LMFDB - Newform orbit 740.2.ca.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [740,2,Mod(19,740)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(740, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([18, 18, 35])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("740.19"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 740 = 2^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	740.ca (of order $36$, degree $12$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$5.90892974957$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{36})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{6} + 1 $ x^12 - x^6 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{36}]$

$q$-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 $\zeta_{36}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \zeta_{36}^{8} + \cdots + \zeta_{36}^{2}) q^{2} - 2 \zeta_{36} q^{4} + ( - \zeta_{36}^{11} - 2 \zeta_{36}^{2}) q^{5} + (2 \zeta_{36}^{9} + \cdots - 2 \zeta_{36}^{3}) q^{8} + 3 \zeta_{36}^{8} q^{9} + \cdots + (7 \zeta_{36}^{10} + 7 \zeta_{36}) q^{98} +O(q^{100}) $ q + (-z^8 - z^5 + z^2) * q^2 - 2z q^4 + (-z^11 - 2z^2) q^5 + (2z^9 + 2z^6 - 2z^3) q^8 + 3z^8 q^9 + (3z^10 + z^7 - 3z^4) * q^10 + (3z^10 + z^7 - z^4 - 3z) * q^13 + 4z^2 q^16 + (-z^10 - z^9 + 4z^6 + z^3 - 4z) * q^17 + (-3z^7 + 3z^4 + 3z) q^18 + (2z^6 + 4z^3 - 2) * q^20 + (4z^7 + 3z^4 - 4z) q^25 + (z^9 + 5z^6 + z^3) q^26 + (-2z^10 + 5z^8 + 5z^7 - 2z^5 + 2z^4 - 5z^2) * q^29 + (-4z^10 - 4z^7 + 4z^4) q^32 + (-5z^11 + 5z^9 + 3z^6 - 5z^3 + 3z^2) q^34 - 6z^9 q^36 + (-6z^10 - z) q^37 + (-6z^11 - 2z^8 + 6z^5) q^40 + (-5z^11 + 4z^8 - 4z^6 + 5z^5 + 5z^3 + 4) q^41 + (-6z^10 + 3z) * q^45 + (7z^8 - 7z^2) * q^49 + (z^9 + 7) * q^50 + (-6z^11 - 2z^8 + 2z^5 + 6z^2) * q^52 + (7z^11 + 4z^8 - 2z^2) q^53 + (7z^10 - 7z^6 + 3z^3 + 3z + 7) * q^58 + (6z^11 + 5z^9 - 5z^2 - 6) q^61 - 8z^3 q^64 + (-z^9 - z^6 + 8z^3 + 4) q^65 + (2z^11 + 2z^10 - 8z^7 - 2z^4 + 8z^2) q^68 + (6z^8 - 6z^5 - 6z^2) q^72 + (3z^9 - 6z^3 + 8) * q^73 + (7z^9 - 5z^6 - 7z^3) q^74 + (-4z^7 - 8z^4 + 4z) q^80 + (9z^10 - 9z^4) * q^81 + (-z^11 - 9z^8 - 9z^7 + z^5 - z^4 + 9z) q^82 + (-2z^11 - 9z^8 + 6z^6 + 2z^5 + 7z^3 - 6) q^85 + (5z^9 - 8z^7 + 5z^4 + 8z - 8) * q^89 + (3z^9 - 9z^6 - 3z^3) q^90 + (4z^11 + 9z^7 - 4z^4 - 9z^2 - 9z) q^97 + (7z^10 + 7z) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{8} + 24 q^{17} - 12 q^{20} + 30 q^{26} + 18 q^{34} + 24 q^{41} + 84 q^{50} + 42 q^{58} - 72 q^{61} + 42 q^{65} + 96 q^{73} - 30 q^{74} - 36 q^{85} - 96 q^{89} - 54 q^{90}+O(q^{100}) $ 12 * q + 12 * q^8 + 24 * q^17 - 12 * q^20 + 30 * q^26 + 18 * q^34 + 24 * q^41 + 84 * q^50 + 42 * q^58 - 72 * q^61 + 42 * q^65 + 96 * q^73 - 30 * q^74 - 36 * q^85 - 96 * q^89 - 54 * q^90

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/740\mathbb{Z}\right)^\times$.

$n$	$261$	$297$	$371$
$\chi(n)$	$\zeta_{36}$	$-1$	$-1$

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} $

19.1

0.984808	−	0.173648i
0.984808	+	0.173648i
0.642788	−	0.766044i
−0.642788	−	0.766044i
0.342020	+	0.939693i
−0.342020	−	0.939693i
0.642788	+	0.766044i
−0.642788	+	0.766044i
−0.984808	−	0.173648i
−0.984808	+	0.173648i
0.342020	−	0.939693i
−0.342020	+	0.939693i

0.123257

1.40883i

−1.96962

0.347296i

−1.53737

1.62373i

−0.732051

−

2.73205i

0.520945

−

2.95442i

−2.47706

−

1.96575i

39.1

0.123257

−

1.40883i

−1.96962

−

0.347296i

−1.53737

−

1.62373i

−0.732051

2.73205i

0.520945

2.95442i

−2.47706

1.96575i

59.1

−0.597672

−

1.28171i

−1.28558

1.53209i

1.33210

1.79597i

2.73205

0.732051i

2.29813

−

1.92836i

1.50575

−

2.78078i

79.1

−1.28171

−

0.597672i

1.28558

1.53209i

−0.637511

−

2.14326i

−0.732051

−

2.73205i

2.29813

1.92836i

−0.463863

3.12807i

239.1

−0.811160

1.15846i

−0.684040

−

1.87939i

0.889301

−

2.05162i

2.73205

0.732051i

−2.81908

−

1.02606i

1.65535

2.69441i

279.1

1.15846

0.811160i

0.684040

1.87939i

2.17488

−

0.519531i

−0.732051

2.73205i

−2.81908

−

1.02606i

2.94092

1.16232i

439.1

−0.597672

1.28171i

−1.28558

−

1.53209i

1.33210

−

1.79597i

2.73205

−

0.732051i

2.29813

1.92836i

1.50575

2.78078i

459.1

−1.28171

0.597672i

1.28558

−

1.53209i

−0.637511

2.14326i

−0.732051

2.73205i

2.29813

−

1.92836i

−0.463863

−

3.12807i

479.1

1.40883

0.123257i

1.96962

0.347296i

−2.22141

0.255652i

2.73205

0.732051i

0.520945

2.95442i

−3.16110

0.0863678i

499.1

1.40883

−

0.123257i

1.96962

−

0.347296i

−2.22141

−

0.255652i

2.73205

−

0.732051i

0.520945

−

2.95442i

−3.16110

−

0.0863678i

579.1

−0.811160

−

1.15846i

−0.684040

1.87939i

0.889301

2.05162i

2.73205

−

0.732051i

−2.81908

1.02606i

1.65535

−

2.69441i

679.1

1.15846

−

0.811160i

0.684040

−

1.87939i

2.17488

0.519531i

−0.732051

−

2.73205i

−2.81908

1.02606i

2.94092

−

1.16232i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
185.ba	odd	36	1	inner
740.ca	even	36	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	740.2.ca.a	✓	12
4.b	odd	2	1	CM	740.2.ca.a	✓	12
5.b	even	2	1		740.2.ca.b	yes	12
20.d	odd	2	1		740.2.ca.b	yes	12
37.i	odd	36	1		740.2.ca.b	yes	12
148.q	even	36	1		740.2.ca.b	yes	12
185.ba	odd	36	1	inner	740.2.ca.a	✓	12
740.ca	even	36	1	inner	740.2.ca.a	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
740.2.ca.a	✓	12	1.a	even	1	1	trivial
740.2.ca.a	✓	12	4.b	odd	2	1	CM
740.2.ca.a	✓	12	185.ba	odd	36	1	inner
740.2.ca.a	✓	12	740.ca	even	36	1	inner
740.2.ca.b	yes	12	5.b	even	2	1
740.2.ca.b	yes	12	20.d	odd	2	1
740.2.ca.b	yes	12	37.i	odd	36	1
740.2.ca.b	yes	12	148.q	even	36	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(740, [\chi])$:

