# Properties

 Label 29.2.e Level 29 Weight 2 Character orbit e Rep. character $$\chi_{29}(4,\cdot)$$ Character field $$\Q(\zeta_{14})$$ Dimension 12 Newform subspaces 1 Sturm bound 5 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 29.e (of order $$14$$ and degree $$6$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$29$$ Character field: $$\Q(\zeta_{14})$$ Newform subspaces: $$1$$ Sturm bound: $$5$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(29, [\chi])$$.

Total New Old
Modular forms 24 24 0
Cusp forms 12 12 0
Eisenstein series 12 12 0

## Trace form

 $$12q - 7q^{2} - 7q^{3} - q^{4} - q^{5} - 3q^{6} - 11q^{7} + 14q^{8} - 3q^{9} + O(q^{10})$$ $$12q - 7q^{2} - 7q^{3} - q^{4} - q^{5} - 3q^{6} - 11q^{7} + 14q^{8} - 3q^{9} - 7q^{10} + 7q^{11} + 9q^{13} - 7q^{14} + 7q^{15} + 9q^{16} + 42q^{18} - 7q^{19} - 11q^{20} - 7q^{21} - 4q^{22} - 5q^{23} - 25q^{24} + 13q^{25} - 21q^{26} - 7q^{27} + 12q^{28} - 15q^{29} + 2q^{30} - 21q^{31} - 17q^{33} - 13q^{34} + 19q^{35} - 40q^{36} + 7q^{37} + 28q^{38} + 21q^{39} + 35q^{40} + 50q^{42} + 7q^{43} + 42q^{44} + 16q^{45} - 7q^{47} - 14q^{48} + 13q^{49} - 28q^{50} + 20q^{51} - 6q^{52} - 10q^{53} - 38q^{54} - 35q^{55} - 21q^{56} - 14q^{57} - 57q^{58} + 44q^{59} - 28q^{60} - 7q^{61} + 37q^{62} - 13q^{63} - 26q^{64} - 6q^{65} + 21q^{66} - 37q^{67} + 14q^{68} + 21q^{69} - 21q^{71} + 35q^{72} + 14q^{73} + 7q^{76} - 7q^{77} + 17q^{78} + 49q^{79} - 6q^{80} + q^{81} + 22q^{82} + 5q^{83} + 21q^{84} + 14q^{85} - 44q^{86} + 15q^{87} - 66q^{88} + 7q^{89} + 28q^{90} - 3q^{91} - 6q^{92} + 19q^{93} + 66q^{94} - 7q^{95} + 30q^{96} + 14q^{97} - 42q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(29, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
29.2.e.a $$12$$ $$0.232$$ 12.0.$$\cdots$$.1 None $$-7$$ $$-7$$ $$-1$$ $$-11$$ $$q+(-1+\beta _{3}+\beta _{7}+\beta _{9}+\beta _{10})q^{2}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 7 T + 27 T^{2} + 70 T^{3} + 130 T^{4} + 161 T^{5} + 84 T^{6} - 147 T^{7} - 401 T^{8} - 301 T^{9} + 646 T^{10} + 2534 T^{11} + 4577 T^{12} + 5068 T^{13} + 2584 T^{14} - 2408 T^{15} - 6416 T^{16} - 4704 T^{17} + 5376 T^{18} + 20608 T^{19} + 33280 T^{20} + 35840 T^{21} + 27648 T^{22} + 14336 T^{23} + 4096 T^{24}$$
$3$ $$1 + 7 T + 29 T^{2} + 91 T^{3} + 244 T^{4} + 546 T^{5} + 1012 T^{6} + 1491 T^{7} + 1497 T^{8} - 259 T^{9} - 5693 T^{10} - 16660 T^{11} - 32900 T^{12} - 49980 T^{13} - 51237 T^{14} - 6993 T^{15} + 121257 T^{16} + 362313 T^{17} + 737748 T^{18} + 1194102 T^{19} + 1600884 T^{20} + 1791153 T^{21} + 1712421 T^{22} + 1240029 T^{23} + 531441 T^{24}$$
$5$ $$1 + T - 11 T^{2} - 11 T^{3} + 66 T^{4} + 76 T^{5} - 174 T^{6} - 69 T^{7} + 399 T^{8} - 1417 T^{9} - 1605 T^{10} + 5700 T^{11} + 14536 T^{12} + 28500 T^{13} - 40125 T^{14} - 177125 T^{15} + 249375 T^{16} - 215625 T^{17} - 2718750 T^{18} + 5937500 T^{19} + 25781250 T^{20} - 21484375 T^{21} - 107421875 T^{22} + 48828125 T^{23} + 244140625 T^{24}$$
$7$ $$1 + 11 T + 47 T^{2} + 123 T^{3} + 458 T^{4} + 2166 T^{5} + 7378 T^{6} + 20319 T^{7} + 61291 T^{8} + 197341 T^{9} + 577023 T^{10} + 1543704 T^{11} + 4058132 T^{12} + 10805928 T^{13} + 28274127 T^{14} + 67687963 T^{15} + 147159691 T^{16} + 341501433 T^{17} + 868014322 T^{18} + 1783794138 T^{19} + 2640278858 T^{20} + 4963493661 T^{21} + 13276336703 T^{22} + 21750594173 T^{23} + 13841287201 T^{24}$$
$11$ $$1 - 7 T + 17 T^{2} + 7 T^{3} - 14 T^{5} - 1448 T^{6} + 5383 T^{7} + 13177 T^{8} - 74137 T^{9} + 167363 T^{10} - 476952 T^{11} + 2425844 T^{12} - 5246472 T^{13} + 20250923 T^{14} - 98676347 T^{15} + 192924457 T^{16} + 866937533 T^{17} - 2565220328 T^{18} - 272820394 T^{19} + 16505633837 T^{21} + 440936218217 T^{22} - 1997181694277 T^{23} + 3138428376721 T^{24}$$
$13$ $$1 - 9 T - 9 T^{2} + 235 T^{3} + 440 T^{4} - 7052 T^{5} + 904 T^{6} + 93021 T^{7} + 10465 T^{8} - 1306603 T^{9} + 1744795 T^{10} + 3749024 T^{11} - 11467928 T^{12} + 48737312 T^{13} + 294870355 T^{14} - 2870606791 T^{15} + 298890865 T^{16} + 34538046153 T^{17} + 4363435336 T^{18} - 442502541884 T^{19} + 358921517240 T^{20} + 2492057352655 T^{21} - 1240726426641 T^{22} - 16129443546333 T^{23} + 23298085122481 T^{24}$$
$17$ $$1 - 133 T^{2} + 8874 T^{4} - 389130 T^{6} + 12426957 T^{8} - 303621745 T^{10} + 5811079504 T^{12} - 87746684305 T^{14} + 1037911875597 T^{16} - 9392652224970 T^{18} + 61902871531434 T^{20} - 268127188759717 T^{22} + 582622237229761 T^{24}$$
$19$ $$1 + 7 T + 31 T^{2} + 189 T^{3} + 1426 T^{4} + 8162 T^{5} + 37516 T^{6} + 183911 T^{7} + 989579 T^{8} + 5075581 T^{9} + 23151601 T^{10} + 103810854 T^{11} + 453309164 T^{12} + 1972406226 T^{13} + 8357727961 T^{14} + 34813410079 T^{15} + 128962924859 T^{16} + 455381843189 T^{17} + 1764973271596 T^{18} + 7295781133718 T^{19} + 24218560896466 T^{20} + 60987974880231 T^{21} + 190063053991831 T^{22} + 815431812287533 T^{23} + 2213314919066161 T^{24}$$
$23$ $$1 + 5 T - 43 T^{2} - 391 T^{3} - 308 T^{4} + 10696 T^{5} + 58590 T^{6} - 14345 T^{7} - 1422557 T^{8} - 5436135 T^{9} + 1966389 T^{10} + 79674994 T^{11} + 420307216 T^{12} + 1832524862 T^{13} + 1040219781 T^{14} - 66141454545 T^{15} - 398089773437 T^{16} - 92329340335 T^{17} + 8673422736510 T^{18} + 36418012981112 T^{19} - 24119783466548 T^{20} - 704250690632033 T^{21} - 1781339982186907 T^{22} + 4764048789569635 T^{23} + 21914624432020321 T^{24}$$
$29$ $$1 + 15 T + 126 T^{2} + 622 T^{3} + 1665 T^{4} - 4109 T^{5} - 52800 T^{6} - 119161 T^{7} + 1400265 T^{8} + 15169958 T^{9} + 89117406 T^{10} + 307667235 T^{11} + 594823321 T^{12}$$
$31$ $$1 + 21 T + 269 T^{2} + 2261 T^{3} + 14784 T^{4} + 77658 T^{5} + 355230 T^{6} + 1212127 T^{7} - 208667 T^{8} - 52988005 T^{9} - 588286643 T^{10} - 4352370134 T^{11} - 26495076348 T^{12} - 134923474154 T^{13} - 565343463923 T^{14} - 1578565656955 T^{15} - 192708356507 T^{16} + 34702166914177 T^{17} + 315267932601630 T^{18} + 2136574586632038 T^{19} + 12609141097527744 T^{20} + 59779985705277131 T^{21} + 220480009197835469 T^{22} + 533578014824501451 T^{23} + 787662783788549761 T^{24}$$
$37$ $$1 - 7 T - 3 T^{2} + 833 T^{3} - 5462 T^{4} - 9800 T^{5} + 411870 T^{6} - 2163693 T^{7} - 6602285 T^{8} + 140542059 T^{9} - 562585377 T^{10} - 2805854072 T^{11} + 36132816832 T^{12} - 103816600664 T^{13} - 770179381113 T^{14} + 7118876914527 T^{15} - 12373745057885 T^{16} - 150039034353201 T^{17} + 1056745736074830 T^{18} - 930332395903400 T^{19} - 19185162777316502 T^{20} + 108258129249299141 T^{21} - 14425753117253547 T^{22} - 1245423352456222891 T^{23} + 6582952005840035281 T^{24}$$
$41$ $$1 - 393 T^{2} + 73710 T^{4} - 8727326 T^{6} + 726625577 T^{8} - 44820888217 T^{10} + 2100378900880 T^{12} - 75343913092777 T^{14} + 2053270217089097 T^{16} - 41455708245189566 T^{18} + 588568838638508910 T^{20} - 5275105108889893593 T^{22} + 22563490300366186081 T^{24}$$
$43$ $$1 - 7 T + 139 T^{2} - 525 T^{3} + 4746 T^{4} + 19698 T^{5} - 227814 T^{6} + 3326155 T^{7} - 15690613 T^{8} + 60933523 T^{9} + 454409879 T^{10} - 6181766668 T^{11} + 56674975708 T^{12} - 265815966724 T^{13} + 840203866271 T^{14} + 4844641613161 T^{15} - 53643083415013 T^{16} + 488972867726665 T^{17} - 1440095001644886 T^{18} + 5354283001585686 T^{19} + 55472198517494346 T^{20} - 263861121266842575 T^{21} + 3003996041546510611 T^{22} - 6505056176298558949 T^{23} + 39959630797262576401 T^{24}$$
$47$ $$1 + 7 T - 9 T^{2} - 497 T^{3} - 5250 T^{4} - 33334 T^{5} - 12592 T^{6} + 1824669 T^{7} + 19098315 T^{8} + 97990109 T^{9} + 127809129 T^{10} - 4616139010 T^{11} - 56042219764 T^{12} - 216958533470 T^{13} + 282330365961 T^{14} + 10173627086707 T^{15} + 93193684837515 T^{16} + 418478724577683 T^{17} - 135731879422768 T^{18} - 16887775097513642 T^{19} - 125009254974245250 T^{20} - 556207845132075199 T^{21} - 473392190122470441 T^{22} + 17305114505588086121 T^{23} +$$$$11\!\cdots\!41$$$$T^{24}$$
