LMFDB - Embedded newform 8464.2.a.v.1.2

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,-2,0,0,0,0,0,-4,0,0,0,-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:	$ 2 $
Twist minimal:	no (minimal twist has level 2116)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-1.00000 q^{3} +3.46410 q^{5} +3.46410 q^{7} -2.00000 q^{9} +3.46410 q^{11} -5.00000 q^{13} -3.46410 q^{15} -3.46410 q^{17} -6.92820 q^{19} -3.46410 q^{21} +7.00000 q^{25} +5.00000 q^{27} -3.00000 q^{29} -1.00000 q^{31} -3.46410 q^{33} +12.0000 q^{35} -3.46410 q^{37} +5.00000 q^{39} -9.00000 q^{41} -6.92820 q^{45} +3.00000 q^{47} +5.00000 q^{49} +3.46410 q^{51} +6.92820 q^{53} +12.0000 q^{55} +6.92820 q^{57} +12.0000 q^{59} -13.8564 q^{61} -6.92820 q^{63} -17.3205 q^{65} -3.46410 q^{67} -9.00000 q^{71} -11.0000 q^{73} -7.00000 q^{75} +12.0000 q^{77} -6.92820 q^{79} +1.00000 q^{81} -10.3923 q^{83} -12.0000 q^{85} +3.00000 q^{87} +6.92820 q^{89} -17.3205 q^{91} +1.00000 q^{93} -24.0000 q^{95} -17.3205 q^{97} -6.92820 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{9} - 10 q^{13} + 14 q^{25} + 10 q^{27} - 6 q^{29} - 2 q^{31} + 24 q^{35} + 10 q^{39} - 18 q^{41} + 6 q^{47} + 10 q^{49} + 24 q^{55} + 24 q^{59} - 18 q^{71} - 22 q^{73} - 14 q^{75} + 24 q^{77}+ \cdots - 48 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^9 - 10 * q^13 + 14 * q^25 + 10 * q^27 - 6 * q^29 - 2 * q^31 + 24 * q^35 + 10 * q^39 - 18 * q^41 + 6 * q^47 + 10 * q^49 + 24 * q^55 + 24 * q^59 - 18 * q^71 - 22 * q^73 - 14 * q^75 + 24 * q^77 + 2 * q^81 - 24 * q^85 + 6 * q^87 + 2 * q^93 - 48 * 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
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	3.46410	1.54919	0.774597	−	0.632456i	$-0.217953\pi$
$5$	3.46410	1.54919	0.774597	+	0.632456i	$0.217953\pi$
$6$	0	0
$7$	3.46410	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	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
$17$	−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
$19$	−6.92820	−1.58944	−0.794719	−	0.606977i	$-0.792382\pi$
$19$	−6.92820	−1.58944	−0.794719	+	0.606977i	$0.792382\pi$
$20$	0	0
$21$	−3.46410	−0.755929
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	7.00000	1.40000
$26$	0	0
$27$	5.00000	0.962250
$28$	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
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	0	0
$33$	−3.46410	−0.603023
$34$	0	0
$35$	12.0000	2.02837
$36$	0	0
$37$	−3.46410	−0.569495	−0.284747	−	0.958603i	$-0.591910\pi$
$37$	−3.46410	−0.569495	−0.284747	+	0.958603i	$0.591910\pi$
$38$	0	0
$39$	5.00000	0.800641
$40$	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
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	−6.92820	−1.03280
$46$	0	0
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	3.46410	0.485071
$52$	0	0
$53$	6.92820	0.951662	0.475831	−	0.879537i	$-0.342147\pi$
$53$	6.92820	0.951662	0.475831	+	0.879537i	$0.342147\pi$
$54$	0	0
$55$	12.0000	1.61808
$56$	0	0
$57$	6.92820	0.917663
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−13.8564	−1.77413	−0.887066	−	0.461644i	$-0.847260\pi$
$61$	−13.8564	−1.77413	−0.887066	+	0.461644i	$0.847260\pi$
$62$	0	0
$63$	−6.92820	−0.872872
$64$	0	0
$65$	−17.3205	−2.14834
$66$	0	0
$67$	−3.46410	−0.423207	−0.211604	−	0.977356i	$-0.567869\pi$
$67$	−3.46410	−0.423207	−0.211604	+	0.977356i	$0.567869\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.00000	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$72$	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
$75$	−7.00000	−0.808290
$76$	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
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.3923	−1.14070	−0.570352	−	0.821401i	$-0.693193\pi$
$83$	−10.3923	−1.14070	−0.570352	+	0.821401i	$0.693193\pi$
$84$	0	0
$85$	−12.0000	−1.30158
$86$	0	0
$87$	3.00000	0.321634
$88$	0	0
$89$	6.92820	0.734388	0.367194	−	0.930144i	$-0.380318\pi$
$89$	6.92820	0.734388	0.367194	+	0.930144i	$0.380318\pi$
$90$	0	0
$91$	−17.3205	−1.81568
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−24.0000	−2.46235
$96$	0	0
$97$	−17.3205	−1.75863	−0.879316	−	0.476240i	$-0.842000\pi$
$97$	−17.3205	−1.75863	−0.879316	+	0.476240i	$0.842000\pi$
$98$	0	0
$99$	−6.92820	−0.696311

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.v.1.2		2
4.3	odd	2		2116.2.a.f.1.2	yes	2
23.22	odd	2	inner	8464.2.a.v.1.1		2
92.91	even	2		2116.2.a.f.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2116.2.a.f.1.1	✓	2	92.91	even	2
2116.2.a.f.1.2	yes	2	4.3	odd	2
8464.2.a.v.1.1		2	23.22	odd	2	inner
8464.2.a.v.1.2		2	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:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.46410 q^{5} +3.46410 q^{7} -2.00000 q^{9} +3.46410 q^{11} -5.00000 q^{13} -3.46410 q^{15} -3.46410 q^{17} -6.92820 q^{19} -3.46410 q^{21} +7.00000 q^{25} +5.00000 q^{27} -3.00000 q^{29} -1.00000 q^{31} -3.46410 q^{33} +12.0000 q^{35} -3.46410 q^{37} +5.00000 q^{39} -9.00000 q^{41} -6.92820 q^{45} +3.00000 q^{47} +5.00000 q^{49} +3.46410 q^{51} +6.92820 q^{53} +12.0000 q^{55} +6.92820 q^{57} +12.0000 q^{59} -13.8564 q^{61} -6.92820 q^{63} -17.3205 q^{65} -3.46410 q^{67} -9.00000 q^{71} -11.0000 q^{73} -7.00000 q^{75} +12.0000 q^{77} -6.92820 q^{79} +1.00000 q^{81} -10.3923 q^{83} -12.0000 q^{85} +3.00000 q^{87} +6.92820 q^{89} -17.3205 q^{91} +1.00000 q^{93} -24.0000 q^{95} -17.3205 q^{97} -6.92820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{9} - 10 q^{13} + 14 q^{25} + 10 q^{27} - 6 q^{29} - 2 q^{31} + 24 q^{35} + 10 q^{39} - 18 q^{41} + 6 q^{47} + 10 q^{49} + 24 q^{55} + 24 q^{59} - 18 q^{71} - 22 q^{73} - 14 q^{75} + 24 q^{77}+ \cdots - 48 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^9 - 10 * q^13 + 14 * q^25 + 10 * q^27 - 6 * q^29 - 2 * q^31 + 24 * q^35 + 10 * q^39 - 18 * q^41 + 6 * q^47 + 10 * q^49 + 24 * q^55 + 24 * q^59 - 18 * q^71 - 22 * q^73 - 14 * q^75 + 24 * q^77 + 2 * q^81 - 24 * q^85 + 6 * q^87 + 2 * q^93 - 48 * q^95

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