LMFDB - Embedded newform 8464.2.a.bg.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 = [2,0,4,0,-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			1.2
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)

$f(q)$	$=$	$q+3.41421 q^{3} -1.00000 q^{5} -4.24264 q^{7} +8.65685 q^{9} +3.41421 q^{11} -1.82843 q^{13} -3.41421 q^{15} +1.17157 q^{17} -2.58579 q^{19} -14.4853 q^{21} -4.00000 q^{25} +19.3137 q^{27} +5.82843 q^{29} +3.17157 q^{31} +11.6569 q^{33} +4.24264 q^{35} -5.65685 q^{37} -6.24264 q^{39} +1.82843 q^{41} +10.4853 q^{43} -8.65685 q^{45} +6.24264 q^{47} +11.0000 q^{49} +4.00000 q^{51} +5.00000 q^{53} -3.41421 q^{55} -8.82843 q^{57} -3.07107 q^{59} +4.17157 q^{61} -36.7279 q^{63} +1.82843 q^{65} -1.65685 q^{67} -11.0711 q^{71} +7.48528 q^{73} -13.6569 q^{75} -14.4853 q^{77} +15.3137 q^{79} +39.9706 q^{81} -8.82843 q^{83} -1.17157 q^{85} +19.8995 q^{87} -7.00000 q^{89} +7.75736 q^{91} +10.8284 q^{93} +2.58579 q^{95} +1.48528 q^{97} +29.5563 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.41421	1.97120	0.985599	−	0.169102i	$-0.0540867\pi$
$3$	3.41421	1.97120	0.985599	+	0.169102i	$0.0540867\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	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
$9$	8.65685	2.88562
$10$	0	0
$11$	3.41421	1.02942	0.514712	−	0.857363i	$-0.327899\pi$
$11$	3.41421	1.02942	0.514712	+	0.857363i	$0.327899\pi$
$12$	0	0
$13$	−1.82843	−0.507114	−0.253557	−	0.967320i	$-0.581601\pi$
$13$	−1.82843	−0.507114	−0.253557	+	0.967320i	$0.581601\pi$
$14$	0	0
$15$	−3.41421	−0.881546
$16$	0	0
$17$	1.17157	0.284148	0.142074	−	0.989856i	$-0.454623\pi$
$17$	1.17157	0.284148	0.142074	+	0.989856i	$0.454623\pi$
$18$	0	0
$19$	−2.58579	−0.593220	−0.296610	−	0.954999i	$-0.595856\pi$
$19$	−2.58579	−0.593220	−0.296610	+	0.954999i	$0.595856\pi$
$20$	0	0
$21$	−14.4853	−3.16095
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	19.3137	3.71692
$28$	0	0
$29$	5.82843	1.08231	0.541156	−	0.840922i	$-0.317987\pi$
$29$	5.82843	1.08231	0.541156	+	0.840922i	$0.317987\pi$
$30$	0	0
$31$	3.17157	0.569631	0.284816	−	0.958582i	$-0.408068\pi$
$31$	3.17157	0.569631	0.284816	+	0.958582i	$0.408068\pi$
$32$	0	0
$33$	11.6569	2.02920
$34$	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
$39$	−6.24264	−0.999623
$40$	0	0
$41$	1.82843	0.285552	0.142776	−	0.989755i	$-0.454397\pi$
$41$	1.82843	0.285552	0.142776	+	0.989755i	$0.454397\pi$
$42$	0	0
$43$	10.4853	1.59899	0.799495	−	0.600672i	$-0.205100\pi$
$43$	10.4853	1.59899	0.799495	+	0.600672i	$0.205100\pi$
$44$	0	0
$45$	−8.65685	−1.29049
$46$	0	0
$47$	6.24264	0.910583	0.455291	−	0.890343i	$-0.349535\pi$
$47$	6.24264	0.910583	0.455291	+	0.890343i	$0.349535\pi$
$48$	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	$-0.388421\pi$
$53$	5.00000	0.686803	0.343401	+	0.939189i	$0.388421\pi$
$54$	0	0
$55$	−3.41421	−0.460372
$56$	0	0
$57$	−8.82843	−1.16935
$58$	0	0
