LMFDB - Newform orbit 18.13.b.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [18,13,Mod(17,18)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(18, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 13, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("18.17");

S:= CuspForms(chi, 13);

N := Newforms(S);

Level:	$N$	$=$	$18 = 2 \cdot 3^{2}$
Weight:	$k$	$=$	$13$
Character orbit:	$[\chi]$	$=$	18.b (of order $2$ , degree $1$ , minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$16.4518887110$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 2$ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of $\beta = \sqrt{-2}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + 32 \beta q^{2} - 2048 q^{4} + 14565 \beta q^{5} + 33572 q^{7} - 65536 \beta q^{8} - 932160 q^{10} + 1211892 \beta q^{11} - 5196976 q^{13} + 1074304 \beta q^{14} + 4194304 q^{16} - 22150593 \beta q^{17} + \cdots - 406854656544 \beta q^{98} +O(q^{100})$ q + 32b q^2 - 2048 * q^4 + 14565b q^5 + 33572 * q^7 - 65536b q^8 - 932160 * q^10 + 1211892b q^11 - 5196976 * q^13 + 1074304b q^14 + 4194304 * q^16 - 22150593b q^17 - 66057424 * q^19 - 29829120b q^20 - 77561088 * q^22 - 121799796b q^23 - 180137825 * q^25 - 166303232b q^26 - 68755456 * q^28 + 286119627b q^29 - 1730932396 * q^31 + 134217728b q^32 + 1417637952 * q^34 + 488976180b q^35 + 4291562342 * q^37 - 2113837568b q^38 + 1909063680 * q^40 + 5956699233b q^41 + 2713386440 * q^43 - 2481954816b q^44 + 7795186944 * q^46 + 5663026140b q^47 - 12714208017 * q^49 - 5764410400b q^50 + 10643406848 * q^52 + 8205930171b q^53 - 35302413960 * q^55 - 2200174592b q^56 - 18311656128 * q^58 - 7634221944b q^59 - 37496538790 * q^61 - 55389836672b q^62 - 8589934592 * q^64 - 75693955440b q^65 + 74662225976 * q^67 + 45364414464b q^68 - 31294475520 * q^70 + 50984072964b q^71 - 72295502128 * q^73 + 137329994944b q^74 + 135285604352 * q^76 + 40685638224b q^77 + 317431332236 * q^79 + 61090037760b q^80 - 381228750912 * q^82 - 86508745428b q^83 + 645246774090 * q^85 + 86828366080b q^86 + 158845108224 * q^88 + 167489026641b q^89 - 174472878272 * q^91 + 249445982208b q^92 - 362433672960 * q^94 - 962126380560b q^95 - 134693309248 * q^97 - 406854656544b q^98
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 4096 q^{4} + 67144 q^{7} - 1864320 q^{10} - 10393952 q^{13} + 8388608 q^{16} - 132114848 q^{19} - 155122176 q^{22} - 360275650 q^{25} - 137510912 q^{28} - 3461864792 q^{31} + 2835275904 q^{34} + 8583124684 q^{37}+ \cdots - 269386618496 q^{97}+O(q^{100})$ 2 * q - 4096 * q^4 + 67144 * q^7 - 1864320 * q^10 - 10393952 * q^13 + 8388608 * q^16 - 132114848 * q^19 - 155122176 * q^22 - 360275650 * q^25 - 137510912 * q^28 - 3461864792 * q^31 + 2835275904 * q^34 + 8583124684 * q^37 + 3818127360 * q^40 + 5426772880 * q^43 + 15590373888 * q^46 - 25428416034 * q^49 + 21286813696 * q^52 - 70604827920 * q^55 - 36623312256 * q^58 - 74993077580 * q^61 - 17179869184 * q^64 + 149324451952 * q^67 - 62588951040 * q^70 - 144591004256 * q^73 + 270571208704 * q^76 + 634862664472 * q^79 - 762457501824 * q^82 + 1290493548180 * q^85 + 317690216448 * q^88 - 348945756544 * q^91 - 724867345920 * q^94 - 269386618496 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/18\mathbb{Z}\right)^\times$ .

$n$	$11$
$\chi(n)$	$-1$

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}

17.1

	−	1.41421i
		1.41421i

−

45.2548i

−2048.00

−

20598.0i

33572.0

92681.9i

−932160.

17.2

45.2548i

−2048.00

20598.0i

33572.0

−

92681.9i

−932160.

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	18.13.b.b	✓	2
3.b	odd	2	1	inner	18.13.b.b	✓	2
4.b	odd	2	1		144.13.e.b		2
9.c	even	3	2		162.13.d.a		4
9.d	odd	6	2		162.13.d.a		4
12.b	even	2	1		144.13.e.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
18.13.b.b	✓	2	1.a	even	1	1	trivial
18.13.b.b	✓	2	3.b	odd	2	1	inner
144.13.e.b		2	4.b	odd	2	1
144.13.e.b		2	12.b	even	2	1
162.13.d.a		4	9.c	even	3	2
162.13.d.a		4	9.d	odd	6	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{2} + 424278450$ T5^2 + 424278450 acting on $S_{13}^{\mathrm{new}}(18, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 2048$ T^2 + 2048
$3$	$T^{2}$ T^2
$5$	$T^{2} + 424278450$ T^2 + 424278450
$7$	$(T - 33572)^{2}$ (T - 33572)^2
$11$	$T^{2} + 2937364439328$ T^2 + 2937364439328
$13$	$(T + 5196976)^{2}$ (T + 5196976)^2
$17$	$T^{2} + 981297540503298$ T^2 + 981297540503298
$19$	$(T + 66057424)^{2}$ (T + 66057424)^2
$23$	$T^{2} + 29\!\cdots\!32$ T^2 + 29670380611283232
$29$	$T^{2} + 16\!\cdots\!58$ T^2 + 163728881909238258
$31$	$(T + 1730932396)^{2}$ (T + 1730932396)^2
$37$	$(T - 4291562342)^{2}$ (T - 4291562342)^2
$41$	$T^{2} + 70\!\cdots\!78$ T^2 + 70964531504845576578
$43$	$(T - 2713386440)^{2}$ (T - 2713386440)^2
$47$	$T^{2} + 64\!\cdots\!00$ T^2 + 64139730124646599200
$53$	$T^{2} + 13\!\cdots\!82$ T^2 + 134674579942656178482
$59$	$T^{2} + 11\!\cdots\!72$ T^2 + 116562689380502278272
$61$	$(T + 37496538790)^{2}$ (T + 37496538790)^2
$67$	$(T - 74662225976)^{2}$ (T - 74662225976)^2
$71$	$T^{2} + 51\!\cdots\!92$ T^2 + 5198751391996951490592
$73$	$(T + 72295502128)^{2}$ (T + 72295502128)^2
$79$	$(T - 317431332236)^{2}$ (T - 317431332236)^2
$83$	$T^{2} + 14\!\cdots\!68$ T^2 + 14967526071053021806368
$89$	$T^{2} + 56\!\cdots\!62$ T^2 + 56105148090299215485762
$97$	$(T + 134693309248)^{2}$ (T + 134693309248)^2
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