Embedded newform 150.3.j.a.11.2

• Label: 150.3.j.a.11.2
• Level: $150$
• Weight: $3$
• Character: 150.11
• Analytic conductor: $4.087$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	150.j (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.08720396540\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(20\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			11.2
Character	\(\chi\)	\(=\)	150.11
Dual form			150.3.j.a.41.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.831254 - 1.14412i) q^{2} +(-2.74953 - 1.20004i) q^{3} +(-0.618034 + 1.90211i) q^{4} +(-3.75827 - 3.29779i) q^{5} +(0.912562 + 4.14334i) q^{6} -3.79877 q^{7} +(2.68999 - 0.874032i) q^{8} +(6.11980 + 6.59909i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 4 q^{3} + 40 q^{4} - 8 q^{7} + 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{4}{5}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.831254	−	1.14412i	−0.415627	−	0.572061i
							\(3\)	−2.74953	−	1.20004i	−0.916509	−	0.400014i
\(5\)	−3.75827	−	3.29779i	−0.751655	−	0.659557i
							\(7\)	−3.79877			−0.542681			−0.271341	−	0.962483i	\(-0.587467\pi\)
−0.271341	+	0.962483i	\(0.587467\pi\)
\(11\)	7.84659	+	10.7999i	0.713327	+	0.981810i	0.999719	+	0.0236977i	\(0.00754390\pi\)
							−0.286392	+	0.958112i	\(0.592456\pi\)
\(13\)	0.230134	+	0.167202i	0.0177026	+	0.0128617i	0.596601	−	0.802538i	\(-0.296517\pi\)
							−0.578899	+	0.815399i	\(0.696517\pi\)
\(17\)	−14.6058	+	4.74571i	−0.859165	+	0.279159i	−0.705280	−	0.708929i	\(-0.749179\pi\)
							−0.153885	+	0.988089i	\(0.549179\pi\)
\(19\)	2.84539	+	8.75720i	0.149757	+	0.460905i	0.997592	−	0.0693551i	\(-0.0220942\pi\)
							−0.847835	+	0.530260i	\(0.822094\pi\)
\(23\)	21.7602	+	29.9503i	0.946094	+	1.30219i	0.953241	+	0.302212i	\(0.0977251\pi\)
							−0.00714672	+	0.999974i	\(0.502275\pi\)
\(29\)	−16.1668	−	5.25293i	−0.557477	−	0.181135i	0.0167084	−	0.999860i	\(-0.494681\pi\)
							−0.574186	+	0.818725i	\(0.694681\pi\)
\(31\)	0.884799	+	2.72313i	0.0285419	+	0.0878430i	0.964313	−	0.264766i	\(-0.0852946\pi\)
							−0.935771	+	0.352609i	\(0.885295\pi\)
\(37\)	−36.8708	−	26.7882i	−0.996509	−	0.724006i	−0.0351725	−	0.999381i	\(-0.511198\pi\)
							−0.961337	+	0.275375i	\(0.911198\pi\)
\(41\)	−28.3234	+	38.9838i	−0.690814	+	0.950824i	−1.00000		0.000177761i	\(-0.999943\pi\)
							0.309186	+	0.951002i	\(0.399943\pi\)
\(43\)	−14.1506			−0.329083			−0.164541	−	0.986370i	\(-0.552614\pi\)
							−0.164541	+	0.986370i	\(0.552614\pi\)
\(47\)	−44.4607	−	14.4461i	−0.945972	−	0.307365i	−0.204894	−	0.978784i	\(-0.565685\pi\)
							−0.741078	+	0.671419i	\(0.765685\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.j.a.11.2	✓	80
3.2	odd	2	inner	150.3.j.a.11.19	yes	80
25.16	even	5	inner	150.3.j.a.41.19	yes	80
75.41	odd	10	inner	150.3.j.a.41.2	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.j.a.11.2	✓	80	1.1	even	1	trivial
150.3.j.a.11.19	yes	80	3.2	odd	2	inner
150.3.j.a.41.2	yes	80	75.41	odd	10	inner
150.3.j.a.41.19	yes	80	25.16	even	5	inner