LMFDB - Newform orbit 2736.2.f.i

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(1025,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2736.1025");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.f (of order $2$, degree $1$, 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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.691798081536.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 14x^{6} - 12x^{5} + 127x^{4} - 144x^{3} - 282x^{2} + 900x + 1350 $ x^8 - 14x^6 - 12x^5 + 127x^4 - 144x^3 - 282x^2 + 900x + 1350
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 684)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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_{2} q^{7}+O(q^{10}) $ q - b3 * q^5 + b2 * q^7	$ q - \beta_{3} q^{5} + \beta_{2} q^{7} + \beta_{6} q^{11} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{17} + (\beta_{2} - \beta_1) q^{19} + ( - \beta_{7} + \beta_{6} + \beta_{3}) q^{23} + (\beta_{2} + 2 \beta_1 - 5) q^{25} + ( - 2 \beta_{7} + \beta_{5} + \beta_{3}) q^{35} + (\beta_{4} - 1) q^{43} + (\beta_{7} + \beta_{6} + \cdots + 2 \beta_{3}) q^{47}+ \cdots + ( - 2 \beta_{7} - \beta_{6} + \cdots - 2 \beta_{3}) q^{95}+O(q^{100}) $ q - b3 * q^5 + b2 * q^7 + b6 * q^11 + (b7 - b5 + b3) * q^17 + (b2 - b1) * q^19 + (-b7 + b6 + b3) * q^23 + (b2 + 2b1 - 5) q^25 + (-2b7 + b5 + b3) q^35 + (b4 - 1) * q^43 + (b7 + b6 + b5 + 2b3) q^47 + (b4 + 4) * q^49 + (-b4 + 2b2 - 2b1 - 3) * q^55 + (-3b2 - 2b1) * q^61 + (-b4 - 5) * q^73 + (-3b7 + 2b5 + b3) * q^77 + (-b7 + b6 - 2b5 + b3) q^83 + (-b4 + 4b2 - 4b1 + 7) * q^85 + (-2b7 - b6 + 2b5 - 2b3) q^95
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 40 q^{25} - 4 q^{43} + 36 q^{49} - 28 q^{55} - 44 q^{73} + 52 q^{85}+O(q^{100}) $ 8 * q - 40 * q^25 - 4 * q^43 + 36 * q^49 - 28 * q^55 - 44 * q^73 + 52 * q^85

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 14x^{6} - 12x^{5} + 127x^{4} - 144x^{3} - 282x^{2} + 900x + 1350 $ x^8 - 14*x^6 - 12*x^5 + 127*x^4 - 144*x^3 - 282*x^2 + 900*x + 1350 :

$\beta_{1}$	$=$	$ ( 12960 \nu^{7} + 23083 \nu^{6} - 32796 \nu^{5} - 597740 \nu^{4} - 196776 \nu^{3} - 859412 \nu^{2} + \cdots - 11605830 ) / 13725555 $ (12960v^7 + 23083v^6 - 32796v^5 - 597740v^4 - 196776v^3 - 859412v^2 - 5392428*v - 11605830) / 13725555
$\beta_{2}$	$=$	$ ( - 24310 \nu^{7} + 8243 \nu^{6} + 544454 \nu^{5} + 90395 \nu^{4} - 5883646 \nu^{3} + \cdots - 38703045 ) / 13725555 $ (-24310v^7 + 8243v^6 + 544454v^5 + 90395v^4 - 5883646v^3 + 1377653v^2 + 27915822*v - 38703045) / 13725555
$\beta_{3}$	$=$	$ ( - 47546 \nu^{7} + 178954 \nu^{6} + 735426 \nu^{5} - 1249773 \nu^{4} - 7482925 \nu^{3} + \cdots - 66840945 ) / 13725555 $ (-47546v^7 + 178954v^6 + 735426v^5 - 1249773v^4 - 7482925v^3 + 25205858v^2 - 16379547*v - 66840945) / 13725555
$\beta_{4}$	$=$	$ ( 16\nu^{7} + 54\nu^{6} - 212\nu^{5} - 567\nu^{4} + 544\nu^{3} - 1830\nu^{2} - 9156\nu + 32190 ) / 3405 $ (16v^7 + 54v^6 - 212v^5 - 567v^4 + 544v^3 - 1830v^2 - 9156*v + 32190) / 3405
$\beta_{5}$	$=$	$ ( -200\nu^{7} + 233\nu^{6} + 1969\nu^{5} - 2560\nu^{4} - 21101\nu^{3} + 71453\nu^{2} - 13578\nu - 147000 ) / 32915 $ (-200v^7 + 233v^6 + 1969v^5 - 2560v^4 - 21101v^3 + 71453v^2 - 13578*v - 147000) / 32915
$\beta_{6}$	$=$	$ ( 108029 \nu^{7} - 113379 \nu^{6} - 1170618 \nu^{5} + 675757 \nu^{4} + 13107106 \nu^{3} + \cdots + 83548605 ) / 13725555 $ (108029v^7 - 113379v^6 - 1170618v^5 + 675757v^4 + 13107106v^3 - 37579915v^2 + 9710091*v + 83548605) / 13725555
$\beta_{7}$	$=$	$ ( 123377 \nu^{7} - 86890 \nu^{6} - 1605973 \nu^{5} - 930214 \nu^{4} + 14155124 \nu^{3} + \cdots + 75574110 ) / 13725555 $ (123377v^7 - 86890v^6 - 1605973v^5 - 930214v^4 + 14155124v^3 - 23614798v^2 + 17283456*v + 75574110) / 13725555

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

$\nu$	$=$	$ ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + 2\beta_{2} - 4\beta_1 ) / 6 $ (b7 + b6 + b5 + b3 + 2b2 - 4b1) / 6
$\nu^{2}$	$=$	$ ( 2\beta_{7} - 4\beta_{6} - 4\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} - \beta _1 + 11 ) / 3 $ (2b7 - 4b6 - 4b5 - b4 + 2b3 - b2 - b1 + 11) / 3
$\nu^{3}$	$=$	$ ( -3\beta_{7} + 33\beta_{6} + 25\beta_{5} - 6\beta_{4} + 15\beta_{3} - 6\beta_{2} - 12\beta _1 + 30 ) / 6 $ (-3b7 + 33b6 + 25b5 - 6b4 + 15b3 - 6b2 - 12*b1 + 30) / 6
$\nu^{4}$	$=$	$ ( 4\beta_{7} - 32\beta_{6} - 52\beta_{5} + \beta_{4} + 16\beta_{3} - 2\beta_{2} - 56\beta _1 - 44 ) / 3 $ (4b7 - 32b6 - 52b5 + b4 + 16b3 - 2b2 - 56b1 - 44) / 3
$\nu^{5}$	$=$	$ ( -49\beta_{7} + 191\beta_{6} + 49\beta_{5} - 100\beta_{4} + 161\beta_{3} - 16\beta_{2} + 248\beta _1 + 1220 ) / 6 $ (-49b7 + 191b6 + 49b5 - 100b4 + 161b3 - 16b2 + 248*b1 + 1220) / 6
$\nu^{6}$	$=$	$ ( -36\beta_{7} + 276\beta_{6} + 56\beta_{5} + 120\beta_{4} + 312\beta_{3} + 120\beta_{2} - 825\beta _1 - 1206 ) / 3 $ (-36b7 + 276b6 + 56b5 + 120b4 + 312b3 + 120b2 - 825*b1 - 1206) / 3
$\nu^{7}$	$=$	$ ( 1009 \beta_{7} - 3065 \beta_{6} - 4607 \beta_{5} - 812 \beta_{4} + 1681 \beta_{3} - 44 \beta_{2} + \cdots + 10612 ) / 6 $ (1009b7 - 3065b6 - 4607b5 - 812b4 + 1681b3 - 44b2 + 2776*b1 + 10612) / 6

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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} $

1025.1

3.04547	+	0.517638i
−1.31342	−	0.517638i
1.31342	+	1.93185i
−3.04547	−	1.93185i
−3.04547	+	1.93185i
1.31342	−	1.93185i
−1.31342	+	0.517638i
3.04547	−	0.517638i

−

4.46919i

−0.418627

1025.2

−

3.95155i

−4.77753

1025.3

−

2.09409i

4.77753

1025.4

−

0.162240i

0.418627

1025.5

0.162240i

0.418627

1025.6

2.09409i

4.77753

1025.7

3.95155i

−4.77753

1025.8

4.46919i

−0.418627

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.b	odd	2	1	CM by $\Q(\sqrt{-19}) $
3.b	odd	2	1	inner
57.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2736.2.f.i		8
3.b	odd	2	1	inner	2736.2.f.i		8
4.b	odd	2	1		684.2.d.a	✓	8
12.b	even	2	1		684.2.d.a	✓	8
19.b	odd	2	1	CM	2736.2.f.i		8
57.d	even	2	1	inner	2736.2.f.i		8
76.d	even	2	1		684.2.d.a	✓	8
228.b	odd	2	1		684.2.d.a	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
684.2.d.a	✓	8	4.b	odd	2	1
684.2.d.a	✓	8	12.b	even	2	1
684.2.d.a	✓	8	76.d	even	2	1
684.2.d.a	✓	8	228.b	odd	2	1
2736.2.f.i		8	1.a	even	1	1	trivial
2736.2.f.i		8	3.b	odd	2	1	inner
2736.2.f.i		8	19.b	odd	2	1	CM
2736.2.f.i		8	57.d	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2736, [\chi])$:

$ T_{5}^{8} + 40T_{5}^{6} + 469T_{5}^{4} + 1380T_{5}^{2} + 36 $

$ T_{7}^{4} - 23T_{7}^{2} + 4 $

$ T_{11}^{8} + 88T_{11}^{6} + 2653T_{11}^{4} + 31548T_{11}^{2} + 125316 $

$ T_{29} $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} $ T^8
$5$	$ T^{8} + 40 T^{6} + \cdots + 36 $ T^8 + 40T^6 + 469T^4 + 1380*T^2 + 36
$7$	$ (T^{4} - 23 T^{2} + 4)^{2} $ (T^4 - 23*T^2 + 4)^2
$11$	$ T^{8} + 88 T^{6} + \cdots + 125316 $ T^8 + 88T^6 + 2653T^4 + 31548*T^2 + 125316
$13$	$ T^{8} $ T^8
$17$	$ T^{8} + 136 T^{6} + \cdots + 412164 $ T^8 + 136T^6 + 6133T^4 + 102612*T^2 + 412164
$19$	$ (T^{2} - 19)^{4} $ (T^2 - 19)^4
$23$	$ (T^{4} + 92 T^{2} + 900)^{2} $ (T^4 + 92*T^2 + 900)^2
$29$	$ T^{8} $ T^8
$31$	$ T^{8} $ T^8
$37$	$ T^{8} $ T^8
$41$	$ T^{8} $ T^8
$43$	$ (T^{2} + T - 128)^{4} $ (T^2 + T - 128)^4
$47$	$ T^{8} + 376 T^{6} + \cdots + 1004004 $ T^8 + 376T^6 + 42973T^4 + 1434252*T^2 + 1004004
$53$	$ T^{8} $ T^8
$59$	$ T^{8} $ T^8
$61$	$ (T^{4} - 347 T^{2} + 26896)^{2} $ (T^4 - 347*T^2 + 26896)^2
$67$	$ T^{8} $ T^8
$71$	$ T^{8} $ T^8
$73$	$ (T^{2} + 11 T - 98)^{4} $ (T^2 + 11*T - 98)^4
$79$	$ T^{8} $ T^8
$83$	$ (T^{4} + 332 T^{2} + 8100)^{2} $ (T^4 + 332*T^2 + 8100)^2
$89$	$ T^{8} $ T^8
$97$	$ T^{8} $ T^8
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Classical	Maass
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{3} q^{5} + \beta_{2} q^{7}+O(q^{10}) \) q - b3 * q^5 + b2 * q^7	\( q - \beta_{3} q^{5} + \beta_{2} q^{7} + \beta_{6} q^{11} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{17} + (\beta_{2} - \beta_1) q^{19} + ( - \beta_{7} + \beta_{6} + \beta_{3}) q^{23} + (\beta_{2} + 2 \beta_1 - 5) q^{25} + ( - 2 \beta_{7} + \beta_{5} + \beta_{3}) q^{35} + (\beta_{4} - 1) q^{43} + (\beta_{7} + \beta_{6} + \cdots + 2 \beta_{3}) q^{47}+ \cdots + ( - 2 \beta_{7} - \beta_{6} + \cdots - 2 \beta_{3}) q^{95}+O(q^{100}) \) q - b3 * q^5 + b2 * q^7 + b6 * q^11 + (b7 - b5 + b3) * q^17 + (b2 - b1) * q^19 + (-b7 + b6 + b3) * q^23 + (b2 + 2b1 - 5) q^25 + (-2b7 + b5 + b3) q^35 + (b4 - 1) * q^43 + (b7 + b6 + b5 + 2b3) q^47 + (b4 + 4) * q^49 + (-b4 + 2b2 - 2b1 - 3) * q^55 + (-3b2 - 2b1) * q^61 + (-b4 - 5) * q^73 + (-3b7 + 2b5 + b3) * q^77 + (-b7 + b6 - 2b5 + b3) q^83 + (-b4 + 4b2 - 4b1 + 7) * q^85 + (-2b7 - b6 + 2b5 - 2b3) q^95
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 40 q^{25} - 4 q^{43} + 36 q^{49} - 28 q^{55} - 44 q^{73} + 52 q^{85}+O(q^{100}) \) 8 * q - 40 * q^25 - 4 * q^43 + 36 * q^49 - 28 * q^55 - 44 * q^73 + 52 * q^85

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