LMFDB - Embedded newform 456.2.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [456,2,Mod(1,456)] 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("456.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(456, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 456 = 2^{3} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	456.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,-1] 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:	$3.64117833217$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.70156$ of defining polynomial
Character	$\chi$	$=$	456.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.70156 q^{5} +1.70156 q^{7} +1.00000 q^{9} -1.70156 q^{11} +6.00000 q^{13} +3.70156 q^{15} +3.70156 q^{17} +1.00000 q^{19} -1.70156 q^{21} +4.00000 q^{23} +8.70156 q^{25} -1.00000 q^{27} +2.00000 q^{29} -3.40312 q^{31} +1.70156 q^{33} -6.29844 q^{35} +9.40312 q^{37} -6.00000 q^{39} +9.40312 q^{41} -9.10469 q^{43} -3.70156 q^{45} +5.70156 q^{47} -4.10469 q^{49} -3.70156 q^{51} -6.00000 q^{53} +6.29844 q^{55} -1.00000 q^{57} -4.00000 q^{59} +7.70156 q^{61} +1.70156 q^{63} -22.2094 q^{65} -12.0000 q^{67} -4.00000 q^{69} +0.298438 q^{73} -8.70156 q^{75} -2.89531 q^{77} +14.8062 q^{79} +1.00000 q^{81} +14.8062 q^{83} -13.7016 q^{85} -2.00000 q^{87} +12.8062 q^{89} +10.2094 q^{91} +3.40312 q^{93} -3.70156 q^{95} -6.00000 q^{97} -1.70156 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - q^{5} - 3 q^{7} + 2 q^{9} + 3 q^{11} + 12 q^{13} + q^{15} + q^{17} + 2 q^{19} + 3 q^{21} + 8 q^{23} + 11 q^{25} - 2 q^{27} + 4 q^{29} + 6 q^{31} - 3 q^{33} - 19 q^{35} + 6 q^{37} - 12 q^{39}+ \cdots + 3 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - q^5 - 3 * q^7 + 2 * q^9 + 3 * q^11 + 12 * q^13 + q^15 + q^17 + 2 * q^19 + 3 * q^21 + 8 * q^23 + 11 * q^25 - 2 * q^27 + 4 * q^29 + 6 * q^31 - 3 * q^33 - 19 * q^35 + 6 * q^37 - 12 * q^39 + 6 * q^41 + q^43 - q^45 + 5 * q^47 + 11 * q^49 - q^51 - 12 * q^53 + 19 * q^55 - 2 * q^57 - 8 * q^59 + 9 * q^61 - 3 * q^63 - 6 * q^65 - 24 * q^67 - 8 * q^69 + 7 * q^73 - 11 * q^75 - 25 * q^77 + 4 * q^79 + 2 * q^81 + 4 * q^83 - 21 * q^85 - 4 * q^87 - 18 * q^91 - 6 * q^93 - 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−3.70156	−1.65539	−0.827694	−	0.561179i	$-0.810348\pi$
$5$	−3.70156	−1.65539	−0.827694	+	0.561179i	$0.810348\pi$
$6$	0	0
$7$	1.70156	0.643130	0.321565	−	0.946888i	$-0.395791\pi$
$7$	1.70156	0.643130	0.321565	+	0.946888i	$0.395791\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.70156	−0.513040	−0.256520	−	0.966539i	$-0.582576\pi$
$11$	−1.70156	−0.513040	−0.256520	+	0.966539i	$0.582576\pi$
$12$	0	0
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	3.70156	0.955739
$16$	0	0
$17$	3.70156	0.897761	0.448880	−	0.893592i	$-0.351823\pi$
$17$	3.70156	0.897761	0.448880	+	0.893592i	$0.351823\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.70156	−0.371311
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	8.70156	1.74031
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−3.40312	−0.611219	−0.305610	−	0.952157i	$-0.598860\pi$
$31$	−3.40312	−0.611219	−0.305610	+	0.952157i	$0.598860\pi$
$32$	0	0
$33$	1.70156	0.296204
$34$	0	0
$35$	−6.29844	−1.06463
$36$	0	0
$37$	9.40312	1.54586	0.772932	−	0.634489i	$-0.218789\pi$
$37$	9.40312	1.54586	0.772932	+	0.634489i	$0.218789\pi$
$38$	0	0
$39$	−6.00000	−0.960769
$40$	0	0
$41$	9.40312	1.46852	0.734261	−	0.678868i	$-0.237529\pi$
$41$	9.40312	1.46852	0.734261	+	0.678868i	$0.237529\pi$
$42$	0	0
$43$	−9.10469	−1.38845	−0.694226	−	0.719757i	$-0.744253\pi$
$43$	−9.10469	−1.38845	−0.694226	+	0.719757i	$0.744253\pi$
$44$	0	0
$45$	−3.70156	−0.551796
$46$	0	0
$47$	5.70156	0.831658	0.415829	−	0.909443i	$-0.363491\pi$
$47$	5.70156	0.831658	0.415829	+	0.909443i	$0.363491\pi$
$48$	0	0
$49$	−4.10469	−0.586384
$50$	0	0
$51$	−3.70156	−0.518322
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	6.29844	0.849281
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	7.70156	0.986084	0.493042	−	0.870006i	$-0.335885\pi$
$61$	7.70156	0.986084	0.493042	+	0.870006i	$0.335885\pi$
$62$	0	0
$63$	1.70156	0.214377
$64$	0	0
$65$	−22.2094	−2.75473
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	0.298438	0.0349295	0.0174648	−	0.999847i	$-0.494441\pi$
$73$	0.298438	0.0349295	0.0174648	+	0.999847i	$0.494441\pi$
$74$	0	0
$75$	−8.70156	−1.00477
$76$	0	0
$77$	−2.89531	−0.329952
$78$	0	0
$79$	14.8062	1.66583	0.832917	−	0.553399i	$-0.186669\pi$
$79$	14.8062	1.66583	0.832917	+	0.553399i	$0.186669\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	14.8062	1.62520	0.812598	−	0.582824i	$-0.198052\pi$
$83$	14.8062	1.62520	0.812598	+	0.582824i	$0.198052\pi$
$84$	0	0
$85$	−13.7016	−1.48614
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	12.8062	1.35746	0.678730	−	0.734388i	$-0.262531\pi$
$89$	12.8062	1.35746	0.678730	+	0.734388i	$0.262531\pi$
$90$	0	0
$91$	10.2094	1.07023
$92$	0	0
$93$	3.40312	0.352888
$94$	0	0
$95$	−3.70156	−0.379772
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	−1.70156	−0.171013

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	456.2.a.e.1.1	✓	2
3.2	odd	2		1368.2.a.l.1.2		2
4.3	odd	2		912.2.a.o.1.1		2
8.3	odd	2		3648.2.a.bn.1.2		2
8.5	even	2		3648.2.a.bs.1.2		2
12.11	even	2		2736.2.a.bb.1.2		2
19.18	odd	2		8664.2.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.a.e.1.1	✓	2	1.1	even	1	trivial
912.2.a.o.1.1		2	4.3	odd	2
1368.2.a.l.1.2		2	3.2	odd	2
2736.2.a.bb.1.2		2	12.11	even	2
3648.2.a.bn.1.2		2	8.3	odd	2
3648.2.a.bs.1.2		2	8.5	even	2
8664.2.a.v.1.1		2	19.18	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:	\( N \)	\(=\)	\( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	456.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.70156 q^{5} +1.70156 q^{7} +1.00000 q^{9} -1.70156 q^{11} +6.00000 q^{13} +3.70156 q^{15} +3.70156 q^{17} +1.00000 q^{19} -1.70156 q^{21} +4.00000 q^{23} +8.70156 q^{25} -1.00000 q^{27} +2.00000 q^{29} -3.40312 q^{31} +1.70156 q^{33} -6.29844 q^{35} +9.40312 q^{37} -6.00000 q^{39} +9.40312 q^{41} -9.10469 q^{43} -3.70156 q^{45} +5.70156 q^{47} -4.10469 q^{49} -3.70156 q^{51} -6.00000 q^{53} +6.29844 q^{55} -1.00000 q^{57} -4.00000 q^{59} +7.70156 q^{61} +1.70156 q^{63} -22.2094 q^{65} -12.0000 q^{67} -4.00000 q^{69} +0.298438 q^{73} -8.70156 q^{75} -2.89531 q^{77} +14.8062 q^{79} +1.00000 q^{81} +14.8062 q^{83} -13.7016 q^{85} -2.00000 q^{87} +12.8062 q^{89} +10.2094 q^{91} +3.40312 q^{93} -3.70156 q^{95} -6.00000 q^{97} -1.70156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - q^{5} - 3 q^{7} + 2 q^{9} + 3 q^{11} + 12 q^{13} + q^{15} + q^{17} + 2 q^{19} + 3 q^{21} + 8 q^{23} + 11 q^{25} - 2 q^{27} + 4 q^{29} + 6 q^{31} - 3 q^{33} - 19 q^{35} + 6 q^{37} - 12 q^{39}+ \cdots + 3 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - q^5 - 3 * q^7 + 2 * q^9 + 3 * q^11 + 12 * q^13 + q^15 + q^17 + 2 * q^19 + 3 * q^21 + 8 * q^23 + 11 * q^25 - 2 * q^27 + 4 * q^29 + 6 * q^31 - 3 * q^33 - 19 * q^35 + 6 * q^37 - 12 * q^39 + 6 * q^41 + q^43 - q^45 + 5 * q^47 + 11 * q^49 - q^51 - 12 * q^53 + 19 * q^55 - 2 * q^57 - 8 * q^59 + 9 * q^61 - 3 * q^63 - 6 * q^65 - 24 * q^67 - 8 * q^69 + 7 * q^73 - 11 * q^75 - 25 * q^77 + 4 * q^79 + 2 * q^81 + 4 * q^83 - 21 * q^85 - 4 * q^87 - 18 * q^91 - 6 * q^93 - q^95 - 12 * q^97 + 3 * q^99

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