LMFDB - Embedded newform 4600.2.a.bc.1.1

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			1.1
Root			$-2.08344$ 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)

$f(q)$	$=$	$q-3.08344 q^{3} +0.555022 q^{7} +6.50759 q^{9} +4.65190 q^{11} -5.02256 q^{13} -1.32867 q^{17} -0.196402 q^{19} -1.71138 q^{21} -1.00000 q^{23} -10.8154 q^{27} -0.812298 q^{29} -2.11145 q^{31} -14.3439 q^{33} +5.64564 q^{37} +15.4868 q^{39} -4.89714 q^{41} -1.66507 q^{43} +9.89310 q^{47} -6.69195 q^{49} +4.09688 q^{51} -2.23261 q^{53} +0.605593 q^{57} +2.43488 q^{59} -5.71138 q^{61} +3.61185 q^{63} -6.91830 q^{67} +3.08344 q^{69} +0.0120411 q^{71} -15.2989 q^{73} +2.58191 q^{77} +10.6351 q^{79} +13.8260 q^{81} +5.64020 q^{83} +2.50467 q^{87} +9.06693 q^{89} -2.78763 q^{91} +6.51051 q^{93} +14.0720 q^{97} +30.2727 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.08344	−1.78022	−0.890112	−	0.455742i	$-0.849374\pi$
$3$	−3.08344	−1.78022	−0.890112	+	0.455742i	$0.849374\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.555022	0.209779	0.104889	−	0.994484i	$-0.466551\pi$
$7$	0.555022	0.209779	0.104889	+	0.994484i	$0.466551\pi$
$8$	0	0
$9$	6.50759	2.16920
$10$	0	0
$11$	4.65190	1.40260	0.701301	−	0.712866i	$-0.252603\pi$
$11$	4.65190	1.40260	0.701301	+	0.712866i	$0.252603\pi$
$12$	0	0
$13$	−5.02256	−1.39301	−0.696504	−	0.717553i	$-0.745262\pi$
$13$	−5.02256	−1.39301	−0.696504	+	0.717553i	$0.745262\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.32867	−0.322251	−0.161125	−	0.986934i	$-0.551512\pi$
$17$	−1.32867	−0.322251	−0.161125	+	0.986934i	$0.551512\pi$
$18$	0	0
$19$	−0.196402	−0.0450577	−0.0225288	−	0.999746i	$-0.507172\pi$
$19$	−0.196402	−0.0450577	−0.0225288	+	0.999746i	$0.507172\pi$
$20$	0	0
$21$	−1.71138	−0.373453
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−10.8154	−2.08143
$28$	0	0
$29$	−0.812298	−0.150840	−0.0754200	−	0.997152i	$-0.524030\pi$
$29$	−0.812298	−0.150840	−0.0754200	+	0.997152i	$0.524030\pi$
$30$	0	0
$31$	−2.11145	−0.379227	−0.189613	−	0.981859i	$-0.560723\pi$
$31$	−2.11145	−0.379227	−0.189613	+	0.981859i	$0.560723\pi$
$32$	0	0
$33$	−14.3439	−2.49694
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.64564	0.928138	0.464069	−	0.885799i	$-0.346389\pi$
$37$	5.64564	0.928138	0.464069	+	0.885799i	$0.346389\pi$
$38$	0	0
$39$	15.4868	2.47987
$40$	0	0
$41$	−4.89714	−0.764804	−0.382402	−	0.923996i	$-0.624903\pi$
$41$	−4.89714	−0.764804	−0.382402	+	0.923996i	$0.624903\pi$
$42$	0	0
$43$	−1.66507	−0.253920	−0.126960	−	0.991908i	$-0.540522\pi$
$43$	−1.66507	−0.253920	−0.126960	+	0.991908i	$0.540522\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.89310	1.44306	0.721528	−	0.692385i	$-0.243440\pi$
$47$	9.89310	1.44306	0.721528	+	0.692385i	$0.243440\pi$
$48$	0	0
$49$	−6.69195	−0.955993
$50$	0	0
$51$	4.09688	0.573678
$52$	0	0
$53$	−2.23261	−0.306673	−0.153336	−	0.988174i	$-0.549002\pi$
$53$	−2.23261	−0.306673	−0.153336	+	0.988174i	$0.549002\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.605593	0.0802127
$58$	0	0
$59$	2.43488	0.316994	0.158497	−	0.987359i	$-0.449335\pi$
$59$	2.43488	0.316994	0.158497	+	0.987359i	$0.449335\pi$
$60$	0	0
$61$	−5.71138	−0.731267	−0.365633	−	0.930759i	$-0.619148\pi$
$61$	−5.71138	−0.731267	−0.365633	+	0.930759i	$0.619148\pi$
$62$	0	0
$63$	3.61185	0.455051
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−6.91830	−0.845205	−0.422602	−	0.906315i	$-0.638883\pi$
$67$	−6.91830	−0.845205	−0.422602	+	0.906315i	$0.638883\pi$
$68$	0	0
$69$	3.08344	0.371202
$70$	0	0
$71$	0.0120411	0.00142902	0.000714510	−	1.00000i	$-0.499773\pi$
$71$	0.0120411	0.00142902	0.000714510		1.00000i	$0.499773\pi$
$72$	0	0
$73$	−15.2989	−1.79061	−0.895303	−	0.445458i	$-0.853041\pi$
$73$	−15.2989	−1.79061	−0.895303	+	0.445458i	$0.853041\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.58191	0.294236
$78$	0	0
$79$	10.6351	1.19654	0.598272	−	0.801293i	$-0.295854\pi$
$79$	10.6351	1.19654	0.598272	+	0.801293i	$0.295854\pi$
$80$	0	0
$81$	13.8260	1.53622
$82$	0	0
$83$	5.64020	0.619092	0.309546	−	0.950884i	$-0.399823\pi$
$83$	5.64020	0.619092	0.309546	+	0.950884i	$0.399823\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.50467	0.268529
$88$	0	0
$89$	9.06693	0.961093	0.480547	−	0.876969i	$-0.340438\pi$
$89$	9.06693	0.961093	0.480547	+	0.876969i	$0.340438\pi$
$90$	0	0
$91$	−2.78763	−0.292223
$92$	0	0
$93$	6.51051	0.675108
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.0720	1.42880	0.714398	−	0.699739i	$-0.246701\pi$
$97$	14.0720	1.42880	0.714398	+	0.699739i	$0.246701\pi$
$98$	0	0
$99$	30.2727	3.04252

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.1.1	✓	5
4.3	odd	2		9200.2.a.cw.1.5		5
5.2	odd	4		4600.2.e.v.4049.10		10
5.3	odd	4		4600.2.e.v.4049.1		10
5.4	even	2		4600.2.a.bg.1.5	yes	5
20.19	odd	2		9200.2.a.cs.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.bc.1.1	✓	5	1.1	even	1	trivial
4600.2.a.bg.1.5	yes	5	5.4	even	2
4600.2.e.v.4049.1		10	5.3	odd	4
4600.2.e.v.4049.10		10	5.2	odd	4
9200.2.a.cs.1.1		5	20.19	odd	2
9200.2.a.cw.1.5		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)

\(f(q)\)	\(=\)	\(q-3.08344 q^{3} +0.555022 q^{7} +6.50759 q^{9} +4.65190 q^{11} -5.02256 q^{13} -1.32867 q^{17} -0.196402 q^{19} -1.71138 q^{21} -1.00000 q^{23} -10.8154 q^{27} -0.812298 q^{29} -2.11145 q^{31} -14.3439 q^{33} +5.64564 q^{37} +15.4868 q^{39} -4.89714 q^{41} -1.66507 q^{43} +9.89310 q^{47} -6.69195 q^{49} +4.09688 q^{51} -2.23261 q^{53} +0.605593 q^{57} +2.43488 q^{59} -5.71138 q^{61} +3.61185 q^{63} -6.91830 q^{67} +3.08344 q^{69} +0.0120411 q^{71} -15.2989 q^{73} +2.58191 q^{77} +10.6351 q^{79} +13.8260 q^{81} +5.64020 q^{83} +2.50467 q^{87} +9.06693 q^{89} -2.78763 q^{91} +6.51051 q^{93} +14.0720 q^{97} +30.2727 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{3} + q^{7} + 4 q^{9} - 4 q^{11} + q^{13} - 5 q^{17} + 4 q^{19} - 6 q^{21} - 5 q^{23} - 6 q^{27} - 11 q^{29} + 4 q^{31} - 13 q^{33} - 6 q^{37} + 31 q^{39} - 8 q^{41} - 3 q^{43} - 2 q^{47} - 2 q^{49}+ \cdots + 37 q^{99}+O(q^{100}) \) 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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