# 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 $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.19775603032$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 12q^{6} + O(q^{10})$$ $$8q + 12q^{6} - 8q^{16} - 24q^{21} - 16q^{31} + 12q^{36} - 48q^{46} - 36q^{51} + 112q^{61} + 36q^{66} - 8q^{76} + 36q^{81} - 12q^{96} + O(q^{100})$$

## 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 Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.2.e.b 8
3.b odd 2 1 inner 150.2.e.b 8
4.b odd 2 1 1200.2.v.l 8
5.b even 2 1 inner 150.2.e.b 8
5.c odd 4 2 inner 150.2.e.b 8
12.b even 2 1 1200.2.v.l 8
15.d odd 2 1 inner 150.2.e.b 8
15.e even 4 2 inner 150.2.e.b 8
20.d odd 2 1 1200.2.v.l 8
20.e even 4 2 1200.2.v.l 8
60.h even 2 1 1200.2.v.l 8
60.l odd 4 2 1200.2.v.l 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.e.b 8 1.a even 1 1 trivial
150.2.e.b 8 3.b odd 2 1 inner
150.2.e.b 8 5.b even 2 1 inner
150.2.e.b 8 5.c odd 4 2 inner
150.2.e.b 8 15.d odd 2 1 inner
150.2.e.b 8 15.e even 4 2 inner
1200.2.v.l 8 4.b odd 2 1
1200.2.v.l 8 12.b even 2 1
1200.2.v.l 8 20.d odd 2 1
1200.2.v.l 8 20.e even 4 2
1200.2.v.l 8 60.h even 2 1
1200.2.v.l 8 60.l odd 4 2

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ $$81 - 9 T^{4} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 144 + T^{4} )^{2}$$
$11$ $$( 27 + T^{2} )^{4}$$
$13$ $$T^{8}$$
$17$ $$( 81 + T^{4} )^{2}$$
$19$ $$( 1 + T^{2} )^{4}$$
$23$ $$( 1296 + T^{4} )^{2}$$
$29$ $$T^{8}$$
$31$ $$( 2 + T )^{8}$$
$37$ $$( 144 + T^{4} )^{2}$$
$41$ $$( 27 + T^{2} )^{4}$$
$43$ $$( 144 + T^{4} )^{2}$$
$47$ $$T^{8}$$
$53$ $$( 1296 + T^{4} )^{2}$$
$59$ $$( -108 + T^{2} )^{4}$$
$61$ $$( -14 + T )^{8}$$
$67$ $$( 729 + T^{4} )^{2}$$
$71$ $$T^{8}$$
$73$ $$( 5625 + T^{4} )^{2}$$
$79$ $$( 196 + T^{2} )^{4}$$
$83$ $$( 81 + T^{4} )^{2}$$
$89$ $$( -243 + T^{2} )^{4}$$
$97$ $$( 2304 + T^{4} )^{2}$$