Embedded newform 2025.2.a.w.1.2

• Label: 2025.2.a.w.1.2
• Level: $2025$
• Weight: $2$
• Character: 2025.1
• Self dual: yes
• Analytic conductor: $16.170$
• Analytic rank: $0$
• Dimension: $4$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2025 = 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2025.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(16.1697064093\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\sqrt{3}, \sqrt{11})\)
Defining polynomial:	\( x^{4} - 7x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 405)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			\(-0.792287\) of defining polynomial
Character	\(\chi\)	\(=\)	2025.1

$q$-expansion

\(f(q)\)	\(=\)	\(q-0.792287 q^{2} -1.37228 q^{4} -3.46410 q^{7} +2.67181 q^{8} -4.37228 q^{11} -5.84096 q^{13} +2.74456 q^{14} +0.627719 q^{16} -0.792287 q^{17} -0.372281 q^{19} +3.46410 q^{22} -1.58457 q^{23} +4.62772 q^{26} +4.75372 q^{28} -5.74456 q^{29} +6.37228 q^{31} -5.84096 q^{32} +0.627719 q^{34} +2.37686 q^{37} +0.294954 q^{38} +4.37228 q^{41} +3.46410 q^{43} +6.00000 q^{44} +1.25544 q^{46} -1.87953 q^{47} +5.00000 q^{49} +8.01544 q^{52} -11.9769 q^{53} -9.25544 q^{56} +4.55134 q^{58} +1.62772 q^{59} +9.37228 q^{61} -5.04868 q^{62} +3.37228 q^{64} -11.6819 q^{67} +1.08724 q^{68} +13.1168 q^{71} +2.37686 q^{73} -1.88316 q^{74} +0.510875 q^{76} +15.1460 q^{77} +6.74456 q^{79} -3.46410 q^{82} -11.9769 q^{83} -2.74456 q^{86} -11.6819 q^{88} -3.00000 q^{89} +20.2337 q^{91} +2.17448 q^{92} +1.48913 q^{94} +1.28962 q^{97} -3.96143 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	4 * q + 6 * q^4 - 6 * q^11 - 12 * q^14 + 14 * q^16 + 10 * q^19 + 30 * q^26 + 14 * q^31 + 14 * q^34 + 6 * q^41 + 24 * q^44 + 28 * q^46 + 20 * q^49 - 60 * q^56 + 18 * q^59 + 26 * q^61 + 2 * q^64 + 18 * q^71 - 42 * q^74 + 48 * q^76 + 4 * q^79 + 12 * q^86 - 12 * q^89 + 12 * q^91 - 40 * q^94

