# Properties

 Label 600.2.k.c Level 600 Weight 2 Character orbit 600.k Analytic conductor 4.791 Analytic rank 0 Dimension 6 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: $$6$$ Coefficient field: 6.0.399424.1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6}\mathstrut -\mathstrut$$ $$2$$ $$x^{5}\mathstrut +\mathstrut$$ $$3$$ $$x^{4}\mathstrut -\mathstrut$$ $$6$$ $$x^{3}\mathstrut +\mathstrut$$ $$6$$ $$x^{2}\mathstrut -\mathstrut$$ $$8$$ $$x\mathstrut +\mathstrut$$ $$8$$:

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

## 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
 1.40680 + 0.144584i 1.40680 − 0.144584i 0.264658 + 1.38923i 0.264658 − 1.38923i −0.671462 + 1.24464i −0.671462 − 1.24464i
−1.40680 0.144584i 1.00000i 1.95819 + 0.406803i 0 0.144584 1.40680i 3.62721 −2.69597 0.855416i −1.00000 0
301.2 −1.40680 + 0.144584i 1.00000i 1.95819 0.406803i 0 0.144584 + 1.40680i 3.62721 −2.69597 + 0.855416i −1.00000 0
301.3 −0.264658 1.38923i 1.00000i −1.85991 + 0.735342i 0 −1.38923 + 0.264658i −0.941367 1.51380 + 2.38923i −1.00000 0
301.4 −0.264658 + 1.38923i 1.00000i −1.85991 0.735342i 0 −1.38923 0.264658i −0.941367 1.51380 2.38923i −1.00000 0
301.5 0.671462 1.24464i 1.00000i −1.09828 1.67146i 0 1.24464 + 0.671462i −4.68585 −2.81783 + 0.244644i −1.00000 0
301.6 0.671462 + 1.24464i 1.00000i −1.09828 + 1.67146i 0 1.24464 0.671462i −4.68585 −2.81783 0.244644i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 301.6 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 $$T_{7}^{3}$$ $$\mathstrut +\mathstrut 2 T_{7}^{2}$$ $$\mathstrut -\mathstrut 16 T_{7}$$ $$\mathstrut -\mathstrut 16$$ acting on $$S_{2}^{\mathrm{new}}(600, [\chi])$$.