LMFDB - Newform orbit 975.1.cj.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,1,Mod(146,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([15, 18, 10]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("975.146");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$975 = 3 \cdot 5^{2} \cdot 13$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	975.cj (of order $30$ , degree $8$ , 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:	$0.486588387317$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{30})$
Coefficient field:	$\Q(\zeta_{60})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1$ x^16 + x^14 - x^10 - x^8 - x^6 + x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$A_{5}$
Projective field:	Galois closure of 5.1.594140625.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $q$ -expansion and trace form are shown below.

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$ .

$n$	$301$	$326$	$352$
$\chi(n)$	$-\zeta_{60}^{10}$	$-1$	$-\zeta_{60}^{18}$

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}

146.1

−0.207912	+	0.978148i
0.207912	−	0.978148i
0.406737	−	0.913545i
−0.406737	+	0.913545i
0.994522	−	0.104528i
−0.994522	+	0.104528i
0.994522	+	0.104528i
−0.994522	−	0.104528i
0.406737	+	0.913545i
−0.406737	−	0.913545i
−0.207912	−	0.978148i
0.207912	+	0.978148i
−0.743145	+	0.669131i
0.743145	−	0.669131i
−0.743145	−	0.669131i
0.743145	+	0.669131i

−0.336408

−

1.58268i

−0.994522

−

0.104528i

−1.47815

0.658114i

−0.951057

−

0.309017i

0.169131

1.60917i

−0.500000

0.866025i

0.587785

0.809017i

0.978148

0.207912i

−0.169131

1.60917i

146.2

0.336408

1.58268i

0.994522

0.104528i

−1.47815

0.658114i

0.951057

0.309017i

0.169131

1.60917i

−0.500000

0.866025i

−0.587785

−

0.809017i

0.978148

0.207912i

−0.169131

1.60917i

191.1

−0.251377

−

0.564602i

0.207912

−

0.978148i

0.413545

−

0.459289i

−0.587785

0.809017i

−0.604528

0.128496i

−0.500000

−

0.866025i

−0.951057

−

0.309017i

−0.913545

−

0.406737i

0.604528

0.128496i

191.2

0.251377

0.564602i

−0.207912

0.978148i

0.413545

−

0.459289i

0.587785

−

0.809017i

−0.604528

0.128496i

−0.500000

−

0.866025i

0.951057

0.309017i

−0.913545

−

0.406737i

0.604528

0.128496i

341.1

−0.614648

−

0.0646021i

−0.743145

−

0.669131i

−0.604528

−

0.128496i

0.587785

−

0.809017i

0.413545

0.459289i

−0.500000

0.866025i

0.951057

0.309017i

0.104528

0.994522i

−0.413545

0.459289i

341.2

0.614648

0.0646021i

0.743145

0.669131i

−0.604528

−

0.128496i

−0.587785

0.809017i

0.413545

0.459289i

−0.500000

0.866025i

−0.951057

−

0.309017i

0.104528

0.994522i

−0.413545

0.459289i

386.1

−0.614648

0.0646021i

−0.743145

0.669131i

−0.604528

0.128496i

0.587785

0.809017i

0.413545

−

0.459289i

−0.500000

−

0.866025i

0.951057

−

0.309017i

0.104528

−

0.994522i

−0.413545

−

0.459289i

386.2

0.614648

−

0.0646021i

0.743145

−

0.669131i

−0.604528

0.128496i

−0.587785

−

0.809017i

0.413545

−

0.459289i

−0.500000

−

0.866025i

−0.951057

0.309017i

0.104528

−

0.994522i

−0.413545

−

0.459289i

536.1

−0.251377

0.564602i

0.207912

0.978148i

0.413545

0.459289i

−0.587785

−

0.809017i

−0.604528

−

0.128496i

−0.500000

0.866025i

−0.951057

0.309017i

−0.913545

0.406737i

0.604528

−

0.128496i

536.2

0.251377

−

0.564602i

−0.207912

−

0.978148i

0.413545

0.459289i

0.587785

0.809017i

−0.604528

−

0.128496i

−0.500000

0.866025i

0.951057

−

0.309017i

−0.913545

0.406737i

0.604528

−

0.128496i

581.1

−0.336408

1.58268i

−0.994522

0.104528i

−1.47815

−

0.658114i

−0.951057

0.309017i

0.169131

−

1.60917i

−0.500000

−

0.866025i

0.587785

−

0.809017i

0.978148

−

0.207912i

−0.169131

−

1.60917i

581.2

0.336408

−

1.58268i

0.994522

−

0.104528i

−1.47815

−

0.658114i

0.951057

−

0.309017i

0.169131

−

1.60917i

−0.500000

−

0.866025i

−0.587785

0.809017i

0.978148

−

0.207912i

−0.169131

−

1.60917i

731.1

−1.20243

−

1.08268i

0.406737

