LMFDB - Newform orbit 280.4.q.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [280,4,Mod(81,280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(280, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

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

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

chi := DirichletCharacter("280.81");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$280 = 2^{3} \cdot 5 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	280.q (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:	$16.5205348016$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
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.

$f(q)$	$=$	$q + (7 \zeta_{6} - 7) q^{3} + 5 \zeta_{6} q^{5} + (21 \zeta_{6} - 7) q^{7} - 22 \zeta_{6} q^{9} + (58 \zeta_{6} - 58) q^{11} + 82 q^{13} - 35 q^{15} + (50 \zeta_{6} - 50) q^{17} - 64 \zeta_{6} q^{19} + ( - 49 \zeta_{6} - 98) q^{21} + \cdots + 1276 q^{99} +O(q^{100})$ q + (7z - 7) q^3 + 5z q^5 + (21z - 7) q^7 - 22z q^9 + (58z - 58) q^11 + 82 * q^13 - 35 * q^15 + (50z - 50) q^17 - 64z q^19 + (-49z - 98) q^21 + 111z q^23 + (25z - 25) q^25 - 35 * q^27 + 103 * q^29 + (-130z + 130) q^31 - 406z q^33 + (70z - 105) q^35 - 376z q^37 + (574z - 574) q^39 - 307 * q^41 - 197 * q^43 + (-110z + 110) q^45 - 120z q^47 + (147z - 392) q^49 - 350z q^51 + (-508z + 508) q^53 - 290 * q^55 + 448 * q^57 + (600z - 600) q^59 + 165z q^61 + (-308z + 462) q^63 + 410z q^65 + (-633z + 633) q^67 - 777 * q^69 + 840 * q^71 + (606z - 606) q^73 - 175z q^75 + (-406z - 812) q^77 + 1316z q^79 + (-839z + 839) q^81 + 61 * q^83 - 250 * q^85 + (721z - 721) q^87 + 187z q^89 + (1722z - 574) q^91 + 910z q^93 + (-320z + 320) q^95 + 406 * q^97 + 1276 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 7 q^{3} + 5 q^{5} + 7 q^{7} - 22 q^{9} - 58 q^{11} + 164 q^{13} - 70 q^{15} - 50 q^{17} - 64 q^{19} - 245 q^{21} + 111 q^{23} - 25 q^{25} - 70 q^{27} + 206 q^{29} + 130 q^{31} - 406 q^{33} - 140 q^{35}+ \cdots + 2552 q^{99}+O(q^{100})$ 2 * q - 7 * q^3 + 5 * q^5 + 7 * q^7 - 22 * q^9 - 58 * q^11 + 164 * q^13 - 70 * q^15 - 50 * q^17 - 64 * q^19 - 245 * q^21 + 111 * q^23 - 25 * q^25 - 70 * q^27 + 206 * q^29 + 130 * q^31 - 406 * q^33 - 140 * q^35 - 376 * q^37 - 574 * q^39 - 614 * q^41 - 394 * q^43 + 110 * q^45 - 120 * q^47 - 637 * q^49 - 350 * q^51 + 508 * q^53 - 580 * q^55 + 896 * q^57 - 600 * q^59 + 165 * q^61 + 616 * q^63 + 410 * q^65 + 633 * q^67 - 1554 * q^69 + 1680 * q^71 - 606 * q^73 - 175 * q^75 - 2030 * q^77 + 1316 * q^79 + 839 * q^81 + 122 * q^83 - 500 * q^85 - 721 * q^87 + 187 * q^89 + 574 * q^91 + 910 * q^93 + 320 * q^95 + 812 * q^97 + 2552 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/280\mathbb{Z}\right)^\times$ .

$n$	$57$	$71$	$141$	$241$
$\chi(n)$	$1$	$1$	$1$	$-\zeta_{6}$

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}

81.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−3.50000

6.06218i

2.50000

4.33013i

3.50000

18.1865i

−11.0000

−

19.0526i

121.1

−3.50000

−

6.06218i

2.50000

−

4.33013i

3.50000

−

18.1865i

−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	280.4.q.a	✓	2
4.b	odd	2	1		560.4.q.f		2
7.c	even	3	1	inner	280.4.q.a	✓	2
7.c	even	3	1		1960.4.a.i		1
7.d	odd	6	1		1960.4.a.c		1
28.g	odd	6	1		560.4.q.f		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
280.4.q.a	✓	2	1.a	even	1	1	trivial
280.4.q.a	✓	2	7.c	even	3	1	inner
560.4.q.f		2	4.b	odd	2	1
560.4.q.f		2	28.g	odd	6	1
1960.4.a.c		1	7.d	odd	6	1
1960.4.a.i		1	7.c	even	3	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{2} + 7T_{3} + 49$ T3^2 + 7*T3 + 49 acting on $S_{4}^{\mathrm{new}}(280, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 7T + 49$ T^2 + 7*T + 49
$5$	$T^{2} - 5T + 25$ T^2 - 5*T + 25
$7$	$T^{2} - 7T + 343$ T^2 - 7*T + 343
$11$	$T^{2} + 58T + 3364$ T^2 + 58*T + 3364
$13$	$(T - 82)^{2}$ (T - 82)^2
$17$	$T^{2} + 50T + 2500$ T^2 + 50*T + 2500
$19$	$T^{2} + 64T + 4096$ T^2 + 64*T + 4096
$23$	$T^{2} - 111T + 12321$ T^2 - 111*T + 12321
$29$	$(T - 103)^{2}$ (T - 103)^2
$31$	$T^{2} - 130T + 16900$ T^2 - 130*T + 16900
$37$	$T^{2} + 376T + 141376$ T^2 + 376*T + 141376
$41$	$(T + 307)^{2}$ (T + 307)^2
$43$	$(T + 197)^{2}$ (T + 197)^2
$47$	$T^{2} + 120T + 14400$ T^2 + 120*T + 14400
$53$	$T^{2} - 508T + 258064$ T^2 - 508*T + 258064
$59$	$T^{2} + 600T + 360000$ T^2 + 600*T + 360000
$61$	$T^{2} - 165T + 27225$ T^2 - 165*T + 27225
$67$	$T^{2} - 633T + 400689$ T^2 - 633*T + 400689
$71$	$(T - 840)^{2}$ (T - 840)^2
$73$	$T^{2} + 606T + 367236$ T^2 + 606*T + 367236
$79$	$T^{2} - 1316 T + 1731856$ T^2 - 1316*T + 1731856
$83$	$(T - 61)^{2}$ (T - 61)^2
$89$	$T^{2} - 187T + 34969$ T^2 - 187*T + 34969
$97$	$(T - 406)^{2}$ (T - 406)^2
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