LMFDB - Newform orbit 63.3.q.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [63,3,Mod(44,63)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 2]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("63.44");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 63 = 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	63.q (of order $6$, degree $2$, 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:	$1.71662566547$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 20x^{10} + 307x^{8} - 1824x^{6} + 8289x^{4} - 1674x^{2} + 324 $ x^12 - 20x^10 + 307x^8 - 1824x^6 + 8289x^4 - 1674*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_1 q^{2} + (\beta_{9} - 3 \beta_{8}) q^{4} - \beta_{6} q^{5} + (4 \beta_{8} - \beta_{5} + \beta_{3} + 1) q^{7} + ( - \beta_{11} - \beta_{10} + \cdots + 2 \beta_1) q^{8}+O(q^{10}) $ q + b1 * q^2 + (b9 - 3b8) q^4 - b6 * q^5 + (4b8 - b5 + b3 + 1) q^7 + (-b11 - b10 - 2b7 + b6 - b4 + 2b1) * q^8	$ q + \beta_1 q^{2} + (\beta_{9} - 3 \beta_{8}) q^{4} - \beta_{6} q^{5} + (4 \beta_{8} - \beta_{5} + \beta_{3} + 1) q^{7} + ( - \beta_{11} - \beta_{10} + \cdots + 2 \beta_1) q^{8}+ \cdots + ( - 2 \beta_{11} + 7 \beta_{7} + \cdots - 7 \beta_1) q^{98}+O(q^{100}) $ q + b1 * q^2 + (b9 - 3b8) q^4 - b6 * q^5 + (4b8 - b5 + b3 + 1) q^7 + (-b11 - b10 - 2b7 + b6 - b4 + 2b1) * q^8 + (-b9 + 4b8 - b5 - 2b2 - 1) * q^10 + (-2b10 + b7) q^11 + (-b5 - b3 + b2 - 4) * q^13 + (b11 + 2b10 + 4b7 - b4 - 5b1) q^14 + (-b9 - 9b8 + 4b5 - b3 + 2b2 - 5) q^16 + (b11 - 7b7) q^17 + (b9 + 3b8 + 2b5 + b3 + b2 + 5) * q^19 + (7b10 + 7b7 + 7b4 - 7b1) * q^20 + (2b5 - b3 - 2b2 + 3) * q^22 + (b6 + 4b4 + 5b1) * q^23 + (-5b9 - 18b8 + 2b5 + 4b2 + 2) * q^25 + (4b6 - 7b4) * q^26 + (-b9 + 24b8 - 2b5 - 5b3 + b2 + 20) q^28 + (4b11 - b10 + b7 - 4b6 - b4 - b1) * q^29 + (4b9 - 2b8 - 3b5 - 6b2 - 3) * q^31 + (-b11 - 7b10 + 2b7) * q^32 + (-b5 + 6b3 + b2 - 46) q^34 + (-4b11 - 7b7 - b6 - 7b4) q^35 + (5b9 - 25b8 - 2b5 + 5b3 - b2 - 27) * q^37 + (-4b11 - 7b10) * q^38 + (-3b9 + 54b8 - 6b5 - 3b3 - 3b2 + 48) q^40 + (-b11 + 7b10 - 7b7 + b6 + 7b4 + 7b1) * q^41 + (-3b5 + 3b3 + 3b2 + 8) q^43 + (-5b6 + 3b4) * q^44 + (6b9 - 27b8 - 3b5 - 6b2 - 3) * q^46 + (-2b6 - 7b4 + 14b1) q^47 + (4b9 + 19b8 + 2b5 + 5b3 - 4b2 + 17) q^49 + (-b11 - 7b10 - 3b7 + b6 - 7b4 + 3b1) * q^50 + (-49b8 + 7b5 + 14b2 + 7) q^52 + (4b11 - 9b10 + 5b7) q^53 + (3b5 - 11b3 - 3b2 + 5) q^55 + (2b11 + 7b10 + 7b7 + 4b6 + 14b1) q^56 + (-5b9 + 20b8 - 6b5 - 5b3 - 3b2 + 14) q^58 + (b11 + 7b10 + 14b7) * q^59 + (-2b9 - 27b8 + 10b5 - 2b3 + 5b2 - 17) q^61 + (5b11 + 14b10 - 14b7 - 5b6 + 14b4 + 14b1) * q^62 + (3b3 + 25) q^64 + (6b6 + 7b4 - 28b1) q^65 + (5b9 - 2b5 - 4b2 - 2) q^67 + (-7b6 - 35b1) * q^68 + (-b9 - 17b8 + 10b5 + 11b3 + 8b2 - 55) * q^70 + (-12b11 + 3b10 + 24b7 + 12b6 + 3b4 - 24b1) * q^71 + (-17b9 + 82b8 + 4b5 + 8b2 + 4) * q^73 + (-2b11 + b10 - 40b7) * q^74 + (7b5 - 7b2 - 42) * q^76 + (7b11 + b10 - 19b7 - b6 - 4b4 + 8b1) * q^77 + (-12b9 + 16b8 - 10b5 - 12b3 - 5b2 + 6) q^79 + (12b11 - 7b10 + 35b7) q^80 + (8b9 - 32b8 - 12b5 + 8b3 - 6b2 - 44) q^82 + (-6b11 - 14b10 + 21b7 + 6b6 - 14b4 - 21b1) * q^83 + (-5b5 - 12b3 + 5b2 - 20) q^85 + (6b6 - 15b4 + 2b1) q^86 + (-9b9 + 33b8) * q^88 + (2b6 + 14b4) * q^89 + (-11b9 + 37b8 + 5b5 - 19b3 - 10b2 + 39) q^91 + (7b11 - 4b10 - 25b7 - 7b6 - 4b4 + 25b1) * q^92 + (12b9 - 111b8 + 5b5 + 10b2 + 5) * q^94 + (-6b11 + 7b10 + 28b7) q^95 + (-2b5 + 19b3 + 2b2 + 41) q^97 + (-2b11 + 7b7 - 11b6 + 21b4 - 7b1) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 16 q^{4} - 2 q^{7}+O(q^{10}) $ 12 * q + 16 * q^4 - 2 * q^7	$ 12 q + 16 q^{4} - 2 q^{7} - 16 q^{10} - 52 q^{13} - 44 q^{16} + 26 q^{19} + 32 q^{22} + 106 q^{25} + 84 q^{28} + 22 q^{31} - 528 q^{34} - 146 q^{37} + 300 q^{40} + 108 q^{43} + 168 q^{46} + 114 q^{49} + 252 q^{52} + 16 q^{55} + 92 q^{58} - 136 q^{61} + 312 q^{64} + 2 q^{67} - 620 q^{70} - 482 q^{73} - 504 q^{76} + 42 q^{79} - 212 q^{82} - 288 q^{85} - 180 q^{88} + 222 q^{91} + 612 q^{94} + 568 q^{97}+O(q^{100}) $ 12 * q + 16 * q^4 - 2 * q^7 - 16 * q^10 - 52 * q^13 - 44 * q^16 + 26 * q^19 + 32 * q^22 + 106 * q^25 + 84 * q^28 + 22 * q^31 - 528 * q^34 - 146 * q^37 + 300 * q^40 + 108 * q^43 + 168 * q^46 + 114 * q^49 + 252 * q^52 + 16 * q^55 + 92 * q^58 - 136 * q^61 + 312 * q^64 + 2 * q^67 - 620 * q^70 - 482 * q^73 - 504 * q^76 + 42 * q^79 - 212 * q^82 - 288 * q^85 - 180 * q^88 + 222 * q^91 + 612 * q^94 + 568 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 20x^{10} + 307x^{8} - 1824x^{6} + 8289x^{4} - 1674x^{2} + 324 $ x^12 - 20*x^10 + 307*x^8 - 1824*x^6 + 8289*x^4 - 1674*x^2 + 324 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( 680\nu^{10} - 10438\nu^{8} - 90901\nu^{6} + 2229417\nu^{4} - 27566757\nu^{2} - 5393898 ) / 15067458 $ (680v^10 - 10438v^8 - 90901v^6 + 2229417v^4 - 27566757*v^2 - 5393898) / 15067458
$\beta_{3}$	$=$	$ ( 400\nu^{10} - 6140\nu^{8} + 94249\nu^{6} - 165780\nu^{4} + 33480\nu^{2} + 17064783 ) / 2511243 $ (400v^10 - 6140v^8 + 94249v^6 - 165780v^4 + 33480*v^2 + 17064783) / 2511243
$\beta_{4}$	$=$	$ ( -4940\nu^{11} + 75829\nu^{9} - 1038413\nu^{7} + 2047383\nu^{5} - 413478\nu^{3} - 142067574\nu ) / 15067458 $ (-4940v^11 + 75829v^9 - 1038413v^7 + 2047383v^5 - 413478v^3 - 142067574v) / 15067458
$\beta_{5}$	$=$	$ ( -6940\nu^{10} + 106529\nu^{8} - 1509658\nu^{6} + 5387526\nu^{4} - 28204551\nu^{2} - 3890862 ) / 15067458 $ (-6940v^10 + 106529v^8 - 1509658v^6 + 5387526v^4 - 28204551*v^2 - 3890862) / 15067458
$\beta_{6}$	$=$	$ ( -7340\nu^{11} + 112669\nu^{9} - 1603907\nu^{7} + 3042063\nu^{5} - 614358\nu^{3} - 289658646\nu ) / 15067458 $ (-7340v^11 + 112669v^9 - 1603907v^7 + 3042063v^5 - 614358v^3 - 289658646v) / 15067458
$\beta_{7}$	$=$	$ ( -9517\nu^{11} + 187940\nu^{9} - 2884879\nu^{7} + 16793514\nu^{5} - 77891733\nu^{3} + 15730578\nu ) / 15067458 $ (-9517v^11 + 187940v^9 - 2884879v^7 + 16793514v^5 - 77891733v^3 + 15730578v) / 15067458
$\beta_{8}$	$=$	$ ( -9517\nu^{10} + 187940\nu^{8} - 2884879\nu^{6} + 16793514\nu^{4} - 77891733\nu^{2} + 663120 ) / 15067458 $ (-9517v^10 + 187940v^8 - 2884879v^6 + 16793514v^4 - 77891733*v^2 + 663120) / 15067458
$\beta_{9}$	$=$	$ ( -9517\nu^{10} + 187940\nu^{8} - 2884879\nu^{6} + 16793514\nu^{4} - 75739239\nu^{2} + 663120 ) / 2152494 $ (-9517v^10 + 187940v^8 - 2884879v^6 + 16793514v^4 - 75739239*v^2 + 663120) / 2152494
$\beta_{10}$	$=$	$ ( - 88370 \nu^{11} + 1775020 \nu^{9} - 27246557 \nu^{7} + 162605637 \nu^{5} - 735657039 \nu^{3} + 148569174 \nu ) / 15067458 $ (-88370v^11 + 1775020v^9 - 27246557v^7 + 162605637v^5 - 735657039v^3 + 148569174v) / 15067458
$\beta_{11}$	$=$	$ ( 60380\nu^{11} - 1205860\nu^{9} + 18509951\nu^{7} - 109848699\nu^{5} + 499768677\nu^{3} - 100930482\nu ) / 5022486 $ (60380v^11 - 1205860v^9 + 18509951v^7 - 109848699v^5 + 499768677v^3 - 100930482v) / 5022486

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{9} - 7\beta_{8} $ b9 - 7*b8
$\nu^{3}$	$=$	$ -\beta_{11} - \beta_{10} - 10\beta_{7} + \beta_{6} - \beta_{4} + 10\beta_1 $ -b11 - b10 - 10b7 + b6 - b4 + 10b1
$\nu^{4}$	$=$	$ 11\beta_{9} - 77\beta_{8} + 4\beta_{5} + 11\beta_{3} + 2\beta_{2} - 73 $ 11b9 - 77b8 + 4b5 + 11b3 + 2*b2 - 73
$\nu^{5}$	$=$	$ -17\beta_{11} - 23\beta_{10} - 110\beta_{7} $ -17b11 - 23b10 - 110*b7
$\nu^{6}$	$=$	$ 40\beta_{5} + 127\beta_{3} - 40\beta_{2} - 867 $ 40b5 + 127b3 - 40*b2 - 867
$\nu^{7}$	$=$	$ -247\beta_{6} + 367\beta_{4} - 1288\beta_1 $ -247b6 + 367b4 - 1288*b1
$\nu^{8}$	$=$	$ -1535\beta_{9} + 11105\beta_{8} - 614\beta_{5} - 1228\beta_{2} - 614 $ -1535b9 + 11105b8 - 614b5 - 1228b2 - 614
$\nu^{9}$	$=$	$ 3377\beta_{11} + 5219\beta_{10} + 15710\beta_{7} - 3377\beta_{6} + 5219\beta_{4} - 15710\beta_1 $ 3377b11 + 5219b10 + 15710b7 - 3377b6 + 5219b4 - 15710b1
$\nu^{10}$	$=$	$ -19087\beta_{9} + 139135\beta_{8} - 17192\beta_{5} - 19087\beta_{3} - 8596\beta_{2} + 121943 $ -19087b9 + 139135b8 - 17192b5 - 19087b3 - 8596*b2 + 121943
$\nu^{11}$	$=$	$ 44875\beta_{11} + 70663\beta_{10} + 196396\beta_{7} $ 44875b11 + 70663b10 + 196396*b7

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$.

$n$	$10$	$29$
$\chi(n)$	$-1 - \beta_{8}$	$-1$

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} $

44.1

−3.11017	−	1.79566i
−2.27490	−	1.31341i
−0.389477	−	0.224865i
0.389477	+	0.224865i
2.27490	+	1.31341i
3.11017	+	1.79566i
−3.11017	+	1.79566i
−2.27490	+	1.31341i
−0.389477	+	0.224865i
0.389477	−	0.224865i
2.27490	−	1.31341i
3.11017	−	1.79566i

−3.11017

−

1.79566i

4.44876

7.70548i

5.33753

3.08163i

0.220803

−

6.99652i

−

17.5885i

−11.0671

−

19.1687i

44.2

−2.27490

−

1.31341i

1.45011

2.51167i

−3.37744

−

1.94997i

−6.97257

0.619118i

2.88892i

5.12223

8.87196i

44.3

−0.389477

−

0.224865i

−1.89887

−

3.28894i

−7.49023

−

4.32449i

6.25176

−

3.14888i

3.50688i

1.94485

3.36858i

44.4

0.389477

0.224865i

−1.89887

−

3.28894i

7.49023

4.32449i

6.25176

−

3.14888i

−

3.50688i

1.94485

3.36858i

44.5

2.27490

1.31341i

1.45011

2.51167i

3.37744

1.94997i

−6.97257

0.619118i

−

2.88892i

5.12223

8.87196i

44.6

3.11017

1.79566i

4.44876

7.70548i

−5.33753

−

3.08163i

0.220803

−

6.99652i

17.5885i

−11.0671

−

19.1687i

53.1

−3.11017

1.79566i

4.44876

−

7.70548i

5.33753

−

3.08163i

0.220803

6.99652i

17.5885i

−11.0671

19.1687i

53.2

−2.27490

1.31341i

1.45011

−

2.51167i

−3.37744

1.94997i

−6.97257

−

0.619118i

−

2.88892i

5.12223

−

8.87196i

53.3

−0.389477

0.224865i

−1.89887

3.28894i

−7.49023

4.32449i

6.25176

3.14888i

−

3.50688i

1.94485

−

3.36858i

53.4

0.389477

−

0.224865i

−1.89887

3.28894i

7.49023

−

4.32449i

6.25176

3.14888i

3.50688i

1.94485

−

3.36858i

53.5

2.27490

−

1.31341i

1.45011

−

2.51167i

3.37744

−

1.94997i

−6.97257

−

0.619118i

2.88892i

5.12223

−

8.87196i

53.6

3.11017

−

1.79566i

4.44876

−

7.70548i

−5.33753

3.08163i

0.220803

6.99652i

−

17.5885i

−11.0671

19.1687i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.c	even	3	1	inner
21.h	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	63.3.q.a	✓	12
3.b	odd	2	1	inner	63.3.q.a	✓	12
4.b	odd	2	1		1008.3.dc.e		12
7.b	odd	2	1		441.3.q.d		12
7.c	even	3	1	inner	63.3.q.a	✓	12
7.c	even	3	1		441.3.b.d		6
7.d	odd	6	1		441.3.b.c		6
7.d	odd	6	1		441.3.q.d		12
12.b	even	2	1		1008.3.dc.e		12
21.c	even	2	1		441.3.q.d		12
21.g	even	6	1		441.3.b.c		6
21.g	even	6	1		441.3.q.d		12
21.h	odd	6	1	inner	63.3.q.a	✓	12
21.h	odd	6	1		441.3.b.d		6
28.g	odd	6	1		1008.3.dc.e		12
84.n	even	6	1		1008.3.dc.e		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
63.3.q.a	✓	12	1.a	even	1	1	trivial
63.3.q.a	✓	12	3.b	odd	2	1	inner
63.3.q.a	✓	12	7.c	even	3	1	inner
63.3.q.a	✓	12	21.h	odd	6	1	inner
441.3.b.c		6	7.d	odd	6	1
441.3.b.c		6	21.g	even	6	1
441.3.b.d		6	7.c	even	3	1
441.3.b.d		6	21.h	odd	6	1
441.3.q.d		12	7.b	odd	2	1
441.3.q.d		12	7.d	odd	6	1
441.3.q.d		12	21.c	even	2	1
441.3.q.d		12	21.g	even	6	1
1008.3.dc.e		12	4.b	odd	2	1
1008.3.dc.e		12	12.b	even	2	1
1008.3.dc.e		12	28.g	odd	6	1
1008.3.dc.e		12	84.n	even	6	1

Hecke kernels

This newform subspace is the entire newspace $S_{3}^{\mathrm{new}}(63, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} - 20 T^{10} + \cdots + 324 $ T^12 - 20T^10 + 307T^8 - 1824T^6 + 8289T^4 - 1674*T^2 + 324
$3$	$ T^{12} $ T^12
$5$	$ T^{12} + \cdots + 1867795524 $ T^12 - 128T^10 + 11827T^8 - 496860T^6 + 15234345T^4 - 196944426*T^2 + 1867795524
$7$	$ (T^{6} + T^{5} + \cdots + 117649)^{2} $ (T^6 + T^5 - 28T^4 + 175T^3 - 1372T^2 + 2401T + 117649)^2
$11$	$ T^{12} + \cdots + 36466485444 $ T^12 - 212T^10 + 32467T^8 - 2263200T^6 + 115191585T^4 - 2382632874*T^2 + 36466485444
$13$	$ (T^{3} + 13 T^{2} + \cdots - 2842)^{4} $ (T^3 + 13T^2 - 238T - 2842)^4
$17$	$ T^{12} + \cdots + 619689727770624 $ T^12 - 996T^10 + 693900T^8 - 247136400T^6 + 64079155728T^4 - 7421170917888*T^2 + 619689727770624
$19$	$ (T^{6} - 13 T^{5} + \cdots + 8076964)^{2} $ (T^6 - 13T^5 + 407T^4 - 2590T^3 + 93590T^2 - 676396*T + 8076964)^2
$23$	$ T^{12} + \cdots + 367505463066624 $ T^12 - 1284T^10 + 1210572T^8 - 524158992T^6 + 167302756368T^4 - 8398259532288*T^2 + 367505463066624
$29$	$ (T^{6} + 2126 T^{4} + \cdots + 64706688)^{2} $ (T^6 + 2126T^4 + 1005201T^2 + 64706688)^2
$31$	$ (T^{6} - 11 T^{5} + \cdots + 115154361)^{2} $ (T^6 - 11T^5 + 1332T^4 + 34783T^3 + 1348480T^2 + 12995241*T + 115154361)^2
$37$	$ (T^{6} + 73 T^{5} + \cdots + 166255236)^{2} $ (T^6 + 73T^5 + 4335T^4 + 98350T^3 + 1929298T^2 - 12816636*T + 166255236)^2
$41$	$ (T^{6} + 3194 T^{4} + \cdots + 6223392)^{2} $ (T^6 + 3194T^4 + 2427264T^2 + 6223392)^2
$43$	$ (T^{3} - 27 T^{2} + \cdots + 26278)^{4} $ (T^3 - 27T^2 - 966T + 26278)^4
$47$	$ T^{12} + \cdots + 82\!\cdots\!04 $ T^12 - 7638T^10 + 43699608T^8 - 106062486264T^6 + 192340370970720T^4 - 42114187122975072*T^2 + 8275765081999754304
$53$	$ T^{12} + \cdots + 15\!\cdots\!44 $ T^12 - 6270T^10 + 32843907T^8 - 40311956286T^6 + 41068415935809T^4 - 804192295523616*T^2 + 15454197345567744
$59$	$ T^{12} + \cdots + 31\!\cdots\!84 $ T^12 - 8178T^10 + 61248555T^8 - 45939351906T^6 + 31251559716657T^4 - 315402508526112*T^2 + 3137179246838784
$61$	$ (T^{6} + 68 T^{5} + \cdots + 13645977856)^{2} $ (T^6 + 68T^5 + 6794T^4 + 86072T^3 + 12652388T^2 + 253490720*T + 13645977856)^2
$67$	$ (T^{6} - T^{5} + \cdots + 19642624)^{2} $ (T^6 - T^5 + 905T^4 - 7960T^3 + 821648T^2 - 4006528T + 19642624)^2
$71$	$ (T^{6} + 26262 T^{4} + \cdots + 95234699592)^{2} $ (T^6 + 26262T^4 + 168833484T^2 + 95234699592)^2
$73$	$ (T^{6} + 241 T^{5} + \cdots + 283799121984)^{2} $ (T^6 + 241T^5 + 47721T^4 + 3562216T^3 + 235717048T^2 - 5519062080*T + 283799121984)^2
$79$	$ (T^{6} - 21 T^{5} + \cdots + 345392465401)^{2} $ (T^6 - 21T^5 + 15252T^4 + 1486433T^3 + 207024000T^2 + 8704439511*T + 345392465401)^2
$83$	$ (T^{6} + 19476 T^{4} + \cdots + 31505922)^{2} $ (T^6 + 19476T^4 + 8434125T^2 + 31505922)^2
$89$	$ T^{12} + \cdots + 12\!\cdots\!44 $ T^12 - 10760T^10 + 81186736T^8 - 302805492864T^6 + 823197815951616T^4 - 1200168141737951232*T^2 + 1203819486222477164544
$97$	$ (T^{3} - 142 T^{2} + \cdots + 125832)^{4} $ (T^3 - 142T^2 - 5495T + 125832)^4
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	63.q (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + (\beta_{9} - 3 \beta_{8}) q^{4} - \beta_{6} q^{5} + (4 \beta_{8} - \beta_{5} + \beta_{3} + 1) q^{7} + ( - \beta_{11} - \beta_{10} + \cdots + 2 \beta_1) q^{8}+O(q^{10}) \) q + b1 * q^2 + (b9 - 3b8) q^4 - b6 * q^5 + (4b8 - b5 + b3 + 1) q^7 + (-b11 - b10 - 2b7 + b6 - b4 + 2b1) * q^8	\( q + \beta_1 q^{2} + (\beta_{9} - 3 \beta_{8}) q^{4} - \beta_{6} q^{5} + (4 \beta_{8} - \beta_{5} + \beta_{3} + 1) q^{7} + ( - \beta_{11} - \beta_{10} + \cdots + 2 \beta_1) q^{8}+ \cdots + ( - 2 \beta_{11} + 7 \beta_{7} + \cdots - 7 \beta_1) q^{98}+O(q^{100}) \) q + b1 * q^2 + (b9 - 3b8) q^4 - b6 * q^5 + (4b8 - b5 + b3 + 1) q^7 + (-b11 - b10 - 2b7 + b6 - b4 + 2b1) * q^8 + (-b9 + 4b8 - b5 - 2b2 - 1) * q^10 + (-2b10 + b7) q^11 + (-b5 - b3 + b2 - 4) * q^13 + (b11 + 2b10 + 4b7 - b4 - 5b1) q^14 + (-b9 - 9b8 + 4b5 - b3 + 2b2 - 5) q^16 + (b11 - 7b7) q^17 + (b9 + 3b8 + 2b5 + b3 + b2 + 5) * q^19 + (7b10 + 7b7 + 7b4 - 7b1) * q^20 + (2b5 - b3 - 2b2 + 3) * q^22 + (b6 + 4b4 + 5b1) * q^23 + (-5b9 - 18b8 + 2b5 + 4b2 + 2) * q^25 + (4b6 - 7b4) * q^26 + (-b9 + 24b8 - 2b5 - 5b3 + b2 + 20) q^28 + (4b11 - b10 + b7 - 4b6 - b4 - b1) * q^29 + (4b9 - 2b8 - 3b5 - 6b2 - 3) * q^31 + (-b11 - 7b10 + 2b7) * q^32 + (-b5 + 6b3 + b2 - 46) q^34 + (-4b11 - 7b7 - b6 - 7b4) q^35 + (5b9 - 25b8 - 2b5 + 5b3 - b2 - 27) * q^37 + (-4b11 - 7b10) * q^38 + (-3b9 + 54b8 - 6b5 - 3b3 - 3b2 + 48) q^40 + (-b11 + 7b10 - 7b7 + b6 + 7b4 + 7b1) * q^41 + (-3b5 + 3b3 + 3b2 + 8) q^43 + (-5b6 + 3b4) * q^44 + (6b9 - 27b8 - 3b5 - 6b2 - 3) * q^46 + (-2b6 - 7b4 + 14b1) q^47 + (4b9 + 19b8 + 2b5 + 5b3 - 4b2 + 17) q^49 + (-b11 - 7b10 - 3b7 + b6 - 7b4 + 3b1) * q^50 + (-49b8 + 7b5 + 14b2 + 7) q^52 + (4b11 - 9b10 + 5b7) q^53 + (3b5 - 11b3 - 3b2 + 5) q^55 + (2b11 + 7b10 + 7b7 + 4b6 + 14b1) q^56 + (-5b9 + 20b8 - 6b5 - 5b3 - 3b2 + 14) q^58 + (b11 + 7b10 + 14b7) * q^59 + (-2b9 - 27b8 + 10b5 - 2b3 + 5b2 - 17) q^61 + (5b11 + 14b10 - 14b7 - 5b6 + 14b4 + 14b1) * q^62 + (3b3 + 25) q^64 + (6b6 + 7b4 - 28b1) q^65 + (5b9 - 2b5 - 4b2 - 2) q^67 + (-7b6 - 35b1) * q^68 + (-b9 - 17b8 + 10b5 + 11b3 + 8b2 - 55) * q^70 + (-12b11 + 3b10 + 24b7 + 12b6 + 3b4 - 24b1) * q^71 + (-17b9 + 82b8 + 4b5 + 8b2 + 4) * q^73 + (-2b11 + b10 - 40b7) * q^74 + (7b5 - 7b2 - 42) * q^76 + (7b11 + b10 - 19b7 - b6 - 4b4 + 8b1) * q^77 + (-12b9 + 16b8 - 10b5 - 12b3 - 5b2 + 6) q^79 + (12b11 - 7b10 + 35b7) q^80 + (8b9 - 32b8 - 12b5 + 8b3 - 6b2 - 44) q^82 + (-6b11 - 14b10 + 21b7 + 6b6 - 14b4 - 21b1) * q^83 + (-5b5 - 12b3 + 5b2 - 20) q^85 + (6b6 - 15b4 + 2b1) q^86 + (-9b9 + 33b8) * q^88 + (2b6 + 14b4) * q^89 + (-11b9 + 37b8 + 5b5 - 19b3 - 10b2 + 39) q^91 + (7b11 - 4b10 - 25b7 - 7b6 - 4b4 + 25b1) * q^92 + (12b9 - 111b8 + 5b5 + 10b2 + 5) * q^94 + (-6b11 + 7b10 + 28b7) q^95 + (-2b5 + 19b3 + 2b2 + 41) q^97 + (-2b11 + 7b7 - 11b6 + 21b4 - 7b1) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 16 q^{4} - 2 q^{7}+O(q^{10}) \) 12 * q + 16 * q^4 - 2 * q^7	\( 12 q + 16 q^{4} - 2 q^{7} - 16 q^{10} - 52 q^{13} - 44 q^{16} + 26 q^{19} + 32 q^{22} + 106 q^{25} + 84 q^{28} + 22 q^{31} - 528 q^{34} - 146 q^{37} + 300 q^{40} + 108 q^{43} + 168 q^{46} + 114 q^{49} + 252 q^{52} + 16 q^{55} + 92 q^{58} - 136 q^{61} + 312 q^{64} + 2 q^{67} - 620 q^{70} - 482 q^{73} - 504 q^{76} + 42 q^{79} - 212 q^{82} - 288 q^{85} - 180 q^{88} + 222 q^{91} + 612 q^{94} + 568 q^{97}+O(q^{100}) \) 12 * q + 16 * q^4 - 2 * q^7 - 16 * q^10 - 52 * q^13 - 44 * q^16 + 26 * q^19 + 32 * q^22 + 106 * q^25 + 84 * q^28 + 22 * q^31 - 528 * q^34 - 146 * q^37 + 300 * q^40 + 108 * q^43 + 168 * q^46 + 114 * q^49 + 252 * q^52 + 16 * q^55 + 92 * q^58 - 136 * q^61 + 312 * q^64 + 2 * q^67 - 620 * q^70 - 482 * q^73 - 504 * q^76 + 42 * q^79 - 212 * q^82 - 288 * q^85 - 180 * q^88 + 222 * q^91 + 612 * q^94 + 568 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	63.3.q.a

Level	$63$
Weight	$3$
Character orbit	63.q
Analytic conductor	$1.717$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.71662566547\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 20x^{10} + 307x^{8} - 1824x^{6} + 8289x^{4} - 1674x^{2} + 324 \) x^12 - 20x^10 + 307x^8 - 1824x^6 + 8289x^4 - 1674*x^2 + 324
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{4}\cdot 3^{5} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( 680\nu^{10} - 10438\nu^{8} - 90901\nu^{6} + 2229417\nu^{4} - 27566757\nu^{2} - 5393898 ) / 15067458 \) (680v^10 - 10438v^8 - 90901v^6 + 2229417v^4 - 27566757*v^2 - 5393898) / 15067458
\(\beta_{3}\)	\(=\)	\( ( 400\nu^{10} - 6140\nu^{8} + 94249\nu^{6} - 165780\nu^{4} + 33480\nu^{2} + 17064783 ) / 2511243 \) (400v^10 - 6140v^8 + 94249v^6 - 165780v^4 + 33480*v^2 + 17064783) / 2511243
\(\beta_{4}\)	\(=\)	\( ( -4940\nu^{11} + 75829\nu^{9} - 1038413\nu^{7} + 2047383\nu^{5} - 413478\nu^{3} - 142067574\nu ) / 15067458 \) (-4940v^11 + 75829v^9 - 1038413v^7 + 2047383v^5 - 413478v^3 - 142067574v) / 15067458
\(\beta_{5}\)	\(=\)	\( ( -6940\nu^{10} + 106529\nu^{8} - 1509658\nu^{6} + 5387526\nu^{4} - 28204551\nu^{2} - 3890862 ) / 15067458 \) (-6940v^10 + 106529v^8 - 1509658v^6 + 5387526v^4 - 28204551*v^2 - 3890862) / 15067458
\(\beta_{6}\)	\(=\)	\( ( -7340\nu^{11} + 112669\nu^{9} - 1603907\nu^{7} + 3042063\nu^{5} - 614358\nu^{3} - 289658646\nu ) / 15067458 \) (-7340v^11 + 112669v^9 - 1603907v^7 + 3042063v^5 - 614358v^3 - 289658646v) / 15067458
\(\beta_{7}\)	\(=\)	\( ( -9517\nu^{11} + 187940\nu^{9} - 2884879\nu^{7} + 16793514\nu^{5} - 77891733\nu^{3} + 15730578\nu ) / 15067458 \) (-9517v^11 + 187940v^9 - 2884879v^7 + 16793514v^5 - 77891733v^3 + 15730578v) / 15067458
\(\beta_{8}\)	\(=\)	\( ( -9517\nu^{10} + 187940\nu^{8} - 2884879\nu^{6} + 16793514\nu^{4} - 77891733\nu^{2} + 663120 ) / 15067458 \) (-9517v^10 + 187940v^8 - 2884879v^6 + 16793514v^4 - 77891733*v^2 + 663120) / 15067458
\(\beta_{9}\)	\(=\)	\( ( -9517\nu^{10} + 187940\nu^{8} - 2884879\nu^{6} + 16793514\nu^{4} - 75739239\nu^{2} + 663120 ) / 2152494 \) (-9517v^10 + 187940v^8 - 2884879v^6 + 16793514v^4 - 75739239*v^2 + 663120) / 2152494
\(\beta_{10}\)	\(=\)	\( ( - 88370 \nu^{11} + 1775020 \nu^{9} - 27246557 \nu^{7} + 162605637 \nu^{5} - 735657039 \nu^{3} + 148569174 \nu ) / 15067458 \) (-88370v^11 + 1775020v^9 - 27246557v^7 + 162605637v^5 - 735657039v^3 + 148569174v) / 15067458
\(\beta_{11}\)	\(=\)	\( ( 60380\nu^{11} - 1205860\nu^{9} + 18509951\nu^{7} - 109848699\nu^{5} + 499768677\nu^{3} - 100930482\nu ) / 5022486 \) (60380v^11 - 1205860v^9 + 18509951v^7 - 109848699v^5 + 499768677v^3 - 100930482v) / 5022486

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{9} - 7\beta_{8} \) b9 - 7*b8
\(\nu^{3}\)	\(=\)	\( -\beta_{11} - \beta_{10} - 10\beta_{7} + \beta_{6} - \beta_{4} + 10\beta_1 \) -b11 - b10 - 10b7 + b6 - b4 + 10b1
\(\nu^{4}\)	\(=\)	\( 11\beta_{9} - 77\beta_{8} + 4\beta_{5} + 11\beta_{3} + 2\beta_{2} - 73 \) 11b9 - 77b8 + 4b5 + 11b3 + 2*b2 - 73
\(\nu^{5}\)	\(=\)	\( -17\beta_{11} - 23\beta_{10} - 110\beta_{7} \) -17b11 - 23b10 - 110*b7
\(\nu^{6}\)	\(=\)	\( 40\beta_{5} + 127\beta_{3} - 40\beta_{2} - 867 \) 40b5 + 127b3 - 40*b2 - 867
\(\nu^{7}\)	\(=\)	\( -247\beta_{6} + 367\beta_{4} - 1288\beta_1 \) -247b6 + 367b4 - 1288*b1
\(\nu^{8}\)	\(=\)	\( -1535\beta_{9} + 11105\beta_{8} - 614\beta_{5} - 1228\beta_{2} - 614 \) -1535b9 + 11105b8 - 614b5 - 1228b2 - 614
\(\nu^{9}\)	\(=\)	\( 3377\beta_{11} + 5219\beta_{10} + 15710\beta_{7} - 3377\beta_{6} + 5219\beta_{4} - 15710\beta_1 \) 3377b11 + 5219b10 + 15710b7 - 3377b6 + 5219b4 - 15710b1
\(\nu^{10}\)	\(=\)	\( -19087\beta_{9} + 139135\beta_{8} - 17192\beta_{5} - 19087\beta_{3} - 8596\beta_{2} + 121943 \) -19087b9 + 139135b8 - 17192b5 - 19087b3 - 8596*b2 + 121943
\(\nu^{11}\)	\(=\)	\( 44875\beta_{11} + 70663\beta_{10} + 196396\beta_{7} \) 44875b11 + 70663b10 + 196396*b7

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 44.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} - 20 T^{10} + \cdots + 324 \) T^12 - 20T^10 + 307T^8 - 1824T^6 + 8289T^4 - 1674*T^2 + 324
$3$	\( T^{12} \) T^12
$5$	\( T^{12} + \cdots + 1867795524 \) T^12 - 128T^10 + 11827T^8 - 496860T^6 + 15234345T^4 - 196944426*T^2 + 1867795524
$7$	\( (T^{6} + T^{5} + \cdots + 117649)^{2} \) (T^6 + T^5 - 28T^4 + 175T^3 - 1372T^2 + 2401T + 117649)^2
$11$	\( T^{12} + \cdots + 36466485444 \) T^12 - 212T^10 + 32467T^8 - 2263200T^6 + 115191585T^4 - 2382632874*T^2 + 36466485444
$13$	\( (T^{3} + 13 T^{2} + \cdots - 2842)^{4} \) (T^3 + 13T^2 - 238T - 2842)^4
$17$	\( T^{12} + \cdots + 619689727770624 \) T^12 - 996T^10 + 693900T^8 - 247136400T^6 + 64079155728T^4 - 7421170917888*T^2 + 619689727770624
$19$	\( (T^{6} - 13 T^{5} + \cdots + 8076964)^{2} \) (T^6 - 13T^5 + 407T^4 - 2590T^3 + 93590T^2 - 676396*T + 8076964)^2
$23$	\( T^{12} + \cdots + 367505463066624 \) T^12 - 1284T^10 + 1210572T^8 - 524158992T^6 + 167302756368T^4 - 8398259532288*T^2 + 367505463066624
$29$	\( (T^{6} + 2126 T^{4} + \cdots + 64706688)^{2} \) (T^6 + 2126T^4 + 1005201T^2 + 64706688)^2
$31$	\( (T^{6} - 11 T^{5} + \cdots + 115154361)^{2} \) (T^6 - 11T^5 + 1332T^4 + 34783T^3 + 1348480T^2 + 12995241*T + 115154361)^2
$37$	\( (T^{6} + 73 T^{5} + \cdots + 166255236)^{2} \) (T^6 + 73T^5 + 4335T^4 + 98350T^3 + 1929298T^2 - 12816636*T + 166255236)^2
$41$	\( (T^{6} + 3194 T^{4} + \cdots + 6223392)^{2} \) (T^6 + 3194T^4 + 2427264T^2 + 6223392)^2
$43$	\( (T^{3} - 27 T^{2} + \cdots + 26278)^{4} \) (T^3 - 27T^2 - 966T + 26278)^4
$47$	\( T^{12} + \cdots + 82\!\cdots\!04 \) T^12 - 7638T^10 + 43699608T^8 - 106062486264T^6 + 192340370970720T^4 - 42114187122975072*T^2 + 8275765081999754304
$53$	\( T^{12} + \cdots + 15\!\cdots\!44 \) T^12 - 6270T^10 + 32843907T^8 - 40311956286T^6 + 41068415935809T^4 - 804192295523616*T^2 + 15454197345567744
$59$	\( T^{12} + \cdots + 31\!\cdots\!84 \) T^12 - 8178T^10 + 61248555T^8 - 45939351906T^6 + 31251559716657T^4 - 315402508526112*T^2 + 3137179246838784
$61$	\( (T^{6} + 68 T^{5} + \cdots + 13645977856)^{2} \) (T^6 + 68T^5 + 6794T^4 + 86072T^3 + 12652388T^2 + 253490720*T + 13645977856)^2
$67$	\( (T^{6} - T^{5} + \cdots + 19642624)^{2} \) (T^6 - T^5 + 905T^4 - 7960T^3 + 821648T^2 - 4006528T + 19642624)^2
$71$	\( (T^{6} + 26262 T^{4} + \cdots + 95234699592)^{2} \) (T^6 + 26262T^4 + 168833484T^2 + 95234699592)^2
$73$	\( (T^{6} + 241 T^{5} + \cdots + 283799121984)^{2} \) (T^6 + 241T^5 + 47721T^4 + 3562216T^3 + 235717048T^2 - 5519062080*T + 283799121984)^2
$79$	\( (T^{6} - 21 T^{5} + \cdots + 345392465401)^{2} \) (T^6 - 21T^5 + 15252T^4 + 1486433T^3 + 207024000T^2 + 8704439511*T + 345392465401)^2
$83$	\( (T^{6} + 19476 T^{4} + \cdots + 31505922)^{2} \) (T^6 + 19476T^4 + 8434125T^2 + 31505922)^2
$89$	\( T^{12} + \cdots + 12\!\cdots\!44 \) T^12 - 10760T^10 + 81186736T^8 - 302805492864T^6 + 823197815951616T^4 - 1200168141737951232*T^2 + 1203819486222477164544
$97$	\( (T^{3} - 142 T^{2} + \cdots + 125832)^{4} \) (T^3 - 142T^2 - 5495T + 125832)^4
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\(n\)	\(10\)	\(29\)
\(\chi(n)\)	\(-1 - \beta_{8}\)	\(-1\)