LMFDB - Embedded newform 4600.2.a.bc.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4600,2,Mod(1,4600)] 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("4600.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-3,0,0,0,1,0,4,0,-4] 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:	$36.7311849298$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.791953.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 7x^{2} + 9x + 2 $ x^5 - 2x^4 - 7x^3 + 7x^2 + 9x + 2
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			1.5
Root			$3.21042$ of defining polynomial
Character	$\chi$	$=$	4600.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.21042 q^{3} -2.22487 q^{7} +1.88594 q^{9} -1.57300 q^{11} +3.96189 q^{13} +0.294907 q^{17} -7.76572 q^{19} -4.91788 q^{21} -1.00000 q^{23} -2.46253 q^{27} -9.29233 q^{29} +9.18913 q^{31} -3.47698 q^{33} -10.5425 q^{37} +8.75744 q^{39} -2.34251 q^{41} +6.67460 q^{43} -1.38007 q^{47} -2.04998 q^{49} +0.651868 q^{51} -11.0395 q^{53} -17.1655 q^{57} -5.09378 q^{59} -8.91788 q^{61} -4.19597 q^{63} -1.12002 q^{67} -2.21042 q^{69} +7.60168 q^{71} +12.8549 q^{73} +3.49971 q^{77} +11.0211 q^{79} -11.1010 q^{81} -13.5257 q^{83} -20.5399 q^{87} +14.3475 q^{89} -8.81468 q^{91} +20.3118 q^{93} -0.199218 q^{97} -2.96658 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{3} + q^{7} + 4 q^{9} - 4 q^{11} + q^{13} - 5 q^{17} + 4 q^{19} - 6 q^{21} - 5 q^{23} - 6 q^{27} - 11 q^{29} + 4 q^{31} - 13 q^{33} - 6 q^{37} + 31 q^{39} - 8 q^{41} - 3 q^{43} - 2 q^{47} - 2 q^{49}+ \cdots + 37 q^{99}+O(q^{100}) $ 5 * q - 3 * q^3 + q^7 + 4 * q^9 - 4 * q^11 + q^13 - 5 * q^17 + 4 * q^19 - 6 * q^21 - 5 * q^23 - 6 * q^27 - 11 * q^29 + 4 * q^31 - 13 * q^33 - 6 * q^37 + 31 * q^39 - 8 * q^41 - 3 * q^43 - 2 * q^47 - 2 * q^49 - 5 * q^51 - 18 * q^53 - 27 * q^57 + 23 * q^59 - 26 * q^61 - 5 * q^63 - 3 * q^67 + 3 * q^69 - 2 * q^71 - 4 * q^73 - 15 * q^77 + 43 * q^79 - 3 * q^81 - 30 * q^83 - 27 * q^87 + 15 * q^89 - 19 * q^91 + 15 * q^93 - 8 * q^97 + 37 * 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
$3$	2.21042	1.27618	0.638092	−	0.769960i	$-0.279724\pi$
$3$	2.21042	1.27618	0.638092	+	0.769960i	$0.279724\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.22487	−0.840920	−0.420460	−	0.907311i	$-0.638131\pi$
$7$	−2.22487	−0.840920	−0.420460	+	0.907311i	$0.638131\pi$
$8$	0	0
$9$	1.88594	0.628648
$10$	0	0
$11$	−1.57300	−0.474276	−0.237138	−	0.971476i	$-0.576209\pi$
$11$	−1.57300	−0.474276	−0.237138	+	0.971476i	$0.576209\pi$
$12$	0	0
$13$	3.96189	1.09883	0.549416	−	0.835549i	$-0.314850\pi$
$13$	3.96189	1.09883	0.549416	+	0.835549i	$0.314850\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.294907	0.0715256	0.0357628	−	0.999360i	$-0.488614\pi$
$17$	0.294907	0.0715256	0.0357628	+	0.999360i	$0.488614\pi$
$18$	0	0
$19$	−7.76572	−1.78158	−0.890789	−	0.454418i	$-0.849847\pi$
$19$	−7.76572	−1.78158	−0.890789	+	0.454418i	$0.849847\pi$
$20$	0	0
$21$	−4.91788	−1.07317
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.46253	−0.473914
$28$	0	0
$29$	−9.29233	−1.72554	−0.862771	−	0.505595i	$-0.831273\pi$
$29$	−9.29233	−1.72554	−0.862771	+	0.505595i	$0.831273\pi$
$30$	0	0
$31$	9.18913	1.65042	0.825208	−	0.564829i	$-0.191058\pi$
$31$	9.18913	1.65042	0.825208	+	0.564829i	$0.191058\pi$
$32$	0	0
$33$	−3.47698	−0.605264
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.5425	−1.73318	−0.866588	−	0.499024i	$-0.833692\pi$
$37$	−10.5425	−1.73318	−0.866588	+	0.499024i	$0.833692\pi$
$38$	0	0
$39$	8.75744	1.40231
$40$	0	0
$41$	−2.34251	−0.365839	−0.182920	−	0.983128i	$-0.558555\pi$
$41$	−2.34251	−0.365839	−0.182920	+	0.983128i	$0.558555\pi$
$42$	0	0
$43$	6.67460	1.01787	0.508933	−	0.860806i	$-0.330040\pi$
$43$	6.67460	1.01787	0.508933	+	0.860806i	$0.330040\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.38007	−0.201304	−0.100652	−	0.994922i	$-0.532093\pi$
$47$	−1.38007	−0.201304	−0.100652	+	0.994922i	$0.532093\pi$
$48$	0	0
$49$	−2.04998	−0.292854
$50$	0	0
$51$	0.651868	0.0912798
$52$	0	0
$53$	−11.0395	−1.51640	−0.758199	−	0.652023i	$-0.773920\pi$
$53$	−11.0395	−1.51640	−0.758199	+	0.652023i	$0.773920\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−17.1655	−2.27362
$58$	0	0
$59$	−5.09378	−0.663154	−0.331577	−	0.943428i	$-0.607581\pi$
$59$	−5.09378	−0.663154	−0.331577	+	0.943428i	$0.607581\pi$
$60$	0	0
$61$	−8.91788	−1.14182	−0.570909	−	0.821014i	$-0.693409\pi$
$61$	−8.91788	−1.14182	−0.570909	+	0.821014i	$0.693409\pi$
$62$	0	0
$63$	−4.19597	−0.528642
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.12002	−0.136832	−0.0684160	−	0.997657i	$-0.521795\pi$
$67$	−1.12002	−0.136832	−0.0684160	+	0.997657i	$0.521795\pi$
$68$	0	0
$69$	−2.21042	−0.266103
$70$	0	0
$71$	7.60168	0.902154	0.451077	−	0.892485i	$-0.351040\pi$
$71$	7.60168	0.902154	0.451077	+	0.892485i	$0.351040\pi$
$72$	0	0
$73$	12.8549	1.50455	0.752276	−	0.658848i	$-0.228956\pi$
$73$	12.8549	1.50455	0.752276	+	0.658848i	$0.228956\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.49971	0.398828
$78$	0	0
$79$	11.0211	1.23997	0.619984	−	0.784614i	$-0.287139\pi$
$79$	11.0211	1.23997	0.619984	+	0.784614i	$0.287139\pi$
$80$	0	0
$81$	−11.1010	−1.23345
$82$	0	0
$83$	−13.5257	−1.48464	−0.742318	−	0.670048i	$-0.766274\pi$
$83$	−13.5257	−1.48464	−0.742318	+	0.670048i	$0.766274\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−20.5399	−2.20211
$88$	0	0
$89$	14.3475	1.52084	0.760418	−	0.649434i	$-0.224994\pi$
$89$	14.3475	1.52084	0.760418	+	0.649434i	$0.224994\pi$
$90$	0	0
$91$	−8.81468	−0.924030
$92$	0	0
$93$	20.3118	2.10624
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.199218	−0.0202275	−0.0101137	−	0.999949i	$-0.503219\pi$
$97$	−0.199218	−0.0202275	−0.0101137	+	0.999949i	$0.503219\pi$
$98$	0	0
$99$	−2.96658	−0.298153

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	4600.2.a.bc.1.5	✓	5
4.3	odd	2		9200.2.a.cw.1.1		5
5.2	odd	4		4600.2.e.v.4049.2		10
5.3	odd	4		4600.2.e.v.4049.9		10
5.4	even	2		4600.2.a.bg.1.1	yes	5
20.19	odd	2		9200.2.a.cs.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.bc.1.5	✓	5	1.1	even	1	trivial
4600.2.a.bg.1.1	yes	5	5.4	even	2
4600.2.e.v.4049.2		10	5.2	odd	4
4600.2.e.v.4049.9		10	5.3	odd	4
9200.2.a.cs.1.5		5	20.19	odd	2
9200.2.a.cw.1.1		5	4.3	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.a (trivial)

\(f(q)\)	\(=\)	\(q+2.21042 q^{3} -2.22487 q^{7} +1.88594 q^{9} -1.57300 q^{11} +3.96189 q^{13} +0.294907 q^{17} -7.76572 q^{19} -4.91788 q^{21} -1.00000 q^{23} -2.46253 q^{27} -9.29233 q^{29} +9.18913 q^{31} -3.47698 q^{33} -10.5425 q^{37} +8.75744 q^{39} -2.34251 q^{41} +6.67460 q^{43} -1.38007 q^{47} -2.04998 q^{49} +0.651868 q^{51} -11.0395 q^{53} -17.1655 q^{57} -5.09378 q^{59} -8.91788 q^{61} -4.19597 q^{63} -1.12002 q^{67} -2.21042 q^{69} +7.60168 q^{71} +12.8549 q^{73} +3.49971 q^{77} +11.0211 q^{79} -11.1010 q^{81} -13.5257 q^{83} -20.5399 q^{87} +14.3475 q^{89} -8.81468 q^{91} +20.3118 q^{93} -0.199218 q^{97} -2.96658 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{3} + q^{7} + 4 q^{9} - 4 q^{11} + q^{13} - 5 q^{17} + 4 q^{19} - 6 q^{21} - 5 q^{23} - 6 q^{27} - 11 q^{29} + 4 q^{31} - 13 q^{33} - 6 q^{37} + 31 q^{39} - 8 q^{41} - 3 q^{43} - 2 q^{47} - 2 q^{49}+ \cdots + 37 q^{99}+O(q^{100}) \) 5 * q - 3 * q^3 + q^7 + 4 * q^9 - 4 * q^11 + q^13 - 5 * q^17 + 4 * q^19 - 6 * q^21 - 5 * q^23 - 6 * q^27 - 11 * q^29 + 4 * q^31 - 13 * q^33 - 6 * q^37 + 31 * q^39 - 8 * q^41 - 3 * q^43 - 2 * q^47 - 2 * q^49 - 5 * q^51 - 18 * q^53 - 27 * q^57 + 23 * q^59 - 26 * q^61 - 5 * q^63 - 3 * q^67 + 3 * q^69 - 2 * q^71 - 4 * q^73 - 15 * q^77 + 43 * q^79 - 3 * q^81 - 30 * q^83 - 27 * q^87 + 15 * q^89 - 19 * q^91 + 15 * q^93 - 8 * q^97 + 37 * q^99

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