LMFDB - Embedded newform 135.4.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [135,4,Mod(1,135)] 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("135.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(135, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	135.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-5] 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:	$7.96525785077$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1772.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 12x + 8 $ x^3 - x^2 - 12*x + 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.654334$ of defining polynomial
Character	$\chi$	$=$	135.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.45876 q^{2} +21.7980 q^{4} -5.00000 q^{5} -11.8065 q^{7} -75.3201 q^{8} +27.2938 q^{10} +56.2376 q^{11} +34.5961 q^{13} +64.4489 q^{14} +236.770 q^{16} -39.2675 q^{17} -146.561 q^{19} -108.990 q^{20} -306.987 q^{22} +23.5777 q^{23} +25.0000 q^{25} -188.851 q^{26} -257.359 q^{28} -161.003 q^{29} -29.5465 q^{31} -689.908 q^{32} +214.352 q^{34} +59.0326 q^{35} -217.688 q^{37} +800.039 q^{38} +376.600 q^{40} -142.290 q^{41} -468.030 q^{43} +1225.87 q^{44} -128.705 q^{46} -394.318 q^{47} -203.606 q^{49} -136.469 q^{50} +754.126 q^{52} +134.780 q^{53} -281.188 q^{55} +889.268 q^{56} +878.875 q^{58} +131.195 q^{59} +259.801 q^{61} +161.287 q^{62} +1871.88 q^{64} -172.980 q^{65} +445.244 q^{67} -855.954 q^{68} -322.245 q^{70} -560.841 q^{71} -88.6681 q^{73} +1188.30 q^{74} -3194.74 q^{76} -663.970 q^{77} +450.342 q^{79} -1183.85 q^{80} +776.726 q^{82} -284.295 q^{83} +196.337 q^{85} +2554.86 q^{86} -4235.82 q^{88} -625.305 q^{89} -408.459 q^{91} +513.948 q^{92} +2152.49 q^{94} +732.804 q^{95} -193.261 q^{97} +1111.44 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 5 q^{2} + 17 q^{4} - 15 q^{5} - 4 q^{7} - 75 q^{8} + 25 q^{10} - 5 q^{11} + 7 q^{13} - 60 q^{14} + 161 q^{16} - 155 q^{17} - 50 q^{19} - 85 q^{20} - 229 q^{22} - 285 q^{23} + 75 q^{25} - 185 q^{26}+ \cdots + 305 q^{98}+O(q^{100}) $ 3 * q - 5 * q^2 + 17 * q^4 - 15 * q^5 - 4 * q^7 - 75 * q^8 + 25 * q^10 - 5 * q^11 + 7 * q^13 - 60 * q^14 + 161 * q^16 - 155 * q^17 - 50 * q^19 - 85 * q^20 - 229 * q^22 - 285 * q^23 + 75 * q^25 - 185 * q^26 - 334 * q^28 - 115 * q^29 - 115 * q^31 - 775 * q^32 + 413 * q^34 + 20 * q^35 - 384 * q^37 + 1150 * q^38 + 375 * q^40 - 580 * q^41 - 797 * q^43 + 1415 * q^44 - 285 * q^46 + 145 * q^47 + 577 * q^49 - 125 * q^50 + 825 * q^52 + 400 * q^53 + 25 * q^55 + 2190 * q^56 - 59 * q^58 - 380 * q^59 - 152 * q^61 + 1005 * q^62 + 2937 * q^64 - 35 * q^65 + 2 * q^67 - 475 * q^68 + 300 * q^70 - 40 * q^71 - 980 * q^73 + 2720 * q^74 - 3276 * q^76 - 1950 * q^77 + 1013 * q^79 - 805 * q^80 + 4 * q^82 - 270 * q^83 + 775 * q^85 + 1555 * q^86 - 5193 * q^88 - 1020 * q^89 - 632 * q^91 + 1215 * q^92 + 3833 * q^94 + 250 * q^95 + 720 * q^97 + 305 * 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 $

$2$	−5.45876	−1.92996	−0.964981	−	0.262320i	$-0.915513\pi$
$2$	−5.45876	−1.92996	−0.964981	+	0.262320i	$0.915513\pi$
$3$	0	0
$3$	0	0
$4$	21.7980	2.72475
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−11.8065	−0.637492	−0.318746	−	0.947840i	$-0.603262\pi$
$7$	−11.8065	−0.637492	−0.318746	+	0.947840i	$0.603262\pi$
$8$	−75.3201	−3.32871
$9$	0	0
$10$	27.2938	0.863105
$11$	56.2376	1.54148	0.770740	−	0.637150i	$-0.219887\pi$
$11$	56.2376	1.54148	0.770740	+	0.637150i	$0.219887\pi$
$12$	0	0
$13$	34.5961	0.738094	0.369047	−	0.929411i	$-0.379684\pi$
$13$	34.5961	0.738094	0.369047	+	0.929411i	$0.379684\pi$
$14$	64.4489	1.23034
$15$	0	0
$16$	236.770	3.69953
$17$	−39.2675	−0.560221	−0.280111	−	0.959968i	$-0.590371\pi$
$17$	−39.2675	−0.560221	−0.280111	+	0.959968i	$0.590371\pi$
$18$	0	0
$19$	−146.561	−1.76965	−0.884825	−	0.465924i	$-0.845722\pi$
$19$	−146.561	−1.76965	−0.884825	+	0.465924i	$0.845722\pi$
$20$	−108.990	−1.21855
$21$	0	0
$22$	−306.987	−2.97500
$23$	23.5777	0.213752	0.106876	−	0.994272i	$-0.465915\pi$
$23$	23.5777	0.213752	0.106876	+	0.994272i	$0.465915\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−188.851	−1.42449
$27$	0	0
$28$	−257.359	−1.73701
$29$	−161.003	−1.03095	−0.515473	−	0.856906i	$-0.672384\pi$
$29$	−161.003	−1.03095	−0.515473	+	0.856906i	$0.672384\pi$
$30$	0	0
$31$	−29.5465	−0.171184	−0.0855921	−	0.996330i	$-0.527278\pi$
$31$	−29.5465	−0.171184	−0.0855921	+	0.996330i	$0.527278\pi$
$32$	−689.908	−3.81124
$33$	0	0
$34$	214.352	1.08121
$35$	59.0326	0.285095
$36$	0	0
$37$	−217.688	−0.967233	−0.483617	−	0.875280i	$-0.660677\pi$
$37$	−217.688	−0.967233	−0.483617	+	0.875280i	$0.660677\pi$
$38$	800.039	3.41536
$39$	0	0
$40$	376.600	1.48864
$41$	−142.290	−0.541999	−0.270999	−	0.962580i	$-0.587354\pi$
$41$	−142.290	−0.541999	−0.270999	+	0.962580i	$0.587354\pi$
$42$	0	0
$43$	−468.030	−1.65986	−0.829929	−	0.557869i	$-0.811619\pi$
$43$	−468.030	−1.65986	−0.829929	+	0.557869i	$0.811619\pi$
$44$	1225.87	4.20015
$45$	0	0
$46$	−128.705	−0.412533
$47$	−394.318	−1.22377	−0.611886	−	0.790946i	$-0.709589\pi$
$47$	−394.318	−1.22377	−0.611886	+	0.790946i	$0.709589\pi$
$48$	0	0
$49$	−203.606	−0.593604
$50$	−136.469	−0.385992
$51$	0	0
$52$	754.126	2.01112
$53$	134.780	0.349311	0.174655	−	0.984630i	$-0.444119\pi$
$53$	134.780	0.349311	0.174655	+	0.984630i	$0.444119\pi$
$54$	0	0
$55$	−281.188	−0.689371
$56$	889.268	2.12203
$57$	0	0
$58$	878.875	1.98969
$59$	131.195	0.289495	0.144747	−	0.989469i	$-0.453763\pi$
$59$	131.195	0.289495	0.144747	+	0.989469i	$0.453763\pi$
$60$	0	0
$61$	259.801	0.545313	0.272657	−	0.962111i	$-0.412098\pi$
$61$	259.801	0.545313	0.272657	+	0.962111i	$0.412098\pi$
$62$	161.287	0.330379
$63$	0	0
$64$	1871.88	3.65602
$65$	−172.980	−0.330086
$66$	0	0
$67$	445.244	0.811869	0.405935	−	0.913902i	$-0.366946\pi$
$67$	445.244	0.811869	0.405935	+	0.913902i	$0.366946\pi$
$68$	−855.954	−1.52647
$69$	0	0
$70$	−322.245	−0.550223
$71$	−560.841	−0.937459	−0.468729	−	0.883342i	$-0.655288\pi$
$71$	−560.841	−0.937459	−0.468729	+	0.883342i	$0.655288\pi$
$72$	0	0
$73$	−88.6681	−0.142162	−0.0710809	−	0.997471i	$-0.522645\pi$
$73$	−88.6681	−0.142162	−0.0710809	+	0.997471i	$0.522645\pi$
$74$	1188.30	1.86672
$75$	0	0
$76$	−3194.74	−4.82186
$77$	−663.970	−0.982681
$78$	0	0
$79$	450.342	0.641360	0.320680	−	0.947188i	$-0.396089\pi$
$79$	450.342	0.641360	0.320680	+	0.947188i	$0.396089\pi$
$80$	−1183.85	−1.65448
$81$	0	0
$82$	776.726	1.04604
$83$	−284.295	−0.375969	−0.187985	−	0.982172i	$-0.560196\pi$
$83$	−284.295	−0.375969	−0.187985	+	0.982172i	$0.560196\pi$
$84$	0	0
$85$	196.337	0.250539
$86$	2554.86	3.20346
$87$	0	0
$88$	−4235.82	−5.13114
$89$	−625.305	−0.744744	−0.372372	−	0.928083i	$-0.621455\pi$
$89$	−625.305	−0.744744	−0.372372	+	0.928083i	$0.621455\pi$
$90$	0	0
$91$	−408.459	−0.470529
$92$	513.948	0.582421
$93$	0	0
$94$	2152.49	2.36183
$95$	732.804	0.791411
$96$	0	0
$97$	−193.261	−0.202296	−0.101148	−	0.994871i	$-0.532252\pi$
$97$	−193.261	−0.202296	−0.101148	+	0.994871i	$0.532252\pi$
$98$	1111.44	1.14563
$99$	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	135.4.a.e.1.1	✓	3
3.2	odd	2		135.4.a.h.1.3	yes	3
4.3	odd	2		2160.4.a.bi.1.2		3
5.2	odd	4		675.4.b.m.649.1		6
5.3	odd	4		675.4.b.m.649.6		6
5.4	even	2		675.4.a.s.1.3		3
9.2	odd	6		405.4.e.q.271.1		6
9.4	even	3		405.4.e.v.136.3		6
9.5	odd	6		405.4.e.q.136.1		6
9.7	even	3		405.4.e.v.271.3		6
12.11	even	2		2160.4.a.bq.1.2		3
15.2	even	4		675.4.b.n.649.6		6
15.8	even	4		675.4.b.n.649.1		6
15.14	odd	2		675.4.a.p.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.e.1.1	✓	3	1.1	even	1	trivial
135.4.a.h.1.3	yes	3	3.2	odd	2
405.4.e.q.136.1		6	9.5	odd	6
405.4.e.q.271.1		6	9.2	odd	6
405.4.e.v.136.3		6	9.4	even	3
405.4.e.v.271.3		6	9.7	even	3
675.4.a.p.1.1		3	15.14	odd	2
675.4.a.s.1.3		3	5.4	even	2
675.4.b.m.649.1		6	5.2	odd	4
675.4.b.m.649.6		6	5.3	odd	4
675.4.b.n.649.1		6	15.8	even	4
675.4.b.n.649.6		6	15.2	even	4
2160.4.a.bi.1.2		3	4.3	odd	2
2160.4.a.bq.1.2		3	12.11	even	2

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

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	135.a (trivial)

\(f(q)\)	\(=\)	\(q-5.45876 q^{2} +21.7980 q^{4} -5.00000 q^{5} -11.8065 q^{7} -75.3201 q^{8} +27.2938 q^{10} +56.2376 q^{11} +34.5961 q^{13} +64.4489 q^{14} +236.770 q^{16} -39.2675 q^{17} -146.561 q^{19} -108.990 q^{20} -306.987 q^{22} +23.5777 q^{23} +25.0000 q^{25} -188.851 q^{26} -257.359 q^{28} -161.003 q^{29} -29.5465 q^{31} -689.908 q^{32} +214.352 q^{34} +59.0326 q^{35} -217.688 q^{37} +800.039 q^{38} +376.600 q^{40} -142.290 q^{41} -468.030 q^{43} +1225.87 q^{44} -128.705 q^{46} -394.318 q^{47} -203.606 q^{49} -136.469 q^{50} +754.126 q^{52} +134.780 q^{53} -281.188 q^{55} +889.268 q^{56} +878.875 q^{58} +131.195 q^{59} +259.801 q^{61} +161.287 q^{62} +1871.88 q^{64} -172.980 q^{65} +445.244 q^{67} -855.954 q^{68} -322.245 q^{70} -560.841 q^{71} -88.6681 q^{73} +1188.30 q^{74} -3194.74 q^{76} -663.970 q^{77} +450.342 q^{79} -1183.85 q^{80} +776.726 q^{82} -284.295 q^{83} +196.337 q^{85} +2554.86 q^{86} -4235.82 q^{88} -625.305 q^{89} -408.459 q^{91} +513.948 q^{92} +2152.49 q^{94} +732.804 q^{95} -193.261 q^{97} +1111.44 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 5 q^{2} + 17 q^{4} - 15 q^{5} - 4 q^{7} - 75 q^{8} + 25 q^{10} - 5 q^{11} + 7 q^{13} - 60 q^{14} + 161 q^{16} - 155 q^{17} - 50 q^{19} - 85 q^{20} - 229 q^{22} - 285 q^{23} + 75 q^{25} - 185 q^{26}+ \cdots + 305 q^{98}+O(q^{100}) \) 3 * q - 5 * q^2 + 17 * q^4 - 15 * q^5 - 4 * q^7 - 75 * q^8 + 25 * q^10 - 5 * q^11 + 7 * q^13 - 60 * q^14 + 161 * q^16 - 155 * q^17 - 50 * q^19 - 85 * q^20 - 229 * q^22 - 285 * q^23 + 75 * q^25 - 185 * q^26 - 334 * q^28 - 115 * q^29 - 115 * q^31 - 775 * q^32 + 413 * q^34 + 20 * q^35 - 384 * q^37 + 1150 * q^38 + 375 * q^40 - 580 * q^41 - 797 * q^43 + 1415 * q^44 - 285 * q^46 + 145 * q^47 + 577 * q^49 - 125 * q^50 + 825 * q^52 + 400 * q^53 + 25 * q^55 + 2190 * q^56 - 59 * q^58 - 380 * q^59 - 152 * q^61 + 1005 * q^62 + 2937 * q^64 - 35 * q^65 + 2 * q^67 - 475 * q^68 + 300 * q^70 - 40 * q^71 - 980 * q^73 + 2720 * q^74 - 3276 * q^76 - 1950 * q^77 + 1013 * q^79 - 805 * q^80 + 4 * q^82 - 270 * q^83 + 775 * q^85 + 1555 * q^86 - 5193 * q^88 - 1020 * q^89 - 632 * q^91 + 1215 * q^92 + 3833 * q^94 + 250 * q^95 + 720 * q^97 + 305 * q^98

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