LMFDB - Newform orbit 336.4.q.k

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,4,Mod(193,336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, 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("336.193"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	336.q (of order $3$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,-9,0,-11] 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:	$19.8246417619$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.9924270768.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - x^{5} + 25x^{4} + 12x^{3} + 582x^{2} - 144x + 36$ x^6 - x^5 + 25x^4 + 12x^3 + 582x^2 - 144x + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$2^{4}\cdot 3$
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$ .

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$1$	$-1 - \beta_{3}$

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}

193.1

−2.27818	−	3.94593i
2.65415	+	4.59712i
0.124036	+	0.214837i
−2.27818	+	3.94593i
2.65415	−	4.59712i
0.124036	−	0.214837i

−1.50000

2.59808i

−8.93660

−

15.4786i

−2.26047

−

18.3818i

−4.50000

−

7.79423i

193.2

−1.50000

2.59808i

−2.78070

−

4.81631i

−9.67799

15.7904i

−4.50000

−

7.79423i

193.3

−1.50000

2.59808i

6.21730

10.7687i

18.4385

−

1.73873i

−4.50000

−

7.79423i

289.1

−1.50000

−

2.59808i

−8.93660

15.4786i

−2.26047

18.3818i

−4.50000

7.79423i

289.2

−1.50000

−

2.59808i

−2.78070

4.81631i

−9.67799

−

15.7904i

−4.50000

7.79423i

289.3

−1.50000

−

2.59808i

6.21730

−

10.7687i

18.4385

1.73873i

−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	336.4.q.k		6
4.b	odd	2	1		21.4.e.b	✓	6
7.c	even	3	1	inner	336.4.q.k		6
7.c	even	3	1		2352.4.a.ci		3
7.d	odd	6	1		2352.4.a.cg		3
12.b	even	2	1		63.4.e.c		6
28.d	even	2	1		147.4.e.n		6
28.f	even	6	1		147.4.a.m		3
28.f	even	6	1		147.4.e.n		6
28.g	odd	6	1		21.4.e.b	✓	6
28.g	odd	6	1		147.4.a.l		3
84.h	odd	2	1		441.4.e.w		6
84.j	odd	6	1		441.4.a.t		3
84.j	odd	6	1		441.4.e.w		6
84.n	even	6	1		63.4.e.c		6
84.n	even	6	1		441.4.a.s		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.4.e.b	✓	6	4.b	odd	2	1
21.4.e.b	✓	6	28.g	odd	6	1
63.4.e.c		6	12.b	even	2	1
63.4.e.c		6	84.n	even	6	1
147.4.a.l		3	28.g	odd	6	1
147.4.a.m		3	28.f	even	6	1
147.4.e.n		6	28.d	even	2	1
147.4.e.n		6	28.f	even	6	1
336.4.q.k		6	1.a	even	1	1	trivial
336.4.q.k		6	7.c	even	3	1	inner
441.4.a.s		3	84.n	even	6	1
441.4.a.t		3	84.j	odd	6	1
441.4.e.w		6	84.h	odd	2	1
441.4.e.w		6	84.j	odd	6	1
2352.4.a.cg		3	7.d	odd	6	1
2352.4.a.ci		3	7.c	even	3	1

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{6} + 11T_{5}^{5} + 313T_{5}^{4} + 360T_{5}^{3} + 50460T_{5}^{2} + 237312T_{5} + 1527696$ T5^6 + 11*T5^5 + 313*T5^4 + 360*T5^3 + 50460*T5^2 + 237312*T5 + 1527696 acting on $S_{4}^{\mathrm{new}}(336, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{6}$ T^6
$3$	$(T^{2} + 3 T + 9)^{3}$ (T^2 + 3*T + 9)^3
$5$	$T^{6} + 11 T^{5} + \cdots + 1527696$ T^6 + 11T^5 + 313T^4 + 360T^3 + 50460T^2 + 237312*T + 1527696
$7$	$T^{6} - 13 T^{5} + \cdots + 40353607$ T^6 - 13T^5 + 236T^4 - 12145T^3 + 80948T^2 - 1529437*T + 40353607
$11$	$T^{6} - 35 T^{5} + \cdots + 91470096$ T^6 - 35T^5 + 2593T^4 + 67008T^3 + 1536684T^2 + 13083552*T + 91470096
$13$	$(T^{3} - 62 T^{2} + \cdots + 18452)^{2}$ (T^3 - 62T^2 + 425T + 18452)^2
$17$	$T^{6} + \cdots + 12745506816$ T^6 + 48T^5 + 4704T^4 + 110592T^3 + 11179008T^2 + 270950400*T + 12745506816
$19$	$T^{6} + \cdots + 54664310416$ T^6 + 202T^5 + 28523T^4 + 2013154T^3 + 103594553T^2 + 2871346924*T + 54664310416
$23$	$T^{6} + \cdots + 2498119335936$ T^6 - 216T^5 + 47328T^4 - 3015936T^3 + 341849088T^2 - 1062125568*T + 2498119335936
$29$	$(T^{3} - 53 T^{2} + \cdots - 824976)^{2}$ (T^3 - 53T^2 - 20472T - 824976)^2
$31$	$T^{6} + 95 T^{5} + \cdots + 139783329$ T^6 + 95T^5 + 19026T^4 - 926449T^3 + 101143186T^2 + 118241823*T + 139783329
$37$	$T^{6} + \cdots + 2415919104$ T^6 + 262T^5 + 54555T^4 + 3593014T^3 + 185622097T^2 + 692502528*T + 2415919104
$41$	$(T^{3} - 244 T^{2} + \cdots - 300384)^{2}$ (T^3 - 244T^2 - 18780T - 300384)^2
$43$	$(T^{3} + 360 T^{2} + \cdots - 18269746)^{2}$ (T^3 + 360T^2 - 72363T - 18269746)^2
$47$	$T^{6} + \cdots + 26205471480384$ T^6 + 210T^5 + 290616T^4 - 62006616T^3 + 59695121376T^2 - 1261946958048*T + 26205471480384
$53$	$T^{6} + \cdots + 11\!\cdots\!64$ T^6 + 393T^5 + 235185T^4 + 34609536T^3 + 19553872752T^2 + 2677964032512*T + 1100208565649664
$59$	$T^{6} + \cdots + 10\!\cdots\!36$ T^6 - 1143T^5 + 1173345T^4 - 353075760T^3 + 132552677808T^2 + 13372818322176*T + 10094008708475136
$61$	$T^{6} + \cdots + 71\!\cdots\!00$ T^6 - 70T^5 + 345800T^4 - 145399000T^3 + 122136980000T^2 - 28850707900000*T + 7162406161000000
$67$	$T^{6} + \cdots + 783608160972004$ T^6 + 628T^5 + 699347T^4 - 247502768T^3 + 75422826113T^2 - 8536829868926*T + 783608160972004
$71$	$(T^{3} + 318 T^{2} + \cdots + 28535976)^{2}$ (T^3 + 318T^2 - 330804T + 28535976)^2
$73$	$T^{6} + \cdots + 20\!\cdots\!24$ T^6 + 988T^5 + 980499T^4 + 282111496T^3 + 141507598609T^2 + 623666998890*T + 20508278645865924
$79$	$T^{6} + \cdots + 37\!\cdots\!69$ T^6 - 861T^5 + 999222T^4 - 165859913T^3 + 233509331958T^2 - 50021533268637*T + 37619060662457569
$83$	$(T^{3} + 519 T^{2} + \cdots - 47916036)^{2}$ (T^3 + 519T^2 - 131616T - 47916036)^2
$89$	$T^{6} + \cdots + 169118164647936$ T^6 + 1766T^5 + 2840836T^4 + 516815808T^3 + 100205551104T^2 - 3614222868480*T + 169118164647936
$97$	$(T^{3} - 19 T^{2} + \cdots + 44776452)^{2}$ (T^3 - 19T^2 - 569600T + 44776452)^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

$f(q)$	$=$	$q + 3 \beta_{3} q^{3} + (\beta_{4} - 4 \beta_{3} + \beta_{2} - 4) q^{5} + (\beta_{5} + \beta_{4} - 2 \beta_{3} + \cdots + 1) q^{7} + ( - 9 \beta_{3} - 9) q^{9} + (\beta_{5} - 2 \beta_{4} + \cdots + \beta_1) q^{11}+ \cdots + ( - 18 \beta_{2} - 9 \beta_1 - 99) q^{99}+O(q^{100})$ q + 3b3 q^3 + (b4 - 4b3 + b2 - 4) q^5 + (b5 + b4 - 2b3 + b2 + b1 + 1) q^7 + (-9b3 - 9) q^9 + (b5 - 2b4 - 11b3 + b1) * q^11 + (2b2 - b1 + 20) q^13 + (-3b2 + 12) q^15 + (b5 - 3b4 + 17b3 + b1) * q^17 + (-2b5 - b4 - 67b3 - b2 - 67) * q^19 + (-3b5 + 9b3 - 3b2 + 6) q^21 + (-7b5 - 3b4 + 73b3 - 3b2 + 73) * q^23 + (-7b5 - 8b4 + 46b3 - 7b1) * q^25 + 27 * q^27 + (-10b2 + 5b1 + 21) * q^29 + (-5b5 - 7b4 + 34b3 - 5b1) * q^31 + (-3b5 + 6b4 + 33b3 + 6b2 + 33) * q^33 + (-7b5 - 3b4 + 151b3 + 14b2 - 10b1 + 61) q^35 + (-5b5 - 4b4 - 86b3 - 4b2 - 86) * q^37 + (3b5 + 6b4 + 60b3 + 3b1) * q^39 + (10b2 + 4b1 + 78) * q^41 + (15b2 - 18b1 - 125) * q^43 + (-9b4 + 36b3) * q^45 + (-25b5 + 3b4 - 71b3 + 3b2 - 71) * q^47 + (-12b5 - 10b4 - 107b3 + 16b2 - 3b1 - 111) q^49 + (-3b5 + 9b4 - 51b3 + 9b2 - 51) * q^51 + (-20b5 - 9b4 + 134b3 - 20b1) * q^53 + (14b2 + 11b1 + 339) * q^55 + (3b2 - 6b1 + 201) * q^57 + (-10b5 - 39b4 - 368b3 - 10b1) * q^59 + (20b5 + 40b4 + 10b3 + 40b2 + 10) * q^61 + (-9b4 - 9b3 - 9b1 - 27) q^63 + (-17b5 + b4 + 157b3 + b2 + 157) * q^65 + (-16b5 + 31b4 + 199b3 - 16b1) * q^67 + (9b2 - 21b1 - 219) * q^69 + (-21b2 + 33b1 - 99) * q^71 + (-31b5 - 8b4 + 332b3 - 31b1) * q^73 + (21b5 + 24b4 - 138b3 + 24b2 - 138) * q^75 + (5b5 - 40b4 - 433b3 - 16b2 + 16b1 - 39) q^77 + (3b5 - 45b4 + 302b3 - 45b2 + 302) * q^79 + 81b3 q^81 + (-33b2 + 6b1 - 162) * q^83 + (-18b2 + 18b1 + 606) * q^85 + (-15b5 - 30b4 + 63b3 - 15b1) * q^87 + (-48b5 - 26b4 - 580b3 - 26b2 - 580) * q^89 + (18b5 - 5b4 + 305b3 + 11b2 + 30b1 + 257) q^91 + (15b5 + 21b4 - 102b3 + 21b2 - 102) * q^93 + (13b5 - 41b4 + 259b3 + 13b1) * q^95 + (-50b2 + b1 + 23) q^97 + (-18b2 - 9b1 - 99) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 9 q^{3} - 11 q^{5} + 13 q^{7} - 27 q^{9} + 35 q^{11} + 124 q^{13} + 66 q^{15} - 48 q^{17} - 202 q^{19} + 3 q^{21} + 216 q^{23} - 130 q^{25} + 162 q^{27} + 106 q^{29} - 95 q^{31} + 105 q^{33} - 56 q^{35}+ \cdots - 630 q^{99}+O(q^{100})$ 6 * q - 9 * q^3 - 11 * q^5 + 13 * q^7 - 27 * q^9 + 35 * q^11 + 124 * q^13 + 66 * q^15 - 48 * q^17 - 202 * q^19 + 3 * q^21 + 216 * q^23 - 130 * q^25 + 162 * q^27 + 106 * q^29 - 95 * q^31 + 105 * q^33 - 56 * q^35 - 262 * q^37 - 186 * q^39 + 488 * q^41 - 720 * q^43 - 99 * q^45 - 210 * q^47 - 303 * q^49 - 144 * q^51 - 393 * q^53 + 2062 * q^55 + 1212 * q^57 + 1143 * q^59 + 70 * q^61 - 126 * q^63 + 472 * q^65 - 628 * q^67 - 1296 * q^69 - 636 * q^71 - 988 * q^73 - 390 * q^75 + 1073 * q^77 + 861 * q^79 - 243 * q^81 - 1038 * q^83 + 3600 * q^85 - 159 * q^87 - 1766 * q^89 + 654 * q^91 - 285 * q^93 - 736 * q^95 + 38 * q^97 - 630 * q^99

$\beta_{1}$	$=$	$( -11\nu^{5} + 275\nu^{4} + 328\nu^{3} + 6402\nu^{2} - 1584\nu + 118833 ) / 7203$ (-11v^5 + 275v^4 + 328v^3 + 6402v^2 - 1584*v + 118833) / 7203
$\beta_{2}$	$=$	$( -13\nu^{5} + 325\nu^{4} - 922\nu^{3} + 7566\nu^{2} - 1872\nu + 118830 ) / 7203$ (-13v^5 + 325v^4 - 922v^3 + 7566v^2 - 1872*v + 118830) / 7203
$\beta_{3}$	$=$	$( 100\nu^{5} - 99\nu^{4} + 2475\nu^{3} + 1825\nu^{2} + 57618\nu - 14256 ) / 14406$ (100v^5 - 99v^4 + 2475v^3 + 1825v^2 + 57618*v - 14256) / 14406
$\beta_{4}$	$=$	$( 801\nu^{5} - 817\nu^{4} + 20425\nu^{3} + 6815\nu^{2} + 475494\nu - 117648 ) / 7203$ (801v^5 - 817v^4 + 20425v^3 + 6815v^2 + 475494*v - 117648) / 7203
$\beta_{5}$	$=$	$( -1676\nu^{5} + 1083\nu^{4} - 41481\nu^{3} - 30587\nu^{2} - 947238\nu - 2514 ) / 14406$ (-1676v^5 + 1083v^4 - 41481v^3 - 30587v^2 - 947238*v - 2514) / 14406

Introduction

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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	336.4.q.k

Level	$336$
Weight	$4$
Character orbit	336.q
Analytic conductor	$19.825$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$

$\nu$	$=$	$( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 1 ) / 4$ (b5 + b4 + b3 + b2 + 1) / 4
$\nu^{2}$	$=$	$( \beta_{5} - \beta_{4} + 33\beta_{3} + \beta_1 ) / 2$ (b5 - b4 + 33*b3 + b1) / 2
$\nu^{3}$	$=$	$( -11\beta_{2} + 13\beta _1 - 33 ) / 2$ (-11b2 + 13b1 - 33) / 2
$\nu^{4}$	$=$	$-17\beta_{5} + 8\beta_{4} - 411\beta_{3} + 8\beta_{2} - 411$ -17b5 + 8b4 - 411b3 + 8b2 - 411
$\nu^{5}$	$=$	$-170\beta_{5} - 127\beta_{4} - 708\beta_{3} - 170\beta_1$ -170b5 - 127b4 - 708b3 - 170b1

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 193.3
Significant digits:
Format: