LMFDB - Embedded newform 5000.2.a.n.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5000,2,Mod(1,5000)] 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("5000.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5000, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5000 = 2^{3} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5000.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,3,0,0,0,2,0,-3,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.9252010106$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.3266578125.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 6x^{6} + 23x^{5} + x^{4} - 45x^{3} + 25x^{2} + 10x - 5 $ x^8 - 3x^7 - 6x^6 + 23x^5 + x^4 - 45x^3 + 25x^2 + 10x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.369714$ of defining polynomial
Character	$\chi$	$=$	5000.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.58658 q^{3} -4.12771 q^{7} +3.69040 q^{9} +3.87939 q^{11} -4.02828 q^{13} +1.06205 q^{17} -5.18213 q^{19} -10.6767 q^{21} +4.11821 q^{23} +1.78578 q^{27} -6.26968 q^{29} -9.36997 q^{31} +10.0344 q^{33} +3.29275 q^{37} -10.4195 q^{39} -3.62436 q^{41} +12.3182 q^{43} -9.91205 q^{47} +10.0380 q^{49} +2.74707 q^{51} -2.88267 q^{53} -13.4040 q^{57} -11.2511 q^{59} -11.5708 q^{61} -15.2329 q^{63} -7.30917 q^{67} +10.6521 q^{69} -6.65838 q^{71} +5.30464 q^{73} -16.0130 q^{77} +7.48354 q^{79} -6.45214 q^{81} +12.2400 q^{83} -16.2170 q^{87} -0.107710 q^{89} +16.6276 q^{91} -24.2362 q^{93} -8.31290 q^{97} +14.3165 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{3} + 2 q^{7} - 3 q^{9} + 5 q^{11} - 2 q^{13} - 7 q^{17} - 10 q^{19} - 14 q^{21} - 2 q^{23} - 6 q^{27} - 10 q^{29} - 27 q^{31} + 3 q^{33} - 10 q^{37} - 18 q^{39} - 16 q^{41} + 4 q^{43} - 8 q^{47}+ \cdots + 6 q^{99}+O(q^{100}) $ 8 * q + 3 * q^3 + 2 * q^7 - 3 * q^9 + 5 * q^11 - 2 * q^13 - 7 * q^17 - 10 * q^19 - 14 * q^21 - 2 * q^23 - 6 * q^27 - 10 * q^29 - 27 * q^31 + 3 * q^33 - 10 * q^37 - 18 * q^39 - 16 * q^41 + 4 * q^43 - 8 * q^47 + 4 * q^49 - 10 * q^51 - 2 * q^53 - 2 * q^57 - 39 * q^59 - 18 * q^61 - 14 * q^63 - 12 * q^67 + 19 * q^69 - 13 * q^71 - 12 * q^73 - 41 * q^77 - 16 * q^79 - 28 * q^81 + 64 * q^83 - 4 * q^87 - 25 * q^89 - 26 * q^91 - 40 * q^93 + 8 * q^97 + 6 * q^99

