Embedded newform 151.2.d.a.64.2

• Label: 151.2.d.a.64.2
• Level: $151$
• Weight: $2$
• Character: 151.64
• Analytic conductor: $1.206$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	151.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.20574107052\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(3\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	x^12 - x^11 + 4*x^10 - 3*x^9 + 16*x^8 + 12*x^7 + 49*x^6 + 56*x^5 + 129*x^4 + 50*x^3 + 53*x^2 - 12*x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			64.2
Root			\(0.0994327 - 0.0722421i\) of defining polynomial
Character	\(\chi\)	\(=\)	151.64
Dual form			151.2.d.a.59.2

$q$-expansion

\(f(q)\)	\(=\)	\(q-0.877094 q^{2} +(-2.41483 + 1.75448i) q^{3} -1.23071 q^{4} +(0.174819 - 0.538037i) q^{5} +(2.11803 - 1.53884i) q^{6} +(0.391956 - 1.20632i) q^{7} +2.83363 q^{8} +(1.82616 - 5.62036i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{2} - 2 q^{3} + 8 q^{4} - 3 q^{5} + 12 q^{6} + 7 q^{7} - 30 q^{8} + 9 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/151\mathbb{Z}\right)^\times\).

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.877094			−0.620199			−0.310100	−	0.950704i	\(-0.600362\pi\)
							−0.310100	+	0.950704i	\(0.600362\pi\)
\(3\)	−2.41483	+	1.75448i	−1.39420	+	1.01295i	−0.398813	+	0.917032i	\(0.630578\pi\)
							−0.995389	+	0.0959155i	\(0.969422\pi\)
\(5\)	0.174819	−	0.538037i	0.0781814	−	0.240617i	−0.904326	−	0.426843i	\(-0.859626\pi\)
							0.982507	+	0.186226i	\(0.0596256\pi\)
\(7\)	0.391956	−	1.20632i	0.148145	−	0.455945i	−0.849257	−	0.527980i	\(-0.822949\pi\)
							0.997402	+	0.0720355i	\(0.0229495\pi\)
\(11\)	−2.00638	−	1.45772i	−0.604946	−	0.439519i	0.242685	−	0.970105i	\(-0.421972\pi\)
							−0.847631	+	0.530586i	\(0.821972\pi\)
\(13\)	2.03734	−	1.48021i	0.565057	−	0.410538i	−0.268250	−	0.963349i	\(-0.586445\pi\)
							0.833306	+	0.552812i	\(0.186445\pi\)
\(17\)	−1.41254	−	4.34736i	−0.342592	−	1.05439i	−0.962860	−	0.270000i	\(-0.912976\pi\)
							0.620268	−	0.784390i	\(-0.287024\pi\)
\(19\)	−4.63265			−1.06280			−0.531402	−	0.847120i	\(-0.678334\pi\)
							−0.531402	+	0.847120i	\(0.678334\pi\)
\(23\)	−0.156135			−0.0325564			−0.0162782	−	0.999868i	\(-0.505182\pi\)
							−0.0162782	+	0.999868i	\(0.505182\pi\)
\(29\)	−6.90420	−	5.01619i	−1.28208	−	0.931484i	−0.282463	−	0.959278i	\(-0.591152\pi\)
							−0.999614	+	0.0277945i	\(0.991152\pi\)
\(31\)	−1.71975	−	5.29286i	−0.308877	−	0.950625i	−0.978202	−	0.207655i	\(-0.933417\pi\)
							0.669325	−	0.742969i	\(-0.266583\pi\)
\(37\)	6.30423	−	4.58029i	1.03641	−	0.752995i	0.0668278	−	0.997765i	\(-0.478712\pi\)
							0.969581	+	0.244769i	\(0.0787122\pi\)
\(41\)	−3.39013	+	2.46307i	−0.529449	+	0.384667i	−0.820152	−	0.572146i	\(-0.806111\pi\)
							0.290702	+	0.956814i	\(0.406111\pi\)
\(43\)	0.656946	+	2.02187i	0.100183	+	0.308333i	0.988570	−	0.150764i	\(-0.0481732\pi\)
							−0.888386	+	0.459096i	\(0.848173\pi\)
\(47\)	7.82829	−	5.68759i	1.14187	−	0.829620i	0.154495	−	0.987994i	\(-0.450625\pi\)
							0.987379	+	0.158374i	\(0.0506251\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	151.2.d.a.64.2	yes	12
151.59	even	5	inner	151.2.d.a.59.2	✓	12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
151.2.d.a.59.2	✓	12	151.59	even	5	inner
151.2.d.a.64.2	yes	12	1.1	even	1	trivial