LMFDB - Newform orbit 42.5.g.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [42,5,Mod(19,42)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(42, 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("42.19");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$N$	$=$	$42 = 2 \cdot 3 \cdot 7$
Weight:	$k$	$=$	$5$
Character orbit:	$[\chi]$	$=$	42.g (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:	$4.34153844952$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2^{2}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/42\mathbb{Z}\right)^\times$ .

$n$	$29$	$31$
$\chi(n)$	$1$	$1 + \beta_{2}$

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

−4.50000

2.59808i

−4.00000

−

6.92820i

−12.2574

−

7.07679i

−

14.6969i

12.1985

−

47.4573i

22.6274

13.5000

−

23.3827i

34.6690

−

20.0162i

19.2

1.41421

−

2.44949i

−4.50000

2.59808i

−4.00000

−

6.92820i

−20.7426

−

11.9758i

14.6969i

−47.1985

−

13.1645i

−22.6274

13.5000

−

23.3827i

−58.6690

33.8726i

31.1

−1.41421

−

2.44949i

−4.50000

−

2.59808i

−4.00000

6.92820i

−12.2574

7.07679i

14.6969i

12.1985

47.4573i

22.6274

13.5000

23.3827i

34.6690

20.0162i

31.2

1.41421

2.44949i

−4.50000

−

2.59808i

−4.00000

6.92820i

−20.7426

11.9758i

−

14.6969i

−47.1985

13.1645i

−22.6274

13.5000

23.3827i

−58.6690

−

33.8726i

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	42.5.g.a	✓	4
3.b	odd	2	1		126.5.n.b		4
4.b	odd	2	1		336.5.bh.d		4
7.b	odd	2	1		294.5.g.c		4
7.c	even	3	1		294.5.c.a		4
7.c	even	3	1		294.5.g.c		4
7.d	odd	6	1	inner	42.5.g.a	✓	4
7.d	odd	6	1		294.5.c.a		4
21.g	even	6	1		126.5.n.b		4
21.g	even	6	1		882.5.c.a		4
21.h	odd	6	1		882.5.c.a		4
28.f	even	6	1		336.5.bh.d		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.5.g.a	✓	4	1.a	even	1	1	trivial
42.5.g.a	✓	4	7.d	odd	6	1	inner
126.5.n.b		4	3.b	odd	2	1
126.5.n.b		4	21.g	even	6	1
294.5.c.a		4	7.c	even	3	1
294.5.c.a		4	7.d	odd	6	1
294.5.g.c		4	7.b	odd	2	1
294.5.g.c		4	7.c	even	3	1
336.5.bh.d		4	4.b	odd	2	1
336.5.bh.d		4	28.f	even	6	1
882.5.c.a		4	21.g	even	6	1
882.5.c.a		4	21.h	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 $S_{5}^{\mathrm{new}}(42, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4} + 8T^{2} + 64$ T^4 + 8*T^2 + 64
$3$	$(T^{2} + 9 T + 27)^{2}$ (T^2 + 9*T + 27)^2
$5$	$T^{4} + 66 T^{3} + \cdots + 114921$ 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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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$f(q)$	$=$	$q + \beta_1 q^{2} + ( - 3 \beta_{2} - 6) q^{3} + 8 \beta_{2} q^{4} + (2 \beta_{3} + 11 \beta_{2} + \cdots - 11) q^{5} + ( - 3 \beta_{3} - 6 \beta_1) q^{6} + (7 \beta_{3} + 35 \beta_{2} - 7 \beta_1) q^{7}+ \cdots + (162 \beta_{3} - 2187) q^{99}+O(q^{100})$ q + b1 * q^2 + (-3b2 - 6) q^3 + 8b2 q^4 + (2b3 + 11b2 + b1 - 11) * q^5 + (-3b3 - 6b1) * q^6 + (7b3 + 35b2 - 7b1) q^7 + 8b3 q^8 + (27b2 + 27) q^9 + (11b3 - 8b2 - 11b1 - 16) q^10 + (6b3 + 81b2 + 6b1) q^11 + (-24b2 + 24) q^12 + (-34b3 + 72b2 - 68b1 + 36) q^13 + (35b3 - 112b2 - 56) * q^14 + (-9b3 + 99) q^15 + (-64b2 - 64) q^16 + (-34b3 - 34b2 + 34b1 - 68) q^17 + (27b3 + 27b1) * q^18 + (-68b3 + 74b2 - 34b1 - 74) q^19 + (-8b3 - 176b2 - 16b1 - 88) q^20 + (-105b2 + 63b1 + 105) * q^21 + (81b3 - 48) q^22 + (156b2 + 306b1 + 156) * q^23 + (-24b3 + 24b1) * q^24 + (-66b3 + 238b2 - 66b1) q^25 + (72b3 - 272b2 + 36b1 + 272) q^26 + (-162b2 - 81) q^27 + (-112b3 - 280b2 - 56b1 - 280) q^28 + (21b3 + 681) q^29 + (72b2 + 99b1 + 72) * q^30 + (87b3 - 631b2 - 87b1 - 1262) q^31 + (-64b3 - 64b1) * q^32 + (-36b3 - 243b2 - 18b1 + 243) q^33 + (-34b3 + 544b2 - 68b1 + 272) q^34 + (-266b3 - 770b2 - 70b1 - 217) q^35 - 