# Properties

 Label 150.2.h.a Level 150 Weight 2 Character orbit 150.h Analytic conductor 1.198 Analytic rank 0 Dimension 8 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 150.h (of order $$10$$ and degree $$4$$)

## Newform invariants

 Self dual: No Analytic conductor: $$1.19775603032$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{20})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{20}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ \zeta_{20} q^{2}$$ $$-\zeta_{20}^{7} q^{3}$$ $$+ \zeta_{20}^{2} q^{4}$$ $$+ ( \zeta_{20} - \zeta_{20}^{2} - \zeta_{20}^{4} - \zeta_{20}^{5} ) q^{5}$$ $$+ ( 1 - \zeta_{20}^{2} + \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{6}$$ $$+ ( -1 + 2 \zeta_{20}^{2} - \zeta_{20}^{5} + 2 \zeta_{20}^{6} ) q^{7}$$ $$+ \zeta_{20}^{3} q^{8}$$ $$-\zeta_{20}^{4} q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \zeta_{20} q^{2}$$ $$-\zeta_{20}^{7} q^{3}$$ $$+ \zeta_{20}^{2} q^{4}$$ $$+ ( \zeta_{20} - \zeta_{20}^{2} - \zeta_{20}^{4} - \zeta_{20}^{5} ) q^{5}$$ $$+ ( 1 - \zeta_{20}^{2} + \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{6}$$ $$+ ( -1 + 2 \zeta_{20}^{2} - \zeta_{20}^{5} + 2 \zeta_{20}^{6} ) q^{7}$$ $$+ \zeta_{20}^{3} q^{8}$$ $$-\zeta_{20}^{4} q^{9}$$ $$+ ( \zeta_{20}^{2} - \zeta_{20}^{3} - \zeta_{20}^{5} - \zeta_{20}^{6} ) q^{10}$$ $$+ ( 1 - \zeta_{20}^{2} - \zeta_{20}^{5} + 2 \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{11}$$ $$+ ( \zeta_{20} - \zeta_{20}^{3} + \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{12}$$ $$+ ( -4 + \zeta_{20} + 2 \zeta_{20}^{2} - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{13}$$ $$+ ( -\zeta_{20} + 2 \zeta_{20}^{3} - \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{14}$$ $$+ ( 1 - 2 \zeta_{20} - 2 \zeta_{20}^{2} + \zeta_{20}^{3} + \zeta_{20}^{4} - \zeta_{20}^{5} - \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{15}$$ $$+ \zeta_{20}^{4} q^{16}$$ $$+ ( 1 + \zeta_{20} + \zeta_{20}^{2} - 2 \zeta_{20}^{3} + \zeta_{20}^{4} + \zeta_{20}^{5} + \zeta_{20}^{6} ) q^{17}$$ $$-\zeta_{20}^{5} q^{18}$$ $$+ ( -2 - 3 \zeta_{20} + \zeta_{20}^{2} + \zeta_{20}^{3} - \zeta_{20}^{4} + \zeta_{20}^{5} + 2 \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{19}$$ $$+ ( \zeta_{20}^{3} - \zeta_{20}^{4} - \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{20}$$ $$+ ( 2 \zeta_{20} - \zeta_{20}^{2} + 2 \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{21}$$ $$+ ( 1 + \zeta_{20} - \zeta_{20}^{2} - \zeta_{20}^{3} + \zeta_{20}^{4} - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{22}$$ $$+ ( 2 + \zeta_{20} - 4 \zeta_{20}^{2} + 6 \zeta_{20}^{4} + 2 \zeta_{20}^{5} - 3 \zeta_{20}^{6} - 2 \zeta_{20}^{7} ) q^{23}$$ $$+ q^{24}$$ $$+ ( -2 - 2 \zeta_{20} + 2 \zeta_{20}^{2} - 4 \zeta_{20}^{5} + \zeta_{20}^{6} + 4 \zeta_{20}^{7} ) q^{25}$$ $$+ ( 1 - 4 \zeta_{20} + 2 \zeta_{20}^{3} + \zeta_{20}^{4} - 2 \zeta_{20}^{5} - \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{26}$$ $$-\zeta_{20} q^{27}$$ $$+ ( -2 + \zeta_{20}^{2} + 2 \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{28}$$ $$+ ( -3 - 2 \zeta_{20}^{2} - 6 \zeta_{20}^{3} - 3 \zeta_{20}^{4} + 6 \zeta_{20}^{5} - 3 \zeta_{20}^{7} ) q^{29}$$ $$+ ( -1 + \zeta_{20} - \zeta_{20}^{2} - 2 \zeta_{20}^{3} + \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{30}$$ $$+ ( 6 - \zeta_{20} - 3 \zeta_{20}^{2} - \zeta_{20}^{3} + 3 \zeta_{20}^{4} + 3 \zeta_{20}^{5} - 6 \zeta_{20}^{6} - 2 \zeta_{20}^{7} ) q^{31}$$ $$+ \zeta_{20}^{5} q^{32}$$ $$+ ( -\zeta_{20} - \zeta_{20}^{2} + 3 \zeta_{20}^{3} - \zeta_{20}^{4} - \zeta_{20}^{5} ) q^{33}$$ $$+ ( \zeta_{20} + \zeta_{20}^{2} + \zeta_{20}^{3} - 2 \zeta_{20}^{4} + \zeta_{20}^{5} + \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{34}$$ $$+ ( 3 - \zeta_{20}^{2} + 3 \zeta_{20}^{3} + \zeta_{20}^{4} - 5 \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{35}$$ $$-\zeta_{20}^{6} q^{36}$$ $$+ ( -4 - 2 \zeta_{20} + 4 \zeta_{20}^{2} - \zeta_{20}^{3} - 2 \zeta_{20}^{4} + \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{37}$$ $$+ ( -2 - 2 \zeta_{20} - \zeta_{20}^{2} + \zeta_{20}^{3} - \zeta_{20}^{4} - \zeta_{20}^{5} + 3 \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{38}$$ $$+ ( 1 - \zeta_{20}^{2} + 2 \zeta_{20}^{5} - \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{39}$$ $$+ ( 1 - \zeta_{20}^{2} + 2 \zeta_{20}^{4} - \zeta_{20}^{5} - \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{40}$$ $$+ ( -\zeta_{20} + 3 \zeta_{20}^{2} - \zeta_{20}^{3} - 5 \zeta_{20}^{4} - \zeta_{20}^{5} + 3 \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{41}$$ $$+ ( 1 + \zeta_{20}^{2} - \zeta_{20}^{3} + \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{42}$$ $$+ ( -2 + 4 \zeta_{20}^{2} + 5 \zeta_{20}^{3} - 3 \zeta_{20}^{4} + \zeta_{20}^{5} + \zeta_{20}^{6} + 5 \zeta_{20}^{7} ) q^{43}$$ $$+ ( -2 + \zeta_{20} + 3 \zeta_{20}^{2} - \zeta_{20}^{3} - 3 \zeta_{20}^{4} + \zeta_{20}^{5} + 2 \zeta_{20}^{6} - 2 \zeta_{20}^{7} ) q^{44}$$ $$+ ( -1 - \zeta_{20} + \zeta_{20}^{2} + \zeta_{20}^{3} - \zeta_{20}^{4} - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{45}$$ $$+ ( 2 + 2 \zeta_{20} - \zeta_{20}^{2} - 4 \zeta_{20}^{3} + 2 \zeta_{20}^{4} + 6 \zeta_{20}^{5} - 3 \zeta_{20}^{7} ) q^{46}$$ $$+ ( 1 + 5 \zeta_{20} - 5 \zeta_{20}^{3} - \zeta_{20}^{4} - 5 \zeta_{20}^{7} ) q^{47}$$ $$+ \zeta_{20} q^{48}$$ $$+ ( -1 + 4 \zeta_{20} - 4 \zeta_{20}^{4} + 2 \zeta_{20}^{5} + 4 \zeta_{20}^{6} - 4 \zeta_{20}^{7} ) q^{49}$$ $$+ ( -4 - 2 \zeta_{20} + 2 \zeta_{20}^{2} + 2 \zeta_{20}^{3} - 4 \zeta_{20}^{4} + \zeta_{20}^{7} ) q^{50}$$ $$+ ( -1 + 2 \zeta_{20} + \zeta_{20}^{4} + \zeta_{20}^{5} - \zeta_{20}^{6} - 2 \zeta_{20}^{7} ) q^{51}$$ $$+ ( -2 + \zeta_{20} - 2 \zeta_{20}^{2} + \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{52}$$ $$+ ( -\zeta_{20} - 3 \zeta_{20}^{2} + \zeta_{20}^{3} + 6 \zeta_{20}^{4} - 6 \zeta_{20}^{6} + 4 \zeta_{20}^{7} ) q^{53}$$ $$-\zeta_{20}^{2} q^{54}$$ $$+ ( 4 - 5 \zeta_{20}^{2} + \zeta_{20}^{3} + 3 \zeta_{20}^{4} - 3 \zeta_{20}^{5} - 3 \zeta_{20}^{6} + 6 \zeta_{20}^{7} ) q^{55}$$ $$+ ( 1 - 2 \zeta_{20} - \zeta_{20}^{2} + \zeta_{20}^{3} + \zeta_{20}^{4} - \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{56}$$ $$+ ( -2 + 4 \zeta_{20}^{2} + \zeta_{20}^{3} - \zeta_{20}^{4} + \zeta_{20}^{5} + 3 \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{57}$$ $$+ ( 3 - 3 \zeta_{20} - 3 \zeta_{20}^{2} - 2 \zeta_{20}^{3} - 3 \zeta_{20}^{4} - 3 \zeta_{20}^{5} + 3 \zeta_{20}^{6} ) q^{58}$$ $$+ ( -2 \zeta_{20}^{2} - 6 \zeta_{20}^{3} + 6 \zeta_{20}^{4} - 6 \zeta_{20}^{5} - 2 \zeta_{20}^{6} ) q^{59}$$ $$+ ( 1 - \zeta_{20} - \zeta_{20}^{3} - \zeta_{20}^{4} ) q^{60}$$ $$+ ( \zeta_{20} - 2 \zeta_{20}^{3} - 5 \zeta_{20}^{5} - 7 \zeta_{20}^{7} ) q^{61}$$ $$+ ( 2 + 6 \zeta_{20} - 3 \zeta_{20}^{2} - 3 \zeta_{20}^{3} + \zeta_{20}^{4} + 3 \zeta_{20}^{5} + \zeta_{20}^{6} - 6 \zeta_{20}^{7} ) q^{62}$$ $$+ ( 2 - \zeta_{20} + \zeta_{20}^{3} + \zeta_{20}^{4} - \zeta_{20}^{5} - 2 \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{63}$$ $$+ \zeta_{20}^{6} q^{64}$$ $$+ ( 3 - 6 \zeta_{20} + 3 \zeta_{20}^{2} + 4 \zeta_{20}^{3} + 3 \zeta_{20}^{4} - 2 \zeta_{20}^{5} - 2 \zeta_{20}^{6} + 3 \zeta_{20}^{7} ) q^{65}$$ $$+ ( -\zeta_{20}^{2} - \zeta_{20}^{3} + 3 \zeta_{20}^{4} - \zeta_{20}^{5} - \zeta_{20}^{6} ) q^{66}$$ $$+ ( 1 - 5 \zeta_{20} + 4 \zeta_{20}^{2} + 6 \zeta_{20}^{3} + 4 \zeta_{20}^{4} - 5 \zeta_{20}^{5} + \zeta_{20}^{6} ) q^{67}$$ $$+ ( -1 + 2 \zeta_{20}^{2} + \zeta_{20}^{3} - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{68}$$ $$+ ( 1 + 2 \zeta_{20} + \zeta_{20}^{2} + \zeta_{20}^{3} - \zeta_{20}^{4} - 4 \zeta_{20}^{5} - \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{69}$$ $$+ ( -2 + 3 \zeta_{20} + 2 \zeta_{20}^{2} - \zeta_{20}^{3} + \zeta_{20}^{4} + \zeta_{20}^{5} + 2 \zeta_{20}^{6} - 5 \zeta_{20}^{7} ) q^{70}$$ $$+ ( 1 - \zeta_{20} + 7 \zeta_{20}^{2} + 3 \zeta_{20}^{3} + \zeta_{20}^{4} - 4 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{71}$$ $$-\zeta_{20}^{7} q^{72}$$ $$+ ( -1 + 4 \zeta_{20} - 3 \zeta_{20}^{2} + 2 \zeta_{20}^{4} + 4 \zeta_{20}^{5} - \zeta_{20}^{6} - 4 \zeta_{20}^{7} ) q^{73}$$ $$+ ( -2 - 4 \zeta_{20} + 4 \zeta_{20}^{3} - 3 \zeta_{20}^{4} - 2 \zeta_{20}^{5} + 3 \zeta_{20}^{6} ) q^{74}$$ $$+ ( -2 + 2 \zeta_{20} - 2 \zeta_{20}^{2} - \zeta_{20}^{3} + 2 \zeta_{20}^{4} + 2 \zeta_{20}^{5} + 2 \zeta_{20}^{6} ) q^{75}$$ $$+ ( -2 - 2 \zeta_{20} - \zeta_{20}^{3} - \zeta_{20}^{4} - \zeta_{20}^{5} + \zeta_{20}^{6} + 3 \zeta_{20}^{7} ) q^{76}$$ $$+ ( -4 + 6 \zeta_{20} - 4 \zeta_{20}^{4} + 2 \zeta_{20}^{5} + 2 \zeta_{20}^{6} - 2 \zeta_{20}^{7} ) q^{77}$$ $$+ ( -2 + \zeta_{20} + 2 \zeta_{20}^{2} - \zeta_{20}^{3} - 2 \zeta_{20}^{4} + 4 \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{78}$$ $$+ ( 6 + 2 \zeta_{20} - 10 \zeta_{20}^{2} + 2 \zeta_{20}^{3} + 6 \zeta_{20}^{4} ) q^{79}$$ $$+ ( 1 + \zeta_{20} - \zeta_{20}^{2} - \zeta_{20}^{3} + \zeta_{20}^{4} + 2 \zeta_{20}^{5} - 2 \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{80}$$ $$+ ( -1 + \zeta_{20}^{2} - \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{81}$$ $$+ ( 1 - 2 \zeta_{20}^{2} + 3 \zeta_{20}^{3} - 5 \zeta_{20}^{5} - 2 \zeta_{20}^{6} + 3 \zeta_{20}^{7} ) q^{82}$$ $$+ ( 7 + \zeta_{20}^{2} + 3 \zeta_{20}^{3} + \zeta_{20}^{4} + 7 \zeta_{20}^{6} ) q^{83}$$ $$+ ( \zeta_{20} + \zeta_{20}^{3} - \zeta_{20}^{4} + \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{84}$$ $$+ ( 2 + 4 \zeta_{20} - 2 \zeta_{20}^{3} - 4 \zeta_{20}^{4} + 3 \zeta_{20}^{5} - 2 \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{85}$$ $$+ ( -5 - 2 \zeta_{20} + 5 \zeta_{20}^{2} + 4 \zeta_{20}^{3} - 3 \zeta_{20}^{5} + 6 \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{86}$$ $$+ ( -6 - 5 \zeta_{20} + 6 \zeta_{20}^{2} + 2 \zeta_{20}^{3} - 3 \zeta_{20}^{4} - 2 \zeta_{20}^{5} + 5 \zeta_{20}^{7} ) q^{87}$$ $$+ ( 2 - 2 \zeta_{20} - \zeta_{20}^{2} + 3 \zeta_{20}^{3} + \zeta_{20}^{4} - 3 \zeta_{20}^{5} - \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{88}$$ $$+ ( -4 + 3 \zeta_{20} + 4 \zeta_{20}^{2} - 6 \zeta_{20}^{3} + \zeta_{20}^{5} - 5 \zeta_{20}^{6} - 5 \zeta_{20}^{7} ) q^{89}$$ $$+ ( -1 - \zeta_{20} + \zeta_{20}^{3} - \zeta_{20}^{5} - \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{90}$$ $$+ ( \zeta_{20} - 7 \zeta_{20}^{2} + 4 \zeta_{20}^{3} - 2 \zeta_{20}^{4} + 4 \zeta_{20}^{5} - 7 \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{91}$$ $$+ ( 3 + 2 \zeta_{20} - \zeta_{20}^{2} - \zeta_{20}^{3} - \zeta_{20}^{4} + 2 \zeta_{20}^{5} + 3 \zeta_{20}^{6} ) q^{92}$$ $$+ ( -2 + 4 \zeta_{20}^{2} - 3 \zeta_{20}^{3} - 3 \zeta_{20}^{4} - 3 \zeta_{20}^{5} + \zeta_{20}^{6} - 3 \zeta_{20}^{7} ) q^{93}$$ $$+ ( 5 + \zeta_{20} - \zeta_{20}^{5} - 5 \zeta_{20}^{6} ) q^{94}$$ $$+ ( 3 + 4 \zeta_{20} + \zeta_{20}^{2} + 2 \zeta_{20}^{3} + 2 \zeta_{20}^{4} + 5 \zeta_{20}^{5} + 4 \zeta_{20}^{6} - 3 \zeta_{20}^{7} ) q^{95}$$ $$+ \zeta_{20}^{2} q^{96}$$ $$+ ( 9 - 3 \zeta_{20}^{2} - 3 \zeta_{20}^{4} - 6 \zeta_{20}^{6} + 2 \zeta_{20}^{7} ) q^{97}$$ $$+ ( 4 - \zeta_{20} + 4 \zeta_{20}^{4} - 4 \zeta_{20}^{5} - 2 \zeta_{20}^{6} + 4 \zeta_{20}^{7} ) q^{98}$$ $$+ ( 2 - 2 \zeta_{20} + \zeta_{20}^{3} - \zeta_{20}^{4} - \zeta_{20}^{5} + \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q$$ $$\mathstrut +\mathstrut 2q^{4}$$ $$\mathstrut +\mathstrut 2q^{6}$$ $$\mathstrut +\mathstrut 2q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$8q$$ $$\mathstrut +\mathstrut 2q^{4}$$ $$\mathstrut +\mathstrut 2q^{6}$$ $$\mathstrut +\mathstrut 2q^{9}$$ $$\mathstrut +\mathstrut 10q^{11}$$ $$\mathstrut -\mathstrut 20q^{13}$$ $$\mathstrut -\mathstrut 2q^{14}$$ $$\mathstrut -\mathstrut 2q^{16}$$ $$\mathstrut +\mathstrut 10q^{17}$$ $$\mathstrut -\mathstrut 8q^{19}$$ $$\mathstrut -\mathstrut 2q^{21}$$ $$\mathstrut -\mathstrut 10q^{23}$$ $$\mathstrut +\mathstrut 8q^{24}$$ $$\mathstrut -\mathstrut 10q^{25}$$ $$\mathstrut +\mathstrut 4q^{26}$$ $$\mathstrut -\mathstrut 10q^{28}$$ $$\mathstrut -\mathstrut 22q^{29}$$ $$\mathstrut -\mathstrut 10q^{30}$$ $$\mathstrut +\mathstrut 24q^{31}$$ $$\mathstrut +\mathstrut 8q^{34}$$ $$\mathstrut +\mathstrut 10q^{35}$$ $$\mathstrut -\mathstrut 2q^{36}$$ $$\mathstrut -\mathstrut 20q^{37}$$ $$\mathstrut -\mathstrut 10q^{38}$$ $$\mathstrut +\mathstrut 4q^{39}$$ $$\mathstrut +\mathstrut 22q^{41}$$ $$\mathstrut +\mathstrut 10q^{42}$$ $$\mathstrut +\mathstrut 10q^{46}$$ $$\mathstrut +\mathstrut 10q^{47}$$ $$\mathstrut +\mathstrut 8q^{49}$$ $$\mathstrut -\mathstrut 20q^{50}$$ $$\mathstrut -\mathstrut 12q^{51}$$ $$\mathstrut -\mathstrut 20q^{52}$$ $$\mathstrut -\mathstrut 30q^{53}$$ $$\mathstrut -\mathstrut 2q^{54}$$ $$\mathstrut +\mathstrut 10q^{55}$$ $$\mathstrut +\mathstrut 2q^{56}$$ $$\mathstrut +\mathstrut 30q^{58}$$ $$\mathstrut -\mathstrut 20q^{59}$$ $$\mathstrut +\mathstrut 10q^{60}$$ $$\mathstrut +\mathstrut 10q^{62}$$ $$\mathstrut +\mathstrut 10q^{63}$$ $$\mathstrut +\mathstrut 2q^{64}$$ $$\mathstrut +\mathstrut 20q^{65}$$ $$\mathstrut -\mathstrut 10q^{66}$$ $$\mathstrut +\mathstrut 10q^{67}$$ $$\mathstrut +\mathstrut 10q^{69}$$ $$\mathstrut -\mathstrut 10q^{70}$$ $$\mathstrut +\mathstrut 20q^{71}$$ $$\mathstrut -\mathstrut 20q^{73}$$ $$\mathstrut -\mathstrut 4q^{74}$$ $$\mathstrut -\mathstrut 20q^{75}$$ $$\mathstrut -\mathstrut 12q^{76}$$ $$\mathstrut -\mathstrut 20q^{77}$$ $$\mathstrut +\mathstrut 16q^{79}$$ $$\mathstrut -\mathstrut 2q^{81}$$ $$\mathstrut +\mathstrut 70q^{83}$$ $$\mathstrut +\mathstrut 2q^{84}$$ $$\mathstrut +\mathstrut 20q^{85}$$ $$\mathstrut -\mathstrut 18q^{86}$$ $$\mathstrut -\mathstrut 30q^{87}$$ $$\mathstrut +\mathstrut 10q^{88}$$ $$\mathstrut -\mathstrut 34q^{89}$$ $$\mathstrut -\mathstrut 10q^{90}$$ $$\mathstrut -\mathstrut 24q^{91}$$ $$\mathstrut +\mathstrut 30q^{92}$$ $$\mathstrut +\mathstrut 30q^{94}$$ $$\mathstrut +\mathstrut 30q^{95}$$ $$\mathstrut +\mathstrut 2q^{96}$$ $$\mathstrut +\mathstrut 60q^{97}$$ $$\mathstrut +\mathstrut 20q^{98}$$ $$\mathstrut +\mathstrut 20q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/150\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$\zeta_{20}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −0.951057 + 0.309017i 0.951057 − 0.309017i −0.951057 − 0.309017i 0.951057 + 0.309017i −0.587785 + 0.809017i 0.587785 − 0.809017i −0.587785 − 0.809017i 0.587785 + 0.809017i
−0.951057 + 0.309017i −0.587785 0.809017i 0.809017 0.587785i −2.06909 + 0.847859i 0.809017 + 0.587785i 4.07768i −0.587785 + 0.809017i −0.309017 + 0.951057i 1.70582 1.44575i
19.2 0.951057 0.309017i 0.587785 + 0.809017i 0.809017 0.587785i −0.166977 + 2.22982i 0.809017 + 0.587785i 2.07768i 0.587785 0.809017i −0.309017 + 0.951057i 0.530249 + 2.17229i
79.1 −0.951057 0.309017i −0.587785 + 0.809017i 0.809017 + 0.587785i −2.06909 0.847859i 0.809017 0.587785i 4.07768i −0.587785 0.809017i −0.309017 0.951057i 1.70582 + 1.44575i
79.2 0.951057 + 0.309017i 0.587785 0.809017i 0.809017 + 0.587785i −0.166977 2.22982i 0.809017 0.587785i 2.07768i 0.587785 + 0.809017i −0.309017 0.951057i 0.530249 2.17229i
109.1 −0.587785 + 0.809017i 0.951057 0.309017i −0.309017 0.951057i 0.530249 + 2.17229i −0.309017 + 0.951057i 0.273457i 0.951057 + 0.309017i 0.809017 0.587785i −2.06909 0.847859i
109.2 0.587785 0.809017i −0.951057 + 0.309017i −0.309017 0.951057i 1.70582 1.44575i −0.309017 + 0.951057i 1.72654i −0.951057 0.309017i 0.809017 0.587785i −0.166977 2.22982i
139.1 −0.587785 0.809017i 0.951057 + 0.309017i −0.309017 + 0.951057i 0.530249 2.17229i −0.309017 0.951057i 0.273457i 0.951057 0.309017i 0.809017 + 0.587785i −2.06909 + 0.847859i
139.2 0.587785 + 0.809017i −0.951057 0.309017i −0.309017 + 0.951057i 1.70582 + 1.44575i −0.309017 0.951057i 1.72654i −0.951057 + 0.309017i 0.809017 + 0.587785i −0.166977 + 2.22982i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 139.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
25.e Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{7}^{8}$$ $$\mathstrut +\mathstrut 24 T_{7}^{6}$$ $$\mathstrut +\mathstrut 136 T_{7}^{4}$$ $$\mathstrut +\mathstrut 224 T_{7}^{2}$$ $$\mathstrut +\mathstrut 16$$ acting on $$S_{2}^{\mathrm{new}}(150, [\chi])$$.