LMFDB - Newform orbit 363.2.e.n

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,2,Mod(124,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 8]))

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

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

chi := DirichletCharacter("363.124");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	363.e (of order $5$, degree $4$, 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:	$2.89856959337$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{5})$
Coefficient field:	16.0.22502537891856000000000000.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 7x^{14} + 45x^{12} + 287x^{10} + 1829x^{8} + 1148x^{6} + 720x^{4} + 448x^{2} + 256 $ x^16 + 7x^14 + 45x^12 + 287x^10 + 1829x^8 + 1148x^6 + 720x^4 + 448*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$-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_{12} q^{2} + \beta_{4} q^{3} + ( - \beta_{15} + \beta_{13} + \cdots + \beta_{3}) q^{4}+ \cdots - \beta_{8} q^{9}+O(q^{10}) $ q - b12 * q^2 + b4 * q^3 + (-b15 + b13 - b10 + b8 - b6 + b3) * q^4 + (-b15 - b13 + b8 - b4 - 1) * q^5 + b1 * q^6 + (-b7 - b5) * q^7 + (-2b14 + b12 + b9 + b5 - b1) q^8 - b8 * q^9	$ q - \beta_{12} q^{2} + \beta_{4} q^{3} + ( - \beta_{15} + \beta_{13} + \cdots + \beta_{3}) q^{4}+ \cdots + (2 \beta_{11} + 5 \beta_{9}) q^{98}+O(q^{100}) $ q - b12 * q^2 + b4 * q^3 + (-b15 + b13 - b10 + b8 - b6 + b3) * q^4 + (-b15 - b13 + b8 - b4 - 1) * q^5 + b1 * q^6 + (-b7 - b5) * q^7 + (-2b14 + b12 + b9 + b5 - b1) q^8 - b8 * q^9 + (2b11 + b9) q^10 + (-b3 + 1) * q^12 + (2b14 + 3b12 + 2b11 - 2b7 + 2b2) q^13 + 2b4 q^14 + (b15 + b13 + b10 - b8 + b6 - b3) * q^15 + (b15 - 3b13 + 3b8 - 3b4 - 3) q^16 + (2b2 - b1) q^17 - b5 * q^18 + (-b14 + b12 + b9 + b5 - b1) * q^19 + (-b10 + 8b8) q^20 + (b11 - b9) * q^21 + 2 * q^23 + (2b14 - b12 + 2b11 - 2b7 + 2b2) * q^24 + (-b6 + 4b4) q^25 + (b15 - 7b13 + b10 - b8 + b6 - b3) q^26 + (-b13 + b8 - b4 - 1) * q^27 - 2b2 q^28 + b5 * q^29 + (2b14 - b12 - b9 - b5 + b1) q^30 + (b10 - 5b8) q^31 + (2b11 - 3b9) * q^32 + (3b3 - 5) q^34 + (-4b14 - 2b12 - 4b11 + 4b7 - 4b2) q^35 + (b6 + b4) * q^36 + 5b13 q^37 + (2b15 - 4b13 + 4b8 - 4b4 - 4) * q^38 + (-2b2 - 3b1) * q^39 + (2b7 + 7b5) * q^40 + (-4b14 - 5b12 - 5b9 - 5b5 + 5b1) q^41 - 2b8 q^42 + (-2b11 + 4b9) * q^43 + (b3 + 1) * q^45 - 2b12 q^46 + (2b6 - 8b4) * q^47 + (-b15 + 3b13 - b10 + b8 - b6 + b3) q^48 + (-b15 + 3b13 - 3b8 + 3b4 + 3) q^49 + (2b2 + 2b1) * q^50 + (-2b7 + b5) q^51 + (6b14 - 3b12 - 3b9 - 3b5 + 3b1) q^52 + (3b10 - 6b8) * q^53 - b9 * q^54 + (-2b3 - 2) q^56 + (b14 - b12 + b11 - b7 + b2) * q^57 + (-b6 - 3b4) q^58 - 6b13 q^59 + (-b15 + 7b13 - 7b8 + 7b4 + 7) q^60 + (3b2 + b1) q^61 + (2b7 - 6b5) * q^62 + (b14 + b12 + b9 + b5 - b1) * q^63 - b10 * q^64 + (-10b11 + 3b9) * q^65 + (-3b3 + 6) q^67 + (-2b14 + 9b12 - 2b11 + 2b7 - 2b2) q^68 + 2b4 q^69 + (2b15 + 2b13 + 2b10 - 2b8 + 2b6 - 2b3) * q^70 + (2b15 + 6b13 - 6b8 + 6b4 + 6) * q^71 + (-2b2 + b1) q^72 + (-3b7 - 5b5) * q^73 + (5b12 + 5b9 + 5b5 - 5b1) * q^74 + (-b10 - 3b8) q^75 + (-2b11 - 6b9) * q^76 + (b3 - 7) * q^78 + (-b14 - 3b12 - b11 + b7 - b2) q^79 + (-3b6 - 5b4) * q^80 + b13 * q^81 + (-b15 + 11b13 - 11b8 + 11b4 + 11) q^82 + (-2b2 + 2b1) * q^83 + 2b7 q^84 + (2b14 + 7b12 + 7b9 + 7b5 - 7b1) q^85 + (-2b10 + 12b8) * q^86 + b9 * q^87 + (b3 - 3) * q^89 + (-2b14 + b12 - 2b11 + 2b7 - 2b2) * q^90 + (2b6 - 10b4) * q^91 + (-2b15 + 2b13 - 2b10 + 2b8 - 2b6 + 2b3) * q^92 + (b15 - 4b13 + 4b8 - 4b4 - 4) q^93 + (-4b2 - 4b1) * q^94 + 4b5 q^95 + (2b14 + 3b12 + 3b9 + 3b5 - 3b1) q^96 + (-4b10 + 3b8) * q^97 + (2b11 + 5b9) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{3} - 6 q^{4} - 2 q^{5} - 4 q^{9}+O(q^{10}) $ 16 * q - 4 * q^3 - 6 * q^4 - 2 * q^5 - 4 * q^9	$ 16 q - 4 q^{3} - 6 q^{4} - 2 q^{5} - 4 q^{9} + 24 q^{12} - 8 q^{14} - 2 q^{15} - 14 q^{16} + 30 q^{20} + 32 q^{23} - 14 q^{25} + 30 q^{26} - 4 q^{27} - 18 q^{31} - 104 q^{34} - 6 q^{36} - 20 q^{37} - 20 q^{38} - 8 q^{42} + 8 q^{45} + 28 q^{47} - 14 q^{48} + 14 q^{49} - 18 q^{53} - 16 q^{56} + 14 q^{58} + 24 q^{59} + 30 q^{60} - 2 q^{64} + 120 q^{67} - 8 q^{69} - 4 q^{70} + 20 q^{71} - 14 q^{75} - 120 q^{78} + 26 q^{80} - 4 q^{81} + 46 q^{82} + 44 q^{86} - 56 q^{89} + 36 q^{91} - 12 q^{92} - 18 q^{93} + 4 q^{97}+O(q^{100}) $ 16 * q - 4 * q^3 - 6 * q^4 - 2 * q^5 - 4 * q^9 + 24 * q^12 - 8 * q^14 - 2 * q^15 - 14 * q^16 + 30 * q^20 + 32 * q^23 - 14 * q^25 + 30 * q^26 - 4 * q^27 - 18 * q^31 - 104 * q^34 - 6 * q^36 - 20 * q^37 - 20 * q^38 - 8 * q^42 + 8 * q^45 + 28 * q^47 - 14 * q^48 + 14 * q^49 - 18 * q^53 - 16 * q^56 + 14 * q^58 + 24 * q^59 + 30 * q^60 - 2 * q^64 + 120 * q^67 - 8 * q^69 - 4 * q^70 + 20 * q^71 - 14 * q^75 - 120 * q^78 + 26 * q^80 - 4 * q^81 + 46 * q^82 + 44 * q^86 - 56 * q^89 + 36 * q^91 - 12 * q^92 - 18 * q^93 + 4 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 7x^{14} + 45x^{12} + 287x^{10} + 1829x^{8} + 1148x^{6} + 720x^{4} + 448x^{2} + 256 $ x^16 + 7*x^14 + 45*x^12 + 287*x^10 + 1829*x^8 + 1148*x^6 + 720*x^4 + 448*x^2 + 256 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( -\nu^{11} + 7997\nu ) / 3658 $ (-v^11 + 7997*v) / 3658
$\beta_{3}$	$=$	$ ( -\nu^{10} + 4339 ) / 1829 $ (-v^10 + 4339) / 1829
$\beta_{4}$	$=$	$ ( -\nu^{12} + 11655\nu^{2} ) / 7316 $ (-v^12 + 11655*v^2) / 7316
$\beta_{5}$	$=$	$ ( \nu^{13} - 11655\nu^{3} ) / 7316 $ (v^13 - 11655*v^3) / 7316
$\beta_{6}$	$=$	$ ( 3\nu^{12} - 27649\nu^{2} ) / 7316 $ (3v^12 - 27649v^2) / 7316
$\beta_{7}$	$=$	$ ( 5\nu^{13} - 50959\nu^{3} ) / 14632 $ (5v^13 - 50959v^3) / 14632
$\beta_{8}$	$=$	$ ( 7\nu^{14} - 74269\nu^{4} ) / 29264 $ (7v^14 - 74269v^4) / 29264
$\beta_{9}$	$=$	$ ( 7\nu^{15} - 74269\nu^{5} ) / 29264 $ (7v^15 - 74269v^5) / 29264
$\beta_{10}$	$=$	$ ( 3\nu^{14} - 31307\nu^{4} ) / 3658 $ (3v^14 - 31307v^4) / 3658
$\beta_{11}$	$=$	$ ( -\nu^{15} + 10475\nu^{5} ) / 1888 $ (-v^15 + 10475*v^5) / 1888
$\beta_{12}$	$=$	$ ( 287 \nu^{15} + 2009 \nu^{13} + 12915 \nu^{11} + 82305 \nu^{9} + 524923 \nu^{7} + 329476 \nu^{5} + \cdots + 128576 \nu ) / 117056 $ (287v^15 + 2009v^13 + 12915v^11 + 82305v^9 + 524923v^7 + 329476v^5 + 206640v^3 + 128576v) / 117056
$\beta_{13}$	$=$	$ ( - 287 \nu^{14} - 2009 \nu^{12} - 12915 \nu^{10} - 82305 \nu^{8} - 524923 \nu^{6} - 329476 \nu^{4} + \cdots - 128576 ) / 117056 $ (-287v^14 - 2009v^12 - 12915v^10 - 82305v^8 - 524923v^6 - 329476v^4 - 206640*v^2 - 128576) / 117056
$\beta_{14}$	$=$	$ ( 1379 \nu^{15} + 8865 \nu^{13} + 56539 \nu^{11} + 360313 \nu^{9} + 2295395 \nu^{7} + \cdots + 50432 \nu ) / 234112 $ (1379v^15 + 8865v^13 + 56539v^11 + 360313v^9 + 2295395v^7 + 141840v^5 + 88256v^3 + 50432v) / 234112
$\beta_{15}$	$=$	$ ( - 749 \nu^{14} - 4815 \nu^{12} - 30709 \nu^{10} - 195703 \nu^{8} - 1245549 \nu^{6} - 77040 \nu^{4} + \cdots - 27392 ) / 117056 $ (-749v^14 - 4815v^12 - 30709v^10 - 195703v^8 - 1245549v^6 - 77040v^4 - 47936*v^2 - 27392) / 117056

