LMFDB - Embedded newform 2025.2.a.w.1.4

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			1.4
Root			$2.52434$ 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)

$f(q)$	$=$	$q+2.52434 q^{2} +4.37228 q^{4} -3.46410 q^{7} +5.98844 q^{8} +1.37228 q^{11} +4.10891 q^{13} -8.74456 q^{14} +6.37228 q^{16} +2.52434 q^{17} +5.37228 q^{19} +3.46410 q^{22} +5.04868 q^{23} +10.3723 q^{26} -15.1460 q^{28} +5.74456 q^{29} +0.627719 q^{31} +4.10891 q^{32} +6.37228 q^{34} -7.57301 q^{37} +13.5615 q^{38} -1.37228 q^{41} +3.46410 q^{43} +6.00000 q^{44} +12.7446 q^{46} -8.51278 q^{47} +5.00000 q^{49} +17.9653 q^{52} -5.34363 q^{53} -20.7446 q^{56} +14.5012 q^{58} +7.37228 q^{59} +3.62772 q^{61} +1.58457 q^{62} -2.37228 q^{64} +8.21782 q^{67} +11.0371 q^{68} -4.11684 q^{71} -7.57301 q^{73} -19.1168 q^{74} +23.4891 q^{76} -4.75372 q^{77} -4.74456 q^{79} -3.46410 q^{82} -5.34363 q^{83} +8.74456 q^{86} +8.21782 q^{88} -3.00000 q^{89} -14.2337 q^{91} +22.0742 q^{92} -21.4891 q^{94} -18.6101 q^{97} +12.6217 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.52434	1.78498	0.892488	−	0.451071i	$-0.148958\pi$
$2$	2.52434	1.78498	0.892488	+	0.451071i	$0.148958\pi$
$3$	0	0
$3$	0	0
$4$	4.37228	2.18614
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.46410	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	5.98844	2.11723
$9$	0	0
$10$	0	0
$11$	1.37228	0.413758	0.206879	−	0.978366i	$-0.433669\pi$
$11$	1.37228	0.413758	0.206879	+	0.978366i	$0.433669\pi$
$12$	0	0
$13$	4.10891	1.13961	0.569804	−	0.821781i	$-0.307019\pi$
$13$	4.10891	1.13961	0.569804	+	0.821781i	$0.307019\pi$
$14$	−8.74456	−2.33708
$15$	0	0
$16$	6.37228	1.59307
$17$	2.52434	0.612242	0.306121	−	0.951993i	$-0.400969\pi$
$17$	2.52434	0.612242	0.306121	+	0.951993i	$0.400969\pi$
$18$	0	0
$19$	5.37228	1.23249	0.616243	−	0.787556i	$-0.288654\pi$
$19$	5.37228	1.23249	0.616243	+	0.787556i	$0.288654\pi$
$20$	0	0
$21$	0	0
$22$	3.46410	0.738549
$23$	5.04868	1.05272	0.526361	−	0.850261i	$-0.323556\pi$
$23$	5.04868	1.05272	0.526361	+	0.850261i	$0.323556\pi$
$24$	0	0
$25$	0	0
$26$	10.3723	2.03417
$27$	0	0
$28$	−15.1460	−2.86233
$29$	5.74456	1.06674	0.533369	−	0.845883i	$-0.320926\pi$
$29$	5.74456	1.06674	0.533369	+	0.845883i	$0.320926\pi$
$30$	0	0
$31$	0.627719	0.112742	0.0563708	−	0.998410i	$-0.482047\pi$
$31$	0.627719	0.112742	0.0563708	+	0.998410i	$0.482047\pi$
$32$	4.10891	0.726360
$33$	0	0
$34$	6.37228	1.09284
$35$	0	0
$36$	0	0
$37$	−7.57301	−1.24500	−0.622498	−	0.782621i	$-0.713882\pi$
$37$	−7.57301	−1.24500	−0.622498	+	0.782621i	$0.713882\pi$
$38$	13.5615	2.19996
$39$	0	0
$40$	0	0
$41$	−1.37228	−0.214314	−0.107157	−	0.994242i	$-0.534175\pi$
$41$	−1.37228	−0.214314	−0.107157	+	0.994242i	$0.534175\pi$
$42$	0	0
$43$	3.46410	0.528271	0.264135	−	0.964486i	$-0.414913\pi$
$43$	3.46410	0.528271	0.264135	+	0.964486i	$0.414913\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	12.7446	1.87908
$47$	−8.51278	−1.24172	−0.620858	−	0.783923i	$-0.713216\pi$
$47$	−8.51278	−1.24172	−0.620858	+	0.783923i	$0.713216\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	0	0
$52$	17.9653	2.49134
$53$	−5.34363	−0.734004	−0.367002	−	0.930220i	$-0.619616\pi$
$53$	−5.34363	−0.734004	−0.367002	+	0.930220i	$0.619616\pi$
$54$	0	0
$55$	0	0
$56$	−20.7446	−2.77211
$57$	0	0
$58$	14.5012	1.90410
$59$	7.37228	0.959789	0.479895	−	0.877326i	$-0.340675\pi$
$59$	7.37228	0.959789	0.479895	+	0.877326i	$0.340675\pi$
$60$	0	0
$61$	3.62772	0.464482	0.232241	−	0.972658i	$-0.425394\pi$
$61$	3.62772	0.464482	0.232241	+	0.972658i	$0.425394\pi$
$62$	1.58457	0.201241
$63$	0	0
$64$	−2.37228	−0.296535
$65$	0	0
$66$	0	0
$67$	8.21782	1.00397	0.501983	−	0.864877i	$-0.332604\pi$
$67$	8.21782	1.00397	0.501983	+	0.864877i	$0.332604\pi$
$68$	11.0371	1.33845
$69$	0	0
$70$	0	0
$71$	−4.11684	−0.488579	−0.244290	−	0.969702i	$-0.578555\pi$
$71$	−4.11684	−0.488579	−0.244290	+	0.969702i	$0.578555\pi$
$72$	0	0
$73$	−7.57301	−0.886354	−0.443177	−	0.896434i	$-0.646149\pi$
$73$	−7.57301	−0.886354	−0.443177	+	0.896434i	$0.646149\pi$
$74$	−19.1168	−2.22229
$75$	0	0
$76$	23.4891	2.69439
$77$	−4.75372	−0.541737
$78$	0	0
$79$	−4.74456	−0.533805	−0.266903	−	0.963724i	$-0.586000\pi$
$79$	−4.74456	−0.533805	−0.266903	+	0.963724i	$0.586000\pi$
$80$	0	0
$81$	0	0
$82$	−3.46410	−0.382546
$83$	−5.34363	−0.586540	−0.293270	−	0.956030i	$-0.594743\pi$
$83$	−5.34363	−0.586540	−0.293270	+	0.956030i	$0.594743\pi$
$84$	0	0
$85$	0	0
$86$	8.74456	0.942950
$87$	0	0
$88$	8.21782	0.876023
$89$	−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
$91$	−14.2337	−1.49210
$92$	22.0742	2.30140
$93$	0	0
$94$	−21.4891	−2.21643
$95$	0	0
$96$	0	0
$97$	−18.6101	−1.88957	−0.944786	−	0.327688i	$-0.893731\pi$
$97$	−18.6101	−1.88957	−0.944786	+	0.327688i	$0.893731\pi$
$98$	12.6217	1.27498
$99$	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.1.4		4
3.2	odd	2		2025.2.a.x.1.1		4
5.2	odd	4		405.2.b.a.244.4	yes	4
5.3	odd	4		405.2.b.a.244.1	✓	4
5.4	even	2	inner	2025.2.a.w.1.1		4
15.2	even	4		405.2.b.b.244.1	yes	4
15.8	even	4		405.2.b.b.244.4	yes	4
15.14	odd	2		2025.2.a.x.1.4		4
45.2	even	12		405.2.j.a.109.1		4
45.7	odd	12		405.2.j.e.109.2		4
45.13	odd	12		405.2.j.e.379.2		4
45.22	odd	12		405.2.j.b.379.1		4
45.23	even	12		405.2.j.a.379.1		4
45.32	even	12		405.2.j.d.379.2		4
45.38	even	12		405.2.j.d.109.2		4
45.43	odd	12		405.2.j.b.109.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.2.b.a.244.1	✓	4	5.3	odd	4
405.2.b.a.244.4	yes	4	5.2	odd	4
405.2.b.b.244.1	yes	4	15.2	even	4
405.2.b.b.244.4	yes	4	15.8	even	4
405.2.j.a.109.1		4	45.2	even	12
405.2.j.a.379.1		4	45.23	even	12
405.2.j.b.109.1		4	45.43	odd	12
405.2.j.b.379.1		4	45.22	odd	12
405.2.j.d.109.2		4	45.38	even	12
405.2.j.d.379.2		4	45.32	even	12
405.2.j.e.109.2		4	45.7	odd	12
405.2.j.e.379.2		4	45.13	odd	12
2025.2.a.w.1.1		4	5.4	even	2	inner
2025.2.a.w.1.4		4	1.1	even	1	trivial
2025.2.a.x.1.1		4	3.2	odd	2
2025.2.a.x.1.4		4	15.14	odd	2

Overview	Random
Universe	Knowledge

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)

\(f(q)\)	\(=\)	\(q+2.52434 q^{2} +4.37228 q^{4} -3.46410 q^{7} +5.98844 q^{8} +1.37228 q^{11} +4.10891 q^{13} -8.74456 q^{14} +6.37228 q^{16} +2.52434 q^{17} +5.37228 q^{19} +3.46410 q^{22} +5.04868 q^{23} +10.3723 q^{26} -15.1460 q^{28} +5.74456 q^{29} +0.627719 q^{31} +4.10891 q^{32} +6.37228 q^{34} -7.57301 q^{37} +13.5615 q^{38} -1.37228 q^{41} +3.46410 q^{43} +6.00000 q^{44} +12.7446 q^{46} -8.51278 q^{47} +5.00000 q^{49} +17.9653 q^{52} -5.34363 q^{53} -20.7446 q^{56} +14.5012 q^{58} +7.37228 q^{59} +3.62772 q^{61} +1.58457 q^{62} -2.37228 q^{64} +8.21782 q^{67} +11.0371 q^{68} -4.11684 q^{71} -7.57301 q^{73} -19.1168 q^{74} +23.4891 q^{76} -4.75372 q^{77} -4.74456 q^{79} -3.46410 q^{82} -5.34363 q^{83} +8.74456 q^{86} +8.21782 q^{88} -3.00000 q^{89} -14.2337 q^{91} +22.0742 q^{92} -21.4891 q^{94} -18.6101 q^{97} +12.6217 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{4} - 6 q^{11} - 12 q^{14} + 14 q^{16} + 10 q^{19} + 30 q^{26} + 14 q^{31} + 14 q^{34} + 6 q^{41} + 24 q^{44} + 28 q^{46} + 20 q^{49} - 60 q^{56} + 18 q^{59} + 26 q^{61} + 2 q^{64} + 18 q^{71}+ \cdots - 40 q^{94}+O(q^{100}) \) 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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