LMFDB - Newform orbit 9800.2.a.cy

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9800,2,Mod(1,9800)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9800.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$78.2533939809$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.239575536.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 12x^{4} + 8x^{3} + 35x^{2} - 15x - 8 $ x^6 - x^5 - 12x^4 + 8x^3 + 35x^2 - 15x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 280)
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 a basis $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_1 q^{3} + (\beta_{2} + 1) q^{9}+O(q^{10}) $ q + b1 * q^3 + (b2 + 1) * q^9	$ q + \beta_1 q^{3} + (\beta_{2} + 1) q^{9} + (\beta_{5} + \beta_1) q^{11} + (\beta_{2} + 1) q^{13} + (\beta_{5} + \beta_{4} - \beta_{3} + \cdots + 1) q^{17}+ \cdots + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + \cdots + 2) q^{99}+O(q^{100}) $ q + b1 * q^3 + (b2 + 1) * q^9 + (b5 + b1) * q^11 + (b2 + 1) * q^13 + (b5 + b4 - b3 - b2 + 1) * q^17 + (b3 + b2 - b1 + 1) * q^19 + (b2 - b1 + 1) * q^23 + (b5 + b3 + b2 + b1 + 1) * q^27 + (2b4 - b3 - b2 + b1) q^29 + (-b5 + 2b4 - 2b3 - b2 - b1 + 1) * q^31 + (-b4 + 2b2 + 2) q^33 + (-b5 + b4 + b3 + 2) * q^37 + (b5 + b3 + b2 + 4b1 + 1) q^39 + (b5 - 2b4 + b2 + b1 - 2) q^41 + (b4 - 2b3 - 3b1) * q^43 + (b4 - b2 + 5) * q^47 + (-b5 - 2b4 - 2b3 + b2 - b1 - 1) * q^51 + (-b4 - b2 - 1) * q^53 + (b5 - b4 + 3b3 + b2 + 4b1 - 5) * q^57 + (-b5 - 2b4 + b2 + b1 + 3) q^59 + (b5 + b3 - 2b2 - 1) q^61 + (b5 + b3 - b2 + b1 + 3) * q^67 + (b5 + b3 + 4b1 - 3) q^69 + (-b5 - b3 - 3b2 + 1) q^71 + (b5 - b4 + 3b3 + b2 + 3) q^73 + (-b5 + 2b4 - b2 + b1 - 3) q^79 + (b5 - 2b4 + 3b3 + b2 + 4b1 - 2) q^81 + (b5 - b4 - b3 + b2 - b1 + 1) * q^83 + (-b5 - 3b4 - b3 + 3b2 - b1 + 5) * q^87 + (b5 + 2b4 - b3 - 2b2 - 4b1 - 1) q^89 + (-b5 - b4 - 3b3 - b2 + 1) q^93 + (-2b5 - 2b3 - 2b2 + 2) q^97 + (-b5 + 2b4 + b3 + 4b1 + 2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{3} + 7 q^{9}+O(q^{10}) $ 6 * q + q^3 + 7 * q^9	$ 6 q + q^{3} + 7 q^{9} + q^{11} + 7 q^{13} + 4 q^{17} + 5 q^{19} + 6 q^{23} + 7 q^{27} - 3 q^{29} + 2 q^{31} + 16 q^{33} + 9 q^{37} + 10 q^{39} - 6 q^{41} - 3 q^{43} + 27 q^{47} - 5 q^{53} - 26 q^{57} + 24 q^{59} - 9 q^{61} + 17 q^{67} - 15 q^{69} + 4 q^{71} + 18 q^{73} - 22 q^{79} - 6 q^{81} + 9 q^{83} + 39 q^{87} - 15 q^{89} + 10 q^{93} + 12 q^{97} + 11 q^{99}+O(q^{100}) $ 6 * q + q^3 + 7 * q^9 + q^11 + 7 * q^13 + 4 * q^17 + 5 * q^19 + 6 * q^23 + 7 * q^27 - 3 * q^29 + 2 * q^31 + 16 * q^33 + 9 * q^37 + 10 * q^39 - 6 * q^41 - 3 * q^43 + 27 * q^47 - 5 * q^53 - 26 * q^57 + 24 * q^59 - 9 * q^61 + 17 * q^67 - 15 * q^69 + 4 * q^71 + 18 * q^73 - 22 * q^79 - 6 * q^81 + 9 * q^83 + 39 * q^87 - 15 * q^89 + 10 * q^93 + 12 * q^97 + 11 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - x^{5} - 12x^{4} + 8x^{3} + 35x^{2} - 15x - 8 $ x^6 - x^5 - 12*x^4 + 8*x^3 + 35*x^2 - 15*x - 8 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - 4 $ v^2 - 4
$\beta_{3}$	$=$	$ ( \nu^{5} - 10\nu^{3} - 4\nu^{2} + 17\nu + 8 ) / 4 $ (v^5 - 10v^3 - 4v^2 + 17*v + 8) / 4
$\beta_{4}$	$=$	$ ( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 14\nu^{2} + 11\nu - 16 ) / 4 $ (v^5 - 2v^4 - 8v^3 + 14v^2 + 11v - 16) / 4
$\beta_{5}$	$=$	$ ( -\nu^{5} + 14\nu^{3} - 45\nu + 4 ) / 4 $ (-v^5 + 14v^3 - 45v + 4) / 4

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} + 4 $ b2 + 4
$\nu^{3}$	$=$	$ \beta_{5} + \beta_{3} + \beta_{2} + 7\beta _1 + 1 $ b5 + b3 + b2 + 7*b1 + 1
$\nu^{4}$	$=$	$ \beta_{5} - 2\beta_{4} + 3\beta_{3} + 10\beta_{2} + 4\beta _1 + 25 $ b5 - 2b4 + 3b3 + 10b2 + 4b1 + 25
$\nu^{5}$	$=$	$ 10\beta_{5} + 14\beta_{3} + 14\beta_{2} + 53\beta _1 + 18 $ 10b5 + 14b3 + 14b2 + 53b1 + 18

Display $\beta_i$ in terms of $\nu^j$

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

−2.54431
−2.03837
−0.319986
0.751428
2.10821
3.04302

−2.54431

3.47349

1.2

−2.03837

1.15495

1.3

−0.319986

−2.89761

1.4

0.751428

−2.43536

1.5

2.10821

1.44457

1.6

3.04302

6.25996

Atkin-Lehner signs

$ p $	Sign
$2$	$-1$
$5$	$-1$
$7$	$-1$

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	9800.2.a.cy		6
5.b	even	2	1		9800.2.a.cw		6
5.c	odd	4	2		1960.2.g.e		12
7.b	odd	2	1		9800.2.a.cv		6
7.d	odd	6	2		1400.2.q.o		12
35.c	odd	2	1		9800.2.a.cx		6
35.f	even	4	2		1960.2.g.f		12
35.i	odd	6	2		1400.2.q.n		12
35.k	even	12	4		280.2.bg.a	✓	24
140.x	odd	12	4		560.2.bw.f		24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
280.2.bg.a	✓	24	35.k	even	12	4
560.2.bw.f		24	140.x	odd	12	4
1400.2.q.n		12	35.i	odd	6	2
1400.2.q.o		12	7.d	odd	6	2
1960.2.g.e		12	5.c	odd	4	2
1960.2.g.f		12	35.f	even	4	2
9800.2.a.cv		6	7.b	odd	2	1
9800.2.a.cw		6	5.b	even	2	1
9800.2.a.cx		6	35.c	odd	2	1
9800.2.a.cy		6	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(9800))$:

$ T_{3}^{6} - T_{3}^{5} - 12T_{3}^{4} + 8T_{3}^{3} + 35T_{3}^{2} - 15T_{3} - 8 $

$ T_{11}^{6} - T_{11}^{5} - 37T_{11}^{4} + 69T_{11}^{3} + 296T_{11}^{2} - 832T_{11} + 512 $

$ T_{13}^{6} - 7T_{13}^{5} - 10T_{13}^{4} + 100T_{13}^{3} - 8T_{13}^{2} - 320T_{13} + 256 $

$ T_{19}^{6} - 5T_{19}^{5} - 47T_{19}^{4} + 301T_{19}^{3} - 94T_{19}^{2} - 1300T_{19} + 568 $

$ T_{23}^{6} - 6T_{23}^{5} - 19T_{23}^{4} + 108T_{23}^{3} + 123T_{23}^{2} - 478T_{23} - 337 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} - T^{5} - 12 T^{4} + \cdots - 8 $ T^6 - T^5 - 12T^4 + 8T^3 + 35T^2 - 15T - 8
$5$	$ T^{6} $ T^6
$7$	$ T^{6} $ T^6
$11$	$ T^{6} - T^{5} + \cdots + 512 $ T^6 - T^5 - 37T^4 + 69T^3 + 296T^2 - 832T + 512
$13$	$ T^{6} - 7 T^{5} + \cdots + 256 $ T^6 - 7T^5 - 10T^4 + 100T^3 - 8T^2 - 320*T + 256
$17$	$ T^{6} - 4 T^{5} + \cdots - 9552 $ T^6 - 4T^5 - 75T^4 + 244T^3 + 1656T^2 - 3552*T - 9552
$19$	$ T^{6} - 5 T^{5} + \cdots + 568 $ T^6 - 5T^5 - 47T^4 + 301T^3 - 94T^2 - 1300*T + 568
$23$	$ T^{6} - 6 T^{5} + \cdots - 337 $ T^6 - 6T^5 - 19T^4 + 108T^3 + 123T^2 - 478*T - 337
$29$	$ T^{6} + 3 T^{5} + \cdots - 7344 $ T^6 + 3T^5 - 105T^4 - 35T^3 + 2652T^2 - 2700*T - 7344
$31$	$ T^{6} - 2 T^{5} + \cdots - 1968 $ T^6 - 2T^5 - 115T^4 + 124T^3 + 2016T^2 + 1728*T - 1968
$37$	$ T^{6} - 9 T^{5} + \cdots - 3456 $ T^6 - 9T^5 - 75T^4 + 547T^3 + 1392T^2 - 8064*T - 3456
$41$	$ T^{6} + 6 T^{5} + \cdots + 2764 $ T^6 + 6T^5 - 100T^4 - 266T^3 + 2871T^2 - 5372*T + 2764
$43$	$ T^{6} + 3 T^{5} + \cdots - 95736 $ T^6 + 3T^5 - 151T^4 - 203T^3 + 6990T^2 + 2292*T - 95736
$47$	$ T^{6} - 27 T^{5} + \cdots - 1392 $ T^6 - 27T^5 + 273T^4 - 1271T^3 + 2604T^2 - 1320*T - 1392
$53$	$ T^{6} + 5 T^{5} + \cdots + 344 $ T^6 + 5T^5 - 77T^4 - 267T^3 + 1030T^2 + 1268*T + 344
$59$	$ T^{6} - 24 T^{5} + \cdots - 124704 $ T^6 - 24T^5 + 93T^4 + 2042T^3 - 23820T^2 + 93384*T - 124704
$61$	$ T^{6} + 9 T^{5} + \cdots - 105806 $ T^6 + 9T^5 - 172T^4 - 1550T^3 + 5289T^2 + 36181*T - 105806
$67$	$ T^{6} - 17 T^{5} + \cdots + 3842 $ T^6 - 17T^5 + 38T^4 + 396T^3 - 1201T^2 - 1883*T + 3842
$71$	$ T^{6} - 4 T^{5} + \cdots - 183296 $ T^6 - 4T^5 - 272T^4 + 1280T^3 + 11984T^2 - 33536*T - 183296
$73$	$ T^{6} - 18 T^{5} + \cdots - 179072 $ T^6 - 18T^5 - 79T^4 + 3998T^3 - 35340T^2 + 130568*T - 179072
$79$	$ T^{6} + 22 T^{5} + \cdots - 272 $ T^6 + 22T^5 + 61T^4 - 872T^3 - 1720T^2 + 1808*T - 272
$83$	$ T^{6} - 9 T^{5} + \cdots - 136 $ T^6 - 9T^5 - 103T^4 + 1389T^3 - 4230T^2 + 1556*T - 136
$89$	$ T^{6} + 15 T^{5} + \cdots + 115554 $ T^6 + 15T^5 - 268T^4 - 4498T^3 - 11367T^2 + 42867*T + 115554
$97$	$ T^{6} - 12 T^{5} + \cdots - 8192 $ T^6 - 12T^5 - 208T^4 + 1600T^3 + 13824T^2 - 14336*T - 8192
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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9800.a (trivial)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) q + b1 * q^3 + (b2 + 1) * q^9	\( q + \beta_1 q^{3} + (\beta_{2} + 1) q^{9} + (\beta_{5} + \beta_1) q^{11} + (\beta_{2} + 1) q^{13} + (\beta_{5} + \beta_{4} - \beta_{3} + \cdots + 1) q^{17}+ \cdots + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + \cdots + 2) q^{99}+O(q^{100}) \) q + b1 * q^3 + (b2 + 1) * q^9 + (b5 + b1) * q^11 + (b2 + 1) * q^13 + (b5 + b4 - b3 - b2 + 1) * q^17 + (b3 + b2 - b1 + 1) * q^19 + (b2 - b1 + 1) * q^23 + (b5 + b3 + b2 + b1 + 1) * q^27 + (2b4 - b3 - b2 + b1) q^29 + (-b5 + 2b4 - 2b3 - b2 - b1 + 1) * q^31 + (-b4 + 2b2 + 2) q^33 + (-b5 + b4 + b3 + 2) * q^37 + (b5 + b3 + b2 + 4b1 + 1) q^39 + (b5 - 2b4 + b2 + b1 - 2) q^41 + (b4 - 2b3 - 3b1) * q^43 + (b4 - b2 + 5) * q^47 + (-b5 - 2b4 - 2b3 + b2 - b1 - 1) * q^51 + (-b4 - b2 - 1) * q^53 + (b5 - b4 + 3b3 + b2 + 4b1 - 5) * q^57 + (-b5 - 2b4 + b2 + b1 + 3) q^59 + (b5 + b3 - 2b2 - 1) q^61 + (b5 + b3 - b2 + b1 + 3) * q^67 + (b5 + b3 + 4b1 - 3) q^69 + (-b5 - b3 - 3b2 + 1) q^71 + (b5 - b4 + 3b3 + b2 + 3) q^73 + (-b5 + 2b4 - b2 + b1 - 3) q^79 + (b5 - 2b4 + 3b3 + b2 + 4b1 - 2) q^81 + (b5 - b4 - b3 + b2 - b1 + 1) * q^83 + (-b5 - 3b4 - b3 + 3b2 - b1 + 5) * q^87 + (b5 + 2b4 - b3 - 2b2 - 4b1 - 1) q^89 + (-b5 - b4 - 3b3 - b2 + 1) q^93 + (-2b5 - 2b3 - 2b2 + 2) q^97 + (-b5 + 2b4 + b3 + 4b1 + 2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{3} + 7 q^{9}+O(q^{10}) \) 6 * q + q^3 + 7 * q^9	\( 6 q + q^{3} + 7 q^{9} + q^{11} + 7 q^{13} + 4 q^{17} + 5 q^{19} + 6 q^{23} + 7 q^{27} - 3 q^{29} + 2 q^{31} + 16 q^{33} + 9 q^{37} + 10 q^{39} - 6 q^{41} - 3 q^{43} + 27 q^{47} - 5 q^{53} - 26 q^{57} + 24 q^{59} - 9 q^{61} + 17 q^{67} - 15 q^{69} + 4 q^{71} + 18 q^{73} - 22 q^{79} - 6 q^{81} + 9 q^{83} + 39 q^{87} - 15 q^{89} + 10 q^{93} + 12 q^{97} + 11 q^{99}+O(q^{100}) \) 6 * q + q^3 + 7 * q^9 + q^11 + 7 * q^13 + 4 * q^17 + 5 * q^19 + 6 * q^23 + 7 * q^27 - 3 * q^29 + 2 * q^31 + 16 * q^33 + 9 * q^37 + 10 * q^39 - 6 * q^41 - 3 * q^43 + 27 * q^47 - 5 * q^53 - 26 * q^57 + 24 * q^59 - 9 * q^61 + 17 * q^67 - 15 * q^69 + 4 * q^71 + 18 * q^73 - 22 * q^79 - 6 * q^81 + 9 * q^83 + 39 * q^87 - 15 * q^89 + 10 * q^93 + 12 * q^97 + 11 * q^99

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