LMFDB - Newform orbit 3960.2.d.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3960,2,Mod(3169,3960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3960, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3960.3169");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3960.d (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:	$31.6207592004$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.47985531136.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 19x^{6} + 91x^{4} + 45x^{2} + 4 $ x^8 + 19x^6 + 91x^4 + 45*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 440)
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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{3} q^{5} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{7}+O(q^{10}) $ q - b3 * q^5 + (-b4 - b3 + b1) * q^7	$ q - \beta_{3} q^{5} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{7} - q^{11} + (\beta_{7} - \beta_{6} + \cdots + \beta_{3}) q^{13}+ \cdots + (3 \beta_{4} + 3 \beta_{3} - 2 \beta_1) q^{97}+O(q^{100}) $ q - b3 * q^5 + (-b4 - b3 + b1) * q^7 - q^11 + (b7 - b6 + b4 + b3) * q^13 + (b7 - b6 + b4 + b3 - b2 - b1) * q^17 + (b7 - b5 + b3 + 3) * q^19 + (-b2 + 2b1) q^23 + (-b7 + b5 - b1 - 1) * q^25 + (-b6 - b5 + b4 + 1) * q^29 + (b7 - b5 + b4 - 3) * q^31 + (-b6 + 2b4 + b3 - 2b1 - 4) * q^35 + (b4 + b3 - b2 + b1) * q^37 + (b7 + b6 - b4 + b3 - 2) * q^41 + (b7 - b6 - b2 + 2b1) q^43 + (-b7 + b6 - 2b4 - 2b3 + b2) * q^47 + (-b6 - b5 + 3b4 - 2b3 - 2) * q^49 + (2b4 + 2b3 - b2 - b1) * q^53 + b3 * q^55 + (b7 + b6 + 2) * q^59 + (b6 + b5 - b4 + 7) * q^61 + (-b7 - b6 - b5 - b4 - b3 + b2 + 2b1 + 1) q^65 + (2b4 + 2b3 - b2 - 2b1) q^67 + (b7 - b5 + 3b4 - 2b3 + 1) * q^71 + (b7 - b6 + 3b4 + 3b3 - 2b1) q^73 + (b4 + b3 - b1) * q^77 + (-2b7 - 2b6 + 4) * q^79 + (-b7 + b6 - b2 - 4b1) q^83 + (-3b7 - 2b4 - b3 - b2 + 2b1 - 4) q^85 + (b6 + b5 - b3 + 5) * q^89 + (-2b7 - 2b6 - 2b4 + 2b3) * q^91 + (b7 - b6 - b5 - 3b4 - b3 - b1 - 3) q^95 + (3b4 + 3b3 - 2b1) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{11} + 18 q^{19} - 2 q^{25} + 6 q^{29} - 30 q^{31} - 30 q^{35} - 20 q^{41} - 18 q^{49} + 12 q^{59} + 58 q^{61} + 8 q^{65} + 2 q^{71} + 40 q^{79} - 26 q^{85} + 42 q^{89} + 8 q^{91} - 28 q^{95}+O(q^{100}) $ 8 * q - 8 * q^11 + 18 * q^19 - 2 * q^25 + 6 * q^29 - 30 * q^31 - 30 * q^35 - 20 * q^41 - 18 * q^49 + 12 * q^59 + 58 * q^61 + 8 * q^65 + 2 * q^71 + 40 * q^79 - 26 * q^85 + 42 * q^89 + 8 * q^91 - 28 * q^95

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} + 19x^{6} + 91x^{4} + 45x^{2} + 4 $ x^8 + 19*x^6 + 91*x^4 + 45*x^2 + 4 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( -\nu^{7} - 18\nu^{5} - 73\nu^{3} + 32\nu ) / 4 $ (-v^7 - 18v^5 - 73v^3 + 32*v) / 4
$\beta_{3}$	$=$	$ ( 2\nu^{7} + \nu^{6} + 38\nu^{5} + 18\nu^{4} + 180\nu^{3} + 77\nu^{2} + 72\nu + 12 ) / 8 $ (2v^7 + v^6 + 38v^5 + 18v^4 + 180v^3 + 77v^2 + 72v + 12) / 8
$\beta_{4}$	$=$	$ ( 2\nu^{7} - \nu^{6} + 38\nu^{5} - 18\nu^{4} + 180\nu^{3} - 77\nu^{2} + 72\nu - 12 ) / 8 $ (2v^7 - v^6 + 38v^5 - 18v^4 + 180v^3 - 77v^2 + 72v - 12) / 8
$\beta_{5}$	$=$	$ ( -3\nu^{7} - \nu^{6} - 56\nu^{5} - 20\nu^{4} - 257\nu^{3} - 95\nu^{2} - 76\nu - 4 ) / 8 $ (-3v^7 - v^6 - 56v^5 - 20v^4 - 257v^3 - 95v^2 - 76v - 4) / 8
$\beta_{6}$	$=$	$ ( 5\nu^{7} + 2\nu^{6} + 94\nu^{5} + 38\nu^{4} + 437\nu^{3} + 180\nu^{2} + 148\nu + 56 ) / 8 $ (5v^7 + 2v^6 + 94v^5 + 38v^4 + 437v^3 + 180v^2 + 148*v + 56) / 8
$\beta_{7}$	$=$	$ ( -5\nu^{7} + 2\nu^{6} - 94\nu^{5} + 38\nu^{4} - 437\nu^{3} + 180\nu^{2} - 148\nu + 56 ) / 8 $ (-5v^7 + 2v^6 - 94v^5 + 38v^4 - 437v^3 + 180v^2 - 148*v + 56) / 8

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{6} + \beta_{5} - \beta_{3} - 5 $ b6 + b5 - b3 - 5
$\nu^{3}$	$=$	$ -\beta_{7} + \beta_{6} - 2\beta_{4} - 2\beta_{3} + \beta_{2} - 9\beta_1 $ -b7 + b6 - 2b4 - 2b3 + b2 - 9*b1
$\nu^{4}$	$=$	$ 2\beta_{7} - 11\beta_{6} - 13\beta_{5} + 4\beta_{4} + 9\beta_{3} + 49 $ 2b7 - 11b6 - 13b5 + 4b4 + 9*b3 + 49
$\nu^{5}$	$=$	$ 17\beta_{7} - 17\beta_{6} + 36\beta_{4} + 36\beta_{3} - 13\beta_{2} + 85\beta_1 $ 17b7 - 17b6 + 36b4 + 36b3 - 13b2 + 85b1
$\nu^{6}$	$=$	$ -36\beta_{7} + 121\beta_{6} + 157\beta_{5} - 76\beta_{4} - 81\beta_{3} - 509 $ -36b7 + 121b6 + 157b5 - 76b4 - 81*b3 - 509
$\nu^{7}$	$=$	$ -233\beta_{7} + 233\beta_{6} - 502\beta_{4} - 502\beta_{3} + 157\beta_{2} - 841\beta_1 $ -233b7 + 233b6 - 502b4 - 502b3 + 157b2 - 841b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3960\mathbb{Z}\right)^\times$.

$n$	$991$	$1981$	$2377$	$2521$	$3521$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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} $

3169.1

	−	2.67673i
		2.67673i
		0.339102i
	−	0.339102i
		3.36007i
	−	3.36007i
		0.655762i
	−	0.655762i

−2.06639

−

0.854430i

−

4.38559i

3169.2

−2.06639

0.854430i

4.38559i

3169.3

−0.422782

−

2.19574i

−

4.05237i

3169.4

−0.422782

2.19574i

4.05237i

3169.5

0.256321

−

2.22133i

−

1.08258i

3169.6

0.256321

2.22133i

1.08258i

3169.7

2.23285

−

0.119978i

0.415806i

3169.8

2.23285

0.119978i

−

0.415806i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3960.2.d.f		8
3.b	odd	2	1		440.2.b.d	✓	8
5.b	even	2	1	inner	3960.2.d.f		8
12.b	even	2	1		880.2.b.j		8
15.d	odd	2	1		440.2.b.d	✓	8
15.e	even	4	1		2200.2.a.x		4
15.e	even	4	1		2200.2.a.y		4
60.h	even	2	1		880.2.b.j		8
60.l	odd	4	1		4400.2.a.cb		4
60.l	odd	4	1		4400.2.a.ce		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
440.2.b.d	✓	8	3.b	odd	2	1
440.2.b.d	✓	8	15.d	odd	2	1
880.2.b.j		8	12.b	even	2	1
880.2.b.j		8	60.h	even	2	1
2200.2.a.x		4	15.e	even	4	1
2200.2.a.y		4	15.e	even	4	1
3960.2.d.f		8	1.a	even	1	1	trivial
3960.2.d.f		8	5.b	even	2	1	inner
4400.2.a.cb		4	60.l	odd	4	1
4400.2.a.ce		4	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 $S_{2}^{\mathrm{new}}(3960, [\chi])$:

$ T_{7}^{8} + 37T_{7}^{6} + 364T_{7}^{4} + 432T_{7}^{2} + 64 $

$ T_{29}^{4} - 3T_{29}^{3} - 38T_{29}^{2} + 32 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} $ T^8
$5$	$ T^{8} + T^{6} + \cdots + 625 $ T^8 + T^6 - 6T^5 - 32T^4 - 30T^3 + 25T^2 + 625
$7$	$ T^{8} + 37 T^{6} + \cdots + 64 $ T^8 + 37T^6 + 364T^4 + 432*T^2 + 64
$11$	$ (T + 1)^{8} $ (T + 1)^8
$13$	$ (T^{2} + 16)^{4} $ (T^2 + 16)^4
$17$	$ T^{8} + 137 T^{6} + \cdots + 1024 $ T^8 + 137T^6 + 6088T^4 + 88080*T^2 + 1024
$19$	$ (T^{4} - 9 T^{3} + \cdots - 128)^{2} $ (T^4 - 9T^3 - 20T^2 + 256*T - 128)^2
$23$	$ T^{8} + 154 T^{6} + \cdots + 732736 $ T^8 + 154T^6 + 7809T^4 + 148532*T^2 + 732736
$29$	$ (T^{4} - 3 T^{3} - 38 T^{2} + 32)^{2} $ (T^4 - 3T^3 - 38T^2 + 32)^2
$31$	$ (T^{4} + 15 T^{3} + \cdots + 16)^{2} $ (T^4 + 15T^3 + 43T^2 - 83*T + 16)^2
$37$	$ T^{8} + 135 T^{6} + \cdots + 21904 $ T^8 + 135T^6 + 4639T^4 + 20201*T^2 + 21904
$41$	$ (T^{4} + 10 T^{3} + \cdots - 1024)^{2} $ (T^4 + 10T^3 - 48T^2 - 576*T - 1024)^2
$43$	$ T^{8} + 192 T^{6} + \cdots + 1290496 $ T^8 + 192T^6 + 9952T^4 + 195584*T^2 + 1290496
$47$	$ T^{8} + 124 T^{6} + \cdots + 409600 $ T^8 + 124T^6 + 5104T^4 + 81984*T^2 + 409600
$53$	$ T^{8} + 177 T^{6} + \cdots + 1638400 $ T^8 + 177T^6 + 9616T^4 + 210944*T^2 + 1638400
$59$	$ (T^{4} - 6 T^{3} - 47 T^{2} + \cdots - 32)^{2} $ (T^4 - 6T^3 - 47T^2 - 76*T - 32)^2
$61$	$ (T^{4} - 29 T^{3} + \cdots + 160)^{2} $ (T^4 - 29T^3 + 274T^2 - 864*T + 160)^2
$67$	$ T^{8} + 202 T^{6} + \cdots + 322624 $ T^8 + 202T^6 + 11601T^4 + 183092*T^2 + 322624
$71$	$ (T^{4} - T^{3} - 137 T^{2} + \cdots - 1336)^{2} $ (T^4 - T^3 - 137T^2 + 865T - 1336)^2
$73$	$ T^{8} + 228 T^{6} + \cdots + 1048576 $ T^8 + 228T^6 + 11776T^4 + 200704*T^2 + 1048576
$79$	$ (T^{4} - 20 T^{3} + \cdots - 19456)^{2} $ (T^4 - 20T^3 - 92T^2 + 3872*T - 19456)^2
$83$	$ T^{8} + 524 T^{6} + \cdots + 38142976 $ T^8 + 524T^6 + 82864T^4 + 3897152*T^2 + 38142976
$89$	$ (T^{4} - 21 T^{3} + \cdots - 346)^{2} $ (T^4 - 21T^3 + 121T^2 - 75*T - 346)^2
$97$	$ T^{8} + 310 T^{6} + \cdots + 262144 $ T^8 + 310T^6 + 28873T^4 + 758928*T^2 + 262144
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3960.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{3} q^{5} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{7}+O(q^{10}) \) q - b3 * q^5 + (-b4 - b3 + b1) * q^7	\( q - \beta_{3} q^{5} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{7} - q^{11} + (\beta_{7} - \beta_{6} + \cdots + \beta_{3}) q^{13}+ \cdots + (3 \beta_{4} + 3 \beta_{3} - 2 \beta_1) q^{97}+O(q^{100}) \) q - b3 * q^5 + (-b4 - b3 + b1) * q^7 - q^11 + (b7 - b6 + b4 + b3) * q^13 + (b7 - b6 + b4 + b3 - b2 - b1) * q^17 + (b7 - b5 + b3 + 3) * q^19 + (-b2 + 2b1) q^23 + (-b7 + b5 - b1 - 1) * q^25 + (-b6 - b5 + b4 + 1) * q^29 + (b7 - b5 + b4 - 3) * q^31 + (-b6 + 2b4 + b3 - 2b1 - 4) * q^35 + (b4 + b3 - b2 + b1) * q^37 + (b7 + b6 - b4 + b3 - 2) * q^41 + (b7 - b6 - b2 + 2b1) q^43 + (-b7 + b6 - 2b4 - 2b3 + b2) * q^47 + (-b6 - b5 + 3b4 - 2b3 - 2) * q^49 + (2b4 + 2b3 - b2 - b1) * q^53 + b3 * q^55 + (b7 + b6 + 2) * q^59 + (b6 + b5 - b4 + 7) * q^61 + (-b7 - b6 - b5 - b4 - b3 + b2 + 2b1 + 1) q^65 + (2b4 + 2b3 - b2 - 2b1) q^67 + (b7 - b5 + 3b4 - 2b3 + 1) * q^71 + (b7 - b6 + 3b4 + 3b3 - 2b1) q^73 + (b4 + b3 - b1) * q^77 + (-2b7 - 2b6 + 4) * q^79 + (-b7 + b6 - b2 - 4b1) q^83 + (-3b7 - 2b4 - b3 - b2 + 2b1 - 4) q^85 + (b6 + b5 - b3 + 5) * q^89 + (-2b7 - 2b6 - 2b4 + 2b3) * q^91 + (b7 - b6 - b5 - 3b4 - b3 - b1 - 3) q^95 + (3b4 + 3b3 - 2b1) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{11} + 18 q^{19} - 2 q^{25} + 6 q^{29} - 30 q^{31} - 30 q^{35} - 20 q^{41} - 18 q^{49} + 12 q^{59} + 58 q^{61} + 8 q^{65} + 2 q^{71} + 40 q^{79} - 26 q^{85} + 42 q^{89} + 8 q^{91} - 28 q^{95}+O(q^{100}) \) 8 * q - 8 * q^11 + 18 * q^19 - 2 * q^25 + 6 * q^29 - 30 * q^31 - 30 * q^35 - 20 * q^41 - 18 * q^49 + 12 * q^59 + 58 * q^61 + 8 * q^65 + 2 * q^71 + 40 * q^79 - 26 * q^85 + 42 * q^89 + 8 * q^91 - 28 * q^95

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