216 * q^36 + (698b2 - 48b1 + 698) * q^37 + (74b3 + 272b2 - 74b1 + 544) q^38 + (306b3 - 324b2 + 306b1) q^39 + (-176b3 - 64b2 - 88b1 + 64) q^40 + (-392b3 + 1036b2 - 784b1 + 518) q^41 + (-105b3 + 504b2 + 105b1) q^42 + (1140b3 - 158) q^43 + (-648b2 - 48b1 - 648) * q^44 + (27b3 - 297b2 - 27b1 - 594) q^45 + (156b3 + 2448b2 + 156b1) q^46 + (680b3 + 1316b2 + 340b1 - 1316) q^47 + (384b2 + 192) q^48 + (-980b3 - 49b2 - 490b1 - 49) q^49 + (238b3 + 528) q^50 + (306b2 - 306b1 + 306) * q^51 + (-272b3 - 288b2 + 272b1 - 576) q^52 + (591b3 + 519b2 + 591b1) q^53 + (-162b3 - 81b1) * q^54 + (-147b3 - 1878b2 - 294b1 - 939) q^55 + (-280b3 + 448b2 - 280b1 + 896) q^56 + (306b3 + 666) q^57 + (-168b2 + 681b1 - 168) * q^58 + (-656b3 - 161b2 + 656b1 - 322) q^59 + (72b3 + 792b2 + 72b1) q^60 + (328b3 - 848b2 + 164b1 + 848) q^61 + (-631b3 - 1392b2 - 1262b1 - 696) q^62 + (-189b3 - 378b1 - 945) * q^63 + 512 * q^64 + (-372b2 + 1014b1 - 372) * q^65 + (-243b3 + 144b2 + 243b1 + 288) q^66 + (486b3 - 7300b2 + 486b1) q^67 + (544b3 - 272b2 + 272b1 + 272) q^68 + (-918b3 - 936b2 - 1836b1 - 468) q^69 + (-770b3 + 1568b2 - 217b1 + 2128) q^70 + (732b3 - 2424) q^71 - 216b1 q^72 + (-500b3 + 3764b2 + 500b1 + 7528) q^73 + (698b3 - 384b2 + 698b1) q^74 + (396b3 - 714b2 + 198b1 + 714) q^75 + (272b3 - 1184b2 + 544b1 - 592) q^76 + (-1134b3 - 3171b2 - 777b1 - 2499) q^77 + (-324b3 - 2448) q^78 + (1987b2 + 2991b1 + 1987) * q^79 + (-64b3 + 704b2 + 64b1 + 1408) q^80 + 729b2 q^81 + (1036b3 - 3136b2 + 518b1 + 3136) q^82 + (-710b3 + 5482b2 - 1420b1 + 2741) q^83 + (504b3 + 1680b2 + 840) * q^84 + (1020b3 + 306) q^85 + (-9120b2 - 158b1 - 9120) * q^86 + (-63b3 - 2043b2 + 63b1 - 4086) q^87 + (-648b3 - 384b2 - 648b1) q^88 + (-2436b3 + 5526b2 - 1218b1 - 5526) q^89 + (-297b3 - 432b2 - 594b1 - 216) q^90 + (-1946b3 + 4452b2 + 434b1 - 2520) q^91 + (2448b3 - 1248) q^92 + (5679b2 + 783b1 + 5679) * q^93 + (1316b3 - 2720b2 - 1316b1 - 5440) q^94 + (900b3 - 1626b2 + 900b1) q^95 + (384b3 + 192b1) * q^96 + (-40b3 - 2026b2 - 80b1 - 1013) q^97 + (-49b3 + 3920b2 - 49b1 + 7840) q^98 + (162b3 - 2187) q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 18 q^{3} - 16 q^{4} - 66 q^{5} - 70 q^{7} + 54 q^{9} - 48 q^{10} - 162 q^{11} + 144 q^{12} + 396 q^{15} - 128 q^{16} - 204 q^{17} - 444 q^{19} + 630 q^{21} - 192 q^{22} + 312 q^{23} - 476 q^{25} + 1632 q^{26}+ \cdots - 8748 q^{99}+O(q^{100})$ 4 * q - 18 * q^3 - 16 * q^4 - 66 * q^5 - 70 * q^7 + 54 * q^9 - 48 * q^10 - 162 * q^11 + 144 * q^12 + 396 * q^15 - 128 * q^16 - 204 * q^17 - 444 * q^19 + 630 * q^21 - 192 * q^22 + 312 * q^23 - 476 * q^25 + 1632 * q^26 - 560 * q^28 + 2724 * q^29 + 144 * q^30 - 3786 * q^31 + 1458 * q^33 + 672 * q^35 - 864 * q^36 + 1396 * q^37 + 1632 * q^38 + 648 * q^39 + 384 * q^40 - 1008 * q^42 - 632 * q^43 - 1296 * q^44 - 1782 * q^45 - 4896 * q^46 - 7896 * q^47 - 98 * q^49 + 2112 * q^50 + 612 * q^51 - 1728 * q^52 - 1038 * q^53 + 2688 * q^56 + 2664 * q^57 - 336 * q^58 - 966 * q^59 - 1584 * q^60 + 5088 * q^61 - 3780 * q^63 + 2048 * q^64 - 744 * q^65 + 864 * q^66 + 14600 * q^67 + 1632 * q^68 + 5376 * q^70 - 9696 * q^71 + 22584 * q^73 + 768 * q^74 + 4284 * q^75 - 3654 * q^77 - 9792 * q^78 + 3974 * q^79 + 4224 * q^80 - 1458 * q^81 + 18816 * q^82 + 1224 * q^85 - 18240 * q^86 - 12258 * q^87 + 768 * q^88 - 33156 * q^89 - 18984 * q^91 - 4992 * q^92 + 11358 * q^93 - 16320 * q^94 + 3252 * q^95 + 23520 * q^98 - 8748 * q^99

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

Introduction

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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	42.5.g.a

Level	$42$
Weight	$5$
Character orbit	42.g
Analytic conductor	$4.342$
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: