Embedded newform 85.2.b.a.69.7

• Label: 85.2.b.a.69.7
• Level: $85$
• Weight: $2$
• Character: 85.69
• Analytic conductor: $0.679$
• Analytic rank: $0$
• Dimension: $8$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 85 = 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	85.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.678728417181\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.619810816.2
Defining polynomial:	\( x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			69.7
Root			\(0.561103 + 0.561103i\) of defining polynomial
Character	\(\chi\)	\(=\)	85.69
Dual form			85.2.b.a.69.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+2.03032i q^{2} +2.37033i q^{3} -2.12221 q^{4} +(1.70032 - 1.45220i) q^{5} -4.81252 q^{6} -5.03032i q^{7} -0.248119i q^{8} -2.61845 q^{9} +(2.94844 + 3.45220i) q^{10} -1.90812 q^{11} -5.03032i q^{12} +1.04155i q^{13} +10.2132 q^{14} +(3.44220 + 4.03032i) q^{15} -3.74065 q^{16} +1.00000i q^{17} -5.31629i q^{18} -3.31999 q^{19} +(-3.60844 + 3.08188i) q^{20} +11.9235 q^{21} -3.87409i q^{22} +0.125912i q^{23} +0.588123 q^{24} +(0.782203 - 4.93844i) q^{25} -2.11468 q^{26} +0.904410i q^{27} +10.6754i q^{28} +5.56441 q^{29} +(-8.18285 + 6.98877i) q^{30} -4.99629 q^{31} -8.09097i q^{32} -4.52285i q^{33} -2.03032 q^{34} +(-7.30506 - 8.55318i) q^{35} +5.55688 q^{36} +1.56441i q^{37} -6.74065i q^{38} -2.46881 q^{39} +(-0.360320 - 0.421883i) q^{40} +4.72064 q^{41} +24.2085i q^{42} +4.46221i q^{43} +4.04941 q^{44} +(-4.45220 + 3.80252i) q^{45} -0.255643 q^{46} -1.04155i q^{47} -8.86656i q^{48} -18.3041 q^{49} +(10.0266 + 1.58812i) q^{50} -2.37033 q^{51} -2.21039i q^{52} +6.48883i q^{53} -1.83624 q^{54} +(-3.24441 + 2.77097i) q^{55} -1.24812 q^{56} -7.86946i q^{57} +11.2975i q^{58} +2.00000 q^{59} +(-7.30506 - 8.55318i) q^{60} +7.14882 q^{61} -10.1441i q^{62} +13.1716i q^{63} +8.94596 q^{64} +(1.51255 + 1.77097i) q^{65} +9.18285 q^{66} -3.28596i q^{67} -2.12221i q^{68} -0.298453 q^{69} +(17.3657 - 14.8316i) q^{70} -5.65629 q^{71} +0.649686i q^{72} -12.5295i q^{73} -3.17625 q^{74} +(11.7057 + 1.85408i) q^{75} +7.04571 q^{76} +9.59843i q^{77} -5.01249i q^{78} +13.3694 q^{79} +(-6.36032 + 5.43219i) q^{80} -9.99908 q^{81} +9.58442i q^{82} -1.14222i q^{83} -25.3041 q^{84} +(1.45220 + 1.70032i) q^{85} -9.05972 q^{86} +13.1895i q^{87} +0.473440i q^{88} -1.64090 q^{89} +(-7.72034 - 9.03941i) q^{90} +5.23934 q^{91} -0.267212i q^{92} -11.8428i q^{93} +2.11468 q^{94} +(-5.64506 + 4.82131i) q^{95} +19.1782 q^{96} -7.29753i q^{97} -37.1633i q^{98} +4.99629 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	8 * q - 8 * q^4 - 2 * q^5 - 8 * q^9 + 6 * q^10 - 4 * q^11 + 12 * q^14 - 8 * q^16 - 8 * q^19 - 2 * q^20 + 24 * q^21 + 12 * q^24 - 12 * q^25 + 8 * q^29 - 16 * q^30 - 24 * q^31 + 4 * q^34 + 44 * q^39 + 22 * q^40 - 12 * q^41 - 32 * q^44 - 22 * q^45 - 8 * q^46 - 16 * q^49 + 44 * q^50 - 8 * q^51 - 20 * q^54 - 8 * q^55 - 8 * q^56 + 16 * q^59 + 12 * q^61 + 48 * q^64 + 20 * q^65 + 24 * q^66 - 8 * q^69 + 40 * q^70 - 20 * q^71 - 40 * q^74 + 4 * q^75 - 24 * q^76 + 24 * q^79 - 26 * q^80 - 8 * q^81 - 72 * q^84 - 2 * q^85 + 40 * q^86 - 48 * q^89 - 74 * q^90 + 36 * q^91 + 4 * q^95 + 16 * q^96 + 24 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/85\mathbb{Z}\right)^\times\).

\(n\)	\(52\)	\(71\)
\(\chi(n)\)	\(-1\)	\(1\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			2.03032i			1.43565i	0.696221	+	0.717827i	\(0.254863\pi\)
							−0.696221	+	0.717827i	\(0.745137\pi\)
\(3\)			2.37033i			1.36851i	0.729244	+	0.684254i	\(0.239872\pi\)
							−0.729244	+	0.684254i	\(0.760128\pi\)
\(5\)	1.70032	−	1.45220i	0.760408	−	0.649446i
							\(7\)		−	5.03032i		−	1.90128i	−0.310291	−	0.950641i	\(-0.600427\pi\)
0.310291	−	0.950641i	\(-0.399573\pi\)
\(11\)	−1.90812			−0.575318			−0.287659	−	0.957733i	\(-0.592877\pi\)
							−0.287659	+	0.957733i	\(0.592877\pi\)
\(13\)			1.04155i			0.288874i	0.989514	+	0.144437i	\(0.0461371\pi\)
							−0.989514	+	0.144437i	\(0.953863\pi\)
\(17\)			1.00000i			0.242536i
							\(19\)	−3.31999			−0.761658			−0.380829	−	0.924645i	\(-0.624361\pi\)
−0.380829	+	0.924645i	\(0.624361\pi\)
\(23\)			0.125912i			0.0262545i	0.999914	+	0.0131273i	\(0.00417866\pi\)
							−0.999914	+	0.0131273i	\(0.995821\pi\)
\(29\)	5.56441			1.03328			0.516642	−	0.856201i	\(-0.327182\pi\)
							0.516642	+	0.856201i	\(0.327182\pi\)
\(31\)	−4.99629			−0.897361			−0.448680	−	0.893692i	\(-0.648106\pi\)
							−0.448680	+	0.893692i	\(0.648106\pi\)
\(37\)			1.56441i			0.257187i	0.991697	+	0.128593i	\(0.0410462\pi\)
							−0.991697	+	0.128593i	\(0.958954\pi\)
\(41\)	4.72064			0.737240			0.368620	−	0.929580i	\(-0.379830\pi\)
							0.368620	+	0.929580i	\(0.379830\pi\)
\(43\)			4.46221i			0.680481i	0.940338	+	0.340240i	\(0.110508\pi\)
							−0.940338	+	0.340240i	\(0.889492\pi\)
\(47\)		−	1.04155i		−	0.151926i	−0.997111	−	0.0759629i	\(-0.975797\pi\)
							0.997111	−	0.0759629i	\(-0.0242031\pi\)
\(53\)			6.48883i			0.891309i	0.895205	+	0.445654i	\(0.147029\pi\)
							−0.895205	+	0.445654i	\(0.852971\pi\)
\(59\)	2.00000			0.260378			0.130189	−	0.991489i	\(-0.458442\pi\)
							0.130189	+	0.991489i	\(0.458442\pi\)
\(61\)	7.14882			0.915313			0.457657	−	0.889129i	\(-0.348689\pi\)
							0.457657	+	0.889129i	\(0.348689\pi\)
\(67\)		−	3.28596i		−	0.401444i	−0.979648	−	0.200722i	\(-0.935671\pi\)
							0.979648	−	0.200722i	\(-0.0643289\pi\)
\(71\)	−5.65629			−0.671278			−0.335639	−	0.941991i	\(-0.608952\pi\)
							−0.335639	+	0.941991i	\(0.608952\pi\)
\(73\)		−	12.5295i		−	1.46646i	−0.679980	−	0.733231i	\(-0.738011\pi\)
							0.679980	−	0.733231i	\(-0.261989\pi\)
\(79\)	13.3694			1.50418			0.752088	−	0.659063i	\(-0.229047\pi\)
							0.752088	+	0.659063i	\(0.229047\pi\)
\(83\)		−	1.14222i		−	0.125375i	−0.998033	−	0.0626874i	\(-0.980033\pi\)
							0.998033	−	0.0626874i	\(-0.0199671\pi\)
\(89\)	−1.64090			−0.173935			−0.0869677	−	0.996211i	\(-0.527718\pi\)
							−0.0869677	+	0.996211i	\(0.527718\pi\)
\(97\)		−	7.29753i		−	0.740952i	−0.928842	−	0.370476i	\(-0.879195\pi\)
							0.928842	−	0.370476i	\(-0.120805\pi\)

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	85.2.b.a.69.7	yes	8
3.2	odd	2		765.2.b.c.154.2		8
4.3	odd	2		1360.2.e.d.1089.2		8
5.2	odd	4		425.2.a.h.1.1		4
5.3	odd	4		425.2.a.g.1.4		4
5.4	even	2	inner	85.2.b.a.69.2	✓	8
15.2	even	4		3825.2.a.bh.1.4		4
15.8	even	4		3825.2.a.bj.1.1		4
15.14	odd	2		765.2.b.c.154.7		8
17.16	even	2		1445.2.b.e.579.7		8
20.3	even	4		6800.2.a.bw.1.3		4
20.7	even	4		6800.2.a.bt.1.2		4
20.19	odd	2		1360.2.e.d.1089.7		8
85.33	odd	4		7225.2.a.v.1.4		4
85.67	odd	4		7225.2.a.w.1.1		4
85.84	even	2		1445.2.b.e.579.2		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.b.a.69.2	✓	8	5.4	even	2	inner
85.2.b.a.69.7	yes	8	1.1	even	1	trivial
425.2.a.g.1.4		4	5.3	odd	4
425.2.a.h.1.1		4	5.2	odd	4
765.2.b.c.154.2		8	3.2	odd	2
765.2.b.c.154.7		8	15.14	odd	2
1360.2.e.d.1089.2		8	4.3	odd	2
1360.2.e.d.1089.7		8	20.19	odd	2
1445.2.b.e.579.2		8	85.84	even	2
1445.2.b.e.579.7		8	17.16	even	2
3825.2.a.bh.1.4		4	15.2	even	4
3825.2.a.bj.1.1		4	15.8	even	4
6800.2.a.bt.1.2		4	20.7	even	4
6800.2.a.bw.1.3		4	20.3	even	4
7225.2.a.v.1.4		4	85.33	odd	4
7225.2.a.w.1.1		4	85.67	odd	4