LMFDB - Newform orbit 900.3.u.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [900,3,Mod(149,900)] 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("900.149"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5, 3])) N = Newforms(chi, 3, names="a")

Level:	$N$	$=$	$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	900.u (of order $6$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$24.5232237924$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.12960000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1$ x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{8}\cdot 3^{2}$
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$ .

$n$	$101$	$451$	$577$
$\chi(n)$	$-\beta_{2}$	$1$	$-1$

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}

149.1

−0.535233	+	0.309017i
1.40126	−	0.809017i
0.535233	−	0.309017i
−1.40126	+	0.809017i
−0.535233	−	0.309017i
1.40126	+	0.809017i
0.535233	+	0.309017i
−1.40126	−	0.809017i

−2.59808

1.50000i

−10.1723

5.87298i

4.50000

−

7.79423i

149.2

−2.59808

1.50000i

3.24410

−

1.87298i

4.50000

−

7.79423i

149.3

2.59808

−

1.50000i

−3.24410

1.87298i

4.50000

−

7.79423i

149.4

2.59808

−

1.50000i

10.1723

−

5.87298i

4.50000

−

7.79423i

749.1

−2.59808

−

1.50000i

−10.1723

−

5.87298i

4.50000

7.79423i

749.2

−2.59808

−

1.50000i

3.24410

1.87298i

4.50000

7.79423i

749.3

2.59808

1.50000i

−3.24410

−

1.87298i

4.50000

7.79423i

749.4

2.59808

1.50000i

10.1723

5.87298i

4.50000

7.79423i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
9.d	odd	6	1	inner
45.h	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	900.3.u.b		8
3.b	odd	2	1		2700.3.u.a		8
5.b	even	2	1	inner	900.3.u.b		8
5.c	odd	4	1		180.3.o.a	✓	4
5.c	odd	4	1		900.3.p.b		4
9.c	even	3	1		2700.3.u.a		8
9.d	odd	6	1	inner	900.3.u.b		8
15.d	odd	2	1		2700.3.u.a		8
15.e	even	4	1		540.3.o.a		4
15.e	even	4	1		2700.3.p.a		4
20.e	even	4	1		720.3.bs.a		4
45.h	odd	6	1	inner	900.3.u.b		8
45.j	even	6	1		2700.3.u.a		8
45.k	odd	12	1		540.3.o.a		4
45.k	odd	12	1		1620.3.g.a		4
45.k	odd	12	1		2700.3.p.a		4
45.l	even	12	1		180.3.o.a	✓	4
45.l	even	12	1		900.3.p.b		4
45.l	even	12	1		1620.3.g.a		4
60.l	odd	4	1		2160.3.bs.a		4
180.v	odd	12	1		720.3.bs.a		4
180.x	even	12	1		2160.3.bs.a		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
180.3.o.a	✓	4	5.c	odd	4	1
180.3.o.a	✓	4	45.l	even	12	1
540.3.o.a		4	15.e	even	4	1
540.3.o.a		4	45.k	odd	12	1
720.3.bs.a		4	20.e	even	4	1
720.3.bs.a		4	180.v	odd	12	1
900.3.p.b		4	5.c	odd	4	1
900.3.p.b		4	45.l	even	12	1
900.3.u.b		8	1.a	even	1	1	trivial
900.3.u.b		8	5.b	even	2	1	inner
900.3.u.b		8	9.d	odd	6	1	inner
900.3.u.b		8	45.h	odd	6	1	inner
1620.3.g.a		4	45.k	odd	12	1
1620.3.g.a		4	45.l	even	12	1
2160.3.bs.a		4	60.l	odd	4	1
2160.3.bs.a		4	180.x	even	12	1
2700.3.p.a		4	15.e	even	4	1
2700.3.p.a		4	45.k	odd	12	1
2700.3.u.a		8	3.b	odd	2	1
2700.3.u.a		8	9.c	even	3	1
2700.3.u.a		8	15.d	odd	2	1
2700.3.u.a		8	45.j	even	6	1

This newform subspace can be constructed as the kernel of the linear operator $T_{7}^{8} - 152T_{7}^{6} + 21168T_{7}^{4} - 294272T_{7}^{2} + 3748096$ T7^8 - 152*T7^6 + 21168*T7^4 - 294272*T7^2 + 3748096 acting on $S_{3}^{\mathrm{new}}(900, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{8}$ T^8
$3$	$(T^{4} - 9 T^{2} + 81)^{2}$ (T^4 - 9*T^2 + 81)^2
$5$	$T^{8}$ T^8
$7$	$T^{8} - 152 T^{6} + \cdots + 3748096$ T^8 - 152T^6 + 21168T^4 - 294272*T^2 + 3748096
$11$	$(T^{4} - 6 T^{3} + \cdots + 31329)^{2}$ (T^4 - 6T^3 - 165T^2 + 1062*T + 31329)^2
$13$	$T^{8} - 320 T^{6} + \cdots + 2560000$ T^8 - 320T^6 + 100800T^4 - 512000*T^2 + 2560000
$17$	$(T^{4} - 366 T^{2} + 31329)^{2}$ (T^4 - 366*T^2 + 31329)^2
$19$	$(T^{2} - 38 T + 301)^{4}$ (T^2 - 38*T + 301)^4
$23$	$T^{8} + \cdots + 203928109056$ T^8 + 1536T^6 + 1907712T^4 + 693633024*T^2 + 203928109056
$29$	$(T^{4} - 60 T^{3} + \cdots + 176400)^{2}$ (T^4 - 60T^3 + 780T^2 + 25200*T + 176400)^2
$31$	$(T^{4} + 4 T^{3} + \cdots + 8620096)^{2}$ (T^4 + 4T^3 + 2952T^2 - 11744*T + 8620096)^2
$37$	$(T^{2} + 196)^{4}$ (T^2 + 196)^4
$41$	$(T^{2} + 45 T + 675)^{4}$ (T^2 + 45*T + 675)^4
$43$	$T^{8} + \cdots + 495504774241$ T^8 - 2162T^6 + 3970323T^4 - 1521877202*T^2 + 495504774241
$47$	$T^{8} + \cdots + 116876510171136$ T^8 + 7296T^6 + 42420672T^4 + 78876647424*T^2 + 116876510171136
$53$	$(T^{4} - 1536 T^{2} + 166464)^{2}$ (T^4 - 1536*T^2 + 166464)^2
$59$	$(T^{4} - 174 T^{3} + \cdots + 5489649)^{2}$ (T^4 - 174T^3 + 12435T^2 - 407682*T + 5489649)^2
$61$	$(T^{4} - 44 T^{3} + \cdots + 3136)^{2}$ (T^4 - 44T^3 + 1992T^2 + 2464*T + 3136)^2
$67$	$T^{8} + \cdots + 54\!\cdots\!61$ T^8 - 17378T^6 + 228189603T^4 - 1282588173218*T^2 + 5447219503488961
$71$	$(T^{4} + 11016 T^{2} + 5143824)^{2}$ (T^4 + 11016*T^2 + 5143824)^2
$73$	$(T^{4} + 3818 T^{2} + 687241)^{2}$ (T^4 + 3818*T^2 + 687241)^2
$79$	$(T^{4} - 160 T^{3} + \cdots + 34339600)^{2}$ (T^4 - 160T^3 + 19740T^2 - 937600*T + 34339600)^2
$83$	$(T^{4} + 8112 T^{2} + 65804544)^{2}$ (T^4 + 8112*T^2 + 65804544)^2
$89$	$(T^{4} + 9216 T^{2} + 1327104)^{2}$ (T^4 + 9216*T^2 + 1327104)^2
$97$	$T^{8} + \cdots + 10\!\cdots\!41$ T^8 - 20282T^6 + 308560203T^4 - 2084975828522*T^2 + 10567700398061041
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

