Embedded newform 403.2.bz.a.38.9

• Label: 403.2.bz.a.38.9
• Level: $403$
• Weight: $2$
• Character: 403.38
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bz (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(36\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			38.9
Character	\(\chi\)	\(=\)	403.38
Dual form			403.2.bz.a.350.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.60645 + 0.521968i) q^{2} +(1.69082 - 0.359396i) q^{3} +(0.690208 - 0.501465i) q^{4} +(-1.67367 - 0.966293i) q^{5} +(-2.52864 + 1.45991i) q^{6} +(0.582041 - 1.30729i) q^{7} +(1.13865 - 1.56721i) q^{8} +(-0.0109161 + 0.00486016i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 16 q^{3} + 60 q^{4} + 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{14}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.60645	+	0.521968i	−1.13593	+	0.369087i	−0.815828	−	0.578294i	\(-0.803719\pi\)
							−0.320106	+	0.947382i	\(0.603719\pi\)
\(3\)	1.69082	−	0.359396i	0.976198	−	0.207497i	0.307923	−	0.951411i	\(-0.400366\pi\)
							0.668275	+	0.743914i	\(0.267033\pi\)
\(5\)	−1.67367	−	0.966293i	−0.748487	−	0.432139i	0.0766598	−	0.997057i	\(-0.475574\pi\)
							−0.825147	+	0.564918i	\(0.808908\pi\)
\(7\)	0.582041	−	1.30729i	0.219991	−	0.494108i	−0.769510	−	0.638635i	\(-0.779499\pi\)
							0.989501	+	0.144527i	\(0.0461661\pi\)
\(11\)	−1.57526	+	0.165566i	−0.474958	+	0.0499201i	−0.338984	−	0.940792i	\(-0.610083\pi\)
							−0.135975	+	0.990712i	\(0.543417\pi\)
\(13\)	−3.56852	−	0.515402i	−0.989730	−	0.142947i
							\(17\)	0.536071	−	5.10038i	0.130016	−	1.23702i	−0.713784	−	0.700366i	\(-0.753020\pi\)
0.843800	−	0.536657i	\(-0.180313\pi\)
\(19\)	0.104618	+	0.0941987i	0.0240011	+	0.0216107i	0.681044	−	0.732243i	\(-0.261526\pi\)
							−0.657043	+	0.753853i	\(0.728193\pi\)
\(23\)	−5.83072	−	4.23627i	−1.21579	−	0.883323i	−0.220046	−	0.975490i	\(-0.570621\pi\)
							−0.995744	+	0.0921668i	\(0.970621\pi\)
\(29\)	0.310073	+	0.954306i	0.0575791	+	0.177210i	0.975710	−	0.219068i	\(-0.0703016\pi\)
							−0.918131	+	0.396278i	\(0.870302\pi\)
\(31\)	5.23746	+	1.88917i	0.940676	+	0.339305i
							\(37\)	−2.92749	+	1.69019i	−0.481277	+	0.277865i	−0.720948	−	0.692989i	\(-0.756294\pi\)
0.239672	+	0.970854i	\(0.422960\pi\)
\(41\)	2.01020	−	9.45725i	0.313940	−	1.47697i	−0.484442	−	0.874823i	\(-0.660977\pi\)
							0.798383	−	0.602150i	\(-0.205689\pi\)
\(43\)	5.25277	−	5.83379i	0.801039	−	0.889644i	−0.194793	−	0.980844i	\(-0.562404\pi\)
							0.995833	+	0.0911999i	\(0.0290702\pi\)
\(47\)	−7.41734	−	2.41004i	−1.08193	−	0.351540i	−0.286807	−	0.957988i	\(-0.592594\pi\)
							−0.795123	+	0.606448i	\(0.792594\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bz.a.38.9	✓	288
13.12	even	2	inner	403.2.bz.a.38.28	yes	288
31.9	even	15	inner	403.2.bz.a.350.28	yes	288
403.350	even	30	inner	403.2.bz.a.350.9	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bz.a.38.9	✓	288	1.1	even	1	trivial
403.2.bz.a.38.28	yes	288	13.12	even	2	inner
403.2.bz.a.350.9	yes	288	403.350	even	30	inner
403.2.bz.a.350.28	yes	288	31.9	even	15	inner