LMFDB - Embedded newform 605.2.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [605,2,Mod(1,605)] 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("605.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(605, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 605 = 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	605.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:	$4.83094932229$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	605.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} +1.73205 q^{7} +1.73205 q^{8} -2.00000 q^{9} +1.73205 q^{10} -1.00000 q^{12} -3.46410 q^{13} -3.00000 q^{14} +1.00000 q^{15} -5.00000 q^{16} +6.92820 q^{17} +3.46410 q^{18} +3.46410 q^{19} -1.00000 q^{20} -1.73205 q^{21} -1.73205 q^{24} +1.00000 q^{25} +6.00000 q^{26} +5.00000 q^{27} +1.73205 q^{28} -1.73205 q^{30} -8.00000 q^{31} +5.19615 q^{32} -12.0000 q^{34} -1.73205 q^{35} -2.00000 q^{36} -8.00000 q^{37} -6.00000 q^{38} +3.46410 q^{39} -1.73205 q^{40} -12.1244 q^{41} +3.00000 q^{42} -8.66025 q^{43} +2.00000 q^{45} +9.00000 q^{47} +5.00000 q^{48} -4.00000 q^{49} -1.73205 q^{50} -6.92820 q^{51} -3.46410 q^{52} +6.00000 q^{53} -8.66025 q^{54} +3.00000 q^{56} -3.46410 q^{57} -12.0000 q^{59} +1.00000 q^{60} -8.66025 q^{61} +13.8564 q^{62} -3.46410 q^{63} +1.00000 q^{64} +3.46410 q^{65} -5.00000 q^{67} +6.92820 q^{68} +3.00000 q^{70} -12.0000 q^{71} -3.46410 q^{72} +13.8564 q^{74} -1.00000 q^{75} +3.46410 q^{76} -6.00000 q^{78} +10.3923 q^{79} +5.00000 q^{80} +1.00000 q^{81} +21.0000 q^{82} -3.46410 q^{83} -1.73205 q^{84} -6.92820 q^{85} +15.0000 q^{86} +3.00000 q^{89} -3.46410 q^{90} -6.00000 q^{91} +8.00000 q^{93} -15.5885 q^{94} -3.46410 q^{95} -5.19615 q^{96} -10.0000 q^{97} +6.92820 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{9} - 2 q^{12} - 6 q^{14} + 2 q^{15} - 10 q^{16} - 2 q^{20} + 2 q^{25} + 12 q^{26} + 10 q^{27} - 16 q^{31} - 24 q^{34} - 4 q^{36} - 16 q^{37} - 12 q^{38} + 6 q^{42}+ \cdots - 20 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^4 - 2 * q^5 - 4 * q^9 - 2 * q^12 - 6 * q^14 + 2 * q^15 - 10 * q^16 - 2 * q^20 + 2 * q^25 + 12 * q^26 + 10 * q^27 - 16 * q^31 - 24 * q^34 - 4 * q^36 - 16 * q^37 - 12 * q^38 + 6 * q^42 + 4 * q^45 + 18 * q^47 + 10 * q^48 - 8 * q^49 + 12 * q^53 + 6 * q^56 - 24 * q^59 + 2 * q^60 + 2 * q^64 - 10 * q^67 + 6 * q^70 - 24 * q^71 - 2 * q^75 - 12 * q^78 + 10 * q^80 + 2 * q^81 + 42 * q^82 + 30 * q^86 + 6 * q^89 - 12 * q^91 + 16 * q^93 - 20 * 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 $

$2$	−1.73205	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.73205	0.707107
$7$	1.73205	0.654654	0.327327	−	0.944911i	$-0.393852\pi$
$7$	1.73205	0.654654	0.327327	+	0.944911i	$0.393852\pi$
$8$	1.73205	0.612372
$9$	−2.00000	−0.666667
$10$	1.73205	0.547723
$11$	0	0
$11$	0	0
$12$	−1.00000	−0.288675
$13$	−3.46410	−0.960769	−0.480384	−	0.877058i	$-0.659503\pi$
$13$	−3.46410	−0.960769	−0.480384	+	0.877058i	$0.659503\pi$
$14$	−3.00000	−0.801784
$15$	1.00000	0.258199
$16$	−5.00000	−1.25000
$17$	6.92820	1.68034	0.840168	−	0.542326i	$-0.182456\pi$
$17$	6.92820	1.68034	0.840168	+	0.542326i	$0.182456\pi$
$18$	3.46410	0.816497
$19$	3.46410	0.794719	0.397360	−	0.917663i	$-0.369927\pi$
$19$	3.46410	0.794719	0.397360	+	0.917663i	$0.369927\pi$
$20$	−1.00000	−0.223607
$21$	−1.73205	−0.377964
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.73205	−0.353553
$25$	1.00000	0.200000
$26$	6.00000	1.17670
$27$	5.00000	0.962250
$28$	1.73205	0.327327
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−1.73205	−0.316228
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	5.19615	0.918559
$33$	0	0
$34$	−12.0000	−2.05798
$35$	−1.73205	−0.292770
$36$	−2.00000	−0.333333
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	−6.00000	−0.973329
$39$	3.46410	0.554700
$40$	−1.73205	−0.273861
$41$	−12.1244	−1.89351	−0.946753	−	0.321960i	$-0.895658\pi$
$41$	−12.1244	−1.89351	−0.946753	+	0.321960i	$0.895658\pi$
$42$	3.00000	0.462910
$43$	−8.66025	−1.32068	−0.660338	−	0.750968i	$-0.729587\pi$
$43$	−8.66025	−1.32068	−0.660338	+	0.750968i	$0.729587\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	5.00000	0.721688
$49$	−4.00000	−0.571429
$50$	−1.73205	−0.244949
$51$	−6.92820	−0.970143
$52$	−3.46410	−0.480384
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−8.66025	−1.17851
$55$	0	0
$56$	3.00000	0.400892
$57$	−3.46410	−0.458831
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	1.00000	0.129099
$61$	−8.66025	−1.10883	−0.554416	−	0.832240i	$-0.687058\pi$
$61$	−8.66025	−1.10883	−0.554416	+	0.832240i	$0.687058\pi$
$62$	13.8564	1.75977
$63$	−3.46410	−0.436436
$64$	1.00000	0.125000
$65$	3.46410	0.429669
$66$	0	0
$67$	−5.00000	−0.610847	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	−5.00000	−0.610847	−0.305424	+	0.952217i	$0.598798\pi$
$68$	6.92820	0.840168
$69$	0	0
$70$	3.00000	0.358569
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	−3.46410	−0.408248
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	13.8564	1.61077
$75$	−1.00000	−0.115470
$76$	3.46410	0.397360
$77$	0	0
$78$	−6.00000	−0.679366
$79$	10.3923	1.16923	0.584613	−	0.811312i	$-0.301246\pi$
$79$	10.3923	1.16923	0.584613	+	0.811312i	$0.301246\pi$
$80$	5.00000	0.559017
$81$	1.00000	0.111111
$82$	21.0000	2.31906
$83$	−3.46410	−0.380235	−0.190117	−	0.981761i	$-0.560887\pi$
$83$	−3.46410	−0.380235	−0.190117	+	0.981761i	$0.560887\pi$
$84$	−1.73205	−0.188982
$85$	−6.92820	−0.751469
$86$	15.0000	1.61749
$87$	0	0
$88$	0	0
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	−3.46410	−0.365148
$91$	−6.00000	−0.628971
$92$	0	0
$93$	8.00000	0.829561
$94$	−15.5885	−1.60783
$95$	−3.46410	−0.355409
$96$	−5.19615	−0.530330
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	6.92820	0.699854
$99$	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	605.2.a.e.1.1	✓	2
3.2	odd	2		5445.2.a.u.1.2		2
4.3	odd	2		9680.2.a.bu.1.1		2
5.4	even	2		3025.2.a.l.1.2		2
11.2	odd	10		605.2.g.i.81.1		8
11.3	even	5		605.2.g.i.251.1		8
