LMFDB - Embedded newform 363.2.a.g.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [363,2,Mod(1,363)] 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("363.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(363, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	363.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$2.89856959337$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	363.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.23607 q^{2} +1.00000 q^{3} +3.00000 q^{4} +2.00000 q^{5} -2.23607 q^{6} +4.47214 q^{7} -2.23607 q^{8} +1.00000 q^{9} -4.47214 q^{10} +3.00000 q^{12} -10.0000 q^{14} +2.00000 q^{15} -1.00000 q^{16} -4.47214 q^{17} -2.23607 q^{18} +4.47214 q^{19} +6.00000 q^{20} +4.47214 q^{21} -4.00000 q^{23} -2.23607 q^{24} -1.00000 q^{25} +1.00000 q^{27} +13.4164 q^{28} -4.47214 q^{29} -4.47214 q^{30} +6.70820 q^{32} +10.0000 q^{34} +8.94427 q^{35} +3.00000 q^{36} +2.00000 q^{37} -10.0000 q^{38} -4.47214 q^{40} +4.47214 q^{41} -10.0000 q^{42} -4.47214 q^{43} +2.00000 q^{45} +8.94427 q^{46} +8.00000 q^{47} -1.00000 q^{48} +13.0000 q^{49} +2.23607 q^{50} -4.47214 q^{51} +6.00000 q^{53} -2.23607 q^{54} -10.0000 q^{56} +4.47214 q^{57} +10.0000 q^{58} +6.00000 q^{60} -8.94427 q^{61} +4.47214 q^{63} -13.0000 q^{64} -12.0000 q^{67} -13.4164 q^{68} -4.00000 q^{69} -20.0000 q^{70} -8.00000 q^{71} -2.23607 q^{72} +8.94427 q^{73} -4.47214 q^{74} -1.00000 q^{75} +13.4164 q^{76} -13.4164 q^{79} -2.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +8.94427 q^{83} +13.4164 q^{84} -8.94427 q^{85} +10.0000 q^{86} -4.47214 q^{87} -14.0000 q^{89} -4.47214 q^{90} -12.0000 q^{92} -17.8885 q^{94} +8.94427 q^{95} +6.70820 q^{96} +2.00000 q^{97} -29.0689 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 6 q^{4} + 4 q^{5} + 2 q^{9} + 6 q^{12} - 20 q^{14} + 4 q^{15} - 2 q^{16} + 12 q^{20} - 8 q^{23} - 2 q^{25} + 2 q^{27} + 20 q^{34} + 6 q^{36} + 4 q^{37} - 20 q^{38} - 20 q^{42} + 4 q^{45}+ \cdots + 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 + 6 * q^4 + 4 * q^5 + 2 * q^9 + 6 * q^12 - 20 * q^14 + 4 * q^15 - 2 * q^16 + 12 * q^20 - 8 * q^23 - 2 * q^25 + 2 * q^27 + 20 * q^34 + 6 * q^36 + 4 * q^37 - 20 * q^38 - 20 * q^42 + 4 * q^45 + 16 * q^47 - 2 * q^48 + 26 * q^49 + 12 * q^53 - 20 * q^56 + 20 * q^58 + 12 * q^60 - 26 * q^64 - 24 * q^67 - 8 * q^69 - 40 * q^70 - 16 * q^71 - 2 * q^75 - 4 * q^80 + 2 * q^81 - 20 * q^82 + 20 * q^86 - 28 * q^89 - 24 * q^92 + 4 * 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 $

$2$	−2.23607	−1.58114	−0.790569	−	0.612372i	$-0.790215\pi$
$2$	−2.23607	−1.58114	−0.790569	+	0.612372i	$0.790215\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	3.00000	1.50000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	−2.23607	−0.912871
$7$	4.47214	1.69031	0.845154	−	0.534522i	$-0.179509\pi$
$7$	4.47214	1.69031	0.845154	+	0.534522i	$0.179509\pi$
$8$	−2.23607	−0.790569
$9$	1.00000	0.333333
$10$	−4.47214	−1.41421
$11$	0	0
$11$	0	0
$12$	3.00000	0.866025
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−10.0000	−2.67261
$15$	2.00000	0.516398
$16$	−1.00000	−0.250000
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	−2.23607	−0.527046
$19$	4.47214	1.02598	0.512989	−	0.858395i	$-0.328538\pi$
$19$	4.47214	1.02598	0.512989	+	0.858395i	$0.328538\pi$
$20$	6.00000	1.34164
$21$	4.47214	0.975900
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−2.23607	−0.456435
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	13.4164	2.53546
$29$	−4.47214	−0.830455	−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214	−0.830455	−0.415227	+	0.909718i	$0.636298\pi$
$30$	−4.47214	−0.816497
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	6.70820	1.18585
$33$	0	0
$34$	10.0000	1.71499
$35$	8.94427	1.51186
$36$	3.00000	0.500000
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−10.0000	−1.62221
$39$	0	0
$40$	−4.47214	−0.707107
$41$	4.47214	0.698430	0.349215	−	0.937043i	$-0.386448\pi$
$41$	4.47214	0.698430	0.349215	+	0.937043i	$0.386448\pi$
$42$	−10.0000	−1.54303
$43$	−4.47214	−0.681994	−0.340997	−	0.940064i	$-0.610765\pi$
$43$	−4.47214	−0.681994	−0.340997	+	0.940064i	$0.610765\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	8.94427	1.31876
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−1.00000	−0.144338
$49$	13.0000	1.85714
$50$	2.23607	0.316228
$51$	−4.47214	−0.626224
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−2.23607	−0.304290
$55$	0	0
$56$	−10.0000	−1.33631
$57$	4.47214	0.592349
$58$	10.0000	1.31306
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	6.00000	0.774597
$61$	−8.94427	−1.14520	−0.572598	−	0.819836i	$-0.694065\pi$
$61$	−8.94427	−1.14520	−0.572598	+	0.819836i	$0.694065\pi$
$62$	0	0
$63$	4.47214	0.563436
$64$	−13.0000	−1.62500
$65$	0	0
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	−13.4164	−1.62698
$69$	−4.00000	−0.481543
$70$	−20.0000	−2.39046
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	−2.23607	−0.263523
$73$	8.94427	1.04685	0.523424	−	0.852072i	$-0.324654\pi$
$73$	8.94427	1.04685	0.523424	+	0.852072i	$0.324654\pi$
$74$	−4.47214	−0.519875
$75$	−1.00000	−0.115470
$76$	13.4164	1.53897
$77$	0	0
$78$	0	0
$79$	−13.4164	−1.50946	−0.754732	−	0.656033i	$-0.772233\pi$
$79$	−13.4164	−1.50946	−0.754732	+	0.656033i	$0.772233\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	−10.0000	−1.10432
$83$	8.94427	0.981761	0.490881	−	0.871227i	$-0.336675\pi$
$83$	8.94427	0.981761	0.490881	+	0.871227i	$0.336675\pi$
$84$	13.4164	1.46385
$85$	−8.94427	−0.970143
$86$	10.0000	1.07833
$87$	−4.47214	−0.479463
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	−4.47214	−0.471405
$91$	0	0
$92$	−12.0000	−1.25109
$93$	0	0
$94$	−17.8885	−1.84506
$95$	8.94427	0.917663
$96$	6.70820	0.684653
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−29.0689	−2.93640
$99$	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	363.2.a.g.1.1	✓	2
3.2	odd	2		1089.2.a.p.1.2		2
4.3	odd	2		5808.2.a.bx.1.1		2
5.4	even	2		9075.2.a.bi.1.2		2
11.2	odd	10		363.2.e.a.202.1		4
11.3	even	5		363.2.e.a.130.1		4
11.4	even	5		363.2.e.a.148.1		4
11.5	even	5		363.2.e.l.124.1		4
11.6	odd	10		363.2.e.a.124.1		4
11.7	odd	10		363.2.e.l.148.1		4
11.8	odd	10		363.2.e.l.130.1		4
11.9	even	5		363.2.e.l.202.1		4
11.10	odd	2	inner	363.2.a.g.1.2	yes	2
33.32	even	2		1089.2.a.p.1.1		2
44.43	even	2		5808.2.a.bx.1.2		2
55.54	odd	2		9075.2.a.bi.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.2.a.g.1.1	✓	2	1.1	even	1	trivial
363.2.a.g.1.2	yes	2	11.10	odd	2	inner
363.2.e.a.124.1		4	11.6	odd	10
363.2.e.a.130.1		4	11.3	even	5
363.2.e.a.148.1		4	11.4	even	5
363.2.e.a.202.1		4	11.2	odd	10
363.2.e.l.124.1		4	11.5	even	5
363.2.e.l.130.1		4	11.8	odd	10
363.2.e.l.148.1		4	11.7	odd	10
363.2.e.l.202.1		4	11.9	even	5
1089.2.a.p.1.1		2	33.32	even	2
1089.2.a.p.1.2		2	3.2	odd	2
5808.2.a.bx.1.1		2	4.3	odd	2
5808.2.a.bx.1.2		2	44.43	even	2
9075.2.a.bi.1.1		2	55.54	odd	2
9075.2.a.bi.1.2		2	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	363.a (trivial)

\(f(q)\)	\(=\)	\(q-2.23607 q^{2} +1.00000 q^{3} +3.00000 q^{4} +2.00000 q^{5} -2.23607 q^{6} +4.47214 q^{7} -2.23607 q^{8} +1.00000 q^{9} -4.47214 q^{10} +3.00000 q^{12} -10.0000 q^{14} +2.00000 q^{15} -1.00000 q^{16} -4.47214 q^{17} -2.23607 q^{18} +4.47214 q^{19} +6.00000 q^{20} +4.47214 q^{21} -4.00000 q^{23} -2.23607 q^{24} -1.00000 q^{25} +1.00000 q^{27} +13.4164 q^{28} -4.47214 q^{29} -4.47214 q^{30} +6.70820 q^{32} +10.0000 q^{34} +8.94427 q^{35} +3.00000 q^{36} +2.00000 q^{37} -10.0000 q^{38} -4.47214 q^{40} +4.47214 q^{41} -10.0000 q^{42} -4.47214 q^{43} +2.00000 q^{45} +8.94427 q^{46} +8.00000 q^{47} -1.00000 q^{48} +13.0000 q^{49} +2.23607 q^{50} -4.47214 q^{51} +6.00000 q^{53} -2.23607 q^{54} -10.0000 q^{56} +4.47214 q^{57} +10.0000 q^{58} +6.00000 q^{60} -8.94427 q^{61} +4.47214 q^{63} -13.0000 q^{64} -12.0000 q^{67} -13.4164 q^{68} -4.00000 q^{69} -20.0000 q^{70} -8.00000 q^{71} -2.23607 q^{72} +8.94427 q^{73} -4.47214 q^{74} -1.00000 q^{75} +13.4164 q^{76} -13.4164 q^{79} -2.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +8.94427 q^{83} +13.4164 q^{84} -8.94427 q^{85} +10.0000 q^{86} -4.47214 q^{87} -14.0000 q^{89} -4.47214 q^{90} -12.0000 q^{92} -17.8885 q^{94} +8.94427 q^{95} +6.70820 q^{96} +2.00000 q^{97} -29.0689 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 6 q^{4} + 4 q^{5} + 2 q^{9} + 6 q^{12} - 20 q^{14} + 4 q^{15} - 2 q^{16} + 12 q^{20} - 8 q^{23} - 2 q^{25} + 2 q^{27} + 20 q^{34} + 6 q^{36} + 4 q^{37} - 20 q^{38} - 20 q^{42} + 4 q^{45}+ \cdots + 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 + 6 * q^4 + 4 * q^5 + 2 * q^9 + 6 * q^12 - 20 * q^14 + 4 * q^15 - 2 * q^16 + 12 * q^20 - 8 * q^23 - 2 * q^25 + 2 * q^27 + 20 * q^34 + 6 * q^36 + 4 * q^37 - 20 * q^38 - 20 * q^42 + 4 * q^45 + 16 * q^47 - 2 * q^48 + 26 * q^49 + 12 * q^53 - 20 * q^56 + 20 * q^58 + 12 * q^60 - 26 * q^64 - 24 * q^67 - 8 * q^69 - 40 * q^70 - 16 * q^71 - 2 * q^75 - 4 * q^80 + 2 * q^81 - 20 * q^82 + 20 * q^86 - 28 * q^89 - 24 * q^92 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more