Properties

 Label 150.2.g.a 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2}$$ $$+ \zeta_{10}^{3} q^{3}$$ $$-\zeta_{10}^{3} q^{4}$$ $$+ ( 1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{5}$$ $$-\zeta_{10}^{2} q^{6}$$ $$+ ( 2 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{7}$$ $$+ \zeta_{10}^{2} q^{8}$$ $$-\zeta_{10} q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2}$$ $$+ \zeta_{10}^{3} q^{3}$$ $$-\zeta_{10}^{3} q^{4}$$ $$+ ( 1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{5}$$ $$-\zeta_{10}^{2} q^{6}$$ $$+ ( 2 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{7}$$ $$+ \zeta_{10}^{2} q^{8}$$ $$-\zeta_{10} q^{9}$$ $$+ ( -2 + 2 \zeta_{10} + \zeta_{10}^{3} ) q^{10}$$ $$+ ( -3 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{11}$$ $$+ \zeta_{10} q^{12}$$ $$+ ( 4 + 4 \zeta_{10}^{2} ) q^{13}$$ $$+ ( -2 + \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{14}$$ $$+ ( 2 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{15}$$ $$-\zeta_{10} q^{16}$$ $$+ ( 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{17}$$ $$+ q^{18}$$ $$+ ( -6 \zeta_{10} + 4 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{19}$$ $$+ ( -2 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{20}$$ $$+ ( -1 + \zeta_{10} + 2 \zeta_{10}^{3} ) q^{21}$$ $$+ ( 1 - \zeta_{10} - 3 \zeta_{10}^{3} ) q^{22}$$ $$+ ( -2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{23}$$ $$- q^{24}$$ $$-5 \zeta_{10}^{3} q^{25}$$ $$+ ( -4 - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{26}$$ $$+ ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{27}$$ $$+ ( 1 - \zeta_{10} - 2 \zeta_{10}^{3} ) q^{28}$$ $$+ ( 4 - 4 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{29}$$ $$+ ( -2 + \zeta_{10} - 2 \zeta_{10}^{2} ) q^{30}$$ $$+ ( \zeta_{10} + 5 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{31}$$ $$+ q^{32}$$ $$+ ( -\zeta_{10} - 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{33}$$ $$+ ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{34}$$ $$+ ( 4 - \zeta_{10} + \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{35}$$ $$+ ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{36}$$ $$-8 \zeta_{10} q^{37}$$ $$+ ( 6 - 4 \zeta_{10} + 6 \zeta_{10}^{2} ) q^{38}$$ $$+ ( -4 + 4 \zeta_{10}^{3} ) q^{39}$$ $$+ ( 2 - \zeta_{10} + 2 \zeta_{10}^{2} ) q^{40}$$ $$+ ( -6 + 4 \zeta_{10} - 6 \zeta_{10}^{2} ) q^{41}$$ $$+ ( -\zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{42}$$ $$+ ( -4 - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{43}$$ $$+ ( \zeta_{10} + 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{44}$$ $$+ ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{45}$$ $$+ ( 4 - 4 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{46}$$ $$+ ( 6 - 6 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{47}$$ $$+ ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{48}$$ $$+ ( -2 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{49}$$ $$+ 5 \zeta_{10}^{2} q^{50}$$ $$+ ( -2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{51}$$ $$+ ( 4 - 4 \zeta_{10}^{3} ) q^{52}$$ $$+ ( -5 + 5 \zeta_{10} - \zeta_{10}^{3} ) q^{53}$$ $$+ \zeta_{10}^{3} q^{54}$$ $$+ ( -5 + 5 \zeta_{10} + 5 \zeta_{10}^{3} ) q^{55}$$ $$+ ( \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{56}$$ $$+ ( 2 + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{57}$$ $$+ ( 4 \zeta_{10} + 2 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{58}$$ $$+ ( 1 + 2 \zeta_{10} + \zeta_{10}^{2} ) q^{59}$$ $$+ ( 1 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{60}$$ $$+ ( -4 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{61}$$ $$+ ( -1 - 5 \zeta_{10} - \zeta_{10}^{2} ) q^{62}$$ $$+ ( -1 - \zeta_{10} - \zeta_{10}^{2} ) q^{63}$$ $$+ ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{64}$$ $$+ ( 12 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{65}$$ $$+ ( 1 + 2 \zeta_{10} + \zeta_{10}^{2} ) q^{66}$$ $$+ ( 4 \zeta_{10} - 8 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{67}$$ $$+ ( 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{68}$$ $$+ ( -4 \zeta_{10} + 2 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{69}$$ $$+ ( -3 + 3 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{70}$$ $$+ ( -4 + 4 \zeta_{10} - 8 \zeta_{10}^{3} ) q^{71}$$ $$-\zeta_{10}^{3} q^{72}$$ $$+ ( -6 + 10 \zeta_{10} - 10 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{73}$$ $$+ 8 q^{74}$$ $$+ 5 \zeta_{10} q^{75}$$ $$+ ( -2 - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{76}$$ $$+ ( -7 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{77}$$ $$+ ( 4 - 4 \zeta_{10} - 4 \zeta_{10}^{3} ) q^{78}$$ $$+ ( -1 + \zeta_{10} + 5 \zeta_{10}^{3} ) q^{79}$$ $$+ ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{80}$$ $$+ \zeta_{10}^{2} q^{81}$$ $$+ ( 2 + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{82}$$ $$+ ( -3 \zeta_{10} + 7 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{83}$$ $$+ ( 1 + \zeta_{10} + \zeta_{10}^{2} ) q^{84}$$ $$+ ( -2 + 6 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{85}$$ $$+ ( 4 + 2 \zeta_{10} - 2 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{86}$$ $$+ ( 4 + 2 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{87}$$ $$+ ( -1 - 2 \zeta_{10} - \zeta_{10}^{2} ) q^{88}$$ $$+ ( -6 + 10 \zeta_{10} - 10 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{89}$$ $$+ ( 1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{90}$$ $$+ ( 8 + 4 \zeta_{10} + 8 \zeta_{10}^{2} ) q^{91}$$ $$+ ( 4 \zeta_{10} - 2 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{92}$$ $$+ ( -6 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{93}$$ $$+ ( 6 \zeta_{10} - 8 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{94}$$ $$+ ( 2 - 16 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{95}$$ $$+ \zeta_{10}^{3} q^{96}$$ $$+ ( 1 - \zeta_{10} + 4 \zeta_{10}^{3} ) q^{97}$$ $$+ ( 2 - 5 \zeta_{10} + 5 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{98}$$ $$+ ( 3 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut q^{2}$$ $$\mathstrut +\mathstrut q^{3}$$ $$\mathstrut -\mathstrut q^{4}$$ $$\mathstrut +\mathstrut 5q^{5}$$ $$\mathstrut +\mathstrut q^{6}$$ $$\mathstrut +\mathstrut 6q^{7}$$ $$\mathstrut -\mathstrut q^{8}$$ $$\mathstrut -\mathstrut q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut q^{2}$$ $$\mathstrut +\mathstrut q^{3}$$ $$\mathstrut -\mathstrut q^{4}$$ $$\mathstrut +\mathstrut 5q^{5}$$ $$\mathstrut +\mathstrut q^{6}$$ $$\mathstrut +\mathstrut 6q^{7}$$ $$\mathstrut -\mathstrut q^{8}$$ $$\mathstrut -\mathstrut q^{9}$$ $$\mathstrut -\mathstrut 5q^{10}$$ $$\mathstrut -\mathstrut 5q^{11}$$ $$\mathstrut +\mathstrut q^{12}$$ $$\mathstrut +\mathstrut 12q^{13}$$ $$\mathstrut -\mathstrut 4q^{14}$$ $$\mathstrut +\mathstrut 5q^{15}$$ $$\mathstrut -\mathstrut q^{16}$$ $$\mathstrut +\mathstrut 6q^{17}$$ $$\mathstrut +\mathstrut 4q^{18}$$ $$\mathstrut -\mathstrut 16q^{19}$$ $$\mathstrut -\mathstrut 5q^{20}$$ $$\mathstrut -\mathstrut q^{21}$$ $$\mathstrut -\mathstrut 10q^{23}$$ $$\mathstrut -\mathstrut 4q^{24}$$ $$\mathstrut -\mathstrut 5q^{25}$$ $$\mathstrut -\mathstrut 8q^{26}$$ $$\mathstrut +\mathstrut q^{27}$$ $$\mathstrut +\mathstrut q^{28}$$ $$\mathstrut +\mathstrut 6q^{29}$$ $$\mathstrut -\mathstrut 5q^{30}$$ $$\mathstrut -\mathstrut 3q^{31}$$ $$\mathstrut +\mathstrut 4q^{32}$$ $$\mathstrut -\mathstrut 4q^{34}$$ $$\mathstrut +\mathstrut 10q^{35}$$ $$\mathstrut -\mathstrut q^{36}$$ $$\mathstrut -\mathstrut 8q^{37}$$ $$\mathstrut +\mathstrut 14q^{38}$$ $$\mathstrut -\mathstrut 12q^{39}$$ $$\mathstrut +\mathstrut 5q^{40}$$ $$\mathstrut -\mathstrut 14q^{41}$$ $$\mathstrut -\mathstrut q^{42}$$ $$\mathstrut -\mathstrut 4q^{43}$$ $$\mathstrut +\mathstrut 10q^{46}$$ $$\mathstrut +\mathstrut 20q^{47}$$ $$\mathstrut +\mathstrut q^{48}$$ $$\mathstrut -\mathstrut 14q^{49}$$ $$\mathstrut -\mathstrut 5q^{50}$$ $$\mathstrut +\mathstrut 4q^{51}$$ $$\mathstrut +\mathstrut 12q^{52}$$ $$\mathstrut -\mathstrut 16q^{53}$$ $$\mathstrut +\mathstrut q^{54}$$ $$\mathstrut -\mathstrut 10q^{55}$$ $$\mathstrut +\mathstrut q^{56}$$ $$\mathstrut -\mathstrut 4q^{57}$$ $$\mathstrut +\mathstrut 6q^{58}$$ $$\mathstrut +\mathstrut 5q^{59}$$ $$\mathstrut -\mathstrut 8q^{62}$$ $$\mathstrut -\mathstrut 4q^{63}$$ $$\mathstrut -\mathstrut q^{64}$$ $$\mathstrut +\mathstrut 40q^{65}$$ $$\mathstrut +\mathstrut 5q^{66}$$ $$\mathstrut +\mathstrut 16q^{67}$$ $$\mathstrut -\mathstrut 4q^{68}$$ $$\mathstrut -\mathstrut 10q^{69}$$ $$\mathstrut -\mathstrut 5q^{70}$$ $$\mathstrut -\mathstrut 20q^{71}$$ $$\mathstrut -\mathstrut q^{72}$$ $$\mathstrut +\mathstrut 2q^{73}$$ $$\mathstrut +\mathstrut 32q^{74}$$ $$\mathstrut +\mathstrut 5q^{75}$$ $$\mathstrut +\mathstrut 4q^{76}$$ $$\mathstrut -\mathstrut 15q^{77}$$ $$\mathstrut +\mathstrut 8q^{78}$$ $$\mathstrut +\mathstrut 2q^{79}$$ $$\mathstrut -\mathstrut q^{81}$$ $$\mathstrut -\mathstrut 4q^{82}$$ $$\mathstrut -\mathstrut 13q^{83}$$ $$\mathstrut +\mathstrut 4q^{84}$$ $$\mathstrut +\mathstrut 16q^{86}$$ $$\mathstrut +\mathstrut 14q^{87}$$ $$\mathstrut -\mathstrut 5q^{88}$$ $$\mathstrut +\mathstrut 2q^{89}$$ $$\mathstrut +\mathstrut 5q^{90}$$ $$\mathstrut +\mathstrut 28q^{91}$$ $$\mathstrut +\mathstrut 10q^{92}$$ $$\mathstrut -\mathstrut 22q^{93}$$ $$\mathstrut +\mathstrut 20q^{94}$$ $$\mathstrut -\mathstrut 10q^{95}$$ $$\mathstrut +\mathstrut q^{96}$$ $$\mathstrut +\mathstrut 7q^{97}$$ $$\mathstrut -\mathstrut 4q^{98}$$ $$\mathstrut +\mathstrut 10q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

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 0.690983 + 2.12663i 0.809017 + 0.587785i 0.381966 −0.809017 0.587785i 0.309017 0.951057i −1.80902 + 1.31433i
61.1 −0.809017 + 0.587785i −0.309017 + 0.951057i 0.309017 0.951057i 1.80902 1.31433i −0.309017 0.951057i 2.61803 0.309017 + 0.951057i −0.809017 0.587785i −0.690983 + 2.12663i
91.1 −0.809017 0.587785i −0.309017 0.951057i 0.309017 + 0.951057i 1.80902 + 1.31433i −0.309017 + 0.951057i 2.61803 0.309017 0.951057i −0.809017 + 0.587785i −0.690983 2.12663i
121.1 0.309017 0.951057i 0.809017 + 0.587785i −0.809017 0.587785i 0.690983 2.12663i 0.809017 0.587785i 0.381966 −0.809017 + 0.587785i 0.309017 + 0.951057i −1.80902 1.31433i
 $$n$$: 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 $$T_{7}^{2}$$ $$\mathstrut -\mathstrut 3 T_{7}$$ $$\mathstrut +\mathstrut 1$$ acting on $$S_{2}^{\mathrm{new}}(150, [\chi])$$.