LMFDB - Newform orbit 81.8.c.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [81,8,Mod(28,81)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("81.28"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	81.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-464] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$25.3031870642$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{10})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 10x^{2} + 100 $ x^4 + 10*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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 basis $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_1 q^{2} + 232 \beta_{2} q^{4} + (16 \beta_{3} + 16 \beta_1) q^{5} + ( - 260 \beta_{2} - 260) q^{7} + 104 \beta_{3} q^{8} - 5760 q^{10} - 320 \beta_1 q^{11} + 6890 \beta_{2} q^{13} + ( - 260 \beta_{3} - 260 \beta_1) q^{14}+ \cdots - 755943 \beta_{3} q^{98}+O(q^{100}) $ q + b1 * q^2 + 232b2 q^4 + (16b3 + 16b1) * q^5 + (-260b2 - 260) q^7 + 104b3 q^8 - 5760 * q^10 - 320b1 q^11 + 6890b2 q^13 + (-260b3 - 260b1) * q^14 + (-7744b2 - 7744) q^16 + 1248b3 q^17 + 33176 * q^19 - 3712b1 q^20 - 115200b2 q^22 + (-1664b3 - 1664b1) * q^23 + (-14035b2 - 14035) q^25 + 6890b3 q^26 + 60320 * q^28 - 7280b1 q^29 + 1508b2 q^31 + (5568b3 + 5568b1) * q^32 + (-449280b2 - 449280) q^34 - 4160b3 q^35 - 380770 * q^37 + 33176b1 q^38 - 599040b2 q^40 + (-4640b3 - 4640b1) * q^41 + (-7640b2 - 7640) q^43 - 74240b3 q^44 + 599040 * q^46 + 29824b1 q^47 - 755943b2 q^49 + (-14035b3 - 14035b1) * q^50 + (-1598480b2 - 1598480) q^52 + 54288b3 q^53 + 1843200 * q^55 + 27040b1 q^56 - 2620800b2 q^58 + (142720b3 + 142720b1) * q^59 + (988858b2 + 988858) q^61 + 1508b3 q^62 - 2995712 * q^64 - 110240b1 q^65 + 3857360b2 q^67 + (-289536b3 - 289536b1) * q^68 + (1497600b2 + 1497600) q^70 + 222720b3 q^71 - 2004730 * q^73 - 380770b1 q^74 + 7696832b2 q^76 + (83200b3 + 83200b1) * q^77 + (-2699684b2 - 2699684) q^79 - 123904b3 q^80 + 1670400 * q^82 + 142912b1 q^83 - 7188480b2 q^85 + (-7640b3 - 7640b1) * q^86 + (11980800b2 + 11980800) q^88 + 408000b3 q^89 + 1791400 * q^91 + 386048b1 q^92 + 10736640b2 q^94 + (530816b3 + 530816b1) * q^95 + (12957490b2 + 12957490) q^97 - 755943b3 q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 464 q^{4} - 520 q^{7} - 23040 q^{10} - 13780 q^{13} - 15488 q^{16} + 132704 q^{19} + 230400 q^{22} - 28070 q^{25} + 241280 q^{28} - 3016 q^{31} - 898560 q^{34} - 1523080 q^{37} + 1198080 q^{40} - 15280 q^{43}+ \cdots + 25914980 q^{97}+O(q^{100}) $ 4 * q - 464 * q^4 - 520 * q^7 - 23040 * q^10 - 13780 * q^13 - 15488 * q^16 + 132704 * q^19 + 230400 * q^22 - 28070 * q^25 + 241280 * q^28 - 3016 * q^31 - 898560 * q^34 - 1523080 * q^37 + 1198080 * q^40 - 15280 * q^43 + 2396160 * q^46 + 1511886 * q^49 - 3196960 * q^52 + 7372800 * q^55 + 5241600 * q^58 + 1977716 * q^61 - 11982848 * q^64 - 7714720 * q^67 + 2995200 * q^70 - 8018920 * q^73 - 15393664 * q^76 - 5399368 * q^79 + 6681600 * q^82 + 14376960 * q^85 + 23961600 * q^88 + 7165600 * q^91 - 21473280 * q^94 + 25914980 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{4} + 10x^{2} + 100 $ x^4 + 10*x^2 + 100 :

$\beta_{1}$	$=$	$ 6\nu $ 6*v
$\beta_{2}$	$=$	$ ( \nu^{2} ) / 10 $ (v^2) / 10
$\beta_{3}$	$=$	$ ( 3\nu^{3} ) / 5 $ (3*v^3) / 5

