LMFDB - Newform orbit 175.4.e.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,4,Mod(51,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("175.51");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$175 = 5^{2} \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	175.e (of order $3$ , degree $2$ , 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:	$10.3253342510$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x + 1$ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 7)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$ -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 a primitive root of unity $\zeta_{6}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + 2 \zeta_{6} q^{2} + ( - 7 \zeta_{6} + 7) q^{3} + ( - 4 \zeta_{6} + 4) q^{4} + 14 q^{6} + ( - 14 \zeta_{6} - 7) q^{7} + 24 q^{8} - 22 \zeta_{6} q^{9} + ( - 5 \zeta_{6} + 5) q^{11} - 28 \zeta_{6} q^{12} + \cdots - 110 q^{99} +O(q^{100})$ q + 2z q^2 + (-7z + 7) q^3 + (-4z + 4) q^4 + 14 * q^6 + (-14z - 7) q^7 + 24 * q^8 - 22z q^9 + (-5z + 5) q^11 - 28z q^12 + 14 * q^13 + (-42z + 28) q^14 + 16z q^16 + (21z - 21) q^17 + (-44z + 44) q^18 - 49z q^19 + (49z - 147) q^21 + 10 * q^22 - 159z q^23 + (-168z + 168) q^24 + 28z q^26 + 35 * q^27 + (28z - 84) q^28 + 58 * q^29 + (147z - 147) q^31 + (-160z + 160) q^32 - 35z q^33 - 42 * q^34 - 88 * q^36 + 219z q^37 + (-98z + 98) q^38 + (-98z + 98) q^39 + 350 * q^41 + (-196z - 98) q^42 + 124 * q^43 - 20z q^44 + (-318z + 318) q^46 + 525z q^47 + 112 * q^48 + (392z - 147) q^49 + 147z q^51 + (-56z + 56) q^52 + (-303z + 303) q^53 + 70z q^54 + (-336z - 168) q^56 - 343 * q^57 + 116z q^58 + (-105z + 105) q^59 + 413z q^61 - 294 * q^62 + (462z - 308) q^63 + 448 * q^64 + (-70z + 70) q^66 + (-415z + 415) q^67 + 84z q^68 - 1113 * q^69 - 432 * q^71 - 528z q^72 + (1113z - 1113) q^73 + (438z - 438) q^74 - 196 * q^76 + (35z - 105) q^77 + 196 * q^78 + 103z q^79 + (-839z + 839) q^81 + 700z q^82 - 1092 * q^83 + (588z - 392) q^84 + 248z q^86 + (-406z + 406) q^87 + (-120z + 120) q^88 + 329z q^89 + (-196z - 98) q^91 - 636 * q^92 + 1029z q^93 + (1050z - 1050) q^94 - 1120z q^96 + 882 * q^97 + (490z - 784) q^98 - 110 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{2} + 7 q^{3} + 4 q^{4} + 28 q^{6} - 28 q^{7} + 48 q^{8} - 22 q^{9} + 5 q^{11} - 28 q^{12} + 28 q^{13} + 14 q^{14} + 16 q^{16} - 21 q^{17} + 44 q^{18} - 49 q^{19} - 245 q^{21} + 20 q^{22} - 159 q^{23}+ \cdots - 220 q^{99}+O(q^{100})$ 2 * q + 2 * q^2 + 7 * q^3 + 4 * q^4 + 28 * q^6 - 28 * q^7 + 48 * q^8 - 22 * q^9 + 5 * q^11 - 28 * q^12 + 28 * q^13 + 14 * q^14 + 16 * q^16 - 21 * q^17 + 44 * q^18 - 49 * q^19 - 245 * q^21 + 20 * q^22 - 159 * q^23 + 168 * q^24 + 28 * q^26 + 70 * q^27 - 140 * q^28 + 116 * q^29 - 147 * q^31 + 160 * q^32 - 35 * q^33 - 84 * q^34 - 176 * q^36 + 219 * q^37 + 98 * q^38 + 98 * q^39 + 700 * q^41 - 392 * q^42 + 248 * q^43 - 20 * q^44 + 318 * q^46 + 525 * q^47 + 224 * q^48 + 98 * q^49 + 147 * q^51 + 56 * q^52 + 303 * q^53 + 70 * q^54 - 672 * q^56 - 686 * q^57 + 116 * q^58 + 105 * q^59 + 413 * q^61 - 588 * q^62 - 154 * q^63 + 896 * q^64 + 70 * q^66 + 415 * q^67 + 84 * q^68 - 2226 * q^69 - 864 * q^71 - 528 * q^72 - 1113 * q^73 - 438 * q^74 - 392 * q^76 - 175 * q^77 + 392 * q^78 + 103 * q^79 + 839 * q^81 + 700 * q^82 - 2184 * q^83 - 196 * q^84 + 248 * q^86 + 406 * q^87 + 120 * q^88 + 329 * q^89 - 392 * q^91 - 1272 * q^92 + 1029 * q^93 - 1050 * q^94 - 1120 * q^96 + 1764 * q^97 - 1078 * q^98 - 220 * q^99

