LMFDB - Embedded newform 4761.2.a.v.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4761,2,Mod(1,4761)] 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("4761.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4761, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4761 = 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4761.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,6,-2,0,-2,0,0,10,8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$38.0167764023$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4761.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607 q^{2} +3.00000 q^{4} -3.23607 q^{5} +1.23607 q^{7} -2.23607 q^{8} +7.23607 q^{10} +4.00000 q^{11} +4.47214 q^{13} -2.76393 q^{14} -1.00000 q^{16} -7.23607 q^{17} -2.76393 q^{19} -9.70820 q^{20} -8.94427 q^{22} +5.47214 q^{25} -10.0000 q^{26} +3.70820 q^{28} +4.47214 q^{29} +2.47214 q^{31} +6.70820 q^{32} +16.1803 q^{34} -4.00000 q^{35} +4.47214 q^{37} +6.18034 q^{38} +7.23607 q^{40} -6.94427 q^{41} -7.70820 q^{43} +12.0000 q^{44} +4.00000 q^{47} -5.47214 q^{49} -12.2361 q^{50} +13.4164 q^{52} -0.763932 q^{53} -12.9443 q^{55} -2.76393 q^{56} -10.0000 q^{58} -12.9443 q^{59} +4.47214 q^{61} -5.52786 q^{62} -13.0000 q^{64} -14.4721 q^{65} -5.23607 q^{67} -21.7082 q^{68} +8.94427 q^{70} +8.00000 q^{71} -10.9443 q^{73} -10.0000 q^{74} -8.29180 q^{76} +4.94427 q^{77} +3.70820 q^{79} +3.23607 q^{80} +15.5279 q^{82} +4.00000 q^{83} +23.4164 q^{85} +17.2361 q^{86} -8.94427 q^{88} +3.23607 q^{89} +5.52786 q^{91} -8.94427 q^{94} +8.94427 q^{95} +0.472136 q^{97} +12.2361 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{4} - 2 q^{5} - 2 q^{7} + 10 q^{10} + 8 q^{11} - 10 q^{14} - 2 q^{16} - 10 q^{17} - 10 q^{19} - 6 q^{20} + 2 q^{25} - 20 q^{26} - 6 q^{28} - 4 q^{31} + 10 q^{34} - 8 q^{35} - 10 q^{38} + 10 q^{40}+ \cdots + 20 q^{98}+O(q^{100}) $ 2 * q + 6 * q^4 - 2 * q^5 - 2 * q^7 + 10 * q^10 + 8 * q^11 - 10 * q^14 - 2 * q^16 - 10 * q^17 - 10 * q^19 - 6 * q^20 + 2 * q^25 - 20 * q^26 - 6 * q^28 - 4 * q^31 + 10 * q^34 - 8 * q^35 - 10 * q^38 + 10 * q^40 + 4 * q^41 - 2 * q^43 + 24 * q^44 + 8 * q^47 - 2 * q^49 - 20 * q^50 - 6 * q^53 - 8 * q^55 - 10 * q^56 - 20 * q^58 - 8 * q^59 - 20 * q^62 - 26 * q^64 - 20 * q^65 - 6 * q^67 - 30 * q^68 + 16 * q^71 - 4 * q^73 - 20 * q^74 - 30 * q^76 - 8 * q^77 - 6 * q^79 + 2 * q^80 + 40 * q^82 + 8 * q^83 + 20 * q^85 + 30 * q^86 + 2 * q^89 + 20 * q^91 - 8 * q^97 + 20 * q^98

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 $

$2$	−2.23607	−1.58114	−0.790569	−	0.612372i	$-0.790215\pi$
$2$	−2.23607	−1.58114	−0.790569	+	0.612372i	$0.790215\pi$
$3$	0	0
$3$	0	0
$4$	3.00000	1.50000
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	0	0
$7$	1.23607	0.467190	0.233595	−	0.972334i	$-0.424951\pi$
$7$	1.23607	0.467190	0.233595	+	0.972334i	$0.424951\pi$
$8$	−2.23607	−0.790569
$9$	0	0
$10$	7.23607	2.28825
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	4.47214	1.24035	0.620174	−	0.784465i	$-0.287062\pi$
$13$	4.47214	1.24035	0.620174	+	0.784465i	$0.287062\pi$
$14$	−2.76393	−0.738692
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−7.23607	−1.75500	−0.877502	−	0.479573i	$-0.840792\pi$
$17$	−7.23607	−1.75500	−0.877502	+	0.479573i	$0.840792\pi$
$18$	0	0
$19$	−2.76393	−0.634089	−0.317045	−	0.948411i	$-0.602691\pi$
$19$	−2.76393	−0.634089	−0.317045	+	0.948411i	$0.602691\pi$
$20$	−9.70820	−2.17082
$21$	0	0
$22$	−8.94427	−1.90693
$23$	0	0
$23$	0	0
$24$	0	0
$25$	5.47214	1.09443
$26$	−10.0000	−1.96116
$27$	0	0
$28$	3.70820	0.700785
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	2.47214	0.444009	0.222004	−	0.975046i	$-0.428740\pi$
$31$	2.47214	0.444009	0.222004	+	0.975046i	$0.428740\pi$
$32$	6.70820	1.18585
$33$	0	0
$34$	16.1803	2.77491
$35$	−4.00000	−0.676123
$36$	0	0
$37$	4.47214	0.735215	0.367607	−	0.929981i	$-0.380177\pi$
$37$	4.47214	0.735215	0.367607	+	0.929981i	$0.380177\pi$
$38$	6.18034	1.00258
$39$	0	0
$40$	7.23607	1.14412
$41$	−6.94427	−1.08451	−0.542257	−	0.840213i	$-0.682430\pi$
$41$	−6.94427	−1.08451	−0.542257	+	0.840213i	$0.682430\pi$
$42$	0	0
$43$	−7.70820	−1.17549	−0.587745	−	0.809046i	$-0.699984\pi$
$43$	−7.70820	−1.17549	−0.587745	+	0.809046i	$0.699984\pi$
$44$	12.0000	1.80907
$45$	0	0
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	−12.2361	−1.73044
$51$	0	0
$52$	13.4164	1.86052
$53$	−0.763932	−0.104934	−0.0524671	−	0.998623i	$-0.516708\pi$
$53$	−0.763932	−0.104934	−0.0524671	+	0.998623i	$0.516708\pi$
$54$	0	0
$55$	−12.9443	−1.74541
$56$	−2.76393	−0.369346
$57$	0	0
$58$	−10.0000	−1.31306
$59$	−12.9443	−1.68520	−0.842600	−	0.538539i	$-0.818976\pi$
$59$	−12.9443	−1.68520	−0.842600	+	0.538539i	$0.818976\pi$
$60$	0	0
$61$	4.47214	0.572598	0.286299	−	0.958140i	$-0.407575\pi$
$61$	4.47214	0.572598	0.286299	+	0.958140i	$0.407575\pi$
$62$	−5.52786	−0.702039
$63$	0	0
$64$	−13.0000	−1.62500
$65$	−14.4721	−1.79505
$66$	0	0
$67$	−5.23607	−0.639688	−0.319844	−	0.947470i	$-0.603630\pi$
$67$	−5.23607	−0.639688	−0.319844	+	0.947470i	$0.603630\pi$
$68$	−21.7082	−2.63251
$69$	0	0
$70$	8.94427	1.06904
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	−10.9443	−1.28093	−0.640465	−	0.767987i	$-0.721258\pi$
$73$	−10.9443	−1.28093	−0.640465	+	0.767987i	$0.721258\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	−8.29180	−0.951134
$77$	4.94427	0.563452
$78$	0	0
$79$	3.70820	0.417206	0.208603	−	0.978000i	$-0.433108\pi$
$79$	3.70820	0.417206	0.208603	+	0.978000i	$0.433108\pi$
$80$	3.23607	0.361803
$81$	0	0
$82$	15.5279	1.71477
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	23.4164	2.53987
$86$	17.2361	1.85861
$87$	0	0
$88$	−8.94427	−0.953463
$89$	3.23607	0.343023	0.171511	−	0.985182i	$-0.445135\pi$
$89$	3.23607	0.343023	0.171511	+	0.985182i	$0.445135\pi$
$90$	0	0
$91$	5.52786	0.579478
$92$	0	0
$93$	0	0
$94$	−8.94427	−0.922531
$95$	8.94427	0.917663
$96$	0	0
$97$	0.472136	0.0479381	0.0239691	−	0.999713i	$-0.492370\pi$
$97$	0.472136	0.0479381	0.0239691	+	0.999713i	$0.492370\pi$
$98$	12.2361	1.23603
$99$	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	4761.2.a.v.1.1		2
3.2	odd	2		1587.2.a.i.1.2		2
23.22	odd	2		207.2.a.c.1.1		2
69.68	even	2		69.2.a.b.1.2	✓	2
92.91	even	2		3312.2.a.bb.1.2		2
115.114	odd	2		5175.2.a.bk.1.2		2
276.275	odd	2		1104.2.a.m.1.1		2
345.68	odd	4		1725.2.b.o.1174.1		4
345.137	odd	4		1725.2.b.o.1174.4		4
345.344	even	2		1725.2.a.ba.1.1		2
483.482	odd	2		3381.2.a.t.1.2		2
552.275	odd	2		4416.2.a.bg.1.2		2
552.413	even	2		4416.2.a.bm.1.2		2
759.758	odd	2		8349.2.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.a.b.1.2	✓	2	69.68	even	2
207.2.a.c.1.1		2	23.22	odd	2
1104.2.a.m.1.1		2	276.275	odd	2
1587.2.a.i.1.2		2	3.2	odd	2
1725.2.a.ba.1.1		2	345.344	even	2
1725.2.b.o.1174.1		4	345.68	odd	4
1725.2.b.o.1174.4		4	345.137	odd	4
3312.2.a.bb.1.2		2	92.91	even	2
3381.2.a.t.1.2		2	483.482	odd	2
4416.2.a.bg.1.2		2	552.275	odd	2
4416.2.a.bm.1.2		2	552.413	even	2
4761.2.a.v.1.1		2	1.1	even	1	trivial
5175.2.a.bk.1.2		2	115.114	odd	2
8349.2.a.i.1.1		2	759.758	odd	2

