LMFDB - Embedded newform 4600.2.a.bc.1.4

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			1.4
Root			$1.72457$ 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)

$f(q)$	$=$	$q+0.724570 q^{3} +2.33840 q^{7} -2.47500 q^{9} -2.62783 q^{11} +4.29631 q^{13} -6.85404 q^{17} +2.87303 q^{19} +1.69433 q^{21} -1.00000 q^{23} -3.96702 q^{27} -5.03711 q^{29} -7.31504 q^{31} -1.90405 q^{33} +9.24142 q^{37} +3.11297 q^{39} -6.95078 q^{41} -7.01520 q^{43} -4.74690 q^{47} -1.53188 q^{49} -4.96623 q^{51} +12.3849 q^{53} +2.08171 q^{57} -2.14073 q^{59} -2.30567 q^{61} -5.78754 q^{63} +1.98376 q^{67} -0.724570 q^{69} -6.87990 q^{71} -13.2518 q^{73} -6.14492 q^{77} +16.6578 q^{79} +4.55062 q^{81} -9.09450 q^{83} -3.64974 q^{87} +0.676383 q^{89} +10.0465 q^{91} -5.30026 q^{93} -15.0759 q^{97} +6.50388 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.724570	0.418331	0.209165	−	0.977880i	$-0.432925\pi$
$3$	0.724570	0.418331	0.209165	+	0.977880i	$0.432925\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.33840	0.883833	0.441916	−	0.897056i	$-0.354299\pi$
$7$	2.33840	0.883833	0.441916	+	0.897056i	$0.354299\pi$
$8$	0	0
$9$	−2.47500	−0.825000
$10$	0	0
$11$	−2.62783	−0.792321	−0.396160	−	0.918181i	$-0.629658\pi$
$11$	−2.62783	−0.792321	−0.396160	+	0.918181i	$0.629658\pi$
$12$	0	0
$13$	4.29631	1.19158	0.595791	−	0.803140i	$-0.296839\pi$
$13$	4.29631	1.19158	0.595791	+	0.803140i	$0.296839\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.85404	−1.66235	−0.831175	−	0.556011i	$-0.812331\pi$
$17$	−6.85404	−1.66235	−0.831175	+	0.556011i	$0.812331\pi$
$18$	0	0
$19$	2.87303	0.659118	0.329559	−	0.944135i	$-0.393100\pi$
$19$	2.87303	0.659118	0.329559	+	0.944135i	$0.393100\pi$
$20$	0	0
$21$	1.69433	0.369734
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3.96702	−0.763453
$28$	0	0
$29$	−5.03711	−0.935368	−0.467684	−	0.883896i	$-0.654912\pi$
$29$	−5.03711	−0.935368	−0.467684	+	0.883896i	$0.654912\pi$
$30$	0	0
$31$	−7.31504	−1.31382	−0.656910	−	0.753969i	$-0.728137\pi$
$31$	−7.31504	−1.31382	−0.656910	+	0.753969i	$0.728137\pi$
$32$	0	0
$33$	−1.90405	−0.331452
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.24142	1.51928	0.759640	−	0.650344i	$-0.225375\pi$
$37$	9.24142	1.51928	0.759640	+	0.650344i	$0.225375\pi$
$38$	0	0
$39$	3.11297	0.498475
$40$	0	0
$41$	−6.95078	−1.08553	−0.542765	−	0.839885i	$-0.682623\pi$
$41$	−6.95078	−1.08553	−0.542765	+	0.839885i	$0.682623\pi$
$42$	0	0
$43$	−7.01520	−1.06981	−0.534904	−	0.844913i	$-0.679652\pi$
$43$	−7.01520	−1.06981	−0.534904	+	0.844913i	$0.679652\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.74690	−0.692406	−0.346203	−	0.938160i	$-0.612529\pi$
$47$	−4.74690	−0.692406	−0.346203	+	0.938160i	$0.612529\pi$
$48$	0	0
$49$	−1.53188	−0.218840
$50$	0	0
$51$	−4.96623	−0.695412
$52$	0	0
$53$	12.3849	1.70120	0.850598	−	0.525817i	$-0.176240\pi$
$53$	12.3849	1.70120	0.850598	+	0.525817i	$0.176240\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.08171	0.275729
$58$	0	0
$59$	−2.14073	−0.278699	−0.139349	−	0.990243i	$-0.544501\pi$
$59$	−2.14073	−0.278699	−0.139349	+	0.990243i	$0.544501\pi$
$60$	0	0
$61$	−2.30567	−0.295210	−0.147605	−	0.989046i	$-0.547156\pi$
$61$	−2.30567	−0.295210	−0.147605	+	0.989046i	$0.547156\pi$
$62$	0	0
$63$	−5.78754	−0.729162
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.98376	0.242355	0.121178	−	0.992631i	$-0.461333\pi$
$67$	1.98376	0.242355	0.121178	+	0.992631i	$0.461333\pi$
$68$	0	0
$69$	−0.724570	−0.0872279
$70$	0	0
$71$	−6.87990	−0.816494	−0.408247	−	0.912871i	$-0.633860\pi$
$71$	−6.87990	−0.816494	−0.408247	+	0.912871i	$0.633860\pi$
$72$	0	0
$73$	−13.2518	−1.55101	−0.775505	−	0.631342i	$-0.782504\pi$
$73$	−13.2518	−1.55101	−0.775505	+	0.631342i	$0.782504\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.14492	−0.700279
$78$	0	0
$79$	16.6578	1.87415	0.937076	−	0.349126i	$-0.113522\pi$
$79$	16.6578	1.87415	0.937076	+	0.349126i	$0.113522\pi$
$80$	0	0
$81$	4.55062	0.505624
$82$	0	0
$83$	−9.09450	−0.998251	−0.499126	−	0.866530i	$-0.666345\pi$
$83$	−9.09450	−0.998251	−0.499126	+	0.866530i	$0.666345\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.64974	−0.391293
$88$	0	0
$89$	0.676383	0.0716965	0.0358482	−	0.999357i	$-0.488587\pi$
$89$	0.676383	0.0716965	0.0358482	+	0.999357i	$0.488587\pi$
$90$	0	0
$91$	10.0465	1.05316
$92$	0	0
$93$	−5.30026	−0.549611
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−15.0759	−1.53073	−0.765365	−	0.643597i	$-0.777441\pi$
$97$	−15.0759	−1.53073	−0.765365	+	0.643597i	$0.777441\pi$
$98$	0	0
$99$	6.50388	0.653664

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.1.4	✓	5
4.3	odd	2		9200.2.a.cw.1.2		5
5.2	odd	4		4600.2.e.v.4049.5		10
5.3	odd	4		4600.2.e.v.4049.6		10
5.4	even	2		4600.2.a.bg.1.2	yes	5
20.19	odd	2		9200.2.a.cs.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.bc.1.4	✓	5	1.1	even	1	trivial
4600.2.a.bg.1.2	yes	5	5.4	even	2
4600.2.e.v.4049.5		10	5.2	odd	4
4600.2.e.v.4049.6		10	5.3	odd	4
9200.2.a.cs.1.4		5	20.19	odd	2
9200.2.a.cw.1.2		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)

\(f(q)\)	\(=\)	\(q+0.724570 q^{3} +2.33840 q^{7} -2.47500 q^{9} -2.62783 q^{11} +4.29631 q^{13} -6.85404 q^{17} +2.87303 q^{19} +1.69433 q^{21} -1.00000 q^{23} -3.96702 q^{27} -5.03711 q^{29} -7.31504 q^{31} -1.90405 q^{33} +9.24142 q^{37} +3.11297 q^{39} -6.95078 q^{41} -7.01520 q^{43} -4.74690 q^{47} -1.53188 q^{49} -4.96623 q^{51} +12.3849 q^{53} +2.08171 q^{57} -2.14073 q^{59} -2.30567 q^{61} -5.78754 q^{63} +1.98376 q^{67} -0.724570 q^{69} -6.87990 q^{71} -13.2518 q^{73} -6.14492 q^{77} +16.6578 q^{79} +4.55062 q^{81} -9.09450 q^{83} -3.64974 q^{87} +0.676383 q^{89} +10.0465 q^{91} -5.30026 q^{93} -15.0759 q^{97} +6.50388 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{3} + q^{7} + 4 q^{9} - 4 q^{11} + q^{13} - 5 q^{17} + 4 q^{19} - 6 q^{21} - 5 q^{23} - 6 q^{27} - 11 q^{29} + 4 q^{31} - 13 q^{33} - 6 q^{37} + 31 q^{39} - 8 q^{41} - 3 q^{43} - 2 q^{47} - 2 q^{49}+ \cdots + 37 q^{99}+O(q^{100}) \) 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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