LMFDB - Embedded newform 4158.2.a.bw.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4158,2,Mod(1,4158)] 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("4158.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4158, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4158 = 2 \cdot 3^{3} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4158.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,0,3,0,0,-3,3,0,0,3,0,1] 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:	$33.2017971604$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3028.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 10x - 6 $ x^3 - 10*x - 6
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			1.3
Root			$3.42788$ of defining polynomial
Character	$\chi$	$=$	4158.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.42788 q^{5} -1.00000 q^{7} +1.00000 q^{8} +3.42788 q^{10} +1.00000 q^{11} +2.10540 q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} +3.42788 q^{20} +1.00000 q^{22} +4.42788 q^{23} +6.75035 q^{25} +2.10540 q^{26} -1.00000 q^{28} -8.17823 q^{29} +8.75035 q^{31} +1.00000 q^{32} +1.00000 q^{34} -3.42788 q^{35} -4.53328 q^{37} +3.42788 q^{40} -5.75035 q^{41} +1.00000 q^{43} +1.00000 q^{44} +4.42788 q^{46} +4.00000 q^{47} +1.00000 q^{49} +6.75035 q^{50} +2.10540 q^{52} +7.64495 q^{53} +3.42788 q^{55} -1.00000 q^{56} -8.17823 q^{58} +4.75035 q^{59} +5.75035 q^{61} +8.75035 q^{62} +1.00000 q^{64} +7.21707 q^{65} +3.78293 q^{67} +1.00000 q^{68} -3.42788 q^{70} -13.0340 q^{71} -15.1782 q^{73} -4.53328 q^{74} -1.00000 q^{77} +3.67753 q^{79} +3.42788 q^{80} -5.75035 q^{82} -14.3565 q^{83} +3.42788 q^{85} +1.00000 q^{86} +1.00000 q^{88} +8.42788 q^{89} -2.10540 q^{91} +4.42788 q^{92} +4.00000 q^{94} +8.28364 q^{97} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 3 q^{7} + 3 q^{8} + 3 q^{11} + q^{13} - 3 q^{14} + 3 q^{16} + 3 q^{17} + 3 q^{22} + 3 q^{23} + 5 q^{25} + q^{26} - 3 q^{28} + q^{29} + 11 q^{31} + 3 q^{32} + 3 q^{34} + 2 q^{37}+ \cdots + 3 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^7 + 3 * q^8 + 3 * q^11 + q^13 - 3 * q^14 + 3 * q^16 + 3 * q^17 + 3 * q^22 + 3 * q^23 + 5 * q^25 + q^26 - 3 * q^28 + q^29 + 11 * q^31 + 3 * q^32 + 3 * q^34 + 2 * q^37 - 2 * q^41 + 3 * q^43 + 3 * q^44 + 3 * q^46 + 12 * q^47 + 3 * q^49 + 5 * q^50 + q^52 + 13 * q^53 - 3 * q^56 + q^58 - q^59 + 2 * q^61 + 11 * q^62 + 3 * q^64 + 22 * q^65 + 11 * q^67 + 3 * q^68 + 7 * q^71 - 20 * q^73 + 2 * q^74 - 3 * q^77 + 16 * q^79 - 2 * q^82 + 8 * q^83 + 3 * q^86 + 3 * q^88 + 15 * q^89 - q^91 + 3 * q^92 + 12 * q^94 - 6 * q^97 + 3 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.42788	1.53299	0.766497	−	0.642248i	$-0.221998\pi$
$5$	3.42788	1.53299	0.766497	+	0.642248i	$0.221998\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	3.42788	1.08399
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.10540	0.583934	0.291967	−	0.956428i	$-0.405690\pi$
$13$	2.10540	0.583934	0.291967	+	0.956428i	$0.405690\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	3.42788	0.766497
$21$	0	0
$22$	1.00000	0.213201
$23$	4.42788	0.923277	0.461638	−	0.887068i	$-0.347262\pi$
$23$	4.42788	0.923277	0.461638	+	0.887068i	$0.347262\pi$
$24$	0	0
$25$	6.75035	1.35007
$26$	2.10540	0.412904
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−8.17823	−1.51866	−0.759330	−	0.650706i	$-0.774473\pi$
$29$	−8.17823	−1.51866	−0.759330	+	0.650706i	$0.774473\pi$
$30$	0	0
$31$	8.75035	1.57161	0.785805	−	0.618474i	$-0.212249\pi$
$31$	8.75035	1.57161	0.785805	+	0.618474i	$0.212249\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.00000	0.171499
$35$	−3.42788	−0.579417
$36$	0	0
$37$	−4.53328	−0.745267	−0.372634	−	0.927979i	$-0.621545\pi$
$37$	−4.53328	−0.745267	−0.372634	+	0.927979i	$0.621545\pi$
$38$	0	0
$39$	0	0
$40$	3.42788	0.541995
$41$	−5.75035	−0.898054	−0.449027	−	0.893518i	$-0.648229\pi$
$41$	−5.75035	−0.898054	−0.449027	+	0.893518i	$0.648229\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	4.42788	0.652855
$47$	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
$49$	1.00000	0.142857
