LMFDB - Newform orbit 2240.2.e.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(2239,2240)]

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([1, 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("2240.2239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.e (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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6x^{14} + 28x^{12} + 16x^{10} - 40x^{8} + 610x^{6} + 1625x^{4} - 524x^{2} + 1444 $ x^16 - 6x^14 + 28x^12 + 16x^10 - 40x^8 + 610x^6 + 1625x^4 - 524*x^2 + 1444
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 140)
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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{7} q^{3} - \beta_{10} q^{5} + (\beta_{8} - \beta_{6}) q^{7} + ( - \beta_{5} + 1) q^{9}+O(q^{10}) $ q - b7 * q^3 - b10 * q^5 + (b8 - b6) * q^7 + (-b5 + 1) * q^9	$ q - \beta_{7} q^{3} - \beta_{10} q^{5} + (\beta_{8} - \beta_{6}) q^{7} + ( - \beta_{5} + 1) q^{9} - \beta_{11} q^{11} + \beta_{14} q^{13} + ( - \beta_{6} - \beta_1) q^{15} + (\beta_{14} - \beta_{10} + \beta_{2}) q^{17} + \beta_{15} q^{19} + (\beta_{10} + \beta_{9} + \cdots + \beta_{2}) q^{21}+ \cdots + ( - 2 \beta_{11} + 2 \beta_1) q^{99}+O(q^{100}) $ q - b7 * q^3 - b10 * q^5 + (b8 - b6) * q^7 + (-b5 + 1) * q^9 - b11 * q^11 + b14 * q^13 + (-b6 - b1) * q^15 + (b14 - b10 + b2) * q^17 + b15 * q^19 + (b10 + b9 + b5 + b2) * q^21 + (2b6 - b4) q^23 + (-b13 + b5) * q^25 + (b8 - 2b7) q^27 + (-b5 + 2) * q^29 + (b15 - b8 + b7 - 2b3) q^31 + (-b14 + b10 - b2) * q^33 + (b11 - b8 + 2b7 - b4 - b3) q^35 + b12 * q^37 - b11 * q^39 + 3b9 q^41 - b4 * q^43 + (-b14 + b9 - b2) * q^45 + (-b8 - 4b7) q^47 + (-2b9 + 2b5 - 1) * q^49 + (-b11 - 2b1) q^51 - 2b13 q^53 + (-b15 - 2b8 - b7) q^55 + b12 * q^57 + (-2b15 - b8 + b7 - 2b3) * q^59 + (b10 - b9 + b2) * q^61 + (2b8 - 2b7 + b6 - b4) * q^63 + (b13 + b12 + 2b5 - 3) q^65 - 2b6 q^67 + (-2b10 - 3b9 - 2b2) q^69 + (-2b11 + 4b1) * q^71 + (2b14 - 2b10 + 2b2) q^73 + (-2b8 - b7 - 2b3) * q^75 + (-b14 - b12 - b10 + b2) * q^77 + (-b11 - 2b1) q^79 + (-4b5 - 1) q^81 + (5b8 - b7) q^83 + (b12 + 3b5 + 2) q^85 + b8 * q^87 - b9 * q^89 + (-b15 - 3b11 + 2b1) * q^91 + (2b13 - b12) q^93 + (-3b11 + b6 - 2b4 + b1) * q^95 + (-3b14 + 3b10 - 3b2) q^97 + (-2b11 + 2b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{9}+O(q^{10}) $ 16 * q + 8 * q^9	$ 16 q + 8 q^{9} + 8 q^{21} + 8 q^{25} + 24 q^{29} - 32 q^{65} - 48 q^{81} + 56 q^{85}+O(q^{100}) $ 16 * q + 8 * q^9 + 8 * q^21 + 8 * q^25 + 24 * q^29 - 32 * q^65 - 48 * q^81 + 56 * q^85

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 6x^{14} + 28x^{12} + 16x^{10} - 40x^{8} + 610x^{6} + 1625x^{4} - 524x^{2} + 1444 $ x^16 - 6*x^14 + 28*x^12 + 16*x^10 - 40*x^8 + 610*x^6 + 1625*x^4 - 524*x^2 + 1444 :

$\beta_{1}$	$=$	$ ( - 84844 \nu^{14} - 2074564 \nu^{12} + 10863016 \nu^{10} - 108971684 \nu^{8} + 259974236 \nu^{6} + \cdots - 140017061 ) / 2498768831 $ (-84844v^14 - 2074564v^12 + 10863016v^10 - 108971684v^8 + 259974236v^6 - 1220553906v^4 - 1954893928*v^2 - 140017061) / 2498768831
$\beta_{2}$	$=$	$ ( 2967085 \nu^{15} + 39930036 \nu^{13} - 191415108 \nu^{11} + 1131088376 \nu^{9} + \cdots + 81935469158 \nu ) / 39980301296 $ (2967085v^15 + 39930036v^13 - 191415108v^11 + 1131088376v^9 + 3306399632v^7 - 3062111390v^5 + 38410007337v^3 + 81935469158v) / 39980301296
$\beta_{3}$	$=$	$ ( 4207655 \nu^{15} - 106481348 \nu^{13} + 576987844 \nu^{11} - 2071902984 \nu^{9} + \cdots - 93332422574 \nu ) / 39980301296 $ (4207655v^15 - 106481348v^13 + 576987844v^11 - 2071902984v^9 - 24668656v^7 - 5606760042v^5 - 12446679765v^3 - 93332422574v) / 39980301296
$\beta_{4}$	$=$	$ ( 274551 \nu^{14} - 2939696 \nu^{12} + 13430352 \nu^{10} - 23555944 \nu^{8} - 84357892 \nu^{6} + \cdots - 3209825990 ) / 768851948 $ (274551v^14 - 2939696v^12 + 13430352v^10 - 23555944v^8 - 84357892v^6 + 137240250v^4 - 124066909*v^2 - 3209825990) / 768851948
$\beta_{5}$	$=$	$ ( 4034 \nu^{14} - 44446 \nu^{12} + 225632 \nu^{10} - 360443 \nu^{8} - 805342 \nu^{6} + \cdots - 13312648 ) / 10116473 $ (4034v^14 - 44446v^12 + 225632v^10 - 360443v^8 - 805342v^6 + 3678039v^4 - 2426116*v^2 - 13312648) / 10116473
$\beta_{6}$	$=$	$ ( 778741 \nu^{14} - 5214580 \nu^{12} + 27045620 \nu^{10} - 25125688 \nu^{8} + 6532248 \nu^{6} + \cdots - 1386720738 ) / 1537703896 $ (778741v^14 - 5214580v^12 + 27045620v^10 - 25125688v^8 + 6532248v^6 + 613266210v^4 - 311886279*v^2 - 1386720738) / 1537703896
$\beta_{7}$	$=$	$ ( - 5290343 \nu^{15} + 30566954 \nu^{13} - 139846998 \nu^{11} - 73573152 \nu^{9} + \cdots - 4522635990 \nu ) / 9995075324 $ (-5290343v^15 + 30566954v^13 - 139846998v^11 - 73573152v^9 - 8711350v^7 - 2721018508v^5 - 5600946403v^3 - 4522635990v) / 9995075324
$\beta_{8}$	$=$	$ ( 4087383 \nu^{15} - 18581871 \nu^{13} + 75636915 \nu^{11} + 248960394 \nu^{9} + \cdots + 6038703550 \nu ) / 4997537662 $ (4087383v^15 - 18581871v^13 + 75636915v^11 + 248960394v^9 - 132623279v^7 + 2173230995v^5 + 11275740801v^3 + 6038703550v) / 4997537662
$\beta_{9}$	$=$	$ ( - 40619 \nu^{15} + 221960 \nu^{13} - 1062732 \nu^{11} - 1111276 \nu^{9} + 645500 \nu^{7} + \cdots - 53518138 \nu ) / 31933148 $ (-40619v^15 + 221960v^13 - 1062732v^11 - 1111276v^9 + 645500v^7 - 26060814v^5 - 81508947v^3 - 53518138v) / 31933148
$\beta_{10}$	$=$	$ ( 51214505 \nu^{15} - 271267084 \nu^{13} + 1118627340 \nu^{11} + 2436041896 \nu^{9} + \cdots - 8462940866 \nu ) / 39980301296 $ (51214505v^15 - 271267084v^13 + 1118627340v^11 + 2436041896v^9 - 4344262192v^7 + 28747923386v^5 + 101267919077v^3 - 8462940866v) / 39980301296
$\beta_{11}$	$=$	$ ( 4924818 \nu^{14} - 28350288 \nu^{12} + 130546916 \nu^{10} + 92658541 \nu^{8} - 56159602 \nu^{6} + \cdots - 774884638 ) / 2498768831 $ (4924818v^14 - 28350288v^12 + 130546916v^10 + 92658541v^8 - 56159602v^6 + 2170744901v^4 + 8291019412*v^2 - 774884638) / 2498768831
$\beta_{12}$	$=$	$ ( 1160909 \nu^{14} - 5994576 \nu^{12} + 20915688 \nu^{10} + 74001744 \nu^{8} - 149350828 \nu^{6} + \cdots - 198617650 ) / 526056596 $ (1160909v^14 - 5994576v^12 + 20915688v^10 + 74001744v^8 - 149350828v^6 + 628515030v^4 + 2016951625*v^2 - 198617650) / 526056596
$\beta_{13}$	$=$	$ ( 2650265 \nu^{14} - 16104476 \nu^{12} + 78559372 \nu^{10} + 16571240 \nu^{8} + 12504832 \nu^{6} + \cdots - 371424458 ) / 1052113192 $ (2650265v^14 - 16104476v^12 + 78559372v^10 + 16571240v^8 + 12504832v^6 + 1528314658v^4 + 5299245757*v^2 - 371424458) / 1052113192
$\beta_{14}$	$=$	$ ( 1659481 \nu^{15} - 10635114 \nu^{13} + 49775130 \nu^{11} + 8859636 \nu^{9} + \cdots - 3111082718 \nu ) / 768851948 $ (1659481v^15 - 10635114v^13 + 49775130v^11 + 8859636v^9 - 99229442v^7 + 974013580v^5 + 2269259769v^3 - 3111082718v) / 768851948
$\beta_{15}$	$=$	$ ( - 3799279 \nu^{15} + 24179876 \nu^{13} - 110686692 \nu^{11} - 54427560 \nu^{9} + \cdots + 7322587014 \nu ) / 1537703896 $ (-3799279v^15 + 24179876v^13 - 110686692v^11 - 54427560v^9 + 324888696v^7 - 2577780150v^5 - 5301496115v^3 + 7322587014v) / 1537703896

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

$\nu$	$=$	$ ( \beta_{15} - \beta_{9} + \beta_{8} - \beta_{7} - 2\beta_{2} ) / 4 $ (b15 - b9 + b8 - b7 - 2*b2) / 4
$\nu^{2}$	$=$	$ ( \beta_{13} - \beta_{11} - \beta_{6} - \beta_{5} + \beta_{4} + 2 ) / 2 $ (b13 - b11 - b6 - b5 + b4 + 2) / 2
$\nu^{3}$	$=$	$ ( \beta_{15} + 2\beta_{14} - 6\beta_{10} + \beta_{9} + 11\beta_{8} + 3\beta_{7} - 4\beta_{2} ) / 4 $ (b15 + 2b14 - 6b10 + b9 + 11b8 + 3b7 - 4*b2) / 4
$\nu^{4}$	$=$	$ ( 4\beta_{13} + 2\beta_{12} - 8\beta_{11} + 4\beta_{6} - 2\beta_{4} + \beta _1 - 5 ) / 2 $ (4b13 + 2b12 - 8b11 + 4b6 - 2*b4 + b1 - 5) / 2
$\nu^{5}$	$=$	$ ( -7\beta_{15} - 4\beta_{14} - 16\beta_{10} + 4\beta_{9} + 17\beta_{8} - 7\beta_{7} - 3\beta_{3} - 5\beta_{2} ) / 2 $ (-7b15 - 4b14 - 16b10 + 4b9 + 17b8 - 7b7 - 3b3 - 5b2) / 2
$\nu^{6}$	$=$	$ ( 9\beta_{13} + 7\beta_{12} - 18\beta_{11} + 3\beta_{6} + 16\beta_{5} - 28\beta_{4} + 16\beta _1 - 92 ) / 2 $ (9b13 + 7b12 - 18b11 + 3b6 + 16b5 - 28b4 + 16*b1 - 92) / 2
$\nu^{7}$	$=$	$ ( -99\beta_{15} - 92\beta_{14} - 56\beta_{10} + 39\beta_{9} + 25\beta_{8} - 85\beta_{7} + 36\beta_{3} + 82\beta_{2} ) / 4 $ (-99b15 - 92b14 - 56b10 + 39b9 + 25b8 - 85b7 + 36b3 + 82b2) / 4
$\nu^{8}$	$=$	$ ( -48\beta_{13} + 72\beta_{11} - 64\beta_{6} + 150\beta_{5} - 136\beta_{4} - 5\beta _1 - 423 ) / 2 $ (-48b13 + 72b11 - 64b6 + 150b5 - 136b4 - 5b1 - 423) / 2
$\nu^{9}$	$=$	$ ( - 341 \beta_{15} - 462 \beta_{14} + 448 \beta_{10} - 145 \beta_{9} - 757 \beta_{8} + \cdots + 534 \beta_{2} ) / 4 $ (-341b15 - 462b14 + 448b10 - 145b9 - 757b8 + 91b7 + 228b3 + 534b2) / 4
$\nu^{10}$	$=$	$ ( -465\beta_{13} - 303\beta_{12} + 957\beta_{11} - 327\beta_{6} + 451\beta_{5} - 210\beta_{4} - 430\beta _1 - 584 ) / 2 $ (-465b13 - 303b12 + 957b11 - 327b6 + 451b5 - 210b4 - 430*b1 - 584) / 2
$\nu^{11}$	$=$	$ ( 152 \beta_{15} - 50 \beta_{14} + 1677 \beta_{10} - 1081 \beta_{9} - 2875 \beta_{8} + \cdots + 1096 \beta_{2} ) / 2 $ (152b15 - 50b14 + 1677b10 - 1081b9 - 2875b8 + 1495b7 + 205b3 + 1096b2) / 2
$\nu^{12}$	$=$	$ ( - 2012 \beta_{13} - 1766 \beta_{12} + 4427 \beta_{11} + 172 \beta_{6} - 1653 \beta_{5} + 2010 \beta_{4} + \cdots + 6211 ) / 2 $ (-2012b13 - 1766b12 + 4427b11 + 172b6 - 1653b5 + 2010b4 - 2785*b1 + 6211) / 2
$\nu^{13}$	$=$	$ ( 10541 \beta_{15} + 11792 \beta_{14} + 11678 \beta_{10} - 8293 \beta_{9} - 19633 \beta_{8} + \cdots + 2676 \beta_{2} ) / 4 $ (10541b15 + 11792b14 + 11678b10 - 8293b9 - 19633b8 + 16121b7 - 4980b3 + 2676b2) / 4
$\nu^{14}$	$=$	$ ( - 1697 \beta_{13} - 2936 \beta_{12} + 4781 \beta_{11} + 10817 \beta_{6} - 22427 \beta_{5} + \cdots + 57296 ) / 2 $ (-1697b13 - 2936b12 + 4781b11 + 10817b6 - 22427b5 + 18575b4 - 4798*b1 + 57296) / 2
$\nu^{15}$	$=$	$ ( 63061 \beta_{15} + 79350 \beta_{14} - 4514 \beta_{10} + 6869 \beta_{9} + 19059 \beta_{8} + \cdots - 38956 \beta_{2} ) / 4 $ (63061b15 + 79350b14 - 4514b10 + 6869b9 + 19059b8 + 10303b7 - 39756b3 - 38956b2) / 4

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} $

2239.1

0.328458	−	1.49331i
1.61596	+	1.02509i
−0.328458	−	1.49331i
−1.61596	+	1.02509i
2.05580	−	0.953651i
−0.744612	−	0.556573i
−2.05580	−	0.953651i
0.744612	−	0.556573i
−0.744612	+	0.556573i
2.05580	+	0.953651i
0.744612	+	0.556573i
−2.05580	+	0.953651i
1.61596	−	1.02509i
0.328458	+	1.49331i
−1.61596	−	1.02509i
−0.328458	+	1.49331i

−

2.13578i

−1.94442

−

1.10418i

−2.35829

1.19935i

−1.56155

2239.2

−

2.13578i

−1.94442

1.10418i

2.35829

1.19935i

−1.56155

2239.3

−

2.13578i

1.94442

−

1.10418i

−2.35829

1.19935i

−1.56155

2239.4

−

2.13578i

1.94442

1.10418i

2.35829

1.19935i

−1.56155

2239.5

−

0.662153i

−1.31119

−

1.81129i

−1.19935

−

2.35829i

2.56155

2239.6

−

0.662153i

−1.31119

1.81129i

1.19935

−

2.35829i

2.56155

2239.7

−

0.662153i

1.31119

−

1.81129i

−1.19935

−

2.35829i

2.56155

2239.8

−

0.662153i

1.31119

1.81129i

1.19935

−

2.35829i

2.56155

2239.9

0.662153i

−1.31119

−

1.81129i

1.19935

2.35829i

2.56155

2239.10

0.662153i

−1.31119

1.81129i

−1.19935

2.35829i

2.56155

2239.11

0.662153i

1.31119

−

1.81129i

1.19935

2.35829i

2.56155

2239.12

0.662153i

1.31119

1.81129i

−1.19935

2.35829i

2.56155

2239.13

2.13578i

−1.94442

−

1.10418i

2.35829

−

1.19935i

−1.56155

2239.14

2.13578i

−1.94442

1.10418i

−2.35829

−

1.19935i

−1.56155

2239.15

2.13578i

1.94442

−

1.10418i

2.35829

−

1.19935i

−1.56155

2239.16

2.13578i

1.94442

1.10418i

−2.35829

−

1.19935i

−1.56155

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
7.b	odd	2	1	inner
20.d	odd	2	1	inner
28.d	even	2	1	inner
35.c	odd	2	1	inner
140.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2240.2.e.f		16
4.b	odd	2	1	inner	2240.2.e.f		16
5.b	even	2	1	inner	2240.2.e.f		16
7.b	odd	2	1	inner	2240.2.e.f		16
8.b	even	2	1		140.2.c.b	✓	16
8.d	odd	2	1		140.2.c.b	✓	16
20.d	odd	2	1	inner	2240.2.e.f		16
28.d	even	2	1	inner	2240.2.e.f		16
35.c	odd	2	1	inner	2240.2.e.f		16
40.e	odd	2	1		140.2.c.b	✓	16
40.f	even	2	1		140.2.c.b	✓	16
40.i	odd	4	2		700.2.g.l		16
40.k	even	4	2		700.2.g.l		16
56.e	even	2	1		140.2.c.b	✓	16
56.h	odd	2	1		140.2.c.b	✓	16
56.j	odd	6	2		980.2.s.f		32
56.k	odd	6	2		980.2.s.f		32
56.m	even	6	2		980.2.s.f		32
56.p	even	6	2		980.2.s.f		32
140.c	even	2	1	inner	2240.2.e.f		16
280.c	odd	2	1		140.2.c.b	✓	16
280.n	even	2	1		140.2.c.b	✓	16
280.s	even	4	2		700.2.g.l		16
280.y	odd	4	2		700.2.g.l		16
280.ba	even	6	2		980.2.s.f		32
280.bf	even	6	2		980.2.s.f		32
280.bi	odd	6	2		980.2.s.f		32
280.bk	odd	6	2		980.2.s.f		32

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
140.2.c.b	✓	16	8.b	even	2	1
140.2.c.b	✓	16	8.d	odd	2	1
140.2.c.b	✓	16	40.e	odd	2	1
140.2.c.b	✓	16	40.f	even	2	1
140.2.c.b	✓	16	56.e	even	2	1
140.2.c.b	✓	16	56.h	odd	2	1
140.2.c.b	✓	16	280.c	odd	2	1
140.2.c.b	✓	16	280.n	even	2	1
700.2.g.l		16	40.i	odd	4	2
700.2.g.l		16	40.k	even	4	2
700.2.g.l		16	280.s	even	4	2
700.2.g.l		16	280.y	odd	4	2
980.2.s.f		32	56.j	odd	6	2
980.2.s.f		32	56.k	odd	6	2
980.2.s.f		32	56.m	even	6	2
980.2.s.f		32	56.p	even	6	2
980.2.s.f		32	280.ba	even	6	2
980.2.s.f		32	280.bf	even	6	2
980.2.s.f		32	280.bi	odd	6	2
980.2.s.f		32	280.bk	odd	6	2
2240.2.e.f		16	1.a	even	1	1	trivial
2240.2.e.f		16	4.b	odd	2	1	inner
2240.2.e.f		16	5.b	even	2	1	inner
2240.2.e.f		16	7.b	odd	2	1	inner
2240.2.e.f		16	20.d	odd	2	1	inner
2240.2.e.f		16	28.d	even	2	1	inner
2240.2.e.f		16	35.c	odd	2	1	inner
2240.2.e.f		16	140.c	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 $S_{2}^{\mathrm{new}}(2240, [\chi])$:

$ T_{3}^{4} + 5T_{3}^{2} + 2 $

$ T_{11}^{4} + 15T_{11}^{2} + 52 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{4} + 5 T^{2} + 2)^{4} $ (T^4 + 5*T^2 + 2)^4
$5$	$ (T^{8} - 2 T^{6} + \cdots + 625)^{2} $ (T^8 - 2T^6 + 34T^4 - 50*T^2 + 625)^2
$7$	$ (T^{8} + 30 T^{4} + 2401)^{2} $ (T^8 + 30*T^4 + 2401)^2
$11$	$ (T^{4} + 15 T^{2} + 52)^{4} $ (T^4 + 15*T^2 + 52)^4
$13$	$ (T^{4} - 23 T^{2} + 26)^{4} $ (T^4 - 23*T^2 + 26)^4
$17$	$ (T^{4} - 29 T^{2} + 104)^{4} $ (T^4 - 29*T^2 + 104)^4
$19$	$ (T^{4} - 38 T^{2} + 208)^{4} $ (T^4 - 38*T^2 + 208)^4
$23$	$ (T^{4} - 40 T^{2} + 128)^{4} $ (T^4 - 40*T^2 + 128)^4
$29$	$ (T^{2} - 3 T - 2)^{8} $ (T^2 - 3*T - 2)^8
$31$	$ (T^{4} - 120 T^{2} + 3328)^{4} $ (T^4 - 120*T^2 + 3328)^4
$37$	$ (T^{4} + 44 T^{2} + 416)^{4} $ (T^4 + 44*T^2 + 416)^4
$41$	$ (T^{2} + 72)^{8} $ (T^2 + 72)^8
$43$	$ (T^{4} - 20 T^{2} + 32)^{4} $ (T^4 - 20*T^2 + 32)^4
$47$	$ (T^{4} + 95 T^{2} + 8)^{4} $ (T^4 + 95*T^2 + 8)^4
$53$	$ (T^{4} + 164 T^{2} + 6656)^{4} $ (T^4 + 164*T^2 + 6656)^4
$59$	$ (T^{4} - 270 T^{2} + 13312)^{4} $ (T^4 - 270*T^2 + 13312)^4
$61$	$ (T^{4} + 42 T^{2} + 16)^{4} $ (T^4 + 42*T^2 + 16)^4
$67$	$ (T^{4} - 28 T^{2} + 128)^{4} $ (T^4 - 28*T^2 + 128)^4
$71$	$ (T^{4} + 236 T^{2} + 13312)^{4} $ (T^4 + 236*T^2 + 13312)^4
$73$	$ (T^{4} - 116 T^{2} + 1664)^{4} $ (T^4 - 116*T^2 + 1664)^4
$79$	$ (T^{4} + 115 T^{2} + 208)^{4} $ (T^4 + 115*T^2 + 208)^4
$83$	$ (T^{4} + 170 T^{2} + 2312)^{4} $ (T^4 + 170*T^2 + 2312)^4
$89$	$ (T^{2} + 8)^{8} $ (T^2 + 8)^8
$97$	$ (T^{4} - 261 T^{2} + 8424)^{4} $ (T^4 - 261*T^2 + 8424)^4
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
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.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{7} q^{3} - \beta_{10} q^{5} + (\beta_{8} - \beta_{6}) q^{7} + ( - \beta_{5} + 1) q^{9}+O(q^{10}) \) q - b7 * q^3 - b10 * q^5 + (b8 - b6) * q^7 + (-b5 + 1) * q^9	\( q - \beta_{7} q^{3} - \beta_{10} q^{5} + (\beta_{8} - \beta_{6}) q^{7} + ( - \beta_{5} + 1) q^{9} - \beta_{11} q^{11} + \beta_{14} q^{13} + ( - \beta_{6} - \beta_1) q^{15} + (\beta_{14} - \beta_{10} + \beta_{2}) q^{17} + \beta_{15} q^{19} + (\beta_{10} + \beta_{9} + \cdots + \beta_{2}) q^{21}+ \cdots + ( - 2 \beta_{11} + 2 \beta_1) q^{99}+O(q^{100}) \) q - b7 * q^3 - b10 * q^5 + (b8 - b6) * q^7 + (-b5 + 1) * q^9 - b11 * q^11 + b14 * q^13 + (-b6 - b1) * q^15 + (b14 - b10 + b2) * q^17 + b15 * q^19 + (b10 + b9 + b5 + b2) * q^21 + (2b6 - b4) q^23 + (-b13 + b5) * q^25 + (b8 - 2b7) q^27 + (-b5 + 2) * q^29 + (b15 - b8 + b7 - 2b3) q^31 + (-b14 + b10 - b2) * q^33 + (b11 - b8 + 2b7 - b4 - b3) q^35 + b12 * q^37 - b11 * q^39 + 3b9 q^41 - b4 * q^43 + (-b14 + b9 - b2) * q^45 + (-b8 - 4b7) q^47 + (-2b9 + 2b5 - 1) * q^49 + (-b11 - 2b1) q^51 - 2b13 q^53 + (-b15 - 2b8 - b7) q^55 + b12 * q^57 + (-2b15 - b8 + b7 - 2b3) * q^59 + (b10 - b9 + b2) * q^61 + (2b8 - 2b7 + b6 - b4) * q^63 + (b13 + b12 + 2b5 - 3) q^65 - 2b6 q^67 + (-2b10 - 3b9 - 2b2) q^69 + (-2b11 + 4b1) * q^71 + (2b14 - 2b10 + 2b2) q^73 + (-2b8 - b7 - 2b3) * q^75 + (-b14 - b12 - b10 + b2) * q^77 + (-b11 - 2b1) q^79 + (-4b5 - 1) q^81 + (5b8 - b7) q^83 + (b12 + 3b5 + 2) q^85 + b8 * q^87 - b9 * q^89 + (-b15 - 3b11 + 2b1) * q^91 + (2b13 - b12) q^93 + (-3b11 + b6 - 2b4 + b1) * q^95 + (-3b14 + 3b10 - 3b2) q^97 + (-2b11 + 2b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{9}+O(q^{10}) \) 16 * q + 8 * q^9	\( 16 q + 8 q^{9} + 8 q^{21} + 8 q^{25} + 24 q^{29} - 32 q^{65} - 48 q^{81} + 56 q^{85}+O(q^{100}) \) 16 * q + 8 * q^9 + 8 * q^21 + 8 * q^25 + 24 * q^29 - 32 * q^65 - 48 * q^81 + 56 * q^85

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

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.e.f

Level	$2240$
Weight	$2$
Character orbit	2240.e
Analytic conductor	$17.886$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(17.8864900528\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 6x^{14} + 28x^{12} + 16x^{10} - 40x^{8} + 610x^{6} + 1625x^{4} - 524x^{2} + 1444 \) x^16 - 6x^14 + 28x^12 + 16x^10 - 40x^8 + 610x^6 + 1625x^4 - 524*x^2 + 1444
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{16} \)
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 84844 \nu^{14} - 2074564 \nu^{12} + 10863016 \nu^{10} - 108971684 \nu^{8} + 259974236 \nu^{6} + \cdots - 140017061 ) / 2498768831 \) (-84844v^14 - 2074564v^12 + 10863016v^10 - 108971684v^8 + 259974236v^6 - 1220553906v^4 - 1954893928*v^2 - 140017061) / 2498768831
\(\beta_{2}\)	\(=\)	\( ( 2967085 \nu^{15} + 39930036 \nu^{13} - 191415108 \nu^{11} + 1131088376 \nu^{9} + \cdots + 81935469158 \nu ) / 39980301296 \) (2967085v^15 + 39930036v^13 - 191415108v^11 + 1131088376v^9 + 3306399632v^7 - 3062111390v^5 + 38410007337v^3 + 81935469158v) / 39980301296
\(\beta_{3}\)	\(=\)	\( ( 4207655 \nu^{15} - 106481348 \nu^{13} + 576987844 \nu^{11} - 2071902984 \nu^{9} + \cdots - 93332422574 \nu ) / 39980301296 \) (4207655v^15 - 106481348v^13 + 576987844v^11 - 2071902984v^9 - 24668656v^7 - 5606760042v^5 - 12446679765v^3 - 93332422574v) / 39980301296
\(\beta_{4}\)	\(=\)	\( ( 274551 \nu^{14} - 2939696 \nu^{12} + 13430352 \nu^{10} - 23555944 \nu^{8} - 84357892 \nu^{6} + \cdots - 3209825990 ) / 768851948 \) (274551v^14 - 2939696v^12 + 13430352v^10 - 23555944v^8 - 84357892v^6 + 137240250v^4 - 124066909*v^2 - 3209825990) / 768851948
\(\beta_{5}\)	\(=\)	\( ( 4034 \nu^{14} - 44446 \nu^{12} + 225632 \nu^{10} - 360443 \nu^{8} - 805342 \nu^{6} + \cdots - 13312648 ) / 10116473 \) (4034v^14 - 44446v^12 + 225632v^10 - 360443v^8 - 805342v^6 + 3678039v^4 - 2426116*v^2 - 13312648) / 10116473
\(\beta_{6}\)	\(=\)	\( ( 778741 \nu^{14} - 5214580 \nu^{12} + 27045620 \nu^{10} - 25125688 \nu^{8} + 6532248 \nu^{6} + \cdots - 1386720738 ) / 1537703896 \) (778741v^14 - 5214580v^12 + 27045620v^10 - 25125688v^8 + 6532248v^6 + 613266210v^4 - 311886279*v^2 - 1386720738) / 1537703896
\(\beta_{7}\)	\(=\)	\( ( - 5290343 \nu^{15} + 30566954 \nu^{13} - 139846998 \nu^{11} - 73573152 \nu^{9} + \cdots - 4522635990 \nu ) / 9995075324 \) (-5290343v^15 + 30566954v^13 - 139846998v^11 - 73573152v^9 - 8711350v^7 - 2721018508v^5 - 5600946403v^3 - 4522635990v) / 9995075324
\(\beta_{8}\)	\(=\)	\( ( 4087383 \nu^{15} - 18581871 \nu^{13} + 75636915 \nu^{11} + 248960394 \nu^{9} + \cdots + 6038703550 \nu ) / 4997537662 \) (4087383v^15 - 18581871v^13 + 75636915v^11 + 248960394v^9 - 132623279v^7 + 2173230995v^5 + 11275740801v^3 + 6038703550v) / 4997537662
\(\beta_{9}\)	\(=\)	\( ( - 40619 \nu^{15} + 221960 \nu^{13} - 1062732 \nu^{11} - 1111276 \nu^{9} + 645500 \nu^{7} + \cdots - 53518138 \nu ) / 31933148 \) (-40619v^15 + 221960v^13 - 1062732v^11 - 1111276v^9 + 645500v^7 - 26060814v^5 - 81508947v^3 - 53518138v) / 31933148
\(\beta_{10}\)	\(=\)	\( ( 51214505 \nu^{15} - 271267084 \nu^{13} + 1118627340 \nu^{11} + 2436041896 \nu^{9} + \cdots - 8462940866 \nu ) / 39980301296 \) (51214505v^15 - 271267084v^13 + 1118627340v^11 + 2436041896v^9 - 4344262192v^7 + 28747923386v^5 + 101267919077v^3 - 8462940866v) / 39980301296
\(\beta_{11}\)	\(=\)	\( ( 4924818 \nu^{14} - 28350288 \nu^{12} + 130546916 \nu^{10} + 92658541 \nu^{8} - 56159602 \nu^{6} + \cdots - 774884638 ) / 2498768831 \) (4924818v^14 - 28350288v^12 + 130546916v^10 + 92658541v^8 - 56159602v^6 + 2170744901v^4 + 8291019412*v^2 - 774884638) / 2498768831
\(\beta_{12}\)	\(=\)	\( ( 1160909 \nu^{14} - 5994576 \nu^{12} + 20915688 \nu^{10} + 74001744 \nu^{8} - 149350828 \nu^{6} + \cdots - 198617650 ) / 526056596 \) (1160909v^14 - 5994576v^12 + 20915688v^10 + 74001744v^8 - 149350828v^6 + 628515030v^4 + 2016951625*v^2 - 198617650) / 526056596
\(\beta_{13}\)	\(=\)	\( ( 2650265 \nu^{14} - 16104476 \nu^{12} + 78559372 \nu^{10} + 16571240 \nu^{8} + 12504832 \nu^{6} + \cdots - 371424458 ) / 1052113192 \) (2650265v^14 - 16104476v^12 + 78559372v^10 + 16571240v^8 + 12504832v^6 + 1528314658v^4 + 5299245757*v^2 - 371424458) / 1052113192
\(\beta_{14}\)	\(=\)	\( ( 1659481 \nu^{15} - 10635114 \nu^{13} + 49775130 \nu^{11} + 8859636 \nu^{9} + \cdots - 3111082718 \nu ) / 768851948 \) (1659481v^15 - 10635114v^13 + 49775130v^11 + 8859636v^9 - 99229442v^7 + 974013580v^5 + 2269259769v^3 - 3111082718v) / 768851948
\(\beta_{15}\)	\(=\)	\( ( - 3799279 \nu^{15} + 24179876 \nu^{13} - 110686692 \nu^{11} - 54427560 \nu^{9} + \cdots + 7322587014 \nu ) / 1537703896 \) (-3799279v^15 + 24179876v^13 - 110686692v^11 - 54427560v^9 + 324888696v^7 - 2577780150v^5 - 5301496115v^3 + 7322587014v) / 1537703896

