# Properties

 Label 600.2.d.f Level 600 Weight 2 Character orbit 600.d Analytic conductor 4.791 Analytic rank 0 Dimension 6 CM No Inner twists 2

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 600.d (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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ 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.

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

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
349.1
 0.264658 − 1.38923i 0.264658 + 1.38923i 1.40680 + 0.144584i 1.40680 − 0.144584i −0.671462 − 1.24464i −0.671462 + 1.24464i
−1.38923 0.264658i 1.00000 1.85991 + 0.735342i 0 −1.38923 0.264658i 0.941367i −2.38923 1.51380i 1.00000 0
349.2 −1.38923 + 0.264658i 1.00000 1.85991 0.735342i 0 −1.38923 + 0.264658i 0.941367i −2.38923 + 1.51380i 1.00000 0
349.3 0.144584 1.40680i 1.00000 −1.95819 0.406803i 0 0.144584 1.40680i 3.62721i −0.855416 + 2.69597i 1.00000 0
349.4 0.144584 + 1.40680i 1.00000 −1.95819 + 0.406803i 0 0.144584 + 1.40680i 3.62721i −0.855416 2.69597i 1.00000 0
349.5 1.24464 0.671462i 1.00000 1.09828 1.67146i 0 1.24464 0.671462i 4.68585i 0.244644 2.81783i 1.00000 0
349.6 1.24464 + 0.671462i 1.00000 1.09828 + 1.67146i 0 1.24464 + 0.671462i 4.68585i 0.244644 + 2.81783i 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 349.6 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
40.f Even 1 no

## Hecke kernels

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

 $$T_{7}^{6}$$ $$\mathstrut +\mathstrut 36 T_{7}^{4}$$ $$\mathstrut +\mathstrut 320 T_{7}^{2}$$ $$\mathstrut +\mathstrut 256$$ $$T_{11}^{6}$$ $$\mathstrut +\mathstrut 64 T_{11}^{4}$$ $$\mathstrut +\mathstrut 1088 T_{11}^{2}$$ $$\mathstrut +\mathstrut 4096$$ $$T_{13}^{3}$$ $$\mathstrut -\mathstrut 28 T_{13}$$ $$\mathstrut +\mathstrut 16$$ $$T_{37}^{3}$$ $$\mathstrut -\mathstrut 4 T_{37}^{2}$$ $$\mathstrut -\mathstrut 60 T_{37}$$ $$\mathstrut +\mathstrut 256$$