LMFDB - Newform orbit 841.2.d.i

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [841,2,Mod(190,841)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("841.190"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(841, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 841 = 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	841.d (of order $7$, degree $6$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,1,-1,1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$6.71541880999$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{7})$
Coefficient field:	12.0.4413675765625.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + 2x^{10} - 3x^{9} + 5x^{8} - 8x^{7} + 13x^{6} + 8x^{5} + 5x^{4} + 3x^{3} + 2x^{2} + x + 1 $ x^12 - x^11 + 2x^10 - 3x^9 + 5x^8 - 8x^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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

$f(q)$	$=$	$ q + \beta_1 q^{2} + \beta_{10} q^{3} + ( - \beta_{5} + \beta_{4}) q^{4} + (2 \beta_{3} + 3 \beta_1) q^{5} + (\beta_{11} + \beta_{10} + \cdots - \beta_1) q^{6} + ( - \beta_{11} + 2 \beta_{10} + \cdots + 1) q^{7}+ \cdots + (\beta_{2} + 3) q^{99}+O(q^{100}) $ q + b1 * q^2 + b10 * q^3 + (-b5 + b4) * q^4 + (2b3 + 3b1) * q^5 + (b11 + b10 + b8 - b6 + b4 - b2 - b1) * q^6 + (-b11 + 2b10 - b9 - b7 + b5 + b3 + 1) q^7 + (b7 - 2b6) q^8 + (2b7 - b6) q^9 + (3b5 + b4) q^10 + (2b9 - b8) q^11 + q^12 + (-b9 + 2b8) q^13 + (2b11 + b10 + b8 - b6 + b4 - b2 - b1) q^14 + (3b11 + b10 + b8 - b6 + b4 - b2 - b1) q^15 - 3b8 q^16 + (b2 - 5) * q^17 + (b9 + b8) * q^18 + (-3b5 + 3b4) * q^19 + (5b7 - 2b6) * q^20 + (-2b7 - b6) q^21 + (b11 + 3b10 + b9 + b7 - b5 - b3 - 1) q^22 + (-2b10 - 2b8 + 2b6 - 2b4 + 2b2 + 2b1) * q^23 + (2b3 - b1) q^24 + (8b5 - 3b4) * q^25 + (-2b11 - 3b10 - 2b9 - 2b7 + 2b5 + 2b3 + 2) * q^26 + (b3 + 4b1) q^27 + (b2 + 3) * q^28 + (-4b2 + 1) q^30 + (-7b3 - 5b1) * q^31 + (5b11 - b10 + 5b9 + 5b7 - 5b5 - 5b3 - 5) q^32 + (-b5 - 3b4) q^33 + (b3 - 6b1) q^34 + (8b11 - b10 - b8 + b6 - b4 + b2 + b1) q^35 + (3b11 - 2b10 + 3b9 + 3b7 - 3b5 - 3b3 - 3) * q^36 + (-b7 + 6b6) q^37 + 3b7 q^38 + (2b5 + 3b4) * q^39 + (8b9 + b8) q^40 + (3b2 + 2) q^41 + (b9 - 3b8) q^42 + (6b11 - 2b10 - 2b8 + 2b6 - 2b4 + 2b2 + 2b1) q^43 + (-b11 + 2b10 + 2b8 - 2b6 + 2b4 - 2b2 - 2b1) * q^44 + (7b9 + 5b8) * q^45 + (2b2 - 2) q^46 + 7b9 q^47 + (-3b5 - 3b4) * q^48 + 2b7 q^49 + (-3b7 + 5b6) * q^50 + (-b11 - 6b10 - b9 - b7 + b5 + b3 + 1) q^51 + (-b11 - b10 - b8 + b6 - b4 + b2 + b1) * q^52 - 2b3 q^53 + (4b5 + 3b4) * q^54 + (-b11 + 7b10 - b9 - b7 + b5 + b3 + 1) q^55 + 5b3 q^56 + 3 * q^57 + (5b2 + 3) q^59 + (2b3 + 3b1) * q^60 + b10 * q^61 + (-5b5 + 2b4) * q^62 + (4b3 + 3b1) * q^63 + (-b11 - 2b10 - 2b8 + 2b6 - 2b4 + 2b2 + 2b1) * q^64 + (-4b11 - 5b10 - 4b9 - 4b7 + 4b5 + 4b3 + 4) * q^65 + (-3b7 - 4b6) * q^66 + (4b7 + 4b6) * q^67 + (4b5 - 5b4) * q^68 + (-2b9 + 2b8) * q^69 + (-7b2 - 1) q^70 + (-8b9 - 4b8) * q^71 + (-4b11 + 3b10 + 3b8 - 3b6 + 3b4 - 3b2 - 3b1) q^72 + (10b11 - 6b10 - 6b8 + 6b6 - 6b4 + 6b2 + 6b1) q^73 + (-6b9 + 5b8) * q^74 + (-5b2 - 3) q^75 + (-6b9 - 3b8) * q^76 - 5b4 q^77 + (3b7 + 5b6) * q^78 + (3b7 - 5b6) * q^79 + (9b11 + 3b10 + 9b9 + 9b7 - 9b5 - 9b3 - 9) * q^80 + (-2b11 + 6b10 + 6b8 - 6b6 + 6b4 - 6b2 - 6b1) q^81 + (3b3 - b1) q^82 + (-5b5 + 8b4) * q^83 + (-b11 + 2b10 - b9 - b7 + b5 + b3 + 1) q^84 + (-7b3 - 16b1) * q^85 + (-4b2 - 2) q^86 + 5b2 q^88 + (b3 + 6b1) q^89 + (-5b11 + 2b10 - 5b9 - 5b7 + 5b5 + 5b3 + 5) * q^90 + (3b5 + 4b4) * q^91 + 2b3 q^92 + (-5b11 + 2b10 + 2b8 - 2b6 + 2b4 - 2b2 - 2b1) q^93 + 7b10 q^94 + (15b7 - 6b6) * q^95 + (b7 - 4b6) q^96 + (-2b5 - 9b4) * q^97 + 2b8 q^98 + (b2 + 3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + q^{2} - q^{3} + q^{4} - q^{5} + 3 q^{6} + 3 q^{9} - 7 q^{10} + 5 q^{11} + 12 q^{12} - 4 q^{13} + 5 q^{14} + 7 q^{15} + 3 q^{16} - 66 q^{17} + q^{18} + 3 q^{19} + 8 q^{20} - 5 q^{21} - 5 q^{22}+ \cdots + 30 q^{99}+O(q^{100}) $ 12 * q + q^2 - q^3 + q^4 - q^5 + 3 * q^6 + 3 * q^9 - 7 * q^10 + 5 * q^11 + 12 * q^12 - 4 * q^13 + 5 * q^14 + 7 * q^15 + 3 * q^16 - 66 * q^17 + q^18 + 3 * q^19 + 8 * q^20 - 5 * q^21 - 5 * q^22 - 2 * q^23 - 5 * q^24 - 13 * q^25 + 7 * q^26 + 2 * q^27 + 30 * q^28 + 36 * q^30 + 9 * q^31 - 9 * q^32 + 5 * q^33 - 8 * q^34 + 15 * q^35 - 4 * q^36 + 4 * q^37 + 6 * q^38 - 7 * q^39 + 15 * q^40 + 6 * q^41 + 5 * q^42 + 10 * q^43 + 9 * q^45 - 36 * q^46 + 14 * q^47 + 9 * q^48 + 4 * q^49 - q^50 + 8 * q^51 - 3 * q^52 + 4 * q^53 - 11 * q^54 - 5 * q^55 - 10 * q^56 + 36 * q^57 + 6 * q^59 - q^60 - q^61 + 8 * q^62 - 5 * q^63 - 4 * q^64 + 13 * q^65 - 10 * q^66 + 12 * q^67 - 3 * q^68 - 6 * q^69 + 30 * q^70 - 12 * q^71 - 5 * q^72 + 14 * q^73 - 17 * q^74 - 6 * q^75 - 9 * q^76 + 5 * q^77 + 11 * q^78 + q^79 - 21 * q^80 + 2 * q^81 - 7 * q^82 + 2 * q^83 - 2 * q^85 - 30 * q^88 + 4 * q^89 + 8 * q^90 - 10 * q^91 - 4 * q^92 - 8 * q^93 - 7 * q^94 + 24 * q^95 - 2 * q^96 + 13 * q^97 - 2 * q^98 + 30 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - x^{11} + 2x^{10} - 3x^{9} + 5x^{8} - 8x^{7} + 13x^{6} + 8x^{5} + 5x^{4} + 3x^{3} + 2x^{2} + x + 1 $ x^12 - x^11 + 2*x^10 - 3*x^9 + 5*x^8 - 8*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^{7} + 8 ) / 13 $ (v^7 + 8) / 13
$\beta_{3}$	$=$	$ ( \nu^{8} + 21\nu ) / 13 $ (v^8 + 21*v) / 13
$\beta_{4}$	$=$	$ ( -\nu^{9} - 21\nu^{2} ) / 13 $ (-v^9 - 21*v^2) / 13
$\beta_{5}$	$=$	$ ( \nu^{9} + 34\nu^{2} ) / 13 $ (v^9 + 34*v^2) / 13
$\beta_{6}$	$=$	$ ( \nu^{10} + 34\nu^{3} ) / 13 $ (v^10 + 34*v^3) / 13
$\beta_{7}$	$=$	$ ( -2\nu^{10} - 55\nu^{3} ) / 13 $ (-2v^10 - 55v^3) / 13
$\beta_{8}$	$=$	$ ( -2\nu^{11} - 55\nu^{4} ) / 13 $ (-2v^11 - 55v^4) / 13
$\beta_{9}$	$=$	$ ( -3\nu^{11} - 89\nu^{4} ) / 13 $ (-3v^11 - 89v^4) / 13
$\beta_{10}$	$=$	$ ( - 3 \nu^{11} + 6 \nu^{10} - 9 \nu^{9} + 15 \nu^{8} - 24 \nu^{7} + 39 \nu^{6} - 65 \nu^{5} + 15 \nu^{4} + \cdots + 3 ) / 13 $ (-3v^11 + 6v^10 - 9v^9 + 15v^8 - 24v^7 + 39v^6 - 65v^5 + 15v^4 + 9v^3 + 6v^2 + 3*v + 3) / 13
$\beta_{11}$	$=$	$ ( 8 \nu^{11} - 8 \nu^{10} + 16 \nu^{9} - 24 \nu^{8} + 40 \nu^{7} - 65 \nu^{6} + 104 \nu^{5} + 64 \nu^{4} + \cdots + 8 ) / 13 $ (8v^11 - 8v^10 + 16v^9 - 24v^8 + 40v^7 - 65v^6 + 104v^5 + 64v^4 + 40v^3 + 24v^2 + 16*v + 8) / 13

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} - 3\beta_{9} - 3\beta_{7} + 3\beta_{5} + 3\beta_{3} + 3 $ -3b11 - 5b10 - 3b9 - 3b7 + 3b5 + 3b3 + 3
$\nu^{6}$	$=$	$ -5\beta_{11} - 8\beta_{10} - 8\beta_{8} + 8\beta_{6} - 8\beta_{4} + 8\beta_{2} + 8\beta_1 $ -5b11 - 8b10 - 8b8 + 8b6 - 8b4 + 8b2 + 8*b1
$\nu^{7}$	$=$	$ 13\beta_{2} - 8 $ 13*b2 - 8
$\nu^{8}$	$=$	$ 13\beta_{3} - 21\beta_1 $ 13b3 - 21b1
$\nu^{9}$	$=$	$ -21\beta_{5} - 34\beta_{4} $ -21b5 - 34b4
$\nu^{10}$	$=$	$ -34\beta_{7} - 55\beta_{6} $ -34b7 - 55b6
$\nu^{11}$	$=$	$ 55\beta_{9} - 89\beta_{8} $ 55b9 - 89b8

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/841\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$\beta_{5}$

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} $

190.1

−0.556829	−	0.268155i
1.45780	+	0.702039i
−0.556829	+	0.268155i
1.45780	−	0.702039i
−1.00883	+	1.26503i
0.385338	−	0.483198i
−0.137526	+	0.602539i
0.360046	−	1.57747i
−0.137526	−	0.602539i
0.360046	+	1.57747i
−1.00883	−	1.26503i
0.385338	+	0.483198i

−0.556829

−

0.268155i

−0.385338

0.483198i

−1.00883

−

1.26503i

−3.47243

−

1.67223i

0.344139

−

0.165729i

−1.39417

1.74823i

0.497572

2.18001i

0.582567

2.55239i

1.48513

1.86230i

190.2

1.45780

0.702039i

1.00883

−

1.26503i

0.385338

0.483198i

2.57146

1.23835i

2.35877

−

1.13592i

1.39417

−

1.74823i

−0.497572

−

2.18001i

0.0849954

0.372389i

2.87930

3.61052i

571.1

−0.556829

0.268155i

−0.385338

−

0.483198i

−1.00883

1.26503i

−3.47243

1.67223i

0.344139

0.165729i

−1.39417

−

1.74823i

0.497572

−

2.18001i

0.582567

−

2.55239i

1.48513

−

1.86230i

571.2

1.45780

−

0.702039i

1.00883

1.26503i

0.385338

−

0.483198i

2.57146

−

1.23835i

2.35877

1.13592i

1.39417

1.74823i

−0.497572

2.18001i

0.0849954

−

0.372389i

2.87930

−

3.61052i

574.1

−1.00883

1.26503i

−0.360046

1.57747i

−0.137526

−

0.602539i

−1.77950

2.23143i

−1.63232

−

2.04686i

−0.497572

2.18001i

−2.01463

−

0.970194i

0.344139

0.165729i

−1.02761

−

4.50225i

574.2

0.385338

−

0.483198i

0.137526

−

0.602539i

0.360046

1.57747i

2.40299

−

3.01326i

−0.238152

−

0.298633i

0.497572

−

2.18001i

2.01463

0.970194i

2.35877

1.13592i

−0.530037

−

2.32225i

605.1

−0.137526

0.602539i

0.556829

−

0.268155i

1.45780

0.702039i

−0.857618

3.75747i

0.0849954

0.372389i

2.01463

−

0.970194i

−1.39417

1.74823i

−1.63232

2.04686i

−2.14608

−

1.03350i

605.2

0.360046

−

1.57747i

−1.45780

0.702039i

−0.556829

−

0.268155i

0.635097

−

2.78254i

0.582567

2.55239i

−2.01463

0.970194i

1.39417

−

1.74823i

−0.238152

0.298633i

−4.16070

−

2.00369i

645.1

−0.137526

−

0.602539i

0.556829

0.268155i

1.45780

−

0.702039i

−0.857618

−

3.75747i

0.0849954

−

0.372389i

2.01463

0.970194i

−1.39417

−

1.74823i

−1.63232

−

2.04686i

−2.14608

1.03350i

645.2

0.360046

1.57747i

−1.45780

−

0.702039i

−0.556829

0.268155i

0.635097

2.78254i

0.582567

−

2.55239i

−2.01463

−

0.970194i

1.39417

1.74823i

−0.238152

−

0.298633i

−4.16070

2.00369i

778.1

−1.00883

−

1.26503i

−0.360046

−

1.57747i

−0.137526

0.602539i

−1.77950

−

2.23143i

−1.63232

2.04686i

−0.497572

−

2.18001i

−2.01463

0.970194i

0.344139

−

0.165729i

−1.02761

4.50225i

778.2

0.385338

0.483198i

0.137526

0.602539i

0.360046

−

1.57747i

2.40299

3.01326i

−0.238152

0.298633i

0.497572

2.18001i

2.01463

−

0.970194i

2.35877

−

1.13592i

−0.530037

2.32225i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
29.d	even	7	5	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	841.2.d.i		12
29.b	even	2	1		841.2.d.g		12
29.c	odd	4	2		841.2.e.j		24
29.d	even	7	1		841.2.a.a	✓	2
29.d	even	7	5	inner	841.2.d.i		12
29.e	even	14	1		841.2.a.c	yes	2
29.e	even	14	5		841.2.d.g		12
29.f	odd	28	2		841.2.b.b		4
29.f	odd	28	10		841.2.e.j		24
87.h	odd	14	1		7569.2.a.d		2
87.j	odd	14	1		7569.2.a.l		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
841.2.a.a	✓	2	29.d	even	7	1
841.2.a.c	yes	2	29.e	even	14	1
841.2.b.b		4	29.f	odd	28	2
841.2.d.g		12	29.b	even	2	1
841.2.d.g		12	29.e	even	14	5
841.2.d.i		12	1.a	even	1	1	trivial
841.2.d.i		12	29.d	even	7	5	inner
841.2.e.j		24	29.c	odd	4	2
841.2.e.j		24	29.f	odd	28	10
7569.2.a.d		2	87.h	odd	14	1
7569.2.a.l		2	87.j	odd	14	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{12} - T_{2}^{11} + 2 T_{2}^{10} - 3 T_{2}^{9} + 5 T_{2}^{8} - 8 T_{2}^{7} + 13 T_{2}^{6} + \cdots + 1 $ T2^12 - T2^11 + 2*T2^10 - 3*T2^9 + 5*T2^8 - 8*T2^7 + 13*T2^6 + 8*T2^5 + 5*T2^4 + 3*T2^3 + 2*T2^2 + T2 + 1 acting on $S_{2}^{\mathrm{new}}(841, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} - T^{11} + \cdots + 1 $ T^12 - T^11 + 2T^10 - 3T^9 + 5T^8 - 8T^7 + 13T^6 + 8T^5 + 5T^4 + 3T^3 + 2*T^2 + T + 1
$3$	$ T^{12} + T^{11} + \cdots + 1 $ T^12 + T^11 + 2T^10 + 3T^9 + 5T^8 + 8T^7 + 13T^6 - 8T^5 + 5T^4 - 3T^3 + 2*T^2 - T + 1
$5$	$ T^{12} + T^{11} + \cdots + 1771561 $ T^12 + T^11 + 12T^10 + 23T^9 + 155T^8 + 408T^7 + 2113T^6 - 4488T^5 + 18755T^4 - 30613T^3 + 175692T^2 - 161051T + 1771561
$7$	$ T^{12} + 5 T^{10} + \cdots + 15625 $ T^12 + 5T^10 + 25T^8 + 125T^6 + 625T^4 + 3125*T^2 + 15625
$11$	$ T^{12} - 5 T^{11} + \cdots + 15625 $ T^12 - 5T^11 + 20T^10 - 75T^9 + 275T^8 - 1000T^7 + 3625T^6 - 5000T^5 + 6875T^4 - 9375T^3 + 12500T^2 - 15625*T + 15625
$13$	$ T^{12} + 4 T^{11} + \cdots + 1 $ T^12 + 4T^11 + 17T^10 + 72T^9 + 305T^8 + 1292T^7 + 5473T^6 - 1292T^5 + 305T^4 - 72T^3 + 17T^2 - 4*T + 1
$17$	$ (T^{2} + 11 T + 29)^{6} $ (T^2 + 11*T + 29)^6
$19$	$ T^{12} - 3 T^{11} + \cdots + 531441 $ T^12 - 3T^11 + 18T^10 - 81T^9 + 405T^8 - 1944T^7 + 9477T^6 + 17496T^5 + 32805T^4 + 59049T^3 + 118098T^2 + 177147*T + 531441
$23$	$ T^{12} + 2 T^{11} + \cdots + 4096 $ T^12 + 2T^11 + 8T^10 + 24T^9 + 80T^8 + 256T^7 + 832T^6 - 1024T^5 + 1280T^4 - 1536T^3 + 2048T^2 - 2048*T + 4096
$29$	$ T^{12} $ T^12
$31$	$ T^{12} - 9 T^{11} + \cdots + 1771561 $ T^12 - 9T^11 + 92T^10 - 927T^9 + 9355T^8 - 94392T^7 + 952433T^6 + 1038312T^5 + 1131955T^4 + 1233837T^3 + 1346972T^2 + 1449459*T + 1771561
$37$	$ T^{12} + \cdots + 4750104241 $ T^12 - 4T^11 + 57T^10 - 392T^9 + 3905T^8 - 31692T^7 + 286873T^6 + 1299372T^5 + 6564305T^4 + 27017032T^3 + 161068377T^2 + 463424804*T + 4750104241
$41$	$ (T^{2} - T - 11)^{6} $ (T^2 - T - 11)^6
$43$	$ T^{12} - 10 T^{11} + \cdots + 64000000 $ T^12 - 10T^11 + 80T^10 - 600T^9 + 4400T^8 - 32000T^7 + 232000T^6 - 640000T^5 + 1760000T^4 - 4800000T^3 + 12800000T^2 - 32000000*T + 64000000
$47$	$ (T^{6} - 7 T^{5} + \cdots + 117649)^{2} $ (T^6 - 7T^5 + 49T^4 - 343T^3 + 2401T^2 - 16807*T + 117649)^2
$53$	$ (T^{6} - 2 T^{5} + 4 T^{4} + \cdots + 64)^{2} $ (T^6 - 2T^5 + 4T^4 - 8T^3 + 16T^2 - 32*T + 64)^2
$59$	$ (T^{2} - T - 31)^{6} $ (T^2 - T - 31)^6
$61$	$ T^{12} + T^{11} + \cdots + 1 $ T^12 + T^11 + 2T^10 + 3T^9 + 5T^8 + 8T^7 + 13T^6 - 8T^5 + 5T^4 - 3T^3 + 2*T^2 - T + 1
$67$	$ T^{12} - 12 T^{11} + \cdots + 16777216 $ T^12 - 12T^11 + 128T^10 - 1344T^9 + 14080T^8 - 147456T^7 + 1544192T^6 - 2359296T^5 + 3604480T^4 - 5505024T^3 + 8388608T^2 - 12582912*T + 16777216
$71$	$ T^{12} + 12 T^{11} + \cdots + 16777216 $ T^12 + 12T^11 + 128T^10 + 1344T^9 + 14080T^8 + 147456T^7 + 1544192T^6 + 2359296T^5 + 3604480T^4 + 5505024T^3 + 8388608T^2 + 12582912*T + 16777216
$73$	$ T^{12} - 14 T^{11} + \cdots + 4096 $ T^12 - 14T^11 + 192T^10 - 2632T^9 + 36080T^8 - 494592T^7 + 6779968T^6 - 1978368T^5 + 577280T^4 - 168448T^3 + 49152T^2 - 14336*T + 4096
$79$	$ T^{12} + \cdots + 887503681 $ T^12 - T^11 + 32T^10 - 63T^9 + 1055T^8 - 3008T^7 + 35713T^6 + 93248T^5 + 1013855T^4 + 1876833T^3 + 29552672T^2 + 28629151T + 887503681
$83$	$ T^{12} + \cdots + 243087455521 $ T^12 - 2T^11 + 83T^10 - 324T^9 + 7205T^8 - 40006T^7 + 649207T^6 + 3160474T^5 + 44966405T^4 + 159744636T^3 + 3232856723T^2 + 6154112798*T + 243087455521
$89$	$ T^{12} + \cdots + 4750104241 $ T^12 - 4T^11 + 57T^10 - 392T^9 + 3905T^8 - 31692T^7 + 286873T^6 + 1299372T^5 + 6564305T^4 + 27017032T^3 + 161068377T^2 + 463424804*T + 4750104241
$97$	$ T^{12} + \cdots + 42180533641 $ T^12 - 13T^11 + 228T^10 - 3731T^9 + 61955T^8 - 1025544T^7 + 16987417T^6 + 60507096T^5 + 215665355T^4 + 766269049T^3 + 2762758308T^2 + 9294015887*T + 42180533641
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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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 841 = 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	841.d (of order \(7\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + \beta_{10} q^{3} + ( - \beta_{5} + \beta_{4}) q^{4} + (2 \beta_{3} + 3 \beta_1) q^{5} + (\beta_{11} + \beta_{10} + \cdots - \beta_1) q^{6} + ( - \beta_{11} + 2 \beta_{10} + \cdots + 1) q^{7}+ \cdots + (\beta_{2} + 3) q^{99}+O(q^{100}) \) q + b1 * q^2 + b10 * q^3 + (-b5 + b4) * q^4 + (2b3 + 3b1) * q^5 + (b11 + b10 + b8 - b6 + b4 - b2 - b1) * q^6 + (-b11 + 2b10 - b9 - b7 + b5 + b3 + 1) q^7 + (b7 - 2b6) q^8 + (2b7 - b6) q^9 + (3b5 + b4) q^10 + (2b9 - b8) q^11 + q^12 + (-b9 + 2b8) q^13 + (2b11 + b10 + b8 - b6 + b4 - b2 - b1) q^14 + (3b11 + b10 + b8 - b6 + b4 - b2 - b1) q^15 - 3b8 q^16 + (b2 - 5) * q^17 + (b9 + b8) * q^18 + (-3b5 + 3b4) * q^19 + (5b7 - 2b6) * q^20 + (-2b7 - b6) q^21 + (b11 + 3b10 + b9 + b7 - b5 - b3 - 1) q^22 + (-2b10 - 2b8 + 2b6 - 2b4 + 2b2 + 2b1) * q^23 + (2b3 - b1) q^24 + (8b5 - 3b4) * q^25 + (-2b11 - 3b10 - 2b9 - 2b7 + 2b5 + 2b3 + 2) * q^26 + (b3 + 4b1) q^27 + (b2 + 3) * q^28 + (-4b2 + 1) q^30 + (-7b3 - 5b1) * q^31 + (5b11 - b10 + 5b9 + 5b7 - 5b5 - 5b3 - 5) q^32 + (-b5 - 3b4) q^33 + (b3 - 6b1) q^34 + (8b11 - b10 - b8 + b6 - b4 + b2 + b1) q^35 + (3b11 - 2b10 + 3b9 + 3b7 - 3b5 - 3b3 - 3) * q^36 + (-b7 + 6b6) q^37 + 3b7 q^38 + (2b5 + 3b4) * q^39 + (8b9 + b8) q^40 + (3b2 + 2) q^41 + (b9 - 3b8) q^42 + (6b11 - 2b10 - 2b8 + 2b6 - 2b4 + 2b2 + 2b1) q^43 + (-b11 + 2b10 + 2b8 - 2b6 + 2b4 - 2b2 - 2b1) * q^44 + (7b9 + 5b8) * q^45 + (2b2 - 2) q^46 + 7b9 q^47 + (-3b5 - 3b4) * q^48 + 2b7 q^49 + (-3b7 + 5b6) * q^50 + (-b11 - 6b10 - b9 - b7 + b5 + b3 + 1) q^51 + (-b11 - b10 - b8 + b6 - b4 + b2 + b1) * q^52 - 2b3 q^53 + (4b5 + 3b4) * q^54 + (-b11 + 7b10 - b9 - b7 + b5 + b3 + 1) q^55 + 5b3 q^56 + 3 * q^57 + (5b2 + 3) q^59 + (2b3 + 3b1) * q^60 + b10 * q^61 + (-5b5 + 2b4) * q^62 + (4b3 + 3b1) * q^63 + (-b11 - 2b10 - 2b8 + 2b6 - 2b4 + 2b2 + 2b1) * q^64 + (-4b11 - 5b10 - 4b9 - 4b7 + 4b5 + 4b3 + 4) * q^65 + (-3b7 - 4b6) * q^66 + (4b7 + 4b6) * q^67 + (4b5 - 5b4) * q^68 + (-2b9 + 2b8) * q^69 + (-7b2 - 1) q^70 + (-8b9 - 4b8) * q^71 + (-4b11 + 3b10 + 3b8 - 3b6 + 3b4 - 3b2 - 3b1) q^72 + (10b11 - 6b10 - 6b8 + 6b6 - 6b4 + 6b2 + 6b1) q^73 + (-6b9 + 5b8) * q^74 + (-5b2 - 3) q^75 + (-6b9 - 3b8) * q^76 - 5b4 q^77 + (3b7 + 5b6) * q^78 + (3b7 - 5b6) * q^79 + (9b11 + 3b10 + 9b9 + 9b7 - 9b5 - 9b3 - 9) * q^80 + (-2b11 + 6b10 + 6b8 - 6b6 + 6b4 - 6b2 - 6b1) q^81 + (3b3 - b1) q^82 + (-5b5 + 8b4) * q^83 + (-b11 + 2b10 - b9 - b7 + b5 + b3 + 1) q^84 + (-7b3 - 16b1) * q^85 + (-4b2 - 2) q^86 + 5b2 q^88 + (b3 + 6b1) q^89 + (-5b11 + 2b10 - 5b9 - 5b7 + 5b5 + 5b3 + 5) * q^90 + (3b5 + 4b4) * q^91 + 2b3 q^92 + (-5b11 + 2b10 + 2b8 - 2b6 + 2b4 - 2b2 - 2b1) q^93 + 7b10 q^94 + (15b7 - 6b6) * q^95 + (b7 - 4b6) q^96 + (-2b5 - 9b4) * q^97 + 2b8 q^98 + (b2 + 3) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{2} - q^{3} + q^{4} - q^{5} + 3 q^{6} + 3 q^{9} - 7 q^{10} + 5 q^{11} + 12 q^{12} - 4 q^{13} + 5 q^{14} + 7 q^{15} + 3 q^{16} - 66 q^{17} + q^{18} + 3 q^{19} + 8 q^{20} - 5 q^{21} - 5 q^{22}+ \cdots + 30 q^{99}+O(q^{100}) \) 12 * q + q^2 - q^3 + q^4 - q^5 + 3 * q^6 + 3 * q^9 - 7 * q^10 + 5 * q^11 + 12 * q^12 - 4 * q^13 + 5 * q^14 + 7 * q^15 + 3 * q^16 - 66 * q^17 + q^18 + 3 * q^19 + 8 * q^20 - 5 * q^21 - 5 * q^22 - 2 * q^23 - 5 * q^24 - 13 * q^25 + 7 * q^26 + 2 * q^27 + 30 * q^28 + 36 * q^30 + 9 * q^31 - 9 * q^32 + 5 * q^33 - 8 * q^34 + 15 * q^35 - 4 * q^36 + 4 * q^37 + 6 * q^38 - 7 * q^39 + 15 * q^40 + 6 * q^41 + 5 * q^42 + 10 * q^43 + 9 * q^45 - 36 * q^46 + 14 * q^47 + 9 * q^48 + 4 * q^49 - q^50 + 8 * q^51 - 3 * q^52 + 4 * q^53 - 11 * q^54 - 5 * q^55 - 10 * q^56 + 36 * q^57 + 6 * q^59 - q^60 - q^61 + 8 * q^62 - 5 * q^63 - 4 * q^64 + 13 * q^65 - 10 * q^66 + 12 * q^67 - 3 * q^68 - 6 * q^69 + 30 * q^70 - 12 * q^71 - 5 * q^72 + 14 * q^73 - 17 * q^74 - 6 * q^75 - 9 * q^76 + 5 * q^77 + 11 * q^78 + q^79 - 21 * q^80 + 2 * q^81 - 7 * q^82 + 2 * q^83 - 2 * q^85 - 30 * q^88 + 4 * q^89 + 8 * q^90 - 10 * q^91 - 4 * q^92 - 8 * q^93 - 7 * q^94 + 24 * q^95 - 2 * q^96 + 13 * q^97 - 2 * q^98 + 30 * 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	841.2.d.i

Level	$841$
Weight	$2$
Character orbit	841.d
Analytic conductor	$6.715$
Analytic rank	$0$
Dimension	$12$
Inner twists	$6$

Self dual:	no
Analytic conductor:	\(6.71541880999\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{7})\)
Coefficient field:	12.0.4413675765625.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - x^{11} + 2x^{10} - 3x^{9} + 5x^{8} - 8x^{7} + 13x^{6} + 8x^{5} + 5x^{4} + 3x^{3} + 2x^{2} + x + 1 \) x^12 - x^11 + 2x^10 - 3x^9 + 5x^8 - 8x^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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( \nu^{7} + 8 ) / 13 \) (v^7 + 8) / 13
\(\beta_{3}\)	\(=\)	\( ( \nu^{8} + 21\nu ) / 13 \) (v^8 + 21*v) / 13
\(\beta_{4}\)	\(=\)	\( ( -\nu^{9} - 21\nu^{2} ) / 13 \) (-v^9 - 21*v^2) / 13
\(\beta_{5}\)	\(=\)	\( ( \nu^{9} + 34\nu^{2} ) / 13 \) (v^9 + 34*v^2) / 13
\(\beta_{6}\)	\(=\)	\( ( \nu^{10} + 34\nu^{3} ) / 13 \) (v^10 + 34*v^3) / 13
\(\beta_{7}\)	\(=\)	\( ( -2\nu^{10} - 55\nu^{3} ) / 13 \) (-2v^10 - 55v^3) / 13
\(\beta_{8}\)	\(=\)	\( ( -2\nu^{11} - 55\nu^{4} ) / 13 \) (-2v^11 - 55v^4) / 13
\(\beta_{9}\)	\(=\)	\( ( -3\nu^{11} - 89\nu^{4} ) / 13 \) (-3v^11 - 89v^4) / 13
\(\beta_{10}\)	\(=\)	\( ( - 3 \nu^{11} + 6 \nu^{10} - 9 \nu^{9} + 15 \nu^{8} - 24 \nu^{7} + 39 \nu^{6} - 65 \nu^{5} + 15 \nu^{4} + \cdots + 3 ) / 13 \) (-3v^11 + 6v^10 - 9v^9 + 15v^8 - 24v^7 + 39v^6 - 65v^5 + 15v^4 + 9v^3 + 6v^2 + 3*v + 3) / 13
\(\beta_{11}\)	\(=\)	\( ( 8 \nu^{11} - 8 \nu^{10} + 16 \nu^{9} - 24 \nu^{8} + 40 \nu^{7} - 65 \nu^{6} + 104 \nu^{5} + 64 \nu^{4} + \cdots + 8 ) / 13 \) (8v^11 - 8v^10 + 16v^9 - 24v^8 + 40v^7 - 65v^6 + 104v^5 + 64v^4 + 40v^3 + 24v^2 + 16*v + 8) / 13

\(\nu\)	\(=\)	\( \beta_1 \) 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
\(\nu^{5}\)	\(=\)	\( -3\beta_{11} - 5\beta_{10} - 3\beta_{9} - 3\beta_{7} + 3\beta_{5} + 3\beta_{3} + 3 \) -3b11 - 5b10 - 3b9 - 3b7 + 3b5 + 3b3 + 3
\(\nu^{6}\)	\(=\)	\( -5\beta_{11} - 8\beta_{10} - 8\beta_{8} + 8\beta_{6} - 8\beta_{4} + 8\beta_{2} + 8\beta_1 \) -5b11 - 8b10 - 8b8 + 8b6 - 8b4 + 8b2 + 8*b1
\(\nu^{7}\)	\(=\)	\( 13\beta_{2} - 8 \) 13*b2 - 8
\(\nu^{8}\)	\(=\)	\( 13\beta_{3} - 21\beta_1 \) 13b3 - 21b1
\(\nu^{9}\)	\(=\)	\( -21\beta_{5} - 34\beta_{4} \) -21b5 - 34b4
\(\nu^{10}\)	\(=\)	\( -34\beta_{7} - 55\beta_{6} \) -34b7 - 55b6
\(\nu^{11}\)	\(=\)	\( 55\beta_{9} - 89\beta_{8} \) 55b9 - 89b8

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 190.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} - T^{11} + \cdots + 1 \) T^12 - T^11 + 2T^10 - 3T^9 + 5T^8 - 8T^7 + 13T^6 + 8T^5 + 5T^4 + 3T^3 + 2*T^2 + T + 1
$3$	\( T^{12} + T^{11} + \cdots + 1 \) T^12 + T^11 + 2T^10 + 3T^9 + 5T^8 + 8T^7 + 13T^6 - 8T^5 + 5T^4 - 3T^3 + 2*T^2 - T + 1
$5$	\( T^{12} + T^{11} + \cdots + 1771561 \) T^12 + T^11 + 12T^10 + 23T^9 + 155T^8 + 408T^7 + 2113T^6 - 4488T^5 + 18755T^4 - 30613T^3 + 175692T^2 - 161051T + 1771561
$7$	\( T^{12} + 5 T^{10} + \cdots + 15625 \) T^12 + 5T^10 + 25T^8 + 125T^6 + 625T^4 + 3125*T^2 + 15625
$11$	\( T^{12} - 5 T^{11} + \cdots + 15625 \) T^12 - 5T^11 + 20T^10 - 75T^9 + 275T^8 - 1000T^7 + 3625T^6 - 5000T^5 + 6875T^4 - 9375T^3 + 12500T^2 - 15625*T + 15625
$13$	\( T^{12} + 4 T^{11} + \cdots + 1 \) T^12 + 4T^11 + 17T^10 + 72T^9 + 305T^8 + 1292T^7 + 5473T^6 - 1292T^5 + 305T^4 - 72T^3 + 17T^2 - 4*T + 1
$17$	\( (T^{2} + 11 T + 29)^{6} \) (T^2 + 11*T + 29)^6
$19$	\( T^{12} - 3 T^{11} + \cdots + 531441 \) T^12 - 3T^11 + 18T^10 - 81T^9 + 405T^8 - 1944T^7 + 9477T^6 + 17496T^5 + 32805T^4 + 59049T^3 + 118098T^2 + 177147*T + 531441
$23$	\( T^{12} + 2 T^{11} + \cdots + 4096 \) T^12 + 2T^11 + 8T^10 + 24T^9 + 80T^8 + 256T^7 + 832T^6 - 1024T^5 + 1280T^4 - 1536T^3 + 2048T^2 - 2048*T + 4096
$29$	\( T^{12} \) T^12
$31$	\( T^{12} - 9 T^{11} + \cdots + 1771561 \) T^12 - 9T^11 + 92T^10 - 927T^9 + 9355T^8 - 94392T^7 + 952433T^6 + 1038312T^5 + 1131955T^4 + 1233837T^3 + 1346972T^2 + 1449459*T + 1771561
$37$	\( T^{12} + \cdots + 4750104241 \) T^12 - 4T^11 + 57T^10 - 392T^9 + 3905T^8 - 31692T^7 + 286873T^6 + 1299372T^5 + 6564305T^4 + 27017032T^3 + 161068377T^2 + 463424804*T + 4750104241
$41$	\( (T^{2} - T - 11)^{6} \) (T^2 - T - 11)^6
$43$	\( T^{12} - 10 T^{11} + \cdots + 64000000 \) T^12 - 10T^11 + 80T^10 - 600T^9 + 4400T^8 - 32000T^7 + 232000T^6 - 640000T^5 + 1760000T^4 - 4800000T^3 + 12800000T^2 - 32000000*T + 64000000
$47$	\( (T^{6} - 7 T^{5} + \cdots + 117649)^{2} \) (T^6 - 7T^5 + 49T^4 - 343T^3 + 2401T^2 - 16807*T + 117649)^2
$53$	\( (T^{6} - 2 T^{5} + 4 T^{4} + \cdots + 64)^{2} \) (T^6 - 2T^5 + 4T^4 - 8T^3 + 16T^2 - 32*T + 64)^2
$59$	\( (T^{2} - T - 31)^{6} \) (T^2 - T - 31)^6
$61$	\( T^{12} + T^{11} + \cdots + 1 \) T^12 + T^11 + 2T^10 + 3T^9 + 5T^8 + 8T^7 + 13T^6 - 8T^5 + 5T^4 - 3T^3 + 2*T^2 - T + 1
$67$	\( T^{12} - 12 T^{11} + \cdots + 16777216 \) T^12 - 12T^11 + 128T^10 - 1344T^9 + 14080T^8 - 147456T^7 + 1544192T^6 - 2359296T^5 + 3604480T^4 - 5505024T^3 + 8388608T^2 - 12582912*T + 16777216
$71$	\( T^{12} + 12 T^{11} + \cdots + 16777216 \) T^12 + 12T^11 + 128T^10 + 1344T^9 + 14080T^8 + 147456T^7 + 1544192T^6 + 2359296T^5 + 3604480T^4 + 5505024T^3 + 8388608T^2 + 12582912*T + 16777216
$73$	\( T^{12} - 14 T^{11} + \cdots + 4096 \) T^12 - 14T^11 + 192T^10 - 2632T^9 + 36080T^8 - 494592T^7 + 6779968T^6 - 1978368T^5 + 577280T^4 - 168448T^3 + 49152T^2 - 14336*T + 4096
$79$	\( T^{12} + \cdots + 887503681 \) T^12 - T^11 + 32T^10 - 63T^9 + 1055T^8 - 3008T^7 + 35713T^6 + 93248T^5 + 1013855T^4 + 1876833T^3 + 29552672T^2 + 28629151T + 887503681
$83$	\( T^{12} + \cdots + 243087455521 \) T^12 - 2T^11 + 83T^10 - 324T^9 + 7205T^8 - 40006T^7 + 649207T^6 + 3160474T^5 + 44966405T^4 + 159744636T^3 + 3232856723T^2 + 6154112798*T + 243087455521
$89$	\( T^{12} + \cdots + 4750104241 \) T^12 - 4T^11 + 57T^10 - 392T^9 + 3905T^8 - 31692T^7 + 286873T^6 + 1299372T^5 + 6564305T^4 + 27017032T^3 + 161068377T^2 + 463424804*T + 4750104241
$97$	\( T^{12} + \cdots + 42180533641 \) T^12 - 13T^11 + 228T^10 - 3731T^9 + 61955T^8 - 1025544T^7 + 16987417T^6 + 60507096T^5 + 215665355T^4 + 766269049T^3 + 2762758308T^2 + 9294015887*T + 42180533641
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\(n\)	\(2\)
\(\chi(n)\)	\(\beta_{5}\)