LMFDB - Embedded newform 3936.2.a.i.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3936,2,Mod(1,3936)] 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("3936.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3936, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3936 = 2^{5} \cdot 3 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3936.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$31.4291182356$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.15188.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7x^{2} + x + 2 $ x^4 - x^3 - 7*x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.490689$ of defining polynomial
Character	$\chi$	$=$	3936.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.92687 q^{5} +1.34166 q^{7} +1.00000 q^{9} -4.41756 q^{11} -0.658339 q^{13} +3.92687 q^{15} +1.43618 q^{17} +1.63972 q^{19} -1.34166 q^{21} +3.24355 q^{23} +10.4203 q^{25} -1.00000 q^{27} +2.02140 q^{29} +6.39258 q^{31} +4.41756 q^{33} -5.26854 q^{35} +6.64249 q^{37} +0.658339 q^{39} -1.00000 q^{41} +0.192634 q^{43} -3.92687 q^{45} -5.00278 q^{47} -5.19994 q^{49} -1.43618 q^{51} -2.39617 q^{53} +17.3472 q^{55} -1.63972 q^{57} +4.83513 q^{59} -6.74060 q^{61} +1.34166 q^{63} +2.58521 q^{65} -1.96276 q^{67} -3.24355 q^{69} +0.552917 q^{71} -3.41120 q^{73} -10.4203 q^{75} -5.92687 q^{77} -0.298055 q^{79} +1.00000 q^{81} -1.66606 q^{83} -5.63972 q^{85} -2.02140 q^{87} -11.7447 q^{89} -0.883267 q^{91} -6.39258 q^{93} -6.43896 q^{95} -7.87873 q^{97} -4.41756 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 4 q^{5} + 6 q^{7} + 4 q^{9} - 3 q^{11} - 2 q^{13} + 4 q^{15} - 3 q^{17} - 6 q^{21} - 4 q^{27} - 13 q^{29} + 9 q^{31} + 3 q^{33} - 10 q^{35} - 7 q^{37} + 2 q^{39} - 4 q^{41} + 5 q^{43} - 4 q^{45}+ \cdots - 3 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 4 * q^5 + 6 * q^7 + 4 * q^9 - 3 * q^11 - 2 * q^13 + 4 * q^15 - 3 * q^17 - 6 * q^21 - 4 * q^27 - 13 * q^29 + 9 * q^31 + 3 * q^33 - 10 * q^35 - 7 * q^37 + 2 * q^39 - 4 * q^41 + 5 * q^43 - 4 * q^45 + 7 * q^47 + 3 * q^51 - 16 * q^53 + 16 * q^55 - 10 * q^59 - 7 * q^61 + 6 * q^63 - 2 * q^65 + 4 * q^67 + 13 * q^71 - 3 * q^73 - 12 * q^77 + 6 * q^79 + 4 * q^81 - 14 * q^83 - 16 * q^85 + 13 * q^87 - 12 * q^89 + 16 * q^91 - 9 * q^93 + 10 * q^95 - 10 * 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
$5$	−3.92687	−1.75615	−0.878076	−	0.478522i	$-0.841173\pi$
$5$	−3.92687	−1.75615	−0.878076	+	0.478522i	$0.841173\pi$
$6$	0	0
$7$	1.34166	0.507100	0.253550	−	0.967322i	$-0.418402\pi$
$7$	1.34166	0.507100	0.253550	+	0.967322i	$0.418402\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.41756	−1.33195	−0.665973	−	0.745976i	$-0.731983\pi$
$11$	−4.41756	−1.33195	−0.665973	+	0.745976i	$0.731983\pi$
$12$	0	0
$13$	−0.658339	−0.182590	−0.0912951	−	0.995824i	$-0.529101\pi$
$13$	−0.658339	−0.182590	−0.0912951	+	0.995824i	$0.529101\pi$
$14$	0	0
$15$	3.92687	1.01391
$16$	0	0
$17$	1.43618	0.348326	0.174163	−	0.984717i	$-0.444278\pi$
$17$	1.43618	0.348326	0.174163	+	0.984717i	$0.444278\pi$
$18$	0	0
$19$	1.63972	0.376177	0.188088	−	0.982152i	$-0.439771\pi$
$19$	1.63972	0.376177	0.188088	+	0.982152i	$0.439771\pi$
$20$	0	0
$21$	−1.34166	−0.292775
$22$	0	0
$23$	3.24355	0.676327	0.338164	−	0.941087i	$-0.390194\pi$
$23$	3.24355	0.676327	0.338164	+	0.941087i	$0.390194\pi$
$24$	0	0
$25$	10.4203	2.08407
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.02140	0.375364	0.187682	−	0.982230i	$-0.439903\pi$
$29$	2.02140	0.375364	0.187682	+	0.982230i	$0.439903\pi$
$30$	0	0
$31$	6.39258	1.14814	0.574070	−	0.818806i	$-0.305364\pi$
$31$	6.39258	1.14814	0.574070	+	0.818806i	$0.305364\pi$
$32$	0	0
$33$	4.41756	0.768999
$34$	0	0
$35$	−5.26854	−0.890545
$36$	0	0
$37$	6.64249	1.09202	0.546009	−	0.837779i	$-0.316146\pi$
$37$	6.64249	1.09202	0.546009	+	0.837779i	$0.316146\pi$
$38$	0	0
$39$	0.658339	0.105419
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	0.192634	0.0293764	0.0146882	−	0.999892i	$-0.495324\pi$
$43$	0.192634	0.0293764	0.0146882	+	0.999892i	$0.495324\pi$
$44$	0	0
$45$	−3.92687	−0.585384
$46$	0	0
$47$	−5.00278	−0.729730	−0.364865	−	0.931060i	$-0.618885\pi$
$47$	−5.00278	−0.729730	−0.364865	+	0.931060i	$0.618885\pi$
$48$	0	0
$49$	−5.19994	−0.742849
$50$	0	0
$51$	−1.43618	−0.201106
$52$	0	0
$53$	−2.39617	−0.329139	−0.164569	−	0.986366i	$-0.552623\pi$
$53$	−2.39617	−0.329139	−0.164569	+	0.986366i	$0.552623\pi$
$54$	0	0
$55$	17.3472	2.33910
$56$	0	0
$57$	−1.63972	−0.217186
$58$	0	0
$59$	4.83513	0.629480	0.314740	−	0.949178i	$-0.398083\pi$
$59$	4.83513	0.629480	0.314740	+	0.949178i	$0.398083\pi$
$60$	0	0
$61$	−6.74060	−0.863046	−0.431523	−	0.902102i	$-0.642024\pi$
$61$	−6.74060	−0.863046	−0.431523	+	0.902102i	$0.642024\pi$
$62$	0	0
$63$	1.34166	0.169033
$64$	0	0
$65$	2.58521	0.320656
$66$	0	0
$67$	−1.96276	−0.239789	−0.119894	−	0.992787i	$-0.538256\pi$
$67$	−1.96276	−0.239789	−0.119894	+	0.992787i	$0.538256\pi$
$68$	0	0
$69$	−3.24355	−0.390478
$70$	0	0
$71$	0.552917	0.0656192	0.0328096	−	0.999462i	$-0.489555\pi$
$71$	0.552917	0.0656192	0.0328096	+	0.999462i	$0.489555\pi$
$72$	0	0
$73$	−3.41120	−0.399251	−0.199625	−	0.979872i	$-0.563973\pi$
$73$	−3.41120	−0.399251	−0.199625	+	0.979872i	$0.563973\pi$
$74$	0	0
$75$	−10.4203	−1.20324
$76$	0	0
$77$	−5.92687	−0.675430
$78$	0	0
$79$	−0.298055	−0.0335338	−0.0167669	−	0.999859i	$-0.505337\pi$
$79$	−0.298055	−0.0335338	−0.0167669	+	0.999859i	$0.505337\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.66606	−0.182874	−0.0914371	−	0.995811i	$-0.529146\pi$
$83$	−1.66606	−0.182874	−0.0914371	+	0.995811i	$0.529146\pi$
$84$	0	0
$85$	−5.63972	−0.611713
$86$	0	0
$87$	−2.02140	−0.216717
$88$	0	0
$89$	−11.7447	−1.24494	−0.622470	−	0.782644i	$-0.713871\pi$
$89$	−11.7447	−1.24494	−0.622470	+	0.782644i	$0.713871\pi$
$90$	0	0
$91$	−0.883267	−0.0925916
$92$	0	0
$93$	−6.39258	−0.662880
$94$	0	0
$95$	−6.43896	−0.660623
$96$	0	0
$97$	−7.87873	−0.799964	−0.399982	−	0.916523i	$-0.630984\pi$
$97$	−7.87873	−0.799964	−0.399982	+	0.916523i	$0.630984\pi$
$98$	0	0
$99$	−4.41756	−0.443982

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	3936.2.a.i.1.1	✓	4
4.3	odd	2		3936.2.a.m.1.1	yes	4
8.3	odd	2		7872.2.a.cf.1.4		4
8.5	even	2		7872.2.a.cj.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3936.2.a.i.1.1	✓	4	1.1	even	1	trivial
3936.2.a.m.1.1	yes	4	4.3	odd	2
7872.2.a.cf.1.4		4	8.3	odd	2
7872.2.a.cj.1.4		4	8.5	even	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:	\( N \)	\(=\)	\( 3936 = 2^{5} \cdot 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3936.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.92687 q^{5} +1.34166 q^{7} +1.00000 q^{9} -4.41756 q^{11} -0.658339 q^{13} +3.92687 q^{15} +1.43618 q^{17} +1.63972 q^{19} -1.34166 q^{21} +3.24355 q^{23} +10.4203 q^{25} -1.00000 q^{27} +2.02140 q^{29} +6.39258 q^{31} +4.41756 q^{33} -5.26854 q^{35} +6.64249 q^{37} +0.658339 q^{39} -1.00000 q^{41} +0.192634 q^{43} -3.92687 q^{45} -5.00278 q^{47} -5.19994 q^{49} -1.43618 q^{51} -2.39617 q^{53} +17.3472 q^{55} -1.63972 q^{57} +4.83513 q^{59} -6.74060 q^{61} +1.34166 q^{63} +2.58521 q^{65} -1.96276 q^{67} -3.24355 q^{69} +0.552917 q^{71} -3.41120 q^{73} -10.4203 q^{75} -5.92687 q^{77} -0.298055 q^{79} +1.00000 q^{81} -1.66606 q^{83} -5.63972 q^{85} -2.02140 q^{87} -11.7447 q^{89} -0.883267 q^{91} -6.39258 q^{93} -6.43896 q^{95} -7.87873 q^{97} -4.41756 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{5} + 6 q^{7} + 4 q^{9} - 3 q^{11} - 2 q^{13} + 4 q^{15} - 3 q^{17} - 6 q^{21} - 4 q^{27} - 13 q^{29} + 9 q^{31} + 3 q^{33} - 10 q^{35} - 7 q^{37} + 2 q^{39} - 4 q^{41} + 5 q^{43} - 4 q^{45}+ \cdots - 3 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^5 + 6 * q^7 + 4 * q^9 - 3 * q^11 - 2 * q^13 + 4 * q^15 - 3 * q^17 - 6 * q^21 - 4 * q^27 - 13 * q^29 + 9 * q^31 + 3 * q^33 - 10 * q^35 - 7 * q^37 + 2 * q^39 - 4 * q^41 + 5 * q^43 - 4 * q^45 + 7 * q^47 + 3 * q^51 - 16 * q^53 + 16 * q^55 - 10 * q^59 - 7 * q^61 + 6 * q^63 - 2 * q^65 + 4 * q^67 + 13 * q^71 - 3 * q^73 - 12 * q^77 + 6 * q^79 + 4 * q^81 - 14 * q^83 - 16 * q^85 + 13 * q^87 - 12 * q^89 + 16 * q^91 - 9 * q^93 + 10 * q^95 - 10 * q^97 - 3 * q^99

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