$ T_{3} $

T3

$ T_{13}^{12} - 32T_{13}^{9} + 10457T_{13}^{6} + 268862T_{13}^{3} + 1771561 $

T13^12 - 32*T13^9 + 10457*T13^6 + 268862*T13^3 + 1771561

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} - 4 T^{9} + \cdots + 64 $ T^12 - 4T^9 + 8T^6 - 32*T^3 + 64
$3$	$ T^{12} $ T^12
$5$	$ T^{12} + 4 T^{9} + \cdots + 15625 $ T^12 + 4T^9 - 109T^6 + 500*T^3 + 15625
$7$	$ T^{12} $ T^12
$11$	$ T^{12} $ T^12
$13$	$ T^{12} - 32 T^{9} + \cdots + 1771561 $ T^12 - 32T^9 + 10457T^6 + 268862*T^3 + 1771561
$17$	$ T^{12} - 24 T^{11} + \cdots + 23648769 $ T^12 - 24T^11 + 333T^10 - 3270T^9 + 23670T^8 - 126954T^7 + 445356T^6 - 565236T^5 - 1917873T^4 + 1700496T^3 + 19603863T^2 + 34750998*T + 23648769
$19$	$ T^{12} $ T^12
$23$	$ T^{12} $ T^12
$29$	$ T^{12} + \cdots + 3758793481 $ T^12 + 154T^9 - 7569T^8 - 60726T^7 + 11858T^6 + 4117536T^5 + 59637369T^4 + 348784046T^3 + 1481896665T^2 + 3723050334*T + 3758793481
$31$	$ T^{12} $ T^12
$37$	$ T^{12} + \cdots + 2565726409 $ T^12 + 396T^9 + 106163T^6 + 20058588*T^3 + 2565726409
$41$	$ T^{12} + \cdots + 23673822769 $ T^12 - 24T^11 + 261T^10 - 3072T^9 + 33690T^8 - 137856T^7 - 1837816T^6 + 27208344T^5 - 70602075T^4 - 404071512T^3 + 4638436635T^2 - 10129109016*T + 23673822769
$43$	$ T^{12} $ T^12
$47$	$ T^{12} $ T^12
$53$	$ T^{12} + \cdots + 2565726409 $ T^12 - 1908T^9 + 1162835T^6 + 96645924*T^3 + 2565726409
$59$	$ T^{12} $ T^12
$61$	$ T^{12} + \cdots + 378743007241 $ T^12 + 72T^11 + 2526T^10 + 55690T^9 + 856095T^8 + 9757872T^7 + 86643203T^6 + 614606736T^5 + 3333455700T^4 + 12966788570T^3 + 42130781211T^2 + 152412703176*T + 378743007241
$67$	$ T^{12} $ T^12
$71$	$ T^{12} $ T^12
$73$	$ (T^{2} - 16 T + 37)^{6} $ (T^2 - 16*T + 37)^6
$79$	$ T^{12} $ T^12
$83$	$ T^{12} $ T^12
$89$	$ T^{12} + \cdots + 997779234321 $ T^12 + 96T^11 + 4374T^10 + 122970T^9 + 2351295T^8 + 32089536T^7 + 323525007T^6 + 2503110132T^5 + 15024407220T^4 + 62460859770T^3 + 127000042479T^2 + 73825888212*T + 997779234321
$97$	$ T^{12} + \cdots + 27566292961 $ T^12 + 2410T^9 - 84681T^8 + 182748T^7 + 2904050T^6 - 257260878T^5 + 8359742949T^4 - 74462780788T^3 + 203738844717T^2 + 30341833188*T + 27566292961