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{6} + 3\beta_{4} $ b6 + 3*b4
$\nu^{3}$	$=$	$ 2\beta_{7} - 5\beta_{5} $ 2b7 - 5b5
$\nu^{4}$	$=$	$ 7\beta_{10} - 24\beta_{8} $ 7b10 - 24b8
$\nu^{5}$	$=$	$ -14\beta_{11} - 31\beta_{9} $ -14b11 - 31b9
$\nu^{6}$	$=$	$ 45\beta_{15} - 107\beta_{13} + 107\beta_{8} - 107\beta_{4} - 107 $ 45b15 - 107b13 + 107b8 - 107b4 - 107
$\nu^{7}$	$=$	$ -90\beta_{14} + 197\beta_{12} + 197\beta_{9} + 197\beta_{5} - 197\beta_1 $ -90b14 + 197b12 + 197b9 + 197b5 - 197*b1
$\nu^{8}$	$=$	$ -287\beta_{15} + 681\beta_{13} - 287\beta_{10} + 287\beta_{8} - 287\beta_{6} + 287\beta_{3} $ -287b15 + 681b13 - 287b10 + 287b8 - 287b6 + 287b3
$\nu^{9}$	$=$	$ 574\beta_{14} - 1255\beta_{12} + 574\beta_{11} - 574\beta_{7} + 574\beta_{2} $ 574b14 - 1255b12 + 574b11 - 574b7 + 574*b2
$\nu^{10}$	$=$	$ -1829\beta_{3} + 4339 $ -1829*b3 + 4339
$\nu^{11}$	$=$	$ -3658\beta_{2} + 7997\beta_1 $ -3658b2 + 7997b1
$\nu^{12}$	$=$	$ 11655\beta_{6} + 27649\beta_{4} $ 11655b6 + 27649b4
$\nu^{13}$	$=$	$ 23310\beta_{7} - 50959\beta_{5} $ 23310b7 - 50959b5
$\nu^{14}$	$=$	$ 74269\beta_{10} - 250456\beta_{8} $ 74269b10 - 250456b8
$\nu^{15}$	$=$	$ -148538\beta_{11} - 324725\beta_{9} $ -148538b11 - 324725b9

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/363\mathbb{Z}\right)^\times$.

$n$	$122$	$244$
$\chi(n)$	$1$	$\beta_{13}$

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} $

124.1

−2.04223	−	1.48377i
−0.640974	−	0.465695i
0.640974	+	0.465695i
2.04223	+	1.48377i
−0.780063	−	2.40079i
−0.244830	−	0.753510i
0.244830	+	0.753510i
0.780063	+	2.40079i
−0.780063	+	2.40079i
−0.244830	+	0.753510i
0.244830	−	0.753510i
0.780063	−	2.40079i
−2.04223	+	1.48377i
−0.640974	+	0.465695i
0.640974	−	0.465695i
2.04223	−	1.48377i

−2.04223

1.48377i

0.309017

0.951057i

1.35111

−

4.15829i

1.91922

1.39439i

−2.04223

−

1.48377i

0.244830

−

0.753510i

1.85053

5.69534i

−0.809017

0.587785i

−5.98844

124.2

−0.640974

0.465695i

0.309017

0.951057i

−0.424058

1.30512i

−2.72823

−

1.98218i

−0.640974

−

0.465695i

0.780063

−

2.40079i

−0.825636

−

2.54105i

−0.809017

0.587785i

2.67181

124.3

0.640974

−

0.465695i

0.309017

0.951057i

−0.424058

1.30512i

−2.72823

−

1.98218i

0.640974

0.465695i

−0.780063

2.40079i

0.825636

2.54105i

−0.809017

0.587785i

−2.67181

124.4

2.04223

−

1.48377i

0.309017

0.951057i

1.35111

−

4.15829i

1.91922

1.39439i

2.04223

1.48377i

−0.244830

0.753510i

−1.85053

−

5.69534i

−0.809017

0.587785i

5.98844

130.1

−0.780063

2.40079i

−0.809017

0.587785i

−3.53725

−

2.56996i

−0.733075

−

2.25617i

−0.780063

−

2.40079i

0.640974

0.465695i

4.84475

−

3.51992i

0.309017

−

0.951057i

5.98844

130.2

−0.244830

0.753510i

−0.809017

0.587785i

1.11020

0.806607i

1.04209

3.20723i

−0.244830

−

0.753510i

2.04223

1.48377i

−2.16154

1.57045i

0.309017

−

0.951057i

−2.67181

130.3

0.244830

−

0.753510i

−0.809017

0.587785i

1.11020

0.806607i

1.04209

3.20723i

0.244830

0.753510i

−2.04223

−

1.48377i

2.16154

−

1.57045i

0.309017

−

0.951057i

2.67181

130.4

0.780063

−

2.40079i

−0.809017

0.587785i

−3.53725

−

2.56996i

−0.733075

−

2.25617i

0.780063

2.40079i

−0.640974

−

0.465695i

−4.84475

3.51992i

0.309017

−

0.951057i

−5.98844

148.1

−0.780063

−

2.40079i

−0.809017

−

0.587785i

−3.53725

2.56996i

−0.733075

2.25617i

−0.780063

2.40079i

0.640974

−

0.465695i

4.84475

3.51992i

0.309017

0.951057i

5.98844

148.2

−0.244830

−

0.753510i

−0.809017

−

0.587785i

1.11020

−

0.806607i

1.04209

−

3.20723i

−0.244830

0.753510i

2.04223

−

1.48377i

−2.16154

−

1.57045i

0.309017

0.951057i

−2.67181

148.3

0.244830

0.753510i

−0.809017

−

0.587785i

1.11020

−

0.806607i

1.04209

−

3.20723i

0.244830

−

0.753510i

−2.04223

1.48377i

2.16154

1.57045i

0.309017

0.951057i

2.67181

148.4

0.780063

2.40079i

−0.809017

−

0.587785i

−3.53725

2.56996i

−0.733075

2.25617i

0.780063

−

2.40079i

−0.640974

0.465695i

−4.84475

−

3.51992i

0.309017

0.951057i

−5.98844

202.1

−2.04223

−

1.48377i

0.309017

−

0.951057i

1.35111

4.15829i

1.91922

−

1.39439i

−2.04223

1.48377i

0.244830

0.753510i

1.85053

−

5.69534i

−0.809017

−

0.587785i

−5.98844

202.2

−0.640974

−

0.465695i

0.309017

−

0.951057i

−0.424058

−

1.30512i

−2.72823

1.98218i

−0.640974

0.465695i

0.780063

2.40079i

−0.825636

2.54105i

−0.809017

−

0.587785i

2.67181

202.3

0.640974

0.465695i

0.309017

−

0.951057i

−0.424058

−

1.30512i

−2.72823

1.98218i

0.640974

−

0.465695i

−0.780063

−

2.40079i

0.825636

−

2.54105i

−0.809017

−

0.587785i

−2.67181

202.4

2.04223

1.48377i

0.309017

−

0.951057i

1.35111

4.15829i

1.91922

−

1.39439i

2.04223

−

1.48377i

−0.244830

−

0.753510i

−1.85053

5.69534i

−0.809017

−

0.587785i

5.98844

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	inner
11.c	even	5	3	inner
11.d	odd	10	3	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	363.2.e.n		16
11.b	odd	2	1	inner	363.2.e.n		16
11.c	even	5	1		363.2.a.j	✓	4
11.c	even	5	3	inner	363.2.e.n		16
11.d	odd	10	1		363.2.a.j	✓	4
11.d	odd	10	3	inner	363.2.e.n		16
33.f	even	10	1		1089.2.a.u		4
33.h	odd	10	1		1089.2.a.u		4
44.g	even	10	1		5808.2.a.ck		4
44.h	odd	10	1		5808.2.a.ck		4
55.h	odd	10	1		9075.2.a.cv		4
55.j	even	10	1		9075.2.a.cv		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
363.2.a.j	✓	4	11.c	even	5	1
363.2.a.j	✓	4	11.d	odd	10	1
363.2.e.n		16	1.a	even	1	1	trivial
363.2.e.n		16	11.b	odd	2	1	inner
363.2.e.n		16	11.c	even	5	3	inner
363.2.e.n		16	11.d	odd	10	3	inner
1089.2.a.u		4	33.f	even	10	1
1089.2.a.u		4	33.h	odd	10	1
5808.2.a.ck		4	44.g	even	10	1
5808.2.a.ck		4	44.h	odd	10	1
9075.2.a.cv		4	55.h	odd	10	1
9075.2.a.cv		4	55.j	even	10	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{16} + 7T_{2}^{14} + 45T_{2}^{12} + 287T_{2}^{10} + 1829T_{2}^{8} + 1148T_{2}^{6} + 720T_{2}^{4} + 448T_{2}^{2} + 256 $ T2^16 + 7*T2^14 + 45*T2^12 + 287*T2^10 + 1829*T2^8 + 1148*T2^6 + 720*T2^4 + 448*T2^2 + 256 acting on $S_{2}^{\mathrm{new}}(363, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} + 7 T^{14} + \cdots + 256 $ T^16 + 7T^14 + 45T^12 + 287T^10 + 1829T^8 + 1148T^6 + 720T^4 + 448*T^2 + 256
$3$	$ (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} $ (T^4 + T^3 + T^2 + T + 1)^4
$5$	$ (T^{8} + T^{7} + 9 T^{6} + \cdots + 4096)^{2} $ (T^8 + T^7 + 9T^6 + 17T^5 + 89T^4 - 136T^3 + 576T^2 - 512T + 4096)^2
$7$	$ T^{16} + 7 T^{14} + \cdots + 256 $ T^16 + 7T^14 + 45T^12 + 287T^10 + 1829T^8 + 1148T^6 + 720T^4 + 448*T^2 + 256
$11$	$ T^{16} $ T^16
$13$	$ T^{16} + \cdots + 110075314176 $ T^16 + 51T^14 + 2025T^12 + 73899T^10 + 2602449T^8 + 42565824T^6 + 671846400T^4 + 9746251776*T^2 + 110075314176
$17$	$ T^{16} + \cdots + 4294967296 $ T^16 + 43T^14 + 1593T^12 + 57491T^10 + 2064305T^8 + 14717696T^6 + 104398848T^4 + 721420288*T^2 + 4294967296
$19$	$ T^{16} + 19 T^{14} + \cdots + 65536 $ T^16 + 19T^14 + 345T^12 + 6251T^10 + 113249T^8 + 100016T^6 + 88320T^4 + 77824*T^2 + 65536
$23$	$ (T - 2)^{16} $ (T - 2)^16
$29$	$ T^{16} + 7 T^{14} + \cdots + 256 $ T^16 + 7T^14 + 45T^12 + 287T^10 + 1829T^8 + 1148T^6 + 720T^4 + 448*T^2 + 256
$31$	$ (T^{8} + 9 T^{7} + \cdots + 20736)^{2} $ (T^8 + 9T^7 + 69T^6 + 513T^5 + 3789T^4 + 6156T^3 + 9936T^2 + 15552*T + 20736)^2
$37$	$ (T^{4} + 5 T^{3} + \cdots + 625)^{4} $ (T^4 + 5T^3 + 25T^2 + 125*T + 625)^4
$41$	$ T^{16} + \cdots + 218340105584896 $ T^16 + 151T^14 + 18957T^12 + 2282063T^10 + 271720805T^8 + 8772250172T^6 + 280115001552T^4 + 8576835573184*T^2 + 218340105584896
$43$	$ (T^{2} - 44)^{8} $ (T^2 - 44)^8
$47$	$ (T^{8} - 14 T^{7} + \cdots + 65536)^{2} $ (T^8 - 14T^7 + 180T^6 - 2296T^5 + 29264T^4 - 36736T^3 + 46080T^2 - 57344*T + 65536)^2
$53$	$ (T^{8} + 9 T^{7} + \cdots + 8503056)^{2} $ (T^8 + 9T^7 + 135T^6 + 1701T^5 + 22599T^4 - 91854T^3 + 393660T^2 - 1417176*T + 8503056)^2
$59$	$ (T^{4} - 6 T^{3} + \cdots + 1296)^{4} $ (T^4 - 6T^3 + 36T^2 - 216*T + 1296)^4
$61$	$ T^{16} + \cdots + 4294967296 $ T^16 + 43T^14 + 1593T^12 + 57491T^10 + 2064305T^8 + 14717696T^6 + 104398848T^4 + 721420288*T^2 + 4294967296
$67$	$ (T^{2} - 15 T - 18)^{8} $ (T^2 - 15*T - 18)^8
$71$	$ (T^{8} - 10 T^{7} + \cdots + 4096)^{2} $ (T^8 - 10T^7 + 108T^6 - 1160T^5 + 12464T^4 + 9280T^3 + 6912T^2 + 5120*T + 4096)^2
$73$	$ T^{16} + \cdots + 457163239653376 $ T^16 + 139T^14 + 14697T^12 + 1400147T^10 + 126661505T^8 + 6474279728T^6 + 314242083072T^4 + 13742580084736*T^2 + 457163239653376
$79$	$ T^{16} + \cdots + 110075314176 $ T^16 + 51T^14 + 2025T^12 + 73899T^10 + 2602449T^8 + 42565824T^6 + 671846400T^4 + 9746251776*T^2 + 110075314176
$83$	$ T^{16} + \cdots + 4294967296 $ T^16 + 76T^14 + 5520T^12 + 400064T^10 + 28991744T^8 + 102416384T^6 + 361758720T^4 + 1275068416*T^2 + 4294967296
$89$	$ (T^{2} + 7 T + 4)^{8} $ (T^2 + 7*T + 4)^8
$97$	$ (T^{8} - 2 T^{7} + \cdots + 294499921)^{2} $ (T^8 - 2T^7 + 135T^6 - 532T^5 + 18749T^4 + 69692T^3 + 2316735T^2 + 4496182*T + 294499921)^2
show more
show less

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:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	363.e (of order \(5\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{12} q^{2} + \beta_{4} q^{3} + ( - \beta_{15} + \beta_{13} + \cdots + \beta_{3}) q^{4}+ \cdots - \beta_{8} q^{9}+O(q^{10}) \) q - b12 * q^2 + b4 * q^3 + (-b15 + b13 - b10 + b8 - b6 + b3) * q^4 + (-b15 - b13 + b8 - b4 - 1) * q^5 + b1 * q^6 + (-b7 - b5) * q^7 + (-2b14 + b12 + b9 + b5 - b1) q^8 - b8 * q^9	\( q - \beta_{12} q^{2} + \beta_{4} q^{3} + ( - \beta_{15} + \beta_{13} + \cdots + \beta_{3}) q^{4}+ \cdots + (2 \beta_{11} + 5 \beta_{9}) q^{98}+O(q^{100}) \) q - b12 * q^2 + b4 * q^3 + (-b15 + b13 - b10 + b8 - b6 + b3) * q^4 + (-b15 - b13 + b8 - b4 - 1) * q^5 + b1 * q^6 + (-b7 - b5) * q^7 + (-2b14 + b12 + b9 + b5 - b1) q^8 - b8 * q^9 + (2b11 + b9) q^10 + (-b3 + 1) * q^12 + (2b14 + 3b12 + 2b11 - 2b7 + 2b2) q^13 + 2b4 q^14 + (b15 + b13 + b10 - b8 + b6 - b3) * q^15 + (b15 - 3b13 + 3b8 - 3b4 - 3) q^16 + (2b2 - b1) q^17 - b5 * q^18 + (-b14 + b12 + b9 + b5 - b1) * q^19 + (-b10 + 8b8) q^20 + (b11 - b9) * q^21 + 2 * q^23 + (2b14 - b12 + 2b11 - 2b7 + 2b2) * q^24 + (-b6 + 4b4) q^25 + (b15 - 7b13 + b10 - b8 + b6 - b3) q^26 + (-b13 + b8 - b4 - 1) * q^27 - 2b2 q^28 + b5 * q^29 + (2b14 - b12 - b9 - b5 + b1) q^30 + (b10 - 5b8) q^31 + (2b11 - 3b9) * q^32 + (3b3 - 5) q^34 + (-4b14 - 2b12 - 4b11 + 4b7 - 4b2) q^35 + (b6 + b4) * q^36 + 5b13 q^37 + (2b15 - 4b13 + 4b8 - 4b4 - 4) * q^38 + (-2b2 - 3b1) * q^39 + (2b7 + 7b5) * q^40 + (-4b14 - 5b12 - 5b9 - 5b5 + 5b1) q^41 - 2b8 q^42 + (-2b11 + 4b9) * q^43 + (b3 + 1) * q^45 - 2b12 q^46 + (2b6 - 8b4) * q^47 + (-b15 + 3b13 - b10 + b8 - b6 + b3) q^48 + (-b15 + 3b13 - 3b8 + 3b4 + 3) q^49 + (2b2 + 2b1) * q^50 + (-2b7 + b5) q^51 + (6b14 - 3b12 - 3b9 - 3b5 + 3b1) q^52 + (3b10 - 6b8) * q^53 - b9 * q^54 + (-2b3 - 2) q^56 + (b14 - b12 + b11 - b7 + b2) * q^57 + (-b6 - 3b4) q^58 - 6b13 q^59 + (-b15 + 7b13 - 7b8 + 7b4 + 7) q^60 + (3b2 + b1) q^61 + (2b7 - 6b5) * q^62 + (b14 + b12 + b9 + b5 - b1) * q^63 - b10 * q^64 + (-10b11 + 3b9) * q^65 + (-3b3 + 6) q^67 + (-2b14 + 9b12 - 2b11 + 2b7 - 2b2) q^68 + 2b4 q^69 + (2b15 + 2b13 + 2b10 - 2b8 + 2b6 - 2b3) * q^70 + (2b15 + 6b13 - 6b8 + 6b4 + 6) * q^71 + (-2b2 + b1) q^72 + (-3b7 - 5b5) * q^73 + (5b12 + 5b9 + 5b5 - 5b1) * q^74 + (-b10 - 3b8) q^75 + (-2b11 - 6b9) * q^76 + (b3 - 7) * q^78 + (-b14 - 3b12 - b11 + b7 - b2) q^79 + (-3b6 - 5b4) * q^80 + b13 * q^81 + (-b15 + 11b13 - 11b8 + 11b4 + 11) q^82 + (-2b2 + 2b1) * q^83 + 2b7 q^84 + (2b14 + 7b12 + 7b9 + 7b5 - 7b1) q^85 + (-2b10 + 12b8) * q^86 + b9 * q^87 + (b3 - 3) * q^89 + (-2b14 + b12 - 2b11 + 2b7 - 2b2) * q^90 + (2b6 - 10b4) * q^91 + (-2b15 + 2b13 - 2b10 + 2b8 - 2b6 + 2b3) * q^92 + (b15 - 4b13 + 4b8 - 4b4 - 4) q^93 + (-4b2 - 4b1) * q^94 + 4b5 q^95 + (2b14 + 3b12 + 3b9 + 3b5 - 3b1) q^96 + (-4b10 + 3b8) * q^97 + (2b11 + 5b9) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{3} - 6 q^{4} - 2 q^{5} - 4 q^{9}+O(q^{10}) \) 16 * q - 4 * q^3 - 6 * q^4 - 2 * q^5 - 4 * q^9	\( 16 q - 4 q^{3} - 6 q^{4} - 2 q^{5} - 4 q^{9} + 24 q^{12} - 8 q^{14} - 2 q^{15} - 14 q^{16} + 30 q^{20} + 32 q^{23} - 14 q^{25} + 30 q^{26} - 4 q^{27} - 18 q^{31} - 104 q^{34} - 6 q^{36} - 20 q^{37} - 20 q^{38} - 8 q^{42} + 8 q^{45} + 28 q^{47} - 14 q^{48} + 14 q^{49} - 18 q^{53} - 16 q^{56} + 14 q^{58} + 24 q^{59} + 30 q^{60} - 2 q^{64} + 120 q^{67} - 8 q^{69} - 4 q^{70} + 20 q^{71} - 14 q^{75} - 120 q^{78} + 26 q^{80} - 4 q^{81} + 46 q^{82} + 44 q^{86} - 56 q^{89} + 36 q^{91} - 12 q^{92} - 18 q^{93} + 4 q^{97}+O(q^{100}) \) 16 * q - 4 * q^3 - 6 * q^4 - 2 * q^5 - 4 * q^9 + 24 * q^12 - 8 * q^14 - 2 * q^15 - 14 * q^16 + 30 * q^20 + 32 * q^23 - 14 * q^25 + 30 * q^26 - 4 * q^27 - 18 * q^31 - 104 * q^34 - 6 * q^36 - 20 * q^37 - 20 * q^38 - 8 * q^42 + 8 * q^45 + 28 * q^47 - 14 * q^48 + 14 * q^49 - 18 * q^53 - 16 * q^56 + 14 * q^58 + 24 * q^59 + 30 * q^60 - 2 * q^64 + 120 * q^67 - 8 * q^69 - 4 * q^70 + 20 * q^71 - 14 * q^75 - 120 * q^78 + 26 * q^80 - 4 * q^81 + 46 * q^82 + 44 * q^86 - 56 * q^89 + 36 * q^91 - 12 * q^92 - 18 * q^93 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	363.2.e.n

Level	$363$
Weight	$2$
Character orbit	363.e
Analytic conductor	$2.899$
Analytic rank	$0$
Dimension	$16$
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(2.89856959337\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{5})\)
Coefficient field:	16.0.22502537891856000000000000.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 7x^{14} + 45x^{12} + 287x^{10} + 1829x^{8} + 1148x^{6} + 720x^{4} + 448x^{2} + 256 \) x^16 + 7x^14 + 45x^12 + 287x^10 + 1829x^8 + 1148x^6 + 720x^4 + 448*x^2 + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( -\nu^{11} + 7997\nu ) / 3658 \) (-v^11 + 7997*v) / 3658
\(\beta_{3}\)	\(=\)	\( ( -\nu^{10} + 4339 ) / 1829 \) (-v^10 + 4339) / 1829
\(\beta_{4}\)	\(=\)	\( ( -\nu^{12} + 11655\nu^{2} ) / 7316 \) (-v^12 + 11655*v^2) / 7316
\(\beta_{5}\)	\(=\)	\( ( \nu^{13} - 11655\nu^{3} ) / 7316 \) (v^13 - 11655*v^3) / 7316
\(\beta_{6}\)	\(=\)	\( ( 3\nu^{12} - 27649\nu^{2} ) / 7316 \) (3v^12 - 27649v^2) / 7316
\(\beta_{7}\)	\(=\)	\( ( 5\nu^{13} - 50959\nu^{3} ) / 14632 \) (5v^13 - 50959v^3) / 14632
\(\beta_{8}\)	\(=\)	\( ( 7\nu^{14} - 74269\nu^{4} ) / 29264 \) (7v^14 - 74269v^4) / 29264
\(\beta_{9}\)	\(=\)	\( ( 7\nu^{15} - 74269\nu^{5} ) / 29264 \) (7v^15 - 74269v^5) / 29264
\(\beta_{10}\)	\(=\)	\( ( 3\nu^{14} - 31307\nu^{4} ) / 3658 \) (3v^14 - 31307v^4) / 3658
\(\beta_{11}\)	\(=\)	\( ( -\nu^{15} + 10475\nu^{5} ) / 1888 \) (-v^15 + 10475*v^5) / 1888
\(\beta_{12}\)	\(=\)	\( ( 287 \nu^{15} + 2009 \nu^{13} + 12915 \nu^{11} + 82305 \nu^{9} + 524923 \nu^{7} + 329476 \nu^{5} + \cdots + 128576 \nu ) / 117056 \) (287v^15 + 2009v^13 + 12915v^11 + 82305v^9 + 524923v^7 + 329476v^5 + 206640v^3 + 128576v) / 117056
\(\beta_{13}\)	\(=\)	\( ( - 287 \nu^{14} - 2009 \nu^{12} - 12915 \nu^{10} - 82305 \nu^{8} - 524923 \nu^{6} - 329476 \nu^{4} + \cdots - 128576 ) / 117056 \) (-287v^14 - 2009v^12 - 12915v^10 - 82305v^8 - 524923v^6 - 329476v^4 - 206640*v^2 - 128576) / 117056
\(\beta_{14}\)	\(=\)	\( ( 1379 \nu^{15} + 8865 \nu^{13} + 56539 \nu^{11} + 360313 \nu^{9} + 2295395 \nu^{7} + \cdots + 50432 \nu ) / 234112 \) (1379v^15 + 8865v^13 + 56539v^11 + 360313v^9 + 2295395v^7 + 141840v^5 + 88256v^3 + 50432v) / 234112
\(\beta_{15}\)	\(=\)	\( ( - 749 \nu^{14} - 4815 \nu^{12} - 30709 \nu^{10} - 195703 \nu^{8} - 1245549 \nu^{6} - 77040 \nu^{4} + \cdots - 27392 ) / 117056 \) (-749v^14 - 4815v^12 - 30709v^10 - 195703v^8 - 1245549v^6 - 77040v^4 - 47936*v^2 - 27392) / 117056

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{6} + 3\beta_{4} \) b6 + 3*b4
\(\nu^{3}\)	\(=\)	\( 2\beta_{7} - 5\beta_{5} \) 2b7 - 5b5
\(\nu^{4}\)	\(=\)	\( 7\beta_{10} - 24\beta_{8} \) 7b10 - 24b8
\(\nu^{5}\)	\(=\)	\( -14\beta_{11} - 31\beta_{9} \) -14b11 - 31b9
\(\nu^{6}\)	\(=\)	\( 45\beta_{15} - 107\beta_{13} + 107\beta_{8} - 107\beta_{4} - 107 \) 45b15 - 107b13 + 107b8 - 107b4 - 107
\(\nu^{7}\)	\(=\)	\( -90\beta_{14} + 197\beta_{12} + 197\beta_{9} + 197\beta_{5} - 197\beta_1 \) -90b14 + 197b12 + 197b9 + 197b5 - 197*b1
\(\nu^{8}\)	\(=\)	\( -287\beta_{15} + 681\beta_{13} - 287\beta_{10} + 287\beta_{8} - 287\beta_{6} + 287\beta_{3} \) -287b15 + 681b13 - 287b10 + 287b8 - 287b6 + 287b3
\(\nu^{9}\)	\(=\)	\( 574\beta_{14} - 1255\beta_{12} + 574\beta_{11} - 574\beta_{7} + 574\beta_{2} \) 574b14 - 1255b12 + 574b11 - 574b7 + 574*b2
\(\nu^{10}\)	\(=\)	\( -1829\beta_{3} + 4339 \) -1829*b3 + 4339
\(\nu^{11}\)	\(=\)	\( -3658\beta_{2} + 7997\beta_1 \) -3658b2 + 7997b1
\(\nu^{12}\)	\(=\)	\( 11655\beta_{6} + 27649\beta_{4} \) 11655b6 + 27649b4
\(\nu^{13}\)	\(=\)	\( 23310\beta_{7} - 50959\beta_{5} \) 23310b7 - 50959b5
\(\nu^{14}\)	\(=\)	\( 74269\beta_{10} - 250456\beta_{8} \) 74269b10 - 250456b8
\(\nu^{15}\)	\(=\)	\( -148538\beta_{11} - 324725\beta_{9} \) -148538b11 - 324725b9

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 124.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} + 7 T^{14} + \cdots + 256 \) T^16 + 7T^14 + 45T^12 + 287T^10 + 1829T^8 + 1148T^6 + 720T^4 + 448*T^2 + 256
$3$	\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \) (T^4 + T^3 + T^2 + T + 1)^4
$5$	\( (T^{8} + T^{7} + 9 T^{6} + \cdots + 4096)^{2} \) (T^8 + T^7 + 9T^6 + 17T^5 + 89T^4 - 136T^3 + 576T^2 - 512T + 4096)^2
$7$	\( T^{16} + 7 T^{14} + \cdots + 256 \) T^16 + 7T^14 + 45T^12 + 287T^10 + 1829T^8 + 1148T^6 + 720T^4 + 448*T^2 + 256
$11$	\( T^{16} \) T^16
$13$	\( T^{16} + \cdots + 110075314176 \) T^16 + 51T^14 + 2025T^12 + 73899T^10 + 2602449T^8 + 42565824T^6 + 671846400T^4 + 9746251776*T^2 + 110075314176
$17$	\( T^{16} + \cdots + 4294967296 \) T^16 + 43T^14 + 1593T^12 + 57491T^10 + 2064305T^8 + 14717696T^6 + 104398848T^4 + 721420288*T^2 + 4294967296
$19$	\( T^{16} + 19 T^{14} + \cdots + 65536 \) T^16 + 19T^14 + 345T^12 + 6251T^10 + 113249T^8 + 100016T^6 + 88320T^4 + 77824*T^2 + 65536
$23$	\( (T - 2)^{16} \) (T - 2)^16
$29$	\( T^{16} + 7 T^{14} + \cdots + 256 \) T^16 + 7T^14 + 45T^12 + 287T^10 + 1829T^8 + 1148T^6 + 720T^4 + 448*T^2 + 256
$31$	\( (T^{8} + 9 T^{7} + \cdots + 20736)^{2} \) (T^8 + 9T^7 + 69T^6 + 513T^5 + 3789T^4 + 6156T^3 + 9936T^2 + 15552*T + 20736)^2
$37$	\( (T^{4} + 5 T^{3} + \cdots + 625)^{4} \) (T^4 + 5T^3 + 25T^2 + 125*T + 625)^4
$41$	\( T^{16} + \cdots + 218340105584896 \) T^16 + 151T^14 + 18957T^12 + 2282063T^10 + 271720805T^8 + 8772250172T^6 + 280115001552T^4 + 8576835573184*T^2 + 218340105584896
$43$	\( (T^{2} - 44)^{8} \) (T^2 - 44)^8
$47$	\( (T^{8} - 14 T^{7} + \cdots + 65536)^{2} \) (T^8 - 14T^7 + 180T^6 - 2296T^5 + 29264T^4 - 36736T^3 + 46080T^2 - 57344*T + 65536)^2
$53$	\( (T^{8} + 9 T^{7} + \cdots + 8503056)^{2} \) (T^8 + 9T^7 + 135T^6 + 1701T^5 + 22599T^4 - 91854T^3 + 393660T^2 - 1417176*T + 8503056)^2
$59$	\( (T^{4} - 6 T^{3} + \cdots + 1296)^{4} \) (T^4 - 6T^3 + 36T^2 - 216*T + 1296)^4
$61$	\( T^{16} + \cdots + 4294967296 \) T^16 + 43T^14 + 1593T^12 + 57491T^10 + 2064305T^8 + 14717696T^6 + 104398848T^4 + 721420288*T^2 + 4294967296
$67$	\( (T^{2} - 15 T - 18)^{8} \) (T^2 - 15*T - 18)^8
$71$	\( (T^{8} - 10 T^{7} + \cdots + 4096)^{2} \) (T^8 - 10T^7 + 108T^6 - 1160T^5 + 12464T^4 + 9280T^3 + 6912T^2 + 5120*T + 4096)^2
$73$	\( T^{16} + \cdots + 457163239653376 \) T^16 + 139T^14 + 14697T^12 + 1400147T^10 + 126661505T^8 + 6474279728T^6 + 314242083072T^4 + 13742580084736*T^2 + 457163239653376
$79$	\( T^{16} + \cdots + 110075314176 \) T^16 + 51T^14 + 2025T^12 + 73899T^10 + 2602449T^8 + 42565824T^6 + 671846400T^4 + 9746251776*T^2 + 110075314176
$83$	\( T^{16} + \cdots + 4294967296 \) T^16 + 76T^14 + 5520T^12 + 400064T^10 + 28991744T^8 + 102416384T^6 + 361758720T^4 + 1275068416*T^2 + 4294967296
$89$	\( (T^{2} + 7 T + 4)^{8} \) (T^2 + 7*T + 4)^8
$97$	\( (T^{8} - 2 T^{7} + \cdots + 294499921)^{2} \) (T^8 - 2T^7 + 135T^6 - 532T^5 + 18749T^4 + 69692T^3 + 2316735T^2 + 4496182*T + 294499921)^2
show more
show less

\(n\)	\(122\)	\(244\)
\(\chi(n)\)	\(1\)	\(\beta_{13}\)