LMFDB - Newform orbit 688.2.o.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [688,2,Mod(351,688)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(688, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 5]))

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

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

chi := DirichletCharacter("688.351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 688 = 2^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	688.o (of order $6$, 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:	$5.49370765906$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
Coefficient field:	6.0.3636603.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 3x^{4} + 4x^{3} + 12x^{2} - 16x + 64 $ x^6 - x^5 + 3x^4 + 4x^3 + 12x^2 - 16x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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_{4} + \beta_{2} + \beta_1) q^{3} + ( - \beta_{4} - \beta_1) q^{7} + ( - \beta_{5} - 3 \beta_{3} - \beta_{2} + \cdots - 3) q^{9}+O(q^{10}) $ q + (-b4 + b2 + b1) * q^3 + (-b4 - b1) * q^7 + (-b5 - 3b3 - b2 - b1 - 3) q^9	$ q + ( - \beta_{4} + \beta_{2} + \beta_1) q^{3} + ( - \beta_{4} - \beta_1) q^{7} + ( - \beta_{5} - 3 \beta_{3} - \beta_{2} + \cdots - 3) q^{9}+ \cdots + (3 \beta_{3} - 3) q^{99}+O(q^{100}) $ q + (-b4 + b2 + b1) * q^3 + (-b4 - b1) * q^7 + (-b5 - 3b3 - b2 - b1 - 3) q^9 + (-b5 + 2b3 - b1 + 1) q^11 + (-b5 + b4 - b2) * q^13 + (-b5 + b4 - b3 - b2 - 1) * q^17 + (-b5 + 3b3 + b2) q^19 + (3b5 - 2b4 - 2b2 - 3b1) * q^21 + (b5 - b4 - 2b3 - b1 - 4) q^23 + (-5b3 - 5) q^25 + (3b5 - b4 - b2 - 3b1 + 6) * q^27 + (2b5 - b4 - 2b3 - 2b2 - b1 + 2) q^29 + (-2b5 + 2b4 + 2b1) q^31 + (b5 - 2b4 + 2b3 - b2 - 2b1 + 4) q^33 + (3b5 - 3b4 + 4b3 - 3b1 + 8) * q^37 + (b4 + b2 + 6) * q^39 + (-2b5 + 2b4 + 2b2 + 2b1 - 1) * q^41 + (-3b5 + 3b4 - 3b3 + b2 + b1 - 3) q^43 + (2b5 + b4 - b2 + 2b1) * q^47 + (b5 + 2b4 - b3 - 3b2 - 2b1) q^49 + (b5 + b4 + b2 - b1 + 6) * q^51 + (2b5 - 3b4 - 4b3 + b2 + 3b1) * q^53 + (-2b5 - 2b2 - 2b1) q^57 + (-2b5 - b4 - 6b3 + b2 - 2b1 - 3) q^59 + (b5 - 2b4 + 2b3 - b2 + b1 - 2) * q^61 + (2b5 + b4 + 6b3 - 3b2 - b1) q^63 + (b5 - b4 + 3b3 - b1 + 6) q^67 + (4b5 + b4 + 2b3 - 4b2 - 5b1 - 2) * q^69 + (-3b5 + 3b4 - 6b3 - 3b2 - 6) * q^71 + (-b5 + 2b4 - b3 + b2 - b1 + 1) q^73 + (5b5 - 5b1) * q^75 + (-b5 + 2b4 + 4b3 + b2 - b1 - 4) * q^77 + (-3b5 - b4 + 3b2 + 4b1) q^79 + (b5 - 4b4 + 3b3 + 3b2 + 4b1) * q^81 + (-2b5 - b4 - 2b3 - 3b2 - b1 - 4) q^83 + (3b5 + 3b1) * q^87 + (-3b5 + 3b4 - 3b3 + 3b1 - 6) * q^89 + (-2b5 - b4 + 3b2 + b1) * q^91 + (-4b5 + 2b4 - 4b3 + 4b2 + 2b1 + 4) q^93 + (-3b5 - b4 - b2 + 3b1 + 5) * q^97 + (3b3 - 3) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{3} - 2 q^{7} - 10 q^{9}+O(q^{10}) $ 6 * q + q^3 - 2 * q^7 - 10 * q^9	$ 6 q + q^{3} - 2 q^{7} - 10 q^{9} + q^{13} - 2 q^{17} - 7 q^{19} - 10 q^{21} - 21 q^{23} - 15 q^{25} + 28 q^{27} + 12 q^{29} + 6 q^{31} + 12 q^{33} + 27 q^{37} + 38 q^{39} + 2 q^{41} - q^{43} - q^{49} + 36 q^{51} + 11 q^{53} - 2 q^{57} - 21 q^{61} - 23 q^{63} + 24 q^{67} - 30 q^{69} - 15 q^{71} + 12 q^{73} - 10 q^{75} - 33 q^{77} + 9 q^{79} - 7 q^{81} - 21 q^{83} - 18 q^{89} + 5 q^{91} + 48 q^{93} + 34 q^{97} - 27 q^{99}+O(q^{100}) $ 6 * q + q^3 - 2 * q^7 - 10 * q^9 + q^13 - 2 * q^17 - 7 * q^19 - 10 * q^21 - 21 * q^23 - 15 * q^25 + 28 * q^27 + 12 * q^29 + 6 * q^31 + 12 * q^33 + 27 * q^37 + 38 * q^39 + 2 * q^41 - q^43 - q^49 + 36 * q^51 + 11 * q^53 - 2 * q^57 - 21 * q^61 - 23 * q^63 + 24 * q^67 - 30 * q^69 - 15 * q^71 + 12 * q^73 - 10 * q^75 - 33 * q^77 + 9 * q^79 - 7 * q^81 - 21 * q^83 - 18 * q^89 + 5 * q^91 + 48 * q^93 + 34 * q^97 - 27 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - x^{5} + 3x^{4} + 4x^{3} + 12x^{2} - 16x + 64 $ x^6 - x^5 + 3*x^4 + 4*x^3 + 12*x^2 - 16*x + 64 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( -\nu^{5} - 5\nu^{4} - \nu^{3} - 10\nu^{2} - 40\nu - 32 ) / 32 $ (-v^5 - 5v^4 - v^3 - 10v^2 - 40*v - 32) / 32
$\beta_{3}$	$=$	$ ( -\nu^{5} + 3\nu^{4} + 7\nu^{3} - 2\nu^{2} + 8\nu + 32 ) / 64 $ (-v^5 + 3v^4 + 7v^3 - 2v^2 + 8v + 32) / 64
$\beta_{4}$	$=$	$ ( -\nu^{5} - \nu^{4} + 3\nu^{3} + 10\nu^{2} - 16\nu + 32 ) / 32 $ (-v^5 - v^4 + 3v^3 + 10v^2 - 16*v + 32) / 32
$\beta_{5}$	$=$	$ ( \nu^{5} - \nu^{4} + 3\nu^{3} + 4\nu^{2} + 12\nu - 16 ) / 16 $ (v^5 - v^4 + 3v^3 + 4v^2 + 12*v - 16) / 16

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ 2\beta_{4} - 2\beta_{3} - \beta_{2} - 2 $ 2b4 - 2b3 - b2 - 2
$\nu^{3}$	$=$	$ 2\beta_{5} + 6\beta_{3} + \beta_{2} - \beta_1 $ 2b5 + 6b3 + b2 - b1
$\nu^{4}$	$=$	$ -2\beta_{5} - 2\beta_{4} + 4\beta_{3} - 4\beta_{2} - 5\beta _1 - 6 $ -2b5 - 2b4 + 4b3 - 4b2 - 5*b1 - 6
$\nu^{5}$	$=$	$ 8\beta_{5} - 10\beta_{4} - 6\beta_{3} - 3\beta_{2} - 14\beta _1 + 18 $ 8b5 - 10b4 - 6b3 - 3b2 - 14*b1 + 18

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/688\mathbb{Z}\right)^\times$.

$n$	$431$	$433$	$517$
$\chi(n)$	$-1$	$1 + \beta_{3}$	$1$

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} $

351.1

−1.55951	−	1.25217i
0.715814	+	1.86751i
1.34370	−	1.48137i
−1.55951	+	1.25217i
0.715814	−	1.86751i
1.34370	+	1.48137i

−1.55951

−

2.70116i

−0.304650

0.527670i

−3.36416

5.82690i

351.2

0.715814

1.23983i

1.25941

−

2.18136i

0.475222

−

0.823108i

351.3

1.34370

2.32736i

−1.95476

3.38574i

−2.11106

3.65646i

639.1

−1.55951

2.70116i

−0.304650

−

0.527670i

−3.36416

−

5.82690i

639.2

0.715814

−

1.23983i

1.25941

2.18136i

0.475222

0.823108i

639.3

1.34370

−

2.32736i

−1.95476

−

3.38574i

−2.11106

−

3.65646i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
172.f	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	688.2.o.e	yes	6
4.b	odd	2	1		688.2.o.d	✓	6
43.d	odd	6	1		688.2.o.d	✓	6
172.f	even	6	1	inner	688.2.o.e	yes	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
688.2.o.d	✓	6	4.b	odd	2	1
688.2.o.d	✓	6	43.d	odd	6	1
688.2.o.e	yes	6	1.a	even	1	1	trivial
688.2.o.e	yes	6	172.f	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(688, [\chi])$:

$ T_{3}^{6} - T_{3}^{5} + 10T_{3}^{4} - 15T_{3}^{3} + 93T_{3}^{2} - 108T_{3} + 144 $

$ T_{5} $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} - T^{5} + \cdots + 144 $ T^6 - T^5 + 10T^4 - 15T^3 + 93T^2 - 108T + 144
$5$	$ T^{6} $ T^6
$7$	$ T^{6} + 2 T^{5} + \cdots + 36 $ T^6 + 2T^5 + 13T^4 - 6T^3 + 93T^2 + 54*T + 36
$11$	$ T^{6} + 32 T^{4} + \cdots + 27 $ T^6 + 32T^4 + 64T^2 + 27
$13$	$ T^{6} - T^{5} + \cdots + 144 $ T^6 - T^5 + 10T^4 - 15T^3 + 93T^2 - 108T + 144
$17$	$ T^{6} + 2 T^{5} + \cdots + 9 $ T^6 + 2T^5 + 12T^4 - 22T^3 + 58T^2 - 24*T + 9
$19$	$ T^{6} + 7 T^{5} + \cdots + 144 $ T^6 + 7T^5 + 43T^4 + 66T^3 + 120T^2 - 72*T + 144
$23$	$ T^{6} + 21 T^{5} + \cdots + 1728 $ T^6 + 21T^5 + 188T^4 + 861T^3 + 2185T^2 + 2952*T + 1728
$29$	$ T^{6} - 12 T^{5} + \cdots + 3888 $ T^6 - 12T^5 + 33T^4 + 180T^3 - 207T^2 - 1620*T + 3888
$31$	$ T^{6} - 6 T^{5} + \cdots + 768 $ T^6 - 6T^5 - 16T^4 + 168T^3 + 880T^2 + 1344*T + 768
$37$	$ T^{6} - 27 T^{5} + \cdots + 199692 $ T^6 - 27T^5 + 252T^4 - 243T^3 - 6885T^2 + 6966*T + 199692
$41$	$ (T^{3} - T^{2} - 41 T + 9)^{2} $ (T^3 - T^2 - 41*T + 9)^2
$43$	$ T^{6} + T^{5} + \cdots + 79507 $ T^6 + T^5 - 36T^4 - 7T^3 - 1548T^2 + 1849T + 79507
$47$	$ T^{6} + 173 T^{4} + \cdots + 78732 $ T^6 + 173T^4 + 8245T^2 + 78732
$53$	$ T^{6} - 11 T^{5} + \cdots + 138384 $ T^6 - 11T^5 + 180T^4 - 95T^3 + 7573T^2 - 21948*T + 138384
$59$	$ T^{6} + 308 T^{4} + \cdots + 143883 $ T^6 + 308T^4 + 19612T^2 + 143883
$61$	$ T^{6} + 21 T^{5} + \cdots + 34992 $ T^6 + 21T^5 + 150T^4 + 63T^3 - 2259T^2 - 972*T + 34992
$67$	$ T^{6} - 24 T^{5} + \cdots + 2883 $ T^6 - 24T^5 + 248T^4 - 1344T^3 + 3880T^2 - 5208*T + 2883
$71$	$ T^{6} + 15 T^{5} + \cdots + 2916 $ T^6 + 15T^5 + 234T^4 - 27T^3 + 891T^2 + 486*T + 2916
$73$	$ T^{6} - 12 T^{5} + \cdots + 25947 $ T^6 - 12T^5 + 18T^4 + 360T^3 - 216T^2 - 8370*T + 25947
$79$	$ T^{6} - 9 T^{5} + \cdots + 15552 $ T^6 - 9T^5 - 70T^4 + 873T^3 + 8761T^2 - 20952*T + 15552
$83$	$ T^{6} + 21 T^{5} + \cdots + 181548 $ T^6 + 21T^5 + 92T^4 - 1155T^3 - 2141T^2 + 40590*T + 181548
$89$	$ T^{6} + 18 T^{5} + \cdots + 177147 $ T^6 + 18T^5 + 72T^4 - 648T^3 - 3078T^2 + 26244*T + 177147
$97$	$ (T^{3} - 17 T^{2} + \cdots + 1268)^{2} $ (T^3 - 17T^2 - 46T + 1268)^2
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Level:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	688.o (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{4} + \beta_{2} + \beta_1) q^{3} + ( - \beta_{4} - \beta_1) q^{7} + ( - \beta_{5} - 3 \beta_{3} - \beta_{2} + \cdots - 3) q^{9}+O(q^{10}) \) q + (-b4 + b2 + b1) * q^3 + (-b4 - b1) * q^7 + (-b5 - 3b3 - b2 - b1 - 3) q^9	\( q + ( - \beta_{4} + \beta_{2} + \beta_1) q^{3} + ( - \beta_{4} - \beta_1) q^{7} + ( - \beta_{5} - 3 \beta_{3} - \beta_{2} + \cdots - 3) q^{9}+ \cdots + (3 \beta_{3} - 3) q^{99}+O(q^{100}) \) q + (-b4 + b2 + b1) * q^3 + (-b4 - b1) * q^7 + (-b5 - 3b3 - b2 - b1 - 3) q^9 + (-b5 + 2b3 - b1 + 1) q^11 + (-b5 + b4 - b2) * q^13 + (-b5 + b4 - b3 - b2 - 1) * q^17 + (-b5 + 3b3 + b2) q^19 + (3b5 - 2b4 - 2b2 - 3b1) * q^21 + (b5 - b4 - 2b3 - b1 - 4) q^23 + (-5b3 - 5) q^25 + (3b5 - b4 - b2 - 3b1 + 6) * q^27 + (2b5 - b4 - 2b3 - 2b2 - b1 + 2) q^29 + (-2b5 + 2b4 + 2b1) q^31 + (b5 - 2b4 + 2b3 - b2 - 2b1 + 4) q^33 + (3b5 - 3b4 + 4b3 - 3b1 + 8) * q^37 + (b4 + b2 + 6) * q^39 + (-2b5 + 2b4 + 2b2 + 2b1 - 1) * q^41 + (-3b5 + 3b4 - 3b3 + b2 + b1 - 3) q^43 + (2b5 + b4 - b2 + 2b1) * q^47 + (b5 + 2b4 - b3 - 3b2 - 2b1) q^49 + (b5 + b4 + b2 - b1 + 6) * q^51 + (2b5 - 3b4 - 4b3 + b2 + 3b1) * q^53 + (-2b5 - 2b2 - 2b1) q^57 + (-2b5 - b4 - 6b3 + b2 - 2b1 - 3) q^59 + (b5 - 2b4 + 2b3 - b2 + b1 - 2) * q^61 + (2b5 + b4 + 6b3 - 3b2 - b1) q^63 + (b5 - b4 + 3b3 - b1 + 6) q^67 + (4b5 + b4 + 2b3 - 4b2 - 5b1 - 2) * q^69 + (-3b5 + 3b4 - 6b3 - 3b2 - 6) * q^71 + (-b5 + 2b4 - b3 + b2 - b1 + 1) q^73 + (5b5 - 5b1) * q^75 + (-b5 + 2b4 + 4b3 + b2 - b1 - 4) * q^77 + (-3b5 - b4 + 3b2 + 4b1) q^79 + (b5 - 4b4 + 3b3 + 3b2 + 4b1) * q^81 + (-2b5 - b4 - 2b3 - 3b2 - b1 - 4) q^83 + (3b5 + 3b1) * q^87 + (-3b5 + 3b4 - 3b3 + 3b1 - 6) * q^89 + (-2b5 - b4 + 3b2 + b1) * q^91 + (-4b5 + 2b4 - 4b3 + 4b2 + 2b1 + 4) q^93 + (-3b5 - b4 - b2 + 3b1 + 5) * q^97 + (3b3 - 3) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{3} - 2 q^{7} - 10 q^{9}+O(q^{10}) \) 6 * q + q^3 - 2 * q^7 - 10 * q^9	\( 6 q + q^{3} - 2 q^{7} - 10 q^{9} + q^{13} - 2 q^{17} - 7 q^{19} - 10 q^{21} - 21 q^{23} - 15 q^{25} + 28 q^{27} + 12 q^{29} + 6 q^{31} + 12 q^{33} + 27 q^{37} + 38 q^{39} + 2 q^{41} - q^{43} - q^{49} + 36 q^{51} + 11 q^{53} - 2 q^{57} - 21 q^{61} - 23 q^{63} + 24 q^{67} - 30 q^{69} - 15 q^{71} + 12 q^{73} - 10 q^{75} - 33 q^{77} + 9 q^{79} - 7 q^{81} - 21 q^{83} - 18 q^{89} + 5 q^{91} + 48 q^{93} + 34 q^{97} - 27 q^{99}+O(q^{100}) \) 6 * q + q^3 - 2 * q^7 - 10 * q^9 + q^13 - 2 * q^17 - 7 * q^19 - 10 * q^21 - 21 * q^23 - 15 * q^25 + 28 * q^27 + 12 * q^29 + 6 * q^31 + 12 * q^33 + 27 * q^37 + 38 * q^39 + 2 * q^41 - q^43 - q^49 + 36 * q^51 + 11 * q^53 - 2 * q^57 - 21 * q^61 - 23 * q^63 + 24 * q^67 - 30 * q^69 - 15 * q^71 + 12 * q^73 - 10 * q^75 - 33 * q^77 + 9 * q^79 - 7 * q^81 - 21 * q^83 - 18 * q^89 + 5 * q^91 + 48 * q^93 + 34 * q^97 - 27 * q^99

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