LMFDB - Embedded newform 8464.2.a.bj.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 = [2,0,4,0,2,0,0,0,6,0,-4,0,2] 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:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
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			$-1.41421$ 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)

$f(q)$	$=$	$q+0.585786 q^{3} +1.00000 q^{5} -4.24264 q^{7} -2.65685 q^{9} -0.585786 q^{11} +3.82843 q^{13} +0.585786 q^{15} -6.82843 q^{17} +5.41421 q^{19} -2.48528 q^{21} -4.00000 q^{25} -3.31371 q^{27} +0.171573 q^{29} +8.82843 q^{31} -0.343146 q^{33} -4.24264 q^{35} -5.65685 q^{37} +2.24264 q^{39} -3.82843 q^{41} +6.48528 q^{43} -2.65685 q^{45} -2.24264 q^{47} +11.0000 q^{49} -4.00000 q^{51} -5.00000 q^{53} -0.585786 q^{55} +3.17157 q^{57} +11.0711 q^{59} -9.82843 q^{61} +11.2721 q^{63} +3.82843 q^{65} -9.65685 q^{67} +3.07107 q^{71} -9.48528 q^{73} -2.34315 q^{75} +2.48528 q^{77} +7.31371 q^{79} +6.02944 q^{81} +3.17157 q^{83} -6.82843 q^{85} +0.100505 q^{87} +7.00000 q^{89} -16.2426 q^{91} +5.17157 q^{93} +5.41421 q^{95} +15.4853 q^{97} +1.55635 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} + 2 q^{5} + 6 q^{9} - 4 q^{11} + 2 q^{13} + 4 q^{15} - 8 q^{17} + 8 q^{19} + 12 q^{21} - 8 q^{25} + 16 q^{27} + 6 q^{29} + 12 q^{31} - 12 q^{33} - 4 q^{39} - 2 q^{41} - 4 q^{43} + 6 q^{45}+ \cdots - 28 q^{99}+O(q^{100}) $ 2 * q + 4 * q^3 + 2 * q^5 + 6 * q^9 - 4 * q^11 + 2 * q^13 + 4 * q^15 - 8 * q^17 + 8 * q^19 + 12 * q^21 - 8 * q^25 + 16 * q^27 + 6 * q^29 + 12 * q^31 - 12 * q^33 - 4 * q^39 - 2 * q^41 - 4 * q^43 + 6 * q^45 + 4 * q^47 + 22 * q^49 - 8 * q^51 - 10 * q^53 - 4 * q^55 + 12 * q^57 + 8 * q^59 - 14 * q^61 + 48 * q^63 + 2 * q^65 - 8 * q^67 - 8 * q^71 - 2 * q^73 - 16 * q^75 - 12 * q^77 - 8 * q^79 + 46 * q^81 + 12 * q^83 - 8 * q^85 + 20 * q^87 + 14 * q^89 - 24 * q^91 + 16 * q^93 + 8 * q^95 + 14 * q^97 - 28 * 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
$3$	0.585786	0.338204	0.169102	−	0.985599i	$-0.445913\pi$
$3$	0.585786	0.338204	0.169102	+	0.985599i	$0.445913\pi$
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	−4.24264	−1.60357	−0.801784	−	0.597614i	$-0.796115\pi$
$7$	−4.24264	−1.60357	−0.801784	+	0.597614i	$0.796115\pi$
$8$	0	0
$9$	−2.65685	−0.885618
$10$	0	0
$11$	−0.585786	−0.176621	−0.0883106	−	0.996093i	$-0.528147\pi$
$11$	−0.585786	−0.176621	−0.0883106	+	0.996093i	$0.528147\pi$
$12$	0	0
$13$	3.82843	1.06181	0.530907	−	0.847430i	$-0.321851\pi$
$13$	3.82843	1.06181	0.530907	+	0.847430i	$0.321851\pi$
