LMFDB - Newform orbit 927.2.c.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [927,2,Mod(926,927)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(927, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("927.926");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 927 = 3^{2} \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	927.c (of order $2$, degree $1$, 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:	$7.40213226737$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 40 x^{18} + 680 x^{16} + 6400 x^{14} + 36400 x^{12} + 128128 x^{10} + 274560 x^{8} + \cdots + 1521 $ x^20 + 40x^18 + 680x^16 + 6400x^14 + 36400x^12 + 128128x^10 + 274560x^8 + 337920x^6 + 211200x^4 + 51200*x^2 + 1521
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 3^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

$f(q)$	$=$	$ q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} - \beta_{6} q^{7} + (\beta_{3} - 2 \beta_1) q^{8}+O(q^{10}) $ q + b1 * q^2 + (b2 - 2) * q^4 - b6 * q^7 + (b3 - 2b1) q^8	$ q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} - \beta_{6} q^{7} + (\beta_{3} - 2 \beta_1) q^{8} + \beta_{9} q^{13} + ( - \beta_{7} - \beta_{5}) q^{14} + (\beta_{4} - 2 \beta_{2} + 4) q^{16} + \beta_{12} q^{17} + ( - \beta_{15} + \beta_{4}) q^{19} + (\beta_{10} - \beta_{5}) q^{23} - 5 q^{25} + (\beta_{14} - \beta_{12} + \beta_{7}) q^{26} + ( - \beta_{13} + \beta_{9} + \cdots + 2 \beta_{6}) q^{28}+ \cdots + (\beta_{17} - 3 \beta_{16} + \cdots + 7 \beta_1) q^{98}+O(q^{100}) $ q + b1 * q^2 + (b2 - 2) * q^4 - b6 * q^7 + (b3 - 2b1) q^8 + b9 * q^13 + (-b7 - b5) * q^14 + (b4 - 2b2 + 4) q^16 + b12 * q^17 + (-b15 + b4) * q^19 + (b10 - b5) * q^23 - 5 * q^25 + (b14 - b12 + b7) * q^26 + (-b13 + b9 - b8 + 2b6) q^28 + b17 * q^29 + (-b16 + b14 - 2b3 + 4b1) * q^32 + (b13 + b11 + b9) * q^34 + (-b17 + b16 + b14 - b3) * q^38 + (-b10 - b5) * q^41 + (-b11 - b8 + b6) * q^46 + (b15 + b4 + 7) * q^49 - 5b1 q^50 + (b15 - b11 - 2b9 - 2b6 - 2b4) q^52 + (-2b16 + b14 - b10 + 2b7 + 4b5) q^56 + (b18 - 2b15 + b4) q^58 + (-b17 + 2b3) q^59 + (-b18 - b2) * q^61 + (b13 - 2b9 - 2b4 + 4b2 - 8) q^64 + (2b16 + b14 - 2b12 + 2b10 - b5) q^68 + (-b18 + 2b15 - b13 - 2b9 - 2b4 + 2b2) * q^76 + (-2b13 + b9) q^79 + (b11 - b8 + 3b6) q^82 + (-b12 - 2b7) q^83 + (b18 + 2b13 + b9 + 3b2) * q^91 + (2b12 - b10 + b7 + 4b5) * q^92 + (b18 - 3b2) q^97 + (b17 - 3b16 + b14 - 3b3 + 7b1) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 40 q^{4}+O(q^{10}) $ 20 * q - 40 * q^4	$ 20 q - 40 q^{4} + 80 q^{16} - 100 q^{25} + 140 q^{49} - 160 q^{64}+O(q^{100}) $ 20 * q - 40 * q^4 + 80 * q^16 - 100 * q^25 + 140 * q^49 - 160 * q^64

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{20} + 40 x^{18} + 680 x^{16} + 6400 x^{14} + 36400 x^{12} + 128128 x^{10} + 274560 x^{8} + \cdots + 1521 $ x^20 + 40*x^18 + 680*x^16 + 6400*x^14 + 36400*x^12 + 128128*x^10 + 274560*x^8 + 337920*x^6 + 211200*x^4 + 51200*x^2 + 1521 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} + 4 $ v^2 + 4
$\beta_{3}$	$=$	$ \nu^{3} + 6\nu $ v^3 + 6*v
$\beta_{4}$	$=$	$ \nu^{4} + 8\nu^{2} + 8 $ v^4 + 8*v^2 + 8
$\beta_{5}$	$=$	$ ( -2\nu^{11} - 35\nu^{9} - 190\nu^{7} - 260\nu^{5} + 400\nu^{3} + 592\nu ) / 65 $ (-2v^11 - 35v^9 - 190v^7 - 260v^5 + 400v^3 + 592v) / 65
$\beta_{6}$	$=$	$ ( \nu^{12} + 24\nu^{10} + 225\nu^{8} + 1040\nu^{6} + 2400\nu^{4} + 2304\nu^{2} + 416 ) / 65 $ (v^12 + 24v^10 + 225v^8 + 1040v^6 + 2400v^4 + 2304*v^2 + 416) / 65
$\beta_{7}$	$=$	$ ( \nu^{13} + 26\nu^{11} + 260\nu^{9} + 1230\nu^{7} + 2660\nu^{5} + 1904\nu^{3} - 176\nu ) / 65 $ (v^13 + 26v^11 + 260v^9 + 1230v^7 + 2660v^5 + 1904v^3 - 176v) / 65
$\beta_{8}$	$=$	$ ( \nu^{10} + 20\nu^{8} + 140\nu^{6} + 400\nu^{4} + 400\nu^{2} + 64 ) / 5 $ (v^10 + 20v^8 + 140v^6 + 400v^4 + 400v^2 + 64) / 5
$\beta_{9}$	$=$	$ ( \nu^{14} + 28\nu^{12} + 308\nu^{10} + 1680\nu^{8} + 4675\nu^{6} + 5924\nu^{4} + 2092\nu^{2} - 208 ) / 65 $ (v^14 + 28v^12 + 308v^10 + 1680v^8 + 4675v^6 + 5924v^4 + 2092v^2 - 208) / 65
$\beta_{10}$	$=$	$ ( \nu^{11} + 24\nu^{9} + 212\nu^{7} + 832\nu^{5} + 1360\nu^{3} + 640\nu ) / 13 $ (v^11 + 24v^9 + 212v^7 + 832v^5 + 1360v^3 + 640*v) / 13
$\beta_{11}$	$=$	$ ( -6\nu^{12} - 144\nu^{10} - 1285\nu^{8} - 5200\nu^{6} - 9200\nu^{4} - 5504\nu^{2} - 416 ) / 65 $ (-6v^12 - 144v^10 - 1285v^8 - 5200v^6 - 9200v^4 - 5504v^2 - 416) / 65
$\beta_{12}$	$=$	$ ( 3\nu^{13} + 78\nu^{11} + 780\nu^{9} + 3755\nu^{7} + 8890\nu^{5} + 9352\nu^{3} + 3112\nu ) / 65 $ (3v^13 + 78v^11 + 780v^9 + 3755v^7 + 8890v^5 + 9352v^3 + 3112*v) / 65
$\beta_{13}$	$=$	$ ( 2\nu^{14} + 56\nu^{12} + 616\nu^{10} + 3360\nu^{8} + 9415\nu^{6} + 12628\nu^{4} + 6524\nu^{2} + 624 ) / 65 $ (2v^14 + 56v^12 + 616v^10 + 3360v^8 + 9415v^6 + 12628v^4 + 6524*v^2 + 624) / 65
$\beta_{14}$	$=$	$ ( \nu^{15} + 30\nu^{13} + 360\nu^{11} + 2200\nu^{9} + 7200\nu^{7} + 12154\nu^{5} + 9540\nu^{3} + 3080\nu ) / 65 $ (v^15 + 30v^13 + 360v^11 + 2200v^9 + 7200v^7 + 12154v^5 + 9540v^3 + 3080*v) / 65
$\beta_{15}$	$=$	$ ( \nu^{16} + 32 \nu^{14} + 416 \nu^{12} + 2816 \nu^{10} + 10560 \nu^{8} + 21504 \nu^{6} + 21518 \nu^{4} + \cdots + 624 ) / 65 $ (v^16 + 32v^14 + 416v^12 + 2816v^10 + 10560v^8 + 21504v^6 + 21518v^4 + 8304*v^2 + 624) / 65
$\beta_{16}$	$=$	$ ( \nu^{15} + 30\nu^{13} + 360\nu^{11} + 2200\nu^{9} + 7200\nu^{7} + 12089\nu^{5} + 8890\nu^{3} + 1780\nu ) / 65 $ (v^15 + 30v^13 + 360v^11 + 2200v^9 + 7200v^7 + 12089v^5 + 8890v^3 + 1780*v) / 65
$\beta_{17}$	$=$	$ ( \nu^{17} + 34 \nu^{15} + 476 \nu^{13} + 3536 \nu^{11} + 14960 \nu^{9} + 35904 \nu^{7} + 45696 \nu^{5} + \cdots + 4574 \nu ) / 65 $ (v^17 + 34v^15 + 476v^13 + 3536v^11 + 14960v^9 + 35904v^7 + 45696v^5 + 26149v^3 + 4574v) / 65
$\beta_{18}$	$=$	$ ( \nu^{18} + 36 \nu^{16} + 540 \nu^{14} + 4368 \nu^{12} + 20592 \nu^{10} + 57024 \nu^{8} + 88704 \nu^{6} + \cdots + 728 ) / 65 $ (v^18 + 36v^16 + 540v^14 + 4368v^12 + 20592v^10 + 57024v^8 + 88704v^6 + 69120v^4 + 20662v^2 + 728) / 65
$\beta_{19}$	$=$	$ ( \nu^{19} + 38 \nu^{17} + 608 \nu^{15} + 5320 \nu^{13} + 27664 \nu^{11} + 86944 \nu^{9} + \cdots + 9876 \nu ) / 65 $ (v^19 + 38v^17 + 608v^15 + 5320v^13 + 27664v^11 + 86944v^9 + 160512v^7 + 160512v^5 + 72960v^3 + 9876*v) / 65

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} - 4 $ b2 - 4
$\nu^{3}$	$=$	$ \beta_{3} - 6\beta_1 $ b3 - 6*b1
$\nu^{4}$	$=$	$ \beta_{4} - 8\beta_{2} + 24 $ b4 - 8*b2 + 24
$\nu^{5}$	$=$	$ -\beta_{16} + \beta_{14} - 10\beta_{3} + 40\beta_1 $ -b16 + b14 - 10b3 + 40b1
$\nu^{6}$	$=$	$ \beta_{13} - 2\beta_{9} - 12\beta_{4} + 60\beta_{2} - 160 $ b13 - 2b9 - 12b4 + 60*b2 - 160
$\nu^{7}$	$=$	$ 14\beta_{16} - 14\beta_{14} + \beta_{12} - 3\beta_{7} + 84\beta_{3} - 280\beta_1 $ 14b16 - 14b14 + b12 - 3b7 + 84b3 - 280*b1
$\nu^{8}$	$=$	$ -16\beta_{13} + \beta_{11} + 32\beta_{9} + 6\beta_{6} + 112\beta_{4} - 448\beta_{2} + 1120 $ -16b13 + b11 + 32b9 + 6b6 + 112b4 - 448*b2 + 1120
$\nu^{9}$	$=$	$ -144\beta_{16} + 144\beta_{14} - 18\beta_{12} + 2\beta_{10} + 54\beta_{7} + 5\beta_{5} - 672\beta_{3} + 2016\beta_1 $ -144b16 + 144b14 - 18b12 + 2b10 + 54b7 + 5b5 - 672b3 + 2016b1
$\nu^{10}$	$=$	$ 180\beta_{13} - 20\beta_{11} - 360\beta_{9} + 5\beta_{8} - 120\beta_{6} - 960\beta_{4} + 3360\beta_{2} - 8064 $ 180b13 - 20b11 - 360b9 + 5b8 - 120b6 - 960b4 + 3360*b2 - 8064
$\nu^{11}$	$=$	$ 1320 \beta_{16} - 1320 \beta_{14} + 220 \beta_{12} - 35 \beta_{10} - 660 \beta_{7} - 120 \beta_{5} + \cdots - 14784 \beta_1 $ 1320b16 - 1320b14 + 220b12 - 35b10 - 660b7 - 120b5 + 5280b3 - 14784b1
$\nu^{12}$	$=$	$ - 1760 \beta_{13} + 255 \beta_{11} + 3520 \beta_{9} - 120 \beta_{8} + 1595 \beta_{6} + 7920 \beta_{4} + \cdots + 59136 $ -1760b13 + 255b11 + 3520b9 - 120b8 + 1595b6 + 7920b4 - 25344*b2 + 59136
$\nu^{13}$	$=$	$ - 11440 \beta_{16} + 11440 \beta_{14} - 2270 \beta_{12} + 390 \beta_{10} + 6875 \beta_{7} + \cdots + 109824 \beta_1 $ -11440b16 + 11440b14 - 2270b12 + 390b10 + 6875b7 + 1820b5 - 41184b3 + 109824b1
$\nu^{14}$	$=$	$ 16045 \beta_{13} - 2660 \beta_{11} - 32025 \beta_{9} + 1820 \beta_{8} - 17780 \beta_{6} - 64064 \beta_{4} + \cdots - 439296 $ 16045b13 - 2660b11 - 32025b9 + 1820b8 - 17780b6 - 64064b4 + 192192*b2 - 439296
$\nu^{15}$	$=$	$ 96154 \beta_{16} - 96089 \beta_{14} + 21300 \beta_{12} - 3500 \beta_{10} - 65850 \beta_{7} + \cdots - 823680 \beta_1 $ 96154b16 - 96089b14 + 21300b12 - 3500b10 - 65850b7 - 22400b5 + 320320b3 - 823680b1
$\nu^{16}$	$=$	$ 65 \beta_{15} - 140704 \beta_{13} + 24800 \beta_{11} + 279328 \beta_{9} - 22400 \beta_{8} + \cdots + 3294720 $ 65b15 - 140704b13 + 24800b11 + 279328b9 - 22400b8 + 180000b6 + 512498b4 - 1464320b2 + 3294720
$\nu^{17}$	$=$	$ 65 \beta_{17} - 794036 \beta_{16} + 791826 \beta_{14} - 188224 \beta_{12} + 27200 \beta_{10} + \cdots + 6223360 \beta_1 $ 65b17 - 794036b16 + 791826b14 - 188224b12 + 27200b10 + 600032b7 + 244800b5 - 2489381b3 + 6223360*b1
$\nu^{18}$	$=$	$ 65 \beta_{18} - 2340 \beta_{15} + 1205844 \beta_{13} - 215424 \beta_{11} - 2371908 \beta_{9} + \cdots - 24893440 $ 65b18 - 2340b15 + 1205844b13 - 215424b11 - 2371908b9 + 244800b8 - 1716864b6 - 4072968b4 + 11202122*b2 - 24893440
$\nu^{19}$	$=$	$ 65 \beta_{19} - 2470 \beta_{17} + 6489336 \beta_{16} - 6444876 \beta_{14} + 1596912 \beta_{12} + \cdots - 47297684 \beta_1 $ 65b19 - 2470b17 + 6489336b16 - 6444876b14 + 1596912b12 - 186048b10 - 5294616b7 - 2480640b5 + 19350398b3 - 47297684b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/927\mathbb{Z}\right)^\times$.

$n$	$722$	$829$
$\chi(n)$	$-1$	$-1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

926.1

	−	2.82236i
	−	2.74145i
	−	2.62699i
	−	2.39220i
	−	2.17447i
	−	1.80877i
	−	1.50910i
	−	1.04830i
	−	0.696015i
	−	0.185206i
		0.185206i
		0.696015i
		1.04830i
		1.50910i
		1.80877i
		2.17447i
		2.39220i
		2.62699i
		2.74145i
		2.82236i

−

2.82236i

−5.96570

−5.22695

11.1926i

926.2

−

2.74145i

−5.51556

3.74435

9.63775i

926.3

−

2.62699i

−4.90107

4.71303

7.62108i

926.4

−

2.39220i

−3.72260

−0.831536

4.12080i

926.5

−

2.17447i

−2.72833

−2.39890

1.58374i

926.6

−

1.80877i

−1.27167

−2.39890

−

1.31739i

926.7

−

1.50910i

−0.277397

−0.831536

−

2.59959i

926.8

−

1.04830i

0.901071

4.71303

−

3.04119i

926.9

−

0.696015i

1.51556

3.74435

−

2.44688i

926.10

−

0.185206i

1.96570

−5.22695

−

0.734472i

926.11

0.185206i

1.96570

−5.22695

0.734472i

926.12

0.696015i

1.51556

3.74435

2.44688i

926.13

1.04830i

0.901071

4.71303

3.04119i

926.14

1.50910i

−0.277397

−0.831536

2.59959i

926.15

1.80877i

−1.27167

−2.39890

1.31739i

926.16

2.17447i

−2.72833

−2.39890

−

1.58374i

926.17

2.39220i

−3.72260

−0.831536

−

4.12080i

926.18

2.62699i

−4.90107

4.71303

−

7.62108i

926.19

2.74145i

−5.51556

3.74435

−

9.63775i

926.20

2.82236i

−5.96570

−5.22695

−

11.1926i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
103.b	odd	2	1	CM by $\Q(\sqrt{-103}) $
3.b	odd	2	1	inner
309.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	927.2.c.b	✓	20
3.b	odd	2	1	inner	927.2.c.b	✓	20
103.b	odd	2	1	CM	927.2.c.b	✓	20
309.c	even	2	1	inner	927.2.c.b	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
927.2.c.b	✓	20	1.a	even	1	1	trivial
927.2.c.b	✓	20	3.b	odd	2	1	inner
927.2.c.b	✓	20	103.b	odd	2	1	CM
927.2.c.b	✓	20	309.c	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{20} + 40 T_{2}^{18} + 680 T_{2}^{16} + 6400 T_{2}^{14} + 36400 T_{2}^{12} + 128128 T_{2}^{10} + \cdots + 1521 $ T2^20 + 40*T2^18 + 680*T2^16 + 6400*T2^14 + 36400*T2^12 + 128128*T2^10 + 274560*T2^8 + 337920*T2^6 + 211200*T2^4 + 51200*T2^2 + 1521 acting on $S_{2}^{\mathrm{new}}(927, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} + 40 T^{18} + \cdots + 1521 $ T^20 + 40T^18 + 680T^16 + 6400T^14 + 36400T^12 + 128128T^10 + 274560T^8 + 337920T^6 + 211200T^4 + 51200*T^2 + 1521
$3$	$ T^{20} $ T^20
$5$	$ T^{20} $ T^20
$7$	$ (T^{5} - 35 T^{3} + \cdots + 184)^{4} $ (T^5 - 35T^3 + 245T + 184)^4
$11$	$ T^{20} $ T^20
$13$	$ (T^{10} - 130 T^{8} + \cdots - 1648)^{2} $ (T^10 - 130T^8 + 5915T^6 - 109850T^4 + 714025T^2 - 1648)^2
$17$	$ T^{20} + \cdots + 2613001124484 $ T^20 + 340T^18 + 49130T^16 + 3930400T^14 + 190010275T^12 + 5685107428T^10 + 103550171010T^8 + 1083294096720T^6 + 5754999888825T^4 + 11858787649700*T^2 + 2613001124484
$19$	$ (T^{5} - 95 T^{3} + \cdots - 2972)^{4} $ (T^5 - 95T^3 + 1805T - 2972)^4
$23$	$ T^{20} + \cdots + 165186704840004 $ T^20 + 460T^18 + 89930T^16 + 9733600T^14 + 636638275T^12 + 25771117372T^10 + 635073963810T^8 + 8988739180080T^6 + 64606562856825T^4 + 180115266146300*T^2 + 165186704840004
$29$	$ T^{20} + \cdots + 16\!\cdots\!36 $ T^20 + 580T^18 + 142970T^16 + 19511200T^14 + 1609064275T^12 + 82126640596T^10 + 2551792047090T^8 + 45539673455760T^6 + 412703290692825T^4 + 1450714597586900*T^2 + 1675354456036836
$31$	$ T^{20} $ T^20
$37$	$ T^{20} $ T^20
$41$	$ T^{20} + \cdots + 53\!\cdots\!04 $ T^20 + 820T^18 + 285770T^16 + 55136800T^14 + 6428606275T^12 + 463888228804T^10 + 20377947193890T^8 + 514151283045840T^6 + 6587563314024825T^4 + 32738193439396100*T^2 + 5329408515828804
$43$	$ T^{20} $ T^20
$47$	$ T^{20} $ T^20
$53$	$ T^{20} $ T^20
$59$	$ T^{20} + \cdots + 44\!\cdots\!16 $ T^20 + 1180T^18 + 591770T^16 + 164303200T^14 + 27566996275T^12 + 2862556893196T^10 + 180954489319890T^8 + 6570039919922160T^6 + 121135111023564825T^4 + 866299581865493900*T^2 + 444478684214983716
$61$	$ (T^{10} - 610 T^{8} + \cdots - 3225467248)^{2} $ (T^10 - 610T^8 + 130235T^6 - 11349050T^4 + 346146025T^2 - 3225467248)^2
$67$	$ T^{20} $ T^20
$71$	$ T^{20} $ T^20
$73$	$ T^{20} $ T^20
$79$	$ (T^{10} - 790 T^{8} + \cdots - 29812732)^{2} $ (T^10 - 790T^8 + 218435T^6 - 24651950T^4 + 973752025T^2 - 29812732)^2
$83$	$ T^{20} + \cdots + 18\!\cdots\!96 $ T^20 + 1660T^18 + 1171130T^16 + 457429600T^14 + 107967680275T^12 + 15771918734572T^10 + 1402574201753010T^8 + 71639174612615280T^6 + 1858141091514708825T^4 + 18694025526754040300*T^2 + 18614868088729872996
$89$	$ T^{20} $ T^20
$97$	$ (T^{10} - 970 T^{8} + \cdots - 4250580928)^{2} $ (T^10 - 970T^8 + 329315T^6 - 45633650T^4 + 2213232025T^2 - 4250580928)^2
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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:	\( N \)	\(=\)	\( 927 = 3^{2} \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	927.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} - \beta_{6} q^{7} + (\beta_{3} - 2 \beta_1) q^{8}+O(q^{10}) \) q + b1 * q^2 + (b2 - 2) * q^4 - b6 * q^7 + (b3 - 2b1) q^8	\( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} - \beta_{6} q^{7} + (\beta_{3} - 2 \beta_1) q^{8} + \beta_{9} q^{13} + ( - \beta_{7} - \beta_{5}) q^{14} + (\beta_{4} - 2 \beta_{2} + 4) q^{16} + \beta_{12} q^{17} + ( - \beta_{15} + \beta_{4}) q^{19} + (\beta_{10} - \beta_{5}) q^{23} - 5 q^{25} + (\beta_{14} - \beta_{12} + \beta_{7}) q^{26} + ( - \beta_{13} + \beta_{9} + \cdots + 2 \beta_{6}) q^{28}+ \cdots + (\beta_{17} - 3 \beta_{16} + \cdots + 7 \beta_1) q^{98}+O(q^{100}) \) q + b1 * q^2 + (b2 - 2) * q^4 - b6 * q^7 + (b3 - 2b1) q^8 + b9 * q^13 + (-b7 - b5) * q^14 + (b4 - 2b2 + 4) q^16 + b12 * q^17 + (-b15 + b4) * q^19 + (b10 - b5) * q^23 - 5 * q^25 + (b14 - b12 + b7) * q^26 + (-b13 + b9 - b8 + 2b6) q^28 + b17 * q^29 + (-b16 + b14 - 2b3 + 4b1) * q^32 + (b13 + b11 + b9) * q^34 + (-b17 + b16 + b14 - b3) * q^38 + (-b10 - b5) * q^41 + (-b11 - b8 + b6) * q^46 + (b15 + b4 + 7) * q^49 - 5b1 q^50 + (b15 - b11 - 2b9 - 2b6 - 2b4) q^52 + (-2b16 + b14 - b10 + 2b7 + 4b5) q^56 + (b18 - 2b15 + b4) q^58 + (-b17 + 2b3) q^59 + (-b18 - b2) * q^61 + (b13 - 2b9 - 2b4 + 4b2 - 8) q^64 + (2b16 + b14 - 2b12 + 2b10 - b5) q^68 + (-b18 + 2b15 - b13 - 2b9 - 2b4 + 2b2) * q^76 + (-2b13 + b9) q^79 + (b11 - b8 + 3b6) q^82 + (-b12 - 2b7) q^83 + (b18 + 2b13 + b9 + 3b2) * q^91 + (2b12 - b10 + b7 + 4b5) * q^92 + (b18 - 3b2) q^97 + (b17 - 3b16 + b14 - 3b3 + 7b1) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 40 q^{4}+O(q^{10}) \) 20 * q - 40 * q^4	\( 20 q - 40 q^{4} + 80 q^{16} - 100 q^{25} + 140 q^{49} - 160 q^{64}+O(q^{100}) \) 20 * q - 40 * q^4 + 80 * q^16 - 100 * q^25 + 140 * q^49 - 160 * q^64

Introduction

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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	927.2.c.b

Level	$927$
Weight	$2$
Character orbit	927.c
Analytic conductor	$7.402$
Analytic rank	$0$
Dimension	$20$
CM discriminant	-103
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(7.40213226737\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} + 40 x^{18} + 680 x^{16} + 6400 x^{14} + 36400 x^{12} + 128128 x^{10} + 274560 x^{8} + \cdots + 1521 \) x^20 + 40x^18 + 680x^16 + 6400x^14 + 36400x^12 + 128128x^10 + 274560x^8 + 337920x^6 + 211200x^4 + 51200*x^2 + 1521
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 3^{9} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( \nu^{2} + 4 \) v^2 + 4
\(\beta_{3}\)	\(=\)	\( \nu^{3} + 6\nu \) v^3 + 6*v
\(\beta_{4}\)	\(=\)	\( \nu^{4} + 8\nu^{2} + 8 \) v^4 + 8*v^2 + 8
\(\beta_{5}\)	\(=\)	\( ( -2\nu^{11} - 35\nu^{9} - 190\nu^{7} - 260\nu^{5} + 400\nu^{3} + 592\nu ) / 65 \) (-2v^11 - 35v^9 - 190v^7 - 260v^5 + 400v^3 + 592v) / 65
\(\beta_{6}\)	\(=\)	\( ( \nu^{12} + 24\nu^{10} + 225\nu^{8} + 1040\nu^{6} + 2400\nu^{4} + 2304\nu^{2} + 416 ) / 65 \) (v^12 + 24v^10 + 225v^8 + 1040v^6 + 2400v^4 + 2304*v^2 + 416) / 65
\(\beta_{7}\)	\(=\)	\( ( \nu^{13} + 26\nu^{11} + 260\nu^{9} + 1230\nu^{7} + 2660\nu^{5} + 1904\nu^{3} - 176\nu ) / 65 \) (v^13 + 26v^11 + 260v^9 + 1230v^7 + 2660v^5 + 1904v^3 - 176v) / 65
\(\beta_{8}\)	\(=\)	\( ( \nu^{10} + 20\nu^{8} + 140\nu^{6} + 400\nu^{4} + 400\nu^{2} + 64 ) / 5 \) (v^10 + 20v^8 + 140v^6 + 400v^4 + 400v^2 + 64) / 5
\(\beta_{9}\)	\(=\)	\( ( \nu^{14} + 28\nu^{12} + 308\nu^{10} + 1680\nu^{8} + 4675\nu^{6} + 5924\nu^{4} + 2092\nu^{2} - 208 ) / 65 \) (v^14 + 28v^12 + 308v^10 + 1680v^8 + 4675v^6 + 5924v^4 + 2092v^2 - 208) / 65
\(\beta_{10}\)	\(=\)	\( ( \nu^{11} + 24\nu^{9} + 212\nu^{7} + 832\nu^{5} + 1360\nu^{3} + 640\nu ) / 13 \) (v^11 + 24v^9 + 212v^7 + 832v^5 + 1360v^3 + 640*v) / 13
\(\beta_{11}\)	\(=\)	\( ( -6\nu^{12} - 144\nu^{10} - 1285\nu^{8} - 5200\nu^{6} - 9200\nu^{4} - 5504\nu^{2} - 416 ) / 65 \) (-6v^12 - 144v^10 - 1285v^8 - 5200v^6 - 9200v^4 - 5504v^2 - 416) / 65
\(\beta_{12}\)	\(=\)	\( ( 3\nu^{13} + 78\nu^{11} + 780\nu^{9} + 3755\nu^{7} + 8890\nu^{5} + 9352\nu^{3} + 3112\nu ) / 65 \) (3v^13 + 78v^11 + 780v^9 + 3755v^7 + 8890v^5 + 9352v^3 + 3112*v) / 65
\(\beta_{13}\)	\(=\)	\( ( 2\nu^{14} + 56\nu^{12} + 616\nu^{10} + 3360\nu^{8} + 9415\nu^{6} + 12628\nu^{4} + 6524\nu^{2} + 624 ) / 65 \) (2v^14 + 56v^12 + 616v^10 + 3360v^8 + 9415v^6 + 12628v^4 + 6524*v^2 + 624) / 65
\(\beta_{14}\)	\(=\)	\( ( \nu^{15} + 30\nu^{13} + 360\nu^{11} + 2200\nu^{9} + 7200\nu^{7} + 12154\nu^{5} + 9540\nu^{3} + 3080\nu ) / 65 \) (v^15 + 30v^13 + 360v^11 + 2200v^9 + 7200v^7 + 12154v^5 + 9540v^3 + 3080*v) / 65
\(\beta_{15}\)	\(=\)	\( ( \nu^{16} + 32 \nu^{14} + 416 \nu^{12} + 2816 \nu^{10} + 10560 \nu^{8} + 21504 \nu^{6} + 21518 \nu^{4} + \cdots + 624 ) / 65 \) (v^16 + 32v^14 + 416v^12 + 2816v^10 + 10560v^8 + 21504v^6 + 21518v^4 + 8304*v^2 + 624) / 65
\(\beta_{16}\)	\(=\)	\( ( \nu^{15} + 30\nu^{13} + 360\nu^{11} + 2200\nu^{9} + 7200\nu^{7} + 12089\nu^{5} + 8890\nu^{3} + 1780\nu ) / 65 \) (v^15 + 30v^13 + 360v^11 + 2200v^9 + 7200v^7 + 12089v^5 + 8890v^3 + 1780*v) / 65
\(\beta_{17}\)	\(=\)	\( ( \nu^{17} + 34 \nu^{15} + 476 \nu^{13} + 3536 \nu^{11} + 14960 \nu^{9} + 35904 \nu^{7} + 45696 \nu^{5} + \cdots + 4574 \nu ) / 65 \) (v^17 + 34v^15 + 476v^13 + 3536v^11 + 14960v^9 + 35904v^7 + 45696v^5 + 26149v^3 + 4574v) / 65
\(\beta_{18}\)	\(=\)	\( ( \nu^{18} + 36 \nu^{16} + 540 \nu^{14} + 4368 \nu^{12} + 20592 \nu^{10} + 57024 \nu^{8} + 88704 \nu^{6} + \cdots + 728 ) / 65 \) (v^18 + 36v^16 + 540v^14 + 4368v^12 + 20592v^10 + 57024v^8 + 88704v^6 + 69120v^4 + 20662v^2 + 728) / 65
\(\beta_{19}\)	\(=\)	\( ( \nu^{19} + 38 \nu^{17} + 608 \nu^{15} + 5320 \nu^{13} + 27664 \nu^{11} + 86944 \nu^{9} + \cdots + 9876 \nu ) / 65 \) (v^19 + 38v^17 + 608v^15 + 5320v^13 + 27664v^11 + 86944v^9 + 160512v^7 + 160512v^5 + 72960v^3 + 9876*v) / 65

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} - 4 \) b2 - 4
\(\nu^{3}\)	\(=\)	\( \beta_{3} - 6\beta_1 \) b3 - 6*b1
\(\nu^{4}\)	\(=\)	\( \beta_{4} - 8\beta_{2} + 24 \) b4 - 8*b2 + 24
\(\nu^{5}\)	\(=\)	\( -\beta_{16} + \beta_{14} - 10\beta_{3} + 40\beta_1 \) -b16 + b14 - 10b3 + 40b1
\(\nu^{6}\)	\(=\)	\( \beta_{13} - 2\beta_{9} - 12\beta_{4} + 60\beta_{2} - 160 \) b13 - 2b9 - 12b4 + 60*b2 - 160
\(\nu^{7}\)	\(=\)	\( 14\beta_{16} - 14\beta_{14} + \beta_{12} - 3\beta_{7} + 84\beta_{3} - 280\beta_1 \) 14b16 - 14b14 + b12 - 3b7 + 84b3 - 280*b1
\(\nu^{8}\)	\(=\)	\( -16\beta_{13} + \beta_{11} + 32\beta_{9} + 6\beta_{6} + 112\beta_{4} - 448\beta_{2} + 1120 \) -16b13 + b11 + 32b9 + 6b6 + 112b4 - 448*b2 + 1120
\(\nu^{9}\)	\(=\)	\( -144\beta_{16} + 144\beta_{14} - 18\beta_{12} + 2\beta_{10} + 54\beta_{7} + 5\beta_{5} - 672\beta_{3} + 2016\beta_1 \) -144b16 + 144b14 - 18b12 + 2b10 + 54b7 + 5b5 - 672b3 + 2016b1
\(\nu^{10}\)	\(=\)	\( 180\beta_{13} - 20\beta_{11} - 360\beta_{9} + 5\beta_{8} - 120\beta_{6} - 960\beta_{4} + 3360\beta_{2} - 8064 \) 180b13 - 20b11 - 360b9 + 5b8 - 120b6 - 960b4 + 3360*b2 - 8064
\(\nu^{11}\)	\(=\)	\( 1320 \beta_{16} - 1320 \beta_{14} + 220 \beta_{12} - 35 \beta_{10} - 660 \beta_{7} - 120 \beta_{5} + \cdots - 14784 \beta_1 \) 1320b16 - 1320b14 + 220b12 - 35b10 - 660b7 - 120b5 + 5280b3 - 14784b1
\(\nu^{12}\)	\(=\)	\( - 1760 \beta_{13} + 255 \beta_{11} + 3520 \beta_{9} - 120 \beta_{8} + 1595 \beta_{6} + 7920 \beta_{4} + \cdots + 59136 \) -1760b13 + 255b11 + 3520b9 - 120b8 + 1595b6 + 7920b4 - 25344*b2 + 59136
\(\nu^{13}\)	\(=\)	\( - 11440 \beta_{16} + 11440 \beta_{14} - 2270 \beta_{12} + 390 \beta_{10} + 6875 \beta_{7} + \cdots + 109824 \beta_1 \) -11440b16 + 11440b14 - 2270b12 + 390b10 + 6875b7 + 1820b5 - 41184b3 + 109824b1
\(\nu^{14}\)	\(=\)	\( 16045 \beta_{13} - 2660 \beta_{11} - 32025 \beta_{9} + 1820 \beta_{8} - 17780 \beta_{6} - 64064 \beta_{4} + \cdots - 439296 \) 16045b13 - 2660b11 - 32025b9 + 1820b8 - 17780b6 - 64064b4 + 192192*b2 - 439296
\(\nu^{15}\)	\(=\)	\( 96154 \beta_{16} - 96089 \beta_{14} + 21300 \beta_{12} - 3500 \beta_{10} - 65850 \beta_{7} + \cdots - 823680 \beta_1 \) 96154b16 - 96089b14 + 21300b12 - 3500b10 - 65850b7 - 22400b5 + 320320b3 - 823680b1
\(\nu^{16}\)	\(=\)	\( 65 \beta_{15} - 140704 \beta_{13} + 24800 \beta_{11} + 279328 \beta_{9} - 22400 \beta_{8} + \cdots + 3294720 \) 65b15 - 140704b13 + 24800b11 + 279328b9 - 22400b8 + 180000b6 + 512498b4 - 1464320b2 + 3294720
\(\nu^{17}\)	\(=\)	\( 65 \beta_{17} - 794036 \beta_{16} + 791826 \beta_{14} - 188224 \beta_{12} + 27200 \beta_{10} + \cdots + 6223360 \beta_1 \) 65b17 - 794036b16 + 791826b14 - 188224b12 + 27200b10 + 600032b7 + 244800b5 - 2489381b3 + 6223360*b1
\(\nu^{18}\)	\(=\)	\( 65 \beta_{18} - 2340 \beta_{15} + 1205844 \beta_{13} - 215424 \beta_{11} - 2371908 \beta_{9} + \cdots - 24893440 \) 65b18 - 2340b15 + 1205844b13 - 215424b11 - 2371908b9 + 244800b8 - 1716864b6 - 4072968b4 + 11202122*b2 - 24893440
\(\nu^{19}\)	\(=\)	\( 65 \beta_{19} - 2470 \beta_{17} + 6489336 \beta_{16} - 6444876 \beta_{14} + 1596912 \beta_{12} + \cdots - 47297684 \beta_1 \) 65b19 - 2470b17 + 6489336b16 - 6444876b14 + 1596912b12 - 186048b10 - 5294616b7 - 2480640b5 + 19350398b3 - 47297684b1

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 926.20
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{20} + 40 T^{18} + \cdots + 1521 \) T^20 + 40T^18 + 680T^16 + 6400T^14 + 36400T^12 + 128128T^10 + 274560T^8 + 337920T^6 + 211200T^4 + 51200*T^2 + 1521
$3$	\( T^{20} \) T^20
$5$	\( T^{20} \) T^20
$7$	\( (T^{5} - 35 T^{3} + \cdots + 184)^{4} \) (T^5 - 35T^3 + 245T + 184)^4
$11$	\( T^{20} \) T^20
$13$	\( (T^{10} - 130 T^{8} + \cdots - 1648)^{2} \) (T^10 - 130T^8 + 5915T^6 - 109850T^4 + 714025T^2 - 1648)^2
$17$	\( T^{20} + \cdots + 2613001124484 \) T^20 + 340T^18 + 49130T^16 + 3930400T^14 + 190010275T^12 + 5685107428T^10 + 103550171010T^8 + 1083294096720T^6 + 5754999888825T^4 + 11858787649700*T^2 + 2613001124484
$19$	\( (T^{5} - 95 T^{3} + \cdots - 2972)^{4} \) (T^5 - 95T^3 + 1805T - 2972)^4
$23$	\( T^{20} + \cdots + 165186704840004 \) T^20 + 460T^18 + 89930T^16 + 9733600T^14 + 636638275T^12 + 25771117372T^10 + 635073963810T^8 + 8988739180080T^6 + 64606562856825T^4 + 180115266146300*T^2 + 165186704840004
$29$	\( T^{20} + \cdots + 16\!\cdots\!36 \) T^20 + 580T^18 + 142970T^16 + 19511200T^14 + 1609064275T^12 + 82126640596T^10 + 2551792047090T^8 + 45539673455760T^6 + 412703290692825T^4 + 1450714597586900*T^2 + 1675354456036836
$31$	\( T^{20} \) T^20
$37$	\( T^{20} \) T^20
$41$	\( T^{20} + \cdots + 53\!\cdots\!04 \) T^20 + 820T^18 + 285770T^16 + 55136800T^14 + 6428606275T^12 + 463888228804T^10 + 20377947193890T^8 + 514151283045840T^6 + 6587563314024825T^4 + 32738193439396100*T^2 + 5329408515828804
$43$	\( T^{20} \) T^20
$47$	\( T^{20} \) T^20
$53$	\( T^{20} \) T^20
$59$	\( T^{20} + \cdots + 44\!\cdots\!16 \) T^20 + 1180T^18 + 591770T^16 + 164303200T^14 + 27566996275T^12 + 2862556893196T^10 + 180954489319890T^8 + 6570039919922160T^6 + 121135111023564825T^4 + 866299581865493900*T^2 + 444478684214983716
$61$	\( (T^{10} - 610 T^{8} + \cdots - 3225467248)^{2} \) (T^10 - 610T^8 + 130235T^6 - 11349050T^4 + 346146025T^2 - 3225467248)^2
$67$	\( T^{20} \) T^20
$71$	\( T^{20} \) T^20
$73$	\( T^{20} \) T^20
$79$	\( (T^{10} - 790 T^{8} + \cdots - 29812732)^{2} \) (T^10 - 790T^8 + 218435T^6 - 24651950T^4 + 973752025T^2 - 29812732)^2
$83$	\( T^{20} + \cdots + 18\!\cdots\!96 \) T^20 + 1660T^18 + 1171130T^16 + 457429600T^14 + 107967680275T^12 + 15771918734572T^10 + 1402574201753010T^8 + 71639174612615280T^6 + 1858141091514708825T^4 + 18694025526754040300*T^2 + 18614868088729872996
$89$	\( T^{20} \) T^20
$97$	\( (T^{10} - 970 T^{8} + \cdots - 4250580928)^{2} \) (T^10 - 970T^8 + 329315T^6 - 45633650T^4 + 2213232025T^2 - 4250580928)^2
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\(n\)	\(722\)	\(829\)
\(\chi(n)\)	\(-1\)	\(-1\)