LMFDB - Embedded newform 2200.2.a.t.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2200,2,Mod(1,2200)] 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("2200.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2200 = 2^{3} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2200.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,0,0,-3,0,6,0,-3] 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:	$17.5670884447$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
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			1.2
Root			$-0.167449$ of defining polynomial
Character	$\chi$	$=$	2200.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.16745 q^{3} -4.97196 q^{7} -1.63706 q^{9} -1.00000 q^{11} +0.665102 q^{13} -6.77647 q^{17} +1.00000 q^{19} +5.80451 q^{21} -2.16745 q^{23} +5.41353 q^{27} +7.97196 q^{29} -8.94392 q^{31} +1.16745 q^{33} +0.139410 q^{37} -0.776472 q^{39} -1.80451 q^{41} -2.80451 q^{43} -0.530387 q^{47} +17.7204 q^{49} +7.91119 q^{51} +6.30216 q^{53} -1.16745 q^{57} +11.4696 q^{59} +10.5810 q^{61} +8.13941 q^{63} -9.60902 q^{67} +2.53039 q^{69} -9.80921 q^{71} +7.02804 q^{73} +4.97196 q^{77} +5.50235 q^{79} -1.40884 q^{81} -13.5810 q^{83} -9.30686 q^{87} -9.58098 q^{89} -3.30686 q^{91} +10.4416 q^{93} +14.7812 q^{97} +1.63706 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{7} + 6 q^{9} - 3 q^{11} + 3 q^{13} + 3 q^{17} + 3 q^{19} + 6 q^{21} - 6 q^{23} - 18 q^{27} + 12 q^{29} - 3 q^{31} + 3 q^{33} - 12 q^{37} + 21 q^{39} + 6 q^{41} + 3 q^{43} - 12 q^{47}+ \cdots - 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^7 + 6 * q^9 - 3 * q^11 + 3 * q^13 + 3 * q^17 + 3 * q^19 + 6 * q^21 - 6 * q^23 - 18 * q^27 + 12 * q^29 - 3 * q^31 + 3 * q^33 - 12 * q^37 + 21 * q^39 + 6 * q^41 + 3 * q^43 - 12 * q^47 + 6 * q^49 - 9 * q^51 + 9 * q^53 - 3 * q^57 + 24 * q^59 - 3 * q^61 + 12 * q^63 - 6 * q^67 + 18 * q^69 - 15 * q^71 + 33 * q^73 + 3 * q^77 + 15 * q^79 + 27 * q^81 - 6 * q^83 - 15 * q^87 + 6 * q^89 + 3 * q^91 + 9 * q^93 + 18 * 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
$3$	−1.16745	−0.674027	−0.337014	−	0.941500i	$-0.609417\pi$
$3$	−1.16745	−0.674027	−0.337014	+	0.941500i	$0.609417\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.97196	−1.87922	−0.939612	−	0.342241i	$-0.888814\pi$
$7$	−4.97196	−1.87922	−0.939612	+	0.342241i	$0.888814\pi$
$8$	0	0
$9$	−1.63706	−0.545687
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.665102	0.184466	0.0922330	−	0.995737i	$-0.470600\pi$
$13$	0.665102	0.184466	0.0922330	+	0.995737i	$0.470600\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.77647	−1.64354	−0.821768	−	0.569822i	$-0.807012\pi$
$17$	−6.77647	−1.64354	−0.821768	+	0.569822i	$0.807012\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	5.80451	1.26665
$22$	0	0
$23$	−2.16745	−0.451944	−0.225972	−	0.974134i	$-0.572556\pi$
$23$	−2.16745	−0.451944	−0.225972	+	0.974134i	$0.572556\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.41353	1.04184
$28$	0	0
$29$	7.97196	1.48036	0.740178	−	0.672411i	$-0.234741\pi$
$29$	7.97196	1.48036	0.740178	+	0.672411i	$0.234741\pi$
$30$	0	0
$31$	−8.94392	−1.60638	−0.803188	−	0.595726i	$-0.796864\pi$
$31$	−8.94392	−1.60638	−0.803188	+	0.595726i	$0.796864\pi$
$32$	0	0
$33$	1.16745	0.203227
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.139410	0.0229189	0.0114594	−	0.999934i	$-0.496352\pi$
$37$	0.139410	0.0229189	0.0114594	+	0.999934i	$0.496352\pi$
$38$	0	0
$39$	−0.776472	−0.124335
$40$	0	0
$41$	−1.80451	−0.281817	−0.140909	−	0.990023i	$-0.545002\pi$
$41$	−1.80451	−0.281817	−0.140909	+	0.990023i	$0.545002\pi$
$42$	0	0
$43$	−2.80451	−0.427684	−0.213842	−	0.976868i	$-0.568598\pi$
$43$	−2.80451	−0.427684	−0.213842	+	0.976868i	$0.568598\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.530387	−0.0773649	−0.0386824	−	0.999252i	$-0.512316\pi$
$47$	−0.530387	−0.0773649	−0.0386824	+	0.999252i	$0.512316\pi$
$48$	0	0
$49$	17.7204	2.53148
$50$	0	0
$51$	7.91119	1.10779
$52$	0	0
$53$	6.30216	0.865669	0.432834	−	0.901473i	$-0.357513\pi$
$53$	6.30216	0.865669	0.432834	+	0.901473i	$0.357513\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.16745	−0.154632
$58$	0	0
$59$	11.4696	1.49322	0.746608	−	0.665264i	$-0.231681\pi$
$59$	11.4696	1.49322	0.746608	+	0.665264i	$0.231681\pi$
$60$	0	0
$61$	10.5810	1.35476	0.677378	−	0.735635i	$-0.263116\pi$
$61$	10.5810	1.35476	0.677378	+	0.735635i	$0.263116\pi$
$62$	0	0
$63$	8.13941	1.02547
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.60902	−1.17393	−0.586965	−	0.809613i	$-0.699677\pi$
$67$	−9.60902	−1.17393	−0.586965	+	0.809613i	$0.699677\pi$
$68$	0	0
$69$	2.53039	0.304623
$70$	0	0
$71$	−9.80921	−1.16414	−0.582069	−	0.813139i	$-0.697757\pi$
$71$	−9.80921	−1.16414	−0.582069	+	0.813139i	$0.697757\pi$
$72$	0	0
$73$	7.02804	0.822570	0.411285	−	0.911507i	$-0.365080\pi$
$73$	7.02804	0.822570	0.411285	+	0.911507i	$0.365080\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.97196	0.566608
$78$	0	0
$79$	5.50235	0.619062	0.309531	−	0.950889i	$-0.399828\pi$
$79$	5.50235	0.619062	0.309531	+	0.950889i	$0.399828\pi$
$80$	0	0
$81$	−1.40884	−0.156538
$82$	0	0
$83$	−13.5810	−1.49071	−0.745353	−	0.666670i	$-0.767719\pi$
$83$	−13.5810	−1.49071	−0.745353	+	0.666670i	$0.767719\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−9.30686	−0.997800
$88$	0	0
$89$	−9.58098	−1.01558	−0.507791	−	0.861480i	$-0.669538\pi$
$89$	−9.58098	−1.01558	−0.507791	+	0.861480i	$0.669538\pi$
$90$	0	0
$91$	−3.30686	−0.346653
$92$	0	0
$93$	10.4416	1.08274
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.7812	1.50080	0.750400	−	0.660984i	$-0.229861\pi$
$97$	14.7812	1.50080	0.750400	+	0.660984i	$0.229861\pi$
$98$	0	0
$99$	1.63706	0.164531

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	2200.2.a.t.1.2	✓	3
4.3	odd	2		4400.2.a.ca.1.2		3
5.2	odd	4		2200.2.b.l.1849.4		6
5.3	odd	4		2200.2.b.l.1849.3		6
5.4	even	2		2200.2.a.w.1.2	yes	3
20.3	even	4		4400.2.b.bc.4049.4		6
20.7	even	4		4400.2.b.bc.4049.3		6
20.19	odd	2		4400.2.a.bx.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2200.2.a.t.1.2	✓	3	1.1	even	1	trivial
2200.2.a.w.1.2	yes	3	5.4	even	2
2200.2.b.l.1849.3		6	5.3	odd	4
2200.2.b.l.1849.4		6	5.2	odd	4
4400.2.a.bx.1.2		3	20.19	odd	2
4400.2.a.ca.1.2		3	4.3	odd	2
4400.2.b.bc.4049.3		6	20.7	even	4
4400.2.b.bc.4049.4		6	20.3	even	4

Overview	Random
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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2200.a (trivial)

\(f(q)\)	\(=\)	\(q-1.16745 q^{3} -4.97196 q^{7} -1.63706 q^{9} -1.00000 q^{11} +0.665102 q^{13} -6.77647 q^{17} +1.00000 q^{19} +5.80451 q^{21} -2.16745 q^{23} +5.41353 q^{27} +7.97196 q^{29} -8.94392 q^{31} +1.16745 q^{33} +0.139410 q^{37} -0.776472 q^{39} -1.80451 q^{41} -2.80451 q^{43} -0.530387 q^{47} +17.7204 q^{49} +7.91119 q^{51} +6.30216 q^{53} -1.16745 q^{57} +11.4696 q^{59} +10.5810 q^{61} +8.13941 q^{63} -9.60902 q^{67} +2.53039 q^{69} -9.80921 q^{71} +7.02804 q^{73} +4.97196 q^{77} +5.50235 q^{79} -1.40884 q^{81} -13.5810 q^{83} -9.30686 q^{87} -9.58098 q^{89} -3.30686 q^{91} +10.4416 q^{93} +14.7812 q^{97} +1.63706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{7} + 6 q^{9} - 3 q^{11} + 3 q^{13} + 3 q^{17} + 3 q^{19} + 6 q^{21} - 6 q^{23} - 18 q^{27} + 12 q^{29} - 3 q^{31} + 3 q^{33} - 12 q^{37} + 21 q^{39} + 6 q^{41} + 3 q^{43} - 12 q^{47}+ \cdots - 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^7 + 6 * q^9 - 3 * q^11 + 3 * q^13 + 3 * q^17 + 3 * q^19 + 6 * q^21 - 6 * q^23 - 18 * q^27 + 12 * q^29 - 3 * q^31 + 3 * q^33 - 12 * q^37 + 21 * q^39 + 6 * q^41 + 3 * q^43 - 12 * q^47 + 6 * q^49 - 9 * q^51 + 9 * q^53 - 3 * q^57 + 24 * q^59 - 3 * q^61 + 12 * q^63 - 6 * q^67 + 18 * q^69 - 15 * q^71 + 33 * q^73 + 3 * q^77 + 15 * q^79 + 27 * q^81 - 6 * q^83 - 15 * q^87 + 6 * q^89 + 3 * q^91 + 9 * q^93 + 18 * q^97 - 6 * q^99

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