LMFDB - Newform orbit 175.5.d.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [175,5,Mod(76,175)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 5, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("175.76"); S:= CuspForms(chi, 5); N := Newforms(S);

Level:	$N$	$=$	$175 = 5^{2} \cdot 7$
Weight:	$k$	$=$	$5$
Character orbit:	$[\chi]$	$=$	175.d (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$18.0897435397$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 5$ x^2 - x - 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$3$
Twist minimal:	yes
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 $\beta = \frac{1}{2}(1 + 3\sqrt{21})$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta q^{2} + (\beta + 31) q^{4} + 49 q^{7} + ( - 16 \beta - 47) q^{8} + 81 q^{9} + (16 \beta + 95) q^{11} - 49 \beta q^{14} + (47 \beta + 256) q^{16} - 81 \beta q^{18} + ( - 111 \beta - 752) q^{22} + \cdots + (1296 \beta + 7695) q^{99} +O(q^{100})$ q - b * q^2 + (b + 31) * q^4 + 49 * q^7 + (-16b - 47) q^8 + 81 * q^9 + (16b + 95) q^11 - 49b q^14 + (47b + 256) q^16 - 81b q^18 + (-111b - 752) q^22 + (-96b + 415) q^23 + (49b + 1519) q^28 + (-144b - 545) q^29 + (-47b - 1457) q^32 + (81b + 2511) q^36 + (304b + 495) q^37 + (464b - 65) q^43 + (607b + 3697) q^44 + (-319b + 4512) q^46 + 2401 * q^49 - 5582 * q^53 + (-784b - 2303) q^56 + (689b + 6768) q^58 + 3969 * q^63 + (752b - 1887) q^64 + (944b - 2945) q^67 + (1216b - 2065) q^71 + (-1296b - 3807) q^72 + (-799b - 14288) q^74 + (784b + 4655) q^77 + (-1504b + 2575) q^79 + 6561 * q^81 + (-399b - 21808) q^86 + (-2528b - 16497) q^88 + (-2657b + 8353) q^92 - 2401b q^98 + (1296b + 7695) q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - q^{2} + 63 q^{4} + 98 q^{7} - 110 q^{8} + 162 q^{9} + 206 q^{11} - 49 q^{14} + 559 q^{16} - 81 q^{18} - 1615 q^{22} + 734 q^{23} + 3087 q^{28} - 1234 q^{29} - 2961 q^{32} + 5103 q^{36} + 1294 q^{37}+ \cdots + 16686 q^{99}+O(q^{100})$ 2 * q - q^2 + 63 * q^4 + 98 * q^7 - 110 * q^8 + 162 * q^9 + 206 * q^11 - 49 * q^14 + 559 * q^16 - 81 * q^18 - 1615 * q^22 + 734 * q^23 + 3087 * q^28 - 1234 * q^29 - 2961 * q^32 + 5103 * q^36 + 1294 * q^37 + 334 * q^43 + 8001 * q^44 + 8705 * q^46 + 4802 * q^49 - 11164 * q^53 - 5390 * q^56 + 14225 * q^58 + 7938 * q^63 - 3022 * q^64 - 4946 * q^67 - 2914 * q^71 - 8910 * q^72 - 29375 * q^74 + 10094 * q^77 + 3646 * q^79 + 13122 * q^81 - 44015 * q^86 - 35522 * q^88 + 14049 * q^92 - 2401 * q^98 + 16686 * 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}

76.1

2.79129
−1.79129

−7.37386

38.3739

49.0000

−164.982

81.0000

76.2

6.37386

24.6261

49.0000

54.9818

81.0000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	175.5.d.b	✓	2
5.b	even	2	1		175.5.d.d	yes	2
5.c	odd	4	2		175.5.c.b		4
7.b	odd	2	1	CM	175.5.d.b	✓	2
35.c	odd	2	1		175.5.d.d	yes	2
35.f	even	4	2		175.5.c.b		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
175.5.c.b		4	5.c	odd	4	2
175.5.c.b		4	35.f	even	4	2
175.5.d.b	✓	2	1.a	even	1	1	trivial
175.5.d.b	✓	2	7.b	odd	2	1	CM
175.5.d.d	yes	2	5.b	even	2	1
175.5.d.d	yes	2	35.c	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{2} + T_{2} - 47$ T2^2 + T2 - 47 acting on $S_{5}^{\mathrm{new}}(175, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + T - 47$ T^2 + T - 47
$3$	$T^{2}$ T^2
$5$	$T^{2}$ T^2
$7$	$(T - 49)^{2}$ (T - 49)^2
$11$	$T^{2} - 206T - 1487$ T^2 - 206*T - 1487
$13$	$T^{2}$ T^2
$17$	$T^{2}$ T^2
$19$	$T^{2}$ T^2
$23$	$T^{2} - 734T - 300767$ T^2 - 734*T - 300767
$29$	$T^{2} + 1234 T - 599087$ T^2 + 1234*T - 599087
$31$	$T^{2}$ T^2
$37$	$T^{2} - 1294 T - 3948047$ T^2 - 1294*T - 3948047
$41$	$T^{2}$ T^2
$43$	$T^{2} - 334 T - 10144847$ T^2 - 334*T - 10144847
$47$	$T^{2}$ T^2
$53$	$(T + 5582)^{2}$ (T + 5582)^2
$59$	$T^{2}$ T^2
$61$	$T^{2}$ T^2
$67$	$T^{2} + 4946 T - 35990447$ T^2 + 4946*T - 35990447
$71$	$T^{2} + 2914 T - 67743647$ T^2 + 2914*T - 67743647
$73$	$T^{2}$ T^2
$79$	$T^{2} - 3646 T - 103556927$ T^2 - 3646*T - 103556927
$83$	$T^{2}$ T^2
$89$	$T^{2}$ T^2
$97$	$T^{2}$ T^2
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