LMFDB - Embedded newform 920.2.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [920,2,Mod(1,920)] 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("920.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	920.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.34623698596$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	920.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155 q^{3} -1.00000 q^{5} +1.56155 q^{7} +3.56155 q^{9} -2.00000 q^{11} +0.561553 q^{13} +2.56155 q^{15} +5.56155 q^{17} -2.00000 q^{19} -4.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.43845 q^{27} +0.123106 q^{29} -8.12311 q^{31} +5.12311 q^{33} -1.56155 q^{35} -3.56155 q^{37} -1.43845 q^{39} -4.12311 q^{41} -10.2462 q^{43} -3.56155 q^{45} +3.68466 q^{47} -4.56155 q^{49} -14.2462 q^{51} +4.43845 q^{53} +2.00000 q^{55} +5.12311 q^{57} -5.56155 q^{59} -9.12311 q^{61} +5.56155 q^{63} -0.561553 q^{65} -11.5616 q^{67} +2.56155 q^{69} -5.00000 q^{71} +3.43845 q^{73} -2.56155 q^{75} -3.12311 q^{77} -9.12311 q^{79} -7.00000 q^{81} -4.68466 q^{83} -5.56155 q^{85} -0.315342 q^{87} +8.00000 q^{89} +0.876894 q^{91} +20.8078 q^{93} +2.00000 q^{95} -3.12311 q^{97} -7.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 2 q^{5} - q^{7} + 3 q^{9} - 4 q^{11} - 3 q^{13} + q^{15} + 7 q^{17} - 4 q^{19} - 8 q^{21} - 2 q^{23} + 2 q^{25} - 7 q^{27} - 8 q^{29} - 8 q^{31} + 2 q^{33} + q^{35} - 3 q^{37} - 7 q^{39}+ \cdots - 6 q^{99}+O(q^{100}) $ 2 * q - q^3 - 2 * q^5 - q^7 + 3 * q^9 - 4 * q^11 - 3 * q^13 + q^15 + 7 * q^17 - 4 * q^19 - 8 * q^21 - 2 * q^23 + 2 * q^25 - 7 * q^27 - 8 * q^29 - 8 * q^31 + 2 * q^33 + q^35 - 3 * q^37 - 7 * q^39 - 4 * q^43 - 3 * q^45 - 5 * q^47 - 5 * q^49 - 12 * q^51 + 13 * q^53 + 4 * q^55 + 2 * q^57 - 7 * q^59 - 10 * q^61 + 7 * q^63 + 3 * q^65 - 19 * q^67 + q^69 - 10 * q^71 + 11 * q^73 - q^75 + 2 * q^77 - 10 * q^79 - 14 * q^81 + 3 * q^83 - 7 * q^85 - 13 * q^87 + 16 * q^89 + 10 * q^91 + 21 * q^93 + 4 * q^95 + 2 * q^97 - 6 * q^99

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	0
$2$	0	0
$3$	−2.56155	−1.47891	−0.739457	−	0.673204i	$-0.764917\pi$
$3$	−2.56155	−1.47891	−0.739457	+	0.673204i	$0.764917\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.56155	0.590211	0.295106	−	0.955465i	$-0.404645\pi$
$7$	1.56155	0.590211	0.295106	+	0.955465i	$0.404645\pi$
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	0.561553	0.155747	0.0778734	−	0.996963i	$-0.475187\pi$
$13$	0.561553	0.155747	0.0778734	+	0.996963i	$0.475187\pi$
$14$	0	0
$15$	2.56155	0.661390
$16$	0	0
$17$	5.56155	1.34887	0.674437	−	0.738332i	$-0.264386\pi$
