LMFDB - Newform orbit 1020.2.bd.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1020,2,Mod(361,1020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1020, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))

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

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

chi := DirichletCharacter("1020.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1020 = 2^{2} \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1020.bd (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$8.14474100617$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 20x^{10} + 152x^{8} + 558x^{6} + 1032x^{4} + 900x^{2} + 289 $ x^12 + 20x^10 + 152x^8 + 558x^6 + 1032x^4 + 900*x^2 + 289
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_1 q^{3} + \beta_1 q^{5} - \beta_{9} q^{7} - \beta_{3} q^{9}+O(q^{10}) $ q + b1 * q^3 + b1 * q^5 - b9 * q^7 - b3 * q^9	$ q + \beta_1 q^{3} + \beta_1 q^{5} - \beta_{9} q^{7} - \beta_{3} q^{9} + ( - \beta_{11} - \beta_{9} + \cdots + \beta_{4}) q^{11}+ \cdots + ( - \beta_{11} + \beta_{7} + \cdots - \beta_{4}) q^{99}+O(q^{100}) $ q + b1 * q^3 + b1 * q^5 - b9 * q^7 - b3 * q^9 + (-b11 - b9 + 2b6 - b5 + b4) q^11 + (-b11 + b8 + b6 - b5 - 1) * q^13 - b3 * q^15 + (-b10 + b7 + b6 - b5 - b4 + b3 - b1) * q^17 + (-b10 + b9 + b7 + b6 - b5 + b4 - b3) * q^19 + b11 * q^21 + (2b11 + b10 - 2b9 - b8 - b6 + 2b5 - b4 + b2) q^23 - b3 * q^25 + b2 * q^27 + (b10 + b8 - 3b7 + b4 - b3 + 1) q^29 + (-b11 + b10 + b8 - 2b7 - b3 + b1 + 1) q^31 + (-b11 + b9 + 2b8 - b7) q^33 + b11 * q^35 + (b11 + b10 + b8 - b7 + b5 + 2b4 - b3 - 2b1 + 1) * q^37 + (-b11 + b10 + b8 - b1) * q^39 + (b11 + 2b10 - 3b9 - 2b8 + 2b6 + b5 + b4 + 2b3 + 2) q^41 + (2b9 + 2b7 - 2b6 - 2b4 + 2b3 + 2b2 + 2b1) q^43 + b2 * q^45 + (-2b11 + b9 - b8 - b7 + b6 - b5 - b2 + b1 + 3) q^47 + (b10 - b9 - b7 + 2b5 - b4 - 2b3 + b2 + b1) * q^49 + (-b11 + b10 - b9 + b8 + b6 + b4 + b3 - b2) * q^51 + (b10 - b9 - b7 + 2b6 + 2b5 + b4 - 3b3 + 2b2 + 2b1) q^53 + (-b11 + b9 + 2b8 - b7) q^55 + (-2b11 - b10 + b9 + b8 + b6 - 2b5 + b4 + b2) * q^57 + (-b10 + 2b6 - 3b5 + 4b4 + 3b3 - b2 - b1) * q^59 + (b11 + b10 - 2b9 - b8 + 3b6 + b5 - b3 + 5b2 - 1) q^61 + b7 * q^63 + (-b11 + b10 + b8 - b1) * q^65 + (3b11 - b9 - 3b8 + b7 - b6 + b5 - 4b2 + 4b1 - 5) * q^67 + (3b11 - b9 - b8 + b7 - b6 + b5 - 1) q^69 + (4b11 - 2b10 - 2b8 - 2b7 + 2b4 - 2b3 - 2b1 + 2) q^71 + (2b10 + 2b8 + b7 - 2b5 + 2b4 + 2b3 - 2) q^73 + b2 * q^75 + (3b10 - 2b9 - 2b7 + b6 + b5 + 3b4 + b3) * q^77 + (2b11 - b10 + b8 - 2b6 + 2b5 - 3b4 + 4b2) q^79 - q^81 + (b9 + b7 - 4b3 - 3b2 - 3b1) q^83 + (-b11 + b10 - b9 + b8 + b6 + b4 + b3 - b2) * q^85 + (b9 + b7 - b6 + 2b5 - 3b4 + b2 + b1) * q^87 + (b11 - 3b8 - 3b2 + 3b1 + 1) q^89 + (b11 + b10 - b8 - b6 + b5 - 3b2) q^91 + (b10 - b6 + 2b5 - 2b4 - b3 + b2 + b1) * q^93 + (-2b11 - b10 + b9 + b8 + b6 - 2b5 + b4 + b2) * q^95 + (-2b7 - 4b3 + 2b1 + 4) q^97 + (-b11 + b7 - b5 - b4) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{7}+O(q^{10}) $ 12 * q + 4 * q^7	$ 12 q + 4 q^{7} + 4 q^{11} - 8 q^{13} + 4 q^{17} + 4 q^{23} + 4 q^{29} + 8 q^{31} + 12 q^{37} + 4 q^{39} + 28 q^{41} + 24 q^{47} + 8 q^{51} - 8 q^{61} + 4 q^{63} + 4 q^{65} - 64 q^{67} - 8 q^{69} + 8 q^{71} - 12 q^{73} + 4 q^{79} - 12 q^{81} + 8 q^{85} - 4 q^{91} + 40 q^{97} + 4 q^{99}+O(q^{100}) $ 12 * q + 4 * q^7 + 4 * q^11 - 8 * q^13 + 4 * q^17 + 4 * q^23 + 4 * q^29 + 8 * q^31 + 12 * q^37 + 4 * q^39 + 28 * q^41 + 24 * q^47 + 8 * q^51 - 8 * q^61 + 4 * q^63 + 4 * q^65 - 64 * q^67 - 8 * q^69 + 8 * q^71 - 12 * q^73 + 4 * q^79 - 12 * q^81 + 8 * q^85 - 4 * q^91 + 40 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} + 20x^{10} + 152x^{8} + 558x^{6} + 1032x^{4} + 900x^{2} + 289 $ x^12 + 20*x^10 + 152*x^8 + 558*x^6 + 1032*x^4 + 900*x^2 + 289 :

$\beta_{1}$	$=$	$ ( 18 \nu^{11} + 17 \nu^{10} + 309 \nu^{9} + 306 \nu^{8} + 1869 \nu^{7} + 2023 \nu^{6} + 4842 \nu^{5} + \cdots + 4267 ) / 408 $ (18v^11 + 17v^10 + 309v^9 + 306v^8 + 1869v^7 + 2023v^6 + 4842v^5 + 6205v^4 + 5061v^3 + 8806v^2 + 1461*v + 4267) / 408
$\beta_{2}$	$=$	$ ( 18 \nu^{11} - 17 \nu^{10} + 309 \nu^{9} - 306 \nu^{8} + 1869 \nu^{7} - 2023 \nu^{6} + 4842 \nu^{5} + \cdots - 4267 ) / 408 $ (18v^11 - 17v^10 + 309v^9 - 306v^8 + 1869v^7 - 2023v^6 + 4842v^5 - 6205v^4 + 5061v^3 - 8806v^2 + 1461*v - 4267) / 408
$\beta_{3}$	$=$	$ ( 10\nu^{11} + 183\nu^{9} + 1214\nu^{7} + 3608\nu^{5} + 4727\nu^{3} + 2132\nu ) / 102 $ (10v^11 + 183v^9 + 1214v^7 + 3608v^5 + 4727v^3 + 2132v) / 102
$\beta_{4}$	$=$	$ ( 18 \nu^{11} + 17 \nu^{10} + 309 \nu^{9} + 408 \nu^{8} + 1869 \nu^{7} + 3553 \nu^{6} + 4842 \nu^{5} + \cdots + 10693 ) / 408 $ (18v^11 + 17v^10 + 309v^9 + 408v^8 + 1869v^7 + 3553v^6 + 4842v^5 + 13549v^4 + 5061v^3 + 21556v^2 + 1869*v + 10693) / 408
$\beta_{5}$	$=$	$ ( \nu^{10} + 3 \nu^{9} + 24 \nu^{8} + 45 \nu^{7} + 209 \nu^{6} + 216 \nu^{5} + 797 \nu^{4} + 363 \nu^{3} + \cdots + 629 ) / 24 $ (v^10 + 3v^9 + 24v^8 + 45v^7 + 209v^6 + 216v^5 + 797v^4 + 363v^3 + 1268v^2 + 141*v + 629) / 24
$\beta_{6}$	$=$	$ ( - \nu^{10} + 3 \nu^{9} - 24 \nu^{8} + 45 \nu^{7} - 209 \nu^{6} + 216 \nu^{5} - 797 \nu^{4} + \cdots - 629 ) / 24 $ (-v^10 + 3v^9 - 24v^8 + 45v^7 - 209v^6 + 216v^5 - 797v^4 + 363v^3 - 1268v^2 + 141*v - 629) / 24
$\beta_{7}$	$=$	$ ( 14 \nu^{11} + 51 \nu^{10} + 297 \nu^{9} + 918 \nu^{8} + 2383 \nu^{7} + 5967 \nu^{6} + 8866 \nu^{5} + \cdots + 8007 ) / 408 $ (14v^11 + 51v^10 + 297v^9 + 918v^8 + 2383v^7 + 5967v^6 + 8866v^5 + 17187v^4 + 14533v^3 + 20910v^2 + 7483*v + 8007) / 408
$\beta_{8}$	$=$	$ ( -\nu^{10} - 18\nu^{8} - 116\nu^{6} - 329\nu^{4} - 410\nu^{2} - 188 ) / 6 $ (-v^10 - 18v^8 - 116v^6 - 329v^4 - 410v^2 - 188) / 6
$\beta_{9}$	$=$	$ ( 14 \nu^{11} - 51 \nu^{10} + 297 \nu^{9} - 918 \nu^{8} + 2383 \nu^{7} - 5967 \nu^{6} + 8866 \nu^{5} + \cdots - 8007 ) / 408 $ (14v^11 - 51v^10 + 297v^9 - 918v^8 + 2383v^7 - 5967v^6 + 8866v^5 - 17187v^4 + 14533v^3 - 20910v^2 + 7483*v - 8007) / 408
$\beta_{10}$	$=$	$ ( 2 \nu^{11} - \nu^{10} + 39 \nu^{9} - 24 \nu^{8} + 283 \nu^{7} - 209 \nu^{6} + 946 \nu^{5} - 797 \nu^{4} + \cdots - 629 ) / 24 $ (2v^11 - v^10 + 39v^9 - 24v^8 + 283v^7 - 209v^6 + 946v^5 - 797v^4 + 1423v^3 - 1268v^2 + 739v - 629) / 24
$\beta_{11}$	$=$	$ ( -2\nu^{10} - 39\nu^{8} - 277\nu^{6} - 874\nu^{4} - 1183\nu^{2} - 529 ) / 12 $ (-2v^10 - 39v^8 - 277v^6 - 874v^4 - 1183*v^2 - 529) / 12

