LMFDB - Newform orbit 98.6.a.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [98,6,Mod(1,98)] 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("98.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 98 = 2 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	98.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-8,0,32,0,0,0,-128,-118] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$15.7176143417$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{46}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 46 $ x^2 - 46
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
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)

Coefficients of the $q$-expansion are expressed in terms of $\beta = 2\sqrt{46}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - 4 q^{2} + \beta q^{3} + 16 q^{4} - 7 \beta q^{5} - 4 \beta q^{6} - 64 q^{8} - 59 q^{9} + 28 \beta q^{10} + 476 q^{11} + 16 \beta q^{12} + 71 \beta q^{13} - 1288 q^{15} + 256 q^{16} - 66 \beta q^{17} + \cdots - 28084 q^{99} +O(q^{100}) $ q - 4 * q^2 + b * q^3 + 16 * q^4 - 7b q^5 - 4b q^6 - 64 * q^8 - 59 * q^9 + 28b q^10 + 476 * q^11 + 16b q^12 + 71b q^13 - 1288 * q^15 + 256 * q^16 - 66b q^17 + 236 * q^18 + 47b q^19 - 112b q^20 - 1904 * q^22 + 3696 * q^23 - 64b q^24 + 5891 * q^25 - 284b q^26 - 302b q^27 + 1394 * q^29 + 5152 * q^30 + 142b q^31 - 1024 * q^32 + 476b q^33 + 264b q^34 - 944 * q^36 + 12090 * q^37 - 188b q^38 + 13064 * q^39 + 448b q^40 + 1122b q^41 + 9724 * q^43 + 7616 * q^44 + 413b q^45 - 14784 * q^46 - 2158b q^47 + 256b q^48 - 23564 * q^50 - 12144 * q^51 + 1136b q^52 + 4310 * q^53 + 1208b q^54 - 3332b q^55 + 8648 * q^57 - 5576 * q^58 - 1537b q^59 - 20608 * q^60 + 685b q^61 - 568b q^62 + 4096 * q^64 - 91448 * q^65 - 1904b q^66 + 20236 * q^67 - 1056b q^68 + 3696b q^69 + 29792 * q^71 + 3776 * q^72 + 832b q^73 - 48360 * q^74 + 5891b q^75 + 752b q^76 - 52256 * q^78 - 33176 * q^79 - 1792b q^80 - 41231 * q^81 - 4488b q^82 - 261b q^83 + 85008 * q^85 - 38896 * q^86 + 1394b q^87 - 30464 * q^88 - 5216b q^89 - 1652b q^90 + 59136 * q^92 + 26128 * q^93 + 8632b q^94 - 60536 * q^95 - 1024b q^96 - 1310b q^97 - 28084 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} + 32 q^{4} - 128 q^{8} - 118 q^{9} + 952 q^{11} - 2576 q^{15} + 512 q^{16} + 472 q^{18} - 3808 q^{22} + 7392 q^{23} + 11782 q^{25} + 2788 q^{29} + 10304 q^{30} - 2048 q^{32} - 1888 q^{36}+ \cdots - 56168 q^{99}+O(q^{100}) $ 2 * q - 8 * q^2 + 32 * q^4 - 128 * q^8 - 118 * q^9 + 952 * q^11 - 2576 * q^15 + 512 * q^16 + 472 * q^18 - 3808 * q^22 + 7392 * q^23 + 11782 * q^25 + 2788 * q^29 + 10304 * q^30 - 2048 * q^32 - 1888 * q^36 + 24180 * q^37 + 26128 * q^39 + 19448 * q^43 + 15232 * q^44 - 29568 * q^46 - 47128 * q^50 - 24288 * q^51 + 8620 * q^53 + 17296 * q^57 - 11152 * q^58 - 41216 * q^60 + 8192 * q^64 - 182896 * q^65 + 40472 * q^67 + 59584 * q^71 + 7552 * q^72 - 96720 * q^74 - 104512 * q^78 - 66352 * q^79 - 82462 * q^81 + 170016 * q^85 - 77792 * q^86 - 60928 * q^88 + 118272 * q^92 + 52256 * q^93 - 121072 * q^95 - 56168 * 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

−6.78233
6.78233

−4.00000

−13.5647

16.0000

94.9526

54.2586

−64.0000

−59.0000

−379.810

1.2

−4.00000

13.5647

16.0000

−94.9526

−54.2586

−64.0000

−59.0000

379.810

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$7$	$ -1 $

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	98.6.a.e	✓	2
3.b	odd	2	1		882.6.a.bo		2
4.b	odd	2	1		784.6.a.y		2
7.b	odd	2	1	inner	98.6.a.e	✓	2
7.c	even	3	2		98.6.c.g		4
7.d	odd	6	2		98.6.c.g		4
21.c	even	2	1		882.6.a.bo		2
28.d	even	2	1		784.6.a.y		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
98.6.a.e	✓	2	1.a	even	1	1	trivial
98.6.a.e	✓	2	7.b	odd	2	1	inner
98.6.c.g		4	7.c	even	3	2
98.6.c.g		4	7.d	odd	6	2
784.6.a.y		2	4.b	odd	2	1
784.6.a.y		2	28.d	even	2	1
882.6.a.bo		2	3.b	odd	2	1
882.6.a.bo		2	21.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{2} - 184 $ T3^2 - 184 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(98))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T + 4)^{2} $ (T + 4)^2
$3$	$ T^{2} - 184 $ T^2 - 184
$5$	$ T^{2} - 9016 $ T^2 - 9016
$7$	$ T^{2} $ T^2
$11$	$ (T - 476)^{2} $ (T - 476)^2
$13$	$ T^{2} - 927544 $ T^2 - 927544
$17$	$ T^{2} - 801504 $ T^2 - 801504
$19$	$ T^{2} - 406456 $ T^2 - 406456
$23$	$ (T - 3696)^{2} $ (T - 3696)^2
$29$	$ (T - 1394)^{2} $ (T - 1394)^2
$31$	$ T^{2} - 3710176 $ T^2 - 3710176
$37$	$ (T - 12090)^{2} $ (T - 12090)^2
$41$	$ T^{2} - 231634656 $ T^2 - 231634656
$43$	$ (T - 9724)^{2} $ (T - 9724)^2
$47$	$ T^{2} - 856881376 $ T^2 - 856881376
$53$	$ (T - 4310)^{2} $ (T - 4310)^2
$59$	$ T^{2} - 434675896 $ T^2 - 434675896
$61$	$ T^{2} - 86337400 $ T^2 - 86337400
$67$	$ (T - 20236)^{2} $ (T - 20236)^2
$71$	$ (T - 29792)^{2} $ (T - 29792)^2
$73$	$ T^{2} - 127369216 $ T^2 - 127369216
$79$	$ (T + 33176)^{2} $ (T + 33176)^2
$83$	$ T^{2} - 12534264 $ T^2 - 12534264
$89$	$ T^{2} - 5006024704 $ T^2 - 5006024704
$97$	$ T^{2} - 315762400 $ T^2 - 315762400
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Level:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	98.a (trivial)

\(f(q)\)	\(=\)	\( q - 4 q^{2} + \beta q^{3} + 16 q^{4} - 7 \beta q^{5} - 4 \beta q^{6} - 64 q^{8} - 59 q^{9} + 28 \beta q^{10} + 476 q^{11} + 16 \beta q^{12} + 71 \beta q^{13} - 1288 q^{15} + 256 q^{16} - 66 \beta q^{17} + \cdots - 28084 q^{99} +O(q^{100}) \) q - 4 * q^2 + b * q^3 + 16 * q^4 - 7b q^5 - 4b q^6 - 64 * q^8 - 59 * q^9 + 28b q^10 + 476 * q^11 + 16b q^12 + 71b q^13 - 1288 * q^15 + 256 * q^16 - 66b q^17 + 236 * q^18 + 47b q^19 - 112b q^20 - 1904 * q^22 + 3696 * q^23 - 64b q^24 + 5891 * q^25 - 284b q^26 - 302b q^27 + 1394 * q^29 + 5152 * q^30 + 142b q^31 - 1024 * q^32 + 476b q^33 + 264b q^34 - 944 * q^36 + 12090 * q^37 - 188b q^38 + 13064 * q^39 + 448b q^40 + 1122b q^41 + 9724 * q^43 + 7616 * q^44 + 413b q^45 - 14784 * q^46 - 2158b q^47 + 256b q^48 - 23564 * q^50 - 12144 * q^51 + 1136b q^52 + 4310 * q^53 + 1208b q^54 - 3332b q^55 + 8648 * q^57 - 5576 * q^58 - 1537b q^59 - 20608 * q^60 + 685b q^61 - 568b q^62 + 4096 * q^64 - 91448 * q^65 - 1904b q^66 + 20236 * q^67 - 1056b q^68 + 3696b q^69 + 29792 * q^71 + 3776 * q^72 + 832b q^73 - 48360 * q^74 + 5891b q^75 + 752b q^76 - 52256 * q^78 - 33176 * q^79 - 1792b q^80 - 41231 * q^81 - 4488b q^82 - 261b q^83 + 85008 * q^85 - 38896 * q^86 + 1394b q^87 - 30464 * q^88 - 5216b q^89 - 1652b q^90 + 59136 * q^92 + 26128 * q^93 + 8632b q^94 - 60536 * q^95 - 1024b q^96 - 1310b q^97 - 28084 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 32 q^{4} - 128 q^{8} - 118 q^{9} + 952 q^{11} - 2576 q^{15} + 512 q^{16} + 472 q^{18} - 3808 q^{22} + 7392 q^{23} + 11782 q^{25} + 2788 q^{29} + 10304 q^{30} - 2048 q^{32} - 1888 q^{36}+ \cdots - 56168 q^{99}+O(q^{100}) \) 2 * q - 8 * q^2 + 32 * q^4 - 128 * q^8 - 118 * q^9 + 952 * q^11 - 2576 * q^15 + 512 * q^16 + 472 * q^18 - 3808 * q^22 + 7392 * q^23 + 11782 * q^25 + 2788 * q^29 + 10304 * q^30 - 2048 * q^32 - 1888 * q^36 + 24180 * q^37 + 26128 * q^39 + 19448 * q^43 + 15232 * q^44 - 29568 * q^46 - 47128 * q^50 - 24288 * q^51 + 8620 * q^53 + 17296 * q^57 - 11152 * q^58 - 41216 * q^60 + 8192 * q^64 - 182896 * q^65 + 40472 * q^67 + 59584 * q^71 + 7552 * q^72 - 96720 * q^74 - 104512 * q^78 - 66352 * q^79 - 82462 * q^81 + 170016 * q^85 - 77792 * q^86 - 60928 * q^88 + 118272 * q^92 + 52256 * q^93 - 121072 * q^95 - 56168 * q^99

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