$53$ $$1 + 10 T + 25 T^{2} - 278 T^{3} + 2512 T^{4} + 20294 T^{5} - 44832 T^{6} - 1725298 T^{7} + 3740845 T^{8} + 15939776 T^{9} - 71501305 T^{10} - 3189845416 T^{11} + 681878948 T^{12} - 169061807048 T^{13} - 200847165745 T^{14} + 2373066031552 T^{15} + 29517066396445 T^{16} - 721511847681914 T^{17} - 993672638135328 T^{18} + 23839587871852078 T^{19} + 156396342313338832 T^{20} - 917334278520992974 T^{21} + 4372186759137826225 T^{22} + 92690359293721915970 T^{23} +$$$$49\!\cdots\!41$$$$T^{24}$$
$59$ $$( 1 - 22 T + 446 T^{2} - 6050 T^{3} + 72311 T^{4} - 688228 T^{5} + 5840260 T^{6} - 40605452 T^{7} + 251714591 T^{8} - 1242542950 T^{9} + 5404343006 T^{10} - 15728334578 T^{11} + 42180533641 T^{12} )^{2}$$
$61$ $$1 + 7 T + 159 T^{2} + 105 T^{3} + 5012 T^{4} - 64120 T^{5} + 241116 T^{6} - 1456693 T^{7} + 53389705 T^{8} + 83034021 T^{9} + 2548004823 T^{10} - 10608027248 T^{11} + 41335695248 T^{12} - 647089662128 T^{13} + 9481125946383 T^{14} + 18847145120601 T^{15} + 739225366466905 T^{16} - 1230317519492593 T^{17} + 12422386584426876 T^{18} - 201512670645666520 T^{19} + 960837052742372372 T^{20} + 1227885339747584805 T^{21} +$$$$11\!\cdots\!59$$$$T^{22} +$$$$30\!\cdots\!27$$$$T^{23} +$$$$26\!\cdots\!21$$$$T^{24}$$
$67$ $$1 + 37 T + 513 T^{2} + 3025 T^{3} - 450 T^{4} - 212780 T^{5} - 3203838 T^{6} - 28238209 T^{7} - 156569483 T^{8} - 254147545 T^{9} + 8082101273 T^{10} + 110845067132 T^{11} + 930583608704 T^{12} + 7426619497844 T^{13} + 36280552614497 T^{14} - 76438178076835 T^{15} - 3155050596840443 T^{16} - 38125114947613363 T^{17} - 289814002211564622 T^{18} - 1289598215380627940 T^{19} - 182730454900488450 T^{20} + 82299766548792214675 T^{21} +$$$$93\!\cdots\!37$$$$T^{22} +$$$$45\!\cdots\!71$$$$T^{23} +$$$$81\!\cdots\!61$$$$T^{24}$$
$71$ $$1 + 21 T + 173 T^{2} + 721 T^{3} - 4778 T^{4} - 187446 T^{5} - 2306210 T^{6} - 16620177 T^{7} - 82259473 T^{8} + 61660641 T^{9} + 7528298205 T^{10} + 91816724300 T^{11} + 804638755620 T^{12} + 6518987425300 T^{13} + 37950151251405 T^{14} + 22069021680951 T^{15} - 2090351487104113 T^{16} - 29986611162215127 T^{17} - 295426155781449410 T^{18} - 1704843893209759386 T^{19} - 3085410372292246058 T^{20} + 33056769018001751351 T^{21} +$$$$56\!\cdots\!73$$$$T^{22} +$$$$48\!\cdots\!91$$$$T^{23} +$$$$16\!\cdots\!41$$$$T^{24}$$
$73$ $$1 - 14 T + 223 T^{2} - 1638 T^{3} + 14902 T^{4} - 24080 T^{5} - 61436 T^{6} + 7903616 T^{7} - 50738367 T^{8} + 656032846 T^{9} - 446390355 T^{10} + 13646495926 T^{11} + 186938570672 T^{12} + 996194202598 T^{13} - 2378814201795 T^{14} + 255207929652382 T^{15} - 1440880374012447 T^{16} + 16384761811580288 T^{17} - 9297369526291004 T^{18} - 266021356339855760 T^{19} + 12017868289405595062 T^{20} - 96431659028142841494 T^{21} +$$$$95\!\cdots\!27$$$$T^{22} -$$$$43\!\cdots\!78$$$$T^{23} +$$$$22\!\cdots\!21$$$$T^{24}$$
$79$ $$1 - 49 T + 1401 T^{2} - 29659 T^{3} + 510220 T^{4} - 7424550 T^{5} + 93952026 T^{6} - 1051207157 T^{7} + 10574913669 T^{8} - 97442137415 T^{9} + 846233526101 T^{10} - 7200350040954 T^{11} + 62681377749836 T^{12} - 568827653235366 T^{13} + 5281343436396341 T^{14} - 48042773988954185 T^{15} + 411893743975557189 T^{16} - 3234623709121447643 T^{17} + 22838558941382835546 T^{18} -$$$$14\!\cdots\!50$$$$T^{19} +$$$$77\!\cdots\!20$$$$T^{20} -$$$$35\!\cdots\!21$$$$T^{21} +$$$$13\!\cdots\!01$$$$T^{22} -$$$$36\!\cdots\!71$$$$T^{23} +$$$$59\!\cdots\!41$$$$T^{24}$$
$83$ $$1 - 5 T + 19 T^{2} - 291 T^{3} + 6964 T^{4} - 112648 T^{5} + 680674 T^{6} - 2772355 T^{7} + 73849055 T^{8} - 851751547 T^{9} + 7543184631 T^{10} - 61831617966 T^{11} + 242863289344 T^{12} - 5132024291178 T^{13} + 51964998922959 T^{14} - 487020461804489 T^{15} + 3504752157736655 T^{16} - 10920419021824265 T^{17} + 222539811702570706 T^{18} - 3056821871879502296 T^{19} + 15684963104616281524 T^{20} - 54399614282854257273 T^{21} +$$$$29\!\cdots\!31$$$$T^{22} -$$$$64\!\cdots\!35$$$$T^{23} +$$$$10\!\cdots\!61$$$$T^{24}$$
$89$ $$1 - 7 T + 319 T^{2} - 2107 T^{3} + 41592 T^{4} - 268912 T^{5} + 2519080 T^{6} - 17316453 T^{7} - 32462455 T^{8} + 662290951 T^{9} - 28551782921 T^{10} + 290592922368 T^{11} - 3660702381280 T^{12} + 25862770090752 T^{13} - 226158672517241 T^{14} + 466894589435519 T^{15} - 2036767175061655 T^{16} - 96696102997814397 T^{17} + 1251935630434035880 T^{18} - 11894336729426494448 T^{19} +$$$$16\!\cdots\!52$$$$T^{20} -$$$$73\!\cdots\!63$$$$T^{21} +$$$$99\!\cdots\!19$$$$T^{22} -$$$$19\!\cdots\!23$$$$T^{23} +$$$$24\!\cdots\!21$$$$T^{24}$$
$97$ $$1 - 14 T + 441 T^{2} - 5068 T^{3} + 83194 T^{4} - 784252 T^{5} + 9216214 T^{6} - 73602242 T^{7} + 720487153 T^{8} - 5423464648 T^{9} + 47440764385 T^{10} - 407534937824 T^{11} + 3645124614984 T^{12} - 39530888968928 T^{13} + 446370152098465 T^{14} - 4949849750684104 T^{15} + 63784209624826993 T^{16} - 632047495732056194 T^{17} + 7676848253434718806 T^{18} - 63366216198529076476 T^{19} +$$$$65\!\cdots\!34$$$$T^{20} -$$$$38\!\cdots\!56$$$$T^{21} +$$$$32\!\cdots\!09$$$$T^{22} -$$$$10\!\cdots\!42$$$$T^{23} +$$$$69\!\cdots\!41$$$$T^{24}$$