Embedded newform 151.2.h.a.29.1

• Label: 151.2.h.a.29.1
• Level: $151$
• Weight: $2$
• Character: 151.29
• Analytic conductor: $1.206$
• Analytic rank: $0$
• Dimension: $220$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	151.h (of order \(25\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.20574107052\)
Analytic rank:	\(0\)
Dimension:	\(220\)
Relative dimension:	\(11\) over \(\Q(\zeta_{25})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{25}]$

Embedding invariants

Embedding label			29.1
Character	\(\chi\)	\(=\)	151.29
Dual form			151.2.h.a.125.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.95484 - 1.42028i) q^{2} +(0.392804 - 2.05915i) q^{3} +(1.18619 + 3.65073i) q^{4} +(-1.81198 - 0.717412i) q^{5} +(-3.69244 + 3.46743i) q^{6} +(-3.21462 + 0.825375i) q^{7} +(1.37286 - 4.22523i) q^{8} +(-1.29648 - 0.513314i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 15 q^{2} - 20 q^{3} - 55 q^{4} - 20 q^{5} - 20 q^{7} + 5 q^{8} - 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{19}{25}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.95484	−	1.42028i	−1.38228	−	1.00429i	−0.996663	−	0.0816310i	\(-0.973987\pi\)
							−0.385621	−	0.922657i	\(-0.626013\pi\)
\(3\)	0.392804	−	2.05915i	0.226786	−	1.18885i	−0.668621	−	0.743603i	\(-0.733115\pi\)
							0.895407	−	0.445249i	\(-0.146885\pi\)
\(5\)	−1.81198	−	0.717412i	−0.810340	−	0.320836i	−0.0738370	−	0.997270i	\(-0.523524\pi\)
							−0.736503	+	0.676434i	\(0.763524\pi\)
\(7\)	−3.21462	+	0.825375i	−1.21501	+	0.311962i	−0.801198	−	0.598399i	\(-0.795804\pi\)
							−0.413815	+	0.910361i	\(0.635804\pi\)
\(11\)	−0.365891	−	1.91807i	−0.110320	−	0.578320i	−0.993923	−	0.110081i	\(-0.964889\pi\)
							0.883602	−	0.468238i	\(-0.155111\pi\)
\(13\)	−1.32513	−	0.728496i	−0.367525	−	0.202048i	0.287280	−	0.957847i	\(-0.407249\pi\)
							−0.654805	+	0.755798i	\(0.727249\pi\)
\(17\)	−3.83969	−	0.985865i	−0.931262	−	0.239107i	−0.247563	−	0.968872i	\(-0.579630\pi\)
							−0.683699	+	0.729764i	\(0.739630\pi\)
\(19\)	2.47416	+	7.61468i	0.567611	+	1.74693i	0.660063	+	0.751210i	\(0.270530\pi\)
							−0.0924520	+	0.995717i	\(0.529470\pi\)
\(23\)	−0.207814	−	0.639587i	−0.0433323	−	0.133363i	0.927050	−	0.374938i	\(-0.122336\pi\)
							−0.970382	+	0.241575i	\(0.922336\pi\)
\(29\)	3.40700	−	7.24024i	0.632664	−	1.34448i	−0.289313	−	0.957234i	\(-0.593427\pi\)
							0.921977	−	0.387244i	\(-0.126573\pi\)
\(31\)	−5.61268	−	6.78457i	−1.00807	−	1.21854i	−0.975710	−	0.219068i	\(-0.929698\pi\)
							−0.0323579	−	0.999476i	\(-0.510302\pi\)
\(37\)	−7.11783	−	3.91306i	−1.17016	−	0.643303i	−0.226436	−	0.974026i	\(-0.572707\pi\)
							−0.943728	+	0.330723i	\(0.892707\pi\)
\(41\)	−5.25549	+	4.93524i	−0.820770	+	0.770754i	−0.976149	−	0.217101i	\(-0.930340\pi\)
							0.155379	+	0.987855i	\(0.450340\pi\)
\(43\)	7.17488	+	1.84220i	1.09416	+	0.280932i	0.752301	−	0.658820i	\(-0.228944\pi\)
							0.341858	+	0.939752i	\(0.388944\pi\)
\(47\)	4.48915	−	4.21560i	0.654810	−	0.614908i	−0.284213	−	0.958761i	\(-0.591732\pi\)
							0.939023	+	0.343853i	\(0.111732\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	151.2.h.a.29.1	✓	220
151.125	even	25	inner	151.2.h.a.125.1	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
151.2.h.a.29.1	✓	220	1.1	even	1	trivial
151.2.h.a.125.1	yes	220	151.125	even	25	inner