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

Level:	\( N \)	\(=\)	\( 4761 = 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4761.a (trivial)

\(f(q)\)	\(=\)	\(q-2.23607 q^{2} +3.00000 q^{4} -3.23607 q^{5} +1.23607 q^{7} -2.23607 q^{8} +7.23607 q^{10} +4.00000 q^{11} +4.47214 q^{13} -2.76393 q^{14} -1.00000 q^{16} -7.23607 q^{17} -2.76393 q^{19} -9.70820 q^{20} -8.94427 q^{22} +5.47214 q^{25} -10.0000 q^{26} +3.70820 q^{28} +4.47214 q^{29} +2.47214 q^{31} +6.70820 q^{32} +16.1803 q^{34} -4.00000 q^{35} +4.47214 q^{37} +6.18034 q^{38} +7.23607 q^{40} -6.94427 q^{41} -7.70820 q^{43} +12.0000 q^{44} +4.00000 q^{47} -5.47214 q^{49} -12.2361 q^{50} +13.4164 q^{52} -0.763932 q^{53} -12.9443 q^{55} -2.76393 q^{56} -10.0000 q^{58} -12.9443 q^{59} +4.47214 q^{61} -5.52786 q^{62} -13.0000 q^{64} -14.4721 q^{65} -5.23607 q^{67} -21.7082 q^{68} +8.94427 q^{70} +8.00000 q^{71} -10.9443 q^{73} -10.0000 q^{74} -8.29180 q^{76} +4.94427 q^{77} +3.70820 q^{79} +3.23607 q^{80} +15.5279 q^{82} +4.00000 q^{83} +23.4164 q^{85} +17.2361 q^{86} -8.94427 q^{88} +3.23607 q^{89} +5.52786 q^{91} -8.94427 q^{94} +8.94427 q^{95} +0.472136 q^{97} +12.2361 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{4} - 2 q^{5} - 2 q^{7} + 10 q^{10} + 8 q^{11} - 10 q^{14} - 2 q^{16} - 10 q^{17} - 10 q^{19} - 6 q^{20} + 2 q^{25} - 20 q^{26} - 6 q^{28} - 4 q^{31} + 10 q^{34} - 8 q^{35} - 10 q^{38} + 10 q^{40}+ \cdots + 20 q^{98}+O(q^{100}) \) 2 * q + 6 * q^4 - 2 * q^5 - 2 * q^7 + 10 * q^10 + 8 * q^11 - 10 * q^14 - 2 * q^16 - 10 * q^17 - 10 * q^19 - 6 * q^20 + 2 * q^25 - 20 * q^26 - 6 * q^28 - 4 * q^31 + 10 * q^34 - 8 * q^35 - 10 * q^38 + 10 * q^40 + 4 * q^41 - 2 * q^43 + 24 * q^44 + 8 * q^47 - 2 * q^49 - 20 * q^50 - 6 * q^53 - 8 * q^55 - 10 * q^56 - 20 * q^58 - 8 * q^59 - 20 * q^62 - 26 * q^64 - 20 * q^65 - 6 * q^67 - 30 * q^68 + 16 * q^71 - 4 * q^73 - 20 * q^74 - 30 * q^76 - 8 * q^77 - 6 * q^79 + 2 * q^80 + 40 * q^82 + 8 * q^83 + 20 * q^85 + 30 * q^86 + 2 * q^89 + 20 * q^91 - 8 * q^97 + 20 * q^98

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