# Properties

 Label 600.2.k.a Level 600 Weight 2 Character orbit 600.k Analytic conductor 4.791 Analytic rank 0 Dimension 2 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 − 1.00000i 1.00000i
−1.00000 1.00000i 1.00000i 2.00000i 0 1.00000 1.00000i −2.00000 2.00000 2.00000i −1.00000 0
301.2 −1.00000 + 1.00000i 1.00000i 2.00000i 0 1.00000 + 1.00000i −2.00000 2.00000 + 2.00000i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 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 $$T_{7}$$ $$\mathstrut +\mathstrut 2$$ acting on $$S_{2}^{\mathrm{new}}(600, [\chi])$$.