LMFDB - Newform orbit 405.4.e.j

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(136,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("405.136");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$405 = 3^{4} \cdot 5$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	405.e (of order $3$ , degree $2$ , not 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:	$23.8957735523$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x + 1$ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$ -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 a primitive root of unity $\zeta_{6}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - 2 \zeta_{6} + 2) q^{2} + 4 \zeta_{6} q^{4} - 5 \zeta_{6} q^{5} + 24 q^{8} - 10 q^{10} + (10 \zeta_{6} - 10) q^{11} + 80 \zeta_{6} q^{13} + ( - 16 \zeta_{6} + 16) q^{16} + 7 q^{17} - 113 q^{19} + \cdots + 686 q^{98} +O(q^{100})$ q + (-2z + 2) q^2 + 4z q^4 - 5z q^5 + 24 * q^8 - 10 * q^10 + (10z - 10) q^11 + 80z q^13 + (-16z + 16) q^16 + 7 * q^17 - 113 * q^19 + (-20z + 20) q^20 + 20z q^22 + 81z q^23 + (25z - 25) q^25 + 160 * q^26 + (-220z + 220) q^29 + 189z q^31 + 160z q^32 + (-14z + 14) q^34 + 170 * q^37 + (226z - 226) q^38 - 120z q^40 + 130z q^41 + (10z - 10) q^43 - 40 * q^44 + 162 * q^46 + (160z - 160) q^47 + 343z q^49 + 50z q^50 + (320z - 320) q^52 + 631 * q^53 + 50 * q^55 - 440z q^58 + 560z q^59 + (229z - 229) q^61 + 378 * q^62 + 448 * q^64 + (-400z + 400) q^65 - 750z q^67 + 28z q^68 + 890 * q^71 - 890 * q^73 + (-340z + 340) q^74 - 452z q^76 + (-27z + 27) q^79 - 80 * q^80 + 260 * q^82 + (429z - 429) q^83 - 35z q^85 + 20z q^86 + (240z - 240) q^88 - 750 * q^89 + (324z - 324) q^92 + 320z q^94 + 565z q^95 + (-1480z + 1480) q^97 + 686 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{2} + 4 q^{4} - 5 q^{5} + 48 q^{8} - 20 q^{10} - 10 q^{11} + 80 q^{13} + 16 q^{16} + 14 q^{17} - 226 q^{19} + 20 q^{20} + 20 q^{22} + 81 q^{23} - 25 q^{25} + 320 q^{26} + 220 q^{29} + 189 q^{31}+ \cdots + 1372 q^{98}+O(q^{100})$ 2 * q + 2 * q^2 + 4 * q^4 - 5 * q^5 + 48 * q^8 - 20 * q^10 - 10 * q^11 + 80 * q^13 + 16 * q^16 + 14 * q^17 - 226 * q^19 + 20 * q^20 + 20 * q^22 + 81 * q^23 - 25 * q^25 + 320 * q^26 + 220 * q^29 + 189 * q^31 + 160 * q^32 + 14 * q^34 + 340 * q^37 - 226 * q^38 - 120 * q^40 + 130 * q^41 - 10 * q^43 - 80 * q^44 + 324 * q^46 - 160 * q^47 + 343 * q^49 + 50 * q^50 - 320 * q^52 + 1262 * q^53 + 100 * q^55 - 440 * q^58 + 560 * q^59 - 229 * q^61 + 756 * q^62 + 896 * q^64 + 400 * q^65 - 750 * q^67 + 28 * q^68 + 1780 * q^71 - 1780 * q^73 + 340 * q^74 - 452 * q^76 + 27 * q^79 - 160 * q^80 + 520 * q^82 - 429 * q^83 - 35 * q^85 + 20 * q^86 - 240 * q^88 - 1500 * q^89 - 324 * q^92 + 320 * q^94 + 565 * q^95 + 1480 * q^97 + 1372 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$ .

$n$	$82$	$326$
$\chi(n)$	$1$	$-\zeta_{6}$

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}

136.1

0.500000	+	0.866025i
0.500000	−	0.866025i

1.00000

−

1.73205i

2.00000

3.46410i

−2.50000

−

4.33013i

24.0000

−10.0000

271.1

1.00000

1.73205i

2.00000

−

3.46410i

−2.50000

4.33013i

24.0000

−10.0000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	405.4.e.j		2
3.b	odd	2	1		405.4.e.e		2
9.c	even	3	1		135.4.a.a	✓	1
9.c	even	3	1	inner	405.4.e.j		2
9.d	odd	6	1		135.4.a.d	yes	1
9.d	odd	6	1		405.4.e.e		2
36.f	odd	6	1		2160.4.a.n		1
36.h	even	6	1		2160.4.a.d		1
45.h	odd	6	1		675.4.a.b		1
45.j	even	6	1		675.4.a.i		1
45.k	odd	12	2		675.4.b.d		2
45.l	even	12	2		675.4.b.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
135.4.a.a	✓	1	9.c	even	3	1
135.4.a.d	yes	1	9.d	odd	6	1
405.4.e.e		2	3.b	odd	2	1
405.4.e.e		2	9.d	odd	6	1
405.4.e.j		2	1.a	even	1	1	trivial
405.4.e.j		2	9.c	even	3	1	inner
675.4.a.b		1	45.h	odd	6	1
675.4.a.i		1	45.j	even	6	1
675.4.b.c		2	45.l	even	12	2
675.4.b.d		2	45.k	odd	12	2
2160.4.a.d		1	36.h	even	6	1
2160.4.a.n		1	36.f	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(405, [\chi])$ :

T_{2}^{2} - 2T_{2} + 4

T_{7}

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} - 2T + 4$ T^2 - 2*T + 4
$3$	$T^{2}$ T^2
$5$	$T^{2} + 5T + 25$ T^2 + 5*T + 25
$7$	$T^{2}$ T^2
$11$	$T^{2} + 10T + 100$ T^2 + 10*T + 100
$13$	$T^{2} - 80T + 6400$ T^2 - 80*T + 6400
$17$	$(T - 7)^{2}$ (T - 7)^2
$19$	$(T + 113)^{2}$ (T + 113)^2
$23$	$T^{2} - 81T + 6561$ T^2 - 81*T + 6561
$29$	$T^{2} - 220T + 48400$ T^2 - 220*T + 48400
$31$	$T^{2} - 189T + 35721$ T^2 - 189*T + 35721
$37$	$(T - 170)^{2}$ (T - 170)^2
$41$	$T^{2} - 130T + 16900$ T^2 - 130*T + 16900
$43$	$T^{2} + 10T + 100$ T^2 + 10*T + 100
$47$	$T^{2} + 160T + 25600$ T^2 + 160*T + 25600
$53$	$(T - 631)^{2}$ (T - 631)^2
$59$	$T^{2} - 560T + 313600$ T^2 - 560*T + 313600
$61$	$T^{2} + 229T + 52441$ T^2 + 229*T + 52441
$67$	$T^{2} + 750T + 562500$ T^2 + 750*T + 562500
$71$	$(T - 890)^{2}$ (T - 890)^2
$73$	$(T + 890)^{2}$ (T + 890)^2
$79$	$T^{2} - 27T + 729$ T^2 - 27*T + 729
$83$	$T^{2} + 429T + 184041$ T^2 + 429*T + 184041
$89$	$(T + 750)^{2}$ (T + 750)^2
$97$	$T^{2} - 1480 T + 2190400$ T^2 - 1480*T + 2190400
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