LMFDB - Newform orbit 175.8.b.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,8,Mod(99,175)]

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, 0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("175.99");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$N$	$=$	$175 = 5^{2} \cdot 7$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	175.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:	$54.6673794597$
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:	$\mathrm{SU}(2)[C_{2}]$

$q$ -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.

$f(q)$	$=$	$q + 6 i q^{2} - 42 i q^{3} + 92 q^{4} + 252 q^{6} - 343 i q^{7} + 1320 i q^{8} + 423 q^{9} - 5568 q^{11} - 3864 i q^{12} - 5152 i q^{13} + 2058 q^{14} + 3856 q^{16} + 13986 i q^{17} + 2538 i q^{18} - 55370 q^{19} + \cdots - 2355264 q^{99} +O(q^{100})$ q + 6i q^2 - 42i q^3 + 92 * q^4 + 252 * q^6 - 343i q^7 + 1320i q^8 + 423 * q^9 - 5568 * q^11 - 3864i q^12 - 5152i q^13 + 2058 * q^14 + 3856 * q^16 + 13986i q^17 + 2538i q^18 - 55370 * q^19 - 14406 * q^21 - 33408i q^22 - 91272i q^23 + 55440 * q^24 + 30912 * q^26 - 109620i q^27 - 31556i q^28 - 41610 * q^29 + 150332 * q^31 + 192096i q^32 + 233856i q^33 - 83916 * q^34 + 38916 * q^36 + 136366i q^37 - 332220i q^38 - 216384 * q^39 - 510258 * q^41 - 86436i q^42 - 172072i q^43 - 512256 * q^44 + 547632 * q^46 + 519036i q^47 - 161952i q^48 - 117649 * q^49 + 587412 * q^51 - 473984i q^52 - 59202i q^53 + 657720 * q^54 + 452760 * q^56 + 2325540i q^57 - 249660i q^58 - 1979250 * q^59 - 2988748 * q^61 + 901992i q^62 - 145089i q^63 - 659008 * q^64 - 1403136 * q^66 - 2409404i q^67 + 1286712i q^68 - 3833424 * q^69 + 1504512 * q^71 + 558360i q^72 - 1821022i q^73 - 818196 * q^74 - 5094040 * q^76 + 1909824i q^77 - 1298304i q^78 + 1669240 * q^79 - 3678939 * q^81 - 3061548i q^82 + 696738i q^83 - 1325352 * q^84 + 1032432 * q^86 + 1747620i q^87 - 7349760i q^88 - 5558490 * q^89 - 1767136 * q^91 - 8397024i q^92 - 6313944i q^93 - 3114216 * q^94 + 8068032 * q^96 - 9876734i q^97 - 705894i q^98 - 2355264 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 184 q^{4} + 504 q^{6} + 846 q^{9} - 11136 q^{11} + 4116 q^{14} + 7712 q^{16} - 110740 q^{19} - 28812 q^{21} + 110880 q^{24} + 61824 q^{26} - 83220 q^{29} + 300664 q^{31} - 167832 q^{34} + 77832 q^{36}+ \cdots - 4710528 q^{99}+O(q^{100})$ 2 * q + 184 * q^4 + 504 * q^6 + 846 * q^9 - 11136 * q^11 + 4116 * q^14 + 7712 * q^16 - 110740 * q^19 - 28812 * q^21 + 110880 * q^24 + 61824 * q^26 - 83220 * q^29 + 300664 * q^31 - 167832 * q^34 + 77832 * q^36 - 432768 * q^39 - 1020516 * q^41 - 1024512 * q^44 + 1095264 * q^46 - 235298 * q^49 + 1174824 * q^51 + 1315440 * q^54 + 905520 * q^56 - 3958500 * q^59 - 5977496 * q^61 - 1318016 * q^64 - 2806272 * q^66 - 7666848 * q^69 + 3009024 * q^71 - 1636392 * q^74 - 10188080 * q^76 + 3338480 * q^79 - 7357878 * q^81 - 2650704 * q^84 + 2064864 * q^86 - 11116980 * q^89 - 3534272 * q^91 - 6228432 * q^94 + 16136064 * q^96 - 4710528 * 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

−

6.00000i

42.0000i

92.0000

252.000

343.000i

−

1320.00i

423.000

99.2

6.00000i

−

42.0000i

92.0000

252.000

−

343.000i

1320.00i

423.000

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	175.8.b.a		2
5.b	even	2	1	inner	175.8.b.a		2
5.c	odd	4	1		7.8.a.a	✓	1
5.c	odd	4	1		175.8.a.a		1
15.e	even	4	1		63.8.a.b		1
20.e	even	4	1		112.8.a.c		1
35.f	even	4	1		49.8.a.b		1
35.k	even	12	2		49.8.c.a		2
35.l	odd	12	2		49.8.c.b		2
40.i	odd	4	1		448.8.a.g		1
40.k	even	4	1		448.8.a.d		1
105.k	odd	4	1		441.8.a.e		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
7.8.a.a	✓	1	5.c	odd	4	1
49.8.a.b		1	35.f	even	4	1
49.8.c.a		2	35.k	even	12	2
49.8.c.b		2	35.l	odd	12	2
63.8.a.b		1	15.e	even	4	1
112.8.a.c		1	20.e	even	4	1
175.8.a.a		1	5.c	odd	4	1
175.8.b.a		2	1.a	even	1	1	trivial
175.8.b.a		2	5.b	even	2	1	inner
441.8.a.e		1	105.k	odd	4	1
448.8.a.d		1	40.k	even	4	1
448.8.a.g		1	40.i	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{2} + 36$ T2^2 + 36 acting on $S_{8}^{\mathrm{new}}(175, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 36$ T^2 + 36
$3$	$T^{2} + 1764$ T^2 + 1764
$5$	$T^{2}$ T^2
$7$	$T^{2} + 117649$ T^2 + 117649
$11$	$(T + 5568)^{2}$ (T + 5568)^2
$13$	$T^{2} + 26543104$ T^2 + 26543104
$17$	$T^{2} + 195608196$ T^2 + 195608196
$19$	$(T + 55370)^{2}$ (T + 55370)^2
$23$	$T^{2} + 8330577984$ T^2 + 8330577984
$29$	$(T + 41610)^{2}$ (T + 41610)^2
$31$	$(T - 150332)^{2}$ (T - 150332)^2
$37$	$T^{2} + 18595685956$ T^2 + 18595685956
$41$	$(T + 510258)^{2}$ (T + 510258)^2
$43$	$T^{2} + 29608773184$ T^2 + 29608773184
$47$	$T^{2} + 269398369296$ T^2 + 269398369296
$53$	$T^{2} + 3504876804$ T^2 + 3504876804
$59$	$(T + 1979250)^{2}$ (T + 1979250)^2
$61$	$(T + 2988748)^{2}$ (T + 2988748)^2
$67$	$T^{2} + 5805227635216$ T^2 + 5805227635216
$71$	$(T - 1504512)^{2}$ (T - 1504512)^2
$73$	$T^{2} + 3316121124484$ T^2 + 3316121124484
$79$	$(T - 1669240)^{2}$ (T - 1669240)^2
$83$	$T^{2} + 485443840644$ T^2 + 485443840644
$89$	$(T + 5558490)^{2}$ (T + 5558490)^2
$97$	$T^{2} + 97549874506756$ T^2 + 97549874506756
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