−

0.913545i

0.169131

1.60917i

−0.951057

0.309017i

−1.47815

0.658114i

−0.500000

0.866025i

0.587785

−

0.809017i

−0.669131

−

0.743145i

1.47815

0.658114i

731.2

1.20243

1.08268i

−0.406737

0.913545i

0.169131

1.60917i

0.951057

−

0.309017i

−1.47815

0.658114i

−0.500000

0.866025i

−0.587785

0.809017i

−0.669131

−

0.743145i

1.47815

0.658114i

971.1

−1.20243

1.08268i

0.406737

0.913545i

0.169131

−

1.60917i

−0.951057

−

0.309017i

−1.47815

−

0.658114i

−0.500000

−

0.866025i

0.587785

0.809017i

−0.669131

0.743145i

1.47815

−

0.658114i

971.2

1.20243

−

1.08268i

−0.406737

−

0.913545i

0.169131

−

1.60917i

0.951057

0.309017i

−1.47815

−

0.658114i

−0.500000

−

0.866025i

−0.587785

−

0.809017i

−0.669131

0.743145i

1.47815

−

0.658114i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
13.c	even	3	1	inner
25.d	even	5	1	inner
39.i	odd	6	1	inner
75.j	odd	10	1	inner
325.y	even	15	1	inner
975.cj	odd	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	975.1.cj.a	✓	16
3.b	odd	2	1	inner	975.1.cj.a	✓	16
13.c	even	3	1	inner	975.1.cj.a	✓	16
25.d	even	5	1	inner	975.1.cj.a	✓	16
39.i	odd	6	1	inner	975.1.cj.a	✓	16
75.j	odd	10	1	inner	975.1.cj.a	✓	16
325.y	even	15	1	inner	975.1.cj.a	✓	16
975.cj	odd	30	1	inner	975.1.cj.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
975.1.cj.a	✓	16	1.a	even	1	1	trivial
975.1.cj.a	✓	16	3.b	odd	2	1	inner
975.1.cj.a	✓	16	13.c	even	3	1	inner
975.1.cj.a	✓	16	25.d	even	5	1	inner
975.1.cj.a	✓	16	39.i	odd	6	1	inner
975.1.cj.a	✓	16	75.j	odd	10	1	inner
975.1.cj.a	✓	16	325.y	even	15	1	inner
975.1.cj.a	✓	16	975.cj	odd	30	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(975, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{16} + 4 T^{14} + \cdots + 1$ T^16 + 4T^14 + 10T^12 + 26T^10 + 39T^8 - 14T^6 - 5T^4 - T^2 + 1
$3$	$T^{16} + T^{14} + \cdots + 1$ T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1
$5$	$(T^{8} - T^{6} + T^{4} + \cdots + 1)^{2}$ (T^8 - T^6 + T^4 - T^2 + 1)^2
$7$	$(T^{2} + T + 1)^{8}$ (T^2 + T + 1)^8
$11$	$T^{16} + 4 T^{14} + \cdots + 1$ T^16 + 4T^14 + 10T^12 + 26T^10 + 39T^8 - 14T^6 - 5T^4 - T^2 + 1
$13$	$(T^{4} - T^{3} + T^{2} + \cdots + 1)^{4}$ (T^4 - T^3 + T^2 - T + 1)^4
$17$	$T^{16} + T^{14} + \cdots + 1$ T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1
$19$	$(T^{8} - T^{7} + T^{5} + \cdots + 1)^{2}$ (T^8 - T^7 + T^5 - T^4 + T^3 - T + 1)^2
$23$	$T^{16} + T^{14} + \cdots + 1$ T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1
$29$	$T^{16}$ T^16
$31$	$T^{16}$ T^16
$37$	$(T^{8} + T^{7} - T^{5} + \cdots + 1)^{2}$ (T^8 + T^7 - T^5 - T^4 - T^3 + T + 1)^2
$41$	$T^{16} + T^{14} + \cdots + 1$ T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1
$43$	$(T^{4} - T^{3} + 2 T^{2} + \cdots + 1)^{4}$ (T^4 - T^3 + 2*T^2 + T + 1)^4
$47$	$(T^{8} - 4 T^{6} + 6 T^{4} + \cdots + 1)^{2}$ (T^8 - 4T^6 + 6T^4 + T^2 + 1)^2
$53$	$(T^{8} + T^{6} + 6 T^{4} + \cdots + 1)^{2}$ (T^8 + T^6 + 6T^4 - 4T^2 + 1)^2
$59$	$T^{16} + T^{14} + \cdots + 1$ T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1
$61$	$(T^{8} - T^{7} + T^{5} + \cdots + 1)^{2}$ (T^8 - T^7 + T^5 - T^4 + T^3 - T + 1)^2
$67$	$(T^{8} - 2 T^{7} - 2 T^{5} + \cdots + 1)^{2}$ (T^8 - 2T^7 - 2T^5 + 9T^4 - 8T^3 + 5T^2 - 3T + 1)^2
$71$	$T^{16}$ T^16
$73$	$(T^{4} - T^{3} + T^{2} + \cdots + 1)^{4}$ (T^4 - T^3 + T^2 - T + 1)^4
$79$	$T^{16}$ T^16
$83$	$(T^{8} - T^{6} + T^{4} + \cdots + 1)^{2}$ (T^8 - T^6 + T^4 - T^2 + 1)^2
$89$	$T^{16} + T^{14} + \cdots + 1$ T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1
$97$	$(T^{8} + 2 T^{7} + 2 T^{5} + \cdots + 1)^{2}$ (T^8 + 2T^7 + 2T^5 + 9T^4 + 8T^3 + 5T^2 + 3T + 1)^2
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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 $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 146.2
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	975.1.cj.a

Level	$975$
Weight	$1$
Character orbit	975.cj
Analytic conductor	$0.487$
Analytic rank	$0$
Dimension	$16$
Projective image	$A_{5}$
CM/RM	no
Inner twists	$8$