LMFDB - Embedded newform 1008.6.a.bf.1.2

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			1.2
Root			$-10.7361$ 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)

$f(q)$	$=$	$q+28.4166 q^{5} -49.0000 q^{7} +424.083 q^{11} +508.500 q^{13} +539.916 q^{17} -2603.00 q^{19} +261.251 q^{23} -2317.50 q^{25} -6879.66 q^{29} -5687.00 q^{31} -1392.41 q^{35} +4909.50 q^{37} +5723.42 q^{41} +1733.99 q^{43} -10147.8 q^{47} +2401.00 q^{49} +31181.5 q^{53} +12051.0 q^{55} -38845.5 q^{59} +13651.0 q^{61} +14449.8 q^{65} +30741.5 q^{67} -45627.9 q^{71} +21753.5 q^{73} -20780.1 q^{77} -32295.5 q^{79} -46637.3 q^{83} +15342.6 q^{85} +63757.4 q^{89} -24916.5 q^{91} -73968.4 q^{95} +115122. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	28.4166	0.508332	0.254166	−	0.967161i	$-0.418199\pi$
$5$	28.4166	0.508332	0.254166	+	0.967161i	$0.418199\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	424.083	1.05674	0.528371	−	0.849013i	$-0.322803\pi$
$11$	424.083	1.05674	0.528371	+	0.849013i	$0.322803\pi$
$12$	0	0
$13$	508.500	0.834511	0.417256	−	0.908789i	$-0.362992\pi$
$13$	508.500	0.834511	0.417256	+	0.908789i	$0.362992\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	539.916	0.453110	0.226555	−	0.973998i	$-0.427254\pi$
$17$	539.916	0.453110	0.226555	+	0.973998i	$0.427254\pi$
$18$	0	0
$19$	−2603.00	−1.65421	−0.827104	−	0.562050i	$-0.810013\pi$
$19$	−2603.00	−1.65421	−0.827104	+	0.562050i	$0.810013\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	261.251	0.102977	0.0514883	−	0.998674i	$-0.483604\pi$
$23$	261.251	0.102977	0.0514883	+	0.998674i	$0.483604\pi$
$24$	0	0
$25$	−2317.50	−0.741599
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6879.66	−1.51905	−0.759525	−	0.650478i	$-0.774569\pi$
$29$	−6879.66	−1.51905	−0.759525	+	0.650478i	$0.774569\pi$
$30$	0	0
$31$	−5687.00	−1.06287	−0.531433	−	0.847100i	$-0.678346\pi$
$31$	−5687.00	−1.06287	−0.531433	+	0.847100i	$0.678346\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1392.41	−0.192131
$36$	0	0
$37$	4909.50	0.589567	0.294783	−	0.955564i	$-0.404752\pi$
$37$	4909.50	0.589567	0.294783	+	0.955564i	$0.404752\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5723.42	0.531736	0.265868	−	0.964009i	$-0.414342\pi$
$41$	5723.42	0.531736	0.265868	+	0.964009i	$0.414342\pi$
$42$	0	0
$43$	1733.99	0.143013	0.0715066	−	0.997440i	$-0.477219\pi$
$43$	1733.99	0.143013	0.0715066	+	0.997440i	$0.477219\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10147.8	−0.670083	−0.335042	−	0.942203i	$-0.608750\pi$
$47$	−10147.8	−0.670083	−0.335042	+	0.942203i	$0.608750\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	31181.5	1.52478	0.762390	−	0.647118i	$-0.224026\pi$
$53$	31181.5	1.52478	0.762390	+	0.647118i	$0.224026\pi$
$54$	0	0
$55$	12051.0	0.537176
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−38845.5	−1.45282	−0.726408	−	0.687264i	$-0.758812\pi$
$59$	−38845.5	−1.45282	−0.726408	+	0.687264i	$0.758812\pi$
$60$	0	0
$61$	13651.0	0.469720	0.234860	−	0.972029i	$-0.424537\pi$
$61$	13651.0	0.469720	0.234860	+	0.972029i	$0.424537\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	14449.8	0.424209
$66$	0	0
$67$	30741.5	0.836639	0.418320	−	0.908300i	$-0.362619\pi$
$67$	30741.5	0.836639	0.418320	+	0.908300i	$0.362619\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−45627.9	−1.07420	−0.537099	−	0.843519i	$-0.680480\pi$
$71$	−45627.9	−1.07420	−0.537099	+	0.843519i	$0.680480\pi$
$72$	0	0
$73$	21753.5	0.477774	0.238887	−	0.971047i	$-0.423218\pi$
$73$	21753.5	0.477774	0.238887	+	0.971047i	$0.423218\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−20780.1	−0.399411
$78$	0	0
$79$	−32295.5	−0.582202	−0.291101	−	0.956692i	$-0.594022\pi$
$79$	−32295.5	−0.582202	−0.291101	+	0.956692i	$0.594022\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−46637.3	−0.743085	−0.371542	−	0.928416i	$-0.621171\pi$
$83$	−46637.3	−0.743085	−0.371542	+	0.928416i	$0.621171\pi$
$84$	0	0
$85$	15342.6	0.230330
$86$	0	0
$87$	0	0
$88$	0	0
$89$	63757.4	0.853209	0.426604	−	0.904438i	$-0.359710\pi$
$89$	63757.4	0.853209	0.426604	+	0.904438i	$0.359710\pi$
$90$	0	0
$91$	−24916.5	−0.315416
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−73968.4	−0.840886
$96$	0	0
$97$	115122.	1.24231	0.621156	−	0.783687i	$-0.286663\pi$
$97$	115122.	1.24231	0.621156	+	0.783687i	$0.286663\pi$
$98$	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.1.2		2
3.2	odd	2		336.6.a.u.1.1		2
4.3	odd	2		252.6.a.e.1.2		2
12.11	even	2		84.6.a.d.1.1	✓	2
84.11	even	6		588.6.i.h.373.2		4
84.23	even	6		588.6.i.h.361.2		4
84.47	odd	6		588.6.i.n.361.1		4
84.59	odd	6		588.6.i.n.373.1		4
84.83	odd	2		588.6.a.g.1.2		2

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

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

\(f(q)\)	\(=\)	\(q+28.4166 q^{5} -49.0000 q^{7} +424.083 q^{11} +508.500 q^{13} +539.916 q^{17} -2603.00 q^{19} +261.251 q^{23} -2317.50 q^{25} -6879.66 q^{29} -5687.00 q^{31} -1392.41 q^{35} +4909.50 q^{37} +5723.42 q^{41} +1733.99 q^{43} -10147.8 q^{47} +2401.00 q^{49} +31181.5 q^{53} +12051.0 q^{55} -38845.5 q^{59} +13651.0 q^{61} +14449.8 q^{65} +30741.5 q^{67} -45627.9 q^{71} +21753.5 q^{73} -20780.1 q^{77} -32295.5 q^{79} -46637.3 q^{83} +15342.6 q^{85} +63757.4 q^{89} -24916.5 q^{91} -73968.4 q^{95} +115122. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 78 q^{5} - 98 q^{7} + 174 q^{11} + 208 q^{13} - 1482 q^{17} - 352 q^{19} + 3354 q^{23} + 5882 q^{25} - 276 q^{29} - 6520 q^{31} + 3822 q^{35} + 13864 q^{37} + 12930 q^{41} - 12712 q^{43} - 28116 q^{47}+ \cdots + 213256 q^{97}+O(q^{100}) \) 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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