LMFDB - Newform orbit 253.3.k.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [253,3,Mod(32,253)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(253, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 10]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("253.32");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 253 = 11 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	253.k (of order $22$, 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:	$6.89375068832$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{22}]$

$q$-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 + (2 \zeta_{22}^{9} - 2 \zeta_{22}^{8} + \cdots - 1) q^{3} + 4 \zeta_{22}^{2} q^{4} + (3 \zeta_{22}^{9} - \zeta_{22}^{7} + \cdots + 3) q^{5} + ( - 5 \zeta_{22}^{9} + \cdots - 5 \zeta_{22}) q^{9} + ( - 11 \zeta_{22}^{9} + 11 \zeta_{22}^{8} + \cdots + 11) q^{11}+ \cdots + ( - 55 \zeta_{22}^{8} + 66 \zeta_{22}^{7} + \cdots - 55) q^{99}+O(q^{100}) $ q + (2z^9 - 2z^8 + 2z^7 + z^5 - 3z^4 + z^3 - z^2 + 3z - 1) q^3 + 4z^2 q^4 + (3z^9 - z^7 - 3z^6 - z^5 + 3z^3 - 3z + 3) * q^5 + (-5z^9 + 6z^8 + 5z^7 - 4z^5 + 5z^3 + 6z^2 - 5z) q^9 + (-11z^9 + 11z^8 - 11z^7 + 11z^6 - 11z^5 + 11z^4 - 11z^3 + 11z^2 - 11z + 11) q^11 + (8z^8 - 4z^7 - 4z^6 - 4z^5 + 4z^4 + 4z^3 + 4z^2 - 8z) * q^12 + (26z^9 - 26z^8 - 10z^6 + 22z^5 - 15z^4 + 14z^3 - 15z^2 + 22z - 10) * q^15 + 16z^4 q^16 + (-4z^9 - 12z^8 - 4z^7 + 12z^5 - 12z^3 + 12z^2 - 12) * q^20 + (22z^9 - 13z^8 + 22z^7 - 22z^6 + 22z^5 - 13z^4 + 13z^3 - 13z^2 + 22z - 13) q^23 + (-23z^9 + 23z^8 - 20z^7 + 20z^6 - 23z^5 + 23z^4 + 26z^2 - 45z + 26) * q^25 + (2z^9 + 32z^8 - 32z^7 - 2z^6 + 25z^4 - 20z^3 - 9z^2 - 20z + 25) * q^27 + (-4z^9 + 4z^8 + 30z^6 - 15z^5 + 15z^4 + 15z^2 - 15z + 30) q^31 + (11z^9 + 11z^8 - 11z^7 + 11z^6 + 11z^5 - 22z^3 + 22) * q^33 + (44z^9 - 24z^8 + 8z^7 - 24z^6 + 44z^5 + 4z^3 - 24z^2 + 24z - 4) * q^36 + (21z^9 - 21z^8 + 44z^7 - 42z^6 - 42z^4 + 44z^3 - 21z^2 + 21z) * q^37 + 44z q^44 + (-40z^9 - 41z^8 + 53z^7 - 53z^4 + 41z^3 + 40z^2 - 62) * q^45 + (-24z^8 - 24z^7 + 13z^6 - 13z^5 + 24z^4 + 24z^3 + 24) * q^47 + (16z^9 - 48z^8 + 16z^7 - 16z^6 + 48z^5 - 16z^4 - 32z^2 + 32z - 32) * q^48 + 49z^8 q^49 + (24z^8 - 24z^7 + z^6 - 48z^5 - 48z^3 + z^2 - 24z + 24) q^53 + (-33z^9 + 66z^8 - 33z^7 + 22z^6 - 66z^5 + 22z^4 - 33z^3 + 66z^2 - 33z) q^55 + (-46z^9 - 15z^8 - 15z^7 - 15z^6 - 46z^5 + 15z^2 - 15z) q^59 + (-104z^9 + 64z^8 - 16z^7 + 44z^6 - 48z^5 + 44z^4 - 16z^3 + 64z^2 - 104z) q^60 + 64z^6 q^64 + (39z^9 - 2z^8 + 39z^7 - 39z^6 + 2z^5 - 39z^4 - 78z^2 + 78z - 78) * q^67 + (5z^9 - 40z^8 + 40z^7 - 40z^6 + 5z^5 - 45z^4 + 44z^3 + 45z - 71) * q^69 + (-15z^9 + 15z^7 - 15z^6 + 15z^4 - 74z^2 - 15z - 74) * q^71 + (81z^9 - 87z^7 + 10z^6 + 74z^5 + 39z^4 - 39z^3 - 74z^2 - 10z + 87) * q^75 + (-64z^9 + 48z^8 + 48z^6 - 96z^5 + 96z^4 - 48z^3 - 48z + 64) q^80 + (71z^9 - 116z^8 + 52z^7 + 9z^6 + 145z^5 - 145z^4 - 9z^3 - 52z^2 + 116z - 71) q^81 + (-90z^9 + 90z^8 - 90z^7 - 45z^5 - 26z^4 - 45z^3 + 45z^2 + 26z + 45) * q^89 + (36z^9 - 36z^8 + 36z^7 + 36z^3 - 52z - 36) q^92 + (45z^9 - 3z^8 + 22z^7 + 15z^6 - 15z^5 - 22z^4 + 3z^3 - 45z^2 + 53) * q^93 + (-102z^9 + 51z^8 - 51z^7 - 51z^5 + 51z^4 - 102z^3 - 124z + 124) q^97 + (-55z^8 + 66z^7 + 55z^6 - 44z^4 + 55z^2 + 66z - 55) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 5 q^{3} - 4 q^{4} + 34 q^{5} - 16 q^{9} + 11 q^{11} - 24 q^{12} + 50 q^{15} - 16 q^{16} - 128 q^{20} + 32 q^{23} + 57 q^{25} + 134 q^{27} + 202 q^{31} + 187 q^{33} + 156 q^{36} + 256 q^{37} + 44 q^{44}+ \cdots - 429 q^{99}+O(q^{100}) $ 10 * q + 5 * q^3 - 4 * q^4 + 34 * q^5 - 16 * q^9 + 11 * q^11 - 24 * q^12 + 50 * q^15 - 16 * q^16 - 128 * q^20 + 32 * q^23 + 57 * q^25 + 134 * q^27 + 202 * q^31 + 187 * q^33 + 156 * q^36 + 256 * q^37 + 44 * q^44 - 512 * q^45 + 214 * q^47 - 96 * q^48 - 49 * q^49 + 70 * q^53 - 374 * q^55 - 107 * q^59 - 504 * q^60 - 64 * q^64 - 464 * q^67 - 446 * q^69 - 681 * q^71 + 914 * q^75 + 192 * q^80 - 31 * q^81 + 97 * q^89 - 268 * q^92 + 640 * q^93 + 708 * q^97 - 429 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/253\mathbb{Z}\right)^\times$.

