LMFDB - Newform orbit 490.4.e.o

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [490,4,Mod(361,490)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(490, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("490.361"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 490 = 2 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	490.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,2,1,-4,5,4,0,-16,26,-10,65] 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:	$28.9109359028$
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:	$\Z[a_1, a_2, a_3]$
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} + ( - \zeta_{6} + 1) q^{3} + (4 \zeta_{6} - 4) q^{4} + 5 \zeta_{6} q^{5} + 2 q^{6} - 8 q^{8} + 26 \zeta_{6} q^{9} + (10 \zeta_{6} - 10) q^{10} + ( - 65 \zeta_{6} + 65) q^{11} + 4 \zeta_{6} q^{12}+ \cdots + 1690 q^{99}+O(q^{100}) $ q + 2z q^2 + (-z + 1) * q^3 + (4z - 4) q^4 + 5z q^5 + 2 * q^6 - 8 * q^8 + 26z q^9 + (10z - 10) q^10 + (-65z + 65) q^11 + 4z q^12 + 13 * q^13 + 5 * q^15 - 16z q^16 + (-73z + 73) q^17 + (52z - 52) q^18 + 142z q^19 - 20 * q^20 + 130 * q^22 - 130z q^23 + (8z - 8) q^24 + (25z - 25) q^25 + 26z q^26 + 53 * q^27 + 111 * q^29 + 10z q^30 + (256z - 256) q^31 + (-32z + 32) q^32 - 65z q^33 + 146 * q^34 - 104 * q^36 + 266z q^37 + (284z - 284) q^38 + (-13z + 13) q^39 - 40z q^40 - 424 * q^41 + 534 * q^43 + 260z q^44 + (130z - 130) q^45 + (-260z + 260) q^46 + 269z q^47 - 16 * q^48 - 50 * q^50 - 73z q^51 + (52z - 52) q^52 + (-132z + 132) q^53 + 106z q^54 + 325 * q^55 + 142 * q^57 + 222z q^58 + (-224z + 224) q^59 + (20z - 20) q^60 + 572z q^61 - 512 * q^62 + 64 * q^64 + 65z q^65 + (-130z + 130) q^66 + (-108z + 108) q^67 + 292z q^68 - 130 * q^69 + 560 * q^71 - 208z q^72 + (586z - 586) q^73 + (532z - 532) q^74 + 25z q^75 - 568 * q^76 + 26 * q^78 - 57z q^79 + (-80z + 80) q^80 + (649z - 649) q^81 - 848z q^82 + 252 * q^83 + 365 * q^85 + 1068z q^86 + (-111z + 111) q^87 + (520z - 520) q^88 + 184z q^89 - 260 * q^90 + 520 * q^92 + 256z q^93 + (538z - 538) q^94 + (710z - 710) q^95 - 32z q^96 - 605 * q^97 + 1690 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + q^{3} - 4 q^{4} + 5 q^{5} + 4 q^{6} - 16 q^{8} + 26 q^{9} - 10 q^{10} + 65 q^{11} + 4 q^{12} + 26 q^{13} + 10 q^{15} - 16 q^{16} + 73 q^{17} - 52 q^{18} + 142 q^{19} - 40 q^{20} + 260 q^{22}+ \cdots + 3380 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + q^3 - 4 * q^4 + 5 * q^5 + 4 * q^6 - 16 * q^8 + 26 * q^9 - 10 * q^10 + 65 * q^11 + 4 * q^12 + 26 * q^13 + 10 * q^15 - 16 * q^16 + 73 * q^17 - 52 * q^18 + 142 * q^19 - 40 * q^20 + 260 * q^22 - 130 * q^23 - 8 * q^24 - 25 * q^25 + 26 * q^26 + 106 * q^27 + 222 * q^29 + 10 * q^30 - 256 * q^31 + 32 * q^32 - 65 * q^33 + 292 * q^34 - 208 * q^36 + 266 * q^37 - 284 * q^38 + 13 * q^39 - 40 * q^40 - 848 * q^41 + 1068 * q^43 + 260 * q^44 - 130 * q^45 + 260 * q^46 + 269 * q^47 - 32 * q^48 - 100 * q^50 - 73 * q^51 - 52 * q^52 + 132 * q^53 + 106 * q^54 + 650 * q^55 + 284 * q^57 + 222 * q^58 + 224 * q^59 - 20 * q^60 + 572 * q^61 - 1024 * q^62 + 128 * q^64 + 65 * q^65 + 130 * q^66 + 108 * q^67 + 292 * q^68 - 260 * q^69 + 1120 * q^71 - 208 * q^72 - 586 * q^73 - 532 * q^74 + 25 * q^75 - 1136 * q^76 + 52 * q^78 - 57 * q^79 + 80 * q^80 - 649 * q^81 - 848 * q^82 + 504 * q^83 + 730 * q^85 + 1068 * q^86 + 111 * q^87 - 520 * q^88 + 184 * q^89 - 520 * q^90 + 1040 * q^92 + 256 * q^93 - 538 * q^94 - 710 * q^95 - 32 * q^96 - 1210 * q^97 + 3380 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/490\mathbb{Z}\right)^\times$.

$n$	$101$	$197$
$\chi(n)$	$-\zeta_{6}$	$1$

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} $

361.1

0.500000	+	0.866025i
0.500000	−	0.866025i

1.00000

1.73205i

0.500000

−

0.866025i

−2.00000

3.46410i

2.50000

4.33013i

2.00000

−8.00000

13.0000

22.5167i

−5.00000

8.66025i

471.1

1.00000

−

1.73205i

0.500000

0.866025i

−2.00000

−

3.46410i

2.50000

−

4.33013i

2.00000

−8.00000

13.0000

−

22.5167i

−5.00000

−

8.66025i

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	490.4.e.o		2
7.b	odd	2	1		490.4.e.n		2
7.c	even	3	1		70.4.a.c	✓	1
7.c	even	3	1	inner	490.4.e.o		2
7.d	odd	6	1		490.4.a.d		1
7.d	odd	6	1		490.4.e.n		2
21.h	odd	6	1		630.4.a.x		1
28.g	odd	6	1		560.4.a.i		1
35.i	odd	6	1		2450.4.a.bc		1
35.j	even	6	1		350.4.a.r		1
35.l	odd	12	2		350.4.c.h		2
56.k	odd	6	1		2240.4.a.r		1
56.p	even	6	1		2240.4.a.v		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.4.a.c	✓	1	7.c	even	3	1
350.4.a.r		1	35.j	even	6	1
350.4.c.h		2	35.l	odd	12	2
490.4.a.d		1	7.d	odd	6	1
490.4.e.n		2	7.b	odd	2	1
490.4.e.n		2	7.d	odd	6	1
490.4.e.o		2	1.a	even	1	1	trivial
490.4.e.o		2	7.c	even	3	1	inner
560.4.a.i		1	28.g	odd	6	1
630.4.a.x		1	21.h	odd	6	1
2240.4.a.r		1	56.k	odd	6	1
2240.4.a.v		1	56.p	even	6	1
2450.4.a.bc		1	35.i	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(490, [\chi])$:

$ T_{3}^{2} - T_{3} + 1 $

T3^2 - T3 + 1

$ T_{11}^{2} - 65T_{11} + 4225 $

T11^2 - 65*T11 + 4225

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} - 2T + 4 $ T^2 - 2*T + 4
$3$	$ T^{2} - T + 1 $ T^2 - T + 1
$5$	$ T^{2} - 5T + 25 $ T^2 - 5*T + 25
$7$	$ T^{2} $ T^2
$11$	$ T^{2} - 65T + 4225 $ T^2 - 65*T + 4225
$13$	$ (T - 13)^{2} $ (T - 13)^2
$17$	$ T^{2} - 73T + 5329 $ T^2 - 73*T + 5329
$19$	$ T^{2} - 142T + 20164 $ T^2 - 142*T + 20164
$23$	$ T^{2} + 130T + 16900 $ T^2 + 130*T + 16900
$29$	$ (T - 111)^{2} $ (T - 111)^2
$31$	$ T^{2} + 256T + 65536 $ T^2 + 256*T + 65536
$37$	$ T^{2} - 266T + 70756 $ T^2 - 266*T + 70756
$41$	$ (T + 424)^{2} $ (T + 424)^2
$43$	$ (T - 534)^{2} $ (T - 534)^2
$47$	$ T^{2} - 269T + 72361 $ T^2 - 269*T + 72361
$53$	$ T^{2} - 132T + 17424 $ T^2 - 132*T + 17424
$59$	$ T^{2} - 224T + 50176 $ T^2 - 224*T + 50176
$61$	$ T^{2} - 572T + 327184 $ T^2 - 572*T + 327184
$67$	$ T^{2} - 108T + 11664 $ T^2 - 108*T + 11664
$71$	$ (T - 560)^{2} $ (T - 560)^2
$73$	$ T^{2} + 586T + 343396 $ T^2 + 586*T + 343396
$79$	$ T^{2} + 57T + 3249 $ T^2 + 57*T + 3249
$83$	$ (T - 252)^{2} $ (T - 252)^2
$89$	$ T^{2} - 184T + 33856 $ T^2 - 184*T + 33856
$97$	$ (T + 605)^{2} $ (T + 605)^2
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Level:	\( N \)	\(=\)	\( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	490.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + 2 \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (4 \zeta_{6} - 4) q^{4} + 5 \zeta_{6} q^{5} + 2 q^{6} - 8 q^{8} + 26 \zeta_{6} q^{9} + (10 \zeta_{6} - 10) q^{10} + ( - 65 \zeta_{6} + 65) q^{11} + 4 \zeta_{6} q^{12}+ \cdots + 1690 q^{99}+O(q^{100}) \) q + 2z q^2 + (-z + 1) * q^3 + (4z - 4) q^4 + 5z q^5 + 2 * q^6 - 8 * q^8 + 26z q^9 + (10z - 10) q^10 + (-65z + 65) q^11 + 4z q^12 + 13 * q^13 + 5 * q^15 - 16z q^16 + (-73z + 73) q^17 + (52z - 52) q^18 + 142z q^19 - 20 * q^20 + 130 * q^22 - 130z q^23 + (8z - 8) q^24 + (25z - 25) q^25 + 26z q^26 + 53 * q^27 + 111 * q^29 + 10z q^30 + (256z - 256) q^31 + (-32z + 32) q^32 - 65z q^33 + 146 * q^34 - 104 * q^36 + 266z q^37 + (284z - 284) q^38 + (-13z + 13) q^39 - 40z q^40 - 424 * q^41 + 534 * q^43 + 260z q^44 + (130z - 130) q^45 + (-260z + 260) q^46 + 269z q^47 - 16 * q^48 - 50 * q^50 - 73z q^51 + (52z - 52) q^52 + (-132z + 132) q^53 + 106z q^54 + 325 * q^55 + 142 * q^57 + 222z q^58 + (-224z + 224) q^59 + (20z - 20) q^60 + 572z q^61 - 512 * q^62 + 64 * q^64 + 65z q^65 + (-130z + 130) q^66 + (-108z + 108) q^67 + 292z q^68 - 130 * q^69 + 560 * q^71 - 208z q^72 + (586z - 586) q^73 + (532z - 532) q^74 + 25z q^75 - 568 * q^76 + 26 * q^78 - 57z q^79 + (-80z + 80) q^80 + (649z - 649) q^81 - 848z q^82 + 252 * q^83 + 365 * q^85 + 1068z q^86 + (-111z + 111) q^87 + (520z - 520) q^88 + 184z q^89 - 260 * q^90 + 520 * q^92 + 256z q^93 + (538z - 538) q^94 + (710z - 710) q^95 - 32z q^96 - 605 * q^97 + 1690 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + q^{3} - 4 q^{4} + 5 q^{5} + 4 q^{6} - 16 q^{8} + 26 q^{9} - 10 q^{10} + 65 q^{11} + 4 q^{12} + 26 q^{13} + 10 q^{15} - 16 q^{16} + 73 q^{17} - 52 q^{18} + 142 q^{19} - 40 q^{20} + 260 q^{22}+ \cdots + 3380 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + q^3 - 4 * q^4 + 5 * q^5 + 4 * q^6 - 16 * q^8 + 26 * q^9 - 10 * q^10 + 65 * q^11 + 4 * q^12 + 26 * q^13 + 10 * q^15 - 16 * q^16 + 73 * q^17 - 52 * q^18 + 142 * q^19 - 40 * q^20 + 260 * q^22 - 130 * q^23 - 8 * q^24 - 25 * q^25 + 26 * q^26 + 106 * q^27 + 222 * q^29 + 10 * q^30 - 256 * q^31 + 32 * q^32 - 65 * q^33 + 292 * q^34 - 208 * q^36 + 266 * q^37 - 284 * q^38 + 13 * q^39 - 40 * q^40 - 848 * q^41 + 1068 * q^43 + 260 * q^44 - 130 * q^45 + 260 * q^46 + 269 * q^47 - 32 * q^48 - 100 * q^50 - 73 * q^51 - 52 * q^52 + 132 * q^53 + 106 * q^54 + 650 * q^55 + 284 * q^57 + 222 * q^58 + 224 * q^59 - 20 * q^60 + 572 * q^61 - 1024 * q^62 + 128 * q^64 + 65 * q^65 + 130 * q^66 + 108 * q^67 + 292 * q^68 - 260 * q^69 + 1120 * q^71 - 208 * q^72 - 586 * q^73 - 532 * q^74 + 25 * q^75 - 1136 * q^76 + 52 * q^78 - 57 * q^79 + 80 * q^80 - 649 * q^81 - 848 * q^82 + 504 * q^83 + 730 * q^85 + 1068 * q^86 + 111 * q^87 - 520 * q^88 + 184 * q^89 - 520 * q^90 + 1040 * q^92 + 256 * q^93 - 538 * q^94 - 710 * q^95 - 32 * q^96 - 1210 * q^97 + 3380 * q^99

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