LMFDB - Embedded newform 3344.2.a.q.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3344,2,Mod(1,3344)] 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("3344.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3344, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,-3,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.14510$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.14510 q^{3} -1.60147 q^{5} -2.89167 q^{7} +1.60147 q^{9} +1.00000 q^{11} +4.89167 q^{13} -3.43531 q^{15} -5.74657 q^{17} +1.00000 q^{19} -6.20293 q^{21} +7.74657 q^{23} -2.43531 q^{25} -3.00000 q^{27} +5.34803 q^{29} +7.43531 q^{31} +2.14510 q^{33} +4.63091 q^{35} -7.49314 q^{37} +10.4931 q^{39} -2.05783 q^{41} +10.6887 q^{43} -2.56469 q^{45} +7.08727 q^{47} +1.36176 q^{49} -12.3270 q^{51} +8.32698 q^{53} -1.60147 q^{55} +2.14510 q^{57} +2.54364 q^{59} +13.4931 q^{61} -4.63091 q^{63} -7.83384 q^{65} +10.3848 q^{67} +16.6172 q^{69} -14.9789 q^{71} -3.74657 q^{73} -5.22399 q^{75} -2.89167 q^{77} +8.69607 q^{79} -11.2397 q^{81} +6.10833 q^{83} +9.20293 q^{85} +11.4721 q^{87} +3.20293 q^{89} -14.1451 q^{91} +15.9495 q^{93} -1.60147 q^{95} -6.98627 q^{97} +1.60147 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 6 q^{7} + 3 q^{9} + 3 q^{11} + 9 q^{15} - 9 q^{17} + 3 q^{19} - 15 q^{21} + 15 q^{23} + 12 q^{25} - 9 q^{27} + 6 q^{29} + 3 q^{31} - 6 q^{37} + 15 q^{39} - 9 q^{41} + 21 q^{43} - 27 q^{45}+ \cdots + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^5 + 6 * q^7 + 3 * q^9 + 3 * q^11 + 9 * q^15 - 9 * q^17 + 3 * q^19 - 15 * q^21 + 15 * q^23 + 12 * q^25 - 9 * q^27 + 6 * q^29 + 3 * q^31 - 6 * q^37 + 15 * q^39 - 9 * q^41 + 21 * q^43 - 27 * q^45 + 12 * q^47 + 27 * q^49 - 3 * q^51 - 9 * q^53 - 3 * q^55 + 3 * q^59 + 24 * q^61 - 6 * q^65 + 3 * q^69 - 21 * q^71 - 3 * q^73 - 36 * q^75 + 6 * q^77 + 6 * q^79 - 9 * q^81 + 33 * q^83 + 24 * q^85 - 6 * q^87 + 6 * q^89 - 36 * q^91 + 36 * q^93 - 3 * q^95 + 12 * q^97 + 3 * 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
$3$	2.14510	1.23848	0.619238	−	0.785204i	$-0.287442\pi$
$3$	2.14510	1.23848	0.619238	+	0.785204i	$0.287442\pi$
$4$	0	0
$5$	−1.60147	−0.716197	−0.358099	−	0.933684i	$-0.616575\pi$
$5$	−1.60147	−0.716197	−0.358099	+	0.933684i	$0.616575\pi$
$6$	0	0
$7$	−2.89167	−1.09295	−0.546474	−	0.837476i	$-0.684030\pi$
$7$	−2.89167	−1.09295	−0.546474	+	0.837476i	$0.684030\pi$
$8$	0	0
$9$	1.60147	0.533822
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.89167	1.35671	0.678353	−	0.734736i	$-0.262694\pi$
$13$	4.89167	1.35671	0.678353	+	0.734736i	$0.262694\pi$
$14$	0	0
$15$	−3.43531	−0.886993
$16$	0	0
$17$	−5.74657	−1.39375	−0.696874	−	0.717194i	$-0.745426\pi$
$17$	−5.74657	−1.39375	−0.696874	+	0.717194i	$0.745426\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−6.20293	−1.35359
$22$	0	0
$23$	7.74657	1.61527	0.807636	−	0.589682i	$-0.200747\pi$
$23$	7.74657	1.61527	0.807636	+	0.589682i	$0.200747\pi$
$24$	0	0
$25$	−2.43531	−0.487062
$26$	0	0
$27$	−3.00000	−0.577350
$28$	0	0
$29$	5.34803	0.993105	0.496552	−	0.868007i	$-0.334599\pi$
$29$	5.34803	0.993105	0.496552	+	0.868007i	$0.334599\pi$
$30$	0	0
$31$	7.43531	1.33542	0.667710	−	0.744421i	$-0.267274\pi$
$31$	7.43531	1.33542	0.667710	+	0.744421i	$0.267274\pi$
$32$	0	0
$33$	2.14510	0.373414
$34$	0	0
$35$	4.63091	0.782767
$36$	0	0
$37$	−7.49314	−1.23186	−0.615932	−	0.787799i	$-0.711220\pi$
$37$	−7.49314	−1.23186	−0.615932	+	0.787799i	$0.711220\pi$
$38$	0	0
$39$	10.4931	1.68025
$40$	0	0
$41$	−2.05783	−0.321379	−0.160689	−	0.987005i	$-0.551372\pi$
$41$	−2.05783	−0.321379	−0.160689	+	0.987005i	$0.551372\pi$
$42$	0	0
$43$	10.6887	1.63002	0.815009	−	0.579449i	$-0.196732\pi$
$43$	10.6887	1.63002	0.815009	+	0.579449i	$0.196732\pi$
$44$	0	0
$45$	−2.56469	−0.382322
$46$	0	0
$47$	7.08727	1.03379	0.516893	−	0.856050i	$-0.327089\pi$
$47$	7.08727	1.03379	0.516893	+	0.856050i	$0.327089\pi$
$48$	0	0
$49$	1.36176	0.194537
$50$	0	0
$51$	−12.3270	−1.72612
$52$	0	0
$53$	8.32698	1.14380	0.571899	−	0.820324i	$-0.306207\pi$
$53$	8.32698	1.14380	0.571899	+	0.820324i	$0.306207\pi$
$54$	0	0
$55$	−1.60147	−0.215942
$56$	0	0
$57$	2.14510	0.284126
$58$	0	0
$59$	2.54364	0.331153	0.165577	−	0.986197i	$-0.447051\pi$
$59$	2.54364	0.331153	0.165577	+	0.986197i	$0.447051\pi$
$60$	0	0
$61$	13.4931	1.72762	0.863810	−	0.503818i	$-0.168072\pi$
$61$	13.4931	1.72762	0.863810	+	0.503818i	$0.168072\pi$
$62$	0	0
$63$	−4.63091	−0.583440
$64$	0	0
$65$	−7.83384	−0.971669
$66$	0	0
$67$	10.3848	1.26871	0.634353	−	0.773043i	$-0.281267\pi$
$67$	10.3848	1.26871	0.634353	+	0.773043i	$0.281267\pi$
$68$	0	0
$69$	16.6172	2.00047
$70$	0	0
$71$	−14.9789	−1.77767	−0.888837	−	0.458224i	$-0.848486\pi$
$71$	−14.9789	−1.77767	−0.888837	+	0.458224i	$0.848486\pi$
$72$	0	0
$73$	−3.74657	−0.438503	−0.219251	−	0.975668i	$-0.570361\pi$
$73$	−3.74657	−0.438503	−0.219251	+	0.975668i	$0.570361\pi$
$74$	0	0
$75$	−5.22399	−0.603214
$76$	0	0
$77$	−2.89167	−0.329536
$78$	0	0
$79$	8.69607	0.978384	0.489192	−	0.872176i	$-0.337292\pi$
$79$	8.69607	0.978384	0.489192	+	0.872176i	$0.337292\pi$
$80$	0	0
$81$	−11.2397	−1.24886
$82$	0	0
$83$	6.10833	0.670476	0.335238	−	0.942133i	$-0.391183\pi$
$83$	6.10833	0.670476	0.335238	+	0.942133i	$0.391183\pi$
$84$	0	0
$85$	9.20293	0.998198
$86$	0	0
$87$	11.4721	1.22994
$88$	0	0
$89$	3.20293	0.339510	0.169755	−	0.985486i	$-0.445702\pi$
$89$	3.20293	0.339510	0.169755	+	0.985486i	$0.445702\pi$
$90$	0	0
$91$	−14.1451	−1.48281
$92$	0	0
$93$	15.9495	1.65389
$94$	0	0
$95$	−1.60147	−0.164307
$96$	0	0
$97$	−6.98627	−0.709349	−0.354674	−	0.934990i	$-0.615408\pi$
$97$	−6.98627	−0.709349	−0.354674	+	0.934990i	$0.615408\pi$
$98$	0	0
$99$	1.60147	0.160953

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	3344.2.a.q.1.3		3
4.3	odd	2		418.2.a.g.1.1	✓	3
12.11	even	2		3762.2.a.bg.1.2		3
44.43	even	2		4598.2.a.bo.1.1		3
76.75	even	2		7942.2.a.bi.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.g.1.1	✓	3	4.3	odd	2
3344.2.a.q.1.3		3	1.1	even	1	trivial
3762.2.a.bg.1.2		3	12.11	even	2
4598.2.a.bo.1.1		3	44.43	even	2
7942.2.a.bi.1.3		3	76.75	even	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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q+2.14510 q^{3} -1.60147 q^{5} -2.89167 q^{7} +1.60147 q^{9} +1.00000 q^{11} +4.89167 q^{13} -3.43531 q^{15} -5.74657 q^{17} +1.00000 q^{19} -6.20293 q^{21} +7.74657 q^{23} -2.43531 q^{25} -3.00000 q^{27} +5.34803 q^{29} +7.43531 q^{31} +2.14510 q^{33} +4.63091 q^{35} -7.49314 q^{37} +10.4931 q^{39} -2.05783 q^{41} +10.6887 q^{43} -2.56469 q^{45} +7.08727 q^{47} +1.36176 q^{49} -12.3270 q^{51} +8.32698 q^{53} -1.60147 q^{55} +2.14510 q^{57} +2.54364 q^{59} +13.4931 q^{61} -4.63091 q^{63} -7.83384 q^{65} +10.3848 q^{67} +16.6172 q^{69} -14.9789 q^{71} -3.74657 q^{73} -5.22399 q^{75} -2.89167 q^{77} +8.69607 q^{79} -11.2397 q^{81} +6.10833 q^{83} +9.20293 q^{85} +11.4721 q^{87} +3.20293 q^{89} -14.1451 q^{91} +15.9495 q^{93} -1.60147 q^{95} -6.98627 q^{97} +1.60147 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 6 q^{7} + 3 q^{9} + 3 q^{11} + 9 q^{15} - 9 q^{17} + 3 q^{19} - 15 q^{21} + 15 q^{23} + 12 q^{25} - 9 q^{27} + 6 q^{29} + 3 q^{31} - 6 q^{37} + 15 q^{39} - 9 q^{41} + 21 q^{43} - 27 q^{45}+ \cdots + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^5 + 6 * q^7 + 3 * q^9 + 3 * q^11 + 9 * q^15 - 9 * q^17 + 3 * q^19 - 15 * q^21 + 15 * q^23 + 12 * q^25 - 9 * q^27 + 6 * q^29 + 3 * q^31 - 6 * q^37 + 15 * q^39 - 9 * q^41 + 21 * q^43 - 27 * q^45 + 12 * q^47 + 27 * q^49 - 3 * q^51 - 9 * q^53 - 3 * q^55 + 3 * q^59 + 24 * q^61 - 6 * q^65 + 3 * q^69 - 21 * q^71 - 3 * q^73 - 36 * q^75 + 6 * q^77 + 6 * q^79 - 9 * q^81 + 33 * q^83 + 24 * q^85 - 6 * q^87 + 6 * q^89 - 36 * q^91 + 36 * q^93 - 3 * q^95 + 12 * q^97 + 3 * q^99

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