LMFDB - Newform orbit 126.5.n.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [126,5,Mod(19,126)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(126, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("126.19");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$N$	$=$	$126 = 2 \cdot 3^{2} \cdot 7$
Weight:	$k$	$=$	$5$
Character orbit:	$[\chi]$	$=$	126.n (of order $6$ , degree $2$ , 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:	$13.0246153486$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 2x^{2} + 4$ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/126\mathbb{Z}\right)^\times$ .

$n$	$29$	$73$
$\chi(n)$	$1$	$1 + \beta_{2}$

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}

19.1

−0.707107	+	1.22474i
0.707107	−	1.22474i
−0.707107	−	1.22474i
0.707107	+	1.22474i

−1.41421

2.44949i

−4.00000

−

6.92820i

20.7426

11.9758i

−47.1985

−

13.1645i

22.6274

−58.6690

33.8726i

19.2

1.41421

−

2.44949i

−4.00000

−

6.92820i

12.2574

7.07679i

12.1985

−

47.4573i

−22.6274

34.6690

−

20.0162i

73.1

−1.41421

−

2.44949i

−4.00000

6.92820i

20.7426

−

11.9758i

−47.1985

13.1645i

22.6274

−58.6690

−

33.8726i

73.2

1.41421

2.44949i

−4.00000

6.92820i

12.2574

−

7.07679i

12.1985

47.4573i

−22.6274

34.6690

20.0162i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	126.5.n.b		4
3.b	odd	2	1		42.5.g.a	✓	4
7.c	even	3	1		882.5.c.a		4
7.d	odd	6	1	inner	126.5.n.b		4
7.d	odd	6	1		882.5.c.a		4
12.b	even	2	1		336.5.bh.d		4
21.c	even	2	1		294.5.g.c		4
21.g	even	6	1		42.5.g.a	✓	4
21.g	even	6	1		294.5.c.a		4
21.h	odd	6	1		294.5.c.a		4
21.h	odd	6	1		294.5.g.c		4
84.j	odd	6	1		336.5.bh.d		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.5.g.a	✓	4	3.b	odd	2	1
42.5.g.a	✓	4	21.g	even	6	1
126.5.n.b		4	1.a	even	1	1	trivial
126.5.n.b		4	7.d	odd	6	1	inner
294.5.c.a		4	21.g	even	6	1
294.5.c.a		4	21.h	odd	6	1
294.5.g.c		4	21.c	even	2	1
294.5.g.c		4	21.h	odd	6	1
336.5.bh.d		4	12.b	even	2	1
336.5.bh.d		4	84.j	odd	6	1
882.5.c.a		4	7.c	even	3	1
882.5.c.a		4	7.d	odd	6	1

