LMFDB - Newform orbit 115.2.g.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,2,Mod(6,115)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 18]))

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

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

chi := DirichletCharacter("115.6");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 115 = 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	115.g (of order $11$, 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:	$0.918279623245$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\Q(\zeta_{22})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

$q$-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 $\zeta_{22}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\zeta_{22}^{7} + \zeta_{22}^{5} + \cdots + \zeta_{22}) q^{2} + ( - \zeta_{22}^{9} + \zeta_{22}^{8} + \cdots + 1) q^{3} + (\zeta_{22}^{9} + \zeta_{22}^{7} + \cdots - 1) q^{4} - \zeta_{22}^{2} q^{5} + \cdots + (2 \zeta_{22}^{9} + \cdots + 2 \zeta_{22}) q^{99} +O(q^{100}) $ q + (z^7 + z^5 - z^4 + z^3 + z) * q^2 + (-z^9 + z^8 - z^7 + z^6 - z^5 + z^4 - z^3 + z^2 - z + 1) * q^3 + (z^9 + z^7 + z^5 - z^4 + 2z^3 - 2z^2 + z - 1) * q^4 - z^2 * q^5 + (z^6 + z^4 - z^3 + z^2 + 1) * q^6 + (z^9 - z^6 + z^5 - z^4 + z) * q^7 + (-z^9 + z^8 - z^7 + z^6 - z^5 + z^4 - z^2 - z - 1) * q^8 + 2z^9 q^9 + (-z^9 - z^7 + z^6 - z^5 - z^3) * q^10 + (z^9 - 2z^8 - 2z^6 + z^5 - z^2 + z) * q^11 + (z^9 + z^7 + z^5 + z^2 - z) * q^12 + (-z^8 - z^7 - z^5 - z^4) * q^13 + (z^7 - z^6 - 2z^4 + 2z^3 + z - 1) * q^14 - z * q^15 + (z^9 - 2z^8 + z^6 + z^5 + z^4 - 2z^2 + z) * q^16 + (-2z^6 - z^5 + z^4 - z^3 + z^2 - z - 2) q^17 + (2z^9 - 2z^8 + 2z^7 - 2z^6 - 2z^4 - 2) q^18 + (-2z^9 + z^8 - 2z^7 + z^4 - 3z^3 + 3z^2 - z) * q^19 + (-z^9 - z^7 + z^6 - 2z^5 + 2z^4 - z^3 + z^2 + 1) * q^20 + (z^8 - z^5 + z^4 - z^3 + 1) * q^21 + (-2z^9 - 2z^8 + z^7 - z^6 + z^5 - z^4 + 2z^3 + 2z^2) * q^22 + (-2z^9 + 2z^8 + 2z^7 + z^6 - 2z^2 - 2) * q^23 + (z^9 - 2z^8 + 2z^7 - 2z^6 + 2z^5 - 2z^4 + 2z^3 - z^2 - 2) * q^24 + z^4 * q^25 + (-3z^9 + z^8 - 3z^7 + z^6 - 3z^5 + 3z^4 - z^3 + 3z^2 - z + 3) q^26 + 5z^8 q^27 + (-z^9 + 2z^8 - 2z^7 + z^6) * q^28 + (-5z^9 + 5z^8 + 4z^6 - 4z^5 + 6z^4 - 4z^3 + 6z^2 - 4z + 4) * q^29 + (-z^8 - z^6 + z^5 - z^4 - z^2) * q^30 + (-2z^9 + z^8 - 3z^7 + 3z^6 - z^5 + 2z^4 - z^2 + z - 1) * q^31 + (2z^9 + 3z^7 - z^6 + 2z^5 - 2z^4 + 2z^3 - 2z^2 + z - 3) * q^32 + (z^8 - 2z^7 - 2z^5 + z^4 - z + 1) * q^33 + (z^9 - 4z^8 - z^7 - 4z^6 + z^5 - z^3 + z^2 - z + 1) * q^34 + (z^8 - z^7 + z^6 - z^3 + 1) * q^35 + (-2z^8 - 2z^6 - 2z^4 - 2z + 2) * q^36 + (z^9 - 6z^4 + 6z^3) * q^37 + (-2z^9 + 2z^4 + 3z^2 - 2z + 3) * q^38 + (-z^7 - z^6 - z^4 - z^3) * q^39 + (-z^5 + 2z^4 + 2z^2 - z) * q^40 + (-z^9 - 4z^8 + 4z^7 + z^6 - z^4 - 2z^3 - 2z^2 - 2z - 1) q^41 + (z^9 - z^8 + z^7 - z^4 - z^3 + z^2 + z) * q^42 + (3z^9 - 7z^8 + 4z^7 - 4z^6 + 6z^5 - 6z^4 + 4z^3 - 4z^2 + 7z - 3) q^43 + (3z^7 - 2z^6 + 4z^5 - z^4 + 4z^3 - 2z^2 + 3z) * q^44 + 2 * q^45 + (3z^8 - 4z^7 + 3z^6 - 3z^5 + z^4 - 5z^3 - 4z^2 - z) * q^46 + (4z^9 - z^7 - z^6 + z^5 + z^4 - 4z^2 - 6) * q^47 + (z^8 - 2z^7 + z^5 + z^4 + z^3 - 2z + 1) * q^48 + (-4z^9 + 3z^8 - 5z^7 + 4z^6 - 4z^5 + 4z^4 - 4z^3 + 5z^2 - 3z + 4) q^49 + (z^9 - z^8 + z^7 + z^5 - 1) * q^50 + (2z^9 - 2z^8 + 2z^7 - 2z^6 - 3z^4 + 3z^3 - 3z^2 + 3z - 3) * q^51 + (3z^6 + 3z^4 + z^3 + 3z^2 + 3) q^52 + (5z^9 + 2z^8 + z^7 + z^6 + z^4 + z^3 + 2z^2 + 5z) * q^53 + (5z^9 - 5z^4 - 5z^2 + 5z - 5) * q^54 + (2z^9 + z^7 - 2z^6 + 2z^5 - z^4 + z^3 - 2z^2 + 2z - 1) q^55 + (-3z^9 + 4z^8 - 3z^7 + 4z^6 - 3z^5 + 4z^4 - 3z^3 - 3z + 3) * q^56 + (-2z^8 + z^7 - 2z^6 + z^3 - 3z^2 + 3z - 1) * q^57 + (8z^9 - z^8 + 7z^7 - z^6 + 8z^5 - 2z^3 - z^2 + z + 2) * q^58 + (2z^7 + 8z^6 + 2z^5 - z + 1) q^59 + (-z^9 - z^7 - z^4 + z^3 + 1) * q^60 + (3z^8 + 2z^7 - 2z^6 - 3z^5 - z) * q^61 + (-3z^9 + 4z^8 - 6z^7 + 10z^6 - 8z^5 + 10z^4 - 6z^3 + 4z^2 - 3z) q^62 + (2z^9 - 2z^8 - 2z^6 + 2z^5 + 2z - 2) q^63 + (5z^9 - 4z^8 + 4z^7 - 5z^6 - 8z^4 + 6z^3 - 8z^2 + 6z - 8) * q^64 + (2z^9 - z^8 + 2z^7 + z^5 - z^4 + z^3 - z^2 + z - 1) * q^65 + (-2z^8 - 2z^7 + z^6 - z^5 + z^4 - z^3 + 2z^2 + 2z) * q^66 + (z^8 - z^7 - 4z^6 - 5z^4 - 4z^2 - z + 1) q^67 + (-4z^9 + z^7 - 4z^6 + 4z^5 - z^4 + 4z^2 + 3) * q^68 + (2z^9 - 4z^8 + 4z^7 + 3z^5 - 2z^4 + 2z^3 - 2z^2 - 2) q^69 + (-z^9 + z^8 + 2z^6 - 2z^5 - z^3 + z^2) * q^70 + (-z^8 - 4z^7 + 3z^6 + z^5 + z^4 + z^3 + 3z^2 - 4z - 1) * q^71 + (-4z^9 + 2z^8 - 2z + 4) q^72 + (2z^9 - 3z^8 + 2z^7 - 4z^5 - z^4 - z^3 + z^2 + z + 4) * q^73 + (z^9 + 5z^8 - 5z^7 - z^6 - z^4 + 6z^3 - 6z^2 + 6z - 1) q^74 + z^3 * q^75 + (-z^9 + 4z^8 - z^6 + 6z^5 - z^4 + 4z^2 - z) q^76 + (2z^9 - 4z^8 + 4z^5 - 2z^4 - z^2 + z - 1) * q^77 + (-3z^9 - 2z^7 - 2z^5 + 2z^2 + 2) * q^78 + (z^9 + 2z^8 + 3z^7 - 2z^6 + 3z^5 + 2z^4 + z^3 - 5z + 5) * q^79 + (2z^9 - 3z^8 + z^7 - 3z^6 + 2z^5 + z^3 - 2z^2 + 2z - 1) * q^80 - z^7 * q^81 + (-6z^9 + z^8 - z^7 - 2z^6 - z^5 + z^4 - 6z^3 - 8z + 8) * q^82 + (-6z^9 - z^7 + 5z^6 - 4z^5 + z^4 - z^3 + 4z^2 - 5z + 1) q^83 + (-z^8 + 2z^7 - 2z^6 + z^5) * q^84 + (2z^8 + z^7 - z^6 + z^5 - z^4 + z^3 + 2z^2) * q^85 + (-6z^9 + 6z^8 - 2z^5 + 7z^3 - 2z) q^86 + (-4z^9 - z^8 + z^7 + 4z^6 + 2z^3 + 2z) * q^87 + (6z^8 - z^5 + 2z^3 - 2z^2 + 1) q^88 + (-11z^9 + 6z^8 - 5z^7 + 4z^6 - 6z^5 + 6z^4 - 4z^3 + 5z^2 - 6z + 11) q^89 + (2z^7 + 2z^5 - 2z^4 + 2z^3 + 2z) q^90 + (-z^9 + z^2 - 2) * q^91 + (-5z^9 + z^8 - 6z^7 + 2z^6 - 9z^5 + 6z^4 - 9z^3 + 4z^2 - z) q^92 + (z^9 - 3z^8 + 2z^7 - 4z^6 + 4z^5 - 2z^4 + 3z^3 - z^2) * q^93 + (-6z^8 - 5z^7 - 8z^5 + z^4 - 8z^3 - 5z - 6) q^94 + (z^9 + z^8 - z^7 + 2z^5 - 2z^4 + z^2 - z - 1) * q^95 + (3z^9 - z^8 + 3z^7 + 2z^5 - z^4 + z^3 - z^2 + z - 2) q^96 + (-z^8 + z^7 + 6z^4 + 4z^2 + 6) * q^97 + (-z^9 + z^8 + 5z^6 - z^5 + 7z^4 - 3z^3 + 7z^2 - z + 5) * q^98 + (2z^9 - 2z^8 + 2z^6 + 2z^5 + 2z^4 - 2z^2 + 2z) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 5 q^{2} + q^{3} - q^{4} + q^{5} + 6 q^{6} + 5 q^{7} - 16 q^{8} + 2 q^{9} - 5 q^{10} + 8 q^{11} + q^{12} - 3 q^{14} - q^{15} + 5 q^{16} - 23 q^{17} - 10 q^{18} - 13 q^{19} + q^{20} + 6 q^{21}+ \cdots + 6 q^{99}+O(q^{100}) $ 10 * q + 5 * q^2 + q^3 - q^4 + q^5 + 6 * q^6 + 5 * q^7 - 16 * q^8 + 2 * q^9 - 5 * q^10 + 8 * q^11 + q^12 - 3 * q^14 - q^15 + 5 * q^16 - 23 * q^17 - 10 * q^18 - 13 * q^19 + q^20 + 6 * q^21 + 4 * q^22 - 21 * q^23 - 6 * q^24 - q^25 + 11 * q^26 - 5 * q^27 - 6 * q^28 + 2 * q^29 + 5 * q^30 - 20 * q^31 - 15 * q^32 + 3 * q^33 + 16 * q^34 + 6 * q^35 + 24 * q^36 + 13 * q^37 + 21 * q^38 - 6 * q^40 - 5 * q^41 + 3 * q^42 + 15 * q^43 + 19 * q^44 + 20 * q^45 - 16 * q^46 - 52 * q^47 + 6 * q^48 + 4 * q^49 - 6 * q^50 - 10 * q^51 + 22 * q^52 + 6 * q^53 - 30 * q^54 + 3 * q^55 + 3 * q^56 + 2 * q^57 + 45 * q^58 + 5 * q^59 + 10 * q^60 - 3 * q^61 - 54 * q^62 - 10 * q^63 - 34 * q^64 - 4 * q^66 + 20 * q^67 + 32 * q^68 - q^69 - 8 * q^70 - 22 * q^71 + 32 * q^72 + 43 * q^73 + q^74 + q^75 - 2 * q^76 + 4 * q^77 + 11 * q^78 + 51 * q^79 + 6 * q^80 - q^81 + 58 * q^82 - 17 * q^83 + 6 * q^84 + q^85 - 9 * q^86 - 2 * q^87 + 7 * q^88 + 57 * q^89 + 10 * q^90 - 22 * q^91 - 43 * q^92 + 20 * q^93 - 81 * q^94 - 9 * q^95 - 7 * q^96 + 52 * q^97 + 24 * q^98 + 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/115\mathbb{Z}\right)^\times$.

$n$	$47$	$51$
$\chi(n)$	$1$	$\zeta_{22}^{4}$

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} $

6.1

0.959493	−	0.281733i
−0.841254	−	0.540641i
−0.415415	−	0.909632i
−0.415415	+	0.909632i
−0.841254	+	0.540641i
0.654861	+	0.755750i
0.142315	−	0.989821i
0.142315	+	0.989821i
0.959493	+	0.281733i
0.654861	−	0.755750i

0.925839

−

2.02730i

0.959493

0.281733i

−1.94306

−

2.24241i

−0.841254

0.540641i

1.45949

−

1.68434i

−0.0125459

0.0872586i

−2.06815

0.607265i

−1.68251

−

1.08128i

0.317178

2.20602i

16.1

1.57028

−

1.81219i

−0.841254

0.540641i

−0.533654

−

3.71165i

−0.415415

−

0.909632i

−0.341254

2.37347i

1.31718

−

0.386758i

−3.52977

−

2.26844i

−0.830830

1.81926i

−2.30075

−

0.675560i

26.1

−0.0125459

−

0.0872586i

−0.415415

0.909632i

1.91153

−

0.561276i

0.654861

−

0.755750i

0.0845850

0.0248364i

−1.30075

0.835939i

−0.146201

−

0.320135i

1.30972

1.51150i

−0.0741615

−

0.0476607i

31.1

−0.0125459

0.0872586i

−0.415415

−

0.909632i

1.91153

0.561276i

0.654861

0.755750i

0.0845850

−

0.0248364i

−1.30075

−

0.835939i

−0.146201

0.320135i

1.30972

−

1.51150i

−0.0741615

0.0476607i

36.1

1.57028

1.81219i

−0.841254

−

0.540641i

−0.533654

3.71165i

−0.415415

0.909632i

−0.341254

−

2.37347i

1.31718

0.386758i

−3.52977

2.26844i

−0.830830

−

1.81926i

−2.30075

0.675560i

41.1

1.31718

0.386758i

0.654861

−

0.755750i

−0.0971309

−

0.0624222i

0.142315

−

0.989821i

1.15486

−

0.742184i

0.925839

2.02730i

−1.90176

−

2.19475i

0.284630

1.97964i

0.570276

−

1.24873i

71.1

−1.30075

−

0.835939i

0.142315

0.989821i

0.162317

0.355426i

0.959493

0.281733i

0.642315

−

1.40647i

1.57028

−

1.81219i

−0.354114

2.46292i

1.91899

−

0.563465i

−1.01255

−

1.16854i

81.1

−1.30075

0.835939i

0.142315

−

0.989821i

0.162317

−

0.355426i

0.959493

−

0.281733i

0.642315

1.40647i

1.57028

1.81219i

−0.354114

−

2.46292i

1.91899

0.563465i

−1.01255

1.16854i

96.1

0.925839

2.02730i

0.959493

−

0.281733i

−1.94306

2.24241i

−0.841254

−

0.540641i

1.45949

1.68434i

−0.0125459

−

0.0872586i

−2.06815

−

0.607265i

−1.68251

1.08128i

0.317178

−

2.20602i

101.1

1.31718

−

0.386758i

0.654861

0.755750i

−0.0971309

0.0624222i

0.142315

0.989821i

1.15486

0.742184i

0.925839

−

2.02730i

−1.90176

2.19475i

0.284630

−

1.97964i

0.570276

1.24873i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.c	even	11	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	115.2.g.a	✓	10
5.b	even	2	1		575.2.k.a		10
5.c	odd	4	2		575.2.p.a		20
23.c	even	11	1	inner	115.2.g.a	✓	10
23.c	even	11	1		2645.2.a.n		5
23.d	odd	22	1		2645.2.a.o		5
115.j	even	22	1		575.2.k.a		10
115.k	odd	44	2		575.2.p.a		20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
115.2.g.a	✓	10	1.a	even	1	1	trivial
115.2.g.a	✓	10	23.c	even	11	1	inner
575.2.k.a		10	5.b	even	2	1
575.2.k.a		10	115.j	even	22	1
575.2.p.a		20	5.c	odd	4	2
575.2.p.a		20	115.k	odd	44	2
2645.2.a.n		5	23.c	even	11	1
2645.2.a.o		5	23.d	odd	22	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{10} - 5T_{2}^{9} + 14T_{2}^{8} - 15T_{2}^{7} - 2T_{2}^{6} + 21T_{2}^{5} + 38T_{2}^{4} - 157T_{2}^{3} + 125T_{2}^{2} + 2T_{2} + 1 $ T2^10 - 5*T2^9 + 14*T2^8 - 15*T2^7 - 2*T2^6 + 21*T2^5 + 38*T2^4 - 157*T2^3 + 125*T2^2 + 2*T2 + 1 acting on $S_{2}^{\mathrm{new}}(115, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{10} - 5 T^{9} + \cdots + 1 $ T^10 - 5T^9 + 14T^8 - 15T^7 - 2T^6 + 21T^5 + 38T^4 - 157T^3 + 125T^2 + 2*T + 1
$3$	$ T^{10} - T^{9} + \cdots + 1 $ T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$5$	$ T^{10} - T^{9} + \cdots + 1 $ T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$7$	$ T^{10} - 5 T^{9} + \cdots + 1 $ T^10 - 5T^9 + 14T^8 - 15T^7 - 2T^6 + 21T^5 + 38T^4 - 157T^3 + 125T^2 + 2*T + 1
$11$	$ T^{10} - 8 T^{9} + \cdots + 7921 $ T^10 - 8T^9 + 31T^8 - 50T^7 - 7T^6 + 353T^5 - 602T^4 + 482T^3 + 2447T^2 - 4539*T + 7921
$13$	$ T^{10} + 11 T^{8} + \cdots + 121 $ T^10 + 11T^8 + 11T^7 - 22T^5 + 121T^4 + 363T^3 + 484T^2 + 363*T + 121
$17$	$ T^{10} + 23 T^{9} + \cdots + 466489 $ T^10 + 23T^9 + 265T^8 + 2014T^7 + 11166T^6 + 46938T^5 + 150844T^4 + 365487T^3 + 642951T^2 + 751983*T + 466489
$19$	$ T^{10} + 13 T^{9} + \cdots + 11881 $ T^10 + 13T^9 + 48T^8 + 85T^7 + 665T^6 + 890T^5 + 4068T^4 + 8037T^3 + 17867T^2 + 20928*T + 11881
$23$	$ T^{10} + 21 T^{9} + \cdots + 6436343 $ T^10 + 21T^9 + 243T^8 + 1913T^7 + 11771T^6 + 60543T^5 + 270733T^4 + 1011977T^3 + 2956581T^2 + 5876661*T + 6436343
$29$	$ T^{10} + \cdots + 127893481 $ T^10 - 2T^9 + 70T^8 - 338T^7 + 4185T^6 + 4599T^5 + 41578T^4 + 348308T^3 - 1971142T^2 + 13491637*T + 127893481
$31$	$ T^{10} + 20 T^{9} + \cdots + 7921 $ T^10 + 20T^9 + 158T^8 + 344T^7 + 27T^6 - 3112T^5 + 3815T^4 - 6068T^3 + 8616T^2 + 445*T + 7921
$37$	$ T^{10} - 13 T^{9} + \cdots + 11485321 $ T^10 - 13T^9 + 169T^8 + 245T^7 - 3185T^6 + 99221T^5 + 265219T^4 + 401273T^3 + 22157281T^2 - 26898493*T + 11485321
$41$	$ T^{10} + \cdots + 193293409 $ T^10 + 5T^9 + 91T^8 + 1082T^7 + 4684T^6 + 37676T^5 + 382255T^4 + 1304768T^3 + 4364782T^2 + 46825304*T + 193293409
$43$	$ T^{10} - 15 T^{9} + \cdots + 4280761 $ T^10 - 15T^9 + 247T^8 - 2946T^7 + 23642T^6 - 113312T^5 + 347593T^4 - 380000T^3 + 115828T^2 - 2031758*T + 4280761
$47$	$ (T^{5} + 26 T^{4} + \cdots - 9811)^{2} $ (T^5 + 26T^4 + 189T^3 - 82T^2 - 4754T - 9811)^2
$53$	$ T^{10} - 6 T^{9} + \cdots + 1408969 $ T^10 - 6T^9 + 179T^8 + 609T^7 + 2693T^6 + 114522T^5 + 118398T^4 - 1311582T^3 + 18111768T^2 + 8541652*T + 1408969
$59$	$ T^{10} + \cdots + 437688241 $ T^10 - 5T^9 + 201T^8 - 587T^7 + 14672T^6 - 2080T^5 + 295674T^4 - 1827862T^3 + 40431593T^2 + 275424965*T + 437688241
$61$	$ T^{10} + 3 T^{9} + \cdots + 436921 $ T^10 + 3T^9 + 75T^8 + 192T^7 + 1126T^6 + 15951T^5 + 71481T^4 + 161258T^3 + 157327T^2 - 564494*T + 436921
$67$	$ T^{10} - 20 T^{9} + \cdots + 25593481 $ T^10 - 20T^9 + 290T^8 - 3765T^7 + 47745T^6 - 430156T^5 + 2427808T^4 - 8328841T^3 + 17666322T^2 - 25628894*T + 25593481
$71$	$ T^{10} + 22 T^{9} + \cdots + 1437601 $ T^10 + 22T^9 + 275T^8 + 3333T^7 + 27962T^6 + 113036T^5 + 275759T^4 + 486783T^3 + 931216T^2 + 1411223*T + 1437601
$73$	$ T^{10} - 43 T^{9} + \cdots + 78623689 $ T^10 - 43T^9 + 870T^8 - 11087T^7 + 99727T^6 - 648020T^5 + 2921326T^4 - 8353971T^3 + 14577157T^2 - 19400996*T + 78623689
$79$	$ T^{10} + \cdots + 246772681 $ T^10 - 51T^9 + 1193T^8 - 17008T^7 + 167896T^6 - 1264064T^5 + 7864278T^4 - 35175355T^3 + 113149199T^2 - 230812337*T + 246772681
$83$	$ T^{10} + 17 T^{9} + \cdots + 10029889 $ T^10 + 17T^9 + 36T^8 + 29T^7 + 3485T^6 - 44078T^5 + 379692T^4 - 1225689T^3 + 2128361T^2 - 494052*T + 10029889
$89$	$ T^{10} - 57 T^{9} + \cdots + 43256929 $ T^10 - 57T^9 + 1544T^8 - 26969T^7 + 336506T^6 - 3006321T^5 + 18714199T^4 - 71841315T^3 + 144725243T^2 - 81350913*T + 43256929
$97$	$ T^{10} - 52 T^{9} + \cdots + 62047129 $ T^10 - 52T^9 + 1318T^8 - 20565T^7 + 217122T^6 - 1552902T^5 + 7342415T^4 - 21752381T^3 + 36209530T^2 - 33760822*T + 62047129
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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:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	115.g (of order \(11\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{22}^{7} + \zeta_{22}^{5} + \cdots + \zeta_{22}) q^{2} + ( - \zeta_{22}^{9} + \zeta_{22}^{8} + \cdots + 1) q^{3} + (\zeta_{22}^{9} + \zeta_{22}^{7} + \cdots - 1) q^{4} - \zeta_{22}^{2} q^{5} + \cdots + (2 \zeta_{22}^{9} + \cdots + 2 \zeta_{22}) q^{99} +O(q^{100}) \) q + (z^7 + z^5 - z^4 + z^3 + z) * q^2 + (-z^9 + z^8 - z^7 + z^6 - z^5 + z^4 - z^3 + z^2 - z + 1) * q^3 + (z^9 + z^7 + z^5 - z^4 + 2z^3 - 2z^2 + z - 1) * q^4 - z^2 * q^5 + (z^6 + z^4 - z^3 + z^2 + 1) * q^6 + (z^9 - z^6 + z^5 - z^4 + z) * q^7 + (-z^9 + z^8 - z^7 + z^6 - z^5 + z^4 - z^2 - z - 1) * q^8 + 2z^9 q^9 + (-z^9 - z^7 + z^6 - z^5 - z^3) * q^10 + (z^9 - 2z^8 - 2z^6 + z^5 - z^2 + z) * q^11 + (z^9 + z^7 + z^5 + z^2 - z) * q^12 + (-z^8 - z^7 - z^5 - z^4) * q^13 + (z^7 - z^6 - 2z^4 + 2z^3 + z - 1) * q^14 - z * q^15 + (z^9 - 2z^8 + z^6 + z^5 + z^4 - 2z^2 + z) * q^16 + (-2z^6 - z^5 + z^4 - z^3 + z^2 - z - 2) q^17 + (2z^9 - 2z^8 + 2z^7 - 2z^6 - 2z^4 - 2) q^18 + (-2z^9 + z^8 - 2z^7 + z^4 - 3z^3 + 3z^2 - z) * q^19 + (-z^9 - z^7 + z^6 - 2z^5 + 2z^4 - z^3 + z^2 + 1) * q^20 + (z^8 - z^5 + z^4 - z^3 + 1) * q^21 + (-2z^9 - 2z^8 + z^7 - z^6 + z^5 - z^4 + 2z^3 + 2z^2) * q^22 + (-2z^9 + 2z^8 + 2z^7 + z^6 - 2z^2 - 2) * q^23 + (z^9 - 2z^8 + 2z^7 - 2z^6 + 2z^5 - 2z^4 + 2z^3 - z^2 - 2) * q^24 + z^4 * q^25 + (-3z^9 + z^8 - 3z^7 + z^6 - 3z^5 + 3z^4 - z^3 + 3z^2 - z + 3) q^26 + 5z^8 q^27 + (-z^9 + 2z^8 - 2z^7 + z^6) * q^28 + (-5z^9 + 5z^8 + 4z^6 - 4z^5 + 6z^4 - 4z^3 + 6z^2 - 4z + 4) * q^29 + (-z^8 - z^6 + z^5 - z^4 - z^2) * q^30 + (-2z^9 + z^8 - 3z^7 + 3z^6 - z^5 + 2z^4 - z^2 + z - 1) * q^31 + (2z^9 + 3z^7 - z^6 + 2z^5 - 2z^4 + 2z^3 - 2z^2 + z - 3) * q^32 + (z^8 - 2z^7 - 2z^5 + z^4 - z + 1) * q^33 + (z^9 - 4z^8 - z^7 - 4z^6 + z^5 - z^3 + z^2 - z + 1) * q^34 + (z^8 - z^7 + z^6 - z^3 + 1) * q^35 + (-2z^8 - 2z^6 - 2z^4 - 2z + 2) * q^36 + (z^9 - 6z^4 + 6z^3) * q^37 + (-2z^9 + 2z^4 + 3z^2 - 2z + 3) * q^38 + (-z^7 - z^6 - z^4 - z^3) * q^39 + (-z^5 + 2z^4 + 2z^2 - z) * q^40 + (-z^9 - 4z^8 + 4z^7 + z^6 - z^4 - 2z^3 - 2z^2 - 2z - 1) q^41 + (z^9 - z^8 + z^7 - z^4 - z^3 + z^2 + z) * q^42 + (3z^9 - 7z^8 + 4z^7 - 4z^6 + 6z^5 - 6z^4 + 4z^3 - 4z^2 + 7z - 3) q^43 + (3z^7 - 2z^6 + 4z^5 - z^4 + 4z^3 - 2z^2 + 3z) * q^44 + 2 * q^45 + (3z^8 - 4z^7 + 3z^6 - 3z^5 + z^4 - 5z^3 - 4z^2 - z) * q^46 + (4z^9 - z^7 - z^6 + z^5 + z^4 - 4z^2 - 6) * q^47 + (z^8 - 2z^7 + z^5 + z^4 + z^3 - 2z + 1) * q^48 + (-4z^9 + 3z^8 - 5z^7 + 4z^6 - 4z^5 + 4z^4 - 4z^3 + 5z^2 - 3z + 4) q^49 + (z^9 - z^8 + z^7 + z^5 - 1) * q^50 + (2z^9 - 2z^8 + 2z^7 - 2z^6 - 3z^4 + 3z^3 - 3z^2 + 3z - 3) * q^51 + (3z^6 + 3z^4 + z^3 + 3z^2 + 3) q^52 + (5z^9 + 2z^8 + z^7 + z^6 + z^4 + z^3 + 2z^2 + 5z) * q^53 + (5z^9 - 5z^4 - 5z^2 + 5z - 5) * q^54 + (2z^9 + z^7 - 2z^6 + 2z^5 - z^4 + z^3 - 2z^2 + 2z - 1) q^55 + (-3z^9 + 4z^8 - 3z^7 + 4z^6 - 3z^5 + 4z^4 - 3z^3 - 3z + 3) * q^56 + (-2z^8 + z^7 - 2z^6 + z^3 - 3z^2 + 3z - 1) * q^57 + (8z^9 - z^8 + 7z^7 - z^6 + 8z^5 - 2z^3 - z^2 + z + 2) * q^58 + (2z^7 + 8z^6 + 2z^5 - z + 1) q^59 + (-z^9 - z^7 - z^4 + z^3 + 1) * q^60 + (3z^8 + 2z^7 - 2z^6 - 3z^5 - z) * q^61 + (-3z^9 + 4z^8 - 6z^7 + 10z^6 - 8z^5 + 10z^4 - 6z^3 + 4z^2 - 3z) q^62 + (2z^9 - 2z^8 - 2z^6 + 2z^5 + 2z - 2) q^63 + (5z^9 - 4z^8 + 4z^7 - 5z^6 - 8z^4 + 6z^3 - 8z^2 + 6z - 8) * q^64 + (2z^9 - z^8 + 2z^7 + z^5 - z^4 + z^3 - z^2 + z - 1) * q^65 + (-2z^8 - 2z^7 + z^6 - z^5 + z^4 - z^3 + 2z^2 + 2z) * q^66 + (z^8 - z^7 - 4z^6 - 5z^4 - 4z^2 - z + 1) q^67 + (-4z^9 + z^7 - 4z^6 + 4z^5 - z^4 + 4z^2 + 3) * q^68 + (2z^9 - 4z^8 + 4z^7 + 3z^5 - 2z^4 + 2z^3 - 2z^2 - 2) q^69 + (-z^9 + z^8 + 2z^6 - 2z^5 - z^3 + z^2) * q^70 + (-z^8 - 4z^7 + 3z^6 + z^5 + z^4 + z^3 + 3z^2 - 4z - 1) * q^71 + (-4z^9 + 2z^8 - 2z + 4) q^72 + (2z^9 - 3z^8 + 2z^7 - 4z^5 - z^4 - z^3 + z^2 + z + 4) * q^73 + (z^9 + 5z^8 - 5z^7 - z^6 - z^4 + 6z^3 - 6z^2 + 6z - 1) q^74 + z^3 * q^75 + (-z^9 + 4z^8 - z^6 + 6z^5 - z^4 + 4z^2 - z) q^76 + (2z^9 - 4z^8 + 4z^5 - 2z^4 - z^2 + z - 1) * q^77 + (-3z^9 - 2z^7 - 2z^5 + 2z^2 + 2) * q^78 + (z^9 + 2z^8 + 3z^7 - 2z^6 + 3z^5 + 2z^4 + z^3 - 5z + 5) * q^79 + (2z^9 - 3z^8 + z^7 - 3z^6 + 2z^5 + z^3 - 2z^2 + 2z - 1) * q^80 - z^7 * q^81 + (-6z^9 + z^8 - z^7 - 2z^6 - z^5 + z^4 - 6z^3 - 8z + 8) * q^82 + (-6z^9 - z^7 + 5z^6 - 4z^5 + z^4 - z^3 + 4z^2 - 5z + 1) q^83 + (-z^8 + 2z^7 - 2z^6 + z^5) * q^84 + (2z^8 + z^7 - z^6 + z^5 - z^4 + z^3 + 2z^2) * q^85 + (-6z^9 + 6z^8 - 2z^5 + 7z^3 - 2z) q^86 + (-4z^9 - z^8 + z^7 + 4z^6 + 2z^3 + 2z) * q^87 + (6z^8 - z^5 + 2z^3 - 2z^2 + 1) q^88 + (-11z^9 + 6z^8 - 5z^7 + 4z^6 - 6z^5 + 6z^4 - 4z^3 + 5z^2 - 6z + 11) q^89 + (2z^7 + 2z^5 - 2z^4 + 2z^3 + 2z) q^90 + (-z^9 + z^2 - 2) * q^91 + (-5z^9 + z^8 - 6z^7 + 2z^6 - 9z^5 + 6z^4 - 9z^3 + 4z^2 - z) q^92 + (z^9 - 3z^8 + 2z^7 - 4z^6 + 4z^5 - 2z^4 + 3z^3 - z^2) * q^93 + (-6z^8 - 5z^7 - 8z^5 + z^4 - 8z^3 - 5z - 6) q^94 + (z^9 + z^8 - z^7 + 2z^5 - 2z^4 + z^2 - z - 1) * q^95 + (3z^9 - z^8 + 3z^7 + 2z^5 - z^4 + z^3 - z^2 + z - 2) q^96 + (-z^8 + z^7 + 6z^4 + 4z^2 + 6) * q^97 + (-z^9 + z^8 + 5z^6 - z^5 + 7z^4 - 3z^3 + 7z^2 - z + 5) * q^98 + (2z^9 - 2z^8 + 2z^6 + 2z^5 + 2z^4 - 2z^2 + 2z) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 5 q^{2} + q^{3} - q^{4} + q^{5} + 6 q^{6} + 5 q^{7} - 16 q^{8} + 2 q^{9} - 5 q^{10} + 8 q^{11} + q^{12} - 3 q^{14} - q^{15} + 5 q^{16} - 23 q^{17} - 10 q^{18} - 13 q^{19} + q^{20} + 6 q^{21}+ \cdots + 6 q^{99}+O(q^{100}) \) 10 * q + 5 * q^2 + q^3 - q^4 + q^5 + 6 * q^6 + 5 * q^7 - 16 * q^8 + 2 * q^9 - 5 * q^10 + 8 * q^11 + q^12 - 3 * q^14 - q^15 + 5 * q^16 - 23 * q^17 - 10 * q^18 - 13 * q^19 + q^20 + 6 * q^21 + 4 * q^22 - 21 * q^23 - 6 * q^24 - q^25 + 11 * q^26 - 5 * q^27 - 6 * q^28 + 2 * q^29 + 5 * q^30 - 20 * q^31 - 15 * q^32 + 3 * q^33 + 16 * q^34 + 6 * q^35 + 24 * q^36 + 13 * q^37 + 21 * q^38 - 6 * q^40 - 5 * q^41 + 3 * q^42 + 15 * q^43 + 19 * q^44 + 20 * q^45 - 16 * q^46 - 52 * q^47 + 6 * q^48 + 4 * q^49 - 6 * q^50 - 10 * q^51 + 22 * q^52 + 6 * q^53 - 30 * q^54 + 3 * q^55 + 3 * q^56 + 2 * q^57 + 45 * q^58 + 5 * q^59 + 10 * q^60 - 3 * q^61 - 54 * q^62 - 10 * q^63 - 34 * q^64 - 4 * q^66 + 20 * q^67 + 32 * q^68 - q^69 - 8 * q^70 - 22 * q^71 + 32 * q^72 + 43 * q^73 + q^74 + q^75 - 2 * q^76 + 4 * q^77 + 11 * q^78 + 51 * q^79 + 6 * q^80 - q^81 + 58 * q^82 - 17 * q^83 + 6 * q^84 + q^85 - 9 * q^86 - 2 * q^87 + 7 * q^88 + 57 * q^89 + 10 * q^90 - 22 * q^91 - 43 * q^92 + 20 * q^93 - 81 * q^94 - 9 * q^95 - 7 * q^96 + 52 * q^97 + 24 * q^98 + 6 * 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	115.2.g.a

Level	$115$
Weight	$2$
Character orbit	115.g
Analytic conductor	$0.918$
Analytic rank	$0$
Dimension	$10$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(0.918279623245\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\Q(\zeta_{22})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 6.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{10} - 5 T^{9} + \cdots + 1 \) T^10 - 5T^9 + 14T^8 - 15T^7 - 2T^6 + 21T^5 + 38T^4 - 157T^3 + 125T^2 + 2*T + 1
$3$	\( T^{10} - T^{9} + \cdots + 1 \) T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$5$	\( T^{10} - T^{9} + \cdots + 1 \) T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$7$	\( T^{10} - 5 T^{9} + \cdots + 1 \) T^10 - 5T^9 + 14T^8 - 15T^7 - 2T^6 + 21T^5 + 38T^4 - 157T^3 + 125T^2 + 2*T + 1
$11$	\( T^{10} - 8 T^{9} + \cdots + 7921 \) T^10 - 8T^9 + 31T^8 - 50T^7 - 7T^6 + 353T^5 - 602T^4 + 482T^3 + 2447T^2 - 4539*T + 7921
$13$	\( T^{10} + 11 T^{8} + \cdots + 121 \) T^10 + 11T^8 + 11T^7 - 22T^5 + 121T^4 + 363T^3 + 484T^2 + 363*T + 121
$17$	\( T^{10} + 23 T^{9} + \cdots + 466489 \) T^10 + 23T^9 + 265T^8 + 2014T^7 + 11166T^6 + 46938T^5 + 150844T^4 + 365487T^3 + 642951T^2 + 751983*T + 466489
$19$	\( T^{10} + 13 T^{9} + \cdots + 11881 \) T^10 + 13T^9 + 48T^8 + 85T^7 + 665T^6 + 890T^5 + 4068T^4 + 8037T^3 + 17867T^2 + 20928*T + 11881
$23$	\( T^{10} + 21 T^{9} + \cdots + 6436343 \) T^10 + 21T^9 + 243T^8 + 1913T^7 + 11771T^6 + 60543T^5 + 270733T^4 + 1011977T^3 + 2956581T^2 + 5876661*T + 6436343
$29$	\( T^{10} + \cdots + 127893481 \) T^10 - 2T^9 + 70T^8 - 338T^7 + 4185T^6 + 4599T^5 + 41578T^4 + 348308T^3 - 1971142T^2 + 13491637*T + 127893481
$31$	\( T^{10} + 20 T^{9} + \cdots + 7921 \) T^10 + 20T^9 + 158T^8 + 344T^7 + 27T^6 - 3112T^5 + 3815T^4 - 6068T^3 + 8616T^2 + 445*T + 7921
$37$	\( T^{10} - 13 T^{9} + \cdots + 11485321 \) T^10 - 13T^9 + 169T^8 + 245T^7 - 3185T^6 + 99221T^5 + 265219T^4 + 401273T^3 + 22157281T^2 - 26898493*T + 11485321
$41$	\( T^{10} + \cdots + 193293409 \) T^10 + 5T^9 + 91T^8 + 1082T^7 + 4684T^6 + 37676T^5 + 382255T^4 + 1304768T^3 + 4364782T^2 + 46825304*T + 193293409
$43$	\( T^{10} - 15 T^{9} + \cdots + 4280761 \) T^10 - 15T^9 + 247T^8 - 2946T^7 + 23642T^6 - 113312T^5 + 347593T^4 - 380000T^3 + 115828T^2 - 2031758*T + 4280761
$47$	\( (T^{5} + 26 T^{4} + \cdots - 9811)^{2} \) (T^5 + 26T^4 + 189T^3 - 82T^2 - 4754T - 9811)^2
$53$	\( T^{10} - 6 T^{9} + \cdots + 1408969 \) T^10 - 6T^9 + 179T^8 + 609T^7 + 2693T^6 + 114522T^5 + 118398T^4 - 1311582T^3 + 18111768T^2 + 8541652*T + 1408969
$59$	\( T^{10} + \cdots + 437688241 \) T^10 - 5T^9 + 201T^8 - 587T^7 + 14672T^6 - 2080T^5 + 295674T^4 - 1827862T^3 + 40431593T^2 + 275424965*T + 437688241
$61$	\( T^{10} + 3 T^{9} + \cdots + 436921 \) T^10 + 3T^9 + 75T^8 + 192T^7 + 1126T^6 + 15951T^5 + 71481T^4 + 161258T^3 + 157327T^2 - 564494*T + 436921
$67$	\( T^{10} - 20 T^{9} + \cdots + 25593481 \) T^10 - 20T^9 + 290T^8 - 3765T^7 + 47745T^6 - 430156T^5 + 2427808T^4 - 8328841T^3 + 17666322T^2 - 25628894*T + 25593481
$71$	\( T^{10} + 22 T^{9} + \cdots + 1437601 \) T^10 + 22T^9 + 275T^8 + 3333T^7 + 27962T^6 + 113036T^5 + 275759T^4 + 486783T^3 + 931216T^2 + 1411223*T + 1437601
$73$	\( T^{10} - 43 T^{9} + \cdots + 78623689 \) T^10 - 43T^9 + 870T^8 - 11087T^7 + 99727T^6 - 648020T^5 + 2921326T^4 - 8353971T^3 + 14577157T^2 - 19400996*T + 78623689
$79$	\( T^{10} + \cdots + 246772681 \) T^10 - 51T^9 + 1193T^8 - 17008T^7 + 167896T^6 - 1264064T^5 + 7864278T^4 - 35175355T^3 + 113149199T^2 - 230812337*T + 246772681
$83$	\( T^{10} + 17 T^{9} + \cdots + 10029889 \) T^10 + 17T^9 + 36T^8 + 29T^7 + 3485T^6 - 44078T^5 + 379692T^4 - 1225689T^3 + 2128361T^2 - 494052*T + 10029889
$89$	\( T^{10} - 57 T^{9} + \cdots + 43256929 \) T^10 - 57T^9 + 1544T^8 - 26969T^7 + 336506T^6 - 3006321T^5 + 18714199T^4 - 71841315T^3 + 144725243T^2 - 81350913*T + 43256929
$97$	\( T^{10} - 52 T^{9} + \cdots + 62047129 \) T^10 - 52T^9 + 1318T^8 - 20565T^7 + 217122T^6 - 1552902T^5 + 7342415T^4 - 21752381T^3 + 36209530T^2 - 33760822*T + 62047129
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\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(1\)	\(\zeta_{22}^{4}\)