# Properties

 Label 600.2.d.a Level 600 Weight 2 Character orbit 600.d 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.d (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.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
349.1
 − 1.00000i 1.00000i
−1.00000 1.00000i −1.00000 2.00000i 0 1.00000 + 1.00000i 2.00000i 2.00000 2.00000i 1.00000 0
349.2 −1.00000 + 1.00000i −1.00000 2.00000i 0 1.00000 1.00000i 2.00000i 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
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 $$S_{2}^{\mathrm{new}}(600, [\chi])$$:

 $$T_{7}^{2}$$ $$\mathstrut +\mathstrut 4$$ $$T_{11}^{2}$$ $$\mathstrut +\mathstrut 16$$ $$T_{13}$$ $$T_{37}$$ $$\mathstrut -\mathstrut 4$$