LMFDB - Newform orbit 675.3.c.r

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,3,Mod(26,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("675.26");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	675.c (of order $2$, degree $1$, 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:	$18.3924178443$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.60217600.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 16x^{4} + 64x^{2} + 4 $ x^6 + 16x^4 + 64x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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^{2} + (\beta_{4} - 2) q^{4} + (\beta_{4} - \beta_{2} - 2) q^{7} + (\beta_{5} + \beta_{3} - 2 \beta_1) q^{8} + (\beta_{5} - 2 \beta_1) q^{11} + ( - 2 \beta_{4} - 3 \beta_{2} + 4) q^{13} + (\beta_{5} - 7 \beta_1) q^{14}+ \cdots + ( - 3 \beta_{5} - 5 \beta_{3} + 22 \beta_1) q^{98}+O(q^{100}) $ q + b1 * q^2 + (b4 - 2) * q^4 + (b4 - b2 - 2) * q^7 + (b5 + b3 - 2b1) q^8 + (b5 - 2b1) q^11 + (-2b4 - 3b2 + 4) * q^13 + (b5 - 7b1) q^14 + (-4b4 - 2b2 + 7) * q^16 + (-b5 - 2b3 + b1) q^17 + (-b4 + 3b2 - 4) q^19 + (-6b4 + b2 + 12) q^22 + (-b5 + b3 - 6b1) q^23 + (-2b5 - 5b3 + 9b1) q^26 + (-7b4 - 3b2 + 34) * q^28 + (b5 + 3b1) q^29 + (3b4 - 4b2 - 7) * q^31 + (-2b3 + 13b1) * q^32 + (9b4 + 5b2 - 12) * q^34 + (-5b4 - 5b2 + 25) * q^37 + (-b5 + 2b3 + 3b1) * q^38 + (b5 - 5b3 - 7b1) * q^41 + (-3b4 - 7b2 - 14) * q^43 + (-2b5 - 5b3 + 29b1) q^44 + (-4b4 - 4b2 + 39) * q^46 + (2b5 - 4b3 - 15b1) q^47 + (-3b4 - 2b2 + 12) * q^49 + (19b4 + b2 - 53) q^52 + (-3b5 + 4b3 + 21b1) q^53 + (-3b5 - 10b3 + 31b1) q^56 + (-b4 + b2 - 18) * q^58 + (-6b5 - 5b3 - 23b1) q^59 + (-9b4 + 7b2 - 13) * q^61 + (3b5 - b3 - 23b1) * q^62 + (b4 - 2b2 - 56) q^64 + (-8b4 - 7b2 + 1) * q^67 + (5b5 + 6b3 - 39b1) q^68 + (6b5 + 13b1) * q^71 + (11b4 + 4b2 + 73) * q^73 + (-5b5 - 10b3 + 40b1) q^74 + (-b4 + 5b2 - 28) q^76 + (2b5 - 5b3 + 36b1) q^77 + (-b4 - 11b2 + 56) q^79 + (-b4 + 16b2 + 27) q^82 + (3b5 + 9b3 + 9b1) q^83 + (-3b5 - 10b3 - 9b1) q^86 + (23b4 + 17b2 - 141) * q^88 + (9b5 + 10b3 + 17b1) q^89 + (26b4 + 9b2 + 13) * q^91 + (-8b5 - 4b3 + 27b1) q^92 + (-15b4 + 14b2 + 78) * q^94 + (-22b4 + 2b2 + 19) * q^97 + (-3b5 - 5b3 + 22b1) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 14 q^{4} - 16 q^{7} + 22 q^{13} + 46 q^{16} - 16 q^{19} + 86 q^{22} + 212 q^{28} - 56 q^{31} - 80 q^{34} + 150 q^{37} - 92 q^{43} + 234 q^{46} + 74 q^{49} - 354 q^{52} - 104 q^{58} - 46 q^{61} - 342 q^{64}+ \cdots + 162 q^{97}+O(q^{100}) $ 6 * q - 14 * q^4 - 16 * q^7 + 22 * q^13 + 46 * q^16 - 16 * q^19 + 86 * q^22 + 212 * q^28 - 56 * q^31 - 80 * q^34 + 150 * q^37 - 92 * q^43 + 234 * q^46 + 74 * q^49 - 354 * q^52 - 104 * q^58 - 46 * q^61 - 342 * q^64 + 8 * q^67 + 424 * q^73 - 156 * q^76 + 316 * q^79 + 196 * q^82 - 858 * q^88 + 44 * q^91 + 526 * q^94 + 162 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} + 16x^{4} + 64x^{2} + 4 $ x^6 + 16*x^4 + 64*x^2 + 4 :

$\beta_{1}$	$=$	$ ( \nu^{3} + 10\nu ) / 2 $ (v^3 + 10*v) / 2
$\beta_{2}$	$=$	$ ( \nu^{4} + 6\nu^{2} - 10 ) / 2 $ (v^4 + 6*v^2 - 10) / 2
$\beta_{3}$	$=$	$ -4\nu^{3} - 31\nu $ -4v^3 - 31v
$\beta_{4}$	$=$	$ \nu^{4} + 9\nu^{2} + 5 $ v^4 + 9*v^2 + 5
$\beta_{5}$	$=$	$ ( 3\nu^{5} + 43\nu^{3} + 148\nu ) / 2 $ (3v^5 + 43v^3 + 148*v) / 2

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

$\nu$	$=$	$ ( \beta_{3} + 8\beta_1 ) / 9 $ (b3 + 8*b1) / 9
$\nu^{2}$	$=$	$ ( \beta_{4} - 2\beta_{2} - 15 ) / 3 $ (b4 - 2*b2 - 15) / 3
$\nu^{3}$	$=$	$ ( -10\beta_{3} - 62\beta_1 ) / 9 $ (-10b3 - 62b1) / 9
$\nu^{4}$	$=$	$ -2\beta_{4} + 6\beta_{2} + 40 $ -2b4 + 6b2 + 40
$\nu^{5}$	$=$	$ ( 6\beta_{5} + 94\beta_{3} + 494\beta_1 ) / 9 $ (6b5 + 94b3 + 494*b1) / 9

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$-1$	$1$

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} $

26.1

	−	2.69399i
	−	2.94600i
	−	0.252000i
		0.252000i
		2.94600i
		2.69399i

−

3.69399i

−9.64560

−9.20921

20.8548i

26.2

−

1.94600i

0.213103

−6.41178

−

8.19868i

26.3

−

1.25200i

2.43250

7.62099

−

8.05349i

26.4

1.25200i

2.43250

7.62099

8.05349i

26.5

1.94600i

0.213103

−6.41178

8.19868i

26.6

3.69399i

−9.64560

−9.20921

−

20.8548i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	675.3.c.r	✓	6
3.b	odd	2	1	inner	675.3.c.r	✓	6
5.b	even	2	1		675.3.c.s	yes	6
5.c	odd	4	1		675.3.d.j		6
5.c	odd	4	1		675.3.d.k		6
15.d	odd	2	1		675.3.c.s	yes	6
15.e	even	4	1		675.3.d.j		6
15.e	even	4	1		675.3.d.k		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
675.3.c.r	✓	6	1.a	even	1	1	trivial
675.3.c.r	✓	6	3.b	odd	2	1	inner
675.3.c.s	yes	6	5.b	even	2	1
675.3.c.s	yes	6	15.d	odd	2	1
675.3.d.j		6	5.c	odd	4	1
675.3.d.j		6	15.e	even	4	1
675.3.d.k		6	5.c	odd	4	1
675.3.d.k		6	15.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(675, [\chi])$:

$ T_{2}^{6} + 19T_{2}^{4} + 79T_{2}^{2} + 81 $

$ T_{7}^{3} + 8T_{7}^{2} - 60T_{7} - 450 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} + 19 T^{4} + \cdots + 81 $ T^6 + 19T^4 + 79T^2 + 81
$3$	$ T^{6} $ T^6
$5$	$ T^{6} $ T^6
$7$	$ (T^{3} + 8 T^{2} + \cdots - 450)^{2} $ (T^3 + 8T^2 - 60T - 450)^2
$11$	$ T^{6} + 499 T^{4} + \cdots + 455625 $ T^6 + 499T^4 + 64075T^2 + 455625
$13$	$ (T^{3} - 11 T^{2} + \cdots + 4475)^{2} $ (T^3 - 11T^2 - 415T + 4475)^2
$17$	$ T^{6} + 952 T^{4} + \cdots + 544644 $ T^6 + 952T^4 + 226828T^2 + 544644
$19$	$ (T^{3} + 8 T^{2} + \cdots + 766)^{2} $ (T^3 + 8T^2 - 352T + 766)^2
$23$	$ T^{6} + 1503 T^{4} + \cdots + 1476225 $ T^6 + 1503T^4 + 425331T^2 + 1476225
$29$	$ T^{6} + 544 T^{4} + \cdots + 202500 $ T^6 + 544T^4 + 28300T^2 + 202500
$31$	$ (T^{3} + 28 T^{2} + \cdots - 20484)^{2} $ (T^3 + 28T^2 - 732T - 20484)^2
$37$	$ (T^{3} - 75 T^{2} + \cdots + 36125)^{2} $ (T^3 - 75T^2 + 75T + 36125)^2
$41$	$ T^{6} + 6904 T^{4} + \cdots + 590490000 $ T^6 + 6904T^4 + 10913200T^2 + 590490000
$43$	$ (T^{3} + 46 T^{2} + \cdots + 7250)^{2} $ (T^3 + 46T^2 - 1300T + 7250)^2
$47$	$ T^{6} + \cdots + 30381535809 $ T^6 + 9931T^4 + 30736039T^2 + 30381535809
$53$	$ T^{6} + \cdots + 153173607876 $ T^6 + 16684T^4 + 89762644T^2 + 153173607876
$59$	$ T^{6} + \cdots + 10115330625 $ T^6 + 21379T^4 + 114772675T^2 + 10115330625
$61$	$ (T^{3} + 23 T^{2} + \cdots + 63331)^{2} $ (T^3 + 23T^2 - 5197T + 63331)^2
$67$	$ (T^{3} - 4 T^{2} + \cdots + 7650)^{2} $ (T^3 - 4T^2 - 4110T + 7650)^2
$71$	$ T^{6} + 16939 T^{4} + \cdots + 14630625 $ T^6 + 16939T^4 + 16469575T^2 + 14630625
$73$	$ (T^{3} - 212 T^{2} + \cdots + 160700)^{2} $ (T^3 - 212T^2 + 9620T + 160700)^2
$79$	$ (T^{3} - 158 T^{2} + \cdots + 141930)^{2} $ (T^3 - 158T^2 + 4056T + 141930)^2
$83$	$ T^{6} + 15804 T^{4} + \cdots + 273042576 $ T^6 + 15804T^4 + 21894624T^2 + 273042576
$89$	$ T^{6} + \cdots + 1117354702500 $ T^6 + 39204T^4 + 417935700T^2 + 1117354702500
$97$	$ (T^{3} - 81 T^{2} + \cdots - 272275)^{2} $ (T^3 - 81T^2 - 18165T - 272275)^2
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Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	675.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + (\beta_{4} - 2) q^{4} + (\beta_{4} - \beta_{2} - 2) q^{7} + (\beta_{5} + \beta_{3} - 2 \beta_1) q^{8} + (\beta_{5} - 2 \beta_1) q^{11} + ( - 2 \beta_{4} - 3 \beta_{2} + 4) q^{13} + (\beta_{5} - 7 \beta_1) q^{14}+ \cdots + ( - 3 \beta_{5} - 5 \beta_{3} + 22 \beta_1) q^{98}+O(q^{100}) \) q + b1 * q^2 + (b4 - 2) * q^4 + (b4 - b2 - 2) * q^7 + (b5 + b3 - 2b1) q^8 + (b5 - 2b1) q^11 + (-2b4 - 3b2 + 4) * q^13 + (b5 - 7b1) q^14 + (-4b4 - 2b2 + 7) * q^16 + (-b5 - 2b3 + b1) q^17 + (-b4 + 3b2 - 4) q^19 + (-6b4 + b2 + 12) q^22 + (-b5 + b3 - 6b1) q^23 + (-2b5 - 5b3 + 9b1) q^26 + (-7b4 - 3b2 + 34) * q^28 + (b5 + 3b1) q^29 + (3b4 - 4b2 - 7) * q^31 + (-2b3 + 13b1) * q^32 + (9b4 + 5b2 - 12) * q^34 + (-5b4 - 5b2 + 25) * q^37 + (-b5 + 2b3 + 3b1) * q^38 + (b5 - 5b3 - 7b1) * q^41 + (-3b4 - 7b2 - 14) * q^43 + (-2b5 - 5b3 + 29b1) q^44 + (-4b4 - 4b2 + 39) * q^46 + (2b5 - 4b3 - 15b1) q^47 + (-3b4 - 2b2 + 12) * q^49 + (19b4 + b2 - 53) q^52 + (-3b5 + 4b3 + 21b1) q^53 + (-3b5 - 10b3 + 31b1) q^56 + (-b4 + b2 - 18) * q^58 + (-6b5 - 5b3 - 23b1) q^59 + (-9b4 + 7b2 - 13) * q^61 + (3b5 - b3 - 23b1) * q^62 + (b4 - 2b2 - 56) q^64 + (-8b4 - 7b2 + 1) * q^67 + (5b5 + 6b3 - 39b1) q^68 + (6b5 + 13b1) * q^71 + (11b4 + 4b2 + 73) * q^73 + (-5b5 - 10b3 + 40b1) q^74 + (-b4 + 5b2 - 28) q^76 + (2b5 - 5b3 + 36b1) q^77 + (-b4 - 11b2 + 56) q^79 + (-b4 + 16b2 + 27) q^82 + (3b5 + 9b3 + 9b1) q^83 + (-3b5 - 10b3 - 9b1) q^86 + (23b4 + 17b2 - 141) * q^88 + (9b5 + 10b3 + 17b1) q^89 + (26b4 + 9b2 + 13) * q^91 + (-8b5 - 4b3 + 27b1) q^92 + (-15b4 + 14b2 + 78) * q^94 + (-22b4 + 2b2 + 19) * q^97 + (-3b5 - 5b3 + 22b1) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 14 q^{4} - 16 q^{7} + 22 q^{13} + 46 q^{16} - 16 q^{19} + 86 q^{22} + 212 q^{28} - 56 q^{31} - 80 q^{34} + 150 q^{37} - 92 q^{43} + 234 q^{46} + 74 q^{49} - 354 q^{52} - 104 q^{58} - 46 q^{61} - 342 q^{64}+ \cdots + 162 q^{97}+O(q^{100}) \) 6 * q - 14 * q^4 - 16 * q^7 + 22 * q^13 + 46 * q^16 - 16 * q^19 + 86 * q^22 + 212 * q^28 - 56 * q^31 - 80 * q^34 + 150 * q^37 - 92 * q^43 + 234 * q^46 + 74 * q^49 - 354 * q^52 - 104 * q^58 - 46 * q^61 - 342 * q^64 + 8 * q^67 + 424 * q^73 - 156 * q^76 + 316 * q^79 + 196 * q^82 - 858 * q^88 + 44 * q^91 + 526 * q^94 + 162 * q^97

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