LMFDB - Newform orbit 1944.2.i.p

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1944,2,Mod(649,1944)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1944, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("1944.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1944 = 2^{3} \cdot 3^{5} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1944.i (of order $3$, degree $2$, not 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:	$15.5229181529$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
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} + \cdots + \beta_{2}) q^{5}+ \cdots + (\beta_{5} - \beta_{3} + \beta_{2} + \cdots + 1) q^{7}+O(q^{10}) $ q + (b5 - b4 - b3 + b2) * q^5 + (b5 - b3 + b2 - b1 + 1) * q^7	$ q + (\beta_{5} - \beta_{4} + \cdots + \beta_{2}) q^{5}+ \cdots + (8 \beta_{5} - 8 \beta_{3} + 8 \beta_{2} + \cdots + 3) q^{97}+O(q^{100}) $ q + (b5 - b4 - b3 + b2) * q^5 + (b5 - b3 + b2 - b1 + 1) * q^7 + (b5 - 2b3 + 2b2 - b1 + 1) * q^11 + (2b5 - 2b4 - 2b3 + 3b2 + b1) * q^13 + (-2b4 - 2b3 + 3) * q^17 + (b4 + 3b3 - 3) q^19 + (-3b5 + 3b4 + 3b3 - 2b2 + 2b1) q^23 + (b5 - 3b1 + 3) q^25 + (b3 - b2 + 2b1 - 2) q^29 + (b5 - b4 - b3 + 3b2 + 2b1) * q^31 + (b3 - 2) * q^35 + (3b4 + 4b3 - 5) * q^37 + (2b5 - 2b4 - 2b3 - 3b2 - b1) * q^41 + 5b5 q^43 + (-b5 + b3 - b2 - 2b1 + 2) q^47 + (b5 - b4 - b3 + 2b2 + 4b1) * q^49 + (3b4 + 5b3 - 2) * q^53 + (2b3 - 3) q^55 + (b5 - b4 - b3 + 5b2 + b1) q^59 + (4b5 - b3 + b2 - 6b1 + 6) * q^61 + (3b5 + 5b1 - 5) * q^65 + (2b5 - 2b4 - 2b3 - b2 + 8b1) * q^67 + (2b4 - 5b3 - 5) * q^71 + (-3b4 + 3b3 - 2) * q^73 + (b5 - b4 - b3 + 4b2 - 4b1) * q^77 + (7b5 - 3b3 + 3b2 - 4b1 + 4) * q^79 + (-8b5 - b3 + b2 - 6b1 + 6) * q^83 + (5b5 - 5b4 - 5b3 + 5b2 + 2b1) q^85 + (-5b4 + 2b3 + 6) * q^89 + (b4 + 4b3 - 4) q^91 + (-4b5 + 4b4 + 4b3 - 6b2 + b1) * q^95 + (8b5 - 8b3 + 8b2 - 3b1 + 3) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{7}+O(q^{10}) $ 6 * q + 3 * q^7	$ 6 q + 3 q^{7} + 3 q^{11} + 3 q^{13} + 18 q^{17} - 18 q^{19} + 6 q^{23} + 9 q^{25} - 6 q^{29} + 6 q^{31} - 12 q^{35} - 30 q^{37} - 3 q^{41} + 6 q^{47} + 12 q^{49} - 12 q^{53} - 18 q^{55} + 3 q^{59} + 18 q^{61} - 15 q^{65} + 24 q^{67} - 30 q^{71} - 12 q^{73} - 12 q^{77} + 12 q^{79} + 18 q^{83} + 6 q^{85} + 36 q^{89} - 24 q^{91} + 3 q^{95} + 9 q^{97}+O(q^{100}) $ 6 * q + 3 * q^7 + 3 * q^11 + 3 * q^13 + 18 * q^17 - 18 * q^19 + 6 * q^23 + 9 * q^25 - 6 * q^29 + 6 * q^31 - 12 * q^35 - 30 * q^37 - 3 * q^41 + 6 * q^47 + 12 * q^49 - 12 * q^53 - 18 * q^55 + 3 * q^59 + 18 * q^61 - 15 * q^65 + 24 * q^67 - 30 * q^71 - 12 * q^73 - 12 * q^77 + 12 * q^79 + 18 * q^83 + 6 * q^85 + 36 * q^89 - 24 * q^91 + 3 * q^95 + 9 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring

$\beta_{1}$	$=$	$ \zeta_{18}^{3} $ v^3
$\beta_{2}$	$=$	$ \zeta_{18}^{5} + \zeta_{18} $ v^5 + v
$\beta_{3}$	$=$	$ -\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} $ -v^4 + v^2 + v
$\beta_{4}$	$=$	$ -\zeta_{18}^{5} + \zeta_{18}^{4} $ -v^5 + v^4
$\beta_{5}$	$=$	$ -\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} $ -v^5 - v^4 + v

Display $\zeta_{18}^j$ in terms of $\beta_i$

$\zeta_{18}$	$=$	$ ( \beta_{5} + \beta_{4} + 2\beta_{2} ) / 3 $ (b5 + b4 + 2*b2) / 3
$\zeta_{18}^{2}$	$=$	$ ( -2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} ) / 3 $ (-2b5 + b4 + 3b3 - b2) / 3
$\zeta_{18}^{3}$	$=$	$ \beta_1 $ b1
$\zeta_{18}^{4}$	$=$	$ ( -\beta_{5} + 2\beta_{4} + \beta_{2} ) / 3 $ (-b5 + 2*b4 + b2) / 3
$\zeta_{18}^{5}$	$=$	$ ( -\beta_{5} - \beta_{4} + \beta_{2} ) / 3 $ (-b5 - b4 + b2) / 3

Display $\beta_i$ in terms of $\zeta_{18}^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1944\mathbb{Z}\right)^\times$.

$n$	$487$	$973$	$1217$
$\chi(n)$	$1$	$1$	$-\beta_{1}$

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} $

649.1

−0.173648	−	0.984808i
0.939693	+	0.342020i
−0.766044	+	0.642788i
−0.173648	+	0.984808i
0.939693	−	0.342020i
−0.766044	−	0.642788i

−0.766044

−

1.32683i

1.26604

−

2.19285i

649.2

−0.173648

−

0.300767i

0.673648

−

1.16679i

649.3

0.939693

1.62760i

−0.439693

0.761570i

1297.1

−0.766044

1.32683i

1.26604

2.19285i

1297.2

−0.173648

0.300767i

0.673648

1.16679i

1297.3

0.939693

−

1.62760i

−0.439693

−

0.761570i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1944.2.i.p		6
3.b	odd	2	1		1944.2.i.o		6
9.c	even	3	1		1944.2.a.o	✓	3
9.c	even	3	1	inner	1944.2.i.p		6
9.d	odd	6	1		1944.2.a.p	yes	3
9.d	odd	6	1		1944.2.i.o		6
36.f	odd	6	1		3888.2.a.bi		3
36.h	even	6	1		3888.2.a.bh		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1944.2.a.o	✓	3	9.c	even	3	1
1944.2.a.p	yes	3	9.d	odd	6	1
1944.2.i.o		6	3.b	odd	2	1
1944.2.i.o		6	9.d	odd	6	1
1944.2.i.p		6	1.a	even	1	1	trivial
1944.2.i.p		6	9.c	even	3	1	inner
3888.2.a.bh		3	36.h	even	6	1
3888.2.a.bi		3	36.f	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1944, [\chi])$:

$ T_{5}^{6} + 3T_{5}^{4} + 2T_{5}^{3} + 9T_{5}^{2} + 3T_{5} + 1 $

$ T_{7}^{6} - 3T_{7}^{5} + 9T_{7}^{4} - 6T_{7}^{3} + 9T_{7}^{2} + 9 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$7$	$ T^{6} - 3 T^{5} + \cdots + 9 $ T^6 - 3T^5 + 9T^4 - 6T^3 + 9T^2 + 9
$11$	$ T^{6} - 3 T^{5} + \cdots + 1 $ T^6 - 3T^5 + 15T^4 + 20T^3 + 33T^2 + 6*T + 1
$13$	$ T^{6} - 3 T^{5} + \cdots + 1369 $ T^6 - 3T^5 + 27T^4 - 20T^3 + 435T^2 - 666*T + 1369
$17$	$ (T^{3} - 9 T^{2} + 15 T + 17)^{2} $ (T^3 - 9T^2 + 15T + 17)^2
$19$	$ (T^{3} + 9 T^{2} + 6 T - 53)^{2} $ (T^3 + 9T^2 + 6T - 53)^2
$23$	$ T^{6} - 6 T^{5} + \cdots + 5041 $ T^6 - 6T^5 + 45T^4 - 88T^3 + 507T^2 - 639*T + 5041
$29$	$ T^{6} + 6 T^{5} + \cdots + 9 $ T^6 + 6T^5 + 27T^4 + 48T^3 + 63T^2 + 27*T + 9
$31$	$ T^{6} - 6 T^{5} + \cdots + 5041 $ T^6 - 6T^5 + 45T^4 - 88T^3 + 507T^2 - 639*T + 5041
$37$	$ (T^{3} + 15 T^{2} + \cdots - 159)^{2} $ (T^3 + 15T^2 + 36T - 159)^2
$41$	$ T^{6} + 3 T^{5} + \cdots + 2601 $ T^6 + 3T^5 + 63T^4 - 264T^3 + 2763T^2 - 2754*T + 2601
$43$	$ T^{6} + 75 T^{4} + \cdots + 15625 $ T^6 + 75T^4 + 250T^3 + 5625T^2 + 9375T + 15625
$47$	$ T^{6} - 6 T^{5} + \cdots + 9 $ T^6 - 6T^5 + 27T^4 - 48T^3 + 63T^2 - 27*T + 9
$53$	$ (T^{3} + 6 T^{2} + \cdots - 269)^{2} $ (T^3 + 6T^2 - 45T - 269)^2
$59$	$ T^{6} - 3 T^{5} + \cdots + 54289 $ T^6 - 3T^5 + 69T^4 - 286T^3 + 4299T^2 - 13980*T + 54289
$61$	$ T^{6} - 18 T^{5} + \cdots + 1369 $ T^6 - 18T^5 + 255T^4 - 1316T^3 + 5427T^2 + 2553*T + 1369
$67$	$ T^{6} - 24 T^{5} + \cdots + 94249 $ T^6 - 24T^5 + 405T^4 - 3490T^3 + 21873T^2 - 52497*T + 94249
$71$	$ (T^{3} + 15 T^{2} + \cdots - 937)^{2} $ (T^3 + 15T^2 - 42T - 937)^2
$73$	$ (T^{3} + 6 T^{2} - 69 T + 89)^{2} $ (T^3 + 6T^2 - 69T + 89)^2
$79$	$ T^{6} - 12 T^{5} + \cdots + 494209 $ T^6 - 12T^5 + 207T^4 - 650T^3 + 12405T^2 - 44289*T + 494209
$83$	$ T^{6} - 18 T^{5} + \cdots + 4068289 $ T^6 - 18T^5 + 435T^4 - 2036T^3 + 48627T^2 - 223887*T + 4068289
$89$	$ (T^{3} - 18 T^{2} + \cdots + 639)^{2} $ (T^3 - 18T^2 - 9T + 639)^2
$97$	$ T^{6} - 9 T^{5} + \cdots + 1125721 $ T^6 - 9T^5 + 246T^4 - 637T^3 + 36774T^2 - 175065*T + 1125721