11.4	even	5		605.2.g.i.511.1		8
11.5	even	5		605.2.g.i.366.2		8
11.6	odd	10		605.2.g.i.366.1		8
11.7	odd	10		605.2.g.i.511.2		8
11.8	odd	10		605.2.g.i.251.2		8
11.9	even	5		605.2.g.i.81.2		8
11.10	odd	2	inner	605.2.a.e.1.2	yes	2
33.32	even	2		5445.2.a.u.1.1		2
44.43	even	2		9680.2.a.bu.1.2		2
55.54	odd	2		3025.2.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.a.e.1.1	✓	2	1.1	even	1	trivial
605.2.a.e.1.2	yes	2	11.10	odd	2	inner
605.2.g.i.81.1		8	11.2	odd	10
605.2.g.i.81.2		8	11.9	even	5
605.2.g.i.251.1		8	11.3	even	5
605.2.g.i.251.2		8	11.8	odd	10
605.2.g.i.366.1		8	11.6	odd	10
605.2.g.i.366.2		8	11.5	even	5
605.2.g.i.511.1		8	11.4	even	5
605.2.g.i.511.2		8	11.7	odd	10
3025.2.a.l.1.1		2	55.54	odd	2
3025.2.a.l.1.2		2	5.4	even	2
5445.2.a.u.1.1		2	33.32	even	2
5445.2.a.u.1.2		2	3.2	odd	2
9680.2.a.bu.1.1		2	4.3	odd	2
9680.2.a.bu.1.2		2	44.43	even	2

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

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} +1.73205 q^{7} +1.73205 q^{8} -2.00000 q^{9} +1.73205 q^{10} -1.00000 q^{12} -3.46410 q^{13} -3.00000 q^{14} +1.00000 q^{15} -5.00000 q^{16} +6.92820 q^{17} +3.46410 q^{18} +3.46410 q^{19} -1.00000 q^{20} -1.73205 q^{21} -1.73205 q^{24} +1.00000 q^{25} +6.00000 q^{26} +5.00000 q^{27} +1.73205 q^{28} -1.73205 q^{30} -8.00000 q^{31} +5.19615 q^{32} -12.0000 q^{34} -1.73205 q^{35} -2.00000 q^{36} -8.00000 q^{37} -6.00000 q^{38} +3.46410 q^{39} -1.73205 q^{40} -12.1244 q^{41} +3.00000 q^{42} -8.66025 q^{43} +2.00000 q^{45} +9.00000 q^{47} +5.00000 q^{48} -4.00000 q^{49} -1.73205 q^{50} -6.92820 q^{51} -3.46410 q^{52} +6.00000 q^{53} -8.66025 q^{54} +3.00000 q^{56} -3.46410 q^{57} -12.0000 q^{59} +1.00000 q^{60} -8.66025 q^{61} +13.8564 q^{62} -3.46410 q^{63} +1.00000 q^{64} +3.46410 q^{65} -5.00000 q^{67} +6.92820 q^{68} +3.00000 q^{70} -12.0000 q^{71} -3.46410 q^{72} +13.8564 q^{74} -1.00000 q^{75} +3.46410 q^{76} -6.00000 q^{78} +10.3923 q^{79} +5.00000 q^{80} +1.00000 q^{81} +21.0000 q^{82} -3.46410 q^{83} -1.73205 q^{84} -6.92820 q^{85} +15.0000 q^{86} +3.00000 q^{89} -3.46410 q^{90} -6.00000 q^{91} +8.00000 q^{93} -15.5885 q^{94} -3.46410 q^{95} -5.19615 q^{96} -10.0000 q^{97} +6.92820 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{9} - 2 q^{12} - 6 q^{14} + 2 q^{15} - 10 q^{16} - 2 q^{20} + 2 q^{25} + 12 q^{26} + 10 q^{27} - 16 q^{31} - 24 q^{34} - 4 q^{36} - 16 q^{37} - 12 q^{38} + 6 q^{42}+ \cdots - 20 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^4 - 2 * q^5 - 4 * q^9 - 2 * q^12 - 6 * q^14 + 2 * q^15 - 10 * q^16 - 2 * q^20 + 2 * q^25 + 12 * q^26 + 10 * q^27 - 16 * q^31 - 24 * q^34 - 4 * q^36 - 16 * q^37 - 12 * q^38 + 6 * q^42 + 4 * q^45 + 18 * q^47 + 10 * q^48 - 8 * q^49 + 12 * q^53 + 6 * q^56 - 24 * q^59 + 2 * q^60 + 2 * q^64 - 10 * q^67 + 6 * q^70 - 24 * q^71 - 2 * q^75 - 12 * q^78 + 10 * q^80 + 2 * q^81 + 42 * q^82 + 30 * q^86 + 6 * q^89 - 12 * q^91 + 16 * q^93 - 20 * q^97

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