Embedded newform 603.2.f.c.238.7

• Label: 603.2.f.c.238.7
• Level: $603$
• Weight: $2$
• Character: 603.238
• Analytic conductor: $4.815$
• Analytic rank: $0$
• Dimension: $128$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 603 = 3^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	603.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.81497924188\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(64\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			238.7
Character	\(\chi\)	\(=\)	603.238
Dual form			603.2.f.c.565.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18873 + 2.05893i) q^{2} +(-1.68608 - 0.396396i) q^{3} +(-1.82614 - 3.16297i) q^{4} +(-1.03610 + 1.79458i) q^{5} +(2.82044 - 3.00033i) q^{6} +1.60807 q^{7} +3.92823 q^{8} +(2.68574 + 1.33671i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + q^{2} - 3 q^{3} - 65 q^{4} + 5 q^{5} + 3 q^{6} - 8 q^{7} - 36 q^{8} + 5 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/603\mathbb{Z}\right)^\times\).

\(n\)	\(136\)	\(470\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18873	+	2.05893i	−0.840557	+	1.45589i	0.0488681	+	0.998805i	\(0.484439\pi\)
							−0.889425	+	0.457082i	\(0.848895\pi\)
\(3\)	−1.68608	−	0.396396i	−0.973460	−	0.228859i
							\(5\)	−1.03610	+	1.79458i	−0.463359	+	0.802561i	−0.999126	−	0.0418046i	\(-0.986689\pi\)
0.535767	+	0.844366i	\(0.320023\pi\)
\(7\)	1.60807			0.607795			0.303897	−	0.952705i	\(-0.401712\pi\)
							0.303897	+	0.952705i	\(0.401712\pi\)
\(11\)	5.86954			1.76973			0.884867	−	0.465844i	\(-0.154249\pi\)
							0.884867	+	0.465844i	\(0.154249\pi\)
\(13\)	−6.05075			−1.67818			−0.839089	−	0.543995i	\(-0.816911\pi\)
							−0.839089	+	0.543995i	\(0.816911\pi\)
\(17\)	2.32438	+	4.02594i	0.563745	+	0.976435i	0.997165	+	0.0752431i	\(0.0239733\pi\)
							−0.433420	+	0.901192i	\(0.642693\pi\)
\(19\)	3.46990	+	6.01005i	0.796050	+	1.37880i	0.922170	+	0.386784i	\(0.126414\pi\)
							−0.126120	+	0.992015i	\(0.540252\pi\)
\(23\)	1.30173			0.271429			0.135715	−	0.990748i	\(-0.456667\pi\)
							0.135715	+	0.990748i	\(0.456667\pi\)
\(29\)	−1.17168			−0.217575			−0.108787	−	0.994065i	\(-0.534697\pi\)
							−0.108787	+	0.994065i	\(0.534697\pi\)
\(31\)	−2.25613	−	3.90772i	−0.405212	−	0.701848i	0.589134	−	0.808035i	\(-0.299469\pi\)
							−0.994346	+	0.106187i	\(0.966136\pi\)
\(37\)	−3.53436	−	6.12169i	−0.581045	−	1.00640i	−0.995356	−	0.0962646i	\(-0.969311\pi\)
							0.414310	−	0.910136i	\(-0.364023\pi\)
\(41\)	2.38442	+	4.12993i	0.372384	+	0.644987i	0.989932	−	0.141546i	\(-0.0452072\pi\)
							−0.617548	+	0.786533i	\(0.711874\pi\)
\(43\)	−0.801782	−	1.38873i	−0.122271	−	0.211779i	0.798392	−	0.602138i	\(-0.205684\pi\)
							−0.920663	+	0.390359i	\(0.872351\pi\)
\(47\)	−5.99274			−0.874131			−0.437066	−	0.899430i	\(-0.643982\pi\)
							−0.437066	+	0.899430i	\(0.643982\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.f.c.238.7	✓	128
9.7	even	3		603.2.h.c.439.58	yes	128
67.29	even	3		603.2.h.c.364.58	yes	128
603.565	even	3	inner	603.2.f.c.565.7	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.f.c.238.7	✓	128	1.1	even	1	trivial
603.2.f.c.565.7	yes	128	603.565	even	3	inner
603.2.h.c.364.58	yes	128	67.29	even	3
603.2.h.c.439.58	yes	128	9.7	even	3