LMFDB - Newform orbit 350.4.c.j

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,4,Mod(99,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("350.99");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$350 = 2 \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	350.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:	$20.6506685020$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 70)
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.

$f(q)$	$=$	$q - 2 i q^{2} + 3 i q^{3} - 4 q^{4} + 6 q^{6} - 7 i q^{7} + 8 i q^{8} + 18 q^{9} - 17 q^{11} - 12 i q^{12} + 81 i q^{13} - 14 q^{14} + 16 q^{16} - 91 i q^{17} - 36 i q^{18} - 102 q^{19} + 21 q^{21} + 34 i q^{22} + \cdots - 306 q^{99} +O(q^{100})$ q - 2i q^2 + 3i q^3 - 4 * q^4 + 6 * q^6 - 7i q^7 + 8i q^8 + 18 * q^9 - 17 * q^11 - 12i q^12 + 81i q^13 - 14 * q^14 + 16 * q^16 - 91i q^17 - 36i q^18 - 102 * q^19 + 21 * q^21 + 34i q^22 + 90i q^23 - 24 * q^24 + 162 * q^26 + 135i q^27 + 28i q^28 + 129 * q^29 + 116 * q^31 - 32i q^32 - 51i q^33 - 182 * q^34 - 72 * q^36 + 314i q^37 + 204i q^38 - 243 * q^39 - 124 * q^41 - 42i q^42 + 434i q^43 + 68 * q^44 + 180 * q^46 + 497i q^47 + 48i q^48 - 49 * q^49 + 273 * q^51 - 324i q^52 + 584i q^53 + 270 * q^54 + 56 * q^56 - 306i q^57 - 258i q^58 + 332 * q^59 + 220 * q^61 - 232i q^62 - 126i q^63 - 64 * q^64 - 102 * q^66 + 384i q^67 + 364i q^68 - 270 * q^69 - 664 * q^71 + 144i q^72 - 230i q^73 + 628 * q^74 + 408 * q^76 + 119i q^77 + 486i q^78 - 361 * q^79 + 81 * q^81 + 248i q^82 - 1172i q^83 - 84 * q^84 + 868 * q^86 + 387i q^87 - 136i q^88 - 40 * q^89 + 567 * q^91 - 360i q^92 + 348i q^93 + 994 * q^94 + 96 * q^96 - 175i q^97 + 98i q^98 - 306 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 8 q^{4} + 12 q^{6} + 36 q^{9} - 34 q^{11} - 28 q^{14} + 32 q^{16} - 204 q^{19} + 42 q^{21} - 48 q^{24} + 324 q^{26} + 258 q^{29} + 232 q^{31} - 364 q^{34} - 144 q^{36} - 486 q^{39} - 248 q^{41} + 136 q^{44}+ \cdots - 612 q^{99}+O(q^{100})$ 2 * q - 8 * q^4 + 12 * q^6 + 36 * q^9 - 34 * q^11 - 28 * q^14 + 32 * q^16 - 204 * q^19 + 42 * q^21 - 48 * q^24 + 324 * q^26 + 258 * q^29 + 232 * q^31 - 364 * q^34 - 144 * q^36 - 486 * q^39 - 248 * q^41 + 136 * q^44 + 360 * q^46 - 98 * q^49 + 546 * q^51 + 540 * q^54 + 112 * q^56 + 664 * q^59 + 440 * q^61 - 128 * q^64 - 204 * q^66 - 540 * q^69 - 1328 * q^71 + 1256 * q^74 + 816 * q^76 - 722 * q^79 + 162 * q^81 - 168 * q^84 + 1736 * q^86 - 80 * q^89 + 1134 * q^91 + 1988 * q^94 + 192 * q^96 - 612 * q^99

Character values

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

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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

−

2.00000i

3.00000i

−4.00000

6.00000

−

7.00000i

8.00000i

18.0000

99.2

2.00000i

−

3.00000i

−4.00000

6.00000

7.00000i

−

8.00000i

18.0000

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	350.4.c.j		2
5.b	even	2	1	inner	350.4.c.j		2
5.c	odd	4	1		70.4.a.b	✓	1
5.c	odd	4	1		350.4.a.t		1
15.e	even	4	1		630.4.a.m		1
20.e	even	4	1		560.4.a.k		1
35.f	even	4	1		490.4.a.f		1
35.f	even	4	1		2450.4.a.ba		1
35.k	even	12	2		490.4.e.l		2
35.l	odd	12	2		490.4.e.p		2
40.i	odd	4	1		2240.4.a.w		1
40.k	even	4	1		2240.4.a.p		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.4.a.b	✓	1	5.c	odd	4	1
350.4.a.t		1	5.c	odd	4	1
350.4.c.j		2	1.a	even	1	1	trivial
350.4.c.j		2	5.b	even	2	1	inner
490.4.a.f		1	35.f	even	4	1
490.4.e.l		2	35.k	even	12	2
490.4.e.p		2	35.l	odd	12	2
560.4.a.k		1	20.e	even	4	1
630.4.a.m		1	15.e	even	4	1
2240.4.a.p		1	40.k	even	4	1
2240.4.a.w		1	40.i	odd	4	1
2450.4.a.ba		1	35.f	even	4	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} + 9

T_{11} + 17

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 4$ T^2 + 4
$3$	$T^{2} + 9$ T^2 + 9
$5$	$T^{2}$ T^2
$7$	$T^{2} + 49$ T^2 + 49
$11$	$(T + 17)^{2}$ (T + 17)^2
$13$	$T^{2} + 6561$ T^2 + 6561
$17$	$T^{2} + 8281$ T^2 + 8281
$19$	$(T + 102)^{2}$ (T + 102)^2
$23$	$T^{2} + 8100$ T^2 + 8100
$29$	$(T - 129)^{2}$ (T - 129)^2
$31$	$(T - 116)^{2}$ (T - 116)^2
$37$	$T^{2} + 98596$ T^2 + 98596
$41$	$(T + 124)^{2}$ (T + 124)^2
$43$	$T^{2} + 188356$ T^2 + 188356
$47$	$T^{2} + 247009$ T^2 + 247009
$53$	$T^{2} + 341056$ T^2 + 341056
$59$	$(T - 332)^{2}$ (T - 332)^2
$61$	$(T - 220)^{2}$ (T - 220)^2
$67$	$T^{2} + 147456$ T^2 + 147456
$71$	$(T + 664)^{2}$ (T + 664)^2
$73$	$T^{2} + 52900$ T^2 + 52900
$79$	$(T + 361)^{2}$ (T + 361)^2
$83$	$T^{2} + 1373584$ T^2 + 1373584
$89$	$(T + 40)^{2}$ (T + 40)^2
$97$	$T^{2} + 30625$ T^2 + 30625
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