LMFDB - Newform orbit 2496.2.m.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,2,Mod(2209,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2496.2209");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2496.m (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:	$19.9306603445$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.162447943996702457856.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} - 15x^{8} - 16x^{4} + 256 $ x^16 - x^12 - 15x^8 - 16x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	yes
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_{2} q^{3} - \beta_{11} q^{5} - \beta_{14} q^{7} - q^{9}+O(q^{10}) $ q - b2 * q^3 - b11 * q^5 - b14 * q^7 - q^9	$ q - \beta_{2} q^{3} - \beta_{11} q^{5} - \beta_{14} q^{7} - q^{9} + \beta_{7} q^{11} + (\beta_{13} - \beta_{11} + \beta_{5}) q^{13} - \beta_{9} q^{15} + ( - \beta_{4} - 1) q^{17} + ( - \beta_{12} - \beta_{7}) q^{19} + \beta_{13} q^{21} - 2 \beta_{6} q^{23} - \beta_{4} q^{25} + \beta_{2} q^{27} + ( - \beta_{5} - \beta_1) q^{29} + 3 \beta_{14} q^{31} - \beta_{10} q^{33} - 2 \beta_{2} q^{35} + (2 \beta_{13} - 2 \beta_{11}) q^{37} + (\beta_{14} - \beta_{9} + \beta_{3}) q^{39} + ( - 6 \beta_{10} + \beta_{8}) q^{41} + ( - \beta_{15} + \beta_{2}) q^{43} + \beta_{11} q^{45} + ( - \beta_{14} - \beta_{9}) q^{47} + ( - \beta_{4} + 2) q^{49} + ( - \beta_{15} + \beta_{2}) q^{51} - 4 \beta_{5} q^{53} + (\beta_{6} - \beta_{3}) q^{55} + (\beta_{10} - \beta_{8}) q^{57} + (2 \beta_{12} - 3 \beta_{7}) q^{59} + ( - \beta_{5} - \beta_1) q^{61} + \beta_{14} q^{63} + ( - 4 \beta_{10} - \beta_{8} + \cdots + 3) q^{65}+ \cdots - \beta_{7} q^{99}+O(q^{100}) $ q - b2 * q^3 - b11 * q^5 - b14 * q^7 - q^9 + b7 * q^11 + (b13 - b11 + b5) * q^13 - b9 * q^15 + (-b4 - 1) * q^17 + (-b12 - b7) * q^19 + b13 * q^21 - 2b6 q^23 - b4 * q^25 + b2 * q^27 + (-b5 - b1) * q^29 + 3b14 q^31 - b10 * q^33 - 2b2 q^35 + (2b13 - 2b11) * q^37 + (b14 - b9 + b3) * q^39 + (-6b10 + b8) q^41 + (-b15 + b2) * q^43 + b11 * q^45 + (-b14 - b9) * q^47 + (-b4 + 2) * q^49 + (-b15 + b2) * q^51 - 4b5 q^53 + (b6 - b3) * q^55 + (b10 - b8) * q^57 + (2b12 - 3b7) * q^59 + (-b5 - b1) * q^61 + b14 * q^63 + (-4b10 - b8 - b4 + 3) q^65 + (3b12 - 5b7) * q^67 + 2b1 q^69 + (-b14 + b9) * q^71 - 4b10 q^73 - b15 * q^75 + (b5 + b1) * q^77 + (2b6 - 4b3) * q^79 + q^81 + (-4b12 + b7) q^83 + (2b13 - 4b11) * q^85 + (-b6 - b3) * q^87 + (-8b10 + b8) q^89 + (-b15 - b12 - 3b7 + 3b2) * q^91 - 3b13 q^93 + 2b6 q^95 + (2b10 + 2b8) * q^97 - b7 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^9	$ 16 q - 16 q^{9} - 16 q^{17} + 32 q^{49} + 48 q^{65} + 16 q^{81}+O(q^{100}) $ 16 * q - 16 * q^9 - 16 * q^17 + 32 * q^49 + 48 * q^65 + 16 * q^81

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - x^{12} - 15x^{8} - 16x^{4} + 256 $ x^16 - x^12 - 15*x^8 - 16*x^4 + 256 :

$\beta_{1}$	$=$	$ ( -\nu^{12} - 15\nu^{8} + 15\nu^{4} + 136 ) / 120 $ (-v^12 - 15v^8 + 15v^4 + 136) / 120
$\beta_{2}$	$=$	$ ( -\nu^{14} - 15\nu^{10} - 33\nu^{6} + 256\nu^{2} ) / 576 $ (-v^14 - 15v^10 - 33v^6 + 256*v^2) / 576
$\beta_{3}$	$=$	$ ( -\nu^{14} - 15\nu^{10} + 95\nu^{6} + 256\nu^{2} ) / 320 $ (-v^14 - 15v^10 + 95v^6 + 256*v^2) / 320
$\beta_{4}$	$=$	$ ( -\nu^{12} + \nu^{8} + 31\nu^{4} + 8 ) / 24 $ (-v^12 + v^8 + 31*v^4 + 8) / 24
$\beta_{5}$	$=$	$ ( -2\nu^{12} + 47 ) / 45 $ (-2*v^12 + 47) / 45
$\beta_{6}$	$=$	$ ( -9\nu^{14} + 25\nu^{10} + 55\nu^{6} + 544\nu^{2} ) / 960 $ (-9v^14 + 25v^10 + 55v^6 + 544v^2) / 960
$\beta_{7}$	$=$	$ ( 4\nu^{15} - 17\nu^{13} - 15\nu^{9} + 255\nu^{5} + 356\nu^{3} + 272\nu ) / 1440 $ (4v^15 - 17v^13 - 15v^9 + 255v^5 + 356v^3 + 272v) / 1440
$\beta_{8}$	$=$	$ ( -19\nu^{15} + 88\nu^{13} + 195\nu^{11} + 360\nu^{9} + 1005\nu^{7} - 360\nu^{5} - 416\nu^{3} - 11968\nu ) / 5760 $ (-19v^15 + 88v^13 + 195v^11 + 360v^9 + 1005v^7 - 360v^5 - 416v^3 - 11968v) / 5760
$\beta_{9}$	$=$	$ ( 5\nu^{15} - 8\nu^{13} - 21\nu^{11} + 72\nu^{9} + 69\nu^{7} - 72\nu^{5} - 224\nu^{3} - 64\nu ) / 1152 $ (5v^15 - 8v^13 - 21v^11 + 72v^9 + 69v^7 - 72v^5 - 224v^3 - 64v) / 1152
$\beta_{10}$	$=$	$ ( -4\nu^{15} - 17\nu^{13} - 15\nu^{9} + 255\nu^{5} - 356\nu^{3} + 272\nu ) / 1440 $ (-4v^15 - 17v^13 - 15v^9 + 255v^5 - 356v^3 + 272v) / 1440
$\beta_{11}$	$=$	$ ( -5\nu^{15} - 8\nu^{13} + 21\nu^{11} + 72\nu^{9} - 69\nu^{7} - 72\nu^{5} + 224\nu^{3} - 64\nu ) / 1152 $ (-5v^15 - 8v^13 + 21v^11 + 72v^9 - 69v^7 - 72v^5 + 224v^3 - 64v) / 1152
$\beta_{12}$	$=$	$ ( -19\nu^{15} - 88\nu^{13} + 195\nu^{11} - 360\nu^{9} + 1005\nu^{7} + 360\nu^{5} - 416\nu^{3} + 11968\nu ) / 5760 $ (-19v^15 - 88v^13 + 195v^11 - 360v^9 + 1005v^7 + 360v^5 - 416v^3 + 11968v) / 5760
$\beta_{13}$	$=$	$ ( -11\nu^{15} + 4\nu^{13} - 21\nu^{11} + 60\nu^{9} + 69\nu^{7} + 132\nu^{5} + 656\nu^{3} + 128\nu ) / 1152 $ (-11v^15 + 4v^13 - 21v^11 + 60v^9 + 69v^7 + 132v^5 + 656v^3 + 128v) / 1152
$\beta_{14}$	$=$	$ ( 11\nu^{15} + 4\nu^{13} + 21\nu^{11} + 60\nu^{9} - 69\nu^{7} + 132\nu^{5} - 656\nu^{3} + 128\nu ) / 1152 $ (11v^15 + 4v^13 + 21v^11 + 60v^9 - 69v^7 + 132v^5 - 656v^3 + 128v) / 1152
$\beta_{15}$	$=$	$ ( 7\nu^{14} + 9\nu^{10} - 57\nu^{6} + 32\nu^{2} ) / 192 $ (7v^14 + 9v^10 - 57v^6 + 32v^2) / 192

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

$\nu$	$=$	$ ( \beta_{14} + \beta_{13} + \beta_{12} - \beta_{10} - \beta_{8} - \beta_{7} ) / 4 $ (b14 + b13 + b12 - b10 - b8 - b7) / 4
$\nu^{2}$	$=$	$ ( \beta_{15} + 3\beta_{6} + \beta_{3} + 3\beta_{2} ) / 4 $ (b15 + 3b6 + b3 + 3b2) / 4
$\nu^{3}$	$=$	$ ( -\beta_{14} + \beta_{13} + \beta_{11} - 5\beta_{10} - \beta_{9} + 5\beta_{7} ) / 4 $ (-b14 + b13 + b11 - 5b10 - b9 + 5b7) / 4
$\nu^{4}$	$=$	$ ( -3\beta_{5} + 3\beta_{4} + \beta _1 + 1 ) / 4 $ (-3b5 + 3b4 + b1 + 1) / 4
$\nu^{5}$	$=$	$ ( 6\beta_{14} + 6\beta_{13} - \beta_{12} - 5\beta_{11} + 6\beta_{10} - 5\beta_{9} + \beta_{8} + 6\beta_{7} ) / 4 $ (6b14 + 6b13 - b12 - 5b11 + 6b10 - 5b9 + b8 + 6b7) / 4
$\nu^{6}$	$=$	$ ( 5\beta_{3} - 9\beta_{2} ) / 2 $ (5b3 - 9b2) / 2
$\nu^{7}$	$=$	$ ( -3\beta_{14} + 3\beta_{13} + 7\beta_{12} - 10\beta_{11} - 3\beta_{10} + 10\beta_{9} + 7\beta_{8} + 3\beta_{7} ) / 4 $ (-3b14 + 3b13 + 7b12 - 10b11 - 3b10 + 10b9 + 7b8 + 3b7) / 4
$\nu^{8}$	$=$	$ ( 3\beta_{5} + 3\beta_{4} - 31\beta _1 + 31 ) / 4 $ (3b5 + 3b4 - 31*b1 + 31) / 4
$\nu^{9}$	$=$	$ ( 17\beta_{14} + 17\beta_{13} + 17\beta_{11} - 5\beta_{10} + 17\beta_{9} - 5\beta_{7} ) / 4 $ (17b14 + 17b13 + 17b11 - 5b10 + 17b9 - 5b7) / 4
$\nu^{10}$	$=$	$ ( 11\beta_{15} + 57\beta_{6} - 11\beta_{3} - 57\beta_{2} ) / 4 $ (11b15 + 57b6 - 11b3 - 57b2) / 4
$\nu^{11}$	$=$	$ ( 22 \beta_{14} - 22 \beta_{13} + 23 \beta_{12} + 45 \beta_{11} - 22 \beta_{10} - 45 \beta_{9} + \cdots + 22 \beta_{7} ) / 4 $ (22b14 - 22b13 + 23b12 + 45b11 - 22b10 - 45b9 + 23b8 + 22b7) / 4
$\nu^{12}$	$=$	$ ( -45\beta_{5} + 47 ) / 2 $ (-45*b5 + 47) / 2
$\nu^{13}$	$=$	$ ( 91\beta_{14} + 91\beta_{13} + \beta_{12} - 90\beta_{11} - 91\beta_{10} - 90\beta_{9} - \beta_{8} - 91\beta_{7} ) / 4 $ (91b14 + 91b13 + b12 - 90b11 - 91b10 - 90b9 - b8 - 91b7) / 4
$\nu^{14}$	$=$	$ ( 91\beta_{15} - 87\beta_{6} + 91\beta_{3} - 87\beta_{2} ) / 4 $ (91b15 - 87b6 + 91b3 - 87b2) / 4
$\nu^{15}$	$=$	$ ( 89\beta_{14} - 89\beta_{13} - 89\beta_{11} - 275\beta_{10} + 89\beta_{9} + 275\beta_{7} ) / 4 $ (89b14 - 89b13 - 89b11 - 275b10 + 89b9 + 275b7) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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} $

2209.1

1.14839	+	0.825348i
−0.825348	−	1.14839i
0.140577	−	1.40721i
1.40721	−	0.140577i
−1.40721	+	0.140577i
−0.140577	+	1.40721i
0.825348	+	1.14839i
−1.14839	−	0.825348i
1.14839	−	0.825348i
−0.825348	+	1.14839i
0.140577	+	1.40721i
1.40721	+	0.140577i
−1.40721	−	0.140577i
−0.140577	−	1.40721i
0.825348	−	1.14839i
−1.14839	+	0.825348i

−

1.00000i

−3.09557

0.646084i

−1.00000

2209.2

−

1.00000i

−3.09557

0.646084i

−1.00000

2209.3

−

1.00000i

−0.646084

3.09557i

−1.00000

2209.4

−

1.00000i

−0.646084

3.09557i

−1.00000

2209.5

−

1.00000i

0.646084

−

3.09557i

−1.00000

2209.6

−

1.00000i

0.646084

−

3.09557i

−1.00000

2209.7

−

1.00000i

3.09557

−

0.646084i

−1.00000

2209.8

−

1.00000i

3.09557

−

0.646084i

−1.00000

2209.9

1.00000i

−3.09557

−

0.646084i

−1.00000

2209.10

1.00000i

−3.09557

−

0.646084i

−1.00000

2209.11

1.00000i

−0.646084

−

3.09557i

−1.00000

2209.12

1.00000i

−0.646084

−

3.09557i

−1.00000

2209.13

1.00000i

0.646084

3.09557i

−1.00000

2209.14

1.00000i

0.646084

3.09557i

−1.00000

2209.15

1.00000i

3.09557

0.646084i

−1.00000

2209.16

1.00000i

3.09557

0.646084i

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
8.b	even	2	1	inner
8.d	odd	2	1	inner
13.b	even	2	1	inner
52.b	odd	2	1	inner
104.e	even	2	1	inner
104.h	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2496.2.m.c	✓	16
4.b	odd	2	1	inner	2496.2.m.c	✓	16
8.b	even	2	1	inner	2496.2.m.c	✓	16
8.d	odd	2	1	inner	2496.2.m.c	✓	16
13.b	even	2	1	inner	2496.2.m.c	✓	16
52.b	odd	2	1	inner	2496.2.m.c	✓	16
104.e	even	2	1	inner	2496.2.m.c	✓	16
104.h	odd	2	1	inner	2496.2.m.c	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2496.2.m.c	✓	16	1.a	even	1	1	trivial
2496.2.m.c	✓	16	4.b	odd	2	1	inner
2496.2.m.c	✓	16	8.b	even	2	1	inner
2496.2.m.c	✓	16	8.d	odd	2	1	inner
2496.2.m.c	✓	16	13.b	even	2	1	inner
2496.2.m.c	✓	16	52.b	odd	2	1	inner
2496.2.m.c	✓	16	104.e	even	2	1	inner
2496.2.m.c	✓	16	104.h	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2496, [\chi])$:

$ T_{5}^{4} - 10T_{5}^{2} + 4 $

$ T_{7}^{4} + 10T_{7}^{2} + 4 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{2} + 1)^{8} $ (T^2 + 1)^8
$5$	$ (T^{4} - 10 T^{2} + 4)^{4} $ (T^4 - 10*T^2 + 4)^4
$7$	$ (T^{4} + 10 T^{2} + 4)^{4} $ (T^4 + 10*T^2 + 4)^4
$11$	$ (T^{2} - 2)^{8} $ (T^2 - 2)^8
$13$	$ (T^{4} + 2 T^{2} + 169)^{4} $ (T^4 + 2*T^2 + 169)^4
$17$	$ (T^{2} + 2 T - 20)^{8} $ (T^2 + 2*T - 20)^8
$19$	$ (T^{4} - 30 T^{2} + 36)^{4} $ (T^4 - 30*T^2 + 36)^4
$23$	$ (T^{2} - 12)^{8} $ (T^2 - 12)^8
$29$	$ (T^{4} + 20 T^{2} + 16)^{4} $ (T^4 + 20*T^2 + 16)^4
$31$	$ (T^{4} + 90 T^{2} + 324)^{4} $ (T^4 + 90*T^2 + 324)^4
$37$	$ (T^{2} - 24)^{8} $ (T^2 - 24)^8
$41$	$ (T^{4} + 190 T^{2} + 5476)^{4} $ (T^4 + 190*T^2 + 5476)^4
$43$	$ (T^{4} + 44 T^{2} + 400)^{4} $ (T^4 + 44*T^2 + 400)^4
$47$	$ (T^{2} + 14)^{8} $ (T^2 + 14)^8
$53$	$ (T^{2} + 112)^{8} $ (T^2 + 112)^8
$59$	$ (T^{4} - 100 T^{2} + 1156)^{4} $ (T^4 - 100*T^2 + 1156)^4
$61$	$ (T^{4} + 20 T^{2} + 16)^{4} $ (T^4 + 20*T^2 + 16)^4
$67$	$ (T^{4} - 238 T^{2} + 4900)^{4} $ (T^4 - 238*T^2 + 4900)^4
$71$	$ (T^{2} + 6)^{8} $ (T^2 + 6)^8
$73$	$ (T^{2} + 32)^{8} $ (T^2 + 32)^8
$79$	$ (T^{4} - 248 T^{2} + 10000)^{4} $ (T^4 - 248*T^2 + 10000)^4
$83$	$ (T^{4} - 340 T^{2} + 27556)^{4} $ (T^4 - 340*T^2 + 27556)^4
$89$	$ (T^{4} + 310 T^{2} + 17956)^{4} $ (T^4 + 310*T^2 + 17956)^4
$97$	$ (T^{4} + 88 T^{2} + 1600)^{4} $ (T^4 + 88*T^2 + 1600)^4
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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:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2496.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{3} - \beta_{11} q^{5} - \beta_{14} q^{7} - q^{9}+O(q^{10}) \) q - b2 * q^3 - b11 * q^5 - b14 * q^7 - q^9	\( q - \beta_{2} q^{3} - \beta_{11} q^{5} - \beta_{14} q^{7} - q^{9} + \beta_{7} q^{11} + (\beta_{13} - \beta_{11} + \beta_{5}) q^{13} - \beta_{9} q^{15} + ( - \beta_{4} - 1) q^{17} + ( - \beta_{12} - \beta_{7}) q^{19} + \beta_{13} q^{21} - 2 \beta_{6} q^{23} - \beta_{4} q^{25} + \beta_{2} q^{27} + ( - \beta_{5} - \beta_1) q^{29} + 3 \beta_{14} q^{31} - \beta_{10} q^{33} - 2 \beta_{2} q^{35} + (2 \beta_{13} - 2 \beta_{11}) q^{37} + (\beta_{14} - \beta_{9} + \beta_{3}) q^{39} + ( - 6 \beta_{10} + \beta_{8}) q^{41} + ( - \beta_{15} + \beta_{2}) q^{43} + \beta_{11} q^{45} + ( - \beta_{14} - \beta_{9}) q^{47} + ( - \beta_{4} + 2) q^{49} + ( - \beta_{15} + \beta_{2}) q^{51} - 4 \beta_{5} q^{53} + (\beta_{6} - \beta_{3}) q^{55} + (\beta_{10} - \beta_{8}) q^{57} + (2 \beta_{12} - 3 \beta_{7}) q^{59} + ( - \beta_{5} - \beta_1) q^{61} + \beta_{14} q^{63} + ( - 4 \beta_{10} - \beta_{8} + \cdots + 3) q^{65}+ \cdots - \beta_{7} q^{99}+O(q^{100}) \) q - b2 * q^3 - b11 * q^5 - b14 * q^7 - q^9 + b7 * q^11 + (b13 - b11 + b5) * q^13 - b9 * q^15 + (-b4 - 1) * q^17 + (-b12 - b7) * q^19 + b13 * q^21 - 2b6 q^23 - b4 * q^25 + b2 * q^27 + (-b5 - b1) * q^29 + 3b14 q^31 - b10 * q^33 - 2b2 q^35 + (2b13 - 2b11) * q^37 + (b14 - b9 + b3) * q^39 + (-6b10 + b8) q^41 + (-b15 + b2) * q^43 + b11 * q^45 + (-b14 - b9) * q^47 + (-b4 + 2) * q^49 + (-b15 + b2) * q^51 - 4b5 q^53 + (b6 - b3) * q^55 + (b10 - b8) * q^57 + (2b12 - 3b7) * q^59 + (-b5 - b1) * q^61 + b14 * q^63 + (-4b10 - b8 - b4 + 3) q^65 + (3b12 - 5b7) * q^67 + 2b1 q^69 + (-b14 + b9) * q^71 - 4b10 q^73 - b15 * q^75 + (b5 + b1) * q^77 + (2b6 - 4b3) * q^79 + q^81 + (-4b12 + b7) q^83 + (2b13 - 4b11) * q^85 + (-b6 - b3) * q^87 + (-8b10 + b8) q^89 + (-b15 - b12 - 3b7 + 3b2) * q^91 - 3b13 q^93 + 2b6 q^95 + (2b10 + 2b8) * q^97 - b7 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^9	\( 16 q - 16 q^{9} - 16 q^{17} + 32 q^{49} + 48 q^{65} + 16 q^{81}+O(q^{100}) \) 16 * q - 16 * q^9 - 16 * q^17 + 32 * q^49 + 48 * q^65 + 16 * q^81

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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	2496.2.m.c

Level	$2496$
Weight	$2$
Character orbit	2496.m
Analytic conductor	$19.931$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(19.9306603445\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	16.0.162447943996702457856.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - x^{12} - 15x^{8} - 16x^{4} + 256 \) x^16 - x^12 - 15x^8 - 16x^4 + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{16} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( -\nu^{12} - 15\nu^{8} + 15\nu^{4} + 136 ) / 120 \) (-v^12 - 15v^8 + 15v^4 + 136) / 120
\(\beta_{2}\)	\(=\)	\( ( -\nu^{14} - 15\nu^{10} - 33\nu^{6} + 256\nu^{2} ) / 576 \) (-v^14 - 15v^10 - 33v^6 + 256*v^2) / 576
\(\beta_{3}\)	\(=\)	\( ( -\nu^{14} - 15\nu^{10} + 95\nu^{6} + 256\nu^{2} ) / 320 \) (-v^14 - 15v^10 + 95v^6 + 256*v^2) / 320
\(\beta_{4}\)	\(=\)	\( ( -\nu^{12} + \nu^{8} + 31\nu^{4} + 8 ) / 24 \) (-v^12 + v^8 + 31*v^4 + 8) / 24
\(\beta_{5}\)	\(=\)	\( ( -2\nu^{12} + 47 ) / 45 \) (-2*v^12 + 47) / 45
\(\beta_{6}\)	\(=\)	\( ( -9\nu^{14} + 25\nu^{10} + 55\nu^{6} + 544\nu^{2} ) / 960 \) (-9v^14 + 25v^10 + 55v^6 + 544v^2) / 960
\(\beta_{7}\)	\(=\)	\( ( 4\nu^{15} - 17\nu^{13} - 15\nu^{9} + 255\nu^{5} + 356\nu^{3} + 272\nu ) / 1440 \) (4v^15 - 17v^13 - 15v^9 + 255v^5 + 356v^3 + 272v) / 1440
\(\beta_{8}\)	\(=\)	\( ( -19\nu^{15} + 88\nu^{13} + 195\nu^{11} + 360\nu^{9} + 1005\nu^{7} - 360\nu^{5} - 416\nu^{3} - 11968\nu ) / 5760 \) (-19v^15 + 88v^13 + 195v^11 + 360v^9 + 1005v^7 - 360v^5 - 416v^3 - 11968v) / 5760
\(\beta_{9}\)	\(=\)	\( ( 5\nu^{15} - 8\nu^{13} - 21\nu^{11} + 72\nu^{9} + 69\nu^{7} - 72\nu^{5} - 224\nu^{3} - 64\nu ) / 1152 \) (5v^15 - 8v^13 - 21v^11 + 72v^9 + 69v^7 - 72v^5 - 224v^3 - 64v) / 1152
\(\beta_{10}\)	\(=\)	\( ( -4\nu^{15} - 17\nu^{13} - 15\nu^{9} + 255\nu^{5} - 356\nu^{3} + 272\nu ) / 1440 \) (-4v^15 - 17v^13 - 15v^9 + 255v^5 - 356v^3 + 272v) / 1440
\(\beta_{11}\)	\(=\)	\( ( -5\nu^{15} - 8\nu^{13} + 21\nu^{11} + 72\nu^{9} - 69\nu^{7} - 72\nu^{5} + 224\nu^{3} - 64\nu ) / 1152 \) (-5v^15 - 8v^13 + 21v^11 + 72v^9 - 69v^7 - 72v^5 + 224v^3 - 64v) / 1152
\(\beta_{12}\)	\(=\)	\( ( -19\nu^{15} - 88\nu^{13} + 195\nu^{11} - 360\nu^{9} + 1005\nu^{7} + 360\nu^{5} - 416\nu^{3} + 11968\nu ) / 5760 \) (-19v^15 - 88v^13 + 195v^11 - 360v^9 + 1005v^7 + 360v^5 - 416v^3 + 11968v) / 5760
\(\beta_{13}\)	\(=\)	\( ( -11\nu^{15} + 4\nu^{13} - 21\nu^{11} + 60\nu^{9} + 69\nu^{7} + 132\nu^{5} + 656\nu^{3} + 128\nu ) / 1152 \) (-11v^15 + 4v^13 - 21v^11 + 60v^9 + 69v^7 + 132v^5 + 656v^3 + 128v) / 1152
\(\beta_{14}\)	\(=\)	\( ( 11\nu^{15} + 4\nu^{13} + 21\nu^{11} + 60\nu^{9} - 69\nu^{7} + 132\nu^{5} - 656\nu^{3} + 128\nu ) / 1152 \) (11v^15 + 4v^13 + 21v^11 + 60v^9 - 69v^7 + 132v^5 - 656v^3 + 128v) / 1152
\(\beta_{15}\)	\(=\)	\( ( 7\nu^{14} + 9\nu^{10} - 57\nu^{6} + 32\nu^{2} ) / 192 \) (7v^14 + 9v^10 - 57v^6 + 32v^2) / 192

\(\nu\)	\(=\)	\( ( \beta_{14} + \beta_{13} + \beta_{12} - \beta_{10} - \beta_{8} - \beta_{7} ) / 4 \) (b14 + b13 + b12 - b10 - b8 - b7) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{15} + 3\beta_{6} + \beta_{3} + 3\beta_{2} ) / 4 \) (b15 + 3b6 + b3 + 3b2) / 4
\(\nu^{3}\)	\(=\)	\( ( -\beta_{14} + \beta_{13} + \beta_{11} - 5\beta_{10} - \beta_{9} + 5\beta_{7} ) / 4 \) (-b14 + b13 + b11 - 5b10 - b9 + 5b7) / 4
\(\nu^{4}\)	\(=\)	\( ( -3\beta_{5} + 3\beta_{4} + \beta _1 + 1 ) / 4 \) (-3b5 + 3b4 + b1 + 1) / 4
\(\nu^{5}\)	\(=\)	\( ( 6\beta_{14} + 6\beta_{13} - \beta_{12} - 5\beta_{11} + 6\beta_{10} - 5\beta_{9} + \beta_{8} + 6\beta_{7} ) / 4 \) (6b14 + 6b13 - b12 - 5b11 + 6b10 - 5b9 + b8 + 6b7) / 4
\(\nu^{6}\)	\(=\)	\( ( 5\beta_{3} - 9\beta_{2} ) / 2 \) (5b3 - 9b2) / 2
\(\nu^{7}\)	\(=\)	\( ( -3\beta_{14} + 3\beta_{13} + 7\beta_{12} - 10\beta_{11} - 3\beta_{10} + 10\beta_{9} + 7\beta_{8} + 3\beta_{7} ) / 4 \) (-3b14 + 3b13 + 7b12 - 10b11 - 3b10 + 10b9 + 7b8 + 3b7) / 4
\(\nu^{8}\)	\(=\)	\( ( 3\beta_{5} + 3\beta_{4} - 31\beta _1 + 31 ) / 4 \) (3b5 + 3b4 - 31*b1 + 31) / 4
\(\nu^{9}\)	\(=\)	\( ( 17\beta_{14} + 17\beta_{13} + 17\beta_{11} - 5\beta_{10} + 17\beta_{9} - 5\beta_{7} ) / 4 \) (17b14 + 17b13 + 17b11 - 5b10 + 17b9 - 5b7) / 4
\(\nu^{10}\)	\(=\)	\( ( 11\beta_{15} + 57\beta_{6} - 11\beta_{3} - 57\beta_{2} ) / 4 \) (11b15 + 57b6 - 11b3 - 57b2) / 4
\(\nu^{11}\)	\(=\)	\( ( 22 \beta_{14} - 22 \beta_{13} + 23 \beta_{12} + 45 \beta_{11} - 22 \beta_{10} - 45 \beta_{9} + \cdots + 22 \beta_{7} ) / 4 \) (22b14 - 22b13 + 23b12 + 45b11 - 22b10 - 45b9 + 23b8 + 22b7) / 4
\(\nu^{12}\)	\(=\)	\( ( -45\beta_{5} + 47 ) / 2 \) (-45*b5 + 47) / 2
\(\nu^{13}\)	\(=\)	\( ( 91\beta_{14} + 91\beta_{13} + \beta_{12} - 90\beta_{11} - 91\beta_{10} - 90\beta_{9} - \beta_{8} - 91\beta_{7} ) / 4 \) (91b14 + 91b13 + b12 - 90b11 - 91b10 - 90b9 - b8 - 91b7) / 4
\(\nu^{14}\)	\(=\)	\( ( 91\beta_{15} - 87\beta_{6} + 91\beta_{3} - 87\beta_{2} ) / 4 \) (91b15 - 87b6 + 91b3 - 87b2) / 4
\(\nu^{15}\)	\(=\)	\( ( 89\beta_{14} - 89\beta_{13} - 89\beta_{11} - 275\beta_{10} + 89\beta_{9} + 275\beta_{7} ) / 4 \) (89b14 - 89b13 - 89b11 - 275b10 + 89b9 + 275b7) / 4

\(n\)	\(703\)	\(769\)	\(833\)	\(1093\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 2209.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{2} + 1)^{8} \) (T^2 + 1)^8
$5$	\( (T^{4} - 10 T^{2} + 4)^{4} \) (T^4 - 10*T^2 + 4)^4
$7$	\( (T^{4} + 10 T^{2} + 4)^{4} \) (T^4 + 10*T^2 + 4)^4
$11$	\( (T^{2} - 2)^{8} \) (T^2 - 2)^8
$13$	\( (T^{4} + 2 T^{2} + 169)^{4} \) (T^4 + 2*T^2 + 169)^4
$17$	\( (T^{2} + 2 T - 20)^{8} \) (T^2 + 2*T - 20)^8
$19$	\( (T^{4} - 30 T^{2} + 36)^{4} \) (T^4 - 30*T^2 + 36)^4
$23$	\( (T^{2} - 12)^{8} \) (T^2 - 12)^8
$29$	\( (T^{4} + 20 T^{2} + 16)^{4} \) (T^4 + 20*T^2 + 16)^4
$31$	\( (T^{4} + 90 T^{2} + 324)^{4} \) (T^4 + 90*T^2 + 324)^4
$37$	\( (T^{2} - 24)^{8} \) (T^2 - 24)^8
$41$	\( (T^{4} + 190 T^{2} + 5476)^{4} \) (T^4 + 190*T^2 + 5476)^4
$43$	\( (T^{4} + 44 T^{2} + 400)^{4} \) (T^4 + 44*T^2 + 400)^4
$47$	\( (T^{2} + 14)^{8} \) (T^2 + 14)^8
$53$	\( (T^{2} + 112)^{8} \) (T^2 + 112)^8
$59$	\( (T^{4} - 100 T^{2} + 1156)^{4} \) (T^4 - 100*T^2 + 1156)^4
$61$	\( (T^{4} + 20 T^{2} + 16)^{4} \) (T^4 + 20*T^2 + 16)^4
$67$	\( (T^{4} - 238 T^{2} + 4900)^{4} \) (T^4 - 238*T^2 + 4900)^4
$71$	\( (T^{2} + 6)^{8} \) (T^2 + 6)^8
$73$	\( (T^{2} + 32)^{8} \) (T^2 + 32)^8
$79$	\( (T^{4} - 248 T^{2} + 10000)^{4} \) (T^4 - 248*T^2 + 10000)^4
$83$	\( (T^{4} - 340 T^{2} + 27556)^{4} \) (T^4 - 340*T^2 + 27556)^4
$89$	\( (T^{4} + 310 T^{2} + 17956)^{4} \) (T^4 + 310*T^2 + 17956)^4
$97$	\( (T^{4} + 88 T^{2} + 1600)^{4} \) (T^4 + 88*T^2 + 1600)^4
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