LMFDB - Embedded newform 252.8.a.e.1.1

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			1.1
Root			$30.2027$ 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)

$f(q)$	$=$	$q-199.216 q^{5} +343.000 q^{7} -1218.97 q^{11} +6449.65 q^{13} -26786.9 q^{17} +14910.9 q^{19} +73500.9 q^{23} -38437.9 q^{25} +106453. q^{29} +66816.7 q^{31} -68331.1 q^{35} -479502. q^{37} +644539. q^{41} +145165. q^{43} -1.01085e6 q^{47} +117649. q^{49} -34208.5 q^{53} +242839. q^{55} +443317. q^{59} -1.14043e6 q^{61} -1.28487e6 q^{65} -4.31928e6 q^{67} -2.54867e6 q^{71} -3.67723e6 q^{73} -418108. q^{77} +8.55564e6 q^{79} +1.79721e6 q^{83} +5.33639e6 q^{85} -5.56317e6 q^{89} +2.21223e6 q^{91} -2.97050e6 q^{95} -1.72057e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−199.216	−0.712737	−0.356369	−	0.934345i	$-0.615985\pi$
$5$	−199.216	−0.712737	−0.356369	+	0.934345i	$0.615985\pi$
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1218.97	−0.276134	−0.138067	−	0.990423i	$-0.544089\pi$
$11$	−1218.97	−0.276134	−0.138067	+	0.990423i	$0.544089\pi$
$12$	0	0
$13$	6449.65	0.814206	0.407103	−	0.913382i	$-0.366539\pi$
$13$	6449.65	0.814206	0.407103	+	0.913382i	$0.366539\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−26786.9	−1.32237	−0.661183	−	0.750225i	$-0.729945\pi$
$17$	−26786.9	−1.32237	−0.661183	+	0.750225i	$0.729945\pi$
$18$	0	0
$19$	14910.9	0.498732	0.249366	−	0.968409i	$-0.419778\pi$
$19$	14910.9	0.498732	0.249366	+	0.968409i	$0.419778\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	73500.9	1.25964	0.629819	−	0.776742i	$-0.283129\pi$
$23$	73500.9	1.25964	0.629819	+	0.776742i	$0.283129\pi$
$24$	0	0
$25$	−38437.9	−0.492005
$26$	0	0
$27$	0	0
$28$	0	0
$29$	106453.	0.810524	0.405262	−	0.914200i	$-0.367180\pi$
$29$	106453.	0.810524	0.405262	+	0.914200i	$0.367180\pi$
$30$	0	0
$31$	66816.7	0.402827	0.201414	−	0.979506i	$-0.435446\pi$
$31$	66816.7	0.402827	0.201414	+	0.979506i	$0.435446\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−68331.1	−0.269389
$36$	0	0
$37$	−479502.	−1.55627	−0.778134	−	0.628099i	$-0.783833\pi$
$37$	−479502.	−1.55627	−0.778134	+	0.628099i	$0.783833\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	644539.	1.46051	0.730257	−	0.683173i	$-0.239401\pi$
$41$	644539.	1.46051	0.730257	+	0.683173i	$0.239401\pi$
$42$	0	0
$43$	145165.	0.278434	0.139217	−	0.990262i	$-0.455541\pi$
$43$	145165.	0.278434	0.139217	+	0.990262i	$0.455541\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.01085e6	−1.42018	−0.710088	−	0.704113i	$-0.751345\pi$
$47$	−1.01085e6	−1.42018	−0.710088	+	0.704113i	$0.751345\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−34208.5	−0.0315623	−0.0157812	−	0.999875i	$-0.505024\pi$
$53$	−34208.5	−0.0315623	−0.0157812	+	0.999875i	$0.505024\pi$
$54$	0	0
$55$	242839.	0.196811
$56$	0	0
$57$	0	0
$58$	0	0
$59$	443317.	0.281017	0.140508	−	0.990079i	$-0.455126\pi$
$59$	443317.	0.281017	0.140508	+	0.990079i	$0.455126\pi$
$60$	0	0
$61$	−1.14043e6	−0.643300	−0.321650	−	0.946859i	$-0.604237\pi$
$61$	−1.14043e6	−0.643300	−0.321650	+	0.946859i	$0.604237\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.28487e6	−0.580315
$66$	0	0
$67$	−4.31928e6	−1.75448	−0.877242	−	0.480048i	$-0.840619\pi$
$67$	−4.31928e6	−1.75448	−0.877242	+	0.480048i	$0.840619\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.54867e6	−0.845103	−0.422551	−	0.906339i	$-0.638865\pi$
$71$	−2.54867e6	−0.845103	−0.422551	+	0.906339i	$0.638865\pi$
$72$	0	0
$73$	−3.67723e6	−1.10635	−0.553173	−	0.833067i	$-0.686583\pi$
$73$	−3.67723e6	−1.10635	−0.553173	+	0.833067i	$0.686583\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−418108.	−0.104369
$78$	0	0
$79$	8.55564e6	1.95235	0.976174	−	0.216987i	$-0.0696230\pi$
$79$	8.55564e6	1.95235	0.976174	+	0.216987i	$0.0696230\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.79721e6	0.345006	0.172503	−	0.985009i	$-0.444815\pi$
$83$	1.79721e6	0.345006	0.172503	+	0.985009i	$0.444815\pi$
$84$	0	0
$85$	5.33639e6	0.942499
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.56317e6	−0.836484	−0.418242	−	0.908336i	$-0.637354\pi$
$89$	−5.56317e6	−0.836484	−0.418242	+	0.908336i	$0.637354\pi$
$90$	0	0
$91$	2.21223e6	0.307741
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.97050e6	−0.355465
$96$	0	0
$97$	−1.72057e7	−1.91413	−0.957064	−	0.289877i	$-0.906386\pi$
$97$	−1.72057e7	−1.91413	−0.957064	+	0.289877i	$0.906386\pi$
$98$	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.1.1		2
3.2	odd	2		28.8.a.a.1.2	✓	2
12.11	even	2		112.8.a.i.1.1		2
21.2	odd	6		196.8.e.d.165.1		4
21.5	even	6		196.8.e.a.165.2		4
21.11	odd	6		196.8.e.d.177.1		4
21.17	even	6		196.8.e.a.177.2		4
21.20	even	2		196.8.a.b.1.1		2
24.5	odd	2		448.8.a.p.1.1		2
24.11	even	2		448.8.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.8.a.a.1.2	✓	2	3.2	odd	2
112.8.a.i.1.1		2	12.11	even	2
196.8.a.b.1.1		2	21.20	even	2
196.8.e.a.165.2		4	21.5	even	6
196.8.e.a.177.2		4	21.17	even	6
196.8.e.d.165.1		4	21.2	odd	6
196.8.e.d.177.1		4	21.11	odd	6
252.8.a.e.1.1		2	1.1	even	1	trivial
448.8.a.n.1.2		2	24.11	even	2
448.8.a.p.1.1		2	24.5	odd	2

Overview	Random
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Classical	Maass
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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)

\(f(q)\)	\(=\)	\(q-199.216 q^{5} +343.000 q^{7} -1218.97 q^{11} +6449.65 q^{13} -26786.9 q^{17} +14910.9 q^{19} +73500.9 q^{23} -38437.9 q^{25} +106453. q^{29} +66816.7 q^{31} -68331.1 q^{35} -479502. q^{37} +644539. q^{41} +145165. q^{43} -1.01085e6 q^{47} +117649. q^{49} -34208.5 q^{53} +242839. q^{55} +443317. q^{59} -1.14043e6 q^{61} -1.28487e6 q^{65} -4.31928e6 q^{67} -2.54867e6 q^{71} -3.67723e6 q^{73} -418108. q^{77} +8.55564e6 q^{79} +1.79721e6 q^{83} +5.33639e6 q^{85} -5.56317e6 q^{89} +2.21223e6 q^{91} -2.97050e6 q^{95} -1.72057e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 42 q^{5} + 686 q^{7} - 7428 q^{11} + 11830 q^{13} - 15792 q^{17} + 26614 q^{19} - 32640 q^{23} - 91846 q^{25} + 158016 q^{29} - 180740 q^{31} - 14406 q^{35} - 45824 q^{37} + 321720 q^{41} + 1023868 q^{43}+ \cdots - 17377472 q^{97}+O(q^{100}) \) 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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