# Properties

 Label 637.2.c.g Level $637$ Weight $2$ Character orbit 637.c Analytic conductor $5.086$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 10 x^{14} + 121 x^{12} + 296 x^{10} + 3468 x^{8} - 1748 x^{6} + 40192 x^{4} - 65056 x^{2} + 228484$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{3}$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{10} q^{2} -\beta_{11} q^{3} + ( -1 + \beta_{2} ) q^{4} -\beta_{6} q^{5} -\beta_{4} q^{6} + ( -\beta_{10} + \beta_{13} - \beta_{14} ) q^{8} + ( 1 + \beta_{2} - \beta_{3} - \beta_{5} ) q^{9} +O(q^{10})$$ $$q + \beta_{10} q^{2} -\beta_{11} q^{3} + ( -1 + \beta_{2} ) q^{4} -\beta_{6} q^{5} -\beta_{4} q^{6} + ( -\beta_{10} + \beta_{13} - \beta_{14} ) q^{8} + ( 1 + \beta_{2} - \beta_{3} - \beta_{5} ) q^{9} + ( -\beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{11} ) q^{10} -\beta_{13} q^{11} + \beta_{12} q^{12} + ( -\beta_{1} - \beta_{7} ) q^{13} + ( \beta_{13} + \beta_{14} ) q^{15} + ( 1 - 2 \beta_{2} - \beta_{3} + \beta_{5} ) q^{16} + ( -\beta_{9} + \beta_{11} + \beta_{12} ) q^{17} + ( -\beta_{14} + \beta_{15} ) q^{18} + ( \beta_{1} + \beta_{4} - \beta_{6} ) q^{19} + ( 2 \beta_{1} + \beta_{4} + \beta_{7} + \beta_{8} ) q^{20} + ( \beta_{2} + \beta_{3} + \beta_{5} ) q^{22} + ( -2 - \beta_{2} + 2 \beta_{3} ) q^{23} + ( 3 \beta_{1} + \beta_{7} + \beta_{8} ) q^{24} + ( -3 - \beta_{2} ) q^{25} + ( \beta_{6} + \beta_{8} + 2 \beta_{9} + \beta_{12} ) q^{26} + ( -\beta_{7} + \beta_{8} + 3 \beta_{9} ) q^{27} + ( 2 - 2 \beta_{2} + \beta_{3} ) q^{29} + ( \beta_{2} - \beta_{3} - 3 \beta_{5} ) q^{30} + ( \beta_{1} - 2 \beta_{4} - \beta_{6} ) q^{31} + ( 4 \beta_{10} - \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{32} + ( -\beta_{1} - \beta_{4} + 2 \beta_{6} - \beta_{7} - \beta_{8} ) q^{33} + ( 2 \beta_{1} + 3 \beta_{4} + \beta_{7} + \beta_{8} ) q^{34} + ( 3 - \beta_{2} + \beta_{5} ) q^{36} + ( -2 \beta_{10} - \beta_{13} ) q^{37} + ( -\beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} - 2 \beta_{12} ) q^{38} + ( 2 + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{10} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{39} + ( -\beta_{7} + \beta_{8} - 3 \beta_{12} ) q^{40} + ( \beta_{1} - 2 \beta_{4} ) q^{41} + ( -2 + \beta_{3} + 2 \beta_{5} ) q^{43} + ( -3 \beta_{10} - \beta_{14} - \beta_{15} ) q^{44} + ( -\beta_{1} + 3 \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{45} + ( -2 \beta_{10} + \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{46} + ( -\beta_{1} - \beta_{4} - \beta_{6} ) q^{47} + ( \beta_{7} - \beta_{8} - 7 \beta_{9} - \beta_{12} ) q^{48} + ( -\beta_{10} - \beta_{13} + \beta_{14} ) q^{50} + ( -4 + \beta_{2} + \beta_{3} + 3 \beta_{5} ) q^{51} + ( 3 \beta_{1} + 2 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{9} - \beta_{11} ) q^{52} + ( -\beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{53} + ( 3 \beta_{1} + 2 \beta_{6} + \beta_{7} + \beta_{8} ) q^{54} + ( \beta_{7} - \beta_{8} - \beta_{9} - 5 \beta_{11} + 2 \beta_{12} ) q^{55} + ( -3 \beta_{10} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{57} + ( 5 \beta_{10} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{58} + ( -2 \beta_{1} - \beta_{4} + 2 \beta_{6} ) q^{59} + ( -\beta_{10} + 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{60} + ( \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{12} ) q^{61} + ( -\beta_{7} + \beta_{8} - \beta_{9} + 5 \beta_{11} + \beta_{12} ) q^{62} + ( -9 + 4 \beta_{2} + \beta_{3} ) q^{64} + ( -1 + 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{10} + \beta_{14} + \beta_{15} ) q^{65} + ( \beta_{7} - \beta_{8} - 3 \beta_{9} + 2 \beta_{12} ) q^{66} + ( 2 \beta_{10} - \beta_{14} - 2 \beta_{15} ) q^{67} + ( \beta_{7} - \beta_{8} - 6 \beta_{9} - 4 \beta_{11} - 3 \beta_{12} ) q^{68} + ( 4 \beta_{9} + 5 \beta_{11} - \beta_{12} ) q^{69} + ( \beta_{10} - \beta_{13} - \beta_{15} ) q^{71} + ( 5 \beta_{10} - \beta_{13} + 2 \beta_{15} ) q^{72} + ( -\beta_{1} + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{73} + ( 6 - \beta_{2} + \beta_{3} + \beta_{5} ) q^{74} + ( 4 \beta_{11} - \beta_{12} ) q^{75} + ( -5 \beta_{1} - 3 \beta_{4} - \beta_{7} - \beta_{8} ) q^{76} + ( 2 + 3 \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{10} - \beta_{14} + \beta_{15} ) q^{78} + ( 2 - \beta_{3} - \beta_{5} ) q^{79} + ( -5 \beta_{1} - 4 \beta_{4} + 2 \beta_{6} ) q^{80} + ( 1 - \beta_{2} + \beta_{3} - 2 \beta_{5} ) q^{81} + ( -3 \beta_{9} + 4 \beta_{11} + \beta_{12} ) q^{82} + ( -\beta_{6} + \beta_{7} + \beta_{8} ) q^{83} + ( \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{85} + ( -3 \beta_{10} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{86} + ( 2 \beta_{9} + \beta_{11} - 2 \beta_{12} ) q^{87} + ( 8 - 2 \beta_{2} + 3 \beta_{5} ) q^{88} + ( -4 \beta_{1} + \beta_{4} + \beta_{6} ) q^{89} + ( 7 \beta_{9} - 5 \beta_{11} - 2 \beta_{12} ) q^{90} + ( -5 \beta_{2} - \beta_{3} - \beta_{5} ) q^{92} + ( 6 \beta_{10} + \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{93} + ( -\beta_{7} + \beta_{8} + 5 \beta_{9} + 3 \beta_{11} + 2 \beta_{12} ) q^{94} + ( -6 - 2 \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{95} + ( -4 \beta_{1} - 2 \beta_{4} - 2 \beta_{6} ) q^{96} + ( \beta_{4} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{97} + ( \beta_{10} - \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 20 q^{4} + 16 q^{9} + O(q^{10})$$ $$16 q - 20 q^{4} + 16 q^{9} + 28 q^{16} - 8 q^{22} - 36 q^{23} - 44 q^{25} + 36 q^{29} + 52 q^{36} + 32 q^{39} - 36 q^{43} - 72 q^{51} + 12 q^{53} - 