# Properties

 Label 14.13.b.a Level $14$ Weight $13$ Character orbit 14.b Analytic conductor $12.796$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$14 = 2 \cdot 7$$ Weight: $$k$$ $$=$$ $$13$$ Character orbit: $$[\chi]$$ $$=$$ 14.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.7959134419$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 154710 x^{6} + 8245426887 x^{4} + 174724076278260 x^{2} + 1264170035276291934$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{26}\cdot 3^{2}\cdot 7^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + \beta_{3} q^{3} + 2048 q^{4} + ( -11 \beta_{3} - \beta_{4} ) q^{5} + ( -5 \beta_{3} - 2 \beta_{4} - \beta_{7} ) q^{6} + ( 24395 + 514 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{7} -2048 \beta_{1} q^{8} + ( -184863 - 1988 \beta_{1} + \beta_{2} + 5 \beta_{4} - 10 \beta_{5} - 5 \beta_{6} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + \beta_{3} q^{3} + 2048 q^{4} + ( -11 \beta_{3} - \beta_{4} ) q^{5} + ( -5 \beta_{3} - 2 \beta_{4} - \beta_{7} ) q^{6} + ( 24395 + 514 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{7} -2048 \beta_{1} q^{8} + ( -184863 - 1988 \beta_{1} + \beta_{2} + 5 \beta_{4} - 10 \beta_{5} - 5 \beta_{6} ) q^{9} + ( 385 \beta_{3} + 10 \beta_{4} - 16 \beta_{6} + 13 \beta_{7} ) q^{10} + ( -26730 + 13508 \beta_{1} - 11 \beta_{2} ) q^{11} + 2048 \beta_{3} q^{12} + ( -2207 \beta_{3} - 290 \beta_{4} + 71 \beta_{6} + 46 \beta_{7} ) q^{13} + ( -1052352 - 24393 \beta_{1} + 16 \beta_{2} + 2269 \beta_{3} - 286 \beta_{4} + 48 \beta_{6} + 57 \beta_{7} ) q^{14} + ( 8235288 + 134504 \beta_{1} - 51 \beta_{2} - 49 \beta_{4} + 98 \beta_{5} + 49 \beta_{6} ) q^{15} + 4194304 q^{16} + ( 14806 \beta_{3} - 1585 \beta_{4} + 933 \beta_{6} - 102 \beta_{7} ) q^{17} + ( 4068480 + 184863 \beta_{1} - 160 \beta_{2} - 64 \beta_{4} + 128 \beta_{5} + 64 \beta_{6} ) q^{18} + ( 12263 \beta_{3} - 1531 \beta_{4} + 91 \beta_{6} + 1562 \beta_{7} ) q^{19} + ( -22528 \beta_{3} - 2048 \beta_{4} ) q^{20} + ( -556248 + 738980 \beta_{1} - 215 \beta_{2} + 88613 \beta_{3} + 237 \beta_{4} - 270 \beta_{5} - 864 \beta_{6} + 726 \beta_{7} ) q^{21} + ( -27667200 + 26510 \beta_{1} + 704 \beta_{4} - 1408 \beta_{5} - 704 \beta_{6} ) q^{22} + ( 19591470 - 589796 \beta_{1} - 535 \beta_{2} - 486 \beta_{4} + 972 \beta_{5} + 486 \beta_{6} ) q^{23} + ( -10240 \beta_{3} - 4096 \beta_{4} - 2048 \beta_{7} ) q^{24} + ( 23904625 - 1602772 \beta_{1} + 1053 \beta_{2} + 485 \beta_{4} - 970 \beta_{5} - 485 \beta_{6} ) q^{25} + ( 42191 \beta_{3} + 5686 \beta_{4} - 7248 \beta_{6} + 1923 \beta_{7} ) q^{26} + ( -296904 \beta_{3} - 6765 \beta_{4} + 6669 \beta_{6} - 6906 \beta_{7} ) q^{27} + ( 49960960 + 1052672 \beta_{1} + 4096 \beta_{3} - 6144 \beta_{4} + 2048 \beta_{5} - 6144 \beta_{6} - 2048 \beta_{7} ) q^{28} + ( 38606706 + 5750456 \beta_{1} + 718 \beta_{2} - 4050 \beta_{4} + 8100 \beta_{5} + 4050 \beta_{6} ) q^{29} + ( -275445888 - 8236112 \beta_{1} + 1568 \beta_{2} + 3264 \beta_{4} - 6528 \beta_{5} - 3264 \beta_{6} ) q^{30} + ( -432366 \beta_{3} + 58173 \beta_{4} + 22383 \beta_{6} - 4926 \beta_{7} ) q^{31} -4194304 \beta_{1} q^{32} + ( 228492 \beta_{3} + 159555 \beta_{4} + 6831 \beta_{6} + 44286 \beta_{7} ) q^{33} + ( 563788 \beta_{3} + 23704 \beta_{4} - 37024 \beta_{6} - 29348 \beta_{7} ) q^{34} + ( -470591856 + 8419184 \beta_{1} + 4369 \beta_{2} - 957181 \beta_{3} - 22343 \beta_{4} - 2106 \beta_{5} + 13251 \beta_{6} - 33072 \beta_{7} ) q^{35} + ( -378599424 - 4071424 \beta_{1} + 2048 \beta_{2} + 10240 \beta_{4} - 20480 \beta_{5} - 10240 \beta_{6} ) q^{36} + ( -405450110 - 27493992 \beta_{1} + 7290 \beta_{2} - 7938 \beta_{4} + 15876 \beta_{5} + 7938 \beta_{6} ) q^{37} + ( -1687457 \beta_{3} - 57850 \beta_{4} - 75936 \beta_{6} + 3219 \beta_{7} ) q^{38} + ( 1690164408 + 15321880 \beta_{1} - 9505 \beta_{2} - 10215 \beta_{4} + 20430 \beta_{5} + 10215 \beta_{6} ) q^{39} + ( 788480 \beta_{3} + 20480 \beta_{4} - 32768 \beta_{6} + 26624 \beta_{7} ) q^{40} + ( 1311530 \beta_{3} - 405416 \beta_{4} + 71370 \beta_{6} - 82908 \beta_{7} ) q^{41} + ( -1513572480 + 551408 \beta_{1} - 4320 \beta_{2} - 1447309 \beta_{3} - 227810 \beta_{4} - 27520 \beta_{5} - 23696 \beta_{6} - 69161 \beta_{7} ) q^{42} + ( 2625787750 + 25211860 \beta_{1} - 10125 \beta_{2} + 154 \beta_{4} - 308 \beta_{5} - 154 \beta_{6} ) q^{43} + ( -54743040 + 27664384 \beta_{1} - 22528 \beta_{2} ) q^{44} + ( 9973623 \beta_{3} + 261318 \beta_{4} - 39771 \beta_{6} + 297978 \beta_{7} ) q^{45} + ( 1208076288 - 19600226 \beta_{1} + 15552 \beta_{2} + 34240 \beta_{4} - 68480 \beta_{5} - 34240 \beta_{6} ) q^{46} + ( -3900530 \beta_{3} - 2395 \beta_{4} + 303327 \beta_{6} + 329922 \beta_{7} ) q^{47} + 4194304 \beta_{3} q^{48} + ( -2407344863 + 17760008 \beta_{1} - 32886 \beta_{2} + 2633372 \beta_{3} - 392189 \beta_{4} + 97580 \beta_{5} + 225659 \beta_{6} - 276262 \beta_{7} ) q^{49} + ( 3282436224 - 23885505 \beta_{1} - 15520 \beta_{2} - 67392 \beta_{4} + 134784 \beta_{5} + 67392 \beta_{6} ) q^{50} + ( -10120729104 + 203423504 \beta_{1} - 45048 \beta_{2} + 35420 \beta_{4} - 70840 \beta_{5} - 35420 \beta_{6} ) q^{51} + ( -4519936 \beta_{3} - 593920 \beta_{4} + 145408 \beta_{6} + 94208 \beta_{7} ) q^{52} + ( 22555704690 + 91069248 \beta_{1} + 19620 \beta_{2} - 19764 \beta_{4} + 39528 \beta_{5} + 19764 \beta_{6} ) q^{53} + ( 12956454 \beta_{3} + 1116156 \beta_{4} + 6048 \beta_{6} + 128238 \beta_{7} ) q^{54} + ( -29824740 \beta_{3} - 1405965 \beta_{4} - 365079 \beta_{6} - 266178 \beta_{7} ) q^{55} + ( -2155216896 - 49956864 \beta_{1} + 32768 \beta_{2} + 4646912 \beta_{3} - 585728 \beta_{4} + 98304 \beta_{6} + 116736 \beta_{7} ) q^{56} + ( -7893245280 - 602885956 \beta_{1} + 69265 \beta_{2} + 145977 \beta_{4} - 291954 \beta_{5} - 145977 \beta_{6} ) q^{57} + ( -11774158080 - 38576146 \beta_{1} + 129600 \beta_{2} - 45952 \beta_{4} + 91904 \beta_{5} + 45952 \beta_{6} ) q^{58} + ( 24324785 \beta_{3} + 1122235 \beta_{4} - 312915 \beta_{6} - 1044570 \beta_{7} ) q^{59} + ( 16865869824 + 275464192 \beta_{1} - 104448 \beta_{2} - 100352 \beta_{4} + 200704 \beta_{5} + 100352 \beta_{6} ) q^{60} + ( -25175257 \beta_{3} + 979649 \beta_{4} - 1826804 \beta_{6} - 605848 \beta_{7} ) q^{61} + ( -10903080 \beta_{3} + 3332880 \beta_{4} + 730272 \beta_{6} - 131208 \beta_{7} ) q^{62} + ( -50338314285 - 267872278 \beta_{1} + 138395 \beta_{2} - 8775096 \beta_{3} + 3237205 \beta_{4} - 494585 \beta_{5} - 1776991 \beta_{6} + 751869 \beta_{7} ) q^{63} + 8589934592 q^{64} + ( -53111271024 - 316013084 \beta_{1} + 181031 \beta_{2} - 14013 \beta_{4} + 28026 \beta_{5} + 14013 \beta_{6} ) q^{65} + ( -114335034 \beta_{3} + 1356828 \beta_{4} + 1026432 \beta_{6} - 271986 \beta_{7} ) q^{66} + ( 46151407430 + 84200692 \beta_{1} + 98415 \beta_{2} + 257686 \beta_{4} - 515372 \beta_{5} - 257686 \beta_{6} ) q^{67} + ( 30322688 \beta_{3} - 3246080 \beta_{4} + 1910784 \beta_{6} - 208896 \beta_{7} ) q^{68} + ( 85692516 \beta_{3} + 4366551 \beta_{4} - 376353 \beta_{6} + 1113198 \beta_{7} ) q^{69} + ( -17242044288 + 470675024 \beta_{1} - 33696 \beta_{2} + 57211817 \beta_{3} + 2904106 \beta_{4} + 559232 \beta_{5} + 734720 \beta_{6} + 448853 \beta_{7} ) q^{70} + ( 71757268038 - 695314076 \beta_{1} - 127648 \beta_{2} - 444285 \beta_{4} + 888570 \beta_{5} + 444285 \beta_{6} ) q^{71} + ( 8332247040 + 378599424 \beta_{1} - 327680 \beta_{2} - 131072 \beta_{4} + 262144 \beta_{5} + 131072 \beta_{6} ) q^{72} + ( 35307944 \beta_{3} - 6670675 \beta_{4} + 1056109 \beta_{6} - 772006 \beta_{7} ) q^{73} + ( 56314642176 + 405627662 \beta_{1} + 254016 \beta_{2} - 466560 \beta_{4} + 933120 \beta_{5} + 466560 \beta_{6} ) q^{74} + ( -61874081 \beta_{3} - 14994921 \beta_{4} + 53217 \beta_{6} - 4754706 \beta_{7} ) q^{75} + ( 25114624 \beta_{3} - 3135488 \beta_{4} + 186368 \beta_{6} + 3198976 \beta_{7} ) q^{76} + ( -9239429430 + 3465388036 \beta_{1} - 484495 \beta_{2} - 156383370 \beta_{3} - 2129061 \beta_{4} + 696762 \beta_{5} + 2953335 \beta_{6} + 3554364 \beta_{7} ) q^{77} + ( -31375105920 - 1690313648 \beta_{1} + 326880 \beta_{2} + 608320 \beta_{4} - 1216640 \beta_{5} - 608320 \beta_{6} ) q^{78} + ( -75978326266 - 2404164060 \beta_{1} - 609930 \beta_{2} + 517905 \beta_{4} - 1035810 \beta_{5} - 517905 \beta_{6} ) q^{79} + ( -46137344 \beta_{3} - 4194304 \beta_{4} ) q^{80} + ( 114881492673 + 3655730820 \beta_{1} - 473985 \beta_{2} + 554295 \beta_{4} - 1108590 \beta_{5} - 554295 \beta_{6} ) q^{81} + ( 238377614 \beta_{3} - 903220 \beta_{4} - 4975520 \beta_{6} - 2531530 \beta_{7} ) q^{82} + ( 203353577 \beta_{3} + 6658870 \beta_{4} + 639090 \beta_{6} + 7113420 \beta_{7} ) q^{83} + ( -1139195904 + 1513431040 \beta_{1} - 440320 \beta_{2} + 181479424 \beta_{3} + 485376 \beta_{4} - 552960 \beta_{5} - 1769472 \beta_{6} + 1486848 \beta_{7} ) q^{84} + ( -30900282576 - 3155464144 \beta_{1} - 717984 \beta_{2} - 1372 \beta_{4} + 2744 \beta_{5} + 1372 \beta_{6} ) q^{85} + ( -51636579840 - 2625990866 \beta_{1} - 4928 \beta_{2} + 648000 \beta_{4} - 1296000 \beta_{5} - 648000 \beta_{6} ) q^{86} + ( 554457270 \beta_{3} - 4650870 \beta_{4} - 6350778 \beta_{6} + 5458692 \beta_{7} ) q^{87} + ( -56662425600 + 54292480 \beta_{1} + 1441792 \beta_{4} - 2883584 \beta_{5} - 1441792 \beta_{6} ) q^{88} + ( -445643820 \beta_{3} - 125727 \beta_{4} + 16869525 \beta_{6} - 574806 \beta_{7} ) q^{89} + ( -541213023 \beta_{3} - 24005718 \beta_{4} - 4717872 \beta_{6} - 7098579 \beta_{7} ) q^{90} + ( 18065927712 + 4598890496 \beta_{1} + 1723923 \beta_{2} - 185973421 \beta_{3} + 1552642 \beta_{4} - 1652950 \beta_{5} + 1772738 \beta_{6} - 6922954 \beta_{7} ) q^{91} + ( 40123330560 - 1207902208 \beta_{1} - 1095680 \beta_{2} - 995328 \beta_{4} + 1990656 \beta_{5} + 995328 \beta_{6} ) q^{92} + ( 286539057360 - 3196820640 \beta_{1} + 1793700 \beta_{2} - 3791712 \beta_{4} + 7583424 \beta_{5} + 3791712 \beta_{6} ) q^{93} + ( -437277188 \beta_{3} + 26206264 \beta_{4} - 15449056 \beta_{6} + 1414732 \beta_{7} ) q^{94} + ( -131705247672 + 12479966984 \beta_{1} - 461771 \beta_{2} - 5520069 \beta_{4} + 11040138 \beta_{5} + 5520069 \beta_{6} ) q^{95} + ( -20971520 \beta_{3} - 8388608 \beta_{4} - 4194304 \beta_{7} ) q^{96} + ( -180748082 \beta_{3} + 47714725 \beta_{4} + 10303211 \beta_{6} - 10704794 \beta_{7} ) q^{97} + ( -36349689600 + 2406882303 \beta_{1} + 1561280 \beta_{2} + 472101322 \beta_{3} + 10026884 \beta_{4} - 4209408 \beta_{5} - 1588608 \beta_{6} - 7616574 \beta_{7} ) q^{98} + ( -225119282058 - 32514314844 \beta_{1} + 4168263 \beta_{2} + 2129490 \beta_{4} - 4258980 \beta_{5} - 2129490 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 16384q^{4} + 195160q^{7} - 1478904q^{9} + O(q^{10})$$ $$8q + 16384q^{4} + 195160q^{7} - 1478904q^{9} - 213840q^{11} - 8418816q^{14} + 65882304q^{15} + 33554432q^{16} + 32547840q^{18} - 4449984q^{21} - 221337600q^{22} + 156731760q^{23} + 191237000q^{25} + 399687680q^{28} + 308853648q^{29} - 2203567104q^{30} - 3764734848q^{35} - 3028795392q^{36} - 3243600880q^{37} + 13521315264q^{39} - 12108579840q^{42} + 21006302000q^{43} - 437944320q^{44} + 9664610304q^{46} - 19258758904q^{49} + 26259489792q^{50} - 80965832832q^{51} + 180445637520q^{53} - 17241735168q^{56} - 63145962240q^{57} - 94193264640q^{58} + 134926958592q^{60} - 402706514280q^{63} + 68719476736q^{64} - 424890168192q^{65} + 369211259440q^{67} - 137936354304q^{70} + 574058144304q^{71} + 66657976320q^{72} + 450517137408q^{74} - 73915435440q^{77} - 251000847360q^{78} - 607826610128q^{79} + 919051941384q^{81} - 9113567232q^{84} - 247202260608q^{85} - 413092638720q^{86} - 453299404800q^{88} + 144527421696q^{91} + 320986644480q^{92} + 2292312458880q^{93} - 1053641981376q^{95} - 290797516800q^{98} - 1800954256464q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 154710 x^{6} + 8245426887 x^{4} + 174724076278260 x^{2} + 1264170035276291934$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-125 \nu^{6} - 8963839 \nu^{4} - 86616988470 \nu^{2} + 824985433394766$$$$)/ 7033679917677$$ $$\beta_{2}$$ $$=$$ $$($$$$38659002 \nu^{6} + 4172748382238 \nu^{4} + 134728750743817140 \nu^{2} + 1313309281764933518688$$$$)/ 218044077447987$$ $$\beta_{3}$$ $$=$$ $$($$$$1774768547 \nu^{7} + 176258268664353 \nu^{5} + 4996545933179551482 \nu^{3} + 42330085042660184769534 \nu$$$$)/$$$$51\!\cdots\!52$$ $$\beta_{4}$$ $$=$$ $$($$$$3573248699 \nu^{7} + 538494861420537 \nu^{5} + 24481269884717511642 \nu^{3} + 294315438819634193998926 \nu$$$$)/$$$$25\!\cdots\!76$$ $$\beta_{5}$$ $$=$$ $$($$$$1886431118 \nu^{7} + 1294978085403 \nu^{6} + 240235006799442 \nu^{5} + 144898251813388725 \nu^{4} + 9831189176412236460 \nu^{3} + 4454151732706627888542 \nu^{2} + 127633302987889994511660 \nu + 32181593594130820913735970$$$$)/$$$$12\!\cdots\!38$$ $$\beta_{6}$$ $$=$$ $$($$$$-1324158591 \nu^{7} - 140815055259077 \nu^{5} - 4947828940310478066 \nu^{3} - 72072591043975261349238 \nu$$$$)/ 86414356597876415892$$ $$\beta_{7}$$ $$=$$ $$($$$$-31567664251 \nu^{7} - 4284780743595945 \nu^{5} - 175256809018488285066 \nu^{3} - 1977787602909975649095918 \nu$$$$)/$$$$51\!\cdots\!52$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-3 \beta_{7} - 4 \beta_{6} - 10 \beta_{4} - 31 \beta_{3}$$$$)/896$$ $$\nu^{2}$$ $$=$$ $$($$$$-8 \beta_{6} - 16 \beta_{5} + 8 \beta_{4} + \beta_{2} + 982 \beta_{1} - 2165940$$$$)/56$$ $$\nu^{3}$$ $$=$$ $$($$$$183441 \beta_{7} + 86196 \beta_{6} + 587382 \beta_{4} + 1283493 \beta_{3}$$$$)/896$$ $$\nu^{4}$$ $$=$$ $$($$$$616590 \beta_{6} + 1233180 \beta_{5} - 616590 \beta_{4} - 68355 \beta_{2} + 11296332 \beta_{1} + 104220624564$$$$)/56$$ $$\nu^{5}$$ $$=$$ $$($$$$-10991030859 \beta_{7} - 2214913356 \beta_{6} - 33479955810 \beta_{4} - 70597450215 \beta_{3}$$$$)/896$$ $$\nu^{6}$$ $$=$$ $$($$$$-38672620650 \beta_{6} - 77345241300 \beta_{5} + 38672620650 \beta_{4} + 4208849811 \beta_{2} - 4641619675248 \beta_{1} - 5603284118354580$$$$)/56$$ $$\nu^{7}$$ $$=$$ $$($$$$646664008432761 \beta_{7} + 72705176403156 \beta_{6} + 1909848334230918 \beta_{4} + 4398967317844845 \beta_{3}$$$$)/896$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/14\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 242.361i 130.480i − 130.480i − 242.361i − 237.947i 149.422i − 149.422i 237.947i
−45.2548 1265.70i 2048.00 23188.3i 57279.1i 109682. + 42556.3i −92681.9 −1.07056e6 1.04938e6i
13.2 −45.2548 102.042i 2048.00 6916.58i 4617.87i −14384.7 + 116766.i −92681.9 521029. 313009.i
13.3 −45.2548 102.042i 2048.00 6916.58i 4617.87i −14384.7 116766.i −92681.9 521029. 313009.i
13.4 −45.2548 1265.70i 2048.00 23188.3i 57279.1i 109682. 42556.3i −92681.9 −1.07056e6 1.04938e6i
13.5 45.2548 1072.00i 2048.00 1140.63i 48513.4i 74038.7 91430.6i 92681.9 −617752. 51619.1i
13.6 45.2548 321.888i 2048.00 17149.5i 14567.0i −71756.5 + 93232.5i 92681.9 427829. 776099.i
13.7 45.2548 321.888i 2048.00 17149.5i 14567.0i −71756.5 93232.5i 92681.9 427829. 776099.i
13.8 45.2548 1072.00i 2048.00 1140.63i 48513.4i 74038.7 + 91430.6i 92681.9 −617752. 51619.1i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 13.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.13.b.a 8
3.b odd 2 1 126.13.c.a 8
4.b odd 2 1 112.13.c.c 8
7.b odd 2 1 inner 14.13.b.a 8
7.c even 3 2 98.13.d.b 16
7.d odd 6 2 98.13.d.b 16
21.c even 2 1 126.13.c.a 8
28.d even 2 1 112.13.c.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.13.b.a 8 1.a even 1 1 trivial
14.13.b.a 8 7.b odd 2 1 inner
98.13.d.b 16 7.c even 3 2
98.13.d.b 16 7.d odd 6 2
112.13.c.c 8 4.b odd 2 1
112.13.c.c 8 28.d even 2 1
126.13.c.a 8 3.b odd 2 1
126.13.c.a 8 21.c even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{13}^{\mathrm{new}}(14, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2048 + T^{2} )^{4}$$
$3$ $$19\!\cdots\!04$$$$+ 212887983685229568 T^{2} + 2155787566560 T^{4} + 2865216 T^{6} + T^{8}$$
$5$ $$98\!\cdots\!00$$$$+$$$$78\!\cdots\!00$$$$T^{2} + 199077441460957152 T^{4} + 880944000 T^{6} + T^{8}$$
$7$ $$36\!\cdots\!01$$$$-$$$$51\!\cdots\!60$$$$T +$$$$54\!\cdots\!52$$$$T^{2} -$$$$55\!\cdots\!80$$$$T^{3} +$$$$54\!\cdots\!66$$$$T^{4} - 4024638607794280 T^{5} + 28673092252 T^{6} - 195160 T^{7} + T^{8}$$
$11$ $$($$$$16\!\cdots\!84$$$$+ 1221010327226622240 T - 9612276413256 T^{2} + 106920 T^{3} + T^{4} )^{2}$$
$13$ $$25\!\cdots\!04$$$$+$$$$11\!\cdots\!36$$$$T^{2} +$$$$15\!\cdots\!60$$$$T^{4} + 67900719168768 T^{6} + T^{8}$$
$17$ $$17\!\cdots\!24$$$$+$$$$43\!\cdots\!04$$$$T^{2} +$$$$25\!\cdots\!60$$$$T^{4} + 3105840619203072 T^{6} + T^{8}$$
$19$ $$55\!\cdots\!84$$$$+$$$$46\!\cdots\!08$$$$T^{2} +$$$$41\!\cdots\!80$$$$T^{4} + 11844400954835136 T^{6} + T^{8}$$
$23$ $$( -$$$$13\!\cdots\!04$$$$+$$$$22\!\cdots\!60$$$$T - 24446073659234184 T^{2} - 78365880 T^{3} + T^{4} )^{2}$$
$29$ $$($$$$26\!\cdots\!16$$$$+$$$$27\!\cdots\!04$$$$T - 813585534009603048 T^{2} - 154426824 T^{3} + T^{4} )^{2}$$
$31$ $$38\!\cdots\!64$$$$+$$$$27\!\cdots\!44$$$$T^{2} +$$$$40\!\cdots\!40$$$$T^{4} + 3999256194825234432 T^{6} + T^{8}$$
$37$ $$( -$$$$16\!\cdots\!44$$$$-$$$$25\!\cdots\!80$$$$T - 9275053808494362216 T^{2} + 1621800440 T^{3} + T^{4} )^{2}$$
$41$ $$10\!\cdots\!84$$$$+$$$$30\!\cdots\!04$$$$T^{2} +$$$$33\!\cdots\!00$$$$T^{4} +$$$$11\!\cdots\!32$$$$T^{6} + T^{8}$$
$43$ $$( -$$$$24\!\cdots\!64$$$$-$$$$16\!\cdots\!00$$$$T + 31223828126804976312 T^{2} - 10503151000 T^{3} + T^{4} )^{2}$$
$47$ $$70\!\cdots\!04$$$$+$$$$51\!\cdots\!68$$$$T^{2} +$$$$10\!\cdots\!60$$$$T^{4} +$$$$58\!\cdots\!16$$$$T^{6} + T^{8}$$
$53$ $$($$$$20\!\cdots\!04$$$$-$$$$41\!\cdots\!80$$$$T +$$$$29\!\cdots\!48$$$$T^{2} - 90222818760 T^{3} + T^{4} )^{2}$$
$59$ $$65\!\cdots\!00$$$$+$$$$10\!\cdots\!00$$$$T^{2} +$$$$14\!\cdots\!00$$$$T^{4} +$$$$69\!\cdots\!00$$$$T^{6} + T^{8}$$
$61$ $$27\!\cdots\!44$$$$+$$$$55\!\cdots\!08$$$$T^{2} +$$$$14\!\cdots\!40$$$$T^{4} +$$$$89\!\cdots\!16$$$$T^{6} + T^{8}$$
$67$ $$($$$$10\!\cdots\!64$$$$-$$$$15\!\cdots\!60$$$$T +$$$$10\!\cdots\!28$$$$T^{2} - 184605629720 T^{3} + T^{4} )^{2}$$
$71$ $$( -$$$$40\!\cdots\!16$$$$+$$$$16\!\cdots\!56$$$$T +$$$$21\!\cdots\!20$$$$T^{2} - 287029072152 T^{3} + T^{4} )^{2}$$
$73$ $$49\!\cdots\!84$$$$+$$$$26\!\cdots\!60$$$$T^{2} +$$$$16\!\cdots\!44$$$$T^{4} +$$$$26\!\cdots\!80$$$$T^{6} + T^{8}$$
$79$ $$( -$$$$12\!\cdots\!64$$$$-$$$$16\!\cdots\!16$$$$T -$$$$31\!\cdots\!64$$$$T^{2} + 303913305064 T^{3} + T^{4} )^{2}$$
$83$ $$49\!\cdots\!64$$$$+$$$$15\!\cdots\!32$$$$T^{2} +$$$$12\!\cdots\!20$$$$T^{4} +$$$$34\!\cdots\!04$$$$T^{6} + T^{8}$$
$89$ $$53\!\cdots\!84$$$$+$$$$85\!\cdots\!76$$$$T^{2} +$$$$48\!\cdots\!60$$$$T^{4} +$$$$11\!\cdots\!08$$$$T^{6} + T^{8}$$
$97$ $$13\!\cdots\!04$$$$+$$$$71\!\cdots\!76$$$$T^{2} +$$$$88\!\cdots\!40$$$$T^{4} +$$$$23\!\cdots\!88$$$$T^{6} + T^{8}$$
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