# Properties

 Label 600.2.m.d Level 600 Weight 2 Character orbit 600.m Analytic conductor 4.791 Analytic rank 0 Dimension 16 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.m (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$4.79102412128$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{11}$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ \beta_{6} q^{3}$$ $$+ \beta_{2} q^{4}$$ $$-\beta_{14} q^{6}$$ $$+ ( \beta_{1} + \beta_{3} + \beta_{7} - \beta_{10} ) q^{7}$$ $$+ \beta_{3} q^{8}$$ $$+ ( -\beta_{2} + \beta_{8} + \beta_{14} - \beta_{15} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ \beta_{6} q^{3}$$ $$+ \beta_{2} q^{4}$$ $$-\beta_{14} q^{6}$$ $$+ ( \beta_{1} + \beta_{3} + \beta_{7} - \beta_{10} ) q^{7}$$ $$+ \beta_{3} q^{8}$$ $$+ ( -\beta_{2} + \beta_{8} + \beta_{14} - \beta_{15} ) q^{9}$$ $$+ ( -\beta_{2} + \beta_{8} - \beta_{9} + \beta_{12} ) q^{11}$$ $$+ ( -\beta_{1} - \beta_{3} - \beta_{7} + \beta_{10} + \beta_{13} ) q^{12}$$ $$+ ( \beta_{1} - \beta_{5} + \beta_{7} + \beta_{13} ) q^{13}$$ $$+ ( 1 + 2 \beta_{8} - \beta_{9} - \beta_{11} + 2 \beta_{12} + \beta_{14} - \beta_{15} ) q^{14}$$ $$+ ( -1 - \beta_{2} + \beta_{8} + 2 \beta_{12} + \beta_{14} ) q^{16}$$ $$+ ( \beta_{1} + \beta_{3} + \beta_{6} - \beta_{10} ) q^{17}$$ $$+ ( \beta_{1} - \beta_{5} + 2 \beta_{7} - \beta_{10} ) q^{18}$$ $$+ ( \beta_{2} - 2 \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} - 2 \beta_{14} ) q^{19}$$ $$+ ( -1 - \beta_{8} + 2 \beta_{9} + \beta_{11} - \beta_{12} - \beta_{14} ) q^{21}$$ $$+ ( -2 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{10} + \beta_{13} ) q^{22}$$ $$+ ( \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{10} ) q^{23}$$ $$+ ( 1 + \beta_{2} - 3 \beta_{8} + \beta_{9} + 2 \beta_{11} - 2 \beta_{12} - \beta_{14} + \beta_{15} ) q^{24}$$ $$+ ( 2 + 2 \beta_{2} - 2 \beta_{8} ) q^{26}$$ $$+ ( 2 \beta_{1} - \beta_{4} - \beta_{7} - 2 \beta_{13} ) q^{27}$$ $$+ ( \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{7} + \beta_{10} + \beta_{13} ) q^{28}$$ $$+ ( -3 \beta_{2} + \beta_{8} + \beta_{12} + 2 \beta_{14} + \beta_{15} ) q^{29}$$ $$+ ( -\beta_{2} + 2 \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} - 2 \beta_{15} ) q^{31}$$ $$+ ( \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{10} - \beta_{13} ) q^{32}$$ $$+ ( 4 \beta_{1} + \beta_{3} - 3 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} - \beta_{13} ) q^{33}$$ $$+ ( 2 \beta_{8} - \beta_{9} + 2 \beta_{12} - \beta_{15} ) q^{34}$$ $$+ ( 2 + \beta_{2} - \beta_{9} - 2 \beta_{11} - \beta_{15} ) q^{36}$$ $$+ ( -\beta_{1} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{13} ) q^{37}$$ $$+ ( -\beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 3 \beta_{7} + \beta_{10} + \beta_{13} ) q^{38}$$ $$+ ( -2 + 2 \beta_{2} - \beta_{8} + \beta_{9} - \beta_{11} - 2 \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{39}$$ $$+ ( -\beta_{2} + \beta_{8} - \beta_{12} - \beta_{15} ) q^{41}$$ $$+ ( 2 \beta_{1} - \beta_{3} - \beta_{4} - 4 \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{13} ) q^{42}$$ $$+ ( -\beta_{6} - \beta_{7} ) q^{43}$$ $$+ ( 4 - 2 \beta_{8} - 2 \beta_{12} + 2 \beta_{15} ) q^{44}$$ $$+ ( -1 - \beta_{11} + 2 \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{46}$$ $$+ ( -5 \beta_{1} + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{10} + 