$17$	5.56155	1.34887	0.674437	+	0.738332i	$0.264386\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.43845	−0.276829
$28$	0	0
$29$	0.123106	0.0228601	0.0114301	−	0.999935i	$-0.496362\pi$
$29$	0.123106	0.0228601	0.0114301	+	0.999935i	$0.496362\pi$
$30$	0	0
$31$	−8.12311	−1.45895	−0.729476	−	0.684006i	$-0.760236\pi$
$31$	−8.12311	−1.45895	−0.729476	+	0.684006i	$0.760236\pi$
$32$	0	0
$33$	5.12311	0.891818
$34$	0	0
$35$	−1.56155	−0.263951
$36$	0	0
$37$	−3.56155	−0.585516	−0.292758	−	0.956187i	$-0.594573\pi$
$37$	−3.56155	−0.585516	−0.292758	+	0.956187i	$0.594573\pi$
$38$	0	0
$39$	−1.43845	−0.230336
$40$	0	0
$41$	−4.12311	−0.643921	−0.321960	−	0.946753i	$-0.604342\pi$
$41$	−4.12311	−0.643921	−0.321960	+	0.946753i	$0.604342\pi$
$42$	0	0
$43$	−10.2462	−1.56253	−0.781266	−	0.624198i	$-0.785426\pi$
$43$	−10.2462	−1.56253	−0.781266	+	0.624198i	$0.785426\pi$
$44$	0	0
$45$	−3.56155	−0.530925
$46$	0	0
$47$	3.68466	0.537463	0.268731	−	0.963215i	$-0.413396\pi$
$47$	3.68466	0.537463	0.268731	+	0.963215i	$0.413396\pi$
$48$	0	0
$49$	−4.56155	−0.651650
$50$	0	0
$51$	−14.2462	−1.99487
$52$	0	0
$53$	4.43845	0.609668	0.304834	−	0.952406i	$-0.401399\pi$
$53$	4.43845	0.609668	0.304834	+	0.952406i	$0.401399\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	5.12311	0.678572
$58$	0	0
$59$	−5.56155	−0.724053	−0.362026	−	0.932168i	$-0.617915\pi$
$59$	−5.56155	−0.724053	−0.362026	+	0.932168i	$0.617915\pi$
$60$	0	0
$61$	−9.12311	−1.16809	−0.584047	−	0.811720i	$-0.698532\pi$
$61$	−9.12311	−1.16809	−0.584047	+	0.811720i	$0.698532\pi$
$62$	0	0
$63$	5.56155	0.700690
$64$	0	0
$65$	−0.561553	−0.0696521
$66$	0	0
$67$	−11.5616	−1.41247	−0.706234	−	0.707978i	$-0.749607\pi$
$67$	−11.5616	−1.41247	−0.706234	+	0.707978i	$0.749607\pi$
$68$	0	0
$69$	2.56155	0.308375
$70$	0	0
$71$	−5.00000	−0.593391	−0.296695	−	0.954972i	$-0.595885\pi$
$71$	−5.00000	−0.593391	−0.296695	+	0.954972i	$0.595885\pi$
$72$	0	0
$73$	3.43845	0.402440	0.201220	−	0.979546i	$-0.435509\pi$
$73$	3.43845	0.402440	0.201220	+	0.979546i	$0.435509\pi$
$74$	0	0
$75$	−2.56155	−0.295783
$76$	0	0
$77$	−3.12311	−0.355911
$78$	0	0
$79$	−9.12311	−1.02643	−0.513215	−	0.858260i	$-0.671546\pi$
$79$	−9.12311	−1.02643	−0.513215	+	0.858260i	$0.671546\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−4.68466	−0.514208	−0.257104	−	0.966384i	$-0.582768\pi$
$83$	−4.68466	−0.514208	−0.257104	+	0.966384i	$0.582768\pi$
$84$	0	0
$85$	−5.56155	−0.603235
$86$	0	0
$87$	−0.315342	−0.0338082
$88$	0	0
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	0.876894	0.0919235
$92$	0	0
$93$	20.8078	2.15766
