LMFDB - Embedded newform 8464.2.a.bh.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8464,2,Mod(1,8464)] 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("8464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,4,0,0,0,0,0,2,0,0,0,-10] 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:	$67.5853802708$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1058)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	8464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{3} -1.73205 q^{5} -3.46410 q^{7} +1.00000 q^{9} +3.46410 q^{11} -5.00000 q^{13} -3.46410 q^{15} +6.92820 q^{17} +3.46410 q^{19} -6.92820 q^{21} -2.00000 q^{25} -4.00000 q^{27} -3.00000 q^{29} +8.00000 q^{31} +6.92820 q^{33} +6.00000 q^{35} -10.0000 q^{39} -9.00000 q^{41} +6.92820 q^{43} -1.73205 q^{45} +6.00000 q^{47} +5.00000 q^{49} +13.8564 q^{51} +1.73205 q^{53} -6.00000 q^{55} +6.92820 q^{57} +6.00000 q^{59} -5.19615 q^{61} -3.46410 q^{63} +8.66025 q^{65} -6.92820 q^{67} -6.00000 q^{71} -11.0000 q^{73} -4.00000 q^{75} -12.0000 q^{77} -6.92820 q^{79} -11.0000 q^{81} -12.0000 q^{85} -6.00000 q^{87} +1.73205 q^{89} +17.3205 q^{91} +16.0000 q^{93} -6.00000 q^{95} -12.1244 q^{97} +3.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} + 2 q^{9} - 10 q^{13} - 4 q^{25} - 8 q^{27} - 6 q^{29} + 16 q^{31} + 12 q^{35} - 20 q^{39} - 18 q^{41} + 12 q^{47} + 10 q^{49} - 12 q^{55} + 12 q^{59} - 12 q^{71} - 22 q^{73} - 8 q^{75} - 24 q^{77}+ \cdots - 12 q^{95}+O(q^{100}) $ 2 * q + 4 * q^3 + 2 * q^9 - 10 * q^13 - 4 * q^25 - 8 * q^27 - 6 * q^29 + 16 * q^31 + 12 * q^35 - 20 * q^39 - 18 * q^41 + 12 * q^47 + 10 * q^49 - 12 * q^55 + 12 * q^59 - 12 * q^71 - 22 * q^73 - 8 * q^75 - 24 * q^77 - 22 * q^81 - 24 * q^85 - 12 * q^87 + 32 * q^93 - 12 * q^95

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
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	0	0
$5$	−1.73205	−0.774597	−0.387298	−	0.921954i	$-0.626592\pi$
$5$	−1.73205	−0.774597	−0.387298	+	0.921954i	$0.626592\pi$
$6$	0	0
$7$	−3.46410	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	0	0
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	−3.46410	−0.894427
$16$	0	0
$17$	6.92820	1.68034	0.840168	−	0.542326i	$-0.182456\pi$
$17$	6.92820	1.68034	0.840168	+	0.542326i	$0.182456\pi$
$18$	0	0
$19$	3.46410	0.794719	0.397360	−	0.917663i	$-0.369927\pi$
$19$	3.46410	0.794719	0.397360	+	0.917663i	$0.369927\pi$
$20$	0	0
$21$	−6.92820	−1.51186
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−2.00000	−0.400000
$26$	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
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	6.92820	1.20605
$34$	0	0
$35$	6.00000	1.01419
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	−10.0000	−1.60128
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	6.92820	1.05654	0.528271	−	0.849076i	$-0.322841\pi$
$43$	6.92820	1.05654	0.528271	+	0.849076i	$0.322841\pi$
$44$	0	0
$45$	−1.73205	−0.258199
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	13.8564	1.94029
$52$	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
$55$	−6.00000	−0.809040
$56$	0	0
$57$	6.92820	0.917663
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−5.19615	−0.665299	−0.332650	−	0.943051i	$-0.607943\pi$
$61$	−5.19615	−0.665299	−0.332650	+	0.943051i	$0.607943\pi$
$62$	0	0
$63$	−3.46410	−0.436436
$64$	0	0
$65$	8.66025	1.07417
$66$	0	0
$67$	−6.92820	−0.846415	−0.423207	−	0.906033i	$-0.639096\pi$
$67$	−6.92820	−0.846415	−0.423207	+	0.906033i	$0.639096\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	−12.0000	−1.36753
$78$	0	0
$79$	−6.92820	−0.779484	−0.389742	−	0.920924i	$-0.627436\pi$
$79$	−6.92820	−0.779484	−0.389742	+	0.920924i	$0.627436\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−12.0000	−1.30158
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	1.73205	0.183597	0.0917985	−	0.995778i	$-0.470738\pi$
$89$	1.73205	0.183597	0.0917985	+	0.995778i	$0.470738\pi$
$90$	0	0
$91$	17.3205	1.81568
$92$	0	0
$93$	16.0000	1.65912
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	−12.1244	−1.23104	−0.615521	−	0.788121i	$-0.711054\pi$
$97$	−12.1244	−1.23104	−0.615521	+	0.788121i	$0.711054\pi$
$98$	0	0
$99$	3.46410	0.348155

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	8464.2.a.bh.1.1		2
4.3	odd	2		1058.2.a.g.1.1	✓	2
12.11	even	2		9522.2.a.t.1.2		2
23.22	odd	2	inner	8464.2.a.bh.1.2		2
92.91	even	2		1058.2.a.g.1.2	yes	2
276.275	odd	2		9522.2.a.t.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1058.2.a.g.1.1	✓	2	4.3	odd	2
1058.2.a.g.1.2	yes	2	92.91	even	2
8464.2.a.bh.1.1		2	1.1	even	1	trivial
8464.2.a.bh.1.2		2	23.22	odd	2	inner
9522.2.a.t.1.1		2	276.275	odd	2
9522.2.a.t.1.2		2	12.11	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:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{3} -1.73205 q^{5} -3.46410 q^{7} +1.00000 q^{9} +3.46410 q^{11} -5.00000 q^{13} -3.46410 q^{15} +6.92820 q^{17} +3.46410 q^{19} -6.92820 q^{21} -2.00000 q^{25} -4.00000 q^{27} -3.00000 q^{29} +8.00000 q^{31} +6.92820 q^{33} +6.00000 q^{35} -10.0000 q^{39} -9.00000 q^{41} +6.92820 q^{43} -1.73205 q^{45} +6.00000 q^{47} +5.00000 q^{49} +13.8564 q^{51} +1.73205 q^{53} -6.00000 q^{55} +6.92820 q^{57} +6.00000 q^{59} -5.19615 q^{61} -3.46410 q^{63} +8.66025 q^{65} -6.92820 q^{67} -6.00000 q^{71} -11.0000 q^{73} -4.00000 q^{75} -12.0000 q^{77} -6.92820 q^{79} -11.0000 q^{81} -12.0000 q^{85} -6.00000 q^{87} +1.73205 q^{89} +17.3205 q^{91} +16.0000 q^{93} -6.00000 q^{95} -12.1244 q^{97} +3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} + 2 q^{9} - 10 q^{13} - 4 q^{25} - 8 q^{27} - 6 q^{29} + 16 q^{31} + 12 q^{35} - 20 q^{39} - 18 q^{41} + 12 q^{47} + 10 q^{49} - 12 q^{55} + 12 q^{59} - 12 q^{71} - 22 q^{73} - 8 q^{75} - 24 q^{77}+ \cdots - 12 q^{95}+O(q^{100}) \) 2 * q + 4 * q^3 + 2 * q^9 - 10 * q^13 - 4 * q^25 - 8 * q^27 - 6 * q^29 + 16 * q^31 + 12 * q^35 - 20 * q^39 - 18 * q^41 + 12 * q^47 + 10 * q^49 - 12 * q^55 + 12 * q^59 - 12 * q^71 - 22 * q^73 - 8 * q^75 - 24 * q^77 - 22 * q^81 - 24 * q^85 - 12 * q^87 + 32 * q^93 - 12 * q^95

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