# Properties

 Label 6422.2.a.bm Level $6422$ Weight $2$ Character orbit 6422.a Self dual yes Analytic conductor $51.280$ Analytic rank $0$ Dimension $14$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6422 = 2 \cdot 13^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6422.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$51.2799281781$$ Analytic rank: $$0$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ Defining polynomial: $$x^{14} - 4 x^{13} - 22 x^{12} + 98 x^{11} + 164 x^{10} - 912 x^{9} - 374 x^{8} + 3996 x^{7} - 817 x^{6} - 8194 x^{5} + 4418 x^{4} + 6430 x^{3} - 4327 x^{2} - 922 x + 358$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 494) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta_{1} q^{3} + q^{4} + \beta_{4} q^{5} -\beta_{1} q^{6} -\beta_{12} q^{7} - q^{8} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q - q^{2} + \beta_{1} q^{3} + q^{4} + \beta_{4} q^{5} -\beta_{1} q^{6} -\beta_{12} q^{7} - q^{8} + ( 1 + \beta_{2} ) q^{9} -\beta_{4} q^{10} + ( -1 - \beta_{11} - \beta_{12} ) q^{11} + \beta_{1} q^{12} + \beta_{12} q^{14} + ( -1 + \beta_{1} + \beta_{4} - \beta_{7} + \beta_{9} - \beta_{11} + \beta_{13} ) q^{15} + q^{16} -\beta_{5} q^{17} + ( -1 - \beta_{2} ) q^{18} - q^{19} + \beta_{4} q^{20} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{6} - \beta_{9} - \beta_{12} ) q^{21} + ( 1 + \beta_{11} + \beta_{12} ) q^{22} + ( 1 - \beta_{1} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{23} -\beta_{1} q^{24} + ( 3 + \beta_{2} + \beta_{6} - \beta_{8} ) q^{25} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{10} + \beta_{13} ) q^{27} -\beta_{12} q^{28} + ( \beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{29} + ( 1 - \beta_{1} - \beta_{4} + \beta_{7} - \beta_{9} + \beta_{11} - \beta_{13} ) q^{30} + ( \beta_{9} + \beta_{11} ) q^{31} - q^{32} + ( -1 - \beta_{1} + \beta_{4} + \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{33} + \beta_{5} q^{34} + ( 1 - \beta_{2} - \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{10} - \beta_{11} ) q^{35} + ( 1 + \beta_{2} ) q^{36} + ( -1 - \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{37} + q^{38} -\beta_{4} q^{40} + ( \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{10} - \beta_{12} ) q^{41} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{6} + \beta_{9} + \beta_{12} ) q^{42} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} - \beta_{8} - \beta_{12} ) q^{43} + ( -1 - \beta_{11} - \beta_{12} ) q^{44} + ( -2 + \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{45} + ( -1 + \beta_{1} + \beta_{7} - \beta_{8} + \beta_{10} ) q^{46} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{9} + \beta_{10} ) q^{47} + \beta_{1} q^{48} + ( 3 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{12} - \beta_{13} ) q^{49} + ( -3 - \beta_{2} - \beta_{6} + \beta_{8} ) q^{50} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{9} - \beta_{10} ) q^{51} + ( \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} + \beta_{13} ) q^{53} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{10} - \beta_{13} ) q^{54} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} ) q^{55} + \beta_{12} q^{56} -\beta_{1} q^{57} + ( -\beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} ) q^{58} + ( -2 + 2 \beta_{1} + \beta_{4} - \beta_{6} + 2 \beta_{7} ) q^{59} + ( -1 + \beta_{1} + \beta_{4} - \beta_{7} + \beta_{9} - \beta_{11} + \beta_{13} ) q^{60} + ( 3 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{61} + ( -\beta_{9} - \beta_{11} ) q^{62} + ( -2 + \beta_{1} - \beta_{4} + \beta_{5} + \beta_{8} ) q^{63} + q^{64} + ( 1 + \beta_{1} - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{66} + ( 1 - 2 \beta_{1} - \beta_{4} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{67} -\beta_{5} q^{68} + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{11} - \beta_{12} + \beta_{13} ) q^{69} + ( -1 + \beta_{2} + \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{10} + \beta_{11} ) q^{70} + ( -4 + 2 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} + \beta_{12} ) q^{71} + ( -1 - \beta_{2} ) q^{72} + ( 3 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{13} ) q^{73} + ( 1 + \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{74} + ( 2 + 4 \beta_{1} + \beta_{4} - \beta_{6} - 3 \beta_{7} + \beta_{9} + \beta_{10} - \beta_{12} ) q^{75} - q^{76} + ( 5 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{11} - \beta_{13} ) q^{77} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{79} + \beta_{4} q^{80} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{10} + 2 \beta_{11} ) q^{81} + ( -\beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{10} + \beta_{12} ) q^{82} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - \beta_{11} - 2 \beta_{12} ) q^{83} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{6} - \beta_{9} - \beta_{12} ) q^{84} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} ) q^{85} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{8} + \beta_{12} ) q^{86} + ( 2 + 3 \beta_{1} + \beta_{3} - \beta_{5} - 2 \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{87} + ( 1 + \beta_{11} + \beta_{12} ) q^{88} + ( -2 - \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{8} - \beta_{10} + \beta_{11} - \beta_{13} ) q^{89} + ( 2 - \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} - \beta_{12} ) q^{90} + ( 1 - \beta_{1} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{92} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{93} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{9} - \beta_{10} ) q^{94} -\beta_{4} q^{95} -\beta_{1} q^{96} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} - 2 \beta_{12} ) q^{97} + ( -3 + \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{12} + \beta_{13} ) q^{98} + ( -2 - 3 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - 3 \beta_{7} + 3 \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14 q - 14 q^{2} + 4 q^{3} + 14 q^{4} + 2 q^{5} - 4 q^{6} - 2 q^{7} - 14 q^{8} + 18 q^{9} + O(q^{10})$$ $$14 q - 14 q^{2} + 4 q^{3} + 14 q^{4} + 2 q^{5} - 4 q^{6} - 2 q^{7} - 14 q^{8} + 18 q^{9} - 2 q^{10} - 10 q^{11} + 4 q^{12} + 2 q^{14} + 4 q^{15} + 14 q^{16} + 2 q^{17} - 18 q^{18} - 14 q^{19} + 2 q^{20} + 18 q^{21} + 10 q^{22} + 12 q^{23} - 4 q^{24} + 44 q^{25} + 10 q^{27} - 2 q^{28} + 4 q^{29} - 4 q^{30} - 4 q^{31} - 14 q^{32} - 12 q^{33} - 2 q^{34} + 14 q^{35} + 18 q^{36} - 18 q^{37} + 14 q^{38} - 2 q^{40} - 6 q^{41} - 18 q^{42} + 28 q^{43} - 10 q^{44} - 8 q^{45} - 12 q^{46} + 20 q^{47} + 4 q^{48} + 28 q^{49} - 44 q^{50} + 10 q^{51} + 12 q^{53} - 10 q^{54} - 2 q^{55} + 2 q^{56} - 4 q^{57} - 4 q^{58} - 16 q^{59} + 4 q^{60} + 30 q^{61} + 4 q^{62} - 28 q^{63} + 14 q^{64} + 12 q^{66} - 2 q^{67} + 2 q^{68} - 42 q^{69} - 14 q^{70} - 44 q^{71} - 18 q^{72} + 36 q^{73} + 18 q^{74} + 46 q^{75} - 14 q^{76} + 68 q^{77} + 34 q^{79} + 2 q^{80} - 6 q^{81} + 6 q^{82} - 2 q^{83} + 18 q^{84} + 30 q^{85} - 28 q^{86} + 52 q^{87} + 10 q^{88} - 42 q^{89} + 8 q^{90} + 12 q^{92} + 12 q^{93} - 20 q^{94} - 2 q^{95} - 4 q^{96} - 40 q^{97} - 28 q^{98} - 12 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} - 4 x^{13} - 22 x^{12} + 98 x^{11} + 164 x^{10} - 912 x^{9} - 374 x^{8} + 3996 x^{7} - 817 x^{6} - 8194 x^{5} + 4418 x^{4} + 6430 x^{3} - 4327 x^{2} - 922 x + 358$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$($$$$-12288 \nu^{13} + 18169 \nu^{12} + 330650 \nu^{11} - 394551 \nu^{10} - 3378153 \nu^{9} + 3164999 \nu^{8} + 16113298 \nu^{7} - 11777770 \nu^{6} - 35391909 \nu^{5} + 21059379 \nu^{4} + 30371059 \nu^{3} - 14001130 \nu^{2} - 5719116 \nu + 991236$$$$)/86614$$ $$\beta_{4}$$ $$=$$ $$($$$$-29665 \nu^{13} + 68184 \nu^{12} + 769316 \nu^{11} - 1597128 \nu^{10} - 7605118 \nu^{9} + 14084748 \nu^{8} + 35348681 \nu^{7} - 58116578 \nu^{6} - 76347991 \nu^{5} + 112156792 \nu^{4} + 64153257 \nu^{3} - 80425246 \nu^{2} - 12158946 \nu + 6286514$$$$)/86614$$ $$\beta_{5}$$ $$=$$ $$($$$$-46556 \nu^{13} + 116698 \nu^{12} + 1206704 \nu^{11} - 2793341 \nu^{10} - 11956519 \nu^{9} + 25284751 \nu^{8} + 55979958 \nu^{7} - 107362733 \nu^{6} - 122850311 \nu^{5} + 212500508 \nu^{4} + 106711973 \nu^{3} - 155423191 \nu^{2} - 24035732 \nu + 11682236$$$$)/86614$$ $$\beta_{6}$$ $$=$$ $$($$$$-121238 \nu^{13} + 289609 \nu^{12} + 3135370 \nu^{11} - 6831637 \nu^{10} - 30930578 \nu^{9} + 60777951 \nu^{8} + 143673826 \nu^{7} - 253328884 \nu^{6} - 310942466 \nu^{5} + 493617047 \nu^{4} + 263101878 \nu^{3} - 357034414 \nu^{2} - 52300506 \nu + 27053054$$$$)/86614$$ $$\beta_{7}$$ $$=$$ $$($$$$-19670 \nu^{13} + 47465 \nu^{12} + 508271 \nu^{11} - 1122043 \nu^{10} - 5009566 \nu^{9} + 10008484 \nu^{8} + 23246303 \nu^{7} - 41841302 \nu^{6} - 50253903 \nu^{5} + 81761376 \nu^{4} + 42491414 \nu^{3} - 59275807 \nu^{2} - 8603972 \nu + 4454632$$$$)/7874$$ $$\beta_{8}$$ $$=$$ $$($$$$-31215 \nu^{13} + 75531 \nu^{12} + 805617 \nu^{11} - 1783686 \nu^{10} - 7930556 \nu^{9} + 15889723 \nu^{8} + 36760018 \nu^{7} - 66324293 \nu^{6} - 79414604 \nu^{5} + 129393474 \nu^{4} + 67202293 \nu^{3} - 93716062 \nu^{2} - 13681108 \nu + 7041860$$$$)/7874$$ $$\beta_{9}$$ $$=$$ $$($$$$-44605 \nu^{13} + 107910 \nu^{12} + 1152434 \nu^{11} - 2551874 \nu^{10} - 11358734 \nu^{9} + 22772210 \nu^{8} + 52724709 \nu^{7} - 95238096 \nu^{6} - 114069083 \nu^{5} + 186127846 \nu^{4} + 96588709 \nu^{3} - 134933996 \nu^{2} - 19617176 \nu + 10141326$$$$)/7874$$ $$\beta_{10}$$ $$=$$ $$($$$$-50702 \nu^{13} + 123330 \nu^{12} + 1307720 \nu^{11} - 2914366 \nu^{10} - 12865611 \nu^{9} + 25980696 \nu^{8} + 59608636 \nu^{7} - 108511602 \nu^{6} - 128760265 \nu^{5} + 211741694 \nu^{4} + 108993933 \nu^{3} - 153326664 \nu^{2} - 22280520 \nu + 11484056$$$$)/7874$$ $$\beta_{11}$$ $$=$$ $$($$$$596839 \nu^{13} - 1447234 \nu^{12} - 15406854 \nu^{11} + 34205757 \nu^{10} + 151712350 \nu^{9} - 305018716 \nu^{8} - 703546583 \nu^{7} + 1274464239 \nu^{6} + 1520957397 \nu^{5} - 2488216335 \nu^{4} - 1288214635 \nu^{3} + 1802243080 \nu^{2} + 263164832 \nu - 134814398$$$$)/86614$$ $$\beta_{12}$$ $$=$$ $$($$$$-654970 \nu^{13} + 1595493 \nu^{12} + 16902234 \nu^{11} - 37743422 \nu^{10} - 166397479 \nu^{9} + 336919665 \nu^{8} + 771594804 \nu^{7} - 1409367343 \nu^{6} - 1668547025 \nu^{5} + 2754459708 \nu^{4} + 1414591507 \nu^{3} - 1997491672 \nu^{2} - 290762638 \nu + 149804500$$$$)/86614$$ $$\beta_{13}$$ $$=$$ $$($$$$850313 \nu^{13} - 2063627 \nu^{12} - 21951921 \nu^{11} + 48790968 \nu^{10} + 216188584 \nu^{9} - 435250479 \nu^{8} - 1002732968 \nu^{7} + 1819361255 \nu^{6} + 2168354150 \nu^{5} - 3553191070 \nu^{4} - 1837117433 \nu^{3} + 2574389004 \nu^{2} + 375317792 \nu - 192947862$$$$)/86614$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{13} + \beta_{10} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_{2} + 7 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{11} + \beta_{10} + \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + 10 \beta_{2} + \beta_{1} + 27$$ $$\nu^{5}$$ $$=$$ $$10 \beta_{13} - \beta_{12} + 3 \beta_{11} + 13 \beta_{10} - \beta_{9} + 14 \beta_{7} + 3 \beta_{6} + 12 \beta_{5} + 8 \beta_{4} + 16 \beta_{2} + 52 \beta_{1} + 11$$ $$\nu^{6}$$ $$=$$ $$-2 \beta_{13} - 5 \beta_{12} + 31 \beta_{11} + 20 \beta_{10} - 3 \beta_{9} + 12 \beta_{8} + 22 \beta_{7} + 4 \beta_{6} + 18 \beta_{5} + 6 \beta_{4} + \beta_{3} + 97 \beta_{2} + 19 \beta_{1} + 224$$ $$\nu^{7}$$ $$=$$ $$80 \beta_{13} - 22 \beta_{12} + 64 \beta_{11} + 145 \beta_{10} - 23 \beta_{9} + 4 \beta_{8} + 166 \beta_{7} + 53 \beta_{6} + 128 \beta_{5} + 45 \beta_{4} + 3 \beta_{3} + 202 \beta_{2} + 406 \beta_{1} + 223$$ $$\nu^{8}$$ $$=$$ $$-34 \beta_{13} - 100 \beta_{12} + 392 \beta_{11} + 291 \beta_{10} - 66 \beta_{9} + 127 \beta_{8} + 341 \beta_{7} + 96 \beta_{6} + 251 \beta_{5} - 11 \beta_{4} + 20 \beta_{3} + 971 \beta_{2} + 257 \beta_{1} + 2069$$ $$\nu^{9}$$ $$=$$ $$587 \beta_{13} - 349 \beta_{12} + 971 \beta_{11} + 1582 \beta_{10} - 363 \beta_{9} + 102 \beta_{8} + 1913 \beta_{7} + 711 \beta_{6} + 1363 \beta_{5} + 125 \beta_{4} + 60 \beta_{3} + 2389 \beta_{2} + 3307 \beta_{1} + 3241$$ $$\nu^{10}$$ $$=$$ $$-426 \beta_{13} - 1459 \beta_{12} + 4657 \beta_{11} + 3773 \beta_{10} - 1047 \beta_{9} + 1341 \beta_{8} + 4631 \beta_{7} + 1580 \beta_{6} + 3207 \beta_{5} - 825 \beta_{4} + 279 \beta_{3} + 10063 \beta_{2} + 3098 \beta_{1} + 20367$$ $$\nu^{11}$$ $$=$$ $$4032 \beta_{13} - 4823 \beta_{12} + 12881 \beta_{11} + 17306 \beta_{10} - 4940 \beta_{9} + 1758 \beta_{8} + 21874 \beta_{7} + 8740 \beta_{6} + 14758 \beta_{5} - 1565 \beta_{4} + 845 \beta_{3} + 27598 \beta_{2} + 28072 \beta_{1} + 41712$$ $$\nu^{12}$$ $$=$$ $$-4890 \beta_{13} - 18955 \beta_{12} + 53935 \beta_{11} + 46265 \beta_{10} - 14571 \beta_{9} + 14385 \beta_{8} + 58835 \beta_{7} + 22248 \beta_{6} + 39237 \beta_{5} - 14879 \beta_{4} + 3393 \beta_{3} + 107196 \beta_{2} + 35660 \beta_{1} + 209179$$ $$\nu^{13}$$ $$=$$ $$25301 \beta_{13} - 61947 \beta_{12} + 159847 \beta_{11} + 190687 \beta_{10} - 62464 \beta_{9} + 25644 \beta_{8} + 249263 \beta_{7} + 103630 \beta_{6} + 162315 \beta_{5} - 40418 \beta_{4} + 10421 \beta_{3} + 315633 \beta_{2} + 248635 \beta_{1} + 507482$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.77993 −2.64536 −2.07634 −2.02205 −1.17869 −0.343703 0.231327 1.30931 1.50159 1.57102 1.58539 2.66506 2.81805 3.36433
−1.00000 −2.77993 1.00000 −0.764778 2.77993 0.951982 −1.00000 4.72802 0.764778
1.2 −1.00000 −2.64536 1.00000 3.69209 2.64536 −4.07315 −1.00000 3.99793 −3.69209
1.3 −1.00000 −2.07634 1.00000 −2.52870 2.07634 −3.77437 −1.00000 1.31117 2.52870
1.4 −1.00000 −2.02205 1.00000 −3.58162 2.02205 −0.649114 −1.00000 1.08868 3.58162
1.5 −1.00000 −1.17869 1.00000 0.970755 1.17869 1.94107 −1.00000 −1.61069 −0.970755
1.6 −1.00000 −0.343703 1.00000 2.06376 0.343703 −1.08834 −1.00000 −2.88187 −2.06376
1.7 −1.00000 0.231327 1.00000 2.62395 −0.231327 2.76976 −1.00000 −2.94649 −2.62395
1.8 −1.00000 1.30931 1.00000 4.16024 −1.30931 4.89391 −1.00000 −1.28570 −4.16024
1.9 −1.00000 1.50159 1.00000 −2.98870 −1.50159 −3.73146 −1.00000 −0.745218 2.98870
1.10 −1.00000 1.57102 1.00000 −2.81950 −1.57102 4.61884 −1.00000 −0.531896 2.81950
1.11 −1.00000 1.58539 1.00000 −2.09051 −1.58539 −3.27454 −1.00000 −0.486543 2.09051
1.12 −1.00000 2.66506 1.00000 2.40281 −2.66506 0.679986 −1.00000 4.10252 −2.40281
