# Properties

 Label 600.2.b.g Level 600 Weight 2 Character orbit 600.b Analytic conductor 4.791 Analytic rank 0 Dimension 12 CM No Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 600.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$4.79102412128$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: 12.0.537291533250985984.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12}\mathstrut -\mathstrut$$ $$5$$ $$x^{10}\mathstrut +\mathstrut$$ $$14$$ $$x^{8}\mathstrut -\mathstrut$$ $$30$$ $$x^{6}\mathstrut +\mathstrut$$ $$56$$ $$x^{4}\mathstrut -\mathstrut$$ $$80$$ $$x^{2}\mathstrut +\mathstrut$$ $$64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - 1$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{9} + \nu^{7} - 2 \nu^{5} + 6 \nu^{3}$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{10} - 5 \nu^{8} + 14 \nu^{6} - 14 \nu^{4} + 40 \nu^{2} - 48$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{10} + 5 \nu^{8} - 14 \nu^{6} + 30 \nu^{4} - 56 \nu^{2} + 64$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{11} + 6 \nu^{10} + \nu^{9} - 22 \nu^{8} - 10 \nu^{7} + 44 \nu^{6} + 22 \nu^{5} - 100 \nu^{4} - 32 \nu^{3} + 160 \nu^{2} + 16 \nu - 160$$$$)/64$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{11} + 5 \nu^{9} - 14 \nu^{7} + 30 \nu^{5} - 24 \nu^{3} + 16 \nu$$$$)/32$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{11} - 6 \nu^{10} + \nu^{9} + 22 \nu^{8} - 10 \nu^{7} - 44 \nu^{6} + 22 \nu^{5} + 100 \nu^{4} - 32 \nu^{3} - 160 \nu^{2} + 16 \nu + 160$$$$)/64$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{11} + 5 \nu^{9} - 14 \nu^{7} + 30 \nu^{5} - 56 \nu^{3} + 80 \nu$$$$)/32$$ $$\beta_{9}$$ $$=$$ $$($$$$-5 \nu^{11} + 2 \nu^{10} + 13 \nu^{9} - 18 \nu^{8} - 26 \nu^{7} + 36 \nu^{6} + 62 \nu^{5} - 76 \nu^{4} - 80 \nu^{3} + 160 \nu^{2} + 80 \nu - 224$$$$)/64$$ $$\beta_{10}$$ $$=$$ $$($$$$-5 \nu^{11} - 2 \nu^{10} + 13 \nu^{9} + 18 \nu^{8} - 26 \nu^{7} - 36 \nu^{6} + 62 \nu^{5} + 76 \nu^{4} - 80 \nu^{3} - 160 \nu^{2} + 80 \nu + 224$$$$)/64$$ $$\beta_{11}$$ $$=$$ $$($$$$\nu^{11} - 3 \nu^{9} + 8 \nu^{7} - 14 \nu^{5} + 20 \nu^{3} - 24 \nu$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{8}\mathstrut -\mathstrut$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$1$$ $$\nu^{3}$$ $$=$$ $$-$$$$\beta_{7}\mathstrut +\mathstrut$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{2}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{1}$$ $$\nu^{5}$$ $$=$$ $$2$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$\beta_{2}$$ $$\nu^{6}$$ $$=$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{1}$$ $$\nu^{7}$$ $$=$$ $$4$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$\beta_{5}$$ $$\nu^{8}$$ $$=$$ $$5$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$5$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$8$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$6$$ $$\nu^{9}$$ $$=$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$7$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$7$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{2}$$ $$\nu^{10}$$ $$=$$ $$11$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$11$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$9$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$9$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$12$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$22$$ $$\nu^{11}$$ $$=$$ $$4$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$7$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$7$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$10$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$5$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$14$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$5$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$6$$ $$\beta_{2}$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$$.

 $$n$$ $$151$$ $$301$$ $$401$$ $$577$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 −1.39298 + 0.244153i −1.39298 − 0.244153i −1.26128 + 0.639662i −1.26128 − 0.639662i −0.847808 + 1.13191i −0.847808 − 1.13191i 0.847808 + 1.13191i 0.847808 − 1.13191i 1.26128 + 0.639662i 1.26128 − 0.639662i 1.39298 + 0.244153i 1.39298 − 0.244153i
−1.39298 0.244153i 1.31310 + 1.12950i 1.88078 + 0.680200i 0 −1.55335 1.89397i 4.34495i −2.45381 1.40670i 0.448458 + 2.96629i 0
251.2 −1.39298 + 0.244153i 1.31310 1.12950i 1.88078 0.680200i 0 −1.55335 + 1.89397i 4.34495i −2.45381 + 1.40670i 0.448458 2.96629i 0
251.3 −1.26128 0.639662i −1.57067 0.730070i 1.18166 + 1.61359i 0 1.51406 + 1.92552i 1.25539i −0.458259 2.79106i 1.93400 + 2.29339i 0
251.4 −1.26128 + 0.639662i −1.57067 + 0.730070i 1.18166 1.61359i 0 1.51406 1.92552i 1.25539i −0.458259 + 2.79106i 1.93400 2.29339i 0
251.5 −0.847808 1.13191i −0.242431 + 1.71500i −0.562443 + 1.91929i 0 2.14676 1.17958i 3.08957i 2.64930 0.990551i −2.88245 0.831539i 0
251.6 −0.847808 + 1.13191i −0.242431 1.71500i −0.562443 1.91929i 0 2.14676 + 1.17958i 3.08957i 2.64930 + 0.990551i −2.88245 + 0.831539i 0
251.7 0.847808 1.13191i −0.242431 + 1.71500i −0.562443 1.91929i 0 1.73569 + 1.72840i 3.08957i −2.64930 0.990551i −2.88245 0.831539i 0
251.8 0.847808 + 1.13191i −0.242431 1.71500i −0.562443 + 1.91929i 0 1.73569 1.72840i 3.08957i −2.64930 + 0.990551i −2.88245 + 0.831539i 0
251.9 1.26128 0.639662i −1.57067 0.730070i 1.18166 1.61359i 0 −2.44805 + 0.0838735i 1.25539i 0.458259 2.79106i 1.93400 + 2.29339i 0
251.10 1.26128 + 0.639662i −1.57067 + 0.730070i 1.18166 + 1.61359i 0 −2.44805 0.0838735i 1.25539i 0.458259 + 2.79106i 1.93400 2.29339i 0
251.11 1.39298 0.244153i 1.31310 + 1.12950i 1.88078 0.680200i 0 2.10489 + 1.25277i 4.34495i 2.45381 1.40670i 0.448458 + 2.96629i 0
251.12 1.39298 + 0.244153i 1.31310 1.12950i 1.88078 + 0.680200i 0 2.10489 1.25277i 4.34495i 2.45381 + 1.40670i 0.448458 2.96629i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.12 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
3.b Odd 1 yes
8.d Odd 1 no
24.f Even 1 no

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(600, [\chi])$$:

 $$T_{7}^{6}$$ $$\mathstrut +\mathstrut 30 T_{7}^{4}$$ $$\mathstrut +\mathstrut 225 T_{7}^{2}$$ $$\mathstrut +\mathstrut 284$$ $$T_{11}^{6}$$ $$\mathstrut +\mathstrut 19 T_{11}^{4}$$ $$\mathstrut +\mathstrut 112 T_{11}^{2}$$ $$\mathstrut +\mathstrut 200$$ $$T_{23}^{6}$$ $$\mathstrut -\mathstrut 104 T_{23}^{4}$$ $$\mathstrut +\mathstrut 2136 T_{23}^{2}$$ $$\mathstrut -\mathstrut 9088$$ $$T_{43}^{3}$$ $$\mathstrut -\mathstrut 10 T_{43}^{2}$$ $$\mathstrut -\mathstrut 29 T_{43}$$ $$\mathstrut +\mathstrut 148$$