Embedded newform 25.6.d.a.11.6

• Label: 25.6.d.a.11.6
• Level: $25$
• Weight: $6$
• Character: 25.11
• Analytic conductor: $4.010$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	25.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.00959549532\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(11\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			11.6
Character	\(\chi\)	\(=\)	25.11
Dual form			25.6.d.a.16.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.208693 + 0.151625i) q^{2} +(1.88200 - 5.79220i) q^{3} +(-9.86798 + 30.3705i) q^{4} +(51.9690 + 20.5968i) q^{5} +(0.485479 + 1.49415i) q^{6} +127.575 q^{7} +(-5.09638 - 15.6850i) q^{8} +(166.583 + 121.030i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 7 q^{2} - q^{3} - 147 q^{4} + 115 q^{5} + 133 q^{6} - 202 q^{7} - 500 q^{8} - 508 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.208693	+	0.151625i	−0.0368921	+	0.0268037i	−0.606079	−	0.795405i	\(-0.707258\pi\)
							0.569186	+	0.822209i	\(0.307258\pi\)
\(3\)	1.88200	−	5.79220i	0.120730	−	0.371570i	−0.872369	−	0.488849i	\(-0.837417\pi\)
							0.993099	+	0.117279i	\(0.0374171\pi\)
\(5\)	51.9690	+	20.5968i	0.929649	+	0.368446i
							\(7\)	127.575			0.984058			0.492029	−	0.870579i	\(-0.336255\pi\)
0.492029	+	0.870579i	\(0.336255\pi\)
\(11\)	−239.217	+	173.802i	−0.596089	+	0.433084i	−0.844489	−	0.535574i	\(-0.820095\pi\)
							0.248400	+	0.968658i	\(0.420095\pi\)
\(13\)	60.8865	+	44.2366i	0.0999224	+	0.0725978i	0.636625	−	0.771174i	\(-0.280330\pi\)
							−0.536702	+	0.843772i	\(0.680330\pi\)
\(17\)	−196.254	−	604.007i	−0.164701	−	0.506897i	0.834313	−	0.551290i	\(-0.185864\pi\)
							−0.999014	+	0.0443937i	\(0.985864\pi\)
\(19\)	−533.686	−	1642.52i	−0.339158	−	1.04382i	−0.964638	−	0.263580i	\(-0.915097\pi\)
							0.625480	−	0.780240i	\(-0.284903\pi\)
\(23\)	−1622.65	+	1178.92i	−0.639595	+	0.464693i	−0.859711	−	0.510780i	\(-0.829356\pi\)
							0.220116	+	0.975474i	\(0.429356\pi\)
\(29\)	1816.24	−	5589.82i	0.401032	−	1.23425i	−0.523131	−	0.852252i	\(-0.675236\pi\)
							0.924163	−	0.381998i	\(-0.124764\pi\)
\(31\)	−80.1130	−	246.562i	−0.0149726	−	0.0460811i	0.943291	−	0.331967i	\(-0.107712\pi\)
							−0.958264	+	0.285886i	\(0.907712\pi\)
\(37\)	12364.9	+	8983.63i	1.48486	+	1.07882i	0.975949	+	0.218000i	\(0.0699532\pi\)
							0.508915	+	0.860817i	\(0.330047\pi\)
\(41\)	−13588.4	−	9872.57i	−1.26244	−	0.917214i	−0.263562	−	0.964643i	\(-0.584897\pi\)
							−0.998875	+	0.0474291i	\(0.984897\pi\)
\(43\)	−13637.8			−1.12479			−0.562396	−	0.826868i	\(-0.690120\pi\)
							−0.562396	+	0.826868i	\(0.690120\pi\)
\(47\)	7979.27	−	24557.7i	0.526888	−	1.62160i	−0.233663	−	0.972318i	\(-0.575071\pi\)
							0.760552	−	0.649278i	\(-0.224929\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.d.a.11.6	✓	44
25.4	even	10		625.6.a.c.1.11		22
25.16	even	5	inner	25.6.d.a.16.6	yes	44
25.21	even	5		625.6.a.d.1.12		22

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.d.a.11.6	✓	44	1.1	even	1	trivial
25.6.d.a.16.6	yes	44	25.16	even	5	inner
625.6.a.c.1.11		22	25.4	even	10
625.6.a.d.1.12		22	25.21	even	5