LMFDB - Embedded newform 912.2.a.o.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [912,2,Mod(1,912)] 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("912.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(912, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,-1,0,3,0,2,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$7.28235666434$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 456)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.70156$ of defining polynomial
Character	$\chi$	$=$	912.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.70156 q^{5} +4.70156 q^{7} +1.00000 q^{9} -4.70156 q^{11} +6.00000 q^{13} +2.70156 q^{15} -2.70156 q^{17} -1.00000 q^{19} +4.70156 q^{21} -4.00000 q^{23} +2.29844 q^{25} +1.00000 q^{27} +2.00000 q^{29} -9.40312 q^{31} -4.70156 q^{33} +12.7016 q^{35} -3.40312 q^{37} +6.00000 q^{39} -3.40312 q^{41} -10.1047 q^{43} +2.70156 q^{45} +0.701562 q^{47} +15.1047 q^{49} -2.70156 q^{51} -6.00000 q^{53} -12.7016 q^{55} -1.00000 q^{57} +4.00000 q^{59} +1.29844 q^{61} +4.70156 q^{63} +16.2094 q^{65} +12.0000 q^{67} -4.00000 q^{69} +6.70156 q^{73} +2.29844 q^{75} -22.1047 q^{77} +10.8062 q^{79} +1.00000 q^{81} +10.8062 q^{83} -7.29844 q^{85} +2.00000 q^{87} -12.8062 q^{89} +28.2094 q^{91} -9.40312 q^{93} -2.70156 q^{95} -6.00000 q^{97} -4.70156 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - q^{5} + 3 q^{7} + 2 q^{9} - 3 q^{11} + 12 q^{13} - q^{15} + q^{17} - 2 q^{19} + 3 q^{21} - 8 q^{23} + 11 q^{25} + 2 q^{27} + 4 q^{29} - 6 q^{31} - 3 q^{33} + 19 q^{35} + 6 q^{37} + 12 q^{39}+ \cdots - 3 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - q^5 + 3 * q^7 + 2 * q^9 - 3 * q^11 + 12 * q^13 - q^15 + q^17 - 2 * q^19 + 3 * q^21 - 8 * q^23 + 11 * q^25 + 2 * q^27 + 4 * q^29 - 6 * q^31 - 3 * q^33 + 19 * q^35 + 6 * q^37 + 12 * q^39 + 6 * q^41 - q^43 - q^45 - 5 * q^47 + 11 * q^49 + q^51 - 12 * q^53 - 19 * q^55 - 2 * q^57 + 8 * q^59 + 9 * q^61 + 3 * q^63 - 6 * q^65 + 24 * q^67 - 8 * q^69 + 7 * q^73 + 11 * q^75 - 25 * q^77 - 4 * q^79 + 2 * q^81 - 4 * q^83 - 21 * q^85 + 4 * q^87 + 18 * q^91 - 6 * q^93 + q^95 - 12 * q^97 - 3 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	2.70156	1.20818	0.604088	−	0.796918i	$-0.293538\pi$
$5$	2.70156	1.20818	0.604088	+	0.796918i	$0.293538\pi$
$6$	0	0
$7$	4.70156	1.77702	0.888512	−	0.458854i	$-0.151740\pi$
$7$	4.70156	1.77702	0.888512	+	0.458854i	$0.151740\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.70156	−1.41757	−0.708787	−	0.705422i	$-0.750757\pi$
$11$	−4.70156	−1.41757	−0.708787	+	0.705422i	$0.750757\pi$
$12$	0	0
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	2.70156	0.697540
$16$	0	0
$17$	−2.70156	−0.655225	−0.327613	−	0.944812i	$-0.606244\pi$
$17$	−2.70156	−0.655225	−0.327613	+	0.944812i	$0.606244\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	4.70156	1.02596
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	2.29844	0.459688
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−9.40312	−1.68885	−0.844425	−	0.535673i	$-0.820058\pi$
$31$	−9.40312	−1.68885	−0.844425	+	0.535673i	$0.820058\pi$
$32$	0	0
$33$	−4.70156	−0.818437
$34$	0	0
$35$	12.7016	2.14696
$36$	0	0
$37$	−3.40312	−0.559470	−0.279735	−	0.960077i	$-0.590247\pi$
$37$	−3.40312	−0.559470	−0.279735	+	0.960077i	$0.590247\pi$
$38$	0	0
$39$	6.00000	0.960769
$40$	0	0
$41$	−3.40312	−0.531479	−0.265739	−	0.964045i	$-0.585616\pi$
$41$	−3.40312	−0.531479	−0.265739	+	0.964045i	$0.585616\pi$
$42$	0	0
$43$	−10.1047	−1.54095	−0.770475	−	0.637470i	$-0.779981\pi$
$43$	−10.1047	−1.54095	−0.770475	+	0.637470i	$0.779981\pi$
$44$	0	0
$45$	2.70156	0.402725
$46$	0	0
$47$	0.701562	0.102333	0.0511667	−	0.998690i	$-0.483706\pi$
$47$	0.701562	0.102333	0.0511667	+	0.998690i	$0.483706\pi$
$48$	0	0
$49$	15.1047	2.15781
$50$	0	0
$51$	−2.70156	−0.378294
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−12.7016	−1.71268
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	1.29844	0.166248	0.0831240	−	0.996539i	$-0.473510\pi$
$61$	1.29844	0.166248	0.0831240	+	0.996539i	$0.473510\pi$
$62$	0	0
$63$	4.70156	0.592341
$64$	0	0
$65$	16.2094	2.01053
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	6.70156	0.784359	0.392179	−	0.919889i	$-0.371721\pi$
$73$	6.70156	0.784359	0.392179	+	0.919889i	$0.371721\pi$
$74$	0	0
$75$	2.29844	0.265401
$76$	0	0
$77$	−22.1047	−2.51906
$78$	0	0
$79$	10.8062	1.21580	0.607899	−	0.794014i	$-0.292013\pi$
$79$	10.8062	1.21580	0.607899	+	0.794014i	$0.292013\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.8062	1.18614	0.593070	−	0.805151i	$-0.297916\pi$
$83$	10.8062	1.18614	0.593070	+	0.805151i	$0.297916\pi$
$84$	0	0
$85$	−7.29844	−0.791627
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	−12.8062	−1.35746	−0.678730	−	0.734388i	$-0.737469\pi$
$89$	−12.8062	−1.35746	−0.678730	+	0.734388i	$0.737469\pi$
$90$	0	0
$91$	28.2094	2.95715
$92$	0	0
$93$	−9.40312	−0.975059
$94$	0	0
$95$	−2.70156	−0.277174
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	−4.70156	−0.472525

