LMFDB - Newform orbit 42.4.e.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [42,4,Mod(25,42)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(42, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("42.25"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$42 = 2 \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	42.e (of order $3$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [4,-4,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$2.47808022024$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{1345})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} + 337x^{2} + 336x + 112896$ x^4 - x^3 + 337x^2 + 336x + 112896
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Character values

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

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

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}

25.1

−8.91856	−	15.4474i
9.41856	+	16.3134i
−8.91856	+	15.4474i
9.41856	−	16.3134i

−1.00000

−

1.73205i

−1.50000

2.59808i

−2.00000

3.46410i

−10.4186

−

18.0455i

6.00000

−18.3371

−

2.59808i

8.00000

−4.50000

−

7.79423i

−20.8371

36.0910i

25.2

−1.00000

−

1.73205i

−1.50000

2.59808i

−2.00000

3.46410i

7.91856

13.7153i

6.00000

18.3371

−

2.59808i

8.00000

−4.50000

−

7.79423i

15.8371

−

27.4307i

37.1

−1.00000

1.73205i

−1.50000

−

2.59808i

−2.00000

−

3.46410i

−10.4186

18.0455i

6.00000

−18.3371

2.59808i

8.00000

−4.50000

7.79423i

−20.8371

−

36.0910i

37.2

−1.00000

1.73205i

−1.50000

−

2.59808i

−2.00000

−

3.46410i

7.91856

−

13.7153i

6.00000

18.3371

2.59808i

8.00000

−4.50000

7.79423i

15.8371

27.4307i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	42.4.e.c	✓	4
3.b	odd	2	1		126.4.g.g		4
4.b	odd	2	1		336.4.q.j		4
7.b	odd	2	1		294.4.e.l		4
7.c	even	3	1	inner	42.4.e.c	✓	4
7.c	even	3	1		294.4.a.n		2
7.d	odd	6	1		294.4.a.m		2
7.d	odd	6	1		294.4.e.l		4
21.c	even	2	1		882.4.g.bf		4
21.g	even	6	1		882.4.a.z		2
21.g	even	6	1		882.4.g.bf		4
21.h	odd	6	1		126.4.g.g		4
21.h	odd	6	1		882.4.a.v		2
28.f	even	6	1		2352.4.a.ca		2
28.g	odd	6	1		336.4.q.j		4
28.g	odd	6	1		2352.4.a.bq		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.4.e.c	✓	4	1.a	even	1	1	trivial
42.4.e.c	✓	4	7.c	even	3	1	inner
126.4.g.g		4	3.b	odd	2	1
126.4.g.g		4	21.h	odd	6	1
294.4.a.m		2	7.d	odd	6	1
294.4.a.n		2	7.c	even	3	1
294.4.e.l		4	7.b	odd	2	1
294.4.e.l		4	7.d	odd	6	1
336.4.q.j		4	4.b	odd	2	1
336.4.q.j		4	28.g	odd	6	1
882.4.a.v		2	21.h	odd	6	1
882.4.a.z		2	21.g	even	6	1
882.4.g.bf		4	21.c	even	2	1
882.4.g.bf		4	21.g	even	6	1
2352.4.a.bq		2	28.g	odd	6	1
2352.4.a.ca		2	28.f	even	6	1

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{4} + 5T_{5}^{3} + 355T_{5}^{2} - 1650T_{5} + 108900$ T5^4 + 5*T5^3 + 355*T5^2 - 1650*T5 + 108900 acting on $S_{4}^{\mathrm{new}}(42, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{2} + 2 T + 4)^{2}$ (T^2 + 2*T + 4)^2
$3$	$(T^{2} + 3 T + 9)^{2}$ (T^2 + 3*T + 9)^2
$5$	$T^{4} + 5 T^{3} + \cdots + 108900$ T^4 + 5T^3 + 355T^2 - 1650*T + 108900
$7$	$T^{4} - 659 T^{2} + 117649$ T^4 - 659*T^2 + 117649
$11$	$T^{4} + 67 T^{3} + \cdots + 617796$ T^4 + 67T^3 + 3703T^2 + 52662*T + 617796
$13$	$(T^{2} - 41 T + 84)^{2}$ (T^2 - 41*T + 84)^2
$17$	$T^{4} - 92 T^{3} + \cdots + 10653696$ T^4 - 92T^3 + 11728T^2 + 300288*T + 10653696
$19$	$T^{4} + 43 T^{3} + \cdots + 6574096$ T^4 + 43T^3 + 4413T^2 - 110252*T + 6574096
$23$	$T^{4} + 148 T^{3} + \cdots + 9216$ T^4 + 148T^3 + 21808T^2 + 14208*T + 9216
$29$	$(T^{2} - 77 T - 39204)^{2}$ (T^2 - 77*T - 39204)^2
$31$	$T^{4} + \cdots + 4389725025$ T^4 - 520T^3 + 204145T^2 - 34452600*T + 4389725025
$37$	$T^{4} + 7 T^{3} + \cdots + 741146176$ T^4 + 7T^3 + 27273T^2 - 190568*T + 741146176
$41$	$(T^{2} + 426 T + 33264)^{2}$ (T^2 + 426*T + 33264)^2
$43$	$(T^{2} + 107 T - 72794)^{2}$ (T^2 + 107*T - 72794)^2
$47$	$T^{4} + \cdots + 1191906576$ T^4 + 576T^3 + 297252T^2 + 19885824*T + 1191906576
$53$	$T^{4} - 243 T^{3} + \cdots + 137733696$ T^4 - 243T^3 + 47313T^2 - 2851848*T + 137733696
$59$	$T^{4} + \cdots + 44160500736$ T^4 - 7T^3 + 210193T^2 + 1471008*T + 44160500736
$61$	$T^{4} + 224 T^{3} + \cdots + 51322896$ T^4 + 224T^3 + 43012T^2 + 1604736*T + 51322896
$67$	$T^{4} + \cdots + 5976217636$ T^4 + 687T^3 + 394663T^2 + 53109222*T + 5976217636
$71$	$(T^{2} - 472 T - 78804)^{2}$ (T^2 - 472*T - 78804)^2
$73$	$T^{4} + \cdots + 39938423716$ T^4 - 921T^3 + 1048087T^2 + 184058166*T + 39938423716
$79$	$T^{4} + \cdots + 2270427201$ T^4 - 526T^3 + 229027T^2 - 25063374*T + 2270427201
$83$	$(T^{2} + 221 T - 197946)^{2}$ (T^2 + 221*T - 197946)^2
$89$	$T^{4} + \cdots + 196582277376$ T^4 + 774T^3 + 1042452T^2 - 343173024*T + 196582277376
$97$	$(T^{2} - 1953 T + 541646)^{2}$ (T^2 - 1953*T + 541646)^2
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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

$f(q)$	$=$	$q - 2 \beta_{2} q^{2} + (3 \beta_{2} - 3) q^{3} + (4 \beta_{2} - 4) q^{4} + ( - 3 \beta_{2} + \beta_1) q^{5} + 6 q^{6} + ( - \beta_{3} - 3 \beta_{2} + 2) q^{7} + 8 q^{8} - 9 \beta_{2} q^{9} + ( - 2 \beta_{3} + 6 \beta_{2} + \cdots - 4) q^{10}+ \cdots + ( - 9 \beta_{3} + 306) q^{99}+O(q^{100})$ q - 2b2 q^2 + (3b2 - 3) q^3 + (4b2 - 4) q^4 + (-3b2 + b1) q^5 + 6 * q^6 + (-b3 - 3b2 + 2) q^7 + 8 * q^8 - 9b2 q^9 + (-2b3 + 6b2 - 2b1 - 4) q^10 + (b3 + 33b2 + b1 - 34) q^11 - 12b2 q^12 + (-b3 + 21) * q^13 + (4b2 - 2b1 - 6) * q^14 + (3b3 + 6) q^15 - 16b2 q^16 + (4b3 - 48b2 + 4b1 + 44) q^17 + (18b2 - 18) q^18 + (-20b2 - 3b1) * q^19 + (4b3 + 8) q^20 + (3b3 + 3b2 + 3b1 + 3) q^21 + (-2b3 + 68) q^22 + (-72b2 - 4b1) * q^23 + (24b2 - 24) q^24 + (-5b3 + 220b2 - 5b1 - 215) q^25 + (-40b2 - 2b1) * q^26 + 27 * q^27 + (4b3 + 4b2 + 4b1 + 4) q^28 + (-11b3 + 44) q^29 + (-18b2 + 6b1) * q^30 + (-2b3 - 259b2 - 2b1 + 261) q^31 + (32b2 - 32) q^32 + (-99b2 - 3b1) * q^33 + (-8b3 - 88) q^34 + (-3b3 + 342b2 - 4b1 - 6) q^35 + 36 * q^36 + (-8b2 + 9b1) * q^37 + (6b3 + 40b2 + 6b1 - 46) q^38 + (3b3 + 60b2 + 3b1 - 63) q^39 + (-24b2 + 8b1) * q^40 + (-6b3 - 210) q^41 + (-6b3 - 18b2 + 12) * q^42 + (15b3 - 61) q^43 + (-132b2 - 4b1) * q^44 + (-9b3 + 27b2 - 9b1 - 18) q^45 + (8b3 + 144b2 + 8b1 - 152) q^46 + (-294b2 + 12b1) * q^47 + 48 * q^48 + (-3b3 + 3b2 - 6b1 + 331) q^49 + (10b3 + 430) q^50 + (144b2 - 12b1) * q^51 + (4b3 + 80b2 + 4b1 - 84) q^52 + (3b3 - 123b2 + 3b1 + 120) q^53 - 54b2 q^54 + (31b3 - 268) q^55 + (-8b3 - 24b2 + 16) * q^56 + (-9b3 + 69) q^57 + (-66b2 - 22b1) * q^58 + (-25b3 + 9b2 - 25b1 + 16) q^59 + (-12b3 + 36b2 - 12b1 - 24) q^60 + (-110b2 - 4b1) * q^61 + (4b3 - 522) q^62 + (18b2 - 9b1 - 27) * q^63 + 64 * q^64 + (276b2 + 18b1) * q^65 + (6b3 + 198b2 + 6b1 - 204) q^66 + (11b3 + 338b2 + 11b1 - 349) q^67 + (192b2 - 16b1) * q^68 + (-12b3 + 228) q^69 + (8b3 - 666b2 + 2b1 + 676) q^70 + (20b3 + 226) q^71 - 72b2 q^72 + (35b3 - 478b2 + 35b1 + 443) q^73 + (-18b3 + 16b2 - 18b1 + 2) q^74 + (-660b2 + 15b1) * q^75 + (-12b3 + 92) q^76 + (32b3 + 369b2 + 35b1 - 302) q^77 + (-6b3 + 126) q^78 + (259b2 + 8b1) * q^79 + (-16b3 + 48b2 - 16b1 - 32) q^80 + (81b2 - 81) q^81 + (432b2 - 12b1) * q^82 + (-25b3 - 98) q^83 + (24b2 - 12b1 - 36) * q^84 + (-56b3 - 1432) q^85 + (92b2 + 30b1) * q^86 + (33b3 + 99b2 + 33b1 - 132) q^87 + (8b3 + 264b2 + 8b1 - 272) q^88 + (-366b2 - 42b1) * q^89 + (18b3 + 36) q^90 + (-22b3 - 60b2 - 3b1 + 378) q^91 + (-16b3 + 304) q^92 + (777b2 + 6b1) * q^93 + (-24b3 + 588b2 - 24b1 - 564) q^94 + (-14b3 - 948b2 - 14b1 + 962) q^95 - 96b2 q^96 + (-35b3 + 994) q^97 + (12b3 - 662b2 + 6b1 - 6) q^98 + (-9b3 + 306) q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{2} - 6 q^{3} - 8 q^{4} - 5 q^{5} + 24 q^{6} + 32 q^{8} - 18 q^{9} - 10 q^{10} - 67 q^{11} - 24 q^{12} + 82 q^{13} - 18 q^{14} + 30 q^{15} - 32 q^{16} + 92 q^{17} - 36 q^{18} - 43 q^{19} + 40 q^{20}+ \cdots + 1206 q^{99}+O(q^{100})$ 4 * q - 4 * q^2 - 6 * q^3 - 8 * q^4 - 5 * q^5 + 24 * q^6 + 32 * q^8 - 18 * q^9 - 10 * q^10 - 67 * q^11 - 24 * q^12 + 82 * q^13 - 18 * q^14 + 30 * q^15 - 32 * q^16 + 92 * q^17 - 36 * q^18 - 43 * q^19 + 40 * q^20 + 27 * q^21 + 268 * q^22 - 148 * q^23 - 48 * q^24 - 435 * q^25 - 82 * q^26 + 108 * q^27 + 36 * q^28 + 154 * q^29 - 30 * q^30 + 520 * q^31 - 64 * q^32 - 201 * q^33 - 368 * q^34 + 650 * q^35 + 144 * q^36 - 7 * q^37 - 86 * q^38 - 123 * q^39 - 40 * q^40 - 852 * q^41 - 214 * q^43 - 268 * q^44 - 45 * q^45 - 296 * q^46 - 576 * q^47 + 192 * q^48 + 1318 * q^49 + 1740 * q^50 + 276 * q^51 - 164 * q^52 + 243 * q^53 - 108 * q^54 - 1010 * q^55 + 258 * q^57 - 154 * q^58 + 7 * q^59 - 60 * q^60 - 224 * q^61 - 2080 * q^62 - 81 * q^63 + 256 * q^64 + 570 * q^65 - 402 * q^66 - 687 * q^67 + 368 * q^68 + 888 * q^69 + 1390 * q^70 + 944 * q^71 - 144 * q^72 + 921 * q^73 - 14 * q^74 - 1305 * q^75 + 344 * q^76 - 371 * q^77 + 492 * q^78 + 526 * q^79 - 80 * q^80 - 162 * q^81 + 852 * q^82 - 442 * q^83 - 108 * q^84 - 5840 * q^85 + 214 * q^86 - 231 * q^87 - 536 * q^88 - 774 * q^89 + 180 * q^90 + 1345 * q^91 + 1184 * q^92 + 1560 * q^93 - 1152 * q^94 + 1910 * q^95 - 192 * q^96 + 3906 * q^97 - 1318 * q^98 + 1206 * q^99

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( -\nu^{3} + 337\nu^{2} - 337\nu + 112896 ) / 113232$ (-v^3 + 337v^2 - 337v + 112896) / 113232
$\beta_{3}$	$=$	$( \nu^{3} + 673 ) / 337$ (v^3 + 673) / 337

Introduction

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Varieties

Fields

Representations

Groups

Database

Properties

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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	42.4.e.c

Level	$42$
Weight	$4$
Character orbit	42.e
Analytic conductor	$2.478$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{3} + 336\beta_{2} + \beta _1 - 337$ b3 + 336*b2 + b1 - 337
$\nu^{3}$	$=$	$337\beta_{3} - 673$ 337*b3 - 673

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