1.13 −1.00000 2.81805 1.00000 3.87767 −2.81805 −2.87057 −1.00000 4.94139 −3.87767
1.14 −1.00000 3.36433 1.00000 −3.01748 −3.36433 1.60598 −1.00000 8.31869 3.01748
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$13$$ $$-1$$
$$19$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6422.2.a.bm 14
13.b even 2 1 6422.2.a.bn 14
13.f odd 12 2 494.2.m.b 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.m.b 28 13.f odd 12 2
6422.2.a.bm 14 1.a even 1 1 trivial
6422.2.a.bn 14 13.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6422))$$:

 $$T_{3}^{14} - \cdots$$ $$T_{5}^{14} - \cdots$$ $$T_{7}^{14} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{14}$$
$3$ $$358 - 922 T - 4327 T^{2} + 6430 T^{3} + 4418 T^{4} - 8194 T^{5} - 817 T^{6} + 3996 T^{7} - 374 T^{8} - 912 T^{9} + 164 T^{10} + 98 T^{11} - 22 T^{12} - 4 T^{13} + T^{14}$$
$5$ $$-276992 - 110848 T + 577856 T^{2} + 129888 T^{3} - 347888 T^{4} - 58424 T^{5} + 97813 T^{6} + 13004 T^{7} - 14806 T^{8} - 1516 T^{9} + 1245 T^{10} + 88 T^{11} - 55 T^{12} - 2 T^{13} + T^{14}$$
$7$ $$-48128 + 26432 T + 179785 T^{2} - 97736 T^{3} - 186813 T^{4} + 87970 T^{5} + 71809 T^{6} - 24912 T^{7} - 13845 T^{8} + 2842 T^{9} + 1357 T^{10} - 130 T^{11} - 61 T^{12} + 2 T^{13} + T^{14}$$
$11$ $$62464 + 164352 T - 2061440 T^{2} - 4188096 T^{3} + 4588768 T^{4} + 2921120 T^{5} - 650314 T^{6} - 473854 T^{7} + 18751 T^{8} + 31358 T^{9} + 1129 T^{10} - 922 T^{11} - 70 T^{12} + 10 T^{13} + T^{14}$$
$13$ $$T^{14}$$
$17$ $$-627200 + 6289920 T + 9202176 T^{2} - 2883168 T^{3} - 4806280 T^{4} + 344244 T^{5} + 965093 T^{6} + 4948 T^{7} - 94262 T^{8} - 3006 T^{9} + 4703 T^{10} + 154 T^{11} - 113 T^{12} - 2 T^{13} + T^{14}$$
$19$ $$( 1 + T )^{14}$$
$23$ $$931408 - 8208944 T - 3520331 T^{2} + 10450054 T^{3} + 2372262 T^{4} - 4723576 T^{5} - 426755 T^{6} + 888754 T^{7} - 9982 T^{8} - 60066 T^{9} + 2950 T^{10} + 1522 T^{11} - 108 T^{12} - 12 T^{13} + T^{14}$$
$29$ $$50879749 - 743165236 T + 201431605 T^{2} + 326900316 T^{3} - 101806975 T^{4} - 38800614 T^{5} + 12635025 T^{6} + 2064370 T^{7} - 677436 T^{8} - 56096 T^{9} + 17682 T^{10} + 762 T^{11} - 217 T^{12} - 4 T^{13} + T^{14}$$
$31$ $$-522788864 - 4379648 T + 324467264 T^{2} - 15495104 T^{3} - 72760864 T^{4} + 5489328 T^{5} + 7808668 T^{6} - 653676 T^{7} - 437827 T^{8} + 32794 T^{9} + 13070 T^{10} - 676 T^{11} - 193 T^{12} + 4 T^{13} + T^{14}$$
$37$ $$-156608 - 20427584 T - 607466656 T^{2} - 166342608 T^{3} + 270219420 T^{4} + 55216980 T^{5} - 29582079 T^{6} - 6762666 T^{7} + 930424 T^{8} + 284234 T^{9} - 2466 T^{10} - 4086 T^{11} - 148 T^{12} + 18 T^{13} + T^{14}$$
$41$ $$-872071616 - 1351582336 T + 1542754496 T^{2} + 647652656 T^{3} - 639338476 T^{4} - 19257280 T^{5} + 64871857 T^{6} - 3279220 T^{7} - 2326495 T^{8} + 129940 T^{9} + 39547 T^{10} - 1584 T^{11} - 324 T^{12} + 6 T^{13} + T^{14}$$
