LMFDB - Newform orbit 350.4.e.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,4,Mod(51,350)] 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("350.51"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 4, names="a")

Level:	$N$	$=$	$350 = 2 \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	350.e (of order $3$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [2,-2,10,-4,0,-40,-28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$20.6506685020$
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:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 70)
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.

$f(q)$	$=$	$q - 2 \zeta_{6} q^{2} + ( - 10 \zeta_{6} + 10) q^{3} + (4 \zeta_{6} - 4) q^{4} - 20 q^{6} + (14 \zeta_{6} - 21) q^{7} + 8 q^{8} - 73 \zeta_{6} q^{9} + (53 \zeta_{6} - 53) q^{11} + 40 \zeta_{6} q^{12} - 25 q^{13} + \cdots + 3869 q^{99} +O(q^{100})$ q - 2z q^2 + (-10z + 10) q^3 + (4z - 4) q^4 - 20 * q^6 + (14z - 21) q^7 + 8 * q^8 - 73z q^9 + (53z - 53) q^11 + 40z q^12 - 25 * q^13 + (14z + 28) q^14 - 16z q^16 + (-14z + 14) q^17 + (146z - 146) q^18 + 95z q^19 + (210z - 70) q^21 + 106 * q^22 + z * q^23 + (-80z + 80) q^24 + 50z q^26 - 460 * q^27 + (-84z + 28) q^28 - 206 * q^29 + (108z - 108) q^31 + (32z - 32) q^32 + 530z q^33 - 28 * q^34 + 292 * q^36 - 57z q^37 + (-190z + 190) q^38 + (250z - 250) q^39 + 243 * q^41 + (-280z + 420) q^42 - 434 * q^43 - 212z q^44 + (-2z + 2) q^46 - 231z q^47 - 160 * q^48 + (-392z + 245) q^49 - 140z q^51 + (-100z + 100) q^52 + (-263z + 263) q^53 + 920z q^54 + (112z - 168) q^56 + 950 * q^57 + 412z q^58 + (24z - 24) q^59 - 116z q^61 + 216 * q^62 + (511z + 1022) q^63 + 64 * q^64 + (-1060z + 1060) q^66 + (204z - 204) q^67 + 56z q^68 + 10 * q^69 + 484 * q^71 - 584z q^72 + (692z - 692) q^73 + (114z - 114) q^74 - 380 * q^76 + (-1113z + 371) q^77 + 500 * q^78 - 466z q^79 + (2629z - 2629) q^81 - 486z q^82 - 228 * q^83 + (-280z - 560) q^84 + 868z q^86 + (2060z - 2060) q^87 + (424z - 424) q^88 + 362z q^89 + (-350z + 525) q^91 - 4 * q^92 + 1080z q^93 + (462z - 462) q^94 + 320z q^96 - 854 * q^97 + (294z - 784) q^98 + 3869 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{2} + 10 q^{3} - 4 q^{4} - 40 q^{6} - 28 q^{7} + 16 q^{8} - 73 q^{9} - 53 q^{11} + 40 q^{12} - 50 q^{13} + 70 q^{14} - 16 q^{16} + 14 q^{17} - 146 q^{18} + 95 q^{19} + 70 q^{21} + 212 q^{22} + q^{23}+ \cdots + 7738 q^{99}+O(q^{100})$ 2 * q - 2 * q^2 + 10 * q^3 - 4 * q^4 - 40 * q^6 - 28 * q^7 + 16 * q^8 - 73 * q^9 - 53 * q^11 + 40 * q^12 - 50 * q^13 + 70 * q^14 - 16 * q^16 + 14 * q^17 - 146 * q^18 + 95 * q^19 + 70 * q^21 + 212 * q^22 + q^23 + 80 * q^24 + 50 * q^26 - 920 * q^27 - 28 * q^28 - 412 * q^29 - 108 * q^31 - 32 * q^32 + 530 * q^33 - 56 * q^34 + 584 * q^36 - 57 * q^37 + 190 * q^38 - 250 * q^39 + 486 * q^41 + 560 * q^42 - 868 * q^43 - 212 * q^44 + 2 * q^46 - 231 * q^47 - 320 * q^48 + 98 * q^49 - 140 * q^51 + 100 * q^52 + 263 * q^53 + 920 * q^54 - 224 * q^56 + 1900 * q^57 + 412 * q^58 - 24 * q^59 - 116 * q^61 + 432 * q^62 + 2555 * q^63 + 128 * q^64 + 1060 * q^66 - 204 * q^67 + 56 * q^68 + 20 * q^69 + 968 * q^71 - 584 * q^72 - 692 * q^73 - 114 * q^74 - 760 * q^76 - 371 * q^77 + 1000 * q^78 - 466 * q^79 - 2629 * q^81 - 486 * q^82 - 456 * q^83 - 1400 * q^84 + 868 * q^86 - 2060 * q^87 - 424 * q^88 + 362 * q^89 + 700 * q^91 - 8 * q^92 + 1080 * q^93 - 462 * q^94 + 320 * q^96 - 1708 * q^97 - 1274 * q^98 + 7738 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$ .

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

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

5.00000

8.66025i

−2.00000

−

3.46410i

−20.0000

−14.0000

−

12.1244i

8.00000

−36.5000

63.2199i

151.1

−1.00000

−

1.73205i

5.00000

−

8.66025i

−2.00000

3.46410i

−20.0000

−14.0000

12.1244i

8.00000

−36.5000

−

63.2199i

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	350.4.e.d		2
5.b	even	2	1		70.4.e.b	✓	2
5.c	odd	4	2		350.4.j.a		4
7.c	even	3	1	inner	350.4.e.d		2
7.c	even	3	1		2450.4.a.v		1
7.d	odd	6	1		2450.4.a.bq		1
15.d	odd	2	1		630.4.k.c		2
20.d	odd	2	1		560.4.q.g		2
35.c	odd	2	1		490.4.e.s		2
35.i	odd	6	1		490.4.a.a		1
35.i	odd	6	1		490.4.e.s		2
35.j	even	6	1		70.4.e.b	✓	2
35.j	even	6	1		490.4.a.h		1
35.l	odd	12	2		350.4.j.a		4
105.o	odd	6	1		630.4.k.c		2
140.p	odd	6	1		560.4.q.g		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.4.e.b	✓	2	5.b	even	2	1
70.4.e.b	✓	2	35.j	even	6	1
350.4.e.d		2	1.a	even	1	1	trivial
350.4.e.d		2	7.c	even	3	1	inner
350.4.j.a		4	5.c	odd	4	2
350.4.j.a		4	35.l	odd	12	2
490.4.a.a		1	35.i	odd	6	1
490.4.a.h		1	35.j	even	6	1
490.4.e.s		2	35.c	odd	2	1
490.4.e.s		2	35.i	odd	6	1
560.4.q.g		2	20.d	odd	2	1
560.4.q.g		2	140.p	odd	6	1
630.4.k.c		2	15.d	odd	2	1
630.4.k.c		2	105.o	odd	6	1
2450.4.a.v		1	7.c	even	3	1
2450.4.a.bq		1	7.d	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(350, [\chi])$ :

T_{3}^{2} - 10T_{3} + 100

T3^2 - 10*T3 + 100

T_{11}^{2} + 53T_{11} + 2809

T11^2 + 53*T11 + 2809

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 2T + 4$ T^2 + 2*T + 4
$3$	$T^{2} - 10T + 100$ T^2 - 10*T + 100
$5$	$T^{2}$ T^2
$7$	$T^{2} + 28T + 343$ T^2 + 28*T + 343
$11$	$T^{2} + 53T + 2809$ T^2 + 53*T + 2809
$13$	$(T + 25)^{2}$ (T + 25)^2
$17$	$T^{2} - 14T + 196$ T^2 - 14*T + 196
$19$	$T^{2} - 95T + 9025$ T^2 - 95*T + 9025
$23$	$T^{2} - T + 1$ T^2 - T + 1
$29$	$(T + 206)^{2}$ (T + 206)^2
$31$	$T^{2} + 108T + 11664$ T^2 + 108*T + 11664
$37$	$T^{2} + 57T + 3249$ T^2 + 57*T + 3249
$41$	$(T - 243)^{2}$ (T - 243)^2
$43$	$(T + 434)^{2}$ (T + 434)^2
$47$	$T^{2} + 231T + 53361$ T^2 + 231*T + 53361
$53$	$T^{2} - 263T + 69169$ T^2 - 263*T + 69169
$59$	$T^{2} + 24T + 576$ T^2 + 24*T + 576
$61$	$T^{2} + 116T + 13456$ T^2 + 116*T + 13456
$67$	$T^{2} + 204T + 41616$ T^2 + 204*T + 41616
$71$	$(T - 484)^{2}$ (T - 484)^2
$73$	$T^{2} + 692T + 478864$ T^2 + 692*T + 478864
$79$	$T^{2} + 466T + 217156$ T^2 + 466*T + 217156
$83$	$(T + 228)^{2}$ (T + 228)^2
$89$	$T^{2} - 362T + 131044$ T^2 - 362*T + 131044
$97$	$(T + 854)^{2}$ (T + 854)^2
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