3982370 \beta_{5} - 46586896 \beta_{6} + 53072875 \beta_{7} + 46175 \beta_{8} + 4037 \beta_{9} + 97930 \beta_{10} + 67950 \beta_{11} - 138850 \beta_{12} - 150600 \beta_{13} ) q^{30} \) \( + ( -79150769198 + 664623628 \beta_{1} + 665232308 \beta_{2} + 7049994 \beta_{3} - 272030 \beta_{4} - 24134770 \beta_{6} - 24521446 \beta_{7} + 14594 \beta_{8} - 969821 \beta_{9} + 90155 \beta_{10} - 103705 \beta_{11} - 168325 \beta_{13} ) q^{31} \) \( + ( -144902304064 - 415922 \beta_{1} + 2091535750 \beta_{2} - 1652882 \beta_{3} - 144902643036 \beta_{4} - 1652882 \beta_{5} + 567550 \beta_{6} + 20154278 \beta_{7} - 244164 \beta_{8} - 1060422 \beta_{9} - 691258 \beta_{10} + 76950 \beta_{11} + 244164 \beta_{12} - 169486 \beta_{13} ) q^{32} \) \( + ( -456122526728 + 136487307 \beta_{1} + 405081 \beta_{2} + 1357704 \beta_{3} + 456122230802 \beta_{4} - 1357704 \beta_{5} + 29806007 \beta_{6} - 673605 \beta_{7} - 307332 \beta_{8} - 571719 \beta_{9} + 1187841 \beta_{10} - 147963 \beta_{11} - 307332 \beta_{12} - 109155 \beta_{13} ) q^{33} \) \( + ( 335620 + 1255626763 \beta_{1} - 1256479383 \beta_{2} - 566325060590 \beta_{4} + 5341546 \beta_{5} - 12509274 \beta_{6} + 13102536 \beta_{7} - 467020 \beta_{9} - 1160242 \beta_{10} + 258500 \beta_{11} - 105118 \beta_{12} - 77120 \beta_{13} ) q^{34} \) \( + ( 496911623907 - 764433041 \beta_{1} + 821688836 \beta_{2} + 7529716 \beta_{3} + 398789235199 \beta_{4} - 3238494 \beta_{5} + 86508693 \beta_{6} - 28791282 \beta_{7} + 212970 \beta_{8} + 212319 \beta_{9} - 199011 \beta_{10} - 390195 \beta_{11} + 388710 \beta_{12} + 127385 \beta_{13} ) q^{35} \) \( + ( 894284660172 - 1297216601 \beta_{1} - 1298505251 \beta_{2} - 19724903 \beta_{3} + 751350 \beta_{4} + 45528692 \beta_{6} + 46051258 \beta_{7} + 199316 \beta_{8} + 2722016 \beta_{9} + 1124850 \beta_{10} + 482700 \beta_{11} + 268650 \beta_{13} ) q^{36} \) \( + ( -644377695811 + 1561787 \beta_{1} - 3479158470 \beta_{2} + 6867657 \beta_{3} - 644376556379 \beta_{4} + 6867657 \beta_{5} - 2011230 \beta_{6} - 99128687 \beta_{7} + 596804 \beta_{8} + 3995372 \beta_{9} + 1608603 \beta_{10} - 422355 \beta_{11} - 596804 \beta_{12} + 569716 \beta_{13} ) q^{37} \) \( + ( -559106541858 + 791032474 \beta_{1} - 1977914 \beta_{2} - 10943806 \beta_{3} + 559107805742 \beta_{4} + 10943806 \beta_{5} - 294054848 \beta_{6} + 2966980 \beta_{7} + 906978 \beta_{8} + 1292656 \beta_{9} - 5658924 \beta_{10} + 631942 \beta_{11} + 906978 \beta_{12} + 714030 \beta_{13} ) q^{38} \) \( + ( -1936710 - 5973554066 \beta_{1} + 5977116926 \beta_{2} - 1198613205830 \beta_{4} - 45704302 \beta_{5} - 98471760 \beta_{6} + 97069184 \beta_{7} - 2055315 \beta_{9} + 6710191 \beta_{10} - 813075 \beta_{11} - 86986 \beta_{12} + 1123635 \beta_{13} ) q^{39} \) \( + ( 2276544325460 + 16479662725 \beta_{1} + 3089068745 \beta_{2} - 6610055 \beta_{3} + 3222134939970 \beta_{4} - 41248375 \beta_{5} + 146259585 \beta_{6} - 601831475 \beta_{7} - 1589850 \beta_{8} - 1193245 \beta_{9} + 86525 \beta_{10} + 1487225 \beta_{11} - 663550 \beta_{12} + 1063075 \beta_{13} ) q^{40} \) \( + ( 2824733727617 + 165709928 \beta_{1} + 164112703 \beta_{2} - 8126686 \beta_{3} - 533475 \beta_{4} + 369754135 \beta_{6} + 373145904 \beta_{7} - 1403106 \beta_{8} - 1938181 \beta_{9} - 9591350 \beta_{10} - 1598825 \beta_{11} + 1065350 \beta_{13} ) q^{41} \) \( + ( -5054893634741 - 1573128 \beta_{1} - 23163113944 \beta_{2} + 41640457 \beta_{3} - 5054893389269 \beta_{4} + 41640457 \beta_{5} + 3840025 \beta_{6} - 37201861 \beta_{7} - 325561 \beta_{8} - 7231753 \beta_{9} + 5621183 \beta_{10} + 1818600 \beta_{11} + 325561 \beta_{12} + 122736 \beta_{13} ) q^{42} \) \( + ( -4503369141745 + 14201307275 \beta_{1} + 4081360 \beta_{2} - 29165750 \beta_{3} + 4503365444805 \beta_{4} + 29165750 \beta_{5} - 341334127 \beta_{6} - 3267420 \beta_{7} - 650110 \beta_{8} - 6890730 \beta_{9} + 7733200 \beta_{10} - 1848470 \beta_{11} - 650110 \beta_{12} - 384420 \beta_{13} ) q^{43} \) \( + ( 3975410 - 27496158942 \beta_{1} + 27488180432 \beta_{2} - 9938066998870 \beta_{4} + 96109256 \beta_{5} - 263758264 \beta_{6} + 268156986 \beta_{7} + 1890790 \beta_{9} - 14295712 \beta_{10} + 2001550 \beta_{11} + 367932 \beta_{12} - 1973860 \beta_{13} ) q^{44} \) \( + ( 9029553967014 + 15386241569 \beta_{1} + 20347796485 \beta_{2} + 47699430 \beta_{3} + 3551456837298 \beta_{4} + 92256241 \beta_{5} + 452060510 \beta_{6} + 586677878 \beta_{7} + 3862650 \beta_{8} + 2665395 \beta_{9} + 375659 \beta_{10} - 3449025 \beta_{11} + 496450 \beta_{12} - 3070050 \beta_{13} ) q^{45} \) \( + ( 3179829210587 - 39654427082 \beta_{1} - 39644096832 \beta_{2} + 45934559 \beta_{3} - 894550 \beta_{4} + 182766440 \beta_{6} + 168871139 \beta_{7} + 4081549 \beta_{8} - 3319951 \beta_{9} + 29446750 \beta_{10} + 3823300 \beta_{11} - 4717850 \beta_{13} ) q^{46} \) \( + ( -201933359770 - 2535740 \beta_{1} - 383507612 \beta_{2} - 168388775 \beta_{3} - 201941179080 \beta_{4} - 168388775 \beta_{5} - 4175720 \beta_{6} - 595144678 \beta_{7} - 2481765 \beta_{8} + 6923550 \beta_{9} - 33971095 \beta_{10} - 5283570 \beta_{11} + 2481765 \beta_{12} - 3909655 \beta_{13} ) q^{47} \) \( + ( -28942054242728 + 15714062924 \beta_{1} - 3482124 \beta_{2} + 213872644 \beta_{3} + 28942062372752 \beta_{4} - 213872644 \beta_{5} + 993857180 \beta_{6} - 9721900 \beta_{7} - 4491112 \beta_{8} + 25712636 \beta_{9} + 10887676 \beta_{10} + 4065012 \beta_{11} - 4491112 \beta_{12} - 4647900 \beta_{13} ) q^{48} \) \( + ( -152625 - 34097154911 \beta_{1} + 34104410636 \beta_{2} - 15183172479984 \beta_{4} - 62901982 \beta_{5} + 287733512 \beta_{6} - 301107425 \beta_{7} + 24250350 \beta_{9} + 3329763 \beta_{10} - 3551550 \beta_{11} + 679662 \beta_{12} - 3398925 \beta_{13} ) q^{49} \) \( + ( 32935198078535 + 1194648780 \beta_{1} + 3671375185 \beta_{2} - 268879815 \beta_{3} + 10418189096495 \beta_{4} + 64225345 \beta_{5} - 3034269695 \beta_{6} + 1091512535 \beta_{7} + 35175 \beta_{8} + 464715 \beta_{9} + 126455 \beta_{10} + 3302700 \beta_{11} + 1070025 \beta_{12} - 1758600 \beta_{13} ) q^{50} \) \( + ( 14899888980814 + 6932816361 \beta_{1} + 6931082601 \beta_{2} + 264691128 \beta_{3} - 3066240 \beta_{4} - 2738480381 \beta_{6} - 2729222705 \beta_{7} - 3408564 \beta_{8} - 10207284 \beta_{9} - 29064960 \beta_{10} - 5466240 \beta_{11} + 2400000 \beta_{13} ) q^{51} \) \( + ( -27824039945010 - 1100890 \beta_{1} - 36197792800 \beta_{2} + 79718125 \beta_{3} - 27824032431070 \beta_{4} + 79718125 \beta_{5} + 6946830 \beta_{6} + 3641963378 \beta_{7} + 6525860 \beta_{8} - 16662550 \beta_{9} + 47398230 \beta_{10} + 8614830 \beta_{11} - 6525860 \beta_{12} + 3756970 \beta_{13} ) q^{52} \) \( + ( -19648739943089 - 20693340354 \beta_{1} + 5630143 \beta_{2} - 337624073 \beta_{3} + 19648725609641 \beta_{4} + 337624073 \beta_{5} + 4270361561 \beta_{6} + 25122880 \beta_{7} + 14882994 \beta_{8} - 39893817 \beta_{9} - 28196042 \beta_{10} - 7166724 \beta_{11} + 14882994 \beta_{12} + 8703305 \beta_{13} ) q^{53} \) \( + ( -10351440 + 104445723972 \beta_{1} - 104441701332 \beta_{2} - 55929844887084 \beta_{4} + 222026604 \beta_{5} + 4174408740 \beta_{6} - 4153651368 \beta_{7} - 63556560 \beta_{9} + 30141588 \beta_{10} + 3164400 \beta_{11} - 2251668 \beta_{12} + 13515840 \beta_{13} ) q^{54} \) \( + ( 23611403480624 - 5134555835 \beta_{1} - 11735062967 \beta_{2} + 488969538 \beta_{3} + 28883804554068 \beta_{4} - 76985650 \beta_{5} + 1501322929 \beta_{6} + 4188981115 \beta_{7} - 22976000 \beta_{8} - 17203788 \beta_{9} - 2360350 \beta_{10} + 5774750 \beta_{11} - 8074250 \beta_{12} + 17948250 \beta_{13} ) q^{55} \) \( + ( 61742136178532 + 152414252100 \beta_{1} + 152362000160 \beta_{2} - 749458420 \beta_{3} + 15997740 \beta_{4} - 1919051516 \beta_{6} - 1877741688 \beta_{7} - 15200832 \beta_{8} + 50919488 \beta_{9} - 62898740 \beta_{10} - 2129360 \beta_{11} + 18127100 \beta_{13} ) q^{56} \) \( + ( -46041056616150 + 31008330 \beta_{1} + 160195054524 \beta_{2} + 160538250 \beta_{3} - 46041025720230 \beta_{4} + 160538250 \beta_{5} - 14993610 \beta_{6} - 4091610348 \beta_{7} - 679170 \beta_{8} + 46114350 \beta_{9} + 60775440 \beta_{10} - 112410 \beta_{11} + 679170 \beta_{12} + 15447960 \beta_{13} ) q^{57} \) \( + ( -70795869243088 - 86171337998 \beta_{1} - 26489454 \beta_{2} + 134694104 \beta_{3} + 70795883577652 \beta_{4} - 134694104 \beta_{5} - 4461363514 \beta_{6} + 22003290 \beta_{7} - 9473772 \beta_{8} - 17269314 \beta_{9} - 60647634 \beta_{10} + 7167282 \beta_{11} - 9473772 \beta_{12} + 12154890 \beta_{13} ) q^{58} \) \( + ( -930360 + 262336291189 \beta_{1} - 262340478579 \beta_{2} - 44890246432432 \beta_{4} - 1110865282 \beta_{5} - 5886836391 \beta_{6} + 5890693411 \beta_{7} - 21633565 \beta_{9} + 7541045 \beta_{10} + 2558875 \beta_{11} - 7308840 \beta_{12} + 3489235 \beta_{13} ) q^{59} \) \( + ( 116603194890482 - 397465296901 \beta_{1} - 243220311019 \beta_{2} - 264751114 \beta_{3} + 88050087568724 \beta_{4} - 1110206934 \beta_{5} + 825572803 \beta_{6} - 11932909977 \beta_{7} + 49369270 \beta_{8} + 37279549 \beta_{9} - 4489521 \beta_{10} - 23239245 \beta_{11} + 30674610 \beta_{12} - 10214715 \beta_{13} ) q^{60} \) \( + ( 48857644711997 - 86546405400 \beta_{1} - 86500229235 \beta_{2} - 272669010 \beta_{3} + 6938235 \beta_{4} + 3282904185 \beta_{6} + 3226077750 \beta_{7} + 56340600 \beta_{8} + 50598105 \beta_{9} + 213653340 \beta_{10} + 33495435 \beta_{11} - 26557200 \beta_{13} ) q^{61} \) \( + ( -67140340417811 - 9182878 \beta_{1} + 497524084322 \beta_{2} + 771522107 \beta_{3} - 67140388052439 \beta_{4} + 771522107 \beta_{5} - 35672275 \beta_{6} + 6983246347 \beta_{7} - 17413911 \beta_{8} + 64941147 \beta_{9} - 228038417 \beta_{10} - 38451750 \beta_{11} + 17413911 \beta_{12} - 23817314 \beta_{13} ) q^{62} \) \( + ( -97261326096798 - 149655263434 \beta_{1} + 6576756 \beta_{2} - 558139241 \beta_{3} + 97261359472332 \beta_{4} + 558139241 \beta_{5} - 12265799004 \beta_{6} - 103357240 \beta_{7} - 40140427 \beta_{8} + 146467511 \beta_{9} + 149886286 \beta_{10} + 16687767 \beta_{11} - 40140427 \beta_{12} - 39952290 \beta_{13} ) q^{63} \) \( + ( 39141700 + 37618248952 \beta_{1} - 37633078452 \beta_{2} - 104358212525316 \beta_{4} + 1245724804 \beta_{5} + 85959528 \beta_{6} - 142456644 \beta_{7} + 241659500 \beta_{9} - 136537984 \beta_{10} - 12156100 \beta_{11} + 31268984 \beta_{12} - 51297800 \beta_{13} ) q^{64} \) \( + ( 36997139319364 - 245251946294 \beta_{1} - 159406348919 \beta_{2} + 723615046 \beta_{3} + 47701999025998 \beta_{4} + 1527890914 \beta_{5} + 20538742463 \beta_{6} + 1244781572 \beta_{7} - 2823510 \beta_{8} - 9699451 \beta_{9} + 36044376 \beta_{10} + 24746185 \beta_{11} - 57514680 \beta_{12} - 53646330 \beta_{13} ) q^{65} \) \( + ( -14205307522730 + 585764484248 \beta_{1} + 585936259568 \beta_{2} + 1516399754 \beta_{3} - 107635320 \beta_{4} + 31828123532 \beta_{6} + 31739227530 \beta_{7} - 68251322 \beta_{8} - 423227282 \beta_{9} - 206051280 \beta_{10} - 75565320 \beta_{11} - 32070000 \beta_{13} ) q^{66} \) \( + ( -12942705313089 - 202631752 \beta_{1} - 238610577997 \beta_{2} - 1373545142 \beta_{3} - 12942803709901 \beta_{4} - 1373545142 \beta_{5} + 263829840 \beta_{6} - 32877690643 \beta_{7} - 6161554 \beta_{8} - 570696532 \beta_{9} + 109749642 \beta_{10} + 104234940 \beta_{11} + 6161554 \beta_{12} - 49198406 \beta_{13} ) q^{67} \) \( + ( -65480338636626 - 569683258252 \beta_{1} + 230512402 \beta_{2} + 1601701103 \beta_{3} + 65480133713654 \beta_{4} - 1601701103 \beta_{5} - 8125916054 \beta_{6} - 74712830 \beta_{7} + 78927516 \beta_{8} - 254150138 \beta_{9} + 330814662 \beta_{10} - 102461486 \beta_{11} + 78927516 \beta_{12} - 25589430 \beta_{13} ) q^{68} \) \( + ( 33211215 - 33036608086 \beta_{1} + 32937044821 \beta_{2} - 137825745090421 \beta_{4} + 1865562148 \beta_{5} - 32020888389 \beta_{6} + 32116011262 \beta_{7} - 99387315 \beta_{9} - 128439658 \beta_{10} + 33176025 \beta_{11} - 4370012 \beta_{12} - 35190 \beta_{13} ) q^{69} \) \( + ( 76060654152062 + 746497513612 \beta_{1} + 61628921660 \beta_{2} - 2108566430 \beta_{3} - 82885349796841 \beta_{4} + 2159626793 \beta_{5} - 38224503555 \beta_{6} - 22633388606 \beta_{7} - 138720150 \beta_{8} - 88049020 \beta_{9} - 43247443 \beta_{10} + 12459650 \beta_{11} - 16072075 \beta_{12} + 46535050 \beta_{13} ) q^{70} \) \( + ( -95366921727398 - 651965026900 \beta_{1} - 652169159380 \beta_{2} + 2445083770 \beta_{3} + 96285130 \beta_{4} - 24182895770 \beta_{6} - 24100937750 \beta_{7} - 37451550 \beta_{8} + 305327515 \beta_{9} + 7674795 \beta_{10} + 42361455 \beta_{11} + 53923675 \beta_{13} ) q^{71} \) \( + ( 183931789279428 + 295126479 \beta_{1} - 842202474861 \beta_{2} - 2847726501 \beta_{3} + 183931954108182 \beta_{4} - 2847726501 \beta_{5} - 461426325 \beta_{6} + 36982568367 \beta_{7} + 118416498 \beta_{8} + 886850529 \beta_{9} + 57180831 \beta_{10} - 130297725 \beta_{11} - 118416498 \beta_{12} + 82414377 \beta_{13} ) q^{72} \) \( + ( 55389673229827 + 1372860704244 \beta_{1} - 558886974 \beta_{2} + 724797654 \beta_{3} - 55389219029683 \beta_{4} - 724797654 \beta_{5} + 39455536920 \beta_{6} + 466909530 \beta_{7} + 30435798 \beta_{8} + 624775596 \beta_{9} - 1130483334 \beta_{10} + 227100072 \beta_{11} + 30435798 \beta_{12} + 104686830 \beta_{13} ) q^{73} \) \( + ( -300270530 - 919809601429 \beta_{1} + 920374791059 \beta_{2} + 352928319317242 \beta_{4} - 4918720388 \beta_{5} + 75478561228 \beta_{6} - 75834473970 \beta_{7} - 273865270 \beta_{9} + 1159616212 \beta_{10} - 132459550 \beta_{11} - 126345072 \beta_{12} + 167810980 \beta_{13} ) q^{74} \) \( + ( -103338685324055 + 384622394535 \beta_{1} + 1861927709920 \beta_{2} - 3087325330 \beta_{3} - 206800244524885 \beta_{4} - 2413991410 \beta_{5} + 24775058635 \beta_{6} + 48037842270 \beta_{7} + 163224100 \beta_{8} + 148443880 \beta_{9} - 88357490 \beta_{10} - 22873350 \beta_{11} + 307863800 \beta_{12} + 130875300 \beta_{13} ) q^{75} \) \( + ( -16462517390116 - 882840151460 \beta_{1} - 883275392440 \beta_{2} - 3640943820 \beta_{3} + 229235980 \beta_{4} - 34480334692 \beta_{6} - 34056984136 \beta_{7} + 240576536 \beta_{8} + 1031286976 \beta_{9} + 195926420 \beta_{10} + 126233480 \beta_{11} + 103002500 \beta_{13} ) q^{76} \) \( + ( 78648735936936 + 342593823 \beta_{1} - 903037819739 \beta_{2} + 2650015038 \beta_{3} + 78649057977534 \beta_{4} + 2650015038 \beta_{5} - 97071975 \beta_{6} - 22518694011 \beta_{7} - 105054774 \beta_{8} + 460219023 \beta_{9} + 477366747 \beta_{10} - 20553225 \beta_{11} + 105054774 \beta_{12} + 161020299 \beta_{13} ) q^{77} \) \( + ( 604936448029129 + 1515344690366 \beta_{1} + 220077882 \beta_{2} - 3440374997 \beta_{3} - 604936816528021 \beta_{4} + 3440374997 \beta_{5} + 16405969919 \beta_{6} - 73164865 \beta_{7} - 185757439 \beta_{8} - 1368018253 \beta_{9} + 144821737 \beta_{10} - 184249446 \beta_{11} - 185757439 \beta_{12} + 148421010 \beta_{13} ) q^{78} \) \( + ( 367796060 - 1013816534784 \beta_{1} + 1012749979124 \beta_{2} + 259164232167084 \beta_{4} + 5348705512 \beta_{5} - 39287009116 \beta_{6} + 40407643528 \beta_{7} - 974474360 \beta_{9} - 1543679252 \beta_{10} + 349379800 \beta_{11} + 90911272 \beta_{12} - 18416260 \beta_{13} ) q^{79} \) \( + ( -1067195056626978 + 3550436131180 \beta_{1} - 564355294446 \beta_{2} + 15086851544 \beta_{3} - 210512893207246 \beta_{4} - 3020128210 \beta_{5} - 66233547748 \beta_{6} + 75297631490 \beta_{7} + 57105600 \beta_{8} - 91823494 \beta_{9} + 217954560 \beta_{10} - 157562350 \beta_{11} - 503276700 \beta_{12} - 52332700 \beta_{13} ) q^{80} \) \( + ( 58368683319624 - 2020667681034 \beta_{1} - 2020356414639 \beta_{2} - 11373335442 \beta_{3} - 236130795 \beta_{4} - 113002270479 \beta_{6} - 113297184054 \beta_{7} - 380773170 \beta_{8} - 1126733355 \beta_{9} - 681548580 \beta_{10} - 198562995 \beta_{11} - 37567800 \beta_{13} ) q^{81} \) \( + ( -21321994024761 - 785498658 \beta_{1} - 2460853724158 \beta_{2} + 8470613137 \beta_{3} - 21322379923749 \beta_{4} + 8470613137 \beta_{5} + 1388395095 \beta_{6} + 105835245817 \beta_{7} - 396246261 \beta_{8} - 2573493423 \beta_{9} + 30754773 \beta_{10} + 399599670 \beta_{11} + 396246261 \beta_{12} - 192949494 \beta_{13} ) q^{82} \) \( + ( 953810209605995 + 2081960536863 \beta_{1} + 937630150 \beta_{2} - 1724659500 \beta_{3} - 953810806781695 \beta_{4} + 1724659500 \beta_{5} + 82934977767 \beta_{6} - 1004770400 \beta_{7} - 25273650 \beta_{8} - 198261700 \beta_{9} + 2282855000 \beta_{10} - 298587850 \beta_{11} - 25273650 \beta_{12} - 340454450 \beta_{13} ) q^{83} \) \( + ( 114694650 - 4265546572070 \beta_{1} + 4266026347520 \beta_{2} + 1718445111011426 \beta_{4} - 8162312200 \beta_{5} + 114857361416 \beta_{6} - 115762517262 \beta_{7} + 2539423950 \beta_{9} - 445327904 \beta_{10} - 297235050 \beta_{11} + 398479004 \beta_{12} - 411929700 \beta_{13} ) q^{84} \) \( + ( -1146174847293398 - 1417884395766 \beta_{1} + 1939072901176 \beta_{2} - 11273888419 \beta_{3} - 206428167408686 \beta_{4} - 10307518319 \beta_{5} + 70753332138 \beta_{6} + 752779618 \beta_{7} - 17045630 \beta_{8} + 53343404 \beta_{9} + 145192714 \beta_{10} + 488405155 \beta_{11} - 152586090 \beta_{12} - 432293915 \beta_{13} ) q^{85} \) \( + ( -1431613084790711 + 2404838847398 \beta_{1} + 2406392687138 \beta_{2} + 16420519159 \beta_{3} - 652903940 \beta_{4} + 146349635614 \beta_{6} + 145560514051 \beta_{7} + 359846097 \beta_{8} - 2049333623 \beta_{9} + 541659540 \beta_{10} - 202436040 \beta_{11} - 450467900 \beta_{13} ) q^{86} \) \( + ( 1237405555436634 - 541562868 \beta_{1} + 2012450200930 \beta_{2} + 9418229162 \beta_{3} + 1237404401078406 \beta_{4} + 9418229162 \beta_{5} - 1580020660 \beta_{6} + 50686312890 \beta_{7} + 931609054 \beta_{8} + 1651253152 \beta_{9} - 3215493482 \beta_{10} - 612795360 \beta_{11} - 931609054 \beta_{12} - 577179114 \beta_{13} ) q^{87} \) \( + ( 1640709050445512 - 9243703152584 \beta_{1} - 1579391144 \beta_{2} - 8869907536 \beta_{3} - 1640706911182968 \beta_{4} + 8869907536 \beta_{5} - 237366252192 \beta_{6} + 345448800 \beta_{7} + 395560328 \beta_{8} + 6353699616 \beta_{9} - 1364968544 \beta_{10} + 1069631272 \beta_{11} + 395560328 \beta_{12} - 559871400 \beta_{13} ) q^{88} \) \( + ( -733060180 + 3848391555230 \beta_{1} - 3846096626550 \beta_{2} + 1941296575394192 \beta_{4} - 2835937060 \beta_{5} - 165139030282 \beta_{6} + 162401413314 \beta_{7} + 2534299030 \beta_{9} + 3327054938 \beta_{10} - 780934250 \beta_{11} - 346940148 \beta_{12} - 47874070 \beta_{13} ) q^{89} \) \( + ( -1561844111557746 - 2553571963939 \beta_{1} - 4319447650224 \beta_{2} - 4101241034 \beta_{3} - 2069538624546147 \beta_{4} + 19748383109 \beta_{5} - 6599200902 \beta_{6} - 274294260643 \beta_{7} - 377244760 \beta_{8} + 112671629 \beta_{9} - 613902194 \beta_{10} - 248113815 \beta_{11} + 1238612195 \beta_{12} + 58578045 \beta_{13} ) q^{90} \) \( + ( -1751378547706866 - 3381307369245 \beta_{1} - 3381863794985 \beta_{2} + 438432020 \beta_{3} + 647278840 \beta_{4} + 151663742529 \beta_{6} + 150820444437 \beta_{7} - 14313052 \beta_{8} + 1882096918 \beta_{9} + 2907101210 \beta_{10} + 692705390 \beta_{11} - 45426550 \beta_{13} ) q^{91} \) \( + ( 2903438400440174 + 847763227 \beta_{1} + 1752827192881 \beta_{2} - 46065288178 \beta_{3} + 2903439203423096 \beta_{4} - 46065288178 \beta_{5} - 308491805 \beta_{6} - 378887925869 \beta_{7} - 182560266 \beta_{8} + 1201035337 \beta_{9} + 1289064663 \beta_{10} - 44780305 \beta_{11} + 182560266 \beta_{12} + 401491461 \beta_{13} ) q^{92} \) \( + ( 1138970693481416 + 6275274759993 \beta_{1} + 1483512183 \beta_{2} + 49150188432 \beta_{3} - 1138973213201714 \beta_{4} - 49150188432 \beta_{5} - 2458021415 \beta_{6} + 817459065 \beta_{7} + 4902984 \beta_{8} - 8143161957 \beta_{9} - 370154997 \beta_{10} - 1259860149 \beta_{11} + 4902984 \beta_{12} + 1036208115 \beta_{13} ) q^{93} \) \( + ( 1660439810 + 11189820513452 \beta_{1} - 11196369162762 \beta_{2} + 52838052784743 \beta_{4} + 28847501039 \beta_{5} - 331769265315 \beta_{6} + 338448810372 \beta_{7} - 10466974010 \beta_{9} - 5988990047 \beta_{10} + 2444104750 \beta_{11} - 1436434133 \beta_{12} + 783664940 \beta_{13} ) q^{94} \) \( + ( -1605817208666210 - 8501620911365 \beta_{1} + 4257048918135 \beta_{2} + 3482333160 \beta_{3} - 1753204070451470 \beta_{4} + 26040670690 \beta_{5} + 159617129005 \beta_{6} - 169781709705 \beta_{7} - 524439450 \beta_{8} - 800538385 \beta_{9} - 607451015 \beta_{10} - 1487212675 \beta_{11} - 546503850 \beta_{12} + 1007107525 \beta_{13} ) q^{95} \) \( + ( -1968775757903480 + 19116253573352 \beta_{1} + 19112545128752 \beta_{2} + 8682945656 \beta_{3} + 1267129800 \beta_{4} + 213003034376 \beta_{6} + 215578048464 \beta_{7} - 1040485712 \beta_{8} + 3981561088 \beta_{9} - 3476082600 \beta_{10} + 46472400 \beta_{11} + 1220657400 \beta_{13} ) q^{96} \) \( + ( -238339782526713 + 2133889776 \beta_{1} + 16003715460520 \beta_{2} - 1674643564 \beta_{3} - 238336559735577 \beta_{4} - 1674643564 \beta_{5} + 2035922910 \beta_{6} - 299413685782 \beta_{7} - 1469515758 \beta_{8} - 990934494 \beta_{9} + 8242770594 \beta_{10} + 1088901360 \beta_{11} + 1469515758 \beta_{12} + 1611395568 \beta_{13} ) q^{97} \) \( + ( 3446763298358533 - 10884980416587 \beta_{1} - 1278305386 \beta_{2} - 48554242929 \beta_{3} - 3446763269978537 \beta_{4} + 48554242929 \beta_{5} - 188500046769 \beta_{6} + 1977636815 \beta_{7} - 536403963 \beta_{8} - 4229420141 \beta_{9} - 4505867591 \beta_{10} + 14189998 \beta_{11} - 536403963 \beta_{12} + 1249925390 \beta_{13} ) q^{98} \) \( + ( -2813954820 + 12576396608230 \beta_{1} - 12569262424360 \beta_{2} + 620716842190734 \beta_{4} + 17559698510 \beta_{5} + 399775866004 \beta_{6} - 403772339168 \beta_{7} + 3864982395 \beta_{9} + 8771948869 \beta_{10} - 2160114525 \beta_{11} + 1830030116 \beta_{12} + 653840295 \beta_{13} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(14q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 7908q^{3} \) \(\mathstrut +\mathstrut 192880q^{5} \) \(\mathstrut +\mathstrut 4417368q^{6} \) \(\mathstrut -\mathstrut 386452q^{7} \) \(\mathstrut -\mathstrut 10493340q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(14q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 7908q^{3} \) \(\mathstrut +\mathstrut 192880q^{5} \) \(\mathstrut +\mathstrut 4417368q^{6} \) \(\mathstrut -\mathstrut 386452q^{7} \) \(\mathstrut -\mathstrut 10493340q^{8} \) \(\mathstrut +\mathstrut 398727830q^{10} \) \(\mathstrut -\mathstrut 539141632q^{11} \) \(\mathstrut +\mathstrut 1973309928q^{12} \) \(\mathstrut -\mathstrut 681358582q^{13} \) \(\mathstrut +\mathstrut 3514452060q^{15} \) \(\mathstrut -\mathstrut 22953579016q^{16} \) \(\mathstrut +\mathstrut 13124182498q^{17} \) \(\mathstrut -\mathstrut 11293583022q^{18} \) \(\mathstrut -\mathstrut 20186820060q^{20} \) \(\mathstrut +\mathstrut 133652286048q^{21} \) \(\mathstrut -\mathstrut 244784660624q^{22} \) \(\mathstrut +\mathstrut 236373052228q^{23} \) \(\mathstrut +\mathstrut 13144083650q^{25} \) \(\mathstrut +\mathstrut 223473840788q^{26} \) \(\mathstrut +\mathstrut 608106053880q^{27} \) \(\mathstrut -\mathstrut 1494176423768q^{28} \) \(\mathstrut +\mathstrut 2252131613760q^{30} \) \(\mathstrut -\mathstrut 1105223074952q^{31} \) \(\mathstrut -\mathstrut 2024535639992q^{32} \) \(\mathstrut -\mathstrut 6385551058704q^{33} \) \(\mathstrut +\mathstrut 6956678093980q^{35} \) \(\mathstrut +\mathstrut 12514339524924q^{36} \) \(\mathstrut -\mathstrut 9027827087002q^{37} \) \(\mathstrut -\mathstrut 7824802789560q^{38} \) \(\mathstrut +\mathstrut 31912538913900q^{40} \) \(\mathstrut +\mathstrut 39543901484288q^{41} \) \(\mathstrut -\mathstrut 70814500818384q^{42} \) \(\mathstrut -\mathstrut 63017457929452q^{43} \) \(\mathstrut +\mathstrut 126481294123110q^{45} \) \(\mathstrut +\mathstrut 44357914058008q^{46} \) \(\mathstrut -\mathstrut 2826187575452q^{47} \) \(\mathstrut -\mathstrut 405160526368272q^{48} \) \(\mathstrut +\mathstrut 461109216233650q^{50} \) \(\mathstrut +\mathstrut 208649050954008q^{51} \) \(\mathstrut -\mathstrut 389623094881012q^{52} \) \(\mathstrut -\mathstrut 275142037498442q^{53} \) \(\mathstrut +\mathstrut 330504874256560q^{55} \) \(\mathstrut +\mathstrut 865011118900080q^{56} \) \(\mathstrut -\mathstrut 644237427525840q^{57} \) \(\mathstrut -\mathstrut 991296507673440q^{58} \) \(\mathstrut +\mathstrut 1631207111049480q^{60} \) \(\mathstrut +\mathstrut 683634528395968q^{61} \) \(\mathstrut -\mathstrut 938995013191864q^{62} \) \(\mathstrut -\mathstrut 1361911146729372q^{63} \) \(\mathstrut +\mathstrut 517066464980110q^{65} \) \(\mathstrut -\mathstrut 196778043917184q^{66} \) \(\mathstrut -\mathstrut 181547742064252q^{67} \) \(\mathstrut -\mathstrut 917821658187868q^{68} \) \(\mathstrut +\mathstrut 1066699466039880q^{70} \) \(\mathstrut -\mathstrut 1337544011098792q^{71} \) \(\mathstrut +\mathstrut 2573199832214340q^{72} \) \(\mathstrut +\mathstrut 778038215296478q^{73} \) \(\mathstrut -\mathstrut 1442550824447700q^{75} \) \(\mathstrut -\mathstrut 233750796900240q^{76} \) \(\mathstrut +\mathstrut 1099376123524976q^{77} \) \(\mathstrut +\mathstrut 8472064951325256q^{78} \) \(\mathstrut -\mathstrut 14934736099860320q^{80} \) \(\mathstrut +\mathstrut 809940307363794q^{81} \) \(\mathstrut -\mathstrut 303814693723184q^{82} \) \(\mathstrut +\mathstrut 13357181209550188q^{83} \) \(\mathstrut -\mathstrut 16045734952112470q^{85} \) \(\mathstrut -\mathstrut 20034048539985352q^{86} \) \(\mathstrut +\mathstrut 17327533268211840q^{87} \) \(\mathstrut +\mathstrut 22952328705949920q^{88} \) \(\mathstrut -\mathstrut 21878453293909290q^{90} \) \(\mathstrut -\mathstrut 24534035720748632q^{91} \) \(\mathstrut +\mathstrut 40652972845631848q^{92} \) \(\mathstrut +\mathstrut 15958372062212256q^{93} \) \(\mathstrut -\mathstrut 22489885958711400q^{95} \) \(\mathstrut -\mathstrut 27488119051061472q^{96} \) \(\mathstrut -\mathstrut 3303551663290402q^{97} \) \(\mathstrut +\mathstrut 48233459501789102q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut +\mathstrut \) \(354217\) \(x^{12}\mathstrut +\mathstrut \) \(47647865256\) \(x^{10}\mathstrut +\mathstrut \) \(3062948254140560\) \(x^{8}\mathstrut +\mathstrut \) \(97982926395671183360\) \(x^{6}\mathstrut +\mathstrut \) \(1450505914836224854327296\) \(x^{4}\mathstrut +\mathstrut \) \(7749363027207208037244731392\) \(x^{2}\mathstrut +\mathstrut \) \(8484029084015876402584471207936\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(19395026881\) \(\nu^{12}\mathstrut -\mathstrut \) \(5101784131984041\) \(\nu^{10}\mathstrut -\mathstrut \) \(433881598453749944040\) \(\nu^{8}\mathstrut -\mathstrut \) \(11915381996069728263743120\) \(\nu^{6}\mathstrut +\mathstrut \) \(7652456798273282271778114560\) \(\nu^{4}\mathstrut +\mathstrut \) \(1806562382916320352001552197156864\) \(\nu^{2}\mathstrut +\mathstrut \) \(135746553974964854480053862400000000\) \(\nu\mathstrut -\mathstrut \) \(2600279260325231550566347833591463936\)\()/\)\(13\!\cdots\!00\)
\(\beta_{2}\)\(=\)\((\)\(-\)\(19395026881\) \(\nu^{12}\mathstrut -\mathstrut \) \(5101784131984041\) \(\nu^{10}\mathstrut -\mathstrut \) \(433881598453749944040\) \(\nu^{8}\mathstrut -\mathstrut \) \(11915381996069728263743120\) \(\nu^{6}\mathstrut +\mathstrut \) \(7652456798273282271778114560\) \(\nu^{4}\mathstrut +\mathstrut \) \(1806562382916320352001552197156864\) \(\nu^{2}\mathstrut -\mathstrut \) \(135746553974964854480053862400000000\) \(\nu\mathstrut -\mathstrut \) \(2600279260325231550566347833591463936\)\()/\)\(13\!\cdots\!00\)
\(\beta_{3}\)\(=\)\((\)\(28058340253264616295893\) \(\nu^{12}\mathstrut +\mathstrut \) \(9195159940417656768107794173\) \(\nu^{10}\mathstrut +\mathstrut \) \(1100344919748515360239952239086120\) \(\nu^{8}\mathstrut +\mathstrut \) \(58957747433427088900053257171606257360\) \(\nu^{6}\mathstrut +\mathstrut \) \(1431263943129674722011255835512755052664320\) \(\nu^{4}\mathstrut +\mathstrut \) \(16335189004828961281575011564803635137637777408\) \(\nu^{2}\mathstrut +\mathstrut \) \(135754668388390846860513301872069607512473870532608\)\()/\)\(91\!\cdots\!00\)
\(\beta_{4}\)\(=\)\((\)\(3487216833586249799\) \(\nu^{13}\mathstrut +\mathstrut \) \(1014557318660728928263839\) \(\nu^{11}\mathstrut +\mathstrut \) \(108110982248511274839701699160\) \(\nu^{9}\mathstrut +\mathstrut \) \(5744514679325030042978957109034480\) \(\nu^{7}\mathstrut +\mathstrut \) \(206115998756370451349625440305946741760\) \(\nu^{5}\mathstrut +\mathstrut \) \(5145297330466188001440381879902353812946944\) \(\nu^{3}\mathstrut +\mathstrut \) \(47578547842678974659740222298405859808765804544\) \(\nu\)\()/\)\(15\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(2405870722536653292448513997831\) \(\nu^{13}\mathstrut +\mathstrut \) \(34199400881884988817744138584935607391\) \(\nu^{11}\mathstrut +\mathstrut \) \(10171305916701520286777867651692978946174040\) \(\nu^{9}\mathstrut +\mathstrut \) \(1111002824927610433379823327134703938659899931120\) \(\nu^{7}\mathstrut +\mathstrut \) \(54043588639842136853049732698769909805295980173757440\) \(\nu^{5}\mathstrut +\mathstrut \) \(1138450654765652769840900940112779429566832438278540558336\) \(\nu^{3}\mathstrut +\mathstrut \) \(7777570297778692971188974709013023099223380279461631844417536\) \(\nu\)\()/\)\(19\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(25\!\cdots\!65\) \(\nu^{13}\mathstrut -\mathstrut \) \(12\!\cdots\!72\) \(\nu^{12}\mathstrut +\mathstrut \) \(11\!\cdots\!65\) \(\nu^{11}\mathstrut -\mathstrut \) \(40\!\cdots\!92\) \(\nu^{10}\mathstrut +\mathstrut \) \(20\!\cdots\!00\) \(\nu^{9}\mathstrut -\mathstrut \) \(49\!\cdots\!80\) \(\nu^{8}\mathstrut +\mathstrut \) \(16\!\cdots\!00\) \(\nu^{7}\mathstrut -\mathstrut \) \(27\!\cdots\!40\) \(\nu^{6}\mathstrut +\mathstrut \) \(63\!\cdots\!00\) \(\nu^{5}\mathstrut -\mathstrut \) \(67\!\cdots\!80\) \(\nu^{4}\mathstrut +\mathstrut \) \(10\!\cdots\!40\) \(\nu^{3}\mathstrut -\mathstrut \) \(68\!\cdots\!32\) \(\nu^{2}\mathstrut +\mathstrut \) \(43\!\cdots\!40\) \(\nu\mathstrut -\mathstrut \) \(19\!\cdots\!32\)\()/\)\(17\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(25\!\cdots\!65\) \(\nu^{13}\mathstrut -\mathstrut \) \(12\!\cdots\!72\) \(\nu^{12}\mathstrut -\mathstrut \) \(11\!\cdots\!65\) \(\nu^{11}\mathstrut -\mathstrut \) \(40\!\cdots\!92\) \(\nu^{10}\mathstrut -\mathstrut \) \(20\!\cdots\!00\) \(\nu^{9}\mathstrut -\mathstrut \) \(49\!\cdots\!80\) \(\nu^{8}\mathstrut -\mathstrut \) \(16\!\cdots\!00\) \(\nu^{7}\mathstrut -\mathstrut \) \(27\!\cdots\!40\) \(\nu^{6}\mathstrut -\mathstrut \) \(63\!\cdots\!00\) \(\nu^{5}\mathstrut -\mathstrut \) \(67\!\cdots\!80\) \(\nu^{4}\mathstrut -\mathstrut \) \(10\!\cdots\!40\) \(\nu^{3}\mathstrut -\mathstrut \) \(68\!\cdots\!32\) \(\nu^{2}\mathstrut -\mathstrut \) \(43\!\cdots\!40\) \(\nu\mathstrut -\mathstrut \) \(19\!\cdots\!32\)\()/\)\(17\!\cdots\!00\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(16\!\cdots\!23\) \(\nu^{13}\mathstrut -\mathstrut \) \(92\!\cdots\!60\) \(\nu^{12}\mathstrut -\mathstrut \) \(53\!\cdots\!03\) \(\nu^{11}\mathstrut -\mathstrut \) \(27\!\cdots\!60\) \(\nu^{10}\mathstrut -\mathstrut \) \(65\!\cdots\!20\) \(\nu^{9}\mathstrut -\mathstrut \) \(28\!\cdots\!00\) \(\nu^{8}\mathstrut -\mathstrut \) \(36\!\cdots\!60\) \(\nu^{7}\mathstrut -\mathstrut \) \(10\!\cdots\!00\) \(\nu^{6}\mathstrut -\mathstrut \) \(91\!\cdots\!20\) \(\nu^{5}\mathstrut -\mathstrut \) \(90\!\cdots\!00\) \(\nu^{4}\mathstrut -\mathstrut \) \(94\!\cdots\!88\) \(\nu^{3}\mathstrut +\mathstrut \) \(11\!\cdots\!40\) \(\nu^{2}\mathstrut -\mathstrut \) \(35\!\cdots\!88\) \(\nu\mathstrut +\mathstrut \) \(49\!\cdots\!40\)\()/\)\(38\!\cdots\!00\)
\(\beta_{9}\)\(=\)\((\)\(16\!\cdots\!53\) \(\nu^{13}\mathstrut -\mathstrut \) \(17\!\cdots\!32\) \(\nu^{12}\mathstrut +\mathstrut \) \(53\!\cdots\!33\) \(\nu^{11}\mathstrut -\mathstrut \) \(58\!\cdots\!52\) \(\nu^{10}\mathstrut +\mathstrut \) \(65\!\cdots\!20\) \(\nu^{9}\mathstrut -\mathstrut \) \(72\!\cdots\!80\) \(\nu^{8}\mathstrut +\mathstrut \) \(36\!\cdots\!60\) \(\nu^{7}\mathstrut -\mathstrut \) \(40\!\cdots\!40\) \(\nu^{6}\mathstrut +\mathstrut \) \(93\!\cdots\!20\) \(\nu^{5}\mathstrut -\mathstrut \) \(10\!\cdots\!80\) \(\nu^{4}\mathstrut +\mathstrut \) \(97\!\cdots\!68\) \(\nu^{3}\mathstrut -\mathstrut \) \(11\!\cdots\!92\) \(\nu^{2}\mathstrut +\mathstrut \) \(36\!\cdots\!68\) \(\nu\mathstrut -\mathstrut \) \(25\!\cdots\!92\)\()/\)\(38\!\cdots\!00\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(88\!\cdots\!97\) \(\nu^{13}\mathstrut +\mathstrut \) \(93\!\cdots\!88\) \(\nu^{12}\mathstrut -\mathstrut \) \(30\!\cdots\!17\) \(\nu^{11}\mathstrut +\mathstrut \) \(29\!\cdots\!68\) \(\nu^{10}\mathstrut -\mathstrut \) \(39\!\cdots\!80\) \(\nu^{9}\mathstrut +\mathstrut \) \(34\!\cdots\!20\) \(\nu^{8}\mathstrut -\mathstrut \) \(23\!\cdots\!40\) \(\nu^{7}\mathstrut +\mathstrut \) \(16\!\cdots\!60\) \(\nu^{6}\mathstrut -\mathstrut \) \(69\!\cdots\!80\) \(\nu^{5}\mathstrut +\mathstrut \) \(32\!\cdots\!20\) \(\nu^{4}\mathstrut -\mathstrut \) \(85\!\cdots\!32\) \(\nu^{3}\mathstrut +\mathstrut \) \(14\!\cdots\!28\) \(\nu^{2}\mathstrut -\mathstrut \) \(24\!\cdots\!32\) \(\nu\mathstrut -\mathstrut \) \(31\!\cdots\!72\)\()/\)\(19\!\cdots\!00\)
\(\beta_{11}\)\(=\)\((\)\(11\!\cdots\!47\) \(\nu^{13}\mathstrut +\mathstrut \) \(11\!\cdots\!68\) \(\nu^{12}\mathstrut +\mathstrut \) \(41\!\cdots\!67\) \(\nu^{11}\mathstrut +\mathstrut \) \(32\!\cdots\!48\) \(\nu^{10}\mathstrut +\mathstrut \) \(59\!\cdots\!80\) \(\nu^{9}\mathstrut +\mathstrut \) \(29\!\cdots\!20\) \(\nu^{8}\mathstrut +\mathstrut \) \(40\!\cdots\!40\) \(\nu^{7}\mathstrut +\mathstrut \) \(66\!\cdots\!60\) \(\nu^{6}\mathstrut +\mathstrut \) \(13\!\cdots\!80\) \(\nu^{5}\mathstrut -\mathstrut \) \(30\!\cdots\!80\) \(\nu^{4}\mathstrut +\mathstrut \) \(19\!\cdots\!32\) \(\nu^{3}\mathstrut -\mathstrut \) \(11\!\cdots\!92\) \(\nu^{2}\mathstrut +\mathstrut \) \(41\!\cdots\!32\) \(\nu\mathstrut -\mathstrut \) \(49\!\cdots\!92\)\()/\)\(19\!\cdots\!00\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(68\!\cdots\!07\) \(\nu^{13}\mathstrut +\mathstrut \) \(17\!\cdots\!60\) \(\nu^{12}\mathstrut -\mathstrut \) \(23\!\cdots\!27\) \(\nu^{11}\mathstrut +\mathstrut \) \(54\!\cdots\!60\) \(\nu^{10}\mathstrut -\mathstrut \) \(29\!\cdots\!80\) \(\nu^{9}\mathstrut +\mathstrut \) \(63\!\cdots\!00\) \(\nu^{8}\mathstrut -\mathstrut \) \(16\!\cdots\!40\) \(\nu^{7}\mathstrut +\mathstrut \) \(31\!\cdots\!00\) \(\nu^{6}\mathstrut -\mathstrut \) \(45\!