LMFDB - Newform orbit 810.4.e.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [810,4,Mod(271,810)] 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("810.271"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(810, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	810.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,-4,-5,0,34,16,0,20,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$47.7915471046$
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:	no (minimal twist has level 270)
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} - 2) q^{2} - 4 \zeta_{6} q^{4} - 5 \zeta_{6} q^{5} + ( - 34 \zeta_{6} + 34) q^{7} + 8 q^{8} + 10 q^{10} + ( - 48 \zeta_{6} + 48) q^{11} + 70 \zeta_{6} q^{13} + 68 \zeta_{6} q^{14} + (16 \zeta_{6} - 16) q^{16} + \cdots + 1626 q^{98} +O(q^{100}) $ q + (2z - 2) q^2 - 4z q^4 - 5z q^5 + (-34z + 34) q^7 + 8 * q^8 + 10 * q^10 + (-48z + 48) q^11 + 70z q^13 + 68z q^14 + (16z - 16) q^16 - 27 * q^17 + 119 * q^19 + (20z - 20) q^20 + 96z q^22 + 51z q^23 + (25z - 25) q^25 - 140 * q^26 - 136 * q^28 + (-30z + 30) q^29 + 133z q^31 - 32z q^32 + (-54z + 54) q^34 - 170 * q^35 + 218 * q^37 + (238z - 238) q^38 - 40z q^40 - 156z q^41 + (-88z + 88) q^43 - 192 * q^44 - 102 * q^46 + (-516z + 516) q^47 - 813z q^49 - 50z q^50 + (-280z + 280) q^52 - 639 * q^53 - 240 * q^55 + (-272z + 272) q^56 + 60z q^58 + 654z q^59 + (461z - 461) q^61 - 266 * q^62 + 64 * q^64 + (-350z + 350) q^65 - 182z q^67 + 108z q^68 + (-340z + 340) q^70 + 900 * q^71 + 704 * q^73 + (436z - 436) q^74 - 476z q^76 - 1632z q^77 + (-1375z + 1375) q^79 + 80 * q^80 + 312 * q^82 + (915z - 915) q^83 + 135z q^85 + 176z q^86 + (-384z + 384) q^88 - 1116 * q^89 + 2380 * q^91 + (-204z + 204) q^92 + 1032z q^94 - 595z q^95 + (-16z + 16) q^97 + 1626 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 4 q^{4} - 5 q^{5} + 34 q^{7} + 16 q^{8} + 20 q^{10} + 48 q^{11} + 70 q^{13} + 68 q^{14} - 16 q^{16} - 54 q^{17} + 238 q^{19} - 20 q^{20} + 96 q^{22} + 51 q^{23} - 25 q^{25} - 280 q^{26}+ \cdots + 3252 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 4 * q^4 - 5 * q^5 + 34 * q^7 + 16 * q^8 + 20 * q^10 + 48 * q^11 + 70 * q^13 + 68 * q^14 - 16 * q^16 - 54 * q^17 + 238 * q^19 - 20 * q^20 + 96 * q^22 + 51 * q^23 - 25 * q^25 - 280 * q^26 - 272 * q^28 + 30 * q^29 + 133 * q^31 - 32 * q^32 + 54 * q^34 - 340 * q^35 + 436 * q^37 - 238 * q^38 - 40 * q^40 - 156 * q^41 + 88 * q^43 - 384 * q^44 - 204 * q^46 + 516 * q^47 - 813 * q^49 - 50 * q^50 + 280 * q^52 - 1278 * q^53 - 480 * q^55 + 272 * q^56 + 60 * q^58 + 654 * q^59 - 461 * q^61 - 532 * q^62 + 128 * q^64 + 350 * q^65 - 182 * q^67 + 108 * q^68 + 340 * q^70 + 1800 * q^71 + 1408 * q^73 - 436 * q^74 - 476 * q^76 - 1632 * q^77 + 1375 * q^79 + 160 * q^80 + 624 * q^82 - 915 * q^83 + 135 * q^85 + 176 * q^86 + 384 * q^88 - 2232 * q^89 + 4760 * q^91 + 204 * q^92 + 1032 * q^94 - 595 * q^95 + 16 * q^97 + 3252 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$1$	$-\zeta_{6}$

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} $

271.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−1.00000

−

1.73205i

−2.00000

3.46410i

−2.50000

4.33013i

17.0000

29.4449i

8.00000

10.0000

541.1

−1.00000

1.73205i

−2.00000

−

3.46410i

−2.50000

−

4.33013i

17.0000

−

29.4449i

8.00000

10.0000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	810.4.e.d		2
3.b	odd	2	1		810.4.e.x		2
9.c	even	3	1		270.4.a.k	yes	1
9.c	even	3	1	inner	810.4.e.d		2
9.d	odd	6	1		270.4.a.a	✓	1
9.d	odd	6	1		810.4.e.x		2
36.f	odd	6	1		2160.4.a.t		1
36.h	even	6	1		2160.4.a.j		1
45.h	odd	6	1		1350.4.a.bb		1
45.j	even	6	1		1350.4.a.n		1
45.k	odd	12	2		1350.4.c.c		2
45.l	even	12	2		1350.4.c.r		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
270.4.a.a	✓	1	9.d	odd	6	1
270.4.a.k	yes	1	9.c	even	3	1
810.4.e.d		2	1.a	even	1	1	trivial
810.4.e.d		2	9.c	even	3	1	inner
810.4.e.x		2	3.b	odd	2	1
810.4.e.x		2	9.d	odd	6	1
1350.4.a.n		1	45.j	even	6	1
1350.4.a.bb		1	45.h	odd	6	1
1350.4.c.c		2	45.k	odd	12	2
1350.4.c.r		2	45.l	even	12	2
2160.4.a.j		1	36.h	even	6	1
2160.4.a.t		1	36.f	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(810, [\chi])$:

$ T_{7}^{2} - 34T_{7} + 1156 $

T7^2 - 34*T7 + 1156

$ T_{11}^{2} - 48T_{11} + 2304 $

T11^2 - 48*T11 + 2304

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 2T + 4 $ T^2 + 2*T + 4
$3$	$ T^{2} $ T^2
$5$	$ T^{2} + 5T + 25 $ T^2 + 5*T + 25
$7$	$ T^{2} - 34T + 1156 $ T^2 - 34*T + 1156
$11$	$ T^{2} - 48T + 2304 $ T^2 - 48*T + 2304
$13$	$ T^{2} - 70T + 4900 $ T^2 - 70*T + 4900
$17$	$ (T + 27)^{2} $ (T + 27)^2
$19$	$ (T - 119)^{2} $ (T - 119)^2
$23$	$ T^{2} - 51T + 2601 $ T^2 - 51*T + 2601
$29$	$ T^{2} - 30T + 900 $ T^2 - 30*T + 900
$31$	$ T^{2} - 133T + 17689 $ T^2 - 133*T + 17689
$37$	$ (T - 218)^{2} $ (T - 218)^2
$41$	$ T^{2} + 156T + 24336 $ T^2 + 156*T + 24336
$43$	$ T^{2} - 88T + 7744 $ T^2 - 88*T + 7744
$47$	$ T^{2} - 516T + 266256 $ T^2 - 516*T + 266256
$53$	$ (T + 639)^{2} $ (T + 639)^2
$59$	$ T^{2} - 654T + 427716 $ T^2 - 654*T + 427716
$61$	$ T^{2} + 461T + 212521 $ T^2 + 461*T + 212521
$67$	$ T^{2} + 182T + 33124 $ T^2 + 182*T + 33124
$71$	$ (T - 900)^{2} $ (T - 900)^2
$73$	$ (T - 704)^{2} $ (T - 704)^2
$79$	$ T^{2} - 1375 T + 1890625 $ T^2 - 1375*T + 1890625
$83$	$ T^{2} + 915T + 837225 $ T^2 + 915*T + 837225
$89$	$ (T + 1116)^{2} $ (T + 1116)^2
$97$	$ T^{2} - 16T + 256 $ T^2 - 16*T + 256
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Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	810.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} - 5 \zeta_{6} q^{5} + ( - 34 \zeta_{6} + 34) q^{7} + 8 q^{8} + 10 q^{10} + ( - 48 \zeta_{6} + 48) q^{11} + 70 \zeta_{6} q^{13} + 68 \zeta_{6} q^{14} + (16 \zeta_{6} - 16) q^{16} + \cdots + 1626 q^{98} +O(q^{100}) \) q + (2z - 2) q^2 - 4z q^4 - 5z q^5 + (-34z + 34) q^7 + 8 * q^8 + 10 * q^10 + (-48z + 48) q^11 + 70z q^13 + 68z q^14 + (16z - 16) q^16 - 27 * q^17 + 119 * q^19 + (20z - 20) q^20 + 96z q^22 + 51z q^23 + (25z - 25) q^25 - 140 * q^26 - 136 * q^28 + (-30z + 30) q^29 + 133z q^31 - 32z q^32 + (-54z + 54) q^34 - 170 * q^35 + 218 * q^37 + (238z - 238) q^38 - 40z q^40 - 156z q^41 + (-88z + 88) q^43 - 192 * q^44 - 102 * q^46 + (-516z + 516) q^47 - 813z q^49 - 50z q^50 + (-280z + 280) q^52 - 639 * q^53 - 240 * q^55 + (-272z + 272) q^56 + 60z q^58 + 654z q^59 + (461z - 461) q^61 - 266 * q^62 + 64 * q^64 + (-350z + 350) q^65 - 182z q^67 + 108z q^68 + (-340z + 340) q^70 + 900 * q^71 + 704 * q^73 + (436z - 436) q^74 - 476z q^76 - 1632z q^77 + (-1375z + 1375) q^79 + 80 * q^80 + 312 * q^82 + (915z - 915) q^83 + 135z q^85 + 176z q^86 + (-384z + 384) q^88 - 1116 * q^89 + 2380 * q^91 + (-204z + 204) q^92 + 1032z q^94 - 595z q^95 + (-16z + 16) q^97 + 1626 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 4 q^{4} - 5 q^{5} + 34 q^{7} + 16 q^{8} + 20 q^{10} + 48 q^{11} + 70 q^{13} + 68 q^{14} - 16 q^{16} - 54 q^{17} + 238 q^{19} - 20 q^{20} + 96 q^{22} + 51 q^{23} - 25 q^{25} - 280 q^{26}+ \cdots + 3252 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 4 * q^4 - 5 * q^5 + 34 * q^7 + 16 * q^8 + 20 * q^10 + 48 * q^11 + 70 * q^13 + 68 * q^14 - 16 * q^16 - 54 * q^17 + 238 * q^19 - 20 * q^20 + 96 * q^22 + 51 * q^23 - 25 * q^25 - 280 * q^26 - 272 * q^28 + 30 * q^29 + 133 * q^31 - 32 * q^32 + 54 * q^34 - 340 * q^35 + 436 * q^37 - 238 * q^38 - 40 * q^40 - 156 * q^41 + 88 * q^43 - 384 * q^44 - 204 * q^46 + 516 * q^47 - 813 * q^49 - 50 * q^50 + 280 * q^52 - 1278 * q^53 - 480 * q^55 + 272 * q^56 + 60 * q^58 + 654 * q^59 - 461 * q^61 - 532 * q^62 + 128 * q^64 + 350 * q^65 - 182 * q^67 + 108 * q^68 + 340 * q^70 + 1800 * q^71 + 1408 * q^73 - 436 * q^74 - 476 * q^76 - 1632 * q^77 + 1375 * q^79 + 160 * q^80 + 624 * q^82 - 915 * q^83 + 135 * q^85 + 176 * q^86 + 384 * q^88 - 2232 * q^89 + 4760 * q^91 + 204 * q^92 + 1032 * q^94 - 595 * q^95 + 16 * q^97 + 3252 * q^98

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