Embedded newform 224.3.v.b.13.13

• Label: 224.3.v.b.13.13
• Level: $224$
• Weight: $3$
• Character: 224.13
• Analytic conductor: $6.104$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	224.v (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.10355792167\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(60\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			13.13
Character	\(\chi\)	\(=\)	224.13
Dual form			224.3.v.b.69.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.46638 + 1.36005i) q^{2} +(-1.65755 - 0.686579i) q^{3} +(0.300514 - 3.98870i) q^{4} +(0.205220 + 0.495445i) q^{5} +(3.36437 - 1.24757i) q^{6} +(-0.193440 + 6.99733i) q^{7} +(4.98417 + 6.25764i) q^{8} +(-4.08789 - 4.08789i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 8 q^{2} - 8 q^{4} - 4 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{7}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.46638	+	1.36005i	−0.733188	+	0.680026i
							\(3\)	−1.65755	−	0.686579i	−0.552516	−	0.228860i	0.0889164	−	0.996039i	\(-0.471660\pi\)
−0.641433	+	0.767179i	\(0.721660\pi\)
\(5\)	0.205220	+	0.495445i	0.0410440	+	0.0990890i	0.943072	−	0.332588i	\(-0.107922\pi\)
							−0.902028	+	0.431677i	\(0.857922\pi\)
\(7\)	−0.193440	+	6.99733i	−0.0276343	+	0.999618i
							\(11\)	−3.44988	−	8.32874i	−0.313625	−	0.757158i	−0.999565	−	0.0294996i	\(-0.990609\pi\)
0.685940	−	0.727658i	\(-0.259391\pi\)
\(13\)	−1.31375	+	3.17168i	−0.101058	+	0.243976i	−0.966320	−	0.257345i	\(-0.917152\pi\)
							0.865262	+	0.501321i	\(0.167152\pi\)
\(17\)	28.1178			1.65399			0.826995	−	0.562210i	\(-0.190049\pi\)
							0.826995	+	0.562210i	\(0.190049\pi\)
\(19\)	12.1154	−	29.2491i	0.637651	−	1.53943i	−0.192149	−	0.981366i	\(-0.561546\pi\)
							0.829801	−	0.558060i	\(-0.188454\pi\)
\(23\)	−11.0067	−	11.0067i	−0.478553	−	0.478553i	0.426115	−	0.904669i	\(-0.359882\pi\)
							−0.904669	+	0.426115i	\(0.859882\pi\)
\(29\)	6.55469	−	15.8244i	0.226024	−	0.545670i	−0.769663	−	0.638451i	\(-0.779576\pi\)
							0.995687	+	0.0927808i	\(0.0295756\pi\)
\(31\)			31.5457i			1.01760i	0.860883	+	0.508802i	\(0.169912\pi\)
							−0.860883	+	0.508802i	\(0.830088\pi\)
\(37\)	39.9500	−	16.5478i	1.07973	−	0.447238i	0.229314	−	0.973352i	\(-0.426352\pi\)
							0.850414	+	0.526114i	\(0.176352\pi\)
\(41\)	53.7822	−	53.7822i	1.31176	−	1.31176i	0.391645	−	0.920116i	\(-0.371906\pi\)
							0.920116	−	0.391645i	\(-0.128094\pi\)
\(43\)	8.17552	+	19.7375i	0.190128	+	0.459011i	0.989984	−	0.141182i	\(-0.0450904\pi\)
							−0.799855	+	0.600193i	\(0.795090\pi\)
\(47\)	−30.9029			−0.657508			−0.328754	−	0.944416i	\(-0.606629\pi\)
							−0.328754	+	0.944416i	\(0.606629\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.v.b.13.13	✓	240
7.6	odd	2	inner	224.3.v.b.13.14	yes	240
32.5	even	8	inner	224.3.v.b.69.14	yes	240
224.69	odd	8	inner	224.3.v.b.69.13	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.v.b.13.13	✓	240	1.1	even	1	trivial
224.3.v.b.13.14	yes	240	7.6	odd	2	inner
224.3.v.b.69.13	yes	240	224.69	odd	8	inner
224.3.v.b.69.14	yes	240	32.5	even	8	inner