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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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	740.ca (of order \(36\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{36}^{8} + \cdots + \zeta_{36}^{2}) q^{2} - 2 \zeta_{36} q^{4} + ( - \zeta_{36}^{11} - 2 \zeta_{36}^{2}) q^{5} + (2 \zeta_{36}^{9} + \cdots - 2 \zeta_{36}^{3}) q^{8} + 3 \zeta_{36}^{8} q^{9} + \cdots + (7 \zeta_{36}^{10} + 7 \zeta_{36}) q^{98} +O(q^{100}) \) q + (-z^8 - z^5 + z^2) * q^2 - 2z q^4 + (-z^11 - 2z^2) q^5 + (2z^9 + 2z^6 - 2z^3) q^8 + 3z^8 q^9 + (3z^10 + z^7 - 3z^4) * q^10 + (3z^10 + z^7 - z^4 - 3z) * q^13 + 4z^2 q^16 + (-z^10 - z^9 + 4z^6 + z^3 - 4z) * q^17 + (-3z^7 + 3z^4 + 3z) q^18 + (2z^6 + 4z^3 - 2) * q^20 + (4z^7 + 3z^4 - 4z) q^25 + (z^9 + 5z^6 + z^3) q^26 + (-2z^10 + 5z^8 + 5z^7 - 2z^5 + 2z^4 - 5z^2) * q^29 + (-4z^10 - 4z^7 + 4z^4) q^32 + (-5z^11 + 5z^9 + 3z^6 - 5z^3 + 3z^2) q^34 - 6z^9 q^36 + (-6z^10 - z) q^37 + (-6z^11 - 2z^8 + 6z^5) q^40 + (-5z^11 + 4z^8 - 4z^6 + 5z^5 + 5z^3 + 4) q^41 + (-6z^10 + 3z) * q^45 + (7z^8 - 7z^2) * q^49 + (z^9 + 7) * q^50 + (-6z^11 - 2z^8 + 2z^5 + 6z^2) * q^52 + (7z^11 + 4z^8 - 2z^2) q^53 + (7z^10 - 7z^6 + 3z^3 + 3z + 7) * q^58 + (6z^11 + 5z^9 - 5z^2 - 6) q^61 - 8z^3 q^64 + (-z^9 - z^6 + 8z^3 + 4) q^65 + (2z^11 + 2z^10 - 8z^7 - 2z^4 + 8z^2) q^68 + (6z^8 - 6z^5 - 6z^2) q^72 + (3z^9 - 6z^3 + 8) * q^73 + (7z^9 - 5z^6 - 7z^3) q^74 + (-4z^7 - 8z^4 + 4z) q^80 + (9z^10 - 9z^4) * q^81 + (-z^11 - 9z^8 - 9z^7 + z^5 - z^4 + 9z) q^82 + (-2z^11 - 9z^8 + 6z^6 + 2z^5 + 7z^3 - 6) q^85 + (5z^9 - 8z^7 + 5z^4 + 8z - 8) * q^89 + (3z^9 - 9z^6 - 3z^3) q^90 + (4z^11 + 9z^7 - 4z^4 - 9z^2 - 9z) q^97 + (7z^10 + 7z) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{8} + 24 q^{17} - 12 q^{20} + 30 q^{26} + 18 q^{34} + 24 q^{41} + 84 q^{50} + 42 q^{58} - 72 q^{61} + 42 q^{65} + 96 q^{73} - 30 q^{74} - 36 q^{85} - 96 q^{89} - 54 q^{90}+O(q^{100}) \) 12 * q + 12 * q^8 + 24 * q^17 - 12 * q^20 + 30 * q^26 + 18 * q^34 + 24 * q^41 + 84 * q^50 + 42 * q^58 - 72 * q^61 + 42 * q^65 + 96 * q^73 - 30 * q^74 - 36 * q^85 - 96 * q^89 - 54 * q^90

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	740.2.ca.a

Level	$740$
Weight	$2$
Character orbit	740.ca
Analytic conductor	$5.909$
Analytic rank	$0$
Dimension	$12$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(5.90892974957\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\Q(\zeta_{36})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - x^{6} + 1 \) x^12 - x^6 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{36}]$

\(n\)	\(261\)	\(297\)	\(371\)
\(\chi(n)\)	\(\zeta_{36}\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 19.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
185.ba	odd	36	1	inner
740.ca	even	36	1	inner

$p$	$F_p(T)$
$2$	\( T^{12} - 4 T^{9} + \cdots + 64 \) T^12 - 4T^9 + 8T^6 - 32*T^3 + 64
$3$	\( T^{12} \) T^12
$5$	\( T^{12} + 4 T^{9} + \cdots + 15625 \) T^12 + 4T^9 - 109T^6 + 500*T^3 + 15625
$7$	\( T^{12} \) T^12
$11$	\( T^{12} \) T^12
$13$	\( T^{12} - 32 T^{9} + \cdots + 1771561 \) T^12 - 32T^9 + 10457T^6 + 268862*T^3 + 1771561
$17$	\( T^{12} - 24 T^{11} + \cdots + 23648769 \) T^12 - 24T^11 + 333T^10 - 3270T^9 + 23670T^8 - 126954T^7 + 445356T^6 - 565236T^5 - 1917873T^4 + 1700496T^3 + 19603863T^2 + 34750998*T + 23648769
$19$	\( T^{12} \) T^12
$23$	\( T^{12} \) T^12
$29$	\( T^{12} + \cdots + 3758793481 \) T^12 + 154T^9 - 7569T^8 - 60726T^7 + 11858T^6 + 4117536T^5 + 59637369T^4 + 348784046T^3 + 1481896665T^2 + 3723050334*T + 3758793481
$31$	\( T^{12} \) T^12
$37$	\( T^{12} + \cdots + 2565726409 \) T^12 + 396T^9 + 106163T^6 + 20058588*T^3 + 2565726409
$41$	\( T^{12} + \cdots + 23673822769 \) T^12 - 24T^11 + 261T^10 - 3072T^9 + 33690T^8 - 137856T^7 - 1837816T^6 + 27208344T^5 - 70602075T^4 - 404071512T^3 + 4638436635T^2 - 10129109016*T + 23673822769
$43$	\( T^{12} \) T^12
$47$	\( T^{12} \) T^12
$53$	\( T^{12} + \cdots + 2565726409 \) T^12 - 1908T^9 + 1162835T^6 + 96645924*T^3 + 2565726409
$59$	\( T^{12} \) T^12
$61$	\( T^{12} + \cdots + 378743007241 \) T^12 + 72T^11 + 2526T^10 + 55690T^9 + 856095T^8 + 9757872T^7 + 86643203T^6 + 614606736T^5 + 3333455700T^4 + 12966788570T^3 + 42130781211T^2 + 152412703176*T + 378743007241
$67$	\( T^{12} \) T^12
$71$	\( T^{12} \) T^12
$73$	\( (T^{2} - 16 T + 37)^{6} \) (T^2 - 16*T + 37)^6
$79$	\( T^{12} \) T^12
$83$	\( T^{12} \) T^12
$89$	\( T^{12} + \cdots + 997779234321 \) T^12 + 96T^11 + 4374T^10 + 122970T^9 + 2351295T^8 + 32089536T^7 + 323525007T^6 + 2503110132T^5 + 15024407220T^4 + 62460859770T^3 + 127000042479T^2 + 73825888212*T + 997779234321
$97$	\( T^{12} + \cdots + 27566292961 \) T^12 + 2410T^9 - 84681T^8 + 182748T^7 + 2904050T^6 - 257260878T^5 + 8359742949T^4 - 74462780788T^3 + 203738844717T^2 + 30341833188*T + 27566292961
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