LMFDB - Newform orbit 245.4.e.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [245,4,Mod(116,245)] 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("245.116"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 245 = 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	245.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,4,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$14.4554679514$
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 5)
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 + 4 \zeta_{6} q^{2} + (2 \zeta_{6} - 2) q^{3} + (8 \zeta_{6} - 8) q^{4} + 5 \zeta_{6} q^{5} - 8 q^{6} + 23 \zeta_{6} q^{9} + (20 \zeta_{6} - 20) q^{10} + (32 \zeta_{6} - 32) q^{11} - 16 \zeta_{6} q^{12} + \cdots - 736 q^{99} +O(q^{100}) $ q + 4z q^2 + (2z - 2) q^3 + (8z - 8) q^4 + 5z q^5 - 8 * q^6 + 23z q^9 + (20z - 20) q^10 + (32z - 32) q^11 - 16z q^12 - 38 * q^13 - 10 * q^15 + 64z q^16 + (26z - 26) q^17 + (92z - 92) q^18 - 100z q^19 - 40 * q^20 - 128 * q^22 + 78z q^23 + (25z - 25) q^25 - 152z q^26 - 100 * q^27 - 50 * q^29 - 40z q^30 + (-108z + 108) q^31 + (256z - 256) q^32 - 64z q^33 - 104 * q^34 - 184 * q^36 - 266z q^37 + (-400z + 400) q^38 + (-76z + 76) q^39 + 22 * q^41 + 442 * q^43 - 256z q^44 + (115z - 115) q^45 + (312z - 312) q^46 + 514z q^47 - 128 * q^48 - 100 * q^50 - 52z q^51 + (-304z + 304) q^52 + (2z - 2) q^53 - 400z q^54 - 160 * q^55 + 200 * q^57 - 200z q^58 + (500z - 500) q^59 + (-80z + 80) q^60 + 518z q^61 + 432 * q^62 - 512 * q^64 - 190z q^65 + (-256z + 256) q^66 + (126z - 126) q^67 - 208z q^68 - 156 * q^69 + 412 * q^71 + (-878z + 878) q^73 + (-1064z + 1064) q^74 - 50z q^75 + 800 * q^76 + 304 * q^78 - 600z q^79 + (320z - 320) q^80 + (421z - 421) q^81 + 88z q^82 + 282 * q^83 - 130 * q^85 + 1768z q^86 + (-100z + 100) q^87 + 150z q^89 - 460 * q^90 - 624 * q^92 + 216z q^93 + (2056z - 2056) q^94 + (-500z + 500) q^95 - 512z q^96 + 386 * q^97 - 736 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - 2 q^{3} - 8 q^{4} + 5 q^{5} - 16 q^{6} + 23 q^{9} - 20 q^{10} - 32 q^{11} - 16 q^{12} - 76 q^{13} - 20 q^{15} + 64 q^{16} - 26 q^{17} - 92 q^{18} - 100 q^{19} - 80 q^{20} - 256 q^{22} + 78 q^{23}+ \cdots - 1472 q^{99}+O(q^{100}) $ 2 * q + 4 * q^2 - 2 * q^3 - 8 * q^4 + 5 * q^5 - 16 * q^6 + 23 * q^9 - 20 * q^10 - 32 * q^11 - 16 * q^12 - 76 * q^13 - 20 * q^15 + 64 * q^16 - 26 * q^17 - 92 * q^18 - 100 * q^19 - 80 * q^20 - 256 * q^22 + 78 * q^23 - 25 * q^25 - 152 * q^26 - 200 * q^27 - 100 * q^29 - 40 * q^30 + 108 * q^31 - 256 * q^32 - 64 * q^33 - 208 * q^34 - 368 * q^36 - 266 * q^37 + 400 * q^38 + 76 * q^39 + 44 * q^41 + 884 * q^43 - 256 * q^44 - 115 * q^45 - 312 * q^46 + 514 * q^47 - 256 * q^48 - 200 * q^50 - 52 * q^51 + 304 * q^52 - 2 * q^53 - 400 * q^54 - 320 * q^55 + 400 * q^57 - 200 * q^58 - 500 * q^59 + 80 * q^60 + 518 * q^61 + 864 * q^62 - 1024 * q^64 - 190 * q^65 + 256 * q^66 - 126 * q^67 - 208 * q^68 - 312 * q^69 + 824 * q^71 + 878 * q^73 + 1064 * q^74 - 50 * q^75 + 1600 * q^76 + 608 * q^78 - 600 * q^79 - 320 * q^80 - 421 * q^81 + 88 * q^82 + 564 * q^83 - 260 * q^85 + 1768 * q^86 + 100 * q^87 + 150 * q^89 - 920 * q^90 - 1248 * q^92 + 216 * q^93 - 2056 * q^94 + 500 * q^95 - 512 * q^96 + 772 * q^97 - 1472 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/245\mathbb{Z}\right)^\times$.

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

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} $

116.1

0.500000	+	0.866025i
0.500000	−	0.866025i

2.00000

3.46410i

−1.00000

1.73205i

−4.00000

6.92820i

2.50000

4.33013i

−8.00000

11.5000

19.9186i

−10.0000

17.3205i

226.1

2.00000

−

3.46410i

−1.00000

−

1.73205i

−4.00000

−

6.92820i

2.50000

−

4.33013i

−8.00000

11.5000

−

19.9186i

−10.0000

−

17.3205i

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	245.4.e.f		2
7.b	odd	2	1		245.4.e.g		2
7.c	even	3	1		5.4.a.a	✓	1
7.c	even	3	1	inner	245.4.e.f		2
7.d	odd	6	1		245.4.a.a		1
7.d	odd	6	1		245.4.e.g		2
21.g	even	6	1		2205.4.a.q		1
21.h	odd	6	1		45.4.a.d		1
28.g	odd	6	1		80.4.a.d		1
35.i	odd	6	1		1225.4.a.k		1
35.j	even	6	1		25.4.a.c		1
35.l	odd	12	2		25.4.b.a		2
56.k	odd	6	1		320.4.a.h		1
56.p	even	6	1		320.4.a.g		1
63.g	even	3	1		405.4.e.l		2
63.h	even	3	1		405.4.e.l		2
63.j	odd	6	1		405.4.e.c		2
63.n	odd	6	1		405.4.e.c		2
77.h	odd	6	1		605.4.a.d		1
84.n	even	6	1		720.4.a.u		1
91.r	even	6	1		845.4.a.b		1
105.o	odd	6	1		225.4.a.b		1
105.x	even	12	2		225.4.b.c		2
112.u	odd	12	2		1280.4.d.l		2
112.w	even	12	2		1280.4.d.e		2
119.j	even	6	1		1445.4.a.a		1
133.r	odd	6	1		1805.4.a.h		1
140.p	odd	6	1		400.4.a.m		1
140.w	even	12	2		400.4.c.k		2
280.bf	even	6	1		1600.4.a.bi		1
280.bi	odd	6	1		1600.4.a.s		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
5.4.a.a	✓	1	7.c	even	3	1
25.4.a.c		1	35.j	even	6	1
25.4.b.a		2	35.l	odd	12	2
45.4.a.d		1	21.h	odd	6	1
80.4.a.d		1	28.g	odd	6	1
225.4.a.b		1	105.o	odd	6	1
225.4.b.c		2	105.x	even	12	2
245.4.a.a		1	7.d	odd	6	1
245.4.e.f		2	1.a	even	1	1	trivial
245.4.e.f		2	7.c	even	3	1	inner
245.4.e.g		2	7.b	odd	2	1
245.4.e.g		2	7.d	odd	6	1
320.4.a.g		1	56.p	even	6	1
320.4.a.h		1	56.k	odd	6	1
400.4.a.m		1	140.p	odd	6	1
400.4.c.k		2	140.w	even	12	2
405.4.e.c		2	63.j	odd	6	1
405.4.e.c		2	63.n	odd	6	1
405.4.e.l		2	63.g	even	3	1
405.4.e.l		2	63.h	even	3	1
605.4.a.d		1	77.h	odd	6	1
720.4.a.u		1	84.n	even	6	1
845.4.a.b		1	91.r	even	6	1
1225.4.a.k		1	35.i	odd	6	1
1280.4.d.e		2	112.w	even	12	2
1280.4.d.l		2	112.u	odd	12	2
1445.4.a.a		1	119.j	even	6	1
1600.4.a.s		1	280.bi	odd	6	1
1600.4.a.bi		1	280.bf	even	6	1
1805.4.a.h		1	133.r	odd	6	1
2205.4.a.q		1	21.g	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(245, [\chi])$:

$ T_{2}^{2} - 4T_{2} + 16 $

T2^2 - 4*T2 + 16

$ T_{3}^{2} + 2T_{3} + 4 $

T3^2 + 2*T3 + 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} - 4T + 16 $ T^2 - 4*T + 16
$3$	$ T^{2} + 2T + 4 $ T^2 + 2*T + 4
$5$	$ T^{2} - 5T + 25 $ T^2 - 5*T + 25
$7$	$ T^{2} $ T^2
$11$	$ T^{2} + 32T + 1024 $ T^2 + 32*T + 1024
$13$	$ (T + 38)^{2} $ (T + 38)^2
$17$	$ T^{2} + 26T + 676 $ T^2 + 26*T + 676
$19$	$ T^{2} + 100T + 10000 $ T^2 + 100*T + 10000
$23$	$ T^{2} - 78T + 6084 $ T^2 - 78*T + 6084
$29$	$ (T + 50)^{2} $ (T + 50)^2
$31$	$ T^{2} - 108T + 11664 $ T^2 - 108*T + 11664
$37$	$ T^{2} + 266T + 70756 $ T^2 + 266*T + 70756
$41$	$ (T - 22)^{2} $ (T - 22)^2
$43$	$ (T - 442)^{2} $ (T - 442)^2
$47$	$ T^{2} - 514T + 264196 $ T^2 - 514*T + 264196
$53$	$ T^{2} + 2T + 4 $ T^2 + 2*T + 4
$59$	$ T^{2} + 500T + 250000 $ T^2 + 500*T + 250000
$61$	$ T^{2} - 518T + 268324 $ T^2 - 518*T + 268324
$67$	$ T^{2} + 126T + 15876 $ T^2 + 126*T + 15876
$71$	$ (T - 412)^{2} $ (T - 412)^2
$73$	$ T^{2} - 878T + 770884 $ T^2 - 878*T + 770884
$79$	$ T^{2} + 600T + 360000 $ T^2 + 600*T + 360000
$83$	$ (T - 282)^{2} $ (T - 282)^2
$89$	$ T^{2} - 150T + 22500 $ T^2 - 150*T + 22500
$97$	$ (T - 386)^{2} $ (T - 386)^2
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Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	245.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + 4 \zeta_{6} q^{2} + (2 \zeta_{6} - 2) q^{3} + (8 \zeta_{6} - 8) q^{4} + 5 \zeta_{6} q^{5} - 8 q^{6} + 23 \zeta_{6} q^{9} + (20 \zeta_{6} - 20) q^{10} + (32 \zeta_{6} - 32) q^{11} - 16 \zeta_{6} q^{12} + \cdots - 736 q^{99} +O(q^{100}) \) q + 4z q^2 + (2z - 2) q^3 + (8z - 8) q^4 + 5z q^5 - 8 * q^6 + 23z q^9 + (20z - 20) q^10 + (32z - 32) q^11 - 16z q^12 - 38 * q^13 - 10 * q^15 + 64z q^16 + (26z - 26) q^17 + (92z - 92) q^18 - 100z q^19 - 40 * q^20 - 128 * q^22 + 78z q^23 + (25z - 25) q^25 - 152z q^26 - 100 * q^27 - 50 * q^29 - 40z q^30 + (-108z + 108) q^31 + (256z - 256) q^32 - 64z q^33 - 104 * q^34 - 184 * q^36 - 266z q^37 + (-400z + 400) q^38 + (-76z + 76) q^39 + 22 * q^41 + 442 * q^43 - 256z q^44 + (115z - 115) q^45 + (312z - 312) q^46 + 514z q^47 - 128 * q^48 - 100 * q^50 - 52z q^51 + (-304z + 304) q^52 + (2z - 2) q^53 - 400z q^54 - 160 * q^55 + 200 * q^57 - 200z q^58 + (500z - 500) q^59 + (-80z + 80) q^60 + 518z q^61 + 432 * q^62 - 512 * q^64 - 190z q^65 + (-256z + 256) q^66 + (126z - 126) q^67 - 208z q^68 - 156 * q^69 + 412 * q^71 + (-878z + 878) q^73 + (-1064z + 1064) q^74 - 50z q^75 + 800 * q^76 + 304 * q^78 - 600z q^79 + (320z - 320) q^80 + (421z - 421) q^81 + 88z q^82 + 282 * q^83 - 130 * q^85 + 1768z q^86 + (-100z + 100) q^87 + 150z q^89 - 460 * q^90 - 624 * q^92 + 216z q^93 + (2056z - 2056) q^94 + (-500z + 500) q^95 - 512z q^96 + 386 * q^97 - 736 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - 2 q^{3} - 8 q^{4} + 5 q^{5} - 16 q^{6} + 23 q^{9} - 20 q^{10} - 32 q^{11} - 16 q^{12} - 76 q^{13} - 20 q^{15} + 64 q^{16} - 26 q^{17} - 92 q^{18} - 100 q^{19} - 80 q^{20} - 256 q^{22} + 78 q^{23}+ \cdots - 1472 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 - 2 * q^3 - 8 * q^4 + 5 * q^5 - 16 * q^6 + 23 * q^9 - 20 * q^10 - 32 * q^11 - 16 * q^12 - 76 * q^13 - 20 * q^15 + 64 * q^16 - 26 * q^17 - 92 * q^18 - 100 * q^19 - 80 * q^20 - 256 * q^22 + 78 * q^23 - 25 * q^25 - 152 * q^26 - 200 * q^27 - 100 * q^29 - 40 * q^30 + 108 * q^31 - 256 * q^32 - 64 * q^33 - 208 * q^34 - 368 * q^36 - 266 * q^37 + 400 * q^38 + 76 * q^39 + 44 * q^41 + 884 * q^43 - 256 * q^44 - 115 * q^45 - 312 * q^46 + 514 * q^47 - 256 * q^48 - 200 * q^50 - 52 * q^51 + 304 * q^52 - 2 * q^53 - 400 * q^54 - 320 * q^55 + 400 * q^57 - 200 * q^58 - 500 * q^59 + 80 * q^60 + 518 * q^61 + 864 * q^62 - 1024 * q^64 - 190 * q^65 + 256 * q^66 - 126 * q^67 - 208 * q^68 - 312 * q^69 + 824 * q^71 + 878 * q^73 + 1064 * q^74 - 50 * q^75 + 1600 * q^76 + 608 * q^78 - 600 * q^79 - 320 * q^80 - 421 * q^81 + 88 * q^82 + 564 * q^83 - 260 * q^85 + 1768 * q^86 + 100 * q^87 + 150 * q^89 - 920 * q^90 - 1248 * q^92 + 216 * q^93 - 2056 * q^94 + 500 * q^95 - 512 * q^96 + 772 * q^97 - 1472 * q^99

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