LMFDB - Newform orbit 1400.2.q.j

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1400,2,Mod(401,1400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1400, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("1400.401");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1400 = 2^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1400.q (of order $3$, degree $2$, minimal)

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:	no
Analytic conductor:	$11.1790562830$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.11337408.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 18x^{4} + 81x^{2} + 12 $ x^6 + 18x^4 + 81x^2 + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
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 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_{5} + \beta_{4} + \beta_{3}) q^{3} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 1) q^{7}+ \cdots + (\beta_{4} - 3 \beta_{2} - \beta_1) q^{9}+O(q^{10}) $ q + (-b5 + b4 + b3) * q^3 + (-b5 + b4 + b3 - b1 - 1) * q^7 + (b4 - 3b2 - b1) q^9	$ q + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{3} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 1) q^{7}+ \cdots + ( - 8 \beta_{4} - 6 \beta_{3} + \cdots + 21) q^{99}+O(q^{100}) $ q + (-b5 + b4 + b3) * q^3 + (-b5 + b4 + b3 - b1 - 1) * q^7 + (b4 - 3b2 - b1) q^9 + (b4 + b2 + 2b1 - 1) q^11 + (2b3 - b1 - 1) q^13 + (-2b2 + 2) q^17 + (-2b5 + b4 - b2 - b1) q^19 + (-b4 - 2b3 - 4b2 - b1 + 2) * q^21 + (-b5 + b4 + b2 - b1) * q^23 + (-3b4 - 3b3 + 6) * q^27 + (2b4 + b1 + 4) q^29 + (2b5 - 2b4 - 2b3 + 4b2 - 4) * q^31 + (4b5 - 6b2) * q^33 + (-2b5 + b4 + 3b2 - b1) * q^37 + (-2b5 + 2b3 - 6b2 - 4b1 + 6) * q^39 + (-2b4 + 2b3 - 2b1 + 3) q^41 + (-b4 - b3 - 4) * q^43 + (-2b5 + 3b4 - 5b2 - 3b1) * q^47 + (-3b4 - 2b3 - 2b2 - 1) q^49 - 2b5 q^51 + (-2b5 + 3b4 + 2b3 - 3b2 + 2b1 + 3) q^53 + (-4b3 + 2b1 - 6) * q^57 + (8b2 - 8) q^59 + (2b5 + b4 + 2b2 - b1) * q^61 + (-2b5 - 3b4 - 2b3 + 9b2 + b1 - 6) * q^63 + (3b5 - 3b4 - 3b3 - 2b2 + 2) * q^67 + (-2b3 + b1) q^69 + (4b5 - 2b4 - 4b3 + 4b1) * q^73 + (2b5 - b4 - 2b3 - b2 - 2b1 + 7) q^77 + (-4b5 + 2b4 + 6b2 - 2b1) * q^79 + (-6b5 + 6b4 + 6b3 + 9b2 - 9) * q^81 + (b4 + b3 + 10) * q^83 + (-b5 + b4 + b3 - 6b2 + 6) q^87 + (2b5 - b4 + b1) q^89 + (-2b5 - 10b2 + b1 + 9) * q^91 + (4b5 - 2b4 + 12b2 + 2b1) * q^93 + 2 * q^97 + (-8b4 - 6b3 - b1 + 21) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{7} - 9 q^{9}+O(q^{10}) $ 6 * q - 6 * q^7 - 9 * q^9	$ 6 q - 6 q^{7} - 9 q^{9} - 3 q^{11} - 6 q^{13} + 6 q^{17} - 3 q^{19} + 3 q^{23} + 36 q^{27} + 24 q^{29} - 12 q^{31} - 18 q^{33} + 9 q^{37} + 18 q^{39} + 18 q^{41} - 24 q^{43} - 15 q^{47} - 12 q^{49} + 9 q^{53} - 36 q^{57} - 24 q^{59} + 6 q^{61} - 9 q^{63} + 6 q^{67} + 39 q^{77} + 18 q^{79} - 27 q^{81} + 60 q^{83} + 18 q^{87} + 24 q^{91} + 36 q^{93} + 12 q^{97} + 126 q^{99}+O(q^{100}) $ 6 * q - 6 * q^7 - 9 * q^9 - 3 * q^11 - 6 * q^13 + 6 * q^17 - 3 * q^19 + 3 * q^23 + 36 * q^27 + 24 * q^29 - 12 * q^31 - 18 * q^33 + 9 * q^37 + 18 * q^39 + 18 * q^41 - 24 * q^43 - 15 * q^47 - 12 * q^49 + 9 * q^53 - 36 * q^57 - 24 * q^59 + 6 * q^61 - 9 * q^63 + 6 * q^67 + 39 * q^77 + 18 * q^79 - 27 * q^81 + 60 * q^83 + 18 * q^87 + 24 * q^91 + 36 * q^93 + 12 * q^97 + 126 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} + 18x^{4} + 81x^{2} + 12 $ x^6 + 18*x^4 + 81*x^2 + 12 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{3} + 9\nu + 2 ) / 4 $ (v^3 + 9*v + 2) / 4
$\beta_{3}$	$=$	$ ( \nu^{2} + \nu + 6 ) / 2 $ (v^2 + v + 6) / 2
$\beta_{4}$	$=$	$ ( \nu^{4} + 9\nu^{2} - 2\nu ) / 4 $ (v^4 + 9v^2 - 2v) / 4
$\beta_{5}$	$=$	$ ( \nu^{5} + \nu^{4} + 15\nu^{3} + 11\nu^{2} + 48\nu + 12 ) / 8 $ (v^5 + v^4 + 15v^3 + 11v^2 + 48*v + 12) / 8

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ 2\beta_{3} - \beta _1 - 6 $ 2*b3 - b1 - 6
$\nu^{3}$	$=$	$ 4\beta_{2} - 9\beta _1 - 2 $ 4b2 - 9b1 - 2
$\nu^{4}$	$=$	$ 4\beta_{4} - 18\beta_{3} + 11\beta _1 + 54 $ 4b4 - 18b3 + 11*b1 + 54
$\nu^{5}$	$=$	$ 8\beta_{5} - 4\beta_{4} - 4\beta_{3} - 60\beta_{2} + 87\beta _1 + 30 $ 8b5 - 4b4 - 4b3 - 60b2 + 87*b1 + 30

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$.

$n$	$351$	$701$	$801$	$1177$
$\chi(n)$	$1$	$1$	$-\beta_{2}$	$1$

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} $

401.1

	−	2.78499i
		3.17656i
	−	0.391571i
		2.78499i
	−	3.17656i
		0.391571i

−1.64497

−

2.84918i

−2.64497

−

0.0641892i

−3.91187

6.77556i

401.2

0.352860

0.611171i

−0.647140

−

2.56539i

1.25098

−

2.16676i

401.3

1.29211

2.23800i

0.292113

2.62958i

−1.83911

3.18543i

1201.1

−1.64497

2.84918i

−2.64497

0.0641892i

−3.91187

−

6.77556i

1201.2

0.352860

−

0.611171i

−0.647140

2.56539i

1.25098

2.16676i

1201.3

1.29211

−

2.23800i

0.292113

−

2.62958i

−1.83911

−

3.18543i

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	1400.2.q.j		6
5.b	even	2	1		280.2.q.e	✓	6
5.c	odd	4	2		1400.2.bh.i		12
7.c	even	3	1	inner	1400.2.q.j		6
7.c	even	3	1		9800.2.a.ce		3
7.d	odd	6	1		9800.2.a.cf		3
15.d	odd	2	1		2520.2.bi.q		6
20.d	odd	2	1		560.2.q.l		6
35.c	odd	2	1		1960.2.q.w		6
35.i	odd	6	1		1960.2.a.v		3
35.i	odd	6	1		1960.2.q.w		6
35.j	even	6	1		280.2.q.e	✓	6
35.j	even	6	1		1960.2.a.w		3
35.l	odd	12	2		1400.2.bh.i		12
105.o	odd	6	1		2520.2.bi.q		6
140.p	odd	6	1		560.2.q.l		6
140.p	odd	6	1		3920.2.a.cc		3
140.s	even	6	1		3920.2.a.cb		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
280.2.q.e	✓	6	5.b	even	2	1
280.2.q.e	✓	6	35.j	even	6	1
560.2.q.l		6	20.d	odd	2	1
560.2.q.l		6	140.p	odd	6	1
1400.2.q.j		6	1.a	even	1	1	trivial
1400.2.q.j		6	7.c	even	3	1	inner
1400.2.bh.i		12	5.c	odd	4	2
1400.2.bh.i		12	35.l	odd	12	2
1960.2.a.v		3	35.i	odd	6	1
1960.2.a.w		3	35.j	even	6	1
1960.2.q.w		6	35.c	odd	2	1
1960.2.q.w		6	35.i	odd	6	1
2520.2.bi.q		6	15.d	odd	2	1
2520.2.bi.q		6	105.o	odd	6	1
3920.2.a.cb		3	140.s	even	6	1
3920.2.a.cc		3	140.p	odd	6	1
9800.2.a.ce		3	7.c	even	3	1
9800.2.a.cf		3	7.d	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1400, [\chi])$:

$ T_{3}^{6} + 9T_{3}^{4} - 12T_{3}^{3} + 81T_{3}^{2} - 54T_{3} + 36 $

$ T_{11}^{6} + 3T_{11}^{5} + 33T_{11}^{4} + 16T_{11}^{3} + 708T_{11}^{2} + 1056T_{11} + 1936 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} + 9 T^{4} + \cdots + 36 $ T^6 + 9T^4 - 12T^3 + 81T^2 - 54T + 36
$5$	$ T^{6} $ T^6
$7$	$ T^{6} + 6 T^{5} + \cdots + 343 $ T^6 + 6T^5 + 24T^4 + 80T^3 + 168T^2 + 294*T + 343
$11$	$ T^{6} + 3 T^{5} + \cdots + 1936 $ T^6 + 3T^5 + 33T^4 + 16T^3 + 708T^2 + 1056*T + 1936
$13$	$ (T^{3} + 3 T^{2} - 24 T - 68)^{2} $ (T^3 + 3T^2 - 24T - 68)^2
$17$	$ (T^{2} - 2 T + 4)^{3} $ (T^2 - 2*T + 4)^3
$19$	$ T^{6} + 3 T^{5} + \cdots + 256 $ T^6 + 3T^5 + 33T^4 - 104T^3 + 528T^2 - 384*T + 256
$23$	$ T^{6} - 3 T^{5} + \cdots + 49 $ T^6 - 3T^5 + 24T^4 + 59T^3 + 204T^2 + 105*T + 49
$29$	$ (T^{3} - 12 T^{2} + \cdots + 26)^{2} $ (T^3 - 12T^2 + 21T + 26)^2
$31$	$ T^{6} + 12 T^{5} + \cdots + 16384 $ T^6 + 12T^5 + 132T^4 + 400T^3 + 1680T^2 - 1536*T + 16384
$37$	$ T^{6} - 9 T^{5} + \cdots + 9216 $ T^6 - 9T^5 + 81T^4 - 192T^3 + 864T^2 + 9216
$41$	$ (T^{3} - 9 T^{2} + \cdots + 381)^{2} $ (T^3 - 9T^2 - 45T + 381)^2
$43$	$ (T^{3} + 12 T^{2} + \cdots + 22)^{2} $ (T^3 + 12T^2 + 39T + 22)^2
$47$	$ T^{6} + 15 T^{5} + \cdots + 2521744 $ T^6 + 15T^5 + 321T^4 + 1736T^3 + 33036T^2 + 152448*T + 2521744
$53$	$ T^{6} - 9 T^{5} + \cdots + 389376 $ T^6 - 9T^5 + 153T^4 - 600T^3 + 10800T^2 - 44928*T + 389376
$59$	$ (T^{2} + 8 T + 64)^{3} $ (T^2 + 8*T + 64)^3
$61$	$ T^{6} - 6 T^{5} + \cdots + 295936 $ T^6 - 6T^5 + 123T^4 - 566T^3 + 10833T^2 - 47328*T + 295936
$67$	$ T^{6} - 6 T^{5} + \cdots + 64 $ T^6 - 6T^5 + 105T^4 + 430T^3 + 4713T^2 + 552*T + 64
$71$	$ T^{6} $ T^6
$73$	$ T^{6} + 108 T^{4} + \cdots + 112896 $ T^6 + 108T^4 - 672T^3 + 11664T^2 - 36288T + 112896
$79$	$ T^{6} - 18 T^{5} + \cdots + 589824 $ T^6 - 18T^5 + 324T^4 - 1536T^3 + 13824T^2 + 589824
$83$	$ (T^{3} - 30 T^{2} + \cdots - 904)^{2} $ (T^3 - 30T^2 + 291T - 904)^2
$89$	$ T^{6} + 27 T^{4} + \cdots + 1764 $ T^6 + 27T^4 + 84T^3 + 729T^2 + 1134T + 1764
$97$	$ (T - 2)^{6} $ (T - 2)^6
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1400.q (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{3} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 1) q^{7}+ \cdots + (\beta_{4} - 3 \beta_{2} - \beta_1) q^{9}+O(q^{10}) \) q + (-b5 + b4 + b3) * q^3 + (-b5 + b4 + b3 - b1 - 1) * q^7 + (b4 - 3b2 - b1) q^9	\( q + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{3} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 1) q^{7}+ \cdots + ( - 8 \beta_{4} - 6 \beta_{3} + \cdots + 21) q^{99}+O(q^{100}) \) q + (-b5 + b4 + b3) * q^3 + (-b5 + b4 + b3 - b1 - 1) * q^7 + (b4 - 3b2 - b1) q^9 + (b4 + b2 + 2b1 - 1) q^11 + (2b3 - b1 - 1) q^13 + (-2b2 + 2) q^17 + (-2b5 + b4 - b2 - b1) q^19 + (-b4 - 2b3 - 4b2 - b1 + 2) * q^21 + (-b5 + b4 + b2 - b1) * q^23 + (-3b4 - 3b3 + 6) * q^27 + (2b4 + b1 + 4) q^29 + (2b5 - 2b4 - 2b3 + 4b2 - 4) * q^31 + (4b5 - 6b2) * q^33 + (-2b5 + b4 + 3b2 - b1) * q^37 + (-2b5 + 2b3 - 6b2 - 4b1 + 6) * q^39 + (-2b4 + 2b3 - 2b1 + 3) q^41 + (-b4 - b3 - 4) * q^43 + (-2b5 + 3b4 - 5b2 - 3b1) * q^47 + (-3b4 - 2b3 - 2b2 - 1) q^49 - 2b5 q^51 + (-2b5 + 3b4 + 2b3 - 3b2 + 2b1 + 3) q^53 + (-4b3 + 2b1 - 6) * q^57 + (8b2 - 8) q^59 + (2b5 + b4 + 2b2 - b1) * q^61 + (-2b5 - 3b4 - 2b3 + 9b2 + b1 - 6) * q^63 + (3b5 - 3b4 - 3b3 - 2b2 + 2) * q^67 + (-2b3 + b1) q^69 + (4b5 - 2b4 - 4b3 + 4b1) * q^73 + (2b5 - b4 - 2b3 - b2 - 2b1 + 7) q^77 + (-4b5 + 2b4 + 6b2 - 2b1) * q^79 + (-6b5 + 6b4 + 6b3 + 9b2 - 9) * q^81 + (b4 + b3 + 10) * q^83 + (-b5 + b4 + b3 - 6b2 + 6) q^87 + (2b5 - b4 + b1) q^89 + (-2b5 - 10b2 + b1 + 9) * q^91 + (4b5 - 2b4 + 12b2 + 2b1) * q^93 + 2 * q^97 + (-8b4 - 6b3 - b1 + 21) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{7} - 9 q^{9}+O(q^{10}) \) 6 * q - 6 * q^7 - 9 * q^9	\( 6 q - 6 q^{7} - 9 q^{9} - 3 q^{11} - 6 q^{13} + 6 q^{17} - 3 q^{19} + 3 q^{23} + 36 q^{27} + 24 q^{29} - 12 q^{31} - 18 q^{33} + 9 q^{37} + 18 q^{39} + 18 q^{41} - 24 q^{43} - 15 q^{47} - 12 q^{49} + 9 q^{53} - 36 q^{57} - 24 q^{59} + 6 q^{61} - 9 q^{63} + 6 q^{67} + 39 q^{77} + 18 q^{79} - 27 q^{81} + 60 q^{83} + 18 q^{87} + 24 q^{91} + 36 q^{93} + 12 q^{97} + 126 q^{99}+O(q^{100}) \) 6 * q - 6 * q^7 - 9 * q^9 - 3 * q^11 - 6 * q^13 + 6 * q^17 - 3 * q^19 + 3 * q^23 + 36 * q^27 + 24 * q^29 - 12 * q^31 - 18 * q^33 + 9 * q^37 + 18 * q^39 + 18 * q^41 - 24 * q^43 - 15 * q^47 - 12 * q^49 + 9 * q^53 - 36 * q^57 - 24 * q^59 + 6 * q^61 - 9 * q^63 + 6 * q^67 + 39 * q^77 + 18 * q^79 - 27 * q^81 + 60 * q^83 + 18 * q^87 + 24 * q^91 + 36 * q^93 + 12 * q^97 + 126 * q^99

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