# Properties

 Label 600.2.k.d Level 600 Weight 2 Character orbit 600.k 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.k (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$4.79102412128$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.214798336.3 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8}\mathstrut -\mathstrut$$ $$2$$ $$x^{7}\mathstrut -\mathstrut$$ $$2$$ $$x^{5}\mathstrut +\mathstrut$$ $$9$$ $$x^{4}\mathstrut -\mathstrut$$ $$4$$ $$x^{3}\mathstrut -\mathstrut$$ $$16$$ $$x\mathstrut +\mathstrut$$ $$16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{4} + 3 \nu^{3} - 6 \nu^{2} - 4 \nu$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{7} + 2 \nu^{6} + 4 \nu^{5} + 18 \nu^{4} - 21 \nu^{3} - 12 \nu^{2} - 20 \nu + 56$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$5 \nu^{7} - 4 \nu^{6} - 4 \nu^{5} - 18 \nu^{4} + 25 \nu^{3} + 10 \nu^{2} + 24 \nu - 64$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{7} - 2 \nu^{6} - 2 \nu^{5} - 10 \nu^{4} + 15 \nu^{3} + 8 \nu^{2} + 10 \nu - 36$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} + 2 \nu^{6} + 4 \nu^{5} + 10 \nu^{4} - 15 \nu^{3} - 12 \nu^{2} - 8 \nu + 36$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$-4 \nu^{7} + 3 \nu^{6} + 4 \nu^{5} + 12 \nu^{4} - 18 \nu^{3} - 9 \nu^{2} - 10 \nu + 44$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$7 \nu^{7} - 4 \nu^{6} - 6 \nu^{5} - 22 \nu^{4} + 31 \nu^{3} + 18 \nu^{2} + 26 \nu - 80$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7}\mathstrut +\mathstrut$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6}\mathstrut -\mathstrut$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$\beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7}\mathstrut +\mathstrut$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$3$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$4$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$\beta_{1}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-$$$$\beta_{7}\mathstrut +\mathstrut$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$7$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$1$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$4$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$8$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$5$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{1}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$7$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$5$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$8$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$11$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$5$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
301.1
 1.41216 − 0.0762223i 1.41216 + 0.0762223i −0.565036 − 1.29643i −0.565036 + 1.29643i −1.08003 − 0.912978i −1.08003 + 0.912978i 1.23291 − 0.692769i 1.23291 + 0.692769i
−1.29150 0.576222i 1.00000i 1.33594 + 1.48838i 0 0.576222 1.29150i −1.97676 −0.867721 2.69204i −1.00000 0
301.2 −1.29150 + 0.576222i 1.00000i 1.33594 1.48838i 0 0.576222 + 1.29150i −1.97676 −0.867721 + 2.69204i −1.00000 0
301.3 −1.16863 0.796431i 1.00000i 0.731395 + 1.86147i 0 −0.796431 + 1.16863i 4.72294 0.627801 2.75787i −1.00000 0
301.4 −1.16863 + 0.796431i 1.00000i 0.731395 1.86147i 0 −0.796431 1.16863i 4.72294 0.627801 + 2.75787i −1.00000 0
301.5 0.0591148 1.41298i 1.00000i −1.99301 0.167056i 0 1.41298 + 0.0591148i 1.33411 −0.353863 + 2.80620i −1.00000 0
301.6 0.0591148 + 1.41298i 1.00000i −1.99301 + 0.167056i 0 1.41298 0.0591148i 1.33411 −0.353863 2.80620i −1.00000 0
301.7 1.40101 0.192769i 1.00000i 1.92568 0.540143i 0 −0.192769 1.40101i −0.0802864 2.59378 1.12796i −1.00000 0
301.8 1.40101 + 0.192769i 1.00000i 1.92568 + 0.540143i 0 −0.192769 + 1.40101i −0.0802864 2.59378 + 1.12796i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 301.8 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
8.b Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{7}^{4}$$ $$\mathstrut -\mathstrut 4 T_{7}^{3}$$ $$\mathstrut -\mathstrut 6 T_{7}^{2}$$ $$\mathstrut +\mathstrut 12 T_{7}$$ $$\mathstrut +\mathstrut 1$$ acting on $$S_{2}^{\mathrm{new}}(600, [\chi])$$.