LMFDB - Newform orbit 650.4.b.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,4,Mod(599,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("650.599");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$650 = 2 \cdot 5^{2} \cdot 13$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	650.b (of order $2$ , degree $1$ , 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:	$38.3512415037$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 1$ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 26)
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 $i = \sqrt{-1}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - 2 i q^{2} + 4 i q^{3} - 4 q^{4} + 8 q^{6} - 20 i q^{7} + 8 i q^{8} + 11 q^{9} - 48 q^{11} - 16 i q^{12} + 13 i q^{13} - 40 q^{14} + 16 q^{16} - 66 i q^{17} - 22 i q^{18} + 16 q^{19} + 80 q^{21} + 96 i q^{22} + \cdots - 528 q^{99} +O(q^{100})$ q - 2i q^2 + 4i q^3 - 4 * q^4 + 8 * q^6 - 20i q^7 + 8i q^8 + 11 * q^9 - 48 * q^11 - 16i q^12 + 13i q^13 - 40 * q^14 + 16 * q^16 - 66i q^17 - 22i q^18 + 16 * q^19 + 80 * q^21 + 96i q^22 + 168i q^23 - 32 * q^24 + 26 * q^26 + 152i q^27 + 80i q^28 - 6 * q^29 + 20 * q^31 - 32i q^32 - 192i q^33 - 132 * q^34 - 44 * q^36 - 254i q^37 - 32i q^38 - 52 * q^39 - 390 * q^41 - 160i q^42 - 124i q^43 + 192 * q^44 + 336 * q^46 + 468i q^47 + 64i q^48 - 57 * q^49 + 264 * q^51 - 52i q^52 + 558i q^53 + 304 * q^54 + 160 * q^56 + 64i q^57 + 12i q^58 + 96 * q^59 - 826 * q^61 - 40i q^62 - 220i q^63 - 64 * q^64 - 384 * q^66 + 160i q^67 + 264i q^68 - 672 * q^69 - 420 * q^71 + 88i q^72 + 362i q^73 - 508 * q^74 - 64 * q^76 + 960i q^77 + 104i q^78 - 776 * q^79 - 311 * q^81 + 780i q^82 - 320 * q^84 - 248 * q^86 - 24i q^87 - 384i q^88 - 1626 * q^89 + 260 * q^91 - 672i q^92 + 80i q^93 + 936 * q^94 + 128 * q^96 + 1294i q^97 + 114i q^98 - 528 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 8 q^{4} + 16 q^{6} + 22 q^{9} - 96 q^{11} - 80 q^{14} + 32 q^{16} + 32 q^{19} + 160 q^{21} - 64 q^{24} + 52 q^{26} - 12 q^{29} + 40 q^{31} - 264 q^{34} - 88 q^{36} - 104 q^{39} - 780 q^{41} + 384 q^{44}+ \cdots - 1056 q^{99}+O(q^{100})$ 2 * q - 8 * q^4 + 16 * q^6 + 22 * q^9 - 96 * q^11 - 80 * q^14 + 32 * q^16 + 32 * q^19 + 160 * q^21 - 64 * q^24 + 52 * q^26 - 12 * q^29 + 40 * q^31 - 264 * q^34 - 88 * q^36 - 104 * q^39 - 780 * q^41 + 384 * q^44 + 672 * q^46 - 114 * q^49 + 528 * q^51 + 608 * q^54 + 320 * q^56 + 192 * q^59 - 1652 * q^61 - 128 * q^64 - 768 * q^66 - 1344 * q^69 - 840 * q^71 - 1016 * q^74 - 128 * q^76 - 1552 * q^79 - 622 * q^81 - 640 * q^84 - 496 * q^86 - 3252 * q^89 + 520 * q^91 + 1872 * q^94 + 256 * q^96 - 1056 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/650\mathbb{Z}\right)^\times$ .

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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}

599.1

		1.00000i
	−	1.00000i

−

2.00000i

4.00000i

−4.00000

8.00000

−

20.0000i

8.00000i

11.0000

599.2

2.00000i

−

4.00000i

−4.00000

8.00000

20.0000i

−

8.00000i

11.0000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	650.4.b.f		2
5.b	even	2	1	inner	650.4.b.f		2
5.c	odd	4	1		26.4.a.c	✓	1
5.c	odd	4	1		650.4.a.b		1
15.e	even	4	1		234.4.a.e		1
20.e	even	4	1		208.4.a.b		1
35.f	even	4	1		1274.4.a.d		1
40.i	odd	4	1		832.4.a.d		1
40.k	even	4	1		832.4.a.o		1
60.l	odd	4	1		1872.4.a.q		1
65.f	even	4	1		338.4.b.d		2
65.h	odd	4	1		338.4.a.c		1
65.k	even	4	1		338.4.b.d		2
65.o	even	12	2		338.4.e.a		4
65.q	odd	12	2		338.4.c.a		2
65.r	odd	12	2		338.4.c.e		2
65.t	even	12	2		338.4.e.a		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
26.4.a.c	✓	1	5.c	odd	4	1
208.4.a.b		1	20.e	even	4	1
234.4.a.e		1	15.e	even	4	1
338.4.a.c		1	65.h	odd	4	1
338.4.b.d		2	65.f	even	4	1
338.4.b.d		2	65.k	even	4	1
338.4.c.a		2	65.q	odd	12	2
338.4.c.e		2	65.r	odd	12	2
338.4.e.a		4	65.o	even	12	2
338.4.e.a		4	65.t	even	12	2
650.4.a.b		1	5.c	odd	4	1
650.4.b.f		2	1.a	even	1	1	trivial
650.4.b.f		2	5.b	even	2	1	inner
832.4.a.d		1	40.i	odd	4	1
832.4.a.o		1	40.k	even	4	1
1274.4.a.d		1	35.f	even	4	1
1872.4.a.q		1	60.l	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(650, [\chi])$ :

T_{3}^{2} + 16

T_{7}^{2} + 400

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 4$ T^2 + 4
$3$	$T^{2} + 16$ T^2 + 16
$5$	$T^{2}$ T^2
$7$	$T^{2} + 400$ T^2 + 400
$11$	$(T + 48)^{2}$ (T + 48)^2
$13$	$T^{2} + 169$ T^2 + 169
$17$	$T^{2} + 4356$ T^2 + 4356
$19$	$(T - 16)^{2}$ (T - 16)^2
$23$	$T^{2} + 28224$ T^2 + 28224
$29$	$(T + 6)^{2}$ (T + 6)^2
$31$	$(T - 20)^{2}$ (T - 20)^2
$37$	$T^{2} + 64516$ T^2 + 64516
$41$	$(T + 390)^{2}$ (T + 390)^2
$43$	$T^{2} + 15376$ T^2 + 15376
$47$	$T^{2} + 219024$ T^2 + 219024
$53$	$T^{2} + 311364$ T^2 + 311364
$59$	$(T - 96)^{2}$ (T - 96)^2
$61$	$(T + 826)^{2}$ (T + 826)^2
$67$	$T^{2} + 25600$ T^2 + 25600
$71$	$(T + 420)^{2}$ (T + 420)^2
$73$	$T^{2} + 131044$ T^2 + 131044
$79$	$(T + 776)^{2}$ (T + 776)^2
$83$	$T^{2}$ T^2
$89$	$(T + 1626)^{2}$ (T + 1626)^2
$97$	$T^{2} + 1674436$ T^2 + 1674436
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