$n$	$24$	$166$
$\chi(n)$	$-1$	$-\zeta_{22}$

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} $

32.1

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

−0.615460

−

0.710278i

3.36501

−

2.16256i

0.262387

1.82494i

1.15513

−

8.03410i

54.1

−2.49223

−

5.45722i

−3.83797

−

1.12693i

6.35626

7.33552i

−17.6763

20.3995i

87.1

−0.615460

0.710278i

3.36501

2.16256i

0.262387

−

1.82494i

1.15513

8.03410i

98.1

2.46714

0.724417i

−2.61944

−

3.02300i

7.26448

−

4.66860i

−2.00930

−

1.29130i

131.1

3.24982

−

2.08853i

−0.569259

−

3.95929i

−2.85172

−

6.24440i

2.46061

−

5.38799i

142.1

2.46714

−

0.724417i

−2.61944

3.02300i

7.26448

4.66860i

−2.00930

1.29130i

164.1

−2.49223

5.45722i

−3.83797

1.12693i

6.35626

−

7.33552i

−17.6763

−

20.3995i

186.1

−0.109264

−

0.759951i

1.66166

3.63853i

5.96859

1.75253i

8.06985

−

2.36952i

197.1

3.24982

2.08853i

−0.569259

3.95929i

−2.85172

6.24440i

2.46061

5.38799i

219.1

−0.109264

0.759951i

1.66166

−

3.63853i

5.96859

−

1.75253i

8.06985

2.36952i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by $\Q(\sqrt{-11}) $
23.c	even	11	1	inner
253.k	odd	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	253.3.k.b	✓	10
11.b	odd	2	1	CM	253.3.k.b	✓	10
23.c	even	11	1	inner	253.3.k.b	✓	10
253.k	odd	22	1	inner	253.3.k.b	✓	10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
253.3.k.b	✓	10	1.a	even	1	1	trivial
253.3.k.b	✓	10	11.b	odd	2	1	CM
253.3.k.b	✓	10	23.c	even	11	1	inner
253.3.k.b	✓	10	253.k	odd	22	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(253, [\chi])$:

$ T_{2} $

$ T_{3}^{10} - 5 T_{3}^{9} + 25 T_{3}^{8} - 224 T_{3}^{7} + 1120 T_{3}^{6} - 2036 T_{3}^{5} + 478 T_{3}^{4} + \cdots + 1849 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{10} $ T^10
$3$	$ T^{10} - 5 T^{9} + \cdots + 1849 $ T^10 - 5T^9 + 25T^8 - 224T^7 + 1120T^6 - 2036T^5 + 478T^4 - 113T^3 + 3535T^2 + 1333*T + 1849
$5$	$ T^{10} - 34 T^{9} + \cdots + 43546801 $ T^10 - 34T^9 + 562T^8 - 5743T^7 + 43517T^6 - 293965T^5 + 1827310T^4 - 8313559T^3 + 21594629T^2 - 29246768*T + 43546801
$7$	$ T^{10} $ T^10
$11$	$ T^{10} + \cdots + 25937424601 $ T^10 - 11T^9 + 121T^8 - 1331T^7 + 14641T^6 - 161051T^5 + 1771561T^4 - 19487171T^3 + 214358881T^2 - 2357947691*T + 25937424601
$13$	$ T^{10} $ T^10
$17$	$ T^{10} $ T^10
$19$	$ T^{10} $ T^10
$23$	$ T^{10} + \cdots + 41426511213649 $ T^10 - 32T^9 + 1420T^8 - 48113T^7 + 1217074T^6 - 32865293T^5 + 643832146T^4 - 13463990033T^3 + 210210962380T^2 - 2505951528992*T + 41426511213649
$29$	$ T^{10} $ T^10
$31$	$ T^{10} + \cdots + 479667335913481 $ T^10 - 202T^9 + 18034T^8 - 934723T^7 + 31422845T^6 - 746848609T^5 + 15069583918T^4 - 251808918091T^3 + 2906274321509T^2 - 20990214545600*T + 479667335913481
$37$	$ T^{10} + \cdots + 71\!\cdots\!89 $ T^10 - 256T^9 + 30193T^8 - 2084923T^7 + 89515276T^6 - 2729896885T^5 + 133185778520T^4 - 8379682574547T^3 + 253058453880556T^2 - 1052242436811075*T + 71532301659084289
$41$	$ T^{10} $ T^10
$43$	$ T^{10} $ T^10
$47$	$ (T^{5} - 107 T^{4} + \cdots - 220576883)^{2} $ (T^5 - 107T^4 - 2500T^3 + 392289T^2 + 810459T - 220576883)^2
$53$	$ T^{10} + \cdots + 37\!\cdots\!49 $ T^10 - 70T^9 + 4900T^8 - 1084576T^7 + 75920320T^6 - 1321839014T^5 + 40227600428T^4 - 5271164190737T^3 + 209914808786835T^2 - 1780799778593333*T + 37898119289918449
$59$	$ T^{10} + \cdots + 84756714608281 $ T^10 + 107T^9 + 11449T^8 + 1799408T^7 + 192536656T^6 + 8338652860T^5 + 246595605710T^4 + 7536783152527T^3 + 122997294514143T^2 - 132987575108315*T + 84756714608281
$61$	$ T^{10} $ T^10
$67$	$ T^{10} + \cdots + 36\!\cdots\!09 $ T^10 + 464T^9 + 100753T^8 + 12997817T^7 + 1041619276T^6 + 54556808855T^5 + 3760439607560T^4 + 396125442189033T^3 + 22255868560183756T^2 + 324656097521170905*T + 36191352967936952809
$71$	$ T^{10} + \cdots + 10\!\cdots\!81 $ T^10 + 681T^9 + 225292T^8 + 47507624T^7 + 7065531628T^6 + 776618009969T^5 + 64453190888065T^4 + 4031618730754790T^3 + 186692904685040920T^2 + 5953026481371517360*T + 105754042648100614681
$73$	$ T^{10} $ T^10
$79$	$ T^{10} $ T^10
$83$	$ T^{10} $ T^10
$89$	$ T^{10} + \cdots + 38\!\cdots\!61 $ T^10 - 97T^9 + 9409T^8 + 3008222T^7 - 291797534T^6 + 126307741240T^5 + 519637253375T^4 - 2936064813843242T^3 + 456805877548865628T^2 + 345361305903416050*T + 381770295383648161
$97$	$ T^{10} + \cdots + 11\!\cdots\!49 $ T^10 - 708T^9 + 295399T^8 - 81899695T^7 + 16555975252T^6 - 2440686424606T^5 + 255470642399515T^4 - 17054733914953465T^3 + 453190885780085308T^2 + 13509024828161589427*T + 113190307161066915649
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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 253 = 11 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	253.k (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\( q + (2 \zeta_{22}^{9} - 2 \zeta_{22}^{8} + \cdots - 1) q^{3} + 4 \zeta_{22}^{2} q^{4} + (3 \zeta_{22}^{9} - \zeta_{22}^{7} + \cdots + 3) q^{5} + ( - 5 \zeta_{22}^{9} + \cdots - 5 \zeta_{22}) q^{9} + ( - 11 \zeta_{22}^{9} + 11 \zeta_{22}^{8} + \cdots + 11) q^{11}+ \cdots + ( - 55 \zeta_{22}^{8} + 66 \zeta_{22}^{7} + \cdots - 55) q^{99}+O(q^{100}) \) q + (2z^9 - 2z^8 + 2z^7 + z^5 - 3z^4 + z^3 - z^2 + 3z - 1) q^3 + 4z^2 q^4 + (3z^9 - z^7 - 3z^6 - z^5 + 3z^3 - 3z + 3) * q^5 + (-5z^9 + 6z^8 + 5z^7 - 4z^5 + 5z^3 + 6z^2 - 5z) q^9 + (-11z^9 + 11z^8 - 11z^7 + 11z^6 - 11z^5 + 11z^4 - 11z^3 + 11z^2 - 11z + 11) q^11 + (8z^8 - 4z^7 - 4z^6 - 4z^5 + 4z^4 + 4z^3 + 4z^2 - 8z) * q^12 + (26z^9 - 26z^8 - 10z^6 + 22z^5 - 15z^4 + 14z^3 - 15z^2 + 22z - 10) * q^15 + 16z^4 q^16 + (-4z^9 - 12z^8 - 4z^7 + 12z^5 - 12z^3 + 12z^2 - 12) * q^20 + (22z^9 - 13z^8 + 22z^7 - 22z^6 + 22z^5 - 13z^4 + 13z^3 - 13z^2 + 22z - 13) q^23 + (-23z^9 + 23z^8 - 20z^7 + 20z^6 - 23z^5 + 23z^4 + 26z^2 - 45z + 26) * q^25 + (2z^9 + 32z^8 - 32z^7 - 2z^6 + 25z^4 - 20z^3 - 9z^2 - 20z + 25) * q^27 + (-4z^9 + 4z^8 + 30z^6 - 15z^5 + 15z^4 + 15z^2 - 15z + 30) q^31 + (11z^9 + 11z^8 - 11z^7 + 11z^6 + 11z^5 - 22z^3 + 22) * q^33 + (44z^9 - 24z^8 + 8z^7 - 24z^6 + 44z^5 + 4z^3 - 24z^2 + 24z - 4) * q^36 + (21z^9 - 21z^8 + 44z^7 - 42z^6 - 42z^4 + 44z^3 - 21z^2 + 21z) * q^37 + 44z q^44 + (-40z^9 - 41z^8 + 53z^7 - 53z^4 + 41z^3 + 40z^2 - 62) * q^45 + (-24z^8 - 24z^7 + 13z^6 - 13z^5 + 24z^4 + 24z^3 + 24) * q^47 + (16z^9 - 48z^8 + 16z^7 - 16z^6 + 48z^5 - 16z^4 - 32z^2 + 32z - 32) * q^48 + 49z^8 q^49 + (24z^8 - 24z^7 + z^6 - 48z^5 - 48z^3 + z^2 - 24z + 24) q^53 + (-33z^9 + 66z^8 - 33z^7 + 22z^6 - 66z^5 + 22z^4 - 33z^3 + 66z^2 - 33z) q^55 + (-46z^9 - 15z^8 - 15z^7 - 15z^6 - 46z^5 + 15z^2 - 15z) q^59 + (-104z^9 + 64z^8 - 16z^7 + 44z^6 - 48z^5 + 44z^4 - 16z^3 + 64z^2 - 104z) q^60 + 64z^6 q^64 + (39z^9 - 2z^8 + 39z^7 - 39z^6 + 2z^5 - 39z^4 - 78z^2 + 78z - 78) * q^67 + (5z^9 - 40z^8 + 40z^7 - 40z^6 + 5z^5 - 45z^4 + 44z^3 + 45z - 71) * q^69 + (-15z^9 + 15z^7 - 15z^6 + 15z^4 - 74z^2 - 15z - 74) * q^71 + (81z^9 - 87z^7 + 10z^6 + 74z^5 + 39z^4 - 39z^3 - 74z^2 - 10z + 87) * q^75 + (-64z^9 + 48z^8 + 48z^6 - 96z^5 + 96z^4 - 48z^3 - 48z + 64) q^80 + (71z^9 - 116z^8 + 52z^7 + 9z^6 + 145z^5 - 145z^4 - 9z^3 - 52z^2 + 116z - 71) q^81 + (-90z^9 + 90z^8 - 90z^7 - 45z^5 - 26z^4 - 45z^3 + 45z^2 + 26z + 45) * q^89 + (36z^9 - 36z^8 + 36z^7 + 36z^3 - 52z - 36) q^92 + (45z^9 - 3z^8 + 22z^7 + 15z^6 - 15z^5 - 22z^4 + 3z^3 - 45z^2 + 53) * q^93 + (-102z^9 + 51z^8 - 51z^7 - 51z^5 + 51z^4 - 102z^3 - 124z + 124) q^97 + (-55z^8 + 66z^7 + 55z^6 - 44z^4 + 55z^2 + 66z - 55) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 5 q^{3} - 4 q^{4} + 34 q^{5} - 16 q^{9} + 11 q^{11} - 24 q^{12} + 50 q^{15} - 16 q^{16} - 128 q^{20} + 32 q^{23} + 57 q^{25} + 134 q^{27} + 202 q^{31} + 187 q^{33} + 156 q^{36} + 256 q^{37} + 44 q^{44}+ \cdots - 429 q^{99}+O(q^{100}) \) 10 * q + 5 * q^3 - 4 * q^4 + 34 * q^5 - 16 * q^9 + 11 * q^11 - 24 * q^12 + 50 * q^15 - 16 * q^16 - 128 * q^20 + 32 * q^23 + 57 * q^25 + 134 * q^27 + 202 * q^31 + 187 * q^33 + 156 * q^36 + 256 * q^37 + 44 * q^44 - 512 * q^45 + 214 * q^47 - 96 * q^48 - 49 * q^49 + 70 * q^53 - 374 * q^55 - 107 * q^59 - 504 * q^60 - 64 * q^64 - 464 * q^67 - 446 * q^69 - 681 * q^71 + 914 * q^75 + 192 * q^80 - 31 * q^81 + 97 * q^89 - 268 * q^92 + 640 * q^93 + 708 * q^97 - 429 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more