$14$	0	0
$15$	0.585786	0.151249
$16$	0	0
$17$	−6.82843	−1.65614	−0.828068	−	0.560627i	$-0.810560\pi$
$17$	−6.82843	−1.65614	−0.828068	+	0.560627i	$0.810560\pi$
$18$	0	0
$19$	5.41421	1.24211	0.621053	−	0.783769i	$-0.286705\pi$
$19$	5.41421	1.24211	0.621053	+	0.783769i	$0.286705\pi$
$20$	0	0
$21$	−2.48528	−0.542333
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−3.31371	−0.637723
$28$	0	0
$29$	0.171573	0.0318603	0.0159301	−	0.999873i	$-0.494929\pi$
$29$	0.171573	0.0318603	0.0159301	+	0.999873i	$0.494929\pi$
$30$	0	0
$31$	8.82843	1.58563	0.792816	−	0.609461i	$-0.208614\pi$
$31$	8.82843	1.58563	0.792816	+	0.609461i	$0.208614\pi$
$32$	0	0
$33$	−0.343146	−0.0597340
$34$	0	0
$35$	−4.24264	−0.717137
$36$	0	0
$37$	−5.65685	−0.929981	−0.464991	−	0.885316i	$-0.653942\pi$
$37$	−5.65685	−0.929981	−0.464991	+	0.885316i	$0.653942\pi$
$38$	0	0
$39$	2.24264	0.359110
$40$	0	0
$41$	−3.82843	−0.597900	−0.298950	−	0.954269i	$-0.596636\pi$
$41$	−3.82843	−0.597900	−0.298950	+	0.954269i	$0.596636\pi$
$42$	0	0
$43$	6.48528	0.988996	0.494498	−	0.869179i	$-0.335352\pi$
$43$	6.48528	0.988996	0.494498	+	0.869179i	$0.335352\pi$
$44$	0	0
$45$	−2.65685	−0.396060
$46$	0	0
$47$	−2.24264	−0.327123	−0.163561	−	0.986533i	$-0.552298\pi$
$47$	−2.24264	−0.327123	−0.163561	+	0.986533i	$0.552298\pi$
$48$	0	0
$49$	11.0000	1.57143
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−5.00000	−0.686803	−0.343401	−	0.939189i	$-0.611579\pi$
$53$	−5.00000	−0.686803	−0.343401	+	0.939189i	$0.611579\pi$
$54$	0	0
$55$	−0.585786	−0.0789874
$56$	0	0
$57$	3.17157	0.420085
$58$	0	0
$59$	11.0711	1.44133	0.720665	−	0.693283i	$-0.243837\pi$
$59$	11.0711	1.44133	0.720665	+	0.693283i	$0.243837\pi$
$60$	0	0
$61$	−9.82843	−1.25840	−0.629201	−	0.777243i	$-0.716618\pi$
$61$	−9.82843	−1.25840	−0.629201	+	0.777243i	$0.716618\pi$
$62$	0	0
$63$	11.2721	1.42015
$64$	0	0
$65$	3.82843	0.474858
$66$	0	0
$67$	−9.65685	−1.17977	−0.589886	−	0.807486i	$-0.700827\pi$
$67$	−9.65685	−1.17977	−0.589886	+	0.807486i	$0.700827\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.07107	0.364469	0.182234	−	0.983255i	$-0.441667\pi$
$71$	3.07107	0.364469	0.182234	+	0.983255i	$0.441667\pi$
$72$	0	0
$73$	−9.48528	−1.11017	−0.555084	−	0.831794i	$-0.687314\pi$
$73$	−9.48528	−1.11017	−0.555084	+	0.831794i	$0.687314\pi$
$74$	0	0
$75$	−2.34315	−0.270563
$76$	0	0
$77$	2.48528	0.283224
$78$	0	0
$79$	7.31371	0.822856	0.411428	−	0.911442i	$-0.365030\pi$
$79$	7.31371	0.822856	0.411428	+	0.911442i	$0.365030\pi$
$80$	0	0
$81$	6.02944	0.669937
$82$	0	0
