LMFDB - Embedded newform 736.2.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [736,2,Mod(1,736)] 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("736.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(736, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 736 = 2^{5} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	736.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,-4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.87698958877$
Analytic rank:	$1$
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:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	736.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421 q^{3} -3.41421 q^{5} +3.41421 q^{7} +2.82843 q^{9} +4.24264 q^{11} +1.82843 q^{13} +8.24264 q^{15} -5.41421 q^{17} -4.82843 q^{19} -8.24264 q^{21} +1.00000 q^{23} +6.65685 q^{25} +0.414214 q^{27} -5.00000 q^{29} +0.757359 q^{31} -10.2426 q^{33} -11.6569 q^{35} -9.07107 q^{37} -4.41421 q^{39} -12.6569 q^{41} -4.00000 q^{43} -9.65685 q^{45} -3.24264 q^{47} +4.65685 q^{49} +13.0711 q^{51} +13.3137 q^{53} -14.4853 q^{55} +11.6569 q^{57} -6.82843 q^{59} +7.65685 q^{61} +9.65685 q^{63} -6.24264 q^{65} +5.89949 q^{67} -2.41421 q^{69} -6.41421 q^{71} +2.17157 q^{73} -16.0711 q^{75} +14.4853 q^{77} -2.48528 q^{79} -9.48528 q^{81} -14.7279 q^{83} +18.4853 q^{85} +12.0711 q^{87} -8.00000 q^{89} +6.24264 q^{91} -1.82843 q^{93} +16.4853 q^{95} -9.89949 q^{97} +12.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{5} + 4 q^{7} - 2 q^{13} + 8 q^{15} - 8 q^{17} - 4 q^{19} - 8 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} - 10 q^{29} + 10 q^{31} - 12 q^{33} - 12 q^{35} - 4 q^{37} - 6 q^{39} - 14 q^{41}+ \cdots + 24 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^5 + 4 * q^7 - 2 * q^13 + 8 * q^15 - 8 * q^17 - 4 * q^19 - 8 * q^21 + 2 * q^23 + 2 * q^25 - 2 * q^27 - 10 * q^29 + 10 * q^31 - 12 * q^33 - 12 * q^35 - 4 * q^37 - 6 * q^39 - 14 * q^41 - 8 * q^43 - 8 * q^45 + 2 * q^47 - 2 * q^49 + 12 * q^51 + 4 * q^53 - 12 * q^55 + 12 * q^57 - 8 * q^59 + 4 * q^61 + 8 * q^63 - 4 * q^65 - 8 * q^67 - 2 * q^69 - 10 * q^71 + 10 * q^73 - 18 * q^75 + 12 * q^77 + 12 * q^79 - 2 * q^81 - 4 * q^83 + 20 * q^85 + 10 * q^87 - 16 * q^89 + 4 * q^91 + 2 * q^93 + 16 * q^95 + 24 * 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
$3$	−2.41421	−1.39385	−0.696923	−	0.717146i	$-0.745448\pi$
$3$	−2.41421	−1.39385	−0.696923	+	0.717146i	$0.745448\pi$
$4$	0	0
$5$	−3.41421	−1.52688	−0.763441	−	0.645877i	$-0.776492\pi$
$5$	−3.41421	−1.52688	−0.763441	+	0.645877i	$0.776492\pi$
$6$	0	0
$7$	3.41421	1.29045	0.645226	−	0.763992i	$-0.276763\pi$
$7$	3.41421	1.29045	0.645226	+	0.763992i	$0.276763\pi$
$8$	0	0
$9$	2.82843	0.942809
$10$	0	0
$11$	4.24264	1.27920	0.639602	−	0.768706i	$-0.279099\pi$
$11$	4.24264	1.27920	0.639602	+	0.768706i	$0.279099\pi$
$12$	0	0
$13$	1.82843	0.507114	0.253557	−	0.967320i	$-0.418399\pi$
$13$	1.82843	0.507114	0.253557	+	0.967320i	$0.418399\pi$
$14$	0	0
$15$	8.24264	2.12824
$16$	0	0
$17$	−5.41421	−1.31314	−0.656570	−	0.754265i	$-0.727993\pi$
$17$	−5.41421	−1.31314	−0.656570	+	0.754265i	$0.727993\pi$
$18$	0	0
$19$	−4.82843	−1.10772	−0.553859	−	0.832611i	$-0.686845\pi$
$19$	−4.82843	−1.10772	−0.553859	+	0.832611i	$0.686845\pi$
$20$	0	0
$21$	−8.24264	−1.79869
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	6.65685	1.33137
$26$	0	0
$27$	0.414214	0.0797154
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	0.757359	0.136026	0.0680129	−	0.997684i	$-0.478334\pi$
$31$	0.757359	0.136026	0.0680129	+	0.997684i	$0.478334\pi$
$32$	0	0
$33$	−10.2426	−1.78301
$34$	0	0
$35$	−11.6569	−1.97037
$36$	0	0
$37$	−9.07107	−1.49127	−0.745637	−	0.666352i	$-0.767855\pi$
$37$	−9.07107	−1.49127	−0.745637	+	0.666352i	$0.767855\pi$
$38$	0	0
$39$	−4.41421	−0.706840
$40$	0	0
$41$	−12.6569	−1.97667	−0.988334	−	0.152300i	$-0.951332\pi$
$41$	−12.6569	−1.97667	−0.988334	+	0.152300i	$0.951332\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	−9.65685	−1.43956
$46$	0	0
$47$	−3.24264	−0.472988	−0.236494	−	0.971633i	$-0.575998\pi$
$47$	−3.24264	−0.472988	−0.236494	+	0.971633i	$0.575998\pi$
$48$	0	0
$49$	4.65685	0.665265
$50$	0	0
$51$	13.0711	1.83032
$52$	0	0
$53$	13.3137	1.82878	0.914389	−	0.404836i	$-0.132671\pi$
$53$	13.3137	1.82878	0.914389	+	0.404836i	$0.132671\pi$
$54$	0	0
$55$	−14.4853	−1.95319
$56$	0	0
$57$	11.6569	1.54399
$58$	0	0
$59$	−6.82843	−0.888985	−0.444493	−	0.895782i	$-0.646616\pi$
$59$	−6.82843	−0.888985	−0.444493	+	0.895782i	$0.646616\pi$
$60$	0	0
$61$	7.65685	0.980360	0.490180	−	0.871621i	$-0.336931\pi$
$61$	7.65685	0.980360	0.490180	+	0.871621i	$0.336931\pi$
$62$	0	0
$63$	9.65685	1.21665
$64$	0	0
$65$	−6.24264	−0.774304
$66$	0	0
$67$	5.89949	0.720738	0.360369	−	0.932810i	$-0.382651\pi$
$67$	5.89949	0.720738	0.360369	+	0.932810i	$0.382651\pi$
$68$	0	0
$69$	−2.41421	−0.290637
$70$	0	0
$71$	−6.41421	−0.761227	−0.380614	−	0.924734i	$-0.624287\pi$
$71$	−6.41421	−0.761227	−0.380614	+	0.924734i	$0.624287\pi$
$72$	0	0
$73$	2.17157	0.254163	0.127082	−	0.991892i	$-0.459439\pi$
$73$	2.17157	0.254163	0.127082	+	0.991892i	$0.459439\pi$
$74$	0	0
$75$	−16.0711	−1.85573
$76$	0	0
$77$	14.4853	1.65075
$78$	0	0
$79$	−2.48528	−0.279616	−0.139808	−	0.990179i	$-0.544649\pi$
$79$	−2.48528	−0.279616	−0.139808	+	0.990179i	$0.544649\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	0	0
$83$	−14.7279	−1.61660	−0.808300	−	0.588771i	$-0.799612\pi$
$83$	−14.7279	−1.61660	−0.808300	+	0.588771i	$0.799612\pi$
$84$	0	0
$85$	18.4853	2.00501
$86$	0	0
$87$	12.0711	1.29415
$88$	0	0
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	0	0
$91$	6.24264	0.654407
$92$	0	0
$93$	−1.82843	−0.189599
$94$	0	0
$95$	16.4853	1.69135
$96$	0	0
$97$	−9.89949	−1.00514	−0.502571	−	0.864536i	$-0.667612\pi$
$97$	−9.89949	−1.00514	−0.502571	+	0.864536i	$0.667612\pi$
$98$	0	0
$99$	12.0000	1.20605

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	736.2.a.a.1.1	✓	2
3.2	odd	2		6624.2.a.t.1.2		2
4.3	odd	2		736.2.a.d.1.2	yes	2
8.3	odd	2		1472.2.a.o.1.1		2
8.5	even	2		1472.2.a.v.1.2		2
12.11	even	2		6624.2.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
736.2.a.a.1.1	✓	2	1.1	even	1	trivial
736.2.a.d.1.2	yes	2	4.3	odd	2
1472.2.a.o.1.1		2	8.3	odd	2
1472.2.a.v.1.2		2	8.5	even	2
6624.2.a.s.1.2		2	12.11	even	2
6624.2.a.t.1.2		2	3.2	odd	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 736 = 2^{5} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	736.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{3} -3.41421 q^{5} +3.41421 q^{7} +2.82843 q^{9} +4.24264 q^{11} +1.82843 q^{13} +8.24264 q^{15} -5.41421 q^{17} -4.82843 q^{19} -8.24264 q^{21} +1.00000 q^{23} +6.65685 q^{25} +0.414214 q^{27} -5.00000 q^{29} +0.757359 q^{31} -10.2426 q^{33} -11.6569 q^{35} -9.07107 q^{37} -4.41421 q^{39} -12.6569 q^{41} -4.00000 q^{43} -9.65685 q^{45} -3.24264 q^{47} +4.65685 q^{49} +13.0711 q^{51} +13.3137 q^{53} -14.4853 q^{55} +11.6569 q^{57} -6.82843 q^{59} +7.65685 q^{61} +9.65685 q^{63} -6.24264 q^{65} +5.89949 q^{67} -2.41421 q^{69} -6.41421 q^{71} +2.17157 q^{73} -16.0711 q^{75} +14.4853 q^{77} -2.48528 q^{79} -9.48528 q^{81} -14.7279 q^{83} +18.4853 q^{85} +12.0711 q^{87} -8.00000 q^{89} +6.24264 q^{91} -1.82843 q^{93} +16.4853 q^{95} -9.89949 q^{97} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{5} + 4 q^{7} - 2 q^{13} + 8 q^{15} - 8 q^{17} - 4 q^{19} - 8 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} - 10 q^{29} + 10 q^{31} - 12 q^{33} - 12 q^{35} - 4 q^{37} - 6 q^{39} - 14 q^{41}+ \cdots + 24 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^5 + 4 * q^7 - 2 * q^13 + 8 * q^15 - 8 * q^17 - 4 * q^19 - 8 * q^21 + 2 * q^23 + 2 * q^25 - 2 * q^27 - 10 * q^29 + 10 * q^31 - 12 * q^33 - 12 * q^35 - 4 * q^37 - 6 * q^39 - 14 * q^41 - 8 * q^43 - 8 * q^45 + 2 * q^47 - 2 * q^49 + 12 * q^51 + 4 * q^53 - 12 * q^55 + 12 * q^57 - 8 * q^59 + 4 * q^61 + 8 * q^63 - 4 * q^65 - 8 * q^67 - 2 * q^69 - 10 * q^71 + 10 * q^73 - 18 * q^75 + 12 * q^77 + 12 * q^79 - 2 * q^81 - 4 * q^83 + 20 * q^85 + 10 * q^87 - 16 * q^89 + 4 * q^91 + 2 * q^93 + 16 * q^95 + 24 * q^99

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