Embedded newform 712.2.t.a.645.6

• Label: 712.2.t.a.645.6
• Level: $712$
• Weight: $2$
• Character: 712.645
• Analytic conductor: $5.685$
• Analytic rank: $0$
• Dimension: $880$
• 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			645.6
Character	\(\chi\)	\(=\)	712.645
Dual form			712.2.t.a.85.6

$q$-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 \(\left(\mathbb{Z}/712\mathbb{Z}\right)^\times\).

\(n\)	\(357\)	\(535\)	\(537\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{3}{22}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
712.2.t.a.85.6	✓	880	712.85	even	22	inner
712.2.t.a.85.29	yes	880	89.85	even	22	inner
712.2.t.a.645.6	yes	880	1.1	even	1	trivial
712.2.t.a.645.29	yes	880	8.5	even	2	inner