$83$	3.17157	0.348125	0.174063	−	0.984735i	$-0.444310\pi$
$83$	3.17157	0.348125	0.174063	+	0.984735i	$0.444310\pi$
$84$	0	0
$85$	−6.82843	−0.740647
$86$	0	0
$87$	0.100505	0.0107753
$88$	0	0
$89$	7.00000	0.741999	0.370999	−	0.928633i	$-0.379015\pi$
$89$	7.00000	0.741999	0.370999	+	0.928633i	$0.379015\pi$
$90$	0	0
$91$	−16.2426	−1.70269
$92$	0	0
$93$	5.17157	0.536267
$94$	0	0
$95$	5.41421	0.555487
$96$	0	0
$97$	15.4853	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$97$	15.4853	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$98$	0	0
$99$	1.55635	0.156419

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.bj.1.1		2
4.3	odd	2		4232.2.a.l.1.2	yes	2
23.22	odd	2		8464.2.a.bg.1.1		2
92.91	even	2		4232.2.a.k.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4232.2.a.k.1.2	✓	2	92.91	even	2
4232.2.a.l.1.2	yes	2	4.3	odd	2
8464.2.a.bg.1.1		2	23.22	odd	2
8464.2.a.bj.1.1		2	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+0.585786 q^{3} +1.00000 q^{5} -4.24264 q^{7} -2.65685 q^{9} -0.585786 q^{11} +3.82843 q^{13} +0.585786 q^{15} -6.82843 q^{17} +5.41421 q^{19} -2.48528 q^{21} -4.00000 q^{25} -3.31371 q^{27} +0.171573 q^{29} +8.82843 q^{31} -0.343146 q^{33} -4.24264 q^{35} -5.65685 q^{37} +2.24264 q^{39} -3.82843 q^{41} +6.48528 q^{43} -2.65685 q^{45} -2.24264 q^{47} +11.0000 q^{49} -4.00000 q^{51} -5.00000 q^{53} -0.585786 q^{55} +3.17157 q^{57} +11.0711 q^{59} -9.82843 q^{61} +11.2721 q^{63} +3.82843 q^{65} -9.65685 q^{67} +3.07107 q^{71} -9.48528 q^{73} -2.34315 q^{75} +2.48528 q^{77} +7.31371 q^{79} +6.02944 q^{81} +3.17157 q^{83} -6.82843 q^{85} +0.100505 q^{87} +7.00000 q^{89} -16.2426 q^{91} +5.17157 q^{93} +5.41421 q^{95} +15.4853 q^{97} +1.55635 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} + 2 q^{5} + 6 q^{9} - 4 q^{11} + 2 q^{13} + 4 q^{15} - 8 q^{17} + 8 q^{19} + 12 q^{21} - 8 q^{25} + 16 q^{27} + 6 q^{29} + 12 q^{31} - 12 q^{33} - 4 q^{39} - 2 q^{41} - 4 q^{43} + 6 q^{45}+ \cdots - 28 q^{99}+O(q^{100}) \) 2 * q + 4 * q^3 + 2 * q^5 + 6 * q^9 - 4 * q^11 + 2 * q^13 + 4 * q^15 - 8 * q^17 + 8 * q^19 + 12 * q^21 - 8 * q^25 + 16 * q^27 + 6 * q^29 + 12 * q^31 - 12 * q^33 - 4 * q^39 - 2 * q^41 - 4 * q^43 + 6 * q^45 + 4 * q^47 + 22 * q^49 - 8 * q^51 - 10 * q^53 - 4 * q^55 + 12 * q^57 + 8 * q^59 - 14 * q^61 + 48 * q^63 + 2 * q^65 - 8 * q^67 - 8 * q^71 - 2 * q^73 - 16 * q^75 - 12 * q^77 - 8 * q^79 + 46 * q^81 + 12 * q^83 - 8 * q^85 + 20 * q^87 + 14 * q^89 - 24 * q^91 + 16 * q^93 + 8 * q^95 + 14 * q^97 - 28 * q^99

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