LMFDB - Embedded newform 800.6.a.y.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [800,6,Mod(1,800)] 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("800.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,228,0,0,0,128] 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:	$128.307055850$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 207x^{4} + 419x^{3} + 5927x^{2} - 6137x - 8328 $ x^6 - 3x^5 - 207x^4 + 419x^3 + 5927x^2 - 6137*x - 8328
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{15}\cdot 5^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.29401$ of defining polynomial
Character	$\chi$	$=$	800.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-11.5880 q^{3} +89.6769 q^{7} -108.718 q^{9} -718.855 q^{11} -871.743 q^{13} -2090.64 q^{17} -476.947 q^{19} -1039.18 q^{21} -1330.77 q^{23} +4075.71 q^{27} +730.313 q^{29} -9788.53 q^{31} +8330.12 q^{33} +6219.08 q^{37} +10101.8 q^{39} +867.570 q^{41} +12946.8 q^{43} -26161.9 q^{47} -8765.06 q^{49} +24226.4 q^{51} -11148.4 q^{53} +5526.87 q^{57} +21407.2 q^{59} +32686.2 q^{61} -9749.45 q^{63} -37209.3 q^{67} +15420.9 q^{69} -35282.1 q^{71} -20909.5 q^{73} -64464.7 q^{77} -21560.4 q^{79} -20811.1 q^{81} -87640.0 q^{83} -8462.89 q^{87} +111553. q^{89} -78175.2 q^{91} +113430. q^{93} -18353.2 q^{97} +78152.2 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 228 q^{9} + 128 q^{13} + 814 q^{17} + 884 q^{21} + 4152 q^{29} - 5058 q^{33} - 348 q^{37} + 10334 q^{41} + 1454 q^{49} + 33924 q^{53} - 38450 q^{57} + 25796 q^{61} + 5796 q^{69} + 59218 q^{73} - 49052 q^{77}+ \cdots - 250052 q^{97}+O(q^{100}) $ 6 * q + 228 * q^9 + 128 * q^13 + 814 * q^17 + 884 * q^21 + 4152 * q^29 - 5058 * q^33 - 348 * q^37 + 10334 * q^41 + 1454 * q^49 + 33924 * q^53 - 38450 * q^57 + 25796 * q^61 + 5796 * q^69 + 59218 * q^73 - 49052 * q^77 + 146998 * q^81 + 83450 * q^89 + 153308 * q^93 - 250052 * 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
$3$	−11.5880	−0.743372	−0.371686	−	0.928358i	$-0.621220\pi$
$3$	−11.5880	−0.743372	−0.371686	+	0.928358i	$0.621220\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	89.6769	0.691728	0.345864	−	0.938285i	$-0.387586\pi$
$7$	89.6769	0.691728	0.345864	+	0.938285i	$0.387586\pi$
$8$	0	0
$9$	−108.718	−0.447397
$10$	0	0
$11$	−718.855	−1.79126	−0.895632	−	0.444795i	$-0.853276\pi$
$11$	−718.855	−1.79126	−0.895632	+	0.444795i	$0.853276\pi$
$12$	0	0
$13$	−871.743	−1.43064	−0.715320	−	0.698797i	$-0.753719\pi$
$13$	−871.743	−1.43064	−0.715320	+	0.698797i	$0.753719\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2090.64	−1.75451	−0.877257	−	0.480021i	$-0.840629\pi$
$17$	−2090.64	−1.75451	−0.877257	+	0.480021i	$0.840629\pi$
$18$	0	0
$19$	−476.947	−0.303100	−0.151550	−	0.988450i	$-0.548426\pi$
$19$	−476.947	−0.303100	−0.151550	+	0.988450i	$0.548426\pi$
$20$	0	0
$21$	−1039.18	−0.514212
$22$	0	0
$23$	−1330.77	−0.524544	−0.262272	−	0.964994i	$-0.584472\pi$
$23$	−1330.77	−0.524544	−0.262272	+	0.964994i	$0.584472\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4075.71	1.07596
$28$	0	0
$29$	730.313	0.161255	0.0806277	−	0.996744i	$-0.474308\pi$
$29$	730.313	0.161255	0.0806277	+	0.996744i	$0.474308\pi$
$30$	0	0
$31$	−9788.53	−1.82942	−0.914710	−	0.404111i	$-0.867581\pi$
$31$	−9788.53	−1.82942	−0.914710	+	0.404111i	$0.867581\pi$
$32$	0	0
$33$	8330.12	1.33158
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6219.08	0.746830	0.373415	−	0.927664i	$-0.378187\pi$
$37$	6219.08	0.746830	0.373415	+	0.927664i	$0.378187\pi$
$38$	0	0
$39$	10101.8	1.06350
$40$	0	0
$41$	867.570	0.0806018	0.0403009	−	0.999188i	$-0.487168\pi$
$41$	867.570	0.0806018	0.0403009	+	0.999188i	$0.487168\pi$
$42$	0	0
$43$	12946.8	1.06780	0.533900	−	0.845548i	$-0.320726\pi$
$43$	12946.8	1.06780	0.533900	+	0.845548i	$0.320726\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−26161.9	−1.72753	−0.863764	−	0.503897i	$-0.831899\pi$
$47$	−26161.9	−1.72753	−0.863764	+	0.503897i	$0.831899\pi$
$48$	0	0
$49$	−8765.06	−0.521512
$50$	0	0
$51$	24226.4	1.30426
$52$	0	0
$53$	−11148.4	−0.545160	−0.272580	−	0.962133i	$-0.587877\pi$
$53$	−11148.4	−0.545160	−0.272580	+	0.962133i	$0.587877\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5526.87	0.225316
$58$	0	0
$59$	21407.2	0.800628	0.400314	−	0.916378i	$-0.368901\pi$
$59$	21407.2	0.800628	0.400314	+	0.916378i	$0.368901\pi$
$60$	0	0
$61$	32686.2	1.12471	0.562354	−	0.826896i	$-0.309896\pi$
$61$	32686.2	1.12471	0.562354	+	0.826896i	$0.309896\pi$
$62$	0	0
$63$	−9749.45	−0.309477
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−37209.3	−1.01266	−0.506331	−	0.862339i	$-0.668998\pi$
$67$	−37209.3	−1.01266	−0.506331	+	0.862339i	$0.668998\pi$
$68$	0	0
$69$	15420.9	0.389932
$70$	0	0
$71$	−35282.1	−0.830633	−0.415316	−	0.909677i	$-0.636329\pi$
$71$	−35282.1	−0.830633	−0.415316	+	0.909677i	$0.636329\pi$
$72$	0	0
$73$	−20909.5	−0.459238	−0.229619	−	0.973281i	$-0.573748\pi$
$73$	−20909.5	−0.459238	−0.229619	+	0.973281i	$0.573748\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−64464.7	−1.23907
$78$	0	0
$79$	−21560.4	−0.388678	−0.194339	−	0.980934i	$-0.562256\pi$
$79$	−21560.4	−0.388678	−0.194339	+	0.980934i	$0.562256\pi$
$80$	0	0
$81$	−20811.1	−0.352438
$82$	0	0
$83$	−87640.0	−1.39639	−0.698195	−	0.715907i	$-0.746013\pi$
$83$	−87640.0	−1.39639	−0.698195	+	0.715907i	$0.746013\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−8462.89	−0.119873
$88$	0	0
$89$	111553.	1.49282	0.746410	−	0.665486i	$-0.231776\pi$
$89$	111553.	1.49282	0.746410	+	0.665486i	$0.231776\pi$
$90$	0	0
$91$	−78175.2	−0.989613
$92$	0	0
$93$	113430.	1.35994
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−18353.2	−0.198053	−0.0990266	−	0.995085i	$-0.531573\pi$
$97$	−18353.2	−0.198053	−0.0990266	+	0.995085i	$0.531573\pi$
$98$	0	0
$99$	78152.2	0.801407

