LMFDB - Embedded newform 3720.2.a.t.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 = [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			1.1
Root			$3.42940$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -2.78290 q^{7} +1.00000 q^{9} -5.21230 q^{11} -0.429398 q^{13} -1.00000 q^{15} +2.09803 q^{17} +1.64650 q^{19} +2.78290 q^{21} +1.54847 q^{23} +1.00000 q^{25} -1.00000 q^{27} -9.19017 q^{29} -1.00000 q^{31} +5.21230 q^{33} -2.78290 q^{35} -2.42940 q^{37} +0.429398 q^{39} +8.66274 q^{41} -8.50529 q^{43} +1.00000 q^{45} +4.95682 q^{47} +0.744526 q^{49} -2.09803 q^{51} +12.2112 q^{53} -5.21230 q^{55} -1.64650 q^{57} +2.33137 q^{59} +11.7176 q^{61} -2.78290 q^{63} -0.429398 q^{65} -5.72240 q^{67} -1.54847 q^{69} -9.07481 q^{71} +1.68596 q^{73} -1.00000 q^{75} +14.5053 q^{77} +10.8761 q^{79} +1.00000 q^{81} -2.99892 q^{83} +2.09803 q^{85} +9.19017 q^{87} +2.21602 q^{89} +1.19497 q^{91} +1.00000 q^{93} +1.64650 q^{95} +11.7597 q^{97} -5.21230 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−2.78290	−1.05184	−0.525918	−	0.850535i	$-0.676278\pi$
$7$	−2.78290	−1.05184	−0.525918	+	0.850535i	$0.676278\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.21230	−1.57157	−0.785783	−	0.618502i	$-0.787740\pi$
$11$	−5.21230	−1.57157	−0.785783	+	0.618502i	$0.787740\pi$
$12$	0	0
$13$	−0.429398	−0.119093	−0.0595467	−	0.998226i	$-0.518966\pi$
$13$	−0.429398	−0.119093	−0.0595467	+	0.998226i	$0.518966\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	2.09803	0.508846	0.254423	−	0.967093i	$-0.418114\pi$
$17$	2.09803	0.508846	0.254423	+	0.967093i	$0.418114\pi$
$18$	0	0
$19$	1.64650	0.377733	0.188866	−	0.982003i	$-0.439519\pi$
$19$	1.64650	0.377733	0.188866	+	0.982003i	$0.439519\pi$
$20$	0	0
$21$	2.78290	0.607278
$22$	0	0
$23$	1.54847	0.322879	0.161439	−	0.986883i	$-0.448386\pi$
$23$	1.54847	0.322879	0.161439	+	0.986883i	$0.448386\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−9.19017	−1.70657	−0.853285	−	0.521444i	$-0.825393\pi$
$29$	−9.19017	−1.70657	−0.853285	+	0.521444i	$0.825393\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	5.21230	0.907344
$34$	0	0
$35$	−2.78290	−0.470396
$36$	0	0
$37$	−2.42940	−0.399391	−0.199695	−	0.979858i	$-0.563995\pi$
$37$	−2.42940	−0.399391	−0.199695	+	0.979858i	$0.563995\pi$
$38$	0	0
$39$	0.429398	0.0687586
$40$	0	0
$41$	8.66274	1.35289	0.676446	−	0.736492i	$-0.263519\pi$
$41$	8.66274	1.35289	0.676446	+	0.736492i	$0.263519\pi$
$42$	0	0
$43$	−8.50529	−1.29705	−0.648523	−	0.761195i	$-0.724613\pi$
$43$	−8.50529	−1.29705	−0.648523	+	0.761195i	$0.724613\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	4.95682	0.723027	0.361513	−	0.932367i	$-0.382260\pi$
$47$	4.95682	0.723027	0.361513	+	0.932367i	$0.382260\pi$
$48$	0	0
$49$	0.744526	0.106361
$50$	0	0
$51$	−2.09803	−0.293783
$52$	0	0
$53$	12.2112	1.67734	0.838669	−	0.544641i	$-0.183334\pi$
$53$	12.2112	1.67734	0.838669	+	0.544641i	$0.183334\pi$
$54$	0	0
$55$	−5.21230	−0.702826
$56$	0	0
$57$	−1.64650	−0.218084
$58$	0	0
$59$	2.33137	0.303519	0.151759	−	0.988417i	$-0.451506\pi$
$59$	2.33137	0.303519	0.151759	+	0.988417i	$0.451506\pi$
$60$	0	0
$61$	11.7176	1.50028	0.750142	−	0.661277i	$-0.229985\pi$
$61$	11.7176	1.50028	0.750142	+	0.661277i	$0.229985\pi$
$62$	0	0
$63$	−2.78290	−0.350612
$64$	0	0
$65$	−0.429398	−0.0532602
$66$	0	0
$67$	−5.72240	−0.699102	−0.349551	−	0.936917i	$-0.613666\pi$
$67$	−5.72240	−0.699102	−0.349551	+	0.936917i	$0.613666\pi$
$68$	0	0
$69$	−1.54847	−0.186414
$70$	0	0
$71$	−9.07481	−1.07698	−0.538491	−	0.842631i	$-0.681005\pi$
$71$	−9.07481	−1.07698	−0.538491	+	0.842631i	$0.681005\pi$
$72$	0	0
$73$	1.68596	0.197326	0.0986631	−	0.995121i	$-0.468543\pi$
$73$	1.68596	0.197326	0.0986631	+	0.995121i	$0.468543\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	14.5053	1.65303
$78$	0	0
$79$	10.8761	1.22366	0.611830	−	0.790990i	$-0.290434\pi$
$79$	10.8761	1.22366	0.611830	+	0.790990i	$0.290434\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.99892	−0.329174	−0.164587	−	0.986363i	$-0.552629\pi$
$83$	−2.99892	−0.329174	−0.164587	+	0.986363i	$0.552629\pi$
$84$	0	0
$85$	2.09803	0.227563
$86$	0	0
$87$	9.19017	0.985289
$88$	0	0
$89$	2.21602	0.234897	0.117449	−	0.993079i	$-0.462528\pi$
$89$	2.21602	0.234897	0.117449	+	0.993079i	$0.462528\pi$
$90$	0	0
$91$	1.19497	0.125267
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	1.64650	0.168927
$96$	0	0
$97$	11.7597	1.19401	0.597007	−	0.802236i	$-0.296356\pi$
$97$	11.7597	1.19401	0.597007	+	0.802236i	$0.296356\pi$
$98$	0	0
$99$	−5.21230	−0.523856

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.1.1	✓	4
4.3	odd	2		7440.2.a.ca.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.t.1.1	✓	4	1.1	even	1	trivial
7440.2.a.ca.1.4		4	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -2.78290 q^{7} +1.00000 q^{9} -5.21230 q^{11} -0.429398 q^{13} -1.00000 q^{15} +2.09803 q^{17} +1.64650 q^{19} +2.78290 q^{21} +1.54847 q^{23} +1.00000 q^{25} -1.00000 q^{27} -9.19017 q^{29} -1.00000 q^{31} +5.21230 q^{33} -2.78290 q^{35} -2.42940 q^{37} +0.429398 q^{39} +8.66274 q^{41} -8.50529 q^{43} +1.00000 q^{45} +4.95682 q^{47} +0.744526 q^{49} -2.09803 q^{51} +12.2112 q^{53} -5.21230 q^{55} -1.64650 q^{57} +2.33137 q^{59} +11.7176 q^{61} -2.78290 q^{63} -0.429398 q^{65} -5.72240 q^{67} -1.54847 q^{69} -9.07481 q^{71} +1.68596 q^{73} -1.00000 q^{75} +14.5053 q^{77} +10.8761 q^{79} +1.00000 q^{81} -2.99892 q^{83} +2.09803 q^{85} +9.19017 q^{87} +2.21602 q^{89} +1.19497 q^{91} +1.00000 q^{93} +1.64650 q^{95} +11.7597 q^{97} -5.21230 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{5} - q^{7} + 4 q^{9} + q^{11} + 10 q^{13} - 4 q^{15} + 12 q^{17} + 5 q^{19} + q^{21} + q^{23} + 4 q^{25} - 4 q^{27} + 2 q^{29} - 4 q^{31} - q^{33} - q^{35} + 2 q^{37} - 10 q^{39}+ \cdots + q^{99}+O(q^{100}) \) 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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