Embedded newform 95.2.e.a.26.1

• Label: 95.2.e.a.26.1
• Level: $95$
• Weight: $2$
• Character: 95.26
• Analytic conductor: $0.759$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 95 = 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	95.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.758578819202\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			26.1
Root			\(0.500000 - 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	95.26
Dual form			95.2.e.a.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{3} +(1.00000 - 1.73205i) q^{4} +(0.500000 + 0.866025i) q^{5} -4.00000 q^{7} +(-0.500000 + 0.866025i) q^{9} +3.00000 q^{11} +4.00000 q^{12} +(-1.00000 + 1.73205i) q^{13} +(-1.00000 + 1.73205i) q^{15} +(-2.00000 - 3.46410i) q^{16} +(-3.00000 - 5.19615i) q^{17} +(-3.50000 + 2.59808i) q^{19} +2.00000 q^{20} +(-4.00000 - 6.92820i) q^{21} +(-0.500000 + 0.866025i) q^{25} +4.00000 q^{27} +(-4.00000 + 6.92820i) q^{28} +(1.50000 - 2.59808i) q^{29} -7.00000 q^{31} +(3.00000 + 5.19615i) q^{33} +(-2.00000 - 3.46410i) q^{35} +(1.00000 + 1.73205i) q^{36} +8.00000 q^{37} -4.00000 q^{39} +(3.00000 + 5.19615i) q^{41} +(2.00000 + 3.46410i) q^{43} +(3.00000 - 5.19615i) q^{44} -1.00000 q^{45} +(-3.00000 + 5.19615i) q^{47} +(4.00000 - 6.92820i) q^{48} +9.00000 q^{49} +(6.00000 - 10.3923i) q^{51} +(2.00000 + 3.46410i) q^{52} +(3.00000 - 5.19615i) q^{53} +(1.50000 + 2.59808i) q^{55} +(-8.00000 - 3.46410i) q^{57} +(7.50000 + 12.9904i) q^{59} +(2.00000 + 3.46410i) q^{60} +(-2.50000 + 4.33013i) q^{61} +(2.00000 - 3.46410i) q^{63} -8.00000 q^{64} -2.00000 q^{65} +(-1.00000 + 1.73205i) q^{67} -12.0000 q^{68} +(1.50000 + 2.59808i) q^{71} +(-4.00000 - 6.92820i) q^{73} -2.00000 q^{75} +(1.00000 + 8.66025i) q^{76} -12.0000 q^{77} +(-2.50000 - 4.33013i) q^{79} +(2.00000 - 3.46410i) q^{80} +(5.50000 + 9.52628i) q^{81} +12.0000 q^{83} -16.0000 q^{84} +(3.00000 - 5.19615i) q^{85} +6.00000 q^{87} +(7.50000 - 12.9904i) q^{89} +(4.00000 - 6.92820i) q^{91} +(-7.00000 - 12.1244i) q^{93} +(-4.00000 - 1.73205i) q^{95} +(-4.00000 - 6.92820i) q^{97} +(-1.50000 + 2.59808i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 2 * q^3 + 2 * q^4 + q^5 - 8 * q^7 - q^9 + 6 * q^11 + 8 * q^12 - 2 * q^13 - 2 * q^15 - 4 * q^16 - 6 * q^17 - 7 * q^19 + 4 * q^20 - 8 * q^21 - q^25 + 8 * q^27 - 8 * q^28 + 3 * q^29 - 14 * q^31 + 6 * q^33 - 4 * q^35 + 2 * q^36 + 16 * q^37 - 8 * q^39 + 6 * q^41 + 4 * q^43 + 6 * q^44 - 2 * q^45 - 6 * q^47 + 8 * q^48 + 18 * q^49 + 12 * q^51 + 4 * q^52 + 6 * q^53 + 3 * q^55 - 16 * q^57 + 15 * q^59 + 4 * q^60 - 5 * q^61 + 4 * q^63 - 16 * q^64 - 4 * q^65 - 2 * q^67 - 24 * q^68 + 3 * q^71 - 8 * q^73 - 4 * q^75 + 2 * q^76 - 24 * q^77 - 5 * q^79 + 4 * q^80 + 11 * q^81 + 24 * q^83 - 32 * q^84 + 6 * q^85 + 12 * q^87 + 15 * q^89 + 8 * q^91 - 14 * q^93 - 8 * q^95 - 8 * q^97 - 3 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(77\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)

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		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(3\)	1.00000	+	1.73205i	0.577350	+	1.00000i	0.995782	+	0.0917517i	\(0.0292466\pi\)
							−0.418432	+	0.908248i	\(0.637420\pi\)
\(5\)	0.500000	+	0.866025i	0.223607	+	0.387298i
							\(7\)	−4.00000			−1.51186			−0.755929	−	0.654654i	\(-0.772814\pi\)
−0.755929	+	0.654654i	\(0.772814\pi\)
\(11\)	3.00000			0.904534			0.452267	−	0.891883i	\(-0.350615\pi\)
							0.452267	+	0.891883i	\(0.350615\pi\)
\(13\)	−1.00000	+	1.73205i	−0.277350	+	0.480384i	−0.970725	−	0.240192i	\(-0.922790\pi\)
							0.693375	+	0.720577i	\(0.256123\pi\)
\(17\)	−3.00000	−	5.19615i	−0.727607	−	1.26025i	−0.957892	−	0.287129i	\(-0.907299\pi\)
							0.230285	−	0.973123i	\(-0.426034\pi\)
\(19\)	−3.50000	+	2.59808i	−0.802955	+	0.596040i
							\(23\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
0.866025	+	0.500000i	\(0.166667\pi\)
\(29\)	1.50000	−	2.59808i	0.278543	−	0.482451i	−0.692480	−	0.721437i	\(-0.743482\pi\)
							0.971023	+	0.238987i	\(0.0768152\pi\)
\(31\)	−7.00000			−1.25724			−0.628619	−	0.777714i	\(-0.716379\pi\)
							−0.628619	+	0.777714i	\(0.716379\pi\)
\(37\)	8.00000			1.31519			0.657596	−	0.753371i	\(-0.271573\pi\)
							0.657596	+	0.753371i	\(0.271573\pi\)
\(41\)	3.00000	+	5.19615i	0.468521	+	0.811503i	0.999353	−	0.0359748i	\(-0.0114536\pi\)
							−0.530831	+	0.847477i	\(0.678120\pi\)
\(43\)	2.00000	+	3.46410i	0.304997	+	0.528271i	0.977261	−	0.212041i	\(-0.0680112\pi\)
							−0.672264	+	0.740312i	\(0.734678\pi\)
\(47\)	−3.00000	+	5.19615i	−0.437595	+	0.757937i	−0.997503	−	0.0706177i	\(-0.977503\pi\)
							0.559908	+	0.828554i	\(0.310836\pi\)
\(53\)	3.00000	−	5.19615i	0.412082	−	0.713746i	−0.583036	−	0.812447i	\(-0.698135\pi\)
							0.995117	+	0.0987002i	\(0.0314685\pi\)
\(59\)	7.50000	+	12.9904i	0.976417	+	1.69120i	0.675178	+	0.737655i	\(0.264067\pi\)
							0.301239	+	0.953549i	\(0.402600\pi\)
\(61\)	−2.50000	+	4.33013i	−0.320092	+	0.554416i	−0.980507	−	0.196485i	\(-0.937047\pi\)
							0.660415	+	0.750901i	\(0.270381\pi\)
\(67\)	−1.00000	+	1.73205i	−0.122169	+	0.211604i	−0.920623	−	0.390453i	\(-0.872318\pi\)
							0.798454	+	0.602056i	\(0.205652\pi\)
\(71\)	1.50000	+	2.59808i	0.178017	+	0.308335i	0.941201	−	0.337846i	\(-0.109698\pi\)
							−0.763184	+	0.646181i	\(0.776365\pi\)
\(73\)	−4.00000	−	6.92820i	−0.468165	−	0.810885i	0.531174	−	0.847263i	\(-0.321751\pi\)
							−0.999338	+	0.0363782i	\(0.988418\pi\)
\(79\)	−2.50000	−	4.33013i	−0.281272	−	0.487177i	0.690426	−	0.723403i	\(-0.257423\pi\)
							−0.971698	+	0.236225i	\(0.924090\pi\)
\(83\)	12.0000			1.31717			0.658586	−	0.752506i	\(-0.271155\pi\)
							0.658586	+	0.752506i	\(0.271155\pi\)
\(89\)	7.50000	−	12.9904i	0.794998	−	1.37698i	−0.127842	−	0.991795i	\(-0.540805\pi\)
							0.922840	−	0.385183i	\(-0.125862\pi\)
\(97\)	−4.00000	−	6.92820i	−0.406138	−	0.703452i	0.588315	−	0.808632i	\(-0.299792\pi\)
							−0.994453	+	0.105180i	\(0.966458\pi\)

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	95.2.e.a.26.1	yes	2
3.2	odd	2		855.2.k.b.406.1		2
4.3	odd	2		1520.2.q.c.881.1		2
5.2	odd	4		475.2.j.a.349.2		4
5.3	odd	4		475.2.j.a.349.1		4
5.4	even	2		475.2.e.b.26.1		2
19.7	even	3		1805.2.a.a.1.1		1
19.11	even	3	inner	95.2.e.a.11.1	✓	2
19.12	odd	6		1805.2.a.b.1.1		1
57.11	odd	6		855.2.k.b.676.1		2
76.11	odd	6		1520.2.q.c.961.1		2
95.49	even	6		475.2.e.b.201.1		2
95.64	even	6		9025.2.a.g.1.1		1
95.68	odd	12		475.2.j.a.49.2		4
95.69	odd	6		9025.2.a.e.1.1		1
95.87	odd	12		475.2.j.a.49.1		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.e.a.11.1	✓	2	19.11	even	3	inner
95.2.e.a.26.1	yes	2	1.1	even	1	trivial
475.2.e.b.26.1		2	5.4	even	2
475.2.e.b.201.1		2	95.49	even	6
475.2.j.a.49.1		4	95.87	odd	12
475.2.j.a.49.2		4	95.68	odd	12
475.2.j.a.349.1		4	5.3	odd	4
475.2.j.a.349.2		4	5.2	odd	4
855.2.k.b.406.1		2	3.2	odd	2
855.2.k.b.676.1		2	57.11	odd	6
1520.2.q.c.881.1		2	4.3	odd	2
1520.2.q.c.961.1		2	76.11	odd	6
1805.2.a.a.1.1		1	19.7	even	3
1805.2.a.b.1.1		1	19.12	odd	6
9025.2.a.e.1.1		1	95.69	odd	6
9025.2.a.g.1.1		1	95.64	even	6