LMFDB - Newform orbit 529.2.c.o

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [529,2,Mod(118,529)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(529, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([16]))

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

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

chi := DirichletCharacter("529.118");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 529 = 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	529.c (of order $11$, degree $10$, 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:	$4.22408626693$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$2$ over $\Q(\zeta_{11})$
Coefficient field:	20.0.54296067514572573056640625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + 2 x^{18} - 3 x^{17} + 5 x^{16} - 8 x^{15} + 13 x^{14} - 21 x^{13} + 34 x^{12} + \cdots + 1 $ x^20 - x^19 + 2x^18 - 3x^17 + 5x^16 - 8x^15 + 13x^14 - 21x^13 + 34x^12 - 55x^11 + 89x^10 + 55x^9 + 34x^8 + 21x^7 + 13x^6 + 8x^5 + 5x^4 + 3x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 23)
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

$q$-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_{16} q^{2} + (\beta_{19} - 2 \beta_{18} + \beta_{17} + \cdots + 1) q^{3}+ \cdots + 2 \beta_{15} q^{9}+O(q^{10}) $ q + b16 * q^2 + (b19 - 2b18 + b17 + 2b16 + b15 - 2b14 - b13 + 2b12 - b11 - 2b10 - b9 - 2b8 - b7 + 2b6 + b5 - 2b4 + b3 + 2b2 + 2b1 + 1) * q^3 + (b11 + b10) * q^4 - 2b8 q^5 + (2b13 + b12) q^6 + (2b19 - 2b18) * q^7 + (-b5 + 2b4) q^8 + 2b15 q^9	$ q + \beta_{16} q^{2} + (\beta_{19} - 2 \beta_{18} + \beta_{17} + \cdots + 1) q^{3}+ \cdots + ( - 8 \beta_{19} + 4 \beta_{18}) q^{99}+O(q^{100}) $ q + b16 * q^2 + (b19 - 2b18 + b17 + 2b16 + b15 - 2b14 - b13 + 2b12 - b11 - 2b10 - b9 - 2b8 - b7 + 2b6 + b5 - 2b4 + b3 + 2b2 + 2b1 + 1) * q^3 + (b11 + b10) * q^4 - 2b8 q^5 + (2b13 + b12) q^6 + (2b19 - 2b18) * q^7 + (-b5 + 2b4) q^8 + 2b15 q^9 + (2b3 - 2b1) * q^10 + (4b7 - 2b6) * q^11 + (-3b7 + b6) q^12 + 3b3 q^13 + 2b15 q^14 + (-4b5 - 2b4) * q^15 - 3b18 q^16 + (-2b13 - 2b12) * q^17 - 2b8 q^18 + 2b11 q^19 + (2b19 + 2b17 + 2b15 - 2b13 - 2b11 - 2b9 - 2b7 + 2b5 + 2b3 + 2) q^20 + (-6b17 - 2b16) * q^21 + (-2b2 - 2) q^22 + 5 * q^24 + (-b17 - 4b16) q^25 + (3b18 - 3b16 + 3b14 - 3b12 + 3b10 + 3b8 - 3b6 + 3b4 - 3b2 - 3b1) * q^26 + (-b11 - 2b10) q^27 + (4b9 + 2b8) * q^28 + 3b13 q^29 + (2b19 + 6b18) * q^30 + (3b5 - 6b4) * q^31 + (5b15 - b14) q^32 + (8b3 + 6b1) * q^33 + (2b7 + 4b6) * q^34 - 4b7 q^35 + (-2b3 - 2b1) * q^36 + 2b14 q^37 + 2b4 q^38 + (-3b19 + 6b18) * q^39 + (4b13 + 2b12) * q^40 + (b9 + 4b8) q^41 + (2b11 + 4b10) * q^42 + (6b17 + 4b16) * q^44 + 4b2 q^45 + (-2b2 - 1) q^47 + (-6b17 + 3b16) * q^48 + (-b19 + 4b18 - b17 - 4b16 - b15 + 4b14 + b13 - 4b12 + b11 + 4b10 + b9 + 4b8 + b7 - 4b6 - b5 + 4b4 - b3 - 4b2 - 4b1 - 1) * q^49 + (4b11 - 3b10) * q^50 + (-2b9 + 6b8) * q^51 + (3b13 - 3b12) * q^52 + (-2b19 - 4b18) * q^53 + (2b5 + b4) q^54 + (-4b15 + 4b14) * q^55 + (-6b3 - 2b1) * q^56 + (-2b7 + 4b6) * q^57 - 3b6 q^58 + (4b3 + 4b1) * q^59 + (2b15 - 4b14) * q^60 + (-2b5 + 8b4) * q^61 + (6b19 + 3b18) * q^62 + (-4b13 + 4b12) * q^63 + (-b9 + 2b8) q^64 - 6b10 q^65 + (-6b19 + 2b18 - 6b17 - 2b16 - 6b15 + 2b14 + 6b13 - 2b12 + 6b11 + 2b10 + 6b9 + 2b8 + 6b7 - 2b6 - 6b5 + 2b4 - 6b3 - 2b2 - 2b1 - 6) q^66 + (-4b17 + 2b16) * q^67 - 2b2 q^68 + 4b2 q^70 + (11b17 + 2b16) * q^71 + (2b19 - 4b18 + 2b17 + 4b16 + 2b15 - 4b14 - 2b13 + 4b12 - 2b11 - 4b10 - 2b9 - 4b8 - 2b7 + 4b6 + 2b5 - 4b4 + 2b3 + 4b2 + 4b1 + 2) q^72 + (-9b11 + 4b10) * q^73 + (2b9 - 2b8) * q^74 + (-9b13 - 2b12) * q^75 + (2b19 - 2b18) * q^76 + (-12b5 + 8b4) * q^77 + (-6b15 - 3b14) * q^78 + (-6b3 - 8b1) * q^79 + (-6b7 - 6b6) * q^80 + 11b7 q^81 + (-4b3 + 3b1) * q^82 + (-10b15 - 2b14) * q^83 + (8b5 - 6b4) * q^84 + (-4b19 - 8b18) * q^85 + (-3b9 - 6b8) * q^87 + (-8b11 - 6b10) * q^88 + (8b19 - 4b18 + 8b17 + 4b16 + 8b15 - 4b14 - 8b13 + 4b12 - 8b11 - 4b10 - 8b9 - 4b8 - 8b7 + 4b6 + 8b5 - 4b4 + 8b3 + 4b2 + 4b1 + 8) q^89 + (4b17 - 4b16) * q^90 + (6b2 + 6) q^91 - 15 * q^93 + (-2b17 + b16) q^94 + (-4b18 + 4b16 - 4b14 + 4b12 - 4b10 - 4b8 + 4b6 - 4b4 + 4b2 + 4b1) * q^95 + (7b11 + 9b10) * q^96 + (-14b9 - 6b8) * q^97 + (-4b13 - 3b12) * q^98 + (-8b19 + 4b18) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + q^{2} + q^{4} + 2 q^{5} + 5 q^{6} - 2 q^{7} - 4 q^{9}+O(q^{10}) $ 20 * q + q^2 + q^4 + 2 * q^5 + 5 * q^6 - 2 * q^7 - 4 * q^9	$ 20 q + q^{2} + q^{4} + 2 q^{5} + 5 q^{6} - 2 q^{7} - 4 q^{9} - 6 q^{10} + 6 q^{11} - 5 q^{12} - 6 q^{13} - 4 q^{14} + 10 q^{15} + 3 q^{16} - 6 q^{17} + 2 q^{18} + 4 q^{19} + 4 q^{20} + 10 q^{21} - 20 q^{22} + 100 q^{24} - 2 q^{25} + 3 q^{26} + 6 q^{28} + 6 q^{29} - 10 q^{30} - 9 q^{32} - 10 q^{33} + 8 q^{34} - 8 q^{35} + 2 q^{36} - 2 q^{37} - 2 q^{38} + 10 q^{40} - 2 q^{41} - 8 q^{44} - 40 q^{45} + 15 q^{48} + 2 q^{49} + 11 q^{50} - 10 q^{51} + 3 q^{52} + 8 q^{53} - 5 q^{54} + 4 q^{55} + 10 q^{56} - 3 q^{58} - 4 q^{59} - 4 q^{61} - 15 q^{62} - 4 q^{63} - 4 q^{64} + 6 q^{65} - 10 q^{66} + 10 q^{67} + 20 q^{68} - 40 q^{70} - 20 q^{71} - 22 q^{73} + 6 q^{74} - 20 q^{75} - 2 q^{76} + 16 q^{77} + 15 q^{78} + 4 q^{79} - 18 q^{80} + 22 q^{81} + 11 q^{82} + 22 q^{83} - 10 q^{84} + 16 q^{85} - 10 q^{88} + 12 q^{89} - 12 q^{90} + 60 q^{91} - 300 q^{93} + 5 q^{94} - 4 q^{95} + 5 q^{96} - 22 q^{97} - 11 q^{98} + 12 q^{99}+O(q^{100}) $ 20 * q + q^2 + q^4 + 2 * q^5 + 5 * q^6 - 2 * q^7 - 4 * q^9 - 6 * q^10 + 6 * q^11 - 5 * q^12 - 6 * q^13 - 4 * q^14 + 10 * q^15 + 3 * q^16 - 6 * q^17 + 2 * q^18 + 4 * q^19 + 4 * q^20 + 10 * q^21 - 20 * q^22 + 100 * q^24 - 2 * q^25 + 3 * q^26 + 6 * q^28 + 6 * q^29 - 10 * q^30 - 9 * q^32 - 10 * q^33 + 8 * q^34 - 8 * q^35 + 2 * q^36 - 2 * q^37 - 2 * q^38 + 10 * q^40 - 2 * q^41 - 8 * q^44 - 40 * q^45 + 15 * q^48 + 2 * q^49 + 11 * q^50 - 10 * q^51 + 3 * q^52 + 8 * q^53 - 5 * q^54 + 4 * q^55 + 10 * q^56 - 3 * q^58 - 4 * q^59 - 4 * q^61 - 15 * q^62 - 4 * q^63 - 4 * q^64 + 6 * q^65 - 10 * q^66 + 10 * q^67 + 20 * q^68 - 40 * q^70 - 20 * q^71 - 22 * q^73 + 6 * q^74 - 20 * q^75 - 2 * q^76 + 16 * q^77 + 15 * q^78 + 4 * q^79 - 18 * q^80 + 22 * q^81 + 11 * q^82 + 22 * q^83 - 10 * q^84 + 16 * q^85 - 10 * q^88 + 12 * q^89 - 12 * q^90 + 60 * q^91 - 300 * q^93 + 5 * q^94 - 4 * q^95 + 5 * q^96 - 22 * q^97 - 11 * q^98 + 12 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{20} - x^{19} + 2 x^{18} - 3 x^{17} + 5 x^{16} - 8 x^{15} + 13 x^{14} - 21 x^{13} + 34 x^{12} + \cdots + 1 $ x^20 - x^19 + 2*x^18 - 3*x^17 + 5*x^16 - 8*x^15 + 13*x^14 - 21*x^13 + 34*x^12 - 55*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{11} + 55 ) / 89 $ (v^11 + 55) / 89
$\beta_{3}$	$=$	$ ( \nu^{12} + 144\nu ) / 89 $ (v^12 + 144*v) / 89
$\beta_{4}$	$=$	$ ( -\nu^{13} - 144\nu^{2} ) / 89 $ (-v^13 - 144*v^2) / 89
$\beta_{5}$	$=$	$ ( \nu^{13} + 233\nu^{2} ) / 89 $ (v^13 + 233*v^2) / 89
$\beta_{6}$	$=$	$ ( \nu^{14} + 233\nu^{3} ) / 89 $ (v^14 + 233*v^3) / 89
$\beta_{7}$	$=$	$ ( -2\nu^{14} - 377\nu^{3} ) / 89 $ (-2v^14 - 377v^3) / 89
$\beta_{8}$	$=$	$ ( -2\nu^{15} - 377\nu^{4} ) / 89 $ (-2v^15 - 377v^4) / 89
$\beta_{9}$	$=$	$ ( -3\nu^{15} - 610\nu^{4} ) / 89 $ (-3v^15 - 610v^4) / 89
$\beta_{10}$	$=$	$ ( -3\nu^{16} - 610\nu^{5} ) / 89 $ (-3v^16 - 610v^5) / 89
$\beta_{11}$	$=$	$ ( -5\nu^{16} - 987\nu^{5} ) / 89 $ (-5v^16 - 987v^5) / 89
$\beta_{12}$	$=$	$ ( 5\nu^{17} + 987\nu^{6} ) / 89 $ (5v^17 + 987v^6) / 89
$\beta_{13}$	$=$	$ ( -8\nu^{17} - 1597\nu^{6} ) / 89 $ (-8v^17 - 1597v^6) / 89
$\beta_{14}$	$=$	$ ( -8\nu^{18} - 1597\nu^{7} ) / 89 $ (-8v^18 - 1597v^7) / 89
$\beta_{15}$	$=$	$ ( 13\nu^{18} + 2584\nu^{7} ) / 89 $ (13v^18 + 2584v^7) / 89
$\beta_{16}$	$=$	$ ( 13\nu^{19} + 2584\nu^{8} ) / 89 $ (13v^19 + 2584v^8) / 89
$\beta_{17}$	$=$	$ ( 21\nu^{19} + 4181\nu^{8} ) / 89 $ (21v^19 + 4181v^8) / 89
$\beta_{18}$	$=$	$ ( 34 \nu^{19} - 34 \nu^{18} + 68 \nu^{17} - 102 \nu^{16} + 170 \nu^{15} - 272 \nu^{14} + 442 \nu^{13} + \cdots + 34 ) / 89 $ (34v^19 - 34v^18 + 68v^17 - 102v^16 + 170v^15 - 272v^14 + 442v^13 - 714v^12 + 1156v^11 - 1869v^10 + 3026v^9 + 1870v^8 + 1156v^7 + 714v^6 + 442v^5 + 272v^4 + 170v^3 + 102v^2 + 68*v + 34) / 89
$\beta_{19}$	$=$	$ ( - 55 \nu^{19} + 55 \nu^{18} - 110 \nu^{17} + 165 \nu^{16} - 275 \nu^{15} + 440 \nu^{14} - 715 \nu^{13} + \cdots - 55 ) / 89 $ (-55v^19 + 55v^18 - 110v^17 + 165v^16 - 275v^15 + 440v^14 - 715v^13 + 1155v^12 - 1870v^11 + 3026v^10 - 4895v^9 - 3025v^8 - 1870v^7 - 1155v^6 - 715v^5 - 440v^4 - 275v^3 - 165v^2 - 110*v - 55) / 89

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{5} + \beta_{4} $ b5 + b4
$\nu^{3}$	$=$	$ \beta_{7} + 2\beta_{6} $ b7 + 2*b6
$\nu^{4}$	$=$	$ -2\beta_{9} + 3\beta_{8} $ -2b9 + 3b8
$\nu^{5}$	$=$	$ 3\beta_{11} - 5\beta_{10} $ 3b11 - 5b10
$\nu^{6}$	$=$	$ -5\beta_{13} - 8\beta_{12} $ -5b13 - 8b12
$\nu^{7}$	$=$	$ -8\beta_{15} - 13\beta_{14} $ -8b15 - 13b14
$\nu^{8}$	$=$	$ 13\beta_{17} - 21\beta_{16} $ 13b17 - 21b16
$\nu^{9}$	$=$	$ 21 \beta_{19} + 34 \beta_{18} + 21 \beta_{17} - 34 \beta_{16} + 21 \beta_{15} + 34 \beta_{14} + \cdots + 21 $ 21b19 + 34b18 + 21b17 - 34b16 + 21b15 + 34b14 - 21b13 - 34b12 - 21b11 + 34b10 - 21b9 + 34b8 - 21b7 - 34b6 + 21b5 + 34b4 + 21b3 - 34b2 - 34*b1 + 21
$\nu^{10}$	$=$	$ 34\beta_{19} + 55\beta_{18} $ 34b19 + 55b18
$\nu^{11}$	$=$	$ 89\beta_{2} - 55 $ 89*b2 - 55
$\nu^{12}$	$=$	$ 89\beta_{3} - 144\beta_1 $ 89b3 - 144b1
$\nu^{13}$	$=$	$ -144\beta_{5} - 233\beta_{4} $ -144b5 - 233b4
$\nu^{14}$	$=$	$ -233\beta_{7} - 377\beta_{6} $ -233b7 - 377b6
$\nu^{15}$	$=$	$ 377\beta_{9} - 610\beta_{8} $ 377b9 - 610b8
$\nu^{16}$	$=$	$ -610\beta_{11} + 987\beta_{10} $ -610b11 + 987b10
$\nu^{17}$	$=$	$ 987\beta_{13} + 1597\beta_{12} $ 987b13 + 1597b12
$\nu^{18}$	$=$	$ 1597\beta_{15} + 2584\beta_{14} $ 1597b15 + 2584b14
$\nu^{19}$	$=$	$ -2584\beta_{17} + 4181\beta_{16} $ -2584b17 + 4181b16

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/529\mathbb{Z}\right)^\times$.

$n$	$5$
$\chi(n)$	$\beta_{17}$

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} $

118.1

0.519923	+	0.334134i
−1.36118	−	0.874775i
0.256741	+	0.562183i
−0.672156	−	1.47182i
−0.592999	−	0.174120i
1.55249	+	0.455853i
1.05959	+	1.22283i
−0.404726	−	0.467079i
−0.592999	+	0.174120i
1.55249	−	0.455853i
1.05959	−	1.22283i
−0.404726	+	0.467079i
0.519923	−	0.334134i
−1.36118	+	0.874775i
0.230270	+	1.60156i
−0.0879554	−	0.611743i
0.230270	−	1.60156i
−0.0879554	+	0.611743i
0.256741	−	0.562183i
−0.672156	+	1.47182i

−0.0879554

−

0.611743i

−0.928896

2.03400i

1.55249

−

0.455853i

−0.809452

0.934158i

1.32599

0.389345i

2.72235

−

1.74955i

−0.928896

−

2.03400i

−1.30972

−

1.51150i

0.642661

0.413013i

118.2

0.230270

1.60156i

0.928896

−

2.03400i

−0.592999

0.174120i

2.11917

−

2.44566i

3.47148

1.01932i

−1.03985

0.668269i

0.928896

2.03400i

−1.30972

−

1.51150i

4.40486

2.83083i

170.1

−0.592999

0.174120i

1.46431

1.68991i

−1.36118

0.874775i

−0.175911

−

1.22349i

−1.16258

−

0.747147i

1.34431

−

2.94363i

1.46431

−

1.68991i

−0.284630

1.97964i

0.317349

0.694897i

170.2

1.55249

−

0.455853i

−1.46431

−

1.68991i

0.519923

−

0.334134i

0.460540

3.20313i

−3.04368

−

1.95606i

−0.513481

1.12437i

−1.46431

1.68991i

−0.284630

1.97964i

2.17514

4.76289i

177.1

−0.404726

0.467079i

−1.88110

1.20891i

0.230270

1.60156i

0.513481

1.12437i

0.196674

−

1.36790i

−3.10498

0.911706i

−1.88110

−

1.20891i

0.830830

−

1.81926i

−0.732987

−

0.215225i

177.2

1.05959

−

1.22283i

1.88110

−

1.20891i

−0.0879554

−

0.611743i

−1.34431

−

2.94363i

0.514900

−

3.58121i

1.18600

−

0.348241i

1.88110

1.20891i

0.830830

−

1.81926i

−5.02397

−

1.47517i

255.1

−1.36118

−

0.874775i

−0.318226

−

2.21331i

0.256741

0.562183i

3.10498

0.911706i

−1.50299

3.29108i

0.809452

−

0.934158i

−0.318226

2.21331i

−1.91899

0.563465i

−3.42890

−

3.95716i

255.2

0.519923

0.334134i

0.318226

2.21331i

−0.672156

−

1.47182i

−1.18600

−

0.348241i

−0.574089

1.25708i

−2.11917

2.44566i

0.318226

−

2.21331i

−1.91899

0.563465i

−0.500269

−

0.577341i

266.1

−0.404726

−

0.467079i

−1.88110

−

1.20891i

0.230270

−

1.60156i

0.513481

−

1.12437i

0.196674

1.36790i

−3.10498

−

0.911706i

−1.88110

1.20891i

0.830830

1.81926i

−0.732987

0.215225i

266.2

1.05959

1.22283i

1.88110

1.20891i

−0.0879554

0.611743i

−1.34431

2.94363i

0.514900

3.58121i

1.18600

0.348241i

1.88110

−

1.20891i

0.830830

1.81926i

−5.02397

1.47517i

334.1

−1.36118

0.874775i

−0.318226

2.21331i

0.256741

−

0.562183i

3.10498

−

0.911706i

−1.50299

−

3.29108i

0.809452

0.934158i

−0.318226

−

2.21331i

−1.91899

−

0.563465i

−3.42890

3.95716i

334.2

0.519923

−

0.334134i

0.318226

−

2.21331i

−0.672156

1.47182i

−1.18600

0.348241i

−0.574089

−

1.25708i

−2.11917

−

2.44566i

0.318226

2.21331i

−1.91899

−

0.563465i

−0.500269

0.577341i

399.1

−0.0879554

0.611743i

−0.928896

−

2.03400i

1.55249

0.455853i

−0.809452

−

0.934158i

1.32599

−

0.389345i

2.72235

1.74955i

−0.928896

2.03400i

−1.30972

1.51150i

0.642661

−

0.413013i

399.2

0.230270

−

1.60156i

0.928896

2.03400i

−0.592999

−

0.174120i

2.11917

2.44566i

3.47148

−

1.01932i

−1.03985

−

0.668269i

0.928896

−

2.03400i

−1.30972

1.51150i

4.40486

−

2.83083i

466.1

−0.672156

1.47182i

−2.14549

−

0.629973i

−0.404726

−

0.467079i

−2.72235

1.74955i

2.36931

−

2.73433i

0.175911

−

1.22349i

−2.14549

0.629973i

1.68251

1.08128i

−0.745170

−

5.18277i

466.2

0.256741

−

0.562183i

2.14549

0.629973i

1.05959

1.22283i

1.03985

−

0.668269i

0.904995

−

1.04442i

−0.460540

3.20313i

2.14549

−

0.629973i

1.68251

1.08128i

−0.108719

−

0.756156i

487.1

−0.672156

−

1.47182i

−2.14549

0.629973i

−0.404726

0.467079i

−2.72235

−

1.74955i

2.36931

2.73433i

0.175911

1.22349i

−2.14549

−

0.629973i

1.68251

−

1.08128i

−0.745170

5.18277i

487.2

0.256741

0.562183i

2.14549

−

0.629973i

1.05959

−

1.22283i

1.03985

0.668269i

0.904995

1.04442i

−0.460540

−

3.20313i

2.14549

0.629973i

1.68251

−

1.08128i

−0.108719

0.756156i

501.1

−0.592999

−

0.174120i

1.46431

−

1.68991i

−1.36118

−

0.874775i

−0.175911

1.22349i

−1.16258

0.747147i

1.34431

2.94363i

1.46431

1.68991i

−0.284630

−

1.97964i

0.317349

−

0.694897i

501.2

1.55249

0.455853i

−1.46431

1.68991i

0.519923

0.334134i

0.460540

−

3.20313i

−3.04368

1.95606i

−0.513481

−

1.12437i

−1.46431

−

1.68991i

−0.284630

−

1.97964i

2.17514

−

4.76289i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.c	even	11	9	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	529.2.c.o		20
23.b	odd	2	1		529.2.c.n		20
23.c	even	11	1		23.2.a.a	✓	2
23.c	even	11	9	inner	529.2.c.o		20
23.d	odd	22	1		529.2.a.a		2
23.d	odd	22	9		529.2.c.n		20
69.g	even	22	1		4761.2.a.w		2
69.h	odd	22	1		207.2.a.d		2
92.g	odd	22	1		368.2.a.h		2
92.h	even	22	1		8464.2.a.bb		2
115.j	even	22	1		575.2.a.f		2
115.k	odd	44	2		575.2.b.d		4
161.l	odd	22	1		1127.2.a.c		2
184.k	odd	22	1		1472.2.a.s		2
184.p	even	22	1		1472.2.a.t		2
253.k	odd	22	1		2783.2.a.c		2
276.o	even	22	1		3312.2.a.ba		2
299.p	even	22	1		3887.2.a.i		2
345.p	odd	22	1		5175.2.a.be		2
391.n	even	22	1		6647.2.a.b		2
437.o	odd	22	1		8303.2.a.e		2
460.n	odd	22	1		9200.2.a.bt		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
23.2.a.a	✓	2	23.c	even	11	1
207.2.a.d		2	69.h	odd	22	1
368.2.a.h		2	92.g	odd	22	1
529.2.a.a		2	23.d	odd	22	1
529.2.c.n		20	23.b	odd	2	1
529.2.c.n		20	23.d	odd	22	9
529.2.c.o		20	1.a	even	1	1	trivial
529.2.c.o		20	23.c	even	11	9	inner
575.2.a.f		2	115.j	even	22	1
575.2.b.d		4	115.k	odd	44	2
1127.2.a.c		2	161.l	odd	22	1
1472.2.a.s		2	184.k	odd	22	1
1472.2.a.t		2	184.p	even	22	1
2783.2.a.c		2	253.k	odd	22	1
3312.2.a.ba		2	276.o	even	22	1
3887.2.a.i		2	299.p	even	22	1
4761.2.a.w		2	69.g	even	22	1
5175.2.a.be		2	345.p	odd	22	1
6647.2.a.b		2	391.n	even	22	1
8303.2.a.e		2	437.o	odd	22	1
8464.2.a.bb		2	92.h	even	22	1
9200.2.a.bt		2	460.n	odd	22	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(529, [\chi])$:

$ T_{2}^{20} - T_{2}^{19} + 2 T_{2}^{18} - 3 T_{2}^{17} + 5 T_{2}^{16} - 8 T_{2}^{15} + 13 T_{2}^{14} + \cdots + 1 $

$ T_{5}^{20} - 2 T_{5}^{19} + 8 T_{5}^{18} - 24 T_{5}^{17} + 80 T_{5}^{16} - 256 T_{5}^{15} + \cdots + 1048576 $

$ T_{7}^{20} + 2 T_{7}^{19} + 8 T_{7}^{18} + 24 T_{7}^{17} + 80 T_{7}^{16} + 256 T_{7}^{15} + \cdots + 1048576 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} - T^{19} + \cdots + 1 $ T^20 - T^19 + 2T^18 - 3T^17 + 5T^16 - 8T^15 + 13T^14 - 21T^13 + 34T^12 - 55T^11 + 89T^10 + 55T^9 + 34T^8 + 21T^7 + 13T^6 + 8T^5 + 5T^4 + 3T^3 + 2*T^2 + T + 1
$3$	$ T^{20} + 5 T^{18} + \cdots + 9765625 $ T^20 + 5T^18 + 25T^16 + 125T^14 + 625T^12 + 3125T^10 + 15625T^8 + 78125T^6 + 390625T^4 + 1953125*T^2 + 9765625
$5$	$ T^{20} - 2 T^{19} + \cdots + 1048576 $ T^20 - 2T^19 + 8T^18 - 24T^17 + 80T^16 - 256T^15 + 832T^14 - 2688T^13 + 8704T^12 - 28160T^11 + 91136T^10 + 112640T^9 + 139264T^8 + 172032T^7 + 212992T^6 + 262144T^5 + 327680T^4 + 393216T^3 + 524288T^2 + 524288*T + 1048576
$7$	$ T^{20} + 2 T^{19} + \cdots + 1048576 $ T^20 + 2T^19 + 8T^18 + 24T^17 + 80T^16 + 256T^15 + 832T^14 + 2688T^13 + 8704T^12 + 28160T^11 + 91136T^10 - 112640T^9 + 139264T^8 - 172032T^7 + 212992T^6 - 262144T^5 + 327680T^4 - 393216T^3 + 524288T^2 - 524288*T + 1048576
$11$	$ T^{20} - 6 T^{19} + \cdots + 1048576 $ T^20 - 6T^19 + 32T^18 - 168T^17 + 880T^16 - 4608T^15 + 24128T^14 - 126336T^13 + 661504T^12 - 3463680T^11 + 18136064T^10 - 13854720T^9 + 10584064T^8 - 8085504T^7 + 6176768T^6 - 4718592T^5 + 3604480T^4 - 2752512T^3 + 2097152T^2 - 1572864*T + 1048576
$13$	$ (T^{10} + 3 T^{9} + \cdots + 59049)^{2} $ (T^10 + 3T^9 + 9T^8 + 27T^7 + 81T^6 + 243T^5 + 729T^4 + 2187T^3 + 6561T^2 + 19683*T + 59049)^2
$17$	$ T^{20} + 6 T^{19} + \cdots + 1048576 $ T^20 + 6T^19 + 32T^18 + 168T^17 + 880T^16 + 4608T^15 + 24128T^14 + 126336T^13 + 661504T^12 + 3463680T^11 + 18136064T^10 + 13854720T^9 + 10584064T^8 + 8085504T^7 + 6176768T^6 + 4718592T^5 + 3604480T^4 + 2752512T^3 + 2097152T^2 + 1572864*T + 1048576
$19$	$ (T^{10} - 2 T^{9} + \cdots + 1024)^{2} $ (T^10 - 2T^9 + 4T^8 - 8T^7 + 16T^6 - 32T^5 + 64T^4 - 128T^3 + 256T^2 - 512*T + 1024)^2
$23$	$ T^{20} $ T^20
$29$	$ (T^{10} - 3 T^{9} + \cdots + 59049)^{2} $ (T^10 - 3T^9 + 9T^8 - 27T^7 + 81T^6 - 243T^5 + 729T^4 - 2187T^3 + 6561T^2 - 19683*T + 59049)^2
$31$	$ T^{20} + \cdots + 34\!\cdots\!25 $ T^20 + 45T^18 + 2025T^16 + 91125T^14 + 4100625T^12 + 184528125T^10 + 8303765625T^8 + 373669453125T^6 + 16815125390625T^4 + 756680642578125*T^2 + 34050628916015625
$37$	$ T^{20} + 2 T^{19} + \cdots + 1048576 $ T^20 + 2T^19 + 8T^18 + 24T^17 + 80T^16 + 256T^15 + 832T^14 + 2688T^13 + 8704T^12 + 28160T^11 + 91136T^10 - 112640T^9 + 139264T^8 - 172032T^7 + 212992T^6 - 262144T^5 + 327680T^4 - 393216T^3 + 524288T^2 - 524288*T + 1048576
$41$	$ T^{20} + \cdots + 6131066257801 $ T^20 + 2T^19 + 23T^18 + 84T^17 + 605T^16 + 2806T^15 + 17107T^14 + 87528T^13 + 500089T^12 + 2663210T^11 + 14828111T^10 - 50600990T^9 + 180532129T^8 - 600354552T^7 + 2229401347T^6 - 6947933794T^5 + 28462758005T^4 - 75085226076T^3 + 390621949943T^2 - 645375395558*T + 6131066257801
$43$	$ T^{20} $ T^20
$47$	$ (T^{2} - 5)^{10} $ (T^2 - 5)^10
$53$	$ T^{20} - 8 T^{19} + \cdots + 1048576 $ T^20 - 8T^19 + 68T^18 - 576T^17 + 4880T^16 - 41344T^15 + 350272T^14 - 2967552T^13 + 25141504T^12 - 213002240T^11 + 1804583936T^10 + 852008960T^9 + 402264064T^8 + 189923328T^7 + 89669632T^6 + 42336256T^5 + 19988480T^4 + 9437184T^3 + 4456448T^2 + 2097152*T + 1048576
$59$	$ T^{20} + \cdots + 1099511627776 $ T^20 + 4T^19 + 32T^18 + 192T^17 + 1280T^16 + 8192T^15 + 53248T^14 + 344064T^13 + 2228224T^12 + 14417920T^11 + 93323264T^10 - 230686720T^9 + 570425344T^8 - 1409286144T^7 + 3489660928T^6 - 8589934592T^5 + 21474836480T^4 - 51539607552T^3 + 137438953472T^2 - 274877906944*T + 1099511627776
$61$	$ T^{20} + \cdots + 64\!\cdots\!76 $ T^20 + 4T^19 + 92T^18 + 672T^17 + 9680T^16 + 89792T^15 + 1094848T^14 + 11203584T^13 + 128022784T^12 + 1363563520T^11 + 15183985664T^10 - 103630827520T^9 + 739459600384T^8 - 4918104489984T^7 + 36526511669248T^6 - 227669894561792T^5 + 1865335308615680T^4 - 9841570752233472T^3 + 102399200445857792T^2 - 338362575386312704*T + 6428888932339941376
$67$	$ T^{20} + \cdots + 10240000000000 $ T^20 - 10T^19 + 80T^18 - 600T^17 + 4400T^16 - 32000T^15 + 232000T^14 - 1680000T^13 + 12160000T^12 - 88000000T^11 + 636800000T^10 - 1760000000T^9 + 4864000000T^8 - 13440000000T^7 + 37120000000T^6 - 102400000000T^5 + 281600000000T^4 - 768000000000T^3 + 2048000000000T^2 - 5120000000000*T + 10240000000000
$71$	$ T^{20} + \cdots + 59\!\cdots\!25 $ T^20 + 20T^19 + 305T^18 + 4200T^17 + 55025T^16 + 701500T^15 + 8802625T^14 + 109410000T^13 + 1351950625T^12 + 16645062500T^11 + 204465940625T^10 + 1581280937500T^9 + 12201354390625T^8 + 93805398750000T^7 + 716979307890625T^6 + 5428073276562500T^5 + 40448431281640625T^4 + 293301664359375000T^3 + 2023432315431640625T^2 + 12604988194492187500*T + 59873693923837890625
$73$	$ T^{20} + \cdots + 11\!\cdots\!01 $ T^20 + 22T^19 + 383T^18 + 6204T^17 + 97805T^16 + 1525106T^15 + 23674027T^14 + 366792888T^13 + 5678366809T^12 + 87877988110T^11 + 1359800690711T^10 + 8875676799110T^9 + 57925019818609T^8 + 377907079299288T^7 + 2463528742904827T^6 + 16029017334678106T^5 + 103821978329530805T^4 + 665152772447189004T^3 + 4147341182555546783T^2 + 24061075999055939822*T + 110462212541120451001
$79$	$ T^{20} + \cdots + 64\!\cdots\!76 $ T^20 - 4T^19 + 92T^18 - 672T^17 + 9680T^16 - 89792T^15 + 1094848T^14 - 11203584T^13 + 128022784T^12 - 1363563520T^11 + 15183985664T^10 + 103630827520T^9 + 739459600384T^8 + 4918104489984T^7 + 36526511669248T^6 + 227669894561792T^5 + 1865335308615680T^4 + 9841570752233472T^3 + 102399200445857792T^2 + 338362575386312704*T + 6428888932339941376
$83$	$ T^{20} + \cdots + 44\!\cdots\!76 $ T^20 - 22T^19 + 368T^18 - 5544T^17 + 79280T^16 - 1101056T^15 + 15026752T^14 - 202866048T^13 + 2719949824T^12 - 36306434560T^11 + 483227380736T^10 - 4211546408960T^9 + 36599644831744T^8 - 316652802859008T^7 + 2720802862415872T^6 - 23125937841504256T^5 + 193157500472852480T^4 - 1566856220788260864T^3 + 12064566802490851328T^2 - 83665148043360468992*T + 441143507864991563776
$89$	$ T^{20} + \cdots + 1099511627776 $ T^20 - 12T^19 + 128T^18 - 1344T^17 + 14080T^16 - 147456T^15 + 1544192T^14 - 16171008T^13 + 169345024T^12 - 1773404160T^11 + 18571329536T^10 - 28374466560T^9 + 43352326144T^8 - 66236448768T^7 + 101200166912T^6 - 154618822656T^5 + 236223201280T^4 - 360777252864T^3 + 549755813888T^2 - 824633720832*T + 1099511627776
$97$	$ T^{20} + \cdots + 64\!\cdots\!76 $ T^20 + 22T^19 + 408T^18 + 7304T^17 + 129680T^16 + 2297856T^15 + 40697152T^14 + 720700288T^13 + 12762422784T^12 + 226000079360T^11 + 4002057614336T^10 + 17176006031360T^9 + 73715754000384T^8 + 316370129625088T^7 + 1357745547722752T^6 + 5826272198393856T^5 + 24989326737735680T^4 + 106968501152251904T^3 + 454118193281630208T^2 + 1860994164624719872*T + 6428888932339941376
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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:	\( N \)	\(=\)	\( 529 = 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	529.c (of order \(11\), degree \(10\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{16} q^{2} + (\beta_{19} - 2 \beta_{18} + \beta_{17} + \cdots + 1) q^{3}+ \cdots + 2 \beta_{15} q^{9}+O(q^{10}) \) q + b16 * q^2 + (b19 - 2b18 + b17 + 2b16 + b15 - 2b14 - b13 + 2b12 - b11 - 2b10 - b9 - 2b8 - b7 + 2b6 + b5 - 2b4 + b3 + 2b2 + 2b1 + 1) * q^3 + (b11 + b10) * q^4 - 2b8 q^5 + (2b13 + b12) q^6 + (2b19 - 2b18) * q^7 + (-b5 + 2b4) q^8 + 2b15 q^9	\( q + \beta_{16} q^{2} + (\beta_{19} - 2 \beta_{18} + \beta_{17} + \cdots + 1) q^{3}+ \cdots + ( - 8 \beta_{19} + 4 \beta_{18}) q^{99}+O(q^{100}) \) q + b16 * q^2 + (b19 - 2b18 + b17 + 2b16 + b15 - 2b14 - b13 + 2b12 - b11 - 2b10 - b9 - 2b8 - b7 + 2b6 + b5 - 2b4 + b3 + 2b2 + 2b1 + 1) * q^3 + (b11 + b10) * q^4 - 2b8 q^5 + (2b13 + b12) q^6 + (2b19 - 2b18) * q^7 + (-b5 + 2b4) q^8 + 2b15 q^9 + (2b3 - 2b1) * q^10 + (4b7 - 2b6) * q^11 + (-3b7 + b6) q^12 + 3b3 q^13 + 2b15 q^14 + (-4b5 - 2b4) * q^15 - 3b18 q^16 + (-2b13 - 2b12) * q^17 - 2b8 q^18 + 2b11 q^19 + (2b19 + 2b17 + 2b15 - 2b13 - 2b11 - 2b9 - 2b7 + 2b5 + 2b3 + 2) q^20 + (-6b17 - 2b16) * q^21 + (-2b2 - 2) q^22 + 5 * q^24 + (-b17 - 4b16) q^25 + (3b18 - 3b16 + 3b14 - 3b12 + 3b10 + 3b8 - 3b6 + 3b4 - 3b2 - 3b1) * q^26 + (-b11 - 2b10) q^27 + (4b9 + 2b8) * q^28 + 3b13 q^29 + (2b19 + 6b18) * q^30 + (3b5 - 6b4) * q^31 + (5b15 - b14) q^32 + (8b3 + 6b1) * q^33 + (2b7 + 4b6) * q^34 - 4b7 q^35 + (-2b3 - 2b1) * q^36 + 2b14 q^37 + 2b4 q^38 + (-3b19 + 6b18) * q^39 + (4b13 + 2b12) * q^40 + (b9 + 4b8) q^41 + (2b11 + 4b10) * q^42 + (6b17 + 4b16) * q^44 + 4b2 q^45 + (-2b2 - 1) q^47 + (-6b17 + 3b16) * q^48 + (-b19 + 4b18 - b17 - 4b16 - b15 + 4b14 + b13 - 4b12 + b11 + 4b10 + b9 + 4b8 + b7 - 4b6 - b5 + 4b4 - b3 - 4b2 - 4b1 - 1) * q^49 + (4b11 - 3b10) * q^50 + (-2b9 + 6b8) * q^51 + (3b13 - 3b12) * q^52 + (-2b19 - 4b18) * q^53 + (2b5 + b4) q^54 + (-4b15 + 4b14) * q^55 + (-6b3 - 2b1) * q^56 + (-2b7 + 4b6) * q^57 - 3b6 q^58 + (4b3 + 4b1) * q^59 + (2b15 - 4b14) * q^60 + (-2b5 + 8b4) * q^61 + (6b19 + 3b18) * q^62 + (-4b13 + 4b12) * q^63 + (-b9 + 2b8) q^64 - 6b10 q^65 + (-6b19 + 2b18 - 6b17 - 2b16 - 6b15 + 2b14 + 6b13 - 2b12 + 6b11 + 2b10 + 6b9 + 2b8 + 6b7 - 2b6 - 6b5 + 2b4 - 6b3 - 2b2 - 2b1 - 6) q^66 + (-4b17 + 2b16) * q^67 - 2b2 q^68 + 4b2 q^70 + (11b17 + 2b16) * q^71 + (2b19 - 4b18 + 2b17 + 4b16 + 2b15 - 4b14 - 2b13 + 4b12 - 2b11 - 4b10 - 2b9 - 4b8 - 2b7 + 4b6 + 2b5 - 4b4 + 2b3 + 4b2 + 4b1 + 2) q^72 + (-9b11 + 4b10) * q^73 + (2b9 - 2b8) * q^74 + (-9b13 - 2b12) * q^75 + (2b19 - 2b18) * q^76 + (-12b5 + 8b4) * q^77 + (-6b15 - 3b14) * q^78 + (-6b3 - 8b1) * q^79 + (-6b7 - 6b6) * q^80 + 11b7 q^81 + (-4b3 + 3b1) * q^82 + (-10b15 - 2b14) * q^83 + (8b5 - 6b4) * q^84 + (-4b19 - 8b18) * q^85 + (-3b9 - 6b8) * q^87 + (-8b11 - 6b10) * q^88 + (8b19 - 4b18 + 8b17 + 4b16 + 8b15 - 4b14 - 8b13 + 4b12 - 8b11 - 4b10 - 8b9 - 4b8 - 8b7 + 4b6 + 8b5 - 4b4 + 8b3 + 4b2 + 4b1 + 8) q^89 + (4b17 - 4b16) * q^90 + (6b2 + 6) q^91 - 15 * q^93 + (-2b17 + b16) q^94 + (-4b18 + 4b16 - 4b14 + 4b12 - 4b10 - 4b8 + 4b6 - 4b4 + 4b2 + 4b1) * q^95 + (7b11 + 9b10) * q^96 + (-14b9 - 6b8) * q^97 + (-4b13 - 3b12) * q^98 + (-8b19 + 4b18) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + q^{2} + q^{4} + 2 q^{5} + 5 q^{6} - 2 q^{7} - 4 q^{9}+O(q^{10}) \) 20 * q + q^2 + q^4 + 2 * q^5 + 5 * q^6 - 2 * q^7 - 4 * q^9	\( 20 q + q^{2} + q^{4} + 2 q^{5} + 5 q^{6} - 2 q^{7} - 4 q^{9} - 6 q^{10} + 6 q^{11} - 5 q^{12} - 6 q^{13} - 4 q^{14} + 10 q^{15} + 3 q^{16} - 6 q^{17} + 2 q^{18} + 4 q^{19} + 4 q^{20} + 10 q^{21} - 20 q^{22} + 100 q^{24} - 2 q^{25} + 3 q^{26} + 6 q^{28} + 6 q^{29} - 10 q^{30} - 9 q^{32} - 10 q^{33} + 8 q^{34} - 8 q^{35} + 2 q^{36} - 2 q^{37} - 2 q^{38} + 10 q^{40} - 2 q^{41} - 8 q^{44} - 40 q^{45} + 15 q^{48} + 2 q^{49} + 11 q^{50} - 10 q^{51} + 3 q^{52} + 8 q^{53} - 5 q^{54} + 4 q^{55} + 10 q^{56} - 3 q^{58} - 4 q^{59} - 4 q^{61} - 15 q^{62} - 4 q^{63} - 4 q^{64} + 6 q^{65} - 10 q^{66} + 10 q^{67} + 20 q^{68} - 40 q^{70} - 20 q^{71} - 22 q^{73} + 6 q^{74} - 20 q^{75} - 2 q^{76} + 16 q^{77} + 15 q^{78} + 4 q^{79} - 18 q^{80} + 22 q^{81} + 11 q^{82} + 22 q^{83} - 10 q^{84} + 16 q^{85} - 10 q^{88} + 12 q^{89} - 12 q^{90} + 60 q^{91} - 300 q^{93} + 5 q^{94} - 4 q^{95} + 5 q^{96} - 22 q^{97} - 11 q^{98} + 12 q^{99}+O(q^{100}) \) 20 * q + q^2 + q^4 + 2 * q^5 + 5 * q^6 - 2 * q^7 - 4 * q^9 - 6 * q^10 + 6 * q^11 - 5 * q^12 - 6 * q^13 - 4 * q^14 + 10 * q^15 + 3 * q^16 - 6 * q^17 + 2 * q^18 + 4 * q^19 + 4 * q^20 + 10 * q^21 - 20 * q^22 + 100 * q^24 - 2 * q^25 + 3 * q^26 + 6 * q^28 + 6 * q^29 - 10 * q^30 - 9 * q^32 - 10 * q^33 + 8 * q^34 - 8 * q^35 + 2 * q^36 - 2 * q^37 - 2 * q^38 + 10 * q^40 - 2 * q^41 - 8 * q^44 - 40 * q^45 + 15 * q^48 + 2 * q^49 + 11 * q^50 - 10 * q^51 + 3 * q^52 + 8 * q^53 - 5 * q^54 + 4 * q^55 + 10 * q^56 - 3 * q^58 - 4 * q^59 - 4 * q^61 - 15 * q^62 - 4 * q^63 - 4 * q^64 + 6 * q^65 - 10 * q^66 + 10 * q^67 + 20 * q^68 - 40 * q^70 - 20 * q^71 - 22 * q^73 + 6 * q^74 - 20 * q^75 - 2 * q^76 + 16 * q^77 + 15 * q^78 + 4 * q^79 - 18 * q^80 + 22 * q^81 + 11 * q^82 + 22 * q^83 - 10 * q^84 + 16 * q^85 - 10 * q^88 + 12 * q^89 - 12 * q^90 + 60 * q^91 - 300 * q^93 + 5 * q^94 - 4 * q^95 + 5 * q^96 - 22 * q^97 - 11 * q^98 + 12 * q^99

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

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Hecke characteristic polynomials

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

Label	529.2.c.o

Level	$529$
Weight	$2$
Character orbit	529.c
Analytic conductor	$4.224$
Analytic rank	$0$
Dimension	$20$
CM	no
Inner twists	$10$

Self dual:	no
Analytic conductor:	\(4.22408626693\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(2\) over \(\Q(\zeta_{11})\)
Coefficient field:	20.0.54296067514572573056640625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - x^{19} + 2 x^{18} - 3 x^{17} + 5 x^{16} - 8 x^{15} + 13 x^{14} - 21 x^{13} + 34 x^{12} + \cdots + 1 \) x^20 - x^19 + 2x^18 - 3x^17 + 5x^16 - 8x^15 + 13x^14 - 21x^13 + 34x^12 - 55x^11 + 89x^10 + 55x^9 + 34x^8 + 21x^7 + 13x^6 + 8x^5 + 5x^4 + 3x^3 + 2*x^2 + x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 23)
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( \nu^{11} + 55 ) / 89 \) (v^11 + 55) / 89
\(\beta_{3}\)	\(=\)	\( ( \nu^{12} + 144\nu ) / 89 \) (v^12 + 144*v) / 89
\(\beta_{4}\)	\(=\)	\( ( -\nu^{13} - 144\nu^{2} ) / 89 \) (-v^13 - 144*v^2) / 89
\(\beta_{5}\)	\(=\)	\( ( \nu^{13} + 233\nu^{2} ) / 89 \) (v^13 + 233*v^2) / 89
\(\beta_{6}\)	\(=\)	\( ( \nu^{14} + 233\nu^{3} ) / 89 \) (v^14 + 233*v^3) / 89
\(\beta_{7}\)	\(=\)	\( ( -2\nu^{14} - 377\nu^{3} ) / 89 \) (-2v^14 - 377v^3) / 89
\(\beta_{8}\)	\(=\)	\( ( -2\nu^{15} - 377\nu^{4} ) / 89 \) (-2v^15 - 377v^4) / 89
\(\beta_{9}\)	\(=\)	\( ( -3\nu^{15} - 610\nu^{4} ) / 89 \) (-3v^15 - 610v^4) / 89
\(\beta_{10}\)	\(=\)	\( ( -3\nu^{16} - 610\nu^{5} ) / 89 \) (-3v^16 - 610v^5) / 89
\(\beta_{11}\)	\(=\)	\( ( -5\nu^{16} - 987\nu^{5} ) / 89 \) (-5v^16 - 987v^5) / 89
\(\beta_{12}\)	\(=\)	\( ( 5\nu^{17} + 987\nu^{6} ) / 89 \) (5v^17 + 987v^6) / 89
\(\beta_{13}\)	\(=\)	\( ( -8\nu^{17} - 1597\nu^{6} ) / 89 \) (-8v^17 - 1597v^6) / 89
\(\beta_{14}\)	\(=\)	\( ( -8\nu^{18} - 1597\nu^{7} ) / 89 \) (-8v^18 - 1597v^7) / 89
\(\beta_{15}\)	\(=\)	\( ( 13\nu^{18} + 2584\nu^{7} ) / 89 \) (13v^18 + 2584v^7) / 89
\(\beta_{16}\)	\(=\)	\( ( 13\nu^{19} + 2584\nu^{8} ) / 89 \) (13v^19 + 2584v^8) / 89
\(\beta_{17}\)	\(=\)	\( ( 21\nu^{19} + 4181\nu^{8} ) / 89 \) (21v^19 + 4181v^8) / 89
\(\beta_{18}\)	\(=\)	\( ( 34 \nu^{19} - 34 \nu^{18} + 68 \nu^{17} - 102 \nu^{16} + 170 \nu^{15} - 272 \nu^{14} + 442 \nu^{13} + \cdots + 34 ) / 89 \) (34v^19 - 34v^18 + 68v^17 - 102v^16 + 170v^15 - 272v^14 + 442v^13 - 714v^12 + 1156v^11 - 1869v^10 + 3026v^9 + 1870v^8 + 1156v^7 + 714v^6 + 442v^5 + 272v^4 + 170v^3 + 102v^2 + 68*v + 34) / 89
\(\beta_{19}\)	\(=\)	\( ( - 55 \nu^{19} + 55 \nu^{18} - 110 \nu^{17} + 165 \nu^{16} - 275 \nu^{15} + 440 \nu^{14} - 715 \nu^{13} + \cdots - 55 ) / 89 \) (-55v^19 + 55v^18 - 110v^17 + 165v^16 - 275v^15 + 440v^14 - 715v^13 + 1155v^12 - 1870v^11 + 3026v^10 - 4895v^9 - 3025v^8 - 1870v^7 - 1155v^6 - 715v^5 - 440v^4 - 275v^3 - 165v^2 - 110*v - 55) / 89

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{5} + \beta_{4} \) b5 + b4
\(\nu^{3}\)	\(=\)	\( \beta_{7} + 2\beta_{6} \) b7 + 2*b6
\(\nu^{4}\)	\(=\)	\( -2\beta_{9} + 3\beta_{8} \) -2b9 + 3b8
\(\nu^{5}\)	\(=\)	\( 3\beta_{11} - 5\beta_{10} \) 3b11 - 5b10
\(\nu^{6}\)	\(=\)	\( -5\beta_{13} - 8\beta_{12} \) -5b13 - 8b12
\(\nu^{7}\)	\(=\)	\( -8\beta_{15} - 13\beta_{14} \) -8b15 - 13b14
\(\nu^{8}\)	\(=\)	\( 13\beta_{17} - 21\beta_{16} \) 13b17 - 21b16
\(\nu^{9}\)	\(=\)	\( 21 \beta_{19} + 34 \beta_{18} + 21 \beta_{17} - 34 \beta_{16} + 21 \beta_{15} + 34 \beta_{14} + \cdots + 21 \) 21b19 + 34b18 + 21b17 - 34b16 + 21b15 + 34b14 - 21b13 - 34b12 - 21b11 + 34b10 - 21b9 + 34b8 - 21b7 - 34b6 + 21b5 + 34b4 + 21b3 - 34b2 - 34*b1 + 21
\(\nu^{10}\)	\(=\)	\( 34\beta_{19} + 55\beta_{18} \) 34b19 + 55b18
\(\nu^{11}\)	\(=\)	\( 89\beta_{2} - 55 \) 89*b2 - 55
\(\nu^{12}\)	\(=\)	\( 89\beta_{3} - 144\beta_1 \) 89b3 - 144b1
\(\nu^{13}\)	\(=\)	\( -144\beta_{5} - 233\beta_{4} \) -144b5 - 233b4
\(\nu^{14}\)	\(=\)	\( -233\beta_{7} - 377\beta_{6} \) -233b7 - 377b6
\(\nu^{15}\)	\(=\)	\( 377\beta_{9} - 610\beta_{8} \) 377b9 - 610b8
\(\nu^{16}\)	\(=\)	\( -610\beta_{11} + 987\beta_{10} \) -610b11 + 987b10
\(\nu^{17}\)	\(=\)	\( 987\beta_{13} + 1597\beta_{12} \) 987b13 + 1597b12
\(\nu^{18}\)	\(=\)	\( 1597\beta_{15} + 2584\beta_{14} \) 1597b15 + 2584b14
\(\nu^{19}\)	\(=\)	\( -2584\beta_{17} + 4181\beta_{16} \) -2584b17 + 4181b16

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 118.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{20} - T^{19} + \cdots + 1 \) T^20 - T^19 + 2T^18 - 3T^17 + 5T^16 - 8T^15 + 13T^14 - 21T^13 + 34T^12 - 55T^11 + 89T^10 + 55T^9 + 34T^8 + 21T^7 + 13T^6 + 8T^5 + 5T^4 + 3T^3 + 2*T^2 + T + 1
$3$	\( T^{20} + 5 T^{18} + \cdots + 9765625 \) T^20 + 5T^18 + 25T^16 + 125T^14 + 625T^12 + 3125T^10 + 15625T^8 + 78125T^6 + 390625T^4 + 1953125*T^2 + 9765625
$5$	\( T^{20} - 2 T^{19} + \cdots + 1048576 \) T^20 - 2T^19 + 8T^18 - 24T^17 + 80T^16 - 256T^15 + 832T^14 - 2688T^13 + 8704T^12 - 28160T^11 + 91136T^10 + 112640T^9 + 139264T^8 + 172032T^7 + 212992T^6 + 262144T^5 + 327680T^4 + 393216T^3 + 524288T^2 + 524288*T + 1048576
$7$	\( T^{20} + 2 T^{19} + \cdots + 1048576 \) T^20 + 2T^19 + 8T^18 + 24T^17 + 80T^16 + 256T^15 + 832T^14 + 2688T^13 + 8704T^12 + 28160T^11 + 91136T^10 - 112640T^9 + 139264T^8 - 172032T^7 + 212992T^6 - 262144T^5 + 327680T^4 - 393216T^3 + 524288T^2 - 524288*T + 1048576
$11$	\( T^{20} - 6 T^{19} + \cdots + 1048576 \) T^20 - 6T^19 + 32T^18 - 168T^17 + 880T^16 - 4608T^15 + 24128T^14 - 126336T^13 + 661504T^12 - 3463680T^11 + 18136064T^10 - 13854720T^9 + 10584064T^8 - 8085504T^7 + 6176768T^6 - 4718592T^5 + 3604480T^4 - 2752512T^3 + 2097152T^2 - 1572864*T + 1048576
$13$	\( (T^{10} + 3 T^{9} + \cdots + 59049)^{2} \) (T^10 + 3T^9 + 9T^8 + 27T^7 + 81T^6 + 243T^5 + 729T^4 + 2187T^3 + 6561T^2 + 19683*T + 59049)^2
$17$	\( T^{20} + 6 T^{19} + \cdots + 1048576 \) T^20 + 6T^19 + 32T^18 + 168T^17 + 880T^16 + 4608T^15 + 24128T^14 + 126336T^13 + 661504T^12 + 3463680T^11 + 18136064T^10 + 13854720T^9 + 10584064T^8 + 8085504T^7 + 6176768T^6 + 4718592T^5 + 3604480T^4 + 2752512T^3 + 2097152T^2 + 1572864*T + 1048576
$19$	\( (T^{10} - 2 T^{9} + \cdots + 1024)^{2} \) (T^10 - 2T^9 + 4T^8 - 8T^7 + 16T^6 - 32T^5 + 64T^4 - 128T^3 + 256T^2 - 512*T + 1024)^2
$23$	\( T^{20} \) T^20
$29$	\( (T^{10} - 3 T^{9} + \cdots + 59049)^{2} \) (T^10 - 3T^9 + 9T^8 - 27T^7 + 81T^6 - 243T^5 + 729T^4 - 2187T^3 + 6561T^2 - 19683*T + 59049)^2
$31$	\( T^{20} + \cdots + 34\!\cdots\!25 \) T^20 + 45T^18 + 2025T^16 + 91125T^14 + 4100625T^12 + 184528125T^10 + 8303765625T^8 + 373669453125T^6 + 16815125390625T^4 + 756680642578125*T^2 + 34050628916015625
$37$	\( T^{20} + 2 T^{19} + \cdots + 1048576 \) T^20 + 2T^19 + 8T^18 + 24T^17 + 80T^16 + 256T^15 + 832T^14 + 2688T^13 + 8704T^12 + 28160T^11 + 91136T^10 - 112640T^9 + 139264T^8 - 172032T^7 + 212992T^6 - 262144T^5 + 327680T^4 - 393216T^3 + 524288T^2 - 524288*T + 1048576
$41$	\( T^{20} + \cdots + 6131066257801 \) T^20 + 2T^19 + 23T^18 + 84T^17 + 605T^16 + 2806T^15 + 17107T^14 + 87528T^13 + 500089T^12 + 2663210T^11 + 14828111T^10 - 50600990T^9 + 180532129T^8 - 600354552T^7 + 2229401347T^6 - 6947933794T^5 + 28462758005T^4 - 75085226076T^3 + 390621949943T^2 - 645375395558*T + 6131066257801
$43$	\( T^{20} \) T^20
$47$	\( (T^{2} - 5)^{10} \) (T^2 - 5)^10
$53$	\( T^{20} - 8 T^{19} + \cdots + 1048576 \) T^20 - 8T^19 + 68T^18 - 576T^17 + 4880T^16 - 41344T^15 + 350272T^14 - 2967552T^13 + 25141504T^12 - 213002240T^11 + 1804583936T^10 + 852008960T^9 + 402264064T^8 + 189923328T^7 + 89669632T^6 + 42336256T^5 + 19988480T^4 + 9437184T^3 + 4456448T^2 + 2097152*T + 1048576
$59$	\( T^{20} + \cdots + 1099511627776 \) T^20 + 4T^19 + 32T^18 + 192T^17 + 1280T^16 + 8192T^15 + 53248T^14 + 344064T^13 + 2228224T^12 + 14417920T^11 + 93323264T^10 - 230686720T^9 + 570425344T^8 - 1409286144T^7 + 3489660928T^6 - 8589934592T^5 + 21474836480T^4 - 51539607552T^3 + 137438953472T^2 - 274877906944*T + 1099511627776
$61$	\( T^{20} + \cdots + 64\!\cdots\!76 \) T^20 + 4T^19 + 92T^18 + 672T^17 + 9680T^16 + 89792T^15 + 1094848T^14 + 11203584T^13 + 128022784T^12 + 1363563520T^11 + 15183985664T^10 - 103630827520T^9 + 739459600384T^8 - 4918104489984T^7 + 36526511669248T^6 - 227669894561792T^5 + 1865335308615680T^4 - 9841570752233472T^3 + 102399200445857792T^2 - 338362575386312704*T + 6428888932339941376
$67$	\( T^{20} + \cdots + 10240000000000 \) T^20 - 10T^19 + 80T^18 - 600T^17 + 4400T^16 - 32000T^15 + 232000T^14 - 1680000T^13 + 12160000T^12 - 88000000T^11 + 636800000T^10 - 1760000000T^9 + 4864000000T^8 - 13440000000T^7 + 37120000000T^6 - 102400000000T^5 + 281600000000T^4 - 768000000000T^3 + 2048000000000T^2 - 5120000000000*T + 10240000000000
$71$	\( T^{20} + \cdots + 59\!\cdots\!25 \) T^20 + 20T^19 + 305T^18 + 4200T^17 + 55025T^16 + 701500T^15 + 8802625T^14 + 109410000T^13 + 1351950625T^12 + 16645062500T^11 + 204465940625T^10 + 1581280937500T^9 + 12201354390625T^8 + 93805398750000T^7 + 716979307890625T^6 + 5428073276562500T^5 + 40448431281640625T^4 + 293301664359375000T^3 + 2023432315431640625T^2 + 12604988194492187500*T + 59873693923837890625
$73$	\( T^{20} + \cdots + 11\!\cdots\!01 \) T^20 + 22T^19 + 383T^18 + 6204T^17 + 97805T^16 + 1525106T^15 + 23674027T^14 + 366792888T^13 + 5678366809T^12 + 87877988110T^11 + 1359800690711T^10 + 8875676799110T^9 + 57925019818609T^8 + 377907079299288T^7 + 2463528742904827T^6 + 16029017334678106T^5 + 103821978329530805T^4 + 665152772447189004T^3 + 4147341182555546783T^2 + 24061075999055939822*T + 110462212541120451001
$79$	\( T^{20} + \cdots + 64\!\cdots\!76 \) T^20 - 4T^19 + 92T^18 - 672T^17 + 9680T^16 - 89792T^15 + 1094848T^14 - 11203584T^13 + 128022784T^12 - 1363563520T^11 + 15183985664T^10 + 103630827520T^9 + 739459600384T^8 + 4918104489984T^7 + 36526511669248T^6 + 227669894561792T^5 + 1865335308615680T^4 + 9841570752233472T^3 + 102399200445857792T^2 + 338362575386312704*T + 6428888932339941376
$83$	\( T^{20} + \cdots + 44\!\cdots\!76 \) T^20 - 22T^19 + 368T^18 - 5544T^17 + 79280T^16 - 1101056T^15 + 15026752T^14 - 202866048T^13 + 2719949824T^12 - 36306434560T^11 + 483227380736T^10 - 4211546408960T^9 + 36599644831744T^8 - 316652802859008T^7 + 2720802862415872T^6 - 23125937841504256T^5 + 193157500472852480T^4 - 1566856220788260864T^3 + 12064566802490851328T^2 - 83665148043360468992*T + 441143507864991563776
$89$	\( T^{20} + \cdots + 1099511627776 \) T^20 - 12T^19 + 128T^18 - 1344T^17 + 14080T^16 - 147456T^15 + 1544192T^14 - 16171008T^13 + 169345024T^12 - 1773404160T^11 + 18571329536T^10 - 28374466560T^9 + 43352326144T^8 - 66236448768T^7 + 101200166912T^6 - 154618822656T^5 + 236223201280T^4 - 360777252864T^3 + 549755813888T^2 - 824633720832*T + 1099511627776
$97$	\( T^{20} + \cdots + 64\!\cdots\!76 \) T^20 + 22T^19 + 408T^18 + 7304T^17 + 129680T^16 + 2297856T^15 + 40697152T^14 + 720700288T^13 + 12762422784T^12 + 226000079360T^11 + 4002057614336T^10 + 17176006031360T^9 + 73715754000384T^8 + 316370129625088T^7 + 1357745547722752T^6 + 5826272198393856T^5 + 24989326737735680T^4 + 106968501152251904T^3 + 454118193281630208T^2 + 1860994164624719872*T + 6428888932339941376
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\(n\)	\(5\)
\(\chi(n)\)	\(\beta_{17}\)