LMFDB - Newform orbit 352.2.s.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [352,2,Mod(79,352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(352, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 5, 1]))

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

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

chi := DirichletCharacter("352.79");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 352 = 2^{5} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	352.s (of order $10$, degree $4$, 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:	$2.81073415115$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	8.0.64000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 $ x^8 - 2x^6 + 4x^4 - 8*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

$q$-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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta_{4} + \beta_{3} - \beta_{2} + \cdots + 1) q^{3}+ \cdots + (2 \beta_{7} + \beta_{6} + \cdots + 2 \beta_1) q^{9}+O(q^{10}) $ q + (b4 + b3 - b2 + b1 + 1) * q^3 + (2b7 + b6 + 3b4 + b2 + 2b1) q^9	$ q + (\beta_{4} + \beta_{3} - \beta_{2} + \cdots + 1) q^{3}+ \cdots + ( - 5 \beta_{7} - 8 \beta_{6} + \cdots - 1) q^{99}+O(q^{100}) $ q + (b4 + b3 - b2 + b1 + 1) * q^3 + (2b7 + b6 + 3b4 + b2 + 2b1) q^9 + (b7 - 3b2) q^11 + (-2b7 + 3b6 + 2b5 - 6b4 + 3b2 + 2b1 - 3) * q^17 + (-3b7 - b4 - 3b3 + 3b1 + 1) q^19 - 5b2 q^25 + (2b7 + 5b6 + 4b5 - 2b3 + 8b2 + b1 - 8) q^27 + (-b6 - 2b5 + b4 - 4b3 - b2 - 4) * q^33 + (-6b6 + 3b4 + 4b3 - 3b2 - 4b1 + 3) q^41 + (3b7 - 5b6 - 3b5 - 5b4 + 3b3) q^43 + 7b6 q^49 + (7b6 - 6b5 + b4 + 5b3 + b2 - 6b1 + 7) * q^51 + (-4b7 - 12b6 + 2b5 + 7b4 - 2b3 - 2b2 + 4b1 + 14) q^57 + (-3b6 + 5b5 - 5b1 + 3) q^59 + (-3b7 - 7b6 + 7b4 + 3b3 + 7) * q^67 + (-6b5 - b4 + 6b3 - b2 - 6b1) q^73 + (-5b6 - 5b5 + 5b4 - 5b3 - 5b2) q^75 + (8b7 + 12b6 + 2b5 - 16b4 + 4b3 + 16b2 - 10b1 - 12) q^81 + (-b7 + b5 - 9b2 - 9) q^83 + (2b7 - 9b6 - 2b5 + 9b4 + 2b3 - 4b1) * q^89 + (5b6 - 6b5 - 5b4 - 6b3 + 5b2) q^97 + (-5b7 - 8b6 + 5b5 - 8b4 - 12b3 + b2 + 2b1 - 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} - 2 q^{9}+O(q^{10}) $ 8 * q + 4 * q^3 - 2 * q^9	$ 8 q + 4 q^{3} - 2 q^{9} - 6 q^{11} + 10 q^{19} - 10 q^{25} - 38 q^{27} - 38 q^{33} + 14 q^{49} + 70 q^{51} + 70 q^{57} + 18 q^{59} + 28 q^{67} - 30 q^{75} - 8 q^{81} - 90 q^{83} - 36 q^{89} + 30 q^{97} - 6 q^{99}+O(q^{100}) $ 8 * q + 4 * q^3 - 2 * q^9 - 6 * q^11 + 10 * q^19 - 10 * q^25 - 38 * q^27 - 38 * q^33 + 14 * q^49 + 70 * q^51 + 70 * q^57 + 18 * q^59 + 28 * q^67 - 30 * q^75 - 8 * q^81 - 90 * q^83 - 36 * q^89 + 30 * q^97 - 6 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 $ x^8 - 2*x^6 + 4*x^4 - 8*x^2 + 16 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{2} ) / 2 $ (v^2) / 2
$\beta_{3}$	$=$	$ ( \nu^{3} ) / 2 $ (v^3) / 2
$\beta_{4}$	$=$	$ ( \nu^{4} ) / 4 $ (v^4) / 4
$\beta_{5}$	$=$	$ ( \nu^{5} ) / 4 $ (v^5) / 4
$\beta_{6}$	$=$	$ ( \nu^{6} ) / 8 $ (v^6) / 8
$\beta_{7}$	$=$	$ ( \nu^{7} ) / 8 $ (v^7) / 8

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ 2\beta_{2} $ 2*b2
$\nu^{3}$	$=$	$ 2\beta_{3} $ 2*b3
$\nu^{4}$	$=$	$ 4\beta_{4} $ 4*b4
$\nu^{5}$	$=$	$ 4\beta_{5} $ 4*b5
$\nu^{6}$	$=$	$ 8\beta_{6} $ 8*b6
$\nu^{7}$	$=$	$ 8\beta_{7} $ 8*b7

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/352\mathbb{Z}\right)^\times$.

$n$	$133$	$287$	$321$
$\chi(n)$	$-1$	$-1$	$1 - \beta_{2} + \beta_{4} - \beta_{6}$

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} $

79.1

−1.34500	+	0.437016i
1.34500	−	0.437016i
0.831254	−	1.14412i
−0.831254	+	1.14412i
0.831254	+	1.14412i
−0.831254	−	1.14412i
−1.34500	−	0.437016i
1.34500	+	0.437016i

−1.67625

1.21787i

0.399565

−

1.22973i

79.2

2.67625

−

1.94441i

2.45454

−

7.55429i

239.1

−0.0137431

−

0.0422971i

2.42545

−

1.76219i

239.2

1.01374

3.11998i

−6.27955

4.56236i

271.1

−0.0137431

0.0422971i

2.42545

1.76219i

271.2

1.01374

−

3.11998i

−6.27955

−

4.56236i

303.1

−1.67625

−

1.21787i

0.399565

1.22973i

303.2

2.67625

1.94441i

2.45454

7.55429i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
11.d	odd	10	1	inner
88.k	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	352.2.s.a		8
4.b	odd	2	1		88.2.k.a	✓	8
8.b	even	2	1		88.2.k.a	✓	8
8.d	odd	2	1	CM	352.2.s.a		8
11.c	even	5	1		3872.2.g.b		8
11.d	odd	10	1	inner	352.2.s.a		8
11.d	odd	10	1		3872.2.g.b		8
12.b	even	2	1		792.2.bp.a		8
24.h	odd	2	1		792.2.bp.a		8
44.c	even	2	1		968.2.k.b		8
44.g	even	10	1		88.2.k.a	✓	8
44.g	even	10	1		968.2.g.a		8
44.g	even	10	1		968.2.k.c		8
44.g	even	10	1		968.2.k.d		8
44.h	odd	10	1		968.2.g.a		8
44.h	odd	10	1		968.2.k.b		8
44.h	odd	10	1		968.2.k.c		8
44.h	odd	10	1		968.2.k.d		8
88.b	odd	2	1		968.2.k.b		8
88.k	even	10	1	inner	352.2.s.a		8
88.k	even	10	1		3872.2.g.b		8
88.l	odd	10	1		3872.2.g.b		8
88.o	even	10	1		968.2.g.a		8
88.o	even	10	1		968.2.k.b		8
88.o	even	10	1		968.2.k.c		8
88.o	even	10	1		968.2.k.d		8
88.p	odd	10	1		88.2.k.a	✓	8
88.p	odd	10	1		968.2.g.a		8
88.p	odd	10	1		968.2.k.c		8
88.p	odd	10	1		968.2.k.d		8
132.n	odd	10	1		792.2.bp.a		8
264.u	even	10	1		792.2.bp.a		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
88.2.k.a	✓	8	4.b	odd	2	1
88.2.k.a	✓	8	8.b	even	2	1
88.2.k.a	✓	8	44.g	even	10	1
88.2.k.a	✓	8	88.p	odd	10	1
352.2.s.a		8	1.a	even	1	1	trivial
352.2.s.a		8	8.d	odd	2	1	CM
352.2.s.a		8	11.d	odd	10	1	inner
352.2.s.a		8	88.k	even	10	1	inner
792.2.bp.a		8	12.b	even	2	1
792.2.bp.a		8	24.h	odd	2	1
792.2.bp.a		8	132.n	odd	10	1
792.2.bp.a		8	264.u	even	10	1
968.2.g.a		8	44.g	even	10	1
968.2.g.a		8	44.h	odd	10	1
968.2.g.a		8	88.o	even	10	1
968.2.g.a		8	88.p	odd	10	1
968.2.k.b		8	44.c	even	2	1
968.2.k.b		8	44.h	odd	10	1
968.2.k.b		8	88.b	odd	2	1
968.2.k.b		8	88.o	even	10	1
968.2.k.c		8	44.g	even	10	1
968.2.k.c		8	44.h	odd	10	1
968.2.k.c		8	88.o	even	10	1
968.2.k.c		8	88.p	odd	10	1
968.2.k.d		8	44.g	even	10	1
968.2.k.d		8	44.h	odd	10	1
968.2.k.d		8	88.o	even	10	1
968.2.k.d		8	88.p	odd	10	1
3872.2.g.b		8	11.c	even	5	1
3872.2.g.b		8	11.d	odd	10	1
3872.2.g.b		8	88.k	even	10	1
3872.2.g.b		8	88.l	odd	10	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} - 4T_{3}^{7} + 12T_{3}^{6} - 2T_{3}^{5} - 10T_{3}^{4} + 52T_{3}^{3} + 507T_{3}^{2} + 14T_{3} + 1 $ T3^8 - 4*T3^7 + 12*T3^6 - 2*T3^5 - 10*T3^4 + 52*T3^3 + 507*T3^2 + 14*T3 + 1 acting on $S_{2}^{\mathrm{new}}(352, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} - 4 T^{7} + \cdots + 1 $ T^8 - 4T^7 + 12T^6 - 2T^5 - 10T^4 + 52T^3 + 507T^2 + 14*T + 1
$5$	$ T^{8} $ T^8
$7$	$ T^{8} $ T^8
$11$	$ T^{8} + 6 T^{7} + \cdots + 14641 $ T^8 + 6T^7 + 25T^6 + 84T^5 + 229T^4 + 924T^3 + 3025T^2 + 7986*T + 14641
$13$	$ T^{8} $ T^8
$17$	$ T^{8} - 32 T^{6} + \cdots + 63001 $ T^8 - 32T^6 + 510T^5 + 1194T^4 - 16320T^3 + 84617T^2 - 128010T + 63001
$19$	$ T^{8} - 10 T^{7} + \cdots + 22201 $ T^8 - 10T^7 + 63T^6 - 120T^5 + 1434T^4 - 13210T^3 + 22592T^2 + 50660*T + 22201
$23$	$ T^{8} $ T^8
$29$	$ T^{8} $ T^8
$31$	$ T^{8} $ T^8
$37$	$ T^{8} $ T^8
$41$	$ T^{8} - 128 T^{6} + \cdots + 2105401 $ T^8 - 128T^6 + 1230T^5 + 6954T^4 - 157440T^3 + 832913T^2 - 1784730T + 2105401
$43$	$ T^{8} + 358 T^{6} + \cdots + 1104601 $ T^8 + 358T^6 + 38939T^4 + 1171742*T^2 + 1104601
$47$	$ T^{8} $ T^8
$53$	$ T^{8} $ T^8
$59$	$ T^{8} - 18 T^{7} + \cdots + 65302561 $ T^8 - 18T^7 + 403T^6 - 3756T^5 + 45430T^4 - 257346T^3 + 2996748T^2 + 23370252*T + 65302561
$61$	$ T^{8} $ T^8
$67$	$ (T^{4} - 14 T^{3} + \cdots - 4799)^{2} $ (T^4 - 14T^3 - 139T^2 + 1946*T - 4799)^2
$71$	$ T^{8} $ T^8
$73$	$ T^{8} - 288 T^{6} + \cdots + 19971961 $ T^8 - 288T^6 - 730T^5 + 31114T^4 + 210240T^3 + 764593T^2 - 3262370T + 19971961
$79$	$ T^{8} $ T^8
$83$	$ T^{8} + 90 T^{7} + \cdots + 972753721 $ T^8 + 90T^7 + 3647T^6 + 87480T^5 + 1371354T^4 + 14501610T^3 + 101878528T^2 + 443507580*T + 972753721
$89$	$ (T^{4} + 18 T^{3} + \cdots + 401)^{2} $ (T^4 + 18T^3 - 121T^2 - 2178*T + 401)^2
$97$	$ T^{8} - 30 T^{7} + \cdots + 73017025 $ T^8 - 30T^7 + 785T^6 - 10700T^5 + 142590T^4 - 1026350T^3 + 16531500T^2 + 57251500*T + 73017025
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 352 = 2^{5} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	352.s (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{4} + \beta_{3} - \beta_{2} + \cdots + 1) q^{3}+ \cdots + (2 \beta_{7} + \beta_{6} + \cdots + 2 \beta_1) q^{9}+O(q^{10}) \) q + (b4 + b3 - b2 + b1 + 1) * q^3 + (2b7 + b6 + 3b4 + b2 + 2b1) q^9	\( q + (\beta_{4} + \beta_{3} - \beta_{2} + \cdots + 1) q^{3}+ \cdots + ( - 5 \beta_{7} - 8 \beta_{6} + \cdots - 1) q^{99}+O(q^{100}) \) q + (b4 + b3 - b2 + b1 + 1) * q^3 + (2b7 + b6 + 3b4 + b2 + 2b1) q^9 + (b7 - 3b2) q^11 + (-2b7 + 3b6 + 2b5 - 6b4 + 3b2 + 2b1 - 3) * q^17 + (-3b7 - b4 - 3b3 + 3b1 + 1) q^19 - 5b2 q^25 + (2b7 + 5b6 + 4b5 - 2b3 + 8b2 + b1 - 8) q^27 + (-b6 - 2b5 + b4 - 4b3 - b2 - 4) * q^33 + (-6b6 + 3b4 + 4b3 - 3b2 - 4b1 + 3) q^41 + (3b7 - 5b6 - 3b5 - 5b4 + 3b3) q^43 + 7b6 q^49 + (7b6 - 6b5 + b4 + 5b3 + b2 - 6b1 + 7) * q^51 + (-4b7 - 12b6 + 2b5 + 7b4 - 2b3 - 2b2 + 4b1 + 14) q^57 + (-3b6 + 5b5 - 5b1 + 3) q^59 + (-3b7 - 7b6 + 7b4 + 3b3 + 7) * q^67 + (-6b5 - b4 + 6b3 - b2 - 6b1) q^73 + (-5b6 - 5b5 + 5b4 - 5b3 - 5b2) q^75 + (8b7 + 12b6 + 2b5 - 16b4 + 4b3 + 16b2 - 10b1 - 12) q^81 + (-b7 + b5 - 9b2 - 9) q^83 + (2b7 - 9b6 - 2b5 + 9b4 + 2b3 - 4b1) * q^89 + (5b6 - 6b5 - 5b4 - 6b3 + 5b2) q^97 + (-5b7 - 8b6 + 5b5 - 8b4 - 12b3 + b2 + 2b1 - 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} - 2 q^{9}+O(q^{10}) \) 8 * q + 4 * q^3 - 2 * q^9	\( 8 q + 4 q^{3} - 2 q^{9} - 6 q^{11} + 10 q^{19} - 10 q^{25} - 38 q^{27} - 38 q^{33} + 14 q^{49} + 70 q^{51} + 70 q^{57} + 18 q^{59} + 28 q^{67} - 30 q^{75} - 8 q^{81} - 90 q^{83} - 36 q^{89} + 30 q^{97} - 6 q^{99}+O(q^{100}) \) 8 * q + 4 * q^3 - 2 * q^9 - 6 * q^11 + 10 * q^19 - 10 * q^25 - 38 * q^27 - 38 * q^33 + 14 * q^49 + 70 * q^51 + 70 * q^57 + 18 * q^59 + 28 * q^67 - 30 * q^75 - 8 * q^81 - 90 * q^83 - 36 * q^89 + 30 * q^97 - 6 * q^99

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