LMFDB - Embedded newform 1008.6.a.bf.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1008,6,Mod(1,1008)] 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("1008.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1008.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-78,0,-98,0,0,0,174,0,208] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$161.666890371$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{505}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 126 $ x^2 - x - 126
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 84)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$11.7361$ of defining polynomial
Character	$\chi$	$=$	1008.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-106.417 q^{5} -49.0000 q^{7} -250.083 q^{11} -300.500 q^{13} -2021.92 q^{17} +2251.00 q^{19} +3092.75 q^{23} +8199.50 q^{25} +6603.66 q^{29} -833.002 q^{31} +5214.41 q^{35} +8954.50 q^{37} +7206.58 q^{41} -14446.0 q^{43} -17968.2 q^{47} +2401.00 q^{49} +15810.5 q^{53} +26613.0 q^{55} -26710.5 q^{59} -26799.0 q^{61} +31978.2 q^{65} +44494.5 q^{67} -20414.1 q^{71} +38742.5 q^{73} +12254.1 q^{77} +67211.5 q^{79} -35850.7 q^{83} +215165. q^{85} -106267. q^{89} +14724.5 q^{91} -239544. q^{95} +98133.5 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 78 q^{5} - 98 q^{7} + 174 q^{11} + 208 q^{13} - 1482 q^{17} - 352 q^{19} + 3354 q^{23} + 5882 q^{25} - 276 q^{29} - 6520 q^{31} + 3822 q^{35} + 13864 q^{37} + 12930 q^{41} - 12712 q^{43} - 28116 q^{47}+ \cdots + 213256 q^{97}+O(q^{100}) $ 2 * q - 78 * q^5 - 98 * q^7 + 174 * q^11 + 208 * q^13 - 1482 * q^17 - 352 * q^19 + 3354 * q^23 + 5882 * q^25 - 276 * q^29 - 6520 * q^31 + 3822 * q^35 + 13864 * q^37 + 12930 * q^41 - 12712 * q^43 - 28116 * q^47 + 4802 * q^49 + 46992 * q^53 + 38664 * q^55 - 65556 * q^59 - 13148 * q^61 + 46428 * q^65 + 75236 * q^67 - 66042 * q^71 + 60496 * q^73 - 8526 * q^77 + 34916 * q^79 - 82488 * q^83 + 230508 * q^85 - 42510 * q^89 - 10192 * q^91 - 313512 * q^95 + 213256 * 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
$5$	−106.417	−1.90364	−0.951819	−	0.306660i	$-0.900789\pi$
$5$	−106.417	−1.90364	−0.951819	+	0.306660i	$0.900789\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−250.083	−0.623164	−0.311582	−	0.950219i	$-0.600859\pi$
$11$	−250.083	−0.623164	−0.311582	+	0.950219i	$0.600859\pi$
$12$	0	0
$13$	−300.500	−0.493158	−0.246579	−	0.969123i	$-0.579306\pi$
$13$	−300.500	−0.493158	−0.246579	+	0.969123i	$0.579306\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2021.92	−1.69684	−0.848420	−	0.529324i	$-0.822446\pi$
$17$	−2021.92	−1.69684	−0.848420	+	0.529324i	$0.822446\pi$
$18$	0	0
$19$	2251.00	1.43051	0.715255	−	0.698863i	$-0.246310\pi$
$19$	2251.00	1.43051	0.715255	+	0.698863i	$0.246310\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3092.75	1.21906	0.609530	−	0.792763i	$-0.291358\pi$
$23$	3092.75	1.21906	0.609530	+	0.792763i	$0.291358\pi$
$24$	0	0
$25$	8199.50	2.62384
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6603.66	1.45811	0.729054	−	0.684456i	$-0.239960\pi$
$29$	6603.66	1.45811	0.729054	+	0.684456i	$0.239960\pi$
$30$	0	0
$31$	−833.002	−0.155683	−0.0778416	−	0.996966i	$-0.524803\pi$
$31$	−833.002	−0.155683	−0.0778416	+	0.996966i	$0.524803\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5214.41	0.719508
$36$	0	0
$37$	8954.50	1.07532	0.537659	−	0.843162i	$-0.319309\pi$
$37$	8954.50	1.07532	0.537659	+	0.843162i	$0.319309\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7206.58	0.669530	0.334765	−	0.942302i	$-0.391343\pi$
$41$	7206.58	0.669530	0.334765	+	0.942302i	$0.391343\pi$
$42$	0	0
$43$	−14446.0	−1.19145	−0.595726	−	0.803188i	$-0.703135\pi$
$43$	−14446.0	−1.19145	−0.595726	+	0.803188i	$0.703135\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−17968.2	−1.18648	−0.593238	−	0.805027i	$-0.702151\pi$