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	−3.12311	−0.317103	−0.158552	−	0.987351i	$-0.550682\pi$
$97$	−3.12311	−0.317103	−0.158552	+	0.987351i	$0.550682\pi$
$98$	0	0
$99$	−7.12311	−0.715899

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	920.2.a.e.1.1	✓	2
3.2	odd	2		8280.2.a.bf.1.2		2
4.3	odd	2		1840.2.a.o.1.2		2
5.2	odd	4		4600.2.e.n.4049.4		4
5.3	odd	4		4600.2.e.n.4049.1		4
5.4	even	2		4600.2.a.t.1.2		2
8.3	odd	2		7360.2.a.bl.1.1		2
8.5	even	2		7360.2.a.bp.1.2		2
20.19	odd	2		9200.2.a.bq.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.e.1.1	✓	2	1.1	even	1	trivial
1840.2.a.o.1.2		2	4.3	odd	2
4600.2.a.t.1.2		2	5.4	even	2
4600.2.e.n.4049.1		4	5.3	odd	4
4600.2.e.n.4049.4		4	5.2	odd	4
7360.2.a.bl.1.1		2	8.3	odd	2
7360.2.a.bp.1.2		2	8.5	even	2
8280.2.a.bf.1.2		2	3.2	odd	2
9200.2.a.bq.1.1		2	20.19	odd	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	920.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56155 q^{3} -1.00000 q^{5} +1.56155 q^{7} +3.56155 q^{9} -2.00000 q^{11} +0.561553 q^{13} +2.56155 q^{15} +5.56155 q^{17} -2.00000 q^{19} -4.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.43845 q^{27} +0.123106 q^{29} -8.12311 q^{31} +5.12311 q^{33} -1.56155 q^{35} -3.56155 q^{37} -1.43845 q^{39} -4.12311 q^{41} -10.2462 q^{43} -3.56155 q^{45} +3.68466 q^{47} -4.56155 q^{49} -14.2462 q^{51} +4.43845 q^{53} +2.00000 q^{55} +5.12311 q^{57} -5.56155 q^{59} -9.12311 q^{61} +5.56155 q^{63} -0.561553 q^{65} -11.5616 q^{67} +2.56155 q^{69} -5.00000 q^{71} +3.43845 q^{73} -2.56155 q^{75} -3.12311 q^{77} -9.12311 q^{79} -7.00000 q^{81} -4.68466 q^{83} -5.56155 q^{85} -0.315342 q^{87} +8.00000 q^{89} +0.876894 q^{91} +20.8078 q^{93} +2.00000 q^{95} -3.12311 q^{97} -7.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 2 q^{5} - q^{7} + 3 q^{9} - 4 q^{11} - 3 q^{13} + q^{15} + 7 q^{17} - 4 q^{19} - 8 q^{21} - 2 q^{23} + 2 q^{25} - 7 q^{27} - 8 q^{29} - 8 q^{31} + 2 q^{33} + q^{35} - 3 q^{37} - 7 q^{39}+ \cdots - 6 q^{99}+O(q^{100}) \) 2 * q - q^3 - 2 * q^5 - q^7 + 3 * q^9 - 4 * q^11 - 3 * q^13 + q^15 + 7 * q^17 - 4 * q^19 - 8 * q^21 - 2 * q^23 + 2 * q^25 - 7 * q^27 - 8 * q^29 - 8 * q^31 + 2 * q^33 + q^35 - 3 * q^37 - 7 * q^39 - 4 * q^43 - 3 * q^45 - 5 * q^47 - 5 * q^49 - 12 * q^51 + 13 * q^53 + 4 * q^55 + 2 * q^57 - 7 * q^59 - 10 * q^61 + 7 * q^63 + 3 * q^65 - 19 * q^67 + q^69 - 10 * q^71 + 11 * q^73 - q^75 + 2 * q^77 - 10 * q^79 - 14 * q^81 + 3 * q^83 - 7 * q^85 - 13 * q^87 + 16 * q^89 + 10 * q^91 + 21 * q^93 + 4 * q^95 + 2 * q^97 - 6 * q^99

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