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_1 ) / 6 $ (b1) / 6
$\nu^{2}$	$=$	$ 10\beta_{2} $ 10*b2
$\nu^{3}$	$=$	$ ( 5\beta_{3} ) / 3 $ (5*b3) / 3

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$\beta_{2}$

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} $

28.1

−1.58114	−	2.73861i
1.58114	+	2.73861i
−1.58114	+	2.73861i
1.58114	−	2.73861i

−9.48683

−

16.4317i

−116.000

200.918i

151.789

−

262.907i

−130.000

−

225.167i

1973.26

−5760.00

28.2

9.48683

16.4317i

−116.000

200.918i

−151.789

262.907i

−130.000

−

225.167i

−1973.26

−5760.00

55.1

−9.48683

16.4317i

−116.000

−

200.918i

151.789

262.907i

−130.000

225.167i

1973.26

−5760.00

55.2

9.48683

−

16.4317i

−116.000

−

200.918i

−151.789

−

262.907i

−130.000

225.167i

−1973.26

−5760.00

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
9.c	even	3	1	inner
9.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	81.8.c.f		4
3.b	odd	2	1	inner	81.8.c.f		4
9.c	even	3	1		9.8.a.b	✓	2
9.c	even	3	1	inner	81.8.c.f		4
9.d	odd	6	1		9.8.a.b	✓	2
9.d	odd	6	1	inner	81.8.c.f		4
36.f	odd	6	1		144.8.a.m		2
36.h	even	6	1		144.8.a.m		2
45.h	odd	6	1		225.8.a.q		2
45.j	even	6	1		225.8.a.q		2
45.k	odd	12	2		225.8.b.k		4
45.l	even	12	2		225.8.b.k		4
63.l	odd	6	1		441.8.a.k		2
63.o	even	6	1		441.8.a.k		2
72.j	odd	6	1		576.8.a.bj		2
72.l	even	6	1		576.8.a.bi		2
72.n	even	6	1		576.8.a.bj		2
72.p	odd	6	1		576.8.a.bi		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
9.8.a.b	✓	2	9.c	even	3	1
9.8.a.b	✓	2	9.d	odd	6	1
81.8.c.f		4	1.a	even	1	1	trivial
81.8.c.f		4	3.b	odd	2	1	inner
81.8.c.f		4	9.c	even	3	1	inner
81.8.c.f		4	9.d	odd	6	1	inner
144.8.a.m		2	36.f	odd	6	1
144.8.a.m		2	36.h	even	6	1
225.8.a.q		2	45.h	odd	6	1
225.8.a.q		2	45.j	even	6	1
225.8.b.k		4	45.k	odd	12	2
225.8.b.k		4	45.l	even	12	2
441.8.a.k		2	63.l	odd	6	1
441.8.a.k		2	63.o	even	6	1
576.8.a.bi		2	72.l	even	6	1
576.8.a.bi		2	72.p	odd	6	1
576.8.a.bj		2	72.j	odd	6	1
576.8.a.bj		2	72.n	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{4} + 360T_{2}^{2} + 129600 $ T2^4 + 360*T2^2 + 129600 acting on $S_{8}^{\mathrm{new}}(81, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} + 360 T^{2} + 129600 $ T^4 + 360*T^2 + 129600
$3$	$ T^{4} $ T^4
$5$	$ T^{4} + \cdots + 8493465600 $ T^4 + 92160*T^2 + 8493465600
$7$	$ (T^{2} + 260 T + 67600)^{2} $ (T^2 + 260*T + 67600)^2
$11$	$ T^{4} + \cdots + 13\!\cdots\!00 $ T^4 + 36864000*T^2 + 1358954496000000
$13$	$ (T^{2} + 6890 T + 47472100)^{2} $ (T^2 + 6890*T + 47472100)^2
$17$	$ (T^{2} - 560701440)^{2} $ (T^2 - 560701440)^2
$19$	$ (T - 33176)^{4} $ (T - 33176)^4
$23$	$ T^{4} + \cdots + 99\!\cdots\!00 $ T^4 + 996802560*T^2 + 993615343622553600
$29$	$ T^{4} + \cdots + 36\!\cdots\!00 $ T^4 + 19079424000*T^2 + 364024420171776000000
$31$	$ (T^{2} + 1508 T + 2274064)^{2} $ (T^2 + 1508*T + 2274064)^2
$37$	$ (T + 380770)^{4} $ (T + 380770)^4
$41$	$ T^{4} + \cdots + 60\!\cdots\!00 $ T^4 + 7750656000*T^2 + 60072668430336000000
$43$	$ (T^{2} + 7640 T + 58369600)^{2} $ (T^2 + 7640*T + 58369600)^2
$47$	$ T^{4} + \cdots + 10\!\cdots\!00 $ T^4 + 320209551360*T^2 + 102534156782172477849600
$53$	$ (T^{2} - 1060987299840)^{2} $ (T^2 - 1060987299840)^2
$59$	$ T^{4} + \cdots + 53\!\cdots\!00 $ T^4 + 7332839424000*T^2 + 53770534018168651776000000
$61$	$ (T^{2} - 988858 T + 977840144164)^{2} $ (T^2 - 988858*T + 977840144164)^2
$67$	$ (T^{2} + \cdots + 14879226169600)^{2} $ (T^2 + 3857360*T + 14879226169600)^2
$71$	$ (T^{2} - 17857511424000)^{2} $ (T^2 - 17857511424000)^2
$73$	$ (T + 2004730)^{4} $ (T + 2004730)^4
$79$	$ (T^{2} + \cdots + 7288293699856)^{2} $ (T^2 + 2699684*T + 7288293699856)^2
$83$	$ T^{4} + \cdots + 54\!\cdots\!00 $ T^4 + 7352582307840*T^2 + 54060466593561780525465600
$89$	$ (T^{2} - 59927040000000)^{2} $ (T^2 - 59927040000000)^2
$97$	$ (T^{2} + \cdots + 167896547100100)^{2} $ (T^2 - 12957490*T + 167896547100100)^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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	81.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + 232 \beta_{2} q^{4} + (16 \beta_{3} + 16 \beta_1) q^{5} + ( - 260 \beta_{2} - 260) q^{7} + 104 \beta_{3} q^{8} - 5760 q^{10} - 320 \beta_1 q^{11} + 6890 \beta_{2} q^{13} + ( - 260 \beta_{3} - 260 \beta_1) q^{14}+ \cdots - 755943 \beta_{3} q^{98}+O(q^{100}) \) q + b1 * q^2 + 232b2 q^4 + (16b3 + 16b1) * q^5 + (-260b2 - 260) q^7 + 104b3 q^8 - 5760 * q^10 - 320b1 q^11 + 6890b2 q^13 + (-260b3 - 260b1) * q^14 + (-7744b2 - 7744) q^16 + 1248b3 q^17 + 33176 * q^19 - 3712b1 q^20 - 115200b2 q^22 + (-1664b3 - 1664b1) * q^23 + (-14035b2 - 14035) q^25 + 6890b3 q^26 + 60320 * q^28 - 7280b1 q^29 + 1508b2 q^31 + (5568b3 + 5568b1) * q^32 + (-449280b2 - 449280) q^34 - 4160b3 q^35 - 380770 * q^37 + 33176b1 q^38 - 599040b2 q^40 + (-4640b3 - 4640b1) * q^41 + (-7640b2 - 7640) q^43 - 74240b3 q^44 + 599040 * q^46 + 29824b1 q^47 - 755943b2 q^49 + (-14035b3 - 14035b1) * q^50 + (-1598480b2 - 1598480) q^52 + 54288b3 q^53 + 1843200 * q^55 + 27040b1 q^56 - 2620800b2 q^58 + (142720b3 + 142720b1) * q^59 + (988858b2 + 988858) q^61 + 1508b3 q^62 - 2995712 * q^64 - 110240b1 q^65 + 3857360b2 q^67 + (-289536b3 - 289536b1) * q^68 + (1497600b2 + 1497600) q^70 + 222720b3 q^71 - 2004730 * q^73 - 380770b1 q^74 + 7696832b2 q^76 + (83200b3 + 83200b1) * q^77 + (-2699684b2 - 2699684) q^79 - 123904b3 q^80 + 1670400 * q^82 + 142912b1 q^83 - 7188480b2 q^85 + (-7640b3 - 7640b1) * q^86 + (11980800b2 + 11980800) q^88 + 408000b3 q^89 + 1791400 * q^91 + 386048b1 q^92 + 10736640b2 q^94 + (530816b3 + 530816b1) * q^95 + (12957490b2 + 12957490) q^97 - 755943b3 q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 464 q^{4} - 520 q^{7} - 23040 q^{10} - 13780 q^{13} - 15488 q^{16} + 132704 q^{19} + 230400 q^{22} - 28070 q^{25} + 241280 q^{28} - 3016 q^{31} - 898560 q^{34} - 1523080 q^{37} + 1198080 q^{40} - 15280 q^{43}+ \cdots + 25914980 q^{97}+O(q^{100}) \) 4 * q - 464 * q^4 - 520 * q^7 - 23040 * q^10 - 13780 * q^13 - 15488 * q^16 + 132704 * q^19 + 230400 * q^22 - 28070 * q^25 + 241280 * q^28 - 3016 * q^31 - 898560 * q^34 - 1523080 * q^37 + 1198080 * q^40 - 15280 * q^43 + 2396160 * q^46 + 1511886 * q^49 - 3196960 * q^52 + 7372800 * q^55 + 5241600 * q^58 + 1977716 * q^61 - 11982848 * q^64 - 7714720 * q^67 + 2995200 * q^70 - 8018920 * q^73 - 15393664 * q^76 - 5399368 * q^79 + 6681600 * q^82 + 14376960 * q^85 + 23961600 * q^88 + 7165600 * q^91 - 21473280 * q^94 + 25914980 * q^97