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

$\nu$	$=$	$ ( \beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{2} - \beta_1 ) / 2 $ (b6 - b5 + 2*b4 - b2 - b1) / 2
$\nu^{2}$	$=$	$ ( 2\beta_{11} - 2\beta_{8} - \beta_{6} + \beta_{5} + \beta_{2} - \beta _1 - 6 ) / 2 $ (2b11 - 2b8 - b6 + b5 + b2 - b1 - 6) / 2
$\nu^{3}$	$=$	$ 2\beta_{10} - \beta_{9} - \beta_{7} - 3\beta_{6} + 3\beta_{5} - 4\beta_{4} - \beta_{3} + 2\beta_{2} + 2\beta_1 $ 2b10 - b9 - b7 - 3b6 + 3b5 - 4b4 - b3 + 2b2 + 2b1
$\nu^{4}$	$=$	$ ( -18\beta_{11} + 2\beta_{9} + 16\beta_{8} - 2\beta_{7} + 9\beta_{6} - 9\beta_{5} - 11\beta_{2} + 11\beta _1 + 28 ) / 2 $ (-18b11 + 2b9 + 16b8 - 2b7 + 9b6 - 9b5 - 11b2 + 11b1 + 28) / 2
$\nu^{5}$	$=$	$ ( - 38 \beta_{10} + 18 \beta_{9} + 18 \beta_{7} + 37 \beta_{6} - 41 \beta_{5} + 40 \beta_{4} + \cdots - 27 \beta_1 ) / 2 $ (-38b10 + 18b9 + 18b7 + 37b6 - 41b5 + 40b4 + 26b3 - 27b2 - 27*b1) / 2
$\nu^{6}$	$=$	$ 72\beta_{11} - 12\beta_{9} - 58\beta_{8} + 12\beta_{7} - 36\beta_{6} + 36\beta_{5} + 44\beta_{2} - 44\beta _1 - 81 $ 72b11 - 12b9 - 58b8 + 12b7 - 36b6 + 36b5 + 44b2 - 44b1 - 81
$\nu^{7}$	$=$	$ ( 300 \beta_{10} - 132 \beta_{9} - 132 \beta_{7} - 245 \beta_{6} + 285 \beta_{5} - 230 \beta_{4} + \cdots + 201 \beta_1 ) / 2 $ (300b10 - 132b9 - 132b7 - 245b6 + 285b5 - 230b4 - 240b3 + 201b2 + 201*b1) / 2
$\nu^{8}$	$=$	$ ( - 1114 \beta_{11} + 216 \beta_{9} + 838 \beta_{8} - 216 \beta_{7} + 553 \beta_{6} - 553 \beta_{5} + \cdots + 1038 ) / 2 $ (-1114b11 + 216b9 + 838b8 - 216b7 + 553b6 - 553b5 - 649b2 + 649b1 + 1038) / 2
$\nu^{9}$	$=$	$ - 1124 \beta_{10} + 463 \beta_{9} + 463 \beta_{7} + 849 \beta_{6} - 997 \beta_{5} + 722 \beta_{4} + \cdots - 754 \beta_1 $ -1124b10 + 463b9 + 463b7 + 849b6 - 997b5 + 722b4 + 985b3 - 754b2 - 754*b1
$\nu^{10}$	$=$	$ ( 8450 \beta_{11} - 1762 \beta_{9} - 6084 \beta_{8} + 1762 \beta_{7} - 4153 \beta_{6} + 4153 \beta_{5} + \cdots - 7020 ) / 2 $ (8450b11 - 1762b9 - 6084b8 + 1762b7 - 4153b6 + 4153b5 + 4683b2 - 4683b1 - 7020) / 2
$\nu^{11}$	$=$	$ ( 16538 \beta_{10} - 6470 \beta_{9} - 6470 \beta_{7} - 12057 \beta_{6} + 14061 \beta_{5} + \cdots + 11259 \beta_1 ) / 2 $ (16538b10 - 6470b9 - 6470b7 - 12057b6 + 14061b5 - 9580b4 - 15330b3 + 11259b2 + 11259*b1) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1020\mathbb{Z}\right)^\times$.

$n$	$241$	$341$	$511$	$817$
$\chi(n)$	$-\beta_{3}$	$1$	$1$	$1$

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} $

361.1

	−	1.10882i
	−	1.86959i
		1.56420i
		0.891578i
		2.70027i
	−	2.17763i
		1.10882i
		1.86959i
	−	1.56420i
	−	0.891578i
	−	2.70027i
		2.17763i

−0.707107

0.707107i

−0.707107

0.707107i

−2.09942

−

2.09942i

−

1.00000i

361.2

−0.707107

0.707107i

−0.707107

0.707107i

1.23078

1.23078i

−

1.00000i

361.3

−0.707107

0.707107i

−0.707107

0.707107i

1.86864

1.86864i

−

1.00000i

361.4

0.707107

−

0.707107i

0.707107

−

0.707107i

−0.983403

−

0.983403i

−

1.00000i

361.5

0.707107

−

0.707107i

0.707107

−

0.707107i

−0.359550

−

0.359550i

−

1.00000i

361.6

0.707107

−

0.707107i

0.707107

−

0.707107i

2.34295

2.34295i

−

1.00000i

421.1

−0.707107

−

0.707107i

−0.707107

−

0.707107i

−2.09942

2.09942i

1.00000i

421.2

−0.707107

−

0.707107i

−0.707107

−

0.707107i

1.23078

−

1.23078i

1.00000i

421.3

−0.707107

−

0.707107i

−0.707107

−

0.707107i

1.86864

−

1.86864i

1.00000i

421.4

0.707107

0.707107i

0.707107

0.707107i

−0.983403

0.983403i

1.00000i

421.5

0.707107

0.707107i

0.707107

0.707107i

−0.359550

0.359550i

1.00000i

421.6

0.707107

0.707107i

0.707107

0.707107i

2.34295

−

2.34295i

1.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.c	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1020.2.bd.b	✓	12
3.b	odd	2	1		3060.2.be.c		12
17.c	even	4	1	inner	1020.2.bd.b	✓	12
51.f	odd	4	1		3060.2.be.c		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1020.2.bd.b	✓	12	1.a	even	1	1	trivial
1020.2.bd.b	✓	12	17.c	even	4	1	inner
3060.2.be.c		12	3.b	odd	2	1
3060.2.be.c		12	51.f	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{12} - 4 T_{7}^{11} + 8 T_{7}^{10} + 4 T_{7}^{9} + 82 T_{7}^{8} - 324 T_{7}^{7} + 648 T_{7}^{6} + \cdots + 1024 $ T7^12 - 4*T7^11 + 8*T7^10 + 4*T7^9 + 82*T7^8 - 324*T7^7 + 648*T7^6 + 276*T7^5 + 193*T7^4 - 1072*T7^3 + 3200*T7^2 + 2560*T7 + 1024 acting on $S_{2}^{\mathrm{new}}(1020, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ (T^{4} + 1)^{3} $ (T^4 + 1)^3
$5$	$ (T^{4} + 1)^{3} $ (T^4 + 1)^3
$7$	$ T^{12} - 4 T^{11} + \cdots + 1024 $ T^12 - 4T^11 + 8T^10 + 4T^9 + 82T^8 - 324T^7 + 648T^6 + 276T^5 + 193T^4 - 1072T^3 + 3200T^2 + 2560*T + 1024
$11$	$ T^{12} - 4 T^{11} + \cdots + 16 $ T^12 - 4T^11 + 8T^10 + 60T^9 + 482T^8 - 1116T^7 + 2408T^6 + 19180T^5 + 73185T^4 + 6776T^3 + 288T^2 - 96*T + 16
$13$	$ (T^{6} + 4 T^{5} - 18 T^{4} + \cdots - 16)^{2} $ (T^6 + 4T^5 - 18T^4 - 64T^3 + 4T^2 + 48*T - 16)^2
$17$	$ T^{12} - 4 T^{11} + \cdots + 24137569 $ T^12 - 4T^11 - 8T^10 + 244T^9 - 673T^8 - 2224T^7 + 24624T^6 - 37808T^5 - 194497T^4 + 1198772T^3 - 668168T^2 - 5679428*T + 24137569
$19$	$ T^{12} + 108 T^{10} + \cdots + 929296 $ T^12 + 108T^10 + 3734T^8 + 52372T^6 + 326945T^4 + 911560*T^2 + 929296
$23$	$ T^{12} - 4 T^{11} + \cdots + 4096 $ T^12 - 4T^11 + 8T^10 + 96T^9 + 4524T^8 - 11744T^7 + 15392T^6 + 7968T^5 + 44304T^4 - 83968T^3 + 73728T^2 - 24576*T + 4096
$29$	$ T^{12} - 4 T^{11} + \cdots + 595984 $ T^12 - 4T^11 + 8T^10 + 28T^9 + 8226T^8 - 33564T^7 + 68840T^6 + 332140T^5 + 1516449T^4 - 1782824T^3 + 1292832T^2 + 1241376*T + 595984
$31$	$ T^{12} - 8 T^{11} + \cdots + 602176 $ T^12 - 8T^11 + 32T^10 + 200T^9 + 1040T^8 - 3712T^7 + 16416T^6 + 107008T^5 + 324928T^4 + 102336T^3 + 2048T^2 + 49664*T + 602176
$37$	$ T^{12} - 12 T^{11} + \cdots + 14227984 $ T^12 - 12T^11 + 72T^10 + 300T^9 + 1218T^8 - 7564T^7 + 48072T^6 + 212924T^5 + 497217T^4 - 284656T^3 + 1548800T^2 + 6638720*T + 14227984
$41$	$ T^{12} - 28 T^{11} + \cdots + 12418576 $ T^12 - 28T^11 + 392T^10 - 2844T^9 + 19826T^8 - 238676T^7 + 2955304T^6 - 21515468T^5 + 92809777T^4 - 197396872T^3 + 226930208T^2 - 75075296*T + 12418576
$43$	$ T^{12} + \cdots + 6338707456 $ T^12 + 280T^10 + 31568T^8 + 1835776T^6 + 58170368T^4 + 953991168*T^2 + 6338707456
$47$	$ (T^{6} - 12 T^{5} + \cdots + 104558)^{2} $ (T^6 - 12T^5 - 96T^4 + 1564T^3 - 1363T^2 - 36944*T + 104558)^2
$53$	$ T^{12} + \cdots + 509585476 $ T^12 + 304T^10 + 31610T^8 + 1371732T^6 + 25681705T^4 + 200872404*T^2 + 509585476
$59$	$ T^{12} + \cdots + 191988736 $ T^12 + 404T^10 + 50668T^8 + 2254416T^6 + 36989200T^4 + 214556416*T^2 + 191988736
$61$	$ T^{12} + \cdots + 3223514176 $ T^12 + 8T^11 + 32T^10 - 40T^9 + 35216T^8 + 308736T^7 + 1343776T^6 - 7884032T^5 + 98008384T^4 + 485631296T^3 + 1400686592T^2 - 3005040128*T + 3223514176