\cdots\!80\) \(\nu^{5}\mathstrut +\mathstrut \) \(61\!\cdots\!00\) \(\nu^{4}\mathstrut -\mathstrut \) \(52\!\cdots\!92\) \(\nu^{3}\mathstrut +\mathstrut \) \(27\!\cdots\!60\) \(\nu^{2}\mathstrut -\mathstrut \) \(14\!\cdots\!92\) \(\nu\mathstrut -\mathstrut \) \(54\!\cdots\!40\)\()/\)\(34\!\cdots\!00\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(11\!\cdots\!97\) \(\nu^{13}\mathstrut -\mathstrut \) \(12\!\cdots\!64\) \(\nu^{12}\mathstrut -\mathstrut \) \(39\!\cdots\!17\) \(\nu^{11}\mathstrut -\mathstrut \) \(41\!\cdots\!04\) \(\nu^{10}\mathstrut -\mathstrut \) \(49\!\cdots\!80\) \(\nu^{9}\mathstrut -\mathstrut \) \(49\!\cdots\!60\) \(\nu^{8}\mathstrut -\mathstrut \) \(28\!\cdots\!40\) \(\nu^{7}\mathstrut -\mathstrut \) \(26\!\cdots\!80\) \(\nu^{6}\mathstrut -\mathstrut \) \(78\!\cdots\!80\) \(\nu^{5}\mathstrut -\mathstrut \) \(63\!\cdots\!60\) \(\nu^{4}\mathstrut -\mathstrut \) \(89\!\cdots\!32\) \(\nu^{3}\mathstrut -\mathstrut \) \(55\!\cdots\!84\) \(\nu^{2}\mathstrut -\mathstrut \) \(28\!\cdots\!32\) \(\nu\mathstrut -\mathstrut \) \(80\!\cdots\!84\)\()/\)\(38\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut -\mathstrut \) \(101206\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(2\) \(\beta_{13}\mathstrut +\mathstrut \) \(6\) \(\beta_{12}\mathstrut -\mathstrut \) \(3\) \(\beta_{11}\mathstrut +\mathstrut \) \(17\) \(\beta_{9}\mathstrut -\mathstrut \) \(163\) \(\beta_{7}\mathstrut +\mathstrut \) \(158\) \(\beta_{6}\mathstrut +\mathstrut \) \(27\) \(\beta_{5}\mathstrut -\mathstrut \) \(755235\) \(\beta_{4}\mathstrut +\mathstrut \) \(85201\) \(\beta_{2}\mathstrut -\mathstrut \) \(85194\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(475\) \(\beta_{13}\mathstrut +\mathstrut \) \(10\) \(\beta_{11}\mathstrut -\mathstrut \) \(1385\) \(\beta_{10}\mathstrut +\mathstrut \) \(1952\) \(\beta_{9}\mathstrut +\mathstrut \) \(22\) \(\beta_{8}\mathstrut -\mathstrut \) \(432342\) \(\beta_{7}\mathstrut -\mathstrut \) \(433779\) \(\beta_{6}\mathstrut +\mathstrut \) \(485\) \(\beta_{4}\mathstrut -\mathstrut \) \(112625\) \(\beta_{3}\mathstrut -\mathstrut \) \(1617550\) \(\beta_{2}\mathstrut -\mathstrut \) \(1616115\) \(\beta_{1}\mathstrut +\mathstrut \) \(8621376925\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(285278\) \(\beta_{13}\mathstrut -\mathstrut \) \(908514\) \(\beta_{12}\mathstrut +\mathstrut \) \(331607\) \(\beta_{11}\mathstrut +\mathstrut \) \(437920\) \(\beta_{10}\mathstrut -\mathstrut \) \(2135933\) \(\beta_{9}\mathstrut +\mathstrut \) \(16184279\) \(\beta_{7}\mathstrut -\mathstrut \) \(15812694\) \(\beta_{6}\mathstrut -\mathstrut \) \(2712503\) \(\beta_{5}\mathstrut +\mathstrut \) \(171441398695\) \(\beta_{4}\mathstrut -\mathstrut \) \(8469019957\) \(\beta_{2}\mathstrut +\mathstrut \) \(8468310414\) \(\beta_{1}\mathstrut +\mathstrut \) \(46329\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(74784975\) \(\beta_{13}\mathstrut -\mathstrut \) \(14462850\) \(\beta_{11}\mathstrut +\mathstrut \) \(166503525\) \(\beta_{10}\mathstrut -\mathstrut \) \(353950176\) \(\beta_{9}\mathstrut -\mathstrut \) \(11421726\) \(\beta_{8}\mathstrut +\mathstrut \) \(57928521510\) \(\beta_{7}\mathstrut +\mathstrut \) \(58149835311\) \(\beta_{6}\mathstrut -\mathstrut \) \(89247825\) \(\beta_{4}\mathstrut +\mathstrut \) \(12321460257\) \(\beta_{3}\mathstrut +\mathstrut \) \(210517673154\) \(\beta_{2}\mathstrut +\mathstrut \) \(210278855379\) \(\beta_{1}\mathstrut -\mathstrut \) \(856970002462529\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(34242288078\) \(\beta_{13}\mathstrut +\mathstrut \) \(112080642834\) \(\beta_{12}\mathstrut -\mathstrut \) \(33898082967\) \(\beta_{11}\mathstrut -\mathstrut \) \(79215175200\) \(\beta_{10}\mathstrut +\mathstrut \) \(238663401213\) \(\beta_{9}\mathstrut -\mathstrut \) \(1232813821143\) \(\beta_{7}\mathstrut +\mathstrut \) \(1208957926998\) \(\beta_{6}\mathstrut +\mathstrut \) \(323461182903\) \(\beta_{5}\mathstrut -\mathstrut \) \(22279026170335335\) \(\beta_{4}\mathstrut +\mathstrut \) \(896995660210549\) \(\beta_{2}\mathstrut -\mathstrut \) \(896928208249726\) \(\beta_{1}\mathstrut +\mathstrut \) \(344205111\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(9193799374575\) \(\beta_{13}\mathstrut +\mathstrut \) \(2657937153090\) \(\beta_{11}\mathstrut -\mathstrut \) \(16949649511365\) \(\beta_{10}\mathstrut +\mathstrut \) \(47171245639392\) \(\beta_{9}\mathstrut +\mathstrut \) \(2422236681822\) \(\beta_{8}\mathstrut -\mathstrut \) \(6910772433083846\) \(\beta_{7}\mathstrut -\mathstrut \) \(6938118130736303\) \(\beta_{6}\mathstrut +\mathstrut \) \(11851736527665\) \(\beta_{4}\mathstrut -\mathstrut \) \(1356281117227057\) \(\beta_{3}\mathstrut -\mathstrut \) \(26175713713898194\) \(\beta_{2}\mathstrut -\mathstrut \) \(26145474378621379\) \(\beta_{1}\mathstrut +\mathstrut \) \(90772766389753538113\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(3933194264251502\) \(\beta_{13}\mathstrut -\mathstrut \) \(13000396234595826\) \(\beta_{12}\mathstrut +\mathstrut \) \(3524276220270663\) \(\beta_{11}\mathstrut +\mathstrut \) \(10702874146267680\) \(\beta_{10}\mathstrut -\mathstrut \) \(26305605717817997\) \(\beta_{9}\mathstrut +\mathstrut \) \(71779326531043975\) \(\beta_{7}\mathstrut -\mathstrut \) \(70273699840576310\) \(\beta_{6}\mathstrut -\mathstrut \) \(41442369069098727\) \(\beta_{5}\mathstrut +\mathstrut \) \(2762708215753090266135\) \(\beta_{4}\mathstrut -\mathstrut \) \(97651901127642897157\) \(\beta_{2}\mathstrut +\mathstrut \) \(97645261493246336670\) \(\beta_{1}\mathstrut -\mathstrut \) \(408918043980839\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(-\)\(1052948945522905375\) \(\beta_{13}\mathstrut -\mathstrut \) \(367770406439726050\) \(\beta_{11}\mathstrut +\mathstrut \) \(1687765210809811925\) \(\beta_{10}\mathstrut -\mathstrut \) \(5700941642795882720\) \(\beta_{9}\mathstrut -\mathstrut \) \(385834641385083070\) \(\beta_{8}\mathstrut +\mathstrut \) \(791026854073043882790\) \(\beta_{7}\mathstrut +\mathstrut \) \(794203765144557955935\) \(\beta_{6}\mathstrut -\mathstrut \) \(1420719351962631425\) \(\beta_{4}\mathstrut +\mathstrut \) \(150088078911600107585\) \(\beta_{3}\mathstrut +\mathstrut \) \(3299017997721991358770\) \(\beta_{2}\mathstrut +\mathstrut \) \(3295491380478982916595\) \(\beta_{1}\mathstrut -\mathstrut \) \(9882945275575547470052497\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-\)\(445137672736373468750\) \(\beta_{13}\mathstrut +\mathstrut \) \(1472991333084148165650\) \(\beta_{12}\mathstrut -\mathstrut \) \(373930962511537659575\) \(\beta_{11}\mathstrut -\mathstrut \) \(1312680501247117933600\) \(\beta_{10}\mathstrut +\mathstrut \) \(2902343578480106853725\) \(\beta_{9}\mathstrut -\mathstrut \) \(1412805011474259202295\) \(\beta_{7}\mathstrut +\mathstrut \) \(1318865784287420920470\) \(\beta_{6}\mathstrut +\mathstrut \) \(5364762517259945324375\) \(\beta_{5}\mathstrut -\mathstrut \) \(346765944965661291350275975\) \(\beta_{4}\mathstrut +\mathstrut \) \(10761253783748626582891861\) \(\beta_{2}\mathstrut -\mathstrut \) \(10760577128533828343381886\) \(\beta_{1}\mathstrut +\mathstrut \) \(71206710224835809175\)\()/2\)
\(\nu^{12}\)\(=\)\((\)\(117433418844927667695375\) \(\beta_{13}\mathstrut +\mathstrut \) \(46169676163856246338050\) \(\beta_{11}\mathstrut -\mathstrut \) \(167621551879358017733925\) \(\beta_{10}\mathstrut +\mathstrut \) \(662573317409753970259680\) \(\beta_{9}\mathstrut +\mathstrut \) \(54330613538474560464030\) \(\beta_{8}\mathstrut -\mathstrut \) \(89049996622019092381205382\) \(\beta_{7}\mathstrut -\mathstrut \) \(89410457815928493698417487\) \(\beta_{6}\mathstrut +\mathstrut \) \(163603095008783914033425\) \(\beta_{4}\mathstrut -\mathstrut \) \(16660031643891816371454801\) \(\beta_{3}\mathstrut -\mathstrut \) \(418539997711910750789792082\) \(\beta_{2}\mathstrut -\mathstrut \) \(418141527779212111540367907\) \(\beta_{1}\mathstrut +\mathstrut \) \(1089201096677296300494893341601\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(50066215558429060509368622\) \(\beta_{13}\mathstrut -\mathstrut \) \(165292257346187230920169266\) \(\beta_{12}\mathstrut +\mathstrut \) \(40197055900864801976473383\) \(\beta_{11}\mathstrut +\mathstrut \) \(154702680418015204542381600\) \(\beta_{10}\mathstrut -\mathstrut \) \(320856029936310647966894637\) \(\beta_{9}\mathstrut -\mathstrut \) \(499538797914258761244455577\) \(\beta_{7}\mathstrut +\mathstrut \) \(504903923829094996763075082\) \(\beta_{6}\mathstrut -\mathstrut \) \(688355281267293989504776647\) \(\beta_{5}\mathstrut +\mathstrut \) \(43804310551151807066444049702135\) \(\beta_{4}\mathstrut -\mathstrut \) \(1192548726917821424236608799141\) \(\beta_{2}\mathstrut +\mathstrut \) \(1192478201965677258891188747614\) \(\beta_{1}\mathstrut -\mathstrut \) \(9869159657564258532895239\)\()/2\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{4}\)

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
330.301i
205.093i
162.000i
38.1160i
88.8004i
233.161i
336.317i
330.301i
205.093i
162.000i
38.1160i
88.8004i
233.161i
336.317i
−330.301 330.301i 4939.05 4939.05i 152661.i 188004. + 342407.i −3.26275e6 5.35472e6 + 5.35472e6i 2.87776e7 2.87776e7i 5.74179e6i 5.09995e7 1.75195e8i
2.2 −205.093 205.093i −8389.67 + 8389.67i 18590.3i −65048.3 + 385171.i 3.44133e6 −4.20104e6 4.20104e6i −9.62824e6 + 9.62824e6i 9.77264e7i 9.23368e7 6.56549e7i
2.3 −162.000 162.000i 380.189 380.189i 13048.1i −70298.4 384247.i −123181. −65928.6 65928.6i −1.27306e7 + 1.27306e7i 4.27576e7i −5.08597e7 + 7.36363e7i
2.4 38.1160 + 38.1160i 7207.82 7207.82i 62630.3i −287350. + 264609.i 549467. −3.21528e6 3.21528e6i 4.88519e6 4.88519e6i 6.08585e7i −2.10385e7 866777.i
2.5 88.8004 + 88.8004i −1526.86 + 1526.86i 49765.0i 383800. + 72701.2i −271171. 2.52190e6 + 2.52190e6i 1.02388e7 1.02388e7i 3.83841e7i 2.76257e7 + 4.05375e7i
2.6 233.161 + 233.161i −4708.07 + 4708.07i 43191.7i −387973. 45440.9i −2.19547e6 −216284. 216284.i 5.20981e6 5.20981e6i 1.28506e6i −7.98650e7 1.01055e8i
2.7 336.317 + 336.317i 6051.53 6051.53i 160682.i 335305. 200395.i 4.07046e6 −371312. 371312.i −3.19992e7 + 3.19992e7i 3.01954e7i 1.80165e8 + 4.53727e7i
3.1 −330.301 + 330.301i 4939.05 + 4939.05i 152661.i 188004. 342407.i −3.26275e6 5.35472e6 5.35472e6i 2.87776e7 + 2.87776e7i 5.74179e6i 5.09995e7 + 1.75195e8i
3.2 −205.093 + 205.093i −8389.67 8389.67i 18590.3i −65048.3 385171.i 3.44133e6 −4.20104e6 + 4.20104e6i −9.62824e6 9.62824e6i 9.77264e7i 9.23368e7 + 6.56549e7i
3.3 −162.000 + 162.000i 380.189 + 380.189i 13048.1i −70298.4 + 384247.i −123181. −65928.6 + 65928.6i −1.27306e7 1.27306e7i 4.27576e7i −5.08597e7 7.36363e7i
3.4 38.1160 38.1160i 7207.82 + 7207.82i 62630.3i −287350. 264609.i 549467. −3.21528e6 + 3.21528e6i 4.88519e6 + 4.88519e6i 6.08585e7i −2.10385e7 + 866777.i
3.5 88.8004 88.8004i −1526.86 1526.86i 49765.0i 383800. 72701.2i −271171. 2.52190e6 2.52190e6i 1.02388e7 + 1.02388e7i 3.83841e7i 2.76257e7 4.05375e7i
3.6 233.161 233.161i −4708.07 4708.07i 43191.7i −387973. + 45440.9i −2.19547e6 −216284. + 216284.i 5.20981e6 + 5.20981e6i 1.28506e6i −7.98650e7 + 1.01055e8i
3.7 336.317 336.317i 6051.53 + 6051.53i 160682.i 335305. + 200395.i 4.07046e6 −371312. + 371312.i −3.19992e7 3.19992e7i 3.01954e7i 1.80165e8 4.53727e7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.7
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{17}^{\mathrm{new}}(5, [\chi])\).