Embedded newform 704.2.m.b.577.1

• Label: 704.2.m.b.577.1
• Level: $704$
• Weight: $2$
• Character: 704.577
• Analytic conductor: $5.621$
• Analytic rank: $0$
• Dimension: $4$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	704.m (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.62146830230\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{10})\)
Defining polynomial:	\( x^{4} - x^{3} + x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			577.1
Root			\(-0.309017 + 0.951057i\) of defining polynomial
Character	\(\chi\)	\(=\)	704.577
Dual form			704.2.m.b.449.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30902 - 0.951057i) q^{3} +(-0.190983 + 0.587785i) q^{5} +(-1.30902 + 0.951057i) q^{7} +(-0.118034 - 0.363271i) q^{9} +(-1.23607 - 3.07768i) q^{11} +(1.80902 + 5.56758i) q^{13} +(0.809017 - 0.587785i) q^{15} +(-0.572949 + 1.76336i) q^{17} +(3.92705 + 2.85317i) q^{19} +2.61803 q^{21} -4.00000 q^{23} +(3.73607 + 2.71441i) q^{25} +(-1.69098 + 5.20431i) q^{27} +(5.92705 - 4.30625i) q^{29} +(0.336881 + 1.03681i) q^{31} +(-1.30902 + 5.20431i) q^{33} +(-0.309017 - 0.951057i) q^{35} +(-7.78115 + 5.65334i) q^{37} +(2.92705 - 9.00854i) q^{39} +(7.78115 + 5.65334i) q^{41} -1.52786 q^{43} +0.236068 q^{45} +(8.54508 + 6.20837i) q^{47} +(-1.35410 + 4.16750i) q^{49} +(2.42705 - 1.76336i) q^{51} +(-0.190983 - 0.587785i) q^{53} +(2.04508 - 0.138757i) q^{55} +(-2.42705 - 7.46969i) q^{57} +(-1.92705 + 1.40008i) q^{59} +(0.572949 - 1.76336i) q^{61} +(0.500000 + 0.363271i) q^{63} -3.61803 q^{65} +14.4721 q^{67} +(5.23607 + 3.80423i) q^{69} +(-1.57295 + 4.84104i) q^{71} +(2.54508 - 1.84911i) q^{73} +(-2.30902 - 7.10642i) q^{75} +(4.54508 + 2.85317i) q^{77} +(-1.19098 - 3.66547i) q^{79} +(6.23607 - 4.53077i) q^{81} +(-2.89919 + 8.92278i) q^{83} +(-0.927051 - 0.673542i) q^{85} -11.8541 q^{87} -4.47214 q^{89} +(-7.66312 - 5.56758i) q^{91} +(0.545085 - 1.67760i) q^{93} +(-2.42705 + 1.76336i) q^{95} +(-3.80902 - 11.7229i) q^{97} +(-0.972136 + 0.812299i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	4 * q - 3 * q^3 - 3 * q^5 - 3 * q^7 + 4 * q^9 + 4 * q^11 + 5 * q^13 + q^15 - 9 * q^17 + 9 * q^19 + 6 * q^21 - 16 * q^23 + 6 * q^25 - 9 * q^27 + 17 * q^29 + 17 * q^31 - 3 * q^33 + q^35 - 11 * q^37 + 5 * q^39 + 11 * q^41 - 24 * q^43 - 8 * q^45 + 23 * q^47 + 8 * q^49 + 3 * q^51 - 3 * q^53 - 3 * q^55 - 3 * q^57 - q^59 + 9 * q^61 + 2 * q^63 - 10 * q^65 + 40 * q^67 + 12 * q^69 - 13 * q^71 - q^73 - 7 * q^75 + 7 * q^77 - 7 * q^79 + 16 * q^81 + 13 * q^83 + 3 * q^85 - 34 * q^87 - 15 * q^91 - 9 * q^93 - 3 * q^95 - 13 * q^97 + 14 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/704\mathbb{Z}\right)^\times\).

\(n\)	\(133\)	\(321\)	\(639\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{5}\right)\)	\(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		0			0
							\(3\)	−1.30902	−	0.951057i	−0.755761	−	0.549093i	0.141846	−	0.989889i	\(-0.454696\pi\)
−0.897607	+	0.440796i	\(0.854696\pi\)
\(5\)	−0.190983	+	0.587785i	−0.0854102	+	0.262866i	−0.984636	−	0.174619i	\(-0.944131\pi\)
							0.899226	+	0.437485i	\(0.144131\pi\)
\(7\)	−1.30902	+	0.951057i	−0.494762	+	0.359466i	−0.807013	−	0.590534i	\(-0.798917\pi\)
							0.312251	+	0.950000i	\(0.398917\pi\)
\(11\)	−1.23607	−	3.07768i	−0.372689	−	0.927957i
							\(13\)	1.80902	+	5.56758i	0.501731	+	1.54417i	0.806198	+	0.591646i	\(0.201522\pi\)
−0.304467	+	0.952523i	\(0.598478\pi\)
\(17\)	−0.572949	+	1.76336i	−0.138961	+	0.427677i	−0.996185	−	0.0872663i	\(-0.972187\pi\)
							0.857224	+	0.514943i	\(0.172187\pi\)
\(19\)	3.92705	+	2.85317i	0.900927	+	0.654562i	0.938704	−	0.344724i	\(-0.112028\pi\)
							−0.0377767	+	0.999286i	\(0.512028\pi\)
\(23\)	−4.00000			−0.834058			−0.417029	−	0.908893i	\(-0.636929\pi\)
							−0.417029	+	0.908893i	\(0.636929\pi\)
\(29\)	5.92705	−	4.30625i	1.10063	−	0.799651i	0.119464	−	0.992839i	\(-0.461882\pi\)
							0.981162	+	0.193187i	\(0.0618825\pi\)
\(31\)	0.336881	+	1.03681i	0.0605056	+	0.186217i	0.976741	−	0.214424i	\(-0.0687874\pi\)
							−0.916235	+	0.400641i	\(0.868787\pi\)
\(37\)	−7.78115	+	5.65334i	−1.27921	+	0.929403i	−0.999529	−	0.0306888i	\(-0.990230\pi\)
							−0.279685	+	0.960092i	\(0.590230\pi\)
\(41\)	7.78115	+	5.65334i	1.21521	+	0.882903i	0.995694	−	0.0927052i	\(-0.0295514\pi\)
							0.219518	+	0.975608i	\(0.429551\pi\)
\(43\)	−1.52786			−0.232997			−0.116499	−	0.993191i	\(-0.537167\pi\)
							−0.116499	+	0.993191i	\(0.537167\pi\)
\(47\)	8.54508	+	6.20837i	1.24643	+	0.905583i	0.998009	−	0.0630690i	\(-0.0200888\pi\)
							0.248420	+	0.968653i	\(0.420089\pi\)
\(53\)	−0.190983	−	0.587785i	−0.0262335	−	0.0807385i	0.937083	−	0.349108i	\(-0.113515\pi\)
							−0.963316	+	0.268369i	\(0.913515\pi\)
\(59\)	−1.92705	+	1.40008i	−0.250881	+	0.182275i	−0.706117	−	0.708095i	\(-0.749555\pi\)
							0.455236	+	0.890371i	\(0.349555\pi\)
\(61\)	0.572949	−	1.76336i	0.0733586	−	0.225775i	−0.907654	−	0.419720i	\(-0.862128\pi\)
							0.981012	+	0.193945i	\(0.0621284\pi\)
\(67\)	14.4721			1.76805			0.884026	−	0.467437i	\(-0.154823\pi\)
							0.884026	+	0.467437i	\(0.154823\pi\)
\(71\)	−1.57295	+	4.84104i	−0.186675	+	0.574526i	−0.999973	−	0.00732101i	\(-0.997670\pi\)
							0.813298	+	0.581847i	\(0.197670\pi\)
\(73\)	2.54508	−	1.84911i	0.297880	−	0.216422i	−0.428799	−	0.903400i	\(-0.641063\pi\)
							0.726678	+	0.686978i	\(0.241063\pi\)
\(79\)	−1.19098	−	3.66547i	−0.133996	−	0.412397i	0.861436	−	0.507865i	\(-0.169565\pi\)
							−0.995433	+	0.0954679i	\(0.969565\pi\)
\(83\)	−2.89919	+	8.92278i	−0.318227	+	0.979402i	0.656179	+	0.754606i	\(0.272172\pi\)
							−0.974406	+	0.224797i	\(0.927828\pi\)
\(89\)	−4.47214			−0.474045			−0.237023	−	0.971504i	\(-0.576172\pi\)
							−0.237023	+	0.971504i	\(0.576172\pi\)
\(97\)	−3.80902	−	11.7229i	−0.386747	−	1.19029i	−0.935205	−	0.354107i	\(-0.884785\pi\)
							0.548458	−	0.836178i	\(-0.315215\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	704.2.m.b.577.1		4
4.3	odd	2		704.2.m.g.577.1		4
8.3	odd	2		176.2.m.a.49.1		4
8.5	even	2		88.2.i.a.49.1	yes	4
11.3	even	5		7744.2.a.cr.1.2		2
11.8	odd	10		7744.2.a.cq.1.2		2
11.9	even	5	inner	704.2.m.b.449.1		4
24.5	odd	2		792.2.r.b.577.1		4
44.3	odd	10		7744.2.a.cb.1.1		2
44.19	even	10		7744.2.a.cc.1.1		2
44.31	odd	10		704.2.m.g.449.1		4
88.3	odd	10		1936.2.a.t.1.2		2
88.5	even	10		968.2.i.d.81.1		4
88.13	odd	10		968.2.i.k.9.1		4
88.19	even	10		1936.2.a.u.1.2		2
88.21	odd	2		968.2.i.k.753.1		4
88.29	odd	10		968.2.i.c.729.1		4
88.37	even	10		968.2.i.d.729.1		4
88.53	even	10		88.2.i.a.9.1	✓	4
88.61	odd	10		968.2.i.c.81.1		4
88.69	even	10		968.2.a.i.1.1		2
88.75	odd	10		176.2.m.a.97.1		4
88.85	odd	10		968.2.a.h.1.1		2
264.53	odd	10		792.2.r.b.361.1		4
264.173	even	10		8712.2.a.bm.1.1		2
264.245	odd	10		8712.2.a.bp.1.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.i.a.9.1	✓	4	88.53	even	10
88.2.i.a.49.1	yes	4	8.5	even	2
176.2.m.a.49.1		4	8.3	odd	2
176.2.m.a.97.1		4	88.75	odd	10
704.2.m.b.449.1		4	11.9	even	5	inner
704.2.m.b.577.1		4	1.1	even	1	trivial
704.2.m.g.449.1		4	44.31	odd	10
704.2.m.g.577.1		4	4.3	odd	2
792.2.r.b.361.1		4	264.53	odd	10
792.2.r.b.577.1		4	24.5	odd	2
968.2.a.h.1.1		2	88.85	odd	10
968.2.a.i.1.1		2	88.69	even	10
968.2.i.c.81.1		4	88.61	odd	10
968.2.i.c.729.1		4	88.29	odd	10
968.2.i.d.81.1		4	88.5	even	10
968.2.i.d.729.1		4	88.37	even	10
968.2.i.k.9.1		4	88.13	odd	10
968.2.i.k.753.1		4	88.21	odd	2
1936.2.a.t.1.2		2	88.3	odd	10
1936.2.a.u.1.2		2	88.19	even	10
7744.2.a.cb.1.1		2	44.3	odd	10
7744.2.a.cc.1.1		2	44.19	even	10
7744.2.a.cq.1.2		2	11.8	odd	10
7744.2.a.cr.1.2		2	11.3	even	5
8712.2.a.bm.1.1		2	264.173	even	10
8712.2.a.bp.1.1		2	264.245	odd	10