LMFDB - Newform orbit 175.7.c.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,7,Mod(174,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("175.174");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$N$	$=$	$175 = 5^{2} \cdot 7$
Weight:	$k$	$=$	$7$
Character orbit:	$[\chi]$	$=$	175.c (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:	$40.2594646335$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 7)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$ -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.

$f(q)$	$=$	$q + 9 i q^{2} - 17 q^{4} - 343 i q^{7} + 423 i q^{8} - 729 q^{9} + 1962 q^{11} + 3087 q^{14} - 4895 q^{16} - 6561 i q^{18} + 17658 i q^{22} + 22734 i q^{23} + 5831 i q^{28} + 21222 q^{29} - 16983 i q^{32} + \cdots - 1430298 q^{99} +O(q^{100})$ q + 9i q^2 - 17 * q^4 - 343i q^7 + 423i q^8 - 729 * q^9 + 1962 * q^11 + 3087 * q^14 - 4895 * q^16 - 6561i q^18 + 17658i q^22 + 22734i q^23 + 5831i q^28 + 21222 * q^29 - 16983i q^32 + 12393 * q^36 + 101194i q^37 + 126614i q^43 - 33354 * q^44 - 204606 * q^46 - 117649 * q^49 - 50346i q^53 + 145089 * q^56 + 190998i q^58 + 250047i q^63 - 160433 * q^64 - 53926i q^67 - 242478 * q^71 - 308367i q^72 - 910746 * q^74 - 672966i q^77 - 929378 * q^79 + 531441 * q^81 - 1139526 * q^86 + 829926i q^88 - 386478i q^92 - 1058841i q^98 - 1430298 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 34 q^{4} - 1458 q^{9} + 3924 q^{11} + 6174 q^{14} - 9790 q^{16} + 42444 q^{29} + 24786 q^{36} - 66708 q^{44} - 409212 q^{46} - 235298 q^{49} + 290178 q^{56} - 320866 q^{64} - 484956 q^{71} - 1821492 q^{74}+ \cdots - 2860596 q^{99}+O(q^{100})$ 2 * q - 34 * q^4 - 1458 * q^9 + 3924 * q^11 + 6174 * q^14 - 9790 * q^16 + 42444 * q^29 + 24786 * q^36 - 66708 * q^44 - 409212 * q^46 - 235298 * q^49 + 290178 * q^56 - 320866 * q^64 - 484956 * q^71 - 1821492 * q^74 - 1858756 * q^79 + 1062882 * q^81 - 2279052 * q^86 - 2860596 * 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}

174.1

	−	1.00000i
		1.00000i

−

9.00000i

−17.0000

343.000i

−

423.000i

−729.000

174.2

9.00000i

−17.0000

−

343.000i

423.000i

−729.000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7})$
5.b	even	2	1	inner
35.c	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	175.7.c.a		2
5.b	even	2	1	inner	175.7.c.a		2
5.c	odd	4	1		7.7.b.a	✓	1
5.c	odd	4	1		175.7.d.a		1
7.b	odd	2	1	CM	175.7.c.a		2
15.e	even	4	1		63.7.d.a		1
20.e	even	4	1		112.7.c.a		1
35.c	odd	2	1	inner	175.7.c.a		2
35.f	even	4	1		7.7.b.a	✓	1
35.f	even	4	1		175.7.d.a		1
35.k	even	12	2		49.7.d.a		2
35.l	odd	12	2		49.7.d.a		2
40.i	odd	4	1		448.7.c.a		1
40.k	even	4	1		448.7.c.b		1
105.k	odd	4	1		63.7.d.a		1
140.j	odd	4	1		112.7.c.a		1
280.s	even	4	1		448.7.c.a		1
280.y	odd	4	1		448.7.c.b		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
7.7.b.a	✓	1	5.c	odd	4	1
7.7.b.a	✓	1	35.f	even	4	1
49.7.d.a		2	35.k	even	12	2
49.7.d.a		2	35.l	odd	12	2
63.7.d.a		1	15.e	even	4	1
63.7.d.a		1	105.k	odd	4	1
112.7.c.a		1	20.e	even	4	1
112.7.c.a		1	140.j	odd	4	1
175.7.c.a		2	1.a	even	1	1	trivial
175.7.c.a		2	5.b	even	2	1	inner
175.7.c.a		2	7.b	odd	2	1	CM
175.7.c.a		2	35.c	odd	2	1	inner
175.7.d.a		1	5.c	odd	4	1
175.7.d.a		1	35.f	even	4	1
448.7.c.a		1	40.i	odd	4	1
448.7.c.a		1	280.s	even	4	1
448.7.c.b		1	40.k	even	4	1
448.7.c.b		1	280.y	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{2} + 81$ T2^2 + 81 acting on $S_{7}^{\mathrm{new}}(175, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 81$ T^2 + 81
$3$	$T^{2}$ T^2
$5$	$T^{2}$ T^2
$7$	$T^{2} + 117649$ T^2 + 117649
$11$	$(T - 1962)^{2}$ (T - 1962)^2
$13$	$T^{2}$ T^2
$17$	$T^{2}$ T^2
$19$	$T^{2}$ T^2
$23$	$T^{2} + 516834756$ T^2 + 516834756
$29$	$(T - 21222)^{2}$ (T - 21222)^2
$31$	$T^{2}$ T^2
$37$	$T^{2} + 10240225636$ T^2 + 10240225636
$41$	$T^{2}$ T^2
$43$	$T^{2} + 16031104996$ T^2 + 16031104996
$47$	$T^{2}$ T^2
$53$	$T^{2} + 2534719716$ T^2 + 2534719716
$59$	$T^{2}$ T^2
$61$	$T^{2}$ T^2
$67$	$T^{2} + 2908013476$ T^2 + 2908013476
$71$	$(T + 242478)^{2}$ (T + 242478)^2
$73$	$T^{2}$ T^2
$79$	$(T + 929378)^{2}$ (T + 929378)^2
$83$	$T^{2}$ T^2
$89$	$T^{2}$ T^2
$97$	$T^{2}$ T^2
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