LMFDB - Newform orbit 245.4.a.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [245,4,Mod(1,245)] 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("245.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 245 = 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	245.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,-6] 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:	yes
Analytic conductor:	$14.4554679514$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$

$=$

$ q + q^{2} - 6 q^{3} - 7 q^{4} + 5 q^{5} - 6 q^{6} - 15 q^{8} + 9 q^{9} + 5 q^{10} - 44 q^{11} + 42 q^{12} - 6 q^{13} - 30 q^{15} + 41 q^{16} + 24 q^{17} + 9 q^{18} + 114 q^{19} - 35 q^{20} - 44 q^{22}+ \cdots - 396 q^{99}+O(q^{100}) $

q + q^2 - 6 * q^3 - 7 * q^4 + 5 * q^5 - 6 * q^6 - 15 * q^8 + 9 * q^9 + 5 * q^10 - 44 * q^11 + 42 * q^12 - 6 * q^13 - 30 * q^15 + 41 * q^16 + 24 * q^17 + 9 * q^18 + 114 * q^19 - 35 * q^20 - 44 * q^22 - 52 * q^23 + 90 * q^24 + 25 * q^25 - 6 * q^26 + 108 * q^27 + 146 * q^29 - 30 * q^30 + 276 * q^31 + 161 * q^32 + 264 * q^33 + 24 * q^34 - 63 * q^36 - 210 * q^37 + 114 * q^38 + 36 * q^39 - 75 * q^40 - 444 * q^41 + 492 * q^43 + 308 * q^44 + 45 * q^45 - 52 * q^46 + 612 * q^47 - 246 * q^48 + 25 * q^50 - 144 * q^51 + 42 * q^52 + 50 * q^53 + 108 * q^54 - 220 * q^55 - 684 * q^57 + 146 * q^58 - 294 * q^59 + 210 * q^60 - 450 * q^61 + 276 * q^62 - 167 * q^64 - 30 * q^65 + 264 * q^66 - 668 * q^67 - 168 * q^68 + 312 * q^69 - 308 * q^71 - 135 * q^72 - 12 * q^73 - 210 * q^74 - 150 * q^75 - 798 * q^76 + 36 * q^78 + 596 * q^79 + 205 * q^80 - 891 * q^81 - 444 * q^82 + 966 * q^83 + 120 * q^85 + 492 * q^86 - 876 * q^87 + 660 * q^88 + 408 * q^89 + 45 * q^90 + 364 * q^92 - 1656 * q^93 + 612 * q^94 + 570 * q^95 - 966 * q^96 + 1200 * q^97 - 396 * q^99

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} $

1.1

1.00000

−6.00000

−7.00000

5.00000

−6.00000

−15.0000

9.00000

5.00000

Atkin-Lehner signs

$ p $	Sign
$5$	$ -1 $
$7$	$ -1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	245.4.a.b	✓	1
3.b	odd	2	1		2205.4.a.k		1
5.b	even	2	1		1225.4.a.g		1
7.b	odd	2	1		245.4.a.c	yes	1
7.c	even	3	2		245.4.e.d		2
7.d	odd	6	2		245.4.e.c		2
21.c	even	2	1		2205.4.a.n		1
35.c	odd	2	1		1225.4.a.f		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
245.4.a.b	✓	1	1.a	even	1	1	trivial
245.4.a.c	yes	1	7.b	odd	2	1
245.4.e.c		2	7.d	odd	6	2
245.4.e.d		2	7.c	even	3	2
1225.4.a.f		1	35.c	odd	2	1
1225.4.a.g		1	5.b	even	2	1
2205.4.a.k		1	3.b	odd	2	1
2205.4.a.n		1	21.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(245))$:

$ T_{2} - 1 $

T2 - 1

$ T_{3} + 6 $

T3 + 6

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 1 $ T - 1
$3$	$ T + 6 $ T + 6
$5$	$ T - 5 $ T - 5
$7$	$ T $ T
$11$	$ T + 44 $ T + 44
$13$	$ T + 6 $ T + 6
$17$	$ T - 24 $ T - 24
$19$	$ T - 114 $ T - 114
$23$	$ T + 52 $ T + 52
$29$	$ T - 146 $ T - 146
$31$	$ T - 276 $ T - 276
$37$	$ T + 210 $ T + 210
$41$	$ T + 444 $ T + 444
$43$	$ T - 492 $ T - 492
$47$	$ T - 612 $ T - 612
$53$	$ T - 50 $ T - 50
$59$	$ T + 294 $ T + 294
$61$	$ T + 450 $ T + 450
$67$	$ T + 668 $ T + 668
$71$	$ T + 308 $ T + 308
$73$	$ T + 12 $ T + 12
$79$	$ T - 596 $ T - 596
$83$	$ T - 966 $ T - 966
$89$	$ T - 408 $ T - 408
$97$	$ T - 1200 $ T - 1200
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Classical	Maass
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Elliptic curves over $\Q$
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Belyi maps

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	245.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
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