Character values

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

$n$	$101$	$127$
$\chi(n)$	$-\zeta_{6}$	$1$

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}

51.1

0.500000	−	0.866025i
0.500000	+	0.866025i

1.00000

−

1.73205i

3.50000

6.06218i

2.00000

3.46410i

14.0000

−14.0000

12.1244i

24.0000

−11.0000

19.0526i

151.1

1.00000

1.73205i

3.50000

−

6.06218i

2.00000

−

3.46410i

14.0000

−14.0000

−

12.1244i

24.0000

−11.0000

−

19.0526i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	175.4.e.a		2
5.b	even	2	1		7.4.c.a	✓	2
5.c	odd	4	2		175.4.k.a		4
7.c	even	3	1	inner	175.4.e.a		2
7.c	even	3	1		1225.4.a.c		1
7.d	odd	6	1		1225.4.a.d		1
15.d	odd	2	1		63.4.e.b		2
20.d	odd	2	1		112.4.i.c		2
35.c	odd	2	1		49.4.c.a		2
35.i	odd	6	1		49.4.a.c		1
35.i	odd	6	1		49.4.c.a		2
35.j	even	6	1		7.4.c.a	✓	2
35.j	even	6	1		49.4.a.d		1
35.l	odd	12	2		175.4.k.a		4
40.e	odd	2	1		448.4.i.a		2
40.f	even	2	1		448.4.i.f		2
105.g	even	2	1		441.4.e.k		2
105.o	odd	6	1		63.4.e.b		2
105.o	odd	6	1		441.4.a.d		1
105.p	even	6	1		441.4.a.e		1
105.p	even	6	1		441.4.e.k		2
140.p	odd	6	1		112.4.i.c		2
140.p	odd	6	1		784.4.a.b		1
140.s	even	6	1		784.4.a.r		1
280.bf	even	6	1		448.4.i.f		2
280.bi	odd	6	1		448.4.i.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
7.4.c.a	✓	2	5.b	even	2	1
7.4.c.a	✓	2	35.j	even	6	1
49.4.a.c		1	35.i	odd	6	1
49.4.a.d		1	35.j	even	6	1
49.4.c.a		2	35.c	odd	2	1
49.4.c.a		2	35.i	odd	6	1
63.4.e.b		2	15.d	odd	2	1
63.4.e.b		2	105.o	odd	6	1
112.4.i.c		2	20.d	odd	2	1
112.4.i.c		2	140.p	odd	6	1
175.4.e.a		2	1.a	even	1	1	trivial
175.4.e.a		2	7.c	even	3	1	inner
175.4.k.a		4	5.c	odd	4	2
175.4.k.a		4	35.l	odd	12	2
441.4.a.d		1	105.o	odd	6	1
441.4.a.e		1	105.p	even	6	1
441.4.e.k		2	105.g	even	2	1
441.4.e.k		2	105.p	even	6	1
448.4.i.a		2	40.e	odd	2	1
448.4.i.a		2	280.bi	odd	6	1
448.4.i.f		2	40.f	even	2	1
448.4.i.f		2	280.bf	even	6	1
784.4.a.b		1	140.p	odd	6	1
784.4.a.r		1	140.s	even	6	1
1225.4.a.c		1	7.c	even	3	1
1225.4.a.d		1	7.d	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{2} - 2T_{2} + 4$ T2^2 - 2*T2 + 4 acting on $S_{4}^{\mathrm{new}}(175, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} - 2T + 4$ T^2 - 2*T + 4
$3$	$T^{2} - 7T + 49$ T^2 - 7*T + 49
$5$	$T^{2}$ T^2
$7$	$T^{2} + 28T + 343$ T^2 + 28*T + 343
$11$	$T^{2} - 5T + 25$ T^2 - 5*T + 25
$13$	$(T - 14)^{2}$ (T - 14)^2
$17$	$T^{2} + 21T + 441$ T^2 + 21*T + 441
$19$	$T^{2} + 49T + 2401$ T^2 + 49*T + 2401
$23$	$T^{2} + 159T + 25281$ T^2 + 159*T + 25281
$29$	$(T - 58)^{2}$ (T - 58)^2
$31$	$T^{2} + 147T + 21609$ T^2 + 147*T + 21609
$37$	$T^{2} - 219T + 47961$ T^2 - 219*T + 47961
$41$	$(T - 350)^{2}$ (T - 350)^2
$43$	$(T - 124)^{2}$ (T - 124)^2
$47$	$T^{2} - 525T + 275625$ T^2 - 525*T + 275625
$53$	$T^{2} - 303T + 91809$ T^2 - 303*T + 91809
$59$	$T^{2} - 105T + 11025$ T^2 - 105*T + 11025
$61$	$T^{2} - 413T + 170569$ T^2 - 413*T + 170569
$67$	$T^{2} - 415T + 172225$ T^2 - 415*T + 172225
$71$	$(T + 432)^{2}$ (T + 432)^2
$73$	$T^{2} + 1113 T + 1238769$ T^2 + 1113*T + 1238769
$79$	$T^{2} - 103T + 10609$ T^2 - 103*T + 10609
$83$	$(T + 1092)^{2}$ (T + 1092)^2
$89$	$T^{2} - 329T + 108241$ T^2 - 329*T + 108241
$97$	$(T - 882)^{2}$ (T - 882)^2
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