LMFDB - Embedded newform 3720.2.a.t.1.3

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 = [4,0,-4,0,4,0,-1] 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:	$0$
Dimension:	$4$
Coefficient field:	4.4.92692.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 8x^{2} + 8x + 9 $ x^4 - 2x^3 - 8x^2 + 8*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.28290$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +0.0644601 q^{7} +1.00000 q^{9} +3.34736 q^{11} +5.28290 q^{13} -1.00000 q^{15} -2.77741 q^{17} -1.21844 q^{19} -0.0644601 q^{21} +3.55897 q^{23} +1.00000 q^{25} -1.00000 q^{27} +3.07128 q^{29} -1.00000 q^{31} -3.34736 q^{33} +0.0644601 q^{35} +3.28290 q^{37} -5.28290 q^{39} +6.98902 q^{41} +5.78423 q^{43} +1.00000 q^{45} -11.3432 q^{47} -6.99584 q^{49} +2.77741 q^{51} +12.5480 q^{53} +3.34736 q^{55} +1.21844 q^{57} +1.49451 q^{59} -11.1316 q^{61} +0.0644601 q^{63} +5.28290 q^{65} +5.71977 q^{67} -3.55897 q^{69} -9.39402 q^{71} -5.18240 q^{73} -1.00000 q^{75} +0.215771 q^{77} -8.25369 q^{79} +1.00000 q^{81} -11.8954 q^{83} -2.77741 q^{85} -3.07128 q^{87} +13.9598 q^{89} +0.340536 q^{91} +1.00000 q^{93} -1.21844 q^{95} +14.1070 q^{97} +3.34736 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{5} - q^{7} + 4 q^{9} + q^{11} + 10 q^{13} - 4 q^{15} + 12 q^{17} + 5 q^{19} + q^{21} + q^{23} + 4 q^{25} - 4 q^{27} + 2 q^{29} - 4 q^{31} - q^{33} - q^{35} + 2 q^{37} - 10 q^{39}+ \cdots + q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^5 - q^7 + 4 * q^9 + q^11 + 10 * q^13 - 4 * q^15 + 12 * q^17 + 5 * q^19 + q^21 + q^23 + 4 * q^25 - 4 * q^27 + 2 * q^29 - 4 * q^31 - q^33 - q^35 + 2 * q^37 - 10 * q^39 + 4 * q^41 - 9 * q^43 + 4 * q^45 + 5 * q^49 - 12 * q^51 + 13 * q^53 + q^55 - 5 * q^57 - 6 * q^59 - q^63 + 10 * q^65 - 8 * q^67 - q^69 - 9 * q^71 + 7 * q^73 - 4 * q^75 + 33 * q^77 + 5 * q^79 + 4 * q^81 + 2 * q^83 + 12 * q^85 - 2 * q^87 + 5 * q^89 - 2 * q^91 + 4 * q^93 + 5 * q^95 + 6 * q^97 + 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.0644601	0.0243636	0.0121818	−	0.999926i	$-0.496122\pi$
$7$	0.0644601	0.0243636	0.0121818	+	0.999926i	$0.496122\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.34736	1.00927	0.504633	−	0.863334i	$-0.331628\pi$
$11$	3.34736	1.00927	0.504633	+	0.863334i	$0.331628\pi$
$12$	0	0
$13$	5.28290	1.46521	0.732606	−	0.680653i	$-0.238304\pi$
$13$	5.28290	1.46521	0.732606	+	0.680653i	$0.238304\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−2.77741	−0.673621	−0.336810	−	0.941573i	$-0.609348\pi$
$17$	−2.77741	−0.673621	−0.336810	+	0.941573i	$0.609348\pi$
$18$	0	0
$19$	−1.21844	−0.279528	−0.139764	−	0.990185i	$-0.544634\pi$
$19$	−1.21844	−0.279528	−0.139764	+	0.990185i	$0.544634\pi$
$20$	0	0
$21$	−0.0644601	−0.0140663
$22$	0	0
$23$	3.55897	0.742097	0.371049	−	0.928613i	$-0.378998\pi$
$23$	3.55897	0.742097	0.371049	+	0.928613i	$0.378998\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.07128	0.570322	0.285161	−	0.958480i	$-0.407953\pi$
$29$	3.07128	0.570322	0.285161	+	0.958480i	$0.407953\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−3.34736	−0.582700
$34$	0	0
$35$	0.0644601	0.0108957
$36$	0	0
$37$	3.28290	0.539705	0.269852	−	0.962902i	$-0.413025\pi$
$37$	3.28290	0.539705	0.269852	+	0.962902i	$0.413025\pi$
$38$	0	0
$39$	−5.28290	−0.845940
$40$	0	0
$41$	6.98902	1.09150	0.545751	−	0.837947i	$-0.316244\pi$
$41$	6.98902	1.09150	0.545751	+	0.837947i	$0.316244\pi$
$42$	0	0
$43$	5.78423	0.882087	0.441043	−	0.897486i	$-0.354609\pi$
$43$	5.78423	0.882087	0.441043	+	0.897486i	$0.354609\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−11.3432	−1.65458	−0.827288	−	0.561778i	$-0.810117\pi$
$47$	−11.3432	−1.65458	−0.827288	+	0.561778i	$0.810117\pi$
$48$	0	0
$49$	−6.99584	−0.999406
$50$	0	0
$51$	2.77741	0.388915
$52$	0	0
$53$	12.5480	1.72360	0.861800	−	0.507248i	$-0.169337\pi$
$53$	12.5480	1.72360	0.861800	+	0.507248i	$0.169337\pi$
$54$	0	0
$55$	3.34736	0.451357
$56$	0	0
$57$	1.21844	0.161386
$58$	0	0
$59$	1.49451	0.194569	0.0972845	−	0.995257i	$-0.468984\pi$
$59$	1.49451	0.194569	0.0972845	+	0.995257i	$0.468984\pi$
$60$	0	0
$61$	−11.1316	−1.42525	−0.712627	−	0.701543i	$-0.752495\pi$
$61$	−11.1316	−1.42525	−0.712627	+	0.701543i	$0.752495\pi$
$62$	0	0
$63$	0.0644601	0.00812121
$64$	0	0
$65$	5.28290	0.655263
$66$	0	0
$67$	5.71977	0.698781	0.349390	−	0.936977i	$-0.386389\pi$
$67$	5.71977	0.698781	0.349390	+	0.936977i	$0.386389\pi$
$68$	0	0
$69$	−3.55897	−0.428450
$70$	0	0
$71$	−9.39402	−1.11487	−0.557433	−	0.830222i	$-0.688213\pi$
$71$	−9.39402	−1.11487	−0.557433	+	0.830222i	$0.688213\pi$
$72$	0	0
$73$	−5.18240	−0.606555	−0.303277	−	0.952902i	$-0.598081\pi$
$73$	−5.18240	−0.606555	−0.303277	+	0.952902i	$0.598081\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	0.215771	0.0245894
$78$	0	0
$79$	−8.25369	−0.928612	−0.464306	−	0.885675i	$-0.653696\pi$
$79$	−8.25369	−0.928612	−0.464306	+	0.885675i	$0.653696\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.8954	−1.30568	−0.652842	−	0.757494i	$-0.726424\pi$
$83$	−11.8954	−1.30568	−0.652842	+	0.757494i	$0.726424\pi$
$84$	0	0
$85$	−2.77741	−0.301252
$86$	0	0
$87$	−3.07128	−0.329276
$88$	0	0
$89$	13.9598	1.47974	0.739869	−	0.672751i	$-0.234888\pi$
$89$	13.9598	1.47974	0.739869	+	0.672751i	$0.234888\pi$
$90$	0	0
$91$	0.340536	0.0356979
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−1.21844	−0.125009
$96$	0	0
$97$	14.1070	1.43235	0.716173	−	0.697923i	$-0.245892\pi$
$97$	14.1070	1.43235	0.716173	+	0.697923i	$0.245892\pi$
$98$	0	0
$99$	3.34736	0.336422

