# Properties

 Label 224.2.p.a Level 224 Weight 2 Character orbit 224.p Analytic conductor 1.789 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 224.p (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.78864900528$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: 16.0.2353561680715186176.2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{20}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{14} q^{3} -\beta_{13} q^{5} -\beta_{9} q^{7} + ( -\beta_{3} + \beta_{8} + \beta_{11} + \beta_{15} ) q^{9} +O(q^{10})$$ $$q -\beta_{14} q^{3} -\beta_{13} q^{5} -\beta_{9} q^{7} + ( -\beta_{3} + \beta_{8} + \beta_{11} + \beta_{15} ) q^{9} + ( \beta_{1} - \beta_{2} + \beta_{6} + \beta_{7} - \beta_{9} + \beta_{14} ) q^{11} + ( -1 + 2 \beta_{3} + \beta_{10} - \beta_{11} ) q^{13} + ( \beta_{2} - \beta_{5} + \beta_{7} + \beta_{9} - \beta_{14} ) q^{15} -\beta_{10} q^{17} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{9} - \beta_{14} ) q^{19} + ( -2 + \beta_{3} + \beta_{8} + \beta_{10} + \beta_{12} ) q^{21} + ( \beta_{1} - \beta_{7} ) q^{23} + ( 2 - 2 \beta_{3} - \beta_{10} + \beta_{11} - \beta_{15} ) q^{25} + ( -\beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{14} ) q^{27} + ( 1 - \beta_{8} ) q^{29} + ( -\beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{31} + ( 2 - \beta_{3} - \beta_{8} + 2 \beta_{13} + \beta_{15} ) q^{33} + ( \beta_{1} - \beta_{2} - \beta_{4} - 3 \beta_{7} + 3 \beta_{14} ) q^{35} + ( -\beta_{3} - \beta_{8} - \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{15} ) q^{37} + ( -\beta_{1} - 4 \beta_{2} - \beta_{4} - \beta_{7} + 2 \beta_{14} ) q^{39} + ( -1 + 2 \beta_{3} + \beta_{10} - \beta_{11} ) q^{41} + ( -2 \beta_{2} + 2 \beta_{5} + 2 \beta_{14} ) q^{43} + ( -1 - \beta_{3} - 2 \beta_{8} - 2 \beta_{10} - \beta_{11} - \beta_{15} ) q^{45} + ( \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} - \beta_{14} ) q^{47} + ( -2 - 2 \beta_{12} - \beta_{15} ) q^{49} + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{9} + \beta_{14} ) q^{51} + ( -1 + \beta_{3} - \beta_{10} + 2 \beta_{11} + \beta_{12} + 2 \beta_{13} ) q^{53} + ( \beta_{1} + 6 \beta_{2} + \beta_{4} - 3 \beta_{7} + 3 \beta_{9} ) q^{55} - q^{57} + ( -2 \beta_{1} + 2 \beta_{7} + 2 \beta_{9} - 3 \beta_{14} ) q^{59} + ( -2 + \beta_{3} + \beta_{8} - \beta_{11} + \beta_{13} - \beta_{15} ) q^{61} + ( -\beta_{2} - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} + 3 \beta_{14} ) q^{63} + ( \beta_{3} + 2 \beta_{10} - \beta_{11} + 4 \beta_{12} + 2 \beta_{13} ) q^{65} + ( -2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} - 2 \beta_{7} + \beta_{9} + \beta_{14} ) q^{67} + ( 2 - 4 \beta_{3} + \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{15} ) q^{69} + ( -3 \beta_{1} + \beta_{2} + 3 \beta_{4} - \beta_{5} + \beta_{7} + \beta_{9} - \beta_{14} ) q^{71} + ( -1 - \beta_{3} - 2 \beta_{8} - \beta_{10} - \beta_{11} - \beta_{15} ) q^{73} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{6} + 3 \beta_{7} + 2 \beta_{9} - \beta_{14} ) q^{75} + ( 5 - 2 \beta_{3} + 2 \beta_{8} + \beta_{10} + 3 \beta_{11} - \beta_{12} + 2 \beta_{15} ) q^{77} + ( -\beta_{1} + 2 \beta_{2} + \beta_{7} + 2 \beta_{14} ) q^{79} + ( -6 + 6 \beta_{3} - \beta_{10} - 2 \beta_{12} - 4 \beta_{13} - 2 \beta_{15} ) q^{81} + ( \beta_{1} - 3 \beta_{2} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} - \beta_{14} ) q^{83} + ( -1 - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{85} + ( \beta_{1} - \beta_{2} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} - 4 \beta_{14} ) q^{87} + ( -6 + 3 \beta_{3} - 2 \beta_{11} - 2 \beta_{13} ) q^{89} + ( 3 \beta_{1} + 2 \beta_{2} + \beta_{5} + 3 \beta_{6} - \beta_{9} + \beta_{14} ) q^{91} + ( \beta_{3} + \beta_{8} + \beta_{11} + 2 \beta_{12} + \beta_{13} + \beta_{15} ) q^{93} + ( 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} + \beta_{6} + 2 \beta_{7} - 4 \beta_{9} + 2 \beta_{14} ) q^{95} + ( 3 - 6 \beta_{3} + \beta_{8} + 2 \beta_{10} - 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{15} ) q^{97} + ( 4 \beta_{1} - 4 \beta_{4} + 2 \beta_{5} - 5 \beta_{7} - 5 \beta_{9} - 2 \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 8q^{9} + O(q^{10})$$ $$16q - 8q^{9} - 24q^{21} + 16q^{25} + 16q^{29} + 24q^{33} - 8q^{37} - 24q^{45} - 32q^{49} - 8q^{53} - 16q^{57} - 24q^{61} + 8q^{65} - 24q^{73} + 64q^{77} - 48q^{81} - 16q^{85} - 72q^{89} + 8q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} + 2 x^{14} + 41 x^{12} - 92 x^{11} + 66 x^{10} - 104 x^{9} + 291 x^{8} - 388 x^{7} + 366 x^{6} - 344 x^{5} + 286 x^{4} - 184 x^{3} + 84 x^{2} - 24 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-4671439980 \nu^{15} - 5019523613 \nu^{14} + 74862020586 \nu^{13} - 4435873025 \nu^{12} - 210602072844 \nu^{11} - 553169578089 \nu^{10} + 1432081245914 \nu^{9} - 85279469075 \nu^{8} + 522318705094 \nu^{7} - 4152490382437 \nu^{6} + 4382872705880 \nu^{5} - 3138092527649 \nu^{4} + 3538225191170 \nu^{3} - 2782602331212 \nu^{2} + 1359834224272 \nu - 353683962760$$$$)/ 20109322702$$ $$\beta_{2}$$ $$=$$ $$($$$$5063424619 \nu^{15} - 34394314873 \nu^{14} + 58272692948 \nu^{13} + 3111527264 \nu^{12} + 197933621407 \nu^{11} - 1046870352209 \nu^{10} + 1297182921138 \nu^{9} - 768732964854 \nu^{8} + 2528765768339 \nu^{7} - 5349204235533 \nu^{6} + 5140695571010 \nu^{5} - 4106878988502 \nu^{4} + 3966533391230 \nu^{3} - 2841292968702 \nu^{2} + 1190351287680 \nu - 314591816094$$$$)/ 20109322702$$ $$\beta_{3}$$ $$=$$ $$($$$$-11175608166 \nu^{15} + 37005033535 \nu^{14} + 2381595084 \nu^{13} + 3936897701 \nu^{12} - 454687961278 \nu^{11} + 715591444243 \nu^{10} - 275560834120 \nu^{9} + 1011875313139 \nu^{8} - 2563606006170 \nu^{7} + 2641228538245 \nu^{6} - 2425489864376 \nu^{5} + 2311962665959 \nu^{4} - 1743124162188 \nu^{3} + 985628049642 \nu^{2} - 363876573956 \nu + 87372551998$$$$)/ 20109322702$$ $$\beta_{4}$$ $$=$$ $$($$$$-1736302763 \nu^{15} + 3368540236 \nu^{14} + 8943298446 \nu^{13} + 179687165 \nu^{12} - 73751071335 \nu^{11} + 11510683964 \nu^{10} + 136916188768 \nu^{9} + 109872186421 \nu^{8} - 223621781697 \nu^{7} - 209166485604 \nu^{6} + 227204090414 \nu^{5} - 65297035181 \nu^{4} + 211706237508 \nu^{3} - 207339126830 \nu^{2} + 