LMFDB - Newform orbit 189.6.a.m

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,6,Mod(1,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("189.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$189 = 3^{3} \cdot 7$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	189.a (trivial)

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:	yes
Analytic conductor:	$30.3125419447$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{5} - x^{4} - 121x^{3} + 265x^{2} + 3580x - 11620$ x^5 - x^4 - 121x^3 + 265x^2 + 3580*x - 11620
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2\cdot 3^{4}$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of a basis $1,\beta_1,\beta_2,\beta_3,\beta_4$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (\beta_1 - 2) q^{2} + ( - \beta_{2} - 3 \beta_1 + 23) q^{4} + ( - \beta_{4} + \beta_{2} + 2 \beta_1 - 22) q^{5} + 49 q^{7} + (\beta_{3} + 5 \beta_{2} + 11 \beta_1 - 129) q^{8} + (6 \beta_{4} - 3 \beta_{3} + \cdots + 121) q^{10}+ \cdots + (2401 \beta_1 - 4802) q^{98}+O(q^{100})$ q + (b1 - 2) * q^2 + (-b2 - 3b1 + 23) q^4 + (-b4 + b2 + 2b1 - 22) q^5 + 49 * q^7 + (b3 + 5b2 + 11b1 - 129) * q^8 + (6b4 - 3b3 - 7b2 - 36b1 + 121) * q^10 + (4b4 - b3 - 2b1 - 150) * q^11 + (3b4 + 6b3 - 5b2 + 79) q^13 + (49b1 - 98) q^14 + (12b4 - 3b3 - 9b2 - 129b1 + 37) * q^16 + (14b4 - 9b3 + 4b2 - 36b1 - 315) * q^17 + (-18b4 + 3b3 + 62b2 - 30b1 - 312) * q^19 + (-40b4 + 13b3 + 96b2 + 182b1 - 1170) * q^20 + (-36b4 + 6b3 + 34b2 - 168b1 + 290) * q^22 + (-7b4 - 15b3 - 23b2 - 118b1 - 193) * q^23 + (33b4 - 9b3 - 85b2 - 396b1 + 827) * q^25 + (54b4 + 23b3 - 101b2 + 149b1 - 167) * q^26 + (-49b2 - 147b1 + 1127) * q^28 + (23b4 + 18b3 - 95b2 - 266b1 - 1061) * q^29 + (-27b4 - 24b3 - 45b2 - 666b1 + 2267) * q^31 + (-108b4 - 5b3 + 83b2 - 93b1 - 2195) * q^32 + (-192b4 + 6b3 + 250b2 - 417b1 - 820) * q^34 + (-49b4 + 49b2 + 98b1 - 1078) q^35 + (63b4 + 33b3 + 213b2 + 108b1 - 3838) * q^37 + (144b4 - 92b3 - 208b2 - 1246b1 - 1668) * q^38 + (204b4 - 54b3 - 530b2 - 1632b1 + 6206) * q^40 + (91b4 + 20b3 - 315b2 - 94b1 - 4364) * q^41 + (204b4 + 57b3 - 170b2 + 348b1 + 2233) * q^43 + (160b4 - 62b3 - 128b2 + 124b1 - 5332) * q^44 + (-138b4 - 21b3 + 443b2 + 351b1 - 5387) * q^46 + (-219b4 + 21b3 + 171b2 - 1316b1 - 5186) * q^47 + 2401 * q^49 + (-306b4 + 133b3 + 845b2 + 2503b1 - 20569) * q^50 + (-144b4 + 63b3 - 85b2 + 1131b1 + 6669) * q^52 + (71b4 + 101b3 - 583b2 + 936b1 - 5605) * q^53 + (144b4 + 3b3 - 104b2 + 1578b1 - 8734) * q^55 + (49b3 + 245b2 + 539b1 - 6321) q^56 + (78b4 + 177b3 + 165b2 + 705b1 - 10725) * q^58 + (42b4 - 10b3 + 430b2 + 1400b1 - 22475) * q^59 + (30b4 - 141b3 - 138b2 + 834b1 + 11396) * q^61 + (-126b4 - 57b3 + 1155b2 + 3833b1 - 38359) * q^62 + (204b4 - 213b3 - 9b2 + 1155b1 - 4007) * q^64 + (-62b4 + 11b3 + 18b2 - 3830b1 - 10410) * q^65 + (-93b4 - 99b3 - 789b2 + 3018b1 - 4633) * q^67 + (776b4 - 334b3 - 907b2 - 2541b1 - 14791) * q^68 + (294b4 - 147b3 - 343b2 - 1764b1 + 5929) * q^70 + (9b4 - 253b3 - 203b2 + 2802b1 - 23587) * q^71 + (-516b4 - 30b3 - 148b2 + 5652b1 - 5586) * q^73 + (18b4 - 21b3 - 1005b2 - 7882b1 + 12575) * q^74 + (-1392b4 + 216b3 + 1950b2 + 3354b1 - 44770) * q^76 + (196b4 - 49b3 - 98b1 - 7350) q^77 + (303b4 + 12b3 + 261b2 + 4566b1 + 4880) * q^79 + (-592b4 + 414b3 + 1312b2 + 10004b1 - 50284) * q^80 + (-306b4 + 537b3 + 597b2 + 630b1 + 7233) * q^82 + (101b4 + 73b3 + 759b2 - 4088b1 - 15160) * q^83 + (165b4 + 87b3 + 67b2 + 11232b1 - 37648) * q^85 + (-540b4 + 692b3 - 536b2 + 3755b1 + 17266) * q^86 + (-552b4 + 132b3 + 764b2 + 1296b1 + 12508) * q^88 + (-1079b4 + 61b3 - 659b2 + 4134b1 + 5167) * q^89 + (147b4 + 294b3 - 245b2 + 3871) q^91 + (800b4 - 281b3 - 495b2 - 8803b1 + 29907) * q^92 + (1566b4 - 567b3 - 103b2 - 5682b1 - 62267) * q^94 + (484b4 - 143b3 - 3836b2 - 9074b1 + 84326) * q^95 + (-648b4 - 447b3 + 664b2 - 2994b1 + 10726) * q^97 + (2401b1 - 4802) q^98
$\operatorname{Tr}(f)(q)$	$=$	$5 q - 9 q^{2} + 113 q^{4} - 111 q^{5} + 245 q^{7} - 639 q^{8} + 588 q^{10} - 744 q^{11} + 406 q^{13} - 441 q^{14} + 89 q^{16} - 1587 q^{17} - 1688 q^{19} - 5844 q^{20} + 1176 q^{22} - 1074 q^{23} + 3890 q^{25}+ \cdots - 21609 q^{98}+O(q^{100})$ 5 * q - 9 * q^2 + 113 * q^4 - 111 * q^5 + 245 * q^7 - 639 * q^8 + 588 * q^10 - 744 * q^11 + 406 * q^13 - 441 * q^14 + 89 * q^16 - 1587 * q^17 - 1688 * q^19 - 5844 * q^20 + 1176 * q^22 - 1074 * q^23 + 3890 * q^25 - 477 * q^26 + 5537 * q^28 - 5430 * q^29 + 10660 * q^31 - 11367 * q^32 - 5151 * q^34 - 5439 * q^35 - 19169 * q^37 - 9090 * q^38 + 30336 * q^40 - 21417 * q^41 + 12091 * q^43 - 26088 * q^44 - 27303 * q^46 - 27855 * q^47 + 12005 * q^49 - 101799 * q^50 + 34273 * q^52 - 26364 * q^53 - 41700 * q^55 - 31311 * q^56 - 52929 * q^58 - 111321 * q^59 + 58012 * q^61 - 189369 * q^62 - 18463 * q^64 - 56022 * q^65 - 19544 * q^67 - 74037 * q^68 + 28812 * q^70 - 114912 * q^71 - 23162 * q^73 + 56034 * q^74 - 225230 * q^76 - 36456 * q^77 + 29311 * q^79 - 243912 * q^80 + 35586 * q^82 - 80445 * q^83 - 176745 * q^85 + 89541 * q^86 + 61968 * q^88 + 28470 * q^89 + 19894 * q^91 + 142827 * q^92 - 313782 * q^94 + 417360 * q^95 + 48676 * q^97 - 21609 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $x^{5} - x^{4} - 121x^{3} + 265x^{2} + 3580x - 11620$ x^5 - x^4 - 121*x^3 + 265*x^2 + 3580*x - 11620 :

$\beta_{1}$	$=$	$( \nu^{4} + 2\nu^{3} - 95\nu^{2} - 100\nu + 1940 ) / 40$ (v^4 + 2v^3 - 95v^2 - 100*v + 1940) / 40
$\beta_{2}$	$=$	$( 3\nu^{2} + 9\nu - 148 ) / 2$ (3v^2 + 9v - 148) / 2
$\beta_{3}$	$=$	$-3\nu^{2} + 9\nu + 144$ -3v^2 + 9v + 144
$\beta_{4}$	$=$	$( 7\nu^{4} + 44\nu^{3} - 665\nu^{2} - 2470\nu + 16480 ) / 40$ (7v^4 + 44v^3 - 665v^2 - 2470v + 16480) / 40

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$( \beta_{3} + 2\beta_{2} + 4 ) / 18$ (b3 + 2*b2 + 4) / 18
$\nu^{2}$	$=$	$( -\beta_{3} + 2\beta_{2} + 292 ) / 6$ (-b3 + 2*b2 + 292) / 6
$\nu^{3}$	$=$	$( 24\beta_{4} + 59\beta_{3} + 118\beta_{2} - 168\beta _1 - 1504 ) / 18$ (24b4 + 59b3 + 118b2 - 168b1 - 1504) / 18
$\nu^{4}$	$=$	$( -16\beta_{4} - 101\beta_{3} + 178\beta_{2} + 352\beta _1 + 17236 ) / 6$ (-16b4 - 101b3 + 178b2 + 352b1 + 17236) / 6

Display $\beta_i$ in terms of $\nu^j$

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}

1.1

−7.33434
5.62037
7.71572
−9.04220
4.04045

−10.3071

74.2371

−107.925

49.0000

−435.344

1112.40

1.2

−8.75089

44.5781

53.4758

49.0000

−110.069

−467.960

1.3

−2.60945

−25.1908

−44.5138

49.0000

149.236

116.156

1.4

5.08011

−6.19245

32.4109

49.0000

−194.022

164.651

1.5

7.58736

25.5681

−44.4480

49.0000

−48.8014

−337.243

Atkin-Lehner signs

$p$	Sign
$3$	$-1$
$7$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	189.6.a.m	✓	5
3.b	odd	2	1		189.6.a.n	yes	5

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
189.6.a.m	✓	5	1.a	even	1	1	trivial
189.6.a.n	yes	5	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{5} + 9T_{2}^{4} - 96T_{2}^{3} - 702T_{2}^{2} + 2412T_{2} + 9072$ T2^5 + 9*T2^4 - 96*T2^3 - 702*T2^2 + 2412*T2 + 9072 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(189))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{5} + 9 T^{4} + \cdots + 9072$ T^5 + 9T^4 - 96T^3 - 702T^2 + 2412T + 9072
$3$	$T^{5}$ T^5
$5$	$T^{5} + 111 T^{4} + \cdots + 370097856$ T^5 + 111T^4 - 3597T^3 - 439767T^2 + 1730196T + 370097856
$7$	$(T - 49)^{5}$ (T - 49)^5
$11$	$T^{5} + \cdots + 212686718976$ T^5 + 744T^4 + 45876T^3 - 26716752T^2 - 739025280T + 212686718976
$13$	$T^{5} + \cdots - 20384169483488$ T^5 - 406T^4 - 1269581T^3 + 630754858T^2 + 5052915872T - 20384169483488
$17$	$T^{5} + \cdots - 19\!\cdots\!57$ T^5 + 1587T^4 - 4059186T^3 - 2790411894T^2 + 5909499623025T - 1945891203605757
$19$	$T^{5} + \cdots - 35990315233856$ T^5 + 1688T^4 - 7820000T^3 - 7048219472T^2 + 17915286546032T - 35990315233856
$23$	$T^{5} + \cdots + 57\!\cdots\!36$ T^5 + 1074T^4 - 9864753T^3 - 6888148758T^2 + 23892400597284T + 5764069634714136
$29$	$T^{5} + \cdots - 93\!\cdots\!56$ T^5 + 5430T^4 - 27883575T^3 - 50756519700T^2 + 199031092226100T - 93649625798792256
$31$	$T^{5} + \cdots - 23\!\cdots\!56$ T^5 - 10660T^4 - 39590495T^3 + 478412292610T^2 + 155147935582400T - 2373474113685493856
$37$	$T^{5} + \cdots - 39\!\cdots\!24$ T^5 + 19169T^4 - 27882365T^3 - 2706086346641T^2 - 19064624044800304T - 39537272093477468324
$41$	$T^{5} + \cdots + 98\!\cdots\!68$ T^5 + 21417T^4 - 56298645T^3 - 2833756416477T^2 - 1274206349962044T + 98458239867720131268
$43$	$T^{5} + \cdots - 27\!\cdots\!19$ T^5 - 12091T^4 - 390823514T^3 + 3470036141278T^2 + 34340792681579225T - 278047656204947530019
$47$	$T^{5} + \cdots - 60\!\cdots\!16$ T^5 + 27855T^4 - 345543297T^3 - 17319833806671T^2 - 181261657936468836T - 604728878406845878116
$53$	$T^{5} + \cdots - 59\!\cdots\!68$ T^5 + 26364T^4 - 975698835T^3 - 35563218968394T^2 - 282672201758187348T - 592005033229328778168
$59$	$T^{5} + \cdots - 26\!\cdots\!55$ T^5 + 111321T^4 + 4274027742T^3 + 63817249740750T^2 + 268093237589792865T - 267729922876928030055
$61$	$T^{5} + \cdots + 62\!\cdots\!48$ T^5 - 58012T^4 + 544464736T^3 + 7751034359920T^2 - 61726866019804816T + 62556125301356328448
$67$	$T^{5} + \cdots + 73\!\cdots\!04$ T^5 + 19544T^4 - 2824910801T^3 - 18400991152400T^2 + 1553779744544072624T + 7384310454274438213504
$71$	$T^{5} + \cdots - 56\!\cdots\!88$ T^5 + 114912T^4 + 2018714427T^3 - 155173066502076T^2 - 5997886804607889024T - 56933865697569577049088
$73$	$T^{5} + \cdots - 50\!\cdots\!00$ T^5 + 23162T^4 - 6040991672T^3 - 69478536734576T^2 + 9041011922676992720T - 505318543608751186400
$79$	$T^{5} + \cdots - 67\!\cdots\!84$ T^5 - 29311T^4 - 3419396273T^3 + 109317398686507T^2 + 2062507861334050184T - 67093923702273296364884
$83$	$T^{5} + \cdots + 36\!\cdots\!16$ T^5 + 80445T^4 - 1200950037T^3 - 127797526450449T^2 + 880176945514274424T + 3622293690796755912816
$89$	$T^{5} + \cdots - 17\!\cdots\!96$ T^5 - 28470T^4 - 13803361557T^3 + 560977651165662T^2 + 39230563538458550196T - 1793080145078882312539896
$97$	$T^{5} + \cdots - 19\!\cdots\!00$ T^5 - 48676T^4 - 10458119708T^3 + 582636439901152T^2 - 526135423323838720T - 196787917440598353459200
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Classical	Maass
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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

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.5
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Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$ -expansion

Embeddings

Atkin-Lehner signs

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	189.6.a.m

Level	$189$
Weight	$6$
Character orbit	189.a
Self dual	yes
Analytic conductor	$30.313$
Analytic rank	$1$
Dimension	$5$
CM	no
Inner twists	$1$