LMFDB - Embedded newform 2025.2.a.w.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2025,2,Mod(1,2025)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2025.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [4,0,0,6,0,0,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$16.1697064093$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{11})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 4 $ x^4 - 7*x^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			1.3
Root			$0.792287$ of defining polynomial
Character	$\chi$	$=$	2025.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

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

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.792287	0.560232	0.280116	−	0.959966i	$-0.409627\pi$
$2$	0.792287	0.560232	0.280116	+	0.959966i	$0.409627\pi$
$3$	0	0
$3$	0	0
$4$	−1.37228	−0.686141
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.46410	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	−2.67181	−0.944629
$9$	0	0
$10$	0	0
$11$	−4.37228	−1.31829	−0.659146	−	0.752015i	$-0.729082\pi$
$11$	−4.37228	−1.31829	−0.659146	+	0.752015i	$0.729082\pi$
$12$	0	0
$13$	5.84096	1.61999	0.809996	−	0.586436i	$-0.199469\pi$
$13$	5.84096	1.61999	0.809996	+	0.586436i	$0.199469\pi$
$14$	2.74456	0.733515
$15$	0	0
$16$	0.627719	0.156930
$17$	0.792287	0.192158	0.0960789	−	0.995374i	$-0.469370\pi$
$17$	0.792287	0.192158	0.0960789	+	0.995374i	$0.469370\pi$
$18$	0	0
$19$	−0.372281	−0.0854072	−0.0427036	−	0.999088i	$-0.513597\pi$
$19$	−0.372281	−0.0854072	−0.0427036	+	0.999088i	$0.513597\pi$
$20$	0	0
$21$	0	0
$22$	−3.46410	−0.738549
$23$	1.58457	0.330407	0.165203	−	0.986260i	$-0.447172\pi$
$23$	1.58457	0.330407	0.165203	+	0.986260i	$0.447172\pi$
$24$	0	0
$25$	0	0
$26$	4.62772	0.907570
$27$	0	0
$28$	−4.75372	−0.898369
$29$	−5.74456	−1.06674	−0.533369	−	0.845883i	$-0.679074\pi$
$29$	−5.74456	−1.06674	−0.533369	+	0.845883i	$0.679074\pi$
$30$	0	0
$31$	6.37228	1.14450	0.572248	−	0.820081i	$-0.306072\pi$
$31$	6.37228	1.14450	0.572248	+	0.820081i	$0.306072\pi$
$32$	5.84096	1.03255
$33$	0	0
$34$	0.627719	0.107653
$35$	0	0
$36$	0	0
$37$	−2.37686	−0.390754	−0.195377	−	0.980728i	$-0.562593\pi$
$37$	−2.37686	−0.390754	−0.195377	+	0.980728i	$0.562593\pi$
$38$	−0.294954	−0.0478478
$39$	0	0
$40$	0	0
$41$	4.37228	0.682836	0.341418	−	0.939912i	$-0.389093\pi$
$41$	4.37228	0.682836	0.341418	+	0.939912i	$0.389093\pi$
$42$	0	0
$43$	−3.46410	−0.528271	−0.264135	−	0.964486i	$-0.585087\pi$
$43$	−3.46410	−0.528271	−0.264135	+	0.964486i	$0.585087\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	1.25544	0.185104
$47$	1.87953	0.274157	0.137079	−	0.990560i	$-0.456229\pi$
$47$	1.87953	0.274157	0.137079	+	0.990560i	$0.456229\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	0	0
$52$	−8.01544	−1.11154
$53$	11.9769	1.64515	0.822575	−	0.568656i	$-0.192536\pi$
$53$	11.9769	1.64515	0.822575	+	0.568656i	$0.192536\pi$
$54$	0	0
$55$	0	0
$56$	−9.25544	−1.23681
$57$	0	0
$58$	−4.55134	−0.597621
$59$	1.62772	0.211911	0.105955	−	0.994371i	$-0.466210\pi$
$59$	1.62772	0.211911	0.105955	+	0.994371i	$0.466210\pi$
$60$	0	0
$61$	9.37228	1.20000	0.599999	−	0.800001i	$-0.295168\pi$
$61$	9.37228	1.20000	0.599999	+	0.800001i	$0.295168\pi$
$62$	5.04868	0.641182
$63$	0	0
$64$	3.37228	0.421535
$65$	0	0
$66$	0	0
$67$	11.6819	1.42717	0.713587	−	0.700566i	$-0.247069\pi$
$67$	11.6819	1.42717	0.713587	+	0.700566i	$0.247069\pi$
$68$	−1.08724	−0.131847
$69$	0	0
$70$	0	0
$71$	13.1168	1.55668	0.778341	−	0.627841i	$-0.216061\pi$
$71$	13.1168	1.55668	0.778341	+	0.627841i	$0.216061\pi$
$72$	0	0
$73$	−2.37686	−0.278191	−0.139095	−	0.990279i	$-0.544419\pi$
$73$	−2.37686	−0.278191	−0.139095	+	0.990279i	$0.544419\pi$
$74$	−1.88316	−0.218912
$75$	0	0
$76$	0.510875	0.0586013
$77$	−15.1460	−1.72605
$78$	0	0
$79$	6.74456	0.758823	0.379411	−	0.925228i	$-0.376127\pi$
$79$	6.74456	0.758823	0.379411	+	0.925228i	$0.376127\pi$
$80$	0	0
$81$	0	0
$82$	3.46410	0.382546
$83$	11.9769	1.31463	0.657317	−	0.753615i	$-0.271691\pi$
$83$	11.9769	1.31463	0.657317	+	0.753615i	$0.271691\pi$
$84$	0	0
$85$	0	0
$86$	−2.74456	−0.295954
$87$	0	0
$88$	11.6819	1.24530
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	20.2337	2.12107
$92$	−2.17448	−0.226705
$93$	0	0
$94$	1.48913	0.153592
$95$	0	0
$96$	0	0
$97$	−1.28962	−0.130941	−0.0654706	−	0.997855i	$-0.520855\pi$
$97$	−1.28962	−0.130941	−0.0654706	+	0.997855i	$0.520855\pi$
$98$	3.96143	0.400165
$99$	0	0

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

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

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

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

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

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