LMFDB - Newform orbit 45.5.g.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [45,5,Mod(28,45)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3])) N = Newforms(chi, 5, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("45.28"); S:= CuspForms(chi, 5); N := Newforms(S);

Level:	$N$	$=$	$45 = 3^{2} \cdot 5$
Weight:	$k$	$=$	$5$
Character orbit:	$[\chi]$	$=$	45.g (of order $4$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [2,10] 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:	no
Analytic conductor:	$4.65164833877$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$ -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.

$f(q)$	$=$	$q + (5 i + 5) q^{2} + 34 i q^{4} - 25 q^{5} + (40 i + 40) q^{7} + (90 i - 90) q^{8} + ( - 125 i - 125) q^{10} + 100 q^{11} + ( - 205 i + 205) q^{13} + 400 i q^{14} - 356 q^{16} + ( - 235 i - 235) q^{17}+ \cdots + (3995 i - 3995) q^{98}+O(q^{100})$ q + (5i + 5) q^2 + 34i q^4 - 25 * q^5 + (40i + 40) q^7 + (90i - 90) q^8 + (-125i - 125) q^10 + 100 * q^11 + (-205i + 205) q^13 + 400i q^14 - 356 * q^16 + (-235i - 235) q^17 + 72i q^19 - 850i q^20 + (500i + 500) q^22 + (340i - 340) q^23 + 625 * q^25 + 2050 * q^26 + (1360i - 1360) q^28 - 450i q^29 + 428 * q^31 + (-340i - 340) q^32 - 2350i q^34 + (-1000i - 1000) q^35 + (-755i - 755) q^37 + (360i - 360) q^38 + (-2250i + 2250) q^40 - 950 * q^41 + (1220i - 1220) q^43 + 3400i q^44 - 3400 * q^46 + (320i + 320) q^47 + 799i q^49 + (3125i + 3125) q^50 + (6970i + 6970) q^52 + (505i - 505) q^53 - 2500 * q^55 - 7200 * q^56 + (-2250i + 2250) q^58 - 6300i q^59 - 3808 * q^61 + (2140i + 2140) q^62 + 2296i q^64 + (5125i - 5125) q^65 + (340i + 340) q^67 + (-7990i + 7990) q^68 - 10000i q^70 + 3400 * q^71 + (-415i + 415) q^73 - 7550i q^74 - 2448 * q^76 + (4000i + 4000) q^77 - 6732i q^79 + 8900 * q^80 + (-4750i - 4750) q^82 + (-680i + 680) q^83 + (5875i + 5875) q^85 - 12200 * q^86 + (9000i - 9000) q^88 - 2250i q^89 + 16400 * q^91 + (-11560i - 11560) q^92 + 3200i q^94 - 1800i q^95 + (1615i + 1615) q^97 + (3995i - 3995) q^98
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 10 q^{2} - 50 q^{5} + 80 q^{7} - 180 q^{8} - 250 q^{10} + 200 q^{11} + 410 q^{13} - 712 q^{16} - 470 q^{17} + 1000 q^{22} - 680 q^{23} + 1250 q^{25} + 4100 q^{26} - 2720 q^{28} + 856 q^{31} - 680 q^{32}+ \cdots - 7990 q^{98}+O(q^{100})$ 2 * q + 10 * q^2 - 50 * q^5 + 80 * q^7 - 180 * q^8 - 250 * q^10 + 200 * q^11 + 410 * q^13 - 712 * q^16 - 470 * q^17 + 1000 * q^22 - 680 * q^23 + 1250 * q^25 + 4100 * q^26 - 2720 * q^28 + 856 * q^31 - 680 * q^32 - 2000 * q^35 - 1510 * q^37 - 720 * q^38 + 4500 * q^40 - 1900 * q^41 - 2440 * q^43 - 6800 * q^46 + 640 * q^47 + 6250 * q^50 + 13940 * q^52 - 1010 * q^53 - 5000 * q^55 - 14400 * q^56 + 4500 * q^58 - 7616 * q^61 + 4280 * q^62 - 10250 * q^65 + 680 * q^67 + 15980 * q^68 + 6800 * q^71 + 830 * q^73 - 4896 * q^76 + 8000 * q^77 + 17800 * q^80 - 9500 * q^82 + 1360 * q^83 + 11750 * q^85 - 24400 * q^86 - 18000 * q^88 + 32800 * q^91 - 23120 * q^92 + 3230 * q^97 - 7990 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$ .

$n$	$11$	$37$
$\chi(n)$	$1$	$i$

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}

28.1

	−	1.00000i
		1.00000i

5.00000

−

5.00000i

−

34.0000i

−25.0000

40.0000

−

40.0000i

−90.0000

−

90.0000i

−125.000

125.000i

37.1

5.00000

5.00000i

34.0000i

−25.0000

40.0000

40.0000i

−90.0000

90.0000i

−125.000

−

125.000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	45.5.g.c	yes	2
3.b	odd	2	1		45.5.g.a	✓	2
5.b	even	2	1		225.5.g.a		2
5.c	odd	4	1	inner	45.5.g.c	yes	2
5.c	odd	4	1		225.5.g.a		2
15.d	odd	2	1		225.5.g.c		2
15.e	even	4	1		45.5.g.a	✓	2
15.e	even	4	1		225.5.g.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
45.5.g.a	✓	2	3.b	odd	2	1
45.5.g.a	✓	2	15.e	even	4	1
45.5.g.c	yes	2	1.a	even	1	1	trivial
45.5.g.c	yes	2	5.c	odd	4	1	inner
225.5.g.a		2	5.b	even	2	1
225.5.g.a		2	5.c	odd	4	1
225.5.g.c		2	15.d	odd	2	1
225.5.g.c		2	15.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{2} - 10T_{2} + 50$ T2^2 - 10*T2 + 50 acting on $S_{5}^{\mathrm{new}}(45, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} - 10T + 50$ T^2 - 10*T + 50
$3$	$T^{2}$ T^2
$5$	$(T + 25)^{2}$ (T + 25)^2
$7$	$T^{2} - 80T + 3200$ T^2 - 80*T + 3200
$11$	$(T - 100)^{2}$ (T - 100)^2
$13$	$T^{2} - 410T + 84050$ T^2 - 410*T + 84050
$17$	$T^{2} + 470T + 110450$ T^2 + 470*T + 110450
$19$	$T^{2} + 5184$ T^2 + 5184
$23$	$T^{2} + 680T + 231200$ T^2 + 680*T + 231200
$29$	$T^{2} + 202500$ T^2 + 202500
$31$	$(T - 428)^{2}$ (T - 428)^2
$37$	$T^{2} + 1510 T + 1140050$ T^2 + 1510*T + 1140050
$41$	$(T + 950)^{2}$ (T + 950)^2
$43$	$T^{2} + 2440 T + 2976800$ T^2 + 2440*T + 2976800
$47$	$T^{2} - 640T + 204800$ T^2 - 640*T + 204800
$53$	$T^{2} + 1010 T + 510050$ T^2 + 1010*T + 510050
$59$	$T^{2} + 39690000$ T^2 + 39690000
$61$	$(T + 3808)^{2}$ (T + 3808)^2
$67$	$T^{2} - 680T + 231200$ T^2 - 680*T + 231200
$71$	$(T - 3400)^{2}$ (T - 3400)^2
$73$	$T^{2} - 830T + 344450$ T^2 - 830*T + 344450
$79$	$T^{2} + 45319824$ T^2 + 45319824
$83$	$T^{2} - 1360 T + 924800$ T^2 - 1360*T + 924800
$89$	$T^{2} + 5062500$ T^2 + 5062500
$97$	$T^{2} - 3230 T + 5216450$ T^2 - 3230*T + 5216450
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