Embedded newform 405.2.e.a.136.1

• Label: 405.2.e.a.136.1
• Level: $405$
• Weight: $2$
• Character: 405.136
• Analytic conductor: $3.234$
• Analytic rank: $1$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.23394128186\)
Analytic rank:	\(1\)
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			136.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	405.136
Dual form			405.2.e.a.271.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(-0.500000 - 0.866025i) q^{5} +2.00000 q^{10} +(-2.50000 + 4.33013i) q^{11} +(-2.00000 - 3.46410i) q^{13} +(2.00000 - 3.46410i) q^{16} -4.00000 q^{17} -5.00000 q^{19} +(-1.00000 + 1.73205i) q^{20} +(-5.00000 - 8.66025i) q^{22} +(-3.00000 - 5.19615i) q^{23} +(-0.500000 + 0.866025i) q^{25} +8.00000 q^{26} +(2.50000 - 4.33013i) q^{29} +(4.50000 + 7.79423i) q^{31} +(4.00000 + 6.92820i) q^{32} +(4.00000 - 6.92820i) q^{34} -10.0000 q^{37} +(5.00000 - 8.66025i) q^{38} +(-3.50000 - 6.06218i) q^{41} +(1.00000 - 1.73205i) q^{43} +10.0000 q^{44} +12.0000 q^{46} +(-1.00000 + 1.73205i) q^{47} +(3.50000 + 6.06218i) q^{49} +(-1.00000 - 1.73205i) q^{50} +(-4.00000 + 6.92820i) q^{52} +8.00000 q^{53} +5.00000 q^{55} +(5.00000 + 8.66025i) q^{58} +(0.500000 + 0.866025i) q^{59} +(1.00000 - 1.73205i) q^{61} -18.0000 q^{62} -8.00000 q^{64} +(-2.00000 + 3.46410i) q^{65} +(-3.00000 - 5.19615i) q^{67} +(4.00000 + 6.92820i) q^{68} +1.00000 q^{71} -8.00000 q^{73} +(10.0000 - 17.3205i) q^{74} +(5.00000 + 8.66025i) q^{76} +(-6.00000 + 10.3923i) q^{79} -4.00000 q^{80} +14.0000 q^{82} +(-3.00000 + 5.19615i) q^{83} +(2.00000 + 3.46410i) q^{85} +(2.00000 + 3.46410i) q^{86} -9.00000 q^{89} +(-6.00000 + 10.3923i) q^{92} +(-2.00000 - 3.46410i) q^{94} +(2.50000 + 4.33013i) q^{95} +(-7.00000 + 12.1244i) q^{97} -14.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 2 * q^2 - 2 * q^4 - q^5 + 4 * q^10 - 5 * q^11 - 4 * q^13 + 4 * q^16 - 8 * q^17 - 10 * q^19 - 2 * q^20 - 10 * q^22 - 6 * q^23 - q^25 + 16 * q^26 + 5 * q^29 + 9 * q^31 + 8 * q^32 + 8 * q^34 - 20 * q^37 + 10 * q^38 - 7 * q^41 + 2 * q^43 + 20 * q^44 + 24 * q^46 - 2 * q^47 + 7 * q^49 - 2 * q^50 - 8 * q^52 + 16 * q^53 + 10 * q^55 + 10 * q^58 + q^59 + 2 * q^61 - 36 * q^62 - 16 * q^64 - 4 * q^65 - 6 * q^67 + 8 * q^68 + 2 * q^71 - 16 * q^73 + 20 * q^74 + 10 * q^76 - 12 * q^79 - 8 * q^80 + 28 * q^82 - 6 * q^83 + 4 * q^85 + 4 * q^86 - 18 * q^89 - 12 * q^92 - 4 * q^94 + 5 * q^95 - 14 * q^97 - 28 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\)	\(82\)	\(326\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

