LMFDB - Embedded newform 136.2.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [136,2,Mod(1,136)] 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("136.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(136, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 136 = 2^{3} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	136.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:	$1.08596546749$
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:	$\Z[a_1, a_2, a_3]$
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	$\chi$	$=$	136.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.23607 q^{3} +2.00000 q^{5} +3.23607 q^{7} +7.47214 q^{9} +3.23607 q^{11} -4.47214 q^{13} -6.47214 q^{15} +1.00000 q^{17} +2.47214 q^{19} -10.4721 q^{21} +3.23607 q^{23} -1.00000 q^{25} -14.4721 q^{27} +2.00000 q^{29} -3.23607 q^{31} -10.4721 q^{33} +6.47214 q^{35} +6.94427 q^{37} +14.4721 q^{39} +2.00000 q^{41} -10.4721 q^{43} +14.9443 q^{45} -4.94427 q^{47} +3.47214 q^{49} -3.23607 q^{51} -2.00000 q^{53} +6.47214 q^{55} -8.00000 q^{57} +5.52786 q^{59} -10.9443 q^{61} +24.1803 q^{63} -8.94427 q^{65} -12.0000 q^{67} -10.4721 q^{69} +4.76393 q^{71} -2.94427 q^{73} +3.23607 q^{75} +10.4721 q^{77} -1.70820 q^{79} +24.4164 q^{81} +10.4721 q^{83} +2.00000 q^{85} -6.47214 q^{87} -16.4721 q^{89} -14.4721 q^{91} +10.4721 q^{93} +4.94427 q^{95} +2.00000 q^{97} +24.1803 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 4 q^{5} + 2 q^{7} + 6 q^{9} + 2 q^{11} - 4 q^{15} + 2 q^{17} - 4 q^{19} - 12 q^{21} + 2 q^{23} - 2 q^{25} - 20 q^{27} + 4 q^{29} - 2 q^{31} - 12 q^{33} + 4 q^{35} - 4 q^{37} + 20 q^{39}+ \cdots + 26 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 4 * q^5 + 2 * q^7 + 6 * q^9 + 2 * q^11 - 4 * q^15 + 2 * q^17 - 4 * q^19 - 12 * q^21 + 2 * q^23 - 2 * q^25 - 20 * q^27 + 4 * q^29 - 2 * q^31 - 12 * q^33 + 4 * q^35 - 4 * q^37 + 20 * q^39 + 4 * q^41 - 12 * q^43 + 12 * q^45 + 8 * q^47 - 2 * q^49 - 2 * q^51 - 4 * q^53 + 4 * q^55 - 16 * q^57 + 20 * q^59 - 4 * q^61 + 26 * q^63 - 24 * q^67 - 12 * q^69 + 14 * q^71 + 12 * q^73 + 2 * q^75 + 12 * q^77 + 10 * q^79 + 22 * q^81 + 12 * q^83 + 4 * q^85 - 4 * q^87 - 24 * q^89 - 20 * q^91 + 12 * q^93 - 8 * q^95 + 4 * q^97 + 26 * 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
$3$	−3.23607	−1.86834	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	−3.23607	−1.86834	−0.934172	+	0.356822i	$0.883860\pi$
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	3.23607	1.22312	0.611559	−	0.791199i	$-0.290543\pi$
$7$	3.23607	1.22312	0.611559	+	0.791199i	$0.290543\pi$
$8$	0	0
$9$	7.47214	2.49071
$10$	0	0
$11$	3.23607	0.975711	0.487856	−	0.872924i	$-0.337779\pi$
$11$	3.23607	0.975711	0.487856	+	0.872924i	$0.337779\pi$
$12$	0	0
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	−6.47214	−1.67110
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	2.47214	0.567147	0.283573	−	0.958951i	$-0.408480\pi$
$19$	2.47214	0.567147	0.283573	+	0.958951i	$0.408480\pi$
$20$	0	0
$21$	−10.4721	−2.28521
$22$	0	0
$23$	3.23607	0.674767	0.337383	−	0.941367i	$-0.390458\pi$
$23$	3.23607	0.674767	0.337383	+	0.941367i	$0.390458\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−14.4721	−2.78516
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−3.23607	−0.581215	−0.290607	−	0.956842i	$-0.593857\pi$
$31$	−3.23607	−0.581215	−0.290607	+	0.956842i	$0.593857\pi$
$32$	0	0
$33$	−10.4721	−1.82296
$34$	0	0
$35$	6.47214	1.09399
$36$	0	0
$37$	6.94427	1.14163	0.570816	−	0.821078i	$-0.306627\pi$
$37$	6.94427	1.14163	0.570816	+	0.821078i	$0.306627\pi$
$38$	0	0
$39$	14.4721	2.31740
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−10.4721	−1.59699	−0.798493	−	0.602004i	$-0.794369\pi$
$43$	−10.4721	−1.59699	−0.798493	+	0.602004i	$0.794369\pi$
$44$	0	0
$45$	14.9443	2.22776
$46$	0	0
$47$	−4.94427	−0.721196	−0.360598	−	0.932721i	$-0.617427\pi$
$47$	−4.94427	−0.721196	−0.360598	+	0.932721i	$0.617427\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	0	0
$51$	−3.23607	−0.453140
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	6.47214	0.872703
$56$	0	0
$57$	−8.00000	−1.05963
$58$	0	0
$59$	5.52786	0.719667	0.359833	−	0.933017i	$-0.382834\pi$
$59$	5.52786	0.719667	0.359833	+	0.933017i	$0.382834\pi$
$60$	0	0
$61$	−10.9443	−1.40127	−0.700635	−	0.713520i	$-0.747100\pi$
$61$	−10.9443	−1.40127	−0.700635	+	0.713520i	$0.747100\pi$
$62$	0	0
$63$	24.1803	3.04644
$64$	0	0
$65$	−8.94427	−1.10940
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	−10.4721	−1.26070
$70$	0	0
$71$	4.76393	0.565375	0.282687	−	0.959212i	$-0.408774\pi$
$71$	4.76393	0.565375	0.282687	+	0.959212i	$0.408774\pi$
$72$	0	0
$73$	−2.94427	−0.344601	−0.172300	−	0.985044i	$-0.555120\pi$
$73$	−2.94427	−0.344601	−0.172300	+	0.985044i	$0.555120\pi$
$74$	0	0
$75$	3.23607	0.373669
$76$	0	0
$77$	10.4721	1.19341
$78$	0	0
$79$	−1.70820	−0.192188	−0.0960940	−	0.995372i	$-0.530635\pi$
$79$	−1.70820	−0.192188	−0.0960940	+	0.995372i	$0.530635\pi$
$80$	0	0
$81$	24.4164	2.71293
$82$	0	0
$83$	10.4721	1.14947	0.574733	−	0.818341i	$-0.305106\pi$
$83$	10.4721	1.14947	0.574733	+	0.818341i	$0.305106\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	−6.47214	−0.693886
$88$	0	0
$89$	−16.4721	−1.74604	−0.873021	−	0.487682i	$-0.837843\pi$
$89$	−16.4721	−1.74604	−0.873021	+	0.487682i	$0.837843\pi$
$90$	0	0
$91$	−14.4721	−1.51709
$92$	0	0
$93$	10.4721	1.08591
$94$	0	0
$95$	4.94427	0.507272
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	24.1803	2.43022

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	136.2.a.c.1.1	✓	2
3.2	odd	2		1224.2.a.i.1.2		2
4.3	odd	2		272.2.a.f.1.2		2
5.2	odd	4		3400.2.e.f.2449.4		4
5.3	odd	4		3400.2.e.f.2449.1		4
5.4	even	2		3400.2.a.i.1.2		2
7.6	odd	2		6664.2.a.i.1.2		2
8.3	odd	2		1088.2.a.o.1.1		2
8.5	even	2		1088.2.a.s.1.2		2
12.11	even	2		2448.2.a.u.1.1		2
17.4	even	4		2312.2.b.g.577.4		4
17.13	even	4		2312.2.b.g.577.1		4
17.16	even	2		2312.2.a.m.1.2		2
20.19	odd	2		6800.2.a.bd.1.1		2
24.5	odd	2		9792.2.a.db.1.2		2
24.11	even	2		9792.2.a.da.1.1		2
68.67	odd	2		4624.2.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.a.c.1.1	✓	2	1.1	even	1	trivial
272.2.a.f.1.2		2	4.3	odd	2
1088.2.a.o.1.1		2	8.3	odd	2
1088.2.a.s.1.2		2	8.5	even	2
1224.2.a.i.1.2		2	3.2	odd	2
2312.2.a.m.1.2		2	17.16	even	2
2312.2.b.g.577.1		4	17.13	even	4
2312.2.b.g.577.4		4	17.4	even	4
2448.2.a.u.1.1		2	12.11	even	2
3400.2.a.i.1.2		2	5.4	even	2
3400.2.e.f.2449.1		4	5.3	odd	4
3400.2.e.f.2449.4		4	5.2	odd	4
4624.2.a.h.1.1		2	68.67	odd	2
6664.2.a.i.1.2		2	7.6	odd	2
6800.2.a.bd.1.1		2	20.19	odd	2
9792.2.a.da.1.1		2	24.11	even	2
9792.2.a.db.1.2		2	24.5	odd	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:	\( N \)	\(=\)	\( 136 = 2^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	136.a (trivial)

\(f(q)\)	\(=\)	\(q-3.23607 q^{3} +2.00000 q^{5} +3.23607 q^{7} +7.47214 q^{9} +3.23607 q^{11} -4.47214 q^{13} -6.47214 q^{15} +1.00000 q^{17} +2.47214 q^{19} -10.4721 q^{21} +3.23607 q^{23} -1.00000 q^{25} -14.4721 q^{27} +2.00000 q^{29} -3.23607 q^{31} -10.4721 q^{33} +6.47214 q^{35} +6.94427 q^{37} +14.4721 q^{39} +2.00000 q^{41} -10.4721 q^{43} +14.9443 q^{45} -4.94427 q^{47} +3.47214 q^{49} -3.23607 q^{51} -2.00000 q^{53} +6.47214 q^{55} -8.00000 q^{57} +5.52786 q^{59} -10.9443 q^{61} +24.1803 q^{63} -8.94427 q^{65} -12.0000 q^{67} -10.4721 q^{69} +4.76393 q^{71} -2.94427 q^{73} +3.23607 q^{75} +10.4721 q^{77} -1.70820 q^{79} +24.4164 q^{81} +10.4721 q^{83} +2.00000 q^{85} -6.47214 q^{87} -16.4721 q^{89} -14.4721 q^{91} +10.4721 q^{93} +4.94427 q^{95} +2.00000 q^{97} +24.1803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 4 q^{5} + 2 q^{7} + 6 q^{9} + 2 q^{11} - 4 q^{15} + 2 q^{17} - 4 q^{19} - 12 q^{21} + 2 q^{23} - 2 q^{25} - 20 q^{27} + 4 q^{29} - 2 q^{31} - 12 q^{33} + 4 q^{35} - 4 q^{37} + 20 q^{39}+ \cdots + 26 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 4 * q^5 + 2 * q^7 + 6 * q^9 + 2 * q^11 - 4 * q^15 + 2 * q^17 - 4 * q^19 - 12 * q^21 + 2 * q^23 - 2 * q^25 - 20 * q^27 + 4 * q^29 - 2 * q^31 - 12 * q^33 + 4 * q^35 - 4 * q^37 + 20 * q^39 + 4 * q^41 - 12 * q^43 + 12 * q^45 + 8 * q^47 - 2 * q^49 - 2 * q^51 - 4 * q^53 + 4 * q^55 - 16 * q^57 + 20 * q^59 - 4 * q^61 + 26 * q^63 - 24 * q^67 - 12 * q^69 + 14 * q^71 + 12 * q^73 + 2 * q^75 + 12 * q^77 + 10 * q^79 + 22 * q^81 + 12 * q^83 + 4 * q^85 - 4 * q^87 - 24 * q^89 - 20 * q^91 + 12 * q^93 - 8 * q^95 + 4 * q^97 + 26 * q^99

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