LMFDB - Newform orbit 522.2.n.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [522,2,Mod(91,522)] 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("522.91"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 522 = 2 \cdot 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	522.n (of order $14$, degree $6$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,2,2,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$4.16819098551$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{14})$
Coefficient field:	$\Q(\zeta_{28})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 58)
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

$q$-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 primitive root of unity $\zeta_{28}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \zeta_{28}^{9} q^{2} - \zeta_{28}^{4} q^{4} + ( - \zeta_{28}^{11} + \cdots + \zeta_{28}) q^{5} + ( - \zeta_{28}^{10} - 2 \zeta_{28}^{9} + \cdots + 1) q^{7} + (\zeta_{28}^{11} - \zeta_{28}^{9} + \cdots - \zeta_{28}) q^{8} + \cdots + ( - 2 \zeta_{28}^{11} + 2 \zeta_{28}^{9} + \cdots - 2) q^{98} +O(q^{100}) $ q - z^9 * q^2 - z^4 * q^4 + (-z^11 + z^7 + z^3 + z^2 + z) * q^5 + (-z^10 - 2z^9 + 2z^7 - z^6 - z^5 + z^4 + z^3 - z^2 - z + 1) * q^7 + (z^11 - z^9 + z^7 - z^5 + z^3 - z) * q^8 + (-z^11 - 2z^10 + z^8 - 2z^6 + z^4 + 1) * q^10 + (-z^11 + 2z^10 + z^9 - z^7 - 2z^6 + z^5 + z) * q^11 + (-2z^11 + z^10 + z^9 - z^7 + z^6 - 2z^3 + 3z^2 + z - 3) q^13 + (z^8 - z^7 - z^6 - z^4 - z^3 + z^2) * q^14 + z^8 * q^16 + (z^10 + 2z^8 - 2z^7 + 2z^6 + z^4) q^17 + (z^11 + 4z^10 - 2z^8 - z^7 + 2z^6 - 2z^5 - 2z^4 + 2z^3 + 2z^2 + z - 2) q^19 + (-z^11 - z^7 - z^6 - z^5 - z) * q^20 + (-z^10 - z^6 + 2z^5 + z^4 - z^2 - 2z + 1) * q^22 + (-2z^10 - 2z^9 + z^7 - z^6 + z^4 - z^3 + 2z + 2) q^23 + (-2z^11 + 4z^9 + 3z^8 - 2z^7 - z^6 + 4z^5 + 2z^4 - z^2 + 2z + 3) q^25 + (-3z^11 + z^10 + 3z^9 - 2z^8 + z^5 - z^4 + z^2 + z - 2) q^26 + (z^10 - z^9 - z^8 + z^7 + z^6 - z^5 - z^4 + 2z^3 - 2z - 1) * q^28 + (2z^11 - 4z^9 + 2z^7 - 3z^6 - 3z^5 - z^4 + 4z^3 - z^2 - z - 1) * q^29 + (-z^11 - 3z^9 + 2z^8 - z^7 - 2z^6 - z^4 - 3z^3 - z^2 + 3z + 3) q^31 + z^3 * q^32 + (-z^11 + z^9 - z^7 + 2z^5 + z^3 - 2z^2 + 3z) q^34 + (-4z^11 - 2z^10 + z^9 + 2z^8 - 2z^7 - 4z^6 + 2z^5 + 4z^4 - 2z^3 - 2z^2 + 3z + 2) * q^35 + (4z^11 + 3z^10 - 3z^9 - 4z^8 + 3z^7 + 2z^6 - 3z^5 + 3z^3 + z^2 - 4z - 4) q^37 + (-3z^10 + 2z^8 + 2z^7 - z^6 + 2z^5 + 2z^4 - 3z^2) * q^38 + (z^10 - z^6 - z^2 - z - 1) * q^40 + (-2z^11 + z^10 - 2z^9 - z^8 + 4z^7 - z^6 - 2z^5 + z^4 - 2z^3) q^41 + (-2z^10 + 2z^9 + z^7 - 4z^5 + z^3 + 2z - 2) * q^43 + (2z^10 - z^7 - z^3 + 2) q^44 + (-z^11 - z^10 - z^9 - z^8 - z^7 + z^6 - z^5 - 3z^4 - z^3 + 2z^2 - 1) * q^46 + (z^11 + 4z^10 + z^9 - 4z^8 - z^7 + 4z^6 - z^5 - 8z^4 + 5z^2 - 3z - 3) * q^47 + (2z^11 - 2z^9 + 2z^8 + 2z^4 - 2z^3 + 2z) * q^49 + (-z^11 - 2z^10 - z^9 - 2z^7 - 2z^6 + 2z^5 + 4z^4 + z^3 - 2z^2 + z + 4) * q^50 + (-z^10 + z^9 + z^7 - 3z^6 + 3z^4 - z^3 - z + 1) * q^52 + (4z^11 - z^10 - z^9 + z^8 - 2z^7 - 2z^5 - 3z^4 + 3z^3 + 2z^2 + z - 3) * q^53 + (2z^11 + 2z^10 - 2z^9 - 4z^8 + 4z^6 + z^5 - z^4 - 4z^3 + 3z^2 + z) q^55 + (z^11 + 2z^8 + z^7 - 2z^6 + z^4 - z^2 + 1) * q^56 + (2z^11 - 3z^10 + 4z^8 + z^7 - 2z^6 - z^5 + z^3 - 2z^2 - 4z + 1) * q^58 + (z^11 + 2z^10 + 4z^9 - 4z^8 - 2z^7 + 4z^6 - 2z^4 - 5z^3 + 4z + 2) * q^59 + (4z^11 - 4z^9 - 2z^6 + 4z^5 - z^4 + 3z^3 - z^2 + 4z - 2) * q^61 + (2z^11 - 4z^9 - 3z^8 + z^7 + 2z^6 - z^5 - 6z^4 + 3z^3 + 2z^2 - 3z - 3) * q^62 + (-z^10 + z^8 - z^6 + z^4 - z^2 + 1) * q^64 + (2z^11 - 3z^10 + 6z^9 - 4z^7 - 2z^6 + 4z^5 + 4z^4 - 2z^3 - 4z^2 + 2) q^65 + (-z^11 - 4z^10 + 3z^9 - 3z^7 - 3z^5 + 3z^3 - 4z^2 - z) * q^67 + (2z^11 - 4z^10 + z^8 - 2z^6 + 2z^4 - 2z^2 + 3) q^68 + (-2z^11 - z^10 + 2z^9 - 2z^8 - 4z^7 - 2z^6 + 2z^5 - z^4 - 2z^3) q^70 + (3z^11 + 3z^10 - 4z^9 + z^8 + 4z^7 + 3z^6 - 5z^5 + 8z^3 + 4z^2 - 4z - 4) q^71 + (3z^10 - z^8 + 2z^5 - z^2 + 3) * q^73 + (-z^11 + z^10 + 4z^9 + 3z^8 + z^6 + 3z^5 - 4z^3 + 2z) q^74 + (z^11 + 2z^9 - 2z^7 - z^5 + 2z^2 + z + 2) q^76 + (-4z^11 + z^8 - z^7 + z^6 + 6z^5 - 6z^3 - z^2 + z - 1) q^77 + (-4z^11 + 6z^9 - 2z^6 + 4z^4 - 6z^3 - 4z^2 + 4z + 4) q^79 + (z^11 + z^10 + z^9 + z^5 - z) * q^80 + (-z^11 + 2z^10 + z^9 - 2z^8 - z^7 + 2z^5 - 4z^4 - 2z^3 + 6z^2 - 4) * q^82 + (-z^11 + 2z^9 + 2z^6 - z^5 + 4z^4 - 3z^3 + 2z^2 + 2z) * q^83 + (7z^11 + z^10 - z^9 - z^8 + 7z^7 + 2z^6 - 3z^4 + 4z^3 + z^2 - 4z + 1) * q^85 + (-3z^10 + 2z^9 + z^8 - z^6 - 2z^5 + 3z^4 - 3) * q^86 + (z^10 - 2z^9 - z^8 + z^6 + 2z^5 - z^4 - 1) * q^88 + (4z^11 + 6z^10 - 2z^9 + 4z^7 + 6z^6 - 2z^4 + 2z^3 + 3z^2 - 2z - 4) q^89 + (-4z^11 + 8z^9 - 2z^8 - 7z^7 + 4z^6 + 3z^5 - 5z^3 + 4z^2 + z - 2) * q^91 + (z^11 + z^10 - 2z^9 - z^8 + 3z^7 - 4z^5 - 2z^4 + 2z^3 - 2z - 2) * q^92 + (3z^11 + 3z^10 - 5z^9 + 8z^7 + z^6 - 4z^5 + z^4 + 4z^3 - z^2 - 4z - 1) q^94 + (7z^11 + z^10 - 4z^9 - z^8 + 5z^7 - 5z^5 + z^4 + 4z^3 - z^2 - 7z) * q^95 + (-6z^11 + 2z^8 + 2z^7 + 4z^6 - 2z^5 + 2z^3 - 4z^2 - 2z - 2) * q^97 + (-2z^11 + 2z^9 - 2z^8 - 2z^7 + 4z^6 + 2z^5 - 4z^4 + 2z^2 + 2z - 2) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2 q^{4} + 2 q^{5} + 4 q^{7} - 26 q^{13} - 2 q^{16} - 2 q^{20} + 4 q^{22} + 16 q^{23} + 22 q^{25} - 14 q^{26} - 4 q^{28} - 18 q^{29} + 28 q^{31} - 4 q^{34} - 4 q^{35} - 28 q^{37} - 22 q^{38} - 14 q^{40}+ \cdots - 28 q^{97}+O(q^{100}) $ 12 * q + 2 * q^4 + 2 * q^5 + 4 * q^7 - 26 * q^13 - 2 * q^16 - 2 * q^20 + 4 * q^22 + 16 * q^23 + 22 * q^25 - 14 * q^26 - 4 * q^28 - 18 * q^29 + 28 * q^31 - 4 * q^34 - 4 * q^35 - 28 * q^37 - 22 * q^38 - 14 * q^40 - 28 * q^43 + 28 * q^44 + 14 * q^47 - 8 * q^49 + 28 * q^50 - 2 * q^52 - 30 * q^53 + 28 * q^55 - 10 * q^58 + 48 * q^59 - 28 * q^61 - 10 * q^62 + 2 * q^64 - 2 * q^65 - 16 * q^67 + 14 * q^68 - 30 * q^71 + 42 * q^73 - 2 * q^74 + 28 * q^76 - 14 * q^77 + 28 * q^79 + 2 * q^80 - 20 * q^82 + 28 * q^85 - 52 * q^86 - 4 * q^88 - 14 * q^89 - 4 * q^91 - 16 * q^92 - 8 * q^94 - 28 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/522\mathbb{Z}\right)^\times$.

$n$	$379$	$407$
$\chi(n)$	$-\zeta_{28}^{4}$	$1$

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} $

91.1

0.781831	−	0.623490i
−0.781831	+	0.623490i
0.781831	+	0.623490i
−0.781831	−	0.623490i
−0.433884	−	0.900969i
0.433884	+	0.900969i
−0.974928	−	0.222521i
0.974928	+	0.222521i
−0.974928	+	0.222521i
0.974928	−	0.222521i
−0.433884	+	0.900969i
0.433884	−	0.900969i

−0.974928

−

0.222521i

0.900969

0.433884i

0.136585

−

0.598418i

−2.59161

1.24805i

−0.781831

−

0.623490i

−0.266321

0.553021i

91.2

0.974928

0.222521i

0.900969

0.433884i

0.308457

−

1.35144i

1.78967

−

0.861862i

0.781831

0.623490i

0.601447

−

1.24892i

109.1

−0.974928

0.222521i

0.900969

−

0.433884i

0.136585

0.598418i

−2.59161

−

1.24805i

−0.781831

0.623490i

−0.266321

−

0.553021i

109.2

0.974928

−

0.222521i

0.900969

−

0.433884i

0.308457

1.35144i

1.78967

0.861862i

0.781831

−

0.623490i

0.601447

1.24892i

325.1

−0.781831

−

0.623490i

0.222521

0.974928i

0.892482

−

1.11914i

0.904459

−

3.96269i

0.433884

−

0.900969i

−1.39554

0.318523i

325.2

0.781831

0.623490i

0.222521

0.974928i

−2.13946

2.68280i

−0.349501

1.53126i

−0.433884

0.900969i

−3.34540

0.763565i

361.1

−0.433884

0.900969i

−0.623490

−

0.781831i

−1.63762

−

0.788637i

0.882702

−

1.10687i

0.974928

−

0.222521i

1.42108

−

1.13327i

361.2

0.433884

−

0.900969i

−0.623490

−

0.781831i

3.43956

1.65640i

1.36428

−

1.71075i

−0.974928

0.222521i

2.98474

−

2.38025i

415.1

−0.433884

−

0.900969i

−0.623490

0.781831i

−1.63762

0.788637i

0.882702

1.10687i

0.974928

0.222521i

1.42108

1.13327i

415.2

0.433884

0.900969i

−0.623490

0.781831i

3.43956

−

1.65640i

1.36428

1.71075i

−0.974928

−

0.222521i

2.98474

2.38025i

469.1

−0.781831

0.623490i

0.222521

−

0.974928i

0.892482

1.11914i

0.904459

3.96269i

0.433884

0.900969i

−1.39554

−

0.318523i

469.2

0.781831

−

0.623490i

0.222521

−

0.974928i

−2.13946

−

2.68280i

−0.349501

−

1.53126i

−0.433884

−

0.900969i

−3.34540

−

0.763565i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
29.e	even	14	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	522.2.n.a		12
3.b	odd	2	1		58.2.e.a	✓	12
12.b	even	2	1		464.2.y.c		12
29.e	even	14	1	inner	522.2.n.a		12
87.h	odd	14	1		58.2.e.a	✓	12
87.h	odd	14	1		1682.2.b.j		12
87.j	odd	14	1		1682.2.b.j		12
87.k	even	28	1		1682.2.a.r		6
87.k	even	28	1		1682.2.a.s		6
348.t	even	14	1		464.2.y.c		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
58.2.e.a	✓	12	3.b	odd	2	1
58.2.e.a	✓	12	87.h	odd	14	1
464.2.y.c		12	12.b	even	2	1
464.2.y.c		12	348.t	even	14	1
522.2.n.a		12	1.a	even	1	1	trivial
522.2.n.a		12	29.e	even	14	1	inner
1682.2.a.r		6	87.k	even	28	1
1682.2.a.s		6	87.k	even	28	1
1682.2.b.j		12	87.h	odd	14	1
1682.2.b.j		12	87.j	odd	14	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{12} - 2 T_{5}^{11} - 4 T_{5}^{10} - 18 T_{5}^{9} + 152 T_{5}^{8} + 64 T_{5}^{7} + 42 T_{5}^{6} + \cdots + 841 $ T5^12 - 2*T5^11 - 4*T5^10 - 18*T5^9 + 152*T5^8 + 64*T5^7 + 42*T5^6 + 596*T5^5 + 775*T5^4 - 732*T5^3 + 2768*T5^2 - 870*T5 + 841 acting on $S_{2}^{\mathrm{new}}(522, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} - T^{10} + \cdots + 1 $ T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$3$	$ T^{12} $ T^12
$5$	$ T^{12} - 2 T^{11} + \cdots + 841 $ T^12 - 2T^11 - 4T^10 - 18T^9 + 152T^8 + 64T^7 + 42T^6 + 596T^5 + 775T^4 - 732T^3 + 2768T^2 - 870*T + 841
$7$	$ T^{12} - 4 T^{11} + \cdots + 12769 $ T^12 - 4T^11 + 19T^10 - 18T^9 - 53T^8 + 158T^7 + 483T^6 - 2798T^5 + 7251T^4 - 12898T^3 + 19879T^2 - 19888*T + 12769
$11$	$ T^{12} - 4 T^{10} + \cdots + 841 $ T^12 - 4T^10 + 226T^8 + 1400T^7 - 1394T^6 - 5600T^5 + 36201T^4 + 71400T^3 + 181046T^2 - 24360*T + 841
$13$	$ T^{12} + 26 T^{11} + \cdots + 38809 $ T^12 + 26T^11 + 353T^10 + 3132T^9 + 19752T^8 + 90630T^7 + 297745T^6 + 663342T^5 + 925314T^4 + 758460T^3 + 481442T^2 + 170602*T + 38809
$17$	$ T^{12} + 94 T^{10} + \cdots + 12769 $ T^12 + 94T^10 + 2613T^8 + 28986T^6 + 129686T^4 + 198140*T^2 + 12769
$19$	$ T^{12} + 26 T^{10} + \cdots + 175561 $ T^12 + 26T^10 + 308T^9 + 1145T^8 - 6720T^7 - 32726T^6 - 30268T^5 + 254494T^4 + 876764T^3 + 1571440T^2 + 926828T + 175561
$23$	$ T^{12} - 16 T^{11} + \cdots + 729 $ T^12 - 16T^11 + 115T^10 - 438T^9 + 1622T^8 - 10856T^7 + 59311T^6 - 119874T^5 + 89136T^4 - 12744T^3 + 36774T^2 + 1944*T + 729
$29$	$ T^{12} + \cdots + 594823321 $ T^12 + 18T^11 + 211T^10 + 1610T^9 + 9687T^8 + 46672T^7 + 241023T^6 + 1353488T^5 + 8146767T^4 + 39266290T^3 + 149236291T^2 + 369200682*T + 594823321
$31$	$ T^{12} + \cdots + 5377435561 $ T^12 - 28T^11 + 346T^10 - 2408T^9 + 11468T^8 - 74662T^7 + 783803T^6 - 6310864T^5 + 32916146T^4 - 137856838T^3 + 679733115T^2 - 2866362128*T + 5377435561
$37$	$ T^{12} + 28 T^{11} + \cdots + 3736489 $ T^12 + 28T^11 + 321T^10 + 1820T^9 + 6763T^8 + 40908T^7 + 197357T^6 + 100996T^5 - 243711T^4 - 140518T^3 + 2498999T^2 - 5845392*T + 3736489
$41$	$ T^{12} + 236 T^{10} + \cdots + 60171049 $ T^12 + 236T^10 + 18526T^8 + 566642T^6 + 7383245T^4 + 37754062*T^2 + 60171049
$43$	$ T^{12} + 28 T^{11} + \cdots + 8637721 $ T^12 + 28T^11 + 255T^10 + 280T^9 - 7243T^8 - 13804T^7 + 148231T^6 - 213108T^5 + 212265T^4 - 1826048T^3 + 3158419T^2 + 2221884*T + 8637721
$47$	$ T^{12} - 14 T^{11} + \cdots + 24571849 $ T^12 - 14T^11 + 131T^10 + 1022T^9 - 20023T^8 + 97342T^7 + 423401T^6 - 7363482T^5 + 43812827T^4 - 140275464T^3 + 252839737T^2 - 140253358*T + 24571849
$53$	$ T^{12} + \cdots + 2202143329 $ T^12 + 30T^11 + 640T^10 + 9524T^9 + 114287T^8 + 1098964T^7 + 8757938T^6 + 55984480T^5 + 277809656T^4 + 1001695258T^3 + 2412823164T^2 + 3431959218*T + 2202143329
$59$	$ (T^{6} - 24 T^{5} + \cdots + 35869)^{2} $ (T^6 - 24T^5 + 65T^4 + 1828T^3 - 12190T^2 + 11160*T + 35869)^2
$61$	$ T^{12} + \cdots + 105805126729 $ T^12 + 28T^11 + 299T^10 + 1120T^9 + 4960T^8 + 89810T^7 - 187699T^6 - 11713338T^5 - 95589478T^4 + 289981622T^3 + 9821016236T^2 - 54059085738*T + 105805126729
$67$	$ T^{12} + \cdots + 2228500849 $ T^12 + 16T^11 + 87T^10 - 1480T^9 - 12707T^8 + 98328T^7 + 2754871T^6 + 27402360T^5 + 173792197T^4 + 586211896T^3 + 1609765447T^2 + 2336180016*T + 2228500849
$71$	$ T^{12} + \cdots + 15150901921 $ T^12 + 30T^11 + 493T^10 + 5072T^9 + 35607T^8 + 116346T^7 + 284809T^6 - 1920682T^5 + 25453433T^4 - 100086598T^3 + 1096223439T^2 - 5851897238*T + 15150901921
$73$	$ T^{12} - 42 T^{11} + \cdots + 5938969 $ T^12 - 42T^11 + 815T^10 - 9492T^9 + 73005T^8 - 387702T^7 + 1449391T^6 - 3807930T^5 + 6853697T^4 - 7918288T^3 + 5685447T^2 - 4947110*T + 5938969
$79$	$ T^{12} + \cdots + 4893282304 $ T^12 - 28T^11 + 412T^10 - 2912T^9 + 22352T^8 - 352128T^7 + 7132992T^6 - 77815808T^5 + 546515712T^4 - 2417945600T^3 + 6275587072T^2 - 8649424896*T + 4893282304
$83$	$ T^{12} - 35 T^{10} + \cdots + 2019241 $ T^12 - 35T^10 + 756T^9 + 5782T^8 - 57232T^7 + 456827T^6 - 1101716T^5 + 6076588T^4 - 11286072T^3 + 10036180T^2 - 5013288T + 2019241
$89$	$ T^{12} + 14 T^{11} + \cdots + 34656769 $ T^12 + 14T^11 + 287T^10 - 448T^9 - 931T^8 - 10206T^7 + 161959T^6 + 595742T^5 - 1875475T^4 - 7241416T^3 + 3255511T^2 + 31071586*T + 34656769
$97$	$ T^{12} + \cdots + 10446475264 $ T^12 + 28T^11 + 332T^10 - 2240T^9 - 108960T^8 - 996352T^7 + 3634112T^6 + 135822848T^5 + 1063453696T^4 + 4113951744T^3 + 9953232896T^2 + 14103068672*T + 10446475264
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	522.n (of order \(14\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{28}^{9} q^{2} - \zeta_{28}^{4} q^{4} + ( - \zeta_{28}^{11} + \cdots + \zeta_{28}) q^{5} + ( - \zeta_{28}^{10} - 2 \zeta_{28}^{9} + \cdots + 1) q^{7} + (\zeta_{28}^{11} - \zeta_{28}^{9} + \cdots - \zeta_{28}) q^{8} + \cdots + ( - 2 \zeta_{28}^{11} + 2 \zeta_{28}^{9} + \cdots - 2) q^{98} +O(q^{100}) \) q - z^9 * q^2 - z^4 * q^4 + (-z^11 + z^7 + z^3 + z^2 + z) * q^5 + (-z^10 - 2z^9 + 2z^7 - z^6 - z^5 + z^4 + z^3 - z^2 - z + 1) * q^7 + (z^11 - z^9 + z^7 - z^5 + z^3 - z) * q^8 + (-z^11 - 2z^10 + z^8 - 2z^6 + z^4 + 1) * q^10 + (-z^11 + 2z^10 + z^9 - z^7 - 2z^6 + z^5 + z) * q^11 + (-2z^11 + z^10 + z^9 - z^7 + z^6 - 2z^3 + 3z^2 + z - 3) q^13 + (z^8 - z^7 - z^6 - z^4 - z^3 + z^2) * q^14 + z^8 * q^16 + (z^10 + 2z^8 - 2z^7 + 2z^6 + z^4) q^17 + (z^11 + 4z^10 - 2z^8 - z^7 + 2z^6 - 2z^5 - 2z^4 + 2z^3 + 2z^2 + z - 2) q^19 + (-z^11 - z^7 - z^6 - z^5 - z) * q^20 + (-z^10 - z^6 + 2z^5 + z^4 - z^2 - 2z + 1) * q^22 + (-2z^10 - 2z^9 + z^7 - z^6 + z^4 - z^3 + 2z + 2) q^23 + (-2z^11 + 4z^9 + 3z^8 - 2z^7 - z^6 + 4z^5 + 2z^4 - z^2 + 2z + 3) q^25 + (-3z^11 + z^10 + 3z^9 - 2z^8 + z^5 - z^4 + z^2 + z - 2) q^26 + (z^10 - z^9 - z^8 + z^7 + z^6 - z^5 - z^4 + 2z^3 - 2z - 1) * q^28 + (2z^11 - 4z^9 + 2z^7 - 3z^6 - 3z^5 - z^4 + 4z^3 - z^2 - z - 1) * q^29 + (-z^11 - 3z^9 + 2z^8 - z^7 - 2z^6 - z^4 - 3z^3 - z^2 + 3z + 3) q^31 + z^3 * q^32 + (-z^11 + z^9 - z^7 + 2z^5 + z^3 - 2z^2 + 3z) q^34 + (-4z^11 - 2z^10 + z^9 + 2z^8 - 2z^7 - 4z^6 + 2z^5 + 4z^4 - 2z^3 - 2z^2 + 3z + 2) * q^35 + (4z^11 + 3z^10 - 3z^9 - 4z^8 + 3z^7 + 2z^6 - 3z^5 + 3z^3 + z^2 - 4z - 4) q^37 + (-3z^10 + 2z^8 + 2z^7 - z^6 + 2z^5 + 2z^4 - 3z^2) * q^38 + (z^10 - z^6 - z^2 - z - 1) * q^40 + (-2z^11 + z^10 - 2z^9 - z^8 + 4z^7 - z^6 - 2z^5 + z^4 - 2z^3) q^41 + (-2z^10 + 2z^9 + z^7 - 4z^5 + z^3 + 2z - 2) * q^43 + (2z^10 - z^7 - z^3 + 2) q^44 + (-z^11 - z^10 - z^9 - z^8 - z^7 + z^6 - z^5 - 3z^4 - z^3 + 2z^2 - 1) * q^46 + (z^11 + 4z^10 + z^9 - 4z^8 - z^7 + 4z^6 - z^5 - 8z^4 + 5z^2 - 3z - 3) * q^47 + (2z^11 - 2z^9 + 2z^8 + 2z^4 - 2z^3 + 2z) * q^49 + (-z^11 - 2z^10 - z^9 - 2z^7 - 2z^6 + 2z^5 + 4z^4 + z^3 - 2z^2 + z + 4) * q^50 + (-z^10 + z^9 + z^7 - 3z^6 + 3z^4 - z^3 - z + 1) * q^52 + (4z^11 - z^10 - z^9 + z^8 - 2z^7 - 2z^5 - 3z^4 + 3z^3 + 2z^2 + z - 3) * q^53 + (2z^11 + 2z^10 - 2z^9 - 4z^8 + 4z^6 + z^5 - z^4 - 4z^3 + 3z^2 + z) q^55 + (z^11 + 2z^8 + z^7 - 2z^6 + z^4 - z^2 + 1) * q^56 + (2z^11 - 3z^10 + 4z^8 + z^7 - 2z^6 - z^5 + z^3 - 2z^2 - 4z + 1) * q^58 + (z^11 + 2z^10 + 4z^9 - 4z^8 - 2z^7 + 4z^6 - 2z^4 - 5z^3 + 4z + 2) * q^59 + (4z^11 - 4z^9 - 2z^6 + 4z^5 - z^4 + 3z^3 - z^2 + 4z - 2) * q^61 + (2z^11 - 4z^9 - 3z^8 + z^7 + 2z^6 - z^5 - 6z^4 + 3z^3 + 2z^2 - 3z - 3) * q^62 + (-z^10 + z^8 - z^6 + z^4 - z^2 + 1) * q^64 + (2z^11 - 3z^10 + 6z^9 - 4z^7 - 2z^6 + 4z^5 + 4z^4 - 2z^3 - 4z^2 + 2) q^65 + (-z^11 - 4z^10 + 3z^9 - 3z^7 - 3z^5 + 3z^3 - 4z^2 - z) * q^67 + (2z^11 - 4z^10 + z^8 - 2z^6 + 2z^4 - 2z^2 + 3) q^68 + (-2z^11 - z^10 + 2z^9 - 2z^8 - 4z^7 - 2z^6 + 2z^5 - z^4 - 2z^3) q^70 + (3z^11 + 3z^10 - 4z^9 + z^8 + 4z^7 + 3z^6 - 5z^5 + 8z^3 + 4z^2 - 4z - 4) q^71 + (3z^10 - z^8 + 2z^5 - z^2 + 3) * q^73 + (-z^11 + z^10 + 4z^9 + 3z^8 + z^6 + 3z^5 - 4z^3 + 2z) q^74 + (z^11 + 2z^9 - 2z^7 - z^5 + 2z^2 + z + 2) q^76 + (-4z^11 + z^8 - z^7 + z^6 + 6z^5 - 6z^3 - z^2 + z - 1) q^77 + (-4z^11 + 6z^9 - 2z^6 + 4z^4 - 6z^3 - 4z^2 + 4z + 4) q^79 + (z^11 + z^10 + z^9 + z^5 - z) * q^80 + (-z^11 + 2z^10 + z^9 - 2z^8 - z^7 + 2z^5 - 4z^4 - 2z^3 + 6z^2 - 4) * q^82 + (-z^11 + 2z^9 + 2z^6 - z^5 + 4z^4 - 3z^3 + 2z^2 + 2z) * q^83 + (7z^11 + z^10 - z^9 - z^8 + 7z^7 + 2z^6 - 3z^4 + 4z^3 + z^2 - 4z + 1) * q^85 + (-3z^10 + 2z^9 + z^8 - z^6 - 2z^5 + 3z^4 - 3) * q^86 + (z^10 - 2z^9 - z^8 + z^6 + 2z^5 - z^4 - 1) * q^88 + (4z^11 + 6z^10 - 2z^9 + 4z^7 + 6z^6 - 2z^4 + 2z^3 + 3z^2 - 2z - 4) q^89 + (-4z^11 + 8z^9 - 2z^8 - 7z^7 + 4z^6 + 3z^5 - 5z^3 + 4z^2 + z - 2) * q^91 + (z^11 + z^10 - 2z^9 - z^8 + 3z^7 - 4z^5 - 2z^4 + 2z^3 - 2z - 2) * q^92 + (3z^11 + 3z^10 - 5z^9 + 8z^7 + z^6 - 4z^5 + z^4 + 4z^3 - z^2 - 4z - 1) q^94 + (7z^11 + z^10 - 4z^9 - z^8 + 5z^7 - 5z^5 + z^4 + 4z^3 - z^2 - 7z) * q^95 + (-6z^11 + 2z^8 + 2z^7 + 4z^6 - 2z^5 + 2z^3 - 4z^2 - 2z - 2) * q^97 + (-2z^11 + 2z^9 - 2z^8 - 2z^7 + 4z^6 + 2z^5 - 4z^4 + 2z^2 + 2z - 2) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{4} + 2 q^{5} + 4 q^{7} - 26 q^{13} - 2 q^{16} - 2 q^{20} + 4 q^{22} + 16 q^{23} + 22 q^{25} - 14 q^{26} - 4 q^{28} - 18 q^{29} + 28 q^{31} - 4 q^{34} - 4 q^{35} - 28 q^{37} - 22 q^{38} - 14 q^{40}+ \cdots - 28 q^{97}+O(q^{100}) \) 12 * q + 2 * q^4 + 2 * q^5 + 4 * q^7 - 26 * q^13 - 2 * q^16 - 2 * q^20 + 4 * q^22 + 16 * q^23 + 22 * q^25 - 14 * q^26 - 4 * q^28 - 18 * q^29 + 28 * q^31 - 4 * q^34 - 4 * q^35 - 28 * q^37 - 22 * q^38 - 14 * q^40 - 28 * q^43 + 28 * q^44 + 14 * q^47 - 8 * q^49 + 28 * q^50 - 2 * q^52 - 30 * q^53 + 28 * q^55 - 10 * q^58 + 48 * q^59 - 28 * q^61 - 10 * q^62 + 2 * q^64 - 2 * q^65 - 16 * q^67 + 14 * q^68 - 30 * q^71 + 42 * q^73 - 2 * q^74 + 28 * q^76 - 14 * q^77 + 28 * q^79 + 2 * q^80 - 20 * q^82 + 28 * q^85 - 52 * q^86 - 4 * q^88 - 14 * q^89 - 4 * q^91 - 16 * q^92 - 8 * q^94 - 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

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Properties

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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	522.2.n.a

Level	$522$
Weight	$2$
Character orbit	522.n
Analytic conductor	$4.168$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(4.16819098551\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{14})\)
Coefficient field:	\(\Q(\zeta_{28})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 58)
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

\(n\):		e.g. 2-40 or 80-90
Embeddings:		e.g. 1-3 or 91.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} - T^{10} + \cdots + 1 \) T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$3$	\( T^{12} \) T^12
$5$	\( T^{12} - 2 T^{11} + \cdots + 841 \) T^12 - 2T^11 - 4T^10 - 18T^9 + 152T^8 + 64T^7 + 42T^6 + 596T^5 + 775T^4 - 732T^3 + 2768T^2 - 870*T + 841
$7$	\( T^{12} - 4 T^{11} + \cdots + 12769 \) T^12 - 4T^11 + 19T^10 - 18T^9 - 53T^8 + 158T^7 + 483T^6 - 2798T^5 + 7251T^4 - 12898T^3 + 19879T^2 - 19888*T + 12769
$11$	\( T^{12} - 4 T^{10} + \cdots + 841 \) T^12 - 4T^10 + 226T^8 + 1400T^7 - 1394T^6 - 5600T^5 + 36201T^4 + 71400T^3 + 181046T^2 - 24360*T + 841
$13$	\( T^{12} + 26 T^{11} + \cdots + 38809 \) T^12 + 26T^11 + 353T^10 + 3132T^9 + 19752T^8 + 90630T^7 + 297745T^6 + 663342T^5 + 925314T^4 + 758460T^3 + 481442T^2 + 170602*T + 38809
$17$	\( T^{12} + 94 T^{10} + \cdots + 12769 \) T^12 + 94T^10 + 2613T^8 + 28986T^6 + 129686T^4 + 198140*T^2 + 12769
$19$	\( T^{12} + 26 T^{10} + \cdots + 175561 \) T^12 + 26T^10 + 308T^9 + 1145T^8 - 6720T^7 - 32726T^6 - 30268T^5 + 254494T^4 + 876764T^3 + 1571440T^2 + 926828T + 175561
$23$	\( T^{12} - 16 T^{11} + \cdots + 729 \) T^12 - 16T^11 + 115T^10 - 438T^9 + 1622T^8 - 10856T^7 + 59311T^6 - 119874T^5 + 89136T^4 - 12744T^3 + 36774T^2 + 1944*T + 729
$29$	\( T^{12} + \cdots + 594823321 \) T^12 + 18T^11 + 211T^10 + 1610T^9 + 9687T^8 + 46672T^7 + 241023T^6 + 1353488T^5 + 8146767T^4 + 39266290T^3 + 149236291T^2 + 369200682*T + 594823321
$31$	\( T^{12} + \cdots + 5377435561 \) T^12 - 28T^11 + 346T^10 - 2408T^9 + 11468T^8 - 74662T^7 + 783803T^6 - 6310864T^5 + 32916146T^4 - 137856838T^3 + 679733115T^2 - 2866362128*T + 5377435561
$37$	\( T^{12} + 28 T^{11} + \cdots + 3736489 \) T^12 + 28T^11 + 321T^10 + 1820T^9 + 6763T^8 + 40908T^7 + 197357T^6 + 100996T^5 - 243711T^4 - 140518T^3 + 2498999T^2 - 5845392*T + 3736489
$41$	\( T^{12} + 236 T^{10} + \cdots + 60171049 \) T^12 + 236T^10 + 18526T^8 + 566642T^6 + 7383245T^4 + 37754062*T^2 + 60171049
$43$	\( T^{12} + 28 T^{11} + \cdots + 8637721 \) T^12 + 28T^11 + 255T^10 + 280T^9 - 7243T^8 - 13804T^7 + 148231T^6 - 213108T^5 + 212265T^4 - 1826048T^3 + 3158419T^2 + 2221884*T + 8637721
$47$	\( T^{12} - 14 T^{11} + \cdots + 24571849 \) T^12 - 14T^11 + 131T^10 + 1022T^9 - 20023T^8 + 97342T^7 + 423401T^6 - 7363482T^5 + 43812827T^4 - 140275464T^3 + 252839737T^2 - 140253358*T + 24571849
$53$	\( T^{12} + \cdots + 2202143329 \) T^12 + 30T^11 + 640T^10 + 9524T^9 + 114287T^8 + 1098964T^7 + 8757938T^6 + 55984480T^5 + 277809656T^4 + 1001695258T^3 + 2412823164T^2 + 3431959218*T + 2202143329
$59$	\( (T^{6} - 24 T^{5} + \cdots + 35869)^{2} \) (T^6 - 24T^5 + 65T^4 + 1828T^3 - 12190T^2 + 11160*T + 35869)^2
$61$	\( T^{12} + \cdots + 105805126729 \) T^12 + 28T^11 + 299T^10 + 1120T^9 + 4960T^8 + 89810T^7 - 187699T^6 - 11713338T^5 - 95589478T^4 + 289981622T^3 + 9821016236T^2 - 54059085738*T + 105805126729
$67$	\( T^{12} + \cdots + 2228500849 \) T^12 + 16T^11 + 87T^10 - 1480T^9 - 12707T^8 + 98328T^7 + 2754871T^6 + 27402360T^5 + 173792197T^4 + 586211896T^3 + 1609765447T^2 + 2336180016*T + 2228500849
$71$	\( T^{12} + \cdots + 15150901921 \) T^12 + 30T^11 + 493T^10 + 5072T^9 + 35607T^8 + 116346T^7 + 284809T^6 - 1920682T^5 + 25453433T^4 - 100086598T^3 + 1096223439T^2 - 5851897238*T + 15150901921
$73$	\( T^{12} - 42 T^{11} + \cdots + 5938969 \) T^12 - 42T^11 + 815T^10 - 9492T^9 + 73005T^8 - 387702T^7 + 1449391T^6 - 3807930T^5 + 6853697T^4 - 7918288T^3 + 5685447T^2 - 4947110*T + 5938969
$79$	\( T^{12} + \cdots + 4893282304 \) T^12 - 28T^11 + 412T^10 - 2912T^9 + 22352T^8 - 352128T^7 + 7132992T^6 - 77815808T^5 + 546515712T^4 - 2417945600T^3 + 6275587072T^2 - 8649424896*T + 4893282304
$83$	\( T^{12} - 35 T^{10} + \cdots + 2019241 \) T^12 - 35T^10 + 756T^9 + 5782T^8 - 57232T^7 + 456827T^6 - 1101716T^5 + 6076588T^4 - 11286072T^3 + 10036180T^2 - 5013288T + 2019241
$89$	\( T^{12} + 14 T^{11} + \cdots + 34656769 \) T^12 + 14T^11 + 287T^10 - 448T^9 - 931T^8 - 10206T^7 + 161959T^6 + 595742T^5 - 1875475T^4 - 7241416T^3 + 3255511T^2 + 31071586*T + 34656769
$97$	\( T^{12} + \cdots + 10446475264 \) T^12 + 28T^11 + 332T^10 - 2240T^9 - 108960T^8 - 996352T^7 + 3634112T^6 + 135822848T^5 + 1063453696T^4 + 4113951744T^3 + 9953232896T^2 + 14103068672*T + 10446475264
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\(n\)	\(379\)	\(407\)
\(\chi(n)\)	\(-\zeta_{28}^{4}\)	\(1\)