LMFDB - Newform orbit 891.2.n.l

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [891,2,Mod(136,891)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(891, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([20, 6])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("891.136"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$891 = 3^{4} \cdot 11$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	891.n (of order $15$ , degree $8$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [96,0,0,8] 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:	no
Analytic conductor:	$7.11467082010$
Analytic rank:	$0$
Dimension:	$96$
Relative dimension:	$12$ over $\Q(\zeta_{15})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

$q$ -expansion

The algebraic $q$ -expansion of this newform has not been computed, but we have computed the trace expansion.

\operatorname{Tr}(f)(q) =

96 q + 8 q^{4} + 14 q^{7} + 64 q^{10} + 14 q^{13} + 4 q^{16} - 16 q^{19} - 16 q^{22} + 20 q^{25} - 36 q^{28} + 4 q^{31} - 84 q^{34} - 24 q^{37} + 106 q^{40} - 84 q^{43} - 76 q^{46} + 54 q^{49} - 52 q^{52}+ \cdots + 66 q^{97}+O(q^{100})

96 * q + 8 * q^4 + 14 * q^7 + 64 * q^10 + 14 * q^13 + 4 * q^16 - 16 * q^19 - 16 * q^22 + 20 * q^25 - 36 * q^28 + 4 * q^31 - 84 * q^34 - 24 * q^37 + 106 * q^40 - 84 * q^43 - 76 * q^46 + 54 * q^49 - 52 * q^52 - 72 * q^55 + 10 * q^58 + 6 * q^61 - 104 * q^64 - 76 * q^67 + 8 * q^70 - 92 * q^73 - 204 * q^76 + 102 * q^79 + 24 * q^82 - 54 * q^85 + 44 * q^91 + 118 * q^94 + 66 * q^97

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label	$a_{2}$				$a_{4}$			$a_{5}$				$a_{7}$			$a_{8}$				$a_{10}$
136.1	−2.33604	+	1.04007i	0	3.03706	−	3.37300i	1.88024	+	0.837138i	0	−3.87187	−	0.822992i	−2.00615	+	6.17430i	0	−5.26300
136.2	−2.26297	+	1.00754i	0	2.76762	−	3.07376i	−2.42411	−	1.07928i	0	−0.234155	−	0.0497713i	−1.63516	+	5.03251i	0	6.57309
136.3	−1.67544	+	0.745953i	0	0.912387	−	1.01331i	−0.0824312	−	0.0367007i	0	4.60859	+	0.979587i	0.360704	−	1.11013i	0	0.165485
136.4	−1.24612	+	0.554810i	0	−0.0932499	+	0.103565i	1.89778	+	0.844946i	0	0.101465	+	0.0215671i	0.901773	−	2.77537i	0	−2.83366
136.5	−1.05062	+	0.467766i	0	−0.453266	+	0.503403i	−3.57276	−	1.59069i	0	−4.48724	−	0.953792i	0.951502	−	2.92842i	0	4.49768
136.6	−0.575085	+	0.256045i	0	−1.07310	+	1.19179i	−1.24862	−	0.555922i	0	0.575145	+	0.122251i	0.701028	−	2.15754i	0	0.860404
136.7	0.575085	−	0.256045i	0	−1.07310	+	1.19179i	1.24862	+	0.555922i	0	0.575145	+	0.122251i	−0.701028	+	2.15754i	0	0.860404
136.8	1.05062	−	0.467766i	0	−0.453266	+	0.503403i	3.57276	+	1.59069i	0	−4.48724	−	0.953792i	−0.951502	+	2.92842i	0	4.49768
136.9	1.24612	−	0.554810i	0	−0.0932499	+	0.103565i	−1.89778	−	0.844946i	0	0.101465	+	0.0215671i	−0.901773	+	2.77537i	0	−2.83366
136.10	1.67544	−	0.745953i	0	0.912387	−	1.01331i	0.0824312	+	0.0367007i	0	4.60859	+	0.979587i	−0.360704	+	1.11013i	0	0.165485
136.11	2.26297	−	1.00754i	0	2.76762	−	3.07376i	2.42411	+	1.07928i	0	−0.234155	−	0.0497713i	1.63516	−	5.03251i	0	6.57309
136.12	2.33604	−	1.04007i	0	3.03706	−	3.37300i	−1.88024	−	0.837138i	0	−3.87187	−	0.822992i	2.00615	−	6.17430i	0	−5.26300
190.1	−2.33604	−	1.04007i	0	3.03706	+	3.37300i	1.88024	−	0.837138i	0	−3.87187	+	0.822992i	−2.00615	−	6.17430i	0	−5.26300
190.2	−2.26297	−	1.00754i	0	2.76762	+	3.07376i	−2.42411	+	1.07928i	0	−0.234155	+	0.0497713i	−1.63516	−	5.03251i	0	6.57309
190.3	−1.67544	−	0.745953i	0	0.912387	+	1.01331i	−0.0824312	+	0.0367007i	0	4.60859	−	0.979587i	0.360704	+	1.11013i	0	0.165485
190.4	−1.24612	−	0.554810i	0	−0.0932499	−	0.103565i	1.89778	−	0.844946i	0	0.101465	−	0.0215671i	0.901773	+	2.77537i	0	−2.83366
190.5	−1.05062	−	0.467766i	0	−0.453266	−	0.503403i	−3.57276	+	1.59069i	0	−4.48724	+	0.953792i	0.951502	+	2.92842i	0	4.49768
190.6	−0.575085	−	0.256045i	0	−1.07310	−	1.19179i	−1.24862	+	0.555922i	0	0.575145	−	0.122251i	0.701028	+	2.15754i	0	0.860404
190.7	0.575085	+	0.256045i	0	−1.07310	−	1.19179i	1.24862	−	0.555922i	0	0.575145	−	0.122251i	−0.701028	−	2.15754i	0	0.860404
190.8	1.05062	+	0.467766i	0	−0.453266	−	0.503403i	3.57276	−	1.59069i	0	−4.48724	+	0.953792i	−0.951502	−	2.92842i	0	4.49768

See all 96 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
9.c	even	3	1	inner
9.d	odd	6	1	inner
11.c	even	5	1	inner
33.h	odd	10	1	inner
99.m	even	15	1	inner
99.n	odd	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	891.2.n.l		96
3.b	odd	2	1	inner	891.2.n.l		96
9.c	even	3	1		891.2.f.g	✓	48
9.c	even	3	1	inner	891.2.n.l		96
9.d	odd	6	1		891.2.f.g	✓	48
9.d	odd	6	1	inner	891.2.n.l		96
11.c	even	5	1	inner	891.2.n.l		96
33.h	odd	10	1	inner	891.2.n.l		96
99.m	even	15	1		891.2.f.g	✓	48
99.m	even	15	1	inner	891.2.n.l		96
99.m	even	15	1		9801.2.a.cr		24
99.n	odd	30	1		891.2.f.g	✓	48
99.n	odd	30	1	inner	891.2.n.l		96
99.n	odd	30	1		9801.2.a.cr		24
99.o	odd	30	1		9801.2.a.cq		24
99.p	even	30	1		9801.2.a.cq		24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
891.2.f.g	✓	48	9.c	even	3	1
891.2.f.g	✓	48	9.d	odd	6	1
891.2.f.g	✓	48	99.m	even	15	1
891.2.f.g	✓	48	99.n	odd	30	1
891.2.n.l		96	1.a	even	1	1	trivial
891.2.n.l		96	3.b	odd	2	1	inner
891.2.n.l		96	9.c	even	3	1	inner
891.2.n.l		96	9.d	odd	6	1	inner
891.2.n.l		96	11.c	even	5	1	inner
891.2.n.l		96	33.h	odd	10	1	inner
891.2.n.l		96	99.m	even	15	1	inner
891.2.n.l		96	99.n	odd	30	1	inner
9801.2.a.cq		24	99.o	odd	30	1
9801.2.a.cq		24	99.p	even	30	1
9801.2.a.cr		24	99.m	even	15	1
9801.2.a.cr		24	99.n	odd	30	1

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{96} - 16 T_{2}^{94} + 76 T_{2}^{92} + 556 T_{2}^{90} - 9664 T_{2}^{88} + 60402 T_{2}^{86} + \cdots + 500246412961$ T2^96 - 16*T2^94 + 76*T2^92 + 556*T2^90 - 9664*T2^88 + 60402*T2^86 - 83074*T2^84 - 2275207*T2^82 + 25611534*T2^80 - 109157270*T2^78 - 146188627*T2^76 + 4516819371*T2^74 - 26265449937*T2^72 + 65477480326*T2^70 + 214207750668*T2^68 - 3115071242584*T2^66 + 14792815998015*T2^64 - 25033775309134*T2^62 - 87196092120030*T2^60 + 748164644600098*T2^58 - 2558301125727668*T2^56 + 2547115071734226*T2^54 + 20519217462503014*T2^52 - 98835965139103055*T2^50 + 135845994355200195*T2^48 + 272438250224945747*T2^46 - 1094799760599436016*T2^44 + 1098105712120937902*T2^42 + 1239493923577846777*T2^40 - 4388139161892728298*T2^38 + 6008272692428586847*T2^36 - 2437848994714344417*T2^34 - 9198042903921451910*T2^32 + 21237357269843649555*T2^30 - 10421810243472118643*T2^28 - 15637579373794916369*T2^26 + 25332734055294860617*T2^24 - 6284023342051447971*T2^22 - 2170304829801009965*T2^20 + 1967375561081577791*T2^18 - 346814812254336202*T2^16 - 31808078946495728*T2^14 + 36091315280747386*T2^12 - 14780314224056900*T2^10 + 3873294385005297*T2^8 - 643562248790934*T2^6 + 83806913214799*T2^4 - 8756394108441*T2^2 + 500246412961 acting on $S_{2}^{\mathrm{new}}(891, [\chi])$ .

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Embeddings:		e.g. 1-3 or 136.12
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