LMFDB - Newform orbit 2240.2.g.p

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(449,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2240.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.g (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:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 20x^{10} + 148x^{8} + 494x^{6} + 708x^{4} + 304x^{2} + 9 $ x^12 + 20x^10 + 148x^8 + 494x^6 + 708x^4 + 304*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 1120)
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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{9} q^{3} - \beta_{10} q^{5} - \beta_{5} q^{7} + ( - \beta_{11} + \beta_{10} - \beta_{3} - 1) q^{9} - \beta_{2} q^{11} + (\beta_{11} + \beta_{10} + \beta_{4}) q^{13} + ( - \beta_{9} + \beta_{5} + \cdots + \beta_1) q^{15}+ \cdots + (2 \beta_{8} + 2 \beta_{7} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) $ q - b9 * q^3 - b10 * q^5 - b5 * q^7 + (-b11 + b10 - b3 - 1) * q^9 - b2 * q^11 + (b11 + b10 + b4) * q^13 + (-b9 + b5 + b2 + b1) * q^15 + (-2b6 + b4) q^17 + (b8 + b7 - 2b1) q^19 - b3 * q^21 + (2b9 - 2b5) * q^23 + (-2b11 + b6 - b3) q^25 + (b9 + b8 - b7 + 2b5) q^27 + (b11 - b10 - b3 + 4) * q^29 + (2b8 + 2b7) * q^31 + (-2b11 - 2b10 + b4) * q^33 + b7 * q^35 + (2b11 + 2b10 - 2b6) q^37 + (2b8 + 2b7 - b2 - 2b1) q^39 + (2b3 + 2) q^41 + (-2b9 - 4b5) * q^43 + (2b11 + b10 - b4 - 4) q^45 + (b9 + 3b8 - 3b7 + 2b5) q^47 - q^49 + (4b8 + 4b7 + b2) * q^51 + (-2b6 + 2b4) * q^53 + (-3b9 - b8 + b7 - 2b1) * q^55 + (-2b11 - 2b10 + 2b6 - 2b4) * q^57 + (-b8 - b7 + 2b1) q^59 + (b11 - b10 + 2b3 - 4) q^61 + (b9 + b8 - b7 + b5) * q^63 + (b11 + b10 - 3b6 - 2b3 + 5) * q^65 + (2b9 + 2b8 - 2b7 + 4b5) * q^67 + (2b11 - 2b10 + 8) * q^69 + 2b1 q^71 + (-2b6 + 2b4) * q^73 + (3b9 - 2b7 + 2b5 + 2b2 + 2b1) q^75 + b4 * q^77 + (-2b8 - 2b7 + b2 + 2b1) q^79 + (-2b11 + 2b10 + 2b3 - 1) q^81 + (-2b9 + 3b8 - 3b7 - 2b5) * q^83 + (b11 + b10 + 2b4 - 5b3 - 2) * q^85 + (-5b9 + b8 - b7 + 6b5) * q^87 - 6 * q^89 + (-b8 - b7 + b2) * q^91 + (4b6 - 4b4) * q^93 + (-b9 + 4b8 + 3b5 - 2b2 + b1) q^95 + (2b6 + b4) q^97 + (2b8 + 2b7 + 2b2 + 4b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 16 q^{9} - 4 q^{21} - 4 q^{25} + 44 q^{29} + 32 q^{41} - 48 q^{45} - 12 q^{49} - 40 q^{61} + 52 q^{65} + 96 q^{69} - 4 q^{81} - 44 q^{85} - 72 q^{89}+O(q^{100}) $ 12 * q - 16 * q^9 - 4 * q^21 - 4 * q^25 + 44 * q^29 + 32 * q^41 - 48 * q^45 - 12 * q^49 - 40 * q^61 + 52 * q^65 + 96 * q^69 - 4 * q^81 - 44 * q^85 - 72 * q^89

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} + 20x^{10} + 148x^{8} + 494x^{6} + 708x^{4} + 304x^{2} + 9 $ x^12 + 20*x^10 + 148*x^8 + 494*x^6 + 708*x^4 + 304*x^2 + 9 :

$\beta_{1}$	$=$	$ ( 4\nu^{10} + 67\nu^{8} + 382\nu^{6} + 874\nu^{4} + 751\nu^{2} + 116 ) / 62 $ (4v^10 + 67v^8 + 382v^6 + 874v^4 + 751*v^2 + 116) / 62
$\beta_{2}$	$=$	$ ( -9\nu^{10} - 143\nu^{8} - 689\nu^{6} - 773\nu^{4} + 1077\nu^{2} + 607 ) / 124 $ (-9v^10 - 143v^8 - 689v^6 - 773v^4 + 1077*v^2 + 607) / 124
$\beta_{3}$	$=$	$ ( 3\nu^{10} + 58\nu^{8} + 395\nu^{6} + 1105\nu^{4} + 1098\nu^{2} + 211 ) / 62 $ (3v^10 + 58v^8 + 395v^6 + 1105v^4 + 1098*v^2 + 211) / 62
$\beta_{4}$	$=$	$ ( \nu^{11} + 71\nu^{9} + 979\nu^{7} + 5039\nu^{5} + 9945\nu^{3} + 5113\nu ) / 186 $ (v^11 + 71v^9 + 979v^7 + 5039v^5 + 9945v^3 + 5113*v) / 186
$\beta_{5}$	$=$	$ ( 11\nu^{11} + 223\nu^{9} + 1655\nu^{7} + 5395\nu^{5} + 7095\nu^{3} + 2303\nu ) / 372 $ (11v^11 + 223v^9 + 1655v^7 + 5395v^5 + 7095v^3 + 2303v) / 372
$\beta_{6}$	$=$	$ ( -7\nu^{11} - 125\nu^{9} - 808\nu^{7} - 2351\nu^{5} - 3213\nu^{3} - 1660\nu ) / 186 $ (-7v^11 - 125v^9 - 808v^7 - 2351v^5 - 3213v^3 - 1660v) / 186
$\beta_{7}$	$=$	$ ( - 16 \nu^{11} - 12 \nu^{10} - 299 \nu^{9} - 201 \nu^{8} - 1993 \nu^{7} - 1146 \nu^{6} - 5666 \nu^{5} + \cdots + 768 ) / 372 $ (-16v^11 - 12v^10 - 299v^9 - 201v^8 - 1993v^7 - 1146v^6 - 5666v^5 - 2436v^4 - 6321v^3 - 951v^2 - 1549*v + 768) / 372
$\beta_{8}$	$=$	$ ( 16 \nu^{11} - 12 \nu^{10} + 299 \nu^{9} - 201 \nu^{8} + 1993 \nu^{7} - 1146 \nu^{6} + 5666 \nu^{5} + \cdots + 768 ) / 372 $ (16v^11 - 12v^10 + 299v^9 - 201v^8 + 1993v^7 - 1146v^6 + 5666v^5 - 2436v^4 + 6321v^3 - 951v^2 + 1549*v + 768) / 372
$\beta_{9}$	$=$	$ ( 25\nu^{11} + 473\nu^{9} + 3271\nu^{7} + 10097\nu^{5} + 13521\nu^{3} + 6367\nu ) / 372 $ (25v^11 + 473v^9 + 3271v^7 + 10097v^5 + 13521v^3 + 6367v) / 372
$\beta_{10}$	$=$	$ ( - 20 \nu^{11} - 15 \nu^{10} - 397 \nu^{9} - 228 \nu^{8} - 2933 \nu^{7} - 1107 \nu^{6} - 9826 \nu^{5} + \cdots - 249 ) / 372 $ (-20v^11 - 15v^10 - 397v^9 - 228v^8 - 2933v^7 - 1107v^6 - 9826v^5 - 1743v^4 - 13923v^3 - 282v^2 - 4889*v - 249) / 372
$\beta_{11}$	$=$	$ ( - 20 \nu^{11} + 15 \nu^{10} - 397 \nu^{9} + 228 \nu^{8} - 2933 \nu^{7} + 1107 \nu^{6} - 9826 \nu^{5} + \cdots + 249 ) / 372 $ (-20v^11 + 15v^10 - 397v^9 + 228v^8 - 2933v^7 + 1107v^6 - 9826v^5 + 1743v^4 - 13923v^3 + 282v^2 - 4889*v + 249) / 372

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

$\nu$	$=$	$ ( \beta_{9} + \beta_{6} - \beta_{5} ) / 2 $ (b9 + b6 - b5) / 2
$\nu^{2}$	$=$	$ ( \beta_{11} - \beta_{10} + \beta_{8} + \beta_{7} + \beta_{3} - \beta _1 - 7 ) / 2 $ (b11 - b10 + b8 + b7 + b3 - b1 - 7) / 2
$\nu^{3}$	$=$	$ ( \beta_{11} + \beta_{10} - 4\beta_{9} - \beta_{8} + \beta_{7} - 6\beta_{6} + 8\beta_{5} ) / 2 $ (b11 + b10 - 4b9 - b8 + b7 - 6b6 + 8*b5) / 2
$\nu^{4}$	$=$	$ ( -7\beta_{11} + 7\beta_{10} - 5\beta_{8} - 5\beta_{7} - 7\beta_{3} + 9\beta _1 + 37 ) / 2 $ (-7b11 + 7b10 - 5b8 - 5b7 - 7b3 + 9b1 + 37) / 2
$\nu^{5}$	$=$	$ ( -7\beta_{11} - 7\beta_{10} + 21\beta_{9} + 11\beta_{8} - 11\beta_{7} + 35\beta_{6} - 61\beta_{5} + 2\beta_{4} ) / 2 $ (-7b11 - 7b10 + 21b9 + 11b8 - 11b7 + 35b6 - 61b5 + 2b4) / 2
$\nu^{6}$	$=$	$ 24\beta_{11} - 24\beta_{10} + 12\beta_{8} + 12\beta_{7} + 23\beta_{3} + 2\beta_{2} - 33\beta _1 - 108 $ 24b11 - 24b10 + 12b8 + 12b7 + 23b3 + 2b2 - 33*b1 - 108
$\nu^{7}$	$=$	$ ( 40 \beta_{11} + 40 \beta_{10} - 121 \beta_{9} - 96 \beta_{8} + 96 \beta_{7} - 213 \beta_{6} + \cdots - 24 \beta_{4} ) / 2 $ (40b11 + 40b10 - 121b9 - 96b8 + 96b7 - 213b6 + 433b5 - 24b4) / 2
$\nu^{8}$	$=$	$ ( -335\beta_{11} + 335\beta_{10} - 115\beta_{8} - 115\beta_{7} - 291\beta_{3} - 56\beta_{2} + 459\beta _1 + 1329 ) / 2 $ (-335b11 + 335b10 - 115b8 - 115b7 - 291b3 - 56b2 + 459*b1 + 1329) / 2
$\nu^{9}$	$=$	$ ( - 209 \beta_{11} - 209 \beta_{10} + 730 \beta_{9} + 765 \beta_{8} - 765 \beta_{7} + \cdots + 220 \beta_{4} ) / 2 $ (-209b11 - 209b10 + 730b9 + 765b8 - 765b7 + 1344b6 - 2974b5 + 220b4) / 2
$\nu^{10}$	$=$	$ ( 2369 \beta_{11} - 2369 \beta_{10} + 539 \beta_{8} + 539 \beta_{7} + 1823 \beta_{3} + 556 \beta_{2} + \cdots - 8461 ) / 2 $ (2369b11 - 2369b10 + 539b8 + 539b7 + 1823b3 + 556b2 - 3133*b1 - 8461) / 2
$\nu^{11}$	$=$	$ ( 1007 \beta_{11} + 1007 \beta_{10} - 4523 \beta_{9} - 5815 \beta_{8} + 5815 \beta_{7} + \cdots - 1830 \beta_{4} ) / 2 $ (1007b11 + 1007b10 - 4523b9 - 5815b8 + 5815b7 - 8705b6 + 20179b5 - 1830b4) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$-1$	$1$	$1$	$1$

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} $

449.1

		1.67709i
		0.178681i
		0.810591i
		2.36682i
		1.97020i
	−	2.64856i
		2.64856i
	−	1.97020i
	−	2.36682i
	−	0.810591i
	−	0.178681i
	−	1.67709i

−

2.85577i

0.649832

−

2.13956i

−

1.00000i

−5.15544

449.2

−

2.85577i

0.649832

2.13956i

−

1.00000i

−5.15544

449.3

−

2.17741i

1.45926

−

1.69427i

1.00000i

−1.74111

449.4

−

2.17741i

1.45926

1.69427i

1.00000i

−1.74111

449.5

−

0.321637i

−2.10909

−

0.742782i

−

1.00000i

2.89655

449.6

−

0.321637i

−2.10909

0.742782i

−

1.00000i

2.89655

449.7

0.321637i

−2.10909

−

0.742782i

1.00000i

2.89655

449.8

0.321637i

−2.10909

0.742782i

1.00000i

2.89655

449.9

2.17741i

1.45926

−

1.69427i

−

1.00000i

−1.74111

449.10

2.17741i

1.45926

1.69427i

−

1.00000i

−1.74111

449.11

2.85577i

0.649832

−

2.13956i

1.00000i

−5.15544

449.12

2.85577i

0.649832

2.13956i

1.00000i

−5.15544

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.b	even	2	1	inner
20.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2240.2.g.p		12
4.b	odd	2	1	inner	2240.2.g.p		12
5.b	even	2	1	inner	2240.2.g.p		12
8.b	even	2	1		1120.2.g.d	✓	12
8.d	odd	2	1		1120.2.g.d	✓	12
20.d	odd	2	1	inner	2240.2.g.p		12
40.e	odd	2	1		1120.2.g.d	✓	12
40.f	even	2	1		1120.2.g.d	✓	12
40.i	odd	4	1		5600.2.a.by		6
40.i	odd	4	1		5600.2.a.bz		6
40.k	even	4	1		5600.2.a.by		6
40.k	even	4	1		5600.2.a.bz		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1120.2.g.d	✓	12	8.b	even	2	1
1120.2.g.d	✓	12	8.d	odd	2	1
1120.2.g.d	✓	12	40.e	odd	2	1
1120.2.g.d	✓	12	40.f	even	2	1
2240.2.g.p		12	1.a	even	1	1	trivial
2240.2.g.p		12	4.b	odd	2	1	inner
2240.2.g.p		12	5.b	even	2	1	inner
2240.2.g.p		12	20.d	odd	2	1	inner
5600.2.a.by		6	40.i	odd	4	1
5600.2.a.by		6	40.k	even	4	1
5600.2.a.bz		6	40.i	odd	4	1
5600.2.a.bz		6	40.k	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2240, [\chi])$:

$ T_{3}^{6} + 13T_{3}^{4} + 40T_{3}^{2} + 4 $

$ T_{11}^{6} - 45T_{11}^{4} + 592T_{11}^{2} - 1856 $

$ T_{19}^{6} - 104T_{19}^{4} + 2708T_{19}^{2} - 4176 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ (T^{6} + 13 T^{4} + 40 T^{2} + 4)^{2} $ (T^6 + 13T^4 + 40T^2 + 4)^2
$5$	$ (T^{6} + T^{4} + 16 T^{3} + \cdots + 125)^{2} $ (T^6 + T^4 + 16T^3 + 5T^2 + 125)^2
$7$	$ (T^{2} + 1)^{6} $ (T^2 + 1)^6
$11$	$ (T^{6} - 45 T^{4} + \cdots - 1856)^{2} $ (T^6 - 45T^4 + 592T^2 - 1856)^2
$13$	$ (T^{6} + 65 T^{4} + \cdots + 116)^{2} $ (T^6 + 65T^4 + 1056T^2 + 116)^2
$17$	$ (T^{6} + 109 T^{4} + \cdots + 37584)^{2} $ (T^6 + 109T^4 + 3608T^2 + 37584)^2
$19$	$ (T^{6} - 104 T^{4} + \cdots - 4176)^{2} $ (T^6 - 104T^4 + 2708T^2 - 4176)^2
$23$	$ (T^{6} + 56 T^{4} + \cdots + 1024)^{2} $ (T^6 + 56T^4 + 656T^2 + 1024)^2
$29$	$ (T^{3} - 11 T^{2} + \cdots + 12)^{4} $ (T^3 - 11T^2 + 16T + 12)^4
$31$	$ (T^{6} - 128 T^{4} + \cdots - 29696)^{2} $ (T^6 - 128T^4 + 4416T^2 - 29696)^2
$37$	$ (T^{6} + 168 T^{4} + \cdots + 118784)^{2} $ (T^6 + 168T^4 + 8080T^2 + 118784)^2
$41$	$ (T^{3} - 8 T^{2} - 4 T + 48)^{4} $ (T^3 - 8T^2 - 4T + 48)^4
$43$	$ (T^{6} + 116 T^{4} + \cdots + 256)^{2} $ (T^6 + 116T^4 + 2048T^2 + 256)^2
$47$	$ (T^{6} + 257 T^{4} + \cdots + 602176)^{2} $ (T^6 + 257T^4 + 21680T^2 + 602176)^2
$53$	$ (T^{6} + 204 T^{4} + \cdots + 1856)^{2} $ (T^6 + 204T^4 + 8176T^2 + 1856)^2
$59$	$ (T^{6} - 104 T^{4} + \cdots - 4176)^{2} $ (T^6 - 104T^4 + 2708T^2 - 4176)^2
$61$	$ (T^{3} + 10 T^{2} + \cdots - 124)^{4} $ (T^3 + 10T^2 + 2T - 124)^4
$67$	$ (T^{6} + 196 T^{4} + \cdots + 65536)^{2} $ (T^6 + 196T^4 + 7168T^2 + 65536)^2
$71$	$ (T^{6} - 104 T^{4} + \cdots - 7424)^{2} $ (T^6 - 104T^4 + 1680T^2 - 7424)^2
$73$	$ (T^{6} + 204 T^{4} + \cdots + 1856)^{2} $ (T^6 + 204T^4 + 8176T^2 + 1856)^2
$79$	$ (T^{6} - 149 T^{4} + \cdots - 464)^{2} $ (T^6 - 149T^4 + 648T^2 - 464)^2
$83$	$ (T^{6} + 372 T^{4} + \cdots + 419904)^{2} $ (T^6 + 372T^4 + 34084T^2 + 419904)^2
$89$	$ (T + 6)^{12} $ (T + 6)^12
$97$	$ (T^{6} + 189 T^{4} + \cdots + 464)^{2} $ (T^6 + 189T^4 + 664T^2 + 464)^2
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{9} q^{3} - \beta_{10} q^{5} - \beta_{5} q^{7} + ( - \beta_{11} + \beta_{10} - \beta_{3} - 1) q^{9} - \beta_{2} q^{11} + (\beta_{11} + \beta_{10} + \beta_{4}) q^{13} + ( - \beta_{9} + \beta_{5} + \cdots + \beta_1) q^{15}+ \cdots + (2 \beta_{8} + 2 \beta_{7} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) \) q - b9 * q^3 - b10 * q^5 - b5 * q^7 + (-b11 + b10 - b3 - 1) * q^9 - b2 * q^11 + (b11 + b10 + b4) * q^13 + (-b9 + b5 + b2 + b1) * q^15 + (-2b6 + b4) q^17 + (b8 + b7 - 2b1) q^19 - b3 * q^21 + (2b9 - 2b5) * q^23 + (-2b11 + b6 - b3) q^25 + (b9 + b8 - b7 + 2b5) q^27 + (b11 - b10 - b3 + 4) * q^29 + (2b8 + 2b7) * q^31 + (-2b11 - 2b10 + b4) * q^33 + b7 * q^35 + (2b11 + 2b10 - 2b6) q^37 + (2b8 + 2b7 - b2 - 2b1) q^39 + (2b3 + 2) q^41 + (-2b9 - 4b5) * q^43 + (2b11 + b10 - b4 - 4) q^45 + (b9 + 3b8 - 3b7 + 2b5) q^47 - q^49 + (4b8 + 4b7 + b2) * q^51 + (-2b6 + 2b4) * q^53 + (-3b9 - b8 + b7 - 2b1) * q^55 + (-2b11 - 2b10 + 2b6 - 2b4) * q^57 + (-b8 - b7 + 2b1) q^59 + (b11 - b10 + 2b3 - 4) q^61 + (b9 + b8 - b7 + b5) * q^63 + (b11 + b10 - 3b6 - 2b3 + 5) * q^65 + (2b9 + 2b8 - 2b7 + 4b5) * q^67 + (2b11 - 2b10 + 8) * q^69 + 2b1 q^71 + (-2b6 + 2b4) * q^73 + (3b9 - 2b7 + 2b5 + 2b2 + 2b1) q^75 + b4 * q^77 + (-2b8 - 2b7 + b2 + 2b1) q^79 + (-2b11 + 2b10 + 2b3 - 1) q^81 + (-2b9 + 3b8 - 3b7 - 2b5) * q^83 + (b11 + b10 + 2b4 - 5b3 - 2) * q^85 + (-5b9 + b8 - b7 + 6b5) * q^87 - 6 * q^89 + (-b8 - b7 + b2) * q^91 + (4b6 - 4b4) * q^93 + (-b9 + 4b8 + 3b5 - 2b2 + b1) q^95 + (2b6 + b4) q^97 + (2b8 + 2b7 + 2b2 + 4b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 16 q^{9} - 4 q^{21} - 4 q^{25} + 44 q^{29} + 32 q^{41} - 48 q^{45} - 12 q^{49} - 40 q^{61} + 52 q^{65} + 96 q^{69} - 4 q^{81} - 44 q^{85} - 72 q^{89}+O(q^{100}) \) 12 * q - 16 * q^9 - 4 * q^21 - 4 * q^25 + 44 * q^29 + 32 * q^41 - 48 * q^45 - 12 * q^49 - 40 * q^61 + 52 * q^65 + 96 * q^69 - 4 * q^81 - 44 * q^85 - 72 * q^89

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Newspace parameters

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Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2240.2.g.p

Level	$2240$
Weight	$2$
Character orbit	2240.g
Analytic conductor	$17.886$
Analytic rank	$0$
Dimension	$12$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(17.8864900528\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} + 20x^{10} + 148x^{8} + 494x^{6} + 708x^{4} + 304x^{2} + 9 \) x^12 + 20x^10 + 148x^8 + 494x^6 + 708x^4 + 304*x^2 + 9
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{6} \)
Twist minimal:	no (minimal twist has level 1120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 4\nu^{10} + 67\nu^{8} + 382\nu^{6} + 874\nu^{4} + 751\nu^{2} + 116 ) / 62 \) (4v^10 + 67v^8 + 382v^6 + 874v^4 + 751*v^2 + 116) / 62
\(\beta_{2}\)	\(=\)	\( ( -9\nu^{10} - 143\nu^{8} - 689\nu^{6} - 773\nu^{4} + 1077\nu^{2} + 607 ) / 124 \) (-9v^10 - 143v^8 - 689v^6 - 773v^4 + 1077*v^2 + 607) / 124
\(\beta_{3}\)	\(=\)	\( ( 3\nu^{10} + 58\nu^{8} + 395\nu^{6} + 1105\nu^{4} + 1098\nu^{2} + 211 ) / 62 \) (3v^10 + 58v^8 + 395v^6 + 1105v^4 + 1098*v^2 + 211) / 62
\(\beta_{4}\)	\(=\)	\( ( \nu^{11} + 71\nu^{9} + 979\nu^{7} + 5039\nu^{5} + 9945\nu^{3} + 5113\nu ) / 186 \) (v^11 + 71v^9 + 979v^7 + 5039v^5 + 9945v^3 + 5113*v) / 186
\(\beta_{5}\)	\(=\)	\( ( 11\nu^{11} + 223\nu^{9} + 1655\nu^{7} + 5395\nu^{5} + 7095\nu^{3} + 2303\nu ) / 372 \) (11v^11 + 223v^9 + 1655v^7 + 5395v^5 + 7095v^3 + 2303v) / 372
\(\beta_{6}\)	\(=\)	\( ( -7\nu^{11} - 125\nu^{9} - 808\nu^{7} - 2351\nu^{5} - 3213\nu^{3} - 1660\nu ) / 186 \) (-7v^11 - 125v^9 - 808v^7 - 2351v^5 - 3213v^3 - 1660v) / 186
\(\beta_{7}\)	\(=\)	\( ( - 16 \nu^{11} - 12 \nu^{10} - 299 \nu^{9} - 201 \nu^{8} - 1993 \nu^{7} - 1146 \nu^{6} - 5666 \nu^{5} + \cdots + 768 ) / 372 \) (-16v^11 - 12v^10 - 299v^9 - 201v^8 - 1993v^7 - 1146v^6 - 5666v^5 - 2436v^4 - 6321v^3 - 951v^2 - 1549*v + 768) / 372
\(\beta_{8}\)	\(=\)	\( ( 16 \nu^{11} - 12 \nu^{10} + 299 \nu^{9} - 201 \nu^{8} + 1993 \nu^{7} - 1146 \nu^{6} + 5666 \nu^{5} + \cdots + 768 ) / 372 \) (16v^11 - 12v^10 + 299v^9 - 201v^8 + 1993v^7 - 1146v^6 + 5666v^5 - 2436v^4 + 6321v^3 - 951v^2 + 1549*v + 768) / 372
\(\beta_{9}\)	\(=\)	\( ( 25\nu^{11} + 473\nu^{9} + 3271\nu^{7} + 10097\nu^{5} + 13521\nu^{3} + 6367\nu ) / 372 \) (25v^11 + 473v^9 + 3271v^7 + 10097v^5 + 13521v^3 + 6367v) / 372
\(\beta_{10}\)	\(=\)	\( ( - 20 \nu^{11} - 15 \nu^{10} - 397 \nu^{9} - 228 \nu^{8} - 2933 \nu^{7} - 1107 \nu^{6} - 9826 \nu^{5} + \cdots - 249 ) / 372 \) (-20v^11 - 15v^10 - 397v^9 - 228v^8 - 2933v^7 - 1107v^6 - 9826v^5 - 1743v^4 - 13923v^3 - 282v^2 - 4889*v - 249) / 372
\(\beta_{11}\)	\(=\)	\( ( - 20 \nu^{11} + 15 \nu^{10} - 397 \nu^{9} + 228 \nu^{8} - 2933 \nu^{7} + 1107 \nu^{6} - 9826 \nu^{5} + \cdots + 249 ) / 372 \) (-20v^11 + 15v^10 - 397v^9 + 228v^8 - 2933v^7 + 1107v^6 - 9826v^5 + 1743v^4 - 13923v^3 + 282v^2 - 4889*v + 249) / 372

