# Properties

 Label 600.2.b.b Level 600 Weight 2 Character orbit 600.b Analytic conductor 4.791 Analytic rank 0 Dimension 4 CM disc. -8 Inner twists 4

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## Newspace parameters

 Level: $$N$$ = $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 600.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$4.79102412128$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$-\beta_{1} q^{2}$$ $$+ ( -1 - \beta_{2} ) q^{3}$$ $$-2 q^{4}$$ $$+ ( -1 + \beta_{1} + \beta_{3} ) q^{6}$$ $$+ 2 \beta_{1} q^{8}$$ $$+ ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$-\beta_{1} q^{2}$$ $$+ ( -1 - \beta_{2} ) q^{3}$$ $$-2 q^{4}$$ $$+ ( -1 + \beta_{1} + \beta_{3} ) q^{6}$$ $$+ 2 \beta_{1} q^{8}$$ $$+ ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{9}$$ $$+ ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{11}$$ $$+ ( 2 + 2 \beta_{2} ) q^{12}$$ $$+ 4 q^{16}$$ $$+ ( 1 - 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{17}$$ $$+ ( -1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{18}$$ $$+ ( 1 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{19}$$ $$+ ( -\beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{22}$$ $$+ ( 2 - 2 \beta_{1} - 2 \beta_{3} ) q^{24}$$ $$+ ( -5 + \beta_{1} ) q^{27}$$ $$-4 \beta_{1} q^{32}$$ $$+ ( -4 - 5 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{33}$$ $$+ ( -6 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{34}$$ $$+ ( -2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{36}$$ $$+ ( 2 - 3 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{38}$$ $$+ ( 1 - 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{41}$$ $$+ 10 q^{43}$$ $$+ ( 2 - 2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{44}$$ $$+ ( -4 - 4 \beta_{2} ) q^{48}$$ $$+ 7 q^{49}$$ $$+ ( 1 + 8 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{51}$$ $$+ ( 2 + 5 \beta_{1} ) q^{54}$$ $$+ ( -8 + 3 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{57}$$ $$-10 \beta_{1} q^{59}$$ $$-8 q^{64}$$ $$+ ( -7 + 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{66}$$ $$+ ( 5 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{67}$$ $$+ ( -2 + 8 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{68}$$ $$+ ( 2 + 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{72}$$ $$+ ( 5 - 2 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} ) q^{73}$$ $$+ ( -2 + 2 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{76}$$ $$+ ( 6 - \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{81}$$ $$+ ( -10 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{82}$$ $$+ ( -3 + 7 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{83}$$ $$-10 \beta_{1} q^{86}$$ $$+ ( 2 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{88}$$ $$+ ( -3 + 8 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{89}$$ $$+ ( -4 + 4 \beta_{1} + 4 \beta_{3} ) q^{96}$$ $$+ 10 q^{97}$$ $$-7 \beta_{1} q^{98}$$ $$+ ( -4 + 7 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 2q^{3}$$ $$\mathstrut -\mathstrut 8q^{4}$$ $$\mathstrut -\mathstrut 4q^{6}$$ $$\mathstrut +\mathstrut 2q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 2q^{3}$$ $$\mathstrut -\mathstrut 8q^{4}$$ $$\mathstrut -\mathstrut 4q^{6}$$ $$\mathstrut +\mathstrut 2q^{9}$$ $$\mathstrut +\mathstrut 4q^{12}$$ $$\mathstrut +\mathstrut 16q^{16}$$ $$\mathstrut -\mathstrut 8q^{18}$$ $$\mathstrut -\mathstrut 4q^{19}$$ $$\mathstrut -\mathstrut 8q^{22}$$ $$\mathstrut +\mathstrut 8q^{24}$$ $$\mathstrut -\mathstrut 20q^{27}$$ $$\mathstrut -\mathstrut 22q^{33}$$ $$\mathstrut -\mathstrut 16q^{34}$$ $$\mathstrut -\mathstrut 4q^{36}$$ $$\mathstrut +\mathstrut 40q^{43}$$ $$\mathstrut -\mathstrut 8q^{48}$$ $$\mathstrut +\mathstrut 28q^{49}$$ $$\mathstrut +\mathstrut 10q^{51}$$ $$\mathstrut +\mathstrut 8q^{54}$$ $$\mathstrut -\mathstrut 34q^{57}$$ $$\mathstrut -\mathstrut 32q^{64}$$ $$\mathstrut -\mathstrut 32q^{66}$$ $$\mathstrut +\mathstrut 28q^{67}$$ $$\mathstrut +\mathstrut 16q^{72}$$ $$\mathstrut +\mathstrut 4q^{73}$$ $$\mathstrut +\mathstrut 8q^{76}$$ $$\mathstrut +\mathstrut 14q^{81}$$ $$\mathstrut -\mathstrut 32q^{82}$$ $$\mathstrut +\mathstrut 16q^{88}$$ $$\mathstrut -\mathstrut 16q^{96}$$ $$\mathstrut +\mathstrut 40q^{97}$$ $$\mathstrut -\mathstrut 26q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 1.22474 + 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i
1.41421i −1.72474 + 0.158919i −2.00000 0 0.224745 + 2.43916i 0 2.82843i 2.94949 0.548188i 0
251.2 1.41421i 0.724745 1.57313i −2.00000 0 −2.22474 1.02494i 0 2.82843i −1.94949 2.28024i 0
251.3 1.41421i −1.72474 0.158919i −2.00000 0 0.224745 2.43916i 0 2.82843i 2.94949 + 0.548188i 0
251.4 1.41421i 0.724745 + 1.57313i −2.00000 0 −2.22474 + 1.02494i 0 2.82843i −1.94949 + 2.28024i 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.d Odd 1 CM by $$\Q(\sqrt{-2})$$ yes
3.b Odd 1 yes
24.f Even 1 yes

## Hecke kernels

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

 $$T_{7}$$ $$T_{11}^{4}$$ $$\mathstrut +\mathstrut 58 T_{11}^{2}$$ $$\mathstrut +\mathstrut 625$$ $$T_{23}$$ $$T_{43}$$ $$\mathstrut -\mathstrut 10$$