$67$	$ (T^{6} + 32 T^{5} + \cdots + 160528)^{2} $ (T^6 + 32T^5 + 266T^4 - 760T^3 - 15772T^2 - 19536*T + 160528)^2
$71$	$ T^{12} + \cdots + 144769024 $ T^12 - 8T^11 + 32T^10 + 704T^9 + 47056T^8 - 293760T^7 + 1092096T^6 + 21374976T^5 + 275405824T^4 - 269287424T^3 + 147095552T^2 + 206372864*T + 144769024
$73$	$ T^{12} + 12 T^{11} + \cdots + 74580496 $ T^12 + 12T^11 + 72T^10 - 916T^9 + 37650T^8 + 287924T^7 + 1163816T^6 - 3619716T^5 + 43094097T^4 + 227293016T^3 + 572843552T^2 + 292311328*T + 74580496
$79$	$ T^{12} + \cdots + 23839360000 $ T^12 - 4T^11 + 8T^10 + 424T^9 + 14308T^8 - 20896T^7 + 59008T^6 + 3165056T^5 + 60050816T^4 + 58851840T^3 + 204800T^2 - 98816000*T + 23839360000
$83$	$ T^{12} + 300 T^{10} + \cdots + 23970816 $ T^12 + 300T^10 + 27796T^8 + 829664T^6 + 9356992T^4 + 33882624*T^2 + 23970816
$89$	$ (T^{6} - 166 T^{4} + \cdots - 16)^{2} $ (T^6 - 166T^4 - 160T^3 + 972T^2 - 368T - 16)^2
$97$	$ T^{12} - 40 T^{11} + \cdots + 65536 $ T^12 - 40T^11 + 800T^10 - 9408T^9 + 69328T^8 - 294784T^7 + 584192T^6 + 247808T^5 + 879616T^4 - 2244608T^3 + 2654208T^2 - 589824*T + 65536
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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:	\( N \)	\(=\)	\( 1020 = 2^{2} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1020.bd (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + \beta_1 q^{5} - \beta_{9} q^{7} - \beta_{3} q^{9}+O(q^{10}) \) q + b1 * q^3 + b1 * q^5 - b9 * q^7 - b3 * q^9	\( q + \beta_1 q^{3} + \beta_1 q^{5} - \beta_{9} q^{7} - \beta_{3} q^{9} + ( - \beta_{11} - \beta_{9} + \cdots + \beta_{4}) q^{11}+ \cdots + ( - \beta_{11} + \beta_{7} + \cdots - \beta_{4}) q^{99}+O(q^{100}) \) q + b1 * q^3 + b1 * q^5 - b9 * q^7 - b3 * q^9 + (-b11 - b9 + 2b6 - b5 + b4) q^11 + (-b11 + b8 + b6 - b5 - 1) * q^13 - b3 * q^15 + (-b10 + b7 + b6 - b5 - b4 + b3 - b1) * q^17 + (-b10 + b9 + b7 + b6 - b5 + b4 - b3) * q^19 + b11 * q^21 + (2b11 + b10 - 2b9 - b8 - b6 + 2b5 - b4 + b2) q^23 - b3 * q^25 + b2 * q^27 + (b10 + b8 - 3b7 + b4 - b3 + 1) q^29 + (-b11 + b10 + b8 - 2b7 - b3 + b1 + 1) q^31 + (-b11 + b9 + 2b8 - b7) q^33 + b11 * q^35 + (b11 + b10 + b8 - b7 + b5 + 2b4 - b3 - 2b1 + 1) * q^37 + (-b11 + b10 + b8 - b1) * q^39 + (b11 + 2b10 - 3b9 - 2b8 + 2b6 + b5 + b4 + 2b3 + 2) q^41 + (2b9 + 2b7 - 2b6 - 2b4 + 2b3 + 2b2 + 2b1) q^43 + b2 * q^45 + (-2b11 + b9 - b8 - b7 + b6 - b5 - b2 + b1 + 3) q^47 + (b10 - b9 - b7 + 2b5 - b4 - 2b3 + b2 + b1) * q^49 + (-b11 + b10 - b9 + b8 + b6 + b4 + b3 - b2) * q^51 + (b10 - b9 - b7 + 2b6 + 2b5 + b4 - 3b3 + 2b2 + 2b1) q^53 + (-b11 + b9 + 2b8 - b7) q^55 + (-2b11 - b10 + b9 + b8 + b6 - 2b5 + b4 + b2) * q^57 + (-b10 + 2b6 - 3b5 + 4b4 + 3b3 - b2 - b1) * q^59 + (b11 + b10 - 2b9 - b8 + 3b6 + b5 - b3 + 5b2 - 1) q^61 + b7 * q^63 + (-b11 + b10 + b8 - b1) * q^65 + (3b11 - b9 - 3b8 + b7 - b6 + b5 - 4b2 + 4b1 - 5) * q^67 + (3b11 - b9 - b8 + b7 - b6 + b5 - 1) q^69 + (4b11 - 2b10 - 2b8 - 2b7 + 2b4 - 2b3 - 2b1 + 2) q^71 + (2b10 + 2b8 + b7 - 2b5 + 2b4 + 2b3 - 2) q^73 + b2 * q^75 + (3b10 - 2b9 - 2b7 + b6 + b5 + 3b4 + b3) * q^77 + (2b11 - b10 + b8 - 2b6 + 2b5 - 3b4 + 4b2) q^79 - q^81 + (b9 + b7 - 4b3 - 3b2 - 3b1) q^83 + (-b11 + b10 - b9 + b8 + b6 + b4 + b3 - b2) * q^85 + (b9 + b7 - b6 + 2b5 - 3b4 + b2 + b1) * q^87 + (b11 - 3b8 - 3b2 + 3b1 + 1) q^89 + (b11 + b10 - b8 - b6 + b5 - 3b2) q^91 + (b10 - b6 + 2b5 - 2b4 - b3 + b2 + b1) * q^93 + (-2b11 - b10 + b9 + b8 + b6 - 2b5 + b4 + b2) * q^95 + (-2b7 - 4b3 + 2b1 + 4) q^97 + (-b11 + b7 - b5 - b4) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{7}+O(q^{10}) \) 12 * q + 4 * q^7	\( 12 q + 4 q^{7} + 4 q^{11} - 8 q^{13} + 4 q^{17} + 4 q^{23} + 4 q^{29} + 8 q^{31} + 12 q^{37} + 4 q^{39} + 28 q^{41} + 24 q^{47} + 8 q^{51} - 8 q^{61} + 4 q^{63} + 4 q^{65} - 64 q^{67} - 8 q^{69} + 8 q^{71} - 12 q^{73} + 4 q^{79} - 12 q^{81} + 8 q^{85} - 4 q^{91} + 40 q^{97} + 4 q^{99}+O(q^{100}) \) 12 * q + 4 * q^7 + 4 * q^11 - 8 * q^13 + 4 * q^17 + 4 * q^23 + 4 * q^29 + 8 * q^31 + 12 * q^37 + 4 * q^39 + 28 * q^41 + 24 * q^47 + 8 * q^51 - 8 * q^61 + 4 * q^63 + 4 * q^65 - 64 * q^67 - 8 * q^69 + 8 * q^71 - 12 * q^73 + 4 * q^79 - 12 * q^81 + 8 * q^85 - 4 * q^91 + 40 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	1020.2.bd.b

Level	$1020$
Weight	$2$
Character orbit	1020.bd
Analytic conductor	$8.145$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(8.14474100617\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} + 20x^{10} + 152x^{8} + 558x^{6} + 1032x^{4} + 900x^{2} + 289 \) x^12 + 20x^10 + 152x^8 + 558x^6 + 1032x^4 + 900*x^2 + 289
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( 18 \nu^{11} + 17 \nu^{10} + 309 \nu^{9} + 306 \nu^{8} + 1869 \nu^{7} + 2023 \nu^{6} + 4842 \nu^{5} + \cdots + 4267 ) / 408 \) (18v^11 + 17v^10 + 309v^9 + 306v^8 + 1869v^7 + 2023v^6 + 4842v^5 + 6205v^4 + 5061v^3 + 8806v^2 + 1461*v + 4267) / 408
\(\beta_{2}\)	\(=\)	\( ( 18 \nu^{11} - 17 \nu^{10} + 309 \nu^{9} - 306 \nu^{8} + 1869 \nu^{7} - 2023 \nu^{6} + 4842 \nu^{5} + \cdots - 4267 ) / 408 \) (18v^11 - 17v^10 + 309v^9 - 306v^8 + 1869v^7 - 2023v^6 + 4842v^5 - 6205v^4 + 5061v^3 - 8806v^2 + 1461*v - 4267) / 408
\(\beta_{3}\)	\(=\)	\( ( 10\nu^{11} + 183\nu^{9} + 1214\nu^{7} + 3608\nu^{5} + 4727\nu^{3} + 2132\nu ) / 102 \) (10v^11 + 183v^9 + 1214v^7 + 3608v^5 + 4727v^3 + 2132v) / 102
\(\beta_{4}\)	\(=\)	\( ( 18 \nu^{11} + 17 \nu^{10} + 309 \nu^{9} + 408 \nu^{8} + 1869 \nu^{7} + 3553 \nu^{6} + 4842 \nu^{5} + \cdots + 10693 ) / 408 \) (18v^11 + 17v^10 + 309v^9 + 408v^8 + 1869v^7 + 3553v^6 + 4842v^5 + 13549v^4 + 5061v^3 + 21556v^2 + 1869*v + 10693) / 408
\(\beta_{5}\)	\(=\)	\( ( \nu^{10} + 3 \nu^{9} + 24 \nu^{8} + 45 \nu^{7} + 209 \nu^{6} + 216 \nu^{5} + 797 \nu^{4} + 363 \nu^{3} + \cdots + 629 ) / 24 \) (v^10 + 3v^9 + 24v^8 + 45v^7 + 209v^6 + 216v^5 + 797v^4 + 363v^3 + 1268v^2 + 141*v + 629) / 24
\(\beta_{6}\)	\(=\)	\( ( - \nu^{10} + 3 \nu^{9} - 24 \nu^{8} + 45 \nu^{7} - 209 \nu^{6} + 216 \nu^{5} - 797 \nu^{4} + \cdots - 629 ) / 24 \) (-v^10 + 3v^9 - 24v^8 + 45v^7 - 209v^6 + 216v^5 - 797v^4 + 363v^3 - 1268v^2 + 141*v - 629) / 24
\(\beta_{7}\)	\(=\)	\( ( 14 \nu^{11} + 51 \nu^{10} + 297 \nu^{9} + 918 \nu^{8} + 2383 \nu^{7} + 5967 \nu^{6} + 8866 \nu^{5} + \cdots + 8007 ) / 408 \) (14v^11 + 51v^10 + 297v^9 + 918v^8 + 2383v^7 + 5967v^6 + 8866v^5 + 17187v^4 + 14533v^3 + 20910v^2 + 7483*v + 8007) / 408
\(\beta_{8}\)	\(=\)	\( ( -\nu^{10} - 18\nu^{8} - 116\nu^{6} - 329\nu^{4} - 410\nu^{2} - 188 ) / 6 \) (-v^10 - 18v^8 - 116v^6 - 329v^4 - 410v^2 - 188) / 6
\(\beta_{9}\)	\(=\)	\( ( 14 \nu^{11} - 51 \nu^{10} + 297 \nu^{9} - 918 \nu^{8} + 2383 \nu^{7} - 5967 \nu^{6} + 8866 \nu^{5} + \cdots - 8007 ) / 408 \) (14v^11 - 51v^10 + 297v^9 - 918v^8 + 2383v^7 - 5967v^6 + 8866v^5 - 17187v^4 + 14533v^3 - 20910v^2 + 7483*v - 8007) / 408
\(\beta_{10}\)	\(=\)	\( ( 2 \nu^{11} - \nu^{10} + 39 \nu^{9} - 24 \nu^{8} + 283 \nu^{7} - 209 \nu^{6} + 946 \nu^{5} - 797 \nu^{4} + \cdots - 629 ) / 24 \) (2v^11 - v^10 + 39v^9 - 24v^8 + 283v^7 - 209v^6 + 946v^5 - 797v^4 + 1423v^3 - 1268v^2 + 739v - 629) / 24
\(\beta_{11}\)	\(=\)	\( ( -2\nu^{10} - 39\nu^{8} - 277\nu^{6} - 874\nu^{4} - 1183\nu^{2} - 529 ) / 12 \) (-2v^10 - 39v^8 - 277v^6 - 874v^4 - 1183*v^2 - 529) / 12

\(\nu\)	\(=\)	\( ( \beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{2} - \beta_1 ) / 2 \) (b6 - b5 + 2*b4 - b2 - b1) / 2
\(\nu^{2}\)	\(=\)	\( ( 2\beta_{11} - 2\beta_{8} - \beta_{6} + \beta_{5} + \beta_{2} - \beta _1 - 6 ) / 2 \) (2b11 - 2b8 - b6 + b5 + b2 - b1 - 6) / 2
\(\nu^{3}\)	\(=\)	\( 2\beta_{10} - \beta_{9} - \beta_{7} - 3\beta_{6} + 3\beta_{5} - 4\beta_{4} - \beta_{3} + 2\beta_{2} + 2\beta_1 \) 2b10 - b9 - b7 - 3b6 + 3b5 - 4b4 - b3 + 2b2 + 2b1
\(\nu^{4}\)	\(=\)	\( ( -18\beta_{11} + 2\beta_{9} + 16\beta_{8} - 2\beta_{7} + 9\beta_{6} - 9\beta_{5} - 11\beta_{2} + 11\beta _1 + 28 ) / 2 \) (-18b11 + 2b9 + 16b8 - 2b7 + 9b6 - 9b5 - 11b2 + 11b1 + 28) / 2
\(\nu^{5}\)	\(=\)	\( ( - 38 \beta_{10} + 18 \beta_{9} + 18 \beta_{7} + 37 \beta_{6} - 41 \beta_{5} + 40 \beta_{4} + \cdots - 27 \beta_1 ) / 2 \) (-38b10 + 18b9 + 18b7 + 37b6 - 41b5 + 40b4 + 26b3 - 27b2 - 27*b1) / 2
\(\nu^{6}\)	\(=\)	\( 72\beta_{11} - 12\beta_{9} - 58\beta_{8} + 12\beta_{7} - 36\beta_{6} + 36\beta_{5} + 44\beta_{2} - 44\beta _1 - 81 \) 72b11 - 12b9 - 58b8 + 12b7 - 36b6 + 36b5 + 44b2 - 44b1 - 81
\(\nu^{7}\)	\(=\)	\( ( 300 \beta_{10} - 132 \beta_{9} - 132 \beta_{7} - 245 \beta_{6} + 285 \beta_{5} - 230 \beta_{4} + \cdots + 201 \beta_1 ) / 2 \) (300b10 - 132b9 - 132b7 - 245b6 + 285b5 - 230b4 - 240b3 + 201b2 + 201*b1) / 2
\(\nu^{8}\)	\(=\)	\( ( - 1114 \beta_{11} + 216 \beta_{9} + 838 \beta_{8} - 216 \beta_{7} + 553 \beta_{6} - 553 \beta_{5} + \cdots + 1038 ) / 2 \) (-1114b11 + 216b9 + 838b8 - 216b7 + 553b6 - 553b5 - 649b2 + 649b1 + 1038) / 2
\(\nu^{9}\)	\(=\)	\( - 1124 \beta_{10} + 463 \beta_{9} + 463 \beta_{7} + 849 \beta_{6} - 997 \beta_{5} + 722 \beta_{4} + \cdots - 754 \beta_1 \) -1124b10 + 463b9 + 463b7 + 849b6 - 997b5 + 722b4 + 985b3 - 754b2 - 754*b1
\(\nu^{10}\)	\(=\)	\( ( 8450 \beta_{11} - 1762 \beta_{9} - 6084 \beta_{8} + 1762 \beta_{7} - 4153 \beta_{6} + 4153 \beta_{5} + \cdots - 7020 ) / 2 \) (8450b11 - 1762b9 - 6084b8 + 1762b7 - 4153b6 + 4153b5 + 4683b2 - 4683b1 - 7020) / 2
\(\nu^{11}\)	\(=\)	\( ( 16538 \beta_{10} - 6470 \beta_{9} - 6470 \beta_{7} - 12057 \beta_{6} + 14061 \beta_{5} + \cdots + 11259 \beta_1 ) / 2 \) (16538b10 - 6470b9 - 6470b7 - 12057b6 + 14061b5 - 9580b4 - 15330b3 + 11259b2 + 11259*b1) / 2

\(n\)	\(241\)	\(341\)	\(511\)	\(817\)
\(\chi(n)\)	\(-\beta_{3}\)	\(1\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 361.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( (T^{4} + 1)^{3} \) (T^4 + 1)^3
$5$	\( (T^{4} + 1)^{3} \) (T^4 + 1)^3
$7$	\( T^{12} - 4 T^{11} + \cdots + 1024 \) T^12 - 4T^11 + 8T^10 + 4T^9 + 82T^8 - 324T^7 + 648T^6 + 276T^5 + 193T^4 - 1072T^3 + 3200T^2 + 2560*T + 1024
$11$	\( T^{12} - 4 T^{11} + \cdots + 16 \) T^12 - 4T^11 + 8T^10 + 60T^9 + 482T^8 - 1116T^7 + 2408T^6 + 19180T^5 + 73185T^4 + 6776T^3 + 288T^2 - 96*T + 16
$13$	\( (T^{6} + 4 T^{5} - 18 T^{4} + \cdots - 16)^{2} \) (T^6 + 4T^5 - 18T^4 - 64T^3 + 4T^2 + 48*T - 16)^2
$17$	\( T^{12} - 4 T^{11} + \cdots + 24137569 \) T^12 - 4T^11 - 8T^10 + 244T^9 - 673T^8 - 2224T^7 + 24624T^6 - 37808T^5 - 194497T^4 + 1198772T^3 - 668168T^2 - 5679428*T + 24137569
$19$	\( T^{12} + 108 T^{10} + \cdots + 929296 \) T^12 + 108T^10 + 3734T^8 + 52372T^6 + 326945T^4 + 911560*T^2 + 929296
$23$	\( T^{12} - 4 T^{11} + \cdots + 4096 \) T^12 - 4T^11 + 8T^10 + 96T^9 + 4524T^8 - 11744T^7 + 15392T^6 + 7968T^5 + 44304T^4 - 83968T^3 + 73728T^2 - 24576*T + 4096
$29$	\( T^{12} - 4 T^{11} + \cdots + 595984 \) T^12 - 4T^11 + 8T^10 + 28T^9 + 8226T^8 - 33564T^7 + 68840T^6 + 332140T^5 + 1516449T^4 - 1782824T^3 + 1292832T^2 + 1241376*T + 595984
$31$	\( T^{12} - 8 T^{11} + \cdots + 602176 \) T^12 - 8T^11 + 32T^10 + 200T^9 + 1040T^8 - 3712T^7 + 16416T^6 + 107008T^5 + 324928T^4 + 102336T^3 + 2048T^2 + 49664*T + 602176
$37$	\( T^{12} - 12 T^{11} + \cdots + 14227984 \) T^12 - 12T^11 + 72T^10 + 300T^9 + 1218T^8 - 7564T^7 + 48072T^6 + 212924T^5 + 497217T^4 - 284656T^3 + 1548800T^2 + 6638720*T + 14227984
$41$	\( T^{12} - 28 T^{11} + \cdots + 12418576 \) T^12 - 28T^11 + 392T^10 - 2844T^9 + 19826T^8 - 238676T^7 + 2955304T^6 - 21515468T^5 + 92809777T^4 - 197396872T^3 + 226930208T^2 - 75075296*T + 12418576
$43$	\( T^{12} + \cdots + 6338707456 \) T^12 + 280T^10 + 31568T^8 + 1835776T^6 + 58170368T^4 + 953991168*T^2 + 6338707456
$47$	\( (T^{6} - 12 T^{5} + \cdots + 104558)^{2} \) (T^6 - 12T^5 - 96T^4 + 1564T^3 - 1363T^2 - 36944*T + 104558)^2
$53$	\( T^{12} + \cdots + 509585476 \) T^12 + 304T^10 + 31610T^8 + 1371732T^6 + 25681705T^4 + 200872404*T^2 + 509585476
$59$	\( T^{12} + \cdots + 191988736 \) T^12 + 404T^10 + 50668T^8 + 2254416T^6 + 36989200T^4 + 214556416*T^2 + 191988736
$61$	\( T^{12} + \cdots + 3223514176 \) T^12 + 8T^11 + 32T^10 - 40T^9 + 35216T^8 + 308736T^7 + 1343776T^6 - 7884032T^5 + 98008384T^4 + 485631296T^3 + 1400686592T^2 - 3005040128*T + 3223514176
$67$	\( (T^{6} + 32 T^{5} + \cdots + 160528)^{2} \) (T^6 + 32T^5 + 266T^4 - 760T^3 - 15772T^2 - 19536*T + 160528)^2
$71$	\( T^{12} + \cdots + 144769024 \) T^12 - 8T^11 + 32T^10 + 704T^9 + 47056T^8 - 293760T^7 + 1092096T^6 + 21374976T^5 + 275405824T^4 - 269287424T^3 + 147095552T^2 + 206372864*T + 144769024
$73$	\( T^{12} + 12 T^{11} + \cdots + 74580496 \) T^12 + 12T^11 + 72T^10 - 916T^9 + 37650T^8 + 287924T^7 + 1163816T^6 - 3619716T^5 + 43094097T^4 + 227293016T^3 + 572843552T^2 + 292311328*T + 74580496
$79$	\( T^{12} + \cdots + 23839360000 \) T^12 - 4T^11 + 8T^10 + 424T^9 + 14308T^8 - 20896T^7 + 59008T^6 + 3165056T^5 + 60050816T^4 + 58851840T^3 + 204800T^2 - 98816000*T + 23839360000
$83$	\( T^{12} + 300 T^{10} + \cdots + 23970816 \) T^12 + 300T^10 + 27796T^8 + 829664T^6 + 9356992T^4 + 33882624*T^2 + 23970816
$89$	\( (T^{6} - 166 T^{4} + \cdots - 16)^{2} \) (T^6 - 166T^4 - 160T^3 + 972T^2 - 368T - 16)^2
$97$	\( T^{12} - 40 T^{11} + \cdots + 65536 \) T^12 - 40T^11 + 800T^10 - 9408T^9 + 69328T^8 - 294784T^7 + 584192T^6 + 247808T^5 + 879616T^4 - 2244608T^3 + 2654208T^2 - 589824*T + 65536
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