$50$	6.75035	0.954644
$51$	0	0
$52$	2.10540	0.291967
$53$	7.64495	1.05011	0.525057	−	0.851067i	$-0.324044\pi$
$53$	7.64495	1.05011	0.525057	+	0.851067i	$0.324044\pi$
$54$	0	0
$55$	3.42788	0.462215
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−8.17823	−1.07385
$59$	4.75035	0.618443	0.309222	−	0.950990i	$-0.399931\pi$
$59$	4.75035	0.618443	0.309222	+	0.950990i	$0.399931\pi$
$60$	0	0
$61$	5.75035	0.736257	0.368129	−	0.929775i	$-0.379999\pi$
$61$	5.75035	0.736257	0.368129	+	0.929775i	$0.379999\pi$
$62$	8.75035	1.11130
$63$	0	0
$64$	1.00000	0.125000
$65$	7.21707	0.895167
$66$	0	0
$67$	3.78293	0.462158	0.231079	−	0.972935i	$-0.425774\pi$
$67$	3.78293	0.462158	0.231079	+	0.972935i	$0.425774\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	−3.42788	−0.409710
$71$	−13.0340	−1.54685	−0.773425	−	0.633888i	$-0.781458\pi$
$71$	−13.0340	−1.54685	−0.773425	+	0.633888i	$0.781458\pi$
$72$	0	0
$73$	−15.1782	−1.77648	−0.888239	−	0.459382i	$-0.848071\pi$
$73$	−15.1782	−1.77648	−0.888239	+	0.459382i	$0.848071\pi$
$74$	−4.53328	−0.526983
$75$	0	0
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	3.67753	0.413754	0.206877	−	0.978367i	$-0.433670\pi$
$79$	3.67753	0.413754	0.206877	+	0.978367i	$0.433670\pi$
$80$	3.42788	0.383249
$81$	0	0
$82$	−5.75035	−0.635020
$83$	−14.3565	−1.57583	−0.787913	−	0.615786i	$-0.788839\pi$
$83$	−14.3565	−1.57583	−0.787913	+	0.615786i	$0.788839\pi$
$84$	0	0
$85$	3.42788	0.371806
$86$	1.00000	0.107833
$87$	0	0
$88$	1.00000	0.106600
$89$	8.42788	0.893353	0.446677	−	0.894695i	$-0.352607\pi$
$89$	8.42788	0.893353	0.446677	+	0.894695i	$0.352607\pi$
$90$	0	0
$91$	−2.10540	−0.220706
$92$	4.42788	0.461638
$93$	0	0
$94$	4.00000	0.412568
$95$	0	0
$96$	0	0
$97$	8.28364	0.841076	0.420538	−	0.907275i	$-0.361841\pi$
$97$	8.28364	0.841076	0.420538	+	0.907275i	$0.361841\pi$
$98$	1.00000	0.101015
$99$	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	4158.2.a.bw.1.3	yes	3
3.2	odd	2		4158.2.a.br.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4158.2.a.br.1.1	✓	3	3.2	odd	2
4158.2.a.bw.1.3	yes	3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4158 = 2 \cdot 3^{3} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4158.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.42788 q^{5} -1.00000 q^{7} +1.00000 q^{8} +3.42788 q^{10} +1.00000 q^{11} +2.10540 q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} +3.42788 q^{20} +1.00000 q^{22} +4.42788 q^{23} +6.75035 q^{25} +2.10540 q^{26} -1.00000 q^{28} -8.17823 q^{29} +8.75035 q^{31} +1.00000 q^{32} +1.00000 q^{34} -3.42788 q^{35} -4.53328 q^{37} +3.42788 q^{40} -5.75035 q^{41} +1.00000 q^{43} +1.00000 q^{44} +4.42788 q^{46} +4.00000 q^{47} +1.00000 q^{49} +6.75035 q^{50} +2.10540 q^{52} +7.64495 q^{53} +3.42788 q^{55} -1.00000 q^{56} -8.17823 q^{58} +4.75035 q^{59} +5.75035 q^{61} +8.75035 q^{62} +1.00000 q^{64} +7.21707 q^{65} +3.78293 q^{67} +1.00000 q^{68} -3.42788 q^{70} -13.0340 q^{71} -15.1782 q^{73} -4.53328 q^{74} -1.00000 q^{77} +3.67753 q^{79} +3.42788 q^{80} -5.75035 q^{82} -14.3565 q^{83} +3.42788 q^{85} +1.00000 q^{86} +1.00000 q^{88} +8.42788 q^{89} -2.10540 q^{91} +4.42788 q^{92} +4.00000 q^{94} +8.28364 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{7} + 3 q^{8} + 3 q^{11} + q^{13} - 3 q^{14} + 3 q^{16} + 3 q^{17} + 3 q^{22} + 3 q^{23} + 5 q^{25} + q^{26} - 3 q^{28} + q^{29} + 11 q^{31} + 3 q^{32} + 3 q^{34} + 2 q^{37}+ \cdots + 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^7 + 3 * q^8 + 3 * q^11 + q^13 - 3 * q^14 + 3 * q^16 + 3 * q^17 + 3 * q^22 + 3 * q^23 + 5 * q^25 + q^26 - 3 * q^28 + q^29 + 11 * q^31 + 3 * q^32 + 3 * q^34 + 2 * q^37 - 2 * q^41 + 3 * q^43 + 3 * q^44 + 3 * q^46 + 12 * q^47 + 3 * q^49 + 5 * q^50 + q^52 + 13 * q^53 - 3 * q^56 + q^58 - q^59 + 2 * q^61 + 11 * q^62 + 3 * q^64 + 22 * q^65 + 11 * q^67 + 3 * q^68 + 7 * q^71 - 20 * q^73 + 2 * q^74 - 3 * q^77 + 16 * q^79 - 2 * q^82 + 8 * q^83 + 3 * q^86 + 3 * q^88 + 15 * q^89 - q^91 + 3 * q^92 + 12 * q^94 - 6 * q^97 + 3 * q^98

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