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{4} - 66T_{5}^{3} + 1791T_{5}^{2} - 22374T_{5} + 114921$ T5^4 - 66*T5^3 + 1791*T5^2 - 22374*T5 + 114921 acting on $S_{5}^{\mathrm{new}}(126, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4} + 8T^{2} + 64$ T^4 + 8*T^2 + 64
$3$	$T^{4}$ T^4
$5$	$T^{4} - 66 T^{3} + \cdots + 114921$ T^4 - 66T^3 + 1791T^2 - 22374*T + 114921
$7$	$T^{4} + 70 T^{3} + \cdots + 5764801$ T^4 + 70T^3 + 2499T^2 + 168070*T + 5764801
$11$	$T^{4} - 162 T^{3} + \cdots + 39350529$ T^4 - 162T^3 + 19971T^2 - 1016226*T + 39350529
$13$	$T^{4} + 63264 T^{2} + 569108736$ T^4 + 63264*T^2 + 569108736
$17$	$T^{4} - 204 T^{3} + \cdots + 589324176$ T^4 - 204T^3 - 10404T^2 + 4952304*T + 589324176
$19$	$T^{4} + 444 T^{3} + \cdots + 128051856$ T^4 + 444T^3 + 54396T^2 - 5024304*T + 128051856
$23$	$T^{4} + \cdots + 525265461504$ T^4 + 312T^3 + 822096T^2 - 226122624*T + 525265461504
$29$	$(T^{2} + 1362 T + 460233)^{2}$ (T^2 + 1362*T + 460233)^2
$31$	$T^{4} + \cdots + 1025818531929$ T^4 + 3786T^3 + 5790759T^2 + 3834563022*T + 1025818531929
$37$	$T^{4} + \cdots + 219747187984$ T^4 - 1396T^3 + 1480044T^2 - 654405712*T + 219747187984
$41$	$T^{4} + \cdots + 8311481425296$ T^4 + 8985816*T^2 + 8311481425296
$43$	$(T^{2} + 316 T - 10371836)^{2}$ (T^2 + 316*T - 10371836)^2
$47$	$T^{4} + \cdots + 5862054484224$ T^4 - 7896T^3 + 23203440T^2 - 19117542528*T + 5862054484224
$53$	$T^{4} + \cdots + 6375054362769$ T^4 - 1038T^3 + 3602331T^2 + 2620832706*T + 6375054362769
$59$	$T^{4} + \cdots + 105068670590601$ T^4 - 966T^3 - 9939249T^2 + 9901790766*T + 105068670590601
$61$	$T^{4} + \cdots + 2285563428864$ T^4 - 5088T^3 + 10141056T^2 - 7692079104*T + 2285563428864
$67$	$T^{4} + \cdots + 26\!\cdots\!24$ T^4 - 14600T^3 + 161759568T^2 - 750446307200*T + 2642004409786624
$71$	$(T^{2} - 4848 T + 1589184)^{2}$ (T^2 - 4848*T + 1589184)^2
$73$	$T^{4} + \cdots + 13\!\cdots\!44$ T^4 - 22584T^3 + 206515440T^2 - 824385739392*T + 1332475433535744
$79$	$T^{4} + \cdots + 45\!\cdots\!41$ T^4 - 3974T^3 + 83413155T^2 + 268723783546*T + 4572529180189441
$83$	$T^{4} + \cdots + 109011202550649$ T^4 + 69275286*T^2 + 109011202550649
$89$	$T^{4} + \cdots + 31\!\cdots\!04$ T^4 - 33156T^3 + 422445564T^2 - 1856916766512*T + 3136610653724304
$97$	$T^{4} + \cdots + 9242250571449$ T^4 + 6233814*T^2 + 9242250571449
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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 $\F_{q}$

$f(q)$	$=$	$q + \beta_1 q^{2} + 8 \beta_{2} q^{4} + (2 \beta_{3} - 11 \beta_{2} + \cdots + 11) q^{5} + ( - 7 \beta_{3} + 35 \beta_{2} + 7 \beta_1) q^{7} + 8 \beta_{3} q^{8} + ( - 11 \beta_{3} - 8 \beta_{2} + \cdots - 16) q^{10}+ \cdots + ( - 49 \beta_{3} - 3920 \beta_{2} + \cdots - 7840) q^{98}+O(q^{100})$ q + b1 * q^2 + 8b2 q^4 + (2b3 - 11b2 + b1 + 11) * q^5 + (-7b3 + 35b2 + 7b1) q^7 + 8b3 q^8 + (-11b3 - 8b2 + 11b1 - 16) q^10 + (6b3 - 81b2 + 6b1) q^11 + (34b3 + 72b2 + 68b1 + 36) q^13 + (35b3 + 112b2 + 56) * q^14 + (-64b2 - 64) q^16 + (-34b3 + 34b2 + 34b1 + 68) q^17 + (68b3 + 74b2 + 34b1 - 74) q^19 + (-8b3 + 176b2 - 16b1 + 88) q^20 + (-81b3 - 48) q^22 + (-156b2 + 306b1 - 156) * q^23 + (66b3 + 238b2 + 66b1) q^25 + (72b3 + 272b2 + 36b1 - 272) q^26 + (112b3 - 280b2 + 56b1 - 280) q^28 + (21b3 - 681) q^29 + (-87b3 - 631b2 + 87b1 - 1262) q^31 + (-64b3 - 64b1) * q^32 + (34b3 + 544b2 + 68b1 + 272) q^34 + (-266b3 + 770b2 - 70b1 + 217) q^35 + (698b2 + 48b1 + 698) * q^37 + (74b3 - 272b2 - 74b1 - 544) q^38 + (176b3 - 64b2 + 88b1 + 64) q^40 + (-392b3 - 1036b2 - 784b1 - 518) q^41 + (-1140b3 - 158) q^43 + (648b2 - 48b1 + 648) * q^44 + (-156b3 + 2448b2 - 156b1) q^46 + (680b3 - 1316b2 + 340b1 + 1316) q^47 + (980b3 - 49b2 + 490b1 - 49) q^49 + (238b3 - 528) q^50 + (272b3 - 288b2 - 272b1 - 576) q^52 + (591b3 - 519b2 + 591b1) q^53 + (147b3 - 1878b2 + 294b1 - 939) q^55 + (-280b3 - 448b2 - 280b1 - 896) q^56 + (-168b2 - 681b1 - 168) * q^58 + (-656b3 + 161b2 + 656b1 + 322) q^59 + (-328b3 - 848b2 - 164b1 + 848) q^61 + (-631b3 + 1392b2 - 1262b1 + 696) q^62 + 512 * q^64 + (372b2 + 1014b1 + 372) * q^65 + (-486b3 - 7300b2 - 486b1) q^67 + (544b3 + 272b2 + 272b1 - 272) q^68 + (770b3 + 1568b2 + 217b1 + 2128) q^70 + (732b3 + 2424) q^71 + (500b3 + 3764b2 - 500b1 + 7528) q^73 + (698b3 + 384b2 + 698b1) q^74 + (-272b3 - 1184b2 - 544b1 - 592) q^76 + (-1134b3 + 3171b2 - 777b1 + 2499) q^77 + (1987b2 - 2991b1 + 1987) * q^79 + (-64b3 - 704b2 + 64b1 - 1408) q^80 + (-1036b3 - 3136b2 - 518b1 + 3136) q^82 + (-710b3 - 5482b2 - 1420b1 - 2741) q^83 + (-1020b3 + 306) q^85 + (9120b2 - 158b1 + 9120) * q^86 + (648b3 - 384b2 + 648b1) q^88 + (-2436b3 - 5526b2 - 1218b1 + 5526) q^89 + (1946b3 + 4452b2 - 434b1 - 2520) q^91 + (2448b3 + 1248) q^92 + (-1316b3 - 2720b2 + 1316b1 - 5440) q^94 + (900b3 + 1626b2 + 900b1) q^95 + (40b3 - 2026b2 + 80b1 - 1013) q^97 + (-49b3 - 3920b2 - 49b1 - 7840) q^98
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 16 q^{4} + 66 q^{5} - 70 q^{7} - 48 q^{10} + 162 q^{11} - 128 q^{16} + 204 q^{17} - 444 q^{19} - 192 q^{22} - 312 q^{23} - 476 q^{25} - 1632 q^{26} - 560 q^{28} - 2724 q^{29} - 3786 q^{31} - 672 q^{35}+ \cdots - 23520 q^{98}+O(q^{100})$ 4 * q - 16 * q^4 + 66 * q^5 - 70 * q^7 - 48 * q^10 + 162 * q^11 - 128 * q^16 + 204 * q^17 - 444 * q^19 - 192 * q^22 - 312 * q^23 - 476 * q^25 - 1632 * q^26 - 560 * q^28 - 2724 * q^29 - 3786 * q^31 - 672 * q^35 + 1396 * q^37 - 1632 * q^38 + 384 * q^40 - 632 * q^43 + 1296 * q^44 - 4896 * q^46 + 7896 * q^47 - 98 * q^49 - 2112 * q^50 - 1728 * q^52 + 1038 * q^53 - 2688 * q^56 - 336 * q^58 + 966 * q^59 + 5088 * q^61 + 2048 * q^64 + 744 * q^65 + 14600 * q^67 - 1632 * q^68 + 5376 * q^70 + 9696 * q^71 + 22584 * q^73 - 768 * q^74 + 3654 * q^77 + 3974 * q^79 - 4224 * q^80 + 18816 * q^82 + 1224 * q^85 + 18240 * q^86 + 768 * q^88 + 33156 * q^89 - 18984 * q^91 + 4992 * q^92 - 16320 * q^94 - 3252 * q^95 - 23520 * q^98

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

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	126.5.n.b

Level	$126$
Weight	$5$
Character orbit	126.n
Analytic conductor	$13.025$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$

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

$n$ :		e.g. 2-40 or 990-1000
Significant digits:
Format: