LMFDB - Embedded newform 252.8.a.e.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [252,8,Mod(1,252)] 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("252.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(252, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	252.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-42,0,686] 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:	$78.7210264220$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3529}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 882 $ x^2 - x - 882
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 28)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-29.2027$ of defining polynomial
Character	$\chi$	$=$	252.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+157.216 q^{5} +343.000 q^{7} -6209.03 q^{11} +5380.35 q^{13} +10994.9 q^{17} +11703.1 q^{19} -106141. q^{23} -53408.1 q^{25} +51562.7 q^{29} -247557. q^{31} +53925.1 q^{35} +433678. q^{37} -322819. q^{41} +878703. q^{43} -655126. q^{47} +117649. q^{49} +444837. q^{53} -976159. q^{55} -2.14545e6 q^{59} +592902. q^{61} +845878. q^{65} +1.72866e6 q^{67} -1.58060e6 q^{71} -4.33164e6 q^{73} -2.12970e6 q^{77} -6.08518e6 q^{79} +8.10357e6 q^{83} +1.72858e6 q^{85} -9.86032e6 q^{89} +1.84546e6 q^{91} +1.83991e6 q^{95} -171786. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 42 q^{5} + 686 q^{7} - 7428 q^{11} + 11830 q^{13} - 15792 q^{17} + 26614 q^{19} - 32640 q^{23} - 91846 q^{25} + 158016 q^{29} - 180740 q^{31} - 14406 q^{35} - 45824 q^{37} + 321720 q^{41} + 1023868 q^{43}+ \cdots - 17377472 q^{97}+O(q^{100}) $ 2 * q - 42 * q^5 + 686 * q^7 - 7428 * q^11 + 11830 * q^13 - 15792 * q^17 + 26614 * q^19 - 32640 * q^23 - 91846 * q^25 + 158016 * q^29 - 180740 * q^31 - 14406 * q^35 - 45824 * q^37 + 321720 * q^41 + 1023868 * q^43 - 1665972 * q^47 + 235298 * q^49 + 410628 * q^53 - 733320 * q^55 - 1702134 * q^59 - 547526 * q^61 - 438996 * q^65 - 2590616 * q^67 - 4129272 * q^71 - 8008868 * q^73 - 2547804 * q^77 + 2470456 * q^79 + 9900786 * q^83 + 7064964 * q^85 - 15423492 * q^89 + 4057690 * q^91 - 1130592 * q^95 - 17377472 * 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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	157.216	0.562474	0.281237	−	0.959638i	$-0.409255\pi$
$5$	157.216	0.562474	0.281237	+	0.959638i	$0.409255\pi$
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−6209.03	−1.40653	−0.703265	−	0.710928i	$-0.748275\pi$
$11$	−6209.03	−1.40653	−0.703265	+	0.710928i	$0.748275\pi$
$12$	0	0
$13$	5380.35	0.679218	0.339609	−	0.940567i	$-0.389705\pi$
$13$	5380.35	0.679218	0.339609	+	0.940567i	$0.389705\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	10994.9	0.542776	0.271388	−	0.962470i	$-0.412517\pi$
$17$	10994.9	0.542776	0.271388	+	0.962470i	$0.412517\pi$
$18$	0	0
$19$	11703.1	0.391437	0.195718	−	0.980660i	$-0.437296\pi$
$19$	11703.1	0.391437	0.195718	+	0.980660i	$0.437296\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−106141.	−1.81901	−0.909506	−	0.415691i	$-0.863540\pi$
$23$	−106141.	−1.81901	−0.909506	+	0.415691i	$0.863540\pi$
$24$	0	0
$25$	−53408.1	−0.683623
$26$	0	0
$27$	0	0
$28$	0	0
$29$	51562.7	0.392593	0.196297	−	0.980545i	$-0.437108\pi$
$29$	51562.7	0.392593	0.196297	+	0.980545i	$0.437108\pi$
$30$	0	0
$31$	−247557.	−1.49248	−0.746240	−	0.665677i	$-0.768143\pi$
$31$	−247557.	−1.49248	−0.746240	+	0.665677i	$0.768143\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	53925.1	0.212595
$36$	0	0
$37$	433678.	1.40754	0.703771	−	0.710427i	$-0.251498\pi$
$37$	433678.	1.40754	0.703771	+	0.710427i	$0.251498\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−322819.	−0.731501	−0.365751	−	0.930713i	$-0.619188\pi$
$41$	−322819.	−0.731501	−0.365751	+	0.930713i	$0.619188\pi$
$42$	0	0
$43$	878703.	1.68540	0.842699	−	0.538385i	$-0.180965\pi$
$43$	878703.	1.68540	0.842699	+	0.538385i	$0.180965\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−655126.	−0.920413	−0.460206	−	0.887812i	$-0.652225\pi$
$47$	−655126.	−0.920413	−0.460206	+	0.887812i	$0.652225\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	444837.	0.410426	0.205213	−	0.978717i	$-0.434211\pi$
$53$	444837.	0.410426	0.205213	+	0.978717i	$0.434211\pi$
$54$	0	0
$55$	−976159.	−0.791136
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.14545e6	−1.35999	−0.679996	−	0.733216i	$-0.738019\pi$
$59$	−2.14545e6	−1.35999	−0.679996	+	0.733216i	$0.738019\pi$
$60$	0	0
$61$	592902.	0.334448	0.167224	−	0.985919i	$-0.446520\pi$
$61$	592902.	0.334448	0.167224	+	0.985919i	$0.446520\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	845878.	0.382042
$66$	0	0
$67$	1.72866e6	0.702180	0.351090	−	0.936342i	$-0.385811\pi$
$67$	1.72866e6	0.702180	0.351090	+	0.936342i	$0.385811\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.58060e6	−0.524105	−0.262052	−	0.965054i	$-0.584399\pi$
$71$	−1.58060e6	−0.524105	−0.262052	+	0.965054i	$0.584399\pi$
$72$	0	0
$73$	−4.33164e6	−1.30323	−0.651617	−	0.758548i	$-0.725909\pi$
$73$	−4.33164e6	−1.30323	−0.651617	+	0.758548i	$0.725909\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.12970e6	−0.531618
$78$	0	0
$79$	−6.08518e6	−1.38860	−0.694302	−	0.719684i	$-0.744287\pi$
$79$	−6.08518e6	−1.38860	−0.694302	+	0.719684i	$0.744287\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.10357e6	1.55562	0.777809	−	0.628500i	$-0.216331\pi$
$83$	8.10357e6	1.55562	0.777809	+	0.628500i	$0.216331\pi$
$84$	0	0
$85$	1.72858e6	0.305297
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.86032e6	−1.48261	−0.741303	−	0.671170i	$-0.765792\pi$
$89$	−9.86032e6	−1.48261	−0.741303	+	0.671170i	$0.765792\pi$
$90$	0	0
$91$	1.84546e6	0.256720
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.83991e6	0.220173
$96$	0	0
$97$	−171786.	−0.0191111	−0.00955555	−	0.999954i	$-0.503042\pi$
$97$	−171786.	−0.0191111	−0.00955555	+	0.999954i	$0.503042\pi$
$98$	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	252.8.a.e.1.2		2
3.2	odd	2		28.8.a.a.1.1	✓	2
12.11	even	2		112.8.a.i.1.2		2
21.2	odd	6		196.8.e.d.165.2		4
21.5	even	6		196.8.e.a.165.1		4
21.11	odd	6		196.8.e.d.177.2		4
21.17	even	6		196.8.e.a.177.1		4
21.20	even	2		196.8.a.b.1.2		2
24.5	odd	2		448.8.a.p.1.2		2
24.11	even	2		448.8.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.8.a.a.1.1	✓	2	3.2	odd	2
112.8.a.i.1.2		2	12.11	even	2
196.8.a.b.1.2		2	21.20	even	2
196.8.e.a.165.1		4	21.5	even	6
196.8.e.a.177.1		4	21.17	even	6
196.8.e.d.165.2		4	21.2	odd	6
196.8.e.d.177.2		4	21.11	odd	6
252.8.a.e.1.2		2	1.1	even	1	trivial
448.8.a.n.1.1		2	24.11	even	2
448.8.a.p.1.2		2	24.5	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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	252.a (trivial)

\(f(q)\)	\(=\)	\(q+157.216 q^{5} +343.000 q^{7} -6209.03 q^{11} +5380.35 q^{13} +10994.9 q^{17} +11703.1 q^{19} -106141. q^{23} -53408.1 q^{25} +51562.7 q^{29} -247557. q^{31} +53925.1 q^{35} +433678. q^{37} -322819. q^{41} +878703. q^{43} -655126. q^{47} +117649. q^{49} +444837. q^{53} -976159. q^{55} -2.14545e6 q^{59} +592902. q^{61} +845878. q^{65} +1.72866e6 q^{67} -1.58060e6 q^{71} -4.33164e6 q^{73} -2.12970e6 q^{77} -6.08518e6 q^{79} +8.10357e6 q^{83} +1.72858e6 q^{85} -9.86032e6 q^{89} +1.84546e6 q^{91} +1.83991e6 q^{95} -171786. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 42 q^{5} + 686 q^{7} - 7428 q^{11} + 11830 q^{13} - 15792 q^{17} + 26614 q^{19} - 32640 q^{23} - 91846 q^{25} + 158016 q^{29} - 180740 q^{31} - 14406 q^{35} - 45824 q^{37} + 321720 q^{41} + 1023868 q^{43}+ \cdots - 17377472 q^{97}+O(q^{100}) \) 2 * q - 42 * q^5 + 686 * q^7 - 7428 * q^11 + 11830 * q^13 - 15792 * q^17 + 26614 * q^19 - 32640 * q^23 - 91846 * q^25 + 158016 * q^29 - 180740 * q^31 - 14406 * q^35 - 45824 * q^37 + 321720 * q^41 + 1023868 * q^43 - 1665972 * q^47 + 235298 * q^49 + 410628 * q^53 - 733320 * q^55 - 1702134 * q^59 - 547526 * q^61 - 438996 * q^65 - 2590616 * q^67 - 4129272 * q^71 - 8008868 * q^73 - 2547804 * q^77 + 2470456 * q^79 + 9900786 * q^83 + 7064964 * q^85 - 15423492 * q^89 + 4057690 * q^91 - 1130592 * q^95 - 17377472 * q^97

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