Embedded newform 245.2.x.a.103.11

• Label: 245.2.x.a.103.11
• Level: $245$
• Weight: $2$
• Character: 245.103
• Analytic conductor: $1.956$
• Analytic rank: $0$
• Dimension: $624$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	245.x (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.95633484952\)
Analytic rank:	\(0\)
Dimension:	\(624\)
Relative dimension:	\(26\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

Embedding invariants

Embedding label			103.11
Character	\(\chi\)	\(=\)	245.103
Dual form			245.2.x.a.157.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.421929 + 0.571694i) q^{2} +(-0.246518 + 0.0466438i) q^{3} +(0.440701 + 1.42872i) q^{4} +(2.22061 + 0.262461i) q^{5} +(0.0773473 - 0.160613i) q^{6} +(2.64300 - 0.120661i) q^{7} +(-2.34406 - 0.820220i) q^{8} +(-2.73403 + 1.07303i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 26 q^{2} - 22 q^{3} - 28 q^{5} - 28 q^{6} - 18 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.421929	+	0.571694i	−0.298349	+	0.404249i	−0.927971	−	0.372653i	\(-0.878448\pi\)
							0.629622	+	0.776902i	\(0.283210\pi\)
\(3\)	−0.246518	+	0.0466438i	−0.142327	+	0.0269298i	−0.256587	−	0.966521i	\(-0.582598\pi\)
							0.114260	+	0.993451i	\(0.463550\pi\)
\(5\)	2.22061	+	0.262461i	0.993088	+	0.117376i
							\(7\)	2.64300	−	0.120661i	0.998960	−	0.0456055i
\(11\)	0.529769	−	1.34983i	0.159731	−	0.406989i								−0.828493	−	0.559999i	\(-0.810801\pi\)
							0.988225	+	0.153010i	\(0.0488967\pi\)
\(13\)	3.28389	+	0.370005i	0.910787	+	0.102621i	0.554908	−	0.831911i	\(-0.312753\pi\)
							0.355878	+	0.934532i	\(0.384182\pi\)
\(17\)	−0.630570	−	0.0235943i	−0.152936	−	0.00572245i	−0.0391863	−	0.999232i	\(-0.512477\pi\)
							−0.113749	+	0.993509i	\(0.536286\pi\)
\(19\)	−1.79095	+	3.10202i	−0.410872	+	0.711652i	−0.994985	−	0.100020i	\(-0.968109\pi\)
							0.584113	+	0.811672i	\(0.301443\pi\)
\(23\)	0.0779271	+	2.08265i	0.0162489	+	0.434262i	0.983551	+	0.180633i	\(0.0578146\pi\)
							−0.967302	+	0.253629i	\(0.918376\pi\)
\(29\)	−0.472800	+	0.107914i	−0.0877968	+	0.0200390i	−0.266194	−	0.963920i	\(-0.585766\pi\)
							0.178397	+	0.983959i	\(0.442909\pi\)
\(31\)	−5.64184	+	3.25732i	−1.01330	+	0.585032i	−0.912158	−	0.409840i	\(-0.865585\pi\)
							−0.101147	+	0.994871i	\(0.532251\pi\)
\(37\)	6.99902	−	3.69909i	1.15063	−	0.608127i	0.220513	−	0.975384i	\(-0.429227\pi\)
							0.930119	+	0.367257i	\(0.119703\pi\)
\(41\)	−4.67387	−	9.70538i	−0.729935	−	1.51573i	−0.852193	−	0.523228i	\(-0.824728\pi\)
							0.122258	−	0.992498i	\(-0.460987\pi\)
\(43\)	−4.24553	−	12.1330i	−0.647437	−	1.85027i	−0.510915	−	0.859631i	\(-0.670693\pi\)
							−0.136522	−	0.990637i	\(-0.543592\pi\)
\(47\)	1.53455	+	1.13255i	0.223837	+	0.165199i	0.699966	−	0.714176i	\(-0.253198\pi\)
							−0.476129	+	0.879376i	\(0.657960\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.x.a.103.11	✓	624
5.2	odd	4	inner	245.2.x.a.152.11	yes	624
49.10	odd	42	inner	245.2.x.a.108.11	yes	624
245.157	even	84	inner	245.2.x.a.157.11	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.x.a.103.11	✓	624	1.1	even	1	trivial
245.2.x.a.108.11	yes	624	49.10	odd	42	inner
245.2.x.a.152.11	yes	624	5.2	odd	4	inner
245.2.x.a.157.11	yes	624	245.157	even	84	inner