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Level:	\( N \)	\(=\)	\( 1944 = 2^{3} \cdot 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1944.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{5} - \beta_{4} + \cdots + \beta_{2}) q^{5}+ \cdots + (\beta_{5} - \beta_{3} + \beta_{2} + \cdots + 1) q^{7}+O(q^{10}) \) q + (b5 - b4 - b3 + b2) * q^5 + (b5 - b3 + b2 - b1 + 1) * q^7	\( q + (\beta_{5} - \beta_{4} + \cdots + \beta_{2}) q^{5}+ \cdots + (8 \beta_{5} - 8 \beta_{3} + 8 \beta_{2} + \cdots + 3) q^{97}+O(q^{100}) \) q + (b5 - b4 - b3 + b2) * q^5 + (b5 - b3 + b2 - b1 + 1) * q^7 + (b5 - 2b3 + 2b2 - b1 + 1) * q^11 + (2b5 - 2b4 - 2b3 + 3b2 + b1) * q^13 + (-2b4 - 2b3 + 3) * q^17 + (b4 + 3b3 - 3) q^19 + (-3b5 + 3b4 + 3b3 - 2b2 + 2b1) q^23 + (b5 - 3b1 + 3) q^25 + (b3 - b2 + 2b1 - 2) q^29 + (b5 - b4 - b3 + 3b2 + 2b1) * q^31 + (b3 - 2) * q^35 + (3b4 + 4b3 - 5) * q^37 + (2b5 - 2b4 - 2b3 - 3b2 - b1) * q^41 + 5b5 q^43 + (-b5 + b3 - b2 - 2b1 + 2) q^47 + (b5 - b4 - b3 + 2b2 + 4b1) * q^49 + (3b4 + 5b3 - 2) * q^53 + (2b3 - 3) q^55 + (b5 - b4 - b3 + 5b2 + b1) q^59 + (4b5 - b3 + b2 - 6b1 + 6) * q^61 + (3b5 + 5b1 - 5) * q^65 + (2b5 - 2b4 - 2b3 - b2 + 8b1) * q^67 + (2b4 - 5b3 - 5) * q^71 + (-3b4 + 3b3 - 2) * q^73 + (b5 - b4 - b3 + 4b2 - 4b1) * q^77 + (7b5 - 3b3 + 3b2 - 4b1 + 4) * q^79 + (-8b5 - b3 + b2 - 6b1 + 6) * q^83 + (5b5 - 5b4 - 5b3 + 5b2 + 2b1) q^85 + (-5b4 + 2b3 + 6) * q^89 + (b4 + 4b3 - 4) q^91 + (-4b5 + 4b4 + 4b3 - 6b2 + b1) * q^95 + (8b5 - 8b3 + 8b2 - 3b1 + 3) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{7}+O(q^{10}) \) 6 * q + 3 * q^7	\( 6 q + 3 q^{7} + 3 q^{11} + 3 q^{13} + 18 q^{17} - 18 q^{19} + 6 q^{23} + 9 q^{25} - 6 q^{29} + 6 q^{31} - 12 q^{35} - 30 q^{37} - 3 q^{41} + 6 q^{47} + 12 q^{49} - 12 q^{53} - 18 q^{55} + 3 q^{59} + 18 q^{61} - 15 q^{65} + 24 q^{67} - 30 q^{71} - 12 q^{73} - 12 q^{77} + 12 q^{79} + 18 q^{83} + 6 q^{85} + 36 q^{89} - 24 q^{91} + 3 q^{95} + 9 q^{97}+O(q^{100}) \) 6 * q + 3 * q^7 + 3 * q^11 + 3 * q^13 + 18 * q^17 - 18 * q^19 + 6 * q^23 + 9 * q^25 - 6 * q^29 + 6 * q^31 - 12 * q^35 - 30 * q^37 - 3 * q^41 + 6 * q^47 + 12 * q^49 - 12 * q^53 - 18 * q^55 + 3 * q^59 + 18 * q^61 - 15 * q^65 + 24 * q^67 - 30 * q^71 - 12 * q^73 - 12 * q^77 + 12 * q^79 + 18 * q^83 + 6 * q^85 + 36 * q^89 - 24 * q^91 + 3 * q^95 + 9 * q^97

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