LMFDB - Newform orbit 175.4.b.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [175,4,Mod(99,175)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("175.99"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")

Level:	$N$	$=$	$175 = 5^{2} \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	175.b (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,14,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	no
Analytic conductor:	$10.3253342510$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 7)
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.

$f(q)$	$=$	$q + i q^{2} - 2 i q^{3} + 7 q^{4} + 2 q^{6} + 7 i q^{7} + 15 i q^{8} + 23 q^{9} - 8 q^{11} - 14 i q^{12} + 28 i q^{13} - 7 q^{14} + 41 q^{16} - 54 i q^{17} + 23 i q^{18} + 110 q^{19} + 14 q^{21} - 8 i q^{22} + \cdots - 184 q^{99} +O(q^{100})$ q + i * q^2 - 2i q^3 + 7 * q^4 + 2 * q^6 + 7i q^7 + 15i q^8 + 23 * q^9 - 8 * q^11 - 14i q^12 + 28i q^13 - 7 * q^14 + 41 * q^16 - 54i q^17 + 23i q^18 + 110 * q^19 + 14 * q^21 - 8i q^22 + 48i q^23 + 30 * q^24 - 28 * q^26 - 100i q^27 + 49i q^28 + 110 * q^29 + 12 * q^31 + 161i q^32 + 16i q^33 + 54 * q^34 + 161 * q^36 + 246i q^37 + 110i q^38 + 56 * q^39 + 182 * q^41 + 14i q^42 + 128i q^43 - 56 * q^44 - 48 * q^46 - 324i q^47 - 82i q^48 - 49 * q^49 - 108 * q^51 + 196i q^52 - 162i q^53 + 100 * q^54 - 105 * q^56 - 220i q^57 + 110i q^58 - 810 * q^59 - 488 * q^61 + 12i q^62 + 161i q^63 + 167 * q^64 - 16 * q^66 - 244i q^67 - 378i q^68 + 96 * q^69 - 768 * q^71 + 345i q^72 - 702i q^73 - 246 * q^74 + 770 * q^76 - 56i q^77 + 56i q^78 - 440 * q^79 + 421 * q^81 + 182i q^82 - 1302i q^83 + 98 * q^84 - 128 * q^86 - 220i q^87 - 120i q^88 - 730 * q^89 - 196 * q^91 + 336i q^92 - 24i q^93 + 324 * q^94 + 322 * q^96 - 294i q^97 - 49i q^98 - 184 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 14 q^{4} + 4 q^{6} + 46 q^{9} - 16 q^{11} - 14 q^{14} + 82 q^{16} + 220 q^{19} + 28 q^{21} + 60 q^{24} - 56 q^{26} + 220 q^{29} + 24 q^{31} + 108 q^{34} + 322 q^{36} + 112 q^{39} + 364 q^{41} - 112 q^{44}+ \cdots - 368 q^{99}+O(q^{100})$ 2 * q + 14 * q^4 + 4 * q^6 + 46 * q^9 - 16 * q^11 - 14 * q^14 + 82 * q^16 + 220 * q^19 + 28 * q^21 + 60 * q^24 - 56 * q^26 + 220 * q^29 + 24 * q^31 + 108 * q^34 + 322 * q^36 + 112 * q^39 + 364 * q^41 - 112 * q^44 - 96 * q^46 - 98 * q^49 - 216 * q^51 + 200 * q^54 - 210 * q^56 - 1620 * q^59 - 976 * q^61 + 334 * q^64 - 32 * q^66 + 192 * q^69 - 1536 * q^71 - 492 * q^74 + 1540 * q^76 - 880 * q^79 + 842 * q^81 + 196 * q^84 - 256 * q^86 - 1460 * q^89 - 392 * q^91 + 648 * q^94 + 644 * q^96 - 368 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$ .

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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}

99.1

	−	1.00000i
		1.00000i

−

1.00000i

2.00000i

7.00000

2.00000

−

7.00000i

−

15.0000i

23.0000

99.2

1.00000i

−

2.00000i

7.00000

2.00000

7.00000i

15.0000i

23.0000

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	175.4.b.b		2
5.b	even	2	1	inner	175.4.b.b		2
5.c	odd	4	1		7.4.a.a	✓	1
5.c	odd	4	1		175.4.a.b		1
15.e	even	4	1		63.4.a.b		1
15.e	even	4	1		1575.4.a.e		1
20.e	even	4	1		112.4.a.f		1
35.f	even	4	1		49.4.a.b		1
35.f	even	4	1		1225.4.a.j		1
35.k	even	12	2		49.4.c.b		2
35.l	odd	12	2		49.4.c.c		2
40.i	odd	4	1		448.4.a.i		1
40.k	even	4	1		448.4.a.e		1
55.e	even	4	1		847.4.a.b		1
60.l	odd	4	1		1008.4.a.c		1
65.h	odd	4	1		1183.4.a.b		1
85.g	odd	4	1		2023.4.a.a		1
105.k	odd	4	1		441.4.a.i		1
105.w	odd	12	2		441.4.e.e		2
105.x	even	12	2		441.4.e.h		2
140.j	odd	4	1		784.4.a.g		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
7.4.a.a	✓	1	5.c	odd	4	1
49.4.a.b		1	35.f	even	4	1
49.4.c.b		2	35.k	even	12	2
49.4.c.c		2	35.l	odd	12	2
63.4.a.b		1	15.e	even	4	1
112.4.a.f		1	20.e	even	4	1
175.4.a.b		1	5.c	odd	4	1
175.4.b.b		2	1.a	even	1	1	trivial
175.4.b.b		2	5.b	even	2	1	inner
441.4.a.i		1	105.k	odd	4	1
441.4.e.e		2	105.w	odd	12	2
441.4.e.h		2	105.x	even	12	2
448.4.a.e		1	40.k	even	4	1
448.4.a.i		1	40.i	odd	4	1
784.4.a.g		1	140.j	odd	4	1
847.4.a.b		1	55.e	even	4	1
1008.4.a.c		1	60.l	odd	4	1
1183.4.a.b		1	65.h	odd	4	1
1225.4.a.j		1	35.f	even	4	1
1575.4.a.e		1	15.e	even	4	1
2023.4.a.a		1	85.g	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(175, [\chi])$ :

T_{2}^{2} + 1

T2^2 + 1

T_{3}^{2} + 4

T3^2 + 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 1$ T^2 + 1
$3$	$T^{2} + 4$ T^2 + 4
$5$	$T^{2}$ T^2
$7$	$T^{2} + 49$ T^2 + 49
$11$	$(T + 8)^{2}$ (T + 8)^2
$13$	$T^{2} + 784$ T^2 + 784
$17$	$T^{2} + 2916$ T^2 + 2916
$19$	$(T - 110)^{2}$ (T - 110)^2
$23$	$T^{2} + 2304$ T^2 + 2304
$29$	$(T - 110)^{2}$ (T - 110)^2
$31$	$(T - 12)^{2}$ (T - 12)^2
$37$	$T^{2} + 60516$ T^2 + 60516
$41$	$(T - 182)^{2}$ (T - 182)^2
$43$	$T^{2} + 16384$ T^2 + 16384
$47$	$T^{2} + 104976$ T^2 + 104976
$53$	$T^{2} + 26244$ T^2 + 26244
$59$	$(T + 810)^{2}$ (T + 810)^2
$61$	$(T + 488)^{2}$ (T + 488)^2
$67$	$T^{2} + 59536$ T^2 + 59536
$71$	$(T + 768)^{2}$ (T + 768)^2
$73$	$T^{2} + 492804$ T^2 + 492804
$79$	$(T + 440)^{2}$ (T + 440)^2
$83$	$T^{2} + 1695204$ T^2 + 1695204
$89$	$(T + 730)^{2}$ (T + 730)^2
$97$	$T^{2} + 86436$ T^2 + 86436
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

$n$ :		e.g. 2-40 or 990-1000
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more