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	800.6.a.y.1.2	yes	6
4.3	odd	2	inner	800.6.a.y.1.5	yes	6
5.2	odd	4		800.6.c.p.449.9		12
5.3	odd	4		800.6.c.p.449.3		12
5.4	even	2		800.6.a.x.1.5	yes	6
20.3	even	4		800.6.c.p.449.10		12
20.7	even	4		800.6.c.p.449.4		12
20.19	odd	2		800.6.a.x.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.6.a.x.1.2	✓	6	20.19	odd	2
800.6.a.x.1.5	yes	6	5.4	even	2
800.6.a.y.1.2	yes	6	1.1	even	1	trivial
800.6.a.y.1.5	yes	6	4.3	odd	2	inner
800.6.c.p.449.3		12	5.3	odd	4
800.6.c.p.449.4		12	20.7	even	4
800.6.c.p.449.9		12	5.2	odd	4
800.6.c.p.449.10		12	20.3	even	4

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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	800.a (trivial)

\(f(q)\)	\(=\)	\(q-11.5880 q^{3} +89.6769 q^{7} -108.718 q^{9} -718.855 q^{11} -871.743 q^{13} -2090.64 q^{17} -476.947 q^{19} -1039.18 q^{21} -1330.77 q^{23} +4075.71 q^{27} +730.313 q^{29} -9788.53 q^{31} +8330.12 q^{33} +6219.08 q^{37} +10101.8 q^{39} +867.570 q^{41} +12946.8 q^{43} -26161.9 q^{47} -8765.06 q^{49} +24226.4 q^{51} -11148.4 q^{53} +5526.87 q^{57} +21407.2 q^{59} +32686.2 q^{61} -9749.45 q^{63} -37209.3 q^{67} +15420.9 q^{69} -35282.1 q^{71} -20909.5 q^{73} -64464.7 q^{77} -21560.4 q^{79} -20811.1 q^{81} -87640.0 q^{83} -8462.89 q^{87} +111553. q^{89} -78175.2 q^{91} +113430. q^{93} -18353.2 q^{97} +78152.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 228 q^{9} + 128 q^{13} + 814 q^{17} + 884 q^{21} + 4152 q^{29} - 5058 q^{33} - 348 q^{37} + 10334 q^{41} + 1454 q^{49} + 33924 q^{53} - 38450 q^{57} + 25796 q^{61} + 5796 q^{69} + 59218 q^{73} - 49052 q^{77}+ \cdots - 250052 q^{97}+O(q^{100}) \) 6 * q + 228 * q^9 + 128 * q^13 + 814 * q^17 + 884 * q^21 + 4152 * q^29 - 5058 * q^33 - 348 * q^37 + 10334 * q^41 + 1454 * q^49 + 33924 * q^53 - 38450 * q^57 + 25796 * q^61 + 5796 * q^69 + 59218 * q^73 - 49052 * q^77 + 146998 * q^81 + 83450 * q^89 + 153308 * q^93 - 250052 * q^97

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