Coefficient data

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

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

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	+	1.73205i	−0.707107	+	1.22474i	0.258819	+	0.965926i	\(0.416667\pi\)
							−0.965926	+	0.258819i	\(0.916667\pi\)
\(3\)		0			0
							\(5\)	−0.500000	−	0.866025i	−0.223607	−	0.387298i
\(7\)		0			0									−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(11\)	−2.50000	+	4.33013i	−0.753778	+	1.30558i	0.192201	+	0.981356i	\(0.438437\pi\)
							−0.945979	+	0.324227i	\(0.894896\pi\)
\(13\)	−2.00000	−	3.46410i	−0.554700	−	0.960769i	−0.997927	−	0.0643593i	\(-0.979500\pi\)
							0.443227	−	0.896410i	\(-0.353834\pi\)
\(17\)	−4.00000			−0.970143			−0.485071	−	0.874475i	\(-0.661206\pi\)
							−0.485071	+	0.874475i	\(0.661206\pi\)
\(19\)	−5.00000			−1.14708			−0.573539	−	0.819178i	\(-0.694430\pi\)
							−0.573539	+	0.819178i	\(0.694430\pi\)
\(23\)	−3.00000	−	5.19615i	−0.625543	−	1.08347i	−0.988436	−	0.151642i	\(-0.951544\pi\)
							0.362892	−	0.931831i	\(-0.381789\pi\)
\(29\)	2.50000	−	4.33013i	0.464238	−	0.804084i	−0.534928	−	0.844897i	\(-0.679661\pi\)
							0.999167	+	0.0408130i	\(0.0129948\pi\)
\(31\)	4.50000	+	7.79423i	0.808224	+	1.39988i	0.914093	+	0.405505i	\(0.132904\pi\)
							−0.105869	+	0.994380i	\(0.533762\pi\)
\(37\)	−10.0000			−1.64399			−0.821995	−	0.569495i	\(-0.807139\pi\)
							−0.821995	+	0.569495i	\(0.807139\pi\)
\(41\)	−3.50000	−	6.06218i	−0.546608	−	0.946753i	−0.998504	−	0.0546823i	\(-0.982585\pi\)
							0.451896	−	0.892071i	\(-0.350748\pi\)
\(43\)	1.00000	−	1.73205i	0.152499	−	0.264135i	−0.779647	−	0.626219i	\(-0.784601\pi\)
							0.932145	+	0.362084i	\(0.117935\pi\)
\(47\)	−1.00000	+	1.73205i	−0.145865	+	0.252646i	−0.929695	−	0.368329i	\(-0.879930\pi\)
							0.783830	+	0.620975i	\(0.213263\pi\)
\(53\)	8.00000			1.09888			0.549442	−	0.835532i	\(-0.314840\pi\)
							0.549442	+	0.835532i	\(0.314840\pi\)
\(59\)	0.500000	+	0.866025i	0.0650945	+	0.112747i	0.896736	−	0.442566i	\(-0.145932\pi\)
							−0.831641	+	0.555313i	\(0.812598\pi\)
\(61\)	1.00000	−	1.73205i	0.128037	−	0.221766i	−0.794879	−	0.606768i	\(-0.792466\pi\)
							0.922916	+	0.385002i	\(0.125799\pi\)
\(67\)	−3.00000	−	5.19615i	−0.366508	−	0.634811i	0.622509	−	0.782613i	\(-0.286114\pi\)
							−0.989017	+	0.147802i	\(0.952780\pi\)
\(71\)	1.00000			0.118678			0.0593391	−	0.998238i	\(-0.481101\pi\)
							0.0593391	+	0.998238i	\(0.481101\pi\)
\(73\)	−8.00000			−0.936329			−0.468165	−	0.883641i	\(-0.655085\pi\)
							−0.468165	+	0.883641i	\(0.655085\pi\)
\(79\)	−6.00000	+	10.3923i	−0.675053	+	1.16923i	0.301401	+	0.953498i	\(0.402546\pi\)
							−0.976453	+	0.215728i	\(0.930788\pi\)
\(83\)	−3.00000	+	5.19615i	−0.329293	+	0.570352i	−0.982372	−	0.186938i	\(-0.940144\pi\)
							0.653079	+	0.757290i	\(0.273477\pi\)
\(89\)	−9.00000			−0.953998			−0.476999	−	0.878904i	\(-0.658275\pi\)
							−0.476999	+	0.878904i	\(0.658275\pi\)
\(97\)	−7.00000	+	12.1244i	−0.710742	+	1.23104i	0.253837	+	0.967247i	\(0.418307\pi\)
							−0.964579	+	0.263795i	\(0.915026\pi\)

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	405.2.e.a.136.1		2
3.2	odd	2		405.2.e.g.136.1		2
9.2	odd	6		405.2.a.a.1.1	✓	1
9.4	even	3	inner	405.2.e.a.271.1		2
9.5	odd	6		405.2.e.g.271.1		2
9.7	even	3		405.2.a.f.1.1	yes	1
36.7	odd	6		6480.2.a.r.1.1		1
36.11	even	6		6480.2.a.f.1.1		1
45.2	even	12		2025.2.b.a.649.1		2
45.7	odd	12		2025.2.b.b.649.2		2
45.29	odd	6		2025.2.a.f.1.1		1
45.34	even	6		2025.2.a.a.1.1		1
45.38	even	12		2025.2.b.a.649.2		2
45.43	odd	12		2025.2.b.b.649.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.2.a.a.1.1	✓	1	9.2	odd	6
405.2.a.f.1.1	yes	1	9.7	even	3
405.2.e.a.136.1		2	1.1	even	1	trivial
405.2.e.a.271.1		2	9.4	even	3	inner
405.2.e.g.136.1		2	3.2	odd	2
405.2.e.g.271.1		2	9.5	odd	6
2025.2.a.a.1.1		1	45.34	even	6
2025.2.a.f.1.1		1	45.29	odd	6
2025.2.b.a.649.1		2	45.2	even	12
2025.2.b.a.649.2		2	45.38	even	12
2025.2.b.b.649.1		2	45.43	odd	12
2025.2.b.b.649.2		2	45.7	odd	12
6480.2.a.f.1.1		1	36.11	even	6
6480.2.a.r.1.1		1	36.7	odd	6