LMFDB - Embedded newform 4232.2.a.n.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4232,2,Mod(1,4232)] 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("4232.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4232, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4232 = 2^{3} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4232.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,4,0,-2] 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:	$33.7926901354$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
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			$1.73205$ of defining polynomial
Character	$\chi$	$=$	4232.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051 q^{3} +0.267949 q^{5} +0.732051 q^{7} -2.46410 q^{9} -1.26795 q^{11} +2.46410 q^{13} +0.196152 q^{15} -3.46410 q^{17} +3.26795 q^{19} +0.535898 q^{21} -4.92820 q^{25} -4.00000 q^{27} -3.00000 q^{29} -3.46410 q^{31} -0.928203 q^{33} +0.196152 q^{35} -2.00000 q^{37} +1.80385 q^{39} +1.53590 q^{41} +3.46410 q^{43} -0.660254 q^{45} -6.19615 q^{47} -6.46410 q^{49} -2.53590 q^{51} +1.73205 q^{53} -0.339746 q^{55} +2.39230 q^{57} +8.19615 q^{59} +1.19615 q^{61} -1.80385 q^{63} +0.660254 q^{65} -13.4641 q^{67} -12.1962 q^{71} +2.46410 q^{73} -3.60770 q^{75} -0.928203 q^{77} -13.4641 q^{79} +4.46410 q^{81} +11.4641 q^{83} -0.928203 q^{85} -2.19615 q^{87} +18.1244 q^{89} +1.80385 q^{91} -2.53590 q^{93} +0.875644 q^{95} -8.12436 q^{97} +3.12436 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9} - 6 q^{11} - 2 q^{13} - 10 q^{15} + 10 q^{19} + 8 q^{21} + 4 q^{25} - 8 q^{27} - 6 q^{29} + 12 q^{33} - 10 q^{35} - 4 q^{37} + 14 q^{39} + 10 q^{41} + 16 q^{45}+ \cdots - 18 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 4 * q^5 - 2 * q^7 + 2 * q^9 - 6 * q^11 - 2 * q^13 - 10 * q^15 + 10 * q^19 + 8 * q^21 + 4 * q^25 - 8 * q^27 - 6 * q^29 + 12 * q^33 - 10 * q^35 - 4 * q^37 + 14 * q^39 + 10 * q^41 + 16 * q^45 - 2 * q^47 - 6 * q^49 - 12 * q^51 - 18 * q^55 - 16 * q^57 + 6 * q^59 - 8 * q^61 - 14 * q^63 - 16 * q^65 - 20 * q^67 - 14 * q^71 - 2 * q^73 - 28 * q^75 + 12 * q^77 - 20 * q^79 + 2 * q^81 + 16 * q^83 + 12 * q^85 + 6 * q^87 + 12 * q^89 + 14 * q^91 - 12 * q^93 + 26 * q^95 + 8 * q^97 - 18 * 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
$3$	0.732051	0.422650	0.211325	−	0.977416i	$-0.432222\pi$
$3$	0.732051	0.422650	0.211325	+	0.977416i	$0.432222\pi$
$4$	0	0
$5$	0.267949	0.119831	0.0599153	−	0.998203i	$-0.480917\pi$
$5$	0.267949	0.119831	0.0599153	+	0.998203i	$0.480917\pi$
$6$	0	0
$7$	0.732051	0.276689	0.138345	−	0.990384i	$-0.455822\pi$
$7$	0.732051	0.276689	0.138345	+	0.990384i	$0.455822\pi$
$8$	0	0
$9$	−2.46410	−0.821367
$10$	0	0
$11$	−1.26795	−0.382301	−0.191151	−	0.981561i	$-0.561222\pi$
$11$	−1.26795	−0.382301	−0.191151	+	0.981561i	$0.561222\pi$
$12$	0	0
$13$	2.46410	0.683419	0.341709	−	0.939806i	$-0.388994\pi$
$13$	2.46410	0.683419	0.341709	+	0.939806i	$0.388994\pi$
$14$	0	0
$15$	0.196152	0.0506463
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	3.26795	0.749719	0.374859	−	0.927082i	$-0.377691\pi$
$19$	3.26795	0.749719	0.374859	+	0.927082i	$0.377691\pi$
$20$	0	0
$21$	0.535898	0.116943
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−4.92820	−0.985641
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	−3.46410	−0.622171	−0.311086	−	0.950382i	$-0.600693\pi$
$31$	−3.46410	−0.622171	−0.311086	+	0.950382i	$0.600693\pi$
$32$	0	0
$33$	−0.928203	−0.161579
$34$	0	0
$35$	0.196152	0.0331558
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	1.80385	0.288847
$40$	0	0
$41$	1.53590	0.239867	0.119934	−	0.992782i	$-0.461732\pi$
$41$	1.53590	0.239867	0.119934	+	0.992782i	$0.461732\pi$
$42$	0	0
$43$	3.46410	0.528271	0.264135	−	0.964486i	$-0.414913\pi$
$43$	3.46410	0.528271	0.264135	+	0.964486i	$0.414913\pi$
$44$	0	0
$45$	−0.660254	−0.0984249
$46$	0	0
$47$	−6.19615	−0.903802	−0.451901	−	0.892068i	$-0.649254\pi$
$47$	−6.19615	−0.903802	−0.451901	+	0.892068i	$0.649254\pi$
$48$	0	0
$49$	−6.46410	−0.923443
$50$	0	0
$51$	−2.53590	−0.355097
$52$	0	0
$53$	1.73205	0.237915	0.118958	−	0.992899i	$-0.462045\pi$
$53$	1.73205	0.237915	0.118958	+	0.992899i	$0.462045\pi$
$54$	0	0
$55$	−0.339746	−0.0458113
$56$	0	0
$57$	2.39230	0.316869
$58$	0	0
$59$	8.19615	1.06705	0.533524	−	0.845785i	$-0.320867\pi$
$59$	8.19615	1.06705	0.533524	+	0.845785i	$0.320867\pi$
$60$	0	0
$61$	1.19615	0.153152	0.0765758	−	0.997064i	$-0.475601\pi$
$61$	1.19615	0.153152	0.0765758	+	0.997064i	$0.475601\pi$
$62$	0	0
$63$	−1.80385	−0.227263
$64$	0	0
$65$	0.660254	0.0818944
$66$	0	0
$67$	−13.4641	−1.64490	−0.822451	−	0.568836i	$-0.807394\pi$
$67$	−13.4641	−1.64490	−0.822451	+	0.568836i	$0.807394\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.1962	−1.44742	−0.723708	−	0.690106i	$-0.757564\pi$
$71$	−12.1962	−1.44742	−0.723708	+	0.690106i	$0.757564\pi$
$72$	0	0
$73$	2.46410	0.288401	0.144201	−	0.989548i	$-0.453939\pi$
$73$	2.46410	0.288401	0.144201	+	0.989548i	$0.453939\pi$
$74$	0	0
$75$	−3.60770	−0.416581
$76$	0	0
$77$	−0.928203	−0.105779
$78$	0	0
$79$	−13.4641	−1.51483	−0.757415	−	0.652934i	$-0.773538\pi$
$79$	−13.4641	−1.51483	−0.757415	+	0.652934i	$0.773538\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	11.4641	1.25835	0.629174	−	0.777264i	$-0.283393\pi$
$83$	11.4641	1.25835	0.629174	+	0.777264i	$0.283393\pi$
$84$	0	0
$85$	−0.928203	−0.100678
$86$	0	0
$87$	−2.19615	−0.235452
$88$	0	0
$89$	18.1244	1.92118	0.960589	−	0.277973i	$-0.0896625\pi$
$89$	18.1244	1.92118	0.960589	+	0.277973i	$0.0896625\pi$
$90$	0	0
$91$	1.80385	0.189095
$92$	0	0
$93$	−2.53590	−0.262960
$94$	0	0
$95$	0.875644	0.0898392
$96$	0	0
$97$	−8.12436	−0.824903	−0.412452	−	0.910979i	$-0.635327\pi$
$97$	−8.12436	−0.824903	−0.412452	+	0.910979i	$0.635327\pi$
$98$	0	0
$99$	3.12436	0.314010

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	4232.2.a.n.1.2	yes	2
4.3	odd	2		8464.2.a.bf.1.1		2
23.22	odd	2		4232.2.a.m.1.2	✓	2
92.91	even	2		8464.2.a.be.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4232.2.a.m.1.2	✓	2	23.22	odd	2
4232.2.a.n.1.2	yes	2	1.1	even	1	trivial
8464.2.a.be.1.1		2	92.91	even	2
8464.2.a.bf.1.1		2	4.3	odd	2

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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:	\( N \)	\(=\)	\( 4232 = 2^{3} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4232.a (trivial)

\(f(q)\)	\(=\)	\(q+0.732051 q^{3} +0.267949 q^{5} +0.732051 q^{7} -2.46410 q^{9} -1.26795 q^{11} +2.46410 q^{13} +0.196152 q^{15} -3.46410 q^{17} +3.26795 q^{19} +0.535898 q^{21} -4.92820 q^{25} -4.00000 q^{27} -3.00000 q^{29} -3.46410 q^{31} -0.928203 q^{33} +0.196152 q^{35} -2.00000 q^{37} +1.80385 q^{39} +1.53590 q^{41} +3.46410 q^{43} -0.660254 q^{45} -6.19615 q^{47} -6.46410 q^{49} -2.53590 q^{51} +1.73205 q^{53} -0.339746 q^{55} +2.39230 q^{57} +8.19615 q^{59} +1.19615 q^{61} -1.80385 q^{63} +0.660254 q^{65} -13.4641 q^{67} -12.1962 q^{71} +2.46410 q^{73} -3.60770 q^{75} -0.928203 q^{77} -13.4641 q^{79} +4.46410 q^{81} +11.4641 q^{83} -0.928203 q^{85} -2.19615 q^{87} +18.1244 q^{89} +1.80385 q^{91} -2.53590 q^{93} +0.875644 q^{95} -8.12436 q^{97} +3.12436 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9} - 6 q^{11} - 2 q^{13} - 10 q^{15} + 10 q^{19} + 8 q^{21} + 4 q^{25} - 8 q^{27} - 6 q^{29} + 12 q^{33} - 10 q^{35} - 4 q^{37} + 14 q^{39} + 10 q^{41} + 16 q^{45}+ \cdots - 18 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 4 * q^5 - 2 * q^7 + 2 * q^9 - 6 * q^11 - 2 * q^13 - 10 * q^15 + 10 * q^19 + 8 * q^21 + 4 * q^25 - 8 * q^27 - 6 * q^29 + 12 * q^33 - 10 * q^35 - 4 * q^37 + 14 * q^39 + 10 * q^41 + 16 * q^45 - 2 * q^47 - 6 * q^49 - 12 * q^51 - 18 * q^55 - 16 * q^57 + 6 * q^59 - 8 * q^61 - 14 * q^63 - 16 * q^65 - 20 * q^67 - 14 * q^71 - 2 * q^73 - 28 * q^75 + 12 * q^77 - 20 * q^79 + 2 * q^81 + 16 * q^83 + 12 * q^85 + 6 * q^87 + 12 * q^89 + 14 * q^91 - 12 * q^93 + 26 * q^95 + 8 * q^97 - 18 * q^99

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