2 \beta_{13} ) q^{47}$$ $$+ ( 2 \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} - 3 \beta_{7} - \beta_{10} - \beta_{13} ) q^{48}$$ $$+ ( 3 - \beta_{2} + 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} + \beta_{12} + 2 \beta_{14} + \beta_{15} ) q^{49}$$ $$+ ( 2 - \beta_{2} + 2 \beta_{9} + \beta_{11} - \beta_{12} - \beta_{15} ) q^{51}$$ $$+ ( 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{52}$$ $$+ ( -\beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{53}$$ $$+ ( -5 + 2 \beta_{8} + \beta_{9} - \beta_{11} - \beta_{15} ) q^{54}$$ $$+ ( 4 - 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{15} ) q^{56}$$ $$+ ( \beta_{1} - \beta_{4} + \beta_{5} + 3 \beta_{7} - 2 \beta_{10} + \beta_{13} ) q^{57}$$ $$+ ( 2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{10} - 3 \beta_{13} ) q^{58}$$ $$+ ( \beta_{2} - \beta_{8} - \beta_{9} + 2 \beta_{11} + \beta_{12} + 2 \beta_{15} ) q^{59}$$ $$+ ( -\beta_{2} - \beta_{8} - 3 \beta_{12} - \beta_{15} ) q^{61}$$ $$+ ( 2 \beta_{1} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} + \beta_{13} ) q^{62}$$ $$+ ( -\beta_{1} - \beta_{3} + \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{10} ) q^{63}$$ $$+ ( -3 - \beta_{2} - \beta_{8} - 2 \beta_{11} - \beta_{14} + 2 \beta_{15} ) q^{64}$$ $$+ ( -4 + 2 \beta_{2} - \beta_{9} - 2 \beta_{11} + 2 \beta_{12} - \beta_{15} ) q^{66}$$ $$+ ( -4 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{10} + 2 \beta_{13} ) q^{67}$$ $$+ ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{7} + 2 \beta_{10} + 2 \beta_{13} ) q^{68}$$ $$+ ( -3 + \beta_{2} - 2 \beta_{8} + 3 \beta_{11} - 2 \beta_{12} - \beta_{14} - \beta_{15} ) q^{69}$$ $$+ ( -4 - 2 \beta_{2} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} ) q^{71}$$ $$+ ( \beta_{3} + 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{13} ) q^{72}$$ $$+ ( -6 \beta_{1} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{13} ) q^{73}$$ $$+ ( -4 - 2 \beta_{2} - 2 \beta_{8} - 2 \beta_{11} + 2 \beta_{14} ) q^{74}$$ $$+ ( -2 + 2 \beta_{2} - 4 \beta_{8} + 4 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} + 2 \beta_{15} ) q^{76}$$ $$+ ( 4 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} - 2 \beta_{10} - 2 \beta_{13} ) q^{77}$$ $$+ ( -2 \beta_{1} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{13} ) q^{78}$$ $$+ ( \beta_{2} - 2 \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} ) q^{79}$$ $$+ ( 1 - 3 \beta_{2} + \beta_{8} - 2 \beta_{9} + 2 \beta_{11} + \beta_{12} + 3 \beta_{15} ) q^{81}$$ $$+ ( -\beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{10} + \beta_{13} ) q^{82}$$ $$+ ( -2 \beta_{1} - 2 \beta_{3} - \beta_{6} - \beta_{7} + 2 \beta_{10} ) q^{83}$$ $$+ ( -8 + 2 \beta_{2} - 4 \beta_{8} + \beta_{9} + 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{84}$$ $$+ ( -1 + \beta_{11} + \beta_{14} ) q^{86}$$ $$+ ( 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{13} ) q^{87}$$ $$+ ( 4 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{10} - 2 \beta_{13} ) q^{88}$$ $$+ ( -2 \beta_{2} + 2 \beta_{8} - 4 \beta_{11} + 2 \beta_{12} - 2 \beta_{15} ) q^{89}$$ $$+ ( 4 + 2 \beta_{2} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{91}$$ $$+ ( -2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{7} + 3 \beta_{10} - \beta_{13} ) q^{92}$$ $$+ ( -\beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} - 3 \beta_{13} ) q^{93}$$ $$+ ( 5 - 4 \beta_{2} - 2 \beta_{9} + \beta_{11} + 2 \beta_{12} + \beta_{14} + 4 \beta_{15} ) q^{94}$$ $$+ ( -5 + \beta_{2} + \beta_{8} - 2 \beta_{9} + 2 \beta_{11} + \beta_{14} ) q^{96}$$ $$+ ( -2 \beta_{1} - 2 \beta_{3} - \beta_{4} - 4 \beta_{6} - 2 \beta_{7} - 2 \beta_{10} ) q^{97}$$ $$+ ( \beta_{1} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - \beta_{10} - \beta_{13} ) q^{98}$$ $$+ ( -2 + \beta_{2} + \beta_{8} + \beta_{9} + 2 \beta_{11} - 3 \beta_{12} - 2 \beta_{14} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q$$ $$\mathstrut -\mathstrut 2q^{4}$$ $$\mathstrut +\mathstrut 2q^{6}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$16q$$ $$\mathstrut -\mathstrut 2q^{4}$$ $$\mathstrut +\mathstrut 2q^{6}$$ $$\mathstrut +\mathstrut 12q^{14}$$ $$\mathstrut -\mathstrut 14q^{16}$$ $$\mathstrut +\mathstrut 8q^{19}$$ $$\mathstrut -\mathstrut 8q^{21}$$ $$\mathstrut +\mathstrut 22q^{24}$$ $$\mathstrut +\mathstrut 32q^{26}$$ $$\mathstrut +\mathstrut 26q^{36}$$ $$\mathstrut -\mathstrut 32q^{39}$$ $$\mathstrut +\mathstrut 60q^{44}$$ $$\mathstrut -\mathstrut 16q^{46}$$ $$\mathstrut +\mathstrut 32q^{49}$$ $$\mathstrut +\mathstrut 40q^{51}$$ $$\mathstrut -\mathstrut 82q^{54}$$ $$\mathstrut +\mathstrut 60q^{56}$$ $$\mathstrut -\mathstrut 50q^{64}$$ $$\mathstrut -\mathstrut 68q^{66}$$ $$\mathstrut -\mathstrut 40q^{69}$$ $$\mathstrut -\mathstrut 48q^{71}$$ $$\mathstrut -\mathstrut 64q^{74}$$ $$\mathstrut -\mathstrut 24q^{76}$$ $$\mathstrut +\mathstrut 16q^{81}$$ $$\mathstrut -\mathstrut 116q^{84}$$ $$\mathstrut -\mathstrut 16q^{86}$$ $$\mathstrut +\mathstrut 48q^{91}$$ $$\mathstrut +\mathstrut 80q^{94}$$ $$\mathstrut -\mathstrut 86q^{96}$$ $$\mathstrut -\mathstrut 32q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16}\mathstrut +\mathstrut$$ $$x^{14}\mathstrut +\mathstrut$$ $$4$$ $$x^{12}\mathstrut +\mathstrut$$ $$12$$ $$x^{10}\mathstrut +\mathstrut$$ $$16$$ $$x^{8}\mathstrut +\mathstrut$$ $$48$$ $$x^{6}\mathstrut +\mathstrut$$ $$64$$ $$x^{4}\mathstrut +\mathstrut$$ $$64$$ $$x^{2}\mathstrut +\mathstrut$$ $$256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{15} + 3 \nu^{13} + 10 \nu^{11} + 8 \nu^{9} + 8 \nu^{7} + 64 \nu^{5} + 64 \nu$$$$)/192$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{9} + \nu^{7} - 2 \nu^{5} - 4 \nu^{3} + 8 \nu$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{15} - 3 \nu^{13} + 2 \nu^{11} - 8 \nu^{9} + 4 \nu^{7} + 8 \nu^{5} - 48 \nu^{3} + 32 \nu$$$$)/192$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{15} - 3 \nu^{13} - 10 \nu^{11} - 20 \nu^{9} - 44 \nu^{7} - 40 \nu^{5} - 144 \nu^{3} - 160 \nu$$$$)/192$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{10} + \nu^{8} - 2 \nu^{6} - 4 \nu^{4} + 8 \nu^{2} - 16$$$$)/16$$ $$\beta_{9}$$ $$=$$ $$($$$$\nu^{14} - 3 \nu^{12} + 4 \nu^{10} + 32 \nu^{8} + 32 \nu^{6} + 112 \nu^{4} + 192 \nu^{2} + 256$$$$)/192$$ $$\beta_{10}$$ $$=$$ $$($$$$-\nu^{15} + 3 \nu^{13} - 4 \nu^{11} + 4 \nu^{9} + 28 \nu^{7} + 56 \nu^{5} + 144 \nu^{3} + 224 \nu$$$$)/192$$ $$\beta_{11}$$ $$=$$ $$($$$$\nu^{14} + 3 \nu^{12} + 4 \nu^{10} + 14 \nu^{8} - 4 \nu^{6} + 40 \nu^{4} + 48 \nu^{2} - 32$$$$)/96$$ $$\beta_{12}$$ $$=$$ $$($$$$-\nu^{14} + 3 \nu^{12} + 8 \nu^{10} + 4 \nu^{8} + 40 \nu^{6} + 128 \nu^{4} + 96 \nu^{2} + 320$$$$)/192$$ $$\beta_{13}$$ $$=$$ $$($$$$-\nu^{15} - \nu^{11} - 2 \nu^{9} - 8 \nu^{7} - 40 \nu^{5} + 32 \nu$$$$)/96$$ $$\beta_{14}$$ $$=$$ $$($$$$\nu^{14} - 3 \nu^{12} - 2 \nu^{10} - 10 \nu^{8} - 28 \nu^{6} - 8 \nu^{4} - 48 \nu^{2} - 128$$$$)/96$$ $$\beta_{15}$$ $$=$$ $$($$$$\nu^{14} + \nu^{12} + 8 \nu^{8} + 16 \nu^{6} + 16 \nu^{4} + 64 \nu^{2}$$$$)/64$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{3}$$ $$\nu^{4}$$ $$=$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$1$$ $$\nu^{5}$$ $$=$$ $$-$$$$\beta_{13}\mathstrut +\mathstrut$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{4}$$ $$\nu^{6}$$ $$=$$ $$2$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$3$$ $$\nu^{7}$$ $$=$$ $$-$$$$\beta_{13}\mathstrut +\mathstrut$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-$$$$2$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$1$$ $$\nu^{9}$$ $$=$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$5$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$11$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$5$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$6$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-$$$$6$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$5$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$12$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$6$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$17$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$11$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$7$$ $$\nu^{11}$$ $$=$$ $$7$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$7$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$9$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$13$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$6$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{1}$$ $$\nu^{12}$$ $$=$$ $$6$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$12$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$10$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$20$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$9$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$15$$ $$\nu^{13}$$ $$=$$ $$17$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$15$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$11$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$33$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$21$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$27$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$6$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$28$$ $$\beta_{1}$$ $$\nu^{14}$$ $$=$$ $$42$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$27$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$12$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$6$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$12$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$17$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$5$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$57$$ $$\nu^{15}$$ $$=$$ $$-$$$$57$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$39$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$55$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$35$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$19$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$22$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$60$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
299.1
 −1.29041 − 0.578647i −1.29041 + 0.578647i −1.15595 − 0.814732i −1.15595 + 0.814732i −0.842022 − 1.13622i −0.842022 + 1.13622i −0.199044 − 1.40014i −0.199044 + 1.40014i 0.199044 − 1.40014i 0.199044 + 1.40014i 0.842022 − 1.13622i 0.842022 + 1.13622i 1.15595 − 0.814732i 1.15595 + 0.814732i 1.29041 − 0.578647i 1.29041 + 0.578647i
−1.29041 0.578647i 1.56044 0.751690i 1.33034 + 1.49339i 0 −2.44857 + 0.0670494i −4.28591 −0.852541 2.69688i 1.86993 2.34593i 0
299.2 −1.29041 + 0.578647i 1.56044 + 0.751690i 1.33034 1.49339i 0 −2.44857 0.0670494i −4.28591 −0.852541 + 2.69688i 1.86993 + 2.34593i 0
299.3 −1.15595 0.814732i −0.887900 + 1.48716i 0.672424 + 1.88357i 0 2.23800 0.995672i −0.797253 0.757320 2.72515i −1.42327 2.64089i 0
299.4 −1.15595 + 0.814732i −0.887900 1.48716i 0.672424 1.88357i 0 2.23800 + 0.995672i −0.797253 0.757320 + 2.72515i −1.42327 + 2.64089i 0
299.5 −0.842022 1.13622i 0.218455 + 1.71822i −0.581998 + 1.91345i 0 1.76833 1.69499i 3.64426 2.66415 0.949886i −2.90455 + 0.750707i 0
299.6 −0.842022 + 1.13622i 0.218455 1.71822i −0.581998 1.91345i 0 1.76833 + 1.69499i 3.64426 2.66415 + 0.949886i −2.90455 0.750707i 0
299.7 −0.199044 1.40014i 1.65195 0.520627i −1.92076 + 0.557378i 0 −1.05776 2.20933i 1.92736 1.16272 + 2.57839i 2.45790 1.72010i 0
299.8 −0.199044 + 1.40014i 1.65195 + 0.520627i −1.92076 0.557378i 0 −1.05776 + 2.20933i 1.92736 1.16272 2.57839i 2.45790 + 1.72010i 0
299.9 0.199044 1.40014i −1.65195 0.520627i −1.92076 0.557378i 0 −1.05776 + 2.20933i −1.92736 −1.16272 + 2.57839i 2.45790 + 1.72010i 0
299.10 0.199044 + 1.40014i −1.65195 + 0.520627i −1.92076 + 0.557378i 0 −1.05776 2.20933i −1.92736 −1.16272 2.57839i 2.45790 1.72010i 0
299.11 0.842022 1.13622i −0.218455 + 1.71822i −0.581998 1.91345i 0 1.76833 + 1.69499i −3.64426 −2.66415 0.949886i −2.90455 0.750707i 0
299.12 0.842022 + 1.13622i −0.218455 1.71822i −0.581998 + 1.91345i 0 1.76833 1.69499i −3.64426 −2.66415 + 0.949886i −2.90455 + 0.750707i 0
299.13 1.15595 0.814732i 0.887900 + 1.48716i 0.672424 1.88357i 0 2.23800 + 0.995672i 0.797253 −0.757320 2.72515i −1.42327 + 2.64089i 0
299.14 1.15595 + 0.814732i 0.887900 1.48716i 0.672424 + 1.88357i 0 2.23800 0.995672i 0.797253 −0.757320 + 2.72515i −1.42327 2.64089i 0
299.15 1.29041 0.578647i −1.56044 0.751690i 1.33034 1.49339i 0 −2.44857 0.0670494i 4.28591 0.852541 2.69688i 1.86993 + 2.34593i 0
299.16 1.29041 + 0.578647i −1.56044 + 0.751690i 1.33034 + 1.49339i 0 −2.44857 + 0.0670494i 4.28591 0.852541 + 2.69688i 1.86993 2.34593i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 299.16 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
5.b Even 1 yes
24.f Even 1 yes
120.m Even 1 yes

## Hecke kernels

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

 $$T_{7}^{8}$$ $$\mathstrut -\mathstrut 36 T_{7}^{6}$$ $$\mathstrut +\mathstrut 384 T_{7}^{4}$$ $$\mathstrut -\mathstrut 1136 T_{7}^{2}$$ $$\mathstrut +\mathstrut 576$$ $$T_{11}^{8}$$ $$\mathstrut +\mathstrut 48 T_{11}^{6}$$ $$\mathstrut +\mathstrut 672 T_{11}^{4}$$ $$\mathstrut +\mathstrut 2560 T_{11}^{2}$$ $$\mathstrut +\mathstrut 256$$ $$T_{29}^{4}$$ $$\mathstrut -\mathstrut 64 T_{29}^{2}$$ $$\mathstrut +\mathstrut 112 T_{29}$$ $$\mathstrut -\mathstrut 48$$