LMFDB - Newform orbit 525.4.d.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [525,4,Mod(274,525)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("525.274"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$525 = 3 \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	525.d (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-16,0,-24,0,0,-18,0,124] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$30.9760027530$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 21)
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.

$f(q)$	$=$	$q + 4 i q^{2} + 3 i q^{3} - 8 q^{4} - 12 q^{6} - 7 i q^{7} - 9 q^{9} + 62 q^{11} - 24 i q^{12} + 62 i q^{13} + 28 q^{14} - 64 q^{16} + 84 i q^{17} - 36 i q^{18} - 100 q^{19} + 21 q^{21} + 248 i q^{22} + \cdots - 558 q^{99} +O(q^{100})$ q + 4i q^2 + 3i q^3 - 8 * q^4 - 12 * q^6 - 7i q^7 - 9 * q^9 + 62 * q^11 - 24i q^12 + 62i q^13 + 28 * q^14 - 64 * q^16 + 84i q^17 - 36i q^18 - 100 * q^19 + 21 * q^21 + 248i q^22 + 42i q^23 - 248 * q^26 - 27i q^27 + 56i q^28 + 10 * q^29 - 48 * q^31 - 256i q^32 + 186i q^33 - 336 * q^34 + 72 * q^36 - 246i q^37 - 400i q^38 - 186 * q^39 - 248 * q^41 + 84i q^42 - 68i q^43 - 496 * q^44 - 168 * q^46 + 324i q^47 - 192i q^48 - 49 * q^49 - 252 * q^51 - 496i q^52 - 258i q^53 + 108 * q^54 - 300i q^57 + 40i q^58 - 120 * q^59 + 622 * q^61 - 192i q^62 + 63i q^63 + 512 * q^64 - 744 * q^66 + 904i q^67 - 672i q^68 - 126 * q^69 - 678 * q^71 + 642i q^73 + 984 * q^74 + 800 * q^76 - 434i q^77 - 744i q^78 - 740 * q^79 + 81 * q^81 - 992i q^82 - 468i q^83 - 168 * q^84 + 272 * q^86 + 30i q^87 - 200 * q^89 + 434 * q^91 - 336i q^92 - 144i q^93 - 1296 * q^94 + 768 * q^96 - 1266i q^97 - 196i q^98 - 558 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 16 q^{4} - 24 q^{6} - 18 q^{9} + 124 q^{11} + 56 q^{14} - 128 q^{16} - 200 q^{19} + 42 q^{21} - 496 q^{26} + 20 q^{29} - 96 q^{31} - 672 q^{34} + 144 q^{36} - 372 q^{39} - 496 q^{41} - 992 q^{44}+ \cdots - 1116 q^{99}+O(q^{100})$ 2 * q - 16 * q^4 - 24 * q^6 - 18 * q^9 + 124 * q^11 + 56 * q^14 - 128 * q^16 - 200 * q^19 + 42 * q^21 - 496 * q^26 + 20 * q^29 - 96 * q^31 - 672 * q^34 + 144 * q^36 - 372 * q^39 - 496 * q^41 - 992 * q^44 - 336 * q^46 - 98 * q^49 - 504 * q^51 + 216 * q^54 - 240 * q^59 + 1244 * q^61 + 1024 * q^64 - 1488 * q^66 - 252 * q^69 - 1356 * q^71 + 1968 * q^74 + 1600 * q^76 - 1480 * q^79 + 162 * q^81 - 336 * q^84 + 544 * q^86 - 400 * q^89 + 868 * q^91 - 2592 * q^94 + 1536 * q^96 - 1116 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$ .

$n$	$127$	$176$	$451$
$\chi(n)$	$-1$	$1$	$1$

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}

274.1

	−	1.00000i
		1.00000i

−

4.00000i

−

3.00000i

−8.00000

−12.0000

7.00000i

−9.00000

274.2

4.00000i

3.00000i

−8.00000

−12.0000

−

7.00000i

−9.00000

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	525.4.d.b		2
5.b	even	2	1	inner	525.4.d.b		2
5.c	odd	4	1		21.4.a.b	✓	1
5.c	odd	4	1		525.4.a.b		1
15.e	even	4	1		63.4.a.a		1
15.e	even	4	1		1575.4.a.k		1
20.e	even	4	1		336.4.a.h		1
35.f	even	4	1		147.4.a.g		1
35.k	even	12	2		147.4.e.b		2
35.l	odd	12	2		147.4.e.c		2
40.i	odd	4	1		1344.4.a.w		1
40.k	even	4	1		1344.4.a.i		1
60.l	odd	4	1		1008.4.a.m		1
105.k	odd	4	1		441.4.a.b		1
105.w	odd	12	2		441.4.e.n		2
105.x	even	12	2		441.4.e.m		2
140.j	odd	4	1		2352.4.a.l		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.4.a.b	✓	1	5.c	odd	4	1
63.4.a.a		1	15.e	even	4	1
147.4.a.g		1	35.f	even	4	1
147.4.e.b		2	35.k	even	12	2
147.4.e.c		2	35.l	odd	12	2
336.4.a.h		1	20.e	even	4	1
441.4.a.b		1	105.k	odd	4	1
441.4.e.m		2	105.x	even	12	2
441.4.e.n		2	105.w	odd	12	2
525.4.a.b		1	5.c	odd	4	1
525.4.d.b		2	1.a	even	1	1	trivial
525.4.d.b		2	5.b	even	2	1	inner
1008.4.a.m		1	60.l	odd	4	1
1344.4.a.i		1	40.k	even	4	1
1344.4.a.w		1	40.i	odd	4	1
1575.4.a.k		1	15.e	even	4	1
2352.4.a.l		1	140.j	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(525, [\chi])$ :

T_{2}^{2} + 16

T2^2 + 16

T_{11} - 62

T11 - 62

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 16$ T^2 + 16
$3$	$T^{2} + 9$ T^2 + 9
$5$	$T^{2}$ T^2
$7$	$T^{2} + 49$ T^2 + 49
$11$	$(T - 62)^{2}$ (T - 62)^2
$13$	$T^{2} + 3844$ T^2 + 3844
$17$	$T^{2} + 7056$ T^2 + 7056
$19$	$(T + 100)^{2}$ (T + 100)^2
$23$	$T^{2} + 1764$ T^2 + 1764
$29$	$(T - 10)^{2}$ (T - 10)^2
$31$	$(T + 48)^{2}$ (T + 48)^2
$37$	$T^{2} + 60516$ T^2 + 60516
$41$	$(T + 248)^{2}$ (T + 248)^2
$43$	$T^{2} + 4624$ T^2 + 4624
$47$	$T^{2} + 104976$ T^2 + 104976
$53$	$T^{2} + 66564$ T^2 + 66564
$59$	$(T + 120)^{2}$ (T + 120)^2
$61$	$(T - 622)^{2}$ (T - 622)^2
$67$	$T^{2} + 817216$ T^2 + 817216
$71$	$(T + 678)^{2}$ (T + 678)^2
$73$	$T^{2} + 412164$ T^2 + 412164
$79$	$(T + 740)^{2}$ (T + 740)^2
$83$	$T^{2} + 219024$ T^2 + 219024
$89$	$(T + 200)^{2}$ (T + 200)^2
$97$	$T^{2} + 1602756$ T^2 + 1602756
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