164 q^{64} - 24 q^{65} + 96 q^{74} + 24 q^{78} + 36 q^{79} + 16 q^{81} + 136 q^{88} + 24 q^{92} - 84 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 10 x^{14} + 121 x^{12} + 296 x^{10} + 3468 x^{8} - 1748 x^{6} + 40192 x^{4} - 65056 x^{2} + 228484$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$127996212 \nu^{14} + 979990851 \nu^{12} + 13620194602 \nu^{10} + 6825369227 \nu^{8} + 476412021848 \nu^{6} - 1697477109286 \nu^{4} + 18907611617666 \nu^{2} - 24635344078600$$$$)/ 29065582596052$$ $$\beta_{2}$$ $$=$$ $$($$$$-127996212 \nu^{14} - 979990851 \nu^{12} - 13620194602 \nu^{10} - 6825369227 \nu^{8} - 476412021848 \nu^{6} + 1697477109286 \nu^{4} - 4374820319640 \nu^{2} + 39168135376626$$$$)/ 14532791298026$$ $$\beta_{3}$$ $$=$$ $$($$$$-306986712 \nu^{14} - 5785250672 \nu^{12} - 51065796556 \nu^{10} - 245977410085 \nu^{8} - 267899653440 \nu^{6} - 4317537291960 \nu^{4} + 9290284007112 \nu^{2} - 47980987815069$$$$)/ 7266395649013$$ $$\beta_{4}$$ $$=$$ $$($$$$-1919925366 \nu^{14} - 6971345277 \nu^{12} - 76432745837 \nu^{10} + 1041868051029 \nu^{8} - 1507742096165 \nu^{6} + 28800814691994 \nu^{4} - 55028459919508 \nu^{2} + 137905300117434$$$$)/ 29065582596052$$ $$\beta_{5}$$ $$=$$ $$($$$$-774992715 \nu^{14} - 9797848728 \nu^{12} - 101725454643 \nu^{10} - 319450784079 \nu^{8} - 1652632053818 \nu^{6} - 629466852193 \nu^{4} - 650215400000 \nu^{2} - 8985354172108$$$$)/ 7266395649013$$ $$\beta_{6}$$ $$=$$ $$($$$$7536716227 \nu^{14} + 51263835016 \nu^{12} + 571971494681 \nu^{10} - 973429379572 \nu^{8} + 15440128011114 \nu^{6} - 44623804952754 \nu^{4} + 205861498545968 \nu^{2} - 200469162311696$$$$)/ 58131165192104$$ $$\beta_{7}$$ $$=$$ $$($$$$14293266565 \nu^{15} + 622324322223 \nu^{14} - 4349411499250 \nu^{13} + 967985226688 \nu^{12} - 41285442593947 \nu^{11} + 4895835168025 \nu^{10} - 429503087786032 \nu^{9} - 525360644247946 \nu^{8} - 273519392828494 \nu^{7} - 293425161053504 \nu^{6} - 7093806687835658 \nu^{5} - 12187826973066802 \nu^{4} + 13968719402934952 \nu^{3} + 3760983663948440 \nu^{2} - 37371933389289288 \nu - 43848384888236764$$$$)/ 13893348480912856$$ $$\beta_{8}$$ $$=$$ $$($$$$-14293266565 \nu^{15} + 622324322223 \nu^{14} + 4349411499250 \nu^{13} + 967985226688 \nu^{12} + 41285442593947 \nu^{11} + 4895835168025 \nu^{10} + 429503087786032 \nu^{9} - 525360644247946 \nu^{8} + 273519392828494 \nu^{7} - 293425161053504 \nu^{6} + 7093806687835658 \nu^{5} - 12187826973066802 \nu^{4} - 13968719402934952 \nu^{3} + 3760983663948440 \nu^{2} + 37371933389289288 \nu - 43848384888236764$$$$)/ 13893348480912856$$ $$\beta_{9}$$ $$=$$ $$($$$$-35037469784 \nu^{15} - 319783603172 \nu^{13} - 4005316030475 \nu^{11} - 7115864546186 \nu^{9} - 119878681965659 \nu^{7} + 175107970404104 \nu^{5} - 1813923014677882 \nu^{3} + 6798316810890078 \nu$$$$)/ 6946674240456428$$ $$\beta_{10}$$ $$=$$ $$($$$$35037469784 \nu^{15} + 319783603172 \nu^{13} + 4005316030475 \nu^{11} + 7115864546186 \nu^{9} + 119878681965659 \nu^{7} - 175107970404104 \nu^{5} + 1813923014677882 \nu^{3} + 148357429566350 \nu$$$$)/ 6946674240456428$$ $$\beta_{11}$$ $$=$$ $$($$$$-213313563659 \nu^{15} - 4532646032618 \nu^{13} - 47876731013081 \nu^{11} - 294174736630566 \nu^{9} - 852548409536598 \nu^{7} - 3507661099590954 \nu^{5} + 3989750050797588 \nu^{3} - 13669488488924920 \nu$$$$)/ 13893348480912856$$ $$\beta_{12}$$ $$=$$ $$($$$$144596254252 \nu^{15} + 1364700202471 \nu^{13} + 16771353642497 \nu^{11} + 33948059485677 \nu^{9} + 485530936606623 \nu^{7} - 469842822901166 \nu^{5} + 4550695896767236 \nu^{3} - 14210388338752134 \nu$$$$)/ 6946674240456428$$ $$\beta_{13}$$ $$=$$ $$($$$$-374090467397 \nu^{15} - 3298265569313 \nu^{13} - 31101270941936 \nu^{11} + 33697049138585 \nu^{9} - 229804479818813 \nu^{7} + 2897226083423780 \nu^{5} + 1476021840137492 \nu^{3} + 9888185130797466 \nu$$$$)/ 6946674240456428$$ $$\beta_{14}$$ $$=$$ $$($$$$-527579471881 \nu^{15} - 4834097351350 \nu^{13} - 49372803625627 \nu^{11} - 11220212948958 \nu^{9} - 727367833913410 \nu^{7} + 2905890788894446 \nu^{5} - 4611355973197588 \nu^{3} + 5079489900389672 \nu$$$$)/ 6946674240456428$$ $$\beta_{15}$$ $$=$$ $$($$$$674575131380 \nu^{15} + 7249765957837 \nu^{13} + 73681722610361 \nu^{11} + 144743314973337 \nu^{9} + 1332704439491341 \nu^{7} + 532755357398390 \nu^{5} + 7184835654673372 \nu^{3} + 20029400440792962 \nu$$$$)/ 6946674240456428$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{10} + \beta_{9}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} - 1$$ $$\nu^{3}$$ $$=$$ $$-\beta_{14} + \beta_{13} - 3 \beta_{12} + \beta_{10} - 7 \beta_{9}$$ $$\nu^{4}$$ $$=$$ $$-4 \beta_{8} - 4 \beta_{7} + \beta_{5} - 8 \beta_{4} - \beta_{3} + 4 \beta_{2} - 16 \beta_{1} - 17$$ $$\nu^{5}$$ $$=$$ $$\beta_{15} - 10 \beta_{14} + 11 \beta_{13} + 20 \beta_{12} + 20 \beta_{11} - 48 \beta_{10} + 24 \beta_{9} + 5 \beta_{8} - 5 \beta_{7}$$ $$\nu^{6}$$ $$=$$ $$14 \beta_{8} + 14 \beta_{7} + 12 \beta_{6} + 20 \beta_{5} + 40 \beta_{4} - 19 \beta_{3} - 120 \beta_{2} + 24 \beta_{1} + 183$$ $$\nu^{7}$$ $$=$$ $$31 \beta_{15} + 199 \beta_{14} - 167 \beta_{13} + 56 \beta_{12} - 14 \beta_{11} + 474 \beta_{10} + 134 \beta_{9} - 7 \beta_{8} + 7 \beta_{7}$$ $$\nu^{8}$$ $$=$$ $$216 \beta_{8} + 216 \beta_{7} + 48 \beta_{6} - 200 \beta_{5} + 496 \beta_{4} + 243 \beta_{3} + 896 \beta_{2} + 912 \beta_{1} - 1109$$ $$\nu^{9}$$ $$=$$ $$-195 \beta_{15} - 843 \beta_{14} + 707 \beta_{13} - 2304 \beta_{12} - 1926 \beta_{11} - 1704 \beta_{10} - 3752 \beta_{9} - 507 \beta_{8} + 507 \beta_{7}$$ $$\nu^{10}$$ $$=$$ $$-3518 \beta_{8} - 3518 \beta_{7} - 1404 \beta_{6} - 128 \beta_{5} - 8220 \beta_{4} - 83 \beta_{3} + 706 \beta_{2} - 12880 \beta_{1} - 1573$$ $$\nu^{11}$$ $$=$$ $$-1321 \beta_{15} - 8971 \beta_{14} + 7659 \beta_{13} + 20394 \beta_{12} + 17754 \beta_{11} - 23030 \beta_{10} + 32922 \beta_{9} + 5005 \beta_{8} - 5005 \beta_{7}$$ $$\nu^{12}$$ $$=$$ $$17740 \beta_{8} + 17740 \beta_{7} + 7368 \beta_{6} + 16706 \beta_{5} + 40600 \beta_{4} - 20311 \beta_{3} - 88098 \beta_{2} + 64736 \beta_{1} + 123603$$ $$\nu^{13}$$ $$=$$ $$27679 \beta_{15} + 165715 \beta_{14} - 143889 \beta_{13} - 17238 \beta_{12} - 14534 \beta_{11} + 421470 \beta_{10} - 29550 \beta_{9} - 4797 \beta_{8} + 4797 \beta_{7}$$ $$\nu^{14}$$ $$=$$ $$121854 \beta_{8} + 121854 \beta_{7} + 45764 \beta_{6} - 164802 \beta_{5} + 282420 \beta_{4} + 208841 \beta_{3} + 903586 \beta_{2} + 456416 \beta_{1} - 1286237$$ $$\nu^{15}$$ $$=$$ $$-163077 \beta_{15} - 994809 \beta_{14} + 868719 \beta_{13} - 1642422 \beta_{12} - 1357890 \beta_{11} - 2587362 \beta_{10} - 2712146 \beta_{9} - 376459 \beta_{8} + 376459 \beta_{7}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/637\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
246.1
 1.41421 − 2.73420i −1.41421 − 2.73420i −1.41421 − 1.65552i 1.41421 − 1.65552i −1.41421 − 1.52340i 1.41421 − 1.52340i 1.41421 − 0.680196i −1.41421 − 0.680196i 1.41421 + 0.680196i −1.41421 + 0.680196i −1.41421 + 1.52340i 1.41421 + 1.52340i −1.41421 + 1.65552i 1.41421 + 1.65552i 1.41421 + 2.73420i −1.41421 + 2.73420i
2.73420i −1.15595 −5.47586 1.87727i 3.16060i 0 9.50370i −1.66378 5.13283
246.2 2.73420i 1.15595 −5.47586 1.87727i 3.16060i 0 9.50370i −1.66378 −5.13283
246.3 1.65552i −0.494977 −0.740740 2.87389i 0.819443i 0 2.08473i −2.75500 4.75778
246.4 1.65552i 0.494977 −0.740740 2.87389i 0.819443i 0 2.08473i −2.75500 −4.75778
246.5 1.52340i −2.99510 −0.320733 2.94606i 4.56272i 0 2.55819i 5.97063 −4.48801
246.6 1.52340i 2.99510 −0.320733 2.94606i 4.56272i 0 2.55819i 5.97063 4.48801
246.7 0.680196i −2.33413 1.53733 3.24613i 1.58766i 0 2.40608i 2.44815 2.20800
246.8 0.680196i 2.33413 1.53733 3.24613i 1.58766i 0 2.40608i 2.44815 −2.20800
246.9 0.680196i −2.33413 1.53733 3.24613i 1.58766i 0 2.40608i 2.44815 2.20800
246.10 0.680196i 2.33413 1.53733 3.24613i 1.58766i 0 2.40608i 2.44815 −2.20800
246.11 1.52340i −2.99510 −0.320733 2.94606i 4.56272i 0 2.55819i 5.97063 −4.48801
246.12 1.52340i 2.99510 −0.320733 2.94606i 4.56272i 0 2.55819i 5.97063 4.48801
246.13 1.65552i −0.494977 −0.740740 2.87389i 0.819443i 0 2.08473i −2.75500 4.75778
246.14 1.65552i 0.494977 −0.740740 2.87389i 0.819443i 0 2.08473i −2.75500 −4.75778
246.15 2.73420i −1.15595 −5.47586 1.87727i 3.16060i 0 9.50370i −1.66378 5.13283
246.16 2.73420i 1.15595 −5.47586 1.87727i 3.16060i 0 9.50370i −1.66378 −5.13283
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 246.16 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
13.b even 2 1 inner
91.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.c.g 16
7.b odd 2 1 inner 637.2.c.g 16
7.c even 3 2 637.2.r.g 32
7.d odd 6 2 637.2.r.g 32
13.b even 2 1 inner 637.2.c.g 16
13.d odd 4 2 8281.2.a.cs 16
91.b odd 2 1 inner 637.2.c.g 16
91.i even 4 2 8281.2.a.cs 16
91.r even 6 2 637.2.r.g 32
91.s odd 6 2 637.2.r.g 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.c.g 16 1.a even 1 1 trivial
637.2.c.g 16 7.b odd 2 1 inner
637.2.c.g 16 13.b even 2 1 inner
637.2.c.g 16 91.b odd 2 1 inner
637.2.r.g 32 7.c even 3 2
637.2.r.g 32 7.d odd 6 2
637.2.r.g 32 91.r even 6 2
637.2.r.g 32 91.s odd 6 2
8281.2.a.cs 16 13.d odd 4 2
8281.2.a.cs 16 91.i even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(637, [\chi])$$:

 $$T_{2}^{8} + 13 T_{2}^{6} + 50 T_{2}^{4} + 68 T_{2}^{2} + 22$$ $$T_{3}^{8} - 16 T_{3}^{6} + 72 T_{3}^{4} - 82 T_{3}^{2} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 22 + 68 T^{2} + 50 T^{4} + 13 T^{6} + T^{8} )^{2}$$
$3$ $$( 16 - 82 T^{2} + 72 T^{4} - 16 T^{6} + T^{8} )^{2}$$
$5$ $$( 2662 + 1637 T^{2} + 347 T^{4} + 31 T^{6} + T^{8} )^{2}$$
$7$ $$T^{16}$$
$11$ $$( 5632 + 3110 T^{2} + 602 T^{4} + 46 T^{6} + T^{8} )^{2}$$
$13$ $$815730721 + 106189798 T^{2} + 4569760 T^{4} - 298454 T^{6} - 47906 T^{8} - 1766 T^{10} + 160 T^{12} + 22 T^{14} + T^{16}$$
$17$ $$( 39204 - 15498 T^{2} + 1908 T^{4} - 78 T^{6} + T^{8} )^{2}$$
$19$ $$( 7128 + 15579 T^{2} + 1863 T^{4} + 75 T^{6} + T^{8} )^{2}$$
$23$ $$( -792 - 441 T - 39 T^{2} + 9 T^{3} + T^{4} )^{4}$$
$29$ $$( -144 + 261 T - 21 T^{2} - 9 T^{3} + T^{4} )^{4}$$
$31$ $$( 1824768 + 223587 T^{2} + 9477 T^{4} + 165 T^{6} + T^{8} )^{2}$$
$37$ $$( 7128 + 16686 T^{2} + 2394 T^{4} + 90 T^{6} + T^{8} )^{2}$$
$41$ $$( 59488 + 26240 T^{2} + 3608 T^{4} + 154 T^{6} + T^{8} )^{2}$$
$43$ $$( 72 - 45 T - 21 T^{2} + 9 T^{3} + T^{4} )^{4}$$
$47$ $$( 1408 + 5699 T^{2} + 2903 T^{4} + 115 T^{6} + T^{8} )^{2}$$
$53$ $$( 2502 + 207 T - 105 T^{2} - 3 T^{3} + T^{4} )^{4}$$
$59$ $$( 1580128 + 267788 T^{2} + 11732 T^{4} + 190 T^{6} + T^{8} )^{2}$$
$61$ $$( 219024 - 79488 T^{2} + 7848 T^{4} - 168 T^{6} + T^{8} )^{2}$$
$67$ $$( 456192 + 374112 T^{2} + 25488 T^{4} + 348 T^{6} + T^{8} )^{2}$$
$71$ $$( 68992 + 50630 T^{2} + 4394 T^{4} + 124 T^{6} + T^{8} )^{2}$$
$73$ $$( 514998 + 167913 T^{2} + 9765 T^{4} + 189 T^{6} + T^{8} )^{2}$$
$79$ $$( -44 + 45 T + 11 T^{2} - 9 T^{3} + T^{4} )^{4}$$
$83$ $$( 25432 + 15227 T^{2} + 2729 T^{4} + 157 T^{6} + T^{8} )^{2}$$
$89$ $$( 1888678 + 1989089 T^{2} + 48173 T^{4} + 385 T^{6} + T^{8} )^{2}$$
$97$ $$( 28741878 + 5114961 T^{2} + 92205 T^{4} + 537 T^{6} + T^{8} )^{2}$$