$f(q)$	$=$	$q + 3 \beta_1 q^{3} + (\beta_{6} + 4 \beta_1) q^{7} - 9 \beta_{2} q^{9} + ( - \beta_{7} - \beta_{5} - \beta_{2} + 1) q^{11} + ( - \beta_{4} + 10 \beta_{3} + 10 \beta_1) q^{13} + ( - \beta_{6} + \beta_{4} + \cdots - 2 \beta_1) q^{17}+ \cdots + ( - 18 \beta_{7} + 9 \beta_{5} + \cdots - 9) q^{99}+O(q^{100})$ q + 3b1 q^3 + (b6 + 4b1) q^7 - 9b2 q^9 + (-b7 - b5 - b2 + 1) * q^11 + (-b4 + 10b3 + 10b1) * q^13 + (-b6 + b4 - b3 - 2b1) q^17 + (b5 + 19) * q^19 + (-3b7 - 12b2) * q^21 + (-4b6 - 2b4 + 4b3 - 4b1) * q^23 - 27b3 q^27 + (2b7 + 2b5 - 10b2 + 10) q^29 + (7b7 + 2b2) * q^31 + (6b6 + 3b4 - 3b3 + 3b1) * q^33 - 14b3 q^37 + (-3b5 + 30) q^39 + (-15b2 - 30) q^41 + (-4b6 + 11b1) * q^43 + (b6 + 2b4 - 68b3 - 34b1) q^47 + (-8b7 - 27b2) * q^49 + (3b7 + 3b5 + 3b2 - 3) q^51 + (b6 - b4 + 14b3 + 28b1) * q^53 + (-3b6 + 57b1) * q^57 + (b7 - 2b5 + 29b2 + 58) * q^59 + (-3b7 + 3b5 + 22b2 + 22) q^61 + (9b6 + 9b4 - 36b3) q^63 + (-12b4 - 7b3 - 7b1) q^67 + (12b7 - 6b5 + 24b2 + 12) q^69 + (-6b7 + 3b5 + 72b2 + 36) q^71 + (3b6 + 3b4 - 37b3) q^73 + (10b6 + 5b4 - 64b3 + 64b1) * q^77 + (3b7 - 3b5 + 80b2 + 80) q^79 + (-81b2 - 81) q^81 + (-104b3 - 52b1) * q^83 + (-12b6 - 6b4 - 30b3 + 30b1) * q^87 + (-8b7 + 4b5 + 48b2 + 24) q^89 + (-14b5 + 100) q^91 + (-21b6 - 21b4 + 6b3) q^93 + (-13b6 + b1) q^97 + (-18b7 + 9b5 - 18b2 - 9) q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 36 q^{9} + 12 q^{11} + 152 q^{19} + 48 q^{21} + 120 q^{29} - 8 q^{31} + 240 q^{39} - 180 q^{41} + 108 q^{49} - 36 q^{51} + 348 q^{59} + 88 q^{61} + 320 q^{79} - 324 q^{81} + 800 q^{91}+O(q^{100})$ 8 * q + 36 * q^9 + 12 * q^11 + 152 * q^19 + 48 * q^21 + 120 * q^29 - 8 * q^31 + 240 * q^39 - 180 * q^41 + 108 * q^49 - 36 * q^51 + 348 * q^59 + 88 * q^61 + 320 * q^79 - 324 * q^81 + 800 * q^91

$\beta_{1}$	$=$	$( \nu^{7} + 13\nu ) / 8$ (v^7 + 13*v) / 8
$\beta_{2}$	$=$	$( -3\nu^{6} + 8\nu^{4} - 24\nu^{2} + 1 ) / 8$ (-3v^6 + 8v^4 - 24*v^2 + 1) / 8
$\beta_{3}$	$=$	$( -3\nu^{7} + 8\nu^{5} - 20\nu^{3} + \nu ) / 4$ (-3v^7 + 8v^5 - 20*v^3 + v) / 4
$\beta_{4}$	$=$	$( -5\nu^{6} + 24\nu^{4} - 56\nu^{2} + 39 ) / 4$ (-5v^6 + 24v^4 - 56*v^2 + 39) / 4
$\beta_{5}$	$=$	$( -5\nu^{7} + 16\nu^{5} - 44\nu^{3} + 31\nu ) / 2$ (-5v^7 + 16v^5 - 44v^3 + 31v) / 2
$\beta_{6}$	$=$	$( -11\nu^{6} + 24\nu^{4} - 56\nu^{2} - 15 ) / 4$ (-11v^6 + 24v^4 - 56*v^2 - 15) / 4
$\beta_{7}$	$=$	$( 13\nu^{7} - 32\nu^{5} + 88\nu^{3} + 25\nu ) / 4$ (13v^7 - 32v^5 + 88v^3 + 25v) / 4

Introduction

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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	900.3.u.b

Level	$900$
Weight	$3$
Character orbit	900.u
Analytic conductor	$24.523$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$

$\nu$	$=$	$( \beta_{7} + \beta_{5} - 6\beta_1 ) / 12$ (b7 + b5 - 6*b1) / 12
$\nu^{2}$	$=$	$( 2\beta_{6} + \beta_{4} - 18\beta_{2} ) / 12$ (2b6 + b4 - 18b2) / 12
$\nu^{3}$	$=$	$( 2\beta_{7} - \beta_{5} + 12\beta_{3} ) / 6$ (2b7 - b5 + 12b3) / 6
$\nu^{4}$	$=$	$( \beta_{6} + 2\beta_{4} - 14\beta_{2} - 14 ) / 4$ (b6 + 2b4 - 14b2 - 14) / 4
$\nu^{5}$	$=$	$( 5\beta_{7} - 10\beta_{5} + 66\beta_{3} + 66\beta_1 ) / 12$ (5b7 - 10b5 + 66b3 + 66b1) / 12
$\nu^{6}$	$=$	$( -2\beta_{6} + 2\beta_{4} - 27 ) / 3$ (-2b6 + 2b4 - 27) / 3
$\nu^{7}$	$=$	$( -13\beta_{7} - 13\beta_{5} + 174\beta_1 ) / 12$ (-13b7 - 13b5 + 174*b1) / 12

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 149.4
Significant digits:
Format: