# Properties

 Label 150.2.e.b Level 150 Weight 2 Character orbit 150.e Analytic conductor 1.198 Analytic rank 0 Dimension 8 CM No Inner twists 8

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: No Analytic conductor: $$1.19775603032$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -\zeta_{24} + \zeta_{24}^{5} ) q^{2}$$ $$+ ( -2 \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{3}$$ $$-\zeta_{24}^{6} q^{4}$$ $$+ ( 2 - \zeta_{24}^{4} ) q^{6}$$ $$+ ( 2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{7}$$ $$+ \zeta_{24}^{3} q^{8}$$ $$+ 3 \zeta_{24}^{2} q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -\zeta_{24} + \zeta_{24}^{5} ) q^{2}$$ $$+ ( -2 \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{3}$$ $$-\zeta_{24}^{6} q^{4}$$ $$+ ( 2 - \zeta_{24}^{4} ) q^{6}$$ $$+ ( 2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{7}$$ $$+ \zeta_{24}^{3} q^{8}$$ $$+ 3 \zeta_{24}^{2} q^{9}$$ $$+ ( -3 + 6 \zeta_{24}^{4} ) q^{11}$$ $$+ ( -\zeta_{24} + 2 \zeta_{24}^{5} ) q^{12}$$ $$+ ( -4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{14}$$ $$- q^{16}$$ $$+ ( 3 \zeta_{24} - 3 \zeta_{24}^{5} ) q^{17}$$ $$+ ( -3 \zeta_{24}^{3} + 3 \zeta_{24}^{7} ) q^{18}$$ $$-\zeta_{24}^{6} q^{19}$$ $$-6 \zeta_{24}^{4} q^{21}$$ $$+ ( -3 \zeta_{24} - 3 \zeta_{24}^{5} ) q^{22}$$ $$+ 6 \zeta_{24}^{3} q^{23}$$ $$+ ( -\zeta_{24}^{2} - \zeta_{24}^{6} ) q^{24}$$ $$+ ( -3 \zeta_{24} - 3 \zeta_{24}^{5} ) q^{27}$$ $$+ ( 2 \zeta_{24}^{3} - 4 \zeta_{24}^{7} ) q^{28}$$ $$-2 q^{31}$$ $$+ ( \zeta_{24} - \zeta_{24}^{5} ) q^{32}$$ $$-9 \zeta_{24}^{7} q^{33}$$ $$+ 3 \zeta_{24}^{6} q^{34}$$ $$+ ( 3 - 3 \zeta_{24}^{4} ) q^{36}$$ $$+ ( -2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{37}$$ $$+ \zeta_{24}^{3} q^{38}$$ $$+ ( 3 - 6 \zeta_{24}^{4} ) q^{41}$$ $$+ 6 \zeta_{24} q^{42}$$ $$+ ( -2 \zeta_{24}^{3} + 4 \zeta_{24}^{7} ) q^{43}$$ $$+ ( 6 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{44}$$ $$-6 q^{46}$$ $$+ ( 2 \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{48}$$ $$+ 5 \zeta_{24}^{6} q^{49}$$ $$+ ( -6 + 3 \zeta_{24}^{4} ) q^{51}$$ $$-6 \zeta_{24}^{3} q^{53}$$ $$+ ( 6 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{54}$$ $$+ ( -2 + 4 \zeta_{24}^{4} ) q^{56}$$ $$+ ( -\zeta_{24} + 2 \zeta_{24}^{5} ) q^{57}$$ $$+ ( -12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{59}$$ $$+ 14 q^{61}$$ $$+ ( 2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{62}$$ $$+ ( 6 \zeta_{24}^{3} + 6 \zeta_{24}^{7} ) q^{63}$$ $$+ \zeta_{24}^{6} q^{64}$$ $$+ 9 \zeta_{24}^{4} q^{66}$$ $$+ ( 3 \zeta_{24} + 3 \zeta_{24}^{5} ) q^{67}$$ $$-3 \zeta_{24}^{3} q^{68}$$ $$+ ( -6 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{69}$$ $$+ 3 \zeta_{24}^{5} q^{72}$$ $$+ ( -5 \zeta_{24}^{3} + 10 \zeta_{24}^{7} ) q^{73}$$ $$+ ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{74}$$ $$- q^{76}$$ $$+ ( -18 \zeta_{24} + 18 \zeta_{24}^{5} ) q^{77}$$ $$-14 \zeta_{24}^{6} q^{79}$$ $$+ 9 \zeta_{24}^{4} q^{81}$$ $$+ ( 3 \zeta_{24} + 3 \zeta_{24}^{5} ) q^{82}$$ $$-3 \zeta_{24}^{3} q^{83}$$ $$+ ( -6 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{84}$$ $$+ ( 2 - 4 \zeta_{24}^{4} ) q^{86}$$ $$+ ( -3 \zeta_{24}^{3} + 6 \zeta_{24}^{7} ) q^{88}$$ $$+ ( 18 \zeta_{24}^{2} - 9 \zeta_{24}^{6} ) q^{89}$$ $$+ ( 6 \zeta_{24} - 6 \zeta_{24}^{5} ) q^{92}$$ $$+ ( 4 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{93}$$ $$+ ( -2 + \zeta_{24}^{4} ) q^{96}$$ $$+ ( 4 \zeta_{24} + 4 \zeta_{24}^{5} ) q^{97}$$ $$-5 \zeta_{24}^{3} q^{98}$$ $$+ ( -9 \zeta_{24}^{2} + 18 \zeta_{24}^{6} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q$$ $$\mathstrut +\mathstrut 12q^{6}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$8q$$ $$\mathstrut +\mathstrut 12q^{6}$$ $$\mathstrut -\mathstrut 8q^{16}$$ $$\mathstrut -\mathstrut 24q^{21}$$ $$\mathstrut -\mathstrut 16q^{31}$$ $$\mathstrut +\mathstrut 12q^{36}$$ $$\mathstrut -\mathstrut 48q^{46}$$ $$\mathstrut -\mathstrut 36q^{51}$$ $$\mathstrut +\mathstrut 112q^{61}$$ $$\mathstrut +\mathstrut 36q^{66}$$ $$\mathstrut -\mathstrut 8q^{76}$$ $$\mathstrut +\mathstrut 36q^{81}$$ $$\mathstrut -\mathstrut 12q^{96}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Character Values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1$$ $$\zeta_{24}^{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 0.965926 + 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i −0.965926 − 0.258819i 0.965926 − 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i −0.965926 + 0.258819i
−0.707107 + 0.707107i −1.67303 0.448288i 1.00000i 0 1.50000 0.866025i 2.44949 + 2.44949i 0.707107 + 0.707107i 2.59808 + 1.50000i 0
107.2 −0.707107 + 0.707107i −0.448288 1.67303i 1.00000i 0 1.50000 + 0.866025i −2.44949 2.44949i 0.707107 + 0.707107i −2.59808 + 1.50000i 0
107.3 0.707107 0.707107i 0.448288 + 1.67303i 1.00000i 0 1.50000 + 0.866025i 2.44949 + 2.44949i −0.707107 0.707107i −2.59808 + 1.50000i 0
107.4 0.707107 0.707107i 1.67303 + 0.448288i 1.00000i 0 1.50000 0.866025i −2.44949 2.44949i −0.707107 0.707107i 2.59808 + 1.50000i 0
143.1 −0.707107 0.707107i −1.67303 + 0.448288i 1.00000i 0 1.50000 + 0.866025i 2.44949 2.44949i 0.707107 0.707107i 2.59808 1.50000i 0
143.2 −0.707107 0.707107i −0.448288 + 1.67303i 1.00000i 0 1.50000 0.866025i −2.44949 + 2.44949i 0.707107 0.707107i −2.59808 1.50000i 0
143.3 0.707107 + 0.707107i 0.448288 1.67303i 1.00000i 0 1.50000 0.866025i 2.44949 2.44949i −0.707107 + 0.707107i −2.59808 1.50000i 0
143.4 0.707107 + 0.707107i 1.67303 0.448288i 1.00000i 0 1.50000 + 0.866025i −2.44949 + 2.44949i −0.707107 + 0.707107i 2.59808 1.50000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 143.4 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.b Even 1 yes
5.c Odd 2 yes
15.d Odd 1 yes
15.e Even 2 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{7}^{4}$$ $$\mathstrut +\mathstrut 144$$ acting on $$S_{2}^{\mathrm{new}}(150, [\chi])$$.