Embedded newform 245.2.s.a.202.5

• Label: 245.2.s.a.202.5
• Level: $245$
• Weight: $2$
• Character: 245.202
• Analytic conductor: $1.956$
• Analytic rank: $0$
• Dimension: $312$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	245.s (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.95633484952\)
Analytic rank:	\(0\)
Dimension:	\(312\)
Relative dimension:	\(26\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			202.5
Character	\(\chi\)	\(=\)	245.202
Dual form			245.2.s.a.188.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.14956 + 0.242197i) q^{2} +(-0.397835 - 1.13695i) q^{3} +(2.61207 - 0.596189i) q^{4} +(-2.20878 + 0.348262i) q^{5} +(1.13053 + 2.34757i) q^{6} +(1.57821 + 2.12350i) q^{7} +(-1.38687 + 0.485286i) q^{8} +(1.21112 - 0.965837i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 10 q^{2} - 14 q^{3} - 14 q^{5} - 56 q^{6} - 18 q^{7} - 6 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.14956	+	0.242197i	−1.51997	+	0.171259i	−0.832119	−	0.554597i	\(-0.812872\pi\)
							−0.687846	+	0.725856i	\(0.741444\pi\)
\(3\)	−0.397835	−	1.13695i	−0.229690	−	0.656416i	−0.999828	−	0.0185389i	\(-0.994099\pi\)
							0.770138	−	0.637877i	\(-0.220187\pi\)
\(5\)	−2.20878	+	0.348262i	−0.987797	+	0.155748i
							\(7\)	1.57821	+	2.12350i	0.596507	+	0.802608i
\(11\)	1.87365	−	2.34948i	0.564926	−	0.708395i								−0.414534	−	0.910034i	\(-0.636055\pi\)
							0.979460	+	0.201639i	\(0.0646268\pi\)
\(13\)	−4.16327	+	0.469088i	−1.15468	+	0.130102i	−0.668470	−	0.743739i	\(-0.733051\pi\)
							−0.486213	+	0.873840i	\(0.661622\pi\)
\(17\)	−3.66309	−	5.82977i	−0.888430	−	1.41393i	−0.910202	−	0.414165i	\(-0.864074\pi\)
							0.0217725	−	0.999763i	\(-0.493069\pi\)
\(19\)	2.08300			0.477873			0.238937	−	0.971035i	\(-0.423201\pi\)
							0.238937	+	0.971035i	\(0.423201\pi\)
\(23\)	−3.48231	−	2.18808i	−0.726113	−	0.456247i	0.117587	−	0.993063i	\(-0.462484\pi\)
							−0.843699	+	0.536816i	\(0.819627\pi\)
\(29\)	−6.24937	−	1.42638i	−1.16048	−	0.264872i	−0.401420	−	0.915894i	\(-0.631483\pi\)
							−0.759059	+	0.651022i	\(0.774340\pi\)
\(31\)		−	7.86172i		−	1.41201i	−0.708208	−	0.706004i	\(-0.750496\pi\)
							0.708208	−	0.706004i	\(-0.249504\pi\)
\(37\)	3.58337	−	2.25158i	0.589102	−	0.370158i	−0.204251	−	0.978918i	\(-0.565476\pi\)
							0.793354	+	0.608761i	\(0.208333\pi\)
\(41\)	1.86698	−	3.87683i	0.291574	−	0.605460i	−0.702800	−	0.711388i	\(-0.748067\pi\)
							0.994374	+	0.105928i	\(0.0337813\pi\)
\(43\)	1.36966	−	3.91427i	0.208872	−	0.596921i	−0.791018	−	0.611793i	\(-0.790448\pi\)
							0.999889	+	0.0148723i	\(0.00473418\pi\)
\(47\)	0.635776	+	5.64267i	0.0927375	+	0.823068i	0.950860	+	0.309622i	\(0.100203\pi\)
							−0.858122	+	0.513446i	\(0.828369\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.s.a.202.5	yes	312
5.3	odd	4	inner	245.2.s.a.153.22	✓	312
49.41	odd	14	inner	245.2.s.a.237.22	yes	312
245.188	even	28	inner	245.2.s.a.188.5	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.s.a.153.22	✓	312	5.3	odd	4	inner
245.2.s.a.188.5	yes	312	245.188	even	28	inner
245.2.s.a.202.5	yes	312	1.1	even	1	trivial
245.2.s.a.237.22	yes	312	49.41	odd	14	inner