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	81.8.c.f

Level	$81$
Weight	$8$
Character orbit	81.c
Analytic conductor	$25.303$
Analytic rank	$0$
Dimension	$4$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(25.3031870642\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{10})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{4} + 10x^{2} + 100 \) x^4 + 10*x^2 + 100
Coefficient ring:	\(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index:	\( 2^{2}\cdot 3^{2} \)
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( 6\nu \) 6*v
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 10 \) (v^2) / 10
\(\beta_{3}\)	\(=\)	\( ( 3\nu^{3} ) / 5 \) (3*v^3) / 5

\(\nu\)	\(=\)	\( ( \beta_1 ) / 6 \) (b1) / 6
\(\nu^{2}\)	\(=\)	\( 10\beta_{2} \) 10*b2
\(\nu^{3}\)	\(=\)	\( ( 5\beta_{3} ) / 3 \) (5*b3) / 3

$p$	$F_p(T)$
$2$	\( T^{4} + 360 T^{2} + 129600 \) T^4 + 360*T^2 + 129600
$3$	\( T^{4} \) T^4
$5$	\( T^{4} + \cdots + 8493465600 \) T^4 + 92160*T^2 + 8493465600
$7$	\( (T^{2} + 260 T + 67600)^{2} \) (T^2 + 260*T + 67600)^2
$11$	\( T^{4} + \cdots + 13\!\cdots\!00 \) T^4 + 36864000*T^2 + 1358954496000000
$13$	\( (T^{2} + 6890 T + 47472100)^{2} \) (T^2 + 6890*T + 47472100)^2
$17$	\( (T^{2} - 560701440)^{2} \) (T^2 - 560701440)^2
$19$	\( (T - 33176)^{4} \) (T - 33176)^4
$23$	\( T^{4} + \cdots + 99\!\cdots\!00 \) T^4 + 996802560*T^2 + 993615343622553600
$29$	\( T^{4} + \cdots + 36\!\cdots\!00 \) T^4 + 19079424000*T^2 + 364024420171776000000
$31$	\( (T^{2} + 1508 T + 2274064)^{2} \) (T^2 + 1508*T + 2274064)^2
$37$	\( (T + 380770)^{4} \) (T + 380770)^4
$41$	\( T^{4} + \cdots + 60\!\cdots\!00 \) T^4 + 7750656000*T^2 + 60072668430336000000
$43$	\( (T^{2} + 7640 T + 58369600)^{2} \) (T^2 + 7640*T + 58369600)^2
$47$	\( T^{4} + \cdots + 10\!\cdots\!00 \) T^4 + 320209551360*T^2 + 102534156782172477849600
$53$	\( (T^{2} - 1060987299840)^{2} \) (T^2 - 1060987299840)^2
$59$	\( T^{4} + \cdots + 53\!\cdots\!00 \) T^4 + 7332839424000*T^2 + 53770534018168651776000000
$61$	\( (T^{2} - 988858 T + 977840144164)^{2} \) (T^2 - 988858*T + 977840144164)^2
$67$	\( (T^{2} + \cdots + 14879226169600)^{2} \) (T^2 + 3857360*T + 14879226169600)^2
$71$	\( (T^{2} - 17857511424000)^{2} \) (T^2 - 17857511424000)^2
$73$	\( (T + 2004730)^{4} \) (T + 2004730)^4
$79$	\( (T^{2} + \cdots + 7288293699856)^{2} \) (T^2 + 2699684*T + 7288293699856)^2
$83$	\( T^{4} + \cdots + 54\!\cdots\!00 \) T^4 + 7352582307840*T^2 + 54060466593561780525465600
$89$	\( (T^{2} - 59927040000000)^{2} \) (T^2 - 59927040000000)^2
$97$	\( (T^{2} + \cdots + 167896547100100)^{2} \) (T^2 - 12957490*T + 167896547100100)^2
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\(n\)	\(2\)
\(\chi(n)\)	\(\beta_{2}\)

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