# Properties

 Label 600.2.b.f Level 600 Weight 2 Character orbit 600.b Analytic conductor 4.791 Analytic rank 0 Dimension 8 CM No Inner twists 2

# 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: $$8$$ Coefficient field: 8.0.1649659456.5 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8}\mathstrut -\mathstrut$$ $$x^{7}\mathstrut -\mathstrut$$ $$2$$ $$x^{5}\mathstrut +\mathstrut$$ $$4$$ $$x^{4}\mathstrut -\mathstrut$$ $$4$$ $$x^{3}\mathstrut -\mathstrut$$ $$8$$ $$x\mathstrut +\mathstrut$$ $$16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + \nu^{5} - 4$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + \nu^{6} + 4 \nu^{2} - 8 \nu - 8$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{6} + 4 \nu^{2} + 8$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + \nu^{6} - 4 \nu^{4} + 4 \nu^{3} + 4 \nu^{2} - 16$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} + 2 \nu^{5} + 4 \nu^{3} - 4 \nu^{2} - 8$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} - 2 \nu^{5} + 4 \nu^{4} + 4 \nu^{2} - 8 \nu + 8$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$1$$ $$\nu^{4}$$ $$=$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$1$$ $$\nu^{5}$$ $$=$$ $$-$$$$\beta_{7}\mathstrut +\mathstrut$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$1$$ $$\nu^{6}$$ $$=$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$3$$ $$\nu^{7}$$ $$=$$ $$-$$$$\beta_{7}\mathstrut +\mathstrut$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$5$$

## 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
 −1.13622 − 0.842022i −1.13622 + 0.842022i −0.578647 − 1.29041i −0.578647 + 1.29041i 0.814732 − 1.15595i 0.814732 + 1.15595i 1.40014 − 0.199044i 1.40014 + 0.199044i
−1.13622 0.842022i 1.71822 0.218455i 0.581998 + 1.91345i 0 −2.13622 1.19857i 3.64426i 0.949886 2.66415i 2.90455 0.750707i 0
251.2 −1.13622 + 0.842022i 1.71822 + 0.218455i 0.581998 1.91345i 0 −2.13622 + 1.19857i 3.64426i 0.949886 + 2.66415i 2.90455 + 0.750707i 0
251.3 −0.578647 1.29041i −0.751690 1.56044i −1.33034 + 1.49339i 0 −1.57865 + 1.87293i 4.28591i 2.69688 + 0.852541i −1.86993 + 2.34593i 0
251.4 −0.578647 + 1.29041i −0.751690 + 1.56044i −1.33034 1.49339i 0 −1.57865 1.87293i 4.28591i 2.69688 0.852541i −1.86993 2.34593i 0
251.5 0.814732 1.15595i −1.48716 + 0.887900i −0.672424 1.88357i 0 −0.185268 + 2.44247i 0.797253i −2.72515 0.757320i 1.42327 2.64089i 0
251.6 0.814732 + 1.15595i −1.48716 0.887900i −0.672424 + 1.88357i 0 −0.185268 2.44247i 0.797253i −2.72515 + 0.757320i 1.42327 + 2.64089i 0
251.7 1.40014 0.199044i 0.520627 1.65195i 1.92076 0.557378i 0 0.400136 2.41659i 1.92736i 2.57839 1.16272i −2.45790 1.72010i 0
251.8 1.40014 + 0.199044i 0.520627 + 1.65195i 1.92076 + 0.557378i 0 0.400136 + 2.41659i 1.92736i 2.57839 + 1.16272i −2.45790 + 1.72010i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.8 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
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 $$S_{2}^{\mathrm{new}}(600, [\chi])$$:

 $$T_{7}^{8}$$ $$\mathstrut +\mathstrut 36 T_{7}^{6}$$ $$\mathstrut +\mathstrut 384 T_{7}^{4}$$ $$\mathstrut +\mathstrut 1136 T_{7}^{2}$$ $$\mathstrut +\mathstrut 576$$ $$T_{11}^{8}$$ $$\mathstrut +\mathstrut 48 T_{11}^{6}$$ $$\mathstrut +\mathstrut 672 T_{11}^{4}$$ $$\mathstrut +\mathstrut 2560 T_{11}^{2}$$ $$\mathstrut +\mathstrut 256$$ $$T_{23}^{4}$$ $$\mathstrut +\mathstrut 2 T_{23}^{3}$$ $$\mathstrut -\mathstrut 44 T_{23}^{2}$$ $$\mathstrut -\mathstrut 188 T_{23}$$ $$\mathstrut -\mathstrut 192$$ $$T_{43}^{4}$$ $$\mathstrut -\mathstrut 12 T_{43}^{2}$$ $$\mathstrut +\mathstrut 4 T_{43}$$ $$\mathstrut +\mathstrut 16$$