LMFDB - Newform orbit 414.2.i.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,2,Mod(55,414)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 10]))

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

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

chi := DirichletCharacter("414.55");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$414 = 2 \cdot 3^{2} \cdot 23$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	414.i (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:	$3.30580664368$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 138)
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.

$f(q)$	$=$	$q + \zeta_{22}^{4} q^{2} + \zeta_{22}^{8} q^{4} + (\zeta_{22}^{9} - 2 \zeta_{22}^{8} + \cdots - 1) q^{5} + ( - \zeta_{22}^{9} + \zeta_{22}^{7} + \cdots - \zeta_{22}) q^{7} - \zeta_{22} q^{8} + ( - \zeta_{22}^{9} - \zeta_{22}^{7} + \cdots - 1) q^{10} + \cdots + (5 \zeta_{22}^{9} + \cdots + 2 \zeta_{22}) q^{98} +O(q^{100})$ q + z^4 * q^2 + z^8 * q^4 + (z^9 - 2z^8 + 2z^7 - z^6 - z^4 - z^2 - 1) * q^5 + (-z^9 + z^7 - z^6 - z^4 + z^3 - z) * q^7 - z * q^8 + (-z^9 - z^7 - z^5 - z^3 + z - 1) * q^10 + (-2z^9 - 2z^5 - z^3 + 1) * q^11 + (-2z^9 + 2z^8 - 3z^7 - 3z^5 + 2z^4 - 2z^3 - 3z + 3) q^13 + (-z^9 + z^6 - 2z^5 + z^4 - z^3 + 2z^2 - z) * q^14 - z^5 * q^16 + (-4z^9 + 4z^8 + z^6 + z^4 + z^3 + z^2 + 1) * q^17 + (2z^9 - 3z^8 + 2z^7 + z^5 + z^3 - z^2 - 1) q^19 + (-z^9 - z^7 + z^5 - z^4 + z^2 + 1) * q^20 + (-2z^9 - z^7 + z^4 + 2z^2) * q^22 + (-z^8 - z^7 + z^6 - 2z^5 + z^4 - 3z^3 - z + 4) * q^23 + (z^8 + z^7 + 2z^6 - 2z^5 + 2z^4 - 2z^3 + 2z^2 + z + 1) q^25 + (-3z^9 + 2z^8 - 2z^7 - 3z^5 + 3z^4 + 2z^2 - 2z + 3) q^26 + (-z^9 + z^6 - z^4 + z^3 + z - 1) * q^28 + (-z^9 + z^8 - 2z^6 - z^5 - 5z^3 - z - 2) * q^29 + (3z^9 - 5z^8 + 4z^7 - 4z^6 + 5z^5 - 3z^4 - 4z^2 + 4z - 4) * q^31 - z^9 * q^32 + (z^9 + 2z^7 + z^5 + z^3 + 3z^2 - 3z - 1) q^34 + (4z^9 - 3z^8 + 3z^7 - 3z^6 + 4z^5 - z^3 + 1) q^35 + (4z^9 + z^7 + 2z^6 + z^4 - z^3 - 2z - 1) q^37 + (z^9 + z^7 - z^6 - z^4 - 2z^2 + 3z - 2) * q^38 + (z^9 - z^8 + z^6 + z^4 + z^2 + 1) * q^40 + (-2z^9 + 5z^8 - 5z^7 + 2z^6 + 3z^4 - z^3 + z^2 - z + 3) q^41 + (-2z^9 - z^8 + 4z^7 - z^6 + 2z^5 - 2z^4 + z^3 - 4z^2 + z + 2) q^43 + (z^8 + 2z^6 + 2z^2 + 1) * q^44 + (-z^9 - 2z^7 - z^6 + 3z^4 + z^3 - z^2 + 2z) q^46 + (-z^9 + z^8 - 2z^7 + 5z^6 - 5z^5 + 2z^4 - z^3 + z^2 + 1) * q^47 + (-3z^9 - 2z^7 + 2z^6 + 3z^5 - 3z^4 - 2z^3 + 2z^2 + 3) q^49 + (3z^5 - z^4 + 2z^3 - 2z^2 + z - 3) q^50 + (-3z^9 + 3z^8 + 2z^6 - 2z^5 + 3z^4 + 3z^2 - 2z + 2) q^52 + (-4z^9 + 2z^7 + 2z^6 - z^5 + 2z^4 + 2z^3 - 4z) * q^53 + (4z^9 + 5z^7 - 3z^6 + 5z^5 - z^4 + z^3 - 5z^2 + 3z - 5) * q^55 + (z^9 - 2z^8 + 2z^7 - z^6 + 2z^5 - 2z^4 + z^3 + z - 1) * q^56 + (-3z^9 + 2z^8 - 7z^7 + 2z^6 - 3z^5 - 2z^3 + 3z^2 - 3z + 2) * q^58 + (6z^9 - 5z^8 + 2z^7 - 4z^6 + 2z^5 - 5z^4 + 6z^3 + z - 1) q^59 + (2z^9 + 2z^8 - z^7 + z^6 - 2z^5 - 2z^4 - z^2 + 7z - 1) q^61 + (z^9 + z^8 - 4z^7 - 4z^3 + z^2 + z) * q^62 + z^2 * q^64 + (8z^9 - 5z^8 + 8z^7 + 8z^5 - 6z^4 + 5z^3 - 5z^2 + 6z - 8) * q^65 + (-z^8 - 6z^7 + 4z^6 - z^5 + z^4 - z^3 + 4z^2 - 6z - 1) * q^67 + (z^9 + z^7 + 3z^6 - 3z^5 - z^4 - z^2 - 2) * q^68 + (z^9 + 3z^8 - 4z^7 + 3z^6 - 3z^5 + 4z^4 - 3z^3 - z^2) * q^70 + (-z^8 - 5z^7 + 3z^6 + 3z^5 + 3z^4 + 3z^3 + 3z^2 - 5z - 1) q^71 + (-6z^9 + 8z^8 - 6z^7 - 9z^5 + 4z^4 - 5z^3 + 5z^2 - 4z + 9) * q^73 + (2z^9 - z^8 + z^7 - 2z^6 - 3z^4 + 2z^3 - 6z^2 + 2z - 3) * q^74 + (-z^9 - z^7 - z^6 + 2z^5 - z^4 - z^3 - z) q^76 + (3z^9 - 3z^8 + z^7 - z^6 + 3z^5 - 3z^4 - 3z^2 + 3z - 3) * q^77 + (z^9 + 6z^8 - 7z^7 + z^6 - 7z^5 + 6z^4 + z^3 - z + 1) * q^79 + (z^9 + z^7 + z^5 + z^3 - 2z^2 + 2z - 1) * q^80 + (2z^9 + z^8 + z^7 - z^6 + z^5 + z^4 + 2z^3 - 3z + 3) q^82 + (9z^9 + 4z^7 - 5z^6 + 6z^5 - 4z^4 + 4z^3 - 6z^2 + 5z - 4) * q^83 + (-3z^8 - z^7 - 10z^6 + 9z^5 - 10z^4 - z^3 - 3z^2) q^85 + (z^9 - z^8 - 3z^6 + 3z^4 - z^3 + 3z^2 - 3) q^86 + (2z^9 - 2z^8 + 2z^7 + 2z^5 - z^4 + 2z^3 - 2z^2 + z - 2) * q^88 + (-z^9 + z^8 - 3z^7 - 4z^6 + z^5 - z^4 + 4z^3 + 3z^2 - z + 1) * q^89 + (-2z^9 - 3z^8 + 6z^7 + 2z^6 - 2z^5 - 6z^4 + 3z^3 + 2z^2 - 8) * q^91 + (-z^9 + 4z^8 + z^5 + z^4 - z^3 + 2z^2 - z + 3) * q^92 + (-3z^8 + 4z^7 - 4z^6 + 5z^5 - 4z^4 + 5z^3 - 4z^2 + 4z - 3) * q^94 + (-2z^9 - z^8 - z^7 + 2z^6 - 7z^5 + 7z^4 - 2z^3 + z^2 + z + 2) q^95 + (-z^9 + z^6 + 3z^4 + 5z^3 - 4z^2 + 5z + 3) * q^97 + (5z^9 - 5z^8 + 2z^5 + z^4 + 2z^3 + z^2 + 2z) q^98
$\operatorname{Tr}(f)(q)$	$=$	$10 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{7} - q^{8} - 13 q^{10} + 5 q^{11} + 13 q^{13} - 9 q^{14} - q^{16} + 9 q^{20} - 6 q^{22} + 32 q^{23} + q^{25} + 13 q^{26} - 9 q^{28} - 27 q^{29} - 8 q^{31} - q^{32}+ \cdots + 14 q^{98}+O(q^{100})$ 10 * q - q^2 - q^4 - 2 * q^5 + 2 * q^7 - q^8 - 13 * q^10 + 5 * q^11 + 13 * q^13 - 9 * q^14 - q^16 + 9 * q^20 - 6 * q^22 + 32 * q^23 + q^25 + 13 * q^26 - 9 * q^28 - 27 * q^29 - 8 * q^31 - q^32 - 11 * q^34 + 26 * q^35 - 11 * q^37 - 11 * q^38 + 9 * q^40 + 10 * q^41 + 34 * q^43 + 5 * q^44 - q^46 - 8 * q^47 + 25 * q^49 - 21 * q^50 + 2 * q^52 - 9 * q^53 - 23 * q^55 + 2 * q^56 - 5 * q^58 + 21 * q^59 - 4 * q^61 - 8 * q^62 - q^64 - 29 * q^65 - 32 * q^67 - 22 * q^68 - 18 * q^70 - 22 * q^71 + 43 * q^73 - 11 * q^74 - 10 * q^77 - 16 * q^79 - 2 * q^80 + 32 * q^82 + 3 * q^83 + 33 * q^85 - 32 * q^86 - 6 * q^88 + 11 * q^89 - 70 * q^91 + 21 * q^92 + 3 * q^94 + 39 * q^97 + 14 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/414\mathbb{Z}\right)^\times$ .

$n$	$47$	$235$
$\chi(n)$	$1$	$\zeta_{22}^{4}$

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}

55.1

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

−0.959493

0.281733i

0.841254

−

0.540641i

0.0651865

0.453382i

−0.134858

0.295298i

−0.654861

0.755750i

−0.190279

−

0.416652i

73.1

0.415415

0.909632i

−0.654861

0.755750i

−2.47672

−

1.59169i

−0.151894

−

1.05645i

−0.959493

−

0.281733i

0.418986

−

2.91411i

127.1

0.841254

−

0.540641i

0.415415

−

0.909632i

−1.78074

0.522874i

−2.54487

−

2.93694i

−0.142315

−

0.989821i

−1.21537

1.40261i

163.1

0.841254

0.540641i

0.415415

0.909632i

−1.78074

−

0.522874i

−2.54487

2.93694i

−0.142315

0.989821i

−1.21537

−

1.40261i

271.1

−0.959493

−

0.281733i

0.841254

0.540641i

0.0651865

−

0.453382i

−0.134858

−

0.295298i

−0.654861

−

0.755750i

−0.190279

0.416652i

289.1

−0.654861

−

0.755750i

−0.142315

0.989821i

1.37787

−

3.01713i

3.66820

1.07708i

0.841254

−

0.540641i

−3.18251

0.934468i

307.1

−0.142315

0.989821i

−0.959493

−

0.281733i

1.81440

2.09393i

0.163423

0.105026i

0.415415

−

0.909632i

−2.33083

1.49793i

325.1

−0.142315

−

0.989821i

−0.959493

0.281733i

1.81440

−

2.09393i

0.163423

−

0.105026i

0.415415

0.909632i

−2.33083

−

1.49793i

361.1

−0.654861

0.755750i

−0.142315

−

0.989821i

1.37787

3.01713i

3.66820

−

1.07708i

0.841254

0.540641i

−3.18251

−

0.934468i

397.1

0.415415

−

0.909632i

−0.654861

−

0.755750i

−2.47672

1.59169i

−0.151894

1.05645i

−0.959493

0.281733i

0.418986

2.91411i

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	414.2.i.a		10
3.b	odd	2	1		138.2.e.d	✓	10
23.c	even	11	1	inner	414.2.i.a		10
23.c	even	11	1		9522.2.a.bx		5
23.d	odd	22	1		9522.2.a.by		5
69.g	even	22	1		3174.2.a.w		5
69.h	odd	22	1		138.2.e.d	✓	10
69.h	odd	22	1		3174.2.a.x		5

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
138.2.e.d	✓	10	3.b	odd	2	1
138.2.e.d	✓	10	69.h	odd	22	1
414.2.i.a		10	1.a	even	1	1	trivial
414.2.i.a		10	23.c	even	11	1	inner
3174.2.a.w		5	69.g	even	22	1
3174.2.a.x		5	69.h	odd	22	1
9522.2.a.bx		5	23.c	even	11	1
9522.2.a.by		5	23.d	odd	22	1

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{10} + 2 T_{5}^{9} + 4 T_{5}^{8} + 41 T_{5}^{7} + 137 T_{5}^{6} + 76 T_{5}^{5} + 460 T_{5}^{4} + \cdots + 529$ T5^10 + 2*T5^9 + 4*T5^8 + 41*T5^7 + 137*T5^6 + 76*T5^5 + 460*T5^4 + 2185*T5^3 + 2324*T5^2 + 138*T5 + 529 acting on $S_{2}^{\mathrm{new}}(414, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{10} + T^{9} + \cdots + 1$ T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$3$	$T^{10}$ T^10
$5$	$T^{10} + 2 T^{9} + \cdots + 529$ T^10 + 2T^9 + 4T^8 + 41T^7 + 137T^6 + 76T^5 + 460T^4 + 2185T^3 + 2324T^2 + 138*T + 529
$7$	$T^{10} - 2 T^{9} + \cdots + 1$ T^10 - 2T^9 - 7T^8 - 41T^7 + 203T^6 + 12T^5 + 262T^4 - 18T^3 + 14T^2 - 6*T + 1
$11$	$T^{10} - 5 T^{9} + \cdots + 529$ T^10 - 5T^9 + 25T^8 - 37T^7 + 20T^6 + 10T^5 - 138T^4 - 1290T^3 + 3689T^2 + 1817*T + 529
$13$	$T^{10} - 13 T^{9} + \cdots + 529$ T^10 - 13T^9 + 125T^8 - 646T^7 + 2854T^6 - 5026T^5 + 5575T^4 - 2020T^3 - 1218T^2 + 1380*T + 529
$17$	$T^{10} + 55 T^{8} + \cdots + 7745089$ T^10 + 55T^8 + 121T^7 + 726T^6 + 4235T^5 + 11979T^4 + 166375T^3 + 58564T^2 - 2693944T + 7745089
$19$	$T^{10} - 33 T^{7} + \cdots + 64009$ T^10 - 33T^7 - 165T^6 - 506T^5 + 2178T^4 + 14399T^3 + 45012T^2 + 66792*T + 64009
$23$	$T^{10} - 32 T^{9} + \cdots + 6436343$ T^10 - 32T^9 + 496T^8 - 4971T^7 + 35982T^6 - 197097T^5 + 827586T^4 - 2629659T^3 + 6034832T^2 - 8954912*T + 6436343
$29$	$T^{10} + 27 T^{9} + \cdots + 7921$ T^10 + 27T^9 + 322T^8 + 2039T^7 + 6532T^6 + 9527T^5 + 9080T^4 - 234253T^3 + 680409T^2 - 62834*T + 7921
$31$	$T^{10} + 8 T^{9} + \cdots + 8300161$ T^10 + 8T^9 + 108T^8 + 699T^7 + 3513T^6 + 8337T^5 - 26298T^4 + 42077T^3 + 1661379T^2 + 5398994*T + 8300161
$37$	$T^{10} + 11 T^{9} + \cdots + 64009$ T^10 + 11T^9 + 66T^8 + 33T^7 + 462T^6 + 583T^5 + 84579T^4 - 285197T^3 + 256157T^2 - 30613*T + 64009
$41$	$T^{10} - 10 T^{9} + \cdots + 1515361$ T^10 - 10T^9 + 23T^8 - 461T^7 + 4940T^6 - 8370T^5 + 48291T^4 - 590523T^3 + 2279850T^2 - 922019*T + 1515361
$43$	$T^{10} - 34 T^{9} + \cdots + 109561$ T^10 - 34T^9 + 595T^8 - 6403T^7 + 46080T^6 - 215634T^5 + 623217T^4 - 769715T^3 + 778922T^2 - 466379*T + 109561
$47$	$(T^{5} + 4 T^{4} + \cdots - 857)^{2}$ (T^5 + 4T^4 - 75T^3 + 28T^2 + 658T - 857)^2
$53$	$T^{10} + 9 T^{9} + \cdots + 8427409$ T^10 + 9T^9 + 103T^8 - 481T^7 + 1765T^6 - 36211T^5 + 340085T^4 - 1481817T^3 + 4374571T^2 - 8828023*T + 8427409
$59$	$T^{10} - 21 T^{9} + \cdots + 978121$ T^10 - 21T^9 + 199T^8 - 2023T^7 + 27677T^6 - 182236T^5 + 194382T^4 + 1009152T^3 + 5151323T^2 - 447028*T + 978121
$61$	$T^{10} + 4 T^{9} + \cdots + 9078169$ T^10 + 4T^9 + 5T^8 + 845T^7 + 21255T^6 - 8810T^5 + 1007054T^4 - 92098T^3 + 4660676T^2 - 13630812*T + 9078169
$67$	$T^{10} + 32 T^{9} + \cdots + 83375161$ T^10 + 32T^9 + 628T^8 + 10438T^7 + 128921T^6 + 1069540T^5 + 6033270T^4 + 22918928T^3 + 58716219T^2 + 95409819*T + 83375161
$71$	$T^{10} + 22 T^{9} + \cdots + 64009$ T^10 + 22T^9 + 198T^8 + 3476T^7 + 32824T^6 - 80564T^5 + 1834723T^4 - 940775T^3 + 31097T^2 + 61226*T + 64009
$73$	$T^{10} - 43 T^{9} + \cdots + 21132409$ T^10 - 43T^9 + 1013T^8 - 13804T^7 + 112740T^6 - 340900T^5 - 1509815T^4 + 14290552T^3 + 45389554T^2 + 15822874*T + 21132409
$79$	$T^{10} + 16 T^{9} + \cdots + 72471169$ T^10 + 16T^9 + 14T^8 + 1665T^7 + 48519T^6 + 439847T^5 + 3749476T^4 - 23797177T^3 + 57311049T^2 - 55606916*T + 72471169
$83$	$T^{10} - 3 T^{9} + \cdots + 16752649$ T^10 - 3T^9 - 79T^8 - 236T^7 + 9860T^6 - 148413T^5 + 1658407T^4 - 5063364T^3 + 19306721T^2 + 22495128*T + 16752649
$89$	$T^{10} + \cdots + 517972081$ T^10 - 11T^9 + 55T^8 - 1034T^7 + 16566T^6 - 117711T^5 + 813241T^4 - 6520206T^3 + 33689667T^2 - 136189856*T + 517972081
$97$	$T^{10} - 39 T^{9} + \cdots + 5650129$ T^10 - 39T^9 + 773T^8 - 10204T^7 + 116972T^6 - 1250688T^5 + 10535838T^4 - 53199165T^3 + 116212529T^2 - 9809879*T + 5650129
show more
show less

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

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 55.1
Significant digits:
Format:

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	414.2.i.a

Level	$414$
Weight	$2$
Character orbit	414.i
Analytic conductor	$3.306$
Analytic rank	$0$
Dimension	$10$
Inner twists	$2$