$43$ $$-53655128576 + 55712529152 T + 10258889664 T^{2} - 17085097088 T^{3} + 1980849936 T^{4} + 1248473392 T^{5} - 323013328 T^{6} - 2649324 T^{7} + 8032953 T^{8} - 606152 T^{9} - 54693 T^{10} + 7952 T^{11} - 45 T^{12} - 28 T^{13} + T^{14}$$
$47$ $$2200916878 + 358761806 T - 1366310225 T^{2} - 89350434 T^{3} + 297573157 T^{4} - 8689962 T^{5} - 28315932 T^{6} + 2860002 T^{7} + 1147875 T^{8} - 185312 T^{9} - 14424 T^{10} + 4016 T^{11} - 90 T^{12} - 20 T^{13} + T^{14}$$
$53$ $$-13585632587 + 22156722204 T - 9251397210 T^{2} - 2365270610 T^{3} + 2548115290 T^{4} - 411145434 T^{5} - 89018270 T^{6} + 27756324 T^{7} + 127312 T^{8} - 566766 T^{9} + 26876 T^{10} + 4532 T^{11} - 318 T^{12} - 12 T^{13} + T^{14}$$
$59$ $$-4563741338 + 5792428590 T - 359029703 T^{2} - 1569556344 T^{3} + 171920614 T^{4} + 185877740 T^{5} - 8377735 T^{6} - 10260292 T^{7} - 36404 T^{8} + 277154 T^{9} + 9412 T^{10} - 3502 T^{11} - 184 T^{12} + 16 T^{13} + T^{14}$$
$61$ $$14608783168 + 61563023424 T - 35585462016 T^{2} - 23874740080 T^{3} + 10647913612 T^{4} + 446752240 T^{5} - 552885635 T^{6} + 29491952 T^{7} + 10103720 T^{8} - 1016356 T^{9} - 59575 T^{10} + 10202 T^{11} - 85 T^{12} - 30 T^{13} + T^{14}$$
$67$ $$-27573248 + 1468369920 T + 300892672 T^{2} - 1228141376 T^{3} - 285371248 T^{4} + 152127144 T^{5} + 37725254 T^{6} - 6488286 T^{7} - 1745789 T^{8} + 116824 T^{9} + 35337 T^{10} - 874 T^{11} - 316 T^{12} + 2 T^{13} + T^{14}$$
$71$ $$8820874624 + 23819127808 T + 12310028352 T^{2} - 2868314400 T^{3} - 3809225880 T^{4} - 864123400 T^{5} + 54806898 T^{6} + 44821022 T^{7} + 4031915 T^{8} - 508812 T^{9} - 103955 T^{10} - 2758 T^{11} + 533 T^{12} + 44 T^{13} + T^{14}$$
$73$ $$27490129 - 240465380 T + 28738397 T^{2} + 319740136 T^{3} + 64982148 T^{4} - 84656576 T^{5} - 23259817 T^{6} + 6840596 T^{7} + 1973793 T^{8} - 204648 T^{9} - 45021 T^{10} + 4564 T^{11} + 226 T^{12} - 36 T^{13} + T^{14}$$
$79$ $$15247371136 + 16792750592 T - 6162248192 T^{2} - 6404019712 T^{3} + 1709907176 T^{4} + 731139144 T^{5} - 242841166 T^{6} - 8984062 T^{7} + 8615635 T^{8} - 505196 T^{9} - 81431 T^{10} + 8772 T^{11} + 70 T^{12} - 34 T^{13} + T^{14}$$
$83$ $$-67167404672 + 186847554944 T - 72412302560 T^{2} - 51795961952 T^{3} + 10591282944 T^{4} + 6524659168 T^{5} + 273176726 T^{6} - 159810706 T^{7} - 14891255 T^{8} + 1250100 T^{9} + 163126 T^{10} - 3416 T^{11} - 681 T^{12} + 2 T^{13} + T^{14}$$
$89$ $$7593700864 + 3420007168 T - 13003435584 T^{2} - 1395735360 T^{3} + 3913364752 T^{4} + 14625488 T^{5} - 272979692 T^{6} - 3698868 T^{7} + 7868533 T^{8} + 305290 T^{9} - 97470 T^{10} - 6658 T^{11} + 347 T^{12} + 42 T^{13} + T^{14}$$
$97$ $$-474544504832 + 520276099072 T + 511191347200 T^{2} - 242449334272 T^{3} - 250308727808 T^{4} - 65546801152 T^{5} - 5782349568 T^{6} + 306125056 T^{7} + 87930432 T^{8} + 3599488 T^{9} - 278064 T^{10} - 24992 T^{11} - 108 T^{12} + 40 T^{13} + T^{14}$$