LMFDB - Embedded newform 729.2.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [729,2,Mod(1,729)] 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("729.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(729, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 729 = 3^{6} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	729.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-3,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.82109430735$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.7459857.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 6x^{4} + 13x^{3} + 12x^{2} - 12x - 8 $ x^6 - 3x^5 - 6x^4 + 13x^3 + 12x^2 - 12*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.17298$ of defining polynomial
Character	$\chi$	$=$	729.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.70506 q^{2} +5.31738 q^{4} +1.67238 q^{5} +0.500591 q^{7} -8.97372 q^{8} -4.52391 q^{10} +1.91772 q^{11} +3.11244 q^{13} -1.35413 q^{14} +13.6397 q^{16} -2.66467 q^{17} +5.79664 q^{19} +8.89270 q^{20} -5.18755 q^{22} +4.64587 q^{23} -2.20313 q^{25} -8.41934 q^{26} +2.66183 q^{28} +2.61507 q^{29} -4.61460 q^{31} -18.9489 q^{32} +7.20811 q^{34} +0.837181 q^{35} -4.85867 q^{37} -15.6803 q^{38} -15.0075 q^{40} +11.5482 q^{41} -9.00434 q^{43} +10.1972 q^{44} -12.5674 q^{46} -6.83224 q^{47} -6.74941 q^{49} +5.95961 q^{50} +16.5500 q^{52} +5.43322 q^{53} +3.20716 q^{55} -4.49216 q^{56} -7.07393 q^{58} +2.19131 q^{59} +6.84034 q^{61} +12.4828 q^{62} +23.9786 q^{64} +5.20519 q^{65} +12.4836 q^{67} -14.1691 q^{68} -2.26463 q^{70} +2.83568 q^{71} +9.93497 q^{73} +13.1430 q^{74} +30.8229 q^{76} +0.959993 q^{77} +5.31121 q^{79} +22.8109 q^{80} -31.2385 q^{82} +2.72815 q^{83} -4.45636 q^{85} +24.3573 q^{86} -17.2091 q^{88} -11.2189 q^{89} +1.55806 q^{91} +24.7038 q^{92} +18.4816 q^{94} +9.69421 q^{95} -6.88914 q^{97} +18.2576 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{2} + 9 q^{4} + 3 q^{5} + 6 q^{7} - 6 q^{8} + 6 q^{10} + 6 q^{11} + 6 q^{13} - 24 q^{14} + 15 q^{16} + 9 q^{17} + 12 q^{19} + 21 q^{20} + 3 q^{22} + 12 q^{23} + 9 q^{25} - 24 q^{26} + 3 q^{28}+ \cdots - 18 q^{98}+O(q^{100}) $ 6 * q - 3 * q^2 + 9 * q^4 + 3 * q^5 + 6 * q^7 - 6 * q^8 + 6 * q^10 + 6 * q^11 + 6 * q^13 - 24 * q^14 + 15 * q^16 + 9 * q^17 + 12 * q^19 + 21 * q^20 + 3 * q^22 + 12 * q^23 + 9 * q^25 - 24 * q^26 + 3 * q^28 - 21 * q^29 + 15 * q^31 - 30 * q^35 + 3 * q^37 - 15 * q^38 + 3 * q^40 + 12 * q^41 + 6 * q^43 + 33 * q^44 - 3 * q^46 + 15 * q^47 + 12 * q^49 + 24 * q^50 + 3 * q^52 + 9 * q^53 + 15 * q^55 - 12 * q^56 - 15 * q^58 - 6 * q^59 + 24 * q^61 + 30 * q^62 + 6 * q^64 + 15 * q^65 + 15 * q^67 - 36 * q^68 - 15 * q^70 + 12 * q^73 - 24 * q^74 + 9 * q^76 - 15 * q^77 + 24 * q^79 + 21 * q^80 - 21 * q^82 + 6 * q^83 - 18 * q^85 + 30 * q^86 - 21 * q^88 + 9 * q^89 + 18 * q^91 - 6 * q^92 - 6 * q^94 + 33 * q^95 - 21 * q^97 - 18 * 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 $

$2$	−2.70506	−1.91277	−0.956385	−	0.292110i	$-0.905643\pi$
$2$	−2.70506	−1.91277	−0.956385	+	0.292110i	$0.905643\pi$
$3$	0	0
$3$	0	0
$4$	5.31738	2.65869
$5$	1.67238	0.747913	0.373957	−	0.927446i	$-0.378001\pi$
$5$	1.67238	0.747913	0.373957	+	0.927446i	$0.378001\pi$
$6$	0	0
$7$	0.500591	0.189206	0.0946028	−	0.995515i	$-0.469842\pi$
$7$	0.500591	0.189206	0.0946028	+	0.995515i	$0.469842\pi$
$8$	−8.97372	−3.17269
$9$	0	0
$10$	−4.52391	−1.43059
$11$	1.91772	0.578214	0.289107	−	0.957297i	$-0.406642\pi$
$11$	1.91772	0.578214	0.289107	+	0.957297i	$0.406642\pi$
$12$	0	0
$13$	3.11244	0.863234	0.431617	−	0.902057i	$-0.357943\pi$
$13$	3.11244	0.863234	0.431617	+	0.902057i	$0.357943\pi$
$14$	−1.35413	−0.361907
$15$	0	0
$16$	13.6397	3.40993
$17$	−2.66467	−0.646278	−0.323139	−	0.946352i	$-0.604738\pi$
$17$	−2.66467	−0.646278	−0.323139	+	0.946352i	$0.604738\pi$
$18$	0	0
$19$	5.79664	1.32984	0.664920	−	0.746915i	$-0.268466\pi$
$19$	5.79664	1.32984	0.664920	+	0.746915i	$0.268466\pi$
$20$	8.89270	1.98847
$21$	0	0
$22$	−5.18755	−1.10599
$23$	4.64587	0.968731	0.484365	−	0.874866i	$-0.339051\pi$
$23$	4.64587	0.968731	0.484365	+	0.874866i	$0.339051\pi$
$24$	0	0
$25$	−2.20313	−0.440626
$26$	−8.41934	−1.65117
$27$	0	0
$28$	2.66183	0.503039
$29$	2.61507	0.485606	0.242803	−	0.970076i	$-0.421933\pi$
$29$	2.61507	0.485606	0.242803	+	0.970076i	$0.421933\pi$
$30$	0	0
$31$	−4.61460	−0.828807	−0.414404	−	0.910093i	$-0.636010\pi$
$31$	−4.61460	−0.828807	−0.414404	+	0.910093i	$0.636010\pi$
$32$	−18.9489	−3.34973
$33$	0	0
$34$	7.20811	1.23618
$35$	0.837181	0.141509
$36$	0	0
$37$	−4.85867	−0.798761	−0.399381	−	0.916785i	$-0.630775\pi$
$37$	−4.85867	−0.798761	−0.399381	+	0.916785i	$0.630775\pi$
$38$	−15.6803	−2.54368
$39$	0	0
$40$	−15.0075	−2.37290
$41$	11.5482	1.80352	0.901760	−	0.432237i	$-0.142276\pi$
$41$	11.5482	1.80352	0.901760	+	0.432237i	$0.142276\pi$
$42$	0	0
$43$	−9.00434	−1.37315	−0.686574	−	0.727060i	$-0.740886\pi$
$43$	−9.00434	−1.37315	−0.686574	+	0.727060i	$0.740886\pi$
$44$	10.1972	1.53729
$45$	0	0
$46$	−12.5674	−1.85296
$47$	−6.83224	−0.996584	−0.498292	−	0.867009i	$-0.666039\pi$
$47$	−6.83224	−0.996584	−0.498292	+	0.867009i	$0.666039\pi$
$48$	0	0
$49$	−6.74941	−0.964201
$50$	5.95961	0.842816
$51$	0	0
$52$	16.5500	2.29507
$53$	5.43322	0.746309	0.373155	−	0.927769i	$-0.378276\pi$
$53$	5.43322	0.746309	0.373155	+	0.927769i	$0.378276\pi$
$54$	0	0
$55$	3.20716	0.432454
$56$	−4.49216	−0.600291
$57$	0	0
$58$	−7.07393	−0.928853
$59$	2.19131	0.285285	0.142642	−	0.989774i	$-0.454440\pi$
$59$	2.19131	0.285285	0.142642	+	0.989774i	$0.454440\pi$
$60$	0	0
$61$	6.84034	0.875815	0.437908	−	0.899020i	$-0.355720\pi$
$61$	6.84034	0.875815	0.437908	+	0.899020i	$0.355720\pi$
$62$	12.4828	1.58532
$63$	0	0
$64$	23.9786	2.99733
$65$	5.20519	0.645624
$66$	0	0
$67$	12.4836	1.52511	0.762557	−	0.646921i	$-0.223944\pi$
$67$	12.4836	1.52511	0.762557	+	0.646921i	$0.223944\pi$
$68$	−14.1691	−1.71825
$69$	0	0
$70$	−2.26463	−0.270675
$71$	2.83568	0.336534	0.168267	−	0.985741i	$-0.446183\pi$
$71$	2.83568	0.336534	0.168267	+	0.985741i	$0.446183\pi$
$72$	0	0
$73$	9.93497	1.16280	0.581400	−	0.813618i	$-0.302505\pi$
$73$	9.93497	1.16280	0.581400	+	0.813618i	$0.302505\pi$
$74$	13.1430	1.52785
$75$	0	0
$76$	30.8229	3.53563
$77$	0.959993	0.109401
$78$	0	0
$79$	5.31121	0.597558	0.298779	−	0.954322i	$-0.403421\pi$
$79$	5.31121	0.597558	0.298779	+	0.954322i	$0.403421\pi$
$80$	22.8109	2.55033
$81$	0	0
$82$	−31.2385	−3.44972
$83$	2.72815	0.299453	0.149727	−	0.988727i	$-0.452161\pi$
$83$	2.72815	0.299453	0.149727	+	0.988727i	$0.452161\pi$
$84$	0	0
$85$	−4.45636	−0.483360
$86$	24.3573	2.62652
$87$	0	0
$88$	−17.2091	−1.83449
$89$	−11.2189	−1.18920	−0.594600	−	0.804021i	$-0.702690\pi$
$89$	−11.2189	−1.18920	−0.594600	+	0.804021i	$0.702690\pi$
$90$	0	0
$91$	1.55806	0.163329
$92$	24.7038	2.57555
$93$	0	0
$94$	18.4816	1.90624
$95$	9.69421	0.994605
$96$	0	0
$97$	−6.88914	−0.699486	−0.349743	−	0.936846i	$-0.613731\pi$
$97$	−6.88914	−0.699486	−0.349743	+	0.936846i	$0.613731\pi$
$98$	18.2576	1.84429
$99$	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	729.2.a.b.1.1	✓	6
3.2	odd	2		729.2.a.e.1.6	yes	6
9.2	odd	6		729.2.c.a.244.1		12
9.4	even	3		729.2.c.d.487.6		12
9.5	odd	6		729.2.c.a.487.1		12
9.7	even	3		729.2.c.d.244.6		12
27.2	odd	18		729.2.e.k.568.1		12
27.4	even	9		729.2.e.j.406.1		12
27.5	odd	18		729.2.e.l.649.2		12
27.7	even	9		729.2.e.j.325.1		12
27.11	odd	18		729.2.e.l.82.2		12
27.13	even	9		729.2.e.t.163.2		12
27.14	odd	18		729.2.e.k.163.1		12
27.16	even	9		729.2.e.s.82.1		12
27.20	odd	18		729.2.e.u.325.2		12
27.22	even	9		729.2.e.s.649.1		12
27.23	odd	18		729.2.e.u.406.2		12
27.25	even	9		729.2.e.t.568.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
729.2.a.b.1.1	✓	6	1.1	even	1	trivial
729.2.a.e.1.6	yes	6	3.2	odd	2
729.2.c.a.244.1		12	9.2	odd	6
729.2.c.a.487.1		12	9.5	odd	6
729.2.c.d.244.6		12	9.7	even	3
729.2.c.d.487.6		12	9.4	even	3
729.2.e.j.325.1		12	27.7	even	9
729.2.e.j.406.1		12	27.4	even	9
729.2.e.k.163.1		12	27.14	odd	18
729.2.e.k.568.1		12	27.2	odd	18
729.2.e.l.82.2		12	27.11	odd	18
729.2.e.l.649.2		12	27.5	odd	18
729.2.e.s.82.1		12	27.16	even	9
729.2.e.s.649.1		12	27.22	even	9
729.2.e.t.163.2		12	27.13	even	9
729.2.e.t.568.2		12	27.25	even	9
729.2.e.u.325.2		12	27.20	odd	18
729.2.e.u.406.2		12	27.23	odd	18

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Level:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	729.a (trivial)

\(f(q)\)	\(=\)	\(q-2.70506 q^{2} +5.31738 q^{4} +1.67238 q^{5} +0.500591 q^{7} -8.97372 q^{8} -4.52391 q^{10} +1.91772 q^{11} +3.11244 q^{13} -1.35413 q^{14} +13.6397 q^{16} -2.66467 q^{17} +5.79664 q^{19} +8.89270 q^{20} -5.18755 q^{22} +4.64587 q^{23} -2.20313 q^{25} -8.41934 q^{26} +2.66183 q^{28} +2.61507 q^{29} -4.61460 q^{31} -18.9489 q^{32} +7.20811 q^{34} +0.837181 q^{35} -4.85867 q^{37} -15.6803 q^{38} -15.0075 q^{40} +11.5482 q^{41} -9.00434 q^{43} +10.1972 q^{44} -12.5674 q^{46} -6.83224 q^{47} -6.74941 q^{49} +5.95961 q^{50} +16.5500 q^{52} +5.43322 q^{53} +3.20716 q^{55} -4.49216 q^{56} -7.07393 q^{58} +2.19131 q^{59} +6.84034 q^{61} +12.4828 q^{62} +23.9786 q^{64} +5.20519 q^{65} +12.4836 q^{67} -14.1691 q^{68} -2.26463 q^{70} +2.83568 q^{71} +9.93497 q^{73} +13.1430 q^{74} +30.8229 q^{76} +0.959993 q^{77} +5.31121 q^{79} +22.8109 q^{80} -31.2385 q^{82} +2.72815 q^{83} -4.45636 q^{85} +24.3573 q^{86} -17.2091 q^{88} -11.2189 q^{89} +1.55806 q^{91} +24.7038 q^{92} +18.4816 q^{94} +9.69421 q^{95} -6.88914 q^{97} +18.2576 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{2} + 9 q^{4} + 3 q^{5} + 6 q^{7} - 6 q^{8} + 6 q^{10} + 6 q^{11} + 6 q^{13} - 24 q^{14} + 15 q^{16} + 9 q^{17} + 12 q^{19} + 21 q^{20} + 3 q^{22} + 12 q^{23} + 9 q^{25} - 24 q^{26} + 3 q^{28}+ \cdots - 18 q^{98}+O(q^{100}) \) 6 * q - 3 * q^2 + 9 * q^4 + 3 * q^5 + 6 * q^7 - 6 * q^8 + 6 * q^10 + 6 * q^11 + 6 * q^13 - 24 * q^14 + 15 * q^16 + 9 * q^17 + 12 * q^19 + 21 * q^20 + 3 * q^22 + 12 * q^23 + 9 * q^25 - 24 * q^26 + 3 * q^28 - 21 * q^29 + 15 * q^31 - 30 * q^35 + 3 * q^37 - 15 * q^38 + 3 * q^40 + 12 * q^41 + 6 * q^43 + 33 * q^44 - 3 * q^46 + 15 * q^47 + 12 * q^49 + 24 * q^50 + 3 * q^52 + 9 * q^53 + 15 * q^55 - 12 * q^56 - 15 * q^58 - 6 * q^59 + 24 * q^61 + 30 * q^62 + 6 * q^64 + 15 * q^65 + 15 * q^67 - 36 * q^68 - 15 * q^70 + 12 * q^73 - 24 * q^74 + 9 * q^76 - 15 * q^77 + 24 * q^79 + 21 * q^80 - 21 * q^82 + 6 * q^83 - 18 * q^85 + 30 * q^86 - 21 * q^88 + 9 * q^89 + 18 * q^91 - 6 * q^92 - 6 * q^94 + 33 * q^95 - 21 * q^97 - 18 * q^98

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