LMFDB - Newform orbit 357.2.c.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [357,2,Mod(356,357)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(357, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("357.356"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 357 = 3 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	357.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.85065935216$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 214x^{10} + 59049 $ x^20 - 214*x^10 + 59049
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{19}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{8} q^{2} + \beta_{18} q^{3} + ( - \beta_{10} - 2) q^{4} - \beta_{4} q^{5} + (\beta_{6} + \beta_{5}) q^{6} + \beta_{6} q^{7} + (\beta_{19} + \beta_{12} - 2 \beta_{8}) q^{8} - \beta_{19} q^{9}+ \cdots + 7 \beta_{8} q^{98}+O(q^{100}) $ q + b8 * q^2 + b18 * q^3 + (-b10 - 2) * q^4 - b4 * q^5 + (b6 + b5) * q^6 + b6 * q^7 + (b19 + b12 - 2b8) q^8 - b19 * q^9 + (-b16 + b11 - b5 - b2) * q^10 + (-2b18 - b13 - b4 - b2) q^12 + (-b18 + b15 - b2) * q^14 + (-b12 + b10 - b1) * q^15 + (2b10 - b9 + b1 + 4) q^16 + (b6 + b5 - b3) * q^17 + (-b14 - b10 + b9) * q^18 + (2b18 + b16 - b6 - b5 + 2b4 + b3 - b2) * q^20 + (b9 - b8) * q^21 + (-b16 + b15 - b11 - 2b6 - 3b5 + b4 + b3 - b2) * q^24 + (b17 + b14 - b9 + b8 + b1 - 5) * q^25 + (b13 + b11) * q^27 + (2b13 - b11 - 2b6 - 2b5 + b4 + b3) q^28 + (-b19 + 2b14 - b10 + 2b8 - b7 - b1) * q^30 + (-2b13 + 2b5 - b4) * q^31 + (-2b19 - b17 - b14 - 2b12 + 4b8 + 2b7 + 1) * q^32 + (-3b18 - b15 - b5 + b3 - b2) q^34 + (b19 - b17 + b14 - b12 + b10) * q^35 + (2b19 - 2b8 - b7 - b1 - 3) * q^36 + (3b18 + b15 - 2b11 - b6 + b5 + b3 + b2) * q^40 + (-b18 - b16 - b15 + 2b2) q^41 + (2b19 + b17 + b12 + b10 - b7 + 3) q^42 + (-b17 - b14 - b9 - b8 + b1) * q^43 + (b16 - b15 + b6 + 2b2) q^45 + (4b18 + 3b13 - b11 - 2b6 + b4 + b3 + 2b2) * q^48 + 7 * q^49 + (-2b19 - 2b14 - b10 - 5b8 + 2b7 + 1) * q^50 + (-b17 - b14 + b9 + b8) * q^51 + (b19 + b17 - b14 + 3b12 - b10) q^53 + (b16 - b15 - b13 + b11 + b5 + b4 - b2) * q^54 + (2b18 - 2b15 + 2b13 + b11 + b5 + 2b3 + 2b2) q^56 + (-b19 + 2b17 + b14 + 3b12 - b10 - b9 - b8 - b7 + 2b1 - 6) q^60 + (3b18 + b16 + b15 + 2b5 - 2b3 + 2b2) * q^61 + (-4b18 - b16 + 2b15 - 2b13 - b11 - b5 - 2b4 - 2b3 - b2) q^62 + (-b13 + b11 + b4 - 2b3) q^63 + (-b17 - b14 - 4b10 + 4b9 - b8 - 4b1 - 8) q^64 + (3b19 + b17 - b14 - b12 + b10) q^67 + (-2b13 - b11 - 2b6 - 3b5 + b4) q^68 + (-3b19 + b17 + 3b14 + b12 - b9 + 2b8 + b1) q^70 + (2b14 + 2b10 - 2b9 - 2b8 - b7 + 2b1 + 6) q^72 + (-3b18 + 3b16 - b15 + 2b5 - 2b3 + 2b2) q^73 + (-5b18 + b13 - b11 + b6 - b4 - 2b3) * q^75 + (b18 - b15 + 2b13 + b11 + 2b6 + 3b5 - b4 + b2) q^80 + (b17 + b14 + 4b8 - b1) q^81 + (-2b13 + 3b11 + 5b6 + 2b5 - b4 - 3b3) q^82 + (b14 - 2b12 + 3b10 - 2b9 + 4b8 + 2b1) q^84 + (-3b19 - b17 + b14 + b12 - 3b10) * q^85 + (-2b19 - 2b17 - 4b12 + b10 + 2b7 + 1) * q^86 + (b18 + 3b15 - 2b13 + b11 + 4b6 + 3b5 - 3b4 - 3b3 - b2) * q^90 + (-2b14 + b12 - 3b10 - 4b8 - b1) q^93 + (-b18 - 2b16 - 4b15 + 2b11 + 4b6 + 3b5 - 2b4 + b3 + 2b2) q^96 + (2b13 + 2b11 - 2b5 + b4 - 2b3) * q^97 + 7b8 q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 40 q^{4} + 80 q^{16} - 100 q^{25} + 10 q^{30} - 50 q^{36} + 70 q^{42} + 140 q^{49} - 110 q^{60} - 160 q^{64} + 130 q^{72}+O(q^{100}) $ 20 * q - 40 * q^4 + 80 * q^16 - 100 * q^25 + 10 * q^30 - 50 * q^36 + 70 * q^42 + 140 * q^49 - 110 * q^60 - 160 * q^64 + 130 * q^72

Basis of coefficient ring in terms of a root $\nu$ of $ x^{20} - 214x^{10} + 59049 $ x^20 - 214*x^10 + 59049 :

$\beta_{1}$	$=$	$ ( -\nu^{16} - 1973\nu^{6} ) / 14580 $ (-v^16 - 1973*v^6) / 14580
$\beta_{2}$	$=$	$ ( \nu^{15} + 27\nu^{11} + 353\nu^{5} - 189\nu ) / 3240 $ (v^15 + 27v^11 + 353v^5 - 189*v) / 3240
$\beta_{3}$	$=$	$ ( \nu^{15} - 3\nu^{13} + 353\nu^{5} - 1059\nu^{3} ) / 3240 $ (v^15 - 3v^13 + 353v^5 - 1059*v^3) / 3240
$\beta_{4}$	$=$	$ ( -\nu^{17} - 81\nu^{13} - 1973\nu^{7} + 15147\nu^{3} ) / 43740 $ (-v^17 - 81v^13 - 1973v^7 + 15147*v^3) / 43740
$\beta_{5}$	$=$	$ ( -\nu^{15} - 3\nu^{13} - 353\nu^{5} - 1059\nu^{3} ) / 3240 $ (-v^15 - 3v^13 - 353v^5 - 1059*v^3) / 3240
$\beta_{6}$	$=$	$ ( -\nu^{15} + 457\nu^{5} ) / 2430 $ (-v^15 + 457*v^5) / 2430
$\beta_{7}$	$=$	$ ( -7\nu^{16} + 18\nu^{14} + 729\nu^{10} + 769\nu^{6} + 6354\nu^{4} - 92583 ) / 29160 $ (-7v^16 + 18v^14 + 729v^10 + 769v^6 + 6354*v^4 - 92583) / 29160
$\beta_{8}$	$=$	$ ( -7\nu^{16} - 9\nu^{14} + 769\nu^{6} - 3177\nu^{4} ) / 29160 $ (-7v^16 - 9v^14 + 769v^6 - 3177v^4) / 29160
$\beta_{9}$	$=$	$ ( -7\nu^{16} - 45\nu^{14} + 769\nu^{6} + 13275\nu^{4} ) / 29160 $ (-7v^16 - 45v^14 + 769v^6 + 13275v^4) / 29160
$\beta_{10}$	$=$	$ ( -23\nu^{18} + 729\nu^{12} - 1639\nu^{8} - 5103\nu^{2} ) / 262440 $ (-23v^18 + 729v^12 - 1639v^8 - 5103v^2) / 262440
$\beta_{11}$	$=$	$ ( -19\nu^{17} + 27\nu^{15} - 162\nu^{13} + 6253\nu^{7} + 9531\nu^{5} + 30294\nu^{3} ) / 87480 $ (-19v^17 + 27v^15 - 162v^13 + 6253v^7 + 9531v^5 + 30294v^3) / 87480
$\beta_{12}$	$=$	$ ( -23\nu^{18} - 729\nu^{12} - 1639\nu^{8} + 267543\nu^{2} ) / 262440 $ (-23v^18 - 729v^12 - 1639v^8 + 267543v^2) / 262440
$\beta_{13}$	$=$	$ ( -7\nu^{17} - 9\nu^{15} + 54\nu^{13} + 769\nu^{7} - 3177\nu^{5} - 10098\nu^{3} ) / 29160 $ (-7v^17 - 9v^15 + 54v^13 + 769v^7 - 3177v^5 - 10098v^3) / 29160
$\beta_{14}$	$=$	$ ( 23\nu^{18} - 63\nu^{16} + 162\nu^{14} + 1639\nu^{8} + 6921\nu^{6} + 57186\nu^{4} + 262440\nu^{2} ) / 262440 $ (23v^18 - 63v^16 + 162v^14 + 1639v^8 + 6921v^6 + 57186v^4 + 262440*v^2) / 262440
$\beta_{15}$	$=$	$ ( -29\nu^{19} + 243\nu^{15} - 13477\nu^{9} + 85779\nu^{5} + 1574640\nu ) / 787320 $ (-29v^19 + 243v^15 - 13477v^9 + 85779v^5 + 1574640*v) / 787320
$\beta_{16}$	$=$	$ ( -29\nu^{19} + 243\nu^{15} - 13477\nu^{9} + 85779\nu^{5} - 787320\nu ) / 787320 $ (-29v^19 + 243v^15 - 13477v^9 + 85779v^5 - 787320*v) / 787320
$\beta_{17}$	$=$	$ ( -23\nu^{18} - 63\nu^{16} + 162\nu^{14} - 1639\nu^{8} + 6921\nu^{6} + 57186\nu^{4} - 262440\nu^{2} ) / 262440 $ (-23v^18 - 63v^16 + 162v^14 - 1639v^8 + 6921v^6 + 57186v^4 - 262440*v^2) / 262440
$\beta_{18}$	$=$	$ ( -\nu^{19} + 214\nu^{9} ) / 19683 $ (-v^19 + 214*v^9) / 19683
$\beta_{19}$	$=$	$ ( \nu^{18} - 214\nu^{8} ) / 6561 $ (v^18 - 214*v^8) / 6561

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

$\nu$	$=$	$ ( -\beta_{16} + \beta_{15} ) / 3 $ (-b16 + b15) / 3
$\nu^{2}$	$=$	$ ( -\beta_{17} + \beta_{14} + \beta_{12} + \beta_{10} ) / 3 $ (-b17 + b14 + b12 + b10) / 3
$\nu^{3}$	$=$	$ ( -\beta_{13} + \beta_{11} - 2\beta_{5} + \beta_{4} - 4\beta_{3} ) / 3 $ (-b13 + b11 - 2b5 + b4 - 4b3) / 3
$\nu^{4}$	$=$	$ ( 2\beta_{17} + 2\beta_{14} + 3\beta_{9} - 7\beta_{8} ) / 3 $ (2b17 + 2b14 + 3b9 - 7b8) / 3
$\nu^{5}$	$=$	$ 3\beta_{6} - 2\beta_{5} + 2\beta_{3} $ 3b6 - 2b5 + 2*b3
$\nu^{6}$	$=$	$ ( \beta_{17} + \beta_{14} + 4\beta_{8} - 21\beta_1 ) / 3 $ (b17 + b14 + 4b8 - 21b1) / 3
$\nu^{7}$	$=$	$ ( -17\beta_{13} + 23\beta_{11} + 20\beta_{5} - 40\beta_{4} - 20\beta_{3} ) / 3 $ (-17b13 + 23b11 + 20b5 - 40b4 - 20*b3) / 3
$\nu^{8}$	$=$	$ ( -69\beta_{19} - 20\beta_{17} + 20\beta_{14} - 40\beta_{12} - 40\beta_{10} ) / 3 $ (-69b19 - 20b17 + 20b14 - 40b12 - 40*b10) / 3
$\nu^{9}$	$=$	$ ( 87\beta_{18} - 80\beta_{16} - 40\beta_{15} - 60\beta_{5} + 60\beta_{3} ) / 3 $ (87b18 - 80b16 - 40b15 - 60b5 + 60*b3) / 3
$\nu^{10}$	$=$	$ -20\beta_{17} - 20\beta_{14} + 40\beta_{7} + 127 $ -20b17 - 20b14 + 40*b7 + 127
$\nu^{11}$	$=$	$ ( -7\beta_{16} + 7\beta_{15} + 180\beta_{5} - 180\beta_{3} + 360\beta_{2} ) / 3 $ (-7b16 + 7b15 + 180b5 - 180b3 + 360*b2) / 3
$\nu^{12}$	$=$	$ ( -187\beta_{17} + 187\beta_{14} - 353\beta_{12} + 727\beta_{10} ) / 3 $ (-187b17 + 187b14 - 353b12 + 727b10) / 3
$\nu^{13}$	$=$	$ ( 353\beta_{13} - 353\beta_{11} - 914\beta_{5} - 353\beta_{4} - 208\beta_{3} ) / 3 $ (353b13 - 353b11 - 914b5 - 353b4 - 208*b3) / 3
$\nu^{14}$	$=$	$ ( 914\beta_{17} + 914\beta_{14} - 1059\beta_{9} - 769\beta_{8} ) / 3 $ (914b17 + 914b14 - 1059b9 - 769b8) / 3
$\nu^{15}$	$=$	$ -1059\beta_{6} - 914\beta_{5} + 914\beta_{3} $ -1059b6 - 914b5 + 914*b3
$\nu^{16}$	$=$	$ ( -1973\beta_{17} - 1973\beta_{14} - 7892\beta_{8} - 2307\beta_1 ) / 3 $ (-1973b17 - 1973b14 - 7892b8 - 2307b1) / 3
$\nu^{17}$	$=$	$ ( -10199\beta_{13} - 1639\beta_{11} + 4280\beta_{5} - 8560\beta_{4} - 4280\beta_{3} ) / 3 $ (-10199b13 - 1639b11 + 4280b5 - 8560b4 - 4280*b3) / 3
$\nu^{18}$	$=$	$ ( 4917\beta_{19} - 4280\beta_{17} + 4280\beta_{14} - 8560\beta_{12} - 8560\beta_{10} ) / 3 $ (4917b19 - 4280b17 + 4280b14 - 8560b12 - 8560*b10) / 3
$\nu^{19}$	$=$	$ ( -40431\beta_{18} - 17120\beta_{16} - 8560\beta_{15} - 12840\beta_{5} + 12840\beta_{3} ) / 3 $ (-40431b18 - 17120b16 - 8560b15 - 12840b5 + 12840*b3) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/357\mathbb{Z}\right)^\times$.

$n$	$52$	$190$	$239$
$\chi(n)$	$-1$	$-1$	$-1$

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} $

356.1

−1.50582	−	0.855861i
1.50582	+	0.855861i
−0.715174	+	1.57751i
0.715174	−	1.57751i
−1.72130	−	0.192695i
1.72130	+	0.192695i
−0.348647	+	1.69660i
0.348647	−	1.69660i
−1.27930	+	1.16765i
1.27930	−	1.16765i
−1.27930	−	1.16765i
1.27930	+	1.16765i
−0.348647	−	1.69660i
0.348647	+	1.69660i
−1.72130	+	0.192695i
1.72130	−	0.192695i
−0.715174	−	1.57751i
0.715174	+	1.57751i
−1.50582	+	0.855861i
1.50582	−	0.855861i

−

2.82435i

−1.50582

0.855861i

−5.97697

1.62447i

2.41725

4.25298i

2.64575

11.2324i

1.53501

−

2.57755i

4.58808

356.2

−

2.82435i

1.50582

−

0.855861i

−5.97697

−

1.62447i

−2.41725

−

4.25298i

−2.64575

11.2324i

1.53501

−

2.57755i

−4.58808

356.3

−

2.37414i

−0.715174

−

1.57751i

−3.63655

4.46472i

−3.74522

1.69793i

−2.64575

3.88541i

−1.97705

2.25638i

10.5999

356.4

−

2.37414i

0.715174

1.57751i

−3.63655

−

4.46472i

3.74522

−

1.69793i

2.64575

3.88541i

−1.97705

2.25638i

−10.5999

356.5

−

2.19576i

−1.72130

0.192695i

−2.82135

−

3.46074i

0.423111

3.77955i

−2.64575

1.80349i

2.92574

−

0.663370i

−7.59896

356.6

−

2.19576i

1.72130

−

0.192695i

−2.82135

3.46074i

−0.423111

−

3.77955i

2.64575

1.80349i

2.92574

−

0.663370i

7.59896

356.7

−

1.01709i

−0.348647

−

1.69660i

0.965529

−

1.13488i

−1.72559

0.354605i

−2.64575

−

3.01621i

−2.75689

1.18303i

−1.15427

356.8

−

1.01709i

0.348647

1.69660i

0.965529

1.13488i

1.72559

−

0.354605i

2.64575

−

3.01621i

−2.75689

1.18303i

1.15427

356.9

−

0.728457i

−1.27930

−

1.16765i

1.46935

−

3.76333i

−0.850581

0.931913i

2.64575

−

2.52727i

0.273200

2.98753i

−2.74142

356.10

−

0.728457i

1.27930

1.16765i

1.46935

3.76333i

0.850581

−

0.931913i

−2.64575

−

2.52727i

0.273200

2.98753i

2.74142

356.11

0.728457i

−1.27930

1.16765i

1.46935

3.76333i

−0.850581

−

0.931913i

2.64575

2.52727i

0.273200

−

2.98753i

−2.74142

356.12

0.728457i

1.27930

−

1.16765i

1.46935

−

3.76333i

0.850581

0.931913i

−2.64575

2.52727i

0.273200

−

2.98753i

2.74142

356.13

1.01709i

−0.348647

1.69660i

0.965529

1.13488i

−1.72559

−

0.354605i

−2.64575

3.01621i

−2.75689

−

1.18303i

−1.15427

356.14

1.01709i

0.348647

−

1.69660i

0.965529

−

1.13488i

1.72559

0.354605i

2.64575

3.01621i

−2.75689

−

1.18303i

1.15427

356.15

2.19576i

−1.72130

−

0.192695i

−2.82135

3.46074i

0.423111

−

3.77955i

−2.64575

−

1.80349i

2.92574

0.663370i

−7.59896

356.16

2.19576i

1.72130

0.192695i

−2.82135

−

3.46074i

−0.423111

3.77955i

2.64575

−

1.80349i

2.92574

0.663370i

7.59896

356.17

2.37414i

−0.715174

1.57751i

−3.63655

−

4.46472i

−3.74522

−

1.69793i

−2.64575

−

3.88541i

−1.97705

−

2.25638i

10.5999

356.18

2.37414i

0.715174

−

1.57751i

−3.63655

4.46472i

3.74522

1.69793i

2.64575

−

3.88541i

−1.97705

−

2.25638i

−10.5999

356.19

2.82435i

−1.50582

−

0.855861i

−5.97697

−

1.62447i

2.41725

−

4.25298i

2.64575

−

11.2324i

1.53501

2.57755i

4.58808

356.20

2.82435i

1.50582

0.855861i

−5.97697

1.62447i

−2.41725

4.25298i

−2.64575

−

11.2324i

1.53501

2.57755i

−4.58808

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
119.d	odd	2	1	CM by $\Q(\sqrt{-119}) $
3.b	odd	2	1	inner
7.b	odd	2	1	inner
17.b	even	2	1	inner
21.c	even	2	1	inner
51.c	odd	2	1	inner
357.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	357.2.c.a	✓	20
3.b	odd	2	1	inner	357.2.c.a	✓	20
7.b	odd	2	1	inner	357.2.c.a	✓	20
17.b	even	2	1	inner	357.2.c.a	✓	20
21.c	even	2	1	inner	357.2.c.a	✓	20
51.c	odd	2	1	inner	357.2.c.a	✓	20
119.d	odd	2	1	CM	357.2.c.a	✓	20
357.c	even	2	1	inner	357.2.c.a	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
357.2.c.a	✓	20	1.a	even	1	1	trivial
357.2.c.a	✓	20	3.b	odd	2	1	inner
357.2.c.a	✓	20	7.b	odd	2	1	inner
357.2.c.a	✓	20	17.b	even	2	1	inner
357.2.c.a	✓	20	21.c	even	2	1	inner
357.2.c.a	✓	20	51.c	odd	2	1	inner
357.2.c.a	✓	20	119.d	odd	2	1	CM
357.2.c.a	✓	20	357.c	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{10} + 20T_{2}^{8} + 140T_{2}^{6} + 400T_{2}^{4} + 400T_{2}^{2} + 119 $ T2^10 + 20*T2^8 + 140*T2^6 + 400*T2^4 + 400*T2^2 + 119 acting on $S_{2}^{\mathrm{new}}(357, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{10} + 20 T^{8} + \cdots + 119)^{2} $ (T^10 + 20T^8 + 140T^6 + 400T^4 + 400T^2 + 119)^2
$3$	$ T^{20} - 214 T^{10} + 59049 $ T^20 - 214*T^10 + 59049
$5$	$ (T^{10} + 50 T^{8} + \cdots + 11492)^{2} $ (T^10 + 50T^8 + 875T^6 + 6250T^4 + 15625T^2 + 11492)^2
$7$	$ (T^{2} - 7)^{10} $ (T^2 - 7)^10
$11$	$ T^{20} $ T^20
$13$	$ T^{20} $ T^20
$17$	$ (T^{2} + 17)^{10} $ (T^2 + 17)^10
$19$	$ T^{20} $ T^20
$23$	$ T^{20} $ T^20
$29$	$ T^{20} $ T^20
$31$	$ (T^{10} - 310 T^{8} + \cdots - 101826172)^{2} $ (T^10 - 310T^8 + 33635T^6 - 1489550T^4 + 23088025T^2 - 101826172)^2
$37$	$ T^{20} $ T^20
$41$	$ (T^{10} + 410 T^{8} + \cdots + 409170212)^{2} $ (T^10 + 410T^8 + 58835T^6 + 3446050T^4 + 70644025T^2 + 409170212)^2
$43$	$ (T^{5} - 215 T^{3} + \cdots + 6484)^{4} $ (T^5 - 215T^3 + 9245T + 6484)^4
$47$	$ T^{20} $ T^20
$53$	$ (T^{10} + 530 T^{8} + \cdots + 669539696)^{2} $ (T^10 + 530T^8 + 98315T^6 + 7443850T^4 + 197262025T^2 + 669539696)^2
$59$	$ T^{20} $ T^20
$61$	$ (T^{10} - 610 T^{8} + \cdots - 878707312)^{2} $ (T^10 - 610T^8 + 130235T^6 - 11349050T^4 + 346146025T^2 - 878707312)^2
$67$	$ (T^{5} - 335 T^{3} + \cdots + 27532)^{4} $ (T^5 - 335T^3 + 22445T + 27532)^4
$71$	$ T^{20} $ T^20
$73$	$ (T^{10} - 730 T^{8} + \cdots - 6754804672)^{2} $ (T^10 - 730T^8 + 186515T^6 - 19450850T^4 + 709956025T^2 - 6754804672)^2
$79$	$ T^{20} $ T^20
$83$	$ T^{20} $ T^20
$89$	$ T^{20} $ T^20
$97$	$ (T^{10} - 970 T^{8} + \cdots - 29255547328)^{2} $ (T^10 - 970T^8 + 329315T^6 - 45633650T^4 + 2213232025T^2 - 29255547328)^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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 357 = 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	357.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{8} q^{2} + \beta_{18} q^{3} + ( - \beta_{10} - 2) q^{4} - \beta_{4} q^{5} + (\beta_{6} + \beta_{5}) q^{6} + \beta_{6} q^{7} + (\beta_{19} + \beta_{12} - 2 \beta_{8}) q^{8} - \beta_{19} q^{9}+ \cdots + 7 \beta_{8} q^{98}+O(q^{100}) \) q + b8 * q^2 + b18 * q^3 + (-b10 - 2) * q^4 - b4 * q^5 + (b6 + b5) * q^6 + b6 * q^7 + (b19 + b12 - 2b8) q^8 - b19 * q^9 + (-b16 + b11 - b5 - b2) * q^10 + (-2b18 - b13 - b4 - b2) q^12 + (-b18 + b15 - b2) * q^14 + (-b12 + b10 - b1) * q^15 + (2b10 - b9 + b1 + 4) q^16 + (b6 + b5 - b3) * q^17 + (-b14 - b10 + b9) * q^18 + (2b18 + b16 - b6 - b5 + 2b4 + b3 - b2) * q^20 + (b9 - b8) * q^21 + (-b16 + b15 - b11 - 2b6 - 3b5 + b4 + b3 - b2) * q^24 + (b17 + b14 - b9 + b8 + b1 - 5) * q^25 + (b13 + b11) * q^27 + (2b13 - b11 - 2b6 - 2b5 + b4 + b3) q^28 + (-b19 + 2b14 - b10 + 2b8 - b7 - b1) * q^30 + (-2b13 + 2b5 - b4) * q^31 + (-2b19 - b17 - b14 - 2b12 + 4b8 + 2b7 + 1) * q^32 + (-3b18 - b15 - b5 + b3 - b2) q^34 + (b19 - b17 + b14 - b12 + b10) * q^35 + (2b19 - 2b8 - b7 - b1 - 3) * q^36 + (3b18 + b15 - 2b11 - b6 + b5 + b3 + b2) * q^40 + (-b18 - b16 - b15 + 2b2) q^41 + (2b19 + b17 + b12 + b10 - b7 + 3) q^42 + (-b17 - b14 - b9 - b8 + b1) * q^43 + (b16 - b15 + b6 + 2b2) q^45 + (4b18 + 3b13 - b11 - 2b6 + b4 + b3 + 2b2) * q^48 + 7 * q^49 + (-2b19 - 2b14 - b10 - 5b8 + 2b7 + 1) * q^50 + (-b17 - b14 + b9 + b8) * q^51 + (b19 + b17 - b14 + 3b12 - b10) q^53 + (b16 - b15 - b13 + b11 + b5 + b4 - b2) * q^54 + (2b18 - 2b15 + 2b13 + b11 + b5 + 2b3 + 2b2) q^56 + (-b19 + 2b17 + b14 + 3b12 - b10 - b9 - b8 - b7 + 2b1 - 6) q^60 + (3b18 + b16 + b15 + 2b5 - 2b3 + 2b2) * q^61 + (-4b18 - b16 + 2b15 - 2b13 - b11 - b5 - 2b4 - 2b3 - b2) q^62 + (-b13 + b11 + b4 - 2b3) q^63 + (-b17 - b14 - 4b10 + 4b9 - b8 - 4b1 - 8) q^64 + (3b19 + b17 - b14 - b12 + b10) q^67 + (-2b13 - b11 - 2b6 - 3b5 + b4) q^68 + (-3b19 + b17 + 3b14 + b12 - b9 + 2b8 + b1) q^70 + (2b14 + 2b10 - 2b9 - 2b8 - b7 + 2b1 + 6) q^72 + (-3b18 + 3b16 - b15 + 2b5 - 2b3 + 2b2) q^73 + (-5b18 + b13 - b11 + b6 - b4 - 2b3) * q^75 + (b18 - b15 + 2b13 + b11 + 2b6 + 3b5 - b4 + b2) q^80 + (b17 + b14 + 4b8 - b1) q^81 + (-2b13 + 3b11 + 5b6 + 2b5 - b4 - 3b3) q^82 + (b14 - 2b12 + 3b10 - 2b9 + 4b8 + 2b1) q^84 + (-3b19 - b17 + b14 + b12 - 3b10) * q^85 + (-2b19 - 2b17 - 4b12 + b10 + 2b7 + 1) * q^86 + (b18 + 3b15 - 2b13 + b11 + 4b6 + 3b5 - 3b4 - 3b3 - b2) * q^90 + (-2b14 + b12 - 3b10 - 4b8 - b1) q^93 + (-b18 - 2b16 - 4b15 + 2b11 + 4b6 + 3b5 - 2b4 + b3 + 2b2) q^96 + (2b13 + 2b11 - 2b5 + b4 - 2b3) * q^97 + 7b8 q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 40 q^{4} + 80 q^{16} - 100 q^{25} + 10 q^{30} - 50 q^{36} + 70 q^{42} + 140 q^{49} - 110 q^{60} - 160 q^{64} + 130 q^{72}+O(q^{100}) \) 20 * q - 40 * q^4 + 80 * q^16 - 100 * q^25 + 10 * q^30 - 50 * q^36 + 70 * q^42 + 140 * q^49 - 110 * q^60 - 160 * q^64 + 130 * q^72

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

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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	357.2.c.a

Level	$357$
Weight	$2$
Character orbit	357.c
Analytic conductor	$2.851$
Analytic rank	$0$
Dimension	$20$
CM discriminant	-119
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(2.85065935216\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - 214x^{10} + 59049 \) x^20 - 214*x^10 + 59049
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2}\cdot 3^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

\(\beta_{1}\)	\(=\)	\( ( -\nu^{16} - 1973\nu^{6} ) / 14580 \) (-v^16 - 1973*v^6) / 14580
\(\beta_{2}\)	\(=\)	\( ( \nu^{15} + 27\nu^{11} + 353\nu^{5} - 189\nu ) / 3240 \) (v^15 + 27v^11 + 353v^5 - 189*v) / 3240
\(\beta_{3}\)	\(=\)	\( ( \nu^{15} - 3\nu^{13} + 353\nu^{5} - 1059\nu^{3} ) / 3240 \) (v^15 - 3v^13 + 353v^5 - 1059*v^3) / 3240
\(\beta_{4}\)	\(=\)	\( ( -\nu^{17} - 81\nu^{13} - 1973\nu^{7} + 15147\nu^{3} ) / 43740 \) (-v^17 - 81v^13 - 1973v^7 + 15147*v^3) / 43740
\(\beta_{5}\)	\(=\)	\( ( -\nu^{15} - 3\nu^{13} - 353\nu^{5} - 1059\nu^{3} ) / 3240 \) (-v^15 - 3v^13 - 353v^5 - 1059*v^3) / 3240
\(\beta_{6}\)	\(=\)	\( ( -\nu^{15} + 457\nu^{5} ) / 2430 \) (-v^15 + 457*v^5) / 2430
\(\beta_{7}\)	\(=\)	\( ( -7\nu^{16} + 18\nu^{14} + 729\nu^{10} + 769\nu^{6} + 6354\nu^{4} - 92583 ) / 29160 \) (-7v^16 + 18v^14 + 729v^10 + 769v^6 + 6354*v^4 - 92583) / 29160
\(\beta_{8}\)	\(=\)	\( ( -7\nu^{16} - 9\nu^{14} + 769\nu^{6} - 3177\nu^{4} ) / 29160 \) (-7v^16 - 9v^14 + 769v^6 - 3177v^4) / 29160
\(\beta_{9}\)	\(=\)	\( ( -7\nu^{16} - 45\nu^{14} + 769\nu^{6} + 13275\nu^{4} ) / 29160 \) (-7v^16 - 45v^14 + 769v^6 + 13275v^4) / 29160
\(\beta_{10}\)	\(=\)	\( ( -23\nu^{18} + 729\nu^{12} - 1639\nu^{8} - 5103\nu^{2} ) / 262440 \) (-23v^18 + 729v^12 - 1639v^8 - 5103v^2) / 262440
\(\beta_{11}\)	\(=\)	\( ( -19\nu^{17} + 27\nu^{15} - 162\nu^{13} + 6253\nu^{7} + 9531\nu^{5} + 30294\nu^{3} ) / 87480 \) (-19v^17 + 27v^15 - 162v^13 + 6253v^7 + 9531v^5 + 30294v^3) / 87480
\(\beta_{12}\)	\(=\)	\( ( -23\nu^{18} - 729\nu^{12} - 1639\nu^{8} + 267543\nu^{2} ) / 262440 \) (-23v^18 - 729v^12 - 1639v^8 + 267543v^2) / 262440
\(\beta_{13}\)	\(=\)	\( ( -7\nu^{17} - 9\nu^{15} + 54\nu^{13} + 769\nu^{7} - 3177\nu^{5} - 10098\nu^{3} ) / 29160 \) (-7v^17 - 9v^15 + 54v^13 + 769v^7 - 3177v^5 - 10098v^3) / 29160
\(\beta_{14}\)	\(=\)	\( ( 23\nu^{18} - 63\nu^{16} + 162\nu^{14} + 1639\nu^{8} + 6921\nu^{6} + 57186\nu^{4} + 262440\nu^{2} ) / 262440 \) (23v^18 - 63v^16 + 162v^14 + 1639v^8 + 6921v^6 + 57186v^4 + 262440*v^2) / 262440
\(\beta_{15}\)	\(=\)	\( ( -29\nu^{19} + 243\nu^{15} - 13477\nu^{9} + 85779\nu^{5} + 1574640\nu ) / 787320 \) (-29v^19 + 243v^15 - 13477v^9 + 85779v^5 + 1574640*v) / 787320
\(\beta_{16}\)	\(=\)	\( ( -29\nu^{19} + 243\nu^{15} - 13477\nu^{9} + 85779\nu^{5} - 787320\nu ) / 787320 \) (-29v^19 + 243v^15 - 13477v^9 + 85779v^5 - 787320*v) / 787320
\(\beta_{17}\)	\(=\)	\( ( -23\nu^{18} - 63\nu^{16} + 162\nu^{14} - 1639\nu^{8} + 6921\nu^{6} + 57186\nu^{4} - 262440\nu^{2} ) / 262440 \) (-23v^18 - 63v^16 + 162v^14 - 1639v^8 + 6921v^6 + 57186v^4 - 262440*v^2) / 262440
\(\beta_{18}\)	\(=\)	\( ( -\nu^{19} + 214\nu^{9} ) / 19683 \) (-v^19 + 214*v^9) / 19683
\(\beta_{19}\)	\(=\)	\( ( \nu^{18} - 214\nu^{8} ) / 6561 \) (v^18 - 214*v^8) / 6561

\(\nu\)	\(=\)	\( ( -\beta_{16} + \beta_{15} ) / 3 \) (-b16 + b15) / 3
\(\nu^{2}\)	\(=\)	\( ( -\beta_{17} + \beta_{14} + \beta_{12} + \beta_{10} ) / 3 \) (-b17 + b14 + b12 + b10) / 3
\(\nu^{3}\)	\(=\)	\( ( -\beta_{13} + \beta_{11} - 2\beta_{5} + \beta_{4} - 4\beta_{3} ) / 3 \) (-b13 + b11 - 2b5 + b4 - 4b3) / 3
\(\nu^{4}\)	\(=\)	\( ( 2\beta_{17} + 2\beta_{14} + 3\beta_{9} - 7\beta_{8} ) / 3 \) (2b17 + 2b14 + 3b9 - 7b8) / 3
\(\nu^{5}\)	\(=\)	\( 3\beta_{6} - 2\beta_{5} + 2\beta_{3} \) 3b6 - 2b5 + 2*b3
\(\nu^{6}\)	\(=\)	\( ( \beta_{17} + \beta_{14} + 4\beta_{8} - 21\beta_1 ) / 3 \) (b17 + b14 + 4b8 - 21b1) / 3
\(\nu^{7}\)	\(=\)	\( ( -17\beta_{13} + 23\beta_{11} + 20\beta_{5} - 40\beta_{4} - 20\beta_{3} ) / 3 \) (-17b13 + 23b11 + 20b5 - 40b4 - 20*b3) / 3
\(\nu^{8}\)	\(=\)	\( ( -69\beta_{19} - 20\beta_{17} + 20\beta_{14} - 40\beta_{12} - 40\beta_{10} ) / 3 \) (-69b19 - 20b17 + 20b14 - 40b12 - 40*b10) / 3
\(\nu^{9}\)	\(=\)	\( ( 87\beta_{18} - 80\beta_{16} - 40\beta_{15} - 60\beta_{5} + 60\beta_{3} ) / 3 \) (87b18 - 80b16 - 40b15 - 60b5 + 60*b3) / 3
\(\nu^{10}\)	\(=\)	\( -20\beta_{17} - 20\beta_{14} + 40\beta_{7} + 127 \) -20b17 - 20b14 + 40*b7 + 127
\(\nu^{11}\)	\(=\)	\( ( -7\beta_{16} + 7\beta_{15} + 180\beta_{5} - 180\beta_{3} + 360\beta_{2} ) / 3 \) (-7b16 + 7b15 + 180b5 - 180b3 + 360*b2) / 3
\(\nu^{12}\)	\(=\)	\( ( -187\beta_{17} + 187\beta_{14} - 353\beta_{12} + 727\beta_{10} ) / 3 \) (-187b17 + 187b14 - 353b12 + 727b10) / 3
\(\nu^{13}\)	\(=\)	\( ( 353\beta_{13} - 353\beta_{11} - 914\beta_{5} - 353\beta_{4} - 208\beta_{3} ) / 3 \) (353b13 - 353b11 - 914b5 - 353b4 - 208*b3) / 3
\(\nu^{14}\)	\(=\)	\( ( 914\beta_{17} + 914\beta_{14} - 1059\beta_{9} - 769\beta_{8} ) / 3 \) (914b17 + 914b14 - 1059b9 - 769b8) / 3
\(\nu^{15}\)	\(=\)	\( -1059\beta_{6} - 914\beta_{5} + 914\beta_{3} \) -1059b6 - 914b5 + 914*b3
\(\nu^{16}\)	\(=\)	\( ( -1973\beta_{17} - 1973\beta_{14} - 7892\beta_{8} - 2307\beta_1 ) / 3 \) (-1973b17 - 1973b14 - 7892b8 - 2307b1) / 3
\(\nu^{17}\)	\(=\)	\( ( -10199\beta_{13} - 1639\beta_{11} + 4280\beta_{5} - 8560\beta_{4} - 4280\beta_{3} ) / 3 \) (-10199b13 - 1639b11 + 4280b5 - 8560b4 - 4280*b3) / 3
\(\nu^{18}\)	\(=\)	\( ( 4917\beta_{19} - 4280\beta_{17} + 4280\beta_{14} - 8560\beta_{12} - 8560\beta_{10} ) / 3 \) (4917b19 - 4280b17 + 4280b14 - 8560b12 - 8560*b10) / 3
\(\nu^{19}\)	\(=\)	\( ( -40431\beta_{18} - 17120\beta_{16} - 8560\beta_{15} - 12840\beta_{5} + 12840\beta_{3} ) / 3 \) (-40431b18 - 17120b16 - 8560b15 - 12840b5 + 12840*b3) / 3

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 356.20
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{10} + 20 T^{8} + \cdots + 119)^{2} \) (T^10 + 20T^8 + 140T^6 + 400T^4 + 400T^2 + 119)^2
$3$	\( T^{20} - 214 T^{10} + 59049 \) T^20 - 214*T^10 + 59049
$5$	\( (T^{10} + 50 T^{8} + \cdots + 11492)^{2} \) (T^10 + 50T^8 + 875T^6 + 6250T^4 + 15625T^2 + 11492)^2
$7$	\( (T^{2} - 7)^{10} \) (T^2 - 7)^10
$11$	\( T^{20} \) T^20
$13$	\( T^{20} \) T^20
$17$	\( (T^{2} + 17)^{10} \) (T^2 + 17)^10
$19$	\( T^{20} \) T^20
$23$	\( T^{20} \) T^20
$29$	\( T^{20} \) T^20
$31$	\( (T^{10} - 310 T^{8} + \cdots - 101826172)^{2} \) (T^10 - 310T^8 + 33635T^6 - 1489550T^4 + 23088025T^2 - 101826172)^2
$37$	\( T^{20} \) T^20
$41$	\( (T^{10} + 410 T^{8} + \cdots + 409170212)^{2} \) (T^10 + 410T^8 + 58835T^6 + 3446050T^4 + 70644025T^2 + 409170212)^2
$43$	\( (T^{5} - 215 T^{3} + \cdots + 6484)^{4} \) (T^5 - 215T^3 + 9245T + 6484)^4
$47$	\( T^{20} \) T^20
$53$	\( (T^{10} + 530 T^{8} + \cdots + 669539696)^{2} \) (T^10 + 530T^8 + 98315T^6 + 7443850T^4 + 197262025T^2 + 669539696)^2
$59$	\( T^{20} \) T^20
$61$	\( (T^{10} - 610 T^{8} + \cdots - 878707312)^{2} \) (T^10 - 610T^8 + 130235T^6 - 11349050T^4 + 346146025T^2 - 878707312)^2
$67$	\( (T^{5} - 335 T^{3} + \cdots + 27532)^{4} \) (T^5 - 335T^3 + 22445T + 27532)^4
$71$	\( T^{20} \) T^20
$73$	\( (T^{10} - 730 T^{8} + \cdots - 6754804672)^{2} \) (T^10 - 730T^8 + 186515T^6 - 19450850T^4 + 709956025T^2 - 6754804672)^2
$79$	\( T^{20} \) T^20
$83$	\( T^{20} \) T^20
$89$	\( T^{20} \) T^20
$97$	\( (T^{10} - 970 T^{8} + \cdots - 29255547328)^{2} \) (T^10 - 970T^8 + 329315T^6 - 45633650T^4 + 2213232025T^2 - 29255547328)^2
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\(n\)	\(52\)	\(190\)	\(239\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)