Embedded newform 231.2.be.a.116.18

• Label: 231.2.be.a.116.18
• Level: $231$
• Weight: $2$
• Character: 231.116
• Analytic conductor: $1.845$
• Analytic rank: $0$
• Dimension: $224$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 231 = 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	231.be (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			116.18
Character	\(\chi\)	\(=\)	231.116
Dual form			231.2.be.a.2.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.590583 - 0.655909i) q^{2} +(-1.65683 + 0.504891i) q^{3} +(0.127629 + 1.21431i) q^{4} +(-0.627181 - 2.95066i) q^{5} +(-0.647333 + 1.38491i) q^{6} +(2.53863 + 0.745226i) q^{7} +(2.29995 + 1.67101i) q^{8} +(2.49017 - 1.67304i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 224 q - 3 q^{3} + 18 q^{4} - 20 q^{6} - 20 q^{7} - 9 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(155\)	\(199\)	\(211\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.590583	−	0.655909i	0.417605	−	0.463798i	−0.497234	−	0.867617i	\(-0.665651\pi\)
							0.914839	+	0.403819i	\(0.132317\pi\)
\(3\)	−1.65683	+	0.504891i	−0.956571	+	0.291499i
							\(5\)	−0.627181	−	2.95066i	−0.280484	−	1.31957i	−0.862360	−	0.506296i	\(-0.831014\pi\)
0.581876	−	0.813278i	\(-0.302319\pi\)
\(7\)	2.53863	+	0.745226i	0.959512	+	0.281669i
							\(11\)	3.06443	−	1.26857i	0.923960	−	0.382490i
\(13\)	1.56018	−	0.506933i	0.432716	−	0.140598i								−0.0845567	−	0.996419i	\(-0.526947\pi\)
							0.517272	+	0.855821i	\(0.326947\pi\)
\(17\)	0.0265497	+	0.0294864i	0.00643924	+	0.00715150i	0.746356	−	0.665547i	\(-0.231802\pi\)
							−0.739917	+	0.672699i	\(0.765135\pi\)
\(19\)	−1.72384	−	0.181182i	−0.395475	−	0.0415661i	−0.0952960	−	0.995449i	\(-0.530380\pi\)
							−0.300179	+	0.953883i	\(0.597046\pi\)
\(23\)	3.55849	−	2.05449i	0.741995	−	0.428391i	−0.0807989	−	0.996730i	\(-0.525747\pi\)
							0.822794	+	0.568339i	\(0.192414\pi\)
\(29\)	−6.87050	+	4.99171i	−1.27582	+	0.926938i	−0.999418	−	0.0341044i	\(-0.989142\pi\)
							−0.276402	+	0.961042i	\(0.589142\pi\)
\(31\)	5.15337	+	1.09538i	0.925573	+	0.196737i	0.645956	−	0.763375i	\(-0.276459\pi\)
							0.279617	+	0.960112i	\(0.409792\pi\)
\(37\)	−8.45969	−	3.76649i	−1.39076	−	0.619208i	−0.431602	−	0.902064i	\(-0.642051\pi\)
							−0.959162	+	0.282856i	\(0.908718\pi\)
\(41\)	−4.13780	−	3.00629i	−0.646216	−	0.469503i	0.215764	−	0.976446i	\(-0.430776\pi\)
							−0.861980	+	0.506942i	\(0.830776\pi\)
\(43\)		−	0.302979i		−	0.0462039i	−0.999733	−	0.0231020i	\(-0.992646\pi\)
							0.999733	−	0.0231020i	\(-0.00735424\pi\)
\(47\)	−0.405927	−	0.0426647i	−0.0592106	−	0.00622328i	0.0748768	−	0.997193i	\(-0.476144\pi\)
							−0.134087	+	0.990970i	\(0.542810\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	231.2.be.a.116.18	yes	224
3.2	odd	2	inner	231.2.be.a.116.11	yes	224
7.2	even	3	inner	231.2.be.a.149.11	yes	224
11.2	odd	10	inner	231.2.be.a.200.18	yes	224
21.2	odd	6	inner	231.2.be.a.149.18	yes	224
33.2	even	10	inner	231.2.be.a.200.11	yes	224
77.2	odd	30	inner	231.2.be.a.2.11	✓	224
231.2	even	30	inner	231.2.be.a.2.18	yes	224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.be.a.2.11	✓	224	77.2	odd	30	inner
231.2.be.a.2.18	yes	224	231.2	even	30	inner
231.2.be.a.116.11	yes	224	3.2	odd	2	inner
231.2.be.a.116.18	yes	224	1.1	even	1	trivial
231.2.be.a.149.11	yes	224	7.2	even	3	inner
231.2.be.a.149.18	yes	224	21.2	odd	6	inner
231.2.be.a.200.11	yes	224	33.2	even	10	inner
231.2.be.a.200.18	yes	224	11.2	odd	10	inner