# Properties

 Label 600.2.r.a Level 600 Weight 2 Character orbit 600.r Analytic conductor 4.791 Analytic rank 0 Dimension 4 CM No Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 600.r (of order $$4$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$4.79102412128$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -1 - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3}$$ $$+ ( -2 + 2 \zeta_{8}^{2} ) q^{7}$$ $$+ ( -2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -1 - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3}$$ $$+ ( -2 + 2 \zeta_{8}^{2} ) q^{7}$$ $$+ ( -2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{9}$$ $$+ ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{11}$$ $$-4 \zeta_{8} q^{17}$$ $$+ 4 \zeta_{8}^{2} q^{19}$$ $$+ ( 4 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{21}$$ $$+ 6 \zeta_{8}^{3} q^{23}$$ $$+ ( -1 + 5 \zeta_{8} + \zeta_{8}^{2} ) q^{27}$$ $$+ ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{29}$$ $$+ 8 q^{31}$$ $$+ ( 4 + 4 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{33}$$ $$+ ( 8 - 8 \zeta_{8}^{2} ) q^{37}$$ $$+ ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{41}$$ $$+ ( -2 - 2 \zeta_{8}^{2} ) q^{43}$$ $$+ 2 \zeta_{8} q^{47}$$ $$-\zeta_{8}^{2} q^{49}$$ $$+ ( -4 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{51}$$ $$+ 8 \zeta_{8}^{3} q^{53}$$ $$+ ( 4 + 4 \zeta_{8} - 4 \zeta_{8}^{2} ) q^{57}$$ $$+ ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{59}$$ $$-6 q^{61}$$ $$+ ( -2 - 2 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{63}$$ $$+ ( -6 + 6 \zeta_{8}^{2} ) q^{67}$$ $$+ ( 6 \zeta_{8} + 6 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{69}$$ $$+ ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{71}$$ $$+ ( 8 + 8 \zeta_{8}^{2} ) q^{73}$$ $$-16 \zeta_{8} q^{77}$$ $$+ ( 7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{81}$$ $$+ 14 \zeta_{8}^{3} q^{83}$$ $$+ ( -4 + 8 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{87}$$ $$+ ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{89}$$ $$+ ( -8 - 8 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{93}$$ $$+ ( -8 + 8 \zeta_{8}^{2} ) q^{97}$$ $$+ ( -4 \zeta_{8} - 16 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 4q^{3}$$ $$\mathstrut -\mathstrut 8q^{7}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 4q^{3}$$ $$\mathstrut -\mathstrut 8q^{7}$$ $$\mathstrut +\mathstrut 16q^{21}$$ $$\mathstrut -\mathstrut 4q^{27}$$ $$\mathstrut +\mathstrut 32q^{31}$$ $$\mathstrut +\mathstrut 16q^{33}$$ $$\mathstrut +\mathstrut 32q^{37}$$ $$\mathstrut -\mathstrut 8q^{43}$$ $$\mathstrut -\mathstrut 16q^{51}$$ $$\mathstrut +\mathstrut 16q^{57}$$ $$\mathstrut -\mathstrut 24q^{61}$$ $$\mathstrut -\mathstrut 8q^{63}$$ $$\mathstrut -\mathstrut 24q^{67}$$ $$\mathstrut +\mathstrut 32q^{73}$$ $$\mathstrut +\mathstrut 28q^{81}$$ $$\mathstrut -\mathstrut 16q^{87}$$ $$\mathstrut -\mathstrut 32q^{93}$$ $$\mathstrut -\mathstrut 32q^{97}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$$.

 $$n$$ $$151$$ $$301$$ $$401$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$-\zeta_{8}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
0 −1.70711 + 0.292893i 0 0 0 −2.00000 2.00000i 0 2.82843 1.00000i 0
257.2 0 −0.292893 + 1.70711i 0 0 0 −2.00000 2.00000i 0 −2.82843 1.00000i 0
593.1 0 −1.70711 0.292893i 0 0 0 −2.00000 + 2.00000i 0 2.82843 + 1.00000i 0
593.2 0 −0.292893 1.70711i 0 0 0 −2.00000 + 2.00000i 0 −2.82843 + 1.00000i 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
3.b Odd 1 yes
5.c Odd 1 yes
15.e Even 1 yes

## Hecke kernels

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

 $$T_{7}^{2}$$ $$\mathstrut +\mathstrut 4 T_{7}$$ $$\mathstrut +\mathstrut 8$$ $$T_{17}^{4}$$ $$\mathstrut +\mathstrut 256$$