LMFDB - Embedded newform 184.2.a.e.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [184,2,Mod(1,184)] 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("184.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(184, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 184 = 2^{3} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	184.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-1,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$1.46924739719$
Analytic rank:	$0$
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			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	184.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{3} +2.00000 q^{5} -0.561553 q^{9} -3.12311 q^{11} +0.438447 q^{13} +3.12311 q^{15} +5.12311 q^{17} -3.12311 q^{19} -1.00000 q^{23} -1.00000 q^{25} -5.56155 q^{27} +3.56155 q^{29} -2.43845 q^{31} -4.87689 q^{33} +8.24621 q^{37} +0.684658 q^{39} -9.80776 q^{41} -8.00000 q^{43} -1.12311 q^{45} -0.684658 q^{47} -7.00000 q^{49} +8.00000 q^{51} +2.00000 q^{53} -6.24621 q^{55} -4.87689 q^{57} +10.2462 q^{59} -4.24621 q^{61} +0.876894 q^{65} +3.12311 q^{67} -1.56155 q^{69} +13.5616 q^{71} -14.6847 q^{73} -1.56155 q^{75} +3.12311 q^{79} -7.00000 q^{81} +14.2462 q^{83} +10.2462 q^{85} +5.56155 q^{87} +11.3693 q^{89} -3.80776 q^{93} -6.24621 q^{95} +11.3693 q^{97} +1.75379 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 4 q^{5} + 3 q^{9} + 2 q^{11} + 5 q^{13} - 2 q^{15} + 2 q^{17} + 2 q^{19} - 2 q^{23} - 2 q^{25} - 7 q^{27} + 3 q^{29} - 9 q^{31} - 18 q^{33} - 11 q^{39} + q^{41} - 16 q^{43} + 6 q^{45} + 11 q^{47}+ \cdots + 20 q^{99}+O(q^{100}) $ 2 * q - q^3 + 4 * q^5 + 3 * q^9 + 2 * q^11 + 5 * q^13 - 2 * q^15 + 2 * q^17 + 2 * q^19 - 2 * q^23 - 2 * q^25 - 7 * q^27 + 3 * q^29 - 9 * q^31 - 18 * q^33 - 11 * q^39 + q^41 - 16 * q^43 + 6 * q^45 + 11 * q^47 - 14 * q^49 + 16 * q^51 + 4 * q^53 + 4 * q^55 - 18 * q^57 + 4 * q^59 + 8 * q^61 + 10 * q^65 - 2 * q^67 + q^69 + 23 * q^71 - 17 * q^73 + q^75 - 2 * q^79 - 14 * q^81 + 12 * q^83 + 4 * q^85 + 7 * q^87 - 2 * q^89 + 13 * q^93 + 4 * q^95 - 2 * q^97 + 20 * 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
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	−3.12311	−0.941652	−0.470826	−	0.882226i	$-0.656044\pi$
$11$	−3.12311	−0.941652	−0.470826	+	0.882226i	$0.656044\pi$
$12$	0	0
$13$	0.438447	0.121603	0.0608017	−	0.998150i	$-0.480634\pi$
$13$	0.438447	0.121603	0.0608017	+	0.998150i	$0.480634\pi$
$14$	0	0
$15$	3.12311	0.806382
$16$	0	0
$17$	5.12311	1.24254	0.621268	−	0.783598i	$-0.286618\pi$
$17$	5.12311	1.24254	0.621268	+	0.783598i	$0.286618\pi$
$18$	0	0
$19$	−3.12311	−0.716490	−0.358245	−	0.933628i	$-0.616625\pi$
$19$	−3.12311	−0.716490	−0.358245	+	0.933628i	$0.616625\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.56155	−1.07032
$28$	0	0
$29$	3.56155	0.661364	0.330682	−	0.943742i	$-0.392721\pi$
$29$	3.56155	0.661364	0.330682	+	0.943742i	$0.392721\pi$
$30$	0	0
$31$	−2.43845	−0.437958	−0.218979	−	0.975730i	$-0.570273\pi$
$31$	−2.43845	−0.437958	−0.218979	+	0.975730i	$0.570273\pi$
$32$	0	0
$33$	−4.87689	−0.848958
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.24621	1.35567	0.677834	−	0.735215i	$-0.262919\pi$
$37$	8.24621	1.35567	0.677834	+	0.735215i	$0.262919\pi$
$38$	0	0
$39$	0.684658	0.109633
$40$	0	0
$41$	−9.80776	−1.53172	−0.765858	−	0.643010i	$-0.777685\pi$
$41$	−9.80776	−1.53172	−0.765858	+	0.643010i	$0.777685\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	−1.12311	−0.167423
$46$	0	0
$47$	−0.684658	−0.0998677	−0.0499338	−	0.998753i	$-0.515901\pi$
$47$	−0.684658	−0.0998677	−0.0499338	+	0.998753i	$0.515901\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	8.00000	1.12022
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	−6.24621	−0.842239
$56$	0	0
$57$	−4.87689	−0.645960
$58$	0	0
$59$	10.2462	1.33394	0.666972	−	0.745083i	$-0.267590\pi$
$59$	10.2462	1.33394	0.666972	+	0.745083i	$0.267590\pi$
$60$	0	0
$61$	−4.24621	−0.543672	−0.271836	−	0.962344i	$-0.587631\pi$
$61$	−4.24621	−0.543672	−0.271836	+	0.962344i	$0.587631\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.876894	0.108765
$66$	0	0
$67$	3.12311	0.381548	0.190774	−	0.981634i	$-0.438900\pi$
$67$	3.12311	0.381548	0.190774	+	0.981634i	$0.438900\pi$
$68$	0	0
$69$	−1.56155	−0.187989
$70$	0	0
$71$	13.5616	1.60946	0.804730	−	0.593641i	$-0.202310\pi$
$71$	13.5616	1.60946	0.804730	+	0.593641i	$0.202310\pi$
$72$	0	0
$73$	−14.6847	−1.71871	−0.859355	−	0.511380i	$-0.829134\pi$
$73$	−14.6847	−1.71871	−0.859355	+	0.511380i	$0.829134\pi$
$74$	0	0
$75$	−1.56155	−0.180313
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3.12311	0.351377	0.175688	−	0.984446i	$-0.443785\pi$
$79$	3.12311	0.351377	0.175688	+	0.984446i	$0.443785\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	14.2462	1.56372	0.781862	−	0.623451i	$-0.214270\pi$
$83$	14.2462	1.56372	0.781862	+	0.623451i	$0.214270\pi$
$84$	0	0
$85$	10.2462	1.11136
$86$	0	0
$87$	5.56155	0.596261
$88$	0	0
$89$	11.3693	1.20515	0.602573	−	0.798064i	$-0.294142\pi$
$89$	11.3693	1.20515	0.602573	+	0.798064i	$0.294142\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−3.80776	−0.394847
$94$	0	0
$95$	−6.24621	−0.640848
$96$	0	0
$97$	11.3693	1.15438	0.577190	−	0.816610i	$-0.304149\pi$
$97$	11.3693	1.15438	0.577190	+	0.816610i	$0.304149\pi$
$98$	0	0
$99$	1.75379	0.176262

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	184.2.a.e.1.2	✓	2
3.2	odd	2		1656.2.a.j.1.2		2
4.3	odd	2		368.2.a.i.1.1		2
5.2	odd	4		4600.2.e.o.4049.2		4
5.3	odd	4		4600.2.e.o.4049.3		4
5.4	even	2		4600.2.a.s.1.1		2
7.6	odd	2		9016.2.a.w.1.1		2
8.3	odd	2		1472.2.a.p.1.2		2
8.5	even	2		1472.2.a.u.1.1		2
12.11	even	2		3312.2.a.t.1.1		2
20.19	odd	2		9200.2.a.br.1.2		2
23.22	odd	2		4232.2.a.o.1.2		2
92.91	even	2		8464.2.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.a.e.1.2	✓	2	1.1	even	1	trivial
368.2.a.i.1.1		2	4.3	odd	2
1472.2.a.p.1.2		2	8.3	odd	2
1472.2.a.u.1.1		2	8.5	even	2
1656.2.a.j.1.2		2	3.2	odd	2
3312.2.a.t.1.1		2	12.11	even	2
4232.2.a.o.1.2		2	23.22	odd	2
4600.2.a.s.1.1		2	5.4	even	2
4600.2.e.o.4049.2		4	5.2	odd	4
4600.2.e.o.4049.3		4	5.3	odd	4
8464.2.a.bd.1.1		2	92.91	even	2
9016.2.a.w.1.1		2	7.6	odd	2
9200.2.a.br.1.2		2	20.19	odd	2

Overview	Random
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Classical	Maass
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 184 = 2^{3} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	184.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{3} +2.00000 q^{5} -0.561553 q^{9} -3.12311 q^{11} +0.438447 q^{13} +3.12311 q^{15} +5.12311 q^{17} -3.12311 q^{19} -1.00000 q^{23} -1.00000 q^{25} -5.56155 q^{27} +3.56155 q^{29} -2.43845 q^{31} -4.87689 q^{33} +8.24621 q^{37} +0.684658 q^{39} -9.80776 q^{41} -8.00000 q^{43} -1.12311 q^{45} -0.684658 q^{47} -7.00000 q^{49} +8.00000 q^{51} +2.00000 q^{53} -6.24621 q^{55} -4.87689 q^{57} +10.2462 q^{59} -4.24621 q^{61} +0.876894 q^{65} +3.12311 q^{67} -1.56155 q^{69} +13.5616 q^{71} -14.6847 q^{73} -1.56155 q^{75} +3.12311 q^{79} -7.00000 q^{81} +14.2462 q^{83} +10.2462 q^{85} +5.56155 q^{87} +11.3693 q^{89} -3.80776 q^{93} -6.24621 q^{95} +11.3693 q^{97} +1.75379 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 4 q^{5} + 3 q^{9} + 2 q^{11} + 5 q^{13} - 2 q^{15} + 2 q^{17} + 2 q^{19} - 2 q^{23} - 2 q^{25} - 7 q^{27} + 3 q^{29} - 9 q^{31} - 18 q^{33} - 11 q^{39} + q^{41} - 16 q^{43} + 6 q^{45} + 11 q^{47}+ \cdots + 20 q^{99}+O(q^{100}) \) 2 * q - q^3 + 4 * q^5 + 3 * q^9 + 2 * q^11 + 5 * q^13 - 2 * q^15 + 2 * q^17 + 2 * q^19 - 2 * q^23 - 2 * q^25 - 7 * q^27 + 3 * q^29 - 9 * q^31 - 18 * q^33 - 11 * q^39 + q^41 - 16 * q^43 + 6 * q^45 + 11 * q^47 - 14 * q^49 + 16 * q^51 + 4 * q^53 + 4 * q^55 - 18 * q^57 + 4 * q^59 + 8 * q^61 + 10 * q^65 - 2 * q^67 + q^69 + 23 * q^71 - 17 * q^73 + q^75 - 2 * q^79 - 14 * q^81 + 12 * q^83 + 4 * q^85 + 7 * q^87 - 2 * q^89 + 13 * q^93 + 4 * q^95 - 2 * q^97 + 20 * q^99

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