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
$3$	2.58658	1.49336	0.746682	−	0.665182i	$-0.231646\pi$
$3$	2.58658	1.49336	0.746682	+	0.665182i	$0.231646\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.12771	−1.56013	−0.780064	−	0.625700i	$-0.784813\pi$
$7$	−4.12771	−1.56013	−0.780064	+	0.625700i	$0.784813\pi$
$8$	0	0
$9$	3.69040	1.23013
$10$	0	0
$11$	3.87939	1.16968	0.584840	−	0.811149i	$-0.301157\pi$
$11$	3.87939	1.16968	0.584840	+	0.811149i	$0.301157\pi$
$12$	0	0
$13$	−4.02828	−1.11724	−0.558622	−	0.829423i	$-0.688670\pi$
$13$	−4.02828	−1.11724	−0.558622	+	0.829423i	$0.688670\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.06205	0.257584	0.128792	−	0.991672i	$-0.458890\pi$
$17$	1.06205	0.257584	0.128792	+	0.991672i	$0.458890\pi$
$18$	0	0
$19$	−5.18213	−1.18886	−0.594432	−	0.804146i	$-0.702623\pi$
$19$	−5.18213	−1.18886	−0.594432	+	0.804146i	$0.702623\pi$
$20$	0	0
$21$	−10.6767	−2.32984
$22$	0	0
$23$	4.11821	0.858706	0.429353	−	0.903137i	$-0.358742\pi$
$23$	4.11821	0.858706	0.429353	+	0.903137i	$0.358742\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.78578	0.343674
$28$	0	0
$29$	−6.26968	−1.16425	−0.582125	−	0.813099i	$-0.697779\pi$
$29$	−6.26968	−1.16425	−0.582125	+	0.813099i	$0.697779\pi$
$30$	0	0
$31$	−9.36997	−1.68290	−0.841448	−	0.540338i	$-0.818296\pi$
$31$	−9.36997	−1.68290	−0.841448	+	0.540338i	$0.818296\pi$
$32$	0	0
$33$	10.0344	1.74676
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.29275	0.541325	0.270662	−	0.962674i	$-0.412757\pi$
$37$	3.29275	0.541325	0.270662	+	0.962674i	$0.412757\pi$
$38$	0	0
$39$	−10.4195	−1.66845
$40$	0	0
$41$	−3.62436	−0.566030	−0.283015	−	0.959115i	$-0.591335\pi$
$41$	−3.62436	−0.566030	−0.283015	+	0.959115i	$0.591335\pi$
$42$	0	0
$43$	12.3182	1.87850	0.939252	−	0.343228i	$-0.111520\pi$
$43$	12.3182	1.87850	0.939252	+	0.343228i	$0.111520\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.91205	−1.44582	−0.722910	−	0.690942i	$-0.757196\pi$
$47$	−9.91205	−1.44582	−0.722910	+	0.690942i	$0.757196\pi$
$48$	0	0
$49$	10.0380	1.43400
$50$	0	0
$51$	2.74707	0.384667
$52$	0	0
$53$	−2.88267	−0.395965	−0.197983	−	0.980206i	$-0.563439\pi$
$53$	−2.88267	−0.395965	−0.197983	+	0.980206i	$0.563439\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−13.4040	−1.77540
$58$	0	0
$59$	−11.2511	−1.46477	−0.732383	−	0.680893i	$-0.761592\pi$
$59$	−11.2511	−1.46477	−0.732383	+	0.680893i	$0.761592\pi$
$60$	0	0
$61$	−11.5708	−1.48149	−0.740744	−	0.671787i	$-0.765527\pi$
$61$	−11.5708	−1.48149	−0.740744	+	0.671787i	$0.765527\pi$
$62$	0	0
$63$	−15.2329	−1.91917
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.30917	−0.892958	−0.446479	−	0.894794i	$-0.647322\pi$
$67$	−7.30917	−0.892958	−0.446479	+	0.894794i	$0.647322\pi$
$68$	0	0
$69$	10.6521	1.28236
$70$	0	0
$71$	−6.65838	−0.790205	−0.395102	−	0.918637i	$-0.629291\pi$
$71$	−6.65838	−0.790205	−0.395102	+	0.918637i	$0.629291\pi$
$72$	0	0
$73$	5.30464	0.620861	0.310431	−	0.950596i	$-0.399527\pi$
$73$	5.30464	0.620861	0.310431	+	0.950596i	$0.399527\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−16.0130	−1.82485
$78$	0	0
$79$	7.48354	0.841964	0.420982	−	0.907069i	$-0.361685\pi$
$79$	7.48354	0.841964	0.420982	+	0.907069i	$0.361685\pi$
$80$	0	0
$81$	−6.45214	−0.716904
$82$	0	0
$83$	12.2400	1.34351	0.671755	−	0.740773i	$-0.265541\pi$
$83$	12.2400	1.34351	0.671755	+	0.740773i	$0.265541\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−16.2170	−1.73865
$88$	0	0
$89$	−0.107710	−0.0114172	−0.00570862	−	0.999984i	$-0.501817\pi$
$89$	−0.107710	−0.0114172	−0.00570862	+	0.999984i	$0.501817\pi$
$90$	0	0
$91$	16.6276	1.74304
$92$	0	0
$93$	−24.2362	−2.51317
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.31290	−0.844047	−0.422024	−	0.906585i	$-0.638680\pi$
$97$	−8.31290	−0.844047	−0.422024	+	0.906585i	$0.638680\pi$
$98$	0	0
$99$	14.3165	1.43886

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	5000.2.a.n.1.8	yes	8
4.3	odd	2		10000.2.a.bg.1.1		8
5.4	even	2		5000.2.a.k.1.1	✓	8
20.19	odd	2		10000.2.a.bl.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5000.2.a.k.1.1	✓	8	5.4	even	2
5000.2.a.n.1.8	yes	8	1.1	even	1	trivial
10000.2.a.bg.1.1		8	4.3	odd	2
10000.2.a.bl.1.8		8	20.19	odd	2

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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:	\( N \)	\(=\)	\( 5000 = 2^{3} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5000.a (trivial)

\(f(q)\)	\(=\)	\(q+2.58658 q^{3} -4.12771 q^{7} +3.69040 q^{9} +3.87939 q^{11} -4.02828 q^{13} +1.06205 q^{17} -5.18213 q^{19} -10.6767 q^{21} +4.11821 q^{23} +1.78578 q^{27} -6.26968 q^{29} -9.36997 q^{31} +10.0344 q^{33} +3.29275 q^{37} -10.4195 q^{39} -3.62436 q^{41} +12.3182 q^{43} -9.91205 q^{47} +10.0380 q^{49} +2.74707 q^{51} -2.88267 q^{53} -13.4040 q^{57} -11.2511 q^{59} -11.5708 q^{61} -15.2329 q^{63} -7.30917 q^{67} +10.6521 q^{69} -6.65838 q^{71} +5.30464 q^{73} -16.0130 q^{77} +7.48354 q^{79} -6.45214 q^{81} +12.2400 q^{83} -16.2170 q^{87} -0.107710 q^{89} +16.6276 q^{91} -24.2362 q^{93} -8.31290 q^{97} +14.3165 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{3} + 2 q^{7} - 3 q^{9} + 5 q^{11} - 2 q^{13} - 7 q^{17} - 10 q^{19} - 14 q^{21} - 2 q^{23} - 6 q^{27} - 10 q^{29} - 27 q^{31} + 3 q^{33} - 10 q^{37} - 18 q^{39} - 16 q^{41} + 4 q^{43} - 8 q^{47}+ \cdots + 6 q^{99}+O(q^{100}) \) 8 * q + 3 * q^3 + 2 * q^7 - 3 * q^9 + 5 * q^11 - 2 * q^13 - 7 * q^17 - 10 * q^19 - 14 * q^21 - 2 * q^23 - 6 * q^27 - 10 * q^29 - 27 * q^31 + 3 * q^33 - 10 * q^37 - 18 * q^39 - 16 * q^41 + 4 * q^43 - 8 * q^47 + 4 * q^49 - 10 * q^51 - 2 * q^53 - 2 * q^57 - 39 * q^59 - 18 * q^61 - 14 * q^63 - 12 * q^67 + 19 * q^69 - 13 * q^71 - 12 * q^73 - 41 * q^77 - 16 * q^79 - 28 * q^81 + 64 * q^83 - 4 * q^87 - 25 * q^89 - 26 * q^91 - 40 * q^93 + 8 * q^97 + 6 * q^99

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