$47$	−17968.2	−1.18648	−0.593238	+	0.805027i	$0.702151\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	15810.5	0.773136	0.386568	−	0.922261i	$-0.373660\pi$
$53$	15810.5	0.773136	0.386568	+	0.922261i	$0.373660\pi$
$54$	0	0
$55$	26613.0	1.18628
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−26710.5	−0.998969	−0.499485	−	0.866323i	$-0.666477\pi$
$59$	−26710.5	−0.998969	−0.499485	+	0.866323i	$0.666477\pi$
$60$	0	0
$61$	−26799.0	−0.922133	−0.461067	−	0.887365i	$-0.652533\pi$
$61$	−26799.0	−0.922133	−0.461067	+	0.887365i	$0.652533\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	31978.2	0.938794
$66$	0	0
$67$	44494.5	1.21093	0.605465	−	0.795872i	$-0.292987\pi$
$67$	44494.5	1.21093	0.605465	+	0.795872i	$0.292987\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−20414.1	−0.480600	−0.240300	−	0.970699i	$-0.577246\pi$
$71$	−20414.1	−0.480600	−0.240300	+	0.970699i	$0.577246\pi$
$72$	0	0
$73$	38742.5	0.850904	0.425452	−	0.904981i	$-0.360115\pi$
$73$	38742.5	0.850904	0.425452	+	0.904981i	$0.360115\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	12254.1	0.235534
$78$	0	0
$79$	67211.5	1.21165	0.605823	−	0.795600i	$-0.292844\pi$
$79$	67211.5	1.21165	0.605823	+	0.795600i	$0.292844\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−35850.7	−0.571218	−0.285609	−	0.958346i	$-0.592196\pi$
$83$	−35850.7	−0.571218	−0.285609	+	0.958346i	$0.592196\pi$
$84$	0	0
$85$	215165.	3.23017
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−106267.	−1.42208	−0.711041	−	0.703150i	$-0.751776\pi$
$89$	−106267.	−1.42208	−0.711041	+	0.703150i	$0.751776\pi$
$90$	0	0
$91$	14724.5	0.186396
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−239544.	−2.72318
$96$	0	0
$97$	98133.5	1.05898	0.529490	−	0.848316i	$-0.322383\pi$
$97$	98133.5	1.05898	0.529490	+	0.848316i	$0.322383\pi$
$98$	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	1008.6.a.bf.1.1		2
3.2	odd	2		336.6.a.u.1.2		2
4.3	odd	2		252.6.a.e.1.1		2
12.11	even	2		84.6.a.d.1.2	✓	2
84.11	even	6		588.6.i.h.373.1		4
84.23	even	6		588.6.i.h.361.1		4
84.47	odd	6		588.6.i.n.361.2		4
84.59	odd	6		588.6.i.n.373.2		4
84.83	odd	2		588.6.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.6.a.d.1.2	✓	2	12.11	even	2
252.6.a.e.1.1		2	4.3	odd	2
336.6.a.u.1.2		2	3.2	odd	2
588.6.a.g.1.1		2	84.83	odd	2
588.6.i.h.361.1		4	84.23	even	6
588.6.i.h.373.1		4	84.11	even	6
588.6.i.n.361.2		4	84.47	odd	6
588.6.i.n.373.2		4	84.59	odd	6
1008.6.a.bf.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1008.a (trivial)

\(f(q)\)	\(=\)	\(q-106.417 q^{5} -49.0000 q^{7} -250.083 q^{11} -300.500 q^{13} -2021.92 q^{17} +2251.00 q^{19} +3092.75 q^{23} +8199.50 q^{25} +6603.66 q^{29} -833.002 q^{31} +5214.41 q^{35} +8954.50 q^{37} +7206.58 q^{41} -14446.0 q^{43} -17968.2 q^{47} +2401.00 q^{49} +15810.5 q^{53} +26613.0 q^{55} -26710.5 q^{59} -26799.0 q^{61} +31978.2 q^{65} +44494.5 q^{67} -20414.1 q^{71} +38742.5 q^{73} +12254.1 q^{77} +67211.5 q^{79} -35850.7 q^{83} +215165. q^{85} -106267. q^{89} +14724.5 q^{91} -239544. q^{95} +98133.5 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 78 q^{5} - 98 q^{7} + 174 q^{11} + 208 q^{13} - 1482 q^{17} - 352 q^{19} + 3354 q^{23} + 5882 q^{25} - 276 q^{29} - 6520 q^{31} + 3822 q^{35} + 13864 q^{37} + 12930 q^{41} - 12712 q^{43} - 28116 q^{47}+ \cdots + 213256 q^{97}+O(q^{100}) \) 2 * q - 78 * q^5 - 98 * q^7 + 174 * q^11 + 208 * q^13 - 1482 * q^17 - 352 * q^19 + 3354 * q^23 + 5882 * q^25 - 276 * q^29 - 6520 * q^31 + 3822 * q^35 + 13864 * q^37 + 12930 * q^41 - 12712 * q^43 - 28116 * q^47 + 4802 * q^49 + 46992 * q^53 + 38664 * q^55 - 65556 * q^59 - 13148 * q^61 + 46428 * q^65 + 75236 * q^67 - 66042 * q^71 + 60496 * q^73 - 8526 * q^77 + 34916 * q^79 - 82488 * q^83 + 230508 * q^85 - 42510 * q^89 - 10192 * q^91 - 313512 * q^95 + 213256 * q^97

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