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.t.1.3	✓	4
4.3	odd	2		7440.2.a.ca.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.t.1.3	✓	4	1.1	even	1	trivial
7440.2.a.ca.1.2		4	4.3	odd	2

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +0.0644601 q^{7} +1.00000 q^{9} +3.34736 q^{11} +5.28290 q^{13} -1.00000 q^{15} -2.77741 q^{17} -1.21844 q^{19} -0.0644601 q^{21} +3.55897 q^{23} +1.00000 q^{25} -1.00000 q^{27} +3.07128 q^{29} -1.00000 q^{31} -3.34736 q^{33} +0.0644601 q^{35} +3.28290 q^{37} -5.28290 q^{39} +6.98902 q^{41} +5.78423 q^{43} +1.00000 q^{45} -11.3432 q^{47} -6.99584 q^{49} +2.77741 q^{51} +12.5480 q^{53} +3.34736 q^{55} +1.21844 q^{57} +1.49451 q^{59} -11.1316 q^{61} +0.0644601 q^{63} +5.28290 q^{65} +5.71977 q^{67} -3.55897 q^{69} -9.39402 q^{71} -5.18240 q^{73} -1.00000 q^{75} +0.215771 q^{77} -8.25369 q^{79} +1.00000 q^{81} -11.8954 q^{83} -2.77741 q^{85} -3.07128 q^{87} +13.9598 q^{89} +0.340536 q^{91} +1.00000 q^{93} -1.21844 q^{95} +14.1070 q^{97} +3.34736 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{5} - q^{7} + 4 q^{9} + q^{11} + 10 q^{13} - 4 q^{15} + 12 q^{17} + 5 q^{19} + q^{21} + q^{23} + 4 q^{25} - 4 q^{27} + 2 q^{29} - 4 q^{31} - q^{33} - q^{35} + 2 q^{37} - 10 q^{39}+ \cdots + q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^5 - q^7 + 4 * q^9 + q^11 + 10 * q^13 - 4 * q^15 + 12 * q^17 + 5 * q^19 + q^21 + q^23 + 4 * q^25 - 4 * q^27 + 2 * q^29 - 4 * q^31 - q^33 - q^35 + 2 * q^37 - 10 * q^39 + 4 * q^41 - 9 * q^43 + 4 * q^45 + 5 * q^49 - 12 * q^51 + 13 * q^53 + q^55 - 5 * q^57 - 6 * q^59 - q^63 + 10 * q^65 - 8 * q^67 - q^69 - 9 * q^71 + 7 * q^73 - 4 * q^75 + 33 * q^77 + 5 * q^79 + 4 * q^81 + 2 * q^83 + 12 * q^85 - 2 * q^87 + 5 * q^89 - 2 * q^91 + 4 * q^93 + 5 * q^95 + 6 * q^97 + q^99

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