# Properties

 Label 150.2.g.b Level 150 Weight 2 Character orbit 150.g Analytic conductor 1.198 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 150.g (of order $$5$$ and degree $$4$$)

## Newform invariants

 Self dual: No Analytic conductor: $$1.19775603032$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2}$$ $$+ \zeta_{10}^{3} q^{3}$$ $$-\zeta_{10}^{3} q^{4}$$ $$+ ( 2 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{5}$$ $$+ \zeta_{10}^{2} q^{6}$$ $$+ 2 q^{7}$$ $$-\zeta_{10}^{2} q^{8}$$ $$-\zeta_{10} q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2}$$ $$+ \zeta_{10}^{3} q^{3}$$ $$-\zeta_{10}^{3} q^{4}$$ $$+ ( 2 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{5}$$ $$+ \zeta_{10}^{2} q^{6}$$ $$+ 2 q^{7}$$ $$-\zeta_{10}^{2} q^{8}$$ $$-\zeta_{10} q^{9}$$ $$+ ( 2 - \zeta_{10} + 2 \zeta_{10}^{2} ) q^{10}$$ $$+ ( 2 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{11}$$ $$+ \zeta_{10} q^{12}$$ $$+ ( -3 + 3 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{13}$$ $$+ ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{14}$$ $$+ ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{15}$$ $$-\zeta_{10} q^{16}$$ $$+ ( -3 \zeta_{10} + 6 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{17}$$ $$- q^{18}$$ $$+ ( -2 \zeta_{10} - 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{19}$$ $$+ ( 1 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{20}$$ $$+ 2 \zeta_{10}^{3} q^{21}$$ $$+ ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{22}$$ $$+ ( -6 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{23}$$ $$+ q^{24}$$ $$+ ( -5 + 5 \zeta_{10} - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{25}$$ $$+ ( -3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{26}$$ $$+ ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{27}$$ $$-2 \zeta_{10}^{3} q^{28}$$ $$+ ( -1 + \zeta_{10} + 3 \zeta_{10}^{3} ) q^{29}$$ $$+ ( -1 - \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{30}$$ $$+ ( -6 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{31}$$ $$- q^{32}$$ $$+ ( -2 \zeta_{10} + 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{33}$$ $$+ ( -3 + 6 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{34}$$ $$+ ( 4 \zeta_{10} - 2 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{35}$$ $$+ ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{36}$$ $$+ ( 3 + 4 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{37}$$ $$+ ( -2 - 4 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{38}$$ $$+ ( 3 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{39}$$ $$+ ( 1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{40}$$ $$+ ( -5 + 3 \zeta_{10} - 5 \zeta_{10}^{2} ) q^{41}$$ $$+ 2 \zeta_{10}^{2} q^{42}$$ $$+ ( 2 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{43}$$ $$+ ( 2 \zeta_{10} - 4 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{44}$$ $$+ ( 2 - 2 \zeta_{10} - \zeta_{10}^{3} ) q^{45}$$ $$+ 6 \zeta_{10}^{3} q^{46}$$ $$+ ( 2 - 2 \zeta_{10} - 8 \zeta_{10}^{3} ) q^{47}$$ $$+ ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{48}$$ $$-3 q^{49}$$ $$+ 5 \zeta_{10}^{3} q^{50}$$ $$+ ( -3 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{51}$$ $$+ ( -3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{52}$$ $$+ ( -9 + 9 \zeta_{10} + 6 \zeta_{10}^{3} ) q^{53}$$ $$-\zeta_{10}^{3} q^{54}$$ $$+ ( 6 - 8 \zeta_{10} + 6 \zeta_{10}^{2} ) q^{55}$$ $$-2 \zeta_{10}^{2} q^{56}$$ $$+ ( 6 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{57}$$ $$+ ( \zeta_{10} + 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{58}$$ $$+ ( 8 - 4 \zeta_{10} + 8 \zeta_{10}^{2} ) q^{59}$$ $$+ ( -2 + 2 \zeta_{10} + \zeta_{10}^{3} ) q^{60}$$ $$+ ( 4 - 7 \zeta_{10} + 7 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{61}$$ $$+ ( -6 - 6 \zeta_{10}^{2} ) q^{62}$$ $$-2 \zeta_{10} q^{63}$$ $$+ ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{64}$$ $$+ ( -3 + 3 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{65}$$ $$+ ( -2 + 4 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{66}$$ $$+ ( 6 \zeta_{10} - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{67}$$ $$+ ( 3 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{68}$$ $$-6 \zeta_{10}^{2} q^{69}$$ $$+ ( 4 - 2 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{70}$$ $$+ ( 10 - 10 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{71}$$ $$+ \zeta_{10}^{3} q^{72}$$ $$+ ( 8 - 5 \zeta_{10} + 5 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{73}$$ $$+ ( 7 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{74}$$ $$-5 \zeta_{10}^{2} q^{75}$$ $$+ ( -6 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{76}$$ $$+ ( 4 - 8 \zeta_{10} + 8 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{77}$$ $$+ ( 3 - 3 \zeta_{10} ) q^{78}$$ $$+ ( 2 - 2 \zeta_{10} - \zeta_{10}^{3} ) q^{80}$$ $$+ \zeta_{10}^{2} q^{81}$$ $$+ ( -2 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{82}$$ $$+ 6 \zeta_{10}^{2} q^{83}$$ $$+ 2 \zeta_{10} q^{84}$$ $$+ ( 3 - 12 \zeta_{10} + 12 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{85}$$ $$+ ( 2 - 2 \zeta_{10}^{3} ) q^{86}$$ $$+ ( -1 - 2 \zeta_{10} - \zeta_{10}^{2} ) q^{87}$$ $$+ ( 2 - 4 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{88}$$ $$+ ( -3 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{89}$$ $$+ ( -2 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{90}$$ $$+ ( -6 + 6 \zeta_{10} - 6 \zeta_{10}^{2} ) q^{91}$$ $$+ 6 \zeta_{10}^{2} q^{92}$$ $$+ ( 6 + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{93}$$ $$+ ( -2 \zeta_{10} - 6 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{94}$$ $$+ ( 10 - 10 \zeta_{10}^{3} ) q^{95}$$ $$-\zeta_{10}^{3} q^{96}$$ $$+ ( 3 - 3 \zeta_{10} + 9 \zeta_{10}^{3} ) q^{97}$$ $$+ ( -3 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{98}$$ $$+ ( -2 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut +\mathstrut q^{2}$$ $$\mathstrut +\mathstrut q^{3}$$ $$\mathstrut -\mathstrut q^{4}$$ $$\mathstrut +\mathstrut 5q^{5}$$ $$\mathstrut -\mathstrut q^{6}$$ $$\mathstrut +\mathstrut 8q^{7}$$ $$\mathstrut +\mathstrut q^{8}$$ $$\mathstrut -\mathstrut q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut +\mathstrut q^{2}$$ $$\mathstrut +\mathstrut q^{3}$$ $$\mathstrut -\mathstrut q^{4}$$ $$\mathstrut +\mathstrut 5q^{5}$$ $$\mathstrut -\mathstrut q^{6}$$ $$\mathstrut +\mathstrut 8q^{7}$$ $$\mathstrut +\mathstrut q^{8}$$ $$\mathstrut -\mathstrut q^{9}$$ $$\mathstrut +\mathstrut 5q^{10}$$ $$\mathstrut -\mathstrut 2q^{11}$$ $$\mathstrut +\mathstrut q^{12}$$ $$\mathstrut -\mathstrut 6q^{13}$$ $$\mathstrut +\mathstrut 2q^{14}$$ $$\mathstrut -\mathstrut q^{16}$$ $$\mathstrut -\mathstrut 12q^{17}$$ $$\mathstrut -\mathstrut 4q^{18}$$ $$\mathstrut +\mathstrut 2q^{21}$$ $$\mathstrut -\mathstrut 8q^{22}$$ $$\mathstrut -\mathstrut 6q^{23}$$ $$\mathstrut +\mathstrut 4q^{24}$$ $$\mathstrut -\mathstrut 5q^{25}$$ $$\mathstrut +\mathstrut 6q^{26}$$ $$\mathstrut +\mathstrut q^{27}$$ $$\mathstrut -\mathstrut 2q^{28}$$ $$\mathstrut -\mathstrut 5q^{30}$$ $$\mathstrut -\mathstrut 12q^{31}$$ $$\mathstrut -\mathstrut 4q^{32}$$ $$\mathstrut -\mathstrut 8q^{33}$$ $$\mathstrut -\mathstrut 3q^{34}$$ $$\mathstrut +\mathstrut 10q^{35}$$ $$\mathstrut -\mathstrut q^{36}$$ $$\mathstrut +\mathstrut 13q^{37}$$ $$\mathstrut -\mathstrut 10q^{38}$$ $$\mathstrut +\mathstrut 6q^{39}$$ $$\mathstrut +\mathstrut 5q^{40}$$ $$\mathstrut -\mathstrut 12q^{41}$$ $$\mathstrut -\mathstrut 2q^{42}$$ $$\mathstrut +\mathstrut 4q^{43}$$ $$\mathstrut +\mathstrut 8q^{44}$$ $$\mathstrut +\mathstrut 5q^{45}$$ $$\mathstrut +\mathstrut 6q^{46}$$ $$\mathstrut -\mathstrut 2q^{47}$$ $$\mathstrut +\mathstrut q^{48}$$ $$\mathstrut -\mathstrut 12q^{49}$$ $$\mathstrut +\mathstrut 5q^{50}$$ $$\mathstrut -\mathstrut 18q^{51}$$ $$\mathstrut -\mathstrut 6q^{52}$$ $$\mathstrut -\mathstrut 21q^{53}$$ $$\mathstrut -\mathstrut q^{54}$$ $$\mathstrut +\mathstrut 10q^{55}$$ $$\mathstrut +\mathstrut 2q^{56}$$ $$\mathstrut +\mathstrut 20q^{57}$$ $$\mathstrut +\mathstrut 20q^{59}$$ $$\mathstrut -\mathstrut 5q^{60}$$ $$\mathstrut -\mathstrut 2q^{61}$$ $$\mathstrut -\mathstrut 18q^{62}$$ $$\mathstrut -\mathstrut 2q^{63}$$ $$\mathstrut -\mathstrut q^{64}$$ $$\mathstrut -\mathstrut 15q^{65}$$ $$\mathstrut -\mathstrut 2q^{66}$$ $$\mathstrut +\mathstrut 18q^{67}$$ $$\mathstrut +\mathstrut 18q^{68}$$ $$\mathstrut +\mathstrut 6q^{69}$$ $$\mathstrut +\mathstrut 10q^{70}$$ $$\mathstrut +\mathstrut 28q^{71}$$ $$\mathstrut +\mathstrut q^{72}$$ $$\mathstrut +\mathstrut 14q^{73}$$ $$\mathstrut +\mathstrut 22q^{74}$$ $$\mathstrut +\mathstrut 5q^{75}$$ $$\mathstrut -\mathstrut 20q^{76}$$ $$\mathstrut -\mathstrut 4q^{77}$$ $$\mathstrut +\mathstrut 9q^{78}$$ $$\mathstrut +\mathstrut 5q^{80}$$ $$\mathstrut -\mathstrut q^{81}$$ $$\mathstrut +\mathstrut 2q^{82}$$ $$\mathstrut -\mathstrut 6q^{83}$$ $$\mathstrut +\mathstrut 2q^{84}$$ $$\mathstrut -\mathstrut 15q^{85}$$ $$\mathstrut +\mathstrut 6q^{86}$$ $$\mathstrut -\mathstrut 5q^{87}$$ $$\mathstrut +\mathstrut 2q^{88}$$ $$\mathstrut -\mathstrut 5q^{89}$$ $$\mathstrut -\mathstrut 5q^{90}$$ $$\mathstrut -\mathstrut 12q^{91}$$ $$\mathstrut -\mathstrut 6q^{92}$$ $$\mathstrut +\mathstrut 12q^{93}$$ $$\mathstrut +\mathstrut 2q^{94}$$ $$\mathstrut +\mathstrut 30q^{95}$$ $$\mathstrut -\mathstrut q^{96}$$ $$\mathstrut +\mathstrut 18q^{97}$$ $$\mathstrut -\mathstrut 3q^{98}$$ $$\mathstrut -\mathstrut 12q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/150\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 −0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 − 0.587785i −0.309017 − 0.951057i
−0.309017 0.951057i 0.809017 0.587785i −0.809017 + 0.587785i 1.80902 + 1.31433i −0.809017 0.587785i 2.00000 0.809017 + 0.587785i 0.309017 0.951057i 0.690983 2.12663i
61.1 0.809017 0.587785i −0.309017 + 0.951057i 0.309017 0.951057i 0.690983 + 2.12663i 0.309017 + 0.951057i 2.00000 −0.309017 0.951057i −0.809017 0.587785i 1.80902 + 1.31433i
91.1 0.809017 + 0.587785i −0.309017 0.951057i 0.309017 + 0.951057i 0.690983 2.12663i 0.309017 0.951057i 2.00000 −0.309017 + 0.951057i −0.809017 + 0.587785i 1.80902 1.31433i
121.1 −0.309017 + 0.951057i 0.809017 + 0.587785i −0.809017 0.587785i 1.80902 1.31433i −0.809017 + 0.587785i 2.00000 0.809017 0.587785i 0.309017 + 0.951057i 0.690983 + 2.12663i
 $$n$$: 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
25.d Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{7}$$ $$\mathstrut -\mathstrut 2$$ acting on $$S_{2}^{\mathrm{new}}(150, [\chi])$$.