\(\nu\)	\(=\)	\( ( \beta_{15} - \beta_{9} + \beta_{8} - \beta_{7} - 2\beta_{2} ) / 4 \) (b15 - b9 + b8 - b7 - 2*b2) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{13} - \beta_{11} - \beta_{6} - \beta_{5} + \beta_{4} + 2 ) / 2 \) (b13 - b11 - b6 - b5 + b4 + 2) / 2
\(\nu^{3}\)	\(=\)	\( ( \beta_{15} + 2\beta_{14} - 6\beta_{10} + \beta_{9} + 11\beta_{8} + 3\beta_{7} - 4\beta_{2} ) / 4 \) (b15 + 2b14 - 6b10 + b9 + 11b8 + 3b7 - 4*b2) / 4
\(\nu^{4}\)	\(=\)	\( ( 4\beta_{13} + 2\beta_{12} - 8\beta_{11} + 4\beta_{6} - 2\beta_{4} + \beta _1 - 5 ) / 2 \) (4b13 + 2b12 - 8b11 + 4b6 - 2*b4 + b1 - 5) / 2
\(\nu^{5}\)	\(=\)	\( ( -7\beta_{15} - 4\beta_{14} - 16\beta_{10} + 4\beta_{9} + 17\beta_{8} - 7\beta_{7} - 3\beta_{3} - 5\beta_{2} ) / 2 \) (-7b15 - 4b14 - 16b10 + 4b9 + 17b8 - 7b7 - 3b3 - 5b2) / 2
\(\nu^{6}\)	\(=\)	\( ( 9\beta_{13} + 7\beta_{12} - 18\beta_{11} + 3\beta_{6} + 16\beta_{5} - 28\beta_{4} + 16\beta _1 - 92 ) / 2 \) (9b13 + 7b12 - 18b11 + 3b6 + 16b5 - 28b4 + 16*b1 - 92) / 2
\(\nu^{7}\)	\(=\)	\( ( -99\beta_{15} - 92\beta_{14} - 56\beta_{10} + 39\beta_{9} + 25\beta_{8} - 85\beta_{7} + 36\beta_{3} + 82\beta_{2} ) / 4 \) (-99b15 - 92b14 - 56b10 + 39b9 + 25b8 - 85b7 + 36b3 + 82b2) / 4
\(\nu^{8}\)	\(=\)	\( ( -48\beta_{13} + 72\beta_{11} - 64\beta_{6} + 150\beta_{5} - 136\beta_{4} - 5\beta _1 - 423 ) / 2 \) (-48b13 + 72b11 - 64b6 + 150b5 - 136b4 - 5b1 - 423) / 2
\(\nu^{9}\)	\(=\)	\( ( - 341 \beta_{15} - 462 \beta_{14} + 448 \beta_{10} - 145 \beta_{9} - 757 \beta_{8} + \cdots + 534 \beta_{2} ) / 4 \) (-341b15 - 462b14 + 448b10 - 145b9 - 757b8 + 91b7 + 228b3 + 534b2) / 4
\(\nu^{10}\)	\(=\)	\( ( -465\beta_{13} - 303\beta_{12} + 957\beta_{11} - 327\beta_{6} + 451\beta_{5} - 210\beta_{4} - 430\beta _1 - 584 ) / 2 \) (-465b13 - 303b12 + 957b11 - 327b6 + 451b5 - 210b4 - 430*b1 - 584) / 2
\(\nu^{11}\)	\(=\)	\( ( 152 \beta_{15} - 50 \beta_{14} + 1677 \beta_{10} - 1081 \beta_{9} - 2875 \beta_{8} + \cdots + 1096 \beta_{2} ) / 2 \) (152b15 - 50b14 + 1677b10 - 1081b9 - 2875b8 + 1495b7 + 205b3 + 1096b2) / 2
\(\nu^{12}\)	\(=\)	\( ( - 2012 \beta_{13} - 1766 \beta_{12} + 4427 \beta_{11} + 172 \beta_{6} - 1653 \beta_{5} + 2010 \beta_{4} + \cdots + 6211 ) / 2 \) (-2012b13 - 1766b12 + 4427b11 + 172b6 - 1653b5 + 2010b4 - 2785*b1 + 6211) / 2
\(\nu^{13}\)	\(=\)	\( ( 10541 \beta_{15} + 11792 \beta_{14} + 11678 \beta_{10} - 8293 \beta_{9} - 19633 \beta_{8} + \cdots + 2676 \beta_{2} ) / 4 \) (10541b15 + 11792b14 + 11678b10 - 8293b9 - 19633b8 + 16121b7 - 4980b3 + 2676b2) / 4
\(\nu^{14}\)	\(=\)	\( ( - 1697 \beta_{13} - 2936 \beta_{12} + 4781 \beta_{11} + 10817 \beta_{6} - 22427 \beta_{5} + \cdots + 57296 ) / 2 \) (-1697b13 - 2936b12 + 4781b11 + 10817b6 - 22427b5 + 18575b4 - 4798*b1 + 57296) / 2
\(\nu^{15}\)	\(=\)	\( ( 63061 \beta_{15} + 79350 \beta_{14} - 4514 \beta_{10} + 6869 \beta_{9} + 19059 \beta_{8} + \cdots - 38956 \beta_{2} ) / 4 \) (63061b15 + 79350b14 - 4514b10 + 6869b9 + 19059b8 + 10303b7 - 39756b3 - 38956b2) / 4

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

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 2239.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{4} + 5 T^{2} + 2)^{4} \) (T^4 + 5*T^2 + 2)^4
$5$	\( (T^{8} - 2 T^{6} + \cdots + 625)^{2} \) (T^8 - 2T^6 + 34T^4 - 50*T^2 + 625)^2
$7$	\( (T^{8} + 30 T^{4} + 2401)^{2} \) (T^8 + 30*T^4 + 2401)^2
$11$	\( (T^{4} + 15 T^{2} + 52)^{4} \) (T^4 + 15*T^2 + 52)^4
$13$	\( (T^{4} - 23 T^{2} + 26)^{4} \) (T^4 - 23*T^2 + 26)^4
$17$	\( (T^{4} - 29 T^{2} + 104)^{4} \) (T^4 - 29*T^2 + 104)^4
$19$	\( (T^{4} - 38 T^{2} + 208)^{4} \) (T^4 - 38*T^2 + 208)^4
$23$	\( (T^{4} - 40 T^{2} + 128)^{4} \) (T^4 - 40*T^2 + 128)^4
$29$	\( (T^{2} - 3 T - 2)^{8} \) (T^2 - 3*T - 2)^8
$31$	\( (T^{4} - 120 T^{2} + 3328)^{4} \) (T^4 - 120*T^2 + 3328)^4
$37$	\( (T^{4} + 44 T^{2} + 416)^{4} \) (T^4 + 44*T^2 + 416)^4
$41$	\( (T^{2} + 72)^{8} \) (T^2 + 72)^8
$43$	\( (T^{4} - 20 T^{2} + 32)^{4} \) (T^4 - 20*T^2 + 32)^4
$47$	\( (T^{4} + 95 T^{2} + 8)^{4} \) (T^4 + 95*T^2 + 8)^4
$53$	\( (T^{4} + 164 T^{2} + 6656)^{4} \) (T^4 + 164*T^2 + 6656)^4
$59$	\( (T^{4} - 270 T^{2} + 13312)^{4} \) (T^4 - 270*T^2 + 13312)^4
$61$	\( (T^{4} + 42 T^{2} + 16)^{4} \) (T^4 + 42*T^2 + 16)^4
$67$	\( (T^{4} - 28 T^{2} + 128)^{4} \) (T^4 - 28*T^2 + 128)^4
$71$	\( (T^{4} + 236 T^{2} + 13312)^{4} \) (T^4 + 236*T^2 + 13312)^4
$73$	\( (T^{4} - 116 T^{2} + 1664)^{4} \) (T^4 - 116*T^2 + 1664)^4
$79$	\( (T^{4} + 115 T^{2} + 208)^{4} \) (T^4 + 115*T^2 + 208)^4
$83$	\( (T^{4} + 170 T^{2} + 2312)^{4} \) (T^4 + 170*T^2 + 2312)^4
$89$	\( (T^{2} + 8)^{8} \) (T^2 + 8)^8
$97$	\( (T^{4} - 261 T^{2} + 8424)^{4} \) (T^4 - 261*T^2 + 8424)^4
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