LMFDB - Embedded newform 2420.4.a.i.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2420,4,Mod(1,2420)] 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("2420.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2420, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2420 = 2^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2420.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,-15,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$142.784622214$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.9192.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 18x + 30 $ x^3 - x^2 - 18*x + 30
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 220)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.67648$ of defining polynomial
Character	$\chi$	$=$	2420.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+6.86946 q^{3} -5.00000 q^{5} -15.2554 q^{7} +20.1894 q^{9} -30.1909 q^{13} -34.3473 q^{15} -74.4793 q^{17} -30.6126 q^{19} -104.796 q^{21} +111.339 q^{23} +25.0000 q^{25} -46.7850 q^{27} -23.5336 q^{29} -272.083 q^{31} +76.2770 q^{35} +292.415 q^{37} -207.395 q^{39} +127.493 q^{41} +466.210 q^{43} -100.947 q^{45} +430.418 q^{47} -110.273 q^{49} -511.632 q^{51} +235.761 q^{53} -210.292 q^{57} +167.162 q^{59} +363.291 q^{61} -307.998 q^{63} +150.954 q^{65} +611.297 q^{67} +764.836 q^{69} -315.466 q^{71} +372.861 q^{73} +171.736 q^{75} +300.829 q^{79} -866.502 q^{81} +1218.18 q^{83} +372.396 q^{85} -161.663 q^{87} +73.6581 q^{89} +460.574 q^{91} -1869.06 q^{93} +153.063 q^{95} -1389.82 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 15 q^{5} + 5 q^{7} + 24 q^{9} - 2 q^{13} + 15 q^{15} - 77 q^{17} - 171 q^{19} - 259 q^{21} + 222 q^{23} + 75 q^{25} + 111 q^{27} - 55 q^{29} + 181 q^{31} - 25 q^{35} + 317 q^{37} - 554 q^{39}+ \cdots - 1980 q^{97}+O(q^{100}) $ 3 * q - 3 * q^3 - 15 * q^5 + 5 * q^7 + 24 * q^9 - 2 * q^13 + 15 * q^15 - 77 * q^17 - 171 * q^19 - 259 * q^21 + 222 * q^23 + 75 * q^25 + 111 * q^27 - 55 * q^29 + 181 * q^31 - 25 * q^35 + 317 * q^37 - 554 * q^39 - 302 * q^41 + 188 * q^43 - 120 * q^45 + 662 * q^47 - 268 * q^49 - 833 * q^51 + 81 * q^53 + 825 * q^57 - 42 * q^59 - 349 * q^61 + 266 * q^63 + 10 * q^65 + 152 * q^67 - 258 * q^69 + 927 * q^71 + 2378 * q^73 - 75 * q^75 + 1146 * q^79 - 1533 * q^81 + 458 * q^83 + 385 * q^85 - 1087 * q^87 - 875 * q^89 + 1982 * q^91 - 3623 * q^93 + 855 * q^95 - 1980 * 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
$3$	6.86946	1.32203	0.661014	−	0.750374i	$-0.270127\pi$
$3$	6.86946	1.32203	0.661014	+	0.750374i	$0.270127\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−15.2554	−0.823715	−0.411857	−	0.911248i	$-0.635120\pi$
$7$	−15.2554	−0.823715	−0.411857	+	0.911248i	$0.635120\pi$
$8$	0	0
$9$	20.1894	0.747756
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−30.1909	−0.644110	−0.322055	−	0.946721i	$-0.604374\pi$
$13$	−30.1909	−0.644110	−0.322055	+	0.946721i	$0.604374\pi$
$14$	0	0
$15$	−34.3473	−0.591229
$16$	0	0
$17$	−74.4793	−1.06258	−0.531290	−	0.847190i	$-0.678293\pi$
$17$	−74.4793	−1.06258	−0.531290	+	0.847190i	$0.678293\pi$
$18$	0	0
$19$	−30.6126	−0.369632	−0.184816	−	0.982773i	$-0.559169\pi$
$19$	−30.6126	−0.369632	−0.184816	+	0.982773i	$0.559169\pi$
$20$	0	0
$21$	−104.796	−1.08897
$22$	0	0
$23$	111.339	1.00938	0.504690	−	0.863301i	$-0.331607\pi$
$23$	111.339	1.00938	0.504690	+	0.863301i	$0.331607\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−46.7850	−0.333473
$28$	0	0
$29$	−23.5336	−0.150692	−0.0753462	−	0.997157i	$-0.524006\pi$
$29$	−23.5336	−0.150692	−0.0753462	+	0.997157i	$0.524006\pi$
$30$	0	0
$31$	−272.083	−1.57637	−0.788185	−	0.615439i	$-0.788979\pi$
$31$	−272.083	−1.57637	−0.788185	+	0.615439i	$0.788979\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	76.2770	0.368376
$36$	0	0
$37$	292.415	1.29926	0.649632	−	0.760249i	$-0.274923\pi$
$37$	292.415	1.29926	0.649632	+	0.760249i	$0.274923\pi$
$38$	0	0
$39$	−207.395	−0.851532
$40$	0	0
$41$	127.493	0.485634	0.242817	−	0.970072i	$-0.421929\pi$
$41$	127.493	0.485634	0.242817	+	0.970072i	$0.421929\pi$
$42$	0	0
$43$	466.210	1.65341	0.826703	−	0.562639i	$-0.190214\pi$
$43$	466.210	1.65341	0.826703	+	0.562639i	$0.190214\pi$
$44$	0	0
$45$	−100.947	−0.334407
$46$	0	0
$47$	430.418	1.33581	0.667903	−	0.744248i	$-0.267192\pi$
$47$	430.418	1.33581	0.667903	+	0.744248i	$0.267192\pi$
$48$	0	0
$49$	−110.273	−0.321494
$50$	0	0
$51$	−511.632	−1.40476
$52$	0	0
$53$	235.761	0.611024	0.305512	−	0.952188i	$-0.401172\pi$
$53$	235.761	0.611024	0.305512	+	0.952188i	$0.401172\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−210.292	−0.488664
$58$	0	0
$59$	167.162	0.368859	0.184429	−	0.982846i	$-0.440956\pi$
$59$	167.162	0.368859	0.184429	+	0.982846i	$0.440956\pi$
$60$	0	0
$61$	363.291	0.762535	0.381267	−	0.924465i	$-0.375488\pi$
$61$	363.291	0.762535	0.381267	+	0.924465i	$0.375488\pi$
$62$	0	0
$63$	−307.998	−0.615938
$64$	0	0
$65$	150.954	0.288055
$66$	0	0
$67$	611.297	1.11465	0.557327	−	0.830293i	$-0.311827\pi$
$67$	611.297	1.11465	0.557327	+	0.830293i	$0.311827\pi$
$68$	0	0
$69$	764.836	1.33443
$70$	0	0
$71$	−315.466	−0.527309	−0.263654	−	0.964617i	$-0.584928\pi$
$71$	−315.466	−0.527309	−0.263654	+	0.964617i	$0.584928\pi$
$72$	0	0
$73$	372.861	0.597810	0.298905	−	0.954283i	$-0.403379\pi$
$73$	372.861	0.597810	0.298905	+	0.954283i	$0.403379\pi$
$74$	0	0
$75$	171.736	0.264405
$76$	0	0
$77$	0	0
$78$	0	0
$79$	300.829	0.428430	0.214215	−	0.976787i	$-0.431281\pi$
$79$	300.829	0.428430	0.214215	+	0.976787i	$0.431281\pi$
$80$	0	0
$81$	−866.502	−1.18862
$82$	0	0
$83$	1218.18	1.61100	0.805498	−	0.592599i	$-0.201898\pi$
$83$	1218.18	1.61100	0.805498	+	0.592599i	$0.201898\pi$
$84$	0	0
$85$	372.396	0.475201
$86$	0	0
$87$	−161.663	−0.199219
$88$	0	0
$89$	73.6581	0.0877275	0.0438637	−	0.999038i	$-0.486033\pi$
$89$	73.6581	0.0877275	0.0438637	+	0.999038i	$0.486033\pi$
$90$	0	0
$91$	460.574	0.530563
$92$	0	0
$93$	−1869.06	−2.08400
$94$	0	0
$95$	153.063	0.165305
$96$	0	0
$97$	−1389.82	−1.45480	−0.727398	−	0.686216i	$-0.759271\pi$
$97$	−1389.82	−1.45480	−0.727398	+	0.686216i	$0.759271\pi$
$98$	0	0
$99$	0	0

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	2420.4.a.i.1.3		3
11.10	odd	2		220.4.a.f.1.3	✓	3
33.32	even	2		1980.4.a.l.1.3		3
44.43	even	2		880.4.a.w.1.1		3
55.32	even	4		1100.4.b.h.749.2		6
55.43	even	4		1100.4.b.h.749.5		6
55.54	odd	2		1100.4.a.i.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.4.a.f.1.3	✓	3	11.10	odd	2
880.4.a.w.1.1		3	44.43	even	2
1100.4.a.i.1.1		3	55.54	odd	2
1100.4.b.h.749.2		6	55.32	even	4
1100.4.b.h.749.5		6	55.43	even	4
1980.4.a.l.1.3		3	33.32	even	2
2420.4.a.i.1.3		3	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 2420 = 2^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2420.a (trivial)

\(f(q)\)	\(=\)	\(q+6.86946 q^{3} -5.00000 q^{5} -15.2554 q^{7} +20.1894 q^{9} -30.1909 q^{13} -34.3473 q^{15} -74.4793 q^{17} -30.6126 q^{19} -104.796 q^{21} +111.339 q^{23} +25.0000 q^{25} -46.7850 q^{27} -23.5336 q^{29} -272.083 q^{31} +76.2770 q^{35} +292.415 q^{37} -207.395 q^{39} +127.493 q^{41} +466.210 q^{43} -100.947 q^{45} +430.418 q^{47} -110.273 q^{49} -511.632 q^{51} +235.761 q^{53} -210.292 q^{57} +167.162 q^{59} +363.291 q^{61} -307.998 q^{63} +150.954 q^{65} +611.297 q^{67} +764.836 q^{69} -315.466 q^{71} +372.861 q^{73} +171.736 q^{75} +300.829 q^{79} -866.502 q^{81} +1218.18 q^{83} +372.396 q^{85} -161.663 q^{87} +73.6581 q^{89} +460.574 q^{91} -1869.06 q^{93} +153.063 q^{95} -1389.82 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 15 q^{5} + 5 q^{7} + 24 q^{9} - 2 q^{13} + 15 q^{15} - 77 q^{17} - 171 q^{19} - 259 q^{21} + 222 q^{23} + 75 q^{25} + 111 q^{27} - 55 q^{29} + 181 q^{31} - 25 q^{35} + 317 q^{37} - 554 q^{39}+ \cdots - 1980 q^{97}+O(q^{100}) \) 3 * q - 3 * q^3 - 15 * q^5 + 5 * q^7 + 24 * q^9 - 2 * q^13 + 15 * q^15 - 77 * q^17 - 171 * q^19 - 259 * q^21 + 222 * q^23 + 75 * q^25 + 111 * q^27 - 55 * q^29 + 181 * q^31 - 25 * q^35 + 317 * q^37 - 554 * q^39 - 302 * q^41 + 188 * q^43 - 120 * q^45 + 662 * q^47 - 268 * q^49 - 833 * q^51 + 81 * q^53 + 825 * q^57 - 42 * q^59 - 349 * q^61 + 266 * q^63 + 10 * q^65 + 152 * q^67 - 258 * q^69 + 927 * q^71 + 2378 * q^73 - 75 * q^75 + 1146 * q^79 - 1533 * q^81 + 458 * q^83 + 385 * q^85 - 1087 * q^87 - 875 * q^89 + 1982 * q^91 - 3623 * q^93 + 855 * q^95 - 1980 * q^97

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