LMFDB - Embedded newform 207.2.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [207,2,Mod(1,207)] 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("207.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(207, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	207.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$1.65290332184$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.23607 q^{2} +3.00000 q^{4} +3.23607 q^{5} -1.23607 q^{7} -2.23607 q^{8} -7.23607 q^{10} -4.00000 q^{11} +4.47214 q^{13} +2.76393 q^{14} -1.00000 q^{16} +7.23607 q^{17} +2.76393 q^{19} +9.70820 q^{20} +8.94427 q^{22} -1.00000 q^{23} +5.47214 q^{25} -10.0000 q^{26} -3.70820 q^{28} +4.47214 q^{29} +2.47214 q^{31} +6.70820 q^{32} -16.1803 q^{34} -4.00000 q^{35} -4.47214 q^{37} -6.18034 q^{38} -7.23607 q^{40} -6.94427 q^{41} +7.70820 q^{43} -12.0000 q^{44} +2.23607 q^{46} +4.00000 q^{47} -5.47214 q^{49} -12.2361 q^{50} +13.4164 q^{52} +0.763932 q^{53} -12.9443 q^{55} +2.76393 q^{56} -10.0000 q^{58} -12.9443 q^{59} -4.47214 q^{61} -5.52786 q^{62} -13.0000 q^{64} +14.4721 q^{65} +5.23607 q^{67} +21.7082 q^{68} +8.94427 q^{70} +8.00000 q^{71} -10.9443 q^{73} +10.0000 q^{74} +8.29180 q^{76} +4.94427 q^{77} -3.70820 q^{79} -3.23607 q^{80} +15.5279 q^{82} -4.00000 q^{83} +23.4164 q^{85} -17.2361 q^{86} +8.94427 q^{88} -3.23607 q^{89} -5.52786 q^{91} -3.00000 q^{92} -8.94427 q^{94} +8.94427 q^{95} -0.472136 q^{97} +12.2361 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{4} + 2 q^{5} + 2 q^{7} - 10 q^{10} - 8 q^{11} + 10 q^{14} - 2 q^{16} + 10 q^{17} + 10 q^{19} + 6 q^{20} - 2 q^{23} + 2 q^{25} - 20 q^{26} + 6 q^{28} - 4 q^{31} - 10 q^{34} - 8 q^{35} + 10 q^{38}+ \cdots + 20 q^{98}+O(q^{100}) $ 2 * q + 6 * q^4 + 2 * q^5 + 2 * q^7 - 10 * q^10 - 8 * q^11 + 10 * q^14 - 2 * q^16 + 10 * q^17 + 10 * q^19 + 6 * q^20 - 2 * q^23 + 2 * q^25 - 20 * q^26 + 6 * q^28 - 4 * q^31 - 10 * q^34 - 8 * q^35 + 10 * q^38 - 10 * q^40 + 4 * q^41 + 2 * q^43 - 24 * q^44 + 8 * q^47 - 2 * q^49 - 20 * q^50 + 6 * q^53 - 8 * q^55 + 10 * q^56 - 20 * q^58 - 8 * q^59 - 20 * q^62 - 26 * q^64 + 20 * q^65 + 6 * q^67 + 30 * q^68 + 16 * q^71 - 4 * q^73 + 20 * q^74 + 30 * q^76 - 8 * q^77 + 6 * q^79 - 2 * q^80 + 40 * q^82 - 8 * q^83 + 20 * q^85 - 30 * q^86 - 2 * q^89 - 20 * q^91 - 6 * q^92 + 8 * q^97 + 20 * q^98

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 $

$2$	−2.23607	−1.58114	−0.790569	−	0.612372i	$-0.790215\pi$
$2$	−2.23607	−1.58114	−0.790569	+	0.612372i	$0.790215\pi$
$3$	0	0
$3$	0	0
$4$	3.00000	1.50000
$5$	3.23607	1.44721	0.723607	−	0.690212i	$-0.242483\pi$
$5$	3.23607	1.44721	0.723607	+	0.690212i	$0.242483\pi$
$6$	0	0
$7$	−1.23607	−0.467190	−0.233595	−	0.972334i	$-0.575049\pi$
$7$	−1.23607	−0.467190	−0.233595	+	0.972334i	$0.575049\pi$
$8$	−2.23607	−0.790569
$9$	0	0
$10$	−7.23607	−2.28825
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	4.47214	1.24035	0.620174	−	0.784465i	$-0.287062\pi$
$13$	4.47214	1.24035	0.620174	+	0.784465i	$0.287062\pi$
$14$	2.76393	0.738692
