Embedded newform 712.2.t.a.85.6

• Label: 712.2.t.a.85.6
• Level: $712$
• Weight: $2$
• Character: 712.85
• Analytic conductor: $5.685$
• Analytic rank: $0$
• Dimension: $880$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 712 = 2^{3} \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	712.t (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.68534862392\)
Analytic rank:	\(0\)
Dimension:	\(880\)
Relative dimension:	\(88\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			85.6
Character	\(\chi\)	\(=\)	712.85
Dual form			712.2.t.a.645.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38825 + 0.269764i) q^{2} +(0.0607019 + 0.132919i) q^{3} +(1.85445 - 0.748998i) q^{4} +(-1.65789 + 0.238369i) q^{5} +(-0.120126 - 0.168149i) q^{6} +(-3.25832 + 0.468475i) q^{7} +(-2.37239 + 1.54006i) q^{8} +(1.95060 - 2.25111i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q - 9 q^{2} - 9 q^{4} - 11 q^{6} - 22 q^{7} - 9 q^{8} - 102 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/712\mathbb{Z}\right)^\times\).

\(n\)	\(357\)	\(535\)	\(537\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{19}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38825	+	0.269764i	−0.981638	+	0.190752i
							\(3\)	0.0607019	+	0.132919i	0.0350463	+	0.0767406i	0.926345	−	0.376675i	\(-0.122933\pi\)
−0.891299	+	0.453416i	\(0.850205\pi\)
\(5\)	−1.65789	+	0.238369i	−0.741432	+	0.106602i	−0.502672	−	0.864477i	\(-0.667650\pi\)
							−0.238759	+	0.971079i	\(0.576741\pi\)
\(7\)	−3.25832	+	0.468475i	−1.23153	+	0.177067i	−0.727188	−	0.686438i	\(-0.759173\pi\)
							−0.504340	+	0.863505i	\(0.668264\pi\)
\(11\)	3.84826	+	0.553296i	1.16029	+	0.166825i	0.695454	−	0.718570i	\(-0.255203\pi\)
							0.464839	+	0.885395i	\(0.346112\pi\)
\(13\)	0.140354	+	0.307332i	0.0389271	+	0.0852384i	0.928097	−	0.372340i	\(-0.121444\pi\)
							−0.889169	+	0.457578i	\(0.848717\pi\)
\(17\)	2.71483	+	1.74472i	0.658444	+	0.423156i	0.826743	−	0.562580i	\(-0.190191\pi\)
							−0.168299	+	0.985736i	\(0.553827\pi\)
\(19\)	−3.33857	+	3.85291i	−0.765920	+	0.883919i	−0.996009	−	0.0892507i	\(-0.971553\pi\)
							0.230089	+	0.973170i	\(0.426098\pi\)
\(23\)	−6.43721	−	5.57787i	−1.34225	−	1.16307i	−0.972226	−	0.234046i	\(-0.924803\pi\)
							−0.370026	−	0.929022i	\(-0.620651\pi\)
\(29\)	0.665026	+	4.62536i	0.123492	+	0.858907i	0.953551	+	0.301231i	\(0.0973976\pi\)
							−0.830059	+	0.557676i	\(0.811693\pi\)
\(31\)	−4.30336	+	3.72888i	−0.772906	+	0.669727i	−0.949222	−	0.314607i	\(-0.898127\pi\)
							0.176316	+	0.984334i	\(0.443582\pi\)
\(37\)	−5.26716			−0.865915			−0.432957	−	0.901414i	\(-0.642530\pi\)
							−0.432957	+	0.901414i	\(0.642530\pi\)
\(41\)	−7.31403	−	3.34021i	−1.14226	−	0.521652i	−0.247804	−	0.968810i	\(-0.579709\pi\)
							−0.894455	+	0.447158i	\(0.852436\pi\)
\(43\)	−1.69852	+	11.8134i	−0.259021	+	1.80153i	0.280814	+	0.959762i	\(0.409396\pi\)
							−0.539835	+	0.841771i	\(0.681513\pi\)
\(47\)	2.90136	−	6.35310i	0.423207	−	0.926695i	−0.571173	−	0.820829i	\(-0.693512\pi\)
							0.994381	−	0.105865i	\(-0.0337611\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	712.2.t.a.85.6	✓	880
8.5	even	2	inner	712.2.t.a.85.29	yes	880
89.22	even	22	inner	712.2.t.a.645.29	yes	880
712.645	even	22	inner	712.2.t.a.645.6	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
712.2.t.a.85.6	✓	880	1.1	even	1	trivial
712.2.t.a.85.29	yes	880	8.5	even	2	inner
712.2.t.a.645.6	yes	880	712.645	even	22	inner
712.2.t.a.645.29	yes	880	89.22	even	22	inner