LMFDB - Newform orbit 3380.2.d.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(1689,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3380.1689");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.d (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:	$26.9894358832$
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} - 7x^{14} + 21x^{12} - 22x^{10} - 26x^{8} - 198x^{6} + 1701x^{4} - 5103x^{2} + 6561 $ x^16 - 7x^14 + 21x^12 - 22x^10 - 26x^8 - 198x^6 + 1701x^4 - 5103*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 260)
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_{6} q^{3} + \beta_{9} q^{5} + \beta_{7} q^{7} + (\beta_{2} - 1) q^{9}+O(q^{10}) $ q + b6 * q^3 + b9 * q^5 + b7 * q^7 + (b2 - 1) * q^9	$ q + \beta_{6} q^{3} + \beta_{9} q^{5} + \beta_{7} q^{7} + (\beta_{2} - 1) q^{9} + ( - \beta_{4} + \beta_{3} + \beta_1) q^{11} + (\beta_{10} - \beta_{5} - \beta_{4}) q^{15} + ( - \beta_{13} - \beta_{12} + \cdots - \beta_{6}) q^{17}+ \cdots + (\beta_{11} - 3 \beta_{10} + \cdots - 4 \beta_1) q^{99}+O(q^{100}) $ q + b6 * q^3 + b9 * q^5 + b7 * q^7 + (b2 - 1) * q^9 + (-b4 + b3 + b1) * q^11 + (b10 - b5 - b4) * q^15 + (-b13 - b12 - b8 - b6) * q^17 + (b11 - b1) * q^19 + (-b4 + b3 + b1) * q^21 + (b8 - b6) * q^23 + (-b14 - b6 - 1) * q^25 + (b15 + b14 - b13 - b12 + b8 - 3b6) q^27 + (b13 - b12 + b2 - 2) * q^29 + (-b10 + b9 + b4 - b3) * q^31 + (-2b10 - 2b9 - 2b7 + b5) q^33 + (b15 + b12 + b8 - b2) * q^35 + (-b7 + b4 + b3) * q^37 + (b10 - b9 + b4 - b3 + b1) * q^41 + (-b15 - b14 + b13 + b12 + 2b8 + b6) q^43 + (-b11 + b10 - b9 - b7 + b5 - b4 + 2b3 + 3b1) * q^45 + (-2b10 - 2b9) * q^47 + (b2 + 1) * q^49 + (-b15 + b14 + 3) * q^51 + (-b15 - b14 + b8) * q^53 + (b15 - b13 - 2b12 - b8 + b6 - b2 - 1) q^55 + (-b10 - b9 - b7 - 4b5 - b4 - b3) q^57 + (-b10 + b9 - b1) * q^59 + (-b13 + b12 - b2 + 4) * q^61 + (-2b10 - 2b9 + b7 + b5) * q^63 + (3b10 + 3b9 - b7 + 2b5 + b4 + b3) q^67 + (b13 - b12 - 2b2 + 3) q^69 + (-b11 + 2b10 - 2b9 - b1) * q^71 + (-b7 + b5 + b4 + b3) * q^73 + (b15 + b14 - 2b13 - 2b12 - b8 - 3b6 - b2 + 5) q^75 + (b15 + b14 - b13 - b12 + b8 + 3b6) q^77 + (b15 - b14 + b2 - 1) * q^79 + (-b15 + b14 + 2b13 - 2b12 - b2 + 4) * q^81 + (-b10 - b9 + 2b7 - b4 - b3) q^83 + (-b10 + 2b7 + b5 + 2b3 - b1) * q^85 + (-b8 - 3b6) q^87 + (-b10 + b9 - b1) * q^89 + (3b10 + 3b9 + 2b7 - 2b5 - b4 - b3) * q^93 + (3b15 + 2b14 - 2b13 - 3b12 - b8 - b6 + 2b2) q^95 + (2b10 + 2b9 + b7 - 2b5) q^97 + (b11 - 3b10 + 3b9 + 2b4 - 2b3 - 4b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 20 q^{9}+O(q^{10}) $ 16 * q - 20 * q^9	$ 16 q - 20 q^{9} - 14 q^{25} - 24 q^{29} + 12 q^{49} + 44 q^{51} - 4 q^{55} + 56 q^{61} + 68 q^{69} + 84 q^{75} - 16 q^{79} + 88 q^{81}+O(q^{100}) $ 16 * q - 20 * q^9 - 14 * q^25 - 24 * q^29 + 12 * q^49 + 44 * q^51 - 4 * q^55 + 56 * q^61 + 68 * q^69 + 84 * q^75 - 16 * q^79 + 88 * q^81

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 7x^{14} + 21x^{12} - 22x^{10} - 26x^{8} - 198x^{6} + 1701x^{4} - 5103x^{2} + 6561 $ x^16 - 7*x^14 + 21*x^12 - 22*x^10 - 26*x^8 - 198*x^6 + 1701*x^4 - 5103*x^2 + 6561 :

$\beta_{1}$	$=$	$ ( -5\nu^{14} + 2\nu^{12} - 63\nu^{10} + 11\nu^{8} + 289\nu^{6} - 69\nu^{4} - 1674\nu^{2} + 16281 ) / 7776 $ (-5v^14 + 2v^12 - 63v^10 + 11v^8 + 289v^6 - 69v^4 - 1674*v^2 + 16281) / 7776
$\beta_{2}$	$=$	$ ( -\nu^{14} + 7\nu^{12} - 21\nu^{10} + 22\nu^{8} + 26\nu^{6} + 198\nu^{4} - 972\nu^{2} + 3645 ) / 729 $ (-v^14 + 7v^12 - 21v^10 + 22v^8 + 26v^6 + 198v^4 - 972v^2 + 3645) / 729
$\beta_{3}$	$=$	$ ( 25 \nu^{15} + 141 \nu^{14} + 518 \nu^{13} - 690 \nu^{12} - 1653 \nu^{11} + 2583 \nu^{10} + \cdots - 636417 ) / 139968 $ (25v^15 + 141v^14 + 518v^13 - 690v^12 - 1653v^11 + 2583v^10 - 1063v^9 - 2211v^8 + 3787v^7 - 5097v^6 - 855v^5 - 18387v^4 - 137214v^3 + 248346v^2 + 276291*v - 636417) / 139968
$\beta_{4}$	$=$	$ ( 25 \nu^{15} - 141 \nu^{14} + 518 \nu^{13} + 690 \nu^{12} - 1653 \nu^{11} - 2583 \nu^{10} + \cdots + 636417 ) / 139968 $ (25v^15 - 141v^14 + 518v^13 + 690v^12 - 1653v^11 - 2583v^10 - 1063v^9 + 2211v^8 + 3787v^7 + 5097v^6 - 855v^5 + 18387v^4 - 137214v^3 - 248346v^2 + 276291*v + 636417) / 139968
$\beta_{5}$	$=$	$ ( 11\nu^{15} - 158\nu^{13} + 393\nu^{11} + 163\nu^{9} + 281\nu^{7} - 10845\nu^{5} + 50382\nu^{3} - 75087\nu ) / 34992 $ (11v^15 - 158v^13 + 393v^11 + 163v^9 + 281v^7 - 10845v^5 + 50382v^3 - 75087v) / 34992
$\beta_{6}$	$=$	$ ( \nu^{15} - 7\nu^{13} + 21\nu^{11} - 22\nu^{9} - 26\nu^{7} - 198\nu^{5} + 1701\nu^{3} - 2916\nu ) / 2187 $ (v^15 - 7v^13 + 21v^11 - 22v^9 - 26v^7 - 198v^5 + 1701v^3 - 2916*v) / 2187
$\beta_{7}$	$=$	$ ( -\nu^{15} + 7\nu^{13} - 21\nu^{11} + 22\nu^{9} + 26\nu^{7} + 198\nu^{5} - 1701\nu^{3} + 7290\nu ) / 2187 $ (-v^15 + 7v^13 - 21v^11 + 22v^9 + 26v^7 + 198v^5 - 1701v^3 + 7290*v) / 2187
$\beta_{8}$	$=$	$ ( 5\nu^{15} - 44\nu^{13} + 6\nu^{11} + 106\nu^{9} - 418\nu^{7} - 1566\nu^{5} + 13041\nu^{3} - 7290\nu ) / 8748 $ (5v^15 - 44v^13 + 6v^11 + 106v^9 - 418v^7 - 1566v^5 + 13041v^3 - 7290v) / 8748
$\beta_{9}$	$=$	$ ( 107 \nu^{15} + 399 \nu^{14} - 686 \nu^{13} - 1902 \nu^{12} + 1401 \nu^{11} + 3357 \nu^{10} + \cdots - 789507 ) / 139968 $ (107v^15 + 399v^14 - 686v^13 - 1902v^12 + 1401v^11 + 3357v^10 + 1075v^9 + 3615v^8 - 5383v^7 - 8835v^6 - 33597v^5 - 87345v^4 + 150174v^3 + 437886v^2 - 285039*v - 789507) / 139968
$\beta_{10}$	$=$	$ ( 107 \nu^{15} - 399 \nu^{14} - 686 \nu^{13} + 1902 \nu^{12} + 1401 \nu^{11} - 3357 \nu^{10} + \cdots + 789507 ) / 139968 $ (107v^15 - 399v^14 - 686v^13 + 1902v^12 + 1401v^11 - 3357v^10 + 1075v^9 - 3615v^8 - 5383v^7 + 8835v^6 - 33597v^5 + 87345v^4 + 150174v^3 - 437886v^2 - 285039*v + 789507) / 139968
$\beta_{11}$	$=$	$ ( -79\nu^{14} + 286\nu^{12} - 573\nu^{10} - 575\nu^{8} - 253\nu^{6} + 21585\nu^{4} - 67230\nu^{2} + 132435 ) / 7776 $ (-79v^14 + 286v^12 - 573v^10 - 575v^8 - 253v^6 + 21585v^4 - 67230*v^2 + 132435) / 7776
$\beta_{12}$	$=$	$ ( 31 \nu^{15} + 36 \nu^{14} - 100 \nu^{13} - 117 \nu^{12} + 156 \nu^{11} + 54 \nu^{10} + 236 \nu^{9} + \cdots - 41553 ) / 17496 $ (31v^15 + 36v^14 - 100v^13 - 117v^12 + 156v^11 + 54v^10 + 236v^9 + 342v^8 + 508v^7 - 990v^6 - 7560v^5 - 11610v^4 + 18225v^3 + 32562v^2 - 37908*v - 41553) / 17496
$\beta_{13}$	$=$	$ ( 31 \nu^{15} - 36 \nu^{14} - 100 \nu^{13} + 117 \nu^{12} + 156 \nu^{11} - 54 \nu^{10} + 236 \nu^{9} + \cdots + 41553 ) / 17496 $ (31v^15 - 36v^14 - 100v^13 + 117v^12 + 156v^11 - 54v^10 + 236v^9 - 342v^8 + 508v^7 + 990v^6 - 7560v^5 + 11610v^4 + 18225v^3 - 32562v^2 - 37908*v + 41553) / 17496
$\beta_{14}$	$=$	$ ( 9 \nu^{15} + 2 \nu^{14} - 36 \nu^{13} + 13 \nu^{12} + 81 \nu^{11} - 66 \nu^{10} + 45 \nu^{9} + \cdots + 10935 ) / 2916 $ (9v^15 + 2v^14 - 36v^13 + 13v^12 + 81v^11 - 66v^10 + 45v^9 - 44v^8 - 99v^7 + 326v^6 - 2079v^5 + 36v^4 + 9072v^3 - 5508v^2 - 18711*v + 10935) / 2916
$\beta_{15}$	$=$	$ ( 9 \nu^{15} - 2 \nu^{14} - 36 \nu^{13} - 13 \nu^{12} + 81 \nu^{11} + 66 \nu^{10} + 45 \nu^{9} + \cdots - 10935 ) / 2916 $ (9v^15 - 2v^14 - 36v^13 - 13v^12 + 81v^11 + 66v^10 + 45v^9 + 44v^8 - 99v^7 - 326v^6 - 2079v^5 - 36v^4 + 9072v^3 + 5508v^2 - 18711*v - 10935) / 2916

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

$\nu$	$=$	$ ( \beta_{7} + \beta_{6} ) / 2 $ (b7 + b6) / 2
$\nu^{2}$	$=$	$ ( -\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 2 ) / 2 $ (-b4 + b3 + b2 + b1 + 2) / 2
$\nu^{3}$	$=$	$ ( \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} - 2\beta_{10} - 2\beta_{9} + \beta_{8} + \beta_{7} + \beta_{5} ) / 2 $ (b15 + b14 - b13 - b12 - 2b10 - 2b9 + b8 + b7 + b5) / 2
$\nu^{4}$	$=$	$ ( - \beta_{15} + \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + \beta_{11} - 3 \beta_{10} + 3 \beta_{9} + \cdots + 1 ) / 2 $ (-b15 + b14 + 2b13 - 2b12 + b11 - 3b10 + 3b9 - b4 + b3 + 2*b2 - b1 + 1) / 2
$\nu^{5}$	$=$	$ ( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - 6 \beta_{10} - 6 \beta_{9} + 3 \beta_{8} + \cdots - 4 \beta_{3} ) / 2 $ (b15 + b14 + b13 + b12 - 6b10 - 6b9 + 3b8 + 3b6 - 7b5 - 4b4 - 4*b3) / 2
$\nu^{6}$	$=$	$ -\beta_{15} + \beta_{14} + 4\beta_{13} - 4\beta_{12} - 2\beta_{11} - 2\beta_{4} + 2\beta_{3} + 3\beta_{2} + 8\beta _1 - 6 $ -b15 + b14 + 4b13 - 4b12 - 2b11 - 2b4 + 2b3 + 3b2 + 8*b1 - 6
$\nu^{7}$	$=$	$ ( - 3 \beta_{15} - 3 \beta_{14} + 13 \beta_{13} + 13 \beta_{12} - 22 \beta_{10} - 22 \beta_{9} + \cdots - 8 \beta_{3} ) / 2 $ (-3b15 - 3b14 + 13b13 + 13b12 - 22b10 - 22b9 + 3b8 - 4b7 + 7b6 + 7b5 - 8b4 - 8b3) / 2
$\nu^{8}$	$=$	$ ( 7 \beta_{15} - 7 \beta_{14} + 14 \beta_{13} - 14 \beta_{12} - 5 \beta_{11} - 13 \beta_{10} + 13 \beta_{9} + \cdots - 35 ) / 2 $ (7b15 - 7b14 + 14b13 - 14b12 - 5b11 - 13b10 + 13b9 + 21b4 - 21b3 + 22b2 - 23*b1 - 35) / 2
$\nu^{9}$	$=$	$ ( 13 \beta_{15} + 13 \beta_{14} + 3 \beta_{13} + 3 \beta_{12} - 6 \beta_{10} - 6 \beta_{9} + \cdots - 32 \beta_{3} ) / 2 $ (13b15 + 13b14 + 3b13 + 3b12 - 6b10 - 6b9 - 11b8 + 35b7 - 96b6 - 13b5 - 32b4 - 32b3) / 2
$\nu^{10}$	$=$	$ ( 24 \beta_{15} - 24 \beta_{14} + 28 \beta_{13} - 28 \beta_{12} - 28 \beta_{11} + 32 \beta_{10} + \cdots + 270 ) / 2 $ (24b15 - 24b14 + 28b13 - 28b12 - 28b11 + 32b10 - 32b9 + 21b4 - 21b3 - 19b2 - 97*b1 + 270) / 2
$\nu^{11}$	$=$	$ ( 40 \beta_{15} + 40 \beta_{14} - 40 \beta_{13} - 40 \beta_{12} + 20 \beta_{10} + 20 \beta_{9} + \cdots - 36 \beta_{3} ) / 2 $ (40b15 + 40b14 - 40b13 - 40b12 + 20b10 + 20b9 - 156b8 + 175b7 + 93b6 + 12b5 - 36b4 - 36b3) / 2
$\nu^{12}$	$=$	$ 16 \beta_{15} - 16 \beta_{14} - 16 \beta_{13} + 16 \beta_{12} - 36 \beta_{11} + 88 \beta_{10} + \cdots + 217 $ 16b15 - 16b14 - 16b13 + 16b12 - 36b11 + 88b10 - 88b9 - 60b4 + 60b3 + 96b2 - 160*b1 + 217
$\nu^{13}$	$=$	$ ( 408 \beta_{15} + 408 \beta_{14} - 472 \beta_{13} - 472 \beta_{12} - 284 \beta_{10} + \cdots + 140 \beta_{3} ) / 2 $ (408b15 + 408b14 - 472b13 - 472b12 - 284b10 - 284b9 - 276b8 + 357b7 - 503b6 + 284b5 + 140b4 + 140b3) / 2
$\nu^{14}$	$=$	$ ( - 376 \beta_{15} + 376 \beta_{14} + 100 \beta_{13} - 100 \beta_{12} + 68 \beta_{11} - 320 \beta_{10} + \cdots + 1830 ) / 2 $ (-376b15 + 376b14 + 100b13 - 100b12 + 68b11 - 320b10 + 320b9 - 149b4 + 149b3 + 349b2 - 1463*b1 + 1830) / 2
$\nu^{15}$	$=$	$ ( 721 \beta_{15} + 721 \beta_{14} - 161 \beta_{13} - 161 \beta_{12} - 898 \beta_{10} - 898 \beta_{9} + \cdots + 32 \beta_{3} ) / 2 $ (721b15 + 721b14 - 161b13 - 161b12 - 898b10 - 898b9 + 73b8 + 705b7 + 480b6 - 1455b5 + 32b4 + 32b3) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$.

$n$	$677$	$1691$	$1861$
$\chi(n)$	$-1$	$1$	$-1$

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} $

1689.1

−0.517063	−	1.65307i
0.517063	−	1.65307i
−1.42836	−	0.979681i
1.42836	−	0.979681i
1.56631	−	0.739379i
−1.56631	−	0.739379i
1.72890	−	0.104392i
−1.72890	−	0.104392i
1.72890	+	0.104392i
−1.72890	+	0.104392i
1.56631	+	0.739379i
−1.56631	+	0.739379i
−1.42836	+	0.979681i
1.42836	+	0.979681i
−0.517063	+	1.65307i
0.517063	+	1.65307i

−

3.30614i

−0.877236

−

2.05681i

−1.03413

−7.93058

1689.2

−

3.30614i

0.877236

2.05681i

1.03413

−7.93058

1689.3

−

1.95936i

−2.16188

−

0.571200i

−2.85673

−0.839100

1689.4

−

1.95936i

2.16188

0.571200i

2.85673

−0.839100

1689.5

−

1.47876i

−0.494086

−

2.18080i

3.13261

0.813273

1689.6

−

1.47876i

0.494086

2.18080i

−3.13261

0.813273

1689.7

−

0.208784i

−1.60081

−

1.56122i

3.45780

2.95641

1689.8

−

0.208784i

1.60081

1.56122i

−3.45780

2.95641

1689.9

0.208784i

−1.60081

1.56122i

3.45780

2.95641

1689.10

0.208784i

1.60081

−

1.56122i

−3.45780

2.95641

1689.11

1.47876i

−0.494086

2.18080i

3.13261

0.813273

1689.12

1.47876i

0.494086

−

2.18080i

−3.13261

0.813273

1689.13

1.95936i

−2.16188

0.571200i

−2.85673

−0.839100

1689.14

1.95936i

2.16188

−

0.571200i

2.85673

−0.839100

1689.15

3.30614i

−0.877236

2.05681i

−1.03413

−7.93058

1689.16

3.30614i

0.877236

−

2.05681i

1.03413

−7.93058

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3380.2.d.d		16
5.b	even	2	1	inner	3380.2.d.d		16
13.b	even	2	1	inner	3380.2.d.d		16
13.c	even	3	1		260.2.z.a	✓	16
13.d	odd	4	2		3380.2.c.e		16
13.e	even	6	1		260.2.z.a	✓	16
39.h	odd	6	1		2340.2.cr.a		16
39.i	odd	6	1		2340.2.cr.a		16
52.i	odd	6	1		1040.2.df.d		16
52.j	odd	6	1		1040.2.df.d		16
65.d	even	2	1	inner	3380.2.d.d		16
65.g	odd	4	2		3380.2.c.e		16
65.l	even	6	1		260.2.z.a	✓	16
65.n	even	6	1		260.2.z.a	✓	16
65.q	odd	12	2		1300.2.y.e		16
65.r	odd	12	2		1300.2.y.e		16
195.x	odd	6	1		2340.2.cr.a		16
195.y	odd	6	1		2340.2.cr.a		16
260.v	odd	6	1		1040.2.df.d		16
260.w	odd	6	1		1040.2.df.d		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
260.2.z.a	✓	16	13.c	even	3	1
260.2.z.a	✓	16	13.e	even	6	1
260.2.z.a	✓	16	65.l	even	6	1
260.2.z.a	✓	16	65.n	even	6	1
1040.2.df.d		16	52.i	odd	6	1
1040.2.df.d		16	52.j	odd	6	1
1040.2.df.d		16	260.v	odd	6	1
1040.2.df.d		16	260.w	odd	6	1
1300.2.y.e		16	65.q	odd	12	2
1300.2.y.e		16	65.r	odd	12	2
2340.2.cr.a		16	39.h	odd	6	1
2340.2.cr.a		16	39.i	odd	6	1
2340.2.cr.a		16	195.x	odd	6	1
2340.2.cr.a		16	195.y	odd	6	1
3380.2.c.e		16	13.d	odd	4	2
3380.2.c.e		16	65.g	odd	4	2
3380.2.d.d		16	1.a	even	1	1	trivial
3380.2.d.d		16	5.b	even	2	1	inner
3380.2.d.d		16	13.b	even	2	1	inner
3380.2.d.d		16	65.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 $S_{2}^{\mathrm{new}}(3380, [\chi])$:

$ T_{3}^{8} + 17T_{3}^{6} + 75T_{3}^{4} + 95T_{3}^{2} + 4 $

$ T_{7}^{8} - 31T_{7}^{6} + 327T_{7}^{4} - 1273T_{7}^{2} + 1024 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} + 17 T^{6} + 75 T^{4} + \cdots + 4)^{2} $ (T^8 + 17T^6 + 75T^4 + 95*T^2 + 4)^2
$5$	$ T^{16} + 7 T^{14} + \cdots + 390625 $ T^16 + 7T^14 + 22T^12 + T^10 - 14T^8 + 25T^6 + 13750T^4 + 109375T^2 + 390625
$7$	$ (T^{8} - 31 T^{6} + \cdots + 1024)^{2} $ (T^8 - 31T^6 + 327T^4 - 1273*T^2 + 1024)^2
$11$	$ (T^{8} + 65 T^{6} + \cdots + 4096)^{2} $ (T^8 + 65T^6 + 1323T^4 + 8531*T^2 + 4096)^2
$13$	$ T^{16} $ T^16
$17$	$ (T^{8} + 80 T^{6} + \cdots + 52441)^{2} $ (T^8 + 80T^6 + 1926T^4 + 17744*T^2 + 52441)^2
$19$	$ (T^{8} + 165 T^{6} + \cdots + 412164)^{2} $ (T^8 + 165T^6 + 9015T^4 + 170271*T^2 + 412164)^2
$23$	$ (T^{8} + 89 T^{6} + \cdots + 1156)^{2} $ (T^8 + 89T^6 + 2091T^4 + 9335*T^2 + 1156)^2
$29$	$ (T^{4} + 6 T^{3} + \cdots - 393)^{4} $ (T^4 + 6T^3 - 48T^2 - 318*T - 393)^4
$31$	$ (T^{8} + 108 T^{6} + \cdots + 2304)^{2} $ (T^8 + 108T^6 + 2688T^4 + 6336*T^2 + 2304)^2
$37$	$ (T^{8} - 160 T^{6} + \cdots + 292681)^{2} $ (T^8 - 160T^6 + 8598T^4 - 160576*T^2 + 292681)^2
$41$	$ (T^{8} + 164 T^{6} + \cdots + 1771561)^{2} $ (T^8 + 164T^6 + 9438T^4 + 221780*T^2 + 1771561)^2
$43$	$ (T^{8} + 237 T^{6} + \cdots + 535824)^{2} $ (T^8 + 237T^6 + 13839T^4 + 174483*T^2 + 535824)^2
$47$	$ (T^{8} - 132 T^{6} + \cdots + 147456)^{2} $ (T^8 - 132T^6 + 4992T^4 - 55296*T^2 + 147456)^2
$53$	$ (T^{8} + 191 T^{6} + \cdots + 30976)^{2} $ (T^8 + 191T^6 + 8004T^4 + 94256*T^2 + 30976)^2
$59$	$ (T^{8} + 65 T^{6} + \cdots + 1936)^{2} $ (T^8 + 65T^6 + 1227T^4 + 5987*T^2 + 1936)^2
$61$	$ (T^{4} - 14 T^{3} + \cdots - 1157)^{4} $ (T^4 - 14T^3 + 12T^2 + 406*T - 1157)^4
$67$	$ (T^{8} - 451 T^{6} + \cdots + 850084)^{2} $ (T^8 - 451T^6 + 59235T^4 - 2128405*T^2 + 850084)^2
$71$	$ (T^{8} + 341 T^{6} + \cdots + 3104644)^{2} $ (T^8 + 341T^6 + 32151T^4 + 699791*T^2 + 3104644)^2
$73$	$ (T^{8} - 133 T^{6} + \cdots + 861184)^{2} $ (T^8 - 133T^6 + 6276T^4 - 123712*T^2 + 861184)^2
$79$	$ (T^{4} + 4 T^{3} + \cdots + 4096)^{4} $ (T^4 + 4T^3 - 156T^2 - 224*T + 4096)^4
$83$	$ (T^{8} - 228 T^{6} + \cdots + 36864)^{2} $ (T^8 - 228T^6 + 9168T^4 - 78336*T^2 + 36864)^2
$89$	$ (T^{8} + 65 T^{6} + \cdots + 1936)^{2} $ (T^8 + 65T^6 + 1227T^4 + 5987*T^2 + 1936)^2
$97$	$ (T^{8} - 331 T^{6} + \cdots + 3020644)^{2} $ (T^8 - 331T^6 + 20931T^4 - 448621*T^2 + 3020644)^2
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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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{6} q^{3} + \beta_{9} q^{5} + \beta_{7} q^{7} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) q + b6 * q^3 + b9 * q^5 + b7 * q^7 + (b2 - 1) * q^9	\( q + \beta_{6} q^{3} + \beta_{9} q^{5} + \beta_{7} q^{7} + (\beta_{2} - 1) q^{9} + ( - \beta_{4} + \beta_{3} + \beta_1) q^{11} + (\beta_{10} - \beta_{5} - \beta_{4}) q^{15} + ( - \beta_{13} - \beta_{12} + \cdots - \beta_{6}) q^{17}+ \cdots + (\beta_{11} - 3 \beta_{10} + \cdots - 4 \beta_1) q^{99}+O(q^{100}) \) q + b6 * q^3 + b9 * q^5 + b7 * q^7 + (b2 - 1) * q^9 + (-b4 + b3 + b1) * q^11 + (b10 - b5 - b4) * q^15 + (-b13 - b12 - b8 - b6) * q^17 + (b11 - b1) * q^19 + (-b4 + b3 + b1) * q^21 + (b8 - b6) * q^23 + (-b14 - b6 - 1) * q^25 + (b15 + b14 - b13 - b12 + b8 - 3b6) q^27 + (b13 - b12 + b2 - 2) * q^29 + (-b10 + b9 + b4 - b3) * q^31 + (-2b10 - 2b9 - 2b7 + b5) q^33 + (b15 + b12 + b8 - b2) * q^35 + (-b7 + b4 + b3) * q^37 + (b10 - b9 + b4 - b3 + b1) * q^41 + (-b15 - b14 + b13 + b12 + 2b8 + b6) q^43 + (-b11 + b10 - b9 - b7 + b5 - b4 + 2b3 + 3b1) * q^45 + (-2b10 - 2b9) * q^47 + (b2 + 1) * q^49 + (-b15 + b14 + 3) * q^51 + (-b15 - b14 + b8) * q^53 + (b15 - b13 - 2b12 - b8 + b6 - b2 - 1) q^55 + (-b10 - b9 - b7 - 4b5 - b4 - b3) q^57 + (-b10 + b9 - b1) * q^59 + (-b13 + b12 - b2 + 4) * q^61 + (-2b10 - 2b9 + b7 + b5) * q^63 + (3b10 + 3b9 - b7 + 2b5 + b4 + b3) q^67 + (b13 - b12 - 2b2 + 3) q^69 + (-b11 + 2b10 - 2b9 - b1) * q^71 + (-b7 + b5 + b4 + b3) * q^73 + (b15 + b14 - 2b13 - 2b12 - b8 - 3b6 - b2 + 5) q^75 + (b15 + b14 - b13 - b12 + b8 + 3b6) q^77 + (b15 - b14 + b2 - 1) * q^79 + (-b15 + b14 + 2b13 - 2b12 - b2 + 4) * q^81 + (-b10 - b9 + 2b7 - b4 - b3) q^83 + (-b10 + 2b7 + b5 + 2b3 - b1) * q^85 + (-b8 - 3b6) q^87 + (-b10 + b9 - b1) * q^89 + (3b10 + 3b9 + 2b7 - 2b5 - b4 - b3) * q^93 + (3b15 + 2b14 - 2b13 - 3b12 - b8 - b6 + 2b2) q^95 + (2b10 + 2b9 + b7 - 2b5) q^97 + (b11 - 3b10 + 3b9 + 2b4 - 2b3 - 4b1) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 20 q^{9}+O(q^{10}) \) 16 * q - 20 * q^9	\( 16 q - 20 q^{9} - 14 q^{25} - 24 q^{29} + 12 q^{49} + 44 q^{51} - 4 q^{55} + 56 q^{61} + 68 q^{69} + 84 q^{75} - 16 q^{79} + 88 q^{81}+O(q^{100}) \) 16 * q - 20 * q^9 - 14 * q^25 - 24 * q^29 + 12 * q^49 + 44 * q^51 - 4 * q^55 + 56 * q^61 + 68 * q^69 + 84 * q^75 - 16 * q^79 + 88 * q^81

Introduction

L-functions

Modular forms

Varieties

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

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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	3380.2.d.d

Level	$3380$
Weight	$2$
Character orbit	3380.d
Analytic conductor	$26.989$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(26.9894358832\)
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:	\( x^{16} - 7x^{14} + 21x^{12} - 22x^{10} - 26x^{8} - 198x^{6} + 1701x^{4} - 5103x^{2} + 6561 \) x^16 - 7x^14 + 21x^12 - 22x^10 - 26x^8 - 198x^6 + 1701x^4 - 5103*x^2 + 6561
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{16} \)
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( -5\nu^{14} + 2\nu^{12} - 63\nu^{10} + 11\nu^{8} + 289\nu^{6} - 69\nu^{4} - 1674\nu^{2} + 16281 ) / 7776 \) (-5v^14 + 2v^12 - 63v^10 + 11v^8 + 289v^6 - 69v^4 - 1674*v^2 + 16281) / 7776
\(\beta_{2}\)	\(=\)	\( ( -\nu^{14} + 7\nu^{12} - 21\nu^{10} + 22\nu^{8} + 26\nu^{6} + 198\nu^{4} - 972\nu^{2} + 3645 ) / 729 \) (-v^14 + 7v^12 - 21v^10 + 22v^8 + 26v^6 + 198v^4 - 972v^2 + 3645) / 729
\(\beta_{3}\)	\(=\)	\( ( 25 \nu^{15} + 141 \nu^{14} + 518 \nu^{13} - 690 \nu^{12} - 1653 \nu^{11} + 2583 \nu^{10} + \cdots - 636417 ) / 139968 \) (25v^15 + 141v^14 + 518v^13 - 690v^12 - 1653v^11 + 2583v^10 - 1063v^9 - 2211v^8 + 3787v^7 - 5097v^6 - 855v^5 - 18387v^4 - 137214v^3 + 248346v^2 + 276291*v - 636417) / 139968
\(\beta_{4}\)	\(=\)	\( ( 25 \nu^{15} - 141 \nu^{14} + 518 \nu^{13} + 690 \nu^{12} - 1653 \nu^{11} - 2583 \nu^{10} + \cdots + 636417 ) / 139968 \) (25v^15 - 141v^14 + 518v^13 + 690v^12 - 1653v^11 - 2583v^10 - 1063v^9 + 2211v^8 + 3787v^7 + 5097v^6 - 855v^5 + 18387v^4 - 137214v^3 - 248346v^2 + 276291*v + 636417) / 139968
\(\beta_{5}\)	\(=\)	\( ( 11\nu^{15} - 158\nu^{13} + 393\nu^{11} + 163\nu^{9} + 281\nu^{7} - 10845\nu^{5} + 50382\nu^{3} - 75087\nu ) / 34992 \) (11v^15 - 158v^13 + 393v^11 + 163v^9 + 281v^7 - 10845v^5 + 50382v^3 - 75087v) / 34992
\(\beta_{6}\)	\(=\)	\( ( \nu^{15} - 7\nu^{13} + 21\nu^{11} - 22\nu^{9} - 26\nu^{7} - 198\nu^{5} + 1701\nu^{3} - 2916\nu ) / 2187 \) (v^15 - 7v^13 + 21v^11 - 22v^9 - 26v^7 - 198v^5 + 1701v^3 - 2916*v) / 2187
\(\beta_{7}\)	\(=\)	\( ( -\nu^{15} + 7\nu^{13} - 21\nu^{11} + 22\nu^{9} + 26\nu^{7} + 198\nu^{5} - 1701\nu^{3} + 7290\nu ) / 2187 \) (-v^15 + 7v^13 - 21v^11 + 22v^9 + 26v^7 + 198v^5 - 1701v^3 + 7290*v) / 2187
\(\beta_{8}\)	\(=\)	\( ( 5\nu^{15} - 44\nu^{13} + 6\nu^{11} + 106\nu^{9} - 418\nu^{7} - 1566\nu^{5} + 13041\nu^{3} - 7290\nu ) / 8748 \) (5v^15 - 44v^13 + 6v^11 + 106v^9 - 418v^7 - 1566v^5 + 13041v^3 - 7290v) / 8748
\(\beta_{9}\)	\(=\)	\( ( 107 \nu^{15} + 399 \nu^{14} - 686 \nu^{13} - 1902 \nu^{12} + 1401 \nu^{11} + 3357 \nu^{10} + \cdots - 789507 ) / 139968 \) (107v^15 + 399v^14 - 686v^13 - 1902v^12 + 1401v^11 + 3357v^10 + 1075v^9 + 3615v^8 - 5383v^7 - 8835v^6 - 33597v^5 - 87345v^4 + 150174v^3 + 437886v^2 - 285039*v - 789507) / 139968
\(\beta_{10}\)	\(=\)	\( ( 107 \nu^{15} - 399 \nu^{14} - 686 \nu^{13} + 1902 \nu^{12} + 1401 \nu^{11} - 3357 \nu^{10} + \cdots + 789507 ) / 139968 \) (107v^15 - 399v^14 - 686v^13 + 1902v^12 + 1401v^11 - 3357v^10 + 1075v^9 - 3615v^8 - 5383v^7 + 8835v^6 - 33597v^5 + 87345v^4 + 150174v^3 - 437886v^2 - 285039*v + 789507) / 139968
\(\beta_{11}\)	\(=\)	\( ( -79\nu^{14} + 286\nu^{12} - 573\nu^{10} - 575\nu^{8} - 253\nu^{6} + 21585\nu^{4} - 67230\nu^{2} + 132435 ) / 7776 \) (-79v^14 + 286v^12 - 573v^10 - 575v^8 - 253v^6 + 21585v^4 - 67230*v^2 + 132435) / 7776
\(\beta_{12}\)	\(=\)	\( ( 31 \nu^{15} + 36 \nu^{14} - 100 \nu^{13} - 117 \nu^{12} + 156 \nu^{11} + 54 \nu^{10} + 236 \nu^{9} + \cdots - 41553 ) / 17496 \) (31v^15 + 36v^14 - 100v^13 - 117v^12 + 156v^11 + 54v^10 + 236v^9 + 342v^8 + 508v^7 - 990v^6 - 7560v^5 - 11610v^4 + 18225v^3 + 32562v^2 - 37908*v - 41553) / 17496
\(\beta_{13}\)	\(=\)	\( ( 31 \nu^{15} - 36 \nu^{14} - 100 \nu^{13} + 117 \nu^{12} + 156 \nu^{11} - 54 \nu^{10} + 236 \nu^{9} + \cdots + 41553 ) / 17496 \) (31v^15 - 36v^14 - 100v^13 + 117v^12 + 156v^11 - 54v^10 + 236v^9 - 342v^8 + 508v^7 + 990v^6 - 7560v^5 + 11610v^4 + 18225v^3 - 32562v^2 - 37908*v + 41553) / 17496
\(\beta_{14}\)	\(=\)	\( ( 9 \nu^{15} + 2 \nu^{14} - 36 \nu^{13} + 13 \nu^{12} + 81 \nu^{11} - 66 \nu^{10} + 45 \nu^{9} + \cdots + 10935 ) / 2916 \) (9v^15 + 2v^14 - 36v^13 + 13v^12 + 81v^11 - 66v^10 + 45v^9 - 44v^8 - 99v^7 + 326v^6 - 2079v^5 + 36v^4 + 9072v^3 - 5508v^2 - 18711*v + 10935) / 2916
\(\beta_{15}\)	\(=\)	\( ( 9 \nu^{15} - 2 \nu^{14} - 36 \nu^{13} - 13 \nu^{12} + 81 \nu^{11} + 66 \nu^{10} + 45 \nu^{9} + \cdots - 10935 ) / 2916 \) (9v^15 - 2v^14 - 36v^13 - 13v^12 + 81v^11 + 66v^10 + 45v^9 + 44v^8 - 99v^7 - 326v^6 - 2079v^5 - 36v^4 + 9072v^3 + 5508v^2 - 18711*v - 10935) / 2916

\(\nu\)	\(=\)	\( ( \beta_{7} + \beta_{6} ) / 2 \) (b7 + b6) / 2
\(\nu^{2}\)	\(=\)	\( ( -\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 2 ) / 2 \) (-b4 + b3 + b2 + b1 + 2) / 2
\(\nu^{3}\)	\(=\)	\( ( \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} - 2\beta_{10} - 2\beta_{9} + \beta_{8} + \beta_{7} + \beta_{5} ) / 2 \) (b15 + b14 - b13 - b12 - 2b10 - 2b9 + b8 + b7 + b5) / 2
\(\nu^{4}\)	\(=\)	\( ( - \beta_{15} + \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + \beta_{11} - 3 \beta_{10} + 3 \beta_{9} + \cdots + 1 ) / 2 \) (-b15 + b14 + 2b13 - 2b12 + b11 - 3b10 + 3b9 - b4 + b3 + 2*b2 - b1 + 1) / 2
\(\nu^{5}\)	\(=\)	\( ( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - 6 \beta_{10} - 6 \beta_{9} + 3 \beta_{8} + \cdots - 4 \beta_{3} ) / 2 \) (b15 + b14 + b13 + b12 - 6b10 - 6b9 + 3b8 + 3b6 - 7b5 - 4b4 - 4*b3) / 2
\(\nu^{6}\)	\(=\)	\( -\beta_{15} + \beta_{14} + 4\beta_{13} - 4\beta_{12} - 2\beta_{11} - 2\beta_{4} + 2\beta_{3} + 3\beta_{2} + 8\beta _1 - 6 \) -b15 + b14 + 4b13 - 4b12 - 2b11 - 2b4 + 2b3 + 3b2 + 8*b1 - 6
\(\nu^{7}\)	\(=\)	\( ( - 3 \beta_{15} - 3 \beta_{14} + 13 \beta_{13} + 13 \beta_{12} - 22 \beta_{10} - 22 \beta_{9} + \cdots - 8 \beta_{3} ) / 2 \) (-3b15 - 3b14 + 13b13 + 13b12 - 22b10 - 22b9 + 3b8 - 4b7 + 7b6 + 7b5 - 8b4 - 8b3) / 2
\(\nu^{8}\)	\(=\)	\( ( 7 \beta_{15} - 7 \beta_{14} + 14 \beta_{13} - 14 \beta_{12} - 5 \beta_{11} - 13 \beta_{10} + 13 \beta_{9} + \cdots - 35 ) / 2 \) (7b15 - 7b14 + 14b13 - 14b12 - 5b11 - 13b10 + 13b9 + 21b4 - 21b3 + 22b2 - 23*b1 - 35) / 2
\(\nu^{9}\)	\(=\)	\( ( 13 \beta_{15} + 13 \beta_{14} + 3 \beta_{13} + 3 \beta_{12} - 6 \beta_{10} - 6 \beta_{9} + \cdots - 32 \beta_{3} ) / 2 \) (13b15 + 13b14 + 3b13 + 3b12 - 6b10 - 6b9 - 11b8 + 35b7 - 96b6 - 13b5 - 32b4 - 32b3) / 2
\(\nu^{10}\)	\(=\)	\( ( 24 \beta_{15} - 24 \beta_{14} + 28 \beta_{13} - 28 \beta_{12} - 28 \beta_{11} + 32 \beta_{10} + \cdots + 270 ) / 2 \) (24b15 - 24b14 + 28b13 - 28b12 - 28b11 + 32b10 - 32b9 + 21b4 - 21b3 - 19b2 - 97*b1 + 270) / 2
\(\nu^{11}\)	\(=\)	\( ( 40 \beta_{15} + 40 \beta_{14} - 40 \beta_{13} - 40 \beta_{12} + 20 \beta_{10} + 20 \beta_{9} + \cdots - 36 \beta_{3} ) / 2 \) (40b15 + 40b14 - 40b13 - 40b12 + 20b10 + 20b9 - 156b8 + 175b7 + 93b6 + 12b5 - 36b4 - 36b3) / 2
\(\nu^{12}\)	\(=\)	\( 16 \beta_{15} - 16 \beta_{14} - 16 \beta_{13} + 16 \beta_{12} - 36 \beta_{11} + 88 \beta_{10} + \cdots + 217 \) 16b15 - 16b14 - 16b13 + 16b12 - 36b11 + 88b10 - 88b9 - 60b4 + 60b3 + 96b2 - 160*b1 + 217
\(\nu^{13}\)	\(=\)	\( ( 408 \beta_{15} + 408 \beta_{14} - 472 \beta_{13} - 472 \beta_{12} - 284 \beta_{10} + \cdots + 140 \beta_{3} ) / 2 \) (408b15 + 408b14 - 472b13 - 472b12 - 284b10 - 284b9 - 276b8 + 357b7 - 503b6 + 284b5 + 140b4 + 140b3) / 2
\(\nu^{14}\)	\(=\)	\( ( - 376 \beta_{15} + 376 \beta_{14} + 100 \beta_{13} - 100 \beta_{12} + 68 \beta_{11} - 320 \beta_{10} + \cdots + 1830 ) / 2 \) (-376b15 + 376b14 + 100b13 - 100b12 + 68b11 - 320b10 + 320b9 - 149b4 + 149b3 + 349b2 - 1463*b1 + 1830) / 2
\(\nu^{15}\)	\(=\)	\( ( 721 \beta_{15} + 721 \beta_{14} - 161 \beta_{13} - 161 \beta_{12} - 898 \beta_{10} - 898 \beta_{9} + \cdots + 32 \beta_{3} ) / 2 \) (721b15 + 721b14 - 161b13 - 161b12 - 898b10 - 898b9 + 73b8 + 705b7 + 480b6 - 1455b5 + 32b4 + 32b3) / 2

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1689.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} + 17 T^{6} + 75 T^{4} + \cdots + 4)^{2} \) (T^8 + 17T^6 + 75T^4 + 95*T^2 + 4)^2
$5$	\( T^{16} + 7 T^{14} + \cdots + 390625 \) T^16 + 7T^14 + 22T^12 + T^10 - 14T^8 + 25T^6 + 13750T^4 + 109375T^2 + 390625
$7$	\( (T^{8} - 31 T^{6} + \cdots + 1024)^{2} \) (T^8 - 31T^6 + 327T^4 - 1273*T^2 + 1024)^2
$11$	\( (T^{8} + 65 T^{6} + \cdots + 4096)^{2} \) (T^8 + 65T^6 + 1323T^4 + 8531*T^2 + 4096)^2
$13$	\( T^{16} \) T^16
$17$	\( (T^{8} + 80 T^{6} + \cdots + 52441)^{2} \) (T^8 + 80T^6 + 1926T^4 + 17744*T^2 + 52441)^2
$19$	\( (T^{8} + 165 T^{6} + \cdots + 412164)^{2} \) (T^8 + 165T^6 + 9015T^4 + 170271*T^2 + 412164)^2
$23$	\( (T^{8} + 89 T^{6} + \cdots + 1156)^{2} \) (T^8 + 89T^6 + 2091T^4 + 9335*T^2 + 1156)^2
$29$	\( (T^{4} + 6 T^{3} + \cdots - 393)^{4} \) (T^4 + 6T^3 - 48T^2 - 318*T - 393)^4
$31$	\( (T^{8} + 108 T^{6} + \cdots + 2304)^{2} \) (T^8 + 108T^6 + 2688T^4 + 6336*T^2 + 2304)^2
$37$	\( (T^{8} - 160 T^{6} + \cdots + 292681)^{2} \) (T^8 - 160T^6 + 8598T^4 - 160576*T^2 + 292681)^2
$41$	\( (T^{8} + 164 T^{6} + \cdots + 1771561)^{2} \) (T^8 + 164T^6 + 9438T^4 + 221780*T^2 + 1771561)^2
$43$	\( (T^{8} + 237 T^{6} + \cdots + 535824)^{2} \) (T^8 + 237T^6 + 13839T^4 + 174483*T^2 + 535824)^2
$47$	\( (T^{8} - 132 T^{6} + \cdots + 147456)^{2} \) (T^8 - 132T^6 + 4992T^4 - 55296*T^2 + 147456)^2
$53$	\( (T^{8} + 191 T^{6} + \cdots + 30976)^{2} \) (T^8 + 191T^6 + 8004T^4 + 94256*T^2 + 30976)^2
$59$	\( (T^{8} + 65 T^{6} + \cdots + 1936)^{2} \) (T^8 + 65T^6 + 1227T^4 + 5987*T^2 + 1936)^2
$61$	\( (T^{4} - 14 T^{3} + \cdots - 1157)^{4} \) (T^4 - 14T^3 + 12T^2 + 406*T - 1157)^4
$67$	\( (T^{8} - 451 T^{6} + \cdots + 850084)^{2} \) (T^8 - 451T^6 + 59235T^4 - 2128405*T^2 + 850084)^2
$71$	\( (T^{8} + 341 T^{6} + \cdots + 3104644)^{2} \) (T^8 + 341T^6 + 32151T^4 + 699791*T^2 + 3104644)^2
$73$	\( (T^{8} - 133 T^{6} + \cdots + 861184)^{2} \) (T^8 - 133T^6 + 6276T^4 - 123712*T^2 + 861184)^2
$79$	\( (T^{4} + 4 T^{3} + \cdots + 4096)^{4} \) (T^4 + 4T^3 - 156T^2 - 224*T + 4096)^4
$83$	\( (T^{8} - 228 T^{6} + \cdots + 36864)^{2} \) (T^8 - 228T^6 + 9168T^4 - 78336*T^2 + 36864)^2
$89$	\( (T^{8} + 65 T^{6} + \cdots + 1936)^{2} \) (T^8 + 65T^6 + 1227T^4 + 5987*T^2 + 1936)^2
$97$	\( (T^{8} - 331 T^{6} + \cdots + 3020644)^{2} \) (T^8 - 331T^6 + 20931T^4 - 448621*T^2 + 3020644)^2
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\(n\)	\(677\)	\(1691\)	\(1861\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)