$15$	0	0
$16$	−1.00000	−0.250000
$17$	7.23607	1.75500	0.877502	−	0.479573i	$-0.159208\pi$
$17$	7.23607	1.75500	0.877502	+	0.479573i	$0.159208\pi$
$18$	0	0
$19$	2.76393	0.634089	0.317045	−	0.948411i	$-0.397309\pi$
$19$	2.76393	0.634089	0.317045	+	0.948411i	$0.397309\pi$
$20$	9.70820	2.17082
$21$	0	0
$22$	8.94427	1.90693
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	5.47214	1.09443
$26$	−10.0000	−1.96116
$27$	0	0
$28$	−3.70820	−0.700785
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	2.47214	0.444009	0.222004	−	0.975046i	$-0.428740\pi$
$31$	2.47214	0.444009	0.222004	+	0.975046i	$0.428740\pi$
$32$	6.70820	1.18585
$33$	0	0
$34$	−16.1803	−2.77491
$35$	−4.00000	−0.676123
$36$	0	0
$37$	−4.47214	−0.735215	−0.367607	−	0.929981i	$-0.619823\pi$
$37$	−4.47214	−0.735215	−0.367607	+	0.929981i	$0.619823\pi$
$38$	−6.18034	−1.00258
$39$	0	0
$40$	−7.23607	−1.14412
$41$	−6.94427	−1.08451	−0.542257	−	0.840213i	$-0.682430\pi$
$41$	−6.94427	−1.08451	−0.542257	+	0.840213i	$0.682430\pi$
$42$	0	0
$43$	7.70820	1.17549	0.587745	−	0.809046i	$-0.300016\pi$
$43$	7.70820	1.17549	0.587745	+	0.809046i	$0.300016\pi$
$44$	−12.0000	−1.80907
$45$	0	0
$46$	2.23607	0.329690
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	−12.2361	−1.73044
$51$	0	0
$52$	13.4164	1.86052
$53$	0.763932	0.104934	0.0524671	−	0.998623i	$-0.483292\pi$
$53$	0.763932	0.104934	0.0524671	+	0.998623i	$0.483292\pi$
$54$	0	0
$55$	−12.9443	−1.74541
$56$	2.76393	0.369346
$57$	0	0
$58$	−10.0000	−1.31306
$59$	−12.9443	−1.68520	−0.842600	−	0.538539i	$-0.818976\pi$
$59$	−12.9443	−1.68520	−0.842600	+	0.538539i	$0.818976\pi$
$60$	0	0
$61$	−4.47214	−0.572598	−0.286299	−	0.958140i	$-0.592425\pi$
$61$	−4.47214	−0.572598	−0.286299	+	0.958140i	$0.592425\pi$
$62$	−5.52786	−0.702039
$63$	0	0
$64$	−13.0000	−1.62500
$65$	14.4721	1.79505
$66$	0	0
$67$	5.23607	0.639688	0.319844	−	0.947470i	$-0.396370\pi$
$67$	5.23607	0.639688	0.319844	+	0.947470i	$0.396370\pi$
$68$	21.7082	2.63251
$69$	0	0
$70$	8.94427	1.06904
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	−10.9443	−1.28093	−0.640465	−	0.767987i	$-0.721258\pi$
$73$	−10.9443	−1.28093	−0.640465	+	0.767987i	$0.721258\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	8.29180	0.951134
$77$	4.94427	0.563452
$78$	0	0
$79$	−3.70820	−0.417206	−0.208603	−	0.978000i	$-0.566892\pi$
$79$	−3.70820	−0.417206	−0.208603	+	0.978000i	$0.566892\pi$
$80$	−3.23607	−0.361803
$81$	0	0
$82$	15.5279	1.71477
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	23.4164	2.53987
$86$	−17.2361	−1.85861
$87$	0	0
$88$	8.94427	0.953463
$89$	−3.23607	−0.343023	−0.171511	−	0.985182i	$-0.554865\pi$
$89$	−3.23607	−0.343023	−0.171511	+	0.985182i	$0.554865\pi$
$90$	0	0
$91$	−5.52786	−0.579478
$92$	−3.00000	−0.312772
$93$	0	0
$94$	−8.94427	−0.922531
$95$	8.94427	0.917663
$96$	0	0
$97$	−0.472136	−0.0479381	−0.0239691	−	0.999713i	$-0.507630\pi$
$97$	−0.472136	−0.0479381	−0.0239691	+	0.999713i	$0.507630\pi$
$98$	12.2361	1.23603
$99$	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	207.2.a.c.1.1		2
3.2	odd	2		69.2.a.b.1.2	✓	2
4.3	odd	2		3312.2.a.bb.1.2		2
5.4	even	2		5175.2.a.bk.1.2		2
12.11	even	2		1104.2.a.m.1.1		2
15.2	even	4		1725.2.b.o.1174.4		4
15.8	even	4		1725.2.b.o.1174.1		4
15.14	odd	2		1725.2.a.ba.1.1		2
21.20	even	2		3381.2.a.t.1.2		2
23.22	odd	2		4761.2.a.v.1.1		2
24.5	odd	2		4416.2.a.bm.1.2		2
24.11	even	2		4416.2.a.bg.1.2		2
33.32	even	2		8349.2.a.i.1.1		2
69.68	even	2		1587.2.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.a.b.1.2	✓	2	3.2	odd	2
207.2.a.c.1.1		2	1.1	even	1	trivial
1104.2.a.m.1.1		2	12.11	even	2
1587.2.a.i.1.2		2	69.68	even	2
1725.2.a.ba.1.1		2	15.14	odd	2
1725.2.b.o.1174.1		4	15.8	even	4
1725.2.b.o.1174.4		4	15.2	even	4
3312.2.a.bb.1.2		2	4.3	odd	2
3381.2.a.t.1.2		2	21.20	even	2
4416.2.a.bg.1.2		2	24.11	even	2
4416.2.a.bm.1.2		2	24.5	odd	2
4761.2.a.v.1.1		2	23.22	odd	2
5175.2.a.bk.1.2		2	5.4	even	2
8349.2.a.i.1.1		2	33.32	even	2

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Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q-2.23607 q^{2} +3.00000 q^{4} +3.23607 q^{5} -1.23607 q^{7} -2.23607 q^{8} -7.23607 q^{10} -4.00000 q^{11} +4.47214 q^{13} +2.76393 q^{14} -1.00000 q^{16} +7.23607 q^{17} +2.76393 q^{19} +9.70820 q^{20} +8.94427 q^{22} -1.00000 q^{23} +5.47214 q^{25} -10.0000 q^{26} -3.70820 q^{28} +4.47214 q^{29} +2.47214 q^{31} +6.70820 q^{32} -16.1803 q^{34} -4.00000 q^{35} -4.47214 q^{37} -6.18034 q^{38} -7.23607 q^{40} -6.94427 q^{41} +7.70820 q^{43} -12.0000 q^{44} +2.23607 q^{46} +4.00000 q^{47} -5.47214 q^{49} -12.2361 q^{50} +13.4164 q^{52} +0.763932 q^{53} -12.9443 q^{55} +2.76393 q^{56} -10.0000 q^{58} -12.9443 q^{59} -4.47214 q^{61} -5.52786 q^{62} -13.0000 q^{64} +14.4721 q^{65} +5.23607 q^{67} +21.7082 q^{68} +8.94427 q^{70} +8.00000 q^{71} -10.9443 q^{73} +10.0000 q^{74} +8.29180 q^{76} +4.94427 q^{77} -3.70820 q^{79} -3.23607 q^{80} +15.5279 q^{82} -4.00000 q^{83} +23.4164 q^{85} -17.2361 q^{86} +8.94427 q^{88} -3.23607 q^{89} -5.52786 q^{91} -3.00000 q^{92} -8.94427 q^{94} +8.94427 q^{95} -0.472136 q^{97} +12.2361 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{4} + 2 q^{5} + 2 q^{7} - 10 q^{10} - 8 q^{11} + 10 q^{14} - 2 q^{16} + 10 q^{17} + 10 q^{19} + 6 q^{20} - 2 q^{23} + 2 q^{25} - 20 q^{26} + 6 q^{28} - 4 q^{31} - 10 q^{34} - 8 q^{35} + 10 q^{38}+ \cdots + 20 q^{98}+O(q^{100}) \) 2 * q + 6 * q^4 + 2 * q^5 + 2 * q^7 - 10 * q^10 - 8 * q^11 + 10 * q^14 - 2 * q^16 + 10 * q^17 + 10 * q^19 + 6 * q^20 - 2 * q^23 + 2 * q^25 - 20 * q^26 + 6 * q^28 - 4 * q^31 - 10 * q^34 - 8 * q^35 + 10 * q^38 - 10 * q^40 + 4 * q^41 + 2 * q^43 - 24 * q^44 + 8 * q^47 - 2 * q^49 - 20 * q^50 + 6 * q^53 - 8 * q^55 + 10 * q^56 - 20 * q^58 - 8 * q^59 - 20 * q^62 - 26 * q^64 + 20 * q^65 + 6 * q^67 + 30 * q^68 + 16 * q^71 - 4 * q^73 + 20 * q^74 + 30 * q^76 - 8 * q^77 + 6 * q^79 - 2 * q^80 + 40 * q^82 - 8 * q^83 + 20 * q^85 - 30 * q^86 - 2 * q^89 - 20 * q^91 - 6 * q^92 + 8 * q^97 + 20 * q^98

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