LMFDB - Newform orbit 11.6.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [11,6,Mod(1,11)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(11, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("11.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$N$	$=$	$11$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	11.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$1.76422201794$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

f(q)

=

q - 4 q^{2} - 15 q^{3} - 16 q^{4} - 19 q^{5} + 60 q^{6} + 10 q^{7} + 192 q^{8} - 18 q^{9} + 76 q^{10} - 121 q^{11} + 240 q^{12} - 1148 q^{13} - 40 q^{14} + 285 q^{15} - 256 q^{16} + 686 q^{17} + 72 q^{18}+ \cdots + 2178 q^{99}+O(q^{100})

q - 4 * q^2 - 15 * q^3 - 16 * q^4 - 19 * q^5 + 60 * q^6 + 10 * q^7 + 192 * q^8 - 18 * q^9 + 76 * q^10 - 121 * q^11 + 240 * q^12 - 1148 * q^13 - 40 * q^14 + 285 * q^15 - 256 * q^16 + 686 * q^17 + 72 * q^18 - 384 * q^19 + 304 * q^20 - 150 * q^21 + 484 * q^22 + 3709 * q^23 - 2880 * q^24 - 2764 * q^25 + 4592 * q^26 + 3915 * q^27 - 160 * q^28 - 5424 * q^29 - 1140 * q^30 - 6443 * q^31 - 5120 * q^32 + 1815 * q^33 - 2744 * q^34 - 190 * q^35 + 288 * q^36 + 12063 * q^37 + 1536 * q^38 + 17220 * q^39 - 3648 * q^40 - 1528 * q^41 + 600 * q^42 - 4026 * q^43 + 1936 * q^44 + 342 * q^45 - 14836 * q^46 + 7168 * q^47 + 3840 * q^48 - 16707 * q^49 + 11056 * q^50 - 10290 * q^51 + 18368 * q^52 - 29862 * q^53 - 15660 * q^54 + 2299 * q^55 + 1920 * q^56 + 5760 * q^57 + 21696 * q^58 - 6461 * q^59 - 4560 * q^60 - 16980 * q^61 + 25772 * q^62 - 180 * q^63 + 28672 * q^64 + 21812 * q^65 - 7260 * q^66 + 29999 * q^67 - 10976 * q^68 - 55635 * q^69 + 760 * q^70 + 31023 * q^71 - 3456 * q^72 + 1924 * q^73 - 48252 * q^74 + 41460 * q^75 + 6144 * q^76 - 1210 * q^77 - 68880 * q^78 + 65138 * q^79 + 4864 * q^80 - 54351 * q^81 + 6112 * q^82 - 102714 * q^83 + 2400 * q^84 - 13034 * q^85 + 16104 * q^86 + 81360 * q^87 - 23232 * q^88 + 17415 * q^89 - 1368 * q^90 - 11480 * q^91 - 59344 * q^92 + 96645 * q^93 - 28672 * q^94 + 7296 * q^95 + 76800 * q^96 + 66905 * q^97 + 66828 * q^98 + 2178 * q^99

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

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

1.1

−4.00000

−15.0000

−16.0000

−19.0000

60.0000

10.0000

192.000

−18.0000

76.0000

Atkin-Lehner signs

$p$	Sign
$11$	$+1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	11.6.a.a	✓	1
3.b	odd	2	1		99.6.a.c		1
4.b	odd	2	1		176.6.a.c		1
5.b	even	2	1		275.6.a.a		1
5.c	odd	4	2		275.6.b.a		2
7.b	odd	2	1		539.6.a.c		1
8.b	even	2	1		704.6.a.h		1
8.d	odd	2	1		704.6.a.c		1
11.b	odd	2	1		121.6.a.b		1
33.d	even	2	1		1089.6.a.c		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
11.6.a.a	✓	1	1.a	even	1	1	trivial
99.6.a.c		1	3.b	odd	2	1
121.6.a.b		1	11.b	odd	2	1
176.6.a.c		1	4.b	odd	2	1
275.6.a.a		1	5.b	even	2	1
275.6.b.a		2	5.c	odd	4	2
539.6.a.c		1	7.b	odd	2	1
704.6.a.c		1	8.d	odd	2	1
704.6.a.h		1	8.b	even	2	1
1089.6.a.c		1	33.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2} + 4$ T2 + 4 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(11))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 4$ T + 4
$3$	$T + 15$ T + 15
$5$	$T + 19$ T + 19
$7$	$T - 10$ T - 10
$11$	$T + 121$ T + 121
$13$	$T + 1148$ T + 1148
$17$	$T - 686$ T - 686
$19$	$T + 384$ T + 384
$23$	$T - 3709$ T - 3709
$29$	$T + 5424$ T + 5424
$31$	$T + 6443$ T + 6443
$37$	$T - 12063$ T - 12063
$41$	$T + 1528$ T + 1528
$43$	$T + 4026$ T + 4026
$47$	$T - 7168$ T - 7168
$53$	$T + 29862$ T + 29862
$59$	$T + 6461$ T + 6461
$61$	$T + 16980$ T + 16980
$67$	$T - 29999$ T - 29999
$71$	$T - 31023$ T - 31023
$73$	$T - 1924$ T - 1924
$79$	$T - 65138$ T - 65138
$83$	$T + 102714$ T + 102714
$89$	$T - 17415$ T - 17415
$97$	$T - 66905$ T - 66905
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