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 \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.792287	−0.560232	−0.280116	−	0.959966i	\(-0.590373\pi\)
			−0.280116	+	0.959966i	\(0.590373\pi\)
\(3\)	0	0
			\(5\)	0	0
\(7\)	−3.46410	−1.30931				−0.654654	−	0.755929i	\(-0.727186\pi\)
			−0.654654	+	0.755929i	\(0.727186\pi\)
\(11\)	−4.37228	−1.31829	−0.659146	−	0.752015i	\(-0.729082\pi\)
			−0.659146	+	0.752015i	\(0.729082\pi\)
\(13\)	−5.84096	−1.61999	−0.809996	−	0.586436i	\(-0.800531\pi\)
			−0.809996	+	0.586436i	\(0.800531\pi\)
\(17\)	−0.792287	−0.192158	−0.0960789	−	0.995374i	\(-0.530630\pi\)
			−0.0960789	+	0.995374i	\(0.530630\pi\)
\(19\)	−0.372281	−0.0854072	−0.0427036	−	0.999088i	\(-0.513597\pi\)
			−0.0427036	+	0.999088i	\(0.513597\pi\)
\(23\)	−1.58457	−0.330407	−0.165203	−	0.986260i	\(-0.552828\pi\)
			−0.165203	+	0.986260i	\(0.552828\pi\)
\(29\)	−5.74456	−1.06674	−0.533369	−	0.845883i	\(-0.679074\pi\)
			−0.533369	+	0.845883i	\(0.679074\pi\)
\(31\)	6.37228	1.14450	0.572248	−	0.820081i	\(-0.306072\pi\)
			0.572248	+	0.820081i	\(0.306072\pi\)
\(37\)	2.37686	0.390754	0.195377	−	0.980728i	\(-0.437407\pi\)
			0.195377	+	0.980728i	\(0.437407\pi\)
\(41\)	4.37228	0.682836	0.341418	−	0.939912i	\(-0.389093\pi\)
			0.341418	+	0.939912i	\(0.389093\pi\)
\(43\)	3.46410	0.528271	0.264135	−	0.964486i	\(-0.414913\pi\)
			0.264135	+	0.964486i	\(0.414913\pi\)
\(47\)	−1.87953	−0.274157	−0.137079	−	0.990560i	\(-0.543771\pi\)
			−0.137079	+	0.990560i	\(0.543771\pi\)
\(53\)	−11.9769	−1.64515	−0.822575	−	0.568656i	\(-0.807464\pi\)
			−0.822575	+	0.568656i	\(0.807464\pi\)
\(59\)	1.62772	0.211911	0.105955	−	0.994371i	\(-0.466210\pi\)
			0.105955	+	0.994371i	\(0.466210\pi\)
\(61\)	9.37228	1.20000	0.599999	−	0.800001i	\(-0.295168\pi\)
			0.599999	+	0.800001i	\(0.295168\pi\)
\(67\)	−11.6819	−1.42717	−0.713587	−	0.700566i	\(-0.752931\pi\)
			−0.713587	+	0.700566i	\(0.752931\pi\)
\(71\)	13.1168	1.55668	0.778341	−	0.627841i	\(-0.216061\pi\)
			0.778341	+	0.627841i	\(0.216061\pi\)
\(73\)	2.37686	0.278191	0.139095	−	0.990279i	\(-0.455581\pi\)
			0.139095	+	0.990279i	\(0.455581\pi\)
\(79\)	6.74456	0.758823	0.379411	−	0.925228i	\(-0.376127\pi\)
			0.379411	+	0.925228i	\(0.376127\pi\)
\(83\)	−11.9769	−1.31463	−0.657317	−	0.753615i	\(-0.728309\pi\)
			−0.657317	+	0.753615i	\(0.728309\pi\)
\(89\)	−3.00000	−0.317999	−0.159000	−	0.987279i	\(-0.550827\pi\)
			−0.159000	+	0.987279i	\(0.550827\pi\)
\(97\)	1.28962	0.130941	0.0654706	−	0.997855i	\(-0.479145\pi\)
			0.0654706	+	0.997855i	\(0.479145\pi\)

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	2025.2.a.w.1.2		4
3.2	odd	2		2025.2.a.x.1.3		4
5.2	odd	4		405.2.b.a.244.2	✓	4
5.3	odd	4		405.2.b.a.244.3	yes	4
5.4	even	2	inner	2025.2.a.w.1.3		4
15.2	even	4		405.2.b.b.244.3	yes	4
15.8	even	4		405.2.b.b.244.2	yes	4
15.14	odd	2		2025.2.a.x.1.2		4
45.2	even	12		405.2.j.a.109.2		4
45.7	odd	12		405.2.j.e.109.1		4
45.13	odd	12		405.2.j.e.379.1		4
45.22	odd	12		405.2.j.b.379.2		4
45.23	even	12		405.2.j.a.379.2		4
45.32	even	12		405.2.j.d.379.1		4
45.38	even	12		405.2.j.d.109.1		4
45.43	odd	12		405.2.j.b.109.2		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.2.b.a.244.2	✓	4	5.2	odd	4
405.2.b.a.244.3	yes	4	5.3	odd	4
405.2.b.b.244.2	yes	4	15.8	even	4
405.2.b.b.244.3	yes	4	15.2	even	4
405.2.j.a.109.2		4	45.2	even	12
405.2.j.a.379.2		4	45.23	even	12
405.2.j.b.109.2		4	45.43	odd	12
405.2.j.b.379.2		4	45.22	odd	12
405.2.j.d.109.1		4	45.38	even	12
405.2.j.d.379.1		4	45.32	even	12
405.2.j.e.109.1		4	45.7	odd	12
405.2.j.e.379.1		4	45.13	odd	12
2025.2.a.w.1.2		4	1.1	even	1	trivial
2025.2.a.w.1.3		4	5.4	even	2	inner
2025.2.a.x.1.2		4	15.14	odd	2
2025.2.a.x.1.3		4	3.2	odd	2