LMFDB - Newform orbit 70.3.l.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [70,3,Mod(23,70)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(70, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([9, 4])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("70.23"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$N$	$=$	$70 = 2 \cdot 5 \cdot 7$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	70.l (of order $12$ , degree $4$ , minimal)

Newform invariants

comment:select newform

sage:traces = [8,4,-4] 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:	$1.90736185052$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	8.0.3317760000.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - 25x^{4} + 625$ x^8 - 25*x^4 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/70\mathbb{Z}\right)^\times$ .

$n$	$31$	$57$
$\chi(n)$	$-1 + \beta_{4}$	$\beta_{6}$

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}

23.1

0.578737	−	2.15988i
−0.578737	+	2.15988i
−2.15988	−	0.578737i
2.15988	+	0.578737i
−2.15988	+	0.578737i
2.15988	−	0.578737i
0.578737	+	2.15988i
−0.578737	−	2.15988i

−0.366025

−

1.36603i

−0.636376

2.37499i

−1.73205

1.00000i

4.96410

−

0.598076i

3.47723

6.99656

−

0.219274i

2.00000

2.00000i

2.55863

1.47723i

−2.63397

−

6.56218i

23.2

−0.366025

−

1.36603i

1.36843

−

5.10704i

−1.73205

1.00000i

4.96410

−

0.598076i

−7.47723

−2.80041

6.41543i

2.00000

2.00000i

−16.4150

−

9.47723i

−2.63397

−

6.56218i

37.1

1.36603

−

0.366025i

−5.10704

−

1.36843i

1.73205

−

1.00000i

−1.96410

−

4.59808i

−7.47723

−6.41543

−

2.80041i

2.00000

−

2.00000i

16.4150

9.47723i

−4.36603

−

5.56218i

37.2

1.36603

−

0.366025i

2.37499

0.636376i

1.73205

−

1.00000i

−1.96410

−

4.59808i

3.47723

0.219274

6.99656i

2.00000

−

2.00000i

−2.55863

−

1.47723i

−4.36603

−

5.56218i

53.1

1.36603

0.366025i

−5.10704

1.36843i

1.73205

1.00000i

−1.96410

4.59808i

−7.47723

−6.41543

2.80041i

2.00000

2.00000i

16.4150

−

9.47723i

−4.36603

5.56218i

53.2

1.36603

0.366025i

2.37499

−

0.636376i

1.73205

1.00000i

−1.96410

4.59808i

3.47723

0.219274

−

6.99656i

2.00000

2.00000i

−2.55863

1.47723i

−4.36603

5.56218i

67.1

−0.366025

1.36603i

−0.636376

−

2.37499i

−1.73205

−

1.00000i

4.96410

0.598076i

3.47723

6.99656

0.219274i

2.00000

−

2.00000i

2.55863

−

1.47723i

−2.63397

6.56218i

67.2

−0.366025

1.36603i

1.36843

5.10704i

−1.73205

−

1.00000i

4.96410

0.598076i

−7.47723

−2.80041

−

6.41543i

2.00000

−

2.00000i

−16.4150

9.47723i

−2.63397

6.56218i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
7.c	even	3	1	inner
35.l	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	70.3.l.a	✓	8
5.b	even	2	1		350.3.p.c		8
5.c	odd	4	1	inner	70.3.l.a	✓	8
5.c	odd	4	1		350.3.p.c		8
7.c	even	3	1	inner	70.3.l.a	✓	8
7.c	even	3	1		490.3.f.l		4
7.d	odd	6	1		490.3.f.e		4
35.j	even	6	1		350.3.p.c		8
35.k	even	12	1		490.3.f.e		4
35.l	odd	12	1	inner	70.3.l.a	✓	8
35.l	odd	12	1		350.3.p.c		8
35.l	odd	12	1		490.3.f.l		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.3.l.a	✓	8	1.a	even	1	1	trivial
70.3.l.a	✓	8	5.c	odd	4	1	inner
70.3.l.a	✓	8	7.c	even	3	1	inner
70.3.l.a	✓	8	35.l	odd	12	1	inner
350.3.p.c		8	5.b	even	2	1
350.3.p.c		8	5.c	odd	4	1
350.3.p.c		8	35.j	even	6	1
350.3.p.c		8	35.l	odd	12	1
490.3.f.e		4	7.d	odd	6	1
490.3.f.e		4	35.k	even	12	1
490.3.f.l		4	7.c	even	3	1
490.3.f.l		4	35.l	odd	12	1

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{8} + 4T_{3}^{7} + 8T_{3}^{6} + 136T_{3}^{5} + 103T_{3}^{4} - 1768T_{3}^{3} + 1352T_{3}^{2} - 8788T_{3} + 28561$ T3^8 + 4*T3^7 + 8*T3^6 + 136*T3^5 + 103*T3^4 - 1768*T3^3 + 1352*T3^2 - 8788*T3 + 28561 acting on $S_{3}^{\mathrm{new}}(70, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2}$ (T^4 - 2T^3 + 2T^2 - 4*T + 4)^2
$3$	$T^{8} + 4 T^{7} + \cdots + 28561$ T^8 + 4T^7 + 8T^6 + 136T^5 + 103T^4 - 1768T^3 + 1352T^2 - 8788*T + 28561
$5$	$(T^{4} - 6 T^{3} + \cdots + 625)^{2}$ (T^4 - 6T^3 + 11T^2 - 150*T + 625)^2
$7$	$T^{8} + 4 T^{7} + \cdots + 5764801$ T^8 + 4T^7 + 8T^6 - 336T^5 - 3577T^4 - 16464T^3 + 19208T^2 + 470596*T + 5764801
$11$	$(T^{4} - 4 T^{3} + \cdots + 676)^{2}$ (T^4 - 4T^3 + 42T^2 + 104*T + 676)^2
$13$	$(T^{4} + 57600)^{2}$ (T^4 + 57600)^2
$17$	$T^{8} + \cdots + 1871773696$ T^8 + 16T^7 + 128T^6 + 8704T^5 + 26368T^4 - 1810432T^3 + 5537792T^2 - 143982592*T + 1871773696
$19$	$T^{8} + \cdots + 5006411536$ T^8 - 548T^6 + 229548T^4 - 38774288*T^2 + 5006411536
$23$	$T^{8} + \cdots + 19356878641$ T^8 + 4T^7 + 8T^6 + 3016T^5 - 133097T^4 - 1124968T^3 + 1113032T^2 - 207580468*T + 19356878641
$29$	$(T^{4} + 2738 T^{2} + 625681)^{2}$ (T^4 + 2738*T^2 + 625681)^2
$31$	$(T^{2} + 16 T + 256)^{4}$ (T^2 + 16*T + 256)^4
$37$	$T^{8} + \cdots + 30601961865216$ T^8 - 144T^7 + 10368T^6 - 815616T^5 + 53192448T^4 - 1918328832T^3 + 57354780672T^2 - 1873589501952*T + 30601961865216
$41$	$(T^{2} + 34 T + 169)^{4}$ (T^2 + 34*T + 169)^4
$43$	$(T^{4} - 60 T^{3} + \cdots + 585225)^{2}$ (T^4 - 60T^3 + 1800T^2 + 45900*T + 585225)^2
$47$	$T^{8} + \cdots + 5226454388736$ T^8 - 72T^7 + 2592T^6 - 404352T^5 + 12270528T^4 + 611380224T^3 + 5925685248T^2 + 248878780416*T + 5226454388736
$53$	$T^{8} + \cdots + 53974440976$ T^8 - 76T^7 + 2888T^6 - 146224T^5 + 5324188T^4 - 70479968T^3 + 670951712T^2 - 8510492768*T + 53974440976
$59$	$T^{8} + \cdots + 562491345610000$ T^8 - 9860T^6 + 73502700T^4 - 233848634000*T^2 + 562491345610000
$61$	$(T^{4} + 162 T^{3} + \cdots + 36978561)^{2}$ (T^4 + 162T^3 + 20163T^2 + 985122*T + 36978561)^2
$67$	$T^{8} + \cdots + 10197605570161$ T^8 - 124T^7 + 7688T^6 - 510136T^5 + 28435063T^4 - 911613032T^3 + 24550620872T^2 - 707612249972*T + 10197605570161
$71$	$(T^{2} - 4 T - 1466)^{4}$ (T^2 - 4*T - 1466)^4
$73$	$T^{8} + \cdots + 11\!\cdots\!56$ T^8 - 32T^7 + 512T^6 - 392192T^5 - 28205312T^4 + 2302951424T^3 + 17653956608T^2 + 6479002075136*T + 1188896880787456
$79$	$T^{8} + \cdots + 302373843210000$ T^8 - 11940T^6 + 125174700T^4 - 207623466000*T^2 + 302373843210000
$83$	$(T^{4} - 84 T^{3} + \cdots + 870489)^{2}$ (T^4 - 84T^3 + 3528T^2 + 78372*T + 870489)^2
$89$	$T^{8} + \cdots + 264028733579521$ T^8 - 9218T^6 + 68722563T^4 - 149782922498*T^2 + 264028733579521
$97$	$(T^{4} + 36 T^{3} + \cdots + 34082244)^{2}$ (T^4 + 36T^3 + 648T^2 - 210168*T + 34082244)^2
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

$f(q)$	$=$	$q + ( - \beta_{4} + \beta_{2} + 1) q^{2} + ( - \beta_{7} + \beta_{6} + \cdots - \beta_{2}) q^{3} + ( - 2 \beta_{6} + 2 \beta_{2}) q^{4} + ( - 3 \beta_{4} - 4 \beta_{2} + 3) q^{5} + ( - 2 \beta_{7} + \beta_{5} + \beta_{3} + \cdots - 2) q^{6}+ \cdots + (8 \beta_{7} - 44 \beta_{6} + \cdots + 4 \beta_1) q^{99}+O(q^{100})$ q + (-b4 + b2 + 1) * q^2 + (-b7 + b6 - b4 + 2b3 - b2) q^3 + (-2b6 + 2b2) * q^4 + (-3b4 - 4b2 + 3) * q^5 + (-2b7 + b5 + b3 + b1 - 2) q^6 + (b6 + 2b5 + 3b4 - 3b2 + b1 - 2) q^7 + (-2b6 + 2) q^8 + (-2b7 + 2b5 - 2b3 + 8b2 - 4b1) q^9 + (b6 - 7b4 - b2) q^10 + (-b7 + 2b5 + 2b4 + 2b3 - b1) q^11 + (-2b5 + 2b4 - 2b2 + 4b1 - 2) * q^12 + (8b7 - 4b3) * q^13 + (2b7 + 6b6 - b5 + b3 - 4b2 + 3b1) * q^14 + (-6b7 + 7b6 - 4b5 + 3b3 - 4b1 + 1) q^15 + (-4b4 + 4) q^16 + (-4b6 - 8b5 - 4b4 + 4b2 + 4b1) q^17 + (4b7 - 8b6 + 8b4 - 8b3 + 8b2) q^18 + (-3b7 + 3b5 - 3b3 + 2b2 - 6b1) q^19 + (-6b6 - 8) q^20 + (6b7 - 4b5 + 23b4 - 4b3 - 2b1 + 1) q^21 + (2b6 + 2b5 + 2b1 + 2) q^22 + (5b7 + b4 + 5b3 + b2 - 1) * q^23 + (-2b7 + 4b6 - 4b5 + 4b3 - 4b2 + 2b1) * q^24 + (24b6 + 7b4 - 24b2) q^25 + (4b7 + 4b5 + 4b3 - 8b1) * q^26 + (-29b6 + 3b5 + 3b1 - 29) q^27 + (-2b7 + 4b6 + 2b4 + 6b3 + 2b2 - 6) q^28 + (12b7 - 17b6 + 6b5 - 6b3 + 6b1) q^29 + (-7b7 + b5 + 6b4 - 7b3 + 8b2 - 2b1 - 6) q^30 - 16b4 q^31 + (-4b6 - 4b4 + 4b2) q^32 + (13b4 + 13b2 - 13) * q^33 + (-8b7 - 8b6 - 4b5 + 4b3 - 4b1) q^34 + (-8b7 - 3b6 - 3b5 + 14b4 - 4b3 + 2b2 + 9b1 + 7) q^35 + (8b7 - 4b5 - 4b3 - 4b1 + 16) * q^36 + (-4b5 - 36b4 + 36b2 + 8b1 + 36) * q^37 + (6b7 - 2b6 + 2b4 - 12b3 + 2b2) q^38 + (-4b7 + 60b6 - 8b5 + 8b3 - 60b2 + 4b1) * q^39 + (2b4 - 14b2 - 2) * q^40 + (4b7 - 2b5 - 2b3 - 2b1 - 17) * q^41 + (23b6 + 4b5 - b4 + b2 - 12b1 + 24) q^42 + (-18b7 - 15b6 + 9b3 + 15) q^43 + (2b7 - 2b5 + 2b3 + 4b2 + 4b1) q^44 + (-2b7 - 24b6 + 28b5 - 32b4 + 4b3 + 24b2 - 14b1) q^45 + (-5b7 + 10b5 + 2b4 + 10b3 - 5b1) q^46 + (-12b5 - 18b4 + 18b2 + 24b1 + 18) * q^47 + (-8b7 + 4b6 + 4b3 - 4) q^48 + (-6b7 + 24b6 + 10b5 - 10b3 - 9b2 - 16b1) * q^49 + (31b6 - 17) q^50 + (-52b4 + 52) q^51 + (16b5 - 8b1) * q^52 + (-4b7 - 19b6 + 19b4 + 8b3 + 19b2) q^53 + (3b7 - 3b5 + 3b3 - 58b2 + 6b1) q^54 + (-14b7 - 8b6 - b5 + 7b3 - b1 + 6) q^55 + (-6b7 + 4b5 + 12b4 + 4b3 + 2b1 - 8) q^56 + (-47b6 + 8b5 + 8b1 - 47) q^57 + (12b7 - 17b4 + 12b3 - 17b2 + 17) * q^58 + (b7 - 70b6 + 2b5 - 2b3 + 70b2 - b1) * q^59 + (8b7 - 2b6 - 12b5 + 14b4 - 16b3 + 2b2 + 6b1) q^60 + (-4b7 - 4b5 + 81b4 - 4b3 + 8b1 - 81) q^61 + (-16b6 - 16) q^62 + (14b6 - 56b4 + 28b3 - 56b2 + 42) * q^63 - 8b6 q^64 + (12b7 - 16b5 + 12b3 + 32b1) * q^65 + 26b4 q^66 + (31b6 + 6b5 + 31b4 - 31b2 - 3b1) q^67 + (-8b7 - 8b4 - 8b3 - 8b2 + 8) * q^68 + (-8b7 + 73b6 - 4b5 + 4b3 - 4b1) q^69 + (b7 + 12b6 - 21b5 - 8b4 - 3b3 + 6b2 + 14b1 + 24) * q^70 + (14b7 - 7b5 - 7b3 - 7b1 + 2) * q^71 + (8b5 - 16b4 + 16b2 - 16b1 + 16) * q^72 + (20b7 - 8b6 + 8b4 - 40b3 + 8b2) q^73 + (-4b7 - 72b6 - 8b5 + 8b3 + 72b2 + 4b1) * q^74 + (7b7 + 24b5 - 31b4 + 7b3 + 17b2 - 48b1 + 31) * q^75 + (12b7 - 6b5 - 6b3 - 6b1 + 4) * q^76 + (16b6 + 4b5 + 27b4 - 27b2 - 12b1 - 11) q^77 + (-16b7 + 60b6 + 8b3 - 60) q^78 + (-13b7 + 13b5 - 13b3 + 30b2 - 26b1) q^79 + (16b6 - 12b4 - 16b2) q^80 + (14b7 - 28b5 + 31b4 - 28b3 + 14b1) q^81 + (4b5 + 17b4 - 17b2 - 8b1 - 17) * q^82 + (22b7 - 21b6 - 11b3 + 21) q^83 + (4b7 - 2b6 + 12b5 - 12b3 + 48b2 - 8b1) * q^84 + (32b7 + 4b6 - 12b5 - 16b3 - 12b1 - 28) q^85 + (-9b7 - 9b5 - 30b4 - 9b3 + 18b1 + 30) q^86 + (107b6 - 58b5 + 107b4 - 107b2 + 29b1) q^87 + (-4b7 - 4b6 + 4b4 + 8b3 + 4b2) q^88 + (12b7 - 12b5 + 12b3 - 17b2 + 24b1) q^89 + (24b7 - 56b6 + 16b5 - 12b3 + 16b1 - 8) q^90 + (-4b7 - 16b5 - 20b4 - 16b3 + 20b1 - 80) q^91 + (2b6 + 10b5 + 10b1 + 2) q^92 + (-16b7 + 16b4 - 16b3 + 16b2 - 16) * q^93 + (-12b7 - 36b6 - 24b5 + 24b3 + 36b2 + 12b1) * q^94 + (-3b7 - 6b6 + 42b5 - 8b4 + 6b3 + 6b2 - 21b1) q^95 + (-4b7 - 4b5 + 8b4 - 4b3 + 8b1 - 8) q^96 + (-9b6 - 20b5 - 20b1 - 9) q^97 + (20b7 + 9b6 + 15b4 - 32b3 + 15b2 - 24) q^98 + (8b7 - 44b6 + 4b5 - 4b3 + 4b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 4 q^{2} - 4 q^{3} + 12 q^{5} - 16 q^{6} - 4 q^{7} + 16 q^{8} - 28 q^{10} + 8 q^{11} - 8 q^{12} + 8 q^{15} + 16 q^{16} - 16 q^{17} + 32 q^{18} - 64 q^{20} + 100 q^{21} + 16 q^{22} - 4 q^{23} + 28 q^{25}+ \cdots - 132 q^{98}+O(q^{100})$ 8 * q + 4 * q^2 - 4 * q^3 + 12 * q^5 - 16 * q^6 - 4 * q^7 + 16 * q^8 - 28 * q^10 + 8 * q^11 - 8 * q^12 + 8 * q^15 + 16 * q^16 - 16 * q^17 + 32 * q^18 - 64 * q^20 + 100 * q^21 + 16 * q^22 - 4 * q^23 + 28 * q^25 - 232 * q^27 - 40 * q^28 - 24 * q^30 - 64 * q^31 - 16 * q^32 - 52 * q^33 + 112 * q^35 + 128 * q^36 + 144 * q^37 + 8 * q^38 - 8 * q^40 - 136 * q^41 + 188 * q^42 + 120 * q^43 - 128 * q^45 + 8 * q^46 + 72 * q^47 - 32 * q^48 - 136 * q^50 + 208 * q^51 + 76 * q^53 + 48 * q^55 - 16 * q^56 - 376 * q^57 + 68 * q^58 + 56 * q^60 - 324 * q^61 - 128 * q^62 + 112 * q^63 + 104 * q^66 + 124 * q^67 + 32 * q^68 + 160 * q^70 + 16 * q^71 + 64 * q^72 + 32 * q^73 + 124 * q^75 + 32 * q^76 + 20 * q^77 - 480 * q^78 - 48 * q^80 + 124 * q^81 - 68 * q^82 + 168 * q^83 - 224 * q^85 + 120 * q^86 + 428 * q^87 + 16 * q^88 - 64 * q^90 - 720 * q^91 + 16 * q^92 - 64 * q^93 - 32 * q^95 - 32 * q^96 - 72 * q^97 - 132 * q^98

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( \nu^{2} ) / 5$ (v^2) / 5
$\beta_{3}$	$=$	$( \nu^{3} ) / 5$ (v^3) / 5
$\beta_{4}$	$=$	$( \nu^{4} ) / 25$ (v^4) / 25
$\beta_{5}$	$=$	$( \nu^{5} ) / 25$ (v^5) / 25
$\beta_{6}$	$=$	$( \nu^{6} ) / 125$ (v^6) / 125
$\beta_{7}$	$=$	$( \nu^{7} ) / 125$ (v^7) / 125

Introduction

L-functions

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	70.3.l.a

Level	$70$
Weight	$3$
Character orbit	70.l
Analytic conductor	$1.907$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$5\beta_{2}$ 5*b2
$\nu^{3}$	$=$	$5\beta_{3}$ 5*b3
$\nu^{4}$	$=$	$25\beta_{4}$ 25*b4
$\nu^{5}$	$=$	$25\beta_{5}$ 25*b5
$\nu^{6}$	$=$	$125\beta_{6}$ 125*b6
$\nu^{7}$	$=$	$125\beta_{7}$ 125*b7

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 23.2
Significant digits:
Format: