LMFDB - Newform orbit 605.8.a.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [605,8,Mod(1,605)] 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("605.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(605, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 605 = 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	605.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,6,-70] 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:	$188.992940418$
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 + 6 q^{2} - 70 q^{3} - 92 q^{4} - 125 q^{5} - 420 q^{6} + 624 q^{7} - 1320 q^{8} + 2713 q^{9} - 750 q^{10} + 6440 q^{12} + 6822 q^{13} + 3744 q^{14} + 8750 q^{15} + 3856 q^{16} + 36246 q^{17} + 16278 q^{18}+ \cdots - 2605002 q^{98}+O(q^{100}) $

q + 6 * q^2 - 70 * q^3 - 92 * q^4 - 125 * q^5 - 420 * q^6 + 624 * q^7 - 1320 * q^8 + 2713 * q^9 - 750 * q^10 + 6440 * q^12 + 6822 * q^13 + 3744 * q^14 + 8750 * q^15 + 3856 * q^16 + 36246 * q^17 + 16278 * q^18 - 34080 * q^19 + 11500 * q^20 - 43680 * q^21 + 16290 * q^23 + 92400 * q^24 + 15625 * q^25 + 40932 * q^26 - 36820 * q^27 - 57408 * q^28 - 158760 * q^29 + 52500 * q^30 + 194620 * q^31 + 192096 * q^32 + 217476 * q^34 - 78000 * q^35 - 249596 * q^36 + 371590 * q^37 - 204480 * q^38 - 477540 * q^39 + 165000 * q^40 - 562980 * q^41 - 262080 * q^42 + 234852 * q^43 - 339125 * q^45 + 97740 * q^46 - 832530 * q^47 - 269920 * q^48 - 434167 * q^49 + 93750 * q^50 - 2537220 * q^51 - 627624 * q^52 - 227010 * q^53 - 220920 * q^54 - 823680 * q^56 + 2385600 * q^57 - 952560 * q^58 - 462624 * q^59 - 805000 * q^60 + 1082760 * q^61 + 1167720 * q^62 + 1692912 * q^63 + 659008 * q^64 - 852750 * q^65 - 2587910 * q^67 - 3334632 * q^68 - 1140300 * q^69 - 468000 * q^70 - 3097992 * q^71 - 3581160 * q^72 + 2722422 * q^73 + 2229540 * q^74 - 1093750 * q^75 + 3135360 * q^76 - 2865240 * q^78 + 211620 * q^79 - 482000 * q^80 - 3355931 * q^81 - 3377880 * q^82 + 5216628 * q^83 + 4018560 * q^84 - 4530750 * q^85 + 1409112 * q^86 + 11113200 * q^87 - 9077130 * q^89 - 2034750 * q^90 + 4256928 * q^91 - 1498680 * q^92 - 13623400 * q^93 - 4995180 * q^94 + 4260000 * q^95 - 13446720 * q^96 + 14734790 * q^97 - 2605002 * q^98

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

6.00000

−70.0000

−92.0000

−125.000

−420.000

624.000

−1320.00

2713.00

−750.000

Atkin-Lehner signs

$ p $	Sign
$5$	$ +1 $
$11$	$ +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	605.8.a.b	yes	1
11.b	odd	2	1		605.8.a.a	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
605.8.a.a	✓	1	11.b	odd	2	1
605.8.a.b	yes	1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2} - 6 $ T2 - 6 acting on $S_{8}^{\mathrm{new}}(\Gamma_0(605))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 6 $ T - 6
$3$	$ T + 70 $ T + 70
$5$	$ T + 125 $ T + 125
$7$	$ T - 624 $ T - 624
$11$	$ T $ T
$13$	$ T - 6822 $ T - 6822
$17$	$ T - 36246 $ T - 36246
$19$	$ T + 34080 $ T + 34080
$23$	$ T - 16290 $ T - 16290
$29$	$ T + 158760 $ T + 158760
$31$	$ T - 194620 $ T - 194620
$37$	$ T - 371590 $ T - 371590
$41$	$ T + 562980 $ T + 562980
$43$	$ T - 234852 $ T - 234852
$47$	$ T + 832530 $ T + 832530
$53$	$ T + 227010 $ T + 227010
$59$	$ T + 462624 $ T + 462624
$61$	$ T - 1082760 $ T - 1082760
$67$	$ T + 2587910 $ T + 2587910
$71$	$ T + 3097992 $ T + 3097992
$73$	$ T - 2722422 $ T - 2722422
$79$	$ T - 211620 $ T - 211620
$83$	$ T - 5216628 $ T - 5216628
$89$	$ T + 9077130 $ T + 9077130
$97$	$ T - 14734790 $ T - 14734790
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	605.a (trivial)

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