LMFDB - Newform orbit 968.2.w.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [968,2,Mod(43,968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(968, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 11, 5]))

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

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

chi := DirichletCharacter("968.43");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 968 = 2^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	968.w (of order $22$, degree $10$, 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:	$7.72951891566$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$2$ over $\Q(\zeta_{22})$
Coefficient field:	20.0.5969915757478328440239161344.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2x^{18} + 4x^{16} - 8x^{14} + 16x^{12} - 32x^{10} + 64x^{8} - 128x^{6} + 256x^{4} - 512x^{2} + 1024 $ x^20 - 2x^18 + 4x^16 - 8x^14 + 16x^12 - 32x^10 + 64x^8 - 128x^6 + 256x^4 - 512*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{22}]$

$q$-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_{5} q^{2} + ( - \beta_{19} + \beta_{17} + \cdots + 2 \beta_1) q^{3}+ \cdots + ( - \beta_{18} + \beta_{16} - \beta_{14} + \cdots + 4) q^{9}+O(q^{10}) $ q - b5 * q^2 + (-b19 + b17 - b15 + b13 + b12 - b11 - b10 + b9 - b7 + b5 - b3 + 2b1) q^3 + 2b10 q^4 + (-b17 + b15 - 2b6 - 2b4) * q^6 - 2b15 q^8 + (-b18 + b16 - b14 + 2b13 + b12 - b10 - 2b9 + b8 - b6 + b4 + 4) * q^9	$ q - \beta_{5} q^{2} + ( - \beta_{19} + \beta_{17} + \cdots + 2 \beta_1) q^{3}+ \cdots + (8 \beta_{18} + 8 \beta_{16} + \cdots + 1) q^{99}+O(q^{100}) $ q - b5 * q^2 + (-b19 + b17 - b15 + b13 + b12 - b11 - b10 + b9 - b7 + b5 - b3 + 2b1) q^3 + 2b10 q^4 + (-b17 + b15 - 2b6 - 2b4) * q^6 - 2b15 q^8 + (-b18 + b16 - b14 + 2b13 + b12 - b10 - 2b9 + b8 - b6 + b4 + 4) * q^9 + (3b18 - b7) q^11 + (-2b18 + 2b16 - 2b14 + 2b12 + 2b11 - 2b10 + 2b9 + 2b8 - 2b6 + 2b4 - 2b2) q^12 + (4b18 - 4b16 + 4b14 - 4b12 + 4b10 - 4b8 + 4b6 - 4b4 + 4b2 - 4) q^16 + (-2b19 - 3b18 + 2b17 - 2b15 + 2b13 - 2b11 + 3b10 + 2b9 + 2b5 - 2b3 + 2b1) q^17 + (-4b18 + 4b14 - b7 - 3b5 - b3) q^18 + (-3b15 - b12 + b4 + 3b1) * q^19 + (2b12 + 3b1) * q^22 + (-4b16 - 4b14 + 2b5 - 2b3) * q^24 - 5b14 q^25 + (-2b19 + 3b17 - 3b15 + 5b14 + 3b13 + 3b12 - 3b11 - 3b10 + 3b9 - 5b8 - 3b7 + 3b5 - 4b3 + 6b1) * q^27 + 4b3 q^32 + (2b19 + 4b17 - 5b8 + b6) q^33 + (-3b15 - 4b12 - 4b4 - 3b1) * q^34 + (-4b19 + 2b12 + 6b10 + 2b8 - 4b1) q^36 + (6b18 + b17 - 6b16 + 6b14 - 6b12 + 6b10 - b9 - 6b8 - 6b4 + 6b2 - 6) * q^38 + (-3b18 + 3b14 + 4b7 + 4b3) * q^41 + (-5b18 - 5b12 + 3b7 - 3b1) * q^43 + (-2b17 - 6b6) * q^44 + (8b19 - 4b17 + 4b15 - 4b13 + 4b11 - 4b10 - 4b9 + 4b8 + 4b7 - 4b5 + 4b3 - 4b1) * q^48 - 7b4 q^49 + 5b19 q^50 + (-b19 - 7b18 - 5b17 + 7b16 - 7b14 + 7b12 + 5b11 - 7b10 + b9 + 14b8 - 6b6 + 7b4 - 7b2 + 8) q^51 + (-5b19 - 3b17 + 3b15 + 5b13 + 2b8 - 6b6 - 6b4 + 2b2) * q^54 + (-5b16 - 7b14 + 2b13 - 4b11 + 4b5 - 2b3 + 7b2 + 5) q^57 + (3b18 - 3b16 + 5b7 + 5b5) * q^59 - 8b8 q^64 + (5b13 - b11 + 4b2 + 8) * q^66 + 14b16 q^67 + (6b18 + 4b17 - 6b16 + 6b14 - 6b12 + 6b10 + 4b9 - 6b8 + 12b6 - 6b4 + 6b2 - 6) q^68 + (-2b17 - 6b15 - 2b13 + 8b6 - 8b2) q^72 + (-6b17 + 6b15 - 6b13 + 6b11 - b10 - 6b9 - b8 + 6b7 - 6b5 + 6b3 - 6b1) q^73 + (-5b15 - 5b13 + 5b4 - 5b2) * q^75 + (2b14 + 6b11 + 6b3 + 2) q^76 + (4b18 + 3b16 + 4b15 - 3b14 + 6b13 + 3b12 - 3b10 - 6b9 + 3b8 - 4b7 - 3b6 - 4b4 + 12) * q^81 + (-3b19 - 8b12 - 8b8 - 3b1) * q^82 - 2b13 q^83 + (5b17 - 6b12 + 6b6 - 5b1) * q^86 + (6b11 - 4) q^88 + (9b16 + 9b12 - 2b5 + 2b1) * q^89 + (4b15 - 4b13 + 8b4 + 8b2) * q^96 + (6b19 + 5b18 - 6b17 - 5b16 + 6b15 + 5b14 - 6b13 - 5b12 + 6b11 - 12b9 - 5b8 + 6b7 + 5b6 - 6b5 - 5b4 + 6b3 + 5b2 - 6b1 - 5) * q^97 + 7b9 q^98 + (8b18 + 8b16 - b14 + b12 - b10 - 7b9 + b8 - 3b7 - b6 + 5b5 + b4 - b2 + 1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{3} + 4 q^{4} + 64 q^{9}+O(q^{10}) $ 20 * q - 4 * q^3 + 4 * q^4 + 64 * q^9	$ 20 q - 4 q^{3} + 4 q^{4} + 64 q^{9} + 6 q^{11} - 36 q^{12} - 8 q^{16} - 4 q^{22} - 10 q^{25} + 8 q^{27} + 12 q^{33} + 16 q^{34} + 4 q^{36} - 24 q^{38} - 12 q^{44} - 16 q^{48} + 14 q^{49} + 22 q^{51} + 110 q^{57} + 12 q^{59} + 16 q^{64} + 168 q^{66} - 28 q^{67} - 20 q^{75} + 44 q^{76} + 220 q^{81} + 32 q^{82} + 24 q^{86} - 80 q^{88} - 36 q^{89} - 20 q^{97} + 6 q^{99}+O(q^{100}) $ 20 * q - 4 * q^3 + 4 * q^4 + 64 * q^9 + 6 * q^11 - 36 * q^12 - 8 * q^16 - 4 * q^22 - 10 * q^25 + 8 * q^27 + 12 * q^33 + 16 * q^34 + 4 * q^36 - 24 * q^38 - 12 * q^44 - 16 * q^48 + 14 * q^49 + 22 * q^51 + 110 * q^57 + 12 * q^59 + 16 * q^64 + 168 * q^66 - 28 * q^67 - 20 * q^75 + 44 * q^76 + 220 * q^81 + 32 * q^82 + 24 * q^86 - 80 * q^88 - 36 * q^89 - 20 * q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{20} - 2x^{18} + 4x^{16} - 8x^{14} + 16x^{12} - 32x^{10} + 64x^{8} - 128x^{6} + 256x^{4} - 512x^{2} + 1024 $ x^20 - 2*x^18 + 4*x^16 - 8*x^14 + 16*x^12 - 32*x^10 + 64*x^8 - 128*x^6 + 256*x^4 - 512*x^2 + 1024 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{2} ) / 2 $ (v^2) / 2
$\beta_{3}$	$=$	$ ( \nu^{3} ) / 2 $ (v^3) / 2
$\beta_{4}$	$=$	$ ( \nu^{4} ) / 4 $ (v^4) / 4
$\beta_{5}$	$=$	$ ( \nu^{5} ) / 4 $ (v^5) / 4
$\beta_{6}$	$=$	$ ( \nu^{6} ) / 8 $ (v^6) / 8
$\beta_{7}$	$=$	$ ( \nu^{7} ) / 8 $ (v^7) / 8
$\beta_{8}$	$=$	$ ( \nu^{8} ) / 16 $ (v^8) / 16
$\beta_{9}$	$=$	$ ( \nu^{9} ) / 16 $ (v^9) / 16
$\beta_{10}$	$=$	$ ( \nu^{10} ) / 32 $ (v^10) / 32
$\beta_{11}$	$=$	$ ( \nu^{11} ) / 32 $ (v^11) / 32
$\beta_{12}$	$=$	$ ( \nu^{12} ) / 64 $ (v^12) / 64
$\beta_{13}$	$=$	$ ( \nu^{13} ) / 64 $ (v^13) / 64
$\beta_{14}$	$=$	$ ( \nu^{14} ) / 128 $ (v^14) / 128
$\beta_{15}$	$=$	$ ( \nu^{15} ) / 128 $ (v^15) / 128
$\beta_{16}$	$=$	$ ( \nu^{16} ) / 256 $ (v^16) / 256
$\beta_{17}$	$=$	$ ( \nu^{17} ) / 256 $ (v^17) / 256
$\beta_{18}$	$=$	$ ( \nu^{18} ) / 512 $ (v^18) / 512
$\beta_{19}$	$=$	$ ( \nu^{19} ) / 512 $ (v^19) / 512

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ 2\beta_{2} $ 2*b2
$\nu^{3}$	$=$	$ 2\beta_{3} $ 2*b3
$\nu^{4}$	$=$	$ 4\beta_{4} $ 4*b4
$\nu^{5}$	$=$	$ 4\beta_{5} $ 4*b5
$\nu^{6}$	$=$	$ 8\beta_{6} $ 8*b6
$\nu^{7}$	$=$	$ 8\beta_{7} $ 8*b7
$\nu^{8}$	$=$	$ 16\beta_{8} $ 16*b8
$\nu^{9}$	$=$	$ 16\beta_{9} $ 16*b9
$\nu^{10}$	$=$	$ 32\beta_{10} $ 32*b10
$\nu^{11}$	$=$	$ 32\beta_{11} $ 32*b11
$\nu^{12}$	$=$	$ 64\beta_{12} $ 64*b12
$\nu^{13}$	$=$	$ 64\beta_{13} $ 64*b13
$\nu^{14}$	$=$	$ 128\beta_{14} $ 128*b14
$\nu^{15}$	$=$	$ 128\beta_{15} $ 128*b15
$\nu^{16}$	$=$	$ 256\beta_{16} $ 256*b16
$\nu^{17}$	$=$	$ 256\beta_{17} $ 256*b17
$\nu^{18}$	$=$	$ 512\beta_{18} $ 512*b18
$\nu^{19}$	$=$	$ 512\beta_{19} $ 512*b19

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/968\mathbb{Z}\right)^\times$.

$n$	$485$	$727$	$849$
$\chi(n)$	$-1$	$-1$	$\beta_{10}$

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} $

43.1

1.39982	+	0.201264i
−1.39982	−	0.201264i
−1.28641	−	0.587486i
1.28641	+	0.587486i
−1.06879	−	0.926113i
1.06879	+	0.926113i
0.764582	+	1.18971i
−0.764582	−	1.18971i
0.398430	+	1.35693i
−0.398430	−	1.35693i
0.398430	−	1.35693i
−0.398430	+	1.35693i
0.764582	−	1.18971i
−0.764582	+	1.18971i
−1.06879	+	0.926113i
1.06879	−	0.926113i
−1.28641	+	0.587486i
1.28641	−	0.587486i
1.39982	−	0.201264i
−1.39982	+	0.201264i

−1.06879

−

0.926113i

2.51501

0.284630

1.97964i

−2.68802

−

2.32918i

1.52916

−

2.37942i

3.32527

43.2

1.06879

0.926113i

−3.08427

0.284630

1.97964i

−3.29644

−

2.85638i

−1.52916

2.37942i

6.51271

131.1

−0.764582

1.18971i

−1.74200

−0.830830

−

1.81926i

1.33190

−

2.07248i

2.79964

0.402527i

0.0345563

131.2

0.764582

−

1.18971i

3.40366

−0.830830

−

1.81926i

2.60237

−

4.04937i

−2.79964

−

0.402527i

8.58489

219.1

−1.28641

−

0.587486i

−3.44730

1.30972

1.51150i

4.43466

2.02524i

−0.796860

−

2.71386i

8.88391

219.2

1.28641

0.587486i

0.827861

1.30972

1.51150i

1.06497

0.486356i

0.796860

2.71386i

−2.31465

307.1

−0.398430

1.35693i

3.21167

−1.68251

−

1.08128i

−1.27963

4.35801i

2.13758

−

1.85223i

7.31483

307.2

0.398430

−

1.35693i

0.153344

−1.68251

−

1.08128i

0.0610968

−

0.208077i

−2.13758

1.85223i

−2.97649

395.1

−1.39982

−

0.201264i

−1.12213

1.91899

0.563465i

1.57077

0.225843i

−2.57283

−

1.17497i

−1.74083

395.2

1.39982

0.201264i

−2.71585

1.91899

0.563465i

−3.80169

−

0.546601i

2.57283

1.17497i

4.37582

571.1

−1.39982

0.201264i

−1.12213

1.91899

−

0.563465i

1.57077

−

0.225843i

−2.57283

1.17497i

−1.74083

571.2

1.39982

−

0.201264i

−2.71585

1.91899

−

0.563465i

−3.80169

0.546601i

2.57283

−

1.17497i

4.37582

659.1

−0.398430

−

1.35693i

3.21167

−1.68251

1.08128i

−1.27963

−

4.35801i

2.13758

1.85223i

7.31483

659.2

0.398430

1.35693i

0.153344

−1.68251

1.08128i

0.0610968

0.208077i

−2.13758

−

1.85223i

−2.97649

747.1

−1.28641

0.587486i

−3.44730

1.30972

−

1.51150i

4.43466

−

2.02524i

−0.796860

2.71386i

8.88391

747.2

1.28641

−

0.587486i

0.827861

1.30972

−

1.51150i

1.06497

−

0.486356i

0.796860

−

2.71386i

−2.31465

835.1

−0.764582

−

1.18971i

−1.74200

−0.830830

1.81926i

1.33190

2.07248i

2.79964

−

0.402527i

0.0345563

835.2

0.764582

1.18971i

3.40366

−0.830830

1.81926i

2.60237

4.04937i

−2.79964

0.402527i

8.58489

923.1

−1.06879

0.926113i

2.51501

0.284630

−

1.97964i

−2.68802

2.32918i

1.52916

2.37942i

3.32527

923.2

1.06879

−

0.926113i

−3.08427

0.284630

−

1.97964i

−3.29644

2.85638i

−1.52916

−

2.37942i

6.51271

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
121.f	odd	22	1	inner
968.w	even	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	968.2.w.a	✓	20
8.d	odd	2	1	CM	968.2.w.a	✓	20
121.f	odd	22	1	inner	968.2.w.a	✓	20
968.w	even	22	1	inner	968.2.w.a	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
968.2.w.a	✓	20	1.a	even	1	1	trivial
968.2.w.a	✓	20	8.d	odd	2	1	CM
968.2.w.a	✓	20	121.f	odd	22	1	inner
968.2.w.a	✓	20	968.w	even	22	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{10} + 2 T_{3}^{9} - 29 T_{3}^{8} - 58 T_{3}^{7} + 280 T_{3}^{6} + 560 T_{3}^{5} - 959 T_{3}^{4} + \cdots - 197 $ T3^10 + 2*T3^9 - 29*T3^8 - 58*T3^7 + 280*T3^6 + 560*T3^5 - 959*T3^4 - 1918*T3^3 + 619*T3^2 + 1238*T3 - 197 acting on $S_{2}^{\mathrm{new}}(968, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} - 2 T^{18} + \cdots + 1024 $ T^20 - 2T^18 + 4T^16 - 8T^14 + 16T^12 - 32T^10 + 64T^8 - 128T^6 + 256T^4 - 512*T^2 + 1024
$3$	$ (T^{10} + 2 T^{9} + \cdots - 197)^{2} $ (T^10 + 2T^9 - 29T^8 - 58T^7 + 280T^6 + 560T^5 - 959T^4 - 1918T^3 + 619T^2 + 1238*T - 197)^2
$5$	$ T^{20} $ T^20
$7$	$ T^{20} $ T^20
$11$	$ T^{20} + \cdots + 25937424601 $ T^20 - 6T^19 + 25T^18 - 84T^17 + 229T^16 - 450T^15 + 181T^14 + 3864T^13 - 25175T^12 + 108546T^11 - 374351T^10 + 1194006T^9 - 3046175T^8 + 5142984T^7 + 2650021T^6 - 72472950T^5 + 405687469T^4 - 1636922364T^3 + 5358972025T^2 - 14147686146*T + 25937424601
$13$	$ T^{20} $ T^20
$17$	$ T^{20} + \cdots + 3433916604889 $ T^20 - 32T^18 + 1024T^16 + 38148T^15 - 32768T^14 - 362406T^13 + 725814T^12 + 15745026T^11 + 665822202T^10 - 503840832T^9 + 13498894740T^8 - 22563018324T^7 + 208019437781T^6 + 252217652874T^5 - 406182351805T^4 + 2296155920916T^3 + 9592400482523T^2 + 10214908786038*T + 3433916604889
$19$	$ T^{20} + \cdots + 4436419650961 $ T^20 - 72T^18 + 5184T^16 + 15884T^15 - 373248T^14 + 119130T^13 + 26355118T^12 - 21059258T^11 - 747876634T^10 + 1516266576T^9 + 12785055684T^8 - 18279593948T^7 - 67083626585T^6 - 716517657350T^5 + 677752084123T^4 + 4736282735756T^3 + 10088346279009T^2 + 6038838216422*T + 4436419650961
$23$	$ T^{20} $ T^20
$29$	$ T^{20} $ T^20
$31$	$ T^{20} $ T^20
$37$	$ T^{20} $ T^20
$41$	$ T^{20} + \cdots + 63\!\cdots\!81 $ T^20 - 128T^18 + 16384T^16 - 2659098T^14 - 22743930T^13 + 96246362T^12 + 1881200376T^11 - 14168186042T^10 - 240793648128T^9 + 2388290426537T^8 + 67410133798194T^7 + 605685301233303T^6 + 3016799343577206T^5 + 9319747461575747T^4 + 18196100913852906T^3 + 21172301802645854T^2 + 12331667893973736T + 6321735306254281
$43$	$ T^{20} + \cdots + 50\!\cdots\!49 $ T^20 - 72T^18 + 11806T^16 + 137170T^15 - 850032T^14 - 9876240T^13 + 332422869T^12 + 6884382560T^11 - 7670100811T^10 - 495675544320T^9 + 1050311863276T^8 + 15970294962490T^7 - 55543750969396T^6 - 1004923663401790T^5 + 9218028387851452T^4 - 31134427119164820T^3 + 61079949328090471T^2 - 75895644240419190*T + 50959927783705249
$47$	$ T^{20} $ T^20
$53$	$ T^{20} $ T^20
$59$	$ T^{20} + \cdots + 30\!\cdots\!01 $ T^20 - 12T^19 + 108T^18 - 864T^17 + 59698T^16 - 1234326T^15 + 12662784T^14 - 107517672T^13 + 2282326005T^12 - 41540390076T^11 + 514503931689T^10 - 4657599841944T^9 + 51556014351880T^8 - 282685548869142T^7 + 881510312673020T^6 - 4160685897180642T^5 + 83496564852543472T^4 + 632615961157305324T^3 + 1537604079172580167T^2 + 2384565518437394970*T + 3060544858646807401
$61$	$ T^{20} $ T^20
$67$	$ (T^{10} + \cdots + 289254654976)^{2} $ (T^10 + 14T^9 + 196T^8 + 2744T^7 + 38416T^6 + 537824T^5 + 7529536T^4 + 105413504T^3 + 1475789056T^2 + 20661046784*T + 289254654976)^2
$71$	$ T^{20} $ T^20
$73$	$ T^{20} + \cdots + 33\!\cdots\!09 $ T^20 - 288T^18 + 82944T^16 + 234476T^15 - 23887872T^14 + 8089422T^13 + 6403420534T^12 - 6236841578T^11 - 885176515222T^10 + 1796210374464T^9 + 58944465469140T^8 - 170927177268188T^7 - 1785268572446531T^6 - 5063874621171938T^5 + 70029439884545155T^4 + 355326430447365212T^3 + 1680641425443521667T^2 - 360889775597097406*T + 333672883023928009
$79$	$ T^{20} $ T^20
$83$	$ T^{20} + \cdots + 1073741824 $ T^20 - 8T^18 + 64T^16 - 512T^14 + 4096T^12 - 32768T^10 + 262144T^8 - 2097152T^6 + 16777216T^4 - 134217728*T^2 + 1073741824
$89$	$ T^{20} + \cdots + 32\!\cdots\!21 $ T^20 + 36T^19 + 972T^18 + 5706T^17 - 109512T^16 - 5791176T^15 - 49535821T^14 + 85985046T^13 + 19145067660T^12 + 235492917894T^11 + 1889879570436T^10 - 8057857287564T^9 - 163895228864971T^8 - 1424647385834976T^7 + 5350056310547321T^6 + 76172695775462808T^5 + 282637070337935146T^4 - 1996797636732317274T^3 + 5384883609493166432T^2 - 9861925928770919988*T + 32802211690363323121
$97$	$ T^{20} + \cdots + 48\!\cdots\!29 $ T^20 + 20T^19 + 300T^18 + 4000T^17 + 50000T^16 + 600000T^15 + 17650794T^14 - 208954270T^13 + 661762374T^12 - 1245283720T^11 - 8476651854T^10 - 184766518540T^9 + 97941947411009T^8 - 7945785433902170T^7 + 212382841724860803T^6 - 2463395964397386310T^5 + 15727147143360669791T^4 - 55815206769859692390T^3 + 103869305724658775414T^2 - 47958633336703040580*T + 48833439206436284929
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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:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	968.w (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{5} q^{2} + ( - \beta_{19} + \beta_{17} + \cdots + 2 \beta_1) q^{3}+ \cdots + ( - \beta_{18} + \beta_{16} - \beta_{14} + \cdots + 4) q^{9}+O(q^{10}) \) q - b5 * q^2 + (-b19 + b17 - b15 + b13 + b12 - b11 - b10 + b9 - b7 + b5 - b3 + 2b1) q^3 + 2b10 q^4 + (-b17 + b15 - 2b6 - 2b4) * q^6 - 2b15 q^8 + (-b18 + b16 - b14 + 2b13 + b12 - b10 - 2b9 + b8 - b6 + b4 + 4) * q^9	\( q - \beta_{5} q^{2} + ( - \beta_{19} + \beta_{17} + \cdots + 2 \beta_1) q^{3}+ \cdots + (8 \beta_{18} + 8 \beta_{16} + \cdots + 1) q^{99}+O(q^{100}) \) q - b5 * q^2 + (-b19 + b17 - b15 + b13 + b12 - b11 - b10 + b9 - b7 + b5 - b3 + 2b1) q^3 + 2b10 q^4 + (-b17 + b15 - 2b6 - 2b4) * q^6 - 2b15 q^8 + (-b18 + b16 - b14 + 2b13 + b12 - b10 - 2b9 + b8 - b6 + b4 + 4) * q^9 + (3b18 - b7) q^11 + (-2b18 + 2b16 - 2b14 + 2b12 + 2b11 - 2b10 + 2b9 + 2b8 - 2b6 + 2b4 - 2b2) q^12 + (4b18 - 4b16 + 4b14 - 4b12 + 4b10 - 4b8 + 4b6 - 4b4 + 4b2 - 4) q^16 + (-2b19 - 3b18 + 2b17 - 2b15 + 2b13 - 2b11 + 3b10 + 2b9 + 2b5 - 2b3 + 2b1) q^17 + (-4b18 + 4b14 - b7 - 3b5 - b3) q^18 + (-3b15 - b12 + b4 + 3b1) * q^19 + (2b12 + 3b1) * q^22 + (-4b16 - 4b14 + 2b5 - 2b3) * q^24 - 5b14 q^25 + (-2b19 + 3b17 - 3b15 + 5b14 + 3b13 + 3b12 - 3b11 - 3b10 + 3b9 - 5b8 - 3b7 + 3b5 - 4b3 + 6b1) * q^27 + 4b3 q^32 + (2b19 + 4b17 - 5b8 + b6) q^33 + (-3b15 - 4b12 - 4b4 - 3b1) * q^34 + (-4b19 + 2b12 + 6b10 + 2b8 - 4b1) q^36 + (6b18 + b17 - 6b16 + 6b14 - 6b12 + 6b10 - b9 - 6b8 - 6b4 + 6b2 - 6) * q^38 + (-3b18 + 3b14 + 4b7 + 4b3) * q^41 + (-5b18 - 5b12 + 3b7 - 3b1) * q^43 + (-2b17 - 6b6) * q^44 + (8b19 - 4b17 + 4b15 - 4b13 + 4b11 - 4b10 - 4b9 + 4b8 + 4b7 - 4b5 + 4b3 - 4b1) * q^48 - 7b4 q^49 + 5b19 q^50 + (-b19 - 7b18 - 5b17 + 7b16 - 7b14 + 7b12 + 5b11 - 7b10 + b9 + 14b8 - 6b6 + 7b4 - 7b2 + 8) q^51 + (-5b19 - 3b17 + 3b15 + 5b13 + 2b8 - 6b6 - 6b4 + 2b2) * q^54 + (-5b16 - 7b14 + 2b13 - 4b11 + 4b5 - 2b3 + 7b2 + 5) q^57 + (3b18 - 3b16 + 5b7 + 5b5) * q^59 - 8b8 q^64 + (5b13 - b11 + 4b2 + 8) * q^66 + 14b16 q^67 + (6b18 + 4b17 - 6b16 + 6b14 - 6b12 + 6b10 + 4b9 - 6b8 + 12b6 - 6b4 + 6b2 - 6) q^68 + (-2b17 - 6b15 - 2b13 + 8b6 - 8b2) q^72 + (-6b17 + 6b15 - 6b13 + 6b11 - b10 - 6b9 - b8 + 6b7 - 6b5 + 6b3 - 6b1) q^73 + (-5b15 - 5b13 + 5b4 - 5b2) * q^75 + (2b14 + 6b11 + 6b3 + 2) q^76 + (4b18 + 3b16 + 4b15 - 3b14 + 6b13 + 3b12 - 3b10 - 6b9 + 3b8 - 4b7 - 3b6 - 4b4 + 12) * q^81 + (-3b19 - 8b12 - 8b8 - 3b1) * q^82 - 2b13 q^83 + (5b17 - 6b12 + 6b6 - 5b1) * q^86 + (6b11 - 4) q^88 + (9b16 + 9b12 - 2b5 + 2b1) * q^89 + (4b15 - 4b13 + 8b4 + 8b2) * q^96 + (6b19 + 5b18 - 6b17 - 5b16 + 6b15 + 5b14 - 6b13 - 5b12 + 6b11 - 12b9 - 5b8 + 6b7 + 5b6 - 6b5 - 5b4 + 6b3 + 5b2 - 6b1 - 5) * q^97 + 7b9 q^98 + (8b18 + 8b16 - b14 + b12 - b10 - 7b9 + b8 - 3b7 - b6 + 5b5 + b4 - b2 + 1) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{3} + 4 q^{4} + 64 q^{9}+O(q^{10}) \) 20 * q - 4 * q^3 + 4 * q^4 + 64 * q^9	\( 20 q - 4 q^{3} + 4 q^{4} + 64 q^{9} + 6 q^{11} - 36 q^{12} - 8 q^{16} - 4 q^{22} - 10 q^{25} + 8 q^{27} + 12 q^{33} + 16 q^{34} + 4 q^{36} - 24 q^{38} - 12 q^{44} - 16 q^{48} + 14 q^{49} + 22 q^{51} + 110 q^{57} + 12 q^{59} + 16 q^{64} + 168 q^{66} - 28 q^{67} - 20 q^{75} + 44 q^{76} + 220 q^{81} + 32 q^{82} + 24 q^{86} - 80 q^{88} - 36 q^{89} - 20 q^{97} + 6 q^{99}+O(q^{100}) \) 20 * q - 4 * q^3 + 4 * q^4 + 64 * q^9 + 6 * q^11 - 36 * q^12 - 8 * q^16 - 4 * q^22 - 10 * q^25 + 8 * q^27 + 12 * q^33 + 16 * q^34 + 4 * q^36 - 24 * q^38 - 12 * q^44 - 16 * q^48 + 14 * q^49 + 22 * q^51 + 110 * q^57 + 12 * q^59 + 16 * q^64 + 168 * q^66 - 28 * q^67 - 20 * q^75 + 44 * q^76 + 220 * q^81 + 32 * q^82 + 24 * q^86 - 80 * q^88 - 36 * q^89 - 20 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	968.2.w.a

Level	$968$
Weight	$2$
Character orbit	968.w
Analytic conductor	$7.730$
Analytic rank	$0$
Dimension	$20$
CM discriminant	-8
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(7.72951891566\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(2\) over \(\Q(\zeta_{22})\)
Coefficient field:	20.0.5969915757478328440239161344.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - 2x^{18} + 4x^{16} - 8x^{14} + 16x^{12} - 32x^{10} + 64x^{8} - 128x^{6} + 256x^{4} - 512x^{2} + 1024 \) x^20 - 2x^18 + 4x^16 - 8x^14 + 16x^12 - 32x^10 + 64x^8 - 128x^6 + 256x^4 - 512*x^2 + 1024
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{22}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 2 \) (v^2) / 2
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} ) / 2 \) (v^3) / 2
\(\beta_{4}\)	\(=\)	\( ( \nu^{4} ) / 4 \) (v^4) / 4
\(\beta_{5}\)	\(=\)	\( ( \nu^{5} ) / 4 \) (v^5) / 4
\(\beta_{6}\)	\(=\)	\( ( \nu^{6} ) / 8 \) (v^6) / 8
\(\beta_{7}\)	\(=\)	\( ( \nu^{7} ) / 8 \) (v^7) / 8
\(\beta_{8}\)	\(=\)	\( ( \nu^{8} ) / 16 \) (v^8) / 16
\(\beta_{9}\)	\(=\)	\( ( \nu^{9} ) / 16 \) (v^9) / 16
\(\beta_{10}\)	\(=\)	\( ( \nu^{10} ) / 32 \) (v^10) / 32
\(\beta_{11}\)	\(=\)	\( ( \nu^{11} ) / 32 \) (v^11) / 32
\(\beta_{12}\)	\(=\)	\( ( \nu^{12} ) / 64 \) (v^12) / 64
\(\beta_{13}\)	\(=\)	\( ( \nu^{13} ) / 64 \) (v^13) / 64
\(\beta_{14}\)	\(=\)	\( ( \nu^{14} ) / 128 \) (v^14) / 128
\(\beta_{15}\)	\(=\)	\( ( \nu^{15} ) / 128 \) (v^15) / 128
\(\beta_{16}\)	\(=\)	\( ( \nu^{16} ) / 256 \) (v^16) / 256
\(\beta_{17}\)	\(=\)	\( ( \nu^{17} ) / 256 \) (v^17) / 256
\(\beta_{18}\)	\(=\)	\( ( \nu^{18} ) / 512 \) (v^18) / 512
\(\beta_{19}\)	\(=\)	\( ( \nu^{19} ) / 512 \) (v^19) / 512

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( 2\beta_{2} \) 2*b2
\(\nu^{3}\)	\(=\)	\( 2\beta_{3} \) 2*b3
\(\nu^{4}\)	\(=\)	\( 4\beta_{4} \) 4*b4
\(\nu^{5}\)	\(=\)	\( 4\beta_{5} \) 4*b5
\(\nu^{6}\)	\(=\)	\( 8\beta_{6} \) 8*b6
\(\nu^{7}\)	\(=\)	\( 8\beta_{7} \) 8*b7
\(\nu^{8}\)	\(=\)	\( 16\beta_{8} \) 16*b8
\(\nu^{9}\)	\(=\)	\( 16\beta_{9} \) 16*b9
\(\nu^{10}\)	\(=\)	\( 32\beta_{10} \) 32*b10
\(\nu^{11}\)	\(=\)	\( 32\beta_{11} \) 32*b11
\(\nu^{12}\)	\(=\)	\( 64\beta_{12} \) 64*b12
\(\nu^{13}\)	\(=\)	\( 64\beta_{13} \) 64*b13
\(\nu^{14}\)	\(=\)	\( 128\beta_{14} \) 128*b14
\(\nu^{15}\)	\(=\)	\( 128\beta_{15} \) 128*b15
\(\nu^{16}\)	\(=\)	\( 256\beta_{16} \) 256*b16
\(\nu^{17}\)	\(=\)	\( 256\beta_{17} \) 256*b17
\(\nu^{18}\)	\(=\)	\( 512\beta_{18} \) 512*b18
\(\nu^{19}\)	\(=\)	\( 512\beta_{19} \) 512*b19

\(n\)	\(485\)	\(727\)	\(849\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(\beta_{10}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 43.2
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
121.f	odd	22	1	inner
968.w	even	22	1	inner

$p$	$F_p(T)$
$2$	\( T^{20} - 2 T^{18} + \cdots + 1024 \) T^20 - 2T^18 + 4T^16 - 8T^14 + 16T^12 - 32T^10 + 64T^8 - 128T^6 + 256T^4 - 512*T^2 + 1024
$3$	\( (T^{10} + 2 T^{9} + \cdots - 197)^{2} \) (T^10 + 2T^9 - 29T^8 - 58T^7 + 280T^6 + 560T^5 - 959T^4 - 1918T^3 + 619T^2 + 1238*T - 197)^2
$5$	\( T^{20} \) T^20
$7$	\( T^{20} \) T^20
$11$	\( T^{20} + \cdots + 25937424601 \) T^20 - 6T^19 + 25T^18 - 84T^17 + 229T^16 - 450T^15 + 181T^14 + 3864T^13 - 25175T^12 + 108546T^11 - 374351T^10 + 1194006T^9 - 3046175T^8 + 5142984T^7 + 2650021T^6 - 72472950T^5 + 405687469T^4 - 1636922364T^3 + 5358972025T^2 - 14147686146*T + 25937424601
$13$	\( T^{20} \) T^20
$17$	\( T^{20} + \cdots + 3433916604889 \) T^20 - 32T^18 + 1024T^16 + 38148T^15 - 32768T^14 - 362406T^13 + 725814T^12 + 15745026T^11 + 665822202T^10 - 503840832T^9 + 13498894740T^8 - 22563018324T^7 + 208019437781T^6 + 252217652874T^5 - 406182351805T^4 + 2296155920916T^3 + 9592400482523T^2 + 10214908786038*T + 3433916604889
$19$	\( T^{20} + \cdots + 4436419650961 \) T^20 - 72T^18 + 5184T^16 + 15884T^15 - 373248T^14 + 119130T^13 + 26355118T^12 - 21059258T^11 - 747876634T^10 + 1516266576T^9 + 12785055684T^8 - 18279593948T^7 - 67083626585T^6 - 716517657350T^5 + 677752084123T^4 + 4736282735756T^3 + 10088346279009T^2 + 6038838216422*T + 4436419650961
$23$	\( T^{20} \) T^20
$29$	\( T^{20} \) T^20
$31$	\( T^{20} \) T^20
$37$	\( T^{20} \) T^20
$41$	\( T^{20} + \cdots + 63\!\cdots\!81 \) T^20 - 128T^18 + 16384T^16 - 2659098T^14 - 22743930T^13 + 96246362T^12 + 1881200376T^11 - 14168186042T^10 - 240793648128T^9 + 2388290426537T^8 + 67410133798194T^7 + 605685301233303T^6 + 3016799343577206T^5 + 9319747461575747T^4 + 18196100913852906T^3 + 21172301802645854T^2 + 12331667893973736T + 6321735306254281
$43$	\( T^{20} + \cdots + 50\!\cdots\!49 \) T^20 - 72T^18 + 11806T^16 + 137170T^15 - 850032T^14 - 9876240T^13 + 332422869T^12 + 6884382560T^11 - 7670100811T^10 - 495675544320T^9 + 1050311863276T^8 + 15970294962490T^7 - 55543750969396T^6 - 1004923663401790T^5 + 9218028387851452T^4 - 31134427119164820T^3 + 61079949328090471T^2 - 75895644240419190*T + 50959927783705249
$47$	\( T^{20} \) T^20
$53$	\( T^{20} \) T^20
$59$	\( T^{20} + \cdots + 30\!\cdots\!01 \) T^20 - 12T^19 + 108T^18 - 864T^17 + 59698T^16 - 1234326T^15 + 12662784T^14 - 107517672T^13 + 2282326005T^12 - 41540390076T^11 + 514503931689T^10 - 4657599841944T^9 + 51556014351880T^8 - 282685548869142T^7 + 881510312673020T^6 - 4160685897180642T^5 + 83496564852543472T^4 + 632615961157305324T^3 + 1537604079172580167T^2 + 2384565518437394970*T + 3060544858646807401
$61$	\( T^{20} \) T^20
$67$	\( (T^{10} + \cdots + 289254654976)^{2} \) (T^10 + 14T^9 + 196T^8 + 2744T^7 + 38416T^6 + 537824T^5 + 7529536T^4 + 105413504T^3 + 1475789056T^2 + 20661046784*T + 289254654976)^2
$71$	\( T^{20} \) T^20
$73$	\( T^{20} + \cdots + 33\!\cdots\!09 \) T^20 - 288T^18 + 82944T^16 + 234476T^15 - 23887872T^14 + 8089422T^13 + 6403420534T^12 - 6236841578T^11 - 885176515222T^10 + 1796210374464T^9 + 58944465469140T^8 - 170927177268188T^7 - 1785268572446531T^6 - 5063874621171938T^5 + 70029439884545155T^4 + 355326430447365212T^3 + 1680641425443521667T^2 - 360889775597097406*T + 333672883023928009
$79$	\( T^{20} \) T^20
$83$	\( T^{20} + \cdots + 1073741824 \) T^20 - 8T^18 + 64T^16 - 512T^14 + 4096T^12 - 32768T^10 + 262144T^8 - 2097152T^6 + 16777216T^4 - 134217728*T^2 + 1073741824
$89$	\( T^{20} + \cdots + 32\!\cdots\!21 \) T^20 + 36T^19 + 972T^18 + 5706T^17 - 109512T^16 - 5791176T^15 - 49535821T^14 + 85985046T^13 + 19145067660T^12 + 235492917894T^11 + 1889879570436T^10 - 8057857287564T^9 - 163895228864971T^8 - 1424647385834976T^7 + 5350056310547321T^6 + 76172695775462808T^5 + 282637070337935146T^4 - 1996797636732317274T^3 + 5384883609493166432T^2 - 9861925928770919988*T + 32802211690363323121
$97$	\( T^{20} + \cdots + 48\!\cdots\!29 \) T^20 + 20T^19 + 300T^18 + 4000T^17 + 50000T^16 + 600000T^15 + 17650794T^14 - 208954270T^13 + 661762374T^12 - 1245283720T^11 - 8476651854T^10 - 184766518540T^9 + 97941947411009T^8 - 7945785433902170T^7 + 212382841724860803T^6 - 2463395964397386310T^5 + 15727147143360669791T^4 - 55815206769859692390T^3 + 103869305724658775414T^2 - 47958633336703040580*T + 48833439206436284929
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