96981591808 \nu - 25579237514$$$$)/ 2872760386$$ $$\beta_{5}$$ $$=$$ $$($$$$12615896138 \nu^{15} - 42612645921 \nu^{14} + 5931103262 \nu^{13} - 19811250885 \nu^{12} + 499479045258 \nu^{11} - 845619686261 \nu^{10} + 604535781354 \nu^{9} - 1376773471743 \nu^{8} + 2866813258228 \nu^{7} - 3654726081713 \nu^{6} + 3946524545552 \nu^{5} - 3264415389805 \nu^{4} + 2631655568822 \nu^{3} - 1752180134328 \nu^{2} + 636854341512 \nu - 179930396744$$$$)/ 20109322702$$ $$\beta_{6}$$ $$=$$ $$($$$$18129728171 \nu^{15} - 67239132782 \nu^{14} + 13728648942 \nu^{13} + 16261528967 \nu^{12} + 743533213171 \nu^{11} - 1459054198710 \nu^{10} + 645301401120 \nu^{9} - 1413868017845 \nu^{8} + 4752183358093 \nu^{7} - 5440849395538 \nu^{6} + 4170972602046 \nu^{5} - 4049499601091 \nu^{4} + 3305520707556 \nu^{3} - 1639495702234 \nu^{2} + 554159019384 \nu - 69968157850$$$$)/ 20109322702$$ $$\beta_{7}$$ $$=$$ $$($$$$2623797969 \nu^{15} - 10304840791 \nu^{14} + 2190403752 \nu^{13} + 6732922644 \nu^{12} + 112107241481 \nu^{11} - 231856184779 \nu^{10} + 61947862130 \nu^{9} - 164982748350 \nu^{8} + 762466069777 \nu^{7} - 766985838727 \nu^{6} + 460781419854 \nu^{5} - 570794974590 \nu^{4} + 435938816234 \nu^{3} - 151480361274 \nu^{2} + 31718599632 \nu + 5890380390$$$$)/ 2872760386$$ $$\beta_{8}$$ $$=$$ $$($$$$139482845 \nu^{15} - 713255513 \nu^{14} + 813740704 \nu^{13} + 30184751 \nu^{12} + 5568358461 \nu^{11} - 19244128631 \nu^{10} + 20009728986 \nu^{9} - 16925651529 \nu^{8} + 51651556669 \nu^{7} - 91857962133 \nu^{6} + 87279176826 \nu^{5} - 72439306253 \nu^{4} + 66687816790 \nu^{3} - 46319052852 \nu^{2} + 18821562600 \nu - 4395910477$$$$)/ 137735087$$ $$\beta_{9}$$ $$=$$ $$($$$$21074697951 \nu^{15} - 104086834189 \nu^{14} + 95566612060 \nu^{13} + 46229985362 \nu^{12} + 871914454327 \nu^{11} - 2751753453829 \nu^{10} + 2154137922350 \nu^{9} - 1835171745948 \nu^{8} + 7683424327567 \nu^{7} - 11727612400677 \nu^{6} + 9418969698846 \nu^{5} - 8639783255096 \nu^{4} + 7778130510050 \nu^{3} - 4550591596230 \nu^{2} + 1735233196936 \nu - 264247625938$$$$)/ 20109322702$$ $$\beta_{10}$$ $$=$$ $$($$$$-22929440334 \nu^{15} + 103630918811 \nu^{14} - 82984027620 \nu^{13} - 16420430373 \nu^{12} - 929634728742 \nu^{11} + 2610374929811 \nu^{10} - 2170386426128 \nu^{9} + 2278929576205 \nu^{8} - 7519139314798 \nu^{7} + 11518339038677 \nu^{6} - 10085004362164 \nu^{5} + 8949442744397 \nu^{4} - 8103799328648 \nu^{3} + 5000860688254 \nu^{2} - 1965241728420 \nu + 438702505064$$$$)/ 20109322702$$ $$\beta_{11}$$ $$=$$ $$($$$$33911516972 \nu^{15} - 108940608997 \nu^{14} - 32201388164 \nu^{13} + 22923719031 \nu^{12} + 1411498051716 \nu^{11} - 2014225248249 \nu^{10} + 64172702604 \nu^{9} - 2520892868363 \nu^{8} + 7621760166076 \nu^{7} - 5993220149479 \nu^{6} + 4268041273800 \nu^{5} - 5045452375491 \nu^{4} + 3122239171348 \nu^{3} - 961405268358 \nu^{2} + 81167549476 \nu + 125770278322$$$$)/ 20109322702$$ $$\beta_{12}$$ $$=$$ $$($$$$-46688945564 \nu^{15} + 219887457941 \nu^{14} - 194424781504 \nu^{13} - 50488884423 \nu^{12} - 1890422689576 \nu^{11} + 5678839565545 \nu^{10} - 4832040026616 \nu^{9} + 4520765331063 \nu^{8} - 15902360689892 \nu^{7} + 25178802231927 \nu^{6} - 21652899765320 \nu^{5} + 18807008211767 \nu^{4} - 17224972534876 \nu^{3} + 10719181869982 \nu^{2} - 4138992241108 \nu + 828605565416$$$$)/ 20109322702$$ $$\beta_{13}$$ $$=$$ $$($$$$-51918966616 \nu^{15} + 156637786453 \nu^{14} + 91292310268 \nu^{13} - 53280838773 \nu^{12} - 2183806181780 \nu^{11} + 2664420234629 \nu^{10} + 895817893144 \nu^{9} + 3375808124601 \nu^{8} - 11013464257600 \nu^{7} + 6196852490403 \nu^{6} - 2797053807220 \nu^{5} + 5190337377625 \nu^{4} - 2414367784536 \nu^{3} - 578312257270 \nu^{2} + 892787498524 \nu - 435752795150$$$$)/ 20109322702$$ $$\beta_{14}$$ $$=$$ $$($$$$-59961413714 \nu^{15} + 200645094063 \nu^{14} + 16512535898 \nu^{13} - 11349732409 \nu^{12} - 2456663221746 \nu^{11} + 3924602278603 \nu^{10} - 1167318358606 \nu^{9} + 4948815345057 \nu^{8} - 13995395422064 \nu^{7} + 13762376176403 \nu^{6} - 11319086764252 \nu^{5} + 11351132808947 \nu^{4} - 8474603814678 \nu^{3} + 4141673676792 \nu^{2} - 1226899096656 \nu + 137646626396$$$$)/ 20109322702$$ $$\beta_{15}$$ $$=$$ $$($$$$36518218171 \nu^{15} - 122459524518 \nu^{14} - 5843623036 \nu^{13} - 322687170 \nu^{12} + 1485454721783 \nu^{11} - 2410210059932 \nu^{10} + 859139333534 \nu^{9} - 3080708284676 \nu^{8} + 8487123321599 \nu^{7} - 8777061742880 \nu^{6} + 7381040079790 \nu^{5} - 7085471800872 \nu^{4} + 5538488457086 \nu^{3} - 2812389425334 \nu^{2} + 818561766836 \nu - 132458831459$$$$)/ 10054661351$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{15} - 2 \beta_{14} - \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{6} + \beta_{4} - 4 \beta_{3} + 3 \beta_{2} + 4$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{14} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} - \beta_{4} + 6 \beta_{3} + 3 \beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{15} - 6 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} - 9 \beta_{11} + \beta_{10} - \beta_{9} - 4 \beta_{8} + 2 \beta_{7} + 3 \beta_{6} + 8 \beta_{5} - \beta_{4} + 5 \beta_{2} - 2 \beta_{1} + 20$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$-5 \beta_{15} - 14 \beta_{14} - 2 \beta_{12} - \beta_{11} + \beta_{10} - 2 \beta_{9} - 6 \beta_{8} - 2 \beta_{7} - 8 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 14 \beta_{3} + 10 \beta_{2} + 2 \beta_{1} - 14$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-18 \beta_{15} - 29 \beta_{14} - 10 \beta_{13} - 11 \beta_{11} - 25 \beta_{10} - 15 \beta_{9} - 7 \beta_{8} - 21 \beta_{7} + 24 \beta_{6} + 9 \beta_{5} - 15 \beta_{4} + 104 \beta_{3} + 27 \beta_{2} + \beta_{1}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-32 \beta_{15} - 38 \beta_{14} - 20 \beta_{13} - 20 \beta_{12} - 44 \beta_{11} + 20 \beta_{10} - 7 \beta_{9} - 12 \beta_{8} + 2 \beta_{7} - \beta_{6} + 42 \beta_{5} + 3 \beta_{4} - 9 \beta_{2} + 8 \beta_{1} - 12$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-79 \beta_{15} - 215 \beta_{14} + 6 \beta_{12} + 68 \beta_{11} - 68 \beta_{10} - 92 \beta_{9} - 11 \beta_{8} - 63 \beta_{7} - 41 \beta_{6} - 63 \beta_{5} + 48 \beta_{4} + 472 \beta_{3} + 60 \beta_{2} + 63 \beta_{1} - 472$$$$)/8$$ $$\nu^{8}$$ $$=$$ $$($$$$-91 \beta_{15} - 24 \beta_{14} - 110 \beta_{13} - 142 \beta_{11} - 40 \beta_{10} - 24 \beta_{9} + 51 \beta_{8} - 56 \beta_{7} + 208 \beta_{6} + 184 \beta_{5} - 24 \beta_{4} + 166 \beta_{3} - 104 \beta_{2} + 24 \beta_{1}$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$-329 \beta_{15} - 480 \beta_{14} - 222 \beta_{13} - 222 \beta_{12} - 177 \beta_{11} + 481 \beta_{10} - 241 \beta_{9} + 152 \beta_{8} + 164 \beta_{7} - 231 \beta_{6} + 326 \beta_{5} + 395 \beta_{4} - 703 \beta_{2} + 472 \beta_{1} - 1816$$$$)/8$$ $$\nu^{10}$$ $$=$$ $$($$$$-14 \beta_{15} - 107 \beta_{14} + 420 \beta_{12} + 512 \beta_{11} - 512 \beta_{10} - 236 \beta_{9} + 498 \beta_{8} - 213 \beta_{7} + 593 \beta_{6} - 213 \beta_{5} + 274 \beta_{4} + 1624 \beta_{3} - 342 \beta_{2} + 213 \beta_{1} - 1624$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$-296 \beta_{15} + 2485 \beta_{14} - 1982 \beta_{13} - 2517 \beta_{11} + 1925 \beta_{10} + 649 \beta_{9} + 2221 \beta_{8} + 793 \beta_{7} + 3158 \beta_{6} + 3807 \beta_{5} + 649 \beta_{4} - 5232 \beta_{3} - 5249 \beta_{2} + 1187 \beta_{1}$$$$)/8$$ $$\nu^{12}$$ $$=$$ $$($$$$1078 \beta_{15} + 264 \beta_{14} + 962 \beta_{13} + 962 \beta_{12} + 3359 \beta_{11} + 1203 \beta_{10} - 788 \beta_{9} + 2281 \beta_{8} + 1416 \beta_{7} - 1284 \beta_{6} - 2176 \beta_{5} + 2700 \beta_{4} - 3356 \beta_{2} + 2072 \beta_{1} - 10066$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$7935 \beta_{15} + 18293 \beta_{14} + 11418 \beta_{12} + 5972 \beta_{11} - 5972 \beta_{10} + 4588 \beta_{9} + 13907 \beta_{8} + 533 \beta_{7} + 19683 \beta_{6} + 533 \beta_{5} + 2456 \beta_{4} - 4632 \beta_{3} - 13172 \beta_{2} - 533 \beta_{1} + 4632$$$$)/8$$ $$\nu^{14}$$ $$=$$ $$($$$$7760 \beta_{15} + 18403 \beta_{14} - 1340 \beta_{13} - 1874 \beta_{11} + 17394 \beta_{10} + 6223 \beta_{9} + 9634 \beta_{8} + 11621 \beta_{7} - 570 \beta_{6} + 5653 \beta_{5} + 6223 \beta_{4} - 49232 \beta_{3} - 23497 \beta_{2} + 5957 \beta_{1}$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$($$$$61371 \beta_{15} + 51324 \beta_{14} + 51026 \beta_{13} + 51026 \beta_{12} + 105043 \beta_{11} - 17699 \beta_{10} + 6773 \beta_{9} + 43672 \beta_{8} + 22984 \beta_{7} - 9689 \beta_{6} - 90770 \beta_{5} + 32673 \beta_{4} - 12605 \beta_{2} + 2916 \beta_{1} - 75680$$$$)/8$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$-1$$ $$\beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 2.07391 + 0.620024i 1.50047 + 0.288947i −1.57391 − 1.48605i 0.224274 − 0.447866i −1.00047 − 1.15497i −0.349168 − 0.778942i 0.275726 − 0.418160i 0.849168 − 0.0870829i 2.07391 − 0.620024i 1.50047 − 0.288947i −1.57391 + 1.48605i 0.224274 + 0.447866i −1.00047 + 1.15497i −0.349168 + 0.778942i 0.275726 + 0.418160i 0.849168 + 0.0870829i
0 −1.60159 + 2.77404i 0 −1.00367 + 0.579471i 0 1.44550 + 2.21597i 0 −3.63019 6.28767i 0
31.2 0 −1.13338 + 1.96307i 0 3.08101 1.77882i 0 0.912798 2.48330i 0 −1.06909 1.85172i 0
31.3 0 −0.376846 + 0.652717i 0 1.00367 0.579471i 0 −0.286555 + 2.63019i 0 1.21597 + 2.10613i 0
31.4 0 −0.0913671 + 0.158252i 0 −3.08101 + 1.77882i 0 −2.64485 + 0.0690906i 0 1.48330 + 2.56916i 0
31.5 0 0.0913671 0.158252i 0 −3.08101 + 1.77882i 0 2.64485 0.0690906i 0 1.48330 + 2.56916i 0
31.6 0 0.376846 0.652717i 0 1.00367 0.579471i 0 0.286555 2.63019i 0 1.21597 + 2.10613i 0
31.7 0 1.13338 1.96307i 0 3.08101 1.77882i 0 −0.912798 + 2.48330i 0 −1.06909 1.85172i 0
31.8 0 1.60159 2.77404i 0 −1.00367 + 0.579471i 0 −1.44550 2.21597i 0 −3.63019 6.28767i 0
159.1 0 −1.60159 2.77404i 0 −1.00367 0.579471i 0 1.44550 2.21597i 0 −3.63019 + 6.28767i 0
159.2 0 −1.13338 1.96307i 0 3.08101 + 1.77882i 0 0.912798 + 2.48330i 0 −1.06909 + 1.85172i 0
159.3 0 −0.376846 0.652717i 0 1.00367 + 0.579471i 0 −0.286555 2.63019i 0 1.21597 2.10613i 0
159.4 0 −0.0913671 0.158252i 0 −3.08101 1.77882i 0 −2.64485 0.0690906i 0 1.48330 2.56916i 0
159.5 0 0.0913671 + 0.158252i 0 −3.08101 1.77882i 0 2.64485 + 0.0690906i 0 1.48330 2.56916i 0
159.6 0 0.376846 + 0.652717i 0 1.00367 + 0.579471i 0 0.286555 + 2.63019i 0 1.21597 2.10613i 0
159.7 0 1.13338 + 1.96307i 0 3.08101 + 1.77882i 0 −0.912798 2.48330i 0 −1.06909 + 1.85172i 0
159.8 0 1.60159 + 2.77404i 0 −1.00367 0.579471i 0 −1.44550 + 2.21597i 0 −3.63019 + 6.28767i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 159.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.d odd 6 1 inner
28.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.p.a 16
3.b odd 2 1 2016.2.cs.b 16
4.b odd 2 1 inner 224.2.p.a 16
7.b odd 2 1 1568.2.p.b 16
7.c even 3 1 1568.2.f.b 16
7.c even 3 1 1568.2.p.b 16
7.d odd 6 1 inner 224.2.p.a 16
7.d odd 6 1 1568.2.f.b 16
8.b even 2 1 448.2.p.e 16
8.d odd 2 1 448.2.p.e 16
12.b even 2 1 2016.2.cs.b 16
21.g even 6 1 2016.2.cs.b 16
28.d even 2 1 1568.2.p.b 16
28.f even 6 1 inner 224.2.p.a 16
28.f even 6 1 1568.2.f.b 16
28.g odd 6 1 1568.2.f.b 16
28.g odd 6 1 1568.2.p.b 16
56.j odd 6 1 448.2.p.e 16
56.j odd 6 1 3136.2.f.j 16
56.k odd 6 1 3136.2.f.j 16
56.m even 6 1 448.2.p.e 16
56.m even 6 1 3136.2.f.j 16
56.p even 6 1 3136.2.f.j 16
84.j odd 6 1 2016.2.cs.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.p.a 16 1.a even 1 1 trivial
224.2.p.a 16 4.b odd 2 1 inner
224.2.p.a 16 7.d odd 6 1 inner
224.2.p.a 16 28.f even 6 1 inner
448.2.p.e 16 8.b even 2 1
448.2.p.e 16 8.d odd 2 1
448.2.p.e 16 56.j odd 6 1
448.2.p.e 16 56.m even 6 1
1568.2.f.b 16 7.c even 3 1
1568.2.f.b 16 7.d odd 6 1
1568.2.f.b 16 28.f even 6 1
1568.2.f.b 16 28.g odd 6 1
1568.2.p.b 16 7.b odd 2 1
1568.2.p.b 16 7.c even 3 1
1568.2.p.b 16 28.d even 2 1
1568.2.p.b 16 28.g odd 6 1
2016.2.cs.b 16 3.b odd 2 1
2016.2.cs.b 16 12.b even 2 1
2016.2.cs.b 16 21.g even 6 1
2016.2.cs.b 16 84.j odd 6 1
3136.2.f.j 16 56.j odd 6 1
3136.2.f.j 16 56.k odd 6 1
3136.2.f.j 16 56.m even 6 1
3136.2.f.j 16 56.p even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(224, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 8 T^{2} + 38 T^{4} - 80 T^{6} - 23 T^{8} + 752 T^{10} - 1162 T^{12} - 4696 T^{14} + 30100 T^{16} - 42264 T^{18} - 94122 T^{20} + 548208 T^{22} - 150903 T^{24} - 4723920 T^{26} + 20194758 T^{28} - 38263752 T^{30} + 43046721 T^{32}$$
$5$ $$( 1 + 6 T^{2} + 9 T^{4} - 138 T^{6} - 796 T^{8} - 3450 T^{10} + 5625 T^{12} + 93750 T^{14} + 390625 T^{16} )^{2}$$
$7$ $$1 + 16 T^{2} + 60 T^{4} - 784 T^{6} - 10426 T^{8} - 38416 T^{10} + 144060 T^{12} + 1882384 T^{14} + 5764801 T^{16}$$
$11$ $$1 + 24 T^{2} + 86 T^{4} - 2064 T^{6} - 12647 T^{8} + 54000 T^{10} - 509434 T^{12} - 4559928 T^{14} + 74796148 T^{16} - 551751288 T^{18} - 7458623194 T^{20} + 95664294000 T^{22} - 2710996768007 T^{24} - 53534844376464 T^{26} + 269904840398006 T^{28} + 9113996005997784 T^{30} + 45949729863572161 T^{32}$$
$13$ $$( 1 - 48 T^{2} + 1308 T^{4} - 24144 T^{6} + 351974 T^{8} - 4080336 T^{10} + 37357788 T^{12} - 231686832 T^{14} + 815730721 T^{16} )^{2}$$
$17$ $$( 1 + 46 T^{2} + 1137 T^{4} - 1392 T^{5} + 19214 T^{6} - 46992 T^{7} + 276452 T^{8} - 798864 T^{9} + 5552846 T^{10} - 6838896 T^{11} + 94963377 T^{12} + 1110328174 T^{14} + 6975757441 T^{16} )^{2}$$
$19$ $$1 - 120 T^{2} + 7878 T^{4} - 351600 T^{6} + 11807689 T^{8} - 315489360 T^{10} + 7073822934 T^{12} - 141586703880 T^{14} + 2708298760020 T^{16} - 51112800100680 T^{18} + 921867678581814 T^{20} - 14842474887326160 T^{22} + 200536630500022249 T^{24} - 2155682896242831600 T^{26} + 17436494932403216358 T^{28} - 95880802293946094520 T^{30} +$$$$28\!\cdots\!81$$$$T^{32}$$
$23$ $$1 + 136 T^{2} + 9622 T^{4} + 484880 T^{6} + 19684601 T^{8} + 683013040 T^{10} + 20851380422 T^{12} + 567119503160 T^{14} + 13792455622804 T^{16} + 300006217171640 T^{18} + 5835071148672902 T^{20} + 101110442574992560 T^{22} + 1541520499173357881 T^{24} + 20086886757274127120 T^{26} +$$$$21\!\cdots\!62$$$$T^{28} +$$$$15\!\cdots\!24$$$$T^{30} +$$$$61\!\cdots\!61$$$$T^{32}$$
$29$ $$( 1 - 4 T + 88 T^{2} - 332 T^{3} + 3534 T^{4} - 9628 T^{5} + 74008 T^{6} - 97556 T^{7} + 707281 T^{8} )^{4}$$
$31$ $$1 - 120 T^{2} + 6726 T^{4} - 224880 T^{6} + 4887337 T^{8} - 66674640 T^{10} + 1194526422 T^{12} - 84694288200 T^{14} + 3829419751284 T^{16} - 81391210960200 T^{18} + 1103170235771862 T^{20} - 59173988429349840 T^{22} + 4168365924253784617 T^{24} -$$$$18\!\cdots\!80$$$$T^{26} +$$$$52\!\cdots\!86$$$$T^{28} -$$$$90\!\cdots\!20$$$$T^{30} +$$$$72\!\cdots\!81$$$$T^{32}$$
$37$ $$( 1 + 4 T - 74 T^{2} - 88 T^{3} + 2945 T^{4} - 5624 T^{5} - 119866 T^{6} + 157628 T^{7} + 4940404 T^{8} + 5832236 T^{9} - 164096554 T^{10} - 284872472 T^{11} + 5519404145 T^{12} - 6102268216 T^{13} - 189863754266 T^{14} + 379727508532 T^{15} + 3512479453921 T^{16} )^{2}$$
$41$ $$( 1 - 272 T^{2} + 34236 T^{4} - 2596336 T^{6} + 129652166 T^{8} - 4364440816 T^{10} + 96742753596 T^{12} - 1292028353552 T^{14} + 7984925229121 T^{16} )^{2}$$
$43$ $$( 1 - 152 T^{2} + 8604 T^{4} - 204136 T^{6} + 3083174 T^{8} - 377447464 T^{10} + 29415363804 T^{12} - 960847183448 T^{14} + 11688200277601 T^{16} )^{2}$$
$47$ $$1 - 200 T^{2} + 18934 T^{4} - 1138000 T^{6} + 52923737 T^{8} - 2454592400 T^{10} + 135007848038 T^{12} - 7743351520600 T^{14} + 394816080594964 T^{16} - 17105063509005400 T^{18} + 658795230921915878 T^{20} - 26458580024526899600 T^{22} +$$$$12\!\cdots\!57$$$$T^{24} -$$$$59\!\cdots\!00$$$$T^{26} +$$$$21\!\cdots\!94$$$$T^{28} -$$$$51\!\cdots\!00$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$( 1 + 4 T - 106 T^{2} - 984 T^{3} + 4577 T^{4} + 69896 T^{5} + 172774 T^{6} - 2132484 T^{7} - 19431692 T^{8} - 113021652 T^{9} + 485322166 T^{10} + 10405906792 T^{11} + 36114731537 T^{12} - 411504365112 T^{13} - 2349422279674 T^{14} + 4698844559348 T^{15} + 62259690411361 T^{16} )^{2}$$
$59$ $$1 - 216 T^{2} + 25302 T^{4} - 1976880 T^{6} + 99819481 T^{8} - 1826074704 T^{10} - 264226437882 T^{12} + 36647033732760 T^{14} - 2664686254766220 T^{16} + 127568324423737560 T^{18} - 3201727133560269402 T^{20} - 77024805483051117264 T^{22} +$$$$14\!\cdots\!01$$$$T^{24} -$$$$10\!\cdots\!80$$$$T^{26} +$$$$45\!\cdots\!62$$$$T^{28} -$$$$13\!\cdots\!76$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$( 1 + 12 T + 182 T^{2} + 1608 T^{3} + 12321 T^{4} + 99864 T^{5} + 781126 T^{6} + 6487236 T^{7} + 58967444 T^{8} + 395721396 T^{9} + 2906569846 T^{10} + 22667230584 T^{11} + 170594606961 T^{12} + 1358110852008 T^{13} + 9376708133702 T^{14} + 37712914032252 T^{15} + 191707312997281 T^{16} )^{2}$$
$67$ $$1 + 248 T^{2} + 30214 T^{4} + 1955248 T^{6} + 42292553 T^{8} - 3785460880 T^{10} - 336164526826 T^{12} - 6739836332504 T^{14} + 264085897769428 T^{16} - 30255125296610456 T^{18} - 6774092055978471946 T^{20} -$$$$34\!\cdots\!20$$$$T^{22} +$$$$17\!\cdots\!73$$$$T^{24} +$$$$35\!\cdots\!52$$$$T^{26} +$$$$24\!\cdots\!54$$$$T^{28} +$$$$91\!\cdots\!92$$$$T^{30} +$$$$16\!\cdots\!81$$$$T^{32}$$
$71$ $$( 1 - 248 T^{2} + 34492 T^{4} - 3463880 T^{6} + 277458886 T^{8} - 17461419080 T^{10} + 876499701052 T^{12} - 31768870412408 T^{14} + 645753531245761 T^{16} )^{2}$$
$73$ $$( 1 + 12 T + 270 T^{2} + 2664 T^{3} + 35289 T^{4} + 261048 T^{5} + 2808174 T^{6} + 17932596 T^{7} + 187438964 T^{8} + 1309079508 T^{9} + 14964759246 T^{10} + 101552109816 T^{11} + 1002145526649 T^{12} + 5522662723752 T^{13} + 40860241098030 T^{14} + 132568782229164 T^{15} + 806460091894081 T^{16} )^{2}$$
$79$ $$1 + 296 T^{2} + 48502 T^{4} + 5442256 T^{6} + 407298713 T^{8} + 12121526192 T^{10} - 1601181207706 T^{12} - 316632003296936 T^{14} - 31211270417534060 T^{16} - 1976100332576177576 T^{18} - 62366137735826524186 T^{20} +$$$$29\!\cdots\!32$$$$T^{22} +$$$$61\!\cdots\!93$$$$T^{24} +$$$$51\!\cdots\!56$$$$T^{26} +$$$$28\!\cdots\!82$$$$T^{28} +$$$$10\!\cdots\!76$$$$T^{30} +$$$$23\!\cdots\!21$$$$T^{32}$$
$83$ $$( 1 + 248 T^{2} + 25404 T^{4} + 1379080 T^{6} + 70811942 T^{8} + 9500482120 T^{10} + 1205631186684 T^{12} + 81081212595512 T^{14} + 2252292232139041 T^{16} )^{2}$$
$89$ $$( 1 + 36 T + 822 T^{2} + 14040 T^{3} + 193953 T^{4} + 2261112 T^{5} + 23442342 T^{6} + 226172268 T^{7} + 2129495396 T^{8} + 20129331852 T^{9} + 185686790982 T^{10} + 1594013865528 T^{11} + 12169045868673 T^{12} + 78400194663960 T^{13} + 408518621169942 T^{14} + 1592328056239044 T^{15} + 3936588805702081 T^{16} )^{2}$$
$97$ $$( 1 - 304 T^{2} + 67644 T^{4} - 9727184 T^{6} + 1110022022 T^{8} - 91523074256 T^{10} + 5988474683964 T^{12} - 253223489498416 T^{14} + 7837433594376961 T^{16} )^{2}$$