Embedded newform 231.2.bc.a.5.6

• Label: 231.2.bc.a.5.6
• Level: $231$
• Weight: $2$
• Character: 231.5
• Analytic conductor: $1.845$
• Analytic rank: $0$
• Dimension: $224$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 231 = 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	231.bc (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.84454428669\)
Analytic rank:	\(0\)
Dimension:	\(224\)
Relative dimension:	\(28\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			5.6
Character	\(\chi\)	\(=\)	231.5
Dual form			231.2.bc.a.185.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.396988 + 1.86768i) q^{2} +(-1.30026 - 1.14426i) q^{3} +(-1.50354 - 0.669417i) q^{4} +(-1.28280 - 1.42469i) q^{5} +(2.65330 - 1.97420i) q^{6} +(2.43431 - 1.03640i) q^{7} +(-0.397497 + 0.547108i) q^{8} +(0.381326 + 2.97567i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 224 q - 9 q^{3} - 30 q^{4} - 16 q^{7} + 3 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/231\mathbb{Z}\right)^\times\).

\(n\)	\(155\)	\(199\)	\(211\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.396988	+	1.86768i	−0.280713	+	1.32065i	0.581270	+	0.813711i	\(0.302556\pi\)
							−0.861983	+	0.506938i	\(0.830777\pi\)
\(3\)	−1.30026	−	1.14426i	−0.750703	−	0.660640i
							\(5\)	−1.28280	−	1.42469i	−0.573685	−	0.637142i	0.384557	−	0.923101i	\(-0.374354\pi\)
−0.958242	+	0.285959i	\(0.907688\pi\)
\(7\)	2.43431	−	1.03640i	0.920084	−	0.391721i
							\(11\)	3.28090	−	0.485474i	0.989229	−	0.146376i
\(13\)	5.98313	−	1.94404i	1.65942	−	0.539179i								0.678671	−	0.734443i	\(-0.262556\pi\)
							0.980751	+	0.195264i	\(0.0625564\pi\)
\(17\)	2.15517	−	0.458096i	0.522706	−	0.111105i	0.0610001	−	0.998138i	\(-0.480571\pi\)
							0.461706	+	0.887033i	\(0.347238\pi\)
\(19\)	1.18799	+	2.66827i	0.272544	+	0.612144i	0.997019	−	0.0771572i	\(-0.0245843\pi\)
							−0.724475	+	0.689301i	\(0.757918\pi\)
\(23\)	−2.38759	−	1.37847i	−0.497847	−	0.287432i	0.229977	−	0.973196i	\(-0.426135\pi\)
							−0.727824	+	0.685764i	\(0.759468\pi\)
\(29\)	−0.173096	−	0.238246i	−0.0321431	−	0.0442412i	0.792643	−	0.609686i	\(-0.208704\pi\)
							−0.824786	+	0.565445i	\(0.808704\pi\)
\(31\)	−3.12594	−	2.81461i	−0.561436	−	0.505519i	0.338837	−	0.940845i	\(-0.389966\pi\)
							−0.900273	+	0.435326i	\(0.856633\pi\)
\(37\)	0.889476	+	8.46280i	0.146229	+	1.39128i	0.783862	+	0.620935i	\(0.213247\pi\)
							−0.637633	+	0.770340i	\(0.720086\pi\)
\(41\)	−3.70133	−	2.68918i	−0.578051	−	0.419979i	0.259970	−	0.965617i	\(-0.416287\pi\)
							−0.838021	+	0.545638i	\(0.816287\pi\)
\(43\)	−4.62662			−0.705553			−0.352777	−	0.935708i	\(-0.614762\pi\)
							−0.352777	+	0.935708i	\(0.614762\pi\)
\(47\)	9.97277	−	4.44016i	1.45468	−	0.647665i	0.481235	−	0.876592i	\(-0.340188\pi\)
							0.973444	+	0.228927i	\(0.0735218\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	231.2.bc.a.5.6	✓	224
3.2	odd	2	inner	231.2.bc.a.5.23	yes	224
7.3	odd	6	inner	231.2.bc.a.38.6	yes	224
11.9	even	5	inner	231.2.bc.a.152.23	yes	224
21.17	even	6	inner	231.2.bc.a.38.23	yes	224
33.20	odd	10	inner	231.2.bc.a.152.6	yes	224
77.31	odd	30	inner	231.2.bc.a.185.23	yes	224
231.185	even	30	inner	231.2.bc.a.185.6	yes	224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.bc.a.5.6	✓	224	1.1	even	1	trivial
231.2.bc.a.5.23	yes	224	3.2	odd	2	inner
231.2.bc.a.38.6	yes	224	7.3	odd	6	inner
231.2.bc.a.38.23	yes	224	21.17	even	6	inner
231.2.bc.a.152.6	yes	224	33.20	odd	10	inner
231.2.bc.a.152.23	yes	224	11.9	even	5	inner
231.2.bc.a.185.6	yes	224	231.185	even	30	inner
231.2.bc.a.185.23	yes	224	77.31	odd	30	inner