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	912.2.a.o.1.2		2
3.2	odd	2		2736.2.a.bb.1.1		2
4.3	odd	2		456.2.a.e.1.2	✓	2
8.3	odd	2		3648.2.a.bs.1.1		2
8.5	even	2		3648.2.a.bn.1.1		2
12.11	even	2		1368.2.a.l.1.1		2
76.75	even	2		8664.2.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.a.e.1.2	✓	2	4.3	odd	2
912.2.a.o.1.2		2	1.1	even	1	trivial
1368.2.a.l.1.1		2	12.11	even	2
2736.2.a.bb.1.1		2	3.2	odd	2
3648.2.a.bn.1.1		2	8.5	even	2
3648.2.a.bs.1.1		2	8.3	odd	2
8664.2.a.v.1.2		2	76.75	even	2

Overview	Random
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Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.70156 q^{5} +4.70156 q^{7} +1.00000 q^{9} -4.70156 q^{11} +6.00000 q^{13} +2.70156 q^{15} -2.70156 q^{17} -1.00000 q^{19} +4.70156 q^{21} -4.00000 q^{23} +2.29844 q^{25} +1.00000 q^{27} +2.00000 q^{29} -9.40312 q^{31} -4.70156 q^{33} +12.7016 q^{35} -3.40312 q^{37} +6.00000 q^{39} -3.40312 q^{41} -10.1047 q^{43} +2.70156 q^{45} +0.701562 q^{47} +15.1047 q^{49} -2.70156 q^{51} -6.00000 q^{53} -12.7016 q^{55} -1.00000 q^{57} +4.00000 q^{59} +1.29844 q^{61} +4.70156 q^{63} +16.2094 q^{65} +12.0000 q^{67} -4.00000 q^{69} +6.70156 q^{73} +2.29844 q^{75} -22.1047 q^{77} +10.8062 q^{79} +1.00000 q^{81} +10.8062 q^{83} -7.29844 q^{85} +2.00000 q^{87} -12.8062 q^{89} +28.2094 q^{91} -9.40312 q^{93} -2.70156 q^{95} -6.00000 q^{97} -4.70156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - q^{5} + 3 q^{7} + 2 q^{9} - 3 q^{11} + 12 q^{13} - q^{15} + q^{17} - 2 q^{19} + 3 q^{21} - 8 q^{23} + 11 q^{25} + 2 q^{27} + 4 q^{29} - 6 q^{31} - 3 q^{33} + 19 q^{35} + 6 q^{37} + 12 q^{39}+ \cdots - 3 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - q^5 + 3 * q^7 + 2 * q^9 - 3 * q^11 + 12 * q^13 - q^15 + q^17 - 2 * q^19 + 3 * q^21 - 8 * q^23 + 11 * q^25 + 2 * q^27 + 4 * q^29 - 6 * q^31 - 3 * q^33 + 19 * q^35 + 6 * q^37 + 12 * q^39 + 6 * q^41 - q^43 - q^45 - 5 * q^47 + 11 * q^49 + q^51 - 12 * q^53 - 19 * q^55 - 2 * q^57 + 8 * q^59 + 9 * q^61 + 3 * q^63 - 6 * q^65 + 24 * q^67 - 8 * q^69 + 7 * q^73 + 11 * q^75 - 25 * q^77 - 4 * q^79 + 2 * q^81 - 4 * q^83 - 21 * q^85 + 4 * q^87 + 18 * q^91 - 6 * q^93 + q^95 - 12 * q^97 - 3 * q^99

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