LMFDB - Newform orbit 400.6.a.z

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [400,6,Mod(1,400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("400.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$N$	$=$	$400 = 2^{4} \cdot 5^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	400.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-4,0,0,0,-148,0,500] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$64.1535279252$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.1595208.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} - 20x^{2} + 33x - 3$ x^4 - x^3 - 20x^2 + 33x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{8}\cdot 5^{2}$
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

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}

1.1

1.64654
3.98753
−4.73066
0.0965878

−28.9338

−146.828

594.165

1.2

−4.69449

−10.2635

−220.962

1.3

5.49000

188.968

−212.860

1.4

24.1383

−179.876

339.657

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$5$	$-1$

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	400.6.a.z		4
4.b	odd	2	1		200.6.a.k		4
5.b	even	2	1		400.6.a.ba		4
5.c	odd	4	2		80.6.c.d		8
15.e	even	4	2		720.6.f.n		8
20.d	odd	2	1		200.6.a.j		4
20.e	even	4	2		40.6.c.a	✓	8
40.i	odd	4	2		320.6.c.i		8
40.k	even	4	2		320.6.c.j		8
60.l	odd	4	2		360.6.f.b		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
40.6.c.a	✓	8	20.e	even	4	2
80.6.c.d		8	5.c	odd	4	2
200.6.a.j		4	20.d	odd	2	1
200.6.a.k		4	4.b	odd	2	1
320.6.c.i		8	40.i	odd	4	2
320.6.c.j		8	40.k	even	4	2
360.6.f.b		8	60.l	odd	4	2
400.6.a.z		4	1.a	even	1	1	trivial
400.6.a.ba		4	5.b	even	2	1
720.6.f.n		8	15.e	even	4	2

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{4} + 4T_{3}^{3} - 728T_{3}^{2} + 432T_{3} + 18000$ T3^4 + 4*T3^3 - 728*T3^2 + 432*T3 + 18000 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(400))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$ T^4
$3$	$T^{4} + 4 T^{3} + \cdots + 18000$ T^4 + 4T^3 - 728T^2 + 432*T + 18000
$5$	$T^{4}$ T^4
$7$	$T^{4} + 148 T^{3} + \cdots - 51222832$ T^4 + 148T^3 - 33912T^2 - 5353360*T - 51222832
$11$	$T^{4} + \cdots + 37397137664$ T^4 - 368T^3 - 468000T^2 + 126641408*T + 37397137664
$13$	$T^{4} + \cdots + 10683053312$ T^4 - 440T^3 - 599136T^2 - 74346112*T + 10683053312
$17$	$T^{4} + \cdots + 22118400000$ T^4 + 672T^3 - 2323200T^2 + 678297600*T + 22118400000
$19$	$T^{4} + \cdots + 1042985883904$ T^4 - 688T^3 - 4596000T^2 - 947468032*T + 1042985883904
$23$	$T^{4} + \cdots - 5643370924592$ T^4 + 4492T^3 + 1681608T^2 - 7543501040*T - 5643370924592
$29$	$T^{4} + \cdots - 97147517576176$ T^4 + 2936T^3 - 23418600T^2 - 102710226464*T - 97147517576176
$31$	$T^{4} + \cdots + 244229603328000$ T^4 + 2112T^3 - 34329600T^2 - 18314035200*T + 244229603328000
$37$	$T^{4} + \cdots - 16\!\cdots\!36$ T^4 - 8792T^3 - 112838880T^2 + 1175662197632*T - 1678428346606336
$41$	$T^{4} + \cdots - 296811236945008$ T^4 - 11800T^3 - 125515464T^2 + 1158289390624*T - 296811236945008
$43$	$T^{4} + \cdots - 74\!\cdots\!72$ T^4 + 48276T^3 + 718858152T^2 + 2811703952880*T - 7418615241277872
$47$	$T^{4} + \cdots + 11\!\cdots\!96$ T^4 + 14724T^3 - 327305784T^2 - 2375278166736*T + 11549492655826896
$53$	$T^{4} + \cdots + 11\!\cdots\!76$ T^4 - 84296T^3 + 2471818656T^2 - 29314195239296*T + 117733297666053376
$59$	$T^{4} + \cdots - 11\!\cdots\!48$ T^4 - 45840T^3 + 204589536T^2 + 3595104117504*T - 11122897833416448
$61$	$T^{4} + \cdots + 31\!\cdots\!00$ T^4 - 61928T^3 + 524629560T^2 + 11624772421600*T + 31896100546906000
$67$	$T^{4} + \cdots + 71\!\cdots\!32$ T^4 + 72700T^3 + 1883104104T^2 + 20158765938512*T + 71865220992337232
$71$	$T^{4} + \cdots + 25\!\cdots\!00$ T^4 - 62816T^3 - 3818130048T^2 + 155979328612352*T + 2599399853623808000
$73$	$T^{4} + \cdots - 15\!\cdots\!32$ T^4 - 133072T^3 + 5132131968T^2 - 18775878179840*T - 1503282909430853632
$79$	$T^{4} + \cdots - 96\!\cdots\!00$ T^4 - 21632T^3 - 5155348992T^2 - 132927966838784*T - 961157665139916800
$83$	$T^{4} + \cdots + 25\!\cdots\!52$ T^4 + 74660T^3 - 9589155864T^2 - 327499484863696*T + 25322368900512073552
$89$	$T^{4} + \cdots - 93\!\cdots\!76$ T^4 - 20952T^3 - 6218073000T^2 - 146924355735648*T - 934147741563457776
$97$	$T^{4} + \cdots - 14\!\cdots\!84$ T^4 - 59456T^3 - 9949895424T^2 + 847593856385024*T - 14794835008591888384
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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

$f(q)$	$=$	$q + ( - \beta_1 - 1) q^{3} + (\beta_{3} - \beta_1 - 37) q^{7} + ( - 2 \beta_{3} - \beta_{2} + \cdots + 125) q^{9} + ( - 2 \beta_{2} - 18 \beta_1 + 92) q^{11} + (3 \beta_{3} - 2 \beta_{2} + 110) q^{13}+ \cdots + ( - 504 \beta_{3} + 426 \beta_{2} + \cdots + 33356) q^{99}+O(q^{100})$ q + (-b1 - 1) * q^3 + (b3 - b1 - 37) * q^7 + (-2b3 - b2 + 7b1 + 125) * q^9 + (-2b2 - 18b1 + 92) * q^11 + (3b3 - 2b2 + 110) * q^13 + (2b3 + 4b2 - 44b1 - 168) q^17 + (-8b3 + 2b2 - 46b1 + 172) q^19 + (6b3 - 3b2 + 149b1 + 248) q^21 + (5b3 - 8b2 - 23b1 - 1123) q^23 + (2b3 + 16b2 - 222b1 - 2038) q^27 + (20b2 - 12b1 - 734) * q^29 + (24b3 + 12b2 - 84b1 - 528) q^31 + (-28b3 - 8b2 - 156b1 + 6716) q^33 + (17b3 + 2b2 - 428b1 + 2198) q^37 + (32b3 + 4b2 + 36b1 - 376) q^39 + (2b3 - 3b2 - 491b1 + 2950) q^41 + (-54b3 - 189b1 - 12069) * q^43 + (-19b3 - 56b2 - 485b1 - 3681) q^47 + (6b3 - 69b2 - 285b1 + 5625) q^49 + (-88b3 - 68b2 + 988b1 + 15600) q^51 + (53b3 + 34b2 + 64b1 + 21074) q^53 + (-164b3 - 40b2 - 572b1 + 17756) q^57 + (-88b3 - 26b2 + 598b1 + 11460) q^59 + (-82b3 + 61b2 - 683b1 + 15482) q^61 + (115b3 + 152b2 - 521b1 - 46573) q^63 + (26b3 + 32b2 + 73b1 - 18175) q^67 + (26b3 + 7b2 + 1103b1 + 9592) q^69 + (-24b3 + 232b2 - 1560b1 + 15704) q^71 + (48b3 + 80b2 - 1212b1 + 33268) q^73 + (440b3 - 56b2 + 1972b1 + 2860) q^77 + (328b3 - 100b2 - 196b1 + 5408) q^79 + (-6b3 - 63b2 + 3257b1 + 51209) q^81 + (166b3 - 352b2 + 1679b1 - 18665) q^83 + (-104b3 - 112b2 + 2526b1 + 3118) q^87 + (96b3 + 300b2 - 564b1 + 5238) q^89 + (440b3 - 200b2 - 2120b1 + 60952) q^91 + (-24b3 - 192b2 + 4608b1 + 26400) q^93 + (-402b3 + 172b2 + 1860b1 + 14864) q^97 + (-504b3 + 426b2 - 5062b1 + 33356) q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{3} - 148 q^{7} + 500 q^{9} + 368 q^{11} + 440 q^{13} - 672 q^{17} + 688 q^{19} + 992 q^{21} - 4492 q^{23} - 8152 q^{27} - 2936 q^{29} - 2112 q^{31} + 26864 q^{33} + 8792 q^{37} - 1504 q^{39}+ \cdots + 133424 q^{99}+O(q^{100})$ 4 * q - 4 * q^3 - 148 * q^7 + 500 * q^9 + 368 * q^11 + 440 * q^13 - 672 * q^17 + 688 * q^19 + 992 * q^21 - 4492 * q^23 - 8152 * q^27 - 2936 * q^29 - 2112 * q^31 + 26864 * q^33 + 8792 * q^37 - 1504 * q^39 + 11800 * q^41 - 48276 * q^43 - 14724 * q^47 + 22500 * q^49 + 62400 * q^51 + 84296 * q^53 + 71024 * q^57 + 45840 * q^59 + 61928 * q^61 - 186292 * q^63 - 72700 * q^67 + 38368 * q^69 + 62816 * q^71 + 133072 * q^73 + 11440 * q^77 + 21632 * q^79 + 204836 * q^81 - 74660 * q^83 + 12472 * q^87 + 20952 * q^89 + 243808 * q^91 + 105600 * q^93 + 59456 * q^97 + 133424 * q^99

$\beta_{1}$	$=$	$-2\nu^{3} + 40\nu - 29$ -2v^3 + 40v - 29
$\beta_{2}$	$=$	$( 22\nu^{3} + 40\nu^{2} - 240\nu - 141 ) / 3$ (22v^3 + 40v^2 - 240*v - 141) / 3
$\beta_{3}$	$=$	$( -8\nu^{3} + 40\nu^{2} + 120\nu - 516 ) / 3$ (-8v^3 + 40v^2 + 120*v - 516) / 3

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