$59$	−3.07107	−0.399819	−0.199909	−	0.979814i	$-0.564065\pi$
$59$	−3.07107	−0.399819	−0.199909	+	0.979814i	$0.564065\pi$
$60$	0	0
$61$	4.17157	0.534115	0.267058	−	0.963681i	$-0.413949\pi$
$61$	4.17157	0.534115	0.267058	+	0.963681i	$0.413949\pi$
$62$	0	0
$63$	−36.7279	−4.62728
$64$	0	0
$65$	1.82843	0.226788
$66$	0	0
$67$	−1.65685	−0.202417	−0.101208	−	0.994865i	$-0.532271\pi$
$67$	−1.65685	−0.202417	−0.101208	+	0.994865i	$0.532271\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.0711	−1.31389	−0.656947	−	0.753937i	$-0.728152\pi$
$71$	−11.0711	−1.31389	−0.656947	+	0.753937i	$0.728152\pi$
$72$	0	0
$73$	7.48528	0.876086	0.438043	−	0.898954i	$-0.355672\pi$
$73$	7.48528	0.876086	0.438043	+	0.898954i	$0.355672\pi$
$74$	0	0
$75$	−13.6569	−1.57696
$76$	0	0
$77$	−14.4853	−1.65075
$78$	0	0
$79$	15.3137	1.72293	0.861463	−	0.507820i	$-0.169548\pi$
$79$	15.3137	1.72293	0.861463	+	0.507820i	$0.169548\pi$
$80$	0	0
$81$	39.9706	4.44117
$82$	0	0
$83$	−8.82843	−0.969046	−0.484523	−	0.874779i	$-0.661007\pi$
$83$	−8.82843	−0.969046	−0.484523	+	0.874779i	$0.661007\pi$
$84$	0	0
$85$	−1.17157	−0.127075
$86$	0	0
$87$	19.8995	2.13345
$88$	0	0
$89$	−7.00000	−0.741999	−0.370999	−	0.928633i	$-0.620985\pi$
$89$	−7.00000	−0.741999	−0.370999	+	0.928633i	$0.620985\pi$
$90$	0	0
$91$	7.75736	0.813192
$92$	0	0
$93$	10.8284	1.12286
$94$	0	0
$95$	2.58579	0.265296
$96$	0	0
$97$	1.48528	0.150807	0.0754037	−	0.997153i	$-0.475975\pi$
$97$	1.48528	0.150807	0.0754037	+	0.997153i	$0.475975\pi$
$98$	0	0
$99$	29.5563	2.97052

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4232.2.a.k.1.1	✓	2	4.3	odd	2
4232.2.a.l.1.1	yes	2	92.91	even	2
8464.2.a.bg.1.2		2	1.1	even	1	trivial
8464.2.a.bj.1.2		2	23.22	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:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q+3.41421 q^{3} -1.00000 q^{5} -4.24264 q^{7} +8.65685 q^{9} +3.41421 q^{11} -1.82843 q^{13} -3.41421 q^{15} +1.17157 q^{17} -2.58579 q^{19} -14.4853 q^{21} -4.00000 q^{25} +19.3137 q^{27} +5.82843 q^{29} +3.17157 q^{31} +11.6569 q^{33} +4.24264 q^{35} -5.65685 q^{37} -6.24264 q^{39} +1.82843 q^{41} +10.4853 q^{43} -8.65685 q^{45} +6.24264 q^{47} +11.0000 q^{49} +4.00000 q^{51} +5.00000 q^{53} -3.41421 q^{55} -8.82843 q^{57} -3.07107 q^{59} +4.17157 q^{61} -36.7279 q^{63} +1.82843 q^{65} -1.65685 q^{67} -11.0711 q^{71} +7.48528 q^{73} -13.6569 q^{75} -14.4853 q^{77} +15.3137 q^{79} +39.9706 q^{81} -8.82843 q^{83} -1.17157 q^{85} +19.8995 q^{87} -7.00000 q^{89} +7.75736 q^{91} +10.8284 q^{93} +2.58579 q^{95} +1.48528 q^{97} +29.5563 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} - 2 q^{5} + 6 q^{9} + 4 q^{11} + 2 q^{13} - 4 q^{15} + 8 q^{17} - 8 q^{19} - 12 q^{21} - 8 q^{25} + 16 q^{27} + 6 q^{29} + 12 q^{31} + 12 q^{33} - 4 q^{39} - 2 q^{41} + 4 q^{43} - 6 q^{45}+ \cdots + 28 q^{99}+O(q^{100}) \) 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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