\(\nu\)	\(=\)	\( ( \beta_{9} + \beta_{6} - \beta_{5} ) / 2 \) (b9 + b6 - b5) / 2
\(\nu^{2}\)	\(=\)	\( ( \beta_{11} - \beta_{10} + \beta_{8} + \beta_{7} + \beta_{3} - \beta _1 - 7 ) / 2 \) (b11 - b10 + b8 + b7 + b3 - b1 - 7) / 2
\(\nu^{3}\)	\(=\)	\( ( \beta_{11} + \beta_{10} - 4\beta_{9} - \beta_{8} + \beta_{7} - 6\beta_{6} + 8\beta_{5} ) / 2 \) (b11 + b10 - 4b9 - b8 + b7 - 6b6 + 8*b5) / 2
\(\nu^{4}\)	\(=\)	\( ( -7\beta_{11} + 7\beta_{10} - 5\beta_{8} - 5\beta_{7} - 7\beta_{3} + 9\beta _1 + 37 ) / 2 \) (-7b11 + 7b10 - 5b8 - 5b7 - 7b3 + 9b1 + 37) / 2
\(\nu^{5}\)	\(=\)	\( ( -7\beta_{11} - 7\beta_{10} + 21\beta_{9} + 11\beta_{8} - 11\beta_{7} + 35\beta_{6} - 61\beta_{5} + 2\beta_{4} ) / 2 \) (-7b11 - 7b10 + 21b9 + 11b8 - 11b7 + 35b6 - 61b5 + 2b4) / 2
\(\nu^{6}\)	\(=\)	\( 24\beta_{11} - 24\beta_{10} + 12\beta_{8} + 12\beta_{7} + 23\beta_{3} + 2\beta_{2} - 33\beta _1 - 108 \) 24b11 - 24b10 + 12b8 + 12b7 + 23b3 + 2b2 - 33*b1 - 108
\(\nu^{7}\)	\(=\)	\( ( 40 \beta_{11} + 40 \beta_{10} - 121 \beta_{9} - 96 \beta_{8} + 96 \beta_{7} - 213 \beta_{6} + \cdots - 24 \beta_{4} ) / 2 \) (40b11 + 40b10 - 121b9 - 96b8 + 96b7 - 213b6 + 433b5 - 24b4) / 2
\(\nu^{8}\)	\(=\)	\( ( -335\beta_{11} + 335\beta_{10} - 115\beta_{8} - 115\beta_{7} - 291\beta_{3} - 56\beta_{2} + 459\beta _1 + 1329 ) / 2 \) (-335b11 + 335b10 - 115b8 - 115b7 - 291b3 - 56b2 + 459*b1 + 1329) / 2
\(\nu^{9}\)	\(=\)	\( ( - 209 \beta_{11} - 209 \beta_{10} + 730 \beta_{9} + 765 \beta_{8} - 765 \beta_{7} + \cdots + 220 \beta_{4} ) / 2 \) (-209b11 - 209b10 + 730b9 + 765b8 - 765b7 + 1344b6 - 2974b5 + 220b4) / 2
\(\nu^{10}\)	\(=\)	\( ( 2369 \beta_{11} - 2369 \beta_{10} + 539 \beta_{8} + 539 \beta_{7} + 1823 \beta_{3} + 556 \beta_{2} + \cdots - 8461 ) / 2 \) (2369b11 - 2369b10 + 539b8 + 539b7 + 1823b3 + 556b2 - 3133*b1 - 8461) / 2
\(\nu^{11}\)	\(=\)	\( ( 1007 \beta_{11} + 1007 \beta_{10} - 4523 \beta_{9} - 5815 \beta_{8} + 5815 \beta_{7} + \cdots - 1830 \beta_{4} ) / 2 \) (1007b11 + 1007b10 - 4523b9 - 5815b8 + 5815b7 - 8705b6 + 20179b5 - 1830b4) / 2

\(n\)	\(897\)	\(1471\)	\(1541\)	\(1921\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 449.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( (T^{6} + 13 T^{4} + 40 T^{2} + 4)^{2} \) (T^6 + 13T^4 + 40T^2 + 4)^2
$5$	\( (T^{6} + T^{4} + 16 T^{3} + \cdots + 125)^{2} \) (T^6 + T^4 + 16T^3 + 5T^2 + 125)^2
$7$	\( (T^{2} + 1)^{6} \) (T^2 + 1)^6
$11$	\( (T^{6} - 45 T^{4} + \cdots - 1856)^{2} \) (T^6 - 45T^4 + 592T^2 - 1856)^2
$13$	\( (T^{6} + 65 T^{4} + \cdots + 116)^{2} \) (T^6 + 65T^4 + 1056T^2 + 116)^2
$17$	\( (T^{6} + 109 T^{4} + \cdots + 37584)^{2} \) (T^6 + 109T^4 + 3608T^2 + 37584)^2
$19$	\( (T^{6} - 104 T^{4} + \cdots - 4176)^{2} \) (T^6 - 104T^4 + 2708T^2 - 4176)^2
$23$	\( (T^{6} + 56 T^{4} + \cdots + 1024)^{2} \) (T^6 + 56T^4 + 656T^2 + 1024)^2
$29$	\( (T^{3} - 11 T^{2} + \cdots + 12)^{4} \) (T^3 - 11T^2 + 16T + 12)^4
$31$	\( (T^{6} - 128 T^{4} + \cdots - 29696)^{2} \) (T^6 - 128T^4 + 4416T^2 - 29696)^2
$37$	\( (T^{6} + 168 T^{4} + \cdots + 118784)^{2} \) (T^6 + 168T^4 + 8080T^2 + 118784)^2
$41$	\( (T^{3} - 8 T^{2} - 4 T + 48)^{4} \) (T^3 - 8T^2 - 4T + 48)^4
$43$	\( (T^{6} + 116 T^{4} + \cdots + 256)^{2} \) (T^6 + 116T^4 + 2048T^2 + 256)^2
$47$	\( (T^{6} + 257 T^{4} + \cdots + 602176)^{2} \) (T^6 + 257T^4 + 21680T^2 + 602176)^2
$53$	\( (T^{6} + 204 T^{4} + \cdots + 1856)^{2} \) (T^6 + 204T^4 + 8176T^2 + 1856)^2
$59$	\( (T^{6} - 104 T^{4} + \cdots - 4176)^{2} \) (T^6 - 104T^4 + 2708T^2 - 4176)^2
$61$	\( (T^{3} + 10 T^{2} + \cdots - 124)^{4} \) (T^3 + 10T^2 + 2T - 124)^4
$67$	\( (T^{6} + 196 T^{4} + \cdots + 65536)^{2} \) (T^6 + 196T^4 + 7168T^2 + 65536)^2
$71$	\( (T^{6} - 104 T^{4} + \cdots - 7424)^{2} \) (T^6 - 104T^4 + 1680T^2 - 7424)^2
$73$	\( (T^{6} + 204 T^{4} + \cdots + 1856)^{2} \) (T^6 + 204T^4 + 8176T^2 + 1856)^2
$79$	\( (T^{6} - 149 T^{4} + \cdots - 464)^{2} \) (T^6 - 149T^4 + 648T^2 - 464)^2
$83$	\( (T^{6} + 372 T^{4} + \cdots + 419904)^{2} \) (T^6 + 372T^4 + 34084T^2 + 419904)^2
$89$	\( (T + 6)^{12} \) (T + 6)^12
$97$	\( (T^{6} + 189 T^{4} + \cdots + 464)^{2} \) (T^6 + 189T^4 + 664T^2 + 464)^2
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