LMFDB - Embedded newform 8464.2.a.bc.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 = [2,0,0,0,6,0,4,0,6,0,0,0,2] 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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4232)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ 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)

$f(q)$	$=$	$q-2.44949 q^{3} +3.00000 q^{5} -0.449490 q^{7} +3.00000 q^{9} -2.44949 q^{11} +5.89898 q^{13} -7.34847 q^{15} +4.89898 q^{17} +3.55051 q^{19} +1.10102 q^{21} +4.00000 q^{25} +7.89898 q^{29} +10.8990 q^{31} +6.00000 q^{33} -1.34847 q^{35} -8.00000 q^{37} -14.4495 q^{39} +7.89898 q^{41} +6.89898 q^{43} +9.00000 q^{45} -1.55051 q^{47} -6.79796 q^{49} -12.0000 q^{51} -8.79796 q^{53} -7.34847 q^{55} -8.69694 q^{57} -5.34847 q^{59} -5.89898 q^{61} -1.34847 q^{63} +17.6969 q^{65} +13.7980 q^{67} +4.44949 q^{71} -1.89898 q^{73} -9.79796 q^{75} +1.10102 q^{77} +12.0000 q^{79} -9.00000 q^{81} +6.89898 q^{83} +14.6969 q^{85} -19.3485 q^{87} -8.79796 q^{89} -2.65153 q^{91} -26.6969 q^{93} +10.6515 q^{95} +1.89898 q^{97} -7.34847 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{5} + 4 q^{7} + 6 q^{9} + 2 q^{13} + 12 q^{19} + 12 q^{21} + 8 q^{25} + 6 q^{29} + 12 q^{31} + 12 q^{33} + 12 q^{35} - 16 q^{37} - 24 q^{39} + 6 q^{41} + 4 q^{43} + 18 q^{45} - 8 q^{47} + 6 q^{49}+ \cdots - 6 q^{97}+O(q^{100}) $ 2 * q + 6 * q^5 + 4 * q^7 + 6 * q^9 + 2 * q^13 + 12 * q^19 + 12 * q^21 + 8 * q^25 + 6 * q^29 + 12 * q^31 + 12 * q^33 + 12 * q^35 - 16 * q^37 - 24 * q^39 + 6 * q^41 + 4 * q^43 + 18 * q^45 - 8 * q^47 + 6 * q^49 - 24 * q^51 + 2 * q^53 + 12 * q^57 + 4 * q^59 - 2 * q^61 + 12 * q^63 + 6 * q^65 + 8 * q^67 + 4 * q^71 + 6 * q^73 + 12 * q^77 + 24 * q^79 - 18 * q^81 + 4 * q^83 - 24 * q^87 + 2 * q^89 - 20 * q^91 - 24 * q^93 + 36 * q^95 - 6 * q^97

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
$3$	−2.44949	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$3$	−2.44949	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$4$	0	0
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	−0.449490	−0.169891	−0.0849456	−	0.996386i	$-0.527072\pi$
$7$	−0.449490	−0.169891	−0.0849456	+	0.996386i	$0.527072\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	−2.44949	−0.738549	−0.369274	−	0.929320i	$-0.620394\pi$
$11$	−2.44949	−0.738549	−0.369274	+	0.929320i	$0.620394\pi$
$12$	0	0
$13$	5.89898	1.63608	0.818041	−	0.575160i	$-0.195060\pi$
$13$	5.89898	1.63608	0.818041	+	0.575160i	$0.195060\pi$
$14$	0	0
$15$	−7.34847	−1.89737
$16$	0	0
$17$	4.89898	1.18818	0.594089	−	0.804400i	$-0.297513\pi$
$17$	4.89898	1.18818	0.594089	+	0.804400i	$0.297513\pi$
$18$	0	0
$19$	3.55051	0.814543	0.407271	−	0.913307i	$-0.366480\pi$
$19$	3.55051	0.814543	0.407271	+	0.913307i	$0.366480\pi$
$20$	0	0
$21$	1.10102	0.240262
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.89898	1.46680	0.733402	−	0.679795i	$-0.237931\pi$
$29$	7.89898	1.46680	0.733402	+	0.679795i	$0.237931\pi$
$30$	0	0
$31$	10.8990	1.95751	0.978757	−	0.205023i	$-0.0657268\pi$
$31$	10.8990	1.95751	0.978757	+	0.205023i	$0.0657268\pi$
$32$	0	0
$33$	6.00000	1.04447
$34$	0	0
$35$	−1.34847	−0.227933
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	−14.4495	−2.31377
$40$	0	0
$41$	7.89898	1.23361	0.616807	−	0.787115i	$-0.288426\pi$
$41$	7.89898	1.23361	0.616807	+	0.787115i	$0.288426\pi$
$42$	0	0
$43$	6.89898	1.05208	0.526042	−	0.850458i	$-0.323675\pi$
$43$	6.89898	1.05208	0.526042	+	0.850458i	$0.323675\pi$
$44$	0	0
$45$	9.00000	1.34164
$46$	0	0
$47$	−1.55051	−0.226165	−0.113083	−	0.993586i	$-0.536072\pi$
$47$	−1.55051	−0.226165	−0.113083	+	0.993586i	$0.536072\pi$
$48$	0	0
$49$	−6.79796	−0.971137
$50$	0	0
$51$	−12.0000	−1.68034
$52$	0	0
$53$	−8.79796	−1.20849	−0.604246	−	0.796798i	$-0.706526\pi$
$53$	−8.79796	−1.20849	−0.604246	+	0.796798i	$0.706526\pi$
$54$	0	0
$55$	−7.34847	−0.990867
$56$	0	0
$57$	−8.69694	−1.15194
$58$	0	0
$59$	−5.34847	−0.696311	−0.348156	−	0.937437i	$-0.613192\pi$
$59$	−5.34847	−0.696311	−0.348156	+	0.937437i	$0.613192\pi$
$60$	0	0
$61$	−5.89898	−0.755287	−0.377643	−	0.925951i	$-0.623265\pi$
$61$	−5.89898	−0.755287	−0.377643	+	0.925951i	$0.623265\pi$
$62$	0	0
$63$	−1.34847	−0.169891
$64$	0	0
$65$	17.6969	2.19504
$66$	0	0
$67$	13.7980	1.68569	0.842844	−	0.538157i	$-0.180879\pi$
$67$	13.7980	1.68569	0.842844	+	0.538157i	$0.180879\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.44949	0.528057	0.264029	−	0.964515i	$-0.414949\pi$
$71$	4.44949	0.528057	0.264029	+	0.964515i	$0.414949\pi$
$72$	0	0
$73$	−1.89898	−0.222259	−0.111129	−	0.993806i	$-0.535447\pi$
$73$	−1.89898	−0.222259	−0.111129	+	0.993806i	$0.535447\pi$
$74$	0	0
$75$	−9.79796	−1.13137
$76$	0	0
$77$	1.10102	0.125473
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	6.89898	0.757261	0.378631	−	0.925548i	$-0.376395\pi$
$83$	6.89898	0.757261	0.378631	+	0.925548i	$0.376395\pi$
$84$	0	0
$85$	14.6969	1.59411
$86$	0	0
$87$	−19.3485	−2.07437
$88$	0	0
$89$	−8.79796	−0.932582	−0.466291	−	0.884631i	$-0.654410\pi$
$89$	−8.79796	−0.932582	−0.466291	+	0.884631i	$0.654410\pi$
$90$	0	0
$91$	−2.65153	−0.277956
$92$	0	0
$93$	−26.6969	−2.76834
$94$	0	0
$95$	10.6515	1.09282
$96$	0	0
$97$	1.89898	0.192812	0.0964061	−	0.995342i	$-0.469265\pi$
$97$	1.89898	0.192812	0.0964061	+	0.995342i	$0.469265\pi$
$98$	0	0
$99$	−7.34847	−0.738549

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.bc.1.1		2
4.3	odd	2		4232.2.a.q.1.2	yes	2
23.22	odd	2		8464.2.a.y.1.1		2
92.91	even	2		4232.2.a.p.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4232.2.a.p.1.2	✓	2	92.91	even	2
4232.2.a.q.1.2	yes	2	4.3	odd	2
8464.2.a.y.1.1		2	23.22	odd	2
8464.2.a.bc.1.1		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)

\(f(q)\)	\(=\)	\(q-2.44949 q^{3} +3.00000 q^{5} -0.449490 q^{7} +3.00000 q^{9} -2.44949 q^{11} +5.89898 q^{13} -7.34847 q^{15} +4.89898 q^{17} +3.55051 q^{19} +1.10102 q^{21} +4.00000 q^{25} +7.89898 q^{29} +10.8990 q^{31} +6.00000 q^{33} -1.34847 q^{35} -8.00000 q^{37} -14.4495 q^{39} +7.89898 q^{41} +6.89898 q^{43} +9.00000 q^{45} -1.55051 q^{47} -6.79796 q^{49} -12.0000 q^{51} -8.79796 q^{53} -7.34847 q^{55} -8.69694 q^{57} -5.34847 q^{59} -5.89898 q^{61} -1.34847 q^{63} +17.6969 q^{65} +13.7980 q^{67} +4.44949 q^{71} -1.89898 q^{73} -9.79796 q^{75} +1.10102 q^{77} +12.0000 q^{79} -9.00000 q^{81} +6.89898 q^{83} +14.6969 q^{85} -19.3485 q^{87} -8.79796 q^{89} -2.65153 q^{91} -26.6969 q^{93} +10.6515 q^{95} +1.89898 q^{97} -7.34847 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{5} + 4 q^{7} + 6 q^{9} + 2 q^{13} + 12 q^{19} + 12 q^{21} + 8 q^{25} + 6 q^{29} + 12 q^{31} + 12 q^{33} + 12 q^{35} - 16 q^{37} - 24 q^{39} + 6 q^{41} + 4 q^{43} + 18 q^{45} - 8 q^{47} + 6 q^{49}+ \cdots - 6 q^{97}+O(q^{100}) \) 2 * q + 6 * q^5 + 4 * q^7 + 6 * q^9 + 2 * q^13 + 12 * q^19 + 12 * q^21 + 8 * q^25 + 6 * q^29 + 12 * q^31 + 12 * q^33 + 12 * q^35 - 16 * q^37 - 24 * q^39 + 6 * q^41 + 4 * q^43 + 18 * q^45 - 8 * q^47 + 6 * q^49 - 24 * q^51 + 2 * q^53 + 12 * q^57 + 4 * q^59 - 2 * q^61 + 12 * q^63 + 6 * q^65 + 8 * q^67 + 4 * q^71 + 6 * q^73 + 12 * q^77 + 24 * q^79 - 18 * q^81 + 4 * q^83 - 24 * q^87 + 2 * q^89 - 20 * q^91 - 24 * q^93 + 36 * q^95 - 6 * q^97

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