LMFDB - Newform orbit 168.4.q.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [168,4,Mod(25,168)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(168, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 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("168.25"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$168 = 2^{3} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	168.q (of order $3$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,-12,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$9.91232088096$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} + 173x^{6} + 9457x^{4} + 168048x^{2} + 746496$ x^8 + 173x^6 + 9457x^4 + 168048*x^2 + 746496
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{6}\cdot 7$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/168\mathbb{Z}\right)^\times$ .

$n$	$73$	$85$	$113$	$127$
$\chi(n)$	$-1 - \beta_{1}$	$1$	$1$	$1$

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}

25.1

		8.67551i
	−	8.34231i
		2.57353i
	−	4.63878i
	−	8.67551i
		8.34231i
	−	2.57353i
		4.63878i

−1.50000

2.59808i

−9.47901

−

16.4181i

12.8033

13.3819i

−4.50000

−

7.79423i

25.2

−1.50000

2.59808i

−0.363171

−

0.629031i

−18.1420

−

3.72380i

−4.50000

−

7.79423i

25.3

−1.50000

2.59808i

−0.0642956

−

0.111363i

−0.866259

−

18.5000i

−4.50000

−

7.79423i

25.4

−1.50000

2.59808i

7.90648

13.6944i

15.2050

10.5739i

−4.50000

−

7.79423i

121.1

−1.50000

−

2.59808i

−9.47901

16.4181i

12.8033

−

13.3819i

−4.50000

7.79423i

121.2

−1.50000

−

2.59808i

−0.363171

0.629031i

−18.1420

3.72380i

−4.50000

7.79423i

121.3

−1.50000

−

2.59808i

−0.0642956

0.111363i

−0.866259

18.5000i

−4.50000

7.79423i

121.4

−1.50000

−

2.59808i

7.90648

−

13.6944i

15.2050

−

10.5739i

−4.50000

7.79423i

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	168.4.q.f	✓	8
3.b	odd	2	1		504.4.s.j		8
4.b	odd	2	1		336.4.q.m		8
7.c	even	3	1	inner	168.4.q.f	✓	8
7.c	even	3	1		1176.4.a.bd		4
7.d	odd	6	1		1176.4.a.ba		4
21.h	odd	6	1		504.4.s.j		8
28.f	even	6	1		2352.4.a.cp		4
28.g	odd	6	1		336.4.q.m		8
28.g	odd	6	1		2352.4.a.cm		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.4.q.f	✓	8	1.a	even	1	1	trivial
168.4.q.f	✓	8	7.c	even	3	1	inner
336.4.q.m		8	4.b	odd	2	1
336.4.q.m		8	28.g	odd	6	1
504.4.s.j		8	3.b	odd	2	1
504.4.s.j		8	21.h	odd	6	1
1176.4.a.ba		4	7.d	odd	6	1
1176.4.a.bd		4	7.c	even	3	1
2352.4.a.cm		4	28.g	odd	6	1
2352.4.a.cp		4	28.f	even	6	1

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{8} + 4T_{5}^{7} + 313T_{5}^{6} - 676T_{5}^{5} + 89261T_{5}^{4} + 76256T_{5}^{3} + 57220T_{5}^{2} + 7168T_{5} + 784$ T5^8 + 4*T5^7 + 313*T5^6 - 676*T5^5 + 89261*T5^4 + 76256*T5^3 + 57220*T5^2 + 7168*T5 + 784 acting on $S_{4}^{\mathrm{new}}(168, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{8}$ T^8
$3$	$(T^{2} + 3 T + 9)^{4}$ (T^2 + 3*T + 9)^4
$5$	$T^{8} + 4 T^{7} + \cdots + 784$ T^8 + 4T^7 + 313T^6 - 676T^5 + 89261T^4 + 76256T^3 + 57220T^2 + 7168*T + 784
$7$	$T^{8} + \cdots + 13841287201$ T^8 - 18T^7 + 84T^6 + 7560T^5 - 128723T^4 + 2593080T^3 + 9882516T^2 - 726364926*T + 13841287201
$11$	$T^{8} + \cdots + 33244957032336$ T^8 + 14T^7 + 5591T^6 + 60302T^5 + 24291005T^4 + 204963188T^3 + 35719311436T^2 - 391593061104*T + 33244957032336
$13$	$(T^{4} - 22 T^{3} + \cdots + 13795008)^{2}$ (T^4 - 22T^3 - 7423T^2 + 84004*T + 13795008)^2
$17$	$T^{8} + \cdots + 19\!\cdots\!64$ T^8 + 96T^7 + 23520T^6 + 2058880T^5 + 413546496T^4 + 33033099264T^3 + 2312485998592T^2 + 75853776224256*T + 1953905916248064
$19$	$T^{8} + \cdots + 30110417391616$ T^8 - 26T^7 + 14911T^6 + 887414T^5 + 190422977T^4 + 3967250612T^3 + 145012515664T^2 - 1419300084992*T + 30110417391616
$23$	$T^{8} + \cdots + 9663676416$ T^8 + 96T^7 + 23520T^6 - 1616512T^5 + 193022976T^4 - 1721407488T^3 + 13395988480T^2 - 11960057856*T + 9663676416
$29$	$(T^{4} + 76 T^{3} + \cdots - 802800)^{2}$ (T^4 + 76T^3 - 52293T^2 - 3696384*T - 802800)^2
$31$	$T^{8} + \cdots + 27\!\cdots\!25$ T^8 + 238T^7 + 135412T^6 + 14770364T^5 + 10357966421T^4 + 1398592677692T^3 + 267850878190996T^2 + 2765633698693390*T + 27234235833160225
$37$	$T^{8} + \cdots + 21\!\cdots\!04$ T^8 + 562T^7 + 299539T^6 + 77367450T^5 + 24055639113T^4 + 4641849317052T^3 + 1238349439157040T^2 + 157703115307512960*T + 21385545645567919104
$41$	$(T^{4} - 428 T^{3} + \cdots - 3866949120)^{2}$ (T^4 - 428T^3 - 41132T^2 + 36803808*T - 3866949120)^2
$43$	$(T^{4} + 258 T^{3} + \cdots - 2654719484)^{2}$ (T^4 + 258T^3 - 179543T^2 - 46672932*T - 2654719484)^2
$47$	$T^{8} + \cdots + 56\!\cdots\!24$ T^8 - 80T^7 + 342648T^6 - 10062848T^5 + 90674636656T^4 - 2395660818432T^3 + 8366652681609600T^2 + 441086629735406592*T + 569614040327552903424
$53$	$T^{8} + \cdots + 25753595040000$ T^8 + 135309T^6 + 6027320T^5 + 18313600281T^4 + 407775320940T^3 + 8395480482400T^2 + 15293721768000T + 25753595040000
$59$	$T^{8} + \cdots + 26\!\cdots\!00$ T^8 + 262T^7 + 345195T^6 - 161800754T^5 + 66418351369T^4 - 11493726649116T^3 + 1541505278215296T^2 - 73352242586819520*T + 2696200518203654400
$61$	$T^{8} + \cdots + 97\!\cdots\!00$ T^8 - 276T^7 + 511872T^6 + 204566560T^5 + 168331971984T^4 + 23812464646272T^3 + 6074779134836224T^2 - 415823608078444800*T + 97291323031404960000
$67$	$T^{8} + \cdots + 70\!\cdots\!24$ T^8 + 150T^7 + 613251T^6 - 138028090T^5 + 318750682533T^4 - 22555081338120T^3 + 16283038431222868T^2 + 655493426514372960*T + 703835778021398427024
$71$	$(T^{4} + 848 T^{3} + \cdots + 1940742864)^{2}$ (T^4 + 848T^3 - 78520T^2 - 46164032*T + 1940742864)^2
$73$	$T^{8} + \cdots + 78\!\cdots\!76$ T^8 - 218T^7 + 79039T^6 - 4410506T^5 + 1942405933T^4 - 55505097884T^3 + 40650589792684T^2 + 1581530253463888*T + 78620725698303376
$79$	$T^{8} + \cdots + 86\!\cdots\!01$ T^8 + 1762T^7 + 2360908T^6 + 1563690004T^5 + 869284996429T^4 + 233736255591028T^3 + 85234608753590332T^2 + 11781219476000650786*T + 8658045721309945280401
$83$	$(T^{4} - 3450 T^{3} + \cdots + 352673538780)^{2}$ (T^4 - 3450T^3 + 4237149T^2 - 2140010896*T + 352673538780)^2
$89$	$T^{8} + \cdots + 20\!\cdots\!76$ T^8 - 344T^7 + 1267972T^6 - 791464416T^5 + 1571613039120T^4 - 713782053251712T^3 + 299556787215937536T^2 - 27178868651391590400*T + 2097325294722956722176
$97$	$(T^{4} + 622 T^{3} + \cdots + 79407506004)^{2}$ (T^4 + 622T^3 - 2800699T^2 - 1902403156*T + 79407506004)^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

$f(q)$	$=$	$q + 3 \beta_1 q^{3} + (\beta_{6} - \beta_1 - 1) q^{5} + (\beta_{5} + 2) q^{7} + ( - 9 \beta_1 - 9) q^{9} + (\beta_{7} - 2 \beta_{6} - \beta_{5} + \cdots + 1) q^{11} + ( - \beta_{5} - \beta_{4} - 4 \beta_{2} + 6) q^{13}+ \cdots + (9 \beta_{5} + 9 \beta_{4} - 18 \beta_{2} + 27) q^{99}+O(q^{100})$ q + 3b1 q^3 + (b6 - b1 - 1) * q^5 + (b5 + 2) * q^7 + (-9b1 - 9) q^9 + (b7 - 2b6 - b5 - b4 + b3 + 2b2 + 4b1 + 1) q^11 + (-b5 - b4 - 4b2 + 6) q^13 + (-3b2 + 3) q^15 + (-2b7 - 3b6 + b4 + 3b3 + 2b2 + 26b1 - 1) q^17 + (4b6 + 3b5 - b4 + b3 - b2 + 5b1 + 5) q^19 + (-3b5 + 3b3 + 9b1) q^21 + (-b7 - 2b6 + 3b5 - b4 - b2 - 26b1 - 26) q^23 + (-b7 - 4b6 - 3b5 - b4 + 9b3 + 2b2 + 33b1 + 1) q^25 + 27 * q^27 + (-b7 - b6 - 7b5 - 2b4 + 3b3 + 6b2 + 4b1 - 16) q^29 + (5b7 + 7b6 - 5b5 - 5b4 + 5b3 - 7b2 + 62b1 + 5) q^31 + (-3b7 + 6b6 - 3b3 - 12b1 - 12) * q^33 + (21b6 + b5 - b3 - 14b2 - 10b1 + 14) q^35 + (b7 - 12b6 + 6b5 - 2b4 + 3b3 - 2b2 - 143b1 - 143) * q^37 + (-3b7 - 12b6 + 3b5 + 3b4 - 3b3 + 12b2 + 15b1 - 3) q^39 + (-2b7 - 2b6 - 8b5 + 2b4 + 6b3 - 14b2 + 8b1 + 110) q^41 + (3b7 + 3b6 + 16b5 + b4 - 9b3 + 15b2 - 12b1 - 71) * q^43 + (-9b6 + 9b2 + 9b1) q^45 + (5b7 - 26b6 - 3b5 + b4 + 4b3 + b2 + 24b1 + 24) q^47 + (b5 - 7b4 + 2b3 + 237b1 + 139) q^49 + (3b7 + 6b6 - 9b5 + 3b4 + 3b2 - 66b1 - 66) * q^51 + (-2b7 + 11b6 + 6b5 + 4b4 - 12b3 - 9b2 - 7b1 - 4) q^53 + (5b7 + 5b6 + 23b5 - 2b4 - 15b3 - 18b2 - 20b1 + 360) q^55 + (-3b7 - 3b6 - 12b5 + 3b4 + 9b3 - 9b2 + 12b1 - 15) q^57 + (-5b7 - 3b6 + 11b5 + 8b4 - 20b3 + 6b2 + 54b1 - 8) q^59 + (4b7 + 38b6 + 6b5 - 2b4 + 6b3 - 2b2 + 68b1 + 68) q^61 + (-9b3 - 27b1 - 18) * q^63 + (-3b7 + 2b6 - 21b5 + 7b4 - 10b3 + 7b2 - 540b1 - 540) q^65 + (9b7 - 5b6 + 5b5 - 2b4 - 26b3 + 12b2 + 21b1 + 2) q^67 + (-3b7 - 3b6 - 9b5 + 6b4 + 9b3 + 9b2 + 12b1 + 75) q^69 + (-b7 - b6 - 7b5 - 2b4 + 3b3 + 31b2 + 4b1 - 209) q^71 + (-b7 - 4b6 - 3b5 - b4 + 9b3 + 2b2 - 49b1 + 1) q^73 + (-3b7 + 6b6 - 18b5 + 6b4 - 9b3 + 6b2 - 75b1 - 75) q^75 + (-7b7 - 21b6 - b5 + 21b4 - 10b3 + 63b2 + 397b1 + 97) * q^77 + (5b7 + 29b6 + 12b5 - 4b4 + 9b3 - 4b2 - 444b1 - 444) q^79 + 81b1 q^81 + (-b7 - b6 - 4b5 + b4 + 3b3 + 27b2 + 4b1 + 864) * q^83 + (4b7 + 4b6 + 18b5 - 2b4 - 12b3 - 94b2 - 16b1 + 356) q^85 + (-6b7 + 18b6 + 12b5 + 9b4 - 21b3 - 15b2 - 69b1 - 9) q^87 + (-8b7 - 42b6 - 24b5 + 8b4 - 16b3 + 8b2 + 94b1 + 94) q^89 + (14b7 - 42b6 + 13b5 + 7b4 - 5b3 - 21b2 + 237b1 - 229) q^91 + (-15b7 - 21b6 - 15b3 - 186b1 - 186) * q^93 + (-4b7 + 67b6 - 10b5 - 3b4 + 31b3 - 74b2 + 520b1 + 3) q^95 + (4b7 + 4b6 + 11b5 - 9b4 - 12b3 + 90b2 - 16b1 - 159) q^97 + (9b5 + 9b4 - 18b2 + 27) q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 12 q^{3} - 4 q^{5} + 18 q^{7} - 36 q^{9} - 14 q^{11} + 44 q^{13} + 24 q^{15} - 96 q^{17} + 26 q^{19} - 36 q^{21} - 96 q^{23} - 110 q^{25} + 216 q^{27} - 152 q^{29} - 238 q^{31} - 42 q^{33} + 152 q^{35}+ \cdots + 252 q^{99}+O(q^{100})$ 8 * q - 12 * q^3 - 4 * q^5 + 18 * q^7 - 36 * q^9 - 14 * q^11 + 44 * q^13 + 24 * q^15 - 96 * q^17 + 26 * q^19 - 36 * q^21 - 96 * q^23 - 110 * q^25 + 216 * q^27 - 152 * q^29 - 238 * q^31 - 42 * q^33 + 152 * q^35 - 562 * q^37 - 66 * q^39 + 856 * q^41 - 516 * q^43 - 36 * q^45 + 80 * q^47 + 156 * q^49 - 288 * q^51 + 2952 * q^55 - 156 * q^57 - 262 * q^59 + 276 * q^61 - 54 * q^63 - 2196 * q^65 - 150 * q^67 + 576 * q^69 - 1696 * q^71 + 218 * q^73 - 330 * q^75 - 764 * q^77 - 1762 * q^79 - 324 * q^81 + 6900 * q^83 + 2904 * q^85 + 228 * q^87 + 344 * q^89 - 2806 * q^91 - 714 * q^93 - 2004 * q^95 - 1244 * q^97 + 252 * q^99

$\beta_{1}$	$=$	$( -\nu^{7} + 691\nu^{5} + 67439\nu^{3} + 869616\nu - 673920 ) / 1347840$ (-v^7 + 691v^5 + 67439v^3 + 869616*v - 673920) / 1347840
$\beta_{2}$	$=$	$( 19\nu^{6} + 911\nu^{4} - 31781\nu^{2} - 208224 ) / 42120$ (19v^6 + 911v^4 - 31781*v^2 - 208224) / 42120
$\beta_{3}$	$=$	$( 283 \nu^{7} + 768 \nu^{6} + 29087 \nu^{5} + 143232 \nu^{4} + 233803 \nu^{3} + 8185728 \nu^{2} + \cdots + 109164672 ) / 4043520$ (283v^7 + 768v^6 + 29087v^5 + 143232v^4 + 233803v^3 + 8185728v^2 - 19439568*v + 109164672) / 4043520
$\beta_{4}$	$=$	$( 135 \nu^{7} - 2384 \nu^{6} + 19035 \nu^{5} - 149776 \nu^{4} + 892215 \nu^{3} + 3526576 \nu^{2} + \cdots + 104194944 ) / 1347840$ (135v^7 - 2384v^6 + 19035v^5 - 149776v^4 + 892215v^3 + 3526576v^2 + 14802480*v + 104194944) / 1347840
$\beta_{5}$	$=$	$( - 27 \nu^{7} - 560 \nu^{6} - 3807 \nu^{5} - 62320 \nu^{4} - 178443 \nu^{3} - 1680944 \nu^{2} + \cdots - 9334656 ) / 269568$ (-27v^7 - 560v^6 - 3807v^5 - 62320v^4 - 178443v^3 - 1680944v^2 - 2960496*v - 9334656) / 269568
$\beta_{6}$	$=$	$( 43\nu^{7} + 57\nu^{6} + 5387\nu^{5} + 2733\nu^{4} + 181903\nu^{3} - 95343\nu^{2} + 1055052\nu - 624672 ) / 252720$ (43v^7 + 57v^6 + 5387v^5 + 2733v^4 + 181903v^3 - 95343v^2 + 1055052*v - 624672) / 252720
$\beta_{7}$	$=$	$( 1087 \nu^{7} - 4272 \nu^{6} + 147443 \nu^{5} - 417648 \nu^{4} + 5991727 \nu^{3} - 7751472 \nu^{2} + \cdots - 12451968 ) / 2021760$ (1087v^7 - 4272v^6 + 147443v^5 - 417648v^4 + 5991727v^3 - 7751472v^2 + 78636528*v - 12451968) / 2021760

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	168.4.q.f

Level	$168$
Weight	$4$
Character orbit	168.q
Analytic conductor	$9.912$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$

$\nu$	$=$	$( 3\beta_{7} - 11\beta_{6} - \beta_{5} - 2\beta_{4} + 5\beta_{3} + 5\beta_{2} - 12\beta _1 - 5 ) / 28$ (3b7 - 11b6 - b5 - 2b4 + 5b3 + 5b2 - 12b1 - 5) / 28
$\nu^{2}$	$=$	$( -3\beta_{7} - 3\beta_{6} + \beta_{5} + 16\beta_{4} + 9\beta_{3} + 51\beta_{2} + 12\beta _1 - 1213 ) / 28$ (-3b7 - 3b6 + b5 + 16b4 + 9b3 + 51b2 + 12b1 - 1213) / 28
$\nu^{3}$	$=$	$( -41\beta_{7} + 155\beta_{6} + 2\beta_{5} + 39\beta_{4} - 115\beta_{3} - 59\beta_{2} - 18\beta _1 - 11 ) / 7$ (-41b7 + 155b6 + 2b5 + 39b4 - 115b3 - 59b2 - 18*b1 - 11) / 7
$\nu^{4}$	$=$	$( 191\beta_{7} + 191\beta_{6} - 241\beta_{5} - 1196\beta_{4} - 573\beta_{3} - 4759\beta_{2} - 764\beta _1 + 77321 ) / 28$ (191b7 + 191b6 - 241b5 - 1196b4 - 573b3 - 4759b2 - 764*b1 + 77321) / 28
$\nu^{5}$	$=$	$( 11113 \beta_{7} - 42185 \beta_{6} + 393 \beta_{5} - 11506 \beta_{4} + 34911 \beta_{3} + 15143 \beta_{2} + \cdots + 32457 ) / 28$ (11113b7 - 42185b6 + 393b5 - 11506b4 + 34911b3 + 15143b2 + 65700*b1 + 32457) / 28
$\nu^{6}$	$=$	$( - 3544 \beta_{7} - 3544 \beta_{6} + 3307 \beta_{5} + 21027 \beta_{4} + 10632 \beta_{3} + 93890 \beta_{2} + \cdots - 1357362 ) / 7$ (-3544b7 - 3544b6 + 3307b5 + 21027b4 + 10632b3 + 93890b2 + 14176*b1 - 1357362) / 7
$\nu^{7}$	$=$	$( - 110295 \beta_{7} + 442367 \beta_{6} - 8363 \beta_{5} + 118658 \beta_{4} - 364337 \beta_{3} + \cdots - 536767 ) / 4$ (-110295b7 + 442367b6 - 8363b5 + 118658b4 - 364337b3 - 157673b2 - 1090260*b1 - 536767) / 4

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 25.4
Significant digits:
Format: