LMFDB - Newform orbit 675.3.c.l

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,3,Mod(26,675)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [2,0,0,6,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$18.3924178443$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 135)
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 $i = \sqrt{-1}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + i q^{2} + 3 q^{4} - 6 q^{7} + 7 i q^{8} + 21 i q^{11} - 15 q^{13} - 6 i q^{14} + 5 q^{16} - 23 i q^{17} - 14 q^{19} - 21 q^{22} - 7 i q^{23} - 15 i q^{26} - 18 q^{28} - 3 i q^{29} - 25 q^{31} + 33 i q^{32} + \cdots - 13 i q^{98} +O(q^{100}) $ q + i * q^2 + 3 * q^4 - 6 * q^7 + 7i q^8 + 21i q^11 - 15 * q^13 - 6i q^14 + 5 * q^16 - 23i q^17 - 14 * q^19 - 21 * q^22 - 7i q^23 - 15i q^26 - 18 * q^28 - 3i q^29 - 25 * q^31 + 33i q^32 + 23 * q^34 - 54 * q^37 - 14i q^38 + 24i q^41 + 15 * q^43 + 63i q^44 + 7 * q^46 + 49i q^47 - 13 * q^49 - 45 * q^52 + 14i q^53 - 42i q^56 + 3 * q^58 + 30i q^59 + 44 * q^61 - 25i q^62 - 13 * q^64 - 66 * q^67 - 69i q^68 + 18i q^71 - 54i q^74 - 42 * q^76 - 126i q^77 + 37 * q^79 - 24 * q^82 + 116i q^83 + 15i q^86 - 147 * q^88 - 126i q^89 + 90 * q^91 - 21i q^92 - 49 * q^94 - 78 * q^97 - 13i q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{4} - 12 q^{7} - 30 q^{13} + 10 q^{16} - 28 q^{19} - 42 q^{22} - 36 q^{28} - 50 q^{31} + 46 q^{34} - 108 q^{37} + 30 q^{43} + 14 q^{46} - 26 q^{49} - 90 q^{52} + 6 q^{58} + 88 q^{61} - 26 q^{64}+ \cdots - 156 q^{97}+O(q^{100}) $ 2 * q + 6 * q^4 - 12 * q^7 - 30 * q^13 + 10 * q^16 - 28 * q^19 - 42 * q^22 - 36 * q^28 - 50 * q^31 + 46 * q^34 - 108 * q^37 + 30 * q^43 + 14 * q^46 - 26 * q^49 - 90 * q^52 + 6 * q^58 + 88 * q^61 - 26 * q^64 - 132 * q^67 - 84 * q^76 + 74 * q^79 - 48 * q^82 - 294 * q^88 + 180 * q^91 - 98 * q^94 - 156 * q^97

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

	−	1.00000i
		1.00000i

−

1.00000i

3.00000

−6.00000

−

7.00000i

26.2

1.00000i

3.00000

−6.00000

7.00000i

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.l		2
3.b	odd	2	1	inner	675.3.c.l		2
5.b	even	2	1		675.3.c.m		2
5.c	odd	4	1		135.3.d.b	✓	2
5.c	odd	4	1		135.3.d.e	yes	2
15.d	odd	2	1		675.3.c.m		2
15.e	even	4	1		135.3.d.b	✓	2
15.e	even	4	1		135.3.d.e	yes	2
20.e	even	4	1		2160.3.c.b		2
20.e	even	4	1		2160.3.c.e		2
45.k	odd	12	2		405.3.h.d		4
45.k	odd	12	2		405.3.h.g		4
45.l	even	12	2		405.3.h.d		4
45.l	even	12	2		405.3.h.g		4
60.l	odd	4	1		2160.3.c.b		2
60.l	odd	4	1		2160.3.c.e		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
135.3.d.b	✓	2	5.c	odd	4	1
135.3.d.b	✓	2	15.e	even	4	1
135.3.d.e	yes	2	5.c	odd	4	1
135.3.d.e	yes	2	15.e	even	4	1
405.3.h.d		4	45.k	odd	12	2
405.3.h.d		4	45.l	even	12	2
405.3.h.g		4	45.k	odd	12	2
405.3.h.g		4	45.l	even	12	2
675.3.c.l		2	1.a	even	1	1	trivial
675.3.c.l		2	3.b	odd	2	1	inner
675.3.c.m		2	5.b	even	2	1
675.3.c.m		2	15.d	odd	2	1
2160.3.c.b		2	20.e	even	4	1
2160.3.c.b		2	60.l	odd	4	1
2160.3.c.e		2	20.e	even	4	1
2160.3.c.e		2	60.l	odd	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}^{2} + 1 $

T2^2 + 1

$ T_{7} + 6 $

T7 + 6

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 1 $ T^2 + 1
$3$	$ T^{2} $ T^2
$5$	$ T^{2} $ T^2
$7$	$ (T + 6)^{2} $ (T + 6)^2
$11$	$ T^{2} + 441 $ T^2 + 441
$13$	$ (T + 15)^{2} $ (T + 15)^2
$17$	$ T^{2} + 529 $ T^2 + 529
$19$	$ (T + 14)^{2} $ (T + 14)^2
$23$	$ T^{2} + 49 $ T^2 + 49
$29$	$ T^{2} + 9 $ T^2 + 9
$31$	$ (T + 25)^{2} $ (T + 25)^2
$37$	$ (T + 54)^{2} $ (T + 54)^2
$41$	$ T^{2} + 576 $ T^2 + 576
$43$	$ (T - 15)^{2} $ (T - 15)^2
$47$	$ T^{2} + 2401 $ T^2 + 2401
$53$	$ T^{2} + 196 $ T^2 + 196
$59$	$ T^{2} + 900 $ T^2 + 900
$61$	$ (T - 44)^{2} $ (T - 44)^2
$67$	$ (T + 66)^{2} $ (T + 66)^2
$71$	$ T^{2} + 324 $ T^2 + 324
$73$	$ T^{2} $ T^2
$79$	$ (T - 37)^{2} $ (T - 37)^2
$83$	$ T^{2} + 13456 $ T^2 + 13456
$89$	$ T^{2} + 15876 $ T^2 + 15876
$97$	$ (T + 78)^{2} $ (T + 78)^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)

\(f(q)\)	\(=\)	\( q + i q^{2} + 3 q^{4} - 6 q^{7} + 7 i q^{8} + 21 i q^{11} - 15 q^{13} - 6 i q^{14} + 5 q^{16} - 23 i q^{17} - 14 q^{19} - 21 q^{22} - 7 i q^{23} - 15 i q^{26} - 18 q^{28} - 3 i q^{29} - 25 q^{31} + 33 i q^{32} + \cdots - 13 i q^{98} +O(q^{100}) \) q + i * q^2 + 3 * q^4 - 6 * q^7 + 7i q^8 + 21i q^11 - 15 * q^13 - 6i q^14 + 5 * q^16 - 23i q^17 - 14 * q^19 - 21 * q^22 - 7i q^23 - 15i q^26 - 18 * q^28 - 3i q^29 - 25 * q^31 + 33i q^32 + 23 * q^34 - 54 * q^37 - 14i q^38 + 24i q^41 + 15 * q^43 + 63i q^44 + 7 * q^46 + 49i q^47 - 13 * q^49 - 45 * q^52 + 14i q^53 - 42i q^56 + 3 * q^58 + 30i q^59 + 44 * q^61 - 25i q^62 - 13 * q^64 - 66 * q^67 - 69i q^68 + 18i q^71 - 54i q^74 - 42 * q^76 - 126i q^77 + 37 * q^79 - 24 * q^82 + 116i q^83 + 15i q^86 - 147 * q^88 - 126i q^89 + 90 * q^91 - 21i q^92 - 49 * q^94 - 78 * q^97 - 13i q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{4} - 12 q^{7} - 30 q^{13} + 10 q^{16} - 28 q^{19} - 42 q^{22} - 36 q^{28} - 50 q^{31} + 46 q^{34} - 108 q^{37} + 30 q^{43} + 14 q^{46} - 26 q^{49} - 90 q^{52} + 6 q^{58} + 88 q^{61} - 26 q^{64}+ \cdots - 156 q^{97}+O(q^{100}) \) 2 * q + 6 * q^4 - 12 * q^7 - 30 * q^13 + 10 * q^16 - 28 * q^19 - 42 * q^22 - 36 * q^28 - 50 * q^31 + 46 * q^34 - 108 * q^37 + 30 * q^43 + 14 * q^46 - 26 * q^49 - 90 * q^52 + 6 * q^58 + 88 * q^61 - 26 * q^64 - 132 * q^67 - 84 * q^76 + 74 * q^79 - 48 * q^82 - 294 * q^88 + 180 * q^91 - 98 * q^94 - 156 * q^97

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