LMFDB - Embedded newform 3720.2.a.i.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3720,2,Mod(1,3720)] 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("3720.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3720.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,-2,0,2] 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:	$29.7043495519$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	3720.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -1.23607 q^{7} +1.00000 q^{9} -3.23607 q^{13} +1.00000 q^{15} +4.47214 q^{17} +6.47214 q^{19} +1.23607 q^{21} -4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.23607 q^{29} -1.00000 q^{31} +1.23607 q^{35} +4.76393 q^{37} +3.23607 q^{39} -6.00000 q^{41} +6.47214 q^{43} -1.00000 q^{45} +8.94427 q^{47} -5.47214 q^{49} -4.47214 q^{51} +8.47214 q^{53} -6.47214 q^{57} +6.76393 q^{59} -12.4721 q^{61} -1.23607 q^{63} +3.23607 q^{65} +3.70820 q^{67} +4.00000 q^{69} -9.23607 q^{71} +12.1803 q^{73} -1.00000 q^{75} -8.94427 q^{79} +1.00000 q^{81} -4.94427 q^{83} -4.47214 q^{85} +7.23607 q^{87} -7.23607 q^{89} +4.00000 q^{91} +1.00000 q^{93} -6.47214 q^{95} -8.47214 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{13} + 2 q^{15} + 4 q^{19} - 2 q^{21} - 8 q^{23} + 2 q^{25} - 2 q^{27} - 10 q^{29} - 2 q^{31} - 2 q^{35} + 14 q^{37} + 2 q^{39} - 12 q^{41} + 4 q^{43} - 2 q^{45}+ \cdots - 8 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 2 * q^13 + 2 * q^15 + 4 * q^19 - 2 * q^21 - 8 * q^23 + 2 * q^25 - 2 * q^27 - 10 * q^29 - 2 * q^31 - 2 * q^35 + 14 * q^37 + 2 * q^39 - 12 * q^41 + 4 * q^43 - 2 * q^45 - 2 * q^49 + 8 * q^53 - 4 * q^57 + 18 * q^59 - 16 * q^61 + 2 * q^63 + 2 * q^65 - 6 * q^67 + 8 * q^69 - 14 * q^71 + 2 * q^73 - 2 * q^75 + 2 * q^81 + 8 * q^83 + 10 * q^87 - 10 * q^89 + 8 * q^91 + 2 * q^93 - 4 * q^95 - 8 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.23607	−0.467190	−0.233595	−	0.972334i	$-0.575049\pi$
$7$	−1.23607	−0.467190	−0.233595	+	0.972334i	$0.575049\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−3.23607	−0.897524	−0.448762	−	0.893651i	$-0.648135\pi$
$13$	−3.23607	−0.897524	−0.448762	+	0.893651i	$0.648135\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	4.47214	1.08465	0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214	1.08465	0.542326	+	0.840168i	$0.317544\pi$
$18$	0	0
$19$	6.47214	1.48481	0.742405	−	0.669951i	$-0.233685\pi$
$19$	6.47214	1.48481	0.742405	+	0.669951i	$0.233685\pi$
$20$	0	0
$21$	1.23607	0.269732
$22$	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
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.23607	−1.34370	−0.671852	−	0.740685i	$-0.734501\pi$
$29$	−7.23607	−1.34370	−0.671852	+	0.740685i	$0.734501\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.23607	0.208934
$36$	0	0
$37$	4.76393	0.783186	0.391593	−	0.920139i	$-0.371924\pi$
$37$	4.76393	0.783186	0.391593	+	0.920139i	$0.371924\pi$
$38$	0	0
$39$	3.23607	0.518186
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	6.47214	0.986991	0.493496	−	0.869748i	$-0.335719\pi$
$43$	6.47214	0.986991	0.493496	+	0.869748i	$0.335719\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	8.94427	1.30466	0.652328	−	0.757937i	$-0.273792\pi$
$47$	8.94427	1.30466	0.652328	+	0.757937i	$0.273792\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	0	0
$51$	−4.47214	−0.626224
$52$	0	0
$53$	8.47214	1.16374	0.581869	−	0.813283i	$-0.302322\pi$
$53$	8.47214	1.16374	0.581869	+	0.813283i	$0.302322\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6.47214	−0.857255
$58$	0	0
$59$	6.76393	0.880589	0.440294	−	0.897853i	$-0.354874\pi$
$59$	6.76393	0.880589	0.440294	+	0.897853i	$0.354874\pi$
$60$	0	0
$61$	−12.4721	−1.59689	−0.798447	−	0.602066i	$-0.794345\pi$
$61$	−12.4721	−1.59689	−0.798447	+	0.602066i	$0.794345\pi$
$62$	0	0
$63$	−1.23607	−0.155730
$64$	0	0
$65$	3.23607	0.401385
$66$	0	0
$67$	3.70820	0.453029	0.226515	−	0.974008i	$-0.427267\pi$
$67$	3.70820	0.453029	0.226515	+	0.974008i	$0.427267\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	−9.23607	−1.09612	−0.548060	−	0.836439i	$-0.684633\pi$
$71$	−9.23607	−1.09612	−0.548060	+	0.836439i	$0.684633\pi$
$72$	0	0
$73$	12.1803	1.42560	0.712800	−	0.701367i	$-0.247427\pi$
$73$	12.1803	1.42560	0.712800	+	0.701367i	$0.247427\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.94427	−1.00631	−0.503155	−	0.864196i	$-0.667827\pi$
$79$	−8.94427	−1.00631	−0.503155	+	0.864196i	$0.667827\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.94427	−0.542704	−0.271352	−	0.962480i	$-0.587471\pi$
$83$	−4.94427	−0.542704	−0.271352	+	0.962480i	$0.587471\pi$
$84$	0	0
$85$	−4.47214	−0.485071
$86$	0	0
$87$	7.23607	0.775788
$88$	0	0
$89$	−7.23607	−0.767022	−0.383511	−	0.923536i	$-0.625285\pi$
$89$	−7.23607	−0.767022	−0.383511	+	0.923536i	$0.625285\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−6.47214	−0.664027
$96$	0	0
$97$	−8.47214	−0.860215	−0.430108	−	0.902778i	$-0.641524\pi$
$97$	−8.47214	−0.860215	−0.430108	+	0.902778i	$0.641524\pi$
$98$	0	0
$99$	0	0

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	3720.2.a.i.1.1	✓	2
4.3	odd	2		7440.2.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.i.1.1	✓	2	1.1	even	1	trivial
7440.2.a.bi.1.2		2	4.3	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:	\( N \)	\(=\)	\( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3720.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -1.23607 q^{7} +1.00000 q^{9} -3.23607 q^{13} +1.00000 q^{15} +4.47214 q^{17} +6.47214 q^{19} +1.23607 q^{21} -4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.23607 q^{29} -1.00000 q^{31} +1.23607 q^{35} +4.76393 q^{37} +3.23607 q^{39} -6.00000 q^{41} +6.47214 q^{43} -1.00000 q^{45} +8.94427 q^{47} -5.47214 q^{49} -4.47214 q^{51} +8.47214 q^{53} -6.47214 q^{57} +6.76393 q^{59} -12.4721 q^{61} -1.23607 q^{63} +3.23607 q^{65} +3.70820 q^{67} +4.00000 q^{69} -9.23607 q^{71} +12.1803 q^{73} -1.00000 q^{75} -8.94427 q^{79} +1.00000 q^{81} -4.94427 q^{83} -4.47214 q^{85} +7.23607 q^{87} -7.23607 q^{89} +4.00000 q^{91} +1.00000 q^{93} -6.47214 q^{95} -8.47214 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{13} + 2 q^{15} + 4 q^{19} - 2 q^{21} - 8 q^{23} + 2 q^{25} - 2 q^{27} - 10 q^{29} - 2 q^{31} - 2 q^{35} + 14 q^{37} + 2 q^{39} - 12 q^{41} + 4 q^{43} - 2 q^{45}+ \cdots - 8 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 2 * q^13 + 2 * q^15 + 4 * q^19 - 2 * q^21 - 8 * q^23 + 2 * q^25 - 2 * q^27 - 10 * q^29 - 2 * q^31 - 2 * q^35 + 14 * q^37 + 2 * q^39 - 12 * q^41 + 4 * q^43 - 2 * q^45 - 2 * q^49 + 8 * q^53 - 4 * q^57 + 18 * q^59 - 16 * q^61 + 2 * q^63 + 2 * q^65 - 6 * q^67 + 8 * q^69 - 14 * q^71 + 2 * q^73 - 2 * q^75 + 2 * q^81 + 8 * q^83 + 10 * q^87 - 10 * q^89 + 8 * q^91 + 2 * q^93 - 4 * q^95 - 8 * q^97

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